We investigate light propagation in the gravitational field of multiple gravitational lenses. Assuming these lenses are sufficiently spaced to prevent interaction, we consider a linear alignment for the
transmitter, lenses, and receiver. Remarkably, in this axially-symmetric configuration, we can solve
the relevant diffraction integrals – result that offers valuable analytical insights. We show that the
point-spread function (PSF) is affected by the number of lenses in the system. Even a single lens is
useful for transmission either it is used as a part of the transmitter or it acts on the receiver’s side.
We show that power transmission via a pair of lenses benefits from light amplification on both ends
of the link. The second lens plays an important role by focusing the signal to a much tighter spot;
but in practical lensing scenarios, that lens changes the structure of the PSF on scales much smaller
than the telescope, so that additional gain due to the presence of the second lens is independent of its
properties and is govern solely by the transmission geometry. While evaluating the signal-to-noise
ratio (SNR) in various transmitting scenarios, we see that a single-lens transmission performs on par
with a pair of lenses. The fact that the second lens amplifies the brightness of the first one, creates a
challenging background for signal reception. Nevertheless, in all the cases considered here, we have
found practically-relevant SNR values. As a result, we were able to demonstrate the feasibility of
establishing interstellar power transmission links relying on gravitational lensing – a finding with
profound implications for applications targeting interstellar power transmission.
This document summarizes a study measuring the transverse beam emittance at the Energy Selection System (ESS) of the KIRAMS-430 superconducting cyclotron. The researchers used a quadrupole variation method, where they varied the magnetic strength of a quadrupole magnet and measured the resulting beam size at a beam profile monitor. They analyzed the measurements using both linear matrix formalism and particle tracking simulations. The results from both analysis methods were consistent with emittances calculated from Monte Carlo simulations within the measurement uncertainties. The study demonstrated the feasibility of using the quadrupole variation method to characterize the beam quality at the ESS ion beamline.
This document presents observations from the VLT X-shooter instrument of two quasars, SDSS J1106+1939 and SDSS J1512+1119. For SDSS J1106+1939, a broad absorption line (BAL) outflow is detected with a kinetic luminosity of at least 10^46 erg/s, which is 5% of the quasar's bolometric luminosity. This outflow has a velocity of ~8000 km/s and is located ~300 pc from the quasar. For SDSS J1512+1119, two separate outflows are detected using the same technique, with distances ranging from 100-2000 pc from the central source. The distances of the outflows
Forming intracluster gas in a galaxy protocluster at a redshift of 2.16Sérgio Sacani
Galaxy clusters are the most massive gravitationally bound structures in the Universe, comprising thousands of galaxies and
pervaded by a diffuse, hot “intracluster medium” (ICM) that dominates the baryonic content of these systems. The formation
and evolution of the ICM across cosmic time1
is thought to be driven by the continuous accretion of matter from the large-scale
filamentary surroundings and dramatic merger events with other clusters or groups. Until now, however, direct observations of
the intracluster gas have been limited only to mature clusters in the latter three-quarters of the history of the Universe, and we
have been lacking a direct view of the hot, thermalized cluster atmosphere at the epoch when the first massive clusters formed.
Here we report the detection (about 6σ) of the thermal Sunyaev-Zeldovich (SZ) effect2
in the direction of a protocluster. In fact,
the SZ signal reveals the ICM thermal energy in a way that is insensitive to cosmological dimming, making it ideal for tracing
the thermal history of cosmic structures3
. This result indicates the presence of a nascent ICM within the Spiderweb protocluster
at redshift z = 2.156, around 10 billion years ago. The amplitude and morphology of the detected signal show that the SZ
effect from the protocluster is lower than expected from dynamical considerations and comparable with that of lower-redshift
group-scale systems, consistent with expectations for a dynamically active progenitor of a local galaxy cluster.
1) Using four laser beams, researchers generated a three-dimensional optical lattice that traps 490nm polystyrene spheres in solution, forming a face-centered orthorhombic crystal structure.
2) The four-beam setup produces a stable periodic potential in all three dimensions that counteracts particle diffusion via radiation pressure balance.
3) Calculations show the four-beam lattice with all beams polarized parallel produces a simple intensity pattern that yields a face-centered orthorhombic crystal structure when the beam angle is 45 degrees.
1) Photonic nanojets can influence the trapping behavior of multiple microspheres that are axially trapped in a focused laser beam. 2) Simulations show that when two microspheres approach each other axially in the beam, they are initially pushed apart by scattering forces but can become drawn together by a connecting photonic nanojet that forms between them. 3) Three microspheres may also become tethered when specific refractive index conditions are met between each neighboring pair.
This document provides an overview of ray optics and matrix optics. It discusses how ray optics can be used to describe the behavior of light when its wavelength is much smaller than the dimensions of objects it interacts with. Ray optics models light as rays that travel in straight lines and bend at interfaces according to Snell's law. Matrix optics uses 2x2 matrices to relate the position and angle of rays entering and exiting optical systems, allowing complex systems to be analyzed. Key concepts covered include image formation by lenses, paraxial approximation, and the ray transfer matrix.
Fusion of Multispectral And Full Polarimetric SAR Images In NSST DomainCSCJournals
Polarimetric SAR (POLSAR) and multispectral images provide different characteristics of the imaged objects. Multispectral provides information about surface material while POLSAR provides information about geometrical and physical properties of the objects. Merging both should resolve many of object recognition problems that exist when they are used separately. Through this paper, we propose a new scheme for image fusion of full polarization radar image (POLSAR) with multispectral optical satellite image (Egyptsat). The proposed scheme is based on Non-Subsampled Shearlet Transform (NSST) and multi-channel Pulse Coupled Neural Network (m-PCNN). We use NSST to decompose images into low frequency and band-pass sub- band coefficients. With respect to low frequency coefficients, a fusion rule is proposed based on local energy and dispersion index. In respect of sub-band coefficients, m-PCNN is used to guide how the fused sub-band coefficients are calculated using image textural information.
The proposed method is applied on three batches of Egyptsat (Red-Green-infra-red) and radarsat2 (C-band full-polarimetric HH-HV and VV-polarization) images. The batches are selected to react differently with different polarization. Visual assessment of the obtained fused image gives excellent information on clarity and delineation of different objects. Quantitative evaluations show the proposed method can superior the other data fusion methods.
Experimental Study of Third Order Nonlinear Absorption in Pure and MG Doped L...ijtsrd
With the discovery of laser in 1960 various nonlinear effect arises, the origin of nonlinear optics lies in the nonlinear response of materials to the incident coherent radiation. Using pulsed Q switched Nd YAG Laser, we have observed the important phenomena i.e. third order nonlinear effect in the LiNbO3 crystals. In present work, I have performed an experiment using simple and sensitive single beam Z scan technique to measure nonlinear absorption in LiNbO3 crystals samples such as pure LiNbO3 crystal, 5mol Mg doped LiNbO3 crystal, and 7mol Mg doped LiNbO3 crystal. Vijay Aithekar | Dr. Vishal Saxana "Experimental Study of Third Order Nonlinear Absorption in Pure and MG Doped Lithium Niobate Crystals" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-3 | Issue-6 , October 2019, URL: https://www.ijtsrd.com/papers/ijtsrd26370.pdf Paper URL: https://www.ijtsrd.com/physics/engineering-physics/26370/experimental-study-of-third-order-nonlinear-absorption-in-pure-and-mg-doped-lithium-niobate-crystals/vijay-aithekar
This document summarizes a study measuring the transverse beam emittance at the Energy Selection System (ESS) of the KIRAMS-430 superconducting cyclotron. The researchers used a quadrupole variation method, where they varied the magnetic strength of a quadrupole magnet and measured the resulting beam size at a beam profile monitor. They analyzed the measurements using both linear matrix formalism and particle tracking simulations. The results from both analysis methods were consistent with emittances calculated from Monte Carlo simulations within the measurement uncertainties. The study demonstrated the feasibility of using the quadrupole variation method to characterize the beam quality at the ESS ion beamline.
This document presents observations from the VLT X-shooter instrument of two quasars, SDSS J1106+1939 and SDSS J1512+1119. For SDSS J1106+1939, a broad absorption line (BAL) outflow is detected with a kinetic luminosity of at least 10^46 erg/s, which is 5% of the quasar's bolometric luminosity. This outflow has a velocity of ~8000 km/s and is located ~300 pc from the quasar. For SDSS J1512+1119, two separate outflows are detected using the same technique, with distances ranging from 100-2000 pc from the central source. The distances of the outflows
Forming intracluster gas in a galaxy protocluster at a redshift of 2.16Sérgio Sacani
Galaxy clusters are the most massive gravitationally bound structures in the Universe, comprising thousands of galaxies and
pervaded by a diffuse, hot “intracluster medium” (ICM) that dominates the baryonic content of these systems. The formation
and evolution of the ICM across cosmic time1
is thought to be driven by the continuous accretion of matter from the large-scale
filamentary surroundings and dramatic merger events with other clusters or groups. Until now, however, direct observations of
the intracluster gas have been limited only to mature clusters in the latter three-quarters of the history of the Universe, and we
have been lacking a direct view of the hot, thermalized cluster atmosphere at the epoch when the first massive clusters formed.
Here we report the detection (about 6σ) of the thermal Sunyaev-Zeldovich (SZ) effect2
in the direction of a protocluster. In fact,
the SZ signal reveals the ICM thermal energy in a way that is insensitive to cosmological dimming, making it ideal for tracing
the thermal history of cosmic structures3
. This result indicates the presence of a nascent ICM within the Spiderweb protocluster
at redshift z = 2.156, around 10 billion years ago. The amplitude and morphology of the detected signal show that the SZ
effect from the protocluster is lower than expected from dynamical considerations and comparable with that of lower-redshift
group-scale systems, consistent with expectations for a dynamically active progenitor of a local galaxy cluster.
1) Using four laser beams, researchers generated a three-dimensional optical lattice that traps 490nm polystyrene spheres in solution, forming a face-centered orthorhombic crystal structure.
2) The four-beam setup produces a stable periodic potential in all three dimensions that counteracts particle diffusion via radiation pressure balance.
3) Calculations show the four-beam lattice with all beams polarized parallel produces a simple intensity pattern that yields a face-centered orthorhombic crystal structure when the beam angle is 45 degrees.
1) Photonic nanojets can influence the trapping behavior of multiple microspheres that are axially trapped in a focused laser beam. 2) Simulations show that when two microspheres approach each other axially in the beam, they are initially pushed apart by scattering forces but can become drawn together by a connecting photonic nanojet that forms between them. 3) Three microspheres may also become tethered when specific refractive index conditions are met between each neighboring pair.
This document provides an overview of ray optics and matrix optics. It discusses how ray optics can be used to describe the behavior of light when its wavelength is much smaller than the dimensions of objects it interacts with. Ray optics models light as rays that travel in straight lines and bend at interfaces according to Snell's law. Matrix optics uses 2x2 matrices to relate the position and angle of rays entering and exiting optical systems, allowing complex systems to be analyzed. Key concepts covered include image formation by lenses, paraxial approximation, and the ray transfer matrix.
Fusion of Multispectral And Full Polarimetric SAR Images In NSST DomainCSCJournals
Polarimetric SAR (POLSAR) and multispectral images provide different characteristics of the imaged objects. Multispectral provides information about surface material while POLSAR provides information about geometrical and physical properties of the objects. Merging both should resolve many of object recognition problems that exist when they are used separately. Through this paper, we propose a new scheme for image fusion of full polarization radar image (POLSAR) with multispectral optical satellite image (Egyptsat). The proposed scheme is based on Non-Subsampled Shearlet Transform (NSST) and multi-channel Pulse Coupled Neural Network (m-PCNN). We use NSST to decompose images into low frequency and band-pass sub- band coefficients. With respect to low frequency coefficients, a fusion rule is proposed based on local energy and dispersion index. In respect of sub-band coefficients, m-PCNN is used to guide how the fused sub-band coefficients are calculated using image textural information.
The proposed method is applied on three batches of Egyptsat (Red-Green-infra-red) and radarsat2 (C-band full-polarimetric HH-HV and VV-polarization) images. The batches are selected to react differently with different polarization. Visual assessment of the obtained fused image gives excellent information on clarity and delineation of different objects. Quantitative evaluations show the proposed method can superior the other data fusion methods.
Experimental Study of Third Order Nonlinear Absorption in Pure and MG Doped L...ijtsrd
With the discovery of laser in 1960 various nonlinear effect arises, the origin of nonlinear optics lies in the nonlinear response of materials to the incident coherent radiation. Using pulsed Q switched Nd YAG Laser, we have observed the important phenomena i.e. third order nonlinear effect in the LiNbO3 crystals. In present work, I have performed an experiment using simple and sensitive single beam Z scan technique to measure nonlinear absorption in LiNbO3 crystals samples such as pure LiNbO3 crystal, 5mol Mg doped LiNbO3 crystal, and 7mol Mg doped LiNbO3 crystal. Vijay Aithekar | Dr. Vishal Saxana "Experimental Study of Third Order Nonlinear Absorption in Pure and MG Doped Lithium Niobate Crystals" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-3 | Issue-6 , October 2019, URL: https://www.ijtsrd.com/papers/ijtsrd26370.pdf Paper URL: https://www.ijtsrd.com/physics/engineering-physics/26370/experimental-study-of-third-order-nonlinear-absorption-in-pure-and-mg-doped-lithium-niobate-crystals/vijay-aithekar
Mapping spiral structure on the far side of the Milky WaySérgio Sacani
Little is known about the portion of the Milky Way lying beyond the Galactic center at distances
of more than 9 kiloparsec from the Sun. These regions are opaque at optical wavelengths
because of absorption by interstellar dust, and distances are very large and hard to measure.
We report a direct trigonometric parallax distance of 20:4þ2:8
2:2 kiloparsec obtained with the Very
Long Baseline Array to a water maser source in a region of active star formation. These
measurements allow us to shed light on Galactic spiral structure by locating the ScutumCentaurus
spiral arm as it passes through the far side of the Milky Way and to validate a
kinematic method for determining distances in this region on the basis of transverse motions.
The document summarizes research conducted on generating and characterizing optical vortices, as well as sorting their orbital angular momentum states. Optical vortices, also known as twisted light beams, have a helical wavefront and carry orbital angular momentum. The research involved using computer-generated holography to create optical vortices with different topological charges, which was verified using a Mach-Zehnder interferometer. Additionally, a Sagnac interferometer was used to separate the even and odd orbital angular momentum states of Laguerre-Gaussian beams.
Detection of an_unindentified_emission_line_in_the_stacked_x_ray_spectrum_of_...Sérgio Sacani
1. Researchers detected a previously unknown emission line in the stacked X-ray spectrum of 73 galaxy clusters observed by XMM-Newton. 2. The line was detected at an energy of 3.55-3.57 keV and was seen independently in subsamples of clusters. 3. The line was also detected in Chandra observations of the Perseus cluster but not in observations of the Virgo cluster. 4. The nature of this line is unclear - it could be a thermal line from an undetected element, or potentially the decay line of a hypothesized dark matter particle called a sterile neutrino. Further observations are needed to determine the origin of the line.
Probing the jet_base_of_blazar_pks1830211_from_the_chromatic_variability_of_i...Sérgio Sacani
This document summarizes ALMA observations of the blazar PKS 1830-211 taken over multiple epochs in 2012. The blazar is lensed by a foreground galaxy, producing two resolved images (NE and SW) separated by 1". The observations were taken at frequencies corresponding to 350-1050 GHz in the blazar rest frame. Analysis of the flux ratio between the two images over time and frequency revealed a remarkable frequency-dependent behavior, implying a "chromatic structure" in the blazar jet. This is interpreted as evidence for a "core-shift effect" caused by plasmon ejection very near the base of the jet. The observations provide a unique probe of activity in the region where plasma acceleration occurs in blazar
X-ray diffraction is a technique used to characterize nanomaterials by analyzing the diffraction patterns produced when X-rays interact with the crystal structure of a material. The document discusses the history, principles, instrumentation, and applications of XRD. It describes how XRD can be used to determine properties like crystallite size, dislocation density, strain, and identify crystalline phases by comparing to known standards. XRD provides a non-destructive way to analyze crystal structures with high accuracy and is suitable for both powder and thin film samples.
Gravitational lensing characteristics of the transparent sunSérgio Sacani
The document models the Sun as a transparent gravitational lens and calculates its lensing characteristics using data from the 2005 Standard Solar Model. It finds that:
1) The Sun's minimum focal length for producing multiple images of distant sources is 23.5 ± 0.1 AU, just beyond the orbit of Uranus.
2) Regions exist where the Sun can produce three images of a distant source along with their associated magnifications.
3) Extremely high magnifications are possible for observers situated such that an unlensed source appears near a three-image caustic.
Expert system of single magnetic lens using JESS in Focused Ion Beamijcsa
This work shows expert system of symmetrical single magnetic lens used in focused ion beam optical system. Java expert system shell(JESS) programming is proposed to build the intelligent agent "MOPTION"for getting an optimum magnetic flux density , and calculate the ion optical trajectory. The combination of such rule based engine and SIMION 8.1 has configured the reconstruction process and compiled the data retrieved by the proposed expert system agent to implement the pole-pieces reconstruction for lens design. The pole pieces reconstruction has been resulted in 3D graph , and under the infinite magnification conditions of the optical path, aberration (spherical / chromatic and total) disks diameters have been obtained and got the values (0.03,0.13 and 0.133) micron (μm) respectively.
Fusion Based Gaussian noise Removal in the Images using Curvelets and Wavelet...CSCJournals
This document presents a fusion-based method for removing Gaussian noise from images using curvelets and wavelets with a Gaussian filter. The proposed method aims to address artifacts that appear when using curvelets alone. It first applies Gaussian filtering, wavelet denoising, and curvelet denoising separately. It then fuses the results of these three approaches to obtain a better denoised image with fewer artifacts. The method is tested on various standard test images and medical images corrupted with white Gaussian noise. Results are evaluated using peak signal-to-noise ratio and weighted peak signal-to-noise ratio, which accounts for human visual sensitivity.
NANO106 is UCSD Department of NanoEngineering's core course on crystallography of materials taught by Prof Shyue Ping Ong. For more information, visit the course wiki at http://nano106.wikispaces.com.
Analytical approach of ordinary frozen wavesAmélia Moreira
This document describes analytical work on frozen waves (FWs) and their potential application in optical trapping and micromanipulation. FWs are constructed from a superposition of Bessel beams with different longitudinal wave numbers that can generate a predetermined longitudinal intensity profile. The generalized Lorenz-Mie theory and integral localized approximation are used to derive expressions for the beam-shape coefficients that describe FWs and allow calculation of their optical properties. Examples of FW intensity profiles are shown to agree with previous works. Optical forces exerted by FWs on spherical particles are also calculated.
The build up_of_the_c_d_halo_of_m87_evidence_for_accretion_in_the_last_gyrSérgio Sacani
Observações recentes obtidas com o Very Large Telescope do ESO mostraram que Messier 87, a galáxia elíptica gigante mais próximo de nós, engoliu uma galáxia inteira de tamanho médio no último bilhão de anos. Uma equipe de astrônomos conseguiu pela primeira vez seguir o movimento de 300 nebulosas planetárias brilhantes, encontrando evidências claras deste evento e encontrando também excesso de radiação emitida pelos restos da vítima completamente desfeita.
Investigation of the bandpass properties of the local impedance of slow wave ...Victor Solntsev
The properties of the local coupling impedance that determines the efficiency of the electron–wave interaction in periodic slow-wave structures are investigated. This impedance is determined (i) through the char- acteristics of the electromagnetic field in a slow-wave structure and (ii) through the parameters of a two-port chain simulating the structure. The continuous behavior of the local coupling impedance in the passbands of slow-wave structures, at the boundaries of the passbands, and beyond the passbands is confirmed with the help of a waveguide–resonator model.
Plasmon-Polaritons And Their Use In Optical Sub-Wavelength. Event Of Copper A...ijrap
The work undertaken in this article concerns the description of the propagation modes of an incident
electromagnetic wave of wavelength λ (the visible spectrum) to its interaction with a structure typical metal
/ dielectric. The study of this interaction process is the measurement of features that are four parameters
associated with longitudinal modes propagating interface. A comparative study between two structures
silver and copper has been established. The characteristic parameters whose behavior is studied in the
visible spectrum are the propagation length, and the length of penetration in rural and dielectric material.
The typical structure of Kretschmann-Raether being used for the diagnosis of structure, analytical study
shows that copper can be used as a guide for photonic transmission. The direction of propagation, the
electromagnetic field associated with the interface modes present evanescent spatial coherence with which
the behavior is justified by a study of the near field. For this, we have given some results on the density of
states of plasmonic modes on a copper-air interface.
Plasmon-Polaritons And Their Use In Optical Sub-Wavelength. Event Of Copper A...ijrap
The work undertaken in this article concerns the description of the propagation modes of an incident
electromagnetic wave of wavelength λ (the visible spectrum) to its interaction with a structure typical metal
/ dielectric. The study of this interaction process is the measurement of features that are four parameters
associated with longitudinal modes propagating interface. A comparative study between two structures
silver and copper has been established. The characteristic parameters whose behavior is studied in the
visible spectrum are the propagation length, and the length of penetration in rural and dielectric material.
The typical structure of Kretschmann-Raether being used for the diagnosis of structure, analytical study
shows that copper can be used as a guide for photonic transmission. The direction of propagation, the
electromagnetic field associated with the interface modes present evanescent spatial coherence with which
the behavior is justified by a study of the near field. For this, we have given some results on the density of
states of plasmonic modes on a copper-air interface
Plasmon-Polaritons And Their Use In Optical Sub-Wavelength. Event Of Copper A...ijrap
The work undertaken in this article concerns the description of the propagation modes of an incident
electromagnetic wave of wavelength λ (the visible spectrum) to its interaction with a structure typical metal
/ dielectric. The study of this interaction process is the measurement of features that are four parameters
associated with longitudinal modes propagating interface. A comparative study between two structures
silver and copper has been established. The characteristic parameters whose behavior is studied in the
visible spectrum are the propagation length, and the length of penetration in rural and dielectric material.
The typical structure of Kretschmann-Raether being used for the diagnosis of structure, analytical study
shows that copper can be used as a guide for photonic transmission. The direction of propagation, the
electromagnetic field associated with the interface modes present evanescent spatial coherence with which
the behavior is justified by a study of the near field. For this, we have given some results on the density of
states of plasmonic modes on a copper-air interface.
An unindetified line_in_xray_spectra_of_the_adromeda_galaxy_and_perseus_galax...Sérgio Sacani
This document summarizes an analysis of X-ray spectra from the Andromeda galaxy and Perseus galaxy cluster observed with the XMM-Newton X-ray observatory. The analysis identified a weak unidentified line at an energy of approximately 3.5 keV in the spectra of both objects. The line strength increases towards the centers of the objects and is stronger in Perseus than in Andromeda. The line properties are consistent with originating from the decay of dark matter particles, though an instrumental or astrophysical source cannot be ruled out based on individual objects. Future detections or non-detections in additional targets could help reveal the nature of this line.
Electron Diffraction Using Transmission Electron MicroscopyLe Scienze Web News
Electron diffraction via the transmission electron microscope is a powerful method for characterizing the structure of materials, including perfect crystals and defect structures. The advantages of elec- tron diffraction over other methods, e.g., x-ray or neutron, arise from the extremely short wavelength (≈2 pm), the strong atomic scattering, and the ability to exam- ine tiny volumes of matter (≈10 nm3). The NIST Materials Science and Engineer- ing Laboratory has a history of discovery and characterization of new structures through electron diffraction, alone or in combination with other diffraction methods. This paper provides a survey of some of this work enabled through electron mi- croscopy.
High resolution alma_observations_of_sdp81_the_innermost_mass_profile_of_the_...Sérgio Sacani
A Campanha de Linha de Base Longa do ALMA produziu uma imagem muito detalhada de uma galáxia distante afetada por lente gravitacional. A imagem mostra uma vista ampliada das regiões de formação estelar na galáxia, com um nível de detalhe nunca antes alcançado numa galáxia tão remota. As novas observações são muito mais detalhadas do que as obtidas pelo Telescópio Espacial Hubble da NASA/ESA e revelam regiões de formação estelar na galáxia equivalentes a versões gigantes da Nebulosa de Orion.
A Campanha de Linha de Base Longa do ALMA produziu algumas observações extraordinárias e coletou informação com um detalhe sem precedentes dos habitantes do Universo próximo e longínquo. Foram feitas observações no final de 2014 no âmbito de uma campanha que pretendeu estudar uma galáxia distante chamada HATLAS J090311.6+003906, também conhecida pelo nome mais simples de SDP.81. A radiação emitida por esta galáxia é “vítima” de um efeito cósmico chamado lente gravitacional. Uma galáxia enorme que se situa entre SDP.81 e o ALMA [1] atua como lente gravitacional, distorcendo a radiação emitida pela galáxia mais distante e criando um exemplo quase perfeito do fenômeno conhecido por Anel de Einstein [2].
Pelo menos sete grupos de cientistas [3] analisaram de forma independente os dados do ALMA sobre SDP.81. Esta profusão de artigos científicos deu-nos informação sem precedentes sobre esta galáxia, revelando detalhes sobre a sua estrutura, conteúdo, movimento e outras características físicas.
O ALMA funciona como um interferômetro, isto é, a rede múltipla de antenas trabalha em sintonia perfeita coletando radiação como se de um único e enorme telescópio virtual se tratasse [4]. Como resultado, estas novas imagens de SDP.81 possuem uma resolução até 6 vezes melhor [5] que as imagens obtidas no infravermelho com o Telescópio Espacial Hubble da NASA/ESA.
This document discusses the propagation and transformation of laser beams, specifically Gaussian beams. It defines key parameters that characterize Gaussian beams, including:
- Beam width, which describes the transverse extent of the beam and evolves as the beam propagates.
- Divergence, which is inversely related to the beam width and describes how the beam spreads as it propagates.
- Radius of curvature, which describes the curvature of the wavefront and depends on the beam width and propagation distance.
- Rayleigh range, which is a characteristic length related to the beam waist width and wavelength, and describes the region near the beam waist where the beam remains relatively collimated.
It explains how these parameters evolve as the
SDSS1335+0728: The awakening of a ∼ 106M⊙ black hole⋆Sérgio Sacani
Context. The early-type galaxy SDSS J133519.91+072807.4 (hereafter SDSS1335+0728), which had exhibited no prior optical variations during the preceding two decades, began showing significant nuclear variability in the Zwicky Transient Facility (ZTF) alert stream from December 2019 (as ZTF19acnskyy). This variability behaviour, coupled with the host-galaxy properties, suggests that SDSS1335+0728 hosts a ∼ 106M⊙ black hole (BH) that is currently in the process of ‘turning on’. Aims. We present a multi-wavelength photometric analysis and spectroscopic follow-up performed with the aim of better understanding the origin of the nuclear variations detected in SDSS1335+0728. Methods. We used archival photometry (from WISE, 2MASS, SDSS, GALEX, eROSITA) and spectroscopic data (from SDSS and LAMOST) to study the state of SDSS1335+0728 prior to December 2019, and new observations from Swift, SOAR/Goodman, VLT/X-shooter, and Keck/LRIS taken after its turn-on to characterise its current state. We analysed the variability of SDSS1335+0728 in the X-ray/UV/optical/mid-infrared range, modelled its spectral energy distribution prior to and after December 2019, and studied the evolution of its UV/optical spectra. Results. From our multi-wavelength photometric analysis, we find that: (a) since 2021, the UV flux (from Swift/UVOT observations) is four times brighter than the flux reported by GALEX in 2004; (b) since June 2022, the mid-infrared flux has risen more than two times, and the W1−W2 WISE colour has become redder; and (c) since February 2024, the source has begun showing X-ray emission. From our spectroscopic follow-up, we see that (i) the narrow emission line ratios are now consistent with a more energetic ionising continuum; (ii) broad emission lines are not detected; and (iii) the [OIII] line increased its flux ∼ 3.6 years after the first ZTF alert, which implies a relatively compact narrow-line-emitting region. Conclusions. We conclude that the variations observed in SDSS1335+0728 could be either explained by a ∼ 106M⊙ AGN that is just turning on or by an exotic tidal disruption event (TDE). If the former is true, SDSS1335+0728 is one of the strongest cases of an AGNobserved in the process of activating. If the latter were found to be the case, it would correspond to the longest and faintest TDE ever observed (or another class of still unknown nuclear transient). Future observations of SDSS1335+0728 are crucial to further understand its behaviour. Key words. galaxies: active– accretion, accretion discs– galaxies: individual: SDSS J133519.91+072807.4
Discovery of An Apparent Red, High-Velocity Type Ia Supernova at 𝐳 = 2.9 wi...Sérgio Sacani
We present the JWST discovery of SN 2023adsy, a transient object located in a host galaxy JADES-GS
+
53.13485
−
27.82088
with a host spectroscopic redshift of
2.903
±
0.007
. The transient was identified in deep James Webb Space Telescope (JWST)/NIRCam imaging from the JWST Advanced Deep Extragalactic Survey (JADES) program. Photometric and spectroscopic followup with NIRCam and NIRSpec, respectively, confirm the redshift and yield UV-NIR light-curve, NIR color, and spectroscopic information all consistent with a Type Ia classification. Despite its classification as a likely SN Ia, SN 2023adsy is both fairly red (
�
(
�
−
�
)
∼
0.9
) despite a host galaxy with low-extinction and has a high Ca II velocity (
19
,
000
±
2
,
000
km/s) compared to the general population of SNe Ia. While these characteristics are consistent with some Ca-rich SNe Ia, particularly SN 2016hnk, SN 2023adsy is intrinsically brighter than the low-
�
Ca-rich population. Although such an object is too red for any low-
�
cosmological sample, we apply a fiducial standardization approach to SN 2023adsy and find that the SN 2023adsy luminosity distance measurement is in excellent agreement (
≲
1
�
) with
Λ
CDM. Therefore unlike low-
�
Ca-rich SNe Ia, SN 2023adsy is standardizable and gives no indication that SN Ia standardized luminosities change significantly with redshift. A larger sample of distant SNe Ia is required to determine if SN Ia population characteristics at high-
�
truly diverge from their low-
�
counterparts, and to confirm that standardized luminosities nevertheless remain constant with redshift.
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Mapping spiral structure on the far side of the Milky WaySérgio Sacani
Little is known about the portion of the Milky Way lying beyond the Galactic center at distances
of more than 9 kiloparsec from the Sun. These regions are opaque at optical wavelengths
because of absorption by interstellar dust, and distances are very large and hard to measure.
We report a direct trigonometric parallax distance of 20:4þ2:8
2:2 kiloparsec obtained with the Very
Long Baseline Array to a water maser source in a region of active star formation. These
measurements allow us to shed light on Galactic spiral structure by locating the ScutumCentaurus
spiral arm as it passes through the far side of the Milky Way and to validate a
kinematic method for determining distances in this region on the basis of transverse motions.
The document summarizes research conducted on generating and characterizing optical vortices, as well as sorting their orbital angular momentum states. Optical vortices, also known as twisted light beams, have a helical wavefront and carry orbital angular momentum. The research involved using computer-generated holography to create optical vortices with different topological charges, which was verified using a Mach-Zehnder interferometer. Additionally, a Sagnac interferometer was used to separate the even and odd orbital angular momentum states of Laguerre-Gaussian beams.
Detection of an_unindentified_emission_line_in_the_stacked_x_ray_spectrum_of_...Sérgio Sacani
1. Researchers detected a previously unknown emission line in the stacked X-ray spectrum of 73 galaxy clusters observed by XMM-Newton. 2. The line was detected at an energy of 3.55-3.57 keV and was seen independently in subsamples of clusters. 3. The line was also detected in Chandra observations of the Perseus cluster but not in observations of the Virgo cluster. 4. The nature of this line is unclear - it could be a thermal line from an undetected element, or potentially the decay line of a hypothesized dark matter particle called a sterile neutrino. Further observations are needed to determine the origin of the line.
Probing the jet_base_of_blazar_pks1830211_from_the_chromatic_variability_of_i...Sérgio Sacani
This document summarizes ALMA observations of the blazar PKS 1830-211 taken over multiple epochs in 2012. The blazar is lensed by a foreground galaxy, producing two resolved images (NE and SW) separated by 1". The observations were taken at frequencies corresponding to 350-1050 GHz in the blazar rest frame. Analysis of the flux ratio between the two images over time and frequency revealed a remarkable frequency-dependent behavior, implying a "chromatic structure" in the blazar jet. This is interpreted as evidence for a "core-shift effect" caused by plasmon ejection very near the base of the jet. The observations provide a unique probe of activity in the region where plasma acceleration occurs in blazar
X-ray diffraction is a technique used to characterize nanomaterials by analyzing the diffraction patterns produced when X-rays interact with the crystal structure of a material. The document discusses the history, principles, instrumentation, and applications of XRD. It describes how XRD can be used to determine properties like crystallite size, dislocation density, strain, and identify crystalline phases by comparing to known standards. XRD provides a non-destructive way to analyze crystal structures with high accuracy and is suitable for both powder and thin film samples.
Gravitational lensing characteristics of the transparent sunSérgio Sacani
The document models the Sun as a transparent gravitational lens and calculates its lensing characteristics using data from the 2005 Standard Solar Model. It finds that:
1) The Sun's minimum focal length for producing multiple images of distant sources is 23.5 ± 0.1 AU, just beyond the orbit of Uranus.
2) Regions exist where the Sun can produce three images of a distant source along with their associated magnifications.
3) Extremely high magnifications are possible for observers situated such that an unlensed source appears near a three-image caustic.
Expert system of single magnetic lens using JESS in Focused Ion Beamijcsa
This work shows expert system of symmetrical single magnetic lens used in focused ion beam optical system. Java expert system shell(JESS) programming is proposed to build the intelligent agent "MOPTION"for getting an optimum magnetic flux density , and calculate the ion optical trajectory. The combination of such rule based engine and SIMION 8.1 has configured the reconstruction process and compiled the data retrieved by the proposed expert system agent to implement the pole-pieces reconstruction for lens design. The pole pieces reconstruction has been resulted in 3D graph , and under the infinite magnification conditions of the optical path, aberration (spherical / chromatic and total) disks diameters have been obtained and got the values (0.03,0.13 and 0.133) micron (μm) respectively.
Fusion Based Gaussian noise Removal in the Images using Curvelets and Wavelet...CSCJournals
This document presents a fusion-based method for removing Gaussian noise from images using curvelets and wavelets with a Gaussian filter. The proposed method aims to address artifacts that appear when using curvelets alone. It first applies Gaussian filtering, wavelet denoising, and curvelet denoising separately. It then fuses the results of these three approaches to obtain a better denoised image with fewer artifacts. The method is tested on various standard test images and medical images corrupted with white Gaussian noise. Results are evaluated using peak signal-to-noise ratio and weighted peak signal-to-noise ratio, which accounts for human visual sensitivity.
NANO106 is UCSD Department of NanoEngineering's core course on crystallography of materials taught by Prof Shyue Ping Ong. For more information, visit the course wiki at http://nano106.wikispaces.com.
Analytical approach of ordinary frozen wavesAmélia Moreira
This document describes analytical work on frozen waves (FWs) and their potential application in optical trapping and micromanipulation. FWs are constructed from a superposition of Bessel beams with different longitudinal wave numbers that can generate a predetermined longitudinal intensity profile. The generalized Lorenz-Mie theory and integral localized approximation are used to derive expressions for the beam-shape coefficients that describe FWs and allow calculation of their optical properties. Examples of FW intensity profiles are shown to agree with previous works. Optical forces exerted by FWs on spherical particles are also calculated.
The build up_of_the_c_d_halo_of_m87_evidence_for_accretion_in_the_last_gyrSérgio Sacani
Observações recentes obtidas com o Very Large Telescope do ESO mostraram que Messier 87, a galáxia elíptica gigante mais próximo de nós, engoliu uma galáxia inteira de tamanho médio no último bilhão de anos. Uma equipe de astrônomos conseguiu pela primeira vez seguir o movimento de 300 nebulosas planetárias brilhantes, encontrando evidências claras deste evento e encontrando também excesso de radiação emitida pelos restos da vítima completamente desfeita.
Investigation of the bandpass properties of the local impedance of slow wave ...Victor Solntsev
The properties of the local coupling impedance that determines the efficiency of the electron–wave interaction in periodic slow-wave structures are investigated. This impedance is determined (i) through the char- acteristics of the electromagnetic field in a slow-wave structure and (ii) through the parameters of a two-port chain simulating the structure. The continuous behavior of the local coupling impedance in the passbands of slow-wave structures, at the boundaries of the passbands, and beyond the passbands is confirmed with the help of a waveguide–resonator model.
Plasmon-Polaritons And Their Use In Optical Sub-Wavelength. Event Of Copper A...ijrap
The work undertaken in this article concerns the description of the propagation modes of an incident
electromagnetic wave of wavelength λ (the visible spectrum) to its interaction with a structure typical metal
/ dielectric. The study of this interaction process is the measurement of features that are four parameters
associated with longitudinal modes propagating interface. A comparative study between two structures
silver and copper has been established. The characteristic parameters whose behavior is studied in the
visible spectrum are the propagation length, and the length of penetration in rural and dielectric material.
The typical structure of Kretschmann-Raether being used for the diagnosis of structure, analytical study
shows that copper can be used as a guide for photonic transmission. The direction of propagation, the
electromagnetic field associated with the interface modes present evanescent spatial coherence with which
the behavior is justified by a study of the near field. For this, we have given some results on the density of
states of plasmonic modes on a copper-air interface.
Plasmon-Polaritons And Their Use In Optical Sub-Wavelength. Event Of Copper A...ijrap
The work undertaken in this article concerns the description of the propagation modes of an incident
electromagnetic wave of wavelength λ (the visible spectrum) to its interaction with a structure typical metal
/ dielectric. The study of this interaction process is the measurement of features that are four parameters
associated with longitudinal modes propagating interface. A comparative study between two structures
silver and copper has been established. The characteristic parameters whose behavior is studied in the
visible spectrum are the propagation length, and the length of penetration in rural and dielectric material.
The typical structure of Kretschmann-Raether being used for the diagnosis of structure, analytical study
shows that copper can be used as a guide for photonic transmission. The direction of propagation, the
electromagnetic field associated with the interface modes present evanescent spatial coherence with which
the behavior is justified by a study of the near field. For this, we have given some results on the density of
states of plasmonic modes on a copper-air interface
Plasmon-Polaritons And Their Use In Optical Sub-Wavelength. Event Of Copper A...ijrap
The work undertaken in this article concerns the description of the propagation modes of an incident
electromagnetic wave of wavelength λ (the visible spectrum) to its interaction with a structure typical metal
/ dielectric. The study of this interaction process is the measurement of features that are four parameters
associated with longitudinal modes propagating interface. A comparative study between two structures
silver and copper has been established. The characteristic parameters whose behavior is studied in the
visible spectrum are the propagation length, and the length of penetration in rural and dielectric material.
The typical structure of Kretschmann-Raether being used for the diagnosis of structure, analytical study
shows that copper can be used as a guide for photonic transmission. The direction of propagation, the
electromagnetic field associated with the interface modes present evanescent spatial coherence with which
the behavior is justified by a study of the near field. For this, we have given some results on the density of
states of plasmonic modes on a copper-air interface.
An unindetified line_in_xray_spectra_of_the_adromeda_galaxy_and_perseus_galax...Sérgio Sacani
This document summarizes an analysis of X-ray spectra from the Andromeda galaxy and Perseus galaxy cluster observed with the XMM-Newton X-ray observatory. The analysis identified a weak unidentified line at an energy of approximately 3.5 keV in the spectra of both objects. The line strength increases towards the centers of the objects and is stronger in Perseus than in Andromeda. The line properties are consistent with originating from the decay of dark matter particles, though an instrumental or astrophysical source cannot be ruled out based on individual objects. Future detections or non-detections in additional targets could help reveal the nature of this line.
Electron Diffraction Using Transmission Electron MicroscopyLe Scienze Web News
Electron diffraction via the transmission electron microscope is a powerful method for characterizing the structure of materials, including perfect crystals and defect structures. The advantages of elec- tron diffraction over other methods, e.g., x-ray or neutron, arise from the extremely short wavelength (≈2 pm), the strong atomic scattering, and the ability to exam- ine tiny volumes of matter (≈10 nm3). The NIST Materials Science and Engineer- ing Laboratory has a history of discovery and characterization of new structures through electron diffraction, alone or in combination with other diffraction methods. This paper provides a survey of some of this work enabled through electron mi- croscopy.
High resolution alma_observations_of_sdp81_the_innermost_mass_profile_of_the_...Sérgio Sacani
A Campanha de Linha de Base Longa do ALMA produziu uma imagem muito detalhada de uma galáxia distante afetada por lente gravitacional. A imagem mostra uma vista ampliada das regiões de formação estelar na galáxia, com um nível de detalhe nunca antes alcançado numa galáxia tão remota. As novas observações são muito mais detalhadas do que as obtidas pelo Telescópio Espacial Hubble da NASA/ESA e revelam regiões de formação estelar na galáxia equivalentes a versões gigantes da Nebulosa de Orion.
A Campanha de Linha de Base Longa do ALMA produziu algumas observações extraordinárias e coletou informação com um detalhe sem precedentes dos habitantes do Universo próximo e longínquo. Foram feitas observações no final de 2014 no âmbito de uma campanha que pretendeu estudar uma galáxia distante chamada HATLAS J090311.6+003906, também conhecida pelo nome mais simples de SDP.81. A radiação emitida por esta galáxia é “vítima” de um efeito cósmico chamado lente gravitacional. Uma galáxia enorme que se situa entre SDP.81 e o ALMA [1] atua como lente gravitacional, distorcendo a radiação emitida pela galáxia mais distante e criando um exemplo quase perfeito do fenômeno conhecido por Anel de Einstein [2].
Pelo menos sete grupos de cientistas [3] analisaram de forma independente os dados do ALMA sobre SDP.81. Esta profusão de artigos científicos deu-nos informação sem precedentes sobre esta galáxia, revelando detalhes sobre a sua estrutura, conteúdo, movimento e outras características físicas.
O ALMA funciona como um interferômetro, isto é, a rede múltipla de antenas trabalha em sintonia perfeita coletando radiação como se de um único e enorme telescópio virtual se tratasse [4]. Como resultado, estas novas imagens de SDP.81 possuem uma resolução até 6 vezes melhor [5] que as imagens obtidas no infravermelho com o Telescópio Espacial Hubble da NASA/ESA.
This document discusses the propagation and transformation of laser beams, specifically Gaussian beams. It defines key parameters that characterize Gaussian beams, including:
- Beam width, which describes the transverse extent of the beam and evolves as the beam propagates.
- Divergence, which is inversely related to the beam width and describes how the beam spreads as it propagates.
- Radius of curvature, which describes the curvature of the wavefront and depends on the beam width and propagation distance.
- Rayleigh range, which is a characteristic length related to the beam waist width and wavelength, and describes the region near the beam waist where the beam remains relatively collimated.
It explains how these parameters evolve as the
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SDSS1335+0728: The awakening of a ∼ 106M⊙ black hole⋆Sérgio Sacani
Context. The early-type galaxy SDSS J133519.91+072807.4 (hereafter SDSS1335+0728), which had exhibited no prior optical variations during the preceding two decades, began showing significant nuclear variability in the Zwicky Transient Facility (ZTF) alert stream from December 2019 (as ZTF19acnskyy). This variability behaviour, coupled with the host-galaxy properties, suggests that SDSS1335+0728 hosts a ∼ 106M⊙ black hole (BH) that is currently in the process of ‘turning on’. Aims. We present a multi-wavelength photometric analysis and spectroscopic follow-up performed with the aim of better understanding the origin of the nuclear variations detected in SDSS1335+0728. Methods. We used archival photometry (from WISE, 2MASS, SDSS, GALEX, eROSITA) and spectroscopic data (from SDSS and LAMOST) to study the state of SDSS1335+0728 prior to December 2019, and new observations from Swift, SOAR/Goodman, VLT/X-shooter, and Keck/LRIS taken after its turn-on to characterise its current state. We analysed the variability of SDSS1335+0728 in the X-ray/UV/optical/mid-infrared range, modelled its spectral energy distribution prior to and after December 2019, and studied the evolution of its UV/optical spectra. Results. From our multi-wavelength photometric analysis, we find that: (a) since 2021, the UV flux (from Swift/UVOT observations) is four times brighter than the flux reported by GALEX in 2004; (b) since June 2022, the mid-infrared flux has risen more than two times, and the W1−W2 WISE colour has become redder; and (c) since February 2024, the source has begun showing X-ray emission. From our spectroscopic follow-up, we see that (i) the narrow emission line ratios are now consistent with a more energetic ionising continuum; (ii) broad emission lines are not detected; and (iii) the [OIII] line increased its flux ∼ 3.6 years after the first ZTF alert, which implies a relatively compact narrow-line-emitting region. Conclusions. We conclude that the variations observed in SDSS1335+0728 could be either explained by a ∼ 106M⊙ AGN that is just turning on or by an exotic tidal disruption event (TDE). If the former is true, SDSS1335+0728 is one of the strongest cases of an AGNobserved in the process of activating. If the latter were found to be the case, it would correspond to the longest and faintest TDE ever observed (or another class of still unknown nuclear transient). Future observations of SDSS1335+0728 are crucial to further understand its behaviour. Key words. galaxies: active– accretion, accretion discs– galaxies: individual: SDSS J133519.91+072807.4
Discovery of An Apparent Red, High-Velocity Type Ia Supernova at 𝐳 = 2.9 wi...Sérgio Sacani
We present the JWST discovery of SN 2023adsy, a transient object located in a host galaxy JADES-GS
+
53.13485
−
27.82088
with a host spectroscopic redshift of
2.903
±
0.007
. The transient was identified in deep James Webb Space Telescope (JWST)/NIRCam imaging from the JWST Advanced Deep Extragalactic Survey (JADES) program. Photometric and spectroscopic followup with NIRCam and NIRSpec, respectively, confirm the redshift and yield UV-NIR light-curve, NIR color, and spectroscopic information all consistent with a Type Ia classification. Despite its classification as a likely SN Ia, SN 2023adsy is both fairly red (
�
(
�
−
�
)
∼
0.9
) despite a host galaxy with low-extinction and has a high Ca II velocity (
19
,
000
±
2
,
000
km/s) compared to the general population of SNe Ia. While these characteristics are consistent with some Ca-rich SNe Ia, particularly SN 2016hnk, SN 2023adsy is intrinsically brighter than the low-
�
Ca-rich population. Although such an object is too red for any low-
�
cosmological sample, we apply a fiducial standardization approach to SN 2023adsy and find that the SN 2023adsy luminosity distance measurement is in excellent agreement (
≲
1
�
) with
Λ
CDM. Therefore unlike low-
�
Ca-rich SNe Ia, SN 2023adsy is standardizable and gives no indication that SN Ia standardized luminosities change significantly with redshift. A larger sample of distant SNe Ia is required to determine if SN Ia population characteristics at high-
�
truly diverge from their low-
�
counterparts, and to confirm that standardized luminosities nevertheless remain constant with redshift.
Evidence of Jet Activity from the Secondary Black Hole in the OJ 287 Binary S...Sérgio Sacani
Wereport the study of a huge optical intraday flare on 2021 November 12 at 2 a.m. UT in the blazar OJ287. In the binary black hole model, it is associated with an impact of the secondary black hole on the accretion disk of the primary. Our multifrequency observing campaign was set up to search for such a signature of the impact based on a prediction made 8 yr earlier. The first I-band results of the flare have already been reported by Kishore et al. (2024). Here we combine these data with our monitoring in the R-band. There is a big change in the R–I spectral index by 1.0 ±0.1 between the normal background and the flare, suggesting a new component of radiation. The polarization variation during the rise of the flare suggests the same. The limits on the source size place it most reasonably in the jet of the secondary BH. We then ask why we have not seen this phenomenon before. We show that OJ287 was never before observed with sufficient sensitivity on the night when the flare should have happened according to the binary model. We also study the probability that this flare is just an oversized example of intraday variability using the Krakow data set of intense monitoring between 2015 and 2023. We find that the occurrence of a flare of this size and rapidity is unlikely. In machine-readable Tables 1 and 2, we give the full orbit-linked historical light curve of OJ287 as well as the dense monitoring sample of Krakow.
Candidate young stellar objects in the S-cluster: Kinematic analysis of a sub...Sérgio Sacani
Context. The observation of several L-band emission sources in the S cluster has led to a rich discussion of their nature. However, a definitive answer to the classification of the dusty objects requires an explanation for the detection of compact Doppler-shifted Brγ emission. The ionized hydrogen in combination with the observation of mid-infrared L-band continuum emission suggests that most of these sources are embedded in a dusty envelope. These embedded sources are part of the S-cluster, and their relationship to the S-stars is still under debate. To date, the question of the origin of these two populations has been vague, although all explanations favor migration processes for the individual cluster members. Aims. This work revisits the S-cluster and its dusty members orbiting the supermassive black hole SgrA* on bound Keplerian orbits from a kinematic perspective. The aim is to explore the Keplerian parameters for patterns that might imply a nonrandom distribution of the sample. Additionally, various analytical aspects are considered to address the nature of the dusty sources. Methods. Based on the photometric analysis, we estimated the individual H−K and K−L colors for the source sample and compared the results to known cluster members. The classification revealed a noticeable contrast between the S-stars and the dusty sources. To fit the flux-density distribution, we utilized the radiative transfer code HYPERION and implemented a young stellar object Class I model. We obtained the position angle from the Keplerian fit results; additionally, we analyzed the distribution of the inclinations and the longitudes of the ascending node. Results. The colors of the dusty sources suggest a stellar nature consistent with the spectral energy distribution in the near and midinfrared domains. Furthermore, the evaporation timescales of dusty and gaseous clumps in the vicinity of SgrA* are much shorter ( 2yr) than the epochs covered by the observations (≈15yr). In addition to the strong evidence for the stellar classification of the D-sources, we also find a clear disk-like pattern following the arrangements of S-stars proposed in the literature. Furthermore, we find a global intrinsic inclination for all dusty sources of 60 ± 20◦, implying a common formation process. Conclusions. The pattern of the dusty sources manifested in the distribution of the position angles, inclinations, and longitudes of the ascending node strongly suggests two different scenarios: the main-sequence stars and the dusty stellar S-cluster sources share a common formation history or migrated with a similar formation channel in the vicinity of SgrA*. Alternatively, the gravitational influence of SgrA* in combination with a massive perturber, such as a putative intermediate mass black hole in the IRS 13 cluster, forces the dusty objects and S-stars to follow a particular orbital arrangement. Key words. stars: black holes– stars: formation– Galaxy: center– galaxies: star formation
JAMES WEBB STUDY THE MASSIVE BLACK HOLE SEEDSSérgio Sacani
The pathway(s) to seeding the massive black holes (MBHs) that exist at the heart of galaxies in the present and distant Universe remains an unsolved problem. Here we categorise, describe and quantitatively discuss the formation pathways of both light and heavy seeds. We emphasise that the most recent computational models suggest that rather than a bimodal-like mass spectrum between light and heavy seeds with light at one end and heavy at the other that instead a continuum exists. Light seeds being more ubiquitous and the heavier seeds becoming less and less abundant due the rarer environmental conditions required for their formation. We therefore examine the different mechanisms that give rise to different seed mass spectrums. We show how and why the mechanisms that produce the heaviest seeds are also among the rarest events in the Universe and are hence extremely unlikely to be the seeds for the vast majority of the MBH population. We quantify, within the limits of the current large uncertainties in the seeding processes, the expected number densities of the seed mass spectrum. We argue that light seeds must be at least 103 to 105 times more numerous than heavy seeds to explain the MBH population as a whole. Based on our current understanding of the seed population this makes heavy seeds (Mseed > 103 M⊙) a significantly more likely pathway given that heavy seeds have an abundance pattern than is close to and likely in excess of 10−4 compared to light seeds. Finally, we examine the current state-of-the-art in numerical calculations and recent observations and plot a path forward for near-future advances in both domains.
Anti-Universe And Emergent Gravity and the Dark UniverseSérgio Sacani
Recent theoretical progress indicates that spacetime and gravity emerge together from the entanglement structure of an underlying microscopic theory. These ideas are best understood in Anti-de Sitter space, where they rely on the area law for entanglement entropy. The extension to de Sitter space requires taking into account the entropy and temperature associated with the cosmological horizon. Using insights from string theory, black hole physics and quantum information theory we argue that the positive dark energy leads to a thermal volume law contribution to the entropy that overtakes the area law precisely at the cosmological horizon. Due to the competition between area and volume law entanglement the microscopic de Sitter states do not thermalise at sub-Hubble scales: they exhibit memory effects in the form of an entropy displacement caused by matter. The emergent laws of gravity contain an additional ‘dark’ gravitational force describing the ‘elastic’ response due to the entropy displacement. We derive an estimate of the strength of this extra force in terms of the baryonic mass, Newton’s constant and the Hubble acceleration scale a0 = cH0, and provide evidence for the fact that this additional ‘dark gravity force’ explains the observed phenomena in galaxies and clusters currently attributed to dark matter.
The binding of cosmological structures by massless topological defectsSérgio Sacani
Assuming spherical symmetry and weak field, it is shown that if one solves the Poisson equation or the Einstein field
equations sourced by a topological defect, i.e. a singularity of a very specific form, the result is a localized gravitational
field capable of driving flat rotation (i.e. Keplerian circular orbits at a constant speed for all radii) of test masses on a thin
spherical shell without any underlying mass. Moreover, a large-scale structure which exploits this solution by assembling
concentrically a number of such topological defects can establish a flat stellar or galactic rotation curve, and can also deflect
light in the same manner as an equipotential (isothermal) sphere. Thus, the need for dark matter or modified gravity theory is
mitigated, at least in part.
EWOCS-I: The catalog of X-ray sources in Westerlund 1 from the Extended Weste...Sérgio Sacani
Context. With a mass exceeding several 104 M⊙ and a rich and dense population of massive stars, supermassive young star clusters
represent the most massive star-forming environment that is dominated by the feedback from massive stars and gravitational interactions
among stars.
Aims. In this paper we present the Extended Westerlund 1 and 2 Open Clusters Survey (EWOCS) project, which aims to investigate
the influence of the starburst environment on the formation of stars and planets, and on the evolution of both low and high mass stars.
The primary targets of this project are Westerlund 1 and 2, the closest supermassive star clusters to the Sun.
Methods. The project is based primarily on recent observations conducted with the Chandra and JWST observatories. Specifically,
the Chandra survey of Westerlund 1 consists of 36 new ACIS-I observations, nearly co-pointed, for a total exposure time of 1 Msec.
Additionally, we included 8 archival Chandra/ACIS-S observations. This paper presents the resulting catalog of X-ray sources within
and around Westerlund 1. Sources were detected by combining various existing methods, and photon extraction and source validation
were carried out using the ACIS-Extract software.
Results. The EWOCS X-ray catalog comprises 5963 validated sources out of the 9420 initially provided to ACIS-Extract, reaching a
photon flux threshold of approximately 2 × 10−8 photons cm−2
s
−1
. The X-ray sources exhibit a highly concentrated spatial distribution,
with 1075 sources located within the central 1 arcmin. We have successfully detected X-ray emissions from 126 out of the 166 known
massive stars of the cluster, and we have collected over 71 000 photons from the magnetar CXO J164710.20-455217.
The debris of the ‘last major merger’ is dynamically youngSérgio Sacani
The Milky Way’s (MW) inner stellar halo contains an [Fe/H]-rich component with highly eccentric orbits, often referred to as the
‘last major merger.’ Hypotheses for the origin of this component include Gaia-Sausage/Enceladus (GSE), where the progenitor
collided with the MW proto-disc 8–11 Gyr ago, and the Virgo Radial Merger (VRM), where the progenitor collided with the
MW disc within the last 3 Gyr. These two scenarios make different predictions about observable structure in local phase space,
because the morphology of debris depends on how long it has had to phase mix. The recently identified phase-space folds in Gaia
DR3 have positive caustic velocities, making them fundamentally different than the phase-mixed chevrons found in simulations
at late times. Roughly 20 per cent of the stars in the prograde local stellar halo are associated with the observed caustics. Based
on a simple phase-mixing model, the observed number of caustics are consistent with a merger that occurred 1–2 Gyr ago.
We also compare the observed phase-space distribution to FIRE-2 Latte simulations of GSE-like mergers, using a quantitative
measurement of phase mixing (2D causticality). The observed local phase-space distribution best matches the simulated data
1–2 Gyr after collision, and certainly not later than 3 Gyr. This is further evidence that the progenitor of the ‘last major merger’
did not collide with the MW proto-disc at early times, as is thought for the GSE, but instead collided with the MW disc within
the last few Gyr, consistent with the body of work surrounding the VRM.
Observation of Io’s Resurfacing via Plume Deposition Using Ground-based Adapt...Sérgio Sacani
Since volcanic activity was first discovered on Io from Voyager images in 1979, changes
on Io’s surface have been monitored from both spacecraft and ground-based telescopes.
Here, we present the highest spatial resolution images of Io ever obtained from a groundbased telescope. These images, acquired by the SHARK-VIS instrument on the Large
Binocular Telescope, show evidence of a major resurfacing event on Io’s trailing hemisphere. When compared to the most recent spacecraft images, the SHARK-VIS images
show that a plume deposit from a powerful eruption at Pillan Patera has covered part
of the long-lived Pele plume deposit. Although this type of resurfacing event may be common on Io, few have been detected due to the rarity of spacecraft visits and the previously low spatial resolution available from Earth-based telescopes. The SHARK-VIS instrument ushers in a new era of high resolution imaging of Io’s surface using adaptive
optics at visible wavelengths.
Earliest Galaxies in the JADES Origins Field: Luminosity Function and Cosmic ...Sérgio Sacani
We characterize the earliest galaxy population in the JADES Origins Field (JOF), the deepest
imaging field observed with JWST. We make use of the ancillary Hubble optical images (5 filters
spanning 0.4−0.9µm) and novel JWST images with 14 filters spanning 0.8−5µm, including 7 mediumband filters, and reaching total exposure times of up to 46 hours per filter. We combine all our data
at > 2.3µm to construct an ultradeep image, reaching as deep as ≈ 31.4 AB mag in the stack and
30.3-31.0 AB mag (5σ, r = 0.1” circular aperture) in individual filters. We measure photometric
redshifts and use robust selection criteria to identify a sample of eight galaxy candidates at redshifts
z = 11.5 − 15. These objects show compact half-light radii of R1/2 ∼ 50 − 200pc, stellar masses of
M⋆ ∼ 107−108M⊙, and star-formation rates of SFR ∼ 0.1−1 M⊙ yr−1
. Our search finds no candidates
at 15 < z < 20, placing upper limits at these redshifts. We develop a forward modeling approach to
infer the properties of the evolving luminosity function without binning in redshift or luminosity that
marginalizes over the photometric redshift uncertainty of our candidate galaxies and incorporates the
impact of non-detections. We find a z = 12 luminosity function in good agreement with prior results,
and that the luminosity function normalization and UV luminosity density decline by a factor of ∼ 2.5
from z = 12 to z = 14. We discuss the possible implications of our results in the context of theoretical
models for evolution of the dark matter halo mass function.
THE IMPORTANCE OF MARTIAN ATMOSPHERE SAMPLE RETURN.Sérgio Sacani
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as high-speed, relatively homogeneous, plasma streams from coronal
holes and slow-speed, highly variable, streams whose source regions are
under debate. A key goal of ESA/NASA’s Solar Orbiter mission is to identify
solar wind sources and understand what drives the complexity seen in the
heliosphere. By combining magnetic feld modelling and spectroscopic
techniques with high-resolution observations and measurements, we show
that the solar wind variability detected in situ by Solar Orbiter in March
2022 is driven by spatio-temporal changes in the magnetic connectivity to
multiple sources in the solar atmosphere. The magnetic feld footpoints
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receiving low insolation is still small, limiting our ability to understand the diversity of the atmospheric
composition and climates of temperate terrestrial planets. We report the discovery of an Earth-sized planet
transiting the nearby (12 pc) inactive M3.0 dwarf Gliese 12 (TOI-6251) with an orbital period (Porb) of 12.76 days.
The planet, Gliese 12 b, was initially identified as a candidate with an ambiguous Porb from TESS data. We
confirmed the transit signal and Porb using ground-based photometry with MuSCAT2 and MuSCAT3, and
validated the planetary nature of the signal using high-resolution images from Gemini/NIRI and Keck/NIRC2 as
well as radial velocity (RV) measurements from the InfraRed Doppler instrument on the Subaru 8.2 m telescope
and from CARMENES on the CAHA 3.5 m telescope. X-ray observations with XMM-Newton showed the host
star is inactive, with an X-ray-to-bolometric luminosity ratio of log 5.7 L L X bol » - . Joint analysis of the light
curves and RV measurements revealed that Gliese 12 b has a radius of 0.96 ± 0.05 R⊕,a3σ mass upper limit of
3.9 M⊕, and an equilibrium temperature of 315 ± 6 K assuming zero albedo. The transmission spectroscopy metric
(TSM) value of Gliese 12 b is close to the TSM values of the TRAPPIST-1 planets, adding Gliese 12 b to the small
list of potentially terrestrial, temperate planets amenable to atmospheric characterization with JWST.
Gliese 12 b, a temperate Earth-sized planet at 12 parsecs discovered with TES...Sérgio Sacani
We report on the discovery of Gliese 12 b, the nearest transiting temperate, Earth-sized planet found to date. Gliese 12 is a
bright (V = 12.6 mag, K = 7.8 mag) metal-poor M4V star only 12.162 ± 0.005 pc away from the Solar system with one of the
lowest stellar activity levels known for M-dwarfs. A planet candidate was detected by TESS based on only 3 transits in sectors
42, 43, and 57, with an ambiguity in the orbital period due to observational gaps. We performed follow-up transit observations
with CHEOPS and ground-based photometry with MINERVA-Australis, SPECULOOS, and Purple Mountain Observatory,
as well as further TESS observations in sector 70. We statistically validate Gliese 12 b as a planet with an orbital period of
12.76144 ± 0.00006 d and a radius of 1.0 ± 0.1 R⊕, resulting in an equilibrium temperature of ∼315 K. Gliese 12 b has excellent
future prospects for precise mass measurement, which may inform how planetary internal structure is affected by the stellar
compositional environment. Gliese 12 b also represents one of the best targets to study whether Earth-like planets orbiting cool
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Within the uncertainties of involved astronomical and biological parameters, the Drake Equation
typically predicts that there should be many exoplanets in our galaxy hosting active, communicative
civilizations (ACCs). These optimistic calculations are however not supported by evidence, which is
often referred to as the Fermi Paradox. Here, we elaborate on this long-standing enigma by showing
the importance of planetary tectonic style for biological evolution. We summarize growing evidence
that a prolonged transition from Mesoproterozoic active single lid tectonics (1.6 to 1.0 Ga) to modern
plate tectonics occurred in the Neoproterozoic Era (1.0 to 0.541 Ga), which dramatically accelerated
emergence and evolution of complex species. We further suggest that both continents and oceans
are required for ACCs because early evolution of simple life must happen in water but late evolution
of advanced life capable of creating technology must happen on land. We resolve the Fermi Paradox
(1) by adding two additional terms to the Drake Equation: foc
(the fraction of habitable exoplanets
with significant continents and oceans) and fpt
(the fraction of habitable exoplanets with significant
continents and oceans that have had plate tectonics operating for at least 0.5 Ga); and (2) by
demonstrating that the product of foc
and fpt
is very small (< 0.00003–0.002). We propose that the lack
of evidence for ACCs reflects the scarcity of long-lived plate tectonics and/or continents and oceans on
exoplanets with primitive life.
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adaptations and resilience to warming and cooling during the Cenozoic. All
life will eventually perish in a runaway greenhouse once absorbed solar
radiation exceeds the emission of thermal radiation in several billions of
years. However, conditions rendering the Earth naturally inhospitable to
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plate tectonics (short-term perturbations are not considered here). In
~250 Myr, all continents will converge to form Earth’s next supercontinent,
Pangea Ultima. A natural consequence of the creation and decay of Pangea
Ultima will be extremes in pCO2 due to changes in volcanic rifting and
outgassing. Here we show that increased pCO2, solar energy (F⨀;
approximately +2.5% W m−2 greater than today) and continentality (larger
range in temperatures away from the ocean) lead to increasing warming
hostile to mammalian life. We assess their impact on mammalian
physiological limits (dry bulb, wet bulb and Humidex heat stress indicators)
as well as a planetary habitability index. Given mammals’ continued survival,
predicted background pCO2 levels of 410–816 ppm combined with increased
F⨀ will probably lead to a climate tipping point and their mass extinction.
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and near-circular orbit (e ≈ 0.02) of VFTS 243 suggest that the progenitor star experienced complete
collapse, with energy-momentum being lost predominantly through neutrinos. VFTS 243 enables us to
constrain the natal kick and neutrino-emission asymmetry during black-hole formation. At 68% confidence
level, the natal kick velocity (mass decrement) is ≲10 km=s (≲1.0M⊙), with a full probability distribution
that peaks when ≈0.3M⊙ were ejected, presumably in neutrinos, and the black hole experienced a natal
kick of 4 km=s. The neutrino-emission asymmetry is ≲4%, with best fit values of ∼0–0.2%. Such a small
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exoplanet as a potential technosignature. Silicon-based photovoltaic cells have high reflectance in the
UV-VIS and in the near-IR, within the wavelength range of a space-based flagship mission concept
like the Habitable Worlds Observatory (HWO). Assuming that only solar energy is used to provide
the 2022 human energy needs with a land cover of ∼ 2.4%, and projecting the future energy demand
assuming various growth-rate scenarios, we assess the detectability with an 8 m HWO-like telescope.
Assuming the most favorable viewing orientation, and focusing on the strong absorption edge in the
ultraviolet-to-visible (0.34 − 0.52 µm), we find that several 100s of hours of observation time is needed
to reach a SNR of 5 for an Earth-like planet around a Sun-like star at 10pc, even with a solar panel
coverage of ∼ 23% land coverage of a future Earth. We discuss the necessity of concepts like Kardeshev
Type I/II civilizations and Dyson spheres, which would aim to harness vast amounts of energy. Even
with much larger populations than today, the total energy use of human civilization would be orders of
magnitude below the threshold for causing direct thermal heating or reaching the scale of a Kardashev
Type I civilization. Any extraterrrestrial civilization that likewise achieves sustainable population
levels may also find a limit on its need to expand, which suggests that a galaxy-spanning civilization
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GBSN - Biochemistry (Unit 6) Chemistry of Proteins
Gravitational lensing for interstellar power transmission
1. arXiv:2310.17578v2
[gr-qc]
31
Oct
2023
Gravitational lensing for interstellar power transmission
Slava G. Turyshev
Jet Propulsion Laboratory, California Institute of Technology,
4800 Oak Grove Drive, Pasadena, CA 91109-0899, USA
(Dated: November 1, 2023)
We investigate light propagation in the gravitational field of multiple gravitational lenses. Assum-
ing these lenses are sufficiently spaced to prevent interaction, we consider a linear alignment for the
transmitter, lenses, and receiver. Remarkably, in this axially-symmetric configuration, we can solve
the relevant diffraction integrals – result that offers valuable analytical insights. We show that the
point-spread function (PSF) is affected by the number of lenses in the system. Even a single lens is
useful for transmission either it is used as a part of the transmitter or it acts on the receiver’s side.
We show that power transmission via a pair of lenses benefits from light amplification on both ends
of the link. The second lens plays an important role by focusing the signal to a much tighter spot;
but in practical lensing scenarios, that lens changes the structure of the PSF on scales much smaller
than the telescope, so that additional gain due to the presence of the second lens is independent of its
properties and is govern solely by the transmission geometry. While evaluating the signal-to-noise
ratio (SNR) in various transmitting scenarios, we see that a single-lens transmission performs on par
with a pair of lenses. The fact that the second lens amplifies the brightness of the first one, creates a
challenging background for signal reception. Nevertheless, in all the cases considered here, we have
found practically-relevant SNR values. As a result, we were able to demonstrate the feasibility of
establishing interstellar power transmission links relying on gravitational lensing – a finding with
profound implications for applications targeting interstellar power transmission.
I. INTRODUCTION
Recently, we have explored the optical properties of the solar gravitational lens (SGL) [1, 2] and have shown that
the SGL is characterized by a significant light amplification and angular resolution. As such, the SGL provides unique
capabilities for direct imaging and spectroscopy of faint targets such as exoplanets in our stellar neighborhood [3, 4].
It can be assumed that pairs of stellar gravitational lenses could facilitate energy transmission across interstellar
distances, utilizing equipment of a scale and power akin to that currently employed for interplanetary communications
[5]. There is a prevailing expectation that such a configuration would benefit from the light amplification by both
lenses, thus, enabling significant increases in the signal-to-noise ratio (SNR) of the transmitted signal [6–11]. However,
a comprehensive analysis of these transmission scenarios remains to be undertaken.
Our objective here is to examine light propagation in multi-lens systems and evaluate the associated light ampli-
fication. To do that, we will rely on the wave-theoretical treatment of the gravitational lensing phenomena and will
use analytical tools that we developed in our prior studies of the SGL [2–4, 12–14]. As such, these methods can be
seamlessly adapted to explore power transmission within the multi-lens configurations.
This paper is organized as follows: In Section II, we present the wave-theoretical tools to describe the propagation
of EM waves in a gravitational field. In Section III, we consider various lensing geometries that involve one and two
gravitational lenses, and derive the relevant light amplification factors. In Section IV, we discuss power transmission
with lensing configurations utilizing both a single lens and a pair of lenses. In Section V we evaluate detection
sensitivity in various cases considered and evaluate the relevant SNRs. Our conclusions are presented in Section VI.
To streamline the discussion, we moved some material to Appendices. Appendix A presents an alternative evaluation
of the diffraction integral in the case of a two-lens transmission. Appendix B presents a path integral formulation.
II. EM WAVES IN A GRAVITATIONAL FIELD
As electromagnetic (EM) wave travels through a gravitational field, interaction with gravity causes the wave to
scatter and diffract [2]. In Ref. [12, 13], while studying the Maxwell equations on the background a weak gravity
space-time, we developed a solution to the Mie problem for the diffraction of the EM waves on a large gravitating
body (see [2, 15]) and found the EM field at an image plane located in any of the optical regions behind the lens.
2. 2
FIG. 1: A lens-centric geometry for power transmission via gravitational lensing showing the transmitter, the lens, and the
receiver. Also shown is the distance from the lens to the transmitter plane, z0, and that from the lens to the receiver plane, z.
A. Diffraction of light
We consider a stellar gravitational lens with the Schwarzschild radius of rg = 2GM/c2
with M being lens’ mass. We
use a cylindrical coordinate system centered at the lens (ρ, φ, z) with its z-coordinate oriented along the wavevector
k, a unit vector in the unperturbed direction of the propagation of the incident wave that originated at the source
positioned at (−z0, x′
). We also introduce a light ray’s impact parameter, b, and coordinates of the receiver (z, x) on
the image plane located in the strong interference region at distance z from the lens. These quantities are given as:
k = (0, 0, 1), b = b(cos φξ, sin φξ, 0), x′
= ρ′
(cos φ′
, sin φ′
, 0), x = ρ(cos φ, sin φ, 0). (1)
With these definitions, the EM field on an image plane takes the following form (see details in [12, 13]):
Eρ
Hρ
=
Hφ
−Eφ
= A(x′
, x)e−iωt
cos φ
sin φ
+ O
r2
g, ρ2
/z2
, (2)
with the remaining components being small, i.e., (Ez, Hz) ∝ O(ρ/z).
Assuming the validity of eikonal and the thin lens approximation, the Fresnel-Kirchhoff diffraction formula yields
the following expression for the wave’s amplitude at the observer (receiver) location
A(x′
, x) = E0
k
iz0z
1
2π
ZZ
d2
b A0(b)eikS(x′
,b,x)
, (3)
where S(x′
, x, b) is the effective path length (eikonal) along a path from the source position (−z0, x′
) to the observer
position (z, x) via a point (0, b) on the lens plane (Fig. 1 shows overall geometry of the gravitational lensing system)
S(x′
, b, x) =
q
(b − x′)2 + z2
0 +
p
(b − x)2 + z2 − ψ(b) =
= z0 + z +
(x − x′
)2
2(z0 + z)
+
z0 + z
2z0z
b −
z0
z0 + z
x −
z
z0 + z
x′
2
− ψ(b) + O
b4
z4
0
,
b4
z4
, (4)
where the last term in this expression is the gravitational phase shift, ψ(b), that is acquired by the EM wave as it
propagates along its geodetic path from the source to the image plane on the background of the gravitational field
with potential, U, that has the form (see discussion in [12, 16]):
ψ(b) =
2
c2
Z z
z0
dzU(b, z) = krg ln 4k2
zz0 + 2rg ln kb + O(Jn), (5)
where Jn, n ∈ 2, 3, 4... are the spherical harmonics coefficients representing the mass distribution inside the stellar
lens [12]. Then, the wave amplitude on the observer plane can be written as
A(x′
, x) = A0(x′
, x)F(x′
, x), (6)
where A0(x, x) is the wave amplitude at the receiver (observer) in the absence of the gravitational potential U:
A0(x′
, x) =
E0
z0 + z
eikS0(x′
,x)
, S0(x′
, x) = z0 + z +
(x − x′
)2
2(z0 + z)
, (7)
3. 3
with S0(x′
, x) is the path length along a straight path from x′
to x. In the case of a monopole lens, the amplification
factor F(x, x) is given by the following form of a diffraction integral
F(x′
, x) =
z0 + z
z0z
keiφG
2πi
ZZ
d2
b A0(b)eikS1(x′
,b,x)
,
S1(x′
, b, x) =
z0 + z
2z0z
b −
z0
z0 + z
x +
z
z0
x′
2
− 2rg ln kb, (8)
where the phase factor is given as φG = krg ln 4k2
zz0 and S1(x′
, x) is the Fermat potential along a path from the
source position x′
to the observer position x via a point b on the lens plane. The first term in S1(x′
, x) is the difference
of the geometric time delay between a straight path from the source to the observer and a deflected path. The second
term is due to the time delay in the gravitational potential of the lens object (i.e., the Shapiro time delay).
In the case of an isolated spherically-symmetric gravitational lens positioned on the optical axis, x′
= 0, collecting
all the relevant terms, we present the amplification factor on the receiver (observer) plane
F(x) =
z0 + z
z0z
keiφG
2πi
ZZ
d2
b A0(b) exp
h
ik
z0 + z
2z0z
b −
z0
z0 + z
x
2
− 2rg ln kb
i
. (9)
We consider the case of two non-interacting thin gravitational lenses. We denote zt to be the transmitter’s distance
from the first lens, z12 is the distance between the lenses, and zr is the distance between the second lens and the
receiver. We assume that the lenses are at a very large distance, z12 ≫ zt, zr, from each other; also zt and zr are in
the focal regions of the respective lenses. This allows us to treat the light propagation independently for each lens.
Furthermore, as we are interested to evaluate the largest light amplification, we consider the most favorable trans-
mission geometry: we assume that all participants – the source (or transmitter), the first and the second monopole
lenses, and the observer (or receiver) – are all situated on the same line – the primary optical axis of the lensing
system. Clearly, any deviation from this axially-symmetric geometry will reduce the energy transmitting efficiency.
In this scheme, the light emitted by the source is diffracted by the gravitational field of the first lens, that focuses
and amplifies light which now becomes the source for the second lens. This EM field encounters the second lens, then
the third lens and so on and ultimately it reaches the observer that is positioned in the focal region of the last lens.
B. Transmission geometry
We observe that there is no formal solution for the scenario in which a transmitter is positioned at a large but finite
distance from a lens, emitting a spherical or Gaussian beam in the direction of the lens. Instead, we will employ the
tools developed for the situation in which it is permissible to approximate the incident waves as plane waves. This
approach uses results obtained from the study of diffraction by a monopole gravitational lens, such as the SGL.
It is known that when an EM wave, originating at infinity, travels near a gravitational body, its wavefront experiences
bending. In general relativity, this deflection angle is θgr = 2rg/b. As a result, a massive body acts as a lens by focusing
the EM radiation (i.e., the light rays intersect the optical axis) at the distance, z, that is determined as
b
z
= θgr ⇒ z =
b2
2rg
. (10)
In [17], we have shown that expression for z from (10) is modified when the light rays are coming from a source
located at a finite distance, z0, from the lens. In this case, defining α = b/z0 and using small angle approximation,
the new expression reads
b
z
= θgr − α ⇒ z =
b2
2rg
1
1 − b2/2rgz0
. (11)
Clearly, the transmitter’s distance with respect to the lens, determines four signal transmission regimes, namely
i). For θgr α or when 0 ≤ z0 b2
/2rg there is no transmission. In this case, the light rays either are completely
absorbed by the lens or are not able to focus, e.g., do not reach the optical axis on the other side behind the lens.
ii). For θgr = α or when z0 = b2
/2rg, the focal distance z is infinite, implying that after passing by the lens the light
rays are collimated, never reaching the optical axis. In this case, there will be shadow behind the lens except for
the presence of the bright sport of Arago that is weakening with distance from the lens [18, 19].
4. 4
iii). For θgr α or when z0 b2
/2rg, the light will begin to focus at a finite distance from the lens, forming all the
regions relevant for the diffraction problem including the shadow, interference region, and that of the geometric
optics (see Fig. 4 in [2]). According to (11), these regions will be formed farther from the lens, compared to (10).
iv). For θgr ≫ α or when z0 ≫ b2
/2rg, the point source is effectively at infinity and, depending on the impact
parameter, the light rays will form all the typical diffraction regions, beginning at the distance given by (10).
As a result, for optimal reception, an observer needs to be positioned in the focal region of a lens at the distance
given by (11). Same logic works when a transmission link includes another lens which is then followed by a receiver.
III. LIGHT AMPLIFICATION
We are now in a position to explore the light propagation in the gravitational field and to describe light amplification
in various lensing configurations, including scenarios with one or two linearly-aligned gravitational lenses.
A. One lens transmission
We consider a stellar monopole gravitational lens with mass M1, Schwarzschild radius rg1 = 2GM1/c2
, and physical
radius R1. We assume that transmitter is positioned in the lens’ focal region and placed on the optical axis, which is
defined to be the line connecting the transmitter, the lens, and the receiver. So, that in (6)–(9) we can set x′
= 0.
Note that there are two distinct architectures to form a transmission link that are determined by where transmitter
and receiver are placed with respect to the lens: i). In one case, the transmitter is positioned close to the lens but
at the distance larger than the beginning of the focal region, namely zt ≥ R2
1/2rg1 = 547.8 (R1/R⊙)2
(M⊙/M1) AU.
(Note that in the case of the Sun, the solar corona increases the distance from where practical transmissions may
occur. Thus, for λ ∼ 1 µm, these ranges are beyond ∼ 650 AU from the Sun.) In this case, the receiver, is placed at
an interstellar distance from the lens, so that zr ≫ zt. ii). In another case, the positions are switched, so that the
transmitter is now placed at an interstellar distance from the lens, so that zt ≫ R2
1/2rg1 , but the receiver is at the
lens’ focal region, zr ≥ R2
1/2rg1 . Our formulation below will cover both of these cases.
Considering diffraction on a single lens, with the help of (9) and taking A0(b) = 1, we determine the amplification
factor of the EM wave at the observer’s plane that is positioned a distance zr from the lens as below
F1GL(x) =
keiφG1
iz̃1
1
2π
ZZ
d2
b1 exp
h
ik
1
2z̃1
b1 −
z̃1
zr
x
2
− 2rg1 ln kb1
i
, where z̃1 =
ztzr
zt + zr
, (12)
where φG1 = krg1 ln 4k2
ztzr. After re-arranging the terms and removing the spherical wave [15], we present (12) as
F1GL(x) =
keiφG1
iz̃1
1
2π
Z ∞
0
b1db1 exp
h
ik
b2
1
2z̃1
− 2rg1 ln kb1
i Z 2π
0
dφξ1 exp
h
− i
k
b1ρ
zr
cos[φξ1 − φ]
i
=
=
keiφG1
iz̃1
Z ∞
0
b1db1J0
k
b1ρ
zr
exp
h
ik
b2
1
2z̃1
− 2rg1 ln kb1
i
. (13)
Considering the four scenarios detailed in Sec. II B: Case i): This case is trivial and does not require a formal
treatment. Case ii): When θ = α, the following condition is met: b2
1/2zt − 2rg1 ln kb1 = 0. For this situation, the
amplification factor (13) simplifies to (see derivation details in [19]):
F0
1GL(x) =
1 +
zr
zt
ei φG1+
kR2
1
2zr
J0
k
R1ρ
zr
, (14)
which has the properties of the bright spot of Arago. This is a faint spot of light with brightness that is proportional
to the ratio (zr/zt). Note that stellar atmospheres will make the edges of a spherical lens softer, thus severely affecting
formation of the spot to the point of washing it out. This scenario is not very useful for power transmission.
Next, we examine the scenarios, Case iii) and Case iv), as highlighted in Sec. II B. These cases are similar and can
be characterized in the same manner. For that, we take the last integral in (13) with the method of stationary phase
[2] to determine the impact parameter for which the phase is stationary:
b1 =
p
2rg1 z̃1, (15)
5. 5
yielding the following result for the amplification factor:
F1GL(x) =
p
2πkrg1 eiϕ1
J0
k
p
2rg1 z̃1
zr
ρ
, (16)
where ϕ1 is given as ϕ1 = φG1 + k rg1 − rg1 ln krg1 − rg1 ln 2kz̃1
− 1
4 π.
To evaluate the light amplification of a single lens, we first determine its point-spread function (PSF). For that,
we use the generic solution for the EM field (2) with solution (7) together with (16), and study the Poynting vector,
S = (c/4π) [ReE × ReH] , that describes the energy flux in the image plane [15]. Normalizing this flux to the time-
averaged value that would be observed if the gravitational field of the first lens were absent, |S0| = (c/8π)E2
0 /(zt+zr)2
,
we determine the PSF of a single lens PSF1GL = |S|/|S0| as below:
PSF1GL(x) = 2πkrg1 J2
0
k
p
2rg1 z̃1
zr
ρ
. (17)
Therefore, the largest value for the light amplification factor of a single lens is realized when ρ = 0, yielding [1, 2]
µ0
1GL = 2πkrg1 ≃ 1.17 × 1011
M1
M⊙
1 µm
λ
. (18)
Normalized
PSF
-600 -400 -200 0 200 400 600
0.0
0.2
0.4
0.6
0.8
1.0
Distance from the optical axis, ρ, [m]
FIG. 2: Normalized PSF of a single lens in the trans-
mitting lens scenario (19). Note that the first zero here
is much farther out from the optical axis.
Nominally, amplification (18) is realized for a telescope with
the diameter that is less then the characteristic pattern of the
Bessel function J0(kρ
p
2rg1 z̃1/zr). In practice, if the telescope
diameter d is larger than that of the first zero of the projected
Airy pattern (17), it would average the light amplification over
that aperture (see discussion in [2].) We shall discuss this fact
when we look at different transmission scenarios below.
Coming back to the transmission scenarios iii) and iv) dis-
cussed in Sec. II B, we note that both of these cases are de-
scribed by the results obtained. The difference is just the
meaning of the distances of transmitter and receiver in these
situations. Both of these distances led to the effective distance
z̃1 in (12). In the case of scenario iii), the transmitter is at the
distance zt ≪ zr. In the case iv), these distances are reversed
with the transmitter now at the large distance compared to the
receiver, zt ≪ zr from it.
With solution (17), we may now consider the two cases with
drastically different positions of the transmitter and the re-
ceiver that were discussed at the beginning of this section.
1. Transmitting lens scenario
In the case, when zt ≪ zr, the effective distance z̃1 from (12) reduces to z̃1 = ztzr/(zt + zr) ≃ zt. Therefore, the
PSF of the transmission with a single lens (17) takes the form
PSFt
1GL(x) ≃ 2πkrg1 J2
0
k
p
2rg1 zt
zr
ρ
. (19)
We shall call this case the transmitting lens scenario, thus there will be superscript {}t
on the relevant quantiles.
Note that in this case, an observer in the focal region of the lens, positioned on the optical axis at zr ≫ zt from it,
will see an Einstein ring around the lens with the radius θt
1 given as (see details in [20]) below:
θt
1 =
p
2rg1 zt
zr
≃ 2.46 × 10−9
M1
M⊙
1
2
zt
650 AU
1
2
10 pc
zr
rad. (20)
We observe that the first zero of the projected Airy pattern in (19) occurs at the distance of ρt
1GL = 2.40483 /(kθt
1) ≃
155.82 m (λ/1 µm)(M⊙/M1)
1
2 (650 AU/zt)
1
2 (zr/10 pc) from the optical axis, which is large and aperture averaging
may not be important. Fig. 2 shows the relevant behavior of this PSF. Therefore, while considering the relevant
transmission links, one may have to use the entire PSF from (19) with its maximal value µ0
1GL given by (18).
6. 6
2. Receiving lens scenario
In the case, when zt ≫ zr, the effective distance z̃1 from (12) reduces to z̃1 = ztzr/(zt + zr) ≃ zr. Therefore, the
PSF of a single lens transmission (17) takes the form
PSFr
1GL(x) ≃ 2πkrg1 J2
0
k
r
2rg1
zr
ρ
. (21)
We shall call this case the receiving lens scenario, thus there will be superscript {}r
on the relevant quantiles.
Note that in this case, an observer in the focal region of the lens, positioned on the optical axis at zr ≫ zt from it,
will see an Einstein ring around the lens with the radius θr
1 given as (see details in [20]):
θr
1 =
r
2rg1
zr
≃ 7.80 × 10−6
M1
M⊙
1
2
650 AU
zr
1
2
rad, (22)
which is much larger than that obtained for the transmitting lens scenario (20).
We observe that, in this case the first zero of the projected Airy pattern (21) occurs at ρr
1GL = 2.40483/(kθr
1) ≃
4.91 cm(λ/1 µm)(M⊙/M1)
1
2 (zr/650 AU)
1
2 , which is very small and needs to be aperture-averaged. Fig. 3 demon-
strates the relevant behavior of the PSF (21) at very short spatial scales of a few cm. To estimate the impact of
the large aperture on the light amplification, we average the result (21) over the aperture of the telescope and, using
approximation for the Bessel functions for large arguments [21], we determine:
hµr
1GLi =
1
π(1
2 d)2
Z 1
2 d
0
Z 2π
0
PSF1GL(x) ρdρdφ = 2πkrg1
J2
0 kθ1
1
2 d
+ J2
1 kθ1
1
2 d
≃
≃
4
p
2rg1 zr
dr
= 3.03 × 109
M1
M⊙
1
2
zr
650 AU
1
2
1 m
dr
, (23)
where we recognize the well-known result obtained in [2]. Compared to (18), the aperture-averaging given by (23)
leads to a reduction in the light amplification by a factor of ≃ 38.46, thus resulting only in 2.6% of the maximal value
suggested by (18). Nevertheless, the overall result is still rather impressive.
B. Two lens transmission
Normalized
PSF
-0.4 -0.2 0.0 0.2 0.4
0.0
0.2
0.4
0.6
0.8
1.0
Distance from the optical axis, ρ, [m]
FIG. 3: Normalized PSFs: showing PSF of a single
lens in the receiving lens scenario (21), with M1 =
M⊙, and that for a double-lens transmission (34), with
M2 = M⊙. Clearly, both of these PSFs are identical.
We consider two lenses, positioned on the same line – the
optical axis – and separated by the distance of z12 from each.
The first lens is with the parameters used in Sec. III A, while
lens 2 is given by the mass, M2, Schwarzschild radius rg2 =
2GM2/c2
, and radius R2. To describe power transmission via
a two-lens system, one can develop a scheme similar to one
in Fig. 1. Next, we extended the expression for the optical
path (4) by modeling the contribution of the second lens to the
overall optical path, including that from its gravitational field.
As a result, we treat the EM wave that already passed by
lens 1 to be the source of light incident on lens 2. For an
observer at the lens 1 plane, the angle subtended by the physical
radius of lens 2 is small R2/z12 ≃ θ1, where θ1 is the angle that
determines the diffraction pattern of the incident field (16).
This angle is explicitly evaluated on the lens 2 plane, so that
distance zr in (16) is replaced by z12, yielding the expression
that is relevant for the transmission scenario discussed here
θ1 =
p
2rg1 z̃1
z12
≃ 2.46 × 10−9
M1
M⊙
1
2
zt
650 AU
1
2
10 pc
z12
rad. (24)
Therefore, as R2/z12 ≃ θ1, this lensing geometry corresponds to the strong interference regime of diffraction on lens
1 (see [2] for description). The analytical description of this process was already addressed in Sec. III A.
7. 7
In the case with two-lens transmission, to describe the complex amplitude of an EM wave that is reaching the
lens plane of the lens 2, we use result (16). Specifically, remembering the structure of the EM field (9), we take the
amplitude of the EM at the second lens to be that given by the EM field after it passed lens 1, namely in (9) we
substitute A0(b) → A1GL(b2), which was obtained from (16) by replacing x → b2, we also bring back the x-dependent
term from (7), so that the entire procedure would yield the relevant one-lens amplification factor:
F1GL(b2) =
p
2πkrg1 exp
h
i
kb2
2
2(zt + z12)
i
J0 kθ1b2
. (25)
As a result, the amplitude of the EM field at the observer plane after lens 2 at zr ≪ z12 from it, following the
structure of (6), has the form similar to that of (6) and is given as below (see (B2)–(B4) for complete sturcture)
A2GL(x) = A02(x)F2GL(x), (26)
where the wave amplitude at the observer, A02(x), is given as
A02(x) =
E0
zt + z12 + zr
eikS02
, S02 = zt + z12 + ϕ̂1, (27)
where ϕ̂1 = krg1 ln[2e(zt + z12)/rg1 ] − 1
4 π, where we collected all the constants phase terms, including the relevant
φG1 term in (8) due to lens 1. The amplification factor F2GL(x) in (26) is given as follows
F2GL(x) =
zt + z12 + zr
(zt + z12)zr
k
2πi
ZZ
d2
b2 F1GL(b2) exp
h
ik
b2
2
2(zt + z12)
+
1
2zr
(b2 − x)2
− 2rg2 ln kb2
i
. (28)
See also Appendix B for a path integral derivation of (28). Next, defining the useful notation
1
z̃2
=
1
zt + z12
+
1
zr
⇒ z̃2 =
(zt + z12)zr
zt + z12 + zr
, (29)
we use expression for F1GL(b2) from (25), remove the spherical wave (as in (13)), we present (28) as below
F2GL(x) =
p
2πkrg1
k
iz̃2
1
2π
Z ∞
0
b2db2 J0 kθ1b2
exp
h
ik
b2
2
2z̃2
− 2rg2 ln kb2
i Z 2π
0
dφξ2 exp
h
− ik
b2ρ
zr
cos[φξ2 − φ]
i
=
=
p
2πkrg1
k
iz̃2
Z ∞
0
b2db2 J0 kθ1b2
J0
k
b2ρ
zr
exp
h
ik
b2
2
2z̃2
− 2rg2 ln kb2
i
. (30)
Similarly to (13), we take the integral over b2 by the method of stationary phase, to determine b2 =
p
2rg2 z̃2 and,
thus, the amplitude of the EM field on the image plane at the distance of zr from the second lens is now given as
F2GL(x) =
p
2πkrg1
p
2πkrg2 eiϕ̂2
J0
k
p
2rg1 z̃1
p
2rg2 z̃2
z12
J0
k
p
2rg2 z̃2
zr
ρ
, (31)
with ϕ̂2 = krg2 ln[2e(zt + z12 + zr)/rg2] − 1
4 π, where we collected all the constant phase terms due to the presence of
lens 2 on the eikonal, including the relevant φG2 term in (8). (See Appendix A for alternative derivation of (30).)
Next, we use solution (31), to determine the PSF of the two-thin-lens system
PSF2GL(x) = (2πkrg1 ) (2πkrg2 )J2
0
k
p
2rg1 z̃1
p
2rg2 z̃2
z12
J2
0
k
p
2rg2 z̃2
zr
ρ
, (32)
where we remind that in the two-lens case the distance to the receiver zr from one-lens case (12) is replaced with z12,
so that the effective distance z̃1 has the form z̃1 = ztz12/(zt + z12).
We observe that the argument of the first Bessel function in this expression is very large and is evaluated to be
k
p
2rg1 z̃1
p
2rg2 z̃2/z12 ≥ 9.86 × 106
(1 µm/λ)(M1/M⊙)
1
2 (M2/M⊙)
1
2 (zt/650 AU)
1
2 (zr/650 AU)
1
2 (10 pc/z12). In this
case, the first Bessel function, J2
0 (kb1b2/z12), can be approximated by using its expression for large arguments [21]:
J2
0
k
p
2rg1 z̃1
p
2rg2 z̃2
z12
=
1
p
2πkrg1
p
2πkrg2
z12
√
z̃1z̃2
1 + sin
φ(z̃1, z̃2)
, with δϕ = 2k
p
2rg1 z̃1
p
2rg2 z̃2
z12
. (33)
8. 8
Considering here the term containing sin[φ(z̃1, z̃2)], we note that at optical wavelengths, this term is a rapidly oscil-
lating function of z̃z and z̃2 that averages to 0. Therefore, the last term in the form of J2
0 (x) expansion in (A2) may
be neglected, allowing us to present the averaged PSF from (32) as below:
hPSF2GL(x)i =
p
2πkrg1
p
2πkrg2
z12
√
z̃1z̃2
J2
0
k
p
2rg2 z̃2
zr
ρ
. (34)
This is our main result for the PSF for a two-lens axially-arranged system of monopole gravitational lenses. Note
that result (34) contains similar mass contributions from each of the lenses at both ends of the transmission link,
rg1 , rg2 . The amplification scales with the factor z12/
√
z̃1z̃2 involving the distance between the lenses, z12, as well as
the distances of both transmitter and receiver with respect to the transmitting and receiving lenses, z̃1, z̃2.
This result yields the maximum amplification factor for the two-lens system:
µ2GL = hPSF2GL(0)i =
p
2πkrg1
p
2πkrg2
z12
√
z̃1z̃2
≃
≃ 3.70 × 1014
M1
M⊙
1
2
M2
M⊙
1
2
1 µm
λ
z12
10 pc
650 AU
zz̃1
1
2
650 AU
zz̃2
1
2
. (35)
Clearly, for lensing architectures involving lenses with similar masses rg2 ≃ rg1 , the overall gain behaves as that of a
single lens (18) scaled with the geometric factor of z12/
√
z̃1z̃2, which provides additional gain for the lensing pair.
Based on (34), an observer in the focal region of lens 2, positioned on the optical axis at the distance of zr ≥ R2
2/2rg2
from it, will see one Einstein ring around the lens with the radius θ2 given as:
θ2 =
p
2rg2 z̃2
zr
≃
r
2rg2
zr
≃ 7.80 × 10−6
M2
M⊙
1
2
650 AU
zr
1
2
rad, (36)
which has the same structure as in the receiving lens case when a single lens is a part of a receiver (22).
However, we note that the estimate (35) may be misleading as it pertains only to the case when the receiver
aperture is smaller than the first zero of the Bessel function J0 kθ2ρ
present in (34), namely ρ2GL = 2.40483/(kθ2) ≃
4.91 × 10−2
(λ/1 µm)(M⊙/M2)
1
2 (zr/650 AU)
1
2 m, as shown in Fig. 3, which for optical wavelengths is not practical
and will be averaged by the aperture, as in (23). Therefore, for a realistic telescope, in context of a two-lens system,
similarly to (23), we develop an aperture-averaged value of the PSF, yielding the appropriate light amplification factor
hµ2GLi =
1
π(1
2 dr)2
Z 1
2 dr
0
Z 2π
0
PSF2GL(x) ρdρdφ =
p
2πkrg1
p
2πkrg2
z12
√
z̃1z̃2
J2
0 kθ2
1
2 dr
+ J2
1 kθ2
1
2t dr
≃
≃
4
p
2rg1 z̃1
dr
z12zr
z̃1z̃2
≃
r
2rg1
zt
4z12
dr
= hµ1GLi
z12
zt
= 9.62 × 1012
M1
M⊙
1
2
650 AU
zt
1
2
1 m
dr
z12
10 pc
. (37)
Note that after averaging, the dependence on the mass of the second lens is absent in (37). The reason behind this
is that the small scale of the diffraction pattern in (34) necessitated the aperture averaging, which caused the second
mass rg2 to drop out. (This is analogous to (21)–(23), where the same procedure also led to removing a factor of
√
rg1 .) This is because the telescope’s size, dr, is much larger than the first zero of the diffraction pattern in (34):
dr ≫ ρ2GL = 2.40483/(kθ2) ≃ 0.38λ
r
zr
2rg2
≃ 4.91 × 10−2
λ
1 µm
M⊙
M2
1
2
zr
650 AU
1
2
m. (38)
Note that when condition (38) in not satisfied and the aperture-averaging may not be relevant (i.e., due to a larger
wavelength, larger receiver distance, etc.), one would have to use the entire PSF with the J2
0 (x) factors included, as
in the case (19). In these cases, the relevant PSFs are (19), (21) for one lens, and either (34) or (A13) for two lenses.
Comparing result (37) with the transmission scenarios involving a single lens (18) and (23), we see that the two-
lens system provides significant additional light amplification captured by the factor (z12/zt). It is 82.5 times more
effective compared to the transmitting lens case (19) and is 3.17 × 103
times more effective than the receiving lens
scenario (23). Clearly, transmission via a pair of lenses benefits from gravitational amplification at both ends of the
transmission link, thus enabling interstellar power transmission with modern-day optical instrumentation.
IV. POWER TRANSMISSION
To assess the effectiveness of power transmission using gravitational lensing, we consider three transmission scenarios
that involve lensing with either a single lens or double lenses. These scenarios differ not only in the number of lenses
used but also in the positions of the transmitter and receiver relative to the lenses.
9. 9
A. Single lens: transmission from its focal region
First, we consider transmission via a single lens with the transmitter positioned in its focal region at a distance
of zt ≥ R2
1/2rg1 from the lens, while the receiver is at interstellar distance of zr ≫ zt. This is a transmitting lens
scenario is the case iii) in Sec. II B and was addressed by (19). We consider transmitter to be a point source.
We assume that transmission is characterized by the beam divergence set by the telescope’s aperture dt yielding
angular resolution of θ0 ≃ λ/dt = 1.00 × 10−6
(λ/1 µm)(1 m/dt) rad. When the signal reaches the receiver at the
distance of (zt +zr) from the transmitter, the beam is expanded to a large spot with the radius of ρ∗ = (zt +zr)(λ/dt).
In addition, while passing by the lens, the light is amplified according to (19). As a result, a telescope with the aperture
dr receives a fraction of the transmit power that is evaluated to be:
Pt
1GL = P0 PSFt
1GL(x)
π(1
2 dr)2
πρ2
∗
= P0
π(1
2 dr)2
π(zt + zr)2
dt
λ
2
2πkrg1 J2
0
k
p
2rg1 zt
zr
ρ
≃
≃ 3.06 × 10−13
P0
1 W
1 µm
λ
2 dt
1 m
2 dr
1 m
210 pc
zr
2 M1
M⊙
W. (39)
For the same transmitting scenario, a free space laser power transmission in the vacuum is described as
Pfree =
P0π(1
2 dr)2
π(zt + zr)2
dt
λ
2
≃ 2.62 × 10−24
P0
1 W
1 µm
λ
2 dt
1 m
2 dr
1 m
210 pc
zr
2
W. (40)
Comparing results (39) and (40), we observe that signal transmission relying on a single gravitational lens amplifies
the received power by Pt
1GL/Pfree ≃ 1.17 × 1011
, as prescribed by (18). We note that, depending on the transmit-
ter’s performance, the power amplification value (39) would have to be adjusted to account for a realistic system’s
throughput. Below, we consider several transmitter implementation approaches relevant to this scenario.
First, from a practical standpoint, we observe that, a single transmitter with the combination of parameters w0 and
λ chosen in (39), will be able to form a beam only with very short impact parameters of ρmult ≃ zt
1
2 θ0 = zt(λ/2dt) =
0.07R⊙ (λ/1 µm)(1 m/dt)(zt/650 AU)(R1/R⊙). The corresponding EM field will be totally absorbed by the lens.
Therefore, the parameter choice (39) would require a special transmitter design that would rely on multiple laser
transmitter heads. Considering the Einstein ring that is formed with the radius of RER =
p
2rg1 zt, one choice would
be to take nt = 2πRER/2ρmult = 2π(dt/λ)
p
2rg1 /zt ≃ 48.98 (dt/1 m)(1 µm/λ) (M1/M⊙)
1
2 (650 AU/zt)
1
2 laser heads
arranged in a conical shape each with the same offset angle of RER/zt =
p
2rg1 /zt ≃ 1.61′′
(M1/M⊙)
1
2 (650 AU/zt)
1
2
from the mean direction to the lens. In effect, such a transmitter will illuminate the Einstein ring on the circumference
of the lens with the signal of total power ntP0 that will then be proceed toward the receiver.
In the scenario with multiple transmitting heads, the signal from nt telescopes illuminates the Einstein ring at RER.
However, only small fraction of this light will be deposited in the ring. That fraction is proportional to the ratio of
the Einstein ring’s area as seen by the receiver, AER0 = π(1
2 dr)2
PSFt
1GL, where PSF is from (19), to the combined area
of diffraction-limited fields projected by nt transmitting telescopes on the circumference of the ring at the lens plane,
ntπ(zt(λ/dt))2
. As the signal propagates toward the receiver, its deviation from the optical axis is controlled by the
gravitational field of lens 1 and grows as zrθt
1, with θt
1 from (20). Thus, the total signal reaching the receiver must be
scaled again by the ratio of the area subtended by the ring with RER to the area resulting from this growth.
Putting this all together, we have
Pt
1GL = ntP0
AER0
ntπ(zt(λ/dt))2
πR2
ER
π(zrθt
1)2
≃ P0 2πkrg1
dt
λ
2 d2
r
4z2
r
J2
0
k
p
2rg1 zt
zr
ρ
, (41)
which, with zt ≪ zr, is identical to (39). Therefore, if such a multiple-heads transmitter can be built, this approach
retains the flexibility of choosing any desirable wavelength to initiate the transmission.
Next, we note that a single transmitter with a small aperture of dsm
t = 6 cm may also be used to illumi-
nate the regions around the lens with the practically-important impact parameters, namely psing = zt(λ/2dsm
t ) ≃
1.17R⊙ (λ/1 µm)(6 cm/dt)(zt/650 AU)(R1/R⊙). In this case, one would be forced to choose very small transmit
aperture or longer wavelengths, thus limiting the flexibility for interstellar communications.
In addition, if implemented at the SGL, such a transmitter will illuminate the area πp2
sing πR2
⊙ resulting in the
situation when a major part of the emitted light field, (R⊙/psing)2
≃ 85.86 %, will be absorbed by the lens and only
small fraction of it, ∼ 14.15 %, will illuminate the region with the Einstein ring. This will reduce the transmission
throughput to only t1tr ∼ 0.188, thus requiring a source with a higher transmit power compared to the case with
multiple transmitters. In any case, the total transmit power in this scenario would have to be adjusted, yielding in
this case the power amplification value of t1trPt
1GL/Pfree ≃ 4.29 × 108
(1 m/dt)(M1/M⊙)
1
2 (zt/650 AU)
1
2 .
10. 10
Finally, going back to the multi-head transmitter design discussed above, we note that, using a transmitter with
nt ∼ 49 laser heads each pointing in different directions may be a bit too complicated and other designs should be
considered. As an example, one may consider a holographic diffusing element1
that can be used to sculpt an outgoing
beam to practically any shape. Such a diffuser may allow for a uniform illumination of the Einstein ring at a specified
angular separation from the lens, thus offering a plausible approach to a transmitter design. In this case, there will
be no significant loss in the optical throughput and the entire power amplification of (39) will be at work.
We observe that each of the three transmission architectures above are different, yielding different optical trans-
mission throughputs, specific noise contributions, and will be characterized by different SNR performances [3, 4]. In
any case, these types of the transmission scenarios may be used to search for the signals before moving to the focal
region of lens 2 (discussed in Sec. IV C) that would be needed to establish a reliable communication infrastructure.
B. Single lens: reception at its focal region
Another single lens transmission scenario, involves transmission from an interstellar distance into the focal region
of a lens (the case iv). in Sec. II B). In this case, transmitter is now positioned at large distance of zt from the lens,
while the receiver is at the focal region of that lens at zr ≥ R2
1/2rg1 ≪ z12 from it, so that in this case zt ≪ zr. This
case is also covered by (23). The only difference from scenario discussed in Sec. IV A is that the laser beam divergence
in this case will naturally result in a large spot size illuminated by the transmitter at the lens plane, thus reducing
the incident power and overall link performance.
We note that a telescope with aperture dr positioned on the optical axis in the focal area of a lens and looking back at
it would see the Einstein ring formed around the lens. The energy deposited in the ring will be the same as that received
by the observer. This means that the effective collecting area of the receiving telescope Atel = π(1
2 dr)2
hµr
1GLi, with
hµr
1GLi from (23), is equal to the area subtended by the observed Einstein ring AER = 2πRERwER, were RER =
p
2rg1 zr.
Equating Atel = AER, allows us to determine the width the ring wER = 1
2 dr, thus establishing the effective area
subtended by the Einstein ring as seen by the receiving telescope in the case of a single lens transmission
AER1 = πdr
p
2rg1 zr. (42)
Clearly, a telescope with the angular resolution of 1.22(λ/dr) = 1.22 × 10−6
(λ/1 µm)(1 m/dr) ≫ dr/2zr = 5.14 ×
10−15
(dr/1 m)(650 AU/zr), will not be able to resolve the width of the ring; however, the length of its circumference
is resolved with nt = 2π
p
2rg1 /zr/(λ/dr) ≃ 48.98 (M1/M⊙)
1
2 (650 AU/zr)
1
2 (1 µm/λ)(dr/1 m) resolution elements.
This information is useful when considering a transmitter’s design or evaluating a receiver’s performance.
To evaluate the effectiveness of such scenario, consider transmitting telescope that illuminates the Einstein ring
around the lens at the radius of RER =
p
2rg1zr. However, because of the diffraction transmitted energy will spread
over a much larger area with the radius of ρ′
∗ = zt(λ/dt) = 443.54R⊙ (λ/1 µm)(zt/10 pc)(1 m/dt)(R1/R⊙). Thus, not
all the energy will be received at the Einstein ring, some of the energy will be lost. By the time the signal reaches
the receiver, its deviation from the optical axis is controlled by gravity changing by zrθr
1, where θr
1 is from (22). As a
result, we estimate the total received power in this case when the single lens acts a part of a receiver as below
Pr
1GL = P0
AER1
π(zt(λ/dt))2
πR2
ER
π(zrθr
1)2
=
P0π(1
2 dr)2
πz2
t
dt
λ
2 4
p
2rg1zr
dr
=
= 7.96 × 10−15
P0
1 W
1 µm
λ
2 dt
1 m
2 dr
1 m
10 pc
zt
2 M1
M⊙
1
2
zr
650 AU
1
2
W, (43)
which is a factor 38.47 times smaller than that obtained in the scenario with transmission from the focal region (39)
discussed in Sec. IV A. As a result, there is some difference between transmitting from the focal region of a lens to
a receiver at large interstellar distance vs transmitting from a large distance into the focal region of the lens. In
addition, there may be different noise sources involved affecting the SNR performance of a transmission link.
C. Two lenses: transmission and reception from/to the focal regions
Now we consider scenario that involves a two-lens power transmission, where transmitter, is positioned in the focal
region of lens 1 at the distance of zt R2
1/2rg1 from it, sends a signal toward lens 2 that is separated from lens 1 by
1 As shown here: https://www.rpcphotonics.com/engineered-diffusers-information.
11. 11
the interstellar distance z12 ≫ zt. After passing by lens 2, the signal is detected by a receiver positioned in the focal
region of lens 2 at the distance of zr R2
2/2rg2 from it. Analytically, this case is described by solution (A16).
The power transmission, as outlined above, is characterized by (39), where the light amplification is from (A16).
In this case, we follow the same approach that we used to derive (39). We may treat the two-lens system in the same
manner as we dealt with a single-lens transmission (39). For that, we realize that when the signal reaches the receiver
at the distance of (zt +z12 +zr), the beam is expanded to a large spot with the radius of ρ∗∗ = (zt +z12 +zr)(λ/dt) ≃
z12(λ/dt). In addition, while passing by the two-lens system, the light is amplified according to (A16).
As a result, a telescope with the aperture dr receives a fraction of the transmit power that is evaluated to be:
P2GL = P0 hµ2GLi
π(1
2 dr)2
πρ2
∗∗
=
P0π(1
2 dr)2
πz2
12
dt
λ
2
r
2rg1
zt
4z12
dr
≃
≃ 2.53 × 10−11
P0
1 W
1 µm
λ
2 dt
1 m
2 dr
1 m
10 pc
z12
2 M1
M⊙
1
2
650 AU
zt
1
2
W, (44)
which clearly demonstrates the benefits of the gravitational amplification provided by the double-lens systems while
yielding the gain of P2GL/Pfree =
p
2rg1 /zt(4z12/dr) ≃ 9.62 × 1012
(M1/M⊙)
1
2 (650 AU/zt)
1
2 (1 m/dr)(z12/10 pc),
which is ∼ (2/π2
)(λ/dr)(z12/
p
2rg1 zt) ≃ 82.50 (λ/1 µm)(1 m/dr)(M⊙/M1)
1
2 (650 AU/zt)
1
2 (z12/10 pc) times higher
than in the case when a singe lens acts as a part of a transmitter, shown by (39), or z12/zt ≃ 3.17 ×
103
(z12/10 pc)(650 AU/zt) times higher than that of a single lens acting as a part of the receiver as seen by (43).
We may consider an alternative derivation for the two-lens transmission link (44). A telescope with aperture dr
would see the two Einstein rings formed around the lens. They are unresolved, but the energy, deposited in the ring
will be the same as that received by the telescope. This means that the effective collecting area of the receiving
telescope Atel = π(1
2 dr)2
hµr
2GLi, with hµr
2GLi is now from (A16), is equal to the area subtended by the observed
Einstein ring AER = 2πRERwER, were RER =
p
2rg1 zt. Equating Atel = AER, allows us to determine the width the
ring wER = 1
2 dr(z12/zt), thus establishing the effective area subtended by the Einstein ring as seen by the receiving
telescope in the case of a two lens transmission
AER2 = πdrz12
r
2rg1
zr
= AER1
z12
zt
, (45)
where AER1 from (42). Therefore, the amount of light captured by the ring increases with distance as (z12/zt), so as
the amplification factor (A16). This factor is the key source of additional amplification for two-lens transmissions.
Again, a telescope with the resolution of 1.22λ/dr = 1.22 × 10−6
(λ/1 µm)(1 m/dr) ≫ (z12/zt)dr/2zr = 1.6 ×
10−11
(dr/1 m)(650 AU/zr)2
(zr/10 pc), will not be able to resolve the width of the Einstein ring; however, the length
of its circumference is resolved with nt = 2π
p
2rg1 /zr/(λ/dr) ≃ 48.98 resolution elements as in the earlier case (42).
We can do that in the manner similar to the approach we developed while developing (41). For that, there are
important steps to consider, namely
• Consider nt transmitting apertures dt positioned at the distance of zt in the focal region of lens 1 (as in the case
of (41)). Assuming that these telescopes coherently illuminate the Einstein ring with the radius of RER around
lens 1. Because of the diffraction, transmitted energy spreads over the area with radius of ρ′
∗ = z12(λ/dt). Thus,
the fraction of the signal that will be deposited in the ring is AER2/π(z12(λ/dt))2
, were AER2 is from (45); also here
we use the amplification factor hµ2GLi from (A16), thus the width of the ring, in the case, is wER = 1
2 dr(z12/zt).
• As the signal propagates toward lens 2, its deviation from the optical axis grows as z12θt
1, where θt
1 is from (24).
By the time is reaches the receiver behind lens 2, its deviation from the optical axis will be changed again by
the amount of zrθ2, where θ2 is from (A6), again requiring an appropriate scaling by πR2
ER′ /π(zrθ2)2
, where
RER′ =
p
2rg2 zr is the radius of the Einstein ring seen by the averaged PSF (A16) around lens 2.
As a result, using the same approach that was used to derive (41), we estimate the total received power as below
P2GL = ntP0
AER2
ntπ(zt(λ/dt))2
πR2
ER
π(z12θt
1)2
πR2
ER′
π(zrθ2)2
≃
P0π(1
2 dr)2
πz2
12
dt
λ
2
r
2rg1
zt
4z12
dr
, (46)
which is identical to (44). As a result, in the case of the two-lens transmission, the total power received is higher than
that for a single-lens transmission. In other words, the second lens adds more power by focusing light, as expected.
12. 12
V. DETECTION SENSITIVITY
Now that we have established the power estimates for the transmission links characteristic for various lensing
architectures involving one and two lenses, we need to consider practical aspects of such transmission links. For this,
can evaluate the contribution of the brightness of the stellar atmospheres to the overall detection sensitivity.
A. Relevant signal estimates
First of all, based on the results for power transmission for various configurations considered, namely (39), for a
lens near the transmitter, (43), for a lens on a receiving end, and (44) for the pair of lenses,, we have the following
estimates for photon flux, Qt
1GL, Qr
1GL, Q2GL, received by the telescope in all three of these cases
Qt
1GL =
λ
hc
Pt
1GL ≃ 1.54 × 106
P0
1 W
1 µm
λ
dt
1 m
2 dr
1 m
210 pc
zr
2 M1
M⊙
phot/s, (47)
Qr
1GL =
λ
hc
Pr
1GL ≃ 4.01 × 104
P0
1 W
1 µm
λ
dt
1 m
2 dr
1 m
10 pc
zt
2 M1
M⊙
1
2
zr
650 AU
1
2
phot/s, (48)
Q2GL =
λ
hc
P2GL ≃ 1.27 × 108
P0
1 W
1 µm
λ
dt
1 m
2 dr
1 m
10 pc
z12
2 M1
M⊙
1
2
650 AU
zt
1
2
phot/s. (49)
Therefore, based on the estimates above, we have significant photon fluxes in each of the three configurations
considered. However, each of the three configurations will have different contributions from the brightness of the
stellar atmospheres along the relevant transmissions links. Below, we will provide estimates for the relevant noise.
B. Dominant noise sources
The Einstein rings that form around the stellar lenses appear on the bright background of the stellar atmospheres
(coronas.) To assess this noise contribution from the stellar atmospheres, we need to estimate their contribution to
the signal received by the receiver. Such signals are different, thus establishing the practically relevant constraints.
Similar to case of imaging with the SGL, we know that the most doming noise sources are those that come for the
lenses themselves, namely the light emitted by them and the brightness of their coronas in the areas where Einstein
rings will appear. Not all that light can be blocked by chronographs and must be accounted for in the sensitivity
analysis. Clearly, there are other sources of noise, especially at small angular separations from the lenses, and they
must be treated which considering realistic transmission scenarios.
Below, we will focus on the main anticipated sources of noise – those from lenses themselves and from their coronas.
1. Single lens: transmission from its focal region
In the case, when the lens is at the transmitter node, the situation is quite different from the one that we considered
in Sec. V B 2. In this case the angular extend of the Einstein ring is given by (20). The angular size of the stellar
lens is R1/zt = 2.26 × 10−9
(R1/R⊙)(10 pc/zt) rad, similar to that of the Einstein ring. A conventional telescope with
a practical diameter will not be able to resolve this star, therefore, the use of a coronagraph is out of the question.
Thus, the light emitted by that stellar lens and received by the telescope will be the noise that must be dealt with.
In other words, the light from the Einstein ring will arrive together with the unattenuated light from the star.
We consider the Sun as a stand-alone for the lens 1. In this case, assuming the Sun’s temperature2
to be T⊙ =
5 772 K, we estimate the solar brightness from Planck’s radiation law:
B⊙ = σT 4
⊙ =
Z ∞
0
Bλ(T⊙)dλ =
Z ∞
0
2hc2
λ5 ehc/λkT⊙ − 1
dλ = 2.0034 × 107 W
m2 sr
, (50)
2 See https://en.wikipedia.org/wiki/Sun
13. 13
where we use a blackbody radiation model with σ as the Stefan-Boltzmann constant, kB the Boltzmann constant.
This yields the total power output of the Sun of L⊙ = πB⊙4πR2
⊙ = 3.828 × 1026
W, which is one nominal solar
luminosity as defined by the IAU.3
When dealing with laser light propagating in the vicinity of the Sun, we need to be concern with the flux within
some bandwidth ∆λ around the laser wavelength λ, assuming we can filter the light that falls outside ∆λ, then
B⊙(λ, ∆λ) = Bλ(T⊙)∆λ =
2hc2
λ5 ehc/λkBT⊙ − 1
∆λ ≃ 1.07 × 105
1 µm
λ
5 ∆λ
10 nm
W
m2 sr
. (51)
We take (51) to derive the luminosity of the Sun within a narrow bandwidth as follows
L⊙(λ, ∆λ) = πB⊙(λ, ∆λ)4πR2
⊙ =
2hc2
λ5 ehc/λkBT⊙ − 1
∆λ ≃ 2.05 × 1024
1 µm
λ
5 ∆λ
10 nm
W. (52)
As a result, in the case when a lens used as a part of the transmitter, the power of the signal received from the
lensing star received by a telescope and the corresponding photon flux are derived from (52) and are given as
Pt ⋆
1GL = L1(λ, ∆λ)
π(1
2 dr)2
πz2
r
= 5.39 × 10−12
L1
L⊙
1 µm
λ
5 ∆λ
10 nm
dr
1 m
210 pc
zr
2
W, (53)
Qt ⋆
1GL =
λ
hc
Pr,st
1GL = 2.71 × 107
L1
L⊙
1 µm
λ
4 ∆λ
10 nm
dr
1 m
210 pc
zr
2
phot/s. (54)
2. Single lens: reception at its focal region
The case of a single lens positioned nearby receiver is the most straightforward one. In this scenario, the angular size
of the Einstein ring is given by (22), which is quite large. The receiving telescope with the diameter of dr resolves the
angular diameter of the stellar lens in (R1/zr)/1.22(λ/dr) = 5.86 (R1/R⊙)(650 AU/zr)(1 µm/λ)(dr/1 m) resolution
elements. Thus, one may consider using a coronagraph to block the stellar light (in our case, this would be the Sun.)
Most of the relevant modeling was already accomplished in the context of our SGL studies. For that we know that
the transmitted signal is seen by the telescope within the annulus that corresponds to the Einstein ring around a lens
at the distance of b =
p
2rg1 z̃1 and within the thickness of wER = 1
2 dr, formed by the transmitted light. However,
the telescope sees a much larger annulus with the thickness of wtel ≃ zr(λ/dr), which is wtel/wER = 2zr(λ/d2
r) =
9.72 × 107
(zr/650 AU)(λ/1 µm)(1 m/dr)2
times thicker, thus receiving the light from a much larger area.
While working on the SGL, we developed an approach to account for the contribution of the solar corona to imaging
with the SGL. We take our Sun as the reference luminosity. In the region occupied by the Einstein ring in the focal
plane of a diffraction-limited telescope, the corona contribution is given after the coronagraph as:
Pcor(λ, ∆λ) = ǫcor π(1
2 dr)2
Z 2π
0
dφ
Z ∞
θ0
θdθ Bcor(θ, λ, ∆λ), (55)
where θ = ρ/zr, θ0 = R⊙/zr, and ǫcor = 0.36 is the fraction of the encircled energy for the solar corona (see [14]).
The surface brightness of the solar corona Bcor may be taken from the Baumbach model [22, 23], or use a more
recent and a bit more conservative model [24] that is given by:
Bcor(θ, λ, ∆λ) =
10−6
· B⊙(λ, ∆λ)
h
3.670
θ0
θ
18
+ 1.939
θ0
θ
7.8
+ 5.51 × 10−2
θ0
θ
2.5i
, (56)
where B⊙(λ, ∆λ) is from (51). Using (56) in (55) and following the approach used in [3, 14], we obtain the power of
the signal received from the corona and the corresponding photon flux
Pcor(λ, ∆λ) ≃ 6.58 × 10−12
1 µm
λ
5 L1
L⊙
∆λ
10 nm
dr
1 m
2650 AU
zr
2.5
W, (57)
Qcor =
λ
hc
Pcor ≃ 3.31 × 107
1 µm
λ
4 L1
L⊙
∆λ
10 nm
dr
1 m
2650 AU
zr
2.5
phot/s. (58)
Although these estimates may need to be refined by analyzing the spectral composition of the corona of a particular
stellar lens and the relevant signal as was recently done in [4], the obtained results provide good initial estimates.
3 See https://www.iau.org/static/resolutions/IAU2015_English.pdf
14. 14
3. Two lenses: transmission and reception from/to the focal regions
In the case a two-lens transmission, the situation changes again. This time, one would have to deal not only with the
light received from lens 2 that falls within the annulus around the Einstein ring as in the case discussed in Sec. V B 2,
but also the light that arrive into this annulus from lens 1 must also be accounted form. However, in this case, the
light from lens 1 will be amplified by the gravitational field of lens 2, providing additional noise background.
Consdier formation of this background light, we note that this process is very similar to imaging of the exoplanets
with the SGL [14]. Except, here we concerned with the formation of another Einstein ring formed around lens 2 at
the same location as the transmitted signal, but this time another ring is formed from the light received from lens 1.
Similar to the SGL, ;ens 2 will focus light while reducing the size of the image compared to the source by a factor
of zr/z12 ∼ 3.15 × 10−4
(zr/650 AU)(10 pc/z12), see [14]. For a lens with physical radius R1, positioned at a distance
of z12 from lens 2, the image of this target at a distance of zr, will be compressed to a cylinder with radius
ρ⋆
1 =
zr
z12
R1 = 219.24
R1
R⊙
zr
650 AU
10 pc
z12
km. (59)
Using (59), we see that a telescope with the aperture dr will resolve this object with Nd linear resolution elements
Nd =
2ρ⋆
1
dr
=
2R1
dr
zr
z12
= 4.39 × 105
R1
R⊙
zr
650 AU
10 pc
z12
1 m
dr
. (60)
Therefore, to account for the source provided by the amplified stellar light, we can use the expressions that we
developed for the SGL to describe the total power collected from a resolve exoplanet. The relevant expression in
this case is Eq. (2) in [3]. Replacing in this expression the surface brightness of the exoplanet Bs with Bcor(θ, λ, ∆λ)
describing the solar surface brightness within the bandwidth of ∆λ around the central wavelength λ from (51), we
have At the same time, the power at the Einstein ring at the detector placed in the focal plane of an optical telescope
is dominated by the blur and is given as
P⋆
2GL(ρ) = ǫblurπB⊙(λ, ∆λ)π(1
2 dr)2 2R1
z12
r
2rg2
zr
µ(ρ), with µ(r) =
2
π
E
h ρ
ρ⋆
1
i
, 0 ≤ ρ/ρ⋆
1 ≤ 1, (61)
where ǫblur = 0.69 is the encircled energy fraction for the light received from the entire resolved lens 1 [14] and E[x]
is the elliptic integral [21]. To evaluate the worst case, below we can take ρ = 0 and thus, µ(0) = 1.
Taking this into account and using B1(λ, ∆λ) from (51) and considering the worst case, when ρ = 0 and thus,
µ(0) = 1, from (61), we have the following estimate for the power at the receiver from the amplified light from lens 1
P⋆
2GL ≃ 6.42 × 10−9
L1
L⊙
1 µm
λ
5 ∆λ
10 nm
dr
1 m
2 R1
R⊙
10 pc
z12
M2
M⊙
1
2
650 AU
zr
1
2
W. (62)
This power corresponds to the enormous photon flux received at the telescope
Q⋆
2GL =
λ
hc
P⋆
2GL = 3.23 × 1010
L1
L⊙
1 µm
λ
4 ∆λ
10 nm
dr
1 m
2 R1
R⊙
10 pc
z12
M2
M⊙
1
2
650 AU
zr
1
2
phot/s. (63)
It is clear that photon flux at such high level will form challenging background for two-lens power transmissions.
C. Interstellar transmission: relevant SNRs
We assume that the contributions of the stellar coronas is removable (e.g., by observing the corona from a slightly
different vantage point) and only stochastic (shot) noise remains, we estimate the resulting SNR as usual
SNR =
QER
√
QER + Qnoise
. (64)
The first case we consider is when transmitter sends a signal directly via its local lens with the receiver being at an
interstellar distance. In this case, we use results (47) and (54), we estimate the relevant SNR as below
SNRt
1GL =
Qt
1GL
p
Qt
1GL + Qt ⋆
1GL
≃ 2.87 × 102
P0
1 W
λ
1 µm
10 nm
∆λ
1
2
dt
1 m
2 dr
1 m
10 pc
zr
M1
M⊙
L⊙
L1
1
2
r
t
1 s
. (65)
15. 15
This represents a very handsome SNR level that shows the feasibility of establishing an interstellar power transmission
link where one would transmit from a focal region behind a lens to a receiver at an interstellar distance.
Next, we consider the case when transmitter sends a signal toward a receiver that uses its local lens to amplify
the signal. In this case, there are two scenarios available: 1) an isolated transmitter is positioned in space with no
significant stellar background, and 2) when transmitter is positioned next to a star, i.e., the situation is similar when
we transmit a signal from a ground-based observatory.
In the first of these two cases, the only background noise to consider is the corona noise from the lensing star next
to the receiver. Using (48) to represent the signal of interest, Qr
1GL, and (58) that for the noise from the stellar corona,
Qcor, we estimate the SNR for the case when a single lens is positioned at the receiving end of the link:
SNR
r [1]
1GL =
Qr
1GL
p
Qr
1GL + Qcor
≃
≃ 6.97
P0
1 W
λ
1 µm
10 nm
∆λ
1
2
dt
1 m
210 pc
zt
2 M1
M⊙
1
2
zr
650 AU
1.75L⊙
L1
1
2
r
t
1 s
. (66)
In the second case, we consider the worst case, when the star is directly behind the transmitter. The light from the
star will also be received and amplified by the lens. In this case, the signal Qr
1GL is still given by (48), but the noise in
this case will be due to corona around the lens Qcor from (58) and the light of the star behind the transmitter that
is amplified by the lens Q⋆
2GL, given by (63). The reslting SNR is given as below
SNR
r [2]
1GL =
Qr
1GL
p
Qr
1GL + Qcor + Q⋆
2GL
≃ 0.22
P0
1 W
λ
1 µm
10 nm
∆λ
1
2
dt
1 m
210 pc
z12
3
2
×
×
M1
M⊙
1
2
M⊙
M2
1
4
650 AU
zt
1
4
L⊙
L1
1
2
R⊙
R1
1
2
r
t
1 s
. (67)
The estimates (66) and (67) also support the feasibility of establishing an interstellar transmission where one would
transmit from an interstellar distance directly into a lens.
Finally, we estimate the SNR for the two-lens transmission. In this case, we use (49) for the signal with the relevant
noise source for the corona around lens 2, Qcor, taken from (58) and the amplified light from lens 1, Q⋆
2GL, given by
(63). As a result we have the following SNR for the two-lens transmission
SNR2GL =
Q2GL
p
Q2GL + Qcor + Q⋆
2GL
≃ 7.05 × 102
P0
1 W
λ
1 µm
10 nm
∆λ
1
2
dt
1 m
210 pc
z12
3
2
×
×
M1
M⊙
1
2
M⊙
M2
1
4
650 AU
zt
1
4
L⊙
L1
1
2
R⊙
R1
1
2
r
t
1 s
, (68)
which is the best among the three scenarios considered. Clearly, the fact that the stellar light from lens 1 is amplified
by lens 2 provides a very strong background. As a result, power transmission with a single lens yields a comparatively
strong SNRs compared to the case of two-lens transmission. One can further increase this SNR by increasing the
transmitted power and/or the size of the transmitting aperture. Nevertheless, analysis presented here demonstrates
the feasibility of using gravitational lensing for establishing practical interstellar power transmission links.
VI. DISCUSSION
We considered the propagation of monochromatic EM waves in the presence of non-interacting monopole gravi-
tational lenses. To do that we examined a scenario in which the source, lenses, and observer align along a single
axis—a situation characterized by axial symmetry. We find that the challenging diffraction integrals can be solved
analytically, offering valuable insights. We observe that the number of Einstein rings formed around each successive
lens doubles, but they are not resolvable from each other. The overall light amplification is determined by the mass
of the first lens, positions of the transmitter/receiver, and is affected by the distances between the lenses. Interesting,
but in the realistic observing scenarios, there is no dependence on the mass of the second lens.
It is remarkable that performance of a transmission link is determined by a single factor z/z0 in front of x′
in (8),
that determines image scaling in different lensing scenarios: 1) Lens acts as a part of a transmitter, when the signal
transmission is done from a distance shorter than that of the receiver, z0 ≪ z. In this case, the image size grows as
z/z0, as in the transmitting lens scenario in Sec. III A 1 or in the transmission part at lens 1 in the two-lens scenario
in Sec. IV C, when the image size grows as z12/zt. 2) Lens acts as a part of a receiver with the transmit/receive
16. 16
positions reversed, z0 ≫ z. In this case, the lens focuses light compressing images by z/z0, as in the receiving lens
scenario of Sec. IV B or in the reception part of the two-lens transmission discussed in Sec. IV C.
This scaling changes the width of the Einstein ring formed at each lens plane and thus the area subtended by the
ring, as seen in (45). We anticipate that including higher order terms ∝ b4
in (4) may improve the fidelity of the
solutions. That can be done following the approach that we have demonstrated here. But until that extension is
developed and evaluated, the solutions that were provided here are sufficient to study various transmission links.
As a result, the two-lens transmission configurations shown by (44), are found to represent the most favorable
transmission architectures, providing access to the largest light amplification of (37) and resulting in the gain of
P2GL/Pfree =
p
2rg1 /zt(4z12/dr) ≃ 9.62 × 1012
(M1/M⊙)
1
2 (650 AU/zt)
1
2 (1 m/dr)(z12/10 pc). Configurations, where a
single lens acts as a part of a transmitter were found to be the second best, providing access to the light amplification
factor of (18), thus yielding the gain of Pt
1GL/Pfree ≃ 2πkrg1 ≃ 1.17 × 1011
(M1/M⊙)(1 µm/λ). Finally, configurations,
where a single lens acts as a part of a receiver, are found to amplify electromagnetic signals in accord to (23), thus
resulting in the gain of Pr
1GL/Pfree = 4
p
2rg1 zr/dr ≃ 3.03 × 109
(M1/M⊙)
1
2 (zr/650 AU)
1
2 (1 m/dr).
Note that although throughout the paper we used λ = 1µm to estimate various values, all our expressions are valid
for shorter wavelengths. In fact, not addressing the technical implementations, for shorter wavelength the gravitational
lensing will be even more robust with higher overall gains. Our models work for longer wavelengths as well, but will
some limits. As we have shown in [25], longer wavelengths, would experience much higher diffraction, so that one
may have to consider using the full diffraction-limited PSFs given by (17) and (A13). Therefore, these wavelengths
were not considered here, as our prior work [25, 26] had shown that such longer wavelengths, say above 30 cm, will
be severely affected by stellar atmospheres to the point of being completely blocked by plasma in their coronas.
Our key results are (34), (37) that present the PSF of a light transmission with a pair of gravitational lenses.
Expression (34) has a similar mass contributions from the lenses located at both the start and end of the transmission
link. The amplification is proportional to the factor z12/
√
z̃1z̃2, where z12 is the distance between the two lenses, while
z̃1, z̃2.being the distances from the transmitter to its respective lens and from the receiver to its lens, respectively.
However, as the size of a realistic telescope will be larger than the diffraction pattern set in (34), the result must be
averaged over the telescope’s aperture. We found that after that averaging, the dependence on the mass of the second
lens is absent in (37). (This is analogous to (21)–(23), where the same procedure also removed a factor of
√
rg1 .)
Conclusively, our analysis underscores the fact that, in terms of light amplification, a multi-lens configuration
outperforms a single-lens system by a significant margin. We observe that all the transmission cases are characterized
by different the PSFs, exhibiting different structures with features from a few cm scales to several 100s of meters.
Notably, in the transmitting lens case the structure within the PSF (19) is large and does not need to be averaged.
However, in the two-lens system the fine structure in the PSF is not going to be directly observed with a reasonable-
size telescope. Instead, this structure is averaged out by a modest, say, a 1-m telescope, yielding the average PSFs as
in the receiving lens scenario (23) and that of the two-lens system (A16).
The feasibility of interstellar transmission became evident when we analyzed the SNRs for various transmission
scenarios (see Sec. V C). Although the two-lens transmission benefits from an impressive combined power amplification
available in the two-lens systems, the two-lens case is severely affected by the much amplified background light coming
from the first lens. Clearly, one would have to include other sources of light in the vicinity of the first lens, but their
presents is not expected to significantly affect the achievable SNR (68), which is quite high. On the other hand, both
single-lens transmission cases, where the lens either close to the transmitter or is near the receiver, also show a very
robust link performance, as evidenced by the corresponding SNR estimates (65) and (66), (67).
Concluding, we would like to mention that the results obtained here, could have a profound effect for applications
aiming at interstellar power transmission. Not only we can look for transmitted signals using modern astronomical
techniques [8–11, 27, 28], we may also transmit such signals with space-based platforms in the focal region of the solar
gravitational lens (SGL) using technologies that are either extant or in active development [29, 30].
Looking ahead, it would be interesting to explore non-axially symmetric setups and also transmission in the presence
of non-spherically symmetric lenses, such as those discussed in [12, 16, 31, 32]. These lenses will have a complex PSF
structure exhibiting contribution of various caustics. Although such lenses will disperse some light toward the cusps of
the caustics [33], with a proper transmission alignment such effects may be reduced. Investigating such configurations
via numerical simulations might shed light on the intricate dynamics in such lensing systems. Nonetheless, our
primary focus should be directed towards deriving a formal solution for the situation where a transmitter, positioned
at a significant yet finite distance from a lens, emits a beam—whether plane, spherical, or Gaussian—towards that
lens. Such a solution may be obtained by following the approach taken in [2] that would put all the related analysis
on a much stronger footing. Work on this matter is currently in progress, and findings will be reported elsewhere.
17. 17
Acknowledgments
We kindly acknowledge discussions with Jason T. Wright who provided us with valuable comments and suggestions
on various topics addressed in this document. This work was performed at the Jet Propulsion Laboratory, California
Institute of Technology, under a contract with the National Aeronautics and Space Administration.
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Appendix A: Alternative derivation for a two-lens transmission
Here we provide an alternative derivation for the diffraction integral (30), which is repeated below, for convenience:
F2GL(x) =
p
2πkrg1
k
iz̃2
Z ∞
0
b2db2 J0 kθ1b2
J0
k
b2ρ
zr
exp
h
ik
b2
2
2z̃2
− 2rg2 ln kb2
i
. (A1)
Clearly, for any impact parameters smaller that the radius of the second lensing star, b2 ≤ R2, the light rays will
be absorbed by that lens. Therefore, the integration over db2 in (30) really starts from R2. In this case, the argument
of the first Bessel function in this expression is very large and is evaluated to be kθ1b2 ≃ kb2
p
2rg1 z̃1/z12 ≥ 1.07 ×
18. 18
107
(1 µm/λ)(M1/M⊙)
1
2 (zt/650 AU)
1
2 (10 pc/z12)(b2/R⊙). In this case, this Bessel function, J0(x) can approximated
by using its expression for large arguments (see [34])
J0 kθ1b2
=
1
√
2πkθ1b2
ei(kθ1b2− π
4 )
+ e−i(kθ1b2− π
4 )
. (A2)
Note that, as we are concerned with the regions on the optical axis after lens 2, where ρ ≈ 0, we do not need to use
a similar approximation for the second Bessel function in (30).
As a result of using approximation (A2), expression (30) takes the form:
F2GL(x) =
p
2πkrg1
k
iz̃2
Z ∞
R2
√
b2db2
√
2πkθ1
J0
k
b2ρ
zr
ei kθ1b2− π
4
+ e−i kθ1b2− π
4
e
ik
b2
2
2z̃2
−2rg2 ln kb2
. (A3)
We evaluate this integral using the method of stationary phase (as was done in [17]). To do that, we see that the
relevant b2-dependent part of the phase in (A3) is of the form
ϕ±(b2) = k
b2
2
2z̃2
± b2θ1 − 2rg2 ln kb2
∓ π
4 . (A4)
The phase is stationary when dϕ±/db2 = 0, which implies
b2
z̃2
−
2rg2
b2
± θ1 = 0. (A5)
As impact parameter is a positive quantity, b2 ≥ 0, this quadratic equation yields two solutions:
b±
2 = z̃2
1
2
q
θ2
1 + 4θ2
2 ∓ θ1
, where θ2 =
r
2rg2
z̃2
≃
r
2rg2
zr
≃ 7.79 × 10−6
M2
M⊙
1
2
650 AU
zr
1
2
rad, (A6)
where θ2 is the radius of the Einstein ring to be formed around the second lens. The impact parameters b±
2 are two
families of impact parameters describing incident and scattered EM waves, corresponding to light rays passing by the
near side and the far side of lens 2, correspondingly (see [17] for details).
Note that, as opposed to the case with one lens (see (16) and (17)), the two solutions b±
2 from (A6) suggest that
there are now two Einstein rings formed around the second lens with the radii θ±
2 = b±
2 /z̃2 given as below:
θ±
2 = 1
2
q
θ2
1 + 4θ2
2 ∓ θ1
. (A7)
Note that, given the fact that nominally θt
1 ≪ θ2 (see (24)) and (A6)), both rings are very close being only
δθ2 = θ−
2 −θ+
2 ≃ θ1 ≃ 2.46×10−9
rad, which is the angle that unresolvable by a conventional telescope. So, nominally
such a telescope would see both of these ring as one, receiving the signal from both of them. Although these rings
may not be resolved from each other, but yet they carry the relevant energy to the receiver.
Following the approach presented in [17], we again use the method of stationary phase (A4) for both solutions in
(A7). Thus, the amplitude of the EM field on the image plane at the distance of zr from the second lens is given as
F2GL(x) =
p
2πkrg1 eiϕ̂2
a+eiϕ+
J0 kθ+
2 ρ
+ a−eiϕ−
J0 kθ−
2 ρ
, (A8)
where ϕ̂2 is the same as in (31). Also, b±
2 are from (A6), with a± and ϕ± as below:
a2
±(θ1, θ2) =
1
2
p
1 + 4θ2
2/θ2
1 ∓ 1
2
p
1 + 4θ2
2/θ2
1
, (A9)
ϕ±(θ1, θ2) = k
rg2 ± 1
4 z̃2θ2
1
q
1 + 4θ2
2/θ2
1 ∓ 1
− 2rg2 ln 1
2 kz̃2θ1
q
1 + 4θ2
2/θ2
1 ∓ 1
. (A10)
Similarly to (17), we use solution (A8), to determine the PSF of the two-thin-lens system
PSF2GL(x) = 2πkrg1
a2
+J2
0 kθ+
2 ρ
+ a2
−J2
0 kθ−
2 ρ
+ 2a+a−J0 kθ+
2 ρ
J0 kθ−
2 ρ
sin[δϕ±]
. (A11)
19. 19
Considering the mixed term in (A11), that contains sin[δϕ±] with δϕ± is given by
δϕ± = ϕ+ − ϕ− = kz̃2
1
2 θ2
1
q
1 + 4θ2
2/θ2
1 − θ2
2 ln
p
1 + 4θ2
2/θ2
1 − 1
p
1 + 4θ2
2/θ2
1 + 1
, (A12)
we note that at optical wavelengths, this term is a rapidly oscillating function of z̃2 that averages to 0. Therefore, the
last term in (A11) may be neglected, allowing us to present the averaged PSF as below:
hPSF2GL(x)i = 2πkrg1
a2
+J2
0 kθ+
2 ρ
+ a2
−J2
0 kθ−
2 ρ
. (A13)
This result yields the maximum amplification factor for the two-lens system which is obtained by setting ρ = 0:
µ2GL = hPSF2GL(0)i = 2πkrg1
a2
+ + a2
−
. (A14)
Using expressions for a+ and a− from (A9) and remembering the single-lens light amplification factor µ1GL = 2πkrg1
from (18) as well as Einstein angles θ1 and θ2 from (A2) and (A6), correspondingly, we present result (A14) as below
µ2GL = µ1GL
1
u
u2
+ 2
√
u2 + 4
, where u =
θ1
θ2
≡
r
M1
M2
√
z̃1z̃2
z12
. (A15)
Considering the case of two lenses of similar masses, we have rg1 ≃ rg2 , z01 ≃ z02 = 650 AU ≪ z12 = 10 pc, and,
thus, u = θ1/θ2 ≃ 3.15 × 10−4
. This result suggests that (u2
+ 2)/(u
√
u2 + 4) ≃ 3.17 × 103
, which implies that light
amplification of symmetric two similar-lens system a factor of 3.17 × 103
larger than that for a single lens (18).
We observe from (A15) that in the case of lenses with uneven masses, light amplification may be larger. Thus,
for exotic cases when θ1 ≫ θ2, the additional amplification factor scales as (u2
+ 2)/(u
√
u2 + 4) ≃ 1 + (θ2/θ1)4
+
O((θ2/θ1)6
), thus, offering very little additional amplification. For another limiting case, for θ1 ≪ θ2, the scaling is
(u2
+ 2)/(u
√
u2 + 4) ≃ θ2/θ1 + 3
8 (θ1/θ2) + O((θ1/θ2)3
), thus, offering a noticeable additional application.
To put this in context, we observe that the masses of stars in the solar neighborhood within 30 pc are within the
range4
of 0.6–2.7 M⊙. Taking the first lens to be our Sun and using (A15), this range of realistic stellar masses implies
additional light amplification factors of 4.10×103
and 1.93×103
, correspondingly. Also, we observe that the direction
of light propagation for the system with uneven masses may be somewhat important.
The estimates above are a bit misleading as they pertain only to the case the apertures that are smaller than
the first zero of the Bessel functions J0 kθ±
2 ρ
present in (A13), namely ρ+
2GL = 2.40483/(kθ+
2 ) ≃ 4.911 cm and
ρ−
2GL = 2.40483/(kθ−
2 ) ≃ 4.909 cm, which is not practical. Given the fact that both of these values are practically the
same as in the receiving lens case, the PSF behavior (21) and (A13) will be the nearly the identical (shown in Fig. 3),
except the latter expression will be θ2/θt
1 = 1/u ≃ 3.17 × 103
times larger.
Therefore, for realistic telescopes, in the context of a two-lens system, similarly to (23) and (37), we need an
aperture-averaged of the PSF, yielding the relevant light amplification factor. After evaluating the requisite integrals
and applying the large argument approximation to the involved Bessel functions, using (A13), we obtain:
hµ2GLi =
1
π(1
2 d)2
Z 1
2 d
0
Z 2π
0
PSF2GL(x) ρdρdφ =
= 2πkrg1
n
a2
+
J2
0 kθ+
2
1
2 d
+ J2
1 kθ+
2
1
2 d
+ a2
−
J2
0 kθ−
2
1
2 d
+ J2
1 kθ−
2
1
2 d
o
≃
≃
8rg1
d
a2
+
θ+
2
+
a2
−
θ−
2
=
r
2rg1
zt
4z12
dr
= hµ1GLi
z12
zt
= 9.62 × 1012
M1
M⊙
1
2
650 AU
zt
1
2
1 m
dr
z12
10 pc
, (A16)
where θ±
2 and a2
± are given by (A7) and (A9), correspondingly. Clearly, this result is identical to (37).
We see again that after averaging, the PSF (A16) does not depend on the mass of lens 2. This is because the size of
the telescope is larger than the first zero of diffraction pattern in (A13), thus satisfying the condition (38). Note that
other parameter choices, especially in the scenarios where the lens is a part of a transmitter, which is characterized by
a much larger scale of the diffraction pattern imprinted in the PSF, like in (19)–(20), does not require the averaging.
4 http://www.solstation.com/stars3/100-gs.htm
20. 20
FIG. 4: A two-lens geometry for power transmission via gravitational lensing showing the transmitter, two lenses, and the
receiver (compare to Fig. 1 for one-lens geometry). Also shown is the distance from the lens 1 to the transmitter plane, z01,
distance between the lenses, as well as that from the lens to the receiver plane, z02.
Appendix B: EM propagation in the presence of two lenses
In Sec. II A we discussed diffraction of light in the presence of one lens. Here we show how the same approach can
be extend to two and more lenses. Clearly, the amplitude of the EM field is given by the same equation (3) where for
each lens i = 1, 2 we will have a different impact parameter bi and the gravitational phase shift, ψ(bi).
In case of two lenses, the effective path length (eikonal) S(x′
, x, b1, b2) (see (4) for one lens) along a path from the
source position (−z0, x′
) to the observer position (z, x) via points (0, b1) and (0, b2) on the lens planes, to the order
of O(b4
1, b4
2)) (similarly to (4)), has the form (Fig. 4 shows overall geometry of the gravitational lensing system):
S(x′
, b1, b2, x) =
q
(b1 − x′)2 + z2
01 +
q
(b1 − b2)2 + z2
12 +
q
(b2 − x)2 + z2
02 − ψ(b1) − ψ(b2) =
= z01 + z12 + z02 +
(b2 − x′
)2
2(z01 + z12)
+
z01 + z12
2z01z12
b1 −
z01
z01 + z12
b2 −
z12
z01 + z12
x′
2
+
(b2 − x)2
2z02
− ψ(b1) − ψ(b2). (B1)
As a result, the wave amplitude on the observer plane is written as below
A(x′
, x) = A0(x′
, x)F2GL(x′
, x), (B2)
where A0(x, x) is the wave amplitude at the observer in the absence of the gravitational potentials U1, U2:
A0(x′
, x) =
E0
(z01 + z12 + z02)
exp
h
ik
z01 + z12 + z02
i
. (B3)
In the case of a pair of isolated monopole gravitational lenses, the amplification factor F2GL(x, x) is given by the
following nested diffraction integral
F2GL(x′
, x) =
(z01 + z12 + z02)
(z01 + z12)z02
keiφG2
2πi
ZZ
d2
b2 exp
h
ik
(b2 − x′
)2
2(z01 + z12)
+
(b2 − x)2
2z02
− 2rg ln kb2
i
×
×
(z01 + z12)
z01z12
keiφG1
2πi
ZZ
d2
b1 exp
h
ik
z01 + z12
2z01z12
b1 −
z01
z01 + z12
b2 +
z12
z01
x′
2
− 2rg ln kb1
i
, (B4)
where the gravitational phase factors are given as φG1 = krg ln 4k2
z01z12 and φG2 = krg ln 4k2
z12z02.
Result (B4) summarizes the logic of the procedure used in this paper. It has the non-vanishing position of the
transmitter x′
6= 0. The same structure will work for extended lenses, i.e., with higher-order multipole moments. We
take these integrals one after another, each time removing the appropriate spherical wave from the integrand. The
integral of (B4) represents the structure of the lensed EM wave amplitude obtained using the scalar theory of light
using the Fresnel–Kirchhoff diffraction formula [15] and has the form of a path integral [35] (see [12] for details.)