Light Field: New opportunities and applications

Light Fields in Ray and Wave Optics

Introduction to Light Fields:

Ramesh Raskar

Wigner Distribution Function to explain Light Fields:
Zhengyun Zhang

Augmenting LF to explain Wigner Distribution Function:
Se Baek Oh

Q&A

Break

Light Fields with Coherent Light:

Anthony Accardi

New Opportunities and Applications:

Raskar and Oh

Q&A:

All

New opportunities and
Applications
Se Baek Oh & Ramesh Raskar

Message
• LF is a very powerful tool to understand
wave-related phenomena
• and potentially design and develop new
systems and applications

3D Optical
Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 3

Outline On

wavefront coding holography 315

rendering
the screen was very large. As expected, we see (Fig. 9) th
Fraunhofer diffraction pattern.

1.1. Double-helix point spread function (DH-PSF)
A DH-PSF system can be implemented by introducing a phase mask in the Fourier plane of an
otherwise standard imaging system. The phase mask is designed such that its transmittance
function generates a rotating pattern in the focal region of a Fourier transform lens [15-18].
Specifically, the DH-PSF exhibits two lobes that spin around the opticalaperture. An animate
Figure 9: Diffraction from a square axis as shown in Fig.
1(a). Note that DH-PSF displays this experiment with of orientation with defocusappears in
of a significant change varying the aperture size over an
gaussian beam rotating PSF
extended depth. In contrast, the standard PSF presents a slowly changing and expanding
plementary material as a video. The distance from the ap
symmetrical pattern throughout the same region [Fig. 1(b)].
the screen is 1 m.
316

317 Double rectangular apertures: Next we created two r
lar apertures and probe them with the AMP. Note that we

3D Optical Fig. 1. Comparison of the (a) DH-PSF and the (b) standard PSF at different axial planes for a
Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future
system with 0.45 numerical aperture (NA) and 633nm wavelength. 4

Augmented LF
light ﬁeld
transformer

WDF LF LF LF LF
negative
radiance
Augmented LF (diffractive)
optical
element
Light
Field
LF propagation LF propagation

3D Optical

Wavefront coding
• ALF of a phase mask(slowly varying ϕ(x))
λ ∂φ
T (x, θ) = δ θ −
2π ∂x
conventional wavefront coding

extended DOF
(w/ deconvolution)

3D Optical

Holography
Recording Reconstruction
object

hologram

3D Optical

Holography
laser
object

object wave

hologram

3D Optical

Holography
laser
object

object wave
reference
wave hologram

3D Optical

Holography
laser
object

object wave
reference
wave hologram

hologram

3D Optical

Holography
laser
object

object wave
reference reference
wave hologram wave

hologram

3D Optical

Holography
laser
object virtual
image

object wave
reference reference
wave hologram wave

hologram

observer
3D Optical

Holography
laser
object virtual
image

object wave real image
reference reference
wave hologram wave

hologram

observer
3D Optical

Holography
• For a point object

recording

reconstruction
3D Optical

Rendering Online Submi

the screen was very large. As expected, we see (Fig. 9) the typical
315 3

• Using virtual
Fraunhofer light sources in photon mapping
diffraction pattern. 3

3

3

3

3

white light 3

3

3

3

3

3

3
rectangular
aperture
screen
Augmented Photon Mapping for Wavefront Transmission Effects
3D Optical
Figure 9: Diffraction S. B. Oha square aperture. An animated version
from et al. (2009)
Se Baek Oh Systems Group of this experiment with varying the aperture sizePresent and Future 9
CVPR 2009 - Light Fields: appears in the sup-

Gaussian Beam
(from a laser pointer)
• Beam from a laser
• a solution of paraxial wave equation
20 mm beam
width

20 m distance

3D Optical

Gaussian Beam
• ALF (and WDF) of the Gaussian Beam is also
Gaussian in x-θ space
θ x

z
x

3D Optical

Gaussian Beam
x-θ space z-x space

20 mm beam 20 m distance
width

3D Optical

Unusual PSF for depth
Double-helix point spread function (DH-PSF)

from defocus
-PSF system can be implemented by introducing a phase mask in the Fourier plane of an
wise standard imaging system. The phase mask is designed such that its transmittance
on generates a rotating pattern in the focal region of a Fourier transform lens [15-18].
fically, the DH-PSF exhibits two lobes that spin around the optical axis as shown in Fig.
Note that DH-PSF displays a significant change of orientation with defocus over an standard PSF DH PSF
ded depth. In contrast, the standard PSF presents a slowly changing and expanding
Defocus circle with distance
metrical pattern throughout the same region [Fig. 1(b)].

1µm 1µm
3D positions

3
5
2
1
4

Fig. 1. Comparison of the (a) DH-PSF and the (b) standard PSF at different axial planes for a
system with 0.45 numerical aperture (NA) and 633nm wavelength.

hile Prof. Rafael Piestun’s group provide valuable insight on wave propagation
analytical solutions for helical beams Courtesy of S. R. P. Pavani
and Univ. of Colorado@Boulder
can be used in photon-unlimited applications [16], they do not provide the high-
ency transfer functions required for photon-limited systems. Hence, we use a design that U. of Colorado@Boulder
nes the helical pattern Optical specific axial range of interest to attain high efficiency
3D
Se Baek Oh standardto aGroup
ms [15]. Unlike Systems CVPR 2009 - Light
and astigmatic PSFs, the DH-PSF concentrates its energy in its
Fields: Present and Future 13

Reference
• “Wave propagation with rotating intensity distributions,” Y.Y. Schechner, R. Piestun,
and J. Shamir, Phys. Rev. E 54: R50–R53 (1996)
Concept • “Wave fields in three dimensions: analysis and synthesis,” R. Piestun, B. Spektor, and
J. Shamir, J. Opt. Soc. Am. A 13:1837-1848 (1996)
• “Propagation-invariant wave fields with finite energy,” R. Piestun,Y.Y. Schechner, and
J. Shamir, J. Opt. Soc. Am. A 17:294-303 (2000)

Implement • "Depth from diffracted rotation," A. Greengard,Y.Y. Schechner, and R. Piestun, Opt.
Lett., 31(2):181-183, (2006)
ation • "High-efficiency rotating point spread functions", S. R. P. Pavani and R. Piestun, Opt.
Express, 16(5):3484-3489, (2008)

• “Three-Dimensional Single-Molecule Fluorescence Imaging Beyond the Diffraction
Limit Using a Double-Helix Point Spread Function,” S. R. P. Pavani, M. A. Thompson,
Microscope J. S. Biteen, S. J. Lord, N. Liu, R. I. Twieg, R. Piestun, and W. E. Moerner, PNAS, 106:
2995, (2009)
• “Three-dimensional localization with nanometer accuracy using a detector-limited
double-helix point spread function system, “ S. R. P. Pavani, A. Greengard, and R.
Piestun, APL (2009) In Press

3D Optical

Gauss-Laguerre mode
2 1/2
w(ˆ) = w0 1 + z
z ˆ
√
U(r, t) = u(r) exp [i(kz − ωt)] 2w0 w
0
Orthogonal basis in the cylindrical coordinate
unm (r) = G(ˆ, z )Rnm (ˆ)Ψm (φ)Zn (ˆ)
ρ ˆ ρ z z0

(0,0): Gaussian beam ρ z
ρ=
ˆ z=
ˆ
w(ˆ)
z z0
w0
G(ˆ, z ) =
ρ ˆ exp −ˆ2 exp iˆ2 z exp (−iψ(ˆ))
ρ ρ ˆ z πw0 2
w(ˆ)
z z0 =
√ |m| λ
|m|
Rnm (ˆ) =
ρ 2ˆ
ρ L(n−|m|)/2 (2ˆ2 )
ρ
Zn (ˆ) = exp {−inψ(ˆ)}
z z
Ψ(φ) = exp(imφ)
ψ(ˆ) = arctan(ˆ) : Gouy phase
z z
3D Optical

Rotating PSF
GL modal plane

• Rotating beams 10

• Superposition along a straight line
n

• Rotation rate related to slope of 5

line
• Both intensity and phase rotate
0

• Maximum rotation rate in Rayleigh -10 -5 0
m
5 10

range
intensity

Courtesy of S. R. P. Pavani
3D Optical

Rotating PSF

Rotating PSF HER-PSF

1.84% 57.01%
Courtesy of S. R. P. Pavani
3D Optical

Conceptually...
y

x
z

3D Optical

Conceptually...
y

x
z

other modes need to be balanced...
3D Optical

WDF (ALF) of (1,1) order
GL modal plane
10
intensity

n
5

0
-10 -5 0 5 10
m

R. Simon and G. S. Agarwal, "Wigner representation of
Laguerre-Gaussian beams", Opt. Lett., 25(18), (2000)
3D Optical

intensity in x-y

y

x

3D Optical

intensity in x-y WDF in θx- θy

y
θy

x θx

3D Optical


y
θy

x θx

WDF in θx- θy

θy

3D Optical θx

WDF in θx- θy

θy
θx

y
θy

x θx

WDF in θx- θy

θy

3D Optical θx

Future direction

• Reﬂectance (e.g. BRDR/BTF) model
• Tomography & Inverse problems
• Beam shaping/phase mask design by ray-
based optimization
• New processing w/ virtual light source
3D Optical

Space of LF representations
Time-frequency representations
Phase space representations
Quasi light ﬁeld

Quasi light ﬁeld

incoherent

coherent

Quasi light ﬁeld

Traditional
light ﬁeld

incoherent

coherent

Quasi light ﬁeld

WDF

Traditional
light ﬁeld

incoherent

coherent

Quasi light ﬁeld

Observable
LF
WDF

Traditional
light ﬁeld

incoherent

coherent

Quasi light ﬁeld

Observable
LF
WDF
Augmented
LF
Traditional
light ﬁeld

incoherent

coherent

Quasi light ﬁeld

Observable
LF
WDF
Augmented
LF
Traditional
light ﬁeld

incoherent
Rihaczek
Distribution
Function

coherent

Quasi light ﬁeld

Other LF
representations

Observable
LF
WDF
Augmented
LF
Other LF
Traditional
representations light ﬁeld

incoherent
Rihaczek
Distribution
Function

coherent

Property of the Representation

Constant along Interference
Non-negativity Coherence Wavelength
rays Cross term

Traditional LF always always only zero no
constant positive incoherent

nearly always any
Observable LF
constant positive coherence any yes
state

only in the positive and
Augmented LF
paraxial region negative any any yes

only in the positive and
WDF
paraxial region negative any any yes

Rihaczek DF no; linear drift complex any any reduced

3D Optical

Beneﬁts & Limitations of the
Representation
Adaptability
Ability to Modeling Simplicity of
to current Near Field Far Field
propagate wave optics computation
pipe line

Traditional
Light Fields x-shear no very simple high no yes

Observable not x-
yes modest low yes yes
Light Fields shear
Augmented
Light Fields x-shear yes modest high no yes

WDF x-shear yes modest low yes yes

better than
Rihaczek DF x-shear yes WDF, not as low no yes
simple as LF
3D Optical

Conclusions
• Wave optics phenomena can be understood
with geometrical ray based representation
• There are many different phase-space
representations
• We hope to inspire researchers in computer
vision/graphics as well as in optics graphics
to develop new tools and algorithms based
on joint exploration of geometric and wave
optics concepts

3D Optical

Light Field: New opportunities and applications

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