2. order to simplify the alignment procedure of the microscope and increase robustness. Wildanger et al.8
proposed
a dispersion based method. Two materials of different refractive indices were combined and introduced in to
the combined beam path of the excitation and depletion wavelengths, transforming the depletion beam into
a vortex but leaving the excitation beam unaffected. Another similar approach was to use a segmented wave
plate (SWP) with combinations of birefringent material place in different direction, the so-called easySTED.9
In easySTED, even the detection light can pass through the SWP although the depletion donut does not have
a uniform distribution. A SWP has also been manufactured for 3D STED.10
The main intrinsic limitation of
these approaches is their very limited bandwidth.
Here we propose a way to both remove the beam splitting required for 3D depletion and enable the generation
of the depletion beam on a broad bandwidth by a beam shaping unit based on conical diffraction (CD).
2. CONICAL DIFFRACTION FOR 3D DEPLETION
2.1 Formalism of conical diffraction
CD is an optical phenomena discovered by Hamilton in 1832 that occurs when a beam propagates along the optic
axis of biaxial crystal (BC).11
CD has been scarcely used during the past 150 years and has been considered
by many a lab curiosity. However some applications arose in the past years and CD has been successfully
used for optical trapping,12
laser mode conversion,13
superresolution imaging14,15
and polarization metrology.16
Generally, CD can be fully described by a dimensionless crystal parameter defined as ρ0 ≡ Al
ω or R0 ≡ Al.
A = 1
n2
(n2 − n1)(n3 − n2) the half cone angle, l the length of the crystal and ω the characteristic beam size
of the input beam. A and l are fixed by the crystal geometry so ω is arguably the simplest way to control ρ0.
After interaction with the crystal, two beams are created with orthogonal circular polarizations. We will refer to
them as B1 and B0, like described previously.11
The emerging beams from the crystal can be described by two
integrals:
B0(R, R0, Z) = k
∞
0
dpPa(p) exp(−
1
2
ikZP2
) × cos(kR0P)J0(kRP) (1)
B1(R, R0, Z) = k
∞
0
dpPa(p) exp(−
1
2
ikZP2
) × sin(kR0P)J1(kRP) (2)
J1 and J0 are Bessel functions of the first kind, k is the wave number, kP = kPx, Py with P << 1 (assuming
paraxiality), and a(p) is the Fourier distribution of the input beam. We remark here that the B0 component has
the same state of polarization as the input beam, whereas B1 is orthogonally polarized. In this article we will
deal with circularly polarized light which enables the simplification of the intensity distribution I = D × D∗
=
|B1 + B0|2
= |B1|2
+ |B0|211
i.e. the two beams will not interfere when propagating after the crystal.
A experimental CD 3D setup without the detection beam path is shown in fig.1. A collimated laser beam
first passes through a quarter-wave plate (QWP) to generate a circular polarization, it is then focused into a BC
that has been cut along one of the optic axes. In practice this is made with a cropped TEM00 beam to achieve a
uniform distribution in Fourier space (reciprocal space). Another lens recollimates the light. The focal lengths
of the focusing and the collimating lenses can be adjust to control both the numerical aperture at the crystal
level and the beam diameter to fill entirely the back focal plane of the microscope objective. The setup is quite
similar to the one previously described,13
except that the PSA after the crystal is removed since both emerging
beams are required for 3D depletion. The excitation is added unto the beam path with a dichroic mirror (DM).
2.2 3D depletion and ”black sphere”
We will now discuss CD 3D STED and what has previously been described as the ”black sphere”.17
The sphere pattern can easily be described by the behavior of the B0 beam emerging from the BC from a
circularly polarized input beam. To analyse B0, a polarization state analyser (PSA) can be placed after the BC
in fig.1 with the same circular polarization state as the input beam.13
We know that |B1 (0, R0, Z) |2
= 0 since
J1 (0) = 0 which is easily confirmed by changing the PSA into allowing the orthogonal circular polarization as
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3. Lens Lens DM
- n
E11__:_,//'Nuu1uuu1
Depl, laser X/4
Crysal
Crystal
exc. laser
Figure 1. Experimental setup of depletion and excitation in CD 3D STED. Only one beam path is needed for the depletion
laser compared to the conventional 3D STED.
Figure 2. Simulation of the xz profile of the three beams of STED microscopy. Bottle beam B0 to deplete in z (a), the
”donut” B1 to deplete in xy (b). The black sphere (c) is the combination of the bottle and the donut or B0 and B1. The
intensity distribution is ≈ 75% to B1 and ≈ 25% to B0.
the incoming beam (fig.2b). B1 retains the ”donut” shape with little variation over ρ0, and can therefore deplete
in XY for conventional 2D STED. B0 depletes in the axial direction, i.e. it will act as a bottle beam. A series
of simulation for several values of ρ0 was made in Matlab c
and the intensity profile in xz along the y = 0 was
examined. Previously it was stated that a bottle beam could be generated for ω << R0 but any specific value
was not given.18
Later, the value for a Gaussian beam was numerical calculated to 0.924.19
However, we find
that for a tophat beam (coherently illuminated pinhole) in Fourier space ρ0 ≈ 0.905 is the value that produces
the intensity distribution |B0|2
of a bottle beam (fig.2a). Removing the analyser results in superposition of both
the B1 and B0 beams and we observe the pattern described as the ”black sphere” (fig.2c). As previously stated,
the beams have opposite polarisation and are circularly polarized, hence they don’t interfere.
The intensity ratio between |B1|2
and |B0|2
for ρ0 = 0.907 is |B1/B0|2
≈ 3.3 as seen in fig.2. This explains the
more intense lobes in the lateral direction rather than in the axial. When considering microscopy the resolution
is then worse in the axial direction compared to the lateral because of the diffraction limit.20
According to
previous papers the smallest focal volume was estimated to be produced when the intensity ratio was 2.3 for the
lateral depletion beam compared to the axial depletion beam, or vice versa.5
The phase in the focal plane of the two beams are plotted in fig.3 and the two PPs used in conventional 3D
STED. As can be seen the phases are similar to that of the PPs or patterns imprinted on an SLM for conventional
STED. B1 (fig.3a) has a spiral shape with a complete −π to π revolution, similar to a beam generated by a 2π PP
(fig.3c). Incident left circular (LC) polarized light generates a vortex with topological charge l = 1 and for right
circular (RC) l = −1, the revolution is in the opposite direction. l = ±1 gives equivalent intensity distributions.
Similarly, for B0 (fig.3b) LC polarized light gives a phase jump of π from the inner region to 0 for the outer one,
and vice versa for RC. The PP is identical (fig.3d). Consequently, either LC or RC polarized light works for CD
3D STED. Comparatively, in this case, the BC acts as a transmissive SLM with 2 patterns imprinted for two
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4. C
Bi
lateral / vortex
Bo
axial
s
o
1
e 0.9
0.8
540 560 580 600 620 640
wavelength in nm
Figure 3. Comparison between phase profile generated by CD in the Fourier plane (a and b) and the one usually used in
STED microscopy (c and d).
Figure 4. Variation of ρ0 for a fixed geometry (Input NA, length) of a LBO crystal.
opposite circular polarizations. Such a SLM solution exists from Abberior Instruments GmbH.7
3. EXPERIMENTAL CONSIDERATIONS
Compared to B1, B0 does not have the bottle beam shape over a large range of ρ0 and hence the ”black sphere”
does not either. The variation of ρ0 for a fixed geometry (input NA, length) of a LBO crystal between 530nm
and 650nm is plotted in fig.4. In this graph the cone angle A was calculated from Sellmeier equations for the
3 refractive indexes of a LBO crystal from Castech Inc. lLBO is set to 1.86mm and the numerical aperture to
NA = 0.0093 which corresponds to ρ0 ≈ 0.907 for coherent illumination at λ = 592nm, a common depletion
wavelength for fluorophores excited at λ = 488nm. As can be seen it exhibits sharp decrease with wavelength
as expected. To determine the spectral tolerance of the black sphere, the acceptable range of λ was estimated
so the intensity in the center of the focal plane was more than 1% of the peak intensity. Variation of intensity
along x for 592nm in the focal plane is shown in fig.5a. A deviation in wavelength of [−18nm, +15nm] was
found to be acceptable, which is also within the accepted spectral range of a laser source with central wavelength
592nm. The difference(error) between the profiles 607-592 and 574-592 are shown fig.5b. The variation is quite
symmetric, the profiles at 574nm(ρ0 ≈ 0.937) and 610nm(ρ0 ≈ 0.879) both exhibit similar profiles. Dispersion
effects like this can readily be compensated by changing the incident beam waist, e.g. through changing focal
length of the lens in the setup or inserting a pinhole. Then, for one crystal with fixed geometry, the 3D depletion
pattern can be generated for a large range of wavelengths. With PPs, a change of wavelength would mean the
change to a new one made for that specific wavelength.
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5. a 1
a 0.8
'tñ
C
0.6
a
N
e; 0.4
E
o
0.01
ó 0
W
-0.04
50 200
Distance in arbitrary units x
50 100 150 200 250
Distance in arbitrary units x
- 610-592
- -.574-592
Figure 5. Variation of intensity along x in the focal plane (top). The error between 592nm with respect to 574nm
(ρ0 ≈ 0.937) and 610 nm (ρ0 ≈ 0.879) (bottom). In the focal spot the error is below 1% of total intensity
The setup is simplified; more compact, homogeneous, less vulnerable to mechanical drift, improved spectral
range compared to PPs and lossless in terms of laser intensity . It relies simply on adjusting the crystal parameter
ρ0 to the correct value in order to produce the suitable intensity distribution. This implies that by adjusting
ω, e.g. through a pinhole or beam expansion, any BC, given it is not too thick and with sufficient width and
height to fit the beam, cut perpendicular to its optic axis can transform a standard laser beam into a ”black
sphere”. The SWP used in 3D easySTED has 8 segments with different orientation and phase retardation. It
is made from 3 simpler SWP and the cemented together, designed for a specific wavelength in mind and is still
only stable over a short range.7
As for aberrations from misalignments, the position of the optic axis of a BC
is invariant. A shift in x or y position has no effect on CD produced from a BC, hence no alignment in lateral
or axial coordinates is required. In contrast, a misalignment of a PP in conventional STED gives aberrations
similar to coma.21
However, the tip and tilt of the BC will change the phase distribution in the pupil plane. The
error is similar to that of a translation of the lateral position of an SLM. Nonetheless the crystal has only these
2 degrees of freedom that can affect the performance of the microscope, while a SLM has 5 (tip,tilt,xyz position)
and a PP 4. The BC and PP have similar tolerances in movement along the optic axis. With a standard mount
the precision of tip or tilt is usually around 50rad which corresponds to an accuracy of CD to ≈ 1/1000 size of
the pupil (for a 60x NA 1.49 objective).
In a range of 577 to 607 nm the deviation of LBO is ∆β ≈ 0.07◦
for LBO. The black sphere pattern is not
limited to superresolution microscopy and could find use in any application that requires a dark focal spot e.g
optical trapping. However, this system currently lacks the versatility and fine tuning of other techniques. In
conventional 3D STED the ratio between the orthogonal components can be tuned to produce an isotropic focal
volume by reducing the axial resolution more than the lateral, which this CD 3D STED cannot do. With circular
diattenuators introduced after the crystal, the intensity ratio could be adjusted to the optimal 2.3 instead of the
fix 3.3 and thus lead to an isotropic focal volume.
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6. 4. CONCLUSIONS AND PERSPECTIVES
We have presented an application of CD to STED microscopy by removing the need for 2 beam paths for 3D
depletion distribution generation.
The system shown here can be made fully achromatic with a cascade of BCs and combinations of prisms so
that any combination of multicolor 3D STED can be performed without having to adjust geometry (beam size,
crystal length etc.). Also, with custom waveplates, the excitation wavelength could be propagated through the
cascade and emerge in a form suitable for excitation, similar to the easySTED approach. The achromatization of
the crystal axis has already been detailed in a soon-to-be-published paper in Optics Letters,22
achromatization
of the conical diffraction parameter ρ0 will shortly follow.
This work is the first step towards the generation of 3D distributions thanks to CD. It is the authors’ belief
that CD will open the way for easier and more robust beam-shaping providing that the phase distribution is
priory known. This kind of assembly could replace many waveplate-based assemblies to simplify optical systems
in different applications.
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