Large laser written liquid crystal devices for spherical aberration correction

We present a new liquid crystal-based device for correction of spherical aberration, which is commonly observed in microscopy and related methods. The device is significantly larger than previous direct laser written aberration correctors, measuring 1 mm across. The device operates in transmission mode for easy integration into the optical path and is capable of continuous greyscale tuning of up to a total amplitude of 2π rad. This device could present a cost effective and simpler to use alternative to traditional wavefront modulation technologies used in adaptive optics.


Introduction
Adaptive optics (AO) is already widely used across the imaging community, in fields as diverse as astronomy, microscopy, and ophthalmology 1 .AO is used to compensate aberrations that detrimentally affect image quality.In this field, aberrations are described as distortions to the propagating wavefront, and these distortions are usually characterized in terms of Zernike polynomials.Such optical aberrations are generally corrected using spatial light modulators (SLMs), deformable mirrors (DMs), or liquid filled deformable phase plates.While these devices are powerful, they are often over specified for applications in microscopy, especially when only one or a few modes need to be corrected.This, combined with their high cost in terms of bulk and expertise, often prevents their adoption into many imaging modes that could otherwise significantly benefit from adaptive optics.
One such example is that of spherical aberration.The lowest order spherical aberration is a fourth order Zernike polynomial ( 4 0 ).The aberration is frequently found in microscopy as it can be introduced by a refractive index mismatch between the sample and the immersion medium of the imaging objective lens 2 .Without adaptive optics, spherical aberrations may be partially corrected using a correction collar, which is incorporated into some highend objective lenses whereby elements in the compound lens are moved mechanically.The complexity of these objectives, however, means that they are expensive to implement.While the spherical aberration mode is radially symmetric, it is nevertheless difficult to correct with other methods because the wavefront does not monotonically increase or decrease from the center of the mode.As such, this mode is most often corrected using an SLM or a DM.Unfortunately, both technologies are costly and difficult to incorporate into many optical systems as they are quite large optical components that are designed to function in a reflective, rather than transmissive, configuration.
Direct laser writing (DLW) in liquid crystals (LC) may offer an alternative path forward.Nematic LCs are a phase of matter wherein the molecules are not spatially fixed but remain rotationally aligned to their neighboring molecules.Thus, when an electric field is applied to a nematic LC, the LC molecules will, on average, attempt to rotate to align with the electric field, allowing for fine control over the effective refractive index of the LC layer.This average orientation of the LC molecules is generally described by the spatially varying director  ⃗ (, , ).Complex spatial modulation patterns are usually implemented in SLMs using pixelated electrodes, which require sophisticated drive electronics.To avoid the need for complex drive electronics, many attempts at harnessing non-pixelated driving for adaptive optics have been made in the past.This was done largely by either shaping the electrode itself, or by curving the substrate of the LC device [3][4][5] .However, while these devices have been versatile for tunable focal or cylindrical lenses, they struggle to create more complex phase patterns, where the phase profile does not monotonically increase or decrease from the center of the device.However, by mixing the LC with a reactive mesogen and photoinitiator, we can selectively polymerize the LC bulk using two-photon initiated polymerization, effectively fixing the orientation of the LC molecules (the director) regardless of a later change in the applied electric field amplitude.Moreover, by laterally varying the height of this fixed polymer bulk, we can in turn vary the retardance to create complex phase patterns in the LC layer.An illustration of this technique is shown in Figure 1, which has been exploited previously to create switchable diffraction gratings, holograms, and other arbitrary patterns [6][7][8][9] .
Recently, in Xu et al. 10 , we were able to further demonstrate that by continuously varying the polymerization depth within the LC layer, we could even successfully recreate Zernike polynomials and generate continuously tunable aberration correction devices.These results were very promising, demonstrating the high versatility and applicability of this technique.A notable drawback in the previous demonstration was that the devices measured only 250 μm in diameter, which was constrained mostly by the fabrication time, rather than any fundamental limitation arising from the manufacturing or device technology.
Figure 1.The laser writing process.First, an empty cell with indium tin oxide electrodes (green) and antiparallel polyamide rubbing (orange) is filled with a LC/monomer mixture (purple).b) a high voltage is applied to the cell.c) The mixture is polymerized with two-photon initiated polymerization, forming a polymer bulk (blue).d) the high voltage is removed, allowing the liquid LC to relax, while the polymerized LC remains in its high voltage state.
In the present paper, we tackle the challenge posed by the commonly occurring spherical aberration by creating a new aberration correction device.The spherical mode corrector device presented is significantly larger than the correctors described in Xu et al. 10 , at 16 times the area, with a large total diameter of 1000 μm, which would allow the device to be much more easily integrated into practical optical paths.Furthermore, a demonstration of devices of this size indicates that the device size is not limited by the manufacturing process.As a result, we expect that the size could readily be increased, potentially even up to several mm diameter for easy integration with commercial optical components.The device operates fully in transmission mode, and so can be included into optical assemblies without the need for beam folding.The device is also extremely easy to use, as it is operated only with a single ac frequency electrode pair tuned between 0 and 10 V.

Simulation and Experiment
Our design methodology closely follows that described in Xu et al. 10 .First, we used a finite difference simulation to determine the effective refractive index of the LC bulk at a variety of voltages, relative to the polymerization depth.This was done by first developing a finite difference solver to calculate the orientation of the LC director by solving the one-elastic constant simplification of the relevant Euler-Lagrange equation where  is the angle of the director  ⃗ relative to the substrate of the LC cell,  the one elastic constant, and Γ(x, y, z) is the spatially varying driving term, related to the electric field  and the parallel and perpendicular dielectric permittivities of the LC mixture at the applied ac frequency (ϵ ∥ and ϵ ⊥ respectively) via Γ(, , ) = (ϵ ∥ −ϵ ⊥ )ϵ 0   (, , ) 2 .Since the electric flux density  = ϵ zz  is constant throughout the LC layer thickness, the spatially varying electric field of the LC can be calculated based on the LC orientation via In such a case, the total phase shift experienced by light transmitted through the device is simply expressed as where (θ, , , ) is the spatially varying refractive index of the birefringent LC bulk, and the refractive index of the polymer bulk is approximated as   since the polymer network is formed at a high voltage locking-in a homeotropic alignment.For a 20 μm thick LC layer, the average refractive index of the device relative to the thickness of the polymerized region and voltage is shown in Figure 2. The maximum polymer thickness of the device that can be created in a single pass of the laser is 8 μm, which is governed by the voxel size of the DLW process.In turn, this is related to the numerical aperture (NA) of the objective lens and the illumination wavelength.This does not, however, account for any diffusion of the polymer network.
Figure 2. Average refractive index of the LC bulk relative to the non-polymerized LC thickness through a 20 μm thick LC layer.The two-photon polymerization laser writing process could achieve a maximum thickness of 8 μm in a single pass, while a minimum polymer thickness of 1 μm was used to ensure uniform director alignment.
Note that at all voltages, the relationship between polymer depth and retardance is expected to be linear, with a higher polymerization depth universally corresponding to a lower retardance.This suggests a highly linear voltage response within the device, and that the device will produce a similar grey-scaled phase profile at all voltages.By determining this relationship, we could then determine the necessary polymerization depth at each point of the LC bulk required to produce the desired wavefront retardance via the relationship where (, ) is the desired phase pattern, (  ) is the polymerization thickness (  ) dependent refractive index at 0 V (where we want the maximum phase profile to be observed), k is the wavenumber, and   is the total thickness of the LC layer.The correction terms   are included to compensate for mechanical tilt and curvature inherent within the device substrate.While these terms proved sufficient for the current writing size, for larger structures additional   terms may need to be included.For a spherical Zernike polynomial device, the ideal polymer height, when no tilt is present, is shown in Figure 3.We prepared a nematic LC mixture consisting of 79 wt.% of the nematic LC, E7 (Synthon Chemicals), 20 wt.% of the reactive mesogen, RM257 (Synthon Chemicals), and 1 wt.%IR819 (Ciba-Geigy) photoinitator.The device was then manufactured by filling a glass cell (sourced from Instec that had an air gap of 20 μm) with this LC solution, with a lead wire soldered to each electrode respectively using indium solder.Laser writing was performed by mounting this device on an Aerotech ANT95XY 2D /ANT95v motorized three axis translation stage before it was polymerized using a Spectra Physics Mai-Tai Titanium-Sapphire laser ( = 780 nm) providing 100 fs pulses at 80 MHz repetition rate, focused through a 0.45 NA lens.Polymerization induced while driving at a high voltage, by applying 100 V RMS square wave with a frequency of 1 kHz using a Tektronix AFG3021 signal generator and an FLC F10AD amplifier.This high applied voltage ensured that the LC bulk was polymerized with a homeotropic director alignment.The laser writing was carried out by writing in a raster pattern.This device was then imaged using polarized optical microscopy (POM), which was achieved with an Olympus BX51 polarizing optical microscope paired with a QImaging R6 Retiga Camera.The magnification was set to 20× using an Olympus LMPLFLN20x objective lens.A narrow band 660 nm filter was applied to the imaging source to ensure a monowavelength intensity response and avoid crosslinking the reactive mesogen after the laser writing process had been completed.

Results and Discussion
The manufactured device is shown in Figure 4, imaged at four different voltages of 0 V, 1.2 V, 2 V, and 10 V, along with the ideal expected transmission at 0V for the device.Note that the transmitted intensity  is related to the retardance of the device via the relation  = sin 2 (∆/2) where Δ is the retardance.As the refractive index of the fast axis is expected to be fixed, this is a good proxy for measuring the laterally variant phase pattern of the spherical Zernike polynomial device.As shown in the figure, the device expresses its maximum amplitude at 0 V, and as the voltage is increased, the amplitude of the phase pattern decreases before ending with a constant retardance at around 10 V. Images are taken at four different voltages of 0 V, 1.2 V, 2.0 V, and 10.0 V and compared to the ideal transmission of such a device, which is the first image shown on the left.The detected intensity is related to the phase shift via  =  2 (/2).
The small features observed in the images are the 20 m spacer beads.
To further confirm the functionality of our device we compared the normalized imaged transmission of the device along the green line in Figure 4 with the simulated transmission of an ideal device.This is shown in Figure 5.The device performance matches well to simulation, which implies that the manufactured height of the polymerized region within the LC layer is very close to the ideal height.Furthermore, this indicates that the measured phase pattern is very close to the ideal spherical phase pattern, indicated by the linear refractive index response illustrated in Figure 2.These results also suggests that the measured maximum magnitude of the spherical corrector matched well to the designed phase shift of 2π rad. Figure 5. Transmission of the spherical correction device placed between crossed polarizers, imaged at 660 nm, and taken at various voltages.The ideal transmission (orange) is compared to the actual measured transmission at the same voltages (pink).The close match between the two sets of lines indicates the accuracy of the manufactured method and suggests a phase match very close to the ideal phase.
Even though this device is sixteen times larger in area than previous devices of this type, it maintains a similar degree of printing accuracy and linear response.We conclude then that our methodology is likely scalable to devices of much larger size and shows that in the future it would be easy to integrate our devices into commercially available optical components and systems.The significantly increased size alleviates the impact of complications commonly seen in adaptive optics, such as beam shrinking or beam folding, improving ease of integration, and encouraging broader adoption of these technologies.

Conclusion
In this paper, we have presented a new DLW based spherical aberration correction device.The device is significantly larger than similar devices from before, measuring 1 mm across, four times the diameter and sixteen times the area of any previous attempts.Despite this increase in size, the device demonstrates similar accuracy to smaller devices reported previously and shows good agreement with simulations, suggesting that the device does indeed provide accurate and near-ideal spherical phase patterns.Comparisons between the ideal and measured transmissions further suggest that the device is able to create a total phase shift of 2π rad at 660 nm.The device operates in transmission for easy integration into the optical path and is operated by a single electrode pair tuned between 0 and 10 V.These findings imply that larger DLW devices could be precisely manufactured to be multiple millimeters in diameter, meaning that they could be integrated into optical systems with simple configurations alongside commercial optical components.

Figure 3 .
Figure 3. Ideal polymer height for a 1mm spherical correction device, as indicated by the color bar on right, shown in μm.The device is written at a high voltage in a 20 μm thick cell with antiparallel rubbing.

Figure 4 .
Figure 4. Monochromatic POM images of the spherical correction device taken at 660 nm through crossed polarizers.Images are taken at four different voltages of 0 V, 1.2 V, 2.0 V, and 10.0 V and compared to the ideal transmission of such a device, which is the first image shown on the left.The detected intensity is related to the phase shift via  =  2 (/2).The small features observed in the images are the 20 m spacer beads.