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Amplitude Zone Plate in Adaptive Optics: Proposal of the Principle

Amplitude Zone Plate in Adaptive Optics: Proposal of the Principle hv photonics Communication Amplitude Zone Plate in Adaptive Optics: Proposal of the Principle Vasily Matkivsky * , Alexsandr Moiseev, Pavel Shilyagin and Grigory Gelikonov Institute of Applied Physics RAS, 46 Ulyanov Street, 603950 Nizhny Novgorod, Russia; aleksandr.moiseev@gmail.com (A.M.); paulo-s@mail.ru (P.S.); gelikon@ufp.appl.sci-nnov.ru (G.G.) * Correspondence: vasilymat@mail.ru Abstract: One of the main elements in hardware-based adaptive optics systems is a deformable mirror. There is quite a large number of such mirrors based on different principles and exhibiting varying performance. They constitute a significant portion of the cost of the final optical devices. In this study, we consider the possibility of replacing an adaptive mirror with the adaptive amplitude Fresnel zone plate, implemented using a digital light-processing matrix. Since such matrices are widely used in mass industry products (light projectors), their costs in large batches are 1–2 orders of magnitude lower than the cost of inexpensive deformable mirrors. Numerical modeling for scanning an optical coherence tomography system with adaptive optics is presented. It is shown that wavefront distor- tions with high spatial frequencies and large amplitudes can be corrected using an amplitude Fresnel zone plate. The results are compared with piezoelectric and microelectromechanical system mirrors. Keywords: optical coherence tomography; aberration compensation; adaptive optics; Fresnel zone plate; diffractive optics 1. Introduction Until now, the main area of application of optical coherence tomography (OCT) sys- Citation: Matkivsky, V.; Moiseev, A.; tems has been the study of the human eye fundus. One of the main factors limiting the Shilyagin, P.; Gelikonov, G. lateral resolution of OCT systems is eye optical aberrations. Adaptive optics made it pos- Amplitude Zone Plate in Adaptive sible to significantly increase the lateral resolution of OCT systems [1]. Recently, several Optics: Proposal of the Principle. methods were developed to compensate for aberration effects in postprocessing [2–8]. Such Photonics 2022, 9, 163. https:// methods also have a number of disadvantages (either they may not work well enough, doi.org/10.3390/photonics9030163 they require large calculations, or both). Compensation for aberrations is of particular Received: 29 November 2021 importance in scanning OCT systems, which, at the moment, have the greatest applicability Accepted: 18 February 2022 in practice [9]. In scanning systems, there is a confocal factor. This means that, at each Published: 7 March 2022 moment in time, only those photons are registered that came from a certain area of the object. The presence of aberrations leads both to a decrease in the illumination of this Publisher’s Note: MDPI stays neutral region (probe beam aberrations) and to the loss of some of the photons when they enter the with regard to jurisdictional claims in receiving system [10]. Therefore, for scanning OCT systems, the use of traditional adaptive published maps and institutional affil- iations. optics (at least the use of wavefront correctors) turns out to be desirable and, possibly, the best option for obtaining high resolution and for the signal-to-noise ratio [11]. On the other hand, the cost of correctors is thousands and tens of thousands of dollars, which leads to an increase in the cost of OCT systems. In this study, we propose the use of adaptive Copyright: © 2022 by the authors. amplitude diffraction lenses. Licensee MDPI, Basel, Switzerland. In addition to adaptive mirrors, it is worth mentioning the class of devices known This article is an open access article as adaptive lenses [12,13]. Most of them can only change their focal length, which is not distributed under the terms and suitable for wavefront correction. However, the phase zone plate has been successfully conditions of the Creative Commons implemented in liquid crystal devices [14–16], which allows for various effects, including Attribution (CC BY) license (https:// compensation of aberrations. It is worth noting that the cost of liquid crystal devices is creativecommons.org/licenses/by/ currently quite high and comparable to the cost of adaptive mirrors. In [17], an adaptive 4.0/). Photonics 2022, 9, 163. https://doi.org/10.3390/photonics9030163 https://www.mdpi.com/journal/photonics Photonics 2022, 9, x FOR PEER REVIEW 2 of 12 Photonics 2022, 9, 163 2 of 12 currently quite high and comparable to the cost of adaptive mirrors. In [17], an adaptive lens based on a silicone elastomer was demonstrated. The first few Zernike aberrations (astigmatism, coma, and spherical aberration of the third order) can be compensated with lens based on a silicone elastomer was demonstrated. The first few Zernike aberrations it. (astigmatism, coma, and spherical aberration of the third order) can be compensated with it. In this study, we analyzed the possibility of using an adaptive Fresnel amplitude In this study, we analyzed the possibility of using an adaptive Fresnel amplitude zone zone plate (AZP) based on a digital light-processing (DLP) matrix. This enabled the fo- plate (AZP) based on a digital light-processing (DLP) matrix. This enabled the focusing cusing of both the probed and the received radiation, and aberrations with large ampli- of both the probed and the received radiation, and aberrations with large amplitudes and tudes and with high spatial frequencies could be compensated. The loss of light (on re- with high spatial frequencies could be compensated. The loss of light (on reflection and flection and diffraction losses, which will be discussed below) on the DLP matrix was diffraction losses, which will be discussed below) on the DLP matrix was compensated by compensated by focusing the radiation when it was detected by the receiving system, focusing the radiation when it was detected by the receiving system, thereby improving thereby improving the overall signal-to-noise ratio and image quality. the overall signal-to-noise ratio and image quality. Amplitude zone Amplitude plates and zone photon plates ic sieves and photonic are main sieves ly usear d for fo e mainly cusin used g short for -focusing wave short-wave radiation or foradiation r obtaining or polychroma for obtaining tic im polychr agesomatic [18–21images ]. In [22] [18 , photonic –21]. In si [22 ev ],es wer photonic e sieves were used instead of used an ar instead ray ofof man icrolen array ses in of micr Sh olenses ack–Hain rtmann Shack–Hartmann sensors. The sensors. authors we The authors re were not aware of any previous work using amplitude zone plates to compensate for aberrations in not aware of any previous work using amplitude zone plates to compensate for aberra- tions in ophthalm ophthalmic ic OCT aOCT t the time at the of time writof ing. writing. 2. Materials and Methods 2. Materials and Methods 2.1. Optical Scheme 2.1. Optical Scheme Consider the object arm of the spectral-domain OCT device (Figure 1). The radiation Consider the object arm of the spectral-domain OCT device (Figure 1). The radiation from the fiber exits the S1 plane and is focused in the S6 plane and is reflected and returned from the fiber exits the S1 plane and is focused in the S6 plane and is reflected and re- to the S1 plane. The fraction of power to the fiber mode was determined. turned to the S1 plane. The fraction of power to the fiber mode was determined. Figure 1. Optical scheme for the deformable mirror/Fresnel plate modeling. Radiation is emitted Figure 1. Optical scheme for the deformable mirror/Fresnel plate modeling. Radiation is emitted from the fiber in the center of the S1 plane. In the case of deformable mirror modeling, a combined from the fiber in the center of the S1 plane. In the case of deformable mirror modeling, a combined phase mask (deformable mirror plus the lens with f2 focal length) is placed in the S2 plane. In the phase mask (deformable mirror plus the lens with f2 focal length) is placed in the S2 plane. In the case case of DLP modeling, a corresponding binary mask is placed in the S2 plane. The combination of of DLP modeling, a corresponding binary mask is placed in the S2 plane. The combination of lenses lenses in the S3 and S4 planes transfers the field from the S2 to the S5 plane. A lens in the S5 plane in the S3 and S4 planes transfers the field from the S2 to the S5 plane. A lens in the S5 plane focuses focuses radiation in the S6 plane. Aberrations are inserted into the S5 plane (marked with a blue dotted line). radiation in the S6 plane. Aberrations are inserted into the S5 plane (marked with a blue dotted line). For simulation, a single-mode fiber with a mode field diameter of 10 microns was For simulation, a single-mode fiber with a mode field diameter of 10 microns was assumed at plane S1. Using the angular spectrum method [18], fields in parallel planes assumed at plane S1. Using the angular spectrum method [18], fields in parallel planes were calculated. Modeling was performed for monochromatic radiation. Let us denote the were calculated. Modeling was performed for monochromatic radiation. Let us denote wavelength as l, the spatial sampling rate dx, the distance between the planes z, and the the wavelength as 𝜆 , the spatial sampling rate 𝑑𝑥 , the distance between the planes 𝑧 , and lz transverse size of the plane L. Then, under the condition < , the equation to calculate the transverse size of the plane 𝐿 . Then, under the condition < , the equation to the field can be expressed as follows: calculate the field can be expressed as follows: E (x, y) = iFT[FT[E (x, y)]FT[h(x, y)]], (1) ( ) ( ) ( ) 𝐸 𝑥, 𝑦 = 𝑖 𝐹𝑇 2 𝐸 𝑥, 𝑦 ∙ 𝑇𝐹 ℎ1 𝑥, 𝑦 , (1) h i ikz e ik 2 2 where h(x, y) = exp x + y is the impulse response function, (x, y) are the trans- where ℎ(𝑥, y ) = exp (𝑥 +𝑦 ) is the impulse response function, (𝑥, y ) are the ilz 2z verse coordinates, k = 2p/l is the wave number, i is an imaginary unit, and FT and iFT transverse coordinates, 𝑘= 2𝜋/𝜆 is the wave number, 𝑖 is an imaginary unit, and lz are forward and inverse Fourier transforms, respectively. In the case when  , instead of and 𝑖𝐹𝑇 are forward and inverse Fourier transforms, respectively. In the case when ≥ Equation (1), the Fresnel transfer function is used: , instead of Equation (1), the Fresnel transfer function is used: E x, y = iFT FT E x, y H k , k , (2) ( ) [ ( )] 2 1 x y 𝐸 (𝑥, 𝑦 ) = 𝑖 𝐹𝑇 𝐸 (𝑥, 𝑦 ) ∙ 𝐻(𝑘 ,𝑘 ) , (2) h  i ikz 2 2 where H k , k = e exp iplz k + k . x y x y 𝐹𝑇 𝐹𝑇 𝐹𝑇 𝜆𝑧 Photonics 2022, 9, 163 3 of 12 The lens, mirror, zone plate, or aberration action were simulated by multiplying the field by the corresponding two-dimensional function (see Section 2.2). In S2, the plane field was multiplied by the function of deformable mirrors of different types or by the DLP matrix. In the case of mirrors, the field in the S2 plane was additionally multiplied by the function of a collimating lens with a focal length of f2. The system of lenses in the S3/S4 planes transferred the field from the S2 plane to the S5 plane with the necessary magnification. In the S5 plane, the field was multiplied by aberration (blue broken line in Figure 1) and lens functions. The aberrated image of the fiber was formed at plane S6. Then, the optical system was mirrored relative to plane S6, and the field was propagated from S6 back to S1. To find the energy returned to the fiber, the scalar product of the mode and the field was calculated as follows: M(x,y)E (x,y) T = R R 2 2 jM(x,y)j jE(x,y)j (3) K = TT where M is the field at the exit from the fiber (mode), E is the complex field in the fiber plane, * denotes the complex conjugation, and K is the fraction of energy returned to the fiber. 2.2. DLP Matrix and Amplitude Fresnel Zone Plate An amplitude zone plate is an amplitude mask that divides the field into zones so that transmittance radiation is focused due to diffraction. Optical aberrations can be corrected with some modifications in the mask, in addition to focusing. As is known, an ideal lens introduces a parabolic factor into the phase front: 2 2 F(x, y) = (x + y ) (4) 2 f 2p where k = / is the wave number, f is the focal length of the lens, and (x, y) are coordinates in the plane of the lens. Supposing that additional distortions are introduced into the wavefront (for example, by the human eye system), then: q(x, y) = F(x, y) + j(x, y) (5) where j(x, y) is the wave aberration function and q(x, y) is the total phase change. Then, using the function q(x, y), the amplitude zone plate as a binary mask can be implemented according to this formula: q(x, y) > np AZP = 1, f or ; AZP = 0, otherwise (6) q x, y < n + 1 p ( ) ( ) where n = 0, 2, 4 . . .. The DLP matrix consists of micromirrors that can assume two positions: rotated through the angle a anda relative to the normal. Accordingly, rays incident on the matrix will be divided into 2 directions. If the optical system is placed in one of the directions, then the reflected light from the DLP can be considered as having passed through the binary amplitude mask. The principle of operation of DLP is illustrated in Section 3.2. Consider the limitations associated with the pixel structure of DLP. We can easily find the equation for the distance Dr between two zones for a lens with a focal length 1 2 f :Dr = / f lr + o Dr . The zone size decreases as the radius (beam/aperture size) increases. For a DLP matrix with a certain element size, the maximum radius of the amplitude mask r can be found. For radius r > r , the Fresnel zone size will be less max max than the DLP element size. The DLP matrix DLP4500NIR from Texas Instruments with a square element size of 7.6 m (with 912 1140 elements) was used in the study. The choice of this matrix was due to its design for infrared radiation, and it is a commercially available Photonics 2022, 9, x FOR PEER REVIEW 4 of 12 Photonics 2022, 9, 163 4 of 12 available device. Using the equation above, the maximum beam diameter was approxi- mately 17 mm, while the DLP size was about 7 × 8.7 mm. In addition, there are two ef- fects: (1) Fresnel zones have circle geometry, while DLP elements are square. This leads device. Using the equation above, the maximum beam diameter was approximately 17 mm, to errors for the radii, including those less than 𝑟 ; (2) in real matrices, the size of the while the DLP size was about 7  8.7 mm. In addition, there are two effects: (1) Fresnel micromirror is about 10% less than the pixel size (there is a gap between the elements). To zones have circle geometry, while DLP elements are square. This leads to errors for the simulate the first effect, the field in the DLP matrix plane was sampled with a sample radii, including those less than r ; (2) in real matrices, the size of the micromirror is about max interval of 𝑑𝑥 equal to the DLP element size. The second effect was not taken into ac- 10% less than the pixel size (there is a gap between the elements). To simulate the first count at this stage. We assume that this will lead to a decrease in radiation intensity by effect, the field in the DLP matrix plane was sampled with a sample interval of dx equal about 20% due to the area of element reduction. to the DLP element size. The second effect was not taken into account at this stage. We assume that this will lead to a decrease in radiation intensity by about 20% due to the area 2.3. Focusing the Beam Using the Fresnel Plate of element reduction. This section presents the results of modeling the focus of a Gaussian beam using a 2.3. Focusing the Beam Using the Fresnel Plate DLP matrix DLP4500NIR from Texas Instruments (the size of a square element was 7.6 This section presents the results of modeling the focus of a Gaussian beam using a μm with 912 × 1140 elements). A Gaussian beam was simulated, which was apertured by DLP matrix DLP4500NIR from Texas Instruments (the size of a square element was 7.6 m a circular diaphragm at a field level of 1/𝑒 and focused by a lens or zone plate with a with 912  1140 elements). A Gaussian beam was simulated, which was apertured by a focal length of 100 mm. The results are shown in Figure 2. The radiation wavelength was circular diaphragm at a field level of 1/e and focused by a lens or zone plate with a focal 850 nm. length of 100 mm. The results are shown in Figure 2. The radiation wavelength was 850 nm. Figure 2. Beam focusing. (a) The modulus of the field passed through the Fresnel zone plate; Figure 2. Beam focusing. (a) The modulus of the field passed through the Fresnel zone plate; (b) point sou (b) point rce sour ince -focu in-focus s profi prlofiles es (x-axis (x-axis isis meas measur ured ed i innmillimeters). millimeters). Orange Orange line—using t line—using the zone he zone plate, b plate, lue—using an ordinary le blue—using an ordinary lens ns of of the thesa same me focal focal leng length; th; ( (cc ,d ,d ))images images of of a a point point sour sou ce rce using using the the lens and the zone plate, respectively; (e,f) images matching images (c,d) but with large field of lens and the zone plate, respectively; (e,f) images matching images (c,d) but with large field of view (1024 px. inste view (1024 px. ad of 32) instead. The of 32). values of The values the color sc of the color ales have been changed for wide “ha scales have been changed for wide “halo” lo” high- lighting highlighting in imag ine ( image f) (the (f va ) (th lues e values of the modu of the modulus lus of the of the fiefield ld themselves re themselves remained mained unchanged). The unchanged). x-axes are mea The x-axes aresured in p measured i in xels, and the pixels, and the siz size e of of the the xx-axis is e -axis is equal qual to the to the y-axis y-axis for a for all images ll images except except (b). Color bars are displayed as field strength in arbitrary units (a.u.). Pixel size is 3.8 μm. (b). Color bars are displayed as field strength in arbitrary units (a.u.). Pixel size is 3.8 m. As can be seen in Figure 2b, the beam focused by the zone plate has the same width As can be seen in Figure 2b, the beam focused by the zone plate has the same width as that focused by a conventional lens but has a field amplitude approximately 3 times as that focused by a conventional lens but has a field amplitude approximately 3 times smaller due to the properties of the amplitude zone plate [23]. As the power depends on smaller due to the properties of the amplitude zone plate [23]. As the power depends on the field amplitude quadratically, about 90% of the energy of the field incident on the plate the field amplitude quadratically, about 90% of the energy of the field incident on the is lost due to reflection and diffraction. In Figure 2f, we can see a wide “halo” around the plate is lost due to reflection and diffraction. In Figure 2f, we can see a wide “halo” point source image focused by the zone plate, which carries a significant amount of energy. around the point source image focused by the zone plate, which carries a significant amount of energy. Photonics 2022, 9, x FOR PEER REVIEW 5 of 12 2.4. Modeling of Deformable Mirrors Photonics 2022, 9, 163 5 of 12 Deformable mirrors differ in the number of actuators and the principle of their op- eration. In general, the greater the number of actuators, the more complex (spatially high frequency) the aberrations that can be modeled. Aberrations of the human eye are usu- 2.4. Modeling of Deformable Mirrors ally described by the number of Zernike polynomials from the 4th to the 8th radial de- Deformable mirrors differ in the number of actuators and the principle of their oper- gree [3,5], i.e., they contain from 12 to 45 orthogonal functions in the Zernike decompo- ation. In general, the greater the number of actuators, the more complex (spatially high sition. The deformable mirror must have at least the same or a greater number of actua- frequency) the aberrations that can be modeled. Aberrations of the human eye are usually tors. The typical number of actuators in deformable mirrors ranges from several tens to described by the number of Zernike polynomials from the 4th to the 8th radial degree [3,5], i.e., sev they eral thou contain sand. As from 12 a ru to 45 le, orthogon their pral ice functions increasein s gre theaZernike tly with the i decomposition. ncrease inThe the number deformable mirror must have at least the same or a greater number of actuators. The typical of actuators. number of actuators in deformable mirrors ranges from several tens to several thousand. Using DLP, a binary amplitude mask can be simulated. The Fresnel zone size is a As a rule, their price increases greatly with the increase in the number of actuators. limitation in this case, which depends on the simulated lens focal length (see Section 2.2.) Using DLP, a binary amplitude mask can be simulated. The Fresnel zone size is a and, in fact, does not depend on the magnitude of the optical aberrations. Below, it will limitation in this case, which depends on the simulated lens focal length (see Section 2.2.) be experimentally shown that with the help of a DLP, not only can aberrations be com- and, in fact, does not depend on the magnitude of the optical aberrations. Below, it pensated/introduced, but a holographic plate can also be realized. will be experimentally shown that with the help of a DLP, not only can aberrations be For mirror modeling, we considered that the aberration function 𝜑 (𝑥, 𝑦 ) was compensated/introduced, but a holographic plate can also be realized. known in the center of eac For mirror modeling, we consider h actuated or. We constructed a new func that the aberration function j(xtion , y) was 𝜓 (𝑥,known 𝑦 ) which was in the center of each actuator. We constructed a new function y(x, y) which was equal equal 𝜑 (𝑥, 𝑦 ) in the centers of the actuators. In the case of the piezoelectric mirror j(x, y) in the centers of the actuators. In the case of the piezoelectric mirror (DMP40/M- (DMP40/M-F01), activation of one actuator deformed the entire mirror, so a cubic spline F01), activation of one actuator deformed the entire mirror, so a cubic spline was applied was applied between the actuators’ centers. In the case of the microelectromechanical between the actuators’ centers. In the case of the microelectromechanical system (MEMS) system (MEMS) mirror (DM140A), according to the manufacturer’s documentation, the mirror (DM140A), according to the manufacturer ’s documentation, the mirror ’s pieces are mirror’s pieces are individually locally sloped, so a linear spline was used. Links to the individually locally sloped, so a linear spline was used. Links to the mirrors’ manufacturer ’s mirrors’ manufacturer’s documentation are provided in the supplementary section. documentation are provided in the supplementary section. ( ) ( ) In Figure 3b,c are shown the functions 𝜑 𝑥, 𝑦 , 𝜓 𝑥, 𝑦 and the difference between In Figure 3b,c are shown the functions j(x, y), y(x, y) and the difference between them. The them. The root-mean-squar root-me e a deviation n-square deviation (RMSD) of the (RM original SD) of function the origin was about al fun7ction radians was with about 7 ra- an amplitude of approximately 35 radians. The RMSD of the difference between j(x, y) dians with an amplitude of approximately 35 radians. The RMSD of the difference be- and y(x, y) was approximately 1 rad. For the demonstration, a large-scale aberration was tween 𝜑 (𝑥, 𝑦 ) and 𝜓 (𝑥, 𝑦 ) was approximately 1 rad. For the demonstration, a taken from [24], corresponding to a numerical aperture of 0.2. large-scale aberration was taken from [24], corresponding to a numerical aperture of 0.2. Figure 3. Piezoelectric mirror modeling. (a) Mirror actuators (according to the documentation from Figure 3. Piezoelectric mirror modeling. (a) Mirror actuators (according to the documentation the manufacturer’s website); (b) initial phase and phase of the mirror obtained by simulation; (c) from the manufacturer ’s website); (b) initial phase and phase of the mirror obtained by simulation; difference between the phases from Figure (b). Color bars are in radians. (c) difference between the phases from Figure (b). Color bars are in radians. The MEMS mirror DM140A was an array of 12 12 elements. The tilt of each element The MEMS mirror DM140A was an array of 12 × 12 elements. The tilt of each ele- could be set individually. As in the previous case, the phase value was found at the center ment could be set individually. As in the previous case, the phase value was found at the of each element, after which the function was constructed using a linear spline. The results center of each element, after which the function was constructed using a linear spline. are shown in Figure 4b. The standard deviation of the function’s difference (Figure 4c) was The results are shown in Figure 4b. The standard deviation of the function’s difference 0.6 rad. (Figure 4c) was 0.6 rad. Photonics Photonics 2022 2022 , 9 , 9 , x FO , 163 R PEER REVIEW 6 of 12 6 of 12 Figure 4. MEMS mirror modeling. (a) Mirror segments (according to the documentation from the Figure 4. MEMS mirror modeling. (a) Mirror segments (according to the documentation from the manufacturer’s website); (b) initial phase and phase of the mirror obtained by simulation; (c) the manufacturer ’s website); (b) initial phase and phase of the mirror obtained by simulation; (c) the difference between the phases from Figure (b). Color bars are in radians. difference between the phases from Figure (b). Color bars are in radians. For modeling, the set of ten aberration functions was generated for a pupil with a For modeling, the set of ten aberration functions was generated for a pupil with a diameter of 5.7 mm (numerical aperture 0.17). For the generation, the Zernike coefficients diameter of 5.7 mm (numerical aperture ≈ 0.17). For the generation, the Zernike coeffi- were used, the mean and variance of which were obtained from the study [25]. They are cients were used, the mean and variance of which were obtained from the study [25]. presented in Appendix A. Ten different aberration functions were generated. They are presented in Appendix A. Ten different aberration functions were generated. 3. Results 3. Results 3.1. Simulation Results Since the Zernike coefficients had a significant variance, the resulting aberrations were 3.1. Simulation Results different. For each of these aberrations, the fraction of energy returned to the fiber was Since the Zernike coefficients had a significant variance, the resulting aberrations calculated (according to Equation (3)). The result was a set of 10 values for each of the were different. For each of these aberrations, the fraction of energy returned to the fiber different aberration compensators (mirrors or zone plate) and without them. For this set, was calculated (according to Equation (3)). The result was a set of 10 values for each of the means and variances were calculated, which are presented in Table 1. the different aberration compensators (mirrors or zone plate) and without them. For this Tset able , th 1. e m Comparison eans anof d va return-to-fiber riances were efficiency calcu for lated differ , which ent aberration-compensation are presented in Tmethods. able 1. With Compensation Without Compensation Table 1. Comparison of return-to-fiber efficiency for different aberration-compensation methods. Piezoelectric mirror 0.80  0.15 MEMS mirr or Wi 0.77 th C  0.07 ompensation 0.016 Without  0.012 Compensation DLP matrix 0.077  0.003 Piezoelectric mirror 0.80 ± 0.15 MEMS mirror 0.77 ± 0.07 0.016 ± 0. 012 The table shows that the mirrors yielded results of comparable quality. With a piezo DLP matrix 0.077 ± 0.003 mirror, slightly better quality was obtained, as well as a greater variance. A MEMS mirror costs much more, but its quality, in this case, was comparable to the cheaper mirror. We The table shows that the mirrors yielded results of comparable quality. With a piezo found that this was due to: (a) the relatively low spatial frequency of aberrations; at a mirror, slightly better quality was obtained, as well as a greater variance. A MEMS mirror higher frequency, 40 piezoelectric actuators might not be sufficient to form a wavefront of costs much more, but its quality, in this case, was comparable to the cheaper mirror. We proper quality; and (b) due to the square geometry of the MEMS mirror, only about 100 of the found th 144 actuators at this participated was due to in : the (a) the wavefr relont atively low spa fitting. tial frequency of aberrations; at a The DLP matrix functioned noticeably worse than the mirrors. At the same time, higher frequency, 40 piezoelectric actuators might not be sufficient to form a wavefront of the value of the returned energy had a relatively low dispersion. Thus, the efficiency proper quality; and (b) due to the square geometry of the MEMS mirror, only about 100 of the DLP matrix weakly depended on the type and scale of the aberrations. With this of the 144 actuators participated in the wavefront fitting. sample and at this pupil size, such a system allowed an approximately five-fold increase in The DLP matrix functioned noticeably worse than the mirrors. At the same time, the the effectiveness of reception. In addition, image defocusing was not taken into account value of the returned energy had a relatively low dispersion. Thus, the efficiency of the during modeling (usually in OCT tomographs, it is compensated physically). However, DLP matrix weakly depended on the type and scale of the aberrations. With this sample in [25], an 80% RMS wavefront error was indicated. Residual defocusing can also be easily and at this pupil size, such a system allowed an approximately five-fold increase in the compensated with the help of the DLP matrix system, which increases its effectiveness. effectiveness of reception. In addition, image defocusing was not taken into account during modeling (usually in OCT tomographs, it is compensated physically). However, in [25], an 80% RMS wavefront error was indicated. Residual defocusing can also be eas- ily compensated with the help of the DLP matrix system, which increases its effective- ness. Photonics 2022, 9, x FOR PEER REVIEW 7 of 12 Photonics 2022, 9, 163 7 of 12 Figure 5 shows a simulation of the effect of wavefront correctors with an eye pupil of Figure 5 shows a simulation of the effect of wavefront correctors with an eye pupil of 7 mm and the aberration obtained from a previous study [24] (the generated amplitude 7 mm and the aberration obtained from a previous study [24] (the generated amplitude zone plate to simulate image 5d is presented in Figure A3). Based on this, we can con- zone plate to simulate image 5d is presented in Figure A3). Based on this, we can conclude clude that the zone plate produces an image of the point with the smallest width, but a that the zone plate produces an image of the point with the smallest width, but a large large amount of energy is lost due to reflection and diffraction. In this case, the MEMS amount of energy is lost due to reflection and diffraction. In this case, the MEMS mirror mirror showed a noticeably better result than the piezoelectric mirror, which was due to showed a noticeably better result than the piezoelectric mirror, which was due to the the presence of high-frequency components in the spatial aberration spectrum (the presence of high-frequency components in the spatial aberration spectrum (the presence of presence of high Zernike polynomials). The amount of energy returned to the fiber for high Zernike polynomials). The amount of energy returned to the fiber for the zone plate, piezo, the zone p and MEMS late, pmirr iezo, ors anis d 0.076, MEMS 0.122, mirr and ors i 0.46, s 0.07 respectively 6, 0.122, and . 0.46, respectively. Figure Figure 5. 5. Residual Residual aberrations and aberrations and calculated calculated field amplitude field amplitude at at the fiber the f end. i(ber e a) Initial nd. ( aberration a) Initial a inberration radians; in radians; ( (b,c) r b esidual ,c) residual aber aberrationsrations for piezo for piezo and MEMS and MEMS mirrors, respectively; mirrors, respectively; ( (d–f) field amplitude d–f) fiat eld ampli- the tude at fiber the f end when iber end when usi using zone plate, ng zone plate, piezo, and MEMS piezo, and correctors, MEMS correctors, respectively (inre arbi spectiv trary units). ely (in arbitrary units). 3.2. Experimental Results To demonstrate the possibility of using a DLP matrix for aberration compensation, an 3.2. Experimental Results experimental apparatus was assembled, shown in Figure 6. For this apparatus, a green laser To demonstrate the possibility of using a DLP matrix for aberration compensation, beam (Melles Griot LGR-025-450) with a wavelength of 544 nm was expanded on a beam an experimental apparatus was assembled, shown in Figure 6. For this apparatus, a green expander and fell on the DLP matrix. A DLP with a driving unit was the commercially laser beam (Melles Griot LGR-025-450) with a wavelength of 544 nm was expanded on a available projector (Philips PicoPix Micro). The diffraction pattern was sent to the HDMI input. The DLP matrix was taken out of the projector box but was driven by projector beam expander and fell on the DLP matrix. A DLP with a driving unit was the commer- electronics. There were no markings on the case of the DLP, but based on its geometry, cially available projector (Philips PicoPix Micro). The diffraction pattern was sent to the it was assumed to be a Texas Instruments DLP230GP with 960  540 square pixels of HDMI input. The DLP matrix was taken out of the projector box but was driven by pro- 5.4  5.4 m. The images obtained using DLP were registered with a Thorlabs DCC1545M jector electronics. There were no markings on the case of the DLP, but based on its ge- camera with 1280  1024 square pixels with a size of 5.2 m. ometry, it was assumed to be a Texas Instruments DLP230GP with 960 × 540 square pix- The DLP matrix is a periodic structure with a specific gap between the micromirrors. els of 5.4 × 5.4 μm. The images obtained using DLP were registered with a Thorlabs Thus, it can be considered as a reflective diffraction grating with a period d equal to the DCC1545M camera with 1280 × 1024 square pixels with a size of 5.2 μm. pixel size. Therefore, the DLP matrix was tilted at an angle so that the diffraction angle was The DLP matrix is a periodic structure with a specific gap between the micromirrors. equal to the reflection angle. Figure 7 presents images of the AZP focused laser beam. There was significant Thus, it can be considered as a reflective diffraction grating with a period 𝑑 equal to the astigmatism in this system, which can be seen in the first column of Figure 7. This was pixel size. Therefore, the DLP matrix was tilted at an angle so that the diffraction angle compensated by modifying the AZP by inserting an astigmatism term (the coefficient before was equal to the reflection angle. the Z term was hand-fitted). The result is shown in the second column in Figure 7. In 1 2 the third and fourth columns of Figure 7, coma 3Z and secondary astigmatism 5Z were 3 4 additionally introduced, respectively, where 3 and 5 are coefficients before the polynomials. We see that the resulting images qualitatively correspond to the images of a point spread function with the coma/astigmatism aberrations. Photonics 2022, 9, x FOR PEER REVIEW 8 of 12 Figure 6. Experimental apparatus. The dashed circle illustrates the principle of DLP matrix opera- tion. Micromirror “0” reflects light to the side, and micromirror “1” reflects light into the cam- era/optical system. Figure 7 presents images of the AZP focused laser beam. There was significant Photonics 2022, 9, 163 8 of 12 Photonics 2022, 9, x FOR PEER REVIEW 8 of 12 astigmatism in this system, which can be seen in the first column of Figure 7. This was compensated by modifying the AZP by inserting an astigmatism term (the coefficient before the 𝑍 term was hand-fitted). The result is shown in the second column in Figure 7. In the third and fourth columns of Figure 7, coma 3𝑍 and secondary astigmatism 5𝑍 were additionally introduced, respectively, where 3 and 5 are coefficients before the polynomials. We see that the resulting images qualitatively correspond to the images of a point spread function with the coma/astigmatism aberrations. The images in Figure 7a–d were obtained at different camera exposures. The brightness of the images was normalized between 0 and 255 for better visibility. The second row presents the intensities along the red lines in the images in the first row. However, in this case, exposure time was fixed (0.2 ms). The third row presents diffrac- Figure 6. Experimental apparatus. The dashed circle illustrates the principle of DLP matrix opera- Figure 6. Experimental apparatus. The dashed circle illustrates the principle of DLP matrix operation. tion masks, corresponding to the images above. tion. Micromirror “0” reflects light to the side, and micromirror “1” reflects light into the cam- Micromirror “0” reflects light to the side, and micromirror “1” reflects light into the camera/optical system. era/optical system. Figure 7 presents images of the AZP focused laser beam. There was significant astigmatism in this system, which can be seen in the first column of Figure 7. This was compensated by modifying the AZP by inserting an astigmatism term (the coefficient before the 𝑍 term was hand-fitted). The result is shown in the second column in Figure 7. In the third and fourth columns of Figure 7, coma 3𝑍 and secondary astigmatism 5𝑍 were additionally introduced, respectively, where 3 and 5 are coefficients before the polynomials. We see that the resulting images qualitatively correspond to the images of a point spread function with the coma/astigmatism aberrations. The images in Figure 7a–d were obtained at different camera exposures. The brightness of the images was normalized between 0 and 255 for better visibility. The second row presents the intensities along the red lines in the images in the first row. However, in this case, exposure time was fixed (0.2 ms). The third row presents diffrac- tion masks, corresponding to the images above. Figure 7. Focusing a laser beam with the AZP. The first row (a–d) shows images of a beam, focused Figure 7. Focusing a laser beam with the AZP. The first row (a–d) shows images of a beam, focused with the AZP patterns from the third row (i–l). In the first column, AZP has a focal length of 150 mm. with the AZP patterns from the third row (i–l). In the first column, AZP has a focal length of 150 In the second column, astigmatism was compensated. In the third and fourth columns, coma and mm. In the second column, astigmatism was compensated. In the third and fourth columns, coma secondary astigmatism were additionally introduced. In the second row (e–h) are graphs of the intensities along the red lines in the first row (at a fixed camera exposure). and secondary astigmatism were additionally introduced. In the second row (e–h) are graphs of the intensities along the red lines in the first row (at a fixed camera exposure). The images in Figure 7a–d were obtained at different camera exposures. The brightness of the images was normalized between 0 and 255 for better visibility. The second row Figure 7. Focusing a laser beam with the AZP. The first row (a–d) shows images of a beam, focused presents the intensities along the red lines in the images in the first row. However, in To show the possibility of phase control using DLP, a special pattern was synthe- with the AZP patterns from the third row (i–l). In the first column, AZP has a focal length of 150 this case, exposure time was fixed (0.2 ms). The third row presents diffraction masks, mm. In the second column, astigmatism was compensated. In the third and fourth columns, coma sized that would pr corr o esponding duce an to ithe mag images e in above. the focus area (which, in fact, is the implementa- and secondary astigmatism were additionally introduced. In the second row (e–h) are graphs of the To show the possibility of phase control using DLP, a special pattern was synthesized tion of a holographic plate). For this, an image (smile) was taken, and the argument of its intensities along the red lines in the first row (at a fixed camera exposure). that would produce an image in the focus area (which, in fact, is the implementation of a holographic plate). For this, an image (smile) was taken, and the argument of its To show the possibility of phase control using DLP, a special pattern was synthe- Fourier transform was found. Then, the lens phase was added to the resulting function sized that would produce an image in the focus area (which, in fact, is the implementa- tion of a holographic plate). For this, an image (smile) was taken, and the argument of its Photonics 2022, 9, x FOR PEER REVIEW 9 of 12 Photonics 2022, 9, 163 9 of 12 Fourier transform was found. Then, the lens phase was added to the resulting function (Equation (4)). Based on the resulting function, using Equation (6), the diffraction pattern (Equation (4)). Based on the resulting function, using Equation (6), the diffraction pattern was synthesized, as shown in Figure 8a. The obtained image is shown in Figure 8b. was synthesized, as shown in Figure 8a. The obtained image is shown in Figure 8b. Figure 8. DLP as a holographic plate. (a) Diffraction pattern; (b) image in focal plane. Figure 8. DLP as a holographic plate. (a) Diffraction pattern; (b) image in focal plane. These experiments demonstrate that it is possible to introduce/correct very significant These experiments demonstrate that it is possible to introduce/correct very signifi- phase distortions with the assistance of DLP. At the same time, the experimental portion cant phase distortions with the assistance of DLP. At the same time, the experimental was performed on a device costing a total of approximately USD 250. portion was performed on a device costing a total of approximately USD 250. 4. Discussion 4. Disc This ussion study provides a theoretical justification for the possibility of using a DLP matrix as an element of an adaptive optics system with spectral-domain OCT devices. Using This study provides a theoretical justification for the possibility of using a DLP ma- a numerical simulation, it was possible to increase the amount of received radiation by trix as an element of an adaptive optics system with spectral-domain OCT devices. Using approximately 5 times (for aberrations of a 5.7 mm eye pupil diameter) compared with a numerical simulation, it was possible to increase the amount of received radiation by the absence of aberration compensation. Numerical modeling showed that aberrations approximately 5 times (for aberrations of a 5.7 mm eye pupil diameter) compared with with large amplitudes can be compensated using an adaptive zone plate. With sufficiently the absence of aberration compensation. Numerical modeling showed that aberrations large aberrations, the efficiency of the DLP matrix becomes comparable to the efficiency with of traditional large am mirr plit ors. ude Thus, s can be compensa in the image in Figur ted us e 5in , DLP g an and adap the tiv piezo e zone mirrp or lat yielded e. With suffi- efficiencies of 0.076 and 0.122, respectively. In addition, the zone plate led to a large loss of ciently large aberrations, the efficiency of the DLP matrix becomes comparable to the ef- light; therefore, it was possible to increase the power of the probe beam without the risk ficiency of traditional mirrors. Thus, in the image in Figure 5, DLP and the piezo mirror of exceeding the permissible norms for the retina. Thus, the efficiency values of received yielded efficiencies of 0.076 and 0.122, respectively. In addition, the zone plate led to a radiation, when using a zone plate, can be doubled. large loss of light; therefore, it was possible to increase the power of the probe beam The larger the aberrations are, the greater the benefit of using a zone plate and the without the risk of exceeding the permissible norms for the retina. Thus, the efficiency smaller the gap compared to traditional deformable mirrors. Taking into account the rather values of received radiation, when using a zone plate, can be doubled. large variance in the aberrations (0.016  0.012 in Table 1), in some cases, it is possible The larger the aberrations are, the greater the benefit of using a zone plate and the without compensation, and in some cases, there is almost no signal and compensation is sm necessary aller the g . Using ap com DLPpalways ared to tra returns dition theasignal l deform to a able certain mirrors. Taking in preset level, since to the account th main e ra- power losses are due to diffraction and not to aberrations. Thus, the system efficiency does ther large variance in the aberrations (0.016 ± 0. 012 in Table 1), in some cases, it is pos- not depend on the presence of aberrations and becomes predictable. sible without compensation, and in some cases, there is almost no signal and compensa- It is also worth mentioning that the DLP matrix has a high speed of operation, which tion is necessary. Using DLP always returns the signal to a certain preset level, since the is important for adaptive optics systems. main power losses are due to diffraction and not to aberrations. Thus, the system effi- From the presented study, it can be concluded that the use of an adaptive zone plate ciency does not depend on the presence of aberrations and becomes predictable. can make it possible to implement a relatively inexpensive system with adaptive optics, It is also worth mentioning that the DLP matrix has a high speed of operation, which which will be especially useful for cases with large aberrations. is important for adaptive optics systems. Author Contributions: Conceptualization, A.M. and G.G.; methodology, V.M., A.M. and P.S.; soft- From the presented study, it can be concluded that the use of an adaptive zone plate ware, V.M.; validation, G.G., V.M. and P.S.; writing—original draft preparation, V.M.; writing—review can make it possible to implement a relatively inexpensive system with adaptive optics, and editing, G.G., A.M. and P.S. All authors have read and agreed to the published version of which will be especially useful for cases with large aberrations. the manuscript. Author Contributions: Conceptualization, A.M. and G.G.; methodology, V.M., A.M., and P.S.; Funding: The study was supported by the WorldClass Research Centre “Photonics Centre” under the software, V.M.; validation, G.G., V.M., and P.S.; writing—original draft preparation, V.M.; writ- financial support of the Ministry of Science and High Education of the Russian Federation (Agreement ing—review and editing, G.G., A.M., and P.S. All authors have read and agreed to the published No. 075-15-2020-906). version of the manuscript. Funding: The study was supported by the WorldClass Research Centre “Photonics Centre” under the financial support of the Ministry of Science and High Education of the Russian Federation (Agreement No. 075-15-2020-906). Photonics 2022, 9, 163 10 of 12 Photonics 2022, 9, x FOR PEER REVIEW 10 of 12 Data Availability Statement: Data from the experimental setup can be found at the link https: Data Availability Statement: data from the experimental setup can be found at the link https://github.com/vasilymat/article7 //github.com/vasilymat/article7. Conflicts of Interest: The authors declare no conflict of interest. The funders had no role in the Conflicts of Interest: The authors declare no conflict of interest. The funders had no role in the design design of the study; in the collection, analyses, or interpretation of data; in the writing of the man- of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or uscript, or in the decision to publish the results. in the decision to publish the results. Appendix A Appendix A In Figure A1, the blue lines represent the means and variances of the Zernike coeffi- In Figure A1, the blue lines represent the means and variances of the Zernike coeffi- cients, corresponding to Figure 3 of [25]. cients, corresponding to Figure 3 of [25]. Figure A1 Figure A1. . GeGenerating nerating sets o sets f Zernik of Zernike e coefficoef cients ficients. . Each oEach f the ten of the sets ten is marked with sets is marked differ- with different- ent-colored circle markers. The blue vertical lines mark the variance value of the corresponding colored circle markers. The blue vertical lines mark the variance value of the corresponding Zernike Zernike coefficient. The x-axis shows the numbers of the coefficients in Noll’s notation; the y-axis coefficient. The x-axis shows the numbers of the coefficients in Noll’s notation; the y-axis shows the shows the values of the coefficients in micrometers. values of the coefficients in micrometers. Using these statistics, we generated 10 sets of Zernike coefficients. The values of the Using these statistics, we generated 10 sets of Zernike coefficients. The values of the generated coefficients are presented in Figure A1 with round markers of various colors. generated coefficients are presented in Figure A1 with round markers of various colors. Afterwards, using the set of coefficients, the set of aberration functions was generated Afterwards, using the set of coefficients, the set of aberration functions was generated (sums of Zernike polynomials with corresponding coefficients). The first nine of them are (sums of Zernike polynomials with corresponding coefficients). The first nine of them are Photonics 2022, 9, x FOR PEER REVIEW shown i n Figure A2. 11 of 12 shown in Figure A2. Figure A2. Generated aberration functions using the Zernike coefficients sets. The first nine indi- Figure A2. Generated aberration functions using the Zernike coefficients sets. The first nine individ- vidual functions. ual functions. Figure A3 shows the amplitude zone plate used to obtain the image in Figure 5d. Figure A3. AZP for aberration related to Figure 5d. Figure A4 presents the photo of a DLP matrix with a binary mask of an aberrated lens. Figure A4. Photo of a binary mask, generated by the DLP matrix. Photonics 2022, 9, x FOR PEER REVIEW 11 of 12 Photonics 2022, 9, x FOR PEER REVIEW 11 of 12 Figure A2. Generated aberration functions using the Zernike coefficients sets. The first nine indi- Photonics 2022, 9, 163 11 of 12 vidual functions. Figure A2. Generated aberration functions using the Zernike coefficients sets. The first nine indi- vidual functions. Figure A3 shows the amplitude zone plate used to obtain the image in Figure 5d. Figure A3 shows the amplitude zone plate used to obtain the image in Figure 5d. Figure A3 shows the amplitude zone plate used to obtain the image in Figure 5d. Figure A3. AZP for aberration related to Figure 5d. Figure A3. AZP for aberration related to Figure 5d. Figure A3. AZP for aberration related to Figure 5d. Figure A4 presents the photo of a DLP matrix with a binary mask of an aberrated Figure A4 presents the photo of a DLP matrix with a binary mask of an aberrated lens. lens. F igure A4 presents the photo of a DLP matrix with a binary mask of an aberrated lens. Figure A4. Photo of a binary mask, generated by the DLP matrix. Figure A4. Photo of a binary mask, generated by the DLP matrix. Figure A4. Photo of a binary mask, generated by the DLP matrix. References 1. Goncharov, A.S.; Iroshnikov, N.G.; Larichev, A.V. Retinal Imaging: Adaptive Optics. In Handbook of Coherent-Domain Optical Methods: Biomedical Diagnostics, Environmental Monitoring, and Materials Science; Tuchin, V.V., Ed.; Springer New York: New York, NY, USA, 2013; pp. 397–434. 2. Kumar, A.; Drexler, W.; Leitgeb, R.A. Subaperture correlation based digital adaptive optics for full field optical coherence tomography. Opt. 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Amplitude Zone Plate in Adaptive Optics: Proposal of the Principle

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hv photonics Communication Amplitude Zone Plate in Adaptive Optics: Proposal of the Principle Vasily Matkivsky * , Alexsandr Moiseev, Pavel Shilyagin and Grigory Gelikonov Institute of Applied Physics RAS, 46 Ulyanov Street, 603950 Nizhny Novgorod, Russia; aleksandr.moiseev@gmail.com (A.M.); paulo-s@mail.ru (P.S.); gelikon@ufp.appl.sci-nnov.ru (G.G.) * Correspondence: vasilymat@mail.ru Abstract: One of the main elements in hardware-based adaptive optics systems is a deformable mirror. There is quite a large number of such mirrors based on different principles and exhibiting varying performance. They constitute a significant portion of the cost of the final optical devices. In this study, we consider the possibility of replacing an adaptive mirror with the adaptive amplitude Fresnel zone plate, implemented using a digital light-processing matrix. Since such matrices are widely used in mass industry products (light projectors), their costs in large batches are 1–2 orders of magnitude lower than the cost of inexpensive deformable mirrors. Numerical modeling for scanning an optical coherence tomography system with adaptive optics is presented. It is shown that wavefront distor- tions with high spatial frequencies and large amplitudes can be corrected using an amplitude Fresnel zone plate. The results are compared with piezoelectric and microelectromechanical system mirrors. Keywords: optical coherence tomography; aberration compensation; adaptive optics; Fresnel zone plate; diffractive optics 1. Introduction Until now, the main area of application of optical coherence tomography (OCT) sys- Citation: Matkivsky, V.; Moiseev, A.; tems has been the study of the human eye fundus. One of the main factors limiting the Shilyagin, P.; Gelikonov, G. lateral resolution of OCT systems is eye optical aberrations. Adaptive optics made it pos- Amplitude Zone Plate in Adaptive sible to significantly increase the lateral resolution of OCT systems [1]. Recently, several Optics: Proposal of the Principle. methods were developed to compensate for aberration effects in postprocessing [2–8]. Such Photonics 2022, 9, 163. https:// methods also have a number of disadvantages (either they may not work well enough, doi.org/10.3390/photonics9030163 they require large calculations, or both). Compensation for aberrations is of particular Received: 29 November 2021 importance in scanning OCT systems, which, at the moment, have the greatest applicability Accepted: 18 February 2022 in practice [9]. In scanning systems, there is a confocal factor. This means that, at each Published: 7 March 2022 moment in time, only those photons are registered that came from a certain area of the object. The presence of aberrations leads both to a decrease in the illumination of this Publisher’s Note: MDPI stays neutral region (probe beam aberrations) and to the loss of some of the photons when they enter the with regard to jurisdictional claims in receiving system [10]. Therefore, for scanning OCT systems, the use of traditional adaptive published maps and institutional affil- iations. optics (at least the use of wavefront correctors) turns out to be desirable and, possibly, the best option for obtaining high resolution and for the signal-to-noise ratio [11]. On the other hand, the cost of correctors is thousands and tens of thousands of dollars, which leads to an increase in the cost of OCT systems. In this study, we propose the use of adaptive Copyright: © 2022 by the authors. amplitude diffraction lenses. Licensee MDPI, Basel, Switzerland. In addition to adaptive mirrors, it is worth mentioning the class of devices known This article is an open access article as adaptive lenses [12,13]. Most of them can only change their focal length, which is not distributed under the terms and suitable for wavefront correction. However, the phase zone plate has been successfully conditions of the Creative Commons implemented in liquid crystal devices [14–16], which allows for various effects, including Attribution (CC BY) license (https:// compensation of aberrations. It is worth noting that the cost of liquid crystal devices is creativecommons.org/licenses/by/ currently quite high and comparable to the cost of adaptive mirrors. In [17], an adaptive 4.0/). Photonics 2022, 9, 163. https://doi.org/10.3390/photonics9030163 https://www.mdpi.com/journal/photonics Photonics 2022, 9, x FOR PEER REVIEW 2 of 12 Photonics 2022, 9, 163 2 of 12 currently quite high and comparable to the cost of adaptive mirrors. In [17], an adaptive lens based on a silicone elastomer was demonstrated. The first few Zernike aberrations (astigmatism, coma, and spherical aberration of the third order) can be compensated with lens based on a silicone elastomer was demonstrated. The first few Zernike aberrations it. (astigmatism, coma, and spherical aberration of the third order) can be compensated with it. In this study, we analyzed the possibility of using an adaptive Fresnel amplitude In this study, we analyzed the possibility of using an adaptive Fresnel amplitude zone zone plate (AZP) based on a digital light-processing (DLP) matrix. This enabled the fo- plate (AZP) based on a digital light-processing (DLP) matrix. This enabled the focusing cusing of both the probed and the received radiation, and aberrations with large ampli- of both the probed and the received radiation, and aberrations with large amplitudes and tudes and with high spatial frequencies could be compensated. The loss of light (on re- with high spatial frequencies could be compensated. The loss of light (on reflection and flection and diffraction losses, which will be discussed below) on the DLP matrix was diffraction losses, which will be discussed below) on the DLP matrix was compensated by compensated by focusing the radiation when it was detected by the receiving system, focusing the radiation when it was detected by the receiving system, thereby improving thereby improving the overall signal-to-noise ratio and image quality. the overall signal-to-noise ratio and image quality. Amplitude zone Amplitude plates and zone photon plates ic sieves and photonic are main sieves ly usear d for fo e mainly cusin used g short for -focusing wave short-wave radiation or foradiation r obtaining or polychroma for obtaining tic im polychr agesomatic [18–21images ]. In [22] [18 , photonic –21]. In si [22 ev ],es wer photonic e sieves were used instead of used an ar instead ray ofof man icrolen array ses in of micr Sh olenses ack–Hain rtmann Shack–Hartmann sensors. The sensors. authors we The authors re were not aware of any previous work using amplitude zone plates to compensate for aberrations in not aware of any previous work using amplitude zone plates to compensate for aberra- tions in ophthalm ophthalmic ic OCT aOCT t the time at the of time writof ing. writing. 2. Materials and Methods 2. Materials and Methods 2.1. Optical Scheme 2.1. Optical Scheme Consider the object arm of the spectral-domain OCT device (Figure 1). The radiation Consider the object arm of the spectral-domain OCT device (Figure 1). The radiation from the fiber exits the S1 plane and is focused in the S6 plane and is reflected and returned from the fiber exits the S1 plane and is focused in the S6 plane and is reflected and re- to the S1 plane. The fraction of power to the fiber mode was determined. turned to the S1 plane. The fraction of power to the fiber mode was determined. Figure 1. Optical scheme for the deformable mirror/Fresnel plate modeling. Radiation is emitted Figure 1. Optical scheme for the deformable mirror/Fresnel plate modeling. Radiation is emitted from the fiber in the center of the S1 plane. In the case of deformable mirror modeling, a combined from the fiber in the center of the S1 plane. In the case of deformable mirror modeling, a combined phase mask (deformable mirror plus the lens with f2 focal length) is placed in the S2 plane. In the phase mask (deformable mirror plus the lens with f2 focal length) is placed in the S2 plane. In the case case of DLP modeling, a corresponding binary mask is placed in the S2 plane. The combination of of DLP modeling, a corresponding binary mask is placed in the S2 plane. The combination of lenses lenses in the S3 and S4 planes transfers the field from the S2 to the S5 plane. A lens in the S5 plane in the S3 and S4 planes transfers the field from the S2 to the S5 plane. A lens in the S5 plane focuses focuses radiation in the S6 plane. Aberrations are inserted into the S5 plane (marked with a blue dotted line). radiation in the S6 plane. Aberrations are inserted into the S5 plane (marked with a blue dotted line). For simulation, a single-mode fiber with a mode field diameter of 10 microns was For simulation, a single-mode fiber with a mode field diameter of 10 microns was assumed at plane S1. Using the angular spectrum method [18], fields in parallel planes assumed at plane S1. Using the angular spectrum method [18], fields in parallel planes were calculated. Modeling was performed for monochromatic radiation. Let us denote the were calculated. Modeling was performed for monochromatic radiation. Let us denote wavelength as l, the spatial sampling rate dx, the distance between the planes z, and the the wavelength as 𝜆 , the spatial sampling rate 𝑑𝑥 , the distance between the planes 𝑧 , and lz transverse size of the plane L. Then, under the condition < , the equation to calculate the transverse size of the plane 𝐿 . Then, under the condition < , the equation to the field can be expressed as follows: calculate the field can be expressed as follows: E (x, y) = iFT[FT[E (x, y)]FT[h(x, y)]], (1) ( ) ( ) ( ) 𝐸 𝑥, 𝑦 = 𝑖 𝐹𝑇 2 𝐸 𝑥, 𝑦 ∙ 𝑇𝐹 ℎ1 𝑥, 𝑦 , (1) h i ikz e ik 2 2 where h(x, y) = exp x + y is the impulse response function, (x, y) are the trans- where ℎ(𝑥, y ) = exp (𝑥 +𝑦 ) is the impulse response function, (𝑥, y ) are the ilz 2z verse coordinates, k = 2p/l is the wave number, i is an imaginary unit, and FT and iFT transverse coordinates, 𝑘= 2𝜋/𝜆 is the wave number, 𝑖 is an imaginary unit, and lz are forward and inverse Fourier transforms, respectively. In the case when  , instead of and 𝑖𝐹𝑇 are forward and inverse Fourier transforms, respectively. In the case when ≥ Equation (1), the Fresnel transfer function is used: , instead of Equation (1), the Fresnel transfer function is used: E x, y = iFT FT E x, y H k , k , (2) ( ) [ ( )] 2 1 x y 𝐸 (𝑥, 𝑦 ) = 𝑖 𝐹𝑇 𝐸 (𝑥, 𝑦 ) ∙ 𝐻(𝑘 ,𝑘 ) , (2) h  i ikz 2 2 where H k , k = e exp iplz k + k . x y x y 𝐹𝑇 𝐹𝑇 𝐹𝑇 𝜆𝑧 Photonics 2022, 9, 163 3 of 12 The lens, mirror, zone plate, or aberration action were simulated by multiplying the field by the corresponding two-dimensional function (see Section 2.2). In S2, the plane field was multiplied by the function of deformable mirrors of different types or by the DLP matrix. In the case of mirrors, the field in the S2 plane was additionally multiplied by the function of a collimating lens with a focal length of f2. The system of lenses in the S3/S4 planes transferred the field from the S2 plane to the S5 plane with the necessary magnification. In the S5 plane, the field was multiplied by aberration (blue broken line in Figure 1) and lens functions. The aberrated image of the fiber was formed at plane S6. Then, the optical system was mirrored relative to plane S6, and the field was propagated from S6 back to S1. To find the energy returned to the fiber, the scalar product of the mode and the field was calculated as follows: M(x,y)E (x,y) T = R R 2 2 jM(x,y)j jE(x,y)j (3) K = TT where M is the field at the exit from the fiber (mode), E is the complex field in the fiber plane, * denotes the complex conjugation, and K is the fraction of energy returned to the fiber. 2.2. DLP Matrix and Amplitude Fresnel Zone Plate An amplitude zone plate is an amplitude mask that divides the field into zones so that transmittance radiation is focused due to diffraction. Optical aberrations can be corrected with some modifications in the mask, in addition to focusing. As is known, an ideal lens introduces a parabolic factor into the phase front: 2 2 F(x, y) = (x + y ) (4) 2 f 2p where k = / is the wave number, f is the focal length of the lens, and (x, y) are coordinates in the plane of the lens. Supposing that additional distortions are introduced into the wavefront (for example, by the human eye system), then: q(x, y) = F(x, y) + j(x, y) (5) where j(x, y) is the wave aberration function and q(x, y) is the total phase change. Then, using the function q(x, y), the amplitude zone plate as a binary mask can be implemented according to this formula: q(x, y) > np AZP = 1, f or ; AZP = 0, otherwise (6) q x, y < n + 1 p ( ) ( ) where n = 0, 2, 4 . . .. The DLP matrix consists of micromirrors that can assume two positions: rotated through the angle a anda relative to the normal. Accordingly, rays incident on the matrix will be divided into 2 directions. If the optical system is placed in one of the directions, then the reflected light from the DLP can be considered as having passed through the binary amplitude mask. The principle of operation of DLP is illustrated in Section 3.2. Consider the limitations associated with the pixel structure of DLP. We can easily find the equation for the distance Dr between two zones for a lens with a focal length 1 2 f :Dr = / f lr + o Dr . The zone size decreases as the radius (beam/aperture size) increases. For a DLP matrix with a certain element size, the maximum radius of the amplitude mask r can be found. For radius r > r , the Fresnel zone size will be less max max than the DLP element size. The DLP matrix DLP4500NIR from Texas Instruments with a square element size of 7.6 m (with 912 1140 elements) was used in the study. The choice of this matrix was due to its design for infrared radiation, and it is a commercially available Photonics 2022, 9, x FOR PEER REVIEW 4 of 12 Photonics 2022, 9, 163 4 of 12 available device. Using the equation above, the maximum beam diameter was approxi- mately 17 mm, while the DLP size was about 7 × 8.7 mm. In addition, there are two ef- fects: (1) Fresnel zones have circle geometry, while DLP elements are square. This leads device. Using the equation above, the maximum beam diameter was approximately 17 mm, to errors for the radii, including those less than 𝑟 ; (2) in real matrices, the size of the while the DLP size was about 7  8.7 mm. In addition, there are two effects: (1) Fresnel micromirror is about 10% less than the pixel size (there is a gap between the elements). To zones have circle geometry, while DLP elements are square. This leads to errors for the simulate the first effect, the field in the DLP matrix plane was sampled with a sample radii, including those less than r ; (2) in real matrices, the size of the micromirror is about max interval of 𝑑𝑥 equal to the DLP element size. The second effect was not taken into ac- 10% less than the pixel size (there is a gap between the elements). To simulate the first count at this stage. We assume that this will lead to a decrease in radiation intensity by effect, the field in the DLP matrix plane was sampled with a sample interval of dx equal about 20% due to the area of element reduction. to the DLP element size. The second effect was not taken into account at this stage. We assume that this will lead to a decrease in radiation intensity by about 20% due to the area 2.3. Focusing the Beam Using the Fresnel Plate of element reduction. This section presents the results of modeling the focus of a Gaussian beam using a 2.3. Focusing the Beam Using the Fresnel Plate DLP matrix DLP4500NIR from Texas Instruments (the size of a square element was 7.6 This section presents the results of modeling the focus of a Gaussian beam using a μm with 912 × 1140 elements). A Gaussian beam was simulated, which was apertured by DLP matrix DLP4500NIR from Texas Instruments (the size of a square element was 7.6 m a circular diaphragm at a field level of 1/𝑒 and focused by a lens or zone plate with a with 912  1140 elements). A Gaussian beam was simulated, which was apertured by a focal length of 100 mm. The results are shown in Figure 2. The radiation wavelength was circular diaphragm at a field level of 1/e and focused by a lens or zone plate with a focal 850 nm. length of 100 mm. The results are shown in Figure 2. The radiation wavelength was 850 nm. Figure 2. Beam focusing. (a) The modulus of the field passed through the Fresnel zone plate; Figure 2. Beam focusing. (a) The modulus of the field passed through the Fresnel zone plate; (b) point sou (b) point rce sour ince -focu in-focus s profi prlofiles es (x-axis (x-axis isis meas measur ured ed i innmillimeters). millimeters). Orange Orange line—using t line—using the zone he zone plate, b plate, lue—using an ordinary le blue—using an ordinary lens ns of of the thesa same me focal focal leng length; th; ( (cc ,d ,d ))images images of of a a point point sour sou ce rce using using the the lens and the zone plate, respectively; (e,f) images matching images (c,d) but with large field of lens and the zone plate, respectively; (e,f) images matching images (c,d) but with large field of view (1024 px. inste view (1024 px. ad of 32) instead. The of 32). values of The values the color sc of the color ales have been changed for wide “ha scales have been changed for wide “halo” lo” high- lighting highlighting in imag ine ( image f) (the (f va ) (th lues e values of the modu of the modulus lus of the of the fiefield ld themselves re themselves remained mained unchanged). The unchanged). x-axes are mea The x-axes aresured in p measured i in xels, and the pixels, and the siz size e of of the the xx-axis is e -axis is equal qual to the to the y-axis y-axis for a for all images ll images except except (b). Color bars are displayed as field strength in arbitrary units (a.u.). Pixel size is 3.8 μm. (b). Color bars are displayed as field strength in arbitrary units (a.u.). Pixel size is 3.8 m. As can be seen in Figure 2b, the beam focused by the zone plate has the same width As can be seen in Figure 2b, the beam focused by the zone plate has the same width as that focused by a conventional lens but has a field amplitude approximately 3 times as that focused by a conventional lens but has a field amplitude approximately 3 times smaller due to the properties of the amplitude zone plate [23]. As the power depends on smaller due to the properties of the amplitude zone plate [23]. As the power depends on the field amplitude quadratically, about 90% of the energy of the field incident on the plate the field amplitude quadratically, about 90% of the energy of the field incident on the is lost due to reflection and diffraction. In Figure 2f, we can see a wide “halo” around the plate is lost due to reflection and diffraction. In Figure 2f, we can see a wide “halo” point source image focused by the zone plate, which carries a significant amount of energy. around the point source image focused by the zone plate, which carries a significant amount of energy. Photonics 2022, 9, x FOR PEER REVIEW 5 of 12 2.4. Modeling of Deformable Mirrors Photonics 2022, 9, 163 5 of 12 Deformable mirrors differ in the number of actuators and the principle of their op- eration. In general, the greater the number of actuators, the more complex (spatially high frequency) the aberrations that can be modeled. Aberrations of the human eye are usu- 2.4. Modeling of Deformable Mirrors ally described by the number of Zernike polynomials from the 4th to the 8th radial de- Deformable mirrors differ in the number of actuators and the principle of their oper- gree [3,5], i.e., they contain from 12 to 45 orthogonal functions in the Zernike decompo- ation. In general, the greater the number of actuators, the more complex (spatially high sition. The deformable mirror must have at least the same or a greater number of actua- frequency) the aberrations that can be modeled. Aberrations of the human eye are usually tors. The typical number of actuators in deformable mirrors ranges from several tens to described by the number of Zernike polynomials from the 4th to the 8th radial degree [3,5], i.e., sev they eral thou contain sand. As from 12 a ru to 45 le, orthogon their pral ice functions increasein s gre theaZernike tly with the i decomposition. ncrease inThe the number deformable mirror must have at least the same or a greater number of actuators. The typical of actuators. number of actuators in deformable mirrors ranges from several tens to several thousand. Using DLP, a binary amplitude mask can be simulated. The Fresnel zone size is a As a rule, their price increases greatly with the increase in the number of actuators. limitation in this case, which depends on the simulated lens focal length (see Section 2.2.) Using DLP, a binary amplitude mask can be simulated. The Fresnel zone size is a and, in fact, does not depend on the magnitude of the optical aberrations. Below, it will limitation in this case, which depends on the simulated lens focal length (see Section 2.2.) be experimentally shown that with the help of a DLP, not only can aberrations be com- and, in fact, does not depend on the magnitude of the optical aberrations. Below, it pensated/introduced, but a holographic plate can also be realized. will be experimentally shown that with the help of a DLP, not only can aberrations be For mirror modeling, we considered that the aberration function 𝜑 (𝑥, 𝑦 ) was compensated/introduced, but a holographic plate can also be realized. known in the center of eac For mirror modeling, we consider h actuated or. We constructed a new func that the aberration function j(xtion , y) was 𝜓 (𝑥,known 𝑦 ) which was in the center of each actuator. We constructed a new function y(x, y) which was equal equal 𝜑 (𝑥, 𝑦 ) in the centers of the actuators. In the case of the piezoelectric mirror j(x, y) in the centers of the actuators. In the case of the piezoelectric mirror (DMP40/M- (DMP40/M-F01), activation of one actuator deformed the entire mirror, so a cubic spline F01), activation of one actuator deformed the entire mirror, so a cubic spline was applied was applied between the actuators’ centers. In the case of the microelectromechanical between the actuators’ centers. In the case of the microelectromechanical system (MEMS) system (MEMS) mirror (DM140A), according to the manufacturer’s documentation, the mirror (DM140A), according to the manufacturer ’s documentation, the mirror ’s pieces are mirror’s pieces are individually locally sloped, so a linear spline was used. Links to the individually locally sloped, so a linear spline was used. Links to the mirrors’ manufacturer ’s mirrors’ manufacturer’s documentation are provided in the supplementary section. documentation are provided in the supplementary section. ( ) ( ) In Figure 3b,c are shown the functions 𝜑 𝑥, 𝑦 , 𝜓 𝑥, 𝑦 and the difference between In Figure 3b,c are shown the functions j(x, y), y(x, y) and the difference between them. The them. The root-mean-squar root-me e a deviation n-square deviation (RMSD) of the (RM original SD) of function the origin was about al fun7ction radians was with about 7 ra- an amplitude of approximately 35 radians. The RMSD of the difference between j(x, y) dians with an amplitude of approximately 35 radians. The RMSD of the difference be- and y(x, y) was approximately 1 rad. For the demonstration, a large-scale aberration was tween 𝜑 (𝑥, 𝑦 ) and 𝜓 (𝑥, 𝑦 ) was approximately 1 rad. For the demonstration, a taken from [24], corresponding to a numerical aperture of 0.2. large-scale aberration was taken from [24], corresponding to a numerical aperture of 0.2. Figure 3. Piezoelectric mirror modeling. (a) Mirror actuators (according to the documentation from Figure 3. Piezoelectric mirror modeling. (a) Mirror actuators (according to the documentation the manufacturer’s website); (b) initial phase and phase of the mirror obtained by simulation; (c) from the manufacturer ’s website); (b) initial phase and phase of the mirror obtained by simulation; difference between the phases from Figure (b). Color bars are in radians. (c) difference between the phases from Figure (b). Color bars are in radians. The MEMS mirror DM140A was an array of 12 12 elements. The tilt of each element The MEMS mirror DM140A was an array of 12 × 12 elements. The tilt of each ele- could be set individually. As in the previous case, the phase value was found at the center ment could be set individually. As in the previous case, the phase value was found at the of each element, after which the function was constructed using a linear spline. The results center of each element, after which the function was constructed using a linear spline. are shown in Figure 4b. The standard deviation of the function’s difference (Figure 4c) was The results are shown in Figure 4b. The standard deviation of the function’s difference 0.6 rad. (Figure 4c) was 0.6 rad. Photonics Photonics 2022 2022 , 9 , 9 , x FO , 163 R PEER REVIEW 6 of 12 6 of 12 Figure 4. MEMS mirror modeling. (a) Mirror segments (according to the documentation from the Figure 4. MEMS mirror modeling. (a) Mirror segments (according to the documentation from the manufacturer’s website); (b) initial phase and phase of the mirror obtained by simulation; (c) the manufacturer ’s website); (b) initial phase and phase of the mirror obtained by simulation; (c) the difference between the phases from Figure (b). Color bars are in radians. difference between the phases from Figure (b). Color bars are in radians. For modeling, the set of ten aberration functions was generated for a pupil with a For modeling, the set of ten aberration functions was generated for a pupil with a diameter of 5.7 mm (numerical aperture 0.17). For the generation, the Zernike coefficients diameter of 5.7 mm (numerical aperture ≈ 0.17). For the generation, the Zernike coeffi- were used, the mean and variance of which were obtained from the study [25]. They are cients were used, the mean and variance of which were obtained from the study [25]. presented in Appendix A. Ten different aberration functions were generated. They are presented in Appendix A. Ten different aberration functions were generated. 3. Results 3. Results 3.1. Simulation Results Since the Zernike coefficients had a significant variance, the resulting aberrations were 3.1. Simulation Results different. For each of these aberrations, the fraction of energy returned to the fiber was Since the Zernike coefficients had a significant variance, the resulting aberrations calculated (according to Equation (3)). The result was a set of 10 values for each of the were different. For each of these aberrations, the fraction of energy returned to the fiber different aberration compensators (mirrors or zone plate) and without them. For this set, was calculated (according to Equation (3)). The result was a set of 10 values for each of the means and variances were calculated, which are presented in Table 1. the different aberration compensators (mirrors or zone plate) and without them. For this Tset able , th 1. e m Comparison eans anof d va return-to-fiber riances were efficiency calcu for lated differ , which ent aberration-compensation are presented in Tmethods. able 1. With Compensation Without Compensation Table 1. Comparison of return-to-fiber efficiency for different aberration-compensation methods. Piezoelectric mirror 0.80  0.15 MEMS mirr or Wi 0.77 th C  0.07 ompensation 0.016 Without  0.012 Compensation DLP matrix 0.077  0.003 Piezoelectric mirror 0.80 ± 0.15 MEMS mirror 0.77 ± 0.07 0.016 ± 0. 012 The table shows that the mirrors yielded results of comparable quality. With a piezo DLP matrix 0.077 ± 0.003 mirror, slightly better quality was obtained, as well as a greater variance. A MEMS mirror costs much more, but its quality, in this case, was comparable to the cheaper mirror. We The table shows that the mirrors yielded results of comparable quality. With a piezo found that this was due to: (a) the relatively low spatial frequency of aberrations; at a mirror, slightly better quality was obtained, as well as a greater variance. A MEMS mirror higher frequency, 40 piezoelectric actuators might not be sufficient to form a wavefront of costs much more, but its quality, in this case, was comparable to the cheaper mirror. We proper quality; and (b) due to the square geometry of the MEMS mirror, only about 100 of the found th 144 actuators at this participated was due to in : the (a) the wavefr relont atively low spa fitting. tial frequency of aberrations; at a The DLP matrix functioned noticeably worse than the mirrors. At the same time, higher frequency, 40 piezoelectric actuators might not be sufficient to form a wavefront of the value of the returned energy had a relatively low dispersion. Thus, the efficiency proper quality; and (b) due to the square geometry of the MEMS mirror, only about 100 of the DLP matrix weakly depended on the type and scale of the aberrations. With this of the 144 actuators participated in the wavefront fitting. sample and at this pupil size, such a system allowed an approximately five-fold increase in The DLP matrix functioned noticeably worse than the mirrors. At the same time, the the effectiveness of reception. In addition, image defocusing was not taken into account value of the returned energy had a relatively low dispersion. Thus, the efficiency of the during modeling (usually in OCT tomographs, it is compensated physically). However, DLP matrix weakly depended on the type and scale of the aberrations. With this sample in [25], an 80% RMS wavefront error was indicated. Residual defocusing can also be easily and at this pupil size, such a system allowed an approximately five-fold increase in the compensated with the help of the DLP matrix system, which increases its effectiveness. effectiveness of reception. In addition, image defocusing was not taken into account during modeling (usually in OCT tomographs, it is compensated physically). However, in [25], an 80% RMS wavefront error was indicated. Residual defocusing can also be eas- ily compensated with the help of the DLP matrix system, which increases its effective- ness. Photonics 2022, 9, x FOR PEER REVIEW 7 of 12 Photonics 2022, 9, 163 7 of 12 Figure 5 shows a simulation of the effect of wavefront correctors with an eye pupil of Figure 5 shows a simulation of the effect of wavefront correctors with an eye pupil of 7 mm and the aberration obtained from a previous study [24] (the generated amplitude 7 mm and the aberration obtained from a previous study [24] (the generated amplitude zone plate to simulate image 5d is presented in Figure A3). Based on this, we can con- zone plate to simulate image 5d is presented in Figure A3). Based on this, we can conclude clude that the zone plate produces an image of the point with the smallest width, but a that the zone plate produces an image of the point with the smallest width, but a large large amount of energy is lost due to reflection and diffraction. In this case, the MEMS amount of energy is lost due to reflection and diffraction. In this case, the MEMS mirror mirror showed a noticeably better result than the piezoelectric mirror, which was due to showed a noticeably better result than the piezoelectric mirror, which was due to the the presence of high-frequency components in the spatial aberration spectrum (the presence of high-frequency components in the spatial aberration spectrum (the presence of presence of high Zernike polynomials). The amount of energy returned to the fiber for high Zernike polynomials). The amount of energy returned to the fiber for the zone plate, piezo, the zone p and MEMS late, pmirr iezo, ors anis d 0.076, MEMS 0.122, mirr and ors i 0.46, s 0.07 respectively 6, 0.122, and . 0.46, respectively. Figure Figure 5. 5. Residual Residual aberrations and aberrations and calculated calculated field amplitude field amplitude at at the fiber the f end. i(ber e a) Initial nd. ( aberration a) Initial a inberration radians; in radians; ( (b,c) r b esidual ,c) residual aber aberrationsrations for piezo for piezo and MEMS and MEMS mirrors, respectively; mirrors, respectively; ( (d–f) field amplitude d–f) fiat eld ampli- the tude at fiber the f end when iber end when usi using zone plate, ng zone plate, piezo, and MEMS piezo, and correctors, MEMS correctors, respectively (inre arbi spectiv trary units). ely (in arbitrary units). 3.2. Experimental Results To demonstrate the possibility of using a DLP matrix for aberration compensation, an 3.2. Experimental Results experimental apparatus was assembled, shown in Figure 6. For this apparatus, a green laser To demonstrate the possibility of using a DLP matrix for aberration compensation, beam (Melles Griot LGR-025-450) with a wavelength of 544 nm was expanded on a beam an experimental apparatus was assembled, shown in Figure 6. For this apparatus, a green expander and fell on the DLP matrix. A DLP with a driving unit was the commercially laser beam (Melles Griot LGR-025-450) with a wavelength of 544 nm was expanded on a available projector (Philips PicoPix Micro). The diffraction pattern was sent to the HDMI input. The DLP matrix was taken out of the projector box but was driven by projector beam expander and fell on the DLP matrix. A DLP with a driving unit was the commer- electronics. There were no markings on the case of the DLP, but based on its geometry, cially available projector (Philips PicoPix Micro). The diffraction pattern was sent to the it was assumed to be a Texas Instruments DLP230GP with 960  540 square pixels of HDMI input. The DLP matrix was taken out of the projector box but was driven by pro- 5.4  5.4 m. The images obtained using DLP were registered with a Thorlabs DCC1545M jector electronics. There were no markings on the case of the DLP, but based on its ge- camera with 1280  1024 square pixels with a size of 5.2 m. ometry, it was assumed to be a Texas Instruments DLP230GP with 960 × 540 square pix- The DLP matrix is a periodic structure with a specific gap between the micromirrors. els of 5.4 × 5.4 μm. The images obtained using DLP were registered with a Thorlabs Thus, it can be considered as a reflective diffraction grating with a period d equal to the DCC1545M camera with 1280 × 1024 square pixels with a size of 5.2 μm. pixel size. Therefore, the DLP matrix was tilted at an angle so that the diffraction angle was The DLP matrix is a periodic structure with a specific gap between the micromirrors. equal to the reflection angle. Figure 7 presents images of the AZP focused laser beam. There was significant Thus, it can be considered as a reflective diffraction grating with a period 𝑑 equal to the astigmatism in this system, which can be seen in the first column of Figure 7. This was pixel size. Therefore, the DLP matrix was tilted at an angle so that the diffraction angle compensated by modifying the AZP by inserting an astigmatism term (the coefficient before was equal to the reflection angle. the Z term was hand-fitted). The result is shown in the second column in Figure 7. In 1 2 the third and fourth columns of Figure 7, coma 3Z and secondary astigmatism 5Z were 3 4 additionally introduced, respectively, where 3 and 5 are coefficients before the polynomials. We see that the resulting images qualitatively correspond to the images of a point spread function with the coma/astigmatism aberrations. Photonics 2022, 9, x FOR PEER REVIEW 8 of 12 Figure 6. Experimental apparatus. The dashed circle illustrates the principle of DLP matrix opera- tion. Micromirror “0” reflects light to the side, and micromirror “1” reflects light into the cam- era/optical system. Figure 7 presents images of the AZP focused laser beam. There was significant Photonics 2022, 9, 163 8 of 12 Photonics 2022, 9, x FOR PEER REVIEW 8 of 12 astigmatism in this system, which can be seen in the first column of Figure 7. This was compensated by modifying the AZP by inserting an astigmatism term (the coefficient before the 𝑍 term was hand-fitted). The result is shown in the second column in Figure 7. In the third and fourth columns of Figure 7, coma 3𝑍 and secondary astigmatism 5𝑍 were additionally introduced, respectively, where 3 and 5 are coefficients before the polynomials. We see that the resulting images qualitatively correspond to the images of a point spread function with the coma/astigmatism aberrations. The images in Figure 7a–d were obtained at different camera exposures. The brightness of the images was normalized between 0 and 255 for better visibility. The second row presents the intensities along the red lines in the images in the first row. However, in this case, exposure time was fixed (0.2 ms). The third row presents diffrac- Figure 6. Experimental apparatus. The dashed circle illustrates the principle of DLP matrix opera- Figure 6. Experimental apparatus. The dashed circle illustrates the principle of DLP matrix operation. tion masks, corresponding to the images above. tion. Micromirror “0” reflects light to the side, and micromirror “1” reflects light into the cam- Micromirror “0” reflects light to the side, and micromirror “1” reflects light into the camera/optical system. era/optical system. Figure 7 presents images of the AZP focused laser beam. There was significant astigmatism in this system, which can be seen in the first column of Figure 7. This was compensated by modifying the AZP by inserting an astigmatism term (the coefficient before the 𝑍 term was hand-fitted). The result is shown in the second column in Figure 7. In the third and fourth columns of Figure 7, coma 3𝑍 and secondary astigmatism 5𝑍 were additionally introduced, respectively, where 3 and 5 are coefficients before the polynomials. We see that the resulting images qualitatively correspond to the images of a point spread function with the coma/astigmatism aberrations. The images in Figure 7a–d were obtained at different camera exposures. The brightness of the images was normalized between 0 and 255 for better visibility. The second row presents the intensities along the red lines in the images in the first row. However, in this case, exposure time was fixed (0.2 ms). The third row presents diffrac- tion masks, corresponding to the images above. Figure 7. Focusing a laser beam with the AZP. The first row (a–d) shows images of a beam, focused Figure 7. Focusing a laser beam with the AZP. The first row (a–d) shows images of a beam, focused with the AZP patterns from the third row (i–l). In the first column, AZP has a focal length of 150 mm. with the AZP patterns from the third row (i–l). In the first column, AZP has a focal length of 150 In the second column, astigmatism was compensated. In the third and fourth columns, coma and mm. In the second column, astigmatism was compensated. In the third and fourth columns, coma secondary astigmatism were additionally introduced. In the second row (e–h) are graphs of the intensities along the red lines in the first row (at a fixed camera exposure). and secondary astigmatism were additionally introduced. In the second row (e–h) are graphs of the intensities along the red lines in the first row (at a fixed camera exposure). The images in Figure 7a–d were obtained at different camera exposures. The brightness of the images was normalized between 0 and 255 for better visibility. The second row Figure 7. Focusing a laser beam with the AZP. The first row (a–d) shows images of a beam, focused presents the intensities along the red lines in the images in the first row. However, in To show the possibility of phase control using DLP, a special pattern was synthe- with the AZP patterns from the third row (i–l). In the first column, AZP has a focal length of 150 this case, exposure time was fixed (0.2 ms). The third row presents diffraction masks, mm. In the second column, astigmatism was compensated. In the third and fourth columns, coma sized that would pr corr o esponding duce an to ithe mag images e in above. the focus area (which, in fact, is the implementa- and secondary astigmatism were additionally introduced. In the second row (e–h) are graphs of the To show the possibility of phase control using DLP, a special pattern was synthesized tion of a holographic plate). For this, an image (smile) was taken, and the argument of its intensities along the red lines in the first row (at a fixed camera exposure). that would produce an image in the focus area (which, in fact, is the implementation of a holographic plate). For this, an image (smile) was taken, and the argument of its To show the possibility of phase control using DLP, a special pattern was synthe- Fourier transform was found. Then, the lens phase was added to the resulting function sized that would produce an image in the focus area (which, in fact, is the implementa- tion of a holographic plate). For this, an image (smile) was taken, and the argument of its Photonics 2022, 9, x FOR PEER REVIEW 9 of 12 Photonics 2022, 9, 163 9 of 12 Fourier transform was found. Then, the lens phase was added to the resulting function (Equation (4)). Based on the resulting function, using Equation (6), the diffraction pattern (Equation (4)). Based on the resulting function, using Equation (6), the diffraction pattern was synthesized, as shown in Figure 8a. The obtained image is shown in Figure 8b. was synthesized, as shown in Figure 8a. The obtained image is shown in Figure 8b. Figure 8. DLP as a holographic plate. (a) Diffraction pattern; (b) image in focal plane. Figure 8. DLP as a holographic plate. (a) Diffraction pattern; (b) image in focal plane. These experiments demonstrate that it is possible to introduce/correct very significant These experiments demonstrate that it is possible to introduce/correct very signifi- phase distortions with the assistance of DLP. At the same time, the experimental portion cant phase distortions with the assistance of DLP. At the same time, the experimental was performed on a device costing a total of approximately USD 250. portion was performed on a device costing a total of approximately USD 250. 4. Discussion 4. Disc This ussion study provides a theoretical justification for the possibility of using a DLP matrix as an element of an adaptive optics system with spectral-domain OCT devices. Using This study provides a theoretical justification for the possibility of using a DLP ma- a numerical simulation, it was possible to increase the amount of received radiation by trix as an element of an adaptive optics system with spectral-domain OCT devices. Using approximately 5 times (for aberrations of a 5.7 mm eye pupil diameter) compared with a numerical simulation, it was possible to increase the amount of received radiation by the absence of aberration compensation. Numerical modeling showed that aberrations approximately 5 times (for aberrations of a 5.7 mm eye pupil diameter) compared with with large amplitudes can be compensated using an adaptive zone plate. With sufficiently the absence of aberration compensation. Numerical modeling showed that aberrations large aberrations, the efficiency of the DLP matrix becomes comparable to the efficiency with of traditional large am mirr plit ors. ude Thus, s can be compensa in the image in Figur ted us e 5in , DLP g an and adap the tiv piezo e zone mirrp or lat yielded e. With suffi- efficiencies of 0.076 and 0.122, respectively. In addition, the zone plate led to a large loss of ciently large aberrations, the efficiency of the DLP matrix becomes comparable to the ef- light; therefore, it was possible to increase the power of the probe beam without the risk ficiency of traditional mirrors. Thus, in the image in Figure 5, DLP and the piezo mirror of exceeding the permissible norms for the retina. Thus, the efficiency values of received yielded efficiencies of 0.076 and 0.122, respectively. In addition, the zone plate led to a radiation, when using a zone plate, can be doubled. large loss of light; therefore, it was possible to increase the power of the probe beam The larger the aberrations are, the greater the benefit of using a zone plate and the without the risk of exceeding the permissible norms for the retina. Thus, the efficiency smaller the gap compared to traditional deformable mirrors. Taking into account the rather values of received radiation, when using a zone plate, can be doubled. large variance in the aberrations (0.016  0.012 in Table 1), in some cases, it is possible The larger the aberrations are, the greater the benefit of using a zone plate and the without compensation, and in some cases, there is almost no signal and compensation is sm necessary aller the g . Using ap com DLPpalways ared to tra returns dition theasignal l deform to a able certain mirrors. Taking in preset level, since to the account th main e ra- power losses are due to diffraction and not to aberrations. Thus, the system efficiency does ther large variance in the aberrations (0.016 ± 0. 012 in Table 1), in some cases, it is pos- not depend on the presence of aberrations and becomes predictable. sible without compensation, and in some cases, there is almost no signal and compensa- It is also worth mentioning that the DLP matrix has a high speed of operation, which tion is necessary. Using DLP always returns the signal to a certain preset level, since the is important for adaptive optics systems. main power losses are due to diffraction and not to aberrations. Thus, the system effi- From the presented study, it can be concluded that the use of an adaptive zone plate ciency does not depend on the presence of aberrations and becomes predictable. can make it possible to implement a relatively inexpensive system with adaptive optics, It is also worth mentioning that the DLP matrix has a high speed of operation, which which will be especially useful for cases with large aberrations. is important for adaptive optics systems. Author Contributions: Conceptualization, A.M. and G.G.; methodology, V.M., A.M. and P.S.; soft- From the presented study, it can be concluded that the use of an adaptive zone plate ware, V.M.; validation, G.G., V.M. and P.S.; writing—original draft preparation, V.M.; writing—review can make it possible to implement a relatively inexpensive system with adaptive optics, and editing, G.G., A.M. and P.S. All authors have read and agreed to the published version of which will be especially useful for cases with large aberrations. the manuscript. Author Contributions: Conceptualization, A.M. and G.G.; methodology, V.M., A.M., and P.S.; Funding: The study was supported by the WorldClass Research Centre “Photonics Centre” under the software, V.M.; validation, G.G., V.M., and P.S.; writing—original draft preparation, V.M.; writ- financial support of the Ministry of Science and High Education of the Russian Federation (Agreement ing—review and editing, G.G., A.M., and P.S. All authors have read and agreed to the published No. 075-15-2020-906). version of the manuscript. Funding: The study was supported by the WorldClass Research Centre “Photonics Centre” under the financial support of the Ministry of Science and High Education of the Russian Federation (Agreement No. 075-15-2020-906). Photonics 2022, 9, 163 10 of 12 Photonics 2022, 9, x FOR PEER REVIEW 10 of 12 Data Availability Statement: Data from the experimental setup can be found at the link https: Data Availability Statement: data from the experimental setup can be found at the link https://github.com/vasilymat/article7 //github.com/vasilymat/article7. Conflicts of Interest: The authors declare no conflict of interest. The funders had no role in the Conflicts of Interest: The authors declare no conflict of interest. The funders had no role in the design design of the study; in the collection, analyses, or interpretation of data; in the writing of the man- of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or uscript, or in the decision to publish the results. in the decision to publish the results. Appendix A Appendix A In Figure A1, the blue lines represent the means and variances of the Zernike coeffi- In Figure A1, the blue lines represent the means and variances of the Zernike coeffi- cients, corresponding to Figure 3 of [25]. cients, corresponding to Figure 3 of [25]. Figure A1 Figure A1. . GeGenerating nerating sets o sets f Zernik of Zernike e coefficoef cients ficients. . Each oEach f the ten of the sets ten is marked with sets is marked differ- with different- ent-colored circle markers. The blue vertical lines mark the variance value of the corresponding colored circle markers. The blue vertical lines mark the variance value of the corresponding Zernike Zernike coefficient. The x-axis shows the numbers of the coefficients in Noll’s notation; the y-axis coefficient. The x-axis shows the numbers of the coefficients in Noll’s notation; the y-axis shows the shows the values of the coefficients in micrometers. values of the coefficients in micrometers. Using these statistics, we generated 10 sets of Zernike coefficients. The values of the Using these statistics, we generated 10 sets of Zernike coefficients. The values of the generated coefficients are presented in Figure A1 with round markers of various colors. generated coefficients are presented in Figure A1 with round markers of various colors. Afterwards, using the set of coefficients, the set of aberration functions was generated Afterwards, using the set of coefficients, the set of aberration functions was generated (sums of Zernike polynomials with corresponding coefficients). The first nine of them are (sums of Zernike polynomials with corresponding coefficients). The first nine of them are Photonics 2022, 9, x FOR PEER REVIEW shown i n Figure A2. 11 of 12 shown in Figure A2. Figure A2. Generated aberration functions using the Zernike coefficients sets. The first nine indi- Figure A2. Generated aberration functions using the Zernike coefficients sets. The first nine individ- vidual functions. ual functions. Figure A3 shows the amplitude zone plate used to obtain the image in Figure 5d. Figure A3. AZP for aberration related to Figure 5d. Figure A4 presents the photo of a DLP matrix with a binary mask of an aberrated lens. Figure A4. Photo of a binary mask, generated by the DLP matrix. Photonics 2022, 9, x FOR PEER REVIEW 11 of 12 Photonics 2022, 9, x FOR PEER REVIEW 11 of 12 Figure A2. Generated aberration functions using the Zernike coefficients sets. The first nine indi- Photonics 2022, 9, 163 11 of 12 vidual functions. Figure A2. Generated aberration functions using the Zernike coefficients sets. The first nine indi- vidual functions. Figure A3 shows the amplitude zone plate used to obtain the image in Figure 5d. Figure A3 shows the amplitude zone plate used to obtain the image in Figure 5d. Figure A3 shows the amplitude zone plate used to obtain the image in Figure 5d. Figure A3. AZP for aberration related to Figure 5d. 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Journal

PhotonicsMultidisciplinary Digital Publishing Institute

Published: Mar 7, 2022

Keywords: optical coherence tomography; aberration compensation; adaptive optics; Fresnel zone plate; diffractive optics

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