Shack Hartmann Sensor Based on a Low-Aperture Off-Axis Diffraction Lens Array
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1 ISSN , Optoelectronics, Instrumentation and Data Processing, 29, Vol. 45, No. 2, pp c Allerton Press, Inc., 29. Original Russian Text c V.P. Lukin, N.N. Botygina, O.N. Emaleev, V.P. Korol kov, L.N. Lavrinova, R.K. Nasyrov, A.G. Poleshchuk, V.V. Cherkashin, 29, published in Avtometriya, 29, Vol. 45, No. 2, pp OPTICAL INFORMATION TECHNOLOGIES Shack Hartmann Sensor Based on a Low-Aperture Off-Axis Diffraction Lens Array V. P. Lukin a,n.n.botygina b,o.n.emaleev b, V. P. Korol kov c,l.n.lavrinova b,r.k.nasyrov c, A. G. Poleshchuk c, and V. V. Cherkashin c a Tomsk State University, pr. Lenina 36, Tomsk,6342 Russia b Zuev Institute of Optics of Atmosphere, Siberian Branch, Russian Academy of Sciences, pr. Akademicheskii, Tomsk, 6342 Russia c Institute of Automatics and Electrometry, Siberian Branch, Russian Academy of Sciences, pr. Akademika Koptyuga, Novosibirsk, 639 Russia lukin@iao.ru Received August 8, 28 Abstract A Shack Hartmann type wavefront sensor designed on the basis of a low-aperture off-axis diffraction lens array is described. Tests experiments showed that the sensor is capable of measuring wavefront slopes at array subapertures of size μm with an error not exceeding 4.8 arcseconds (.5 pixel), which corresponds to the standard deviation (SD) (SD =.7λ) at the reconstructed wavefront. Advantages of the array used in the sensor and the technology of its manufacture are discussed. The sensor tested in the experiments can be used to measure atmospheric turbulence parameters and as an element of adaptive optical imaging systems. DOI:.33/S Key words: Shack Hartmann sensor, low-aperture off-axis diffraction lens array, atmospheric turbulence, adaptive optics. INTRODUCTION A wavefront sensor is a key element of adaptive optic system that gives information on the structure of heterogeneities in the optical wave propagation channel, thus providing effective operation of the executive correcting element of the correction system. At present, adaptive optical systems most often use Shack Hartmann sensors, which is suitable for measuring both coherent and incoherent radiations. A sensor of this type consists of a microlens array and a video camera with high spatial and time resolutions placed in the plane of image recording. From measured displacements of the centroid of the images formed by individual array elements, the local wavefront slopes at the entrance subapertures of the array are calculated and the wavefront radiation incident on the microlens array is reconstructed. The adaptive optical systems compensating the wavefront aberrations due to atmospheric turbulence, as a rule, use microlens arrays. The parameters of the Shack Hartmann sensor and its array are calculated on the basis of the general principles of designing Hartmann sensors [] with the influence of atmospheric turbulence on the radiation characteristics taken into account [2 4]. Existing technologies of microlens arrays are imperfect; in particular, the parameters of microlenses with a spherical surface shape are poorly reproducible [5]. In this connection, it is natural to employ diffraction optics which provides better reproducibility of the key parameters [6]. In the present paper, we propose a new type of sensor which uses off-axis diffraction optical elements. 6
2 62 LUKIN et al. CALCULATION OF MICROLENS ARRAY PARAMETERS Let us analyze microlens array parameters that provide a match, on the one hand, to a real optical system, and on the other hand, to the video camera matrix size, which is the sensitive element of the adaptive system itself. Because the entrance apertures of the real optical systems forming images of remote objects far exceed the size of the photosensitive matrix of video cameras, to match the sizes, it is necessary to use additional optical elements. For simplicity of the analysis, we estimate the parameters of the microlens array of a Shack Hartmann sensor measuring the aberrations of a planar wavefront propagating along a horizontal atmospheric path. The calculation will be made for a telescopic system forming an exit pupil. We assume that the microlens array is placed in the exit pupil of the optical system and the plane of the photosensitive matrix is made coincident with the focal plane of the microlenses. In this device, the microlens array should be not larger than the video camera matrix. In the case of close packing of the array elements, the linear field in the space of images should be limited by the individual element size. The criterion determining the sizes of the subapertures [3, 4] measuring local slopes of the radiation wavefront passing through a turbulent layer is the Fried parameter r. For the entrance subaperture diameter smaller than the Fried parameter, wavefront aberrations lead to negligibly small distortions of the diffraction image. In this case, the local wavefront slopes can be determined from the displacement of the centroid of the image in a coordinate system attached to the reference wavefront. The angular field 2ω of an individual microlens array element in the space of images should be estimated taking into account the angular displacement of the image caused by atmospheric turbulence. The dispersion of the angles of arrival σα 2 at the entrance subaperture can be written as [2 4] σα 2 =.345λ2 r 5/3 (D sub ) /3, () where λ is the radiation wavelength and D sub is the diameter of the entrance subaperture. Then, the angular field 2ω of the optical system in the object space is determined by the maximum angle of arrival 5σ α. From the theory of optical systems, it is known that tan ω =Γtanω, where Γ is the angular magnification of the telescopic system. For local wavefront slopes due to atmospheric turbulence, the slope can be replaced by the angle. Taking into account the diffraction on a microlens of diameter 2α ml,weobtain the following formula for the parameters of an individual array element: α ml λ λ [ ( Dsub ) 5/6 ] =Γ 5σ α +.22 =Γ , (2) f ml 2α ml D sub r where f ml is the focal length of the microlens. Let the diameter of the entrance aperture of an adaptive optical system D sub be equal to mm (typical sizes of the photosensitive matrices of video cameras are 5 mm), typical value of the Fried parameter r for an optical system of visible range be equal to 4 mm (LCn 2 =3 m /3,whereL is the atmospheric path length and Cn 2 is the structural constant of the refractive index of the atmosphere). Then, for a wavelength λ =.63 μm and a subaperture size D sub = mm, we obtain α ml /f ml =.4.2 and 2α ml =..5 mm, i.e., for experiments with laser radiation propagating along horizontal atmospheric paths, Shack Hartmann sensor requires a microlens array with low-aperture elements. To record the image angular displacement in the range from fractions to 2 3 radii of diffraction images, one needs long-focus microlens arrays. The elements in the array should be close packed. This is convincingly proved by a numerical experiment in [7, 8], which showed that the adaptive correction efficiency decreases markedly with decreasing diameter of the transparent region of an individual array element. The dependence of its efficiency (phase linkage) on the intensity of turbulent aberrations is shown in Fig. : curve corresponds to an ideal sensor, curve 2 to a sensor with 64 subapertures at a radius ratio of an individual element to the transparent region equal to 2, curve 3 to a sensor with 64 subapertures at a radius ratio of an individual element to the transparent region equal to 7.5, and curve 4 to a sensor without adaptive control. IMPLEMENTATION OF A LOW-APERTURE MICROLENS ARRAY The fabrication of fully filled long-focus microlens arrays involves serious difficulties. The photoresist melting technology, which provides the highest-quality lens arrays, cannot be used for the following reasons. First, it is limited by a ratio of the lens deflection to diameter not less than : 2, and second, it does not provide % filling because molten photoresist drops should not merge. Direct laser writing on a photoresist OPTOELECTRONICS, INSTRUMENTATION AND DATA PROCESSING Vol. 45 No. 2 29
3 SHACK HARTMANN SENSOR BASED ON A LOW-APERTURE 63 J Fig.. r +st order _st order CCD CCD2 Fig. 2. Fig. 3. material at large element sizes gives both local profile roughness and depth (and, hence, focus) deviation from lens to lens, which reaches %. In the present paper, we propose a new approach to the problem of designing low-aperture microlens arrays and Shack Hartmann sensors, which involves the replacement of an array lenslet by an off-axis fragment of a diffraction lens with the same focus but of much greater diameter. The distance from the optical axis of the diffraction lens to the fragment is chosen such that the diffracted focused beam does not overlap any of the th order diffraction beams from the other lenses. Thus, in this sensor, the charge coupled device (CCD) camera should be displaced from the optical axis of the system (Fig. 2). The off-axis lens array can also be implemented as both a multilevel diffraction element and as a binary element. In the latter case, the diffraction efficiency (in the +st order) of the lens-focused light is 4%, but this disadvantage is compensated by the following advantages: high reproducibility of focuses in the array, % filling of the array subapertures, and the possibility of creating two sensor channels with different apertures. An additional digital camera (CCD2) can be placed in the st diffraction order. In this case, the divergent st order is focused onto the camera by an additional lens. The image scale in the second channel is determined by its focus. The accuracy of wavefrontimaging by this array depends largely on the pixel size of the recording system. The problem is that the off-axis lens microstructure differs only slightly from a linear grating. A fragment of an off-axis lens calculated for the 5th diffraction to facilitate the visual perception of the microstructure OPTOELECTRONICS, INSTRUMENTATION AND DATA PROCESSING Vol. 45 No. 2 29
4 64 LUKIN et al Fig. 4. Fig. 5. curvature is shown in Fig. 3. The microstructure of a real-axis lens array was calculated for the +st diffraction order with a pixel size of.25 μm, which corresponds to an accuracy of the formed wavefront not worse than λ/4 (the period of the diffraction zones in the off-axis lens in the range of 6 7 μm). Array writing was made on a CLWS-3IAE circular laser writing system using a thermochemical technology. The structure of the written chromium mask was transferred onto a quartz substrate by ion etching. A Shack Hartmann sensor was implemented on the basis of the produced array with a microlens matrix cell of size μm and a numerical aperture of an individual element of.5. The intensity distribution in the focusing plane of the +st diffraction order is given in Fig. 4. Parasitic diffraction orders are caused by diffraction at the square subapertures of the array cells. However, the intensity in them is very low since the photograph is made with a large overexposure to determine the structure of the parasitic exposure. TEST OF THE SHACK HARTMANN SENSOR WITH AN OFF-AXIS DIFFRACTION LENS ARRAY In the test experiment, the reference distortion was an optical wedge which was placed in a collimated laser beam with a radiation wavelength of.63 μm ahead of the lens array. Figure 5 shows a diagram of the experimental setup, which consists of an optical wedge (apex angle 6 ± ; surface accuracy λ/4) (), an off-axis diffraction lens array (8 8 microlenses with a numerical aperture of.5; a square subaperture size of μm) (2), a Dalsa DS 4-3k262 video camera [Dalsa, Inc. Canada; pixels ( pixel = μm); 262 frames/sec] (3), and a computer with a Coreco PC-DIG-L frame grabber board (Coreco Imaging, Canada) (4). The rotation of the wedge around the optical axis of the system leads to the displacement of the diffraction images in the focal plane of the array microlenses. Image recording for each position of the optical wedge is implemented by the video camera. The plane of the camera CCD matrix coincides with the focal plane of the microlens array. A specially developed software package is used to determine the coordinates of the centroid of diffraction images and calculate the local slopes of the radiation wavefront incident on the lens array. In the calculation of the centroid coordinates, the influence of the noise component of light signals is reduced by using a moving analysis window algorithm. The algorithm consists of the following steps: (a) The coordinates of the light intensity maximum of each diffraction image are determined in the Hartmann pattern; (b) Around the maximum point of each image, an analysis window is created whose size is equal to the diameter of the first dark ring of the diffraction pattern; (c) The coordinates of the centroid of the image are calculated within the analysis window. Thus, there is truncation of the dark and low-light elements of the CCD matrix, whose signal contains a large fraction of the noise component. The position of the center of the analysis window of each diffraction images is determined in a coordinate system attached to the CCD matrix. The coordinates of the centroids of such images calculated in the analysis window are referred to the same coordinate system. OPTOELECTRONICS, INSTRUMENTATION AND DATA PROCESSING Vol. 45 No. 2 29
5 SHACK HARTMANN SENSOR BASED ON A LOW-APERTURE 65 Fig. 6. (a) (b) (c) (d) Fig. 7. OPTOELECTRONICS, INSTRUMENTATION AND DATA PROCESSING Vol. 45 No. 2 29
6 66 LUKIN et al. _2.7 (a) _2.7 (b) _2. _2. _. _..8 _.7 _ _.7 _...8 Fig. 8. _2.7 _2. _..8 _.7 _...8 Fig. 9. Typical recorded Hartmann patterns are presented in Fig. 6. Figure 7 gives the light intensity distributions in the first (a and c) and second (b and d) selected diffraction images. In Fig. 6 and 7, the coordinates of the images are given in the pixels of the video camera matrix. The displacement trajectories of the centroids of the first (a) and second (b) selected images during rotation of the wedge around the optical axis are shown in Fig. 8. The coordinates of the centroid were calculated in an analysis window of size 2 2 pixels. The displacements of the centroids in pixels are plotted on the coordinate axes in Fig. 8. The angular scale of the pixel, which is determined by the element size of the camera CCD matrix and the focal length of the lens array, is equal to The point with the coordinates (, ) corresponds to the initial position of the optical wedge. The angle of the optical wedge is equal to 6 ±, and, hence, the slope of the radiation wave front after passage through the wedge is 3 ± 5. It is evident from the figure that, for a small total slope of the wavefront close to the angular size of the pixel, the recording system watches the displacement of the centroid by fractions of a pixel. The displacement trajectories of the centroids of all 64 diffraction images produced by the lens array during rotation of the optical wedge by 36 around the optical axis are made coincident in Fig. 9. The discrepancy of the trajectories is due to errors in the fabrication of the optical elements, aberrations of the laser radiation wavefront, and the noise of the recording system. OPTOELECTRONICS, INSTRUMENTATION AND DATA PROCESSING Vol. 45 No. 2 29
7 SHACK HARTMANN SENSOR BASED ON A LOW-APERTURE 67 Mean displacement along the X axis, pixel.5..5 _.5 _. (a) Mean displacement along the Y axis, pixel _.5 _. _.5 _2. (b) Angular position of wedge Angular position of wedge Fig.. Figure shows the mean displacements of the centroids of all diffraction images of the Hartmann pattern for each fixed position of the optical wedge on the X axis (a) and the Y axis (b). Each subsequent position differs from the previous one in the rotation of the optical wedge around the optical axis of the system by an angle equal to 3. The vertical segments in the plots show the standard deviations of the displacements. It follows from the plots that the measurement error (standard deviation) for any position of the wedge does not exceed.5 pixels. TEST OF THE WAVEFRONT RECONSTRUCTION ALGORITHM AND ITS SOFTWARE IMPLEMENTATION If the radiation wavefront at the entrance aperture of the sensor for the initial wedge position is assumed to be the reference one, its aberrations for another position of the wedge can be described by the wave aberration function W (ρ) in the form of an expansion in the basic Zernicke polynomials Z i (ρ) inacircleof unit radius: N W (ρ) = c i Z i (ρ), (3) i= where ρ = {X, Y } (X and Y are the coordinates normalized by the radius of the entrance aperture). The values of the derivative W N ρ = Z i c i (4) ρ i= at the points related to centers of the entrance subapertures are proportional to the measured displacements of the centroids of the diffraction images. Substituting the measured values of W/ ρ into the left side of Eq. (4), it is possible to estimate the expansion coefficients c i. The accuracy of the estimation increases if the number of such W/ ρ exceeds the number of these coefficients, and the estimation is performed using the least-squares minimization of the c i of the deviations of the measured derivatives and those calculated at the same points by formula (4). The algorithm of wavefront reconstruction described above was software implemented in the Shack Hartmann sensor and used in a test experiment. Table gives the first 4 Zernike polynomials. In an experiment with ideal elements for any position of the wedge, the reconstructed wavefront should have only tilts, i.e., the expansion should contain only the first two polynomials. Figures a and b give the results of wavefront reconstruction from 64 measurements W/ X (curves ) and W/ Y (curves 2) for expansion in two Zernike polynomials and the first 4 Zernike polynomials, respectively. From a comparison of Fig. and a, it follows that the change in the expansion coefficients c and c 2 during rotation of the optical wedge corresponds to the change in the mean displacements of the centroid of the diffraction images. The absolute values of the slopes of the wavefront calculated from the mean displacement of the centroid of the images and in terms of the wave aberration function (3) almost coincide: Δ x f = W (X =,Y =) R = 2c λ R, where R is the radius of the entrance aperture. Δȳ f = W (X =,Y =) R = 2c 2λ R, OPTOELECTRONICS, INSTRUMENTATION AND DATA PROCESSING Vol. 45 No. 2 29
8 68 LUKIN et al. Table Polynomial number Zernike polynomials Z i(x, Y ) Aberrations 2X Tilt 2 2Y Tilt XY Astigmatism 4 3(2X 2 +2Y 2 ) Defocusing 5 6(X 2 Y 2 ) Astigmatism 6 2 2(3X 2 Y Y 3 ) Trefoil 7 2 2(3X 2 Y +3Y 3 2Y ) Coma 8 2 2(3X 3 +3XY 2 2X) Coma 9 2 2(X 3 3XY 2 ) Trefoil 4 (X 3 Y XY 3 ) Quatrefoil 2 (4X 3 Y +4XY 3 3XY ) Astigmatism 2 5(6X 4 +2X 2 Y 2 +6Y 4 6X 2 6Y 2 +) Spherical 3 (4X 4 4Y 4 3X 2 +3Y 2 ) Astigmatism 4 (X 4 6X 2 Y 2 + Y 4 ) Quatrefoil C j (l).4 (a) C j (l).4 (b).2.2 _.2 _.4 _ _.2 _.4 _ _.8 _.8 _. _. Fig.. In real experiments, the radiation wavefront incident on the entrance aperture of the sensor is not planar. In addition, in wavefront reconstruction, the error of wavefront imaging by individual array elements and the noise of image recording and centroid calculation will be perceived as wavefront aberrations. Reconstruction of the more detailed structure of the wavefront makes it possible to verify the operation of the developed sensor with increasing number of Zernike polynomials. The results of wavefront reconstruction by expansion of the wave aberration function in 4 polynomials for rotation of the optical wedge by 8 (SD =.48λ) (angular position equal to 6 in Fig. ) are presented in Figs. 2a and 2b. Table 2 gives the expansion coefficients c i for this position of the wedge. The spread in the displacements of the diffraction images of the Hartmann pattern during wavefront measurement in the test experiment leads to a departure of the reconstructed wavefront from planarity. Since for a constant number of measurements, the accuracy of adjustment of the expansion coefficients increases with decreasing number of polynomials, the wavefront reconstructed from 28 measurements by two polynomials can be taken to be the reference one the deviations of the wavefront from planarity can be calculated as a function of the number of Zernike polynomials (N Z ). The calculation results are shown in Fig. 3. The deviations were calculated at the points taken at the sensor entrance aperture at a step of R/255 (R is the aperture radius). For the reconstruction of the wavefront by 4 polynomials, the SDs are in the interval (.48.6)λ and increase as the wavefront slope decreases from 62. to 8.4. For wavefront reconstruction by 8 polynomials, the interval is displaced insignificantly (.5.65)λ and practically does not change with increasing number of polynomials. OPTOELECTRONICS, INSTRUMENTATION AND DATA PROCESSING Vol. 45 No. 2 29
9 SHACK HARTMANN SENSOR BASED ON A LOW-APERTURE 69 (a) Y W(l).5..5 (b) _ X _.5 _ _.5 _...5 _. _.5 Fig. 2. Table 2 Expansion coefficient Value Expansion coefficient Value c c c c c c c c c c c c c c SD(l) N Z Fig. 3. OPTOELECTRONICS, INSTRUMENTATION AND DATA PROCESSING Vol. 45 No. 2 29
10 7 LUKIN et al. CONCLUSIONS A new type of Shack Hartmann wavefront sensor based on off-axis diffraction optics was constructed. The elements and software of such a sensor used to measure aberrations of the laser radiation wavefront propagating along horizontal atmospheric paths were tested in an experiment. Use of a low-aperture off-axis diffraction lens array with a high quality of wavefront imaging and high reproducibility of the parameters of the individual array elements allow the sensor to record wavefront slopes with high angular resolutions ( ). The total error of the measurement system (imaging, recording of a Hartmann pattern, and calculation of the angular displacement of the centroid of the image) does not exceed 4.8 (.5 pixel in the recording plane), which yields a standard deviation of the reconstructed wavefront from planarity not exceeding.7λ. The developed software package allows real-time measurements of the displacement of the centroids of images and wavefront reconstruction at the sensor entrance aperture. The Shack Hartmann sensor tested in the experiment can be used as a measuring instrument to determine atmospheric turbulence parameters and as an element of adaptive optical imaging systems. ACKNOWLEDGMENTS This work was supported by the Siberian Branch of the Russian Academy of Sciences (Complex Integration Project No. 8 Development of Methods for High-Precision Astroclimatic Observations to Provide the Operation of Adaptive Systems) and the Presidium of the Russian Academy of Sciences [Program of Basic Research No. 6 Environment Under Changing Climate Conditions: Extreme Natural Phenomena and Catastrophes (Part 3); project Development of Adaptive Systems and Instruments for Measuring Atmospheric Turbulence Parameters for Improving Solar Astronomical Observations]. REFERENCES. E. A. Vitrichenko, Methods of Studying Astronomical Optics (Nauka, Moscow, 98) [in Russian]. 2. A. S. Gurvich, A. I. Kon, V. L. Mironov, and S. S. Khmelevtsov, Laser Radiation in the Turbulent Atmosphere (Nauka, Moscow, 976) [in Russian]. 3. V. P. Lukin, Atmospheric Adaptive Optics (Nauka, Novosibirsk, 986) [in Russian]. 4. V. E. Zuev, V. A. Banakh, and V. V. Pokasov, Optics of Turbulent Atmosphere (Gidrometeoizdat, Leningrad, 988) [in Russian]. 5. V. G. Taranenko and O. I. Shinin, Adaptive Optics in Devices and Facilities (TsNIIATOMINFORM, Moscow, 25) [in Russian]. 6. V. P. Korolkov, A. G. Poleshchuk, R. K. Nasyrov, et al., Application of Off-Axis Microlens Array in Low Aperture Shack Hartmann Sensor, in Proc. 4 Int. Forum EXPO-Holography (Holographiya-Servis, Moscow, 27), pp F. Kanev, V. Lukin, and N. Makenova, Limitations of adaptive control efficiency due to singular points in the wavefront of a laser beam, Proc. SPIE 4884, (23). 8. F. Kanev, V. Lukin, and N. Makenova, Recording the Phase Profile of Coherent Radiation and Adaptive Control by a Laser Beam in the Presence of Singular Points at the Wavefront, Opt. Atmos. Okeana 5 (), 8 26 (22). OPTOELECTRONICS, INSTRUMENTATION AND DATA PROCESSING Vol. 45 No. 2 29
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