## Abstract

A method to expand the dynamic range for a Shack–Hartmann wavefront sensor (SHWFS) is proposed. An SHWFS consists of a microlens array and an image sensor, and it has been widely used to measure the wavefront aberration of a lightwave in various fields. However, a very large aberrated wave cannot be correctly measured due to the finite dynamic range that depends on the diameter of each microlens. The proposed method enables an SHWFS to measure wavefronts with larger aberrations by applying holography and pattern matching technologies. For measurement of a spherical wave, the proposed method is compared with a conventional one by numerical simulations and optical experiments. Their results confirm the performance of the proposed method.

© 2015 Optical Society of America

## 1. INTRODUCTION

An adaptive optics system plays an important role in correcting a wavefront aberration caused by various disturbances in the fields of astronomy [1–4], communication [5,6], and medicine [7–10]. The system consists of two parts: a wavefront sensor and a wavefront compensator. The wavefront sensor is the device that measures the wavefront aberration of a lightwave, and it is also applied to measure the surface profiles of optical elements, such as an aspherical lens. A Shack–Hartmann wavefront sensor (SHWFS) is used to measure a wavefront or a phase gradient distribution in these applications. A general SHWFS consists of a microlens array and an image sensor, such as a charge coupled device (CCD) or a complementary metal oxide semiconductor (CMOS) sensor, as shown in Fig. 1. An incident wave passes through the microlens array, and the spot map is obtained by the image sensor. Local phase gradients of the incident wave are measured from the centroid displacement of each spot on the map.

Figure 2 shows the calculation principle of a local phase gradient. If the local incident wave to a microlens is an untilted plane wavefront, the spot is located on the optical axis or the center of the detection area. This point is a reference position. On the other hand, if the local incident wave is a tilted wavefront, the spot moves corresponding to the local phase gradient. The local phase gradient is calculated by the centroid displacement of the spot from the reference position and the focal length of a microlens. A local phase gradient in the $x$ direction is given by [11]

where $\mathrm{\Phi}$, $\mathrm{\Delta}{x}_{c}$, and $f$ denote the phase of an incident wave, the centroid displacement of a spot from the reference position in the $x$ direction, and the focal length of a microlens, respectively. The local phase gradient in the $y$ direction can also be calculated in the same way. After calculating these processes for every spot, the wavefront is obtained by the aberration approximation with Zernike polynomials [12].An SHWFS is more tolerant of both a vibration of the system and a temporal variation of the measured wavefront than an interferometric measurement. However, a wavefront that has large phase gradients beyond the dynamic range of the sensor cannot be accurately measured. The dynamic range is one of the important factors in determining the performance of an SHWFS. The limitation of the dynamic range is determined by the size of the sensor area allocated to each microlens. The area is called the detection area. Various techniques to expand the dynamic range have been proposed [13–18]. Although the methods with a time-sequential approach [13–15] can expand the dynamic range, these cannot be applied to certain momentary wavefronts. Software approaches [16–18] can be applied to certain momentary wavefronts. However, analyses in these methods require complicated algorithms and long processing time. If every spot corresponds to each microlens correctly, the dynamic range can be expanded. However, each spot cannot be easily discriminated from the other because their shapes are almost same.

In this paper, we propose a holographic Shack–Hartmann wavefront sensor (H-SHWFS) to overcome the above problems. In the H-SHWFS, a microhologram array, which consists of independent small phase holograms, is used instead of a microlens array. It is easily realized through the use of the phase-modulated spatial light modulator (P-SLM). Discriminable patterns are reconstructed on the image sensor by the microhologram array, which enables these patterns to correspond to each detection area. In fact, methods to expand the dynamic range using linear spots with different orientations generated by astigmatic microlenses have been proposed [19,20]. Our proposed method is also based on a similar concept. These patterns move in proportion as each local phase gradient of an incident wave, just like in a conventional SHWFS. In addition, the H-SHWFS is combined with the correlation peak displacement detection method. Although measurement for only a coherent and monochromatic wave is ensured and a higher intensity wave than that of a conventional SHWFS is required, the proposed method enables the SHWFS to expand its dynamic range with single shot and simple analysis in a static condition. The performance of the proposed method is confirmed by numerical simulations and optical experiments.

## 2. H-SHWFS

#### A. Basic Principle

An SHWFS using the phase distribution of a microlens array displayed on a P-SLM has been researched [21]. This technique has the advantages that the focal length and the diameter of each microlens can be easily changed. Additionally, an arbitrary spatial phase modulation can be given to the incident wave. The proposed H-SHWFS is realized by displaying a microhologram array on the P-SLM instead of a microlens array. The microhologram array consists of small phase holograms as shown in Fig. 3. In the method, different patterns are reconstructed from the microhologram array on each detection area. Figure 3 shows the case in which different patterns are reconstructed on $3\times 3$ detection areas. The patterns are obtained on the reconstruction plane determined during the design process of the microhologram array, and move corresponding to the phase gradient of an incident wave. The phase gradient distribution is calculated from the displacement of the patterns, just like in a conventional SHWFS. Owing to the discriminable patterns, the detection area can be expanded to adjacent areas.

#### B. Correlation Peak Displacement Detection Method

In the H-SHWFS, the centroid displacements of the reconstructed pattern cannot be easily calculated because their shapes are complicated. To solve the problem, the correlation peak displacement detection (CPDD) method is applied to the H-SHWFS. As used in Ref. [22], the approach with cross correlation is effective for detecting the displacement of a certain pattern. Figure 4 shows the principle of the CPDD method. Cross correlations between the reconstructed image including nine different reconstructed patterns and template images are easily calculated by fast Fourier transform (FFT)-based calculations, which are given by

where $R(x,y)$, ${T}_{n}(x,y)$, and * denote the reconstructed image, an $n$th ($1\le n\le 9$) template image, and he complex conjugate, respectively. Each template image involves only a pattern. Also, ⋆, FFT[], and IFFT[] are cross correlation, fast Fourier transform, and inverse fast Fourier transform operators, respectively. As a result of the calculations, correlation maps are obtained. On the maps, the point where the high-intensity correlation signal arises shows that the reconstructed pattern is consistent with the template image. On the other hand, the signals hardly arise at the other points owing to low correlation. The detection area can be expanded to the eight adjacent areas because only a correlation peak exists within the $3\times 3$ areas. Additionally, the conventional spot centroid detection algorithm is similarly available for detecting the correlation peak point because the shape of the correlation signal is quite similar to the spot generated by a microlens. In this case, the dynamic range of the H-SHWFS is 3 times larger than that of the conventional SHWFS. In the CPDD method, it takes more time to measure a wavefront than in a conventional SHWFS, and high processing power is required because multiple FFT-based cross correlations need to be calculated for the analysis. However, the analysis is expected to accelerate by calculating the cross correlations with a graphics processing unit (GPU).#### C. Design of the Microhologram Array

A microhologram array is expressed as a phase-modulated pattern to use the incident wave energy effectively and can be easily realized through the use of the P-SLM. The microholograms that constitute the microhologram array are designed as phase-only Fourier transform holograms. Thus, intensity patterns generated by the microhologram array are reconstructed on the Fourier plane. Each pattern should be appropriately selected because the measurement accuracy depends on the intensity of correlation signals. For example, in the case of patterns as shown in Fig. 4, undesirable correlation signals arise around the correlation peak, as shown in Fig. 5. These signals might cause measurement error. Therefore, low-correlated patterns should be selected to reduce undesirable signals. Additionally, the following design conditions are also introduced. To ensure the intensity of the reconstructed pattern in the case of a low-intensity incident wave, a pattern has to be expressed in as few pixels as possible. In this study, the patterns which consist of $6\times 6$ pixels including six bright pixels were used, which reduces the disadvantage of the proposed method. A pattern consists of binary intensity pixels in cases when the intensity of a reconstructed pattern is less than that of noise, including dark current noise, undesirable stray light, and so on. High-intensity pixels did not appear side by side to avoid degradation of the reconstructed patterns caused by interpixel interference. Selected patterns that satisfy these conditions are shown in Fig. 6. Each microhologram is designed from these patterns and is arranged in $3\times 3$ areas. Additionally, the phase distribution of a lens whose focal length can be arbitrarily set is added to these holograms to provide a lensless Fourier transform effect. The unit microhologram array shown in Fig. 7(a) is generated by the above procedures. The reconstructed patterns as shown in Fig. 7(b) are obtained on the focal plane by numerical propagation. In fact, the same patterns appear periodically on the focal plane because the microhologram array has many unit microhologram arrays for wavefront measurement.

## 3. EVALUATION

#### A. Wavefront Measurement Simulation

The performance of the proposed method was compared with that of a conventional one by numerical simulations. A spherical wave whose curvature radius is 350 mm is assumed. The microhologram array, which consists of an arrangement of many unit microhologram arrays, was used. Simulation parameters are shown in Table 1. A microlens array in the conventional method has similar parameters. Smaller microholograms are desirable to improve the measurement accuracy by increasing the spatial resolution. Therefore, the measurement accuracy depends on the pixel size of the available SLM. Correlation maps are obtained from each cross-correlation calculation corresponding to nine reconstructed patterns. One of nine correlation maps and its magnified view of the red square area are shown in Figs. 8(a) and 8(b), respectively. The red cross in Fig. 8(b) denotes a reference position for calculation of the peak displacement. Figure 9 shows magnified phase gradient distributions of size $1152\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{\mu m}\times 1152\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{\mu m}$ on the $x\u2013y$ plane. Each correlation peak moves, corresponding to the phase gradient of the spherical wave, and the correlation peaks on outer areas are beyond the conventional detection area. In the conventional method, these areas cause measurement errors, as shown in Fig. 9(b), in the process of calculating the phase gradient distribution. In contrast, the phase gradient distribution as shown in Fig. 9(a) is correctly calculated in the proposed method because the peak exists within the expanded detection area. The measurement results of the proposed and the conventional methods are shown in Fig. 10. These were obtained by approximating each calculated phase gradient distribution by Zernike polynomials. The spherical wave is nearly correctly obtained in the proposed method. In contrast, the result in the conventional method is definitely incorrect. These are quantitatively evaluated by a root mean square error (RMSE). The RMSEs were 0.504 rad in the proposed method and 23.724 rad in the conventional one. These results show the expansion of the dynamic range and high-accuracy wavefront measurement with the H-SHWFS.

#### B. Optical Experiment

The wavefront measurement using the proposed method is experimentally confirmed. The H-SHWFS is realized by the optical setup shown in Fig. 11. The P-SLM and the image sensor have $792\times 600$ pixels and $1280\times 960$ pixels, respectively. Their pixel sizes are 20 μm and 4.65 μm, respectively. To show the potential of the H-SHWFS, we introduced a wavefront with a large phase gradient. Namely, a spherical wave whose curvature radius is about 350 mm from a point source is used. Furthermore, the source was placed away from the optical axis. Therefore, its outer area with a large phase gradient is measured so that the difference of the dynamic range between the proposed method and the conventional one is indicated under the condition of the measurable area restricted by the image sensor size. Other parameters are similar to the numerical simulations described in Section 3.A. It is difficult to realize an arrangement that in which the image sensor is close to the SLM plane because a reflection-type SLM is used. Therefore, the image sensor is arranged at a distance of 30 mm from the conjugate SLM plane (broken line in Fig. 11) by a $4\text{-}f$ configuration. The reconstructed image and one of its correlation maps, as shown in Fig. 12, are experimentally obtained, as is the case with the simulation. Figure 13 shows the phase gradient distributions calculated by the proposed and the conventional methods in sizes of $1152\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{\mu m}\times 1152\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{\mu m}$ on the $x\u2013y$ plane. The upper phase gradient obtained by the conventional method is definitely incorrect. On the other hand, that obtained by the proposed method could be measured all over the area. Wavefronts approximated by Zernike polynomials are also shown in Fig. 14. Although the wrong wavefront was obtained by the conventional method due to phase gradient errors, the spherical wave is correctly measured by the proposed method. From these results, the expansion of the dynamic range by the proposed method was verified.

## 4. CONCLUSION

A H-SHWFS in which discriminable patterns are generated and their displacements are detected by the CPDD method is proposed. Although the light source is restricted and extra calculations are required in the analysis, the proposed method enables the SHWFS to expand its dynamic range with a single shot and simple analysis in a static condition. The performance of the proposed method is confirmed by numerical simulations and optical experiments.

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