## Abstract

The measurement ability of the conventional Phase diversity wavefront sensor (C-PD WFS) is limited by the accuracy and dynamic range of CCD cameras. In this letter, a modified Phase diversity wavefront sensor based on a diffraction grating (G-PD WFS) is proposed. We build a corresponding experimental setup to compare the measurement accuracy of the G-PD WFS and the C-GPDWFS under the same experimental conditions. The experimental results show that the measurement ability of G-PD WFS is improved obviously, especially for the wavefront aberration with larger amplitude.

© 2012 OSA

## 1. Introduction

Wavefront sensor (WFS) is one of the key parts in adaptive optics system, and its measurement accuracy decides the wavefront controlling ability of the system [1]. The phase diversity wavefront sensor (PD WFS) captures two images, which are on the conventional focal plane and on a diversity plane with a known defocused distance. The images are used to estimate unknown wavefront aberration and object by phase diversity reconstructing algorithm (PD RA). Since the PD WFS was proposed firstly by Gonsalves and Chidlaw in 1979, simple requirement and high accuracy make PD WFS into an attractive candidate as the WFS in adaptive optics [2]. During the past decades, many authors have used the PD WFS to estimate wavefront aberrations and improve image quality [3–6]. In 1999 and 2000, Blanchard *et al* described a novel PD WFS comprising a distorted diffraction grating and a single camera, so that series of focal and defocused images can be collected by one camera [4, 5]. In 2004, Campbell *et al* introduced a quadratically distorted diffraction grating to image multiple object planes onto a single camera, and proposed an algorithm by a pair of images to reconstruct the wavefront aberration [6].

When the PD WFS works under the monochromatic point source, the wavefront aberrations, especially the high-spatial-frequency components, could cause the energy spread around on the focal and defocused planes. The dispersed intensity distribution contains the wavefront phase information with high-spatial-frequency, and some intensity information will be submerged by the CCD camera’s noise. Then, the measurement accuracy of the intensity images will be degraded, so the measurement ability of the conventional phase diversity wavefront sensor (C-PDWFS) is limited. In other words, if we could increase the precision of the collected images’ information, the accuracy of the reconstructed wavefront must be improved. We employed a modified phase diversity wavefront sensor with a diffraction grating (G-PD WFS) to improve the measurement ability. The diffraction grating is placed before the imaging lens, and it split the incidence beam into a number of beams with different light energy. The different diffraction order spots are output to the CCD camera. The image stitching technology is employed to calculate the intensity distribution on the focus and defocus plane with two or more diffraction spots. The calculated spots contain more information in the lower light level part of original image, which are submersed in noise. In this case, the random wavefront phase can be reconstructed from the new images, and the measurement accuracy can be improved.

In this paper, we build an experimental setup to study the performance of G-PD WFS and compare with C-PD WFS.

## 2. Principle of the grating phase diversity wavefront sensor

When an object is illuminated with spatially coherent monochromatic light and imaged by a linear shift-invariant system, the image can be approximated by the Fourier transform. Assume that the noise at each detector element is modeled as an additive random variable. The intensity distribution can be represented by Eq. (1).

_{$$\begin{array}{l}{I}_{f}={{I}^{\prime}}_{f}+n={\left|FFT\left\{p\cdot \mathrm{exp}i\phi \right\}\right|}^{2}+n\\ {I}_{d}={{I}^{\prime}}_{d}+n={\left|FFT\left\{p\cdot \mathrm{exp}i\left[\phi +{\phi}_{d}\right]\right\}\right|}^{2}+n\end{array}$$}

*p*is the amplitude of generalized pupil function;

*φ*is the wavefront aberration to be measured;

*φ*is the defocused phase function;

_{d}*FFT*{·} is Fourier transform;

*n*is the additive detector noise;

*I’*and

_{f}*I’*are the intensity distribution on focal and defocused plan without detector noise;

_{d}*I*and

_{f}*I*are intensity distributions captured by the CCD camera.

_{d}In this letter, we adopt the phase diversity reconstructed algorithm (PD RA) which is presented by Naoshi Baba and Kohta Nutoh in 2001, and the error metric can be written as Eq. (2) [7].

_{$$E={\displaystyle \sum \left[{\left|{I}_{f}-{{I}^{\prime}}_{f}\right|}^{2}+{\left|{I}_{d}-{{I}^{\prime}}_{d}\right|}^{2}\right]}$$}

The optical layout of G-PD WFS is shown in Fig. 1 .

In experiment, an one-dimensional diffraction grating is employed to divide the incidence beam into a number of beams with the same phase and different intensities. An imaging lens images beams to the focal and defocused planes. A set of diffraction spots, which have different intensities but have the same distribution, are formed. Finally, these images are captured by the CCD cameras.

The shape of the one-dimensional diffraction grating is rectangle with 45*mm* width and 60*mm* length, and its transmission function can be written as Eq. (3).

_{$$t(r)=\left[\frac{1}{2}+\frac{m}{2}\mathrm{cos}(2\pi {f}_{0}\cdot x)\right]rect(\frac{x}{2{L}_{x}})rect(\frac{y}{2{L}_{y}})$$}

*L*,

_{x}*L*are the width and length of grating;

_{y}*rect*(

*x*/2

*L*)

_{x}*rect*(

*y*/2

*L*) is a two-dimensional rectangular function with a shape function of grating;

_{y}*f*is the grating-spatial-frequency and

_{0}*m*is the ratio of transmission function’s peak to valley.

According to the Fraunhofer diffraction formula, the intensity distribution *I _{g}* on the image plane can be approximately as Eq. (4).

_{$${I}_{g}\approx {\eta}_{0}\cdot I+{\eta}_{\pm 1}[I(u-\lambda f\cdot {f}_{0},v)+I(u+\lambda f\cdot {f}_{0},v)]$$}

*u*,

*v*) is the Cartesian coordinate on the image plane;

*λ*is the wavelength;

*f*is the focal length of the imaging lens;

*I*is the intensity distribution without the diffraction grating;

*f*is the grating-spatial-frequency;

_{0}*η*is the ratio of 0th order diffraction spot’s energy to total energy;

_{0}*η*

_{± 1}is the ratio of ± 1st order diffraction spot’s energy to total energy. From the Eq. (3), the shape of 0th and ± 1st order spots are the same, while the spots’ center position and energy are different. The ratio of 0th order spot peak value to ± 1st order depends on grating parameters, and denoted by

*η*(

*η*=

*η*/

_{0}*η*

_{± 1}).

Figure 2
shows the G-PD WFS’ focused image with a uniform plane wavefront. The central and brighter spot is the 0th order diffraction spot, denoted by *I _{g0}*, and the bilateral spots are the ± 1st order diffraction spots. We add the ± 1st order diffraction spots together, then multiplied by

*η*/2, and the result denoted by

*I*. Comparing the

_{g1}*I*and

_{g0}*I*, Some pixels of the

_{g1}*I*are saturated, so these pixels can’t exactly express the intensity distribution. But the pixels in the surrounding part of the

_{g0}*I*are clearer than

_{g0}*I*, and these pixels contain many details of intensity distribution. We use the corresponding pixels in

_{g1}*I*to replace the saturation pixels in

_{g1}*I*, and the new image is denoted by

_{g0}*I*. Obviously,

_{g}*I*contains more detailed information.

_{g}## 3. Experiment

#### 3.1 Experimental setup

The experimental setup, which is used to validate the measurement ability of G-PD WFS, is shown in Fig. 3 . If the diffraction grating is moved out, this experimental setup can be used as the C-PD WFS.

The He-Ne laser beam (wavelength: 632.8 *nm*) is focused by a 40 × microscope objective onto a 10*μm* diameter pinhole, which is a point source in this system. The diameter of circle iris is 4.6*mm*. The focal length of L1, L2, L3 are 200*mm* and the L4 is 250*mm*.

The LC-SLM can be actuated to introduce a wavefront aberration into the system. The coherent point source is collimated by L1 and incident on the LC-SLM. The reflected light from the LC-SLM contains the wavefront aberration. The incident beam is divided into a number of beams by the diffraction grating, and then imaged onto the CCD camera by lens L4. The CCD camera (Cascade 650) is a 16 bits camera, with a 653 × 492 array of 7.4 *μm* pixels. Its exposure time for each image are set by integrating until the signal in the brightest pixel of ± 1st order diffraction spots is nearly 50000.

The defocus is the chosen phase diversity function, and the relationship between the diversity distance (*d*) and peak-to-valley (*PV*) of defocused aberration in wavefront is given by Eq. (5).

_{$${\text{\Phi}}_{d}^{PV}=\raisebox{1ex}{$d$}\!\left/ \!\raisebox{-1ex}{$8\lambda {({F}^{\#})}^{2}$}\right.$$}

*λ*is the wavelength;

*F*

^{#}is the image space f-number which is 45 is this experiment setup. In our experiment, the CCD camera is fixed on the focal plane. A defocused aberration is added on the LC-SLM to capture the defocused image. The pupil sampling size is calculated by focal length, wavelength, physical size of pupil and CCD camera’s pixel.

Conditional upon the physical size of photosensitive plane, only three diffraction spots (0th order diffraction spot and ± 1st order diffraction spots) can be captured. The energy of 0th order and ± 1st order diffraction spots is stronger than other order spots. Considering the capability of the CCD camera (such as the value of the CCD camera’s gain and the exposure time) and the incident intensity, we decide to collect the intensity image of the 0th order and ± 1st order diffraction spots. The ratio *η* is determined by the parameters of the diffraction grating. In this experimental setup, the ratio *η* is 16, which is obtained by a laser power meter. The energy of the 0th order diffraction spot and ± 1st order diffraction spots are measured by the laser power meter separately. Then, the ratio *η* could be computed.

According to the image processing method of G-PD WFS, the intensity distribution can be obtained from collection data. For processing in PD RA, each intensity distribution is background subtracted.

#### 3.2 Experimental results

In order to validate the usefulness of the diffraction grating for measurement ability, two experiments are performed. The conditions for these two experiments are the same, as following:

- (1) The images are both cropped to 256 × 256 array. The size of the discrete matrix of the pupil plane is 256 × 256 array. The sample space of phase distribution is 66 × 66 in a circular field.

- (3) A defocused aberration, with 0.25
*λ*root-mean-square (*RMS*) is adopted for capturing the defocused image. - (4) Estimated wavefront aberration, denoted by
*Ф*, can be obtained by the PD RA. In this paper, the measurement ability is evaluated by the root-mean-squared error (*RMSE*) between*Ф*and_{0}*Ф*, and it can be computed by Eq. (6).

*N*

^{2}is the number of image pixels,

*∆Ф*is the residual wavefront between

*Ф*and

*Ф*.

_{0}The focal and defocused image, which are captured by CCD camera with the diffraction grating, are shown in Fig. 5(*a*)
and Fig. 5(*b*). The intensity distributions, which are calculated by image processing method, are shown in Fig. 5(*c*) and Fig. 5(*d*). The images, which are captured by CCD camera when the diffraction grating is moved out, are shown in Fig. 6(*a*)
and Fig. 6(*b*). Figure 5(*c*), Fig. 5(*d*) and Fig. 6 only show the central regions (128 × 128 pixels) in logarithmic scale, and their peak value is normalized to 1 in each image. Applying the PD RA, the measured wavefront (Fig. 4(*a*), for 0.3*λ RMS* and 2.16*λ PV*) can be estimated based on the focal and defocused images.

Figure 7 shows the estimate wavefront aberrations and residual wavefront with respect to G-PD WFS and C-PD WFS, and Table 1 lists the associated parameters.

Figure 7(*a*) is the estimated wavefront of G-PDWFS, denoted by *Ф _{G}*. Figure 7(

*b*) is the residual aberration between

*Ф*and

_{G}*Ф*, denoted by

_{0}*∆Ф*. Figure 7(

_{G}*c*) is the estimated wavefront of C-PD WFS, denoted by

*Ф*. Figure 7(

_{C}*d*) is the residual wavefront between

*Ф*and

_{C}*Ф*, denoted by

_{0}*∆Ф*. Figure 7(

_{C}*e*) is the 65 order Zernike coefficients of

*Ф*and

_{G}*Ф*. Figure 7(

_{0}*f*) is the 65 order Zernike coefficients of

*Ф*and

_{C}*Ф*. Figure 7(

_{0}*g*) is the differences between the Zernike coefficients of

*Ф*and estimated wavefront

_{0}*Ф*and

_{G}*Ф*.

_{C}As can be seen from the estimated wavefront *Ф _{G}* shown in Fig. 7(

*a*), it is much better approximate to

*Ф*. Comparing the residual aberrations shown in Fig. 7(

_{0}*b*) and Fig. 7(

*d*),

*∆Ф*is much smaller. According to the experimental results in Fig. 7 and Table 1, as expected, the measurement accuracy of G-PD WFS is improved with the diffraction grating, and it is much better than that of C-PD WFS.

_{G}## 4. Conclusion

In conclusion, a modified PD WFS with the diffraction grating is proposed to improve measurement accuracy. The basic principle of the proposed method and the processing of PD RA are described. The experimental setup of G-PD WFS is also illustrated in detail. It is shown that the diffraction grating is in front of the image lens, and the structural is simple. Under the same conditions, the experiments are done to validate the measurement accuracy of G-PD WFS and compare with C-PD WFS. According to the experimental results, it can be seen that the modified wavefront technology has better measurement accuracy, especially for the wavefront aberration with larger amplitude. It can be use as the WFS in an adaptive optics system.

## References and links

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**4. **P. M. Blanchard and A. H. Greenaway, “Simultaneous multiplane imaging with a distorted diffraction grating,” Appl. Opt. **38**(32), 6692–6699 (1999). [CrossRef] [PubMed]

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