Experimental study of a modified phase diversity with a diffraction grating

doi:10.1364/OE.20.012059

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Experimental study of a modified phase diversity with a diffraction grating

Qun Luo, Linhai Huang, Naiting Gu, and Changhui Rao »View Author Affiliations

Optics Express, Vol. 20, Issue 11, pp. 12059-12066 (2012)
http://dx.doi.org/10.1364/OE.20.012059

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▸ Article Outline
– Abstract
1. Introduction
2. Principle of the grating phase diversity wavefront sensor
3. Experiment
4. Conclusion
– References and links

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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.

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

J. W. Hardy, “Adaptive optics: a progress review,” Proc. SPIE 1542, 2–17 (1991). [CrossRef]

]. 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

R. A. Gonsalves and R. C. Hidlaw, “Wavefront sensing by phase retrieval,” in Applications of Digital Image Processing, Proc. SPIE 207, 32–39 (1979).

]. During the past decades, many authors have used the PD WFS to estimate wavefront aberrations and improve image quality [3

J. H. Seldin, R. G. Paxman, V. G. Zarifis, L. Benson, and R. E. Stone, “Closed-loop wavefront sensing for a sparse-aperture, phased-array telescope using broadband phase diversity,” in Imaging Technology and Telescopes, SPIE 4091, 48–63 (2000).

–6

H. I. Campbell, S. Zhang, A. H. Greenaway, and S. Restaino, “Generalized phase diversity for wave-front sensing,” Opt. Lett. 29(23), 2707–2709 (2004). [CrossRef] [PubMed]

]. 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

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

, 5

P. M. Blanchard, D. J. Fisher, S. C. Woods, and A. H. Greenaway, “Phase diversity wavefront sensing with a distorted diffraction grating,” Appl. Opt. 39(35), 6649–6655 (2000). [CrossRef] [PubMed]

]. 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

H. I. Campbell, S. Zhang, A. H. Greenaway, and S. Restaino, “Generalized phase diversity for wave-front sensing,” Opt. Lett. 29(23), 2707–2709 (2004). [CrossRef] [PubMed]

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^{'}}_{f} + n = {| F F T {p \cdot \exp i φ} |}^{2} + n \\ I_{d} = {I^{'}}_{d} + n = {| F F T {p \cdot \exp i [φ + φ_{d}]} |}^{2} + n \end{array}

(1)

where p is the amplitude of generalized pupil function; φ is the wavefront aberration to be measured; φ_d is the defocused phase function; FFT{·} is Fourier transform; n is the additive detector noise; I’_f and I’_d are the intensity distribution on focal and defocused plan without detector noise; I_f and I_d are intensity distributions captured by the CCD camera.

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

N. Baba and K. Mutoh, “Measurement of telescope aberrations through atmospheric turbulence by use of phase diversity,” Appl. Opt. 40(4), 544–552 (2001). [CrossRef] [PubMed]

E = \sum [{| I_{f} - {I^{'}}_{f} |}^{2} + {| I_{d} - {I^{'}}_{d} |}^{2}]

(2)

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

Fig. 1 Schematic diagram of the G-PD WFS. d: defocused distance, B.S.: beam splitter

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 45mm width and 60mm length, and its transmission function can be written as Eq. (3).

t (r) = [\frac{1}{2} + \frac{m}{2} \cos (2 π f_{0} \cdot x)] r e c t (\frac{x}{2 L_{x}}) r e c t (\frac{y}{2 L_{y}})

(3)

where L_x, L_y are the width and length of grating; rect(x/2L_x)rect(y/2L_y) is a two-dimensional rectangular function with a shape function of grating; f₀ is the grating-spatial-frequency and 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 η_{0} \cdot I + η_{\pm 1} [I (u - λ f \cdot f_{0}, v) + I (u + λ f \cdot f_{0}, v)]

(4)

where (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; f₀ is the grating-spatial-frequency; η₀ is the ratio of 0th order diffraction spot’s energy to total energy; 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 = η₀/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_g1. Comparing the I_g0 and I_g1, Some pixels of the I_g0 are saturated, so these pixels can’t exactly express the intensity distribution. But the pixels in the surrounding part of the I_g0 are clearer than I_g1, and these pixels contain many details of intensity distribution. We use the corresponding pixels in I_g1 to replace the saturation pixels in I_g0, and the new image is denoted by I_g. Obviously, I_g contains more detailed information.

Fig. 2 Focal image

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.

Fig. 3 Experimental setup of G-PD WFS. Lenses are denoted by L prefix, M: reflective mirror, S.F.: spatial filter, Aperture: circle iris, B.S.: beam splitter, LC-SLM: liquid-crystal spatial light modulator, Grating: one-dimensional diffraction grating

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

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).

Φ_{d}^{P V} = \frac{d}{8 λ {(F^{#})}^{2}}

(5)

where 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.
(2) The wavefront aberration under test: A random wavefront aberration (Fig. 4(a) ) is created by the first 15 order Zernike polynomials, denoted by Ф₀. The Ф₀ is 0.3λ RMS, and is 2.16λ PV. The coefficients of Zernike polynomials are shown in Fig. 4(b).
Fig. 4 Random wavefront aberration Ф₀. (a) Phase distribution, (b) The coefficients of Zernike polynomials

(3) A defocused aberration, with 0.25λ 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 Ф₀ and Ф, and it can be computed by Eq. (6).
$R M S E = \sqrt{\sum {(Δ Φ)}_{ij}^{2} / N^{2}}$

(6)

where N² is the number of image pixels, ∆Ф is the residual wavefront between Ф and Ф₀.

Fig. 4 Random wavefront aberration Ф₀. (a) Phase distribution, (b) The coefficients of Zernike polynomials

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 0.3λ RMS and 2.16λ PV) can be estimated based on the focal and defocused images.

Fig. 5 Experimental images of G-PD WFS. (a) Captured focal image, (b) Captured defocused image, (c) Calculated focal image, (d) Calculated defocused image

Fig. 6 Experimental images of C-PD WFS. (a) Focal image, (b) Defocused image

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.

Fig. 7 Experimental results. (a) The estimated wavefront of G-PD WFS, (b) The residual aberration ∆Ф_G, (c) The estimated wavefront of C-PD WFS. (d) The residual aberration ∆Ф_C, (e) The Zernike coefficients of Ф_G and Ф₀, (f) The Zernike coefficients of Ф_C and Ф₀, (g) The Zernike coefficients of ∆Ф_G and ∆Ф_C

Table 1 Experimental Results of G-PD WFS and C-PD WFS

		G-PD WFS	C-PD WFS
Measured Wavefront	RMS(λ)	0.3
Measured Wavefront	PV(λ)	2.16
Estimated Wavefront	RMS(λ)	0.289	0.213
Estimated Wavefront	PV(λ)	2.08	1.50
Residual Wavefront	RMSE(λ)	0.0308	0.106
Residual Wavefront	PV(λ)	0.220	0.891

Figure 7(a) is the estimated wavefront of G-PDWFS, denoted by Ф_G. Figure 7(b) is the residual aberration between Ф_G and Ф₀, denoted by ∆Ф_G. Figure 7(c) is the estimated wavefront of C-PD WFS, denoted by Ф_C. Figure 7(d) is the residual wavefront between Ф_C and Ф₀, denoted by ∆Ф_C. Figure 7(e) is the 65 order Zernike coefficients of Ф_G and Ф₀. Figure 7(f) is the 65 order Zernike coefficients of Ф_C and Ф₀. Figure 7(g) is the differences between the Zernike coefficients of Ф₀ and estimated wavefront Ф_G and Ф_C.

As can be seen from the estimated wavefront Ф_G shown in Fig. 7(a), it is much better approximate to Ф₀. Comparing the residual aberrations shown in Fig. 7(b) and Fig. 7(d), ∆Ф_G 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.

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

1.	J. W. Hardy, “Adaptive optics: a progress review,” Proc. SPIE 1542, 2–17 (1991). [CrossRef]
2.	R. A. Gonsalves and R. C. Hidlaw, “Wavefront sensing by phase retrieval,” in Applications of Digital Image Processing, Proc. SPIE 207, 32–39 (1979).
3.	J. H. Seldin, R. G. Paxman, V. G. Zarifis, L. Benson, and R. E. Stone, “Closed-loop wavefront sensing for a sparse-aperture, phased-array telescope using broadband phase diversity,” in Imaging Technology and Telescopes, SPIE 4091, 48–63 (2000).
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]
5.	P. M. Blanchard, D. J. Fisher, S. C. Woods, and A. H. Greenaway, “Phase diversity wavefront sensing with a distorted diffraction grating,” Appl. Opt. 39(35), 6649–6655 (2000). [CrossRef] [PubMed]
6.	H. I. Campbell, S. Zhang, A. H. Greenaway, and S. Restaino, “Generalized phase diversity for wave-front sensing,” Opt. Lett. 29(23), 2707–2709 (2004). [CrossRef] [PubMed]
7.	N. Baba and K. Mutoh, “Measurement of telescope aberrations through atmospheric turbulence by use of phase diversity,” Appl. Opt. 40(4), 544–552 (2001). [CrossRef] [PubMed]

OCIS Codes
(010.1080) Atmospheric and oceanic optics : Active or adaptive optics
(010.7350) Atmospheric and oceanic optics : Wave-front sensing

ToC Category:
Adaptive Optics

History
Original Manuscript: March 21, 2012
Revised Manuscript: April 18, 2012
Manuscript Accepted: April 19, 2012
Published: May 11, 2012

Citation
Qun Luo, Linhai Huang, Naiting Gu, and Changhui Rao, "Experimental study of a modified phase diversity with a diffraction grating," Opt. Express 20, 12059-12066 (2012)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-11-12059

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References

J. W. Hardy, “Adaptive optics: a progress review,” Proc. SPIE1542, 2–17 (1991). [CrossRef]
R. A. Gonsalves and R. C. Hidlaw, “Wavefront sensing by phase retrieval,” in Applications of Digital Image Processing, Proc. SPIE207, 32–39 (1979).
J. H. Seldin, R. G. Paxman, V. G. Zarifis, L. Benson, and R. E. Stone, “Closed-loop wavefront sensing for a sparse-aperture, phased-array telescope using broadband phase diversity,” in Imaging Technology and Telescopes, SPIE4091, 48–63 (2000).
P. M. Blanchard and A. H. Greenaway, “Simultaneous multiplane imaging with a distorted diffraction grating,” Appl. Opt.38(32), 6692–6699 (1999). [CrossRef] [PubMed]
P. M. Blanchard, D. J. Fisher, S. C. Woods, and A. H. Greenaway, “Phase diversity wavefront sensing with a distorted diffraction grating,” Appl. Opt.39(35), 6649–6655 (2000). [CrossRef] [PubMed]
H. I. Campbell, S. Zhang, A. H. Greenaway, and S. Restaino, “Generalized phase diversity for wave-front sensing,” Opt. Lett.29(23), 2707–2709 (2004). [CrossRef] [PubMed]
N. Baba and K. Mutoh, “Measurement of telescope aberrations through atmospheric turbulence by use of phase diversity,” Appl. Opt.40(4), 544–552 (2001). [CrossRef] [PubMed]

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