## Optical imaging with phase-coded aperture |

Optics Express, Vol. 19, Issue 5, pp. 4294-4300 (2011)

http://dx.doi.org/10.1364/OE.19.004294

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### Abstract

Experimental results are shown for an integrated computational imaging system with a phase-coded aperture. A spatial light modulator works as a phase screen that diffracts light from a point object into a uniformly redundant array (URA). Excellent imaging results are achieved after correlation processing. The system has the same depth of field as a diffraction-limited lens. Potential applications are discussed.

© 2011 Optical Society of America

## 1. Introduction

1. W. Chi and N. George, “Phase-coded aperture for optical imaging,” Opt. Commun. **282**, 2110–2117 (2009). [CrossRef]

2. R. H. Dicke, “Scatter-hole cameras for X-rays and Gamma rays,” Astrophys. J. **153**, L101 (1968). [CrossRef]

5. F. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. **21**, 2758–2769 (1982). [CrossRef] [PubMed]

## 2. Linear system theory for integrated imaging

*i*(

*x,y*) can be expressed as a function of object

*o*(

*x,y*) and point spread function (PSF)

*h*(

*x,y*) in the following convolution form,

*h*(

*x,y*) can be any realizable function. More specifically,

*h*(

*x,y*) need not be a delta-like function for sharp imaging, as is seen below.

*L*{•} which, when applied to

*h*(

*x,y*), yields the following result, where

*f*(

_{δ}*x,y*) is a Dirac delta-like function such as an Airy disk function that is spatially separated from the function

*g*(

*x,y*). We can add a linear digital processing system by applying the optical image to the linear operator

*L*{•}. The result is

*f*(

_{δ}*x,y*) is the PSF of the overall system including both optics and image processing.

*h*(

*x,y*) which (i) is realizable with optical system and (ii) can transform to a delta-like function under some linear operators. In application, the optical system generally has further constraints on optical material, size, number of elements etc. so the function must be further required to be realizable with the optical system satisfying these constraints.

*L*{•} can take many forms. For example, it can be an identity operator, a differential operator, or a correlation operator.

*h*(

*x,y*) is a delta-like Airy disk, and

*L*{•} is an identity operator. Another example is an integrated system where the image is blurred by lens aberrations and a subsequent linear deconvolution algorithm is applied to the intermediate image to yield a sharp picture.

1. W. Chi and N. George, “Phase-coded aperture for optical imaging,” Opt. Commun. **282**, 2110–2117 (2009). [CrossRef]

*h*(

*x,y*) is a bl-URA, and linear operator

*L*{•} is a correlation operator. The main purpose of this article is to present experimental results of such a system which we call a phase-coded-aperture imaging system.

*t*(

*x,y*) is a URA and

*b*(

*x,y*) is a bandlimited function. A phase retrieval method is used to calculate the phase function that yields such a bl-URA. The linear operator is defined as where ⊗ is a correlation operator and

*t*(

_{R}*x, y*) is a repeated URA with mean removed, i.e., in which

*D*,

_{x}*D*are the sizes of URA in X and Y directions, respectively; and

_{y}*t*̄ is the mean value of URA

*t*(

*x,y*).

*x*) = max{1 – |

*x*|,0} ;

*C*is a constant with exact value determined by URA; and Δ

*, Δ*

_{x}*are the pixel size of URA in X and Y directions, respectively. Equation (7) consists of an array of delta-like functions as in which a normalization constant is omitted.*

_{y}*i*(

*x,y*) with

*t*(

_{R}*x, y*), we can recover a sharp image with the overall PSF

*f*(

_{δ}*x,y*) shown in Eq. (8).

## 3. Experimental setup

*π*, as shown in Fig. 1, where black means 0 and white means 2

*π*. This phase modulated wave is then reflected by the beam splitter and received by a detector array. The phase profile is calculated using a Fresnel domain phase retrieval method. Parameters of the setup are the following: the spatial modulator has a pixel size of 8

*μm*and a fill factor of 85% ; the detector array has a pixel size of 13

*μm*and pixel number of 1024×1024; the distance between object plane and phase screen is 1275

*mm*; the equivalent free space distance between phase screen and detector array is 204.4

*mm*; and the square aperture in front of the phase screen has a dimension of 5.5

*mm*×5.5

*mm*. In this paper as a proof of principle to our theoretical concept [1

1. W. Chi and N. George, “Phase-coded aperture for optical imaging,” Opt. Commun. **282**, 2110–2117 (2009). [CrossRef]

## 4. Experimental results

*θ*is

*θ*=

*D*/(2

*L*), where

*L*is the distance between the phase plate and the detector.

*mm*, the bl-URA has a size of 6.3

*mm*. From Fig. 3 one can observe the overall PSFs due to the mismatch of URA size to bl-URA at detector. When the URA size used for correlation processing is changed to 6.2

*mm*, significant artifacts appeared in the combined PSF after digital processing. This can result in a poor imaging result especially for extended objects. Sometimes, resampling and interpolation of the URA are required before correlation processing. Studying the effect of interpolation on the artifacts is beyond the scope of this paper.

*mm*. The intermediate image in Fig. 4a is processed linearly using the same correlation method, and the recovery result is shown in Fig. 4b. In Fig. 4a we observe a general feature of the intermediate image for an extended object. The intermediate image has an overall envelop shape that is bright in the center and the intensity slowly drops to zero at the edge. There are also small scale intensity variations which reflect the details of the object. In Fig. 4a one can see regions of the CCD where detector response is low. This causes a big intensity drop (with specks in the image, some are indicated by the arrows) in comparison to the envelop. Despite this, a good recovery is still achieved as shown in Fig. 4b. In the image recovery, one can simply change these low values to bigger ones such that the envelop of the intermediate image is smooth. The exact value of these small regions has little effect on the recovery. In comparison with conventional imaging system, a small region of dead pixels in conventional lens system would cause a complete lose of image in that section.

*mm*to 1000

*mm*. This corresponds to defocus amounts up to 1.6

*λ*. In the recovery the same repeated URA pattern is used for correlation processing of all the intermediate images at different distances. The depth of field of the coded aperture system is similar to that of a diffraction limited lens. i.e., the images within ±

*λ*/4 defocus provide good quality. Interestingly, the defocused image quality degradation takes a different form in comparison to that with a conventional diffraction limited lens. For a coded aperture system, if there exists large defocus, then one observes the repetition of objects over the whole scene. The intensity of these repetitions increases as the defocus amount becomes larger.

*mm*and 1100mm. The distance of 1100mm corresponds to a defocus of

*λ*. Two differences are noticed: (i) the fine details of the PSF are different; (ii) the full sizes of the PSFs are different. The size of the PSF for a focused distance is 6.3

*mm*, while the size of the defocused PSF is 6.42

*mm*. (The sizes of PSFs are found by correlating the PSFs with repeated URA of different scales. The size of the URA which yields the best recovered point object is considered as the size of the PSF of the optical system.)

*mm*is used to recover the object located at the defocused distance of 1100

*mm*. We see a significant improvement of image quality.

## 5. Concluding remarks

**282**, 2110–2117 (2009). [CrossRef]

*λ*/4 defocus is acceptable.

## Acknowledgments

## References and links

1. | W. Chi and N. George, “Phase-coded aperture for optical imaging,” Opt. Commun. |

2. | R. H. Dicke, “Scatter-hole cameras for X-rays and Gamma rays,” Astrophys. J. |

3. | E. E. Fenimore and T. M. Cannon, “Coded aperture imaging with uniformly redundant array,” Appl. Opt. |

4. | R. G. Simpson and H. H. Barrett, “Coded aperture imaging,” in |

5. | F. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. |

6. | J. C. Dainty and F. R. Fienup, “Phase retrieval and image reconstruction for astronomy,” in |

7. | D. P. Casasent and T. Clark (Ed.), “Adaptive Coded Aperture Imaging and Non-imaging Sensors,” Proc. SPIE |

8. | D. P. Casasent and S. Rogers (Ed.), “Adaptive Coded Aperture Imaging and Non-imaging Sensors II,” Proc. SPIE |

**OCIS Codes**

(100.0100) Image processing : Image processing

(110.0110) Imaging systems : Imaging systems

(110.1758) Imaging systems : Computational imaging

(110.7348) Imaging systems : Wavefront encoding

**ToC Category:**

Imaging Systems

**History**

Original Manuscript: November 23, 2010

Revised Manuscript: February 11, 2011

Manuscript Accepted: February 13, 2011

Published: February 18, 2011

**Citation**

Wanli Chi and Nicholas George, "Optical imaging with phase-coded aperture," Opt. Express **19**, 4294-4300 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-5-4294

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### References

- W. Chi, and N. George, “Phase-coded aperture for optical imaging,” Opt. Commun. 282, 2110–2117 (2009). [CrossRef]
- R. H. Dicke, “Scatter-hole cameras for X-rays and Gamma rays,” Astrophys. J. 153, L101 (1968). [CrossRef]
- E. E. Fenimore, and T. M. Cannon, “Coded aperture imaging with uniformly redundant array,” Appl. Opt. 17, 337–347 (1978). [CrossRef] [PubMed]
- R. G. Simpson, and H. H. Barrett, “Coded aperture imaging,” in Imaging in Diagnositc Medicine, S. Nudel-man, (Ed.) (Plenum, New York, 1980).
- F. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2769 (1982). [CrossRef] [PubMed]
- J. C. Dainty, and F. R. Fienup, “Phase retrieval and image reconstruction for astronomy,” in Image Recovery: Theory and Application, H. Stark, (Ed.) (Academic, Florida, 1987).
- D. P. Casasent, and T. Clark, eds., “Adaptive Coded Aperture Imaging and Non-imaging Sensors,” Proc. SPIE 6714 (2007).
- D. P. Casasent, and S. Rogers, eds., “Adaptive Coded Aperture Imaging and Non-imaging Sensors II,” Proc. SPIE 7096 (2008).

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