Synthetic aperture digital holography by movement of object

∗ Yu Z ha ng , Xiaoxu Lü1, Yinlong Luo, Liyun Zhong, Canlin She Laser Institute of Kunming University of Science and Technology, Kunming, Yunnan, P.R.china, 650051

ABSTRACT

Synthetic aperture technique can be used to improve the reconstruction image resolution of digital hologram. In this paper, a synthetic aperture digital holographic method is proposed. In the method, the recording of synthetic aperture digital hologram is carried out by means of object movement in one direction, and the reconstruction is achieved by cross-correlation synthesis technique. Compared to the digital holography without synthetic aperture technique, only a one-dimension movable bracket is added in the experimental setup. The recorded object fixed on the movable bracket is placed in Mach-Zehnder interferometer and moved along horizontal direction. For the purpose of aperture synthesis reconstructing, four sub-digital holograms have been recorded by a CCD camera. While recording, superposition area of about half pixels of single digital hologram size should be assured between the adjacent holograms in order to ensure that the sub-digital holograms are synthesized accurately. The experimental result shows that every horizontal line of the reconstruction image is distinct in the synthetic aperture digital hologram, but the transverse is unidentifiable in the images reconstructed from the single digital holograms. It can be seen obviously that compared to sub-digital hologram the reconstruction image resolution of synthetic aperture digital hologram is improved.

Keywords: Holography, digital holography, hologram, aperture, synthetic aperture, reconstruction, resolution, CCD camera, cross-correlation, intensity correlation.

1. INTRODUCTION

In recent years, along with the development of computer technology and highly sensitive and real-time CCD camera, a great step forward has been made in holography1-5. But for the small amount of pixels, the effective recording area is much minor compared to the traditional holographic recording material. This leads to low resolution of reconstruction image in digital holography. For little aberration optical system, the theoretical equation of resolution can be expressed as ∆θ = λ 1.22 / D (1)

Where D, λ and ∆θ represent the diameter of entrance pupil, wavelength of the homogeneous light and the minimum angular resolution of the optical system respectively.

∗ E-mail address: [email protected]. 1E-mail address: [email protected]; phone: +86(0871)3304093; fax: +86(0871)3304093.

Holography, Diffractive Optics, and Applications II, edited by Yunlong Sheng, 581 Dahsiung Hsu, Chongxiu Yu, Byoungho Lee, Proceedings of SPIE Vol. 5636 (SPIE, Bellingham, WA, 2005) · 0277-786X/05/$15 · doi: 10.1117/12.573772 From the equation (1), it can be seen obviously that when wavelength is chosen, the resolution of optical system is dependent on the size of its transmission aperture D without regard to the environment noise. Therefore, in digital holography when monochromatic coherent source is selected, the resolution is mainly determined by the diameter of CCD camera. But for a variety of limiting conditions, it is impossible to increase the CCD pupil infinitely. To improve the reconstruction image resolution of digital hologram without adding extra main equipment, the method to increase the equivalent pupil of CCD by synthetic aperture technique is proposed in this paper. Recently, several reports about synthetic aperture digital holography have appeared and some preliminary study achievements in resolution improvement of reconstructed image are showed in these reports. According to Le. Lierc and M. Gross6, the synthetic aperture used in phase-shifting digital holography is acquired by moving a two-dimension detector and the method is demonstrated both theoretically and experimentally. In the report of Jürgen H.Massig7, the synthetic aperture digital hologram is reconstructed from camera recordings at nine different positions. Binet and Colineau8 propose an active optical synthetic aperture-imaging system in which a phase-step setup is required and the aperture synthesis is executed in one direction by means of rotation of object. A synthetic aperture obtained by object translation in horizontal direction is proposed in this paper. No phase shifters are required, which results in a less complex setup. In this method, the illumination object wave front and the reference arriving at CCD target surface is unchanged all along; this will overcome the inhomogeneity of the approximate plane wave.

2. THE DISTRIBUTION OF COMPLEX OPTICAL FIELD ON RECORDING AND RECONSTRUCTION PLANE IN SYNTHETIC APERTURE DIGITAL HOLOGRAPHY

The coordination system schematic of the optical arrangement is shown in Fig.1. Here and in what follows, the

object and CCD plane are represented by x0 y0 and x y respectively and the distance between recording object and CCD camera is represented by character z. If complex amplitude of homogeneous plane reference light and diffracted field

distribution of an arbitrary object point (x0, y0) on CCD target surface are represented by R(x, y) and O(x0, y0 ；x, y)

Reference beam

Beam splitter CCD Object y z x

Z Computer x0 y0 xy

Fig.1. Coordination system of object and hologram

respectively, then the complex amplitude of the digital hologram in the CCD plane can be written as

582 Proc. of SPIE Vol. 5636 = + 2 I(x, y) |O(xO , yO ,x, y) R(x, y)| (2) = 2 + 2 + * + * |O(xO , yO ,x, y)| | R(x, y)| O(xO , yO,x, y)R (x, y) O (xO, yO, x, y)R(x, y) Where * is the complex-conjugate operator and I(x, y) represents the complex amplitude of optical field in CCD plane. Let it be supposed that the recording positions are 1、2、3、…….N. During the movement of object and sub-digital holograms recording, we choose the object center when it is in the first position as the origin of x0y0 coordinate and the ideal image point of object center in CCD plane in every recording position as the xy coordinate origin. It is easy to know that the zero point of xy plane is shifting with the movement of the object. When the recording object is in position n, an arbitrary object and hologram point coordinate are written as (xon, yon) and (x, y) respectively. Then the sub-digital hologram recorded in position n can be represented as follow

= 2 + 2 + * + * ∈ In (x, y) |O(xon, yon;x, y)| | R(x, y)| O(xon, yon;x, y)R (x, y) O (xon, yon;x, y)R(x, y) x, y Sn (3)

Where Sn is the area of sub-digital hologram in position n. Then the synthetic aperture digital hologram combined by sub-digital holograms can be expressed as

N N = = [ 2 + 2 + * + * ] I(x, y) ∑In (x, y) ∑ |O(xon, yon; x, y)| | R(x, y)| O(xon, yon;x, y)R (x, y) O (xon, yon;x, y)R(x, y) n=1 n=1

x, y∈(S1∪S2∪S3．．．．．．∪Sn） (4) If the simulated reconstruction reference beam of sub-digital hologram and synthetic aperture digital hologram is the same as the recording plane reference wave ( Cn(x, y)=C(x, y)=R(x, y)), then the complex amplitude distribution of reconstruction optical field of the sub-digital hologram recorded in position n can be expressed as = An (x, y) I n (x, y)C n (x, y) = [ 2 + 2 + * + | O(xon , yon , x, y) | | R(x, y) | O(xon , yon , x, y)R (x, y) * ] ∈ O (xon , yon , x, y)R(x, y) C(x, y) x, y Sn (5) = 2 + 2 + | O(xon , yon , x, y) | R(x, y) | R(x, y) | R(x, y) 2 + * 2 | R(x, y) | O(xon , yon , x, y) O (xon , yon , x, y)R (x, y)

The reconstruction optical field of synthetic aperture digital hologram is A(x, y) = I(x, y)C(x, y)  N  = [ 2 + 2 + * + * ] ∑|O(xon, yon;x, y)| | R(x, y)| O(xon, yon; x, y)R (x, y) O (xon, yon;x, y)R(x, y) R(x, y) n=1  N = { 2 + 2 + 2 + * 2 } ∑ |O(xon, yon; x, y)| R(x, y) | R(x, y)| R(x, y) O(xon, yon; x, y)| R(x, y)| O (xon, yon; x, y)R (x, y) n=1 N = ∑An (x, y) n=1

x, y∈(S1∪S2∪S3．．．．．．∪Sn) (6) From equation (6), a conclusion can be reached: if a plane reference beam same as the recording reference beam is

Proc. of SPIE Vol. 5636 583 chosen to reconstruct the sub-digital holograms and the synthetic aperture hologram, the superposition of the sub-reconstruction optical fields is just the complex amplitude distribution of the synthetic aperture digital hologram reconstruction optical field.

3. THE RECORDING AND RECONSTRUCTION METHOD OF SYNTHETIC APERTURE DIGITAL HOLOGRAPHY

3.1 Recording The fundamental light path of synthetic aperture digital holography is the same as the ordinary digital holography. The only difference is while recording each sub-digital hologram, the relative position of digital hologram and CCD camera must be changed. If CCD camera is moved parallel to the hologram plane and the arrangement of optical system hold the line, the Sub-hologram 1 Sub-hologram 2 whole hologram will not be changed but the part of (a) Two adjacent sub-holograms hologram on the CCD target surface will be varied. The mentioned reports6-8 about synthetic aperture digital holography are all based on this. When the illumination and the reference beam are plane, on any plane perpendicular to the light propagation Template n direction, the intensity and phase are both constant n1 2 value. Therefore, if the recording object is moved in (b) Template taking and recognizing such a plan and CCD camera is positioned perpendicular to the hologram plan, the relative intensity and phase distribution of the whole digital hologram will not change. Also, the part of hologram on the CCD target surface will be different from the 1 n1 n1+512- n2 one without movement. Therefore, the synthetic aperture digital holography is equally realized by (c) Synthesis of sub-hologram 1 and sub-hologram 2 translation of the recording object. Fig.2. The synthesis course of two adjacent sub-holograms

3.2 Intensity correlation synthesis and Reconstruction When carrying out correlation calculation, we know that the correlation coefficient represents the similarity of the two calculation parts. Under ideal condition, when the correlation coefficient is 1, it can be considered that the two calculation parts are identical. But in practice, the two parts will be thought as the same even the coefficient is much less than 1. The correlation coefficient can be figured out by following equation9

∞ = * − rxy (m) ∑ x(n)y (n m) (7) n=−∞

Where rxy is the coefficient of variable x and y.

584 Proc. of SPIE Vol. 5636 In this paper, the synthetic aperture digital hologram is reconstructed by intensity correlation synthesis method. For the sake of synthesis precision, a superposition of about half a sub-digital hologram size between two adjacent sub-holograms must be assured. The whole reconstruction course can be carried out by two steps. Firstly, a computer program to accomplish the sub-holograms synthesis should be finished. The combination course sketch of two adjacent 512x512 pixel size sub-holograms which have relative position change in horizontal direction is shown in Fig.2. The part with shadow represents the superposition area of the two adjacent sub-holograms in Fig.2. The specific operation is as follow: take out part of the superposition from sub-digital hologram 1 as a template, then recognize in the adjacent sub-digital hologram 2; when the correlation coefficient reaches the maximum peak value, the calculation part can be considered same as the template; then the pixel ordination n1 and n2 as shown in Fig.2(b) are obtained, according to which we synthesize sub-digital hologram1 and sub-digital hologram2; after the synthesis operation, a synthetic aperture digital hologram of 512×(n1+512- n2) represented in Fig.2(c) is acquired, the 1~ n1 column of which is taking from sub-digital hologram 1 of 1~ n1 column and (n1+1) ~ (n1+512- n2) is (n2+1) ~ 512 column of sub-digital hologram 2. All the recording sub-digital holograms are synthesized in turn by the same method. Secondly, the synthetic aperture digital hologram is reconstructed by the ordinary digital reconstruction method.

4. EXPERIMENT

4.1 Experiment setup The schematic of our experimental setup is shown in Fig.3. The whole optical system is a Mach-Zehnder interferometer and the recorded object fixed on a movable platform is placed in it. The part in the dashed line frame is the part of synthetic aperture and the arrow direction in the frame detonates the moving direction of the recorded object. For the precision control of the object moving in horizontal direction, Michelson interferometer is used to serve as a

M2

Attenuator CCD

He-Ne BS1 L1 Pinhole E BS2 E E E M1 Synthetic aperture Computer M3 Object

Fig.3. Experimental setup for digital holography with synthetic aperture mobile platform. The beam from He-Ne laser (633nm wave length) pass through beam expander and pinhole filter then is collimated by collimator lens L1. Then the plane wave is split into two beams by beam splitter

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(a) Sub-digital hologram 1 and the (b) Sub-digital hologram 2 and the (c) Sub-digital hologram 3 and the

corresponding reconstruction image corresponding reconstruction image corresponding reconstruction image

(e) Sub-hologram 4 and the corresponding reconstruction image

Fig.4. Four recording sub-digital holograms and the corresponding

reconstruction images reconstructed by the same reference beam as

the recording reference plane wave

586 Proc. of SPIE Vol. 5636

Fig.5. Synthetic aperture and the corresponding reconstructed image by the same reference beam as the recording reference plane wave

BS1, one of which is reflected by reflecting mirror M3 and illuminates the transparent recording object forming the object wave, another acts as plane reference beam after passing through the attenuator which is used to adjust the intensity proportion of object and reference wave. The reference beam is reflected by reflector M2 and converged with object wave by beam splitter BS2 and form interference pattern field. The part of interference pattern field with size of 512×512 pixels is recorded by CCD camera, then a sub-digital hologram is obtained and transferred to computer. The type of CCD camera is MTV—1802CB and the pixel size of it is 0.010mm×0.0108mm. Four sub-digital holograms and their corresponding reconstruction images are shown in Fig.4 (a), (b), (c) and (d) respectively. The pixel size of a sub-digital hologram in Fig.4 is 512×512, but after they are combined by intensity correlation, a synthetic aperture digital hologram with size of 512 x639 pixels is obtained. Fig.5 shows the synthetic aperture digital hologram and its reconstruction image. When synthesizing sub-digital hologram 1~4, the correlation coefficient of the adjacent sub-digital holograms are 0.5560, 0.5737 and 0.8632 in turn. It can be seen that the maximum correlation coefficient is 0.8632 which is much less than the theory perfect correlation coefficient 1. The minor correlation coefficient is probably due to the instability of the experimental system, but from the reconstruction result we discover it doesn’t influence the reconstruction result much. It can be seen from Fig.4 that the transverse of the reconstructed images from the four sub-digital holograms are almost unidentifiable and we can hardly recognize from these images what the recorded object is. But in Fig.5, we can make out the letter “E” clearly and the structures in horizontal and vertical direction orientation are both distinct. Compared the reconstruction image from synthetic aperture digital hologram to those images from sub-digital holograms, it is obvious that the resolution is improved especially in horizontal direction.

5. CONCLUSION

The recording of synthetic aperture digital hologram can be achieved by moving object. It can be seen from the reconstruction result, the quality of reconstructed image is improved obviously by use of synthetic aperture technique. But there also exist some problems in synthetic aperture digital holography. Proportional to the increase of equivalent aperture, the area of zero order also becomes larger which can be seen from the reconstruction result in Fig.4 (e) and

Proc. of SPIE Vol. 5636 587 Fig.5. Moreover, when the equivalent aperture is large to a certain extent, the condition of equation (1) will not be fulfilled, and then the reconstructed image is probably distorted. The problem can be solved by diminishing the included angle between reference and object beam, but this will lead to separation problem of the three images and phase-step setup must be introduced. So the quantity of sub-digital holograms is not the more and the better but proper as required.

This work is supported by the National Nature Science Foundation of P.R. China (No.60277032) and the Nature Science Fund of Yunnan Province under grants 2002F0030M and 2001F0026M and Technology monitored by Xiaoxu Lü.

REFERENCE

1. J.W.Goodman, Introduction to Fourier Optics, Chap.5, McGrawHill, New York,1996. 2. U. Schnars, “Direct phase determination in hologram interferometry with use of digitally recorded holograms”, J. Opt. Soc. Am, A 11, 2011-2015,1994. 3. W. T. Cathey, Optical Information Processing and Holography, Chap.9, Wiley, New York, 1974. 4. R. J. Collier, C. B. Burckhardt, L.H.Lin, Optical Holography, Chap.20.2, Academic, New York,1971. 5. M. A. Kronrod, N. S. Merzlyakov, and L. P. Yaroslavski, “Reconstruction of holograms with a computer”, Sov. Phys. Tech. Phys, 17,333-334, 1972. 6. F.Le.Lierc, M.Gross, L.Collot, “Synthetic-aperture experiment in the visible with on-axis digital heterodyne holography”, Opt.Lett, 26 (20), 1550-1553, 2001. 7. Jürgen H.Massig, “Digital off-axis holography with a synthetic-aperture”, Opt.Lett, 27(24), 2179-2181, 2001. 8. R.Binet, J.Colineau, J.C.Lehureau. “Short-range synthetic aperture imaging at 633nm by digital holography”. Appl.Opt., 41(23), 4775-4782, 2002. 9. Peiqing Cheng, Digital Signal Processing Course, Chap.4, Tsinghua publishing house, Beijing, 2001.

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