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Optics Express

  • Editor: Michael Duncan
  • Vol. 14, Iss. 23 — Nov. 13, 2006
  • pp: 11103–11112
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Optical implementation of iterative fractional Fourier transform algorithm

Joonku Hahn, Hwi Kim, and Byoungho Lee  »View Author Affiliations


Optics Express, Vol. 14, Issue 23, pp. 11103-11112 (2006)
http://dx.doi.org/10.1364/OE.14.011103


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Abstract

An optical implementation of iterative fractional Fourier transform algorithm is proposed and demonstrated. In the proposed implementation, the phase-shifting digital holography technique and the phase-type spatial light modulator are adopted for the measurement and the modulation of complex optical fields, respectively. With the devised iterative fractional Fourier transform system, we demonstrate two-dimensional intensity distribution synthesis in the fractional Fourier domain and three-dimensional intensity distribution synthesis simultaneously forming desired intensity distributions at several multi-focal planes.

© 2006 Optical Society of America

1. Introduction

Fig. 1. Iterative FRFT algorithm.

The optical implementation of the iterative FRFT algorithm includes measuring and reconstructing the optical field as well as the optical implementation of a FRFT. The FRFT theory has been actively researched in the optical information processing fields [10–13

10. H. M. Ozaktas and D. Mendlovic, “Fractional Fourier Optics,” J. Opt. Soc. Am. A 12, 743–751 (1995). [CrossRef]

]. The firmly established operator approach originated from the FRFT theory gives more systematic management and insightful understanding of general complex optical information processing system that the simple use of the generalized Fresnel transform cannot provide [14–17

14. D. Mendlovic, Y. Bitran, R. G. Dorsch, C. Ferreira, J. Garcia, and H. M. Ozaktaz, “Anamorphic fractional Fourier transform: optical implementation and applications,” Appl. Opt. 34, 7451–7456 (1995). [CrossRef] [PubMed]

]. The phase-shifting holography technique is the most important dynamic optical field measurement techniques using charge-coupled devices (CCDs) [18

18. I. Yamaguchi and T. Zhang, “Phase-shifting digital holography,” Opt. Lett. 22, 1268–1270 (1997). [CrossRef] [PubMed]

]. With the phase-shifting holography technique, the wave-front of diffraction fields generated in optical information processing systems can be accurately measured and additionally the reconstruction of measured optical field is possible using spatial light modulators (SLMs) with the electrical controllability.

This paper is organized as follows. In Sec. 2, the optical implementation of FRFT is described. In Sec. 3, the optical implementation of iterative FRFT algorithm is proposed. In Sec. 4, experimental results are presented and discussed. In Sec. 5, conclusion and perspective are given.

2. Optical implementation of fractional Fourier transforms

The scheme of the iterative FRFT algorithm is composed of several parts. The forward FRFT and its inverse FRFT are basically necessary. The constraint functions at the input and output planes should also be devised so that they can be optically implemented. In this section, the optical implementation of FRFT is discussed.

At first we should choose an implementation form of FRFT appropriate for our objective. The ath order two-dimensional fractional Fourier transform is defined by

Fα(u,v)=Kα(u,v,u,v)G(u,v)dudv,
(1)

where the integral kernel is defined by,

fora2m,Ka(u,v,u,v)=[1jcot(πa2)]exp(jπ[cot(πa2)(u2+v2)2csc(πa2)(uu+vv)+cot(πa2)(u2+v2)]),
(2a)
fora=4m,Ka(u,v,u,v)=δ(uu)δ(vv),
(2b)
fora=4m±2,Ka(u,v,u,v)=δ(u+u)δ(v+v),
(2c)

where m is an integer. The important properties of the fractional Fourier transform are the associative and the communicative properties as

Fa1[Fa2(f)]=Fa2[Fa1(f)]=Fa1+a2(f).
(2d)

In general, paraxial optical system can be described with the well-defined FRFT. The ath order two-dimensional FRFT in optical system is defined by the linear integral transform

F(x,y)=h(x,y,x,y)G(x,y)dxdy,
(3a)

where the transform kernel is given by

h(x,y,x,y)=csc(aπ2)s2Mejπ2exp(jπ(x2+y2)λR)exp(jπs2[cot(πa2)(x2+y2)M22csc(πa2)M(xx+yy)+cot(πa2)(x2+y2)]).
(3b)

The above kernel may be found in Ref. 13

13. H. M. Ozaktas, M. A. Kutay, and Z. Zalevsky, The Fractional Fourier Transform with Applications in Optics and Signal Processing (John Wiley and Sons Ltd., New York, 2001).

, and it maps a function G(x′, y′)=(1/s)Ĝ(x/s, y/s) into [1/(sM)]exp((x 2+y 2)/(λR))Ĝa (x/(sM)), y/(sM)) where Ĝa (u,v) is the ath order two-dimensional fractional Fourier transform of Ĝ(u,v).

Let us consider the optical system shown in Fig. 2. This is an optical implementation of the cascade of the ath order two-dimensional FRFT and its complementary (2-a)th order FRFT. Form Eq. (2d) we see that the overall system is an FRFT with the order of 2 because a+(2-a)=2. Then, from Eq. (2c) we see that this overall system is just inversing the image in transverse coordinates from G(x 1, y 1) to F(x 3, y 3), which should be true because the overall system is a 4-f imaging system. A diverging spherical wave is incident on the filter denoted by G(x 1, y 1) and the optical field on the filter is Fourier-transformed to P(x 2, y 2) that is given by

P(x2,y2)=jλf[exp(jπ(x12+y12)λR)G(x1,y1)]exp(j2πλf(x2x1+y2y1))dx1dy1
=jλfexp(jπ(x22+y22)λR)exp(jπλ[(x12+y12)R2f(x2x1+y2y1)+(x22+y22)R])G(x1,y1)dx1dy1,
(4)

where, R is the radius of the spherical phase, with which the input optical field G(x 1, y 1) is wrapped.

Fig. 2. Optical implementation of the ath order and its complementary (2-a)th order two-dimensional FRFT. The incident optical field may be diverging, converging or normally incident to the input plane.

Since we are of interest in the intensity distribution of the output optical field P(x 2, y 2), the extra quadratic phase term exp(-(x22+y22)/λR) can be neglected in Eq. (4). Comparing Eqs. (3b) and (4), we can estimate the transform order and the scaling factor of the corresponding FRFT, respectively, as

a=2πarccos(fR),
(5a)
s4=λ2R2f2R2f2.
(5b)

The second section of the total optical system in Fig. 2 indicated by the (2-a)th order fractional Fourier transform is similarly analyzed. To construct the (2-a)th order fractional Fourier transform, the converging phase factor exp((x22+y22)/λR′) must be multiplied by the optical field P(x 2, y 2), where -R′ is the converging radius of curvature (R′>0). Then the optical field is Fourier-transformed to F(x 3, y 3). The second stage is equivalent to the first stage except the sign change of the incident spherical phase factor. Then the transform order of the second stage can be found just by changing R with -R′:

2a=2πarccos(fR),
(6a)
s4=λ2R2f2R2f2.
(6b)

As pointed out earlier, the overall system of Fig. 2 is a 4-f imaging system, i.e., the image F(x 3, y 3) is the same as G(x 1, y 1) in Fig. 2 except the coordinate inversion. Hence the (2-a)th order FRFT, which is called a complementary transform for the ath order FRFT, can be considered as the inverse transform of the ath order FRFT. We can control the FRFT order a by adjusting the curvature R of the incident spherical field as can be seen in Eq. (5a). With changing the sign of the curvature of the incident spherical wave, we can realize the forward FRFT and its inverse FRFT in the same FRFT optic setup. This is an important factor to be made use of to optically implement the iterative FRFT algorithm.

3. Optical implementation of iterative fractional Fourier transform algorithm

In this section, the proposed optical implementation of the iterative FRFT algorithm is elucidated. The functional parts of the implemented system and the algorithmic equipments necessary for the iterative FRFT can be separately explained.

Figure 3 shows the schematic of the implemented system. As indicated in Fig. 3, the system is composed of several functional parts of (a) the field measurement part using the phase shifting holography technique, (b) the 1st beam path with a negative lens for the ath order FRFT, (c) the 2nd beam path with a positive lens for the (2-a)th order FRFT, (d) the 4-f imaging system for matching the scaling factors between the CCD and the SLM, and (e) LabView-based system control unit.

Coherent Verdi 5W Nd:YAG laser with wavelength of 532 nm is used as a light source. The field measurement part is composed of the piezo stage XYZ-38 of Piezosystem Jena and the CCD (KODAK MegaPLUS ES1.0/MV with 8 bit resolution) with the pixel size of 9µm. The optical field distribution is measured by the phase-shifting holography technique.

The 1st and 2nd beam paths share an SLM composed of the liquid crystal device (SONY LCX016AL-6) with the pixel size of 24µm and two polarizers placed before and after the liquid crystal device. To optimally control the phase modulation of SLM in the full range of 2π, the rotation angles of the former and latter polarizers to the reference axis parallel with the SLM are tuned as 330° at the input side and 10° at the output side, respectively. Mechanical shutters (SIGMA KOKI 65GR) are used to switch the 1st and 2nd beam paths automatically. The 1st beam path and the 2nd beam path are corresponding to the ath order FRFT and the (2-a)th order FRFT, respectively. In the 1st beam path a negative lens (denoted by NL in Fig. 3) is placed to wrap the phase profile on the SLM with the spherical phase of the radius R, and on the other side, in the second beam path, a positive lens (PL) is placed to provide the spherical phase of the radius -R′. A pattern mask describing the target intensity distribution is placed on the 2nd beam path (the inverse transform).

Fig. 3. Optical implementation of the iterative FRFT algorithm.

Especially, the optical field information measured by the CCD is transferred to the SLM after signal processing of the phase-shifting digital holography. Here the scale of the field distribution is magnified because the pixel size of the SLM (24µm) is bigger than that of CCD (9µm). However, matching the scaling factors between the CCD and the SLM is necessary to correctly carry out the forward and the inverse FRFTs. The scaling factor mismatch will result in very complicated FRFT relation [5

5. T. Shirai and T. H. Barnes, “Adaptive restoration of a partially coherent blurred image using an all-optical feedback interferometer with a liquid-crystal device,” J. Opt. Soc. Am. A 19, 369–377 (2002). [CrossRef]

]. To avoid this difficulty, a 4-f imaging optic setup with magnification feasibility is employed for matching the scaling factors of the SLM and the CCD. By the 4-f imaging optics, the optical field represented on the SLM is adjusted to fit the scale of the CCD.

Basically, the algorithmic flow chart of the iterative FRFT algorithm follows that presented in Fig. 1. However, in the optical implementation, additional steps of adding and subtracting spherical phase profile from the measured phase profile have to be inserted at every iteration step as shown in Fig. 4. At the first step, the initial phase profile in the input plane is obtained by the inverse transform from the patterned mask representing the signal domain, where the inverse transform is implemented by the (2-a)th order FRFT. At the following steps, the forward and the successive inverse transforms are conducted iteratively. The beam paths are selected by mechanical shutters to implement the ath order FRFT (1st beam path) and the (2-a)th order FRFT (2nd beam path) alternatively. That is, the ath order FRFT and the (2-a)th order FRFT in Fig. 1 are split into time sequential steps along the 1st and 2nd beam paths.

Fig. 4. Flow chart of the iterative FRFT algorithm.

Between the steps the spherical phases are subtracted from the measured phase profiles, and at the following step the conjugate spherical phases are added to the phase profile encoded on the SLM, since the spherical phase with the small radius compared with the pixel size of the encoded phase profile cannot be correctly represented by the SLM. This problem results from a discrete sampling of the spherical phase.

A soft clipping method is devised to realize the soft constraint in the output plane. For this, the inverse transform is conducted twice as shown in Fig. 5. First, the optical field on the signal area selectively filtered by the patterned mask is inversely transformed and next, the optical field on both the signal and noise areas is inversely transformed. Two inversely transformed field distributions are summed with specific weight factors. The ratio of the weight factors is controlled to be decreased as the iteration progresses.

With this system, the DOE phase profile generating a wanted two-dimensional intensity distribution at the ath order FRFT domain can be attained through this iteration procedure. With additional control of the spherical curvature of the incident spherical wave, we can obtain different intensity distributions at different FRFT domains. Multiplexing several DOE phase profiles enable simultaneous generation of intensity distributions at several FRFT domains with different FRFT orders. In other words, two-dimensional intensity distribution synthesis in the fractional Fourier domain and three-dimensional intensity distribution synthesis simultaneously forming desired intensity distributions at several multi-focal planes can be realized in real time with the devised iterative FRFT system.

Fig. 5. Soft clipping in the iterative FRFT algorithm.

Regarding the three-dimensional multi-focal intensity distribution synthesis, we can see that when the DOE with the phase profile obtained by the proposed system at the ath order FRFT domain is illuminated by a plane wave, the diffraction pattern is formed at the defocused plane. In Fig. 6, the transform of the wave wrapped with diverging spherical phase (denoted by dotted lines) to the focal plane is formulated as

P(x2,y2)=jλf[exp(jπ(x12+y12)λR)G(x1,y1)]exp(j2πλf(x2x1+y2y1))dx1dy1,
(7a)

and the transform of the plane input wave (denoted by the solid line) modulated by G(x 1, y 1) to a Δd distance from the focal plane is formulated as

F(x2,y2)=jλf{exp[jπλf(Δdf)(x12+y12)]G(x1,y1)}exp(j2πλf(x2x1+y2y1))dx1dy1.
(7b)

By comparing the integrands of Eqs. (7a) and (7b), the relation between the radius R and the defocus Δd is given by

Δd=f2R.
(7c)

Therefore, with several positive and negative lens pairs of radius R and - R, we can design DOE phase profiles to generate diffraction images at distinguished image planes. Also, the obtained DOE phase profiles can be multiplexed to form three-dimensional multi-focal intensity distribution at several distinguished planes.

4. Experimental results

Fig. 6. The relationship for the two transforms: input with the spherical radius R (dotted lines) and the plane wave input with the corresponding defocus Δd.
Fig. 7. Improvement of diffraction images as the number of iteration increases in the iterative FRFT algorithm for the two conditions of the radius of the spherical phase: (a) R=∞ (0.24 MB movie) and (b) R=62mm (0.13 MB movie).

In our another experiment, four DOE phase profiles are designed with different patterned masks being applied with different spherical phases as shown in Fig. 8. Based on the principle of Fig. 6, by multiplexing obtained DOE phase profiles, the overall DOE can form desired intensity distributions at several different longitudinal locations simultaneously as shown in Fig. 9. Since the diffraction images are detected after the incident beam passes through the lens, the converging spherical phases are applied to designed DOE phase profiles. In Fig. 9, we see that diffraction images are changed with the change in the image-capturing location of the CCD.

5. Conclusion

With the devised iterative FRFT algorithm, we can design the DOE phase profile to generate two-dimensional intensity distribution in a certain defocused plane at a time without modeling the real system. The proposed technique is implemented in the almost fully optical way and we can simplify the processes such as measuring the real system parameters and encoding the DOE phase profile into the SLM. With the FRFT by an incident spherical phase profile, we show that the FRFT order at the lens focal plane is controllable and the amount of the defocus has a clear relationship with the FRFT order at the lens focal plane. We successfully realized an optical implementation of iterative FRFT algorithm with the aid of wave-optical engineering technologies like the SLM and the phase-shifting digital holography technique.

Fig. 8. Multiplexing four DOE phase profiles (a) the patterned masks and (b) the applied spherical phase profiles.
Fig. 9. (0.88 MB movie) Diffraction images of the multiplexed DOE captured by a CCD as the capturing location is changed.

With the FRFT language, the forward and the inverse transform are clearly defined and the optical implementation is also manageable. A noticeable engineering point in this work is the 4-f imaging part for matching the scaling factors of the CCD and the SLM. By adjusting the spherical phase curvature, we can obtain the DOE phase profiles to generate diffraction images at several defocused planes. The experimental feasibility of multi-focal image synthesis using the simple DOE multiplexing method opens the possibility of shaping three-dimensional volumetric intensity distribution synthesis in volumetric region by using a fully optical implementation setup.

In short, we studied the optical implementation of an iterative algorithm and the convergence of the algorithm. Based on this work we will improve the adaptive optical field synthesis algorithm compensating the distributed aberration in a real time.

Acknowledgment

This work was supported by Samsung SDI., Ltd.

References and links

1.

J. Turunen and F. Wyrowski, Diffractive Optics for Industrial and Commercial Applications (John Wiley and Sons Ltd., New York, 1997).

2.

B. Kress and P. Meyrueis, Digital Diffractive Optics: An Introduction to Planar Diffraction Optics and Related Technology (John Wiley and Sons Ltd., New York, 2000).

3.

H. Kim, K. Choi, and B. Lee, “Diffractive optic synthesis and analysis of light fields and recent applications,” Jpn. J. Appl. Phys. 45, 6555–6575 (2006). [CrossRef]

4.

V. A. Soifer, V. V. Kotlyar, and L. Doskolovich, Iterative Methods for Diffractive Optical Elements Computation (Tayor & Francis Ltd., London, 1997).

5.

T. Shirai and T. H. Barnes, “Adaptive restoration of a partially coherent blurred image using an all-optical feedback interferometer with a liquid-crystal device,” J. Opt. Soc. Am. A 19, 369–377 (2002). [CrossRef]

6.

J. Hahn, H. Kim, K. Choi, and B. Lee, “Real-time digital holographic beam-shaping system with a genetic feedback tuning loop,” Appl. Opt. 45, 915–924 (2006). [CrossRef] [PubMed]

7.

H. Kim and B. Lee, “Optimal nonmonotonic convergence of the iterative Fourier-transform algorithm,” Opt. Lett. 30, 296–298 (2005). [CrossRef] [PubMed]

8.

H. Kim, B. Yang, and B. Lee, “Iterative Fourier transform algorithm with regularization for the optimal design of diffractive optical elements,” J. Opt. Soc. Am. A 12, 2353–2365 (2004). [CrossRef]

9.

K. Choi, H. Kim, and B. Lee, “Synthetic phase holograms for auto-stereoscopic image display using a modified IFTA,” Opt. Express 12, 2454–2462 (2004). [CrossRef] [PubMed]

10.

H. M. Ozaktas and D. Mendlovic, “Fractional Fourier Optics,” J. Opt. Soc. Am. A 12, 743–751 (1995). [CrossRef]

11.

A. Sahin, H. M. Ozaktas, and D. Mendlovic, “Optical implementation of the two-dimensional fractional Fourier transform with different orders in the two dimensions,” Opt. Commun. 120, 134–138 (1995). [CrossRef]

12.

A. Sahin, H. M. Ozaktas, and D. Mendlovic, “Optical implementations of two-dimensional fractional Fourier transforms and linear canonical transforms with arbitrary parameters,” Appl. Opt. 37, 2130–2141 (1998). [CrossRef]

13.

H. M. Ozaktas, M. A. Kutay, and Z. Zalevsky, The Fractional Fourier Transform with Applications in Optics and Signal Processing (John Wiley and Sons Ltd., New York, 2001).

14.

D. Mendlovic, Y. Bitran, R. G. Dorsch, C. Ferreira, J. Garcia, and H. M. Ozaktaz, “Anamorphic fractional Fourier transform: optical implementation and applications,” Appl. Opt. 34, 7451–7456 (1995). [CrossRef] [PubMed]

15.

T. Kim and T-C. Poon, “Three-dimensional matching by use of phase-only holographic information and the Wigner distribution,” J. Opt. Soc. Am. A 12, 2520–2528 (2000). [CrossRef]

16.

P. Andrés, W. D. Furlan, G. Saavedra, and A. W. Lohmann, “Variable fractional Fourier processor: a simple implementation,” J. Opt. Soc. Am. A 14, 853–858 (1997). [CrossRef]

17.

E. Tajahuerce, G. Saavedra, W. D. Furlan, E. E. Sicre, and P. Andrés, “White-light optical implementation of the fractional Fourier transform with adjustable order control,” Appl. Opt. 39, 238–245 (2000). [CrossRef]

18.

I. Yamaguchi and T. Zhang, “Phase-shifting digital holography,” Opt. Lett. 22, 1268–1270 (1997). [CrossRef] [PubMed]

OCIS Codes
(070.2580) Fourier optics and signal processing : Paraxial wave optics
(090.1760) Holography : Computer holography
(230.6120) Optical devices : Spatial light modulators

ToC Category:
Fourier Optics and Optical Signal Processing

History
Original Manuscript: September 5, 2006
Revised Manuscript: October 31, 2006
Manuscript Accepted: October 31, 2006
Published: November 13, 2006

Citation
Joonku Hahn, Hwi Kim, and Byoungho Lee, "Optical implementation of iterative fractional Fourier transform algorithm," Opt. Express 14, 11103-11112 (2006)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-23-11103


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References

  1. J. Turunen and F. Wyrowski, Diffractive Optics for Industrial and Commercial Applications (John Wiley and Sons Ltd., New York, 1997).
  2. B. Kress and P. Meyrueis, Digital Diffractive Optics: An Introduction to Planar Diffraction Optics and Related Technology (John Wiley and Sons Ltd., New York, 2000).
  3. H. Kim, K. Choi, and B. Lee, "Diffractive optic synthesis and analysis of light fields and recent applications," Jpn. J. Appl. Phys. 45, 6555-6575 (2006). [CrossRef]
  4. V. A. Soifer, V. V. Kotlyar, and L. Doskolovich, Iterative Methods for Diffractive Optical Elements Computation (Tayor & Francis Ltd., London, 1997).
  5. T. Shirai and T. H. Barnes, "Adaptive restoration of a partially coherent blurred image using an all-optical feedback interferometer with a liquid-crystal device," J. Opt. Soc. Am. A 19, 369-377 (2002). [CrossRef]
  6. J. Hahn, H. Kim, K. Choi, and B. Lee, "Real-time digital holographic beam-shaping system with a genetic feedback tuning loop," Appl. Opt. 45, 915-924 (2006). [CrossRef] [PubMed]
  7. H. Kim and B. Lee, "Optimal nonmonotonic convergence of the iterative Fourier-transform algorithm," Opt. Lett. 30, 296-298 (2005). [CrossRef] [PubMed]
  8. H. Kim, B. Yang, and B. Lee, "Iterative Fourier transform algorithm with regularization for the optimal design of diffractive optical elements," J. Opt. Soc. Am. A 12, 2353-2365 (2004). [CrossRef]
  9. K. Choi, H. Kim, and B. Lee, "Synthetic phase holograms for auto-stereoscopic image display using a modified IFTA," Opt. Express 12, 2454-2462 (2004). [CrossRef] [PubMed]
  10. H. M. Ozaktas and D. Mendlovic, "Fractional Fourier Optics," J. Opt. Soc. Am. A 12, 743-751 (1995). [CrossRef]
  11. A. Sahin, H. M. Ozaktas, and D. Mendlovic, "Optical implementation of the two-dimensional fractional Fourier transform with different orders in the two dimensions," Opt. Commun. 120, 134-138 (1995). [CrossRef]
  12. A. Sahin, H. M. Ozaktas, and D. Mendlovic, "Optical implementations of two-dimensional fractional Fourier transforms and linear canonical transforms with arbitrary parameters," Appl. Opt. 37, 2130-2141 (1998). [CrossRef]
  13. H. M. Ozaktas, M. A. Kutay, and Z. Zalevsky, The Fractional Fourier Transform with Applications in Optics and Signal Processing (John Wiley and Sons Ltd., New York, 2001).
  14. D. Mendlovic, Y. Bitran, R. G. Dorsch, C. Ferreira, J. Garcia, and H. M. Ozaktaz, "Anamorphic fractional Fourier transform: optical implementation and applications," Appl. Opt. 34, 7451-7456 (1995). [CrossRef] [PubMed]
  15. T. Kim and T-C. Poon, " Three-dimensional matching by use of phase-only holographic information and the Wigner distribution," J. Opt. Soc. Am. A 12, 2520-2528 (2000). [CrossRef]
  16. P. Andrés, W. D. Furlan, G. Saavedra, and A. W. Lohmann, "Variable fractional Fourier processor: a simple implementation," J. Opt. Soc. Am. A 14, 853-858 (1997). [CrossRef]
  17. E. Tajahuerce, G. Saavedra, W. D. Furlan, E. E. Sicre, and P. Andrés, "White-light optical implementation of the fractional Fourier transform with adjustable order control," Appl. Opt. 39, 238-245 (2000). [CrossRef]
  18. I. Yamaguchi and T. Zhang, "Phase-shifting digital holography," Opt. Lett. 22, 1268-1270 (1997). [CrossRef] [PubMed]

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