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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 18, Iss. 13 — Jun. 21, 2010
  • pp: 13536–13541
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Quantitative phase retrieval of a complex-valued object using variable function orders in the fractional Fourier domain

Wen Chen and Xudong Chen  »View Author Affiliations


Optics Express, Vol. 18, Issue 13, pp. 13536-13541 (2010)
http://dx.doi.org/10.1364/OE.18.013536


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Abstract

We propose a novel and effective method to quantitatively recover a complex-valued object from diffraction intensity maps recorded in the fractional Fourier domain. A wavefront modulation is introduced in the wave path, and several diffraction intensity maps are recorded through variable function orders in the fractional Fourier transform. A new phase retrieval algorithm is then proposed, and advantages of the proposed algorithm are also discussed. A proof-of-principle study is presented to show the feasibility and effectiveness of the proposed method.

© 2010 OSA

1. Introduction

Phase retrieval technique plays an important role in the structural determination of micro or nano-objects [1

1. J. M. Zuo, I. Vartanyants, M. Gao, R. Zhang, and L. A. Nagahara, “Atomic resolution imaging of a carbon nanotube from diffraction intensities,” Science 300(5624), 1419–1421 (2003). [CrossRef] [PubMed]

3

3. D. Gabor, “A new microscopic principle,” Nature 161(4098), 777–778 (1948). [CrossRef] [PubMed]

]. Because only intensity distribution can be recorded by a charge-coupled device (CCD) camera, the phase information is lost. Since holography was introduced by Gabor [3

3. D. Gabor, “A new microscopic principle,” Nature 161(4098), 777–778 (1948). [CrossRef] [PubMed]

], it has been widely applied to various fields [4

4. C. J. Mann, L. Yu, C. Lo, and M. K. Kim, “High-resolution quantitative phase-contrast microscopy by digital holography,” Opt. Express 13, 8693–8698 (2005), http://www.opticsinfobase.org/oe/abstract.cfm? URI = oe-13–22–8693.

6

6. W. Chen, C. Quan, and C. J. Tay, “Measurement of curvature and twist of a deformed object using digital holography,” Appl. Opt. 47(15), 2874–2881 (2008). [CrossRef] [PubMed]

], such as biological microscopy [4

4. C. J. Mann, L. Yu, C. Lo, and M. K. Kim, “High-resolution quantitative phase-contrast microscopy by digital holography,” Opt. Express 13, 8693–8698 (2005), http://www.opticsinfobase.org/oe/abstract.cfm? URI = oe-13–22–8693.

] and metrology [5

5. W. Chen, C. Quan, and C. J. Tay, “Extended depth of focus in a particle field measurement using a single-shot digital hologram,” Appl. Phys. Lett. 95(20), 201103 (2009). [CrossRef]

,6

6. W. Chen, C. Quan, and C. J. Tay, “Measurement of curvature and twist of a deformed object using digital holography,” Appl. Opt. 47(15), 2874–2881 (2008). [CrossRef] [PubMed]

]. Two main types of holography, i.e., off-axis digital holography [4

4. C. J. Mann, L. Yu, C. Lo, and M. K. Kim, “High-resolution quantitative phase-contrast microscopy by digital holography,” Opt. Express 13, 8693–8698 (2005), http://www.opticsinfobase.org/oe/abstract.cfm? URI = oe-13–22–8693.

] and phase-shifting digital holography [7

7. I. Yamaguchi and T. Zhang, “Phase-shifting digital holography,” Opt. Lett. 22(16), 1268–1270 (1997). [CrossRef] [PubMed]

], have been developed. Since complex amplitude can be extracted by a numerical reconstruction algorithm in digital holography, phase information is directly determined by using an arc-tangent operation. However, the interferometric method requires a relatively complex optical system, such as a reference wave, temporal coherence and a stable mechanical setup. Recently, non-interferometric method [1

1. J. M. Zuo, I. Vartanyants, M. Gao, R. Zhang, and L. A. Nagahara, “Atomic resolution imaging of a carbon nanotube from diffraction intensities,” Science 300(5624), 1419–1421 (2003). [CrossRef] [PubMed]

] is considered as a promising alternative for phase retrieval, and has attracted much attention in various fields, such as material [1

1. J. M. Zuo, I. Vartanyants, M. Gao, R. Zhang, and L. A. Nagahara, “Atomic resolution imaging of a carbon nanotube from diffraction intensities,” Science 300(5624), 1419–1421 (2003). [CrossRef] [PubMed]

,2

2. M. A. Pfeifer, G. J. Williams, I. A. Vartanyants, R. Harder, and I. K. Robinson, “Three-dimensional mapping of a deformation field inside a nanocrystal,” Nature 442(7098), 63–66 (2006). [CrossRef] [PubMed]

].

Hybrid input-output method which was partially derived based on Gerchberg-Saxton algorithm [8

8. R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttg.) 35, 237–246 (1972).

] was proposed by Fienup [9

9. J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21(15), 2758–2769 (1982). [CrossRef] [PubMed]

], and it has been commonly used in the non-interferometric phase retrieval. When the test object is real or non-negative, it is easy to apply certain support constraints to extract the phase information from the recorded diffraction intensity. However, support constraints in the conventional hybrid input-output method [9

9. J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21(15), 2758–2769 (1982). [CrossRef] [PubMed]

] might be inapplicable for the study of complex-valued objects, and stagnation problem is still a great challenge. It is demonstrated that a stronger support constraint is usually required for a complex-valued object than that for a real or non-negative object. Miao et al. [10

10. J. Miao, P. Charalambous, J. Kirz, and D. Sayre, “Extending the methodology of X-ray crystallography to allow imaging of micrometer-sized non-crystalline specimens,” Nature 400(6742), 342–344 (1999). [CrossRef]

] proposed an oversampling phasing method to extract complex-valued objects. Liu et al. [11

11. Y. J. Liu, B. Chen, E. R. Li, J. Y. Wang, A. Marcelli, S. W. Wilkins, H. Ming, Y. C. Tian, K. A. Nugent, P. P. Zhu, and Z. Y. Wu, “Phase retrieval in X-ray imaging based on using structured illumination,” Phys. Rev. A 78(2), 023817 (2008). [CrossRef]

] proposed a phase contrast imaging technique based on a combination of structured illumination and an optimized hybrid input-output algorithm for the extraction of complex-valued objects. In this paper, we propose a novel method with variable function orders in the fractional Fourier transform (FRFT) for the extraction of a complex-valued object. A wavefront modulation is introduced in the wave path, and several diffraction intensity maps are recorded. A new phase retrieval algorithm is then proposed to recover the complex-valued object.

2. Theoretical analysis

Figure 1
Fig. 1 A schematic experimental setup.
shows a schematic experimental arrangement. A collimated plane wave is used to illuminate a test complex-valued object, and a diffraction intensity distribution is recorded by a CCD camera. A series of intensity distributions can be recorded, when the function order in the FRFT is modified correspondingly. Wave propagation between the object plane and the phase-only mask plane can be described by the diffraction principle.
F(x,y)=jλ++C(ξ,η)exp(j   kρ)ρ       dξ   dη,
(1)
where C(ξ,η) denotes the original complex-valued object, j=1, λ is the wavelength of a light source, wave number k=2π/λ, ρ is the distance from a point in the object plane to a point in the phase-only mask plane[ρ=(xξ)2+(yη)2+z2], and z denotes the axial distance between the object plane and the phase-only mask plane.

Equation (1) can be simplified with a paraxial or small-angle approximation, and can also be implemented by a convolution method. In this study, angular spectrum algorithm [5

5. W. Chen, C. Quan, and C. J. Tay, “Extended depth of focus in a particle field measurement using a single-shot digital hologram,” Appl. Phys. Lett. 95(20), 201103 (2009). [CrossRef]

,12

12. J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996).

] is used, and the complex amplitude just before the phase-only mask plane is described by
F(x,y)=FT1[C(ξ,η)   ¯   T(fξ,fη)],
(2)
where fξ and fη are spatial frequencies, FT1 denotes two-dimensional (2D) inverse Fourier transform, C(ξ,η)   ¯denotes a 2D Fourier transform of C(ξ,η)   ,and T(fξ,fη)is a transfer function described by
T(fξ,fη)=exp{   (j2π   z/λ)           [1(λ   fξ)2(λ   fη)2]1/2}                       if       (fξ2+fη2)1/2<(1/λ)   .
(3)
An advantage of angular spectrum algorithm is that unlike Fresnel approximation algorithm [12

12. J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996).

], no minimum distance is required. It is assumed in this study that fξ and fη are always within the above bandwidth. A discrete representation of Eq. (3) is expressed as
T(g,h)=exp(j   k   z   {1[λ   (gG/2)/(GΔξ)]2[λ   (hH/2)/(HΔη)]2}1/2),
(4)
where g=0,   1,     ...,     G1, h=0,   1,     ...,     H1, G and H denote pixel numbers, and Δξ and Δη are equal to the pixel sizes of CCD camera.

The object wave F(x,y) is multiplied by a known phase-only mask M(x,y), and then propagates through a fractional Fourier domain. In practice, a spatial light modulator can be used to display the phase-only mask M(x,y). For the sake of brevity, an one-dimensional FRFT [13

13. A. W. Lohmann, “Image rotation, Wigner rotation, and the fractional Fourier transform,” J. Opt. Soc. Am. A 10(10), 2181–2186 (1993). [CrossRef]

16

16. M. G. Ertosun, H. Atlı, H. M. Ozaktas, and B. Barshan, “Complex signal recovery from multiple fractional Fourier-transform intensities,” Appl. Opt. 44(23), 4902–4908 (2005). [CrossRef] [PubMed]

] is described, and the FRFT with an order a can be defined as
FRFTa[F(x)M(x)]=+F(x)M(x)     Pa(μa,x)   dx,
(5)
where Pa(μa,x)={Rexp{jπ[μa2cot(aπ/2)+x2cot(aπ/2)2μaxcsc(aπ/2)]}ifa2mδ(μax)ifa=4mδ(μa+x)ifa=4m±2, m denotes an integer, and R=1j   cot(aπ/2). Similarly, the 2D FRFT can be defined, and an intensity map is expressed as
I(μ,ν)=|FRFTa,b[F(x,y)M(x,y)]|2,
(6)
where for simplicity the value of b is equal to a in this study. The function order can be modified through the adjustment of the distance and the focal length in the FRFT domain [13

13. A. W. Lohmann, “Image rotation, Wigner rotation, and the fractional Fourier transform,” J. Opt. Soc. Am. A 10(10), 2181–2186 (1993). [CrossRef]

16

16. M. G. Ertosun, H. Atlı, H. M. Ozaktas, and B. Barshan, “Complex signal recovery from multiple fractional Fourier-transform intensities,” Appl. Opt. 44(23), 4902–4908 (2005). [CrossRef] [PubMed]

], thus a series of diffraction intensity maps are recorded correspondingly.

Here, we propose a new iterative algorithm using three recorded diffraction intensity maps [I1(μ,ν),     I2(μ,ν)     and     I3(μ,ν)] to recover the original complex-valued object, and a flow chart for the proposed method is shown in Fig. 2
Fig. 2 A flow chart of the proposed quantitative phase retrieval algorithm.
. The proposed phase retrieval algorithm proceeds as follows:

  • (1) Assume an initial random or constant distribution Fn(x,y) (iteration number n = 1, 2, 3…) just before the phase-only mask M(x,y);
  • (2) Multiply by the phase-only maskM(x,y): Fn'(x,y)=Fn(x,y)     M(x,y);
  • (3) Propagate to CCD plane with function order of a1: On1(μ,v)=FRFTa1,a1[Fn'(x,y)]   ;
  • (4) Apply a support constraint in the CCD plane with the square root of the recorded intensity distribution: On1(μ,ν)¯=[I1(μ,ν)]1/2[On1(μ,ν)/|On1(μ,ν)|]     ;
  • (5) Numerically propagate back to the phase mask plane and multiply byM*(x,y): Fn1(x,y)¯=FRFTa1,a1[On1(μ,ν)¯]       M*(x,y), and asterisk denotes complex conjugate;
  • (6) Using Fn1(x,y)¯, repeat steps 2-5 with corresponding parameter modifications (i.e., function order and recorded intensity map), and obtain Fn2(x,y)¯;
  • (7) Using Fn2(x,y)¯, further repeat steps 2-5 with corresponding parameter modifications, and obtain Fn3(x,y)¯;
  • (8) Calculate the measurement error by using the following formulae:

         [   |Fn3(x,y)¯||Fn(x,y)|]2σ         (if     n=1);     [   |Fn3(x,y)¯||Fn13(x,y)¯|]2σ       (if     n>1).

  • (9) When the condition is met, the complex-valued object can be recovered by using the angular spectrum algorithm with an inverse distance (-z).

To evaluate the accuracy of the recovered object, sum-squared error (SSE) is calculated by
SSE={[|C(ξ,η)||C'(ξ,η)|]   2}/{[|C(ξ,η)|]2},
(7)
where C'(ξ,η) denotes a recovered complex-valued object in the object plane.

3. Results and discussion

We carry out a numerical experiment as shown in Fig. 1 to demonstrate the feasibility and effectiveness of the proposed method. The pixel size of CCD camera is 4.65     μm, and the pixel number is512×512. The light source wavelength is 632.8 nm, and the distance between the object plane and phase-only mask plane is 20 mm. The phase mask is a map randomly distributed in a range of 0 to2π. According to the definition of FRFT [13

13. A. W. Lohmann, “Image rotation, Wigner rotation, and the fractional Fourier transform,” J. Opt. Soc. Am. A 10(10), 2181–2186 (1993). [CrossRef]

16

16. M. G. Ertosun, H. Atlı, H. M. Ozaktas, and B. Barshan, “Complex signal recovery from multiple fractional Fourier-transform intensities,” Appl. Opt. 44(23), 4902–4908 (2005). [CrossRef] [PubMed]

], function orders of 0.25, 0.50 and 0.75 are generated in the three recordings, respectively. Original object amplitude (Barbara) and phase (Phantom) are shown in Figs. 3(a)
Fig. 3 (a) Original object amplitude; (b) original object phase; (c) a typical one of recorded diffraction intensity maps; (d) recovered object amplitude; (e) recovered object phase; and (f) a comparison along the dashed line between Figs. 3(b) and 3(e).
and 3(b), and a typical one of recorded diffraction intensity maps is shown in Fig. 3(c). With the proposed phase retrieval algorithm, object amplitude and phase are recovered as shown in Figs. 3(d) and 3(e). The phase-contrast map is directly extracted by an arc-tangent operation from the recovered complex-valued object. The number of iterations is 63 to ensure that measurement error is not larger than a preset threshold of σ=0.001. As the threshold increases, the number of iterations will reduce correspondingly and the quality of reconstructed images might degrade. To present the quality of the recovered phase map, a comparison along the dashed line [as indicated in Fig. 3(b)] between Figs. 3(b) and 3(e) is shown in Fig. 3(f). It can be seen that a high-quality phase map is recovered, and the only difference from the original phase map is just a constant offset. Figure 4(a)
Fig. 4 Relationships between the iteration number and SSE using (a) 3 and (b) 4 recordings.
shows a relationship between the number of iterations and SSE. It can be seen that the SSE decreases rapidly just after several iterations.

Since the recordings are usually disturbed by noise, the recorded diffraction intensity maps with different levels of noise are also studied. Figures 5(a)
Fig. 5 Relationships between the iteration number and SSE using (a) 3 and (b) 4 recordings with noise.
and 5(b) show the relationships between the number of iterations and SSE using three and four noisy intensity maps, respectively. Random noise with a signal-to-noise ratio of 20 is added to all intensity maps, and the number of iterations is set as 300. The fast convergence rate is also achieved, and the proposed method still can effectively extract the information about the complex-valued object but probably with a relatively low quality of the extracted phase-contrast map and an increase of the number of iterations. In this case, an average algorithm can be applied to a series of recovered phase maps using different initial estimates.

4. Conclusions

In this paper, we have proposed a novel and effective method to quantitatively recover a complex-valued object from the diffraction intensity maps. A series of diffraction intensity maps are recorded by using variable function orders in the FRFT, and a new phase retrieval algorithm is then proposed. It is illustrated that the recovered phase and amplitude images are of high quality. In addition, a fast convergence rate is achieved, and no support constraint is required in the phase mask. The proposed method can also be applied in the extended FRFT domain [17

17. J. Hua, L. Liu, and G. Li, “Extended fractional Fourier transforms,” J. Opt. Soc. Am. A 14(12), 3316–3322 (1997). [CrossRef]

] and find some other applications, such as in the virtual optics.

Acknowledgements

This work was supported by the Singapore Temasek Defence Systems Institute under grant TDSI/09-001/1A, and the authors are grateful to the reviewers for their valuable comments.

References and links

1.

J. M. Zuo, I. Vartanyants, M. Gao, R. Zhang, and L. A. Nagahara, “Atomic resolution imaging of a carbon nanotube from diffraction intensities,” Science 300(5624), 1419–1421 (2003). [CrossRef] [PubMed]

2.

M. A. Pfeifer, G. J. Williams, I. A. Vartanyants, R. Harder, and I. K. Robinson, “Three-dimensional mapping of a deformation field inside a nanocrystal,” Nature 442(7098), 63–66 (2006). [CrossRef] [PubMed]

3.

D. Gabor, “A new microscopic principle,” Nature 161(4098), 777–778 (1948). [CrossRef] [PubMed]

4.

C. J. Mann, L. Yu, C. Lo, and M. K. Kim, “High-resolution quantitative phase-contrast microscopy by digital holography,” Opt. Express 13, 8693–8698 (2005), http://www.opticsinfobase.org/oe/abstract.cfm? URI = oe-13–22–8693.

5.

W. Chen, C. Quan, and C. J. Tay, “Extended depth of focus in a particle field measurement using a single-shot digital hologram,” Appl. Phys. Lett. 95(20), 201103 (2009). [CrossRef]

6.

W. Chen, C. Quan, and C. J. Tay, “Measurement of curvature and twist of a deformed object using digital holography,” Appl. Opt. 47(15), 2874–2881 (2008). [CrossRef] [PubMed]

7.

I. Yamaguchi and T. Zhang, “Phase-shifting digital holography,” Opt. Lett. 22(16), 1268–1270 (1997). [CrossRef] [PubMed]

8.

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttg.) 35, 237–246 (1972).

9.

J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21(15), 2758–2769 (1982). [CrossRef] [PubMed]

10.

J. Miao, P. Charalambous, J. Kirz, and D. Sayre, “Extending the methodology of X-ray crystallography to allow imaging of micrometer-sized non-crystalline specimens,” Nature 400(6742), 342–344 (1999). [CrossRef]

11.

Y. J. Liu, B. Chen, E. R. Li, J. Y. Wang, A. Marcelli, S. W. Wilkins, H. Ming, Y. C. Tian, K. A. Nugent, P. P. Zhu, and Z. Y. Wu, “Phase retrieval in X-ray imaging based on using structured illumination,” Phys. Rev. A 78(2), 023817 (2008). [CrossRef]

12.

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996).

13.

A. W. Lohmann, “Image rotation, Wigner rotation, and the fractional Fourier transform,” J. Opt. Soc. Am. A 10(10), 2181–2186 (1993). [CrossRef]

14.

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

15.

M. G. Ertosun, H. Atlı, H. M. Ozaktas, and B. Barshan, “Complex signal recovery from two fractional Fourier transform intensities: order and noise dependence,” Opt. Commun. 244(1-6), 61–70 (2005). [CrossRef]

16.

M. G. Ertosun, H. Atlı, H. M. Ozaktas, and B. Barshan, “Complex signal recovery from multiple fractional Fourier-transform intensities,” Appl. Opt. 44(23), 4902–4908 (2005). [CrossRef] [PubMed]

17.

J. Hua, L. Liu, and G. Li, “Extended fractional Fourier transforms,” J. Opt. Soc. Am. A 14(12), 3316–3322 (1997). [CrossRef]

OCIS Codes
(100.2000) Image processing : Digital image processing
(100.3190) Image processing : Inverse problems
(100.5070) Image processing : Phase retrieval
(070.2575) Fourier optics and signal processing : Fractional Fourier transforms
(070.7345) Fourier optics and signal processing : Wave propagation

ToC Category:
Image Processing

History
Original Manuscript: May 7, 2010
Revised Manuscript: May 25, 2010
Manuscript Accepted: May 26, 2010
Published: June 8, 2010

Citation
Wen Chen and Xudong Chen, "Quantitative phase retrieval of a complex-valued object using variable function orders in the fractional Fourier domain," Opt. Express 18, 13536-13541 (2010)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-13-13536


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References

  1. J. M. Zuo, I. Vartanyants, M. Gao, R. Zhang, and L. A. Nagahara, “Atomic resolution imaging of a carbon nanotube from diffraction intensities,” Science 300(5624), 1419–1421 (2003). [CrossRef] [PubMed]
  2. M. A. Pfeifer, G. J. Williams, I. A. Vartanyants, R. Harder, and I. K. Robinson, “Three-dimensional mapping of a deformation field inside a nanocrystal,” Nature 442(7098), 63–66 (2006). [CrossRef] [PubMed]
  3. D. Gabor, “A new microscopic principle,” Nature 161(4098), 777–778 (1948). [CrossRef] [PubMed]
  4. C. J. Mann, L. Yu, C. Lo, and M. K. Kim, “High-resolution quantitative phase-contrast microscopy by digital holography,” Opt. Express 13, 8693–8698 (2005), http://www.opticsinfobase.org/oe/abstract.cfm ? URI = oe-13–22–8693.
  5. W. Chen, C. Quan, and C. J. Tay, “Extended depth of focus in a particle field measurement using a single-shot digital hologram,” Appl. Phys. Lett. 95(20), 201103 (2009). [CrossRef]
  6. W. Chen, C. Quan, and C. J. Tay, “Measurement of curvature and twist of a deformed object using digital holography,” Appl. Opt. 47(15), 2874–2881 (2008). [CrossRef] [PubMed]
  7. I. Yamaguchi and T. Zhang, “Phase-shifting digital holography,” Opt. Lett. 22(16), 1268–1270 (1997). [CrossRef] [PubMed]
  8. R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttg.) 35, 237–246 (1972).
  9. J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21(15), 2758–2769 (1982). [CrossRef] [PubMed]
  10. J. Miao, P. Charalambous, J. Kirz, and D. Sayre, “Extending the methodology of X-ray crystallography to allow imaging of micrometer-sized non-crystalline specimens,” Nature 400(6742), 342–344 (1999). [CrossRef]
  11. Y. J. Liu, B. Chen, E. R. Li, J. Y. Wang, A. Marcelli, S. W. Wilkins, H. Ming, Y. C. Tian, K. A. Nugent, P. P. Zhu, and Z. Y. Wu, “Phase retrieval in X-ray imaging based on using structured illumination,” Phys. Rev. A 78(2), 023817 (2008). [CrossRef]
  12. J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996).
  13. A. W. Lohmann, “Image rotation, Wigner rotation, and the fractional Fourier transform,” J. Opt. Soc. Am. A 10(10), 2181–2186 (1993). [CrossRef]
  14. H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, 2001).
  15. M. G. Ertosun, H. Atlı, H. M. Ozaktas, and B. Barshan, “Complex signal recovery from two fractional Fourier transform intensities: order and noise dependence,” Opt. Commun. 244(1-6), 61–70 (2005). [CrossRef]
  16. M. G. Ertosun, H. Atlı, H. M. Ozaktas, and B. Barshan, “Complex signal recovery from multiple fractional Fourier-transform intensities,” Appl. Opt. 44(23), 4902–4908 (2005). [CrossRef] [PubMed]
  17. J. Hua, L. Liu, and G. Li, “Extended fractional Fourier transforms,” J. Opt. Soc. Am. A 14(12), 3316–3322 (1997). [CrossRef]

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