Abstract

We propose a novel phase retrieval algorithm in Hankel (or called Fourier-Bessel) transform domains by using Monte-Carlo method. Based on the proposed algorithm, we investigate the generation of Gaussian-like beams, such as hollow Gaussian beam, Bessel-Gaussian beam and Laguerre-Gaussian beam, with double phase filtering operations. The phase distributions of filters are determined by employing the proposed phase retrieval algorithm. The advantage of the method is that the total energy of the beam is conservative. Numerical simulations are shown to demonstrate the validity of the scheme.

© 2008 Optical Society of America

1. Introduction

In recent years, the application and the generation of dark hollow beam (DHB), the intensity of which shows the formation of ring or multiple rings, have been became a focused point in the field of optical physics [1, 2, 3, 4, 5, 6], for the reasons that the DHBs can be used in the fields of optical atom trapping, cold atom guidance and particle controlling. Hollow Gaussian beam (HGB)[4] is one of these hollow beams. In our previous work, we have reported a scheme to generate HGBs with spatial filtering on amplitude [5]. Mathematically this method can generate the exact HGBs, however, the operation of amplitude filtering reduces the total energy of output beam. This drawback will eventually limit the practical application of this scheme as optical tweezers. Therefore, it is obviously significant and necessary to investigate the schemes to generate DHBs without energy loss.

Phase-only filtering will be introduced into the design of new optical scheme because this kind of operator can conserve the total energy of beam in propagation process under paraxial approximation. In the procedure of beam transformation, the information of input and output is known, but detail internal relations are unknown, which is an optical inverse problem. As we known, Gerchberg-Saxton (G-S) phase retrieval algorithm had been applied in beam shaping [7, 8, 9], X-ray system [10, 11, 12], the coherent electron imaging of a carbon nanotube [13] and information processing [15, 16, 17, 18, 19, 20, 23, 22, 23, 24]. Yang-Gu algorithm [25] is an improved version of G-S algorithm and was used to the phase retrieval algorithm in fractional Fourier transform domain [26]. However, to our best knowledge, there have been only few investigations on phase retrieval algorithm in Hankel transform (HT) domain. Moreover, the phase retrieval algorithm among multiple planes is rarely reported and required very high precision between two adjacent planes. In our specific issue of generating HGBs, single phase-only filtering operation cannot complete the task, therefore phase retrieval between multiple planes must be involved. In this article we propose a novel phase retrieval algorithm between three planes. Based on this proposed algorithm, we propose a scheme to generate Gaussian-like beams from a fundamental Gaussian beam with double phase-only filtering both in the input plane and the HT domain. The phase filtering can be implemented by phase-type spatial light modulators (SLMs) and their phase distributions can be calculated with the phase retrieval algorithm. A significant advantage of this method is that the energy is held between input and output planes in this optical system. Moreover, an arbitrary beam with central symmetric amplitude and phase distributions can be generated with this system.

The rest of the paper is organized in the following sequence. In Section 2, the proposed phase retrieval algorithm is introduced. In Section 3, some numerical results are given to demonstrate the validity of the algorithm. Concluding remarks are summarized in the final section.

2. Phase retrieval algorithm

Figure 1 illustrates an optical set-up to generate HGBs, in which a double phase filtering process is performed in a classical 4 f optical system. Two phase-only filters, exp[iΔϕ 1(x,y)]

and exp[iΔϕ 2(u, ν)], which can be displayed by spatial light modulators SLM1 and SLM2, are located at the input plane P1 and the Fourier spectrum plane P2, respectively. The incidence Gaussian beam with fundamental mode on the input plane, E(x, y), can be generally expressed as

E(x,y)=G(z)exp[x2+y2ω2(z)]exp[ikx2+y22R(z)],

where G(z), ω(z) and R(z) are the functions depending on the variable z. k=2π/λ is wave vector. For simplicity, we only consider an incidence Gaussian beam at its waist with minimum spot size ω 0 and a constant phase ϕ1. Therefore the amplitude of the incidence beam A 1(x, y) can be expressed as

A1(x,y)=G10exp(x2+y2ω02),

where ω 0 is the waist of the Gaussian beam and G 10 is a constant. In such cease the desired output beam E3(x′, y′) will also be at its waist, with the amplitude A3(x′, y′) and and a constant phase profile ϕ 3(x′, y′) at the plane P3, which can be expressed as [4, 5]

A3(x,y)=G30(x2+y2ω02)pexp(x2+y2ω02),
ϕ3(x,y)=C0,

where p is the order of the HGB.

 

Fig. 1. Schematic Illustration of the optical setup for the generation of hollow Gaussian beams based on phase retrieval algorithm.

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As we have known, the amplitude and phase distributions of Gaussian beam, Hollow-Gaussian beam, Laguerre-Gaussian beam, Bessel-Gaussian beam and etc, are central symmetric about z direction on the transverse planes. Therefore, in the following analysis we express the amplitude and phase distributions of beam under the polar coordinate system. All the functions of amplitude and phase at the three planes are only radial coordinates dependent, denoted by r, ρ, r′, respectively. From this symmetry, we can express the propagation of light wave in the 4 f optical system shown as Fig. 1 by use of the Hankel transform (HT) as follows

A2(ρ)exp[iϕ2(ρ)]=H{A1(r)exp[iϕ1(r)]},

and

A3(r)exp[iϕ3(r)]=H{A2(ρ)exp[iφ2(ρ)]},

where ϕ 1(r)=ϕ1ϕ 1(r) and ϕ2(ρ)=ϕ 2(ρ)+Δϕ2(ρ), and ϕ2(ρ) is the phase distribution at the plane P2 just after the phase modulation of SLM2. denotes HT, which of a function f(r) is expressed as

H[f(ρ)]=2π0f(r)J0(2πρr)dr,

where J 0 is the zeroth order Bessel function of the first kind. From the known A3(r′) and ϕ 3(r′) in Eq. (3), we can compute A 2(ρ) and ϕ2(ρ) by using HT as following

A2(ρ)=H{A3(r)exp[iϕ3(r)]},
φ2(ρ)=arg{H{A3(r)exp[iϕ3(r)]}A2(ρ)},

where the function ‘arg’ denotes the operation of taking phase angle of a complex number. In Eq. (7), HT is used for the inverse HT because the inverse transform of HT is itself yet.

In general case, the following equation

A2(ρ)exp[iϕ2(ρ)]=H{A1(r)exp[iϕ1(r)]},

is not satisfied directly, the phase profiles ϕ 2(ρ) and ϕ 1(r) are in the black box with known external conditions in the optical system. It is a typical optical inverse problem. Getting analytic solution of Eq. (6) is very difficult from known amplitudes A 1(r) and A2(ρ) only. In fact, however, we can obtain the numerical solution by employing a phase retrieval algorithm. As we know, Yang-Gu phase retrieval algorithm [25] is best one in all iterative phase retrieval algorithms and it can be used in non-unitary transform domain. However, we have tested G-S algorithm and Yang-Gu algorithm for solving Eq. (6) and found out that they cannot deal with this problem because both them do not achieve convergence behavior in this case. Moreover, the error of the amplitude A 2(ρ) will affect the precision of A3(r′) in the transform of next step. Thereby, a convergence phase retrieval with high precision is required.

Here we introduce the idea of Monte-Carlo model to design an iterative phase retrieval algorithm, which is shown in Fig. 2. In the initialization of the process, ϕ 1, 0(r) is chosen randomly from the interval [-π, π]. The amplitudes A 1(r) and A 2(ρ) are known and computed possibly, respectively. Other parameters will be pre-defined. The particular iterative process will be continued according to following several steps: (1) Multiple values of the phase ϕ 1(r) and ϕ 1, k(k=1, 2, …, M), which can introduce parallel computation mechanism into this algorithm, will be generated by adding a random disturbing value ϕ r, k(r) located in the interval [-0.5, 0.5]c 0εt. The corresponding complex amplitude A 1, k(r)exp[iϕ′1, k(r)] can be obtained. (2) The HT of A1, k(r)exp[iϕ′1, k(r)] is calculated and get complex amplitudes A2, k(ρ)exp[i ϕ′2, k(ρ)]. To express the error between A2, k(ρ) and A 2(ρ), a function of mean square error (MSE) is defined as

MSE=mse(A2,k,A2)=ΣA2,kA22L,

where L denotes the total number of sample point at ρ-direction. (3) Here a minimum value of these MSE, mse(A′2,k,A2)min will be found out to be regarded as an optimized result and record corresponding value of ϕ′1, k(r), which will substitute ϕ 1, 0 when returning to the step (1). If the minimum value satisfies the condition mse(A′2, k, A2)min>εt, the procedure will return to the step (1). The condition n-n 0≤Δn is used for checking and controlling the magnitude of random vectors ϕr, k(r). When n-n0>Δn, the convergence closed to correct value of ϕ 1(r) is considered as stop and εt is decreased to continue the convergence. If mse(A′2, k, A2)min<εt and εt>ε0, the parameters εt and ϕ 1, 0(r) are renewed with better values mse(A′2, k, A2)min and ϕ1, k(r), respectively, and label parameters n 0 and n are also updated. If εt<ε0, we close this recurrence process and consider that pre-defined precision has been achieved. According to the characteristic of this algorithm, it can be refereed as an optimized Monte-Carlo phase retrieval algorithm. The conventional phase retrieval is to explore correct phase information. In this paper, however, the purpose of phase retrieval is to obtain more approximative amplitude A 2(ρ).

After the acquirement of the corresponding recovered phases ϕr1(r) and ϕr2(ρ), the modulation phase profiles on SLM1 and SLM2 can be directly calculated as

Δϕ1=ϕ1r(r)φ1,Δϕ2=φ2(ρ)ϕ2r(ρ).

Optically these phase distributions can be displayed on the phase-type SLMs. Alternatively, according to Eq. 8 we can fabricate two phase masks like Fresnel zone plate by use of femtosecond laser technology to replace two SLMs.

 

Fig. 2. Flow chart of the proposed phase retrieval algorithm in Hankel transform domains.

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Fig. 3. The numerical result of the proposed phase retrieval algorithm. Here the output beam is the second-order hollow Gaussian beam.

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Fig. 4. The intensity distribution of the hollow Gaussian beam: (a) the theoretical result directly from the analytical expression, (b) the result generated by the phase retrieval algorithm and double phase filtering.

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3. Numerical simulation

Figure 3 gives an example of numerical simulation with this algorithm to obtain the secondorder hollow Gaussian beam. The values of all parameters are given as following context: c 1=0.99, c 0=0.2, Δn=4, ε 0=1×10-4, the initial value of εt is equal to 1, M=100, the sample point number and the interval of r (ρ and r′) are equal to 256 and [0, 4]×ω 0, respectively. A group of random vectors is adopted as the initial value of phase ϕ 1(r). Under the condition limited with above mentioned parameters, the total iterative number is 68673 times (This number is also marked with underline and bold in Table 1). The analytical values of amplitudes A 2(ρ) and A3(r′) are depicted for the comparison with their recovered values and are marked by dashed and solid lines, respectively. The recovered phases ϕ1 (r) and ϕ2 (ρ) are given in the second row sub-figures in Fig. 3. The precision of the recovered phases ϕ1 (r) and ϕ2(ρ) and hence the corresponding amplitudes A 2(ρ) and A 3(r′) can be increased by choosing a smaller value of ε 0. Figure 4 gives the comparison of the transverse profiles of the HGBs calculated with analytical expression and generated by the phase retrieval algorithm. The results demonstrate well the effectiveness of the proposed algorithm.

It is worth to point out that choosing a proper coefficient c0 εt of ϕr, k(r) can increase the convergence speed of this phase retrieval algorithm. In our algorithm the value of ε t is given by the minimum value of the MSE and it will decrease with the convergence accordingly. Such a variable coefficient εt can enable a high precision phase retrieval results with a reasonable convergence speed. The coefficients c 0 and c 1 can also affect the convergence speed. We have also numerically calculated the influence of the coefficients on the iterative steps. The results are shown in the Table 1.

Tables Icon

Table 1. The total iterative number under different parameters c0 and c1 and other parameters is the same in Fig.3.

 

Fig. 5. Convergence speed curves of G-S, Yang-Gu and the proposed algorithms.

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With the same initial values used in Fig. 3, three simulations with G-S, Yang-Gu and the proposed algorithms are tested. Figure 5 shows a comparison of convergence speed between G-S, Yang-Gu and the proposed algorithms, in which the values of MSE are calculated from the analytical and recovered values of amplitude A 2(ρ). From Fig. 5, one can see that the G-S and Yang-Gu algorithm can not converge to an approximative value of A 2(ρ), while the proposed algorithm can converge quickly to a desired value. That demonstrates the proposed algorithm is only effective in above mentioned three methods in the issue of generating HGBs. We have also demonstrated that the proposed algorithm is effective to generate other Gaussian-like beams, such as hollow Gaussian beam, Bessel-Gaussian beam and Laguerre-Gaussian beam, by changing the distributions of A 3(r′) and ϕ 3(r′).

4. Conclusion

We have proposed a novel phase retrieval algorithm based on Monte-Carlo structure in Hankel transform domains. This algorithm can be also expanded into other transform domains to deal with general phase retrieval problem from two amplitude (or intensity) images and it is more effective than the G-S and Yang-Gu algorithm in the behaviors of the convergence. By use of this phase retrieval algorithm, we have designed an optical system with phase-only filtering to generate hollow Gaussian beam or other central symmetric beams. Numerical simulations have demonstrated its validity.

Acknowledgments

This work was supported by the National Natural Science Foundation of China under Grant No. 10674038 and 10604042, National Basic Research Program of China under Grant 2006CB302901, China Postdoctoral Science Foundation (20080430913), and development program for outstanding young teachers in Harbin Institute of Technology, HITQNJS. 2008. 027. Zhengjun Liu wishes to acknowledge Prof. Amilia Torre, who is working in the ENEA Center, Roman, Italy, for her help in obtaining the references on Hankel transform and some useful discussions.

References and links

1. T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, “Novel optical trap of atoms with a doughnut beam,” Phys. Rev. Lett. 78, 4713–4716 (1997). [CrossRef]  

2. H. Ito, T. Nakata, K. Sakaki, M. Ohtsu, K. I. Lee, and W. Jhe, “Laser spectroscopy of atoms guided by evanescent waves in micron-sized hollow optical fibers,” Phys. Rev. Lett. 76, 4500 (1996). [CrossRef]   [PubMed]  

3. Y. K. Wu, J. Li, and J. Wu, “Anomalous hollow electron beams in a storage ring,” Phys. Rev. Lett. 94, 134802 (2005). [CrossRef]   [PubMed]  

4. Y. Cai, X. Lu, and Q. Lin, “Hollow Gaussian beams and their propagation properties,” Opt. Lett. 28, 1084 (2003). [CrossRef]   [PubMed]  

5. Z. Liu, H. Zhao, J. Liu, J. Lin, M. A. Ahmad, and S. Liu, “Generation of hollow Gaussian beams by spatial filtering,” Opt. Lett. 32, 2076 (2007). [CrossRef]   [PubMed]  

6. Y. Zhao, J. S. Edgar, G. D. M. Jeffries, D. McGloin, and D. T. Chiu, “Spin-to-orbital angular momentum conversion in a strongly focused optical beam,” Phys. Rev. Lett. 99, 073901 (2007). [CrossRef]   [PubMed]  

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

8. Z. Zalevsky, D. Mendlovic, and R. G. Dorsch, “Gerchberg-Saxton algorithm applied in the fractional Fourieror the Fresnel domain,” Opt. Lett. 21, 842–844 (1996). [CrossRef]   [PubMed]  

9. D. F. McAlister, M. Beck, L. Clarke, A. Mayer, and M. G. Raymer, “Optical phase retrieval by phase-space tomography and fractional-order Fourier transforms,” Opt. Lett. 20, 1181–1183 (1995). [CrossRef]   [PubMed]  

10. K. A. Nugent, T. E. Gureyev, D. F. Cookson, D. Paganin, and Z. Barena, “Quantitative Phase Imaging Using Hard X Rays,” Phys. Rev. Lett. 77, 2961–2964 (1996). [CrossRef]   [PubMed]  

11. J. Miao, T. Ishikawa, B. Johnson, E. H. Anderson, B. Lai, and K. O. Hodgson, “High resolution 3D X-ray diffraction microscopy,” Phys. Rev. Lett. 89, 088303 (2002). [CrossRef]   [PubMed]  

12. F. Meng, H. Liu, and X. Wu, “An iterative phase retrieval algorithm for in-line x-ray phase imaging,” Opt. Express 15, 8383–8390 (2007), http://www.opticsinfobase.org/abstract.cfm?URI=oe-15-13-8383. [CrossRef]   [PubMed]  

13. 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, 1419 (2003). [CrossRef]   [PubMed]  

14. R. Borghi, F. Gori, and M. Santarsiero, “Phase and amplitude retrieval in ghost diffraction from field-correlation measurements,” Phys. Rev. Lett. 96, 183901 (2006). [CrossRef]   [PubMed]  

15. A. Anand, G. Pedrini, W. Osten, and P. Almoro, “Wavefront sensing with random amplitude mask and phase retrieval,” Opt. Lett. 32, 1584–1586 (2007). [CrossRef]   [PubMed]  

16. P. Almoro, G. Pedrini, and W. Osten, “Wavefront sensing with random amplitude mask and phase retrieval,” Opt. Lett. 32, 733–735 (2007). [CrossRef]   [PubMed]  

17. A. K. Ellerbee and J. A. Izatt, “Phase retrieval in low-coherence interferometric microscopy,” Opt. Lett. 32, 388–390 (2007). [CrossRef]   [PubMed]  

18. J. Zhong and J. Weng, “Phase retrieval of optical fringe patterns from the ridge of a wavelet transform,” Opt. Lett. 30, 2560–2562 (2005). [CrossRef]   [PubMed]  

19. J. S. Wu, U. Weierstall, J. C. H. Spence, and C. T. Koch, “Iterative phase retrieval without support,” Opt. Lett. 29, 2737 (2004). [CrossRef]   [PubMed]  

20. N. Nakajima, “Phase and amplitude retrieval in ghost diffraction from field-correlation measurements,” Phys. Rev. Lett. 98, 223901 (2007). [CrossRef]   [PubMed]  

21. F. Zhang, G. Pedrini, and W. Osten, “Phase retrieval of arbitrary complex-valued fields through aperture-plane modulation,” Phys. Rev. A 75, 043805 (2007). [CrossRef]  

22. T. Latychevskaia and H.-W. Fink, “Solution to the twin image problem in holography,” Phys. Rev. Lett. 98, 233901 (2007). [CrossRef]   [PubMed]  

23. Y. Zhang, G. Pedrini, W. Osten, and H. Tiziani, “Whole optical wave field reconstruction from double or multi in-line holograms by phase retrieval algorithm,” Opt. Express 11, 3234–3241 (2003), http://www.opticsinfobase.org/abstract.cfm?URI=oe-11-24-3234. [CrossRef]   [PubMed]  

24. Z. Liu and S. Liu, “Double image encryption based on iterative fractional Fourier transform,” Opt. Commun. 272, 324–329 (2007). [CrossRef]  

25. G.-Z. Yang, B.-Z. Dong, B.-Y. Gu, J.-Y. Zhuang, and O. K. Ersoy, “Gerchberg-Saxton and Yang-Gu algorithms for phase retrieval in a nonunitary transform system: a comparison,” Appl. Opt. 33, 209 (1994). [CrossRef]   [PubMed]  

26. B. Dong, Y. Zhang, B. Gu, and G. Yang, “Numerical investigation of phase retrieval in a fractional Fourier transform,” J. Opt. Soc. Am. A 14, 2709–2714 (1997). [CrossRef]  

References

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  1. T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, "Novel optical trap of atoms with a doughnut beam," Phys. Rev. Lett. 78, 4713-4716 (1997).
    [CrossRef]
  2. H. Ito, T. Nakata, K. Sakaki, M. Ohtsu, K. I. Lee, and W. Jhe, "Laser spectroscopy of atoms guided by evanescent waves in micron-sized hollow optical fibers," Phys. Rev. Lett. 76, 4500 (1996).
    [CrossRef] [PubMed]
  3. Y. K. Wu, J. Li, and J. Wu, "Anomalous hollow electron beams in a storage ring," Phys. Rev. Lett. 94, 134802 (2005).
    [CrossRef] [PubMed]
  4. Y. Cai, X. Lu, and Q. Lin, "Hollow Gaussian beams and their propagation properties," Opt. Lett. 28, 1084 (2003).
    [CrossRef] [PubMed]
  5. Z. Liu, H. Zhao, J. Liu, J. Lin, M. A. Ahmad, and S. Liu, "Generation of hollow Gaussian beams by spatial filtering," Opt. Lett. 32, 2076 (2007).
    [CrossRef] [PubMed]
  6. Y. Zhao, J. S. Edgar, G. D. M. Jeffries, D. McGloin, and D. T. Chiu, "Spin-to-orbital angular momentum conversion in a strongly focused optical beam," Phys. Rev. Lett. 99, 073901 (2007).
    [CrossRef] [PubMed]
  7. R. W. Gerchberg and W. O. Saxton, "A practical algorithm for the determination of phase from image and diffraction plane pictures," Optik 35, 237-246 (1972).
  8. Z. Zalevsky, D. Mendlovic, and R. G. Dorsch, "Gerchberg-Saxton algorithm applied in the fractional Fourieror the Fresnel domain," Opt. Lett. 21, 842-844 (1996).
    [CrossRef] [PubMed]
  9. D. F. McAlister, M. Beck, L. Clarke, A. Mayer, and M. G. Raymer, "Optical phase retrieval by phase-space tomography and fractional-order Fourier transforms," Opt. Lett. 20, 1181-1183 (1995).
    [CrossRef] [PubMed]
  10. K. A. Nugent, T. E. Gureyev, D. F. Cookson, D. Paganin, and Z. Barena, "Quantitative Phase Imaging using Hard X Rays," Phys. Rev. Lett. 77, 2961-2964 (1996).
    [CrossRef] [PubMed]
  11. J. Miao T. Ishikawa, B. Johnson, E. H. Anderson, B. Lai, and K. O. Hodgson, "High resolution 3D X-ray diffraction microscopy," Phys. Rev. Lett. 89, 088303 (2002).
    [CrossRef] [PubMed]
  12. F. Meng, H. Liu, and X. Wu, "An iterative phase retrieval algorithm for in-line x-ray phase imaging," Opt. Express 15, 8383-8390 (2007), http://www.opticsinfobase.org/abstract.cfm?URI=oe-15-13-8383.
    [CrossRef] [PubMed]
  13. 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, 1419 (2003).
    [CrossRef] [PubMed]
  14. R. Borghi, F. Gori, and M. Santarsiero, "Phase and amplitude retrieval in ghost diffraction from field-correlation measurements," Phys. Rev. Lett. 96, 183901 (2006).
    [CrossRef] [PubMed]
  15. A. Anand, G. Pedrini, W. Osten, and P. Almoro, "Wavefront sensing with random amplitude mask and phase retrieval," Opt. Lett. 32, 1584-1586 (2007).
    [CrossRef] [PubMed]
  16. P. Almoro, G. Pedrini, and W. Osten, "Wavefront sensing with random amplitude mask and phase retrieval," Opt. Lett. 32, 733-735 (2007).
    [CrossRef] [PubMed]
  17. A. K. Ellerbee and J. A. Izatt, "Phase retrieval in low-coherence interferometric microscopy," Opt. Lett. 32, 388-390 (2007).
    [CrossRef] [PubMed]
  18. J. Zhong and J. Weng, "Phase retrieval of optical fringe patterns from the ridge of a wavelet transform," Opt. Lett. 30, 2560-2562 (2005).
    [CrossRef] [PubMed]
  19. J. S. Wu, U. Weierstall, J. C. H. Spence, and C. T. Koch, "Iterative phase retrieval without support," Opt. Lett. 29, 2737 (2004).
    [CrossRef] [PubMed]
  20. N. Nakajima, "Phase and amplitude retrieval in ghost diffraction from field-correlation measurements," Phys. Rev. Lett. 98, 223901 (2007).
    [CrossRef] [PubMed]
  21. F. Zhang, G. Pedrini, and W. Osten, "Phase retrieval of arbitrary complex-valued fields through aperture-plane modulation," Phys. Rev. A 75, 043805 (2007).
    [CrossRef]
  22. T. Latychevskaia and H.-W. Fink, "Solution to the twin image problem in holography," Phys. Rev. Lett. 98, 233901 (2007).
    [CrossRef] [PubMed]
  23. Y. Zhang, G. Pedrini, W. Osten, and H. Tiziani, "Whole optical wave field reconstruction from double or multi in-line holograms by phase retrieval algorithm," Opt. Express 11, 3234-3241 (2003), http://www.opticsinfobase.org/abstract.cfm?URI=oe-11-24-3234.
    [CrossRef] [PubMed]
  24. Z. Liu and S. Liu, "Double image encryption based on iterative fractional Fourier transform," Opt. Commun. 272, 324-329 (2007).
    [CrossRef]
  25. G.-Z. Yang, B.-Z. Dong, B.-Y. Gu, J.-Y. Zhuang, and O. K. Ersoy, "Gerchberg-Saxton and Yang-Gu algorithms for phase retrieval in a nonunitary transform system: a comparison," Appl. Opt. 33, 209 (1994).
    [CrossRef] [PubMed]
  26. B. Dong, Y. Zhang, B. Gu, and G. Yang, "Numerical investigation of phase retrieval in a fractional Fourier transform," J. Opt. Soc. Am. A 14, 2709-2714 (1997).
    [CrossRef]

2007 (10)

Z. Liu, H. Zhao, J. Liu, J. Lin, M. A. Ahmad, and S. Liu, "Generation of hollow Gaussian beams by spatial filtering," Opt. Lett. 32, 2076 (2007).
[CrossRef] [PubMed]

Y. Zhao, J. S. Edgar, G. D. M. Jeffries, D. McGloin, and D. T. Chiu, "Spin-to-orbital angular momentum conversion in a strongly focused optical beam," Phys. Rev. Lett. 99, 073901 (2007).
[CrossRef] [PubMed]

F. Meng, H. Liu, and X. Wu, "An iterative phase retrieval algorithm for in-line x-ray phase imaging," Opt. Express 15, 8383-8390 (2007), http://www.opticsinfobase.org/abstract.cfm?URI=oe-15-13-8383.
[CrossRef] [PubMed]

A. Anand, G. Pedrini, W. Osten, and P. Almoro, "Wavefront sensing with random amplitude mask and phase retrieval," Opt. Lett. 32, 1584-1586 (2007).
[CrossRef] [PubMed]

P. Almoro, G. Pedrini, and W. Osten, "Wavefront sensing with random amplitude mask and phase retrieval," Opt. Lett. 32, 733-735 (2007).
[CrossRef] [PubMed]

A. K. Ellerbee and J. A. Izatt, "Phase retrieval in low-coherence interferometric microscopy," Opt. Lett. 32, 388-390 (2007).
[CrossRef] [PubMed]

N. Nakajima, "Phase and amplitude retrieval in ghost diffraction from field-correlation measurements," Phys. Rev. Lett. 98, 223901 (2007).
[CrossRef] [PubMed]

F. Zhang, G. Pedrini, and W. Osten, "Phase retrieval of arbitrary complex-valued fields through aperture-plane modulation," Phys. Rev. A 75, 043805 (2007).
[CrossRef]

T. Latychevskaia and H.-W. Fink, "Solution to the twin image problem in holography," Phys. Rev. Lett. 98, 233901 (2007).
[CrossRef] [PubMed]

Z. Liu and S. Liu, "Double image encryption based on iterative fractional Fourier transform," Opt. Commun. 272, 324-329 (2007).
[CrossRef]

2006 (1)

R. Borghi, F. Gori, and M. Santarsiero, "Phase and amplitude retrieval in ghost diffraction from field-correlation measurements," Phys. Rev. Lett. 96, 183901 (2006).
[CrossRef] [PubMed]

2005 (2)

J. Zhong and J. Weng, "Phase retrieval of optical fringe patterns from the ridge of a wavelet transform," Opt. Lett. 30, 2560-2562 (2005).
[CrossRef] [PubMed]

Y. K. Wu, J. Li, and J. Wu, "Anomalous hollow electron beams in a storage ring," Phys. Rev. Lett. 94, 134802 (2005).
[CrossRef] [PubMed]

2004 (1)

2003 (3)

2002 (1)

J. Miao T. Ishikawa, B. Johnson, E. H. Anderson, B. Lai, and K. O. Hodgson, "High resolution 3D X-ray diffraction microscopy," Phys. Rev. Lett. 89, 088303 (2002).
[CrossRef] [PubMed]

1997 (2)

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, "Novel optical trap of atoms with a doughnut beam," Phys. Rev. Lett. 78, 4713-4716 (1997).
[CrossRef]

B. Dong, Y. Zhang, B. Gu, and G. Yang, "Numerical investigation of phase retrieval in a fractional Fourier transform," J. Opt. Soc. Am. A 14, 2709-2714 (1997).
[CrossRef]

1996 (3)

H. Ito, T. Nakata, K. Sakaki, M. Ohtsu, K. I. Lee, and W. Jhe, "Laser spectroscopy of atoms guided by evanescent waves in micron-sized hollow optical fibers," Phys. Rev. Lett. 76, 4500 (1996).
[CrossRef] [PubMed]

Z. Zalevsky, D. Mendlovic, and R. G. Dorsch, "Gerchberg-Saxton algorithm applied in the fractional Fourieror the Fresnel domain," Opt. Lett. 21, 842-844 (1996).
[CrossRef] [PubMed]

K. A. Nugent, T. E. Gureyev, D. F. Cookson, D. Paganin, and Z. Barena, "Quantitative Phase Imaging using Hard X Rays," Phys. Rev. Lett. 77, 2961-2964 (1996).
[CrossRef] [PubMed]

1995 (1)

1994 (1)

1972 (1)

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

Ahmad, M. A.

Almoro, P.

Anand, A.

Barena, Z.

K. A. Nugent, T. E. Gureyev, D. F. Cookson, D. Paganin, and Z. Barena, "Quantitative Phase Imaging using Hard X Rays," Phys. Rev. Lett. 77, 2961-2964 (1996).
[CrossRef] [PubMed]

Beck, M.

Borghi, R.

R. Borghi, F. Gori, and M. Santarsiero, "Phase and amplitude retrieval in ghost diffraction from field-correlation measurements," Phys. Rev. Lett. 96, 183901 (2006).
[CrossRef] [PubMed]

Cai, Y.

Chiu, D. T.

Y. Zhao, J. S. Edgar, G. D. M. Jeffries, D. McGloin, and D. T. Chiu, "Spin-to-orbital angular momentum conversion in a strongly focused optical beam," Phys. Rev. Lett. 99, 073901 (2007).
[CrossRef] [PubMed]

Clarke, L.

Cookson, D. F.

K. A. Nugent, T. E. Gureyev, D. F. Cookson, D. Paganin, and Z. Barena, "Quantitative Phase Imaging using Hard X Rays," Phys. Rev. Lett. 77, 2961-2964 (1996).
[CrossRef] [PubMed]

Dong, B.

Dong, B.-Z.

Dorsch, R. G.

Edgar, J. S.

Y. Zhao, J. S. Edgar, G. D. M. Jeffries, D. McGloin, and D. T. Chiu, "Spin-to-orbital angular momentum conversion in a strongly focused optical beam," Phys. Rev. Lett. 99, 073901 (2007).
[CrossRef] [PubMed]

Ellerbee, A. K.

Ersoy, O. K.

Fink, H.-W.

T. Latychevskaia and H.-W. Fink, "Solution to the twin image problem in holography," Phys. Rev. Lett. 98, 233901 (2007).
[CrossRef] [PubMed]

Gerchberg, R. W.

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

Gori, F.

R. Borghi, F. Gori, and M. Santarsiero, "Phase and amplitude retrieval in ghost diffraction from field-correlation measurements," Phys. Rev. Lett. 96, 183901 (2006).
[CrossRef] [PubMed]

Gu, B.

Gu, B.-Y.

Gureyev, T. E.

K. A. Nugent, T. E. Gureyev, D. F. Cookson, D. Paganin, and Z. Barena, "Quantitative Phase Imaging using Hard X Rays," Phys. Rev. Lett. 77, 2961-2964 (1996).
[CrossRef] [PubMed]

Hirano, T.

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, "Novel optical trap of atoms with a doughnut beam," Phys. Rev. Lett. 78, 4713-4716 (1997).
[CrossRef]

Ito, H.

H. Ito, T. Nakata, K. Sakaki, M. Ohtsu, K. I. Lee, and W. Jhe, "Laser spectroscopy of atoms guided by evanescent waves in micron-sized hollow optical fibers," Phys. Rev. Lett. 76, 4500 (1996).
[CrossRef] [PubMed]

Izatt, J. A.

Jeffries, G. D. M.

Y. Zhao, J. S. Edgar, G. D. M. Jeffries, D. McGloin, and D. T. Chiu, "Spin-to-orbital angular momentum conversion in a strongly focused optical beam," Phys. Rev. Lett. 99, 073901 (2007).
[CrossRef] [PubMed]

Koch, C. T.

Kuga, T.

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, "Novel optical trap of atoms with a doughnut beam," Phys. Rev. Lett. 78, 4713-4716 (1997).
[CrossRef]

Latychevskaia, T.

T. Latychevskaia and H.-W. Fink, "Solution to the twin image problem in holography," Phys. Rev. Lett. 98, 233901 (2007).
[CrossRef] [PubMed]

Lee, K. I.

H. Ito, T. Nakata, K. Sakaki, M. Ohtsu, K. I. Lee, and W. Jhe, "Laser spectroscopy of atoms guided by evanescent waves in micron-sized hollow optical fibers," Phys. Rev. Lett. 76, 4500 (1996).
[CrossRef] [PubMed]

Li, J.

Y. K. Wu, J. Li, and J. Wu, "Anomalous hollow electron beams in a storage ring," Phys. Rev. Lett. 94, 134802 (2005).
[CrossRef] [PubMed]

Lin, J.

Lin, Q.

Liu, H.

Liu, J.

Liu, S.

Z. Liu, H. Zhao, J. Liu, J. Lin, M. A. Ahmad, and S. Liu, "Generation of hollow Gaussian beams by spatial filtering," Opt. Lett. 32, 2076 (2007).
[CrossRef] [PubMed]

Z. Liu and S. Liu, "Double image encryption based on iterative fractional Fourier transform," Opt. Commun. 272, 324-329 (2007).
[CrossRef]

Liu, Z.

Z. Liu and S. Liu, "Double image encryption based on iterative fractional Fourier transform," Opt. Commun. 272, 324-329 (2007).
[CrossRef]

Z. Liu, H. Zhao, J. Liu, J. Lin, M. A. Ahmad, and S. Liu, "Generation of hollow Gaussian beams by spatial filtering," Opt. Lett. 32, 2076 (2007).
[CrossRef] [PubMed]

Lu, X.

Mayer, A.

McAlister, D. F.

McGloin, D.

Y. Zhao, J. S. Edgar, G. D. M. Jeffries, D. McGloin, and D. T. Chiu, "Spin-to-orbital angular momentum conversion in a strongly focused optical beam," Phys. Rev. Lett. 99, 073901 (2007).
[CrossRef] [PubMed]

Mendlovic, D.

Meng, F.

Nakajima, N.

N. Nakajima, "Phase and amplitude retrieval in ghost diffraction from field-correlation measurements," Phys. Rev. Lett. 98, 223901 (2007).
[CrossRef] [PubMed]

Nakata, T.

H. Ito, T. Nakata, K. Sakaki, M. Ohtsu, K. I. Lee, and W. Jhe, "Laser spectroscopy of atoms guided by evanescent waves in micron-sized hollow optical fibers," Phys. Rev. Lett. 76, 4500 (1996).
[CrossRef] [PubMed]

Nugent, K. A.

K. A. Nugent, T. E. Gureyev, D. F. Cookson, D. Paganin, and Z. Barena, "Quantitative Phase Imaging using Hard X Rays," Phys. Rev. Lett. 77, 2961-2964 (1996).
[CrossRef] [PubMed]

Ohtsu, M.

H. Ito, T. Nakata, K. Sakaki, M. Ohtsu, K. I. Lee, and W. Jhe, "Laser spectroscopy of atoms guided by evanescent waves in micron-sized hollow optical fibers," Phys. Rev. Lett. 76, 4500 (1996).
[CrossRef] [PubMed]

Osten, W.

Paganin, D.

K. A. Nugent, T. E. Gureyev, D. F. Cookson, D. Paganin, and Z. Barena, "Quantitative Phase Imaging using Hard X Rays," Phys. Rev. Lett. 77, 2961-2964 (1996).
[CrossRef] [PubMed]

Pedrini, G.

Raymer, M. G.

Sakaki, K.

H. Ito, T. Nakata, K. Sakaki, M. Ohtsu, K. I. Lee, and W. Jhe, "Laser spectroscopy of atoms guided by evanescent waves in micron-sized hollow optical fibers," Phys. Rev. Lett. 76, 4500 (1996).
[CrossRef] [PubMed]

Santarsiero, M.

R. Borghi, F. Gori, and M. Santarsiero, "Phase and amplitude retrieval in ghost diffraction from field-correlation measurements," Phys. Rev. Lett. 96, 183901 (2006).
[CrossRef] [PubMed]

Sasada, H.

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, "Novel optical trap of atoms with a doughnut beam," Phys. Rev. Lett. 78, 4713-4716 (1997).
[CrossRef]

Saxton, W. O.

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

Shimizu, Y.

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, "Novel optical trap of atoms with a doughnut beam," Phys. Rev. Lett. 78, 4713-4716 (1997).
[CrossRef]

Shiokawa, N.

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, "Novel optical trap of atoms with a doughnut beam," Phys. Rev. Lett. 78, 4713-4716 (1997).
[CrossRef]

Spence, J. C. H.

Tiziani, H.

Torii, Y.

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, "Novel optical trap of atoms with a doughnut beam," Phys. Rev. Lett. 78, 4713-4716 (1997).
[CrossRef]

Weierstall, U.

Weng, J.

Wu, J.

Y. K. Wu, J. Li, and J. Wu, "Anomalous hollow electron beams in a storage ring," Phys. Rev. Lett. 94, 134802 (2005).
[CrossRef] [PubMed]

Wu, J. S.

Wu, X.

Wu, Y. K.

Y. K. Wu, J. Li, and J. Wu, "Anomalous hollow electron beams in a storage ring," Phys. Rev. Lett. 94, 134802 (2005).
[CrossRef] [PubMed]

Yang, G.

Yang, G.-Z.

Zalevsky, Z.

Zhang, F.

F. Zhang, G. Pedrini, and W. Osten, "Phase retrieval of arbitrary complex-valued fields through aperture-plane modulation," Phys. Rev. A 75, 043805 (2007).
[CrossRef]

Zhang, Y.

Zhao, H.

Zhao, Y.

Y. Zhao, J. S. Edgar, G. D. M. Jeffries, D. McGloin, and D. T. Chiu, "Spin-to-orbital angular momentum conversion in a strongly focused optical beam," Phys. Rev. Lett. 99, 073901 (2007).
[CrossRef] [PubMed]

Zhong, J.

Zhuang, J.-Y.

Appl. Opt. (1)

J. Opt. Soc. Am. A (1)

Opt. Commun. (1)

Z. Liu and S. Liu, "Double image encryption based on iterative fractional Fourier transform," Opt. Commun. 272, 324-329 (2007).
[CrossRef]

Opt. Express (2)

Opt. Lett. (9)

A. Anand, G. Pedrini, W. Osten, and P. Almoro, "Wavefront sensing with random amplitude mask and phase retrieval," Opt. Lett. 32, 1584-1586 (2007).
[CrossRef] [PubMed]

P. Almoro, G. Pedrini, and W. Osten, "Wavefront sensing with random amplitude mask and phase retrieval," Opt. Lett. 32, 733-735 (2007).
[CrossRef] [PubMed]

A. K. Ellerbee and J. A. Izatt, "Phase retrieval in low-coherence interferometric microscopy," Opt. Lett. 32, 388-390 (2007).
[CrossRef] [PubMed]

J. Zhong and J. Weng, "Phase retrieval of optical fringe patterns from the ridge of a wavelet transform," Opt. Lett. 30, 2560-2562 (2005).
[CrossRef] [PubMed]

J. S. Wu, U. Weierstall, J. C. H. Spence, and C. T. Koch, "Iterative phase retrieval without support," Opt. Lett. 29, 2737 (2004).
[CrossRef] [PubMed]

Y. Cai, X. Lu, and Q. Lin, "Hollow Gaussian beams and their propagation properties," Opt. Lett. 28, 1084 (2003).
[CrossRef] [PubMed]

Z. Liu, H. Zhao, J. Liu, J. Lin, M. A. Ahmad, and S. Liu, "Generation of hollow Gaussian beams by spatial filtering," Opt. Lett. 32, 2076 (2007).
[CrossRef] [PubMed]

Z. Zalevsky, D. Mendlovic, and R. G. Dorsch, "Gerchberg-Saxton algorithm applied in the fractional Fourieror the Fresnel domain," Opt. Lett. 21, 842-844 (1996).
[CrossRef] [PubMed]

D. F. McAlister, M. Beck, L. Clarke, A. Mayer, and M. G. Raymer, "Optical phase retrieval by phase-space tomography and fractional-order Fourier transforms," Opt. Lett. 20, 1181-1183 (1995).
[CrossRef] [PubMed]

Optik (1)

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

Phys. Rev. A (1)

F. Zhang, G. Pedrini, and W. Osten, "Phase retrieval of arbitrary complex-valued fields through aperture-plane modulation," Phys. Rev. A 75, 043805 (2007).
[CrossRef]

Phys. Rev. Lett. (9)

T. Latychevskaia and H.-W. Fink, "Solution to the twin image problem in holography," Phys. Rev. Lett. 98, 233901 (2007).
[CrossRef] [PubMed]

R. Borghi, F. Gori, and M. Santarsiero, "Phase and amplitude retrieval in ghost diffraction from field-correlation measurements," Phys. Rev. Lett. 96, 183901 (2006).
[CrossRef] [PubMed]

N. Nakajima, "Phase and amplitude retrieval in ghost diffraction from field-correlation measurements," Phys. Rev. Lett. 98, 223901 (2007).
[CrossRef] [PubMed]

K. A. Nugent, T. E. Gureyev, D. F. Cookson, D. Paganin, and Z. Barena, "Quantitative Phase Imaging using Hard X Rays," Phys. Rev. Lett. 77, 2961-2964 (1996).
[CrossRef] [PubMed]

J. Miao T. Ishikawa, B. Johnson, E. H. Anderson, B. Lai, and K. O. Hodgson, "High resolution 3D X-ray diffraction microscopy," Phys. Rev. Lett. 89, 088303 (2002).
[CrossRef] [PubMed]

Y. Zhao, J. S. Edgar, G. D. M. Jeffries, D. McGloin, and D. T. Chiu, "Spin-to-orbital angular momentum conversion in a strongly focused optical beam," Phys. Rev. Lett. 99, 073901 (2007).
[CrossRef] [PubMed]

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, "Novel optical trap of atoms with a doughnut beam," Phys. Rev. Lett. 78, 4713-4716 (1997).
[CrossRef]

H. Ito, T. Nakata, K. Sakaki, M. Ohtsu, K. I. Lee, and W. Jhe, "Laser spectroscopy of atoms guided by evanescent waves in micron-sized hollow optical fibers," Phys. Rev. Lett. 76, 4500 (1996).
[CrossRef] [PubMed]

Y. K. Wu, J. Li, and J. Wu, "Anomalous hollow electron beams in a storage ring," Phys. Rev. Lett. 94, 134802 (2005).
[CrossRef] [PubMed]

Science (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, 1419 (2003).
[CrossRef] [PubMed]

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Figures (5)

Fig. 1.
Fig. 1.

Schematic Illustration of the optical setup for the generation of hollow Gaussian beams based on phase retrieval algorithm.

Fig. 2.
Fig. 2.

Flow chart of the proposed phase retrieval algorithm in Hankel transform domains.

Fig. 3.
Fig. 3.

The numerical result of the proposed phase retrieval algorithm. Here the output beam is the second-order hollow Gaussian beam.

Fig. 4.
Fig. 4.

The intensity distribution of the hollow Gaussian beam: (a) the theoretical result directly from the analytical expression, (b) the result generated by the phase retrieval algorithm and double phase filtering.

Fig. 5.
Fig. 5.

Convergence speed curves of G-S, Yang-Gu and the proposed algorithms.

Tables (1)

Tables Icon

Table 1. The total iterative number under different parameters c0 and c1 and other parameters is the same in Fig.3.

Equations (12)

Equations on this page are rendered with MathJax. Learn more.

E ( x , y ) = G ( z ) exp [ x 2 + y 2 ω 2 ( z ) ] exp [ ik x 2 + y 2 2 R ( z ) ] ,
A 1 ( x , y ) = G 10 exp ( x 2 + y 2 ω 0 2 ) ,
A 3 ( x , y ) = G 30 ( x 2 + y 2 ω 0 2 ) p exp ( x 2 + y 2 ω 0 2 ) ,
ϕ 3 ( x , y ) = C 0 ,
A 2 ( ρ ) exp [ i ϕ 2 ( ρ ) ] = H { A 1 ( r ) exp [ i ϕ 1 ( r ) ] } ,
A 3 ( r ) exp [ i ϕ 3 ( r ) ] = H { A 2 ( ρ ) exp [ i φ 2 ( ρ ) ] } ,
H [ f ( ρ ) ] = 2 π 0 f ( r ) J 0 ( 2 π ρ r ) d r ,
A 2 ( ρ ) = H { A 3 ( r ) exp [ i ϕ 3 ( r ) ] } ,
φ 2 ( ρ ) = arg { H { A 3 ( r ) exp [ i ϕ 3 ( r ) ] } A 2 ( ρ ) } ,
A 2 ( ρ ) exp [ i ϕ 2 ( ρ ) ] = H { A 1 ( r ) exp [ i ϕ 1 ( r ) ] } ,
MSE = mse ( A 2 , k , A 2 ) = Σ A 2 , k A 2 2 L ,
Δ ϕ 1 = ϕ 1 r ( r ) φ 1 , Δ ϕ 2 = φ 2 ( ρ ) ϕ 2 r ( ρ ) .

Metrics