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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 17, Iss. 8 — Apr. 13, 2009
  • pp: 6311–6319
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Approach to improve beam quality of inter-satellite optical communication system based on diffractive optical elements

Liying Tan, Jianjie Yu, Jing Ma, Yuqiang Yang, Mi Li, Yijun Jiang, Jianfeng Liu, and Qiqi Han  »View Author Affiliations


Optics Express, Vol. 17, Issue 8, pp. 6311-6319 (2009)
http://dx.doi.org/10.1364/OE.17.006311


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Abstract

For inter-satellite optical communication transmitter with reflective telescope of two-mirrors on axis, a large mount of the transmitted energy will be blocked by central obscuration of the secondary mirror. In this paper, a novel scheme based on diffractive optical element (DOE) is introduced to avoid it. This scheme includes one diffractive beam shaper and another diffractive phase corrector, which can diffract the obscured part of transmitted beam into the domain unobscured by the secondary mirror. The proposed approach is firstly researched with a fixed obscuration ratio of 1/4. Numerical simulation shows that the emission efficiency of new figuration is 99.99%; the beam divergence from the novel inter-satellite optical communication transmitter is unchanged; and the peak intensity of receiver plane is increased about 31% compared with the typical configuration. Then the intensity patterns of receiver plane are analyzed with various obscuration ratio, the corresponding numerical modelling reveals that the intensity patterns with various obscuration ratio are nearly identical, but the amplify of relative peak intensity is getting down with the growth of obscuration ratio. This work can improve the beam quality of inter-satellite optical communication system without affecting any other functionality.

© 2009 Optical Society of America

1. Introduction

As a kind of line-of-sight wireless transmission utilizing light carrier, inter-satellite optical communication has became a subject of research interest since the advent of the laser. Comparing with a RF or microwave system of same size, it could be much more efficient and could provide data rate and capability of a high orders of magnitude [1

1. M. Katzman, ed., Laser Satellite Communications, (Englewood Cliffs, N.J., Prentice-Hall, 1987).

]. An afocal or focal reflective optical telescope referred as a beam expander was commonly used in inter-satellite optical communication or ground-to-satellite optical communication system [2-5

2. P. Henneberg and H. Schubert, “A new telescope concept for space communication,” Proc. SPIE 1218, 153–159 (1990). [CrossRef]

], which made an action in compressing beam divergence, reducing the field of view as well as increasing receiving energy. For inter-satellite optical communication transmitter with reflective optical telescope of two-mirrors on axis, unfortunately, a considerable amount of the beam energy is lost for central obscuration of secondary mirror. Accordingly, emission efficiency of inter-satellite optical communication transmitter and beam power density in far-field will be both decreased.

Different approaches have been proposed to avoid the secondary mirror obscuration. Generally, they can be classified into two groups according to whether secondary mirror positioned on axis or not. One can find that many existing optical elements or devices have been introduced in the former group, such as axicon optical element [6

6. W. N. Peters and A. M. Ledger, “Techniques for matching laser TEM00 mode to obstructed circular aperture,” Appl. Opt . 9, 1435–1442 (1970). [CrossRef] [PubMed]

,7

7. B. J. Klein and J. J. Degnan,“Optical antenna gain 1 : transmitting antennas,” Appl. Opt . 13, 2134–2141 (1974). [CrossRef] [PubMed]

], dual-secondary mirror [8

8. O. D. Christy, “Dual-secondary mirror Cassegrain optical system,” U. S. Patent 4,439,012, Mar 27 (1984).

], cone reflecting mirror [9

9. X. L. Kong and P. M. Hao, “New method to remove central shade for reflecting laser beam expander,” Chin. J. Quantum Electron . 19, 205–209 (2002).

], prism beam slier and beam-splitter/beam combiner [10

10. H. Hemmati and N. Page, “Approaches for efficient coupling of lasers to telescope with secondary mirror and baffle obscuration,” Proc. SPIE 4635, 288–294 (2002). [CrossRef]

], etc. Low transmission efficiency is mutual defect of these schemes, and most of them have extraordinary drawbacks such as overfull optical elements, low flexibility or difficult to manufacture. In the latter case, the secondary mirror on-axis is replaced by more than one off-axis fold or turning mirrors, and the off-axis tertiary reflective surfaces will be structured to avoid central obscuration [11

11. C. W. Chen, “Re-imaging optical system including refractive and diffractive optical elements,” U. S. Patent 5,287,218, Feb 15 (1994).

,12

12. R. N. Smartt and E. W. Cross, “Advances in spherical-mirror telescopic systems design: application to large-aperature solar coronagraphs,” Opt. Eng . 41, 2055–2058 (2002). [CrossRef]

]. However, they are not suitable for space applications for strict assembly, lower stability, and larger size. So, how to improve transmitted beam quality with intrinsic bulk, mass, and power consumption is becoming a topic of interest. Here, we represent a method adopting two DOEs, which rearrange the transmitted beam with Gaussian intensity profile into a hollow Gaussian beam [13

13. 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, 2076–2078 (2007). [CrossRef] [PubMed]

,14

14. Z. Liu, J. Dai, X. Sun, and S. Liu, “Generation of hollow Gaussian beam by phase-only filtering,” Opt. Express 16, 19926–19933 (2008). [CrossRef] [PubMed]

] to avoid the secondary mirror obscuration.

The structure of this paper is arranged as follows: a basic configuration of the system containing two DOEs for improving beam quality of inter-satellite optical communication transmitter is described in Section 2. The design of DOE and beam transformation within inter-satellite optical communication system are addressed in Section 3. In Section 4, we present the numerical modelling of DOE for beam shaping and corresponding far-field pattern. At last section, a summarization is given.

2. Improved inter-satellite optical communication transmitter configuration

The typical simplified scheme of inter-satellite optical communication transmitter is illustrated in Fig. 1, which includes: (1) laser source; (2) beam collimator; (3) reflective optical antenna. As shown in Fig. 1, when the transmitted beam emitting from fiber-coupled laser diode (FLD) has been collimated, a monochromatic plane-wave with Gaussian intensity profile will be achieved in plane P1. Successively, light beams will arrive at the secondary mirror perpendicularly and then be reflected onto primary mirror. Finally, the solid Gaussian beam is degenerated into a hollow Gaussian beam (HGB) in plane P3 (i.e., the exit pupil plane) for the central obscuration of secondary mirror. Obviously, when the beam leaves from plane P3, lots of the beam energy has been lost .

Fig. 1. The typical simplified scheme of inter-satellite optical communication transmitter.

Comparing with the typical scheme, we introduce an improved configuration of inter-satellite optical communication transmitter (as shown in Fig. 2) which can avoid energy loss by adding two DOEs. These two elements are placed between collimator and optical antenna along axis, and they act as a beam shaper and a phase corrector, respectively. Different from the typical scheme, in this configuration, the central portion of collimated Gaussian beams in plane P1 will be diffracted to annular domain beneath solid blue line in plane P2, which is the focal plane of diffractive beam shaper. Afterwards, the transmitted beam will be rearranged to a plane wave by diffractive phase corrector positioning in plane P2. Thus a HGB is achieved prior to secondary mirror. Since an afocal reflective telescope is usually considered as a linear beam expander, consequently a magnified HGB will be obtained in plane P3. Obviously, the intensity of each point of the magnified HGB in Fig. 2 is larger than that of HGB in Fig. 1, and the transmitted beam won’t be blocked by secondary mirror when a proper inner diameter of HGB in Fig. 2 is selected.

3. Design of DOE and beam transformation

3.1. Design of DOE

DOEs mentioned above are classified as pure-phase elements, which can be described by phase functions [15

15. W. B. Veldkamp and T. J. McHugh, “Binary optics,” Sci. Am. May , 50-55 (1992).

], so the phase profiles of both beam shaper and phase corrector must be precisely determined to obtain a lossless output beam. In this paper, only design scheme of DOE for beam shaping is demonstrated, which enables beams with Gaussian intensity profile to hollow Gaussian intensity profile. Actually, the design for phase corrector is very simple, which can be reached only across a subtraction of phase functions between plane-wave and actual wave-front of plane P2.

Fig. 2. Improved configuration of inter-satellite optical communication transmitter.

The design of DOE is similar to phase retrieval from intensity measurements. In fact, the intensity distribution of laser beam in plane P1 and the required hollow Gaussian intensity distribution in plane P2 are all calculable or measurable, and hence the generation problem of diffractive beam shaper can be referred as a phase reconstruction from two known intensity information. Many optimization algorithms have been developed to compute phase profiles of DOE [16-19

16. R. W. Gerchberg and W. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane picture,” Optik 35, 247–246 (1972).

], here GS optimization algorithm [16

16. R. W. Gerchberg and W. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane picture,” Optik 35, 247–246 (1972).

] will be proposed for it. The iterative GS algorithm depends on a Fourier Transform relation between the waves in plane P1 and plane P2, Fig. 3 shows this iterative process.

Fig. 3. Flow chart of GS algorithm.

As shown in Fig. 3, GS algorithm is an infinite loop of forward and backward transforms from input plane P1 to reconstruction plane P2, the known sampled amplitude function A and B are assigned to these two planes, respectively.To begin, a random phase distribution φ 0 is generated to serve as the initial phase estimate and combined with the corresponding sampled amplitude function A to form input wave function Ein(x 1,y 1). Then, this synthesized complex discrete function is done by means of the Fast Fourier Transform (FFT) algorithm. The phase portion ϕ′ of the complex wave function Eout′(x 2,y 2) resulting from this transformation are computed and reserved, and it is combined with the corresponding sampled amplitude function B. This new complex function Eout(x 2,y 2) is then done by Inverse Fast Fourier Transform (IFFT), the phases φ′ of the sample points are calculated and combined with the known sampled amplitude function A to form a new estimate of the complex sampled input plane, and the iteration process is repeated.

The phase mapping of two planes evolves through the iterations [16

16. R. W. Gerchberg and W. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane picture,” Optik 35, 247–246 (1972).

, 20

20. B. Lin and N. C. Gallagher, “Convergence of a spectrum shaping algorithm,” Appl. Opt . 13, 2470–2471 (1974). [CrossRef]

] and eventually comes to a stable position or stopped on the condition:

SSE=(Ein)B2dx2dy2B2dx2dy2<ζ,
(1)

where SSE is the sum of squared error between the desired and computed amplitude function of the reconstruction plane, represents FFT, ζ is an infinitesimal. Finally, the phase mapping representing the DOE data will be obtained across a subtraction between calculated phase mapping and initial phase of input plane-wave.

3.2. Beam transformation

As mentioned above, when the transmitted beam has been transformed to a HGB in plane P2, the phase pattern of this HGB is modified by a phase corrector placed in this plane and form a new plane wave. Subsequently, this new plane wave with hollow Gaussian intensity pattern is magnified by optical antenna, and the emission efficiency of inter-satellite optical communication transmitter is increased greatly for the absence of the secondary mirror obscuration. In order to evaluate the performance of the improved scheme, we define the emission efficiency of inter-satellite optical communication transmitter as:

ηE=E'E×100%,
(2)

where E represents the total beam energy emitting from light source, and E′ represents the beam energy emitting from the exit pupil pupil.

The HGB leave from inter-satellite optical communication transmitter and transmit to opposite inter-satellite optical communication receiver apart from ten-thousands of kilometers via free space, and the complex amplitude function of receiver plane can be computed by Fraunhofer diffractive formula [21

21. J. W. Goodman, Introduction to Fourier Optics, Second Edition, (New York, McGraw-Hill, 2006).

]:

E4(x4,y4)=ejkzejk2z(x32+y32)jλzE3(x3,y3)exp[jkz(x3x4+y3y4)]dx3dy3
=ejkzejk2z(x32+y32)jλz.{E3(x3,y3)}fx=x4λz,fy=y4λz,
(3)

where k = 2π/λ represents wavenumber, z is transmitted distance, fx , fy are spatial frequency of receiver plane along x-direction and y-direction, respectively.

The amplitudes are proportional to the square roots of the measured intensities, thus the intensity function of far-field (i.e. the plane of inter-satellite optical communication receiver) can be described as:

I4(x4,y4)=(1λz)2·{E3(x3,y3)}2.
(4)

In order to check whether beam quality has been improved with the improved scheme, the far-field pattern of beams is analyzed, and a relative peak energy increment is used to evaluate beam quality, which is denoted as ηP:

ηP=I4maxI4maxI4max×100%,
(5)

where I 4max and I 4max′ are the peak energy of far field with and without DOE, respectively.

4. Numerical simulations

In the design of inter-satellite optical communication transmitter, the complex wave function of exit pupil plane is usually known as design requirements, which includes the diameter of exit pupil, beam waist or beam divergence. Only when these parameters have been given, we can determine the magnification of optical antenna, which is defined as the diameter ratio between primary mirror and secondary mirror. The reciprocal of magnification is referred as obscuration ratio, which is denoted as T. Obviously, the inner diameter of HGB in exit pupil plane varies with T, so the intensity patterns of far-field are also varied with T. This variation prompts us to investigate the far-field intensity patterns with various obscuration ratio.

The complex wave function with rotational symmetry in exit pupil plane can be written as:

E3(r3)=C(r3)·exp().
(6)

In Eq. (5), the phase and amplitude of complex function can be obtained by phase corrector and beam shaper, respectively. The amplitude function is defined as:

C(r3)={exp(r32/ω032),ifr30r3r3m0,otherwise,
(7)

where r 3m, ω 03, and r 30 = r 3m · T represent the outer radius, beam waist, and inner radius of required HGB profile, respectively.

Some parameters are made in the following section: wavelength λ = 800 nm; the radius of exit pupil plane r 3m = 10 mm, corresponding beam waist ω 03 = r 3m/√2; the transmitted distance between inter-satellite optical communication transmitter and inter-satellite optical communication receiver z = 5 × 107 m.

4.1. Design of DOE with fixed obscuration ratio

Considering a typical obscuration ratio of T = 1/4, when the amplitude function of exit pupil plane has been given, the amplitude function of plane P2 and plane P1 in Fig. 2 can be determined by

B(r2)={c2·exp(r22/ω022),ifr20r2r2m0,otherwise,
(8)

where r 2m = r 3m · T,ω 02 = ω 03 · T, and r 20 = r 30 · T represent the outer radius, beam waist, and inner radius of desired HGB profile in plane P2, respectively. c 2 is an amplitude profile coefficient, which makes a relation of conservation of energy between the waves of plane P2 and P3:

A(r1)=c1·exp(r12/ω012),
(9)

where r 1 ∈ [0,r 1m], and r 1m = r 2m is the maximal spot radius of plane P1 in Fig. 2, ω 01 = r 1m/√2 is the corresponding beam waist. The action of c1 is similar to c 2, which ensures the energy conservation between plane P1 and P2.

Fig. 4. The beam transformation in inter-satellite optical communication transmitter.

The phase profiles of DOE are shown in Fig. 5, which represent an arbitrary radial distribution and they have been quantized to offer a convenience for manufacturing. Fig. 5(a) is the phase function of beam shaper, and Fig. 5(b) is the phase function of phase corrector.

Fig. 5. Quantized phase distribution of DOEs (n=16). (a): beam shaper; (b): phase corrector

Figure 6 represents the intensity patterns of receiver plane, which is about 50,000 kilometers apart from exit pupil plane. As shown in Fig. 6, the intensity pattern in whole receiver plane is a Gaussian-like distribution, and the central energy valley of near field is found to disappear in the receiver plane. Compared with the typical inter-satellite optical communication transmitter, the intensity pattern of whole receiver plane and the beam power density are all grown with the introduction of DOEs within the new scheme, and the corresponding ηP is 31.5%. Furthermore, these two distribution curves are kept with a same width of main lobe, which indicates that the PAT (pointing, acquisition, and tracking) mechanism of inter-satellite optical communication system won’t be influenced.

Fig. 6. Beam intensity distribution of the receiver plane.

4.2. Design of DOE with varying obscuration ratio

In practice, the obscuration ratio will vary with design requirements or other limitations. Assuming that we still take the diameter and beam waist of exit pupil plane as the known preconditions, and the same light source in each configuration of inter-satellite optical communication transmitter is adopted. When obscuration ratio changes, the inner diameter of HGB in plane P3, the amplitude distribution in plane P2 and P1 are all changed. In order to satisfy energy conservation theorem, it is necessary to modify amplitude functions of plane P2 and P1 accordingly.

Fig. 7. The intensity distribution of receiver plane with various obscuration ratio.

Figure 7 presents the intensity distribution in far-field with obscuration ratio equal to 1/4, 1/6, 1/8, and 1/10, respectively. As shown in Fig. 7, the intensity patterns of far-field change little with the variation of obscuration ratio, which indicates that the improved configuration can be suitable for various optical antennas. However, further comparison reveals that the smaller obscuration ratio, the lower side lobe of the intensity patterns. Obviously, This difference comes from the variation of peak valley width and amplitude coefficient in exit pupil plane, and they don’t affect inter-satellite optical communication system.

Fig. 8. The relative intensity distribution of receiver plane with various obscuration ratio.

Figure 8 shows the normalized intensity in far field with DOE and without DOE. The relative increment of peak energy will be reduced with the rising of obscuration ratio T. When T equal to 1/4, 1/6, 1/8, and 1/10, the corresponding ηP are 31.5%, 13.7%, 8.1%, and 5.5%, respectively.

5. Conclusion

A novel scheme for improving beam quality of inter-satellite optical communication system was presented. Comparing to other methods, this scheme adds only two diffractive lens into optical path along optical axis, and a lossless output beam was obtained with the same mass, volume, and power consumption. The corresponding far field pattern were analyzed and discussed for fixed and varying obscuration ratio, respectively. Numerical modelling with a fixed obscuration ratio of 1/4 demonstrated that the beam quality of inter-satellite optical communication system could be increased greatly. The results of numerical simulation with varying obscuration ratio revealed that the intensity patterns of far-field change little with the variation of obscuration ratio, and the improved configuration can be suitable for various optical antenna.

Although, the realization of the method presented in this paper may be troubled by some uncertain factors. Whereas, this work can offer a useful reference for optimal design of optical telescope centrally obscured, and can also be a guidance for other applications of DOE in satellite optical communication system or other modern optical systems.

References and links

1.

M. Katzman, ed., Laser Satellite Communications, (Englewood Cliffs, N.J., Prentice-Hall, 1987).

2.

P. Henneberg and H. Schubert, “A new telescope concept for space communication,” Proc. SPIE 1218, 153–159 (1990). [CrossRef]

3.

A. Yamamoto, T. Hori, T. Shimizu, and K. Nakagawa, “Japanese first optical inter-orbit communications engineering test satellite (OICETS),” Proc. SPIE 2210, 30–38 (1994). [CrossRef]

4.

H. Hemmati and N. Page, “Preliminary opto-mechanical design for the X2000 transceiver,” Proc. SPIE 3615, 206–211 (1999). [CrossRef]

5.

M. Knapek, J. Horwath, N. Perlot, and B. Wilkerson, “The DLR ground station in the optical payload experiment (STROPEX) - results of the atmospheric measurement instruments,” Proc. SPIE 6304, 63041U (2006). [CrossRef]

6.

W. N. Peters and A. M. Ledger, “Techniques for matching laser TEM00 mode to obstructed circular aperture,” Appl. Opt . 9, 1435–1442 (1970). [CrossRef] [PubMed]

7.

B. J. Klein and J. J. Degnan,“Optical antenna gain 1 : transmitting antennas,” Appl. Opt . 13, 2134–2141 (1974). [CrossRef] [PubMed]

8.

O. D. Christy, “Dual-secondary mirror Cassegrain optical system,” U. S. Patent 4,439,012, Mar 27 (1984).

9.

X. L. Kong and P. M. Hao, “New method to remove central shade for reflecting laser beam expander,” Chin. J. Quantum Electron . 19, 205–209 (2002).

10.

H. Hemmati and N. Page, “Approaches for efficient coupling of lasers to telescope with secondary mirror and baffle obscuration,” Proc. SPIE 4635, 288–294 (2002). [CrossRef]

11.

C. W. Chen, “Re-imaging optical system including refractive and diffractive optical elements,” U. S. Patent 5,287,218, Feb 15 (1994).

12.

R. N. Smartt and E. W. Cross, “Advances in spherical-mirror telescopic systems design: application to large-aperature solar coronagraphs,” Opt. Eng . 41, 2055–2058 (2002). [CrossRef]

13.

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, 2076–2078 (2007). [CrossRef] [PubMed]

14.

Z. Liu, J. Dai, X. Sun, and S. Liu, “Generation of hollow Gaussian beam by phase-only filtering,” Opt. Express 16, 19926–19933 (2008). [CrossRef] [PubMed]

15.

W. B. Veldkamp and T. J. McHugh, “Binary optics,” Sci. Am. May , 50-55 (1992).

16.

R. W. Gerchberg and W. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane picture,” Optik 35, 247–246 (1972).

17.

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

18.

G. R. Brady and J. R. Fienup, “Nonlinear optimization algorithm for retrieving the full complex pupil function,” Opt. Express 14, 474–486 (2006). [CrossRef] [PubMed]

19.

C. Rydberg and J. Bengtsson, “Numerical algorithm for the retrieval of coherent beams from transverse intensity measurements,” Opt. Express 15, 13613–13623 (2007). [CrossRef] [PubMed]

20.

B. Lin and N. C. Gallagher, “Convergence of a spectrum shaping algorithm,” Appl. Opt . 13, 2470–2471 (1974). [CrossRef]

21.

J. W. Goodman, Introduction to Fourier Optics, Second Edition, (New York, McGraw-Hill, 2006).

OCIS Codes
(050.1970) Diffraction and gratings : Diffractive optics
(060.4510) Fiber optics and optical communications : Optical communications

ToC Category:
Fiber Optics and Optical Communications

History
Original Manuscript: December 4, 2008
Revised Manuscript: February 25, 2009
Manuscript Accepted: March 20, 2009
Published: April 2, 2009

Citation
Liying Tan, Jianjie Yu, Jing Ma, Yuqiang Yang, Mi Li, Yijun Jiang, Jianfeng Liu, and Qiqi Han, "Approach to improve beam quality of inter-satellite optical communication system based on diffractive optical elements," Opt. Express 17, 6311-6319 (2009)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-8-6311


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References

  1. M. Katzman, ed., Laser Satellite Communications, (Englewood Cliffs, N.J., Prentice-Hall, 1987).
  2. P. Henneberg and H. Schubert, "A new telescope concept for space communication," Proc. SPIE 1218, 153-159 (1990). [CrossRef]
  3. A. Yamamoto, T. Hori, T. Shimizu, and K. Nakagawa, "Japanese first optical inter-orbit communications engineering test satellite (OICETS)," Proc. SPIE 2210, 30-38 (1994). [CrossRef]
  4. H. Hemmati and N. Page, "Preliminary opto-mechanical design for the X2000 transceiver," Proc. SPIE 3615, 206-211 (1999). [CrossRef]
  5. M. Knapek, J. Horwath, N. Perlot, and B. Wilkerson, "The DLR ground station in the optical payload experiment (STROPEX) - results of the atmospheric measurement instruments," Proc. SPIE 6304, 63041U (2006). [CrossRef]
  6. W. N. Peters and A. M. Ledger, "Techniques for matching laser TEM00 mode to obstructed circular aperture," Appl. Opt. 9, 1435-1442 (1970). [CrossRef] [PubMed]
  7. B. J. Klein and J. J. Degnan, "Optical antenna gain 1: transmitting antennas," Appl. Opt. 13, 2134-2141 (1974). [CrossRef] [PubMed]
  8. O. D. Christy, "Dual-secondary mirror Cassegrain optical system," U. S. Patent 4,439,012, Mar. 27 (1984).
  9. X. L. Kong and P. M. Hao, "New method to remove central shade for reflecting laser beam expander," Chin. J. Quantum Electron. 19, 205-209 (2002).
  10. H. Hemmati and N. Page, "Approaches for efficient coupling of lasers to telescope with secondary mirror and baffle obscuration," Proc. SPIE 4635, 288-294 (2002). [CrossRef]
  11. C. W. Chen, "Re-imaging optical system including refractive and diffractive optical elements," U. S. Patent 5,287,218, Feb. 15 (1994).
  12. R. N. Smartt and E. W. Cross, "Advances in spherical-mirror telescopic systems design: application to largeaperature solar coronagraphs," Opt. Eng. 41, 2055-2058 (2002). [CrossRef]
  13. 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, 2076-2078 (2007). [CrossRef] [PubMed]
  14. Z. Liu, J. Dai, X. Sun, and S. Liu, "Generation of hollow Gaussian beam by phase-only filtering," Opt. Express 16, 19926-19933 (2008). [CrossRef] [PubMed]
  15. W. B. Veldkamp and T. J. McHugh, "Binary optics," Sci. Am. 266, 50-55 (1992).
  16. R.W. Gerchberg and W. Saxton, "A practical algorithm for the determination of phase from image and diffraction plane picture," Optik 35, 227-246 (1972).
  17. J. R. Fienup, "Phase retrieval algorithms: a comparison," Appl. Opt. 15, 2758-2769 (1982). [CrossRef]
  18. G. R. Brady and J. R. Fienup, "Nonlinear optimization algorithm for retrieving the full complex pupil function," Opt. Express 14, 474-486 (2006). [CrossRef] [PubMed]
  19. C. Rydberg and J. Bengtsson, "Numerical algorithm for the retrieval of coherent beams from transverse intensity measurements," Opt. Express 15, 13613-13623 (2007). [CrossRef] [PubMed]
  20. B. Lin and N. C. Gallagher, "Convergence of a spectrum shaping algorithm," Appl. Opt. 13, 2470-2471 (1974). [CrossRef]
  21. J. W. Goodman, Introduction to Fourier Optics, Second Edition, (New York, McGraw-Hill, 2006).

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