Abstract

We investigate the performance and capability of a holographic modal wavefront sensor (HMWS) that is based on a multiplexed phase computer-generated hologram (MPCGH). The theoretical treatments of the HMWS are presented with scalar diffraction approximations and Fourier analysis. Several MPCGHs have been designed with different linear carrier frequencies, by using of the multiplexed coding scheme we have proposed, and by coding some common Zernike modes. The numerical simulation is carried out to investigate the performance of the HMWS to detect particular aberration mode(s), by considering the effect of different carrier frequency selections and the capability of coding a large number of modes. The results exhibit the expected characteristics of a corresponding symmetric spot pair, and indicate that the wavefront distorted by a particular Zernike mode(s) can be retrieved immediately through solving the amplitude of each mode coded in MPCGHs through the response curves of the HMWS.

© 2011 Optical Society of America

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References

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    [CrossRef]
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2010 (2)

2009 (2)

2008 (2)

R. Bhatt, S. K. Mishra, D. Mohan, and A. K. Gupta, “Direct amplitude detection of Zernike modes by computer-generated holographic wavefront sensor: modeling and simulation,” Opt. Lasers Eng. 46, 428–439 (2008).
[CrossRef]

F. Ghebremichael, G. P. Andersen, and K. S. Gurley, “Holography-based wavefront sensing,” Appl. Opt. 47, A62–A70(2008).
[CrossRef] [PubMed]

2007 (2)

2005 (2)

M. Booth, T. Wilson, H. B. Sun, T. Ota, and S. Kawata, “Methods for the characterization of deformable membrane mirrors,” Appl. Opt. 44, 5131–5139 (2005).
[CrossRef] [PubMed]

J. J. Zhong, D. G. Leyva, A. Corbett, L. Diaz-Santanal, and T. D. Wilkinson, “Mirror-mode sensing with a holographic modal wavefront sensor,” Proc. SPIE 6018, 60181I(2005).
[CrossRef]

2003 (1)

M. J. Booth, “Direct measurement of Zernike aberration modes with a modal wavefront sensor,” Proc. SPIE 5162, 79–90 (2003).
[CrossRef]

2002 (1)

R. Ragazzoni, E. Diolaiti, and E. Vernet, “A pyramid wavefront sensor with no dynamic modulation,” Opt. Commun. 208, 51–60 (2002).
[CrossRef]

2001 (1)

B. C. Platt and R. Shack, “History and principles of Shack-Hartmann wavefront sensing,” J. Refractive Surg. 17, s453–s457 (2001).

2000 (1)

1995 (1)

1988 (1)

Andersen, G. P.

G. P. Andersen, L. Dussan, F. Ghebremichael, and K. Chen, “Holographic wavefront sensor,” Opt. Eng. 48, 085801(2009).
[CrossRef]

F. Ghebremichael, G. P. Andersen, and K. S. Gurley, “Holography-based wavefront sensing,” Appl. Opt. 47, A62–A70(2008).
[CrossRef] [PubMed]

Bengtsson, J.

Bhatt, R.

S. K. Mishra, R. Bhatt, and D. Mohan, “Differential modal Zernike wavefront sensor employing a computer-generated hologram: a proposal,” Appl. Opt. 48, 6458–6465(2009).
[CrossRef] [PubMed]

R. Bhatt, S. K. Mishra, D. Mohan, and A. K. Gupta, “Direct amplitude detection of Zernike modes by computer-generated holographic wavefront sensor: modeling and simulation,” Opt. Lasers Eng. 46, 428–439 (2008).
[CrossRef]

Booth, M.

Booth, M. J.

Chen, K.

G. P. Andersen, L. Dussan, F. Ghebremichael, and K. Chen, “Holographic wavefront sensor,” Opt. Eng. 48, 085801(2009).
[CrossRef]

Corbett, A.

J. J. Zhong, D. G. Leyva, A. Corbett, L. Diaz-Santanal, and T. D. Wilkinson, “Mirror-mode sensing with a holographic modal wavefront sensor,” Proc. SPIE 6018, 60181I(2005).
[CrossRef]

Dai, G.-M.

Diaz-Santanal, L.

J. J. Zhong, D. G. Leyva, A. Corbett, L. Diaz-Santanal, and T. D. Wilkinson, “Mirror-mode sensing with a holographic modal wavefront sensor,” Proc. SPIE 6018, 60181I(2005).
[CrossRef]

Diolaiti, E.

R. Ragazzoni, E. Diolaiti, and E. Vernet, “A pyramid wavefront sensor with no dynamic modulation,” Opt. Commun. 208, 51–60 (2002).
[CrossRef]

Dussan, L.

G. P. Andersen, L. Dussan, F. Ghebremichael, and K. Chen, “Holographic wavefront sensor,” Opt. Eng. 48, 085801(2009).
[CrossRef]

Ghebremichael, F.

G. P. Andersen, L. Dussan, F. Ghebremichael, and K. Chen, “Holographic wavefront sensor,” Opt. Eng. 48, 085801(2009).
[CrossRef]

F. Ghebremichael, G. P. Andersen, and K. S. Gurley, “Holography-based wavefront sensing,” Appl. Opt. 47, A62–A70(2008).
[CrossRef] [PubMed]

Gupta, A. K.

R. Bhatt, S. K. Mishra, D. Mohan, and A. K. Gupta, “Direct amplitude detection of Zernike modes by computer-generated holographic wavefront sensor: modeling and simulation,” Opt. Lasers Eng. 46, 428–439 (2008).
[CrossRef]

Gurley, K. S.

Haist, T.

Huang, S.

Jiang, Z.

Kawata, S.

Leyva, D. G.

J. J. Zhong, D. G. Leyva, A. Corbett, L. Diaz-Santanal, and T. D. Wilkinson, “Mirror-mode sensing with a holographic modal wavefront sensor,” Proc. SPIE 6018, 60181I(2005).
[CrossRef]

Liu, C.

Lloyd-Hart, M.

N. M. Milton and M. Lloyd-Hart, “Disk harmonic functions for adaptive optics simulations,” in Adaptive Optics: Analysis and Methods/Computational Optical Sensing and Imaging/Information Photonics/Signal Recovery and Synthesis Topical Meetings, Technical Digest (CD) (Optical Society of America, 2005), paper AWA3.
[PubMed]

Ma, H.

Malacara, D.

D. Malacara, Optical Shop Testing, 2nd ed. (Wiley, 1992).

Milewski, D. E. G.

Milton, N. M.

N. M. Milton and M. Lloyd-Hart, “Disk harmonic functions for adaptive optics simulations,” in Adaptive Optics: Analysis and Methods/Computational Optical Sensing and Imaging/Information Photonics/Signal Recovery and Synthesis Topical Meetings, Technical Digest (CD) (Optical Society of America, 2005), paper AWA3.
[PubMed]

Mishra, S. K.

S. K. Mishra, R. Bhatt, and D. Mohan, “Differential modal Zernike wavefront sensor employing a computer-generated hologram: a proposal,” Appl. Opt. 48, 6458–6465(2009).
[CrossRef] [PubMed]

R. Bhatt, S. K. Mishra, D. Mohan, and A. K. Gupta, “Direct amplitude detection of Zernike modes by computer-generated holographic wavefront sensor: modeling and simulation,” Opt. Lasers Eng. 46, 428–439 (2008).
[CrossRef]

Mohan, D.

S. K. Mishra, R. Bhatt, and D. Mohan, “Differential modal Zernike wavefront sensor employing a computer-generated hologram: a proposal,” Appl. Opt. 48, 6458–6465(2009).
[CrossRef] [PubMed]

R. Bhatt, S. K. Mishra, D. Mohan, and A. K. Gupta, “Direct amplitude detection of Zernike modes by computer-generated holographic wavefront sensor: modeling and simulation,” Opt. Lasers Eng. 46, 428–439 (2008).
[CrossRef]

Neil, M. A. A.

Osten, W.

Ota, T.

Platt, B. C.

B. C. Platt and R. Shack, “History and principles of Shack-Hartmann wavefront sensing,” J. Refractive Surg. 17, s453–s457 (2001).

Ragazzoni, R.

R. Ragazzoni, E. Diolaiti, and E. Vernet, “A pyramid wavefront sensor with no dynamic modulation,” Opt. Commun. 208, 51–60 (2002).
[CrossRef]

Roddier, F.

Shack, R.

B. C. Platt and R. Shack, “History and principles of Shack-Hartmann wavefront sensing,” J. Refractive Surg. 17, s453–s457 (2001).

Sun, H. B.

Tyson, R. K.

R. K. Tyson, Principles of Adaptive Optics (Academic, 1991), pp. 213–255.

Vernet, E.

R. Ragazzoni, E. Diolaiti, and E. Vernet, “A pyramid wavefront sensor with no dynamic modulation,” Opt. Commun. 208, 51–60 (2002).
[CrossRef]

Warber, M.

Wilkinson, T. D.

J. J. Zhong, D. G. Leyva, A. Corbett, L. Diaz-Santanal, and T. D. Wilkinson, “Mirror-mode sensing with a holographic modal wavefront sensor,” Proc. SPIE 6018, 60181I(2005).
[CrossRef]

Wilson, T.

Xi, F.

Zhong, J. J.

J. J. Zhong, D. G. Leyva, A. Corbett, L. Diaz-Santanal, and T. D. Wilkinson, “Mirror-mode sensing with a holographic modal wavefront sensor,” Proc. SPIE 6018, 60181I(2005).
[CrossRef]

Zwick, S.

Appl. Opt. (7)

J. Opt. Soc. Am. A (2)

J. Refractive Surg. (1)

B. C. Platt and R. Shack, “History and principles of Shack-Hartmann wavefront sensing,” J. Refractive Surg. 17, s453–s457 (2001).

Opt. Commun. (1)

R. Ragazzoni, E. Diolaiti, and E. Vernet, “A pyramid wavefront sensor with no dynamic modulation,” Opt. Commun. 208, 51–60 (2002).
[CrossRef]

Opt. Eng. (1)

G. P. Andersen, L. Dussan, F. Ghebremichael, and K. Chen, “Holographic wavefront sensor,” Opt. Eng. 48, 085801(2009).
[CrossRef]

Opt. Lasers Eng. (1)

R. Bhatt, S. K. Mishra, D. Mohan, and A. K. Gupta, “Direct amplitude detection of Zernike modes by computer-generated holographic wavefront sensor: modeling and simulation,” Opt. Lasers Eng. 46, 428–439 (2008).
[CrossRef]

Opt. Lett. (1)

Proc. SPIE (2)

J. J. Zhong, D. G. Leyva, A. Corbett, L. Diaz-Santanal, and T. D. Wilkinson, “Mirror-mode sensing with a holographic modal wavefront sensor,” Proc. SPIE 6018, 60181I(2005).
[CrossRef]

M. J. Booth, “Direct measurement of Zernike aberration modes with a modal wavefront sensor,” Proc. SPIE 5162, 79–90 (2003).
[CrossRef]

Other (3)

N. M. Milton and M. Lloyd-Hart, “Disk harmonic functions for adaptive optics simulations,” in Adaptive Optics: Analysis and Methods/Computational Optical Sensing and Imaging/Information Photonics/Signal Recovery and Synthesis Topical Meetings, Technical Digest (CD) (Optical Society of America, 2005), paper AWA3.
[PubMed]

D. Malacara, Optical Shop Testing, 2nd ed. (Wiley, 1992).

R. K. Tyson, Principles of Adaptive Optics (Academic, 1991), pp. 213–255.

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Figures (11)

Fig. 1
Fig. 1

Schematic diagram of the HMWS using an MPCGH. An aberrated input light beam illuminates the assembly of the MPCGH and a Fourier lens, and then it is received by a detector.

Fig. 2
Fig. 2

Plots of coefficient-squared ratios, i.e., the coefficient square of diffraction orders denoted by Eqs. (7, 8, 9), versus the modulation depth m.

Fig. 3
Fig. 3

MPCGHs encoded with four Zernike aberration modes: Z 2 0 , Z 2 2 , Z 3 1 , and Z 4 0 . The amplitude of the modes was set as 1 rad , with different distribution approaches of carrier frequencies: (a) circular, (b) column, (c) square, (d) quadrant, and (e) on-axis (black represents 0, white represents π).

Fig. 4
Fig. 4

MPCGHs coded with different numbers of Zernike aberration modes, (a) 10 Zernike aberration modes ( Z 4 Z 13 ), (b) 16 Zernike aberration modes ( Z 4 Z 19 ), and (c) 31 Zernike aberration modes ( Z 4 Z 34 ) (black represents 0, white represents π).

Fig. 5
Fig. 5

Simulated diffraction order distributions for the five MPCGHs listed in Figs. 3a, 3b, 3c, 3d, 3e with a standard plane wave as the input illumination wave, i.e., no Zernike aberration modes in the test wave: (a) circle pattern, (b) column pattern, (c) square pattern, (d) quadrant pattern, and (e) on-axis pattern.

Fig. 6
Fig. 6

Simulated diffraction order distributions for the three MPCGHs listed in Figs. 4a, 4b, 4c with a standard plane wave as the input illumination wave, i.e., no Zernike aberration modes in the test wave: (a) 10 spot pairs, (b) 16 spot pairs, and (c) 31 spot pairs. Note that the top row presents gray images, and the bottom row presents corresponding contour plots with normalized intensity distribution.

Fig. 7
Fig. 7

(a)–(e) Normalized intensity differential signals of the symmetric spot pairs in the focal plane as the function of amplitude of a single Zernike mode Z 2 0 sensed with the MPCGHs shown in Figs. 3a, 3b, 3c, 3d, 3e. Note that the other three coded modes also present slight responses accordingly in the five carrier frequency approaches. (f) Comparison of sensitivities of the four coded Zernike modes in the five carrier frequency approaches.

Fig. 8
Fig. 8

Sensing multiple Zernike modes by using of MPCGHs shown in Figs. 3a, 3b, 3c, 3d, 3e. (a) Test phase screen comprising multiple Zernike modes. (b)–(f) Recovered phase (top row) and residual error (bottom row) corresponding to the MPCGHs shown in Figs. 3a, 3b, 3c, 3d, 3e respectively (unit: radians).

Fig. 9
Fig. 9

Direct integral decomposition of test phase shown in Fig. 8a in order to compare with the results of Figs. 8b, 8c, 8d, 8e, 8f. (a) The components of four coded Zernike modes in the test phase; (b) the residual error between the decomposition result and the test phase (units: radians).

Fig. 10
Fig. 10

(a)–(c) Normalized intensity differential signals of the symmetric spot pairs in the focal plane as the function of amplitude of a single Zernike mode Z 4 sensed with the MPCGHs shown in Figs. 4a, 4b, 4c. Note that other coded modes also present slight responses near the horizontal line accordingly in the three MPCGHs, which are denoted uniformly with a bundle of almost overlapped blue lines.

Fig. 11
Fig. 11

(a)–(c) Normalized intensity differential signals of the symmetric spot pairs in the focal plane as the function of amplitude of two Zernike modes Z 4 , Z 5 sensed with MPCGHs shown in Figs. 4a, 4b, 4c. Note that other coded modes also present slight responses near the horizontal line accordingly in the three MPCGHs, which are denoted uniformly with a bundle of almost overlapped blue lines.

Tables (2)

Tables Icon

Table 1 Four Zernike Modes to be Coded for the Investigation of Carrier Frequency Distribution

Tables Icon

Table 2 Retrieving the Amplitudes of Coded Zernike Modes by Inputting Different Amplitudes of Single Mode Z 2 0 by Using an MPCGH Coded with the Circle Carrier Frequency Approach Shown in Fig. 3a

Equations (12)

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t N ( ρ , θ ) = exp [ j m N k cos ( α k φ k ( ρ , θ ) + τ k ( ρ , θ ) ) ] ,
exp ( j z cos ϕ ) = J 0 ( z ) + 2 s = 1 j s J s ( z ) cos s ϕ ,
k { J 0 ( m N ) + s = 1 j s J s ( m N ) ( exp [ j s ( α k φ k + τ k ) ] + exp [ j s ( α k φ k + τ k ) ] ) } ,
U d = FT { U } * ( k ( * ) [ J 0 ( m N ) δ ( f ) + s = 1 j s J s ( m N ) ( V s , k + + V s , k ) ] ) ,
V s , k ± = FT { exp ( ± j s α k φ k ) } * δ ( f s τ ˜ k ) ,
U d 1 = [ J 0 ( m 2 ) ] 2 Y zero + j J 0 ( m 2 ) J 1 ( m 2 ) Y linear [ J 1 ( m 2 ) ] 2 Y cross ,
Y zero = FT { U } δ ( f ) = FT { exp ( j β q φ q ) } δ ( f ) ,
Y linear = FT { exp [ j ( α 1 φ 1 + β q φ q ) ] } * δ ( f τ ˜ 1 ) + FT { exp [ j ( α 1 φ 1 β q φ q ) ] } * δ ( f + τ ˜ 1 ) + FT { exp [ j ( α 2 φ 2 + β q φ q ) ] } * δ ( f τ ˜ 2 ) + FT { exp [ j ( α 2 φ 2 β q φ q ) ] } * δ ( f + τ ˜ 2 ) ,
Y cross = FT { exp [ j ( α 1 φ 1 + α 2 φ 2 + β q φ q ) ] } * δ [ f ( τ ˜ 1 + τ ˜ 2 ) ] + FT { exp [ j ( α 1 φ 1 + α 2 φ 2 β q φ q ) ] } * δ [ f + ( τ ˜ 1 + τ ˜ 2 ) ] + FT { exp [ j ( α 1 φ 1 α 2 φ 2 β q φ q ) ] } * δ [ f + ( τ ˜ 1 τ ˜ 2 ) ] + FT { exp [ j ( α 1 φ 1 α 2 φ 2 + β q φ q ) ] } * δ [ f ( τ ˜ 1 τ ˜ 2 ) ] .
U 1 ± = FT { exp [ j ( β q φ q ± α 1 φ 1 ) ] } * δ ( f τ ˜ 1 ) .
I 1 ± ( v ) = | Ω exp [ j ( β q φ q ± α 1 φ 1 ) ] exp [ j 2 π λ f 0 ( v λ f 0 τ ˜ 1 ) · r ] d Ω | 2 ,
I 1 ± ( ± λ f 0 τ ˜ 1 ) = | Ω exp [ j ( β q φ q ± α 1 φ 1 ) ] d Ω | 2 .

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