Abstract

We propose an easy method for unknown wavefront reconstruction by a typical Fourier transform system based on an amplitude-only spatial light modulator (SLM). The Fourier transform of the diffraction intensity pattern on the backfocal plane is the autocorrelation of the wavefront-transmitted SLM. Two intensity patterns were recorded when we set the transmittance of one corner pixel on the SLM as 0 and 1, separately, under the condition that the transmittances of the other pixels equal 1. The complex amplitude of the wavefront-transmitted SLM can be extracted from the Fourier transform of the diffraction intensity pattern subtraction without complicated calculations. The simulation results certifying our theories are also given. Our method provides us with an easy way for detecting the wavefront with high speed and high quality in real time without the reference wave.

© 2010 Optical Society of America

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2010 (1)

2009 (2)

C.-S. Guo, L.-L. Lu, and H.-T. Wang, Opt. Lett. 34, 3686(2009).
[CrossRef] [PubMed]

C.-S. Guo, S.-J. Yue, and G.-X. Wei, Appl. Phys. Lett. 94, 231104 (2009).
[CrossRef]

2008 (2)

G. C. G. Berkhout and M. W. Beijersbergen, Phys. Rev. Lett. 101, 100801 (2008).
[CrossRef] [PubMed]

P. Thibault, M. Dierolf, A. Menzel, O. Bunk, C. David, and F. Pfeiffer, Science 321, 379 (2008).
[CrossRef] [PubMed]

2007 (3)

J. M. Rodenburg, A. C. Hurst, A. G. Cullis, B. R. Dobson, F. Pfeiffer, O. Bunk, C. David, K. Jefimovs, and I. Johnson, Phys. Rev. Lett. 98, 034801 (2007).
[CrossRef] [PubMed]

X. F. Xu, L. Z. Cai, Y. R. Wang, X. F. Meng, H. Zhang, G. Y. Dong, and X. X. Shen, Opt. Commun. 273, 54 (2007).
[CrossRef]

X. F. Xu, L. Z. Cai, Y. R. Wang, X. L. Yang, X. F. Meng, G. Y. Dong, X. X. Shen, and H. Zhang, Appl. Phys. Lett. 90, 121124-1 (2007).

2004 (4)

H. M. L. Faulkner and J. M. Rodenburg, Phys. Rev. Lett. 93, 023903 (2004).
[CrossRef] [PubMed]

S. Eisebitt, J. Lüning, W. F. Schlotter, M. Lörgen1, O. Hellwig, W. Eberhardt, and J. Stöhr, Nature 432, 885(2004).
[CrossRef] [PubMed]

L. Z. Cai, Q. Liu, and X. L. Yang, Opt. Lett. 29, 183 (2004).
[CrossRef] [PubMed]

M. Liebling, T. Blu, and M. Unser, J. Opt. Soc. Am. A 21, 367 (2004).
[CrossRef]

2002 (1)

2000 (1)

K. Buse and M. Luennemann, Phys. Rev. Lett. 85, 3385 (2000).
[CrossRef] [PubMed]

1997 (1)

Beijersbergen, M. W.

G. C. G. Berkhout and M. W. Beijersbergen, Phys. Rev. Lett. 101, 100801 (2008).
[CrossRef] [PubMed]

Berkhout, G. C. G.

G. C. G. Berkhout and M. W. Beijersbergen, Phys. Rev. Lett. 101, 100801 (2008).
[CrossRef] [PubMed]

Blu, T.

Bunk, O.

P. Thibault, M. Dierolf, A. Menzel, O. Bunk, C. David, and F. Pfeiffer, Science 321, 379 (2008).
[CrossRef] [PubMed]

J. M. Rodenburg, A. C. Hurst, A. G. Cullis, B. R. Dobson, F. Pfeiffer, O. Bunk, C. David, K. Jefimovs, and I. Johnson, Phys. Rev. Lett. 98, 034801 (2007).
[CrossRef] [PubMed]

Buse, K.

K. Buse and M. Luennemann, Phys. Rev. Lett. 85, 3385 (2000).
[CrossRef] [PubMed]

Cai, L. Z.

X. F. Xu, L. Z. Cai, Y. R. Wang, X. L. Yang, X. F. Meng, G. Y. Dong, X. X. Shen, and H. Zhang, Appl. Phys. Lett. 90, 121124-1 (2007).

X. F. Xu, L. Z. Cai, Y. R. Wang, X. F. Meng, H. Zhang, G. Y. Dong, and X. X. Shen, Opt. Commun. 273, 54 (2007).
[CrossRef]

L. Z. Cai, Q. Liu, and X. L. Yang, Opt. Lett. 29, 183 (2004).
[CrossRef] [PubMed]

Cullis, A. G.

J. M. Rodenburg, A. C. Hurst, A. G. Cullis, B. R. Dobson, F. Pfeiffer, O. Bunk, C. David, K. Jefimovs, and I. Johnson, Phys. Rev. Lett. 98, 034801 (2007).
[CrossRef] [PubMed]

David, C.

P. Thibault, M. Dierolf, A. Menzel, O. Bunk, C. David, and F. Pfeiffer, Science 321, 379 (2008).
[CrossRef] [PubMed]

J. M. Rodenburg, A. C. Hurst, A. G. Cullis, B. R. Dobson, F. Pfeiffer, O. Bunk, C. David, K. Jefimovs, and I. Johnson, Phys. Rev. Lett. 98, 034801 (2007).
[CrossRef] [PubMed]

Dierolf, M.

P. Thibault, M. Dierolf, A. Menzel, O. Bunk, C. David, and F. Pfeiffer, Science 321, 379 (2008).
[CrossRef] [PubMed]

Dobson, B. R.

J. M. Rodenburg, A. C. Hurst, A. G. Cullis, B. R. Dobson, F. Pfeiffer, O. Bunk, C. David, K. Jefimovs, and I. Johnson, Phys. Rev. Lett. 98, 034801 (2007).
[CrossRef] [PubMed]

Dong, G. Y.

X. F. Xu, L. Z. Cai, Y. R. Wang, X. F. Meng, H. Zhang, G. Y. Dong, and X. X. Shen, Opt. Commun. 273, 54 (2007).
[CrossRef]

X. F. Xu, L. Z. Cai, Y. R. Wang, X. L. Yang, X. F. Meng, G. Y. Dong, X. X. Shen, and H. Zhang, Appl. Phys. Lett. 90, 121124-1 (2007).

Eberhardt, W.

S. Eisebitt, J. Lüning, W. F. Schlotter, M. Lörgen1, O. Hellwig, W. Eberhardt, and J. Stöhr, Nature 432, 885(2004).
[CrossRef] [PubMed]

Eisebitt, S.

S. Eisebitt, J. Lüning, W. F. Schlotter, M. Lörgen1, O. Hellwig, W. Eberhardt, and J. Stöhr, Nature 432, 885(2004).
[CrossRef] [PubMed]

Faulkner, H. M. L.

H. M. L. Faulkner and J. M. Rodenburg, Phys. Rev. Lett. 93, 023903 (2004).
[CrossRef] [PubMed]

Guo, C. S.

Guo, C.-S.

Hellwig, O.

S. Eisebitt, J. Lüning, W. F. Schlotter, M. Lörgen1, O. Hellwig, W. Eberhardt, and J. Stöhr, Nature 432, 885(2004).
[CrossRef] [PubMed]

Hurst, A. C.

J. M. Rodenburg, A. C. Hurst, A. G. Cullis, B. R. Dobson, F. Pfeiffer, O. Bunk, C. David, K. Jefimovs, and I. Johnson, Phys. Rev. Lett. 98, 034801 (2007).
[CrossRef] [PubMed]

Jefimovs, K.

J. M. Rodenburg, A. C. Hurst, A. G. Cullis, B. R. Dobson, F. Pfeiffer, O. Bunk, C. David, K. Jefimovs, and I. Johnson, Phys. Rev. Lett. 98, 034801 (2007).
[CrossRef] [PubMed]

Johnson, I.

J. M. Rodenburg, A. C. Hurst, A. G. Cullis, B. R. Dobson, F. Pfeiffer, O. Bunk, C. David, K. Jefimovs, and I. Johnson, Phys. Rev. Lett. 98, 034801 (2007).
[CrossRef] [PubMed]

Liang, K.

Liao, J.

Liebling, M.

Liu, Q.

Lörgen1, M.

S. Eisebitt, J. Lüning, W. F. Schlotter, M. Lörgen1, O. Hellwig, W. Eberhardt, and J. Stöhr, Nature 432, 885(2004).
[CrossRef] [PubMed]

Lu, L.-L.

Luennemann, M.

K. Buse and M. Luennemann, Phys. Rev. Lett. 85, 3385 (2000).
[CrossRef] [PubMed]

Lüning, J.

S. Eisebitt, J. Lüning, W. F. Schlotter, M. Lörgen1, O. Hellwig, W. Eberhardt, and J. Stöhr, Nature 432, 885(2004).
[CrossRef] [PubMed]

Meng, X. F.

X. F. Xu, L. Z. Cai, Y. R. Wang, X. F. Meng, H. Zhang, G. Y. Dong, and X. X. Shen, Opt. Commun. 273, 54 (2007).
[CrossRef]

X. F. Xu, L. Z. Cai, Y. R. Wang, X. L. Yang, X. F. Meng, G. Y. Dong, X. X. Shen, and H. Zhang, Appl. Phys. Lett. 90, 121124-1 (2007).

Menzel, A.

P. Thibault, M. Dierolf, A. Menzel, O. Bunk, C. David, and F. Pfeiffer, Science 321, 379 (2008).
[CrossRef] [PubMed]

Pfeiffer, F.

P. Thibault, M. Dierolf, A. Menzel, O. Bunk, C. David, and F. Pfeiffer, Science 321, 379 (2008).
[CrossRef] [PubMed]

J. M. Rodenburg, A. C. Hurst, A. G. Cullis, B. R. Dobson, F. Pfeiffer, O. Bunk, C. David, K. Jefimovs, and I. Johnson, Phys. Rev. Lett. 98, 034801 (2007).
[CrossRef] [PubMed]

Rodenburg, J. M.

J. M. Rodenburg, A. C. Hurst, A. G. Cullis, B. R. Dobson, F. Pfeiffer, O. Bunk, C. David, K. Jefimovs, and I. Johnson, Phys. Rev. Lett. 98, 034801 (2007).
[CrossRef] [PubMed]

H. M. L. Faulkner and J. M. Rodenburg, Phys. Rev. Lett. 93, 023903 (2004).
[CrossRef] [PubMed]

Schlotter, W. F.

S. Eisebitt, J. Lüning, W. F. Schlotter, M. Lörgen1, O. Hellwig, W. Eberhardt, and J. Stöhr, Nature 432, 885(2004).
[CrossRef] [PubMed]

Shen, X. X.

X. F. Xu, L. Z. Cai, Y. R. Wang, X. F. Meng, H. Zhang, G. Y. Dong, and X. X. Shen, Opt. Commun. 273, 54 (2007).
[CrossRef]

X. F. Xu, L. Z. Cai, Y. R. Wang, X. L. Yang, X. F. Meng, G. Y. Dong, X. X. Shen, and H. Zhang, Appl. Phys. Lett. 90, 121124-1 (2007).

Stöhr, J.

S. Eisebitt, J. Lüning, W. F. Schlotter, M. Lörgen1, O. Hellwig, W. Eberhardt, and J. Stöhr, Nature 432, 885(2004).
[CrossRef] [PubMed]

Thibault, P.

P. Thibault, M. Dierolf, A. Menzel, O. Bunk, C. David, and F. Pfeiffer, Science 321, 379 (2008).
[CrossRef] [PubMed]

Unser, M.

Wang, H. T.

Wang, H.-T.

Wang, Y. R.

X. F. Xu, L. Z. Cai, Y. R. Wang, X. L. Yang, X. F. Meng, G. Y. Dong, X. X. Shen, and H. Zhang, Appl. Phys. Lett. 90, 121124-1 (2007).

X. F. Xu, L. Z. Cai, Y. R. Wang, X. F. Meng, H. Zhang, G. Y. Dong, and X. X. Shen, Opt. Commun. 273, 54 (2007).
[CrossRef]

Wei, G.-X.

C.-S. Guo, S.-J. Yue, and G.-X. Wei, Appl. Phys. Lett. 94, 231104 (2009).
[CrossRef]

Xu, X. F.

X. F. Xu, L. Z. Cai, Y. R. Wang, X. F. Meng, H. Zhang, G. Y. Dong, and X. X. Shen, Opt. Commun. 273, 54 (2007).
[CrossRef]

X. F. Xu, L. Z. Cai, Y. R. Wang, X. L. Yang, X. F. Meng, G. Y. Dong, X. X. Shen, and H. Zhang, Appl. Phys. Lett. 90, 121124-1 (2007).

Yamaguchi, I.

Yang, X. L.

X. F. Xu, L. Z. Cai, Y. R. Wang, X. L. Yang, X. F. Meng, G. Y. Dong, X. X. Shen, and H. Zhang, Appl. Phys. Lett. 90, 121124-1 (2007).

L. Z. Cai, Q. Liu, and X. L. Yang, Opt. Lett. 29, 183 (2004).
[CrossRef] [PubMed]

Yue, S.-J.

C.-S. Guo, S.-J. Yue, and G.-X. Wei, Appl. Phys. Lett. 94, 231104 (2009).
[CrossRef]

Zhang, H.

X. F. Xu, L. Z. Cai, Y. R. Wang, X. F. Meng, H. Zhang, G. Y. Dong, and X. X. Shen, Opt. Commun. 273, 54 (2007).
[CrossRef]

X. F. Xu, L. Z. Cai, Y. R. Wang, X. L. Yang, X. F. Meng, G. Y. Dong, X. X. Shen, and H. Zhang, Appl. Phys. Lett. 90, 121124-1 (2007).

Zhang, L.

Zhang, T.

Zhang, X.-T.

Zhu, Y. Y.

Appl. Phys. Lett. (2)

C.-S. Guo, S.-J. Yue, and G.-X. Wei, Appl. Phys. Lett. 94, 231104 (2009).
[CrossRef]

X. F. Xu, L. Z. Cai, Y. R. Wang, X. L. Yang, X. F. Meng, G. Y. Dong, X. X. Shen, and H. Zhang, Appl. Phys. Lett. 90, 121124-1 (2007).

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

Nature (1)

S. Eisebitt, J. Lüning, W. F. Schlotter, M. Lörgen1, O. Hellwig, W. Eberhardt, and J. Stöhr, Nature 432, 885(2004).
[CrossRef] [PubMed]

Opt. Commun. (1)

X. F. Xu, L. Z. Cai, Y. R. Wang, X. F. Meng, H. Zhang, G. Y. Dong, and X. X. Shen, Opt. Commun. 273, 54 (2007).
[CrossRef]

Opt. Lett. (5)

Phys. Rev. Lett. (4)

G. C. G. Berkhout and M. W. Beijersbergen, Phys. Rev. Lett. 101, 100801 (2008).
[CrossRef] [PubMed]

H. M. L. Faulkner and J. M. Rodenburg, Phys. Rev. Lett. 93, 023903 (2004).
[CrossRef] [PubMed]

J. M. Rodenburg, A. C. Hurst, A. G. Cullis, B. R. Dobson, F. Pfeiffer, O. Bunk, C. David, K. Jefimovs, and I. Johnson, Phys. Rev. Lett. 98, 034801 (2007).
[CrossRef] [PubMed]

K. Buse and M. Luennemann, Phys. Rev. Lett. 85, 3385 (2000).
[CrossRef] [PubMed]

Science (1)

P. Thibault, M. Dierolf, A. Menzel, O. Bunk, C. David, and F. Pfeiffer, Science 321, 379 (2008).
[CrossRef] [PubMed]

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Figures (4)

Fig. 1
Fig. 1

Typical Fourier transform system.

Fig. 2
Fig. 2

Ideal simulation results. (a) Amplitude distribution of the primary wavefront. (b) Phase distribution of the primary wavefront. (c) Recorded intensity pattern under the condition of the transmittance of upper-left corner pixel equal to 0. (d) Recorded intensity pattern under the condition of the transmittance of upper-left corner pixel equal to 1. (e) Amplitude distribution of the field in Eq. (7) by use of Figs. 2c, 2d. (f) Phase distribution of the field in Eq. (7) by use of Figs. 2c, 2d. (g) Amplitude distribution of the extracted wavefront. (h) Phase distribution of the extracted wavefront when the phase of upper-left corner pixel is equal to 0. (i) Amplitude differences between the primary and reconstructed wavefronts. (j) Phase differences between the primary and reconstructed wavefronts.

Fig. 3
Fig. 3

Schematic of the experiment setup for detecting the object before the SLM.

Fig. 4
Fig. 4

Simulation results of taking the grayscale of the CCD into account. (a) Original object. (b) Amplitude of reconstructed object under the condition of taking the upper-left corner pixel as the reference pixel. (c) Amplitude of reconstructed object under the condition of taking the upper-left corner 2 × 2 pixel array as the reference pixel.

Equations (7)

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χ ( μ , v ) = FT [ u ( x , y ) ] ,
I ( μ , v ) = χ ( μ , v ) · χ * ( μ , v ) ,
P ( x , y ) = FT [ I ( μ , v ) ] = FT [ χ ( μ , v ) ] FT [ χ * ( μ , v ) ] = u ( x , y ) u * ( x , y ) = u ( x , y ) u ( x , y ) ,
I 1 = | FT ( u 2 ) | 2 , I 2 = | FT ( u 1 + u 2 ) | 2 .
ϑ = FT ( I 2 I 1 ) = FT ( I 2 ) FT ( I 1 ) = u 1 ( x , y ) u 2 ( x , y ) + u 2 ( x , y ) u 1 ( x , y ) + u 1 ( x , y ) u 1 ( x , y ) ,
ϑ = a u 2 * ( x x 1 , y y 1 ) + u 2 ( x 1 x , y 1 y ) a * + | a | 2 δ ( 0 , 0 ) ,
ϑ = a u * ( x x 1 , y y 1 ) + u ( x 1 x , y 1 y ) a * | a | 2 δ ( 0 , 0 ) = a u * ( x x 1 , y y 1 ) + u 2 ( x 1 x , y 1 y ) a * = a u 2 * ( x x 1 , y y 1 ) + u ( x 1 x , y 1 y ) a * .

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