Abstract

We present a local Gaussian beam decomposition method for calculating the scalar diffraction field due to a two-dimensional field specified on a curved surface. We write the three-dimensional field as a sum of Gaussian beams that propagate toward different directions and whose waist positions are taken at discrete points on the curved surface. The discrete positions of the beam waists are obtained by sampling the curved surface such that transversal components of the positions form a regular grid. The modulated Gaussian window functions corresponding to Gaussian beams are placed on the transversal planes that pass through the discrete beam-waist position. The coefficients of the Gaussian beams are found by solving the linear system of equations where the columns of the system matrix represent the field patterns that the Gaussian beams produce on the given curved surface. As a result of using local beams in the expansion, we end up with sparse system matrices. The sparsity of the system matrices provides important advantages in terms of computational complexity and memory allocation while solving the system of linear equations.

© 2013 Optical Society of America

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  21. E. Ulusoy, L. Onural, and H. M. Ozaktas, “Synthesis of three-dimensional light fields with binary spatial light modulators,” J. Opt. Soc. Am. A 28, 1211–1223 (2011).
    [CrossRef]
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2012 (1)

2011 (4)

L. Onural, “Exact solution for scalar diffraction between tilted and translated planes using impulse functions over a surface,” J. Opt. Soc. Am. A 28, 290–295 (2011).
[CrossRef]

E. Ulusoy, L. Onural, and H. M. Ozaktas, “Synthesis of three-dimensional light fields with binary spatial light modulators,” J. Opt. Soc. Am. A 28, 1211–1223 (2011).
[CrossRef]

G. B. Esmer, L. Onural, and H. M. Ozaktas, “Exact diffraction calculation from fields specified over arbitrary curved surfaces,” Opt. Commun. 284, 5537–5548 (2011).
[CrossRef]

T. A. Davis, “Algorithm 915, SuiteSparseQR: multifrontal multithreaded rank-revealing sparse QR factorization,” ACM Trans. Math. Softw. 38, 1–22 (2011).
[CrossRef]

2008 (1)

2005 (1)

2000 (1)

1994 (1)

1992 (2)

J. R. Gilbert, C. Moler, and R. Schreiber, “Sparse matrices in MATLAB: design and implementation,” SIAM J. Matrix Anal. Appl. 13, 333–356 (1992).
[CrossRef]

M. Lucente, “Optimization of hologram computation for real-time display,” Proc. SPIE 1667, 32–43 (1992).
[CrossRef]

1981 (1)

A. J. E. M. Janssen, “Gabor representation of generalized functions,” J. Math. Anal. Appl. 83, 377–394 (1981).
[CrossRef]

1977 (1)

I. S. Duff, “A survey of sparse matrix research,” Proc. IEEE 65, 500–535 (1977).
[CrossRef]

1976 (1)

1966 (1)

J. P. Waters, “Holographic image synthesis utilizing theoretical methods,” Appl. Phys. Lett. 9, 405–407 (1966).
[CrossRef]

1946 (1)

D. Gabor, “Theory of communication,” J. Inst. Electr. Eng. 93, 429–457 (1946).

Ahrenberg, L.

L. Ahrenberg, “Methods for transform, analysis and rendering of complete light representations,” Ph.D. thesis (Max-Planck-Institut für Informatik, 2010).

Bastiaans, M. J.

M. J. Bastiaans, “Gabor’s signal expansion and the Zak transform,” Appl. Opt. 33, 5241–5255 (1994).
[CrossRef]

M. J. Bastiaans, “Oversampling in Gabor’s signal expansion by an integer factor,” in International Symposium on Time-Frequency and Time-Scale Analysis (IEEE, 1994), pp. 280–283.

Davis, T. A.

T. A. Davis, “Algorithm 915, SuiteSparseQR: multifrontal multithreaded rank-revealing sparse QR factorization,” ACM Trans. Math. Softw. 38, 1–22 (2011).
[CrossRef]

Duff, I. S.

I. S. Duff, “A survey of sparse matrix research,” Proc. IEEE 65, 500–535 (1977).
[CrossRef]

Esmer, G. B.

G. B. Esmer, L. Onural, and H. M. Ozaktas, “Exact diffraction calculation from fields specified over arbitrary curved surfaces,” Opt. Commun. 284, 5537–5548 (2011).
[CrossRef]

G. B. Esmer, “Calculation of scalar optical diffraction field from its distributed samples over the space,” Ph.D. thesis (Bilkent University, 2010).

Flandrin, P.

P. Flandrin, Time-Frequency/Time-Scale Analysis (Academic, 1999).

Gabor, D.

D. Gabor, “Theory of communication,” J. Inst. Electr. Eng. 93, 429–457 (1946).

Gilbert, J. R.

J. R. Gilbert, C. Moler, and R. Schreiber, “Sparse matrices in MATLAB: design and implementation,” SIAM J. Matrix Anal. Appl. 13, 333–356 (1992).
[CrossRef]

Goodman, J. W.

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

Hanák, I.

Janda, M.

Janssen, A. J. E. M.

A. J. E. M. Janssen, “Gabor representation of generalized functions,” J. Math. Anal. Appl. 83, 377–394 (1981).
[CrossRef]

Lucente, M.

M. Lucente, “Optimization of hologram computation for real-time display,” Proc. SPIE 1667, 32–43 (1992).
[CrossRef]

Matsushima, K.

Moler, C.

J. R. Gilbert, C. Moler, and R. Schreiber, “Sparse matrices in MATLAB: design and implementation,” SIAM J. Matrix Anal. Appl. 13, 333–356 (1992).
[CrossRef]

Onural, L.

Ozaktas, H. M.

E. Ulusoy, L. Onural, and H. M. Ozaktas, “Synthesis of three-dimensional light fields with binary spatial light modulators,” J. Opt. Soc. Am. A 28, 1211–1223 (2011).
[CrossRef]

G. B. Esmer, L. Onural, and H. M. Ozaktas, “Exact diffraction calculation from fields specified over arbitrary curved surfaces,” Opt. Commun. 284, 5537–5548 (2011).
[CrossRef]

Sahin, E.

Schreiber, R.

J. R. Gilbert, C. Moler, and R. Schreiber, “Sparse matrices in MATLAB: design and implementation,” SIAM J. Matrix Anal. Appl. 13, 333–356 (1992).
[CrossRef]

Stewart, G. W.

G. W. Stewart, Matrix Algorithms (Society for Industrial and Applied Mathematics, 1998).

Takai, M.

Ulusoy, E.

Waters, J. P.

J. P. Waters, “Holographic image synthesis utilizing theoretical methods,” Appl. Phys. Lett. 9, 405–407 (1966).
[CrossRef]

Yatagai, T.

ACM Trans. Math. Softw. (1)

T. A. Davis, “Algorithm 915, SuiteSparseQR: multifrontal multithreaded rank-revealing sparse QR factorization,” ACM Trans. Math. Softw. 38, 1–22 (2011).
[CrossRef]

Appl. Opt. (4)

Appl. Phys. Lett. (1)

J. P. Waters, “Holographic image synthesis utilizing theoretical methods,” Appl. Phys. Lett. 9, 405–407 (1966).
[CrossRef]

J. Inst. Electr. Eng. (1)

D. Gabor, “Theory of communication,” J. Inst. Electr. Eng. 93, 429–457 (1946).

J. Math. Anal. Appl. (1)

A. J. E. M. Janssen, “Gabor representation of generalized functions,” J. Math. Anal. Appl. 83, 377–394 (1981).
[CrossRef]

J. Opt. Soc. Am. A (4)

Opt. Commun. (1)

G. B. Esmer, L. Onural, and H. M. Ozaktas, “Exact diffraction calculation from fields specified over arbitrary curved surfaces,” Opt. Commun. 284, 5537–5548 (2011).
[CrossRef]

Proc. IEEE (1)

I. S. Duff, “A survey of sparse matrix research,” Proc. IEEE 65, 500–535 (1977).
[CrossRef]

Proc. SPIE (1)

M. Lucente, “Optimization of hologram computation for real-time display,” Proc. SPIE 1667, 32–43 (1992).
[CrossRef]

SIAM J. Matrix Anal. Appl. (1)

J. R. Gilbert, C. Moler, and R. Schreiber, “Sparse matrices in MATLAB: design and implementation,” SIAM J. Matrix Anal. Appl. 13, 333–356 (1992).
[CrossRef]

Other (6)

G. W. Stewart, Matrix Algorithms (Society for Industrial and Applied Mathematics, 1998).

G. B. Esmer, “Calculation of scalar optical diffraction field from its distributed samples over the space,” Ph.D. thesis (Bilkent University, 2010).

M. J. Bastiaans, “Oversampling in Gabor’s signal expansion by an integer factor,” in International Symposium on Time-Frequency and Time-Scale Analysis (IEEE, 1994), pp. 280–283.

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

L. Ahrenberg, “Methods for transform, analysis and rendering of complete light representations,” Ph.D. thesis (Max-Planck-Institut für Informatik, 2010).

P. Flandrin, Time-Frequency/Time-Scale Analysis (Academic, 1999).

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