## Calculation of the scalar diffraction field from curved surfaces by decomposing the three-dimensional field into a sum of Gaussian beams |

JOSA A, Vol. 30, Issue 3, pp. 527-536 (2013)

http://dx.doi.org/10.1364/JOSAA.30.000527

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### 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

**OCIS Codes**

(090.0090) Holography : Holography

(090.1760) Holography : Computer holography

(090.1995) Holography : Digital holography

(070.7345) Fourier optics and signal processing : Wave propagation

**ToC Category:**

Fourier Optics and Signal Processing

**History**

Original Manuscript: December 19, 2012

Revised Manuscript: January 28, 2013

Manuscript Accepted: February 1, 2013

Published: February 28, 2013

**Citation**

Erdem Şahin and Levent Onural, "Calculation of the scalar diffraction field from curved surfaces by decomposing the three-dimensional field into a sum of Gaussian beams," J. Opt. Soc. Am. A **30**, 527-536 (2013)

http://www.opticsinfobase.org/josaa/abstract.cfm?URI=josaa-30-3-527

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### References

- J. P. Waters, “Holographic image synthesis utilizing theoretical methods,” Appl. Phys. Lett. 9, 405–407 (1966). [CrossRef]
- T. Yatagai, “Stereoscopic approach to 3-d display using computer-generated holograms,” Appl. Opt. 15, 2722–2729 (1976). [CrossRef]
- K. Matsushima and M. Takai, “Recurrence formulas for fast creation of synthetic three-dimensional holograms,” Appl. Opt. 39, 6587–6594 (2000). [CrossRef]
- M. Lucente, “Optimization of hologram computation for real-time display,” Proc. SPIE 1667, 32–43 (1992). [CrossRef]
- K. Matsushima, “Computer-generated holograms for three-dimensional surface objects with shade and texture,” Appl. Opt. 44, 4607–4614 (2005). [CrossRef]
- M. Janda, I. Hanák, and L. Onural, “Hologram synthesis for photorealistic reconstruction,” J. Opt. Soc. Am. A 25, 3083–3096 (2008). [CrossRef]
- L. Ahrenberg, “Methods for transform, analysis and rendering of complete light representations,” Ph.D. thesis (Max-Planck-Institut für Informatik, 2010).
- E. Şahin and L. Onural, “Scalar diffraction field calculation from curved surfaces via Gaussian beam decomposition,” J. Opt. Soc. Am. A 29, 1459–1469 (2012). [CrossRef]
- G. B. Esmer, “Calculation of scalar optical diffraction field from its distributed samples over the space,” Ph.D. thesis (Bilkent University, 2010).
- 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]
- J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996).
- 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]
- P. Flandrin, Time-Frequency/Time-Scale Analysis (Academic, 1999).
- D. Gabor, “Theory of communication,” J. Inst. Electr. Eng. 93, 429–457 (1946).
- 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.
- A. J. E. M. Janssen, “Gabor representation of generalized functions,” J. Math. Anal. Appl. 83, 377–394 (1981). [CrossRef]
- T. A. Davis, “Algorithm 915, SuiteSparseQR: multifrontal multithreaded rank-revealing sparse QR factorization,” ACM Trans. Math. Softw. 38, 1–22 (2011). [CrossRef]
- I. S. Duff, “A survey of sparse matrix research,” Proc. IEEE 65, 500–535 (1977). [CrossRef]
- 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]
- 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. W. Stewart, Matrix Algorithms (Society for Industrial and Applied Mathematics, 1998).

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