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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References

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  1. J. P. Waters, “Holographic image synthesis utilizing theoretical methods,” Appl. Phys. Lett. 9, 405–407 (1966).
    [CrossRef]
  2. T. Yatagai, “Stereoscopic approach to 3-d display using computer-generated holograms,” Appl. Opt. 15, 2722–2729 (1976).
    [CrossRef]
  3. K. Matsushima and M. Takai, “Recurrence formulas for fast creation of synthetic three-dimensional holograms,” Appl. Opt. 39, 6587–6594 (2000).
    [CrossRef]
  4. M. Lucente, “Optimization of hologram computation for real-time display,” Proc. SPIE 1667, 32–43 (1992).
    [CrossRef]
  5. K. Matsushima, “Computer-generated holograms for three-dimensional surface objects with shade and texture,” Appl. Opt. 44, 4607–4614 (2005).
    [CrossRef]
  6. M. Janda, I. Hanák, and L. Onural, “Hologram synthesis for photorealistic reconstruction,” J. Opt. Soc. Am. A 25, 3083–3096 (2008).
    [CrossRef]
  7. L. Ahrenberg, “Methods for transform, analysis and rendering of complete light representations,” Ph.D. thesis (Max-Planck-Institut für Informatik, 2010).
  8. 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]
  9. G. B. Esmer, “Calculation of scalar optical diffraction field from its distributed samples over the space,” Ph.D. thesis (Bilkent University, 2010).
  10. 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]
  11. J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996).
  12. 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]
  13. P. Flandrin, Time-Frequency/Time-Scale Analysis (Academic, 1999).
  14. D. Gabor, “Theory of communication,” J. Inst. Electr. Eng. 93, 429–457 (1946).
  15. M. J. Bastiaans, “Gabor’s signal expansion and the Zak transform,” Appl. Opt. 33, 5241–5255 (1994).
    [CrossRef]
  16. 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.
  17. A. J. E. M. Janssen, “Gabor representation of generalized functions,” J. Math. Anal. Appl. 83, 377–394 (1981).
    [CrossRef]
  18. T. A. Davis, “Algorithm 915, SuiteSparseQR: multifrontal multithreaded rank-revealing sparse QR factorization,” ACM Trans. Math. Softw. 38, 1–22 (2011).
    [CrossRef]
  19. I. S. Duff, “A survey of sparse matrix research,” Proc. IEEE 65, 500–535 (1977).
    [CrossRef]
  20. 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]
  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]
  22. G. W. Stewart, Matrix Algorithms (Society for Industrial and Applied Mathematics, 1998).

2012 (1)

2011 (4)

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]

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]

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

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.

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]

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]

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)

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.

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

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

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

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

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

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Figures (12)

Fig. 1.
Fig. 1.

Setup of the proposed Gaussian beam decomposition method (a 2D pattern is shown for the sake of simplicity).

Fig. 2.
Fig. 2.

Periodic 2D simulation setup.

Fig. 3.
Fig. 3.

Ratios of the complexities of the operations QRPWD, SQRPWD, and QRGBD to the complexity of SQRGBD (note that the result shown for SQRGBD is “1”).

Fig. 4.
Fig. 4.

Sparsity of the system matrices for the proposed Gaussian beam decomposition method.

Fig. 5.
Fig. 5.

Absolute value of the system matrix G, which has size 2400×2000 and is formed by the application of the proposed Gaussian beam decomposition method to the curved line shown in Fig. 2.

Fig. 6.
Fig. 6.

Real part of the system matrix G, which has size 2400×2000 and is formed by the application of the PWD method to the curved line shown in Fig. 2. (Note that the absolute values of all matrix entries are 1.) Note the difference with respect to Fig. 5 in terms of sparsity.

Fig. 7.
Fig. 7.

One period of the 2D periodic curved surface.

Fig. 8.
Fig. 8.

One period of the 2D periodic object. (Gilles Tran © 2007 www.oyonale.com, used under the Creative Commons Attribution license.)

Fig. 9.
Fig. 9.

Absolute value of the system matrix G, which has size 11664×9472 and is formed by the application of the proposed Gaussian beam decomposition method for the curved surface given in Fig. 7.

Fig. 10.
Fig. 10.

Real part of the system matrix G, which has size 11664×9475 and is formed by the application of the PWD method for the curved surface given in Fig. 7. (The absolute values of all matrix entries are 1.) Note the random noiselike appearance, which is a consequence of the structure of the surface given in Fig. 7.

Fig. 11.
Fig. 11.

Absolute value of the system matrix G, which has size 3973×2809 and is formed by the application of the proposed Gaussian beam decomposition method for the object given in Fig. 8.

Fig. 12.
Fig. 12.

Real part of the system matrix G, which has size 3973×2819 and is formed by the application of the PWD method for the object given in Fig. 8. (The absolute values of all matrix entries are 1.)

Equations (25)

Equations on this page are rendered with MathJax. Learn more.

u(x)=BA(fx,fy)exp{j2π(fxTx)}dfxdfy,
A(fx,fy)=u0(x,y)exp{j2π(fxx+fyy)}dxdy,
P(fx,fy)=p(x,y)exp{j2π(fxx+fyy)}dxdy.
u0(x,y)=a(ξ,η,fx,fy)g(xξ,yη)×exp{j2π(fxx+fyy)}dξdηdfxdfy,
a(ξ,η,fx,fy)=u0(x,y)g*(xξ,yη)exp{j2π(fxx+fyy)}dxdy.
u0(x,y)=mnklamnklg(xmX,ynY)exp{j2π(kFxx+lFyy)},
amnkl=u0(x,y)w*(xmX,ynY)exp{j2π(kFxx+lFyy)}dxdy.
u(x)=mnklamnklgmnkl(xpmn).
u(r)=BA(fx,fy)exp{j2πfxTr}dfxdfy,
u(r)=mnklamnklgmnkl(rpmn),
u(x)=mnklamnklgmnkl(xsmn),
u(r)=mnklamnklgmnkl(rsmn).
u(x)=mnkla^mnklgmnkl(xsmn),
a^mnkl=argminamnkl{S|u(r)mnklamnklgmnkl(rsmn)|2dS}.
u(x)=mkamkgmk(xsm),
u(r)=mkamkgmk(rsm),
gmk(xsm)=1Nl=N2N21G^mk(lXp)exp{j2π[lXpx+1λ2l2Xp2(zζm)]},
G^mk(fx)=F{g(xmX)exp(j2πkFxx)}=cπσexp{π2σ2(fxkFx)2}×exp{j2π(fxkFx)mX}.
u(rn)=m=M2M21k=K2K21amkgmk(rnsm),
U=Ga,
G=[g11(rs11)|g12(rs12)||gMK(rsMK)]
a=[a11,a12,,aMK]T.
a^=[a^11,a^12,,a^MK]T=argmina{n=N2N21|u(rn)mkamkgmk(rnsm)|2},
u(x)=m=M2M21k=K2K21a^mkgmk(xsm).
u(x,z)=1Nk=N2N21A^kexp{j2π(kXpx+1λ2k2Xp2z)},

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