Abstract

A fast calculation method for computer generation of spherical holograms in proposed. This method is based on wave propagation defined in spectral domain and in spherical coordinates. The spherical wave spectrum and transfer function were derived from boundary value solutions to the scalar wave equation. It is a spectral propagation formula analogous to angular spectrum formula in cartesian coordinates. A numerical method to evaluate the derived formula is suggested, which uses only N(logN)2 operations for calculations on N sampling points. Simulation results are presented to verify the correctness of the proposed method. A spherical hologram for a spherical object was generated and reconstructed successfully using the proposed method.

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  1. C. Gmachl, F. Capasso, E. E. Narimanov, J. U. Nockel, A. D. Stone, J. Faist, D. L. Sivco, and A. Y. Cho, “High power directional emission from microlasers with chaotic resonators,” Science280, 1544–1545 (1998).
    [CrossRef]
  2. N. M. Lawandy and R. M. Balachandran “Random laser?,” Nature373, 204, (1995).
    [CrossRef]
  3. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968).
  4. T. Tommasi and B. Bianco, “Computer-generated holograms of tilted planes by a spatial frequency approach,” J. Opt. Soc. Am. A10, 299–305 (1993), http://www.opticsinfobase.org/josaa/abstract.cfm?URI=josaa-10-2-299 .
    [CrossRef]
  5. Y. Sakamoto and M. Tobise, “Computer generated cylindrical hologram,” in Practical Holography XIX: Materials and ApplicationsTung H. Jeong and Hans I. Bjelkhagen, eds., Proc. SPIE5742, 267–274 (2005).
  6. A. Kashiwagi and Y. Sakamoto, “A fast calculation method of cylindrical computer-generated holograms which perform image-reconstruction of volume data,” in Adaptive Optics: Analysis and Methods/Computational Optical Sensing and Imaging/Information Photonics/Signal Recovery and Synthesis Topical Meetings on CD-ROM OSA Technical Digest (CD) (Optical Society of America, 2007), paper DWB7, http://www.opticsinfobase.org/abstract.cfm?URI=DH-2007-DWB7 .
  7. T. Yamaguchi, T. Fujii, and H. Yoshikawa, “Fast calculation method for computer-generated cylindrical holograms,” Appl. Opt.47, D63–D70 (2008), http://www.opticsinfobase.org/ao/abstract.cfm?URI=ao-47-19-D63 .
    [CrossRef] [PubMed]
  8. Y. Sando, M. Itoh, and T. Yatagai, “Fast calculation method for cylindrical computer-generated holograms,” Opt.Express13, 1418–1423 (2005), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-13-5-1418 .
    [CrossRef] [PubMed]
  9. B. J. Jackin and T. Yatagai, “Fast calculation method for computer-generated cylindrical hologram based on wave propagation in spectral domain,” Opt.Express18, 25546–25555 (2010), http://www.opticsinfobase.org/oe/abstract.cfm?uri=oe-18-25-25546 .
    [CrossRef] [PubMed]
  10. B. J. Jackin and T. Yatagai, “360° reconstruction of a 3D object using cylindrical computer generated holography,” Appl. Opt.50, H147–H152 (2011), http://www.opticsinfobase.org/ao/abstract.cfm?URI=ao-50-34-H147 .
    [CrossRef] [PubMed]
  11. M. L. Tachiki, Y. Sando, M. Itoh, and T. Yatagai, “Fast calculation method for spherical computer-generated holograms,” Appl. Opt.45, 3527–3533 (2006), http://www.opticsinfobase.org/ao/abstract.cfm?URI=ao-45-15-3527 .
    [CrossRef] [PubMed]
  12. R. J. Chien and B. K. Alpert, “A fast spherical filter with uniform resolution,” J. Comput. Phys.136, 580–584 (1997).
    [CrossRef]
  13. J. R. Driscoll and D. M Healy, “Computing Fourier transfroms and convolutions on the sphere,” Adv. Appl. Math.15, 201–250 (1994).
    [CrossRef]
  14. N. N. Lebedev, Special Functions and their Applications (Prentice Hall, 1965).
  15. G. B. Arfken and H. J. Weber, Mathematical Method for Physicist (Academic Press, 2001).
  16. D. M. Healy, D. Rockmore, P. J. Kostelec, and S. Moore, “FFTs for the 2-sphere - improvements and variations,” J. Fourier. Anal. Appl.9, No.4, 341–385(1998).
    [CrossRef]
  17. P. N. Swarztrauber, “On computing the points and weights for Gauss-Legendre quadrature,” SIAM. J. Sci. Computing24. Issue. 3, 945–954(2002).
    [CrossRef]
  18. M. Wieczorek, SHTools, http://shtools.ipgp.fr/ .
  19. Z. G. Horvath, “Beyond the beam: A history of multidimensional lasers,” Opt. Photonics News23. No. 7/8, 36–41(2012).
    [CrossRef]

2012

Z. G. Horvath, “Beyond the beam: A history of multidimensional lasers,” Opt. Photonics News23. No. 7/8, 36–41(2012).
[CrossRef]

2011

2010

B. J. Jackin and T. Yatagai, “Fast calculation method for computer-generated cylindrical hologram based on wave propagation in spectral domain,” Opt.Express18, 25546–25555 (2010), http://www.opticsinfobase.org/oe/abstract.cfm?uri=oe-18-25-25546 .
[CrossRef] [PubMed]

2008

2006

2005

Y. Sando, M. Itoh, and T. Yatagai, “Fast calculation method for cylindrical computer-generated holograms,” Opt.Express13, 1418–1423 (2005), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-13-5-1418 .
[CrossRef] [PubMed]

Y. Sakamoto and M. Tobise, “Computer generated cylindrical hologram,” in Practical Holography XIX: Materials and ApplicationsTung H. Jeong and Hans I. Bjelkhagen, eds., Proc. SPIE5742, 267–274 (2005).

2002

P. N. Swarztrauber, “On computing the points and weights for Gauss-Legendre quadrature,” SIAM. J. Sci. Computing24. Issue. 3, 945–954(2002).
[CrossRef]

1998

D. M. Healy, D. Rockmore, P. J. Kostelec, and S. Moore, “FFTs for the 2-sphere - improvements and variations,” J. Fourier. Anal. Appl.9, No.4, 341–385(1998).
[CrossRef]

C. Gmachl, F. Capasso, E. E. Narimanov, J. U. Nockel, A. D. Stone, J. Faist, D. L. Sivco, and A. Y. Cho, “High power directional emission from microlasers with chaotic resonators,” Science280, 1544–1545 (1998).
[CrossRef]

1997

R. J. Chien and B. K. Alpert, “A fast spherical filter with uniform resolution,” J. Comput. Phys.136, 580–584 (1997).
[CrossRef]

1995

N. M. Lawandy and R. M. Balachandran “Random laser?,” Nature373, 204, (1995).
[CrossRef]

1994

J. R. Driscoll and D. M Healy, “Computing Fourier transfroms and convolutions on the sphere,” Adv. Appl. Math.15, 201–250 (1994).
[CrossRef]

1993

Alpert, B. K.

R. J. Chien and B. K. Alpert, “A fast spherical filter with uniform resolution,” J. Comput. Phys.136, 580–584 (1997).
[CrossRef]

Arfken, G. B.

G. B. Arfken and H. J. Weber, Mathematical Method for Physicist (Academic Press, 2001).

Balachandran, R. M.

N. M. Lawandy and R. M. Balachandran “Random laser?,” Nature373, 204, (1995).
[CrossRef]

Bianco, B.

Capasso, F.

C. Gmachl, F. Capasso, E. E. Narimanov, J. U. Nockel, A. D. Stone, J. Faist, D. L. Sivco, and A. Y. Cho, “High power directional emission from microlasers with chaotic resonators,” Science280, 1544–1545 (1998).
[CrossRef]

Chien, R. J.

R. J. Chien and B. K. Alpert, “A fast spherical filter with uniform resolution,” J. Comput. Phys.136, 580–584 (1997).
[CrossRef]

Cho, A. Y.

C. Gmachl, F. Capasso, E. E. Narimanov, J. U. Nockel, A. D. Stone, J. Faist, D. L. Sivco, and A. Y. Cho, “High power directional emission from microlasers with chaotic resonators,” Science280, 1544–1545 (1998).
[CrossRef]

Driscoll, J. R.

J. R. Driscoll and D. M Healy, “Computing Fourier transfroms and convolutions on the sphere,” Adv. Appl. Math.15, 201–250 (1994).
[CrossRef]

Faist, J.

C. Gmachl, F. Capasso, E. E. Narimanov, J. U. Nockel, A. D. Stone, J. Faist, D. L. Sivco, and A. Y. Cho, “High power directional emission from microlasers with chaotic resonators,” Science280, 1544–1545 (1998).
[CrossRef]

Fujii, T.

Gmachl, C.

C. Gmachl, F. Capasso, E. E. Narimanov, J. U. Nockel, A. D. Stone, J. Faist, D. L. Sivco, and A. Y. Cho, “High power directional emission from microlasers with chaotic resonators,” Science280, 1544–1545 (1998).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968).

Healy, D. M

J. R. Driscoll and D. M Healy, “Computing Fourier transfroms and convolutions on the sphere,” Adv. Appl. Math.15, 201–250 (1994).
[CrossRef]

Healy, D. M.

D. M. Healy, D. Rockmore, P. J. Kostelec, and S. Moore, “FFTs for the 2-sphere - improvements and variations,” J. Fourier. Anal. Appl.9, No.4, 341–385(1998).
[CrossRef]

Horvath, Z. G.

Z. G. Horvath, “Beyond the beam: A history of multidimensional lasers,” Opt. Photonics News23. No. 7/8, 36–41(2012).
[CrossRef]

Itoh, M.

M. L. Tachiki, Y. Sando, M. Itoh, and T. Yatagai, “Fast calculation method for spherical computer-generated holograms,” Appl. Opt.45, 3527–3533 (2006), http://www.opticsinfobase.org/ao/abstract.cfm?URI=ao-45-15-3527 .
[CrossRef] [PubMed]

Y. Sando, M. Itoh, and T. Yatagai, “Fast calculation method for cylindrical computer-generated holograms,” Opt.Express13, 1418–1423 (2005), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-13-5-1418 .
[CrossRef] [PubMed]

Jackin, B. J.

B. J. Jackin and T. Yatagai, “360° reconstruction of a 3D object using cylindrical computer generated holography,” Appl. Opt.50, H147–H152 (2011), http://www.opticsinfobase.org/ao/abstract.cfm?URI=ao-50-34-H147 .
[CrossRef] [PubMed]

B. J. Jackin and T. Yatagai, “Fast calculation method for computer-generated cylindrical hologram based on wave propagation in spectral domain,” Opt.Express18, 25546–25555 (2010), http://www.opticsinfobase.org/oe/abstract.cfm?uri=oe-18-25-25546 .
[CrossRef] [PubMed]

Kostelec, P. J.

D. M. Healy, D. Rockmore, P. J. Kostelec, and S. Moore, “FFTs for the 2-sphere - improvements and variations,” J. Fourier. Anal. Appl.9, No.4, 341–385(1998).
[CrossRef]

Lawandy, N. M.

N. M. Lawandy and R. M. Balachandran “Random laser?,” Nature373, 204, (1995).
[CrossRef]

Lebedev, N. N.

N. N. Lebedev, Special Functions and their Applications (Prentice Hall, 1965).

Moore, S.

D. M. Healy, D. Rockmore, P. J. Kostelec, and S. Moore, “FFTs for the 2-sphere - improvements and variations,” J. Fourier. Anal. Appl.9, No.4, 341–385(1998).
[CrossRef]

Narimanov, E. E.

C. Gmachl, F. Capasso, E. E. Narimanov, J. U. Nockel, A. D. Stone, J. Faist, D. L. Sivco, and A. Y. Cho, “High power directional emission from microlasers with chaotic resonators,” Science280, 1544–1545 (1998).
[CrossRef]

Nockel, J. U.

C. Gmachl, F. Capasso, E. E. Narimanov, J. U. Nockel, A. D. Stone, J. Faist, D. L. Sivco, and A. Y. Cho, “High power directional emission from microlasers with chaotic resonators,” Science280, 1544–1545 (1998).
[CrossRef]

Rockmore, D.

D. M. Healy, D. Rockmore, P. J. Kostelec, and S. Moore, “FFTs for the 2-sphere - improvements and variations,” J. Fourier. Anal. Appl.9, No.4, 341–385(1998).
[CrossRef]

Sakamoto, Y.

Y. Sakamoto and M. Tobise, “Computer generated cylindrical hologram,” in Practical Holography XIX: Materials and ApplicationsTung H. Jeong and Hans I. Bjelkhagen, eds., Proc. SPIE5742, 267–274 (2005).

Sando, Y.

M. L. Tachiki, Y. Sando, M. Itoh, and T. Yatagai, “Fast calculation method for spherical computer-generated holograms,” Appl. Opt.45, 3527–3533 (2006), http://www.opticsinfobase.org/ao/abstract.cfm?URI=ao-45-15-3527 .
[CrossRef] [PubMed]

Y. Sando, M. Itoh, and T. Yatagai, “Fast calculation method for cylindrical computer-generated holograms,” Opt.Express13, 1418–1423 (2005), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-13-5-1418 .
[CrossRef] [PubMed]

Sivco, D. L.

C. Gmachl, F. Capasso, E. E. Narimanov, J. U. Nockel, A. D. Stone, J. Faist, D. L. Sivco, and A. Y. Cho, “High power directional emission from microlasers with chaotic resonators,” Science280, 1544–1545 (1998).
[CrossRef]

Stone, A. D.

C. Gmachl, F. Capasso, E. E. Narimanov, J. U. Nockel, A. D. Stone, J. Faist, D. L. Sivco, and A. Y. Cho, “High power directional emission from microlasers with chaotic resonators,” Science280, 1544–1545 (1998).
[CrossRef]

Swarztrauber, P. N.

P. N. Swarztrauber, “On computing the points and weights for Gauss-Legendre quadrature,” SIAM. J. Sci. Computing24. Issue. 3, 945–954(2002).
[CrossRef]

Tachiki, M. L.

Tobise, M.

Y. Sakamoto and M. Tobise, “Computer generated cylindrical hologram,” in Practical Holography XIX: Materials and ApplicationsTung H. Jeong and Hans I. Bjelkhagen, eds., Proc. SPIE5742, 267–274 (2005).

Tommasi, T.

Weber, H. J.

G. B. Arfken and H. J. Weber, Mathematical Method for Physicist (Academic Press, 2001).

Yamaguchi, T.

Yatagai, T.

B. J. Jackin and T. Yatagai, “360° reconstruction of a 3D object using cylindrical computer generated holography,” Appl. Opt.50, H147–H152 (2011), http://www.opticsinfobase.org/ao/abstract.cfm?URI=ao-50-34-H147 .
[CrossRef] [PubMed]

B. J. Jackin and T. Yatagai, “Fast calculation method for computer-generated cylindrical hologram based on wave propagation in spectral domain,” Opt.Express18, 25546–25555 (2010), http://www.opticsinfobase.org/oe/abstract.cfm?uri=oe-18-25-25546 .
[CrossRef] [PubMed]

M. L. Tachiki, Y. Sando, M. Itoh, and T. Yatagai, “Fast calculation method for spherical computer-generated holograms,” Appl. Opt.45, 3527–3533 (2006), http://www.opticsinfobase.org/ao/abstract.cfm?URI=ao-45-15-3527 .
[CrossRef] [PubMed]

Y. Sando, M. Itoh, and T. Yatagai, “Fast calculation method for cylindrical computer-generated holograms,” Opt.Express13, 1418–1423 (2005), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-13-5-1418 .
[CrossRef] [PubMed]

Yoshikawa, H.

Adv. Appl. Math.

J. R. Driscoll and D. M Healy, “Computing Fourier transfroms and convolutions on the sphere,” Adv. Appl. Math.15, 201–250 (1994).
[CrossRef]

Appl. Opt.

J. Comput. Phys.

R. J. Chien and B. K. Alpert, “A fast spherical filter with uniform resolution,” J. Comput. Phys.136, 580–584 (1997).
[CrossRef]

J. Fourier. Anal. Appl.

D. M. Healy, D. Rockmore, P. J. Kostelec, and S. Moore, “FFTs for the 2-sphere - improvements and variations,” J. Fourier. Anal. Appl.9, No.4, 341–385(1998).
[CrossRef]

J. Opt. Soc. Am. A

Nature

N. M. Lawandy and R. M. Balachandran “Random laser?,” Nature373, 204, (1995).
[CrossRef]

Opt. Photonics News

Z. G. Horvath, “Beyond the beam: A history of multidimensional lasers,” Opt. Photonics News23. No. 7/8, 36–41(2012).
[CrossRef]

Opt.Express

Y. Sando, M. Itoh, and T. Yatagai, “Fast calculation method for cylindrical computer-generated holograms,” Opt.Express13, 1418–1423 (2005), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-13-5-1418 .
[CrossRef] [PubMed]

B. J. Jackin and T. Yatagai, “Fast calculation method for computer-generated cylindrical hologram based on wave propagation in spectral domain,” Opt.Express18, 25546–25555 (2010), http://www.opticsinfobase.org/oe/abstract.cfm?uri=oe-18-25-25546 .
[CrossRef] [PubMed]

Practical Holography XIX: Materials and Applications

Y. Sakamoto and M. Tobise, “Computer generated cylindrical hologram,” in Practical Holography XIX: Materials and ApplicationsTung H. Jeong and Hans I. Bjelkhagen, eds., Proc. SPIE5742, 267–274 (2005).

Science

C. Gmachl, F. Capasso, E. E. Narimanov, J. U. Nockel, A. D. Stone, J. Faist, D. L. Sivco, and A. Y. Cho, “High power directional emission from microlasers with chaotic resonators,” Science280, 1544–1545 (1998).
[CrossRef]

SIAM. J. Sci. Computing

P. N. Swarztrauber, “On computing the points and weights for Gauss-Legendre quadrature,” SIAM. J. Sci. Computing24. Issue. 3, 945–954(2002).
[CrossRef]

Other

M. Wieczorek, SHTools, http://shtools.ipgp.fr/ .

N. N. Lebedev, Special Functions and their Applications (Prentice Hall, 1965).

G. B. Arfken and H. J. Weber, Mathematical Method for Physicist (Academic Press, 2001).

A. Kashiwagi and Y. Sakamoto, “A fast calculation method of cylindrical computer-generated holograms which perform image-reconstruction of volume data,” in Adaptive Optics: Analysis and Methods/Computational Optical Sensing and Imaging/Information Photonics/Signal Recovery and Synthesis Topical Meetings on CD-ROM OSA Technical Digest (CD) (Optical Society of America, 2007), paper DWB7, http://www.opticsinfobase.org/abstract.cfm?URI=DH-2007-DWB7 .

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968).

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Figures (9)

Fig. 1
Fig. 1

Geometry of the problem.

Fig. 2
Fig. 2

Plot of phase(in radians) of the transfer function for increasing order (n).

Fig. 3
Fig. 3

Object.

Fig. 4
Fig. 4

Computed hologram(intensity) using a)proposed method and b)direct integration.

Fig. 5
Fig. 5

Computed hologram(intensity) for wavelength a) 150μm, b) 200μm, c) 250μm, d) 300μm, e) 350μm, f) 400μm.

Fig. 6
Fig. 6

Object with point sources at a)(θ = −π/6, θ = π/6), b)(θ = −π/8, θ = −π/8), c) (θ = −π/16, θ = π/16), d) (θ = −π/32, θ = π/32) and their corresponding hologram pattern(intensity).

Fig. 7
Fig. 7

Object.

Fig. 8
Fig. 8

Hologram(intensity).

Fig. 9
Fig. 9

Reconstruction.

Tables (1)

Tables Icon

Table 1 Comparision of calculation speed

Equations (38)

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

2 u ε μ 2 u t 2 = 0
2 = 1 r 2 r ( r 2 r ) + 1 r 2 sin θ θ ( sin θ θ ) + 1 r 2 sin 2 θ 2 ϕ 2
1 r 2 r ( r 2 u r ) + 1 r 2 sin θ θ ( sin θ u θ ) + 1 r 2 sin 2 θ 2 u ϕ 2 1 c 2 2 u t 2 = 0
u ( r , θ , ϕ , t ) = R ( r ) Θ ( θ ) Φ ( ϕ ) T ( t )
d 2 Φ d ϕ 2 + m 2 Φ = 0
1 sin θ d d θ ( sin θ d Θ d θ ) + [ n ( n + 1 ) m 2 sin 2 θ ] Θ = 0
1 r 2 d d r ( r 2 d R d r ) + k 2 R n ( n + 1 ) r 2 R = 0
1 c 2 d 2 T d t 2 + k 2 T = 0
Φ ( ϕ ) = Φ 1 e im ϕ + Φ 2 e im ϕ
Θ ( θ ) = Θ 1 P n m ( cos θ ) + Θ 2 Q n m ( cos θ )
R ( r ) = R 1 h n ( 1 ) ( k r ) + R 2 h n ( 2 ) ( k r )
T ( ω ) = T 1 e i ω t + T 2 e i ω t
Y n m ( θ , ϕ ) ( 1 ) m ( 2 n + 1 ) ( n m ) ! 4 π ( n + m ) ! P n m cos ( θ ) e im ϕ
P ¯ n m = ( 2 n + 1 ) ( n m ) ! 4 π ( n + m ) ! P n m ( cos θ )
Y n m ( θ , ϕ ) = P ¯ n m ( cos θ ) e im ϕ
u ( r , θ , ϕ , ω ) = n = 0 m = n n A m n ( ω ) h n ( k r ) Y n m ( θ , ϕ )
A m n = 1 h n ( k a ) π 2 π 2 0 2 π u ( a , θ , ϕ ) Y n m ( θ , ϕ ) * sin ( θ ) d θ d ϕ
u ( r , θ , ϕ ) = n = 0 m = n n Y n m ( θ , ϕ ) ( [ π 2 π 2 0 2 π u ( a , θ , ϕ ) Y n m ( θ , ϕ ) * d Ω ] h n ( k r ) h n ( k a ) )
u ( x , y , z ) = 1 4 π 2 e i ( k x x + k y y ) d k x d k y ( [ u ( x , y , 0 ) e i ( k x x + k y y ) d x d y ] e i k z z )
U m n ( a ) = u ( a , θ , ϕ ) Y n m ( θ , ϕ ) * d Ω
U m n ( r ) = h n ( k r ) h n ( k a ) U m n ( a )
u ( r , θ , ϕ ) = n = 0 m = n n U n m ( r ) Y n m ( θ , ϕ )
h n ( 1 ) ( x ) = ( i ) n + 1 e i x i x
u ( r , θ , ϕ ) = ISHT [ SHT ( u ( a , θ , ϕ ) ) × T F s ]
u ( x , y , z ) = IFFT [ FFT ( u ( x , y , 0 ) ) × T F c ]
U m n ( r ) = π / 2 π / 2 ( π π u ( r , θ , ϕ ) e im ϕ d ϕ ) P ¯ n m ( cos θ ) d θ
U m ( θ ) = π π u ( r , θ , ϕ ) e i m ϕ d θ
= 1 I i = 1 I u ( r , θ , ϕ i ) e im ϕ i
U n m = π / 2 π / 2 U m ( θ j ) P ¯ n m ( cos θ ) sin θ d θ
= j = | m | N U m ( θ j ) P ¯ n m ( cos θ j ) w j
u ( θ , ϕ ) = m = N N ( n = | m | N U n m P n m ( cos θ ) ) e im ϕ
U m ( θ ) = n = | m | N U n m P ¯ n m ( cos θ )
u ( θ , ϕ ) = m = N N U m ( θ ) e im ϕ
AmplitudeHologram = | ( ISHT [ SHT ( Object ) × T F ] ) + ISHT [ SHT ( Reference ) × T F ] | 2
H ( r , θ , ϕ ) = O ( θ , ϕ ) exp ( i k L ) L d x d y
L = r 2 + a 2 2 r a [ sin ( θ ) sin ( θ ) + cos ( θ ) cos ( θ ) cos ( ϕ ϕ ) ]
Hologram = | H object ( r , θ , ϕ ) + H reference ( r , θ , ϕ ) | 2
Reconstruction = | ISHT [ SHT ( Hologram × Conjugate [ Reference ] ) × T F ] | 2

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