Abstract

The author derives Latta’s ray-tracing equations for holograms of arbitrary thickness [Appl. Opt. 10, 2698 (1971) ] from Welford’s vector ray-tracing equation for holograms of arbitrary shape [Opt. Commun. 14, 322 (1975) ]. The derivation follows Welford’s original approach but accounts for changes in the shape and thickness of the recording medium between construction and reconstruction.

© 2008 Optical Society of America

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References

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  1. W. T. Welford, “A vector raytracing equation for hologram lenses of arbitrary shape,” Opt. Commun. 14, 322-323 (1975).
    [CrossRef]
  2. J. L. Latta, “Computer-based analysis of holography using ray tracing,” Appl. Opt. 10, 2698-2710 (1971).
    [CrossRef] [PubMed]
  3. S. A. Benton and V. M. Bove, Jr., Holographic Imaging (Wiley, 2008), p. 52.
  4. H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909-2947 (1968).
  5. G. Montemezzani and M. Zgonik, “Light diffraction at mixed phase and absorption gratings in anisotropic media for arbitrary geometries,” Phys. Rev. E 35, 1035-1047 (1997).
    [CrossRef]
  6. J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996), pp. 333-336.
  7. A. Belendez, I. Pascual, and A. Fimia, “Effective holographic grating model to analyze thick holograms,” Proc. SPIE 1507, 268-276 (1991).
    [CrossRef]
  8. M. A. Murison and M. C. Noecker, “Ray tracing of and optical path across a holographic optical element,” SAO Technical Memorandum TM93-04 (Smithsonian Astrophysical Observatory, 1993).

1997

G. Montemezzani and M. Zgonik, “Light diffraction at mixed phase and absorption gratings in anisotropic media for arbitrary geometries,” Phys. Rev. E 35, 1035-1047 (1997).
[CrossRef]

1991

A. Belendez, I. Pascual, and A. Fimia, “Effective holographic grating model to analyze thick holograms,” Proc. SPIE 1507, 268-276 (1991).
[CrossRef]

1975

W. T. Welford, “A vector raytracing equation for hologram lenses of arbitrary shape,” Opt. Commun. 14, 322-323 (1975).
[CrossRef]

1971

1968

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909-2947 (1968).

Appl. Opt.

Bell Syst. Tech. J.

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909-2947 (1968).

Opt. Commun.

W. T. Welford, “A vector raytracing equation for hologram lenses of arbitrary shape,” Opt. Commun. 14, 322-323 (1975).
[CrossRef]

Phys. Rev. E

G. Montemezzani and M. Zgonik, “Light diffraction at mixed phase and absorption gratings in anisotropic media for arbitrary geometries,” Phys. Rev. E 35, 1035-1047 (1997).
[CrossRef]

Proc. SPIE

A. Belendez, I. Pascual, and A. Fimia, “Effective holographic grating model to analyze thick holograms,” Proc. SPIE 1507, 268-276 (1991).
[CrossRef]

Other

M. A. Murison and M. C. Noecker, “Ray tracing of and optical path across a holographic optical element,” SAO Technical Memorandum TM93-04 (Smithsonian Astrophysical Observatory, 1993).

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996), pp. 333-336.

S. A. Benton and V. M. Bove, Jr., Holographic Imaging (Wiley, 2008), p. 52.

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Figures (3)

Fig. 1
Fig. 1

Construction geometry: rays o ̂ and r ̂ emanating from source points O and R intersect at point P in the volume of a holographic recording medium. An interference fringe plane is formed along the bisector b ̂ to the two rays. The plane also contains a fringe vector f ̂ such that f ̂ , b ̂ , and ( o ̂ r ̂ ) are perpendicular to each other.

Fig. 2
Fig. 2

Constructive interference: a point P offset slightly from point P may be the site of constructive interference if the optical path difference is equal to the construction wavelength [see Eq. (7)].

Fig. 3
Fig. 3

Magnification: if the holographic recording medium shrinks or swells in one or more directions, the fringe vectors are magnified as well. Pictured here is a pair of fringe vectors before and after shrinkage of the recording medium. As the fringe vectors tilt ( f ̂ , f ̂ 2 to f ̂ , f ̂ 2 ), the interfringe distance along s shrinks to the interfringe distance along s . Although the postshrinkage thickness of the recording medium is not shown, the dashed line marks the effective thickness [7] between points on a fringe vector and corresponding points on the magnified fringe vector in the direction of the magnification scaling vector.

Equations (29)

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n ̂ × ( o ̂ r ̂ ) = m λ λ n ̂ × ( o ̂ r ̂ ) .
sin ( θ o ) sin ( θ r ) = m λ λ [ sin ( θ o ) sin ( θ r ) ] .
x o x r = m λ λ ( x o x r ) M x ,
y o y r = m λ λ ( y o y r ) M y ,
z o z r = m λ λ ( z o z r ) M z .
cos ( θ o ) cos ( θ r ) = m λ λ cos ( θ o ) cos ( θ r ) M z .
o ̂ r ̂ = m λ λ ( o ̂ r ̂ ) . M ,
f ̂ = b ̂ × s ̂ ,
D = ( OP RP ) ( OP RP ) = ( OP OP ) + ( RP RP ) p o ̂ p r ̂ p s .
p s λ .
p s = s p s ̂ = s σ ,
σ = λ s .
f = f ̂ . M ,
f ̂ = b ̂ × s ̂ ,
σ = m λ s .
s ̂ = s . M s . M .
σ = s = ( s . M ) s ̂ = ( s . M ) ( s . M ) s . M = s 2 s . M = s s ̂ . M = σ s ̂ . M .
b ̂ × s ̂ = f ̂ . M f ̂ . M .
b ̂ × s ̂ = s s ( b ̂ × s ̂ ) . M ( b ̂ × s ̂ ) . M ,
b ̂ × s ̂ = σ λ ( b ̂ × s ) . M ( b ̂ × s ̂ ) . M .
b ̂ × s = s σ λ ( b ̂ × s ) . M ( b ̂ × s ̂ ) . M ,
b ̂ × s = m λ σ λ σ ( b ̂ × s ) . M ( b ̂ × s ̂ ) . M .
b ̂ × s = m λ λ ( b ̂ × s ) . M ( b ̂ × s ̂ ) . M s ̂ . M .
s ̂ = s . M s . M .
s ̂ = 1 s s . M s ̂ . M
s ̂ = σ λ s . M s ̂ . M .
s = s σ λ s . M s ̂ . M
s = m λ σ λ σ s . M s ̂ . M .
s = m λ λ s . M .

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