Abstract

The application of a ray-tracing methodology to holography is presented. Emphasis is placed on establishing a very general foundation from which to build a general computer-based implementation. As few restrictions as possible are placed on the recording and reconstruction geometry. The necessary equations are established from the construction and reconstruction parameters of the hologram. The aberrations are defined following H. H. Hopkins, and these aberration specification techniques are compared with those used previously to analyze holography. Representative of the flexibility of the ray-tracing approach, two examples are considered. The first compares the answers between a wavefront matching and the ray-tracing analysis in the case of aberration balancing to compensate for chromatic aberrations. The results are very close and establish the basic utility of aberration balancing. Further indicative of the power of a ray tracing, a thick media analysis is included in the computer programs. This section is then used to perform a study of the effects of hologram emulsion shrinkage and methods for compensation. The results of compensating such holograms are to introduce aberrations, and these are considered in both reflection and transmission holograms.

© 1971 Optical Society of America

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References

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  1. C. W. Helstrom, J. Opt. Soc. Am. 56, 433 (1966).
    [CrossRef]
  2. A. Offner, J. Opt. Soc. Am. 56, 1509 (1966).
    [CrossRef]
  3. I. A. Abramowitz, J. M. Ballantyne, J. Opt. Soc. Am. 57, 1522 (1967).
    [CrossRef]
  4. I. A. Abramowitz, Appl. Opt. 8, 403 (1969).
    [CrossRef] [PubMed]
  5. I. A. Abramowitz, “Design of Holographic Systems by Ray Tracing,” Ph.D. Thesis, Cornell University, Ithaca, New York, 1968 (U. Microfilms, 68–4656).
  6. Hereafter, Refs. 7 and 8 to Parts I and II of the article “Computer-Based Analysis of Hologram Imagery and Aberrations” will simply be I and II.
  7. J. N. Latta, Appl. Opt. 10, 599 (1971).
    [CrossRef] [PubMed]
  8. J. N. Latta, Appl. Opt. 10, 609 (1971).
    [CrossRef] [PubMed]
  9. H. Kogelnik, Bell Syst. Tech. J. 48, 2909 (1969).
  10. H. M. Smith, Principles of Holography (Wiley-Interscience, New York, 1969).
  11. E. G. Ramberg, RCA Rev. 27, 467 (1966).
  12. H. H. Hopkins, Wave Theory of Aberrations (Oxford U. P., Oxford, England, 1950).
  13. H. H. Hopkins, M. J. Yzuel, Optica Acta 17, 157 (1970).
    [CrossRef]
  14. B. Carnahan, J. O. Wilkes, Numerical Methods, Optimization Techniques and Simulation for Engineers (The University of Michigan Engineering Summer Conference Notes, Ann Arbor, Michigan, August1970).
  15. R. Hooke, T. A. Jeeves, J. Assoc. Comp. Mach. 8, 212 (1961).
    [CrossRef]
  16. D. H. R. Vilkomersom, D. Bostwick, Appl. Opt. 6, 1270 (1969).
    [CrossRef]

1971 (2)

1970 (1)

H. H. Hopkins, M. J. Yzuel, Optica Acta 17, 157 (1970).
[CrossRef]

1969 (3)

1967 (1)

1966 (3)

1961 (1)

R. Hooke, T. A. Jeeves, J. Assoc. Comp. Mach. 8, 212 (1961).
[CrossRef]

Abramowitz, I. A.

I. A. Abramowitz, Appl. Opt. 8, 403 (1969).
[CrossRef] [PubMed]

I. A. Abramowitz, J. M. Ballantyne, J. Opt. Soc. Am. 57, 1522 (1967).
[CrossRef]

I. A. Abramowitz, “Design of Holographic Systems by Ray Tracing,” Ph.D. Thesis, Cornell University, Ithaca, New York, 1968 (U. Microfilms, 68–4656).

Ballantyne, J. M.

Bostwick, D.

Carnahan, B.

B. Carnahan, J. O. Wilkes, Numerical Methods, Optimization Techniques and Simulation for Engineers (The University of Michigan Engineering Summer Conference Notes, Ann Arbor, Michigan, August1970).

Helstrom, C. W.

Hooke, R.

R. Hooke, T. A. Jeeves, J. Assoc. Comp. Mach. 8, 212 (1961).
[CrossRef]

Hopkins, H. H.

H. H. Hopkins, M. J. Yzuel, Optica Acta 17, 157 (1970).
[CrossRef]

H. H. Hopkins, Wave Theory of Aberrations (Oxford U. P., Oxford, England, 1950).

Jeeves, T. A.

R. Hooke, T. A. Jeeves, J. Assoc. Comp. Mach. 8, 212 (1961).
[CrossRef]

Kogelnik, H.

H. Kogelnik, Bell Syst. Tech. J. 48, 2909 (1969).

Latta, J. N.

Offner, A.

Ramberg, E. G.

E. G. Ramberg, RCA Rev. 27, 467 (1966).

Smith, H. M.

H. M. Smith, Principles of Holography (Wiley-Interscience, New York, 1969).

Vilkomersom, D. H. R.

Wilkes, J. O.

B. Carnahan, J. O. Wilkes, Numerical Methods, Optimization Techniques and Simulation for Engineers (The University of Michigan Engineering Summer Conference Notes, Ann Arbor, Michigan, August1970).

Yzuel, M. J.

H. H. Hopkins, M. J. Yzuel, Optica Acta 17, 157 (1970).
[CrossRef]

Appl. Opt. (4)

Bell Syst. Tech. J. (1)

H. Kogelnik, Bell Syst. Tech. J. 48, 2909 (1969).

J. Assoc. Comp. Mach. (1)

R. Hooke, T. A. Jeeves, J. Assoc. Comp. Mach. 8, 212 (1961).
[CrossRef]

J. Opt. Soc. Am. (3)

Optica Acta (1)

H. H. Hopkins, M. J. Yzuel, Optica Acta 17, 157 (1970).
[CrossRef]

RCA Rev. (1)

E. G. Ramberg, RCA Rev. 27, 467 (1966).

Other (5)

H. H. Hopkins, Wave Theory of Aberrations (Oxford U. P., Oxford, England, 1950).

H. M. Smith, Principles of Holography (Wiley-Interscience, New York, 1969).

B. Carnahan, J. O. Wilkes, Numerical Methods, Optimization Techniques and Simulation for Engineers (The University of Michigan Engineering Summer Conference Notes, Ann Arbor, Michigan, August1970).

I. A. Abramowitz, “Design of Holographic Systems by Ray Tracing,” Ph.D. Thesis, Cornell University, Ithaca, New York, 1968 (U. Microfilms, 68–4656).

Hereafter, Refs. 7 and 8 to Parts I and II of the article “Computer-Based Analysis of Hologram Imagery and Aberrations” will simply be I and II.

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Figures (18)

Fig. 1
Fig. 1

Hologram recording geometry for the ray-tracing analysis.

Fig. 2
Fig. 2

Hologram reconstruction geometry for the ray-tracing analysis.

Fig. 3
Fig. 3

Hologram grating formation by an object and reference beam ray.

Fig. 4
Fig. 4

Hologram grating interaction with a reconstruction beam ray and the resulting image beam ray.

Fig. 5
Fig. 5

Vector scaling in the x, y, z directions by the multiplicative scaling factors Mx, My, Mz.

Fig. 6
Fig. 6

Hologram phase during the construction process for the first and nth object and reference beam rays.

Fig. 7
Fig. 7

Hologram phase, Φ, during the reconstruction process for the first and nth image beam rays.

Fig. 8
Fig. 8

Determination of the wavefront aberration, D1, on an image ray given by the point x1, y1, z1, and direction cosines 1I, mI, rI. The distance D1 is with respect to a reference sphere with center x0, y0, z0, and radius R.

Fig. 9
Fig. 9

A comparison of the aberration specifications for the wavefront matching analysis, Wh, and the ray-tracing analysis, Ws.

Fig. 10
Fig. 10

Detail drawing of the wavefront aberration specifications for wavefront matching and ray tracing.

Fig. 11
Fig. 11

Comparison of the wavefront matching and ray-tracing aberrations along the y axis for a hologram geometry determined by the aberration balancing technique discussed in II. The parameters are the same as those listed in Fig. 7 of II, except in the case of the image location as determined by the ray-tracing analysis: RIV = 0.299992 m, αIV = 9.99898°.

Fig. 12
Fig. 12

Comparison of the wavefront matching and ray-tracing aberrations along the x axis for a hologram geometry determined by the aberration balancing technique discussed in II. The parameters are the same as those indicated in Fig. 11.

Fig. 13
Fig. 13

Hologram grating parameters for a grating vector, K, confined to the xz plane. The incoming reconstruction beam C is incident at an angle γ. The grating period is Γ, and the grating vector K is at an angle ξ with respect to the z axis.

Fig. 14
Fig. 14

The Q factor and hologram ray efficiency vs the hologram x coordinate for the hologram geometry implementing the aberration balancing technique discussed in Fig. 11. The hologram and media parameters are: n 1 = n 2 = n 3 = n 4 = n B = 1 , n Δ = 0.02 , α Δ = 0 , D = 15 μ , and M Z = 1.

Fig. 15
Fig. 15

The construction and reconstruction geometry for a transmission hologram that undergoes shrinkage in the z direction by the factor Mz. To compensate for this shrinkage, the reconstruction beam is incident at the angle αC.

Fig. 16
Fig. 16

The ray efficiency vs hologram x coordinate for a transmission hologram after shrinkage and after shrinkage compensation. The hologram parameters are β = 0° for all beams; λC = λ0 = 6328Å; R0 = 30 cm, α0 = 30°; RR → ∞, αR = 0°; RC → ∞, αC = 0° (no shrinkage compensation), αC = 2.43° (with shrinkage compensation); n1 = n2 = n3 = n4 = 1.0; nA = nB = 1.5; nΔ = 0.02, αΔ = 0; Mz = 0.85, D = 15 μ; XMAX = 5 cm.

Fig. 17
Fig. 17

The total wavefront deviation from the reference sphere, ΔG, vs the object beam angle α0 for a transmission and reflection hologram subject to 15% shrinkage and compensation for shrinkage. The parameters for the transmission hologram are the same as those shown in Fig. 16, except 10° α 0 45° , 0.874° α C 3.29° , where αC varies over the range indicated to compensate for the change in the central Bragg angle as a function of α0. The parameters for the reflection hologram are the same as those shown in Fig. 16, except 10° α 0 45° , - 1.11° α C - 3.06° , 5381 A ˚ λ C 5423 A ˚ , where αC and λC vary over the range indicated to compensate for the change in central Bragg angle as a function of α0.

Fig. 18
Fig. 18

The construction and reconstruction geometry for a reflection hologram that undergoes shrinkage in the z direction by the factor Mz. To compensate for this shrinkage, the reconstruction beam is incident at the angle αC.

Equations (62)

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O = l O i + m O j + n O k ,
R = l R i + m R j + n R k ,
V 1 = R × O = l 1 i + m 1 j + n 1 k .
l 2 = l O + l R ,
m 2 = m O + m R ,
n 2 = n O + n R .
V 2 = l 2 i + m 2 j + n 2 k .
V 3 = V 1 × V 2 = l 3 i + m 3 j + n 3 k .
d x = λ O / ( l O - l R ) ,
d y = λ O / ( m O - m R ) ,
d z = λ O / ( n O - n R ) ,
Λ = d x / l 3 ,
d x = l 3 Λ ,
d y = m 3 Λ ,
d z = n 3 Λ .
C = l C i + m C j + n C k .
I = l I i + m I j + n I k .
l I = l C ± ( λ C / d x ) ,
m I = m C ± ( λ C / d y ) ,
n I = ± ( 1 - l I 2 - n I 2 ) 1 2 ,
Before magnification ( V ) x / l = y / m = z / n .
After magnification ( V ) M x x / l = M y y / m = M z z / n .
l 2 + m 2 + n 2 = 1.
l = ± { 1 / [ ( n M z / l M x ) 2 + ( m M y / l M x ) 2 + 1 ] 1 2 } ,
m = m ( M y / M x ) ( l / l ) ,
n = n ( M z / M x ) ( l / l ) ,
l = l ( M x / M y ) ( m / m ) ,
m = ± { 1 / [ ( n M z / m M y ) 2 + ( l M x / m M y ) 2 + 1 ] 1 2 } ,
n = n ( M z / M y ) ( m / m ) ,
l = l ( M x / M z ) ( n / n ) ,
m = m ( M y / M z ) ( n / n ) ,
n = ± { 1 / [ ( l M x / n M z ) 2 + ( m M y / n M z ) 2 + 1 ] 1 2 } .
SIGN ( l ) = SIGN ( l ) , ( M x > 0 ) , SIGN ( m ) = SIGN ( m ) , ( M y > 0 ) , SIGN ( n ) = SIGN ( n ) , ( M z > 0 ) .
V 1 = l 1 i + m 1 j + n 1 k ,
V 2 = l 2 i + m 2 j + n 2 k .
V 3 = V 1 × V 2 = l 3 i + m 3 j + n 3 k .
Λ = d x / l 3 ,
d x = M x d x .
( n 1 R O 1 - n 1 R R 1 ) / λ O ,
( n 1 R O n - n 1 R R n ) / λ O .
Φ i = n 3 R C i λ C ± n 1 R O - n 1 R R i λ O | i = 1 & n ,
( x - x 1 ) / l I = ( y - y 1 ) / m I = ( z - z 1 ) / n I ,
( x - x O ) 2 + ( y - y O ) 2 + ( z - z O ) 2 = R 2 .
( 1 + m 2 l 2 + n 2 l 2 ) x 2 + [ - 2 x O - 2 m 2 l 2 x 1 + 2 m l ( y 1 - y O ) - 2 n 2 l 2 x 1 + 2 n l ( z 1 - z O ) ] x + x O 2 + m 2 l 2 x 1 2 - 2 m l ( y 1 - y O ) + ( y 1 - y O ) 2 + n 2 2 x 1 2 - 2 n l ( z 1 - z O ) x 1 + ( z 1 - z O ) 2 - R 2 = 0 ,
( 1 + l 2 m 2 + n 2 m 2 ) y 2 + [ - 2 y O - 2 l 2 m 2 y 1 + 2 l m ( x 1 - x O ) - 2 n 2 m 2 y 1 + 2 n m ( z 1 - z O ) ] y + y O 2 + l 2 m 2 y 1 2 - 2 l m ( x 1 - x O ) y 1 + ( x 1 - x O ) 2 + n 2 m 2 y 1 2 - 2 n m ( z 1 - z O ) y 1 + ( z 1 - z O ) 2 - R 2 = 0 ,
( 1 + l 2 n 2 + m 2 n 2 ) z 2 + [ - 2 z O - 2 l 2 n 2 z 1 + 2 l n ( x 1 - x O ) - 2 m 2 n 2 z 1 + 2 m n ( y 1 - y O ) ] z + z O 2 + l 2 n 2 z 1 2 - 2 l n ( x 1 - x O ) z 1 + ( x - x O ) 2 + m 2 n 2 z 1 2 - 2 m n ( y 1 - y O ) z 1 + ( y 1 - y O ) 2 - R = 0.
x = x + ,
y = y - ,
z = z + .
( x - x 1 ) / ( x - x 1 ) = ( y - y 1 ) / ( y - y 1 ) = ( z - z 1 ) / ( z - z 1 ) .
K = K V 3 ,
K = 2 π / Λ ,
cos ( ξ - γ B ) = K / 2 β ,
β = 2 π n / λ C ,
γ B = ξ ± cos - 1 ( K / 2 β ) .
Q = M z D K 2 / β .
K / 2 β = 2.3098
λ C / 2 Λ = 2.3098.
λ C = ( 1.9616 / 2.3098 ) λ O ,
n 1 = n 2 = n 3 = n 4 = n B = 1 , n Δ = 0.02 , α Δ = 0 , D = 15 μ , and M Z = 1.
10° α 0 45° , 0.874° α C 3.29° ,
10° α 0 45° , - 1.11° α C - 3.06° , 5381 A ˚ λ C 5423 A ˚ ,

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