Abstract

This paper examines the aberrations in holographic systems characterized by a ratio of reconstruction wavelength to recording wavelength of the order of 10−3. The study is restricted to those systems using a plane reference beam and with the hologram scaled by a factor greater than the wavelength ratio. An inline hologram geometry is examined where the reconstruction beam is placed such that the image of a selected object point on the optical axis has zero aberrations. The aberrations for object points off axis then are calculated. For off-axis holograms, a technique for balancing spherical aberration and astigmatism has been modified to apply to scaled holograms. Some general rules for the optimum recording geometry are developed. The aberrations for the balanced off-axis system are calculated and compared to those of the in-line system.

© 1971 Optical Society of America

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References

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  1. E. N. Leith, J. Upatnieks, K. A. Haines, J. Opt. Soc. Amer. 55, 981 (1965).
  2. R. W. Meier, J. Opt. Soc. Amer. 55, 987 (1965).
  3. E. B. Champagne, J. Opt. Soc. Amer. 57, 51 (1967); E. B. Champagne, A Qualitative and Quantitative Study of Holographic Imaging, Ph.D. Thesis, Ohio State University, Ohio, July1967; University Microfilms, #67-10876).
    [CrossRef]
  4. A. F. Metherell, H. M. A. El-Sum, L. Larmore, Acoustical Holography (Plenum Press, New York, 1969), Vol. 1.
  5. A. F. Metherell, L. Larmore, Acoustical Holography, (Plenum Press, New York, 1970), Vol. 2.
    [CrossRef]
  6. We refer to M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1958), p. 204: “In ordinary instruments the wave aberrations may be as much as 40 or 50 wavelengths, but in instruments used for more precise work (such as astronomical telescopes or microscopes) they must be reduced to a much smaller value, only a fraction of a wavelength.”
  7. J. N. Latta, J. Opt. Soc. Amer. 60, 715A (1970).

1970 (1)

J. N. Latta, J. Opt. Soc. Amer. 60, 715A (1970).

1967 (1)

E. B. Champagne, J. Opt. Soc. Amer. 57, 51 (1967); E. B. Champagne, A Qualitative and Quantitative Study of Holographic Imaging, Ph.D. Thesis, Ohio State University, Ohio, July1967; University Microfilms, #67-10876).
[CrossRef]

1965 (2)

E. N. Leith, J. Upatnieks, K. A. Haines, J. Opt. Soc. Amer. 55, 981 (1965).

R. W. Meier, J. Opt. Soc. Amer. 55, 987 (1965).

Born, M.

We refer to M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1958), p. 204: “In ordinary instruments the wave aberrations may be as much as 40 or 50 wavelengths, but in instruments used for more precise work (such as astronomical telescopes or microscopes) they must be reduced to a much smaller value, only a fraction of a wavelength.”

Champagne, E. B.

E. B. Champagne, J. Opt. Soc. Amer. 57, 51 (1967); E. B. Champagne, A Qualitative and Quantitative Study of Holographic Imaging, Ph.D. Thesis, Ohio State University, Ohio, July1967; University Microfilms, #67-10876).
[CrossRef]

El-Sum, H. M. A.

A. F. Metherell, H. M. A. El-Sum, L. Larmore, Acoustical Holography (Plenum Press, New York, 1969), Vol. 1.

Haines, K. A.

E. N. Leith, J. Upatnieks, K. A. Haines, J. Opt. Soc. Amer. 55, 981 (1965).

Larmore, L.

A. F. Metherell, H. M. A. El-Sum, L. Larmore, Acoustical Holography (Plenum Press, New York, 1969), Vol. 1.

A. F. Metherell, L. Larmore, Acoustical Holography, (Plenum Press, New York, 1970), Vol. 2.
[CrossRef]

Latta, J. N.

J. N. Latta, J. Opt. Soc. Amer. 60, 715A (1970).

Leith, E. N.

E. N. Leith, J. Upatnieks, K. A. Haines, J. Opt. Soc. Amer. 55, 981 (1965).

Meier, R. W.

R. W. Meier, J. Opt. Soc. Amer. 55, 987 (1965).

Metherell, A. F.

A. F. Metherell, L. Larmore, Acoustical Holography, (Plenum Press, New York, 1970), Vol. 2.
[CrossRef]

A. F. Metherell, H. M. A. El-Sum, L. Larmore, Acoustical Holography (Plenum Press, New York, 1969), Vol. 1.

Upatnieks, J.

E. N. Leith, J. Upatnieks, K. A. Haines, J. Opt. Soc. Amer. 55, 981 (1965).

Wolf, E.

We refer to M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1958), p. 204: “In ordinary instruments the wave aberrations may be as much as 40 or 50 wavelengths, but in instruments used for more precise work (such as astronomical telescopes or microscopes) they must be reduced to a much smaller value, only a fraction of a wavelength.”

J. Opt. Soc. Amer. (4)

E. N. Leith, J. Upatnieks, K. A. Haines, J. Opt. Soc. Amer. 55, 981 (1965).

R. W. Meier, J. Opt. Soc. Amer. 55, 987 (1965).

E. B. Champagne, J. Opt. Soc. Amer. 57, 51 (1967); E. B. Champagne, A Qualitative and Quantitative Study of Holographic Imaging, Ph.D. Thesis, Ohio State University, Ohio, July1967; University Microfilms, #67-10876).
[CrossRef]

J. N. Latta, J. Opt. Soc. Amer. 60, 715A (1970).

Other (3)

A. F. Metherell, H. M. A. El-Sum, L. Larmore, Acoustical Holography (Plenum Press, New York, 1969), Vol. 1.

A. F. Metherell, L. Larmore, Acoustical Holography, (Plenum Press, New York, 1970), Vol. 2.
[CrossRef]

We refer to M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1958), p. 204: “In ordinary instruments the wave aberrations may be as much as 40 or 50 wavelengths, but in instruments used for more precise work (such as astronomical telescopes or microscopes) they must be reduced to a much smaller value, only a fraction of a wavelength.”

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Figures (11)

Fig. 1
Fig. 1

Coordinate geometry of an arbitrary point source, Q at xq, yq, zq, in front of a hologram in the xy plane.

Fig. 2
Fig. 2

Magnitude of the wave aberration; |Δ|, vs αo, λo = 3.75 × 10−4 m, λc = 6.328 × 10−7 m, m = 0.05. Hologram diameter at recording = 1.0 m. Reference beam is plane wave at αr = βr = 0.0°, Ro = 0.5 m, βo = 0.0°, Rc = −0.0446 m, αc = βc = 0.0°.

Fig. 3
Fig. 3

Magnitude of the wave aberration, |Δ|, versus R0, λ0 = 3.75 × 10−4 m, λc = 6.328 × 10−7 m, m = 0.05. Hologram diameter at recording = 1.0 m. Reference beam is plane wave at αr = βr = 0.0°, αo = βo = 0.0° Rc = −0.0446 m, αc = βc = 0.0°.

Fig. 4
Fig. 4

Magnitude of the wave aberration, |Δ|, for the compensated object point vs αo. For each value of αo, the reconstruction geometry is calculated according to the aberration balancing technique. The wave aberration is then calculated using the new values of αo, Rc and αc: λo = 3.75 × 10−4 m, λc = 6.328 × 10−7 m, m = 0.05. Hologram diameter at recording = 1.0 m. Reference beam is plane wave at αr = 45.0°, βr = 0.0°, Ro = 0.5 m, βo = 0.0°.

Fig. 5
Fig. 5

Magnitude of the wave aberration, |Δ|, for the compensated object point vs αr. For each value of αr, the reconstruction geometry is calculated according to the aberration balancing technique. The wave aberration is then calculated using the new values of αr, Rc, and αc: λo = 3.75 × 10−4 m, λc = 6.328 × 10−7 m, m = 0.05. Hologram diameter at recording = 1.0 m, Ro = 0.5 m, αo = −45.0°, βo = 0.0°. Reference beam is a plane wave with βr = 0.0°.

Fig. 6
Fig. 6

Magnitude of the wave aberration, |Δ|, vs Ro. The reconstruction geometry is calculated for the object point—Ro = 0.5 m, αo = 75.0°, βo = 0.0°—according to the aberration balancing technique. When m = 0.05, Rc = −0.0448 m, αc = −29.59°, βc = 0.0°. When m = 0.01, Rc = −0.0120 m, αc = −24.46°, βc = 0.0°. For each value of m, Rc and αc, the wave aberration is calculated as a function of Ro, with αo = 75.0° and βo = 0.0°, λo = 3.75 × 10−4 m, λc = 6.328 × 10−7 m. Hologram diameter at recording = 1.0 m. Reference beam is plane wave at αr = −10.0°, βr = 0.0°.

Fig. 7
Fig. 7

Magnitude of the wave aberration, |Δ|, vs αo. The reconstruction geometry is calculated for the object point Ro = 0.5 m, αo = 75.0°, βo = 0.0° according to the aberration balancing technique. When m = 0.05, Rc = −0.0448 m, αc = −29.59°, βc = 0.0°. When m = 0.01, Rc = −0.0102 m, αc = −24.46°, βc = 0.0°. For each value of m, Rc, and αc the wave aberration is calculated as a function of αo, with Ro = 0.5 m and βo = 0.0°. λo = 3.75 × 10−4 m, λc = 6.328 × 10−7 m. Reference beam is plane wave at αr = −10.0°, βr = 0.0°. Hologram diameter at recording = 1.0 m.]

Fig. 8
Fig. 8

Magnitude of the wave aberration |Δ|, vs βo. The reconstruction geometry is calculated for the object point Ro = 0.5 m, αo = 75.0°, βo = 0.0° according to the aberration balancing technique. When m = 0.05, Rc = −0.0448 m, αc = −29.59°, βc = 0.0°. When m = 0.01, Rc = −0.0102 m, αc = −24.46°, βc = 0.0°. For each value of m, Rc and αc, the wave aberration is calculated as a function of βo, with Ro = 0.5 m, and α0 = 75.0°. λ0 = 3.75 × 10−4 m, λc = 6.328 × 10−7 m. Hologram diameter at recording = 1.0 m. Reference beam is a plane wave at αr = −10.0°, βr = 0.0°.

Fig. 9
Fig. 9

Magnitude of the wave aberration |Δ|, vs Ro. The reconstruction geometry is calculated for the object point Ro = 0.5 m, αo = 45.0°, βo = 0.0° according to the aberration balancing technique. Rc = −0.0469 m, αc = −18.24°, βc = 0.0°. Rc = −0.049 m, αc = −18.24°, βc = 0.0°. The wave aberration is calculated as a function of Ro with αo = 45.0° and βo = 0.0°. λ0 = 3.75 × 10−4 m, λc = 6.328 × 10−7 m, m = 0.05. Hologram diameter at recording = 1.0 m. Reference beam is a plane wave at αr = −25.0°, βr = 0.0°.

Fig. 10
Fig. 10

Magnitude of the wave aberration |Δ|, vs αo. The reconstruction geometry is calculated for the object point Ro = 0.5 m, αo = 45.0°, βo = 0.0° according to the aberration balancing technique. Rc = −0.0469 m, αc = −18.24°, βc = 0.0°. The wave aberration is calculated as a function of αo with Ro = 0.5m, βo = 0.0°. λo = 3.75 × 10−4 m, λc = 6.328 × 10−7 m, m = 0.05. Hologram diameter at recording = 1.0 m. Reference beam is a plane wave at αr = −25.0°, βr = 0.0°.

Fig. 11
Fig. 11

Magnitude of the wave aberration |Δ|, vs βo. The reconstruction geometry is calculated for the object point Ro = 0.5 m, αo = 45.0°, βo = 0.0° according to the aberration balancing technique. Rc = −0.0469 m, αc = −18.24°, βc = 0.0°. The wave aberration is calculated as a function of αo with Ro = 0.5m, αo = 45.0°. λo = 3.75 × 10−4 m, λc = 6.328 × 10−7 m, m = 0.05. Hologram diameter at recording = 1.0 m. Reference beam is a plane wave at αr = −25.0°, βr = 0.0°.

Equations (23)

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λ c = 6.328 × 10 - 7 m ;             λ o = 3.750 × 10 - 4 m ;             μ = 0.00169 ; diameter of unscaled hologram = 1.0 m ;             scaling factor : m = 0.05.
1 R i = 1 R c ± μ m 2 ( 1 R o - 1 R r )
sin α i = sin α c ± ( μ / m ) ( sin α o - sin α r )
cos α i sin β i = cos α c sin β c ± ( μ / m ) ( cos α o sin β o - cos α r sin β r )
+ 1 / λ c [ - 1 8 ρ 4 S spherical aberration , + 1 2 ρ 2 ( ρ C x cos ϕ + ρ C y sin ϕ ) coma , - 1 2 ( ρ 2 A x cos 2 ϕ + ρ 2 A y sin 2 ϕ ) + 2 ρ 2 A x y sin ϕ cos ϕ ) ] astigmatism ,
S = 1 R c 3 ± μ m 4 ( 1 R o 3 - 1 R r 3 ) - 1 R i 3 ,
C x = sin α c R c 2 ± μ m 3 ( sin α o R o 2 - sin α r R r 2 ) - sin α i R i 2 ,
C y = cos α c sin β c R c 2 ± μ m 3 ( cos α o sin β o R o 2 - cos α r sin β r R r 2 ) - cos α i sin β i R i 2
A x = sin 2 α c R c ± μ m 2 ± μ m 2 ( sin 2 α o R o - sin 2 α r R r ) - sin 2 α i R i
A y = cos 2 α c sin 2 β c R c ± μ m 2 ( cos 2 α o sin 2 β o R o - cos 2 α r sin 2 β r R r ) - cos 2 α i sin 2 β i R i
A x y = sin α c cos α c sin β c R c ± μ m 2 ( sin α o cos α o sin β o R o - sin α r cos α r sin β r R r ) - sin α i cos α i sin β i R i ,
R c 2 [ ± μ m 4 1 R o 3 - μ 3 m 1 R o 3 ] + R c [ - 3 μ 2 m 4 R o 2 ] + [ 3 μ m 2 R o ] = 0.
α r = α c = 45.0°             β r = β c = 0.0° ,
R o = 0.5 m             α o = 0.0°             β o = 0.0° .
R o = 1.0 m             α o = 45.0°             β o = 0.0° ,
α r = α c = 0.0°             β r = β c = 0.0° .
C x = [ sin α c R c 2 - sin α i R i 2 ± μ m 3 ( sin 2 α o R o - sin 2 α r R r ) ] = 0
- 1 8 ρ 4 max [ 1 R c 3 - 1 R i 3 ± μ m 4 ( 1 R o 3 - 1 R r 3 ) ] = 1 2 ρ 2 max [ sin 2 α c R c - sin 2 α i R i ± μ m 2 ( sin 2 α o R o - sin 2 α r R r ) ] .
sin α c = ( G + 2 F G R c + G F 2 R c 2 - D R c 2 ) / ( 2 F R c - F 2 R c 2 ) ,
R c 4 ( - ρ 2 max 4 B F 4 + ρ 2 max 4 F 7 + D 2 F - E F 4 ) + R c 3 [ 7 4 ρ 2 max F 6 - ρ 2 max F 3 B + 2 G F 2 D - F 4 G 2 - 4 E F 3 ] + R c 2 [ 19 4 ρ 2 max F 5 - ρ 2 max B F 2 + 2 G F D - 4 G 2 F 3 - 4 E F 2 ] + R o [ 6 ρ 2 max F 4 - 6 G 2 F 2 ] + [ 3 ρ 2 max F 3 - 3 G 2 F ] = 0 ,
B = ± μ / m 4 [ 1 / ( R o 3 ) - 1 / ( R r ) 3 ] D = ± μ / m 3 [ ( sin α o ) / ( R o 2 ) - ( sin α r ) / ( R r 2 ) ] E = ± μ / m 2 [ ( sin 2 α o ) / R o - ( sin 2 α r ) / R r ] F = ± μ / m 2 ( 1 / R o - 1 / R r ) G = ± μ / m ( sin α o - sin α r ) ,
α o = 75.0°             R o = 0.5 m             α r = - 10.0° .
R c = - 0.0448 m             α c = - 29.59°             R i = - 0.0423 m α i = - 32.16°

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