Abstract

This paper reports an analytical study to determine the stability requirements for a holographic system intended for the NASA Space Shuttle. The primary emphasis is on optimization of the holographic geometry to minimize the effect of film variations between hologram exposure in space and wave-front reconstruction on the ground. It is shown that it is extremely important to use a balanced geometry (equal reference and object beam angles) and, if possible, a specific object beam angle.

© 1983 Optical Society of America

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References

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  1. R. Wuerker, L. O. Heflinger, J. V. Flannery, A. Kassel, A. M. Rollauer, “Holography on the NASA Space Shuttle,” CICESE, Ensenada, Mex., 4–8 Aug. (1980), paper TH1-1.
  2. Rayleigh, Philos. Mag. 8, 403 (1879).
  3. A. Marechal, Rev. Opt. 26, 257 (1947).

1947 (1)

A. Marechal, Rev. Opt. 26, 257 (1947).

1879 (1)

Rayleigh, Philos. Mag. 8, 403 (1879).

Flannery, J. V.

R. Wuerker, L. O. Heflinger, J. V. Flannery, A. Kassel, A. M. Rollauer, “Holography on the NASA Space Shuttle,” CICESE, Ensenada, Mex., 4–8 Aug. (1980), paper TH1-1.

Heflinger, L. O.

R. Wuerker, L. O. Heflinger, J. V. Flannery, A. Kassel, A. M. Rollauer, “Holography on the NASA Space Shuttle,” CICESE, Ensenada, Mex., 4–8 Aug. (1980), paper TH1-1.

Kassel, A.

R. Wuerker, L. O. Heflinger, J. V. Flannery, A. Kassel, A. M. Rollauer, “Holography on the NASA Space Shuttle,” CICESE, Ensenada, Mex., 4–8 Aug. (1980), paper TH1-1.

Marechal, A.

A. Marechal, Rev. Opt. 26, 257 (1947).

Rayleigh,

Rayleigh, Philos. Mag. 8, 403 (1879).

Rollauer, A. M.

R. Wuerker, L. O. Heflinger, J. V. Flannery, A. Kassel, A. M. Rollauer, “Holography on the NASA Space Shuttle,” CICESE, Ensenada, Mex., 4–8 Aug. (1980), paper TH1-1.

Wuerker, R.

R. Wuerker, L. O. Heflinger, J. V. Flannery, A. Kassel, A. M. Rollauer, “Holography on the NASA Space Shuttle,” CICESE, Ensenada, Mex., 4–8 Aug. (1980), paper TH1-1.

Philos. Mag. (1)

Rayleigh, Philos. Mag. 8, 403 (1879).

Rev. Opt. (1)

A. Marechal, Rev. Opt. 26, 257 (1947).

Other (1)

R. Wuerker, L. O. Heflinger, J. V. Flannery, A. Kassel, A. M. Rollauer, “Holography on the NASA Space Shuttle,” CICESE, Ensenada, Mex., 4–8 Aug. (1980), paper TH1-1.

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Figures (4)

Fig. 1
Fig. 1

Geometry for the analysis of the effect of film variations on the holographic image.

Fig. 2
Fig. 2

Tolerance to out-of-plane film errors in centimeters as a function of the reconstruction beam angle for object beam angles of 0, 5, 10, and 15°.

Fig. 3
Fig. 3

Tolerance to in-plane film errors in centimeters as a function of object beam angle.

Fig. 4
Fig. 4

Error in direction of reconstructed plane wave ɛ as a function of the error in direction of the reconstruction beam θ for various object and reference beam angles.

Equations (28)

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p ( Δ , z ) = [ 1 2 π σ Δ σ z ( 1 - ρ 2 ) 1 / 2 ] × exp [ - σ z 2 Δ 2 - 2 σ z σ Δ ρ Δ z + σ Δ 2 z 2 2 σ Δ 2 σ z 2 ( 1 - ρ 2 ) ] ,
ϕ ( x , Δ , z ) = 2 π λ { ( x + Δ ) sin ψ r + z cos ψ r + [ ( x + Δ - x o ) 2 + ( z - z o ) 2 ] 1 / 2 } .
ϕ ( x , Δ , z ) = 2 π λ [ x ( sin ψ r - sin ψ o ) + z ( cos ψ r - cos ψ o ) + Δ ( sin ψ r - sin ψ o ) + x 2 + Δ 2 + z 2 2 r o + x Δ r o + r o ] ,
m ϕ = E ( ϕ ) = 2 π λ [ x ( sin ψ r - sin ψ o ) + x 2 2 r o + r o + σ Δ 2 + σ z 2 2 r o ] .
V ( ϕ ) = 4 π 2 λ 2 [ σ Δ 4 + 2 σ Δ 2 σ z 2 ρ + σ z 4 2 r o 2 + σ z 2 ( cos ψ r - cos ψ o ) 2 + σ Δ 2 ( x r o + sin ψ r - sin ψ o ) 2 + 2 σ Δ σ z ρ ( x r o + sin ψ r - sin ψ o ) · ( cos ψ r - cos ψ o ) ] .
ψ r = - ψ o = - sin - 1 ( x 2 r o ) ,
V ( ϕ ) / MIN = 4 π 2 λ 2 ( σ Δ 4 + 2 σ z 2 ϕ z 2 ρ + σ z 4 2 r o 2 ) .
ψ r = - ψ o = - sin - 1 ( A 4 r o ) .
V ( ϕ ) = 4 π 2 λ 2 { ( σ Δ 2 ± σ z 2 ) 2 2 r o 2 + [ σ z ( cos ψ r - cos ψ o ) ± σ Δ ( A 2 r o + sin ψ r - sin ψ o ) ] 2 }             for ρ = ± 1 ,
V ( ϕ ) = 4 π 2 λ 2 { ( σ Δ 4 ± σ z 4 ) 2 2 r o 2 + [ σ z 2 ( cos ψ r - cos ψ o ) 2 ± σ Δ 2 ( A 2 r o + sin ψ r - sin ψ o ) 2 ] }             for ρ = 0.
δ R = 1.22 λ F ,
3 σ ϕ = 3 V ( ϕ ) π / 2.
σ ϕ π / 6.
σ ϕ 2 π 13.5 = π 6.75 .
[ σ Δ 4 + 2 σ Δ 2 σ z 2 ρ + σ z 4 2 r o + σ Δ 2 ( A 2 r o - 2 sin ψ o ) 2 ] 1 / 2 λ 12 .
V ( ϕ ) = 4 π 2 λ 2 { σ z 4 2 r o + σ z 2 ( cos ψ r - cos ψ o ) 2 } .
( σ z 4 2 r o + σ z 2 ( cos ψ r - cos ψ o ) 2 ) 1 / 2 λ 6 2 .
V ( ϕ ) = 4 π 2 λ 2 { σ Δ 4 2 r o 2 + σ Δ 2 ( A 2 r o - 2 sin ψ o ) 2 } ,
[ σ Δ 4 2 r o 2 + σ Δ 2 ( A 2 r o - 2 sin ψ o ) ] 1 / 2 λ 6 2 .
ϕ ( x ) = - 2 π x λ ( sin ψ r - sin ψ o ) ,
ϕ i ( x ) = 2 π λ x ( sin ψ a - sin ψ r + sin ψ o ) , sin ψ i ( x ) = sin ψ a - sin ψ r + sin ψ o
ψ a = ψ r + θ ,             ψ i = ψ o + θ + ɛ .
ɛ = sin θ ( cos ψ r - cos ψ o ) + ( 1 - cos θ ) ( sin ψ o - sin ψ r ) cos ψ o cos θ - sin ψ o sin θ .
ɛ = 2 ( 1 - cos θ ) sin ψ o cos ( ψ o + θ ) .
ɛ = θ 2 sin ψ cos ( ψ o + θ ) .
ɛ r o 1.22 λ r o A             or             ɛ 1.22 λ A .
sin ψ i = ( 1 + ρ ) ( sin ψ a - sin ψ r + sin ψ o ) = ( 1 + ρ ) sin ψ o for ψ a = ψ r .
ɛ ρ tan ψ o .

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