Abstract

Numerical values are shown for the aberrations that are induced by a wavelength shift from construction to reconstruction. The holograms discussed in this paper may be constructed and reconstructed from a plane wave or point source. Four types of hologram geometries have been studied: in-line, off-axis, near-image plane, and lensless-fourier transform. The aberrations are first considered where the reconstruction beam geometry is the same as that needed to reconstruct an ideal image if no wavelength shift is present. The change in wavelength, however, causes aberrations and these are discussed as a function of the various hologram geometries and parameters. A considerable reduction in the aberrations can be realized in off-axis holograms by a technique of aberration balancing. In one example the total aberrations were reduced to approximately a Rayleigh limit of λ/4 in a hologram that was recorded at 4880 Å and reconstructed at 6328 Å. The minimum aberrations possible using the balancing process are discussed with respect to various hologram parameters.

© 1971 Optical Society of America

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References

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  1. I. A. Abramowitz, J. M. Ballantyne, J. Opt. Soc. Amer. 57, 1522 (1967).
    [CrossRef]
  2. I. A. Abramowitz, Appl. Opt. 8, 403 (1969).
    [CrossRef] [PubMed]
  3. I. A. Abramowitz, Design of Holographic Systems by Ray Tracing, Ph.D. Thesis, Cornell University, 1968 (U. Microfilms, 68-4656).
  4. E. B. Champagne, N. G. Massey, Appl. Opt. 8, 1879 (1969).
    [CrossRef] [PubMed]

1969 (2)

1967 (1)

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

Abramowitz, I. A.

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

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

I. A. Abramowitz, Design of Holographic Systems by Ray Tracing, Ph.D. Thesis, Cornell University, 1968 (U. Microfilms, 68-4656).

Ballantyne, J. M.

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

Champagne, E. B.

Massey, N. G.

Appl. Opt. (2)

J. Opt. Soc. Amer. (1)

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

Other (1)

I. A. Abramowitz, Design of Holographic Systems by Ray Tracing, Ph.D. Thesis, Cornell University, 1968 (U. Microfilms, 68-4656).

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Figures (14)

Fig. 1
Fig. 1

Magnitude of the spherical aberration wavefront deviation |ΔS| vs the wavelength deviation of the reconstruction beam, ΔλC, from the construction wavelength, λO, for the in-line hologram. The image is real. Curve (1) corresponds to the hologram geometry: λO = 6328 Å; RO = 0.20 m, αO = 0°; RR → ∞, αR = 0°; λC = λO ± |ΔλC|; RC → ∞, αC = 0°; and f/1.0. The other curves are the same as (1) with the exception of the f# as noted.

Fig. 2
Fig. 2

Magnitude of the wavefront deviation from the gaussian sphere, |ΔT|, at ϕ = 90° vs the wavelength deviation of the reconstruction beam, ΔλC, from the construction wavelength, λO, for the off-axis hologram. The image is real. Curve (1) corresponds to the hologram geometry: λO = 6328 Å; RO = 0.20 m, αO = 10.0°; RR → ∞, αR = −10.0°; λC = λO ± |ΔλC|; RC → ∞, αC = 10.0°; and f/1.0. The other curves are the same as (1) with the exception of the f# as noted.

Fig. 3
Fig. 3

Magnitude of the wavefront deviation from the gaussian sphere, |ΔT|, at ϕ = 90° vs the wavelength deviation of the reconstruction beam, ΔλC, from the construction wavelength, λO, for the off-axis hologram. The image is real. Curve (1) corresponds to the hologram geometry: λO = 6328 Å; RO = 0.20 m, αO = 25.0°; RR → ∞, αR = −25.0°; λC = λO ± |ΔλC|; RC → ∞, αC = 25.0°; and f/1.0. The other curves are the same as (1) with the exception of the f# as noted.

Fig. 4
Fig. 4

Magnitude of the wavefront deviation from the gaussian sphere, |ΔT|, at ϕ = 90° vs the wavelength deviation of the reconstruction beam, ΔλC, from the construction wavelength, λO, for the off-axis hologram. The image is real. Curve (1) corresponds to the hologram geometry: λO = 6328 Å; RO = 0.01 m, αO = 10.0°; RR → ∞, αR = −10.0°; λC = λO ± |ΔλC|; RC → ∞, αC = 10.0°; and f/1.0. The other curves are the same as (1) with the exception of the f# as noted.

Fig. 5
Fig. 5

Magnitude of the astigmatic wavefront deviation, |ΔA|, vs the wavelength deviation of the reconstruction beam, ΔλC, from the construction wavelength, λO, for the lensless-fourier transform hologram. The image is virtual and found by using the + sign in the gaussian image relations. The numbered curves correspond as follows: (1) λO = 6328 Å; RO = 0.1, αO = −25.0°; RR = 0.1, αR = 25.0°; λC = λO ± |ΔλC|; RC = 0.1, αC = 25.0°; |XMAX| = 0.05. (2) → (1) except αO = −10°, αR = 10°, αC = 10°. (3) → (1) except αO = −5°, αR = 5°, αC = 5°. (4) → (1) except αO = −2°, αR = 2°, αC = 1°. (5) → (1) except αO = −1°, αR = 1°, αC = 1°.

Fig. 6
Fig. 6

Hologram aberrations: spherical, ΔS, coma, ΔC, astigmatism, ΔA, and total, ΔT, at ϕ = 90° vs hologram pupil coordinate (X-direction). The image is virtual. λO = 4880 Å; RO = 0.30 m, αO = 10.0°; RR → ∞, αR = −10.0°; λC = 6328 Å; RC → ∞, αC = −10.0°; RIV = 0.23135 m, αIV = 16.063°; f/4.627.

Fig. 7
Fig. 7

Hologram aberrations: spherical, ΔS, astigmatism, ΔA, and total, ΔT, at ϕ = 90° vs hologram pupil coordinate (X-direction) when aberrations are balanced for the virtual image by the wavelength compensation technique. λO = 4880 Å; RR = 2.463 m, αR = −5.841°; RO = 0.33596 m, αO = 9.558°; λC = 6328 Å; RC → ∞, αC = −10.0°. RIV = 0.30 m, αIV = + 10.0°; |XMAX| = 0.05 m.

Fig. 8
Fig. 8

Magnitude of the wavefront deviation from the gaussian sphere, |ΔG| vs the angle of the reconstruction beam. Comparison of the chromatic aberrations caused by a wavelength shift from λO = 4880 Å to λC = 6328 Å when (1) the hologram is reconstructed with no modification in the construction or reconstruction beams to compensate for wavelength shift, (2) the curvature and angle of the real-image beam is matched to the object beam while the reconstruction beam remains a plane wave. The reference beam is no longer a plane wave but found to satisfy the gaussian image relations for the three other beams, and (3) the aberrations are balanced as discussed in the text to minimize chromatic aberrations. The numbered curves correspond as follows: (1) λO = 4880 Å; RO = 0.01 m, αO = αC; RR → ∞, αR = −αC; λC = 6328 Å; RC → ∞, RIR = 0.00771 m; |XMAX| = 0.02 m, f/3.86. (2) λO = 4880 Å; RO = 0.01 m, αO = αC; λC = 6328 Å; RC → ∞; RIR = −0.01 m, αIR = −αC; |XMAX| = 0.02 m, f/5.0. (3) λO = 4880 Å; RO, αO, RC, αC → found by aberration balancing technique; λC = 6328 Å; RC → ∞; RIR = − 0.01, αIR = − αC; |XMAX| = 0.02 m, f/5.0.

Fig. 9
Fig. 9

Magnitude of the wavefront deviation from the gaussian sphere, |ΔG|, vs the angle of the reconstruction and real-image beams. The wavelength shift is from 4880 Å to 6328 Å and the aberrations have been decreased using the aberration-balancing technique. Curve (1) corresponds to the following hologram geometry: λO = 4880 Å; RO, αO, RC, αC → found by the aberration balancing technique; λC = 6328 Å; RC → ∞; RIR = 0.1 m, αIR = −αC; f/3.0. The other curves are the same as (1) with the exception of RIR and f# as noted.

Fig. 10
Fig. 10

Wavefront deviation, Δ, at ϕ = 90° and X = |XMAX| vs deviation from the real-image reconstruction angle for the off-axis hologram. λO = λC = 6328 Å; RO = 0.20 m, αO = 5.0°; RR → ∞, αR = −5.0°; RC → ∞, αC = 5.0° ± |ΔαC|; f/5.0.

Fig. 11
Fig. 11

Wavefront deviation, Δ, at X = |XMAX| and ϕ = 90° for ΔA, ΔS, and ΔC vs deviation from the real image reconstruction angle. The hologram has been compensated for chromatic aberrations by the technique discussed in the text. λO = 4880 Å; RO = 0.22535 m, αO = 4.845°; RR = 1.7189 m, αR = −2.864°; λC = 6328 Å; RC → ∞ αC = 5.0° ± |ΔαC|; RIR = −0.20 m; f/5.0.

Fig. 12
Fig. 12

The hologram reconstruction geometry that is used to determine variations of total wavefront deviation, ΔG, and Bragg angle deviation, ΔαBr, as a function of the reconstruction angle αC and real-image beam angle αIR. This geometry is used in Fig. 13.

Fig. 13
Fig. 13

Magnitude of the wavefront deviation from the gaussian sphere, |ΔG|, and magnitude of the deviation of the angle of the reconstruction beam at the center of the hologram from the Bragg angle, |ΔαBr|, vs the angles of the reconstruction beams, αC and αIR for the hologram subject to the balancing technique discussed in the text. The hologram parameters are: λO = 4880 Å; RR, αR, RO, αO → found by the aberration balancing technique; λC = 6328 Å; RC → ∞, RIR = −0.1 m; f/5.0.

Fig. 14
Fig. 14

Magnitude of the wavefront deviation from the gaussian sphere, |ΔG|, and the magnitude of the deviation of the angle of the reconstruction beam at the center of the hologram from the Bragg angle, |ΔαBr|, vs the construction wavelength, λO. The real image is subject to the balancing technique discussed in the text. The hologram parameters are: (1)–(1′) RR, αR, RO, αO → found by the aberration balancing technique; λC = 6328 Å; RC → ∞, αC = 5.0°; RIR = 0.1 m, αIR = −5.0°; f/5.0, (1) → |ΔG|, (1′) → |ΔαBr|; (2)–(2′) → (1)–(1′) except αC = 0°, αIR = −10.0°; (3)–(3′) → (1)–(1′) except αC = −17.5°, αIR = −27.5°.

Equations (14)

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MAX Δ S = - MAX Δ A ,
Δ C = 0 ,             all x , y hologram surface .
Δ MAX = 1 λ C [ - 1 8 ( ρ m 4 S ) + 1 2 ( ρ m 3 C X ) - 1 2 ( ρ m 2 A X ) ] .
1 R 1 = 1 R I R - 1 R C ,
S C = sin α I R - sin α C ,
1 R 2 = 1 R C 3 - 1 R I R 3 ,
1 R 3 = sin α C R C 2 - sin α I R R I R 2 ,
1 R 4 = sin 2 α C R C - sin 2 α I R R I R .
- 1 4 ρ m 2 [ 1 R 2 - μ ( 1 R O 3 - 1 R R 3 ) ] = 1 R 4 - μ ( sin 2 α O R O - sin 2 α R R R ) ,
( 1 / R 3 ) - μ ( sin α O R O 2 - sin α R R R 2 ) = 0 ,
( 1 / R O ) = ( 1 / R R ) - 1 / ( μ R 1 ) ,
sin α O = sin α R - ( S C / μ ) .
sin α O = ( R O 2 μ 2 R 1 2 ) / R C + S C μ 2 R 1 2 + 2 S C R 1 μ R O + S C R O 2 - ( 2 μ 2 R 1 R O - R O 2 μ ) ;
( - ρ m 2 μ 2 4 R 2 - ρ m 2 4 R 1 3 - μ 2 R 4 - μ 4 R 1 3 R 3 2 ) R O 4 + { - ρ m 2 [ ( 7 4 ) μ R 1 2 + μ 3 R 1 R 2 ] - 4 μ 3 R 1 R 4 + 2 μ 3 R 1 2 S C R 3 + S C 2 μ } × R O 3 + [ - ( 19 4 ) ρ m 2 μ 2 R 1 - ρ m 2 μ 4 R 1 2 R 2 - 4 μ 4 R 1 2 R 4 + 2 S C μ 4 R 1 3 R 3 + 4 S C 2 R 1 μ 2 ] R O 2 + ( - 6 ρ m 2 μ 3 + 6 S C 2 μ 3 R 1 2 ) R O - 3 ρ m 2 μ 4 R 1 + 3 μ 4 S C 2 R 1 3 = 0.

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