Abstract

When holograms are used as optical elements two of their properties are of interest: dispersion and aberrations. This paper uses a general hologram analysis computer program based on ray tracing to analyze these properties. Numerical examples are given for both single and multiple element systems where the goal is to reduce the lateral dispersion. A significant reduction in this dispersion is accomplished with the introduction of a plane grating adjacent to the hologram. The problem of reducing the longitudinal dispersion is more complex, and it is studied in the paper by minimizing the variation in focus distance as a function of the reconstruction wavelength for two- and three-element on-axis hologram systems. To optimize the reduction in dispersion and design a system with low aberrations, the hologram ray tracing programs were modified to be driven by an optimization program. These techniques realized a three-element on-axis f/3.84 system with aberrations and dispersions less than 0.32λc and 87 μ, respectively, when operated over a band from 4385 Å to 7060 Å.

© 1972 Optical Society of America

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References

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  1. J. N. Latta, Appl. Opt. 10, 2698 (1971).
    [CrossRef] [PubMed]
  2. G. A. Korn, T. M. Korn, Mathematical Handbook for Scientists and Engineers (McGraw-Hill, New York, 1968).
  3. C. B. Burckhardt, Bell Syst. Tech. J. 45, 1841 (1966).
  4. D. J. De Bitteto, Appl. Phys. Lett. 9, 417 (1966).
    [CrossRef]
  5. E. B. Champagne, J. Opt. Soc. Am. 57, 51 (1967).
    [CrossRef]
  6. E. B. Champagne, “A Qualitative and Quantitative Study of Holographic Imaging,” Ph.D. Thesis, Ohio State University, Columbus, Ohio (July1967) (U. Microfilms 67-10876).
  7. R. W. Meier, J. Opt. Soc. Am. 57, 895 (1967).
    [CrossRef]
  8. J. N. Latta, Appl. Opt. 10, 599 (1971).
    [CrossRef] [PubMed]
  9. J. N. Latta, Appl. Opt. 10, 609 (1971).
    [CrossRef] [PubMed]
  10. R. Hooke, T. A. Jeeves, J. Assoc. Comput. Mach. 8, 212 (1961).
    [CrossRef]
  11. Because of the complexity of the equations needed to solve for the variables of interest in the three-element problem, we have used extensively the algebraic manipulation language REDUCE2.12 Such a tool is an invaluable aid in manipulating the long expressions and taking the derivatives, etc.
  12. A. C. Hern, REDUCE2 Users Manual, Standard Artificial Intelligence Project Memo. AIM-133 (October1970).

1971 (3)

1967 (2)

1966 (2)

C. B. Burckhardt, Bell Syst. Tech. J. 45, 1841 (1966).

D. J. De Bitteto, Appl. Phys. Lett. 9, 417 (1966).
[CrossRef]

1961 (1)

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

Burckhardt, C. B.

C. B. Burckhardt, Bell Syst. Tech. J. 45, 1841 (1966).

Champagne, E. B.

E. B. Champagne, J. Opt. Soc. Am. 57, 51 (1967).
[CrossRef]

E. B. Champagne, “A Qualitative and Quantitative Study of Holographic Imaging,” Ph.D. Thesis, Ohio State University, Columbus, Ohio (July1967) (U. Microfilms 67-10876).

De Bitteto, D. J.

D. J. De Bitteto, Appl. Phys. Lett. 9, 417 (1966).
[CrossRef]

Hern, A. C.

A. C. Hern, REDUCE2 Users Manual, Standard Artificial Intelligence Project Memo. AIM-133 (October1970).

Hooke, R.

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

Jeeves, T. A.

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

Korn, G. A.

G. A. Korn, T. M. Korn, Mathematical Handbook for Scientists and Engineers (McGraw-Hill, New York, 1968).

Korn, T. M.

G. A. Korn, T. M. Korn, Mathematical Handbook for Scientists and Engineers (McGraw-Hill, New York, 1968).

Latta, J. N.

Meier, R. W.

Appl. Opt. (3)

Appl. Phys. Lett. (1)

D. J. De Bitteto, Appl. Phys. Lett. 9, 417 (1966).
[CrossRef]

Bell Syst. Tech. J. (1)

C. B. Burckhardt, Bell Syst. Tech. J. 45, 1841 (1966).

J. Assoc. Comput. Mach. (1)

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

J. Opt. Soc. Am. (2)

Other (4)

G. A. Korn, T. M. Korn, Mathematical Handbook for Scientists and Engineers (McGraw-Hill, New York, 1968).

E. B. Champagne, “A Qualitative and Quantitative Study of Holographic Imaging,” Ph.D. Thesis, Ohio State University, Columbus, Ohio (July1967) (U. Microfilms 67-10876).

Because of the complexity of the equations needed to solve for the variables of interest in the three-element problem, we have used extensively the algebraic manipulation language REDUCE2.12 Such a tool is an invaluable aid in manipulating the long expressions and taking the derivatives, etc.

A. C. Hern, REDUCE2 Users Manual, Standard Artificial Intelligence Project Memo. AIM-133 (October1970).

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Figures (18)

Fig. 1
Fig. 1

Example of a multiple hologram element geometry that can be analyzed by the hologram ray tracing programs.

Fig. 2
Fig. 2

The combination of a hologram grating (1) and a zone plate hologram (2) used to compensate for the lateral dispersion.

Fig. 3
Fig. 3

The longitudinal dispersion and aberrations of an in-line hologram.

Fig. 4
Fig. 4

The longitudinal and lateral dispersion and aberrations of an off-axis hologram.

Fig. 5
Fig. 5

The combination of a hologram zone plate and grating to reduce the lateral dispersion.

Fig. 6
Fig. 6

The combination of a hologram grating and zone plate to reduce the lateral dispersion.

Fig. 7
Fig. 7

The combination of a hologram grating and zone plate to appear as an in-line hologram and reduce the lateral dispersion.

Fig. 8
Fig. 8

The combination of two holograms designed with aberration balancing and a hologram grating to reduce lateral dispersion and aberrations.

Fig. 9
Fig. 9

The basic two- and three-element geometries used for designing low dispersion hologram optical element systems.

Fig. 10
Fig. 10

The focus distance, RI and RI2, of a single and two-element system vs μ for the broadband system.

Fig. 11
Fig. 11

Block diagram of the relationship between the optimization program and the ray tracking program.

Fig. 12
Fig. 12

Ray paths for the two-element broadband hologram lens when the object is at infinity and normal to the first hologram.

Fig. 13
Fig. 13

Magnitude of the wavefront deviation from the Gaussian sphere, |ΔG|, vs μ for the narrow band (NB) and broadband (BB) geometries when αC1 = 0.

Fig. 14
Fig. 14

The partial derivative of RI3 with respect to μ of a three-element hologram system vs μ. The parameters are A12 = A23 = 5 cm and μ1 = 0.8, μ2 = 1.1.

Fig. 15
Fig. 15

The focus distance, RI3, of the three-element system vs μ. The limits of the broadband system are shown in the outer dashed lines while the narrow-band system is confined to the inner dashed lines.

Fig. 16
Fig. 16

Ray paths for the three-element broadband hologram lens when the object is at infinity and normal to the first hologram.

Fig. 17
Fig. 17

Magnitude of the wavefront deviation from the Gaussian sphere, |ΔG|, vs μ for the narrow-band (NB) and broadband geometries (BB) when αC1 = 0°.

Fig. 18
Fig. 18

Magnitude of the naximum wavefront deviation from the Gaussian sphere, |ΔG|, vs ±αC for both the NB and BB geometries.

Tables (2)

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Table I Two-Element Hologram Lens Parameters

Tables Icon

Table II Three-Element Hologram Lens Parameters

Equations (27)

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sin α I 2 = sin α C 2 + μ ( sin α O 2 - sin α R 2 ) ,
sin α I 1 = sin α C 1 + μ ( sin α O 1 - sin α R 1 ) ,
α O 1 = α R 2 , α R 1 = α O 2 ,
sin α I 2 = sin α C 1 + μ ( sin α O 1 - sin α R 1 ) + μ ( sin α O 2 - sin α R 2 ) ,             sin α I 2 = sin α C 1 ,
1 / R I 1 = μ / f 1 ,
R C 2 = R I 1 ,
1 / R I 2 = ( 1 / R C 2 ) + ( μ / f 2 ) ,
1 / f 1 = ± [ ( 1 / R O 1 ) - ( 1 / R R 1 ) ] ,
1 / f 2 = ± [ ( 1 / R O 2 ) - ( 1 / R R 2 ) ] .
R I 2 = f 2 f 1 / [ μ ( f 2 + f 1 ) ] .
R C 3 = R I 2 ,
1 / R I 3 = ( 1 / R C 3 ) + ( μ / f 3 ) ,
R I 3 = f 3 f 2 f 1 / [ μ ( f 3 f 2 + f 3 f 1 + f 2 f 1 ) ] .
R I 2 μ = - f 2 f 1 ( f 2 + f 1 ) μ 2 ( f 2 2 + 2 f 2 f 1 + f 1 2 ) ,
R I 3 μ = - f 3 f 2 f 1 ( f 3 f 2 + f 3 f 1 + f 2 f 1 ) μ 2 ( f 3 2 f 2 2 + 2 f 3 2 f 2 f 1 + f 3 2 f 1 2 + 2 f 3 f 2 2 f 1 + 2 f 3 f 2 f 1 2 + f 2 2 f 1 2 ) .
R C 2 = R I 1 + A 12 .
R I 2 = μ f 2 A 12 + f 2 f 1 μ 2 A 12 + μ ( f 2 + f 1 ) ,
R I 2 μ = - μ 2 f 2 A 12 2 - 2 μ f 2 f 1 A 12 - f 2 f 1 ( f 2 + f 1 ) μ 4 A 12 2 + 2 μ 3 A 12 ( f 2 + f 1 ) + μ 2 ( f 2 2 + 2 f 2 f 1 + f 1 2 ) .
( μ - μ 1 ) ( μ - Q ) = μ 2 + μ ( - μ 1 - Q ) + μ 1 Q ,
- μ 1 - Q = ( 2 f 1 / A 12 ) ,
μ 1 Q = [ f 1 ( f 2 + f 1 ) ] / A 12 2 ,
f 2 = - [ ( f 1 2 + μ 1 2 A 12 2 + 2 f 1 μ 1 A 12 ) / f 1 ] .
f 1 3 + 3 μ 1 A 12 f 1 2 + μ 1 2 A 12 ( 3 A 12 - R I 2 ) f 1 + μ 1 3 A 12 2 ( A 12 - R I 2 ) = 0.
R C 3 = R I 2 + A 23 .
R I 3 = μ 2 f 3 A 23 A 12 + μ ( f 3 A 23 f 2 + f 3 A 23 f 1 + f 3 f 2 A 12 ) + f 3 f 2 f 1 . μ 3 A 23 A 12 + μ 2 ( f 3 A 12 + A 23 f 2 + A 23 f 1 + f 2 A 12 ) + μ ( f 3 f 2 + f 3 f 1 + f 2 f 1 ) .
R I 3 μ = - μ 4 f 3 A 23 2 A 12 2 + μ 3 ( - 2 f 3 A 23 2 f 2 A 12 - 2 f 3 A 23 2 A 12 f 1 - 2 f 3 A 23 f 2 A 12 2 ) + μ 2 ( - f 3 2 f 2 A 12 2 - f 3 A 23 2 f 2 2 - 2 f 3 A 23 2 f 2 f 1 - f 3 A 23 2 f 1 2 - 2 f 3 A 23 f 2 2 A 12 - 4 f 3 A 23 f 2 A 12 f 1 - f 3 f 2 2 A 12 2 ) + μ ( - 2 f 3 2 f 2 A 12 f 1 - 2 f 3 A 23 f 2 2 f 1 - 2 f 3 A 23 f 2 f 1 2 - 2 f 3 f 2 2 A 12 f 1 ) - f 3 2 f 2 2 f 1 - f 3 2 f 2 f 1 2 - f 3 f 2 2 f 1 2 μ 6 A 23 2 A 12 2 + μ 5 ( 2 f 3 A 23 A 12 2 + 2 A 23 2 f 2 A 12 + 2 A 23 2 A 12 f 1 + 2 A 23 f 2 A 12 2 ) + μ 4 ( f 3 2 A 12 2 + 4 f 3 A 23 f 2 A 12 + 4 f 3 A 23 A 12 f 1 + 2 f 3 f 2 A 12 2 + A 23 2 f 2 2 + 2 A 23 2 f 2 f 1 + A 23 2 f 1 2 + 2 A 23 f 2 2 A 12 + 4 A 23 f 2 A 12 f 1 + f 2 2 A 12 2 ) + μ 3 ( 2 f 3 2 f 2 A 12 + 2 f 3 2 A 12 f 1 + 2 f 3 A 23 f 2 2 + 4 f 3 A 23 f 2 f 1 + 2 f 3 A 23 f 1 2 + 2 f 3 f 2 2 A 12 + 4 f 3 f 2 A 12 f 1 + 2 A 23 f 2 2 f 1 + 2 A 23 f 2 f 1 2 + 2 f 2 2 A 12 f 1 ) + μ 2 ( f 3 2 f 2 2 + 2 f 3 2 f 2 f 1 + f 3 2 f 1 2 + 2 f 3 f 2 2 f 1 + 2 f 3 f 2 f 1 2 + f 2 2 f 1 2 ) ,
( μ - μ 1 ) ( μ - μ 2 ) ( A μ 2 + B μ + C μ ) = μ 4 A + μ 3 ( - A μ 2 - A μ 1 + B ) + μ 2 ( A μ 2 μ 1 - B μ 2 - B μ 1 + C ) + μ ( B μ 2 μ 1 - C μ 2 - C μ 1 ) + C μ 2 μ 1 .

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