Abstract

Numerical values are shown for the aberrations that are present in holograms which may be constructed and reconstructed from a plane wave or point source. This study centers on four types of hologram geometries: in-line, off-axis, near-image plane, and lensless-fourier transform. The aberrations illustrated in this paper have been caused by a spatial deviation of the reconstruction beam from the location where either an ideal aberration free virtual or real image is reconstructed. In addition to the above analysis of aberrations a close parallelism between the bending characteristics of lenses and the construction geometry of holograms is noted. This analogy between lenses and holograms illustrates the flexibility with which the aberrations of a hologram may be modified by changes in the recording geometry.

© 1971 Optical Society of America

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  1. The major portion of this program was written while the author was at the Bell Telephone Laboratories, Murray Hill, N.J.
  2. J. A. Armstrong, IBM J. Res. Devel. 9, 171 (1965).
    [CrossRef]
  3. M. Marquet, H. Royer, Comp. Rend. 260, 6051 (1965).
  4. R. W. Meier, J. Opt. Soc. Amer. 55, 987 (1965).
  5. E. N. Leith, J. Upatnieks, K. A. Haines, J. Opt. Soc. Amer. 55, 981 (1965).
  6. A. Offner, J. Opt. Soc. Amer. 56, 1509 (1966).
    [CrossRef]
  7. E. B. Champagne, J. Opt. Soc. Amer. 57, 51 (1967).
    [CrossRef]
  8. E. B. Champagne, A Qualitative and Quantitative Study of Holographic Imaging, Ph.D. Thesis, Ohio State University, July1967 (U. Microfilms, 67-10876).
  9. K. A. Haines, The Analysis and Application of Hologram Interferometry, Ph.D. Thesis, University of Michigan, January1967 (U. Microfilms, 67-8264).
  10. I. A. Abramowitz, J. M. Ballantyne, J. Opt. Soc. Amer. 57, 1522 (1967).
    [CrossRef]
  11. Iu. Uder, Eesti nsv Teaduste Akadeemia, Toimetised, Fuusiku-Matemaatika 17, 190 (1968).
  12. I. A. Abramowitz, Appl. Opt. 8, 403 (1969).
    [CrossRef] [PubMed]
  13. I. A. Abramowitz, Design of Holographic Systems by Ray Tracing, Ph.D. Thesis, Cornell University, 1968 (U. Microfilms, 68-4656).
  14. V. I. Mandrosov, Opt. Spectrosc. 26, 254 (1969).
  15. The unsymmetrical nature of hologram aberrations was first pointed out by Offner6 and brought to the author’s attention by A. Vander Lugt. The characteristics of nonrotationally symmetric optical systems have been discussed by Barakat and Houston.16
  16. R. Barakat, A. Houston, Opt. Acta 13, 1 (1966).
    [CrossRef]
  17. E. B. Champagne, N. G. Massey, Appl. Opt. 8, 1879 (1969).
    [CrossRef] [PubMed]
  18. J. T. Winthrop, C. R. Worthington, Phys. Lett. 15, 124 (1965).
    [CrossRef]
  19. J. T. Winthrop, C. R. Worthington, Phys. Lett. 21, 413 (1966).
    [CrossRef]
  20. G. W. Stroke, Appl. Phys. Lett. 6, 201 (1965).
    [CrossRef]
  21. D. Gabor, Proc. Royal Soc. London, Ser. A. 197, 454 (1949).
    [CrossRef]
  22. G. B. Brandt, Appl. Opt. 8, 1421 (1969).
    [CrossRef] [PubMed]
  23. K. A. Stetson, Optik 29, 520 (1969).
  24. A. E. Conrady, Applied Optics and Optical Design, Parts 1 and 2 (Dover Publications, New York, 1957 and 1960).
  25. F. A. Jenkins, H. E. White, Fundamentals of Optics (McGraw-Hill, New York, 1957).
  26. W. J. Smith, Modern Optical Engineering (McGraw-Hill, New York, 1966).
  27. T. Smith, Proc. Phys. Soc. London 27, 485 (1915). Also: National Physical Laboratory Collected Researches 13, 179 (1916).
    [CrossRef]
  28. I. C. Gardner, Scientific Papers of the Bureau of Standards 22, 73 (1927).
    [CrossRef]
  29. B. K. Johnson, Optical Design and Lens Computation (Hatton Press, London, 1948).
  30. H. H. Jopkins, Wave Theory of Aberrations (Oxford University Press, London, 1950).
  31. A. Marechal, in Proc. Symp. on Optical Image Evaluation (1961), NBS Circ. 526 (U.S. GPO, Washington, D.C., 1954), p. 9.
  32. A. Marechal, M. Francon, Diffraction Structure des Images (Editions of La Revue d’Optique, Paris, 1960).
  33. H. H. Hopkins, Opt. Acta 13, 343 (1966).
    [CrossRef]
  34. A. V. Lenskii, Sov. J. Opt. Technol. 36, 259 (1969).
  35. H. H. Hopkins, Proc. Phys. Soc. London 69B, 562 (1956).
  36. H. H. Hopkins, Proc. Phys. Soc. London 79, 889 (1962).
    [CrossRef]
  37. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).
  38. H. Kogelnik, Bell System Tech. J. 48, 2909 (1969).

1969 (7)

K. A. Stetson, Optik 29, 520 (1969).

A. V. Lenskii, Sov. J. Opt. Technol. 36, 259 (1969).

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

V. I. Mandrosov, Opt. Spectrosc. 26, 254 (1969).

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

G. B. Brandt, Appl. Opt. 8, 1421 (1969).
[CrossRef] [PubMed]

E. B. Champagne, N. G. Massey, Appl. Opt. 8, 1879 (1969).
[CrossRef] [PubMed]

1968 (1)

Iu. Uder, Eesti nsv Teaduste Akadeemia, Toimetised, Fuusiku-Matemaatika 17, 190 (1968).

1967 (2)

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

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

1966 (4)

A. Offner, J. Opt. Soc. Amer. 56, 1509 (1966).
[CrossRef]

H. H. Hopkins, Opt. Acta 13, 343 (1966).
[CrossRef]

R. Barakat, A. Houston, Opt. Acta 13, 1 (1966).
[CrossRef]

J. T. Winthrop, C. R. Worthington, Phys. Lett. 21, 413 (1966).
[CrossRef]

1965 (6)

G. W. Stroke, Appl. Phys. Lett. 6, 201 (1965).
[CrossRef]

J. T. Winthrop, C. R. Worthington, Phys. Lett. 15, 124 (1965).
[CrossRef]

J. A. Armstrong, IBM J. Res. Devel. 9, 171 (1965).
[CrossRef]

M. Marquet, H. Royer, Comp. Rend. 260, 6051 (1965).

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

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

1962 (1)

H. H. Hopkins, Proc. Phys. Soc. London 79, 889 (1962).
[CrossRef]

1956 (1)

H. H. Hopkins, Proc. Phys. Soc. London 69B, 562 (1956).

1949 (1)

D. Gabor, Proc. Royal Soc. London, Ser. A. 197, 454 (1949).
[CrossRef]

1927 (1)

I. C. Gardner, Scientific Papers of the Bureau of Standards 22, 73 (1927).
[CrossRef]

1915 (1)

T. Smith, Proc. Phys. Soc. London 27, 485 (1915). Also: National Physical Laboratory Collected Researches 13, 179 (1916).
[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).

Armstrong, J. A.

J. A. Armstrong, IBM J. Res. Devel. 9, 171 (1965).
[CrossRef]

Ballantyne, J. M.

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

Barakat, R.

R. Barakat, A. Houston, Opt. Acta 13, 1 (1966).
[CrossRef]

Brandt, G. B.

Champagne, E. B.

E. B. Champagne, N. G. Massey, Appl. Opt. 8, 1879 (1969).
[CrossRef] [PubMed]

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

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

Conrady, A. E.

A. E. Conrady, Applied Optics and Optical Design, Parts 1 and 2 (Dover Publications, New York, 1957 and 1960).

Francon, M.

A. Marechal, M. Francon, Diffraction Structure des Images (Editions of La Revue d’Optique, Paris, 1960).

Gabor, D.

D. Gabor, Proc. Royal Soc. London, Ser. A. 197, 454 (1949).
[CrossRef]

Gardner, I. C.

I. C. Gardner, Scientific Papers of the Bureau of Standards 22, 73 (1927).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

Haines, K. A.

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

K. A. Haines, The Analysis and Application of Hologram Interferometry, Ph.D. Thesis, University of Michigan, January1967 (U. Microfilms, 67-8264).

Hopkins, H. H.

H. H. Hopkins, Opt. Acta 13, 343 (1966).
[CrossRef]

H. H. Hopkins, Proc. Phys. Soc. London 79, 889 (1962).
[CrossRef]

H. H. Hopkins, Proc. Phys. Soc. London 69B, 562 (1956).

Houston, A.

R. Barakat, A. Houston, Opt. Acta 13, 1 (1966).
[CrossRef]

Jenkins, F. A.

F. A. Jenkins, H. E. White, Fundamentals of Optics (McGraw-Hill, New York, 1957).

Johnson, B. K.

B. K. Johnson, Optical Design and Lens Computation (Hatton Press, London, 1948).

Jopkins, H. H.

H. H. Jopkins, Wave Theory of Aberrations (Oxford University Press, London, 1950).

Kogelnik, H.

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

Leith, E. N.

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

Lenskii, A. V.

A. V. Lenskii, Sov. J. Opt. Technol. 36, 259 (1969).

Mandrosov, V. I.

V. I. Mandrosov, Opt. Spectrosc. 26, 254 (1969).

Marechal, A.

A. Marechal, M. Francon, Diffraction Structure des Images (Editions of La Revue d’Optique, Paris, 1960).

A. Marechal, in Proc. Symp. on Optical Image Evaluation (1961), NBS Circ. 526 (U.S. GPO, Washington, D.C., 1954), p. 9.

Marquet, M.

M. Marquet, H. Royer, Comp. Rend. 260, 6051 (1965).

Massey, N. G.

Meier, R. W.

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

Offner, A.

A. Offner, J. Opt. Soc. Amer. 56, 1509 (1966).
[CrossRef]

Royer, H.

M. Marquet, H. Royer, Comp. Rend. 260, 6051 (1965).

Smith, T.

T. Smith, Proc. Phys. Soc. London 27, 485 (1915). Also: National Physical Laboratory Collected Researches 13, 179 (1916).
[CrossRef]

Smith, W. J.

W. J. Smith, Modern Optical Engineering (McGraw-Hill, New York, 1966).

Stetson, K. A.

K. A. Stetson, Optik 29, 520 (1969).

Stroke, G. W.

G. W. Stroke, Appl. Phys. Lett. 6, 201 (1965).
[CrossRef]

Uder, Iu.

Iu. Uder, Eesti nsv Teaduste Akadeemia, Toimetised, Fuusiku-Matemaatika 17, 190 (1968).

Upatnieks, J.

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

White, H. E.

F. A. Jenkins, H. E. White, Fundamentals of Optics (McGraw-Hill, New York, 1957).

Winthrop, J. T.

J. T. Winthrop, C. R. Worthington, Phys. Lett. 21, 413 (1966).
[CrossRef]

J. T. Winthrop, C. R. Worthington, Phys. Lett. 15, 124 (1965).
[CrossRef]

Worthington, C. R.

J. T. Winthrop, C. R. Worthington, Phys. Lett. 21, 413 (1966).
[CrossRef]

J. T. Winthrop, C. R. Worthington, Phys. Lett. 15, 124 (1965).
[CrossRef]

Appl. Opt. (3)

Appl. Phys. Lett. (1)

G. W. Stroke, Appl. Phys. Lett. 6, 201 (1965).
[CrossRef]

Bell System Tech. J. (1)

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

Comp. Rend. (1)

M. Marquet, H. Royer, Comp. Rend. 260, 6051 (1965).

Eesti nsv Teaduste Akadeemia, Toimetised, Fuusiku-Matemaatika (1)

Iu. Uder, Eesti nsv Teaduste Akadeemia, Toimetised, Fuusiku-Matemaatika 17, 190 (1968).

IBM J. Res. Devel. (1)

J. A. Armstrong, IBM J. Res. Devel. 9, 171 (1965).
[CrossRef]

J. Opt. Soc. Amer. (5)

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

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

A. Offner, J. Opt. Soc. Amer. 56, 1509 (1966).
[CrossRef]

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

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

Opt. Acta (2)

H. H. Hopkins, Opt. Acta 13, 343 (1966).
[CrossRef]

R. Barakat, A. Houston, Opt. Acta 13, 1 (1966).
[CrossRef]

Opt. Spectrosc. (1)

V. I. Mandrosov, Opt. Spectrosc. 26, 254 (1969).

Optik (1)

K. A. Stetson, Optik 29, 520 (1969).

Phys. Lett. (2)

J. T. Winthrop, C. R. Worthington, Phys. Lett. 15, 124 (1965).
[CrossRef]

J. T. Winthrop, C. R. Worthington, Phys. Lett. 21, 413 (1966).
[CrossRef]

Proc. Phys. Soc. London (3)

T. Smith, Proc. Phys. Soc. London 27, 485 (1915). Also: National Physical Laboratory Collected Researches 13, 179 (1916).
[CrossRef]

H. H. Hopkins, Proc. Phys. Soc. London 69B, 562 (1956).

H. H. Hopkins, Proc. Phys. Soc. London 79, 889 (1962).
[CrossRef]

Proc. Royal Soc. London, Ser. A. (1)

D. Gabor, Proc. Royal Soc. London, Ser. A. 197, 454 (1949).
[CrossRef]

Scientific Papers of the Bureau of Standards (1)

I. C. Gardner, Scientific Papers of the Bureau of Standards 22, 73 (1927).
[CrossRef]

Sov. J. Opt. Technol. (1)

A. V. Lenskii, Sov. J. Opt. Technol. 36, 259 (1969).

Other (13)

B. K. Johnson, Optical Design and Lens Computation (Hatton Press, London, 1948).

H. H. Jopkins, Wave Theory of Aberrations (Oxford University Press, London, 1950).

A. Marechal, in Proc. Symp. on Optical Image Evaluation (1961), NBS Circ. 526 (U.S. GPO, Washington, D.C., 1954), p. 9.

A. Marechal, M. Francon, Diffraction Structure des Images (Editions of La Revue d’Optique, Paris, 1960).

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

The unsymmetrical nature of hologram aberrations was first pointed out by Offner6 and brought to the author’s attention by A. Vander Lugt. The characteristics of nonrotationally symmetric optical systems have been discussed by Barakat and Houston.16

A. E. Conrady, Applied Optics and Optical Design, Parts 1 and 2 (Dover Publications, New York, 1957 and 1960).

F. A. Jenkins, H. E. White, Fundamentals of Optics (McGraw-Hill, New York, 1957).

W. J. Smith, Modern Optical Engineering (McGraw-Hill, New York, 1966).

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

K. A. Haines, The Analysis and Application of Hologram Interferometry, Ph.D. Thesis, University of Michigan, January1967 (U. Microfilms, 67-8264).

The major portion of this program was written while the author was at the Bell Telephone Laboratories, Murray Hill, N.J.

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 (13)

Fig. 1
Fig. 1

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

Fig. 2
Fig. 2

Magnitude of the wavefront deviation from the gaussian sphere, |ΔT|, at ϕ = 90° vs the deviation of the reconstruction beam incident angle, |ΔαC|, from the reference beam angle αR for the off-axis and near-image plane hologram. The image is virtual. The curve numbered (1) corresponds to the following hologram geometry: λO = λC = 6328 Å; RO = 0.1 m, αO = −25.0°; RR → ∞, αR = 25.0°; RC → ∞, αC = 25.0° ± |ΔαC|; and f/1.0 The remaining curves are the same as (1) with the exception of RO and f# as noted.

Fig. 3
Fig. 3

Magnitude of the wavefront deviation from the gaussian sphere, |ΔT|, at ϕ = 90° vs the deviation of the reconstruction beam incident angle, |ΔαC|, from the reference beam angle, αR, for the off-axis and near-image plane holograms. The image is virtual. Each of the numbered curves corresponds as follows: (1) λO = λC = 6328 Å; αO = −40.0°; RR → ∞, αR = 40.0°; RC → ∞, αC = 40.0° − |ΔαC|; and f/1.0. (1′) → (1) except αC = 40.0° + |ΔαC|. (2) → (1) except αO = −25.0°, αR = 25.0°; αC = 25.0° − |ΔαC|. (2′) → (2) except αC = 25.0° + |ΔαC|. (3) → (1) except αO = −10°, αR = 10.0°; αC = 10.0° ± |ΔαC|. Each family of curves corresponds to a different value of RO as indicated.

Fig. 4
Fig. 4

Magnitude of the wavefront deviation from the gaussian sphere, |ΔT|, at ϕ = 90° vs hologram f# for the off-axis hologram. The image is virtual. Each of the numbered curves corresponds as follows: (1) λO = λC = 6328 Å; RO = 0.5 m, αO = −25.0°; RR → ∞, αR = 25.0°; RC → ∞, αC = 25.0° + ΔαC; and ΔαC = 0.01°. (2) → (1) except ΔαC = 0.1°. (3) → (1) except ΔαC = 0.25°. (4) → (1) except ΔαC = 0.5°. (5) → (1) except ΔαC = 1.0°. (6) → (1) except ΔαC = 2.0°. (7) → (1) except ΔαC = 4.0°. (8) → (1) except ΔαC = 6.0°.

Fig. 5
Fig. 5

Magnitude of the wavefront deviation from the gaussian sphere, |ΔT|, at ϕ = 90° vs hologram f# for the in-line hologram. The image is virtual. Each of the numbered curves corresponds to the same parameters as in Fig. 4 except that αO = αR = 0°, αC = +ΔαC.

Fig. 6
Fig. 6

Magnitude of the spherical aberration wavefront deviation, |ΔS|, vs the reconstruction beam radial distance to the hologram center, RC. The image is real. The curves correspond to the following geometry: λO = λC = 6328 Å; αO = 0°; RR → ∞, αR = 0°, αC = 0°; and f/1.0, where RO is denoted on each curve.

Fig. 7
Fig. 7

Magnitude of the astigmatic wavefront deviation, |ΔA|, vs the deviation of the reconstruction angle, |ΔαC|, from the reference beam angle for the LLFT hologram. The image is virtual and found by using the + sign in the gaussian image relations. Each of the numbered curves corresponds as follows: (1) λC = λO = 6328 Å; RO = 0.1 m, αO = −25.0°; RR = 0.1 m, αR = 25.0°; RC = 0.1 m, αC = 25.0° ± |ΔαC|, + ΔαC → −ΔA, − Δαc → ΔA. (2) → (1) except αO = −10°, αR = 10°, αC = 10°, ± |ΔαC|. (3) → (1) except αO = −5°, αR = 5°, αC = 5° ± |ΔαC|. (4) → (1) except αO = −2°, αR = 2°, αC = 2° ± |ΔαC|. (5) → (1) except αO = −1°, αR = 1°, αC = 1° ± |ΔαC|. The value of |XMAX| for the two families of curves is denoted directly below each family.

Fig. 8
Fig. 8

Magnitude of the comatic wavefront deviation, |ΔC|, vs the reconstruction beam radial distance to the hologram center, RC, for the LLFT hologram. The image is virtual and found by using the + sign in the gaussian image relations. Each of the numbered curves corresponds as follows: (1) λO = λC = 6328 Å; RO = 0.1 m, αO = −25.0°; RR = 0.1 m, αR = 25.0°, αC = 25.0°. (2) → (1) except αO = −10°, αR = αC = 10°. (3) → (1) except αO = −5°, αR = αC = 5°. (4) → (1) except αO = −2°, αR = αC = 2°. (5) → (1) except αO = −1°, αR = αC = 1°. The value of |XMAX| for the two families of curves is denoted directly below each family.

Fig. 9
Fig. 9

Comparison of the construction geometries of on-axis holograms and thin lenses with the same q factor.

Figure 10
Figure 10

Wavefront deviation, Δ, at ϕ = 90° and X = |XMAX| vs the hologram q factor for the in-line hologram. The focal length of the hologram is constant. λO = λC = 6328 Å; αR = 0°, αO = 0°; αC → ∞, αC = 4°; RIR = −0.20 m, αIR = 4.0°; and f/5.0.

Fig. 11
Fig. 11

Wavefront deviation, Δ, at ϕ = 90° and X = |XMAX| vs the hologram q factor for the off-axis hologram. The focal length of the hologram is constant. λO = λC = 6328 Å; αR = −5.0°, αO = 5.0°; αC → ∞, αC = 9.0°; RIR = −0.20 m, αIR = −1.024; and f/5.0.

Fig. 12
Fig. 12

Wavefront deviation, Δ, at ϕ = 90° and X = |XMAX| vs the hologram q factor for the off-axis hologram. The focal length of the hologram is constant. λO = λC = 6328 Å; αR = −7.5°, αO = 2.5°; αC → ∞, αC = 6.5°; RIR = −0.20 m, αIR = −3.494°; and f/5.0

Fig. 13
Fig. 13

Wavefront deviation, Δ, at ϕ = 90° and X = |XMAX| vs the hologram q factor for the off-axis hologram. The focal length of the hologram is constant. λO = λC = 6328 Å; αR = −7.5°, αO = 2.5°; RC → ∞, αC = −2.5°; RIR = −0.20 m, αIR −7.50; and f/5.0.

Equations (28)

Equations on this page are rendered with MathJax. Learn more.

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 ) ,
where             μ = ( λ C / λ O ) ,
Δ = Δ S + Δ C + Δ A ,
where             Δ S = - ( 1 / 8 λ C ) ( x 2 + y 2 ) 2 S ,
Δ C = ( 1 / 2 λ C ) ( x 2 + y 2 ) ( x C x + y C y ) ,
Δ A = - ( 1 / 2 λ C ) ( x 2 A x + y 2 A y + x y A x y ) ,
S = 1 R C 3 - 1 R I 3 ± μ m 4 ( 1 R O 3 - 1 R R 3 ) ,
C x = sin α C R C 2 - sin α I R I 2 ± μ m 3 ( sin α O R O 2 - sin α R R R 2 ) ,
C y = cos α C sin β C R C 2 - cos α I sin β I R I 2 ± μ m 3 ( cos α O sin β O R O 2 - cos α R sin β R R R 2 ) ,
A x = 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 ) ,
A y = cos 2 α C sin 2 β C R C - cos 2 α I sin 2 β I R I ± μ m 2 ( cos 2 α O sin 2 β O R O - cos 2 α R sin 2 β R R R ) ,
A x y = sin α C cos α C sin β C R C - sin α I cos α I sin β I R I ± 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 ) .
1 f = μ m 2 ( 1 R O - 1 R R ) .
f # = f / D ,
Δ = 1 2 λ C [ x 3 ( 1 - sin α I R I 2 ± μ m 2 sin α O R O 2 ) - x 2 ( - sin 2 α I R I ± μ m 2 sin 2 α O R O ) ] .
Δ S = - 1 8 λ C x 4 ( 1 R C 3 - 1 R I 3 ± μ m 4 1 R O 3 ) ,
q = ( r 2 + r 1 ) / ( r 2 - r 1 ) ,
1 f = ( n - 1 ) ( 1 r 1 - 1 r 2 ) .
q = ( R R + R O ) / ( R R - R O ) .
R O = ( μ / m 2 ) 2 f / ( 1 + q ) ;
R R = ( μ / m 2 ) 2 f / ( q - 1 ) .
S = ± 1 f 3 [ 3 4 ( m μ ) 2 q 2 + 1 4 ( m μ ) 2 - 1 ] ;
C x = - sin α C f 2 ± 1 f 2 { 1 4 ( m μ ) ( sin α O - sin α R ) q 2 + 1 2 ( m μ ) × ( sin α O + sin α R ) q + [ 1 4 ( m μ ) - μ m ] ( sin α O - sin α R ) } ;
A x = ± 1 2 f [ ( sin 2 α O - sin 2 α R ) q - ( sin 2 α R + sin 2 α O ) ] [ sin α C ± ( μ / m ) ( sin α O - sin α R ) ] 2 .
C x = - ( sin α C ) / f 2 ,
A x = ( sin 2 α C ) / f ,

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