Abstract

This paper presents the methodology for the design of double-layer antireflection (AR) coatings for grazing incidence angles for a given state of polarization. The designs are based on commonly used thin-film materials with the optical constants that can be realized by using standard evaporation techniques. The performance of some AR stacks has been computed, and the effect on spectral reflectance with variation in the thickness of the high-index layer, angle of incidence, and the refractive indices of the materials used for the inner and outer layers has been studied with a view to selecting a suitable design that gives the lowest reflectance over the widest wavelength range. The AR stacks show a reflectance of <0.5% over most parts of the visible and near-infrared regions of the spectrum.

© 1992 Optical Society of America

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References

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  1. S. A. Myers, “An improved line narrowing technique for a dye laser excited by a nitrogen laser,” Opt. Commun. 4, 187–189 (1971).
    [CrossRef]
  2. E. D. Stokes, F. B. Dunnings, R. F. Stebbings, G. K. Walters, R. D. Rundel, “A high efficiency dye laser tunable from UV to the IR,” Opt. Commun. 5, 267–270 (1972).
    [CrossRef]
  3. A. Bernhardt, P. Rassmussen, “Design criteria and operating characteristics of a single-mode pulsed dye laser,” Appl. Phys. B 26, 141–146 (1981).
    [CrossRef]
  4. L. G. Nair, “Dye lasers,” Prog. Quantum Electron. 7, 153–268 (1982).
    [CrossRef]
  5. F. J. Duarte, J. A. Piper, “Comparison of prism-expander and grazing-incidence grating cavities for copper laser pumped dye lasers,” Appl. Opt. 21, 2782–2786 (1982).
    [CrossRef] [PubMed]
  6. R. Kingslake, Applied Optics and Optical Engineering (Academic, New York, 1965).
  7. F. J. Duarte, J. A. Piper, “A double-prism beam expander for pulsed dye lasers,” Opt. Commun. 35, 100–104 (1980).
    [CrossRef]
  8. J. R. M. Barr, “Achromatic prism beam expanders,” Opt. Commun. 51, 41–46 (1984).
    [CrossRef]
  9. R. J. Niefer, J. B. Atkinson, “The design of achromatic prism beam expanders for pulsed dye lasers,” Opt. Commun. 67, 139–143 (1988).
    [CrossRef]
  10. D. C. Hanna, P. A. Karkkainen, R. Wyatt, “A simple beam expander for frequency narrowing of dye lasers,” Opt. Quantum Electron. 7, 115–119 (1975).
    [CrossRef]
  11. T. Kasyga, T. Suzuki, K. Shimoda, “A prism anamorphic system for Gaussian beam expander,” Appl. Phys. 17, 131–136 (1978).
    [CrossRef]
  12. F. J. Duarte, J. A. Piper, “Dispersion theory of multiple-prism beam expanders for pulsed dye lasers,” Opt. Commun. 43, 303–307 (1982).
    [CrossRef]
  13. J. C. Monga, “Anti-reflection coatings for grazing incidence angles,” J. Mod. Opt. 36, 381–387 (1989).
    [CrossRef]
  14. H. A. Macleod, Thin Film Optical FiltersHilger, Bristol, UK, 1986), p. 78.
  15. L. G. Nair, “A double-wavelength nitrogen-laser-pumped dye laser,” Appl. Phys. 20, 97–99 (1979).
    [CrossRef]
  16. B. E. Newnam, D. H. Gill, G. Faulkner, “Influence of standing-wave fields on the laser damage resistance of dielectric films,” in Laser-Induced Damage in Optical Materials: 1975, Natl. Bur. Stand. U.S. Spec. Publ. 435 (U.S. Government Printing Office, Washington, D.C., 1976), p. 254.
  17. T. W. Walker, A. H. Guenther, P. E. Nielsen, “Pulsed laser-induced damage to thin-film optical coatings. Part I: experimental,” IEEE J. Quantum Electron. QE-17, 2041–2052 (1981).
    [CrossRef]
  18. F. Stetter, R. Esselborn, N. Harder, M. Friz, P. Tolles, “New materials for optical thin-films,” Appl. Opt. 15, 2315–2317 (1976).
    [CrossRef] [PubMed]
  19. H. W. Lehman, K. Frick, “Optimizing deposition parameters of electron beam evaporated TiO2 films,” Appl. Opt. 27, 4920–4924 (1988).
    [CrossRef]
  20. J. A. Dobrowolski, P. D. Grant, R. Simpson, A. J. Waldorf, “Investigation of the evaporation process conditions on the optical constants of zirconia films,” Appl. Opt. 28, 3997–4005 (1989).
    [CrossRef] [PubMed]
  21. J. M. Bennett, E. Pelletier, G. Albrand, J. P. Borgogno, B. Lazarides, C. K. Carniglia, R. A. Schmell, T. H. Allen, T. Tuttle-Hart, K. H. Guenther, A. Saxer, “Comparison of the properties of titanium dioxide films prepared by various techniques,” Appl. Opt. 28, 3303–3317 (1989).
    [CrossRef] [PubMed]

1989 (3)

1988 (2)

R. J. Niefer, J. B. Atkinson, “The design of achromatic prism beam expanders for pulsed dye lasers,” Opt. Commun. 67, 139–143 (1988).
[CrossRef]

H. W. Lehman, K. Frick, “Optimizing deposition parameters of electron beam evaporated TiO2 films,” Appl. Opt. 27, 4920–4924 (1988).
[CrossRef]

1984 (1)

J. R. M. Barr, “Achromatic prism beam expanders,” Opt. Commun. 51, 41–46 (1984).
[CrossRef]

1982 (3)

L. G. Nair, “Dye lasers,” Prog. Quantum Electron. 7, 153–268 (1982).
[CrossRef]

F. J. Duarte, J. A. Piper, “Comparison of prism-expander and grazing-incidence grating cavities for copper laser pumped dye lasers,” Appl. Opt. 21, 2782–2786 (1982).
[CrossRef] [PubMed]

F. J. Duarte, J. A. Piper, “Dispersion theory of multiple-prism beam expanders for pulsed dye lasers,” Opt. Commun. 43, 303–307 (1982).
[CrossRef]

1981 (2)

T. W. Walker, A. H. Guenther, P. E. Nielsen, “Pulsed laser-induced damage to thin-film optical coatings. Part I: experimental,” IEEE J. Quantum Electron. QE-17, 2041–2052 (1981).
[CrossRef]

A. Bernhardt, P. Rassmussen, “Design criteria and operating characteristics of a single-mode pulsed dye laser,” Appl. Phys. B 26, 141–146 (1981).
[CrossRef]

1980 (1)

F. J. Duarte, J. A. Piper, “A double-prism beam expander for pulsed dye lasers,” Opt. Commun. 35, 100–104 (1980).
[CrossRef]

1979 (1)

L. G. Nair, “A double-wavelength nitrogen-laser-pumped dye laser,” Appl. Phys. 20, 97–99 (1979).
[CrossRef]

1978 (1)

T. Kasyga, T. Suzuki, K. Shimoda, “A prism anamorphic system for Gaussian beam expander,” Appl. Phys. 17, 131–136 (1978).
[CrossRef]

1976 (1)

1975 (1)

D. C. Hanna, P. A. Karkkainen, R. Wyatt, “A simple beam expander for frequency narrowing of dye lasers,” Opt. Quantum Electron. 7, 115–119 (1975).
[CrossRef]

1972 (1)

E. D. Stokes, F. B. Dunnings, R. F. Stebbings, G. K. Walters, R. D. Rundel, “A high efficiency dye laser tunable from UV to the IR,” Opt. Commun. 5, 267–270 (1972).
[CrossRef]

1971 (1)

S. A. Myers, “An improved line narrowing technique for a dye laser excited by a nitrogen laser,” Opt. Commun. 4, 187–189 (1971).
[CrossRef]

Albrand, G.

Allen, T. H.

Atkinson, J. B.

R. J. Niefer, J. B. Atkinson, “The design of achromatic prism beam expanders for pulsed dye lasers,” Opt. Commun. 67, 139–143 (1988).
[CrossRef]

Barr, J. R. M.

J. R. M. Barr, “Achromatic prism beam expanders,” Opt. Commun. 51, 41–46 (1984).
[CrossRef]

Bennett, J. M.

Bernhardt, A.

A. Bernhardt, P. Rassmussen, “Design criteria and operating characteristics of a single-mode pulsed dye laser,” Appl. Phys. B 26, 141–146 (1981).
[CrossRef]

Borgogno, J. P.

Carniglia, C. K.

Dobrowolski, J. A.

Duarte, F. J.

F. J. Duarte, J. A. Piper, “Comparison of prism-expander and grazing-incidence grating cavities for copper laser pumped dye lasers,” Appl. Opt. 21, 2782–2786 (1982).
[CrossRef] [PubMed]

F. J. Duarte, J. A. Piper, “Dispersion theory of multiple-prism beam expanders for pulsed dye lasers,” Opt. Commun. 43, 303–307 (1982).
[CrossRef]

F. J. Duarte, J. A. Piper, “A double-prism beam expander for pulsed dye lasers,” Opt. Commun. 35, 100–104 (1980).
[CrossRef]

Dunnings, F. B.

E. D. Stokes, F. B. Dunnings, R. F. Stebbings, G. K. Walters, R. D. Rundel, “A high efficiency dye laser tunable from UV to the IR,” Opt. Commun. 5, 267–270 (1972).
[CrossRef]

Esselborn, R.

Faulkner, G.

B. E. Newnam, D. H. Gill, G. Faulkner, “Influence of standing-wave fields on the laser damage resistance of dielectric films,” in Laser-Induced Damage in Optical Materials: 1975, Natl. Bur. Stand. U.S. Spec. Publ. 435 (U.S. Government Printing Office, Washington, D.C., 1976), p. 254.

Frick, K.

Friz, M.

Gill, D. H.

B. E. Newnam, D. H. Gill, G. Faulkner, “Influence of standing-wave fields on the laser damage resistance of dielectric films,” in Laser-Induced Damage in Optical Materials: 1975, Natl. Bur. Stand. U.S. Spec. Publ. 435 (U.S. Government Printing Office, Washington, D.C., 1976), p. 254.

Grant, P. D.

Guenther, A. H.

T. W. Walker, A. H. Guenther, P. E. Nielsen, “Pulsed laser-induced damage to thin-film optical coatings. Part I: experimental,” IEEE J. Quantum Electron. QE-17, 2041–2052 (1981).
[CrossRef]

Guenther, K. H.

Hanna, D. C.

D. C. Hanna, P. A. Karkkainen, R. Wyatt, “A simple beam expander for frequency narrowing of dye lasers,” Opt. Quantum Electron. 7, 115–119 (1975).
[CrossRef]

Harder, N.

Karkkainen, P. A.

D. C. Hanna, P. A. Karkkainen, R. Wyatt, “A simple beam expander for frequency narrowing of dye lasers,” Opt. Quantum Electron. 7, 115–119 (1975).
[CrossRef]

Kasyga, T.

T. Kasyga, T. Suzuki, K. Shimoda, “A prism anamorphic system for Gaussian beam expander,” Appl. Phys. 17, 131–136 (1978).
[CrossRef]

Kingslake, R.

R. Kingslake, Applied Optics and Optical Engineering (Academic, New York, 1965).

Lazarides, B.

Lehman, H. W.

Macleod, H. A.

H. A. Macleod, Thin Film Optical FiltersHilger, Bristol, UK, 1986), p. 78.

Monga, J. C.

J. C. Monga, “Anti-reflection coatings for grazing incidence angles,” J. Mod. Opt. 36, 381–387 (1989).
[CrossRef]

Myers, S. A.

S. A. Myers, “An improved line narrowing technique for a dye laser excited by a nitrogen laser,” Opt. Commun. 4, 187–189 (1971).
[CrossRef]

Nair, L. G.

L. G. Nair, “Dye lasers,” Prog. Quantum Electron. 7, 153–268 (1982).
[CrossRef]

L. G. Nair, “A double-wavelength nitrogen-laser-pumped dye laser,” Appl. Phys. 20, 97–99 (1979).
[CrossRef]

Newnam, B. E.

B. E. Newnam, D. H. Gill, G. Faulkner, “Influence of standing-wave fields on the laser damage resistance of dielectric films,” in Laser-Induced Damage in Optical Materials: 1975, Natl. Bur. Stand. U.S. Spec. Publ. 435 (U.S. Government Printing Office, Washington, D.C., 1976), p. 254.

Niefer, R. J.

R. J. Niefer, J. B. Atkinson, “The design of achromatic prism beam expanders for pulsed dye lasers,” Opt. Commun. 67, 139–143 (1988).
[CrossRef]

Nielsen, P. E.

T. W. Walker, A. H. Guenther, P. E. Nielsen, “Pulsed laser-induced damage to thin-film optical coatings. Part I: experimental,” IEEE J. Quantum Electron. QE-17, 2041–2052 (1981).
[CrossRef]

Pelletier, E.

Piper, J. A.

F. J. Duarte, J. A. Piper, “Dispersion theory of multiple-prism beam expanders for pulsed dye lasers,” Opt. Commun. 43, 303–307 (1982).
[CrossRef]

F. J. Duarte, J. A. Piper, “Comparison of prism-expander and grazing-incidence grating cavities for copper laser pumped dye lasers,” Appl. Opt. 21, 2782–2786 (1982).
[CrossRef] [PubMed]

F. J. Duarte, J. A. Piper, “A double-prism beam expander for pulsed dye lasers,” Opt. Commun. 35, 100–104 (1980).
[CrossRef]

Rassmussen, P.

A. Bernhardt, P. Rassmussen, “Design criteria and operating characteristics of a single-mode pulsed dye laser,” Appl. Phys. B 26, 141–146 (1981).
[CrossRef]

Rundel, R. D.

E. D. Stokes, F. B. Dunnings, R. F. Stebbings, G. K. Walters, R. D. Rundel, “A high efficiency dye laser tunable from UV to the IR,” Opt. Commun. 5, 267–270 (1972).
[CrossRef]

Saxer, A.

Schmell, R. A.

Shimoda, K.

T. Kasyga, T. Suzuki, K. Shimoda, “A prism anamorphic system for Gaussian beam expander,” Appl. Phys. 17, 131–136 (1978).
[CrossRef]

Simpson, R.

Stebbings, R. F.

E. D. Stokes, F. B. Dunnings, R. F. Stebbings, G. K. Walters, R. D. Rundel, “A high efficiency dye laser tunable from UV to the IR,” Opt. Commun. 5, 267–270 (1972).
[CrossRef]

Stetter, F.

Stokes, E. D.

E. D. Stokes, F. B. Dunnings, R. F. Stebbings, G. K. Walters, R. D. Rundel, “A high efficiency dye laser tunable from UV to the IR,” Opt. Commun. 5, 267–270 (1972).
[CrossRef]

Suzuki, T.

T. Kasyga, T. Suzuki, K. Shimoda, “A prism anamorphic system for Gaussian beam expander,” Appl. Phys. 17, 131–136 (1978).
[CrossRef]

Tolles, P.

Tuttle-Hart, T.

Waldorf, A. J.

Walker, T. W.

T. W. Walker, A. H. Guenther, P. E. Nielsen, “Pulsed laser-induced damage to thin-film optical coatings. Part I: experimental,” IEEE J. Quantum Electron. QE-17, 2041–2052 (1981).
[CrossRef]

Walters, G. K.

E. D. Stokes, F. B. Dunnings, R. F. Stebbings, G. K. Walters, R. D. Rundel, “A high efficiency dye laser tunable from UV to the IR,” Opt. Commun. 5, 267–270 (1972).
[CrossRef]

Wyatt, R.

D. C. Hanna, P. A. Karkkainen, R. Wyatt, “A simple beam expander for frequency narrowing of dye lasers,” Opt. Quantum Electron. 7, 115–119 (1975).
[CrossRef]

Appl. Opt. (5)

Appl. Phys. (2)

T. Kasyga, T. Suzuki, K. Shimoda, “A prism anamorphic system for Gaussian beam expander,” Appl. Phys. 17, 131–136 (1978).
[CrossRef]

L. G. Nair, “A double-wavelength nitrogen-laser-pumped dye laser,” Appl. Phys. 20, 97–99 (1979).
[CrossRef]

Appl. Phys. B (1)

A. Bernhardt, P. Rassmussen, “Design criteria and operating characteristics of a single-mode pulsed dye laser,” Appl. Phys. B 26, 141–146 (1981).
[CrossRef]

IEEE J. Quantum Electron. (1)

T. W. Walker, A. H. Guenther, P. E. Nielsen, “Pulsed laser-induced damage to thin-film optical coatings. Part I: experimental,” IEEE J. Quantum Electron. QE-17, 2041–2052 (1981).
[CrossRef]

J. Mod. Opt. (1)

J. C. Monga, “Anti-reflection coatings for grazing incidence angles,” J. Mod. Opt. 36, 381–387 (1989).
[CrossRef]

Opt. Commun. (6)

F. J. Duarte, J. A. Piper, “Dispersion theory of multiple-prism beam expanders for pulsed dye lasers,” Opt. Commun. 43, 303–307 (1982).
[CrossRef]

S. A. Myers, “An improved line narrowing technique for a dye laser excited by a nitrogen laser,” Opt. Commun. 4, 187–189 (1971).
[CrossRef]

E. D. Stokes, F. B. Dunnings, R. F. Stebbings, G. K. Walters, R. D. Rundel, “A high efficiency dye laser tunable from UV to the IR,” Opt. Commun. 5, 267–270 (1972).
[CrossRef]

F. J. Duarte, J. A. Piper, “A double-prism beam expander for pulsed dye lasers,” Opt. Commun. 35, 100–104 (1980).
[CrossRef]

J. R. M. Barr, “Achromatic prism beam expanders,” Opt. Commun. 51, 41–46 (1984).
[CrossRef]

R. J. Niefer, J. B. Atkinson, “The design of achromatic prism beam expanders for pulsed dye lasers,” Opt. Commun. 67, 139–143 (1988).
[CrossRef]

Opt. Quantum Electron. (1)

D. C. Hanna, P. A. Karkkainen, R. Wyatt, “A simple beam expander for frequency narrowing of dye lasers,” Opt. Quantum Electron. 7, 115–119 (1975).
[CrossRef]

Prog. Quantum Electron. (1)

L. G. Nair, “Dye lasers,” Prog. Quantum Electron. 7, 153–268 (1982).
[CrossRef]

Other (3)

R. Kingslake, Applied Optics and Optical Engineering (Academic, New York, 1965).

H. A. Macleod, Thin Film Optical FiltersHilger, Bristol, UK, 1986), p. 78.

B. E. Newnam, D. H. Gill, G. Faulkner, “Influence of standing-wave fields on the laser damage resistance of dielectric films,” in Laser-Induced Damage in Optical Materials: 1975, Natl. Bur. Stand. U.S. Spec. Publ. 435 (U.S. Government Printing Office, Washington, D.C., 1976), p. 254.

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Figures (16)

Fig. 1
Fig. 1

Schematic of a double-layer AR coating showing the various parameters used in the text.

Fig. 2
Fig. 2

Diagram showing the values of the effective refractive indices η1 and η2 for double-layer AR coatings on glass (nm = 1.52) for p-polarized radiation incident at 70°. Suitable layer thicknesses of materials (shaded regions) can produce zero reflectance.

Fig. 3
Fig. 3

Spectral reflectance of double-layer AR coatings on glass (nm = 1.52) for p-polarized radiation incident at 70°: A, air(0.264L, 0.628H)sub; B, air(2.362L, 1.564H)sub with n1 = 1.4.5, n2 = 2.30.

Fig. 4
Fig. 4

Spectral reflectance of double-layer AR coatings on glass (nm = 1.52) for p-polarized radiation incident at 70°: A, air(0.246H, 0.939L)sub; B, air(1.946H, 1.292L)sub with n1 = 2.30, n2 = 2.12.

Fig. 5
Fig. 5

Spectral reflectance of double-layer AR coatings on glass (nm = 1.52) for p-polarized radiation incident at 70°: A, air(1.091H, 2.114L)sub; B, air(1.14H, 0.521L)sub with n1 = 2.12, n2 = 1.45.

Fig. 6
Fig. 6

Spectral reflectance of double-layer AR coatings on glass (nm = 1.52) for p-polarized radiation incident at 70° for different materials as the outer layer. The layer thicknesses for each combination of materials were adjusted to produce zero reflectance. n2 = 1.92 for all the stacks. A, air(0.843H, 0.475L)sub with n1 = 2.20; B, air(0.767H, 0.713L)sub with n1 = 2.30; C, air(0.786H, 0.878L)sub with n1 = 2.40; D, air(0.966H, 1.073L)sub with n1 = 2.50.

Fig. 7
Fig. 7

Spectral reflectance of double-layer AR coatings on glass (nm = 1.52) for p-polarized radiation incident at 70° for different materials as the inner layer. The layer thicknesses for each combination of materials were adjusted to produce zero reflectance. n1 = 2.30 for all the stacks. A, air(1.004H, 0.955L)sub with n2 = 1.74; B, air(0.793H, 0.718L)sub with n2 = 1.92; C, air(0.642H, 0.725L)sub with n2 = 2.0; D, air(0.246H, 0.94L)sub with n2 = 2.12.

Fig. 8
Fig. 8

Spectral reflectance of double-layer AR coatings on glass (nm = 1.52) for p-polarized radiation incident at 70° for different materials as the inner layer. The layer thicknesses for each combination of materials were adjusted to produce zero reflectance. n1 = 2.50 for all the stacks. A, air(0.966H, 1.073L)sub with n2 = 1.92; B, air(0.69H, 0.961L)sub with n2 = 2.00; C, air(0.228H, 1.028L)sub with n2 = 2.12.

Fig. 9
Fig. 9

Spectral reflectance of double-layer AR coatings on glass (nm = 1.52) for p-polarized radiation incident at 70° by using the same set of materials but interchangeably as the inner and outer layers. The layer thicknesses were adjusted to produce zero reflectance. A, air(1.004H, 0.955L)sub with n1 = 2.30 and n2 = 1.74; B, air(0.323L, 0.581H)sub with n1 = 1.74 and n2 = 2.30.

Fig. 10
Fig. 10

Equireflectance contours of a double-layer AR coating on glass (nm = 1.52) for p-polarized radiation incident at 70°. The AR stack is air(0.264L, 0.628H)sub with n1 = 1.45 and n2 = 2.30.

Fig. 11
Fig. 11

Equireflectance contours of a double-layer AR coating on glass (nm = 1.52) for p-polarized radiation incident at 70°. The AR stack is air(2.362L, 1.564H)sub with n1 = 1.45 and n2 = 2.30.

Fig. 12
Fig. 12

Equireflectance contours of a double-layer AR coating on glass (nm = 1.52) for p-polarized radiation incident at 70°. The AR stack is sub(0.955L, 1.004H)air with n1 = 2.3 and n2 = 1.74.

Fig. 13
Fig. 13

Equireflectance contours of a double-layer AR coating on glass (nm = 1.52) for p-polarized radiation incident at 70°. The AR stack is sub(2.114L, 1.091H)air with n1 = 2.3 and n2 = 1.45.

Fig. 14
Fig. 14

Diagram showing the values of the effective refractive indices η1 and η2 for a double-layer AR coating on glass (nm = 1.52) for s-polarized radiation incident at 70°. Suitable layer thicknesses of materials (shaded region) can produce zero reflectance.

Fig. 15
Fig. 15

Spectral reflectance of double-layer AR coatings on glass (nm = 1.52) on s-polarized radiation incident at 70° for different materials as the outer layer. The layer thicknesses for each combination of materials were adjusted to produce zero reflectance. n2 = 2.3 for both stacks. A, air(1.558L, 0.661H)sub with n1 = 1.38; B, air(1.396L, 0.92H)sub with n1 = 1.45.

Fig. 16
Fig. 16

Variation of spectral reflectance with incident angle of a double-layer AR coating on a fused silica substrate that was designed to give zero reflectance at 77° for s-polarized radiation. The AR stack is air(1.49L, 0.925H)sub with nm = 1.463, n1 = 1.38, and n2 = 2.40: curve A, 77°; curve B, 79°; curve C, 81°.

Equations (8)

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[ B C ] = [ cos ϕ 1 i η 1 sin ϕ 1 i η 1 sin ϕ 1 cos ϕ 1 ] [ cos ϕ 2 i η 2 sin ϕ 2 i η 2 sin ϕ 2 cos ϕ 2 ] [ 1 η m ] ,
Y = C B = ( η m cos ϕ 1 cos ϕ 2 η 1 η m η 2 sin ϕ 1 sin ϕ 2 ) + i ( η 1 sin ϕ 1 cos ϕ 2 + η 2 cos ϕ 1 sin ϕ 2 ) ( cos ϕ 1 cos ϕ 2 η 2 η 1 sin ϕ 1 sin ϕ 2 ) + i ( η m η 2 cos ϕ 1 sin ϕ 2 + η m η 1 sin ϕ 1 cos ϕ 2 ) .
( η 2 η m 2 ) cos ϕ 1 cos ϕ 2 + ( η 1 2 η m 2 η 2 η 2 η 2 η 1 ) sin ϕ 1 sin ϕ 2 = 0 ,
( η 2 η 2 η 2 ) cos ϕ 1 sin ϕ 2 + ( η 2 η 1 η 1 ) sin ϕ 1 cos ϕ 2 = 0 .
tan 2 ϕ 1 = η 1 2 ( η m 2 η 2 ) ( η 2 2 η 2 ) ( η 1 2 η m 2 η 2 2 η 2 ) ( η 2 η 1 2 ) .
tan 2 ϕ 2 = η 2 2 ( η m 2 η 2 ) ( η 2 η 1 2 ) ( η 1 2 η m 2 η 2 2 η 2 ) ( η 2 2 η 2 ) .
Region I 1 . 884 < η 1 < 2 . 378 , η 2 > 2 . 378 , Region II 2 . 378 < η 1 < 1 . 229 η 2 , 1 . 935 < η 2 < 2 . 378 , Region III 1 . 229 η 2 < η 1 < 2 . 378 , 1 . 884 < η 2 < 1 . 935 .
n 2 = η 2 + ( η 4 4 η 2 sin 2 θ 0 ) 1 / 2 2 .

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