Abstract

The authors’ experience with the application of the needle optimization technique to the design of optical coating is summarized. A physical interpretation of the technique is provided, and its main features are identified. Guidelines on the application of the needle optimization technique to various types of design problems are given.

© 1996 Optical Society of America

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References

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  1. P. Baumeister, “Design of multilayer filters by successive approximations,” J. Opt. Soc. Am. 48, 955–958 (1958).
    [CrossRef]
  2. J. A. Dobrowolski, “Computer design of optical coatings,” Thin Solid Films 163, 97–110 (1988).
    [CrossRef]
  3. J. A. Dobrowolski, R. A. Kemp, “Refinement of optical multilayer systems with different optimization procedures,” Appl. Opt. 29, 2876–2893 (1990).
    [CrossRef] [PubMed]
  4. L. Li, J. A. Dobrowolski, “Computation speed of different optical thin-film synthesis methods,” Appl. Opt. 31, 3790–3799 (1992).
    [CrossRef] [PubMed]
  5. A. V. Tikhonravov, “Synthesis of optical coatings using optimality conditions,” Vestn. Mosk. Univ. Fiz. Astronomiya 23, 91–93 (1982).
  6. A. N. Baskakov, A. V. Tikhonravov, “Synthesis of two-component optical coatings,” Opt. Spectrosc. 56, 915–919 (1984).
  7. A. V. Tikhonravov, M. K. Trubetskov, “Thin film coating design using second order optimization methods,” in Thin Films for Optical Systems, K. Guenther, ed., Proc. SPIE1782, 156–164 (1992).
  8. A. V. Tikhonravov, M. K. Trubetskov, “Development of the needle optimization technique and new features of OptiLayer design software,” in Optical Interference Coatings, F. Abelès, ed., Proc. SPIE2253, 10–20 (1994).
  9. A. V. Tikhonravov, “On the optimality of thin film optical coating design,” in Optical Thin Film and Applications, R. Herrmann ed., Proc. SPIE1270, 28–35 (1990).
  10. A. V. Tikhonravov, “Some theoretical aspects of thin film optics and their applications,” Appl. Opt. 32, 5417–5426 (1993).
    [CrossRef] [PubMed]
  11. A. N. Tikhonov, A. V. Tikhonravov, M. K. Trubetskov, “Second order optimization methods in the synthesis of multilayer coatings,” J. Comput. Math. Math. Phys. 33, 1339–1352 (1993).
  12. Sh. Furman, A. V. Tikhonravov, Basics of Optics of Multilayer Systems (Editions Frontieres, Gifsur-Yvette, France, 1992).
  13. W. H. Southwell, “Using apodization function to reduce sidelobes in rugate filters,” Appl. Opt. 28, 5091–5094 (1989).
    [CrossRef] [PubMed]
  14. B. G. Bovard, “Rugate filter design: the modified Fourier transform technique,” Appl. Opt. 29, 24–30 (1990).
    [CrossRef] [PubMed]
  15. P. G. Verly, J. A. Dobrowolski, “Iterative correction procedure for optical thin film synthesis with the Fourier transform method,” Appl. Opt. 29, 3672–3684 (1990).
    [CrossRef] [PubMed]
  16. H. Fabricius, “Gradient-index filter: designing filters with step skirts, high reflection, and quintic matching layers,” Appl. Opt. 31, 5191–5196 (1992).
    [CrossRef] [PubMed]
  17. J. Allen, B. Harrington, “Digitized rugate filters for laser application,” in Inhomogeneous and Quasi-Inhomogeneous Optical Coatings, J. A. Dobrowolski, P. G. Verly, eds., Proc. SPIE2046, 126–131 (1993).
  18. W. E. Johnson, R. L. Crane, “Color neutral rugate filters,” in Inhomogeneous and Quasi-Inhomogeneous Optical Coatings, J. A. Dobrowolski, P. G. Verly, eds., Proc. SPIE2046, 132–140 (1993).
  19. J. B. Adolph, R. W. Bertram, K. L. Yan, P. Zhou, R. de Leon, “Design and fabrication of multiline inhomogeneous rejection filters,” in Inhomogeneous and Quasi-Inhomogeneous Optical Coatings, J. A. Dobrowolski, P. G. Verly, eds., Proc. SPIE2046, 141–146 (1993).
  20. A. Thelen, “Design of a hot mirror—contest results,” in Optical Interference Coatings, Vol. 17 of OSA Technical Digest Series (Optical Society of America, Washington, D. C., 1995), pp. 2–10.
  21. D. Gray, ed., American Institute of Physics Handbook (American Institute of Physics, New York, 1972), pp. 6–143.
  22. A. G. Sveshnikov, A. V. Tikhonravov, S. A. Yanshin, “Synthesis of optical coating at oblique incidence,” Zh. Vychisl. Mat. Mat. Fiz. 23, 929–936 (1983).
  23. A. Thelen, Design of Optical Interference Coatings, (McGraw-Hill, New York, 1989), Chap. 9.
  24. V. R. Costich, “Reduction of polarization effects in interference coatings,” Appl. Opt. 9, 866–870 (1970).
    [CrossRef] [PubMed]
  25. A. Thelen, “Nonpolarizing interference filters inside a glass cube,” Appl. Opt. 15, 2983–2985 (1976).
    [CrossRef] [PubMed]
  26. A. Thelen, “Nonpolarizing edge filters,” J. Opt. Soc. Am. 71, 309–314 (1981).
    [CrossRef]
  27. Z. Knittl, H. Houserkova, “Equivalent layers in oblique incidence: the problem of unsplit admittance and depolarization of partial reflectors,” Appl. Opt. 21, 2055–2068 (1982).
    [CrossRef] [PubMed]
  28. C. M. de Sterke, C. J. van der Laan, H. J. Frankena, “Nonpolarizing beamsplitter design,” Appl. Opt. 23, 595–601 (1983).
    [CrossRef]
  29. A. Thelen, “Nonpolarizing edge filters. Part 2,” Appl. Opt. 23, 3541–3543 (1984).
    [CrossRef] [PubMed]
  30. M. Gilo, “Design of nonpolarizing beam splitter inside a glass cube,” Appl. Opt. 31, 5345–5349 (1992).
    [CrossRef] [PubMed]

1993

A. N. Tikhonov, A. V. Tikhonravov, M. K. Trubetskov, “Second order optimization methods in the synthesis of multilayer coatings,” J. Comput. Math. Math. Phys. 33, 1339–1352 (1993).

A. V. Tikhonravov, “Some theoretical aspects of thin film optics and their applications,” Appl. Opt. 32, 5417–5426 (1993).
[CrossRef] [PubMed]

1992

1990

1989

1988

J. A. Dobrowolski, “Computer design of optical coatings,” Thin Solid Films 163, 97–110 (1988).
[CrossRef]

1984

A. N. Baskakov, A. V. Tikhonravov, “Synthesis of two-component optical coatings,” Opt. Spectrosc. 56, 915–919 (1984).

A. Thelen, “Nonpolarizing edge filters. Part 2,” Appl. Opt. 23, 3541–3543 (1984).
[CrossRef] [PubMed]

1983

A. G. Sveshnikov, A. V. Tikhonravov, S. A. Yanshin, “Synthesis of optical coating at oblique incidence,” Zh. Vychisl. Mat. Mat. Fiz. 23, 929–936 (1983).

C. M. de Sterke, C. J. van der Laan, H. J. Frankena, “Nonpolarizing beamsplitter design,” Appl. Opt. 23, 595–601 (1983).
[CrossRef]

1982

A. V. Tikhonravov, “Synthesis of optical coatings using optimality conditions,” Vestn. Mosk. Univ. Fiz. Astronomiya 23, 91–93 (1982).

Z. Knittl, H. Houserkova, “Equivalent layers in oblique incidence: the problem of unsplit admittance and depolarization of partial reflectors,” Appl. Opt. 21, 2055–2068 (1982).
[CrossRef] [PubMed]

1981

1976

1970

1958

Adolph, J. B.

J. B. Adolph, R. W. Bertram, K. L. Yan, P. Zhou, R. de Leon, “Design and fabrication of multiline inhomogeneous rejection filters,” in Inhomogeneous and Quasi-Inhomogeneous Optical Coatings, J. A. Dobrowolski, P. G. Verly, eds., Proc. SPIE2046, 141–146 (1993).

Allen, J.

J. Allen, B. Harrington, “Digitized rugate filters for laser application,” in Inhomogeneous and Quasi-Inhomogeneous Optical Coatings, J. A. Dobrowolski, P. G. Verly, eds., Proc. SPIE2046, 126–131 (1993).

Baskakov, A. N.

A. N. Baskakov, A. V. Tikhonravov, “Synthesis of two-component optical coatings,” Opt. Spectrosc. 56, 915–919 (1984).

Baumeister, P.

Bertram, R. W.

J. B. Adolph, R. W. Bertram, K. L. Yan, P. Zhou, R. de Leon, “Design and fabrication of multiline inhomogeneous rejection filters,” in Inhomogeneous and Quasi-Inhomogeneous Optical Coatings, J. A. Dobrowolski, P. G. Verly, eds., Proc. SPIE2046, 141–146 (1993).

Bovard, B. G.

Costich, V. R.

Crane, R. L.

W. E. Johnson, R. L. Crane, “Color neutral rugate filters,” in Inhomogeneous and Quasi-Inhomogeneous Optical Coatings, J. A. Dobrowolski, P. G. Verly, eds., Proc. SPIE2046, 132–140 (1993).

de Leon, R.

J. B. Adolph, R. W. Bertram, K. L. Yan, P. Zhou, R. de Leon, “Design and fabrication of multiline inhomogeneous rejection filters,” in Inhomogeneous and Quasi-Inhomogeneous Optical Coatings, J. A. Dobrowolski, P. G. Verly, eds., Proc. SPIE2046, 141–146 (1993).

de Sterke, C. M.

C. M. de Sterke, C. J. van der Laan, H. J. Frankena, “Nonpolarizing beamsplitter design,” Appl. Opt. 23, 595–601 (1983).
[CrossRef]

Dobrowolski, J. A.

Fabricius, H.

Frankena, H. J.

C. M. de Sterke, C. J. van der Laan, H. J. Frankena, “Nonpolarizing beamsplitter design,” Appl. Opt. 23, 595–601 (1983).
[CrossRef]

Gilo, M.

Harrington, B.

J. Allen, B. Harrington, “Digitized rugate filters for laser application,” in Inhomogeneous and Quasi-Inhomogeneous Optical Coatings, J. A. Dobrowolski, P. G. Verly, eds., Proc. SPIE2046, 126–131 (1993).

Houserkova, H.

Johnson, W. E.

W. E. Johnson, R. L. Crane, “Color neutral rugate filters,” in Inhomogeneous and Quasi-Inhomogeneous Optical Coatings, J. A. Dobrowolski, P. G. Verly, eds., Proc. SPIE2046, 132–140 (1993).

Kemp, R. A.

Knittl, Z.

Li, L.

Southwell, W. H.

Sveshnikov, A. G.

A. G. Sveshnikov, A. V. Tikhonravov, S. A. Yanshin, “Synthesis of optical coating at oblique incidence,” Zh. Vychisl. Mat. Mat. Fiz. 23, 929–936 (1983).

Thelen, A.

A. Thelen, “Nonpolarizing edge filters. Part 2,” Appl. Opt. 23, 3541–3543 (1984).
[CrossRef] [PubMed]

A. Thelen, “Nonpolarizing edge filters,” J. Opt. Soc. Am. 71, 309–314 (1981).
[CrossRef]

A. Thelen, “Nonpolarizing interference filters inside a glass cube,” Appl. Opt. 15, 2983–2985 (1976).
[CrossRef] [PubMed]

A. Thelen, Design of Optical Interference Coatings, (McGraw-Hill, New York, 1989), Chap. 9.

A. Thelen, “Design of a hot mirror—contest results,” in Optical Interference Coatings, Vol. 17 of OSA Technical Digest Series (Optical Society of America, Washington, D. C., 1995), pp. 2–10.

Tikhonov, A. N.

A. N. Tikhonov, A. V. Tikhonravov, M. K. Trubetskov, “Second order optimization methods in the synthesis of multilayer coatings,” J. Comput. Math. Math. Phys. 33, 1339–1352 (1993).

Tikhonravov, A. V.

A. N. Tikhonov, A. V. Tikhonravov, M. K. Trubetskov, “Second order optimization methods in the synthesis of multilayer coatings,” J. Comput. Math. Math. Phys. 33, 1339–1352 (1993).

A. V. Tikhonravov, “Some theoretical aspects of thin film optics and their applications,” Appl. Opt. 32, 5417–5426 (1993).
[CrossRef] [PubMed]

A. N. Baskakov, A. V. Tikhonravov, “Synthesis of two-component optical coatings,” Opt. Spectrosc. 56, 915–919 (1984).

A. G. Sveshnikov, A. V. Tikhonravov, S. A. Yanshin, “Synthesis of optical coating at oblique incidence,” Zh. Vychisl. Mat. Mat. Fiz. 23, 929–936 (1983).

A. V. Tikhonravov, “Synthesis of optical coatings using optimality conditions,” Vestn. Mosk. Univ. Fiz. Astronomiya 23, 91–93 (1982).

A. V. Tikhonravov, M. K. Trubetskov, “Development of the needle optimization technique and new features of OptiLayer design software,” in Optical Interference Coatings, F. Abelès, ed., Proc. SPIE2253, 10–20 (1994).

A. V. Tikhonravov, “On the optimality of thin film optical coating design,” in Optical Thin Film and Applications, R. Herrmann ed., Proc. SPIE1270, 28–35 (1990).

A. V. Tikhonravov, M. K. Trubetskov, “Thin film coating design using second order optimization methods,” in Thin Films for Optical Systems, K. Guenther, ed., Proc. SPIE1782, 156–164 (1992).

Trubetskov, M. K.

A. N. Tikhonov, A. V. Tikhonravov, M. K. Trubetskov, “Second order optimization methods in the synthesis of multilayer coatings,” J. Comput. Math. Math. Phys. 33, 1339–1352 (1993).

A. V. Tikhonravov, M. K. Trubetskov, “Thin film coating design using second order optimization methods,” in Thin Films for Optical Systems, K. Guenther, ed., Proc. SPIE1782, 156–164 (1992).

A. V. Tikhonravov, M. K. Trubetskov, “Development of the needle optimization technique and new features of OptiLayer design software,” in Optical Interference Coatings, F. Abelès, ed., Proc. SPIE2253, 10–20 (1994).

van der Laan, C. J.

C. M. de Sterke, C. J. van der Laan, H. J. Frankena, “Nonpolarizing beamsplitter design,” Appl. Opt. 23, 595–601 (1983).
[CrossRef]

Verly, P. G.

Yan, K. L.

J. B. Adolph, R. W. Bertram, K. L. Yan, P. Zhou, R. de Leon, “Design and fabrication of multiline inhomogeneous rejection filters,” in Inhomogeneous and Quasi-Inhomogeneous Optical Coatings, J. A. Dobrowolski, P. G. Verly, eds., Proc. SPIE2046, 141–146 (1993).

Yanshin, S. A.

A. G. Sveshnikov, A. V. Tikhonravov, S. A. Yanshin, “Synthesis of optical coating at oblique incidence,” Zh. Vychisl. Mat. Mat. Fiz. 23, 929–936 (1983).

Zhou, P.

J. B. Adolph, R. W. Bertram, K. L. Yan, P. Zhou, R. de Leon, “Design and fabrication of multiline inhomogeneous rejection filters,” in Inhomogeneous and Quasi-Inhomogeneous Optical Coatings, J. A. Dobrowolski, P. G. Verly, eds., Proc. SPIE2046, 141–146 (1993).

Appl. Opt.

C. M. de Sterke, C. J. van der Laan, H. J. Frankena, “Nonpolarizing beamsplitter design,” Appl. Opt. 23, 595–601 (1983).
[CrossRef]

V. R. Costich, “Reduction of polarization effects in interference coatings,” Appl. Opt. 9, 866–870 (1970).
[CrossRef] [PubMed]

A. Thelen, “Nonpolarizing interference filters inside a glass cube,” Appl. Opt. 15, 2983–2985 (1976).
[CrossRef] [PubMed]

Z. Knittl, H. Houserkova, “Equivalent layers in oblique incidence: the problem of unsplit admittance and depolarization of partial reflectors,” Appl. Opt. 21, 2055–2068 (1982).
[CrossRef] [PubMed]

A. Thelen, “Nonpolarizing edge filters. Part 2,” Appl. Opt. 23, 3541–3543 (1984).
[CrossRef] [PubMed]

W. H. Southwell, “Using apodization function to reduce sidelobes in rugate filters,” Appl. Opt. 28, 5091–5094 (1989).
[CrossRef] [PubMed]

B. G. Bovard, “Rugate filter design: the modified Fourier transform technique,” Appl. Opt. 29, 24–30 (1990).
[CrossRef] [PubMed]

J. A. Dobrowolski, R. A. Kemp, “Refinement of optical multilayer systems with different optimization procedures,” Appl. Opt. 29, 2876–2893 (1990).
[CrossRef] [PubMed]

P. G. Verly, J. A. Dobrowolski, “Iterative correction procedure for optical thin film synthesis with the Fourier transform method,” Appl. Opt. 29, 3672–3684 (1990).
[CrossRef] [PubMed]

L. Li, J. A. Dobrowolski, “Computation speed of different optical thin-film synthesis methods,” Appl. Opt. 31, 3790–3799 (1992).
[CrossRef] [PubMed]

H. Fabricius, “Gradient-index filter: designing filters with step skirts, high reflection, and quintic matching layers,” Appl. Opt. 31, 5191–5196 (1992).
[CrossRef] [PubMed]

M. Gilo, “Design of nonpolarizing beam splitter inside a glass cube,” Appl. Opt. 31, 5345–5349 (1992).
[CrossRef] [PubMed]

A. V. Tikhonravov, “Some theoretical aspects of thin film optics and their applications,” Appl. Opt. 32, 5417–5426 (1993).
[CrossRef] [PubMed]

J. Comput. Math. Math. Phys.

A. N. Tikhonov, A. V. Tikhonravov, M. K. Trubetskov, “Second order optimization methods in the synthesis of multilayer coatings,” J. Comput. Math. Math. Phys. 33, 1339–1352 (1993).

J. Opt. Soc. Am.

Opt. Spectrosc.

A. N. Baskakov, A. V. Tikhonravov, “Synthesis of two-component optical coatings,” Opt. Spectrosc. 56, 915–919 (1984).

Thin Solid Films

J. A. Dobrowolski, “Computer design of optical coatings,” Thin Solid Films 163, 97–110 (1988).
[CrossRef]

Vestn. Mosk. Univ. Fiz. Astronomiya

A. V. Tikhonravov, “Synthesis of optical coatings using optimality conditions,” Vestn. Mosk. Univ. Fiz. Astronomiya 23, 91–93 (1982).

Zh. Vychisl. Mat. Mat. Fiz.

A. G. Sveshnikov, A. V. Tikhonravov, S. A. Yanshin, “Synthesis of optical coating at oblique incidence,” Zh. Vychisl. Mat. Mat. Fiz. 23, 929–936 (1983).

Other

A. Thelen, Design of Optical Interference Coatings, (McGraw-Hill, New York, 1989), Chap. 9.

A. V. Tikhonravov, M. K. Trubetskov, “Thin film coating design using second order optimization methods,” in Thin Films for Optical Systems, K. Guenther, ed., Proc. SPIE1782, 156–164 (1992).

A. V. Tikhonravov, M. K. Trubetskov, “Development of the needle optimization technique and new features of OptiLayer design software,” in Optical Interference Coatings, F. Abelès, ed., Proc. SPIE2253, 10–20 (1994).

A. V. Tikhonravov, “On the optimality of thin film optical coating design,” in Optical Thin Film and Applications, R. Herrmann ed., Proc. SPIE1270, 28–35 (1990).

Sh. Furman, A. V. Tikhonravov, Basics of Optics of Multilayer Systems (Editions Frontieres, Gifsur-Yvette, France, 1992).

J. Allen, B. Harrington, “Digitized rugate filters for laser application,” in Inhomogeneous and Quasi-Inhomogeneous Optical Coatings, J. A. Dobrowolski, P. G. Verly, eds., Proc. SPIE2046, 126–131 (1993).

W. E. Johnson, R. L. Crane, “Color neutral rugate filters,” in Inhomogeneous and Quasi-Inhomogeneous Optical Coatings, J. A. Dobrowolski, P. G. Verly, eds., Proc. SPIE2046, 132–140 (1993).

J. B. Adolph, R. W. Bertram, K. L. Yan, P. Zhou, R. de Leon, “Design and fabrication of multiline inhomogeneous rejection filters,” in Inhomogeneous and Quasi-Inhomogeneous Optical Coatings, J. A. Dobrowolski, P. G. Verly, eds., Proc. SPIE2046, 141–146 (1993).

A. Thelen, “Design of a hot mirror—contest results,” in Optical Interference Coatings, Vol. 17 of OSA Technical Digest Series (Optical Society of America, Washington, D. C., 1995), pp. 2–10.

D. Gray, ed., American Institute of Physics Handbook (American Institute of Physics, New York, 1972), pp. 6–143.

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Figures (20)

Fig. 1
Fig. 1

Transformation of the optical coating that is due to the insertion of new layers: solid lines in A, refractive-index profile before the insertion; solid lines in B, refractive-index profile after the insertion; dashed curve in A, function P(z), the key function of the needle optimization technique that determines the thicknesses and insertion points; δ1, δ2, …, thicknesses of new layers.

Fig. 2
Fig. 2

General scheme of the synthesis procedure based on the needle optimization technique.

Fig. 3
Fig. 3

Reflectance of a 40-layer antireflection coating obtained with the one-layer starting design, 24H. The displayed range of reflectance is from 0 to 0.05 in absolute values.

Fig. 4
Fig. 4

Transmittance of a 34-layer neutral beam splitter obtained with the same starting design as in Fig. 3. The displayed range of transmittance is from 0.49 to 0.51 in absolute values.

Fig. 5
Fig. 5

Reflectance of a 29-layer high-reflection coating obtained with the same starting design as in Fig. 3. The displayed range of reflectance is from 0 to 1 in absolute values.

Fig. 6
Fig. 6

Transmittance of a 31-layer ramp coating obtained with the same starting design as in Fig. 3. The displayed range of transmittance is from 0 to 1 in absolute values.

Fig. 7
Fig. 7

Transmittance of a 27-layer X filter obtained with the same starting design as in Fig. 3. The displayed range of transmittance is from 0 to 1 in absolute values.

Fig. 8
Fig. 8

Transmittance of a 41-layer X filter obtained with the one-layer starting design, 40H. The displayed range of transmittance is from 0 to 1 in absolute values.

Fig. 9
Fig. 9

Transmittance of a 56-layer mirror with the suppressed secondary reflection zone. The displayed range of transmittance is from 0 to 1 in absolute values.

Fig. 10
Fig. 10

Transmittance of a 100-layer filter with two 40-nm high-reflection zones at 440 and 580 nm. The displayed range of transmittance is from 0 to 1 in absolute values.

Fig. 11
Fig. 11

Transmittance of a 99-layer filter with two 40-nm high-reflection zones at 540 and 690 nm. The displayed range of transmittance is from 0 to 1 in absolute values.

Fig. 12
Fig. 12

Reflectance of a 39-layer metal-dielectric edge filter for the spectral band from 400 to 1600 nm. The displayed range of reflectance is from 0 to 1 in absolute values.

Fig. 13
Fig. 13

Effective refractive indices for s- and p-polarized light, ns and np, respectively (na = 1.52, θa = 45°).

Fig. 14
Fig. 14

Reflectance of a 26-layer short-wave-pass filter for s-polarized light at a 45° angle of incidence. The displayed range of reflectance is from 0 to 1 in absolute values.

Fig. 15
Fig. 15

Reflectance of a 49-layer short-wave-pass filter for p-polarized light at a 45° angle of incidence. The displayed range of reflectance is from 0 to 1 in absolute values.

Fig. 16
Fig. 16

49-layer nonpolarizing short-wave-pass filter at a 45° angle of incidence. The displayed range of s and p transmittances is from 0 to 1 in absolute values.

Fig. 17
Fig. 17

79-layer nonpolarizing long-wave-pass filter at a 45° angle of incidence. The displayed range of s and p transmittances is from 0 to 1 in absolute values.

Fig. 18
Fig. 18

Wideband nonpolarizing neutral beam splitter immersed in glass with n = 1.52. The angle of incidence is 45° in glass. The displayed range of s and p transmittances is from 0 to 1 in absolute values.

Fig. 19
Fig. 19

Wide spectral band and wide angular range antireflection coating. Reflectance curves for s- and p-polarized light and for angles of incidence 0°, 30°, 45°, 60°, and 70° are displayed in the range from 0% to 20%.

Fig. 20
Fig. 20

High-reflection coating with a 90° differential phase shift at a 45° angle of incidence. The displayed range of reflectances is from 0.95 to 1 in absolute values (left scale). The differential phase shift is displayed in the range from 85° to 95° (right scale).

Equations (10)

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

δ F = P 1 ( z , n ) δ + P 2 ( z , n ) δ 2 + .
P ( z ) = min 1 < j < J P j ( z , n j ) .
S 0.1103 H 0.6563 L 0.3192 H 0.4200 L 0.4930 H 0.1747 L 2.5677 H 0.3436 L 0.4085 H 2.7275 L 0.4579 H 0.3477 L 1.4849 H 0.2372 L 0.6685 H 0.8768 L 0.1467 H 1.4584 L 0.4524 H 0.3814 L 1.4966 H 0.2381 L 0.6603 H 0.8837 L 0.1135 H 1.5485 L 0.4524 H 0.3498 L 1.6062 H 0.2349 L 0.6093 H 0.8696 L 0.0901 H 1.6217 L 0.4350 H 0.3535 L 1.5733 H 0.1939 L 0.6211 H 1.3048 L A .
S 0.0711 H 0.8142 L 0.2413 H 0.7928 L 0.4858 H 0.5604 L 0.8002 H 0.4178 L 0.7959 H 0.5991 L 0.3728 H 2.9718 L 0.3797 H 0.4021 L 2.9052 H 0.3120 L 0.6005 H 0.5519 L 0.6480 H 0.3230 L 2.6713 H 0.4777 L 0.3422 H 2.0391 L 0.4654 H 0.3096 L 2.5050 H 0.9640 L 0.5068 H 1.1436 L 1.2897 H 1.4506 L 1.7634 H 0.4154 L A .
S 0.8926 H 0.9898 L 0.9927 H 0.9607 L 0.8611 H 0.7616 L 0.8590 H 0.9563 L 0.9850 H 0.9801 L 0.9338 H 0.8197 L 0.7938 H 0.9187 L 0.9994 H 1.0552 L 1.2505 H 1.4073 L 1.1121 H 1.0855 L 1.1558 H 1.4139 L 1.1923 H 1.0962 L 1.1288 H 1.4949 L 1.2206 H 1.0852 L 1.1455 H A .
S 0.0980 H 0.7182 L 0.4055 H 0.4722 L 1.0244 H 0.2855 L 0.7949 H 0.8614 L 0.3311 H 0.8507 L 0.8898 H 0.3578 L 0.7922 H 0.9370 L 0.4367 H 0.6559 L 0.9555 H 0.5975 L 0.4923 H 0.9518 L 0.8196 H 0.4004 L 0.8714 H 0.9406 L 0.5694 H 0.6332 L 0.9345 H 0.9234 L 0.9071 H 0.9613 L 1.0538 H A .
S 0.4524 H 0.3206 L 0.5520 H 1.3012 L 0.2443 H 0.3393 L 1.2612 H 1.6863 L 1.2118 H 1.2345 L 0.3495 H 0.2040 L 1.2583 H 1.4485 L 1.5397 H 1.0529 L 1.0278 H 0.5734 L 0.4477 H 1.0117 L 1.8081 H 1.1005 L 0.9664 H 0.8521 L 0.8101 H 0.8959 L 0.6966 H A .
MF = { 1 N j = 1 N [ f ( λ j ) - f ^ j Δ f j ] 2 } 1 / 2 ,
n eff s = ( n 2 - n a 2 sin 2 θ a ) 1 / 2 ,
n eff p = n 2 / ( n 2 - n a 2 sin 2 θ a ) 1 / 2 .

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