Abstract

Some novel concepts of designing antireflection (AR) coatings with equivalent layers are presented. As an introduction, essential papers concerning thin-film optics and AR designs are cited, and the AR problem and a previously introduced AR-hard design type are discussed. Based on the known matrix formalism, a potential AR region, an equivalent stack index, and an equivalent substrate index are defined to use the theory of stop-band suppression as a starting point for the design of broadband AR coatings. The known multicycle AR design type is identified as a typical solution to the AR problem if the presented approach is used.

© 2006 Optical Society of America

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  1. J. Strong, 'On a method of decreasing the reflection from nonmetallic substances,' J. Opt. Soc. Am. 26, 73-74 (1936).
    [CrossRef]
  2. W. Geffken, 'Überzug aus mindestens drei Schichten von verschiedener Brechungszahl für einen nicht-metallischen Gegenstand zur Verminderung von dessen Oberflächenreflexion,' German patent 758,767 (19 July 1940).
  3. A. Herpin, 'Calcul du pouvoir réflecteur d'un systèm stratifiè quelconque,' Acad. Sci. Paris , C. R. 225, 182-183 (1947).
  4. F. Abelés, 'Recherches sur la propagation des ondes électromagnétiques sinusoïdales dans les milieux stratifiés. Application aux couches minces,' Ann. Phys. 5, 596-640, 706-782 (1950).
  5. L. I. Epstein, 'The design of optical filters,' J. Opt. Soc. Am. 42, 806-810 (1952).
    [CrossRef]
  6. H. Pohlack, 'Die Synthese optischer Interferenzschichtsysteme mit vorgegebenen Eigenschaften,' in Jenaer Jahrbuch 1952 (1952), pp. 181-221.
  7. E. Delano, 'Fourier synthesis of multilayer filters,' J. Opt. Soc. Am. 57, 1529-1533 (1967).
    [CrossRef]
  8. L. Young, 'Synthesis of multiple antireflection films over a prescribed frequency band,' J. Opt. Soc. Am. 51, 967-974 (1961).
    [CrossRef]
  9. J. S. Seeley, 'Synthesis of interference filters,' Proc. Phys. Soc. London 78, 998-1008 (1961).
    [CrossRef]
  10. P. H. Berning, 'Use of equivalent films in the design of infrared multilayer antireflection coatings,' J. Opt. Soc. Am. 52, 431-436 (1962).
    [CrossRef]
  11. F. C. Rock, 'Antireflection coating and assembly having synthesized layer of index of refraction,' U.S. patent 3,432,225 (4 May 1964).
  12. A. Musset and A. Thelen, 'Multilayer antireflection coatings,' in Progress in Optics, E.Wolf, ed. (North-Holland, 1970), Vol. 8, pp. 203-237.
  13. A. Thelen, 'Multilayer filters with wide transmittance bands,' J. Opt. Soc. Am. 53, 1266-1270 (1963).
    [CrossRef]
  14. A. Thelen, 'Multilayer filters with wide transmittance bands, part II,' J. Opt. Soc. Am. 63, 65-68 (1973).
    [CrossRef]
  15. M. C. Ohmer, 'Design of three-layer equivalent films,' J. Opt. Soc. Am. 68, 137-139 (1978).
    [CrossRef]
  16. W. H. Southwell, 'Coating design using very thin high- and low-index layers,' Appl. Opt. 24, 457-3460 (1985).
    [CrossRef] [PubMed]
  17. H. Anders and R. Eichinger, 'Die optische Wirkung und die praktische Bedeutung inhomogener Schichten,' Appl. Opt. 4, 899-905 (1965).
    [CrossRef]
  18. R. Jacobsson and J. O. Martensson, 'Evaporated inhomogeneous thin films,' Appl. Opt. 5, 29-34 (1966).
    [CrossRef] [PubMed]
  19. J. A. Aguilera, J. Aguilera, P. Baumeister, A. Bloom, D. Coursen, J. A. Dobrowolski, F. T. Goldstein, D. E. Gustafson, and R. A. Kemp, 'Antireflection coatings for germanium IR optics: a comparison of numerical design methods,' Appl. Opt. 27, 2832-2840 (1988).
    [CrossRef] [PubMed]
  20. A. Thelen and R. Langfeld, 'Coating design contest: antireflection coating for lenses used with normal and infrared photographic film,' in Thin Film for Optical Systems, K.H.Guenther, ed., Proc. SPIE 1782, 551-601 (1993).
  21. P. G. Verly, J. A. Dobrowolski, and R. R. Willey, 'Fourier-transform method for the design of wideband anti-reflection coatings,' Appl. Opt. 31, 3836-3846 (1992).
    [CrossRef] [PubMed]
  22. P. G. Verly, A. V. Tikhonravov, and M. K. Trubetskov, 'Efficient refinement algorithm for the synthesis of inhomogeneous coatings,' Appl. Opt. 36, 1487-1495 (1997).
    [CrossRef] [PubMed]
  23. A. V. Tikhonravov and J. A. Dobrowolski, 'Quasi-optimal synthesis for antireflection coatings: a new method,' Appl. Opt. 32, 4265-4275 (1993).
    [CrossRef] [PubMed]
  24. J. A. Dobrowolski, A. V. Tikhonravov, M. K. Trubetskov, B. T. Sullivan, and P. G. Verly, 'Optimal single-band normal-incidence antireflection coatings,' Appl. Opt. 35, 644-658 (1996).
    [CrossRef] [PubMed]
  25. R. R. Willey, 'Predicting achievable design performance of broadband antireflection coatings,' Appl. Opt. 32, 5447-5451 (1993).
    [CrossRef] [PubMed]
  26. A. V. Tikhonravov, M. K. Trubetskov, and G. W. DeBell, 'Application of the needle optimization technique to the design of optical coatings,' Appl. Opt. 35, 5493-5508 (1996).
    [CrossRef] [PubMed]
  27. U. B. Schallenberg, S. Jakobs, and N. Kaiser, 'Analytical design of multicycle broadband AR coatings,' in Advances in Optical Interference Coatings, C.Amra and A.Macleod, eds., Proc. SPIE 3738, 230-238 (1999).
  28. U. Schallenberg, U. Schulz, and N. Kaiser, 'Multicycle AR coatings: a theoretical approach,' in Advances in Optical Thin Films, C.Amra, N.Kaiser, and H.A.Macleod, eds., Proc. SPIE 5250, 357-366 (2004).
  29. J. A. Dobrowolski, D. Poitras, P. Ma, H. Vakil, and M. Acree, 'Toward perfect antireflection coatings: numerical investigation,' Appl. Opt. 41, 3075-3083 (2002).
    [CrossRef] [PubMed]
  30. U. Schulz, U. B. Schallenberg, and N. Kaiser, 'Antireflection coating design for plastic optics,' Appl. Opt. 41, 3107-3110 (2002).
    [CrossRef] [PubMed]
  31. U. Schulz, U. B. Schallenberg, and N. Kaiser, 'Symmetrical periods in antireflective coatings for plastic optics,' Appl. Opt. 42, 1346-1351 (2003).
    [CrossRef] [PubMed]
  32. U. Schulz, N. Kaiser, and U. Schallenberg, 'Reflection reducing coating,' U.S. Patent 6,645,608 (11 November 2003).
  33. A. Macleod, Thin-Film Optical Filters, 3rd ed. (Institute of Physics, 2001).
    [CrossRef]
  34. L. I. Epstein, 'Improvements in heat-reflecting filters,' J. Opt. Soc. Am. 45, 360-362 (1955).
    [CrossRef]
  35. A. Thelen, Design of Optical Interference Coatings (McGraw-Hill, 1989).
  36. U. Schulz, N. Kaiser, and U. B. Schallenberg, 'AR-hard broadband antireflective coatings generated by a controlled needle-optimization technique,' in Optical Interference Coatings on CD-ROM (Optical Society of America, 2004), Paper TuB2.

2003 (1)

2002 (2)

1997 (1)

1996 (2)

1993 (2)

1992 (1)

1988 (1)

1985 (1)

1978 (1)

1973 (1)

1967 (1)

1966 (1)

1965 (1)

1963 (1)

1962 (1)

1961 (2)

1955 (1)

1952 (2)

H. Pohlack, 'Die Synthese optischer Interferenzschichtsysteme mit vorgegebenen Eigenschaften,' in Jenaer Jahrbuch 1952 (1952), pp. 181-221.

L. I. Epstein, 'The design of optical filters,' J. Opt. Soc. Am. 42, 806-810 (1952).
[CrossRef]

1950 (1)

F. Abelés, 'Recherches sur la propagation des ondes électromagnétiques sinusoïdales dans les milieux stratifiés. Application aux couches minces,' Ann. Phys. 5, 596-640, 706-782 (1950).

1947 (1)

A. Herpin, 'Calcul du pouvoir réflecteur d'un systèm stratifiè quelconque,' Acad. Sci. Paris , C. R. 225, 182-183 (1947).

1936 (1)

Abelés, F.

F. Abelés, 'Recherches sur la propagation des ondes électromagnétiques sinusoïdales dans les milieux stratifiés. Application aux couches minces,' Ann. Phys. 5, 596-640, 706-782 (1950).

Acree, M.

Aguilera, J.

Aguilera, J. A.

Anders, H.

Baumeister, P.

Berning, P. H.

Bloom, A.

Coursen, D.

DeBell, G. W.

Delano, E.

Dobrowolski, J. A.

Eichinger, R.

Epstein, L. I.

Geffken, W.

W. Geffken, 'Überzug aus mindestens drei Schichten von verschiedener Brechungszahl für einen nicht-metallischen Gegenstand zur Verminderung von dessen Oberflächenreflexion,' German patent 758,767 (19 July 1940).

Goldstein, F. T.

Gustafson, D. E.

Herpin, A.

A. Herpin, 'Calcul du pouvoir réflecteur d'un systèm stratifiè quelconque,' Acad. Sci. Paris , C. R. 225, 182-183 (1947).

Jacobsson, R.

Jakobs, S.

U. B. Schallenberg, S. Jakobs, and N. Kaiser, 'Analytical design of multicycle broadband AR coatings,' in Advances in Optical Interference Coatings, C.Amra and A.Macleod, eds., Proc. SPIE 3738, 230-238 (1999).

Kaiser, N.

U. Schulz, U. B. Schallenberg, and N. Kaiser, 'Symmetrical periods in antireflective coatings for plastic optics,' Appl. Opt. 42, 1346-1351 (2003).
[CrossRef] [PubMed]

U. Schulz, U. B. Schallenberg, and N. Kaiser, 'Antireflection coating design for plastic optics,' Appl. Opt. 41, 3107-3110 (2002).
[CrossRef] [PubMed]

U. Schulz, N. Kaiser, and U. B. Schallenberg, 'AR-hard broadband antireflective coatings generated by a controlled needle-optimization technique,' in Optical Interference Coatings on CD-ROM (Optical Society of America, 2004), Paper TuB2.

U. B. Schallenberg, S. Jakobs, and N. Kaiser, 'Analytical design of multicycle broadband AR coatings,' in Advances in Optical Interference Coatings, C.Amra and A.Macleod, eds., Proc. SPIE 3738, 230-238 (1999).

U. Schallenberg, U. Schulz, and N. Kaiser, 'Multicycle AR coatings: a theoretical approach,' in Advances in Optical Thin Films, C.Amra, N.Kaiser, and H.A.Macleod, eds., Proc. SPIE 5250, 357-366 (2004).

U. Schulz, N. Kaiser, and U. Schallenberg, 'Reflection reducing coating,' U.S. Patent 6,645,608 (11 November 2003).

Kemp, R. A.

Langfeld, R.

A. Thelen and R. Langfeld, 'Coating design contest: antireflection coating for lenses used with normal and infrared photographic film,' in Thin Film for Optical Systems, K.H.Guenther, ed., Proc. SPIE 1782, 551-601 (1993).

Ma, P.

Macleod, A.

A. Macleod, Thin-Film Optical Filters, 3rd ed. (Institute of Physics, 2001).
[CrossRef]

Martensson, J. O.

Musset, A.

A. Musset and A. Thelen, 'Multilayer antireflection coatings,' in Progress in Optics, E.Wolf, ed. (North-Holland, 1970), Vol. 8, pp. 203-237.

Ohmer, M. C.

Pohlack, H.

H. Pohlack, 'Die Synthese optischer Interferenzschichtsysteme mit vorgegebenen Eigenschaften,' in Jenaer Jahrbuch 1952 (1952), pp. 181-221.

Poitras, D.

Rock, F. C.

F. C. Rock, 'Antireflection coating and assembly having synthesized layer of index of refraction,' U.S. patent 3,432,225 (4 May 1964).

Schallenberg, U.

U. Schulz, N. Kaiser, and U. Schallenberg, 'Reflection reducing coating,' U.S. Patent 6,645,608 (11 November 2003).

U. Schallenberg, U. Schulz, and N. Kaiser, 'Multicycle AR coatings: a theoretical approach,' in Advances in Optical Thin Films, C.Amra, N.Kaiser, and H.A.Macleod, eds., Proc. SPIE 5250, 357-366 (2004).

Schallenberg, U. B.

U. Schulz, U. B. Schallenberg, and N. Kaiser, 'Symmetrical periods in antireflective coatings for plastic optics,' Appl. Opt. 42, 1346-1351 (2003).
[CrossRef] [PubMed]

U. Schulz, U. B. Schallenberg, and N. Kaiser, 'Antireflection coating design for plastic optics,' Appl. Opt. 41, 3107-3110 (2002).
[CrossRef] [PubMed]

U. Schulz, N. Kaiser, and U. B. Schallenberg, 'AR-hard broadband antireflective coatings generated by a controlled needle-optimization technique,' in Optical Interference Coatings on CD-ROM (Optical Society of America, 2004), Paper TuB2.

U. B. Schallenberg, S. Jakobs, and N. Kaiser, 'Analytical design of multicycle broadband AR coatings,' in Advances in Optical Interference Coatings, C.Amra and A.Macleod, eds., Proc. SPIE 3738, 230-238 (1999).

Schulz, U.

U. Schulz, U. B. Schallenberg, and N. Kaiser, 'Symmetrical periods in antireflective coatings for plastic optics,' Appl. Opt. 42, 1346-1351 (2003).
[CrossRef] [PubMed]

U. Schulz, U. B. Schallenberg, and N. Kaiser, 'Antireflection coating design for plastic optics,' Appl. Opt. 41, 3107-3110 (2002).
[CrossRef] [PubMed]

U. Schallenberg, U. Schulz, and N. Kaiser, 'Multicycle AR coatings: a theoretical approach,' in Advances in Optical Thin Films, C.Amra, N.Kaiser, and H.A.Macleod, eds., Proc. SPIE 5250, 357-366 (2004).

U. Schulz, N. Kaiser, and U. Schallenberg, 'Reflection reducing coating,' U.S. Patent 6,645,608 (11 November 2003).

U. Schulz, N. Kaiser, and U. B. Schallenberg, 'AR-hard broadband antireflective coatings generated by a controlled needle-optimization technique,' in Optical Interference Coatings on CD-ROM (Optical Society of America, 2004), Paper TuB2.

Seeley, J. S.

J. S. Seeley, 'Synthesis of interference filters,' Proc. Phys. Soc. London 78, 998-1008 (1961).
[CrossRef]

Southwell, W. H.

Strong, J.

Sullivan, B. T.

Thelen, A.

A. Thelen, 'Multilayer filters with wide transmittance bands, part II,' J. Opt. Soc. Am. 63, 65-68 (1973).
[CrossRef]

A. Thelen, 'Multilayer filters with wide transmittance bands,' J. Opt. Soc. Am. 53, 1266-1270 (1963).
[CrossRef]

A. Thelen, Design of Optical Interference Coatings (McGraw-Hill, 1989).

A. Musset and A. Thelen, 'Multilayer antireflection coatings,' in Progress in Optics, E.Wolf, ed. (North-Holland, 1970), Vol. 8, pp. 203-237.

A. Thelen and R. Langfeld, 'Coating design contest: antireflection coating for lenses used with normal and infrared photographic film,' in Thin Film for Optical Systems, K.H.Guenther, ed., Proc. SPIE 1782, 551-601 (1993).

Tikhonravov, A. V.

Trubetskov, M. K.

Vakil, H.

Verly, P. G.

Willey, R. R.

Young, L.

Acad. Sci. Paris (1)

A. Herpin, 'Calcul du pouvoir réflecteur d'un systèm stratifiè quelconque,' Acad. Sci. Paris , C. R. 225, 182-183 (1947).

Ann. Phys. (1)

F. Abelés, 'Recherches sur la propagation des ondes électromagnétiques sinusoïdales dans les milieux stratifiés. Application aux couches minces,' Ann. Phys. 5, 596-640, 706-782 (1950).

Appl. Opt. (13)

J. A. Dobrowolski, D. Poitras, P. Ma, H. Vakil, and M. Acree, 'Toward perfect antireflection coatings: numerical investigation,' Appl. Opt. 41, 3075-3083 (2002).
[CrossRef] [PubMed]

U. Schulz, U. B. Schallenberg, and N. Kaiser, 'Antireflection coating design for plastic optics,' Appl. Opt. 41, 3107-3110 (2002).
[CrossRef] [PubMed]

U. Schulz, U. B. Schallenberg, and N. Kaiser, 'Symmetrical periods in antireflective coatings for plastic optics,' Appl. Opt. 42, 1346-1351 (2003).
[CrossRef] [PubMed]

H. Anders and R. Eichinger, 'Die optische Wirkung und die praktische Bedeutung inhomogener Schichten,' Appl. Opt. 4, 899-905 (1965).
[CrossRef]

R. Jacobsson and J. O. Martensson, 'Evaporated inhomogeneous thin films,' Appl. Opt. 5, 29-34 (1966).
[CrossRef] [PubMed]

W. H. Southwell, 'Coating design using very thin high- and low-index layers,' Appl. Opt. 24, 457-3460 (1985).
[CrossRef] [PubMed]

J. A. Aguilera, J. Aguilera, P. Baumeister, A. Bloom, D. Coursen, J. A. Dobrowolski, F. T. Goldstein, D. E. Gustafson, and R. A. Kemp, 'Antireflection coatings for germanium IR optics: a comparison of numerical design methods,' Appl. Opt. 27, 2832-2840 (1988).
[CrossRef] [PubMed]

P. G. Verly, J. A. Dobrowolski, and R. R. Willey, 'Fourier-transform method for the design of wideband anti-reflection coatings,' Appl. Opt. 31, 3836-3846 (1992).
[CrossRef] [PubMed]

A. V. Tikhonravov and J. A. Dobrowolski, 'Quasi-optimal synthesis for antireflection coatings: a new method,' Appl. Opt. 32, 4265-4275 (1993).
[CrossRef] [PubMed]

R. R. Willey, 'Predicting achievable design performance of broadband antireflection coatings,' Appl. Opt. 32, 5447-5451 (1993).
[CrossRef] [PubMed]

P. G. Verly, A. V. Tikhonravov, and M. K. Trubetskov, 'Efficient refinement algorithm for the synthesis of inhomogeneous coatings,' Appl. Opt. 36, 1487-1495 (1997).
[CrossRef] [PubMed]

J. A. Dobrowolski, A. V. Tikhonravov, M. K. Trubetskov, B. T. Sullivan, and P. G. Verly, 'Optimal single-band normal-incidence antireflection coatings,' Appl. Opt. 35, 644-658 (1996).
[CrossRef] [PubMed]

A. V. Tikhonravov, M. K. Trubetskov, and G. W. DeBell, 'Application of the needle optimization technique to the design of optical coatings,' Appl. Opt. 35, 5493-5508 (1996).
[CrossRef] [PubMed]

J. Opt. Soc. Am. (9)

Proc. Phys. Soc. London (1)

J. S. Seeley, 'Synthesis of interference filters,' Proc. Phys. Soc. London 78, 998-1008 (1961).
[CrossRef]

Other (11)

F. C. Rock, 'Antireflection coating and assembly having synthesized layer of index of refraction,' U.S. patent 3,432,225 (4 May 1964).

A. Musset and A. Thelen, 'Multilayer antireflection coatings,' in Progress in Optics, E.Wolf, ed. (North-Holland, 1970), Vol. 8, pp. 203-237.

A. Thelen and R. Langfeld, 'Coating design contest: antireflection coating for lenses used with normal and infrared photographic film,' in Thin Film for Optical Systems, K.H.Guenther, ed., Proc. SPIE 1782, 551-601 (1993).

U. B. Schallenberg, S. Jakobs, and N. Kaiser, 'Analytical design of multicycle broadband AR coatings,' in Advances in Optical Interference Coatings, C.Amra and A.Macleod, eds., Proc. SPIE 3738, 230-238 (1999).

U. Schallenberg, U. Schulz, and N. Kaiser, 'Multicycle AR coatings: a theoretical approach,' in Advances in Optical Thin Films, C.Amra, N.Kaiser, and H.A.Macleod, eds., Proc. SPIE 5250, 357-366 (2004).

U. Schulz, N. Kaiser, and U. Schallenberg, 'Reflection reducing coating,' U.S. Patent 6,645,608 (11 November 2003).

A. Macleod, Thin-Film Optical Filters, 3rd ed. (Institute of Physics, 2001).
[CrossRef]

A. Thelen, Design of Optical Interference Coatings (McGraw-Hill, 1989).

U. Schulz, N. Kaiser, and U. B. Schallenberg, 'AR-hard broadband antireflective coatings generated by a controlled needle-optimization technique,' in Optical Interference Coatings on CD-ROM (Optical Society of America, 2004), Paper TuB2.

H. Pohlack, 'Die Synthese optischer Interferenzschichtsysteme mit vorgegebenen Eigenschaften,' in Jenaer Jahrbuch 1952 (1952), pp. 181-221.

W. Geffken, 'Überzug aus mindestens drei Schichten von verschiedener Brechungszahl für einen nicht-metallischen Gegenstand zur Verminderung von dessen Oberflächenreflexion,' German patent 758,767 (19 July 1940).

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Figures (8)

Fig. 1
Fig. 1

AR-hard example with (a) step-index profile, (b) index profile, (c) spectral performance, and (d) thickness characteristic at 650 nm.[31]

Fig. 2
Fig. 2

Equivalent stack index versus normalized wave number of different QW stacks: ET , symmetrical trilayer; EQ , asymmetrical four layer; and EF , asymmetrical five layer.[28]

Fig. 3
Fig. 3

Illustration of passband, potential AR region, required AR region, reflectance R, and ESI versus normalized wave number, three-layer ABC QWS with refractive indices nS = 1.52, n 0 = 1, nA = 1.788, nB = 2.132, and nC = 1.46.

Fig. 4
Fig. 4

Spectral performance of a heat reflector with second- and third-order stop-band suppression. The straight line shows the reflectance of the uncoated substrate (Thelen, Ref. 13).

Fig. 5
Fig. 5

Spectral performance of QW stacks with different orders of stop-band suppression: (a) second-order suppression (ABBA)[4] with n A = 1.45, n B = 1.95; (b) second- and third-order suppression (ABCBA)[4] with n A = 1.45, n B = 1.95, n C = 2.35; (c) second–fourth-order suppression (ABCCBA)[4] with n A = 1.45, n B = 1.85, n C = 2.35; and (d) second–fifth-order suppression (ABCDCBA)[4] with n A = 1.45, n B = 1.72, n C = 2.12, n D = 2.35 (Thelen, Ref. 35).

Fig. 6
Fig. 6

ESI and reflectance R versus the normalized wave number, the symmetrical five-layer ABCBA QWS, with refractive indices nS = 1.52, n 0 = 1, n A = 1.46, n B = 1.954, and n C = 2.35.

Fig. 7
Fig. 7

ESS versus the normalized wave number, the asymmetrical five-layer ABCDA QWS with refractive indices n A = 1.46, n B = 1.68, n C = 2.35, and n D = 2.273.

Fig. 8
Fig. 8

ESI, ESS, and reflectance R versus the normalized wave number, the asymmetrical five-layer ABCDA QWS with refractive indices nS = 1.52, n 0 = 1, n A = 1.46, n B = 1.68, n C = 2.35, n D = 2.273.

Tables (2)

Tables Icon

Table 1 Selection of Essential Papers Concerning AR Designs and Thin-Film Optics in General

Tables Icon

Table 2 Selection of Essential Papers Concerning AR Designs and Thin-Film Software

Equations (12)

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

BW AR = λ 2 / λ 1 .
λ 0 = 2 λ 1 λ 2 / ( λ 1 + λ 2 ) .
n A i = n L    2 E i ,
n 1 d 1 = n 2 d 2 = n 3 d 3 = = n p d p = λ 0 / 4 ,
ϕ = 2 π / λ × n d = π / 2 × λ 0 / λ .
( M 11 i M 12 i M 21 M 22 ) = j = 1 p ( cos ϕ i  sin   ϕ / n j i n j sin ϕ cos ϕ )  .
R = ( n 0 M 11 n s M 22 ) 2 + ( n 0 n S M 12 M 21 ) 2 ( n 0 M 11 + n s M 22 ) 2 + ( n 0 n S M 12 + M 21 ) 2 .
E ( ϕ ) = ( M 21 M 12 ) 1 / 2 = ( a 1 cos p 1 ϕ + a 2 cos p 3 ϕ + a 3 cos p 5     ϕ + b 1 cos p 1 ϕ + b 2 cos p 3 ϕ + b 3 cos p 5     ϕ + ) 1 / 2 ,
S = n S M 22 M 11 = n S ( c 1 cos p ϕ + c 2 cos p 2 ϕ + c 3 cos p 4 ϕ + ) d 1 cos p ϕ + d 2 cos p 2 ϕ + d 3 cos p 4 ϕ + ,
R = M 11           2 ( n 0 S ) 2 + M 12           2 ( n 0 n S E 2 ) 2 M 11           2 ( n 0 + S ) 2 + M 12           2 ( n 0 n S + E 2 ) 2 ,
R = ( n 0 S ) 2 ( n 0 + S ) 2 at M 12 = 0 ,
R = ( n 0 n S E 2 ) 2 ( n 0 n S + E 2 ) 2 at M 11 = 0.

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