Abstract

The principal aspects of rugate filter theory are reviewed and expanded to show how the Fourier-transform technique can be used to design rugate filters that fulfill many optical coating functions.

© 1993 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. H. Sankur, W. J. Gunning, J. F. DeNatale, “Intrinsic stress and structural properties of mixed composition thin films,” Appl. Opt. 27, 1564–1567 (1988).
    [Crossref] [PubMed]
  2. R. Bertram, M. F. Ouellette, P. Y. Tse, “Inhomogeneous optical coatings: an experimental study of new approach,” Appl. Opt. 28, 2935–2939 (1989).
    [Crossref] [PubMed]
  3. E. P. Donovan, D. V. Vechten, A. D. F. Kahn, C. A. Carosella, G. K. Hubler, “Near infrared rugate filter fabrication by ion beam assisted deposition of Si(1−x) Nx films,” Appl. Opt. 28, 2940–2944 (1989).
    [Crossref] [PubMed]
  4. W. J. Gunning, R. L. Hall, F. J. Woodberry, W. H. Southwell, N. S. Gluck, “Codeposition of continuous composition rugate filters,” Appl. Opt. 28, 2945–2948 (1989).
    [Crossref] [PubMed]
  5. J. P. Heuer, J. P. Eblen, R. L. Hall, W. J. Gunning, “Scale-up considerations for codeposited gradient index optical thin film filters,” in Optical Interference Coatings, Vol. 15 of 1992 OSA Technical Digest Series (Optical Society of America, Washington D.C., 1992), pp. 122–124.
  6. H. Sankur, W. Southwell, R. Hall, W. J. Gunning, “Rugate filter deposition by the OMVPE technique,” in Optical Interference Coatings, Vol. 15 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), pp. 125–127.
  7. R. L. Hall, “Gradient index bandpass filters,” in Optical Interference Coatings, Vol. 15 of 1992 OSA Technical Digest Series (Optical Society of America, Washington D.C., 1992), pp. 116–118.
  8. R. R. Willey, “Rugate broadband antireflection coating design,” in Current Developments in Optical Engineering and Commercial Optics, R. E. Fischer, H. M. Pollicove, W. J. Smith, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1168, 224–228 (1989).
  9. R. R. Willey, P. G. Verly, J. A. Dobrowolski, “Design of wideband antireflection coating with the Fourier transform method,” in Optical Thin Films and Applications, R. Herrmann, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1270, 36–44 (1990).
  10. P. G. Verly, J. A. Dobrowolski, R. R. Willey, “Fourier-transform method for the design of wideband antireflection coatings,” Appl. Opt. 31, 3836–3846 (1992).
    [Crossref] [PubMed]
  11. E. Delano, “Fourier synthesis of multilayer filters,” J. Opt. Soc. Am. 57, 1529–1533 (1967).
    [Crossref]
  12. B. G. Bovard, “Derivation of a matrix describing a rugate dielectric thin film,” Appl. Opt. 27, 1998–2005 (1988).
    [Crossref] [PubMed]
  13. L. Sossi, “On the theory of the synthesis of multilayer dielectric light filters,” Eesti NSV Tead. Akad. Toim. Füüs. Mat. 25, 171–176 (1974).
  14. L. Sossi, “A method for the synthesis of multilayer dielectric interference coatings,” Eesti NSV Tead. Akad. Toim. Füüs. Mat. 23, 229–237 (1974).
  15. J. A. Dobrowolski, D. Lowe, “Optical thin film synthesis program based on the use of Fourier transform,” Appl. Opt. 17, 3039–3050 (1978).
    [Crossref] [PubMed]
  16. B. G. Bovard, Angle of incidence effect on multiline rugate filters, in Laser Safety, Eyesafe Laser Systems, and Laser Eye Protection, P. K. Galoff, D. H. Sliney, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1207, 218–229 (1990).
  17. P. G. Verly, J. A. Dobrowolski, W. J. Wild, R. L. Burton, “Synthesis of high rejection filters with the Fourier transform method,” Appl. Opt. 28, 2864–2875 (1989).
    [Crossref] [PubMed]
  18. B. G. Bovard, “Rugate filter design: the modified Fourier transform technique,” Appl. Opt. 29, 24–30 (1990).
    [Crossref] [PubMed]
  19. W. H. Southwell, “Spectral response calculations of rugate filters using coupled wave theory,” J. Opt. Soc. Am. A 5, 1558–1564 (1989).
    [Crossref]
  20. J. F. Kaiser, Digital Filters (Wiley, New York, 1966).
  21. W. H. Southwell, “Using apodization functions to reduce sidelobes in rugate filters,” Appl. Opt. 28, 5091–5094 (1989).
    [Crossref] [PubMed]
  22. W. H. Southwell, R. L. Hall, “Rugate filter sidelobe suppression using quintic and rugated quintic matching layers,” Appl. Opt. 28, 2949–2951 (1989).
    [Crossref] [PubMed]

1992 (1)

1990 (1)

1989 (7)

1988 (2)

1978 (1)

1974 (2)

L. Sossi, “On the theory of the synthesis of multilayer dielectric light filters,” Eesti NSV Tead. Akad. Toim. Füüs. Mat. 25, 171–176 (1974).

L. Sossi, “A method for the synthesis of multilayer dielectric interference coatings,” Eesti NSV Tead. Akad. Toim. Füüs. Mat. 23, 229–237 (1974).

1967 (1)

Bertram, R.

Bovard, B. G.

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

B. G. Bovard, “Derivation of a matrix describing a rugate dielectric thin film,” Appl. Opt. 27, 1998–2005 (1988).
[Crossref] [PubMed]

B. G. Bovard, Angle of incidence effect on multiline rugate filters, in Laser Safety, Eyesafe Laser Systems, and Laser Eye Protection, P. K. Galoff, D. H. Sliney, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1207, 218–229 (1990).

Burton, R. L.

Carosella, C. A.

Delano, E.

DeNatale, J. F.

Dobrowolski, J. A.

Donovan, E. P.

Eblen, J. P.

J. P. Heuer, J. P. Eblen, R. L. Hall, W. J. Gunning, “Scale-up considerations for codeposited gradient index optical thin film filters,” in Optical Interference Coatings, Vol. 15 of 1992 OSA Technical Digest Series (Optical Society of America, Washington D.C., 1992), pp. 122–124.

Gluck, N. S.

Gunning, W. J.

W. J. Gunning, R. L. Hall, F. J. Woodberry, W. H. Southwell, N. S. Gluck, “Codeposition of continuous composition rugate filters,” Appl. Opt. 28, 2945–2948 (1989).
[Crossref] [PubMed]

H. Sankur, W. J. Gunning, J. F. DeNatale, “Intrinsic stress and structural properties of mixed composition thin films,” Appl. Opt. 27, 1564–1567 (1988).
[Crossref] [PubMed]

J. P. Heuer, J. P. Eblen, R. L. Hall, W. J. Gunning, “Scale-up considerations for codeposited gradient index optical thin film filters,” in Optical Interference Coatings, Vol. 15 of 1992 OSA Technical Digest Series (Optical Society of America, Washington D.C., 1992), pp. 122–124.

H. Sankur, W. Southwell, R. Hall, W. J. Gunning, “Rugate filter deposition by the OMVPE technique,” in Optical Interference Coatings, Vol. 15 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), pp. 125–127.

Hall, R.

H. Sankur, W. Southwell, R. Hall, W. J. Gunning, “Rugate filter deposition by the OMVPE technique,” in Optical Interference Coatings, Vol. 15 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), pp. 125–127.

Hall, R. L.

W. J. Gunning, R. L. Hall, F. J. Woodberry, W. H. Southwell, N. S. Gluck, “Codeposition of continuous composition rugate filters,” Appl. Opt. 28, 2945–2948 (1989).
[Crossref] [PubMed]

W. H. Southwell, R. L. Hall, “Rugate filter sidelobe suppression using quintic and rugated quintic matching layers,” Appl. Opt. 28, 2949–2951 (1989).
[Crossref] [PubMed]

R. L. Hall, “Gradient index bandpass filters,” in Optical Interference Coatings, Vol. 15 of 1992 OSA Technical Digest Series (Optical Society of America, Washington D.C., 1992), pp. 116–118.

J. P. Heuer, J. P. Eblen, R. L. Hall, W. J. Gunning, “Scale-up considerations for codeposited gradient index optical thin film filters,” in Optical Interference Coatings, Vol. 15 of 1992 OSA Technical Digest Series (Optical Society of America, Washington D.C., 1992), pp. 122–124.

Heuer, J. P.

J. P. Heuer, J. P. Eblen, R. L. Hall, W. J. Gunning, “Scale-up considerations for codeposited gradient index optical thin film filters,” in Optical Interference Coatings, Vol. 15 of 1992 OSA Technical Digest Series (Optical Society of America, Washington D.C., 1992), pp. 122–124.

Hubler, G. K.

Kahn, A. D. F.

Kaiser, J. F.

J. F. Kaiser, Digital Filters (Wiley, New York, 1966).

Lowe, D.

Ouellette, M. F.

Sankur, H.

H. Sankur, W. J. Gunning, J. F. DeNatale, “Intrinsic stress and structural properties of mixed composition thin films,” Appl. Opt. 27, 1564–1567 (1988).
[Crossref] [PubMed]

H. Sankur, W. Southwell, R. Hall, W. J. Gunning, “Rugate filter deposition by the OMVPE technique,” in Optical Interference Coatings, Vol. 15 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), pp. 125–127.

Sossi, L.

L. Sossi, “On the theory of the synthesis of multilayer dielectric light filters,” Eesti NSV Tead. Akad. Toim. Füüs. Mat. 25, 171–176 (1974).

L. Sossi, “A method for the synthesis of multilayer dielectric interference coatings,” Eesti NSV Tead. Akad. Toim. Füüs. Mat. 23, 229–237 (1974).

Southwell, W.

H. Sankur, W. Southwell, R. Hall, W. J. Gunning, “Rugate filter deposition by the OMVPE technique,” in Optical Interference Coatings, Vol. 15 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), pp. 125–127.

Southwell, W. H.

Tse, P. Y.

Vechten, D. V.

Verly, P. G.

Wild, W. J.

Willey, R. R.

P. G. Verly, J. A. Dobrowolski, R. R. Willey, “Fourier-transform method for the design of wideband antireflection coatings,” Appl. Opt. 31, 3836–3846 (1992).
[Crossref] [PubMed]

R. R. Willey, “Rugate broadband antireflection coating design,” in Current Developments in Optical Engineering and Commercial Optics, R. E. Fischer, H. M. Pollicove, W. J. Smith, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1168, 224–228 (1989).

R. R. Willey, P. G. Verly, J. A. Dobrowolski, “Design of wideband antireflection coating with the Fourier transform method,” in Optical Thin Films and Applications, R. Herrmann, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1270, 36–44 (1990).

Woodberry, F. J.

Appl. Opt. (11)

H. Sankur, W. J. Gunning, J. F. DeNatale, “Intrinsic stress and structural properties of mixed composition thin films,” Appl. Opt. 27, 1564–1567 (1988).
[Crossref] [PubMed]

R. Bertram, M. F. Ouellette, P. Y. Tse, “Inhomogeneous optical coatings: an experimental study of new approach,” Appl. Opt. 28, 2935–2939 (1989).
[Crossref] [PubMed]

E. P. Donovan, D. V. Vechten, A. D. F. Kahn, C. A. Carosella, G. K. Hubler, “Near infrared rugate filter fabrication by ion beam assisted deposition of Si(1−x) Nx films,” Appl. Opt. 28, 2940–2944 (1989).
[Crossref] [PubMed]

W. J. Gunning, R. L. Hall, F. J. Woodberry, W. H. Southwell, N. S. Gluck, “Codeposition of continuous composition rugate filters,” Appl. Opt. 28, 2945–2948 (1989).
[Crossref] [PubMed]

P. G. Verly, J. A. Dobrowolski, R. R. Willey, “Fourier-transform method for the design of wideband antireflection coatings,” Appl. Opt. 31, 3836–3846 (1992).
[Crossref] [PubMed]

B. G. Bovard, “Derivation of a matrix describing a rugate dielectric thin film,” Appl. Opt. 27, 1998–2005 (1988).
[Crossref] [PubMed]

P. G. Verly, J. A. Dobrowolski, W. J. Wild, R. L. Burton, “Synthesis of high rejection filters with the Fourier transform method,” Appl. Opt. 28, 2864–2875 (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, D. Lowe, “Optical thin film synthesis program based on the use of Fourier transform,” Appl. Opt. 17, 3039–3050 (1978).
[Crossref] [PubMed]

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

W. H. Southwell, R. L. Hall, “Rugate filter sidelobe suppression using quintic and rugated quintic matching layers,” Appl. Opt. 28, 2949–2951 (1989).
[Crossref] [PubMed]

Eesti NSV Tead. Akad. Toim. Füüs. Mat. (2)

L. Sossi, “On the theory of the synthesis of multilayer dielectric light filters,” Eesti NSV Tead. Akad. Toim. Füüs. Mat. 25, 171–176 (1974).

L. Sossi, “A method for the synthesis of multilayer dielectric interference coatings,” Eesti NSV Tead. Akad. Toim. Füüs. Mat. 23, 229–237 (1974).

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (1)

Other (7)

J. F. Kaiser, Digital Filters (Wiley, New York, 1966).

J. P. Heuer, J. P. Eblen, R. L. Hall, W. J. Gunning, “Scale-up considerations for codeposited gradient index optical thin film filters,” in Optical Interference Coatings, Vol. 15 of 1992 OSA Technical Digest Series (Optical Society of America, Washington D.C., 1992), pp. 122–124.

H. Sankur, W. Southwell, R. Hall, W. J. Gunning, “Rugate filter deposition by the OMVPE technique,” in Optical Interference Coatings, Vol. 15 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), pp. 125–127.

R. L. Hall, “Gradient index bandpass filters,” in Optical Interference Coatings, Vol. 15 of 1992 OSA Technical Digest Series (Optical Society of America, Washington D.C., 1992), pp. 116–118.

R. R. Willey, “Rugate broadband antireflection coating design,” in Current Developments in Optical Engineering and Commercial Optics, R. E. Fischer, H. M. Pollicove, W. J. Smith, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1168, 224–228 (1989).

R. R. Willey, P. G. Verly, J. A. Dobrowolski, “Design of wideband antireflection coating with the Fourier transform method,” in Optical Thin Films and Applications, R. Herrmann, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1270, 36–44 (1990).

B. G. Bovard, Angle of incidence effect on multiline rugate filters, in Laser Safety, Eyesafe Laser Systems, and Laser Eye Protection, P. K. Galoff, D. H. Sliney, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1207, 218–229 (1990).

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (18)

Fig. 1
Fig. 1

Optical admittance profile of an inhomogeneous layer versus optical phase thickness. The figure illustrates how extending the layer removes the optical admittance contrasts at the outermost boundaries.

Fig. 2
Fig. 2

Reflectance spectrum of a 90% reflecting rugate mirror with matching outer media (thicker curve) and Fourier transform of the corresponding sA function (thinner curve); ηm = 1.45, ηm = 2.0, xA = 3551.72 nm, ΦA = 0°.

Fig. 3
Fig. 3

Phase of the amplitude reflection coefficient of a 90% reflecting rugate mirror with matching outer media (thicker curve) and Fourier transform of the corresponding sA function (thinner curve); ηm = 1.45, ηM = 2.0, xA = 3551.72 nm, ΦA = 0°.

Fig. 4
Fig. 4

Influence of β on the reflectance spectra of 90% reflecting rugate mirrors centered at 1 μm. The sidelobes decrease as β spans the values 0, 1, 3, and 5.

Fig. 5
Fig. 5

Comparison of apodizations with a Kaiser window (thicker curve) and a Bartlett window (thinner curve). The optical thickness is 7119.8 nm for both designs (90% reflecting rugate mirror at 1 μm).

Fig. 6
Fig. 6

Reflectance spectrum of a rugate mirror (R = 99%). The phase ΦA = 197.85° is chosen to optimize the peak reflectance with minimum optical thickness.

Fig. 7
Fig. 7

Reflectance spectrum of a double-line rugate mirror designed to reflect 99% at 800 and 1000 nm. Actual peak reflectances are 98.67% and 98.75%. Outer media were chosen to match the layer outermost indices.

Fig. 8
Fig. 8

Refractive-index profile of a double-line rugate mirror designed to reflect 99% at 800 and 1000 nm.

Fig. 9
Fig. 9

Reflectance spectrum of a broadband rugate mirror. The outermost indices match those of the outer media to show oscillations that are due to the rectangular truncation and the use of shifted structures.

Fig. 10
Fig. 10

Reflectance spectra of a rugate V coat (thin curve) compared with a regular V coat (thick curve).

Fig. 11
Fig. 11

Refractive-index profile of a rugate V coat (thin curve) compared with a regular V coat (thick curve).

Fig. 12
Fig. 12

Reflectance spectra of three quarter-wave rugate V coats. As the optical thickness increases from 750 to 1250 and 1750 nm, the antireflection band narrows.

Fig. 13
Fig. 13

Refractive-index profile of a half-wave-thick rugate anti-reflection V coat.

Fig. 14
Fig. 14

Reflectance spectrum of a half-wave-thick rugate antireflection V coat.

Fig. 15
Fig. 15

Refractive-index profile of a bandpass rugate filter. xA = 2500 nm, λA = 1000 nm, ΦA = 90°, ηM = 2.0, ηm = 1.45, ηi = 1.0, ηs = 1.52, hA = −hB = −5000 nm.

Fig. 16
Fig. 16

Transmittance spectrum of a bandpass rugate filter. xA = 2500 nm, λA = 1000 nm, ΦA = 90°, ηM = 2.0, ηm = 1.45, ηi = 1.0, ηs = 1.52, hA =−hB = −5000 nm. One peak is positioned at 1000 nm, as designed, the second is positioned near 890 nm, and the Fourier-transform theory predicts a peak at 909 nm. Note that the Fourier-transform theory predicts peaks in the ripple region correctly.

Fig. 17
Fig. 17

Refractive-index profile of a narrow-band rugate filter. xA = 2500 nm, λA = 1000 nm, ΦA = ΦB = 0°, ηM = 2.0, ηm = 1.45, ηi = 1.0, ηs = 1.52, hA = −hB = −5000 nm.

Fig. 18
Fig. 18

Reflectance spectrum of a double-band rugate filter. xA = 2500 nm, λA = 1000 nm, ΦA = ΦB = 0° (n = 0), ηM = 2.0, ηm = 1.45, ηi = 1.0, ηs = 1.52, hA = −hB = −5000 nm (p = 5). Fourier-transform theory predicts a peak at 952 nm (q = 10) and a peak at 1053 nm (q = 9).

Tables (6)

Tables Icon

Table 1 Comparison between the Reflectances Obtained by the Exact (RA Exact) and Approximate (RA Approx.) Optical Thickness Calculation Methodsa

Tables Icon

Table 2 Influence of β on the Continuous Fourier Component of the Kaiser Apodization Windowa

Tables Icon

Table 3 Optical Thickness of Several Truncated Single-Line Rugate Mirrors Designed to have Minimum Optical Thicknessa

Tables Icon

Table 4 Key Parameters of the Broadband Rugate Mirrora

Tables Icon

Table 5 Phase Factors and Minimum and Maximum Optical Admittances for Three Quarter-Wave Rugate Antireflection Coatings

Tables Icon

Table 6 Results Obtained in the Case of Narrow-Band Filtersa

Equations (81)

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

[ E ( 0 ) H ( 0 ) ] = [ L ( β 0 ) [ η ( β 0 ) η ( 0 ) ] 1 / 2 i G ( β 0 ) [ η ( β 0 ) η ( 0 ) ] 1 / 2 i K ( β 0 ) [ η ( β 0 ) η ( 0 ) ] 1 / 2 F ( β 0 ) [ η ( 0 ) η ( β 0 ) ] 1 / 2 ] ( 1 η s ) ,
β = 2 π λ 0 z n ( u ) cos θ ( u ) d u ,
F ( β ) = f ( β ) cos β + k ( β ) sin β , K ( β ) = f ( β ) cos β k ( β ) cos β , G ( β ) = l ( β ) cos β + g ( β ) cos β , L ( β ) = l ( β ) cos β g ( β ) sin β ,
f A ( β 0 ) = m = 0 m = C m ( β 0 ; ζ A ) , g A ( β 0 ) = m = 0 m = S m ( β 0 ; ζ A ) , l A ( β 0 ) = m = 0 m = ( 1 ) m C m ( β 0 ; ζ A ) , k A ( β 0 ) = m = 0 m = ( 1 ) m + 1 S m ( β 0 ; ζ A ) ,
f B ( β 0 ) = [ m = 0 m = C 2 m ( 0 ; ζ B ) ] [ 1 + m = 0 m = C 2 m + 1 ( β 0 ; ζ B ) ] + [ m = 0 m = S 2 m ( 0 ; ζ B ) ] [ m = 0 m = S 2 m + 1 ( β 0 ; ζ B ) ] [ m = 0 m = C 2 m ( 0 ; ζ B ) ] 2 + [ m = 0 m = S 2 m ( 0 ; ζ B ) ] 2 g B ( β 0 ) = [ m = 0 m = S 2 m ( 0 ; ζ B ) ] [ 1 + m = 0 m = C 2 m + 1 ( β 0 ; ζ B ) ] + [ m = 0 m = C 2 m ( 0 ; ζ B ) ] [ m = 0 m = S 2 m + 1 ( β 0 ; ζ B ) ] [ m = 0 m = C 2 m ( 0 ; ζ B ) ] 2 + [ m = 0 m = S 2 m ( 0 ; ζ B ) ] 2 l B ( β 0 ) = [ m = 0 m = S 2 m ( 0 ; ζ B ) ] [ m = 0 m = S 2 m + 1 ( β 0 ; ζ B ) ] + [ m = 0 m = C 2 m ( 0 ; ζ B ) ] [ 1 m = 0 m = C 2 m + 1 ( β 0 ; ζ B ) ] [ m = 0 m = C 2 m ( 0 ; ζ B ) ] 2 + [ m = 0 m = S 2 m ( 0 ; ζ B ) ] 2 k B ( β 0 ) = [ m = 0 m = C 2 m ( 0 ; ζ B ) ] [ m = 0 m = S 2 m + 1 ( β 0 ; ζ B ) ] + [ m = 0 m = S 2 m ( 0 ; ζ B ) ] [ 1 m = 0 m = C 2 m + 1 ( β 0 ; ζ B ) ] [ m = 0 m = C 2 m ( 0 ; ζ B ) ] 2 + [ m = 0 m = S 2 m ( 0 ; ζ B ) ] 2
C m ( β ; ξ ) = ζ m β ζ m 1 β 1 ζ m 2 β 2 ζ 1 β m 1 r ( β 1 ) r ( β 2 ) r ( β m ) × cos 2 ( β m β m 1 + β 1 ) d β 1 d β m , S m ( β ; ξ ) = ζ m β ζ m 1 β 1 ζ m 2 β 2 ζ 1 β m 1 r ( β 1 ) r ( β 2 ) r ( β m ) × sin 2 ( β m β m 1 + β 1 ) d β 1 d β m ,
r ( β ) = η ( β ) 2 η ( β ) .
r ( β ) = { r ( β ) } function + 1 2 n = 1 n = N ln ( η n + η n ) δ ( β β n ) ,
r ( β ) = r old ( β ) + 1 2 ln η ( 0 ) η i δ ( β ) + 1 2 ln η s η i ( β 0 ) δ ( β β 0 ) .
R = ( f l ) 2 + ( g + k ) 2 ( f + l ) 2 + ( g k ) 2 , T = 4 ( f + l ) 2 + ( g k ) 2 , sin ψ r = 2 ( f g + k l ) ( f + l ) 2 + ( k g ) 2 cos ψ r = ( l 2 f 2 ) + ( g 2 k 2 ) ( f + l ) 2 + ( k g ) 2 .
Q ( β 0 ) exp [ i Φ ( β 0 ) ] = 0 β 0 r ( β 1 ) exp [ 2 i β 1 ) d β 1 ,
Q = R ,
Q = ( R / T ) 1 / 2 ,
Q = [ 1 2 ( 1 T T ) ] 1 / 2 ,
Q = ( 1 T T ) 1 / 2 .
Φ = ψ r + ( 2 m + 1 ) π ,
π σ x = β ( β 0 / 2 ) ,
Q ( σ ) exp [ i Φ ( σ ) ] = + r ( x ) exp ( 2 i π σ x ) d x , r ( x ) = + Q ( σ ) exp [ i Φ ( σ ) ] exp ( 2 i π σ x ) d σ ,
Φ = ψ r + 2 π σ x 0 + ( 2 m + 1 ) π .
ν = 2 σ 0 π sin 1 ( cosh r sin π σ 2 σ 0 ) ,
r = ½ ln ( η H / η L ) .
Q ( ν ) exp [ i Φ ( ν ) ] = + r ( x ) exp ( 2 i π ν x ) d x , Q = ( R / T ) 1 / 2 , ψ r = Φ 2 π ν x 0 + ( 2 m + 1 ) π ,
Q = 1 2 log 1 + R 1 R
r ( x ) = R A ( x ) + r B ( x ) ,
Q ( σ ) exp [ i Φ ( σ ) ] = Q A ( σ ) exp [ i Φ A ( σ ) ] + Q B ( σ ) exp [ i Φ B ( σ ) ] ,
Q 2 ( σ ) = Q A 2 ( σ ) + Q B 2 ( σ ) = 2 Q A ( σ ) Q B ( σ ) cos [ Φ A ( σ ) Φ B ( σ ) ]
η ( x ) = K η A ( x ) η B ( x ) ,
r ( x ) = r A ( x + h A ) + r B ( x + h B ) ,
Q ( σ ) exp [ i Φ ( σ ) ] = Q A ( σ ) exp [ i Φ A ( σ ) + 2 i π σ h A ] + Q B ( σ ) exp [ i Φ B ( σ ) + 2 i π σ h B ] ,
Q 2 = Q A 2 + Q B 2 + 2 Q A Q B × cos | ( Φ A 2 π σ h A ) ( Φ B 2 π σ h B ) | .
Q A ( σ ) = Q A σ A [ δ ( σ σ A ) + δ ( σ + σ A ) ] ,
Φ A ( σ ) = Φ A σ A [ δ ( σ σ A ) δ ( σ + σ A ) ] .
η A ( x ) = η A ( 0 ) exp 2 Q A π [ sin ( 2 π σ A x Φ A ) sin ( Φ A ) ] .
ln η M η m = 4 Q A π ,
s A ( x ) = r A ( x ) Π ( x 2 x A ) .
Q ¯ A ( σ ) exp i Φ ¯ A ( σ ) = Q A ( σ ) A exp ( i Φ A ) sin 2 π ( σ σ A ) x A π ( σ σ A ) + Q A σ A exp ( i Φ A ) sin 2 π ( σ + σ A ) x A π ( σ + σ A ) ,
Q ¯ A ( σ A ) exp [ i Φ ¯ A ( σ A ) ] = 2 Q A σ A x A [ exp ( i Φ A ) + exp ( i Φ A ) sinc 4 π σ A x A ] 2 Q A σ A x A exp ( i Φ A ) Q ¯ A ( σ A ) exp i Φ ¯ A ( σ A ) = 2 Q A σ A x A [ exp ( i Φ A ) + exp ( i Φ A ) sinc 4 π σ A x A ] 2 Q A σ A x A exp ( i Φ A ) .
x A Q ¯ A 2 Q A σ A .
x A = Q ¯ A 2 Q A σ A [ sin 2 Φ ¯ A ( 1 sinc 4 π σ A x A ) 2 + cos 2 Φ ¯ A ( 1 + sinc 4 π σ A x A ) 2 ] 1 / 2 ,
Q ¯ A = 1 2 ln 1 + R A 1 R A ,
R A = [ 1 ( η M / η m ) m π / 2 1 + ( η M / η m ) m π / 2 ] 2 .
η A ( x ) = η A ( 0 ) + η A ( 0 ) 2 Q A π [ sin ( 2 π σ A x + Φ A ) sin Φ A ] ,
n A ( u ) = n A ( 0 ) exp 2 Q A π × [ sin ( 4 π u λ A 2 π x A λ A + Φ A ) sin Φ A ] .
Δ λ λ = 1 2 ln η M η m .
s A ( x ) = r A ( x ) w ( x 2 x A ) ,
Q ¯ A ( σ A ) exp i Φ ¯ A ( σ ) = 2 Q A σ A x A [ exp ( i Φ A ) W ( σ σ A ) + exp ( i Φ A ) W ( σ + σ A ) ] ,
Q ¯ A ( σ A ) exp i Φ ¯ A ( σ A ) = 2 Q A σ A x A [ exp ( i Φ A ) W ( 0 ) + exp ( i Φ A ) W ( 2 σ A ) ] 2 Q A σ A x A exp ( i Φ A ) W ( 0 ) ] Q ¯ A ( σ A ) exp i Φ ¯ A ( σ A ) = 2 Q A σ A x A [ exp ( i Φ A ) W ( 2 σ A ) + exp ( i Φ A ) W ( 0 ) ] 2 Q A σ A x A exp ( i Φ A ) W ( 0 ) ] .
x A Q ¯ A 2 Q A σ A | W ( 0 ) | .
R A = [ 1 ( η M / η m ) m π | W | ( 0 ) | / 2 1 + ( η M / η m ) m π | W | ( 0 ) | / 2 ] .
w ( x ) = ( 1 2 | x | ) Π ( x ) ,
w ( x ) = I 0 [ β ( 1 4 x 2 ) 1 / 2 ] I 0 ( β ) Π ( x ) ,
W ( 0 ) = sinh β β I 0 ( β ) .
r B ( x ) = 1 2 ln η η i δ ( x + x A ) + 1 2 ln η s η + δ ( x x A ) ,
Q B ( σ ) exp [ i Φ B ( σ ) ] = r cos ( 2 π σ x A ) + i r + sin ( 2 π σ x A ) ,
r = 1 2 ln η η s η + η i , r + = 1 2 ln η + η η i η s ,
η = η M exp 2 Q A π [ sin ( Φ A 2 π σ A x A ) 1 ] , η + = η M exp 2 Q A π [ sin ( Φ A + 2 π σ A x A ) 1 ] .
Q ( σ ) exp [ i Φ ( σ ) ] = Q ¯ A exp [ i Φ ¯ A ( σ ) ] + Q B exp [ i Φ B ( σ ) ] .
x A = 1 2 Q A σ A [ ( Q cos Φ 1 2 ln η s η i cos 2 π σ A x A 1 sinc 4 π σ A x A ) 2 + ( Q sin Φ 1 2 ln η m η M η i η s sin 2 π σ A x A 1 + sinc 4 π σ A x A ) 2 ] 1 / 2 ,
Q ( σ ) exp [ i Φ ( σ ) ] = Q A ( σ ) exp [ i Φ A ( σ ) ] + Q B ( σ ) exp [ i Φ B ( σ ) ] ,
η A ( x ) = η A ( 0 ) exp 2 Q A π [ sin ( 2 π σ A x + Φ A ) sin ( Φ A ) ] ,
η B ( x ) = η B ( 0 ) exp 2 Q B π [ sin ( 2 π σ B x + Φ B ) sin ( Φ B ) ] .
η ( x ) = η A ( 0 ) η B ( 0 ) exp { Q ¯ A π x A σ A + [ sin ( 2 π σ A x + Φ A ) sin ( Φ A ) ] + Q ¯ B π x B σ B + [ sin ( 2 π σ B x + Φ B ) sin ( Φ B ) ] } ,
x 0 = 2 π ln ( η M / η m ) ( Q ¯ A σ A + Q ¯ B σ B ) ,
ln η M , A η m , A = Q ¯ A ln η M / η m σ A ( Q ¯ A / σ A + Q ¯ B / σ B ) , ln η M , B η m , B = Q ¯ B ln η M / η m σ B ( Q ¯ A / σ A + Q ¯ B / σ B ) .
n ( x ) = η ( 0 ) η A ( h A ) η B ( h B ) exp ( 2 Q A π { sin [ 2 π σ A ( x + h A ) + Φ A ] sin Φ A } Π ( x + h A 2 x A ) ) exp ( 2 Q B π { sin [ 2 π σ B ( x + h B ) + Φ B ] sin Φ B } Π ( x + h B 2 x B ) ) ,
Q 2 ( σ ) = Q ¯ A 2 ( σ ) + Q ¯ B 2 ( σ ) + 2 Q ¯ A ( σ ) Q ¯ B ( σ ) × cos [ ( Φ ¯ A + 2 π σ h A ) ( Φ ¯ B + 2 π σ h B ) ] .
x A h A = x B h B .
x A = 2 Q ¯ A π σ A ln ( η M / η m ) , x B = 2 Q ¯ B π σ B ln ( η M / η m ) ,
Φ ¯ A = Φ A , Φ ¯ B = Φ B .
Φ A Φ B = 2 m π 2 ( σ A + σ B ) ln ( η M / η m ) ( Q ¯ A σ A + Q ¯ B σ B ) ,
2 m + 1 2 Q A cos Φ A = 0 , 2 m + 1 2 Q A sin Φ A + ( 1 ) m 2 ln η M η m η i η s = 0 ,
( ± 2 m + 1 4 π + 1 ) ln η M ( ± 2 m + 1 4 π 1 ) ln η m = ln η i η s ,
ln η M η m = 2 m π ln η s η i .
Φ A Φ B = 2 π σ ( h B h A ) + ( 2 m + 1 ) π ,
s A ( x ) = r A ( x ) Π ( x 2 x A ) , s B ( x ) = r B ( x ) Π ( x 2 x B )
s ( x ) = s A ( x + h A ) + s B ( x + h B )
Q ( σ ) exp [ i Φ ( σ ) ] = exp ( + 2 i π σ h A ) Q ¯ A ( σ ) exp [ i Φ ¯ A ( σ ) ] + exp ( + 2 i π σ h B ) Q ¯ B ( σ ) exp [ i Φ ¯ B ( σ ) ] .
h A = h B , σ A = σ B , Q A = Q B , x A = x B , Φ A = Φ B ,
Q ( σ ) exp [ i Φ ( σ ) ] = 4 Q A σ A x A { sinc [ 2 π ( σ σ A ) x A ] cos ( Φ A + 2 π σ h A ) + sinc [ 2 π ( σ + σ A ) x A ] cos ( Φ A 2 π σ h A ) } .
Q s ( σ / cos θ * ) exp [ i Φ s ( σ / cos θ * ) ] = Q ( σ ) exp [ i Φ ( σ ) ] * + [ 1 + t ( x ) ] exp ( 2 i π σ x ) d x , Q p ( σ / cos θ * ) exp [ i Φ p ( σ / cos θ * ) ] = Q ( σ ) exp [ i Φ ( σ ) ] * + [ 1 t ( x ) ] exp ( 2 i π σ x ) d x ,
t ( x ) = n i 2 sin 2 θ i n 2 ( x ) n i 2 sin 2 θ i ,

Metrics