Abstract

Fast computer algorithms for calculating the reflection coefficients of inhomogeneous films are developed. The applications of such algorithms are aimed at real-time monitoring and control of the growth of gradient index optical coatings. Inhomogeneous films with a refractive index varying along the direction normal to the surface are generally treated by dividing the film into a large number of homogeneous layers and using an iterative or transfer-matrix method to calculate reflectivities. We develop alternative methods. We show that the elements of the transfer matrix and the reflection coefficients can be written in terms of single, double, and higher-dimensional integrals over the films. In most cases it is sufficient to include only the first term, i.e., the single integrals, and the calculation of reflection coefficients is then between 1 and 2 orders of magnitude faster than that for the transfer-matrix and iteration methods. This gives the possibility of real-time determination of optical parameters of inhomogeneous layers for growth monitoring and opens up feedback control.

© 1997 Optical Society of America

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References

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  1. W. E. Johnson, R. L. Crane, “Introduction to rugate filter technology,” in Inhomogeneous and Quasi-Inhomogeneous Optical Coatings, J. A. Dobrowolski, P. G. Verly, eds., Proc. SPIE2046, 88–108 (1993).
    [CrossRef]
  2. J. F. Hall, “Reflection coefficient of optically inhomogeneous layers,” J. Opt. Soc. Am. 48, 654–657 (1958).
    [CrossRef]
  3. R. Jacobssen, “Light reflection from films of continuously varying refractive index,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1975), Vol. 5, pp. 249–286.
  4. J. M. Vigoureux, “Polynomial formulation of reflection and transmission by stratified planar structures,” J. Opt. Soc. Am. A 8, 1697–1701 (1991).
    [CrossRef]
  5. B. G. Bovard, “Derivation of a matrix describing a rugate dielectric thin film,” Appl. Opt. 27, 1998–2005 (1988).
    [CrossRef] [PubMed]
  6. L. Li, Y. Yen, “Wideband monitoring and measuring system for optical coatings,” Appl. Opt. 28, 2889–2894 (1989).
    [CrossRef] [PubMed]
  7. B. T. Sullivan, J. A. Dobrowolski, “Deposition error compensation for optical multilayer coatings,” Appl. Opt. 31, 3821–3835 (1995).
    [CrossRef]
  8. H. H. Bauer, E. Nüssler, “In situ optical multichannel spectrometer system,” in Optical Interference Coatings, F. Abeles, ed., Proc. SPIE2253, 423–431 (1994).
    [CrossRef]
  9. D. E. Aspnes, “Minimal-data approaches for determining outer-layer dielectric responses of films from kinetic reflectometric and ellipsometric measurements,” J. Opt. Soc. Am. A 10, 974–983 (1993).
    [CrossRef]
  10. Sangbo Kim, R. W. Collins, “Optical characterization of continuous compositional gradients in thin films by real time spectroscopic ellipsometry,” Appl. Phys. Lett. 67, 3010–3012 (1995).
    [CrossRef]
  11. W. M. Duncan, S. A. Henck, “In situ spectral ellipsometry for real-time measurement and control,” Appl. Surf. Sci. 63, 9–16 (1993).
    [CrossRef]
  12. I. F. Wu, J. B. Dottelis, M. Dagenais, “Real-time in situ ellipsometric control of antireflection coatings for semiconductor laser amplifiers using SiOx,” J. Vac. Sci. Technol. A 11, 2398–2405 (1993).
    [CrossRef]
  13. M. Kildemo, B. Drévillon, “Real time monitoring of the growth of transparent thin films by spectroscopic ellipsometry,” Appl. Phys. Lett. 67, 918–920 (1995).
    [CrossRef]
  14. M. Kildemo, B. Drévillon, “Real time monitoring of the growth of transparent thin films by spectroscopic ellipsometry,” Rev. Sci. Instrum. 67, 1956–1960 (1996).
    [CrossRef]
  15. M. Kildemo, B. Drévillon, “Real time control of the growth of silicon alloy multilayers by multiwavelength ellipsometry,” Thin Solid Films (to be published).
  16. M. Abramowitz, I. A. Stegun, eds., Handbook of Mathematical Functions (Dover, New York, 1965).
  17. J. R. Wait, Electromagnetic Waves in Stratified Media (Pergamon, New York, 1962).
  18. M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1970).
  19. R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977).
  20. J. Lekner, Theory of Reflection (Nijhoff, Norwell, Mass., 1987).
  21. F. Abeles, “Optical properties of inhomogeneous films,” Natl. Bur. Stand. (U.S.), Misc. Publ. 256, 41–58 (1964).
  22. J. Lekner, “Exact reflection amplitude for the Rayleigh profile,” Physica A 116, 235–247 (1982).
    [CrossRef]
  23. W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes (Cambridge U. Press, Cambridge, 1986).
  24. E. D. Palik, Handbook of Optical Constants of Solids (Academic, New York, 1985).
  25. B. Drévillon, “Phase modulated ellipsometry from the ultraviolet to the infrared: in situ applications to the growth of semiconductors,” Prog. Cryst. Growth Charact. Matter 27, 1–87 (1993).
  26. M. Kildemo, S. Deniau, P. Bulkin, B. Drévillon, O. Hunderi, “Real time monitoring by multiwavelength ellipsometry of the growth of silicon alloy multilayers and gradient index structures,” in Optical Systems Design and Production II, I. Reed, ed., Proc. SPIE2776, 84–95 (1996).
    [CrossRef]

1996 (1)

M. Kildemo, B. Drévillon, “Real time monitoring of the growth of transparent thin films by spectroscopic ellipsometry,” Rev. Sci. Instrum. 67, 1956–1960 (1996).
[CrossRef]

1995 (3)

M. Kildemo, B. Drévillon, “Real time monitoring of the growth of transparent thin films by spectroscopic ellipsometry,” Appl. Phys. Lett. 67, 918–920 (1995).
[CrossRef]

B. T. Sullivan, J. A. Dobrowolski, “Deposition error compensation for optical multilayer coatings,” Appl. Opt. 31, 3821–3835 (1995).
[CrossRef]

Sangbo Kim, R. W. Collins, “Optical characterization of continuous compositional gradients in thin films by real time spectroscopic ellipsometry,” Appl. Phys. Lett. 67, 3010–3012 (1995).
[CrossRef]

1993 (4)

W. M. Duncan, S. A. Henck, “In situ spectral ellipsometry for real-time measurement and control,” Appl. Surf. Sci. 63, 9–16 (1993).
[CrossRef]

I. F. Wu, J. B. Dottelis, M. Dagenais, “Real-time in situ ellipsometric control of antireflection coatings for semiconductor laser amplifiers using SiOx,” J. Vac. Sci. Technol. A 11, 2398–2405 (1993).
[CrossRef]

D. E. Aspnes, “Minimal-data approaches for determining outer-layer dielectric responses of films from kinetic reflectometric and ellipsometric measurements,” J. Opt. Soc. Am. A 10, 974–983 (1993).
[CrossRef]

B. Drévillon, “Phase modulated ellipsometry from the ultraviolet to the infrared: in situ applications to the growth of semiconductors,” Prog. Cryst. Growth Charact. Matter 27, 1–87 (1993).

1991 (1)

1989 (1)

1988 (1)

1982 (1)

J. Lekner, “Exact reflection amplitude for the Rayleigh profile,” Physica A 116, 235–247 (1982).
[CrossRef]

1964 (1)

F. Abeles, “Optical properties of inhomogeneous films,” Natl. Bur. Stand. (U.S.), Misc. Publ. 256, 41–58 (1964).

1958 (1)

Abeles, F.

F. Abeles, “Optical properties of inhomogeneous films,” Natl. Bur. Stand. (U.S.), Misc. Publ. 256, 41–58 (1964).

Aspnes, D. E.

Azzam, R. M. A.

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977).

Bashara, N. M.

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977).

Bauer, H. H.

H. H. Bauer, E. Nüssler, “In situ optical multichannel spectrometer system,” in Optical Interference Coatings, F. Abeles, ed., Proc. SPIE2253, 423–431 (1994).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1970).

Bovard, B. G.

Bulkin, P.

M. Kildemo, S. Deniau, P. Bulkin, B. Drévillon, O. Hunderi, “Real time monitoring by multiwavelength ellipsometry of the growth of silicon alloy multilayers and gradient index structures,” in Optical Systems Design and Production II, I. Reed, ed., Proc. SPIE2776, 84–95 (1996).
[CrossRef]

Collins, R. W.

Sangbo Kim, R. W. Collins, “Optical characterization of continuous compositional gradients in thin films by real time spectroscopic ellipsometry,” Appl. Phys. Lett. 67, 3010–3012 (1995).
[CrossRef]

Crane, R. L.

W. E. Johnson, R. L. Crane, “Introduction to rugate filter technology,” in Inhomogeneous and Quasi-Inhomogeneous Optical Coatings, J. A. Dobrowolski, P. G. Verly, eds., Proc. SPIE2046, 88–108 (1993).
[CrossRef]

Dagenais, M.

I. F. Wu, J. B. Dottelis, M. Dagenais, “Real-time in situ ellipsometric control of antireflection coatings for semiconductor laser amplifiers using SiOx,” J. Vac. Sci. Technol. A 11, 2398–2405 (1993).
[CrossRef]

Deniau, S.

M. Kildemo, S. Deniau, P. Bulkin, B. Drévillon, O. Hunderi, “Real time monitoring by multiwavelength ellipsometry of the growth of silicon alloy multilayers and gradient index structures,” in Optical Systems Design and Production II, I. Reed, ed., Proc. SPIE2776, 84–95 (1996).
[CrossRef]

Dobrowolski, J. A.

Dottelis, J. B.

I. F. Wu, J. B. Dottelis, M. Dagenais, “Real-time in situ ellipsometric control of antireflection coatings for semiconductor laser amplifiers using SiOx,” J. Vac. Sci. Technol. A 11, 2398–2405 (1993).
[CrossRef]

Drévillon, B.

M. Kildemo, B. Drévillon, “Real time monitoring of the growth of transparent thin films by spectroscopic ellipsometry,” Rev. Sci. Instrum. 67, 1956–1960 (1996).
[CrossRef]

M. Kildemo, B. Drévillon, “Real time monitoring of the growth of transparent thin films by spectroscopic ellipsometry,” Appl. Phys. Lett. 67, 918–920 (1995).
[CrossRef]

B. Drévillon, “Phase modulated ellipsometry from the ultraviolet to the infrared: in situ applications to the growth of semiconductors,” Prog. Cryst. Growth Charact. Matter 27, 1–87 (1993).

M. Kildemo, S. Deniau, P. Bulkin, B. Drévillon, O. Hunderi, “Real time monitoring by multiwavelength ellipsometry of the growth of silicon alloy multilayers and gradient index structures,” in Optical Systems Design and Production II, I. Reed, ed., Proc. SPIE2776, 84–95 (1996).
[CrossRef]

M. Kildemo, B. Drévillon, “Real time control of the growth of silicon alloy multilayers by multiwavelength ellipsometry,” Thin Solid Films (to be published).

Duncan, W. M.

W. M. Duncan, S. A. Henck, “In situ spectral ellipsometry for real-time measurement and control,” Appl. Surf. Sci. 63, 9–16 (1993).
[CrossRef]

Flannery, B. P.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes (Cambridge U. Press, Cambridge, 1986).

Hall, J. F.

Henck, S. A.

W. M. Duncan, S. A. Henck, “In situ spectral ellipsometry for real-time measurement and control,” Appl. Surf. Sci. 63, 9–16 (1993).
[CrossRef]

Hunderi, O.

M. Kildemo, S. Deniau, P. Bulkin, B. Drévillon, O. Hunderi, “Real time monitoring by multiwavelength ellipsometry of the growth of silicon alloy multilayers and gradient index structures,” in Optical Systems Design and Production II, I. Reed, ed., Proc. SPIE2776, 84–95 (1996).
[CrossRef]

Jacobssen, R.

R. Jacobssen, “Light reflection from films of continuously varying refractive index,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1975), Vol. 5, pp. 249–286.

Johnson, W. E.

W. E. Johnson, R. L. Crane, “Introduction to rugate filter technology,” in Inhomogeneous and Quasi-Inhomogeneous Optical Coatings, J. A. Dobrowolski, P. G. Verly, eds., Proc. SPIE2046, 88–108 (1993).
[CrossRef]

Kildemo, M.

M. Kildemo, B. Drévillon, “Real time monitoring of the growth of transparent thin films by spectroscopic ellipsometry,” Rev. Sci. Instrum. 67, 1956–1960 (1996).
[CrossRef]

M. Kildemo, B. Drévillon, “Real time monitoring of the growth of transparent thin films by spectroscopic ellipsometry,” Appl. Phys. Lett. 67, 918–920 (1995).
[CrossRef]

M. Kildemo, B. Drévillon, “Real time control of the growth of silicon alloy multilayers by multiwavelength ellipsometry,” Thin Solid Films (to be published).

M. Kildemo, S. Deniau, P. Bulkin, B. Drévillon, O. Hunderi, “Real time monitoring by multiwavelength ellipsometry of the growth of silicon alloy multilayers and gradient index structures,” in Optical Systems Design and Production II, I. Reed, ed., Proc. SPIE2776, 84–95 (1996).
[CrossRef]

Kim, Sangbo

Sangbo Kim, R. W. Collins, “Optical characterization of continuous compositional gradients in thin films by real time spectroscopic ellipsometry,” Appl. Phys. Lett. 67, 3010–3012 (1995).
[CrossRef]

Lekner, J.

J. Lekner, “Exact reflection amplitude for the Rayleigh profile,” Physica A 116, 235–247 (1982).
[CrossRef]

J. Lekner, Theory of Reflection (Nijhoff, Norwell, Mass., 1987).

Li, L.

Nüssler, E.

H. H. Bauer, E. Nüssler, “In situ optical multichannel spectrometer system,” in Optical Interference Coatings, F. Abeles, ed., Proc. SPIE2253, 423–431 (1994).
[CrossRef]

Palik, E. D.

E. D. Palik, Handbook of Optical Constants of Solids (Academic, New York, 1985).

Press, W. H.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes (Cambridge U. Press, Cambridge, 1986).

Sullivan, B. T.

Teukolsky, S. A.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes (Cambridge U. Press, Cambridge, 1986).

Vetterling, W. T.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes (Cambridge U. Press, Cambridge, 1986).

Vigoureux, J. M.

Wait, J. R.

J. R. Wait, Electromagnetic Waves in Stratified Media (Pergamon, New York, 1962).

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1970).

Wu, I. F.

I. F. Wu, J. B. Dottelis, M. Dagenais, “Real-time in situ ellipsometric control of antireflection coatings for semiconductor laser amplifiers using SiOx,” J. Vac. Sci. Technol. A 11, 2398–2405 (1993).
[CrossRef]

Yen, Y.

Appl. Opt. (3)

Appl. Phys. Lett. (2)

M. Kildemo, B. Drévillon, “Real time monitoring of the growth of transparent thin films by spectroscopic ellipsometry,” Appl. Phys. Lett. 67, 918–920 (1995).
[CrossRef]

Sangbo Kim, R. W. Collins, “Optical characterization of continuous compositional gradients in thin films by real time spectroscopic ellipsometry,” Appl. Phys. Lett. 67, 3010–3012 (1995).
[CrossRef]

Appl. Surf. Sci. (1)

W. M. Duncan, S. A. Henck, “In situ spectral ellipsometry for real-time measurement and control,” Appl. Surf. Sci. 63, 9–16 (1993).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (2)

J. Vac. Sci. Technol. A (1)

I. F. Wu, J. B. Dottelis, M. Dagenais, “Real-time in situ ellipsometric control of antireflection coatings for semiconductor laser amplifiers using SiOx,” J. Vac. Sci. Technol. A 11, 2398–2405 (1993).
[CrossRef]

Natl. Bur. Stand. (U.S.), Misc. Publ. (1)

F. Abeles, “Optical properties of inhomogeneous films,” Natl. Bur. Stand. (U.S.), Misc. Publ. 256, 41–58 (1964).

Physica A (1)

J. Lekner, “Exact reflection amplitude for the Rayleigh profile,” Physica A 116, 235–247 (1982).
[CrossRef]

Prog. Cryst. Growth Charact. Matter (1)

B. Drévillon, “Phase modulated ellipsometry from the ultraviolet to the infrared: in situ applications to the growth of semiconductors,” Prog. Cryst. Growth Charact. Matter 27, 1–87 (1993).

Rev. Sci. Instrum. (1)

M. Kildemo, B. Drévillon, “Real time monitoring of the growth of transparent thin films by spectroscopic ellipsometry,” Rev. Sci. Instrum. 67, 1956–1960 (1996).
[CrossRef]

Other (12)

M. Kildemo, B. Drévillon, “Real time control of the growth of silicon alloy multilayers by multiwavelength ellipsometry,” Thin Solid Films (to be published).

M. Abramowitz, I. A. Stegun, eds., Handbook of Mathematical Functions (Dover, New York, 1965).

J. R. Wait, Electromagnetic Waves in Stratified Media (Pergamon, New York, 1962).

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1970).

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977).

J. Lekner, Theory of Reflection (Nijhoff, Norwell, Mass., 1987).

W. E. Johnson, R. L. Crane, “Introduction to rugate filter technology,” in Inhomogeneous and Quasi-Inhomogeneous Optical Coatings, J. A. Dobrowolski, P. G. Verly, eds., Proc. SPIE2046, 88–108 (1993).
[CrossRef]

R. Jacobssen, “Light reflection from films of continuously varying refractive index,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1975), Vol. 5, pp. 249–286.

H. H. Bauer, E. Nüssler, “In situ optical multichannel spectrometer system,” in Optical Interference Coatings, F. Abeles, ed., Proc. SPIE2253, 423–431 (1994).
[CrossRef]

M. Kildemo, S. Deniau, P. Bulkin, B. Drévillon, O. Hunderi, “Real time monitoring by multiwavelength ellipsometry of the growth of silicon alloy multilayers and gradient index structures,” in Optical Systems Design and Production II, I. Reed, ed., Proc. SPIE2776, 84–95 (1996).
[CrossRef]

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes (Cambridge U. Press, Cambridge, 1986).

E. D. Palik, Handbook of Optical Constants of Solids (Academic, New York, 1985).

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Figures (9)

Fig. 1
Fig. 1

Refractive index profile of a general inhomogeneous medium with discrete index variations. The inhomogeneity is perpendicular to the surface.

Fig. 2
Fig. 2

Summary of the basic reflection processes and the resulting phases or optical thicknesses.

Fig. 3
Fig. 3

(a) Real and imaginary parts of the single and double integrals as a function of d/λ. The photon energy is E=1.5 eV, and the profile is a linear variation in index, n=A+Bz, with A=1.5 and B=5µm-1. The ambient medium is vacuum and the substrate is c-Si, as described in the text. (b) Double and triple integrals for the same profile.

Fig. 4
Fig. 4

Calculated reflectivity at normal incidence for the profile n=A+Bz, with A=1.5, B=5µm-1, and total thickness 0.2 µm. The dashed curve is calculated with use of the WKBJ approximation, while the solid curve is calculated with the use of Eq. (17). The difference between first- and higher-order integral approximations is not visible is this plot.

Fig. 5
Fig. 5

Difference in reflectivity between the integral approximations and the exact iterative recursive Fresnel method for the linear profile used in Fig. 4.

Fig. 6
Fig. 6

Difference between the exact and the single-integral approximation for the calculated phase-modulated ellipsometry parameter Is. The angle of incidence is 70°, and the linear profile is given by n=1.5+Bz. The ambient is vacuum and the substrate is c-Si, as above. The inhomogeneity B is varied from 0 to 5 µm-1.

Fig. 7
Fig. 7

Same as Fig. 6, but for the calculated phase-modulated ellipsometry parameter Ic.

Fig. 8
Fig. 8

Rugate filter with matched surrounding layers. The refractive index of the ambient and the substrate is 2.0, and the film index is given by n=2.0[1+0.05 sin(ωz)]. The resonance wavelength is λ0=550 nm, ω=(4πn0)/λ0 (with n0 =2.0), the number of periods is 10, and the total thickness is 1375 nm. The reflectivity is calculated with the exact method and single-, double-, and triple-integral approximations.

Fig. 9
Fig. 9

Illustrative rugate filter with a step in the refractive index. The ambient is vacuum, the index of the first film is given by n=1.5+0.1 sin(ωz), and the index of the second film is n =2.0+0.1 sin(ωz). The substrate index is 2.0. The resonance wavelengths are λ1=412.5 nm and λ2=550 nm, respectively, and ω=(4πn1)/λ1=(4πn2)/λ2, with n1=1.5 and n2 =2.0. Each period has a thickness of 137.5 nm, and there are 20 periods for each step. The reflectivity is calculated with the exact method and the single-integral method describing one matrix for one period, as described in the text.

Equations (49)

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ri,i+1s,p=ηi-ηi+1ηi+ηi+1,
ηi=ni cos ϕi=i-a sin2 ϕ0,spolarizationnicos ϕi=ii-a sin2 ϕ0,ppolarization.
rˆi=ri+rˆi+1 exp(i2βi)1+rirˆi+1 exp(i2βi),
βi=2πλi-a sin2 ϕ0di.
rˆ(2)=r0,1+r1,2 exp(i2β1)+r0,1r1,2r2,3 exp(i2β2)+r2,3 exp[i2(β1+β2)]1+r0,1r1,2 exp(i2β1)+r1,2r2,3 exp(i2β2)+r0,1r2,3 exp[2i(β1+β2)].
r0,1+r1,2 exp(i2β1)+r2,3 exp[i2(β1+β2)]+r3,4 exp[i2(β1+β2+β3)]+r4,5 exp[2i(β1+β2+β3+β4)]+r0,1r4,5{r1,2 exp[i2(β2+β3+β4)]+r2,3 exp[i2(β3+β4)]+r3,4 exp(i2β4)}+r0,1{r1,2r2,3 exp(i2β2)+r2,3r3,4 exp(i2β3)+r1,2r3,4 exp[2i(β2+β3)]}+r4,5{r1,2r2,3 exp[2i(β1+β3+β4)]+r1,2r3,4 exp[2i(β1+β4)]+r2,3r3,4 exp[2i(β1+β2+β4)]}+r1,2r2,3r3,4 exp[2i(β1+β3)]+r0,1r4,5{r1,2r2,3r3,4 exp[2i(β1+β4)]},
ri,i+1=Δη(zi)2η(zi).
I1=limN j=1N-1rj,j+1 expik=1j2βk=limN j=1N-1Δηj2ηjexpik=1j2βk=0d12η(z)dη(z)dzexp[2iβ(z)]dz,
β(z)=2πλ0z(ζ)-a sin2 φ0 dζ,
βd=β(d).
I1=0d 12η(z)dη(z)dzexp[2iβ(z)]dz,
I2=0d12η(z)dη(z)dzexp[-2iβ(z)]dz.
DI1=0d0y 14η(y)η(z)dη(y)dydη(z)dz×exp(2i[β(y)-β(z)])dzdy,
DI2=0d0y 14η(y)η(z)dη(y)dydη(z)dz×exp(-2i[β(y)-β(z)])dzdy.
I1K=0d 12η(y)dη(y)dy{exp[2iβ(y)]}I2K-1 dy,
K=2, 3, ,
I2K=0d 12η(y)dη(y)dy{exp[-2iβ(y)]}I1K-1 dy,
K=2, 3,.
I2K=I1K*.
r^=r0,1+I1+r0,1rNI2exp(i2βd)+r0,1DI1+rNDI2exp(i2βd)+TI1+r0,1rNTI2exp(i2βd)++rNexp(i2βd)1+r0,1I1+rNI2exp(i2βd)+DI1+r0,1rNDI2exp(i2βd)+r0,1TI1+rNTI2exp(i2βd)++r0,1rNexp(i2βd)
rˆ=r0,1+I1+r0,1rNI2 exp(i2βd)+rN exp(i2βd)1+r0,1I1+rNI2 exp(i2βd)+r0,1rN exp(i2βd),
rˆ=r0,1+rN exp(i2βd)1+r0,1rN exp(i2βd).
M˜=cos β-iη-1 sin β-iη sin βcos β,
M˜tot=M˜1M˜2M˜N,
M˜i=B˜iλ˜iB˜i-1.
M˜tot=B˜1λ˜1B˜1-1B˜2λ˜2B˜2-1B˜3λ˜3B˜3-1B˜Nλ˜NB˜N-1.
λ˜=exp(-iβ)00exp(iβ),
B˜=121/ηη-1/ηη.
B˜i-1B˜i+1=1Δηi2ηiΔηi2ηi1=I˜+δ˜i.
M˜tot=B˜1λ˜1(I˜+δ˜1)λ˜2(I˜+δ˜2)λ˜3(I˜+δ˜3)λ˜NB˜N-1=B˜1S˜MB˜N-1.
S˜M=[exp(-iβd)](1+DI1)[exp(-iβd)]I1[exp(iβd)]I2[exp(iβd)](1+DI2).
M˜tot=η(d)η(0) (cos βd+A11)-1η(0)η(d) (i sin βd+A12)-η(0)η(d)(i sin βd+A21)η(0)η(d) (cos βd+A22),
A11=-[exp(-iβd)](I1-DI1)-[exp(iβd)](I2-DI2),
A12=[exp(-iβd)](I1-DI1)-[exp(iβd)](I2-DI2),
A21=-[exp(-iβd)](I1+DI1)+[exp(iβd)](I2+DI2),
A22=[exp(-iβd)](I1+DI1)+[exp(iβd)](I2+DI2).
d2Edz2+q2(z)E(z)=0,spolarization,
d2Edz2+ddzln(z)-a sin2 φ0(z)dEdz+q2(z)E(z)=0,ppolarization,
q(z)=2πλ(z)-a sin2 φ0.
χ(z)=E(z)Z0H(z),
χ(0)=M˜χ(z),
M˜=v(z)-v(z)γs,p-γs,pu(z)u(z),
γs=1iq(z),
γp=(z)iq(z).
γs,p dEdz=H(z)
ρ=rˆprˆs=(tan Ψ)exp(iΔ).
Is=2 Im(rsrp*)|rs|2+|rp|2=sin(2Ψ)sin Δ,
Ic=2 Re(rsrp*)|rs|2+|rp|2=sin(2Ψ)cos Δ.
M=M120M220.

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