Abstract

A new microscope is described that is capable of measuring the polarization characteristics of materials in normal-incidence reflection with a demonstrated lateral resolution of 4  μm. The instrument measures eight parameters of the sample Mueller matrix, which can be related to the diattenuation, retardation, circular diattenuation, direction of the principal axis, and the polarization factor. With proper calibration, the eight elements of the sample Mueller matrix can be determined to better than 0.001–0.002 for small values. Examples are given for aluminum, rutile (TiO2), and calcite (CaCO3).

© 2006 Optical Society of America

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References

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  1. R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, 1977).
  2. M. Schubert, B. Rheinlander, J. A. Woollam, B. Johs, and C. M. Herzinger, "Extension of rotating-analyzer ellipsometry to generalized ellipsometry: determination of the dielectric function tensor from uniaxial TiO2," J. Opt. Soc. Am. A 13, 875-883 (1996).
    [CrossRef]
  3. M. Schubert, "Generalized ellipsometry and complex optical systems," Thin Solid Films 313-314, 323-332 (1998).
    [CrossRef]
  4. P. S. Hauge, "Mueller matrix ellipsometry with imperfect compensators," J. Opt. Soc. Am. 68, 1519-1528 (1978).
    [CrossRef]
  5. D. H. Goldstein, "Mueller matrix dual-rotating retarder polarimeter," Appl. Opt. 31, 6676-6683 (1992).
    [CrossRef] [PubMed]
  6. D. H. Goldstein and R. A. Chipman, "Error analysis of a Mueller matrix polarimeter," J. Opt. Soc. Am. A 7, 693-700 (1990).
    [CrossRef]
  7. C. Chen, I. An, G. M. Ferreira, N. J. Podraza, J. A. Zapien, and R. W. Collins, "Multichannel Mueller matrix ellipsometer based on the dual rotating compensator principle," Thin Solid Films 455-5614-23 (2004).
    [CrossRef]
  8. G. E. Jellison, Jr. and F. A. Modine, "Two-modulator generalized ellipsometry: experiment and calibration," Appl. Opt. 36, 8184-8189 (1997).
    [CrossRef]
  9. G. E. Jellison, Jr. and F. A. Modine, "Two-modulator generalized ellipsometry: theory," Appl. Opt. 36, 8190-8198 (1997).
    [CrossRef]
  10. G. E. Jellison, Jr., F. A. Modine, and L. A. Boatner, "The measurement of the optical functions of uniaxial materials using two-modulator generalized ellipsometry: rutile (TiO2)," Opt. Lett 22, 1808-1810 (1997).
    [CrossRef]
  11. G. E. Jellison, Jr. and L. A. Boatner, "Optical functions of uniaxial ZnO determined by generalized ellipsometry," Phys. Rev. B 58, 3586-3589 (1998).
    [CrossRef]
  12. G. E. Jellison, Jr., F. A. Modine, and L. A. Boatner, "Measurements of the optical functions of uniaxial crystals using two-modulator generalized ellipsometry (2-MGE)," in Optical Diagnostic Methods for Inorganic Transmissive Materials, R. V. Datla and L. M. Hanssen, eds., Proc. SPIE 3425, 232-238 (1998).
    [CrossRef]
  13. G. E. Jellison, Jr., C. O. Griffiths, and D. E. Holcomb, "Electric field-induced birefringence in LiNbO3 measured by generalized transmission ellipsometry," Appl. Phys. Lett. 81, 1222-1224 (2002).
    [CrossRef]
  14. G. E. Jellison, Jr., C. O. Griffiths, D. E. Holcomb, and C. M. Rouleau, "Transmission two-modulator generalized ellipsometry measurements," Appl. Opt. 41, 6555-6566 (2002).
    [CrossRef] [PubMed]
  15. G. E. Jellison, Jr., C. O. Griffiths, D. E. Holcomb, and C. M. Rouleau, "Characterization of linear diattenuator and retarders using a two-modulator generalized ellipsometer," in Polarization Measurement, Analysis, and Applications V, D. H. Goldstein and D. B. Chenault, eds., Proc. SPIE 4819, 9-19 (2002).
    [CrossRef]
  16. D. W. Stevens, "A perpendicular-incidence microellipsometer," Surf. Sci 96, 174-201 (1980).
    [CrossRef]
  17. E. D. Palik, ed., Handbook of Optical Constants of Solids (Academic, 1985).
  18. V. V. Filippov, A. Yu. Tronin, and A. F. Konstantinova, "Ellipsometry of anisotropic media," Crystallogr. Rep. 39, 313-335 (1994).
  19. R. A. Chipman, "Polarimetry," in Handbook of Optics, Vol. II, 2nd ed. (McGraw-Hill, 1995), Chap. 22.
  20. D. G. M. Anderson and R. Barakat, "Necessary and sufficient conditions for a Mueller matrix to be derivable from a Jones matrix," J. Opt. Soc. Am. A 11, 2305-2319 (1994).
    [CrossRef]
  21. T. C. Oakberg, "Modulated interference effects: use of photoelastic modulators with lasers," Opt. Eng. 34, 1545-1550 (1995).
    [CrossRef]
  22. G. E. Jellison, Jr., "Windows in ellipsometry measurements," Appl. Opt. 38, 4784-4789 (1999).
    [CrossRef]
  23. J. M. Bennett, "Polarization," in Handbook of Optics, Vol. I, 2nd ed. (McGraw-Hill, 1995), Chap. 5.

2004

C. Chen, I. An, G. M. Ferreira, N. J. Podraza, J. A. Zapien, and R. W. Collins, "Multichannel Mueller matrix ellipsometer based on the dual rotating compensator principle," Thin Solid Films 455-5614-23 (2004).
[CrossRef]

2002

G. E. Jellison, Jr., C. O. Griffiths, and D. E. Holcomb, "Electric field-induced birefringence in LiNbO3 measured by generalized transmission ellipsometry," Appl. Phys. Lett. 81, 1222-1224 (2002).
[CrossRef]

G. E. Jellison, Jr., C. O. Griffiths, D. E. Holcomb, and C. M. Rouleau, "Transmission two-modulator generalized ellipsometry measurements," Appl. Opt. 41, 6555-6566 (2002).
[CrossRef] [PubMed]

G. E. Jellison, Jr., C. O. Griffiths, D. E. Holcomb, and C. M. Rouleau, "Characterization of linear diattenuator and retarders using a two-modulator generalized ellipsometer," in Polarization Measurement, Analysis, and Applications V, D. H. Goldstein and D. B. Chenault, eds., Proc. SPIE 4819, 9-19 (2002).
[CrossRef]

1999

1998

G. E. Jellison, Jr. and L. A. Boatner, "Optical functions of uniaxial ZnO determined by generalized ellipsometry," Phys. Rev. B 58, 3586-3589 (1998).
[CrossRef]

G. E. Jellison, Jr., F. A. Modine, and L. A. Boatner, "Measurements of the optical functions of uniaxial crystals using two-modulator generalized ellipsometry (2-MGE)," in Optical Diagnostic Methods for Inorganic Transmissive Materials, R. V. Datla and L. M. Hanssen, eds., Proc. SPIE 3425, 232-238 (1998).
[CrossRef]

M. Schubert, "Generalized ellipsometry and complex optical systems," Thin Solid Films 313-314, 323-332 (1998).
[CrossRef]

1997

G. E. Jellison, Jr. and F. A. Modine, "Two-modulator generalized ellipsometry: experiment and calibration," Appl. Opt. 36, 8184-8189 (1997).
[CrossRef]

G. E. Jellison, Jr. and F. A. Modine, "Two-modulator generalized ellipsometry: theory," Appl. Opt. 36, 8190-8198 (1997).
[CrossRef]

G. E. Jellison, Jr., F. A. Modine, and L. A. Boatner, "The measurement of the optical functions of uniaxial materials using two-modulator generalized ellipsometry: rutile (TiO2)," Opt. Lett 22, 1808-1810 (1997).
[CrossRef]

1996

1995

T. C. Oakberg, "Modulated interference effects: use of photoelastic modulators with lasers," Opt. Eng. 34, 1545-1550 (1995).
[CrossRef]

1994

V. V. Filippov, A. Yu. Tronin, and A. F. Konstantinova, "Ellipsometry of anisotropic media," Crystallogr. Rep. 39, 313-335 (1994).

D. G. M. Anderson and R. Barakat, "Necessary and sufficient conditions for a Mueller matrix to be derivable from a Jones matrix," J. Opt. Soc. Am. A 11, 2305-2319 (1994).
[CrossRef]

1992

1990

1980

D. W. Stevens, "A perpendicular-incidence microellipsometer," Surf. Sci 96, 174-201 (1980).
[CrossRef]

1978

An, I.

C. Chen, I. An, G. M. Ferreira, N. J. Podraza, J. A. Zapien, and R. W. Collins, "Multichannel Mueller matrix ellipsometer based on the dual rotating compensator principle," Thin Solid Films 455-5614-23 (2004).
[CrossRef]

Anderson, D. G. M.

Azzam, R. M. A.

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

Barakat, R.

Bashara, N. M.

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

Bennett, J. M.

J. M. Bennett, "Polarization," in Handbook of Optics, Vol. I, 2nd ed. (McGraw-Hill, 1995), Chap. 5.

Boatner, L. A.

G. E. Jellison, Jr. and L. A. Boatner, "Optical functions of uniaxial ZnO determined by generalized ellipsometry," Phys. Rev. B 58, 3586-3589 (1998).
[CrossRef]

G. E. Jellison, Jr., F. A. Modine, and L. A. Boatner, "Measurements of the optical functions of uniaxial crystals using two-modulator generalized ellipsometry (2-MGE)," in Optical Diagnostic Methods for Inorganic Transmissive Materials, R. V. Datla and L. M. Hanssen, eds., Proc. SPIE 3425, 232-238 (1998).
[CrossRef]

G. E. Jellison, Jr., F. A. Modine, and L. A. Boatner, "The measurement of the optical functions of uniaxial materials using two-modulator generalized ellipsometry: rutile (TiO2)," Opt. Lett 22, 1808-1810 (1997).
[CrossRef]

Chen, C.

C. Chen, I. An, G. M. Ferreira, N. J. Podraza, J. A. Zapien, and R. W. Collins, "Multichannel Mueller matrix ellipsometer based on the dual rotating compensator principle," Thin Solid Films 455-5614-23 (2004).
[CrossRef]

Chipman, R. A.

D. H. Goldstein and R. A. Chipman, "Error analysis of a Mueller matrix polarimeter," J. Opt. Soc. Am. A 7, 693-700 (1990).
[CrossRef]

R. A. Chipman, "Polarimetry," in Handbook of Optics, Vol. II, 2nd ed. (McGraw-Hill, 1995), Chap. 22.

Collins, R. W.

C. Chen, I. An, G. M. Ferreira, N. J. Podraza, J. A. Zapien, and R. W. Collins, "Multichannel Mueller matrix ellipsometer based on the dual rotating compensator principle," Thin Solid Films 455-5614-23 (2004).
[CrossRef]

Ferreira, G. M.

C. Chen, I. An, G. M. Ferreira, N. J. Podraza, J. A. Zapien, and R. W. Collins, "Multichannel Mueller matrix ellipsometer based on the dual rotating compensator principle," Thin Solid Films 455-5614-23 (2004).
[CrossRef]

Filippov, V. V.

V. V. Filippov, A. Yu. Tronin, and A. F. Konstantinova, "Ellipsometry of anisotropic media," Crystallogr. Rep. 39, 313-335 (1994).

Goldstein, D. H.

Griffiths, C. O.

G. E. Jellison, Jr., C. O. Griffiths, and D. E. Holcomb, "Electric field-induced birefringence in LiNbO3 measured by generalized transmission ellipsometry," Appl. Phys. Lett. 81, 1222-1224 (2002).
[CrossRef]

G. E. Jellison, Jr., C. O. Griffiths, D. E. Holcomb, and C. M. Rouleau, "Transmission two-modulator generalized ellipsometry measurements," Appl. Opt. 41, 6555-6566 (2002).
[CrossRef] [PubMed]

G. E. Jellison, Jr., C. O. Griffiths, D. E. Holcomb, and C. M. Rouleau, "Characterization of linear diattenuator and retarders using a two-modulator generalized ellipsometer," in Polarization Measurement, Analysis, and Applications V, D. H. Goldstein and D. B. Chenault, eds., Proc. SPIE 4819, 9-19 (2002).
[CrossRef]

Hauge, P. S.

Herzinger, C. M.

Holcomb, D. E.

G. E. Jellison, Jr., C. O. Griffiths, D. E. Holcomb, and C. M. Rouleau, "Transmission two-modulator generalized ellipsometry measurements," Appl. Opt. 41, 6555-6566 (2002).
[CrossRef] [PubMed]

G. E. Jellison, Jr., C. O. Griffiths, and D. E. Holcomb, "Electric field-induced birefringence in LiNbO3 measured by generalized transmission ellipsometry," Appl. Phys. Lett. 81, 1222-1224 (2002).
[CrossRef]

G. E. Jellison, Jr., C. O. Griffiths, D. E. Holcomb, and C. M. Rouleau, "Characterization of linear diattenuator and retarders using a two-modulator generalized ellipsometer," in Polarization Measurement, Analysis, and Applications V, D. H. Goldstein and D. B. Chenault, eds., Proc. SPIE 4819, 9-19 (2002).
[CrossRef]

Jellison, G. E.

G. E. Jellison, Jr., C. O. Griffiths, D. E. Holcomb, and C. M. Rouleau, "Characterization of linear diattenuator and retarders using a two-modulator generalized ellipsometer," in Polarization Measurement, Analysis, and Applications V, D. H. Goldstein and D. B. Chenault, eds., Proc. SPIE 4819, 9-19 (2002).
[CrossRef]

G. E. Jellison, Jr., C. O. Griffiths, D. E. Holcomb, and C. M. Rouleau, "Transmission two-modulator generalized ellipsometry measurements," Appl. Opt. 41, 6555-6566 (2002).
[CrossRef] [PubMed]

G. E. Jellison, Jr., C. O. Griffiths, and D. E. Holcomb, "Electric field-induced birefringence in LiNbO3 measured by generalized transmission ellipsometry," Appl. Phys. Lett. 81, 1222-1224 (2002).
[CrossRef]

G. E. Jellison, Jr., "Windows in ellipsometry measurements," Appl. Opt. 38, 4784-4789 (1999).
[CrossRef]

G. E. Jellison, Jr., F. A. Modine, and L. A. Boatner, "Measurements of the optical functions of uniaxial crystals using two-modulator generalized ellipsometry (2-MGE)," in Optical Diagnostic Methods for Inorganic Transmissive Materials, R. V. Datla and L. M. Hanssen, eds., Proc. SPIE 3425, 232-238 (1998).
[CrossRef]

G. E. Jellison, Jr. and L. A. Boatner, "Optical functions of uniaxial ZnO determined by generalized ellipsometry," Phys. Rev. B 58, 3586-3589 (1998).
[CrossRef]

G. E. Jellison, Jr., F. A. Modine, and L. A. Boatner, "The measurement of the optical functions of uniaxial materials using two-modulator generalized ellipsometry: rutile (TiO2)," Opt. Lett 22, 1808-1810 (1997).
[CrossRef]

G. E. Jellison, Jr. and F. A. Modine, "Two-modulator generalized ellipsometry: experiment and calibration," Appl. Opt. 36, 8184-8189 (1997).
[CrossRef]

G. E. Jellison, Jr. and F. A. Modine, "Two-modulator generalized ellipsometry: theory," Appl. Opt. 36, 8190-8198 (1997).
[CrossRef]

Johs, B.

Konstantinova, A. F.

V. V. Filippov, A. Yu. Tronin, and A. F. Konstantinova, "Ellipsometry of anisotropic media," Crystallogr. Rep. 39, 313-335 (1994).

Modine, F. A.

G. E. Jellison, Jr., F. A. Modine, and L. A. Boatner, "Measurements of the optical functions of uniaxial crystals using two-modulator generalized ellipsometry (2-MGE)," in Optical Diagnostic Methods for Inorganic Transmissive Materials, R. V. Datla and L. M. Hanssen, eds., Proc. SPIE 3425, 232-238 (1998).
[CrossRef]

G. E. Jellison, Jr. and F. A. Modine, "Two-modulator generalized ellipsometry: theory," Appl. Opt. 36, 8190-8198 (1997).
[CrossRef]

G. E. Jellison, Jr. and F. A. Modine, "Two-modulator generalized ellipsometry: experiment and calibration," Appl. Opt. 36, 8184-8189 (1997).
[CrossRef]

G. E. Jellison, Jr., F. A. Modine, and L. A. Boatner, "The measurement of the optical functions of uniaxial materials using two-modulator generalized ellipsometry: rutile (TiO2)," Opt. Lett 22, 1808-1810 (1997).
[CrossRef]

Oakberg, T. C.

T. C. Oakberg, "Modulated interference effects: use of photoelastic modulators with lasers," Opt. Eng. 34, 1545-1550 (1995).
[CrossRef]

Palik, E. D.

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

Podraza, N. J.

C. Chen, I. An, G. M. Ferreira, N. J. Podraza, J. A. Zapien, and R. W. Collins, "Multichannel Mueller matrix ellipsometer based on the dual rotating compensator principle," Thin Solid Films 455-5614-23 (2004).
[CrossRef]

Rheinlander, B.

Rouleau, C. M.

G. E. Jellison, Jr., C. O. Griffiths, D. E. Holcomb, and C. M. Rouleau, "Transmission two-modulator generalized ellipsometry measurements," Appl. Opt. 41, 6555-6566 (2002).
[CrossRef] [PubMed]

G. E. Jellison, Jr., C. O. Griffiths, D. E. Holcomb, and C. M. Rouleau, "Characterization of linear diattenuator and retarders using a two-modulator generalized ellipsometer," in Polarization Measurement, Analysis, and Applications V, D. H. Goldstein and D. B. Chenault, eds., Proc. SPIE 4819, 9-19 (2002).
[CrossRef]

Schubert, M.

Stevens, D. W.

D. W. Stevens, "A perpendicular-incidence microellipsometer," Surf. Sci 96, 174-201 (1980).
[CrossRef]

Tronin, A. Yu.

V. V. Filippov, A. Yu. Tronin, and A. F. Konstantinova, "Ellipsometry of anisotropic media," Crystallogr. Rep. 39, 313-335 (1994).

Woollam, J. A.

Zapien, J. A.

C. Chen, I. An, G. M. Ferreira, N. J. Podraza, J. A. Zapien, and R. W. Collins, "Multichannel Mueller matrix ellipsometer based on the dual rotating compensator principle," Thin Solid Films 455-5614-23 (2004).
[CrossRef]

Appl. Opt.

Appl. Phys. Lett.

G. E. Jellison, Jr., C. O. Griffiths, and D. E. Holcomb, "Electric field-induced birefringence in LiNbO3 measured by generalized transmission ellipsometry," Appl. Phys. Lett. 81, 1222-1224 (2002).
[CrossRef]

Crystallogr. Rep.

V. V. Filippov, A. Yu. Tronin, and A. F. Konstantinova, "Ellipsometry of anisotropic media," Crystallogr. Rep. 39, 313-335 (1994).

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Eng.

T. C. Oakberg, "Modulated interference effects: use of photoelastic modulators with lasers," Opt. Eng. 34, 1545-1550 (1995).
[CrossRef]

Opt. Lett

G. E. Jellison, Jr., F. A. Modine, and L. A. Boatner, "The measurement of the optical functions of uniaxial materials using two-modulator generalized ellipsometry: rutile (TiO2)," Opt. Lett 22, 1808-1810 (1997).
[CrossRef]

Phys. Rev. B

G. E. Jellison, Jr. and L. A. Boatner, "Optical functions of uniaxial ZnO determined by generalized ellipsometry," Phys. Rev. B 58, 3586-3589 (1998).
[CrossRef]

Proc. SPIE

G. E. Jellison, Jr., F. A. Modine, and L. A. Boatner, "Measurements of the optical functions of uniaxial crystals using two-modulator generalized ellipsometry (2-MGE)," in Optical Diagnostic Methods for Inorganic Transmissive Materials, R. V. Datla and L. M. Hanssen, eds., Proc. SPIE 3425, 232-238 (1998).
[CrossRef]

G. E. Jellison, Jr., C. O. Griffiths, D. E. Holcomb, and C. M. Rouleau, "Characterization of linear diattenuator and retarders using a two-modulator generalized ellipsometer," in Polarization Measurement, Analysis, and Applications V, D. H. Goldstein and D. B. Chenault, eds., Proc. SPIE 4819, 9-19 (2002).
[CrossRef]

Surf. Sci

D. W. Stevens, "A perpendicular-incidence microellipsometer," Surf. Sci 96, 174-201 (1980).
[CrossRef]

Thin Solid Films

C. Chen, I. An, G. M. Ferreira, N. J. Podraza, J. A. Zapien, and R. W. Collins, "Multichannel Mueller matrix ellipsometer based on the dual rotating compensator principle," Thin Solid Films 455-5614-23 (2004).
[CrossRef]

M. Schubert, "Generalized ellipsometry and complex optical systems," Thin Solid Films 313-314, 323-332 (1998).
[CrossRef]

Other

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

R. A. Chipman, "Polarimetry," in Handbook of Optics, Vol. II, 2nd ed. (McGraw-Hill, 1995), Chap. 22.

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

J. M. Bennett, "Polarization," in Handbook of Optics, Vol. I, 2nd ed. (McGraw-Hill, 1995), Chap. 5.

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Figures (3)

Fig. 1
Fig. 1

Schematic diagram of the reflection 2-MGEM.

Fig. 2
Fig. 2

Detector and CCD camera configuration of the 2-MGEM.

Fig. 3
Fig. 3

The diattenuation (left) and degree of polarization (right) of a sample of rutile (TiO2) discussed in the text. The diattenuation varies from 0.07 (dark) to 0.10 (white), while the degree of polarization β varies from 0.92 (dark) to 1.01 (white). The dimension of each frame is 500   μm × 500   μm , where each pixel measures 5   μm × 5   μm.

Tables (3)

Tables Icon

Table 1 Values of N and S for Various Metals at Different Values of Surface Roughness and Angles of Incidence a

Tables Icon

Table 2 Summary of the Measurements Made on Rutile (TiO2) Where the c Axis Lies in the Surface Plane a

Tables Icon

Table 3 Summary of the Measurements Made on Calcite (CaCO3)

Equations (52)

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

M = [ 1 N 0 0 N 1 0 0 0 0 C S 0 0 S C ] .
ρ = r p r s = tan ( ψ ) exp ( i Δ ) ,
r o = η o 1 η o + 1 ,
r e = η e 1 η e + 1 ,
η e = ε o ε e [ ε o + ( ε e ε o ) cos 2 ( θ ) ] .
r pp = r e cos 2 ( γ ) + r o sin 2 ( γ ) ,
r ss = [ r e sin 2 ( γ ) + r o cos 2 ( γ ) ] ,
r ps = sin ( γ ) cos ( γ ) ( r e r o ) = r s p .
M J = A ( J J * ) A 1 ,
A = [ 1 0 0 1 1 0 0 1 0 1 1 0 0 i i 0 ] .
r d = r o r e r o + r e = r dr + i r di ,
M J = 1 R d + 1 [ R d + 1 - 2 cos ( 2 γ ) r dr - 2 sin ( 2 γ ) r dr 0 - 2 cos ( 2 γ ) r dr cos ( 4 γ ) R d + 1 sin ( 4 γ ) R d 2 sin ( 2 γ ) r di 2 sin ( 2 γ ) r dr sin ( 4 γ ) R d cos ( 4 γ ) R d - 1 2 cos ( 2 γ ) r di 0 2 sin ( 2 γ ) r di - 2 cos ( 2 γ ) r di R d - 1 ] ,
N = 2 r dr R d + 1 ,
S = 2 r di R d + 1 ,
C = R d 1 R d + 1 ,
M J = [ 1 C 2 g N S 2 g N 0 C 2 g N C 2 g 2 S 2 g 2 C C 2 g S 2 g ( 1 + C ) S 2 g S S 2 g N C 2 g S 2 g ( 1 + C ) S 2 g 2 + C 2 g 2 C C 2 g S 0 S 2 g S C 2 g S C ] .
M J = R ( γ ) M R ( γ ) = [ 1 0 0 0 0 C 2 g S 2 g 0 0 S 2 g C 2 g 0 0 0 0 1 ] [ 1 N 0 0 N 1 0 0 0 0 C S 0 0 S C ] [ 1 0 0 0 0 C 2 g S 2 g 0 0 S 2 g C 2 g 0 0 0 0 1 ]  =
M T = R ( γ ) M R ( γ ) =
M T = [ 1 C 2 g N S 2 g N 0 C 2 g N C 2 g 2 + S 2 g 2 C C 2 g S 2 g ( 1 - C ) S 2 g S S 2 g N C 2 g S 2 g ( 1 C ) S 2 g 2 + C 2 g 2 C C 2 g S 0 S 2 g S C 2 g S C ] .
δ = 2 π d Δ η / λ ,
N = R o R e R o + R e ,
M = [ 1 m 12 m 13 0 m 12 m 22 m 23 m 24 F m 13 F m 23 m 33 m 34 0 F m 24 m 34 m 44 ] ,
I ( t ) = I dc + I X 0 X 0 + I Y 0 Y 0 + I X 1 X 1 + I Y 1 Y 1 + I X 0 X 1 X 0 X 1 + I X 0 Y 1 X 0 Y 1 + I Y 0 X 1 Y 0 X 1 + I Y 0 Y 1 Y 0 Y 1 ,
X 0 = sin [ A m 0 sin ( ω m 0 t ) ] ,
Y 0 = cos [ A m 0 sin ( ω m 0 t ) ] ,
X 1 = sin [ A m 1 sin ( ω m 1 t ) ] ,
Y 1 = cos [ A m 1 sin ( ω m 1 t ) ] .
M sample + perturbations = [ 1 I Y 0 I X 0 I Y 1 I Y 0 Y 1 I X 0 Y 1 I X 1 I Y 0 X 1 I X 0 X 1 ] .
β 2 = I Y 0 2 + I Y 1 2 + I X 0 X 1 2 + I X 0 Y 1 2 + I Y 0 X 1 2 .
M Almirror = [ 1 A Al 0 0 A Al 1 0 0 0 0 1 D Al 0 0 D Al 1 ] ,
M mirror = [ 1 N 0 0 N 1 0 0 0 0 1 S 0 0 S 1 ] ,
M 0 = [ 1 A 0 B 0 0 A 0 1 0 C 0 B 0 0 1 D 0 0 C 0 D 0 1 ] ,
M composite = [ 1 A t B t 0 A t 1 0 C t B t 0 1 D t 0 C t D t 1 ] ,
A t = ( A 0 + A 1 ) ,
B t = B 0 + B 1 ,
C t = C 0 + C 1 ,
D t = D 0 + D 1 .
I X 0 = 0 ,
I Y 0 = 2 S m 01 ε b 1 + S m 0 A t C m 0 B t C m 01 J 0 ( A m 1 ) ,
I X 1 = 0 ,
I Y 1 = 2 S m 01 ε b 0 + S m 1 A t + C m 0 B t C m 01 J 0 ( A m 0 ) ,
I X 0 X 1 = 1 ,
I X 0 Y 1 = S m 1 C t + C m 01 δ m 0 + δ m 1 + C m 1 D t ,
I Y 0 X 1 = S m 0 C t + C m 01 δ m 1 + δ m 0 + C m 0 D t ,
I Y 0 Y 1 = C m 01 .
S m 0 = sin ( 2 θ m 0 ) ,
C m 0 = cos ( 2 θ m 0 ) ,
S m 1 = sin ( 2 θ m 1 ) ,
C m 1 = cos ( 2 θ m 1 ) .
S m 01 = sin [ 2 ( θ m 0 θ m 1 ) ] ,
C m 01 = cos [ 2 ( θ m 0 θ m 1 ) ] .
[ 1 0.0000 ± 0.0011 0.0003 ± 0.0012 0.0003 ± 0.0011 0.9969 ± 0.0050 0.0013 ± 0.0020 0.0010 ± 0.0020 0.9939 ± 0.0067 0.0004 ± 0.0023 1.0013 ± 0.0037 ] ,

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