Abstract

The polarization-dependent transmission of a basic anisotropic feedback system (Fabry–Perot resonator) is mathematically modeled by means of the Jones-matrix formalism. Detailed numerical simulations of the resonance case are performed. Small phase anisotropies as well as small polarization-dependent losses of the resonator components can be extremely amplified by resonant feedback. The amplification factors depend on the magnitudes of amplitude and phase anisotropy and their mutual interactions as well as on the polarization-independent system parameters (forward transmission, system feedback). However, for higher phase anisotropies, saturation effects occur and, therefore, the anisotropy amplification factors decrease. Our experimental investigations applying anisotropic Fabry–Perot resonators in different ellipsometer systems confirm the predicted amplification of phase and loss anisotropies in resonance operation.

© 2007 Optical Society of America

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References

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  1. R. Brunetton and J. Monin, "Faraday multipass rotator for use in accuracy polarimetric or ellipsometric devices," Appl. Opt. 26, 3158-3160 (1987).
    [CrossRef] [PubMed]
  2. M. Born and E. Wolf, Principles of Optics (Pergamon, 1980), Chap. 1.6, p. 62.
  3. A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley, 1984), Chap. 8.2.2, p. 292.
  4. F. Maystre and R. Dandliker, "Polarimetric fiber optical sensor with high sensitivity using a Fabry-Perot structure," Appl. Opt. 28, 1995-2000 (1989).
    [CrossRef] [PubMed]
  5. G. R. Boyer, B. F. Lamouroux, B. S. Prade, and J. Y. Vinet, "Elastooptical antenna for detection of gravitational radiation," Appl. Opt. 19, 382-385 (1980).
    [CrossRef] [PubMed]
  6. W. Holzapfel, S. Neuschaefer-Rube, and U. Neuschaefer-Rube, "Intracavity transmission ellipsometry for optical anisotropic components," Appl. Opt. 32, 6022-6031 (1993).
    [CrossRef] [PubMed]
  7. W. Holzapfel, S. Neuschaefer-Rube, and U. Neuschaefer-Rube, "The photoelastic microellipsometer--a new tool for high resolution force vector measurements," in Polarization Analysis and Applications to Device Technology, T. Yoshisawa and J. Yokota, eds., Proc. SPIE 2873, 176-179 (1996).
  8. W. Holzapfel, S. Neuschaefer-Rube, and M. Kobusch, "High resolution, very broadband force measurements by solid laser transducers," Measurement 28, 277-291 (2000).
    [CrossRef]
  9. W. Holzapfel, U. Neuschaefer-Rube, and M. Kobusch, "Opto-mechatronic frequency response of high-performance force-sensing microlasers," in Optomechatronic Systems II, H. S. Cho, ed., Proc. SPIE 4564, 290-301 (2001).
  10. B. C. Jones and H. Hurwitz, "A new calculus for the treatment of optical systems 2. Proof of three general equivalence theorems," J. Opt. Soc. Am. 41, 493-503 (1941).
  11. H. de Lang, "Polarization properties of optical resonators passive and active," Ph.D. dissertation (University of Utrecht, 1966).
  12. T. Yoshino, "Polarization properties of internal-mirror He-Ne lasers at 6328 Å," Jpn. J. Appl. Phys. 11, 263-265 (1972).
    [CrossRef]
  13. T. Yoshino, "Reflection anisotropy of 6328 Å laser mirrors," Jpn. J. Appl. Phys. 18, 1503-1507 (1979).
    [CrossRef]
  14. W. M. Doyle and M. B. White, "Properties of an anisotropic Fabry-Perot resonator," J. Opt. Soc. Am. 55, 1221-1225 (1965).
    [CrossRef]
  15. P. S. Thoecaris and E. E. Gdoutos, Matrix Theory of Photoelasticity (Springer-Verlag, 1979).
  16. H. Wolf, Spannungsoptik, 2nd ed. (Springer-Verlag, 1976), Vol. 1, Chap. 1.3, p. 79.
  17. J. C. Braasch and W. Holzapfel, "Frequency stabilization of monomode semiconductor lasers to birefringent resonators," Electron. Lett. 28, 849-851 (1992).
    [CrossRef]
  18. R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light, 1st ed. (North Holland, 1979), Chap. 1.7, p. 29 and Chap. 3.2, p. 156.
  19. W. Holzapfel and U. Riß, "Computer-based high resolution transmission ellipsometry," Appl. Opt. 26, 145-152 (1987).
    [CrossRef] [PubMed]
  20. In our experimental setup (Fig. 8), a force-induced amplitude anisotropy could not be measured with sufficient resolution and without damaging the test etalons. The effect of stress-induced amplitude anisotropy has been experimentally proved for the first time by Ref. , applying high-resolution transmission-ellipsometry (Ref. 19) for a cylindrical, nonmirrored sample of BK7 (ø = 10 mm, d = 5 mm). However, the measured amplitude anisotropy was very small (DeltaxiM < 0.0002, applied force F = 10 N).
  21. U. Riß, "Zur Transmissionsellipsometrie optisch anisotroper Komponenten und Systeme," Ph.D. dissertation D34 (University of Kassel, 1988), p. 172.

2001 (1)

W. Holzapfel, U. Neuschaefer-Rube, and M. Kobusch, "Opto-mechatronic frequency response of high-performance force-sensing microlasers," in Optomechatronic Systems II, H. S. Cho, ed., Proc. SPIE 4564, 290-301 (2001).

2000 (1)

W. Holzapfel, S. Neuschaefer-Rube, and M. Kobusch, "High resolution, very broadband force measurements by solid laser transducers," Measurement 28, 277-291 (2000).
[CrossRef]

1996 (1)

W. Holzapfel, S. Neuschaefer-Rube, and U. Neuschaefer-Rube, "The photoelastic microellipsometer--a new tool for high resolution force vector measurements," in Polarization Analysis and Applications to Device Technology, T. Yoshisawa and J. Yokota, eds., Proc. SPIE 2873, 176-179 (1996).

1993 (1)

1992 (1)

J. C. Braasch and W. Holzapfel, "Frequency stabilization of monomode semiconductor lasers to birefringent resonators," Electron. Lett. 28, 849-851 (1992).
[CrossRef]

1989 (1)

1987 (2)

1980 (1)

1979 (1)

T. Yoshino, "Reflection anisotropy of 6328 Å laser mirrors," Jpn. J. Appl. Phys. 18, 1503-1507 (1979).
[CrossRef]

1972 (1)

T. Yoshino, "Polarization properties of internal-mirror He-Ne lasers at 6328 Å," Jpn. J. Appl. Phys. 11, 263-265 (1972).
[CrossRef]

1965 (1)

1941 (1)

B. C. Jones and H. Hurwitz, "A new calculus for the treatment of optical systems 2. Proof of three general equivalence theorems," J. Opt. Soc. Am. 41, 493-503 (1941).

Azzam, R. M. A.

R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light, 1st ed. (North Holland, 1979), Chap. 1.7, p. 29 and Chap. 3.2, p. 156.

Bashara, N. M.

R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light, 1st ed. (North Holland, 1979), Chap. 1.7, p. 29 and Chap. 3.2, p. 156.

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon, 1980), Chap. 1.6, p. 62.

Boyer, G. R.

Braasch, J. C.

J. C. Braasch and W. Holzapfel, "Frequency stabilization of monomode semiconductor lasers to birefringent resonators," Electron. Lett. 28, 849-851 (1992).
[CrossRef]

Brunetton, R.

Dandliker, R.

de Lang, H.

H. de Lang, "Polarization properties of optical resonators passive and active," Ph.D. dissertation (University of Utrecht, 1966).

Doyle, W. M.

Gdoutos, E. E.

P. S. Thoecaris and E. E. Gdoutos, Matrix Theory of Photoelasticity (Springer-Verlag, 1979).

Holzapfel, W.

W. Holzapfel, U. Neuschaefer-Rube, and M. Kobusch, "Opto-mechatronic frequency response of high-performance force-sensing microlasers," in Optomechatronic Systems II, H. S. Cho, ed., Proc. SPIE 4564, 290-301 (2001).

W. Holzapfel, S. Neuschaefer-Rube, and M. Kobusch, "High resolution, very broadband force measurements by solid laser transducers," Measurement 28, 277-291 (2000).
[CrossRef]

W. Holzapfel, S. Neuschaefer-Rube, and U. Neuschaefer-Rube, "The photoelastic microellipsometer--a new tool for high resolution force vector measurements," in Polarization Analysis and Applications to Device Technology, T. Yoshisawa and J. Yokota, eds., Proc. SPIE 2873, 176-179 (1996).

W. Holzapfel, S. Neuschaefer-Rube, and U. Neuschaefer-Rube, "Intracavity transmission ellipsometry for optical anisotropic components," Appl. Opt. 32, 6022-6031 (1993).
[CrossRef] [PubMed]

J. C. Braasch and W. Holzapfel, "Frequency stabilization of monomode semiconductor lasers to birefringent resonators," Electron. Lett. 28, 849-851 (1992).
[CrossRef]

W. Holzapfel and U. Riß, "Computer-based high resolution transmission ellipsometry," Appl. Opt. 26, 145-152 (1987).
[CrossRef] [PubMed]

Hurwitz, H.

B. C. Jones and H. Hurwitz, "A new calculus for the treatment of optical systems 2. Proof of three general equivalence theorems," J. Opt. Soc. Am. 41, 493-503 (1941).

Jones, B. C.

B. C. Jones and H. Hurwitz, "A new calculus for the treatment of optical systems 2. Proof of three general equivalence theorems," J. Opt. Soc. Am. 41, 493-503 (1941).

Kobusch, M.

W. Holzapfel, U. Neuschaefer-Rube, and M. Kobusch, "Opto-mechatronic frequency response of high-performance force-sensing microlasers," in Optomechatronic Systems II, H. S. Cho, ed., Proc. SPIE 4564, 290-301 (2001).

W. Holzapfel, S. Neuschaefer-Rube, and M. Kobusch, "High resolution, very broadband force measurements by solid laser transducers," Measurement 28, 277-291 (2000).
[CrossRef]

Lamouroux, B. F.

Maystre, F.

Monin, J.

Neuschaefer-Rube, S.

W. Holzapfel, S. Neuschaefer-Rube, and M. Kobusch, "High resolution, very broadband force measurements by solid laser transducers," Measurement 28, 277-291 (2000).
[CrossRef]

W. Holzapfel, S. Neuschaefer-Rube, and U. Neuschaefer-Rube, "The photoelastic microellipsometer--a new tool for high resolution force vector measurements," in Polarization Analysis and Applications to Device Technology, T. Yoshisawa and J. Yokota, eds., Proc. SPIE 2873, 176-179 (1996).

W. Holzapfel, S. Neuschaefer-Rube, and U. Neuschaefer-Rube, "Intracavity transmission ellipsometry for optical anisotropic components," Appl. Opt. 32, 6022-6031 (1993).
[CrossRef] [PubMed]

Neuschaefer-Rube, U.

W. Holzapfel, U. Neuschaefer-Rube, and M. Kobusch, "Opto-mechatronic frequency response of high-performance force-sensing microlasers," in Optomechatronic Systems II, H. S. Cho, ed., Proc. SPIE 4564, 290-301 (2001).

W. Holzapfel, S. Neuschaefer-Rube, and U. Neuschaefer-Rube, "The photoelastic microellipsometer--a new tool for high resolution force vector measurements," in Polarization Analysis and Applications to Device Technology, T. Yoshisawa and J. Yokota, eds., Proc. SPIE 2873, 176-179 (1996).

W. Holzapfel, S. Neuschaefer-Rube, and U. Neuschaefer-Rube, "Intracavity transmission ellipsometry for optical anisotropic components," Appl. Opt. 32, 6022-6031 (1993).
[CrossRef] [PubMed]

Prade, B. S.

Riß, U.

W. Holzapfel and U. Riß, "Computer-based high resolution transmission ellipsometry," Appl. Opt. 26, 145-152 (1987).
[CrossRef] [PubMed]

U. Riß, "Zur Transmissionsellipsometrie optisch anisotroper Komponenten und Systeme," Ph.D. dissertation D34 (University of Kassel, 1988), p. 172.

Thoecaris, P. S.

P. S. Thoecaris and E. E. Gdoutos, Matrix Theory of Photoelasticity (Springer-Verlag, 1979).

Vinet, J. Y.

White, M. B.

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Pergamon, 1980), Chap. 1.6, p. 62.

Wolf, H.

H. Wolf, Spannungsoptik, 2nd ed. (Springer-Verlag, 1976), Vol. 1, Chap. 1.3, p. 79.

Yariv, A.

A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley, 1984), Chap. 8.2.2, p. 292.

Yeh, P.

A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley, 1984), Chap. 8.2.2, p. 292.

Yoshino, T.

T. Yoshino, "Reflection anisotropy of 6328 Å laser mirrors," Jpn. J. Appl. Phys. 18, 1503-1507 (1979).
[CrossRef]

T. Yoshino, "Polarization properties of internal-mirror He-Ne lasers at 6328 Å," Jpn. J. Appl. Phys. 11, 263-265 (1972).
[CrossRef]

Appl. Opt. (5)

Electron. Lett. (1)

J. C. Braasch and W. Holzapfel, "Frequency stabilization of monomode semiconductor lasers to birefringent resonators," Electron. Lett. 28, 849-851 (1992).
[CrossRef]

J. Opt. Soc. Am. (2)

W. M. Doyle and M. B. White, "Properties of an anisotropic Fabry-Perot resonator," J. Opt. Soc. Am. 55, 1221-1225 (1965).
[CrossRef]

B. C. Jones and H. Hurwitz, "A new calculus for the treatment of optical systems 2. Proof of three general equivalence theorems," J. Opt. Soc. Am. 41, 493-503 (1941).

Jpn. J. Appl. Phys. (2)

T. Yoshino, "Polarization properties of internal-mirror He-Ne lasers at 6328 Å," Jpn. J. Appl. Phys. 11, 263-265 (1972).
[CrossRef]

T. Yoshino, "Reflection anisotropy of 6328 Å laser mirrors," Jpn. J. Appl. Phys. 18, 1503-1507 (1979).
[CrossRef]

Measurement (1)

W. Holzapfel, S. Neuschaefer-Rube, and M. Kobusch, "High resolution, very broadband force measurements by solid laser transducers," Measurement 28, 277-291 (2000).
[CrossRef]

Proc. SPIE (1)

W. Holzapfel, S. Neuschaefer-Rube, and U. Neuschaefer-Rube, "The photoelastic microellipsometer--a new tool for high resolution force vector measurements," in Polarization Analysis and Applications to Device Technology, T. Yoshisawa and J. Yokota, eds., Proc. SPIE 2873, 176-179 (1996).

Other (9)

W. Holzapfel, U. Neuschaefer-Rube, and M. Kobusch, "Opto-mechatronic frequency response of high-performance force-sensing microlasers," in Optomechatronic Systems II, H. S. Cho, ed., Proc. SPIE 4564, 290-301 (2001).

H. de Lang, "Polarization properties of optical resonators passive and active," Ph.D. dissertation (University of Utrecht, 1966).

M. Born and E. Wolf, Principles of Optics (Pergamon, 1980), Chap. 1.6, p. 62.

A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley, 1984), Chap. 8.2.2, p. 292.

P. S. Thoecaris and E. E. Gdoutos, Matrix Theory of Photoelasticity (Springer-Verlag, 1979).

H. Wolf, Spannungsoptik, 2nd ed. (Springer-Verlag, 1976), Vol. 1, Chap. 1.3, p. 79.

R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light, 1st ed. (North Holland, 1979), Chap. 1.7, p. 29 and Chap. 3.2, p. 156.

In our experimental setup (Fig. 8), a force-induced amplitude anisotropy could not be measured with sufficient resolution and without damaging the test etalons. The effect of stress-induced amplitude anisotropy has been experimentally proved for the first time by Ref. , applying high-resolution transmission-ellipsometry (Ref. 19) for a cylindrical, nonmirrored sample of BK7 (ø = 10 mm, d = 5 mm). However, the measured amplitude anisotropy was very small (DeltaxiM < 0.0002, applied force F = 10 N).

U. Riß, "Zur Transmissionsellipsometrie optisch anisotroper Komponenten und Systeme," Ph.D. dissertation D34 (University of Kassel, 1988), p. 172.

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Figures (13)

Fig. 1
Fig. 1

Anisotropic transmission of an optical feedback system (idealized FP resonator). (a) Schematic, (b) block diagram (B, branching point; S, summing point).

Fig. 2
Fig. 2

Resulting phase difference Δ Res of the FP resonator as a function of the phase difference Δ M of the anisotropic material for different system reflection factors R.

Fig. 3
Fig. 3

Amplification factor V as a function of phase difference Δ M of the material for different system reflection factors R.

Fig. 4
Fig. 4

Amplification factor V as a function of the system reflection factor R for different transmission factors T M .

Fig. 5
Fig. 5

Dependence of the amplification factor V on the polarization-dependent attenuation ξ M of the material ( T M = 0.9 ) for different R.

Fig. 6
Fig. 6

Polarization-dependent attenuation ξ Res of the resonator as a function of the polarization-dependent attenuation ξ M of the material ( T M = 0.9 ) for different R.

Fig. 7
Fig. 7

Polarization-dependent attenuation ξ Res of the resonator as a function of the phase difference Δ M ( T M = 0.9 , ξ M = 0.01 ) for different R.

Fig. 8
Fig. 8

Experimental setup (PSA ellipsometer) for phase anisotropy measurement of birefringent resonators. Birefringence is induced by force F using the photoelastic effect.

Fig. 9
Fig. 9

Transmitted intensities in the eigenpolarizations of a birefringent resonator (lower traces) and their fivefold amplified difference (top trace), ( R = 78.1 % , D = 25   mm , F = 5.9 N).

Fig. 10
Fig. 10

(Color online) Measured relative phase shift Δ Res in the transmission of a birefringent resonator (dots) and their corresponding theoretical values (straight lines) for various reflectivities.

Fig. 11
Fig. 11

Experimental setup (WME) for amplitude anisotropy measurement of dichroitic resonators. Dichroism is generated by the tilted glass plate. (Thickness 1   mm , refractive index n G = 1.49 ).

Fig. 12
Fig. 12

(Color online) Measured polarization-dependent attenuation ξ Res versus material attenuation ξ M of a dichroitic resonator.

Fig. 13
Fig. 13

Simplified block diagram of the FP resonator.

Equations (76)

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E out = T Res · E in .
T O L = R 2 · M 0 · R 1 R 1 · R 1 T · M 0 T · R 2 T R 2 = T F · T B ,   
T F R 2 · M 0 · R 1 ,
T B R 1 R 1 T · M 0 T · R 2 T R 2 = R 1 T F T R 2 .
T C L = ( I T O L ) 1 · T F , w i t h I = identity   matrix .
T Res = T 2 · T C L · T 1 .
T F R 2 · M 0 · R 1 = V β · P · V α · R o t ( φ F ) .
P = R o t ( φ P ) T P [ exp ( ξ P / 2 ) 0 0 exp ( ξ P / 2 ) ] R o t ( φ P ) ,
V i = R o t ( φ i ) exp ( j δ i ) [ exp ( j Δ i / 2 ) 0 0 exp ( j Δ i / 2 ) ] R o t ( φ i ) , i = α , β ,
R o t ( φ i ) = [ cos   φ i sin   φ i sin   φ i cos   φ i ] i = α , β , P , F .
T O L = V β · P · V α R 1 V α T · P T · V β T R 2 .
C = V C ( δ C , Δ C ) P C ( T C , ξ C ) = R o t ( φ C ) T C   exp ( j δ C ) [ exp [ j ( Δ C j ξ C ) / 2 ] 0 0 exp [ j ( Δ C j ξ C ) / 2 ] ] R o t ( φ C ) , C : T 1 , T 2 , M 0 , R 1 R 1 , o r R 2 R 2 .
T F R 2 · M 0 · R 1 = R o t ( φ M ) T M  exp ( j δ M ) [ exp [ j ( Δ M j ξ M ) / 2 ] 0 0 exp [ j ( Δ M j ξ M ) / 2 ] ] R o t ( φ M ) = M ,
Δ M = Δ 2 + Δ M 0 + Δ 1 ,
ξ M = ξ 2 + ξ M 0 + ξ 1 ,
δ M = δ 2 + δ M 0 + δ 1 ,
T M = T M 0 .
R i = I , i = 1 , 2.
T i = T i I , i = 1 , 2.
T O L = R 1 R 2 T M   exp ( j 2 δ M ) [ exp [ j ( Δ M j ξ M ) ] 0 0 exp [ j ( Δ M j ξ M ) ] ] .
T Res = X Res [ exp [ j ( Δ Res j ξ Res ) ] 0 0 exp [ j ( Δ Res j ξ Res ) ] ] ,
X Res = T 1 T M T 2  exp ( j δ M ) 1 2 R 1 R 2 T M   exp ( j 2 δ M ) cos ( Δ M j ξ M ) + R 1 R 2 T M 2   exp ( j 4 δ M ) ,
Δ Res = arctan ( R 1 R 2 T M   exp ( ξ M ) sin ( 2 δ M Δ M ) 1 R 1 R 2 T M   exp ( ξ M ) cos ( 2 δ M Δ M ) ) arctan ( R 1 R 2 T M   exp ( ξ M ) sin ( 2 δ M + Δ M ) 1 R 1 R 2 T M   exp ( ξ M ) cos ( 2 δ M + Δ M ) ) + Δ M ,
ξ Res = 1 2   ln ( 1 2 R 1 R 2 T M   exp ( ξ M ) cos ( 2 δ M Δ M ) + R 1 R 2 T M 2   exp ( 2 ξ M ) 1 2 R 1 R 2 T M   exp ( ξ M ) cos ( 2 δ M + Δ M ) + R 1 R 2 T M 2   exp ( 2 ξ M ) ) + ξ M .
Δ Res = V Δ M ,
V 0 1 + R 1 R ,
Δ M = 16 C 0 λ D F ,
T Sum = T BS · T Res .
[ E out x E out y ] = T Sum · E in = X Sum [ exp [ j ( Δ Sum j ξ Sum ) / 2 ] 0 0 exp [ j ( Δ Sum j ξ Sum ] / 2 ] ] [ 1 1 ] .
Δ Res = Δ Sum Δ BS ,
ξ Res = ξ Sum ξ BS .
χ out = E out y E out x = tan   θ + j   tan   ε 1 j   tan   θ   tan   ε .
E out y E out x = χ out = e ξ Sum e j Δ Sum .
| Δ BS | = 1.78 ° ,
ξ BS = 0.0965.
Δ M = ( 0.155 ° / N ) F ,
V 78.1 % = 8.171 ,
V 49.1 % = 2.754.
V 78.1 % * = 7.935 ,
V 49.1 % * = 2.921.
T π T σ = exp ( ξ M ) ,
T M = T σ T π .
   δ M = π ,
   Δ M = 0.
ξ Res = ln ( T π T σ 1 R T σ 1 R T π ) .
E out = T F · ( E in + T B · E out ) .
E out = ( I T F · T B ) 1 · T F · E in ,
E out = T C L · E in
T O L = T F R 1 T F T R 2 = V β · P · V α · R o t ( φ F ) R 1 R o t ( φ F ) V α T · P T · V β t R 2 ,
i = 1 , 2.
( V β · P · V α · R o t ( φ F ) ) T = R o t ( φ F ) · V α T · P T · V β T .
A = R 1 R 2 T M   exp ( j 2 δ M ) ,
B = T M exp ( j δ M ) ,
α = j ( Δ M j ξ M ) ,
T C L = ( I A [ exp ( j α ) 0 0 exp ( j α ) ] ) 1 B [ exp ( j α / 2 ) 0 0 exp ( j α / 2 ) ] .
T C L = ( I A [ exp ( j α ) 0 0 exp ( j α ) ] ) 1 B [ exp ( j α / 2 ) 0 0 exp ( j α / 2 ) ] = 1 1 2 A   cos   α + A 2 [ 1 A   cos   α + j   sin   α 1 2 A   cos   α + A 2 0 0 1 A   cos   α + j   sin   α 1 2 A   cos   α + A 2 ] B [ exp ( j α / 2 ) 0 0 exp ( j α / 2 ) ] = 1 1 2 A   cos   α + A 2 [ exp [ j   arctan ( sin   α 1 A   cos   α ) ] 0 0 exp [ j   arctan ( sin   α 1 A   cos   α ) ] ] B [ exp ( j α / 2 ) 0 0 exp ( j α / 2 ) ] .
T C L = 1 1 2 A   cos   α + A 2 · [ exp [ j 2 ln ( 1 A e j α 1 A e j α ) ] 0 0 exp [ j 2  ln ( 1 A e j α 1 A e j α ) ] ] B [ exp ( j α / 2 ) 0 0 exp ( j α / 2 ) ] ,
T C L = 1 1 2 A   cos   α + A 2 [ exp [ j ( Δ j ξ ) / 2 ] 0 0 exp [ j ( Δ j ξ ) / 2 ] ] B [ exp ( j α / 2 ) 0 0 exp ( j α / 2 ) ] = X C L · [ exp [ j ( Δ + Δ M j ( ξ + ξ M ) ) / 2 ] 0 0 exp [ j ( Δ + Δ M j ( ξ + ξ M ) ) / 2 ] ] ,
X C L = T M e i δ M 1 2 R 1 R 2 T M   exp ( j 2 δ M ) cos ( Δ M j ξ M ) + R 1 R 2 T M 2   exp ( j 4 δ M ) ,
Δ = arctan ( R 1 R 2 T M   exp ( ξ M ) sin ( 2 δ M Δ M ) 1 R 1 R 2 T M   exp ( ξ M ) cos ( 2 δ M Δ M ) ) arctan ( R 1 R 2 T M   exp ( ξ M ) sin ( 2 δ M Δ M ) 1 R 1 R 2 T M   exp ( ξ M ) cos ( 2 δ M Δ M ) ) ,
ξ = 1 2   ln ( 1 2 R 1 R 2 T M   exp ( ξ M ) cos ( 2 δ M Δ M ) + R 1 R 2 T M 2   exp ( 2 ξ M ) 1 2 R 1 R 2 T M   exp ( ξ M ) cos ( 2 δ M + Δ M ) + R 1 R 2 T M 2   exp ( 2 ξ M ) ) .
X Res = T 1 T 2 X C L ,
Δ Res = Δ C L = Δ + Δ M ,
ξ Res = ξ C L = ξ + ξ M .
sin ( 2 δ M Δ M ) Δ M , sin ( 2 δ M + Δ M ) Δ M ,
cos ( 2 δ M Δ M ) = cos ( 2 δ M + Δ M ) 1.
Δ Res = arctan ( R Δ M 1 R ) arctan ( R Δ M 1 R ) + Δ M .
arctan ( R Δ M 1 R ) R Δ M 1 R .
Δ Res = 1 + R 1 R Δ M ,
t π AG = 2 n A   cos ( Θ 1 ) n G   cos ( Θ 1 ) + n A   cos ( Θ 2 ) ,
t π GA = 2 n G   cos ( Θ 2 ) n A   cos ( Θ 2 ) + n G   cos ( Θ 1 ) ,
Θ 2 = arcsin ( n A n G   sin ( Θ 1 ) ) .
T π = t π AG 2 t π GA 2 .
t σ AG = 2 n A   cos ( Θ 1 ) n A   cos ( Θ 1 ) + n G   cos ( Θ 2 ) .
t σ GA = 2 n G   cos ( Θ 2 ) n G   cos ( Θ 2 ) + n A   cos ( Θ 1 ) ,
T σ = t σ AG 2 t σ GA 2 .

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