Abstract

The complex refractive index of materials at infrared wavelengths is often determined when absorption measurements are made at selected wavelengths, and then the Kramers–Kronig relationship is used to calculate the real part of the index. Because many organic materials are highly absorbing in the infrared, absorption measurements require a short path length. We report on the use of an attenuated total internal reflection (TIR) method in combination with an infrared Mueller matrix spectropolarimeter to measure the Mueller matrix spectrum of samples from 3 to 14 µm. From the elements of the Mueller matrix the complex refractive index is determined for materials whose TIR interfaces are eigenstates of s and p polarization. The calculated index for water compares well with data found in the literature.

© 2001 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. K. K. Chittur, “FTIR/ATR for protein adsorption to biomaterial surfaces,” Biomaterials 19, 357–369 (1998).
    [CrossRef] [PubMed]
  2. N. J. Harrick, Internal Reflection Spectroscopy, 1st ed. (Interscience, New York, 1967).
  3. R. M. A. Azzam, “Differential reflection phase shift under conditions of attenuated internal reflection,” J. Opt. Soc. Am. 16, 1700–1702 (1999).
    [CrossRef]
  4. R. M. A. Azzam, “Correlation of Fresnel’s interface reflection coefficients of external and internal reflection at the same angle of incidence for dielectric-dielectric interfaces,” in Polarization: Measurement, Analysis and Remote Sensing II, D. H. Goldstein, D. B. Chenault, eds., Proc. SPIE3754, 88–98 (1999).
  5. R. M. A. Azzam, “Photopolarimetric measurement of the Mueller matrix by Fourier analysis of a single detected signal,” Opt. Lett. 2, 148–150 (1978).
    [CrossRef] [PubMed]
  6. D. B. Chenault, R. A. Chipman, “Infrared achromatic retarder,” U.S. patent4,961,634 (9Oct.1990).
  7. D. B. Chenault, R. A. Chipman, “Measurements of linear diattenuation and linear retardance spectra with a rotating sample spectropolarimeter,” Appl. Opt. 32, 3513–3519 (1993).
    [CrossRef] [PubMed]
  8. R. A. Chipman, “Polarimetry,” in Handbook of Optics, M. Bass, ed. (McGraw-Hill, New York, 1995), Vol. 2, Chap. 22.
  9. D. H. Goldstein, R. A. Chipman, “Error analysis of Mueller matrix polarimeters,” J. Opt. Soc. Am. 7, 693–700 (1990).
    [CrossRef]
  10. W. N. Hansen, “Expanded formulas of attenuated total reflection and the derivation of absorption rules for single and multiple ATR spectrometer cells,” Spectrochim. Acta 21, 815–833 (1965).
    [CrossRef]
  11. D. J. Ahn, E. I. Franses, “Orientations of chain axes and transition moments in Langmuir-Blodgett mono-layers determined by polarized FTIR-ATR spectroscopy,” J. Phys. Chem. 96, 9952–9958 (1992).
    [CrossRef]
  12. E. Collett, Polarized Light, Fundamentals and Applications (Marcel Decker, New York, 1992).
  13. T. W. Nee, S. M. F. Nee, “Infrared polarization signatures for targets,” in Targets and Backgrounds: Characterization and Representation, W. R. Watkins, D. Clement, eds., Proc. SPIE2469, 231–241 (1995).
    [CrossRef]
  14. W. J. Tropf, M. E. Thomas, T. J. Harris, “Properties of crystals and glasses,” in Handbook of Optics, M. Bass, ed. (McGraw-Hill, New York, 1995), Vol. 2, Chap. 33, Table 22.
  15. E. D. Palik, Handbook of Optical Constants of Solids (Academic, Orlando, Fla., 1985).
  16. L. L. Deibler, “Infrared polarimetry using attenuated total reflection,” Ph.D. dissertation (University of Alabama in Huntsville, Huntsville, Ala., 2001).
  17. D. H. Goldstein, R. A. Chipman, D. B. Chenault, “Infrared spectropolarimetry,” Opt. Eng. 28(2), 120–125 (1989).
  18. D. B. Chenault, J. L. Pezzaniti, R. A. Chipman, “Mueller matrix algorithms,” in Polarization Analysis and Measurement, D. H. Goldstein, R. A. Chipman, eds., Proc. SPIE1746, 231–246 (1992).
    [CrossRef]
  19. G. M. Hale, M. R. Querry, “Optical constants of water in the 200-nm to 200-µm wavelength region,” Appl. Opt. 12, 555–562 (1973).
    [CrossRef] [PubMed]
  20. E. A. Sornsin, R. A. Chipman, “Alignment and calibration of an infrared achromatic retarder using FTIR Mueller matrix spectropolarimetry,” in Polarization: Measurement, Analysis, and Remote Sensing, D. H. Goldstein, R. A. Chipman, eds., Proc. SPIE3121, 28–34 (1997).
  21. P. H. Axelsen, M. J. Citra, “Orientational order determination by internal reflection infrared spectroscopy,” Prog. Biophys. Mol. Biol. 66(3), 227–253 (1996).
    [CrossRef]

1999

R. M. A. Azzam, “Differential reflection phase shift under conditions of attenuated internal reflection,” J. Opt. Soc. Am. 16, 1700–1702 (1999).
[CrossRef]

1998

K. K. Chittur, “FTIR/ATR for protein adsorption to biomaterial surfaces,” Biomaterials 19, 357–369 (1998).
[CrossRef] [PubMed]

1996

P. H. Axelsen, M. J. Citra, “Orientational order determination by internal reflection infrared spectroscopy,” Prog. Biophys. Mol. Biol. 66(3), 227–253 (1996).
[CrossRef]

1993

1992

D. J. Ahn, E. I. Franses, “Orientations of chain axes and transition moments in Langmuir-Blodgett mono-layers determined by polarized FTIR-ATR spectroscopy,” J. Phys. Chem. 96, 9952–9958 (1992).
[CrossRef]

1990

1989

D. H. Goldstein, R. A. Chipman, D. B. Chenault, “Infrared spectropolarimetry,” Opt. Eng. 28(2), 120–125 (1989).

1978

1973

1965

W. N. Hansen, “Expanded formulas of attenuated total reflection and the derivation of absorption rules for single and multiple ATR spectrometer cells,” Spectrochim. Acta 21, 815–833 (1965).
[CrossRef]

Ahn, D. J.

D. J. Ahn, E. I. Franses, “Orientations of chain axes and transition moments in Langmuir-Blodgett mono-layers determined by polarized FTIR-ATR spectroscopy,” J. Phys. Chem. 96, 9952–9958 (1992).
[CrossRef]

Axelsen, P. H.

P. H. Axelsen, M. J. Citra, “Orientational order determination by internal reflection infrared spectroscopy,” Prog. Biophys. Mol. Biol. 66(3), 227–253 (1996).
[CrossRef]

Azzam, R. M. A.

R. M. A. Azzam, “Differential reflection phase shift under conditions of attenuated internal reflection,” J. Opt. Soc. Am. 16, 1700–1702 (1999).
[CrossRef]

R. M. A. Azzam, “Photopolarimetric measurement of the Mueller matrix by Fourier analysis of a single detected signal,” Opt. Lett. 2, 148–150 (1978).
[CrossRef] [PubMed]

R. M. A. Azzam, “Correlation of Fresnel’s interface reflection coefficients of external and internal reflection at the same angle of incidence for dielectric-dielectric interfaces,” in Polarization: Measurement, Analysis and Remote Sensing II, D. H. Goldstein, D. B. Chenault, eds., Proc. SPIE3754, 88–98 (1999).

Chenault, D. B.

D. B. Chenault, R. A. Chipman, “Measurements of linear diattenuation and linear retardance spectra with a rotating sample spectropolarimeter,” Appl. Opt. 32, 3513–3519 (1993).
[CrossRef] [PubMed]

D. H. Goldstein, R. A. Chipman, D. B. Chenault, “Infrared spectropolarimetry,” Opt. Eng. 28(2), 120–125 (1989).

D. B. Chenault, J. L. Pezzaniti, R. A. Chipman, “Mueller matrix algorithms,” in Polarization Analysis and Measurement, D. H. Goldstein, R. A. Chipman, eds., Proc. SPIE1746, 231–246 (1992).
[CrossRef]

D. B. Chenault, R. A. Chipman, “Infrared achromatic retarder,” U.S. patent4,961,634 (9Oct.1990).

Chipman, R. A.

D. B. Chenault, R. A. Chipman, “Measurements of linear diattenuation and linear retardance spectra with a rotating sample spectropolarimeter,” Appl. Opt. 32, 3513–3519 (1993).
[CrossRef] [PubMed]

D. H. Goldstein, R. A. Chipman, “Error analysis of Mueller matrix polarimeters,” J. Opt. Soc. Am. 7, 693–700 (1990).
[CrossRef]

D. H. Goldstein, R. A. Chipman, D. B. Chenault, “Infrared spectropolarimetry,” Opt. Eng. 28(2), 120–125 (1989).

D. B. Chenault, J. L. Pezzaniti, R. A. Chipman, “Mueller matrix algorithms,” in Polarization Analysis and Measurement, D. H. Goldstein, R. A. Chipman, eds., Proc. SPIE1746, 231–246 (1992).
[CrossRef]

D. B. Chenault, R. A. Chipman, “Infrared achromatic retarder,” U.S. patent4,961,634 (9Oct.1990).

R. A. Chipman, “Polarimetry,” in Handbook of Optics, M. Bass, ed. (McGraw-Hill, New York, 1995), Vol. 2, Chap. 22.

E. A. Sornsin, R. A. Chipman, “Alignment and calibration of an infrared achromatic retarder using FTIR Mueller matrix spectropolarimetry,” in Polarization: Measurement, Analysis, and Remote Sensing, D. H. Goldstein, R. A. Chipman, eds., Proc. SPIE3121, 28–34 (1997).

Chittur, K. K.

K. K. Chittur, “FTIR/ATR for protein adsorption to biomaterial surfaces,” Biomaterials 19, 357–369 (1998).
[CrossRef] [PubMed]

Citra, M. J.

P. H. Axelsen, M. J. Citra, “Orientational order determination by internal reflection infrared spectroscopy,” Prog. Biophys. Mol. Biol. 66(3), 227–253 (1996).
[CrossRef]

Collett, E.

E. Collett, Polarized Light, Fundamentals and Applications (Marcel Decker, New York, 1992).

Deibler, L. L.

L. L. Deibler, “Infrared polarimetry using attenuated total reflection,” Ph.D. dissertation (University of Alabama in Huntsville, Huntsville, Ala., 2001).

Franses, E. I.

D. J. Ahn, E. I. Franses, “Orientations of chain axes and transition moments in Langmuir-Blodgett mono-layers determined by polarized FTIR-ATR spectroscopy,” J. Phys. Chem. 96, 9952–9958 (1992).
[CrossRef]

Goldstein, D. H.

D. H. Goldstein, R. A. Chipman, “Error analysis of Mueller matrix polarimeters,” J. Opt. Soc. Am. 7, 693–700 (1990).
[CrossRef]

D. H. Goldstein, R. A. Chipman, D. B. Chenault, “Infrared spectropolarimetry,” Opt. Eng. 28(2), 120–125 (1989).

Hale, G. M.

Hansen, W. N.

W. N. Hansen, “Expanded formulas of attenuated total reflection and the derivation of absorption rules for single and multiple ATR spectrometer cells,” Spectrochim. Acta 21, 815–833 (1965).
[CrossRef]

Harrick, N. J.

N. J. Harrick, Internal Reflection Spectroscopy, 1st ed. (Interscience, New York, 1967).

Harris, T. J.

W. J. Tropf, M. E. Thomas, T. J. Harris, “Properties of crystals and glasses,” in Handbook of Optics, M. Bass, ed. (McGraw-Hill, New York, 1995), Vol. 2, Chap. 33, Table 22.

Nee, S. M. F.

T. W. Nee, S. M. F. Nee, “Infrared polarization signatures for targets,” in Targets and Backgrounds: Characterization and Representation, W. R. Watkins, D. Clement, eds., Proc. SPIE2469, 231–241 (1995).
[CrossRef]

Nee, T. W.

T. W. Nee, S. M. F. Nee, “Infrared polarization signatures for targets,” in Targets and Backgrounds: Characterization and Representation, W. R. Watkins, D. Clement, eds., Proc. SPIE2469, 231–241 (1995).
[CrossRef]

Palik, E. D.

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

Pezzaniti, J. L.

D. B. Chenault, J. L. Pezzaniti, R. A. Chipman, “Mueller matrix algorithms,” in Polarization Analysis and Measurement, D. H. Goldstein, R. A. Chipman, eds., Proc. SPIE1746, 231–246 (1992).
[CrossRef]

Querry, M. R.

Sornsin, E. A.

E. A. Sornsin, R. A. Chipman, “Alignment and calibration of an infrared achromatic retarder using FTIR Mueller matrix spectropolarimetry,” in Polarization: Measurement, Analysis, and Remote Sensing, D. H. Goldstein, R. A. Chipman, eds., Proc. SPIE3121, 28–34 (1997).

Thomas, M. E.

W. J. Tropf, M. E. Thomas, T. J. Harris, “Properties of crystals and glasses,” in Handbook of Optics, M. Bass, ed. (McGraw-Hill, New York, 1995), Vol. 2, Chap. 33, Table 22.

Tropf, W. J.

W. J. Tropf, M. E. Thomas, T. J. Harris, “Properties of crystals and glasses,” in Handbook of Optics, M. Bass, ed. (McGraw-Hill, New York, 1995), Vol. 2, Chap. 33, Table 22.

Appl. Opt.

Biomaterials

K. K. Chittur, “FTIR/ATR for protein adsorption to biomaterial surfaces,” Biomaterials 19, 357–369 (1998).
[CrossRef] [PubMed]

J. Opt. Soc. Am.

R. M. A. Azzam, “Differential reflection phase shift under conditions of attenuated internal reflection,” J. Opt. Soc. Am. 16, 1700–1702 (1999).
[CrossRef]

D. H. Goldstein, R. A. Chipman, “Error analysis of Mueller matrix polarimeters,” J. Opt. Soc. Am. 7, 693–700 (1990).
[CrossRef]

J. Phys. Chem.

D. J. Ahn, E. I. Franses, “Orientations of chain axes and transition moments in Langmuir-Blodgett mono-layers determined by polarized FTIR-ATR spectroscopy,” J. Phys. Chem. 96, 9952–9958 (1992).
[CrossRef]

Opt. Eng.

D. H. Goldstein, R. A. Chipman, D. B. Chenault, “Infrared spectropolarimetry,” Opt. Eng. 28(2), 120–125 (1989).

Opt. Lett.

Prog. Biophys. Mol. Biol.

P. H. Axelsen, M. J. Citra, “Orientational order determination by internal reflection infrared spectroscopy,” Prog. Biophys. Mol. Biol. 66(3), 227–253 (1996).
[CrossRef]

Spectrochim. Acta

W. N. Hansen, “Expanded formulas of attenuated total reflection and the derivation of absorption rules for single and multiple ATR spectrometer cells,” Spectrochim. Acta 21, 815–833 (1965).
[CrossRef]

Other

D. B. Chenault, J. L. Pezzaniti, R. A. Chipman, “Mueller matrix algorithms,” in Polarization Analysis and Measurement, D. H. Goldstein, R. A. Chipman, eds., Proc. SPIE1746, 231–246 (1992).
[CrossRef]

E. Collett, Polarized Light, Fundamentals and Applications (Marcel Decker, New York, 1992).

T. W. Nee, S. M. F. Nee, “Infrared polarization signatures for targets,” in Targets and Backgrounds: Characterization and Representation, W. R. Watkins, D. Clement, eds., Proc. SPIE2469, 231–241 (1995).
[CrossRef]

W. J. Tropf, M. E. Thomas, T. J. Harris, “Properties of crystals and glasses,” in Handbook of Optics, M. Bass, ed. (McGraw-Hill, New York, 1995), Vol. 2, Chap. 33, Table 22.

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

L. L. Deibler, “Infrared polarimetry using attenuated total reflection,” Ph.D. dissertation (University of Alabama in Huntsville, Huntsville, Ala., 2001).

D. B. Chenault, R. A. Chipman, “Infrared achromatic retarder,” U.S. patent4,961,634 (9Oct.1990).

R. A. Chipman, “Polarimetry,” in Handbook of Optics, M. Bass, ed. (McGraw-Hill, New York, 1995), Vol. 2, Chap. 22.

R. M. A. Azzam, “Correlation of Fresnel’s interface reflection coefficients of external and internal reflection at the same angle of incidence for dielectric-dielectric interfaces,” in Polarization: Measurement, Analysis and Remote Sensing II, D. H. Goldstein, D. B. Chenault, eds., Proc. SPIE3754, 88–98 (1999).

N. J. Harrick, Internal Reflection Spectroscopy, 1st ed. (Interscience, New York, 1967).

E. A. Sornsin, R. A. Chipman, “Alignment and calibration of an infrared achromatic retarder using FTIR Mueller matrix spectropolarimetry,” in Polarization: Measurement, Analysis, and Remote Sensing, D. H. Goldstein, R. A. Chipman, eds., Proc. SPIE3121, 28–34 (1997).

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 (7)

Fig. 1
Fig. 1

ATR device consisting of two gold mirrors to deviate the infrared beam through the ZnSe prism resulting in a 45° incidence angle at the prism top surface. The ATR was positioned in the chamber of a FTIR spectropolarimeter to measure the Mueller matrix of the sample placed on the surface of the ATR.

Fig. 2
Fig. 2

Real part of the refractive index for gold-plated mirrors. The solid curve shows the values we measured for the mirrors used on our ATR along with the values determined by Palik (triangles) polished gold in the infrared.

Fig. 3
Fig. 3

Imaginary part of the refractive index for gold-plated mirrors. The solid curve shows the values we measured for the mirrors used on our ATR along with the values determined by Palik (triangles) for polished gold in the infrared.

Fig. 4
Fig. 4

Real part of the refractive index of distilled water measured with our ATR device using the UAH spectropolarimeter as compared with data compiled by Hale and Querry (triangles). The measured data consist of 128 FTIR scans for each of the 60 polarization states. The sinusoidal behavior at short wavelengths is a result of the misalignment of the cadmium sulfide and cadmium selenide plates of the achromatic retarder.

Fig. 5
Fig. 5

Imaginary part of the refractive index of distilled water measured with our ATR device using the UAH spectropolarimeter as compared with data compiled by Hale and Querry (triangles). The measured data consist of 128 FTIR scans for each of the 60 polarization states. The sinusoidal behavior at short wavelengths is a result of the misalignment of the CdS and CdSe plates of the achromatic retarder.

Fig. 6
Fig. 6

Change in the calculated value of the real part of the refractive index, when we use a baseline of 1.3 for the real part of the refractive index, as the measured value of all the Mueller matrix elements increases by 10%.

Fig. 7
Fig. 7

Change in the calculated value of the imaginary part of the refractive index, when we use a baseline of 0.015 for the imaginary part of refractive index, as the measured value of all the Mueller matrix elements increases by 10%.

Equations (60)

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

S1=EsEs*+EpEp*, S2=EsEs*-EpEp*, S3=EsEp*+EpEs*, S4=iEsEp*-EpEs*,
S=MS.
Mmeasured=MmirrorMprism faceMprismsample×Mprism faceMmirror.
Rs=cos θ-η cos ϕcos θ+η cos ϕ,
Rp=η cos θ-cos ϕη cos θ+cos ϕ,
η=nˆ2nˆ1,
D=η cos ϕ+cos θη* cos ϕ*+cos θ×cos ϕ+η cos θcos ϕ*+η* cos θ,
m13=m14=m23=m24=m31=m32=m41=m42=0,
m11=m22=η2-1η*2-1cos ϕ cos ϕ* cos2 θ+ηη*cos2 θ-cos2 ϕcos2 θ-cos2 ϕ*,
m12=m21=ηη*2-1cos2 ϕ-cos2 θcos ϕ*+η*η2-1cos2 ϕ*-cos2 θcos ϕcos θ,
m33=m44=-η2-1η*2-1cos ϕ* cos ϕ cos2 θ+ηη*cos2 θ-cos2 ϕ*cos2 θ-cos2 ϕ,
m34=-m43=-ηη*2-1cos2 ϕ-cos2 θcos ϕ*+η*η2-1cos2 ϕ*-cos2 θcos ϕcos θ.
M=1Dm11m1200m21m220000m33m3400m43m44.
m11TIR=m22TIR=1,
m12TIR=m21TIR=0,
m33TIR=m44TIR=n12+2n22-2n12-n22cos 2θ+n12 cos 4θ2n12-n22-n12+n22cos 2θ,
m34TIR=-m43TIR=4n1 cos θ sin2 θn12 sin2 θ-n221/2n12-n22-n12+n22cos 2θ.
MTIR=1000010000m33TIRm34TIR00m43TIRm44TIR.
nZnSe=4.2980149λ2λ2-0.19206302+0.62776557λ2λ2-0.378782602+2.8955633λ2λ2-46.9945952+11/2.
TZnSe=1-nZnSe-12nZnSe+12.
ηˆ=n+ik.
Mgold=1Dgm11gm12g00m21gm22g0000m33gm34g00m43gm44g,
m11g=n2+k2+1-4n2+n2+k22+n2+k2cos2 θcos2 θ,
m12g=2nn2+k2-1cos θ sin2 θ,
m33g=n2+k2-1+4k2+n2+k22-n2+k2cos2 θcos2 θ,
m34g=-2kn2+k2+1cos θ sin2 θ,
Dg=cos2 θ+n2+k2+2n cos θ1+n2 cos2 θ+k2 cos2 θ+2n cos θ.
t2Dg2m11g2+m12g22m11gm12g02m11gm12gm11g2+m12g2000m33TIRm33g2-m34g2-2m33gm34gm34TIR00m34TIRm33g2-m34g2+2m33gm34gm33TIR00-m34TIRm33g2-m34g2-2m33gm34gm33TIRm33TIRm33g2-m34g2-2m33gm34gm34TIR.
12m11gm12gm11g2+m12g202m11gm12gm11g2+m12g21000m33TIRm33g2-m34g2-2m33gm34gm34TIRm11g2+m12g200m34TIRm33g2-m34g2+2m33gm34gm33TIRm11g2+m12g200-m34TIRm33g2-m34g2-2m33gm34gm33TIRm11g2+m12g2m33TIRm33g2-m34g2-2m33gm34gm34TIRm11g2+m12g2.
m11=m22;  m12=m21;  m33=m44; m34=-m43.
m12=2m11gm12gm11g2+m12g2.
ξ=m33TIRm34-m34TIRm332m33TIR2+m34TIR2=m33gm34g.
m12=2ηnk+1-4n2+ηnk2+ηnk cos2 θcos2 θ2nηnk-1cos θ sin2 θ2ηnk+1-4n2+ηnk2+ηnk cos2 θcos2 θ2+2nηnk-1cos θ sin2 θ2,
ξ=ηnk-1+4k2+ηnk2-ηnk cos2 θcos2 θ-2kηnk+1cos θ sin2 θηnk+1-4k2+ηnk2+ηnk cos2 θcos2θ2+2nηnk-1cos θ sin2 θ2,
ηnk=n2+k2.
kmirror=7.444+21.47λ-1+1.449λ+0.6608λ2-0.03256λ3,
nmirror=-0.6651+0.2456λ2-0.01035λ3.
MATR=1DATRm11ATRm12ATR00m21ATRm22ATR0000m33ATRm34ATR00m43ATRm44ATR.
nˆ2=n2+ik2,
cos ϕ=n22+2in2k2-k22-n12 sin2 θ1/2n2+ik2.
m11ATR=m22ATR=n22+k222n12 cos4 θ+A4n12+A2n14 cos2 θ+4A2k2n2n12 cos2 θ sin 2α+4A2k22-n22n12 cos2 θ sin2 α+A2n22+k222 cos2 θ+2A4n12 cos2 α sin2 α,
m12ATR=m21ATR=2A2k2n2n12 cos α+n22+k222+n12k22-n22sin αcos2 θ+A22k2n2 cos α+(k22-n22+n12sin α)cos θ,
m33ATR=m44ATR=n22+k222n12 cos4 θ+A4n12+A24k2n2n12 sin 2α-4k22-n22n12 cos2 α-n22+k222-n14cos2 θ,
m34ATR=-m43ATR=-2An1(2k2n2n12 sin α+n12n22-k22-n22+k222cos αcos2 θ+A2k22-n22+n12cos α-2k2n2 sin α)cos θ,
DATR=A2n12+4Ak2n2n1 cos α cos θ+n22+k222 cos θ-2An22-k22n1 cos αcos θ+A2+n12 cos2 θ-2An1 sin α cos θ.
m11=m22;  m12=m21;  m33=m44; m34=-m43,
m12+m212=m12ATRm11ATR,
m34+m432=m34ATRm11ATR,
m33+m442=m33ATRm11ATR,
nˆ2=n2+ik2,
cos ϕ=n22+2in2k2-k22-n12 sin2 θ1/2n2+ik2.
a+ib=A expiα=n12 sin2 θ-n22+k22-2in2k21/2.
a2+b2=a+iba-ib=A2=n12 sin2 θ-n22+k222+4n22k221/2.
a+ib2=a2-b2+2iab=n12 sin2 θ-n22+k22-2in2k2,
a2-b2=n12 sin2 θ-n22+k22,
ab=-n2k2.
a=12n12 sin2 θ-n22+k222+4n22k221/2+n12 sin2 θ-n22+k221/2,
b=12n12 sin2 θ-n22+k222+4n22k221/2-n12 sin2 θ-n22+k221/2,
α=tan-1ba.
cos ϕ=iA expiαn2+ik2=An2+ik2i cos α-sin α,

Metrics