Abstract

Mueller matrices for normal transmission of light through a birefringent slab are formulated to analyze retardation and depolarization. A finite wave band, wedge slab, and microroughness may cause a spread in retardance, which in turn produces depolarization. The spectra of depolarization, cross-polarized transmittance, and retardance by rotating-analyzer ellipsometry are simulated for the quasi-monochromatic effect with a finite bandwidth. These spectra agree excellently with the measured spectra for a sapphire slab. The depolarization spectrum simulated for the wedge effect fits the measured spectrum in the long-wave region but is too small in the short-wave region. The depolarization simulated for incoherent multiple reflections demonstrates the oscillating structure, which is small compared with the measured depolarization. The finite bandwidth effect contributes more than the other effects to the measured depolarization of a sapphire slab.

© 2000 Optical Society of America

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References

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  1. N. G. Theofanous, A. T. Arapoyianni, “Effect of multiple reflections on retardation-based electro-optic measurements,” J. Opt. Soc. Am. A 8, 1746–1754 (1991).
    [CrossRef]
  2. S.-M. F. Nee, “Birefringence characterization using transmission ellipsometry,” in Polarization Analysis and Measurement, D. H. Goldstein, R. A. Chipman, eds., Proc. SPIE1746, 269–280 (1992).
    [CrossRef]
  3. S.-M. F. Nee, T. Cole, “Effects of depolarization of optical components on null ellipsometry,” Thin Solid Films 313–314, 90–96 (1998).
    [CrossRef]
  4. S.-M. F. Nee, “Effects of incoherent scattering on ellipsometry,” in Polarization Analysis and Measurement, D. H. Goldstein, R. A. Chipman, eds., Proc. SPIE1746, 119–127 (1992).
    [CrossRef]
  5. S.-M. F. Nee, “Effects of near-specular scattering on polarimetry,” in Polarization Analysis and Measurement II, D. H. Goldstein, David B. Chenault, eds., Proc. SPIE2265, 304–313 (1994).
    [CrossRef]
  6. S.-M. F. Nee, “Polarization of specular reflection and near-specular scattering by a rough surface,” Appl. Opt. 35, 3570–3582 (1996).
    [CrossRef]
  7. S. F. Nee, C. Yoo, T. Cole, D. Burge, “Characterization of imperfect polarizers under imperfect conditions,” Appl. Opt. 37, 57–64 (1998).
    [CrossRef]
  8. S.-M. F. Nee, “Error analysis of null ellipsometry with depolarization,” Appl. Opt. 38, 5388–5398 (1999).
    [CrossRef]
  9. S.-M. F. Nee, “Polarization measurement,” in The Measurement, Instrumentation, and Sensors Handbook, J. G. Webster, ed. (CRC Press, Boca Raton, Fl., 1999), Chap. 60.
  10. G. E. Jellison, D. H. Lowndes, “Time-resolved ellipsometry,” Appl. Opt. 24, 2948–2955 (1985).
    [CrossRef] [PubMed]
  11. G. E. Jellison, J. W. McCamy, “Sample depolarization effects from thin films of ZnS on GaAs as measured by spectroscopic ellipsometry,” Appl. Phys. Lett. 61, 512–514 (1992).
    [CrossRef]
  12. U. Rossow, “Depolarization/mixed polarization corrections of ellipsometry spectra,” Thin Solid Films 313–314, 97–101 (1998).
    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
  15. M. Kildemo, P. Bulkin, B. Drevillon, O. Hunderi, “Real-time control by multiwavelength ellipsometry of plasma-deposited multilayers on glass by use of an incoherent-reflection model,” Appl. Opt. 36, 6352–6359 (1997).
    [CrossRef]
  16. U. Richter, “Application of the degree of polarization to film thickness gradients,” Thin Solid Films 313–314, 102–107 (1998).
    [CrossRef]
  17. M. Kildemo, R. Ossikovski, M. Stchakovsky, “Measurement of the absorption edge of thick transparent substrates using the incoherent reflection model and spectroscopic UV-visible–near-IR ellipsometry,” Thin Solid Films 313–314, 108–113 (1998).
    [CrossRef]
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    [CrossRef] [PubMed]
  21. F. Gervais, “Aluminum oxide,” in Handbook of Optical Constants of Solids II, E. D. Palik, ed., (Academic, Boston, 1991), pp. 761–775.

1999 (1)

1998 (5)

U. Rossow, “Depolarization/mixed polarization corrections of ellipsometry spectra,” Thin Solid Films 313–314, 97–101 (1998).
[CrossRef]

U. Richter, “Application of the degree of polarization to film thickness gradients,” Thin Solid Films 313–314, 102–107 (1998).
[CrossRef]

M. Kildemo, R. Ossikovski, M. Stchakovsky, “Measurement of the absorption edge of thick transparent substrates using the incoherent reflection model and spectroscopic UV-visible–near-IR ellipsometry,” Thin Solid Films 313–314, 108–113 (1998).
[CrossRef]

S.-M. F. Nee, T. Cole, “Effects of depolarization of optical components on null ellipsometry,” Thin Solid Films 313–314, 90–96 (1998).
[CrossRef]

S. F. Nee, C. Yoo, T. Cole, D. Burge, “Characterization of imperfect polarizers under imperfect conditions,” Appl. Opt. 37, 57–64 (1998).
[CrossRef]

1997 (2)

1996 (1)

1992 (3)

1991 (1)

1987 (1)

1985 (1)

1981 (1)

Arapoyianni, A. T.

Bulkin, P.

Burge, D.

S. F. Nee, C. Yoo, T. Cole, D. Burge, “Characterization of imperfect polarizers under imperfect conditions,” Appl. Opt. 37, 57–64 (1998).
[CrossRef]

Cole, T.

S. F. Nee, C. Yoo, T. Cole, D. Burge, “Characterization of imperfect polarizers under imperfect conditions,” Appl. Opt. 37, 57–64 (1998).
[CrossRef]

S.-M. F. Nee, T. Cole, “Effects of depolarization of optical components on null ellipsometry,” Thin Solid Films 313–314, 90–96 (1998).
[CrossRef]

Drevillon, B.

Forcht, K.

Fry, E. S.

Gervais, F.

F. Gervais, “Aluminum oxide,” in Handbook of Optical Constants of Solids II, E. D. Palik, ed., (Academic, Boston, 1991), pp. 761–775.

Gombert, A.

Graf, W.

Hunderi, O.

Jellison, G. E.

G. E. Jellison, J. W. McCamy, “Sample depolarization effects from thin films of ZnS on GaAs as measured by spectroscopic ellipsometry,” Appl. Phys. Lett. 61, 512–514 (1992).
[CrossRef]

G. E. Jellison, D. H. Lowndes, “Time-resolved ellipsometry,” Appl. Opt. 24, 2948–2955 (1985).
[CrossRef] [PubMed]

Joerger, R.

Kattawar, G. W.

Kildemo, M.

M. Kildemo, R. Ossikovski, M. Stchakovsky, “Measurement of the absorption edge of thick transparent substrates using the incoherent reflection model and spectroscopic UV-visible–near-IR ellipsometry,” Thin Solid Films 313–314, 108–113 (1998).
[CrossRef]

M. Kildemo, P. Bulkin, B. Drevillon, O. Hunderi, “Real-time control by multiwavelength ellipsometry of plasma-deposited multilayers on glass by use of an incoherent-reflection model,” Appl. Opt. 36, 6352–6359 (1997).
[CrossRef]

Kim, K.

Köhl, M.

Kostinski, A. B.

Lowndes, D. H.

Mandel, L.

McCamy, J. W.

G. E. Jellison, J. W. McCamy, “Sample depolarization effects from thin films of ZnS on GaAs as measured by spectroscopic ellipsometry,” Appl. Phys. Lett. 61, 512–514 (1992).
[CrossRef]

Nee, S. F.

S. F. Nee, C. Yoo, T. Cole, D. Burge, “Characterization of imperfect polarizers under imperfect conditions,” Appl. Opt. 37, 57–64 (1998).
[CrossRef]

Nee, S.-M. F.

S.-M. F. Nee, “Error analysis of null ellipsometry with depolarization,” Appl. Opt. 38, 5388–5398 (1999).
[CrossRef]

S.-M. F. Nee, T. Cole, “Effects of depolarization of optical components on null ellipsometry,” Thin Solid Films 313–314, 90–96 (1998).
[CrossRef]

S.-M. F. Nee, “Polarization of specular reflection and near-specular scattering by a rough surface,” Appl. Opt. 35, 3570–3582 (1996).
[CrossRef]

S.-M. F. Nee, “Polarization measurement,” in The Measurement, Instrumentation, and Sensors Handbook, J. G. Webster, ed. (CRC Press, Boca Raton, Fl., 1999), Chap. 60.

S.-M. F. Nee, “Effects of near-specular scattering on polarimetry,” in Polarization Analysis and Measurement II, D. H. Goldstein, David B. Chenault, eds., Proc. SPIE2265, 304–313 (1994).
[CrossRef]

S.-M. F. Nee, “Effects of incoherent scattering on ellipsometry,” in Polarization Analysis and Measurement, D. H. Goldstein, R. A. Chipman, eds., Proc. SPIE1746, 119–127 (1992).
[CrossRef]

S.-M. F. Nee, “Birefringence characterization using transmission ellipsometry,” in Polarization Analysis and Measurement, D. H. Goldstein, R. A. Chipman, eds., Proc. SPIE1746, 269–280 (1992).
[CrossRef]

Ossikovski, R.

M. Kildemo, R. Ossikovski, M. Stchakovsky, “Measurement of the absorption edge of thick transparent substrates using the incoherent reflection model and spectroscopic UV-visible–near-IR ellipsometry,” Thin Solid Films 313–314, 108–113 (1998).
[CrossRef]

Richter, U.

U. Richter, “Application of the degree of polarization to film thickness gradients,” Thin Solid Films 313–314, 102–107 (1998).
[CrossRef]

Röseler, A.

Rossow, U.

U. Rossow, “Depolarization/mixed polarization corrections of ellipsometry spectra,” Thin Solid Films 313–314, 97–101 (1998).
[CrossRef]

Stchakovsky, M.

M. Kildemo, R. Ossikovski, M. Stchakovsky, “Measurement of the absorption edge of thick transparent substrates using the incoherent reflection model and spectroscopic UV-visible–near-IR ellipsometry,” Thin Solid Films 313–314, 108–113 (1998).
[CrossRef]

Theofanous, N. G.

Wolf, E.

Yoo, C.

S. F. Nee, C. Yoo, T. Cole, D. Burge, “Characterization of imperfect polarizers under imperfect conditions,” Appl. Opt. 37, 57–64 (1998).
[CrossRef]

Appl. Opt. (8)

Appl. Phys. Lett. (1)

G. E. Jellison, J. W. McCamy, “Sample depolarization effects from thin films of ZnS on GaAs as measured by spectroscopic ellipsometry,” Appl. Phys. Lett. 61, 512–514 (1992).
[CrossRef]

J. Opt. Soc. Am. A (3)

Thin Solid Films (4)

U. Rossow, “Depolarization/mixed polarization corrections of ellipsometry spectra,” Thin Solid Films 313–314, 97–101 (1998).
[CrossRef]

U. Richter, “Application of the degree of polarization to film thickness gradients,” Thin Solid Films 313–314, 102–107 (1998).
[CrossRef]

M. Kildemo, R. Ossikovski, M. Stchakovsky, “Measurement of the absorption edge of thick transparent substrates using the incoherent reflection model and spectroscopic UV-visible–near-IR ellipsometry,” Thin Solid Films 313–314, 108–113 (1998).
[CrossRef]

S.-M. F. Nee, T. Cole, “Effects of depolarization of optical components on null ellipsometry,” Thin Solid Films 313–314, 90–96 (1998).
[CrossRef]

Other (5)

S.-M. F. Nee, “Effects of incoherent scattering on ellipsometry,” in Polarization Analysis and Measurement, D. H. Goldstein, R. A. Chipman, eds., Proc. SPIE1746, 119–127 (1992).
[CrossRef]

S.-M. F. Nee, “Effects of near-specular scattering on polarimetry,” in Polarization Analysis and Measurement II, D. H. Goldstein, David B. Chenault, eds., Proc. SPIE2265, 304–313 (1994).
[CrossRef]

S.-M. F. Nee, “Polarization measurement,” in The Measurement, Instrumentation, and Sensors Handbook, J. G. Webster, ed. (CRC Press, Boca Raton, Fl., 1999), Chap. 60.

F. Gervais, “Aluminum oxide,” in Handbook of Optical Constants of Solids II, E. D. Palik, ed., (Academic, Boston, 1991), pp. 761–775.

S.-M. F. Nee, “Birefringence characterization using transmission ellipsometry,” in Polarization Analysis and Measurement, D. H. Goldstein, R. A. Chipman, eds., Proc. SPIE1746, 269–280 (1992).
[CrossRef]

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Figures (6)

Fig. 1
Fig. 1

Depolarization spectra for 0.84-mm-thick sapphire slab measured using null ellipsometry (NE), RAE, and cross-polarized transmittance (T).

Fig. 2
Fig. 2

Retardance spectra for sapphire slab. Discrete symbol, data measured by RAE; solid curve, simulated retardance for quasi-monochromatic effect with δλ=18 nm.

Fig. 3
Fig. 3

Number of harmonics m versus 1/λ for occurrence of λ at Δ=90° and extrema of measured curve in Fig. 2.

Fig. 4
Fig. 4

Depolarization spectrum simulated for incoherent multiple reflections. This spectrum does not look like those of Fig. 1.

Fig. 5
Fig. 5

Depolarization spectra for 0.84-mm-thick sapphire slab. Solid curve is simulated for quasi-monochromatic effect with δ λ=18 nm, dashed curve is simulated for wedge effect with δd/d=0.04, and Δ represents measured RAE data.

Fig. 6
Fig. 6

Spectra for cross-polarized transmittance. Solid line is simulated for quasi-monochromatic effect with δ λ=16.5 nm, and discrete symbols represent measured transmittance in different regions.

Equations (61)

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M=T1Px00Px1-2Dv0000PyPz00-PzPy,
Px=-Pcos 2Ψ,
Py=Psin 2Ψ cos Δ,
Pz=Psin 2Ψ sin Δ.
P=(Px2+Py2+Pz2)1/21,
D1-P=Du+Dv,
i, j=14Mij24 M112.
M11=M22=(tx tx*+ty ty*)/2,
M12=M21=(tx tx*-ty ty*)/2,
M33=M44=(tx ty*+ty tx*)/2,
M34=-M43=(tx ty*-ty tx*)/2i.
tx=Txexp(iβx),ty=Tyexp(iβy),
βx,y=2πdnx,y/λ, nx,y=n±n/2,
Tx,y=4(n±n/2)(n+1±n/2)2=To±T,
To=4n(n+1)2,T-2n(n-1)(n+1)3.
T11=T22=(Tx2+Ty2)/216n2/(n+1)4,
T12=T21=(Tx2-Ty2)/2-16nn(n-1)/(n+1)5,
T33=T44=TxTycos Δ16n2cos Δ/(n+1)4,
T34=-T43=TxTysin Δ16n2sin Δ/(n+1)4,
Δ=βx-βy=2πnd/λ,
β=(βx+βy)/2=2πnd/λ.
T=T1a00a10000b cos Δb sin Δ00-b sin Δb cos Δ;
T=T1116n2/(n+1)4,
a=Px=T12/T11-n(n-1)/n(n+1),
b=(T332+T342)1/2T11=2TxTyTx2+Ty21-n2(n-1)22n2(n+1)2.
δΔ=2πndλδdd+δnn-δλλ.
T
=T1a00a10000b sinc(δΔ/2)cos Δb sinc(δΔ/2)sin Δ00-b sinc(δΔ/2)sin Δb sinc(δΔ/2)cos Δ.
 
P=[b2sinc2(δΔ/2)+a2]1/2=sinc(δΔ/2)+a2[1-sinc2(δΔ/2)]2 sinc(δΔ/2),
D=1-sinc(δΔ/2).
tx=Txexp(iβx)1-Rxexp(i2βx),ty=Tyexp(iβy)1-Ryexp(i2βy),
βx,y=β±Δ/2,
Rx,y=(n-1±n/2)2(n+1±n/2)2=1-Tx,y=Ro±R,
Ro=(n-1)2(n+1)2,R2n(n-1)(n+1)3.
M11=M22=To1+Ro [1+2Rocos 2βcos Δ-2R(1-2Ro)sin 2βsin Δ],
M12=M21=-2RoTo1+Ro sin 2βsin Δ-2R(1+Ro)2 [1-(1-2Ro-Ro2)cos 2βcos Δ],
M33=M44=To2(1-Ro2)cos Δ+2Rocos 2β(cos 2Δ-Ro2)-2Rsin 2βsin 2Δ1-2Ro2cos 2Δ+Ro4,
M34=-M43=To2(1+Ro2)sin Δ+2Rocos 2βsin 2Δ+2Rsin 2β(cos 2Δ-Ro2)1-2Ro2cos 2Δ+Ro4.
M11=M22=1-Ro1+Ro=2nn2+1,
M12=M21=-2R(1+Ro)2=-n(n2-1)(n2+1)2,
M33=M44=To2(1-Ro2)cos Δ1+Ro4-2Ro2cos 2Δ,
M34=-M43=To2(1+Ro2)sin Δ1+Ro4-2Ro2cos 2Δ.
tan Δ*=M34/M33=tan Δ(1+Ro2)/(1-Ro2),
Δ*Δ+Ro2sin 2Δ.
D2Ro2sin2 Δ(1-Ro2)2.
I=B(1+cos 2ωt)(1+Pxcos 2ωt±Pysin 2ωt),
cos 2Ψ=Pcos 2Ψ=-Px,
sin 2Ψcos Δ=Psin 2Ψ cos Δ=Py.
cos Δ=Pcos Δ=(1-D)cos Δ.
sin ΔdΔdλ=Psin Δ dΔdλ+cos Δ dDdλ.
D=1-P=1-|cos Δ|extrema.
m=Δ(λ)/π=0.4878+11.6077/λ+0.7388/λ2.
n(λ)=1.7453+0.00691/λ+0.003872/λ2.
δΔλ=-π(11.6077+1.4776/λ)δλ/λ2,
δΔd=π(0.4878+11.6077/λ+0.7388/λ2)δd/d.
I=B(1+Dp+Da-Psin 2Ψ cos Δ),
T*(λ)=I(P=45°, A=-45°, Sample)I(P=45°, A=45°, NoSample).
Ts(λ)=T(λ)(1-Pcos Δ).
m=Δ/π=0.52+11.6/λ+0.76/λ2,
δΔ=-π(11.6+1.52/λ)δ λ/λ2.

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