Abstract

A design procedure is presented for a near-optimal, single-layer-coated prism beam splitter that serves as the key optical element of the division-of-amplitude photopolarimeter (DOAP). For given film and substrate refractive indices, the angle of incidence and film thickness are selected such that the ellipsometric differential phase shifts in reflection and transmission Δr and Δt differ by ±π/2, and the normalized determinant of the instrument matrix is maximized. The best results are obtained by using high-index films on low-index substrates. This is illustrated by examples of ZnS and GaP films on silica prisms in the visible and Si, Ge, and PbTe films on Irtran 1 substrates in the infrared. A 16° Si-prism DOAP beam splitter at the 1.55-μm lightwave-communications wavelength is also presented. It uses a 163-nm SiO2 coating on the entrance face to satisfy the optimum delta condition at 73° incidence, and the determinant of the instrument matrix is 78.23% of its theoretical maximum. The exit face of the Si prism is antireflection coated with a 208-nm Si3N4 film.

© 2005 Optical Society of America

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References

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  1. R. M. A. Azzam, “Ellipsometry,” in Handbook of Optics, 2nd ed., M. Bass, E. W. van Stryland, D. R. Williams, W. L. Wolfe, eds. (McGraw-Hill, New York, 1995), Vol. II, Chap. 27.
  2. R. A. Chipman, “Polarimetry,” in Handbook of Optics, 2nd ed., M. Bass, E. W. van Stryland, D. R. Williams, W. L. Wolfe, eds. (McGraw-Hill, New York, 1995), Vol. II, Chap. 22.
  3. P. S. Hauge, “Recent developments in instrumentation in ellipsometry,” Surf. Sci. 96, 108–140 (1980).
    [CrossRef]
  4. R. M. A. Azzam, “Division-of-amplitude photopolarimeter (DOAP) for the simultaneous measurement of all four Stokes parameters of light,” Opt. Acta 29, 685–689 (1982).
    [CrossRef]
  5. R. M. A. Azzam, “Beam splitters for the division-of-amplitude photopolarimeter,” Opt. Acta 32, 1407–1412 (1985).
    [CrossRef]
  6. R. M. A. Azzam, A. G. Lopez, “Accurate calibration of the four-detector photopolarimeter with imperfect polarizing optical elements,” J. Opt. Soc. Am. A 6, 1513–1521 (1989).
    [CrossRef]
  7. K. Brudzewski, “Static Stokes ellipsometer: general analysis,” J. Mod. Opt. 38, 889–896 (1991).
    [CrossRef]
  8. S. Krishnan, “Calibration, properties, and applications of the division-of-amplitude photopolarimeter at 632.8 and 1523 nm,” J. Opt. Soc. Am. A 9, 1615–1622 (1992).
    [CrossRef]
  9. F. Delplancke, “Automated high-speed Mueller matrix scatterometer,” Appl. Opt. 36, 5388–5395 (1997).
    [CrossRef] [PubMed]
  10. E. Compain, B. Drevillon, “Broadband division-of-amplitude polarimeter based on uncoated prisms,” Appl. Opt. 37, 5938–5946 (1998).
    [CrossRef]
  11. R. M. A. Azzam, “Recent developments of division-of-amplitude photopolarimeters,” in Proceedings of the International Symposium on Polarization Analysis and Applications to Device Technology, T. Yoshizawa, H. Yokota, eds., Proc. SPIE2873, 1–4 (1996).
    [CrossRef]
  12. R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1987).
  13. R. M. A. Azzam, A. De, “Optimal beam splitters for the division-of-amplitude photopolarimeter,” J. Opt. Soc. Am. A 20, 955–958 (2003).
    [CrossRef]
  14. W. L. Wolfe, “Properties of optical materials,” in Handbook of Optics, W. D. Driscoll, W. Vaughan, eds. (McGraw-Hill, New York, 1978), Sec. 7.
  15. W. J. Tropf, M. T. Thomas, T. J. Harris, “Properties of crystals and glasses,” in Handbook of Optics, M. Bass, E. W. van Stryland, D. R. Williams, W. L. Wolfe, eds. (McGraw-Hill, New York, 1995), Vol. II, Chap. 33.
  16. G. Eisenstein, L. W. Stulz, “High quality antireflection coatings on laser facets by sputtered silicon nitride,” Appl. Opt. 23, 161–164 (1984).
    [CrossRef] [PubMed]

2003 (1)

1998 (1)

1997 (1)

1992 (1)

1991 (1)

K. Brudzewski, “Static Stokes ellipsometer: general analysis,” J. Mod. Opt. 38, 889–896 (1991).
[CrossRef]

1989 (1)

1985 (1)

R. M. A. Azzam, “Beam splitters for the division-of-amplitude photopolarimeter,” Opt. Acta 32, 1407–1412 (1985).
[CrossRef]

1984 (1)

1982 (1)

R. M. A. Azzam, “Division-of-amplitude photopolarimeter (DOAP) for the simultaneous measurement of all four Stokes parameters of light,” Opt. Acta 29, 685–689 (1982).
[CrossRef]

1980 (1)

P. S. Hauge, “Recent developments in instrumentation in ellipsometry,” Surf. Sci. 96, 108–140 (1980).
[CrossRef]

Azzam, R. M. A.

R. M. A. Azzam, A. De, “Optimal beam splitters for the division-of-amplitude photopolarimeter,” J. Opt. Soc. Am. A 20, 955–958 (2003).
[CrossRef]

R. M. A. Azzam, A. G. Lopez, “Accurate calibration of the four-detector photopolarimeter with imperfect polarizing optical elements,” J. Opt. Soc. Am. A 6, 1513–1521 (1989).
[CrossRef]

R. M. A. Azzam, “Beam splitters for the division-of-amplitude photopolarimeter,” Opt. Acta 32, 1407–1412 (1985).
[CrossRef]

R. M. A. Azzam, “Division-of-amplitude photopolarimeter (DOAP) for the simultaneous measurement of all four Stokes parameters of light,” Opt. Acta 29, 685–689 (1982).
[CrossRef]

R. M. A. Azzam, “Ellipsometry,” in Handbook of Optics, 2nd ed., M. Bass, E. W. van Stryland, D. R. Williams, W. L. Wolfe, eds. (McGraw-Hill, New York, 1995), Vol. II, Chap. 27.

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1987).

R. M. A. Azzam, “Recent developments of division-of-amplitude photopolarimeters,” in Proceedings of the International Symposium on Polarization Analysis and Applications to Device Technology, T. Yoshizawa, H. Yokota, eds., Proc. SPIE2873, 1–4 (1996).
[CrossRef]

Bashara, N. M.

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1987).

Brudzewski, K.

K. Brudzewski, “Static Stokes ellipsometer: general analysis,” J. Mod. Opt. 38, 889–896 (1991).
[CrossRef]

Chipman, R. A.

R. A. Chipman, “Polarimetry,” in Handbook of Optics, 2nd ed., M. Bass, E. W. van Stryland, D. R. Williams, W. L. Wolfe, eds. (McGraw-Hill, New York, 1995), Vol. II, Chap. 22.

Compain, E.

De, A.

Delplancke, F.

Drevillon, B.

Eisenstein, G.

Harris, T. J.

W. J. Tropf, M. T. Thomas, T. J. Harris, “Properties of crystals and glasses,” in Handbook of Optics, M. Bass, E. W. van Stryland, D. R. Williams, W. L. Wolfe, eds. (McGraw-Hill, New York, 1995), Vol. II, Chap. 33.

Hauge, P. S.

P. S. Hauge, “Recent developments in instrumentation in ellipsometry,” Surf. Sci. 96, 108–140 (1980).
[CrossRef]

Krishnan, S.

Lopez, A. G.

Stulz, L. W.

Thomas, M. T.

W. J. Tropf, M. T. Thomas, T. J. Harris, “Properties of crystals and glasses,” in Handbook of Optics, M. Bass, E. W. van Stryland, D. R. Williams, W. L. Wolfe, eds. (McGraw-Hill, New York, 1995), Vol. II, Chap. 33.

Tropf, W. J.

W. J. Tropf, M. T. Thomas, T. J. Harris, “Properties of crystals and glasses,” in Handbook of Optics, M. Bass, E. W. van Stryland, D. R. Williams, W. L. Wolfe, eds. (McGraw-Hill, New York, 1995), Vol. II, Chap. 33.

Wolfe, W. L.

W. L. Wolfe, “Properties of optical materials,” in Handbook of Optics, W. D. Driscoll, W. Vaughan, eds. (McGraw-Hill, New York, 1978), Sec. 7.

Appl. Opt. (3)

J. Mod. Opt. (1)

K. Brudzewski, “Static Stokes ellipsometer: general analysis,” J. Mod. Opt. 38, 889–896 (1991).
[CrossRef]

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

Opt. Acta (2)

R. M. A. Azzam, “Division-of-amplitude photopolarimeter (DOAP) for the simultaneous measurement of all four Stokes parameters of light,” Opt. Acta 29, 685–689 (1982).
[CrossRef]

R. M. A. Azzam, “Beam splitters for the division-of-amplitude photopolarimeter,” Opt. Acta 32, 1407–1412 (1985).
[CrossRef]

Surf. Sci. (1)

P. S. Hauge, “Recent developments in instrumentation in ellipsometry,” Surf. Sci. 96, 108–140 (1980).
[CrossRef]

Other (6)

R. M. A. Azzam, “Ellipsometry,” in Handbook of Optics, 2nd ed., M. Bass, E. W. van Stryland, D. R. Williams, W. L. Wolfe, eds. (McGraw-Hill, New York, 1995), Vol. II, Chap. 27.

R. A. Chipman, “Polarimetry,” in Handbook of Optics, 2nd ed., M. Bass, E. W. van Stryland, D. R. Williams, W. L. Wolfe, eds. (McGraw-Hill, New York, 1995), Vol. II, Chap. 22.

R. M. A. Azzam, “Recent developments of division-of-amplitude photopolarimeters,” in Proceedings of the International Symposium on Polarization Analysis and Applications to Device Technology, T. Yoshizawa, H. Yokota, eds., Proc. SPIE2873, 1–4 (1996).
[CrossRef]

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1987).

W. L. Wolfe, “Properties of optical materials,” in Handbook of Optics, W. D. Driscoll, W. Vaughan, eds. (McGraw-Hill, New York, 1978), Sec. 7.

W. J. Tropf, M. T. Thomas, T. J. Harris, “Properties of crystals and glasses,” in Handbook of Optics, M. Bass, E. W. van Stryland, D. R. Williams, W. L. Wolfe, eds. (McGraw-Hill, New York, 1995), Vol. II, Chap. 33.

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Figures (8)

Fig. 1
Fig. 1

Division-of-amplitude photopolarimeter (DOAP). BS, beam splitter to be designed; WP1, WP2, Wollaston prisms; D0, D1, D2 D3, linear photodetectors that generate output electrical signals i0, i1, i2, and i3, respectively; p and s linear polarization directions parallel and perpendicular to the plane of incidence at BS, respectively.

Fig. 2
Fig. 2

Prism BS for DOAP. The incident collimated light beam i, whose polarization is to be measured, strikes the coated, beam-splitting entrance face of the prism at an angle ϕ. The prism angle α equals the angle of refraction in the prism, so that the transmitted beam is normal to the (antireflection-coated) exit face of the prism. The angular separation between the reflected (r) and the transmitted (t) beams is π − (ϕ + α).

Fig. 3
Fig. 3

Normalized determinant, |det A|norm, and unpolarized-light reflectance R plotted as functions of the angle of incidence ϕ for BSs that consist of a Ge film (n1 = 4.00) on an Irtran 1 substrate (n2 = 1.30) at wavelength λ = 6.75 μm such that the phase condition of Eq. (6) is exactly satisfied at each angle. The best design is obtained at ϕ = 78°, where (|det A|norm)max = 0.9583, and the remaining optimum conditions of Eqs. (7) and (8) are nearly satisfied.

Fig. 4
Fig. 4

Normalized determinant, |det A|norm as a function of angle of incidence ϕ for a DOAP BS that consists of a 131.3-nm Ge thin film on an Irtran 1 substrate at wavelength λ = 6.75 μm. Note that |det A|norm remains near its maximum for 76 ≤ ϕ ≤ 80°.

Fig. 5
Fig. 5

Normalized determinant |det A|norm as a function of wavelength λ for a DOAP BS that consists of a 131.3-nm Ge thin film on an Irtran 1 substrate at an angle of incidence ϕ = 78°. Note that |det A|norm increases as λ increases in the spectral range 5.75 ≤ λ ≤ 7.75 μm.

Fig. 6
Fig. 6

Normalized determinant |det A|norm as a function of ϕ for five different film–substrate systems: (a) ZnS (2.352)/SiO2 (1.457) at λ = 633 nm, (b) GaP (3.308)/SiO2 (1.457) at λ = 633 nm, (c) Si (3.432)/SiO2 (1.419) at λ = 3 μm, (d) Ge (4.00)/Irtran 1 (1.30) at λ = 6.75 μm, and (e) PbTe (5.655)/Irtran 1 (1.227) at λ = 9 μm. The refractive indices of all materials are obtained from Refs. 14 and 15. As the film refractive index n1 increases, the peak of |det A|norm increases, and its location is shifted toward higher angles.

Fig. 7
Fig. 7

Normalized determinant, |det A|norm as a function of angle of incidence ϕ for a DOAP BS that consists of a 163-nm SiO2 thin film on a Si substrate at wavelength λ = 1.55 μm. |det A|norm increases monotonically with ϕ in the range 71 ≤ ϕ ≤ 75°.

Fig. 8
Fig. 8

Normalized determinant, |det A|norm as a function of wavelength λ for a DOAP BS that consists of a 163-nm SiO2 thin film on a Si substrate at an angle of incidence ϕ = 73°. |det A|norm increases monotonically with λ in the spectral range 1.30 ≤ λ ≤ 1.55 μm.

Equations (36)

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S = [ S 0 S 1 S 2 S 3 ] t ,
I = [ i 0 i 1 i 2 i 3 ] t ,
I = AS .
S = A - 1 I ,
det A = ( R T ) 2 sin 2 ψ r sin 2 ψ t ( cos 2 ψ r - cos 2 ψ t ) × sin ( Δ r - Δ t ) .
Δ r - Δ t = ± π / 2 ,
R = T = 0.5 ,
( ψ r , ψ t ) = ( 1 2 arccos ( ± 1 / 3 ) , 90 ° - ψ r ) = ( 27.368 ° , 62.632 ° ) or ( 62.632 ° , 27.368 ° ) .
det A max = 3 / 36 = 0.0481.
det A max = det A / det A max ,
R ν = ( r 01 ν + r 12 ν X ) / ( 1 + r 01 ν r 12 ν X ) , T ν = ( t 01 ν t 12 ν X 1 / 2 ) / ( 1 + r 01 ν r 12 ν X ) .
X = exp ( - j θ ) ,
θ = 2 π d / D , D = ( λ / 2 ) ( n 1 2 - sin 2 ϕ ) - 1 / 2 .
ρ r = R p / R s = tan ψ r exp ( j Δ r ) , ρ t = T p / T s = tan ψ t exp ( j Δ t ) .
Γ = ρ r / ρ t = Γ exp ( j γ ) , γ = Δ r - Δ t .
Γ = b ( 1 + B p X ) / ( 1 + B s X ) ,
b = ( r 01 p / r 01 s ) ( t 01 s / t 01 p ) ( t 12 s / t 12 p ) , B p = ( r 12 p / r 01 p ) , B s = ( r 12 s / r 01 s ) .
tan γ = ( B p - B s ) sin θ / [ ( 1 + B p B s ) + ( B p + B s ) cos θ ] .
cos θ = - ( 1 + B p B s ) / ( B p + B s ) .
cos θ = f ( ϕ , n 1 , n 2 ) = - ( r 01 p r 01 s + r 12 p r 12 s ) / ( r 01 p r 12 s + r 01 s r 12 p ) .
ϕ 1 = ϕ B 02 = arctan n 2 .
r 01 p = - r 12 p .
cos θ = + 1 ;
r 01 p = r 12 p .
cos θ = - 1 ;
n 1 4 cos ϕ 2 ( n 2 2 - sin 2 ϕ 2 ) 1 / 2 = n 2 2 ( n 1 2 - sin 2 ϕ 2 ) .
R p = 0 ,             ρ r = 0 ,             ψ r = 0 ,             det A = 0.
α = 48.80 ° , θ = 54.3027 ° , Δ r - Δ t = - 90 ° , R = 0.4916 , T = 0.5084 , ψ r = 23.537 ° , ψ t = 65.599 ° , det A norm = 0.9583.
d / D = θ / 2 π = 0.1508.
d = 131.26 nm .
r 01 s = r 12 s .
cos θ = - 1 ;
R s = 0 ,             ρ r = ,             ψ r = 90 ° ,             det A = 0
tan 2 ϕ 1 = ( n 2 2 - n 1 4 ) / ( n 1 2 - 1 ) 2 .
ϕ 2 = ϕ B 02 = arctan n 2 .
α = 15.96 ° , θ = 81.817 ° , d / D = 0.2273 , Δ r - Δ t = 90 ° , R = 0.363 , T = 0.637 , ψ r = 26.955 ° , ψ t = 54.809 ° .

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