Abstract

The four-detector photopolarimeter (FDP) is analyzed for an arbitrary spatial configuration and any reflection characteristics (ri, ψi, Δi,) of the first three detectors. The instrument matrix A, which relates the output signal vector I to the input Stokes vector S by I = AS, and its determinant are derived explicitly. The essential condition that A be nonsingular (det A ≠ 0) is satisfied in general with uncoated absorbing detector surfaces, assuming that the plane of incidence (POI) is rotated between successive reflections by other than 90°. Therefore no special coatings on the detectors are required, and a thin dielectric (e.g., thermal oxide) layer would suffice. The differential reflection phase shift Δ is unrestricted for the first and third detectors and has optimum values of ±90° for the second. The optimum rotation angles of the POI are ±45° and ±135°. The optimum values of the surface parameter ψ are 27.37°, 22.5° or 67.5°, and 0 or 90° for the first, second, and third reflections, respectively. The following topics are also considered: (1) the partition of energy among detectors, (2) the effect of tilting the last detector, (3) operation of the FDP over a broadband spectral range, (4) choice of the light-beam path, and (5) calibration.

© 1988 Optical Society of America

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References

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  1. See, for example, M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1975), p. 554.
  2. R. M. A. Azzam, “Arrangement of four photodetectors for measuring the state of polarization of light,” Opt. Lett. 10, 309–311 (1985);U.S. Patent4,681,450 (July21, 1987).
    [CrossRef] [PubMed]
  3. R. M. A. Azzam, E. Masetti, I. M. Elminyawi, F. G. Grosz, “Construction, calibration, and testing of a four-detector photopolarimeter,” Rev. Sci. Instrum. 59, 84–88 (1988).
    [CrossRef]
  4. R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1987).
  5. R. M. A. Azzam, “Contours of constant principal angle and constant principal azimuth in the complex ∊ plane,” J. Opt. Soc. Am. 71, 1523–1528 (1981).
    [CrossRef]
  6. R. M. A. Azzam, “Stationary property of normal-incidence reflection from isotropic surfaces,” J. Opt. Soc. Am. 72, 1187–1189 (1982).
    [CrossRef]
  7. R. M. A. Azzam, “In-line light-saving photopolarimeter and its fiber-optic analog,” Opt. Lett. 12, 558–560 (1987).
    [CrossRef] [PubMed]
  8. A. J. Warnecke, P. J. LoPresti, “Refractive index dispersion in semiconductor-related thin films,” IBM J. Res. Dev. 17, 256–262 (1973).
    [CrossRef]
  9. D. E. Aspnes, A. A. Studna, “Dielectric functions and optical parameters of Si, Ge, GaP, GaAs, GaSb, InP, InAs, and InSb from 1.5 to 6.0 eV,” Phys. Rev. B 27, 985–1009 (1983).
    [CrossRef]
  10. R. M. A. Azzam, E. Masetti, F. G. Grosz, I. Elminyawi, “Four-detector photopolarimeter: first experimental results,” J. Opt. Soc. Am. A 4(13), P78 (1987).
  11. See, for example, Ref. 4, Sec. 1.8.
  12. P. S. Hauge, “Mueller matrix ellipsometry with imperfect compensators,” J. Opt. Soc. Am. 68, 1519–1528 (1978).
    [CrossRef]
  13. See, for example, Ref. 4, Sec. 3.9.1.

1988 (1)

R. M. A. Azzam, E. Masetti, I. M. Elminyawi, F. G. Grosz, “Construction, calibration, and testing of a four-detector photopolarimeter,” Rev. Sci. Instrum. 59, 84–88 (1988).
[CrossRef]

1987 (2)

R. M. A. Azzam, “In-line light-saving photopolarimeter and its fiber-optic analog,” Opt. Lett. 12, 558–560 (1987).
[CrossRef] [PubMed]

R. M. A. Azzam, E. Masetti, F. G. Grosz, I. Elminyawi, “Four-detector photopolarimeter: first experimental results,” J. Opt. Soc. Am. A 4(13), P78 (1987).

1985 (1)

1983 (1)

D. E. Aspnes, A. A. Studna, “Dielectric functions and optical parameters of Si, Ge, GaP, GaAs, GaSb, InP, InAs, and InSb from 1.5 to 6.0 eV,” Phys. Rev. B 27, 985–1009 (1983).
[CrossRef]

1982 (1)

1981 (1)

1978 (1)

1973 (1)

A. J. Warnecke, P. J. LoPresti, “Refractive index dispersion in semiconductor-related thin films,” IBM J. Res. Dev. 17, 256–262 (1973).
[CrossRef]

Aspnes, D. E.

D. E. Aspnes, A. A. Studna, “Dielectric functions and optical parameters of Si, Ge, GaP, GaAs, GaSb, InP, InAs, and InSb from 1.5 to 6.0 eV,” Phys. Rev. B 27, 985–1009 (1983).
[CrossRef]

Azzam, R. M. A.

Bashara, N. M.

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

Born, M.

See, for example, M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1975), p. 554.

Elminyawi, I.

R. M. A. Azzam, E. Masetti, F. G. Grosz, I. Elminyawi, “Four-detector photopolarimeter: first experimental results,” J. Opt. Soc. Am. A 4(13), P78 (1987).

Elminyawi, I. M.

R. M. A. Azzam, E. Masetti, I. M. Elminyawi, F. G. Grosz, “Construction, calibration, and testing of a four-detector photopolarimeter,” Rev. Sci. Instrum. 59, 84–88 (1988).
[CrossRef]

Grosz, F. G.

R. M. A. Azzam, E. Masetti, I. M. Elminyawi, F. G. Grosz, “Construction, calibration, and testing of a four-detector photopolarimeter,” Rev. Sci. Instrum. 59, 84–88 (1988).
[CrossRef]

R. M. A. Azzam, E. Masetti, F. G. Grosz, I. Elminyawi, “Four-detector photopolarimeter: first experimental results,” J. Opt. Soc. Am. A 4(13), P78 (1987).

Hauge, P. S.

LoPresti, P. J.

A. J. Warnecke, P. J. LoPresti, “Refractive index dispersion in semiconductor-related thin films,” IBM J. Res. Dev. 17, 256–262 (1973).
[CrossRef]

Masetti, E.

R. M. A. Azzam, E. Masetti, I. M. Elminyawi, F. G. Grosz, “Construction, calibration, and testing of a four-detector photopolarimeter,” Rev. Sci. Instrum. 59, 84–88 (1988).
[CrossRef]

R. M. A. Azzam, E. Masetti, F. G. Grosz, I. Elminyawi, “Four-detector photopolarimeter: first experimental results,” J. Opt. Soc. Am. A 4(13), P78 (1987).

Studna, A. A.

D. E. Aspnes, A. A. Studna, “Dielectric functions and optical parameters of Si, Ge, GaP, GaAs, GaSb, InP, InAs, and InSb from 1.5 to 6.0 eV,” Phys. Rev. B 27, 985–1009 (1983).
[CrossRef]

Warnecke, A. J.

A. J. Warnecke, P. J. LoPresti, “Refractive index dispersion in semiconductor-related thin films,” IBM J. Res. Dev. 17, 256–262 (1973).
[CrossRef]

Wolf, E.

See, for example, M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1975), p. 554.

IBM J. Res. Dev. (1)

A. J. Warnecke, P. J. LoPresti, “Refractive index dispersion in semiconductor-related thin films,” IBM J. Res. Dev. 17, 256–262 (1973).
[CrossRef]

J. Opt. Soc. Am. (3)

J. Opt. Soc. Am. A (1)

R. M. A. Azzam, E. Masetti, F. G. Grosz, I. Elminyawi, “Four-detector photopolarimeter: first experimental results,” J. Opt. Soc. Am. A 4(13), P78 (1987).

Opt. Lett. (2)

Phys. Rev. B (1)

D. E. Aspnes, A. A. Studna, “Dielectric functions and optical parameters of Si, Ge, GaP, GaAs, GaSb, InP, InAs, and InSb from 1.5 to 6.0 eV,” Phys. Rev. B 27, 985–1009 (1983).
[CrossRef]

Rev. Sci. Instrum. (1)

R. M. A. Azzam, E. Masetti, I. M. Elminyawi, F. G. Grosz, “Construction, calibration, and testing of a four-detector photopolarimeter,” Rev. Sci. Instrum. 59, 84–88 (1988).
[CrossRef]

Other (4)

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

See, for example, Ref. 4, Sec. 1.8.

See, for example, Ref. 4, Sec. 3.9.1.

See, for example, M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1975), p. 554.

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Figures (11)

Fig. 1
Fig. 1

Diagram of the FDP. The surfaces of photodetectors D0, D1, and D2 are partially specularly reflecting, whereas that of D3 is substantially totally absorbing. The four output currents i0, i1, i2, and i3 determine the input Stokes parameters S0, S1, S2, and S3. pn is a transverse reference polarization direction parallel to the nth plane of incidence. α1 and α2 are the rotation angles of the plane of incidence.

Fig. 2
Fig. 2

The quantity Q(λ) [Eq. (36)], which is 1/8 of the normalized determinant of the instrument matrix A, versus the wavelength λ for a FDP that uses identical Si detectors D0, D1, and D2, each of which is coated by a thin (10-nm) SiO2 film and each of which reflects light at the same angle of incidence ϕ. The four curves correspond to ϕ = 50°, 60°, 70°, 80°. The optical properties of SiO2 and Si are taken from Refs. 8 and 9, respectively. Notice that Q(λ) ≠ 0, and hence A is nonsingular over the entire 207–826-nm spectral range.

Fig. 3
Fig. 3

The unpolarized-light spectral reflectance of a Si detector coated by a 10-nm SiO2 film at four angles of incidence ϕ = 50°, 60°, 70°, 80°. The optical properties of SiO2 and Si are taken from Refs. 8 and 9, respectively.

Fig. 4
Fig. 4

Same as in Fig. 2 except that the thickness of the SiO2 film is now 200 nm. The oscillatory behavior of Q(λ), a consequence of interference in the optically thick film, causes the instrument matrix to be singular (det A = 0) at a number of discrete wavelengths.

Fig. 5
Fig. 5

Same as in Fig. 3, except that the SiO2 film is now 200 nm thick.

Fig. 6
Fig. 6

A general light path IOPQR in the FDP. The meaning of the two indicated cones appears in the text.

Fig. 7
Fig. 7

A simple symmetrical light path IOPQR for the FDP. This path is determined completely by the distance 2a and the two angles β1 and β2.

Fig. 8
Fig. 8

Angle of incidence ϕ0 = ϕ2 at the first and third detectors (positioned at O and Q) for the light path of Fig. 7 plotted as a function of the angle β2, with β1 as a parameter, where β1 = 15°, 30°, 45°, 60°, 75°, as marked by each curve.

Fig. 9
Fig. 9

Angle of incidence ϕ1 at the second detector (positioned at P) for the light path of Fig. 7 plotted as a function of β2 with β1 = 15°, 30°, 45°, 60°, 75°, as marked by each curve.

Fig. 10
Fig. 10

Rotation angle α1 = α2 between successive planes of incidence for the light path of Fig. 7 plotted versus the angle β2 with β1 = 15°, 30°, 45°, 60°, 75°, as marked by each curve.

Fig. 11
Fig. 11

The optimum calibration polarization states C1 C2, C3, and C4 are the vertices of a tetrahedron (maximum-volume pyramid) inscribed inside the Poincaré sphere.

Equations (69)

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I = AS
S = A 1 I .
S = S , S ( 0 ) = M 0 S , S ( 1 ) = M 1 R 1 M 0 S , S ( 2 ) = M 2 R 2 M 1 R 1 M 0 S ,
M l = r l [ 1 cos 2 ψ l 0 0 cos 2 ψ l 1 0 0 0 0 sin 2 ψ l cos Δ l sin 2 ψ l sin Δ l 0 0 sin 2 ψ l sin Δ l sin 2 ψ l cos Δ l ]
R l = [ 1 0 0 0 0 cos 2 α l sin 2 α l 0 0 sin 2 α l cos 2 α l 0 0 0 0 1 ]
Γ = [ 1 0 0 0 ] .
L = [ S 0 S 0 ( 0 ) S 0 ( 1 ) S 0 ( 2 ) ] .
L = FS ,
F = [ F 0 F 1 F 2 F 3 ] = [ Γ Γ M 0 Γ M 1 R 1 M 0 Γ M 2 R 2 M 1 R 1 M 0 ] .
F = [ 1 0 0 0 f 10 f 11 0 0 f 20 f 21 f 22 f 23 f 30 f 31 f 32 f 33 ] ,
f 10 = r 0 ,
f 11 = r 0 cos 2 ψ 0 ,
f 20 = r 0 r 1 ( 1 + cos 2 ψ 0 cos 2 ψ 1 cos 2 α 1 ) ,
f 21 = r 0 r 1 ( cos 2 ψ 0 + cos 2 ψ 1 cos 2 α 1 ) ,
f 22 = r 0 r 1 ( sin 2 ψ 0 cos Δ 0 cos 2 ψ 1 sin 2 α 1 ) ,
f 23 = r 0 r 1 ( sin 2 ψ 0 sin Δ 0 cos 2 ψ 1 sin 2 α 1 ) ,
f 30 = r 0 r 1 r 2 ( 1 + cos 2 ψ 0 cos 2 ψ 1 cos 2 α 1 + cos 2 ψ 1 cos 2 ψ 2 cos 2 α 2 + cos 2 ψ 0 cos 2 ψ 2 cos 2 α 1 cos 2 α 2 cos 2 ψ 0 sin 2 ψ 1 cos Δ 1 cos 2 ψ 2 sin 2 α 1 sin 2 α 2 ) ,
f 31 = r 0 r 1 r 2 ( cos 2 ψ 0 + cos 2 ψ 1 cos 2 α 1 + cos 2 ψ 0 cos 2 ψ 1 cos 2 ψ 2 cos 2 α 2 + cos 2 ψ 2 cos 2 α 1 cos 2 α 2 sin 2 ψ 1 cos Δ 1 cos 2 ψ 2 sin 2 α 1 sin 2 α 2 ) ,
f 32 = r 0 r 1 r 2 ( sin 2 ψ 0 cos Δ 0 cos 2 ψ 1 sin 2 α 1 + sin 2 ψ 0 cos Δ 0 cos 2 ψ 2 sin 2 α 1 cos 2 α 2 + sin 2 ψ 0 cos Δ 0 sin 2 ψ 1 cos Δ 1 cos 2 ψ 2 cos 2 α 1 sin 2 α 2 sin 2 ψ 0 sin Δ 0 sin 2 ψ 1 sin Δ 1 cos 2 ψ 2 sin 2 α 2 ) ,
f 33 = r 0 r 1 r 2 ( sin 2 ψ 0 sin Δ 0 cos 2 ψ 1 sin 2 α 1 + sin 2 ψ 0 sin Δ 0 cos 2 ψ 2 sin 2 α 1 cos 2 α 2 + sin 2 ψ 0 cos Δ 0 sin 2 ψ 1 sin Δ 1 cos 2 ψ 2 sin 2 α 2 + sin 2 ψ 0 sin Δ 0 sin 2 ψ 1 cos Δ 1 cos 2 ψ 2 cos 2 α 1 sin 2 α 2 ) .
i 0 = k 0 ( S 0 S 0 ( 0 ) ) , i 1 = k 1 ( S 0 ( 0 ) S 0 ( 1 ) ) , i 2 = k 2 ( S 0 ( 1 ) S 0 ( 2 ) ) , i 3 = k 3 S 0 ( 2 ) ,
I = KDL ,
K = [ k 0 0 0 0 0 k 1 0 0 0 0 k 2 0 0 0 0 k 3 ] ,
D = [ 1 1 0 0 0 1 1 0 0 0 1 1 0 0 0 1 ] ,
I = KDFS .
A = KDF ,
det A = ( det K ) ( det D ) ( det F ) ,
det K = k 0 k 1 k 2 k 3 , det D = 1 , det F = f 11 ( f 22 f 33 f 23 f 32 ) ,
det A = ( k 0 k 1 k 2 k 3 ) f 11 ( f 22 f 33 f 23 f 32 ) .
det A = ( k 0 k 1 k 2 k 3 ) ( r 0 3 r 1 2 r 2 ) ( sin 2 α 1 sin 2 α 2 ) × ( sin 2 2 ψ 0 cos 2 ψ 0 sin 2 ψ 1 cos 2 ψ 1 cos 2 ψ 2 ) ( sin Δ 1 ) .
ψ 0 = 0 or π / 2 ,
ψ 0 = π / 4 ,
ψ 1 = 0 or π / 2 ,
ψ 1 = π / 4 ,
ψ 2 = π / 4 .
Δ 1 = 0 or π .
( det A ) n = det A / ( k 0 k 1 k 2 k 3 ) ( r 0 3 r 1 2 r 2 ) .
α 1 , α 2 = ± 45 ° or ± 135 °
d d ψ 0 ( sin 2 2 ψ 0 cos 2 ψ 0 ) = 0 ,
ψ 0 = ½ cos 1 1 / 3 = 27.37 ° ,
ψ 1 = π / 8 = 22.5 °
ψ 1 = 3 π / 8 = 67.5 °
ψ 2 = 0 or 90 ° .
Δ 1 = ± 90 ° ,
( det A ) n max = 3 .
r 0 3 = r 1 2 = r 2 .
S = [ 1 0 0 0 ] t .
L u = [ 1 f 10 f 20 f 30 ] t ,
V = D L u = [ ( 1 f 10 ) ( f 10 f 20 ) ( f 20 f 30 ) f 30 ] t ,
1 f 10 = 1 / 4 , f 10 f 20 = 1 / 4 , f 20 f 30 = 1 / 4 , f 30 = 1 / 4 .
f 10 = 3 / 4 ,
f 20 = 1 / 2 ,
f 30 = 1 / 4 .
r 0 = 3 / 4 , or 75 % .
r 1 = 2 / 3 , or 66.67 % .
r 2 = 1 / 2 , or 50 % .
± 8 ( det A ) n = Q = sin 3 ( 4 ψ ) sin Δ ,
ϕ 0 = ϕ 2 , α 1 = α 2 .
cos 2 ϕ 0 = tan β 1 / ( 1 + tan 2 β 1 sec 2 β 2 ) 1 / 2 ,
cos 2 ϕ 1 = ( 1 + tan 2 β 1 sec 2 β 2 ) / ( 1 + tan 2 β 1 sec 2 β 2 ) ,
cos α 1 = cos β 2 / ( 1 + tan 2 β 1 tan 2 β 2 ) 1 / 2 .
= AS ,
= [ I 1 I 2 I 3 I 4 ] ,
S = [ S 1 S 2 S 3 S 4 ] .
A = S 1 .
det A = det / det S .
det 0
V max = 8 3 / 27 .
V max / V = 8 3 / 9 = 1.54

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