Abstract

Optimal optical parameters of the beam splitter that is used in the division-of-amplitude photopolarimeter are determined. These are (1) 50%–50% split ratio of the all-dielectric beam splitter, (2) differential phase shifts in reflection and transmission Δr and Δt that differ by ±π/2, and (3) ellipsometric parameters (ψr, ψt)=(27.368°, 62.632°) or (62.632°, 27.368°). It is also shown that for any nonabsorbing beam splitter that splits incident unpolarized light equally, the relationship ψr+ψt=π/2 is always satisfied.

© 2003 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, ed. (McGraw-Hill, New York, 1995), Vol. 2, Chap. 27.
  2. R. A. Chipman, “Polarimetry,” in Handbook of Optics, 2nd ed., M. Bass, ed. (McGraw-Hill, New York, 1995), Vol. 2, 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. K. Brudzewski, “Static Stokes ellipsometer: general analysis,” J. Mod. Opt. 38, 889–896 (1991).
    [CrossRef]
  7. 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]
  8. F. Delplancke, “Automated high-speed Mueller matrix scatterometer,” Appl. Opt. 36, 5388–5395 (1997).
    [CrossRef] [PubMed]
  9. E. Compain, B. Drevillon, “Broadband division-of-amplitude polarimeter based on uncoated prisms,” Appl. Opt. 37, 5938–5946 (1998).
    [CrossRef]
  10. R. M. A. Azzam, “Mueller-matrix ellipsometry: a review,” in Polarization: Measurement, Analysis, and Remote Sensing, D. H. Goldstein, R. A. Chipman, eds., Proc. SPIE3121, 396–405 (1997).
    [CrossRef]
  11. R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1987).
  12. For a thick uncoated dielectric slab BS, Δrand Δtassume the trivial values of 0 or ±π, and condition (6c) is violated; hence detA=0.Therefore thin-film coatings are required for the BS of DOAP.
  13. According to Eq. (4), a given error δIin the measured signal vector Iresults in a corresponding error δSof the derived Stokes vector S, which is given by δS=A-1δI.Because A-1is proportional to 1/detA,maximization of |detA|is consistent with reduction of the error δS.
  14. Equation (13a) has an infinite number of solutions, and it is by convention11that the ellipsometric angle ψ is confined to the first quadrant, 0≤ψ≤π/2.
  15. R. M. A. Azzam, “Arrangement of four photodetectors for measuring the state of polarization of light,” Opt. Lett. 10, 309–311 (1985).
    [CrossRef] [PubMed]
  16. R. M. A. Azzam, I. M. Elminyawi, A. M. El-Saba, “General analysis and optimization of the four-detector photopolarimeter,” J. Opt. Soc. Am. A 5, 681–689 (1988).
    [CrossRef]
  17. R. M. A. Azzam, E. Masetti, I. M. Elminyawi, F. B. Grosz, “Construction, calibration, and testing of a four-detector photopolarimeter,” Rev. Sci. Instrum. 59, 84–88 (1988).
    [CrossRef]
  18. A second optimum value of ψ0=62.63°was inadvertently missed in Eq. (26a) of Ref. 16.
  19. D. S. Sabatke, M. R. Descour, E. Dereniak, W. C. Sweatt, S. A. Kemme, G. S. Phipps, “Optimization of retardance for a complete Stokes polarimeter,” Opt. Lett. 25, 802–804 (2000).
    [CrossRef]
  20. J. S. Tyo, “Design of optimal polarimeters: maximization of signal-to-noise ratio and minimization of systematic error,” Appl. Opt. 41, 619–630 (2002).
    [CrossRef] [PubMed]
  21. Since the determinant of the instrument matrix appears in the denominator of various condition numbers, it is not surprising that the criterion of the maximum determinant approximately corresponds to the minimum condition number.
  22. A. De, R. M. A. Azzam, “Optimal coated silicon membrane beam splitters for the division-of-amplitude photopolarimeter (DOAP),” presented at the Annual Meeting of the Optical Society of America, Orlando, Florida, September 29–October 3, 2002, paper WZ3.

2002 (1)

2000 (1)

1998 (1)

1997 (1)

1992 (1)

1991 (1)

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

1988 (2)

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

R. M. A. Azzam, I. M. Elminyawi, A. M. El-Saba, “General analysis and optimization of the four-detector photopolarimeter,” J. Opt. Soc. Am. A 5, 681–689 (1988).
[CrossRef]

1985 (2)

R. M. A. Azzam, “Arrangement of four photodetectors for measuring the state of polarization of light,” Opt. Lett. 10, 309–311 (1985).
[CrossRef] [PubMed]

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

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, I. M. Elminyawi, A. M. El-Saba, “General analysis and optimization of the four-detector photopolarimeter,” J. Opt. Soc. Am. A 5, 681–689 (1988).
[CrossRef]

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

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

R. M. A. Azzam, “Arrangement of four photodetectors for measuring the state of polarization of light,” Opt. Lett. 10, 309–311 (1985).
[CrossRef] [PubMed]

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, ed. (McGraw-Hill, New York, 1995), Vol. 2, Chap. 27.

R. M. A. Azzam, “Mueller-matrix ellipsometry: a review,” in Polarization: Measurement, Analysis, and Remote Sensing, D. H. Goldstein, R. A. Chipman, eds., Proc. SPIE3121, 396–405 (1997).
[CrossRef]

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

A. De, R. M. A. Azzam, “Optimal coated silicon membrane beam splitters for the division-of-amplitude photopolarimeter (DOAP),” presented at the Annual Meeting of the Optical Society of America, Orlando, Florida, September 29–October 3, 2002, paper WZ3.

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, ed. (McGraw-Hill, New York, 1995), Vol. 2, Chap. 22.

Compain, E.

De, A.

A. De, R. M. A. Azzam, “Optimal coated silicon membrane beam splitters for the division-of-amplitude photopolarimeter (DOAP),” presented at the Annual Meeting of the Optical Society of America, Orlando, Florida, September 29–October 3, 2002, paper WZ3.

Delplancke, F.

Dereniak, E.

Descour, M. R.

Drevillon, B.

Elminyawi, I. M.

R. M. A. Azzam, I. M. Elminyawi, A. M. El-Saba, “General analysis and optimization of the four-detector photopolarimeter,” J. Opt. Soc. Am. A 5, 681–689 (1988).
[CrossRef]

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

El-Saba, A. M.

Grosz, F. B.

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

Hauge, P. S.

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

Kemme, S. A.

Krishnan, S.

Masetti, E.

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

Phipps, G. S.

Sabatke, D. S.

Sweatt, W. C.

Tyo, J. S.

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 (2)

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]

Opt. Lett. (2)

Rev. Sci. Instrum. (1)

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

Surf. Sci. (1)

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

Other (10)

A second optimum value of ψ0=62.63°was inadvertently missed in Eq. (26a) of Ref. 16.

Since the determinant of the instrument matrix appears in the denominator of various condition numbers, it is not surprising that the criterion of the maximum determinant approximately corresponds to the minimum condition number.

A. De, R. M. A. Azzam, “Optimal coated silicon membrane beam splitters for the division-of-amplitude photopolarimeter (DOAP),” presented at the Annual Meeting of the Optical Society of America, Orlando, Florida, September 29–October 3, 2002, paper WZ3.

R. M. A. Azzam, “Ellipsometry,” in Handbook of Optics, 2nd ed., M. Bass, ed. (McGraw-Hill, New York, 1995), Vol. 2, Chap. 27.

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

R. M. A. Azzam, “Mueller-matrix ellipsometry: a review,” in Polarization: Measurement, Analysis, and Remote Sensing, D. H. Goldstein, R. A. Chipman, eds., Proc. SPIE3121, 396–405 (1997).
[CrossRef]

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

For a thick uncoated dielectric slab BS, Δrand Δtassume the trivial values of 0 or ±π, and condition (6c) is violated; hence detA=0.Therefore thin-film coatings are required for the BS of DOAP.

According to Eq. (4), a given error δIin the measured signal vector Iresults in a corresponding error δSof the derived Stokes vector S, which is given by δS=A-1δI.Because A-1is proportional to 1/detA,maximization of |detA|is consistent with reduction of the error δS.

Equation (13a) has an infinite number of solutions, and it is by convention11that the ellipsometric angle ψ is confined to the first quadrant, 0≤ψ≤π/2.

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Figures (3)

Fig. 1
Fig. 1

Schematic depiction of the DOAP. BS, the beam splitter to be optimized; WP1 and WP2, Wollaston prisms; D 0 , D 1 , D 2 , and D 3 , linear photodetectors that produce output electrical signals i 0 , i 1 , i 2 , and i 3 , respectively. p and s, the linear polarization directions parallel and perpendicular, respectively, to the plane of incidence at the beam splitter.

Fig. 2
Fig. 2

(a) Three-dimensional plot of Q [Eq. (11)] as a function of ψ r and ψ t . (b) Family of constant-Q contours in the ψ r   ψ t plane. Note that Q ( ψ r ,   ψ t ) = - Q ( ψ t ,   ψ r ) , a condition that follows directly from Eq. (11).

Fig. 3
Fig. 3

Close-up view of the constant-Q contours in the immediate neighborhood of the maximum.

Equations (26)

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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 = ( RT ) 2 sin   2 ψ r sin   2 ψ t ( cos   2 ψ r - cos   2 ψ t ) sin ( Δ r - Δ t ) .
ψ r , ψ t 0 or π / 2 ,
ψ r ψ t ,
Δ r - Δ t 0 or π ,
Δ r - Δ t = ± π / 2 .
T = 1 - R ,
P = ( RT ) 2 = R 2 ( 1 - R ) 2 ,
R = T = 0.5 .
Q = sin   2 ψ r sin   2 ψ t ( cos   2 ψ r - cos   2 ψ t ) = 1 2 ( sin   4 ψ r sin   2 ψ t - sin   2 ψ r sin   4 ψ t ) .
Q / ψ r = 0 = 2   cos   4 ψ r sin   2 ψ t - cos   2 ψ r sin   4 ψ t
Q / ψ t = 0 = sin   4 ψ r cos   2 ψ t - 2   sin   2 ψ r cos   4 ψ t .
cos   4 ψ r = cos   4 ψ t ,
cos   4 ψ r = cos   2 ψ t cos   2 ψ r ,
ψ r + ψ t = π / 2 .
cos 2   2 ψ r = 1 / 3 .
( ψ r , ψ t )
= ( 27.368 ° , 62.632 ° ) and ( 62.632 ° , 27.368 ° ) ,
Q max = 4 / 3 3 ) = 0.7698 ,
tan 2   ψ t = T p / T s ,
tan 2   ψ r = R p / R s = ( 1 - T p ) / ( 1 - T s ) ,
T = ( T p + T s ) / 2 = 1 / 2 .
tan 2   ψ r tan 2   ψ t = 1 .

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