Abstract

Polarization phase-shifting interferometry is an established technique in optical metrology. In the present study it is shown that, by use of this technique, not only is it possible to realize any discrete magnitude of a predetermined phase difference (from 0 to 2π) between two light beams but also phase-modulated periodic optical signals can be generated simply by rotation of a polarizer or a retarder or both placed at the input of a conventional two-beam interferometer. Some representative linear and nonlinear periodic polarization-induced phase-modulated optical signals are shown. A linear phase modulation of 0–2π with constant output intensity is obtained in some cases. The Poincaré sphere representation is introduced as a convenient tool for visualizing the dynamics involved in the generation of polarization-phase-modulated waveforms and as a possible aid to intelligent modification of the generated waveform as required. This all-optical technique of continuous and periodic phase variation should prove useful for introducing phase modulation without the need for electro-optic devices.

© 2001 Optical Society of America

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References

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  1. J. E. Greivenkamp, J. H. Bruning, “Phase shifting interferometry,” in Optical Shop Testing, D. Malacara, ed. (Wiley, New York, 1992), p. 501.
  2. G. Jin, N. Bao, P. S. Chung, “Applications of a novel phase shift method using a computer controlled polarization mechanism,” Opt. Eng. 33, 2733–2737 (1994).
    [CrossRef]
  3. K. Bhattacharya, A. Basuray, A. K. Chakraborty, “Photoelastic testing using a birefringence-sensitive interferometer,” Opt. Commun. 109, 380–386 (1994).
    [CrossRef]
  4. Y. Otani, T. Shimada, T. Yoshizawa, M. Umeda, “Two-dimensional birefringence measurement using the phase shifting technique,” Opt. Eng. 33, 1604–1609 (1994).
    [CrossRef]
  5. K. Bhattacharya, A. K. Chakraborty, A. Ghosh, “Simulation of effects of phase and amplitude coatings on the lens aperture with polarization masks,” J. Opt. Soc. Am. A 11, 586–592 (1994).
    [CrossRef]
  6. G. E. Sommargren, “Up/down frequency shifter for optical heterodyne interferometry,” J. Opt. Soc. Am. 65, 960–961 (1975).
    [CrossRef]
  7. R. M. A. Azzam, “Polarization Michelson interferometer as a global polarization state generator and for the measurement of the coherent and spectral properties of quasimonochromatic light,” Rev. Sci. Instrum. 64, 2834–2837 (1993).
    [CrossRef]
  8. H. F. Hazebroek, A. A. Holscher, “Interferometric ellipsometry,” J. Phys. E 6, 822–826 (1973).
    [CrossRef]
  9. H. F. Hazebroek, A. A. Holscher, “Automated laser interferometric ellipsometry and precision reflectometry,” J. Phys. E 16, 654–661 (1983).
    [CrossRef]
  10. R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1987).
  11. W. A. Shurcliff, Polarized Light (Harvard U. Press, Cambridge, Mass., 1962), App. 2, p. 170.
  12. E. Collet, Polarized Light: Fundamentals and Applications (Marcel Dekker, New York, 1993), p. 219.
  13. S. Pancharatnum, “Generalized theory of interference and its applications. I. Coherent pencils,” Proc. Ind. Acad. Sci. 44, 254 (1956); in Collected Works of S. Pancharatnum (Oxford U. Press, London, UK, 1975), pp. 77–92.

1994 (4)

G. Jin, N. Bao, P. S. Chung, “Applications of a novel phase shift method using a computer controlled polarization mechanism,” Opt. Eng. 33, 2733–2737 (1994).
[CrossRef]

K. Bhattacharya, A. Basuray, A. K. Chakraborty, “Photoelastic testing using a birefringence-sensitive interferometer,” Opt. Commun. 109, 380–386 (1994).
[CrossRef]

Y. Otani, T. Shimada, T. Yoshizawa, M. Umeda, “Two-dimensional birefringence measurement using the phase shifting technique,” Opt. Eng. 33, 1604–1609 (1994).
[CrossRef]

K. Bhattacharya, A. K. Chakraborty, A. Ghosh, “Simulation of effects of phase and amplitude coatings on the lens aperture with polarization masks,” J. Opt. Soc. Am. A 11, 586–592 (1994).
[CrossRef]

1993 (1)

R. M. A. Azzam, “Polarization Michelson interferometer as a global polarization state generator and for the measurement of the coherent and spectral properties of quasimonochromatic light,” Rev. Sci. Instrum. 64, 2834–2837 (1993).
[CrossRef]

1983 (1)

H. F. Hazebroek, A. A. Holscher, “Automated laser interferometric ellipsometry and precision reflectometry,” J. Phys. E 16, 654–661 (1983).
[CrossRef]

1975 (1)

1973 (1)

H. F. Hazebroek, A. A. Holscher, “Interferometric ellipsometry,” J. Phys. E 6, 822–826 (1973).
[CrossRef]

1956 (1)

S. Pancharatnum, “Generalized theory of interference and its applications. I. Coherent pencils,” Proc. Ind. Acad. Sci. 44, 254 (1956); in Collected Works of S. Pancharatnum (Oxford U. Press, London, UK, 1975), pp. 77–92.

Azzam, R. M. A.

R. M. A. Azzam, “Polarization Michelson interferometer as a global polarization state generator and for the measurement of the coherent and spectral properties of quasimonochromatic light,” Rev. Sci. Instrum. 64, 2834–2837 (1993).
[CrossRef]

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

Bao, N.

G. Jin, N. Bao, P. S. Chung, “Applications of a novel phase shift method using a computer controlled polarization mechanism,” Opt. Eng. 33, 2733–2737 (1994).
[CrossRef]

Bashara, N. M.

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

Basuray, A.

K. Bhattacharya, A. Basuray, A. K. Chakraborty, “Photoelastic testing using a birefringence-sensitive interferometer,” Opt. Commun. 109, 380–386 (1994).
[CrossRef]

Bhattacharya, K.

K. Bhattacharya, A. Basuray, A. K. Chakraborty, “Photoelastic testing using a birefringence-sensitive interferometer,” Opt. Commun. 109, 380–386 (1994).
[CrossRef]

K. Bhattacharya, A. K. Chakraborty, A. Ghosh, “Simulation of effects of phase and amplitude coatings on the lens aperture with polarization masks,” J. Opt. Soc. Am. A 11, 586–592 (1994).
[CrossRef]

Bruning, J. H.

J. E. Greivenkamp, J. H. Bruning, “Phase shifting interferometry,” in Optical Shop Testing, D. Malacara, ed. (Wiley, New York, 1992), p. 501.

Chakraborty, A. K.

K. Bhattacharya, A. K. Chakraborty, A. Ghosh, “Simulation of effects of phase and amplitude coatings on the lens aperture with polarization masks,” J. Opt. Soc. Am. A 11, 586–592 (1994).
[CrossRef]

K. Bhattacharya, A. Basuray, A. K. Chakraborty, “Photoelastic testing using a birefringence-sensitive interferometer,” Opt. Commun. 109, 380–386 (1994).
[CrossRef]

Chung, P. S.

G. Jin, N. Bao, P. S. Chung, “Applications of a novel phase shift method using a computer controlled polarization mechanism,” Opt. Eng. 33, 2733–2737 (1994).
[CrossRef]

Collet, E.

E. Collet, Polarized Light: Fundamentals and Applications (Marcel Dekker, New York, 1993), p. 219.

Ghosh, A.

Greivenkamp, J. E.

J. E. Greivenkamp, J. H. Bruning, “Phase shifting interferometry,” in Optical Shop Testing, D. Malacara, ed. (Wiley, New York, 1992), p. 501.

Hazebroek, H. F.

H. F. Hazebroek, A. A. Holscher, “Automated laser interferometric ellipsometry and precision reflectometry,” J. Phys. E 16, 654–661 (1983).
[CrossRef]

H. F. Hazebroek, A. A. Holscher, “Interferometric ellipsometry,” J. Phys. E 6, 822–826 (1973).
[CrossRef]

Holscher, A. A.

H. F. Hazebroek, A. A. Holscher, “Automated laser interferometric ellipsometry and precision reflectometry,” J. Phys. E 16, 654–661 (1983).
[CrossRef]

H. F. Hazebroek, A. A. Holscher, “Interferometric ellipsometry,” J. Phys. E 6, 822–826 (1973).
[CrossRef]

Jin, G.

G. Jin, N. Bao, P. S. Chung, “Applications of a novel phase shift method using a computer controlled polarization mechanism,” Opt. Eng. 33, 2733–2737 (1994).
[CrossRef]

Otani, Y.

Y. Otani, T. Shimada, T. Yoshizawa, M. Umeda, “Two-dimensional birefringence measurement using the phase shifting technique,” Opt. Eng. 33, 1604–1609 (1994).
[CrossRef]

Pancharatnum, S.

S. Pancharatnum, “Generalized theory of interference and its applications. I. Coherent pencils,” Proc. Ind. Acad. Sci. 44, 254 (1956); in Collected Works of S. Pancharatnum (Oxford U. Press, London, UK, 1975), pp. 77–92.

Shimada, T.

Y. Otani, T. Shimada, T. Yoshizawa, M. Umeda, “Two-dimensional birefringence measurement using the phase shifting technique,” Opt. Eng. 33, 1604–1609 (1994).
[CrossRef]

Shurcliff, W. A.

W. A. Shurcliff, Polarized Light (Harvard U. Press, Cambridge, Mass., 1962), App. 2, p. 170.

Sommargren, G. E.

Umeda, M.

Y. Otani, T. Shimada, T. Yoshizawa, M. Umeda, “Two-dimensional birefringence measurement using the phase shifting technique,” Opt. Eng. 33, 1604–1609 (1994).
[CrossRef]

Yoshizawa, T.

Y. Otani, T. Shimada, T. Yoshizawa, M. Umeda, “Two-dimensional birefringence measurement using the phase shifting technique,” Opt. Eng. 33, 1604–1609 (1994).
[CrossRef]

J. Opt. Soc. Am. (1)

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

J. Phys. E (2)

H. F. Hazebroek, A. A. Holscher, “Interferometric ellipsometry,” J. Phys. E 6, 822–826 (1973).
[CrossRef]

H. F. Hazebroek, A. A. Holscher, “Automated laser interferometric ellipsometry and precision reflectometry,” J. Phys. E 16, 654–661 (1983).
[CrossRef]

Opt. Commun. (1)

K. Bhattacharya, A. Basuray, A. K. Chakraborty, “Photoelastic testing using a birefringence-sensitive interferometer,” Opt. Commun. 109, 380–386 (1994).
[CrossRef]

Opt. Eng. (2)

Y. Otani, T. Shimada, T. Yoshizawa, M. Umeda, “Two-dimensional birefringence measurement using the phase shifting technique,” Opt. Eng. 33, 1604–1609 (1994).
[CrossRef]

G. Jin, N. Bao, P. S. Chung, “Applications of a novel phase shift method using a computer controlled polarization mechanism,” Opt. Eng. 33, 2733–2737 (1994).
[CrossRef]

Proc. Ind. Acad. Sci. (1)

S. Pancharatnum, “Generalized theory of interference and its applications. I. Coherent pencils,” Proc. Ind. Acad. Sci. 44, 254 (1956); in Collected Works of S. Pancharatnum (Oxford U. Press, London, UK, 1975), pp. 77–92.

Rev. Sci. Instrum. (1)

R. M. A. Azzam, “Polarization Michelson interferometer as a global polarization state generator and for the measurement of the coherent and spectral properties of quasimonochromatic light,” Rev. Sci. Instrum. 64, 2834–2837 (1993).
[CrossRef]

Other (4)

J. E. Greivenkamp, J. H. Bruning, “Phase shifting interferometry,” in Optical Shop Testing, D. Malacara, ed. (Wiley, New York, 1992), p. 501.

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

W. A. Shurcliff, Polarized Light (Harvard U. Press, Cambridge, Mass., 1962), App. 2, p. 170.

E. Collet, Polarized Light: Fundamentals and Applications (Marcel Dekker, New York, 1993), p. 219.

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Figures (10)

Fig. 1
Fig. 1

Schematic of a phase-shifting Mach–Zehnder interferometer. P(θ), P(α), P(β), and P(γ) are linear polarizers with transmission axes oriented at θ, α, β, and γ, respectively. W(ρ, δ) is a retarder with its fast axis oriented along ρ and retardance δ.

Fig. 2
Fig. 2

Loci on the Poincaré sphere that are due to rotation of polarizer P(θ) for several magnitudes of retardance δ of the retarder W(ρ, δ). The retarder is oriented such that ρ = 0°.

Fig. 3
Fig. 3

Loci on the Poincaré sphere that are due to rotation of the retarder for several magnitudes of retardance δ of the retarder W(ρ, δ). The retarder is oriented such that ρ = 0°.

Fig. 4
Fig. 4

Poincaré sphere representation of polarization-induced phase difference. The area of the spherical triangle P(α)–PP(β) is proportional to the phase difference Δ′ introduced between the two beams.

Fig. 5
Fig. 5

Variation of PIPD and relative amplitude τ with rotation of polarizer P(θ) for ρ = 45°, α = 0°, β = 90°, and γ = 45° (curve 1) and ρ = 0°, α = 135°, β = 45°, and γ = 90° (curve 2). The retarder is a quarter-wave plate.

Fig. 6
Fig. 6

Variation of PIPD and relative amplitude τ with rotation of the retarder (quarter-wave plate) for θ = 45°, α = 0°, β = 90°, and γ = 45°.

Fig. 7
Fig. 7

PIPD and τ as functions of δ for ρ = 0°, θ = 45°, α = 0°, β = 90°, and γ = 45°.

Fig. 8
Fig. 8

Loci on the Poincaré sphere that are due to simultaneous rotation of P(θ) and W(ρ, δ) for three values of ρ–θ, as shown. Rather than the absolute values of ρ and θ, it is their difference that determines the locus on the Poincaré sphere.

Fig. 9
Fig. 9

Variation of PIPD as a function of ρ for rotation of the polarizer P(θ)–W(ρ, 90°) (quarter-wave plate) combination as a whole, for some specific values of ρ–θ with α = 0°, β = 90°, and γ = 45°.

Fig. 10
Fig. 10

Relative amplitude τ as a function of ρ for the curves shown in Fig. 9.

Equations (21)

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J=ab expiΔ,
J=Wρ, δJθ,
Jθ=cos θsin θ
Wρ, δ=cos2 ρ expiδ/2+sin2 ρ exp-iδ/22i cos ρ sin ρ sinδ/22i cos ρ sin ρ sinδ/2cos2 ρ exp-iδ/2+sin2 ρ expiδ/2.
a=cos ρ1-cos δcosρ-θ+cos θ cos δ2+sin δ sin ρ sinθ-ρ21/2,
b=sin ρ1-cos δcosρ-θ+sin θ cos δ2+sin δ cos ρ sinρ-θ21/2,
Δ=tan-1sin δ cos ρ sinρ-θsin ρ1-cos δcosρ-θ+sin θ cos δ-tan-1sin δ sin ρ sinθ-ρcos ρ1-cos δcosρ-θ+cos θ cos δ.
Jα=PαJ=cos2 αcos α sin αcos α sin αsin2 αab expiΔ
=a cos α+b expiΔsin αcos αsin α.
Jα=R1 expiΔ1cos αsin α,
R1=a cos α+b cos Δ sin α2+b sin Δ sin α21/2,
Δ1=tan-1b sin Δ sin αa cos α+b cos Δ sin α.
Jβ=a cos β+b expiΔsin βcos βsin β=R2 expiΔ2cos βsin β,
R2=a cos β+cos Δ sin β2+b sin Δ sin β21/2,
Δ2=tan-1b sin Δ sin βa cos β+b cos Δ sin β.
Jγ=PγJα+Jβ=PγR1 expiΔ1cos α+R2 expiΔ2cos βR1 expiΔ1sin α+R2 expiΔ2sin β
=R1 cosα-γexpiΔ1cos γsin γ+R2 cosβ-γexpiΔ2cos γsin γ.
Δ=tan-1ab sin Δ sinα-βa2 cos α cos β+b2 sin α sin β+ab sinα+βcos Δ,
τ=R1 cosα-γ/R2 cosβ-γ=a cos α+b cos Δ sin α2+b sin Δ sin α21/2 cosα-γa cos β+b cos Δ sin δ2+b sin Δ sin β21/2 cosβ-γ
Δ=Δ,
τ=a/b.

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