Abstract

A novel technique that measures the linear birefringence of crystal quartz within the configuration of a Soliel-Babinet compensator (SBC) is proposed. A characteristic of this technique is that phase retardation introduced by quartz is amplitude modulation (AM) instead of phase modulation (PM). The linear birefringence is measured regardless of the azimuth angle of the SBC and the orientation of the linear polarization laser beam. Compared with the single-wedge method, the SBC is similar to a parallel plate that allows for a wider range of refractive index of the test material to be measured. This proposed method uses a conventional amplitude demodulation method in conjunction with an optical heterodyne technique and a bandpass filter to produce a better signal-to-noise ratio. Although the SBC configuration is more complex than a single element, the independence of azimuth angle and the orientation of the linear polarized laser beam can enhance the sensitivity of the linear birefringence measurement.

© 2003 Optical Society of America

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References

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  1. R. M. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1980).
  2. M. Kujawinska, “Spatial phase measurement methods,” in Interferogram Analysis, D. W. Robinson, G. T. Reid, eds. (Institute of Physics, London, 1993), pp. 141–193.
  3. K. Schoenenberger, B. W. Colston, D. J. Maitland, L. B. D. Silva, M. J. Everett, “Mapping of birefringence and thermal damage in tissue by use of polarization-sensitive optical coherence tomography,” Appl. Opt. 37, 6026–6036 (1998).
    [CrossRef]
  4. Y. Shindo, R. Takigaura, “An improved highly sensitive instrument for measuring optical birefringence,” Polym. Commun. 25, 378–382 (1984).
  5. L. M. Bernard, O. D. D. Soares, “Birefringence measurements by double speckle photography,” Appl. Opt. 26, 769–771 (1987).
  6. S. Nakadate, “High precision retardation measurement using phase detection of Young’s fringes,” Appl. Opt. 29, 242–246 (1990).
    [CrossRef] [PubMed]
  7. K. Cho, S. P. Bush, B. David, L. Mazzoni, C. C. Davis, “Linear magnetic birefringence measurements of Faraday materials,” Phys. Rev. B 43, 965–971 (1991).
    [CrossRef]
  8. L. Yi, Z. Yi, S. Xiyu, S. Liaanke, “Birefringence measurement by the self-compensation method,” Appl. Opt. 31, 2968–2969 (1992).
    [CrossRef] [PubMed]
  9. Y. Otani, T. Shimada, T. Yushizawa, N. Umeda, “Two-dimensional birefringence measurement using the phase shifting technique,” Opt. Eng. 33, 1604–1609 (1994).
    [CrossRef]
  10. M. S. El-Bahrawi, N. N. Nagib, S. A. Khodier, H. M. Sidki, “Birefringence of muscovite mica,” Opt. Laser Technol. 30, 411–415 (1998).
    [CrossRef]
  11. Y. Zhang, S. Zhang, Y. Han, X. Xu, “Method for the measurement of retardation of wave plates based on laser frequency-splitting technology,” Opt. Eng. 40, 1071–1075 (2001).
    [CrossRef]
  12. E. A. West, M. H. Smith, “Polarization errors associated with birefringence waveplates,” in Polarization Analysis and Measurement II, D. H. Goldstein, D. B. Chenault, eds., Proc. SPIE2265, 260–271 (1994).
    [CrossRef]
  13. F. Cecelja, R. Jung, B. Balachandran, M. Berwick, “Electro-optic sensors: a precise in-situ alignment of quarter-wave plate,” in Fiber Optic and Laser Sensors XIII, R. P. DePaula, J. W. Berthold, eds., Proc. SPIE2510, 205–211 (1995).
    [CrossRef]
  14. H. K. Teng, C. Chou, C. N. Chang, C. W. Lyu, Y. C. Huang, “Linear birefringence measurement with a differential-phase optical heterodyne polarimeter,” Jpn. J. Appl. Phys. Part 1 41, 3140–3144 (2002).
    [CrossRef]
  15. C. Chou, C. W. Lyu, L. C. Peng, “Polarized differential-phase laser scanning microscope,” Appl. Opt. 40, 95–99 (2001).
    [CrossRef]
  16. M. Bass, E. W. Van Stryland, D. R. Williams, eds., Handbook of Optics, 2nd ed. (Optical Society of America, Washington, D.C., 1995).
  17. J. R. Taylor, Introduction to Error Analysis: the Study of Uncertainties in Physical Measurement (University Science, Mill Valley, Calif., 1997), pp. 137–146.

2002 (1)

H. K. Teng, C. Chou, C. N. Chang, C. W. Lyu, Y. C. Huang, “Linear birefringence measurement with a differential-phase optical heterodyne polarimeter,” Jpn. J. Appl. Phys. Part 1 41, 3140–3144 (2002).
[CrossRef]

2001 (2)

C. Chou, C. W. Lyu, L. C. Peng, “Polarized differential-phase laser scanning microscope,” Appl. Opt. 40, 95–99 (2001).
[CrossRef]

Y. Zhang, S. Zhang, Y. Han, X. Xu, “Method for the measurement of retardation of wave plates based on laser frequency-splitting technology,” Opt. Eng. 40, 1071–1075 (2001).
[CrossRef]

1998 (2)

1994 (1)

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

1992 (1)

1991 (1)

K. Cho, S. P. Bush, B. David, L. Mazzoni, C. C. Davis, “Linear magnetic birefringence measurements of Faraday materials,” Phys. Rev. B 43, 965–971 (1991).
[CrossRef]

1990 (1)

1987 (1)

1984 (1)

Y. Shindo, R. Takigaura, “An improved highly sensitive instrument for measuring optical birefringence,” Polym. Commun. 25, 378–382 (1984).

Azzam, R. M.

R. M. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1980).

Balachandran, B.

F. Cecelja, R. Jung, B. Balachandran, M. Berwick, “Electro-optic sensors: a precise in-situ alignment of quarter-wave plate,” in Fiber Optic and Laser Sensors XIII, R. P. DePaula, J. W. Berthold, eds., Proc. SPIE2510, 205–211 (1995).
[CrossRef]

Bashara, N. M.

R. M. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1980).

Bernard, L. M.

Berwick, M.

F. Cecelja, R. Jung, B. Balachandran, M. Berwick, “Electro-optic sensors: a precise in-situ alignment of quarter-wave plate,” in Fiber Optic and Laser Sensors XIII, R. P. DePaula, J. W. Berthold, eds., Proc. SPIE2510, 205–211 (1995).
[CrossRef]

Bush, S. P.

K. Cho, S. P. Bush, B. David, L. Mazzoni, C. C. Davis, “Linear magnetic birefringence measurements of Faraday materials,” Phys. Rev. B 43, 965–971 (1991).
[CrossRef]

Cecelja, F.

F. Cecelja, R. Jung, B. Balachandran, M. Berwick, “Electro-optic sensors: a precise in-situ alignment of quarter-wave plate,” in Fiber Optic and Laser Sensors XIII, R. P. DePaula, J. W. Berthold, eds., Proc. SPIE2510, 205–211 (1995).
[CrossRef]

Chang, C. N.

H. K. Teng, C. Chou, C. N. Chang, C. W. Lyu, Y. C. Huang, “Linear birefringence measurement with a differential-phase optical heterodyne polarimeter,” Jpn. J. Appl. Phys. Part 1 41, 3140–3144 (2002).
[CrossRef]

Cho, K.

K. Cho, S. P. Bush, B. David, L. Mazzoni, C. C. Davis, “Linear magnetic birefringence measurements of Faraday materials,” Phys. Rev. B 43, 965–971 (1991).
[CrossRef]

Chou, C.

H. K. Teng, C. Chou, C. N. Chang, C. W. Lyu, Y. C. Huang, “Linear birefringence measurement with a differential-phase optical heterodyne polarimeter,” Jpn. J. Appl. Phys. Part 1 41, 3140–3144 (2002).
[CrossRef]

C. Chou, C. W. Lyu, L. C. Peng, “Polarized differential-phase laser scanning microscope,” Appl. Opt. 40, 95–99 (2001).
[CrossRef]

Colston, B. W.

David, B.

K. Cho, S. P. Bush, B. David, L. Mazzoni, C. C. Davis, “Linear magnetic birefringence measurements of Faraday materials,” Phys. Rev. B 43, 965–971 (1991).
[CrossRef]

Davis, C. C.

K. Cho, S. P. Bush, B. David, L. Mazzoni, C. C. Davis, “Linear magnetic birefringence measurements of Faraday materials,” Phys. Rev. B 43, 965–971 (1991).
[CrossRef]

El-Bahrawi, M. S.

M. S. El-Bahrawi, N. N. Nagib, S. A. Khodier, H. M. Sidki, “Birefringence of muscovite mica,” Opt. Laser Technol. 30, 411–415 (1998).
[CrossRef]

Everett, M. J.

Han, Y.

Y. Zhang, S. Zhang, Y. Han, X. Xu, “Method for the measurement of retardation of wave plates based on laser frequency-splitting technology,” Opt. Eng. 40, 1071–1075 (2001).
[CrossRef]

Huang, Y. C.

H. K. Teng, C. Chou, C. N. Chang, C. W. Lyu, Y. C. Huang, “Linear birefringence measurement with a differential-phase optical heterodyne polarimeter,” Jpn. J. Appl. Phys. Part 1 41, 3140–3144 (2002).
[CrossRef]

Jung, R.

F. Cecelja, R. Jung, B. Balachandran, M. Berwick, “Electro-optic sensors: a precise in-situ alignment of quarter-wave plate,” in Fiber Optic and Laser Sensors XIII, R. P. DePaula, J. W. Berthold, eds., Proc. SPIE2510, 205–211 (1995).
[CrossRef]

Khodier, S. A.

M. S. El-Bahrawi, N. N. Nagib, S. A. Khodier, H. M. Sidki, “Birefringence of muscovite mica,” Opt. Laser Technol. 30, 411–415 (1998).
[CrossRef]

Kujawinska, M.

M. Kujawinska, “Spatial phase measurement methods,” in Interferogram Analysis, D. W. Robinson, G. T. Reid, eds. (Institute of Physics, London, 1993), pp. 141–193.

Liaanke, S.

Lyu, C. W.

H. K. Teng, C. Chou, C. N. Chang, C. W. Lyu, Y. C. Huang, “Linear birefringence measurement with a differential-phase optical heterodyne polarimeter,” Jpn. J. Appl. Phys. Part 1 41, 3140–3144 (2002).
[CrossRef]

C. Chou, C. W. Lyu, L. C. Peng, “Polarized differential-phase laser scanning microscope,” Appl. Opt. 40, 95–99 (2001).
[CrossRef]

Maitland, D. J.

Mazzoni, L.

K. Cho, S. P. Bush, B. David, L. Mazzoni, C. C. Davis, “Linear magnetic birefringence measurements of Faraday materials,” Phys. Rev. B 43, 965–971 (1991).
[CrossRef]

Nagib, N. N.

M. S. El-Bahrawi, N. N. Nagib, S. A. Khodier, H. M. Sidki, “Birefringence of muscovite mica,” Opt. Laser Technol. 30, 411–415 (1998).
[CrossRef]

Nakadate, S.

Otani, Y.

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

Peng, L. C.

Schoenenberger, K.

Shimada, T.

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

Shindo, Y.

Y. Shindo, R. Takigaura, “An improved highly sensitive instrument for measuring optical birefringence,” Polym. Commun. 25, 378–382 (1984).

Sidki, H. M.

M. S. El-Bahrawi, N. N. Nagib, S. A. Khodier, H. M. Sidki, “Birefringence of muscovite mica,” Opt. Laser Technol. 30, 411–415 (1998).
[CrossRef]

Silva, L. B. D.

Smith, M. H.

E. A. West, M. H. Smith, “Polarization errors associated with birefringence waveplates,” in Polarization Analysis and Measurement II, D. H. Goldstein, D. B. Chenault, eds., Proc. SPIE2265, 260–271 (1994).
[CrossRef]

Soares, O. D. D.

Takigaura, R.

Y. Shindo, R. Takigaura, “An improved highly sensitive instrument for measuring optical birefringence,” Polym. Commun. 25, 378–382 (1984).

Taylor, J. R.

J. R. Taylor, Introduction to Error Analysis: the Study of Uncertainties in Physical Measurement (University Science, Mill Valley, Calif., 1997), pp. 137–146.

Teng, H. K.

H. K. Teng, C. Chou, C. N. Chang, C. W. Lyu, Y. C. Huang, “Linear birefringence measurement with a differential-phase optical heterodyne polarimeter,” Jpn. J. Appl. Phys. Part 1 41, 3140–3144 (2002).
[CrossRef]

Umeda, N.

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

West, E. A.

E. A. West, M. H. Smith, “Polarization errors associated with birefringence waveplates,” in Polarization Analysis and Measurement II, D. H. Goldstein, D. B. Chenault, eds., Proc. SPIE2265, 260–271 (1994).
[CrossRef]

Xiyu, S.

Xu, X.

Y. Zhang, S. Zhang, Y. Han, X. Xu, “Method for the measurement of retardation of wave plates based on laser frequency-splitting technology,” Opt. Eng. 40, 1071–1075 (2001).
[CrossRef]

Yi, L.

Yi, Z.

Yushizawa, T.

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

Zhang, S.

Y. Zhang, S. Zhang, Y. Han, X. Xu, “Method for the measurement of retardation of wave plates based on laser frequency-splitting technology,” Opt. Eng. 40, 1071–1075 (2001).
[CrossRef]

Zhang, Y.

Y. Zhang, S. Zhang, Y. Han, X. Xu, “Method for the measurement of retardation of wave plates based on laser frequency-splitting technology,” Opt. Eng. 40, 1071–1075 (2001).
[CrossRef]

Appl. Opt. (5)

Jpn. J. Appl. Phys. Part 1 (1)

H. K. Teng, C. Chou, C. N. Chang, C. W. Lyu, Y. C. Huang, “Linear birefringence measurement with a differential-phase optical heterodyne polarimeter,” Jpn. J. Appl. Phys. Part 1 41, 3140–3144 (2002).
[CrossRef]

Opt. Eng. (2)

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

Y. Zhang, S. Zhang, Y. Han, X. Xu, “Method for the measurement of retardation of wave plates based on laser frequency-splitting technology,” Opt. Eng. 40, 1071–1075 (2001).
[CrossRef]

Opt. Laser Technol. (1)

M. S. El-Bahrawi, N. N. Nagib, S. A. Khodier, H. M. Sidki, “Birefringence of muscovite mica,” Opt. Laser Technol. 30, 411–415 (1998).
[CrossRef]

Phys. Rev. B (1)

K. Cho, S. P. Bush, B. David, L. Mazzoni, C. C. Davis, “Linear magnetic birefringence measurements of Faraday materials,” Phys. Rev. B 43, 965–971 (1991).
[CrossRef]

Polym. Commun. (1)

Y. Shindo, R. Takigaura, “An improved highly sensitive instrument for measuring optical birefringence,” Polym. Commun. 25, 378–382 (1984).

Other (6)

E. A. West, M. H. Smith, “Polarization errors associated with birefringence waveplates,” in Polarization Analysis and Measurement II, D. H. Goldstein, D. B. Chenault, eds., Proc. SPIE2265, 260–271 (1994).
[CrossRef]

F. Cecelja, R. Jung, B. Balachandran, M. Berwick, “Electro-optic sensors: a precise in-situ alignment of quarter-wave plate,” in Fiber Optic and Laser Sensors XIII, R. P. DePaula, J. W. Berthold, eds., Proc. SPIE2510, 205–211 (1995).
[CrossRef]

M. Bass, E. W. Van Stryland, D. R. Williams, eds., Handbook of Optics, 2nd ed. (Optical Society of America, Washington, D.C., 1995).

J. R. Taylor, Introduction to Error Analysis: the Study of Uncertainties in Physical Measurement (University Science, Mill Valley, Calif., 1997), pp. 137–146.

R. M. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1980).

M. Kujawinska, “Spatial phase measurement methods,” in Interferogram Analysis, D. W. Robinson, G. T. Reid, eds. (Institute of Physics, London, 1993), pp. 141–193.

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Figures (7)

Fig. 1
Fig. 1

(a) Schematic diagram of the SBC that we used for this study. (b) Demonstration of the laboratory coordinate (x, y) system to the principal axes f 1, s 1 and f 2, s 2 of an individual wedge in the SBC.

Fig. 2
Fig. 2

Optical arrangement of the proposed method: HWP, half-wave plate; BS1, BS2, beam splitters; PBS, polarized beam splitter; AOM1, AOM2, acousto-optic modulators; M1, M2, plane mirrors; LP, linear polarizer; D p , detector; PREAMP/BPF, preamplifier and bandpass filter; DVM, digital voltmeter; PC, personal computer.

Fig. 3
Fig. 3

Amplitude of the heterodyne signal demodulated by a DVM at each step of translation. Azimuth angle β ≈ 30°.

Fig. 4
Fig. 4

Same as Fig. 3 when β ≈ -20° and P polarization of the signal beam propagates through the SBC.

Fig. 5
Fig. 5

Calculation of (a) linear birefringence B versus Φ d and (b) the ratio of uncertainty to the linear birefringence δB/ B versus Φ d . The S component of the signal beam is incident upon the SBC. The azimuth angle is β ≈ 30° and the inclined angle is θ ≈ π/4. The B and the δB/ B obtained in (a) and (b) clearly show larger variations as Φ d approaches 180° and 540°.

Fig. 6
Fig. 6

Periodic profile of V AM versus Φ d as a function of polarization leakage η. S polarization of the signal beam passes through the SBC at β = 30° and θ = 45°.

Fig. 7
Fig. 7

Theoretical calculations of the amplitude of the heterodyne signal versus Φ d as a function of deviation angle ε at (a) Ψ = 0° and (b) Ψ = 55°. The extreme values shifted from Nπ as functions of ε and Ψ.

Equations (34)

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EP=E1 cosθexp-iω0+ω1t,
ES=E1 sinθexp-iω0+ω1t,
ESf=E1 sinθsinβexp-iω0+ω1t-Φ1,
ESs=E1 sinθcosβexp-iω0+ω1t-Φ2,
Φ1=2πλned1+nod2,
Φ2=2πλnod1+ned2,
B=ne-no.
EP1=12 E1sinθsin2βexpiΦ1-expiΦ2×exp-iω0+ω1t+φ1,
ES1=E1sinθsin2βexpiΦ1+cos2βexpiΦ2×exp-iω0+ω1t+φ1.
EP2=E2 cosθexp-iω0+ω2t-φ2,
ES2=E2 sinθexp-iω0+ω2t-φ2,
ISP=|EP1+EP2|2 =IP1+IP2-I0 sin2θsin2β×sinΦd2sinΔωt+ξ1,
ISS=|ES1+ES2|2 =IS1+IS2+2I0 sin2θcos2Φd2+sin2Φd2cos22β1/2 sinΔωt+ξ2.
ξ1=Φ1+Φ2/2+φd,
ξ2=ξ1+tan-1cos2βtanΦd/2,
Φd=2πλ BΔd.
IPP=IP1+IP2+2I0 cos2θcos2Φd2+sin2Φd2×cos22β1/2 sinΔωt+ξ2,
IPS=IS1+IS2-I0 sin2θsin2βsinΦd2×sinΔωt+ξ1.
IΔωt=I0 sin2θsin2βsinΦd2sinΔωt+ξ1.
VAM=|IΔωt|=V0 sinΦd2,
V0=I0 sin2θsin2β.
B=λπdsin-1VAMV0
B=λΔd2π
B=λΔd2π=0.00908.
δB=Bd δd2+BVAM δVAM21/2λπdtanΦd/2δVAMVAM,
ES=E1 sinθexp-iω1t,
EP=ηE1 cosθexp-iω1t,
Ip,ηΔωt=2I0 sinΦd22η cos2θcotΦd/22+2η cosθcos2β+12×sin2βsin2θ21/2×sinΔωt+ξη.
IP,εΔωt=-I0 sin2θMε, ΦdsinΔωt+ξε,
Mε, Φd=14sin22εcosΦd2-cosΦd2+Ψ22+cosεsin2β-εsinΦd2-sinεcos2β-εsinΦd2+Ψ221/2,
Φd=Nπ-tan-1B sinΨ+C sinΨ/2A+B cosΨ+C cosΨ/2
A=12cos2εsin22β-ε-18sin22ε,
B=12sin2εcos22β-ε,
C=14sin2εsin4β-2ε-18sin22ε.

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