Abstract

The two-wave-plate compensator (TWC) technique is introduced for single-point retardation measurements. The TWC method uses a known wave plate together with a wave plate of unknown retardation and produces a linearly polarized output that allows a null of intensity to be detected. The TWC method is compared both theoretically and experimentally with the existing Brace–Köhler and Sénarmont methods. The resolution of the TWC is shown to be 0.02 nm. TWC enables the measurement of a sample retardation with as little as 0.13% error and thus is more accurate than either the Brace–Köhler or the Sénarmont method.

© 2004 Optical Society of America

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2003 (4)

S. Y. Cheng, K. S. Chiang, H. P. Chan, “Birefringence in benzocyclobutene strip optical waveguides,” IEEE Photon. Technol. Lett. 15, 700–702 (2003).
[CrossRef]

M. Huang, “Stress effects on the performance of optical waveguides,” Intl. J. Solids Struct. 40, 1615–1632 (2003).
[CrossRef]

L. G. Peralta, A. A. Bernussi, H. Temkin, M. M. Borhani, D. E. Doucette, “Silicon-dioxide waveguides with low birefringence,” IEEE J. Quantum Electron. 39, 874–879 (2003).
[CrossRef]

X. Zhao, C. Li, Y. Z. Xu, “Stress-induced birefringence control in optical planar waveguides,” Opt. Lett. 28, 564–566 (2003).
[CrossRef] [PubMed]

2002 (6)

2001 (3)

2000 (2)

A. Kilian, J. Kirchhof, B. Kuhlow, G. Przyrembel, W. Wischmann, “Birefringence free planar optical waveguide made by flame hydrolysis deposition (FHD) through tailoring of the overcladding,” J. Lightwave Technol. 18, 193–198 (2000).
[CrossRef]

C. S. Kim, Y. Han, B. H. Lee, W. T. Han, U. C. Paek, Y. Chung, “Induction of the refractive index change in B-doped optical fibers through relaxation of the mechanical stress,” Opt. Commun. 185, 337–342 (2000).
[CrossRef]

1999 (2)

B. Wang, T. Oakberg, “A new instrument for measuring both magnitude and angle of low level linear birefringence,” Rev. Sci. Instrum. 70, 3847–3854 (1999).
[CrossRef]

K. Katoh, K. Hammar, P. Smith, R. Oldenbourg, “Birefringence imaging directly reveals architectural dynamics of filamentous actin in living growth cones,” Mol. Biol. Cell 10, 197–210 (1999).
[CrossRef] [PubMed]

1998 (2)

R. Oldenbourg, E. D. Salmon, P. T. Tran, “Birefringence of single and bundled microtubules,” Biophys. J. 74, 645–654 (1998).
[CrossRef] [PubMed]

A. Ajovalasit, S. Baronne, G. Petrucci, “A review of automated methods for the collection and analysis of photoelastic data,” J. Strain Anal. 33, 75–91 (1998).
[CrossRef]

1997 (2)

E. D. Fabrizio, M. Baciocchi, M. Gentili, L. Grella, “Microphotonic devices fabricated by silicon micromachining techniques,” J. Appl. Phys. 36, 7757–7762 (1997).

D. A. B. Miller, “Physical reasons for optical interconnection,” Intl. J. Opt. 11, 155–168 (1997).

1995 (2)

R. Oldenbourg, G. Mei, “New polarized light microscope with precision universal compensator,” J. Microsc. 180, 140–147 (1995).
[CrossRef] [PubMed]

A. Ajovalasit, S. Baronne, G. Petrucci, “Automated photoelasticity in white light: influences of quarter-wave plates,” J. Strain Anal. 30, 29–34 (1995).
[CrossRef]

1986 (1)

K. Kitamura, N. Lyi, S. Kimura, “Growth-induced optical anisotropy of epitaxial garnet films grown on (110)-oriented substrates,” J. Appl. Phys. 60, 1486–1489 (1986).
[CrossRef]

1985 (1)

A. Redner, “Photoelastic measurements by means of computer-assisted spectral-contents analysis,” Exp. Mech. 25, 148–153 (1985).
[CrossRef]

1983 (2)

K. Kitamura, S. Kimura, Y. Miyazawa, Y. Mori, O. Kamada, “Stress-birefringence associated with facets of rare-earth garnets grown from the melt; a model and measurement of stress-birefringence observed in thin sections,” J. Cryst. Growth 62, 351–359 (1983).
[CrossRef]

K. Kitamura, Y. Miyazawa, Y. Mori, S. Kimura, M. Higuchi, “Origin of difference in lattice spacings between on- and off-facet regions of rare-earth garnets grown from the melt,” J. Cryst. Growth 64, 207–216 (1983).
[CrossRef]

1982 (1)

1980 (2)

G. W. Scherer, “Stress-induced index profile distortion in optical waveguides,” Appl. Opt. 19, 2000–2006 (1980).
[CrossRef] [PubMed]

G. W. Scherer, “Stress-optical effects in optical waveguides,” J. Non-Cryst. Solids 38, 201–204 (1980).
[CrossRef]

1979 (1)

G. W. Scherer, “Thermal stress in a cylinder: application to optical waveguide blanks,” J. Non-Cryst. Solids 34, 223–238 (1979).
[CrossRef]

1975 (1)

U. C. Paek, C. R. Kurkjian, “Calculation of cooling rate and induced stresses in drawing of optical fibers,” J. Am. Ceram. Soc. 58, 330–335 (1975).
[CrossRef]

1970 (1)

D. A. Krohn, “Determination of axial stress in clad glass fibers,” J. Am. Ceram. Soc. 53, 505–507 (1970).
[CrossRef]

1963 (1)

R. D. Allen, L. I. Rebhun, “Photoelectric measurement of small fluctuating retardations in weakly birefringent, light-scattering biological objects,” Exp. Cell Res. 29, 583–592 (1963).
[CrossRef] [PubMed]

1950 (1)

M. M. Swann, J. M. Mitchison, “Refinements in polarized light microscopy,” J. Exper. Biol. 27, 226–237 (1950).

Ahan, T. J.

Ahn, T. J.

Ajovalasit, A.

A. Ajovalasit, S. Baronne, G. Petrucci, “A review of automated methods for the collection and analysis of photoelastic data,” J. Strain Anal. 33, 75–91 (1998).
[CrossRef]

A. Ajovalasit, S. Baronne, G. Petrucci, “Automated photoelasticity in white light: influences of quarter-wave plates,” J. Strain Anal. 30, 29–34 (1995).
[CrossRef]

Allen, R. D.

R. D. Allen, L. I. Rebhun, “Photoelectric measurement of small fluctuating retardations in weakly birefringent, light-scattering biological objects,” Exp. Cell Res. 29, 583–592 (1963).
[CrossRef] [PubMed]

Baciocchi, M.

E. D. Fabrizio, M. Baciocchi, M. Gentili, L. Grella, “Microphotonic devices fabricated by silicon micromachining techniques,” J. Appl. Phys. 36, 7757–7762 (1997).

Baronne, S.

A. Ajovalasit, S. Baronne, G. Petrucci, “A review of automated methods for the collection and analysis of photoelastic data,” J. Strain Anal. 33, 75–91 (1998).
[CrossRef]

A. Ajovalasit, S. Baronne, G. Petrucci, “Automated photoelasticity in white light: influences of quarter-wave plates,” J. Strain Anal. 30, 29–34 (1995).
[CrossRef]

Bernussi, A. A.

L. G. Peralta, A. A. Bernussi, H. Temkin, M. M. Borhani, D. E. Doucette, “Silicon-dioxide waveguides with low birefringence,” IEEE J. Quantum Electron. 39, 874–879 (2003).
[CrossRef]

Borhani, M. M.

L. G. Peralta, A. A. Bernussi, H. Temkin, M. M. Borhani, D. E. Doucette, “Silicon-dioxide waveguides with low birefringence,” IEEE J. Quantum Electron. 39, 874–879 (2003).
[CrossRef]

Cense, B.

Chan, H. P.

S. Y. Cheng, K. S. Chiang, H. P. Chan, “Birefringence in benzocyclobutene strip optical waveguides,” IEEE Photon. Technol. Lett. 15, 700–702 (2003).
[CrossRef]

Chen, T. C.

Cheng, S. Y.

S. Y. Cheng, K. S. Chiang, H. P. Chan, “Birefringence in benzocyclobutene strip optical waveguides,” IEEE Photon. Technol. Lett. 15, 700–702 (2003).
[CrossRef]

Chiang, K. S.

S. Y. Cheng, K. S. Chiang, H. P. Chan, “Birefringence in benzocyclobutene strip optical waveguides,” IEEE Photon. Technol. Lett. 15, 700–702 (2003).
[CrossRef]

Chu, P. L.

Chung, Y.

Dossou, K.

Doucette, D. E.

L. G. Peralta, A. A. Bernussi, H. Temkin, M. M. Borhani, D. E. Doucette, “Silicon-dioxide waveguides with low birefringence,” IEEE J. Quantum Electron. 39, 874–879 (2003).
[CrossRef]

Fabrizio, E. D.

E. D. Fabrizio, M. Baciocchi, M. Gentili, L. Grella, “Microphotonic devices fabricated by silicon micromachining techniques,” J. Appl. Phys. 36, 7757–7762 (1997).

Fontaine, M.

Gaylord, T. K.

C. C. Montarou, T. K. Gaylord, “Single-point two-waveplate compensator for optical retardation, thickness, and refractive index measurement,” U.S. utility patent60/506,381 (24September2004).

Gentili, M.

E. D. Fabrizio, M. Baciocchi, M. Gentili, L. Grella, “Microphotonic devices fabricated by silicon micromachining techniques,” J. Appl. Phys. 36, 7757–7762 (1997).

Grella, L.

E. D. Fabrizio, M. Baciocchi, M. Gentili, L. Grella, “Microphotonic devices fabricated by silicon micromachining techniques,” J. Appl. Phys. 36, 7757–7762 (1997).

Hammar, K.

K. Katoh, K. Hammar, P. Smith, R. Oldenbourg, “Birefringence imaging directly reveals architectural dynamics of filamentous actin in living growth cones,” Mol. Biol. Cell 10, 197–210 (1999).
[CrossRef] [PubMed]

Han, W. T.

Han, Y.

C. S. Kim, Y. Han, B. H. Lee, W. T. Han, U. C. Paek, Y. Chung, “Induction of the refractive index change in B-doped optical fibers through relaxation of the mechanical stress,” Opt. Commun. 185, 337–342 (2000).
[CrossRef]

Hartshorne, N. H.

N. H. Hartshorne, A. Stuart, Crystals And The Polarizing Microscope, A Handbook For Chemists And Others (Edward Arnold, London, 1960).

Higuchi, M.

K. Kitamura, Y. Miyazawa, Y. Mori, S. Kimura, M. Higuchi, “Origin of difference in lattice spacings between on- and off-facet regions of rare-earth garnets grown from the melt,” J. Cryst. Growth 64, 207–216 (1983).
[CrossRef]

Houston, J. E.

J. E. Houston, “The effect of stress on the nanomechanical properties of Au surfaces,” in Structure and Evolution of Surfaces, R. Cammarata, ed., Proc. Mater. Res. Soc.440, 177–187 (1997).

Huang, M.

M. Huang, “Stress effects on the performance of optical waveguides,” Intl. J. Solids Struct. 40, 1615–1632 (2003).
[CrossRef]

Kamada, O.

K. Kitamura, S. Kimura, Y. Miyazawa, Y. Mori, O. Kamada, “Stress-birefringence associated with facets of rare-earth garnets grown from the melt; a model and measurement of stress-birefringence observed in thin sections,” J. Cryst. Growth 62, 351–359 (1983).
[CrossRef]

Katoh, K.

K. Katoh, K. Hammar, P. Smith, R. Oldenbourg, “Birefringence imaging directly reveals architectural dynamics of filamentous actin in living growth cones,” Mol. Biol. Cell 10, 197–210 (1999).
[CrossRef] [PubMed]

Kilian, A.

Kim, B. H.

Kim, C. S.

C. S. Kim, Y. Han, B. H. Lee, W. T. Han, U. C. Paek, Y. Chung, “Induction of the refractive index change in B-doped optical fibers through relaxation of the mechanical stress,” Opt. Commun. 185, 337–342 (2000).
[CrossRef]

Kim, D. Y.

Kim, Y. H.

Kimura, S.

K. Kitamura, N. Lyi, S. Kimura, “Growth-induced optical anisotropy of epitaxial garnet films grown on (110)-oriented substrates,” J. Appl. Phys. 60, 1486–1489 (1986).
[CrossRef]

K. Kitamura, S. Kimura, Y. Miyazawa, Y. Mori, O. Kamada, “Stress-birefringence associated with facets of rare-earth garnets grown from the melt; a model and measurement of stress-birefringence observed in thin sections,” J. Cryst. Growth 62, 351–359 (1983).
[CrossRef]

K. Kitamura, Y. Miyazawa, Y. Mori, S. Kimura, M. Higuchi, “Origin of difference in lattice spacings between on- and off-facet regions of rare-earth garnets grown from the melt,” J. Cryst. Growth 64, 207–216 (1983).
[CrossRef]

Kirchhof, J.

Kitamura, K.

K. Kitamura, N. Lyi, S. Kimura, “Growth-induced optical anisotropy of epitaxial garnet films grown on (110)-oriented substrates,” J. Appl. Phys. 60, 1486–1489 (1986).
[CrossRef]

K. Kitamura, S. Kimura, Y. Miyazawa, Y. Mori, O. Kamada, “Stress-birefringence associated with facets of rare-earth garnets grown from the melt; a model and measurement of stress-birefringence observed in thin sections,” J. Cryst. Growth 62, 351–359 (1983).
[CrossRef]

K. Kitamura, Y. Miyazawa, Y. Mori, S. Kimura, M. Higuchi, “Origin of difference in lattice spacings between on- and off-facet regions of rare-earth garnets grown from the melt,” J. Cryst. Growth 64, 207–216 (1983).
[CrossRef]

Krohn, D. A.

D. A. Krohn, “Determination of axial stress in clad glass fibers,” J. Am. Ceram. Soc. 53, 505–507 (1970).
[CrossRef]

Kuhlow, B.

Kurkjian, C. R.

U. C. Paek, C. R. Kurkjian, “Calculation of cooling rate and induced stresses in drawing of optical fibers,” J. Am. Ceram. Soc. 58, 330–335 (1975).
[CrossRef]

LaRochelle, S.

Lee, B. H.

Li, C.

Lyi, N.

K. Kitamura, N. Lyi, S. Kimura, “Growth-induced optical anisotropy of epitaxial garnet films grown on (110)-oriented substrates,” J. Appl. Phys. 60, 1486–1489 (1986).
[CrossRef]

Mei, G.

R. Oldenbourg, G. Mei, “New polarized light microscope with precision universal compensator,” J. Microsc. 180, 140–147 (1995).
[CrossRef] [PubMed]

Miller, D. A. B.

D. A. B. Miller, “Physical reasons for optical interconnection,” Intl. J. Opt. 11, 155–168 (1997).

Mitchison, J. M.

M. M. Swann, J. M. Mitchison, “Refinements in polarized light microscopy,” J. Exper. Biol. 27, 226–237 (1950).

Miyazawa, Y.

K. Kitamura, S. Kimura, Y. Miyazawa, Y. Mori, O. Kamada, “Stress-birefringence associated with facets of rare-earth garnets grown from the melt; a model and measurement of stress-birefringence observed in thin sections,” J. Cryst. Growth 62, 351–359 (1983).
[CrossRef]

K. Kitamura, Y. Miyazawa, Y. Mori, S. Kimura, M. Higuchi, “Origin of difference in lattice spacings between on- and off-facet regions of rare-earth garnets grown from the melt,” J. Cryst. Growth 64, 207–216 (1983).
[CrossRef]

Montarou, C. C.

C. C. Montarou, T. K. Gaylord, “Single-point two-waveplate compensator for optical retardation, thickness, and refractive index measurement,” U.S. utility patent60/506,381 (24September2004).

Mori, Y.

K. Kitamura, S. Kimura, Y. Miyazawa, Y. Mori, O. Kamada, “Stress-birefringence associated with facets of rare-earth garnets grown from the melt; a model and measurement of stress-birefringence observed in thin sections,” J. Cryst. Growth 62, 351–359 (1983).
[CrossRef]

K. Kitamura, Y. Miyazawa, Y. Mori, S. Kimura, M. Higuchi, “Origin of difference in lattice spacings between on- and off-facet regions of rare-earth garnets grown from the melt,” J. Cryst. Growth 64, 207–216 (1983).
[CrossRef]

Oakberg, T.

B. Wang, T. Oakberg, “A new instrument for measuring both magnitude and angle of low level linear birefringence,” Rev. Sci. Instrum. 70, 3847–3854 (1999).
[CrossRef]

Oakberg, T. C.

T. C. Oakberg, “Measurement of low-level strain birefringence in optical elements using a photoelastic modulator,” in Polarization Analysis and Applications to Device, T. Yoshizawa, H. Yokota, eds., Proc. SPIE2873, 17–20 (1996).

Oldenbourg, R.

K. Katoh, K. Hammar, P. Smith, R. Oldenbourg, “Birefringence imaging directly reveals architectural dynamics of filamentous actin in living growth cones,” Mol. Biol. Cell 10, 197–210 (1999).
[CrossRef] [PubMed]

R. Oldenbourg, E. D. Salmon, P. T. Tran, “Birefringence of single and bundled microtubules,” Biophys. J. 74, 645–654 (1998).
[CrossRef] [PubMed]

R. Oldenbourg, G. Mei, “New polarized light microscope with precision universal compensator,” J. Microsc. 180, 140–147 (1995).
[CrossRef] [PubMed]

Paek, U. C.

Park, Y.

Peralta, L. G.

L. G. Peralta, A. A. Bernussi, H. Temkin, M. M. Borhani, D. E. Doucette, “Silicon-dioxide waveguides with low birefringence,” IEEE J. Quantum Electron. 39, 874–879 (2003).
[CrossRef]

Petrucci, G.

A. Ajovalasit, S. Baronne, G. Petrucci, “A review of automated methods for the collection and analysis of photoelastic data,” J. Strain Anal. 33, 75–91 (1998).
[CrossRef]

A. Ajovalasit, S. Baronne, G. Petrucci, “Automated photoelasticity in white light: influences of quarter-wave plates,” J. Strain Anal. 30, 29–34 (1995).
[CrossRef]

Ponader, C. W.

D. J. Wissuchek, C. W. Ponader, J. J. Price, “Analysis of residual stress in optical fiber,” in Optical Fiber Reliability and Testing, Proc. SPIE3848, 34–43 (1999).
[CrossRef]

Price, J. J.

D. J. Wissuchek, C. W. Ponader, J. J. Price, “Analysis of residual stress in optical fiber,” in Optical Fiber Reliability and Testing, Proc. SPIE3848, 34–43 (1999).
[CrossRef]

Przyrembel, G.

Rebhun, L. I.

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G. S. Smith, An Introduction to Classical Electromagnetic Radiation (Cambridge University, New York, 1997).

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K. Katoh, K. Hammar, P. Smith, R. Oldenbourg, “Birefringence imaging directly reveals architectural dynamics of filamentous actin in living growth cones,” Mol. Biol. Cell 10, 197–210 (1999).
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Stuart, A.

N. H. Hartshorne, A. Stuart, Crystals And The Polarizing Microscope, A Handbook For Chemists And Others (Edward Arnold, London, 1960).

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L. G. Peralta, A. A. Bernussi, H. Temkin, M. M. Borhani, D. E. Doucette, “Silicon-dioxide waveguides with low birefringence,” IEEE J. Quantum Electron. 39, 874–879 (2003).
[CrossRef]

Tran, P. T.

R. Oldenbourg, E. D. Salmon, P. T. Tran, “Birefringence of single and bundled microtubules,” Biophys. J. 74, 645–654 (1998).
[CrossRef] [PubMed]

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B. Wang, “Linear birefringence measurement instrument using two photoelastic modulators,” Opt. Eng. 41, 981–987 (2002).
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B. Wang, “Accuracy assessment of a linear birefringence measurement system using a Soleil–Babinet compensator,” Rev. Sci. Instrum. 72, 4066–4070 (2001).
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B. Wang, T. Oakberg, “A new instrument for measuring both magnitude and angle of low level linear birefringence,” Rev. Sci. Instrum. 70, 3847–3854 (1999).
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B. Wang, “An improved method for measuring low-level linear birefringence in optical materials,” in Inorganic Optical Materials, A. J. Marker, ed., Proc. SPIE3424, 120–124 (1998).
[CrossRef]

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Wischmann, W.

Wissuchek, D. J.

D. J. Wissuchek, C. W. Ponader, J. J. Price, “Analysis of residual stress in optical fiber,” in Optical Fiber Reliability and Testing, Proc. SPIE3848, 34–43 (1999).
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Appl. Opt. (4)

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[CrossRef] [PubMed]

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R. D. Allen, L. I. Rebhun, “Photoelectric measurement of small fluctuating retardations in weakly birefringent, light-scattering biological objects,” Exp. Cell Res. 29, 583–592 (1963).
[CrossRef] [PubMed]

Exp. Mech. (1)

A. Redner, “Photoelastic measurements by means of computer-assisted spectral-contents analysis,” Exp. Mech. 25, 148–153 (1985).
[CrossRef]

IEEE J. Quantum Electron. (1)

L. G. Peralta, A. A. Bernussi, H. Temkin, M. M. Borhani, D. E. Doucette, “Silicon-dioxide waveguides with low birefringence,” IEEE J. Quantum Electron. 39, 874–879 (2003).
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[CrossRef]

J. Exper. Biol. (1)

M. M. Swann, J. M. Mitchison, “Refinements in polarized light microscopy,” J. Exper. Biol. 27, 226–237 (1950).

J. Lightwave Technol. (2)

J. Microsc. (1)

R. Oldenbourg, G. Mei, “New polarized light microscope with precision universal compensator,” J. Microsc. 180, 140–147 (1995).
[CrossRef] [PubMed]

J. Non-Cryst. Solids (2)

G. W. Scherer, “Stress-optical effects in optical waveguides,” J. Non-Cryst. Solids 38, 201–204 (1980).
[CrossRef]

G. W. Scherer, “Thermal stress in a cylinder: application to optical waveguide blanks,” J. Non-Cryst. Solids 34, 223–238 (1979).
[CrossRef]

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[CrossRef]

Mol. Biol. Cell (1)

K. Katoh, K. Hammar, P. Smith, R. Oldenbourg, “Birefringence imaging directly reveals architectural dynamics of filamentous actin in living growth cones,” Mol. Biol. Cell 10, 197–210 (1999).
[CrossRef] [PubMed]

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C. S. Kim, Y. Han, B. H. Lee, W. T. Han, U. C. Paek, Y. Chung, “Induction of the refractive index change in B-doped optical fibers through relaxation of the mechanical stress,” Opt. Commun. 185, 337–342 (2000).
[CrossRef]

Opt. Eng. (1)

B. Wang, “Linear birefringence measurement instrument using two photoelastic modulators,” Opt. Eng. 41, 981–987 (2002).
[CrossRef]

Opt. Lett. (5)

Rev. Sci. Instrum. (2)

B. Wang, “Accuracy assessment of a linear birefringence measurement system using a Soleil–Babinet compensator,” Rev. Sci. Instrum. 72, 4066–4070 (2001).
[CrossRef]

B. Wang, T. Oakberg, “A new instrument for measuring both magnitude and angle of low level linear birefringence,” Rev. Sci. Instrum. 70, 3847–3854 (1999).
[CrossRef]

Other (8)

N. H. Hartshorne, A. Stuart, Crystals And The Polarizing Microscope, A Handbook For Chemists And Others (Edward Arnold, London, 1960).

G. S. Smith, An Introduction to Classical Electromagnetic Radiation (Cambridge University, New York, 1997).

B. Wang, “An improved method for measuring low-level linear birefringence in optical materials,” in Inorganic Optical Materials, A. J. Marker, ed., Proc. SPIE3424, 120–124 (1998).
[CrossRef]

C. C. Montarou, T. K. Gaylord, “Single-point two-waveplate compensator for optical retardation, thickness, and refractive index measurement,” U.S. utility patent60/506,381 (24September2004).

D. J. Wissuchek, C. W. Ponader, J. J. Price, “Analysis of residual stress in optical fiber,” in Optical Fiber Reliability and Testing, Proc. SPIE3848, 34–43 (1999).
[CrossRef]

T. C. Oakberg, “Measurement of low-level strain birefringence in optical elements using a photoelastic modulator,” in Polarization Analysis and Applications to Device, T. Yoshizawa, H. Yokota, eds., Proc. SPIE2873, 17–20 (1996).

A. Redner, “Photoelastic measurements of residual stresses for NDE,” in Photomechanics and Speckle Metrology, F.-P. Chiang, ed., Proc. SPIE814, 16–19 (1987).
[CrossRef]

J. E. Houston, “The effect of stress on the nanomechanical properties of Au surfaces,” in Structure and Evolution of Surfaces, R. Cammarata, ed., Proc. Mater. Res. Soc.440, 177–187 (1997).

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Figures (22)

Fig. 1
Fig. 1

Wave plate 1 and wave plate 2 with phase retardations equal to ϕ1 and ϕ2 are placed between crossed polarizers. Wave plate 1 and wave plate 2 slow-axis angles with respect to the first polarizer transmission direction are θ1 and θ2, respectively. The Jones rotation matrices are R1) and R2), respectively, and the Jones transmission matrices are T1) and T2).

Fig. 2
Fig. 2

Axes x P and x A are the polarization transmission direction of the polarizers. The wave plates’ slow axes are x 1 and x 2, their fast axes are y 1 and y 2. The wave plates’ slow-axes angles with respect to the polarizer transmission direction are θ1 and θ2.

Fig. 3
Fig. 3

In a two-wave-plate system, the first sample of phase retardation ϕ1 corresponding to λ/18 is at 45° from extinction, and the compensator of phase retardation ϕ2 corresponding to λ/10 is rotated. The intensity is plotted as a function of the compensator slow-axis angle θ2. The solid curve represents the exact intensity; the dotted curve represents the intensity calculated using a small-retardation approximation.

Fig. 4
Fig. 4

Normalized intensity minima for Brace–Köhler compensator applicability range between crossed polarizers. The retardation-based intensity minima are calculated for sample and compensator retardations ranging from 0 to λ. It is assumed that the wave plates are between crossed polarizers. The white region represents retardations for which there are no retardation-based minima.

Fig. 5
Fig. 5

Normalized intensity minima for Brace–Köhler compensator applicability range between parallel polarizers. The retardation-based intensity minima are calculated for sample and compensator retardations ranging from 0 to λ. It is assumed that the wave plates are between parallel polarizers. The white region represents retardations for which there are no retardation-based minima.

Fig. 6
Fig. 6

Retardation-based minima between crossed and parallel polarizers: (a) Retardation based minima between parallel polarizers are superimposed onto those between crossed polarizers. (b) Retardation-based minima between crossed polarizers are superimposed onto those between parallel polarizers.

Fig. 7
Fig. 7

Normalized transmitted intensity for the limiting case for the existence of retardation-based minima. Shown is the compensator phase retardation ϕ2 corresponding to λ/10. The retardation limit of the sample is calculated using Eq. (11) with θ2 equal to -45°. For this limiting case, the retardation-based intensity minimum merges with the nonretardation-based maximum at -45°.

Fig. 8
Fig. 8

Resolvability of the Brace–Köhler compensator technique. (a)–(e) Transmitted intensity for various values of sample phase retardation ϕ1. The compensator phase retardation ϕ2 corresponds to λ/10. The input intensity is equal to 15 mW.

Fig. 9
Fig. 9

Resolution of the Brace–Köhler compensator technique as a function of sample retardation and angular resolution. (a) Compensator retardation equals λ/10. (b) Compensator retardation equals λ/30.

Fig. 10
Fig. 10

Error in Brace–Köhler compensator method for sample and compensator retardations ranging from 0 to λ/8. The error is determined taking into account the measurement angular uncertainty that is due to the sensitivity of the detector and taking into account the small-retardation approximation. Only 22.125% of the total number of error data in the plot are <1%.

Fig. 11
Fig. 11

Magnitude of the angle θ2 producing linearly polarized light is calculated for sample and compensator retardations ranging from 0 to λ. (a) The linearly polarizing angle is calculated when the compensator is rotated. The white region represents retardations for which no linearly polarized output is produced as the compensator is rotated. (b) The linearly polarizing angle is calculated when the sample is rotated. The white region represents retardations for which no linearly polarized output is produced as the sample is rotated.

Fig. 12
Fig. 12

Semiminor, semimajor axes, and the ellipticity of the output light are plotted as a function of the sample slow-axis angle.

Fig. 13
Fig. 13

Output light polarization states are represented for various sample slow-axes angles in the polarizer system of axes x P and y P (Fig. 2). The sample phase retardation ϕsamp corresponds to 0.15λ, and the compensator phase retardation ϕcomp corresponds to 0.45λ. Linearly polarized light is produced for a sample slow-axis angle equal to 6.83°.

Fig. 14
Fig. 14

Semiminor, semimajor axes, and the ellipticity of the output light are plotted as a function of the compensator slow-axis angle.

Fig. 15
Fig. 15

Output light polarization states are represented for various compensator slow-axes angles in the polarizer system of axes x P and y P (Fig. 2). The sample phase retardation ϕsamp corresponds to 0.15λ, and the compensator phase retardation ϕcomp corresponds to 0.45λ. No linearly polarized light is produced when the compensator is rotated.

Fig. 16
Fig. 16

Flow chart representing the experimental procedure to determine whether the sample or the compensator needs to be rotated to produce linearly polarized light. The flow chart is based on having the analyzer transmission direction parallel to the output ellipse semiminor axis. Sample (compensator) and analyzer are successively rotated to observe the variations of the transmitted intensity.

Fig. 17
Fig. 17

Flow chart representing the experimental procedure to measure retardation using the TWC method.

Fig. 18
Fig. 18

Intensity transmitted along the analyzer transmission direction is plotted for a sample phase retardation corresponding to 0.15λ and a compensator phase retardation corresponding to 0.1502λ. The input intensity is 15 mW. In one case, the sample is rotated and no extinction is obtained (upper curve with one global minimum). In the other case, the compensator is rotated and extinction is obtained (lower curve with two global minima and one local maximum).

Fig. 19
Fig. 19

Resolution of the TWC technique as a function of sample retardation and angular resolution. (a) Compensator retardation equals λ/10. (b) Compensator retardation equals λ/30.

Fig. 20
Fig. 20

Measurement error in TWC for any given pair of sample and compensator retardations. The error is calculated based on the measurement angular uncertainty owing to the sensitivity limit of the experiment.

Fig. 21
Fig. 21

Error in TWC method for sample and compensator retardations ranging from 0 to λ/8. The error is determined taking into account the measurement angular uncertainty owing to the sensitivity of the detector. 71.41% of the total number of error data in the plot are <1%.

Fig. 22
Fig. 22

Experimental configuration used to measure small retardations with the TWC, the Brace–Köhler compensator, and the Sénarmont compensator techniques. The light source is a He-Ne laser: D, diaphragm; P, Glan-Thompson polarizer; S, sample; C, compensator; A, Glan-Thompson analyzer; PD, photodetector.

Tables (5)

Tables Icon

Table 1 Semiaxes Lengths, Ellipticity, and Semiminor Axis Angle of the Output Light Polarization Ellipse as a Function of the Sample Slow-Axis Angle θ2 a

Tables Icon

Table 2 Semiaxes Lengths, Ellipticity, and Semiminor Axis Angle of the Output Light Polarization Ellipse as a Function of the Compensator Slow-Axis Angle θ2 a

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Table 3 Karl Lambrecht Wave Plates’ Retardations at λ = 632.8 nm Measured Using Sénarmont Compensator Technique

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Table 4 Comparison between the Brace–Köhler, TWC, and Sénarmont Compensator Techniques

Tables Icon

Table 5 Further Comparison between the Brace–Köhler, TWC, and Sénarmont Compensator Techniques

Equations (44)

Equations on this page are rendered with MathJax. Learn more.

RS=-RC sin2θC,
RS=- RCsin2θS.
εA= sin θ2cos θ2-cos θ2sin θ2100expjϕ2× cosθ2-θ1sinθ2-θ1-sinθ2-θ1cosθ2-θ1100expjϕ1×cos θ1sin θ1-sin θ1cos θ110.
IBK=sin2 2θ2 cos ϕ1 sin2ϕ22+ 12sin 2θ2 sin ϕ1 sin ϕ2+sin2ϕ12,
IBK=-sin2 2θ2 cos ϕ1 sin2ϕ22- 12sin 2θ2×sin ϕ1 sin ϕ2+cos2ϕ12.
IBK+IBK=1.
IAPX=ϕ12+ ϕ22sin 2θ22,
ϕ1=-ϕ2 sin 2θ2.
IBKθ2=cos 2θ22 sin 2θ2 cos ϕ11-cos ϕ2+sin ϕ1 sin ϕ2.
INRBθ2=±45°=sin2ϕ1±ϕ22.
sin 2θ2= sin ϕ1 sin ϕ22 cos ϕ1cos ϕ2-1.
sin ϕ1 sin ϕ22 cos ϕ1cos ϕ2-11.
sin2θ2= sin2θ2+180°=sin290°-θ2=sin2-90°-θ2.
IRB=sin2ϕ12- sin2 ϕ1 sin2 ϕ216 cos ϕ1 sin2ϕ2/2.
0sin2ϕ12- sin2 ϕ1 sin2 ϕ216 cos ϕ1 sin2ϕ22sin2ϕ1+ϕ22.
INRBθ2=±45°=cos2ϕ1±ϕ22,
IRB=cos2ϕ12+ sin2 ϕ1 sin2 ϕ216 cos ϕ1 sin2ϕ22.
0cos2ϕ12+ sin2 ϕ1 sin2 ϕ216 cos ϕ1 sin2ϕ22cos2ϕ1+ϕ22
ϕL1=arctan2 1-cos ϕ2sin ϕ2.
sin ϕ1 sin ϕ22 cos ϕ1cos ϕ2-1<0.999,
sin2θc= sin ϕs1 sin ϕc2 cos ϕs1cos ϕc-1.
tan ϕs2= 2cos ϕc-1sin2θc+2Δθsin ϕc.
ε2=Tϕ2Rθ2ab expj π2,
ε2=a1 expjδ1a2 expjδ2+ϕ2,
a1=a2 cos2 θ2+b2 sin2 θ21/2,
a2= a2 sin2 θ2+b2 cos2 θ2 1/2,
δ1=arctanb sin θ2a cos θ2,
δ2=arctanb cos θ2-a sin θ2.
δ2+ϕ2=δ1+kπ,
sin 2θ2=- tan ϕ1tan ϕ2.
- tan ϕ1tan ϕ21.
I1θ2=±45°=Io cosϕ1±ϕ222,
I2θ2=±45°=Io sin ϕ1±ϕ22 2.
Io cos ϕ1±ϕ22 2>Imin,
Io sin ϕ1±ϕ22 2>Imin.
2 arcsinIminIo<ϕ1-ϕ2<2 arccosIminIo,
2 arcsinIminIo<ϕ1+ϕ2<2 arccosIminIo.
tan ϕs2=-tan ϕc sin2θc+2Δθ,
ε=c1 expjβ1c2 expjβ2,
ωt1=- 12arctanc22 sin2β2-β1c12+c22 cos2β2-β1,
ωt2=ωt1+90°.
S1θ2=±45°=cosϕ1±ϕ22,
S2θ2=±45°=sinϕ1±ϕ22.
Isθ2=±45°=Si ±45° 2Io,

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