Abstract

The effects of changes in temperature, wavelength, and direction of propagation (angle of incidence) on the retardance of zero-order, multiple-order, compound zero-order, and temperature-compensated waveplates are described in detail. A disagreement in the literature regarding the properties of a compound zero-order waveplate is resolved by showing that with respect to temperature and wavelength it behaves like a true zero-order waveplate, but with respect to angle of incidence it behaves like a multiple-order waveplate. A previously proposed temperature-compensated design is shown to suffer from the same directional limitations. A new design for a retarder consisting of one element of a positive uniaxial crystal and one element of a negative uniaxial crystal is proposed. The retardance of such a waveplate would be much less sensitive to the direction of propagation, but somewhat more sensitive to temperature, than a typical compound zero-order waveplate.

© 1988 Optical Society of America

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References

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  1. R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North Holand, Amsterdam, 1977); D. Chandler-Horowitz, “Semiconductor Measurement Technology: Analytic Analysis of Ellipsometric Errors,” Natl. Bur. Stand. (U.S.) Spec. Publ. 400–78 (1986).
  2. G. W. Day, P. D. Hale, M. Deeter, T. E. Milner, D. Conrad, S. M. Etzel, “Limits to the Precision of Electro-Optic and Magneto-Optic Sensors,” Natl. Bur. Stand. (U.S.) Tech. Note 1307 (1987); also published as G. W. Day, P. D. Hale, M. Deeter, T. E. Milner, D. Conrad, S. M. Etzel, Electric Power Research Institute Report 5431, Vol. 1 (Sept.1987).
  3. M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1984).
  4. J. M. Bennett, H. E. Bennett, “Polarization,” in Handbook of Optics, W. G. Driscoll, W. Vaughan, Eds. (McGraw-Hill, New York, 1978).
  5. B. N. Grechushnikov, “Quartz Circular Polarizers,” Opt. Spectrosc. 12, 69 (1962).
  6. Newport Corp. 18235 Mt. Baldy Circle, Fountain Valley, CA 92708, Catalog No. 100, p. M-71.
  7. Karl Lambrecht Corp., Chicago IL 60618, “Waveplates, Rhombs, Rotators, and Depolarizers” (1986), p. 2. Catalog states that a single (multiple-order) quartz plate is “somewhat unsuitable” and that a double cemented quartz plate is “suitable” with respect to angle sensitivity.
  8. D. E. Gray, Ed., American Institute of Physics Handbook (McGraw-Hill, New York, 1972), pp. 6–27, 4–138.
  9. T. Toyoda, M. Yabe, “The Temperature Dependence of the Refractive Indices of Fused Silica and Crystal Quartz,” J. Phys. D 16, L97 (1983).
    [CrossRef]
  10. M. J. Dodge, “Refractive Properties of Magnesium Fluoride,” Appl. Opt. 23, 1980 (1984).
    [CrossRef] [PubMed]
  11. A. Feldman, D. Horowitz, R. M. Waxler, M. J. Dodge, “Optical Materials Characterization,” Natl. Bur. Stand. (U.S.) Tech. Note 993 (1979).
  12. D. A. Holmes, “Exact Theory of Retardation Plates,” J. Opt. Soc. Am. 54, 1115 (1964).
    [CrossRef]
  13. E. O. Ammann, “Temperature Compensated Birefringent Networks,” U.S. Patent3,529,885 (1970).
  14. T. Kimura, M. Saruwatari, “Temperature Compensation of Birefringent Optical Filters,” Proc. IEEE 59, 1273 (1971).
    [CrossRef]
  15. M. A. Jeppesen, “Some Optical Thermo-Optical, and Piezo-Optical Properties of Synthetic Sapphire,” J. Opt. Soc. Am. 48, 629 (1958).
    [CrossRef]

1987 (1)

G. W. Day, P. D. Hale, M. Deeter, T. E. Milner, D. Conrad, S. M. Etzel, “Limits to the Precision of Electro-Optic and Magneto-Optic Sensors,” Natl. Bur. Stand. (U.S.) Tech. Note 1307 (1987); also published as G. W. Day, P. D. Hale, M. Deeter, T. E. Milner, D. Conrad, S. M. Etzel, Electric Power Research Institute Report 5431, Vol. 1 (Sept.1987).

1984 (1)

1983 (1)

T. Toyoda, M. Yabe, “The Temperature Dependence of the Refractive Indices of Fused Silica and Crystal Quartz,” J. Phys. D 16, L97 (1983).
[CrossRef]

1979 (1)

A. Feldman, D. Horowitz, R. M. Waxler, M. J. Dodge, “Optical Materials Characterization,” Natl. Bur. Stand. (U.S.) Tech. Note 993 (1979).

1971 (1)

T. Kimura, M. Saruwatari, “Temperature Compensation of Birefringent Optical Filters,” Proc. IEEE 59, 1273 (1971).
[CrossRef]

1964 (1)

1962 (1)

B. N. Grechushnikov, “Quartz Circular Polarizers,” Opt. Spectrosc. 12, 69 (1962).

1958 (1)

Ammann, E. O.

E. O. Ammann, “Temperature Compensated Birefringent Networks,” U.S. Patent3,529,885 (1970).

Azzam, R. M. A.

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North Holand, Amsterdam, 1977); D. Chandler-Horowitz, “Semiconductor Measurement Technology: Analytic Analysis of Ellipsometric Errors,” Natl. Bur. Stand. (U.S.) Spec. Publ. 400–78 (1986).

Bashara, N. M.

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North Holand, Amsterdam, 1977); D. Chandler-Horowitz, “Semiconductor Measurement Technology: Analytic Analysis of Ellipsometric Errors,” Natl. Bur. Stand. (U.S.) Spec. Publ. 400–78 (1986).

Bennett, H. E.

J. M. Bennett, H. E. Bennett, “Polarization,” in Handbook of Optics, W. G. Driscoll, W. Vaughan, Eds. (McGraw-Hill, New York, 1978).

Bennett, J. M.

J. M. Bennett, H. E. Bennett, “Polarization,” in Handbook of Optics, W. G. Driscoll, W. Vaughan, Eds. (McGraw-Hill, New York, 1978).

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1984).

Conrad, D.

G. W. Day, P. D. Hale, M. Deeter, T. E. Milner, D. Conrad, S. M. Etzel, “Limits to the Precision of Electro-Optic and Magneto-Optic Sensors,” Natl. Bur. Stand. (U.S.) Tech. Note 1307 (1987); also published as G. W. Day, P. D. Hale, M. Deeter, T. E. Milner, D. Conrad, S. M. Etzel, Electric Power Research Institute Report 5431, Vol. 1 (Sept.1987).

Day, G. W.

G. W. Day, P. D. Hale, M. Deeter, T. E. Milner, D. Conrad, S. M. Etzel, “Limits to the Precision of Electro-Optic and Magneto-Optic Sensors,” Natl. Bur. Stand. (U.S.) Tech. Note 1307 (1987); also published as G. W. Day, P. D. Hale, M. Deeter, T. E. Milner, D. Conrad, S. M. Etzel, Electric Power Research Institute Report 5431, Vol. 1 (Sept.1987).

Deeter, M.

G. W. Day, P. D. Hale, M. Deeter, T. E. Milner, D. Conrad, S. M. Etzel, “Limits to the Precision of Electro-Optic and Magneto-Optic Sensors,” Natl. Bur. Stand. (U.S.) Tech. Note 1307 (1987); also published as G. W. Day, P. D. Hale, M. Deeter, T. E. Milner, D. Conrad, S. M. Etzel, Electric Power Research Institute Report 5431, Vol. 1 (Sept.1987).

Dodge, M. J.

M. J. Dodge, “Refractive Properties of Magnesium Fluoride,” Appl. Opt. 23, 1980 (1984).
[CrossRef] [PubMed]

A. Feldman, D. Horowitz, R. M. Waxler, M. J. Dodge, “Optical Materials Characterization,” Natl. Bur. Stand. (U.S.) Tech. Note 993 (1979).

Etzel, S. M.

G. W. Day, P. D. Hale, M. Deeter, T. E. Milner, D. Conrad, S. M. Etzel, “Limits to the Precision of Electro-Optic and Magneto-Optic Sensors,” Natl. Bur. Stand. (U.S.) Tech. Note 1307 (1987); also published as G. W. Day, P. D. Hale, M. Deeter, T. E. Milner, D. Conrad, S. M. Etzel, Electric Power Research Institute Report 5431, Vol. 1 (Sept.1987).

Feldman, A.

A. Feldman, D. Horowitz, R. M. Waxler, M. J. Dodge, “Optical Materials Characterization,” Natl. Bur. Stand. (U.S.) Tech. Note 993 (1979).

Grechushnikov, B. N.

B. N. Grechushnikov, “Quartz Circular Polarizers,” Opt. Spectrosc. 12, 69 (1962).

Hale, P. D.

G. W. Day, P. D. Hale, M. Deeter, T. E. Milner, D. Conrad, S. M. Etzel, “Limits to the Precision of Electro-Optic and Magneto-Optic Sensors,” Natl. Bur. Stand. (U.S.) Tech. Note 1307 (1987); also published as G. W. Day, P. D. Hale, M. Deeter, T. E. Milner, D. Conrad, S. M. Etzel, Electric Power Research Institute Report 5431, Vol. 1 (Sept.1987).

Holmes, D. A.

Horowitz, D.

A. Feldman, D. Horowitz, R. M. Waxler, M. J. Dodge, “Optical Materials Characterization,” Natl. Bur. Stand. (U.S.) Tech. Note 993 (1979).

Jeppesen, M. A.

Kimura, T.

T. Kimura, M. Saruwatari, “Temperature Compensation of Birefringent Optical Filters,” Proc. IEEE 59, 1273 (1971).
[CrossRef]

Milner, T. E.

G. W. Day, P. D. Hale, M. Deeter, T. E. Milner, D. Conrad, S. M. Etzel, “Limits to the Precision of Electro-Optic and Magneto-Optic Sensors,” Natl. Bur. Stand. (U.S.) Tech. Note 1307 (1987); also published as G. W. Day, P. D. Hale, M. Deeter, T. E. Milner, D. Conrad, S. M. Etzel, Electric Power Research Institute Report 5431, Vol. 1 (Sept.1987).

Saruwatari, M.

T. Kimura, M. Saruwatari, “Temperature Compensation of Birefringent Optical Filters,” Proc. IEEE 59, 1273 (1971).
[CrossRef]

Toyoda, T.

T. Toyoda, M. Yabe, “The Temperature Dependence of the Refractive Indices of Fused Silica and Crystal Quartz,” J. Phys. D 16, L97 (1983).
[CrossRef]

Waxler, R. M.

A. Feldman, D. Horowitz, R. M. Waxler, M. J. Dodge, “Optical Materials Characterization,” Natl. Bur. Stand. (U.S.) Tech. Note 993 (1979).

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1984).

Yabe, M.

T. Toyoda, M. Yabe, “The Temperature Dependence of the Refractive Indices of Fused Silica and Crystal Quartz,” J. Phys. D 16, L97 (1983).
[CrossRef]

Appl. Opt. (1)

J. Opt. Soc. Am. (2)

J. Phys. D (1)

T. Toyoda, M. Yabe, “The Temperature Dependence of the Refractive Indices of Fused Silica and Crystal Quartz,” J. Phys. D 16, L97 (1983).
[CrossRef]

Natl. Bur. Stand. (U.S.) Tech. Note (1)

A. Feldman, D. Horowitz, R. M. Waxler, M. J. Dodge, “Optical Materials Characterization,” Natl. Bur. Stand. (U.S.) Tech. Note 993 (1979).

Natl. Bur. Stand. (U.S.) Tech. Note (1)

G. W. Day, P. D. Hale, M. Deeter, T. E. Milner, D. Conrad, S. M. Etzel, “Limits to the Precision of Electro-Optic and Magneto-Optic Sensors,” Natl. Bur. Stand. (U.S.) Tech. Note 1307 (1987); also published as G. W. Day, P. D. Hale, M. Deeter, T. E. Milner, D. Conrad, S. M. Etzel, Electric Power Research Institute Report 5431, Vol. 1 (Sept.1987).

Opt. Spectrosc. (1)

B. N. Grechushnikov, “Quartz Circular Polarizers,” Opt. Spectrosc. 12, 69 (1962).

Proc. IEEE (1)

T. Kimura, M. Saruwatari, “Temperature Compensation of Birefringent Optical Filters,” Proc. IEEE 59, 1273 (1971).
[CrossRef]

Other (7)

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North Holand, Amsterdam, 1977); D. Chandler-Horowitz, “Semiconductor Measurement Technology: Analytic Analysis of Ellipsometric Errors,” Natl. Bur. Stand. (U.S.) Spec. Publ. 400–78 (1986).

Newport Corp. 18235 Mt. Baldy Circle, Fountain Valley, CA 92708, Catalog No. 100, p. M-71.

Karl Lambrecht Corp., Chicago IL 60618, “Waveplates, Rhombs, Rotators, and Depolarizers” (1986), p. 2. Catalog states that a single (multiple-order) quartz plate is “somewhat unsuitable” and that a double cemented quartz plate is “suitable” with respect to angle sensitivity.

D. E. Gray, Ed., American Institute of Physics Handbook (McGraw-Hill, New York, 1972), pp. 6–27, 4–138.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1984).

J. M. Bennett, H. E. Bennett, “Polarization,” in Handbook of Optics, W. G. Driscoll, W. Vaughan, Eds. (McGraw-Hill, New York, 1978).

E. O. Ammann, “Temperature Compensated Birefringent Networks,” U.S. Patent3,529,885 (1970).

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Figures (7)

Fig. 1
Fig. 1

Coordinate system used for a single birefringent waveplate.

Fig. 2
Fig. 2

Coherence effects in a 100-order quartz quarterwave plate at 633-nm wavelength. Solid curves are for coherent light; dashed curves are for incoherent light: (a) uncoated waveplate; (b) same waveplate with 0.25% AR coating.

Fig. 3
Fig. 3

Amplitude transmittance ratio (extraordinary component to ordinary component) for an uncoated 100-order quarterwave plate as a function of temperature.

Fig. 4
Fig. 4

Variation of retardance with temperature in a true zero-order quartz quarterwave plate (~30 μm thick): a, incoherent light, b, with AR coating (R = 0.25%) in coherent light, and b, without coatings in coherent light.

Fig. 5
Fig. 5

Coordinate system used for the second plate in a compound waveplate. The origin of this coordinate system is at the exit point of the first plate.

Fig. 6
Fig. 6

Computed (solid) and measured (dots) values of retardance of a 6.3-mm thick quartz compound quarterwave plate at 633-nm wavelength.

Fig. 7
Fig. 7

Design for a temperature-compensated quarterwave plate using quartz and magnesium fluoride. The two quartz elements represent one compound zero-order waveplate and the magnesium fluoride elements another. The fast axes of the two compound waveplates are orthogonal; the retardances are in degrees.

Tables (2)

Tables Icon

Table I Temperature and Wavelength Dependence of Some Properties of Quartz and Magnesium Fluoride

Tables Icon

Table II Some Properties of Quartz and Sapphire at 633 nm

Equations (55)

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

δ = k B h ,
δ = k B ρ sin 2 θ ,
δ = k B h / cos θ i .
δ = k B h cos θ i .
C = ρ sin 2 θ = x 2 + y 2 ( x 2 + y 2 + z 2 ) 1 / 2 .
x 2 + h 2 = h 2 cos 2 χ ,
z 2 + h 2 = h 2 cos 2 ψ ,
x 2 + h 2 + z 2 = h 2 ( 1 - sin 2 χ sin 2 ψ ) cos 2 ψ cos 2 χ .
δ = k B C = k B h cos ψ cos χ ( 1 - sin 2 χ sin 2 ψ ) 1 / 2 .
d δ d T = d ( k B C ) d T = k B C [ 1 C d C d T + 1 B d B d T ] = k B C [ 1 h d h d T + 1 B d B d T ] ,
γ 1 δ d δ d T = 1 h d h d T + 1 B d B d T ,
1 δ d δ d λ = 1 B d B d λ - 1 λ .
δ = tan - 1 [ N e tan k e h - N o tan k o h 1 + N e N o tan k e h tan k o h ] ,
E e E o = E i e E i o [ cos 2 k o h + N o 2 sin 2 k o h cos 2 k e h + N e 2 sin 2 k e h ] ,
N j = n j 2 + 1 2 n j , k j = 2 π λ n j , j = e , o ,
δ = k B ( h 1 - h 2 ) ,
z 1 h 1 = x 2 h 2 and x 1 h 1 = - z 2 h 2 .
δ = k B ( C 1 - C 2 ) ,
C 1 = x 1 2 + h 1 2 ( x 1 2 + h 1 2 + z 1 2 ) 1 / 2 ,
C 2 = ( h 2 h 1 ) z 1 2 + h 1 2 ( x 1 2 + h 1 2 + z 1 2 ) 1 / 2 .
δ = k B h 1 cos ψ cos χ - h 2 cos χ cos ψ ( 1 - sin 2 χ sin 2 ψ ) 1 / 2
δ = k B ( h 1 / cos χ - h 2 cos χ ) ,
δ k B [ ( h 1 - h 2 ) + χ 2 ( h 1 + h 2 ) / 2 ] .
δ = k B ( h 1 cos ψ - h 2 / cos ψ ) ,
δ k B [ ( h 1 - h 2 ) - ψ 2 ( h 2 + h 2 ) / 2 ] .
1 δ d δ d T = 1 B d B d T + d h 1 d T cos 2 ψ - d h 2 d T cos 2 χ h 1 cos 2 ψ - h 2 cos 2 χ .
d h 1 d T = α h 1 and d h 2 d T = α h 2 ,
1 δ d δ d T = 1 B d B d T + 1 h d h d T
δ = tan - 1 [ N tan Δ 1 - N tan Δ 2 1 + N 2 tan Δ 1 tan Δ 2 ] ,
E 1 E 2 = E 1 i E 2 i [ 1 - 2 r r cos2 Δ 2 + ( r r ) 2 1 - 2 r r cos 2 Δ 1 + ( r r ) 2 ] ,
δ 1 - δ 2 = π / 2 + 2 m π ,
γ 1 δ 1 - γ 2 δ 2 = 0 ,
δ 1 = ( π 2 + 2 m π ) γ 2 γ 2 - γ 1 ,
δ 2 = ( π 2 + 2 m π ) γ 1 γ 2 - γ 1 .
δ 1 + δ 2 = π / 2 + 2 m π ,
γ 1 δ 1 - γ 2 δ 2 = 0 ,
δ 1 = ( π 2 + 2 m π ) γ 2 γ 2 - γ 1 ,
δ 2 = ( π 2 + 2 m π ) γ 1 γ 2 - γ 1 .
δ = k ( B 1 C 1 + B 2 C 2 ) .
z 1 h 1 = z 2 h 2 and x 1 h 1 = x 2 h 2 .
C 1 = ( x 1 2 + h 1 2 ) ( x 1 2 + h 1 2 + z 1 2 ) 1 / 2 ,
C 2 = h 2 h 1 ( x 1 2 + h 1 2 ) ( x 1 2 + h 1 2 + z 1 2 ) 1 / 2 .
δ = k ( B 1 h 1 + B 2 h 2 ) cos ψ cos χ ( 1 - sin 2 χ sin 2 ψ ) 1 / 2 .
k ( B 1 h 1 + B 2 h 2 ) 2 π ,
ψ 2 = n 1 ψ 2 n 2 and χ 2 = n 1 χ 1 n 2 ,
δ = δ 1 + δ 2 = k B 1 h 1 cos ψ 1 cos χ 1 ( 1 - sin 2 χ 1 sin 2 ψ 1 ) 1 / 2 + k B 2 h 2 cos ( n 1 n 2 ψ 1 ) cos ( n 1 n 2 χ 1 ) [ 1 - sin 2 ( n 1 n 2 χ 1 ) sin 2 ( n 1 n 2 ψ 1 ) ] 1 / 2 .
δ = k [ B 1 h 1 cos ψ 1 + B 2 h 2 cos ( n 1 n 2 ψ 1 ) ] ,
δ δ o - 1 2 δ o ψ 1 2 - 1 2 ( n 1 2 - n 2 2 n 2 2 ) k B 2 h 2 ψ 1 2 ,
δ = ( π / 2 ) - 10.9 ψ 1 2 ,
δ = ( π / 2 ) - 26.1 ψ 2 .
δ = ( π / 2 ) - 79 ψ 1 2 ,
δ = ( π / 2 ) - 244 ψ 2 ,
δ = ( π / 2 ) - 89 ψ 1 2 ,
δ = ( π / 2 ) - 213 ψ 2 .
d δ d T = δ 1 γ 1 + δ 2 γ 2 = 8.7 × 10 - 4 ,

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