Abstract

The high-accuracy universal polarimeter (HAUP) method for simultaneous measurement of birefringence and optical activity in crystals is used when multiple reflections at the plate faces cause modulated variation of the optical parameters. We applied the HAUP method to obtain the birefringence and gyration tensors of α-quartz as a function of temperature in the range 25–100 °C at λ=632.8 nm. Results for birefringence Δn=ne-no and nonnull gyration tensor components g11, g22=g11, and g33, with samples cut in different crystallography planes, are consistent with crystal symmetry and in good agreement with other reported values. For example, the temperature coefficient d(Δn)/dt for a cut perpendicular to the optical axis is -1.06×10-6 (°C)-1, g11=(5.9±0.4)×10-5, and g33=-(10.1±0.2)×10-5 at 24 °C.

© 1998 Optical Society of America

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  1. J. Kobayashi and Y. Uesu, J. Appl. Crystallogr. 16, 204–211 (1983).
    [CrossRef]
  2. J. Kobayashi, Y. Uesu, and H. Takehara, J. Appl. Crystallogr. 16, 212–219 (1983).
    [CrossRef]
  3. J. Kobayashi, H. Kumoni, and K. Saito, J. Appl. Crystallogr. 19, 337–381 (1986).
    [CrossRef]
  4. J. R. L. Moxon and A. R. Renshaw, J. Phys. Condens. Matter 2, 6807–6836 (1990).
    [CrossRef]
  5. D. A. Holmes, J. Opt. Soc. Am. 54, 1115–1120 (1964).
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  6. H. Melle, Optik (Stuttgart) 72, 157–164 (1986).
  7. G. D. Landry and T. A. Maldonado, Appl. Opt. 35, 5870–5879 (1996).
    [CrossRef] [PubMed]
  8. K. Zander, J. Moser, and H. Melle, Optik (Stuttgart) 70, 6–13 (1985).
  9. T. C. Oakberg, Opt. Eng. 34, 1545–1550 (1995).
    [CrossRef]
  10. G. D. Landry and T. A. Maldonado, J. Opt. Soc. Am. A 13, 1737–1748 (1996).
    [CrossRef]
  11. A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley, New York, 1984), Chap. 4, pp. 94–103.
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    [CrossRef] [PubMed]
  14. A. Miller, Phys. Rev. B 8, 5902–5908 (1973).
    [CrossRef]
  15. R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1989), Chap. 1, p. 27.
  16. Ref. 15, Chap. 1, p. 64.
  17. A. J. Rogers, Proc. R. Soc. London Ser. A 353, 177–192 (1977).
    [CrossRef]
  18. F. J. Micheli, American Institute of Physics Handbook (McGraw-Hill, New York, 1972), pp. 6–27.
  19. I. I. Vishnevskii, N. L. Kireeva, M. L. Litvin, and Yu. I. Sazonov, Inorg. Mater. 26, 1255–1258 (1990).
  20. J. P. Bachheimer and G. Dolino, Phys. Rev. B 11, 3195–3205 (1974).
    [CrossRef]
  21. F. A. Modine, R. W. Major, and E. Sonder, Appl. Opt. 14, 757–760 (1975).
    [CrossRef] [PubMed]
  22. J. Kobayashi, T. Asahi, S. Takahashi, and A. M. Glazer, J. Appl. Crystallogr. 21, 479–484 (1988).
    [CrossRef]
  23. J. Kobayashi, M. Takada, N. Hosogaya, and T. Someya, Ferroelectr. Lett. Sect. 8, 145–152 (1988).
    [CrossRef]
  24. H. Horinaka, K. Tomii, H. Sonomura, and T. Miyauchi, Jpn. J. Appl. Phys. 24, 755–760 (1985).
    [CrossRef]
  25. G. Szivessy and Cl. Munster, Ann. Phys. (Leipzig) 20, 703–726 (1934).
    [CrossRef]

1996 (2)

1995 (1)

T. C. Oakberg, Opt. Eng. 34, 1545–1550 (1995).
[CrossRef]

1990 (2)

J. R. L. Moxon and A. R. Renshaw, J. Phys. Condens. Matter 2, 6807–6836 (1990).
[CrossRef]

I. I. Vishnevskii, N. L. Kireeva, M. L. Litvin, and Yu. I. Sazonov, Inorg. Mater. 26, 1255–1258 (1990).

1989 (1)

1988 (2)

J. Kobayashi, T. Asahi, S. Takahashi, and A. M. Glazer, J. Appl. Crystallogr. 21, 479–484 (1988).
[CrossRef]

J. Kobayashi, M. Takada, N. Hosogaya, and T. Someya, Ferroelectr. Lett. Sect. 8, 145–152 (1988).
[CrossRef]

1986 (2)

J. Kobayashi, H. Kumoni, and K. Saito, J. Appl. Crystallogr. 19, 337–381 (1986).
[CrossRef]

H. Melle, Optik (Stuttgart) 72, 157–164 (1986).

1985 (2)

K. Zander, J. Moser, and H. Melle, Optik (Stuttgart) 70, 6–13 (1985).

H. Horinaka, K. Tomii, H. Sonomura, and T. Miyauchi, Jpn. J. Appl. Phys. 24, 755–760 (1985).
[CrossRef]

1983 (2)

J. Kobayashi and Y. Uesu, J. Appl. Crystallogr. 16, 204–211 (1983).
[CrossRef]

J. Kobayashi, Y. Uesu, and H. Takehara, J. Appl. Crystallogr. 16, 212–219 (1983).
[CrossRef]

1977 (1)

A. J. Rogers, Proc. R. Soc. London Ser. A 353, 177–192 (1977).
[CrossRef]

1975 (1)

1974 (1)

J. P. Bachheimer and G. Dolino, Phys. Rev. B 11, 3195–3205 (1974).
[CrossRef]

1973 (1)

A. Miller, Phys. Rev. B 8, 5902–5908 (1973).
[CrossRef]

1964 (1)

1934 (1)

G. Szivessy and Cl. Munster, Ann. Phys. (Leipzig) 20, 703–726 (1934).
[CrossRef]

Asahi, T.

J. Kobayashi, T. Asahi, S. Takahashi, and A. M. Glazer, J. Appl. Crystallogr. 21, 479–484 (1988).
[CrossRef]

Bachheimer, J. P.

J. P. Bachheimer and G. Dolino, Phys. Rev. B 11, 3195–3205 (1974).
[CrossRef]

Dolino, G.

J. P. Bachheimer and G. Dolino, Phys. Rev. B 11, 3195–3205 (1974).
[CrossRef]

Gaylord, T. K.

Glazer, A. M.

J. Kobayashi, T. Asahi, S. Takahashi, and A. M. Glazer, J. Appl. Crystallogr. 21, 479–484 (1988).
[CrossRef]

Holmes, D. A.

Horinaka, H.

H. Horinaka, K. Tomii, H. Sonomura, and T. Miyauchi, Jpn. J. Appl. Phys. 24, 755–760 (1985).
[CrossRef]

Hosogaya, N.

J. Kobayashi, M. Takada, N. Hosogaya, and T. Someya, Ferroelectr. Lett. Sect. 8, 145–152 (1988).
[CrossRef]

Kireeva, N. L.

I. I. Vishnevskii, N. L. Kireeva, M. L. Litvin, and Yu. I. Sazonov, Inorg. Mater. 26, 1255–1258 (1990).

Kobayashi, J.

J. Kobayashi, M. Takada, N. Hosogaya, and T. Someya, Ferroelectr. Lett. Sect. 8, 145–152 (1988).
[CrossRef]

J. Kobayashi, T. Asahi, S. Takahashi, and A. M. Glazer, J. Appl. Crystallogr. 21, 479–484 (1988).
[CrossRef]

J. Kobayashi, H. Kumoni, and K. Saito, J. Appl. Crystallogr. 19, 337–381 (1986).
[CrossRef]

J. Kobayashi and Y. Uesu, J. Appl. Crystallogr. 16, 204–211 (1983).
[CrossRef]

J. Kobayashi, Y. Uesu, and H. Takehara, J. Appl. Crystallogr. 16, 212–219 (1983).
[CrossRef]

Kumoni, H.

J. Kobayashi, H. Kumoni, and K. Saito, J. Appl. Crystallogr. 19, 337–381 (1986).
[CrossRef]

Landry, G. D.

Litvin, M. L.

I. I. Vishnevskii, N. L. Kireeva, M. L. Litvin, and Yu. I. Sazonov, Inorg. Mater. 26, 1255–1258 (1990).

Major, R. W.

Maldonado, T. A.

Melle, H.

H. Melle, Optik (Stuttgart) 72, 157–164 (1986).

K. Zander, J. Moser, and H. Melle, Optik (Stuttgart) 70, 6–13 (1985).

Miller, A.

A. Miller, Phys. Rev. B 8, 5902–5908 (1973).
[CrossRef]

Miyauchi, T.

H. Horinaka, K. Tomii, H. Sonomura, and T. Miyauchi, Jpn. J. Appl. Phys. 24, 755–760 (1985).
[CrossRef]

Modine, F. A.

Moser, J.

K. Zander, J. Moser, and H. Melle, Optik (Stuttgart) 70, 6–13 (1985).

Moxon, J. R. L.

J. R. L. Moxon and A. R. Renshaw, J. Phys. Condens. Matter 2, 6807–6836 (1990).
[CrossRef]

Munster, Cl.

G. Szivessy and Cl. Munster, Ann. Phys. (Leipzig) 20, 703–726 (1934).
[CrossRef]

Oakberg, T. C.

T. C. Oakberg, Opt. Eng. 34, 1545–1550 (1995).
[CrossRef]

Renshaw, A. R.

J. R. L. Moxon and A. R. Renshaw, J. Phys. Condens. Matter 2, 6807–6836 (1990).
[CrossRef]

Rogers, A. J.

A. J. Rogers, Proc. R. Soc. London Ser. A 353, 177–192 (1977).
[CrossRef]

Saito, K.

J. Kobayashi, H. Kumoni, and K. Saito, J. Appl. Crystallogr. 19, 337–381 (1986).
[CrossRef]

Sazonov, Yu. I.

I. I. Vishnevskii, N. L. Kireeva, M. L. Litvin, and Yu. I. Sazonov, Inorg. Mater. 26, 1255–1258 (1990).

Someya, T.

J. Kobayashi, M. Takada, N. Hosogaya, and T. Someya, Ferroelectr. Lett. Sect. 8, 145–152 (1988).
[CrossRef]

Sonder, E.

Sonomura, H.

H. Horinaka, K. Tomii, H. Sonomura, and T. Miyauchi, Jpn. J. Appl. Phys. 24, 755–760 (1985).
[CrossRef]

Szivessy, G.

G. Szivessy and Cl. Munster, Ann. Phys. (Leipzig) 20, 703–726 (1934).
[CrossRef]

Takada, M.

J. Kobayashi, M. Takada, N. Hosogaya, and T. Someya, Ferroelectr. Lett. Sect. 8, 145–152 (1988).
[CrossRef]

Takahashi, S.

J. Kobayashi, T. Asahi, S. Takahashi, and A. M. Glazer, J. Appl. Crystallogr. 21, 479–484 (1988).
[CrossRef]

Takehara, H.

J. Kobayashi, Y. Uesu, and H. Takehara, J. Appl. Crystallogr. 16, 212–219 (1983).
[CrossRef]

Tomii, K.

H. Horinaka, K. Tomii, H. Sonomura, and T. Miyauchi, Jpn. J. Appl. Phys. 24, 755–760 (1985).
[CrossRef]

Uesu, Y.

J. Kobayashi, Y. Uesu, and H. Takehara, J. Appl. Crystallogr. 16, 212–219 (1983).
[CrossRef]

J. Kobayashi and Y. Uesu, J. Appl. Crystallogr. 16, 204–211 (1983).
[CrossRef]

Vishnevskii, I. I.

I. I. Vishnevskii, N. L. Kireeva, M. L. Litvin, and Yu. I. Sazonov, Inorg. Mater. 26, 1255–1258 (1990).

Zander, K.

K. Zander, J. Moser, and H. Melle, Optik (Stuttgart) 70, 6–13 (1985).

Ann. Phys. (Leipzig) (1)

G. Szivessy and Cl. Munster, Ann. Phys. (Leipzig) 20, 703–726 (1934).
[CrossRef]

Appl. Opt. (3)

Ferroelectr. Lett. Sect. (1)

J. Kobayashi, M. Takada, N. Hosogaya, and T. Someya, Ferroelectr. Lett. Sect. 8, 145–152 (1988).
[CrossRef]

Inorg. Mater. (1)

I. I. Vishnevskii, N. L. Kireeva, M. L. Litvin, and Yu. I. Sazonov, Inorg. Mater. 26, 1255–1258 (1990).

J. Appl. Crystallogr. (4)

J. Kobayashi, T. Asahi, S. Takahashi, and A. M. Glazer, J. Appl. Crystallogr. 21, 479–484 (1988).
[CrossRef]

J. Kobayashi and Y. Uesu, J. Appl. Crystallogr. 16, 204–211 (1983).
[CrossRef]

J. Kobayashi, Y. Uesu, and H. Takehara, J. Appl. Crystallogr. 16, 212–219 (1983).
[CrossRef]

J. Kobayashi, H. Kumoni, and K. Saito, J. Appl. Crystallogr. 19, 337–381 (1986).
[CrossRef]

J. Opt. Soc. Am. (1)

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

J. Phys. Condens. Matter (1)

J. R. L. Moxon and A. R. Renshaw, J. Phys. Condens. Matter 2, 6807–6836 (1990).
[CrossRef]

Jpn. J. Appl. Phys. (1)

H. Horinaka, K. Tomii, H. Sonomura, and T. Miyauchi, Jpn. J. Appl. Phys. 24, 755–760 (1985).
[CrossRef]

Opt. Eng. (1)

T. C. Oakberg, Opt. Eng. 34, 1545–1550 (1995).
[CrossRef]

Optik (Stuttgart) (2)

H. Melle, Optik (Stuttgart) 72, 157–164 (1986).

K. Zander, J. Moser, and H. Melle, Optik (Stuttgart) 70, 6–13 (1985).

Phys. Rev. B (2)

A. Miller, Phys. Rev. B 8, 5902–5908 (1973).
[CrossRef]

J. P. Bachheimer and G. Dolino, Phys. Rev. B 11, 3195–3205 (1974).
[CrossRef]

Proc. R. Soc. London Ser. A (1)

A. J. Rogers, Proc. R. Soc. London Ser. A 353, 177–192 (1977).
[CrossRef]

Other (5)

F. J. Micheli, American Institute of Physics Handbook (McGraw-Hill, New York, 1972), pp. 6–27.

A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley, New York, 1984), Chap. 4, pp. 94–103.

J. F. Nye, Physical Properties of Crystals (Oxford U. Press, Oxford, UK, 1985), Chap. 14.

R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1989), Chap. 1, p. 27.

Ref. 15, Chap. 1, p. 64.

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Figures (5)

Fig. 1
Fig. 1

(a) Block diagram of the HAUP: 1, He–Ne laser; 2, beam splitter; 3, polarizer driver; 4, sample holder; 5, analyzer driver; 6, 7, photomultipliers; 8, temperature controller. (b) Schematic representation of the azimuth Y of linearly polarized incident light propagated through polarizer P, the deflecting angle Y+θ of the crystal fast axis S, and the azimuth Φ of analyzer A, all with respect to the reference OX axis.

Fig. 2
Fig. 2

Relationship between the experimental coefficients of transmitted light and temperature for (100) plane levorotatory quartz: (a), (b), and (c), respectively, represent Cexp, Bexp, and Dexp fitted coefficients of the experimental data in accordance with Eqs. (38), (36), and (40).

Fig. 3
Fig. 3

Relationship between experimental phase change and temperature for levorotatory quartz at 632.8 nm for the (100) plane.

Fig. 4
Fig. 4

Temperature dependence of the gyration coefficients for (010), (100), and (001) planes for 632.8-nm wavelength: □, ○, and ⋄ correspond to g22, g11, and g33 components according to (010), (100), and (001) planes, respectively; Δ, ×, and the solid line are the values of Szivessy and Munster,25 Horinaka,24 and Kobayashi,22 respectively. The dotted line is the value corresponding to G=(1/2)(g11+g33).

Fig. 5
Fig. 5

Temperature dependence of the g33 coefficient for α-quartz.

Tables (1)

Tables Icon

Table 1 Temperature Dependence of the Birefringence along a Direction Perpendicular to the Optical Axis of α-Quartz

Equations (50)

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

k=tan(β/2),
tan β=ΔG/ΔB,
G=gijlilj,
Γ(θ, Y, p, q)=A+Bθ+C(θ2+θY)+DY+Y2,
A=A0+(p+q)2+4[k2-k(p-q)-pq]sin2(Δ/2),
B=2(p+q)sin Δ,
C=4 sin2(Δ/2)
D=2(p-k)sin Δ
θ0=(Γ/θ)Y=0=0
θ0=-(1/2)(p+q)cotan(Δ/2),
Γ(θ, Y, p, q)=A+C(θ2+θY)+DY+Y2,
A=A-(B2/4C)+[D-(B/2)-(C/4)]δY+δY2,
B=0,
C=C=4 sin2(Δ/2),
D=D-(B/2)+[2-(C/2)]δY=(γ-2k)sin Δ+2δY cos2(Δ/2),
θ0=-(B/2C)-(1/2)δY+Ψ=-(1/2)(p+q)cot(Δ/2)-(1/2)δY+Ψ.
JF=Man×RT(Y+θ)×T×M×T×R(Y+θ)×Ji,
T=ts00tp,T=ts00tp.
Ji=cos Y cos p-i sin Y sin psin Y cos p+i cos Y sin p.
Man=(1+cos 2Φ cos 2q)(sin 2Φ cos 2q-i sin 2q)(sin 2Φ cos 2q+i sin 2q)(1-cos 2Φ cos 2q),
M=exp(iφ)exp[i(Δ/2)]2 k sin(Δ/2)-2k sin(Δ/2)exp[-i(Δ/2)],
S=ν=1(N)ν-1,
Ω=rs00-rp,
N=r02 exp(i2φ)M112-i2 sin(Δ/2)M12i2 sin(Δ/2)M21M222,
S=1-r02 exp(i2φ)(M222+M122)-i2r02 exp(i2φ)sin(Δ/2)M12i2r02 exp i2φ sin(Δ/2)M211-r02 exp(i2φ)(M112+M212).
S111-r02 exp(i2φ)M222,
S221-r02 exp(i2φ)M112.
M11=S11M11+S12M21S11M11=M11[1-r02 exp(i2φ)M222],
M12=S11M12+S12M22M12[1-r02 exp(i2φ)M222-i2 sin(Δ/2)r02 exp(i2φ)M22],
M21=S21M11+S22M21M21[1-r02 exp(i2φ)M112+i2 sin(Δ/2)r02 exp(i2φ)M11],
M22=S21M12+S22M22S22M22=M22[1-r02 exp(i2φ)M112].
M11=cos(Δ/2)(1-r2)+i sin(Δ/2)(1+r2),
M12=2k sin(Δ/2)(1-r2),
M21=-2k sin(Δ/2)(1-r2),
M22=cos(Δ/2)(1-r2)-i sin(Δ/2)(1+r2),
r2=r02 cos 2φ
=[(nˆ-1)/(nˆ+1)]2 cos(4πnˆd/λ),
Γ=JFJF*.
Bexp=B×(1+2r2 cos Δ)=2(p+q)sin Δ(1+2r2 cos Δ),
Cexp=C×[1+4r2 cos2(Δ/2)]=4 sin2(Δ/2)[1+4r2 cos2(Δ/2)],
Dexp=D×(1+2r2 cos Δ)=2(p-k)sin Δ(1+2r2 cos Δ),
Dexp=D×[1-4r2 sin2(Δ/2)]+2r2(γ-2k)sin Δ=[(γ-2k)sin Δ+2δY cos2(Δ/2)]×[1-4r2 sin2(Δ/2)]+2r2(γ-2k)sin Δ.
Δexp-Δ(1/2)(Ky+Kx-2)cos[(βy+βx)d]×sin[(βy-βx)d]
ΔexpΔ+2r2 sin Δ,
KxKy=(n¯2+1)/2n¯,
βx+βy=(2π/λ)(nx+ny)(4π/λ)n¯=2φ/d,
βy-βx=(2π/λ)Δn
Dexp(90°)=-[(γ+2k)sin Δ-2δY cos2(Δ/2)]×[1-4r2 sin2(Δ/2)]-2r2(γ+2k)sin Δ,
Dexp-Dexp(90°)=-4k sin Δ(1+2r2 cos Δ).
G45=(1/2)(g11+g33).

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