Abstract

The ordinary and extraordinary refractive index of two samples of sapphire, which differed in the way each was grown, was measured. The measurements were made over a wavelength range of 477–701 nm and a temperature range of 20–295 K. A three-term Sellmeier dispersion equation was fit to the data to permit refractive-index interpolation within several parts in 104. The data of index versus temperature were fit to a model and the results of dn/dT versus temperature are given along with certain physical constants that were extracted from the model.

© 1993 Optical Society of America

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  1. F. Schmid, D. Viechnicki, “Growth of sapphire disks from the melt by a gradient furnace technique,” J. Am. Ceram. Soc. 53, 528–529 (1970).
    [CrossRef]
  2. F. Schmid, D. Viechnicki, “A new approach to high temperature crystal growth from the melt,” Solid State Technol.45–48 (September1973).
  3. H. E. LaBelle, “EFG, the invention and application to sapphire growth,” J. Cryst. Growth 50, (9) 8–17 (1980).
    [CrossRef]
  4. A. C. DeFranzo, B. G. Pazol, C. E. Wheeler, K. A. McCarthy, “Index of refraction measurements at low temperatures,” Rev. Sci. Instrum. 62, 1214 (1990).
    [CrossRef]
  5. HEM sapphire purchased from Crystal Systems, Inc., Salem, Mass.
  6. EFG sapphire supplied by Saphikon, Inc., Milford, N.H.
  7. M. Born, E. Wolf, Principles of Optics, 2nd ed. (Pergamon, Oxford, 1964), Chap. 2.
  8. I. H. Malitson, “Refraction and dispersion of synthetic sapphire,” J. Opt. Soc. Am. 52, 1377–1379 (1962).
    [CrossRef]
  9. igor Graphing and Data Analysis, Wavemetrics, P.O. Box 2088, Lake Oswego, Ore. 97035.
  10. G. N. Ramachandran, “Thermo-optic behavior of solids. I. Diamond,” Proc. Indian Acad. Sci. Sect. A 25, 266–279 (1947);G. N. Ramachandran, “Thermo-optic behavior of solids. II. Fused quartz,” Proc. Indian Acad. Sci. Sect. A 25, 280–285 (1947);G. N. Ramachandran, “Thermo-optic behavior of solids. III. Fluorspar,” Proc. Indian Acad. Sci. Sect. A 25, 286–295 (1947).
  11. T. S. Moss, Photoconductivity (Butterworth, London, 1952).
  12. E. Antoncik, “On the theory of the temperature dependence of the refractive index of homopolar crystals,” Czech. J. Phys. 6, 209–216 (1956).
    [CrossRef]
  13. P. Y. Yu, M. Cardona, “Temperature coefficient of the refractive index of diamond-and-zinc-blende-type semiconductors,” Phys. Rev. B 2, 3193–3197 (1970).
    [CrossRef]
  14. S. C. Yu, “Calculation of electronic band energies in the presence of electron-phonon interaction,” Ph.D. dissertation (Harvard University, Cambridge, Mass., 1964).
  15. J. Van Vechten, “Quantum dielectric theory of electronegativity in covalent systems,” Phys. Rev. 182, 891–905 (1969).
    [CrossRef]
  16. D. Penn, “Wave-number-dependent dielectric function of semiconductors,” Phys. Rev. 128, 2093–2097 (1962).
    [CrossRef]
  17. Y. Tsay, B. Bendow, S. Mitra, “Theory of the temperature derivative of the refractive index in transparent crystals,” Phys. Rev. B 8, 2688–2696 (1973).
    [CrossRef]
  18. J. M. Ziman, Principles of the Theory of Solids (Cambridge U. Press, Cambridge, 1964).
  19. E. Burstein, “Properties of transparent materials,” in Lattice Dynamics,R. F. Wallis, ed. (Pergamon, London, 1965).
  20. Y. F. Tsay, S. S. Mitra, J. F. Vetelino, “Debye–Waller factor and temperature dependence of band gaps of zinc blende type semiconductors,” in Proceedings of the Eleventh International Conference on the Physics of Semiconductors (Polish Science Publishers, Warsaw, 1972).
  21. Y. F. Tsay, G. Gong, S. S. Mitra, J. F. Vetelino, “Temperature dependence of energy gaps of some III–V semiconductors,” Phys. Rev. B 6, 2330–2336 (1972).
    [CrossRef]
  22. M. L. Cohen, V. Heine, Solid State Physics (Academic, New York, 1970), Vol. 24.
    [CrossRef]
  23. The temperature dependence of the plasma energy was neglected based on trial and error polynomial fits to the index versus temperature data. Simple polynomials with integer powers of temperature to represent the plasma energy were used. The most consistent fits to the data were found to contain only a constant term. A more thorough temperature dependence for the plasma energy may exist that could fit the data just as well, but this was not investigated.
  24. D. E. Gray, ed., American Institute of Physics Handbook, 3rd ed. (McGraw-Hill, New York, 1963), p. 9–20.
  25. F. Seitz, D. Turnbull, eds., Solid State Physics, Vol. 6 of Advances in Research and Applications (Academic, New York, 1958), p. 285.
    [CrossRef]
  26. N. A. Kulagin, L. A. Litvinov, “The defects and spectral properties of sapphire grown by melting methods,” Cryst. Res. Technol. 20, 1667–1672 (1985).
    [CrossRef]

1990

A. C. DeFranzo, B. G. Pazol, C. E. Wheeler, K. A. McCarthy, “Index of refraction measurements at low temperatures,” Rev. Sci. Instrum. 62, 1214 (1990).
[CrossRef]

1985

N. A. Kulagin, L. A. Litvinov, “The defects and spectral properties of sapphire grown by melting methods,” Cryst. Res. Technol. 20, 1667–1672 (1985).
[CrossRef]

1980

H. E. LaBelle, “EFG, the invention and application to sapphire growth,” J. Cryst. Growth 50, (9) 8–17 (1980).
[CrossRef]

1973

F. Schmid, D. Viechnicki, “A new approach to high temperature crystal growth from the melt,” Solid State Technol.45–48 (September1973).

Y. Tsay, B. Bendow, S. Mitra, “Theory of the temperature derivative of the refractive index in transparent crystals,” Phys. Rev. B 8, 2688–2696 (1973).
[CrossRef]

1972

Y. F. Tsay, G. Gong, S. S. Mitra, J. F. Vetelino, “Temperature dependence of energy gaps of some III–V semiconductors,” Phys. Rev. B 6, 2330–2336 (1972).
[CrossRef]

1970

F. Schmid, D. Viechnicki, “Growth of sapphire disks from the melt by a gradient furnace technique,” J. Am. Ceram. Soc. 53, 528–529 (1970).
[CrossRef]

P. Y. Yu, M. Cardona, “Temperature coefficient of the refractive index of diamond-and-zinc-blende-type semiconductors,” Phys. Rev. B 2, 3193–3197 (1970).
[CrossRef]

1969

J. Van Vechten, “Quantum dielectric theory of electronegativity in covalent systems,” Phys. Rev. 182, 891–905 (1969).
[CrossRef]

1962

D. Penn, “Wave-number-dependent dielectric function of semiconductors,” Phys. Rev. 128, 2093–2097 (1962).
[CrossRef]

I. H. Malitson, “Refraction and dispersion of synthetic sapphire,” J. Opt. Soc. Am. 52, 1377–1379 (1962).
[CrossRef]

1956

E. Antoncik, “On the theory of the temperature dependence of the refractive index of homopolar crystals,” Czech. J. Phys. 6, 209–216 (1956).
[CrossRef]

1947

G. N. Ramachandran, “Thermo-optic behavior of solids. I. Diamond,” Proc. Indian Acad. Sci. Sect. A 25, 266–279 (1947);G. N. Ramachandran, “Thermo-optic behavior of solids. II. Fused quartz,” Proc. Indian Acad. Sci. Sect. A 25, 280–285 (1947);G. N. Ramachandran, “Thermo-optic behavior of solids. III. Fluorspar,” Proc. Indian Acad. Sci. Sect. A 25, 286–295 (1947).

Antoncik, E.

E. Antoncik, “On the theory of the temperature dependence of the refractive index of homopolar crystals,” Czech. J. Phys. 6, 209–216 (1956).
[CrossRef]

Bendow, B.

Y. Tsay, B. Bendow, S. Mitra, “Theory of the temperature derivative of the refractive index in transparent crystals,” Phys. Rev. B 8, 2688–2696 (1973).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics, 2nd ed. (Pergamon, Oxford, 1964), Chap. 2.

Burstein, E.

E. Burstein, “Properties of transparent materials,” in Lattice Dynamics,R. F. Wallis, ed. (Pergamon, London, 1965).

Cardona, M.

P. Y. Yu, M. Cardona, “Temperature coefficient of the refractive index of diamond-and-zinc-blende-type semiconductors,” Phys. Rev. B 2, 3193–3197 (1970).
[CrossRef]

Cohen, M. L.

M. L. Cohen, V. Heine, Solid State Physics (Academic, New York, 1970), Vol. 24.
[CrossRef]

DeFranzo, A. C.

A. C. DeFranzo, B. G. Pazol, C. E. Wheeler, K. A. McCarthy, “Index of refraction measurements at low temperatures,” Rev. Sci. Instrum. 62, 1214 (1990).
[CrossRef]

Gong, G.

Y. F. Tsay, G. Gong, S. S. Mitra, J. F. Vetelino, “Temperature dependence of energy gaps of some III–V semiconductors,” Phys. Rev. B 6, 2330–2336 (1972).
[CrossRef]

Heine, V.

M. L. Cohen, V. Heine, Solid State Physics (Academic, New York, 1970), Vol. 24.
[CrossRef]

Kulagin, N. A.

N. A. Kulagin, L. A. Litvinov, “The defects and spectral properties of sapphire grown by melting methods,” Cryst. Res. Technol. 20, 1667–1672 (1985).
[CrossRef]

LaBelle, H. E.

H. E. LaBelle, “EFG, the invention and application to sapphire growth,” J. Cryst. Growth 50, (9) 8–17 (1980).
[CrossRef]

Litvinov, L. A.

N. A. Kulagin, L. A. Litvinov, “The defects and spectral properties of sapphire grown by melting methods,” Cryst. Res. Technol. 20, 1667–1672 (1985).
[CrossRef]

Malitson, I. H.

McCarthy, K. A.

A. C. DeFranzo, B. G. Pazol, C. E. Wheeler, K. A. McCarthy, “Index of refraction measurements at low temperatures,” Rev. Sci. Instrum. 62, 1214 (1990).
[CrossRef]

Mitra, S.

Y. Tsay, B. Bendow, S. Mitra, “Theory of the temperature derivative of the refractive index in transparent crystals,” Phys. Rev. B 8, 2688–2696 (1973).
[CrossRef]

Mitra, S. S.

Y. F. Tsay, G. Gong, S. S. Mitra, J. F. Vetelino, “Temperature dependence of energy gaps of some III–V semiconductors,” Phys. Rev. B 6, 2330–2336 (1972).
[CrossRef]

Y. F. Tsay, S. S. Mitra, J. F. Vetelino, “Debye–Waller factor and temperature dependence of band gaps of zinc blende type semiconductors,” in Proceedings of the Eleventh International Conference on the Physics of Semiconductors (Polish Science Publishers, Warsaw, 1972).

Moss, T. S.

T. S. Moss, Photoconductivity (Butterworth, London, 1952).

Pazol, B. G.

A. C. DeFranzo, B. G. Pazol, C. E. Wheeler, K. A. McCarthy, “Index of refraction measurements at low temperatures,” Rev. Sci. Instrum. 62, 1214 (1990).
[CrossRef]

Penn, D.

D. Penn, “Wave-number-dependent dielectric function of semiconductors,” Phys. Rev. 128, 2093–2097 (1962).
[CrossRef]

Ramachandran, G. N.

G. N. Ramachandran, “Thermo-optic behavior of solids. I. Diamond,” Proc. Indian Acad. Sci. Sect. A 25, 266–279 (1947);G. N. Ramachandran, “Thermo-optic behavior of solids. II. Fused quartz,” Proc. Indian Acad. Sci. Sect. A 25, 280–285 (1947);G. N. Ramachandran, “Thermo-optic behavior of solids. III. Fluorspar,” Proc. Indian Acad. Sci. Sect. A 25, 286–295 (1947).

Schmid, F.

F. Schmid, D. Viechnicki, “A new approach to high temperature crystal growth from the melt,” Solid State Technol.45–48 (September1973).

F. Schmid, D. Viechnicki, “Growth of sapphire disks from the melt by a gradient furnace technique,” J. Am. Ceram. Soc. 53, 528–529 (1970).
[CrossRef]

Tsay, Y.

Y. Tsay, B. Bendow, S. Mitra, “Theory of the temperature derivative of the refractive index in transparent crystals,” Phys. Rev. B 8, 2688–2696 (1973).
[CrossRef]

Tsay, Y. F.

Y. F. Tsay, G. Gong, S. S. Mitra, J. F. Vetelino, “Temperature dependence of energy gaps of some III–V semiconductors,” Phys. Rev. B 6, 2330–2336 (1972).
[CrossRef]

Y. F. Tsay, S. S. Mitra, J. F. Vetelino, “Debye–Waller factor and temperature dependence of band gaps of zinc blende type semiconductors,” in Proceedings of the Eleventh International Conference on the Physics of Semiconductors (Polish Science Publishers, Warsaw, 1972).

Van Vechten, J.

J. Van Vechten, “Quantum dielectric theory of electronegativity in covalent systems,” Phys. Rev. 182, 891–905 (1969).
[CrossRef]

Vetelino, J. F.

Y. F. Tsay, G. Gong, S. S. Mitra, J. F. Vetelino, “Temperature dependence of energy gaps of some III–V semiconductors,” Phys. Rev. B 6, 2330–2336 (1972).
[CrossRef]

Y. F. Tsay, S. S. Mitra, J. F. Vetelino, “Debye–Waller factor and temperature dependence of band gaps of zinc blende type semiconductors,” in Proceedings of the Eleventh International Conference on the Physics of Semiconductors (Polish Science Publishers, Warsaw, 1972).

Viechnicki, D.

F. Schmid, D. Viechnicki, “A new approach to high temperature crystal growth from the melt,” Solid State Technol.45–48 (September1973).

F. Schmid, D. Viechnicki, “Growth of sapphire disks from the melt by a gradient furnace technique,” J. Am. Ceram. Soc. 53, 528–529 (1970).
[CrossRef]

Wheeler, C. E.

A. C. DeFranzo, B. G. Pazol, C. E. Wheeler, K. A. McCarthy, “Index of refraction measurements at low temperatures,” Rev. Sci. Instrum. 62, 1214 (1990).
[CrossRef]

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 2nd ed. (Pergamon, Oxford, 1964), Chap. 2.

Yu, P. Y.

P. Y. Yu, M. Cardona, “Temperature coefficient of the refractive index of diamond-and-zinc-blende-type semiconductors,” Phys. Rev. B 2, 3193–3197 (1970).
[CrossRef]

Yu, S. C.

S. C. Yu, “Calculation of electronic band energies in the presence of electron-phonon interaction,” Ph.D. dissertation (Harvard University, Cambridge, Mass., 1964).

Ziman, J. M.

J. M. Ziman, Principles of the Theory of Solids (Cambridge U. Press, Cambridge, 1964).

Cryst. Res. Technol.

N. A. Kulagin, L. A. Litvinov, “The defects and spectral properties of sapphire grown by melting methods,” Cryst. Res. Technol. 20, 1667–1672 (1985).
[CrossRef]

Czech. J. Phys.

E. Antoncik, “On the theory of the temperature dependence of the refractive index of homopolar crystals,” Czech. J. Phys. 6, 209–216 (1956).
[CrossRef]

J. Am. Ceram. Soc.

F. Schmid, D. Viechnicki, “Growth of sapphire disks from the melt by a gradient furnace technique,” J. Am. Ceram. Soc. 53, 528–529 (1970).
[CrossRef]

J. Cryst. Growth

H. E. LaBelle, “EFG, the invention and application to sapphire growth,” J. Cryst. Growth 50, (9) 8–17 (1980).
[CrossRef]

J. Opt. Soc. Am.

Phys. Rev.

J. Van Vechten, “Quantum dielectric theory of electronegativity in covalent systems,” Phys. Rev. 182, 891–905 (1969).
[CrossRef]

D. Penn, “Wave-number-dependent dielectric function of semiconductors,” Phys. Rev. 128, 2093–2097 (1962).
[CrossRef]

Phys. Rev. B

Y. Tsay, B. Bendow, S. Mitra, “Theory of the temperature derivative of the refractive index in transparent crystals,” Phys. Rev. B 8, 2688–2696 (1973).
[CrossRef]

P. Y. Yu, M. Cardona, “Temperature coefficient of the refractive index of diamond-and-zinc-blende-type semiconductors,” Phys. Rev. B 2, 3193–3197 (1970).
[CrossRef]

Y. F. Tsay, G. Gong, S. S. Mitra, J. F. Vetelino, “Temperature dependence of energy gaps of some III–V semiconductors,” Phys. Rev. B 6, 2330–2336 (1972).
[CrossRef]

Proc. Indian Acad. Sci. Sect. A

G. N. Ramachandran, “Thermo-optic behavior of solids. I. Diamond,” Proc. Indian Acad. Sci. Sect. A 25, 266–279 (1947);G. N. Ramachandran, “Thermo-optic behavior of solids. II. Fused quartz,” Proc. Indian Acad. Sci. Sect. A 25, 280–285 (1947);G. N. Ramachandran, “Thermo-optic behavior of solids. III. Fluorspar,” Proc. Indian Acad. Sci. Sect. A 25, 286–295 (1947).

Rev. Sci. Instrum.

A. C. DeFranzo, B. G. Pazol, C. E. Wheeler, K. A. McCarthy, “Index of refraction measurements at low temperatures,” Rev. Sci. Instrum. 62, 1214 (1990).
[CrossRef]

Solid State Technol.

F. Schmid, D. Viechnicki, “A new approach to high temperature crystal growth from the melt,” Solid State Technol.45–48 (September1973).

Other

igor Graphing and Data Analysis, Wavemetrics, P.O. Box 2088, Lake Oswego, Ore. 97035.

HEM sapphire purchased from Crystal Systems, Inc., Salem, Mass.

EFG sapphire supplied by Saphikon, Inc., Milford, N.H.

M. Born, E. Wolf, Principles of Optics, 2nd ed. (Pergamon, Oxford, 1964), Chap. 2.

T. S. Moss, Photoconductivity (Butterworth, London, 1952).

S. C. Yu, “Calculation of electronic band energies in the presence of electron-phonon interaction,” Ph.D. dissertation (Harvard University, Cambridge, Mass., 1964).

J. M. Ziman, Principles of the Theory of Solids (Cambridge U. Press, Cambridge, 1964).

E. Burstein, “Properties of transparent materials,” in Lattice Dynamics,R. F. Wallis, ed. (Pergamon, London, 1965).

Y. F. Tsay, S. S. Mitra, J. F. Vetelino, “Debye–Waller factor and temperature dependence of band gaps of zinc blende type semiconductors,” in Proceedings of the Eleventh International Conference on the Physics of Semiconductors (Polish Science Publishers, Warsaw, 1972).

M. L. Cohen, V. Heine, Solid State Physics (Academic, New York, 1970), Vol. 24.
[CrossRef]

The temperature dependence of the plasma energy was neglected based on trial and error polynomial fits to the index versus temperature data. Simple polynomials with integer powers of temperature to represent the plasma energy were used. The most consistent fits to the data were found to contain only a constant term. A more thorough temperature dependence for the plasma energy may exist that could fit the data just as well, but this was not investigated.

D. E. Gray, ed., American Institute of Physics Handbook, 3rd ed. (McGraw-Hill, New York, 1963), p. 9–20.

F. Seitz, D. Turnbull, eds., Solid State Physics, Vol. 6 of Advances in Research and Applications (Academic, New York, 1958), p. 285.
[CrossRef]

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Figures (22)

Fig. 1
Fig. 1

Sketch of the prism geometry.

Fig. 2
Fig. 2

Dispersion curves for HEM and EFG samples at T = 20 K. The curves are fits to the data.

Fig. 3
Fig. 3

Dispersion curves for HEM and EFG samples at T = 50 K. The curves are fits to the data.

Fig. 4
Fig. 4

Dispersion curves for HEM and EFG samples at T = 77 K. The curves are fits to the data.

Fig. 5
Fig. 5

Dispersion curves for HEM and EFG samples at T = 120 K. The curves are fits to the data.

Fig. 6
Fig. 6

Dispersion curves for HEM and EFG samples at T = 170 K. The curves are fits to the data.

Fig. 7
Fig. 7

Dispersion curves for HEM and EFG samples at T = 200 K. The curves are fits to the data.

Fig. 8
Fig. 8

Dispersion curves for HEM and EFG samples at T = 220 K. The curves are fits to the data.

Fig. 9
Fig. 9

Dispersion curves for HEM and EFG samples at T = 240 K. The curves are fits to the data.

Fig. 10
Fig. 10

Dispersion curves for HEM and EFG samples at T = 260 K. The curves are fits to the data.

Fig. 11
Fig. 11

Dispersion curves for HEM and EFG samples at T = 280 K. The curves are fits to the data.

Fig. 12
Fig. 12

Dispersion curves for HEM and EFG samples at T = 295 K. The curves are fits to the data.

Fig. 13
Fig. 13

Ordinary and extraordinary indices versus temperature at 477 nm for HEM and EFG samples. The curves are fits to the data.

Fig. 14
Fig. 14

Ordinary and extraordinary indices versus temperature at 501 nm for HEM and EFG samples. The curves are fits to the data.

Fig. 15
Fig. 15

Ordinary and extraordinary indices versus temperature at 552 nm for HEM and EFG samples. The curves are fits to the data.

Fig. 16
Fig. 16

Ordinary and extraordinary indices versus temperature at 603 nm for HEM and EFG samples. The curves are fits to the data.

Fig. 17
Fig. 17

Ordinary and extraordinary indices versus temperature at 654 nm for HEM and EFG samples. The curves are fits to the data.

Fig. 18
Fig. 18

Ordinary and extraordinary indices versus temperature at 701 nm for HEM and EFG samples. The curves are fits to the data.

Fig. 19
Fig. 19

dn/dT versus temperature for the ordinary index of the HEM sample.

Fig. 20
Fig. 20

dn/dT versus temperature for the extraordinary index of the HEM sample.

Fig. 21
Fig. 21

dn/dT versus temperature for the ordinary index of the EFG sample.

Fig. 22
Fig. 22

dn/dT versus temperature for the extraordinary index of the EFG sample.

Tables (12)

Tables Icon

Table 1 Ordinary and Extraordinary Index Values at Different Wavelengths and Temperatures for HEM-Grown Sapphirea

Tables Icon

Table 2 Ordinary and Extraordinary Index Values at Different Wavelengths and Temperatures for EFG-Grown Sapphirea

Tables Icon

Table 3 Difference between the Ordinary and Extraordinary Indices for HEM- and EFG-Grown Sapphire at Different Wavelengths and Temperaturesa

Tables Icon

Table 4 Fit Coefficients to a Three-Term Sellmeier Equation for Ordinary (O) and Extraordinary (E) Indices of HEM Sapphire for Each Temperature

Tables Icon

Table 5 Fit Coefficients to a Three-Term Sellmeier Equation for Ordinary (O) and Extraordinary (E) Indices of EFG Sapphire for Each Temperature

Tables Icon

Table 6 Fit Coefficients to Eq. (12) for Ordinary (O) and Extraordinary (E) Indices for HEM Sapphire at Each Wavelength

Tables Icon

Table 7 Fit Coefficients to Eq. (12) for Ordinary (O) and Extraordinary (E) Indices for EFG Sapphire at Each Wavelength

Tables Icon

Table 8 dn/dT Values for Ordinary (O) and Extraordinary (E) Indices at Different Wavelengths and Temperatures for HEM-Grown Sapphire

Tables Icon

Table 9 dn/dT Values for Ordinary (O) and Extraordinary (E) Indices at Different Wavelengths and Tempratures for EFG-Grown Sapphire

Tables Icon

Table 10 Average Gap Energies for Ordinary (O) and Extraordinary (E) Indices for HEM- and EFG-Grown Sapphire

Tables Icon

Table 11 Average Values of Coefficients C and D Used in Eq. (13) to Compute dEg/dT at 295 and 0 Ka

Tables Icon

Table 12 Average Values of Coefficient A Used in Eq. (14) to find the Plasma Energy Associated with Ordinary (O) and Extraordinary (E) Indices of the HEM and EFG Samples

Equations (18)

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

n 2 = 1 + 0 A i λ 2 λ 2 λ i 2 .
4 π χ e ( ω ) = ω pe 2 ( ω g 2 ω 2 ) for ( ω < ω g ) .
4 π χ l ( ω ) = ω pl ω o 2 ω 2 .
ω pl 2 = 4 π N e * 2 μ = ( ɛ 0 ɛ ) ω o 2 ,
ɛ = n 2 = 1 + 4 π ( χ e + χ l )
d n d T = ( d n d T ) e + ( d n d T ) l .
( n T ) p = n 2 1 2 n [ 3 α 2 ω g ( d ω g d T ) p ( 1 ω 2 ω g 2 ) 1 ] .
T [ ln ( χ e ) ] = 1 4 π [ 3 α 2 ω g ( d ω g d T ) p ( 1 ω 2 ω g 2 ) 1 ] ,
E g = ω g ,
E pe = ω pe ,
E = ω .
n 2 = 1 + E pe 2 E g 2 E 2 .
E g = B + C T + D T 3 ,
E pe 2 = A .
d E g d T = C + 3 D T 2 .
C ¯ ( eV / K ) O
D ¯ ( eV / K ) 3 O
A ¯ ( eV / K ) 2 O

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