Abstract

In principle, the real n and imaginary k parts of the complex refractive index of a crystal can be determined by measuring, when possible, its transmission and reflection spectra. We show that it is possible to derive the frequency dependence of n and k in a simple way by using exact expressions for the interference-free transmittance and reflectance to describe low-resolution data. We also derive n and k data starting from one high-resolution spectrum. As an example, we have measured the room-temperature transmittance and reflectance of MgO in the far-infrared region. Reliable n and k data are then obtained in the frequency region 50–250 cm−1.

© 1992 Optical Society of America

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References

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  1. M. Born, E. Wolf, Principles of Optics (Macmillan, New York, 1964).
  2. F. Abeles, Optical Properties of Solids (North-Holland, Amsterdam, 1972), pp. 95–100.
  3. O. S. Heavens, Optical Properties of Thin Solid Films (Dover, New York, 1965), pp. 77–79.
  4. E. V. Loewenstein, D. R. Smith, “Optical constants of far infrared materials. I: Analysis of channeled spectra an application to Mylar,” Appl. Opt. 10, 577–583 (1971).
    [CrossRef] [PubMed]
  5. T. R. Yang, S. Perkowitz, G. L. Carr, R. C. Budhani, G. P. Williams, C. J. Hirschmugl, “Infrared properties of single crystal MgO, a substrate for high temperature superconducting films,” Appl. Opt. 29, 332–333 (1990).
    [CrossRef] [PubMed]
  6. J. R. Jasperse, A. Kahan, J. N. Plendl, “Temperature dependence of infrared dispersion in the ionic crystals LiF and MgO,” Phys. Rev. 146, 526–531 (1966).
    [CrossRef]
  7. P. Calvani, M. Capizzi, S. Lupi, P. Maselli, E. Agostinelli, “High Tc superconducting thin films: an analysis of reflectance spectra,” Physica C 180, 116–119 (1991).
    [CrossRef]
  8. F. Gao, G. L. Carr, C. D. Porter, D. B. Tanner, S. Etemad, T. Venkatesan, A. Inam, B. Dutta, X. D. Wu, G. P. Williams, C. J. Hirschmugl, “Far infrared transmittance and reflectance studies of oriented YBaCuO thin films,” Phys. Rev. B 43, 10383–10389 (1991).
    [CrossRef]
  9. A. Deneuville, D. Tanner, P. H. Holloway, “Optical constants of ZnSe in the far infrared,” Phys. Rev. B 43, 6544–6550 (1991).
    [CrossRef]

1991 (3)

P. Calvani, M. Capizzi, S. Lupi, P. Maselli, E. Agostinelli, “High Tc superconducting thin films: an analysis of reflectance spectra,” Physica C 180, 116–119 (1991).
[CrossRef]

F. Gao, G. L. Carr, C. D. Porter, D. B. Tanner, S. Etemad, T. Venkatesan, A. Inam, B. Dutta, X. D. Wu, G. P. Williams, C. J. Hirschmugl, “Far infrared transmittance and reflectance studies of oriented YBaCuO thin films,” Phys. Rev. B 43, 10383–10389 (1991).
[CrossRef]

A. Deneuville, D. Tanner, P. H. Holloway, “Optical constants of ZnSe in the far infrared,” Phys. Rev. B 43, 6544–6550 (1991).
[CrossRef]

1990 (1)

1971 (1)

1966 (1)

J. R. Jasperse, A. Kahan, J. N. Plendl, “Temperature dependence of infrared dispersion in the ionic crystals LiF and MgO,” Phys. Rev. 146, 526–531 (1966).
[CrossRef]

Abeles, F.

F. Abeles, Optical Properties of Solids (North-Holland, Amsterdam, 1972), pp. 95–100.

Agostinelli, E.

P. Calvani, M. Capizzi, S. Lupi, P. Maselli, E. Agostinelli, “High Tc superconducting thin films: an analysis of reflectance spectra,” Physica C 180, 116–119 (1991).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics (Macmillan, New York, 1964).

Budhani, R. C.

Calvani, P.

P. Calvani, M. Capizzi, S. Lupi, P. Maselli, E. Agostinelli, “High Tc superconducting thin films: an analysis of reflectance spectra,” Physica C 180, 116–119 (1991).
[CrossRef]

Capizzi, M.

P. Calvani, M. Capizzi, S. Lupi, P. Maselli, E. Agostinelli, “High Tc superconducting thin films: an analysis of reflectance spectra,” Physica C 180, 116–119 (1991).
[CrossRef]

Carr, G. L.

F. Gao, G. L. Carr, C. D. Porter, D. B. Tanner, S. Etemad, T. Venkatesan, A. Inam, B. Dutta, X. D. Wu, G. P. Williams, C. J. Hirschmugl, “Far infrared transmittance and reflectance studies of oriented YBaCuO thin films,” Phys. Rev. B 43, 10383–10389 (1991).
[CrossRef]

T. R. Yang, S. Perkowitz, G. L. Carr, R. C. Budhani, G. P. Williams, C. J. Hirschmugl, “Infrared properties of single crystal MgO, a substrate for high temperature superconducting films,” Appl. Opt. 29, 332–333 (1990).
[CrossRef] [PubMed]

Deneuville, A.

A. Deneuville, D. Tanner, P. H. Holloway, “Optical constants of ZnSe in the far infrared,” Phys. Rev. B 43, 6544–6550 (1991).
[CrossRef]

Dutta, B.

F. Gao, G. L. Carr, C. D. Porter, D. B. Tanner, S. Etemad, T. Venkatesan, A. Inam, B. Dutta, X. D. Wu, G. P. Williams, C. J. Hirschmugl, “Far infrared transmittance and reflectance studies of oriented YBaCuO thin films,” Phys. Rev. B 43, 10383–10389 (1991).
[CrossRef]

Etemad, S.

F. Gao, G. L. Carr, C. D. Porter, D. B. Tanner, S. Etemad, T. Venkatesan, A. Inam, B. Dutta, X. D. Wu, G. P. Williams, C. J. Hirschmugl, “Far infrared transmittance and reflectance studies of oriented YBaCuO thin films,” Phys. Rev. B 43, 10383–10389 (1991).
[CrossRef]

Gao, F.

F. Gao, G. L. Carr, C. D. Porter, D. B. Tanner, S. Etemad, T. Venkatesan, A. Inam, B. Dutta, X. D. Wu, G. P. Williams, C. J. Hirschmugl, “Far infrared transmittance and reflectance studies of oriented YBaCuO thin films,” Phys. Rev. B 43, 10383–10389 (1991).
[CrossRef]

Heavens, O. S.

O. S. Heavens, Optical Properties of Thin Solid Films (Dover, New York, 1965), pp. 77–79.

Hirschmugl, C. J.

F. Gao, G. L. Carr, C. D. Porter, D. B. Tanner, S. Etemad, T. Venkatesan, A. Inam, B. Dutta, X. D. Wu, G. P. Williams, C. J. Hirschmugl, “Far infrared transmittance and reflectance studies of oriented YBaCuO thin films,” Phys. Rev. B 43, 10383–10389 (1991).
[CrossRef]

T. R. Yang, S. Perkowitz, G. L. Carr, R. C. Budhani, G. P. Williams, C. J. Hirschmugl, “Infrared properties of single crystal MgO, a substrate for high temperature superconducting films,” Appl. Opt. 29, 332–333 (1990).
[CrossRef] [PubMed]

Holloway, P. H.

A. Deneuville, D. Tanner, P. H. Holloway, “Optical constants of ZnSe in the far infrared,” Phys. Rev. B 43, 6544–6550 (1991).
[CrossRef]

Inam, A.

F. Gao, G. L. Carr, C. D. Porter, D. B. Tanner, S. Etemad, T. Venkatesan, A. Inam, B. Dutta, X. D. Wu, G. P. Williams, C. J. Hirschmugl, “Far infrared transmittance and reflectance studies of oriented YBaCuO thin films,” Phys. Rev. B 43, 10383–10389 (1991).
[CrossRef]

Jasperse, J. R.

J. R. Jasperse, A. Kahan, J. N. Plendl, “Temperature dependence of infrared dispersion in the ionic crystals LiF and MgO,” Phys. Rev. 146, 526–531 (1966).
[CrossRef]

Kahan, A.

J. R. Jasperse, A. Kahan, J. N. Plendl, “Temperature dependence of infrared dispersion in the ionic crystals LiF and MgO,” Phys. Rev. 146, 526–531 (1966).
[CrossRef]

Loewenstein, E. V.

Lupi, S.

P. Calvani, M. Capizzi, S. Lupi, P. Maselli, E. Agostinelli, “High Tc superconducting thin films: an analysis of reflectance spectra,” Physica C 180, 116–119 (1991).
[CrossRef]

Maselli, P.

P. Calvani, M. Capizzi, S. Lupi, P. Maselli, E. Agostinelli, “High Tc superconducting thin films: an analysis of reflectance spectra,” Physica C 180, 116–119 (1991).
[CrossRef]

Perkowitz, S.

Plendl, J. N.

J. R. Jasperse, A. Kahan, J. N. Plendl, “Temperature dependence of infrared dispersion in the ionic crystals LiF and MgO,” Phys. Rev. 146, 526–531 (1966).
[CrossRef]

Porter, C. D.

F. Gao, G. L. Carr, C. D. Porter, D. B. Tanner, S. Etemad, T. Venkatesan, A. Inam, B. Dutta, X. D. Wu, G. P. Williams, C. J. Hirschmugl, “Far infrared transmittance and reflectance studies of oriented YBaCuO thin films,” Phys. Rev. B 43, 10383–10389 (1991).
[CrossRef]

Smith, D. R.

Tanner, D.

A. Deneuville, D. Tanner, P. H. Holloway, “Optical constants of ZnSe in the far infrared,” Phys. Rev. B 43, 6544–6550 (1991).
[CrossRef]

Tanner, D. B.

F. Gao, G. L. Carr, C. D. Porter, D. B. Tanner, S. Etemad, T. Venkatesan, A. Inam, B. Dutta, X. D. Wu, G. P. Williams, C. J. Hirschmugl, “Far infrared transmittance and reflectance studies of oriented YBaCuO thin films,” Phys. Rev. B 43, 10383–10389 (1991).
[CrossRef]

Venkatesan, T.

F. Gao, G. L. Carr, C. D. Porter, D. B. Tanner, S. Etemad, T. Venkatesan, A. Inam, B. Dutta, X. D. Wu, G. P. Williams, C. J. Hirschmugl, “Far infrared transmittance and reflectance studies of oriented YBaCuO thin films,” Phys. Rev. B 43, 10383–10389 (1991).
[CrossRef]

Williams, G. P.

F. Gao, G. L. Carr, C. D. Porter, D. B. Tanner, S. Etemad, T. Venkatesan, A. Inam, B. Dutta, X. D. Wu, G. P. Williams, C. J. Hirschmugl, “Far infrared transmittance and reflectance studies of oriented YBaCuO thin films,” Phys. Rev. B 43, 10383–10389 (1991).
[CrossRef]

T. R. Yang, S. Perkowitz, G. L. Carr, R. C. Budhani, G. P. Williams, C. J. Hirschmugl, “Infrared properties of single crystal MgO, a substrate for high temperature superconducting films,” Appl. Opt. 29, 332–333 (1990).
[CrossRef] [PubMed]

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Macmillan, New York, 1964).

Wu, X. D.

F. Gao, G. L. Carr, C. D. Porter, D. B. Tanner, S. Etemad, T. Venkatesan, A. Inam, B. Dutta, X. D. Wu, G. P. Williams, C. J. Hirschmugl, “Far infrared transmittance and reflectance studies of oriented YBaCuO thin films,” Phys. Rev. B 43, 10383–10389 (1991).
[CrossRef]

Yang, T. R.

Appl. Opt. (2)

Phys. Rev. (1)

J. R. Jasperse, A. Kahan, J. N. Plendl, “Temperature dependence of infrared dispersion in the ionic crystals LiF and MgO,” Phys. Rev. 146, 526–531 (1966).
[CrossRef]

Phys. Rev. B (2)

F. Gao, G. L. Carr, C. D. Porter, D. B. Tanner, S. Etemad, T. Venkatesan, A. Inam, B. Dutta, X. D. Wu, G. P. Williams, C. J. Hirschmugl, “Far infrared transmittance and reflectance studies of oriented YBaCuO thin films,” Phys. Rev. B 43, 10383–10389 (1991).
[CrossRef]

A. Deneuville, D. Tanner, P. H. Holloway, “Optical constants of ZnSe in the far infrared,” Phys. Rev. B 43, 6544–6550 (1991).
[CrossRef]

Physica C (1)

P. Calvani, M. Capizzi, S. Lupi, P. Maselli, E. Agostinelli, “High Tc superconducting thin films: an analysis of reflectance spectra,” Physica C 180, 116–119 (1991).
[CrossRef]

Other (3)

M. Born, E. Wolf, Principles of Optics (Macmillan, New York, 1964).

F. Abeles, Optical Properties of Solids (North-Holland, Amsterdam, 1972), pp. 95–100.

O. S. Heavens, Optical Properties of Thin Solid Films (Dover, New York, 1965), pp. 77–79.

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Figures (3)

Fig. 1
Fig. 1

Low-resolution (10 cm−1) transmittance T0(ν) and reflectance R0(ν) of the 0.5-mm-thick MgO crystal. The solid curve in the T0(ν) plot represents experimental data multipled by 20.

Fig. 2
Fig. 2

High-resolution (0.6 cm−1) transmittance T(ν) and reflectance R(ν) (solid curves), superimposed on the low-resolution data of Fig. 1.

Fig. 3
Fig. 3

n and k frequency dependence as derived from reflectance and transmittance data: n(open circles) and k(open squares) from low-resolution T0(ν) and R0(ν); dashed curve, n, and k from high-resolution T(ν); solid curve, n, and k from high-resolution R(ν).

Equations (17)

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r 01 = 1 - n + i k 1 + n - i k ,
R ˜ = r 01 2 = ( 1 - n ) 2 + k 2 ( 1 + n ) 2 + k 2 .
T = [ 1 - R ˜ 2 - 2 Re ( r 01 2 ) ] D [ A + B cos 2 φ + C sin 2 φ ] ,
R = H + F cos 2 φ A + B cos 2 φ + C sin 2 φ ,
A = 1 + R ˜ 2 D 2 ,             B = - 2 D Re ( r 01 2 ) , C = 2 D lm ( r 01 ) 2 ,             H = R ˜ ( 1 + D 2 ) , F = 2 R ˜ D ,             φ = 2 π ν n d ,             D = exp ( - α d ) .
T = [ 1 + R ˜ 2 - 2 Re ( r 01 2 ) ] D 1 - R ˜ 2 D 2 × [ 1 + N = 1 D N R ˜ N 2 cos 2 N ( θ - φ ) ] .
R = R ˜ ( 1 + D 2 ) 1 - R ˜ 2 D 2 - 2 R ˜ D cos 2 φ 1 - R ˜ 2 D 2 + R ˜ ( 1 + D 2 ) 1 - R ˜ 2 D 2 N = 1 D N R ˜ N 2 cos 2 N ( θ - φ ) - 2 D R ˜ 1 - R ˜ 2 D 2 N = 1 D N R ˜ N × { cos [ 2 N θ - 2 ( N + 1 ) φ ] + cos [ 2 N θ - 2 ( 2 N - 1 ) φ ] } ,
θ = arctan [ lm ( r 01 ) Re ( r 01 ) ] .
T 0 = [ 1 + R ˜ 2 - 2 Re ( r 01 2 ) ] D 1 + R ˜ 2 D 2 .
R 0 = R ˜ ( 1 + D 2 ) 1 - R ˜ 2 D 2 - 2 R ˜ 2 D 2 1 - R ˜ 2 D 2 cos 2 θ ,
T 0 = ( 1 - R ˜ ) 2 D 1 - R ˜ 2 D 2 + 4 [ lm ( r 01 ) ] 2 D 1 - R ˜ 2 D 2 ,
R 0 = R ˜ [ 1 + ( 1 - R ˜ ) 2 D 2 1 - R ˜ 2 D 2 ] + 4 R ˜ [ lm ( r 01 ) ] 2 D 2 1 - R ˜ 2 D 2 .
T 0 = ( 1 - R ˜ ) 2 D 1 - R ˜ 2 D 2 ,
R 0 = R ˜ [ 1 + ( 1 - R ˜ ) 2 D 2 1 - R ˜ 2 D 2 ] = R ˜ [ 1 + T 0 D ] .
R 0 = 1 ± [ T 0 D ( 1 + T 0 D - D 2 ) ] 1 / 2 ,
T 1 = 2 T 0 R ˜ D ,
D 2 - [ T 0 ( 1 - a 2 ) + 2 a ] D + a 2 = 0 ,

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