Abstract

Measurements of the optical properties, and thus the optical constants, of metals at submillimeter wavelengths are almost nonexistent. We used a nonresonant cavity to measure at ambient temperature the angle averaged absorptance spectra P(ω) of gold, nickel, and lead in the 30–300-cm−1 wave-number region. The real part of the normalized surface impedance spectrum z(ω) = r(ω) + ix(ω) was determined from P(ω). The r(ω) spectrum was combined with previous measurements by others at higher frequencies, and Kramers-Kronig analyses of the resultant r(ω) spectra provided (ω) = 1(ω) + i∊2(ω) and N(ω) = n(ω) + ik(ω) for gold and nickel in the 35–15,000-cm−1 region and for lead in the 15–15,000-cm−1 region. We also derived an exact analytical expression for P(ω) of a metal.

© 1987 Optical Society of America

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References

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  1. E. D. Palik, Handbook of Optical Constants of Solids (Academic, New York, 1985).
  2. J. H. Weaver, C. Krafka, D. W. Lynch, E. E. Koch, Physics Data, Optical Properties of Metals, Part 1: The Transition Metals and Physics Data, Optical Properties of Metals, Part 2: The Noble Metals, Aluminum, Scandium, Yttrium, the Lanthanides, and the Actinides (Fachinformationszentrum, 7514 Eggenatein-Leopoldshafen 2, Karlsruhe, F.R.G., 1981).
  3. F. E. Pinkerton, A. J. Sievers, “Quantitative FIR Absorptivity Measurements of Metals with Dual Nonresonant Cavities,” Infrared Phys. 22, 377 (1982).
    [CrossRef]
  4. F. E. Pinkerton, A. J. Sievers, M. B. Maple, B. C. Sales, “Enhanced Far-Infrared Absorption in CdPd3 and YbCuSi2 Experiment,” Phys. Rev. B 29, 609 (1984).
    [CrossRef]
  5. G. Brandli, A. J. Sievers, “Absolute Measurement of the Far-Infrared Surface Resistance of Pb,” Phys. Rev. B 5, 3550 (1972).
    [CrossRef]
  6. W. E. Lamb, “Theory of a Microwave Spectroscope,” Phys. Rev. 70, 308 (1946).
    [CrossRef]
  7. W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes: the Art of Scientific Computing (Cambridge U.P., New York, 1986).
  8. M. A. Ordal, R. J. Bell, R. W. Alexander, L. L. Long, M. R. Querry, “Optical Properties of Fourteen Metals in the Infrared and Far Infrared: Al, Co, Cu, Au, Fe, Pb, Mo, Ni, Pd, Pt, Ag, Ti, V, and W,” Appl. Opt. 24, 4493 (1985).
    [CrossRef] [PubMed]
  9. A. I. Golovashkin, G. P. Motulevich, “Optical Properties of Lead in the Visible and Infrared Spectral Ranges,” Sov. Phys. JETP 26, 881 (1968).
  10. R. J. Bell, M. A. Ordal, R. W. Alexander, “Equations Linking Different Sets of Optical Properties for Nonmagnetic Materials,” Appl. Opt. 24, 3680 (1985).
    [CrossRef] [PubMed]
  11. M. R. Spiegel, Mathematical Handbook of Formulas and Tables, Schaum’s Outline Series (McGraw-Hill, New York, 1968), p. 71; G. A. Korn, T. M. Korn, Mathematical Handbook of Scientists and Engineers (McGraw-Hill, New York, 1968), pp. 931–2.
  12. A. J. Sievers, “Thermal Radiation from Metal Surfaces,” J. Opt. Soc. Am. 68, 1505 (1978).
    [CrossRef]

1985 (2)

1984 (1)

F. E. Pinkerton, A. J. Sievers, M. B. Maple, B. C. Sales, “Enhanced Far-Infrared Absorption in CdPd3 and YbCuSi2 Experiment,” Phys. Rev. B 29, 609 (1984).
[CrossRef]

1982 (1)

F. E. Pinkerton, A. J. Sievers, “Quantitative FIR Absorptivity Measurements of Metals with Dual Nonresonant Cavities,” Infrared Phys. 22, 377 (1982).
[CrossRef]

1978 (1)

1972 (1)

G. Brandli, A. J. Sievers, “Absolute Measurement of the Far-Infrared Surface Resistance of Pb,” Phys. Rev. B 5, 3550 (1972).
[CrossRef]

1968 (1)

A. I. Golovashkin, G. P. Motulevich, “Optical Properties of Lead in the Visible and Infrared Spectral Ranges,” Sov. Phys. JETP 26, 881 (1968).

1946 (1)

W. E. Lamb, “Theory of a Microwave Spectroscope,” Phys. Rev. 70, 308 (1946).
[CrossRef]

Alexander, R. W.

Bell, R. J.

Brandli, G.

G. Brandli, A. J. Sievers, “Absolute Measurement of the Far-Infrared Surface Resistance of Pb,” Phys. Rev. B 5, 3550 (1972).
[CrossRef]

Flannery, B. P.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes: the Art of Scientific Computing (Cambridge U.P., New York, 1986).

Golovashkin, A. I.

A. I. Golovashkin, G. P. Motulevich, “Optical Properties of Lead in the Visible and Infrared Spectral Ranges,” Sov. Phys. JETP 26, 881 (1968).

Koch, E. E.

J. H. Weaver, C. Krafka, D. W. Lynch, E. E. Koch, Physics Data, Optical Properties of Metals, Part 1: The Transition Metals and Physics Data, Optical Properties of Metals, Part 2: The Noble Metals, Aluminum, Scandium, Yttrium, the Lanthanides, and the Actinides (Fachinformationszentrum, 7514 Eggenatein-Leopoldshafen 2, Karlsruhe, F.R.G., 1981).

Krafka, C.

J. H. Weaver, C. Krafka, D. W. Lynch, E. E. Koch, Physics Data, Optical Properties of Metals, Part 1: The Transition Metals and Physics Data, Optical Properties of Metals, Part 2: The Noble Metals, Aluminum, Scandium, Yttrium, the Lanthanides, and the Actinides (Fachinformationszentrum, 7514 Eggenatein-Leopoldshafen 2, Karlsruhe, F.R.G., 1981).

Lamb, W. E.

W. E. Lamb, “Theory of a Microwave Spectroscope,” Phys. Rev. 70, 308 (1946).
[CrossRef]

Long, L. L.

Lynch, D. W.

J. H. Weaver, C. Krafka, D. W. Lynch, E. E. Koch, Physics Data, Optical Properties of Metals, Part 1: The Transition Metals and Physics Data, Optical Properties of Metals, Part 2: The Noble Metals, Aluminum, Scandium, Yttrium, the Lanthanides, and the Actinides (Fachinformationszentrum, 7514 Eggenatein-Leopoldshafen 2, Karlsruhe, F.R.G., 1981).

Maple, M. B.

F. E. Pinkerton, A. J. Sievers, M. B. Maple, B. C. Sales, “Enhanced Far-Infrared Absorption in CdPd3 and YbCuSi2 Experiment,” Phys. Rev. B 29, 609 (1984).
[CrossRef]

Motulevich, G. P.

A. I. Golovashkin, G. P. Motulevich, “Optical Properties of Lead in the Visible and Infrared Spectral Ranges,” Sov. Phys. JETP 26, 881 (1968).

Ordal, M. A.

Palik, E. D.

E. D. Palik, Handbook of Optical Constants of Solids (Academic, New York, 1985).

Pinkerton, F. E.

F. E. Pinkerton, A. J. Sievers, M. B. Maple, B. C. Sales, “Enhanced Far-Infrared Absorption in CdPd3 and YbCuSi2 Experiment,” Phys. Rev. B 29, 609 (1984).
[CrossRef]

F. E. Pinkerton, A. J. Sievers, “Quantitative FIR Absorptivity Measurements of Metals with Dual Nonresonant Cavities,” Infrared Phys. 22, 377 (1982).
[CrossRef]

Press, W. H.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes: the Art of Scientific Computing (Cambridge U.P., New York, 1986).

Querry, M. R.

Sales, B. C.

F. E. Pinkerton, A. J. Sievers, M. B. Maple, B. C. Sales, “Enhanced Far-Infrared Absorption in CdPd3 and YbCuSi2 Experiment,” Phys. Rev. B 29, 609 (1984).
[CrossRef]

Sievers, A. J.

F. E. Pinkerton, A. J. Sievers, M. B. Maple, B. C. Sales, “Enhanced Far-Infrared Absorption in CdPd3 and YbCuSi2 Experiment,” Phys. Rev. B 29, 609 (1984).
[CrossRef]

F. E. Pinkerton, A. J. Sievers, “Quantitative FIR Absorptivity Measurements of Metals with Dual Nonresonant Cavities,” Infrared Phys. 22, 377 (1982).
[CrossRef]

A. J. Sievers, “Thermal Radiation from Metal Surfaces,” J. Opt. Soc. Am. 68, 1505 (1978).
[CrossRef]

G. Brandli, A. J. Sievers, “Absolute Measurement of the Far-Infrared Surface Resistance of Pb,” Phys. Rev. B 5, 3550 (1972).
[CrossRef]

Spiegel, M. R.

M. R. Spiegel, Mathematical Handbook of Formulas and Tables, Schaum’s Outline Series (McGraw-Hill, New York, 1968), p. 71; G. A. Korn, T. M. Korn, Mathematical Handbook of Scientists and Engineers (McGraw-Hill, New York, 1968), pp. 931–2.

Teukolsky, S. A.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes: the Art of Scientific Computing (Cambridge U.P., New York, 1986).

Vetterling, W. T.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes: the Art of Scientific Computing (Cambridge U.P., New York, 1986).

Weaver, J. H.

J. H. Weaver, C. Krafka, D. W. Lynch, E. E. Koch, Physics Data, Optical Properties of Metals, Part 1: The Transition Metals and Physics Data, Optical Properties of Metals, Part 2: The Noble Metals, Aluminum, Scandium, Yttrium, the Lanthanides, and the Actinides (Fachinformationszentrum, 7514 Eggenatein-Leopoldshafen 2, Karlsruhe, F.R.G., 1981).

Appl. Opt. (2)

Infrared Phys. (1)

F. E. Pinkerton, A. J. Sievers, “Quantitative FIR Absorptivity Measurements of Metals with Dual Nonresonant Cavities,” Infrared Phys. 22, 377 (1982).
[CrossRef]

J. Opt. Soc. Am. (1)

Phys. Rev. (1)

W. E. Lamb, “Theory of a Microwave Spectroscope,” Phys. Rev. 70, 308 (1946).
[CrossRef]

Phys. Rev. B (2)

F. E. Pinkerton, A. J. Sievers, M. B. Maple, B. C. Sales, “Enhanced Far-Infrared Absorption in CdPd3 and YbCuSi2 Experiment,” Phys. Rev. B 29, 609 (1984).
[CrossRef]

G. Brandli, A. J. Sievers, “Absolute Measurement of the Far-Infrared Surface Resistance of Pb,” Phys. Rev. B 5, 3550 (1972).
[CrossRef]

Sov. Phys. JETP (1)

A. I. Golovashkin, G. P. Motulevich, “Optical Properties of Lead in the Visible and Infrared Spectral Ranges,” Sov. Phys. JETP 26, 881 (1968).

Other (4)

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes: the Art of Scientific Computing (Cambridge U.P., New York, 1986).

E. D. Palik, Handbook of Optical Constants of Solids (Academic, New York, 1985).

J. H. Weaver, C. Krafka, D. W. Lynch, E. E. Koch, Physics Data, Optical Properties of Metals, Part 1: The Transition Metals and Physics Data, Optical Properties of Metals, Part 2: The Noble Metals, Aluminum, Scandium, Yttrium, the Lanthanides, and the Actinides (Fachinformationszentrum, 7514 Eggenatein-Leopoldshafen 2, Karlsruhe, F.R.G., 1981).

M. R. Spiegel, Mathematical Handbook of Formulas and Tables, Schaum’s Outline Series (McGraw-Hill, New York, 1968), p. 71; G. A. Korn, T. M. Korn, Mathematical Handbook of Scientists and Engineers (McGraw-Hill, New York, 1968), pp. 931–2.

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Figures (6)

Fig. 1
Fig. 1

Normalized surface resistance of Au: solid circles, nonresonant cavity results; dashed line, Lynch and Hunter (tabulated in Ref. 1); dash–dot line, Motulevich et al. (tabulated in Ref. 2); solid line, Brandli and Sievers.5

Fig. 2
Fig. 2

Normalized surface resistance of Ni: solid circles, nonresonant cavity results; dash–dot line, Lynch and Hunter (tabulated in Ref. 1); dash line, average of the results in Ref. 7 with previously unpublished measurements.

Fig. 3
Fig. 3

Normalized surface resistance of Pb: solid circles, nonresonant cavity results; dash–dot line, Motulevich et al.9; solid line, Brandli and Sievers.5

Fig. 4
Fig. 4

Kramers-Kronig results for −1 and 2 of Au: solid line, results obtained by combining nonresonant cavity measurements with the results of Motulevich et al.9; dash line for Lynch and Hunter (tabulated in Ref. 1); dash–dot line for Motulevich et al.9

Fig. 5
Fig. 5

Kramers-Kronig results for −1 and 2 of Ni: solid line, results obtained by combining nonresonant cavity measurements with the Kramers-Kronig results obtained from our reflectance measurements (with the values of Lynch and Hunter from Ref. 1 used above 20,000 cm1); dash–dot line for Lynch and Hunter (tabulated in Ref. 1); and dash line for the Kramers-Kronig results from averaging the reflectance in Ref. 8 with previously unpublished reflectance measurements.

Fig. 6
Fig. 6

Kramers-Kronig results for −1 and 2 of Pb: solid line, results obtained by combining nonresonant cavity measurements with the results of Motulevich et al.9; dash–dot line for the results of Motulevich et al.9

Tables (3)

Tables Icon

Table I Optical Constants of Au Obtained from Kramers-Kronig Analysis of the Normalized Surface Resistance

Tables Icon

Table II Optical Constants of Ni Obtained from Kramers-Kronig Analysis of the Normalized Surface Resistance

Tables Icon

Table III Optical Constants of Pb Obtained from Kramers-Kronig Analysis of the Normalized Surface Resistance

Equations (14)

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I s ( ω ) I r ( ω ) - [ 2 S 1 + ( S 2 + S 3 ) P r ( ω ) ] + S 4 P r ( ω ) [ 2 S 1 + ( S 2 + S 3 ) P r ( ω ) ] + S 4 P s ( ω ) = 0 ,
A p ( η ) = 1 - R p ( η ) = 4 η ( 1 r ) η 2 + 2 η + r ( 1 + ξ 2 ) , A s ( η ) = 1 - R s ( η ) = 4 η r ( 1 + ξ 2 ) η 2 + 2 η + 1 r .
4 y 2 d y a y 2 + b y + c .
y 2 d y a y 2 + b y + c = y a - b 2 a 2 ln ( a y 2 + b y + c ) + b 2 - 2 a c 2 a 2 d y a y 2 + b y + c ,
d y a y 2 + b y + c = { 2 4 a c - b 2 arctan ( 2 a y + b 4 a c - b 2 ) , 1 b 2 - 4 a c ln ( 2 a y + b - b 2 - 4 a c 2 a y + b + b 2 - 4 a c ) .
0 1 A ( η ) η d η = 4 [ η a - b 2 a 2 ln ( a η 2 + b η + c ) + ( b 2 - 2 a c a 2 d ) tan - 1 ( 2 a η + b d ) ] | η = 0 η = 1 = 4 { 1 a - 1 2 a 2 ln ( a + b + c c ) + ( b 2 - 2 a c a 2 d ) [ tan - 1 ( 2 a + b d ) - tan - 1 ( b d ) ] } = 4 { 1 a - 1 2 a 2 ln ( a + b + c c ) + ( b 2 - 2 a c a 2 d ) tan - 1 [ 2 a d d 2 + b ( 2 a + b ) ] } ,
tan - 1 ( A ) - tan - 1 ( B ) = tan - 1 ( A - B 1 + A B ) .
a c = ( 1 r ) [ r ( 1 + ξ 2 ) ] = 1 + ξ 2 , b 2 - 2 a c = 2 2 - 2 ( 1 + ξ 2 ) = 2 ( 1 - ξ 2 ) .
b 2 a 2 = r 2 ,             2 a b = 4 r , 2 a d = 4 ξ r ,             b 2 - 2 a c a 2 d = ( r 2 ξ ) ( 1 - ξ 2 ) , a + b + c c = 1 + 2 r + r 2 ( 1 + ξ 2 ) r 2 ( 1 + ξ 2 ) ,             2 a d d 2 + b ( 2 a + b ) = ξ 1 + r ( 1 + ξ 2 ) .
0 1 A p ( η ) η d η = 4 { r - r 2 ln [ 1 + 2 r + r 2 ( 1 + ξ 2 ) r 2 ( 1 + ξ 2 ) ] + r 2 ( 1 - ξ 2 ) ξ tan - 1 ( ξ 1 + r + r ξ 2 ) } .
b 2 a 2 = 1 r 2 ( 1 + ξ 2 ) 2 ,             2 a b = 4 r ( 1 + ξ 2 ) , b 2 - 2 a c a 2 d = ( 1 - ξ 2 ) ξ r 2 ( 1 + ξ 2 ) 2 , 2 a d = 4 ξ r ( 1 + ξ 2 ) ,             a + b + c c = 1 + 2 r + r 2 ( 1 + ξ 2 ) , 2 a d d 2 + b ( 2 a + b ) = r ξ 1 + r .
0 1 A s ( η ) η d η = 4 { 1 r ( 1 + ξ 2 ) - 1 r 2 ( 1 + ξ 2 ) 2 × ln [ 1 + 2 r + r 2 ( 1 + ξ 2 ) ] + ( 1 - ξ 2 ) ξ r 2 ( 1 + ξ 2 ) 2 tan - 1 ( r ξ 1 + r ) } .
P ( ω ) 4 r ( [ 1 + 1 r 2 ( 1 + ξ 2 ) ] - [ r + 1 r 3 ( 1 + ξ 2 ) 2 ] × ln [ 1 + 2 r + r 2 ( 1 + ξ 2 ) ] + r ln [ r 2 ( 1 + ξ 2 ) ] + ( 1 - ξ 2 ) ξ { r tan - 1 [ ξ 1 + r ( 1 + ξ 2 ) ] + 1 r 3 ( 1 + ξ 2 ) 2 tan - 1 ( r ξ 1 + r ) } ) .
r ( ω ) = 3 16 P ( ω ) [ 1 + 1.6 P ( ω ) ] .

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