Abstract

We present models for the optical functions of 11 metals used as mirrors and contacts in optoelectronic and optical devices: noble metals (Ag, Au, Cu), aluminum, beryllium, and transition metals (Cr, Ni, Pd, Pt, Ti, W). We used two simple phenomenological models, the Lorentz–Drude (LD) and the Brendel–Bormann (BB), to interpret both the free-electron and the interband parts of the dielectric response of metals in a wide spectral range from 0.1 to 6 eV. Our results show that the BB model was needed to describe appropriately the interband absorption in noble metals, while for Al, Be, and the transition metals both models exhibit good agreement with the experimental data. A comparison with measurements on surface normal structures confirmed that the reflectance and the phase change on reflection from semiconductor–metal interfaces (including the case of metallic multilayers) can be accurately described by use of the proposed models for the optical functions of metallic films and the matrix method for multilayer calculations.

© 1998 Optical Society of America

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1997 (2)

A. B. Djurišić, J. M. Elazar, A. D. Rakić, “Modeling the optical constants of solids using genetic algorithms with parameter space size adjustment,” Opt. Commun. 134, 407–414 (1997).
[CrossRef]

A. B. Djurišić, A. D. Rakić, J. M. Elazar, “Modeling the optical constants of solids using acceptance-probability-controlled simulated annealing with an adaptive move generation procedure,” Phys. Rev. E 55, 4797–4803 (1997).
[CrossRef]

1996 (5)

T. Baba, R. Watanabe, K. Asano, F. Koyama, K. Iga, “Theoretical and experimental estimations of photon recycling effect in light emitting devices with a metal mirror,” Jpn. J. Appl. Phys. 35(1A), 97–100 (1996).
[CrossRef]

G. Du, K. A. Stair, G. Devane, J. Zhang, R. P. H. Chang, C. W. White, X. Li, Z. Wang, Y. Liu, “Vertical-cavity surface-emitting laser with a thin metal mirror fabricated by double implantation using a tungsten wire mask,” Semicond. Sci. Technol. 11, 1734–1736 (1996).
[CrossRef]

C. M. Herzinger, P. G. Snyder, F. G. Celii, Y. C. Kao, D. Chow, B. Johs, J. A. Woollam, “Studies of thin strained InAs, AlAs, and AlSb layers by spectroscopic ellipsometry,” J. Appl. Phys. 79, 2663–2674 (1996).
[CrossRef]

A. D. Rakić, M. L. Majewski, “Modeling the optical dielectric function of GaAs and AlAs: extension of Adachi’s model,” J. Appl. Phys. 80, 5909–5914 (1996).
[CrossRef]

D. I. Babić, R. P. Mirin, E. L. Hu, J. E. Bowers, “Characterization of metal mirrors on GaAs,” Electron. Lett. 32, 319–320 (1996).
[CrossRef]

1995 (9)

A. D. Rakić, J. M. Elazar, A. B. Djurišić, “Acceptance-probability-controlled simulated annealing: a method for modeling the optical constants of solids,” Phys. Rev. E 52, 6862–6867 (1995).
[CrossRef]

S. T. Wilkinson, N. M. Jokerst, R. P. Leavitt, “Resonant-cavity-enhanced thin-film AlGaAs/GaAs/AlGaAs LED’s with metal mirrors,” Appl. Opt. 34, 8298–8302 (1995).
[CrossRef] [PubMed]

A. D. Rakić, “Algorithm for the determination of intrinsic optical constants of metal films: application to aluminum,” Appl. Opt. 34, 4755–4767 (1995).
[CrossRef] [PubMed]

C. M. Herzinger, H. Yao, P. G. Snyder, F. G. Celii, Y. C. Kao, B. Johs, J. A. Woollam, “Determination of AlAs optical constants by variable-angle spectroscopic ellipsometry and a multisample analysis,” J. Appl. Phys. 77, 4677–4687 (1995).
[CrossRef]

M. Schubert, V. Gottschalch, C. M. Herzinger, H. Yao, P. G. Snyder, J. A. Woollam, “Optical-constants of GaxIn1-xP lattice-matched to GaAs,” J. Appl. Phys. 77, 3416–3419 (1995).
[CrossRef]

M. S. Ünlü, S. Strite, “Resonant cavity enhanced photonic devices,” J. Appl. Phys. 78, 607–639 (1995).
[CrossRef]

G. M. Smith, D. V. Forbes, R. M. Lammert, J. J. Coleman, “Metalization to asymetric cladding separate confinement heterostructure lasers,” Appl. Phys. Lett. 67, 3847–3849 (1995).
[CrossRef]

E. Hadji, J. Bleuse, N. Magnes, J. L. Pautrat, “3.2-μm infrared resonant cavity light emitting diode,” Appl. Phys. Lett. 67, 2591–2593 (1995).
[CrossRef]

D. J. Nash, J. R. Sambles, “Surface plasmon-polariton study of the optical dielectric function of copper,” J. Mod. Opt. 42, 1639–1647 (1995).
[CrossRef]

1994 (2)

K.-H. Lee, K. J. Chang, “First-principles study of the optical properties and the dielectric response of Al,” Phys. Rev. B 49, 2362–2367 (1994).
[CrossRef]

C. H. Wu, P. S. Zory, M. A. Emanuel, “Contact reflectivity effects on thin p-clad InGaAs single quantum-well lasers,” IEEE Photon. Technol. Lett. 6, 1427–1429 (1994).
[CrossRef]

1993 (5)

N. E. J. Hunt, E. F. Schubert, R. F. Kopf, D. L. Sivco, A. Y. Cho, G. J. Zydzik, “Increased fiber communications bandwidth from a resonant cavity light emitting diode emitting at λ = 940 nm,” Appl. Phys. Lett. 63, 2600–2602 (1993).
[CrossRef]

B. Corbett, L. Considine, S. Walsh, W. M. Kelly, “Resonant cavity light emitting diode and detector using epitaxial liftoff,” IEEE Photon. Technol. Lett. 5, 1041–1043 (1993).
[CrossRef]

B. Corbett, L. Considine, S. Walsh, W. M. Kelly, “Narrow bandwidth long wavelength resonant cavity photodiodes,” Electron. Lett. 29, 2148–2149 (1993).
[CrossRef]

H. V. Nguyen, I. An, R. W. Collins, “Evolution of the optical functions of thin-film aluminum: A real-time spectroscopic ellipsometry study,” Phys. Rev. B 47, 3947–3965 (1993).
[CrossRef]

C. C. Kim, J. W. Garland, P. M. Raccah, “Modeling the optical dielectric function of the alloy system AlxGa1-xAs,” Phys. Rev. B 47, 1876–1888 (1993).
[CrossRef]

1992 (4)

K. Iga, “Surface emitting lasers,” Opt. Quant. Electron. 24(2), s97–s104 (1992).
[CrossRef]

C. C. Kim, J. W. Garland, H. Abad, P. M. Raccah, “Modeling the optical dielectric function of semiconductors: extension of the critical-point parabolic-band approximation,” Phys. Rev. B 45, 11,749–11,767 (1992).
[CrossRef]

R. Brendel, D. Bormann, “An infrared dielectric function model for amorphous solids,” J. Appl. Phys. 71, 1–6 (1992).
[CrossRef]

E. F. Schubert, Y.-H. Wang, A. Y. Cho, L.-W. Tu, G. J. Zydzik, “Resonant cavity light-emitting diode,” Appl. Phys. Lett. 60, 921–923 (1992).
[CrossRef]

1991 (2)

R. S. Geels, S. W. Corzine, L. A. Coldren, “InGaAs vertical-cavity surface-emitting lasers,” IEEE J. Quantum Electron. 27, 1359–1367 (1991).
[CrossRef]

C. J. Chang-Hasnain, J. B. Harbison, G. Hasnain, A. C. Von Lehmen, L. T. Florez, N. G. Stoffel, “Dynamic, polarization, and transverse mode characteristics of vertical cavity surface emitting lasers,” IEEE J. Quantum Electron. 27, 1402–1409 (1991).
[CrossRef]

1990 (4)

H. J. Luo, P. S. Zory, “Distributed feedback coupling coefficient in diode lasers with metallized gratings,” IEEE Photon. Technol. Lett. 2, 614–616 (1990).
[CrossRef]

L. Yang, M. C. Wu, K. Tai, T. Tanbun-Ek, R. A. Logan, “InGaAsP(1.3-μm)/InP vertical-cavity surface-emitting laser grown by metalorganic vapor phase epitaxy,” Appl. Phys. Lett. 56, 889–891 (1990).
[CrossRef]

M. I. Marković, A. D. Rakić, “Determination of optical properties of aluminum including electron reradiation in the Lorentz–Drude model,” Opt. Laser Technol. 22, 394–398 (1990).
[CrossRef]

M. I. Marković, A. D. Rakić, “Determination of the reflection coefficients of laser light of wavelengths λ ∈ (0.22 μm, 200 μm) from the surface of aluminum using the Lorentz–Drude model,” Appl. Opt. 29, 3479–3483 (1990).
[CrossRef]

1985 (1)

1984 (1)

M. Erman, J. B. Theeten, P. Chambon, S. M. Kelso, D. E. Aspnes, “Optical properties and damage analysis of GaAs single crystals partly amorphized by ion implantation,” J. Appl. Phys. 56, 2664–2671 (1984).
[CrossRef]

1983 (2)

1975 (6)

H. J. Hagemann, W. Gudat, C. Kunz, “Optical constants from the far infrared to the x-ray region: Mg, Al, Cu, Ag, Au, Bi, C, and Al2O3,” J. Opt. Soc. Am. 65, 742–744 (1975).
[CrossRef]

J. H. Weaver, R. L. Benbow, “Low-energy intraband absorption in Pd,” Phys. Rev. B 12, 3509–3510 (1975).
[CrossRef]

J. H. Weaver, “Optical properties of Rh, Pd, Ir, and Pt,” Phys. Rev. B 11, 1416–1425 (1975).
[CrossRef]

J. H. Weaver, C. G. Olson, D. W. Lynch, “Optical properties of crystalline tungsten,” Phys. Rev. B 12, 1293–1297 (1975).
[CrossRef]

D. W. Lynch, C. G. Olson, J. H. Weaver, “Optical properties of Ti, Zr, and Hf from 0.15 to 30 eV,” Phys. Rev. B 11, 3617–3624 (1975).
[CrossRef]

P. Winsemius, H. P. Langkeek, F. F. van Kampen, “Structure dependence of the optical properties of Cu, Ag and Au,” Physica 79B, 529–546 (1975).

1974 (2)

N. V. Smith, “Photoemission spectra and band structures of d-band metals. III. Model band calculations on Rh, Pd, Ag, Ir, Pt, and Au,” Phys. Rev. B 9, 1365–1376 (1974).
[CrossRef]

P. B. Johnson, R. W. Christy, “Optical constants of transition metals: Ti, V, Cr, Mn, Fe, Co, Ni, and Pd,” Phys. Rev. B 9, 5056–5070 (1974).
[CrossRef]

1972 (2)

P. B. Johnson, R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1972).
[CrossRef]

A. Seignac, S. Robin, “Optical properties of thin films of Pt in the far ultraviolet,” Solid State Commun. 11, 217–219 (1972).
[CrossRef]

1971 (1)

N. W. Ashcroft, K. Sturm, “Interband absorption and the optical properties of polyvalent metals,” Phys. Rev. B 3, 1898–1910 (1971).
[CrossRef]

1970 (3)

C. J. Powell, “Analysis of optical and inelastic-electron-scattering data II. Application to Al,” J. Opt. Soc. Am. 60, 78–93 (1970).
[CrossRef]

M. L. Thèye, “Investigation of the optical properties of Au by means of thin semitransparent films,” Phys. Rev. B 2, 3060–3078 (1970).
[CrossRef]

L. W. Bos, D. W. Lynch, “Optical properties of antiferromagnetic chromium and dilute Cr-Mn and Cr-Re alloys,” Phys. Rev. B 2, 4567–4577 (1970).
[CrossRef]

1969 (1)

R. Haensel, K. Radler, B. Sonntag, C. Kunz, “Optical absorption measurements of tantalum, tungsten, rhenium and platinum in the extreme ultraviolet,” Solid State Commun. 7, 1495–1497 (1969).
[CrossRef]

1968 (3)

A. Y.-C. Yu, W. E. Spicer, G. Hass, “Optical properties of platinum,” Phys. Rev. 171, 834–835 (1968).
[CrossRef]

R. Haensel, C. Kunz, T. Sasaki, B. Sonntag, “Absorption measurements of copper, silver, tin, gold, and bismuth in the far ultraviolet,” Appl. Opt. 7, 301–306 (1968).
[CrossRef] [PubMed]

M. M. Kirillova, M. M. Noskov, “Optical properties of chromium,” Phys. Met. Metallogr. 26, 189–192 (1968).

1965 (2)

B. Dold, R. Mecke, “Optische Eigenschaften von Edelmetallen, Übergangsmetallen und deren Legierungen im Infrarot (1. Teil),” Optik 22, 435–446 (1965).

L. R. Canfield, G. Hass, “Reflectance and optical constants of evaporated copper and silver in the vacuum ultraviolet from 1000 to 2000 Å,” J. Opt. Soc. Am. 55, 61–64 (1965).
[CrossRef]

1964 (1)

M. M. Kirillova, B. A. Charikov, “Study of the optical properties of transition metals,” Opt. Spectrosc. USSR 17, 134–135 (1964).

1963 (2)

M. M. Kirillova, B. A. Charikov, “Optical properties of titanium in the quantum transition range,” Phys. Met. Metallogr. 15, 138–139 (1963).

H. Ehrenreich, H. R. Philipp, B. Segall, “Optical properties of aluminum,” Phys. Rev. 132, 1918–1928 (1963).
[CrossRef]

1962 (2)

H. Ehrenreich, H. R. Philipp, “Optical properties of Ag and Cu,” Phys. Rev. 128, 1622–1629 (1962).
[CrossRef]

G. A. Bolotin, A. N. Voloshinskii, M. M. Kirillova, M. M. Noskov, A. V. Sokolov, B. A. Charikov, “Optical properties of titanium and vanadium in the infrared range of the spectrum,” Phys. Met. Metallogr. 13, 24–31 (1962).

Abad, H.

C. C. Kim, J. W. Garland, H. Abad, P. M. Raccah, “Modeling the optical dielectric function of semiconductors: extension of the critical-point parabolic-band approximation,” Phys. Rev. B 45, 11,749–11,767 (1992).
[CrossRef]

Alexander, R. W.

An, I.

H. V. Nguyen, I. An, R. W. Collins, “Evolution of the optical functions of thin-film aluminum: A real-time spectroscopic ellipsometry study,” Phys. Rev. B 47, 3947–3965 (1993).
[CrossRef]

Arakawa, E. T.

E. T. Arakawa, T. A. Callcott, Y.-C. Chang, “Beryllium (Be),” in Handbook of Optical Constants of Solids II, E. D. Palik, ed. (Academic, San Diego, Calif., 1991), pp. 421–433.

Asano, K.

T. Baba, R. Watanabe, K. Asano, F. Koyama, K. Iga, “Theoretical and experimental estimations of photon recycling effect in light emitting devices with a metal mirror,” Jpn. J. Appl. Phys. 35(1A), 97–100 (1996).
[CrossRef]

Ashcroft, N. W.

N. W. Ashcroft, K. Sturm, “Interband absorption and the optical properties of polyvalent metals,” Phys. Rev. B 3, 1898–1910 (1971).
[CrossRef]

Aspnes, D. E.

M. Erman, J. B. Theeten, P. Chambon, S. M. Kelso, D. E. Aspnes, “Optical properties and damage analysis of GaAs single crystals partly amorphized by ion implantation,” J. Appl. Phys. 56, 2664–2671 (1984).
[CrossRef]

Baba, T.

T. Baba, R. Watanabe, K. Asano, F. Koyama, K. Iga, “Theoretical and experimental estimations of photon recycling effect in light emitting devices with a metal mirror,” Jpn. J. Appl. Phys. 35(1A), 97–100 (1996).
[CrossRef]

Babic, D. I.

D. I. Babić, R. P. Mirin, E. L. Hu, J. E. Bowers, “Characterization of metal mirrors on GaAs,” Electron. Lett. 32, 319–320 (1996).
[CrossRef]

Bell, R. J.

Bell, R. R.

Bell, S. E.

Benbow, R. L.

J. H. Weaver, R. L. Benbow, “Low-energy intraband absorption in Pd,” Phys. Rev. B 12, 3509–3510 (1975).
[CrossRef]

Bleuse, J.

E. Hadji, J. Bleuse, N. Magnes, J. L. Pautrat, “3.2-μm infrared resonant cavity light emitting diode,” Appl. Phys. Lett. 67, 2591–2593 (1995).
[CrossRef]

Bolotin, G. A.

G. A. Bolotin, A. N. Voloshinskii, M. M. Kirillova, M. M. Noskov, A. V. Sokolov, B. A. Charikov, “Optical properties of titanium and vanadium in the infrared range of the spectrum,” Phys. Met. Metallogr. 13, 24–31 (1962).

Borghesi, A.

A. Borghesi, A. Piaggi, “Palladium (Pd),” in Handbook of Optical Constants of Solids II, E. D. Palik, ed. (Academic, San Diego, Calif., 1991), pp. 469–476.

Bormann, D.

R. Brendel, D. Bormann, “An infrared dielectric function model for amorphous solids,” J. Appl. Phys. 71, 1–6 (1992).
[CrossRef]

Bos, L. W.

L. W. Bos, D. W. Lynch, “Optical properties of antiferromagnetic chromium and dilute Cr-Mn and Cr-Re alloys,” Phys. Rev. B 2, 4567–4577 (1970).
[CrossRef]

Bowers, J. E.

D. I. Babić, R. P. Mirin, E. L. Hu, J. E. Bowers, “Characterization of metal mirrors on GaAs,” Electron. Lett. 32, 319–320 (1996).
[CrossRef]

Brendel, R.

R. Brendel, D. Bormann, “An infrared dielectric function model for amorphous solids,” J. Appl. Phys. 71, 1–6 (1992).
[CrossRef]

Callcott, T. A.

E. T. Arakawa, T. A. Callcott, Y.-C. Chang, “Beryllium (Be),” in Handbook of Optical Constants of Solids II, E. D. Palik, ed. (Academic, San Diego, Calif., 1991), pp. 421–433.

Canfield, L. R.

Celii, F. G.

C. M. Herzinger, P. G. Snyder, F. G. Celii, Y. C. Kao, D. Chow, B. Johs, J. A. Woollam, “Studies of thin strained InAs, AlAs, and AlSb layers by spectroscopic ellipsometry,” J. Appl. Phys. 79, 2663–2674 (1996).
[CrossRef]

C. M. Herzinger, H. Yao, P. G. Snyder, F. G. Celii, Y. C. Kao, B. Johs, J. A. Woollam, “Determination of AlAs optical constants by variable-angle spectroscopic ellipsometry and a multisample analysis,” J. Appl. Phys. 77, 4677–4687 (1995).
[CrossRef]

Chambon, P.

M. Erman, J. B. Theeten, P. Chambon, S. M. Kelso, D. E. Aspnes, “Optical properties and damage analysis of GaAs single crystals partly amorphized by ion implantation,” J. Appl. Phys. 56, 2664–2671 (1984).
[CrossRef]

Chang, K. J.

K.-H. Lee, K. J. Chang, “First-principles study of the optical properties and the dielectric response of Al,” Phys. Rev. B 49, 2362–2367 (1994).
[CrossRef]

Chang, R. P. H.

G. Du, K. A. Stair, G. Devane, J. Zhang, R. P. H. Chang, C. W. White, X. Li, Z. Wang, Y. Liu, “Vertical-cavity surface-emitting laser with a thin metal mirror fabricated by double implantation using a tungsten wire mask,” Semicond. Sci. Technol. 11, 1734–1736 (1996).
[CrossRef]

Chang, Y.-C.

E. T. Arakawa, T. A. Callcott, Y.-C. Chang, “Beryllium (Be),” in Handbook of Optical Constants of Solids II, E. D. Palik, ed. (Academic, San Diego, Calif., 1991), pp. 421–433.

Chang-Hasnain, C. J.

C. J. Chang-Hasnain, J. B. Harbison, G. Hasnain, A. C. Von Lehmen, L. T. Florez, N. G. Stoffel, “Dynamic, polarization, and transverse mode characteristics of vertical cavity surface emitting lasers,” IEEE J. Quantum Electron. 27, 1402–1409 (1991).
[CrossRef]

Charikov, B. A.

M. M. Kirillova, B. A. Charikov, “Study of the optical properties of transition metals,” Opt. Spectrosc. USSR 17, 134–135 (1964).

M. M. Kirillova, B. A. Charikov, “Optical properties of titanium in the quantum transition range,” Phys. Met. Metallogr. 15, 138–139 (1963).

G. A. Bolotin, A. N. Voloshinskii, M. M. Kirillova, M. M. Noskov, A. V. Sokolov, B. A. Charikov, “Optical properties of titanium and vanadium in the infrared range of the spectrum,” Phys. Met. Metallogr. 13, 24–31 (1962).

Cho, A. Y.

N. E. J. Hunt, E. F. Schubert, R. F. Kopf, D. L. Sivco, A. Y. Cho, G. J. Zydzik, “Increased fiber communications bandwidth from a resonant cavity light emitting diode emitting at λ = 940 nm,” Appl. Phys. Lett. 63, 2600–2602 (1993).
[CrossRef]

E. F. Schubert, Y.-H. Wang, A. Y. Cho, L.-W. Tu, G. J. Zydzik, “Resonant cavity light-emitting diode,” Appl. Phys. Lett. 60, 921–923 (1992).
[CrossRef]

Chow, D.

C. M. Herzinger, P. G. Snyder, F. G. Celii, Y. C. Kao, D. Chow, B. Johs, J. A. Woollam, “Studies of thin strained InAs, AlAs, and AlSb layers by spectroscopic ellipsometry,” J. Appl. Phys. 79, 2663–2674 (1996).
[CrossRef]

Christy, R. W.

P. B. Johnson, R. W. Christy, “Optical constants of transition metals: Ti, V, Cr, Mn, Fe, Co, Ni, and Pd,” Phys. Rev. B 9, 5056–5070 (1974).
[CrossRef]

P. B. Johnson, R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1972).
[CrossRef]

Coldren, L. A.

R. S. Geels, S. W. Corzine, L. A. Coldren, “InGaAs vertical-cavity surface-emitting lasers,” IEEE J. Quantum Electron. 27, 1359–1367 (1991).
[CrossRef]

Coleman, J. J.

G. M. Smith, D. V. Forbes, R. M. Lammert, J. J. Coleman, “Metalization to asymetric cladding separate confinement heterostructure lasers,” Appl. Phys. Lett. 67, 3847–3849 (1995).
[CrossRef]

Collins, R. W.

H. V. Nguyen, I. An, R. W. Collins, “Evolution of the optical functions of thin-film aluminum: A real-time spectroscopic ellipsometry study,” Phys. Rev. B 47, 3947–3965 (1993).
[CrossRef]

Considine, L.

B. Corbett, L. Considine, S. Walsh, W. M. Kelly, “Resonant cavity light emitting diode and detector using epitaxial liftoff,” IEEE Photon. Technol. Lett. 5, 1041–1043 (1993).
[CrossRef]

B. Corbett, L. Considine, S. Walsh, W. M. Kelly, “Narrow bandwidth long wavelength resonant cavity photodiodes,” Electron. Lett. 29, 2148–2149 (1993).
[CrossRef]

Corbett, B.

B. Corbett, L. Considine, S. Walsh, W. M. Kelly, “Resonant cavity light emitting diode and detector using epitaxial liftoff,” IEEE Photon. Technol. Lett. 5, 1041–1043 (1993).
[CrossRef]

B. Corbett, L. Considine, S. Walsh, W. M. Kelly, “Narrow bandwidth long wavelength resonant cavity photodiodes,” Electron. Lett. 29, 2148–2149 (1993).
[CrossRef]

Corzine, S. W.

R. S. Geels, S. W. Corzine, L. A. Coldren, “InGaAs vertical-cavity surface-emitting lasers,” IEEE J. Quantum Electron. 27, 1359–1367 (1991).
[CrossRef]

Devane, G.

G. Du, K. A. Stair, G. Devane, J. Zhang, R. P. H. Chang, C. W. White, X. Li, Z. Wang, Y. Liu, “Vertical-cavity surface-emitting laser with a thin metal mirror fabricated by double implantation using a tungsten wire mask,” Semicond. Sci. Technol. 11, 1734–1736 (1996).
[CrossRef]

Djurišic, A. B.

A. B. Djurišić, J. M. Elazar, A. D. Rakić, “Modeling the optical constants of solids using genetic algorithms with parameter space size adjustment,” Opt. Commun. 134, 407–414 (1997).
[CrossRef]

A. B. Djurišić, A. D. Rakić, J. M. Elazar, “Modeling the optical constants of solids using acceptance-probability-controlled simulated annealing with an adaptive move generation procedure,” Phys. Rev. E 55, 4797–4803 (1997).
[CrossRef]

A. D. Rakić, J. M. Elazar, A. B. Djurišić, “Acceptance-probability-controlled simulated annealing: a method for modeling the optical constants of solids,” Phys. Rev. E 52, 6862–6867 (1995).
[CrossRef]

Dold, B.

B. Dold, R. Mecke, “Optische Eigenschaften von Edelmetallen, Übergangsmetallen und deren Legierungen im Infrarot (1. Teil),” Optik 22, 435–446 (1965).

Du, G.

G. Du, K. A. Stair, G. Devane, J. Zhang, R. P. H. Chang, C. W. White, X. Li, Z. Wang, Y. Liu, “Vertical-cavity surface-emitting laser with a thin metal mirror fabricated by double implantation using a tungsten wire mask,” Semicond. Sci. Technol. 11, 1734–1736 (1996).
[CrossRef]

Ehrenreich, H.

H. Ehrenreich, H. R. Philipp, B. Segall, “Optical properties of aluminum,” Phys. Rev. 132, 1918–1928 (1963).
[CrossRef]

H. Ehrenreich, H. R. Philipp, “Optical properties of Ag and Cu,” Phys. Rev. 128, 1622–1629 (1962).
[CrossRef]

Elazar, J. M.

A. B. Djurišić, A. D. Rakić, J. M. Elazar, “Modeling the optical constants of solids using acceptance-probability-controlled simulated annealing with an adaptive move generation procedure,” Phys. Rev. E 55, 4797–4803 (1997).
[CrossRef]

A. B. Djurišić, J. M. Elazar, A. D. Rakić, “Modeling the optical constants of solids using genetic algorithms with parameter space size adjustment,” Opt. Commun. 134, 407–414 (1997).
[CrossRef]

A. D. Rakić, J. M. Elazar, A. B. Djurišić, “Acceptance-probability-controlled simulated annealing: a method for modeling the optical constants of solids,” Phys. Rev. E 52, 6862–6867 (1995).
[CrossRef]

Emanuel, M. A.

C. H. Wu, P. S. Zory, M. A. Emanuel, “Contact reflectivity effects on thin p-clad InGaAs single quantum-well lasers,” IEEE Photon. Technol. Lett. 6, 1427–1429 (1994).
[CrossRef]

Erman, M.

M. Erman, J. B. Theeten, P. Chambon, S. M. Kelso, D. E. Aspnes, “Optical properties and damage analysis of GaAs single crystals partly amorphized by ion implantation,” J. Appl. Phys. 56, 2664–2671 (1984).
[CrossRef]

Florez, L. T.

C. J. Chang-Hasnain, J. B. Harbison, G. Hasnain, A. C. Von Lehmen, L. T. Florez, N. G. Stoffel, “Dynamic, polarization, and transverse mode characteristics of vertical cavity surface emitting lasers,” IEEE J. Quantum Electron. 27, 1402–1409 (1991).
[CrossRef]

Foiles, C. L.

C. L. Foiles, “Optical properties of pure metals and binary alloys,” in Landolt-Börnstein, Group III: Crystal and Solid State Physics, K.-H. Hellwege, O. Madelung, eds., Vol. 15b of New Series (Springer-Verlag, Berlin, 1985), Chap. 4, pp. 210–489.

Forbes, D. V.

G. M. Smith, D. V. Forbes, R. M. Lammert, J. J. Coleman, “Metalization to asymetric cladding separate confinement heterostructure lasers,” Appl. Phys. Lett. 67, 3847–3849 (1995).
[CrossRef]

Garland, J. W.

C. C. Kim, J. W. Garland, P. M. Raccah, “Modeling the optical dielectric function of the alloy system AlxGa1-xAs,” Phys. Rev. B 47, 1876–1888 (1993).
[CrossRef]

C. C. Kim, J. W. Garland, H. Abad, P. M. Raccah, “Modeling the optical dielectric function of semiconductors: extension of the critical-point parabolic-band approximation,” Phys. Rev. B 45, 11,749–11,767 (1992).
[CrossRef]

Geels, R. S.

R. S. Geels, S. W. Corzine, L. A. Coldren, “InGaAs vertical-cavity surface-emitting lasers,” IEEE J. Quantum Electron. 27, 1359–1367 (1991).
[CrossRef]

Gottschalch, V.

M. Schubert, V. Gottschalch, C. M. Herzinger, H. Yao, P. G. Snyder, J. A. Woollam, “Optical-constants of GaxIn1-xP lattice-matched to GaAs,” J. Appl. Phys. 77, 3416–3419 (1995).
[CrossRef]

Gudat, W.

Hadji, E.

E. Hadji, J. Bleuse, N. Magnes, J. L. Pautrat, “3.2-μm infrared resonant cavity light emitting diode,” Appl. Phys. Lett. 67, 2591–2593 (1995).
[CrossRef]

Haensel, R.

R. Haensel, K. Radler, B. Sonntag, C. Kunz, “Optical absorption measurements of tantalum, tungsten, rhenium and platinum in the extreme ultraviolet,” Solid State Commun. 7, 1495–1497 (1969).
[CrossRef]

R. Haensel, C. Kunz, T. Sasaki, B. Sonntag, “Absorption measurements of copper, silver, tin, gold, and bismuth in the far ultraviolet,” Appl. Opt. 7, 301–306 (1968).
[CrossRef] [PubMed]

Hagemann, H. J.

Harbison, J. B.

C. J. Chang-Hasnain, J. B. Harbison, G. Hasnain, A. C. Von Lehmen, L. T. Florez, N. G. Stoffel, “Dynamic, polarization, and transverse mode characteristics of vertical cavity surface emitting lasers,” IEEE J. Quantum Electron. 27, 1402–1409 (1991).
[CrossRef]

Hasnain, G.

C. J. Chang-Hasnain, J. B. Harbison, G. Hasnain, A. C. Von Lehmen, L. T. Florez, N. G. Stoffel, “Dynamic, polarization, and transverse mode characteristics of vertical cavity surface emitting lasers,” IEEE J. Quantum Electron. 27, 1402–1409 (1991).
[CrossRef]

Hass, G.

A. Y.-C. Yu, W. E. Spicer, G. Hass, “Optical properties of platinum,” Phys. Rev. 171, 834–835 (1968).
[CrossRef]

L. R. Canfield, G. Hass, “Reflectance and optical constants of evaporated copper and silver in the vacuum ultraviolet from 1000 to 2000 Å,” J. Opt. Soc. Am. 55, 61–64 (1965).
[CrossRef]

G. Hass, W. R. Hunter, “New developments in vacuum-ultraviolet reflecting coatings for space astronomy,” in Space Optics, B. J. Thompson, R. R. Shanon, eds. (National Academy of Sciences, Washington, D.C., 1974), pp. 525–553.

Herzinger, C. M.

C. M. Herzinger, P. G. Snyder, F. G. Celii, Y. C. Kao, D. Chow, B. Johs, J. A. Woollam, “Studies of thin strained InAs, AlAs, and AlSb layers by spectroscopic ellipsometry,” J. Appl. Phys. 79, 2663–2674 (1996).
[CrossRef]

M. Schubert, V. Gottschalch, C. M. Herzinger, H. Yao, P. G. Snyder, J. A. Woollam, “Optical-constants of GaxIn1-xP lattice-matched to GaAs,” J. Appl. Phys. 77, 3416–3419 (1995).
[CrossRef]

C. M. Herzinger, H. Yao, P. G. Snyder, F. G. Celii, Y. C. Kao, B. Johs, J. A. Woollam, “Determination of AlAs optical constants by variable-angle spectroscopic ellipsometry and a multisample analysis,” J. Appl. Phys. 77, 4677–4687 (1995).
[CrossRef]

Hu, E. L.

D. I. Babić, R. P. Mirin, E. L. Hu, J. E. Bowers, “Characterization of metal mirrors on GaAs,” Electron. Lett. 32, 319–320 (1996).
[CrossRef]

Hunt, N. E. J.

N. E. J. Hunt, E. F. Schubert, R. F. Kopf, D. L. Sivco, A. Y. Cho, G. J. Zydzik, “Increased fiber communications bandwidth from a resonant cavity light emitting diode emitting at λ = 940 nm,” Appl. Phys. Lett. 63, 2600–2602 (1993).
[CrossRef]

Hunter, W. R.

D. W. Lynch, W. R. Hunter, “An introduction to the data for several metals,” in Handbook of Optical Constants of Solids II, E. D. Palik, ed. (Academic, San Diego, Calif., 1991) pp. 341–419.

D. W. Lynch, W. R. Hunter, “Comments on the optical constants of metals and an introduction to the data for several metals,” in Handbook of Optical Constants of Solids, E. D. Palik, ed. (Academic, Orlando, Fla., 1985), pp. 275–367.

G. Hass, W. R. Hunter, “New developments in vacuum-ultraviolet reflecting coatings for space astronomy,” in Space Optics, B. J. Thompson, R. R. Shanon, eds. (National Academy of Sciences, Washington, D.C., 1974), pp. 525–553.

Iga, K.

T. Baba, R. Watanabe, K. Asano, F. Koyama, K. Iga, “Theoretical and experimental estimations of photon recycling effect in light emitting devices with a metal mirror,” Jpn. J. Appl. Phys. 35(1A), 97–100 (1996).
[CrossRef]

K. Iga, “Surface emitting lasers,” Opt. Quant. Electron. 24(2), s97–s104 (1992).
[CrossRef]

Johnson, P. B.

P. B. Johnson, R. W. Christy, “Optical constants of transition metals: Ti, V, Cr, Mn, Fe, Co, Ni, and Pd,” Phys. Rev. B 9, 5056–5070 (1974).
[CrossRef]

P. B. Johnson, R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1972).
[CrossRef]

Johs, B.

C. M. Herzinger, P. G. Snyder, F. G. Celii, Y. C. Kao, D. Chow, B. Johs, J. A. Woollam, “Studies of thin strained InAs, AlAs, and AlSb layers by spectroscopic ellipsometry,” J. Appl. Phys. 79, 2663–2674 (1996).
[CrossRef]

C. M. Herzinger, H. Yao, P. G. Snyder, F. G. Celii, Y. C. Kao, B. Johs, J. A. Woollam, “Determination of AlAs optical constants by variable-angle spectroscopic ellipsometry and a multisample analysis,” J. Appl. Phys. 77, 4677–4687 (1995).
[CrossRef]

Jokerst, N. M.

Kao, Y. C.

C. M. Herzinger, P. G. Snyder, F. G. Celii, Y. C. Kao, D. Chow, B. Johs, J. A. Woollam, “Studies of thin strained InAs, AlAs, and AlSb layers by spectroscopic ellipsometry,” J. Appl. Phys. 79, 2663–2674 (1996).
[CrossRef]

C. M. Herzinger, H. Yao, P. G. Snyder, F. G. Celii, Y. C. Kao, B. Johs, J. A. Woollam, “Determination of AlAs optical constants by variable-angle spectroscopic ellipsometry and a multisample analysis,” J. Appl. Phys. 77, 4677–4687 (1995).
[CrossRef]

Katz, A.

A. Katz, “Physical and chemical deposition of metals as ohmic contacts to InP and related materials,” in Handbook of Compound Semiconductors, P. H. Holloway, G. E. McGuire, eds. (Noyes Publications, Park Ridge, N.J., 1995), pp. 170–250.

Kelly, W. M.

B. Corbett, L. Considine, S. Walsh, W. M. Kelly, “Resonant cavity light emitting diode and detector using epitaxial liftoff,” IEEE Photon. Technol. Lett. 5, 1041–1043 (1993).
[CrossRef]

B. Corbett, L. Considine, S. Walsh, W. M. Kelly, “Narrow bandwidth long wavelength resonant cavity photodiodes,” Electron. Lett. 29, 2148–2149 (1993).
[CrossRef]

Kelso, S. M.

M. Erman, J. B. Theeten, P. Chambon, S. M. Kelso, D. E. Aspnes, “Optical properties and damage analysis of GaAs single crystals partly amorphized by ion implantation,” J. Appl. Phys. 56, 2664–2671 (1984).
[CrossRef]

Kim, C. C.

C. C. Kim, J. W. Garland, P. M. Raccah, “Modeling the optical dielectric function of the alloy system AlxGa1-xAs,” Phys. Rev. B 47, 1876–1888 (1993).
[CrossRef]

C. C. Kim, J. W. Garland, H. Abad, P. M. Raccah, “Modeling the optical dielectric function of semiconductors: extension of the critical-point parabolic-band approximation,” Phys. Rev. B 45, 11,749–11,767 (1992).
[CrossRef]

Kirillova, M. M.

M. M. Kirillova, M. M. Noskov, “Optical properties of chromium,” Phys. Met. Metallogr. 26, 189–192 (1968).

M. M. Kirillova, B. A. Charikov, “Study of the optical properties of transition metals,” Opt. Spectrosc. USSR 17, 134–135 (1964).

M. M. Kirillova, B. A. Charikov, “Optical properties of titanium in the quantum transition range,” Phys. Met. Metallogr. 15, 138–139 (1963).

G. A. Bolotin, A. N. Voloshinskii, M. M. Kirillova, M. M. Noskov, A. V. Sokolov, B. A. Charikov, “Optical properties of titanium and vanadium in the infrared range of the spectrum,” Phys. Met. Metallogr. 13, 24–31 (1962).

Knittl, Z.

Z. Knittl, Optics of Thin Films (Wiley, New York, 1976).

Kopf, R. F.

N. E. J. Hunt, E. F. Schubert, R. F. Kopf, D. L. Sivco, A. Y. Cho, G. J. Zydzik, “Increased fiber communications bandwidth from a resonant cavity light emitting diode emitting at λ = 940 nm,” Appl. Phys. Lett. 63, 2600–2602 (1993).
[CrossRef]

Koyama, F.

T. Baba, R. Watanabe, K. Asano, F. Koyama, K. Iga, “Theoretical and experimental estimations of photon recycling effect in light emitting devices with a metal mirror,” Jpn. J. Appl. Phys. 35(1A), 97–100 (1996).
[CrossRef]

Kunz, C.

Lammert, R. M.

G. M. Smith, D. V. Forbes, R. M. Lammert, J. J. Coleman, “Metalization to asymetric cladding separate confinement heterostructure lasers,” Appl. Phys. Lett. 67, 3847–3849 (1995).
[CrossRef]

Langkeek, H. P.

P. Winsemius, H. P. Langkeek, F. F. van Kampen, “Structure dependence of the optical properties of Cu, Ag and Au,” Physica 79B, 529–546 (1975).

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M. Schubert, V. Gottschalch, C. M. Herzinger, H. Yao, P. G. Snyder, J. A. Woollam, “Optical-constants of GaxIn1-xP lattice-matched to GaAs,” J. Appl. Phys. 77, 3416–3419 (1995).
[CrossRef]

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[CrossRef]

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L. Yang, M. C. Wu, K. Tai, T. Tanbun-Ek, R. A. Logan, “InGaAsP(1.3-μm)/InP vertical-cavity surface-emitting laser grown by metalorganic vapor phase epitaxy,” Appl. Phys. Lett. 56, 889–891 (1990).
[CrossRef]

Yao, H.

C. M. Herzinger, H. Yao, P. G. Snyder, F. G. Celii, Y. C. Kao, B. Johs, J. A. Woollam, “Determination of AlAs optical constants by variable-angle spectroscopic ellipsometry and a multisample analysis,” J. Appl. Phys. 77, 4677–4687 (1995).
[CrossRef]

M. Schubert, V. Gottschalch, C. M. Herzinger, H. Yao, P. G. Snyder, J. A. Woollam, “Optical-constants of GaxIn1-xP lattice-matched to GaAs,” J. Appl. Phys. 77, 3416–3419 (1995).
[CrossRef]

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G. Du, K. A. Stair, G. Devane, J. Zhang, R. P. H. Chang, C. W. White, X. Li, Z. Wang, Y. Liu, “Vertical-cavity surface-emitting laser with a thin metal mirror fabricated by double implantation using a tungsten wire mask,” Semicond. Sci. Technol. 11, 1734–1736 (1996).
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C. H. Wu, P. S. Zory, M. A. Emanuel, “Contact reflectivity effects on thin p-clad InGaAs single quantum-well lasers,” IEEE Photon. Technol. Lett. 6, 1427–1429 (1994).
[CrossRef]

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[CrossRef]

Zydzik, G. J.

N. E. J. Hunt, E. F. Schubert, R. F. Kopf, D. L. Sivco, A. Y. Cho, G. J. Zydzik, “Increased fiber communications bandwidth from a resonant cavity light emitting diode emitting at λ = 940 nm,” Appl. Phys. Lett. 63, 2600–2602 (1993).
[CrossRef]

E. F. Schubert, Y.-H. Wang, A. Y. Cho, L.-W. Tu, G. J. Zydzik, “Resonant cavity light-emitting diode,” Appl. Phys. Lett. 60, 921–923 (1992).
[CrossRef]

Appl. Opt. (6)

Appl. Phys. Lett. (5)

E. Hadji, J. Bleuse, N. Magnes, J. L. Pautrat, “3.2-μm infrared resonant cavity light emitting diode,” Appl. Phys. Lett. 67, 2591–2593 (1995).
[CrossRef]

N. E. J. Hunt, E. F. Schubert, R. F. Kopf, D. L. Sivco, A. Y. Cho, G. J. Zydzik, “Increased fiber communications bandwidth from a resonant cavity light emitting diode emitting at λ = 940 nm,” Appl. Phys. Lett. 63, 2600–2602 (1993).
[CrossRef]

E. F. Schubert, Y.-H. Wang, A. Y. Cho, L.-W. Tu, G. J. Zydzik, “Resonant cavity light-emitting diode,” Appl. Phys. Lett. 60, 921–923 (1992).
[CrossRef]

L. Yang, M. C. Wu, K. Tai, T. Tanbun-Ek, R. A. Logan, “InGaAsP(1.3-μm)/InP vertical-cavity surface-emitting laser grown by metalorganic vapor phase epitaxy,” Appl. Phys. Lett. 56, 889–891 (1990).
[CrossRef]

G. M. Smith, D. V. Forbes, R. M. Lammert, J. J. Coleman, “Metalization to asymetric cladding separate confinement heterostructure lasers,” Appl. Phys. Lett. 67, 3847–3849 (1995).
[CrossRef]

Electron. Lett. (2)

B. Corbett, L. Considine, S. Walsh, W. M. Kelly, “Narrow bandwidth long wavelength resonant cavity photodiodes,” Electron. Lett. 29, 2148–2149 (1993).
[CrossRef]

D. I. Babić, R. P. Mirin, E. L. Hu, J. E. Bowers, “Characterization of metal mirrors on GaAs,” Electron. Lett. 32, 319–320 (1996).
[CrossRef]

IEEE J. Quantum Electron. (2)

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C. J. Chang-Hasnain, J. B. Harbison, G. Hasnain, A. C. Von Lehmen, L. T. Florez, N. G. Stoffel, “Dynamic, polarization, and transverse mode characteristics of vertical cavity surface emitting lasers,” IEEE J. Quantum Electron. 27, 1402–1409 (1991).
[CrossRef]

IEEE Photon. Technol. Lett. (3)

B. Corbett, L. Considine, S. Walsh, W. M. Kelly, “Resonant cavity light emitting diode and detector using epitaxial liftoff,” IEEE Photon. Technol. Lett. 5, 1041–1043 (1993).
[CrossRef]

C. H. Wu, P. S. Zory, M. A. Emanuel, “Contact reflectivity effects on thin p-clad InGaAs single quantum-well lasers,” IEEE Photon. Technol. Lett. 6, 1427–1429 (1994).
[CrossRef]

H. J. Luo, P. S. Zory, “Distributed feedback coupling coefficient in diode lasers with metallized gratings,” IEEE Photon. Technol. Lett. 2, 614–616 (1990).
[CrossRef]

J. Appl. Phys. (7)

M. S. Ünlü, S. Strite, “Resonant cavity enhanced photonic devices,” J. Appl. Phys. 78, 607–639 (1995).
[CrossRef]

R. Brendel, D. Bormann, “An infrared dielectric function model for amorphous solids,” J. Appl. Phys. 71, 1–6 (1992).
[CrossRef]

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Optik (1)

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Figures (16)

Fig. 1
Fig. 1

Real and imaginary parts of the optical dielectric function of Ag: solid curves, values that we calculated using the BB model; dashed curves, the LD model. Also shown are the selected experimental data points from Dold and Mecke,46 Winsemius et al.,47 and Leveque et al.48

Fig. 2
Fig. 2

Real and imaginary parts of the optical dielectric function of Au: solid curves, values that we calculated using the BB model; dashed curves, the LD model. Also shown are selected experimental data points from Dold and Mecke46 and Thèye.49

Fig. 3
Fig. 3

Real and imaginary parts of the optical dielectric function of Cu: solid curves, values calculated by use of the BB model; dashed curves, LD model. Also shown are tabulated data from Hagemann et al.,50 Ordal et al.,26 and Nash and Sambles.54

Fig. 4
Fig. 4

Real and imaginary parts of the optical dielectric function of Al: solid curves, values that we calculated using the BB model; dashed curves, the LD model. Also shown are selected data from Rakić.18

Fig. 5
Fig. 5

Real and imaginary parts of the optical dielectric function of Be versus photon energy (open circles, data from Arakawa et al.57; solid curves, the BB model; dashed curves, the LD model).

Fig. 6
Fig. 6

Real and imaginary parts of the optical dielectric function of Cr: solid curves, values that we calculated using the BB model; dashed curves, the LD model. Also shown are tabulated data from Bos and Lynch58 and Kirillova and Noskov.60

Fig. 7
Fig. 7

Real and imaginary parts of the optical dielectric function of Ni: solid curves, values that we calculated using the BB model; dashed curves, the LD model. Also shown are tabulated data from Lynch and Hunter.45

Fig. 8
Fig. 8

Real and imaginary parts of the optical dielectric function of Pd: solid curves, values that we calculated using the BB model; dashed curves, the LD model. Also shown are tabulated data from Weaver and Benbow62 and Johnson and Christy.61

Fig. 9
Fig. 9

Real and imaginary parts of the optical dielectric function of Pt versus photon energy (open circles, data from Weaver64; solid curves, BB model; dashed curves, LD model).

Fig. 10
Fig. 10

Real and imaginary parts of the optical dielectric function of Ti: experimental data points, Kirillova and Charikov,72,73 Bolotin et al.,74 Johnson and Christy,61 and Lynch et al.70

Fig. 11
Fig. 11

Real and imaginary parts of the optical dielectric function of Ti: solid curves, values that we calculated using the BB model; dashed curves, the LD model. Also shown are tabulated data from Kirillova and Charikov72,73 and Bolotin et al.74

Fig. 12
Fig. 12

Real and imaginary parts of the optical dielectric function of W versus photon energy (open circles, data from Weaver et al.75; solid curves, BB model; dashed curves, LD model).

Fig. 13
Fig. 13

Calculated reflectivities for four GaAs–metal interfaces: GaAs/100 nm Ag, GaAs/100 nm Au, GaAs/100 nm Al, and GaAs/100 nm Pt.

Fig. 14
Fig. 14

Calculated phase changes on reflection for four GaAs/metal interfaces: GaAs/100 nm Ag, GaAs/100 nm Au, GaAs/100 nm Al, and GaAs/100 nm Pt.

Fig. 15
Fig. 15

DBR mirror terminating on (a) air, (b) perfectly conductive metal, and (c) real metal.

Fig. 16
Fig. 16

Measured and calculated reflectivity of metal clad (air–GaAs–Ti/Au) and dielectric clad (air–GaAs–wax) F–P cavities: reflectance that we calculated using the transfer matrix method, solid curves; experimental curves from Ref. 77, dashed curves.

Tables (3)

Tables Icon

Table 1 Values of the Plasma Frequencies ℏωp (eV)

Tables Icon

Table 2 Values of the LD Model Parameters

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Table 3 Values of the BB Model Parameters

Equations (25)

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ˆ r ω = ˆ r f ω + ˆ r b ω ,
ˆ r f ω = 1 - Ω p 2 ω ω - i Γ 0 .
ˆ r b ω = j = 1 k f j ω p 2 ω j 2 - ω 2 + i ω Γ j ,
χ j ω = 1 2 π σ j - + exp - x - ω j 2 2 σ j 2 × f j ω p 2 x 2 - ω 2 + i ω Γ j d x .
χ j = i π   f j ω p 2 2 2 a j σ j w a j - ω j 2 σ j + w a j + ω j 2 σ j ,
w z = e - z 2   erfc - iz Im z > 0 ,
erfc z = 2 π z + exp - t 2 d t .
a j = ω 2 1 + Γ k / ω 2 1 / 2 + 1 1 / 2 , a j = ω 2 1 + Γ k / ω 2 1 / 2 - 1 1 / 2 .
U 1 / 2 ,   1 / 2 ,   z 2 = π e z 2   erfc z ,
χ j = if j ω p 2 2 2 a j σ j U 1 / 2 ,   1 / 2 ,   - a j - ω j 2 σ j 2 + U 1 / 2 ,   1 / 2 ,   - a j + ω j 2 σ j 2 .
ˆ r ω = 1 - Ω p 2 ω ω - i Γ 0 + j = 1 k   χ j ω ,
n = 1 2 r 1 2 + r 2 2 1 / 2 + r 1 1 / 2 , k = 1 2 r 1 2 + r 2 2 1 / 2 - r 1 1 / 2 .
R = n 0 - n 1 2 + k 1 2 n 0 + n 1 2 + k 1 2 ,
φ = arctan 2 n 0 k 1 n 0 2 - n 1 2 - k 1 2
χ 2 = i = 1 i = N r 1 ω i - r 1 exp ω i r 1 exp ω i + r 2 ω i - r 2 exp ω i r 2 exp ω i 2 .
δ = φ 2 = 2 π λ 0   nd ,
d = φ π λ 0 4 n = φ π QWOT .
χ j ω = 1 2 π σ j - + exp - x - ω j 2 2 σ j 2 × f j ω p 2 x 2 - ω 2 + i ω Γ j d x .
χ j ω = - f j ω p 2 2 π σ j - + exp - x - ω j 2 2 σ j 2 a j 2 - x 2 d x ,
χ j ω = - f j ω p 2 2 2 π a j σ j - + exp - x - ω j 2 2 σ j 2 a j - x d x + - + exp - x - ω j 2 2 σ j 2 a j + x d x .
I 1 = - + exp - x - ω j 2 2 σ j 2 a j - ω j - x - ω j d x .
I 1 = - + e - t 2 z 1 - t d t = π i   e - z 1 2   erfc - iz 1 = π i   w a j - ω j 2 σ j .
I 2 = - + exp - x - ω j 2 2 σ j 2 a j - ω j + x - ω j d x
I 2 = t = - t = + e - t 2 z 2 + t d t = t = - t = + e - t 2 z 2 - t d t = π i   e - z 2 2   erfc - iz 2 = π i   w a j + ω j 2 σ j .
χ j = i π   f j ω p 2 2 2 a j σ j w a j - ω j 2 σ j + w a j + ω j 2 σ j .

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