Abstract

The linear and nonlinear optical properties of Ag/Au bilayer metallic thin films with a total thickness of around 20 nm and with different Ag/Au mass-thickness ratios were studied. This study shows that the spectral dispersion of the effective refractive index of bilayer films can be tuned by controlling the mass-thickness ratio between Au and Ag. Improvement of the figure-of-merit for potential plasmonic applications and linear optical filters in the visible spectral range are reported and discussed. The nonlinear optical properties of bilayer metal films studied using femtosecond white-light continuum pump-probe experiments are also shown to be tunable with this ratio. The nonlinear change of optical path length is extracted from the pump-probe data and agrees with simulated values derived from a combination of the two-temperature model, describing the ultrafast electron heating dynamics, and a physical model that describes the dielectric permittivity of Au as a function of electron and lattice temperature.

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    [CrossRef] [PubMed]

2011

M. A. Swillam, N. Rotenberg, and H. M. van Driel, “All-optical ultrafast control of beaming through a single sub-wavelength aperture in a metal film,” Opt. Express 19(8), 7856–7864 (2011).
[CrossRef] [PubMed]

G. A. Wurtz, R. Pollard, W. Hendren, G. P. Wiederrecht, D. J. Gosztola, V. A. Podolskiy, and A. V. Zayats, “Designed ultrafast optical nonlinearity in a plasmonic nanorod metamaterial enhanced by nonlocality,” Nat. Nanotechnol. 6(2), 107–111 (2011).
[CrossRef] [PubMed]

J. Y. Zhang, L. Wang, S. Krishna, M. Sheik-Bahae, and S. R. J. Brueck, “Saturation of the second harmonic generation from GaAs-filled metallic hole arrays by nonlinear absorption,” Phys. Rev. B 83(16), 165438 (2011).
[CrossRef]

M. Thomschke, S. Hofmann, S. Olthof, M. Anderson, H. Kleemann, M. Schober, B. Lussem, and K. Leo, “Improvement of voltage and charge balance in inverted top-emitting organic electroluminescent diodes comprising doped transport layers by thermal annealing,” Appl. Phys. Lett. 98(8), 083304 (2011).
[CrossRef]

2010

D. T. Owens, C. Fuentes-Hernandez, J. M. Hales, J. W. Perry, and B. Kippelen, “A comprehensive analysis of the contributions to the nonlinear optical properties of thin Ag films,” J. Appl. Phys. 107(12), 123114 (2010).
[CrossRef]

N. Rotenberg, M. Betz, and H. M. van Driel, “Ultrafast all-optical coupling of light to surface plasmon polaritons on plain metal surfaces,” Phys. Rev. Lett. 105(1), 017402 (2010).
[CrossRef] [PubMed]

S. Y. Ryu, C. H. Lee, I. S. Oh, S. Y. Song, K. H. Hwang, H. S. Hwang, M. H. Han, B. H. Hwang, H. K. Baik, Y. S. Kim, and J. Y. Lee, “Efficient inverted top-emitting organic light emitting diodes with transparent and surface-modified multilayer anodes,” Electrochem. Solid-State Lett. 13(5), J43–J46 (2010).
[CrossRef]

D. T. Owens, C. Fuentes-Hernandez, J. M. Hales, J. W. Perry, and B. Kippelen, “Nonlinear optical properties of induced transmission filters,” Opt. Express 18(18), 19101–19113 (2010).
[CrossRef] [PubMed]

P. E. Hopkins, “Influence of Inter- and Intraband Transitions to Electron Temperature Decay in Noble Metals After Short-Pulsed Laser Heating,” J. Heat Transfer 132(12), 122402 (2010).
[CrossRef]

M. G. Blaber, M. D. Arnold, and M. J. Ford, “A review of the optical properties of alloys and intermetallics for plasmonics,” J. Phys. Condens. Matter 22(14), 143201 (2010).
[CrossRef] [PubMed]

2009

H. Tóháti, A. Sipos, G. Szekeres, A. Mathesz, A. Szalai, P. Jojart, J. Budai, C. Vass, A. Kohazi-Kis, M. Csete, and Z. Bor, “Surface plasmon scattering on polymer-bimetal layer covered fused silica gratings generated by laser induced backside wet etching,” Appl. Surf. Sci. 255(10), 5130–5137 (2009).
[CrossRef]

2008

Z. Lin, L. V. Zhigilei, and V. Celli, “Electron-phonon coupling and electron heat capacity of metals under conditions of strong electron-phonon nonequilibrium,” Phys. Rev. B 77(7), 075133 (2008).
[CrossRef]

2007

B. H. Ong, X. C. Yuan, and S. C. Tjin, “Bimetallic silver-gold film waveguide surface plasmon resonance sensor,” Fiber Integr. Opt. 26(4), 229–240 (2007).
[CrossRef]

N. Rotenberg, A. D. Bristow, M. Pfeiffer, M. Betz, and H. M. van Driel, “Nonlinear absorption in Au films: Role of thermal effects,” Phys. Rev. B 75(15), 155426 (2007).
[CrossRef]

2006

H. T. Chen, W. J. Padilla, J. M. O. Zide, A. C. Gossard, A. J. Taylor, and R. D. Averitt, “Active terahertz metamaterial devices,” Nature 444(7119), 597–600 (2006).
[CrossRef] [PubMed]

P. G. Etchegoin, E. C. Le Ru, and M. Meyer, “An analytic model for the optical properties of gold,” J. Chem. Phys. 125(16), 164705 (2006).
[CrossRef] [PubMed]

2003

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424(6950), 824–830 (2003).
[CrossRef] [PubMed]

2001

C. Voisin, N. Del Fatti, D. Christofilos, and F. Vallee, “Ultrafast electron dynamics and optical nonlinearities in metal nanoparticles,” J. Phys. Chem. B 105(12), 2264–2280 (2001).
[CrossRef]

1998

M. J. Bloemer and M. Scalora, “Transmissive properties of Ag/MgF2 photonic band gaps,” Appl. Phys. Lett. 72(14), 1676–1678 (1998).
[CrossRef]

1995

J. Y. Bigot, J. Y. Merle, O. Cregut, and A. Daunois, “Electron dynamics in copper metallic nanoparticles probed with femtosecond optical pulses,” Phys. Rev. Lett. 75(25), 4702–4705 (1995).
[CrossRef] [PubMed]

1994

T. Q. Qiu, T. Juhasz, C. Suarez, W. E. Bron, and C. L. Tien, “Femtosecond laser heating of multi-layer metals—II. Experiments,” Int. J. Heat Mass Transfer 37(17), 2799–2808 (1994).
[CrossRef]

T. Q. Qiu and C. L. Tien, “Femtosecond laser heating of multi-layer metals—I. Analysis,” Int. J. Heat Mass Transfer 37(17), 2789–2797 (1994).
[CrossRef]

1974

S. I. Anisimov, B. L. Kapeliovich, and T. L Perel'Man, “Electron emission from metal surfaces exposed to ultrashort laser pulses,” Sov. Phys. JETP 39, 375–377 (1974).

1967

W. J. Scouler, “Temperature-modulated reflectance of gold from 2 to 10 ev,” Phys. Rev. Lett. 18(12), 445–448 (1967).
[CrossRef]

Anderson, M.

M. Thomschke, S. Hofmann, S. Olthof, M. Anderson, H. Kleemann, M. Schober, B. Lussem, and K. Leo, “Improvement of voltage and charge balance in inverted top-emitting organic electroluminescent diodes comprising doped transport layers by thermal annealing,” Appl. Phys. Lett. 98(8), 083304 (2011).
[CrossRef]

Anisimov, S. I.

S. I. Anisimov, B. L. Kapeliovich, and T. L Perel'Man, “Electron emission from metal surfaces exposed to ultrashort laser pulses,” Sov. Phys. JETP 39, 375–377 (1974).

Arnold, M. D.

M. G. Blaber, M. D. Arnold, and M. J. Ford, “A review of the optical properties of alloys and intermetallics for plasmonics,” J. Phys. Condens. Matter 22(14), 143201 (2010).
[CrossRef] [PubMed]

Averitt, R. D.

H. T. Chen, W. J. Padilla, J. M. O. Zide, A. C. Gossard, A. J. Taylor, and R. D. Averitt, “Active terahertz metamaterial devices,” Nature 444(7119), 597–600 (2006).
[CrossRef] [PubMed]

Baik, H. K.

S. Y. Ryu, C. H. Lee, I. S. Oh, S. Y. Song, K. H. Hwang, H. S. Hwang, M. H. Han, B. H. Hwang, H. K. Baik, Y. S. Kim, and J. Y. Lee, “Efficient inverted top-emitting organic light emitting diodes with transparent and surface-modified multilayer anodes,” Electrochem. Solid-State Lett. 13(5), J43–J46 (2010).
[CrossRef]

Barnes, W. L.

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424(6950), 824–830 (2003).
[CrossRef] [PubMed]

Betz, M.

N. Rotenberg, M. Betz, and H. M. van Driel, “Ultrafast all-optical coupling of light to surface plasmon polaritons on plain metal surfaces,” Phys. Rev. Lett. 105(1), 017402 (2010).
[CrossRef] [PubMed]

N. Rotenberg, A. D. Bristow, M. Pfeiffer, M. Betz, and H. M. van Driel, “Nonlinear absorption in Au films: Role of thermal effects,” Phys. Rev. B 75(15), 155426 (2007).
[CrossRef]

Bigot, J. Y.

J. Y. Bigot, J. Y. Merle, O. Cregut, and A. Daunois, “Electron dynamics in copper metallic nanoparticles probed with femtosecond optical pulses,” Phys. Rev. Lett. 75(25), 4702–4705 (1995).
[CrossRef] [PubMed]

Blaber, M. G.

M. G. Blaber, M. D. Arnold, and M. J. Ford, “A review of the optical properties of alloys and intermetallics for plasmonics,” J. Phys. Condens. Matter 22(14), 143201 (2010).
[CrossRef] [PubMed]

Bloemer, M. J.

M. J. Bloemer and M. Scalora, “Transmissive properties of Ag/MgF2 photonic band gaps,” Appl. Phys. Lett. 72(14), 1676–1678 (1998).
[CrossRef]

Bor, Z.

H. Tóháti, A. Sipos, G. Szekeres, A. Mathesz, A. Szalai, P. Jojart, J. Budai, C. Vass, A. Kohazi-Kis, M. Csete, and Z. Bor, “Surface plasmon scattering on polymer-bimetal layer covered fused silica gratings generated by laser induced backside wet etching,” Appl. Surf. Sci. 255(10), 5130–5137 (2009).
[CrossRef]

Bristow, A. D.

N. Rotenberg, A. D. Bristow, M. Pfeiffer, M. Betz, and H. M. van Driel, “Nonlinear absorption in Au films: Role of thermal effects,” Phys. Rev. B 75(15), 155426 (2007).
[CrossRef]

Bron, W. E.

T. Q. Qiu, T. Juhasz, C. Suarez, W. E. Bron, and C. L. Tien, “Femtosecond laser heating of multi-layer metals—II. Experiments,” Int. J. Heat Mass Transfer 37(17), 2799–2808 (1994).
[CrossRef]

Brueck, S. R. J.

J. Y. Zhang, L. Wang, S. Krishna, M. Sheik-Bahae, and S. R. J. Brueck, “Saturation of the second harmonic generation from GaAs-filled metallic hole arrays by nonlinear absorption,” Phys. Rev. B 83(16), 165438 (2011).
[CrossRef]

Budai, J.

H. Tóháti, A. Sipos, G. Szekeres, A. Mathesz, A. Szalai, P. Jojart, J. Budai, C. Vass, A. Kohazi-Kis, M. Csete, and Z. Bor, “Surface plasmon scattering on polymer-bimetal layer covered fused silica gratings generated by laser induced backside wet etching,” Appl. Surf. Sci. 255(10), 5130–5137 (2009).
[CrossRef]

Celli, V.

Z. Lin, L. V. Zhigilei, and V. Celli, “Electron-phonon coupling and electron heat capacity of metals under conditions of strong electron-phonon nonequilibrium,” Phys. Rev. B 77(7), 075133 (2008).
[CrossRef]

Chen, H. T.

H. T. Chen, W. J. Padilla, J. M. O. Zide, A. C. Gossard, A. J. Taylor, and R. D. Averitt, “Active terahertz metamaterial devices,” Nature 444(7119), 597–600 (2006).
[CrossRef] [PubMed]

Christofilos, D.

C. Voisin, N. Del Fatti, D. Christofilos, and F. Vallee, “Ultrafast electron dynamics and optical nonlinearities in metal nanoparticles,” J. Phys. Chem. B 105(12), 2264–2280 (2001).
[CrossRef]

Cregut, O.

J. Y. Bigot, J. Y. Merle, O. Cregut, and A. Daunois, “Electron dynamics in copper metallic nanoparticles probed with femtosecond optical pulses,” Phys. Rev. Lett. 75(25), 4702–4705 (1995).
[CrossRef] [PubMed]

Csete, M.

H. Tóháti, A. Sipos, G. Szekeres, A. Mathesz, A. Szalai, P. Jojart, J. Budai, C. Vass, A. Kohazi-Kis, M. Csete, and Z. Bor, “Surface plasmon scattering on polymer-bimetal layer covered fused silica gratings generated by laser induced backside wet etching,” Appl. Surf. Sci. 255(10), 5130–5137 (2009).
[CrossRef]

Daunois, A.

J. Y. Bigot, J. Y. Merle, O. Cregut, and A. Daunois, “Electron dynamics in copper metallic nanoparticles probed with femtosecond optical pulses,” Phys. Rev. Lett. 75(25), 4702–4705 (1995).
[CrossRef] [PubMed]

Del Fatti, N.

C. Voisin, N. Del Fatti, D. Christofilos, and F. Vallee, “Ultrafast electron dynamics and optical nonlinearities in metal nanoparticles,” J. Phys. Chem. B 105(12), 2264–2280 (2001).
[CrossRef]

Dereux, A.

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424(6950), 824–830 (2003).
[CrossRef] [PubMed]

Ebbesen, T. W.

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424(6950), 824–830 (2003).
[CrossRef] [PubMed]

Etchegoin, P. G.

P. G. Etchegoin, E. C. Le Ru, and M. Meyer, “An analytic model for the optical properties of gold,” J. Chem. Phys. 125(16), 164705 (2006).
[CrossRef] [PubMed]

Ford, M. J.

M. G. Blaber, M. D. Arnold, and M. J. Ford, “A review of the optical properties of alloys and intermetallics for plasmonics,” J. Phys. Condens. Matter 22(14), 143201 (2010).
[CrossRef] [PubMed]

Fuentes-Hernandez, C.

D. T. Owens, C. Fuentes-Hernandez, J. M. Hales, J. W. Perry, and B. Kippelen, “Nonlinear optical properties of induced transmission filters,” Opt. Express 18(18), 19101–19113 (2010).
[CrossRef] [PubMed]

D. T. Owens, C. Fuentes-Hernandez, J. M. Hales, J. W. Perry, and B. Kippelen, “A comprehensive analysis of the contributions to the nonlinear optical properties of thin Ag films,” J. Appl. Phys. 107(12), 123114 (2010).
[CrossRef]

Gossard, A. C.

H. T. Chen, W. J. Padilla, J. M. O. Zide, A. C. Gossard, A. J. Taylor, and R. D. Averitt, “Active terahertz metamaterial devices,” Nature 444(7119), 597–600 (2006).
[CrossRef] [PubMed]

Gosztola, D. J.

G. A. Wurtz, R. Pollard, W. Hendren, G. P. Wiederrecht, D. J. Gosztola, V. A. Podolskiy, and A. V. Zayats, “Designed ultrafast optical nonlinearity in a plasmonic nanorod metamaterial enhanced by nonlocality,” Nat. Nanotechnol. 6(2), 107–111 (2011).
[CrossRef] [PubMed]

Hales, J. M.

D. T. Owens, C. Fuentes-Hernandez, J. M. Hales, J. W. Perry, and B. Kippelen, “Nonlinear optical properties of induced transmission filters,” Opt. Express 18(18), 19101–19113 (2010).
[CrossRef] [PubMed]

D. T. Owens, C. Fuentes-Hernandez, J. M. Hales, J. W. Perry, and B. Kippelen, “A comprehensive analysis of the contributions to the nonlinear optical properties of thin Ag films,” J. Appl. Phys. 107(12), 123114 (2010).
[CrossRef]

Han, M. H.

S. Y. Ryu, C. H. Lee, I. S. Oh, S. Y. Song, K. H. Hwang, H. S. Hwang, M. H. Han, B. H. Hwang, H. K. Baik, Y. S. Kim, and J. Y. Lee, “Efficient inverted top-emitting organic light emitting diodes with transparent and surface-modified multilayer anodes,” Electrochem. Solid-State Lett. 13(5), J43–J46 (2010).
[CrossRef]

Hendren, W.

G. A. Wurtz, R. Pollard, W. Hendren, G. P. Wiederrecht, D. J. Gosztola, V. A. Podolskiy, and A. V. Zayats, “Designed ultrafast optical nonlinearity in a plasmonic nanorod metamaterial enhanced by nonlocality,” Nat. Nanotechnol. 6(2), 107–111 (2011).
[CrossRef] [PubMed]

Hofmann, S.

M. Thomschke, S. Hofmann, S. Olthof, M. Anderson, H. Kleemann, M. Schober, B. Lussem, and K. Leo, “Improvement of voltage and charge balance in inverted top-emitting organic electroluminescent diodes comprising doped transport layers by thermal annealing,” Appl. Phys. Lett. 98(8), 083304 (2011).
[CrossRef]

Hopkins, P. E.

P. E. Hopkins, “Influence of Inter- and Intraband Transitions to Electron Temperature Decay in Noble Metals After Short-Pulsed Laser Heating,” J. Heat Transfer 132(12), 122402 (2010).
[CrossRef]

Hwang, B. H.

S. Y. Ryu, C. H. Lee, I. S. Oh, S. Y. Song, K. H. Hwang, H. S. Hwang, M. H. Han, B. H. Hwang, H. K. Baik, Y. S. Kim, and J. Y. Lee, “Efficient inverted top-emitting organic light emitting diodes with transparent and surface-modified multilayer anodes,” Electrochem. Solid-State Lett. 13(5), J43–J46 (2010).
[CrossRef]

Hwang, H. S.

S. Y. Ryu, C. H. Lee, I. S. Oh, S. Y. Song, K. H. Hwang, H. S. Hwang, M. H. Han, B. H. Hwang, H. K. Baik, Y. S. Kim, and J. Y. Lee, “Efficient inverted top-emitting organic light emitting diodes with transparent and surface-modified multilayer anodes,” Electrochem. Solid-State Lett. 13(5), J43–J46 (2010).
[CrossRef]

Hwang, K. H.

S. Y. Ryu, C. H. Lee, I. S. Oh, S. Y. Song, K. H. Hwang, H. S. Hwang, M. H. Han, B. H. Hwang, H. K. Baik, Y. S. Kim, and J. Y. Lee, “Efficient inverted top-emitting organic light emitting diodes with transparent and surface-modified multilayer anodes,” Electrochem. Solid-State Lett. 13(5), J43–J46 (2010).
[CrossRef]

Jojart, P.

H. Tóháti, A. Sipos, G. Szekeres, A. Mathesz, A. Szalai, P. Jojart, J. Budai, C. Vass, A. Kohazi-Kis, M. Csete, and Z. Bor, “Surface plasmon scattering on polymer-bimetal layer covered fused silica gratings generated by laser induced backside wet etching,” Appl. Surf. Sci. 255(10), 5130–5137 (2009).
[CrossRef]

Juhasz, T.

T. Q. Qiu, T. Juhasz, C. Suarez, W. E. Bron, and C. L. Tien, “Femtosecond laser heating of multi-layer metals—II. Experiments,” Int. J. Heat Mass Transfer 37(17), 2799–2808 (1994).
[CrossRef]

Kapeliovich, B. L.

S. I. Anisimov, B. L. Kapeliovich, and T. L Perel'Man, “Electron emission from metal surfaces exposed to ultrashort laser pulses,” Sov. Phys. JETP 39, 375–377 (1974).

Kim, Y. S.

S. Y. Ryu, C. H. Lee, I. S. Oh, S. Y. Song, K. H. Hwang, H. S. Hwang, M. H. Han, B. H. Hwang, H. K. Baik, Y. S. Kim, and J. Y. Lee, “Efficient inverted top-emitting organic light emitting diodes with transparent and surface-modified multilayer anodes,” Electrochem. Solid-State Lett. 13(5), J43–J46 (2010).
[CrossRef]

Kippelen, B.

D. T. Owens, C. Fuentes-Hernandez, J. M. Hales, J. W. Perry, and B. Kippelen, “A comprehensive analysis of the contributions to the nonlinear optical properties of thin Ag films,” J. Appl. Phys. 107(12), 123114 (2010).
[CrossRef]

D. T. Owens, C. Fuentes-Hernandez, J. M. Hales, J. W. Perry, and B. Kippelen, “Nonlinear optical properties of induced transmission filters,” Opt. Express 18(18), 19101–19113 (2010).
[CrossRef] [PubMed]

Kleemann, H.

M. Thomschke, S. Hofmann, S. Olthof, M. Anderson, H. Kleemann, M. Schober, B. Lussem, and K. Leo, “Improvement of voltage and charge balance in inverted top-emitting organic electroluminescent diodes comprising doped transport layers by thermal annealing,” Appl. Phys. Lett. 98(8), 083304 (2011).
[CrossRef]

Kohazi-Kis, A.

H. Tóháti, A. Sipos, G. Szekeres, A. Mathesz, A. Szalai, P. Jojart, J. Budai, C. Vass, A. Kohazi-Kis, M. Csete, and Z. Bor, “Surface plasmon scattering on polymer-bimetal layer covered fused silica gratings generated by laser induced backside wet etching,” Appl. Surf. Sci. 255(10), 5130–5137 (2009).
[CrossRef]

Krishna, S.

J. Y. Zhang, L. Wang, S. Krishna, M. Sheik-Bahae, and S. R. J. Brueck, “Saturation of the second harmonic generation from GaAs-filled metallic hole arrays by nonlinear absorption,” Phys. Rev. B 83(16), 165438 (2011).
[CrossRef]

Le Ru, E. C.

P. G. Etchegoin, E. C. Le Ru, and M. Meyer, “An analytic model for the optical properties of gold,” J. Chem. Phys. 125(16), 164705 (2006).
[CrossRef] [PubMed]

Lee, C. H.

S. Y. Ryu, C. H. Lee, I. S. Oh, S. Y. Song, K. H. Hwang, H. S. Hwang, M. H. Han, B. H. Hwang, H. K. Baik, Y. S. Kim, and J. Y. Lee, “Efficient inverted top-emitting organic light emitting diodes with transparent and surface-modified multilayer anodes,” Electrochem. Solid-State Lett. 13(5), J43–J46 (2010).
[CrossRef]

Lee, J. Y.

S. Y. Ryu, C. H. Lee, I. S. Oh, S. Y. Song, K. H. Hwang, H. S. Hwang, M. H. Han, B. H. Hwang, H. K. Baik, Y. S. Kim, and J. Y. Lee, “Efficient inverted top-emitting organic light emitting diodes with transparent and surface-modified multilayer anodes,” Electrochem. Solid-State Lett. 13(5), J43–J46 (2010).
[CrossRef]

Leo, K.

M. Thomschke, S. Hofmann, S. Olthof, M. Anderson, H. Kleemann, M. Schober, B. Lussem, and K. Leo, “Improvement of voltage and charge balance in inverted top-emitting organic electroluminescent diodes comprising doped transport layers by thermal annealing,” Appl. Phys. Lett. 98(8), 083304 (2011).
[CrossRef]

Lin, Z.

Z. Lin, L. V. Zhigilei, and V. Celli, “Electron-phonon coupling and electron heat capacity of metals under conditions of strong electron-phonon nonequilibrium,” Phys. Rev. B 77(7), 075133 (2008).
[CrossRef]

Lussem, B.

M. Thomschke, S. Hofmann, S. Olthof, M. Anderson, H. Kleemann, M. Schober, B. Lussem, and K. Leo, “Improvement of voltage and charge balance in inverted top-emitting organic electroluminescent diodes comprising doped transport layers by thermal annealing,” Appl. Phys. Lett. 98(8), 083304 (2011).
[CrossRef]

Mathesz, A.

H. Tóháti, A. Sipos, G. Szekeres, A. Mathesz, A. Szalai, P. Jojart, J. Budai, C. Vass, A. Kohazi-Kis, M. Csete, and Z. Bor, “Surface plasmon scattering on polymer-bimetal layer covered fused silica gratings generated by laser induced backside wet etching,” Appl. Surf. Sci. 255(10), 5130–5137 (2009).
[CrossRef]

Merle, J. Y.

J. Y. Bigot, J. Y. Merle, O. Cregut, and A. Daunois, “Electron dynamics in copper metallic nanoparticles probed with femtosecond optical pulses,” Phys. Rev. Lett. 75(25), 4702–4705 (1995).
[CrossRef] [PubMed]

Meyer, M.

P. G. Etchegoin, E. C. Le Ru, and M. Meyer, “An analytic model for the optical properties of gold,” J. Chem. Phys. 125(16), 164705 (2006).
[CrossRef] [PubMed]

Oh, I. S.

S. Y. Ryu, C. H. Lee, I. S. Oh, S. Y. Song, K. H. Hwang, H. S. Hwang, M. H. Han, B. H. Hwang, H. K. Baik, Y. S. Kim, and J. Y. Lee, “Efficient inverted top-emitting organic light emitting diodes with transparent and surface-modified multilayer anodes,” Electrochem. Solid-State Lett. 13(5), J43–J46 (2010).
[CrossRef]

Olthof, S.

M. Thomschke, S. Hofmann, S. Olthof, M. Anderson, H. Kleemann, M. Schober, B. Lussem, and K. Leo, “Improvement of voltage and charge balance in inverted top-emitting organic electroluminescent diodes comprising doped transport layers by thermal annealing,” Appl. Phys. Lett. 98(8), 083304 (2011).
[CrossRef]

Ong, B. H.

B. H. Ong, X. C. Yuan, and S. C. Tjin, “Bimetallic silver-gold film waveguide surface plasmon resonance sensor,” Fiber Integr. Opt. 26(4), 229–240 (2007).
[CrossRef]

Owens, D. T.

D. T. Owens, C. Fuentes-Hernandez, J. M. Hales, J. W. Perry, and B. Kippelen, “A comprehensive analysis of the contributions to the nonlinear optical properties of thin Ag films,” J. Appl. Phys. 107(12), 123114 (2010).
[CrossRef]

D. T. Owens, C. Fuentes-Hernandez, J. M. Hales, J. W. Perry, and B. Kippelen, “Nonlinear optical properties of induced transmission filters,” Opt. Express 18(18), 19101–19113 (2010).
[CrossRef] [PubMed]

Padilla, W. J.

H. T. Chen, W. J. Padilla, J. M. O. Zide, A. C. Gossard, A. J. Taylor, and R. D. Averitt, “Active terahertz metamaterial devices,” Nature 444(7119), 597–600 (2006).
[CrossRef] [PubMed]

Perel'Man, T. L

S. I. Anisimov, B. L. Kapeliovich, and T. L Perel'Man, “Electron emission from metal surfaces exposed to ultrashort laser pulses,” Sov. Phys. JETP 39, 375–377 (1974).

Perry, J. W.

D. T. Owens, C. Fuentes-Hernandez, J. M. Hales, J. W. Perry, and B. Kippelen, “Nonlinear optical properties of induced transmission filters,” Opt. Express 18(18), 19101–19113 (2010).
[CrossRef] [PubMed]

D. T. Owens, C. Fuentes-Hernandez, J. M. Hales, J. W. Perry, and B. Kippelen, “A comprehensive analysis of the contributions to the nonlinear optical properties of thin Ag films,” J. Appl. Phys. 107(12), 123114 (2010).
[CrossRef]

Pfeiffer, M.

N. Rotenberg, A. D. Bristow, M. Pfeiffer, M. Betz, and H. M. van Driel, “Nonlinear absorption in Au films: Role of thermal effects,” Phys. Rev. B 75(15), 155426 (2007).
[CrossRef]

Podolskiy, V. A.

G. A. Wurtz, R. Pollard, W. Hendren, G. P. Wiederrecht, D. J. Gosztola, V. A. Podolskiy, and A. V. Zayats, “Designed ultrafast optical nonlinearity in a plasmonic nanorod metamaterial enhanced by nonlocality,” Nat. Nanotechnol. 6(2), 107–111 (2011).
[CrossRef] [PubMed]

Pollard, R.

G. A. Wurtz, R. Pollard, W. Hendren, G. P. Wiederrecht, D. J. Gosztola, V. A. Podolskiy, and A. V. Zayats, “Designed ultrafast optical nonlinearity in a plasmonic nanorod metamaterial enhanced by nonlocality,” Nat. Nanotechnol. 6(2), 107–111 (2011).
[CrossRef] [PubMed]

Qiu, T. Q.

T. Q. Qiu, T. Juhasz, C. Suarez, W. E. Bron, and C. L. Tien, “Femtosecond laser heating of multi-layer metals—II. Experiments,” Int. J. Heat Mass Transfer 37(17), 2799–2808 (1994).
[CrossRef]

T. Q. Qiu and C. L. Tien, “Femtosecond laser heating of multi-layer metals—I. Analysis,” Int. J. Heat Mass Transfer 37(17), 2789–2797 (1994).
[CrossRef]

Rotenberg, N.

M. A. Swillam, N. Rotenberg, and H. M. van Driel, “All-optical ultrafast control of beaming through a single sub-wavelength aperture in a metal film,” Opt. Express 19(8), 7856–7864 (2011).
[CrossRef] [PubMed]

N. Rotenberg, M. Betz, and H. M. van Driel, “Ultrafast all-optical coupling of light to surface plasmon polaritons on plain metal surfaces,” Phys. Rev. Lett. 105(1), 017402 (2010).
[CrossRef] [PubMed]

N. Rotenberg, A. D. Bristow, M. Pfeiffer, M. Betz, and H. M. van Driel, “Nonlinear absorption in Au films: Role of thermal effects,” Phys. Rev. B 75(15), 155426 (2007).
[CrossRef]

Ryu, S. Y.

S. Y. Ryu, C. H. Lee, I. S. Oh, S. Y. Song, K. H. Hwang, H. S. Hwang, M. H. Han, B. H. Hwang, H. K. Baik, Y. S. Kim, and J. Y. Lee, “Efficient inverted top-emitting organic light emitting diodes with transparent and surface-modified multilayer anodes,” Electrochem. Solid-State Lett. 13(5), J43–J46 (2010).
[CrossRef]

Scalora, M.

M. J. Bloemer and M. Scalora, “Transmissive properties of Ag/MgF2 photonic band gaps,” Appl. Phys. Lett. 72(14), 1676–1678 (1998).
[CrossRef]

Schober, M.

M. Thomschke, S. Hofmann, S. Olthof, M. Anderson, H. Kleemann, M. Schober, B. Lussem, and K. Leo, “Improvement of voltage and charge balance in inverted top-emitting organic electroluminescent diodes comprising doped transport layers by thermal annealing,” Appl. Phys. Lett. 98(8), 083304 (2011).
[CrossRef]

Scouler, W. J.

W. J. Scouler, “Temperature-modulated reflectance of gold from 2 to 10 ev,” Phys. Rev. Lett. 18(12), 445–448 (1967).
[CrossRef]

Sheik-Bahae, M.

J. Y. Zhang, L. Wang, S. Krishna, M. Sheik-Bahae, and S. R. J. Brueck, “Saturation of the second harmonic generation from GaAs-filled metallic hole arrays by nonlinear absorption,” Phys. Rev. B 83(16), 165438 (2011).
[CrossRef]

Sipos, A.

H. Tóháti, A. Sipos, G. Szekeres, A. Mathesz, A. Szalai, P. Jojart, J. Budai, C. Vass, A. Kohazi-Kis, M. Csete, and Z. Bor, “Surface plasmon scattering on polymer-bimetal layer covered fused silica gratings generated by laser induced backside wet etching,” Appl. Surf. Sci. 255(10), 5130–5137 (2009).
[CrossRef]

Song, S. Y.

S. Y. Ryu, C. H. Lee, I. S. Oh, S. Y. Song, K. H. Hwang, H. S. Hwang, M. H. Han, B. H. Hwang, H. K. Baik, Y. S. Kim, and J. Y. Lee, “Efficient inverted top-emitting organic light emitting diodes with transparent and surface-modified multilayer anodes,” Electrochem. Solid-State Lett. 13(5), J43–J46 (2010).
[CrossRef]

Suarez, C.

T. Q. Qiu, T. Juhasz, C. Suarez, W. E. Bron, and C. L. Tien, “Femtosecond laser heating of multi-layer metals—II. Experiments,” Int. J. Heat Mass Transfer 37(17), 2799–2808 (1994).
[CrossRef]

Swillam, M. A.

Szalai, A.

H. Tóháti, A. Sipos, G. Szekeres, A. Mathesz, A. Szalai, P. Jojart, J. Budai, C. Vass, A. Kohazi-Kis, M. Csete, and Z. Bor, “Surface plasmon scattering on polymer-bimetal layer covered fused silica gratings generated by laser induced backside wet etching,” Appl. Surf. Sci. 255(10), 5130–5137 (2009).
[CrossRef]

Szekeres, G.

H. Tóháti, A. Sipos, G. Szekeres, A. Mathesz, A. Szalai, P. Jojart, J. Budai, C. Vass, A. Kohazi-Kis, M. Csete, and Z. Bor, “Surface plasmon scattering on polymer-bimetal layer covered fused silica gratings generated by laser induced backside wet etching,” Appl. Surf. Sci. 255(10), 5130–5137 (2009).
[CrossRef]

Taylor, A. J.

H. T. Chen, W. J. Padilla, J. M. O. Zide, A. C. Gossard, A. J. Taylor, and R. D. Averitt, “Active terahertz metamaterial devices,” Nature 444(7119), 597–600 (2006).
[CrossRef] [PubMed]

Thomschke, M.

M. Thomschke, S. Hofmann, S. Olthof, M. Anderson, H. Kleemann, M. Schober, B. Lussem, and K. Leo, “Improvement of voltage and charge balance in inverted top-emitting organic electroluminescent diodes comprising doped transport layers by thermal annealing,” Appl. Phys. Lett. 98(8), 083304 (2011).
[CrossRef]

Tien, C. L.

T. Q. Qiu, T. Juhasz, C. Suarez, W. E. Bron, and C. L. Tien, “Femtosecond laser heating of multi-layer metals—II. Experiments,” Int. J. Heat Mass Transfer 37(17), 2799–2808 (1994).
[CrossRef]

T. Q. Qiu and C. L. Tien, “Femtosecond laser heating of multi-layer metals—I. Analysis,” Int. J. Heat Mass Transfer 37(17), 2789–2797 (1994).
[CrossRef]

Tjin, S. C.

B. H. Ong, X. C. Yuan, and S. C. Tjin, “Bimetallic silver-gold film waveguide surface plasmon resonance sensor,” Fiber Integr. Opt. 26(4), 229–240 (2007).
[CrossRef]

Tóháti, H.

H. Tóháti, A. Sipos, G. Szekeres, A. Mathesz, A. Szalai, P. Jojart, J. Budai, C. Vass, A. Kohazi-Kis, M. Csete, and Z. Bor, “Surface plasmon scattering on polymer-bimetal layer covered fused silica gratings generated by laser induced backside wet etching,” Appl. Surf. Sci. 255(10), 5130–5137 (2009).
[CrossRef]

Vallee, F.

C. Voisin, N. Del Fatti, D. Christofilos, and F. Vallee, “Ultrafast electron dynamics and optical nonlinearities in metal nanoparticles,” J. Phys. Chem. B 105(12), 2264–2280 (2001).
[CrossRef]

van Driel, H. M.

M. A. Swillam, N. Rotenberg, and H. M. van Driel, “All-optical ultrafast control of beaming through a single sub-wavelength aperture in a metal film,” Opt. Express 19(8), 7856–7864 (2011).
[CrossRef] [PubMed]

N. Rotenberg, M. Betz, and H. M. van Driel, “Ultrafast all-optical coupling of light to surface plasmon polaritons on plain metal surfaces,” Phys. Rev. Lett. 105(1), 017402 (2010).
[CrossRef] [PubMed]

N. Rotenberg, A. D. Bristow, M. Pfeiffer, M. Betz, and H. M. van Driel, “Nonlinear absorption in Au films: Role of thermal effects,” Phys. Rev. B 75(15), 155426 (2007).
[CrossRef]

Vass, C.

H. Tóháti, A. Sipos, G. Szekeres, A. Mathesz, A. Szalai, P. Jojart, J. Budai, C. Vass, A. Kohazi-Kis, M. Csete, and Z. Bor, “Surface plasmon scattering on polymer-bimetal layer covered fused silica gratings generated by laser induced backside wet etching,” Appl. Surf. Sci. 255(10), 5130–5137 (2009).
[CrossRef]

Voisin, C.

C. Voisin, N. Del Fatti, D. Christofilos, and F. Vallee, “Ultrafast electron dynamics and optical nonlinearities in metal nanoparticles,” J. Phys. Chem. B 105(12), 2264–2280 (2001).
[CrossRef]

Wang, L.

J. Y. Zhang, L. Wang, S. Krishna, M. Sheik-Bahae, and S. R. J. Brueck, “Saturation of the second harmonic generation from GaAs-filled metallic hole arrays by nonlinear absorption,” Phys. Rev. B 83(16), 165438 (2011).
[CrossRef]

Wiederrecht, G. P.

G. A. Wurtz, R. Pollard, W. Hendren, G. P. Wiederrecht, D. J. Gosztola, V. A. Podolskiy, and A. V. Zayats, “Designed ultrafast optical nonlinearity in a plasmonic nanorod metamaterial enhanced by nonlocality,” Nat. Nanotechnol. 6(2), 107–111 (2011).
[CrossRef] [PubMed]

Wurtz, G. A.

G. A. Wurtz, R. Pollard, W. Hendren, G. P. Wiederrecht, D. J. Gosztola, V. A. Podolskiy, and A. V. Zayats, “Designed ultrafast optical nonlinearity in a plasmonic nanorod metamaterial enhanced by nonlocality,” Nat. Nanotechnol. 6(2), 107–111 (2011).
[CrossRef] [PubMed]

Yuan, X. C.

B. H. Ong, X. C. Yuan, and S. C. Tjin, “Bimetallic silver-gold film waveguide surface plasmon resonance sensor,” Fiber Integr. Opt. 26(4), 229–240 (2007).
[CrossRef]

Zayats, A. V.

G. A. Wurtz, R. Pollard, W. Hendren, G. P. Wiederrecht, D. J. Gosztola, V. A. Podolskiy, and A. V. Zayats, “Designed ultrafast optical nonlinearity in a plasmonic nanorod metamaterial enhanced by nonlocality,” Nat. Nanotechnol. 6(2), 107–111 (2011).
[CrossRef] [PubMed]

Zhang, J. Y.

J. Y. Zhang, L. Wang, S. Krishna, M. Sheik-Bahae, and S. R. J. Brueck, “Saturation of the second harmonic generation from GaAs-filled metallic hole arrays by nonlinear absorption,” Phys. Rev. B 83(16), 165438 (2011).
[CrossRef]

Zhigilei, L. V.

Z. Lin, L. V. Zhigilei, and V. Celli, “Electron-phonon coupling and electron heat capacity of metals under conditions of strong electron-phonon nonequilibrium,” Phys. Rev. B 77(7), 075133 (2008).
[CrossRef]

Zide, J. M. O.

H. T. Chen, W. J. Padilla, J. M. O. Zide, A. C. Gossard, A. J. Taylor, and R. D. Averitt, “Active terahertz metamaterial devices,” Nature 444(7119), 597–600 (2006).
[CrossRef] [PubMed]

Appl. Phys. Lett.

M. J. Bloemer and M. Scalora, “Transmissive properties of Ag/MgF2 photonic band gaps,” Appl. Phys. Lett. 72(14), 1676–1678 (1998).
[CrossRef]

M. Thomschke, S. Hofmann, S. Olthof, M. Anderson, H. Kleemann, M. Schober, B. Lussem, and K. Leo, “Improvement of voltage and charge balance in inverted top-emitting organic electroluminescent diodes comprising doped transport layers by thermal annealing,” Appl. Phys. Lett. 98(8), 083304 (2011).
[CrossRef]

Appl. Surf. Sci.

H. Tóháti, A. Sipos, G. Szekeres, A. Mathesz, A. Szalai, P. Jojart, J. Budai, C. Vass, A. Kohazi-Kis, M. Csete, and Z. Bor, “Surface plasmon scattering on polymer-bimetal layer covered fused silica gratings generated by laser induced backside wet etching,” Appl. Surf. Sci. 255(10), 5130–5137 (2009).
[CrossRef]

Electrochem. Solid-State Lett.

S. Y. Ryu, C. H. Lee, I. S. Oh, S. Y. Song, K. H. Hwang, H. S. Hwang, M. H. Han, B. H. Hwang, H. K. Baik, Y. S. Kim, and J. Y. Lee, “Efficient inverted top-emitting organic light emitting diodes with transparent and surface-modified multilayer anodes,” Electrochem. Solid-State Lett. 13(5), J43–J46 (2010).
[CrossRef]

Fiber Integr. Opt.

B. H. Ong, X. C. Yuan, and S. C. Tjin, “Bimetallic silver-gold film waveguide surface plasmon resonance sensor,” Fiber Integr. Opt. 26(4), 229–240 (2007).
[CrossRef]

Int. J. Heat Mass Transfer

T. Q. Qiu, T. Juhasz, C. Suarez, W. E. Bron, and C. L. Tien, “Femtosecond laser heating of multi-layer metals—II. Experiments,” Int. J. Heat Mass Transfer 37(17), 2799–2808 (1994).
[CrossRef]

T. Q. Qiu and C. L. Tien, “Femtosecond laser heating of multi-layer metals—I. Analysis,” Int. J. Heat Mass Transfer 37(17), 2789–2797 (1994).
[CrossRef]

J. Appl. Phys.

D. T. Owens, C. Fuentes-Hernandez, J. M. Hales, J. W. Perry, and B. Kippelen, “A comprehensive analysis of the contributions to the nonlinear optical properties of thin Ag films,” J. Appl. Phys. 107(12), 123114 (2010).
[CrossRef]

J. Chem. Phys.

P. G. Etchegoin, E. C. Le Ru, and M. Meyer, “An analytic model for the optical properties of gold,” J. Chem. Phys. 125(16), 164705 (2006).
[CrossRef] [PubMed]

J. Heat Transfer

P. E. Hopkins, “Influence of Inter- and Intraband Transitions to Electron Temperature Decay in Noble Metals After Short-Pulsed Laser Heating,” J. Heat Transfer 132(12), 122402 (2010).
[CrossRef]

J. Phys. Chem. B

C. Voisin, N. Del Fatti, D. Christofilos, and F. Vallee, “Ultrafast electron dynamics and optical nonlinearities in metal nanoparticles,” J. Phys. Chem. B 105(12), 2264–2280 (2001).
[CrossRef]

J. Phys. Condens. Matter

M. G. Blaber, M. D. Arnold, and M. J. Ford, “A review of the optical properties of alloys and intermetallics for plasmonics,” J. Phys. Condens. Matter 22(14), 143201 (2010).
[CrossRef] [PubMed]

Nat. Nanotechnol.

G. A. Wurtz, R. Pollard, W. Hendren, G. P. Wiederrecht, D. J. Gosztola, V. A. Podolskiy, and A. V. Zayats, “Designed ultrafast optical nonlinearity in a plasmonic nanorod metamaterial enhanced by nonlocality,” Nat. Nanotechnol. 6(2), 107–111 (2011).
[CrossRef] [PubMed]

Nature

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424(6950), 824–830 (2003).
[CrossRef] [PubMed]

H. T. Chen, W. J. Padilla, J. M. O. Zide, A. C. Gossard, A. J. Taylor, and R. D. Averitt, “Active terahertz metamaterial devices,” Nature 444(7119), 597–600 (2006).
[CrossRef] [PubMed]

Opt. Express

Phys. Rev. B

N. Rotenberg, A. D. Bristow, M. Pfeiffer, M. Betz, and H. M. van Driel, “Nonlinear absorption in Au films: Role of thermal effects,” Phys. Rev. B 75(15), 155426 (2007).
[CrossRef]

J. Y. Zhang, L. Wang, S. Krishna, M. Sheik-Bahae, and S. R. J. Brueck, “Saturation of the second harmonic generation from GaAs-filled metallic hole arrays by nonlinear absorption,” Phys. Rev. B 83(16), 165438 (2011).
[CrossRef]

Z. Lin, L. V. Zhigilei, and V. Celli, “Electron-phonon coupling and electron heat capacity of metals under conditions of strong electron-phonon nonequilibrium,” Phys. Rev. B 77(7), 075133 (2008).
[CrossRef]

Phys. Rev. Lett.

J. Y. Bigot, J. Y. Merle, O. Cregut, and A. Daunois, “Electron dynamics in copper metallic nanoparticles probed with femtosecond optical pulses,” Phys. Rev. Lett. 75(25), 4702–4705 (1995).
[CrossRef] [PubMed]

W. J. Scouler, “Temperature-modulated reflectance of gold from 2 to 10 ev,” Phys. Rev. Lett. 18(12), 445–448 (1967).
[CrossRef]

N. Rotenberg, M. Betz, and H. M. van Driel, “Ultrafast all-optical coupling of light to surface plasmon polaritons on plain metal surfaces,” Phys. Rev. Lett. 105(1), 017402 (2010).
[CrossRef] [PubMed]

Sov. Phys. JETP

S. I. Anisimov, B. L. Kapeliovich, and T. L Perel'Man, “Electron emission from metal surfaces exposed to ultrashort laser pulses,” Sov. Phys. JETP 39, 375–377 (1974).

Other

H. A. Macleod, “The induced-transmission filter,” in Thin-Film Optical Filters, 3rd ed. (Institute of Physics Publishing, London, 2001).

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Figures (5)

Fig. 1
Fig. 1

Comparison of measured (symbols) and simulated (linear model 1: thin lines, linear model 2: thick lines) transmittance (T) (blue), reflectance (R) (green) and absorptance (A) (red) spectra in the visible range for fabricated samples R1, S1, S2, S3 and R2.

Fig. 2
Fig. 2

(a) Real and (b) imaginary effective refractive index values of bilayer Ag/Au metal thin films: M1, M2 and M3; Au and Ag are shown as reference. (c) Quality factor spectra for localized surface plasmon and (d) quality factor spectra for surface plasmon polariton. (e) Simulated maximum potential transmittance spectra.

Fig. 3
Fig. 3

(a), (b) Spectral dependence of transmittance and reflectance changes (ΔT(λ, tpeak) and ΔR(λ, tpeak)) measured from WLC pump probe experiment (solid line) and simulation by two-temperature model (dashed line) of samples R1, S1, S2 and S3 with a pump fluence of 25 J/m2; (c), (d) experimental (solid line) and simulated (dashed line) temporal dependence of the normalized absolute transmittance and reflectance changes probed at 520 nm. The excitation wavelength in all cases was 560 nm.

Fig. 4
Fig. 4

Comparison of measured (symbols) and simulated (solid lines) complex dielectric permittivity spectra in visible range for fabricated samples R1. The dielectric permittivity of Au was modeled as a set of explicit equations (Eqs. (3)-(5)) and compared with measured ellipsometric data.

Fig. 5
Fig. 5

Spectral dependence of the (a) real and (b) imaginary optical path length change (ΔOPL) pumped at 560 nm with a fluence of 25 J/m2 for Au, M1, M2 and M3. Solid line is the extracted ΔOPL = ΔNeffdT and the dashed line is the simulated ΔOPL = ΔNsimdAu probed at the peak transmittance delay time; (c) and (d) Absorbed pump fluence dependence of extracted ΔOPL(520 nm) at the peak transmittance change delay time.

Tables (2)

Tables Icon

Table 1 Constant Coefficients of a Au Dielectric Permittivity Model for Intraband Transition Term (Eq. (4))

Tables Icon

Table 2 Constant Coefficients of a Au Dielectric Permittivity Model of the First and Second Interband Transition Term (Eq. (5)), Respectively

Equations (6)

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ψ= ( ( n 2 k 2 2nk(Z/X)) ( n 2 + k 2 ) ( sin 2 αcos h 2 β+ cos 2 αsin h 2 β)+( cos 2 αcos h 2 β+ sin 2 αsin h 2 β) + 1 X (nsinhβcoshβ+kcosαsinα)+ X 2 + Z 2 X( n 2 + k 2 ) (nsinhβcoshβkcosαsinα) ) 1 with parametersα= 2πnd/λandβ=2πkd/λ.XandZare defined as X= [ ( n 2 + k 2 )(nsinhβcoshβ+ksinαcosα) (nsinhβcoshβksinαcosα) n 2 k 2 ( sin 2 αcos h 2 β+ cos 2 αsin h 2 β) 2 (nsinhβcoshβksinαcosα) 2 ] 1 2 Z= nk( sin 2 αcos h 2 β+ cos 2 αsin h 2 β) (nsinhβcoshβksinαcosα)
C e ( T e ) d T e dt =G( T e T l )+P(t) C l d T l dt =G( T e T l )
ε( ω, T l ( t ), T e ( t ) )= ε intra ( ω, T l ( t ), T e ( t ) )+ ε inter ( ω, T e ( t ) )
ε intra ( ω, T l ( t ), T e ( t ) )=1+ ε b ω p 2 ω 2 +iγω with parameters, γ[ T l (t), T e (t) ]= γ 0 + γ 1 × T l (t)+ γ 2 × T e 2 (t)+ γ 3 × ω 2
ε inter ( ω, T e ( t ) )= j=1 M κ (j) 0 dx x E g(j) x 2 [ 1 F (j) ( x, T e (t) ) ] 2 ω 2 x 2 γ ee(j) 2 2iω γ ee(j) ( 2 ω 2 x 2 γ ee(j) 2 ) 2 +4 2 ω 2 γ ee(j) 2 with parameters, F (j) (x, T e (t))= ( 1+exp( x E f(j) k B T e (t) ) ) 1 , E f(j) = E fd(j) ( 1 π 2 12 ( k B T e (t) E fd(j) ) 2 ), and γ ee(j) [ T e (t)]=( γ a(j) T e 2 (t)+ γ b(j) ω 2 )
ΔT= T n eff Δ n eff + T k eff Δ k eff ΔR= R n eff Δ n eff + R k eff Δ k eff

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