Abstract

We derive analytical formulas for the modulation of the reflectance and transmittance of light normally incident on a multilayer thin-film structure whose refractive indices are perturbed by an ultrashort optical pulse. The formulas, expressed in compact form, should prove useful for analysis of a wide range of ultrashort time-scale experiments on multilayers as well as longer time-scale photoacoustic and photothermal experiments based on optical probing. We demonstrate our method by the analysis of the modulated reflectance variation of a SiO2/Cr structure in which picosecond acoustic pulses have been optically excited.

© 2002 Optical Society of America

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  6. P. Basséras, S. M. Gracewski, G. W. Wicks, and R. J. D. Miller, “Optical generation of high-frequency acoustic waves in GaAs/AlxGa1−xAs periodic multilayer structures,” J. Appl. Phys. 75, 2761–2767 (1994).
    [CrossRef]
  7. A. Bartels, T. Dekorsy, and H. Kurz, “Coherent zone-folded longitudinal acoustic phonons in semiconductor superlattices: excitation and detection,” Phys. Rev. Lett. 82, 1044–1047 (1999).
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  8. K. Mizoguchi, M. Hase, S. Nakashima, and M. Nakayama, “Observation of coherent folded acoustic phonons propagating in a GaAs/AlAs superlattice by two-color pump-probe spectroscopy,” Phys. Rev. B 60, 8262–8266 (1999).
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    [CrossRef]
  32. H. Hu, X. Wang, and X. Xu, “Generalized theory of the photoacoustic effect in a multilayer material,” J. Appl. Phys. 86, 3953–3958 (1999).
    [CrossRef]
  33. J. A. Batista, A. M. Mansanares, E. C. da Silva, C. C. Vaz, and L. C. M. Miranda, “Contrast enhancement in the detection of defects in transparent layered structures: the use of optothermal interference technique in solar cell investigation,” J. Appl. Phys. 88, 5079–5086 (2000).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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2002 (1)

T. Saito, O. Matsuda, and O. B. Wright, “Ultrafast acoustic phonon pulse generation in chromium,” Physica B 316–317, 304–307 (2002).
[CrossRef]

2000 (4)

R. L. Hartman, “Green dyadic calculation for inhomogeneous optical media,” J. Opt. Soc. Am. A 17, 1067–1076 (2000).
[CrossRef]

R. Ziebold, T. Witte, M. Hübner, and R. G. Ulbrich, “Direct observation of Fermi-pressure-driven electron-hole plasma expansion in GaAs on a picosecond time scale,” Phys. Rev. B 61, 16610–16618 (2000).
[CrossRef]

J. A. Batista, A. M. Mansanares, E. C. da Silva, C. C. Vaz, and L. C. M. Miranda, “Contrast enhancement in the detection of defects in transparent layered structures: the use of optothermal interference technique in solar cell investigation,” J. Appl. Phys. 88, 5079–5086 (2000).
[CrossRef]

N. D. Fatti, C. Voisin, M. Achermann, S. Tzortzakis, D. Christofilos, and F. Vallée, “Nonequilibrium electron dynamics in noble metals,” Phys. Rev. B 61, 16956–16966 (2000).
[CrossRef]

1999 (6)

A. Bartels, T. Dekorsy, and H. Kurz, “Coherent zone-folded longitudinal acoustic phonons in semiconductor superlattices: excitation and detection,” Phys. Rev. Lett. 82, 1044–1047 (1999).
[CrossRef]

K. Mizoguchi, M. Hase, S. Nakashima, and M. Nakayama, “Observation of coherent folded acoustic phonons propagating in a GaAs/AlAs superlattice by two-color pump-probe spectroscopy,” Phys. Rev. B 60, 8262–8266 (1999).
[CrossRef]

H. Hu, X. Wang, and X. Xu, “Generalized theory of the photoacoustic effect in a multilayer material,” J. Appl. Phys. 86, 3953–3958 (1999).
[CrossRef]

A. N. Smith, J. L. Hostetler, and P. M. Norris, “Nonequilibrium heating in metal films: an analytical and numerical analysis,” Numer. Heat Transfer, Part A 35, 859–873 (1999).
[CrossRef]

D. H. Hurley and O. B. Wright, “Detection of ultrafast phenomena by use of a modified Sagnac interferometer,” Opt. Lett. 24, 1305–1307 (1999).
[CrossRef]

G. Caviglia and A. Morro, “Reflection and transmission of electromagnetic waves in planarly stratified media,” Nuovo Cimento B 114, 885–901 (1999).

1996 (3)

T. Elperin and G. Rudin, “Thermoelasticity problem for a multilayer coating-substrate assembly irradiated by a laser beam,” Int. Commun. Heat Mass Transfer 23, 133–142 (1996).
[CrossRef]

B. Perrin, B. Bonello, J. C. Jeannet, and E. Romatet, “Interferometric detection of hypersound waves in modulated structures,” Prog. Nat. Sci. Suppl. 6, S444–S448 (1996).

V. E. Gusev, “Laser hypersonics in fundamental and applied research,” Acustica Suppl. 82, S37–S45 (1996).

1995 (2)

O. B. Wright and V. E. Gusev, “Ultrafast generation of acoustic waves in copper,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 42, 331–338 (1995).
[CrossRef]

O. B. Wright, “Laser picosecond acoustics in double-layer transparent films,” Opt. Lett. 20, 632–634 (1995).
[CrossRef] [PubMed]

1994 (3)

O. B. Wright, “Ultrafast nonequilibrium stress generation in gold and silver,” Phys. Rev. B 49, 9985–9988 (1994).
[CrossRef]

W. Chen, Y. Lu, H. J. Maris, and G. Xiao, “Picosecond ultrasonic study of localized phonon surface modes in Al/Ag superlattices,” Phys. Rev. B 50, 14506–14515 (1994).
[CrossRef]

P. Basséras, S. M. Gracewski, G. W. Wicks, and R. J. D. Miller, “Optical generation of high-frequency acoustic waves in GaAs/AlxGa1−xAs periodic multilayer structures,” J. Appl. Phys. 75, 2761–2767 (1994).
[CrossRef]

1993 (4)

J. A. Moon and J. Tauc, “Interference effects in pump-probe spectroscopy of thin films,” J. Appl. Phys. 73, 4571–4578 (1993).
[CrossRef]

H. E. Elsayed-Ali and T. Juhasz, “Femtosecond time-resolved thermomodulation of thin gold films with different crystal structures,” Phys. Rev. B 47, 13599–13610 (1993).
[CrossRef]

G. Chen and C. L. Tien, “Internal reflection effects on transient photothermal reflectance,” J. Appl. Phys. 73, 3461–3466 (1993).
[CrossRef]

T. Q. Qiu and C. L. Tien, “Heat transfer mechanisms during short-pulse laser heating of metals,” J. Heat Transfer 115, 835–841 (1993).
[CrossRef]

1992 (2)

O. B. Wright, “Thickness and sound velocity measurement in thin transparent films with laser picosecond acoustics,” J. Appl. Phys. 71, 1617–1629 (1992).
[CrossRef]

O. B. Wright and K. Kawashima, “Coherent phonon detection from ultrafast surface vibrations,” Phys. Rev. Lett. 69, 1668–1671 (1992).
[CrossRef] [PubMed]

1990 (1)

S. D. Brorson, A. Kazeroonian, J. S. Moodera, D. W. Face, T. K. Cheng, E. P. Ippen, M. S. Dresselhaus, and G. Dresselhaus, “Femtosecond room-temperature measurement of the electron-phonon coupling constant λ in metallic superconductors,” Phys. Rev. Lett. 64, 2172–2175 (1990).
[CrossRef] [PubMed]

1988 (1)

A. Miklós and A. Lörincz, “Transient thermoreflectance of thin metal films in the picosecond regime,” J. Appl. Phys. 63, 2391–2395 (1988).
[CrossRef]

1986 (3)

C. Thomsen, H. T. Grahn, H. J. Maris, and J. Tauc, “Surface generation and detection of phonons by picosecond light pulses,” Phys. Rev. B 34, 4129–4138 (1986).
[CrossRef]

H. G. Walther, E. Welsch, and J. Opfermann, “Calculation and measurement of the absorption in multilayer films by means of photoacoustics,” Thin Solid Films 142, 27–35 (1986).
[CrossRef]

C. A. Paddock and G. L. Eesley, “Transient thermoreflectance from metal films,” Opt. Lett. 11, 273–275 (1986).
[CrossRef] [PubMed]

1969 (1)

D. E. Aspnes and A. Frova, “Influence of spatially dependent perturbations on modulated reflectance and absorption of solids,” Solid State Commun. 7, 155–159 (1969).
[CrossRef]

1967 (2)

V. K. Subashiev and A. A. Kukharskii, “The reflection coefficient of optically inhomogeneous solids,” Phys. Status Solidi 23, 447–452 (1967).
[CrossRef]

R. W. Dixon, “Photoelastic properties of selected materials and their relevance for applications to acoustic light modulators and scanners,” J. Appl. Phys. 38, 5149–5153 (1967).
[CrossRef]

Achermann, M.

N. D. Fatti, C. Voisin, M. Achermann, S. Tzortzakis, D. Christofilos, and F. Vallée, “Nonequilibrium electron dynamics in noble metals,” Phys. Rev. B 61, 16956–16966 (2000).
[CrossRef]

Aspnes, D. E.

D. E. Aspnes and A. Frova, “Influence of spatially dependent perturbations on modulated reflectance and absorption of solids,” Solid State Commun. 7, 155–159 (1969).
[CrossRef]

Bartels, A.

A. Bartels, T. Dekorsy, and H. Kurz, “Coherent zone-folded longitudinal acoustic phonons in semiconductor superlattices: excitation and detection,” Phys. Rev. Lett. 82, 1044–1047 (1999).
[CrossRef]

Basséras, P.

P. Basséras, S. M. Gracewski, G. W. Wicks, and R. J. D. Miller, “Optical generation of high-frequency acoustic waves in GaAs/AlxGa1−xAs periodic multilayer structures,” J. Appl. Phys. 75, 2761–2767 (1994).
[CrossRef]

Batista, J. A.

J. A. Batista, A. M. Mansanares, E. C. da Silva, C. C. Vaz, and L. C. M. Miranda, “Contrast enhancement in the detection of defects in transparent layered structures: the use of optothermal interference technique in solar cell investigation,” J. Appl. Phys. 88, 5079–5086 (2000).
[CrossRef]

Bonello, B.

B. Perrin, B. Bonello, J. C. Jeannet, and E. Romatet, “Interferometric detection of hypersound waves in modulated structures,” Prog. Nat. Sci. Suppl. 6, S444–S448 (1996).

Brorson, S. D.

S. D. Brorson, A. Kazeroonian, J. S. Moodera, D. W. Face, T. K. Cheng, E. P. Ippen, M. S. Dresselhaus, and G. Dresselhaus, “Femtosecond room-temperature measurement of the electron-phonon coupling constant λ in metallic superconductors,” Phys. Rev. Lett. 64, 2172–2175 (1990).
[CrossRef] [PubMed]

Caviglia, G.

G. Caviglia and A. Morro, “Reflection and transmission of electromagnetic waves in planarly stratified media,” Nuovo Cimento B 114, 885–901 (1999).

Chen, G.

G. Chen and C. L. Tien, “Internal reflection effects on transient photothermal reflectance,” J. Appl. Phys. 73, 3461–3466 (1993).
[CrossRef]

Chen, W.

W. Chen, Y. Lu, H. J. Maris, and G. Xiao, “Picosecond ultrasonic study of localized phonon surface modes in Al/Ag superlattices,” Phys. Rev. B 50, 14506–14515 (1994).
[CrossRef]

Cheng, T. K.

S. D. Brorson, A. Kazeroonian, J. S. Moodera, D. W. Face, T. K. Cheng, E. P. Ippen, M. S. Dresselhaus, and G. Dresselhaus, “Femtosecond room-temperature measurement of the electron-phonon coupling constant λ in metallic superconductors,” Phys. Rev. Lett. 64, 2172–2175 (1990).
[CrossRef] [PubMed]

Christofilos, D.

N. D. Fatti, C. Voisin, M. Achermann, S. Tzortzakis, D. Christofilos, and F. Vallée, “Nonequilibrium electron dynamics in noble metals,” Phys. Rev. B 61, 16956–16966 (2000).
[CrossRef]

da Silva, E. C.

J. A. Batista, A. M. Mansanares, E. C. da Silva, C. C. Vaz, and L. C. M. Miranda, “Contrast enhancement in the detection of defects in transparent layered structures: the use of optothermal interference technique in solar cell investigation,” J. Appl. Phys. 88, 5079–5086 (2000).
[CrossRef]

Dekorsy, T.

A. Bartels, T. Dekorsy, and H. Kurz, “Coherent zone-folded longitudinal acoustic phonons in semiconductor superlattices: excitation and detection,” Phys. Rev. Lett. 82, 1044–1047 (1999).
[CrossRef]

Dixon, R. W.

R. W. Dixon, “Photoelastic properties of selected materials and their relevance for applications to acoustic light modulators and scanners,” J. Appl. Phys. 38, 5149–5153 (1967).
[CrossRef]

Dresselhaus, G.

S. D. Brorson, A. Kazeroonian, J. S. Moodera, D. W. Face, T. K. Cheng, E. P. Ippen, M. S. Dresselhaus, and G. Dresselhaus, “Femtosecond room-temperature measurement of the electron-phonon coupling constant λ in metallic superconductors,” Phys. Rev. Lett. 64, 2172–2175 (1990).
[CrossRef] [PubMed]

Dresselhaus, M. S.

S. D. Brorson, A. Kazeroonian, J. S. Moodera, D. W. Face, T. K. Cheng, E. P. Ippen, M. S. Dresselhaus, and G. Dresselhaus, “Femtosecond room-temperature measurement of the electron-phonon coupling constant λ in metallic superconductors,” Phys. Rev. Lett. 64, 2172–2175 (1990).
[CrossRef] [PubMed]

Eesley, G. L.

Elperin, T.

T. Elperin and G. Rudin, “Thermoelasticity problem for a multilayer coating-substrate assembly irradiated by a laser beam,” Int. Commun. Heat Mass Transfer 23, 133–142 (1996).
[CrossRef]

Elsayed-Ali, H. E.

H. E. Elsayed-Ali and T. Juhasz, “Femtosecond time-resolved thermomodulation of thin gold films with different crystal structures,” Phys. Rev. B 47, 13599–13610 (1993).
[CrossRef]

Face, D. W.

S. D. Brorson, A. Kazeroonian, J. S. Moodera, D. W. Face, T. K. Cheng, E. P. Ippen, M. S. Dresselhaus, and G. Dresselhaus, “Femtosecond room-temperature measurement of the electron-phonon coupling constant λ in metallic superconductors,” Phys. Rev. Lett. 64, 2172–2175 (1990).
[CrossRef] [PubMed]

Fatti, N. D.

N. D. Fatti, C. Voisin, M. Achermann, S. Tzortzakis, D. Christofilos, and F. Vallée, “Nonequilibrium electron dynamics in noble metals,” Phys. Rev. B 61, 16956–16966 (2000).
[CrossRef]

Frova, A.

D. E. Aspnes and A. Frova, “Influence of spatially dependent perturbations on modulated reflectance and absorption of solids,” Solid State Commun. 7, 155–159 (1969).
[CrossRef]

Gracewski, S. M.

P. Basséras, S. M. Gracewski, G. W. Wicks, and R. J. D. Miller, “Optical generation of high-frequency acoustic waves in GaAs/AlxGa1−xAs periodic multilayer structures,” J. Appl. Phys. 75, 2761–2767 (1994).
[CrossRef]

Grahn, H. T.

C. Thomsen, H. T. Grahn, H. J. Maris, and J. Tauc, “Surface generation and detection of phonons by picosecond light pulses,” Phys. Rev. B 34, 4129–4138 (1986).
[CrossRef]

Gusev, V. E.

V. E. Gusev, “Laser hypersonics in fundamental and applied research,” Acustica Suppl. 82, S37–S45 (1996).

O. B. Wright and V. E. Gusev, “Ultrafast generation of acoustic waves in copper,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 42, 331–338 (1995).
[CrossRef]

Hartman, R. L.

Hase, M.

K. Mizoguchi, M. Hase, S. Nakashima, and M. Nakayama, “Observation of coherent folded acoustic phonons propagating in a GaAs/AlAs superlattice by two-color pump-probe spectroscopy,” Phys. Rev. B 60, 8262–8266 (1999).
[CrossRef]

Hostetler, J. L.

A. N. Smith, J. L. Hostetler, and P. M. Norris, “Nonequilibrium heating in metal films: an analytical and numerical analysis,” Numer. Heat Transfer, Part A 35, 859–873 (1999).
[CrossRef]

Hu, H.

H. Hu, X. Wang, and X. Xu, “Generalized theory of the photoacoustic effect in a multilayer material,” J. Appl. Phys. 86, 3953–3958 (1999).
[CrossRef]

Hübner, M.

R. Ziebold, T. Witte, M. Hübner, and R. G. Ulbrich, “Direct observation of Fermi-pressure-driven electron-hole plasma expansion in GaAs on a picosecond time scale,” Phys. Rev. B 61, 16610–16618 (2000).
[CrossRef]

Hurley, D. H.

Ippen, E. P.

S. D. Brorson, A. Kazeroonian, J. S. Moodera, D. W. Face, T. K. Cheng, E. P. Ippen, M. S. Dresselhaus, and G. Dresselhaus, “Femtosecond room-temperature measurement of the electron-phonon coupling constant λ in metallic superconductors,” Phys. Rev. Lett. 64, 2172–2175 (1990).
[CrossRef] [PubMed]

Jeannet, J. C.

B. Perrin, B. Bonello, J. C. Jeannet, and E. Romatet, “Interferometric detection of hypersound waves in modulated structures,” Prog. Nat. Sci. Suppl. 6, S444–S448 (1996).

Juhasz, T.

H. E. Elsayed-Ali and T. Juhasz, “Femtosecond time-resolved thermomodulation of thin gold films with different crystal structures,” Phys. Rev. B 47, 13599–13610 (1993).
[CrossRef]

Kawashima, K.

O. B. Wright and K. Kawashima, “Coherent phonon detection from ultrafast surface vibrations,” Phys. Rev. Lett. 69, 1668–1671 (1992).
[CrossRef] [PubMed]

Kazeroonian, A.

S. D. Brorson, A. Kazeroonian, J. S. Moodera, D. W. Face, T. K. Cheng, E. P. Ippen, M. S. Dresselhaus, and G. Dresselhaus, “Femtosecond room-temperature measurement of the electron-phonon coupling constant λ in metallic superconductors,” Phys. Rev. Lett. 64, 2172–2175 (1990).
[CrossRef] [PubMed]

Kukharskii, A. A.

V. K. Subashiev and A. A. Kukharskii, “The reflection coefficient of optically inhomogeneous solids,” Phys. Status Solidi 23, 447–452 (1967).
[CrossRef]

Kurz, H.

A. Bartels, T. Dekorsy, and H. Kurz, “Coherent zone-folded longitudinal acoustic phonons in semiconductor superlattices: excitation and detection,” Phys. Rev. Lett. 82, 1044–1047 (1999).
[CrossRef]

Lörincz, A.

A. Miklós and A. Lörincz, “Transient thermoreflectance of thin metal films in the picosecond regime,” J. Appl. Phys. 63, 2391–2395 (1988).
[CrossRef]

Lu, Y.

W. Chen, Y. Lu, H. J. Maris, and G. Xiao, “Picosecond ultrasonic study of localized phonon surface modes in Al/Ag superlattices,” Phys. Rev. B 50, 14506–14515 (1994).
[CrossRef]

Mansanares, A. M.

J. A. Batista, A. M. Mansanares, E. C. da Silva, C. C. Vaz, and L. C. M. Miranda, “Contrast enhancement in the detection of defects in transparent layered structures: the use of optothermal interference technique in solar cell investigation,” J. Appl. Phys. 88, 5079–5086 (2000).
[CrossRef]

Maris, H. J.

W. Chen, Y. Lu, H. J. Maris, and G. Xiao, “Picosecond ultrasonic study of localized phonon surface modes in Al/Ag superlattices,” Phys. Rev. B 50, 14506–14515 (1994).
[CrossRef]

C. Thomsen, H. T. Grahn, H. J. Maris, and J. Tauc, “Surface generation and detection of phonons by picosecond light pulses,” Phys. Rev. B 34, 4129–4138 (1986).
[CrossRef]

Matsuda, O.

T. Saito, O. Matsuda, and O. B. Wright, “Ultrafast acoustic phonon pulse generation in chromium,” Physica B 316–317, 304–307 (2002).
[CrossRef]

Miklós, A.

A. Miklós and A. Lörincz, “Transient thermoreflectance of thin metal films in the picosecond regime,” J. Appl. Phys. 63, 2391–2395 (1988).
[CrossRef]

Miller, R. J. D.

P. Basséras, S. M. Gracewski, G. W. Wicks, and R. J. D. Miller, “Optical generation of high-frequency acoustic waves in GaAs/AlxGa1−xAs periodic multilayer structures,” J. Appl. Phys. 75, 2761–2767 (1994).
[CrossRef]

Miranda, L. C. M.

J. A. Batista, A. M. Mansanares, E. C. da Silva, C. C. Vaz, and L. C. M. Miranda, “Contrast enhancement in the detection of defects in transparent layered structures: the use of optothermal interference technique in solar cell investigation,” J. Appl. Phys. 88, 5079–5086 (2000).
[CrossRef]

Mizoguchi, K.

K. Mizoguchi, M. Hase, S. Nakashima, and M. Nakayama, “Observation of coherent folded acoustic phonons propagating in a GaAs/AlAs superlattice by two-color pump-probe spectroscopy,” Phys. Rev. B 60, 8262–8266 (1999).
[CrossRef]

Moodera, J. S.

S. D. Brorson, A. Kazeroonian, J. S. Moodera, D. W. Face, T. K. Cheng, E. P. Ippen, M. S. Dresselhaus, and G. Dresselhaus, “Femtosecond room-temperature measurement of the electron-phonon coupling constant λ in metallic superconductors,” Phys. Rev. Lett. 64, 2172–2175 (1990).
[CrossRef] [PubMed]

Moon, J. A.

J. A. Moon and J. Tauc, “Interference effects in pump-probe spectroscopy of thin films,” J. Appl. Phys. 73, 4571–4578 (1993).
[CrossRef]

Morro, A.

G. Caviglia and A. Morro, “Reflection and transmission of electromagnetic waves in planarly stratified media,” Nuovo Cimento B 114, 885–901 (1999).

Nakashima, S.

K. Mizoguchi, M. Hase, S. Nakashima, and M. Nakayama, “Observation of coherent folded acoustic phonons propagating in a GaAs/AlAs superlattice by two-color pump-probe spectroscopy,” Phys. Rev. B 60, 8262–8266 (1999).
[CrossRef]

Nakayama, M.

K. Mizoguchi, M. Hase, S. Nakashima, and M. Nakayama, “Observation of coherent folded acoustic phonons propagating in a GaAs/AlAs superlattice by two-color pump-probe spectroscopy,” Phys. Rev. B 60, 8262–8266 (1999).
[CrossRef]

Norris, P. M.

A. N. Smith, J. L. Hostetler, and P. M. Norris, “Nonequilibrium heating in metal films: an analytical and numerical analysis,” Numer. Heat Transfer, Part A 35, 859–873 (1999).
[CrossRef]

Opfermann, J.

H. G. Walther, E. Welsch, and J. Opfermann, “Calculation and measurement of the absorption in multilayer films by means of photoacoustics,” Thin Solid Films 142, 27–35 (1986).
[CrossRef]

Paddock, C. A.

Perrin, B.

B. Perrin, B. Bonello, J. C. Jeannet, and E. Romatet, “Interferometric detection of hypersound waves in modulated structures,” Prog. Nat. Sci. Suppl. 6, S444–S448 (1996).

Qiu, T. Q.

T. Q. Qiu and C. L. Tien, “Heat transfer mechanisms during short-pulse laser heating of metals,” J. Heat Transfer 115, 835–841 (1993).
[CrossRef]

Romatet, E.

B. Perrin, B. Bonello, J. C. Jeannet, and E. Romatet, “Interferometric detection of hypersound waves in modulated structures,” Prog. Nat. Sci. Suppl. 6, S444–S448 (1996).

Rudin, G.

T. Elperin and G. Rudin, “Thermoelasticity problem for a multilayer coating-substrate assembly irradiated by a laser beam,” Int. Commun. Heat Mass Transfer 23, 133–142 (1996).
[CrossRef]

Saito, T.

T. Saito, O. Matsuda, and O. B. Wright, “Ultrafast acoustic phonon pulse generation in chromium,” Physica B 316–317, 304–307 (2002).
[CrossRef]

Smith, A. N.

A. N. Smith, J. L. Hostetler, and P. M. Norris, “Nonequilibrium heating in metal films: an analytical and numerical analysis,” Numer. Heat Transfer, Part A 35, 859–873 (1999).
[CrossRef]

Subashiev, V. K.

V. K. Subashiev and A. A. Kukharskii, “The reflection coefficient of optically inhomogeneous solids,” Phys. Status Solidi 23, 447–452 (1967).
[CrossRef]

Tauc, J.

J. A. Moon and J. Tauc, “Interference effects in pump-probe spectroscopy of thin films,” J. Appl. Phys. 73, 4571–4578 (1993).
[CrossRef]

C. Thomsen, H. T. Grahn, H. J. Maris, and J. Tauc, “Surface generation and detection of phonons by picosecond light pulses,” Phys. Rev. B 34, 4129–4138 (1986).
[CrossRef]

Thomsen, C.

C. Thomsen, H. T. Grahn, H. J. Maris, and J. Tauc, “Surface generation and detection of phonons by picosecond light pulses,” Phys. Rev. B 34, 4129–4138 (1986).
[CrossRef]

Tien, C. L.

T. Q. Qiu and C. L. Tien, “Heat transfer mechanisms during short-pulse laser heating of metals,” J. Heat Transfer 115, 835–841 (1993).
[CrossRef]

G. Chen and C. L. Tien, “Internal reflection effects on transient photothermal reflectance,” J. Appl. Phys. 73, 3461–3466 (1993).
[CrossRef]

Tzortzakis, S.

N. D. Fatti, C. Voisin, M. Achermann, S. Tzortzakis, D. Christofilos, and F. Vallée, “Nonequilibrium electron dynamics in noble metals,” Phys. Rev. B 61, 16956–16966 (2000).
[CrossRef]

Ulbrich, R. G.

R. Ziebold, T. Witte, M. Hübner, and R. G. Ulbrich, “Direct observation of Fermi-pressure-driven electron-hole plasma expansion in GaAs on a picosecond time scale,” Phys. Rev. B 61, 16610–16618 (2000).
[CrossRef]

Vallée, F.

N. D. Fatti, C. Voisin, M. Achermann, S. Tzortzakis, D. Christofilos, and F. Vallée, “Nonequilibrium electron dynamics in noble metals,” Phys. Rev. B 61, 16956–16966 (2000).
[CrossRef]

Vaz, C. C.

J. A. Batista, A. M. Mansanares, E. C. da Silva, C. C. Vaz, and L. C. M. Miranda, “Contrast enhancement in the detection of defects in transparent layered structures: the use of optothermal interference technique in solar cell investigation,” J. Appl. Phys. 88, 5079–5086 (2000).
[CrossRef]

Voisin, C.

N. D. Fatti, C. Voisin, M. Achermann, S. Tzortzakis, D. Christofilos, and F. Vallée, “Nonequilibrium electron dynamics in noble metals,” Phys. Rev. B 61, 16956–16966 (2000).
[CrossRef]

Walther, H. G.

H. G. Walther, E. Welsch, and J. Opfermann, “Calculation and measurement of the absorption in multilayer films by means of photoacoustics,” Thin Solid Films 142, 27–35 (1986).
[CrossRef]

Wang, X.

H. Hu, X. Wang, and X. Xu, “Generalized theory of the photoacoustic effect in a multilayer material,” J. Appl. Phys. 86, 3953–3958 (1999).
[CrossRef]

Welsch, E.

H. G. Walther, E. Welsch, and J. Opfermann, “Calculation and measurement of the absorption in multilayer films by means of photoacoustics,” Thin Solid Films 142, 27–35 (1986).
[CrossRef]

Wicks, G. W.

P. Basséras, S. M. Gracewski, G. W. Wicks, and R. J. D. Miller, “Optical generation of high-frequency acoustic waves in GaAs/AlxGa1−xAs periodic multilayer structures,” J. Appl. Phys. 75, 2761–2767 (1994).
[CrossRef]

Witte, T.

R. Ziebold, T. Witte, M. Hübner, and R. G. Ulbrich, “Direct observation of Fermi-pressure-driven electron-hole plasma expansion in GaAs on a picosecond time scale,” Phys. Rev. B 61, 16610–16618 (2000).
[CrossRef]

Wright, O. B.

T. Saito, O. Matsuda, and O. B. Wright, “Ultrafast acoustic phonon pulse generation in chromium,” Physica B 316–317, 304–307 (2002).
[CrossRef]

D. H. Hurley and O. B. Wright, “Detection of ultrafast phenomena by use of a modified Sagnac interferometer,” Opt. Lett. 24, 1305–1307 (1999).
[CrossRef]

O. B. Wright, “Laser picosecond acoustics in double-layer transparent films,” Opt. Lett. 20, 632–634 (1995).
[CrossRef] [PubMed]

O. B. Wright and V. E. Gusev, “Ultrafast generation of acoustic waves in copper,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 42, 331–338 (1995).
[CrossRef]

O. B. Wright, “Ultrafast nonequilibrium stress generation in gold and silver,” Phys. Rev. B 49, 9985–9988 (1994).
[CrossRef]

O. B. Wright and K. Kawashima, “Coherent phonon detection from ultrafast surface vibrations,” Phys. Rev. Lett. 69, 1668–1671 (1992).
[CrossRef] [PubMed]

O. B. Wright, “Thickness and sound velocity measurement in thin transparent films with laser picosecond acoustics,” J. Appl. Phys. 71, 1617–1629 (1992).
[CrossRef]

Xiao, G.

W. Chen, Y. Lu, H. J. Maris, and G. Xiao, “Picosecond ultrasonic study of localized phonon surface modes in Al/Ag superlattices,” Phys. Rev. B 50, 14506–14515 (1994).
[CrossRef]

Xu, X.

H. Hu, X. Wang, and X. Xu, “Generalized theory of the photoacoustic effect in a multilayer material,” J. Appl. Phys. 86, 3953–3958 (1999).
[CrossRef]

Ziebold, R.

R. Ziebold, T. Witte, M. Hübner, and R. G. Ulbrich, “Direct observation of Fermi-pressure-driven electron-hole plasma expansion in GaAs on a picosecond time scale,” Phys. Rev. B 61, 16610–16618 (2000).
[CrossRef]

Acustica Suppl. (1)

V. E. Gusev, “Laser hypersonics in fundamental and applied research,” Acustica Suppl. 82, S37–S45 (1996).

IEEE Trans. Ultrason. Ferroelectr. Freq. Control (1)

O. B. Wright and V. E. Gusev, “Ultrafast generation of acoustic waves in copper,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 42, 331–338 (1995).
[CrossRef]

Int. Commun. Heat Mass Transfer (1)

T. Elperin and G. Rudin, “Thermoelasticity problem for a multilayer coating-substrate assembly irradiated by a laser beam,” Int. Commun. Heat Mass Transfer 23, 133–142 (1996).
[CrossRef]

J. Appl. Phys. (8)

H. Hu, X. Wang, and X. Xu, “Generalized theory of the photoacoustic effect in a multilayer material,” J. Appl. Phys. 86, 3953–3958 (1999).
[CrossRef]

J. A. Batista, A. M. Mansanares, E. C. da Silva, C. C. Vaz, and L. C. M. Miranda, “Contrast enhancement in the detection of defects in transparent layered structures: the use of optothermal interference technique in solar cell investigation,” J. Appl. Phys. 88, 5079–5086 (2000).
[CrossRef]

R. W. Dixon, “Photoelastic properties of selected materials and their relevance for applications to acoustic light modulators and scanners,” J. Appl. Phys. 38, 5149–5153 (1967).
[CrossRef]

A. Miklós and A. Lörincz, “Transient thermoreflectance of thin metal films in the picosecond regime,” J. Appl. Phys. 63, 2391–2395 (1988).
[CrossRef]

P. Basséras, S. M. Gracewski, G. W. Wicks, and R. J. D. Miller, “Optical generation of high-frequency acoustic waves in GaAs/AlxGa1−xAs periodic multilayer structures,” J. Appl. Phys. 75, 2761–2767 (1994).
[CrossRef]

J. A. Moon and J. Tauc, “Interference effects in pump-probe spectroscopy of thin films,” J. Appl. Phys. 73, 4571–4578 (1993).
[CrossRef]

O. B. Wright, “Thickness and sound velocity measurement in thin transparent films with laser picosecond acoustics,” J. Appl. Phys. 71, 1617–1629 (1992).
[CrossRef]

G. Chen and C. L. Tien, “Internal reflection effects on transient photothermal reflectance,” J. Appl. Phys. 73, 3461–3466 (1993).
[CrossRef]

J. Heat Transfer (1)

T. Q. Qiu and C. L. Tien, “Heat transfer mechanisms during short-pulse laser heating of metals,” J. Heat Transfer 115, 835–841 (1993).
[CrossRef]

J. Opt. Soc. Am. A (1)

Numer. Heat Transfer, Part A (1)

A. N. Smith, J. L. Hostetler, and P. M. Norris, “Nonequilibrium heating in metal films: an analytical and numerical analysis,” Numer. Heat Transfer, Part A 35, 859–873 (1999).
[CrossRef]

Nuovo Cimento B (1)

G. Caviglia and A. Morro, “Reflection and transmission of electromagnetic waves in planarly stratified media,” Nuovo Cimento B 114, 885–901 (1999).

Opt. Lett. (3)

Phys. Rev. B (7)

R. Ziebold, T. Witte, M. Hübner, and R. G. Ulbrich, “Direct observation of Fermi-pressure-driven electron-hole plasma expansion in GaAs on a picosecond time scale,” Phys. Rev. B 61, 16610–16618 (2000).
[CrossRef]

W. Chen, Y. Lu, H. J. Maris, and G. Xiao, “Picosecond ultrasonic study of localized phonon surface modes in Al/Ag superlattices,” Phys. Rev. B 50, 14506–14515 (1994).
[CrossRef]

C. Thomsen, H. T. Grahn, H. J. Maris, and J. Tauc, “Surface generation and detection of phonons by picosecond light pulses,” Phys. Rev. B 34, 4129–4138 (1986).
[CrossRef]

K. Mizoguchi, M. Hase, S. Nakashima, and M. Nakayama, “Observation of coherent folded acoustic phonons propagating in a GaAs/AlAs superlattice by two-color pump-probe spectroscopy,” Phys. Rev. B 60, 8262–8266 (1999).
[CrossRef]

H. E. Elsayed-Ali and T. Juhasz, “Femtosecond time-resolved thermomodulation of thin gold films with different crystal structures,” Phys. Rev. B 47, 13599–13610 (1993).
[CrossRef]

N. D. Fatti, C. Voisin, M. Achermann, S. Tzortzakis, D. Christofilos, and F. Vallée, “Nonequilibrium electron dynamics in noble metals,” Phys. Rev. B 61, 16956–16966 (2000).
[CrossRef]

O. B. Wright, “Ultrafast nonequilibrium stress generation in gold and silver,” Phys. Rev. B 49, 9985–9988 (1994).
[CrossRef]

Phys. Rev. Lett. (3)

A. Bartels, T. Dekorsy, and H. Kurz, “Coherent zone-folded longitudinal acoustic phonons in semiconductor superlattices: excitation and detection,” Phys. Rev. Lett. 82, 1044–1047 (1999).
[CrossRef]

O. B. Wright and K. Kawashima, “Coherent phonon detection from ultrafast surface vibrations,” Phys. Rev. Lett. 69, 1668–1671 (1992).
[CrossRef] [PubMed]

S. D. Brorson, A. Kazeroonian, J. S. Moodera, D. W. Face, T. K. Cheng, E. P. Ippen, M. S. Dresselhaus, and G. Dresselhaus, “Femtosecond room-temperature measurement of the electron-phonon coupling constant λ in metallic superconductors,” Phys. Rev. Lett. 64, 2172–2175 (1990).
[CrossRef] [PubMed]

Phys. Status Solidi (1)

V. K. Subashiev and A. A. Kukharskii, “The reflection coefficient of optically inhomogeneous solids,” Phys. Status Solidi 23, 447–452 (1967).
[CrossRef]

Physica B (1)

T. Saito, O. Matsuda, and O. B. Wright, “Ultrafast acoustic phonon pulse generation in chromium,” Physica B 316–317, 304–307 (2002).
[CrossRef]

Prog. Nat. Sci. Suppl. (1)

B. Perrin, B. Bonello, J. C. Jeannet, and E. Romatet, “Interferometric detection of hypersound waves in modulated structures,” Prog. Nat. Sci. Suppl. 6, S444–S448 (1996).

Solid State Commun. (1)

D. E. Aspnes and A. Frova, “Influence of spatially dependent perturbations on modulated reflectance and absorption of solids,” Solid State Commun. 7, 155–159 (1969).
[CrossRef]

Thin Solid Films (1)

H. G. Walther, E. Welsch, and J. Opfermann, “Calculation and measurement of the absorption in multilayer films by means of photoacoustics,” Thin Solid Films 142, 27–35 (1986).
[CrossRef]

Other (10)

R. Jacobsson, “Light reflection from films of continuously varying refractive index,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1965), Vol. 5, Chap. 5, pp. 247–286.

J. Shah, Ultrafast Spectroscopy of Semiconductors and Semiconductor Nanostructures (Springer, Heidelberg, Germany, 1996).

O. L. Anderson, “Determination and some uses of isotropic elastic constants of polycrystalline aggregates using single-crystal data,” in Physical Acoustics, W. P. Mason, ed. (Academic, New York, 1965), Vol. 3B, Chap. 2, pp. 43–95.

J. F. Nye, Physical Properties of Crystals (Oxford U. Press, Oxford, UK, 1957).

B. A. Auld, Acoustic Fields and Waves in Solids, 2nd ed. (Krieger, Malabar, Fla., 1990).

F. Abelès, “Optics of thin films,” in Advanced Optical Techniques, A. C. S. V. Heel, ed. (Wiley, New York, 1967), Chap. 5, pp. 143–188.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, Cambridge, UK, 1999).

E. D. Palik, ed., Handbook of Optical Constants of Solids (Academic, Boston, 1998).

D. R. Lide, ed., CRC Handbook of Chemistry and Physics, 79th ed. (CRC Press, Boca Raton, Fla., 1998).

G. W. C. Kaye and T. H. Laby, Tables of Physical and Chemical Constants, 16th ed. (Longmans, London, 1995).

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Figures (2)

Fig. 1
Fig. 1

N layers on a substrate. The incident light comes from z<0. Each layer is assumed to be optically isotropic in its equilibrium state.

Fig. 2
Fig. 2

Change in reflectance ρ (lower solid curve) and phase δϕ (upper solid curve) obtained with a a-SiO2-Cr double layer on a fused-silica substrate by use of a modified Sagnac interferometer detection scheme. The fitted thermal background signals (smooth curves) are shown superimposed on the raw data. The theoretical curves are in good agreement with the experimental data after the subtraction of the thermal background. Note that ρ and δϕ are plotted on the same vertical scale (or differing by a factor of 2 where indicated).

Tables (1)

Tables Icon

Table 1 Parameters Used for the Simulation of the Photoacoustic Signalsa

Equations (107)

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

(2-graddiv)E(x, t)=μ0 2t2 D(x, t).
(2-graddiv+k2˜)E(x)=0,
2/z20002/z20000+k2˜E(z)=0.
h(z)=(n)forzn-1<z<zn.
b(z)=P12(n)η(z)forzn-1<z<zn,
if(z)
=(n)-(n+1)foru(zn)>0,zn<z<zn+u(zn)(n+1)-(n)foru(zn)<0,zn+u(zn)<z<zn,0otherwise,
u(z)=-zη(z)dz
2z2+k2h(z)E0(z)=0,
2z2+k2h(z)G(z, z)=-δ(z-z).
E(z)=E0(z)+-k2ih(z)E(z)G(z, z)dzE0(z)+-k2ih(z)E0(z)G(z, z)dz+k4ih(z)ih(z)E0(z)G(z, z)×G(z, z)dzdz+ .
an exp(iknζn)+bn exp(-iknζn),
ζnz-zn-1forn1,zforn=0,
kn(n)k(Nn+iKn)k,
G(z, z)=i2k0a0 exp(-ik0z)E0(z)
forz>z,z<0.
E(z)E0(z)+z+k2[b(z)+if (z)]E0(z)G(z, z)dz=E0(z)+z+k2b(z)E0(z)G(z, z)dz+n=0Nznzn+u(zn)k2[(n)-(n+1)]E0(z)×G(z, z)dz=a0 exp(ik0z)+b0 exp(-ik0z)+ik2 exp(-ik0z)2k0a0 z0P12(0)η(z)[a0 exp(ik0z)+b0 exp(-ik0z)]2dz+n=1N+10dnP12(n)η(z+zn-1)[an exp(iknz)+bn exp(-iknz)]2dz+n=1N+1(an+bn)2[(n-1)-(n)]u(zn-1),
δrr=ik22k0a0b0 z0P12(0)η(z)[a0 exp(ik0z)+b0 exp(-ik0z)]2dz+n=1N+10dnP12(n)η(z+zn-1)[an exp(iknz)+bn exp(-iknz)]2dz+n=1N+1(an+bn)2[(n-1)-(n)]u(zn-1),
δrr=2ik1- 0+Δ(z)exp(2ik1z)dz+2iku(0)=4ikn˜1-n˜2 0+Δn˜(z)exp(2ikn˜z)dz+2iku(0),
δrr=ik2a0b0 0dΔ(z)[a1 exp(ik1z)+b1 exp(-ik1z)]2dz+0Δ(z+d)a22 exp(2ik2z)dz+(a1+b1)2(1-1)u(0)+a22(1-2)u(d),
a0=(k+k1)(k1+k2)+(k-k1)(k1-k2)exp(2ik1d),
b0=(k-k1)(k1+k2)+(k+k1)(k1-k2)exp(2ik1d),
a1=2k(k1+k2),
b1=2k(k1-k2)exp(2ik1d),
a2=4kk1 exp(ik1d).
G(z, z)=iτ exp[ikN+1(z-zN)]2k0E0(z; a0=0, b0=1)forz<z,z>zN,
E(z)E0(z)+ikτ2 exp[ikN+1(z-zN)]×-0P12(0)η(z)[a0 exp(ik0z)+b0 exp(-ik0z)]exp(-ik0z)dz+n=1N+10dnP12(n)η(z+zn-1)[an exp(iknz)+bn exp(-iknz)][an exp(iknz)+bn exp(-iknz)]dz+n=1N+1(an+bn)(an+bn)[(n-1)-(n)]u(zn-1).
δττ=ik2a0 -0P12(0)η(z)[a0 exp(ik0z)+b0 exp(-ik0z)]exp(-ik0z)dz+n=1N+10dnP12(n)η(z+zn-1)[an exp(iknz)+bn exp(-iknz)][an exp(iknz)+bn exp(-iknz)]dz+n=1N+1(an+bn)(an+bn)[(n-1)-(n)]u(zn-1).
δττ=ik2(k+k1)0zΔ(z)k1-k2k1 exp(2ik1z)+k1+k2k1dz+(1-1)u(0).
δττ=ik22k1 0zΔ(z)k1-kk1+k exp(2ik1z)+1dz+i(k-k1)u(0).
δττ=ik2a0 0dΔ(z)[a1 exp(ik1z)+b1 exp(-ik1z)]×[a1 exp(ik1z)+b1 exp(-ik1z)]dz+0zΔ(z+d)a2[a2 exp(2ik2z)+b2]dz+(a1+b1)(a1+b1)(1-1)u(0)+a2(a2+b2)(1-2)u(d),
a1=-k-k12k1,
b1=k+k12k1,
a2=-(k+k1)(k1-k2)exp(-ik1d)+(k-k1)(k1+k2)exp(ik1d)4k1k2,
b2=(k-k1)(k1-k2)exp(ik1d)+(k+k1)(k1+k2)exp(-ik1d)4k1k2.
σzz=3 1-ν1+νBηzz-3BβΔT(z, t),
ρ 2ut2=σzzz,
u(z, t)=+zηzz(z, t)dz,
ρv2=3 1-ν1+νB.
ΔT(z, t)=ΔT(z)θ(t).
0=ρv2ηA-3BβΔT(z),
uA(z)=+zηA(z)dz,
ηzz(z, t)=ηA(z)+ηB(z, t),
u(z, t)=uA(z)+uB(z, t).
σzz=ρv2ηB,
ρ 2uBt2=σzzz,
uB(z, t)=+zηB(z, t)dz,
ηB(z, t)=f(z-vt)+g(z+vt).
ηB(z, t=0)=f(z)+g(z)=-ηA(z),
η˙B(z, t=0)=-vdf(z)dz-dg(z)dz=0.
f(z)=g(z)=-ηA(z)/2.
ΔT(z)=Q(z)/C,
Q˙(z)=-div S¯=0ωNnKn[|an|2 exp(-2Knkζn)+|bn|2 exp(2Knkζn)+anbn* exp(2iNnkζn)+an*bn exp(-2iNnkζn)],
Q˙(z)=0ωμnNnKn[|an|2 exp(-2μnKnkζn)+|bn|2 exp(2μnKnkζn)+anbn* exp(2iNnkζn)+an*bn exp(-2iNnkζn)].
ηAn(z)=2kFβnμnNnKnCn|a0|2 1+νn1-νn×[|an|2 exp(-2μnKnkζn)+|bn|2 exp(2μnKnkζn)+anbn* exp(2iNnkζn)+an*bn exp(-2iNnkζn)],
η(z, tj)=f(z-vtj)+g(z+vtj)Fj(z)+Gj(z).
Fj+1(z)=Fj(z-vΔt),
Gj+1(z)=Gj(z+vΔt),
Fj+1I(zi)=FjI(zi-vIΔt),
Gj+1I(zi)=ZII-ZIZII+ZIFjI(zi-vIΔt)+vIIvI 2ZIIZII+ZIGjII(zi+vIIΔt),
Fj+1II(zi)=ZI-ZIIZII+ZIGjII(zi+vIIΔt)+vIvII 2ZIZII+ZIFjI(zi-vIΔt),
Gj+1II(zi)=GjII(zi+vIIΔt),
Δ(z, t)=ηzzP12000P12000P11+(sΔT+γΔn)×100010001,
ΔBij=πijklσkl=pijrsηrs,pijrs=πijklcklrs,
(B+ΔB)ij(+Δ)jk=(+Δ)ij(B+ΔB)jk=δik.
Δij=-iλΔBλμμj,
ΔBij=-BiλΔλμBμj.
Δpq=-piΔBijjq=-pijqpijrsηrsPpqrsηrs.
Ppqrs=-pijqpijrs.
P11=P22=P33,
P12=P21=P23=P32=P31=P13,
P44=P55=P66=12(P11-P12),
Δxx=Δyy=P12ηzz=-2p12ηzzb.
Δn˜=P122n˜ηzz=-n˜32p12ηzz.
dn˜dη=dNdη+i dKdη=-n˜32p12,
ddη=P12=2n˜dNdη+i dKdη,
(z)=[z-u(z)]+Δ(z)=[z-u(z)]+P12η(z),
xx(z)=(0) forz<z0,(1) forz>z0.
xx(z)=(0)+P12(0)η(z)forz<z0+u(z0),(1)+P12(1)η(z)forz>z0+u(z0).
ih=b+if.
b(z)=P12(n)η(z).
if(z)=[P12(n)-P12(n+1)]η(z)+[(n)-(n+1)]foru(zn)>0,zn<z<zn+u(zn),[P12(n+1)-P12(n)]η(z)+[(n+1)-(n)]foru(zn)<0,zn+u(zn)<z<zn,0otherwise.
if(z)=(n)-(n+1)foru(zn)>0,zn<z<zn+u(zn),(n+1)-(n)foru(zn)<0,zn+u(zn)<z<zn,0otherwise.
H(z)=1iμ0ω curl E(z),
M0a0b0=M1a1b1,
MnQnanbn=Mn+1an+1bn+1,
Mn11kn-kn,
Qnexp(ikndn)00exp(-ikndn).
a0b0=MaN+1bN+1,
MM0-1j=1NMjQj-1Mj-1MN+1.
lim+0[G(z, z-)-G(z, z+)]=0.
-1=lim+0 z=z-z=z+2z2+k2h(z)G(z, z)dz=lim0 zG(z, z)z=z-z=z+.
G(z, z)
=g1(z)E0(z; a0=0, b0=1)forz<z,g2(z)E0(z; a0=1, bN+1=0)forz>z,
G(z, z)2z2+k2h(z)G(z, z)dz-G(z, z)2z2+k2h(z)G(z, z)dz
=G(z, z)-G(z, z)=0,
g1(z)=i2k0[exp(ik0z)+r exp(-ik0z)],
g2(z)=i2k0 exp(-ik0z).
G(z, z)
=i2k0a0 exp(-ik0z)E0(z)forz<z<0,i2k0a0 exp(-ik0z)E0(z)forz>z, z<0.
G(z, z)
=i2k0a0 exp(-ik0z)E0(z)forz<z<0,i2k0a0 exp(-ik0z)E0(z)forz>z, z<0
G(z, z)
=g1(z)E0(z; a0=0, b0=1)forz<z,g2(z)E0(z; aN+1=1, bN+1=0)forz>z,
g1(z)=i exp[ikN+1(z-zN)]2k0M11,
g2(z)=i{M11 exp[-ikN+1(z-zN)]-M12 exp[ikN+1(z-zN)]}2kN+1M11.
M11M22-M12M21=kN+1/k0

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