Abstract

Ultra short pulse lasers with sub-picosecond pulse duration can generate ultrasonic acoustic pulses with wavelengths in the sub-micrometer range. The reflectance modulation induced by the acoustic pulses can be detected by a probe laser. Once a method of calculating the acoustic pulse propagation and the reflectance modulation is established, it will be possible to extract information about the internal structures of a medium from measurements of the time-sequential reflectance modulation by fitting the calculations to experimental results. Thus, a multi-slicing matrix method of calculating the reflectance modulation with an arbitrary incident angle of the probe laser for s-polarization is proposed here. The method based on the Abeles transfer matrix method applicable to stratified media is mathematically validated and is shown to reproduce the analytical formula derived using the Born approximation. The method only uses matrix multiplications, which makes the calculation algorithm simple and is easy to code with matrix manipulation software. The thickness of a stratified layer of an amorphous carbon stacked on a substrate of silicon is demonstrated to be measured with the method.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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  1. C. Thomsen, H. T. Grahn, H. J. Maris, and J. Tauc, “Surface generation and detection of phonons by picoseconds light pulses,” Phys. Rev. B 34(6), 4129–4138 (1986).
    [Crossref]
  2. O. B. Wright and K. Kawashima, “Coherent phonon detection from ultrafast surface vibrations,” Phys. Rev. Lett. 69(11), 1668–1671 (1992).
    [Crossref]
  3. 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).
  4. D. H. Hurley and O. B. Wright, “Detection of ultrafast phenomena by use of a modified Sagnac interferometer,” Opt. Lett. 24(18), 1305–1307 (1999).
    [Crossref]
  5. W. Chen, Y. Lu, H. J. Maris, and G. Xiao, “Picosecond ultra-sonic study of localized phonon surface modes in Al/Ag superlattices,” Phys. Rev. B 50(19), 14506–14515 (1994).
    [Crossref]
  6. P. Basséras, S. M. Gracewski, G. W. Wicks, and R. J. D. Miller, “Optical generation of high-frequency acoustic waves in GaAs/AlxGa12xAs periodic multilayer structures,” J. Appl. Phys. 75(6), 2761–2768 (1994).
    [Crossref]
  7. A. Bartels, T. Dekors, and H. Kurz, “Coherent zone-folded longitudinal acoustic phonons in semiconductor superlattices: excitation and detection,” Phys. Rev. Lett. 82(5), 1044–1047 (1999).
    [Crossref]
  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(11), 8262–8266 (1999).
    [Crossref]
  9. H. E. Elsayed-Ali and T. Juhasz, “Femtosecond timeresolved thermomodulation of thin gold films with different crystal structures,” Phys. Rev. B 47(20), 13599–13610 (1993).
    [Crossref]
  10. A. Miklós and A. Lörincz, “Transient thermoreflectance of thin metal films in the picosecond regime,” J. Appl. Phys. 63(7), 2391–2395 (1988).
    [Crossref]
  11. O. B. Wright and V. E. Gusev, “Ultrafast generation of acoustic waves in copper,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 42(3), 331–338 (1995).
    [Crossref]
  12. O. B. Wright, “Ultrafast nonequilibrium stress generation in gold and silver,” Phys. Rev. B 49(14), 9985–9988 (1994).
    [Crossref]
  13. O. B. Wright, “Thickness and sound velocity measurement in thin transparent films with laser picosecond acoustics,” J. Appl. Phys. 71(4), 1617–1629 (1992).
    [Crossref]
  14. O. B. Wright, “Laser picoseconds acoustics in double-layer transparent films,” Opt. Lett. 20(6), 632–634 (1995).
    [Crossref]
  15. M. Kouyate, T. Pezeril, V. Gusev, and O. Matsuda, “Theory for optical detection of picosecond shear acoustic gratings”,” J. Opt. Soc. Am. B 33(12), 2634–2648 (2016).
    [Crossref]
  16. K. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propag. 14(3), 302–307 (1966).
    [Crossref]
  17. H. Ohno, “A numerical method to reconstruct internal structures with a laser ultrasonic technique,” Proc. SPIE 10726, 1072612 (2018).
    [Crossref]
  18. O. Matsuda and O. B. Wright, “Reflection and transmission of light in multilayers perturbed by picoseconds strain pulse propagation,” J. Opt. Soc. Am. B 19(12), 3028–3041 (2002).
    [Crossref]
  19. L. Nevot, “Caractérisation des surfaces par réflexion rasante de rayons X. Application à l'étude du polissage de quelques verres silicates,” Rev. Phys. Appl. 15(3), 761–779 (1980).
    [Crossref]
  20. F. Abeles, “Recherches sur la Propagation des Ondes Electromagnetiques Sinusoidales dans les Milieux Stratifies. Application aux Couches Minces,” Ann. Phys. 12(5), 596–640 (1950).
    [Crossref]
  21. F. Abeles, “Sur la Propagation des Ondes Electromagnetiques dans les Milieux Stratifies,” Ann. Phys. 12(3), 504–520 (1948).
    [Crossref]
  22. M. Born and E. Wolf, Principles of Optics (Cambridge University Press, 1999).
  23. K. Ohta and H. Ishida, “Matrix formalism for calculation of electric field intensity of light in stratified multilayered films,” Appl. Opt. 29(13), 1952–1959 (1990).
    [Crossref]
  24. R. Jacobsson, “Light reflection films of continuously varying refractive index,” Prog. Opt. 5, 247–286 (1966).
    [Crossref]
  25. Python, https://www.python.org/

2018 (1)

H. Ohno, “A numerical method to reconstruct internal structures with a laser ultrasonic technique,” Proc. SPIE 10726, 1072612 (2018).
[Crossref]

2016 (1)

2002 (1)

1999 (3)

D. H. Hurley and O. B. Wright, “Detection of ultrafast phenomena by use of a modified Sagnac interferometer,” Opt. Lett. 24(18), 1305–1307 (1999).
[Crossref]

A. Bartels, T. Dekors, and H. Kurz, “Coherent zone-folded longitudinal acoustic phonons in semiconductor superlattices: excitation and detection,” Phys. Rev. Lett. 82(5), 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(11), 8262–8266 (1999).
[Crossref]

1996 (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).

1995 (2)

O. B. Wright, “Laser picoseconds acoustics in double-layer transparent films,” Opt. Lett. 20(6), 632–634 (1995).
[Crossref]

O. B. Wright and V. E. Gusev, “Ultrafast generation of acoustic waves in copper,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 42(3), 331–338 (1995).
[Crossref]

1994 (3)

O. B. Wright, “Ultrafast nonequilibrium stress generation in gold and silver,” Phys. Rev. B 49(14), 9985–9988 (1994).
[Crossref]

W. Chen, Y. Lu, H. J. Maris, and G. Xiao, “Picosecond ultra-sonic study of localized phonon surface modes in Al/Ag superlattices,” Phys. Rev. B 50(19), 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/AlxGa12xAs periodic multilayer structures,” J. Appl. Phys. 75(6), 2761–2768 (1994).
[Crossref]

1993 (1)

H. E. Elsayed-Ali and T. Juhasz, “Femtosecond timeresolved thermomodulation of thin gold films with different crystal structures,” Phys. Rev. B 47(20), 13599–13610 (1993).
[Crossref]

1992 (2)

O. B. Wright and K. Kawashima, “Coherent phonon detection from ultrafast surface vibrations,” Phys. Rev. Lett. 69(11), 1668–1671 (1992).
[Crossref]

O. B. Wright, “Thickness and sound velocity measurement in thin transparent films with laser picosecond acoustics,” J. Appl. Phys. 71(4), 1617–1629 (1992).
[Crossref]

1990 (1)

1988 (1)

A. Miklós and A. Lörincz, “Transient thermoreflectance of thin metal films in the picosecond regime,” J. Appl. Phys. 63(7), 2391–2395 (1988).
[Crossref]

1986 (1)

C. Thomsen, H. T. Grahn, H. J. Maris, and J. Tauc, “Surface generation and detection of phonons by picoseconds light pulses,” Phys. Rev. B 34(6), 4129–4138 (1986).
[Crossref]

1980 (1)

L. Nevot, “Caractérisation des surfaces par réflexion rasante de rayons X. Application à l'étude du polissage de quelques verres silicates,” Rev. Phys. Appl. 15(3), 761–779 (1980).
[Crossref]

1966 (2)

R. Jacobsson, “Light reflection films of continuously varying refractive index,” Prog. Opt. 5, 247–286 (1966).
[Crossref]

K. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propag. 14(3), 302–307 (1966).
[Crossref]

1950 (1)

F. Abeles, “Recherches sur la Propagation des Ondes Electromagnetiques Sinusoidales dans les Milieux Stratifies. Application aux Couches Minces,” Ann. Phys. 12(5), 596–640 (1950).
[Crossref]

1948 (1)

F. Abeles, “Sur la Propagation des Ondes Electromagnetiques dans les Milieux Stratifies,” Ann. Phys. 12(3), 504–520 (1948).
[Crossref]

Abeles, F.

F. Abeles, “Recherches sur la Propagation des Ondes Electromagnetiques Sinusoidales dans les Milieux Stratifies. Application aux Couches Minces,” Ann. Phys. 12(5), 596–640 (1950).
[Crossref]

F. Abeles, “Sur la Propagation des Ondes Electromagnetiques dans les Milieux Stratifies,” Ann. Phys. 12(3), 504–520 (1948).
[Crossref]

Bartels, A.

A. Bartels, T. Dekors, and H. Kurz, “Coherent zone-folded longitudinal acoustic phonons in semiconductor superlattices: excitation and detection,” Phys. Rev. Lett. 82(5), 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/AlxGa12xAs periodic multilayer structures,” J. Appl. Phys. 75(6), 2761–2768 (1994).
[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).

Born, M.

M. Born and E. Wolf, Principles of Optics (Cambridge University Press, 1999).

Chen, W.

W. Chen, Y. Lu, H. J. Maris, and G. Xiao, “Picosecond ultra-sonic study of localized phonon surface modes in Al/Ag superlattices,” Phys. Rev. B 50(19), 14506–14515 (1994).
[Crossref]

Dekors, T.

A. Bartels, T. Dekors, and H. Kurz, “Coherent zone-folded longitudinal acoustic phonons in semiconductor superlattices: excitation and detection,” Phys. Rev. Lett. 82(5), 1044–1047 (1999).
[Crossref]

Elsayed-Ali, H. E.

H. E. Elsayed-Ali and T. Juhasz, “Femtosecond timeresolved thermomodulation of thin gold films with different crystal structures,” Phys. Rev. B 47(20), 13599–13610 (1993).
[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/AlxGa12xAs periodic multilayer structures,” J. Appl. Phys. 75(6), 2761–2768 (1994).
[Crossref]

Grahn, H. T.

C. Thomsen, H. T. Grahn, H. J. Maris, and J. Tauc, “Surface generation and detection of phonons by picoseconds light pulses,” Phys. Rev. B 34(6), 4129–4138 (1986).
[Crossref]

Gusev, V.

Gusev, V. E.

O. B. Wright and V. E. Gusev, “Ultrafast generation of acoustic waves in copper,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 42(3), 331–338 (1995).
[Crossref]

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(11), 8262–8266 (1999).
[Crossref]

Hurley, D. H.

Ishida, H.

Jacobsson, R.

R. Jacobsson, “Light reflection films of continuously varying refractive index,” Prog. Opt. 5, 247–286 (1966).
[Crossref]

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 timeresolved thermomodulation of thin gold films with different crystal structures,” Phys. Rev. B 47(20), 13599–13610 (1993).
[Crossref]

Kawashima, K.

O. B. Wright and K. Kawashima, “Coherent phonon detection from ultrafast surface vibrations,” Phys. Rev. Lett. 69(11), 1668–1671 (1992).
[Crossref]

Kouyate, M.

Kurz, H.

A. Bartels, T. Dekors, and H. Kurz, “Coherent zone-folded longitudinal acoustic phonons in semiconductor superlattices: excitation and detection,” Phys. Rev. Lett. 82(5), 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(7), 2391–2395 (1988).
[Crossref]

Lu, Y.

W. Chen, Y. Lu, H. J. Maris, and G. Xiao, “Picosecond ultra-sonic study of localized phonon surface modes in Al/Ag superlattices,” Phys. Rev. B 50(19), 14506–14515 (1994).
[Crossref]

Maris, H. J.

W. Chen, Y. Lu, H. J. Maris, and G. Xiao, “Picosecond ultra-sonic study of localized phonon surface modes in Al/Ag superlattices,” Phys. Rev. B 50(19), 14506–14515 (1994).
[Crossref]

C. Thomsen, H. T. Grahn, H. J. Maris, and J. Tauc, “Surface generation and detection of phonons by picoseconds light pulses,” Phys. Rev. B 34(6), 4129–4138 (1986).
[Crossref]

Matsuda, O.

Miklós, A.

A. Miklós and A. Lörincz, “Transient thermoreflectance of thin metal films in the picosecond regime,” J. Appl. Phys. 63(7), 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/AlxGa12xAs periodic multilayer structures,” J. Appl. Phys. 75(6), 2761–2768 (1994).
[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(11), 8262–8266 (1999).
[Crossref]

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(11), 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(11), 8262–8266 (1999).
[Crossref]

Nevot, L.

L. Nevot, “Caractérisation des surfaces par réflexion rasante de rayons X. Application à l'étude du polissage de quelques verres silicates,” Rev. Phys. Appl. 15(3), 761–779 (1980).
[Crossref]

Ohno, H.

H. Ohno, “A numerical method to reconstruct internal structures with a laser ultrasonic technique,” Proc. SPIE 10726, 1072612 (2018).
[Crossref]

Ohta, K.

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).

Pezeril, T.

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).

Tauc, J.

C. Thomsen, H. T. Grahn, H. J. Maris, and J. Tauc, “Surface generation and detection of phonons by picoseconds light pulses,” Phys. Rev. B 34(6), 4129–4138 (1986).
[Crossref]

Thomsen, C.

C. Thomsen, H. T. Grahn, H. J. Maris, and J. Tauc, “Surface generation and detection of phonons by picoseconds light pulses,” Phys. Rev. B 34(6), 4129–4138 (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/AlxGa12xAs periodic multilayer structures,” J. Appl. Phys. 75(6), 2761–2768 (1994).
[Crossref]

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Cambridge University Press, 1999).

Wright, O. B.

O. Matsuda and O. B. Wright, “Reflection and transmission of light in multilayers perturbed by picoseconds strain pulse propagation,” J. Opt. Soc. Am. B 19(12), 3028–3041 (2002).
[Crossref]

D. H. Hurley and O. B. Wright, “Detection of ultrafast phenomena by use of a modified Sagnac interferometer,” Opt. Lett. 24(18), 1305–1307 (1999).
[Crossref]

O. B. Wright and V. E. Gusev, “Ultrafast generation of acoustic waves in copper,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 42(3), 331–338 (1995).
[Crossref]

O. B. Wright, “Laser picoseconds acoustics in double-layer transparent films,” Opt. Lett. 20(6), 632–634 (1995).
[Crossref]

O. B. Wright, “Ultrafast nonequilibrium stress generation in gold and silver,” Phys. Rev. B 49(14), 9985–9988 (1994).
[Crossref]

O. B. Wright, “Thickness and sound velocity measurement in thin transparent films with laser picosecond acoustics,” J. Appl. Phys. 71(4), 1617–1629 (1992).
[Crossref]

O. B. Wright and K. Kawashima, “Coherent phonon detection from ultrafast surface vibrations,” Phys. Rev. Lett. 69(11), 1668–1671 (1992).
[Crossref]

Xiao, G.

W. Chen, Y. Lu, H. J. Maris, and G. Xiao, “Picosecond ultra-sonic study of localized phonon surface modes in Al/Ag superlattices,” Phys. Rev. B 50(19), 14506–14515 (1994).
[Crossref]

Yee, K.

K. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propag. 14(3), 302–307 (1966).
[Crossref]

Ann. Phys. (2)

F. Abeles, “Recherches sur la Propagation des Ondes Electromagnetiques Sinusoidales dans les Milieux Stratifies. Application aux Couches Minces,” Ann. Phys. 12(5), 596–640 (1950).
[Crossref]

F. Abeles, “Sur la Propagation des Ondes Electromagnetiques dans les Milieux Stratifies,” Ann. Phys. 12(3), 504–520 (1948).
[Crossref]

Appl. Opt. (1)

IEEE Trans. Antennas Propag. (1)

K. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propag. 14(3), 302–307 (1966).
[Crossref]

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(3), 331–338 (1995).
[Crossref]

J. Appl. Phys. (3)

A. Miklós and A. Lörincz, “Transient thermoreflectance of thin metal films in the picosecond regime,” J. Appl. Phys. 63(7), 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/AlxGa12xAs periodic multilayer structures,” J. Appl. Phys. 75(6), 2761–2768 (1994).
[Crossref]

O. B. Wright, “Thickness and sound velocity measurement in thin transparent films with laser picosecond acoustics,” J. Appl. Phys. 71(4), 1617–1629 (1992).
[Crossref]

J. Opt. Soc. Am. B (2)

Opt. Lett. (2)

Phys. Rev. B (5)

C. Thomsen, H. T. Grahn, H. J. Maris, and J. Tauc, “Surface generation and detection of phonons by picoseconds light pulses,” Phys. Rev. B 34(6), 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(11), 8262–8266 (1999).
[Crossref]

H. E. Elsayed-Ali and T. Juhasz, “Femtosecond timeresolved thermomodulation of thin gold films with different crystal structures,” Phys. Rev. B 47(20), 13599–13610 (1993).
[Crossref]

W. Chen, Y. Lu, H. J. Maris, and G. Xiao, “Picosecond ultra-sonic study of localized phonon surface modes in Al/Ag superlattices,” Phys. Rev. B 50(19), 14506–14515 (1994).
[Crossref]

O. B. Wright, “Ultrafast nonequilibrium stress generation in gold and silver,” Phys. Rev. B 49(14), 9985–9988 (1994).
[Crossref]

Phys. Rev. Lett. (2)

A. Bartels, T. Dekors, and H. Kurz, “Coherent zone-folded longitudinal acoustic phonons in semiconductor superlattices: excitation and detection,” Phys. Rev. Lett. 82(5), 1044–1047 (1999).
[Crossref]

O. B. Wright and K. Kawashima, “Coherent phonon detection from ultrafast surface vibrations,” Phys. Rev. Lett. 69(11), 1668–1671 (1992).
[Crossref]

Proc. SPIE (1)

H. Ohno, “A numerical method to reconstruct internal structures with a laser ultrasonic technique,” Proc. SPIE 10726, 1072612 (2018).
[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).

Prog. Opt. (1)

R. Jacobsson, “Light reflection films of continuously varying refractive index,” Prog. Opt. 5, 247–286 (1966).
[Crossref]

Rev. Phys. Appl. (1)

L. Nevot, “Caractérisation des surfaces par réflexion rasante de rayons X. Application à l'étude du polissage de quelques verres silicates,” Rev. Phys. Appl. 15(3), 761–779 (1980).
[Crossref]

Other (2)

Python, https://www.python.org/

M. Born and E. Wolf, Principles of Optics (Cambridge University Press, 1999).

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Figures (3)

Fig. 1.
Fig. 1. Cross-sectional view of a stratified medium on the x-z plane where the top surface of the medium is set to z = 0 plane before the elastic displacement induced by acoustic pulse. The incident light comes from the z < Z0. The l-th layer (l = 1, 2, …, L) is located in the region from z = Zl-1 to z = Zl with the relative permittivity of ${\bar{\varepsilon }_l}$ before the modulation induced by the acoustic pulse.
Fig. 2.
Fig. 2. Cross-sectional view of a stratified medium on x-z plane where the top surface of the medium is set to z = 0 plane before the elastic displacement induced by acoustic pulse. The incident light comes from the z < z0. The medium consists of multi-slicing films, with each film having the same infinitesimal thicknesses of Δz. Each film is located in the region of ${z_{m - 1}} < z \le {z_m}$ with zm-zm-1=Δz (m = 1, …, N) where the permittivity of each film at z = zm is set to ɛm defined as ɛ(zm).
Fig. 3.
Fig. 3. Reflectance modulation calculated by the multi-slicing matrix method with respect to the time duration is plotted with a black solid line. An experiment with respect to time duration is plotted with a gray solid line. The infinitesimal thickness Δz of each film in the multi-slicing matrix method is set to sufficiently smaller than the light penetration length, which satisfies Eq. (26).

Tables (1)

Tables Icon

Table 1. Specifications for amorphous carbon and fitting parameters

Equations (94)

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

× E = μ 0 H t ,
× H = ε a ( ε ¯ E ) t ,
( ε ¯ E ) = 0 ,
B = 0 ,
q = z ( e z ( Re ( E ) × Re ( H ) ) ) ,
ρ t ( u z t ) = z σ z z .
σ z z = 3 Λ 1 ν 1 + ν u z z 3 Λ γ ρ C 0 Δ t q d t ,
ρ 2 η z z t 2 = 3 Λ 1 ν 1 + ν 2 η z z z 2 3 Λ γ ρ C 0 Δ t 2 q z 2 d t ,
η z z = u z z .
Δ n = K η z z .
Δ ε = ( n ¯ l + Δ n ) 2 n ¯ l 2 2 n ¯ l Δ n = 2 ε ¯ l K η z z .
× E = i ω μ 0 H ,
× H = i ω ε ( z ) ε a E ,
( ε ( z ) E ) = 0 ,
H = 0 ,
ε ( z ) = n 2 ( z ) .
E = [ E x ( z ) E y ( z ) E z ( z ) ] exp ( i k x x i ω t ) ,
H = [ H x ( z ) H y ( z ) H z ( z ) ] exp ( i k x x i ω t ) .
k x = k 0 sin θ ,
k 0 = ω ε a μ 0 .
E = 1 ε ( z ) ε ( z ) z E z ( z ) exp ( i k x x i ω t ) .
Δ E + ε ( z ) k 0 2 E = ( E ) = [ i k x 1 ε ( z ) ε ( z ) z E z 0 z ( 1 ε ( z ) ε ( z ) z E z ) ] exp ( i k x x i ω t ) .
ε ( z ) = ε ¯ + Δ ε ( z ) .
ε ( z ) = ε ¯ l + Δ ε ( z m ) + ( Δ ε ( z ) Δ ε ( z m ) ) = ε ¯ l + Δ ε m + Δ ε ( z m ) Δ z + O ( Δ z 2 ) ,
Δ E + ε ¯ l k 0 2 E = Δ ε m k 0 2 E Δ ε ( z m ) Δ z k 0 2 E [ i k x Δ ε ε E z 0 ( Δ ε ε E z ) ] exp ( i k x x i ω t ) .
Δ z Δ ε m Δ ε ( z m ) .
Δ E + ( ε ¯ l + Δ ε m ) k 0 2 E = Δ E + ε m k 0 2 E = Δ ε ( z m ) Δ z k 0 2 E 0.
Δ z Δ ε m 2 Δ ε ( z m ) ,
E = A exp ( i k m z ( z z m 1 ) ) ,
E = B exp ( i k m z ( z m z ) ) ,
k m z = k 0 ε m sin 2 θ .
[ E x E y E z ] = [ E x E s E z ] = [ 0 a s m exp ( i k m z ( z z m 1 ) ) + b s m exp ( i k m z ( z m z ) ) 0 ] .
[ H x H y H z ] = [ H s H y H z ] = 1 i ω μ 0 [ i k m z ( a s m exp ( i k m z ( z z m 1 ) ) b s m exp ( i k m z ( z m z ) ) ) 0 i k x ( a s m exp ( i k m z ( z z m 1 ) ) + b s m exp ( i k m z ( z m z ) ) ) ] .
[ E s ( z m 1 ) H s ( z m 1 ) / Y 0 ] = [ 1 exp ( i k m z Δ z ) β m β m exp ( i k m z Δ z ) ] [ a s m b s m ] ,
[ E s ( z m ) H s ( z m ) / Y 0 ] = [ exp ( i k m z Δ z ) 1 β m exp ( i k m z Δ z ) β m ] [ a s m b s m ] ,
Y 0 = k 0 ω μ 0 ,
β m = k m z k 0 = ε m sin 2 θ .
[ E s ( z m 1 ) H s ( z m 1 ) / Y 0 ] = M s ( z m 1 , z m ) [ E s ( z m ) H s ( z m ) / Y 0 ] ,
M s ( z m 1 , z m ) = [ cos ( k m z Δ z ) i β m 1 sin ( k m z Δ z ) i β m sin ( k m z Δ z ) cos ( k m z Δ z ) ] .
[ E s ( z 0 ) H s ( z 0 ) / Y 0 ] = M s ( z 0 , z 1 ) M s ( z N 1 , z N ) [ E s ( z N ) H s ( z N ) / Y 0 ] .
u l = Z l Z L η z z ( z ) d z .
ε ( z ) = ε ¯ l + Δ ε ( z ) ,
Z l 1 = Z ¯ l 1 + u l 1 < z Z ¯ l + u l = Z l ,
ε m = ε ¯ l + Δ ε ( z m ) = ε ¯ l + Δ ε m ,
z m 1 < z z m ,
Δ z = z m z m 1 = Z L Z 0 N .
[ E x E y E z ] = [ E x E s E z ] = [ 0 a s 0 exp ( i k 0 z z ) + b s 0 exp ( i k 0 z z ) 0 ] .
[ H x H y H z ] = [ H s H y H z ] = 1 i ω μ 0 [ i k 0 z ( a s 0 exp ( i k 0 z z ) b s 0 exp ( i k 0 z z ) ) 0 i k x ( a s 0 exp ( i k 0 z z ) + b s 0 exp ( i k 0 z z ) ) ] .
[ E s ( u 0 ) H s ( u 0 ) / Y 0 ] = [ exp ( i k 0 z u 0 ) exp ( i k 0 z u 0 ) β ¯ 0 exp ( i k 0 z u 0 ) β ¯ 0 exp ( i k 0 z u 0 ) ] [ a s 0 b s 0 ] .
[ E x E y E z ] = [ E x E s E z ] = [ 0 a s N + 1 exp ( i k 0 β ¯ L + 1 ( z Z L ) ) 0 ] .
[ E s ( Z L ) H s ( Z L ) / Y 0 ] = [ 1 β ¯ L + 1 ] a s N + 1 .
[ a s 0 / a s N + 1 b s 0 / a s N + 1 ] = 1 2 [ ζ 1 β ¯ 0 1 ζ 1 ζ β ¯ 0 1 ζ ] M s ( z 0 , z 1 ) M s ( z N 1 , z N ) [ 1 β ¯ L + 1 ] ,
ζ = exp ( i k 0 z u 0 ) .
[ a ¯ s 0 / a ¯ s L + 1 b ¯ s 0 / a ¯ s L + 1 ] = 1 2 [ 1 β ¯ 0 1 1 β ¯ 0 1 ] M ¯ s ( Z ¯ 0 , Z ¯ 1 ) M ¯ s ( Z ¯ L 1 , Z ¯ L ) [ 1 β ¯ L + 1 ] ,
Δ r s = b s 0 / a s N + 1 a s 0 / a s N + 1 b ¯ s 0 / a ¯ s L + 1 a ¯ s 0 / a ¯ s L + 1 = b s 0 a s 0 b ¯ s 0 a ¯ s 0 ,
Δ r s = ζ 2 c s 1 t M s ( z 0 , z 1 ) M s ( z N 1 , z N ) f s c s 0 t M s ( z 0 , z 1 ) M s ( z N 1 , z N ) f s c s 1 t M ¯ s ( Z ¯ 0 , Z ¯ 1 ) M ¯ s ( Z ¯ L 1 , Z ¯ L ) f s c s 0 t M ¯ s ( Z ¯ 0 , Z 1 ) M ¯ s ( Z ¯ L 1 , Z ¯ L ) f s ,
[ c s 0 t c s 1 t ] = 1 2 [ ( 1 β ¯ 0 1 ) ( 1 β ¯ 0 1 ) ] ,
f s = [ 1 β ¯ L + 1 ] .
[ a s 0 / a s N + 1 b s 0 / a s N + 1 ] = 1 2 [ ζ 1 ζ 1 β ¯ 0 1 ζ ζ β ¯ 0 1 ] M s ( Z 0 , z 1 ) M s ( z N 1 , Z 1 ) [ 1 β ¯ 1 ] 1 2 [ 1 i k 0 z u 0 ( 1 i k 0 z u 0 ) β ¯ 0 1 1 + i k 0 z u 0 ( 1 + i k 0 z u 0 ) β ¯ 0 1 ] M s | Δ ε = 0 ( Z 0 , z 1 ) M s | Δ ε = 0 ( z N 1 , Z 1 ) [ 1 β ¯ 1 ] + 1 2 [ 1 β ¯ 0 1 1 β ¯ 0 1 ] ( j = 1 N ε j ( M ¯ s ( Z ¯ 0 , z 1 ) M ¯ s ( z N 1 , Z ¯ 1 ) ) Δ ε j ) [ 1 β ¯ 1 ] + O ( Δ ε 2 ) ,
[ a s 0 / a s N + 1 b s 0 / a s N + 1 ] = 1 2 [ 1 i k 0 z u 0 ( 1 i k 0 z u 0 ) β ¯ 0 1 1 + i k 0 z u 0 ( 1 + i k 0 z u 0 ) β ¯ 0 1 ] M s | Δ ε = 0 ( Z 0 , z 1 ) M s | Δ ε = 0 ( z N 1 , Z 1 ) [ 1 β ¯ 1 ] + 1 2 Δ z [ 1 β ¯ 0 1 1 β ¯ 0 1 ] × ( j = 1 N M ¯ s ( Z ¯ 0 , z 1 ) M ¯ s ( z j 1 , z j ) X s M ¯ s ( z j , z j + 1 ) M ¯ s ( z N 1 , Z ¯ 1 ) Δ ε j ) [ 1 β ¯ 1 ] + O ( Δ ε Δ z 2 ) + O ( Δ ε 2 ) ,
X s = i k 0 [ 0 0 1 0 ] .
β m β ¯ 1 + Δ ε m 2 ε ¯ 1 sin 2 θ + O ( Δ ε 2 ) = β ¯ 1 + Δ ε m 2 β ¯ 1 + O ( Δ ε 2 ) .
M s | Δ ε = 0 ( z i , z i + 1 ) M s | Δ ε = 0 ( z j 1 , z j ) = [ cos ( k 0 β ¯ 1 ( z j z i ) ) i β ¯ 1 1 sin ( k 0 β ¯ 1 ( z j z i ) ) i β ¯ 1 sin ( k 0 β ¯ 1 ( z j z i ) ) cos ( k 0 β ¯ 1 ( z j z i ) ) ] = M s | Δ ε = 0 ( z i , z j ) .
[ a s 0 / a s N + 1 b s 0 / a s N + 1 ] = 1 2 [ 1 i k 0 z u 0 ( 1 i k 0 z u 0 ) β ¯ 0 1 1 + i k 0 z u 0 ( 1 + i k 0 z u 0 ) β ¯ 0 1 ] M s | Δ ε = 0 ( Z 0 , Z 1 ) [ 1 β ¯ 1 ] + 1 2 Δ z [ 1 β ¯ 0 1 1 β ¯ 0 1 ] ( j = 1 N M ¯ s ( 0 , z j ) X s M ¯ s ( z j , Z ¯ 1 ) Δ ε j ) [ 1 β ¯ 1 ] .
[ a s 0 / a s N + 1 b s 0 / a s N + 1 ] = exp ( i k 0 β ¯ 1 ( Z 1 Z 0 ) ) 2 [ 1 i k 0 z u 0 ( 1 i k 0 z u 0 ) β ¯ 0 1 1 + i k 0 z u 0 ( 1 + i k 0 z u 0 ) β ¯ 0 1 ] [ 1 β ¯ 1 ] + i k 0 Δ z 2 [ 1 β ¯ 0 1 1 β ¯ 0 1 ] ( j = 1 N [ i β ¯ 1 1 sin ( k 0 β ¯ 1 z j ) cos ( k 0 β ¯ 1 z j ) ] exp ( i k 0 β ¯ 1 ( Z ¯ 1 z j ) ) Δ ε j ) .
Δ r s = 2 i k 0 z u 0 β ¯ 0 β ¯ 1 β ¯ 0 + β ¯ 1 + j = 1 N ( i k 0 exp ( 2 i k 0 β ¯ 1 z j ) 2 β ¯ 0 ( β ¯ 0 + β ¯ 1 ) 2 Δ ε j Δ z ) .
r ¯ = b ¯ a ¯ = β ¯ 0 β ¯ 1 β ¯ 0 + β ¯ 1 .
Δ r s = 2 i k 0 z u 0 r ¯ + t ¯ t ~ ¯ i k 0 2 2 k ¯ 1 z j = 1 N ( exp ( 2 i k ¯ 1 z z j ) Δ ε j Δ z ) .
k 0 z = k 0 β ¯ 0 = k 0 ε ¯ 0 sin 2 θ = k 0 1 sin 2 θ ,
k ¯ 1 z = k 0 β ¯ 1 = k 0 ε ¯ 1 sin 2 θ ,
t ¯ = 2 k 0 z k 0 z + k ¯ 1 z ,
t ~ ¯ = 2 k ¯ 1 z k 0 z + k ¯ 1 z .
Δ r s = 2 i k 0 z u 0 r ¯ + t ¯ t ~ ¯ i k 0 2 2 k ¯ 1 z 0 exp ( 2 i k ¯ 1 z z ) Δ ε ( z ) d z .
Δ T = q ρ C = Q Δ t i n ρ C z a exp ( z z a ) ,
z a = λ 4 π Im ( n a ) ,
P = 3 Λ γ Δ T .
η z z = G 0 ( exp ( z z a ) 1 2 exp ( z + V t z a ) 1 2 exp ( | z V t | z a ) sgn ( z V t ) ) .
Δ z Δ ε m Δ ε ( z m ) = z a ,
Δ ε = 2 ε ¯ l K η z z ,
k m z = k 0 β m ,
β m = ε m sin 2 θ = ε ¯ l + Δ ε m sin 2 θ ,
Δ r s = ζ 2 c s 1 t M s ( z 0 , z 1 ) M s ( z N 1 , z N ) f s c s 0 t M s ( z 0 , z 1 ) M s ( z N 1 , z N ) f s c s 1 t M ¯ s ( Z ¯ 0 , Z ¯ 1 ) M ¯ s ( Z ¯ L 1 , Z ¯ L ) f s c s 0 t M ¯ s ( Z ¯ 0 , Z 1 ) M ¯ s ( Z ¯ L 1 , Z ¯ L ) f s ,
ζ = exp ( i k 0 z u 0 ) .
c s 0 = 1 2 [ 1 β ¯ 0 1 ] ,
c s 1 = 1 2 [ 1 β ¯ 0 1 ] ,
f s = [ 1 β ¯ L + 1 ] .
M s ( z m 1 , z m ) = [ cos ( k m z Δ z ) i β m 1 sin ( k m z Δ z ) i β m sin ( k m z Δ z ) cos ( k m z Δ z ) ] ,
Δ z = z m z m 1 = Z L Z 0 N .
M ¯ s ( Z ¯ l 1 , Z ¯ l ) = [ cos ( k ¯ l z ( Z ¯ l Z ¯ l 1 ) ) i β ¯ l 1 sin ( k ¯ l z ( Z ¯ l Z ¯ l 1 ) ) i β ¯ l sin ( k ¯ l z ( Z ¯ l Z ¯ l 1 ) ) cos ( k ¯ l z ( Z ¯ l Z ¯ l 1 ) ) ] .
Δ r s = ζ 2 G 1 G 2 G 3 G 4 ,
G 1 = [ 1 2 β ¯ 0 1 2 ] M s ( z 0 , z 1 ) M s ( z m 1 , z m ) M s ( z N 1 , z N ) [ 1 β ¯ L + 1 ] ,
G 2 = [ 1 2 β ¯ 0 1 2 ] M s ( z 0 , z 1 ) M s ( z m 1 , z m ) M s ( z N 1 , z N ) [ 1 β ¯ L + 1 ] ,
G 3 = [ 1 2 β ¯ 0 1 2 ] M ¯ s ( Z ¯ 0 , Z ¯ 1 ) M ¯ s ( Z ¯ l 1 , Z ¯ l ) M ¯ s ( Z ¯ L 1 , Z ¯ L ) [ 1 β ¯ L + 1 ] ,
G 4 = [ 1 2 β ¯ 0 1 2 ] M ¯ s ( Z ¯ 0 , Z ¯ 1 ) M ¯ s ( Z ¯ l 1 , Z ¯ l ) M ¯ s ( Z ¯ L 1 , Z ¯ L ) [ 1 β ¯ L + 1 ] .

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