Abstract

This paper presents experiments and in-depth analysis of the imaging of surface acoustic waves by means of the photoelastic effect. Gigahertz surface acoustic waves, generated by optical pump pulses in a thin gold film on a glass substrate, are imaged in the time domain by monitoring ultrafast changes in optical reflectivity. We demonstrate how images of the in-plane acoustic shear strain component can be obtained by measurements with two different optical probe pulse polarizations incident from the substrate side.

© 2010 Optical Society of America

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  1. R. Adler, A. Korpel, and P. Desmares, “An instrument for making surface waves visible,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 15, 157–161 (1968).
  2. J. V. Knuuttila, P. T. Tikka, and M. M. Salomaa, “Scanning Michelson interferometer for imaging surface acoustic wave fields,” Opt. Lett. 25, 613–615 (2000).
    [CrossRef]
  3. J. E. Graebner, B. P. Barber, P. L. Gammel, and D. S. Greywall, “Dynamic visualization of subangstrom high-frequency surface vibrations,” Appl. Phys. Lett. 78, 159–161 (2001).
    [CrossRef]
  4. J. L. Blackshire, S. Sathish, B. D. Duncan, and M. Millard, “Real-time, frequency-translated holographic visualization of surface acoustic wave interactions with surface-breaking defects,” Opt. Lett. 27, 1025–1027 (2002).
    [CrossRef]
  5. A. Miyamoto, S. Matsuda, S. Wakana, and A. Ito, “Optical observation technique for SH-type surface acoustic wave,” Electron. Commun. Jpn., Part 2: Electron. 87, 1295–1301 (2004).
    [CrossRef]
  6. K. Kokkonen and M. Kaivola, “Scanning heterodyne laser interferometer for phase-sensitive absolute-amplitude measurements of surface vibrations,” Appl. Phys. Lett. 92, 063502 (2008).
    [CrossRef]
  7. T. Fujikura, O. Matsuda, D. M. Profunser, O. B. Wright, J. Masson, and S. Ballandras, “Real-time imaging of acoustic waves on a bulk acoustic resonator,” Appl. Phys. Lett. 93, 261101 (2008).
    [CrossRef]
  8. M. Clark, S. D. Sharples, and M. G. Somekh, “Diffractive acoustic elements for laser ultrasonics,” J. Acoust. Soc. Am. 107, 3179–3185 (2000).
    [CrossRef] [PubMed]
  9. Y. Sugawara, O. B. Wright, O. Matsuda, M. Takigahira, Y. Tanaka, S. Tamura, and V. E. Gusev, “Watching ripples on crystals,” Phys. Rev. Lett. 88, 185504 (2002).
    [CrossRef] [PubMed]
  10. J. A. Scales and A. E. Malcolm, “Laser characterization of ultrasonic wave propagation in random media,” Phys. Rev. E 67, 046618 (2003).
    [CrossRef]
  11. A. A. Maznev, A. M. Lomonosov, P. Hess, and A. A. Kolomenskii, “Anisotropic effects in surface acoustic wave propagation from a point source in a crystal,” Eur. Phys. J. B 35, 429–439 (2003).
    [CrossRef]
  12. T. Tachizaki, T. Muroya, O. Matsuda, Y. Sugawara, D. H. Hurley, and O. B. Wright, “Scanning ultrafast sagnac interferometry for imaging two-dimensional surface wave propagation,” Rev. Sci. Instrum. 77, 043713 (2006).
    [CrossRef]
  13. D. M. Profunser, O. B. Wright, and O. Matsuda, “Imaging ripples on phononic crystals reveals acoustic band structure and Bloch harmonics,” Phys. Rev. Lett. 97, 055502 (2006).
    [CrossRef] [PubMed]
  14. D. M. Profunser, E. Muramoto, O. Matsuda, and O. B. Wright, “Dynamic visualization of surface acoustic waves on a two-dimensional phononic crystal,” Phys. Rev. B 80, 014301 (2009).
    [CrossRef]
  15. M. M. Frocht, Photoelasticity (Wiley, 1957), Vols. 1 and 2.
  16. C. P. Burger, “Photoelasticity,” in Handbook on Experimental Mechanics (Second Revised Edition), A.S.Kobayashi, ed. (VCH, 1993), Chap. 5, pp. 165–266.
  17. W. F. Riley and J. W. Dally, “A photoelastic analysis of stress wave propagation in a layered model,” Geophysics 31, 881–899 (1966).
    [CrossRef]
  18. J. W. Dally, “An introduction to dynamic photoelasticity,” Exp. Mech. 20, 409–416 (1980).
    [CrossRef]
  19. Y. H. Nam and S. S. Lee, “A quantitative evaluation of elastic wave in solid by strobscopic photoelasticity,” J. Sound Vib. 259, 1199–1207 (2003).
    [CrossRef]
  20. W.-C. Wang and Y.-H. Tsai, “Digital dynamic photoelastic and numerical stress analyses of a strip,” J. Vib. Control 12, 927–938 (2006).
    [CrossRef]
  21. R. Hayasi, Y. Masuda, S. Hashimoto, and S. Kuriyama, “Analysis of geometric effects on stress wave propagation in epoxy resins of plate-like structure by dynamic photoelasticity combined with strain gauge,” Jpn. J. Appl. Phys. 47, 4676–4681 (2008).
    [CrossRef]
  22. H. Yamazaki, O. Matsuda, and O. B. Wright, “Surface phonon imaging through the photoelastic effect,” Phys. Status Solidi C 1, 2991–2994 (2004).
    [CrossRef]
  23. M. Garfinkel, J. J. Tiemann, and W. E. Engeler, “Piezoreflectivity of the noble metals,” Phys. Rev. 148, 695–706 (1966).
    [CrossRef]
  24. This is supported by measurements made by probing from the film side of our sample, showing a response ∼10 times smaller. Other measurements also indicate a negligible photoelastic coefficient P12 at our probe wavelength. The photoelastic constant of fused silica in the visible at this wavelength is approximately P12=−1.3.
  25. G. W. C. Kaye and T. H. Laby, Tables of Physical and Chemical Constants, 16th ed. (Longman, 1995).
  26. The SSLW ring in fact overlaps in Fig. with a weaker second RW ring, but the former ring dominates.
  27. 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]
  28. B. Perrin, B. Bonello, J. C. Jeannet, and E. Romatet, “Interferometric detection of hypersound waves in modulated structures,” Prog. Nat. Sci. S6, S444–S448 (1996).
  29. V. E. Gusev, “Laser hypersonics in fundamental and applied research,” Acust. Acta Acust. 82, S37–S45 (1996).
  30. O. Matsuda and O. B. Wright, “Reflection and transmission of light in multilayers perturbed by picosecond strain pulse propagation,” J. Opt. Soc. Am. B 19, 3028–3041 (2002).
    [CrossRef]
  31. O. Matsuda, O. B. Wright, D. H. Hurley, V. E. Gusev, and K. Shimizu, “Coherent shear phonon generation and detection with picosecond laser acoustics,” Phys. Rev. B 77, 224110 (2008).
    [CrossRef]
  32. N. Chigarev, C. Rossignol, and B. Audoin, “Surface displacement measured by beam distortion detection technique: Application to picosecond ultrasonics,” Rev. Sci. Instrum. 77, 114901 (2006).
    [CrossRef]
  33. C. Matteï, X. Jia, and G. Quentin, “Measurement of Rayleigh wave strains inside a transparent solid by optical interferometry,” Acta Acust. 2, 65–67 (1994).
  34. M. Tomoda, O. Matsuda, and O. B. Wright, “Tomographic reconstruction of picosecond acoustic strain propagation,” Appl. Phys. Lett. 90, 041114 (2007).
    [CrossRef]
  35. 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]
  36. 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]

2009

D. M. Profunser, E. Muramoto, O. Matsuda, and O. B. Wright, “Dynamic visualization of surface acoustic waves on a two-dimensional phononic crystal,” Phys. Rev. B 80, 014301 (2009).
[CrossRef]

2008

R. Hayasi, Y. Masuda, S. Hashimoto, and S. Kuriyama, “Analysis of geometric effects on stress wave propagation in epoxy resins of plate-like structure by dynamic photoelasticity combined with strain gauge,” Jpn. J. Appl. Phys. 47, 4676–4681 (2008).
[CrossRef]

K. Kokkonen and M. Kaivola, “Scanning heterodyne laser interferometer for phase-sensitive absolute-amplitude measurements of surface vibrations,” Appl. Phys. Lett. 92, 063502 (2008).
[CrossRef]

T. Fujikura, O. Matsuda, D. M. Profunser, O. B. Wright, J. Masson, and S. Ballandras, “Real-time imaging of acoustic waves on a bulk acoustic resonator,” Appl. Phys. Lett. 93, 261101 (2008).
[CrossRef]

O. Matsuda, O. B. Wright, D. H. Hurley, V. E. Gusev, and K. Shimizu, “Coherent shear phonon generation and detection with picosecond laser acoustics,” Phys. Rev. B 77, 224110 (2008).
[CrossRef]

2007

M. Tomoda, O. Matsuda, and O. B. Wright, “Tomographic reconstruction of picosecond acoustic strain propagation,” Appl. Phys. Lett. 90, 041114 (2007).
[CrossRef]

2006

N. Chigarev, C. Rossignol, and B. Audoin, “Surface displacement measured by beam distortion detection technique: Application to picosecond ultrasonics,” Rev. Sci. Instrum. 77, 114901 (2006).
[CrossRef]

W.-C. Wang and Y.-H. Tsai, “Digital dynamic photoelastic and numerical stress analyses of a strip,” J. Vib. Control 12, 927–938 (2006).
[CrossRef]

T. Tachizaki, T. Muroya, O. Matsuda, Y. Sugawara, D. H. Hurley, and O. B. Wright, “Scanning ultrafast sagnac interferometry for imaging two-dimensional surface wave propagation,” Rev. Sci. Instrum. 77, 043713 (2006).
[CrossRef]

D. M. Profunser, O. B. Wright, and O. Matsuda, “Imaging ripples on phononic crystals reveals acoustic band structure and Bloch harmonics,” Phys. Rev. Lett. 97, 055502 (2006).
[CrossRef] [PubMed]

2004

A. Miyamoto, S. Matsuda, S. Wakana, and A. Ito, “Optical observation technique for SH-type surface acoustic wave,” Electron. Commun. Jpn., Part 2: Electron. 87, 1295–1301 (2004).
[CrossRef]

H. Yamazaki, O. Matsuda, and O. B. Wright, “Surface phonon imaging through the photoelastic effect,” Phys. Status Solidi C 1, 2991–2994 (2004).
[CrossRef]

2003

Y. H. Nam and S. S. Lee, “A quantitative evaluation of elastic wave in solid by strobscopic photoelasticity,” J. Sound Vib. 259, 1199–1207 (2003).
[CrossRef]

J. A. Scales and A. E. Malcolm, “Laser characterization of ultrasonic wave propagation in random media,” Phys. Rev. E 67, 046618 (2003).
[CrossRef]

A. A. Maznev, A. M. Lomonosov, P. Hess, and A. A. Kolomenskii, “Anisotropic effects in surface acoustic wave propagation from a point source in a crystal,” Eur. Phys. J. B 35, 429–439 (2003).
[CrossRef]

2002

2001

J. E. Graebner, B. P. Barber, P. L. Gammel, and D. S. Greywall, “Dynamic visualization of subangstrom high-frequency surface vibrations,” Appl. Phys. Lett. 78, 159–161 (2001).
[CrossRef]

2000

J. V. Knuuttila, P. T. Tikka, and M. M. Salomaa, “Scanning Michelson interferometer for imaging surface acoustic wave fields,” Opt. Lett. 25, 613–615 (2000).
[CrossRef]

M. Clark, S. D. Sharples, and M. G. Somekh, “Diffractive acoustic elements for laser ultrasonics,” J. Acoust. Soc. Am. 107, 3179–3185 (2000).
[CrossRef] [PubMed]

1999

1996

B. Perrin, B. Bonello, J. C. Jeannet, and E. Romatet, “Interferometric detection of hypersound waves in modulated structures,” Prog. Nat. Sci. S6, S444–S448 (1996).

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

1995

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

1994

C. Matteï, X. Jia, and G. Quentin, “Measurement of Rayleigh wave strains inside a transparent solid by optical interferometry,” Acta Acust. 2, 65–67 (1994).

1993

C. P. Burger, “Photoelasticity,” in Handbook on Experimental Mechanics (Second Revised Edition), A.S.Kobayashi, ed. (VCH, 1993), Chap. 5, pp. 165–266.

1986

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]

1980

J. W. Dally, “An introduction to dynamic photoelasticity,” Exp. Mech. 20, 409–416 (1980).
[CrossRef]

1968

R. Adler, A. Korpel, and P. Desmares, “An instrument for making surface waves visible,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 15, 157–161 (1968).

1967

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]

1966

M. Garfinkel, J. J. Tiemann, and W. E. Engeler, “Piezoreflectivity of the noble metals,” Phys. Rev. 148, 695–706 (1966).
[CrossRef]

W. F. Riley and J. W. Dally, “A photoelastic analysis of stress wave propagation in a layered model,” Geophysics 31, 881–899 (1966).
[CrossRef]

1957

M. M. Frocht, Photoelasticity (Wiley, 1957), Vols. 1 and 2.

Adler, R.

R. Adler, A. Korpel, and P. Desmares, “An instrument for making surface waves visible,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 15, 157–161 (1968).

Audoin, B.

N. Chigarev, C. Rossignol, and B. Audoin, “Surface displacement measured by beam distortion detection technique: Application to picosecond ultrasonics,” Rev. Sci. Instrum. 77, 114901 (2006).
[CrossRef]

Ballandras, S.

T. Fujikura, O. Matsuda, D. M. Profunser, O. B. Wright, J. Masson, and S. Ballandras, “Real-time imaging of acoustic waves on a bulk acoustic resonator,” Appl. Phys. Lett. 93, 261101 (2008).
[CrossRef]

Barber, B. P.

J. E. Graebner, B. P. Barber, P. L. Gammel, and D. S. Greywall, “Dynamic visualization of subangstrom high-frequency surface vibrations,” Appl. Phys. Lett. 78, 159–161 (2001).
[CrossRef]

Blackshire, J. L.

Bonello, B.

B. Perrin, B. Bonello, J. C. Jeannet, and E. Romatet, “Interferometric detection of hypersound waves in modulated structures,” Prog. Nat. Sci. S6, S444–S448 (1996).

Burger, C. P.

C. P. Burger, “Photoelasticity,” in Handbook on Experimental Mechanics (Second Revised Edition), A.S.Kobayashi, ed. (VCH, 1993), Chap. 5, pp. 165–266.

Chigarev, N.

N. Chigarev, C. Rossignol, and B. Audoin, “Surface displacement measured by beam distortion detection technique: Application to picosecond ultrasonics,” Rev. Sci. Instrum. 77, 114901 (2006).
[CrossRef]

Clark, M.

M. Clark, S. D. Sharples, and M. G. Somekh, “Diffractive acoustic elements for laser ultrasonics,” J. Acoust. Soc. Am. 107, 3179–3185 (2000).
[CrossRef] [PubMed]

Dally, J. W.

J. W. Dally, “An introduction to dynamic photoelasticity,” Exp. Mech. 20, 409–416 (1980).
[CrossRef]

W. F. Riley and J. W. Dally, “A photoelastic analysis of stress wave propagation in a layered model,” Geophysics 31, 881–899 (1966).
[CrossRef]

Desmares, P.

R. Adler, A. Korpel, and P. Desmares, “An instrument for making surface waves visible,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 15, 157–161 (1968).

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]

Duncan, B. D.

Engeler, W. E.

M. Garfinkel, J. J. Tiemann, and W. E. Engeler, “Piezoreflectivity of the noble metals,” Phys. Rev. 148, 695–706 (1966).
[CrossRef]

Frocht, M. M.

M. M. Frocht, Photoelasticity (Wiley, 1957), Vols. 1 and 2.

Fujikura, T.

T. Fujikura, O. Matsuda, D. M. Profunser, O. B. Wright, J. Masson, and S. Ballandras, “Real-time imaging of acoustic waves on a bulk acoustic resonator,” Appl. Phys. Lett. 93, 261101 (2008).
[CrossRef]

Gammel, P. L.

J. E. Graebner, B. P. Barber, P. L. Gammel, and D. S. Greywall, “Dynamic visualization of subangstrom high-frequency surface vibrations,” Appl. Phys. Lett. 78, 159–161 (2001).
[CrossRef]

Garfinkel, M.

M. Garfinkel, J. J. Tiemann, and W. E. Engeler, “Piezoreflectivity of the noble metals,” Phys. Rev. 148, 695–706 (1966).
[CrossRef]

Graebner, J. E.

J. E. Graebner, B. P. Barber, P. L. Gammel, and D. S. Greywall, “Dynamic visualization of subangstrom high-frequency surface vibrations,” Appl. Phys. Lett. 78, 159–161 (2001).
[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]

Greywall, D. S.

J. E. Graebner, B. P. Barber, P. L. Gammel, and D. S. Greywall, “Dynamic visualization of subangstrom high-frequency surface vibrations,” Appl. Phys. Lett. 78, 159–161 (2001).
[CrossRef]

Gusev, V. E.

O. Matsuda, O. B. Wright, D. H. Hurley, V. E. Gusev, and K. Shimizu, “Coherent shear phonon generation and detection with picosecond laser acoustics,” Phys. Rev. B 77, 224110 (2008).
[CrossRef]

Y. Sugawara, O. B. Wright, O. Matsuda, M. Takigahira, Y. Tanaka, S. Tamura, and V. E. Gusev, “Watching ripples on crystals,” Phys. Rev. Lett. 88, 185504 (2002).
[CrossRef] [PubMed]

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

Hashimoto, S.

R. Hayasi, Y. Masuda, S. Hashimoto, and S. Kuriyama, “Analysis of geometric effects on stress wave propagation in epoxy resins of plate-like structure by dynamic photoelasticity combined with strain gauge,” Jpn. J. Appl. Phys. 47, 4676–4681 (2008).
[CrossRef]

Hayasi, R.

R. Hayasi, Y. Masuda, S. Hashimoto, and S. Kuriyama, “Analysis of geometric effects on stress wave propagation in epoxy resins of plate-like structure by dynamic photoelasticity combined with strain gauge,” Jpn. J. Appl. Phys. 47, 4676–4681 (2008).
[CrossRef]

Hess, P.

A. A. Maznev, A. M. Lomonosov, P. Hess, and A. A. Kolomenskii, “Anisotropic effects in surface acoustic wave propagation from a point source in a crystal,” Eur. Phys. J. B 35, 429–439 (2003).
[CrossRef]

Hurley, D. H.

O. Matsuda, O. B. Wright, D. H. Hurley, V. E. Gusev, and K. Shimizu, “Coherent shear phonon generation and detection with picosecond laser acoustics,” Phys. Rev. B 77, 224110 (2008).
[CrossRef]

T. Tachizaki, T. Muroya, O. Matsuda, Y. Sugawara, D. H. Hurley, and O. B. Wright, “Scanning ultrafast sagnac interferometry for imaging two-dimensional surface wave propagation,” Rev. Sci. Instrum. 77, 043713 (2006).
[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]

Ito, A.

A. Miyamoto, S. Matsuda, S. Wakana, and A. Ito, “Optical observation technique for SH-type surface acoustic wave,” Electron. Commun. Jpn., Part 2: Electron. 87, 1295–1301 (2004).
[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. S6, S444–S448 (1996).

Jia, X.

C. Matteï, X. Jia, and G. Quentin, “Measurement of Rayleigh wave strains inside a transparent solid by optical interferometry,” Acta Acust. 2, 65–67 (1994).

Kaivola, M.

K. Kokkonen and M. Kaivola, “Scanning heterodyne laser interferometer for phase-sensitive absolute-amplitude measurements of surface vibrations,” Appl. Phys. Lett. 92, 063502 (2008).
[CrossRef]

Kaye, G. W. C.

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

Knuuttila, J. V.

Kokkonen, K.

K. Kokkonen and M. Kaivola, “Scanning heterodyne laser interferometer for phase-sensitive absolute-amplitude measurements of surface vibrations,” Appl. Phys. Lett. 92, 063502 (2008).
[CrossRef]

Kolomenskii, A. A.

A. A. Maznev, A. M. Lomonosov, P. Hess, and A. A. Kolomenskii, “Anisotropic effects in surface acoustic wave propagation from a point source in a crystal,” Eur. Phys. J. B 35, 429–439 (2003).
[CrossRef]

Korpel, A.

R. Adler, A. Korpel, and P. Desmares, “An instrument for making surface waves visible,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 15, 157–161 (1968).

Kuriyama, S.

R. Hayasi, Y. Masuda, S. Hashimoto, and S. Kuriyama, “Analysis of geometric effects on stress wave propagation in epoxy resins of plate-like structure by dynamic photoelasticity combined with strain gauge,” Jpn. J. Appl. Phys. 47, 4676–4681 (2008).
[CrossRef]

Laby, T. H.

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

Lee, S. S.

Y. H. Nam and S. S. Lee, “A quantitative evaluation of elastic wave in solid by strobscopic photoelasticity,” J. Sound Vib. 259, 1199–1207 (2003).
[CrossRef]

Lomonosov, A. M.

A. A. Maznev, A. M. Lomonosov, P. Hess, and A. A. Kolomenskii, “Anisotropic effects in surface acoustic wave propagation from a point source in a crystal,” Eur. Phys. J. B 35, 429–439 (2003).
[CrossRef]

Malcolm, A. E.

J. A. Scales and A. E. Malcolm, “Laser characterization of ultrasonic wave propagation in random media,” Phys. Rev. E 67, 046618 (2003).
[CrossRef]

Maris, H. J.

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]

Masson, J.

T. Fujikura, O. Matsuda, D. M. Profunser, O. B. Wright, J. Masson, and S. Ballandras, “Real-time imaging of acoustic waves on a bulk acoustic resonator,” Appl. Phys. Lett. 93, 261101 (2008).
[CrossRef]

Masuda, Y.

R. Hayasi, Y. Masuda, S. Hashimoto, and S. Kuriyama, “Analysis of geometric effects on stress wave propagation in epoxy resins of plate-like structure by dynamic photoelasticity combined with strain gauge,” Jpn. J. Appl. Phys. 47, 4676–4681 (2008).
[CrossRef]

Matsuda, O.

D. M. Profunser, E. Muramoto, O. Matsuda, and O. B. Wright, “Dynamic visualization of surface acoustic waves on a two-dimensional phononic crystal,” Phys. Rev. B 80, 014301 (2009).
[CrossRef]

O. Matsuda, O. B. Wright, D. H. Hurley, V. E. Gusev, and K. Shimizu, “Coherent shear phonon generation and detection with picosecond laser acoustics,” Phys. Rev. B 77, 224110 (2008).
[CrossRef]

T. Fujikura, O. Matsuda, D. M. Profunser, O. B. Wright, J. Masson, and S. Ballandras, “Real-time imaging of acoustic waves on a bulk acoustic resonator,” Appl. Phys. Lett. 93, 261101 (2008).
[CrossRef]

M. Tomoda, O. Matsuda, and O. B. Wright, “Tomographic reconstruction of picosecond acoustic strain propagation,” Appl. Phys. Lett. 90, 041114 (2007).
[CrossRef]

T. Tachizaki, T. Muroya, O. Matsuda, Y. Sugawara, D. H. Hurley, and O. B. Wright, “Scanning ultrafast sagnac interferometry for imaging two-dimensional surface wave propagation,” Rev. Sci. Instrum. 77, 043713 (2006).
[CrossRef]

D. M. Profunser, O. B. Wright, and O. Matsuda, “Imaging ripples on phononic crystals reveals acoustic band structure and Bloch harmonics,” Phys. Rev. Lett. 97, 055502 (2006).
[CrossRef] [PubMed]

H. Yamazaki, O. Matsuda, and O. B. Wright, “Surface phonon imaging through the photoelastic effect,” Phys. Status Solidi C 1, 2991–2994 (2004).
[CrossRef]

Y. Sugawara, O. B. Wright, O. Matsuda, M. Takigahira, Y. Tanaka, S. Tamura, and V. E. Gusev, “Watching ripples on crystals,” Phys. Rev. Lett. 88, 185504 (2002).
[CrossRef] [PubMed]

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

Matsuda, S.

A. Miyamoto, S. Matsuda, S. Wakana, and A. Ito, “Optical observation technique for SH-type surface acoustic wave,” Electron. Commun. Jpn., Part 2: Electron. 87, 1295–1301 (2004).
[CrossRef]

Matteï, C.

C. Matteï, X. Jia, and G. Quentin, “Measurement of Rayleigh wave strains inside a transparent solid by optical interferometry,” Acta Acust. 2, 65–67 (1994).

Maznev, A. A.

A. A. Maznev, A. M. Lomonosov, P. Hess, and A. A. Kolomenskii, “Anisotropic effects in surface acoustic wave propagation from a point source in a crystal,” Eur. Phys. J. B 35, 429–439 (2003).
[CrossRef]

Millard, M.

Miyamoto, A.

A. Miyamoto, S. Matsuda, S. Wakana, and A. Ito, “Optical observation technique for SH-type surface acoustic wave,” Electron. Commun. Jpn., Part 2: Electron. 87, 1295–1301 (2004).
[CrossRef]

Muramoto, E.

D. M. Profunser, E. Muramoto, O. Matsuda, and O. B. Wright, “Dynamic visualization of surface acoustic waves on a two-dimensional phononic crystal,” Phys. Rev. B 80, 014301 (2009).
[CrossRef]

Muroya, T.

T. Tachizaki, T. Muroya, O. Matsuda, Y. Sugawara, D. H. Hurley, and O. B. Wright, “Scanning ultrafast sagnac interferometry for imaging two-dimensional surface wave propagation,” Rev. Sci. Instrum. 77, 043713 (2006).
[CrossRef]

Nam, Y. H.

Y. H. Nam and S. S. Lee, “A quantitative evaluation of elastic wave in solid by strobscopic photoelasticity,” J. Sound Vib. 259, 1199–1207 (2003).
[CrossRef]

Perrin, B.

B. Perrin, B. Bonello, J. C. Jeannet, and E. Romatet, “Interferometric detection of hypersound waves in modulated structures,” Prog. Nat. Sci. S6, S444–S448 (1996).

Profunser, D. M.

D. M. Profunser, E. Muramoto, O. Matsuda, and O. B. Wright, “Dynamic visualization of surface acoustic waves on a two-dimensional phononic crystal,” Phys. Rev. B 80, 014301 (2009).
[CrossRef]

T. Fujikura, O. Matsuda, D. M. Profunser, O. B. Wright, J. Masson, and S. Ballandras, “Real-time imaging of acoustic waves on a bulk acoustic resonator,” Appl. Phys. Lett. 93, 261101 (2008).
[CrossRef]

D. M. Profunser, O. B. Wright, and O. Matsuda, “Imaging ripples on phononic crystals reveals acoustic band structure and Bloch harmonics,” Phys. Rev. Lett. 97, 055502 (2006).
[CrossRef] [PubMed]

Quentin, G.

C. Matteï, X. Jia, and G. Quentin, “Measurement of Rayleigh wave strains inside a transparent solid by optical interferometry,” Acta Acust. 2, 65–67 (1994).

Riley, W. F.

W. F. Riley and J. W. Dally, “A photoelastic analysis of stress wave propagation in a layered model,” Geophysics 31, 881–899 (1966).
[CrossRef]

Romatet, E.

B. Perrin, B. Bonello, J. C. Jeannet, and E. Romatet, “Interferometric detection of hypersound waves in modulated structures,” Prog. Nat. Sci. S6, S444–S448 (1996).

Rossignol, C.

N. Chigarev, C. Rossignol, and B. Audoin, “Surface displacement measured by beam distortion detection technique: Application to picosecond ultrasonics,” Rev. Sci. Instrum. 77, 114901 (2006).
[CrossRef]

Salomaa, M. M.

Sathish, S.

Scales, J. A.

J. A. Scales and A. E. Malcolm, “Laser characterization of ultrasonic wave propagation in random media,” Phys. Rev. E 67, 046618 (2003).
[CrossRef]

Sharples, S. D.

M. Clark, S. D. Sharples, and M. G. Somekh, “Diffractive acoustic elements for laser ultrasonics,” J. Acoust. Soc. Am. 107, 3179–3185 (2000).
[CrossRef] [PubMed]

Shimizu, K.

O. Matsuda, O. B. Wright, D. H. Hurley, V. E. Gusev, and K. Shimizu, “Coherent shear phonon generation and detection with picosecond laser acoustics,” Phys. Rev. B 77, 224110 (2008).
[CrossRef]

Somekh, M. G.

M. Clark, S. D. Sharples, and M. G. Somekh, “Diffractive acoustic elements for laser ultrasonics,” J. Acoust. Soc. Am. 107, 3179–3185 (2000).
[CrossRef] [PubMed]

Sugawara, Y.

T. Tachizaki, T. Muroya, O. Matsuda, Y. Sugawara, D. H. Hurley, and O. B. Wright, “Scanning ultrafast sagnac interferometry for imaging two-dimensional surface wave propagation,” Rev. Sci. Instrum. 77, 043713 (2006).
[CrossRef]

Y. Sugawara, O. B. Wright, O. Matsuda, M. Takigahira, Y. Tanaka, S. Tamura, and V. E. Gusev, “Watching ripples on crystals,” Phys. Rev. Lett. 88, 185504 (2002).
[CrossRef] [PubMed]

Tachizaki, T.

T. Tachizaki, T. Muroya, O. Matsuda, Y. Sugawara, D. H. Hurley, and O. B. Wright, “Scanning ultrafast sagnac interferometry for imaging two-dimensional surface wave propagation,” Rev. Sci. Instrum. 77, 043713 (2006).
[CrossRef]

Takigahira, M.

Y. Sugawara, O. B. Wright, O. Matsuda, M. Takigahira, Y. Tanaka, S. Tamura, and V. E. Gusev, “Watching ripples on crystals,” Phys. Rev. Lett. 88, 185504 (2002).
[CrossRef] [PubMed]

Tamura, S.

Y. Sugawara, O. B. Wright, O. Matsuda, M. Takigahira, Y. Tanaka, S. Tamura, and V. E. Gusev, “Watching ripples on crystals,” Phys. Rev. Lett. 88, 185504 (2002).
[CrossRef] [PubMed]

Tanaka, Y.

Y. Sugawara, O. B. Wright, O. Matsuda, M. Takigahira, Y. Tanaka, S. Tamura, and V. E. Gusev, “Watching ripples on crystals,” Phys. Rev. Lett. 88, 185504 (2002).
[CrossRef] [PubMed]

Tauc, J.

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]

Tiemann, J. J.

M. Garfinkel, J. J. Tiemann, and W. E. Engeler, “Piezoreflectivity of the noble metals,” Phys. Rev. 148, 695–706 (1966).
[CrossRef]

Tikka, P. T.

Tomoda, M.

M. Tomoda, O. Matsuda, and O. B. Wright, “Tomographic reconstruction of picosecond acoustic strain propagation,” Appl. Phys. Lett. 90, 041114 (2007).
[CrossRef]

Tsai, Y. -H.

W.-C. Wang and Y.-H. Tsai, “Digital dynamic photoelastic and numerical stress analyses of a strip,” J. Vib. Control 12, 927–938 (2006).
[CrossRef]

Wakana, S.

A. Miyamoto, S. Matsuda, S. Wakana, and A. Ito, “Optical observation technique for SH-type surface acoustic wave,” Electron. Commun. Jpn., Part 2: Electron. 87, 1295–1301 (2004).
[CrossRef]

Wang, W. -C.

W.-C. Wang and Y.-H. Tsai, “Digital dynamic photoelastic and numerical stress analyses of a strip,” J. Vib. Control 12, 927–938 (2006).
[CrossRef]

Wright, O. B.

D. M. Profunser, E. Muramoto, O. Matsuda, and O. B. Wright, “Dynamic visualization of surface acoustic waves on a two-dimensional phononic crystal,” Phys. Rev. B 80, 014301 (2009).
[CrossRef]

O. Matsuda, O. B. Wright, D. H. Hurley, V. E. Gusev, and K. Shimizu, “Coherent shear phonon generation and detection with picosecond laser acoustics,” Phys. Rev. B 77, 224110 (2008).
[CrossRef]

T. Fujikura, O. Matsuda, D. M. Profunser, O. B. Wright, J. Masson, and S. Ballandras, “Real-time imaging of acoustic waves on a bulk acoustic resonator,” Appl. Phys. Lett. 93, 261101 (2008).
[CrossRef]

M. Tomoda, O. Matsuda, and O. B. Wright, “Tomographic reconstruction of picosecond acoustic strain propagation,” Appl. Phys. Lett. 90, 041114 (2007).
[CrossRef]

D. M. Profunser, O. B. Wright, and O. Matsuda, “Imaging ripples on phononic crystals reveals acoustic band structure and Bloch harmonics,” Phys. Rev. Lett. 97, 055502 (2006).
[CrossRef] [PubMed]

T. Tachizaki, T. Muroya, O. Matsuda, Y. Sugawara, D. H. Hurley, and O. B. Wright, “Scanning ultrafast sagnac interferometry for imaging two-dimensional surface wave propagation,” Rev. Sci. Instrum. 77, 043713 (2006).
[CrossRef]

H. Yamazaki, O. Matsuda, and O. B. Wright, “Surface phonon imaging through the photoelastic effect,” Phys. Status Solidi C 1, 2991–2994 (2004).
[CrossRef]

Y. Sugawara, O. B. Wright, O. Matsuda, M. Takigahira, Y. Tanaka, S. Tamura, and V. E. Gusev, “Watching ripples on crystals,” Phys. Rev. Lett. 88, 185504 (2002).
[CrossRef] [PubMed]

O. Matsuda and O. B. Wright, “Reflection and transmission of light in multilayers perturbed by picosecond strain pulse propagation,” J. Opt. Soc. Am. B 19, 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, 1305–1307 (1999).
[CrossRef]

Yamazaki, H.

H. Yamazaki, O. Matsuda, and O. B. Wright, “Surface phonon imaging through the photoelastic effect,” Phys. Status Solidi C 1, 2991–2994 (2004).
[CrossRef]

Acta Acust.

C. Matteï, X. Jia, and G. Quentin, “Measurement of Rayleigh wave strains inside a transparent solid by optical interferometry,” Acta Acust. 2, 65–67 (1994).

Acust. Acta Acust.

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

Appl. Phys. Lett.

M. Tomoda, O. Matsuda, and O. B. Wright, “Tomographic reconstruction of picosecond acoustic strain propagation,” Appl. Phys. Lett. 90, 041114 (2007).
[CrossRef]

J. E. Graebner, B. P. Barber, P. L. Gammel, and D. S. Greywall, “Dynamic visualization of subangstrom high-frequency surface vibrations,” Appl. Phys. Lett. 78, 159–161 (2001).
[CrossRef]

K. Kokkonen and M. Kaivola, “Scanning heterodyne laser interferometer for phase-sensitive absolute-amplitude measurements of surface vibrations,” Appl. Phys. Lett. 92, 063502 (2008).
[CrossRef]

T. Fujikura, O. Matsuda, D. M. Profunser, O. B. Wright, J. Masson, and S. Ballandras, “Real-time imaging of acoustic waves on a bulk acoustic resonator,” Appl. Phys. Lett. 93, 261101 (2008).
[CrossRef]

Electron. Commun. Jpn., Part 2: Electron.

A. Miyamoto, S. Matsuda, S. Wakana, and A. Ito, “Optical observation technique for SH-type surface acoustic wave,” Electron. Commun. Jpn., Part 2: Electron. 87, 1295–1301 (2004).
[CrossRef]

Eur. Phys. J. B

A. A. Maznev, A. M. Lomonosov, P. Hess, and A. A. Kolomenskii, “Anisotropic effects in surface acoustic wave propagation from a point source in a crystal,” Eur. Phys. J. B 35, 429–439 (2003).
[CrossRef]

Exp. Mech.

J. W. Dally, “An introduction to dynamic photoelasticity,” Exp. Mech. 20, 409–416 (1980).
[CrossRef]

Geophysics

W. F. Riley and J. W. Dally, “A photoelastic analysis of stress wave propagation in a layered model,” Geophysics 31, 881–899 (1966).
[CrossRef]

IEEE Trans. Ultrason. Ferroelectr. Freq. Control

R. Adler, A. Korpel, and P. Desmares, “An instrument for making surface waves visible,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 15, 157–161 (1968).

J. Acoust. Soc. Am.

M. Clark, S. D. Sharples, and M. G. Somekh, “Diffractive acoustic elements for laser ultrasonics,” J. Acoust. Soc. Am. 107, 3179–3185 (2000).
[CrossRef] [PubMed]

J. Appl. Phys.

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]

J. Opt. Soc. Am. B

J. Sound Vib.

Y. H. Nam and S. S. Lee, “A quantitative evaluation of elastic wave in solid by strobscopic photoelasticity,” J. Sound Vib. 259, 1199–1207 (2003).
[CrossRef]

J. Vib. Control

W.-C. Wang and Y.-H. Tsai, “Digital dynamic photoelastic and numerical stress analyses of a strip,” J. Vib. Control 12, 927–938 (2006).
[CrossRef]

Jpn. J. Appl. Phys.

R. Hayasi, Y. Masuda, S. Hashimoto, and S. Kuriyama, “Analysis of geometric effects on stress wave propagation in epoxy resins of plate-like structure by dynamic photoelasticity combined with strain gauge,” Jpn. J. Appl. Phys. 47, 4676–4681 (2008).
[CrossRef]

Opt. Lett.

Phys. Rev.

M. Garfinkel, J. J. Tiemann, and W. E. Engeler, “Piezoreflectivity of the noble metals,” Phys. Rev. 148, 695–706 (1966).
[CrossRef]

Phys. Rev. B

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]

O. Matsuda, O. B. Wright, D. H. Hurley, V. E. Gusev, and K. Shimizu, “Coherent shear phonon generation and detection with picosecond laser acoustics,” Phys. Rev. B 77, 224110 (2008).
[CrossRef]

D. M. Profunser, E. Muramoto, O. Matsuda, and O. B. Wright, “Dynamic visualization of surface acoustic waves on a two-dimensional phononic crystal,” Phys. Rev. B 80, 014301 (2009).
[CrossRef]

Phys. Rev. E

J. A. Scales and A. E. Malcolm, “Laser characterization of ultrasonic wave propagation in random media,” Phys. Rev. E 67, 046618 (2003).
[CrossRef]

Phys. Rev. Lett.

Y. Sugawara, O. B. Wright, O. Matsuda, M. Takigahira, Y. Tanaka, S. Tamura, and V. E. Gusev, “Watching ripples on crystals,” Phys. Rev. Lett. 88, 185504 (2002).
[CrossRef] [PubMed]

D. M. Profunser, O. B. Wright, and O. Matsuda, “Imaging ripples on phononic crystals reveals acoustic band structure and Bloch harmonics,” Phys. Rev. Lett. 97, 055502 (2006).
[CrossRef] [PubMed]

Phys. Status Solidi C

H. Yamazaki, O. Matsuda, and O. B. Wright, “Surface phonon imaging through the photoelastic effect,” Phys. Status Solidi C 1, 2991–2994 (2004).
[CrossRef]

Prog. Nat. Sci.

B. Perrin, B. Bonello, J. C. Jeannet, and E. Romatet, “Interferometric detection of hypersound waves in modulated structures,” Prog. Nat. Sci. S6, S444–S448 (1996).

Rev. Sci. Instrum.

N. Chigarev, C. Rossignol, and B. Audoin, “Surface displacement measured by beam distortion detection technique: Application to picosecond ultrasonics,” Rev. Sci. Instrum. 77, 114901 (2006).
[CrossRef]

T. Tachizaki, T. Muroya, O. Matsuda, Y. Sugawara, D. H. Hurley, and O. B. Wright, “Scanning ultrafast sagnac interferometry for imaging two-dimensional surface wave propagation,” Rev. Sci. Instrum. 77, 043713 (2006).
[CrossRef]

Other

M. M. Frocht, Photoelasticity (Wiley, 1957), Vols. 1 and 2.

C. P. Burger, “Photoelasticity,” in Handbook on Experimental Mechanics (Second Revised Edition), A.S.Kobayashi, ed. (VCH, 1993), Chap. 5, pp. 165–266.

This is supported by measurements made by probing from the film side of our sample, showing a response ∼10 times smaller. Other measurements also indicate a negligible photoelastic coefficient P12 at our probe wavelength. The photoelastic constant of fused silica in the visible at this wavelength is approximately P12=−1.3.

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

The SSLW ring in fact overlaps in Fig. with a weaker second RW ring, but the former ring dominates.

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Figures (3)

Fig. 1
Fig. 1

(a) Schematic diagram of the experimental setup for SAW imaging through the photoelastic effect: pol., polarizer; QWP, quarter wave plate; NPBS, non-polarizing beam splitter. (b) Polarization configurations for the measurement and the definitions of the polarizations R, L, X, and Y. The x, y, and z axes are also shown. LCP and RCP, left and right circular polarization.

Fig. 2
Fig. 2

Images of the optical intensity change Δ I at a pump-probe delay time of 11.7 ns obtained with various polarization configurations for the probe light. The suffix notation A-B denotes the light polarization that is incident (A) and the polarization that is detected (B). A may be either L or R: left or right circular polarization. B may be either X or Y: linear polarization along x or y axis. The imaged area is 140 μ m × 140 μ m . The outer ring represents the SSLW and the inner ring represents the RW.

Fig. 3
Fig. 3

Images of the difference or sum of Δ I of Fig. 2. (a), (b) Difference images as defined at the top of each figure. These are related to the image of η x y . (c), (d) Sum images. These are related to η x x , η y y , and η z z . The imaged area is 140 μ m × 140 μ m and the pump-probe delay time is 11.7 ns. The outer ring represents the SSLW and the inner ring represents the RW.

Equations (45)

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( 2 grad   div + k 2 ε ( r , t ) ) E ( r , t ) = 0 ,
{ L ( z ) + k 2 ε ( z ) } E ( z ) = 0 ,
L ( z ) ( 2 / z 2 0 0 0 2 / z 2 0 0 0 0 ) ,
ε ( z ) ε h ( z ) + ε i h ( z ) ,
ε h ( z ) { ε ( 0 ) I for   z < 0 ε ( 1 ) I for   z > 0 , }
{ L ( z ) + k 2 ε h ( z ) } E 0 ( z ) = 0 ,
{ L ( z ) + k 2 ε h ( z ) } G ( z , z ) = δ ( z z ) I .
E = E 0 ( z ) + k 2 G ( z , z ) ε i h ( z ) E ( z ) d z .
E E 0 ( z ) + k 2 G ( z , z ) ε i h ( z ) E 0 ( z ) d z .
E 0 = { ( e i k ( 0 ) z + r 0 e i k ( 0 ) z ) ( E x 0 E y 0 0 ) for   z < 0 t 0 e i k ( 1 ) z ( E x 0 E y 0 0 ) for   z > 0 , }
r 0 k ( 0 ) k ( 1 ) k ( 0 ) + k ( 1 ) ,     t 0 2 k ( 0 ) k ( 0 ) + k ( 1 ) ,
G ( z , z ) = g ( z , z ) ( 1 0 0 0 1 0 0 0 0 ) ,
g ( z , z ) i e i k ( 0 ) z 2 k ( 0 ) × { e i k ( 0 ) z + r 0 e i k ( 0 ) z for   z < 0 t 0 e i k ( 1 ) z for   z > 0. }
E ( z ) = ( e i k ( 0 ) z + r 0 e i k ( 0 ) z ) ( E x 0 E y 0 0 ) + e i k ( 0 ) z ( E x 0 Z 1 + E y 0 Z 6 E x 0 Z 6 + E y 0 Z 2 0 ) + 2 i k ( 0 ) u z ( 0 ) r 0 e i k ( 0 ) z ( E x 0 E y 0 0 ) ,
Z I i k 2 2 k ( 0 ) { z 0 ( e i k ( 0 ) z + r 0 e i k ( 0 ) z ) 2 ε pe , I ( z ) d z + 0 ( t 0 e i k ( 1 ) z ) 2 ε pe , I ( z ) d z } ,
( Δ R R ) X - U = 2   Re ( Z 1 r 0 ) .
( Δ R R ) Y - U = 2   Re ( Z 2 r 0 ) .
( Δ R R ) 45 - X = 2   Re ( Z 1 + Z 6 r 0 ) .
( Δ R R ) 45 - Y = 2   Re ( Z 6 + Z 2 r 0 ) .
( Δ R R ) 45 ¯ - X = 2   Re ( Z 1 Z 6 r 0 ) .
( Δ R R ) 45 ¯ - Y = 2   Re ( Z 2 Z 6 r 0 ) .
( Δ R R ) L - X = 2   Re ( Z 1 + i Z 6 r 0 ) .
( Δ R R ) L - Y = 2   Re ( Z 2 i Z 6 r 0 ) .
( Δ R R ) R - X = 2   Re ( Z 1 i Z 6 r 0 ) .
( Δ R R ) R - Y = 2   Re ( Z 2 + i Z 6 r 0 ) .
4   Re ( Z 6 r 0 ) = ( Δ R R ) 45 - X ( Δ R R ) 45 ¯ - X = ( Δ R R ) 45 - Y ( Δ R R ) 45 ¯ - Y ,
4   Im ( Z 6 r 0 ) = ( Δ R R ) R - X ( Δ R R ) L - X = ( Δ R R ) L - Y ( Δ R R ) R - Y .
4   Re ( Z 1 + Z 2 r 0 ) = ( Δ R R ) 45 - X + ( Δ R R ) 45 ¯ - Y = ( Δ R R ) 45 - Y + ( Δ R R ) 45 ¯ - X
= ( Δ R R ) R - X + ( Δ R R ) R - Y = ( Δ R R ) L - X + ( Δ R R ) L - Y .
ε pe , 1 = P 11 η 1 + P 12 ( η 2 + η 3 ) ,
ε pe , 2 = P 11 η 2 + P 12 ( η 3 + η 1 ) ,
ε pe , 3 = P 11 η 3 + P 12 ( η 1 + η 2 ) ,
ε pe , I = P 44 η I     ( I = 4 , 5 , 6 ) ,
η 1 = u r r cos 2 θ + u r r sin 2 θ ,
η 2 = u r r sin 2 θ + u r r cos 2 θ ,
η 3 = u z z ,
η 4 = ( u r z + u z r ) sin   θ ,
η 5 = ( u r z + u z r ) cos   θ ,
η 6 = 2 ( u r r u r r ) cos   θ   sin   θ .
ε pe , 1 = ( u r r P 11 + u r r P 12 ) cos 2 θ + ( u r r P 12 + u r r P 11 ) sin 2 θ + u z z P 12 ,
ε pe , 2 = ( u r r P 11 + u r r P 12 ) sin 2 θ + ( u r r P 12 + u r r P 11 ) cos 2 θ + u z z P 12 ,
ε pe , 6 = 2 ( u r r u r r ) cos   θ   sin   θ .
ε pe , 1 + ε pe , 2 = ( u r r + u r r ) ( P 11 + P 12 ) + 2 u z z P 12 .
u r r u r r = f ( r ) ,
u r ( r ) = r f ( r ) r d r + C r ,

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