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Laser picosecond acoustics in a two-layer structure with oblique probe light incidence
Ultrasonics 42 (2004) 653–656 www.elsevier.com/locate/ultras
Laser picosecond acoustics in a two-layer structure with oblique probe light incidence O. Matsuda *, O.B. Wright Department of Applied Physics, Faculty of Engineering, Hokkaido University, Sapporo 060-8628, Japan
Abstract A theory for the analysis of experiments involving laser picosecond acoustics with obliquely incident probe light in a two-layer structure is outlined. The reflectance and phase changes of the reflected light are calculated with a theory that takes into account the effects of multiple optical reflections. The sample consists of a single partially transparent layer on a substrate, both with arbitrary optical constants. We discuss the conditions in which one may discriminate between components of the optical modulation of a probe beam arising from the photoelastic effect and from the displacement of the sample interfaces induced by the acoustic strain. 2004 Elsevier B.V. All rights reserved. Keywords: Laser ultrasonics; Multilayer; Interferometer; Oblique probe incidence
1. Introduction Laser picosecond acoustics has been increasingly used to evaluate thin film properties. Ultrashort light pulses absorbed in a sample generate picosecond acoustic pulses that propagate in the sample according to its structure and elastic properties. Delayed ultrashort probe light pulses are used to detect the acoustic pulse propagation through the transient modulation in reflectance or transmittance caused by the photoelastic effect [1] or by surface and interface displacements in the sample [2,3]. In addition to the possibility of extracting structural information such as film thicknesses from such measurements, the technique also allows the study of various properties connected with the interactions between photons, excited carriers and phonons. The ultrafast diffusion of photoexcited carriers in metals [4] and semiconductors [5], for example, have been investigated by this technique. In such studies a precise knowledge of the generated acoustic strain pulse shape is desirable. This pulse shape is directly related to the time-dependent surface displacement, and can be accessed through the optical phase variation as measured by a probe beam
*
Corresponding author. Tel./fax: +81-11-7067190. E-mail address: [email protected] (O. Matsuda).
0041-624X/$ - see front matter 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.ultras.2004.01.052
using interferometric detection [3,6]. The data analysis used for the extraction of the acoustic strain pulse shape is relatively easy for materials exhibiting weak photoelastic effects or for the case in which the acoustic pulse length is much longer than the optical penetration depth of the probe light. However, it is in general not straightforward to derive the acoustic strain pulse shape since the surface displacement contribution to the optical phase change is superimposed on the photoelastic contribution, the latter depending in a complex way on the spatiotemporal modulation of the optical constants. Although the use of normally incident probe light results in a relatively simple theoretical analysis (even in multilayer samples [7]), we have previously shown theoretically that the use of obliquely incident light combined with interferometric detection allows the discrimination of the surface displacement and photoelastic contributions for the case of opaque samples [8]. In this paper we extend the analysis for laser picosecond acoustics with oblique probe light incidence to a more complex sample geometry: an isotropic two-layer system. The complex transient reflectance change caused by the inhomogeneous modulation of the optical constants and by surface and interface displacements induced by the acoustic strain pulse is treated in a rigorous way. The theory can be applied to substrates coated with a thin transparent film, for example. We demonstrate that it is possible to discriminate between the surface
654
O. Matsuda, O.B. Wright / Ultrasonics 42 (2004) 653–656
displacement and photoelastic contributions under certain conditions.
the probe light with angular frequency x can be expressed as Eðz; tÞ expfiðkx x xtÞg, and the quantity Eðz; tÞ should satisfy ½Lðkx Þ þ k 2 feh ðzÞ þ eih ðz; tÞgEðz; tÞ ¼ 0;
2. Theory Consider a sample consisting of a layer of thickness d on a substrate that occupies the z > 0 region (Fig. 1). The z < 0 region is considered to be a vacuum. Each layer is macroscopically homogeneous, allowing the permittivity tensor to be described by a staircase-like function that we define as eh ðzÞ. The acoustic pulses in the sample are generated by an optical pump light pulse that is absorbed in it. In laser picosecond acoustics the diameter of the irradiated region is usually of the order of 10 lm or greater, and is much larger than the typical depth of interest in the sample K 1 lm. The generated acoustic pulse thus propagates along the z direction, and the strain distribution depends only on z and time t. The acoustic strain modulates the permittivity tensor spatiotemporally through the photoelastic effect and also creates time-dependent surface and interface displacements. These transient perturbations of the permittivity tensor are described as eih ðz; tÞ, and result in a change in the complex optical reflectance. Our object here is to calculate the reflectance change for arbitrary probe light incident angle and polarization. To this end we solve the appropriate electromagnetic wave equation with an inhomogeneously perturbed permittivity tensor eh ðzÞ þ eih ðz; tÞ. We take the x-axis parallel to the plane of incidence as in Fig. 1. Spatial invariance for translations in the x direction ensures that the x component of the wave vector (kx ) is conserved. Because of the relatively low frequency of the acoustic perturbation (<1 THz) compared with the frequency of the probe light, we can regard the problem as a quasistatic one. Under these assumptions the electric field of
reflected light
incident light vacuum
film
substrate
ð1Þ
where k ¼ x=c is the wave vector in vacuum and the operator matrix Lðkx Þ is given by 1 0 o2 o 0 ik x B oz2 oz C C B 2 C B o 2 : Lðkx Þ B 0 kx 0 C 2 C B oz A @ o 0 kx2 ikx oz This equation can be solved by the use of a 3 · 3 Green’s function matrix [9] that satisfies ½Lðkx Þ þ k 2 eh ðzÞGðz; z0 Þ ¼ dðz z0 ÞI;
ð2Þ
where I is the identity matrix, and by the use of the function E 0 ðzÞ that is the solution of the homogeneous equation ½Lðkx Þ þ k 2 eh ðzÞE 0 ðzÞ ¼ 0:
ð3Þ
The resulting solution can be expressed as Eðz; tÞ ¼ E 0 ðzÞ þ ’ E 0 ðzÞ þ
Z
1
k 2 Gðz; z0 Þeih ðz0 ; tÞEðz0 Þ dz0
1 Z 1
k 2 Gðz; z0 Þeih ðz0 ; tÞE 0 ðz0 Þ dz0 :
ð4Þ
1
The above procedure can be applied to any arbitrary multilayer. Here we concentrate on a two-layer system subject to an inhomogeneously perturbed permittivity tensor and to strain-induced surface and interface displacements. The equilibrium permittivity tensor eh in the absence of any perturbation is given by 8 < e0 ¼ I ðz < 0Þ; ð0 6 z < dÞ; eh ðzÞ ¼ e1 ð5Þ : e2 ðd 6 zÞ: The displacement of the film surface, uð0; tÞ, and of the interface between the film and substrate, uðd; tÞ, can equally well be described in terms of a change in the permittivity tensor, ed , by [7] 8 e2 e1 ðd þ uðd; tÞ < z < dÞ; > > > > > < e1 e2 ðd < z < d þ uðd; tÞÞ; ð6Þ ed ðz; tÞ ¼ e1 I ðuð0; tÞ < z < 0Þ; > > > I e1 ð0 < z < uð0; tÞÞ; > > : 0 elsewhere: The overall perturbation is given by eih ðz; tÞ ¼ epe ðz; tÞ þ ed ðz; tÞ;
Fig. 1. The incident light comes from z < 0. All incident, reflected, and transmitted light has common kx .
ð7Þ
where epe is the change caused by the spatiotemporal strain distribution.
O. Matsuda, O.B. Wright / Ultrasonics 42 (2004) 653–656
For small epe ðz; tÞ and displacement uðz; tÞ, Eq. (4) is given by Z 1 k 2 Gðz; z0 Þepe ðz0 ; tÞE 0 ðz0 Þ dz0 Eðz; tÞ ’ E 0 ðzÞ þ 1
þ k 2 uð0; tÞGðz; þ0ÞðI e1 ÞE 0 ð0Þ þ k 2 uðd; tÞGðz; d þ 0Þðe1 e2 ÞE 0 ðd 0Þ:
ð8Þ
The limiting values from the right and left should be distinguished because the functions may have a discontinuity at the surface or interface. The appropriate Green’s function Gðz; z0 Þ is calculated from the unperturbed solution E 0 . Here we consider the case in which both film and substrate are isotropic. The permittivity tensor of Eq. (5) is then reduced to a scalar quantity. The two independent polarizations of E 0 , s and p, are described by ðlÞ
ðlÞ ðl;þÞ kj fj
E 0 ðzÞ ¼ aj ej
e
ðlÞ ðl;Þ kj fj
þ bj e j
e
;
ð9Þ
where l ¼ s; p denotes the polarization and j denotes the particular layer: vacuum (j ¼ 0, z < 0), film (j ¼ 1, 0 6 z < d), and substrate (j ¼ 2, d 6 z). The quantities fj denote z except for the case f2 ¼ z d. The quantity kj is the z component of the wave vector in the jth layer: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð10Þ kj ¼ ej k 2 kx2 : The polarization vectors are given by ðs;þÞ
ðs;Þ
ej
¼ ej
ðp;þÞ ej
¼
ðp;Þ
¼
ej
¼ ð0; 1; 0Þ;
! kj kx pffiffiffiffi ; 0; pffiffiffiffi ; ej k ej k ! kj kx pffiffiffiffi ; 0; pffiffiffiffi : ej k ej k
ð11Þ
The electric field amplitudes are given by ðsÞ
a0 ¼ ðk0 k1 Þðk1 k2 Þe2ik1 d þ ðk0 þ k1 Þðk1 þ k2 Þ; ðsÞ
b0 ¼ ðk0 þ k1 Þðk1 k2 Þe2ik1 d þ ðk0 k1 Þðk1 þ k2 Þ; ðsÞ a1 ðsÞ b1 ðsÞ a2 ðpÞ a0
¼ 2k0 ðk1 k2 Þe2ik1 d ;
ðpÞ
ðpÞ a2
and for z < 0, d < z0 by ik2 0 ðpÞ Gxx ðz; z0 Þ ¼ pffiffiffiffi ðpÞ a2 eik2 ðz dÞ eik0 z ; 2 2 e 2 k a0 kx Gxx ðz; z0 Þ; k2 i 0 ðsÞ Gyy ðz; z0 Þ ¼ a eik2 ðz dÞ eik0 z ðsÞ 2 2k0 a0 Gxz ðz; z0 Þ ¼
ð14Þ
kx Gxx ðz; z0 Þ; k0 kx Gzz ðz; z0 Þ ¼ Gxz ðz; z0 Þ: k0
Gzx ðz; z0 Þ ¼
The above formulation is enough to calculate the reflectance change caused by any perturbation given by epe ðz; tÞ. Here we concentrate on the perturbation caused by a longitudinal acoustic wave. The perturbation epe in an isotropic material depends on the strain gzz ðz; tÞ through the photoelastic effect: 0 1 ðjÞ 0 0 P12 B C ðjÞ epe ðz; tÞ ¼ @ 0 P12 ð15Þ 0 Agzz ðz; tÞ; ðjÞ 0 0 P11 ðjÞ
¼ 4k0 k1 eik1 d ; ¼ ðe1 k0 k1 Þðe2 k1 e1 k2 Þe2ik1 d
ð12Þ
ðpÞ
b1
kx Gxx ðz; z0 Þ; k0 kx Gzz ðz; z0 Þ ¼ Gxz ðz; z0 Þ; k0 Gzx ðz; z0 Þ ¼
ðjÞ
b0 ¼ ðe1 k0 þ k1 Þðe2 k1 e1 k2 Þe2ik1 d ðpÞ
ð13Þ
where P12 and P11 are components of the photoelastic tensor, depending on the layer number j. The complex reflectance change for s polarized light is given by
¼ 2k0 ðk1 þ k2 Þ;
þ ðe1 k0 þ k1 Þðe2 k1 þ e1 k2 Þ;
a1
ik1 0 0 ðpÞ ðpÞ Gxx ðz; z0 Þ ¼ pffiffiffiffi ðpÞ a1 eik1 z þ b1 eik1 z eik0 z ; 2 e 1 k 2 a0 ikx 0 0 ðpÞ ðpÞ Gxz ðz; z0 Þ ¼ pffiffiffiffi ðpÞ a1 eik1 z b1 eik1 z eik0 z ; 2 e 1 k 2 a0 i ðsÞ ik1 z0 ðsÞ ik1 z0 þ b eik0 z ; Gyy ðz; z0 Þ ¼ a e e 1 1 ðsÞ 2k0 a0
655
ðe1 k0 k1 Þðe2 k1 þ e1 k2 Þ; pffiffiffiffi ¼ 2 e1 k0 ðe2 k1 þ e1 k2 Þ; pffiffiffiffi ¼ 2 e1 k0 ðe2 k1 e1 k2 Þe2ik1 d ; pffiffiffiffi ¼ 4e1 e2 k0 k1 eik1 d :
The non-zero components of the Green’s function are given [10], for z < 0; 0 < z0 < d, by
drðsÞ ik 2 ¼ ðsÞ ðsÞ ðsÞ r 2k0 a0 b0 Z ð1Þ P12
d
0 0 2 ðsÞ ðsÞ gðz0 ; tÞ a1 eik1 z þ b1 eik1 z dz0
0
ð2Þ
þ P12
Z
1
0 2 ðsÞ gðz0 þ d; tÞ a2 eik2 z dz0
0
2 ðsÞ ðsÞ þ uð0; tÞð1 e1 Þ a1 þ b1 2 ðsÞ þ uðd; tÞðe1 e2 Þ a2 ;
ð16Þ
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O. Matsuda, O.B. Wright / Ultrasonics 42 (2004) 653–656
whereas that for p polarized light is given by drðpÞ i ¼ ðpÞ ðpÞ rðpÞ 2k0 a0 b0 ( ð1Þ Z d k12 P12 0 0 2 ðpÞ ðpÞ gðz0 ; tÞ a1 eik1 z þ b1 eik1 z dz0 e1 0 Z ð1Þ d k2P 0 0 2 ðpÞ ðpÞ x 11 gðz0 ; tÞ a1 eik1 z b1 eik1 z dz0 e1 0 ð2Þ Z 1 2 ð2Þ k2 P12 kx2 P11 0 2 ðpÞ þ gðz0 þ d; tÞ a2 eik2 z dz0 e2 02 2 k ðpÞ ðpÞ þ uð0; tÞð1 e1 Þ 1 a1 þ b1 e1 2 ðpÞ ðpÞ kx2 a1 b1 ) 2 k2 kx2 ðpÞ 2 þ uðd; tÞðe1 e2 Þ ; ð17Þ a2 e2 e1 ðlÞ
ðlÞ
where rðlÞ ¼ b0 =a0 is the reflectance for the unperturbed sample. The right hand sides of Eqs. (16) and (17) can be regarded as linear combinations of six unknown timedependent functions: Z d a1 ðtÞ ¼ gðz0 ; tÞ cos 2k1 z0 dz0 ; 0 Z d gðz0 ; tÞ sin 2k1 z0 dz0 ; b1 ðtÞ ¼ 0
c1 ðtÞ ¼ uð0; tÞ; Z 1 a2 ðtÞ ¼ gðz0 þ d; tÞ cos 2k2 z0 dz0 ; 0 Z 1 b2 ðtÞ ¼ gðz0 þ d; tÞ sin 2k2 z0 dz0 ;
ð18Þ
reflectance data. One example of this occurs when the film is transparent, in which case the acoustic pulse is generated only in the opaque substrate. In addition the spatial extent of the acoustic pulse should be much smaller than d. Then we expect a2 and b2 to be zero and c2 to be constant during the time in which the acoustic pulse is travelling in the film layer. In this special case one may directly deduce uð0; tÞ and thus the acoustic strain pulse shape from the experimental results.
3. Summary In conclusion, we have presented an analytical treatment for the problem of calculating the reflectance variation for light incident at an oblique angle on an isotropic two-layer sample whose permittivity tensor is inhomogeneously perturbed, and whose surface and interface are displaced by a propagating acoustic strain wave. It is shown that the displacement and photoelastic contributions to the relative reflectance changes can be separated under certain conditions provided that experimental results are obtained using both s and p polarized probe light.
Acknowledgements This work was partially supported by a Grant-in-Aid for Scientific Research from the Ministry of Education, Science, Sports, and Culture (Japan) and by the Izumi Science and Technology Foundation.
References
0
c2 ðtÞ ¼ uðd; tÞ: The interferometric transient reflectance measurement provides four independent linear combinations, namely the real and imaginary parts of the reflectance change for s and p polarized probe light. The coefficients appearing in these linear combinations consist of photoelastic constants, permittivities, wave vectors, incident angle, etc. and are all obtainable from independent measurements. If we know at least two of the six unknown functions by some other means, it is possible to determine the remaining four from the measured transient
[1] C. Thomsen, H.T. Grahn, H.J. Maris, J. Tauc, Phys. Rev. B 34 (1986) 4129. [2] O.B. Wright, K. Kawashima, Phys. Rev. Lett. 69 (1992) 1668. [3] B. Perrin, B. Bonello, J.C. Jeannet, E. Romatet, Prog. Natur. Sci. S6 (1996) S444. [4] O.B. Wright, Phys. Rev. B 49 (1994) 9985. [5] O.B. Wright, B. Perrin, O. Matsuda, V.E. Gusev, Phys. Rev. B 64 (2001) R081202. [6] D.H. Hurley, O.B. Wright, Opt. Lett. 24 (1999) 1305. [7] O. Matsuda, O.B. Wright, J. Opt. Soc. Am. B 19 (2002) 3028. [8] O. Matsuda, O.B. Wright, Rev. Sci. Instrum. 74 (2003) 895. [9] A.A. Maradudin, D.L. Mills, Phys. Rev. B 11 (1975) 1392. [10] O. Matsuda, unpublished.
Laser picosecond acoustics in a two-layer structure with oblique probe light incidence O. Matsuda *, O.B. Wright Department of Applied Physics, Faculty of Engineering, Hokkaido University, Sapporo 060-8628, Japan
Abstract A theory for the analysis of experiments involving laser picosecond acoustics with obliquely incident probe light in a two-layer structure is outlined. The reflectance and phase changes of the reflected light are calculated with a theory that takes into account the effects of multiple optical reflections. The sample consists of a single partially transparent layer on a substrate, both with arbitrary optical constants. We discuss the conditions in which one may discriminate between components of the optical modulation of a probe beam arising from the photoelastic effect and from the displacement of the sample interfaces induced by the acoustic strain. 2004 Elsevier B.V. All rights reserved. Keywords: Laser ultrasonics; Multilayer; Interferometer; Oblique probe incidence
1. Introduction Laser picosecond acoustics has been increasingly used to evaluate thin film properties. Ultrashort light pulses absorbed in a sample generate picosecond acoustic pulses that propagate in the sample according to its structure and elastic properties. Delayed ultrashort probe light pulses are used to detect the acoustic pulse propagation through the transient modulation in reflectance or transmittance caused by the photoelastic effect [1] or by surface and interface displacements in the sample [2,3]. In addition to the possibility of extracting structural information such as film thicknesses from such measurements, the technique also allows the study of various properties connected with the interactions between photons, excited carriers and phonons. The ultrafast diffusion of photoexcited carriers in metals [4] and semiconductors [5], for example, have been investigated by this technique. In such studies a precise knowledge of the generated acoustic strain pulse shape is desirable. This pulse shape is directly related to the time-dependent surface displacement, and can be accessed through the optical phase variation as measured by a probe beam
*
Corresponding author. Tel./fax: +81-11-7067190. E-mail address: [email protected] (O. Matsuda).
0041-624X/$ - see front matter 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.ultras.2004.01.052
using interferometric detection [3,6]. The data analysis used for the extraction of the acoustic strain pulse shape is relatively easy for materials exhibiting weak photoelastic effects or for the case in which the acoustic pulse length is much longer than the optical penetration depth of the probe light. However, it is in general not straightforward to derive the acoustic strain pulse shape since the surface displacement contribution to the optical phase change is superimposed on the photoelastic contribution, the latter depending in a complex way on the spatiotemporal modulation of the optical constants. Although the use of normally incident probe light results in a relatively simple theoretical analysis (even in multilayer samples [7]), we have previously shown theoretically that the use of obliquely incident light combined with interferometric detection allows the discrimination of the surface displacement and photoelastic contributions for the case of opaque samples [8]. In this paper we extend the analysis for laser picosecond acoustics with oblique probe light incidence to a more complex sample geometry: an isotropic two-layer system. The complex transient reflectance change caused by the inhomogeneous modulation of the optical constants and by surface and interface displacements induced by the acoustic strain pulse is treated in a rigorous way. The theory can be applied to substrates coated with a thin transparent film, for example. We demonstrate that it is possible to discriminate between the surface
654
O. Matsuda, O.B. Wright / Ultrasonics 42 (2004) 653–656
displacement and photoelastic contributions under certain conditions.
the probe light with angular frequency x can be expressed as Eðz; tÞ expfiðkx x xtÞg, and the quantity Eðz; tÞ should satisfy ½Lðkx Þ þ k 2 feh ðzÞ þ eih ðz; tÞgEðz; tÞ ¼ 0;
2. Theory Consider a sample consisting of a layer of thickness d on a substrate that occupies the z > 0 region (Fig. 1). The z < 0 region is considered to be a vacuum. Each layer is macroscopically homogeneous, allowing the permittivity tensor to be described by a staircase-like function that we define as eh ðzÞ. The acoustic pulses in the sample are generated by an optical pump light pulse that is absorbed in it. In laser picosecond acoustics the diameter of the irradiated region is usually of the order of 10 lm or greater, and is much larger than the typical depth of interest in the sample K 1 lm. The generated acoustic pulse thus propagates along the z direction, and the strain distribution depends only on z and time t. The acoustic strain modulates the permittivity tensor spatiotemporally through the photoelastic effect and also creates time-dependent surface and interface displacements. These transient perturbations of the permittivity tensor are described as eih ðz; tÞ, and result in a change in the complex optical reflectance. Our object here is to calculate the reflectance change for arbitrary probe light incident angle and polarization. To this end we solve the appropriate electromagnetic wave equation with an inhomogeneously perturbed permittivity tensor eh ðzÞ þ eih ðz; tÞ. We take the x-axis parallel to the plane of incidence as in Fig. 1. Spatial invariance for translations in the x direction ensures that the x component of the wave vector (kx ) is conserved. Because of the relatively low frequency of the acoustic perturbation (<1 THz) compared with the frequency of the probe light, we can regard the problem as a quasistatic one. Under these assumptions the electric field of
reflected light
incident light vacuum
film
substrate
ð1Þ
where k ¼ x=c is the wave vector in vacuum and the operator matrix Lðkx Þ is given by 1 0 o2 o 0 ik x B oz2 oz C C B 2 C B o 2 : Lðkx Þ B 0 kx 0 C 2 C B oz A @ o 0 kx2 ikx oz This equation can be solved by the use of a 3 · 3 Green’s function matrix [9] that satisfies ½Lðkx Þ þ k 2 eh ðzÞGðz; z0 Þ ¼ dðz z0 ÞI;
ð2Þ
where I is the identity matrix, and by the use of the function E 0 ðzÞ that is the solution of the homogeneous equation ½Lðkx Þ þ k 2 eh ðzÞE 0 ðzÞ ¼ 0:
ð3Þ
The resulting solution can be expressed as Eðz; tÞ ¼ E 0 ðzÞ þ ’ E 0 ðzÞ þ
Z
1
k 2 Gðz; z0 Þeih ðz0 ; tÞEðz0 Þ dz0
1 Z 1
k 2 Gðz; z0 Þeih ðz0 ; tÞE 0 ðz0 Þ dz0 :
ð4Þ
1
The above procedure can be applied to any arbitrary multilayer. Here we concentrate on a two-layer system subject to an inhomogeneously perturbed permittivity tensor and to strain-induced surface and interface displacements. The equilibrium permittivity tensor eh in the absence of any perturbation is given by 8 < e0 ¼ I ðz < 0Þ; ð0 6 z < dÞ; eh ðzÞ ¼ e1 ð5Þ : e2 ðd 6 zÞ: The displacement of the film surface, uð0; tÞ, and of the interface between the film and substrate, uðd; tÞ, can equally well be described in terms of a change in the permittivity tensor, ed , by [7] 8 e2 e1 ðd þ uðd; tÞ < z < dÞ; > > > > > < e1 e2 ðd < z < d þ uðd; tÞÞ; ð6Þ ed ðz; tÞ ¼ e1 I ðuð0; tÞ < z < 0Þ; > > > I e1 ð0 < z < uð0; tÞÞ; > > : 0 elsewhere: The overall perturbation is given by eih ðz; tÞ ¼ epe ðz; tÞ þ ed ðz; tÞ;
Fig. 1. The incident light comes from z < 0. All incident, reflected, and transmitted light has common kx .
ð7Þ
where epe is the change caused by the spatiotemporal strain distribution.
O. Matsuda, O.B. Wright / Ultrasonics 42 (2004) 653–656
For small epe ðz; tÞ and displacement uðz; tÞ, Eq. (4) is given by Z 1 k 2 Gðz; z0 Þepe ðz0 ; tÞE 0 ðz0 Þ dz0 Eðz; tÞ ’ E 0 ðzÞ þ 1
þ k 2 uð0; tÞGðz; þ0ÞðI e1 ÞE 0 ð0Þ þ k 2 uðd; tÞGðz; d þ 0Þðe1 e2 ÞE 0 ðd 0Þ:
ð8Þ
The limiting values from the right and left should be distinguished because the functions may have a discontinuity at the surface or interface. The appropriate Green’s function Gðz; z0 Þ is calculated from the unperturbed solution E 0 . Here we consider the case in which both film and substrate are isotropic. The permittivity tensor of Eq. (5) is then reduced to a scalar quantity. The two independent polarizations of E 0 , s and p, are described by ðlÞ
ðlÞ ðl;þÞ kj fj
E 0 ðzÞ ¼ aj ej
e
ðlÞ ðl;Þ kj fj
þ bj e j
e
;
ð9Þ
where l ¼ s; p denotes the polarization and j denotes the particular layer: vacuum (j ¼ 0, z < 0), film (j ¼ 1, 0 6 z < d), and substrate (j ¼ 2, d 6 z). The quantities fj denote z except for the case f2 ¼ z d. The quantity kj is the z component of the wave vector in the jth layer: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð10Þ kj ¼ ej k 2 kx2 : The polarization vectors are given by ðs;þÞ
ðs;Þ
ej
¼ ej
ðp;þÞ ej
¼
ðp;Þ
¼
ej
¼ ð0; 1; 0Þ;
! kj kx pffiffiffiffi ; 0; pffiffiffiffi ; ej k ej k ! kj kx pffiffiffiffi ; 0; pffiffiffiffi : ej k ej k
ð11Þ
The electric field amplitudes are given by ðsÞ
a0 ¼ ðk0 k1 Þðk1 k2 Þe2ik1 d þ ðk0 þ k1 Þðk1 þ k2 Þ; ðsÞ
b0 ¼ ðk0 þ k1 Þðk1 k2 Þe2ik1 d þ ðk0 k1 Þðk1 þ k2 Þ; ðsÞ a1 ðsÞ b1 ðsÞ a2 ðpÞ a0
¼ 2k0 ðk1 k2 Þe2ik1 d ;
ðpÞ
ðpÞ a2
and for z < 0, d < z0 by ik2 0 ðpÞ Gxx ðz; z0 Þ ¼ pffiffiffiffi ðpÞ a2 eik2 ðz dÞ eik0 z ; 2 2 e 2 k a0 kx Gxx ðz; z0 Þ; k2 i 0 ðsÞ Gyy ðz; z0 Þ ¼ a eik2 ðz dÞ eik0 z ðsÞ 2 2k0 a0 Gxz ðz; z0 Þ ¼
ð14Þ
kx Gxx ðz; z0 Þ; k0 kx Gzz ðz; z0 Þ ¼ Gxz ðz; z0 Þ: k0
Gzx ðz; z0 Þ ¼
The above formulation is enough to calculate the reflectance change caused by any perturbation given by epe ðz; tÞ. Here we concentrate on the perturbation caused by a longitudinal acoustic wave. The perturbation epe in an isotropic material depends on the strain gzz ðz; tÞ through the photoelastic effect: 0 1 ðjÞ 0 0 P12 B C ðjÞ epe ðz; tÞ ¼ @ 0 P12 ð15Þ 0 Agzz ðz; tÞ; ðjÞ 0 0 P11 ðjÞ
¼ 4k0 k1 eik1 d ; ¼ ðe1 k0 k1 Þðe2 k1 e1 k2 Þe2ik1 d
ð12Þ
ðpÞ
b1
kx Gxx ðz; z0 Þ; k0 kx Gzz ðz; z0 Þ ¼ Gxz ðz; z0 Þ; k0 Gzx ðz; z0 Þ ¼
ðjÞ
b0 ¼ ðe1 k0 þ k1 Þðe2 k1 e1 k2 Þe2ik1 d ðpÞ
ð13Þ
where P12 and P11 are components of the photoelastic tensor, depending on the layer number j. The complex reflectance change for s polarized light is given by
¼ 2k0 ðk1 þ k2 Þ;
þ ðe1 k0 þ k1 Þðe2 k1 þ e1 k2 Þ;
a1
ik1 0 0 ðpÞ ðpÞ Gxx ðz; z0 Þ ¼ pffiffiffiffi ðpÞ a1 eik1 z þ b1 eik1 z eik0 z ; 2 e 1 k 2 a0 ikx 0 0 ðpÞ ðpÞ Gxz ðz; z0 Þ ¼ pffiffiffiffi ðpÞ a1 eik1 z b1 eik1 z eik0 z ; 2 e 1 k 2 a0 i ðsÞ ik1 z0 ðsÞ ik1 z0 þ b eik0 z ; Gyy ðz; z0 Þ ¼ a e e 1 1 ðsÞ 2k0 a0
655
ðe1 k0 k1 Þðe2 k1 þ e1 k2 Þ; pffiffiffiffi ¼ 2 e1 k0 ðe2 k1 þ e1 k2 Þ; pffiffiffiffi ¼ 2 e1 k0 ðe2 k1 e1 k2 Þe2ik1 d ; pffiffiffiffi ¼ 4e1 e2 k0 k1 eik1 d :
The non-zero components of the Green’s function are given [10], for z < 0; 0 < z0 < d, by
drðsÞ ik 2 ¼ ðsÞ ðsÞ ðsÞ r 2k0 a0 b0 Z ð1Þ P12
d
0 0 2 ðsÞ ðsÞ gðz0 ; tÞ a1 eik1 z þ b1 eik1 z dz0
0
ð2Þ
þ P12
Z
1
0 2 ðsÞ gðz0 þ d; tÞ a2 eik2 z dz0
0
2 ðsÞ ðsÞ þ uð0; tÞð1 e1 Þ a1 þ b1 2 ðsÞ þ uðd; tÞðe1 e2 Þ a2 ;
ð16Þ
656
O. Matsuda, O.B. Wright / Ultrasonics 42 (2004) 653–656
whereas that for p polarized light is given by drðpÞ i ¼ ðpÞ ðpÞ rðpÞ 2k0 a0 b0 ( ð1Þ Z d k12 P12 0 0 2 ðpÞ ðpÞ gðz0 ; tÞ a1 eik1 z þ b1 eik1 z dz0 e1 0 Z ð1Þ d k2P 0 0 2 ðpÞ ðpÞ x 11 gðz0 ; tÞ a1 eik1 z b1 eik1 z dz0 e1 0 ð2Þ Z 1 2 ð2Þ k2 P12 kx2 P11 0 2 ðpÞ þ gðz0 þ d; tÞ a2 eik2 z dz0 e2 02 2 k ðpÞ ðpÞ þ uð0; tÞð1 e1 Þ 1 a1 þ b1 e1 2 ðpÞ ðpÞ kx2 a1 b1 ) 2 k2 kx2 ðpÞ 2 þ uðd; tÞðe1 e2 Þ ; ð17Þ a2 e2 e1 ðlÞ
ðlÞ
where rðlÞ ¼ b0 =a0 is the reflectance for the unperturbed sample. The right hand sides of Eqs. (16) and (17) can be regarded as linear combinations of six unknown timedependent functions: Z d a1 ðtÞ ¼ gðz0 ; tÞ cos 2k1 z0 dz0 ; 0 Z d gðz0 ; tÞ sin 2k1 z0 dz0 ; b1 ðtÞ ¼ 0
c1 ðtÞ ¼ uð0; tÞ; Z 1 a2 ðtÞ ¼ gðz0 þ d; tÞ cos 2k2 z0 dz0 ; 0 Z 1 b2 ðtÞ ¼ gðz0 þ d; tÞ sin 2k2 z0 dz0 ;
ð18Þ
reflectance data. One example of this occurs when the film is transparent, in which case the acoustic pulse is generated only in the opaque substrate. In addition the spatial extent of the acoustic pulse should be much smaller than d. Then we expect a2 and b2 to be zero and c2 to be constant during the time in which the acoustic pulse is travelling in the film layer. In this special case one may directly deduce uð0; tÞ and thus the acoustic strain pulse shape from the experimental results.
3. Summary In conclusion, we have presented an analytical treatment for the problem of calculating the reflectance variation for light incident at an oblique angle on an isotropic two-layer sample whose permittivity tensor is inhomogeneously perturbed, and whose surface and interface are displaced by a propagating acoustic strain wave. It is shown that the displacement and photoelastic contributions to the relative reflectance changes can be separated under certain conditions provided that experimental results are obtained using both s and p polarized probe light.
Acknowledgements This work was partially supported by a Grant-in-Aid for Scientific Research from the Ministry of Education, Science, Sports, and Culture (Japan) and by the Izumi Science and Technology Foundation.
References
0
c2 ðtÞ ¼ uðd; tÞ: The interferometric transient reflectance measurement provides four independent linear combinations, namely the real and imaginary parts of the reflectance change for s and p polarized probe light. The coefficients appearing in these linear combinations consist of photoelastic constants, permittivities, wave vectors, incident angle, etc. and are all obtainable from independent measurements. If we know at least two of the six unknown functions by some other means, it is possible to determine the remaining four from the measured transient
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