## Jones matrix formalism for the theory of picosecond shear acoustic pulse detection

Optics Express, Vol. 18, Issue 7, pp. 6767-6778 (2010)

http://dx.doi.org/10.1364/OE.18.006767

Acrobat PDF (277 KB)

### Abstract

A theoretical analysis of the transient optical reflectivity of a sample by a normalized Jones matrix is presented. The off-diagonal components of the normalized matrix are identified with the complex rotation of the polarization ellipse. Transient optical polarimetry is a relevant technique to detect shear acoustic strain pulses propagating normally to the surface of an optically isotropic sample. Moreover, polarimetry has a selective sensitivity to shear waves, as this technique cannot detect longitudinal waves that propagate normally to the sample surface.

© 2010 OSA

## 1. Introduction

1. Opaque film metrology. http://www.rudolphtech.com/TechnologyOverview_TechnologyOpaqueFilms.aspx.

2. MetaPULSE System, http://www.rudolphtech.com/MetrologyProduct_ProductMetaPULSE.aspx.

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

8. M. Nikoonahad, S. Lee, and H. Wang, “Picosecond photoacoustics using common-path interferometry,” Appl. Phys. Lett. **76**(4), 514–516 (2000). [CrossRef]

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

10. N. Chigarev, C. Rossignol, and B. Audoin, “Surface displacement measured by beam distortion detection technique: Application to picosecond ultrasonics,” Rev. Sci. Instrum. **77**(11), 114901 (2006). [CrossRef]

11. D. H. Hurley, O. B. Wright, O. Matsuda, V. E. Gusev, and O. V. Kolosov, “Laser picosecond acoustics in isotropic and anisotropic materials,” Ultrasonics **38**(1-8), 470–474 (2000). [CrossRef] [PubMed]

17. D. Mounier, E. Morosov, P. Ruello, M. Edely, P. Babilotte, C. Mechri, J.-M. Breteau, and V. Gusev, “Application of transient femtosecond polarimetry/ellipsometry technique in picosecond laser ultrasonics,” J. Phys.: Conference Series **92**, 012179 (2007). [CrossRef]

13. T. Pezeril, P. Ruello, S. Gougeon, N. Chigarev, D. Mounier, J.-M. Breteau, P. Picart, and V. Gusev, “Generation and detection of plane coherent shear picosecond acoustic pulses by lasers: Experiment and theory,” Phys. Rev. B **75**(17), 174307 (2007). [CrossRef]

15. T. Dehoux, N. Chigarev, C. Rossignol, and B. Audoin, “Three-dimensional elasto-optical interaction for reflectometric detection of diffracted acoustic fields in picosecond ultrasonics,” Phys. Rev. B **76**(2), 024311 (2007). [CrossRef]

^{iθ}. The experimental results obtained with transient reflectometry are generally expressed as the relative variations of the reflectance ΔR/R = 2 Δρ/ρ whereas in transient interferometry the transient phase signal Δθ is measured. In the case of an optically anisotropic sample, the representation of the ultrafast optical response by one reflection coefficient r for each possible state of linear polarization,

*i.e.*p or s, e is no longer sufficient because of the coupling that occurs between polarization states during reflection. For a probe beam at oblique incidence, there are two eigenstates of polarization: in the p-polarized optical mode the electric field lies in the plane of incidence whereas the electric field is perpendicular to the plane of incidence for the s-polarized mode. Thus, in addition to the reflection coefficients r

_{pp}and r

_{ss}for p- and s-polarized probe, the optical properties of the sample must be described by the scattering coefficients r

_{sp}and r

_{ps}which characterize the amplitude of the optical fields which are scattered from p to s polarization and

*vice versa*. The four coefficients r

_{pp}, r

_{ss}, r

_{sp}and r

_{ps}are the components of the reflection Jones matrix of the sample. The advantage of the reflection matrix formalism is to handle at once the polarization effects via the scattering coefficients r

_{sp}and r

_{ps}as well as the variations in amplitude of the reflected field represented by the coefficients r

_{pp}and r

_{ss}.

_{sp}and r

_{ps}may be non zero in the presence of shear waves. Indeed, acousto-optic modulators or deflectors that involve shear acoustic waves are able to switch the polarization from p at the input to s polarization at the output, and

*vice versa*[18

18. C.-H. Chang, R. K. Heilmann, M. L. Schattenburg, and P. Glenn, “Design of a double-pass shear mode acousto-optic modulator,” Rev. Sci. Instrum. **79**(3), 033104 (2008). [CrossRef] [PubMed]

19. Application of acousto-optic devices for spectral imaging system. http://www.goochandhousego.com/files/Technical%20Essay%20AO%20Devices%20for%20spectral%20imaging%20systems.pdf.

*i.e.*perfectly reflecting sample, so the Jones matrix formalism is sufficient to properly deal with the problem [20]. The case of diffusing samples would require the more general Mueller matrix formalism. Section 2 will introduce the transient reflection matrix which is a normalized Jones reflection matrix to describe the transient optical properties of the sample. The relationship of its off-diagonal components with the complex rotation of polarization will be pointed out. Then, the relevance of transient polarimetry techniques to detect shear waves in optically isotropic samples will clearly appear. In section 3, the transient reflection matrix will be calculated in the case of a cubic crystal to predict the polarimetric signals that are expected to be measured. Then the optimal experimental conditions to probe shear acoustic waves will be presented.

## 2. The transient reflection matrix

_{a}=λ/(4πn”) of the substrate, where

_{0}. The plane of incidence of the optical probe beam is the ZX-plane. The incidence angle is

*ϕ*(Fig. 1).

_{ps}= r

_{sp}= 0 and therefore the TRM matrix simplifies as follows:Since the shear waves couples the p and s polarization states [18

18. C.-H. Chang, R. K. Heilmann, M. L. Schattenburg, and P. Glenn, “Design of a double-pass shear mode acousto-optic modulator,” Rev. Sci. Instrum. **79**(3), 033104 (2008). [CrossRef] [PubMed]

19. Application of acousto-optic devices for spectral imaging system. http://www.goochandhousego.com/files/Technical%20Essay%20AO%20Devices%20for%20spectral%20imaging%20systems.pdf.

_{sp}and Δr

_{ps}components are non-zero. So it is not possible to treat the p and s polarizations as independent optical modes. In order to go further, we must express the Δ

**R**⋅

**R**

^{−1}matrix in terms of the amplitude and polarization parameters of the reflected field. We consider that the incident field is not modulated, so the Jones vector

**E**

_{i}can be regarded as a constant vector. Thus, the variations of the reflected field

**E**

_{r}is only due to the perturbation of the sample, hence Δ

**E**

_{r}= Δ

**R**⋅

**E**

_{i}=(Δ

**R**⋅

**R**

^{−1})⋅

**E**

_{r}. The reflected field

**E**

_{r}can be written in the factorized form

**E**

_{r}=A⋅

**J**[20], where the scalar A and the vector

**J**are respectively the complex amplitude and the normalized Jones vector of the field. The complex amplitude can be expressed as: A=A

_{0}.exp(i

*α*), where A

_{0}and

*α*are the magnitude and the phase. The Jones vector

**J**represents a field of unit intensity (

**J**⋅

**J*** = 1), so the intensity of the reflected field is expressed in terms of the p and s components of the Jones vector

**E**

_{r}as: I

_{r}=

**E**

_{r}⋅

**E**

_{r}* = E

_{rp}E

_{rp}* + E

_{rs}E

_{rs}* = |A|

^{2}, where the symbol * represents the complex conjugate.

**J**can be expressed in terms of the two angular parameters ψ and χ which define the polarization ellipse (Fig. 2 ) [20]:We consider that the unperturbed reflected field is either p or s-polarized, corresponding to the normalized Jones vectors

**J**

_{p}and

**J**

_{s}, respectively. The power series expansions of the perturbation vectors Δ

**J**

_{p}and Δ

**J**

_{s}to first order in the vicinity of the p and s polarization states, i. e. around the parameters (ψ,χ)

_{p}=(0,0) and (ψ,χ)

_{s}=(π/2,0) respectively, are:The subscript p or s means that the unperturbed reflected field

**E**

_{r}is p- or s-polarized. As the perturbation of the reflected field is: Δ

**E**

_{r}=ΔA⋅

**J+**A⋅Δ

**J**, the TRM can be expressed in terms of the amplitude and polarization parameters as follows:The remarkable property of the TRM is the separation of the amplitude and the polarization parameters in the matrix.

21. L. Guidoni, E. Beaurepaire, and J.-Y. Bigot, “Magneto-optics in the Ultrafast Regime: Thermalization of Spin Populations in Ferromagnetic Films,” Phys. Rev. Lett. **89**, 017401–1 (2002). [CrossRef] [PubMed]

22. J. Li, M.-S. Lee, W. He, B. Redeker, A. Remhof, E. Amaladass, C. Hassel, and T. Eimüller, “Magnetic imaging with femtosecond temporal resolution,” Rev. Sci. Instrum. **80**(7), 073703 (2009). [CrossRef] [PubMed]

23. K. J. Weingarten, M. J. W. Rodwell, and D. M. Bloom, “Picosecond Optical Sampling of GaAs Integrated Circuits,” IEEE J. Quantum Electron. **24**(2), 198–220 (1988). [CrossRef]

24. J. Warnock, D. D. Awschalom, and M. W. Shafer, “Orientational behavior of molecular liquids in restricted geometries,” Phys. Rev. B **34**(1), 475–478 (1986). [CrossRef]

14. O. Matsuda, O. B. Wright, D. H. Hurley, V. Gusev, and K. Shimizu, “Coherent shear phonon generation and detection with picosecond laser acoustics,” Phys. Rev. B **77**(22), 224110 (2008). [CrossRef]

25. O. Matsuda, O. B. Wright, D. H. Hurley, V. E. Gusev, and K. Shimizu, “Coherent shear phonon generation and detection with ultrashort optical pulses,” Phys. Rev. Lett. **93**(9), 095501 (2004). [CrossRef] [PubMed]

14. O. Matsuda, O. B. Wright, D. H. Hurley, V. Gusev, and K. Shimizu, “Coherent shear phonon generation and detection with picosecond laser acoustics,” Phys. Rev. B **77**(22), 224110 (2008). [CrossRef]

25. O. Matsuda, O. B. Wright, D. H. Hurley, V. E. Gusev, and K. Shimizu, “Coherent shear phonon generation and detection with ultrashort optical pulses,” Phys. Rev. Lett. **93**(9), 095501 (2004). [CrossRef] [PubMed]

_{rp}of the reflected field. Thus the Jones vector for the incident field polarized at 45° can be written in the form

**E**

_{i}= (1,1)

^{T}, where the superscript T denotes the transposition of the row vector (1,1). The reflected field is

**E**

_{r}=(r

_{pp}, r

_{ss})

^{T}, if r

_{ps}=r

_{sp}=0, according to Eq. (1). Indeed the last assumption is almost valid for the SiO

_{2}/Zn sample of [25

25. O. Matsuda, O. B. Wright, D. H. Hurley, V. E. Gusev, and K. Shimizu, “Coherent shear phonon generation and detection with ultrashort optical pulses,” Phys. Rev. Lett. **93**(9), 095501 (2004). [CrossRef] [PubMed]

_{p}=2 Re(ΔE

_{rp}/E

_{rp})=2 (ΔA

_{0}/A

_{0})

_{p}+ 2 Δψ

_{p}. The configuration p-u (p-polarized incident probe and no analyzer) of Ref [14

14. O. Matsuda, O. B. Wright, D. H. Hurley, V. Gusev, and K. Shimizu, “Coherent shear phonon generation and detection with picosecond laser acoustics,” Phys. Rev. B **77**(22), 224110 (2008). [CrossRef]

**93**(9), 095501 (2004). [CrossRef] [PubMed]

_{u}=2 (ΔA

_{0}/A

_{0})

_{p}. The shear pulses were detected in the transparent SiO

_{2}film with the 45-p configuration but not with the p-u configuration. Thus, the shear wave signal is included in the rotation of polarization Δψ

_{p}only, whereas the longitudinal wave signal is contained in the term 2 (ΔA

_{0}/A

_{0})

_{p}. The Jones matrix formalism clearly shows that the shear wave that propagates in the SiO

_{2}film induces a rotation of polarization of the reflected probe. Hence, the opportunity to detect selectively shear waves would be provided by a direct measurement of the rotation of polarization, i.e. by transient polarimetry.

16. D. Mounier, E. Morozov, P. Ruello, J.-M. Breteau, P. Picart, and V. Gusev, “Detection of shear picosecond acoustic pulses by transient femtosecond polarimetry,” Eur. Phys. J. Spec. Top. **153**(1), 243–246 (2008). [CrossRef]

17. D. Mounier, E. Morosov, P. Ruello, M. Edely, P. Babilotte, C. Mechri, J.-M. Breteau, and V. Gusev, “Application of transient femtosecond polarimetry/ellipsometry technique in picosecond laser ultrasonics,” J. Phys.: Conference Series **92**, 012179 (2007). [CrossRef]

16. D. Mounier, E. Morozov, P. Ruello, J.-M. Breteau, P. Picart, and V. Gusev, “Detection of shear picosecond acoustic pulses by transient femtosecond polarimetry,” Eur. Phys. J. Spec. Top. **153**(1), 243–246 (2008). [CrossRef]

21. L. Guidoni, E. Beaurepaire, and J.-Y. Bigot, “Magneto-optics in the Ultrafast Regime: Thermalization of Spin Populations in Ferromagnetic Films,” Phys. Rev. Lett. **89**, 017401–1 (2002). [CrossRef] [PubMed]

24. J. Warnock, D. D. Awschalom, and M. W. Shafer, “Orientational behavior of molecular liquids in restricted geometries,” Phys. Rev. B **34**(1), 475–478 (1986). [CrossRef]

## 3. Shear wave detection

**matrix of the sample perturbed by an arbitrary strain profile, we start by calculating the response to a delta-strain pulse.**

_{R}26. T. C. Zhu, H. J. Maris, and J. Tauc, “Attenuation of longitudinal-acoustic phonons in amorphous SiO_{2} at frequencies up to 440 GHz,” Phys. Rev. B **44**(9), 4281–4289 (1991). [CrossRef]

*i.e.*they are quasi-transverse (QT) or quasi-longitudinal (QL) [27]. The displacement vectors of the QT and QL waves have both transverse and longitudinal components. However, when the waves propagate in a direction parallel to a symmetry axis of the crystal, the QT and QL waves degenerate as pure transverse and longitudinal waves. In the following, we will assume that the Z-axis coincides with the [001] crystal axis of the cubic crystal. This assumption implies that the acoustic velocity of shear waves is independent of the direction of the transverse displacement vector

**u**

_{T}in the XY-plane. This velocity is

_{44}is a component of the elastic tensor and

*ρ*is the density of the medium.

_{ijkl}which governs the optical detection. This tensor is defined by the relationship:

_{11}and p

_{12}[28]. For a cubic crystal which belongs to one of the following crystal classes: 432, m3m, and 4̄3m, the three required independent components are p

_{11}, p

_{12}and p

_{44}. We suppose that the [001] crystal axis coincides with the Z-axis whereas the [100] crystal axis forms an angle

*γ*with the X-axis. The strain tensor of a pure transverse wave that propagates along the Z-axis is defined by the two components: S

_{4}=∂u

_{Y}/∂z and S

_{5}=∂u

_{X}/∂z, where u

_{X}= u

_{T}cos

*φ*and u

_{Y}= u

_{T}sin

*φ*are the components of the transverse displacement vector

**u**

_{T}in the XY-plane. The angle

*φ*characterizes the direction of the displacement vector

**u**

_{T}relative to the plane of incidence. Using formulas for the rotation of the photoelastic tensor [28], one can express the tensor relationship:

*γ*. Hence, p

_{44}is the only photoelastic coefficient involved in the detection of shear waves that propagate in a <100> direction.

**R**of the reflection matrix due to a delta strain pulse located within the substrate at a depth z (Fig. 1). To explain the mechanism of detection of a delta strain perturbation, we follow the same reasoning as in Ref [3

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

**T**

_{0}. After propagation over a distance z, characterized by the propagation matrix

**P**

_{0}, the optical wave interacts with the δ-strain pulse. The discontinuity of the permittivity tensor at the z-plane induces a partial reflection characterized by the δ

**R**

_{T}reflection matrix. Then, the reflected field propagates towards the z=0 interface, characterized by the propagation matrix

**P**

_{1}. Finally, the transmission of the field into the transparent medium n

_{0}is characterized by the transmission matrix

**T**

_{1}. Therefore the Δ

**R**perturbation matrix is expressed as a product of Jones matrices as follows:The

**P**

_{0}and

**P**

_{1}matrices are each equal to the identity matrix multiplied by the phase factor

_{z}= k n cos

*ϕ*is degenerated for both p and s polarizations. So, the same phase factor is applied to both optical modes.

**R**

_{T}reflection matrix has to be evaluated at the δ-discontinuity. We consider a shear strain of magnitude S

_{T}(z)=∂u

_{T}/∂z, where u

_{T}is the magnitude of the transverse displacement in the XY-plane. The magnitude of the delta-like S

_{T}(z) strain at z is such that

_{T}represents here the magnitude of the step-like shear displacement localized at a depth z within the sample. The δ

**R**

_{T}matrix can be calculated using a 4 x 4 matrix formalism [20,29

29. D. W. Berreman, “Optics of stratified and Anisotropic Media: 4 x 4- matrix Formulation,” J. Opt. Soc. Am. **62**(4), 502–510 (1972). [CrossRef]

30. P. Yeh, “Optics of anisotropic layered media: a new 4 x 4 matrix algebra,” Surf. Sci. **96**(1-3), 41–53 (1980). [CrossRef]

**R**

_{T}matrix is calculated for the limit e→0. The result of the calculation, exact to the first order in u

_{T}, is:where

**)**

_{R}_{Tδ},

*i.e.*the matrix (Δ

**)**

_{R}_{T}for a delta-like shear strain pulse, we need the expression of the diagonal matrices:

**T**

_{0}=[t

_{0p}, t

_{0s}],

**T**

_{1}=[t

_{1p}, t

_{1s}], and the unperturbed diagonal reflection matrix

**R**=[r

_{p}, r

_{s}], where:

_{0p}= 2n

_{0}cos

*ϕ*

_{0}/ (n

_{0}cos

*ϕ*+ n cos

*ϕ*

_{0}), t

_{0s}= 2n

_{0}cos

*ϕ*

_{0}/ (n

_{0}cos

*ϕ*

_{0}+ n cos

*ϕ*), t

_{1p}= 2n cos

*ϕ*/ (n

_{0}cos

*ϕ*+ n cos

*ϕ*

_{0}), t

_{1s}= 2n cos

*ϕ*/ (n

_{0}cos

*ϕ*

_{0}+ n cos

*ϕ*). r

_{p}= (n

_{0}cos

*ϕ*- n cos

*ϕ*

_{0}) / (n

_{0}cos

*ϕ*+ n cos

*ϕ*

_{0}), r

_{s}= (n

_{0}cos

*ϕ*

_{0}- n cos

*ϕ*) / (n

_{0}cos

*ϕ*

_{0}+ n cos

*ϕ*).Equation (10) leads to:where

_{p,s}and

*θ*

_{p,s}are respectively the modulus and the phase of h

_{p,s}components. If the shear acoustic strain cannot be considered as a delta pulse, then the transient (Δ

**)**

_{R}_{T}matrix for an arbitrary shear strain profile S

_{T}(z, t), which may depend on the time t, is calculated by the following expression:In particular, Eq. (13) can be applied to calculate the (Δ

**)**

_{R}_{T}matrix which is associated to a homogenous shear strain in the substrate. The result is in agreement with what was presented in Ref [31].

**)**

_{R}_{T}matrix are zero. This result means that the shear waves have no effect on the amplitude of the reflected field. Therefore shear plane waves cannot be detected by transient reflectometry or interferometry techniques, independently of the choice of the probe incidence angle and the probe polarization. In consequence, the only way to detect a shear plane wave -not a QT wave - that propagates normally to the surface in an optically isotropic medium is to use a detection configuration which is sensitive to the rotation of polarization of the reflected probe. On the contrary, the detection of QT waves is possible with transient reflectometry or interferometry, with a probe beam at normal incidence [13

13. T. Pezeril, P. Ruello, S. Gougeon, N. Chigarev, D. Mounier, J.-M. Breteau, P. Picart, and V. Gusev, “Generation and detection of plane coherent shear picosecond acoustic pulses by lasers: Experiment and theory,” Phys. Rev. B **75**(17), 174307 (2007). [CrossRef]

15. T. Dehoux, N. Chigarev, C. Rossignol, and B. Audoin, “Three-dimensional elasto-optical interaction for reflectometric detection of diffracted acoustic fields in picosecond ultrasonics,” Phys. Rev. B **76**(2), 024311 (2007). [CrossRef]

_{T}coefficient, which determines the detection sensitivity, shows that a shear wave is not detected if the transverse displacement

**u**

_{T}is parallel to the plane of incidence, i.e. for

*φ*=0°, but the sensitivity reaches a maximum for

*φ*=90°,

*i.e.*when the displacement vector

**u**

_{T}is perpendicular to the plane of incidence. In addition, the sensitivity vanishes for normal incidence (

*ϕ*

_{0}=0° and

*ϕ*=0°), it is thus essential to probe shear waves at oblique incidence.

*ϕ*= (n

^{2}- n

_{0}

^{2}sin

^{2}

*ϕ*

_{0})

^{1/2}= ν’ - iν”, as n

_{0}sin

*ϕ*

_{0}= n sin

*ϕ*, with in order to ensure the damping of the optical field in the opaque medium for z>0.

_{T}, for both p and s polarizations, induced by a shear delta strain pulse within a Gallium Arsenide (GaAs) substrate with (100) orientation in air (n

_{0}=1). The Δψ and Δχ polarimetric signals of Eq. (14) are represented in Fig. 3 . The transverse displacement is supposed to be perpendicular to the plane of incidence. The refractive index of GaAs at the probe wavelength 800 nm (E=1.552 eV) is n=3.684-0.092i [32]. The photoelastic coefficient p

_{44}= −0.46-0.21i for GaAs at 800 nm is calculated from the piezo-optical constant π

_{44}=(1.5+0.5i).10

^{−9}Pa

^{−1}of Ref [33

33. P. Etchegoin, J. Kircher, M. Cardona, C. Grein, and E. Bustarret, “Piezo-optics of GaAs,” Phys. Rev. B **46**(23), 15139–15149 (1992). [CrossRef]

_{44}= 59.44 GPa [32] by the relationship: p

_{44}= – (π

_{44}C

_{44}) / n

^{4}. Equation (14) shows that each polarimetric signal oscillates in function of z with the spatial pseudo-period Λ = λ /(2ν′) . The oscillations are damped with a characteristic length L = λ / (4πν″), which depends both on the refractive index of the material and the incidence angle. For either p or s polarizations, the Δψ and Δχ oscillations have the same magnitude H but are phase-shifted by π/2. As a shear pulse propagates with a velocity c

_{T}, the polarimetric signals oscillate at the Brillouin frequency: f

_{B}= c

_{T}/ Λ = (2c

_{T}ν′) / λ, and the oscillations decay with the characteristic damping time τ = L / c

_{T}. When a shear pulse is reflected at the free surface z=0, a change of sign occurs in the displacement vector

**u**

_{T}. Consequently, the signs of Δψ and Δχ are changed after reflection. This results in a step in the polarimetric signals recorded as a function of time. Figure 3 shows that, at z = 0, Δχ signal has a greater magnitude than the Δψ signal. Hence, the step of the shear pulse echo at the free surface would be greater with the Δχ signal. depends on If the phase

*θ*

_{p,s}, is zero or 180°, then the amplitude of the step Δψ signal would be zero and the echo would not be detected. On the contrary, the Δχ signal, phase shifted by π/2, would lead to a step of maximum amplitude. It is therefore more advantageous to measure both the complementary signals: Δψ and Δχ.

*ϕ*

_{0}. For a p-polarized probe, a resonance occurs at the Brewster angle (ϕ

_{0}≈74.8°). Nevertheless, probing the shear waves with p polarization will be impossible around the Brewster angle, because of the exceedingly small reflectance of the substrate (Fig. 4b). In conclusion, it would be preferable to probe the shear pulse with s polarization at an incidence angle around ϕ

_{0}≈56° for which the magnitude of the Brillouin oscillations reaches a maximum. At the optimum angle of incidence ϕ

_{0}≈56°, the Brillouin frequency and the decay time, are respectively 30.0 GHz and 202 ps, since a shear wave velocity along a <100> crystallographic axis of GaAs is c

_{T}= 3343.5 m/s.

**R**

_{L}reflection matrix for a longitudinal delta-strain pulse, corresponding to a step-like displacement u

_{L}, located within the substrate at depth z, is calculated. Here, the only non zero component of the strain tensor is S

_{3}=∂u

_{Z}/∂z. Hence, Eq. (9) shows that the photoelastic constants involved in the detection process are p

_{11}and p

_{12}. The calculation leads to:where

**R**

_{T}matrix in Eq. (10) by δ

**R**

_{L,}, one can calculate the (Δ

**)**

_{R}_{L}matrix for the longitudinal strain pulse. As all the matrices in Eq. (10) are diagonal, the (Δ

**)**

_{R}_{L}matrix is also diagonal. Consequently, the longitudinal waves do not induce any polarimetric signal, as the off-diagonal components of the (Δ

**)**

_{R}_{L}matrix are zero. This property is of practical importance if an echo of a shear pulse has to be detected with polarimetry at the same time as an echo of a longitudinal pulse. The diagonal components of the (Δ

**)**

_{R}_{L}matrix calculated by Eqs. (10) and (13), provided that the subscript T is replaced by L, is in agreement with Eqs. (15) and (16) of [34

34. O. Matsuda and O. B. Wright, “Laser picosecond acoustics with oblique probe light incidence,” Rev. Sci. Instrum. **74**(1), 895–897 (2003). [CrossRef]

**)**

_{R}_{θ}matrix, where the subscript θ denotes the temperature, must be diagonal in the case of homogeneous heating along the XY-plane. Hence, thermal transients can be in principle cancelled in polarimetric measurements.

## 4. Conclusion

## Acknowledgements

## References and Links

1. | Opaque film metrology. http://www.rudolphtech.com/TechnologyOverview_TechnologyOpaqueFilms.aspx. |

2. | MetaPULSE System, http://www.rudolphtech.com/MetrologyProduct_ProductMetaPULSE.aspx. |

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5. | B. Perrin, C. Rossignol, B. Bonello, and J.-C. Jeannet, “Interferometic detection in picosecond ultrasonics,” Physica B |

6. | D. H. Hurley and O. B. Wright, “Detection of ultrafast phenomena by use of a modified Sagnac interferometer,” Opt. Lett. |

7. | 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. |

8. | M. Nikoonahad, S. Lee, and H. Wang, “Picosecond photoacoustics using common-path interferometry,” Appl. Phys. Lett. |

9. | O. B. Wright and K. Kawashima, “Coherent phonon detection from ultrafast surface vibrations,” Phys. Rev. Lett. |

10. | N. Chigarev, C. Rossignol, and B. Audoin, “Surface displacement measured by beam distortion detection technique: Application to picosecond ultrasonics,” Rev. Sci. Instrum. |

11. | D. H. Hurley, O. B. Wright, O. Matsuda, V. E. Gusev, and O. V. Kolosov, “Laser picosecond acoustics in isotropic and anisotropic materials,” Ultrasonics |

12. | T. Pezeril, C. Klieber, S. Andrieu, and K. A. Nelson, “Optical generation of gigahertz-frequency shear acoustic waves in liquid glycerol,” Phys. Rev. Lett. |

13. | T. Pezeril, P. Ruello, S. Gougeon, N. Chigarev, D. Mounier, J.-M. Breteau, P. Picart, and V. Gusev, “Generation and detection of plane coherent shear picosecond acoustic pulses by lasers: Experiment and theory,” Phys. Rev. B |

14. | O. Matsuda, O. B. Wright, D. H. Hurley, V. Gusev, and K. Shimizu, “Coherent shear phonon generation and detection with picosecond laser acoustics,” Phys. Rev. B |

15. | T. Dehoux, N. Chigarev, C. Rossignol, and B. Audoin, “Three-dimensional elasto-optical interaction for reflectometric detection of diffracted acoustic fields in picosecond ultrasonics,” Phys. Rev. B |

16. | D. Mounier, E. Morozov, P. Ruello, J.-M. Breteau, P. Picart, and V. Gusev, “Detection of shear picosecond acoustic pulses by transient femtosecond polarimetry,” Eur. Phys. J. Spec. Top. |

17. | D. Mounier, E. Morosov, P. Ruello, M. Edely, P. Babilotte, C. Mechri, J.-M. Breteau, and V. Gusev, “Application of transient femtosecond polarimetry/ellipsometry technique in picosecond laser ultrasonics,” J. Phys.: Conference Series |

18. | C.-H. Chang, R. K. Heilmann, M. L. Schattenburg, and P. Glenn, “Design of a double-pass shear mode acousto-optic modulator,” Rev. Sci. Instrum. |

19. | Application of acousto-optic devices for spectral imaging system. http://www.goochandhousego.com/files/Technical%20Essay%20AO%20Devices%20for%20spectral%20imaging%20systems.pdf. |

20. | R. M. A. Azzam, and N. M. Bashara, |

21. | L. Guidoni, E. Beaurepaire, and J.-Y. Bigot, “Magneto-optics in the Ultrafast Regime: Thermalization of Spin Populations in Ferromagnetic Films,” Phys. Rev. Lett. |

22. | J. Li, M.-S. Lee, W. He, B. Redeker, A. Remhof, E. Amaladass, C. Hassel, and T. Eimüller, “Magnetic imaging with femtosecond temporal resolution,” Rev. Sci. Instrum. |

23. | K. J. Weingarten, M. J. W. Rodwell, and D. M. Bloom, “Picosecond Optical Sampling of GaAs Integrated Circuits,” IEEE J. Quantum Electron. |

24. | J. Warnock, D. D. Awschalom, and M. W. Shafer, “Orientational behavior of molecular liquids in restricted geometries,” Phys. Rev. B |

25. | O. Matsuda, O. B. Wright, D. H. Hurley, V. E. Gusev, and K. Shimizu, “Coherent shear phonon generation and detection with ultrashort optical pulses,” Phys. Rev. Lett. |

26. | T. C. Zhu, H. J. Maris, and J. Tauc, “Attenuation of longitudinal-acoustic phonons in amorphous SiO |

27. | B. A. Auld, |

28. | J. F. Nye, |

29. | D. W. Berreman, “Optics of stratified and Anisotropic Media: 4 x 4- matrix Formulation,” J. Opt. Soc. Am. |

30. | P. Yeh, “Optics of anisotropic layered media: a new 4 x 4 matrix algebra,” Surf. Sci. |

31. | O. Matsuda and O. B. Wright, “Theory of Detection of Shear Strain Pulses with Laser Picosecond Acoustics,” Anal. Sci. |

32. | CRC Hand Book of Physics and Chemistry (2004). |

33. | P. Etchegoin, J. Kircher, M. Cardona, C. Grein, and E. Bustarret, “Piezo-optics of GaAs,” Phys. Rev. B |

34. | O. Matsuda and O. B. Wright, “Laser picosecond acoustics with oblique probe light incidence,” Rev. Sci. Instrum. |

**OCIS Codes**

(120.0120) Instrumentation, measurement, and metrology : Instrumentation, measurement, and metrology

(120.2130) Instrumentation, measurement, and metrology : Ellipsometry and polarimetry

(120.4630) Instrumentation, measurement, and metrology : Optical inspection

(320.0320) Ultrafast optics : Ultrafast optics

(320.5390) Ultrafast optics : Picosecond phenomena

(120.4880) Instrumentation, measurement, and metrology : Optomechanics

**ToC Category:**

Ultrafast Optics

**History**

Original Manuscript: December 22, 2009

Revised Manuscript: February 5, 2010

Manuscript Accepted: February 8, 2010

Published: March 17, 2010

**Citation**

Denis Mounier, Pascal Picart, Philippe Babilotte, Pascal Ruello, Jean-Marc Breteau, Thomas Pézeril, Gwenaëlle Vaudel, Mansour Kouyaté, and Vitalyi Gusev, "Jones matrix formalism for the theory of picosecond shear acoustic pulse detection," Opt. Express **18**, 6767-6778 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-7-6767

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### References

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- D. H. Hurley, O. B. Wright, O. Matsuda, V. E. Gusev, and O. V. Kolosov, “Laser picosecond acoustics in isotropic and anisotropic materials,” Ultrasonics 38(1-8), 470–474 (2000). [CrossRef] [PubMed]
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- T. Pezeril, P. Ruello, S. Gougeon, N. Chigarev, D. Mounier, J.-M. Breteau, P. Picart, and V. Gusev, “Generation and detection of plane coherent shear picosecond acoustic pulses by lasers: Experiment and theory,” Phys. Rev. B 75(17), 174307 (2007). [CrossRef]
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- T. Dehoux, N. Chigarev, C. Rossignol, and B. Audoin, “Three-dimensional elasto-optical interaction for reflectometric detection of diffracted acoustic fields in picosecond ultrasonics,” Phys. Rev. B 76(2), 024311 (2007). [CrossRef]
- D. Mounier, E. Morozov, P. Ruello, J.-M. Breteau, P. Picart, and V. Gusev, “Detection of shear picosecond acoustic pulses by transient femtosecond polarimetry,” Eur. Phys. J. Spec. Top. 153(1), 243–246 (2008). [CrossRef]
- D. Mounier, E. Morosov, P. Ruello, M. Edely, P. Babilotte, C. Mechri, J.-M. Breteau, and V. Gusev, “Application of transient femtosecond polarimetry/ellipsometry technique in picosecond laser ultrasonics,” J. Phys.: Conference Series 92, 012179 (2007). [CrossRef]
- C.-H. Chang, R. K. Heilmann, M. L. Schattenburg, and P. Glenn, “Design of a double-pass shear mode acousto-optic modulator,” Rev. Sci. Instrum. 79(3), 033104 (2008). [CrossRef] [PubMed]
- Application of acousto-optic devices for spectral imaging system. http://www.goochandhousego.com/files/Technical%20Essay%20AO%20Devices%20for%20spectral%20imaging%20systems.pdf .
- R. M. A. Azzam, and N. M. Bashara, Ellipsometry and Polarized Light (North Holland, Elsevier, Amsterdam–London–New York–Tokyo, 1987).
- L. Guidoni, E. Beaurepaire, and J.-Y. Bigot, “Magneto-optics in the Ultrafast Regime: Thermalization of Spin Populations in Ferromagnetic Films,” Phys. Rev. Lett. 89, 017401–1 (2002). [CrossRef] [PubMed]
- J. Li, M.-S. Lee, W. He, B. Redeker, A. Remhof, E. Amaladass, C. Hassel, and T. Eimüller, “Magnetic imaging with femtosecond temporal resolution,” Rev. Sci. Instrum. 80(7), 073703 (2009). [CrossRef] [PubMed]
- K. J. Weingarten, M. J. W. Rodwell, and D. M. Bloom, “Picosecond Optical Sampling of GaAs Integrated Circuits,” IEEE J. Quantum Electron. 24(2), 198–220 (1988). [CrossRef]
- J. Warnock, D. D. Awschalom, and M. W. Shafer, “Orientational behavior of molecular liquids in restricted geometries,” Phys. Rev. B 34(1), 475–478 (1986). [CrossRef]
- O. Matsuda, O. B. Wright, D. H. Hurley, V. E. Gusev, and K. Shimizu, “Coherent shear phonon generation and detection with ultrashort optical pulses,” Phys. Rev. Lett. 93(9), 095501 (2004). [CrossRef] [PubMed]
- T. C. Zhu, H. J. Maris, and J. Tauc, “Attenuation of longitudinal-acoustic phonons in amorphous SiO2 at frequencies up to 440 GHz,” Phys. Rev. B 44(9), 4281–4289 (1991). [CrossRef]
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- P. Yeh, “Optics of anisotropic layered media: a new 4 x 4 matrix algebra,” Surf. Sci. 96(1-3), 41–53 (1980). [CrossRef]
- O. Matsuda and O. B. Wright, “Theory of Detection of Shear Strain Pulses with Laser Picosecond Acoustics,” Anal. Sci. 17(Special Issue), s216–s218 (2001).
- CRC Hand Book of Physics and Chemistry (2004).
- P. Etchegoin, J. Kircher, M. Cardona, C. Grein, and E. Bustarret, “Piezo-optics of GaAs,” Phys. Rev. B 46(23), 15139–15149 (1992). [CrossRef]
- O. Matsuda and O. B. Wright, “Laser picosecond acoustics with oblique probe light incidence,” Rev. Sci. Instrum. 74(1), 895–897 (2003). [CrossRef]

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