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Abstract
Pseudoreflection geometry Brillouin spectroscopy can be used to probe acoustic wave dispersion approximately along the surface normal of a material system while avoiding the difficulties associated with specularly reflected light encountered in an ideal reflection configuration. As an example of its application, we show analytically that it can be used to determine both the refractive index and bulk acoustic mode velocities of opticallyisotropic nonmetallic materials and confirm the utility of the approach via a series of experiments on fused quartz, gallium phosphide, water, and porous silicon films.
I. INTRODUCTION
II. THEORY
A. PseudoReflection Brillouin Scattering
B. Scattering from an Elastically Homogeneous Material
C. Error Analysis
III. EXPERIMENTAL DETAILS
IV. RESULTS AND DISCUSSION
V. CONCLUSIONS
Key Topics
 Refractive index
 24.0
 Light scattering
 20.0
 Brillouin scattering
 19.0
 Silicon
 17.0
 Quartz
 12.0
Figures
Brillouin scattering geometries: (a) pseudoreflection and (b) 180°backscattering. The wavevectors of the probed acoustic wave, external (internal) incident light, and external (internal) scattered light are q, ( ), and ( ), respectively. The propagation directions of the acoustic wave, external (internal) incident light, and external (internal) scattered light are defined relative to the surface normal by angles α, θ_{ i } (Θ_{ i }), and θ_{ s } (Θ_{ s }), respectively. All angles are between 0 and π/2.
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Brillouin scattering geometries: (a) pseudoreflection and (b) 180°backscattering. The wavevectors of the probed acoustic wave, external (internal) incident light, and external (internal) scattered light are q, ( ), and ( ), respectively. The propagation directions of the acoustic wave, external (internal) incident light, and external (internal) scattered light are defined relative to the surface normal by angles α, θ_{ i } (Θ_{ i }), and θ_{ s } (Θ_{ s }), respectively. All angles are between 0 and π/2.
Dependence of α_{ PR } on θ_{ m } and refractive index for (a) θ_{ d } = 3° and (b) θ_{ d } = 5°. Here, θ_{ d } ⩽ θ_{ m } ⩽ (π/2 − θ_{ d }) ensures that both θ_{ i } and θ_{ s } lie between 0 and π/2.
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Dependence of α_{ PR } on θ_{ m } and refractive index for (a) θ_{ d } = 3° and (b) θ_{ d } = 5°. Here, θ_{ d } ⩽ θ_{ m } ⩽ (π/2 − θ_{ d }) ensures that both θ_{ i } and θ_{ s } lie between 0 and π/2.
Dependence of error in the sound velocity on angle of propagation within the (001)plane of GaP. The dashed (solid) curve corresponds to the calculated error in the velocity of the slow transverse (longitudinal) mode. Calculations were determined using equations (3.1), (3.2), and (3.3) in ref. ^{ 24 } together with the sample parameters therein. Since the velocity of the fast transverse mode is independent of propagation direction within this crystal plane, the expected error is zero.
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Dependence of error in the sound velocity on angle of propagation within the (001)plane of GaP. The dashed (solid) curve corresponds to the calculated error in the velocity of the slow transverse (longitudinal) mode. Calculations were determined using equations (3.1), (3.2), and (3.3) in ref. ^{ 24 } together with the sample parameters therein. Since the velocity of the fast transverse mode is independent of propagation direction within this crystal plane, the expected error is zero.
Brillouin spectra of (a) fused quartz, (b) (100)oriented GaP, (c) deionized water, and (d) a 58 % porous silicon film for different values of θ_{ m }. Peaks due to transverse (T) and longitudinal (L) bulk acoustic modes were observed. The Rayleigh peak at ∼0 GHz was shuttered to prevent photomultiplier saturation. The arrow indicates the base of this peak which is outside the shuttered region. In the case of water, the spectrum collected at θ_{ m } = 63° was smoothed using 3point adjacentaveraging. For clarity, only the Stokes portions of the GaP spectra are shown. The inset is a closeup of the longitudinal and transverse mode peaks collected from GaP at θ_{ m } = 0°. For all spectra shown, the collected scattered light was unpolarized.
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Brillouin spectra of (a) fused quartz, (b) (100)oriented GaP, (c) deionized water, and (d) a 58 % porous silicon film for different values of θ_{ m }. Peaks due to transverse (T) and longitudinal (L) bulk acoustic modes were observed. The Rayleigh peak at ∼0 GHz was shuttered to prevent photomultiplier saturation. The arrow indicates the base of this peak which is outside the shuttered region. In the case of water, the spectrum collected at θ_{ m } = 63° was smoothed using 3point adjacentaveraging. For clarity, only the Stokes portions of the GaP spectra are shown. The inset is a closeup of the longitudinal and transverse mode peaks collected from GaP at θ_{ m } = 0°. For all spectra shown, the collected scattered light was unpolarized.
Acoustic wave frequency versus k _{ m } for (a) fused quartz, (b) GaP, (c) water, and (d) 50 % (diamonds) and 58 % (circles) porous silicon. Filled (open) symbols: experimental longitudinal (transverse) mode data; dashed lines: best fits of Eq. (18) . The shown fits include the data collected using the 180°backscattering geometry. Fits with and without 180°backscattering geometry data had adjusted R ^{2} values greater than 0.99. The vertical error bars are smaller than the symbols, while the horizontal error bars correspond to the experimental random error obtained from Eq. (23) using the refractive index values obtained using alternate methods.
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Acoustic wave frequency versus k _{ m } for (a) fused quartz, (b) GaP, (c) water, and (d) 50 % (diamonds) and 58 % (circles) porous silicon. Filled (open) symbols: experimental longitudinal (transverse) mode data; dashed lines: best fits of Eq. (18) . The shown fits include the data collected using the 180°backscattering geometry. Fits with and without 180°backscattering geometry data had adjusted R ^{2} values greater than 0.99. The vertical error bars are smaller than the symbols, while the horizontal error bars correspond to the experimental random error obtained from Eq. (23) using the refractive index values obtained using alternate methods.
Acoustic wave frequency versus q _{ PR } for fused quartz. Filled (open) symbols: experimental longitudinal (transverse) mode data; dashed lines: dispersion curves deduced from obtained V _{ L } and V _{ T } values. The refractive index used to compute the q _{ PR } values was that obtained from the fit of Eq. (18) to the corresponding ν versus k _{ m } data shown in Fig. 5(a) . The vertical error bars are smaller than the symbols.
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Acoustic wave frequency versus q _{ PR } for fused quartz. Filled (open) symbols: experimental longitudinal (transverse) mode data; dashed lines: dispersion curves deduced from obtained V _{ L } and V _{ T } values. The refractive index used to compute the q _{ PR } values was that obtained from the fit of Eq. (18) to the corresponding ν versus k _{ m } data shown in Fig. 5(a) . The vertical error bars are smaller than the symbols.
Tables
Refractive index and bulk acoustic velocities for fused quartz, GaP, water, and porous silicon (πSi) determined using pseudoreflection Brillouin spectroscopy. Values in parentheses correspond to fits done using both the pseudoreflection and 180°backscattering geometry data. Previously published values of these parameters are also given. Note: The n ^{ alt } values for porous silicon were deduced from gravimetricallydetermined porosities using the twocomponent Bruggeman effective medium model. The corresponding ( ) value were obtained using n ^{ alt } with earlier 180°backscattering Brillouin measurements done on similar samples.
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Refractive index and bulk acoustic velocities for fused quartz, GaP, water, and porous silicon (πSi) determined using pseudoreflection Brillouin spectroscopy. Values in parentheses correspond to fits done using both the pseudoreflection and 180°backscattering geometry data. Previously published values of these parameters are also given. Note: The n ^{ alt } values for porous silicon were deduced from gravimetricallydetermined porosities using the twocomponent Bruggeman effective medium model. The corresponding ( ) value were obtained using n ^{ alt } with earlier 180°backscattering Brillouin measurements done on similar samples.
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Abstract
Pseudoreflection geometry Brillouin spectroscopy can be used to probe acoustic wave dispersion approximately along the surface normal of a material system while avoiding the difficulties associated with specularly reflected light encountered in an ideal reflection configuration. As an example of its application, we show analytically that it can be used to determine both the refractive index and bulk acoustic mode velocities of opticallyisotropic nonmetallic materials and confirm the utility of the approach via a series of experiments on fused quartz, gallium phosphide, water, and porous silicon films.
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