Goos–Hänchen shift for coupled vibrational modes in a semiconductor structure

We present a theoretical investigation of the Goös–Hanchen shift (GHS) experienced by acoustic and optical vibrational modes reflected and transmitted from the surfaces of a semiconductor thin film sandwiched between two semi-infinite media. Our study focuses on the impact of the incident angle on the GHS, considering the coupling between longitudinal and transverse modes. For acoustic vibrations, our findings reveal that the GHS can reach magnitudes up to seven times larger than the thickness of the thin film and up to 20 times larger than the incident wavelength. Besides, it is shown that this significant amplification of the GHS highlights the strong influence of the incident angle and the frequency of the modes involved. In the case of optical vibrations, we observe even more pronounced GHS values, exceeding 30 times the incident wavelength. This demonstrates the potential of GHS in acoustical systems, which opens up possibilities for applications in the design of acoustic devices.


Introduction
The behaviour of optical, acoustic, and matter waves can be described by solving a wave equation with corresponding boundary conditions.Despite their differences, wave phenomena exhibit similar characteristics.For instance, many of the properties observed in geometrical optics also manifest themselves in acoustic and matter waves.One of such properties is * Authors to whom any correspondence should be addressed.
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the reflection of a wave from an interface at an angle of incidence close to the critical value, resulting in a lateral shift of the reflected wave.This phenomenon is known as the Goos-Hänchen effect [1].
The Goös-Hanchen shift (GHS) has recently achieved significant interest due to its potential applications, such as optical differential operations and image edge detection [2], as well as refractive index sensors [3] and plasmonic biosensors [4].For example, biosensor applications include the design of a structure based on two-dimensional graphene and transition metal dichalcogenides to greatly enhance the GHS, improving the performance of plasmonic biosensors [5].Additionally, high sensitive label-free optical sensors based on GHS effect have been studied by using a single chirped laser pulse [6].It has also been shown that the sensitivity of bimetallic sensors based on the graphene-hexagonal boron nitride heterostructure can be enhanced by increasing the GHS in the infrared band [7].Likewise, giant GHS has been predicted theoretically [8][9][10][11] and demonstrated experimentally [12][13][14][15].In a recent experiment, Sreekanth et al demonstrated a thirtyfold enhancement in GHS at the Brewster angle of a nanophotonic cavity operating at a given wavelength [12].Zheng et al achieved a giant GHS by using magnetic dipole quasi-bound states in the continuum in dielectric metasurfaces [11].Photonic heterostructures composed of two types of dielectric one-dimensional photonic crystals with symmetric unit cells can be used to significantly increase GHS, presenting a new promising method for design high-performance and sensitive sensors [9].The GHS at the interface of topological materials, such as the topological edge states of a magnetic photonic crystal [16] and silicene [17] has also been recently investigated.
The GHS effect is not limited to light waves.Researchers have made significant advances using the GHS effect to solve many acoustic problems, which have aided our understanding of wave propagation.These advances have contributed to the development of new techniques to control and manipulate sound waves in different systems.For example, Zeng et al derived an analytic expression for the GHS occurring in acoustic metamaterial heterostructures [18], while FuPing et al and De Leo et al carried out the analysis on seismic waves [19,20].In addition, models of lateral displacement of acoustic wave scattering at a fluid-solid interface have revealed an acoustic analog of the GHS effect in optics [21].Villegas et al investigated the relationship between tunneling of optical vibrational modes with transverse horizontal polarisation impinging on a semiconductor heterostructure together with the GHS [8].Further research has revealed that the lateral shift of an optical beam undergoing Brillouin light scattering on an acoustic wave in the total internal reflection geometry depends on the polarization (longitudinal or transverse) of the acoustic wave [22].Experimentally, acoustic GHS must be taken into account to design acoustic lenses, such as a flattened hemispherical acoustic Luneburg lens [23].
In this paper, we study the lateral shift experienced by both acoustic and optical vibrational modes upon reflection and transmission on both sides of a semiconductor layer sandwiched between two semi-infinite media.More specifically, we assess the impact of lateral shift of coupled longitudinal and transverse acoustic waves.These modes are excited by a longitudinal wave incident on the surface of a semiconductor layer at an oblique angle.To the best of our knowledge, this study is the first exploration of GHS in coupled vibrational modes.We compare the GHS with the length of the system and with the incident wavelength, and identify giant GHS in the structure.
We study the behaviour of the GHS in isotopically controlled 70 (Ge) layers.Over the past two decades, isotopically controlled semiconductors have garnered significant attention.These studies are particularly relevant because the nuclear mass difference directly impacts the phonon frequencies and related properties in crystalline solids, including lattice constant, thermal conductivity, electronic band gaps, and localized vibrational modes [24].For example, molecular dynamics simulations have shown that isotope mixing can effectively reduce the thermal conductivities of rectangular graphene nanoribbons [25].Ultrafast studies of phonon transport in isotopically controlled semiconductor nanostructures have confirmed that isotopic modulation significantly decreases thermal conductivity beyond expectations in the coherent phonon transport regime [26].Furthermore, Raman scattering measurements in germanium nanocrystals have exposed a rich multiple-peak structure, in addition to the well-known Ge-Ge peak at approximately 300 cm −1 of bulk Ge [27].However, Raman spectroscopy analysis of 12 C/ 13 C graphene isotope superlattices demonstrates a different behaviour.While the Raman spectrum corresponds to the sum of the bulk 12 C and 13 C contributions for large periods, the formation of mixed 12 C/ 13 C modes is observed for Raman processes involving two phonons for small periods [28].Moreover, in situ Raman spectroelectrochemistry has been adopted to study isotopically engineered graphene systems.This approach provides valuable information about the mutual interactions between graphene layers, their directly adjacent environment (e.g.substrate or electrolyte), and their extended environment separated by additional graphene layers [29].Furthermore, the gradual introduction of heavier isotopes in the chemical vapor deposition growth of MoS 2 produces a systematic variation of the lattice phonon energy that modifies the transient and steady-state spontaneous photoluminescence spectra [30].
This work is organized as follows.Section 2 provides a concise overview of the general formalism utilized to describe the Goos-Hänchen effect.This includes an explanation of the stationary-phase approximation method, as well as the implementation of the long-wavelength continuum phenomenological model to describe vibrational modes in semiconductor heterostructures.The formalism established there will be used in section 3 to calculate the GHS for acoustic vibrational modes.In section 4, we focus on the analysis of the GHS for optical vibrational modes.We present numerical results for both longitudinal and transverse waves in a single-layer structure, demonstrating the validity of the analytical treatment presented in section 2. Finally, in section 5, we summarize the key findings of this study and draw our conclusions based on the obtained results.

Goos-Hänchen effect: general formalism
Figure 1 shows a schematic representation of the oblique incident problem addressed in this paper.The problem involves a semiconductor layer (spacer medium II) with thickness h sandwiched between two semi-infinite media, labelled as I and III, respectively.The thickness h is determined by the number of atomic layers comprising the spacer medium.
In our analysis, we assume a longitudinal (L) wave with an amplitude A i L incident at an angle θ i onto the interface between media I and II, as depicted in figure 1.It is worth noting that for the elastic isotropic materials under investigation (Ge isotopes), the propagation of oblique waves through the structure leads to non-degenerate transverse vibrational modes.Specifically, one of the transverse modes, polarized parallel to the interfaces and called shear mode (SH mode), remains decoupled from the other modes (transverse (SV) and longitudinal (L) modes).The SV and L modes are polarized in the sagittal plane (the plane defined by the wave vector and the unitary vector perpendicular to the interfaces), and are coupled to each other.According to figure 1, at the interface z = 0, the amplitudes of longitudinal and transverse reflected waves are A r L and A r T , respectively, with angles α r L and α r T .Additionally, at z = h, waves with amplitudes A t L and A t T , and angles α t L and α t T , respectively, are transmitted.The reflected and refracted angles are calculated using Snell's law.It is important to note that the wave propagation is confined into the yz-plane.
To investigate the Goos-Hänchen shift, we model the impinging displacement field, denoted by u i L (r, t), as a twodimensional wave packet propagating within the yz-plane.The incident wave packet can be written as [31], where r = (y, z) is the position vector, k = (k y , k z ) is the wave vector, and e L is the polarization of the wave.We assume that the momentum distribution, denoted as ϕ L (k), follows a Gaussian profile.The transmitted packet can be expressed as where t l (k) = |t l (k)| e iα t l represents the amplitude of the transmitted wave, l ∈ (L, T).The total phase of the transmitted mode can be written as, If the width of the Gaussian energy distribution is sufficiently narrow, resembling a collimated beam, the transmission coefficients exhibit approximately constant magnitudes within the range of the integral.As result, the wave packet remains undistorted and does not undergo any reshaping [32].When the stationary phase method is applied, the peak of the transmitted wave packet is located at the point where the phase of the transmitted wave packet becomes stationary.Therefore, both the partial derivatives of Ψ t l in equation ( 3) with respect to the incident angle (θ i ) and with respect to the frequency (ω) must vanish, Let us assume that the peak of the transmitted packet within medium III is located at r = (∆y t l , h).By substituting equation (3) into equations ( 4) and ( 5), we obtain the following expression for the GHS of the transmitted packet, Similarly, the reflected wave packet u r l can be expressed as where r l (k) = |r l (k)| e iα r l represents the amplitude of the reflected wave.The corresponding total phase of the reflected modes is given by We assume that the peak of the reflected packet within medium I is located at r = (∆y r l , 0).Similarly to the previous case, the GHS for the reflected packet can be written as In this work, acoustic and optical vibrations in semiconductor material are described within the framework of the long-wavelength continuum phenomenological model [33,34].This model provides a suitable description for nanostructures such as those studied here, whose dimensions are in the order of a few atomic layers [35].Within this model, the propagation of acoustic and optical waves can be described by the following equation of motion Here, u represents the displacement field, ρ is the mass density, ω Γ is the phonon frequency at the center of the Brillouin zone, and σ is the stress tensor.The components of the stress tensor can be expressed as (11) where β L and β T are two parameters that describe the dispersion relations of the longitudinal and transverse oscillations.For optical branches, the dispersion relation in the neighbourhood of the Γ point of the Brillouin zone is given by the parabolic function For acoustic branches, the dispersion relation is a linear function of the form ω(k) = v l k l , where v l represents the sound velocity.
The reflection coefficient (R l = j r l /j i ) and transmission coefficient (T l = j t l /j i ) can be defined as the ratios of the reflected and transmitted fluxes to the incident flux, respectively [34].The reflected flux is given by j r l = ρωk l (k and the transmitted flux is given by To calculate j r l and j t l , the parameters corresponding to media I and III must be considered.The incident flux is given by , where A i L = 1.A r l and A t l are calculated using the transfer matrix method.

Goos-Hänchen shift for acoustic vibrational modes
In this section, we apply the previously presented formalism to calculate the GHS for the system described earlier.The longitudinal P-waves and transverse SV-waves, which are polarized in the yz-plane, are coupled at the interfaces between different layers in the system.However, SH-waves, with polarization normal to the yz-plane, do not interact with the P-and SVwaves and can be considered separately.Here, we focus on coupled P-SV waves and exclude SH-waves, which has been studied in a previous work [8].
Let us assume that a longitudinal wave with an amplitude u i = A i L e i(kyy+kzz) is incident on the interface between media I and II, where The total scattered field is a combination of leftward-and rightward-propagating waves, including both propagating and evanescent modes.The wavenumber of an evanescent mode is purely imaginary.Furthermore, the wave vector components along the interface are conserved.
The components of the displacement field in regions I, II, and III are solutions of equation ( 10) satisfying the boundary conditions at the interface, i.e. the continuity of the displacement field and the stress tensor [35].For simplicity, we assume identical media for regions I and III.Under this simplification, the z component of the wave vectors in regions I and III can be written as the refracted angle in medium II becomes sin 2 (θ i ) becomes purely imaginary, and the corresponding longitudinal mode in medium II is evanescent.The second critical angle θ c2 occurs when the refracted transverse angle becomes α T2 = π 2 .Consequently, sin 2 (θ i ) is imaginary and its corresponding transverse mode is evanescent.Moreover, we can derive an additional angle from the condition k T1 = 0, which can be written as This condition indicates that, when θ i > θ c3 , both the reflected and transmitted transverse modes are evanescent.Numerical simulations were performed for media I and III consisting of 74 (Ge), and a spacer layer II composed of four atomic layers of 70 (Ge) with thickness h = 5.7 Å.
The velocities of the longitudinal and transverse acoustic waves, normalized to the speed of light, travelling through 74 (Ge) are v L1 = 2.35 × 10 −6 and v T1 = 1.32 × 10 −6 , respectively.Similarly, in 70 (Ge), the normalized velocities are v L2 = 2.41 × 10 −6 and v T2 = 1.36 × 10 −6 for longitudinal and transverse waves, respectively.Because both materials exhibit linear dispersion relations for acoustic modes, the frequency in the center of the Brillouin zone is zero, ω L = ω T = 0.
The approximate value for the first critical angle between 74 (Ge) and 70 (Ge) is θc 1 ≈ 76.76 • .However, when attempting to calculate the second critical angle, θ c2 , between the two materials, it is found that such an angle does not exist because sin(θ c2 ) > 1.This means that refracted angles are smaller than incident angles when the wave velocity in the second material is slower than in the first material.On the other hand, refracted angles are larger than incident angles when the wave velocities in the second material are greater than in the first material.Since v L1 > v T1 , the third critical angle θ c3 is complex.
The expressions for the Goos-Hänchen shift for transmitted (equation ( 6)) and reflected waves (equation ( 9)) can be simplified to the following expressions by considering the linear dispersion relation for acoustic vibrational modes, It is worth stressing that when the angle of incidence is equal to the critical angle θ c1 , total internal reflection of the longitudinal mode occurs.Therefore, the amplitude of the wave in region II (see figure 1) decays exponentially in the z direction and propagates in the y direction up to a distance ∆y r L (GHS in reflection).The energy tunneled through medium II emerges at the point (h, ∆y r L ) as a longitudinal wave transmitted in medium III, where ∆y t L is the GHS in reflection.Figure 2 shows the GHS calculated from equations ( 15) and ( 16) as a function of wavelength (λ) and incidence angle (θ i ), for θ i greater than the first critical angle θ c1 ≈ 76.76 • .Figure 2(a) represents the GHS for transmitted acoustic waves (∆y t L ), while figure 2(b) depicts the GHS for reflected acoustic modes (∆y r L ).Both ∆y t L and ∆y r L are normalized to the layer thickness h.It is observed that both ∆y t L and ∆y r L increase when we increase the incident angle.The contour map of figure 2 exhibit a non-monotonic behaviour, as a function of λ, which is more pronounced in the case of ∆y r L .From equations ( 15) and ( 16), it is inferred that the GHS is proportional to the phase variation with respect to the angle of incidence θ i and inversely proportional to cos(θ i ).Due to the weak dependence of the transmitted and reflected phases on θ i , an increase of θ i implies an increase of the GHS observed for ∆y t L and ∆y r L in figures 2(a) and (b), respectively.Remarkably, the GHS is found to be more than six times larger than the structural thickness and even exceeds the wavelength within the spectral window depicted in figure 2.
Figure 3 displays selected cuts from the contour plots of figure 2 at specific incident angles: (a) θ i = 76.76 • , (b) θ i = 78 • , (c) θ i = 80.0 • , and (d) θ i = 84.0• .In particular, figure 3(a) presents the GHS at the first critical angle, θ i = θ c1 .It can be observed that both the transmitted (black line) and reflected (red line) acoustic modes exhibit a similar trend, characterized by a maximum value of the GHS.A similar behaviour is observed in figure 3(b) for a slightly larger angle θ i = 78 • .However, if the incident angle increases further, such as θ i = 80 • and θ i = 84 • (see figures 3(c) and (d)), the peak disappears, and the GHS increases monotonically with respect to λ until it reaches a saturation value.It is noteworthy that as θ i becomes larger, the transmitted GHS (∆y t L ) becomes similar to the reflected GHS (∆y r L ).In fact, both curves are practically indistinguishable at θ i = 84 • .Note that, as expected, the GHS varies significantly when the wavelength is of the order of the scattering material's thickness, while it remains practically constant for λ ≫ h.This is because the incident wave is unable to resolve the inhomogeneities it encounters in the transmitting medium.
The differences observed between ∆y t L and ∆y r L are attributed to the coupling with transverse modes.Figures 3(a)-(c) show that ∆y t L and ∆y r L intersect at a value of λ, denoted as λ c .The following values are obtained: λ c /h = 2.58 and ∆y L /h = 6.44 for θ i = 76.76 • ; λ c /h = 2.60 and ∆y L /h = 7.57 for θ i = 78.0• ; and λ c /h = 2.62 and ∆y L /h = 10.38 for θ i = 80.0 • .It is worth noting that λ c increases as θ i increases.Furthermore, for these angles, near the first critical angle, ∆y r L > ∆y t L when λ < λ c , while ∆y r L < ∆y t L when λ > λ c .However, for larger angles, as is shown in figure 3 L for all values of λ, as mentioned before.Although for simpler systems it is possible to provide a rigorous analytical demonstration of the equality of tunneling times for transmitted and reflected wave packets [36,37], the numerical verification that the phase difference equals the constant term π/2 allows us to ensure that this property also holds, under the conditions analyzed here, when considering coupled longitudinal and transverse modes.
So far, we have compared the GHS with the layer thickness.However, it is also valuable to consider its comparison with the wavelength.Figure 4 presents the transmitted GHS normalized to λ.It can be observed that ∆y t L /λ increases when we increase the incident angle and decrease the wavelength, and that a GHS is obtained almost 20 times larger than the wavelength.
The GHS as a function of the transmission (t L ) and reflection (r L ) amplitudes from equations ( 6) and ( 9) can be written as, where ℜ and ℑ denote real part and imaginary part, respectively.It can be seen, from equations ( 17) and (18), that the behaviour of the GHS is influenced by the transmission and reflection coefficients.Figure 5 displays the reflection and  transmission coefficients R L , R T , T L , and T T as a function of λ and θ i .These coefficients satisfy the identity T L + R L + T T + R T = 1, if, as in this work, absorption is not considered.
In the post-critical region where θ i > θ c1 , longitudinal modes become evanescent.Consequently, T L grows as θ i decreases and λ increases, approaching rapidly to the case of perfect transmission (T L = 1) as shown in figure 5(a).However, ∆y t L increases when we increase the angle of incidence and decrease the wavelength.Since the coefficients T T and R T for transverse modes are very small (see figures 5(c) and (d)), we have R L ≈ 1 − T L .Despite the relatively small values of T T and R T , the coupling between longitudinal (L) and transverse (T) modes leads to the observed differences between the transmitted and reflected GHS depicted in figures 2 and 3.This coupling effect plays a crucial role in the behaviour of the GHS and contributes to the distinctive features observed in the plots.

Goos-Hänchen shift for optical vibrational modes
In this section, we study the GHS for optical vibrations.Similar to the previous section, we examine the case of an optical longitudinal wave incident on a 70 (Ge) layer sandwiched between two isotropic half-spaces of 74 (Ge) (see figure 1).At the Γ point of the Brillouin zone, the 74 (Ge) and 70 (Ge) atoms exhibit a frequency value of approximately ω T1 = ω L1 ≈ 301.5 cm −1 and ω T2 = ω L2 ≈ 309.4 cm −1 ), respectively.The parameters β 2 L = 2.8×10 −12 and β 2 T = 4×10 −12 , normalized to the speed of light, are employed for both isotopes, as reported in [34].
In this case, the relevant components of the wavevector in media I and III can be expressed as follows: According to the procedure described in section 3, the critical angles can be obtained as a function of the frequency.In particular, from the Snell's law, we obtain two critical angles.The first critical angle, denoted as θ c1 , is defined as Therefore, for θ i > θ c1 , the longitudinal mode becomes evanescent because is imaginary.The second critical angle, denoted as θ c2 , is given by For θ i > θ c2 , the transverse mode becomes evanescent because, is also imaginary.Additionally, it should be noted that k T1 has only real values for incident angles less than the critical angle θ c3 , given by For incident angles greater than θ c3 , k T1 becomes imaginary, indicating the presence of an evanescent wave.
In figure 6, the critical angles θ c1 (black line), θ c2 (red line), and θ c3 (green line) are shown as a function of the wavelength in the spectral window from λ/h = 0.79 to λ/h = 2.67.It is worth noting that λ can be computed in terms of the frequency In this figure, the θ i − λ/h-plane is divided in five different regions.Region R 1 corresponds to all the values of λ/h and θ i bellow θ c2 .In this region the wavenumbers are real, and both longitudinal and transverse modes propagate in the structure.In region R 2 , defined by the intersection of the different critical angles the wavenumber k T2 becomes imaginary, indicating that the transverse mode is evanescent within the layer.k T1 and k T2 are both imaginary in R 3 .Therefore, the transverse modes are evanescent within and outside the layer, while the longitudinal modes are propagating.In region R 4 , k L2 , k T2 , and k T1 are imaginary, indicating that the transverse and the longitudinal modes are nonpropagating within and outside the layer.Finally, in region R 5 , k L2 and k T2 are imaginary and only evanescent modes are present within the layer.
Because the GHS is observed for evanescent modes, the optical transverse modes experience GHS in regions R 2 , R 3 , R 4 , and R 5 .Meanwhile, the optical longitudinal modes experience GHS in regions R 4 and R 5 only.Figures 7  and 8 illustrate the GHS for the L and T optical modes, respectively.
In figure 7(a), we present the plot of ∆y t L /h as a function of the wavelength for five different incident angles in the region R 4 .We observe that ∆y t L exhibits values approximately one order of magnitude larger than h when λ is approximately 0.8h.Furthermore, larger values of ∆y t L are obtained as the incidence angle θ i increases.However, ∆y t L gradually decreases if the wavelength increases.It should be noted that the most substantial enhancements of ∆y t L occur when the wavelength λ is close to the thickness of the layer h.Consequently, the enhancement of ∆y t L with respect to λ is comparable to the enhancement with respect to h.For completeness, we also present in figure 7(b) the plot of ∆y t L /λ.We find a remarkable giant GHS, as large as 30 times the value of λ for the smallest wavelengths and largest incidence angles.An intriguing and noteworthy aspect of the GHS is that the transmitted and reflected GHS are equal for longitudinal modes in region R 4 .The physical character of this result is associated with the behaviour of the reflected (α r L ) and transmitted (α t L ) phases, displayed in figure 7(c).Through a numerical calculation (not shown here), it has been determined that the phase difference ∆α(λ, θ i ) = α t L − α r L is approximately ±π/2.Consequently, α t L and α r L have almost the same slope (∂α t L /∂θ i ≈ ∂α r L /∂θ i ).If these slope values are substituted into equations ( 6) and ( 9), we find that ∆y t L ≈ ∆y r L .In summary, the behaviour of the phase difference between the transmitted and reflected longitudinal optical modes, shown in figure 7(c), allows us to ensure that there are regions within the plane (θ i , λ/h) where, jointly, the general principles of temporal reversal invariance, spatial reversal invariance, and conservation of energy density flux are satisfied, similar to the one-dimensional and two-dimensional problems described above.
Figure 8 illustrates the GHS for transverse modes at some representative angles within regions from R 2 to R 5 , as described in figure 6.The upper panel focuses on the region R 4 , while the middle panel depicts curves starting in the region R 3 and entering R 4 as λ increases (see figure 6).Similarly, the lower panel includes regions R 2 and R 5 .In the left panels, the GHS is normalized to λ, while in the right panels the corresponding GHS curves are normalized to h.
In contrast to the behaviour observed for longitudinal modes in figure 7, the GHS exhibits distinct characteristics for transmitted and reflected transverse modes in region R 4 .Specifically, we observe that the GHS for transmitted modes (∆y t T ) is larger than the GHS for reflected modes (∆y r T ).Within region R 4 , both ∆y t T and ∆y r T show a monotonically decaying trend, as depicted in figure 8(a).From figure 8(d) it can be seen that the GHS for transmitted modes normalized to h decreases until certain value of λ/h and then grows slightly, this growth being more pronounced for the angle of 85 • .On the other hand, ∆y r T /h and ∆y r T /λ, depicted in dashed lines in figures 8(a) and (d), show a similar behaviour.Moreover, within this region, as the angle of incidence increases the GHS for reflected and transmitted transverse modes also increases.However, in regions R 3 and R 4 , these quantities exhibit non-monotonic behaviour as functions of both λ and θ i , as shown in figures 8(b) and (e).These unusual trends arise due to the coupling of L and T modes.Although T modes are evanescent in both regions R 3 and R 4 , they are coupled with L modes, which propagate in R 3 and decay in region R 4 .The trends in the curves change at the boundary between these two regions, where λ/h is between 1.25 and 1.50.On the left side of this boundary, in region R 3 , the GHS increases with θ i .Conversely, in region R 4 , on the right side of the boundary, the GHS shows the opposite behaviour.Notably, for the smallest values of the wavelength (0.79 < λ/h < 0.90) and an incidence angle of θ i = 65 • , the GHS of the reflected waves takes negative values, indicating an intriguing behaviour in this specific regime.Since phase changes depend on the interaction between the modes, they can be positive or negative.These phase changes are associated with the existence of GHS of both signs.For example, in a hybrid surface plasmon resonance sensor structure based on Cu-BlueP/TMDC-graphene, Z-shaped to Lorentzian-like phase changes produce GHS shifts from positive to negative [38].For our system, a numerical study of the behaviour of the phase associated with the reflected transverse mode, in the region considered above, shows an ascending convex curve and as a consequence the corresponding GHS is negative.
Finally, in regions R 2 and R 5 , analyzed in figures 8(c) and (f), the GHS exhibits similar trends to those observed in region R 4 .
For completeness, figure 9 presents the reflection and transmission coefficients for optical modes at oblique incidence.These coefficients provide insight into the energy conversion in different scattering channels.In particular, the coefficients exhibit an abrupt change in slope at the critical angle (θ c3 = 56.79• ).For high incident angles in region R 4 (θ i > 73 • ), the longitudinal mode becomes evanescent.Consequently, almost all of the energy is reflected (R L ≈ 1, see figure 9(b)).However, the evanescent longitudinal wave is still capable of excite longitudinal vibrations in the region z > h, resulting in nonzero transmission (T L ̸ = 0, see figure 9(a)).Furthermore, it can be observed that in the interval 60 • < θ i < 70 • , for regions R 3 and R 4 , the coefficient R T consistently exceeds the coefficient T T (see figures 9(c) and (d)) regardless of the incident angle and the frequency.In this interval, the transverse mode is evanescent.As a result, the frustrated total internal reflection typically exhibits a larger ∆y t T /h compared to ∆y r T /h.

Conclusions
We studied the GHS of both acoustic and optical vibrational modes in the 74 (Ge)-70 (Ge)-74 (Ge) structure.Our numerical results highlight the significant dependence of the GHS on the incident angle, the size of the system, and the wavelength of the incident vibrational mode.An interesting feature of the GHS for acoustic vibrations is that for large angles of incidence (greater than 84 • ) ∆y r L is approximately equal to ∆y t L , for all the wavelengths considered.However, for lower angles, ∆y r L and ∆y t L are equal only for certain values of the incident wavelength, λ c .This characteristic wavelength determines the range of wavelengths at which the GHS for reflected modes is greater or less than the GHS for transmitted modes.In the case of transverse optical modes, the transmitted GHS is greater than the reflected one for all incidence angles in the entire wavelength range considered.Furthermore, in general, it is observed that increasing the angle of incidence increases the GHS for both the transmitted and reflected modes; however, there is a range of wavelengths where the GHS decreases with increasing the incidence angle, which is associated with the coupling of the L and T modes.On the other hand, in the case of longitudinal optical modes, an approximately equal GHS is observed for transmitted and reflected waves, as a consequence of an approximately constant phase difference between them.This result is similar to that reported previously in [8] for SH-modes.It is worth noting that the incident angle not only affects the behaviour of the modes but also influences the magnitude of the GHS.Additionally, the magnitude of the GHS for optical and acoustic modes can exceed several times the incident wavelength or the system length obtaining giant GHS.These findings demonstrate the capacity of this type of systems that, taking advantage of the Goos-Hänchen effect, could be used to design acoustic devices.The properties of vibrational modes studied in this work may be relevant for future research on phonon transport properties in semiconductor heterostructures.partially supported by VIEP-BUAP, and Conacyt, México under Grant A1-S23120.

Figure 1 .
Figure 1.Schematic representation of the structure.The incident longitudinal L-wave forms an angle θ i ≡ α i L with the z-axis.Longitudinal (L) and transverse (T) waves are reflected in medium I forming angles α r L and α r T with the z-axis, respectively.Additionally, L and T waves are transmitted to medium III forming angles α t L and α t T with the z-axis, respectively.

Figure 2 .
Figure 2. GHS generated by the structure illustrated in figure 1 as a function of the wavelength (λ) and the incidence angle (θ i ) for (a) transmitted (∆y t L ) and (b) reflected (∆y r L ) longitudinal waves.∆y t L , ∆y r L , and λ are normalized to the layer thickness h.

Figure 3 .
Figure 3. Transmitted (∆y t L ) and reflected (∆y r L ) longitudinal GHS as a function of wavelength (λ) for incident angles (a) θ i = 76.76 • , (b) θ i = 78 • , (c) θ i = 80.0 • , and (d) θ i = 84.0• .∆y t L , ∆y r L , and λ are normalized to the layer thickness h.The inset in (c) is a zoom of the intersection between ∆y t L and ∆y r L .

Figure 4 .
Figure 4. Normalized GHS ∆y t L /λ as a function of λ/h and θ i .

Figure 5 .
Figure 5. Transmission and reflection coefficients for longitudinal and transverse acoustic modes as a function of the normalized wavelength λ/h and the incidence angle θ i .

Figure 8 .
Figure 8. GHS is shown as a function of λ for transmitted (solid lines) and reflected (dashed lines) transverse modes at various angles in regions (a) R 4 , (b) R 3 , R 4 , and (c) R 2 , R 5 .GHS is normalized to λ, while lambda in the horizontal axis is normalized to h.Panels (d), (e), and (f) display the h-normalized GHS of (a), (b), and (c) respectively.

Figure 9 .
Figure 9.The transmission and reflection coefficients, T and R, respectively as a function of the incident angle and the normalized wavelength for both longitudinal and transverse optical modes.