Refraction of space-time wave packets: III. Experiments at oblique incidence

Snell’s law governs the change in the propagation direction of a monochromatic plane wave incident onto a planar interface between two optical media Saleh and Teich (2019). If $n_{1}$ and $n_{2}$ are the refractive indices of the two media, and $\phi_{1}$ and $\phi_{2}$ are the propagation angles with respect to the normal to the interface for the incident and refracted waves, respectively, then Snell’s law dictates that $n_{1}\sin{\phi_{1}}\!=\!n_{2}\sin{\phi_{2}}$. Although this result, strictly speaking, applies to only monochromatic plane waves, nevertheless its utility is typically extended in practice to conventional pulsed beams where it can provide an adequate approximation, especially for narrowband paraxial fields in which the spatial and temporal degrees of freedom (DoFs) are uncoupled. Crucially, this entails that the group velocity of the transmitted wave packet depends solely on the local optical properties of the second medium, and is independent of the incident angle. In other words, no ‘memory’ of the incident wave packet is retained as far as the group velocity of the transmitted wave packet is concerned.

In contrast to conventional wave packets, space-time (ST) wave packets Kondakci and Abouraddy (2016); Parker and Alonso (2016) are a unique class of pulsed optical beams in which the spatial and temporal DoFs are non-separable, and are instead inextricably intertwined Reivelt and Saari (2003); Kiselev (2007); Turunen and Friberg (2010); Hernández-Figueroa et al. (2014). ST wave packets are endowed with a precise spatio-temporal structure in which each spatial frequency is tightly associated with a single wavelength (or temporal frequency) Donnelly and Ziolkowski (1993); Saari and Reivelt (2004); Longhi (2004); Kondakci et al. (2019); Hall et al. (2021). Uniquely, this spatio-temporal structure determines the group velocity of the ST wave packet Salo and Salomaa (2001); Recami et al. (2003); Valtna et al. (2007); Zamboni-Rached and Recami (2008); Kondakci and Abouraddy (2017); Efremidis (2017); Wong and Kaminer (2017); Kondakci and Abouraddy (2019); Bhaduri et al. (2019a). Consequently, the rearrangement of the field structure upon refraction at a planar interface leads to a change in the group velocity of the transmitted wave packet that depends on the group velocity of the incident wave packet. Because of this, ST wave packets – in contrast to conventional wave packets – retain a ‘memory’ of the incident wave packet, which is the basis for the fascinating refractive phenomena exhibited by ST wave packets at normal and oblique incidence Bhaduri et al. (2020).

In paper (I) of this series, we presented a theoretical study of the refraction of ST wave packets at normal and oblique incidence on a planar interface between two non-dispersive, homogeneous, isotropic dielectrics Yessenov et al. (2021). At normal incidence, the above-described memory effect is the basis for group-velocity invariance, anomalous refraction, and group-velocity inversion Bhaduri et al. (2020). These phenomena extend to oblique incidence, although the conditions for their realization change with incident angle. Crucially, a new refractive phenomenon emerges at oblique incidence: the group velocity of the transmitted wave packet changes with the incident angle when all else is held fixed. Indeed, the group velocity increases with incident angle upon refraction into a higher-index medium when the incident wave packet is subluminal, and it decreases when the wave packet is superluminal Bhaduri et al. (2020); Yessenov et al. (2021).

In paper (II), we provided experimental confirmation of the predicted phenomena at normal incidence Allende Motz et al. (2021a). Here we examine experimentally in detail the refraction of ST wave packets at oblique incidence at a planar interface between two non-dispersive, homogeneous, isotropic dielectrics using the interferometric measurement strategy described in paper (II). After a brief overview of the basic law of refraction governing baseband ST wave packets at oblique incidence, we confirm the predicted dependence at oblique incidence of the group velocity for the transmitted wave packet on that of the incident wave packet and on the incidence angle. We then verify the predicted changes that occur at oblique incidence in the conditions required to realize group-velocity invariance and group-velocity inversion. Finally, we examine a proposal that makes use of these unique characteristics to realize blind synchronization of multiple receivers via pulses emitted from a single transmitter with no a priori knowledge of the receiver positions except that they are situated at the same depth below an interface between two media. Building on our recent observation of isochronous ST wave packets that traverse a planar slab at a fixed group delay independently of the angle of incidence (despite the change in distance traversed) Allende Motz et al. (2021b), we report here a proof-of-principle realization of the blind synchronization scheme.

A ST wave packet is a pulsed beam having finite spatial and temporal bandwidths in which each spatial frequency $k_{x}$ is associated with a single temporal frequency $\omega$; here $x$ and $z$ are the transverse and axial coordinates, respectively, and $k_{x}$ and $k_{z}$ are the corresponding components of the wave vector. We identify the propagation direction with the $z$-axis, we assume a spatial spectrum that is symmetric around $k_{x}\!=\!0$, and we assign the temporal frequency $\omega_{\mathrm{o}}$ to the spatial frequency $k_{x}\!=\!0$. The spectral support domain for a ‘baseband’ ST wave packet Yessenov et al. (2019a) in a non-dispersive medium of index $n$ lies at the intersection of the light-cone $k_{x}^{2}+k_{z}^{2}\!=\!n^{2}(\tfrac{\omega}{c})^{2}$ with a tilted plane $\omega\!=\!\omega_{\mathrm{o}}+(k_{z}-nk_{\mathrm{o}})c\tan{\theta}$ that is parallel to the $k_{x}$-axis and makes an angle $\theta$ (the spectral tilt angle) with respect to the $k_{z}$-axis, where $k_{\mathrm{o}}\!=\!\omega_{\mathrm{o}}/c$ and $c$ is the speed of light in vacuum Yessenov et al. (2019b, a, c). This construction results in a propagation-invariant wave packet transported rigidly at a group velocity $\widetilde{v}\!=\!c\tan{\theta}\!=\!c/\widetilde{n}$, where $\widetilde{n}\!=\!\cot{\theta}$ is the group index. In other words, the group velocity is determined by an internal DoF (the spectral tilt angle $\theta$) that is related to the spatio-temporal field structure. Note that the ST wave packet reverts to a plane-wave pulse when $\widetilde{n}\!=\!n$. Within this framework, $\omega_{\mathrm{o}}$ is either the minimum temporal frequency when the wave packet is superluminal ($\widetilde{v}\!>\!c/n$), or the maximum temporal fewquency when the wave packet is subluminal ($\widetilde{v}\!<\!c/n$) Yessenov et al. (2019a).

When the ST wave packet is normally incident on the interface, the wave vector associated with $\omega\!=\!\omega_{\mathrm{o}}$ is perpendicular to the interface [Fig. 1(a)], whereas all other temporal frequencies $\omega$ are not. Two quantities are conserved across the interface: the transverse momentum $k_{x}$ and the temporal frequency $\omega$. If the refractive indices of the media are $n_{1}$ and $n_{2}$, and the group indices of the incident and transmitted wave packets are $\widetilde{n}_{1}$ and $\widetilde{n}_{2}$, respectively, then it can be shown that:

$$n_{1}(n_{1}-\widetilde{n}_{1})=n_{2}(n_{2}-\widetilde{n}_{2}).$$

(1)

This law of refraction for ST wave packets was derived in paper (I) Yessenov et al. (2021) and explored experimentally in paper (II) Allende Motz et al. (2021a); see also Ref. Bhaduri et al. (2020). The quantity $n(n-\widetilde{n})$ is related to the curvature of the spatio-temporal spectrum $\tfrac{\omega-\omega_{\mathrm{o}}}{\omega_{\mathrm{o}}}\!=\!\tfrac{k_{x}^{2}/k_{\mathrm{o}}^{2}}{2n(n-\widetilde{n})}$ when projected onto the $(k_{x},\tfrac{\omega}{c})$-plane. We have denoted this newly identified refractive invariant the ‘spectral curvature’. The law of refraction in Eq. 1 thus expresses the invariance of the spectral curvature across the planar interface at normal incidence.

At oblique incidence, the wave vector associated with $\omega_{\mathrm{o}}$ is incident at an angle $\phi_{1}$ with respect to the normal to the interface [Fig. 1(b)]. The group indices of the incident and transmitted ST wave packets satisfy a modified law Bhaduri et al. (2020); Yessenov et al. (2021):

$$n_{1}(n_{1}-\widetilde{n}_{1})\cos^{2}{\phi_{1}}=n_{2}(n_{2}-\widetilde{n}_{2})\cos^{2}{\phi_{2}},$$

(2)

where $\phi_{2}$ is obtained from Snell’s law $n_{1}\sin{\phi_{1}}\!=\!n_{2}\sin{\phi_{2}}$. Equation 2 indicates the invariance of a new spectral curvature $n(n-\widetilde{n})\cos^{2}{\phi}$ at oblique incidence. The two curves corresponding to Eq. 1 and Eq. 2 intersect at the luminal point $\widetilde{n}_{1}\!=\!n_{1}$ and $\widetilde{n}_{2}\!=\!n_{2}$, where the ST wave packets revert to plane-wave pulses [Fig. 3], and Eq. 2 reverts back to Eq. 1 at normal incidence $\phi_{1}\!=\!\phi_{2}\!=\!0$.

The rationale for the modified oblique-incidence spectral curvature can be elucidated by reference to Fig. 1(b) and Fig. 2. The spectral support domain of the incident ST wave packet in the coordinate system $(x,z)$ – which is aligned with the propagation direction of the wave packet – lies at the intersection of the light-cone with the plane $\mathcal{P}_{\mathrm{B}}(\theta)$ [Fig. 2(a)]. At normal incidence, the projection of this spectrum onto the $(k_{x},\tfrac{\omega}{c})$-plane is invariant across the interface Bhaduri et al. (2019a, 2020). At oblique incidence, the transverse wave number $k_{x}$ is not parallel to the interface, and therefore is not conserved. However, in the coordinate system $(x^{\prime},z^{\prime})$ that is aligned with the interface rather than with the propagation direction of the ST wave packet, the transverse wave number $k_{x}^{\prime}\!=\!k_{x}\cos{\phi}-k_{z}\sin{\phi}$ is conserved. In this coordinate system, the spectral support domain is rotated an angle $\phi_{1}$ around the $\tfrac{\omega}{c}$-axis [Fig. 2(b)], and the projection of this new spectrum onto the $(k_{x}^{\prime},\tfrac{\omega}{c})$-plane is invariant across the interface [Fig. 2(c)]. It can be shown that $k_{x}\cos{\phi}$ is invariant to first order (in the small-angle approximation $\Delta k_{x}\!\ll\!k_{\mathrm{o}}$), and the new invariant spectral curvature at oblique incidence is $n(n-\widetilde{n})\cos^{2}{\phi}$. Because the opening angle of the light-cone changes from $\arctan{(n_{1})}$ in the first medium to $\arctan{(n_{2})}$ in the second, the invariance of the spectral projection onto the $(k_{x}^{\prime},\tfrac{\omega}{c})$-plane produces a new spectral support domain for the transmitted wave packet on the surface of the light-cone [Fig. 2(d)]. A rotation through an angle $\phi_{2}$ (rather than $\phi_{1}$) around the $\tfrac{\omega}{c}$-axis returns the spectral support domain to the $(x,z)$ coordinate system aligned with the propagation direction of the transmitted wave packet [Fig. 2(e)].

To verify the law of refraction in Eq. 2 for baseband ST wave packets at oblique incidence, we make use of the interferometric procedure outlined in paper (II); see also Kondakci and Abouraddy (2019); Bhaduri et al. (2019a). The planar slabs used in our measurements are placed in the common path of the co-aligned ST wave packet and the reference pulse, and we tilt the sample with respect to the incident wave packets. We plot the measurement results in Fig. 3. Here we fix the angle of incidence $\phi_{1}$ and sweep the spectral tilt angle $\theta_{1}$ of the ST wave packet incident from free space onto a 5-mm-thick layer of sapphire. For each value of $\theta_{1}$, we measure the group delay incurred by the wave packet across the layer, from which we estimate $\theta_{2}$. The measurements are carried out at normal incidence $\phi_{1}\!=\!0$ and at oblique incidence $\phi_{1}\!=\!20^{\circ}$. The measurements verify for the first time the change in the law of refraction for ST wave packets with respect to that at normal incidence, which was demonstrated in Bhaduri et al. (2020). It is clear that the two curves for normal and oblique incidence in Fig. 3 intersect at the point corresponding to the luminal condition $(\theta_{1},\theta_{2})\!=\!(\theta_{1}^{\mathrm{L}},\theta_{2}^{\mathrm{L}})$. The curve for oblique incidence is shifted below that for normal incidence in the superluminal regime ($\theta_{1}\!>\!\theta_{1}^{\mathrm{L}}$ and $\widetilde{v}\!>\!c/n$) and above it in the subluminal regime ($\theta_{1}\!<\!\theta_{1}^{\mathrm{L}}$ and $\widetilde{v}\!<\!c/n$). We next explore the far-reaching ramifications of this change.

Figure 3 yields the relationship between the group indices $\widetilde{n}_{1}\!=\!\cot{\theta_{1}}$ and $\widetilde{n}_{2}\!=\!\cot{\theta_{2}}$ for the incident and refracted ST wave packets, respectively, at a fixed incident angle $\phi_{1}$. We next examine the change in the group index $\widetilde{n}_{2}$ for the transmitted wave packet as we vary the incidence angle $\phi_{1}$ while holding fixed the group index $\widetilde{n}_{1}$ for the incident wave packet. In Fig. 4(a), we plot the calculated $\widetilde{n}_{2}$ while varying both $\widetilde{n}_{1}$ and $\phi_{1}$ for incidence from free space $n_{1}\!=\!1$ onto sapphire $n_{2}\!=\!1.76$. This calculation highlights the above-mentioned ‘memory’ effect. The group index $\widetilde{n}_{2}$ of the transmitted wave packet depends not only the refractive index of the second medium $n_{2}$, but also on the characteristics of the incident wave packet: its group index $\widetilde{n}_{1}$ and incident angle $\phi_{1}$. The luminal condition for the incident ST wave packet is $\widetilde{n}_{1}\!=\!n_{1}\!=\!1$, in which case this wave packet is simply a plane-wave pulse, and the refracted wave packet is also a luminal plane-wave pulse $\widetilde{n}_{2}\!=\!n_{2}\!=\!1.76$. For subluminal incidence ST wave packets $\widetilde{n}_{1}\!>\!n_{1}$, $\widetilde{n}_{2}$ decreases with $\phi_{2}$; i.e., the group velocity of the transmitted subluminal wave packet increases with $\phi_{1}$ in the higher-index medium. Conversely, for superluminal incident ST wave packets $\widetilde{n}_{1}\!<\!n_{1}$, $\widetilde{n}_{2}$ increases with $\phi_{1}$; i.e., the group velocity of the transmitted superluminal wave packet increases with $\phi_{1}$ in the higher-index medium. In the case of a conventional wave packet whose spatial and temporal DoFs are separable, $\widetilde{n}_{2}$ is independent of $\phi_{1}$.

This behavior can be understood on the basis of Eq. 2. Substituting for $\phi_{2}$ in terms of $\phi_{1}$ from Snell’s law yields:

$$\frac{\widetilde{n}_{2}(\phi_{1})}{n_{2}}=1+\left(\frac{\widetilde{n}_{1}}{n_{1}}-1\right)\eta(\phi_{1}),$$

(3)

where the factor $\eta(\phi_{1})$ depends on the indices $n_{1}$ and $n_{2}$ through

$$\eta(\phi_{1})=\frac{\cos^{2}{\phi_{1}}}{(\frac{n_{2}}{n_{1}})^{2}-\sin^{2}{\phi_{1}}}.$$

(4)

The behavior of $\eta(\phi_{1})$ depends on the ratio $n_{1}/n_{2}$. When $n_{1}\!<\!n_{2}$, $\eta(\phi_{1})$ drops monotonically from its initial value to $0$ at $\phi_{1}\!=\!90^{\circ}$. Because the term $(\tfrac{\widetilde{n}_{1}}{n_{1}}-1)$ is positive in the subluminal regime $\widetilde{n}_{1}\!>\!n_{1}$, the decrease of $\eta(\phi_{1})$ with $\phi_{1}$ results in a decrease of $\widetilde{n}_{2}(\phi_{1})$ with $\phi_{1}$. In the superluminal regimes, the term $(\tfrac{\widetilde{n}_{1}}{n_{1}}-1)$ is negative, and $\widetilde{n}_{2}(\phi_{1})$ increase with $\phi_{1}$. On the other hand, when $n_{1}\!>\!n_{2}$, $\eta(\phi_{1})$ increases monotonically with $\phi_{1}$ and reaches a singularity when $\sin{\phi_{1}}\!=\!n_{2}/n_{1}$, corresponding to the critical angle (total internal reflection) at the interface. Consequently, the opposite trends for $\widetilde{n}_{2}$ with $\phi_{1}$ ensue.

Our measurements confirm this predicted behavior. In Fig. 4(b,c) we plot the change in the refracted group index $\widetilde{n}_{2}$ with $\phi_{1}$ with respect to that at normal incidence, $\Delta\widetilde{n}_{2}\!=\!\widetilde{n}_{2}(\phi_{1})-\widetilde{n}_{2}(0)$, for incidence from free space onto two materials: MgF₂ in Fig. 4(b) and sapphire in Fig. 4(c). For each material, we carry out measurements with ST wave packets synthesized in free space in the subluminal ($\theta_{1}\!<\!45^{\circ}$) and superluminal ($\theta_{1}\!>\!45^{\circ}$) regimes. We note several general observations about the results that are expected from our analysis. First, $\widetilde{n}_{2}(\phi_{1})$ deviates monotonically from the normal-incidence value $\widetilde{n}_{2}(0)$. Second, the group index drops at oblique incidence $\widetilde{n}_{2}(\phi_{1})\!<\!\widetilde{n}_{2}(0)$ in the subluminal regime, and increases $\widetilde{n}_{2}(\phi_{1})\!>\!\widetilde{n}_{2}(0)$ in the superluminal regime. That is, the group velocity of a refracted subluminal ST wave packet increases with incident angle in the higher-index medium. Conversely, the group velocity of a refracted superluminal ST wave packet decreases with $\phi_{1}$. These features are critical for the blind synchronization scheme we investigate below.

The law of refraction at normal incidence (Eq. 1) predicts three phenomena that occur for any pair of media: (1) group velocity invariance $\widetilde{v}_{2}\!=\!\widetilde{v}_{1}$, which occurs when $\widetilde{n}_{1}\!=\!\widetilde{n}_{\mathrm{th}}\!=\!n_{1}+n_{2}$; (2) anomalous refraction whereby $\widetilde{v}_{2}\!>\!\widetilde{v}_{1}$ when $n_{2}\!>\!n_{1}$, which occurs when $\widetilde{n}_{1}\!>\!\widetilde{n}_{\mathrm{th}}$; and (3) group-velocity inversion $\widetilde{v}_{2}\!=\!-\widetilde{v}_{1}$, which occurs when $\widetilde{n}_{1}\!=\!n_{1}-n_{2}$; see Fig. 3. These three phenomena were confirmed experimentally in Ref. Bhaduri et al. (2020) and in more detail in paper (II) Allende Motz et al. (2021a). Despite the difference between the law of refraction at oblique incidence (Eq. 2) from that at normal incidence (Eq. 1), these three phenomena are still realizable, albeit with modifications to the conditions to be satisfied.

First, group-velocity invariance occurs at oblique incidence when Yessenov et al. (2021):

$$\widetilde{n}_{1}=\widetilde{n}_{\mathrm{th}}(\phi_{1})=\frac{n_{1}+n_{2}}{1+\frac{n_{1}}{n_{2}}\sin^{2}{\phi_{1}}},$$

(5)

where $\widetilde{n}_{\mathrm{th}}(\phi_{1})\!<\!\widetilde{n}_{\mathrm{th}}(0)\!=\!n_{1}+n_{2}$. We plot in Fig. 5 measurements of the group delay incurred upon traversing equal lengths ($L\!=\!5$ mm) of free space ($n_{1}\!=\!1$) and MgF₂ ($n_{2}\!=\!1.38$). At normal incidence $\widetilde{n}_{\mathrm{th}}(0)\!=\!2.38$ corresponding to $\theta_{1}\!\approx\!22.8^{\circ}$ [Fig. 5(a)], and at oblique incidence $\phi_{1}\!=\!30^{\circ}$ we have $\widetilde{n}_{\mathrm{th}}(30^{\circ})\!=\!1.86$, corresponding to $\theta\!\approx\!28.2^{\circ}$ [Fig. 5(b)]. In both cases the ST wave packets accrue approximately the same delay in equal lengths of free space and MgF₂. Whereas anomalous refraction occurs at normal incidence when $\theta_{1}\!<\!22.8^{\circ}$, this regime is expanded for oblique incidence to $\theta_{1}\!<\!28.2^{\circ}$ at $\phi_{1}\!=\!30^{\circ}$.

Second, group-velocity inversion occurs at oblique incidence when Yessenov et al. (2021):

$$\widetilde{n}_{1}(\phi_{1})=\frac{n_{1}-n_{2}}{1-\frac{n_{1}}{n_{2}}\sin^{2}{\phi_{1}}},$$

(6)

where $\widetilde{n}_{1}(\phi_{1})\!<\!\widetilde{n}_{1}(0)\!=\!n_{1}-n_{2}$ when $n_{2}\!>\!n_{1}$. We plot in Fig. 6 the measured group delay incurred upon traversing equal lengths ($L\!=\!5$ mm) of sapphire and MgF₂ separately, and after traversing a bilayer of the two media. The required group index for a ST wave packet in free space that experiences group-velocity invariance in sapphire and MgF₂ is $\widetilde{n}_{0}\!\approx\!-1.43$ ($\theta_{0}\!\approx\!145^{\circ}$); see Fig. 6(a). At oblique incidence ($\phi_{1}\!=\!30^{\circ}$), the required free-space group index to realize group-velocity inversion is $\widetilde{n}_{0}\approx\!-2.25$ ($\theta_{0}\!\approx\!156^{\circ}$); see Fig. 6(b). In both cases, the group delays measured in the individual layers are approximately equal in magnitude but opposite in sign. Consequently, the group delay in the bilayer almost vanishes as shown.

The proof-of-principle experimental demonstration of blind synchronization described here extends our recent report on isochronous ST wave packets Allende Motz et al. (2021b). These are wave packets that traverse a material layer after accruing a fixed group delay independently of the angle of incidence, despite traveling larger distances in the layer ($L/\cos{\phi_{2}}$, where $L$ is the layer thickness). This can be seen as a special case of blind synchronization where we set $d_{1}\!=\!0$. An increase in group velocity with incident angle for isochronous ST wave packets is required to compensate for the increase in propagation distance within the layer. This is achieved, as we have done here for blind synchronization, by employing subluminal incident ST wave packets. However, the blind synchronization scenario is more stringent because the necessary increase in group velocity with incident angle is larger: this change must counterbalance the increase in propagation distance within both media. As such, an incident ST wave packet with larger $\widetilde{n}_{1}$ (deeper within the subluminal regime) is required for blind synchronization compared to isochronous ST wave packets. Pursuing the concept of blind synchronization further requires increasing the propagation distance over which this effect is realized by reducing the so-called ‘spectral uncertainty’ Yessenov et al. (2019b); Bhaduri et al. (2019b); Kondakci et al. (2019), which is the unavoidable ‘fuzziness’ in the association between spatial and temporal frequencies in the wave packet spectrum. In general, reducing the spectral uncertainty to increase the propagation distance requires increasing the system’s numerical aperture.

The work reported in this paper sequence, in addition to Bhaduri et al. (2020), has examined the refraction of ST wave packets in the simplest scenario: at a planar interface between two non-dispersive, homogeneous, isotropic dielectrics. Future work will be directed to an exploration of other important configurations. Paramount amongst these are the cases of dispersive and anisotropic media, which will be critical as a prelude to studying nonlinear interactions involving ST wave packets. Additionally, it will be useful to study the refraction of ST wave packets in which both transverse dimensions are included (ST needles rather than ST sheets), and ST wave packets in which a parameter is controlled along the propagation axis, such as wave packets that accelerate or decelerate Yessenov and Abouraddy (2020), or are endowed with axial spectral encoding Allende Motz et al. (2021c).

In conclusion, we have verified experimentally the law of refraction for ST wave packets obliquely incident on a planar interface between two non-dispersive, homogeneous, isotropic dielectrics (Eq. 2). This law was verified first by changing the group velocity of the incident ST wave packet at a fixed angle of incidence, and, second, by changing the angle of incidence while holding the group velocity of the incident wave packet fixed. Our measurements reveal that the group velocity of the transmitted wave packet increases with the incident angle in the subluminal regime (when the index of the second medium is higher than that of the first); conversely, the group velocity of the transmitted wave packet decreases with incident angle in the superluminal regime. Furthermore, we observed the shift that occurs in the conditions for group-velocity invariance and group-velocity inversion at oblique incidence with respect to those at normal incidence. These findings made possible the first observation of blind synchronization, whereby ST wave packets produced from a fixed source (Tx) are arranged to arrive simultaneously at different receivers (Rx) at a priori unknown positions, except that they are located at a fixed depth beyond an interface.

U.S. Office of Naval Research (ONR) contract N00014-17-1-2458.