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

Murat Yessenov,1 Alyssa M. Allende Motz,1 Basanta Bhaduri,1 and Ayman F. Abouraddy1, ∗

1CREOL, The College of Optics & Photonics, University of Central Florida, Orlando, FL 32816, USA

The refraction of space-time (ST) wave packets at planar interfaces between non-dispersive, ho-mogeneous, isotropic dielectrics exhibit fascinating phenomena, even at normal incidence. Examplesof such refractive phenomena include group-velocity invariance across the interface, anomalous re-fraction, and group-velocity inversion. Crucial differences emerge at oblique incidence with respectto the results established at normal incidence. For example, the group velocity of the refracted STwave packet can be tuned simply by changing the angle of incidence. In paper (III) of this sequence,we present experimental verification of the refractive phenomena exhibited by ST wave packets atoblique incidence that were predicted in paper (I). We also examine a proposal for ‘blind synchro-nization’ whereby identical ST wave packets arrive simultaneously at different receivers without apriori knowledge of their locations except that they are all located at the same depth beyond aninterface between two media. A first proof-of-principle experimental demonstration of this effect isprovided.

I. INTRODUCTION

Snell’s law governs the change in the propagation di-rection of a monochromatic plane wave incident ontoa planar interface between two optical media [1]. Ifn1 and n2 are the refractive indices of the two media,and φ1 and φ2 are the propagation angles with respectto the normal to the interface for the incident and re-fracted waves, respectively, then Snell’s law dictates thatn1 sinφ1 =n2 sinφ2. Although this result, strictly speak-ing, applies to only monochromatic plane waves, nev-ertheless its utility is typically extended in practice toconventional pulsed beams where it can provide an ade-quate approximation, especially for narrowband paraxialfields in which the spatial and temporal degrees of free-dom (DoFs) are uncoupled. Crucially, this entails thatthe group velocity of the transmitted wave packet de-pends solely on the local optical properties of the secondmedium, and is independent of the incident angle. Inother words, no ‘memory’ of the incident wave packet isretained as far as the group velocity of the transmittedwave packet is concerned.

In contrast to conventional wave packets, space-time(ST) wave packets [2, 3] are a unique class of pulsed op-tical beams in which the spatial and temporal DoFs arenon-separable, and are instead inextricably intertwined[4–7]. ST wave packets are endowed with a precise spatio-temporal structure in which each spatial frequency istightly associated with a single wavelength (or temporalfrequency) [8–12]. Uniquely, this spatio-temporal struc-ture determines the group velocity of the ST wave packet[13–21]. Consequently, the rearrangement of the fieldstructure upon refraction at a planar interface leads toa change in the group velocity of the transmitted wavepacket that depends on the group velocity of the incidentwave packet. Because of this, ST wave packets – in con-trast to conventional wave packets – retain a ‘memory’

∗ corresponding author: raddy@creol.ucf.edu

of the incident wave packet, which is the basis for thefascinating refractive phenomena exhibited by ST wavepackets at normal and oblique incidence [22].

In paper (I) of this series, we presented a theoreticalstudy of the refraction of ST wave packets at normal andoblique incidence on a planar interface between two non-dispersive, homogeneous, isotropic dielectrics [23]. Atnormal incidence, the above-described memory effect isthe basis for group-velocity invariance, anomalous refrac-tion, and group-velocity inversion [22]. These phenomenaextend to oblique incidence, although the conditions fortheir realization change with incident angle. Crucially,a new refractive phenomenon emerges at oblique inci-dence: the group velocity of the transmitted wave packetchanges with the incident angle when all else is held fixed.Indeed, the group velocity increases with incident angleupon refraction into a higher-index medium when the in-cident wave packet is subluminal, and it decreases whenthe wave packet is superluminal [22, 23].

In paper (II), we provided experimental confirmationof the predicted phenomena at normal incidence [24].Here we examine experimentally in detail the refractionof ST wave packets at oblique incidence at a planar inter-face between two non-dispersive, homogeneous, isotropicdielectrics using the interferometric measurement strat-egy described in paper (II). After a brief overview ofthe basic law of refraction governing baseband ST wavepackets at oblique incidence, we confirm the predicteddependence at oblique incidence of the group velocityfor the transmitted wave packet on that of the incidentwave packet and on the incidence angle. We then verifythe predicted changes that occur at oblique incidence inthe conditions required to realize group-velocity invari-ance and group-velocity inversion. Finally, we examine aproposal that makes use of these unique characteristicsto realize blind synchronization of multiple receivers viapulses emitted from a single transmitter with no a pri-ori knowledge of the receiver positions except that theyare situated at the same depth below an interface be-tween two media. Building on our recent observation ofisochronous ST wave packets that traverse a planar slab

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FIG. 1: (a) Normal and (b) oblique incidence of a STwave packet on a planar interface.

at a fixed group delay independently of the angle of inci-dence (despite the change in distance traversed) [25], wereport here a proof-of-principle realization of the blindsynchronization scheme.

II. LAW OF REFRACTION FOR SPACE-TIMEWAVE PACKETS AT OBLIQUE INCIDENCE

A ST wave packet is a pulsed beam having finite spatialand temporal bandwidths in which each spatial frequencykx is associated with a single temporal frequency ω; herex and z are the transverse and axial coordinates, respec-tively, and kx and kz are the corresponding componentsof the wave vector. We identify the propagation direc-tion with the z-axis, we assume a spatial spectrum thatis symmetric around kx =0, and we assign the temporalfrequency ωo to the spatial frequency kx = 0. The spec-tral support domain for a ‘baseband’ ST wave packet[26] in a non-dispersive medium of index n lies at theintersection of the light-cone k2x + k2z = n2(ω

c )2 with atilted plane ω=ωo + (kz − nko)c tan θ that is parallel tothe kx-axis and makes an angle θ (the spectral tilt an-gle) with respect to the kz-axis, where ko = ωo/c and cis the speed of light in vacuum [26–28]. This construc-tion results in a propagation-invariant wave packet trans-ported rigidly at a group velocity v=c tan θ=c/n, wheren= cot θ is the group index. In other words, the groupvelocity is determined by an internal DoF (the spectraltilt angle θ) that is related to the spatio-temporal fieldstructure. Note that the ST wave packet reverts to aplane-wave pulse when n = n. Within this framework,ωo is either the minimum temporal frequency when thewave packet is superluminal (v > c/n), or the maximumtemporal fewquency when the wave packet is subluminal(v<c/n) [26].

When the ST wave packet is normally incident on theinterface, the wave vector associated with ω=ωo is per-pendicular to the interface [Fig. 1(a)], whereas all other

temporal frequencies ω are not. Two quantities are con-served across the interface: the transverse momentum kxand the temporal frequency ω. If the refractive indicesof the media are n1 and n2, and the group indices of theincident and transmitted wave packets are n1 and n2,respectively, then it can be shown that:

n1(n1 − n1) = n2(n2 − n2). (1)

This law of refraction for ST wave packets was derivedin paper (I) [23] and explored experimentally in paper(II) [24]; see also Ref. [22]. The quantity n(n − n) isrelated to the curvature of the spatio-temporal spectrumω−ωo

ωo=

k2x/k

2o

2n(n−n) when projected onto the (kx,ωc )-plane.

We have denoted this newly identified refractive invariantthe ‘spectral curvature’. The law of refraction in Eq. 1thus expresses the invariance of the spectral curvatureacross the planar interface at normal incidence.

At oblique incidence, the wave vector associated withωo is incident at an angle φ1 with respect to the normal tothe interface [Fig. 1(b)]. The group indices of the incidentand transmitted ST wave packets satisfy a modified law[22, 23]:

n1(n1 − n1) cos2 φ1 = n2(n2 − n2) cos2 φ2, (2)

where φ2 is obtained from Snell’s law n1 sinφ1 =n2 sinφ2.Equation 2 indicates the invariance of a new spectralcurvature n(n − n) cos2 φ at oblique incidence. The twocurves corresponding to Eq. 1 and Eq. 2 intersect at theluminal point n1 = n1 and n2 = n2, where the ST wavepackets revert to plane-wave pulses [Fig. 3], and Eq. 2reverts back to Eq. 1 at normal incidence φ1 =φ2 =0.

The rationale for the modified oblique-incidence spec-tral curvature can be elucidated by reference to Fig. 1(b)and Fig. 2. The spectral support domain of the inci-dent ST wave packet in the coordinate system (x, z) –which is aligned with the propagation direction of thewave packet – lies at the intersection of the light-conewith the plane PB(θ) [Fig. 2(a)]. At normal incidence,the projection of this spectrum onto the (kx,

ωc )-plane

is invariant across the interface [21, 22]. At oblique in-cidence, the transverse wave number kx is not parallelto the interface, and therefore is not conserved. How-ever, in the coordinate system (x′, z′) that is alignedwith the interface rather than with the propagation direc-tion of the ST wave packet, the transverse wave numberk′x = kx cosφ − kz sinφ is conserved. In this coordinatesystem, the spectral support domain is rotated an an-gle φ1 around the ω

c -axis [Fig. 2(b)], and the projectionof this new spectrum onto the (k′x,

ωc )-plane is invariant

across the interface [Fig. 2(c)]. It can be shown thatkx cosφ is invariant to first order (in the small-angle ap-proximation ∆kx� ko), and the new invariant spectralcurvature at oblique incidence is n(n − n) cos2 φ. Be-cause the opening angle of the light-cone changes fromarctan (n1) in the first medium to arctan (n2) in the sec-ond, the invariance of the spectral projection onto the(k′x,

ωc )-plane produces a new spectral support domain

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FIG. 2: (a) Representation of the spectral support domain of a ST wave packet in a medium of refractive index n1in the (x, z) coordinate system that is aligned with the propagation direction of the ST wave packet. The spectrum

lies at the intersection of the light-cone with a plane that is parallel to the kx-axis and makes an angle θ1 withrespect to the kz-axis in (kx, kz,

ωc ) space. The z-axis is aligned with the plane-wave component at ω=ωo and

kx =0. (b) The spectral support domain at oblique incidence represented in the coordinate system (x′, z′) that isaligned with the interface but not with the propagation direction of the wave packet. The spectral support domainis rotated by φ1 (the incident angle) with respect to the ω

c -axis in (k′x, k′z,

ωc ) space. (c) The spectral projection onto

the (k′x,ωc )-plane from (b) is invariant across the interface at oblique incidence (dotted parabolic curve centered at

k′x 6=0). At normal incidence, the spectral projection onto the (kx,ωc )-plane from (a) is invariant across the interface

(solid parabolic curve centered at kx =0). (d) Spectral support domain of the transmitted ST wave packet afteroblique incidence. (e) Representation of the spectral support domain of a ST wave packet in a medium of refractive

index n2. The new spectral tilt angle θ2 is related to θ1 via Eq. 2.

for the transmitted wave packet on the surface of thelight-cone [Fig. 2(d)]. A rotation through an angle φ2(rather than φ1) around the ω

c -axis returns the spectralsupport domain to the (x, z) coordinate system alignedwith the propagation direction of the transmitted wavepacket [Fig. 2(e)].

III. CONFIRMING THE LAW OFREFRACTION AT OBLIQUE INCIDENCE

To verify the law of refraction in Eq. 2 for basebandST wave packets at oblique incidence, we make use of theinterferometric procedure outlined in paper (II); see also[20, 21]. The planar slabs used in our measurements areplaced in the common path of the co-aligned ST wavepacket and the reference pulse, and we tilt the samplewith respect to the incident wave packets. We plot themeasurement results in Fig. 3. Here we fix the angleof incidence φ1 and sweep the spectral tilt angle θ1 ofthe ST wave packet incident from free space onto a 5-mm-thick layer of sapphire. For each value of θ1, wemeasure the group delay incurred by the wave packetacross the layer, from which we estimate θ2. The mea-surements are carried out at normal incidence φ1 =0 andat oblique incidence φ1 = 20◦. The measurements ver-ify for the first time the change in the law of refractionfor ST wave packets with respect to that at normal in-

cidence, which was demonstrated in [22]. It is clear thatthe two curves for normal and oblique incidence in Fig. 3intersect at the point corresponding to the luminal con-dition (θ1, θ2)=(θL1 , θ

L2 ). The curve for oblique incidence

is shifted below that for normal incidence in the super-luminal regime (θ1 > θL1 and v > c/n) and above it inthe subluminal regime (θ1 < θL1 and v < c/n). We nextexplore the far-reaching ramifications of this change.

IV. DEPENDENCE OF THE GROUPVELOCITY ON THE ANGLE OF INCIDENCE

Figure 3 yields the relationship between the group in-dices n1 = cot θ1 and n2 = cot θ2 for the incident andrefracted ST wave packets, respectively, at a fixed inci-dent angle φ1. We next examine the change in the groupindex n2 for the transmitted wave packet as we vary theincidence angle φ1 while holding fixed the group indexn1 for the incident wave packet. In Fig. 4(a), we plotthe calculated n2 while varying both n1 and φ1 for in-cidence from free space n1 = 1 onto sapphire n2 = 1.76.This calculation highlights the above-mentioned ‘mem-ory’ effect. The group index n2 of the transmitted wavepacket depends not only the refractive index of the secondmedium n2, but also on the characteristics of the incidentwave packet: its group index n1 and incident angle φ1.The luminal condition for the incident ST wave packet is

4

FIG. 3: Verification of the law of refraction for ST wavepackets at oblique incidence (Eq. 2) for φ1 =20◦,

compared to normal incidence (Eq. 1) φ1 =0◦. Theinset highlights the transition from normal refraction to

anomalous refraction. We also identify the positionsalong the curves corresponding to group-velocity

invariance v= v1 and group-velocity inversion v2 =−v1.

n1 = n1 = 1, in which case this wave packet is simply aplane-wave pulse, and the refracted wave packet is alsoa luminal plane-wave pulse n2 = n2 = 1.76. For sublu-minal incidence ST wave packets n1 > n1, n2 decreaseswith φ2; i.e., the group velocity of the transmitted sublu-minal wave packet increases with φ1 in the higher-indexmedium. Conversely, for superluminal incident ST wavepackets n1 < n1, n2 increases with φ1; i.e., the groupvelocity of the transmitted superluminal wave packet in-creases with φ1 in the higher-index medium. In the caseof a conventional wave packet whose spatial and temporalDoFs are separable, n2 is independent of φ1.

This behavior can be understood on the basis of Eq. 2.Substituting for φ2 in terms of φ1 from Snell’s law yields:

n2(φ1)

n2= 1 +

(n1n1− 1

)η(φ1), (3)

where the factor η(φ1) depends on the indices n1 and n2through

η(φ1) =cos2 φ1

(n2

n1)2 − sin2 φ1

. (4)

The behavior of η(φ1) depends on the ratio n1/n2. Whenn1<n2, η(φ1) drops monotonically from its initial value

to 0 at φ1 = 90◦. Because the term ( n1

n1− 1) is positive

in the subluminal regime n1>n1, the decrease of η(φ1)

FIG. 4: (a) Calculated group index n2 of thetransmitted ST wave packet as a function of the angleof incidence φ1 and the group index n1 of the incidentST wave packet from free space (n1 =1) onto sapphire

(n2 =1.76). The dashed line is the locus of luminalplane-wave pulses whose spatial and temporal DoFs areseparable; here n2 =n2 =1.76 is independent of φ1. (b)Measured change in n2 at oblique incidence onto MgF2

(n2 =1.38) from free space with respect to normalincidence, ∆n2 = n2(φ1)− n2(0). Measurements arecarried out for a ST wave packet in the subluminalregime θ1 =30◦ (v=0.58c) and in the superluminalregime θ1 =109◦ (v=−2.9c). (c) Same as (a) for

incidence from free space onto sapphire.

with φ1 results in a decrease of n2(φ1) with φ1. In the

superluminal regimes, the term ( n1

n1− 1) is negative, and

5

n2(φ1) increase with φ1. On the other hand, when n1>n2, η(φ1) increases monotonically with φ1 and reachesa singularity when sinφ1 = n2/n1, corresponding to thecritical angle (total internal reflection) at the interface.Consequently, the opposite trends for n2 with φ1 ensue.

Our measurements confirm this predicted behavior. InFig. 4(b,c) we plot the change in the refracted groupindex n2 with φ1 with respect to that at normal inci-dence, ∆n2 = n2(φ1) − n2(0), for incidence from freespace onto two materials: MgF2 in Fig. 4(b) and sap-phire in Fig. 4(c). For each material, we carry out mea-surements with ST wave packets synthesized in free spacein the subluminal (θ1<45◦) and superluminal (θ1>45◦)regimes. We note several general observations about theresults that are expected from our analysis. First, n2(φ1)deviates monotonically from the normal-incidence valuen2(0). Second, the group index drops at oblique inci-dence n2(φ1)< n2(0) in the subluminal regime, and in-creases n2(φ1)>n2(0) in the superluminal regime. Thatis, the group velocity of a refracted subluminal ST wavepacket increases with incident angle in the higher-indexmedium. Conversely, the group velocity of a refractedsuperluminal ST wave packet decreases with φ1. Thesefeatures are critical for the blind synchronization schemewe investigate below.

V. IMPACT OF THE ANGLE OF INCIDENCEON GROUP-VELOCITY INVARIANCE AND

INVERSION

The law of refraction at normal incidence (Eq. 1) pre-dicts three phenomena that occur for any pair of media:(1) group velocity invariance v2 = v1, which occurs whenn1 = nth = n1 + n2; (2) anomalous refraction wherebyv2 > v1 when n2 >n1, which occurs when n1 > nth; and(3) group-velocity inversion v2 =−v1, which occurs whenn1 = n1 − n2; see Fig. 3. These three phenomena wereconfirmed experimentally in Ref. [22] and in more de-tail in paper (II) [24]. Despite the difference between thelaw of refraction at oblique incidence (Eq. 2) from that atnormal incidence (Eq. 1), these three phenomena are stillrealizable, albeit with modifications to the conditions tobe satisfied.

First, group-velocity invariance occurs at oblique inci-dence when [23]:

n1 = nth(φ1) =n1 + n2

1 + n1

n2sin2 φ1

, (5)

where nth(φ1) < nth(0) = n1 + n2. We plot in Fig. 5measurements of the group delay incurred upon travers-ing equal lengths (L= 5 mm) of free space (n1 = 1) andMgF2 (n2 = 1.38). At normal incidence nth(0) = 2.38corresponding to θ1≈22.8◦ [Fig. 5(a)], and at oblique in-cidence φ1 = 30◦ we have nth(30◦) = 1.86, correspondingto θ≈28.2◦ [Fig. 5(b)]. In both cases the ST wave pack-ets accrue approximately the same delay in equal lengthsof free space and MgF2. Whereas anomalous refraction

FIG. 5: Impact of oblique incidence on the condition forrealizing group-velocity invariance. (a) Normal

incidence, φ1 =0. We plot the on-axis x=0 input-pulseprofile I(z=0; τ) on the left, and the output pulse

profiles I(L; τ) after traversing L=5 mm of free space(dotted curve) and MgF2 (solid curve) at normal

incidence for a ST wave packet with θ1 =22.8◦

(v=0.42c) synthesized in free space. (b) Same as (a) foroblique incidence at φ1 =30◦. Here the group delaystraversing equal lengths of free space and MgF2 areequal but have a smaller overall delay than in (a),

despite the longer distance traveled.

occurs at normal incidence when θ1<22.8◦, this regime isexpanded for oblique incidence to θ1<28.2◦ at φ1 =30◦.

Second, group-velocity inversion occurs at oblique in-cidence when [23]:

n1(φ1) =n1 − n2

1− n1

n2sin2 φ1

, (6)

where n1(φ1)<n1(0)=n1− n2 when n2>n1. We plot inFig. 6 the measured group delay incurred upon travers-ing equal lengths (L = 5 mm) of sapphire and MgF2

separately, and after traversing a bilayer of the two me-dia. The required group index for a ST wave packetin free space that experiences group-velocity invariancein sapphire and MgF2 is n0 ≈ −1.43 (θ0 ≈ 145◦); seeFig. 6(a). At oblique incidence (φ1 = 30◦), the requiredfree-space group index to realize group-velocity inversionis n0 ≈−2.25 (θ0 ≈ 156◦); see Fig. 6(b). In both cases,the group delays measured in the individual layers areapproximately equal in magnitude but opposite in sign.Consequently, the group delay in the bilayer almost van-ishes as shown.

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FIG. 6: Impact of oblique incidence on group-velocityinversion. (a) At normal incidence (φ1 =0), a ST wavepacket with θ0 =145◦ traverses a 5-mm-thick layer of

MgF2 with a negative group delay (dashed curve;v1 =−2.6c and θ1 =111◦ in MgF2) and a 5-mm-thicklayer of sapphire with a positive group delay (dotted

curve; v2 =2.6c and θ1 =69◦ in sapphire), correspondingto group delay inversion. When the ST wave packet is

incident from free space on a bilayer of MgF2 andsapphire, the total group delay is zero (solid curve),

corresponding to group-delay cancellation. (b) Same as(a) for oblique incidence at φ1 =30◦. The incident STwave packet from free space has θ0 =156◦. The delaysincurred by the two pulses in the individual layers inlarger than at normal incidence because the distancetraveled is larger at oblique incidence and the wave

packet is in the normal refraction regime.

VI. BLIND SYNCHRONIZATION

A. Concept of blind synchronization using ST wavepackets

Consider the configuration shown in Fig. 7. A trans-mitter (Tx) in the first medium at a height d1 above theinterface directs an ST wave packet at different anglesto receivers (Rx) at different locations but at the samedepth d2 beneath the interface in the second medium.The optical delay in the first medium from Tx to theinterface is τ1(φ1) = τ1(0)/ cosφ1, where τ1(0) = d1n1/cis the group delay at normal incidence. The group ve-locity of the ST wave packet is the same in all direc-tions independently of φ1 because the medium is homo-geneous and isotropic. Therefore, the group delay τ1(φ1)increases with φ1. The group delay in the second mediumis τ2(φ1)=d2n2(φ1)/(c cosφ2), where n2(φ1) is the groupindex of the transmitted wave packet, determined fromEq. 2. In contrast to the incident wave packet, the group

FIG. 7: Concept of blind synchronization. We depictthe configuration for the blind synchronization of

multiple receivers (Rx) receiving pulses from atransmitter (Tx). All the receivers are at the same

depth d2 beneath the interface, but their locations areotherwise not known a priori. When using conventionalwave packets, the group delay increases as the physical

path length from Tx to Rx increases, which occurswhen the incident angle φ1 at the interface increases.

By employing ST wave packets instead, the group delayaccrued along the different pathways from Tx to Rx can

be held constant because the group velocity of thetransmitted wave packet increases with φ1, therebycompensating for the longer path length covered.

velocity of the transmitted wave packet depends on φ1.The relative group delay between obliquely and nor-

mally incident ST wave packets in the first medium is:

∆τ1 = τ1(φ1)− τ1(0) =d1cn1

(1

cosφ1− 1

), (7)

and in the second medium is:

∆τ2 = τ2(φ1)− τ2(0) =d2c

(n2(φ1)

cosφ2− n2(0)

). (8)

We define the total relative group delay ∆τ=∆τ1+∆τ2 =τ(φ1) − τ(0), where τ(φ1) = τ1(φ1) + τ2(φ1) is the totalgroup delay incurred from Tx to Rx across the interface.Blind synchronization requires ∆τ = 0 independently ofφ1, such that the ST wave packet arrives simultaneouslyat all receivers at the depth d2 despite the different phys-ical distances covered along the different trajectories.

We first ascertain whether realizing ∆τ = 0 is physi-cally feasible. Note that ∆τ1 in the first medium is alwayspositive, because the group delay along the longer oblique

7

path is always larger than that along the shorter normalpath. Blind synchronization therefore requires that ∆τ2in the second medium be negative. That is, the groupvelocity of the transmitted wave packet must increase atoblique incidence to compensate for the longer propaga-tion distances in both media. In other words, a neces-sary (but not sufficient) condition is that v2(φ1)>v2(0).From the analysis and measurements reported above, thisscenario occurs for subluminal ST wave packets whenn2 > n1. We next consider whether the blind synchro-nization condition can be met quantitatively.

FIG. 8: Plots of the relative group delay ∆τ(normalized with respect to d1/c) as a function of the

incident angle φ1 and the group index n1 of the incidentST wave packet: (a) n1 =1, n2 =1.76, and d2/d1 =5;

and (b) n1 =1, n2 =2, and d2/d1 =10.

B. Calculated group delay of ST wave packetsbetween transmitter and receivers

The total relative group delay ∆τ=∆τ1 +∆τ2 is givenby:

∆τ(φ1)

d1/c=n2

d2d1

{(1

cosφ1− 1

)+

(n1n1−1

)(η(φ1)

cosφ2− η(0)

)}+ n1

(1

cosφ1−1

). (9)

where we made use of the law of refraction in Eq. 2, andwe normalized ∆τ with respect to d1/c. We plot in Fig. 8calculations for ∆τ as a function of the incident angle φ1and the group index of the incident wave packet n1 intwo different configurations. In the first configuration wehave n1 = 1, n2 = 1.76, and d2/d1 = 5; and in the secondconfiguration we have n1 = 1, n2 = 2, and d2/d1 = 10. Inboth cases we find the contour ∆τ=0 occurs at a specificgroup index n1 over a range of incident angles. Such STwave packets arrive simultaneously at receivers locatedat the same depth within this angular range despite fol-lowing paths of very different lengths, thereby realizingblind synchronization. We find in general that meetingthe requirement for blind synchronization is more favor-able when d2/d1 and/or n2/n1 increase.

For the sake of comparison, we plot in Fig. 9(a) therelative group delay ∆τ for ST wave packets having dif-ferent group indices alongside that for a conventionalwave packet whose spatial and temporal DoFs are sepa-rable. The conventional wave packet propagates in thefirst medium at a group velocity c/n1 and in the secondat c/n2 (assuming non-dispersive media). The relativegroup delay is given by:

∆τconv(φ1)

d1/c= n1

(1

cosφ1− 1

)+ n2

d2d1

(1

cosφ2− 1

).

(10)The relative group delay ∆τ is always positive for conven-tional pulses because the pulses traveling longer distancesinevitably accrues larger group delays. In contrast, vary-ing n1 for ST wave packets changes ∆τ from positivevalues (similarly to a conventional pulse where longerpaths incur longer delays) to anomalously negative val-ues (longer paths incur shorter delays) passing throughthe desired target of blind synchronization ∆τ = 0. InFig. 9(b) we highlight this transition from ∆τ > 0 to∆τ <0 as n1 is changed in the vicinity of n1 =3.

C. Blind synchronization

We plot in Fig. 10 results for a proof-of-principle testof the blind synchronization effect. The overall systemconfiguration is illustrated in Fig. 10(a). We considerthe configuration in Fig. 8(a) for propagation from freespace (n1 = 1) to sapphire (n2 = 1.76) with the prop-agation distances at normal incidence having the ratio

8

FIG. 9: (a) Comparison of the relative group delay ∆τ(normalized with respect to d1/c) of ST wave packets

(solid curves; Eq. 9) from Fig. 8(a) at n1 =1.46, 3 and 4with ∆τ of a traditional pulse (dashed curve; Eq. 10).

(b) Same as (a), but with a reduced range for ∆τ alongthe vertical axis.

d2/d1 = 5; we use here d1 = 1 mm and d2 = 5 mm. Wemeasure the group delay of the ST wave packet in freespace [Fig. 10(b)] and in the sapphire layer [Fig. 10(c)]separately. The separate delays for the oblique-incidencepaths are measured by tilting the sapphire layer and infree space by extending the distance d1→d1/ cosφ1 overwhich the delay is measured.

We synthesize a subluminal ST wave packet in freespace with n1 =3 (θ1 =18◦). According to Fig. 8(a), blindsynchronization ∆τ=0 is realized by such a wave packetover a range of incidence angles φ1 =0→ ±35◦. We plotin Fig. 10(b) the measured group delay in free space overdistances extending from d1 = 1 mm (corresponding toφ1 = 0◦) to d1 = 1.2 mm (corresponding to φ1 = 35◦),with d1(φ1) = d1/ cosφ1. We next plot in Fig. 10(c) themeasured group delay in the sapphire layer as we increasethe angle of incidence. Here, the group delay decreaseswith the increases distance traveled in the layer at largerφ1. As shown in Fig. 4, this is a general feature at obliqueincidence for subluminal ST wave packets when n2>n1.For this particular ST wave packet with n1 = 3, the de-crease in delay in the layer (increase in group velocity)with φ1 is sufficient to counterbalance the increase in de-

FIG. 10: Proof-of-principle realization of blindsynchronization. (a) Schematic of the experimental

configuration. (b) Measured group delay accrued over1 mm of free space by a ST wave packet with n1 =3.

The delay increases with φ1. (c) Measured group delayover 5 mm of sapphire for the ST wave packet from (b).Here, the group delay decreases with φ1. (d) Sum of the

group delays from (b) and (c), showing anapproximately constant value despite the different path

lengths associated with different incident angles φ1.

lay in free space and the increase in propagation distancein the sapphire layer, such that the total delays in freespace and the layer are approximately equal [Fig. 10(d)]for all trajectories corresponding to incident angles fromφ1 =0◦ to φ1 =35◦.

9

VII. DISCUSSION AND CONCLUSIONS

The proof-of-principle experimental demonstration ofblind synchronization described here extends our recentreport on isochronous ST wave packets [25]. These arewave packets that traverse a material layer after accru-ing a fixed group delay independently of the angle ofincidence, despite traveling larger distances in the layer(L/ cosφ2, where L is the layer thickness). This canbe seen as a special case of blind synchronization wherewe set d1 = 0. An increase in group velocity with inci-dent angle for isochronous ST wave packets is requiredto compensate for the increase in propagation distancewithin the layer. This is achieved, as we have done herefor blind synchronization, by employing subluminal in-cident ST wave packets. However, the blind synchro-nization scenario is more stringent because the necessaryincrease in group velocity with incident angle is larger:this change must counterbalance the increase in propaga-tion distance within both media. As such, an incident STwave packet with larger n1 (deeper within the subluminalregime) is required for blind synchronization comparedto isochronous ST wave packets. Pursuing the conceptof blind synchronization further requires increasing thepropagation distance over which this effect is realized byreducing the so-called ‘spectral uncertainty’ [11, 27, 29],which is the unavoidable ‘fuzziness’ in the associationbetween spatial and temporal frequencies in the wavepacket spectrum. In general, reducing the spectral un-certainty to increase the propagation distance requiresincreasing the system’s numerical aperture.

The work reported in this paper sequence, in additionto [22], has examined the refraction of ST wave packets inthe simplest scenario: at a planar interface between twonon-dispersive, homogeneous, isotropic dielectrics. Fu-ture work will be directed to an exploration of other im-portant configurations. Paramount amongst these arethe cases of dispersive and anisotropic media, which will

be critical as a prelude to studying nonlinear interactionsinvolving ST wave packets. Additionally, it will be usefulto study the refraction of ST wave packets in which bothtransverse dimensions are included (ST needles ratherthan ST sheets), and ST wave packets in which a pa-rameter is controlled along the propagation axis, suchas wave packets that accelerate or decelerate [30], or areendowed with axial spectral encoding [31].

In conclusion, we have verified experimentally the lawof refraction for ST wave packets obliquely incident ona planar interface between two non-dispersive, homoge-neous, isotropic dielectrics (Eq. 2). This law was verifiedfirst by changing the group velocity of the incident STwave packet at a fixed angle of incidence, and, second, bychanging the angle of incidence while holding the groupvelocity of the incident wave packet fixed. Our measure-ments reveal that the group velocity of the transmittedwave packet increases with the incident angle in the sub-luminal regime (when the index of the second mediumis higher than that of the first); conversely, the groupvelocity of the transmitted wave packet decreases withincident angle in the superluminal regime. Furthermore,we observed the shift that occurs in the conditions forgroup-velocity invariance and group-velocity inversion atoblique incidence with respect to those at normal inci-dence. These findings made possible the first observa-tion of blind synchronization, whereby ST wave packetsproduced from a fixed source (Tx) are arranged to arrivesimultaneously at different receivers (Rx) at a priori un-known positions, except that they are located at a fixeddepth beyond an interface.

FUNDING

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

Disclosures. The authors declare no conflicts of inter-est.

[1] B. E. A. Saleh and M. C. Teich, Fundamentals of Pho-tonics (Wiley, 2019).

[2] H. E. Kondakci and A. F. Abouraddy, Opt. Express 24,28659 (2016).

[3] K. J. Parker and M. A. Alonso, Opt. Express 24, 28669(2016).

[4] K. Reivelt and P. Saari, arxiv:physics/0309079 (2003).[5] A. P. Kiselev, Opt. Spectrosc. 102, 603 (2007).[6] J. Turunen and A. T. Friberg, Prog. Opt. 54, 1 (2010).[7] H. E. Hernandez-Figueroa, E. Recami, and M. Zamboni-

Rached, eds., Non-diffracting Waves (Wiley-VCH, 2014).[8] R. Donnelly and R. W. Ziolkowski, Proc. R. Soc. Lond.

A 440, 541 (1993).[9] P. Saari and K. Reivelt, Phys. Rev. E 69, 036612 (2004).

[10] S. Longhi, Opt. Express 12, 935 (2004).[11] H. E. Kondakci, M. A. Alonso, and A. F. Abouraddy,

Opt. Lett. 44, 2645 (2019).[12] L. A. Hall, M. Yessenov, and A. F. Abouraddy, Opt.

Lett. 46, 1672 (2021).[13] J. Salo and M. M. Salomaa, J. Opt. A 3, 366 (2001).[14] E. Recami, M. Zamboni-Rached, K. Z. Nobrega, and

C. A. Dartora, IEEE J. Sel. Top. Quantum Electron. 9,59 (2003).

[15] H. Valtna, K. Reivelt, and P. Saari, Opt. Commun. 278,1 (2007).

[16] M. Zamboni-Rached and E. Recami, Phys. Rev. A 77,033824 (2008).

[17] H. E. Kondakci and A. F. Abouraddy, Nat. Photon. 11,733 (2017).

[18] N. K. Efremidis, Opt. Lett. 42, 5038 (2017).[19] L. J. Wong and I. Kaminer, ACS Photon. 4, 2257 (2017).[20] H. E. Kondakci and A. F. Abouraddy, Nat. Commun.

10

10, 929 (2019).[21] B. Bhaduri, M. Yessenov, and A. F. Abouraddy, Optica

6, 139 (2019).[22] B. Bhaduri, M. Yessenov, and A. F. Abouraddy, Nat.

Photon. 14, 416 (2020).[23] M. Yessenov, B. Bhaduri, and A. F. Abouraddy, accom-

panying paper (2021).[24] A. M. Allende Motz, M. Yessenov, B. Bhaduri, and A. F.

Abouraddy, accompanying paper (2021).[25] A. M. Allende Motz, M. Yessenov, and A. F. Abouraddy,

arXiv:2102.10505 (2021).[26] M. Yessenov, B. Bhaduri, H. E. Kondakci, and A. F.

Abouraddy, Phys. Rev. A 99, 023856 (2019).[27] M. Yessenov, B. Bhaduri, L. Mach, D. Mardani, H. E.

Kondakci, M. A. Alonso, G. A. Atia, and A. F.Abouraddy, Opt. Express 27, 12443 (2019).

[28] M. Yessenov, B. Bhaduri, H. E. Kondakci, and A. F.Abouraddy, Opt. Photon. News 30, 34 (2019).

[29] B. Bhaduri, M. Yessenov, D. Reyes, J. Pena, M. Meem,S. R. Fairchild, R. Menon, M. C. Richardson, and A. F.Abouraddy, Opt. Lett. 44, 2073 (2019).

[30] M. Yessenov and A. F. Abouraddy, Phys. Rev. Lett. 125,233901 (2020).

[31] A. M. Allende Motz, M. Yessenov, and A. F. Abouraddy,Phys. Rev. Appl. 15, 024067 (2021).