Isochronous space-time wave packets
IIsochronous space-time wave packets
Alyssa A. Allende Motz, Murat Yessenov, and Ayman F. Abouraddy ∗ CREOL, The College of Optics & Photonics, University of Central Florida, Orlando, FL 32816, USA
The group delay incurred by an optical wave packet depends on its path length. Therefore, whena wave packet is obliquely incident on a planar homogeneous slab, the group delay upon traversingit inevitably increases with the angle of incidence. Here we confirm the existence of isochronous‘space-time’ (ST) wave packets: pulsed beams whose spatio-temporal structure enables them totraverse the layer with a fixed group delay over a wide span of incident angles. This unique behaviorstems from the dependence of the group velocity of a refracted ST wave packet on its angle ofincidence. Isochronous ST wave packets are observed in slabs of optical materials with indices inthe range from 1.38 to 2.5 for angles up to 50 ◦ away from normal incidence. Fermat demonstrated in 1662 that a minimal principleunderpins Snell’s law; namely, the refracted trajectorycorresponds to a minimum optical path length [1]. Iden-tifying minimal principles is now a commonplace strategythroughout the physical sciences. A different strategy isto search for isochronous (or equal-time) configurations.One example is Huygens’ isochronous pendulum, whichis built on the discovery that a particle sliding on a cy-cloid (or tautochrone curve) returns after a fixed timeinterval independently of point of release [2, 3]. How-ever, this strategy has had less impact in optics. In gen-eral, the group delay accrued by a pulse in a homoge-neous isotropic medium is proportional to the distancetraveled, and no isochronous configurations arise exceptalong equal-length paths.We show here that the refraction of a recently devel-oped family of pulsed beams denoted ‘space-time’ (ST)wave packets [4–7] offers a unique configuration, wherebythe group delay incurred by the refracted wave packettraversing a homogeneous, isotropic dielectric slab re-mains fixed with increasing incident angle despite theincrease in path length. We thus uncover a new opticalrefractory phenomenon: the existence of isochronous
STwave packets. This counter-intuitive phenomenon stemsfrom the unusual behavior of ST wave packets when re-fracting across a planar interface, as verified recently in[8]. Specifically, the group velocity of certain ST wavepackets increases with increasing incident angle. Theisochronous condition is reached when the increase in therefracted group velocity at any incident angle compen-sates for the larger associated path length in the slab.Consequently, the wave packet emerges from the slab ata fixed delay over a large span of incident angles. Weconfirm these predictions experimentally in three ma-terials over a wide range of refractive indices: MgF ( n ≈ . n ≈ . n ≈ . decreases . Thisrare example of an optical isochronous configuration isa critical experimental step towards realizing blind syn- ∗ corresponding author: [email protected] FIG. 1: a) Concept of isochronous ST wave packets. Aconventional wave packet incurs a larger group delaytraversing a layer at oblique incidence than at normalincidence. In contrast, an isochronous ST wave packetmaintains the same group delay independently of theincident angle. (b) The path length d across the layer atoblique incidence increases with the incident angle φ .(c) Calculated oblique-incidence delay τ across asapphire layer normalized to the normal-incidence delay τ o . (d) The delay τ ST for a ST wave packet is theproduct of the delay τ for a conventional wave packetand a factor η ( φ ) dictated by the group index (cid:101) n . Atthe isochronous condition, τ ST is independent of φ .chronization of remote clocks as proposed in [8].Consider the scenario depicted in Fig. 1(a) where awave packet is incident onto a planar layer of a non-dispersive material of refractive index n and thickness a r X i v : . [ phy s i c s . op ti c s ] F e b L at an angle φ with respect to the layer normal froma medium of refractive index n . At normal incidence,the group delay upon traversing the layer is τ o = n L/c ,where c is the speed of light in vacuum. At obliqueincidence the group delay increases monotonically with φ , τ ( φ ) = τ o / cos φ , because of the larger path length d > L [Fig. 1(b)]; where n sin φ = n sin φ . We posethe following question: is it possible to synthesize anisochronous optical wave packet, one for which the groupdelay at oblique incidence τ ( φ ) is independent of φ , sothat the delay incurred is fixed despite the longer prop-agation distance [Fig 1(a)]? Realizing this isochronouscondition requires that the group velocity (cid:101) v in the layerincreases with φ , (cid:101) v ( φ ) ∝ / cos φ , to counterbalancethe longer propagation distance d ( φ ) ∝ cos φ [Fig 1(b)],so that τ ( φ ) = d ( φ ) / (cid:101) v ( φ ) is constant [Fig 1(c)]. Forconventional wave packets, oblique incidence does not impact (cid:101) v . However, it was recently shown that refrac-tion of ST wave packets follows an unusual rule whereby (cid:101) v depends not only on n , but also on n , the groupvelocity of the incident wave packet (cid:101) v , and the angle ofincidence φ . This unique behavior offers the possibilityof an isochronous configuration.The key distinguishing characteristic of ST wave pack-ets is that each spatial frequency is associated with a sin-gle wavelength [9–13] so as to realize non-differentiableangular dispersion [14], which enables realizing wavepackets endowed with propagation invariance [15–17]and tunable group velocities [18–20]. Specifically, thespatio-temporal spectrum of a ST wave packet in a non-dispersive medium of index n lies at the intersection ofthe light cone k x + k z = n ( ωc ) with a spectral plane P ( θ )defined by the equation ω = ω o +( k z − nk o ) c tan θ , which isparallel to the k x -axis and makes an angle θ (the spectraltilt angle) with respect to the k z -axis [6, 9]. Here x and z are the transverse and axial coordinates, respectively; k x and k z are the corresponding components of the wavevector; ω is the angular frequency; and ω o is a fixed fre-quency whose corresponding free-space wave number is k o = ω o c . For simplicity, but without loss of generality, wehold the field uniform along the transverse y -coordinate( k y = 0). This configuration imposes a precise associ-ation between each k x with a single ω . The ST wavepacket is propagation invariant : it is transposed rigidlyin the medium without diffraction or dispersion [6, 21–23] at a group velocity (cid:101) v = c tan θ = c/ (cid:101) n ( (cid:101) n = c/ (cid:101) v = cot θ is the group index), which is determined by the wave-packet spatio-temporal spectral structure [12]. The STwave packet is subluminal when cot θ > n and superlumi-nal when cot θ < n . Assuming paraxial (∆ k x (cid:28) k o ) andnarrowband (∆ ω (cid:28) ω o ) conditions, the conic section atthe intersection of the light-cone with P ( θ ) in the vicinityof k x = 0 can be approximated by a parabola: ωω o = 1 + 1 n ( n − (cid:101) n ) k x k . (1)This association between the spatial and temporal fre-quencies ( k x and ω , respectively) can be achieved by the FIG. 2: (a) Calculated group index (cid:101) n of the refractedST wave packet and the normalized delay τ ST /τ o (Eq. 3) as a function of the incident wave packet groupindex (cid:101) n and the incident angle φ , the lattercorresponding to Fig. 1(c). The calculations areperformed for a planar layer of sapphire n = 1 . n = 1. (b)Calculated isochronous condition (cid:101) n (iso)1 and theassociated angular acceptance range ∆ φ for differentmaterials, assuming incidence from free space n = 1 anda delay tolerance of δτ /τ o = 0 . , sapphire, and ZnSe) areidentified. (c) Calculated (cid:101) n (iso)1 and ∆ φ for differentmaterials while varying the delay tolerance δτ /τ o .pulse shaper developed in [6, 20]; see Fig. 3(a) below.The refraction of a ST wave packet across a planar in-terface between two transparent, isotropic, homogeneous,non-dispersive dielectrics of refractive indices n and n is governed by [8]: n ( n − (cid:101) n ) cos φ = n ( n − (cid:101) n ) cos φ . (2)where (cid:101) n and (cid:101) n are the group indices for the incidentand transmitted wave packets, respectively, and φ and φ are their angles with respect to the normal to the in-terface. That is, (cid:101) n depends on the refractive indices of both media, and also on (cid:101) n and φ . This relationship re-flects the invariance of the so-called ‘spectral curvature’of the ST wave packet, n ( n − (cid:101) n ) cos φ . This new opti-cal invariant results from combining the conservation oftransverse momentum and energy ( k x and ω ) across aplanar interface for ST wave packets in which the spatialand temporal degrees of freedom are inextricably linked[8].If the second medium is a slab of thickness L , then thegroup delay incurred by the ST wave packet traversing itis: τ ST ( φ ) = (cid:101) n dc = τ o cos φ (cid:26) n cos φ n cos φ (cid:18) (cid:101) n n − (cid:19)(cid:27) , (3)where τ o = n (cid:96)/c is the group delay for a conventionalwave packet at normal incidence. Intuitively we expect τ ST ( φ ) to increase with φ because of the increase in thetraveled distance. This monotonic trend is however coun-terbalanced by the term in the parentheses that decreaseswith φ in the subluminal regime (cid:101) n > n , whereupon thegroup velocity of the refracted ST wave packet increaseswith φ [8].We plot in Fig. 2(a) the group index (cid:101) n of the trans-mitted wave packet traversing a planar interface from freespace ( n = 1) to sapphire ( n = 1 .
76 at a wavelength of λ = 800 nm) while varying the group index (cid:101) n of the inci-dent wave packet and its incident angle φ . When (cid:101) n = 1(a plane-wave pulse [12]), (cid:101) n = n independently of φ asusual, and the group delay τ = (cid:101) n d/c increases monotoni-cally with φ because of the increased path length acrossthe slab. For all other values of (cid:101) n , the refracted groupindex (cid:101) n varies with φ . We require that (cid:101) n decrease with φ to counterbalance the increased path length. It is clearin Fig. 2(a), that this condition occurs when (cid:101) n > n ; i.e.,in the subluminal regime. When we plot τ ST normalizedwith respect to τ o in Fig. 2(a), we observe that a flat iso-delay contour occurs at a particular value of (cid:101) n = (cid:101) n , iso .This ST wave packet encounters a constant delay as φ varies over a broad span.We plot in Fig. 2(b) the calculated group index (cid:101) n , iso for the incident ST wave packet that satisfies theisochronous condition in a medium of index n (assumingincidence from free space n = 1). We define this condi-tion as follows: for a given n , we calculate the normal-ized group delay τ ST ( φ ; (cid:101) n ) /τ o . For each value of (cid:101) n , wedetermine the maximum incident angle φ that maintainsthe delay τ ST ( φ ; (cid:101) n ) within the range τ ST (0; (cid:101) n ) ± δτ .The group index (cid:101) n for the incident wave packet thatyields the maximum incident angle within this range forthe delay is denoted (cid:101) n , iso . We plot the calculated (cid:101) n , iso for δτ /τ o = 0 .
1% and the corresponding isochronous an-gular range ∆ φ . Of course, the value of (cid:101) n , iso and ∆ φ depend on the choice of δτ . We plot both (cid:101) n , iso and∆ φ as we vary the delay tolerance δτ /τ o in Fig. 2(c)for different materials. It is clear that relaxing the de-lay tolerance increases the angular range ∆ φ over whichthe isochronous condition is maintained. Furthermore, relaxing δτ reduces (cid:101) n , iso , which approaches n = 1. Thismakes synthesizing the isochronous ST wave packet moreconvenient [12].To measure the group delay τ ST ( φ ) for planar sam-ples, we make use of the interferometric arrangementin Fig. 3(a), which is based on our previous work in[8, 19, 20]. We employ 100-fs pulses from a mode-lockedTi:sapphire laser at a central wavelength of 800 nm di-rected to a two-path interferometer. In one path, weplace the setup for synthesizing the ST wave packets inwhich the spectrum of the laser pulses is resolved with adiffraction grating (1200 lines/mm), and the first diffrac-tion order is collimated by a cylindrical lens (focal length50 cm) and directed to a reflective phase-only spatial lightmodulator (SLM; Hamamatsu X10468-02) that impartsa two-dimensional phase distribution to the spectrally re-solved wave front. This phase distribution is designed toassign a specific spatial frequency k x to each wavelength λ according to the constraint in Eq. 1. Tuning the spec-tral tilt angle θ of the ST wave packet to change itsgroup index (cid:101) n = cot θ is achieved by sculpting the SLMphase [12]. A delay line τ is placed in the path of thereference arm of the interferometer. The wave packetshaper spectrally filters the initial pulses to a bandwidthof ∆ λ ≈ . τ [Fig. 3(b)]. Thegroup index (cid:101) n of the synthesized ST wave packet is ver-ified in two ways. First, the differential group delay be-tween the ST wave packet (traveling at c/ (cid:101) n ) and the ref-erence pulse (traveling at c ) is measured after placing thedetector at two different axial positions, from which wecan estimate (cid:101) n . Second, we measure the spatio-temporalspectrum in the ( k x , λ )-plane [Fig. 3(c)] using a combina-tion of grating and Fourier-transforming lens (not shownin the setup for simplicity) [19].The isochronous condition is verified in planar samplesof MgF , sapphire, and ZnSe all of thickness L = 5 mm,which are placed in the common path of the ST wavepacket and reference pulse after the interferometer, asshown in Fig. 3(a). Each sample is rotated about the y -axis in increments of 5 ◦ , resulting in an increase in pathlength from 5 mm to ≈ , 5.6 mm in sap-phire, and 5.25 mm in ZnSe. The group delay τ ST ( φ )is measured for each incident angle φ , and the resultsfor the three layers are plotted in Fig. 4(b-d). In eachcase, we plot results for three group indices of the inci-dent ST wave packets: (cid:101) n = (cid:101) n , iso where τ ST ( φ ) is inde-pendent of φ over an extended range of incident angles, (cid:101) n < (cid:101) n , iso where τ ST ( φ ) increases with φ as expected,and (cid:101) n > (cid:101) n , iso where τ ST ( φ ) anomalously drops with φ . We compare the results in each case to those for aconventional wave packet (plane-wave pulse) correspond-ing to (cid:101) n = 1 ( θ = 45 ◦ ) obtained by idling the SLM inFig. 3(a). The case of ZnSe is particularly striking wherethe isochronous condition is maintained over the largestspan of incident angles despite the changes in the delay τ ST ( φ ) with φ .FIG. 3: (a) Schematic of the setup for synthesizing andcharacterizing isochronous ST wave packets. BS: Beamsplitter, G: grating, L cyl : cylindrical lens, SLM: spatiallight modulator, DL: delay line. (b) Thespatio-temporal intensity profiles for conventional andisochronous ST wave packets (the latter is measured)and (c) their corresponding spatio-temporal spectrum.We hypothesized in [8] that the appropriate ST wavepackets can be utilized to blindly synchronize a transmit-ter with multiple remote receivers at different unknownlocations at the same depth beyond an interface betweentwo media. In such a configuration, the sum of the de-lays in the two media must be invariant with angle ofincidence at the interface. Our results here regardingisochronous ST wave packets are a significant step to-wards such a goal by demonstrating the invariant delayin the second medium (the slab here) with angle of inci-dence. The maximum distance over which the ST wavepackets can be used (the maximum thickness of the slabhere) is limited by the so-called ‘spectral uncertainty’ δλ , which is the unavoidable finite bandwidth associatedwith each spatial frequency [13]. In our experiments here, δλ ∼
25 pm, as determined by the spectral resolutionof the grating in Fig. 3(a). The maximum propagationdistance exceeds the 5-mm-thickness of the slabs usedhere. This is confirmed in Fig. 4(e), where only mini-mal changes are observed in the spatio-temporal profileof the wave packet after traversing the sapphire slab atall values of φ of interest.In conclusion, we have demonstrated for the first time,to the best of our knowledge, isochronous optical wavepackets: pulsed beams that incur the same group delayafter traversing a dielectric slab at any incident angle de-spite the different path lengths. In our realization, STwave packets synthesized in free space with a particulargroup index can satisfy this condition and maintain aninvariant group delay over a wide range of incident an-gles. Isochronous ST wave packets may have applicationsin clock-synchronization, in free-space optical communi-cations, and in nonlinear optics. Finally, our work here is FIG. 4: (a) Illustration of the path across a planar layerwith φ . (b) Measured group delay τ with φ for MgF .The three curves correspond to different spectral tiltangles θ for the incident wave packet from free space,compared to a conventional pulse. The dots are datapoints. (c) Same as (b) for sapphire. (d) Same as (b)for ZnSe. A small modification is made to Eq. 2 in thiscase to account for dispersion in ZnSe. (e) Measuredspatio-temporal profiles I ( x, τ ) for the isochronous wavepackets after traversing the Sapphire layer at φ = 0 ◦ ,25 ◦ , and 50 ◦ .based on the refraction of propagation-invariant ST wavepacket in non-dispersive media. It will be interesting toextend this work to dispersive materials [24, 25], and tostudy the refraction of recently developed ST wave pack-ets that undergo controllable axial evolution, such as ac-celerating or decelerating wave packets [26], and those en-dowed with axial spectral encoding [27] or group-velocitydispersion in free space [28]. FUNDING
U.S. Office of Naval Research (ONR) contract N00014-17-1-2458 and ONR MURI contract N00014-20-1-2789.
Disclosures.
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