Abstract
Optical wave packets that are localized in space and time, but nevertheless overcome diffraction and travel rigidly in free space, are a long soughtafter field structure with applications ranging from microscopy and remote sensing, to nonlinear and quantum optics. However, synthesizing such wave packets requires introducing nondifferentiable angular dispersion with high spectral precision in two transverse dimensions, a capability that has eluded optics to date. Here, we describe an experimental strategy capable of sculpting the spatiotemporal spectrum of a generic pulsed beam by introducing arbitrary radial chirp via twodimensional conformal coordinate transformations of the spectrally resolved field. This procedure yields propagationinvariant ‘spacetime’ wave packets localized in all dimensions, with tunable group velocity in the range from 0.7c to 1.8c in free space, and endowed with prescribed orbital angular momentum. By providing unprecedented flexibility in sculpting the threedimensional structure of pulsed optical fields, our experimental strategy promises to be a versatile platform for the emerging enterprise of spacetime optics.
Similar content being viewed by others
Introduction
Creating spatiotemporally localized optical wave packets that overcome diffraction and propagate rigidly in free space has been a longstanding yet elusive goal in optics. Such wave packets can have applications ranging from remote optical sensing and biological imaging, to nonlinear and quantum optics. To date, this challenge has been addressed via nonlinear optical effects that sustain solitons^{1}, waveguiding structures^{2}, or by exploiting particularly shaped waveforms such as BesselAiry wave packets in linear dispersive media^{3}. Propagation invariance in a linear nondispersive medium necessitates inculcating a precise spatiotemporal spectral structure into the field by introducing angular dispersion (AD)^{4,5}; i.e., associating each wavelength with a single propagation direction^{6,7}. Examples of such wave packets date back to Brittingham’s focuswave mode^{8}, Xwaves^{9,10}, and more recently the general class of ‘spacetime’ (ST) wave packets^{11,12,13,14,15,16,17,18,19}. The challenge of producing the AD necessary for propagationinvariant wave packets localized in all dimensions (referred to hereon as 3D ST wave packets) is twofold. First, the AD must be inculcated in two transverse dimensions rather than in one as typically realized via gratings or prisms^{4,5}. Second, nondifferentiable AD is required^{20}; i.e., it is necessary that the derivative of the wavelengthdependent propagation angle not be defined at some wavelength^{21,22} – a field configuration that cannot be directly produced with conventional optical components. Consequently, with the exception of Xwaves that are ADfree, no propagationinvariant optical wave packets that are localized in all dimensions have been observed in free space^{7}.
The challenge of introducing arbitrary AD into a generic pulsed beam along one transverse dimension has been recently addressed by constructing a universal AD synthesizer^{23}. This experimental strategy has enabled the realization of ST wave packets in the form of light sheets^{16} (referred to hereon as 2D ST wave packets), which exhibit a broad host of soughtafter effects, such as longdistance propagation invariance^{24}, tunable group velocities^{13,25,26,27,28,29}, anomalous refraction at planar interfaces^{30}, and the spacetime Talbot effect^{31}. Although this arrangement produces nondifferentiable AD with high spectral resolution, these features cannot be extended to both transverse dimensions. Crucially, the centerpiece of this configuration is a spatial light modulator that modifies the temporal spectrum along one dimension, leaving only one dimension to manipulate the field spatially – a limitation that is shared by other recently investigated spatiotemporal field structures^{32,33,34,35,36,37,38,39}. Therefore, the fundamental challenge of producing nondifferentiable AD encompassing both transverse dimensions remains outstanding.
Here, we demonstrate a spatiotemporal modulation strategy that efficiently produces arbitrary yet precise AD in two transverse dimensions, and thus yields ST wave packets localized in all dimensions – while preserving all the key attributes of its reduceddimension counterpart. This modulation scheme is implemented in three stages. In the first stage, the spectrum of a generic planewave pulse is spatially resolved along one dimension after a doublepass through a volume chirped Bragg grating. In the second stage, a spectral transformation ‘reshuffles’ the wavelengths into a prescribed sequence. In the third stage, a logpolartoCartesian conformal coordinate transformation converts the spatial locus of each wavelength from a line into a circle^{40,41}. A lens finally converts the spectrally resolved wave front into a 3D ST wave packet localized in all three dimensions. Utilizing this approach, we produce 3D ST wave packets with ≈30 μm transverse beam width and ≈6 ps pulse width that propagate for over 50 mm. Moreover, by modulating the spatiotemporal spectral structure, we realize group velocities extending from the subluminal to the superluminal regimes over the range from 0.7c to 1.8c (c is the speed of light in vacuum). Furthermore, by providing access to both transverse dimensions in a ST wave packet, new degrees of freedom of the optical field can be accessed, such as orbital angular momentum (OAM)^{42,43,44}. Specifically, by encoding a helical phase structure in the spatiotemporal spectrum, we demonstrate propagationinvariant pulsed OAM wave packets with controllable group velocity in free space, which we refer to as STOAM wave packets. In addition to the propagationinvariance and arbitrary group velocities of STOAM wave packets, their underlying spatiotemporal structure may lead to variations of some of the recently uncovered behaviors of conventional OAM pulses, such as the tradeoff between the topological charge and pulse duration^{43,45,46}. Such 3D ST wave packets that are fully localized in all dimensions have potential uses in areas such as freespace optical communications, imaging, and nonlinear optics.
Results
Theory of 3D spacetime wave packets
A useful conceptual tool for understanding the characteristics of ST wave packets and the requirements for their synthesis is to visualize their spectral support domain on the surface of the light cone. The lightcone is the geometric representation of the freespace dispersion relationship \({k}_{x}^{2}+{k}_{y}^{2}+{k}_{z}^{2}\,=\,{(\frac{\omega }{c})}^{2}\), where ω is the temporal frequency, c is the speed of light in vacuum, (k_{x}, k_{y}, k_{z}) are the components of the wave vector in the Cartesian coordinate system (x, y, z), x and y are the transverse coordinate, and z is the axial coordinate. Although this relationship corresponds to the surface of a fourdimensional hypercone, a useful representation follows from initially restricting our attention to azimuthally symmetric fields in which k_{x} and k_{y} are combined into a radial wave number \({k}_{r}\,=\,\sqrt{{k}_{x}^{2}+{k}_{y}^{2}}\), so that the lightcone can be then visualized in \(({k}_{r},{k}_{z},\frac{\omega }{c})\)space (Fig. 1). The spectral support domain for 3D ST wave packets is restricted to the conic section at the intersection of the lightcone with a spectral plane that is parallel to the k_{r}axis and makes an angle θ (the spectral tilt angle) with the k_{z}axis, which is given by the equation \({{\Omega }}\,=\,({k}_{z}{k}_{{{{{{{{\rm{o}}}}}}}}})c\tan \theta\); here Ω = ω − ω_{o}, ω_{o} is a carrier frequency, and k_{o} = ω_{o}/c. It can be readily shown that such a construction in the narrowband paraxial regime results in a propagationinvariant 3D ST wave packet \(E(r,\, z;t)\,=\,{e}^{i({k}_{{{{{{{{\rm{o}}}}}}}}}z{\omega }_{{{{{{{{\rm{o}}}}}}}}}t)}\psi (r,\, z;t)\), where the slowly varying envelope ψ(r, z; t) travels rigidly at a group velocity \(\widetilde{v}\,=\,c\tan \theta\), \(\psi (r,z;t)\,=\,\psi (r,0;tz/\widetilde{v})\), where \(\psi (r,0;t)\,=\,\int \,d{k}_{r}\,\,{k}_{r}\widetilde{\psi }({k}_{r}){J}_{0}({k}_{r}r){e}^{i{{\Omega }}t}\), and \(\widetilde{\psi }({k}_{r})\) is the spectrum. Here k_{r} and Ω are no longer independent variables, but are instead related via the particular spectral trajectory on the lightcone (Supplementary Note 1). Although this spectral trajectory is a conic section whose kind is determined by the spectral tilt angle θ, it can nevertheless be approximated in the narrowband paraxial regime by a parabola in the vicinity of k_{r} = 0:
where \(\widetilde{n}\,=\,\cot \theta\) is the wavepacket group index in free space. By setting \({k}_{r}\,=\,k\sin \varphi (\omega )\), where φ(ω) is the propagation angle for ω as shown in Fig. 1a, we have \(\varphi (\omega )\,\approx \,\eta \sqrt{\frac{{{\Omega }}}{{\omega }_{{{{{{{{\rm{o}}}}}}}}}}}\), which is not differentiable at Ω = 0^{20,23}; here \(\widetilde{n}\,=\,1\frac{\sigma }{2}{\eta }^{2}\), σ = 1 in the superluminal regime, and σ = − 1 in the subluminal regime. In other words, nondifferentiable AD is required to produce a propagationinvariant ST wave packet. This result is similar to that for ST lightsheets^{16} except that the transverse coordinate x is now replaced with the radial coordinate r.
The representation in Fig. 1 is particularly useful in identifying a path towards synthesizing 3D ST wave packets. When 45° < θ < 90°, the ST wave packet is superluminal \(\widetilde{v}\, > \,c\), Ω is positive, and ω_{o} is the minimum allowable frequency in the spectrum. When viewed in \(({k}_{x},{k}_{y},\frac{\omega }{c})\)space, the wavelengths are arranged in concentric circles, with long wavelengths (low frequencies) at the center, and shorter wavelengths (higher frequencies) extending outward. On the other hand, when 0° < θ < 45°, the ST wave packet is subluminal \(\widetilde{v}\, < \,c\), Ω is negative, and ω_{o} is the maximum allowable frequency in the spectrum. The wavelengths are again arranged in concentric circles in \(({k}_{x},{k}_{y},\frac{\omega }{c})\)space – but in the opposite order: short wavelengths are close to the center and longer wavelengths extend outward. For both subluminal and superluminal 3D ST wave packets, each ω is associated with a single radial spatial frequency k_{r}(ω), and is related to it via the relationship in Eq. (1). This representation indicates the need for arranging the wavelengths in concentric circles with squareroot radial chirp, and then converting the spatial spectrum into physical space via a spherical lens. Moreover, adding a spectral phase factor e^{iℓχ}, where ℓ is an integer and χ is the azimuthal angle in spectral space, produces OAM in physical space (Supplementary Note 1B).
Closedform expressions can be obtained for 3D ST wave packets by applying Lorentz boosts to an appropriate initial field^{47,48,49,50}. For example, starting with a monochromatic beam E_{o}(r, z; t), a subluminal 3D ST wave packet at a group velocity \(\widetilde{v}\) is obtained by the Lorentz boost \(E(r,\, z;t)\,=\,{E}_{{{{{{{{\rm{o}}}}}}}}}(r,\frac{z\widetilde{v}t}{\sqrt{1{\beta }^{2}}};\frac{t\widetilde{v}z/{c}^{2}}{\sqrt{1{\beta }^{2}}})\), where \(\beta \,=\,\frac{\widetilde{v}}{c}\) is the Lorentz factor. On the other hand, closedform expressions for superluminal 3D ST wave packets can be obtained by applying a Lorentz boost to the ‘needle beam’ in^{12}. The timeaveraged intensity is \(I(r,\varphi,z)\,=\,2\pi {k}_{{{{{{{{\rm{o}}}}}}}}}^{2}(1\widetilde{n})\int d{k}_{r}{k}_{r}^{2}\widetilde{\psi }({k}_{r}){}^{2}{J}_{\ell }^{2}({k}_{r}r)\), which is independent of φ even if the field is endowed with OAM. In the case of 2D ST lightsheets, the timeaveraged intensity separates into a sum of a constant background pedestal and a spatially localized feature at the center^{16}. A similar decomposition is not possible for 3D ST wave packets. However, using the asymptotic form for Bessel functions that is valid far from r = 0, we have:
where the first term is a pedestal decaying at a rate of \(\frac{1}{r}\), and the second term tends to be localized closer to the beam center. In the vicinity of r = 0, the two terms merge and cannot be separated. The spatiotemporal intensity profile of such a 3D ST wave packet is depicted in Fig. 1c: two conic field structures emanate from the wavepacket center, such that the profile is Xshaped in any (meridional plane containing the optical axis, and the intensity profile is circularly symmetric in any transverse plane.
Synthesizing ST wave packets localized in all dimensions
Central to converting a generic pulsed beam into a ST wave packet localized in all dimensions is the construction of an optical scheme that can associate each wavelength λ with a particular azimuthally symmetric spatial frequency k_{r}(λ) and arrange the wavelengths in concentric circles with the order prescribed in Eq. (1) (Fig. 2a). This system realizes two functionalities, producing a particular wavelength sequence, and changing the coordinate system, which are implemented in succession via the threestage strategy outlined in Fig. 2b. In the first stage, the spectrum of a planewave pulse is resolved along one spatial dimension. At this point, the field is endowed with linear spatial chirp and the wavelengths are arranged in a fixed sequence. The second stage rearranges the wavelengths in a new prescribed sequence. This spectral transformation is tunable; that is, a wide range of spectral structures can be obtained from a fixed input. In the third stage, a 2D conformal transformation converts the coordinate system to map the rectilinear chirp into a radial chirp; i.e., lines corresponding to different wavelengths at the input are converted into circles at the output^{40,41}. Because the spectral transformation in the second stage is tunable, the 2D coordinate transformation can be held fixed. In this way, we obtain arbitrary (including nondifferentiable) AD in two dimensions.
The layout of the experimental setup is depicted in Fig. 3. We start off in the first stage with pulses from a Ti:sapphire laser (pulse width ≈ 100 fs and bandwidth ≈10 nm at a central wavelength of ≈800 nm). Because a flatphase front is critical for successfully implementing the subsequent transformations, the use of conventional surface gratings is precluded, and we utilize instead a doublepass configuration through a volume chirped Bragg grating (CBG). The CBG resolves the spectrum horizontally along the xaxis and introduces linear spatial chirp so that x_{1}(λ) = α(λ − λ_{o}); where α is the linear spatial chirp rate^{51}, λ_{o} is a fixed wavelength, and the bandwidth utilized is Δλ ≈ 0.3 nm. It is crucial that this task be achieved with high spectral resolution. Previous studies have shown that the critical parameter determining the propagation distance of ST wave packets is the ’spectral uncertainty’ δλ, which is the finite spectral uncertainty in the association between spatial and temporal frequencies^{52}. Our measurements indicate that the optimal spectral uncertainty after the CBG arrangement is δλ ~ 35 pm, which is achieved for a 2mm input beam width (Supplementary Fig. 11).
The second stage of the synthesis strategy is a 1D spatial transformation along the xaxis to rearrange the wavelength sequence, thereby implementing a spectral transformation. Specifically, each wavelength λ is transposed from x_{1}(λ) at the input via a logarithmic mapping to \({x}_{2}(\lambda )\,=\,A\ln (\frac{{x}_{1}(\lambda )}{B})\) at the output. This transformation is realized via two phase patterns implemented by a pair of spatial light modulators (SLMs) to enable tuning the transformation parameters A and B. This particular ‘reshuffling’ of the wavelength sequence precompensates the exponentiation included in the subsequent coordinate transformation. By tuning the value of B, we can vary the group velocity \(\widetilde{v}\) over the subluminal and superluminal regimes (Supplementary Table 1).
In the third stage we perform a logpolartoCartesian coordinate transformation: (x_{2}, y_{2}) → (r, φ) via the 2D mapping: \(r(\lambda )\,=\,C\exp (\frac{{x}_{2}(\lambda )}{D})\) and \(\varphi=\frac{{y}_{2}}{D}\)^{40,41}. The exponentiation here is precompensated by the logarithmic mapping in the 1D spectral transformation, and the wavelength at position x_{2}(λ) at the input is converted into a circle of radius \(r(\lambda )\,\propto \,{(\lambda {\lambda }_{{{{{{{{\rm{o}}}}}}}}})}^{A/D}\) at the output. This 2D coordinate transformation was developed decades ago^{40,41}, and was recently revived as a methodology for sorting OAM modes^{53,54}. We operate the system in reverse (linestocircles, rather than the more typical circlestolines^{53}), and we make use of a polychromatic field (rather than monochromatic field). The exponent of the chirp rate depends only on the ratio \(\frac{A}{D}\), so that setting D = 2A yields \(r(\lambda )\,\propto \,\sqrt{\lambda {\lambda }_{{{{{{{{\rm{o}}}}}}}}}}\) in accordance with Eq. (1). The wavelengths are arranged with squareroot radial chirp, thereby realizing the required nondifferentiable AD. Finally, a spherical converging lens of focal length f generates the 3D ST wave packets in physical space, equivalently mapping \(r\to \,{k}_{r}\,=\,k\frac{r}{f}\).
The 2D coordinate transformation is performed with two different embodiments: using a pair of diamondmachined refractive phase plates^{54}, and using a pair of diffractive phase plates^{55}, which yielded similar performance. Because both of these realizations are stationary, the values of C and D are fixed. The data reported in Fig. 4 through Fig. 7 made use of the refractive phase plates with C = 4.77 mm and D = 1 mm. Moreover, fixing the value of D entails in turn fixing the value of A to maintain A = D/2. The group velocity \(\widetilde{v}\,=\,c/\widetilde{n}\) is tuned over the subluminal and superluminal regimes by varying B, whereby \(\widetilde{n}\,\approx \,1\frac{2.24}{B}\), with B in units of mm [Supplementary Note 2].
This experimental strategy provides two pathways for introducing OAM into the 3D ST wave packet. One may utilize a conventional spiral phase plate to imprint an OAM order ℓ after the 2D coordinate transformation and before the final Fouriertransforming lens. Another approach, which we implemented here, is to add at the output of the 1D spectral transformation a linear phase distribution along y extending from 0 to 2πℓ, which is subsequently wrapped around the azimuthal direction after traversing the 2D coordinate transformation, thereby realizing OAM of order ℓ^{55}.
For the sake of benchmarking, we also synthesized pulsed Bessel beams with separable spatiotemporal spectrum by circumventing the spectral analysis and 1D spectral transformation, and sending the input laser pulses directly to the 2D coordinate transformation. To match the temporal bandwidth of the pulsed Bessel beams to that of the 3D ST wave packets, we spectrally filter Δλ = 0.3 nm from the input spectrum via a planar FabryPérot cavity.
Characterizing 3D ST wave packets
To verify the structure of the synthesized 3D ST wave packet, we characterize the field in four distinct domains: (1) the spatiotemporal spectrum to verify the squareroot radial chirp (Fig. 4]; (2) the timeaveraged intensity to confirm diffractionfree propagation along z (Fig. 5); (3) timeresolved intensity measurements to reconstruct the wavepacket spatiotemporal profile and estimate the group velocity (Fig. 6); and (4) complexfield measurements to resolve the spiral phase of the STOAM wave packets (Fig. 7).
Spectraldomain characterization
We measure the spatiotemporal spectrum by scanning a singlemode fiber connected to an optical spectrum analyzer across the spectrally resolved field profile. We scan the fiber along x_{1} after the spectral analysis stage and verify the linear spatial chirp (Supplementary Fig. 10), and then scan the fiber along x_{2} after the 1D spectral transformation to confirm the implemented change in spatial chirp. The measurement is repeated for superluminal (B = 10 mm, \(\widetilde{v}\,\approx \,1.37c\)) and subluminal (B = −10 mm, \(\widetilde{v}\,\approx \,0.83c\)) wave packets, both with temporal bandwidth Δλ ≈ 0.3 nm, pulse width of ~ 6 ps, and λ_{o} = 796.1 nm. After the 2D coordinate transformation, the spectrum is arranged radially along an annulus rather than a rectilinear domain, as shown in Fig. 4a. By calibrating the conversion x_{2} → r engendered by the 2D coordinate transformation, and combining with the measured spatial chirp x_{2}(λ) at its input, we obtain the radial chirp k_{r}(λ) as shown in Fig. 4b (Supplementary Fig. 15). We find at each radial position a narrow spectrum (δλ ≈ 50 pm) whose central wavelength λ_{c} shifts quadratically with r, but with differently signed curvature for the superluminal and subluminal cases (Fig. 4c).
Propagationinvariance of the intensity distribution
The timeaveraged intensity profile I(x, y, z) ∝ ∫dt∣E(x, y, z; t)∣^{2} is captured by scanning a CCD camera along the propagation axis z after the Fourier transforming lens (Fig. 3). For each wave packet, we plot in Fig. 5 the intensity distribution (at a fixed axial plane z = 30 mm) in transverse and meridional planes. As a point of reference, we start with a pulsed Bessel beam whose spatiotemporal spectrum is separable, where the spatial bandwidth is Δk_{r} = 0.02 rad/μm and is centered at k_{r} ≈ 0.06 rad/μm (Fig. 5a). Here, the full temporal bandwidth Δλ is associated with each spatial frequency k_{r}. The finite spatial bandwidth Δk_{r} renders the propagation distance finite^{56}, and we observe a Bessel beam comprising a main lobe of width Δr ≈ 30 μm (FWHM) accompanied by several side lobes, which propagates for a distance \({L}_{\max }\,\approx\, 50\) mm. For comparison, the Rayleigh range of a Gaussian beam with a similar size and central wavelength is z_{R} ≈ 1 mm. By further increasing Δk_{r} to 0.07 rad/μm while remaining centered at k_{r} ≈ 0.06 rad/μm as shown in Fig. 5b, the axial propagation distance is reduced proportionately to \({L}_{\max }\,\approx\, 15\) mm, and the side lobes are diminished.
Now, rather than the separable spatiotemporal spectra for pulsed Bessel beams (Fig. 5a, b), we utilize the structured spatiotemporal spectra associated with 3D ST wave packets in which each k_{r} is associated with a single λ (Fig. 4), whose spatial bandwidths are all Δk_{r} = 0.07 rad/μm centered at k_{r} ≈ 0.06 rad/μm, similarly to the pulsed Bessel beam in Fig. 5b. Despite the large spatial bandwidth, the onetoone correspondence between k_{r} and λ curtails the effect of diffraction, leading to an increase in the propagation distance (Fig. 5c–e). The subluminal 3D ST wave packet (\(\widetilde{v}\,=\,0.83c\)) in Fig. 5c propagates for \({L}_{\max }\,\approx\, 60\) mm, which is a 4 × improvement compared with the separable Bessel beam and a 60 × improvement compared with a Gaussian beam of the same spatial bandwidth. We observe a similar behavior for a superluminal 3D ST wave packet (\(\widetilde{v}\,=\,1.37c\)) in Fig. 5d, and a superluminal STOAM wave packet (\(\widetilde{v}\,=\,1.16c\)) with ℓ = 1 in Fig. 5e.
Reconstructing the spatiotemporal profile and measuring the group velocity
The spatiotemporal intensity profile I(x, y, z; t) = ∣E(x, y, z; t)∣^{2} of the 3D ST wave packet is reconstructed by placing the synthesizer (Fig. 3) in one arm of a MachZehnder interferometer, while the initial 100fs planewave pulses from the laser traverse an optical delay line τ in the reference arm (Fig. 6a). By scanning τ we reconstruct the spatiotemporal intensity profile in a meridional plane from the visibility of spatiallyresolved interference fringes recorded by a CCD camera when the 3D ST wave packet and the reference pulse overlap in space and time. The reconstructed timeresolved intensity profile I(0, y, z; t) of the 3D ST wave packets corresponding to those in Fig. 5c–e are plotted in Fig. 6b–d at multiple axial planes, which reveal clearly the expected Xshaped profile that remains invariant over the propagation distance \({L}_{\max }\). In all cases, the onaxis pulse width, taken as the FWHM of I(0, 0, 0; t), is Δt ≈ 6 ps. The spatiotemporal intensity profile of the superluminal STOAM wave packet with ℓ = 1 in Fig. 6d reveals a similar Xshaped profile, but with a central null instead of a peak, as expected from the helical phase structure associated with the OAM mode.
A subtle distinction emerges between the subluminal and superluminal wave packets regarding the axial evolution of their spatiotemporal profile. It can be shown that in presence of finite spectral uncertainty δλ, the realized ST wave packet can be separated into the product of an ideal ST wave packet traveling indefinitely at \(\widetilde{v}\) and a long ‘pilot envelope’ traveling at c. The finite propagation distance \({L}_{\max }\) is then a consequence of temporal walkoff between the ST wave packet and the pilot envelope^{52}. For subluminal ST wave packets, this results initially in a ‘clipping’ of the leading edge of the wave packet in (Fig. 6b at z = 20 mm), and ultimately a clipping of the trailing edge of the ST wave packet as the faster pilot envelope catches up with it (Fig. 6b at z = 40 mm). The opposite behavior occurs for the superluminal ST wave packet in Fig. 6c, d.
This experimental methodology also enables us to estimate the group velocity \(\widetilde{v}\)^{28,30}. After displacing the CCD camera until the interference fringes are lost due to the mismatch between \(\widetilde{v}\,=\,c\tan \theta\) for the ST wave packets and the reference pulses traveling at \(\widetilde{v}\,=\,c\), we restore the interference by inserting a delay Δt (Fig. 6e), which allows us to estimate \(\widetilde{v}\) for the 3D ST wave packet. By tuning B, we record a broad span of group velocities in the range from \(\widetilde{v}\ \approx \ 0.7c\) to \(\widetilde{v}\ \approx \ 1.8c\) in free space (Fig. 6f). The continuous tunability of the group velocity of 3D ST wave packets over the subluminal and superluminal ranges allows them to be exploited in applications previously proposed for ST lightsheets, such as for constructing inline optical delay lines for alloptical communications^{29}, whereby the localization of 3D ST wave packets in both transverse dimensions can provide a significant advantage with regards to efficiently coupling into optical fibers.
Field amplitude and phase measurements
Lastly, we modify the measurement system in Fig. 6a by adding a small relative angle between the propagation directions of the 3D ST wave packets and the reference pulses, and make use of offaxis digital holography^{57} to reconstruct the amplitude ∣ψ(x, y, z; τ)∣ and phase ϕ(x, y, z; τ) of their complex field envelope ψ(x, y, z; t) = ∣ψ(x, y, z; t)∣e^{iϕ(x, y, z; t)} (Supplementary Note 3D). We reconstruct the complex field at a fixed axial plane z = 30 mm for the time delays: τ = −5, 0, and 5 ps (Fig. 7). First, we plot the results for ∣ψ(x, y, z; τ)∣ and phase ϕ(x, y, z; τ) for a superluminal 3D ST wave packet (\(\widetilde{v}=1.1c\)) with no OAM (ℓ = 0). At the pulse center τ = 0, the field is localized on the optical axis, whereas at τ = ±5 ps the field spreads away from the center (Fig. 7a)]. For τ ≠ 0 we find a spherical transverse phase distribution that is almost flat at τ = 0, similar to what one finds during the axial evolution of a Gaussian beam in space through its focal plane^{14}.
After adding the OAM mode ℓ = 1 to the field structure, a similar overall behavior is observed for the superluminal STOAM wave packet except for two significant features. First, a dip is observed onaxis in Fig. 7b, in lieu of the central peak in Fig. 7a, as a result of the phase singularity associated with the OAM mode. Second, the phase at the wavepacket center ϕ(x, y, z; 0) at z = 30 mm is almost flat, while a helical phase front corresponding to OAM of order ℓ = 1 emerges as we move away from τ = 0. Finally, we plot in Fig. 7c, d isoamplitude surface contours (0.6 × and 0.15 × the maximum amplitude \({I}_{\max }\)) for the two 3D ST wave packets in Fig. 7a, b. We find a closed surface in Fig. 7c when ℓ = 0, and a doughnut structure in Fig. 7d when ℓ = 1 for the first contour \(I\,=\,0.6{I}_{\max }\) that captures the structure of the wavepacket center. The second contour for \(I\,=\,0.15{I}_{\max }\) captures the conic structure emanating from the wavepacket center that is responsible for the characteristic Xshaped spatiotemporal profile of all propagationinvariant wave packets in the paraxial regime.
Discussion
We have demonstrated a general procedure for spatiotemporal spectral modulation of pulsed optical fields that is capable of synthesizing 3D ST wave packets localized in all dimensions. At the heart of our experimental methodology lies the ability to sculpt the angular dispersion of a generic optical pulse in two transverse dimensions. Crucially, this approach produces the nondifferentiable angulardispersion necessary for propagation invariance^{21}. Because such a capability has proven elusive to date, ADfree Xwaves have been the sole class of 3D propagationinvariant wave packets conclusively produced in free space. Unfortunately, Xwaves can exhibit only minuscule changes in the group velocity with respect to c (typically \({{\Delta }}\widetilde{v}\, {\sim} \,0.001c\)) in the paraxial regime, and only superluminal group velocities are supported. Furthermore, ultrashort pulses of width < 20 fs are required to observe a clear Xshaped profile^{10}, and OAMcarrying Xwaves have not been realized to date. Even more stringent requirements are necessary for producing focuswave modes, and consequently they have not been synthesized in three dimensions to date. By realizing instead propagationinvariant 3D ST wave packets, an unprecedented tunable span of group velocities has been realized, clear Xshaped profiles are observed with pulse widths in the picosecond regime, and they outperformed spectrally separable pulsed Bessel beams of the same spatial bandwidth with respect to their propagation distance and transverse sidelobe structure. In addition, we demonstrated propagationinvariant STOAM wave packets with tunable group velocity in free space.
Further optimization of the experimental layout is possible. We made use of four phase patterns to produce the target spatiotemporal spectral structure. It is conceivable that this spectral modulation scheme can be performed with only three phase patterns, or perhaps even fewer. Excitingly, a new theoretical proposal suggests that a single nonlocal nanophotonic structure can produce 3D ST wave packets through a process of spatiotemporal spectral filtering^{58}. This theoretical proposal indicates the role nanophotonics is poised to play in reducing the complexity of the synthesis system, potentially without recourse to filtering strategies.
Finally, efforts in the near future will be directed to reducing the spectral uncertainty δλ and concomitantly approaching θ → 45° to increase the propagation length to the kilometer range^{24}. The experimental procedure presented here can in principle be extended to the synthesis of other exotic variants of ST wave packet, such as abruptly focusing needle pulses^{59} among other possibilities^{19,60}. With access to 3D ST wave packets, previous work on guided ST modes in planar waveguides^{61} can be extended to conventional singlemode and multimode waveguides^{62}, and potentially to optical fibers^{63,64,65,66}. Moreover, the localization in both transverse dimensions provided by 3D ST wave packets opens new avenues for nonlinear optics by increasing the intensity with respect to 2D ST wave packets, for introducing topological features such as spin texture in momentum space^{58}, and for the exploration of spatiotemporal vortices and polarization singularities^{67}. Our findings point therefore to profound new opportunities provided by the emerging field of spacetime optics^{58,61,62,68,69,70,71}.
Methods
The 2D transformation used to construct the 3D STWP can be implemented by making use of diffractive optics^{53,55,72,73} or refractive optics^{54}. We exploited both types of phase plates in our experiments to imprint the desired phase profiles: diamondedged refractive phase plates^{54} and analog diffractive phase plates^{55}.
Refractive phase plates
The refractive optical elements used in our experiments are similar to those outlined by Lavery et al. in^{54}, in which the transformation parameters are C = 4.77 mm, \(D\,=\,\frac{3.2}{\pi }\approx \,1\) mm, and d_{2} = 310 mm. Each phase plate is made of the polymer PMMA (Poly methyl methacrylate) with accurately manufactured height profiles Z_{1}(x_{3}, y_{3}) and Z_{2}(x_{4}, y_{4}) to imprint the required phase profiles. The phase encountered by light at a wavelength λ traversing a height Z of a material of refractive index n – with respect to the phase encountered over the same distance in vacuum – is given by Φ = 2π(n − 1)Z/λ. Thus, the height profile of the first element is \({Z}_{1}({x}_{3},\ {y}_{3})=\frac{\lambda }{2\pi (n1)}{{{\Phi }}}_{3}({x}_{3},\ {y}_{3})\) (Supplementary Fig. 14a) and that of the profile of the second element is \({Z}_{2}({x}_{4},\ {y}_{4})=\frac{\lambda }{2\pi (n1)}{{{\Phi }}}_{4}({x}_{4},\ {y}_{4})\) (Supplementary Fig. 14b). Note that each surface height is wavelengthindependent, and dispersion effects in the material manifest themselves as a change in the focal length d_{2} of the integrated lens for different wavelengths. Hence, in the experiment the system can be tuned to a specific wavelength by changing the distance between the two elements.
The elements were diamondmachined using a Natotech, 3axis (X,Z,C) ultra precision lathe (UPL) in combination with a Nanotech NFTS6000 fast tool servo (FTS) system. The machined PMMA surfaces had a radius of 5.64 mm, angular spacing 1°, radial spacing of 5 μm, a spindle speed of 500 RPM, a roughing feed rate 5 mm/minute with a cut depth of 20 μm, and a finishing feed rate of 1 mm/minute with a cut depth of 10 μm^{74}. The total sag height difference for each part was relatively small (≈115 μm for surface 1 and ≈144 μm for surface 2). The transmission efficiency of the combination of the elements is ≈85%.
Diffractive phase plates
The diffractive phase plates were fabricated in fused silica using Clemson University facilities. The fabrication process is outlined in^{75}, which involves writing a binary phase grating on a stepper mask with an electronbeam and subsequently transferring this analog mask into a fused silica substrate with projection lithography. The phase grating period is designed to be larger than the cutoff period of the projection stepper for higher diffraction orders, so only the zerothorder diffracted light from the stepper can be transmitted. The transmission coefficient of the stepper light is then a function of the duty cycle of the electronbeampatterned binary phase grating. The spatial intensity distribution of light in the wafer plane can be controlled with a spatial duty cycle function, which then exposes the Iline resist with a spatially varying analog intensity profile. This allows fabrication of analog diffractive optics with a single exposure from the stepper rather than binary 2^{n} diffractive optics, resulting in highefficiency optics. The transmission efficiency of the combination of the two faces is ≈ 92%.
The design parameters for the analog diffractive phase plates are chosen as follows: \(D\,=\,\frac{7}{\pi }\,\approx \,2.2\) mm, C = 6 mm, λ_{o} = 798 nm, and d_{2} = 225 mm. These design parameters were optimized so the paraxial approximation remains valid over the desired transformation range of 5 mm.
Data availability
The data that support the plots within this paper and other findings of this study are available from the corresponding author upon reasonable request.
Code availability
The software code used for data acquisition and data analysis are available from the corresponding author upon reasonable request.
References
Malomed, B. A., Mihalache, D., Wise, F. & Torner, L. Spatiotemporal optical solitons. J. Opt. B 7, R53–R72 (2005).
SalehM, B.E.A. & Teich, C. Principles of Photonics (Wiley, 2007)
Chong, A., Renninger, W. H., Christodoulides, D. N. & Wise, F. W. AiryBessel wave packets as versatile linear light bullets. Nat. Photon. 4, 103–106 (2010).
Fülöp, J.A., & Hebling, J. Applications of tiltedpulsefront excitation, in Recent Optical and Photonic Technologies (ed. K. Y. Kim) (InTech, 2010).
Torres, J. P., Hendrych, M. & Valencia, A. Angular dispersion: an enabling tool in nonlinear and quantum optics. Adv. Opt. Photon. 2, 319–369 (2010).
Donnelly, R. & Ziolkowski, R. W. Designing localized waves. Proc. R. Soc. Lond. A 440, 541–565 (1993).
Turunen, J. & Friberg, A. T. Propagationinvariant optical fields. Prog. Opt. 54, 1–88 (2010).
Brittingham, J. N. Focus wave modes in homogeneous maxwell’s equations: Transverse electric mode. J. Appl. Phys. 54, 1179–1189 (1983).
Saari, P. & Reivelt, K. Evidence of Xshaped propagationinvariant localized light waves. Phys. Rev. Lett. 79, 4135–4138 (1997).
Grunwald, R. et al. Generation and characterization of spatially and temporally localized fewcycle optical wave packets. Phys. Rev. A 67, 063820 (2003).
Kondakci, H. E. & Abouraddy, A. F. Diffractionfree pulsed optical beams via spacetime correlations. Opt. Express 24, 28659–28668 (2016).
Parker, K. J. & Alonso, M. A. The longitudinal isophase condition and needle pulses. Opt. Express 24, 28669–28677 (2016).
Wong, L. J. & Kaminer, I. Ultrashort tiltedpulsefront pulses and nonparaxial tiltedphasefront beams. ACS Photon. 4, 2257–2264 (2017).
Porras, M. A. Gaussian beams diffracting in time. Opt. Lett. 42, 4679–4682 (2017).
Efremidis, N. K. Spatiotemporal diffractionfree pulsed beams in freespace of the Airy and Bessel type. Opt. Lett. 42, 5038–5041 (2017).
Kondakci, H. E. & Abouraddy, A. F. Diffractionfree spacetime beams. Nat. Photon. 11, 733–740 (2017).
Yessenov, M., Bhaduri, B., Kondakci, H. E. & Abouraddy, A. F. Classification of propagationinvariant spacetime lightsheets in free space: Theory and experiments. Phys. Rev. A 99, 023856 (2019).
Yessenov, M., Bhaduri, B., Kondakci, H. E. & Abouraddy, A. F. Weaving the rainbow: Spacetime optical wave packets. Opt. Photon. N. 30, 34–41 (2019).
Wong, L. J. Propagationinvariant spacetime caustics of light. Opt. Express 29, 30682 (2021).
Hall, L. A. & Abouraddy, A. F. Consequences of nondifferentiable angular dispersion in optics: tilted pulse fronts versus spacetime wave packets. Opt. Express 30, 4817–4832 (2022).
Yessenov, M., Hall, L. A. & Abouraddy, A. F. Engineering the optical vacuum: Arbitrary magnitude, sign, and order of dispersion in free space using spacetime wave packets. ACS Photon. 8, 2274–2284 (2021).
Hall, L. A., Yessenov, M. & Abouraddy, A. F. Space–time wave packets violate the universal relationship between angular dispersion and pulsefront tilt. Opt. Lett. 46, 1672–1675 (2021).
Hall, L.A., & Abouraddy, F. A universal angulardispersion synthesizer, arXiv:2109.13987 (2021).
Bhaduri, B. et al. Broadband spacetime wave packets propagating 70 m. Opt. Lett. 44, 2073–2076 (2019).
Salo, J. & Salomaa, M. M. Diffractionfree pulses at arbitrary speeds. J. Opt. A 3, 366–373 (2001).
Valtna, H., Reivelt, K. & Saari, P. Methods for generating wideband localized waves of superluminal group velocity. Opt. Commun. 278, 1–7 (2007).
ZamboniRached, M. & Recami, E. Subluminal wave bullets: Exact localized subluminal solutions to the wave equations. Phys. Rev. A 77, 033824 (2008).
Kondakci, H. E. & Abouraddy, A. F. Optical spacetime wave packets of arbitrary group velocity in free space. Nat. Commun. 10, 929 (2019).
Yessenov, M., Bhaduri, B., Delfyett, P. J. & Abouraddy, A. F. Freespace optical delay line using spacetime wave packets. Nat. Commun. 11, 5782 (2020).
Bhaduri, B., Yessenov, M. & Abouraddy, A. F. Anomalous refraction of optical spacetime wave packets. Nat. Photon. 14, 416–421 (2020).
Hall, L. A., Yessenov, M., Ponomarenko, S. A. & Abouraddy, A. F. The spacetime Talbot effect. APL Photon. 6, 056105 (2021).
Vaughan, J. C., Feurer, T. & Nelson, K. A. Automated spatiotemporal diffraction of ultrashort laser pulses. Opt. Lett. 28, 2408–2410 (2003).
Jhajj, N. et al. Spatiotemporal optical vortices. Phys. Rev. X 6, 031037 (2016).
Hancock, S. W., Zahedpour, S., Goffin, A. & Milchberg, H. M. Freespace propagation of spatiotemporal optical vortices. Optica 6, 1547–1553 (2019).
Chong, A., Wan, C., Chen, J. & Zhan, Q. Generation of spatiotemporal optical vortices with controllable transverse orbital angular momentum. Nat. Photon. 14, 350–354 (2020).
Cao, Q. et al. Sculpturing spatiotemporal wavepackets with chirped pulses. Photon. Res. 9, 2261–2264 (2021).
Wan, C., Cao, Q., Chen, J., Chong, A., & Zhan, Q. Photonic toroidal vortex, arXiv:2109.02833 (2021).
Hancock, S. W., Zahedpour, S. & Milchberg, H. M. Secondharmonic generation of spatiotemporal optical vortices and conservation of orbital angular momentum. Optica 8, 594–597 (2021).
Hancock, S. W., Zahedpour, S. & Milchberg, H. M. Mode structure and orbital angular momentum of spatiotemporal optical vortex pulses. Phys. Rev. Lett. 127, 193901 (2021).
Bryngdahl, O. Geometrical transformations in optics. J. Opt. Soc. Am. A 64, 1092–1099 (1974).
Hossack, W. J., Darling, A. M. & Dahdouh, A. Coordinate transformations with multiple computergenerated optical elements. J. Mod. Opt. 34, 1235–1250 (1987).
Allen, L., Beijersbergen, M. W., Spreeuw, R. J. C. & Woerdman, J. P. Orbital angular momentum of light and the transformation of LaguerreGaussian laser modes. Phys. Rev. A 45, 8185–8189 (1992).
Ornigotti, M., Conti, C. & Szameit, A. Effect of orbital angular momentum on nondiffracting ultrashort optical pulses. Phys. Rev. Lett. 115, 100401 (2015).
Porras, M. A. & GarcíaÁlvarez, Raúl Broadband x waves with orbital angular momentum. Phys. Rev. A 105, 013509 (2022).
Porras, M. A. Upper bound to the orbital angular momentum carried by an ultrashort pulse. Phys. Rev. Lett. 122, 123904 (2019).
Porras, M. A. & Conti, C. Couplings between the temporal and orbital angular momentum degrees of freedom in ultrafast optical vortices. Phys. Rev. A 101, 063803 (2020).
Bélanger, P. A. Lorentz transformation of packetlike solutions of the homogeneouswave equation. J. Opt. Soc. Am. A 3, 541–542 (1986).
Saari, P. & Reivelt, K. Generation and classification of localized waves by Lorentz transformations in Fourier space. Phys. Rev. E 69, 036612 (2004).
Longhi, S. Gaussian pulsed beams with arbitrary speed. Opt. Express 12, 935–940 (2004).
Kondakci, H. E. & Abouraddy, A. F. Airy wavepackets accelerating in spacetime. Phys. Rev. Lett. 120, 163901 (2018).
Glebov, L. B. et al. Volumechirped Bragg gratings: monolithic components for stretching and compression of ultrashort laser pulses. Opt. Eng. 53, 051514 (2014).
Yessenov, M. et al. What is the maximum differential group delay achievable by a spacetime wave packet in free space? Opt. Express 27, 12443–12457 (2019).
Berkhout, G. C. G., Lavery, M. P. J., Courtial, J., Beijersbergen, M. W. & Padgett, M. J. Efficient sorting of orbital angular momentum states of light. Phys. Rev. Lett. 105, 153601 (2010).
Lavery, M. P. J. et al. Refractive elements for the measurement of the orbital angular momentum of a single photon. Opt. Express 20, 2110–2115 (2012).
Li, W. & Johnson, E. G. Rapidly tunable orbital angular momentum (OAM) system for higher order Bessel beams integrated in time (HOBBIT). Opt. Express 27, 3920–3934 (2019).
Durnin, J., Miceli, J. J. & Eberly, J. H. Diffractionfree beams. Phys. Rev. Lett. 58, 1499–1501 (1987).
SánchezOrtiga, E., Doblas, A., Saavedra, G., MartínezCorral, M. & GarciaSucerquia, J. Offaxis digital holographic microscopy: practical design parameters for operating at diffraction limit. Appl. Opt. 53, 2058–2066 (2014).
Guo, C., Xiao, M., Orenstein, M. & Fan, S. Structured 3D linear space–time light bullets by nonlocal nanophotonics. Light.: Sci. Appl 10, 1–15 (2021).
Wong, L. J. & Kaminer, I. Abruptly focusing and defocusing needles of light and closedform electromagnetic wavepackets. ACS Photon. 4, 1131–1137 (2017).
Wong, L. J., Christodoulides, D. N. & Kaminer, I. The complex charge paradigm: A new approach for designing electromagnetic wavepackets. Adv. Sci. 7, 1903377 (2020).
Shiri, A., Yessenov, M., Webster, S., Schepler, K. L. & Abouraddy, A. F. Hybrid guided spacetime optical modes in unpatterned films. Nat. Commun. 11, 6273 (2020).
Guo, C. & Fan, S. Generation of guided spacetime wave packets using multilevel indirect photonic transitions in integrated photonics. Phys. Rev. Res. 3, 033161 (2021).
Ruano, P. N., Robson, C. W. & Ornigotti, M. Localized waves carrying orbital angular momentum in optical fibers. J. Opt. 23, 075603 (2021).
Béjot, P. & Kibler, B. Spatiotemporal helicon wavepackets. ACS Photon. 8, 2345–2354 (2021).
Kibler, B. & Béjot, P. Discretized conical waves in multimode optical fibers. Phys. Rev. Lett. 126, 023902 (2021).
P., BéjotB., Kibler, Quadrics for structuring spacetime wavepackets, arXiv:2202.00407 (2022).
Bliokh, K. Y. & Nori, F. Spatiotemporal vortex beams and angular momentum. Phys. Rev. A 86, 033824 (2012).
SainteMarie, A., Gobert, O. & Quéré, F. Controlling the velocity of ultrashort light pulses in vacuum through spatiotemporal couplings. Optica 4, 1298–1304 (2017).
Froula, D. H. et al. Spatiotemporal control of laser intensity. Nat. Photon. 12, 262–265 (2018).
Shaltout, A. M. Spatiotemporal light control with frequencygradient metasurfaces. Science 365, 374–377 (2019).
Zdagkas, A., Shen, Y., Papasimakis, N., & Zheludev, N.I. Observation of toroidal pulses of light, arXiv:2102.03636 (2021)
Lavery, M. P. J., Berkhout, G. C. G., Courtial, J. & Padgett, M. J. Measurement of the light orbital angular momentum spectrum using an optical geometric transformation. J. Opt. 13, 064006 (2011).
Berkhout, G. C. G., Lavery, M. P. J., Padgett, M. J. & Beijersbergen, M. W. Measuring orbital angular momentum superpositions of light by mode transformation. Opt. Lett. 36, 1863–1865 (2011).
Dow, T. A., Miller, M. H. & Falter, P. J. Application of a fast tool servo for diamond turning of nonrotationally symmetric surfaces. Precis. Eng. 13, 243–250 (1991).
Sung, J. W., Hockel, H., Brown, J. D. & Johnson, E. G. Development of a twodimensional phasegrating mask for fabrication of an analogresist profile. Appl. Opt. 45, 33–43 (2006).
Acknowledgements
We thank OptiGrate Company for making volume Bragg gratings, and Dr. Peter J. Delfyett and Dr. Ivan Divliansky for lending equipment. We thank L. A. Hall, A. Shiri, K. L. Schepler, L. Mach, M. G. Vazimali, I. Hatipoglu, and M. Eshaghi for useful discussions. M.Y. and A.F.A. were supported by the U.S. Office of Naval Research (ONR) under contracts N000141712458, N000141912192, and N000142012789. J.F. and E.G.J. were supported by ONR contract N000142012558. M.A.A. was funded by the Excellence Initiative of Aix Marseille University – A*MIDEX, a French ‘Investissements d’Avenir’ programme.
Author information
Authors and Affiliations
Contributions
A.F.A. and M.Y. developed the concept. M.Y. designed the experiments, carried out the measurements, and analyzed the data. Z.C. and M.P.J.L. designed and manufactured the diamondmachined refractive elements for the 2D coordinate transformation. J.F and E.G.J. designed and fabricated the analog diffractive phase plates for the 2D coordinate transformation. M.A.A. and A.F.A. developed the theoretical aspects. A.F.A. supervised the research. All the authors contributed to writing the paper.
Corresponding authors
Ethics declarations
Competing interests
The authors declare no competing interests.
Peer review
Peer review information
Nature Communications thanks Liang Jie Wong and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.
Additional information
Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Supplementary information
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Yessenov, M., Free, J., Chen, Z. et al. Spacetime wave packets localized in all dimensions. Nat Commun 13, 4573 (2022). https://doi.org/10.1038/s41467022322400
Received:
Accepted:
Published:
DOI: https://doi.org/10.1038/s41467022322400
This article is cited by

Timeofflight resolved stimulated Raman scattering microscopy using counterpropagating ultraslow Bessel light bullets generation
Light: Science & Applications (2024)

Nondiffracting supertoroidal pulses and optical “Kármán vortex streets”
Nature Communications (2024)

Observation of optical de Broglie–Mackinnon wave packets
Nature Physics (2023)

The propagation speed of optical speckle
Scientific Reports (2023)

Broadband control of topological–spectral correlations in space–time beams
Nature Photonics (2023)
Comments
By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.