Observation of space-time surface plasmon polaritons

Domain: physics
Published: 2025-12-01
Editorial outcome: minor_revision

Abstract
Surface plasmon polaritons (SPPs) at metal-dielectric interfaces provide strong out-of-plane confinement enabling nano-scale sensing and imaging, yet diffraction causes spatial delocalization. Conventional strategies to combat diffraction through spatial structuring are inapplicable to dimensionally restricted SPPs, except for nonlocalized cosine plasmons and Airy plasmons that follow curved trajectories. Here we demonstrate space-time SPPs (ST-SPPs), ultrashort (16-fs) diffraction-free SPPs that propagate rectilinearly via precise sculpting of their spatiotemporal spectra. By synthesizing a spatio-temporally structured field in free space and coupling the field to an axially invariant ST-SPP at a metal-dielectric surface, we control the ST-SPP group velocity and propagation characteristics. Time-resolved two-photon fluorescence microscopy reconstructs the surface-bound field in space and time, verifying the predicted spatiotemporal wavefront and diffraction-free propagation. Our work opens new avenues for combining spatiotemporally structured light with the field-localization associated with nanophotonics, and may thus enable novel applications in surface-enhanced sensing and nonlinear optical interactions.

Introduction
Surface plasmon polaritons (SPPs) are optical surface waves that propagate along metal-dielectric interfaces1,2. Because SPPs are strongly localized orthogonally to the interface, they are suitable for a host of photonic applications ranging from biosensing3 and tweezers4, to nanofocusing5,6, modulators7–9, and harvesting solar energy10,11, among other fascinating possibilities12–15. This strong field confinement of the SPP at the interface is marred by free in-plane propagation, resulting in diffractive spreading of its transverse spatial profile. In addition, SPPs are dispersive, so that the temporal (axial) profile of a pulsed SPP undergoes dispersive spreading (Fig. 1a). Producing diffraction-free, tightly focused, ultrashort pulsed SPPs would yield strong field localization in all dimensions that is maintained over extended propagation distances, which can help enhance many SPP applications.

Axially maintained, in-plane confinement can be provided by surface-patterned waveguide structures16,17, or by hyperbolic metamaterials that require strongly anisotropic materials18–21. In free space, so-called diffraction-free optical beams can maintain their spatial profile axially over considerable distances if their profile conforms to particular functional distributions, e.g., Bessel beams22. However, the one-dimensional (1D) transverse nature of SPPs is a restrictive condition with regard to achieving diffraction-free propagation. Indeed, there are no monochromatic diffraction-free light sheets–which are localized along one transverse dimension and uniform along the other – that travel in a straight line23. Conventional diffraction-free beams require two transverse dimensions for their viability; so that even a light-sheet that conforms to a Bessel function in 1D is diffractive, in contrast to its 2D counterpart. There are only two exceptions: the cosine wave that is not localized, and the Airy beam that does not travel along a straight line23. Both of these monochromatic field structures have been implemented as SPPs: the cosine plasmon24,25 and the Airy plasmon26,27. Therefore, no diffraction-free SPP traveling along a straight path has been realized to date. This remains a fundamental challenge that cannot be overcome by means of spatially structuring the excitation, which is a consequence of the dimensionality of SPPs as surface waves, along with the monochromaticity of the excitation. Whereas the first restriction cannot be avoided, the second can be readily circumvented by sculpting a pulsed SPP wave packet.

Recently, free-space diffraction-free light sheets that travel along a straight line have been finally realized using a class of pulsed beams known as space-time wave packets (STWPs), which are diffraction-free without imposing a particular 1D transverse spatial profile28. Key to this unique feature is lifting the monochromaticity constraint, and then introducing a strict spectral association between the spatial and temporal frequencies undergirding the field spatiotemporal structure, thereby yielding unique characteristics such as propagation invariance in linear media28–31, tunable group velocities independently of the medium refractive index30,32, self-healing upon traversing opaque obstacles33, and anomalous refraction34. We have proposed a pulsed SPP beam (or SPP wave packet) that adapts the characteristic structure underpinning freely propagating STWPs to the particular setting of surface waves35. Because STWPs are the only propagation-invariant light sheets, they offer the potential for finally yielding diffraction-free SPPs that propagate along a straight line. By restricting the spatiotemporal spectrum of the SPP to the intersection of the SPP light-cone with a tilted spectral plane, one obtains a space-time SPP wave packet (ST-SPP) that inherits many of the unique propagation characteristics of STWPs; in particular, ST-SPPs propagate in a straight line without diffraction or dispersion at a metal-dielectric interface at a group velocity that may be tuned above (superluminal) or below (subluminal) that of a conventional SPP wave packet35,36; see Fig. 1b. However, experimentally realizing an ST-SPP and ascertaining its propagation characteristics offers a host of challenges. Because the typical decay length due to ohmic losses on a metal surface is limited to tens of microns37–39, unambiguously monitoring the propagation of a pulsed SPP necessitates utilizing a broadband ultrashort pulse of width <20 fs, which then raises the question of coupling a broadband spatiotemporally structured optical field to a surface-bound field40.

Here we report the experimental realization of ST-SPPs at a metal-dielectric interface, which is made possible by resolving the multidimensional experimental challenge outlined above. First, we make use of ultrashort pulses of width 16-fs pulses (110-nm bandwidth), which correspond to an SPP of axial extent ≈3.6 μm, thereby enabling the monitoring of their propagation over a short distance, and thus enabling us to confirm the tunability of the ST-SPP group velocity above and below that of an SPP as determined by its intrinsic dispersion. Second, we make use of a nano-slit milled into the metal surface for efficient broadband coupling of STWPs from free space to an ST-SPP (predicted theoretically35,41 and demonstrated with minimally structured fields42), while maintaining the structure of the target spatiotemporal spectrum. Third, the use of ultrashort pulses plays a crucial role in the phase-sensitive detection of the spatially and temporally resolved surface-bound field through two-photon fluorescence excited by a reference pulse interfering with the evanescent tail of the SPP at the metal surface. We thus reconstruct the surface-bound field for a variety of ST-SPPs, in addition to conventional SPP wave packets, and verify the details of their spatiotemporal structure, including a predicted signature tilt in their phase front with respect to the propagation direction. Furthermore, by exciting conventional SPP wave packets and ST-SPPs of the same bandwidth, we confirm the diffractive spreading of the former and the diffraction-free propagation of the latter. These results establish a new form of SPP wave packets that offers unprecedented control over their propagation characteristics. This may enable novel applications in enhanced surface nonlinear optical interactions mediated by the tunable group velocity of ST-SPPs, in generating radiation through accelerating ST-SPPs43, and launching isolated propagating topological SPP spin textures13,44,45 or SPPs endowed with transverse orbital angular momentum46,47.

Results
Theoretical formulation
The conceptual formulation of ST-SPPs is best understood by visualizing their spectral representation on the surface of the light-cone. For (2+1)D pulsed optical fields restricted to one transverse dimension x in addition to the axial dimension z, the light-cone is the geometric representation of the dispersion relationship involving the transverse wavenumber kx (or spatial frequency), the longitudinal wavenumber kz, and the temporal frequency ω. In free space, the light-cone is k2x+k2z=(ω/c)2, and that for an SPP at a metal-dielectric interface is k2x+k2z=k2SPP(ω); where c is the speed of light in vacuum, kSPP=ω/c sqrt(ϵmϵd/(ϵm+ϵd)), and ϵm and ϵd are the relative permittivities of the metal and dielectric, respectively (Fig. 1a). In free space, where a (2+1)D pulsed beam (or wave packet) takes the form of a light-sheet that is uniform along y and localized along x, its spectral support is a two-dimensional (2D) region on the surface of the free-space light-cone (Fig. 1a). Such a light-sheet undergoes diffractive spreading with free propagation. A conventional SPP wave packet is a (2+1)D surface wave with finite transverse-spatial extent along x in addition to surface localization along y, whose spectral support in turn corresponds to a 2D region on the surface of the SPP light-cone (Fig. 1a)35,48. This conventional SPP wave packet undergoes both diffractive spreading in space and dispersive spreading in time.

Ideal STWPs are pulsed beams that propagate rigidly in linear media without diffraction or dispersion, and whose propagation characteristics can be tuned largely independently of the material parameters23,28. Rather than the 2D spectral support on the free-space light-cone associated with conventional pulsed light sheets, the spatiotemporal spectrum of an STWP is restricted to a 1D curve (a conic section) at the intersection of the light-cone with a plane that is parallel to the kx-axis and makes an angle θ with the kz-axis (Fig. 1b)28. This plane, ω−ωo=(kz−ko)c tanθ, results in a straight-line spectral projection onto the (kz, ω/c)-plane, which indicates a fixed group velocity ev=c tanθ and the absence of dispersion to all orders49; here ωo is the fixed carrier frequency, and ko=ωo/c is its associated wave number. Moreover, the one-to-one association between ω and |kx| guarantees diffraction-free propagation of the time-averaged intensity23. Similarly for ST-SPPs, their spectral support as shown in Fig. 1b is also a 1D curve at the intersection of the SPP light-cone with the plane ω−ωo=(kz−ko)evSPP tanθ, where evSPP is the group velocity of a plane-wave pulsed SPP at ω=ωo, and ko is the SPP wavenumber at ωo35. The spectral projection onto the (kz, ω/c)-plane for the ST-SPP is also a straight line, indicating that – in addition to being diffraction-free – ST-SPPs propagate dispersion-free at a group velocity ev=evSPP tanθ despite the intrinsic GVD associated with freely propagating SPPs (Fig. 1b)35.

We show in Fig. 2 (first row) the spectral support for a conventional SPP wave packet on the SPP light-cone (Fig. 2a), its spectral projections onto the (kx, ω/c) and (kz, ω/c) planes (Fig. 2b, c), along with the out-of-plane field profile (Fig. 2d). We assume a separable Gaussian spectrum in space and time, and take into account the temporal bandwidth Δλ≈110 nm (FWHM) utilized in our experiments (Methods). Crucially, the phase front for this conventional SPP wave packet is orthogonal to the propagation axis z (Fig. 2d).

Producing an ideal ST-SPP with a bandwidth Δλ=110 nm is prohibitive, as can be understood by considering a so-called ‘striped ST-SPP’ whose spatial profile results simply from the interference of a pair of plane waves with fixed spatial frequencies ±kx (i.e., the field takes the transverse form cos kx x)42. Consequently, a striped ST-SPP is separable with respect to space and time, and it thus does not offer the possibility of tuning the group velocity (indeed, its group velocity is ev≈evSPP; Methods). We plot in Fig. 2a–c (second row) the spectral support for a striped ST-SPP, which is the intersection of iso-|kx| planes with the SPP light-cone. The out-of-plane field is separable, extended and periodic along x (with period λx≡2π/kx=10 μm), and its phase front is orthogonal to the z-axis, leading to a lattice-like spatial distribution at any instant t (Fig. 2d).

A broadband ST-SPP is composed of a multiplicity of cosine SPPs with different kx, each of which is associated with a prescribed ω, such that kz is maintained in a linear relationship with ω. However, maintaining this linear relationship between kz and ω over a large temporal bandwidth Δω is prohibitive for SPPs because the curvature of the SPP light-line dictates a large accompanying spatial bandwidth Δkx. Indeed, even if ev it remains within 1% of evSPP, we still need Δkx≈0.23 ko (with the parameters used in our experiments below) to maintain the spatiotemporal spectrum of an ideal ST-SPP over the large bandwidth Δλ≈110 nm used here. Because our experimental scheme conserves the spatial bandwidth Δkx from free space to the surface-bound field, a large numerical aperture is required. We define kx(λ)=2π/λ sinφ(λ) in free space, where λ is the free-space wavelength, φ(λ) is the propagation angle with the z-axis, and λx=λ/sinφ(λ) is the transverse period of the field along x. Our system is limited to a maximum angular acceptance φmax≈±5°, a numerical aperture (NA) of ≈0.09, which restricts the minimum spatial feature size to ≈9 μm which falls considerably short of the requirement for an ideal ST-SPP over Δλ=110 nm. To address this challenge, we have implemented a compromise regarding the structure of the realized ST-SPP with respect to an ideal ST-SPP: (1) we maintain the one-to-one correspondence between |kx| and ω that is necessary for diffraction-free propagation; (2) we maintain control over the group velocity ev of the ST-SPP away from that for a conventional SPP evSPP in both the subluminal (ev<evSPP) and superluminal (ev>evSPP) regimes; however (3) we allow the spectral projection onto the (kz, ω/c)-plane to be curved such that it lies in the (kz, ω/c)-plane between the SPP light-line and the linear spectral projection for an ideal ST-SPP (Fig. 2c). Consequently, we can operate with the available system NA, but at the cost of vouchsafing the complete elimination of GVD. This is a minimal disadvantage in light of the short SPP propagation distance limited by ohmic losses.

The spectral support of the realized ST-SPP is still a 1D curve on the surface of the SPP light-cone, but this curve results from the intersection of the SPP light-cone with a curved planar surface (rather than a plane) designed to yield a V-shaped spectral projection onto the (kx, ω/c)-plane50. For a subluminal ST-SPP (ev<evSPP), this projection takes the form ω−ωU=c|kx|tanα, where ωU=ωo+Δω/2 is the maximum frequency, ω<ωU, α is the angle with the kx-axis, and tanα<0. For a superluminal ST-SPP (ev>evSPP), we have ω−ωL=c|kx|tanα, where ωL=ωo−Δω/2 is the minimum frequency, ω>ωL, and tanα>0. In both cases, ωo is the central frequency, so that ωL−Δω<ω<ωo+Δω. This curved surface remains parallel to the kx-axis, but its projection onto the (kz, ω/c)-plane is no longer a straight line and is instead a curve that remains closer to the SPP light-line than in the case of the ideal ST-SPP (to remain within the system NA). The analytical relationship between ST-SPP group velocity ev and the angle α is derived in Methods. The GVD coefficient for a conventional SPP wave packet in this case is 2.5×103 fs2/mm, and is 2.5×103, 2.3×102, and 2.0×102 fs2/mm for a striped ST-SPP (with λx = 10 μm), a subluminal ST-SPP (with α = −64.5°), and a superluminal ST-SPP (with α = 64.5°), respectively.

We plot in Fig. 2 the spectral representation for the subluminal ST-SPP (third row) and the superluminal ST-SPP (fourth row). Of particular interest is the spatial profile of the out-of-plane field for both ST-SPPs, which is no longer separable as a result of the non-separability of their spatiotemporal spectra. Instead, we observe an X-shaped profile reminiscent of free-space STWPs49,51. Crucially, the direction of the phase fronts varies across the wave front. In the vicinity of the center x = 0, the phase front is orthogonal to the propagation axis. However, along the branches of the X-shaped profile, the phase fronts are tilted with respect to the z-axis. Crucially, the sign of this phase-front tilt switches between the subluminal and superluminal regimes, and the magnitude of the tilt angle increases with the deviation of ev from evSPP52.

Experimental arrangement
We sketch in Fig. 3a the optical arrangement for synthesizing free-space STWPs, launching them into ST-SPPs on a metal-dielectric interface via scattering from a nano-slit, and observing the propagation of surface-bound ST-SPPs via spatially and temporally resolved two-photon fluorescence produced from the interference of the ST-SPP with a free-space reference pulse. We start with pulses from a Ti:sapphire laser oscillator of bandwidth Δλ≈110 nm (FWHM), center wavelength λo≈800 nm, and a 10-fs transform-limited pulse duration (FWHM). However, the measured pulse width at the sample surface is ΔT≈16 fs (FWHM) because of residual chirp in the optical system.

Synthesis of ST-SPPs. The STWPs are synthesized from the pulsed laser via spatiotemporal spectral phase modulation as established in refs. 28,49. The pulse spectrum is spatially resolved by a grating (300 lines/mm), the first diffraction order is selected and collimated using a cylindrical lens, which then impinges on a reflective, phase-only spatial light modulator (SLM); see Fig. 3a. This SLM imparts a 2D phase distribution to the incident wave front to deflect each wavelength λ by angles ± φ(λ) with respect to the z-axis, where sin{φ(λ)}=±(1−λc/λ)cotα. Here λc is the minimum (λ>λc) or the maximum (λ<λc) wavelength in the superluminal or subluminal regimes, respectively. The reflected phase-modulated wave front is directed to a second grating identical to the first, where upon the pulse is reconstituted to produce the STWP. For simplicity, the synthesis system in Fig. 3a is depicted as retro-reflecting from the SLM (whereupon the second grating coincides with the first); see Methods for details.

Sample preparation and launching ST-SPPs onto a metal-dielectric interface. The sample is a 100-nm-thick Ag film deposited on a silicon substrate. A crucial experimental challenge is to couple a broadband free-space STWP to a surface-bound ST-SPP at the metal-dielectric interface. We have recently examined theoretically an alternative approach to launching free-space STWPs into ST-SPPs that relies on scattering from a nano-slit in the metal surface (Fig. 3b)41. This coupling strategy retains a high efficiency over the bandwidth used here, and is thus preferable to conventional approaches for SPP coupling (e.g., gratings or evanescent prism-coupling) that modify the transverse wavenumber or do not operate over large bandwidths. Such a nano-slit conserves the transverse wavenumber kfreex=kSPPx=kx, where kfreex and kSPPx are the transverse wavenumbers for the free-space field and the launched SPP on either side of the nano-slit, respectively, so that the axial wavenumber is kz=± sqrt(k2SPP−k2x).

Two sets of 170-nm-wide, 100-nm-deep nano-slits were milled into the Ag film (Fig. 3c, e). One set of nano-slits has a lateral length of 200 μm and is used to launch ST-SPPs onto the Ag surface when illuminated with a free-space STWP (Fig. 3c). When illuminated with a conventional pulsed beam of large transverse extent along x, a conventional SPP wave packet is launched with a transverse width equal to that of the incident field (Fig. 3d). The second set of nano-slits has a reduced lateral length of 5 μm (Fig. 3e), and is used to launch conventional SPPs with a 5-μm-wide spatial profile (Fig. 3f). We have experimentally confirmed the efficacy of the nano-slit coupling methodology with striped ST-SPPs42, the spatiotemporally separable building blocks of ST-SPPs that comprise a single spatial frequency. Our results here confirm the theoretical predictions in ref. 41 regarding the broadband coupling of STWPs to ST-SPPs. Finally, the sample is coated by a 30-nm-thick dye-doped PMMA film to form a two-photon fluorescent layer.

Spatiotemporal characterization of the ST-SPPs. A portion of the initial laser pulse (20% by power) is split off to serve as a reference pulse, while the remainder (80%) is directed to the above-described STWP synthesis system (Fig. 3a). The synthesized STWP that excites the ST-SPP is incident obliquely at an angle ψex=−45° with respect to the sample normal, and is focused via a combination of spherical mirror and cylindrical lens onto the nano-slit, which launches it onto the metal surface (Fig. 3b; Methods)42. The reference pulse traverses an optical delay line τ and is then focused onto the metal surface away from the nano-slit to interact with the surface-bound SPP. The interference beat pattern of the SPP and the reference pulse excites two-photon fluorescence from the polymer layer55. Because we aim at detecting exclusively the interference of the reference pulse with the SPP, after eliminating any interference between the incident STWP excitation and the excited ST-SPP, we make use of the backward-coupled SPP (z>0) rather than the stronger forward-coupled SPP (z<0); see Fig. 3b.

Observation of SPPs and ST-SPPs
Time-resolved measurements. We show in Fig. 3d the fluorescence profile of a conventional SPP wave packet with a large transverse spatial profile excited by irradiating the nano-slit in Fig. 3c with a conventional pump pulse after setting the SLM phase to 0 everywhere; i.e., φ(λ)=0. The beat profile to the left of the nano-slit (z<0) results from self-interference of the launched SPP and the incident free-space excitation, which is thus stationary and independent of τ because the reference pulse is not involved. The two-photon fluorescence beat profile to the right is delimited by the irradiation area of the reference pulse on the sample surface (green dashed circle; focal spot 30×60 μm2). The delay τ is adjusted to correspond to the propagating SPP wave packet reaching z~30 μm. We define t as the time incurred by the surface-bound SPP (that travels at a group velocity ev) to propagate from the nano-slit to the observation location. The time t will be different from the free-space delay τ placed in the path of the focused reference pulse (that travels at a group velocity c/sinψref along the metal surface) to reach the same location and produce the interference beat pattern; see Methods for the conversion between the delay τ and the time t.

We are now in a position to compare the time-resolved measurements of conventional SPP wave packets and ST-SPPs. We first investigate a conventional SPP wave packet where the initial laser pulses are focused onto the metal surface at the location of a 5-μm-long nano-slit (Fig. 3e). A conventional SPP wave packet is thus launched at the metal surface with a flat intensity profile, 5-μm transverse spatial width, and 16-fs temporal pulse width. The recorded two-photon fluorescence beat profile resulting from the launched SPP interfering with the reference pulse (focused spot size 60×15 μm2) is shown in Fig. 3f. The intensity drops due to ohmic losses and diffractive spreading, and only a faint arc-shaped beat pattern is visible when the SPP wave packet reaches z~40 μm. From this intensity profile we extract the SPP field distribution U(x, z; τ) plotted in the bottom panel of Fig. 3f at fixed delay τ≈80 fs (Methods); see Supplementary Movies 1 and 2.

In Fig. 4 we plot the experimentally recorded and the computed two-photon fluorescence beat profiles at a fixed delay τ (Methods) for a conventional SPP of width 5 μm (Fig. 4a, b); a striped ST-SPP with transverse period ≈ 10 μm (Fig. 4c, d); a subluminal ST-SPP with α=−64.5° (Fig. 4e, f); and a superluminal ST-SPP with α=64.5° (Fig. 4g, h). The differences in the wave packet center position and the extent of the interference envelopes are attributed to uncertainties in the time origin, spatial inhomogeneity in the fluorescent film, and variations in the intensity distribution of the probe beam. Typical time-resolved measurements of ST-SPPs are provided in Supplementary Movies 3–6. Computed spatiotemporal profiles capture all the key features of their measured counterparts. Crucially, the different phase front structures predicted in Fig. 2d are clearly observable, as depicted in the insets of the corresponding panels in Fig. 4. Whereas the conventional SPP has a flat phase front orthogonal to the z-axis (Fig. 4a, b, and Fig. 2d, first row), the striped ST-SPP has a checkered lattice-like structure (Fig. 4c, d, and Fig. 2d, second row). The phase front for the subluminal ST-SPP away from its center is tilted with respect to the z-axis (Fig. 4e, f, and Fig. 2d, third row). The sign of this phase-front tilt is reversed for the superluminal ST-SPP (Fig. 4g, h, and Fig. 2d, fourth row). The remaining quantitative differences between the computed and measured profiles in Fig. 4 are attributed to the residual chirp in the laser pulses, additional chirp encumbered in the STWP synthesis system, and spectral dissipation at the metal surface.

We have thus confirmed that the nano-slit is capable of launching free-space STWPs onto the metal surface as ST-SPPs, and that our experimental configuration can capture the time-resolved structure of the surface-bound, propagating ST-SPPs. We proceed to utilize this capability to verify two key features of ST-SPPs: their diffraction-free propagation, and the controllability of their group velocity (ev) above and below that of a conventional SPP (evSPP).

Diffraction-free propagation of ST-SPPs. To verify the diffraction-free behavior of the ST-SPPs, we evaluated the intensity profiles I(x, z)=∫dτ|Ubeat(x, z; τ)|2 at different axial positions by time-integration of the extracted field profiles U(x, z; t) obtained by removing background intensity from the two-photon fluorescence beat profiles (Methods). We plot in Fig. 5 the time-averaged intensity profiles for the striped ST-SPP (Fig. 5a), the subluminal ST-SPP (Fig. 5b), and the superluminal ST-SPP (Fig. 5c). For each case, we plot the intensity profiles obtained after focusing the reference pulse at two different axial positions along the sample surface: at z≈35 μm and z≈85 μm measured from the location of the nano-slit.

In contrast to a conventional SPP wave packet whose profile diffracts rapidly with propagation (Fig. 3f), the profiles of the ST-SPPs remain unchanged over the same propagation distance. In the case of the striped ST-SPP, this is to be expected because it contains only a single transverse wavenumber (i.e., an extended cosine wave profile). The limited transverse profile is a result of the narrow reference pulse irradiation spot utilized at the sample surface. The intensity profiles in Fig. 5a demonstrate that the striped ST-SPP maintains the same profile over a range of ≈5× the Rayleigh range (zR≈18.2 μm; Methods). The transverse profiles of the subluminal (Fig. 5b) and superluminal (Fig. 5c) ST-SPPs are also maintained over the distance examined, which is over ≈4× Rayleigh lengths (the Rayleigh length is zR≈21 μm). The reduced visibility in these time-averaged profiles compared to the time-resolved measurements in Fig. 4e, g is attributed to the integration over the full spatiotemporal field, which includes a broadband pedestal that partially obscures the localized structures28,49. In all cases, the integrated cross-section along x plotted in the right panel is reasonably fitted by the (superposition) of Gaussian functions (representing the focused spot of the reference pulse) and the square of the sinusoidal-Gaussian function (representing the ST-SPP). The propagation distance here is limited by the ohmic losses. It is expected that the diffraction-free behavior extends significantly beyond 4zR, which can become manifest by increasing the NA and reducing the width Δx of the ST-SPP (reducing zR significantly below the loss-restricted propagation distance).

Tuning of the group velocity of ST-SPPs. A unique feature of ST-SPPs is the possibility of tuning their group velocity by modifying only the structure of their spatiotemporal spectrum – without changing the sample itself. The group velocities of the ST-SPPs were estimated by examining a sequence of time-resolved two-photon fluorescence beat profiles. All group velocity measurements were performed at a fixed location on the sample to eliminate variability arising from spatial inhomogeneity in the fluorescent film. Additionally, the delay time τ was adjusted such that the center of the ST-SPP wave packet appeared at z=0 when t=0. An example is shown in Fig. 6a corresponding to the striped ST-SPP. For different values of the relative delay τ, we extract the field structure U(x, z; τ) and then fit (in the least-square sense) a sinusoidally modulated Gaussian function to the axial field U(0, z; τ) at each τ (Fig. 6a). We take the peak of this fitted function to be the axial center-of-weight zo of the propagating ST-SPP. The delay τ was converted into propagation time t of the ST-SPP on the metal surface by factoring in that the surface-bound ST-SPP travels at a group velocity ev, whereas the reference pulse travels at c/sinψref along the surface (Methods). We plot in Fig. 6b the measured axial positions zo of the striped ST-SPP with real time t, and the estimated slope yields the group velocity ev=(2.22±0.004)×108 m/s (0.74c), which is consistent with the group velocity of a conventional SPP evSPP=2.22×108 m/s obtained from the first derivative of the dispersion curve of the SPP mode at the PMMA-coated Ag surface.

We repeated the procedure for the subluminal and superluminal ST-SPPs, and the estimated group velocities are ev−=(2.18±0.02)×108 m/s (≈0.73c) and ev+=(2.28±0.03)×108 m/s (≈0.76c), respectively. The relative relationships of the measured group velocities ev−<evSPP<ev+ are thus consistent with theoretical expectations and with the wavefront tilts in the two-photon fluorescence beat patterns in Fig. 4. We plot in Fig. 6c the measured group velocities compared to ev calculated as a function of the opening angle α after taking into consideration the uncertainty resulting from the finite accuracy of determining the thickness (30 ± 2.5 nm) of the fluoresecent PMMA layer.

Methods
The calculated profiles of the out-of-plane electric fields for the SPPs and ST-SPPs plotted in Fig. 2d assume a 100-nm-thick Ag film on a Si substrate, with a 30-nm-thick PMMA coating clad with free space. The SPP dispersion curve is calculated using the dielectric function for Ag proposed by Rakic et al.62. The measured refractive index of PMMA is n≈1.53 at the operating wavelength. We take the pulse spectrum to be E~(ω), where E(t)=∫dωE~(ω)e−iωt. We assume a Gaussian profile in time:

E(t)=Eo e−iωo t exp[−t2/2(ΔT)2], (1)

where ΔT=10 fs is the FWHM of the intensity profile, and ωo/2π=375 THz.

In general, the out-of-plane field is given by:

E(x, z; t)=∫∫dkx dω ψe(kx, ω) ei(kx x+kz z−iωt), (2)

where ψe(kx, ω) is the spatiotemporal spectrum, which is the Fourier transform of the initial field E(x, 0; t). The first row in Fig. 2 corresponds to a conventional SPP wave packet, where ψe(kx, ω) is separable, ψe(kx, ω)=ψe(kx)Ee(ω); here ψe(kx) is the spatial spectrum of the excited SPP and assumed also to have a Gaussian profile.

The second row in Fig. 2 corresponds to a striped ST-SPP. The spatiotemporal spectrum is once again separable ψe(kx, ω)={δ(kx−kxo)+δ(kx+kxo)}Ee(ω). The electric-field distribution is thus given by:

E(x, z; t)∝cos(kxo x) ∫dω E~(ω) ei(kz(ω)z−ωt), (3)

where the spatial frequency kxo is fixed and is independent of ω, so that the transverse spatial DoF is separable with respect to the axial spatial and temporal DoFs. Consequently, the striped ST-SPP is diffraction-free (the transverse spatial profile does not change with propagation), but it is not localized.

The third and fourth rows in Fig. 2 correspond to subluminal and superluminal ST-SPPs, respectively. The spatiotemporal spectrum is no longer separable, in which ω and |kx| are in one-to-one correspondence after selecting a value for the angle α. The field distribution for the ST-SPP is given by:

E(x, z; t)∝∫dω E~(ω) cos{kx(ω)x} ei{kz(ω)z−ωt}, (4)

which is not separable with respect to the spatial and temporal DoFs. In all cases we have kz(ω)=sqrt(k2SPP(ω)−k2x(ω)), and we take the same spectral profile Ee(ω) used above for the conventional SPP, the striped ST-SPPs, and the ST-SPPs.

Relationship between the spectral tilt angle θ and the opening angle α
We define the axial group velocity ev of the ST-SPP in the usual way as ev=[d(kz)/dω|ωo]−1. For the V-shaped spectra realized here (Fig. 2a, b, third and fourth rows), tuning the opening angle α varies θ, and thus in turn ev. For subluminal ST-SPPs we have ω−ωU=c|kx|tanα, where ωU=ωo+Δω/2 is the maximum frequency, ω<ωU, and tanα<0 (−90° < α < 0). The axial wave number is kz(ω)=sqrt(k2SPP(ω)−k2x(ω)), and we make use of a Taylor expansion for kSPP(ω) as kSPP≈ko+evSPP Ω + 1/2 k2 Ω2, where evSPP and k2 are the group velocity and GVD coefficient for a plane-wave pulsed SPP, respectively, and k2x(ω)=(Ω/c−Δω/2tanα)2. We can thus write kz=A+2BΩ+CΩ2, where:

A=ko2−(Δω/2tanα)2, B=evSPP ko + Δω/2c tan2α, C=ko k2 + ev2SPP/c2−1/tan2α, and Δk=Δω/c. We now have the following expression for the group velocity:

ev=[d(kz)/dΩ|Ω=0]−1 = A/B; (5)

Defining a group index ne=c/ev=cotθ for the ST-SPP and neSPP=c/evSPP for a conventional SPP wave packet, we have:

ne=neSPP [1+ (Δk/ko)2 / 8tan2α] / (1+ Δk/ko / 2tan2α), (6)

so that ne>neSPP (ev<evSPP) as expected for a subluminal ST-SPP.

For the superluminal ST-SPP we have ω−ωL=c|kx|tanα, where ωL=ωo−Δω/2 is the lower temporal frequency in the spectrum, ω>ωL, and tanα>0 (0 < α < 90°). The coefficients A and C are identical to those for the subluminal ST-SPP, and B=evSPP ko − Δω/2c tan2α, so that the group index for the ST-SPP is:

ne=neSPP [1+ (Δk/ko)2 / 8tan2α] / (1− Δk/ko / 2tan2α), (7)

and to first order we have ne≈neSPP(1− Δk/ko / 2tan2α), ne<neSPP (ev>evSPP) as expected for a superluminal ST-SPP.

Synthesizing free-space STWPs
The light source used in our experiments is a custom-built Ti:sapphire laser oscillator with a transform-limited pulse duration of 10 fs, center wavelength λo=800 nm, FWHM-bandwidth Δλ≈110 nm, a repetition rate 90 MHz, and average power ≈400 mW. The spectrum is measured via a spectrometer (Ocean Optics HR4000) after coupling to an optical fiber with diameter 200 μm. Because of residual chirp accumulated through various optical components such as lenses, gratings, and the SLM, the pulses are not transform-limited, and instead have a pulse duration of ΔT≈16 fs at the sample surface. The pulse width is estimated by a home-built auto-corrrelator55.

A portion of the laser power (20%) is reserved for subsequent use as a reference pulse. The remainder of the laser power (80%) is directed by a beam splitter to an optical system for synthesizing STWPs built along the lines described in ref. 28. The spectrum of the femtosecond pulse is spatially resolved by a grating (300 lines/mm), collimated by a cylindrical lens Lc (focal length f = 250 mm), and directed to a 2D phase-only, reflective SLM (Hamamatsu X13138-07). Each wavelength λ occupies a column of the SLM, along which the SLM imparts a spatial phase distribution ±2πsin{φ(λ)}x. The overall 2D phase distribution imparted to the wave front corresponds to each wavelength being deflected by a prescribed propagation angle ±φ(λ) to yield the target relationship between kx and λ. The reflected phase-modulated, spectrally resolved wave front is directed to a second grating (identical to the first), where upon the pulse is reconstituted to produce an STWP, which is then directed to the sample to excite an ST-SPP.

Sample fabrication
The sample was prepared by first depositing a 100-nm-thick Ag layer on a Si substrate via sputtering. The nano-slit structure was then milled into the Ag surface using a focused ion beam. The dimensions of the nano-slit are as follows: width w=170 nm, depth h=100 nm into the Ag layer, and extending across the sample by a length L=200 μm (Fig. 3c). These were used to excite the ST-SPPs, including the striped ST-SPPs, and the subluminal and superluminal ST-SPPs (Figs. 4, 6, and 5). This nano-slit was also used to excite the conventional plane-wave SPP wave packet (Fig. 3d). A second set of nano-slits was prepared with the same dimensions w and h, but with length L=5 μm (Fig. 1e). These nano-slits were used to excite conventional SPP wave packets with 5-μm-wide transverse profiles (Fig. 3f).

The full sample area was spin-coated with a 30-nm-thick poly(methyl methacrylate) (PMMA) layer doped with a laser dye (coumarin 343). The thickness of the coating was measured by spectral ellipsometry (Uvisel plus), from which we estimated an uncertainty in the coating thickness of ~5 nm. This in turn results in an uncertainty in estimating the SPP group velocity of ~0.1×108 m/s.

SPP detection
Two wave packets overlap and interfere in the fluorescent polymer layer: the free-space incident reference pulse and the surface-bound SPP. The interference of these two wave packets produces a beat profile that excites two-photon fluorescence. The two-photon-fluorescence emission from the sample surface is collected with an objective lens (M Plan Apo SL 20X, Mitutoyo) equipped with a band-pass filter (475–495 nm), and imaged by a CCD camera (Roper Scientific, CoolSnap HQ2). The two-photon fluorescence signal is expected at a wavelength ≈490 nm from the dye used.

In this study, we chose a field of view of approximately 150×150 μm, which allowed simultaneous imaging of both the light-coupler and the full extent of the ST-SPP wave packet, while maintaining the spatial resolution required for accurate phase retrieval. As a result, the observation range was primarily limited to approximately 100 μm from the slit, despite the ST-SPPs propagating beyond this distance.

Calculating the SPP intensity profile
The intensity distribution I(x, z; τ) on the sample surface detected by the CCD camera is the time-average of the fourth power of the total electric field55:

I(x, z; τ)=∫ dt |ESP P(x, z; t)+Efree(x, z; t+τ)|2 2; (8)

Here, a proportionality constant is ignored because we are only interested in the relative intensity. The term ESPP and Efree are the scalar field components of the excited SPP and the free-space fields, respectively, that overlap in the PMMA film. The interference of the two fields produces a spatial pattern having a spatial beat wavelength λbeat given by:

λbeat = 2π / |kz(ωo, kx) − ko sinψref| (9)

where kz(ωo, kx) is the SPP axial wavenumber at ω=ωo, and ko sinψref is the in-plane axial wavenumber of the incident free-space field of the reference pulse. Similarly, the interference beat between the pump-probe pulse pair is λlight = 2π / |kx−ko sinψex|, where ko sinψex is the in-plane axial wavenumber of the incident free-space field of the pump pulse. Therefore, the intensity, phase, and spatial distribution of the SPP wave packet can be extracted from the observed beat pattern through appropriate analysis. However, it is important to note that the beat pattern is not a direct representation of the SPP field itself. In particular, due to the interference-based nature of the detection method, the spatial frequency of the beat does not directly correspond to the SPP wave vector, and the detected signal decays more rapidly than the actual SPP field because it scales with the fourth power of the electric field amplitude. When employing ψex and ψref with the same signs to observe forward-coupling SPP, λlight ≈ λSPP if the values of ψex and ψref are comparable, making differentiation between them difficult. Consequently, we employed a contrasting configuration in which the signs of ψex and ψref are opposite of each other, such that λlight is significantly less than λSPP, thereby facilitating their delineation. The interference beats caused by this pump-probe pulse pair can be seen as the intensity distribution moving from right to left at the beginning of Supplementary Movies 1 and 3.

Intensity profiles in Fig. 4 were calculated from the two-photon fluorescence signal resulting from the interference of the ST-SPP and the incident free-space conventional reference pulse. The intensity profile was calculated using Eq. (8) and assuming that both the ST-SPP and reference pulses are of width ≈ 16 fs as determined experimentally.

Removing the background
To extract the field distribution U(x, z; τ) from the measured two-photon fluorescence, we need to experimentally estimate the background constants. This is achieved by changing the delay τ in the path of the reference pulse by 2.7 fs, which corresponds to moving the excitation-reference interference beat profile by the laser carrier wavelength laser λo (a phase shift of 2π rad), and carrying out the time-resolved measurements. To obtain the background intensity, we first obtain the time-resolved field U(x, z; τ) from the interference beat profiles, and then obtain their average U¯(x, z; τ) evaluated over an optical cycle:

U¯(x, z; τ)= 1/m ∑i=1m U(x, z; τ+(i−2)×2.7); (10)

In our work here, we have used m=2 or 4 according to the exposure time used to avoid bleaching the fluorescence layer during laser irradiation. This averaging eliminates the excitation-reference oscillatory beat profile and extracts only the delay-independent background. This procedure assumes that minimal spatial change occurs in the SPP profile while advancing τ by 2.7 fs. By subtracting the averaged profile U¯(x, z; τ) from the time-resolved profiles U(x, z; τ) we obtain the background-free profile of the pump-probe beat pattern at τ, and Ubeat(x, z; τ)=U(x, z; τ)−U¯(x, z; τ). This procedure yields the field beat profile displayed in the bottom panel of Figs. 3f, 6a, and Supplementary Movies 2 and 4.

Rayleigh range
The intensity distribution of the microscopic fluorescence beat images obtained via two-photon fluorescence are not proportional to the electric field intensity. We estimate the beam waist of the striped ST-SPP by obtaining λx by fitting the striped ST-SPP to a sinusoidal distribution along x. For a striped ST-SPP with λx ≈ 9.4 μm, the beam waist is 2.15 μm, corresponding to a Rayleigh range zR ≈ 18.2 μm at a wavelength of 800 nm.

Group-velocity calculation
The two-photon fluorescence beat pattern is formed by the spatial overlap of the SPP propagating at a group velocity ev and the reference pulse traveling along the sample surface at a group velocity ev=c/sinψref, where ψref is the reference-pulse incident angle. We need to convert the external relative delay τ between the excitation and reference pulses to the propagation time t of the surface-bound SPP. As the SPP travels after a time t a distance z1=evt, whereas the delayed reference pulse travels a distance z2=c/sinψref (t−τ). For the two wave packets to meet at z1=z2=z, and thus produce the two-photon fluorescence beat profile observed in the micrographs, we have t=τ/(1−ev sinψref/c). This conversion between t and τ was used in plotting Fig. 4 and 6b.

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "This work reports on the synthesis and detection of ST-SPPs in different coupling regimes. The unique spatiotemporal structure of these waves allows them to travel long distances without diffraction. This work builds on the authors' previous reports of STWPSs and striped ST-SPPs (their refs. 28 and 42). The main innovation here is the synthesis of ST-SPPs which may tuned into striped, subluminal, and superluminal regimes, as they show experimentally. The present results and the detailed analysis and modeling provided are noteworthy, but it would be helpful to more clearly draw a comparison between the previous work (ref. 42) and this report since similar theory, methods, and experimental configuration are used to generate the striped ST-SPPs. A few other comments below.\nSince the reference pulse co-propagates with the SPP as shown in Fig. 3d, I would expect the beat and self interference frequencies to be different but they appear very close in the figure. Can the authors comment on this?\nThe ST-SPP images recorded near the trench (Fig. 4) agree with the simulated field profiles. For verifying diffraction free propagation, it is not clear why 85 um was chosen as the appropriate distance. There is a trade-off between dissipation and intensity, but farther away from the trench would make this result more convincing since the propagation length for a NIR SPP in Ag is at least twice that distance, in theory. A second point is that the profiles shown in Fig. 5 seem to limited by the spot size of the reference beam. As such, if the ST-SPP diffracts beyond this area the signal will fade and give the appearance of diffraction free propagation. The concept of diffraction free propagation is supported by the theory provided, however, it is unclear to what extent the spatial convolution with the reference beam influences the Fig. 5 profiles.\nThe ST-SPPs possess complex phase structure which could be exploited or may make some applications more difficult. Can the authors comment on how the phase profiles will influence some of the nanophotonic applications they propose such as surface-enhanced sensing?\nOther comments:\n-Consider moving the last sentence of the Fig. 2 caption to the beginning of the caption to help guide the reader.\n-In Fig. 3b , the red excitation arrow is difficult to see since it aligns with the trench.\n-The text on page 10 refers to Fig. 3g but there is no such label for the field profile U(x,z;t).\n-Consider making the labels c, d, g, and h bold in Fig. 4 like the others.", "reviewer_3": "The authors of the manuscript “observation of space-time surface plasmon polaritons” have experimentally prepared a non-diffracting space-time surface plasmon polariton (ST-SPP) wavepacket by coupling a precisely tailored optical field having a resricted 1D curvature in the spectral representation. Using two photon fluorecence microscopy equiped with interferometry technique, they have unambiguously imaged the propagation of such ST-SPP on silver surfaces over tens of microns and hundreds of femtosecond. and reconstructed its spatiotemporal field. Finally, the capability of tuning, despite the small range, of the group velocity of the ST-SPP wavepacket was shown. I find the result to be of interesting for the plasmon community, and the manuscript is well prepared. I therefore suggest the authors to address the following points before I can recommend it for publication.\nQ1. Since the excitation light for SPP fields needs spatiotemporal modulation of the wavefront, it unncessarily inherent some spatiotemporal chirp of the excitation pulse. Could the authors discuss a bit of the effect of chripping, so that what would be the ultimate limit for the spatial confinement in the lateral dimension (kz in Fig. 1) of the ST-SPP. Will the angle of excitation matter in generation of the striped ST-SPP?\nQ2. The striped ST-SPP in Fig. 4c seem to have interference features extending upto 65 microns, much longer than that of the conventional SPPs in Fig. 4a, although they are measured a the same time, and their respective simulations have similar spatial distribution. So what is the cause of this “potential” enlongation of the interference contrast?\nQ3. Related to Q1, the lateral confinement (now becomes x-axis, however), is not much different in Fig. 4a and c. So what is the advantage of having striped ST-SPP in this case? I assue by having ST-SPP, one already reduces the intensity of the excitation light, and the coupling efficiency, possibly.\nQ4. The superluminal and subluminal propagation seem to have unsatisfying contrast in the experiments. Can the authors comment in a bit more detail why this occurs? This is rather critical in a sense that whether further tuning of the group velocity is feasible for ST-SPP. For example, only limited tuning range was experimentally demonstrated, desipte the theroetical curve has a broader capability of tuning. If coupling efficiency reduces significantly for SPPs of different velocities, then it kind of loses the advantange as an information carrier."}, "author_response": "Responses to Reviewer 1\nReviewer’s comment\nThis work reports on the synthesis and detection of ST-SPPs in different coupling regimes.\nThe unique spatiotemporal structure of these waves allows them to travel long distances\nwithout diffraction. This work builds on the authors' previous reports of STWPSs and\nstriped ST-SPPs (their refs. 28 and 42). The main innovation here is the synthesis of ST-\nSPPs which may tuned into striped, subluminal, and superluminal regimes, as they show\nexperimentally. The present results and the detailed analysis and modeling provided are\nnoteworthy,\nAuthors’ response:\nWe appreciate the reviewer for the positive evaluation of our manuscript.\nbut it would be helpful to more clearly draw a comparison between the previous work (ref.\n42) and this report since similar theory, methods, and experimental configuration are used\nto generate the striped ST-SPPs. A few other comments below.\nAuthors’ response:\nWe thank reviewer for raising this issue. Ref. [42] is positioned as a preliminary demonstration\nof a limited case of ST-SPPs within the scope of this work. Accordingly, this work and Ref. [42]\nshare certain attributes, while introducing several significant differences. Below we outline the\ncommon attributes and differences between the previous work [42] and this work.\nCommon attributes between this work and previous work [42]:\n- Both works exploit the same femtosecond laser source.\n- Same coupling technique via nano-slit is used to excite the ST-SPP in this work and in\nthe previous work.\nNovelty of this work:\n(1) Fundamental differences in target / theory\n- The theory in this work is a generic for ST-SPP with controllable group velocity and finite\nspatial bandwidth, whereas in ref. [42] it was simplified to the case of single 𝑘 .\n𝑥\n- The ST-SPP here is non-separable in space and time, whereas in ref. [42] it is separable.\n- The group velocity is tunable across superluminal and subluminal regimes.\n(2) Advances in experimental setup\n- Direct observation of the spatiotemporal evolution of the ST-SPP wave packet during\npropagation was enabled by incorporating a pump–probe interferometer–based imaging\nsystem.\n- Newly designed custom phase masks were implemented on the SLM.\n- A silver film was employed instead of gold, enhancing the SPP propagation length.\n(3) New observations enabled by these improvements\n- The full spatiotemporal evolution of ST-SPPs was captured far beyond the slit and into\nthe far field, whereas Ref. [42] observed only the region immediately adjacent to the slit\nand only at the moment just after excitation.\n- Group velocity tunability across superluminal and subluminal regimes was demonstrated,\nconsistent with the generalized theory.\n- We demonstrated that the diffraction-free behavior of ST-SPPs is maintained over a\ndistance of 100 μm, exceeding the typical propagation length.\n1\n- Transverse spatial localization, which was absent in ref. 42, is demonstrated in this work\nthrough the generation of ST-SPPs with finite spatial bandwidth, resulting in three-\ndimensional localization.\nWe have also included the differences in the Discussion section of the revised manuscript. The\ncorresponding additions are provided below.\nPage 13 (Discussion section)\nThis work builds upon our previous demonstration of striped ST-SPPs [42], sharing the\nfemtosecond laser source and a single-nano-slit light–SPP coupling strategy. While retaining\nthese foundations, the present study significantly extends the experimental framework to access\na broader class of ST-SPPs [35]. Specifically, whereas the prior work treated separable ST-SPPs\nassociated with a single transverse spatial frequency 𝑘 , the current setup accommodates\n𝑥\nspatiotemporal spectra with finite spatial bandwidths, thereby enabling the excitation of non-\nseparable ST-SPPs with tunable group velocity.\nBy incorporating an interferometric detection scheme, we can now obtain time-resolved\nmeasurements of the evolving ST-SPP field—rather than the time-averaged intensity profiles\nused previously. This capability allows direct evaluation of both the spatiotemporal structure of\nthe wave packet and its propagation characteristics.\nSince the reference pulse co-propagates with the SPP as shown in Fig. 3d, I would expect\nthe beat and self interference frequencies to be different but they appear very close in the\nfigure. Can the authors comment on this?\nAuthors’ response:\nWe thank the reviewer for the constructive comment. In our experimental setup, both λ and\nbeat\nλ arise from the interference between the in-plane wavevector of the surface-propagating SPP\nlight\nand the in-plane projection of an obliquely incident optical pulse. Specifically, the interference\npattern enclosed in the green circle results from right-propagating SPP (+𝑘 ) and the reference\n𝑆𝑃𝑃\npulse, whereas the interference pattern enclosed in the orange bounded area results from left-\npropagating SPP (−𝑘 ) with the excitation pulse. The incident angle for the reference and\n𝑆𝑃𝑃\nexcitation pulses are equal in magnitude but opposite in sign: 𝜓 =−𝜓 .\nex ref\nTherefore, beat wavelengths for two interference patterns are given by\n2𝜋\n𝜆 = , (green circled area)\nbeat |𝑘𝑆𝑃𝑃− 𝑘𝑜sin𝜓ref |\n2𝜋\n𝜆 = . (orange circled area)\nlight |−𝑘𝑆𝑃𝑃− 𝑘𝑜sin𝜓ex |\nConsequently, 𝜆 = 𝜆 . We have updated Fig. 3d in the revised manuscript to avoid any\nbeat light\nambiguity.\nFor verifying diffraction free propagation, it is not clear why 85 um was chosen as the\nappropriate distance. There is a trade-off between dissipation and intensity, but farther\naway from the trench would make this result more convincing since the propagation length\nfor a NIR SPP in Ag is at least twice that distance, in theory.\nAuthors’ response:\nWe thank the reviewer for raising an excellent point. We limit our observation length to 60 𝜇m\nin Fig. 4 as the axial length of the interference beat between ST-SPP and the reference pulse is\napproximately 30 𝜇m for a fixed delay line position, as can be clearly observed in Fig. 6a. We\ndo observe propagation of ST-SPP at the 100 𝜇m distance from the slit by adding a delay in Fig.\n5, second column.\nAlthough the reviewer is right pointing out that near-infrared SPPs on silver can exhibit\npropagation lengths exceeding 100 μm under ideal conditions, in our experimental setup, a\nPMMA/dye overlayer was added to enable two-photon fluorescence imaging [41,55]. This\noverlayer alters the dielectric environment and increases modal losses, which in turn shortens the\neffective propagation length in our case. This reduction becomes particularly relevant in two-\nphoton fluorescence microscopy, where the detected signal scales with the fourth power of the\nelectric field amplitude. Therefore, we limited our analysis to within ~100 μm from the excitation\npoint, where both amplitude and phase information could still be reliably extracted.\nIn addition, the observation range in our experiment was primarily limited by the field of view of\nthe microscope. At the magnification used to extract reliable phase information, the field of view\nwas approximately 150 × 150 μm. To simultaneously capture both the slit and the center of the\nwavepacket within this area, the practical observation range was limited to around 100 μm.\nThe corresponding additions to the Method section are provided below.\nPage 24 (SPP detection section)\nIn this study, we chose a field of view of approximately 150 × 150 μm, which allowed\nsimultaneous imaging of both the light-SPP coupler and the full extent of the ST-SPP wave packet,\nwhile maintaining the spatial resolution required for accurate phase retrieval. As a result, the\nobservation range was primarily limited to approximately 100 μm from the slit, despite the ST-\nSPPs propagating beyond this distance.\nPage 24 (Calculating the SPP intensity profile section)\nTherefore, the intensity, phase, and spatial distribution of the SPP wave packet can be extracted\nfrom the observed beat pattern through appropriate analysis. However, it is important to note that\nthe beat pattern is not a direct representation of the SPP field itself. In particular, due to the\ninterference-based nature of the detection method, the spatial frequency of the beat does not\ndirectly correspond to the SPP wavevector, and the detected signal decays more rapidly than the\nactual SPP field because it scales with the fourth power of the electric field amplitude.\nreference beam. As such, if the ST-SPP diffracts beyond this area the signal will fade and\ngive the appearance of diffraction free propagation. The concept of diffraction free\npropagation is supported by the theory provided, however, it is unclear to what extent the\nspatial convolution with the reference beam influences the Fig. 5 profiles.\nAuthors’ response:\nWe appreciate the reviewer’s constructive comment. We acknowledge that the finite size of the\nreference beam acts as a spatial detection window, potentially obscuring diffraction effects\nbeyond the illuminated area. The reviewer is correct in pointing out this limitation of the current\nmeasurement scheme and that the theory predicts longer propagation distances. Due to the\nattenuation constraints described earlier, it was necessary to spatially confine the probe beam,\nparticularly at locations farther from the light-SPP coupler, in order to maintain detectable signal\nlevels in our current setup. However, by reference to Fig. 3f we show a conventional SPP of 5 μm\nwidth and observe clearly its diffraction, which proves that the finite size of the reference pulse\ndoes not represent an impediment to observe diffraction at 5 μm spatial scales.\nWe are currently working to expand the probe beam region in order to further extend the\nobservable propagation range and to address the limitation of the current measurement scheme.\nThe ST-SPPs possess complex phase structure which could be exploited or may make some\napplications more difficult. Can the authors comment on how the phase profiles will\ninfluence some of the nanophotonic applications they propose such as surface-enhanced\nsensing?\nAuthors’ response:\nThe reviewer is correct pointing out that ST-SPP in general have more sophisticated waveforms\nthan conventional SPPs. This stems from non-separable spatiotemporal spectrum and intensity\nprofile. Nevertheless, this does not necessarily entail that ST-SPP has a complex phase structure.\nBecause each frequency component corresponds to a cosine wave [52], the phase alternates\nsimply between 0 and π across the spectrum, rather than forming a more complicated phase\nstructure. In fact, in our experiments we have maintained the phase flat across the entire\nspatiotemporal spectrum except for this 0–π alternation. We have corrected the wording in a few\nplaces in the revised manuscript (captions in the supplementary movies), which might have\npreviously given the incorrect impression that the ST-SPPs possess complex phase structure\nbeyond this simple alternation.\nThe reviewer nevertheless makes a good point, which is that the phase is an added degree of\nfreedom that can be manipulated. For example, we have demonstrated previously that simply\nadding a 𝜋-step phase between the two halves of the spectrum results in a dark STWP in which\na dip exists at the center rather than a peak. We expect that using such an STWP to excite an ST-\nSPP would result in a dark plasmon, in which a diffraction-free dip propagates in a metal—\ndielectric surface. Many applications can be envisioned for dark SPPs in addition to the ST-SPPs\nwe have produced here, which we intend to investigate in the future.\nOther comments:\na) Consider moving the last sentence of the Fig. 2 caption to the beginning of the\ncaption to help guide the reader.\nAuthors’ response:\nWe agree with the reviewer, and we have implemented the suggested changes.\nb) In Fig. 3b, the red excitation arrow is difficult to see since it aligns with the trench.\nWe agree with the reviewer, and we have implemented the suggested changes.\nc) The text on page 10 refers to Fig. 3g but there is no such label for the field profile\nU(x,z;t).\nAuthors’ response:\nWe thank the reviewer for catching this error. We have corrected it.\nd) Consider making the labels c, d, g, and h bold in Fig. 4 like the others.\nAuthors’ response:\nWe would like to kindly mention that the labels are already in bold. However, we have moved\nthem to the left for clarity.\nWe thank the reviewer for his very close reading our manuscript and providing thoughtful\nsuggestions. We have revised the manuscript to implement the suggestions in figure captions,\nfigure labels, and the text as indicated to improve clarity and consistency throughout the\nmanuscript.\nReviewer’s comment\nThe authors of the manuscript “observation of space-time surface plasmon polaritons” have\nexperimentally prepared a non-diffracting space-time surface plasmon polariton (ST-SPP)\nwavepacket by coupling a precisely tailored optical field having a resricted 1D curvature in\nthe spectral representation. Using two photon fluorecence microscopy equiped with\ninterferometry technique, they have unambiguously imaged the propagation of such ST-\nSPP on silver surfaces over tens of microns and hundreds of femtosecond. and reconstructed\nits spatiotemporal field. Finally, the capability of tuning, despite the small range, of the\ngroup velocity of the ST-SPP wavepacket was shown. I find the result to be of interesting\nfor the plasmon community, and the manuscript is well prepared.\nAuthors’ response:\nWe thank the reviewer for taking time to review our work and the high evaluation of our\nmanuscript.\nI therefore suggest the authors to address the following points before I can recommend it\nfor publication.\nQ1. Since the excitation light for SPP fields needs spatiotemporal modulation of the\nwavefront, it unncessarily inherent some spatiotemporal chirp of the excitation pulse. Could\nthe authors discuss a bit of the effect of chripping, so that what would be the ultimate limit\nfor the spatial confinement in the lateral dimension (kz in Fig. 1) of the ST-SPP.\nAuthors’ response:\nFrom our previous experimental and theoretical research in STWPs in free space, there is no\nspatiotemporal chirp in the wavefront [28，49]. Therefore, the ultimate limit of the spatial\nconfinement along 𝑧 is determined by the transform-limited pulse width associated with the\nspectral bandwidth effectively coupled to the SPP. In our experiment, however, there is some\nresidual chirp that elongates the pulse from 10-fs to 16-fs stemming from the ultrashort nature of\nthe laser source passing through various optical elements, such as lenses, gratings and SLM.\nFurthermore, any limiting of the bandwidth, either due to the nano-slit coupling or absorption at\nthe silver-PMMA interface will further elongate the pulse and extend its length along 𝑧. In\nprinciple, this can be compensated through pre-compensation of dispersion using a chirped mirror\nor programmable phase control via the SLM. We have added the following sentence to clarify\nthis point.\nPage 22-23(Synthesizing free-space STWPs section)\nBecause of residual chirp in the optical system accumulated through various optical components\nsuch as lenses, gratings, and the SLM, the pulses are not transform-limited, and instead have a\npulse duration of Δ 𝑇 ≈16 fs at the sample surface.\nWill the angle of excitation matter in generation of the striped ST-SPP?\nAuthors’ response:\nBy reference to our previous theoretical analysis studying the coupling mechanism of ST-SPP\n[41] and previous work done by our group and others in ref. [54, R1, R2], the influence of the\nexcitation angle on the generation of ST-SPPs is very small and can be considered negligible\nunder our experimental conditions.\n6\n[R1] Buckanie, N. M.; Kirschbaum, P.; Sindermann, S.; Meyer zu Heringdorf, F.-J. Interaction\nof Light and Surface Plasmon Polaritons in Ag Islands Studied by Nonlinear Photoemission\nMicroscopy. Ultramicroscopy 2013, 130, 49–53. https://doi.org/10.1016/j.ultramic.2013.02.003.\n[R2] Wang, B.; Aigouy, L.; Bourhis, E.; Gierak, J.; Hugonin, J. P.; Lalanne, P. Efficient\nGeneration of Surface Plasmon by Single-Nanoslit Illumination under Highly Oblique Incidence.\nAppl. Phys. Lett. 2009, 94, 011114. https://doi.org/10.1063/1.3068747.\nQ2. The striped ST-SPP in Fig. 4c seem to have interference features extending upto 65\nmicrons, much longer than that of the conventional SPPs in Fig. 4a, although they are\nmeasured a the same time, and their respective simulations have similar spatial distribution.\nSo what is the cause of this “potential” enlongation of the interference contrast?\nAuthors’ response:\nWe appreciate the reviewer’s comment. There are combination of two effects here.\nFirst, each fluorescent image (Figs. 4a, 4c, 4f, and 4g) was acquired in a separate measurement.\nSince the fluorescent film experienced photobleaching during the time-resolved measurements,\nwe had to use different regions of the sample to obtain the static images shown in these figures.\nAs a result, slight variations in the experimental environment, such as local film thickness, led to\nsmall differences in effective group velocity for the measurements shown in Fig. 4.\nSecond, to capture a clear spatial distribution in Fig. 4, we used a relatively strong laser intensity\nand a long exposure time for each fluorescent image. Under these conditions, it was not feasible\nto acquire multiple images for time-origin correction, and the reference pulse delay was adjusted\nwith limited timing resolution. The effective delay time in Fig. 4c is therefore estimated to be\napproximately 75 fs, with an uncertainty of about ±5 fs.\nThese combined effects resulted in an apparent shift of the interference envelope toward longer z\nin Fig. 4c, giving the impression that the interference contrast extends farther than in Fig. 4a.\nWe note that the group velocity measurements were performed at a fixed position, with the laser\nintensity carefully adjusted to minimize photobleaching. In addition, the time origin was corrected\nby fitting the experimental data, allowing us to compensate for the timing offset. Therefore, the\nabove variations do not affect the velocity analysis.\nPage11 (Results section)\nThe differences in the wave packet center position and the extent of the interference envelopes\nare attributed to uncertainties in the time origin, spatial inhomogeneity in the fluorescent film,\nand variations in the intensity distribution of the probe beam.\nPage12 (Results section)\nAll group velocity measurements were performed at a fixed location on the sample to eliminate\nvariability arising from spatial inhomogeneity in the fluorescent film. Additionally, the delay time\n𝜏 was adjusted such that the center of the ST-SPP wave packet appeared at 𝑧 =0 when 𝑡 =0.\nQ3. Related to Q1, the lateral confinement (now becomes x-axis, however), is not much\ndifferent in Fig. 4a and c. So what is the advantage of having striped ST-SPP in this case? I\nassume by having ST-SPP, one already reduces the intensity of the excitation light, and the\ncoupling efficiency, possibly.\nAuthors’ response:\nReviewer raises a crucial point. We have different spot sizes of the reference pulse in Fig. 4a-d\ncompared to Fig. 4e-h. We listed them in the captions to the supplementary movies but did not\nlist them in the caption to the Fig. 4, which we have corrected in the revised manuscript. The\n7\nspatial width along x in Fig. 4a and Fig. 4c are limited by the reference pulse spot size, which was\n60×15 𝜇m2. Therefore in Fig. 4a and Fig. 4c the width along x was limited by the reference pulse\nalthough striped ST-SPP extends significantly beyond that, whereas the feature size of the striped\nST-SPP is only 5 𝜇m. In the case of Fig. 4e and Fig. 4g for the subluminal and superluminal ST-\nSPPs we changed the reference pulse spot size to 30×50 𝜇m2. We extended the spot size along x\nin order to capture the tilt in the wavefront.\nQ4. The superluminal and subluminal propagation seem to have unsatisfying contrast in\nthe experiments. Can the authors comment in a bit more detail why this occurs?\nAuthors’ response:\nWe assume the reviewer is referring to plots in Fig. 5. Here we are plotting the time-averaged\nintensities of ST-SPP in contrast to the time-resolved measurements in Fig. 4, which have a better\ncontrast. The reason is that in time-averaged measurements there is a background pedestal on the\nspatial profile that does not exist in time-resolved measurements; see ref. [28]. This background\nterm does not exist for striped ST-SPPs because the spatiotemporal profile is separable. We have\nadded a sentence clarifying this point on page 12 of the revised manuscript. The corresponding\nadditions to the Discussion section are provided below.\nPage12 (Results section)\nThe reduced visibility in these time-averaged profiles compared to the time-resolved\nmeasurements in Fig. 4e,g is attributed to the integration over the full spatiotemporal field, which\nincludes a broadband pedestal that partially obscures the localized structures.\nThis is rather critical in a sense that whether further tuning of the group velocity is feasible\nfor ST-SPP.\nAuthors’ response:\nThe pedestal height is fixed and independent of the group velocity for ST-SPPs.\nFor example, only limited tuning range was experimentally demonstrated, desipte the\ntheroetical curve has a broader capability of tuning.\nAuthors’ response:\nThe reviewer is correct that pointing out the theoretical curve extends beyond our capabilities in\ntunability. In our experiment the limitation in the group velocity we can achieve is enforced by\nthe numerical aperture (NA≈0.09) of the system and the spectral bandwidth exploited (≈110\nnm). The group velocity can be further tuned by either increasing NA or decreasing the bandwidth.\nAs mentioned in the introduction we selected a large bandwidth and hence short pulse to resolve\nthe propagation dynamics within the plasmon decay length.\nIf coupling efficiency reduces significantly for SPPs of different velocities, then it kind of\nloses the advantange as an information carrier.\nAuthors’ response:\nThe coupling efficiency is independent of the group velocity. As mentioned above, the range of\ngroup velocity tunability is dependent not on the coupling efficiency, but the numerical aperture\nof the system and the spectral bandwidth of the source. We have added a comment on page 13 of\nthe revised manuscript to clarify this issue. Indeed, as shown in our previous study [41] and a\nrelated conference publication [R3], we have confirmed through calculations that tuning the group\nvelocity in fact does not affect the coupling efficiency as shown in the figure below (Fig. R1). To\nclarify this point, we have added the following text to page 13 of the revised manuscript:\nThe demonstrated tuning range of 𝑣̃ was limited by the numerical aperture of our optical system\n(NA≈0.09), given the broad spectral bandwidth (∼110 nm) required to ensure sufficient\ntemporal resolution for resolving the propagation dynamics. However, the coupling efficiency is\nnot significantly affected by the group velocity, and simulations have confirmed that ST-SPPs\nwith a broader group velocity range can be excited [41]. Accordingly, broader tunability of 𝑣̃\ncould be achieved by increasing the NA or reducing the spectral bandwidth.\nFig. R1. ST-SPPs with a broad range of group velocities, as presented in Ref. [R3]. The\nsuperluminal case (1×109 m/s) and subluminal case (1×108 m/s) demonstrate that efficient\nexcitation is maintained across widely varying group velocities.\n[R3] N. Ichiji, M. Yessenov, K. L. Schepler, A. F. Abouraddy, and A. Kubo, “Propagation analysis of\nspace-time surface plasmon polariton wave packet excited by light irradiation on nanobeam structure”,\nProceedings of SPIE 11999 1199904-1-1199904-7 (2022), https://doi.org/10.1117/12.2607690\nWe thank the reviewer for his very close reading our manuscript and providing suggestions. We\nhave revised the manuscript to implement the suggestions in figure captions, figure labels, and\nthe text as indicated to improve clarity and consistency throughout the manuscript.\n9"}, {"round": 2, "reviewer_comments": {"reviewer_1": "The revised manuscript and comments received by the authors address all of my questions and concerns. I enthusiastically support its publication at this time.", "reviewer_3": "The authors have largely revised the manuscript based on my comments and concerns, which addresses most of the issues. The reported technique is indeed a good alternative for imaging nanoscale fields, and certainly deserve further investigation and broad interest. I think the manuscript is now in good shape for publication."}, "author_response": ""}]

Decision letter