Determining conic sections for the spatio-temporal spectra

The intersection of the light-cone \(k_x^2 + k_z^2 = ({\textstyle{\omega \over c}})^2\) with the spectral hyperplane \({\cal P}(\theta )\) described by the equation \({\textstyle{\omega \over c}} = k_{\mathrm{o}} + (k_z - k_{\mathrm{o}})\,{\mathrm{tan}}\,\theta\) is a conic section: an ellipse (0° < θ < 45° or 135° < θ < 180°), a tangential line (θ = 45°), a hyperbola (45° < θ < 135°), or a parabola (θ = 135°). In all cases v g = ctanθ. The projection onto the \((k_x,{\textstyle{\omega \over c}})\) plane, which the basis for our experimental synthesis procedure, is in all cases a conic section given by

$$\frac{1}{{k_1^2}}\left( {{\textstyle{\omega \over c}}{\kern 1pt} {\kern 1pt} \pm {\kern 1pt} {\kern 1pt} k_2} \right)^2 \pm \frac{{k_x^2}}{{k_3^2}} = 1,$$ (2)

where k 1 , k 2 , and k 3 are positive-valued constants: \({\textstyle{{k_1} \over {k_{\mathrm{o}}}}} = \left| {{\textstyle{{{\mathrm{tan}}\theta } \over {1 + {\mathrm{tan}}\theta }}}} \right|\), \({\textstyle{{k_2} \over {k_{\mathrm{o}}}}} = {\textstyle{1 \over {|1 + {\mathrm{tan}}\theta |}}}\), and \({\textstyle{{k_3} \over {k_{\mathrm{o}}}}} = \sqrt {|{\textstyle{{1 + {\mathrm{tan}}\theta } \over {1 - {\mathrm{tan}}\theta }}}|}\). The signs in the equation are (−, +) in the range 0 < θ < 45° (an ellipse), (−,−) in the range 45° < θ < 90°, and (+, −) in the range 90° < θ < 135°.

In the paraxial limit where \(k_x^{{\mathrm{max}}} \ll k_{\mathrm{o}}\), the conic section in the vicinity of k x = 0 can be approximated by a section of a parabola,

$$\frac{\omega }{{\omega _{\mathrm{o}}}} = 1 + f(\theta )\frac{{k_x^2}}{{2k_{\mathrm{o}}^2}},$$ (3)

whose curvature is determined by θ through the function f(θ) given by

$$f(\theta ) = \frac{{{\mathrm{tan}}\,\theta }}{{{\mathrm{tan}}\,\theta - 1}}.$$ (4)

Spatially resolved interference to obtain intensity profiles

We take the ST wave packet to be \(E(x,z,t) = e^{i(k_{\mathrm{o}}z - \omega _{\mathrm{o}}t)}\psi (x,z - v_{\mathrm{g}}t)\) as provided in Eq. (1), and that of the reference plane wave pulse to be \(E_{\mathrm{r}} = e^{i(k_{\mathrm{o}}z - \omega _{\mathrm{o}}t)}\psi _{\mathrm{r}}(z - ct)\). We have dropped the x-dependence of the reference and ψ r (z) is a slowly varying envelope. Superposing the two fields in the interferometer after delaying the reference by τ results in a new field ∝E(x, z, t) + E r (x, z, t − τ), whose time-average I(x, τ) is recorded at the output,

$$I(x,\tau ) \propto {\int} {\mathrm{d}}t\left| {E(x,z,t) + E_{\mathrm{r}}(x,z,t - \tau )} \right|^2.$$ (5)

We make use of the following representations of the fields for the ST wave packet and the reference pulse:

$$\begin{array}{*{20}{l}} {E(x,z,t)} \hfill & = \hfill & {e^{i(k_{\mathrm{o}}z - \omega _{\mathrm{o}}t)}{\int} {\mathrm{d}}k_x\tilde \psi (k_x)e^{ik_xx}e^{ - i(\omega - \omega _{\mathrm{o}})(t - z/v_{\mathrm{g}})}} \hfill \\ {} \hfill & = \hfill & {e^{i(k_{\mathrm{o}}z - \omega _{\mathrm{o}}t)}\psi \left( {x,t - z/v_{\mathrm{g}}} \right),} \hfill \end{array}$$ (6)

$$\begin{array}{*{20}{l}} {E_{\mathrm{r}}(x,z,t)} \hfill & = \hfill & {e^{i(k_{\mathrm{o}}z - \omega _{\mathrm{o}}t)}{\int} {\mathrm{d}}\omega \tilde \psi _{\mathrm{r}}(\omega - \omega _{\mathrm{o}})e^{i(\omega - \omega _{\mathrm{o}})(t - z/c)}} \hfill \\ {} \hfill & = \hfill & {e^{i(k_{\mathrm{o}}z - \omega _{\mathrm{o}}t)}\psi _{\mathrm{r}}\left( {x,t - z/c} \right).} \hfill \end{array}$$ (7)

We set the plane of the detector at z = 0 (CCD 1 in our experiment; see Supplementary Fig. 1), from which we obtain the spatio-temporal interferogram

$$I(x,\tau ) \propto I_{{\mathrm{ST}}}(x) + I_{\mathrm{r}} + 2|R(x,\tau )|{\mathrm{cos}}(\omega _{\mathrm{o}}\tau - \varphi _{\mathrm{R}}(x,\tau )),$$ (8)

where

$$I_{{\mathrm{ST}}}(x) = {\int} {\mathrm{d}}t\left| {\psi (x,t)} \right|^2 = {\int} {\mathrm{d}}k_x\left| {\tilde \psi (k_x)} \right|^2(1 + {\mathrm{cos}}2k_xx),$$ (9)

$$I_{\mathrm{r}} = {\int} {\mathrm{d}}t\left| {\psi _{\mathrm{r}}(t)} \right|^2 = {\int} {\mathrm{d}}\omega \left| {\tilde \psi (\omega )} \right|^2,$$ (10)

where we have made the simplifying assumption that the spatial spectrum of the ST wave packet is an even function, \(\tilde \psi (k_x) = \tilde \psi ( - k_x)\). This assumption is applicable to our experiment and does not result in any loss of generality. Note that I ST (x) corresponds to the time-averaged transverse spatial intensity profile of the ST wave packet, as would be registered by a charge-coupled device (CCD), for example, in the absence of an interferometer. Similarly, I r is equal to the time-averaged reference pulse and represents constant background term. Note that z could be set at an arbitrary value because both the reference pulse and the ST wave packet are propagation invariant.

The cross-correlation function \(R(x,\tau ) = |R(x,\tau )|e^{i\varphi _{\mathrm{R}}(x,\tau )}\) is given by

$$R(x,\tau ) = {\int} {\mathrm{d}}t\,\psi (x,t)\psi _{\mathrm{r}}^ \ast (t - \tau ).$$ (11)

Taking the integral over time t produces

$$R(x,\tau ) = {\int} {\mathrm{d}}k_x\tilde \psi (k_x)\tilde \psi _{\mathrm{r}}^ \ast (\omega )e^{ik_xx}e^{i(\omega - \omega _{\mathrm{r}})\tau },$$ (12)

where ω is no longer an independent variable, but is correlate to the spatial frequency k x through the spatio-temporal curve at the intersection of the light-cone with the hyperspectral plane \({\cal P}(\theta )\). Because the reference pulse is significantly shorter that the ST wave packet, the spectral width of \(\tilde \psi _{\mathrm{r}}\) is larger than that of \(\tilde \psi\), so that one can ignore it, while retaining its amplitude,

$$R(x,\tau ) \approx \left| {\tilde \psi _{\mathrm{r}}(\omega _{\mathrm{o}})} \right|{\int} {\mathrm{d}}k_x\tilde \psi (k_x)e^{ik_xx}e^{i(\omega - \omega _{\mathrm{o}})\tau } = \left| {\tilde \psi _{\mathrm{r}}(\omega _{\mathrm{o}})} \right|\psi (x,\tau ).$$ (13)

Note that the spectral function \(\tilde \psi (k_x)\) of the ST wave packet determines the coherence length of the observed spatio-temporal interferogram, which we thus expect to be on the order of the temporal width of the ST wave packet itself.

The visibility of the spatially resolved interference fringes (Supplementary Fig. 2) is given by

$$

u (x,\tau ) = \frac{{2|R(x,\tau )|}}{{I_{ST}(x) + I_{\mathrm{r}}}}.$$ (14)

The squared visibility is then given by

$$

u ^2(x,\tau ) \approx \frac{{4\left| {\tilde \psi _{\mathrm{r}}(\omega _{\mathrm{o}})} \right|^2\left| {\psi (x,\tau )} \right|^2}}{({I_{ST}(x) + I_{\mathrm{r}}})^{2}} \propto \left| {\psi (x,\tau )} \right|^2,$$ (15)

where the last approximation requires that we can ignore I ST (x) with respect to the constant background term I r stemming from the reference pulse.

Synthesis of ST wave packets

The input pulsed plane wave is produced by expanding the horizontally polarized pulses from a Ti:sapphire laser (Tsunami, Spectra Physics) having a bandwidth of ~8.5 nm centered on a wavelength of 800 nm, corresponding to pulses having a width of ~100 fs. A diffraction grating having a ruling of 1200 lines/mm and area 25 × 25 mm2 in reflection mode (Newport 10HG1200-800-1) is used to spread the pulse spectrum in space and the second diffraction order is selected to increase the spectral resolving power, resulting in an estimated spectral uncertainty of δλ ≈ 24 pm. After spreading the full spectral bandwidth of the pulse in space, the width size of the SLM (≈16 mm) acts as a spectral filter, thus reducing the bandwidth of the ST wave packet below the initial available bandwidth and minimizing the impact of any residual chirping in the input pulse. An aperture A can be used to further reduce the temporal bandwidth when needed. The spectrum is collimated using a cylindrical lens L 1−y of focal length f = 50 cm in a 2f configuration before impinging on the SLM. The SLM imparts a 2D phase modulation to the wave front that introduces controllable spatio-temporal spectral correlations. The retro-reflected wave from is then directed through the lens L 1−y back to the grating G, whereupon the ST wave packet is formed once the temporal/spatial frequencies are superposed; see Supplementary Fig. 1. Details of the synthesis procedure are described elsewhere20,43,47,48.

Spectral analysis of ST wave packets

To obtain the spatio-temporal spectrum \(|\tilde E(k_x,\lambda )|^2\) plotted in Fig. 3a in the main text, we place a beam splitter BS 2 within the ST synthesis system to sample a portion of the field retro-reflected from the SLM after passing through the lens L 1−y . The field is directed through a spherical lens L 4−s of focal length f = 7.5 cm to a CCD camera (CCD 2 ); see Supplementary Fig. 1. The distances are selected such that the field from the SLM undergoes a 4f configuration along the direction of the spread spectrum (such that the wavelengths remain separated at the plane of CCD 2 ), while undergoing a 2f system along the orthogonal direction, thus mapping each spatial frequency k x to a point.

Reference pulse preparation

The reference pulse is obtained from the initial pulsed beam before entering the ST wave packet synthesis stage via a beam splitter BS 1 . The beam power is adjusted using a neutral density filter, and the spatial profile is enlarged by adding a spatial filtering system consisting of two lenses and a pinhole of diameter 30 μm. The spherical lenses are L 5−s of focal length f = 50 cm and L 6−s of focal length f = 10 cm, and they are arranged such that the pinhole lies at the Fourier plane. The spatially filtered pulsed reference then traverses an optical delay line before being brought together with the ST wave packet.

Beam analysis

The ST wave packet is imaged from the plane of the grating G to an output plane via a telescope system comprising two cylindrical lenses L 2−x and L 3−x of focal lengths 40 cm and 10 cm, respectively, arranged in a 4f system. This system introduced a demagnification by a factor 4×, which modifies the spatial spectrum of the ST wave packet. The phase pattern displayed by the SLM is adjusted to pre-compensate for this modification. The ST wave packet and the reference pulse are then combined into a common path via a beam splitter BS 3 . A CCD camera (CCD 1 ) records the interference pattern resulting from the overlap of the ST wave packet and reference pulse, which takes place only when the two pulses overlap also in time; see Supplementary Fig. 2.

Group velocity measurements

Moving CCD 1 a distance Δz introduces an extra common distance in the path of both beams. However, since the ST wave packet travels at a group velocity v g and the reference pulse at c, a relative group delay of \(\Delta \tau = \Delta z({\textstyle{1 \over c}} - {\textstyle{1 \over {v_{\mathrm{g}}}}})\) is introduced and the interference at CCD 1 is lost if \({\mathrm{\Delta }}\tau \gg {\mathrm{\Delta }}T\), where ΔT is the width of the ST wave packet in time. The delay line in the path of the reference pulse is then adjusted to introduce a delay τ = Δτ to regain the interference. In the subluminal case v g < c, the reference pulse advances beyond the ST wave packet, and the interference is regained by increasing the delay traversed by the reference pulse with respect to the original position of the delay line. In the superluminal case v g > c, the ST wave packet advances beyond the reference pulse, and the interference is regained by reducing the delay traversed by the reference pulse with respect to the original position of the delay line. When v g takes on negative values, the delay traversed by the reference pulse must be reduced even further. Of course, in the luminal case the visibility is not lost by introducing any extra common path distance Δz. See Supplementary Fig. 3 for a graphical depiction.

From this, the group velocity is given by

$$v_{\mathrm{g}} = \frac{{{\mathrm{\Delta }}z}}{{{\mathrm{\Delta }}z/c - {\mathrm{\Delta }}\tau }}.$$ (16)