It is well known that the focusing of femtosecond laser pulses with even slightly tilted pulse fronts leads to an increase of the tilt angle during propagation towards the focus, a reversal of the tilt after the focus and a pronounced impact on the field distribution in the focus. In particular, the influence of pulse-front tilts and spatio-temporal couplings on the focus of high-power lasers has attracted more and more interest in recent years as several groups have either directly observed the impact of pulse-front tilts in laser–matter interactions or exploited pulse-front tilted lasers to optimize the interaction. As has been shown, for example, spatio-temporal couplings hamper reaching maximum intensity in the focus of petawatt-class laser pulses^[1^], limit the efficiency or introduce a detuning in higher-harmonic generation^[2^,3^], impact the particle pointing direction in laser particle acceleration setups^[4^,5^], are utilized in nonlinear and quantum optics^[6^] as well as to generate attosecond light pulses^[7^] and are fundamental to the simultaneous spatial and temporal focusing geometries used in ultra-short laser pulse material processing^[8^,9^].

In addition, exact knowledge of pulse-front tilt angles resulting from spatio-temporal couplings is required in traveling wave geometries, where pulse-front tilts are exploited to maximize the overlap of a moving target with a laser pulse^[10^–14^], in the generation of THz-wave pulses, where pulse-front tilts are exploited to match the group velocity of the pump light pulse and the phase velocity of the THz radiation^[15^,16^], in laser plasma accelerators, where spatio-temporal couplings can be used to control the particle pointing direction^[17^–19^], and in laser writing, where the pulse-front tilt can be exploited to control directional asymmetries in written structures^[20^]. These applications exploiting pulse-front tilts rely on dedicated dispersion management and diagnostics in the laser system in order to control the pulse’s tilt angle at the target point of interaction.

Today several techniques exist to diagnose pulse-front tilt and other pulse parameters, such as duration, along the beamline of a high-power laser up to the focus^[21^–26^]. Yet, it is not clear from the theory which tilt angle and pulse duration are to be expected while the laser pulse propagates from the last focusing mirror into the focus. The existing theory focuses on the calculation of tilt angles before focusing, where the laser is well collimated, or directly at the focus position^[10^,27^–32^], either directly or indirectly through the usage of approximations, and is not applicable at distances of the order of the Rayleigh length or more from the focus. Since Rayleigh lengths in tightly focusing geometries can be as short as tens of micrometers, this is a significant shortcoming.

As we present in this paper, the tilt and duration of femtosecond pulses can significantly evolve over these distances, resulting in deviations of pulse parameters at the actual laser–matter interaction point compared to initial expectations. Important typical affected pulse parameters are, for example, the maximum intensity on the target, created plasma density or charge separation in the target, laser depletion length in the target and spatio-temporal overlap with an evolving target region. That is, even if the dispersion properties are known before focusing, they may not be known at the interaction point, so that correlations between pulse parameters and observations in the laser–matter interaction cannot be understood. These kinds of issues become particularly relevant in applications where targets may not be reliably aligned with an accuracy smaller than the Rayleigh length^[33^] or where the laser–matter interaction already starts before the laser pulse reaches its focus as, for example, in scenarios where the laser focus is within a gas jet^[34^–36^]. Particularly in the latter, spatio-temporal couplings present at the start of the interaction may significantly impact the laser’s evolution in the target medium.

Here we derive for the first time analytic expressions providing the tilt, duration and width of a focused laser pulse under the influence of spatial, angular and group-delay dispersion. These expressions are valid along the whole propagation distance from the focusing off-axis parabola (OAP) into the focus. They not only allow quantifying the parameters of a pulse with dispersion in the surroundings of the laser–matter interaction region, but also provide understanding of the spatio-temporal couplings in real focused laser pulses. Specifically for high-power lasers, where pulse parameters cannot be measured in the vicinity of the focus, these formulas facilitate estimating pulse properties in the interaction region from dispersion measurements before the final focusing mirror. Since dispersions in the laser pulse exist in experiments, for example, originating from misalignment of laser system components or imperfect optics, the presented results are particularly relevant when relating laser pulse parameters to observations from the laser–matter interaction, for example, via simulations, as they allow one to adequately model the laser pulse in the interaction region.

Figure 1 sketches a typical situation encountered in experiments, where a laser pulse with angular dispersion and consequential spatial dispersion, $\mathrm{AD}_{\mathrm{in}}$ and $\mathrm{SD}_{\mathrm{in}}$ respectively, is focused at an OAP. During propagation to the focus the pulse-front rotates, spatial dispersion increases and the pulse duration increases.

For the derivation of the focused pulse parameters during propagation, the problem is split into two work items, allowing one to base the calculation on a combination of geometrical optics and wave optics^[37^–40^].

Firstly, the electric field of a defocusing laser pulse with known dispersion in the focus is calculated using the Fresnel diffraction integral (Ref. [41], p.636). This yields analytical relations for the change of dispersion quantities and laser parameters during propagation. Our results exceed previously published findings in that they are valid along the whole propagation path from the focusing mirror to the focus and beyond.

Secondly, the in-focus values of spatial, angular and group-delay dispersion are analytically derived from the respective quantities just before focusing at the OAP by a ray tracing approach. The expressions we derive for in-focus second- and third-order dispersion values exceed typical analysis performed with Kostenbauder ray-pulse matrices^[42^].

Figure 2 provides an overview of the geometry underlying the analytic calculations in the two steps. It visualizes important quantities used throughout the derivations.

Our derivation of a laser pulse’s tilt angle, duration and width from given spatial, angular and higher-order dispersion starts by modeling the laser’s scalar electric field distribution in frequency space $\widehat{E}$ in the focal plane and propagating this to an arbitrary distance $z$ from the focus using the Fresnel diffraction integral. We assume that the initial dispersion is present only along one axis in the transverse plane. This allows for a 2D formulation of laser propagation, where $x$ is the transverse direction and $z$ is the laser propagation axis, on the basis of cylindrical waves in the following. This work can be extended to three dimensions by treating the other transverse direction ($y$) with cylindrical waves analogously.

We assume the laser frequency spectrum and transverse profile to be Gaussian in the focus:

$$\begin{align*}\widehat{E}\left(x,z=0,\Omega \right)&={\epsilon}_{\Omega}{\epsilon}_x{e}^{- i\varphi},\\{\epsilon}_{\Omega}\left(\Omega \right)&={e}^{-\frac{\tau_0^2}{4}{\left(\Omega -{\Omega}_0\right)}^2},\\ {\epsilon}_x(x)&={e}^{-\frac{{\left(x-{x}_0\right)}^2}{w_0^2}},\end{align*}$$ 

$$\begin{align}\mathrm{SD}:= {\left.\frac{\mathrm{d}{x}_{\mathrm{c}}}{\mathrm{d}\Omega}\right|}_{\Omega ={\Omega}_0}.\end{align}$$ (1)

The laser’s spectral phase $\varphi \left(x,z=0,\Omega \right)$ in the focus is defined by the existence of angular dispersion in the focus $\mathrm{AD}_{\mathrm{foc}}$. Angular dispersion manifests in the divergence of propagation directions between frequencies, where the propagation directions of frequency $\Omega$ and the central laser frequency ${\Omega}_0$ enclose the angle $\theta =\theta \left(\Omega \right)$. Similar to $\mathrm{SD}$, $\mathrm{AD}$ is defined as the coefficient of the linear term in the expansion of $\theta$ with respect to frequency:

$$\begin{align}\mathrm{AD}:= {\left.\frac{\mathrm{d}\theta }{\mathrm{d}\Omega}\right|}_{\Omega ={\Omega}_0}={\theta}^{\prime }.\end{align}$$ (2)

We deduce the laser’s initial spectral phase $\varphi$ from the spectral phase $\phi$ of a plane wave of frequency $\Omega$ propagating at an angle $\theta$ with respect to the $z$-axis:

$$\begin{align*}\phi \left(x,z,\Omega \right)=\frac{\Omega}{c}\left(-x\sin \theta +z\cos \theta \right).\end{align*}$$ 

Expanding this about $\Omega \approx {\Omega}_0$ and evaluating at the focus position $z=0$, the laser’s initial spectral phase $\varphi$ is obtained. Up to the third order it reads, cf. Appendix A.1,

$$\begin{align*}&\varphi \left(x,z=0,\Omega \right)\approx -\frac{x}{c}{\Omega}_0{\theta}^{\prime}\left(\Omega -{\Omega}_0\right)-\frac{1}{2}\frac{x}{c}\left(2{\theta}^{\prime }+{\Omega}_0{\theta}^{{\prime\prime}}\right){\left(\Omega -{\Omega}_0\right)}^2\\& \qquad -\frac{1}{6}\frac{x}{c}\left(3{\theta}^{{\prime\prime} }+{\Omega}_0{\theta^{\prime\prime\prime} }-{\Omega}_0{\theta^{\prime}}^3\right){\left(\Omega -{\Omega}_0\right)}^3\\& \qquad +\frac{1}{2}{\mathrm{GDD}}_{\mathrm{foc}}{\left(\Omega -{\Omega}_0\right)}^2+\frac{1}{6}{\mathrm{TOD}}_{\mathrm{foc}}{\left(\Omega -{\Omega}_0\right)}^3\\& \quad =: -\alpha \frac{x}{w_0}+\frac{1}{2}{\mathrm{GDD}}_{\mathrm{foc}}{\left(\Omega -{\Omega}_0\right)}^2+\frac{1}{6}{\mathrm{TOD}}_{\mathrm{foc}}{\left(\Omega -{\Omega}_0\right)}^3,\end{align*}$$ 

$$\begin{align}\alpha \left(\Omega \right)&=\frac{w_0}{c}\left[{\Omega}_0{\theta}^{\prime}\left(\Omega -{\Omega}_0\right)+\frac{1}{2}\left(2{\theta}^{\prime }+{\Omega}_0{\theta}^{{\prime\prime}}\right){\left(\Omega -{\Omega}_0\right)}^2\right. \nonumber\\& \qquad\quad \left.+\frac{1}{6}\left(3{\theta}^{{\prime\prime} }+{\Omega}_0{\theta^{\prime\prime\prime} }-{\Omega}_0{\theta^{\prime}}^3\right){\left(\Omega -{\Omega}_0\right)}^3\right].\end{align}$$ (3)

The quantity $\alpha /{w}_0$ can be regarded as the series expansion of a frequency’s wave vector $x$-component, ${k}_x=-\left(\Omega /c\right)\sin \theta \left(\Omega \right)\approx -\alpha \left(\Omega \right)/{w}_0$.

The expansion of the spectral phase in the focus above includes values $\mathrm{GDD}_{\mathrm{foc}}$ and $\mathrm{TOD}_{\mathrm{foc}}$ at $z=0$ for group-delay dispersion and third-order dispersion in the focus, respectively. Generally, group-delay dispersion $\mathrm{GDD}$ and third-order dispersion $\mathrm{TOD}$ are defined as follows:

$$\begin{align}\mathrm{GDD}:= {\left.\frac{{\mathrm{d}}^2\varphi }{\mathrm{d}{\Omega}^2}\right|}_{\Omega ={\Omega}_0},\end{align}$$ (4)

$$\begin{align}\mathrm{TOD}:= {\left.\frac{{\mathrm{d}}^3\varphi }{\mathrm{d}{\Omega}^3}\right|}_{\Omega ={\Omega}_0},\end{align}$$ (5)

The field distribution outside the focus is obtained by propagating the initial field with the Fresnel diffraction integral for cylindrical waves^[41^,43^], cf. Appendix A.2,

$$\begin{align} &\widehat{E}\left(x,z,\Omega \right)=\sqrt{\frac{\Omega}{2\pi c}}\frac{e^{-i\left(\frac{\Omega}{c}z-\frac{\pi }{4}\right)}}{\sqrt{z}}\underset{-\infty }{\overset{\infty }{\int }}\widehat{E}\left(\xi, z=0,\Omega \right){e}^{-i\frac{\Omega}{2 cz}{\left(x-\xi \right)}^2}\kern0.1em \mathrm{d}\xi \nonumber\\&\quad =\sqrt{\frac{\Omega}{2\pi c}}\frac{e^{-i\left(\frac{\Omega}{c}z-\frac{\pi }{4}\right)}}{\sqrt{z}}{\epsilon}_{\Omega}{e}^{-i\frac{1}{2}{\mathrm{GDD}}_{\mathrm{foc}}{\left(\Omega -{\Omega}_0\right)}^2}{e}^{-i\frac{1}{6}{\mathrm{TOD}}_{\mathrm{foc}}{\left(\Omega -{\Omega}_0\right)}^3}\nonumber\\& \qquad \times\underset{-\infty }{\overset{\infty }{\int }}{\epsilon}_x\left(\xi \right){e}^{i\alpha \frac{\xi }{w_0}}{e}^{-i\frac{\Omega}{2 cz}{\left(x-\xi \right)}^2}\kern0.1em \mathrm{d}\xi \nonumber\\&\quad ={\epsilon}_{\Omega}{\left(1+\frac{z^2}{z_{\mathrm{R}}^2}\right)}^{-1/4}{e}^{-{\left[x-\left({x}_0-\frac{c}{\Omega_0{w}_0}\alpha z\right)\right]}^2\left[\frac{1}{w_0^2\left(1+{z}^2/{z}_{\mathrm{R}}^2\right)}+i\frac{\Omega}{2c}\frac{z}{\left({z}^2+{z}_{\mathrm{R}}^2\right)}\right]}\nonumber\\& \qquad \times {e}^{-i\frac{\Omega}{c}z}{e}^{i\alpha \frac{x}{w_0}}{e}^{i\frac{\alpha^2}{4}\frac{z}{z_{\mathrm{R}}}}{e}^{i\frac{1}{2}\arctan \frac{z}{z_{\mathrm{R}}}}{e}^{-i\frac{1}{2}{\mathrm{GDD}}_{\mathrm{foc}}{\left(\Omega -{\Omega}_0\right)}^2}\nonumber\\& \qquad \times {e}^{-i\frac{1}{6}{\mathrm{TOD}}_{\mathrm{foc}}{\left(\Omega -{\Omega}_0\right)}^3},\end{align}$$ (6)

$$\begin{align}w(z)={w}_0\sqrt{1+\frac{z^2}{z_{\mathrm{R}}^2}},\end{align}$$ (7)

$$\begin{align}R(z)=z\left(1+\frac{z_{\mathrm{R}}^2}{z^2}\right).\end{align}$$ (8)

While these expressions for $w$ and $R$ are frequency dependent in general, we set ${z}_{\mathrm{R}}\approx \pi {w}_0^2/{\lambda}_0$ to good approximation in Equation (6) for the following calculations. Equation (6) is a well-known result^[44^]. See Appendix A for details of this and the following derivations.

As is evident from the proportionality of the laser’s Gaussian transverse profile center ${x}_{\mathrm{c}}\left(z,\Omega \right)={x}_0-\alpha zc/\left({\Omega}_0{w}_0\right)$ to $\alpha$ in Equation (6), a frequency’s spatial distribution center is subject to higher-order dispersion. From this, the scaling of spatial dispersion with distance from the focus can be derived using Equation (1), which reads to the first order as

$$\begin{align*}\mathrm{SD}(z)={\mathrm{SD}}_{\mathrm{foc}}-{\mathrm{AD}}_{\mathrm{foc}}z.\end{align*}$$ 

Furthermore, Equation (6) allows identifying the advancement of higher-order dispersion with distance from the focus by performing the respective number of derivatives of the spectral phase $\varphi \left(x,z,\Omega \right)=- \mathrm{Arg}\left[\widehat{E}\left(x,z,\Omega \right)\right]$ with respect to $\Omega$ and evaluating at $\Omega ={\Omega}_0$. Accordingly, advancements of $\mathrm{GDD}$ and $\mathrm{TOD}$ with $z$ are obtained using Equations (4) and (5), respectively, cf. Appendix A.3,

$$\begin{align}\mathrm{GDD}(z)={\mathrm{GDD}}_{\mathrm{foc}}+4\frac{x}{w}{\beta}_3{\beta}_5+2{\Omega}_0\left(2\frac{x}{w}{\beta}_4+{\beta}_3^2\right){\beta}_5-2{\beta}_6,\end{align}$$ (9)

$$\begin{align}\mathrm{TOD}(z)&={\mathrm{TOD}}_{\mathrm{foc}}+12\frac{x}{w}{\beta}_4{\beta}_5+6{\beta}_3^2{\beta}_5+12{\Omega}_0\frac{x}{w}{\beta}_5{\delta}_1\nonumber\\& \quad +12{\Omega}_0{\beta}_3{\beta}_4{\beta}_5-6{\delta}_2,\end{align}$$ (10)

$$\begin{align*}{\delta}_1&=\frac{z}{6w{\Omega}_0}\left(3{\theta}^{{\prime\prime} }+{\Omega}_0{\theta^{\prime\prime\prime} }-{\Omega}_0{\theta^{\prime}}^3-{x^{\prime\prime\prime}}_0\right),\\[2pt]{\delta}_2&=\frac{1}{2c}\left[{\theta}^{\prime}\left(2{\theta}^{\prime }+{\Omega}_0{\theta}^{{\prime\prime}}\right)z+\frac{1}{3}\left(3{\theta}^{{\prime\prime} }+{\Omega}_0{\theta^{\prime\prime\prime} }-{\Omega}_0{\theta^{\prime}}^3\right)x\right],\end{align*}$$ 

Equations (9) and (10) are more complex than those typically used^[44^] and exhibit a variation over the transverse pulse profile either due to angular dispersion or the combination of spatial dispersion and diffraction or both. Moreover, even along the laser propagation axis ($x=0$) spatial dispersion contributes to group-delay dispersion:

$$\begin{align}{\left. \mathrm{GDD}(z)\right|}_{x=0}={\mathrm{GDD}}_{\mathrm{foc}}+\frac{\Omega_0}{c}\frac{\mathrm{SD}{(z)}^2}{R}-\frac{\Omega_0}{c}{\mathrm{AD}}_{\mathrm{foc}}^2z.\end{align}$$ (11)

This contribution compensates phase run-up outside the focus for off-axis traveling frequencies by taking phase front curvature into account. Phase run-up outside the focus originates from the term proportional to $\mathrm{AD}_{\mathrm{foc}}^2z$, which itself represents a correction of phase due to a corrected traveling distance for off-axis traveling frequencies. Since this traveling distance correction is based upon a plane wave assumption, it is only valid near the focus, where $z\ll {z}_{\mathrm{R}}$, and the correction by the term $\propto {\mathrm{SD}}^2/R$ is necessary. Far from the focus, where $z\gg {z}_{\mathrm{R}}$ and $R(z)\approx z$, the two corrections cancel each other out in the case of vanishing spatial dispersion in the focus: ${\left. \mathrm{GDD}\left(z\gg {z}_{\mathrm{R}}\right)\right|}_{x=0,{\mathrm{SD}}_{\mathrm{foc}}=0}={\mathrm{GDD}}_{\mathrm{foc}}$.

The above form of the initial field in the focus $E\left(x,z=0,\Omega \right)$ assumes that all frequencies focus at the same position $z=0$ along the central frequency’s propagation direction. This holds as long as the phase fronts of the expanded laser pulse, which is focused by the OAP, are flat. Typically this requires keeping the distance between the last telescope in the laser system and the OAP well below the Rayleigh range of the expanded laser pulse. If this is not the case, chromatic aberration will occur and further distort the pulse, as has been studied for focusing by a lens^[45^].

The field distribution in the time–space domain is obtained by Fourier transforming the above field distribution in the frequency domain (Equation (6)) to the time domain:

$$\begin{align*}E\left(x,z,t\right)=\frac{1}{2\pi}\int \widehat{E}\left(x,z,\Omega \right){e}^{i\Omega t}\mathrm{d}\Omega .\end{align*}$$ 

The result presented in the following allows for the first time to read off analytical relations for the scaling of pulse-front tilt and pulse duration valid in the close vicinity, as well as far from the focus.

In order to perform the Fourier transform, the in-focus transverse distribution center ${x}_0$ of a frequency is expanded up to the second order, ${x}_0\approx {x}_0^{\prime}\left(\Omega -{\Omega}_0\right)+\frac{1}{2}{x}_0^{{\prime\prime} }{\left(\Omega -{\Omega}_0\right)}^2$, and the definition in Equation (3) of $\alpha$ is inserted in the complex argument of Equation (6), allowing one to order terms in powers of $\Omega -{\Omega}_0$. Neglecting every contribution of the third order and higher, cf. Equations (98)–(129) in Appendix A.3,

$$\begin{align} &\widehat{E}\left(x,z,\Omega \right)={\left(1+\frac{z^2}{z_{\mathrm{R}}^2}\right)}^{-1/4}{e}^{-\frac{x^2}{w^2}}{e}^{-i{\Omega}_0\frac{x^2}{2 cR}}{e}^{-i\frac{\Omega_0}{c}z}{e}^{i\frac{1}{2}\arctan \frac{z}{z_{\mathrm{R}}}}\nonumber\\& \quad \times {e}^{-\left[\left({\beta}_1+2\frac{x}{w}{\beta}_4+{\beta}_3^2\right)+i\left(\frac{1}{2}{\mathrm{GDD}}_{\mathrm{foc}}+2\frac{x}{w}{\beta}_3{\beta}_5+\left(2\frac{x}{w}{\beta}_4+{\beta}_3^2\right){\Omega}_0{\beta}_5-{\beta}_6\right)\right]{\left(\Omega -{\Omega}_0\right)}^2}\nonumber\\& \quad \times {e}^{-\left[2\frac{x}{w}{\beta}_3+i\left({\beta}_2+\frac{x^2}{w^2}{\beta}_5+2{\Omega}_0\frac{x}{w}{\beta}_3{\beta}_5\right)\right]\left(\Omega -{\Omega}_0\right)},\end{align}$$ (12)

where

$$\begin{align} {\beta}_1&=\frac{\tau_0^2}{4}\nonumber,\\{\beta}_2&=\frac{z}{c}-{\Omega}_0{\theta}^{\prime}\frac{x}{c}\nonumber,\\{\beta}_3&=-\frac{\mathrm{SD}(z)}{w}\nonumber,\\{\beta}_4&=\frac{1}{2w}\left(2{\theta}^{\prime}\frac{z}{\Omega_0}+{\Omega}_0{\theta}^{{\prime\prime}}\frac{z}{\Omega_0}-{x}_0^{{\prime\prime}}\right)\nonumber,\\{\beta}_5&=\frac{w^2}{2 cR}\nonumber,\\{\beta}_6&=\frac{1}{2c}\left[{\Omega}_0{\theta^{\prime}}^2z+\left(2{\theta}^{\prime }+{\Omega}_0{\theta}^{{\prime\prime}}\right)x\right],\end{align}$$ (13)

Further defining

$$\begin{align*} {\gamma}_1&=1+8\frac{x}{w}\frac{\beta_4}{\tau_0^2}+4\frac{\beta_3^2}{\tau_0^2},\\{\gamma}_2&=\left[\frac{1}{2}{\mathrm{GDD}}_{\mathrm{foc}}+2\frac{x}{w}{\beta}_3{\beta}_5+\left(2\frac{x}{w}{\beta}_4+{\beta}_3^2\right){\Omega}_0{\beta}_5-{\beta}_6\right]\frac{4}{\tau_0^2}\\&= \mathrm{GDD}(z)\frac{2}{\tau_0^2},\\{\gamma}_3&=-2\frac{x}{w}\frac{\beta_3}{\tau_0}=2\frac{\mathrm{SD}(z)}{w^2{\tau}_0}x,\\{\gamma}_4&=\left(t-{\beta}_2-\frac{x^2}{w^2}{\beta}_5-2{\Omega}_0\frac{x}{w}{\beta}_3{\beta}_5\right)\frac{1}{\tau_0},\end{align*}$$ 

$$\begin{align} E\left(x,z,t\right)&=\frac{1}{\tau_0\sqrt{\pi }}{\left[\left(1+\frac{z^2}{z_{\mathrm{R}}^2}\right)\left({\gamma}_1^2+{\gamma}_2^2\right)\right]}^{-1/4}{e}^{i{\Omega}_0\left(t-\frac{z}{c}-\frac{x^2}{2 cR}\right)}\nonumber\\& \quad \times {e}^{i\frac{1}{2}\left(\arctan \frac{z}{z_{\mathrm{R}}}-\arctan \frac{\gamma_2}{\gamma_1}\right)} {e}^{-\frac{x^2}{w^2{\gamma}_1}\left(1+8\frac{x}{w}{\beta}_4/{\tau}_0^2\right)}\nonumber\\& \quad \times {e}^{-\frac{{\left[{\tau}_0{\gamma}_4-\frac{\left({\tau}_0{\gamma}_3\right)\left({\tau}_0^2{\gamma}_2\right)}{\tau_0^2{\gamma}_1}\right]}^2}{\tau_0^2\left({\gamma}_1+{\gamma}_2^2/{\gamma}_1\right)}}{e}^{i\frac{\left({\gamma}_4^2-{\gamma}_3^2\right){\gamma}_2+2{\gamma}_3{\gamma}_4{\gamma}_1}{{\gamma}_1^2+{\gamma}_2^2}},\nonumber\\[-14pt] \end{align}$$ (14)

The only problematic term with respect to the requirement ${\gamma}_1>0$ is the middle term in ${\gamma}_1$ being proportional to ${\beta}_4$, which also appears in the nominator of the exponent of the transverse profile scaling as ${e}^{-{x}^2}$ in Equation (14). In general, this term cannot be neglected and its contribution can become significant in certain regimes, for example, for pulses with a duration of the order of only a few femtoseconds or shorter. These regimes demand to verify ${\gamma}_1>0$ when using the analytic expression for the total field or those for the pulse’s spatio-temporal properties further below.

There is, however, the ‘long pulse’ regime where the term proportional to ${\beta}_4$ can be neglected and ${\gamma}_1$ remains positive always. In this regime, $\left|8{\beta}_4/{\tau}_0^2\right|\ll 1$, which can be rewritten as $8\pi \left|{\Omega}_0{\mathrm{AD}}_{\mathrm{foc}}\right|\left({w}_0/{\lambda}_0\right){\left({\Omega}_0{\tau}_0\right)}^{-2}\ll 1$, with $\left|{\Omega}_0{\mathrm{AD}}_{\mathrm{foc}}\right|=\left|\tan {\psi}_{\mathrm{tilt},{\mathrm{AD}}_{\mathrm{foc}}}\right|$ representing pulse-front tilt in the focus due to $\mathrm{AD}_{\mathrm{foc}}$ alone. That is, the middle term proportional to ${\beta}_4$ will not be of relevance in ${\gamma}_1$ as long as the $\mathrm{AD}_{\mathrm{foc}}$ induced angle of pulse-front tilt ${\psi}_{\mathrm{tilt},{\mathrm{AD}}_{\mathrm{foc}}}$ satisfies $\left|\tan {\psi}_{\mathrm{tilt},{\mathrm{AD}}_{\mathrm{foc}}}\right|\ll {\lambda}_0{\left({\Omega}_0{\tau}_0\right)}^2/\left(8\pi {w}_0\right)$, meaning that the tilt angle needs to be of the order of the ratio of the pulse duration (measured in number of laser oscillations) over the pulse width (measured in wavelengths) and provided that the pulse duration extends over several laser oscillations. Exemplarily, for a ${\lambda}_0=0.8$ μm, ${\tau}_{\mathrm{FWHM},\mathrm{I}}=30$ fs (${\tau}_0=25.5$ fs) pulse with a focal width of $\pi {w}_0\equiv 60{\lambda}_0$, the ‘long pulse’ regime is reached if ${\psi}_{\mathrm{tilt},{\mathrm{AD}}_{\mathrm{foc}}}\ll {82}^{\circ }$, and the requirement relaxes further for smaller focal spot diameters.

To our knowledge, the term $8\left(x/w\right)\left({\beta}_4/{\tau}_0^2\right)$ in ${\gamma}_1$ has not been taken into account in previous analysis of spatio-temporal couplings and its appearance outside the long pulse regime could only be recognized from the fully analytic treatment presented here.

While expressions for typically interesting intensity-related pulse parameters are derived from the time–space domain field (Equation (14)) in the following, it has several more areas of applicability. For example, one can derive the phase-related wavefront rotation^[29^] or feed the field into self-consistent simulations of pulse propagation or laser–matter interaction.

From Equation (14) the duration $T$ and width $W$ of the propagating Gaussian laser pulse with spatial, angular and group-delay dispersion in the focus are readily identified. These are the denominators of the fractions in the exponents of the last and next to last real exponential:

$$\begin{align}{\tau}^2&={\tau}_0^2{\gamma}_1={\tau}_0^2\left[1+8\frac{x}{w}\frac{\beta_4}{\tau_0^2}+4\frac{\mathrm{SD}{(z)}^2}{w^2{\tau}_0^2}\right], \end{align}$$ (15)

$$\begin{align}{T}^2&={\tau}_0^2\left({\gamma}_1+\frac{\gamma_2^2}{\gamma_1}\right)={\tau}^2+4\frac{\mathrm{GDD}{(z)}^2}{\tau^2},\end{align}$$ (16)

$$\begin{align}{W}^2&={w}^2\frac{\tau^2}{\tau_0^2+8\frac{x}{w}{\beta}_4}.\end{align}$$ (17)

However, $W$ generally is not a typical Gaussian pulse width, as it still depends on the transverse coordinate $x$. This rather shows that the transverse envelope does not keep a Gaussian shape during propagation but evolves to something more complex. (By setting $x=W$ in Equation (17) and solving the resulting cubic equation in $W/w$ it is possible to yield a value for $W$ that corresponds to its original meaning for a Gaussian beam, that is, as the distance from the pulse center along the transverse direction where the intensity reduces to $1/{e}^2$ compared to its center value.) Yet, these deviations from a Gaussian profile are not of relevance in the long pulse regime where the term proportional to ${\beta}_4$ can be neglected. Then $W$ quantifies the width of a normal Gaussian transverse profile. That is, the laser pulse keeps a Gaussian transverse profile and the spatio-temporal couplings do not alter the transverse profile to something more complex during propagation.

The expression for pulse duration (Equation (16)) is structurally equal to previously published results^[44^], but comprises more complex expressions for $\tau$ and $\mathrm{GDD}(z)$, Equations (15) and (9), respectively. In the long pulse regime, $\tau$ assumes the well-known form ${\tau}^2={\tau}_0^2+4 \mathrm{SD}{(z)}^2/{w}^2$, and represents pulse elongation due to spatial dispersion alone. In cases where pulse elongation takes place via group velocity dispersion in a dispersive material, the proportion of spatial dispersion is zero and $\tau ={\tau}_0$.

From the numerator of the exponent of the last real exponential in Equation (14) the time delay ${t}_0$ of the pulse maximum can be identified:

$$\begin{align*}{\tau}_0{\gamma}_4-\frac{\left({\tau}_0{\gamma}_3\right)\left({\tau}_0^2{\gamma}_2\right)}{\tau_0^2{\gamma}_1}=: t-{t}_0,\end{align*}$$ 

$$\begin{align*}{t}_0=\frac{z}{c}-{\Omega}_0{\mathrm{AD}}_{\mathrm{foc}}\frac{x}{c}+\frac{x^2}{2 cR}-\frac{\Omega_0}{c}\frac{\mathrm{SD}(z)}{R}x+4\frac{\mathrm{SD}(z)}{w^2}\frac{\mathrm{GDD}(z)}{\tau^2}x.\end{align*}$$ 

The time delay is directly connected to the pulse-front tilt by

$$\begin{align*}\tan {\psi}_{\mathrm{tilt}}={\left.\frac{\mathrm{d}\left({ct}_0\right)}{\mathrm{d}x}\right|}_{x=0},\end{align*}$$ 

$$\begin{align}\tan {\psi}_{\mathrm{tilt}}=-{\Omega}_0{\mathrm{AD}}_{\mathrm{foc}}-{\Omega}_0\frac{\mathrm{SD}(z)}{R}+4c\frac{\mathrm{SD}(z)}{w^2}{\left[\frac{\mathrm{GDD}(z)}{\tau^2}\right]}_{x=0}.\end{align}$$ (18)

In this expression, the first term represents a constant base value of pulse-front tilt due to angular dispersion, which is the true value of pulse-front tilt in the center of the focal plane^[27^]. The remaining terms represent deviations from the focal plane center value due to radial offset of the point of evaluation or pulse propagation.

The second term is zero in the focus, but non-zero outside. For a specific frequency, it represents an effective angle of propagation due to increasing SD during propagation, just as AD represents an angle of propagation. It can be the major source of pulse-front tilt outside the focus, as observed for the setups in the next section. Its derivation is a main result of this work.

The structure of the third term is in line with previous findings^[44^]. However, the definition for ${\left. \mathrm{GDD}(z)\right|}_{x=0}$ is extended in this work by the contribution of spatial dispersion, that is, the term proportional to $\mathrm{SD}^2/R$ in Equation (11).

Note that the definition of pulse-front tilt is not unique. The above definition is with respect to time delay ${t}_0$ of the pulse maximum along the transverse direction at some position $z$, but pulse-front tilt can be defined with respect to longitudinal spatial offset ${z}_0$ between the pulse maximum and pulse center along the transverse direction at some time $t$, too. The relation between the two definitions is

$$\begin{align*}\tan {\alpha}_{\mathrm{tilt}}={\left.\frac{\mathrm{d}\left({z}_0\right)}{\mathrm{d}x}\right|}_{x=0}=-\tan {\psi}_{\mathrm{tilt}}.\end{align*}$$ 

Using the above formulas to estimate pulse properties during propagation of a tightly focused laser pulse requires knowledge about the dispersion in the focus. Usually, these dispersion properties in the focus are unknown but estimated from the dispersion properties before the focusing mirror, where these can be measured. Using a ray tracing approach, dispersion parameters in the focus are derived in the following from the known dispersion parameters before focusing, which couple during reflection at the focusing mirror. We denote parameters before focusing with subscript ‘in’, and parameters in the focus with subscript ‘foc,coupl’. The in-focus dispersion values derived in this section will be used in the next section as input for the in-focus dispersion values in the pulse parameter formulas, Equations (18) and (16), where the latter are denoted with subscript ‘foc’.

We will assume focusing of the laser pulse at an OAP, as is standard for high-power laser systems. The pulse has only first-order contributions ${x}_{\mathrm{in}}^{\prime }$ and ${\theta}_{\mathrm{in}}^{\prime }$ to spatial and angular dispersion, respectively, before focusing. Group-delay dispersion before focusing ${\mathrm{GDD}}_{\mathrm{in}}$ is not explicitly taken into account as it does not evolve, but can simply be added to the in-focus value of group-delay dispersion $\mathrm{GDD}_{\mathrm{foc},\mathrm{coupl}}$, that is, ${\mathrm{GDD}}_{\mathrm{foc}}={\mathrm{GDD}}_{\mathrm{foc},\mathrm{coupl}}+{\mathrm{GDD}}_{\mathrm{in}}$, and similar for ${\mathrm{TOD}}_{\mathrm{foc}}$. Obtaining estimates for dispersion-coupling induced in-focus values of spatial dispersion ${\mathrm{SD}}_{\mathrm{foc},\mathrm{coupl}}$, angular dispersion ${\mathrm{AD}}_{\mathrm{foc},\mathrm{coupl}}$, group-delay dispersion ${\mathrm{GDD}}_{\mathrm{foc},\mathrm{coupl}}$ and third-order dispersion ${\mathrm{TOD}}_{\mathrm{foc},\mathrm{coupl}}$ relies on analytic tracing of rays representing the propagation of the center of a frequency’s transverse spatial distribution. Figure 3 sketches sample rays and defines all quantities used in the following derivation of dispersion properties in the focus.

In the focus, the rays of frequency $\Omega$ and ${\Omega}_0$ enclose the propagation angle $\theta$ being required to calculate angular dispersion by Equation (2). The propagation angle is determined from the difference between the angles enclosed by the OAP’s optical axis and the deflected rays of $\Omega$ and ${\Omega}_0$. Since there is angular dispersion already present before deflection at the mirror, the angle enclosed by the deflected ray of frequency $\Omega$ and the OAP’s optical axis is ${\psi}_{\mathrm{defl}}-{\theta}_{\mathrm{in}}$, which leads to

$$\begin{align*}\theta \left(\Omega \right)={\psi}_{\mathrm{defl}}-{\theta}_{\mathrm{in}}-{\psi}_{\mathrm{defl},0}.\end{align*}$$ 

The deflection angle of frequency $\Omega$ is given by

$$\begin{align}{\psi}_{\mathrm{defl}}=2\delta,\end{align}$$ (19)

$$\begin{align}\delta =\arctan \frac{\xi }{2f}.\end{align}$$ (20)

The position of incidence is obtained by computing the intersection point between the ray and the mirror surface, that is, by equating

$$\begin{align*}\left({z}_{\mathrm{ray}}+\frac{\xi_0^2}{4f}\right)\tan {\theta}_{\mathrm{in}}=\xi -\left({\xi}_0-{x}_{\mathrm{in}}\right) \quad\mathrm{and}\quad {z}_{\mathrm{OAP}}=-\frac{\xi^2}{4f},\end{align*}$$ 

$$\begin{align*} 0&=\frac{\tan {\theta}_{\mathrm{in}}}{4f}{\xi}^2+\xi -\left({\xi}_0-{x}_{\mathrm{in}}\right)-\frac{\tan {\theta}_{\mathrm{in}}}{4f}{\xi}_0^2\\\iff 0&=a\frac{\xi^2}{f^2}+b\frac{\xi }{f}+c,\end{align*}$$ 

$$\begin{align*}a&=\frac{\tan {\theta}_{\mathrm{in}}}{4},\\b&=1,\\c&=-\frac{\xi_0-{x}_{\mathrm{in}}}{f}-\frac{\tan {\theta}_{\mathrm{in}}}{4}\frac{\xi_0^2}{f^2}. \end{align*}$$ 

This quadratic equation in $\xi /f$ has the solution

$$\begin{align} \kern-12pt\xi &=2f\frac{-1+\sqrt{1-\tan {\theta}_{\mathrm{in}}c}}{\tan {\theta}_{\mathrm{in}}}\nonumber\\&\approx p+\left(q-\frac{p^2}{4f}\right){\theta}_{\mathrm{in}},\ \mathrm{where}\;p={\xi}_0-{x}_{\mathrm{in}}\;\mathrm{and}\;q=\frac{\xi_0^2}{4f},\end{align}$$ (21)

$$\begin{align*}{\xi}_0={f}_{\mathrm{eff},0}\sin {\psi}_{\mathrm{defl},0}.\end{align*}$$ 

With the above solution for the incidence point on the parabola surface, angular dispersion in the focus can be calculated:

$$\begin{align}\mathrm{AD}_{\mathrm{foc},\mathrm{coupl}}&=\frac{\mathrm{d}}{\mathrm{d}\Omega}{\left[2\arctan \left(\frac{\xi }{2f}\right)-{\theta}_{\mathrm{in}}-{\psi}_{\mathrm{d}\mathrm{efl},0}\right]}_{\Omega ={\Omega}_0}\nonumber\\ &=-\frac{1}{f_{\mathrm{eff},0}}{x}_{\mathrm{in}}^{\prime }-{\theta}_{\mathrm{in}}^{\prime }.\end{align}$$ (22)

Calculating spatial dispersion according to Equation (1) requires one to determine the spatial offset ${x}_0$ of frequency $\Omega$ in the focal plane. According to Figure 3, the spatial offset ${x}_0$ can be determined from ${\tilde{x}}_0$:

$$\begin{align*}{x}_0=\frac{{\tilde{x}}_0}{\cos \theta },\end{align*}$$ 

$$\begin{align}\mathrm{SD}_{\mathrm{foc},\mathrm{coupl}}=\frac{\mathrm{d}}{\mathrm{d}\Omega}{\left(\frac{f_{\mathrm{eff}}\sin {\theta}_{\mathrm{in}}}{\cos \theta}\right)}_{\Omega ={\Omega}_0}={f}_{\mathrm{eff},0}{\theta}_{\mathrm{in}}^{\prime }.\end{align}$$ (23)

Calculating group-delay dispersion according to Equation (4) requires one to determine the phase advance of every frequency from the incidence plane to the focal plane, which can be calculated from a frequency’s optical path length. The path of a ray starts where its phase front intersects with the incidence position of the central frequency ray on the mirror surface and it ends where its phase front intersects with the focus (see Figure 3). The path length of a frequency $\Omega$ is divided into two sections, ${L}_{\mathrm{in}}$ and ${L}_{\mathrm{foc}}$. The former is the distance from the starting point until the ray intersects with the parabola surface, while the latter is the distance from the parabola surface until the focal plane. The phase advance is

$$\begin{align}\varphi \left(\Omega \right)=\frac{\Omega}{c}\left({L}_{\mathrm{in}}+{L}_{\mathrm{foc}}\right),\end{align}$$ (24)

$$\begin{align*}{L}_{\mathrm{in}}\left(\Omega \right)=-{x}_{\mathrm{in}}\sin {\theta}_{\mathrm{in}}+\frac{\xi_0^2-{\xi}^2}{4f\cos {\theta}_{\mathrm{in}}},\kern1em {L}_{\mathrm{foc}}\left(\Omega \right)={f}_{\mathrm{eff}}\cos {\theta}_{\mathrm{in}}, \end{align*}$$ 

$$\begin{align}\mathrm{GDD}_{\mathrm{foc},\mathrm{coupl}}={\left.\frac{{\mathrm{d}}^2\varphi }{\mathrm{d}{\Omega}^2}\right|}_{\Omega ={\Omega}_0}=-\frac{\Omega_0}{c}\left({f}_{\mathrm{eff},0}{\theta_{\mathrm{in}}^{\prime}}^2+2{\theta}_{\mathrm{in}}^{\prime }{x}_{\mathrm{in}}^{\prime}\right).\end{align}$$ (25)

For future real and numerical experiments the value of third-order dispersion in the focus can be of interest. It is evaluated by applying Equation (5) on the phase advance (Equation (24)):

$$\begin{align}\mathrm{TOD}_{\mathrm{foc},\mathrm{coupl}}={\left.\frac{{\mathrm{d}}^3\varphi }{\mathrm{d}{\Omega}^3}\right|}_{\Omega ={\Omega}_0}=3\frac{\theta_{\mathrm{in}}^{\prime }}{c}\left({\Omega}_0{\xi}_0\frac{\theta_{\mathrm{in}}^{\prime }{x}_{\mathrm{in}}^{\prime }}{f}-{f}_{\mathrm{eff},0}{\theta}_{\mathrm{in}}^{\prime }-2{x}_{\mathrm{in}}^{\prime}\right).\end{align}$$ (26)

In exemplary long and short focal range setups, pulse-front tilt and pulse duration during propagation of a focusing pulse through its focus are presented in the following. As is shown, pulse-front tilts can become several tens of degrees large in the close vicinity of a couple of tens of micrometers around the focus. Pulse-front tilts of this order were observed in previous numerical experiments^[8^], but could not be fully analytically explained.

The laser pulse is focused at an OAP, and for the calculation we assume that dispersion parameters before reflection at the OAP, that is, angular dispersion $\mathrm{AD}_{\mathrm{in}}$ and spatial dispersion $\mathrm{SD}_{\mathrm{in}}$, are known. From these, the dispersion values in the focus are deduced by Equations (22), (23) and (25) using ${\theta}_{\mathrm{in}}=\mathrm{AD}_{\mathrm{in}}$ and ${x}_{\mathrm{in}}^{\prime}=\mathrm{SD}_{\mathrm{in}}$. The in-focus dispersion values $\mathrm{AD}_{\mathrm{foc},\mathrm{coupl}}$, $\mathrm{SD}_{\mathrm{foc},\mathrm{coupl}}$ and $\mathrm{GDD}_{\mathrm{foc},\mathrm{coupl}}$, respectively, are then plugged into Equations (16) and (18) in order to determine pulse duration and tilt, respectively, during propagation.

All setups will use a laser pulse with a central wavelength ${\lambda}_0=0.8$ μm, duration ${\tau}_{\mathrm{FWHM},\mathrm{I}}=30$ fs and width ${D}_{\mathrm{in}}=\pi {w}_{\mathrm{in}}=100$ mm (99% power transmission through an aperture of this diameter for Gaussian beams) before focusing.

This setup’s OAP has ${f}_{\mathrm{eff},0}/{D}_{\mathrm{in}}=2.5$ ($=f/\#$), focusing the incident pulse to a width ${w}_{\mathrm{FWHM},\mathrm{I}}=2.35$ μm and resulting in a Rayleigh length ${z}_{\mathrm{R}}=16$ μm.

Figure 4 visualizes pulse-front tilt and pulse duration in the course of propagation through the focus for angular dispersion values before focusing ranging from $\mathrm{AD}_{\mathrm{in}}=5\times {10}^{-3}$ to 1 μrad/nm without spatial dispersion before focusing, that is, $\mathrm{SD}_{\mathrm{in}}=0$. While small values of $\mathrm{AD}_{\mathrm{in}}$ below ${10}^{-2}$ μrad/nm result in a maximum pulse-front tilt of $-{3.6}^{\circ }$ at a Rayleigh length before the focus, higher values such as $0.25$ μrad/nm result in a large maximum pulse-front tilt of $-{51}^{\circ }$ at about the same position. Angular dispersion before focusing of 1 μrad/nm results in an even larger maximum pulse-front tilt of $-{61}^{\circ }$ at $3.5$ Rayleigh lengths before the focus. Generally, it can be observed that larger values of $\mathrm{AD}_{\mathrm{in}}$ result in larger maximum pulse-front tilt farther away from the focus. In all of these examples, group-delay dispersion in the focus $\mathrm{GDD}_{\mathrm{foc},\mathrm{coupl}}$ due to $\mathrm{AD}_{\mathrm{in}}$ is negligible, as it is only $-0.2$ fs${}^2$ for the largest $\mathrm{AD}_{\mathrm{in}}$. To ease comparison to the numerical results shown later, the in-focus value $\mathrm{GDD}_{\mathrm{foc}}$ is therefore set to zero in the calculations.

The major source of these large pulse-front tilts is the appearance of spatial dispersion, that is, the term $-{\Omega}_0 \mathrm{SD}/R$ in Equation (18). In this term, $\mathrm{SD}$ and $R$ together define a maximum propagation angle, which at the same time is the maximum angle enclosed by a phase front and the laser propagation axis. Just as for $-{\Omega}_0{\theta}^{\prime }$, this angle leads to a maximum time delay along the transverse direction and therefore pulse-front tilt. This pulse-front tilt caused by spatial dispersion varies during propagation due to the varying radius of curvature $R$. It reaches its maximum at a Rayleigh length from the focus, where the respective time delay is largest due to $R$ being smallest, and it vanishes in the focus where $R$ is infinite such that there is no time delay.

The third term in Equation (18) constitutes a damping of the leading second term. For larger angular dispersion before focusing it provides for the shift of maximum pulse-front tilt to positions beyond the Rayleigh length, which is the position where the second term peaks.

In contrast to pulse-front tilt, the increase of pulse duration is only relevant for the two largest $\mathrm{AD}_{\mathrm{in}}$ values, with a maximum of $3.5{\tau}_0$ in the focus for $\mathrm{AD}_{\mathrm{in}}=1$ μrad/nm.

The source of pulse elongation is again spatial dispersion, described by Equation (15) alone since $\mathrm{GDD}$ is assumed to vanish in the focus. Spatial dispersion leads to a loss of overlap between spatial distributions of frequencies, which thins out the local spectrum. This effect is largest in the focus where the spatial frequency distributions are smallest, and thus the local spectrum is smallest and the pulse duration is longest.

The above setup neglects $\mathrm{SD}_{\mathrm{in}}$, that is, spatial dispersion generated by angular dispersion during propagation from the laser system’s compressor until the OAP. Assuming 10 m distance from the compressor until the OAP, spatial dispersion before focusing at the OAP due to propagation with angular dispersion is $\mathrm{SD}_{\mathrm{in}}=-\mathrm{AD}_{\mathrm{in}}\cdot 10\ \mathrm{m}={-}0.05,{-}0.1,{-}0.25,{-}0.5,{-}1,{-}2.5,{-}5,{-}10$ μm/nm. This spatial dispersion before focusing will not influence spatial dispersion in the focus, according to Equation (23), but it will increase angular dispersion, and therefore pulse-front tilt, in the focus. Pulse-front tilts in the focus are ${\psi}_{\mathrm{tilt}}=\mathrm{0.01}^{\circ},\mathrm{0.02}^{\circ},\mathrm{0.04}^{\circ},\mathrm{0.09}^{\circ},\mathrm{0.18}^{\circ},\mathrm{0.45}^{\circ},\mathrm{0.90}^{\circ},\mathrm{1.79}^{\circ}$, respectively. Since these are still small, the overall picture of the scaling remains equal compared to the setup with $\mathrm{SD}_{\mathrm{in}}=0$. In particular, maximum values of pulse-front tilt and duration do not change.

This setup’s OAP has ${f}_{\mathrm{eff},0}/{D}_{\mathrm{in}}=250$, focusing the incident pulse to a width ${w}_{\mathrm{FWHM},\mathrm{I}}=19$ μm and resulting in a Rayleigh length ${z}_{\mathrm{R}}=1.0$ mm.

Figure 5 visualizes pulse-front tilt and pulse duration in the course of propagation through the focus for the same range of angular dispersion values before focusing as for the short focal range setup and without spatial dispersion before focusing. Due to equal laser parameters, the scaling is qualitatively equal to the short focal range setup. Only the maximum value of pulse-front tilt is reduced, since the radius of the pulse-front curvature scales quadratically in the focal distance while spatial dispersion scales linearly for equal laser parameters before focusing. In total, this results in less time delay between frequencies along the transverse direction, reducing pulse-front tilt.

Pulse duration in focus remains equal between long and short focal range setups, as the ratio of spatial dispersion and width in focus, which determines pulse elongation, is independent of focal length.

The considerations for group-delay dispersion in the focus and spatial dispersion before focusing outlined for the short focal range setup can be identically applied to this long focal range setup.

The obtained pulse-front tilt and pulse duration of the short focal range setup are cross-checked by numerically Fourier transforming the propagated pulse in Fourier space (Equation (6)) for the setup with $\mathrm{AD}_{\mathrm{in}}=1$ μrad/nm and measuring pulse-front tilt and pulse duration from the obtained time–space domain spatio-temporal intensity envelope on a grid. This intensity envelope is obtained from the complex field distribution by taking the absolute square. Taking from this 2D intensity distribution two 1D intensity distributions at constant transverse positions ${x}_{\mathrm{center}}$ and ${x}_{\mathrm{out}}$ allows for measuring pulse-front tilt. We chose ${x}_{\mathrm{center}}=0$ and ${x}_{\mathrm{out}}={w}_0$. By determining the respective times ${t}_{\mathrm{center}}$ and ${t}_{\mathrm{out}}$ at which the intensity reaches its maximum along these two 1D intensity distributions, the pulse-front tilt angle can be approximated by

$$\begin{align}\tan {\psi}_{\mathrm{tilt},\mathrm{num}}=\frac{c\left({t}_{\mathrm{center}}-{t}_{\mathrm{out}}\right)}{x_{\mathrm{center}}-{x}_{\mathrm{out}}}.\end{align}$$ (27)

Pulse duration is measured by the least square fit of a Gaussian curve ${I}_0\exp \left[-{\left(t-{t}_{\mathrm{center}}\right)}^2/\left(2{\sigma}^{2}\right)\right]$ to the 1D intensity distribution at ${x}_{\mathrm{center}}$, where ${I}_0$ equals the maximum of the intensity distribution. The fit determines ${\sigma}_{I,t}$, which is related to the pulse duration of the field by $T=2{\sigma}_{I,t}$.

Figure 6 visualizes intensity envelope distributions at different distances $z$ from the focus together with measured and predicted contours for the pulse front as well as measured values of pulse-front tilt and duration. Agreement between measured and predicted values can be observed, from which we conclude successful verification of the analytic formulas derived in this work.

The remaining differences between measured and predicted values originate from finite sampling of the intensity distribution along the $t$-axis. The arrival time of the intensity maximum at some $x$ can only be determined with an uncertainty about the size of the time sampling step, which results in an uncertainty on the pulse-front tilt angle. It is of the order of one degree or less in our setup.

Note, since the pulse’s width in the focus is significantly smaller than its length, the visible envelope ellipse is not aligned with the drawn contour of the pulse front. However, for each $x$ the highest intensity is indeed on this contour, which just defines this contour as the pulse front. Further note that the difference in the analytically calculated absolute value of the pulse-front tilt angle between $z=-{z}_{\mathrm{R}}$ and $z={z}_{\mathrm{R}}$ originates from the fact that the position where the pulse-front tilt vanishes is slightly behind the focus at $z>0$.

We presented analytical expressions allowing one to evaluate the electric field, width, duration and tilt of dispersive, tightly focused, short pulse, Gaussian lasers in the vicinity and far from their focus in the time–space domain, which was not possible before. With the help of these expressions we were able to link the appearance of large pulse-front tilts of several tens of degrees, observed within a few Rayleigh lengths distance from the focus of a laser pulse featuring only weak angular dispersion, to the accompanying spatial dispersion. Numerical evaluation of the tilt and duration of Gaussian pulses propagated in simulations verified the predictions provided by the analytic expressions, which proves their applicability.

The possibility of generating large pulse-front tilts in the vicinity of the laser’s focus with moderate to low pulse elongation is thereby interesting on its own, as generating and utilizing pulses with large pulse-front tilts becomes simpler in ‘out-of-focus’ interaction geometries without the cost of large pulse elongation usually connected to large pulse-front tilt.

Moreover, the presented analytic expressions of the dispersion variation during propagation or of the full electric field can be of general use, for example, to simply estimate pulse properties at any position along the beamline of a given laser system, or to study the interaction of these pulses with other fields or matter in complex geometries and with correct phase contributions analytically or in simulations.