The interplay between the frequency chirping of a broadband laser pulse and the longitudinal chromatic aberration of a focusing optic introduces the superluminal or subluminal behavior to a laser focus. In this paper, we present an analytic expression for an electric field describing a superluminal or subluminal femtosecond laser focus with orbital angular momentum. The analytic expression for a superluminal or subluminal laser focus is obtained through a diffraction integral, in which the focal length is replaced by a time-dependent focal length under the paraxial approximation, and the Fourier transformation. The speed and pulse duration of a laser focus are determined by the total group delay dispersion and a chromaticity parameter defined by the longitudinal chromatic aberration of a dispersive focusing optic. It is shown that it is possible to generate a several femtosecond superluminal orbital angular momentum laser focus in the focal region.
1. Introduction
The femtosecond (fs) high-power laser based on the chirped-pulse amplification technique [1] increases the focused laser intensity to the relativistic and ultra-relativistic regimes [2,3]. It enables ones to study the generation of high-energy secondary sources (such as electrons/positrons, protons/neutrons, photons, etc.) [4–6], laboratory astrophysics [7], and strong-field quantum electrodynamics (SF QED) [8,9]. Recently, two physical aspects, the superluminal behavior of the laser focus [10–12] and the orbital angular momentum (OAM) [13–16], have drawn much interest in the laser-plasma interaction study.
Modern discussion on the superluminal behavior of an electromagnetic radiation dates back as early as 1900s [17]. The formation of the superluminal electromagnetic radiation and its aspect in the laser-plasma interaction have been discussed in [10–12,18–25]. For the ultrashort high-power laser pulse, the superluminal or subluminal motion of a focused laser intensity (laser focus) is obtained by introducing a longitudinal chromatic aberration (LCA) to a frequency-chirped (shortly, chirped hereafter) laser pulse. It is well known that the moving direction and speed of a superluminal or subluminal laser focus are determined by the interplay between the group delay dispersion (GDD) and the LCA of a dispersive focusing optic (Fig. 1). Thus, it is of fundamental interest to have a mathematical formula for the superluminal or subluminal laser focus for further investigation in laser-plasma interaction studies.
In this paper, an analytic expression has been derived to describe a spatio-temporal electric field distribution for a superluminal or subluminal laser focus with OAM. A chirped laser pulse is described by a spectral phase with GDD and higher-order dispersions (third-order dispersion and higher). Since the laser pulse is chirped, the frequency-dependent focal length of a focusing optic becomes a time-dependent focal length. Then, the spatio-temporal electric field distribution of a laser focus can be analytically calculated through the diffraction integral and Fourier transformation method. The time-dependent focal length is plugged into the diffraction integral in order to impose the superluminal/subluminal property to a laser focus.
The center of mass (CoM) speed of the laser focus is considered as the speed of the laser focus. The CoM speed shows how the speed of the laser focus is related to the GDD, chromaticity parameter, and the f-number of a focusing optic. The mathematical expression for the speed of the laser focus reveals two important features: One is that there is a maximum speed for the laser focus at given GDD and LCA parameters. The other is that the pulse duration of the laser focus can be controlled by both the pulse broadening due to GDD and the pulse broadening/shortening effect due to LCA. In addition, the expression for the CoM speed provides information on the formation of a stationary laser focus (previously interpreted as an infinite speed laser focus in [10,11]) and the generation of superluminal several fs laser focus. All mathematical derivations are performed for the OAM electric field. Therefore, the field expressions presented is directly applied to the generation of a super-intense superluminal or subluminal fs OAM laser focus. The analytical expression for the super-intense superluminal subluminal OAM fs laser focus can be used to study SF-QED in the superluminal/subluminal regime, where the interaction between electron spin and photon OAM is important [26–28].
2. Description of an electric field for a chirped OAM laser pulse
2.1 Spectral phase and group delay dispersion
Let us assume that an OAM fs laser pulse passes through a dispersive optic (e.g., thick optical material, a grating or prism pair), which introduces the GDD in the pulse. The mathematical form of an OAM electric field that freely propagates along the z-axis before the optic is given by [13],
(1)$$\begin{aligned} E(\rho;\omega) & = E_{nl}(\rho) \exp \left[ i\omega \left(t - \frac{z}{c} \right) +i \Phi_H \left(\phi; \omega\right) \right] \\ & = E_0 \rho^{|l|} \exp \left( -\frac{\rho^2}{2 \rho_0^2} \right) L_n ^{l} \left( \frac{\rho^2}{\rho_0^2} \right) \exp \left[ i \omega \left( t - \frac{z}{c} \right) + i \Phi_H \left( \phi; \omega \right) \right]. \end{aligned}$$
where $E_0$ is the peak field strength, $\rho$ the radial distance in cylindrical coordinates, $\rho _0$ the Gaussian width, and $L_n ^{l} \left ( \rho ^2/\rho _0^2 \right )$ the generalized Laguerre polynomial with a radial order of $n$ and an azimuthal order of $l$. The spatial distribution function, $E_{nl}(\rho )$, which determines the $n$-th radial and $l$-th azimuthal mode is in general defined as $E_0 \rho ^{|l|} \exp \left ( -\rho ^2/2 \rho _0^2 \right ) L_n ^{l} \left ( \rho ^2 /\rho _0^2 \right )$. Through the paper, we consider only the lowest radial order, so $n=0$, $L_0 ^{l} \left ( \rho ^2 /\rho _0^2 \right )=1$, and $E_{0l}(\rho )=E_0 \rho ^{|l|} \exp \left ( -\rho ^2 / 2 \rho _0^2 \right )$.In Eq. (1), the OAM characteristic of the beam is imposed by the helical phase shift, $\Phi _H = l \phi$, where $\phi$ is the azimuthal angle and $l$ the topological charge (TC), which means the winding of helical phase shift over one wavelength. For a broadband spectrum, the helical phase can be modified as
(2)$$\Phi_H \left(\phi;\omega \right) = \frac{\omega}{\omega_0}l\phi,$$
where $\omega _0$ is the reference angular frequency which yields the integer $l$ for TC. A general discussion on the OAM beam with a non-integer topological charge can be found in [29]. When the OAM fs laser pulse passes through the optic, the spectral phase, $\Phi (\omega )$, is affected by the dispersion of the optic, and at a certain frequency of $\omega _0$ the spectral phase can be expressed by the Taylor series expansion as [30], (3)$$\begin{aligned} \Phi(\omega) & = \sum_{n = 0}^{\infty} \frac{\beta_n}{n!} \left( \omega -\omega_0 \right)^n \\ & = \Phi \left(\omega_0 \right) + \left.\frac{\partial \Phi}{\partial \omega} \right\vert_{\omega = \omega_0} \left( \omega -\omega_0 \right) + \frac{1}{2!} \left.\frac{\partial^2 \Phi}{\partial \omega^2} \right\vert_{\omega = \omega_0} \left( \omega -\omega_0 \right)^2 + \cdots. \end{aligned}$$
On the right-hand side of Eq. (3), the second derivative, $\beta _2 = \left .\partial ^2 \Phi / \partial \omega ^2 \right \vert _{\omega =\omega _0}$, is known as GDD, which is responsible for the linear frequency-chirping of an ultrashort laser pulse and the pulse broadening in time. Other derivatives, such as $\beta _3$, $\beta _4$, and so on, are known as third-order dispersion (TOD), fourth-order dispersion (FOD), and so so. These are responsible for non-linear frequency chirping and the deformation of the laser temporal shape.When an OAM fs laser pulse passes through a dispersive optic, the final expression for the spectral phase is given by the sum of Eqs. (2) and (3) as,
(4)$$\Phi(\omega) = \sum_{n = 0}^{\infty} \frac{\beta_n}{n!} \left( \omega -\omega_0 \right)^n + \frac{\omega}{\omega_0} l \phi.$$
The GDD and TOD in the spectral phase are the most dominant terms in the laser pulse. By ignoring higher-order terms, the total spectral phase of the OAM fs laser pulse can be simplified to (5)$$\Phi(\omega) \approx \frac{\beta_2}{2} \left( \omega -\omega_0 \right)^2 + \frac{\beta_3}{6} \left( \omega -\omega_0 \right)^3 + \frac{\omega}{\omega_0} l \phi,$$
and the chirped electric field with an OAM attribute becomes (6)$$E (\rho;\omega) = E_{0l} (\rho) \exp \left[ i\omega \left(t - \frac{z}{c} \right) +i \frac{\beta_2}{2} \left( \omega -\omega_0 \right)^2 +i \frac{\beta_3}{6} \left( \omega -\omega_0 \right)^3 + i \frac{\omega}{\omega_0} l \phi \right].$$
Through this paper it is understood that a positive $\beta _2$ means a positively-chirped pulse.2.2 Incident electric field in time
Assuming that the laser pulse has a Gaussian spectrum, $G(\omega )=\exp \left [ - \left ( \omega - \omega _0 \right )^2 / 2 \Delta \omega ^2 \right ]$, in the frequency domain, the laser pulse profile in time can be calculated by the Fourier transformation as
(7)$$E (\rho, t) = E_{0l} (\rho) \int_{-\infty}^{\infty} d \omega \exp \left[ i\omega \left(t - \frac{z}{c} \right) + i \Phi \left( \omega \right) \right] \exp \left[ - \frac{\left( \omega - \omega_0 \right)^2}{2 \Delta \omega^2} \right].$$
Here, $\Delta \omega$ is the Gaussian width of the spectrum. Now, by introducing new frequency variable, $\tilde{\omega }$, defined as $\omega -\omega _0$, Eq. (7) can be re-written as, (8)$$E (\rho, t) = E_{0l} (\rho) \exp \left( i \omega_0 T \right) \int_{-\infty}^{\infty} d \tilde{\omega} \exp \left( i \delta_3 \tilde{\omega}^3 - \frac{\tilde{\omega}^2}{\tilde{\alpha}} + i \tilde{\omega}T \right).$$
Here, we define $\delta _3$ as $\beta _3 /6$, $T$ as $t-z/c-l\phi /\omega _0$, and a complex parameter, $\tilde{\alpha }$, as $2\Delta \omega ^2 / \left ( 1 - i D_2 \right )$ with the definition of $D_2 = \beta _2 \Delta \omega ^2$. After performing the integration as described in Sec. I of Supplement 1, a generalized form of electric field in time for a chirped OAM laser pulse is given by Eqs. (S12) and (S13) of Supplement 1. When there is no TOD ($\delta _3 =0$) in the laser pulse, then only $m=0$ term in Eq. (S13) in Supplement 1 survives and we obtain (9)$$\begin{aligned} E(\rho,t) &= \sqrt{\pi \tilde{\alpha}} E_{0l} (\rho) \exp \left( -\tilde{T}^2 \right) \exp\left( i\omega_0 T \right) \\ &= \sqrt{2 \pi} \frac{\Delta \omega E_{0l}(\rho)}{\sqrt[4]{1+D_2^2}} \exp \left[ -\frac{\Delta \omega^2 T^2}{2 \left( 1+D_2^2 \right)} \right] \exp \left[ i \omega_0 T + i \frac{D_2 \Delta^2 T^2}{2 \left( 1+D_2^2 \right)} + i \frac{1}{2} \tan^{{-}1} D_2 \right]. \end{aligned}$$
It should be noted that the asymptote of $\tan ^{-1}D_2$ is $\pm \pi /2$ as $\beta _2 \to \pm \infty$.Now, consider that the incident laser field has GDD and TOD in its spectral phase. For the weakly nonlinear case ($|\tilde{\alpha }^3 \delta _3^2| \ll 1$), we take the first two terms ($m=0$ and $m' =1$) from Eq. (S13) in Supplement 1 and we have
(10)$$ E(\rho,t) \approx \sqrt{ \tilde{\alpha}} E_{0l} \left( \rho \right) \exp \left( -\tilde{T}^2 \right) \exp\left( i\omega_0 T \right) \left[ \left( -\frac{1}{2} \right) ! - 2^6 \left( \tilde{\alpha}^3 \delta_3^2 \right)^{1/2} \tilde{T} \frac{(3/2)!}{8 \cdot 3!} L_1^{1/2} \left( \tilde{T}^2 \right) \right]. $$
After a straightforward calculation with Eq. (10) provided in Sec. II of Supplement 1, we obtain the electric field as, (11)$$\begin{aligned} E(\rho,t) \approx & \frac{\sqrt{2 \pi} \Delta \omega E_{0l}(\rho)}{\sqrt[4]{1+D_2^2}} \exp \left[ - \frac{\Delta \omega^2 T^2}{2 \left( 1 + D_2^2 \right)} \right] \\ & \times \exp \left[ i \left(\omega_0 T + \frac{D_2 \Delta \omega^2 T^2}{2 \left( 1 + D_2^2 \right)} - \frac{D_3 D_2^3 \Delta \omega^3 T^3}{\left( 1 + D_2^2 \right)^3} + \frac{\tan^{{-}1}D_2}{2} \right) \right], \end{aligned}$$
where $D_3 = \delta _3 \Delta \omega ^3 = \beta _3 \Delta \omega ^3 /6$ . From Eqs. (9) and (11), the chirped frequency, $\omega (t)$, can be obtained by differentiating the spectral phase with respect to time as, (12)$$\omega (t) = \omega_0 + \tilde{\omega} (t) = \omega_0 + \frac{D_2 \Delta \omega^2}{1+D_2^2}t,$$
or (13)$$\omega (t) = \omega_0 + \tilde{\omega} (t) = \omega_0 + \frac{D_2 \Delta \omega^2}{1+D_2^2}t - \frac{3D_3 D_2^3 \Delta \omega^3}{\left( 1+D_2^2 \right) ^3}t^2 .$$
Since the pulse duration of a laser pulse is mainly determined by GDD, the full-width at half maximum (FWHM) pulse duration, $\tau _{\mathrm {FWHM}}$, for a stretched laser pulse is easily obtained as (14)$$\tau_{\mathrm{FWHM}} = \frac{2}{\Delta \omega} \sqrt{\left( 1 + D_2^2 \right) \ln 2},$$
yielding the time-bandwidth product of $\omega _{\mathrm {FWHM}}\tau _{\mathrm {FWHM}} = 4 \ln 2 \sqrt { 1 + D_2^2 }$ as in [30]. Here, the FWHM spectral bandwidth, $\omega _{\mathrm {FWHM}}$, is defined as $2 \Delta \omega \sqrt {\ln 2}$. Equation (12) will be mostly used to determine the time-dependent focal length of a focusing optic due to LCA.3. Calculation of the focused field: flying laser focus with an arbitrary velocity
3.1 Highly-dispersive focusing optic and time-dependent focal length
The fs high-power laser pulse is usually focused by a reflective optic such as a parabolic mirror, to avoid inducing the dispersion effect. However, we consider a dispersive focusing optic (refractive lens) that induces a frequency-dependent focus shift known as LCA. This optic also induces an additional GDD in the laser pulse. Let us first consider the LCA induced by a dispersive focusing lens. According to Sellmiere’s equation [31], the refractive index, $n$, is in general a function of wavelength as
(15)$$n^2 (\lambda) = 1+\frac{B_1 \lambda^2}{\lambda^2 - C_1} + \frac{B_2 \lambda^2}{\lambda^2 - C_2} +\frac{B_3 \lambda^2}{\lambda^2 - C_3},$$
where $B_i$ and $C_i$ are material constants. By treating the change of the refractive index as the perturbation, the refractive index of a focusing optic can be then modeled by, (16)$$n(\omega) = n_0 + \Delta n(\omega ),$$
where, $n_0 = n(\omega _0)$, $|\Delta n(\omega )| \ll n_0$, and $\Delta n (\omega _0 ) =0$. The $\Delta n ( \omega )$ determines the dispersion property of the focusing optic. Under the thin lens approximation, the focal length of a lens is given by (17)$$\frac{1}{f} \approx (n-1) \left( \frac{1}{R_1} - \frac{1}{R_2} \right),$$
where $n$ is the refractive index of the lens, $R_1$ and $R_2$ are the radii of curvature for the front and back surfaces of the lens, respectively. By inserting Eq. (16) into Eq. (17), the frequency-dependent focal length can be modeled as, (18)$$\frac{1}{f( \omega )} = \left[ n( \omega_0 ) -1 \right] \left( \frac{1}{R_1} - \frac{1}{R_2} \right) + \Delta n (\omega) \left( \frac{1}{R_1} - \frac{1}{R_2} \right) = \frac{1}{f_0} + \frac{1}{f'(\omega)},$$
where $f_0$ is the focal length at $\omega _0$ and $f'(\omega _0)=\infty$. Since $f'(\omega ) \gg 1$, the focus shift, $\Delta f (\omega )$, known as LCA can be approximated as (19)$$\Delta f (\omega)=f(\omega) - f(\omega_0) \approx{-} \frac{f^2 (\omega_0)}{f' (\omega)}.$$
Figure 2 shows the focus shift, $\Delta f(\omega )$, with respect to the frequency. In the figure, the leftmost frequency corresponds to a wavelength of 850 nm and the rightmost frequency does 750 nm. The change of focus depending on the frequency is shown for typical BK7 and highly-dispersive (low Abbe number) SF66 glass substrates. An f-number of 10 (for example, beam diameter of 40 mm and focal length of 400 mm) is assumed for the Zemax calculation. The focus changes almost linearly in the wavelength range from 725 to 875 nm.According to [32], the dioptric power, $D_p$, of a focusing optic which is the reciprocal of the focal length can be expressed by the Zernike coefficient, $c_2^0$, for the defocus of the wavefront as
(20)$$D_p = \frac{1}{f'} = \frac{4\sqrt{3}c_2^0}{{\rho'_0}^2},$$
where $\rho '_0$ is an effective beam radius of which intensity becomes zero. For a circular flat-top beam profile, the effective beam radius is the same as the beam radius, $\rho _0$. However, for a Gaussian beam profile, it is reasonable to take $2\rho _0$ as the effective beam radius since the intensity decreases to 1.8$\%$ of its maximum value at $2\rho _0$. The linear dependency of the focus shift on the frequency ensures that the dioptric power given by Eq. (20) holds the linear dependency on the frequency. This fact enables ones to re-write the Zernike coefficient, $c_2^0$, as a linear function of frequency, $c_2^0 (\omega ) = c_d \left ( \omega - \omega _0 \right )$. So, the focal length induced by the frequency change is written as, (21)$$\frac{1}{f'(\omega)} = \frac{4\sqrt{3}}{{\rho'_0}^2} c_d \left( \omega - \omega_0 \right).$$
Here, $c_d$ is the chromaticity parameter which means the changing rate of defocus per unit frequency shift. By combining Eqs. (19) and (21), the chromaticity parameter has a form of $c_d \approx - \left (1/4\sqrt {3} \right ) \left ({\rho '_0}^2 / f_0^2 \right ) \left (\Delta f (\omega ) / \Delta \omega \right )$. Due to the definition of the f-number, $F_N = f_0 / 2 \rho '_0$, the chromaticity parameter, $c_d$, can be re-written as, (22)$$c_d \approx{-} \frac{1}{16\sqrt{3}F_N^2} \frac{\Delta f (\omega )}{\Delta \omega}.$$
The chromaticity parameters for the BK7 and SF66 materials are shown in Fig. 2. The chromaticity parameter in the normal dispersion range is positive since $\Delta f /\Delta \omega < 0$, and it becomes negative in the abnormal dispersion range where $\Delta f/ \Delta \omega >0$.After inserting Eqs. (21) and (22) into Eq. (18), the frequency-dependent focal length can be expressed as,
(23)$$f(\omega) \approx f_0 \left[ 1 - 8 \sqrt{3} F_N \frac{c_d}{\rho'_0} \left( \omega - \omega_0 \right) \right].$$
As shown in Eqs. (12) and (13), the frequency is a function of time for the chirped laser pulse. Therefore, the frequency-dependent focal length can be re-expressed by a time-dependent focal length as, (24)$$f(t) \approx f_0 \left[ 1 - \Omega \left( c_d, D_2 \right) t \right].$$
with the definition of the changing rate of the focus, $\Omega \left ( c_d, D_2 \right )$, per unit time as, (25)$$\Omega \left( c_d, D_2 \right) = 8 \sqrt{3} F_N \frac{c_d}{\rho'_0} \frac{D_2 \Delta \omega^2}{1+D_2^2}.$$
Let us now consider the GDD experienced when a laser pulse propagates through a focusing lens. Since the optical phase, $\Phi _{\Delta } (x,y)$, depending on the thickness, $\Delta (x,y)$, of the lens is given by $\Phi _{\Delta } = nk\Delta (x,y)$, the GDD by the lens is written as, (26)$$\frac{d^2 \Phi_{\Delta}}{d \omega^2} = \left( \frac{\omega}{c} \frac{d^2 n}{d \omega^2} + \frac{2}{c} \frac{dn}{d \omega} \right) \Delta (x,y) = k^{\prime\prime} \Delta (x,y),$$
with the definition of $k''= \left (\lambda ^3 / 2 \pi c^2 \right ) \left (d^2 n /d \lambda ^2 \right )$. Under the thin lens approximation, a thickness function, $\Delta (x,y)$, of the thin lens is given by [33], (27)$$\Delta (x,y) \approx \Delta_0 - \frac{x^2 + y^2}{2} \left( \frac{1}{R_1} - \frac{1}{R_2} \right) = \Delta_0 - \frac{\rho^2}{2} \left( \frac{1}{R_1} - \frac{1}{R_2} \right),$$
and then an additional spectral phase by the GDD of the lens material is given by (28)$$\Phi_{\Delta} (\omega) = k^{\prime\prime} \Delta (x,y) \left( \omega - \omega_0 \right)^2 = k^{\prime\prime} \left[ \Delta_0 - \frac{\rho^2}{2} \left( \frac{1}{R_1} - \frac{1}{R_2} \right) \right] \left( \omega - \omega_0 \right)^2 .$$
In Eqs. (27) and (28), $\Delta _0$ is the center thickness of the lens. The total spectral phase of the laser pulse can be calculated by adding Eq. (28) to the phase in Eq. (6) as, (29)$$\Phi (\omega) \approx \frac{\left(2k^{\prime\prime} \Delta_0 + \beta_2 \right)}{2} \left( \omega - \omega_0 \right)^2 - \frac{k^{\prime\prime}\rho^2}{2} \left( \frac{1}{R_1} - \frac{1}{R_2} \right) \left( \omega - \omega_0 \right)^2 + \frac{\omega}{\omega_0} l \phi.$$
The first term on the right-hand side of Eq. (29) means the total GDD of the laser pulse after the lens and the second term indicates the radially-varying GDD due to the thickness change. Since the focal length is given by Eq. (18), the spectral phase the laser pulse can be re-written as, (30)$$\Phi (\omega) \approx \frac{1}{2} \left[ \beta_2^T - \frac{k^{\prime\prime} \rho^2}{\left( n_0 - 1 \right) f_0} \right] \left( \omega - \omega_0 \right)^2 + \frac{\omega}{\omega_0} l \phi.$$
Here, $\beta _2^T$ is used to express the total GDD given by $2k'' \Delta _0 + \beta _2$. Two cases can be considered for the dispersive focusing optic: one is a conventional singlet lens and the other is a specially-designed flat diffractive lens which has a large amount of GDD as in [11]. In the first case, the full expression for the spectral phase expressed by Eq. (30) should be used for the calculation of field distribution. However, in the second case, there is no radially-varying spectral phase term, i.e., $(1/2) \left ( k'' \rho ^2 \left ( \omega - \omega _0\right )^2 / \left ( n_0 - 1 \right ) f_0 \right ) = 0$, so a relatively simple expression with no radially-varying spectral phase term can be used to calculate the field distribution. As can be seen in the following subsection, the radially-varying spectral phase term modifies the spatial field distribution and provides a general solution of the field.3.2 Focused electric field in the focal region
The focused electric field distribution can be calculated through the diffraction integral. According to [33], under the paraxial approximation, the electric field in the focal region is given by the Fresnel integral as,
(31)$$dE(r,\theta,\phi;\omega) = \frac{ik}{2\pi} E_s (r_s,\theta_s,\phi_s;\omega) \frac{\exp \left({-}i\vec{k}\cdot \vec{u}\right)}{|\vec{u}|} dA_s,$$
where $E(r,\theta ,\phi ;\omega )$ is the electric field in the focal region, $E_s \left ( r_s, \theta _s, \phi _s; \omega \right )$ the incident electric field on the focusing optic, $\vec {u}$ the displacement vector between a source point on the incident field and an observation point in the focal region, $A_s$ the surface enclosed by the focusing optic. So, the infinitesimal area, $dA_s$, on the focusing optic is given by $r_s^2 \sin \left ( \pi - \theta _s\right ) d \left ( \pi - \theta _s \right ) d \phi _s$. The range of polar angle, $\theta _s$, is $\left [ \theta _{s,\mathrm {min}},\pi \right ]$. The displacement vector, $\vec {u}$, is expressed as $\vec {r} - \vec {r}_s$. As shown in Fig. 3, $\vec {r}_s$ is the displacement vector from the origin to the source point and $\vec {r}$ the displacement vector from the origin to the observation point. In the following, it is shown how the time-dependent focal length is introduced to the Fresnel integral and how the field distribution of the flying focus is calculated with the time-dependent focal length.Since the amplitude of the electric field varies slowly than the phase, we write
(32)$$|\vec{u}| \approx r_s \approx f_0, \quad\mathrm{and}\quad \vec{k}\cdot\vec{u} \approx k \sqrt{r_s^2 + r^2 - 2r_s r \cos \gamma} \approx kr_s - kr \cos \gamma,$$
with the directional cosine of $\cos \gamma = \cos \theta \cos \theta _s + \sin \theta \sin \theta _s \cos \left ( \phi _s - \phi \right )$. The minimum value ($\theta _{s,\mathrm {min}}$) for $\theta _s$ determines the opening angle of the focusing optic. Under the paraxial approximation, the minimum angle satisfies the condition of $\pi - \theta _{s,\mathrm {min}} \ll 1$. By using the supplementary angle, $\vartheta _s \equiv \pi - \theta _s$, the infinitesimal area, $dA_s$, and the direction cosine can be approximated as (33)$$dA_s \approx f_0^2 \vartheta_s d \vartheta_s d \phi_s, \quad\mathrm{and}\quad \cos \gamma \approx{-}\cos \theta + \vartheta_s \sin \theta \cos \left( \phi_s - \phi \right).$$
The distance to the source, $|r_s|$, can be approximated by the focal length of $f_0$, so, the argument, $kr_s$, shown in Eq. (32) can be expressed as, (34)$$kr_s \approx k f_0 \approx \frac{\omega}{c} f(t) + \frac{\omega}{c} f_0 \Omega t.$$
Now, under the paraxial approximation, we replace the radius, $\rho$, and the beam radius, $\rho _0$, expressed in cylindrical coordinates with $\rho = r_s \sin \left ( \pi - \theta _s \right ) \approx f_0 \vartheta _s$ and $\rho _0 = r_s \sin \left ( \pi - \theta _0 \right ) \approx f_0 \vartheta _0$. Then, the spatial distribution of the incident electric field and the spectral phase can be approximated as, (35)$$E_{0l}(\rho) \approx E_{0l} \left( \vartheta_s \right) \approx E_0 f_0^{|l|} \vartheta_s^{|l|} \exp \left( - \frac{\vartheta_s^2}{2 \vartheta_0^2} \right),$$
and (36)$$\Phi (\omega) \approx \frac{\beta_2^T}{2} \left( \omega - \omega_0 \right)^2 + \frac{\omega}{\omega_0} l \phi - \frac{k^{\prime\prime}_r}{2} \left( \omega - \omega_0 \right)^2 \vartheta_s^2.$$
Here, $\vartheta _0 = \pi - \theta _0$ and $k''f_0/\left ( n_0 - 1 \right )$ is replaced by $k''_r$. Again, considering the Gaussian spectrum, the spatio-temporal electric field for the chirped OAM laser pulse can be obtained by inserting above equations into Eq. (31) and by performing the Fourier transformation in the frequency domain as, (37)$$E(r,\theta,\phi;t) \approx i \frac{f_0^{|l|+1}E_0}{2 \pi c} \int_{-\infty}^{\infty} d \omega \tilde{F}(\omega),$$
with the definition of a complex function, $\tilde{F}(\omega )$, given by, (38)$$\begin{aligned} \tilde{F} (\omega) = &\omega \exp \left[ -\frac{\left( \omega - \omega_0 \right)^2}{2 \Delta \omega^2} \right] \exp \left\{{-}i \left[ \frac{\omega}{c} f(t) + \frac{\omega}{c} f_0 \Omega t \right] \right\} \\ & \times \exp \left[ i \omega \left( t - \frac{r \cos \theta}{c} \right) + i \frac{\beta_2^T}{2} \left( \omega - \omega_0 \right)^2 \right] \cdot I_{\theta\phi} (\omega), \end{aligned}$$
and the definite integral, $I_{\theta \phi }(\omega )$, of (39)$$\begin{aligned} I_{\theta\phi}(\omega) = & \int_{0}^{\theta_{s,\mathrm{min}}} \int_{0}^{2\pi} d \vartheta_s d \phi_s \vartheta_s^{|l|} \exp \left[ - \frac{\vartheta_s^2}{2\vartheta_0^2} -i \frac{k^{\prime\prime}_r \left( \omega - \omega_0 \right)^2}{2} \vartheta_s^2 \right] \\ & \times \exp \left[ i \frac{\omega}{c} r \vartheta_s \sin \theta \cos \left( \phi_s - \phi \right) \right] \exp \left( i \frac{\omega}{\omega_0} l \phi_s \right). \end{aligned}$$
From the Jacobi-Anger identity of $e^{ix\cos \phi } = \sum _{p=-\infty }^{\infty } i^p J_p (x) e^{ip\phi }$, the following identities of (40)$$\int_0^{2\pi} \exp(ip\phi) \exp(ix\cos \phi) d \phi = 2\pi i^p J_p (x),$$
and (41)$$\int_0^{2\pi} \exp(ip\phi) \exp \left[ ix\cos \left(\phi - \phi' \right) \right] d \phi = 2\pi i^p \exp \left( i p \phi' \right) J_p (x),$$
can be obtained. The integration of $I_{\theta \phi }(\omega )$ in the azimuthal angle, $\phi _s$, over $[0, 2 \pi ]$ yields (42)$$\begin{aligned} I_{\theta \phi}(\omega) &= 2 \pi i^l e^{il \phi} \\ & \times \sum_{m=0}^{\infty} \int_{0}^{\theta_{s,\mathrm{min}}} \frac{\vartheta_s^{|l|+2m+1}}{m!} \left[{-}i \frac{k^{\prime\prime}_r \left(\omega -\omega_0 \right)^2}{2} \right]^m \exp \left( - \frac{\vartheta_s^2}{2 \vartheta_0^2} \right) J_l \left( \frac{\omega}{c} r \vartheta_s \sin \theta \right) d \vartheta_s \end{aligned}.$$
Due to the symmetry of the Gaussian spectrum, it is reasonable to replace $\omega l \phi / \omega _0$ with $l \phi$ since $\int _{-\infty }^{\infty } d \omega \exp \left [ - \left ( \omega - \omega _0 \right )^2 / 2 \Delta \omega ^2 \right ] \exp (i \omega l \phi / \omega _0 ) \approx \exp (i l \phi ) \int _{-\infty }^{\infty } d \omega \exp \left [ - \left ( \omega - \omega _0 \right )^2 / 2 \Delta \omega ^2 \right ]$ when $\omega _0 \gg \Delta \omega$. The Taylor’s series expansion of $\exp \left ( -i a_0 \vartheta _s^2 \right ) = \sum _{m=0}^{\infty } \vartheta _s^{2m} \left ( -i a_0 \right )^m / m!$ is used in obtaining Eq. (42). Considering $\exp \left ( - \vartheta _s^2/2\vartheta _0^2 \right ) \ll 1$ when $\vartheta _s \gg \vartheta _0$, the domain $[0, \theta _{s,\mathrm {min}}]$ for the integral in Eq. (42) can be replaced by $[0,\infty ]$. Then, the integral identity of $\int _{0}^{\infty } x^{2 \mu + \nu +1} e^{-x^2} J_{\nu } \left ( 2x \sqrt {z} \right ) dx = \left ( \mu !/2 \right ) e^{-z} z^{\nu /2} L_{\mu }^{\nu } (z)$ [34] can be applied to obtain (43)$$\begin{aligned} I_{\theta \phi}(\omega) &\approx 4 \pi i^l \vartheta_0^{|l|+2} \left( \frac{\omega}{c} R_{\sin} \right)^{|l|} \exp \left( -\frac{1}{2} \frac{\omega^2}{c^2} R_{\sin}^2 \right) e^{il \phi} \\ & \times \sum_{m=0}^{\infty} ({-}i)^m \left( k^{\prime\prime}_r \vartheta_0^2 \right)^m \left( \omega - \omega_0 \right)^{2m} L_m^{|l|} \left( \frac{1}{2} \frac{\omega^2}{c^2} R_{\sin}^2 \right), \end{aligned}$$
with the definition of $R_{\sin }=r \vartheta _0 \sin \theta$. Then, from the definition of the f-number, $F_N$, given by the effective beam radius ($\rho '_0$), it is reasonable to consider $\vartheta _0$ as $1/4F_N$ for the Laguerre-Gaussian beam profile. When a flat and diffractive focusing optic with no radially-varying GDD is considered ($k''_r=0$), Eq. (43) is reduced to (44)$$I_{\theta \phi}(\omega) \approx \frac{4 \pi i^l}{\left( 4F_N \right)^{|l|+2}} \left( \frac{\omega}{c} R_{\sin} \right)^{|l|} \exp \left( -\frac{1}{2} \frac{\omega^2}{c^2} R_{\sin}^2 \right) e^{il \phi}.$$
The general solution with the radially-varying GDD has been discussed in Sec. III of Supplement 1. However, even with the radially-varying GDD, Eq. (42) can be approximated to Eq. (44) again under the paraxial approximation satisfying the condition of $k''_r \left ( \omega - \omega _0 \right )^2 / 16 F_N^2 \ll 1$.Now, by using the relationship of $\omega = \omega _0 + \tilde{\omega }$, Eq. (34) can be re-written as,
(45)$$kr_s \approx \frac{\omega_0}{c} f_0 + \frac{\tilde{\omega}}{c} f_0 + \frac{\tilde{\omega}}{c} f_0 \Omega t - 8 \sqrt{3}F_N \frac{f_0}{c} \frac{c_d}{\rho'_0} \tilde{\omega}^2.$$
Then, Eq. (46) can be written with a new parameter, $T'=t-f_0 \Omega t/c -r \cos \theta /c -f_0 /c$, as follows: (46)$$\begin{aligned} \tilde{F}(\tilde{\omega}) = &\frac{4 \pi i^l}{\left( 4F_N \right)^{|l|+2}} \left( \omega_0 + \tilde{\omega} \right)^{|l|+1} \left( \frac{R_{\sin}}{c} \right)^{|l|} \exp \left( -\frac{1}{2} \frac{\omega_0^2}{c^2} R_{\sin}^2 \right) e^{i \varphi(t) } \\ & \times \exp \left[ i \delta_3 \tilde{\omega}^3 - \frac{1}{2} \left( \frac{1}{\Delta \omega^2} + \frac{R_{\sin}^2}{c^2} - i \beta_2^T - i \frac{2f_0}{c} \frac{1+D_2^2}{D_2 \Delta \omega^2} \Omega \right) \tilde{\omega}^2 + i \tilde{\omega} \left( T' +i \frac{\omega_0 R_{\sin}^2}{c^2} \right) \right] \\ & \times \sum_{m=0}^{\infty} ({-}i)^m \left( \frac{ k^{\prime\prime}_r}{16F_N^2} \right)^m \tilde{\omega}^{2m} L_m^{|l|} \left[ \frac{1}{2} \left( \omega_0 + \tilde{\omega} \right)^2 \frac{R_{\sin}^2}{c^2} \right]. \end{aligned}$$
Here, $\varphi (t)$ is $\varphi _0(t) + l\phi = \omega _0 \left ( t - f_0 /c - r \cos \theta /c \right ) + l \phi$, and $8 \sqrt {3} F_N c_d / \rho '_0$ is replaced by $\Omega \left ( 1+D_2^2 \right )/D_2 \Delta \omega ^2$. Since $\omega _0 \gg \tilde{\omega }$ in the spectral range, we replace $\left ( \omega _0 + \tilde{\omega }\right )^{|l|+1}$ by $\omega _0^{|l|+1}$, and introduce new complex parameters, $\tilde{\alpha }'$ and $\tilde{T}'$, defined by, (47)$$\frac{1}{\tilde{\alpha}'}=\frac{1}{2} \left( \frac{1}{\Delta \omega^2} + \frac{R_{\sin}^2}{c^2} - i \beta_2^T - i \frac{2f_0}{c} \frac{1+D_2^2}{D_2 \Delta \omega^2} \Omega \right), \quad\mathrm{and}\quad \tilde{T}' = T' + i \frac{\omega_0 R_{\sin}^2}{c^2},$$
then Eq. (46) can be simplified as, (48)$$\begin{aligned} \tilde{F}(\tilde{\omega}) = & \frac{4 \pi i^l}{\left( 4F_N \right)^{|l|+2}} \omega_0^{|l|+1} \left( \frac{R_{\sin}}{c} \right)^{|l|} \exp \left( -\frac{1}{2} \frac{\omega_0^2}{c^2} R_{\sin}^2 \right) \exp \left( - \frac{\tilde{\omega}^2}{\tilde{\alpha}'} + i \tilde{\omega} \tilde{T}' \right) e^{i \varphi(t)} \\ & \times \sum_{p=0}^{\infty} \frac{(i \delta_3)^p}{p!} \tilde{\omega}^{3p} \sum_{q=0}^{\infty} ({-}i)^q \left( \frac{k^{\prime\prime}_r}{16F_N^2} \right)^q \tilde{\omega}^{2q} L_q^{l} \left[ \frac{1}{2} \left( \omega_0 + \tilde{\omega} \right)^2 \frac{R_{\sin}^2}{c^2} \right] \end{aligned},$$
with $\exp \left ( i \delta _3 \tilde{\omega }^3 \right ) = \sum _{p=0}^{\infty } \left ( i \delta _3 \right )^p \tilde{\omega }^{3p}/p!$. As a simple case, a flat and diffractive focusing optic with no radially-varying GDD is considered. In this case, since only $q=0$ term survives, the spatio-temporal electric field distribution can be calculated from (49)$$\begin{aligned} &E(r,\theta,\phi;t) \approx i\frac{f_0^{|l|+1}E_0}{2 \pi c} \int_{-\infty}^{\infty} d \tilde{\omega} \tilde{F} (\tilde{\omega}) \\ &= i^{l+1} \frac{2k_0 f_0^{|l|+1}E_0 e^{i \varphi (t)}}{\left( 4F_N \right)^{|l|+2}} LG_0^{l} \left( -\frac{\omega_0^2 R_{\sin}^2}{2 c^2} \right) \int_{-\infty}^{\infty} d \tilde{\omega} \sum_{p=0}^{\infty} \frac{\left( i \delta_3 \right)^p}{p!} \tilde{\omega}^{3p} \exp \left( -\frac{\tilde{\omega}^2}{\tilde{\alpha}'} + i \tilde{T}' \tilde{\omega} \right). \end{aligned}$$
Here, the expression for the Laguerre-Gaussian beam mode, (50)$$LG_0^{l} \left( -\frac{1}{2} \frac{\omega_0^2}{c^2} R_{\sin}^2 \right) = \left( \frac{\omega_0}{c} R_{\sin} \right)^{|l|} \exp \left( - \frac{1}{2} \frac{\omega_0^2}{c^2} R_{\sin}^2 \right),$$
is used in Eq. (49). After taking a similar mathematical procedure in Sec. I of Supplement 1, the final and general expression for the spatio-temporal electric field distribution of a superluminal/subluminal OAM fs laser focus can be expressed in terms of Laguerre polynomial functions as (51)$$E(r,\theta,\phi;t) \approx i^{l+1} \left(-\frac{1}{2}\right)! \frac{2\sqrt{\tilde{\alpha}'}k_0 f_0^{|l|+1} E_0 e^{i\varphi (t)}}{\left( 4 F_N \right)^{|l|+2}} LG_0^{l} \left( -\frac{\omega_0^2 R_{\sin}^2}{2c^2} \right) \exp \left( - \tilde{\tau}^2 \right).$$
In Eq. (51), a complex variable, $\tilde {\tau }$, is defined as (52)$$\tilde{\tau} = \frac{\sqrt{\tilde{\alpha}'} \tilde{T}'}{2} \approx \frac{\sqrt{2} \Delta \omega \sqrt{T^{'2} + \omega_0^2 R_{\sin}^4 /c^4}}{2 \sqrt[4]{\left( 1+ R_{\sin}^2 \Delta \omega^{2} / c^{2} \right)^{2} + \left( D_2^{T} + D_{\mathrm{LCA}} \right)^{2}}} \exp \left( i \frac{\pi}{4} +i \tan^{{-}1} \frac{\omega_0 R_{\sin}^2}{c^{2} T'} \right).$$
Here, $D_2^T$ and $D_{\mathrm {LCA}}$ are defined as $\beta _2^T \Delta \omega ^2$ and $\left ( 2f_0/c \right ) \cdot \left [ \left (1+D_2^2 \right ) /D_2 \right ] \cdot \Omega$, respectively. For a linearly-chirped ($\delta _3 = 0$) OAM fs laser focus, the spatio-temporal electric field can be expressed by the first term ($m=0$) as, (53)$$\begin{aligned} E&(r,\theta,\phi;t) \approx i^{l+1} \frac{2 k_0 f_0^{|l|+1} E_0 }{\left( 4 F_N \right)^{|l|+2}} \frac{\sqrt{2 \pi} \Delta \omega}{\sqrt[4]{\left( 1 + R_{\sin}^2 \Delta \omega^2 / c^2 \right)^2 + \left( D_2^T + D_{\mathrm{LCA}}\right)^2}} \\ &\times LG_0^{|l|} \left(-\frac{\omega_0^2 R_{\sin}^2}{2c^2} \right) e^{il\phi} \exp \left[ i \varphi_0 (t) + i \frac{1}{2} \tan^{{-}1} \left( \frac{D_2^T + D_{\mathrm{LCA}}}{1+R_{\sin}^2 \Delta \omega^2 / c^2} \right) \right] \\ & \times \exp \left\{ -\frac{\Delta \omega^2 \left[ \left( T^{'2} - \omega_0^2 R_{\sin}^4 / c^4 \right) \left( 1+ R_{\sin}^2 \Delta \omega^2 /c^2 \right) + 2T' \omega_0 \left( R_{\sin}^2 /c^2\right) \left(D_2^T + D_{\mathrm{LCA}} \right) \right]}{2 \left[ \left( 1+R_{\sin}^2 \Delta \omega^2 /c^2 \right)^2 + \left( D_2^T + D_{\mathrm{LCA}} \right)^2 \right]} \right\} \\ & \times \exp \left\{{-}i\frac{\Delta \omega^2 \left[ \left( T^{'2} - \omega_0^2 R_{\sin}^4 / c^4 \right)\left(D_2^T + D_{\mathrm{LCA}} \right) - 2T' \omega_0 \left( R_{\sin}^2 /c^2\right) \left( 1+ R_{\sin}^2 \Delta \omega^2 /c^2 \right) \right]}{2 \left[ \left( 1+R_{\sin}^2 \Delta \omega^2 /c^2 \right)^2 + \left( D_2^T + D_{\mathrm{LCA}} \right)^2 \right]} \right\}. \end{aligned}$$
From Eq. (53), it is evident that the FWHM pulse duration, $\tau _{\mathrm {FWHM}}$, for a laser focus is given by (54)$$\tau_{\mathrm{FWHM}} = \frac{2}{\Delta \omega} \sqrt{\left[ \left( 1 + R_{\sin}^2 \Delta \omega^2/c^2 \right)^2 + \left( D_2^T + D_{\mathrm{LCA}} \right)^2 \right] \ln 2}.$$
and it can be controlled by GDD ($D_2^T$) and LCA ($D_{\mathrm {LCA}}$) parameters.Now, it is interesting to calculate the speed of laser focus intensity. The center-of-mass (CoM) speed of the laser focus intensity can be considered as the speed of the laser focus. Since the laser focus propagates along the z-axis, the CoM of the laser focus can be calculated by,
(55)$$z_{\mathrm{CM}} = \frac{\int_{-\infty}^{\infty} z E^2 (r, \theta, \phi;t) dz}{\int_{-\infty}^{\infty} E^2 (r, \theta, \phi;t) dz}.$$
After calculating Eq. (55) with $z=r \cos \theta$, the location, $z_{\mathrm {CM}}$, of CoM of the laser focus becomes (56)$$z_{\mathrm{CM}} = ct-f_0 \Omega t -f_0 - \zeta_0 c,$$
where $\zeta _0$ is a constant defined as, (57)$$\zeta_0 = \frac{\omega_0}{\Delta \omega^2} \frac{R_{\sin}^2}{c^2} \left( D_2^T +D_{\mathrm{LCA}} \right).$$
Its time derivative defining the speed, $v_{z,\mathrm {CM}}$, of the CoM becomes (58)$$v_{z,\mathrm{CM}}=\frac{dz_{\mathrm{CM}}}{dt} = c \left(1- \frac{f_0}{c} \Omega \right) = c \left[ 1 - 16 \sqrt{3} F_N^2 \frac{\Delta \omega^2}{c} \frac{c_d D_2^T}{1+ \left( D_2^T \right)^2} \right].$$
Equation (58) shows how the CoM speed becomes superluminal ($|v_{z,\mathrm {CM}}|>c$) or subluminal ($|v_{z,\mathrm {CM}}|<c$), depending on the chromaticity ($c_d$) and GDD ($D_2^T$) parameters. It is obvious, from Eq. (58), that in general the normal dispersion ($c_d > 0$) focusing optic generates a superluminal laser focus with a negatively-chirped ($D_2^T$ or $\beta _2^T <0$) laser pulse and the abnormal one ($c_d < 0$) does with a positively chirped ($D_2^T$ or $\beta _2^T >0$) laser pulse. At a given chromaticity parameter, the CoM speed has the maximum value of $v_{z,\mathrm {CM,max}} = c \left ( 1 - 8 \sqrt {3} F_N^2 c_d \Delta \omega ^2 /c \right )$ at $D_2^T = \pm 1$ (e.g., $\beta _2^T = \pm 1/\Delta \omega ^2$) in the forward and backward directions. In the range of $-1< D_2^T <1$, the speed decreases from the maximum value and becomes zero (stationary) when $16\sqrt {3} F_N^2 c_d D_2^T \Delta \omega ^2 / c \left [ 1 + \left ( D_2^T \right )^2 \right ]=1$. In two extreme cases ($|D_2^T| \gg 1$ and $|D_2^T| \ll 1$), the speed of the laser focus can be expressed as (59)$$v_{z,\mathrm{CM}} \approx \begin{cases} c \left( 1 - 16 \sqrt{3} F_N^2 \frac{c_d}{D_2^T} \frac{\Delta \omega^2}{c} \right), & \text{for $|D_2^T| \gg 1$} \\ c \left( 1 - 16 \sqrt{3} F_N^2 c_d D_2^T \frac{\Delta \omega^2}{c} \right), & \text{for $|D_2^T| \ll 1$} \end{cases}$$
Since the minimum pulse duration is obtained at $D_2^T = -D_{\mathrm {LCA}}$ from Eq. (54), the speed of the laser focus with the minimum pulse duration is given by, (60)$$v_{z,\mathrm{CM},\mathrm{min}} = c \left(1 + 16 \sqrt{3} F_N^2 \frac{\Delta \omega^2}{c} \frac{c_d D_{\mathrm{LCA}}}{1 + D_{\mathrm{LCA}}^2} \right).$$
Figure 4 shows the CoM speed and the pulse duration as a function of the total GDD ($\beta _2^T$). An f-number of 10 was used to ensure the paraxial approximation. An input laser pulse has a spectral bandwidth of 150 nm, yielding a transform-limited pulse duration of 4 fs. A Gaussian beam profile with $l=0$ is used. The chromaticity parameters, $c_d$, of $\sim \pm$4.5$\times$10$^{-16}$ $\mu$m$\cdot$s are chosen to show how the CoM speed and pulse duration change with GDD. The magnitude of the chromaticity parameter corresponds to one-third of $c_d$ for the low dispersion N-FA51A substrate. The maximum speed of $\sim \pm$410$\times$c is obtained at $D_2^T = \pm 1$ condition. Under this condition, the pulse duration of the stretched pulse becomes $\sim$6.4 ps with a stretching factor of $\sim$1630. The minimum pulse duration condition is satisfied at GDDs of $\pm$8360 fs$^2$. In this case, due to the combination of the pulse broadening due to GDD and the pulse shortening effect due to LCA, the pulse duration is restored as short ($\sim$4 fs) as the transform-limited pulse duration. The CoM speed of 1.5$\times$c is obtained, providing a superluminal fs laser focus.Two main differences can be found when comparing Eq. (58) with previous results [10,11]. First, in the previous results, the speed of laser focus becomes zero when the LCA of a focusing optic becomes zero ($\alpha =0$ in Eq. (3) of [10] or $dz/d \lambda =0$ in Eq. (1) of [11]). These results sound contradictory to the fact that the speed of a laser focus should propagate at a speed of c when it is focused by a parabolic mirror with no LCA. However, the speed of laser focus given by Eq. (58) becomes a speed of light (c) when LCA becomes zero ($c_d =0$). This result complies with the speed of the laser focus obtained with the parabolic mirror. Second, previous results in [10,11] claim that the speed of the laser focus becomes infinite under certain GDD and LCA conditions, resulting in a wide intensity spread along the propagation direction. However, the speed given by Eq. (58) shows that there are a maximum speed and a zero (stationary) speed. The condition for the infinite speed is exactly the same as the condition for the stationary focus under $|D_2^T| \gg 1$ in our case. Since the CoM speed is considered as the speed of the laser focus, the nomenclature of "infinite speed" denoting the wide intensity spread with a large GDD represents a stationary focus in our case.
It is interesting to form a superluminal OAM fs laser focus for further investigation on SF-QED in the superluminal regime, where the interaction between electron spin and photon OAM matters. Figure 5 shows snapshots of the electric field for a superluminal OAM ($l=1$) fs laser focus propagating along the z-axis. A chromaticity parameter, $c_d$, of $\sim +$1.36$\times$10$^{-15}$ $\mu$m$\cdot$s (assuming the use of N-FK51A focusing optic) and a total GDD of -25090 fs$^2$ are chosen for the minimal pulse broadening to maintain the pulse duration as short as the transform-limited pulse duration. The speed of the focus is calculated to be 1.5$\times$c and the superluminal OAM fs laser focus propagates 13.5 $\mu$m in 30 fs.
4. Conclusion
The analytical formula for the electric field distribution of the superluminal/subluminal OAM ultrashort laser focus has been derived under the paraxial approximation condition. The formula shows that the speed and the pulse duration of the laser focus are functions of chromaticity and GDD parameters. It has been generally thought that a superluminal or subluminal laser pulse should have a longer pulse duration more than 100 $\times$ transform-limited pulse duration. In this paper, due to the combined effect between the pulse broadening by GDD and the pulse shortening by LCA, it is shown that the generation of a superluminal fs laser focus is possible. The analytical formula is applied to generate a super-intense superluminal OAM fs laser focus and it can be used to study SF-QED phenomenon, in which the interaction between electron spin angular momentum and photon OAM matters.
Funding
European Regional Development Fund (CZ.02.1.01/0.0/0.0/15 003/0000449).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
Supplemental document
See Supplement 1 for supporting content.
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