Exact solutions for the electromagnetic fields of a flying focus

The intensity peak of a “flying” focus travels at a programmable velocity over many Rayleigh ranges while maintaining a near-constant profile. Assessing the extent to which these features can enhance laser-based applications requires an accurate description of the electromagnetic fields. Here we present exact analytical solutions to Maxwells equations for the electromagnetic fields of a constant-velocity flying focus, generalized for arbitrary polarization and orbital angular momentum. The approach combines the complex source-point method, which transforms multipole solutions into beamlike solutions, with the Lorentz invariance of Maxwell's equations. Propagating the fields backward in space reveals the space-time profile that an optical assembly must produce to realize these fields in the laboratory. Comparisons with simpler paraxial solutions provide conditions for their reliable use when modeling a flying focus.

Figure 1

A schematic of the theoretical approach. (a) The approach starts with a multipole spherical (top) or hyperbolic (bottom) solution. (b) A displacement of the coordinate $z$ (top) or $t$ (bottom) into its complex plane transforms the multipole solution into a beamlike solution with a stationary focus in space (top) or time (bottom). (c) A Lorentz transformation of either beamlike solution from a frame of reference in which the foci appear stationary to the laboratory frame produces the exact electromagnetic fields of a flying focus.

Figure 2

A subluminal focus with ${\beta }_I=0.5$, $n=\ell =0$, and ${\kappa }^{\prime }{w}_0=20$. [(a)–(f)] Cross sections of the electromagnetic field amplitudes at the location of the moving focus $z^{\prime }={\beta }_I{t}^{\prime }$. Here $\widetilde{E^{\prime }}={\langle E^{\prime 2}\rangle }^{1/2}$, where $\langle \rangle$ denotes a cycle average. The amplitudes are normalized to the $\max \left(\widetilde{E_x^{\prime }}\right)$. (g) The longitudinal component of the cycle-averaged Poynting vector $S_z^{\prime }$ at $\rho =0$, showing the motion of the focus. The dashed black line demarcates the trajectory $z^{\prime }=t^{\prime }$ for reference.

Figure 3

A backward focus carrying orbital angular momentum with ${\beta }_I=-0.99$, $n=\ell =1$, and ${\kappa }^{\prime }{w}_0=20$. [(a)–(f)] Cross sections of the electromagnetic field amplitudes at the location of the moving focus $z^{\prime }={\beta }_I{t}^{\prime }$. Here $\widetilde{E^{\prime }}={\langle E^{\prime 2}\rangle }^{1/2}$, where $\langle \rangle$ denotes a cycle average. The amplitudes are normalized to the $\max \left(\widetilde{E_x^{\prime }}\right)$. (g) The longitudinal component of the cycle-averaged Poynting vector $S_z^{\prime }$ at $\rho =0$, showing the motion of the focus. The dashed black line demarcates the trajectory $z^{\prime }=t^{\prime }$ for reference.

Figure 4

A circularly polarized superluminal focus with ${\beta }_I=2$, $n=\ell =0$, and ${\kappa }^{\prime }{w}_0=20$. [(a)–(f)] Cross sections of the electromagnetic field amplitudes at the location of the moving focus $z^{\prime }={\beta }_I{t}^{\prime }$. Here $\widetilde{E^{\prime }}={\langle E^{\prime 2}\rangle }^{1/2}$, where $\langle \rangle$ denotes a cycle average. The amplitudes are normalized to the $\max \left(\widetilde{E_x^{\prime }}\right)$. (g) The longitudinal component of the cycle-averaged Poynting vector $S_z^{\prime }$ at $\rho =0$, showing the motion of the focus. The dashed black line demarcates the trajectory $z^{\prime }=t^{\prime }$ for reference.

Figure 5

A pulsed subluminal focus constructed from a superposition of modal solutions with ${\beta }_I=0.5$, $n=\ell =0$, ${\kappa }_0^{\prime }{w}_0=20$, and $T=10Z_R^{\prime }$ ($L=10Z_R^{\prime }$). (a) The longitudinal component of the cycle-averaged Poynting vector $S_z^{\prime }$ at $\rho =0$, showing the motion and finite extent of the focus [cf. Fig. 2]. The dashed black lines mark the full width at half maximum of the pulse ${\widehat{g}}_{00}$, which travels at the speed of light. [(b)–(d)] The profile of the longitudinal Poynting vector in the $x-z^{\prime }$ plane at $t^{\prime }=-10Z_R^{\prime }$, 0, and $10Z_R^{\prime }$. The right and left edges of the focus at $t^{\prime }=-10Z_R^{\prime }$ and $10Z_R^{\prime }$ are clipped by the front and rear edges of the pulse, respectively, leading to the asymmetric profile in $z^{\prime }$.

Figure 6

Propagating the fields backward to $z^{\prime }=-150Z_R^{\prime }$ provides (a) the slow phase ${\mathrm{\Theta }}_s$, (b) the corresponding time-dependent focal length $f$, and (c) the amplitude required to create a flying focus pulse with ${\beta }_I=0.5$, $n=\ell =0$, ${\kappa }_0^{\prime }{w}_0=20$, and $L=10Z_R^{\prime }$ (the same parameters as in Fig. 4). In panels (a) and (c) the dashed lines illustrate the parabolic shape of ${\mathrm{\Theta }}_s$ and the on-axis amplitude, respectively. The transverse dimension is normalized to the spot size of a standard Gaussian beam at the same location: $w=w_0{[1+{(z^{\prime }/Z_R^{\prime })}^2]}^{1/2}$. More generally, a flying focus can be created by using a lens with focal length that depends linearly on time: $f\left(t^{\prime }\right)=f_0+f_1{t}^{\prime }$, where $f_1$ is given by Eq. (22).

Figure 7

One minus the projection integral [Eq. (29)] as a function of the focal velocity ${\beta }_I$ for different values of spot size $w_0$. The paraxial solutions are an excellent approximation to the exact solutions everywhere except for a small interval around ${\beta }_I=1$.