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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 Maxwell's 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 beam-like 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.

experienced by the pulse in the Kerr lens provides the bandwidth necessary to support the duration of the intensity peak. The "achromatic" flying focus combines an axiparabola [7] with a radial echelon to control the focal location and relative timing of each annulus, respectively [5]. As the annuli come in and out of focus, they interfere to form an intensity peak with a duration equal to that of the initial pulse.
The programmable velocity v I and extended focal range L of a flying focus offer new approaches to realizing or optimizing laser-based applications. The intensity peak of a flying focus pulse can travel slower than the group velocity; faster than the group velocity, i.e., superluminally; or backward with respect to the phase fronts of the pulse. Superluminal intensity peaks have been proposed to overcome dephasing and wave breaking in laser wakefield acceleration [5,[8][9][10] and to increase the rate of frequency upshifting in photon acceleration [11]. Backward intensity peaks can facilitate the formation of long plasma channels by mitigating ionization refraction [12,13] and may improve the performance of Raman amplifiers by ensuring quasi-stationary plasma conditions [14]. The motion of a backward intensity peak against its phase fronts also allows for longer interaction lengths in fundamental studies of nonlinear Compton scattering and radiation reaction, which can amplify observable signatures of these processes [15,16]. Further, a backward intensity peak can ponderomotively accelerate electrons to relativistic momenta in the backward direction, providing unprecedented control over the electron trajectory and greatly enhancing the radiation properties in nonlinear Thomson scattering [17,18].
Assessing the extent to which a flying focus can enable or enhance these applications requires an accurate description of the electromagnetic fields. With the exception of the special case v I = −c [15,16], all of the aforementioned applications were modeled using approximate solutions for the electromagnetic fields of flying focus pulses. In the case of conventional pulses with stationary foci, improving the accuracy of approximate solutions has been found to impact models of phenomena ranging from direct laser acceleration to optical trapping [19][20][21][22][23][24][25]. Methods for obtaining accurate solutions to Maxwell's equations for conventional laser pulses come in three forms: a "Lax"-like series expansion in which corrections to paraxial fields can be calculated recursively [26][27][28][29]; series expansions of exact spectral integrals for each field component [21,24,30]; and the complex source-point method (CSPM), which exploits the invariance of Maxwell's equations under a translation in the complex plane to transform multipole solutions into beam-like solutions [31][32][33][34][35][36][37][38]. Of these three, the CSPM is unique in its ability to provide closed-form solutions that exactly satisfy Maxwell's equations. As a result, the solutions can be Lorentz transformed without introducing additional error.
In this article, we derive exact solutions to Maxwell's equations for the electromagnetic fields of a constant-velocity flying focus pulse. The approach combines the CSPM with a Lorentz transformation from a frame in which the focus is stationary to a frame in which the focus is moving. The vector solutions are inherently non-paraxial, can have arbitrary polarization, and are generalized to higher-order radial and orbital angular momentum modes.
Subluminal (|v I | < c) and superluminal (|v I | > c) solutions are constructed from multipole spherical and hyperbolic wave solutions, respectively. Propagating the fields backward in space reveals that each solution corresponds to a pulse that was focused by a lens with a time-dependent focal length. Thus, these solutions can be experimentally realized using the flying focus X. For a wide range of parameters, the differences between the exact solutions and simpler paraxial solutions are small, justifying the use of paraxial solutions for theoretical or computational studies of flying focus applications in many regimes. These solutions also compliment the larger body of work on arbitrary velocity autofocusing and nondiffracting waves, e.g. Refs. [39][40][41].
The remainder of this article is organized as follows. In Sec. II, multipole solutions to Maxwell's equations are derived and then transformed into beam-like solutions using the CSPM. Guided by the insights of Refs. [42] and [43], the exact fields of subluminal and superluminal foci are found by Lorentz transforming the beam-like solutions. Section III presents explicit modal solutions for the four-potential of moving foci, including an example with orbital angular momentum. Section IV describes the procedure for constructing pulsed solutions from a superposition of modal solutions and determines the optical assembly required to produce the pulsed fields in an experiment. Section V compares the exact solutions to paraxial solutions and provides conditions for the reliable use of paraxial solutions when modeling a flying focus. Section VI summarizes the results.

II. LORENTZ TRANSFORMATIONS OF COMPLEX SOURCE POINT FIELDS
In vacuum, the electromagnetic fields, the four-potential in the Lorenz gauge, and the Hertz vectors, all satisfy the homogeneous wave equation. Consider a scalar solution u(x, t) to the wave equation, where the speed of light c = 1. The solution can be written as a superposition of modal solutions with explicit harmonic dependence in either time or space: u(x, t) = u κ (x, t) dκ, where u κ (x, t) = 1 2 S(x ⊥ , z)e −iκt + c.c. or u κ (x, t) = 1 2 H(x ⊥ , t)e iκz + c.c.. Substituting these into the wave equation yields H(x ⊥ , t) = n n =−n α n (κ)k n (κR)P n (cos ϕ)e i θ , where α n (κ) is a weighting factor, j n is the n th spherical Bessel function of the first kind, k n is the modified spherical Bessel function of the second kind, P n is the associated Legendre polynomial, is the azimuthal mode number, R = (ρ 2 + z 2 ) 1/2 , R = (ρ 2 − t 2 ) 1/2 , ρ = |x ⊥ |, whereê ·m = 0 andê ×m =ẑ [36]. With this configuration,ẑ andê determine the predominate directions of propagation and electric-field polarization, respectively.
A spherical or hyperbolic solution u κ (x, t) remains a solution to the homogeneous wave equation under a coordinate translation along the real or imaginary axis. Displacing a coordinate into its complex plane transforms a multipole spherical or hyperbolic wave into a beam-like wave, in which the phase fronts pass through the origin instead of originating from it [ Fig. 1(b)] [31][32][33][34][35][36][37][38]. This is the CSPM. For the spherical solutions, a beam-like wave is generated by transforming the axial coordinate as z → z − iZ R , such that S(x ⊥ , z) → S(x ⊥ , z − iZ R ). For hyperbolic solutions, a beam-like wave is generated by transforming In the paraxial limit, i.e., when |z − iZ R | or |t − iZ R | ρ, the minimum spot size of the beam-like wave is given by w 0 = (2Z R /κ) 1/2 , thus Z R = 1 2 κw 2 0 corresponds to the Rayleigh range. When working with beam-like waves, it is convenient to introduce the complex beam parameter q. In the context of spherical and hyperbolic solutions, q(z) = z − 1 2 iκw 2 0 and q(t) = t − 1 2 iκw 2 0 , respectively.
Electromagnetic fields that satisfy Maxwell's equations in one frame of reference satisfy Maxwell's equations in all other inertial reference frames. Therefore, there exists a frame of reference in which the focus appears to be moving at a velocity v I = cβ I . In the context of a flying focus, this frame with a moving focus is the laboratory frame.
When performing a Lorentz transformation from the stationary frame to the laboratory frame, it is convenient to work with the four-potential A µ = (Φ, A). The four-potential in the stationary frame can be calculated from the Hertz vectors in Eq. (5) as follows [44]: Because the Hertz vectors are formulated in the Lorenz gauge [45], the condition ∇·A+∂ t Φ = 0 is automatically satisfied. Further, the relationship Π m =ẑ × Π e implies that A z = −Φ, The four-potential in the laboratory frame (denoted by a prime ) is given by Note that in the laboratory frame, the stationary frame appears to be moving at a velocity −β. The definition of this velocity and the mapping between (x, t) and (x , t ) depend on whether the focal velocity is subluminal |β I | < 1 or superluminal |β I | > 1.
Upon Lorentz transforming to the laboratory frame, the spherical solutions describe foci that move at subluminal velocities |β I | < 1. In this case, β = β I and the coordinates transform as where γ < = (1 − β 2 I ) −1/2 and κ = γ < (1 + β I )κ is the laboratory frame value of κ. The hyperbolic solutions describe foci that move at superluminal velocities |β I | > 1. Clearly a Lorentz transformation using a |β| > 1 would be unphysical. Nevertheless, a superluminal focus can be achieved by Lorentz transforming the hyperbolic solutions using β = β −1 I , such that where, in this case, where Z R = 1 2 κ w 2 0 is the Rayleigh range in the laboratory frame and β I can take any value other than 1. The duration is identical to that of an intensity peak produced by a lens with a focal length that depends linearly on time, as in the flying focus X [4].
Once the potentials have been calculated using Eqs. (5) -(9), the electromagnetic fields can be found in the usual way: E = −∇ Φ − ∂ t A and B = ∇ × A . As will be shown below, all six components of the resulting fields have nonzero values. Further, the transverse components of the electric and magnetic field that are perpendicular toê andm, respectively, are equal-a symmetry which is frequently sought-after in solutions to Maxwell's equations [24,28,37,46]. The relative amplitudes of the field components scale as For circular polarization, these two scalings are sufficient. Linear polarization has the additional scaling When β I = 0, these scalings reduce to those of a stationary focus. In the limit as β I → 1, the duration τ becomes shorter than 2π/κ , and the components of the electric and magnetic fields orthogonal toê andm vanish.

III. EXPLICIT SOLUTIONS
This section presents expressions for the four-potential of arbitrary-velocity subluminal and superluminal foci. The expressions exactly satisfy the wave equation. In each example, the four-potential is derived from a single modal solution u κ . Explicit expressions for each component of the electromagnetic field can be found directly from the four-potential, but are unwieldy and provide little additional insight. Instead, the structure of the electromagnetic fields is highlighted by figures.

A. Subluminal focus
Expressions for the four-potential of subluminal foci are obtained from the spherical solutions. As a first example, consider the lowest order radial and azimuthal mode (n = = 0) in Eq. (3a). Upon choosing the polarization vectorê =x, the electric and magnetic Hertz vectors are given by and Π m =ẑ × Π e , respectively. After applying the CSPM, Eq. (6), and a Lorentz transformation to Eq. (13), one finds the laboratory frame four-potential A y = 0, and A z = −Φ , where κ = γ < (1 − β I )κ and R = ρ 2 + q 2 (z) with q(z) and t given by Eq. (8). For a stationary focus with β I = 0, the fields derived from Eq. (14) are identical to those in Ref. [37]. As expected from Eqs. (8) and (10), the peak of S z travels at the velocity β I and has a duration τ = Z R . For comparison, the dashed black line demarcates the speed of light trajectory z = t .
A moving focus carrying orbital angular momentum can be described by any solution with | | > 0. The spherical solution with = n = 1 will be used as an example. The electric and magnetic Hertz vectors for this mode are given by and Π m =ẑ × Π e , whereê =x has been chosen for the polarization vector. Following the same procedure described above provides A y = 0, and A z = −Φ .  andẑ × Π e , respectively. Here circular polarization, i.e.,ê = 1 √ 2 (x + iŷ), has been chosen to demonstrate the generality of the solutions to describe polarizations other than linear.

C. Luminal focus
Expressions for the four-potential of luminal foci can be found by taking the limit of the subluminal and superluminal solutions as |β I | → 1. The limit as |β I | → 1 is identical from above and below, and the sub-and superluminal solutions reduce to the same expression.
When β I = −1, the transverse vector potential exactly satisfies a paraxial wave equation, and thus can be expressed as a Laguerre-Gaussian mode (see Sec. V). Taking the limit of the linearly polarized n = = 0 sub or superluminal solution as β I → 1 yields and Φ = A z = A y = 0. Equation (19) is independent of the transverse coordinates, i.e., it is a plane wave with E x = B y and all other field components equal to zero [see Eqs. (11) and (12)].
As β I → 1, the complex coordinate introduced in the CSPM approaches the real axis.
The absence of the imaginary term in Eqs. (8) and (9)  To begin, note that an A x derived from a single modal solution always takes the form of c.c. for superluminal foci. The space-time location of the maximum of |f n | is insensitive to κ (and is fully independent of κ in the paraxial limit). As a result, the choice α n (κ) = a 0 g n (κ) max|f n | (20) ensures that a pulse with a subluminal or superluminal focus has a temporal profileĝ n (t) = g n (κ)e −iκt dκ or longitudinal profileĝ n (z) = g n (κ)e iκz dκ, respectively. Further, if |ĝ n (0)| = 1, then the transverse vector potential in the laboratory frame will have a max-imum amplitude of a 0 . To ensure that the moving focus has a near-constant profile and maximum Poynting vector over its entire trajectory,ĝ n should have a near-flattop profile.
The pulsed fields generated by a superposition of modal solutions exhibit a moving focus over a finite duration and spatial extent. The length L over which the focus persists, i.e., the focal range, is determined by the transit time ∆T of the focus through the entire pulse duration T . In vacuum, the pulse propagates at its group velocity c = 1, such that ∆T = Over the interval ∆T , the focus travels a distance L = |β I |∆T , providing Equation (21) demonstrates that the focal range L increases with T , approaches T as |β I | → ∞, and is much greater than T when β I ≈ 1. In addition, Eq. (21) can be combined with Eq. (10) to show that τ /T = Z R /L. Note that every wavenumber or frequency κ within the superposition will have a different Rayleigh length, 1 2 κ w 2 0 . For this section, the quantity Z R is defined in terms of the central frequency or wavenumber κ 0 , such that Z R = 1 2 κ 0 w 2 0 .  The pulsed electromagnetic fields can be propagated backward in space to determine the amplitude and phase that an optical assembly must produce to realize these fields in an experiment. When discussing the phase, it is convenient to define a "slow" phase Θ s which excludes the contribution from the carrier frequency, i.e., Θ s ≡ Θ + κ 0 t , where Θ is the total phase. Figure 6(a) shows the slow phase of the transverse electric field at z = −150Z R for the same parameters of Fig. 5. The slow phase decreases linearly in time and has a transverse profile that is nearly parabolic. This is equivalent to the phase imparted by a lens with a focal length that depends linearly on time, which is consistent with Eq. (10) and the flying focus X. The time-dependent focal length f (t ) can be extracted from the phase by equating Θ s = −κ 0 ρ 2 /2f (t ) [ Fig. 6(b)]. In general, one can show that where Higher-order contributions to f (t ), e.g., f 2 t 2 , drop as 1/z 2 or faster and thus are negligible in the near-field of the optical assembly.
The amplitude of the transverse electric field is plotted in Fig. 6(c). The transverse profile is nearly Gaussian with small deviations due to non-paraxiality. The temporal profile is nearly super-Gaussian, consistent withĝ n (t) ∝ exp[−(t/T ) 4 ], but has an observable taper at earlier times. The later time slices of the pulse are farther from their focus and therefore have slightly larger spot sizes than the earlier time slices. In the case of a superluminal focus, the taper would be reversed, i.e., the earlier time slices would be closer to their focus and have smaller spot sizes. From conservation of power, the change in spot size throughout the pulse is accompanied by a change in amplitude. Specifically, to leading order in Z R /z . Far from the central focus z = 0, the variation in amplitude is negligible and can be ignored for experimental purposes.
Note that all quantities in Eq. (24) are written in the laboratory frame. Upon performing the Galilean change of variables ξ = z − β I t and η = z − t , the modal solution for A ⊥ can be expressed as A ⊥ (x , t ) = 1 2 a ⊥ (x ⊥ , ξ )e iκ η ê + c.c., where the envelope a ⊥ satisfies to the paraxial wave equation (25), this condition can be reexpressed as Inequality (27) reveals that the paraxial approximation is accurate when β I ≈ −1 or when the variations of a ⊥ with respect to ξ are slow compared to the frequency. Under these conditions, the solutions to Eq. (25) can be approximated as Laguerre-Gaussian modes (LG p ), such that where L | | p is a generalized Laguerre polynomial, w(ξ ) = w 0 [1 + (ξ /ξ 0 ) 2 ] 1/2 , ξ 0 = |1 − β I |Z R , e =x has been chosen for the polarization vector, and the superscript P distinguishes paraxial solutions A P x from exact solutions A x .
In the special case of β I = −1, Eq. (25) is identical to Eq. (26), and the Laguerre-Gaussian modes are exact solutions [16]. For p = n = = 0, one can take the limit of the solutions in Sec. III as β I → −1 from above and below to show that A x = A P x . However, for n > 0 or | | > 0, there is not a one-to-one mapping between the exact solutions generated from Eq. In the more general case of |β I | = 1, the Laguerre-Gaussian modes are only approximate solutions. For n = = 0, the exact solutions derived in Secs. II and III approach the LG 00 mode in the paraxial limit. For n > 0, the exact solutions approach a superposition of multiple LG modes with the same value. The one-to-one correspondence when p = n = = 0 allows for direct comparison of the exact and paraxial solutions.
The similarity of the exact and paraxial solutions can be quantified using the projection integral: where denotes an average over the rapidly varying phase and A x and A P x are evaluated in the focal plane z = β I t . As defined, Γ depends only on β I and κ w 0 . The values of Γ ∈ [0, 1]. A value of Γ = 1 indicates that the vector potentials are identical, while a value of Γ = 0 indicates that they are orthogonal. The quantity A x A x was chosen for the projection because it is Lorentz invariant (recall from Sec. II that A µ A µ = A 2 ⊥ ). Figure 7 displays 1 − Γ as a function of β I for several κ w 0 values. Consistent with inequality (27), the paraxial solution provides an excellent approximation to the exact solution everywhere except for a small interval around β I = 1. The width of this interval narrows as κ w 0 increases, which reflects the departure from paraxiality when the field is more tightly focused. As expected, Γ → 1 as β I → −1.

VI. SUMMARY AND CONCLUSIONS
The flying focus belongs to a broader class of optical techniques for controlling the spacetime structure of laser pulses that also includes laser smoothing [47,48], light sheets [49][50][51], and spatiotemporal optical vortices [52][53][54]. Each of these has unique features that can lead to a deeper understanding of the fundamental properties of light or advance laser-based applications. The flying focus provides a cylindrically symmetric, programmable-velocity intensity peak that can travel distances far greater than a Rayleigh range while maintaining a near-constant profile [1,2,4,5]. These particular features can enable or enhance the broad range of applications that requires velocity matching and a high-intensity interaction over an extended distance. Assessing the extent to which a flying focus can improve these applications requires an accurate description of the electromagnetic fields.
Motivated by this requirement, this paper has presented a theoretical method for obtaining exact, closed form solutions for the electromagnetic fields of a constant-velocity flying focus. The method consists of three steps. In the first step, one finds multipole spherical or hyperbolic solutions to the wave equation that satisfy appropriate boundary conditions. These solutions are then converted into beam-like solutions using the complex source point method, i.e., by displacing one of the coordinates into its complex plane. The spherical and hyperbolic beam-like solutions have stationary foci in space and time, respectively. In the final step, the beam-like solutions are Lorentz boosted into a frame in which the foci appear to be moving. The spherical solutions produce subluminal foci, while the hyperbolic solutions produce superluminal foci. The method produces all six components of the electromagnetic field, does not impose a paraxial approximation, and is generalized for arbitrary orbital angular momentum and polarization.
Explicit expressions for the exact four-potential were provided to illustrate the structure of the solutions in select examples. The procedure for constructing pulsed solutions was then described. The pulsed solutions revealed that the peak of the Poynting vector travels at the focal velocity, whether it is positive or negative, while the sign of the Poynting vector always matches the direction of pulse propagation. Propagating the pulsed solutions backward in space demonstrated that the solutions describe a laser pulse focused by a lens with a focal length that depends linearly on time. As a result, these solutions may be generated in practice using a technique like the flying focus X [4]. Alternatively, a time-ordered sequence of pulses each with a different focal length could be used as in the ASTRL technique [6].
Finally, it was shown that even in tight focusing geometries, approximate paraxial solutions for the flying focus [18] can be accurate as long as the focal velocity is not too close to the speed of light.