¸ois Bacher
email: francois.bacher@univ-lille.fr.

HEAT DIFFUSIONS ON HOLOMORPHIC FOLIATIONS WITH NON-DEGENERATE SINGULARITIES

Keywords: Singular holomorphic foliation, Leafwise Poincaré metric, Heat diffusions, Directed positive harmonic current, Harmonic measure

Consider a Brody hyperbolic foliation with non-degenerate singularities on a compact complex manifold. We show that the leafwise heat diffusions and the abstract heat diffusions coincide. In particular, this will imply that the abstract heat diffusions are unique.

INTRODUCTION

The global dynamics of Riemann surface laminations have recently received much attention. Much progress has been focused on developing an adapted ergodic theory. This could be a powerful tool to understand the global behaviour of Riemann surface laminations. The case of the projective space is maybe the most typical, since polynomial vector fields can be compactified naturally into foliations on P n . Lins Neto and Soares in [START_REF] Neto | Algebraic solutions of one-dimensional foliations[END_REF] and Jouanolou in [START_REF] Jouanolou | Équations de Pfaff algébriques[END_REF] have shown that a generic foliation F of a given degree d ě 2 on P n has only non-degenerate singularities. By a result of Lins Neto [START_REF] Neto | Uniformization and the Poincaré metric on the leaves of a foliation by curves[END_REF] and Glutsyuk [START_REF] Glutsyuk | Hyperbolicity of the leaves of a generic one-dimensional holomorphic foliation on a nonsingular projective algebraic variety[END_REF], a foliation having only non-degenerate singularities is necessarily hyperbolic, and even Brody hyperbolic in the sense of [START_REF] Dinh | Entropy for hyperbolic Riemann surface laminations II[END_REF]. Therefore, many authors have studied hyperbolic Riemann surface laminations with singularities and in particular hyperbolic singular holomorphic foliations. We present briefly the results of some recent work and refer the reader to the surveys [START_REF] Dinh | Some open problems on holomorphic foliation theory[END_REF][START_REF] Fornaess | Riemann surface laminations with singularities[END_REF][START_REF] Nguyên | Ergodic theory for Riemann surface laminations: a survey[END_REF][START_REF] Nguyên | Ergodic theorems for laminations and foliations: recent results and perspectives[END_REF] for more details.

To set up an ergodic theory, one needs at least some notions of time, time average, space average (i.e. measure), invariant and ergodic measure. It turns out that for foliations, it will also be convenient to use currents instead of measures and forms instead of functions. In the case of hyperbolic foliations, the Poincaré distance in a universal covering can be seen as a canonical time. As an analogue to invariant measures, it is natural to consider directed closed currents. This has led to various results and dynamical methods (see for example Rebelo [START_REF] Rebelo | On closed currents invariant by holomorphic foliations[END_REF]), but a large class of foliations do not have any. That is why Garnett has considered the weaker notions of harmonic currents and harmonic measures in [START_REF] Garnett | Foliations, the ergodic theorem and Brownian motion[END_REF]. This approach has led to strong results, see for example the survey [START_REF] Fornaess | Riemann surface laminations with singularities[END_REF] of Fornaess and Sibony or the unique ergodicity result of Dinh, Nguyên, Sibony in [START_REF] Dinh | Unique ergodicity for foliations on compact Kähler surfaces[END_REF]. The notion of quasi-harmonic measure was also introduced by Nguyên in [START_REF] Nguyên | Oseledec multiplicative ergodic theorem for laminations[END_REF] to establish an Oseledec multiplicative ergodic theorem for laminations. Then, the natural processes to take the average of a function on the leaves (that is, on the orbits) becomes the heat diffusions. Two approaches to define such diffusions have been tried and both have given a series of ergodic theorems. It is natural to wonder whether these two approaches lead to the same object, that is, whether both heat diffusions coincide. Though the equations they satisfy look similar, it is not clear and not proven if this is true in general. In his survey [START_REF] Nguyên | Ergodic theorems for laminations and foliations: recent results and perspectives[END_REF], Nguyên has worked on this question. His result is the following. Theorem 1.1 [START_REF] Nguyên | Ergodic theorems for laminations and foliations: recent results and perspectives[END_REF]Corollary 5.23]). Let F " pM, L , Eq be a Brody hyperbolic compact singular holomorphic foliation. Suppose that all the singularities of F are linearizable and hyperbolic. Then, for every harmonic measure that does not give mass to any leaf, the abstract heat diffusions and the leafwise heat diffusions coincide. In particular, the abstract heat diffusions are unique, i.e., they do not depend on the considered harmonic measures.

He mentions as a question (see [START_REF] Nguyên | Ergodic theorems for laminations and foliations: recent results and perspectives[END_REF]Problem 5.24]) to find better sufficient conditions for the two heat diffusions to coincide. Our main result gives an improvement of Theorem 1.1.

Theorem 1.2. Let F " pM, L , Eq be a Brody hyperbolic compact singular holomorphic foliation. Suppose that all the singularities of F are non-degenerate. Then, for every harmonic measure, the abstract heat diffusions coincide with the leafwise heat diffusions. In particular, the abstract heat diffusions are unique, i.e., they are independent of the harmonic measures.

Let us explain the method of our proof. When we study non-degenerate singularities, we encounter more singular objects than in the case of linearizable weakly hyperbolic singularities. In our context, the current associated to the harmonic measure may have non-vanishing Lelong numbers on the singularities. Indeed, Chen has shown in [START_REF] Chen | Directed harmonic currents near non-hyperbolic linearized singularities[END_REF] that the Lelong number of a directed harmonic current on a non-hyperbolic singularity may be positive. On the other hand, Nguyên has shown in [START_REF] Viêt-Anh | Singular holomorphic foliations by curves. III: Zero Lelong numbers[END_REF] that it cannot if the singularity is linearizable and weakly hyperbolic. To deal with this new difficulty, we have to control the heat diffusions near the singularities. More precisely, given an initial heat distribution far from the singular set, we show that it essentially does not reach the neighbourhoods of a non-degenerate singularity in finite time (see Lemma 6.3). This result relies on estimates of the Poincaré metric near the singularities.

The article is organized as follows. In Section 2, we introduce the leafwise Poincaré metric g P . We specify the type of singularities we will consider. In the case where a foliation F " pM, L , Eq is also endowed with a Riemannian metric g M , we define the function η. It is a quantitative way to compare g P and g M so that η 2 g P " 4g M . This is the main tool we use to estimate the diffusions near the singularities. In Section 3, we describe the heat kernel and the leafwise heat diffusions, as well as their behaviours with respect to a uniformization of a leaf. In Section 4, we explain the link between harmonic measures and harmonic currents. In our context, we show that the Poincaré mass of a directed harmonic current is finite. In Section 5, we recall the construction of the abstract heat diffusions. We state some intermediate lemmas of [START_REF] Dinh | Heat equation and ergodic theorems for Riemann surface laminations[END_REF] to show its existence, because we will need them for our proof. Section 6 is devoted to the proof of Theorem 1.2. We state and prove an abstract criterion, analogous to [START_REF] Nguyên | Ergodic theorems for laminations and foliations: recent results and perspectives[END_REF]Theorem 5.17] in Theorem 6.1. In most of the article, we follow the method and notations of [START_REF] Nguyên | Ergodic theorems for laminations and foliations: recent results and perspectives[END_REF].

Notations. Throughout this paper, we will denote by D the unit disk of C and B the unit ball in C k . We denote by rD the disk of radius r in C and rB the ball of radius r in C k .

When we consider the hyperbolic distance in D, we denote by D R the disk of hyperbolic radius R in D, so that rD " D R for r " e R ´1 e R `1 or R " ln 1`r 1´r . If F " pM, L , Eq is a singular holomorphic foliation and x P M zE, we denote by L x the leaf of F through x. Moreover, if L x is hyperbolic, we denote by ϕ x : D Ñ L x a uniformization of L x such that ϕ x p0q " x.

Given a Hermitian metric g M on a complex manifold M , we denote by d M p¨, ¨q the distance induced by g M . Similarly, we denote by d P p¨, ¨q the Poincaré distance if one is given a Poincaré metric denoted g P .

Recall that d c " i 2π `B ´B˘s o that dd c " i π BB. Acknowledgments. The author is supported by the Labex CEMPI (ANR-11-LABX-0007-01) and by the project QuaSiDy (ANR-21-CE40-0016).

LEAFWISE POINCARÉ METRIC

Let F " pM, L , Eq be a singular holomorphic foliation. Fix g M a Hermitian metric on M . Define (2.1) ηpxq " sup

! }α 1 p0q} g M | α : D Ñ L x holomorphic such that αp0q " x ) ,
where }v} g M is the norm of v P T x L x with respect to the Hermitian metric g M . That is,

}v} g M " a g M,x pv, vq. Definition 2.1. A leaf L of F is called hyperbolic if it is uniformized by the Poincaré disk D.
The foliation F is called hyperbolic if all the leaves of F are hyperbolic.

We will be mostly interested in the case of hyperbolic foliations. Using Schwarz' Lemma, it is quite clear that L x is hyperbolic iff ηpxq ă 8. In that case, ηpxq " }ϕ 1

x p0q} g M , where ϕ x : D Ñ L x is a uniformization of L x such that ϕ x p0q " x. By pushing forward the Poincaré metric on D by ϕ x , we obtain the so-called Poincaré metric g P on the leaf L x . Since ϕ x can be chosen up to pre-composition by a rotation, and since the biholomorphisms of D are isometries of pD, g P q, the Poincaré metric is intrinsically defined. It is quite clear that we have

(2.2) η 2 g P " 4g M .
We recall from [7, Definition 3.1] the following.

Definition 2.2. We say that F is Brody hyperbolic if there exists a positive constant c 0 such that η ă c 0 on M zE.

Let us define the type of singularities we will study.

Definition 2.3. Let F " pM, L , Eq be a singular holomorphic foliation. Near a singularity a P E, there exists a vector field X defining F . In coordinates pz 1 , . . . , z n q centered at a, we can write

Xpzq " n ÿ j"1 F j pzq B Bz j .
The functions F j can be developed as a power series F j " ř αPN n c α,j z α . The 1-jet of X at a is defined in the chart pU, zq as X 1 " ř n j"1 ř |α|ď1 c α,j z α B Bz j . See for example [13, Chapter I] for more details. If the 1-jet of X has an isolated singularity at a, we say that a is a non-degenerate singularity of F .

The following result is a direct consequence of a local study of non-degenerate vector fields in [START_REF] Neto | Hermitian metrics inducing the Poincaré metric, in the leaves of a singular holomorphic foliation by curves[END_REF]. We denote by log ‹ " 1 `|log| a log-type function and by d M px, Eq the distance from x P M to the singular set E with respect to the Hermitian metric g M . Proposition 2.4 (Dinh-Nguyên-Sibony). Let F " pM, L , Eq be a Brody hyperbolic compact singular holomorphic foliation. Suppose that all the singularities of F are nondegenerate. Then, there exists a constant c ą 1 such that the Poincaré metric of F satisfies

c ´1d M px, Eq log ‹ d M px, Eq ď ηpxq ď cd M px, Eq log ‹ d M px, Eq, x P M zE.
See [2, Proposition 4.1] for a proof, which is inspired by [7, Proposition 3.3].

LEAFWISE HEAT DIFFUSION OPERATORS

We will now define one of the two heat diffusion operators we are interested in. Let F " pM, L , Eq be a hyperbolic singular holomorphic foliation. The leafwise Poincaré metric g P gives rise to its associated Laplacian ∆ P on leaves. More precisely, for f P C 2 pDq or for f P C 2 pL x q, ∆ P is defined by the following formula.

p∆ P f q g P " πdd c f " iBBf on D or L x .
For any fixed x P M zE, we study the heat equation on L x (3.1) Bppx, y, tq Bt " ∆ P,y ppx, y, tq, lim tÑ0 `ppx, y, tq " δ x pyq, y P L x , t P R `. Here, we have denoted by δ x the Dirac mass in x and by ∆ P,y the Laplacian ∆ P with respect to the variable y. The limit is taken in the sense of distribution.

The heat kernel on L x , denoted by ppx, y, tq, is defined as the smallest positive solution of the heat equation (3.1) (see Chavel [START_REF] Chavel | Eigenvalues in Riemannian geometry[END_REF]). By diffusing ppx, y, tq, we get a one-parameter family of operators tD t | t P R `u defined by This family is a semi-group of positive contractions of L 8 pM zEq (see (3.3)). Its elements are called the leafwise heat diffusion operators.

(3.3) D 0 " id; D t 1 " 1 and D t`s " D t ˝Ds , for t, s P R `,
where 1 is the function identically equal to 1.

The same results hold on D itself. We denote by p D pθ, ζ, tq the heat kernel on D. The heat diffusions on a leaf L x are then related to the one on D by the formula (3.4) D t pf ˝ϕx q " pD t f q ˝ϕx , on D, for t P R `, f P L 8 pL x q .

See [18, Proposition 2.7] for a proof.

DIRECTED CURRENTS AND HARMONIC MEASURES

We suppose that the reader is already familiar with the notion of currents, positive currents, harmonic currents. For a more detailed exposition, see [START_REF] Nguyên | Ergodic theorems for laminations and foliations: recent results and perspectives[END_REF]. For a new example, see [START_REF] Alkateeb | Examples of harmonic foliated currents and singular Levi-flats on the projective plane[END_REF]. Let F " pM, L , Eq be a hyperbolic singular holomorphic foliation. A directed harmonic current on F can be decomposed leafwise and transversally using the following lemma. Its proof was done in [6, Proposition 2.3] (see also [START_REF] Nguyên | Ergodic theorems for laminations and foliations: recent results and perspectives[END_REF]Proposition 2.5]). We will need it to define the abstract heat diffusions. Lemma 4.1. Let T be a directed harmonic current on F . Consider U » D ˆT a flow box the coordinates of which can be extended to a neighbourhood of its closure in M . Then, there is a positive Radon measure ν on T and for ν-almost every t P T a harmonic function h t on D such that ' The mass ş T }h t } L 1 pDq dνptq is finite; ' For α P D 1,1 pF q compactly supported in U,

T pαq " ż T ˆżD h t pyqαpy, tq ˙dνptq.
If moreover T is positive, then for ν-almost every t P T, h t is positive on D.

We have a notion of harmonicity for measures, characterizing their behaviour with respect to the Laplacians on the leaves. Definition 4.2. Let ∆ P be the aggregate of the leafwise Laplacians on the leaves of F . A finite positive Borel measure µ on M zE is called harmonic if

ż M ∆ P f dµ " 0, f P D pF q ,
where D pF q " D 0,0 pF q denotes the space of test functions.

Harmonic measures and directed harmonic currents are strongly linked. More precisely, if T is a directed harmonic current and if the measure (4.1)

µ " T ^gP on M zE, is finite, then µ is harmonic. Conversely, if µ is a harmonic measure, then there exists a directed harmonic current T such that (4.1). See [START_REF] Dinh | Heat equation and ergodic theorems for Riemann surface laminations[END_REF] for an implicit proof. In any case, we call the total mass of µ defined by (4.1) the Poincaré mass of T . It may be infinite near singularities. We will need the following results. The first one is a weak version of a theorem by Skoda in [START_REF] Skoda | Prolongement des courants, positifs, fermés de masse finie. (French) [Extension of closed, positive currents of finite mass[END_REF]. The second is due to Dinh, Nguyên and Sibony in [START_REF] Dinh | Heat equation and ergodic theorems for Riemann surface laminations[END_REF].

Proposition 4.3. Let T be a positive dd c -closed p0, 0q-current on a ball r 0 B and β " dd c }z} 2 be the standard Kähler form. Then, the quantity r ´2 }T ^β} rB is bounded for r P p0, r 1 s, r 1 ă r 0 .

Here, }T ^β} A denotes the mass of T ^β as a positive measure on a measurable set A.

Proposition 4.4. Let F " pM, L , Eq be a singular holomorphic foliation with isolated singularities and T be a directed positive harmonic current on M zE. Then, T has locally finite mass near any singularity a P E and can be extended through a into a positive dd cclosed current.

The following criterion for the Poincaré mass of a dd c -closed current to be finite will be very important to prove Theroem 1.2. Proposition 4.5. Let F " pM, L , Eq be a singular holomorphic foliation such that E is composed of isolated singularities. Suppose that in a neighbourhood of a P E, the Poincaré metric of F satisfies ηpxq ě cd M px, aq log ‹ d M px, aq for a constant c ą 0. Then, any directed positive harmonic current on F has locally finite Poincaré mass near a.

Proof. The proof was implicitly done in [START_REF] Dinh | Heat equation and ergodic theorems for Riemann surface laminations[END_REF]Proposition 4.2]. Basically, we work in coordinates and with the standard Kähler form β " dd c }z} 2 of C k . Propositions 4.4 and 4.3 ensure that }T ^β} rB ď Cr 2 . Then, we use (2.2) and the proof ends with the following integration by parts. Let F " pM, L , Eq be a hyperbolic singular holomorphic foliation and µ be a harmonic measure on F . We want to have a solution for an abstract heat equation. It will be given by Hille-Yosida's theorem. First recall some notions and facts of functional analysis. The reader can find proofs and exposition in Brezis [START_REF] Brezis | Analyse fonctionnelle. Théorie et applications[END_REF].

Let L be a Hilbert space and x¨, ¨y be its scalar product. A linear operator A : DompAq Ñ L, for DompAq Ă L, is called monotone if xAu, uy ě 0, u P DompAq. It is called maximal monotone if moreover for f P L, there exists u P DompAq such that Au `u " f . If A is maximal monotone, then DompAq is dense in L and the graph of A is closed. A semi-group of contractions is a family Sptq : L Ñ L, t P R `satisfying ' Spt `t1 q " Sptq ˝Spt 1 q, t, t 1 P R `, ' }Sptq} ď 1, t P R `. A maximal monotone operator A gives rise to a semi-group of contractions having A as infinitesimal generator by the following result. Theorem 5.1 (Hille-Yosida). Let A be a maximal monotone operator on a Hilbert space L. The equation

(5.1)
Bupt, ¨q Bt `Aupt, ¨q " 0, and up0, ¨q " u 0 , for (5.2) u P C 1 pR `, Lq X C 0 pR `, DompAqq and u 0 P DompAq, has a unique solution u. Moreover, u is given by a semi-group of contractions Sptq : L Ñ L, t P R `acting on u 0 , i.e. upt, ¨q " Sptqu 0 .

So, one will be able to solve the heat equation if one can show that the operator ´∆P is maximal monotone. This work was done in [6, Proposition 5.6] (see also [START_REF] Nguyên | Ergodic theorems for laminations and foliations: recent results and perspectives[END_REF]Proposition 5.8]) and has led to the following result. Proposition 5.2. Let F " pM, L , Eq be a singular holomorphic foliation endowed with its Poincaré metric g P . If µ is a harmonic measure on F , then ´∆P is maximal monotone on L 2 pµq.

In particular, there exists a semi-group of contractions Sptq, t P R `such that for u 0 P Dom p´∆ P q, upt, ¨q " Sptqu 0 satisfies (5.2) and (5.1). The Sptq are called the abstract heat diffusion operators.

The proof of [START_REF] Dinh | Heat equation and ergodic theorems for Riemann surface laminations[END_REF] relies on the study of bilinear forms defined through the operator ∆ P and another Laplace type operator ∆P . We recall the definition of these bilinear forms and some of their properties we will need. Let T be a directed harmonic current associated to µ by ( 4 Here, the gradient is defined with respect to the Poincaré metric so that x∇u, ξy g P " dupξq, for u P D pF q and ξ tangent to the leaf. Note that |∇u| " |du| P . The gradient can be extended to an operator of domain H 1 pµq onto L 2 pµq. The space H 1 pµq is the completion of D pF q for the norm

}u} H 1 pµq " ´}u} 2 L 2 pµq `}∇u} 2 L 2 pµq ¯1 2 .
We have Dom p´∆ P q Ă H 1 pµq. A priori, the definition of ∆P depends on the choice of flow boxes, but the uniqueness of the functions h t and the current T ensure that it is a global operator. Define the bilinear forms qpu, vq " ´ż p∆ P uqv dµ, qpu, vq " ´ż p ∆P uqv dµ, for u, v P D pF q . We summarize the properties of q and q we will need in Section 6 in the following statement. The reader can find proofs in [6, Lemmas 5.4 and 5.5] (see also [START_REF] Nguyên | Ergodic theorems for laminations and foliations: recent results and perspectives[END_REF]Lemmas 5.4 and 5.5, Remark 5.7]).

Proposition 5.3. Let µ be a harmonic measure. The bilinear forms q and q can be extended continuously to H 1 pµq ˆH1 pµq. Moreover, (1) For u P H 1 pµq, qpu, uq " qpu, uq.

(2) For u P Dom p´∆ P q and v P H 1 pµq, we have qpu, vq " ż x∇u, ∇vy g P dµ " ż iBu ^Bv ^T.

COINCIDENCE OF THE HEAT DIFFUSIONS : PROOF OF THEOREM 1.2

The aim of this section is to compare the diffusion operators Sptq defined in Section 5 and D t defined in Section 3, for any t P R `. We begin by stating a very abstract criterion for both heat diffusions to be equal. Theorem 6.1. Let F " pM, L , Eq be a singular holomorphic foliation with isolated singularities. Consider on M the leafwise Poincaré metric g P and a Hermitian metric g M . Denote by η the ratio function defined by (2.1). Let µ be a harmonic measure on F . Assume that (H1) η is locally bounded from above on M zE and in the neighbourhood of any a P E, ηpxq ď F a pd M px, aqq for some function F a : R ˚Ñ R ˚such that F a is continuous and

› › L 8 " 0, belongs to H 1 pµq.
Then, the abstract heat diffusions operators and the leafwise heat diffusions operators coincide. Remark 6.2. This theorem is analogous to a result by Nguyên [START_REF] Nguyên | Ergodic theorems for laminations and foliations: recent results and perspectives[END_REF]Theorem 5.17]. In comparison, our hypothesis (H1) is stronger than his, and our hypothesis (H2) is weaker than his. Actually, he uses the first one in a context where he has far better. Therefore, he is forced to suppose quite specific singularities and currents in order to satisfy the second one. Namely, he has to work with hyperbolic singularities and a current that does not give mass to any leaf to have a vanishing Lelong number. Here, we have relaxed the second hypothesis to apply the theorem to more singular measures, and strengthened the first one in consequence. In particular, our assumption (H1) allows us to obtain the result of Lemma 6.3 below.

Before proving Theorem 6.1, we will need two lemmas. The first one is new and will be a crucial ingredient of our proof. The second one can be found in [START_REF] Nguyên | Ergodic theorems for laminations and foliations: recent results and perspectives[END_REF]. Lemma 6.3. We keep the notations of Theorem 6.1 and suppose hypothesis (H1). Let u 0 P D pF q and upt, ¨q " D t u 0 . Then, for t ą 0 and a P E, lim εÑ0 › › u td M px,aqăεu pt, ¨q› › L 8 " 0.

Proof. Let a P E and δ be such that there exists a Hermitian chart U around a of radius δ. Without loss of generality, we can assume that g M is the standard Kähler metric on U » δB. Shrinking the chart if necessary, we also suppose that δ ă d M psupp u 0 , aq. Fix t ą 0, take ε ă δ and x P εB. For y P L x X supp u 0 , we have by (2.2),

d P px, yq ě 2 ż δ ε dt F a ptq " R ε Ñ εÑ0 `8,
since F a is not integrable near 0. It follows that ϕ ´1 x psupp u 0 q Ă DzD Rε . In other words, in the uniformization of L x centered at x, the support of u 0 is sent uniformly to the boundary when x goes to the singularity. By (3.4) See [START_REF] Nguyên | Ergodic theorems for laminations and foliations: recent results and perspectives[END_REF]Facts (5.15), (5.16) and (5.17)] for a proof relying on a precise estimate of the heat kernel.

Proof of Theorem 6.1. We follow the proof of [START_REF] Nguyên | Ergodic theorems for laminations and foliations: recent results and perspectives[END_REF]Theorem 5.17]. Let u 0 P Dom p´∆ P q and t P R `. We have to show that (6.1) D t u 0 " Sptqu 0 .

Since the operators D t and Sptq are positive contractions and D pF q is dense in Dom p´∆ P q, it is enough to show (6.1) for a non-negative u 0 P D pF q . Take such a u 0 . Denote by upt, ¨q " D t u 0 and U pt, ¨q " Sptqu 0 . By Lemmas 6.3 and 6.4 and hypothesis (H2), upt, ¨q, U pt, ¨q and ũpt, ¨q " U pt, ¨q´upt, ¨q belong to H 1 pµq for a fixed t. Morevoer, they are C 1 `R˚, L 2 pµq ˘. The heat equations (5.1) and ( 2) in Lemma 6.4 satisfied by Sptq and D t imply that

1 2 B Bt
´}ũpt, ¨q} 2 L 2 pµq ¯" x∆ P ũpt, ¨q, ũpt, ¨qy " ´q pũpt, ¨q, ũpt, ¨qq .

By an approximation of ũpt, ¨q by elements of D pF q and Proposition 5.3, we get q pũpt, ¨q, ũpt, ¨qq " q pũpt, ¨q, ũpt, ¨qq ě 0 (see [20, p. 50-51]). Hence, }ũpt, ¨q} 2 L 2 pµq is decreasing and goes to 0 for t Ñ 0 by the boundary conditions. This implies that it is identically zero and U pt, ¨q " upt, ¨q. □ Proof of Theorem 1.2. We are inspired by the proof of [START_REF] Nguyên | Ergodic theorems for laminations and foliations: recent results and perspectives[END_REF]Proposition 5.22]. Let T be a directed positive harmonic current on F . By Proposition 2.4, we have (6.2) c ´1d M px, Eq log ‹ d M px, Eq ď ηpxq ď cd M px, Eq log ‹ d M px, Eq, x P M zE.

Then, Proposition 4.5 ensures that the measure µ associated to T by (4.1) is finite. Hence, it is a harmonic measure. We want to apply Theorem 6. We need to show that u belongs to H 1 pµq. Using the compactness of M , we can find a finite open covering U " pU p q pPI by (1) flow boxes U p » D ˆT such that d M pU p , Eq ą 0,

(2) Hermitian charts U a » B for a P E with U a X E " tau.

Since µ is finite, using a partition of the unity, we can suppose that u is compactly supported in a single U p or U a .

Case [START_REF] Alkateeb | Examples of harmonic foliated currents and singular Levi-flats on the projective plane[END_REF]. We need to find u ε P D pF q such that }u ´uε } H 1 pµq ă ε. We will follow the three following steps. (ii) Let v be a bounded continuous function such that }du 1 ´v} L 2 pmq ă ε and w be a bounded continuous function such that }u 1 p0, tq ´wptq} L 2 pν 1 q ă ε. Here, m and ν 1 are measures respectively on D ˆT and T that our computation will specify.

Recall that we can find such v and w if m and ν 1 are Radon measures by Luzin's Theorem. Define u 2 pz, tq " wptq `ż 1 0 vpτ z, tq ¨zdτ.

(iii) Let u ε " u 3 be the leafwise convolution of u 2 with χ p1q δ . Note first that by (6.2) and (2.2), there is a constant C ą 1 such that C ´1 }dz} 2 ď g P ď C }dz} 2 on U p . It follows that the first and third steps will give }u 1 ´u} H 1 pµq ă ε 3 and }u 3 ´u2 } H 1 pµq ă ε 3 , for sufficiently small δ. Moreover, it is clear that u 2 is continuous. Thus, u 3 P D pF q . So, it remains to prove that }u 1 ´u2 } H 1 pµq ă ε 3 . Since u 1 is leafwise smooth, we have u 1 pz, tq " u 1 p0, tq `ż 1 0 du 1 pτ z, tq ¨zdτ everywhere on pz, tq P D ˆT. Recall that µ is given by T ^gP and T by Lemma 4.1.

Decompose u 1 ´u2 into the transversal to 0 part and the integral part, and apply the Cauchy-Schwarz inequality to the second part. We get }u 1 ´u2 } L 2 pµq ď }w ´u1 p0, ¨q} L 2 pν 1 q `}v ´du 1 } L 2 pm 1 q , where ν 1 ptq " }h t } L 1 pg P q νptq and m 1 " λ ˚pLeb ˆµq with λ : pτ, z, tq Þ Ñ pτ z, tq. Moreover, du 2 " v and since C ´1 }dz} 2 ď g P ď C }dz} , we have

}∇u 1 ´∇u 2 } L 2 pµq ď C }v ´du 1 } L 2 pµq .
It is clear that m " m 1 `µ is finite, hence is a Radon measure. So is ν 1 by Lemma 4.1. Choose v a bounded continuous function such that }du 1 ´v} L 2 pmq ă αε and w a bounded continuous function such that }u 1 p0, ¨q ´w} L 2 pν 1 q ă αε, for α ď 1 3 p4 `C2 q ´1{2 . Then }u 2 ´u1 } H 1 pµq ă ε 3 and therefore, }u ´uε } H 1 pµq ă ε. This concludes Case (1). Case [START_REF] Bacher | Poincaré metric of holomorphic foliations with non-degenerate singularities[END_REF]. We work in coordinates U a » B Ă C k centered at a. Let χ : C k Ñ r0, 1s be a smooth function such that χpzq " 1 on 1 2 B and χpzq " 0 on C k zB. For ε P p0, 1q, consider the function v ε pzq " `1 ´χ `ε´1 z ˘˘upzq, z P B.

By construction, we have v ε " 0 on ε 2 B and v ε satisfies (6.3). By the same arguments as in Case (1), we get that v ε P H 1 pµq. So, it will be sufficient to prove that lim εÑ0 }u ´vε } H 1 pµq " 0. Moreover, u ´vε is supported in εB and |u ´vε | ď |u| everywhere. Since µ ptauq " 0, Up to multiplying by constants, the right hand side is lower than the sum of these three terms Recall that |du| "

|du| P η
" |∇u| η . By (6.2) and the equivalence of g M and the standard norm, we get

I 2 ď
C }u} L 8 log ‹ pcεq }|du| P } L 8 ε ´2 }T ^β} εB .

Then, Proposition 4.3 implies that I 2 Ñ 0 when ε Ñ 0. Finally,

I 3 ď › › u td M px,aqăεu › › 2 L 8 ε ´2
}T ^β} εB . Thus, Proposition 4.3 and (6.3) ensure that I 3 Ñ 0 when ε Ñ 0. It follows that }u ´vε } H 1 pµq goes to 0 when ε Ñ 0. Hence, u P H 1 pµq and Theorem 1.2 is a consequence of Theorem 6.1. □ Remark 6.5. In fact, I 1 is estimated exactly the same way as [20, p. 56-57] and I 2 similarly. The new point here is the estimate on I 3 which is based on Lemma 6.3.

(3. 2 )

 2 D t f pxq " ż Lx ppx, y, tqf pyqg P pyq,x P M zE, f P L 8 pM zEq .

2 " }T ^β} rB r 2 ln 2 prq `ż r 0 }T ^β} ρB ˆ2 ρ 3 ln 2 ρ `2 ρ 3 ln 3

 203

  (i) Let χ p1q : C Ñ r0, 1s be a smooth function supported in D such that ş C χ p1q d Leb " 1. Define χ p1q δ pzq " δ ´2χpδzq for δ P p0, 1q. Let u 1 be the leafwise convolution of u with χ p1q δ .

  ż M |u ´vε | 2 dµ ď }u} 2 adress the gradient-part of the H 1 pµq-norm. We have ż M |∇ pu ´vε q| 2 dµ " ż εB iB `upzqχ `ε´1 z ˘˘^B `upzqχ `ε´1 z ˘˘^T pzq.

2 Bż εBz ε 2 B 1 " ż εB |∇u| 2

 2212 |u| |du| T ^β, I 3 " ε ´2 |u| 2 T ^β,where β " dd c }z} 2 is the standard Kähler metric. Since µ ptauq " 0, we obtainI dµ ď }|du| P } 2 µ pεBq Ñ εÑ0 0.

  .1). By Lemma 4.1, we can write in a given flow box U » D ˆT where the h t , for t P T, are positive harmonic functions on D and ν is a Radon measure. Define ∆P u " ∆ P u `@h ´1 t ∇h t , ∇u D

	T pαq "	ż	T ˆżD	h t pyqαpy, tq ˙dνptq,
				g P	.

  DzD Rε pyqp D p0, y, tqg P pyq, where 1 DzD Rε denotes the characteristic function of DzD Rε . This bound is independent on x such that d M px, aq ă ε. By Lebesgue's dominated convergence theorem and the fact that R ε Ñ This We keep the notations and hypothesis of Lemma 6.3. Suppose moreover that u 0 is positive. Then u is measurable and satisfies (1) (a) }up¨, ¨q} L 8 ă 8, (b) For t ą 0, }|dupt, ¨q| P } L 8 ă 8, (c) For t ą 0, }∆ P upt, ¨q} L 8 ă 8;(2) u P C 1 `R˚, L 2 pµq ˘and for t P R ˚,Bupt,¨q

	Lemma 6.4. Bt ´∆P upt, ¨q " 0; (3) lim tÑ0 upt, ¨q " u 0 in L 2 pµq.
		and (3.2), we get
	ż
	D t u 0 pxq " upt, xq "	p D p0, y, tqu 0 pϕ x pyqq g P pyq,
	D
		ż
	|upt, xq| ď }u 0 } L 8
		implies the desired
	conclusion.	□

D 1 εÑ0 `8, the right hand side tends to 0 when ε goes to 0.

  1. Hypothesis (H1) is clear by (6.2). It remains to check hypothesis (H2). Let u be a measurable function on M zE such that (6.3) }u} L 8 ă 8, }|du| P } L 8 ă 8, }∆ P u} L 8 ă 8, lim

	εÑ0	› › u td M px,aqăεu	› ›	L 8 " 0, a P E.


Spencer W Jolly

Focused fields of ultrashort radially polarized laser pulses having low-order spatiotemporal couplings

We present a detailed study of the fields and propagation characteristics around the focus of ultrashort radially polarized laser beams (RPLBs) having low-order spatiotemporal couplings (STCs). The three STCs considered are the focusing of the different frequencies to different positions along the longitudinal coordinate, the focusing of the frequencies to different positions along one transverse coordinate, and the beam waist or Rayleigh range having a dependence on frequency. The STCs considered are deemed low-order because they are primarily linear in frequency. The combination of a low-order vector beam, ultrashort pulse duration, and the three different STCs shows promise for exotic applications in dielectric or charged particle manipulation and potentially other strong-field phenomena. The STCs presented are all developed in a standard frequency-domain model where each case involves a different chromatic term. We present the results unique to the vector nature of the RPLBs and compare them to the linearly polarized cases, opening up opportunities for control of the electric field around the focus with the additional element of polarization.

I. INTRODUCTION

Radially polarized laser beams (RPLBs), a unique solution to Maxwell's equations with primarily radial polarization, are a low-order vector beam (having spatially varying polarization) that can have a very tight focus [START_REF] Quabis | Focusing light to a tighter spot[END_REF][START_REF] Dorn | Sharper Focus for a Radially Polarized Light Beam[END_REF] and have applications in microscopy [START_REF] Sheppard | Annular pupils, radial polarization, and superresolution[END_REF][START_REF] Lu | Coherent anti-Stokes Raman scattering microscopy using tightly focused radially polarized light[END_REF][START_REF] Kozawa | Lateral resolution enhancement of laser scanning microscopy by a higher-order radially polarized mode beam[END_REF] and manipulation of microparticles [START_REF] Zhan | Trapping metallic rayleigh particles with radial polarization[END_REF][START_REF] Kawauchi | Calculation of optical trapping forces on a dielectric sphere in the ray optics regime produced by a radially polarized laser beam[END_REF][START_REF] Kozawa | Optical trapping of micrometer-sized dielectric particles by cylindrical vector beams[END_REF][START_REF] Huang | Optical trapping of gold nanoparticles by cylindrical vector beam[END_REF] for example. Ultrashort RPLBs require additional modeling due to their broad bandwidth and can be applied to particle acceleration [START_REF] Varin | Acceleration of electrons from rest to GeV energies by ultrashort transverse magnetic laser pulses in free space[END_REF][START_REF] Fortin | Direct-field electron acceleration with ultrafast radially polarized laser beams: Scaling laws and optimization[END_REF][START_REF] Wong | Direct acceleration of an electron in infinite vacuum by a pulsed radially-polarized laser beam[END_REF][START_REF] Payeur | Generation of a beam of fast electrons by tightly focusing a radially polarized ultrashort laser pulse[END_REF][START_REF] Carbajo | Direct longitudinal laser acceleration of electrons in free space[END_REF][START_REF] Wong | Laser-induced linear-field particle acceleration in free space[END_REF] and potentially high-harmonic generation and other realms of high-field nano-optics [START_REF] Dombi | Strongfield nano-optics[END_REF]. Combing these ultrashort RPLBs with spatiotemporal couplings (STCs), aberrations that lead to unseparable space-time or space-frequency electric fields [START_REF] Akturk | Spatio-temporal couplings in ultrashort laser pulses[END_REF], results in interesting properties. In this paper we present frequency-domain models to describe RPLBs having different STCs and investigate potentially valuable phenomena unique to the vector nature of RPLBs.

We first describe the standard ultrashort RPLB in frequency space, and describe its basic properties. Then we compare certain properties of the RPLB when it has three different low-order STCs (i.e., generally linear in frequency dependence). The three STCs considered are as follows: the focusing of the different frequencies to different positions along the longitudinal coordinate, longitudinal chromatism; the focusing of the frequencies to different positions along one transverse coordinate, spatial chirp; and the beam waist or Rayleigh range having a power-law frequency dependence; referred to as frequency-dependent beam parameters. * spencer.jolly@vub.be In the case of each STC there are general effects around the focus, which often involve an increase in the integrated beam size or an increase in the pulse duration (and equivalently a decrease in the intensity), similar to with linearly polarized pulses [START_REF] Bourassin-Bouchet | Duration of ultrashort pulses in the presence of spatio-temporal coupling[END_REF]. But in addition to those general effects there are more nuanced effects in each case, which can involve the pulse temporal profile, spatial intensity profile, or its behavior during propagation away from the focus. In the case of longitudinal chromatism there is an effective increase in the Rayleigh range due to the longitudinal separation of the frequencies, and an additional effect when combined with temporal chirp where the velocity of the intensity peak can be different than the speed of light c, called the flying focus. For spatial chirp there can be wavefront rotation, where the wavefront in focus points in different directions over time, and there can also be a tilt in the arrival time of the pulse when combined with temporal chirp. In the case of the frequencyvarying beam parameters we consider the evolution of the carrier-envelope offset phase, which differs from the Gouy phase in a nontrivial fashion. Indeed all of the above phenomena have been described and measured experimentally for linearly polarized ultrashort laser pulses. In the case of the RPLBs which we describe theoretically here, there is first the fact that the model for the fields is different, resulting in different precise descriptions for each phenomenon. But there is also the crucial fact that RPLBs are vector beams and the different polarization components of the electric field not only have different basic properties that influence how to look at the effect of STCs on each polarization, but the different components are also described via different equations resulting in different behavior. In each case we develop the model for the RPLB with STCs using the same frequency-space model as for the standard ultrashort RPLB, but clearly explain the modified or added terms which result in the fundamental changes in beam properties or propagation effects. Because the STCs add nontrivial space-frequency terms in our model we do not have the ability to solve for the fields in time, but can present results in time based on our model using numerical Fourier transforms.

II. STANDARD ULTRASHORT RPLB

In the following model we use pulses that have Gaussian spatial and spectral or temporal profiles, with characteristic widths s 0 and τ 0 , respectively, at a central wavelength λ 0 (ω 0 = 2π c/λ 0 ) and the Rayleigh range z R = ω 0 s 2 0 /2c. The fields of the focused ultrashort RPLBs are modeled in the frequency domain as in Refs. [START_REF] Sainte-Marie | Controlling the velocity of ultrashort light pulses in vacuum through spatiotemporal couplings[END_REF][START_REF] Jolly | Influence of longitudinal chromatism on vacuum acceleration by intense radially polarized laser beams[END_REF][START_REF] Jolly | On the importance of frequency-dependent beam parameters for vacuum acceleration with few-cycle radiallypolarized laser beams[END_REF] using the proper form for the longitudinal fields including only paraxial terms. With A ω = exp(-δω 2 / ω 2 ), ω = 2/τ 0 , and δω = (ω -ω 0 ) we have the solutions for the radial electric field E r , longitudinal electric field E z , and azimuthal magnetic field B θ ,

Ẽr (ω) = A ρC 2 , ( 1
)
Ẽz (ω) = A 2 [S 2 -ρ 2 S 3 ], (2)
Bθ (ω) = Ẽr (ω) c , ( 3
)
where the tilde denotes frequency space, ρ = r/s 0 and = s 0 /z R , and

A = ω 0 2c 8P 0 π 0 c √ 2A ω ω e -r 2 /s 2 , ( 4
)
C n = s 0 s n e i(ψ+nψ G ) , ( 5
)
S n = s 0 s n e i(ψ+nψ G +π/2) , ( 6
)
ψ = 0 - ωr 2 2cR - ωz c -ψ spec , ( 7
)
ψ spec = φ 2 δω 2 2 + φ 3 δω 3 6 + φ 4 δω 4 24 + • • • , ( 8
)
ψ G = tan -1 z z R , ( 9
)
s = s 0 1 + z z R 2 , ( 10
)
R = z + z 2 R z . ( 11
)
where P 0 is the peak power of the pulse (in this case with zero STCs), 0 is the permittivity of free space, and c is the speed of light in vacuum, however, for the rest of this work we will present normalized fields since we are modeling their properties in free space only. The spectral phase parameters φ 2 , φ 3 , and φ 4 are the group-delay dispersion, third-order dispersion, and fourth-order dispersion, respectively, included for completeness, but in our presented phenomena we will only show results with nonzero φ 2 . The fields above are the complex fields, where for all visualizations we show the real part of the fields in time corresponding to the physically relevant quantities.

Of course, in the case of the equations above, the fields in time can be easily calculated. This is because the only frequency dependence is in the Gaussian spectral profile A ω and the term ωz/c in the phase, so the temporal profile is simply a Gaussian envelope traveling in z at the speed of light. However, for the rest of this work we will build off of this standard case to consider pulses with nontrivial chromatic terms, which cannot be expressed simply in time, so we leave these fields in terms of frequency for reference. Because it is often the electric fields that are most important, B θ will not generally be included in visualizations, except for in the case of spatial chirp where calculating the magnetic field requires a different coordinate transformation due to the breaking of cylindrical symmetry. It must also be noted that modeling an azimuthally polarized laser beam requires only a simple transformation of these equations, so the work in the rest of this paper will also apply.

Figure 1 shows the basic characteristics of the standard ultrashort RPLB for a wavelength of 800 nm that we will use throughout this paper. These characteristics include the longitudinal field E z being strongly localized on-axis and the transverse field E r being zero on-axis, shown in Figs. 1(a) and 1(b), respectively, with both having cylindrical symmetry. As already mentioned, since this description does not yet contain STCs, the fields are trivially extended to having an ultrashort envelope that maintains these features in the focus, shown in Figs. 1(c) and 1(d). Both the monochromatic RPLB and the ultrashort RPLB become more complicated as they propagate away from the focus, shown for the ultrashort case in Figs. 1(e) and 1(f), still without STCs.

Note as well that the nonparaxial form could be easily constructed by adding terms of higher order in as in Ref. [START_REF] Salamin | Fields of a radially polarized Gaussian laser beam beyond the paraxial approximation[END_REF], for example. This will also be the case for all of the models used in the following sections with STCs added, but we write only the paraxial forms for simplicity in the main text to emphasize the chromatic terms. For a more detailed discussion of the nonparaxial description see the Appendix. An accurate nonparaxial description is generally believed to be necessary when the beam waist approaches the wavelength, but for highly sensitive interactions such as high-field particle acceleration a nonparaxial description becomes important even at larger beam waists [START_REF] Marceau | Validity of the paraxial approximation for electron acceleration with radially polarized laser beams[END_REF].

III. LONGITUDINAL CHROMATISM

Longitudinal chromatism (LC) is a form of spatiotemporal coupling that is when the different spectral components of an ultrashort pulse are separated longitudinally, that is, along the direction of propagation z. In the case in this section, we model LC at the focus of an ultrashort RPLB. In fact, LC is the result of two equivalent phenomenon on the collimated beam before focusing, pulse-front curvature (PFC) and chromatic curvature (CC). PFC is when the arrival time of the ultrashort pulse varies quadratically with the radius. Due to the fundamental relationship between time and frequency in light waves (via the Fourier transform), PFC is equivalent to the radius of curvature of the pulse depending linearly on frequency, which we refer to as chromatic curvature (CC). Because of this equivalence, we refer to this phenomenon on the collimated beam as CC/PFC, which is denoted by α [fs/m 2 ].

Because the manifestation of the near-field couplings in focus depends on the focusing geometry, the corresponding near-field beam radius can be chosen as s i with a focal length of f , making s 0 = 2c f /ω 0 s i . However, if the collimated beam were smaller and the focal length shorter so as to have the same beam waist and Rayleigh range, then the relative effect of a certain CC/PFC α would be different. So, in fact, it is the quantity τ p = αs 2 i [see Fig. 2(a)] that can properly parametrize the effect of the CC or PFC coupling regardless of the actual focusing geometry. So it is this τ p that we use to parametrize the LC in-focus (resulting from CC or PFC on the collimated beam) for the rest of this section. The LC produces a frequency-dependent longitudinal waist position z 0 of

z 0 = z R τ p δω, ( 12
)
where δω = (ω -ω 0 ), i.e., describing linear LC. The fields around the focus z = 0 are as before in Eqs. (1) to [START_REF] Kozawa | Optical trapping of micrometer-sized dielectric particles by cylindrical vector beams[END_REF], but with modified terms

ψ G = tan -1 z -z 0 z R , ( 13
)
s = s 0 1 + z -z 0 z R 2 , ( 14
) R = (z -z 0 ) + z 2 R (z -z 0 ) . ( 15
)
We must note that the waist s 0 and Rayleigh range z R being constant values relies on the pulses being longer than few-cycle as mentioned earlier, but in the case of LC it also relies on the difference in focal lengths between the extreme frequencies of the pulse being negligible. This is equivalent to the assumption that the extended Rayleigh range z R e = τ p ωz R is negligible compared to the focal length f :

τ p ωz R f or τ p /τ 0 f /2z R .
If this were not the case, then the focal length would also need to be treated as dependent on frequency, resulting in a frequency dependence of the waist and Rayleigh range independent of the assumption of the pulse duration. After calculating the fields in frequency space with the stated assumptions, they must be inverse Fourier transformed to time.

The basic results of the model of an ultrashort RPLB with LC is shown in Fig. 3 for one value of τ p = 60 fs in the focus at z = 0. The pulse retains cylindrical symmetry, but both E z and E r have much longer-tailed transverse distributions, shown in Figs. 3(a In addition to the effect on the transverse profile of both polarization components, there is also a significant change in the temporal profile. As shown in Figs. 3(c) and 3(d) the temporal intensity profiles of both E z and E r are no longer symmetric in time, and have significant interference effects at the outer portions of the transverse distributions. The asymmetry and interferences are common characteristics of LC, and in this case are manifested in both polarization components, while each component maintains the basic characteristics of the ultrashort RPLB (E z is localized on-axis and E r is zero on-axis).

Another feature of the ultrashort RPLB with LC is an extended Rayleigh range, defined earlier as z R e = τ p ωz R . This can be simply reasoned from the fact that different colors have their waist position at different z positions. So, although the waist is larger due to the LC, the pulse retains that slightly larger waist within an increased longitudinal distance z R e , larger than the basic Rayleigh range by both the amount of LC τ p and the bandwidth of the pulse ω.

This increased Rayleigh range, effectively beating diffraction, has already been shown for linearly polarized pulses [START_REF] Froula | Spatiotemporal control of laser intensity[END_REF]. We confirm in Fig. 4 that it is also valid for ultrashort RPLBs having LC for both E z and E r . Comparing Figs. 4(a (c) and 4(d), of course, is the temporal behavior of the electric field at each z position, as it is only showing the integrated intensity. Because the LC separates the colors then at the different points within the extended Rayleigh range the central wavelength will evolve, which can be thought of as the price paid for "beating" diffraction. Two-dimensional STC wave packets [START_REF] Kondakci | Diffraction-free pulsed optical beams via space-time correlations[END_REF][START_REF] Kondakci | Diffraction-free spacetime light sheets[END_REF] have been shown to remain localized for orders of magnitude beyond the diffraction limit without separating the frequencies [START_REF] Bhaduri | Meters-long propagation of diffraction-free space-time light-sheets[END_REF][START_REF] Bhaduri | Broadband space-time wave packets propagating 70 m[END_REF], but have other limitations and are not vector beams, so can be considered a uniquely different phenomenon.

Flying focus effect

Longitudinal chromatism combined with linear temporal chirp (or quadratic spectral phase φ 2 ) results in the intensity of the ultrashort laser pulse traveling at velocities significantly different than c, referred to as the flying focus [START_REF] Sainte-Marie | Controlling the velocity of ultrashort light pulses in vacuum through spatiotemporal couplings[END_REF][START_REF] Palastro | Ionization waves of arbitrary velocity driven by a flying focus[END_REF]. This has been implemented with either diffractive optics [START_REF] Froula | Spatiotemporal control of laser intensity[END_REF] or specially produced lens doublets [START_REF] Jolly | Controlling the velocity of a femtosecond laser pulse using refractive lenses[END_REF] to produce the LC, and has also been used to create ionization waves in a plasma at such superluminal and even negative velocities [START_REF] Turnbull | Ionization Waves of Arbitrary Velocity[END_REF][START_REF] Franke | Measurement and control of large diameter ionization waves of arbitrary velocity[END_REF]. Theoretical work shows potential applications in laser-plasma Raman amplification [START_REF] Turnbull | Raman Amplification with a Flying Focus[END_REF], plasma-based photon acceleration [START_REF] Howard | Photon Acceleration in a Flying Focus[END_REF], and vacuum electron acceleration [START_REF] Ramsey | Vacuum acceleration of electrons in a dynamic laser pulse[END_REF]. All of the past work has dealt with scalar electric fields, so we discuss and simulate the same effect with the fields of an ultrashort RPLB.

The velocity of the intensity peak in the flying focus scenario v f f has been shown to obey a simple formula [START_REF] Sainte-Marie | Controlling the velocity of ultrashort light pulses in vacuum through spatiotemporal couplings[END_REF][START_REF] Palastro | Ionization waves of arbitrary velocity driven by a flying focus[END_REF][START_REF] Jolly | Controlling the velocity of a femtosecond laser pulse using refractive lenses[END_REF]

v f f c = 1 1 + cφ 2 τ p z R . ( 16
)
This simple formula is derived based on only the varying central frequency along the longitudinal direction and concurrent varying arrival time for the different frequencies resulting from the spectral phase. We expect that it holds similarly for the RPLB having chirp and LC since the relevant phase terms are the same.

The results for the combination of LC and chirp are shown in Fig. 5 for E z only, with r = 0, confirming that the flying focus phenomenon occurs as well with ultrashort RPLBs. The maps of the on-axis intensity profile for z positions through the focus is shown for E z with a single value of τ p = 60 fs and three chirp values in Figs. 5(a) and 5(c). These specific combinations of LC and chirp produce negative superluminal, positive superluminal, and positive subluminal v f f in Figs. 5(a), 5(b) and 5(c), respectively. The dependence of v f f on the chirp from the simulations agrees very well with that predicted from the simple relationship in Eq. ( 16), shown in Fig. 5(d). The transition chirp where the velocity transitions from purely positive to negative occurs at roughly φ 2 = -τ p z R /c, the same as for linear polarization. The peak intensity of the longitudinal field decreases significantly already with the LC, and decreases further with added chirp [Fig. 5(e)], which is qualitatively similar to the intensity reduction for linear polarization [START_REF] Sainte-Marie | Controlling the velocity of ultrashort light pulses in vacuum through spatiotemporal couplings[END_REF]. The situation of LC and chirp in the transverse polarization, E r , is more complicated both because the field is only nonzero off-axis, and because as the pulse diffracts away from the focus the position of maximum intensity is increasingly off-axis. However, as already shown in Fig. 4(d), within the extended Rayleigh range the position of maximum intensity is relatively constant. Therefore we look at the velocity of the intensity peak of E r at r = s 0 (ρ = 1). The results for the same nonzero chirps as for E z is shown for E r in Fig. 6.

The result of the analysis of the flying focus off-axis with E r is that it is only slightly different than that of E z , both in terms of the quantitative results and the qualitative dependence on φ 2 . This is due to the additional phase term dependent on r in Eq. ( 7), the curvature term, that due to the transformation of z to zz 0 (ω) introduces a nontrivial frequency dependence when r = 0. When comparing the simulated temporal intensity profiles of E z at r = 0 in Figs. 5(a) to 5(c) and for E r at r = s 0 in Figs. 6(a) to 6(c), the results are therefore slightly different. However, this is most apparent around the transition chirp. For example with φ 2 = -15 000 fs 2 the velocity is -5.93c for E z on-axis and -7.4c for E r off-axis. The difference in the dependence on the chirp is seen clearly in Fig. 6(d) near the transition chirp of -12 500 fs 2 when compared to Fig. 5(d). It is interesting that the phase term due to curvature is in the phase both for E z and E r , meaning that as r increases the velocity changes for E z as well, which we confirmed via numerical calculations, but this difference is less important for E z because the intensity decreases rapidly off-axis.

As a last observation, both for E z and E r , producing a changing velocity (i.e., accelerating, decelerating, or nonmonotonic) of the intensity peak is also possible with higher orders of spectral phase φ 3 , φ 4 , and so on as already predicted for linear polarization [START_REF] Sainte-Marie | Controlling the velocity of ultrashort light pulses in vacuum through spatiotemporal couplings[END_REF].

It is important again to emphasize the difference from STC wave packets [START_REF] Kondakci | Diffraction-free spacetime light sheets[END_REF] that have been shown as well to have tunable velocities [START_REF] Kondakci | Optical space-time wave packets having arbitrary group velocities in free space[END_REF][START_REF] Bhaduri | Space-time wave packets that travel in optical materials at the speed of light in vacuum[END_REF][START_REF] Yessenov | Accelerating and Decelerating Space-Time Optical Wave Packets in Free Space[END_REF]. The flying focus shown here, besides being with a cylindrical vector beam, is limited to a region around the focus defined by z e R and the different frequencies of the initial laser pulse are separated longitudinally within that region. Therefore, the flying focus for both linear polarization and with ultrashort RPLBs is most relevant to phenomena that are purely intensity-dependent, otherwise this separation of the frequencies must be taken into account. The tunable velocity STC wave packets, shown so far only for linear polarization, do not have the separation of frequencies and can have localized propagation orders of magnitude longer than as predicted by diffraction, but for reasons already discussed can be considered significantly different than the case discussed in this paper. Besides these STC wave packets, more complex examples with programmable group velocities involving Bessel beams have recently been described [START_REF] Li | Optical wave-packet with nearlyprogrammable group velocities[END_REF], which could also theoretically be adapted to RPLBs.

IV. SPATIAL CHIRP

In this section we model spatial chirp in the focus of an RPLB. Spatial chirp around the focus is due to angular dispersion (AD) in the near field, where different frequencies have different pointing directions, or equivalently the pulse front of the beam is tilted from the wavefront (pulse front tilt, PFT), see Fig. 7(a). Because these are equivalent, it can be referred to as AD-PFT to be unambiguous. When focused the different pointing directions of different frequencies result in the frequencies being spatially separated in the focal region, as in Fig. 7(b), along the axis where there was AD-PFT in the near field. This transverse spatial separation of frequencies is exactly spatial chirp (SC).

As it was with LC, the in-focus manifestation of the nearfield AD-PFT, represented by the tilt angle η = dt/dx [fs/m], depends on the focusing geometry. So, similar to the case of LC it is the time delay at the outer edge of the near field (with beam size s i ) due to AD-PFT τ t = ηs i [see Fig. 7(a)] that can properly parametrize the effect of the near-field AD-PFT coupling regardless of the actual focusing geometry. So it is this τ t that we use to parametrize the SC in-focus (resulting from AD-PFT on the collimated beam) for the rest of this section. Linear SC is described by a best-focus position x 0 of

x 0 = s 0 τ t δω/2, ( 17
)
which is clearly linear in frequency and proportional to τ t representing the AD-PFT.

The key difference between the modeling of SC is that there is no longer cylindrical symmetry as in the previous section. This is because with AD-PFT on the collimated beam, the different frequencies are focused to different positions along one transverse axis, chosen to be x here [see Fig. 7(b)]. This not only affects the spectral content at any given position, but it affects the direction of the fields, which depends nontrivially on frequency.

The implication is that the initial fields Ẽr , Ẽz , and Bθ must be transformed to Cartesian coordinates, and all fields Ẽx , Ẽy , Ẽz , Bx , and By ( Bz = 0 still) must be calculated at all x, y, and z positions. However, because the frequencies are shifted a different amount along x in the focal plane, the coordinate transformation is also frequency dependent. This concept is sketched in Fig. 7(c) for Ẽr and in Fig. 7(d) for Bθ . The results are Ẽx (ω) = Ẽr (ω)

xx 0 r = Ẽr (ω)

1 + y x-x 0 2 , ( 18
)
Ẽy (ω) = Ẽr (ω) y r = Ẽr (ω) 1 + x-x 0 y 2 , ( 19
)
Bx (ω) = Bθ (ω) -y r = -Bθ (ω) 1 + x-x 0 y 2 , ( 20
)
By (ω) = Bθ (ω) x -x 0 r = Bθ (ω) 1 + y x-x 0 2 , ( 21
)
where r = (xx 0 ) 2 + y 2 . The Ẽz field does not require any transformation, but it is still distributed along x according to x 0 . As before we use pulses that have Gaussian spatial and temporal profiles, with characteristic widths s 0 and τ 0 , respectively, at a central wavelength λ 0 (ω 0 = 2π c/λ 0 ). The fields of the focused RPLB with SC are modeled similarly in the frequency domain as in Refs. [START_REF] Sainte-Marie | Controlling the velocity of ultrashort light pulses in vacuum through spatiotemporal couplings[END_REF][START_REF] Jolly | Influence of longitudinal chromatism on vacuum acceleration by intense radially polarized laser beams[END_REF] and the previous sections. We first must simply have the new frequency-dependent coordinate ρ = r /s 0 = (xx 0 ) 2 + y 2 /s 0 . However, a more complicated step is to apply the transformation to Cartesian coordinates from Eqs. ( 18) to (21) as follows: where C n and S n are as in Eqs. ( 5) and ( 6), with the modified r replacing r in the amplitude and phase

Ẽx (ω) = A SC ⎛ ⎝ ρ C 2 1 + y x-x 0 2 ⎞ ⎠ , ( 22
)
Ẽy (ω) = A SC ⎛ ⎝ ρ C 2 1 + x-x 0 y 2 ⎞ ⎠ , ( 23
)
Ẽz (ω) = A SC 2 [S 2 -ρ 2 S 3 ], (24)
Bx (ω) = A SC c ⎛ ⎝ -ρ C 2 1 + x-x 0 y 2 ⎞ ⎠ , ( 25
)
By (ω) = A SC c ⎛ ⎝ ρ C 2 1 + y x-x 0 2 ⎞ ⎠ , (26)
A SC = ω 0 2c 8P 0 π 0 c √ 2A ω ω e -r 2 /s 2 , ( 27
)
ψ = 0 - ωr 2 2cR - ωz c -ψ spec . ( 28
)
After calculating the fields in frequency space they must be inverse Fourier transformed to time as before.

The first most basic result is shown in Fig. 8 via the fluence of the different polarization components with a single value of τ t at the focus z = 0. The simulated data are shown to replicate what would be seen on a camera when rejecting orthogonal polarizations in Figs. 8(a) and 8(b), and shows only the longitudinal component in Fig. 8(c). The total intensity shown in Fig. 8(d) replicates what would be seen on a standard camera with no analysis optics.

The qualitative results of Fig. 8 agree with our expectations that with SC along x, E y and E z will be simply elongated along x. However, nontrivially, E x is extended along x but also loses its zero at x = y = 0, seen in Fig. 8(a). Therefore, not only is the total electric field in Fig. 8(d) extended along x, but it has more nuanced asymmetries and no longer a polarization singularity on axis at x = y = 0.

However, beyond the simple asymmetry, there are effects on the temporal evolution of the electric field and the intensity that have potential applications and require a closer look.

A. Wavefront rotation

Wavefront rotation is a phenomenon with many tions with linearly polarized laser pulses [START_REF] Quéré | Applications of ultrafast wavefront rotation in highly nonlinear optics[END_REF] including the attosecond lighthouse technique for generating isolated attosecond pulses [START_REF] Vincenti | Attosecond Lighthouses: How to use Spatiotemporally Coupled Light Fields to Generate Isolated Attosecond Pulses[END_REF][START_REF] Wheeler | Attosecond lighthouses from plasma mirrors[END_REF][START_REF] Kim | Photonic streaking of attosecond pulse trains[END_REF][START_REF] Auguste | Attosecond lighthouses in gases: A theoretical and numerical study[END_REF]. The wavefront rotation, defined as a changing wavefront plane in time at one point in a laser pulses propagation, is a result of SC in a focus.

Mathematically, if we follow the definition of Ref. [START_REF] Auguste | Attosecond lighthouses in gases: A theoretical and numerical study[END_REF], the angle of the wavefront is β = [∂φ(x, t )/∂x](c/ω 0 ) and the rate of change of this angle, the WFR velocity v r = dβ/dt [rad/s], for a linearly polarized pulse is

v (lp) r = cτ t 2ω 0 s 0 ω 2 1 + (τ t ω/2) 2 , ( 29
)
with z = y = 0 and using the notation of this paper.

It can be shown that this WFR velocity can be calculated directly from the fields in frequency space using the mean frequency ω (which depends on the transverse coordinate x in the case of SC) such that v r = (∂ ω/∂x)(c/ω 0 ). This results in exactly the same relationship for RPLBs as that in Ref. [START_REF] Auguste | Attosecond lighthouses in gases: A theoretical and numerical study[END_REF] and Eq. ( 29) for linearly polarized pulses.

Being able to calculate this WFR velocity using only frequency-space intuition is helpful for this work since we model the ultrashort RPLB field in frequency space. However, additional terms in the spectral amplitude for E z and E r mean that the WFR velocity will have a transverse dependence and the meaning of it will be more complicated due to the minima in the intensity. Rather than doing this calculation we visualize the WFR.

Figure 9 shows the resulting WFR in the case of E z and E x for three values of τ t . One can see complexities in both E z and E x . For example, the WFR in E x reverses sign near the axis for both τ t = 5 fs and 15 fs, but disappears at larger values (see right of Fig. 9). The sign of the WFR also briefly changes near the off-axis intensity minima of E z when τ t = 5 fs and 15 fs, but because it is so local it manifests rather as dislocations in the electric field. Away from the axis for E x and away from the intensity minima for E z the WFR velocity is roughly uniform corresponding to Eq. ( 29). Note as well the on-axis field and amplitude is significantly modified, with a longer pulse for both E z and E x as expected, but also the temporal structure in E x .

B. Pulse-front tilt

A beam that has linear SC combined with linear temporal chirp (quadratic spectral phase or group-delay dispersion, GDD, φ 2 ) also has pulse-front tilt [START_REF] Akturk | Pulse-front tilt caused by spatial and temporal chirp[END_REF]. This is simply due to the fact that the frequencies are distributed along one spatial axis and the arrival time is also linearly dependent on frequency. This is pulse-front tilt of a different nature than AD-PFT on the collimated beam as described earlier, and since we are now looking in the focus the transverse dimension is rather s 0 .

The central frequency of a linearly polarized pulse has the same behavior as in the previous section such that ω(x) = ω 0 + xζ . The temporal phase is, in general, not linearly dependent on the GDD φ 2 , but if we assume that the chirp is large,

φ 2 ω 2 2, then ω(t ) = ω 0 + t/φ 2 . This results in a pulse-front tilt η = dt/dx = φ 2 ζ [s/m] of η = τ t φ 2 2s 0 ω 2 1 + (τ t ω/2) 2 , ( 30
)
which for linearly polarized pulses is accurate with the stated assumption of large chirp. However, as was the case with WFR, the PFT is also influenced by the more complicated off-axis spectral envelope of the RPLB such that Eq. ( 30) is not fully accurate. Figure 10 shows the developing tilt at z = 0 for a single value of τ t = 30 fs as the chirp is increased, for both |E z | 2 and |E x | 2 . The agreement with Eq. ( 30) improves as the chirp is increased, seen clearly comparing Figs. 10(a) and 10(d). The agreement for |E z | 2 is still not great when φ 2 = 600 fs 2 , Fig. 10(d), however, for |E x | 2 the agreement is very good despite the more complicated temporal intensity profile. Note counterintuitively that the peak intensity of E z is not decreasing as chirp is added in the left column of Fig. 10, and is, in fact, slightly increasing. We confirm that beyond 600 fs 2 the peak intensity does begin to decrease for the specific parameters of Fig. 10, and that for larger values of τ t the peak intensity is maximum at a larger chirp. This interesting result potentially allows for designed aberrations with a partially mitigated intensity decrease.

V. FREQUENCY-DEPENDENT BEAM PARAMETERS

In the previous two models the beam width and Rayleigh range have been approximated as constant values related to the central frequency and the pulse duration was such that the RPLB with no STCs had many cycles. However, diffraction itself and especially tight focusing is a chromatic phenomenon, and therefore the waist and Rayleigh range are in reality frequency-dependent properties: z R = ωs 2 0 /2c is a fixed relationship where either z R or s 0 can have frequency dependence according to the specifics of the physical situation. This phenomenon was known early on to effect the reshaping of single-cycle pulses through the focus [START_REF] Christov | Propagation of femtosecond light pulses[END_REF][START_REF] Ziolkowski | Propagation characteristics of ultrawide-bandwidth pulsed gaussian beams[END_REF][START_REF] Feng | Gouy shift and temporal reshaping of focused single-cycle electromagnetic pulses[END_REF][START_REF] Porras | Diffraction effects in few-cycle optical pulses[END_REF]. 30). The intensity in all panels is normalized to the max intensity of E z without any STC.

The detailed effect of a nonuniform spatiospectral beam width on the phase in the focus of ultrashort pulses (the so-called focal-phase) has been explored in-depth with linearly polarized pulses [START_REF] Porras | Characterization of the electric field of focused pulsed gaussian beams for phase-sensitive interactions with matter[END_REF][START_REF] Porras | Carrier-envelope phase shift of few-cycle pulses along the focus of lenses and mirrors beyond the nonreshaping pulse approximation: the effect of pulse chirp[END_REF][START_REF] Hoff | Tracing the phase of focused broadband laser pulses[END_REF][START_REF] Porras | Three-dimensional carrier-envelope-phase map of focused few-cycle pulsed gaussian beams[END_REF] and shown to have an effect on photoelectron production driven by such ultrashort laser pulses [START_REF] Hoff | Using the focal phase to control attosecond processes[END_REF][START_REF] Zhang | Observing the Importance of the Phase-Volume Effect for Few-Cycle Light-Matter Interactions[END_REF]. A model with RPLBs, similar to that presented here, was used to show that the effect is significant as well in vacuum laser acceleration [START_REF] Jolly | On the importance of frequency-dependent beam parameters for vacuum acceleration with few-cycle radiallypolarized laser beams[END_REF].

The basics of the model are first that Eqs. ( 1) to [START_REF] Fortin | Direct-field electron acceleration with ultrafast radially polarized laser beams: Scaling laws and optimization[END_REF] for the standard RPLB are modified with frequency-dependent Rayleigh range z R (ω) and waist s 0 (ω). To encapsulate the frequency dependence we use the "Porras factor" g 0 :

g 0 = - dz R (ω) dω ω 0 ω 0 z R (ω 0 ) . ( 31
)
We use the reference parameter of the beam waist at the central frequency s 0 (ω 0 ) = s 00 (and the corresponding Rayleigh range z R0 ), which results in the equations for the beam parameters

z R (ω) = z R0 ω 0 ω g 0 = ω 0 s 2 00 2c ω 0 ω g 0 , ( 32
)
s 0 (ω) = s 00 ω 0 ω g 0 +1 2 . ( 33
)
This is shown for three frequencies in the case of g 0 = -1, 0, and 1 in Fig. 11.

In order for the effects of the frequency-dependent parameters to be significant, the temporal duration must approach just a few optical cycles. In this regime the Gaussian spectral envelope is no longer strictly valid. We use a Poisson-like spectrum as follows:

A ω (ω) = 2 π γ ω ω 0 γ +1 1 ω e -γ ω/ω 0 (γ + 1) , ( 34
)
where γ is the parameter that now represents the pulse duration, with τ 0 = γ √ 4 (1/(γ +1)) -1/ω 0 √ 2 (1/e 2 intensity width) [START_REF] Caron | Free-space propagation of ultrashort pulses: Space-time couplings in gaussian pulse beams[END_REF], and (. . .) is the Gamma function for the purely real argument γ + 1.

Using these frequency-dependent parameters [Eqs. ( 32) and ( 33)] and Poisson-like spectral envelope [Eq. [START_REF] Howard | Photon Acceleration in a Flying Focus[END_REF]] in place of the relevant terms in Eqs. ( 1) to [START_REF] Fortin | Direct-field electron acceleration with ultrafast radially polarized laser beams: Scaling laws and optimization[END_REF], and Fourier transforming to time, produces the fields without the common frequency-independent approximation.

The main result of properly modeling the frequencydependent beam parameters is that the carrier-envelope phase (CEP) varies through the focus in a way nontrivially different than the standard Guoy phase. The key difference with the RPLB is that this behavior is not the same for the different polarization components. The CEP evolution through the focus is calculated using the relation = φ| ω 0 -ω 0 ∂φ ∂ω | ω 0 (the difference between the pulse-front and the wavefront) where φ is the total phase such that

r (r, z) = g 0 [2 -2r 2 /s 2 ] z z R0 + z R0 z -2 tan -1 z z R0 , ( 35
) z (r = 0, z) = r (r = 0, z). ( 36
)
Note that s is the usual frequency-independent value as in Eq. ( 10) and the last term in Eq. ( 35) is simply double the frequency-independent Gouy phase as in Eq. [START_REF] Huang | Optical trapping of gold nanoparticles by cylindrical vector beam[END_REF]. Due to the complexity of the phase for E z (r = 0, z), the expression for the CEP evolution is not solved for explicitly in that case except for when r = 0, where it is, in fact, equal to that of E r .

Besides the CEP evolution through the focus, the central frequency also evolves. This is easy to visualize since, as depicted in Fig. 11, the different colors are focused to different waists depending on the g 0 value. Regardless of the assumed integrated spectral envelope A ω , the central frequency must evolve through the focus since the colors that are larger outside of the focus are the smallest within the focus and vice versa. The on-axis (r = 0) CEP evolution, identical for E r and E z as in Eq. ( 36), is shown in Fig. 12 (top), and the central frequency evolution for E z is shown in Fig. 12 (bottom) for τ 0 = 3.5 fs (γ = 35). It is important to note that the CEP evolution does not change with the pulse duration, it only has a greater significance as the pulse duration decreases, but the central frequency evolution does depend strongly on the pulse duration [START_REF] Porras | Characterization of the electric field of focused pulsed gaussian beams for phase-sensitive interactions with matter[END_REF].

These on-axis results in Fig. 12 were already reported in the context of the application to vacuum acceleration of electrons [START_REF] Jolly | On the importance of frequency-dependent beam parameters for vacuum acceleration with few-cycle radiallypolarized laser beams[END_REF]. However, the off-axis CEP evolution has not yet been reported and is shown here in Fig. 13 for E z and E r for both g 0 = -1 and +1. The plotted results for E r are simply that of Eq. [START_REF] Ramsey | Vacuum acceleration of electrons in a dynamic laser pulse[END_REF], while the results for E z are calculated via the numerical differentiation of the phase of the field.

The first result to note is that the difference between the phase for varying g 0 clearly extends to off-axis positions as well both for E z and E r , most clear when comparing off-axis positions in Figs. 13(b) to 13(d). The second notable result is that for E z there are significant discontinuities as r increases. Specifically at r/s 00 = 1 and z = 0 the discontinuity is due to local reshaping of the field as shown in Fig. 13(e), which exposes a weakness of the simple method to calculate the CEP change. One could extend the analysis away from the implicit nonreshaping assumption as in Ref. [START_REF] Porras | Carrier-envelope phase shift of few-cycle pulses along the focus of lenses and mirrors beyond the nonreshaping pulse approximation: the effect of pulse chirp[END_REF], but in this specific region E z becomes very small and is therefore less important, and due to the very significant reshaping the meaning of the CEP is less clear. We therefore leave a more nuanced analysis to a further work.

VI. CONCLUSION

We presented the fields of ultrashort radially polarized laser pulses when tightly focused and having three different low-order spatiotemporal couplings: longitudinal chromatism, spatial chirp, and frequency-dependent beam parameters. We presented results mostly directly in the focus, but also included some propagation characteristics. This was presented for all three cases using a model where the effect of the STCs was developed by adding frequency dependence to parameters of the description in frequency space and confirming the existence of important effects in the time domain via numerical Fourier transformation. The lack of an analytical description directly in the time domain is an important aspect of ongoing work, and we hope that drawing upon the descriptions of other fundamental ultrashort pulses that have interesting space-time properties such as tilted beams [START_REF] Wong | Ultrashort tilted-pulse-front pulses and nonparaxial tilted-phase-front beams[END_REF], the flying donut [START_REF] Ziolkowski | Localized transmission of electromagnetic energy[END_REF][START_REF] Hellwarth | Focused one-cycle electromagnetic pulses[END_REF][START_REF] Zdagkas | Space-time nonseparable pulses: Constructing isodiffracting donut pulses from plane waves and single-cycle pulses[END_REF], or the previously mentioned space-time light sheets [START_REF] Kondakci | Diffraction-free spacetime light sheets[END_REF] may address this issue. Still, we believe that developing this model and recipe for describing low-order STCs and showing the powerful effects that take place in the focus of ultrashort radially polarized laser beams is useful without the time domain equations.

Further complexities can be imagined beyond what was presented in this paper. The first possibility of course is including higher-order STCs, such as LC or SC that are nonlinearly dependent on the frequency or frequency-dependent beam width that cannot be described by one parameter, or indeed a simple combination of the presented STCs. Or, for example, combining the presented STCs with more realistic near-field profiles for high-power laser beams such as the super-Gaussian or flattened-Gaussian [START_REF] Bagini | Propagation of axially symmetric flattened gaussian beams[END_REF], or modeling higher-order vector beams, or using the Richards-Wolf formalism [START_REF] Wolf | Electromagnetic diffraction in optical systems -I. An integral representation of the image field[END_REF][START_REF] Richards | Electromagnetic diffraction in optical systems, II. Structure of the image field in an aplanatic system[END_REF] to model very nonparaxial beams or beams diffracting from an aperture, important for the Guoy phase of RPLBs [START_REF] Pelchat-Voyer | Clarifications on the Gouy phase of radially polarized laser beams[END_REF]. Additionally, there may be nuanced behavior of either the phase or the electric field when including a more detailed analysis of the RPLBs outside of the focus. The models presented here are general and provide the framework for investigating these more complex situations.

We believe that the combination of vector beams and STCs will become of increasing interest for ultrafast optical applications as the knowledge and understanding becomes more advanced. The combination of the models and analysis presented in this work and the further development of methods to control and characterize STCs [START_REF] Jolly | Spatio-temporal characterization of ultrashort laser beams: a tutorial[END_REF] (also for vector beams [START_REF] Alonso | Complete spatiotemporal and polarization characterization of ultrafast vector beams[END_REF]) will enable this advancement.

ACKNOWLEDGMENTS

The author would like to acknowledge Fabien Quéré for helpful discussions and Martin Virte for input on the manuscript.

APPENDIX: NONPARAXIAL DESCRIPTION

Here we outline the description for nonparaxial beams according to the method used in Ref. [START_REF] Salamin | Fields of a radially polarized Gaussian laser beam beyond the paraxial approximation[END_REF] to add to the paraxial equations and results presented in the main text. The chosen method, without STCs, expands the wave equation to higher orders of the generally small parameter = s 0 /z R = λ 0 /π s 0 to have more accurate descriptions of the fields, improving upon Eqs. ( 1) to ( 3), with all of the same properties unchanged from Eqs. ( 5) to [START_REF] Fortin | Direct-field electron acceleration with ultrafast radially polarized laser beams: Scaling laws and optimization[END_REF]

(A3)

The only additional correction is on the beam power, of the form where P 0 is the physical power of the beam, equal to the power in the nonparaxial case (without STCs). It must be noted that while this description is more accurate in describing the fields for moderate values of , the convergence is not well understood as becomes comparable to 1 since it is based on an expansion in as a small parameter. When longitudinal chromatism (LC) is present z is replaced everywhere with zz 0 (ω) which effects the same terms ψ G , s, and R as in Eqs. [START_REF] Payeur | Generation of a beam of fast electrons by tightly focusing a radially polarized ultrashort laser pulse[END_REF] to [START_REF] Wong | Laser-induced linear-field particle acceleration in free space[END_REF]. This results in both the amplitude and phase of all C n and S n being frequency dependent, with the only difference being that in the nonparaxial case there are many more terms of higher n.

A np = ω 0 2c 8P np π 0 c √ 2A ω ω e -r
In the case where spatial chirp (SC) is present the modifications are also as before in the paraxial case, effecting the amplitude of all of the fields as in Eq. ( 27) and the phase as in Eq. ( 28) due to the modified r , and also requiring a frequencydependent coordinate transformation as in Eqs. [START_REF] Bourassin-Bouchet | Duration of ultrashort pulses in the presence of spatio-temporal coupling[END_REF] to [START_REF] Jolly | On the importance of frequency-dependent beam parameters for vacuum acceleration with few-cycle radiallypolarized laser beams[END_REF]. The main difference then for the nonparaxial case is the higher orders of ρ that now need to be described by the frequencyvarying ρ , and as for LC the phase effects are present as well in the higher orders of C n and S n .

For the third and final STC described in the main text, frequency-varying beam parameters z R and s 0 need to be replaced by their frequency-dependent form depending on the g 0 value as in Eqs. ( 32) and [START_REF] Turnbull | Raman Amplification with a Flying Focus[END_REF]. This introduces frequency dependence in , ρ, s, ψ G , and R which in sum effect both the phase and amplitude. And again, the primary additional complexity in the nonparaxial case is the higher orders present of , ρ, C n , and S n .

In summary, the nonparaxial equations have been presented in this Appendix and a recipe for adapting each of the three STCs presented in the main text to the nonparaxial form has been given here. However, this could only be in a general matter since in all cases both the amplitude and phase develop frequency dependence that is of much higher order in the nonparaxial form and therefore nontrivial. It must be noted that for the parameters presented in the main text (λ 0 = 800 nm, s 0 = 4 μm, = 0.0637) the nonparaxial results are qualitatively the same, confirmed by numerical calculations, and the fields and intensity are very similar quantitatively. Only for extremely sensitive interactions such as vacuum electron acceleration would the nonparaxial fields be important with such focusing [START_REF] Marceau | Validity of the paraxial approximation for electron acceleration with radially polarized laser beams[END_REF]. Indeed for tighter focusing (larger ) the nonparaxial form becomes more important, and would begin to significantly effect especially the field-based effects presented in the main text, specifically wavefront rotation and the changing CEP through the focal volume.

FIG. 1 .

 1 FIG. 1. Standard ultrashort RPLB in-focus with λ 0 = 800 nm, τ 0 = 15 fs, and s 0 = 4 μm. The integrated E z (a) is strongly localized on-axis, where the integrated E r (b) has a zero on-axis. The E z and E r fields in time, (c) and (d), respectively, show similar characteristics, but as they diffract they change [shown at z = 5z R in (e) and (f), respectively]. The fluence in (a) and (b) is normalized to the max fluence of E z , and the colorscale in each of panels (c)-(f) is relative to the maximum in that panel.

FIG. 2 .

 2 FIG. 2. Basic concept of longitudinal chromatism (LC) in the focus. A beam in the near field (a) before being focused can have pulse-front curvature (PFC) where the arrival time depends quadratically on the radius, and equivalently the frequencies have a linearly varying convergence. When focused the frequencies focus to different longitudinal positions and produce LC, shown on the right in (a). The scenario relevant for modeling the RPLB is shown schematically in (b), now with multiple components of the field.

  ) and3(b). This is seen very clearly when comparing Figs.3(a) and 3(b) directly to Figs. 1(a) and 3(b), where the only difference is the LC.

FIG. 3 .

 3 FIG. 3. Basic results with longitudinal chromatism τ p = 60 fs (λ 0 = 800 nm, τ 0 = 15 fs, s 0 = 4 μm). The integrated E z and E r fields in-focus, (a) and (b), respectively, show larger tails as r increases. The E z and E r fields in time, (c) and (d), respectively, now have a time asymmetry and fringes at larger r characteristic of chromaticity, but the qualitative characteristics stay the same. The fluence in (a) and (b) is normalized to the max fluence of E z without any LC and the intensity in (c) and (d) is normalized to the max intensity of E z without any LC.

  FIG. 4. Increased Rayleigh range due to the longitudinal chromatism of both the longitudinal field E z , (a) with no LC and (c) with τ p = 60 fs, and transverse field E r , (c) with no LC and (d) with τ p = 60 fs (τ 0 = 15 fs, s 0 = 4 μm).

and 4 (

 4 b) without LC to Figs. 4(c) and 4(d) with τ p = 60 fs the Rayleigh range is roughly 2τ p /τ 0 = 8 times larger in the latter case. What is not shown in Figs. 4

FIG. 5 .

 5 FIG. 5. Flying focus effect with the longitudinal field E z at r = 0. An RPLB with an LC of τ p = 60 fs (τ 0 = 15 fs, s 0 = 4 μm) is simulated with varying linear chirp (φ 2 ) to produce an intensity envelope that travels at velocities much different than c (a)-(c). The dependence of the intensity peak velocity v f f on chirp (d) in the analytical equation Eq. (16) (black dashed line) agrees with the full simulations (red solid line). The intensity decreases (e) from the case with no STCs with different values of τ p and with increasing chirp.

FIG. 6 .

 6 FIG.6. Flying focus effect with the transverse field E r at the offaxis position r = s 0 . An RPLB with an LC of τ p = 60 fs (τ 0 = 15 fs, s 0 = 4 μm) is simulated with varying linear chirp (φ 2 ) to produce an (a)-(c) intensity envelope that travels at velocities much different than c, slightly different than those for E z on-axis. The dependence on φ 2 (d) is slightly different in this case as well (red solid line), diverging more from the simple prediction (black dashed line) based on Eq. (16).

FIG. 7 .

 7 FIG. 7. Basic concept of spatial chirp (SC) in the focus. A beam in the near field (a) before being focused can have pulse-front tilt (PFT) where the arrival time depends linearly on one spatial coordinate, and equivalently the frequencies have a linearly varying pointing direction, or angular dispersion (AD). When focused (b) the frequencies separate along the same axis as the AD/PFT and produce spatial chirp (SC). Due to the breaking of cylindrical symmetry both E r and B θ require a frequency-dependent transformation to Cartesian coordinates, shown in (c) and (d), respectively.

FIG. 8 .

 8 FIG. 8. The intensity of the different electric field components (a) |E x | 2 (b) |E y | 2 and (c) |E z | 2 are shown with an SC in the x direction of τ t = 30 fs (λ 0 = 800 nm, τ 0 = 15 fs, s 0 = 4 μm). These show the asymmetry and elongation along x due to the SC. The total intensity (d), which one would see on a standard camera, is also elongated along x and is asymmetric. The fluence in all panels is normalized to the max fluence of E z without any STC.

FIG. 9 .

 9 FIG. 9. Varying wavefront rotation (WFR) with different spatial chirp values, parametrized by τ t , of (a) 5 fs, (b) 15 fs, and (c) 30 fs (τ 0 = 15 fs, s 0 = 4 μm). The WFR is clearly visible and increasing in E z (left), but is less clear in E x (right), discussed in more detail in the text.

FIG. 10 .

 10 FIG. 10. Adding linear chirp (nonzero φ 2 ) to an RPLB having SC can lead to in-focus pulse-front tilt of a different nature than standard AD/PFT on a collimated beam. This is shown with τ t = 30 fs on all beams and an increasing chirp of (a) 75 fs 2 , (b) 150 fs 2 , (c) 300 fs 2 , and (c) 600 fs 2 (τ 0 = 15 fs, s 0 = 4 μm). The PFT manifests in both |E z | 2 (left) and |E x | 2 (right). The white dashed lines are the PFT predicted by Eq. (30). The intensity in all panels is normalized to the max intensity of E z without any STC.

FIG. 11 .

 11 FIG.11. The basic concept of g 0 shown with three frequencies and three g 0 values. When g 0 = -1 (left) all frequencies have the same beam size in the focus, where in contrast when g 0 = 1 (right) all three frequencies diverge with the same opening angle outside of the focus. g 0 = 0 (middle) is an intermediate situation.

FIG. 12 .

 12 FIG.12. On-axis CEP and central frequency evolution through the focus with a duration of 3.5 fs (γ = 35). The CEP (top) varies widely, showing a steep slope at g 0 = -2 and an inflection when g 0 1. The central frequency (bottom) also varies through the focus due to the reshaping of the frequency envelope as different frequencies are focused more tightly. This figure is identical to one presented in Ref.[START_REF] Jolly | On the importance of frequency-dependent beam parameters for vacuum acceleration with few-cycle radiallypolarized laser beams[END_REF].

FIG. 13 .

 13 FIG. 13. Off-axis CEP evolution through the focus. The CEP for both (a), (c) E z and (b), (d) E z are shown through the focus for onand off-axis positions, with a g 0 of (a), (b) -1 and (c), (d) +1. The discontinuities in the CEP of E z are due to pulse reshaping around r/s 00 = 1 and z = 0, shown in (e) for a pulse duration of 3.5 fs (γ = 35) with g 0 = 1 at four z/z R0 positions of -1/4, -1/8, +1/8, +1/4 from left to right, with the field (blue solid line) and the amplitude (orange dashed line).

  . Specifically,Ẽr = A np [ρC 2 ] + 3 -ρC 3 2 + ρ 3 C 4 -ρ 5 C 5 4

	+ 5 -	3ρC 4 8	-	3ρ 3 C 5 8	+	17ρ 5 C 6 16	-	3ρ 7 C 7 8	+	ρ 9 C 8 32	,
												(A1)
		Ẽz = A np		2 [S 2 -ρ 2 S 3 ]
			+ 4 S 3 2	+	ρ 2 S 4 2	-	5ρ 4 S 5 4	+	ρ 6 S 6 4	,	(A2)
	Bθ =	A np c	[ρC 2 ] + 3 ρC 3 2	+	ρ 3 C 4 2	-	ρ 5 C 5 4
		+ 5 3ρC 4 8	+	3p 3 C 5 8	+	3ρ 5 C 6 16	-	ρ 7 C 7 4	+	ρ 9 C 8 32	.