## Abstract

We introduce Bessel-Gauss beam enhancement cavities that may circumvent the major obstacles to more efficient cavity-enhanced high-field physics such as high-harmonic generation. The basic properties of Bessel-Gauss beams are reviewed and their transformation properties through simple optical systems (consisting of spherical and conical elements) are presented. A general Bessel-Gauss cavity design strategy is outlined, and a particular geometry, the confocal Bessel-Gauss cavity, is analyzed in detail. We numerically simulate the confocal Bessel-Gauss cavity and present an example cavity with 300 MHz repetition rate supporting an effective waist of 33 μm at the focus and an intensity ratio from the focus to the cavity mirror surfaces of 1.5 *×* 10^{4}.

©2012 Optical Society of America

## 1. Introduction

Interest in strong-field physics has exploded over the past several decades. High-harmonic generation (HHG) has provided a route to compact, coherent sources of short-wavelength light in the extreme-ultraviolet (XUV) and soft x-ray regime, attosecond science is pushing temporal resolution to the atomic scale, and new applications are constantly appearing. To reach the necessary intensities (> 10^{13} W/cm^{2}) for strong-field physics, complex amplifier systems are generally required. Such amplifiers can readily produce millijoule pulses of tens of femtoseconds in duration, however only after reducing the seed oscillator’s repetition rate from near 100 MHz down to the kHz regime. In recent years femtosecond enhancement cavities have emerged as an alternative route to achieving the high-intensities necessary for strong-field physics with the additional advantage of maintaining the driving oscillator's high repetition rate [1–6].

Inside femtosecond enhancement cavities, pulses are enhanced through constructive interference. After a single round-trip through the cavity, a pulse constructively interferes, adds to, the next pulse in the optical pulse train. Femtosecond enhancement cavities have been demonstrated with intra-cavity average powers of kilowatts and peak intensities > 10^{14} W/cm^{2} [3,4,7]. Intra-cavity HHG (i.e. cavity-enhanced HHG) has been demonstrated with average powers per harmonic in the XUV spectral region of tens of microwatts [3,4]. Additionally, the high-repetition rates of these sources has allowed for spectroscopic studies in this hard-to-reach spectral region and branded them XUV frequency combs [4].

Cavity enhancement techniques for high-intensity applications are not without limitations. These limitations are primarily out-coupling (e.g. efficiently coupling the high-harmonics out of the cavity in cavity-enhanced HHG) and intensity gain (i.e. ensuring an intensity ratio from focus to mirrors such that high-intensities can be reached at the focus while not damaging the cavity mirrors). To date, the high-intensity enhancement cavity of choice has been the bow-tie Gaussian cavity. For cavity-enhanced HHG, the harmonics are generated collinearly with the driving light, so special out-coupling optics have been necessary. A sapphire plate, placed near the cavity focus, can provide a small Fresnel reflection to high-harmonics while passing the driving light at Brewster’s angle [1–3]. However, the Fresnel reflection yields relatively low out-coupling efficiencies of < 20% [1,2,8], nonlinearities in the plate (due to the high-drive intensities) decrease the cavity enhancement [8], and the plate can be damaged by high XUV powers [3]. Alternatively, an XUV grating etched onto a highly-reflective cavity mirror can diffract high-harmonics out of the cavity while avoiding problematic nonlinearities; however, out-coupling efficiencies are limited to ~10% [4,5]. Out-coupling the harmonics through a small hole in a cavity mirror has also been explored [6,8,9]. Superposing several higher-order modes of the bow-tie cavity, an intra-cavity mode with low-intensity near the hole can be formed [8,9]; however, such techniques have been limited to small holes of diameter ≈100 μm [6,8,9]. As for intensity gain, scaling studies of the bow-tie cavity have been performed and shown mirror damage thresholds of state-of-the-art ion-beam sputtered dielectric mirrors up to nearly 10^{11} W/cm^{2} [7]. In high-intensity bow-tie enhancement cavities, the mode waist on the cavity mirrors has been restricted to around 1 mm in size [1–7], so to go beyond the achieved intra-cavity peak intensities and reach those required for futuristic cavity-enhanced applications like electron-acceleration or inverse-Compton scattering, a new cavity design, allowing higher intensity ratios from focus to mirrors, may be necessary.

In this work, an alternative high-intensity enhancement cavity design, based on Bessel-Gauss type beams [10–12], is presented. These cavities allow for large (> 1 mm) diameter holes in the cavity mirrors as well as centimeter size effective mode diameters on the cavity mirrors. These Bessel-Gauss type cavities allow efficient out-coupling as well as increased intensity gain for future cavity-enhanced high-intensity physics. The paper is organized as follows: in Section 2 the basic properties of Bessel-Gauss beams are presented. The derivation and definition of the different Bessel-Gauss beam types are quickly reviewed, the focal properties are discussed, and the transformation properties of Bessel-Gauss beams through spherical and conical optical elements are presented. In Section 3 Bessel-Gauss cavities are introduced. A general cavity design procedure is presented, a specific geometry (the confocal Bessel-Gauss cavity) is considered in detail analytically and numerically, and potential challenges in realizing high-intensity Bessel-Gauss enhancement cavities are discussed. In Section 4, we conclude this work.

## 2. Bessel-Gauss beams

#### 2.1 Bessel-Gauss beams from decentered Gaussians

Bessel-Gauss beams are exact solutions to the paraxial wave equation and provide a physically realizable approximation to the non-diffracting Bessel beam [10]. In the following, we will take an intuitive approach to Bessel-Gauss beams and derive them from decentered Gaussian beams (this subsection parallels earlier work [11]).

Consider, in the *z =* 0 plane of a cylindrical coordinate system (*r, θ, z*), a field of the form:

*u*(

*r, θ, z =*0) resembles a Gaussian beam of waist

*w*whose central wavevector has a component of magnitude

_{0}*β*in the

*z =*0 plane inclined at an angle

*γ*to the

*x*-axis (illustrated in Fig. 1(a) ). Defining ${q}_{0}=-i{z}_{0}=-ik{w}_{0}^{2}/2$ and propagating

*u*(

*r, θ, z =*0), a Gaussian-like beam is found [11]:

*z*-dependent phase terms (e.g. $\mathrm{exp}(ikz)$) for simplicity. The beam

*u*(

*r, θ, z*) defined by Eq. (2) closely resembles a Gaussian beam propagating at an angle $\phi ={\mathrm{sin}}^{-1}(\beta /k)$ to the

*z*-axis as shown in Fig. 1(a). The

*q*-parameter for this beam transforms like that for an on-axis Gaussian beam. The center of the beam, i.e. the intensity maximum, follows

*r*(

_{c}*z*) at an angle

*φ*to the

*z*-axis. We refer to the type of beam in Eq. (2) as the decentered Gaussian beam (or decentered beam for short) [13].

Consider the different decentered Gaussian beams produced as we let *γ*, the inclination angle of *β* with respect to the *x*-axis, vary. We see the central wavevectors of the different decentered Gaussian beams trace out the surface of a cone with semi-aperture angle *φ*. Superposing these different decentered Gaussian beams, we obtain:

*J*, the zero order Bessel function of the first kind (see pp. 140 of Ref [14].),

_{0}*A*is a constant, and

_{0}*q*(

*z*) and

*r*(

_{c}*z*) are given in Eq. (3). In Eq. (4) we have a beam that at the focus (

*z =*0) resembles a Gaussian modulating a Bessel function. This is the Bessel-Gauss beam (called the BG beam from here on).

We can generalize the BG beam by superposing decentered Gaussian beams with centers lying on a circle of radius *r _{0}* (illustrated in Fig. 1(b)). The central wavevectors of these decentered Gaussians make up the surface of a frustum (i.e. a truncated cone) with semi-aperture angle

*φ*. This generalization amounts to letting ${r}_{c}(z)\to {r}_{0}+{r}_{c}(z)$. This form of beam is known as the generalized Bessel-Gauss beam (called the gBG beam from here on),:

*U*(

_{gBG}*r, z*), with a Bessel function of a complex argument, may not easily reveal the essential properties and behaviors of the gBG beam, the beam can intuitively be understood by recalling that it is a superposition of physically-intuitive decentered Gaussian beams. The

*r-z*plane cross-section of a gBG beam, consisting of intersecting decentered Gaussian beams, is illustrated in Fig. 2(a) , and the amplitude is plotted for a specific gBG beam in Fig. 2(d).

There are two special cases of the gBG beam that are of interest [11]. Firstly, the BG beam corresponds to a gBG beam with *r _{0} =* 0. The

*r-z*plane cross-section of a BG beam is illustrated in Fig. 2(b), and the amplitude is plotted for a specific BG beam in Fig. 2(e). Secondly, consider a gBG beam with

*β =*0 and

*r*0. This is the modified Bessel-Gauss beam (mBG beam from here on) and is a superposition of decentered Gaussian beams lying along the surface of a cylinder with radius

_{0}≠*r*. The

_{0}*r-z*plane cross-section of a mBG beam is illustrated in Fig. 2(c), and the amplitude is plotted for a specific mBG beam in Fig. 2(f).

#### 2.2 Bessel-Gauss beam focal properties and intensity gain

Here, we summarize the focal properties of BG beams essential for our purposes. (We discuss only BG beams as our cavity designs will consist of BG beams at the foci.) As already described, at the focal plane the BG beam takes the form of a Gaussian component modulating a Bessel function (an *r-θ* plane cross-section of the BG beam amplitude at the focus is plotted in Fig. 3(a)
). From Eq. (4), ${U}_{BG}(r,z=0)={A}_{0}\mathrm{exp}(-{r}^{2}/{w}_{0}^{2}){J}_{0}(\beta r)$. The peak intensity of a BG beam at its focus, *I _{P}^{foc}*, can then readily be found (using the integral in Eq. (2).3) of Ref [10].):

*P*is the beam power,

*I*is the zero-order modified Bessel function of the first kind, ${w}_{eff}^{foc}$ is the effective beam waist of the BG beam at the focus, and

_{0}*w*2.4/

_{B}=*β*is the approximate waist of the Bessel component (i.e. the first zero of

*J*(

_{0}*βr*), illustrated in Fig. 3(a)). The approximate form of ${w}_{eff}^{foc}$given in Eq. (7) follows from inspecting the argument of the Bessel function in Eq. (7). We see ${\beta}^{2}{w}_{0}^{2}/4={\left(\phi /{\phi}_{G}\right)}^{2}$where $\phi \approx \beta /k$is the semi-aperture angle of the BG beam (as already described) and ${\phi}_{G}=2/k{w}_{0}$is the divergence angle of the component decentered Gaussian beams. Since we are interested in BG beams that result in an annular (i.e. donut) shape far from the focus, we must have

*φ*<<

_{G}*φ*i.e the Gaussian components must diverge slower than their peak intensity axes spread apart. So, for the regime of interest ${\beta}^{2}{w}_{0}^{2}/4={\left(\phi /{\phi}_{G}\right)}^{2}>>1$, and the asymptotic expansion of

*I*(pp. 116 of Ref [14].) yields the approximate form of ${w}_{eff}^{foc}$.

_{0}As mentioned, far from the focus, the BG beams of interest resemble an annular shape (the BG beam amplitude far from the focus is plotted in Fig. 3(c)). The amplitude of the BG beam far from the focus can be approximated as an annulus with Gaussian cross-section i.e. ${U}_{BG}(r,z)\approx \left({B}_{0}/\sqrt{r}\right)\mathrm{exp}(-(r-{r}_{c}(z){)}^{2}/w{(z)}^{2})$ where $w(z)={w}_{0}\sqrt{1+{(z/{z}_{0})}^{2}}$, *z >> z _{0}*, and

*B*is a constant [12]. Using this expression, the peak-intensity of the BG beam far from the focus at position

_{0}*z*can be approximated:

*r*(

_{c}*z*) is as in Eq. (3) i.e. the peak-intensity axes of the component decentered Gaussian beams (illustrated in Fig. 3(c)), and

*w*(

*z*) is as defined above i.e. the waist of the component decentered Gaussian beams at

*z*(illustrated in Fig. 3(c)).

We can now put together a simple expression for the intensity gain of a BG beam. Let us re-iterate that we define intensity gain as the ratio of the peak intensity at the focus to the peak intensity at the position *z*, so *I _{g}*(

*z*)

*= I*where

_{P}^{foc}/I_{P}^{FF}(z)*I*(

_{g}*z*) is the intensity gain. As defined above, intensity gain is obviously a parameter of relevance for high-intensity enhancement cavities. For a Gaussian beam, we readily see that ${I}_{g}^{G}(z)=\pi w{(z)}^{2}/\pi {w}_{0}^{2}\approx {(z/{z}_{0})}^{2}$where

*z*>>

*z*. Combining Eq. (7) and Eq. (8) we find the intensity gain for a BG beam:

_{0}Recalling that for the beams of interest *φ >> φ _{G}* and comparing the intensity gain expressions in Eq. (9), we see that the BG beam's intensity gain can exceed that of the Gaussian beam by orders of magnitude. In Fig. 3(b), the intensity gain of a Gaussian beam with

*w*30 μm is compared to that of BG beams with Gaussian component

_{0}=*w*30 μm and semi-aperture angles

_{0}=*φ*of 1°, 2°, 3°, and 4°. In Fig. 3(b), the green curves represent exact numerical calculations and the orange curves are based on the approximate form in Eq. (9). From Fig. 3(b) we see that our approximate expression is very accurate far from the focus (

*z >> z*). Additionally, we see that for the reasonable parameters plotted, the intensity gain of a BG beam may far exceed that of a normal Gaussian, and the BG beam may allow cavity geometries with intensity gains far exceeding those of bow-tie Gaussian cavities.

_{0}#### 2.2 Generalized Bessel-Gauss beams through optical elements

We consider now the transformation of gBG beams by spherical and conical optical elements. Spherical optical elements are those that impart a quadratic spatial phase to a wavefront e.g. a thin lens or a spherical mirror. Conical optical elements give a linear (i.e. ~*iαkr*) spatial phase to wavefronts e.g. transmitting or reflecting axicons. The importance of these elements in manipulating gBG beams will be discussed.

Consider the spatial phase, i.e. the *r*-dependent phase, of a gBG beam at plane *z = L*, *U _{gBG}*(

*r, z = L*). Denoting this phase by ${\varphi}_{gBG}(r)$ we find (from Eq. (5)):

*R*(

*L*)

*= L + z*

_{0}^{2}

*/ L*. The Gaussian part of the gBG beam gives a quadratic phase while the Bessel part contributes the last-term in Eq. (10). For a large class of gBG beams, we can accurately approximate (as shown and discussed in the Appendix) the last term in Eq. (10):

*z = L*, changes the linear part of an incident gBG beam's spatial phase. The overall functional form of this phase remains unchanged however, and to account for the new linear part of the spatial phase, the gBG beam transforms to a new gBG beam with altered parameters (

*q*,

_{0}'*r*,

_{0}'*β'*). From a straightforward calculation we determine these altered parameters; they are included in Eq. (12). A spherical element, with spatial phase ${\varphi}_{sph}(r)=-ik{r}^{2}/2f$ at

*z = L*, changes the quadratic part of the gBG beam's spatial phase while leaving the overall functional form unchanged. Similarly, a gBG beam transforms after a spherical element into another gBG beam with new parameters (

*q*,

_{0}'*r*,

_{0}'*β'*). This transformation has been previously described in detail [15]. After conical or spherical optical elements the gBG beam parameters transform as:

*z' = z - L*i.e.

*z'*is the distance to the optical element at

*z = L*.

Considering the gBG transformation properties in Eq. (12) and Eq. (13), we can formulate an intuitive picture of gBG beam propagation through conical and spherical optical elements. Propagation through such elements can be compactly summarized as follows:

- (1) Through conical optical elements, (
*a*) the Gaussian component i.e.*q*-parameter of a gBG beam is unaffected; and (*b*) the peak-intensity axes of the decentered component beams follow the trajectories of meridional rays through the element. - (2) Through spherical optical elements, (
*a*) the Gaussian component i.e.*q*-parameter transforms like that of an on-axis Gaussian beam; and (*b*) the peak-intensity axes of the decentered Gaussian component beams follow the trajectories of meridional rays.

These basic propagation rules are demonstrated and tested through three examples illustrated in Fig. 4
. In the first example (illustrated in Fig. 4(a)-(c)), consider a spherical mirror with radius of curvature *R =* 20 cm at position *z = R =* 20 cm and an incident mBG beam of wavelength *λ =* 1 μm, Gaussian component waist *w _{0} =* 300 μm, and

*r*1 mm (the focal plane is

_{0}=*z = R*/2

*=*10 cm as shown in Fig. 4(b)). The mBG beam propagates, reflects from the mirror, and transforms to a new gBG type beam. From Eq. (13) and the above discussion, we expect (

*a*) the Gaussian component waist of the new gBG beam to be

*w*= 106 μm (as for an on-axis Gaussian) and (

_{0}' = λf/ πw_{0}*b*) the mBG beam to transform into a BG beam with its focus at

*z = R/2*(meridional rays parallel to the optical axis transform to meridional rays intersecting the optical axis at the focus). In Fig. 4(b) we plot an

*r-z*plane cross-section of the numerically simulated amplitude in this scenario and observe the expected behavior. In Fig. 4(c) we plot cross-sections in the

*r*direction of the spatial amplitude and phase of the field at the end of propagation and see our numerical (blue) simulation agrees to a high degree of accuracy with our analytical prediction from Eq. (13) (red-dashed). The wave-propagation software used for numerical simulation will be discussed in the next section

For the second example (illustrated in Fig. 4(d)-(f)), the same mBG beam from above propagates through the same geometry and reflects from a reflecting axicon of apex angle *α =* 0.57°. In this example we expect the mBG beam to become a gBG beam (we do not expect the Gaussian waist of the component decentered beams to occur at their intersection point). In Fig. 4(e) an *r-z* plane cross-section of a numerical simulation of the amplitude is plotted, and we observe the expected behavior. In Fig. 4(f) an *r* direction cross-section of the field's spatial amplitude and phase at the end of propagation are plotted, and our numerical (blue) simulation agrees well with our analytical prediction from Eq. (12) (red-dashed).

Finally, the third example (illustrated in Fig. 4(g)-(i)) contains a hybrid conical-spherical optic. The optic is a general toroidal optical element i.e. a spherical and a conical element separated by zero distance. The same mBG beam from the prior two examples propagates through the same geometry and reflects from this toroidal element. Recall that a conical optical element adjusts the tilt parameters of a gBG beam while leaving the Gaussian parameters alone (i.e. the conical element affects only the peak-intensity axes of the decentered component beams), and a spherical optical element adjusts all the parameters of a gBG beam. Therefore, by combining a conical and spherical element into a general toroidal optic, the tilt parameters (i.e. *r _{0}* and

*β*) and Gaussian parameter (i.e.

*q*-parameter) of a gBG beam can be independently adjusted by one optical element. Our final example illustrates this as the toroidal element consists of a spherical part of radius of curvature

*R =*20 cm and a conical part with tilt such that the focus (i.e. the point of intersection for the decentered component beams) will lie at exactly

*z =*2

*R*/3. In Fig. 4(h) an

*r-z*plane cross-section of a numerical simulation of the amplitude is plotted, and again, we observe the expected behavior. Figure 4(i) shows an

*r*direction cross-section of the field's spatial amplitude and phase at the end of propagation, and our numerical (blue) simulation agrees well with our analytical prediction from Eq. (12) and Eq. (13) (red-dashed).

## 3. Bessel-Gauss beam enhancement cavities

In the following, we build upon the properties of Bessel-Gauss beams derived and discussed in the previous section to analyze Bessel-Gauss beam enhancement cavities. Prior work has explored Bessel-Gauss beam cavities with axicons and flat mirrors [16], axicons and curved mirrors [17,18], and general phase-conjugating optics [19]; however, this past work has focused on Bessel-Gauss cavities for use as laser resonators. In the following section, we extend this body of work on Bessel-Gauss cavities to enhancement cavities. We outline a different general approach to designing gBG beam cavities, discuss in detail a particular novel gBG cavity i.e. the confocal BG cavity, and comment on future challenges in realizing high-intensity gBG cavities.

#### 3.1 Bessel-Gauss cavity design strategy

Consider an enhancement cavity composed of two spherical and two flat mirrors supporting a Gaussian beam solution as illustrated in Fig. 5(a)
. The Gaussian beam is re-imaged as it traverses the cavity i.e. *q*(*z +* 2*L*) *= q*(*z*) where 2*L* is the round-trip cavity length. Additionally, for small-angles, the Gaussian beam's peak intensity axis follows that of a ray through the system.

For an enhancement cavity to support a gBG mode, the intra-cavity gBG beam's Gaussian parameter (i.e. *q*-parameter) and tilt parameters (i.e. *r _{0}* and

*β*) must repeat after every round-trip. (Recall that the gBG beam's

*q*-parameter is associated with the Gaussian properties (e.g. waist) of the component decentered Gaussian beams, and the tilt parameters are associated with the peak-intensity axes of the component decentered beams.) Consider the

*r-z*plane cross-section of our conventional Gaussian cavity. If we revolve this cross-section about its central axis (as illustrated in Fig. 5(b)), the tilted flat mirrors become conical optical elements, and the spherical mirrors become toroidal optical elements (these elements can be imagined as different sections of one complex, segmented mirror structure as illustrated in Fig. 5(b)). Recalling the transformation properties of gBG beams, we see this cylindrically symmetric cavity structure supports a gBG mode that is composed of decentered component beams that closely resemble the Gaussian mode of the conventional Gaussian cavity. This link between conventional Gaussian cavities and Bessel-Gauss cavities is a powerful one. It allows us to directly generate gBG cavity designs from well-known Gaussian ones (albeit the gBG cavities may demand sophisticated mirror structures that are non-trivial to fabricate). In this initial discussion, we restrict our focus to gBG cavities that require only spherical mirrors (in particular, the confocal BG cavity). Before embarking on this discussion, we include a brief description of the mode-solver we use to numerically analyze the confocal BG cavity.

#### 3.1 Numerical simulations

Our cavity mode solver is based on the scattering matrix method for optical systems and is designed for cylindrically symmetric cavity geometries [20]. Cylindrically symmetric cavity modes are represented as *N*-dimensional column vectors (the radial coordinate is discretized into *N* points). Each optical element composing the cavity, including lengths of dielectric or vacuum, is described by a 2*N ×* 2*N* scattering matrix. (Scattering matrices for optical systems are generally 2 *×* 2 matrices relating incoming waves to outgoing ones [20,21]; here, each radial point has its own scattering matrix and lumping all the points together, we represent each element as a 2*N ×* 2*N* scattering matrix). Lengths of dielectric or vacuum have block diagonal scattering matrices where each block is a matrix describing propagation. Using the exact matrix representation of the quasi-discrete Hankel transform (denoted here as *F*) [22,23], each propagation block can be written as ${P}_{\lambda ,z}={F}^{-1}\mathrm{exp}\left(ikz\sqrt{1-{(\lambda \nu )}^{2}}\right)F$where *λ* is the wavelength, *k* is the wavevector, ν is the spatial frequency, and *z* is the propagation length. Propagation amounts to transforming the wavefront to the spatial frequency domain, weighting each spatial frequency by the correct phase factor for propagation, and transforming back to the spatial domain (the matrices *P _{λ,z}* were used for propagation in the simulations in Fig. 4). After forming scattering matrices for each individual cavity component, these matrices can be composed to form a scattering matrix of the complete cavity system [20]. The entire cavity can then be represented by a single 2

*N ×*2

*N*dimensional matrix. The cavity modes correspond to the eigenvectors of this matrix and can be found by any standard numerical eigenvalue solver. An obvious advantage of our mode solver is the ability to immediately solve for all the higher order modes of a cavity system. This does come with the disadvantage of having to store and manipulate a possibly large 2

*N ×*2

*N*matrix; for all simulations in this paper however, the modes were solved for on a desktop computer with a radial step-size of < 1 μm in a matter of minutes.

#### 3.2 Confocal Bessel-Gauss cavity

Possible cavity arrangements based on two spherical mirrors are the confocal cavity (*L = R* where *L* is the mirror separation and *R* is the mirror radius of curvature) and the concentric cavity (*L =* 2*R*). In the following we will discuss the confocal cavity and show it supports BG type modes.

The confocal cavity is degenerate i.e. every other Hermite-Gaussian mode of the confocal cavity shares the same resonance frequency. These modes can then simultaneously resonate in the cavity and superpose to form different field profiles. To restrict the cavity to operate only in a single BG type mode, we consider patterning the cavity mirrors in an annular (i.e. donut) shape. The annulus is highly reflective (reflectivity *R _{H}*) and has average radius

*r*and thickness Δ

_{avg}*r*(illustrated in Fig. 6(a) ); the rest of the mirror surface has a low reflectivity (

*R*). The highly reflective annular pattern yields low-loss to only a single BG mode: the BG mode composed of minimally-divergent decentered Gaussian beams (illustrated in Fig. 6(b)). From the cavity center to the mirror surface (a distance of

_{L}*R*/2) there exist minimally divergent decentered Gaussian beams. These beams have a waist ${w}_{0,\mathrm{min}}=\sqrt{\lambda R/2\pi}$at the cavity center, and ${w}_{\mathrm{min}}=\sqrt{2}{w}_{0,\mathrm{min}}$at the mirror surfaces. All other decentered Gaussians and higher-order decentered Hermite-Gaussians have a larger waist at the mirror surface. Therefore, if the width of the patterned annulus is chosen to be small enough (i.e. Δ

*r*~3

*w*), then only the BG mode composed of decentered Gaussians with waist

*w*

_{0,min}will have low-loss. This method of single-mode selection is analogous to inserting an iris in a laser resonator to restrict the output to the fundamental Gaussian mode. The average radius of the highly reflective annulus determines the tilt angle of each decentered component beam and accordingly of the BG mode, $\phi ={\mathrm{tan}}^{-1}\left(2{r}_{avg}/R\right)$.

Using our cavity mode-solver, we simulate an example patterned-mirror confocal cavity (shown in Fig. 7
). This cavity's patterned mirrors have parameters: *r _{avg} =* 8 mm, Δ

*r =*3.1

*w*

_{min}

*=*1.2 mm,

*R*1, and

_{H}=*R*0.1. The mirror radius of curvature is

_{L}=*R =*50 cm and spacing is

*L =*49.97 cm. The cavity is simulated at wavelength

*λ =*1 μm. From the

*r-z*plane cross-section plot of the mode amplitude in Fig. 7(a), we see, as expected, the cavity mode resembles a BG beam through one pass of the cavity (through the focus) and transforms at the cavity mirror to a mBG beam for the return trip. In Fig. 7(b) and 7(c), we plot the intensity in the radial direction at the cavity mirror and at the focus, respectively (labeled in Fig. 7(a)). From these plots we see our numerical simulation (blue) agrees well with the analytically expected mode (red-dashed). Additionally, normalizing the peak intensity at the cavity mirror, we see the peak intensity at the focus is

*I*1.5

_{g}=*×*10

^{4}(this is the intensity gain). We also see the effective waist at the focus is

*w*33 μm. From our mode-solver, we find the loss of the fundamental mode plotted in Fig. 7 is < 0.0011%, and the loss of the next higher-order mode is > 2.5% (note this is exclusively diffraction-loss as

_{eff}=*R*1). These losses can be fine tuned by adjusting Δ

_{H}=*r*. Additionally, we note that although we simulate a continuous-wave cavity, the patterned mirror confocal cavity supports a wide bandwidth. Simulating the example cavity above at

*λ =*950 nm and

*λ =*1050 nm, we find the fundamental mode has < 0.0015% loss and the next higher-order mode has > 1.7% loss. Finally, we should note for our example cavity

*L ≠ R*; this is due to a non-paraxial propagation effect. For even modest tilt angles (for this cavity, $\phi ={\mathrm{tan}}^{-1}\left(2{r}_{avg}/R\right)=1.8\xb0$), non-paraxial propagation leads to small spatial phase shifts, and to maintain low-loss modes we must have

*L = R*cos

*φ*.

The example cavity above shows virtually no intensity on the optical axis at the cavity mirrors. With millimeter-sized holes at the centers of the cavity mirrors, the modes are unaffected. The above cavity, which corresponds to a repetition rate of *f _{R} =* 300 MHz, provides near-perfect out-coupling for intra-cavity HHG. Additionally, with its high-intensity gain, this cavity may support peak intensities at the focus approaching 10

^{15}W/cm

^{2}without damage to the cavity mirrors. We can use our analytical understanding of the example cavity above and our mode-solver to see how the properties of the patterned mirror confocal cavity scale as we shift the cavity's geometry. In particular, we are interested in how the intensity gain,

*I*, and effective waist,

_{g}*w*, scale with varying repetition rate and

_{eff}*r*. The results of an analytical and numerical scaling are given in Fig. 8 where we plot

_{avg}*I*and

_{g}*w*of the simulated example cavity above (red dot) and other numerically simulated cavity geometries (black dots) and the analytical scaling results for

_{eff}*I*and

_{g}*w*using numerical integration (green) and the approximate expressions from section 2.2 (orange-dashed).

_{eff}From the scaling results, we see that the approximate and exact analytical expressions agree well with each other and with the numerically simulated cavities. All numerically simulated modes have fundamental mode loss < 0.0016% and higher-order mode loss > 2.5%. A limitation of the patterned-mirror confocal BG cavity is also apparent. As the repetition rate grows so does the intensity gain, and so shrinks the effective waist. This is due to the connection between the Gaussian component of the BG mode and the repetition rate (connected through ${w}_{0,\mathrm{min}}=\sqrt{\lambda R/2\pi}$). For lower repetition rates, the Gaussian component is large. The intensity gain can still be made high and the effective waist small by making a very tight Bessel focus (i.e. small *w _{B}*) by increasing

*r*; however non-paraxial effects ultimately limit

_{avg}*r*, and the patterned-mirror confocal BG cavity is likely best suited for higher repetition rates.

_{avg}#### 3.3 Future challenges

There are two clear possible future challenges of gBG cavities: stability and mirror surface variations. When considering cavities with only spherical mirrors, the requirements to support a gBG type mode lead directly to the confocal and concentric cavity (both of which, as conventional Gaussian cavities, lie on the stability boundary). Performing a stability analysis with our cavity mode-solver on the example cavity discussed previously, we find that for a range of Δ*L =* 50 μm about the cavity length *L =* 49.97 cm, the fundamental mode loss can be kept < 0.002% while the next higher-order mode loss > 2.4%. This relatively narrow stability regime may make realization of the patterned-mirror confocal cavity challenging. However, we note that gBG type cavities with more sophisticated mirror structures (not restricted to only spherical mirrors) can easily avoid these stability regime boundary issues.

The challenges associated with mirror surface variations may prove more difficult to remedy. Consider a cavity geometry supporting a gBG type mode; the mode is composed of decentered Gaussian component beams. The cavity mirrors have some surface variations associated with the manufacturing process. If one localized region of the mirror surface varies e.g. the local radius of curvature, designed to be *R*, is actually *R +* δ*R*, then the decentered Gaussian component beam situated in this region may not be resonant in the cavity. Across the entire mirror, depending on surface variations, only a subset of the entire family of decentered component beams may resonate and, accordingly, the entire gBG beam may not resonate. This problem is associated with azimuthal degeneracy. Returning to our derivation of gBG beams in Section 2, if we vary the amplitudes of the component decentered Gaussian beams as we superpose them, we can produce azimuthal modulation in the final gBG beam and a higher-order (azimuthal) gBG beam [11]. Returning to our general Bessel-Gauss cavity design strategy, we see that such higher-order azimuthal gBG beams are also modes of gBG cavities. Therefore, mirror surface variations in a gBG cavity may prefer a particular higher-order azimuthal gBG beam (or superposition of such beams) over the fundamental mode. Issues and restrictions associated with mirror surface variations will be analyzed in detail in future work.

## 4. Conclusion

In this work, we introduced an alternative high-intensity enhancement cavity design, based on Bessel-Gauss type beams. We derived the basic properties of Bessel-Gauss beams including the focal properties and the transformation properties through spherical and conical optical elements. We presented a general Bessel-Gauss cavity design procedure and showed a specific example cavity, the patterned-mirror confocal Bessel-Gauss cavity, that has an effective waist of 33 μm, an intensity gain of 1.5 *×* 10^{4}, a repetition rate of 300 MHz, and supports holes in the cavity mirrors of millimeters in size. This cavity may be suitable for intra-cavity HHG as well as other cavity-enhanced strong-field physics applications and may support peak intensities of nearly 10^{15} W/cm^{2} without damage to the cavity mirrors. Finally, we discussed possible challenges to future implementations of Bessel-Gauss enhancement cavities.

## Appendix

We seek to show that Eq. (10) can be approximated by Eq. (11). (A similar approximation has previously been used for BG beams [17].) First, we write the second term in Eq. (10) as:

*u*and

*v*are given by:

*u + iv| >>*1, we can use the asymptotic form of the Bessel function (pp. 114 of Ref [14].) and find:

*v*|

*>*3 (tanh(3) ~.995). Therefore, when |

*v*|

*>>*1, Eq. (17) is an accurate approximation. Inspecting Eq. (16), we see a variety of different conditions can lead to |

*v*|

*>>*1, and a large class of gBG type beams have a phase term accurately approximated by Eq. (17). Two particular cases of this class are BG-like beams, i.e.

*r*is very small, and mBG-like beams, i.e.

_{0}*β*is very small. To see this, note that we are primarily interested in the region where there is significant intensity, i.e.

*r ≈r*(

_{c}*L*). Plugging

*r ≈r*(

_{c}*L*) into Eq. (16) for these two beam types we find:

*v*|

*>>*1, and for a mBG-like beam with ${r}_{0}^{2}/{w}_{0}^{2}>>1$and $L/{z}_{0}$not too large, |

*v*|

*>>*1. Plugging Eq. (17) into Eq. (10) produces Eq. (11) (up to a constant phase offset).

## Acknowledgments

This work was supported by AFOSR grant FA9550-10-1-0063 and the Center for Free-Electron Laser Science at DESY, Hamburg. William Putnam also acknowledges support from the NSF graduate research fellowships program.

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