## Bessel-Gauss beam enhancement cavities for high-intensity applications |

Optics Express, Vol. 20, Issue 22, pp. 24429-24443 (2012)

http://dx.doi.org/10.1364/OE.20.024429

Acrobat PDF (1425 KB)

### 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 OSA

## 1. Introduction

^{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

1. R. J. Jones, K. D. Moll, M. J. Thorpe, and J. Ye, “Phase-coherent frequency combs in the vacuum ultraviolet via high-harmonic generation inside a femtosecond enhancement cavity,” Phys. Rev. Lett. **94**(19), 193201 (2005). [CrossRef] [PubMed]

^{14}W/cm

^{2}[3

3. J. Lee, D. R. Carlson, and R. J. Jones, “Optimizing intracavity high harmonic generation for XUV fs frequency combs,” Opt. Express **19**(23), 23315–23326 (2011). [CrossRef] [PubMed]

4. A. Cingöz, D. C. Yost, T. K. Allison, A. Ruehl, M. E. Fermann, I. Hartl, and J. Ye, “Direct frequency comb spectroscopy in the extreme ultraviolet,” Nature **482**(7383), 68–71 (2012). [CrossRef] [PubMed]

7. I. Pupeza, T. Eidam, J. Rauschenberger, B. Bernhardt, A. Ozawa, E. Fill, A. Apolonski, T. Udem, J. Limpert, Z. A. Alahmed, A. M. Azzeer, A. Tünnermann, T. W. Hänsch, and F. Krausz, “Power scaling of a high-repetition-rate enhancement cavity,” Opt. Lett. **35**(12), 2052–2054 (2010). [CrossRef] [PubMed]

3. J. Lee, D. R. Carlson, and R. J. Jones, “Optimizing intracavity high harmonic generation for XUV fs frequency combs,” Opt. Express **19**(23), 23315–23326 (2011). [CrossRef] [PubMed]

4. A. Cingöz, D. C. Yost, T. K. Allison, A. Ruehl, M. E. Fermann, I. Hartl, and J. Ye, “Direct frequency comb spectroscopy in the extreme ultraviolet,” Nature **482**(7383), 68–71 (2012). [CrossRef] [PubMed]

4. A. Cingöz, D. C. Yost, T. K. Allison, A. Ruehl, M. E. Fermann, I. Hartl, and J. Ye, “Direct frequency comb spectroscopy in the extreme ultraviolet,” Nature **482**(7383), 68–71 (2012). [CrossRef] [PubMed]

1. R. J. Jones, K. D. Moll, M. J. Thorpe, and J. Ye, “Phase-coherent frequency combs in the vacuum ultraviolet via high-harmonic generation inside a femtosecond enhancement cavity,” Phys. Rev. Lett. **94**(19), 193201 (2005). [CrossRef] [PubMed]

3. J. Lee, D. R. Carlson, and R. J. Jones, “Optimizing intracavity high harmonic generation for XUV fs frequency combs,” Opt. Express **19**(23), 23315–23326 (2011). [CrossRef] [PubMed]

1. R. J. Jones, K. D. Moll, M. J. Thorpe, and J. Ye, “Phase-coherent frequency combs in the vacuum ultraviolet via high-harmonic generation inside a femtosecond enhancement cavity,” Phys. Rev. Lett. **94**(19), 193201 (2005). [CrossRef] [PubMed]

2. C. Gohle, T. Udem, M. Herrmann, J. Rauschenberger, R. Holzwarth, H. A. Schuessler, F. Krausz, and T. W. Hänsch, “A frequency comb in the extreme ultraviolet,” Nature **436**(7048), 234–237 (2005). [CrossRef] [PubMed]

8. K. D. Moll, R. J. Jones, and J. Ye, “Output coupling methods for cavity-based high-harmonic generation,” Opt. Express **14**(18), 8189–8197 (2006). [CrossRef] [PubMed]

8. K. D. Moll, R. J. Jones, and J. Ye, “Output coupling methods for cavity-based high-harmonic generation,” Opt. Express **14**(18), 8189–8197 (2006). [CrossRef] [PubMed]

**19**(23), 23315–23326 (2011). [CrossRef] [PubMed]

**482**(7383), 68–71 (2012). [CrossRef] [PubMed]

5. D. C. Yost, T. R. Schibli, and J. Ye, “Efficient output coupling of intracavity high-harmonic generation,” Opt. Lett. **33**(10), 1099–1101 (2008). [CrossRef] [PubMed]

8. K. D. Moll, R. J. Jones, and J. Ye, “Output coupling methods for cavity-based high-harmonic generation,” Opt. Express **14**(18), 8189–8197 (2006). [CrossRef] [PubMed]

9. areJ. Weitenberg, P. Rußbüldt, T. Eidam, and I. Pupeza, “Transverse mode tailoring in a quasi-imaging high-finesse femtosecond enhancement cavity,” Opt. Express **19**(10), 9551–9561 (2011). [CrossRef] [PubMed]

**14**(18), 8189–8197 (2006). [CrossRef] [PubMed]

9. areJ. Weitenberg, P. Rußbüldt, T. Eidam, and I. Pupeza, “Transverse mode tailoring in a quasi-imaging high-finesse femtosecond enhancement cavity,” Opt. Express **19**(10), 9551–9561 (2011). [CrossRef] [PubMed]

**14**(18), 8189–8197 (2006). [CrossRef] [PubMed]

9. areJ. Weitenberg, P. Rußbüldt, T. Eidam, and I. Pupeza, “Transverse mode tailoring in a quasi-imaging high-finesse femtosecond enhancement cavity,” Opt. Express **19**(10), 9551–9561 (2011). [CrossRef] [PubMed]

^{11}W/cm

^{2}[7

7. I. Pupeza, T. Eidam, J. Rauschenberger, B. Bernhardt, A. Ozawa, E. Fill, A. Apolonski, T. Udem, J. Limpert, Z. A. Alahmed, A. M. Azzeer, A. Tünnermann, T. W. Hänsch, and F. Krausz, “Power scaling of a high-repetition-rate enhancement cavity,” Opt. Lett. **35**(12), 2052–2054 (2010). [CrossRef] [PubMed]

**94**(19), 193201 (2005). [CrossRef] [PubMed]

7. I. Pupeza, T. Eidam, J. Rauschenberger, B. Bernhardt, A. Ozawa, E. Fill, A. Apolonski, T. Udem, J. Limpert, Z. A. Alahmed, A. M. Azzeer, A. Tünnermann, T. W. Hänsch, and F. Krausz, “Power scaling of a high-repetition-rate enhancement cavity,” Opt. Lett. **35**(12), 2052–2054 (2010). [CrossRef] [PubMed]

10. F. Gori, G. Guattari, and C. Padovani, “Bessel-Gauss beams,” Opt. Commun. **64**(6), 491–495 (1987). [CrossRef]

12. C. J. R. Sheppard and T. Wilson, “Gaussian-beam theory of lenses with annular aperture,” IEE J Microwaves, Opt. Acoust. **2**(4), 105–112 (1978). [CrossRef]

## 2. Bessel-Gauss beams

### 2.1 Bessel-Gauss beams from decentered Gaussians

10. F. Gori, G. Guattari, and C. Padovani, “Bessel-Gauss beams,” Opt. Commun. **64**(6), 491–495 (1987). [CrossRef]

*z =*0 plane of a cylindrical coordinate system (

*r, θ, z*), a field of the form:The field

*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

*u*(

*r, θ, z =*0), a Gaussian-like beam is found [11]: In Eq. (2) and in the following, we neglect exclusively

*z*-dependent phase terms (e.g.

*u*(

*r, θ, z*) defined by Eq. (2) closely resembles a Gaussian beam propagating at an angle

*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

13. A. A. Al-Rashed and B. E. A. Saleh, “Decentered Gaussian beams,” Appl. Opt. **34**(30), 6819–6825 (1995). [CrossRef] [PubMed]

*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:where we have used the integral representation of

*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).

*r*(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

_{0}*φ*. This generalization amounts to letting

*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).

*r*0. The

_{0}=*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

*r-θ*plane cross-section of the BG beam amplitude at the focus is plotted in Fig. 3(a) ). From Eq. (4),

*I*, can then readily be found (using the integral in Eq. (2).3) of Ref [10

_{P}^{foc}10. F. Gori, G. Guattari, and C. Padovani, “Bessel-Gauss beams,” Opt. Commun. **64**(6), 491–495 (1987). [CrossRef]

*P*is the beam power,

*I*is the zero-order modified Bessel function of the first kind,

_{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

*φ*<<

_{G}*φ*i.e the Gaussian components must diverge slower than their peak intensity axes spread apart. So, for the regime of interest

*I*(pp. 116 of Ref [14].) yields the approximate form of

_{0}*z >> z*, and

_{0}*B*is a constant [12

_{0}12. C. J. R. Sheppard and T. Wilson, “Gaussian-beam theory of lenses with annular aperture,” IEE J Microwaves, Opt. Acoust. **2**(4), 105–112 (1978). [CrossRef]

*z*can be approximated:where

*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)).

*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

*z*>>

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

_{0}*φ >> φ*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

_{G}*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

*iαkr*) spatial phase to wavefronts e.g. transmitting or reflecting axicons. The importance of these elements in manipulating gBG beams will be discussed.

*r*-dependent phase, of a gBG beam at plane

*z = L*,

*U*(

_{gBG}*r, z = L*). Denoting this phase by

*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):The spatial phase of the gBG beam is well-approximated as the sum of a quadratic part and a linear part. A conical optical element, with spatial phase

*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

*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

15. M. Santarsiero, “Propagation of generalized Bessel-Gauss beams through ABCD optical systems,” Opt. Commun. **132**(1-2), 1–7 (1996). [CrossRef]

15. M. Santarsiero, “Propagation of generalized Bessel-Gauss beams through ABCD optical systems,” Opt. Commun. **132**(1-2), 1–7 (1996). [CrossRef]

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

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

*z = L*.

- (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.

*R =*20 cm at position

*z = R =*20 cm and an incident mBG beam of wavelength

*λ =*1 μm, Gaussian component waist

*w*300 μm, and

_{0}=*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

*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).

*r*and

_{0}*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

16. J. Rogel-Salazar, G. H. C. New, and S. Chávez-Cerda, “Bessel-Gauss beam optical resonator,” Opt. Commun. **190**(1-6), 117–122 (2001). [CrossRef]

17. A. N. Khilo, E. G. Katranji, and A. A. Ryzhevich, “Axicon-based Bessel resonator: analytical description and experiment,” J. Opt. Soc. Am. A **18**(8), 1986–1992 (2001). [CrossRef] [PubMed]

18. J. C. Gutiérrez-Vega, R. Rodríguez-Masegosa, and S. Chávez-Cerda, “Bessel-Gauss resonator with spherical output mirror: geometrical- and wave-optics analysis,” J. Opt. Soc. Am. A **20**(11), 2113–2122 (2003). [CrossRef] [PubMed]

19. P. Pääkkönen and J. Turunen, “Resonators with Bessel-Gauss modes,” Opt. Commun. **156**(4-6), 359–366 (1998). [CrossRef]

### 3.1 Bessel-Gauss cavity design strategy

*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.

*q*-parameter) and tilt parameters (i.e.

*r*and

_{0}*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

*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

22. L. Yu, M. Huang, M. Chen, W. Chen, W. Huang, and Z. Zhu, “Quasi-discrete Hankel transform,” Opt. Lett. **23**(6), 409–411 (1998). [CrossRef] [PubMed]

23. M. Guizar-Sicairos and J. C. Gutiérrez-Vega, “Computation of quasi-discrete Hankel transforms of integer order for propagating optical wave fields,” J. Opt. Soc. Am. A **21**(1), 53–58 (2004). [CrossRef] [PubMed]

*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*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

_{λ,z}*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

*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.

*r*8 mm, Δ

_{avg}=*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,

*L = R*cos

*φ*.

*f*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

_{R}=^{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}*w*) by increasing

_{B}*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

*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.

*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

*×*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

17. A. N. Khilo, E. G. Katranji, and A. A. Ryzhevich, “Axicon-based Bessel resonator: analytical description and experiment,” J. Opt. Soc. Am. A **18**(8), 1986–1992 (2001). [CrossRef] [PubMed]

*u*and

*v*are given by: The relation

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

*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: From the above, we see that for a BG-like beam with

*v*|

*>>*1, and for a mBG-like beam with

*v*|

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

## Acknowledgments

## References and links

1. | R. J. Jones, K. D. Moll, M. J. Thorpe, and J. Ye, “Phase-coherent frequency combs in the vacuum ultraviolet via high-harmonic generation inside a femtosecond enhancement cavity,” Phys. Rev. Lett. |

2. | C. Gohle, T. Udem, M. Herrmann, J. Rauschenberger, R. Holzwarth, H. A. Schuessler, F. Krausz, and T. W. Hänsch, “A frequency comb in the extreme ultraviolet,” Nature |

3. | J. Lee, D. R. Carlson, and R. J. Jones, “Optimizing intracavity high harmonic generation for XUV fs frequency combs,” Opt. Express |

4. | A. Cingöz, D. C. Yost, T. K. Allison, A. Ruehl, M. E. Fermann, I. Hartl, and J. Ye, “Direct frequency comb spectroscopy in the extreme ultraviolet,” Nature |

5. | D. C. Yost, T. R. Schibli, and J. Ye, “Efficient output coupling of intracavity high-harmonic generation,” Opt. Lett. |

6. | S. Holzberger, I. Pupeza, D. Esser, J. Weitenberg, H. Carstens, T. Eidam, P. Russbüldt, J. Limpert, T. Udem, A. Tünnermann, T. Hänsch, F. Krausz, and E. Fill, “Sub-25 nm high-harmonic generation with a 78-MHz repetition rate enhancement cavity,” QELS 2012, Postdeadline Paper QTh5B.7. |

7. | I. Pupeza, T. Eidam, J. Rauschenberger, B. Bernhardt, A. Ozawa, E. Fill, A. Apolonski, T. Udem, J. Limpert, Z. A. Alahmed, A. M. Azzeer, A. Tünnermann, T. W. Hänsch, and F. Krausz, “Power scaling of a high-repetition-rate enhancement cavity,” Opt. Lett. |

8. | K. D. Moll, R. J. Jones, and J. Ye, “Output coupling methods for cavity-based high-harmonic generation,” Opt. Express |

9. | areJ. Weitenberg, P. Rußbüldt, T. Eidam, and I. Pupeza, “Transverse mode tailoring in a quasi-imaging high-finesse femtosecond enhancement cavity,” Opt. Express |

10. | F. Gori, G. Guattari, and C. Padovani, “Bessel-Gauss beams,” Opt. Commun. |

11. | V. Bagini, F. Frezza, M. Santarsiero, G. Schettini, and G. Schirripa Spagnolo, “Generalized Bessel-Gauss beams,” J. Mod. Opt. |

12. | C. J. R. Sheppard and T. Wilson, “Gaussian-beam theory of lenses with annular aperture,” IEE J Microwaves, Opt. Acoust. |

13. | A. A. Al-Rashed and B. E. A. Saleh, “Decentered Gaussian beams,” Appl. Opt. |

14. | J. D. Jackson, |

15. | M. Santarsiero, “Propagation of generalized Bessel-Gauss beams through ABCD optical systems,” Opt. Commun. |

16. | J. Rogel-Salazar, G. H. C. New, and S. Chávez-Cerda, “Bessel-Gauss beam optical resonator,” Opt. Commun. |

17. | A. N. Khilo, E. G. Katranji, and A. A. Ryzhevich, “Axicon-based Bessel resonator: analytical description and experiment,” J. Opt. Soc. Am. A |

18. | J. C. Gutiérrez-Vega, R. Rodríguez-Masegosa, and S. Chávez-Cerda, “Bessel-Gauss resonator with spherical output mirror: geometrical- and wave-optics analysis,” J. Opt. Soc. Am. A |

19. | P. Pääkkönen and J. Turunen, “Resonators with Bessel-Gauss modes,” Opt. Commun. |

20. | G. Abram, “High intensity femtosecond enhancement cavities,” M. Eng Thesis, MIT (2009). |

21. | H. A. Haus, |

22. | L. Yu, M. Huang, M. Chen, W. Chen, W. Huang, and Z. Zhu, “Quasi-discrete Hankel transform,” Opt. Lett. |

23. | M. Guizar-Sicairos and J. C. Gutiérrez-Vega, “Computation of quasi-discrete Hankel transforms of integer order for propagating optical wave fields,” J. Opt. Soc. Am. A |

**OCIS Codes**

(140.7240) Lasers and laser optics : UV, EUV, and X-ray lasers

(020.2649) Atomic and molecular physics : Strong field laser physics

(070.5753) Fourier optics and signal processing : Resonators

**ToC Category:**

Lasers and Laser Optics

**History**

Original Manuscript: August 16, 2012

Manuscript Accepted: October 1, 2012

Published: October 11, 2012

**Citation**

William P. Putnam, Damian N. Schimpf, Gilberto Abram, and Franz X. Kärtner, "Bessel-Gauss beam enhancement cavities for high-intensity applications," Opt. Express **20**, 24429-24443 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-22-24429

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### References

- R. J. Jones, K. D. Moll, M. J. Thorpe, and J. Ye, “Phase-coherent frequency combs in the vacuum ultraviolet via high-harmonic generation inside a femtosecond enhancement cavity,” Phys. Rev. Lett.94(19), 193201 (2005). [CrossRef] [PubMed]
- C. Gohle, T. Udem, M. Herrmann, J. Rauschenberger, R. Holzwarth, H. A. Schuessler, F. Krausz, and T. W. Hänsch, “A frequency comb in the extreme ultraviolet,” Nature436(7048), 234–237 (2005). [CrossRef] [PubMed]
- J. Lee, D. R. Carlson, and R. J. Jones, “Optimizing intracavity high harmonic generation for XUV fs frequency combs,” Opt. Express19(23), 23315–23326 (2011). [CrossRef] [PubMed]
- A. Cingöz, D. C. Yost, T. K. Allison, A. Ruehl, M. E. Fermann, I. Hartl, and J. Ye, “Direct frequency comb spectroscopy in the extreme ultraviolet,” Nature482(7383), 68–71 (2012). [CrossRef] [PubMed]
- D. C. Yost, T. R. Schibli, and J. Ye, “Efficient output coupling of intracavity high-harmonic generation,” Opt. Lett.33(10), 1099–1101 (2008). [CrossRef] [PubMed]
- S. Holzberger, I. Pupeza, D. Esser, J. Weitenberg, H. Carstens, T. Eidam, P. Russbüldt, J. Limpert, T. Udem, A. Tünnermann, T. Hänsch, F. Krausz, and E. Fill, “Sub-25 nm high-harmonic generation with a 78-MHz repetition rate enhancement cavity,” QELS 2012, Postdeadline Paper QTh5B.7.
- I. Pupeza, T. Eidam, J. Rauschenberger, B. Bernhardt, A. Ozawa, E. Fill, A. Apolonski, T. Udem, J. Limpert, Z. A. Alahmed, A. M. Azzeer, A. Tünnermann, T. W. Hänsch, and F. Krausz, “Power scaling of a high-repetition-rate enhancement cavity,” Opt. Lett.35(12), 2052–2054 (2010). [CrossRef] [PubMed]
- K. D. Moll, R. J. Jones, and J. Ye, “Output coupling methods for cavity-based high-harmonic generation,” Opt. Express14(18), 8189–8197 (2006). [CrossRef] [PubMed]
- areJ. Weitenberg, P. Rußbüldt, T. Eidam, and I. Pupeza, “Transverse mode tailoring in a quasi-imaging high-finesse femtosecond enhancement cavity,” Opt. Express19(10), 9551–9561 (2011). [CrossRef] [PubMed]
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- V. Bagini, F. Frezza, M. Santarsiero, G. Schettini, and G. Schirripa Spagnolo, “Generalized Bessel-Gauss beams,” J. Mod. Opt.43, 1155–1166 (1996).
- C. J. R. Sheppard and T. Wilson, “Gaussian-beam theory of lenses with annular aperture,” IEE J Microwaves, Opt. Acoust.2(4), 105–112 (1978). [CrossRef]
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- J. D. Jackson, Classical Electrodynamics (John Wiley & Sons, 1999), Chap. 3.
- M. Santarsiero, “Propagation of generalized Bessel-Gauss beams through ABCD optical systems,” Opt. Commun.132(1-2), 1–7 (1996). [CrossRef]
- J. Rogel-Salazar, G. H. C. New, and S. Chávez-Cerda, “Bessel-Gauss beam optical resonator,” Opt. Commun.190(1-6), 117–122 (2001). [CrossRef]
- A. N. Khilo, E. G. Katranji, and A. A. Ryzhevich, “Axicon-based Bessel resonator: analytical description and experiment,” J. Opt. Soc. Am. A18(8), 1986–1992 (2001). [CrossRef] [PubMed]
- J. C. Gutiérrez-Vega, R. Rodríguez-Masegosa, and S. Chávez-Cerda, “Bessel-Gauss resonator with spherical output mirror: geometrical- and wave-optics analysis,” J. Opt. Soc. Am. A20(11), 2113–2122 (2003). [CrossRef] [PubMed]
- P. Pääkkönen and J. Turunen, “Resonators with Bessel-Gauss modes,” Opt. Commun.156(4-6), 359–366 (1998). [CrossRef]
- G. Abram, “High intensity femtosecond enhancement cavities,” M. Eng Thesis, MIT (2009).
- H. A. Haus, Waves and Fields in Optoelectronics (CBLS, 2004), Chap. 3.
- L. Yu, M. Huang, M. Chen, W. Chen, W. Huang, and Z. Zhu, “Quasi-discrete Hankel transform,” Opt. Lett.23(6), 409–411 (1998). [CrossRef] [PubMed]
- M. Guizar-Sicairos and J. C. Gutiérrez-Vega, “Computation of quasi-discrete Hankel transforms of integer order for propagating optical wave fields,” J. Opt. Soc. Am. A21(1), 53–58 (2004). [CrossRef] [PubMed]

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