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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 20, Iss. 22 — Oct. 22, 2012
  • pp: 24429–24443
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Bessel-Gauss beam enhancement cavities for high-intensity applications

William P. Putnam, Damian N. Schimpf, Gilberto Abram, and Franz X. Kärtner  »View Author Affiliations


Optics Express, Vol. 20, Issue 22, pp. 24429-24443 (2012)
http://dx.doi.org/10.1364/OE.20.024429


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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 × 104.

© 2012 OSA

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 (> 1013 W/cm2) 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]

6

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.

].

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

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

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]

]. However, the Fresnel reflection yields relatively low out-coupling efficiencies of < 20% [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]

,2

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

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]

], nonlinearities in the plate (due to the high-drive intensities) decrease the cavity enhancement [8

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]

], and the plate can be damaged by high XUV powers [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]

]. 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

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]

,5

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]

]. Out-coupling the harmonics through a small hole in a cavity mirror has also been explored [6

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.

,8

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

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]

]. Superposing several higher-order modes of the bow-tie cavity, an intra-cavity mode with low-intensity near the hole can be formed [8

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

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]

]; however, such techniques have been limited to small holes of diameter ≈100 μm [6

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.

,8

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

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]

]. 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 1011 W/cm2 [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]

]. In high-intensity bow-tie enhancement cavities, the mode waist on the cavity mirrors has been restricted to around 1 mm in size [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]

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]

], 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

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

12

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]

], 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

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

]. 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

11. V. Bagini, F. Frezza, M. Santarsiero, G. Schettini, and G. Schirripa Spagnolo, “Generalized Bessel-Gauss beams,” J. Mod. Opt. 43, 1155–1166 (1996).

]).

Consider, in the z = 0 plane of a cylindrical coordinate system (r, θ, z), a field of the form:
u(r,θ,z=0)=a0exp(r2/w02)exp(iβrcos(θγ))
(1)
The field u(r, θ, z = 0) resembles a Gaussian beam of waist w0 whose central wavevector has a component of magnitude β in the z = 0 plane inclined at an angle γ to the x-axis (illustrated in Fig. 1(a)
Fig. 1 Decentered Gaussians and Bessel-Gauss beam construction. (a) A single decentered Gaussian component for constructing a Bessel-Gauss beam (b) A decentered component for constructing a generalized Bessel-Gauss beam.
). Defining q0=iz0=ikw02/2 and propagating u(r, θ, z = 0), a Gaussian-like beam is found [11

11. V. Bagini, F. Frezza, M. Santarsiero, G. Schettini, and G. Schirripa Spagnolo, “Generalized Bessel-Gauss beams,” J. Mod. Opt. 43, 1155–1166 (1996).

]:
u(r,θ,z)=a0q0q(z)exp(ik2q(z)(r2+rc2(z)2rrc(z)cos(θγ)))exp(iβrcos(θγ))
(2)
q(z)=q0+z;rc(z)=(β/k)z
(3)
In Eq. (2) and in the following, we neglect exclusively z-dependent phase terms (e.g. exp(ikz)) for simplicity. The beam u(r, θ, z) defined by Eq. (2) closely resembles a Gaussian beam propagating at an angle φ=sin1(β/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 rc(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]

].

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:
UBG(r,z)=02πu(r,θ,z)dγ=A0q0q(z)exp(ik2q(z)(r2+rc2(z)))J0(βrkrc(z)q(z)r)
(4)
where we have used the integral representation of J0, the zero order Bessel function of the first kind (see pp. 140 of Ref [14

14. J. D. Jackson, Classical Electrodynamics (John Wiley & Sons, 1999), Chap. 3.

].), A0 is a constant, and q(z) and rc(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 r0 (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 rc(z)r0+rc(z). This form of beam is known as the generalized Bessel-Gauss beam (called the gBG beam from here on),:
UgBG(r,z)=A0q0q(z)exp(ik2q(z)(r2+rc2(z)))J0(βrkrc(z)q(z)r)
(5)
q(z)=q0+z;rc(z)=r0+(β/k)z
(6)
Although the expression for UgBG(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)
Fig. 2 Bessel-Gauss beam types. (a) - (c) Illustrations of the r-z plane cross-section of gBG, BG, and mBG beams respectively. (d) - (f) Plots of the amplitude in the r-z plane for gBG (λ = 1 μm, w0 = 200 μm, φ = 0.21°, r0 = 0.25 mm), BG (λ = 1 μm, w0 = 200 μm, φ = 0.29°). and mBG (λ = 1 μm, w0 = 200 μm, r0 = 1 mm) beams respectively.
, 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

11. V. Bagini, F. Frezza, M. Santarsiero, G. Schettini, and G. Schirripa Spagnolo, “Generalized Bessel-Gauss beams,” J. Mod. Opt. 43, 1155–1166 (1996).

]. Firstly, the BG beam corresponds to a gBG beam with r0 = 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 r0 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 r0. The 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)
Fig. 3 BG beam and intensity gain. (a) Plot of amplitude cross-section in the z = 0 plane of a BG beam with λ = 1 μm, w0 = 200 μm, and semi-aperture angle φ = 0.29°. Cross-section of the focus in the y-direction is on the right with 2wB labeled. (b) Plot of approximate (orange dashed) and exact (solid green) intensity gain of BG beams with λ = 1 μm, w0 = 30 μm, and semi-aperture angles φ of 1°, 2°, 3°, and 4° at distance z. The intensity gain of a Gaussian beam with λ = 1 μm and w0 = 30 μm (blue curve) is also included. (c) Plot of amplitude cross-section in the z = 20 cm plane of the BG beam from plot (a). Cross-section in the y-direction is included on the right with w and rc labeled.
). From Eq. (4), UBG(r,z=0)=A0exp(r2/w02)J0(βr). The peak intensity of a BG beam at its focus, IPfoc, can then readily be found (using the integral in Eq. (2).3) of Ref [10

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

].):
IPfoc=2Pπ(wefffoc)2,wefffoc=w0eβ2w02/4I0(β2w02/4)w0wB3
(7)
where P is the beam power, I0 is the zero-order modified Bessel function of the first kind, wefffoc is the effective beam waist of the BG beam at the focus, and wB = 2.4/β is the approximate waist of the Bessel component (i.e. the first zero of J0(βr), illustrated in Fig. 3(a)). The approximate form of wefffocgiven in Eq. (7) follows from inspecting the argument of the Bessel function in Eq. (7). We see β2w02/4=(φ/φG)2where φβ/kis the semi-aperture angle of the BG beam (as already described) and φG=2/kw0is 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 φG << φ i.e the Gaussian components must diverge slower than their peak intensity axes spread apart. So, for the regime of interest β2w02/4=(φ/φG)2>>1, and the asymptotic expansion of I0 (pp. 116 of Ref [14

14. J. D. Jackson, Classical Electrodynamics (John Wiley & Sons, 1999), Chap. 3.

].) yields the approximate form of wefffoc.

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. UBG(r,z)(B0/r)exp((rrc(z))2/w(z)2) where w(z)=w01+(z/z0)2, z >> z0, and B0 is a constant [12

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]

]. Using this expression, the peak-intensity of the BG beam far from the focus at position z can be approximated:
IPFF(z)=2Pπ(weffFF(z))2,weffFF(z)22πw(z)rc(z)
(8)
where rc(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 Ig(z) = IPfoc/IPFF(z) where Ig(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 IgG(z)=πw(z)2/πw02(z/z0)2where z >> z0. Combining Eq. (7) and Eq. (8) we find the intensity gain for a BG beam:

Bessel-GaussBeam:IgBG(z)C(φφG)2(zz0)2,C=122π2.412.5GaussianBeam:IgG(z)(zz0)2
(9)

Recalling that for the beams of interest φ >> φG 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 w0 = 30 μm is compared to that of BG beams with Gaussian component w0 = 30 μm and semi-aperture angles 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 >> z0). 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.

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, UgBG(r, z = L). Denoting this phase by ϕgBG(r) we find (from Eq. (5)):
ϕgBG(r)=ik2R(L)r2+iarg(J0(βrkrc(L)q(L)r))
(10)
where we have expanded the Gaussian term in the conventional way so that R(L) = L + z02 / 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):
ϕgBG(r)ik2R(L)r2+((Lz0)(r0z0)βk)ik1+(L/z0)2r
(11)
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 ϕcon(r)=iαkr at 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 (q0', r0', β'). From a straightforward calculation we determine these altered parameters; they are included in Eq. (12). A spherical element, with spatial phase ϕsph(r)=ikr2/2f 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 (q0', r0', 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]

]. After conical or spherical optical elements the gBG beam parameters transform as:
Conical(ϕcon=iαkr):q0'=q(L);r0'=rc(L);β=βkα
(12)
Spherical(ϕsph=ikr2/2f):q0'=q(L)q(L)/f+1;r0'=rc(L);β'=βkrc(L)f
(13)
The gBG beam, after a conical or spherical element, can then be written (up to a constant phase factor [15

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

]) in the standard form of Eq. (5) and Eq. (6) with the substitutions q0q0', r0r0', ββ', and zz'where 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.

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. r0 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 = 2R/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

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)
Fig. 5 Conventional Gaussian cavities and gBG cavities. (a) Conventional Gaussian cavity (b) Revolving the conventional Gaussian cavity about its central axis (dashed), we obtain a cavity supporting gBG modes.
. The Gaussian beam is re-imaged as it traverses the cavity i.e. q(z + 2L) = q(z) where 2L 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.

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

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

]. 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 2N × 2N scattering matrix. (Scattering matrices for optical systems are generally 2 × 2 matrices relating incoming waves to outgoing ones [20

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

,21

21. H. A. Haus, Waves and Fields in Optoelectronics (CBLS, 2004), Chap. 3.

]; here, each radial point has its own scattering matrix and lumping all the points together, we represent each element as a 2N × 2N 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

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]

], each propagation block can be written as Pλ,z=F1exp(ikz1(λν)2)Fwhere 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 Pλ,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

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

]. The entire cavity can then be represented by a single 2N × 2N 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 2N × 2N 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 = 2R). 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 RH) and has average radius ravg and thickness Δr (illustrated in Fig. 6(a)
Fig. 6 Single-mode selection in the confocal BG cavity. (a) Cavity mirror with patterned annular (donut-shaped) region of high-reflectivity. (b) Cross-section of patterned cavity mirror with incident beams.
); the rest of the mirror surface has a low reflectivity (RL). 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 R/2) there exist minimally divergent decentered Gaussian beams. These beams have a waist w0,min=λR/2πat the cavity center, and wmin=2w0,minat 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. Δr ~3w), then only the BG mode composed of decentered Gaussians with waist w0,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, φ=tan1(2ravg/R).

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 fR = 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 1015 W/cm2 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, Ig, and effective waist, weff, scale with varying repetition rate and ravg. The results of an analytical and numerical scaling are given in Fig. 8
Fig. 8 Patterned mirror confocal cavity scaling. (a) Intensity gain, Ig, scaling with repetition rate (i.e. cavity length and mirror radius of curvature). (b) Effective waist, weff, scaling with repetition rate (i.e. cavity length and mirror radius of curvature). For all cavities in these plots Δr = 3.1wmin.
where we plot Ig and weff of the simulated example cavity above (red dot) and other numerically simulated cavity geometries (black dots) and the analytical scaling results for Ig and weff using numerical integration (green) and the approximate expressions from section 2.2 (orange-dashed).

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 w0,min=λR/2π). 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 wB) by increasing ravg; however non-paraxial effects ultimately limit ravg, and the patterned-mirror confocal BG cavity is likely best suited for higher repetition rates.

3.3 Future challenges

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 + δR, 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

11. V. Bagini, F. Frezza, M. Santarsiero, G. Schettini, and G. Schirripa Spagnolo, “Generalized Bessel-Gauss beams,” J. Mod. Opt. 43, 1155–1166 (1996).

]. 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 × 104, 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 1015 W/cm2 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

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]

].) First, we write the second term in Eq. (10) as:
arg(J0(βrkrc(L)q(L)r))=arg(J0(u+iv))
(14)
where u and v are given by:
u=βr1+(L/z0)2kr1+(L/z0)2(Lz0)(r0z0)
(15)
v=βr1+(L/z0)2(Lz0)kr1+(L/z0)2(r0z0)
(16)
The relation rc(L)=r0+(β/k)L has been used in the above (from Eq. (6)). Returning to Eq. (14), with |u + iv| >> 1, we can use the asymptotic form of the Bessel function (pp. 114 of Ref [14

14. J. D. Jackson, Classical Electrodynamics (John Wiley & Sons, 1999), Chap. 3.

].) and find:
arg(J0(u+iv))arg(cos(u+ivπ/4))=tan1(tan(uπ/4)tanhv)u+π/4=((Lz0)(r0z0)βk)k1+(L/z0)2r+π4
(17)
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. r0 is very small, and mBG-like beams, i.e. β is very small. To see this, note that we are primarily interested in the region where there is significant intensity, i.e. r ≈rc(L). Plugging r ≈rc(L) into Eq. (16) for these two beam types we find:
BG-likeBeam:v=β2w024(2(L/z0)21+(L/z0)2)
(18)
mBG-likeBeam:v=r02w02(21+(L/z0)2)
(19)
From the above, we see that for a BG-like beam with β2w02/4=(φ/φG)2>>1and L/z0not too small, |v| >> 1, and for a mBG-like beam with r02/w02>>1and L/z0not 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.

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. 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]

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]

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]

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. 35(12), 2052–2054 (2010). [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]

10.

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

11.

V. Bagini, F. Frezza, M. Santarsiero, G. Schettini, and G. Schirripa Spagnolo, “Generalized Bessel-Gauss beams,” J. Mod. Opt. 43, 1155–1166 (1996).

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]

13.

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

14.

J. D. Jackson, Classical Electrodynamics (John Wiley & Sons, 1999), Chap. 3.

15.

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

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]

20.

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

21.

H. A. Haus, Waves and Fields in Optoelectronics (CBLS, 2004), Chap. 3.

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]

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

  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,” Nature436(7048), 234–237 (2005). [CrossRef] [PubMed]
  3. 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]
  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,” Nature482(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]
  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.35(12), 2052–2054 (2010). [CrossRef] [PubMed]
  8. 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]
  9. 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]
  10. F. Gori, G. Guattari, and C. Padovani, “Bessel-Gauss beams,” Opt. Commun.64(6), 491–495 (1987). [CrossRef]
  11. V. Bagini, F. Frezza, M. Santarsiero, G. Schettini, and G. Schirripa Spagnolo, “Generalized Bessel-Gauss beams,” J. Mod. Opt.43, 1155–1166 (1996).
  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]
  13. A. A. Al-Rashed and B. E. A. Saleh, “Decentered Gaussian beams,” Appl. Opt.34(30), 6819–6825 (1995). [CrossRef] [PubMed]
  14. J. D. Jackson, Classical Electrodynamics (John Wiley & Sons, 1999), Chap. 3.
  15. M. Santarsiero, “Propagation of generalized Bessel-Gauss beams through ABCD optical systems,” Opt. Commun.132(1-2), 1–7 (1996). [CrossRef]
  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. A18(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. A20(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]
  20. G. Abram, “High intensity femtosecond enhancement cavities,” M. Eng Thesis, MIT (2009).
  21. H. A. Haus, Waves and Fields in Optoelectronics (CBLS, 2004), Chap. 3.
  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. A21(1), 53–58 (2004). [CrossRef] [PubMed]

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