## Abstract

We demonstrate a high-finesse femtosecond enhancement cavity with an on-axis obstacle. By inserting a wire with a width of 5% of the fundamental mode diameter, the finesse of F = 3400 is only slightly reduced to F = 3000. The low loss is due to the degeneracy of transverse modes, which allows for exciting a circulating field distribution avoiding the obstacle. We call this condition quasi-imaging. The concept could be used for output coupling of intracavity-generated higher-order harmonics through an on-axis opening in one of the cavity mirrors.

© 2011 OSA

## 1. Introduction

The generation of high harmonics (HHG) of femtosecond radiation in a gas makes possible table-top coherent light sources with wavelengths down to a few nanometers. To reach the required peak intensities on the order of 10^{14} W/cm^{2} for the nonlinear conversion at MHz repetition rates, as well as to increase the yield of the highly inefficient conversion process, resonant enhancement in a passive cavity can be employed [1–10]. In the infrared range, a power enhancement of three orders of magnitude has been demonstrated [1]. High repetition rates (several tens of MHz and more) are desirable for several applications, including high precision spectroscopy in the XUV range [2] and coincidence spectroscopy of correlated electron dynamics [3]. The intracavity-generated high harmonics propagate along the resonator optical axis. Their output coupling from the cavity is a major challenge and has been subject to intensive research over the past few years [6–10]. An established method relies on a thin plate, placed behind the focus at Brewster’s angle for the fundamental radiation [2,4,5]. The intracavity-generated XUV radiation experiences total reflexion at the plate and is thus separated from the fundamental beam. However, due to strong absorption of the evanescent field, the output coupling efficiency is typically 0.1-0.2 for wavelengths ~60 nm and significantly decreases for shorter wavelengths. For wavelengths around 10 nm this method is not suitable because the efficiency is typically <10^{−3}. Moreover, nonlinearities introduced by the plate limit the peak power circulating in the cavity. A different output coupling method uses a grating, manufactured in the top coating layer of a cavity mirror behind the focus [6,7]. While the XUV radiation is diffracted by the grating, the optical element acts as a highly reflecting mirror for the fundamental beam. The output coupling efficiency is typically <0.2 and also decreases for smaller wavelengths. Recently, it has been shown that the nanostructure leads to increased nonlinear effects on the mirror surface induced by the fundamental radiation [7], which might limit the scaling of intracavity peak power. Moreover, the fact that the harmonics are spatially dispersed might constitute a drawback for some applications. In a third approach, the output coupling is achieved geometrically through an on-axis hole or slit in the mirror behind the focus. The smaller divergence angle of the high harmonics due to the shorter wavelength makes this approach possible. Because there is no interaction of the XUV radiation with any optical element, a geometrical access allows for efficient output coupling of short wavelengths (e.g. sub-10 nm). Moreover, no dispersion or nonlinearities are introduced by the output coupling mechanism. If the fundamental mode of the cavity is used, an on-axis hole has to be very small to allow for a high enhancement [8]. Approaches allowing for a broad hole or slit at small loss for the fundamental radiation have been proposed by employing either transverse modes with vanishing field on the optical axis [8], an imaging resonator [9] or by using non-collinear high harmonic generation [8,10].

In this paper we experimentally demonstrate the possibility of geometrical on-axis access to a high-finesse enhancement cavity by a technique which we call quasi-imaging. By particular choice of the resonator geometry, higher-order transverse modes can be simultaneously resonant with the fundamental mode. In our case, the Gauss-Hermite modes *GH*
_{0,0}, G*H*
_{4,0} and *GH*
_{8,0} can be excited simultaneously. If an obstacle, e.g. a wire or a slit in a resonator mirror, is introduced into the beam path, these modes arrange to avoid the obstacle. We call this quasi-imaging, because only a subset of the transverse modes is involved, whereas in an imaging resonator all transverse modes are simultaneously resonant.

The manuscript is structured as follows: Section 2 contains an introduction to the concept of quasi-imaging, in Section 3 we present the experimental realization of a quasi-imaging resonator, Section 4 gives a brief discussion and Section 5 concludes the paper.

## 2. Quasi-imaging

#### 2.1 Definition of a quasi-imaging resonator

For a stable spherical resonator in paraxial approximation, one set of eigen-modes, i.e. field distributions which are reproduced after one resonator round trip, are the Gauss-Hermite *GH _{n}*

_{,}

*modes. When an obstacle is placed in the beam path, these modes are in general no longer the resonator eigen-modes. However, if a set of*

_{m}*GH*modes is simultaneously resonant, new eigen-modes can be constructed as a combination thereof, which avoid the obstacle and have small diffraction loss. We call this condition quasi-imaging. Quasi-imaging is related to imaging, because the hole in a field distribution is reproduced after a resonator round trip. However, quasi-imaging can be achieved in a stable resonator (i.e. with a distinguished eigen-q-parameter, or equivalently for |

*A*+

*D*| < 2 with the elements of the beam transfer matrix for one resonator round trip

*M*= [[

*A*,

*B*],[

*C*,

*D*]]), which is not the case for imaging. The on-axis phase

*ϕ*acquired at a resonator round trip by a

*GH*

_{n}_{,}

*mode with eigen-q-parameter (*

_{m}*q*and

_{x}*q*) and wave number

_{y}*k*is given by

*L*is the resonator length (length of a ring resonator or twice the length of a linear resonator) and

*ψ*is a phase which we will refer to as the Gouy parameter. This phase is due to the Gouy effect and is defined by the resonator geometry [11]. The Gouy parameter and the eigen-q-parameter can be different for the two transverse directions in Cartesian geometry. The stability range of the resonator is determined by the condition 0 <

_{x,y}*ψ*< π or π <

*ψ*< 2π. Modes with mode number difference Δ

*n*are simultaneously resonant if the Gouy parameter is

*ψ*= 2π

_{x}*h*/Δ

*n*with an integer

*h*, because their round-trip phases are equal mod(2π). The simplest case, in which a zero of the electric field at the position of the obstacle is achieved, is the combination of the fundamental mode with the next resonant transverse mode. In the middle of the stability range, where the Gouy parameter is

*ψ*= π/2 or 3π/2, the mode number difference for simultaneously resonant modes is Δ

*n*= 4. The corresponding mode combination is shown in Fig. 1 . It constitutes a new eigen-mode of the resonator with a sufficiently small obstacle on the optical axis. The obstacle leads to a coupling between the modes [12].

Further resonant *GH* modes can contribute to the field distribution. This can change the spatial overlap with the incident field, the width of the on-axis region of small intensity, and loss at apertures limiting the transverse extent of the field distribution. Such apertures can be used to suppress the contribution of higher-order *GH* modes.

If a resonator has cylindrical symmetry, the Gouy parameters in the transverse directions are equal, i.e. *ψ _{x}* =

*ψ*, and the Gauss-Laguerre modes

_{y}*GL*with radial and azimuth mode index

^{l}_{p}*p*and

*l*can be used instead of the

*GH*modes as an orthonormal basis of eigen-modes. Various mode number differences and mode combinations can be chosen. A detailed theoretical description exceeds the scope of this paper and will be provided in [13].

#### 2.2 Geometrical on-axis access to a bow-tie ring resonator by quasi-imaging

The simplest setup of a ring resonator, as it is e.g. commonly used for HHG, is a resonator with two focusing mirrors with radius of curvature *R _{C}* and resonator length

*L*(see Fig. 2 ). The distance

*d*of the focusing mirrors determines the Gouy parameters

*ψ*and

_{x}*ψ*in the transverse directions. The angle of incidence

_{y}*α*on the curved mirrors causes different Gouy parameters in the transverse directions, i.e.

*ψ*≠

_{x}*ψ*. Therefore, the condition for quasi-imaging can be achieved only in one transverse direction. In the middle of the stability range, for

_{y}*ψ*= 3π/2, the modes

_{y}*GH*

_{0,0}and

*GH*

_{0,4}are simultaneously resonant and can be combined to a “slit mode” avoiding an on-axis slit in the output coupling mirror. The on-axis intensity oscillates along the propagation direction, with zero on the optical axis in the focal plane and an intensity maximum one Rayleigh length from the focus (see Fig. 2).

#### 2.3 Adjustment of transverse mode degeneracy

Quasi-imaging relies on a degeneracy of transverse modes, which are simultaneously excited to form a field distribution avoiding an obstacle. A degeneracy can be easily adjusted by interpreting the scan pattern, i.e. the transverse mode resonances at a scan of the oscillator cavity length, for the enhancement cavity without an obstacle. Figure 3
shows a schematic diagram of such a scan pattern for a femtosecond bow-tie ring resonator. The abscissa shows the varied oscillator cavity length multiplied with the wave number and the ordinate shows the corresponding intracavity power. Resonances of the fundamental mode (*GH*
_{0,0}) appear with a period of 2π, corresponding to a change in cavity length by one wavelength λ. Because higher-order transverse modes acquire an additional phase (Eq. (1)), they are resonant at smaller frequencies than the fundamental mode, i.e. at longer oscillator cavity lengths. As can be easily seen, the Gouy parameter in tangential direction *ψ _{x}* is larger than in sagittal direction

*ψ*for a bow-tie ring resonator. This difference between the Gouy parameters becomes visible in a resonator with finesse

_{y}*F*, if the resonance width 2π/

*F*is smaller than the Gouy parameter difference

*ψ*−

_{x}*ψ*. The position in the stability range can be read from the position of the resonances, the Gouy parameter being the distance between the 0-resonance and 1-resonance. In practice, the transverse modes can be easily identified by the number of adjacent resonances and by tilting of the incident beam and thereby changing its spatial overlap with the cavity transverse modes. Mode

_{y}*GH*

_{4,0}acquires an additional phase compared to the fundamental mode, which is four times the Gouy parameter

*ψ*. Therefore, the two modes are simultaneously resonant if

_{x}*ψ*= 3π/2 holds. In the situation shown in Fig. 3,

_{x}*ψ*has to be increased (by increasing the distance

_{x}*d*between the curved mirrors) in order to achieve the degeneracy.

The main resonance is the resonance for which all comb lines of the incident frequency comb simultaneously coincide with a cavity resonance. The side resonances drop in height and increase in width (width not indicated in Fig. 3) depending on the cavity finesse and spectral bandwidth, because different spectral parts of the frequency comb are on resonance for slightly different cavity lengths.

## 3. Experimental realization

#### 3.1 Experimental setup

In order to demonstrate quasi-imaging, we employed the experimental setup shown in Fig. 4
, and described in [1]. The Yb-based fiber laser system is described in [14]. The laser emits almost Fourier-limited τ = 200 fs pulses (τ∙Δν = 0.39) with a central wavelength λ = 1040 nm and bandwidth Δλ = 7 nm. The repetition rate is ν* _{rep}* = 78 MHz. Throughout the experiments presented in this paper we used an incident average power of

*P*= 1.4 W. Two spherical lenses were used to match the laser output mode to the enhancement cavity fundamental mode. The passive cavity of length

_{in}*L*= 3.84 m consists of eight mirrors. Mirrors M2-M8 are highly reflective (

*R*= 0.99995) over the entire laser bandwidth and M1 is the input coupler (

_{M}*R*= 0.9986). Mirrors M5 and M6 are spherically curved (radius of curvature

_{IC}*R*= 150 mm), all other mirrors are plane. This yields a Gaussian waist radius of

_{C}*w*

_{0}= 22 µm in the middle of the stability range. All cavity mirrors exhibit low dispersion. In order to minimize round-trip dispersion, as well as to avoid nonlinearities in air, the cavity is placed in a vacuum chamber.

For resonant enhancement, the incident laser frequency comb modes have to spectrally overlap with the cavity resonances. As discussed in [3], due to the relatively narrow optical bandwidth of the femtosecond radiation, an active locking of the frequency comb to the cavity resonances can be realized by controlling only one of the two comb parameters, once they are brought in vicinity of optimum spectral overlap. The latter is performed by manually adjusting the enhancement cavity length and the carrier-envelope offset (CEO) phase by using a pair of wedges in the oscillator cavity. The locking electronics drives a piezoelectric transducer (PZT) varying the laser oscillator cavity length. The error signal for the locking is generated with the Hänsch-Couillaud scheme [15] from the signal reflected by the cavity input coupler. The only difference towards the setup described in [1] is a pinhole, enabling the spatial selection of the reflected beam profile for the generation of the error signal. This is particularly useful if the incident transverse mode does not match the excited cavity mode. The intracavity beam leaking through one of the highly reflecting cavity mirrors (M8) with a transmission of 1.65·10^{−6} is characterized using a power meter and an optical spectrum analyzer. The circulating power *P _{circ}* is calculated from the power

*P*leaking through this mirror according to

_{leak}*P*=

_{circ}*P*/1.65·10

_{leak}^{−6}.

A copper wire was placed vertically on a motor-driven translation stage to function as an obstacle on the optical axis of the resonator. The position of the wire along the optical axis determines the positions of the on-axis intensity minima and maxima in the cavity. In order to yield the situation shown in Fig. 2 offering a geometrical access to the focus, the wire would have to be placed in front of the curved mirror behind the focus (M6). However, this position is not relevant for the demonstration of quasi-imaging. This is because the Gouy parameter *ψ* = 3π/2 acquired at one round trip starting at the wire is independent of its position. Therefore, the convenient implementation of the wire (400 mm before mirror M5) as well as the vertical orientation is chosen for good accessibility. The plane of the wire is imaged 1:1 with a lens (*f* = 400 mm) on a CCD camera (see Fig. 4(b)). This allows for adjusting the wire to the optical axis and observing the beam profile at that position. To record the beam profile at different positions in the cavity the camera can be moved on a rail. The 100 µm width of the wire equals 5% of the Gaussian beam diameter of 2*w* = 2·0.99 mm at that position. An additional vertical aperture is placed in the cavity at the same longitudinal position and with a distance from the optical axis of roughly three times the Gaussian beam radius. This aperture serves as a spatial filter, i.e. it suppresses contributions of higher-order transverse modes other than *GH*
_{0,0} and *GH*
_{4,0} to the circulating field.

A variation of the position in the stability range is achieved by displacing mirror M5 and compensating the change in the resonator length by displacing mirror M7 (see Fig. 4(a)). The cavity alignment is not significantly affected.

Throughout the experiments presented here, the q-parameter of the incident beam was adjusted for maximum spatial overlap with the fundamental mode of the enhancement cavity without the wire. The spatial overlap with higher-order transverse cavity modes is non-zero. We determined the spatial overlap with the fundamental mode by analysis of a scan pattern. We scanned the oscillator cavity length over more than one free spectral range and recorded the intracavity power measured with a photodiode placed behind mirror M8 and connected to an oscilloscope. For this experiment, we replaced mirrors M1 (the input coupler) and M8 by mirrors with a reflectivity of *R _{M}* = 0.99. With the resulting finesse

*F*= 100π and with a scan time

*t*= 7 ms for one free spectral range, the scan time over one resonance width

_{fsr}*t*=

_{res}*t*/

_{fsr}*F*= 22 µs is about 17 times larger than the cavity build-up time

*t*= (

_{bu}*F*/π)/ν

*= 1.3 µs. This is long enough to allow for a full build-up of the intracavity radiation for each transverse mode, and short enough to avoid distortions by mechanical vibrations. We assume a linear behavior of the scanning PZT. Then the intracavity power integrated over the illumination time of a transverse mode or a group of transverse modes is proportional to the spatial overlap of the incident beam with the respective mode or group of modes. Transverse modes with mode order higher than 4 were not measurable and therefore not considered. In this manner, the spatial overlap of the incident beam with the fundamental mode was estimated to be*

_{rep}*U*= 0.94.

#### 3.2 Enhancement of the cavity modes GH_{0,0} and GH_{1,0}

In two first experiments, we locked the cavity without the wire to the fundamental mode (*GH*
_{0,0}) and, subsequently, to the *GH*
_{1,0} mode. The purpose of these experiments was to characterize the empty cavity, in particular the round-trip loss and impedance matching. Also, the elaborate evaluation of finesse and overlap from enhancement and coupling was tested for consistency.

With the fundamental mode we achieved stable locking over several minutes with a circulating average power of *P _{circ}* = 1950 W. This corresponds to a power enhancement of

*P*/

_{circ}*P*= 1400. The circulating power fluctuates in time and repeatedly drops to ~0.8 of the maximal value. We attribute this strong fluctuation to the relatively slow PZT used for locking (measured mechanical bandwidth <20 kHz) in conjunction with the high cavity finesse. This was confirmed by decreasing the cavity finesse, which led to a significant decrease of the fluctuations. Furthermore, this is not a fundamental limitation of cavity enhancement, since a larger locking bandwidth can be achieved by advanced design of PZT-actuated mirrors [16]. The time-averaged circulating power

_{in}*P*was measured with a power meter. We assume that the maximal power level

_{circ}*P*is determined by the loss and overlap, and the dropping is due to a deviation from resonance. This maximum value is

_{circ}^{max}*P*= 1.1·

_{circ}^{max}*P*(compare Fig. 5(a) ). The corresponding maximum level of the enhancement, which was used to calculate the loss and the overlap, is

_{circ}*E*=

*P*/

_{circ}^{max}*P*= 1540. The coupling ratio

_{in}*K*of the power coupled into the cavity to the incident power was determined from the drop of the signal reflected from the input coupler compared to the unlocked case. This signal was measured with a photodiode. The maximum level of the coupling was determined to be

*K*= 0.70. From these values the round trip loss factor is calculated to be

*R*= 1 −

*K*/

*E*= 0.99955. It follows for the cavity finesse

*F*= 2π/(1 −

*R*) = 3400. The difference between the measured enhancement

_{IC}R*E*= 1540 and the enhancement expected from the loss

*E*= (1 −

_{R}*R*)/(1 − (

_{IC}*R*)

_{IC}R^{1/2})

^{2}= 1630 can be attributed to an incomplete spatial overlap of

*U*=

*E*/

*E*= 0.94, since a complete spectral overlap was confirmed by measuring the incident and circulating spectra. This spatial overlap matches the value estimated from the scan. The pulse duration was measured to be τ = 200 fs.

_{R}We then locked the cavity without the wire to the *GH*
_{1,0} mode. Because the *GH* modes form an orthogonal set of modes, the spatial overlap of an incident *GH*
_{0,0} mode with the circulating *GH*
_{1,0} mode vanishes, unless the incident beam has a tilt or transverse offset with respect to the cavity optical axis. In the experiment, the q-parameter was left the same as before and the beam was tilted horizontally to optimize the spatial overlap. The maximum spatial overlap for an incident Gaussian beam with eigen-q-parameter is easily calculated to be *U* = 1/*e* = 0.37 for a tilt angle of γ = 2/(*k*·*w*) or an offset of Δ*x* = *w* with the wave number *k* and Gaussian beam radius *w*. The pinhole in the locking scheme was used to select one of the two lobes in the reflected signal. The CEO phase of the incident pulse train had to be adjusted for maximum spectral overlap of the frequency comb modes with the cavity resonances corresponding to the *GH*
_{1,0} mode. This is due to the fact that the cavity resonances for mode order *n* = 1 are shifted by the Gouy parameter *ψ _{x}* compared to the fundamental mode (see Fig. 3). A power enhancement of

*P*/

_{circ}*P*= 550,

_{in}*P*= 1.1·

_{circ}^{max}*P*and a coupling of

_{circ}*K*= 0.28 were measured. The finesse is determined to be

*F*= 3400, which is the same as for the fundamental mode. The spatial overlap is

*U*= 0.37, which matches the value calculated for an incident Gaussian beam. The smaller power enhancement compared to the fundamental mode is due to the reduced spatial overlap alone, the resonator loss and therefore the finesse is identical in both cases. As in the case of the fundamental mode, the entire spectrum was coupled to the

*GH*

_{1,0}mode and evenly enhanced.

#### 3.3 Quasi-imaging: enhancement of slit modes with an on-axis obstacle

In the final experiment the incident beam was aligned to the cavity optical axis and the CEO phase was adjusted for maximum spectral overlap with the cavity resonances of the fundamental mode. The wire was placed as an obstacle on the optical axis, which can be done with high precision by monitoring the image of the wire while the cavity is not locked (see Fig. 4(b)). In order to achieve the transverse mode degeneracy, we adjusted the position in the stability range by changing the distance *d* between the curved mirrors by displacing mirror M5 and compensating the change in the resonator length by displacing mirror M7. The position in the stability range can be read from the scan pattern of the cavity with the wire removed from the beam path, as is discussed in Section 2.3. Figure 5(b) shows a recorded scan pattern.

In the locked case, different field distributions with different numbers of lobes were observed, depending on the position in the stability range (see Fig. 6
). Field distributions with more than the four lobes for the simple slit mode depicted in Fig. 1(a) involve the contribution of higher-order transverse modes (*GH*
_{8,0} and *GH*
_{12,0}). In order to suppress these contributions, a vertical aperture was placed in the cavity and moved towards the optical axis until additional lobes besides the four were suppressed. This was the case for a distance from the optical axis of about three times the Gaussian beam radius (see Fig. 7(a)
).

As in the previous experiment, we used the pinhole in the locking scheme to select one of the lobes for the generation of the error signal. We went through a region in the stability range around the degeneracy by varying the distance *d* by some tens of µm. Figure 8(a)
shows the enhancement as a function of the detuning from the quasi-imaging condition in terms of the Gouy parameter *ψ _{x}* and of the distance

*d*between the curved mirrors. We have derived a theoretical model for the detuning curve. However, the model exceeds the scope of this paper and will be described in [13]. The experimentally measured width (FWHM) of the detuning curve is Δ

*δ*= 17 µm. On the one hand, this is a small fraction of the entire stability range of 3.2 mm. On the other hand, it is so broad that the quasi-imaging can be easily adjusted manually and does not demand for an active control of the distance

*d*.

For maximum power enhancement *P _{circ}*/

*P*= 330 at circulating power

_{in}*P*= 460 W, a maximum level of the circulating power

_{circ}*P*= 1.08·

_{circ}^{max}*P*(see Fig. 5(a)) and a coupling of

_{circ}*K*= 0.24 were measured. The finesse is determined to be

*F*= 3000, which represents only a small decrease compared to the cavity without the wire

*F*= 3400. The additional round-trip loss at the wire amounts to 2.3·10

^{−4}. With the wire the enhancement expected from the loss is

*E*= 1290. This represents a reduction by 0.79 compared to the cavity without the wire. However, the smaller power enhancement compared to the fundamental mode is mainly due to the reduced spatial overlap. Figure 7 shows the measured intensity distribution of the enhanced field at the position of the wire and at a position of maximum intensity on the optical axis. A fit of a mode combination of

_{R}*GH*

_{0,0},

*GH*

_{4,0}and

*GH*

_{8,0}with complex coefficients to the intensity distribution yields a power fraction of |

*c*

_{0}|

^{2}= 0.25 in the fundamental mode. The power fraction in the

*GH*

_{8,0}mode is only |

*c*

_{8}|

^{2}= 0.03, because it is suppressed by the aperture. The power fraction of |

*c*

_{0}|

^{2}= 0.25 defines the expected spatial overlap with an incident Gaussian beam with the eigen-q-parameter. The spatial overlap calculated from enhancement and coupling is

*U*= 0.27, which is in good agreement with the value of |

*c*

_{0}|

^{2}. The entire spectrum was coupled to the slit mode in the cavity and evenly enhanced (see Fig. 8(b)).

For a thicker wire with 220 µm width (which equals 11% of the Gaussian beam diameter) placed at the same position we measured a power enhancement *P _{circ}*/

*P*> 300. Table 1 summarizes the results for the different resonator modes.

_{in}The power enhancement in the experiment is limited by the imperfect spatial overlap of *U* = 0.27 for the simple slit mode. The spatial overlap is expected to reach 0.44 for an incident Gaussian beam with an adapted q-parameter [13]. A cylindrical telescope has to be used then for mode matching. Theoretically, the spatial overlap can reach unity if the incident field is adequately shaped. Beam shaping can e.g. be achieved with free form optics. The increase of the spatial overlap is a problem which can be solved outside the enhancement cavity.

## 4. Further discussion

The experiment presented here shows that it is possible to combine simultaneously resonant transverse modes in a stable degenerate cavity to form a field distribution reproducing an on-axis hole or slit at every round trip. A hole or slit is also reproduced in an imaging resonator. However, unlike imaging, quasi-imaging can be achieved in a stable resonator. In the simple case of a ring resonator with two focusing mirrors and a constant length, only one parameter, namely the distance of the focusing mirrors, has to be adjusted to achieve quasi-imaging. An imaging resonator is understood to be telecentrically imaging with magnification ± 1, because only then a field distribution can be reproduced after one resonator round trip. This can be achieved in a symmetrical confocal resonator, for which the length has to equal the radius of curvature [9]. If one wants to choose the resonator length and the radius of curvature independently, further curved mirrors have to be added, leading to additional distances which have to be adjusted [17]. An imaging resonator is not stable in the sense that it does not possess a distinguished eigen-q-parameter. Instead, any q-parameter is reproduced after a resonator round trip. Both concepts, imaging and quasi-imaging, are promising as geometrical output coupling methods for intracavity-generated XUV radiation, offering the prospect of average and peak power scalability in enhancement cavities for HHG.

In order to achieve a geometrical on-axis access to the cavity, the obstacle can be an opening in a cavity mirror. This is preferably a curved mirror, if an access to the focus is desired. The fabrication of a standard quarter-inch-thick mirror with a hole or slit with dimensions on the order of 100 µm is challenging, nevertheless first experiments are promising. Fabrication of such mirrors is currently being pursued by direct laser drilling with a frequency doubled ns-laser and a sub-ns-laser [18] and by in-volume selective laser etching (ISLE).

The on-axis access to the cavity could be used for output coupling of high harmonics generated in a gas jet near the focus. The suitability of the field distribution for efficient high harmonic generation still has to be shown. As can be seen from Fig. 2, the field distribution changes its shape rapidly in the region around the focus. Therefore, the coherence length is somewhat reduced compared to the fundamental mode with the same Rayleigh length. The field one Rayleigh length from the focus has a central lobe on the optical axis containing 0.45 of the power. This central lobe has an elliptical shape, as can be seen in Fig. 7(b). In the focal plane the two central lobes contain 0.59 of the power. These values hold for the simple slit mode of *GH*
_{0,0} and *GH*
_{4,0} alone. Contributions of higher-order transverse modes can change the shape of the field. This can be manipulated by apertures, by the shape of the incident field and by the position in the stability range. The interaction region for HHG could be limited to one of these two regions. The lobes in the focal plane are in phase, which means that the harmonic signal will exhibit a maximum on the optical axis in the far field. In contrast, the *GH*
_{1,0} mode, as well as other *GH* or *GL* modes with vanishing intensity on the optical axis, are unfavourable for HHG, because the lobes oscillate with opposite phase and the harmonic signal vanishes on the optical axis [8].

The geometrical on-axis access to a high-finesse cavity could also be used for other applications besides output coupling of high harmonics, e.g. for Thomson back-scattering at electrons for the generation of coherent X-radiation [19]. Here, a cavity can be used to enhance the fs-radiation which is back-scattered at a high-energy electron beam. A geometrical access could be used for output coupling of the X-radiation and/or the input of the electron beam.

The rapidly changing shape of the intensity distribution could be advantageous for other applications. E.g. by the combination of two *GL* modes, an intensity minimum in the focus can be formed, which is strongly localized in radial as well as in axial direction. Such a field distribution, possibly enhanced in a cavity, can serve as a dipole trap for neutral atoms [20]. Moreover, such field distributions can also be excited in an active resonator by the use of a degeneracy of transverse modes and adapted apertures.

## 5. Conclusion

We demonstrated quasi-imaging in a high-finesse femtosecond enhancement cavity. This is a stable cavity with a degeneracy of transverse modes and an obstacle in the beam path. The obstacle exacts a hole in the field distribution. This hole is reproduced after a resonator round trip due to the mode degeneracy. Therefore, only small loss is introduced by the obstacle. The concept allows for a geometrical on-axis access to the cavity, which could be used as a dispersion-free and peak as well as average power scalable method for output coupling of intracavity-generated high harmonics with increasing efficiency for decreasing wavelengths. In order to achieve quasi-imaging, the distance of the curved mirrors in the bow-tie ring cavity has to be adjusted with a precision of a few µm, which is only a small fraction of the stability range but still can be attained manually.

We excited a field distribution avoiding a wire on the optical axis and at a different position exhibiting an on-axis intensity maximum. Compared to the fundamental mode with *F* = 3400, the cavity finesse was only slightly reduced to *F* = 3000 by the insertion of the wire with a width equal to 5% of the Gaussian beam diameter. This corresponds to an additional round-trip loss of 2.3·10^{−4}. The decrease of the power enhancement from *P _{circ}*/

*P*= 1400 to

_{in}*P*/

_{circ}*P*= 330 is mainly due to the imperfect spatial overlap, which could be increased by shaping the incident beam.

_{in}## Acknowledgments

This work was supported within the cooperation project KORONA between the Max-Planck-Institut für Quantenoptik (MPQ) and Fraunhofer-Institut für Lasertechnik (ILT). We thank the fiber laser group at the Institute of Applied Physics at Friedrich-Schiller-University Jena for providing the laser source for the experiments.

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