Geometrical on-axis access to high-finesse resonators by quasi-imaging: a theoretical description

If an obstacle is placed in the beam path of a stable resonator the GH or GL modes are no longer the resonator eigen-modes, unless the transverse extension of the obstacle is sufficiently small and it is placed at a zero of the electric field [22]. The new eigen-modes of the resonator with an obstacle are in general afflicted with diffraction loss. However, in the case of a quasi-imaging resonator, where a set of GH or GL modes is degenerate, new eigen-modes can be simply constructed as a linear combination thereof. If such a linear combination avoids the obstacle it can be considered a new eigen-mode with small diffraction loss. In order to set the field exactly to zero on a finite area, the number of modes would have to be infinite. Therefore, the diffraction loss cannot reach zero. It can, however, for a suitable obstacle be very small for the combination of just a few modes.

Of particular interest is an on-axis obstacle that is e.g. formed by a hole or slit in a resonator mirror. Mode combinations of GH modes avoiding a slit-shaped obstacle we call 'slit modes', mode combinations of GL modes avoiding a hole we call 'hole modes'. Depending on the geometry of the resonator and the Gouy parameter, i.e. mode number difference, numerous mode combinations avoiding an on-axis obstacle can be constructed. The simplest fashion to achieve a vanishing on-axis field at the position of the obstacle is to combine the fundamental mode with the next resonant even transverse mode. For ψ_E = π/2 or 3π/2 it is Δn = 4 and Δp = 2 and the mode combinations are GH_0,0 & GH_4,0 and GL⁰₀ & GL⁰₂ (figures 2(a), (b)). For the slit mode we assume a Gaussian distribution in the direction parallel to the slit.

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For ψ_E = π/3 or 5π/3 it is Δn = 6 and Δp = 3 and the mode combinations are GH_0,0 & GH_6,0 and GL⁰₀ & GL⁰₃ (figures 2(c), (d)). For ψ_E = 2π/3 or 4π/3 it is Δn = 3, however, the GH_3,0 mode is odd and cannot yield zero on the optical axis in combination with the fundamental mode. For the sake of completeness, note that in the case of an imaging resonator with ψ_E = 0 or π all respectively all even/odd modes are simultaneously resonant and the simplest mode combinations avoiding an on-axis obstacle are GH_0,0 & GH_2,0 and GL⁰₀ & GL⁰₁ (figures 2(g), (h)). The difference between an imaging resonator and a quasi-imaging stable resonator will be discussed in more detail in section 7.

The mode combinations yield zero intensity on the optical axis at the position of the obstacle. Upon propagation in the resonator the modes acquire a different phase, so that the on-axis intensity of the mode combination oscillates between zero and a value I_max. For two combined transverse modes GH_n1,m(x,y), GH_n2,m(x,y) with coefficients c₁, c₂ the on-axis intensity is I(z) = P|c₁GH_n1,m(0,0) exp(i(n₁ + ½)ψ_x(z)) + c₂GH_n1,m(0,0) exp(i(n₂ + ½)ψ_x(z))|². With c₁ GH_n1,m(0,0) = –c₂ GH_n2,m(0,0) by construction and c₁² + c₂² = 1, this yields I(z) = I_maxsin²((Δn/2)ψ_x(z)) with I_max = 4P|c₁ GH_n1,m(0)|² = 4P|c₂ GH_n2,m(0)|², assuming m even. The on-axis intensity is I(z) = I_max sin²(Δp ψ_r(z)) in the case of cylindrical symmetry. The intensity course changes if additional transverse modes are involved. For the case that the obstacle is a hole or slit in a focusing mirror behind a resonator-internal focus, the on-axis intensity in the focal plane is maximal for the case ψ_E = π/3 and is zero for the case ψ_E = π/2. In the latter case, the on-axis intensity is maximal one Rayleigh length before and behind the focus. This follows from (4), which yields ψ = π/4 for propagating the distance z_R from the waist position and ψ ≈ π/2 for propagating many Rayleigh lengths to the focusing mirror.

Single GH_n,m modes with odd order n or m and GL^l_p modes with azimuth mode order l > 0 exhibit zero intensity on the optical axis and can avoid a small on-axis obstacle (as demonstrated for GH_1,0 in [22]). However, the width of the intensity minimum can be enlarged by combination with suitable additional modes. Two simple cases in Cartesian and cylindrical geometry, respectively, are the combination of GH_1,0 & GH_5,0 and of GL²₀ & GL²₂ (figures 2(e), (f)). If an on-axis maximum of the field distribution in the focus is desired, phase masks can be applied (figure 2) as suggested in [22]. In order to minimize loss due to the phase masks, the second phase mask should be in the image plane of the first. This is only approximately given for the two focusing mirrors of a bow-tie resonator in the middle of the stability range (section 4). Therefore, the first phase mask should be on a plane mirror before the first focusing mirror and the second one on the focusing mirror with a slit or hole following the focus.

The diffraction loss due to an obstacle can be estimated by the following consideration. The power transmission for a normalized field u at the position of an obstacle is T = ∫∫_Ω dξ|u|², with the integration area Ω omitting the obstacle. Note that T is the power reflection if the obstacle is a slit of hole in a mirror. The normalized truncated field then is u_T = u/T^1/2 in Ω and 0 elsewhere. The spatial overlap U_obstacle of this truncated field with the original field is

$$\begin{eqnarray}\begin{array}{lllllllllllllll} {{U}_{{\rm obstacle}}}\left( u,\Omega \right) & = & {{\left| \mathop \iint \limits_{I{{R}^{2}}} {\rm d}\xi uu_{T}^{*} \right|}^{2}}={{\left| \mathop \iint \limits_{\Omega } {\rm d}\xi u\frac{{{u}^{*}}}{\sqrt{T}} \right|}^{2}} \\ {} & = & {{\left| \frac{1}{\sqrt{T}}\mathop \iint \limits_{\Omega } {\rm d}\xi {{\left| u \right|}^{2}} \right|}^{2}}=T. \\ \end{array}\end{eqnarray} \tag{ 8 }$$

Therefore, the same loss as at the obstacle can be expected at the propagation of the truncated field, and the diffraction loss per resonator round trip is estimated to be 1 − T². This assumes that the part of the truncated field not overlapping with the original field is lost at the resonator round trip, which is not necessarily true in particular in the case of transverse mode degeneracy. A rigorous model is given in section 6. For a non-degenerate resonator this is expected to be a very good estimation, which has been validated in [23]. In order to damp the part of the circulating field that is diffracted into higher order modes at the obstacle, appropriate apertures can be included in the resonator. The correspondingly estimated round trip loss is shown for different modes as a function of the obstacle size in figure 3. The calculation suggests that for eigen-modes of a quasi-imaging resonator compared to simple GH or GL modes, a much broader slit or hole is possible at small loss.

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Besides the spatial overlap U_obstacle describing the diffraction loss at the obstacle, another spatial overlap has to be introduced: The spatial overlap U_IC of the circulating field u_circ in the resonator (mode or mode combination) and the impinging field u_in (typically a Gaussian beam), which indicates the spatial overlap at the input coupling mirror of an enhancement resonator (section 5)

$$\begin{eqnarray}&&{{U}_{IC}}={{\left| \mathop \iint \limits_{I{{R}^{2}}} {\rm d}\xi {{u}_{{\rm circ}}}u_{{\rm in}}^{*} \right|}^{2}}.\end{eqnarray} \tag{ 9 }$$

An incomplete overlap proportionally diminishes the power enhancement.

Table 1 summarizes some properties of different GH and GL modes and mode combinations.

Table 1.  Properties of different modes and mode combinations. Spatial overlap U_IC (q_E) of the circulating field with an impinging Gaussian beam with the same q-parameter and spatial overlap U_IC (q_max) with a Gaussian beam with the q-parameter adapted for maximal overlap. The values in parentheses hold if a phase mask is used (step of π on the optical axis for GH_1,0 and GH_1,0 & GH_5,0, sinusoidal around the optical axis for GL²₀ and GL²₀ & GL²₂). Hole radius or half slit width a divided by the Gaussian beam radius w for a loss of 1% and 0.1%, allowing for a power enhancement of 100 and 1000 (with negligible other losses and impedance matched) referred to the spatially and spectrally overlapping part of the incident beam. The loss is estimated by the square of the transmission T through the obstacle. Power P₀ in central lobe, maximal intensity I₀ in central lobe; distance x₁ from the optical axis to the intensity maximum at the obstacle, power P₁ in ring around the hole or in the two lobes next to the slit, maximal intensity I₁ in the ring around the hole or in the lobes next to the slit. The power is indicated relative to the total power P, the intensity relative to the on-axis intensity of the fundamental mode I = 2P/(πw²) at the same longitudinal position.

Geometry	Cartesian						Cylindrical
Gouy parameter ψE			$\pi$	π/2	$\pi /2$	$\pi /3$			$\pi$	π/2	$\pi /2$	$\pi /3$
GL/GH mode or mode combination	$G{{H}_{0,0}}$	$G{{H}_{1,0}}$	$G{{H}_{0,0}}$ & $G{{H}_{2,0}}$	$G{{H}_{0,0}}$ & $G{{H}_{4,0}}$	$G{{H}_{1,0}}$ & $G{{H}_{5,0}}$	$G{{H}_{0,0}}$ & $G{{H}_{6,0}}$	$GL_{0}^{0}$	$GL_{0}^{2}$	$GL_{0}^{0}$ & $GL_{1}^{0}$	$GL_{0}^{0}$ & $GL_{2}^{0}$	$GL_{0}^{2}$ & $GL_{2}^{2}$	$GL_{0}^{0}$ & $GL_{3}^{0}$
Overlap UIC (qE)	1	0	0.333	0.273	0	0.238	1	0	0.500	0.500	0	0.500
Overlap UIC (qmax)	1	(0.827)	0.690	0.442	(0.819)	0.328	1	(0.844)	0.844	0.593	(0.559)	0.500
a/w for T2 = 0.99	0.003	0.134	0.321	0.254	0.403	0.222	0.050	0.411	0.411	0.326	0.626	0.285
a/w for T2 = 0.999	0.0003	0.062	0.199	0.157	0.282	0.137	0.016	0.274	0.274	0.217	0.481	0.190
At the position of the maximal on-axis intensity: field in the central lobe
Power P0/P	1	—	0.704	0.449	—	0.332	1	—	0.865	0.594	—	0.463
Intensity I0/I	1	—	1.333	1.091	—	0.952	1	—	2	2	—	2
At the position of the obstacle: field in a ring around the hole or in the two lobes next to the slit
Distance x1/w	—	0.707	1	0.707	0.959	0.590	—	1	1	0.765	1.199	0.656
Power P1/P	—	1	1	0.588	0.861	0.473	—	1	1	0.762	0.966	0.689
Intensity I1/I	—	0.736	0.722	0.713	0.916	0.693	—	0.271	0.271	0.425	0.301	0.335

The discussed mode combinations of two GH or GL modes are eigen-modes of a resonator in which these modes are simultaneously resonant and with an obstacle on the optical axis which is sufficiently small so that the diffraction loss is small. By adding higher-order resonant GH or GL modes, further eigen-modes of the quasi-imaging resonator can be constructed. For a quasi-imaging resonator with mode number difference Δn = 4, the simplest slit mode containing the fundamental mode is

$$\begin{eqnarray}&&{{a}_{1}}\left( x \right)=\sqrt{\frac{3}{11}}G{{H}_{0,0}}\left( x \right)-\sqrt{\frac{8}{11}}G{{H}_{4,0}}\left( x \right).\end{eqnarray} \tag{ 10 }$$

We therefore call it 'simple slit mode'. Another slit mode, that also vanishes on the optical axis and is orthogonal to the first, can be constructed by adding the next resonant mode GH_8,0

$$\begin{eqnarray}\begin{array}{lllllllllllllll} \begin{array}{lllllllllllllll} {{a}_{2}}\left( x \right) & = & \sqrt{\frac{280}{2321}}G{{H}_{0,0}}\left( x \right)+\sqrt{\frac{105}{2321}}G{{H}_{4,0}}\left( x \right) \\ {} & {} & -\sqrt{\frac{1936}{2321}}G{{H}_{8,0}}\left( x \right). \\ \end{array} \\ \end{array}\end{eqnarray} \tag{ 11 }$$

Another slit mode, orthogonal to the two others is

$$\begin{eqnarray}\begin{array}{lllllllllllllll} {{a}_{3}}\left( x \right) & = & \sqrt{\frac{29568}{404909}}G{{H}_{0,0}}\left( x \right)+\sqrt{\frac{11088}{404909}}G{{H}_{4,0}}\left( x \right) \\ {} & {} & +\sqrt{\frac{8085}{404909}}G{{H}_{8,0}}\left( x \right) \\ {} & {} & -\sqrt{\frac{356168}{404909}}G{{H}_{12,0}}\left( x \right). \\ \end{array}\end{eqnarray} \tag{ 12 }$$

These eigen-modes of a quasi-imaging resonator are visualized in figures 4(a), (b). The contribution of the higher-order slit modes to the circulating field changes the spatial overlap with the impinging field distribution and the width and depth of sharpness of the intensity minimum. Higher order slit modes are transversally more expanded and more sensitive to loss at additional apertures limiting the transverse extent of the field distribution. They change their shape at propagation more rapidly, because GH modes of higher order are involved.

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Figures 4(c), (d) shows the simple slit mode a₁ complemented with a contribution of a₂ that yields a maximal slit width (vanishing d²E/dx²| _x × 0) and a maximal overlap with a Gaussian beam with the same q-parameter. It is descriptive that these two conditions are opponent, because the field of the Gaussian beam is maximal on the optical axis.

A common and simple resonator design for an enhancement resonator is a (symmetric) bow-tie ring resonator with two curved focusing mirrors with radius of curvature R_C and resonator length L (figure 5) [4, 5, 7, 8, 22, 23, 28, 36]. Confined by the curved mirrors this resonator has a long arm and a short arm with a small beam waist and a distance of the curved mirrors d. For a constant resonator length (e.g. determined by the repetition rate of the impinging fs pulse train), the resonator is stable for the range of the distance of the curved mirrors d₁ < d < d₂. The Gouy parameter of the resonator is

$$\begin{eqnarray}&&{{\psi }_{E}}=\pi +{\rm arccos} \left( \frac{2{{R}_{C}}L-2dL+2{{d}^{2}}-R_{C}^{2}}{R_{C}^{2}} \right).\end{eqnarray} \tag{ 13 }$$

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Quasi-imaging with Gouy parameter ψ_E = 3π/2 is achieved in the middle of the stability range for d = d_m.

$$\begin{eqnarray}\begin{array}{lllllllllllllll} {{d}_{1}} & = & {{R}_{C}}, \\ {{d}_{m}} & = & \frac{1}{2}L-\frac{1}{2}\sqrt{{{L}^{2}}+2R_{C}^{2}-4{{R}_{C}}L}, \\ {{d}_{2}} & = & \frac{1}{2}L-\frac{1}{2}\sqrt{{{L}^{2}}-4{{R}_{C}}L}. \\ \end{array}\end{eqnarray} \tag{ 14 }$$

With this condition met (d = d_m), there are only two design parameters of the resonator, L and R_C. The Rayleigh lengths z_R in the short (+) and in the long (–) arm are

$$\begin{eqnarray}&&{{z}_{R}}=\frac{R_{C}^{2}}{2L-4{{R}_{C}}\pm 2\sqrt{{{L}^{2}}+2R_{C}^{2}-4{{R}_{C}}L}}.\end{eqnarray} \tag{ 15 }$$

The beam waist radius in the short arm is w₀, which is in first approximation proportional to R_C. In order to achieve a desired beam waist radius, the radius of curvature R_C has to be chosen such that:

$$\begin{eqnarray}\begin{array}{lllllllllllllll} {{w}_{0}} & = & \sqrt{\frac{{{z}_{R}}\lambda }{\pi }}=\sqrt{\frac{L\lambda }{\pi }}\left( \frac{{{R}_{C}}}{2L}+O\left( \frac{R_{C}^{2}}{{{L}^{2}}} \right) \right), \\ {{R}_{C}} & = & \frac{4\pi w_{0}^{2}}{\lambda }\left( \sqrt{\frac{1}{2}+\frac{L\lambda }{4\pi w_{0}^{2}}}-1 \right) \\ {} & = & 2{{w}_{0}}\sqrt{\frac{\pi L}{\lambda }}+O\left( \frac{w_{0}^{2}}{L\lambda } \right), \\ \end{array}\end{eqnarray} \tag{ 16 }$$

with wavelength λ. The beam radius on the curved mirrors is

$$\begin{eqnarray}&&{{w}_{1}}=\sqrt{\frac{\lambda }{\pi }\left( L-{{R}_{C}} \right)}=\sqrt{\frac{L\lambda }{\pi }}\left( 1+O\left( \frac{{{R}_{C}}}{L} \right) \right),\end{eqnarray} \tag{ 17 }$$

i.e. the beam cross section is in first approximation proportional to the resonator length and independent of the radius of curvature. A detuning δψ from quasi-imaging (ψ_E = 3π/2) in terms of the Gouy parameter ψ_E is given by a detuning δ d in terms of the distance d by

$$\begin{eqnarray}&&\delta \psi =\frac{2}{R_{C}^{2}}\sqrt{{{L}^{2}}+2R_{C}^{2}-4{{R}_{C}}L}\delta d+O\left( \frac{\delta {{d}^{2}}}{R_{C}^{2}} \right).\end{eqnarray} \tag{ 18 }$$

For the design of an enhancement resonator for nonlinear processes the ratio of the maximal intensity at the focus and on the focusing mirrors is of importance, because the latter is limited by damage thresholds. For the simple slit mode with Δn = 4 this ratio is the same as for the fundamental mode with the same q-parameter. The intensity can be a factor 1.5 larger on a mirror in the long arm of a bow-tie resonator (if located in the middle of the long arm).

The discussion in this section did so far not include the angle of incidence α on the curved mirrors. This angle leads to an effective radius of curvature of the focusing mirrors R_C,t = R_C cos(α) and R_C,s = R_C /cos(α) in tangential (i.e. plane of beam path) and the sagittal transverse direction, respectively. This leads to an ellipticity of the resonator eigen-mode and a different Gouy parameter for the two transverse directions. Even though the resulting ellipticity  = w_0,s/w_0,t in the beam waist is small for a small angle of incidence ( ≈ 1 + α² in the short arm and  ≈ 1 + (L/R_C − 1) α² in the long arm for the middle of the stability range), the difference in the Gouy parameter Δψ_E = ψ_E,s − ψ_E,t ≈ 2(L/R_C − 1) α² is significant in a high-finesse resonator. The degeneracy of transverse modes with the same sum of the two transverse mode order numbers n + m = const. is lifted and the resonances of these modes can be clearly distinguished [7]. Therefore, quasi-imaging can in this setup possibly be achieved only in one transverse direction. In the other direction, the field distribution is that of the fundamental mode (or another single transverse mode). If quasi-imaging is adjusted in the sagittal instead of tangential direction, the angle of incidence can be smaller, because the field is transversally more expanded in the direction of quasi-imaging. Figure 5 shows a sketch of such a quasi-imaging ring resonator. The discussions in the following sections refer to this situation.

The degeneracy of the transverse modes with n + m = const can be restored, i.e. the Gouy parameters in the transverse directions can be matched, by different means. Further astigmatic elements like a plate at Brewster's angle can be introduced into the resonator. This, however, introduces dispersion and nonlinearity. Off-axis parabolic mirrors can be used instead of spherical mirrors, avoiding the astigmatism. Moreover, a defocusing mirror in the long arm close to a focusing mirror allows for matching the Gouy parameters in the transverse directions, if the angle of incidence is properly chosen. However, all these setups go along with increased alignment sensitivity. Undoubtedly, a resonator with quasi-imaging in both directions has potential advantages: the hole mode allows for a larger opening, exhibits a larger overlap with an impinging Gaussian beam and a larger power fraction in the central lobe compared to the slit mode (table 1). But then, also e.g. the accessibility of the intensity lobes for a gas nozzle for HHG has to be considered. This is better provided for a slit mode, with a Gaussian intensity distribution in tangential direction. Moreover, the alignment of the resonator axis to a slit in a mirror is crucial only in one direction.

This section introduces an analytical model based on mode expansion of the circulating field, which allows for describing the sensitivity against detuning from the degeneracy and against aberrations. The sensitivity against detuning is of crucial importance for the experimental implementation of a quasi-imaging resonator. These calculations can just as well be done by using the method of Fox and Li [10], which gives the same results. However, by the mode expansion the prominence of the Gouy parameter and the transverse modes is more obvious.

In section 3 an orthogonal set of slit modes is introduced (10)–(12), which allows for describing the field distribution in a quasi-imaging resonator. These slit modes assume that all contributing GH modes are simultaneously resonant. In order to model the performance of the resonator for a detuning from degeneracy, a different approach has to be chosen. The circulating field can be expanded by the GH modes, which are close to resonance. This is a limitation of generality only in so far, as non-resonant modes are neglected. If several GH modes propagating independently in the resonator have to be simultaneously enhanced, their round-trip phase difference Δn δψ determined by the mode number difference Δn and the detuning δψ against degeneracy has to be smaller than the width of the resonance curve. The resonance curve for a GH mode in a resonator with Finesse F = 2π/(1 − R_ICR) is P(ϕ) = (1 − R_IC)/(1 + R_ICR − 2(R_ICR)^1/2cos(ϕ)) with the power loss factor R of the resonator, the input coupler reflectivity R_IC and the round-trip phase ϕ (figure 7(a)). The full width at half maximum of this curve is Δϕ = 2π/F. If in a resonator there is only loss acting evenly on the amplitude of the electric field (like the reflectivity of mirrors), the transverse modes propagate independently in the resonator. An impinging beam will be projected on the orthogonal set of GH modes, which will be enhanced (or suppressed) differently depending on their round trip phase ϕ [20, 24]. If two modes with mode number difference Δn and with their resonances shifted due to a detuning δψ against the degeneracy should be equally enhanced, both modes have to be off resonance by half the phase difference Δn δψ/2. The power enhancement of this mode combination as a function of the detuning then is

$$\begin{eqnarray}&&P\left( \delta \psi \right)={{U}_{IC}}\frac{1-{{R}_{IC}}}{1+{{R}_{IC}}R-2\sqrt{{{R}_{IC}}R}{\rm cos} \left( \Delta n\delta \psi /2 \right)},\end{eqnarray} \tag{ 19 }$$

with the spatial overlap U_IC. The width of this detuning curve is Δψ = (2/Δn)(2π/F). This situation is depicted in figure 7(b).

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If an obstacle is placed in the resonator the GH modes are coupled at the obstacle. The above-mentioned consideration is no longer valid and the detuning curve P(δψ) becomes broader. The projection of one GH mode onto another is non-zero if the field is set to zero on a part of the integration area due to an obstacle. This shall be discussed for the quasi-imaging resonator with the modes GH_0,0, GH_4,0 and GH_8,0. In this model the field is described by the expansion for GH modes (which are not eigen-modes of the resonator with the obstacle), because for these modes the Gouy phase is well-defined as a phase additionally acquired on the optical axis according to the Gouy parameter of the resonator. The circulating field is described by the complex coefficients c₀, c₄, c₈ for the (normalized) modes, which are abbreviated by u₀, u₄, u₈. The coupling between the modes is described by a matrix T with elements t_j,l = 2∫_x = a..A u_j(x) u_l^*(x) dx, j, l = 0, 4, 8 [26]. Here a is the half width of the slit and A the distance from the optical axis of an additional aperture, which can be applied to restrict contributions of mode u₈ or higher. T is real and symmetric for GH modes with the same q-parameter. T is the identity matrix for a vanishing obstacle and aperture, but in the presence of an obstacle, the diagonal elements are smaller than unity and the non-diagonal elements differ from zero. By this coupling a power transfer between the modes occurs. This is necessary for instance, if the impinging beam overlaps with the fundamental mode u₀ alone, the circulating field, however, is composed of u₀ and u₄ in order to minimize the loss at a slit. The modes acquire a different phase at a round trip due to a detuning δψ against degeneracy according to the mode order, ψ_l,l = exp(i l δψ), l = 0, 4, 8. The coefficients describing the circulating field c_l for an impinging field b_l can be determined from the postulation of a stationary solution [11, 22]

$$\begin{eqnarray}\begin{array}{lllllllllllllll} {{r}_{IC}}S\left[ \begin{array}{ccccccccccccccc} {{t}_{0,0}} & {{t}_{0,4}} & {{t}_{0,8}} \\ {{t}_{4,0}} & {{t}_{4,4}} & {{t}_{4,8}} \\ {{t}_{8,0}} & {{t}_{8,4}} & {{t}_{8,8}} \\ \end{array} \right]Sr{{{\rm e}}^{{\rm i}\phi }}\left[ \begin{array}{ccccccccccccccc} 1 & 0 & 0 \\ 0 & {{{\rm e}}^{{\rm i}4\delta \psi }} & 0 \\ 0 & 0 & {{{\rm e}}^{{\rm i}8\delta \psi }} \\ \end{array} \right]\left[ \begin{array}{ccccccccccccccc} {{c}_{0}} \\ {{c}_{4}} \\ {{c}_{8}} \\ \end{array} \right] \\ \quad +{\rm i}\sqrt{1-r_{IC}^{2}}\ \;\left[ \begin{array}{ccccccccccccccc} {{b}_{0}} \\ {{b}_{4}} \\ {{b}_{8}} \\ \end{array} \right]=\left[ \begin{array}{ccccccccccccccc} {{c}_{0}} \\ {{c}_{4}} \\ {{c}_{8}} \\ \end{array} \right] \\ \end{array}\end{eqnarray} \tag{ 20 }$$

The round trip is described by the coupling T at the slit (and aperture), the phase δψ imprinted due to detuning, the amplitude loss factor r and the reflection at the input coupler with reflection coefficient r_IC. The matrix S (given below) accounts for potential aberrations at the focusing mirrors. The input coupling is given by the overlap of the impinging beam u_in with the resonator modes b_l = ∫u_l(x) u_in^*(x) dx, l = 0, 4, 8 and the transmission through the input coupler with transmission coefficient t_IC = ${{(1-{{r}_{IC}}^{2})}^{1/2}}$. The round trip phase ϕ is chosen for maximal power enhancement P = P_circ/P_in = Σ_l|c_l|². This derivation is analogous to the derivation of the resonance curve in figure 7(a), which is valid for a single resonator mode.

Wave-front aberrations in an enhancement resonator can limit the enhancement. This limitation is considerably more severe in case of a circulating slit mode or hole mode compared to the fundamental mode due to their larger transverse extent. The reflection on a curved mirror (radius of curvature R_C) placed under an angle of incidence α yields astigmatism, spherical aberration, coma and further aberration terms. Astigmatism, i.e. different effective radii of curvature in the two transverse directions, does not represent an aberration in this context, because it does not yield a deviation from a Gaussian beam but only produces different eigen-parameters for the transverse directions (section 4). Aberrations, i.e. deviations from a parabolic wave front, have basically two effects: a coupling between transverse modes and different phase shifts for the transverse modes. The latter effect can be relevant even if the coupling between the transverse modes is weak. From the additional mode-dependent phase shift it follows that not all modes of a degeneracy (in our case u₀, u₄, u₈...) are resonant for exactly the same Gouy parameter. Therefore, the detuning curve can show multiple maxima with different modes being simultaneously resonant. The effect of the aberration on the field in the resonator is described by a matrix S with elements

$$\begin{eqnarray*}&&{{s}_{j,l}}=\iint {{u}_{j}}(x,y){\rm exp} ({\rm i}\Delta \phi (x,y)u_{l}^{*}(x{{,}_{~}}y)dxdy\end{eqnarray*}$$

Here Δϕ (x, y) is the difference between the imprinted phase at reflection from the mirror and a parabolic phase, suitably chosen to remove a focusing effect or tilt, thereby considering the net aberration. As criterion for the proper removal of focusing and tilt we minimize the rms wave-front difference weighted with the intensity profile of the fundamental mode u₀, which is equivalent to maximizing |s_0,0|. Moreover, an offset Δϕ (0, 0) is chosen to yield zero phase shift for the fundamental mode.

For spherical aberrations alone (vanishing angle of incidence α) the phase difference at reflection on a mirror taking into account terms up to the fourth order is Δϕ (r) = −Δϕ_w (r⁴/w⁴ − 2r²/w² + 1/2), where the parameter Δϕ_w = kw⁴/(4R_C³) with the Gaussian beam radius w on the mirror indicates the strength of the aberration. With (16) and (17) it follows Δϕ_w ≈ (1/16) (λ/π)^5/2L^1/2/w₀³. This means that the strength of the aberration Δϕ_w strongly depends on the beam waist radius w₀. Expanding the phase shift imprinted due to spherical aberration in Δϕ_w we find in first approximation arg(s_0,0) = 0, arg(s_4,4) = (9/2) Δϕ_w and arg(s_8,8) = 21Δϕ_w. A finite angle of incidence α additionally leads to coma, which adds to the phase shift. The phase shift increases by a factor 1/(1 − α²)^3/2 for transverse modes in sagittal direction and by a factor (1 + 4α²)/(1 − α²)^7/2 for transverse modes in tangential direction. That is, the effect is stronger in tangential direction.

The absolute values of the matrix elements describe the coupling strength and loss due to the aberration. For vanishing angle of incidence they are in first approximation given by |s_0,0| = 1 − (1/8) Δϕ_w², |s_4,4| = 1 − (689/32) Δϕ_w² and |s_8,8| = 1 − (4413/16) Δϕ_w². Again, coma adds to the deviation from unity and this increase is stronger for transverse modes in tangential direction compared to the sagittal direction. For small angles of incidence the effect of coma is small and it will be neglected in the following.

Estimating |s_l,l| as a loss factor for mode u_l, the loss for u₄ is considerably greater than for the fundamental mode u₀ (the factor being 170), as can be expected from the greater transverse extent. The limitation of the finesse due to spherical aberration for a quasi-imaging resonator is therefore considerably more critical than for operation in the fundamental mode.

Besides the aberrations of spherical surfaces also the deformation of surfaces can produce aberrations and lead to a coupling of transverse modes. This can be surface roughness [27], thermally induced deformations [28] or surface deformations caused by the processing of holes or slits [29].

Figure 8 shows the calculated power enhancement as a function of detuning from quasi-imaging together with experimental data, which are taken from [7]. Model and measurement are in excellent agreement. The finesse at the maximum of the curve is F = 3000, power enhancement is limited by the spatial overlap of U_IC = 0.27 of the impinging Gaussian beam with the circulating simple slit mode. Also the detuning curve according to (19) is drawn in figure 8, showing that the coupling of the modes at the obstacle leads to a significant broader detuning curve. It is sufficiently broad that it does not demand for an active control. Then again, this is only a small fraction of the stability range of d₂ − d₁ = 3.2 mm.

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In the experiment [7] an aperture was used to suppress contributions of transverse modes higher than u₄. A distance of the aperture of A = 2.9 w from the resonator axis is sufficient to damp mode u₈ and higher, and does not introduce significant loss to u₀ and u₄. Figure 9 shows results of a calculation with the same parameters as in figure 8, contrasting the situation with and without the aperture. With the aperture omitted, higher-order transverse modes can contribute to the circulating field. These contributions appear at a position in the stability range which is detuned compared to the degeneracy of the modes u₀ and u₄ (δψ = 0), thereby compensating for the greater phase which these modes acquire due to spherical aberration. They lead to a structured detuning curve with maxima and minima (figure 9(c)). For a given detuning there can be more than one maximum of the enhancement at varied round-trip phase, corresponding to different field distributions. The simulation yields three distinguished areas of high enhancement in a space spanned by detuning δψ and round-trip phase ϕ (figure 9(a)). This follows from the inclusion of the modes u₀, u₄, u₈, u₁₂ in the simulation, which can be combined to the three slit modes a₁, a₂, a₃ (10)−(12). By inserting the aperture the contributions of higher transverse modes can be reduced to only a small portion (figure 9(d)).

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We performed calculations to investigate, which half slit width a related to the beam radius w at the position of the slit can be tolerated to allow for a desired enhancement of the simple slit mode. We assume the impinging beam to be shaped for complete spatial overlap with the simple slit mode. To show the qualitative effects, a round-trip loss factor R = 0.999 is assumed, which accounts for the imperfect reflectivity of the resonator mirrors. Spherical aberration of Δϕ_w = 0.5 mrad per focusing mirror are assumed (as a reasonable value for a bow-tie enhancement resonator). An aperture suppresses contributions of higher transverse modes. Figure 10(a) shows the calculated power enhancement P_circ/P_in versus slit width for different input coupler reflectivities R_IC. The inlay of figure 10(a) shows the spatial overlap of the circulating field with the simple slit mode, which is close to unity in a large range of slit widths.

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For very broad slits, the circulating field arranges in a way which yields a decreasing overlap with the slit mode. Surprisingly, also for very small slits the spatial overlap decreases, therefore deviating from the impinging field distribution, and also the enhancement is smaller compared to a half slit width of a/w ∼ 0.05. This is due to the spherical aberration, which lets the contributing modes u₀ and u₄ run out of phase. A finite slit width with a stronger coupling between these modes helps to avoid this. Also included in figure 10(a) (dashed lines) is the simple calculation of the enhancement according to P(a/w) = (1 − R_IC)/(1 − (R_IC R T(a/w))^1/2)² with the loss due to the slit estimated by the power transmission T(a/w) of the simple slit mode with Gaussian beam diameter 2w through the obstacle with width 2a (section 3). The estimation of the loss by squaring this loss factor in order to account for the diffraction loss at propagation (which is valid in a non-degenerate resonator) leads to an overestimated loss here. The circulating field in the degenerate resonator can arrange to minimize loss, which includes small contributions of higher transverse modes despite the aperture and does not imply that the field vanishes completely on the resonator axis (as assumed for the simple slit mode). For a broad slit the intensity at the edge of the slit can be reduced at the expense of a non-vanishing intensity on-axis. Therefore, the enhancement for broad slits is larger than calculated with this simple loss estimation. Figure 10(b) shows the intensity profile at the position of the slit for an exemplary situation, which is close to impedance-matched and allows for an enhancement of P_circ/P_in = 220 at a half slit width of a/w = 0.25. It can be seen from the inlay of figure 10(b), that the intensity is not exactly zero on the resonator axis. The contribution of u₈ is |c₈|²/P_circ = 0.01 and the spatial overlap of the circulating field with the simple slit mode is U_IC = 0.98. The width of the detuning curve is Δψ = 25 mrad. 0.16% of the circulating power (35% of the incident power) is transmitted through the slit. The gray lines indicate the position of the slit and the aperture. Significantly broader slits at a large enhancement are possible if large contributions of higher transverse modes are allowed (compare figure 4(c)).

The on-axis access to a high-finesse resonator can be used for output coupling of high harmonics generated in a gas jet near the focus. We recently demonstrated the suitability of the field distribution for efficient high-order harmonic generation, achieving an output-coupled power of 11 μW at 61 nm (17th harmonic) [8]. Figure 11 shows the intensity and phase distribution of the simple slit mode around the beam waist. Because the slit mode changes its shape rapidly around the focal plane (see also figure 5) the intensity course is steeper than for the fundamental mode at equal Rayleigh length. The slit mode exhibits on-axis lobes around one Rayleigh length before and behind the focus and two central lobes in the focal plane. The lobes in the focal plane contain 0.59 of the power and exhibit a plane wave front with a slope of the phase dϕ/dz = 3/z_R, which is three times larger than for the fundamental mode with the same q-parameter (figure 11(c)). The two lobes are in phase, which means that the harmonic signal will exhibit a maximum on the optical axis in the far field, which can pass through a slit [8]. The on-axis intensity is maximal at a distance ±z_R/2^1/2 from the focal plane. This maximum is closer to the focus than one Rayleigh length where the contributing modes interfere constructively on-axis (compare figure 2(a)), because the beam cross section increases with distance from the focus. The central lobe resembles a Gaussian profile with radius w = 0.53·w₀ containing 0.42 of the power. The curvature of the intensity course in propagation direction is with −½ (d²I /dz²) (1/I₀) = 5/z_R² larger than for the fundamental mode in the focal plane with −½ (d²I /dz²) (1/I₀) = 1/z_R² (figure 11(a)). The slope of the phase is dϕ/dz = 2/z_R, which is two times larger than for the fundamental mode with the same q-parameter in the focal plane (figure 11(c)). The curvature in x-direction of the phase front evolves similarly to a Gaussian beam in the sense that it is basically negative before the on-axis intensity maximum and positive behind (figure 11(d)). The position of vanishing on-axis curvature is at z = ±0.82 z_R, which is just somewhat shifted away from the position of the intensity maximum at z = ±0.71 z_R. This means that the on-axis lobe is similar to a fundamental mode and favourable conditions for HHG are expected. Calculations employing the strong field approximation model by Lewenstein et al [32] suggest that the output coupling efficiency for harmonics generated with the simple slit mode can be very large. For harmonics with order 15–19 it is between 0.6 and 0.8 for a feasible half slit width of a/w = 0.25 (figure 10(b)) [8].

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The intensity lobe before or behind the focus or the two lobes around the focus can be readily accessed by a gas nozzle from the y-direction, which is perpendicular to the drawing plane in figure 11(b), because the intensity profile is Gaussian in this direction. Also, multiple gas jets could be employed, thereby achieving quasi-phase-matching between e.g. the on-axis lobes before and behind the focus [33].

The intensity and phase distribution around the focus can be changed by the contribution of higher-order slit modes, as well as by tailoring the ellipticity and astigmatism of the resonator mode. It remains to be investigated theoretically and experimentally, how the circulating field can be tailored by the shape and position of apertures, the impinging field distribution and the position in the stability range. The possibility of tailoring the circulating field distribution in an enhancement resonator for HHG might open new prospects for increasing the conversion efficiency by improving phase-matching.

A (quasi-)imaging enhancement resonator is closely related to a non-collinear setup, with two Gaussian beams intersecting in the focal plane under a small angle (figure 12(a)). High harmonics generated in the focus can be coupled out through a gap between the focusing mirrors. To this end, the two beams can be enhanced using two appropriately arranged stable enhancement resonators [34]. Also, a single stable enhancement resonator with length equal to two times the laser resonator length and two pulses circulating can be applied [22]. The resonator geometry has to be arranged for the pulses to overlap in time, and the two halves of the resonator have to be stabilized such that the two pulses will maintain constructive interference on the optical axis. This non-collinear setup can be simplified by using a degenerate resonator and a beam with an offset to the optical axis that oscillates transversally while it retraces its path after N = 2π K/ψ_E,x round trips with the Gouy parameter ψ_E,x and an integer K [35]. The resonator degeneracy and synchronization of the pulses can be adjusted by a single parameter. In case of an imaging enhancement resonator the overlap and synchronization of two pulses circulating in a resonator with length equal to the laser resonator length is then inherently linked to the condition of the degeneracy. The high harmonics would be coupled out through a slit in the mirror behind the focus. In order to realize the situation in figure 12(a) the resonator has to be imaging (magnification –1) with the beam retracing its path every other round trip.

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The simplest case of a stable resonator is shown in figure 12(b), with the beam retracing its path after N = 4 round trips, like for a bow-tie resonator in the middle of the stability range. For a non-collinear setup, a small angle between the two beams is preferable, because it decreases the number of interference fringes in the focus, which determines the divergence of the high harmonics. The simple mode combinations shown in figures 2(g) and figure 2(a) correspond to the situations in figures 12(a) and (b) with the smallest possible non-collinear angle. In this sense, an imaging or quasi-imaging enhancement resonator can be regarded as a perfect non-collinear setup with smallest non-collinear angle and inherently synchronized non-collinear beams.

The conceptually simplest concept for geometrical output coupling is a circulating fundamental mode and a small circular hole in the resonator mirror following the focus. Recently, geometrical output coupling of high harmonics based on this concept was demonstrated for the first time [23]. The hole has to be very small compared to the fundamental mode beam diameter, in order to allow for a high enhancement (figure 3), which limits the output coupling efficiency. However, this limitation is not so strong for harmonics of high order, i.e. short wavelengths, because the far field divergence angle of the generated harmonics decreases with decreasing wavelength. The output coupling efficiency is estimated to greater than 0.5 for the highest harmonic generated (91st harmonic, corresponding to 11.45 nm) in [23], with the hole allowing for a power enhancement of 250. The output coupling efficiency decreases to less than 0.1 for wavelengths around 70 nm. Therefore, HHG with a slit or hole mode in a quasi-imaging resonator, which allows for a larger opening, is especially beneficial for harmonics of moderate order (e.g. 17th). Furthermore, the small intensity at the edges of the slit or hole in a mirror is expected to be advantageous, because a high intensity can lead to damage. For both methods the manufacturing of mirrors with small-size holes or slits is essential, which was recently demonstrated with inverse laser drilling by Esser et al [29]. In comparing these methods it has to be considered that the exploitation of a mode degeneracy for quasi-imaging and therefore the predefinition of a position in the stability range of the enhancement resonator represents a restraint for the resonator design (see section 4). It is e.g. not possible to move towards the border of the stability range, in order to achieve larger mode cross sections for the scaling to higher circulating power, as is discussed and demonstrated in [28, 36].

Another promising method for geometrical output coupling of high harmonics is a Bessel−Gauss beam enhancement resonator with an on-axis intensity maximum in the focus and an annular intensity distribution in the far field, which allows for a large opening in a resonator mirror [31, 37, 38]. The suitability of this field distribution for efficient HHG still has to be investigated. A constraint might be the large and radially symmetric extent of this field distribution around the focus, which impedes the accessibility for a gas nozzle. Furthermore, Bessel−Gauss beam resonators seem to be very sensitive to surface imperfections of the resonator mirrors [38].