1 Introduction

Laser beams from Ti:Sapphire chains relying on the technique of chirped-pulse amplification (CPA) [1], commonly used for the production of high peak power femtosecond pulses, often suffer from poor spatial quality. Aberrations resulting from anisotropic thermal dissipation in the amplification media [2], self-phase modulation during propagation of such intense pulses in air or in materials [3], intracavity beam distortions [4] and imperfections in optical components involve a deterioration of both wavefront and transverse intensity profile.

A common technique to recover a good spatial beam quality is to focus the laser through a pinhole. The lens used to focus the beam performs a Fourier transform of the field in the plane of the pinhole where there is a one-to-one mapping between transverse position and spatial frequencies. This allows one to filter the high spatial frequencies, blocked by the pinhole [5]. However, the beam keeps its low spatial frequencies distortions and a significant amount of energy may be lost. Moreover, unless a specific conic pinhole is used [6], any misalignment of the laser beam can permanently damage the pinhole. Other or complementary techniques include the use of a saturable absorber [7] or a deformable mirror [8] and diffraction from Bragg gratings [9]. Moreover, active filtering can be achieved through nonlinear processes such as second harmonic generation via a nonlinear crystal [10], plasma mirror generation [11] or cross-polarized wave generation [12]. It should be noticed, however, that these active filters are not usable with stretched pulses in a CPA laser system—which is the case considered here.

A scheme based on the propagation of an electromagnetic wave in a cylindrical dielectric waveguide has been considered. In the community of ultrashort lasers, studies on the propagation into such waveguides have been initiated by the post-compression technique used to generate sub-10 fs pulses, where a mJ-level femtosecond laser beam propagates over a short distance in a capillary filled with gas [13]. The electromagnetic field of the laser beam can then be expressed as a linear superposition of the modes specific to this waveguide. By choosing appropriate parameters, the beam couples preferentially into the \(\hbox{LP}_{01}\) (\(\hbox{EH}_{11}\)) mode, which is very similar to the \(\hbox {TEM}_{00}\) one in free space, and other guided modes suffer from higher attenuation. Therefore, if the capillary is sufficiently long and has a sufficiently small core diameter, modal filtering is made on the laser beam. Thus, a setup aimed at spatial quality improvement of a laser beam can be designed by propagation in a carefully chosen waveguide. Modal filtering techniques have already been studied in astronomy for the purpose of nulling interferometers [14, 15], however designed for low-power lasers, while high-energy (hundreds of mJ) pulses are considered in the application presented hereafter.

A modal filtering setup has been implemented at the LUCA facility of CEA Saclay (France), a Ti:Sapphire CPA laser source. The purpose of the article is to provide a comprehensive characterization of modal filtering and to give a demonstration of its practical prominence for numerous applications. The outline of the article is the following. In the first part, we present the principle of the modal filtering and the properties of the modes (coupling, structure, propagation) and then compare the theoretical performances of our setup with those of a “classical” spatial filtering stage. In the second part, dedicated to the experimental study, we spatially and temporally characterize the LUCA laser beam in front and behind the spatial filter in order to (experimentally) demonstrate the beam improvement. We then compare measurements with simulations and theoretical expectations. Finally, as an application, we provide measurements and simulations of high-order harmonic generation (HHG) in a rare gas, a highly nonlinear process whose enhancement confirms the efficiency of modal filtering.

2 Theory

2.1 Modal filtering setup and mode structure

The basic layout of the system is shown in Fig. 1. Its main component is the hollow fiber, a cylindrical dielectric waveguide in silica, of core radius \(a=125~\upmu\hbox{m}\) and length \(L=30~\hbox{cm}\).

Fig. 1
figure 1

Modal filtering setup. The laser beam is focused by lens 1 at the entrance of the fiber. After propagation within the fiber, the beam propagates in free space and is collimated by lens 2. An iris, whose use will be described later on in the text, is placed at the back focal plane of lens 2. The beam may then pass through different optical systems for further applications. A control loop has been implemented in order to optimize the beam pointing at the fiber entrance

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The general solutions of the propagation equations of electromagnetic waves in a cylindrical waveguide were addressed by Stratton [16]. In 1961, Snitzer [17] studied the characteristics of cylindrical metallic and dielectric waveguides, and in 1964 Marcatili [18] was the first to give the expressions of the modes of a hollow dielectric cylinder when the core radius is much larger than the wavelength. In [19], Degnan proposed to call \(\hbox{LP}_{\rm nm}\) the linearly polarized modes, which we are dealing with in this study. The set of eigenmodes can be expressed, in cylindrical coordinates \((r,\theta ,z)\), through the following expression:

$$\begin{aligned} e_{\rm nm}(r,\theta ,z)&= A_{\rm nm} e^{i(\beta _{\rm nm}z-\omega t)} e^{-\alpha _{\rm nm} z} J_n\left( \frac{u_{\rm nm} r}{a}\right) e^{\hbox{in} \theta }\nonumber \\& \quad (n \, \epsilon \, \mathbb {Z}, m \geqslant 1, r \leqslant a), \end{aligned}$$
(1)

where \(J_n\) is the Bessel function of the first kind and \(u_{\rm nm}\) its mth root. \(\omega =\frac{2\pi c}{\lambda }\) is the pulse angular frequency, \(\lambda\) being the wavelength and c the speed of light. The energy normalization \(A_{\rm nm}\), the attenuation coefficient \(\alpha _{\rm nm}\) and the propagation constant \(\beta _{\rm nm}\) are given, respectively, by:

$$A_{\rm nm}= \frac{1}{a \sqrt{\pi} J_{n+1}(u_{\rm nm})},$$
(2)
$$\alpha_{\rm nm}= \left(\frac{u_{\rm nm}}{k} \right)^2 \frac{1}{a^3} \frac{\varepsilon +1}{2\sqrt{\varepsilon-1}},$$
(3)
$$\beta _{\rm nm}= k \left( 1-\frac{1}{2}\left( \frac{u_{\rm nm}}{k a} \right)^2 \right),$$
(4)

where \(k=\frac{\omega }{c}\) is the wave number and \(\varepsilon\) is the dielectric permittivity (we consider here real valued \(\varepsilon\) and \(\frac{\varepsilon +1}{2\sqrt{\varepsilon -1}}\) conditions which are fulfilled for dielectric waveguides). The mode with smaller attenuation (smaller \(\alpha _{\rm nm}\)) is called the fundamental mode of the waveguide.

The laser beam is chosen as linearly polarized along the vertical axis, characterized by the unit vector \(\mathbf{u }_y\). The electric field at the input of the hollow fiber can be written as:

$$\begin{aligned} {\mathbf{E}}(r,\theta ,z=0,t)&= E(r,\theta ,z=0,t) \cdot {\mathbf{u}_{y}} = \nonumber \\& \quad \sum_{n=-\infty }^{+\infty} \sum_{m=1}^{+\infty }\eta _{\rm nm} A_{\rm nm}e^{-i\omega t} J_n\left( \frac{u_{\rm nm} r}{a}\right) e^{i n \theta } \cdot {\mathbf{u}_{y}}, \end{aligned}$$
(5)

with

$$\begin{aligned}\eta _{\rm nm} &= \int _0^a \int _0^{2\pi } E^*(r,\theta ) A_{\rm nm} J_n\left( \frac{u_{\rm nm} r}{a}\right) e^{i n \theta } r \, \hbox{d}r \, \hbox{d}\theta \nonumber \\ & \qquad (n \epsilon {\mathbb {Z}}, m \geqslant 1).\end{aligned}$$
(6)

The field at the output of the fiber can be calculated analytically when the \(\eta _{\rm nm}\) coefficients have been calculated, i.e., when the input laser field is known:

$$\begin{aligned} \mathbf{E }(r,\theta ,z=L,t)&= E(r,\theta ,z=L,t) \cdot \mathbf{u }_y = \nonumber \\& \quad \sum _{n=-\infty }^{+\infty } \sum _{m=1}^{+\infty } \eta _{\rm nm} A_{\rm nm}e^{i(\beta _{\rm nm}L-\omega t)} e^{-\alpha _{\rm nm}L} J_n\left( \frac{u_{\rm nm} r}{a}\right) e^{i n \theta } \cdot {\mathbf{u}_{y}}. \end{aligned}$$
(7)

Moreover, the fraction of input energy coupled in the different modes is given by:

$$P_{\rm nm} = \frac{ |\eta _{\rm nm}|^2}{\int _0^{+\infty } \int _0^{2\pi } E^*(r,\theta ) E(r,\theta ) r, \hbox {d}r \, \hbox{d}\theta }.$$
(8)

Another way to write the electric field, with appropriate \(c_{\rm nm}\) coefficients, is the following:

$$\begin{aligned} \mathbf{E }(r,\theta ,z,t) = \sum _{n=0}^{+\infty } \sum _{m=1}^{+\infty } [c_{\rm nm}^{\rm even}\cos (n\theta )+c_{\rm nm}^{\rm odd}\sin (n\theta )] \times \nonumber \\ J_n\left( \frac{u_{\rm nm} r}{a}\right) e^{-\alpha _{\rm nm}z} e^{i(\beta _{\rm nm}z-\omega t)} \cdot {\mathbf{u}_{y}}, \end{aligned}$$
(9)

or

$$\begin{aligned} \mathbf{E }(r,\theta ,z,t) = \sum _{n=0}^{+\infty } \sum _{m=1}^{+\infty } \left[c_{\rm nm}^{\rm even}LP_{\rm nm}^{\rm even}+c_{\rm nm}^{\rm odd}LP_{\rm nm}^{\rm odd}\right] \times \nonumber \\ e^{-\alpha _{\rm nm}z} e^{i(\beta _{\rm nm}z-\omega t)} \cdot \mathbf{u }_y, \end{aligned}$$
(10)

where the superscripts \(^{\rm even}\) and \(^{\rm odd}\) refer to the cosine or sine azimuthal dependence of the field, respectively. The family of \(\hbox{LP}_{\rm nm}\) modes shown in Fig. 2 is very similar to the Laguerre–Gauss modes in free space.

Fig. 2
figure 2

Transverse intensity distribution of the first \(\hbox{LP}_{\rm nm}\) modes

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Both in near- and far-field (as will be detailed hereafter), the \(\hbox{LP}_{01}\) mode is quite close to a \(\hbox{TEM}_{00}\) one. That is the reason why we concentrate on this mode and wish to favor it in our system. To this aim, the first step is to maximize \(P_{01}\). We will consider two kinds of laser beams with circular symmetry, focused at the entrance of the fiber: a Gaussian beam and a top-hat one. Those are, in fact, two limiting cases encountered at the output of our amplifiers.

In the front focal plane of lens 1 (see Fig. 1), the Gaussian beam has a waist \(w_0\) and the top-hat one is characterized by its radius \(a_0\), outside which there is no energy. In the back focal plane of the same lens, the waist of the Gaussian beam becomes \(w_1=\frac{\lambda f_1}{\pi w_0}\) and the propagation of the top-hat beam results in an Airy disk whose radius at first zero is \(a_1=\frac{u_{11} f_1}{a_0 k}\). Using Eq. 8, Fig. 3 shows the fraction of energy coupled into \(\hbox{LP}_{01}\), i.e., \(P_{01}\), as a function of the beam size. For the Gaussian beam, a maximum of \(\approx\)98 % is found for \(w_{1} \approx 0.645a,\) whereas a maximum of \(\approx\)83 % is obtained for \(a_1\approx 1.05 a\) in the second case.

Fig. 3
figure 3

Left \(P_{01}\) as a function of \(\frac{w_1}{a}\) for a Gaussian beam focused at the fiber entrance; right \(P_{01}\) as a function of \(\frac{a_1}{a}\) for a top-hat beam focused at the fiber entrance

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We note that, if we can consider that the energy outside the core (\(r>a\)) is lost, the coupling in the fiber also plays the role of a classical spatial filter, suppressing the spatial frequencies above a given value. The geometry of the tips of the hollow fiber can be improved in this aim.

The good coupling in the fundamental mode must be associated with a sufficiently long fiber for efficient modal filtering. Table 1 gives the attenuation of the modes calculated from Eq. 3 for the fiber used here. Even if \(\hbox{LP}_{01}\) is the fundamental fiber mode, one can see that other modes such as \(\hbox{LP}_{02}\) or \(\hbox{LP}_{11}\) also have quite low attenuations, so that one should take care of not coupling too much energy in these modes.

Table 1 Loss in \(\%\) of the \(\hbox{LP}_{\rm nm}\) modes after 30 cm of propagation in a hollow fiber in silica, of core radius \(a=125~\upmu\hbox{m}\)
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Let us consider some temporal effects likely to occur during the propagation along the fiber. The relative phase delay \(\Delta t_{\varphi } = (\beta _{\rm nm}-\beta _{01})\frac{L}{\omega }\) and group delay \(\Delta t_{\rm g} = \left(\frac{\hbox{d}\beta _{\rm nm}}{\hbox{d}\omega }-\frac{\hbox{d}\beta _{01}}{\hbox{d}\omega }\right) L\) are given in Table 2a, b. For sufficiently short pulses, the consequence of these delays is to delay higher-order modes compared with \(\hbox{LP}_{01}\). This point can be interesting for highly nonlinear phenomena because high-order modes with insufficient energy would be in this case temporally filtered. Table 2c shows the group delay dispersion of the different modes. According to such values, the \(\hbox{LP}_{01}\) mode will not temporally spread even for sub-10 \(fs\) pulses. However, the dispersion in higher-order modes might not be negligible.

Table 2 Phase delay (a), group delay (b) and group delay dispersion (c) of the \(\hbox{LP}_{\rm nm}\) modes after 30 cm of propagation in a hollow fiber in silica of core radius \(a=125~\upmu\hbox{m}\)
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At the output of the fiber, the electric field propagates in free space and lens 2 (see Fig. 1) is used to collimate the beam. It is then important to know the behavior of the beam, and especially of the mode \(\hbox{LP}_{01}\), both in the near- and the far-field. In the near-field, as was already discussed by Degnan [19], a small modulation appears in the central part of the \(\hbox{LP}_{01}\) mode for planes where the ratio of its radius versus \(\sqrt{(}\lambda )\) is close to an integer value. This phenomenon is also encountered with a Gaussian beam truncated by a circular aperture. The expression of the field in the back focal plane of lens 2 can be obtained analytically [20]:

$$\begin{aligned} LP_{01}^{\rm far field}(r) = 2 u_{01} a \sqrt{\pi } \frac{J_0 \left( \frac{2\pi a}{\lambda f_2} r \right) }{u_{01}^2 - \left( \frac{2\pi a}{\lambda f_2} r \right) ^2}, \end{aligned}$$
(11)

\(f_2\) being the focal length of lens 2. The mode shape is still circularly symmetric, the mode having no azimuthal dependence. Figure 4 underlines the similarity between the \(\hbox{TEM}_{00}\) and \(\hbox{LP}_{01}\) modes both in near- (Fig. 4a) and far-field (Fig. 4b).

Fig. 4
figure 4

Comparison of amplitude profiles of a Gaussian beam of parameter \(w_0=0.645a\) (dashed curves) and the \(\hbox{LP}_{01}\) mode (full line) in the fiber (a) and at the back focal plane of lens 2 for \(f_2=1~\hbox{m}\) (b)

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The shape of the \(\hbox{LP}_{01}\) mode at the output of the fiber [Fig. 4a (full line)] is responsible for rings surrounding the main lobe of the energy distribution in the far-field. These rings can be seen on the profile of \(\hbox{LP}_{01}\) given in Fig. 4b (full line). In order to suppress this energy in the spatial pedestal, an iris (see Fig. 1) of aperture diameter \(\frac{u_{01}\lambda f_2}{\pi a}\), corresponding to the first zero of the energy distribution of the \(\hbox{LP}_{01}\) mode in the back focal plane of lens 2 (see Eq. 11), is used. This is, after the good coupling into the mode of interest (\(\hbox{LP}_{01}\)) and the higher attenuation of other modes through the fiber, the “third step” of the modal filtering system. However, the total field is also composed of higher-order fiber modes. Hence, a very important point here is that those modes exhibit a higher divergence, so that they are also filtered by the iris. This can be seen from the calculated values of the \(M^2\) factor [21] for each \(\hbox{LP}_{0m}\) mode and the corresponding transmission through the iris (see Table 3). It shows that most of the contribution of modes having \(m \geqslant 3\) will be canceled due to the iris.

Table 3 First line: modes; second line: corresponding \(M^2\) factor being the ratio between the calculated mode divergence and the divergence of a Gaussian beam with same standard deviation of its intensity distribution at the focus; third line: according to the \(M^2\), proportion of energy transmitted through the iris closed at a diameter \(\frac{u_{01}\lambda f_2}{\pi a}\)
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These results demonstrate that part of the efficiency of the modal filtering is linked to the spatial filtering in the far-field, behind the hollow fiber. It can also be noticed that, due to the filtering of the higher-order modes, the aperture size of the iris also improves the quality of the beam in the back focal plane of a lens that would be placed downstream on the beamline (see Fig. 1), which is a significant asset for experiments.

2.2 Efficiency of modal filtering versus classical spatial filtering

So far, we have presented three different processes leading to the filtering of the beam (the suppression of the high spatial frequencies at the input of the hollow fiber, the maximum coupling in the fundamental mode and the spatial filtering in the far-field after the hollow fiber). We now discuss one particular case illustrating the potential superiority of the “modal filtering” over conventional spatial filtering, which does not allow one to cancel low-frequency modulations (in amplitude or phase), except at the expense of energy and increase of the \(M^2\) value. In order to be able to make calculations analytically, we consider the spatial filtering of a Gaussian beam with balanced astigmatism. The electric field in the front focal plane of lens 1 is then written [22, 23]:

$$E(r,\theta ) = E_0 e^{-\frac{r^2}{w_0^2}} e^{-i\frac{\pi r^2}{\lambda R} \cos (2\theta )},$$
(12)

where \(R\) is the wavefront curvature along \(Ox\) or \(Oy\). The waist along \(Ox\) (or \(Oy\)), noted \(w_x\) (or \(w_y\)) is located at a distance dz from the front focal plane of the lens 1, where the beam size is \(w_0\). dz is given by:

$$\begin{aligned} dz = \frac{R}{1+\gamma }, \end{aligned}$$
(13)

and the ratio between the waist sizes by:

$$\begin{aligned} \frac{w_y}{w_x} = \sqrt{1+4\gamma }, \end{aligned}$$
(14)

with \(\gamma =\frac{\lambda ^2 R^2}{\pi w_0^4}\). The same value of \(\gamma\) is obtained in the back focal plane of the lens.

We first consider conventional spatial filtering. If we suppose that we use a Gaussian filter with a transmission \(T(r,\kappa ) = e^{-\kappa \frac{r^2}{w_1^2}}\) in the back focal plane of the lens, then the ratio will transform to:

$$\begin{aligned} \frac{w_y}{w_x} = \sqrt{1+4\kappa ^4\gamma }. \end{aligned}$$
(15)

A reduction of this ratio from a value of 2 to a value of 1.15 gives a transmission of 22.3 % for the filter. For a classical filter using a pinhole, numerical calculations have to be done. The same ratio of the waists is obtained with a filter transmission of 65.7 %. The truncation of the beam in the back focal plane of the lens increases its divergence. Furthermore, the beam profile does not remain Gaussian and there is a non-negligible part of the energy in a spatial pedestal.

We now consider the modal filter with the parameters used for the experiment, i.e., \(a=125~\upmu\hbox{m}\) and \(L=30~\hbox{cm}\). The value of \(a\) gives an optimum waist of \(80.6~\upmu\hbox{m}\) at the entrance of the fiber. The radius of the wavefront curvature \(R\) is then \(\approx\)2.95 cm to obtain parameters similar to those discussed with classical spatial filtering. In this case, a calculation shows that 83.7 % of the beam energy in the back focal plane of lens 1 is coupled in the \(\hbox{LP}_{01}\) mode. The determination of which modes give the main contributions to the spatial phase is more interesting.

This is illustrated, in this particular case, in Fig. 5. One can see that the main contribution to the phase in the inner part of the beam comes from the \((n=\pm 2, 0; m=1)\) modes (there is no contribution from modes with odd values of \(n\)). To obtain a relatively good fit of the phase, modes \((n=\pm 4,\pm 2,0; m=1,2)\) have to be considered. Low attenuations, given in Table 1, show that, for the parameters considered here, the spatial phase will not be strongly corrected. Figure 6 shows the intensity and phase at the input and output of the fiber, respectively. Only attenuation coefficients have been taken into account here (see Eqs. 1, 3).

Fig. 5
figure 5

Astigmatic Gaussian beam intensity and phase (along the \(Ox\) direction) and reconstructed intensity and phase at the input of the hollow fiber. The modes taken into account (see Eq. 1) for the different curves are specified by the value of \(n\) and \(m\). It can be noticed that a high-quality beam reconstruction can be obtained by considering modes with \(-4 \leqslant n \leqslant 4\) and \(1 \leqslant m \leqslant 15\). Black lines input. Light gray lines \(-2 \leqslant n \leqslant 2\); \(m = 1\). Gray lines \(-2 \leqslant n \leqslant 2\), \(1 \leqslant m \leqslant 3\). Dark gray lines \(-2 \leqslant n \leqslant 2\), \(1 \leqslant m \leqslant 15\). Dashed lines \(-4 \leqslant n \leqslant 4\), \(1 \leqslant m \leqslant 15\)

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Fig. 6
figure 6

Astigmatic Gaussian beam intensity and phase (along the \(Ox\) direction) and reconstructed intensity and phase (\(-4 \leqslant n \leqslant 4\); \(1 \leqslant m \leqslant 15\)) at the input and at the output of the hollow fiber used in the experimental part of this paper. Black line input, light gray line reconstructed input, dark gray line reconstructed output

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In this particular case, the advantage of modal filtering does not seem very important for the chosen parameters. We can then decrease the hollow fiber radius and increase its length to obtain filtering of the low-order modes (attenuation factor scales as \(a^{-3} \times L\)). The radius of curvature of the wavefront at the entrance of the hollow fiber scales as \(a^2\). To decrease the ratio of the waists below 1.15, as was considered with a classical filter, one has to decrease the radius of curvature of the wavefront by a factor of \(\approx\)3 between the input and the output of the fiber. We thus consider a fiber of \(1~\hbox{m}\) length and of \(75~\upmu\hbox{m}\) inner radius. \(83~\%\) of the energy is coupled in the fundamental mode. The result of the calculation is shown in Fig. 7.

Fig. 7
figure 7

Astigmatic Gaussian beam intensity and phase (along the \(Ox\) direction) and reconstructed intensity and phase (\(-4 \leqslant n \leqslant 4\); \(1 \leqslant m \leqslant 15\)) at the input and at the output of the hollow fiber whose parameters are chosen to reduce wavefront radius of curvature by a factor of 3 (see text for details). Black line input, light gray line reconstructed input, dark gray line reconstructed output

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Only 42 % of the energy is transmitted at the output of the fiber, but the beam quality is better than that obtained with classical spatial filtering when reducing the diameter of the pinhole. The use of a longer fiber would, however, emphasize the temporal effects discussed in Table 2. As was mentioned there, if the pulse duration is sufficiently short to avoid temporal overlap between the different modes, nonlinear effects (like HHG for instance) will filter the beam temporally. Another solution that has been chosen in our experiment to better filter astigmatism is, as for a classical spatial filter, to increase the ratio \(\frac{w_1}{a}\) at the entrance of the fiber. In this case, the situation is better than for a classical filter for two reasons: First, the hollow fiber can withstand a higher energy; second, the beam profile at the output of the fiber will be nearly unaffected (except for lower transmission and reduction of the astigmatism) as high-order modes in which energy is coupled are attenuated. To be completely fair, it has to be noticed that efficient spatial filtering of a beam with astigmatism can be done with a system which compensates this aberration (with the use of a spherical mirror working off-axis, for example [24]). Nevertheless, it is less easy to implement and requires the knowledge of the aberration to compensate.

This theoretical part, which introduced the functioning of the system and its potential assets—as compared to classical spatial filtering—enables us to present our experimental results hereafter.

3 Experiment

3.1 Methods of characterization

Fig. 8
figure 8

Schematic of the setup with the diagnostics used for the beam’s characterization. The full line represents the direct beam path: the stretched (\(\approx\)200 ps) amplified (up to \(\approx\)150 mJ) Ti:Sapphire laser beam enter the modal filtering setup, composed of the lens 1, the hollow fiber, the lens 2 and an iris. The beam is then time-compressed and passes through another iris before being focused by lens 3 and used for a HHG experiment. The dashed lines represent derivations that have been set for different diagnostics: beam analyzer, wavefront sensor and SPIDER apparatus (see text for details)

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The overall setup implemented on the LUCA source is shown in Fig. 8. The hollow fiber is placed in a vacuum chamber at a pressure of the order of \(10^{-3}~\hbox{mbar}\). The whole modal filtering setup (see Fig. 1) is placed in front of the compressor. This configuration is used in order to reduce nonlinear effects due to the propagation of the beam in air and in the entrance window of the vacuum chamber. The latter choice is also motivated by a slower degradation of the input of the hollow fiber and a better tolerance to a slight misalignment with the uncompressed beam. The mean lifetime of a hollow fiber in our conditions (repetition rate 20 Hz, energy per pulse \(<\)150 mJ, duration \(\approx\)200 ps, central wavelength \(\lambda \approx 795\) nm, fiber core radius \(a \approx125~\upmu\hbox{m}\)) is about 1,250 h.

Three different beam characterizations have been performed:

  • Measurements of intensity patterns around the waist of the focused beam, in front of and behind the hollow fiber. These measurements allow us to obtain a direct image of the transverse shape of the beam and information on its main aberrations, such as astigmatism. Moreover, the collected data enable the calculation of the \(M^2\) factor. The measurements were done using a beam analyzer and the \(M^2\) calculated following the standard technique that considers the measurements of the second moment widths of the beam [25, 26], as in the theoretical part (see Table 3). The second moment width, corresponding to four times the standard deviation \(\sigma _r\) of the transverse intensity distribution at a given position \(z\) along the propagation axis, is the beam diameter definition used in the following. For a Gaussian beam, it matches the parameter \(w\) (\(2w = 4\sigma _r\), including 86.5 % of the beam energy). The obtained results, in particular the \(M^2\) values, can then be compared with theoretical expectations.

  • Measurements of the spatial phase and amplitude with a Shack–Hartmann sensor. Indeed, beam analyzer measurements lack information about the spatial phase (at least if no iterative reconstruction algorithm is used [27]), which is important for the knowledge of the beam aberrations but is also required, among other things, for the calculation of the \(\eta _{\rm nm}\) coefficients (see Eq. 6). We used a Haso 64 from Imagine Optic™ with a pupil of \(1.5 \times 1.5~\hbox{cm}\) and a discretization step of \(186~\upmu\hbox{m}\). The complete determination of the electric field in one plane provided by such an analyzer allowed us to calculate the electric field in any other plane using the Fresnel diffraction theory [28]. We measured the wavefront and intensity distributions with the Shack–Hartmann sensor at three different positions: at the front focal plane of lens 1 and lens 3 and at the back focal plane of lens 2 (see Fig. 8). In every case, the tilt and curvature (quadratic phase term due to beam focusing) of the wavefront were removed before data analysis. One has to note that the sensor has a threshold at \(\frac{1}{e^2}\) of the maximum fluence: Below this value, no information is collected. This impacts on the accuracy of the simulation of the experimental results due to incomplete knowledge of the input beam.

  • Spectrum and spectral phase measurements with the SPIDER technique [29] after the stage of pulse compression, providing information on the spectral and temporal properties of the femtosecond pulses. The purpose of these measurements was to verify whether, or not, the temporal effects discussed in Table 2 have a consequence on the spectrum and the temporal intensity distribution of the femtosecond pulses.

3.2 Characterization of the modal spatial filter

Fig. 9
figure 9

Spot of the LUCA laser beam at focus, measured by the beam analyzer in front of (a) and behind (b) the modal filtering setup. For (a), the measurement has been done at the focus of a lens of focal length \(500~\hbox{mm}\). For (b), the measurement has been done by imaging the output of the fiber by means of a 4f-system. A non-negligible amount of energy at high spatial frequencies can be seen in (a). (\(4\sigma _r \approx 200~\upmu\hbox{m}\))

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The beam analyzer measurements at the waist of the beam are shown in Fig. 9 . In front of the hollow fiber, the beam is elliptic and “twisted” along the longitudinal axis (this is typical of a general astigmatic beam). Behind the hollow fiber, astigmatism and high spatial frequencies are mainly suppressed (see the profiles in Fig. 9b) and the transverse beam profile is nearly Gaussian. As pointed out in the theoretical part, an optimization of the beam quality can be obtained by the use of an iris placed in the back focal plane of the lens 2 (see Fig. 1 ), with a diameter corresponding to the first zero of the \(\hbox{LP}_{01}\) mode. The beam diameter at the entrance of the hollow fiber is estimated to be \(285~\upmu\hbox{m}\) (1.5 times larger than the one measured in Fig. 9 with a lens of focal length 1.5 times smaller than the one of lens 1). Experimentally, the choice of lens 1 focal length was made, among some discrete values available, on the maximum transmission criteria. The beam diameter obtained is larger than the core diameter of the hollow fiber. It was demonstrated in Sect. 2.1 that, in the case of a Gaussian beam, the optimum diameter for a maximum coupling in \(\hbox{LP}_{01}\) mode (see Fig. 3) is given by 0.645 times the hollow fiber diameter, so it should correspond to \({\approx}161~\upmu\hbox{m}\) for a Gaussian beam. The difference is due to the presence of energy in high spatial frequencies, and to wavefront and amplitude distortions. In Sect. 2.2, we saw that the hollow fiber used does not correct astigmatism efficiently in the case of a Gaussian beam with optimum coupling. It was mentioned nevertheless that there were two ways to increase the correction: using a smaller core and longer hollow fiber or using a beam with a diameter larger than the optimum one at the entrance of the hollow fiber. This second solution was used here.

Fig. 10
figure 10

Evolution of the beam diameter (geometrical mean of diameters measured in \(x\) and \(y\) directions) along the propagation axis, in front of (left) and behind (right) the hollow fiber. Crosses correspond to experimental values and full-line curves to their interpolations. The dashed curve represents the diameter of a Gaussian beam of same size at focus

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Figure 10 shows the evolution of the diameter of the measured beam spots around the waist and that of a Gaussian beam with the same width at focus. The ratio of their divergences corresponds to the \(M^2\) of the beam. It is equal to 2.1 in front of the fiber and 1.4 behind it. As a comparison, the \(M^2\) of \(\hbox{LP}_{01}\) is \(1.12\). This difference underlines the presence of higher-order modes in the output beam. The diameter of the laser beam at the output of the hollow fiber is estimated to be \(210~\upmu\hbox{m}\).

Fig. 11
figure 11

Shack–Hartmann measurements: intensity and wavefront at the front focal plane of lens 1 (left) and the back focal plane of lens 2 (right). (\(4\sigma _r \approx 8{-}10~\hbox{mm}\))

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As was previously mentioned, the knowledge of the intensity beam distribution at the entrance of the hollow fiber is not sufficient to calculate the electric field at the output of the hollow fiber. Thus, we now present characterizations of both intensity and wavefront (equivalent to amplitude and spatial phase) of the beam with a Shack–Hartmann sensor. In the front focal plane of lens 1, the beam intensity distribution has no cylindrical symmetry and the wavefront is distorted (left side of Fig. 11). The transverse intensity distribution is roughly super-Gaussian with three hot regions, and the wavefront contains astigmatism and higher-order phase distortions. After filtering (right side of Fig. 11), the wavefront is relatively flat and the intensity distribution nearly Gaussian, with a quasi-perfect cylindrical symmetry.

From Table 4, it can be seen that the peak-to-valley and root-mean square (RMS) values are divided by a factor of \(\approx\)5 after propagation in the hollow fiber, giving values of \(\frac{\lambda }{10}\) peak-to-valley and \(\frac{\lambda }{58}\) RMS.

Table 4 Measurements of the wavefront amplitude of the laser beam, in front of and behind the fiber
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From these results, we made a simulation of the modal filtering using the theory discussed in Sect. 2.1. The wavefront and intensity measurements in the front focal plane of lens 1 (Fig. 11, left panel) allow us to calculate the electric field at the input of the fiber (Fig. 12, left panel) by performing a two-dimensional (2D) Fourier transform of the measured field. This field is then decomposed on the hollow fiber modes by means of Eq. 6. Each mode is propagated through the hollow fiber by taking into account its attenuation coefficient (see Eq. 3). The field at the output of the fiber results from the summation of all modes after propagation into the fiber (see Eq. 7) (Fig. 12, right panel). Finally, by performing 2D Fourier transform, the intensity and wavefront of the beam are obtained in the back focal plane of lens 2 (Fig. 13), whose result is in quite good agreement with the measurements (Fig. 11, right panel).

As discussed in Sect. 3.1, the detection threshold of the Shack–Hartmann device has non-negligible effect on the accuracy of the reconstructed fields. It is indeed very difficult to estimate the error in a quantitative way. It has also to be noticed that a smoothing of the edge was done on the intensity distribution of the Shack–Hartmann measurements (Fig. 11, left panel), in order to avoid diffraction by artificial sharp edges.

Fig. 12
figure 12

Simulations of beam propagation. Left fiber input (back focal plane of lens 1); right fiber output. The wavefront is displayed only where the intensity is higher than \(\frac{1}{e^2}\) of the maximum intensity. (\(4\sigma _r \approx 250~\upmu\hbox{m}\))

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Fig. 13
figure 13

Simulation of beam propagation: back focal plane of lens 2. The wavefront is displayed only where the intensity is higher than \(\frac{1}{e^2}\) of the maximum intensity. (\(4\sigma _r \approx 10~\hbox{mm}\))

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A \(M^2\) of 2 in front of the hollow fiber is inferred from simulations of beam propagation. This value is in good agreement with the 2.1 value deduced from measurements with the beam analyzer. At the output of the hollow fiber, the \(M^2\) value obtained is 1.3 which is again in good agreement with the \(1.4\) value found from beam analyzer measurements (Fig. 10). Experimentally, a transmission of 78 % has been found after optimization. This value is lower than the value obtained by considering the energy coupled in the \(\hbox{LP}_{01}\) mode and its attenuation in the fiber (\(\approx\)90 %) and cannot be compared with a sufficient accuracy to a calculated value due to the fact that the complete beam is not measured with the Shack–Hartmann sensor. Imperfect alignment and roughness of the inner wall of the hollow fiber can also decrease the transmission obtained experimentally.

Finally, one has to note that in the reconstructions, the temporal term of the phase has not been taken into account. This does not have a huge consequence since most of the energy couples into the fundamental mode. Moreover, as will be discussed hereafter, no spectrotemporal deterioration, potentially caused by the fiber stage, has been detected.

3.3 Characterization behind the compressor

After the fiber stage and the passage through the iris of the modal filtering system, the temporally stretched pulses are compressed down to the femtosecond level by a set of gratings before reaching lens 3 (see Fig. 8). At a picosecond level, the temporal effects pointed out in Table 2 do not have any consequence. However, they may not be negligible for the compressed beam. This is why we performed SPIDER characterizations. Three measurements have been done, for energy values of 4, 18 and 42 mJ after compression, respectively. For the two first values, pulse durations are in the range 40–50 fs and have a nearly Gaussian temporal profile. These values are identical to those obtained without the fiber stage. No spectral modulations, signature of the presence of delayed modes or nonlinear broadening by self-phase modulation, are observed. The fact that the spectrotemporal properties are not distorted (as compared to the case without fiber) is the sign that only a low amount of high-order modes is coupled. The last measurement at 42 mJ gives a longer pulse (about 65 fs) and a non-Gaussian profile. This is attributed to nonlinear effects during the propagation after compression, which could, however, be overcome by enlarging the beam.

The results of measurements carried out with the Shack–Hartmann sensor after the stage of pulse compression are presented in Fig. 14. Imperfections of the optics (lenses, gratings of the compressor, mirrors which send the beam to the sensor) cause intensity and phase modulations on the beam. Nevertheless, the beam quality is far better with the modal filtering setup inserted before the pulse compression stage than without it. Wavefront amplitudes are given in Table 5: the RMS value is improved from \(\frac{\lambda }{5.5}\) to \(\frac{\lambda }{14.5}\) with modal filtering.

Fig. 14
figure 14

Shack–Hartmann measurements: intensity and wavefront without (left) and with (right) the hollow fiber. In both cases, the beam mean intensity is similar (\(\approx\)15TW/cm2). (\(4\sigma _r \approx 10~\hbox{mm}\))

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Table 5 Measurements of the wavefront amplitude of the laser beam after pulse compression, without and with the fiber
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The beam is then focused by lens 3. A Fresnel diffraction calculation provides the field at the back focal plane of lens 3 (Fig. 15) from the electric field deduced from the Shack–Hartmann measurement behind the compressor (Fig. 14). A clear improvement is found for the filtered beam. In the standard setup (i.e., without modal filter), the wavefront exhibits an important deformation and the intensity distribution includes a non-negligible amount of high spatial frequencies (Fig. 15, left panel). On the other hand, the filtered beam is Gaussian-like at focus (Fig. 15, right panel). Considering only the beam part above \(\frac{1}{e^2}\) of the maximum intensity, the RMS amplitude of the wavefront is decreased by a factor \(\approx\)5 (from \(\frac{\lambda }{9}\) to \(\frac{\lambda }{48}\), see Table 5). The significant improvement at this point of the beamline is very important as will be shown in the HHG experiment discussed below.

Fig. 15
figure 15

Propagated electric field: intensity and wavefront at the back focal plane of lens 3, without (left) and with (right) modal filtering. The wavefront is displayed only where the intensity is higher than \(\frac{1}{e^2}\) of the maximum intensity. (\(4\sigma _r \approx 400~\upmu\hbox{m}\))

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3.4 Application to high-order harmonic generation

High-order harmonic generation, as a highly nonlinear phenomenon, is a very interesting application to illustrate the efficiency of the filtering setup. Homogeneous intensity distribution and low wavefront distortions of the driving laser are crucial for the efficient and coherent macroscopic construction of the harmonic beam. This is especially true in the so-called loose-focusing geometry [30, 31], where the interaction occurs over a long distance (several centimeters), as compared to the wavelength of the fundamental beam (795 nm).

In [32], a general improvement of HHG was already noticed after propagation of the driving field in a capillary. This was, however, in a tight-focusing configuration. More generally, several methods exist for improving the phase-matching in HHG [33–35]. In our case, we first noted a significant improvement of the spatial quality of the generated extreme-ultraviolet beam (Fig. 16). Indeed, reducing the aperture diameter of the iris placed in front of lens 3 (see Fig. 8) on the unfiltered beam for improving the generation [36] is clearly not sufficient in our case.

In addition, the stability and the harmonic conversion efficiency are enhanced: Without modal filtering, the use of 30 mJ of infrared energy (measured after the last iris) allows obtaining \(3.3 \times 10^7\) harmonic photons per shot (measured on the charge-coupled-device (CCD) camera at 32 nm wavelength, i.e., the 25th harmonic of the driving laser) while \(5.7 \times 10^7\) photons per shot are collected with only 8.5 mJ energy of the driving laser when using the modal filtering stage. In other words, the harmonic conversion efficiency is \(\approx\)6 times higher with the filtered beam, making negligible the drawback of the loss of \(\approx\)20–30 % of energy within the fiber stage. This is partly explained, in the non-filtered infrared beam (Fig. 15, left panel), by the presence of a larger number of high spatial frequencies where there is not enough intensity for generating high-order harmonics. Moreover, the stability of the intensity and of the wavefront of the filtered beam is a great advantage for the macroscopic effects occurring all along the generating medium. The results presented correspond to optimal conditions of generation (aperture size of the iris, gas pressure, focus position with respect to the cell, energy of the driving laser) for which a trade-off was found between spatial quality and intensity of the harmonic signal.

Fig. 16
figure 16

Far-field transverse intensity distributions of the twenty-fifth harmonic (32 nm) of the fundamental wavelength of LUCA (795 nm), measured without (left) and with modal filtering (right). In both cases, the iris placed in front of lens 3 was closed (respectively to a diameter of 25 and 19.5 mm) so as to filter the outer part of the laser beam (principally for the case without modal filtering) and to improve the generation conditions. Harmonics were generated in a gas cell filled with argon at a backing pressure of \(\approx\)10 mbar. Intensity distributions are measured on a CCD camera placed behind a monochromator

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Finally, HHG experimental data have been compared with simulation results from a 3D model in cartesian coordinates which 2D version in cylindrical geometry was already extensively described elsewhere (for more details about the code see, e.g., [37]). Briefly, the coupled propagation equations on the infrared and harmonic fields are solved in the paraxial \(\left(\frac{\partial ^{2}}{\partial z^{2}} \ll k\frac{\partial }{\partial z}\right)\) and slowing varying envelope \(\left(\frac{\partial ^{2}}{\partial t^{2}} \ll \omega \frac{\partial }{\partial t}\right)\) approximations. The space and time varying ionization rate is calculated using Ammosov–Delone–Kraïnov tunneling model [38], and the dipole moment component at the harmonic frequency is calculated in the strong-field-approximation [39]. In both cases (with and without modal filtering system), parameters similar to the experiment presented in Fig. 16 have been used (gas pressure, energy of the infrared beam, aperture size of the iris through which we numerically propagated the laser field toward the gas cell, etc.). To save computational time, the simulations have been performed on a 500-\(\upmu\hbox{m}\) gas medium length, but the far-field harmonic intensity distributions shown in Fig. 17 are in good agreement with those measured (Fig. 16).

Fig. 17
figure 17

HHG simulations: far-field transverse intensity distributions of the twenty-fifth harmonic (32 nm) without (left) and with modal filtering (right). The simulations have been performed for a propagation length of \(500~\upmu\hbox{m}\), starting \(750~\upmu\hbox{m}\) in front of the focus of the source beam. The transverse intensity distribution of the source beams that have been used at the input of each simulation are shown on the top-right of each figure and correspond to the conditions described in the caption of Fig. 16

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4 Conclusion

The optimization of the spatial quality of a Ti:Sapphire laser source by means of modal filtering was studied both theoretically (in the general case, i.e., without cylindrical symmetry assumption) and experimentally. This filtering technique can be divided into three different steps: the spatial filtering of the high spatial frequencies at the input of the hollow fiber (as in a classical spatial filter), the mode-dependent attenuation (modal filtering) in the hollow fiber and the mode-selective attenuation by an iris in the back focal plane of a collimating lens. After analyzing theoretically the behavior of the ”modal” filter and the differences with a classical spatial filter for low-order aberration (astigmatism), we presented an experimental study with our CPA Ti:Sapphire laser system. The filter was used in front of the compressor, on the temporally stretched pulse. Measurement of amplitude and phase of the electric field allowed comparison of experimental results with theoretical predictions which proved to be in a good agreement. The results show that the beam wavefront quality as well as beam intensity distribution was strongly improved. Furthermore, no modification of the temporal intensity distribution of the compressed beam was observed in the presence of modal filtering. An additional benefit of the setup is the pointing stability. The overall enhancement of HHG with our experimental setup underlines this filter attractiveness for users applications. On account of its unique features, easy maintenance and robustness, this kind of spatial filtering is currently used on all the experimental beamlines of the CPA Ti:Sapphire LUCA laser facility.