Method and assembly for a nonlinear spectral processing of a pulsed laser beam

Abstract

The invention relates to a method and an assembly for a nonlinear spectral processing of a pulsed laser beam (3) in a periodic multipass assembly. According to the method, prior to coupling the laser beam into the multipass assembly, the transversal intensity distribution of the input laser beam (3) is transformed into an elliptical virtual eigenmode of the multipass assembly and is coupled into the multipass assembly with said intensity distribution. The elliptical virtual eigenmode has a transversal intensity distribution with two main axes, the length of which differ by at least a factor of 2 and which rotate about the propagation direction during the propagation of the laser beam (3). The method and the assembly allow a more compact design of the multipass assembly while using higher pulse energies.

Description

[0001] Method and arrangement for the nonlinear spectral broadening of a pulsed laser beam Technical field of application The present invention relates to a method for the nonlinear spectral broadening of a pulsed laser beam in a periodic multipass arrangement comprising a plurality of propagation sections, in each of which a medium or element with nonlinear properties suitable for generating self-phase modulation of the laser beam coupled into the multipass arrangement is present at one or more positions, as well as at least one further optical element in the form of a curved mirror or a lens. The invention also relates to an arrangement for carrying out the method. Pulsed laser radiation has a wide variety of applications in science and industry. The pulse duration, pulse energy, and average power all play a role.The pulse duration of a laser beam source is limited by the gain bandwidth of the laser medium or amplifier medium. However, the choice of medium also limits other parameters of the laser radiation due to the material properties, in particular the achievable average power through thermal management. To achieve certain laser radiation parameters, the combination of a laser beam source with a longer pulse duration and subsequent compression of the pulses may be helpful. This applies, for example, to ultrashort pulses from Yb-based laser beam sources, which enable a high average power. The nonlinear spectral broadening of laser pulses in a nonlinear multipass arrangement with subsequent compression (e.g., from ~1 ps to 30 fs) has proven particularly successful for high average powers and pulse energies.Examples of applications requiring short pulses, high average powers, and high pulse energies are primarily found in secondary sources, e.g., X-ray sources, inverse Compton scattering, electron and proton acceleration, neutron sources, high harmonic generation (HHG), but also in other areas such as large-area material processing. Nonlinear spectral broadening of laser pulses through self-phase modulation in gases or dielectrics, with subsequent removal of the resulting spectral chirp, is a standard method for reducing the pulse duration of ultrashort laser pulses. In addition to waveguides such as fibers or capillaries, free propagation in multipass arrangements with a nonlinear medium or element has proven successful in achieving the necessary propagation path and the necessary high intensity in the nonlinear medium and in preserving the transverse mode profile.A multipass arrangement consists of a periodic optical arrangement whose smallest possible unit is represented by a propagation section containing at least one lens or a mirror and a non-linear optical element or medium. In a particularly advantageous embodiment of the periodic optical arrangement, there are also several consecutive unit cells that are identical in their optical effect, consisting of one or more propagation sections that are delimited by an entrance and exit plane. For such a multipass arrangement, as in an optical resonator, eigenmodes are defined whose transverse field distribution is exactly reproduced after passing through the unit cell. The non-linear pulse compression in multipass arrangements is suitable due to the free, i.e. transverse (e.g.by apertures) and the associated insensitivity to pointing and deviations in the beam profile, particularly for high pulse energies and medium powers. For pulse energies > 1 mJ, multipass arrangements can also be designed much more compactly than waveguides or capillaries thanks to a folded beam path. The geometry and minimum size (i.e. length and transverse dimensions of the optics) of multipass arrangements for pulse compression are determined not only by the desired spectral broadening factor and thus the number of passes through the nonlinear medium, but also by the damage threshold of the optical components and the ionization threshold of the nonlinear medium. In order to make the multipass arrangement as small as possible, the beam cross sections should be as small as possible, i.e. just large enough that the damage threshold and ionization are just below the point of reaching.As a result, the size of a specific multipass arrangement is determined by the pulse energy, and to scale the energy to higher energies, the dimensions of the multipass arrangement must be increased proportionally. For example, if one considers the special case of a gas-filled Herriott cell as a multipass arrangement, this means that for a given pulse energy, the beam cross-section of the beam waist in the center of the cell is determined by the ionization threshold, and the beam cross-section on the mirrors of the Herriott cell is determined by the damage threshold, thus determining the radius of curvature of the cell mirrors and their spacing, or the Gouy parameter of the Herriott cell. For typical gases, mirrors, pulse durations, and wavelengths, this is achieved for a nearly concentric cell and with a practical number of unit cells or revolutions.If one wishes to increase the pulse energy for this state with minimal size, this can be achieved simply by proportionally increasing the length of the Herriott cell (or the unit cells of the multipass arrangement) and the area of ​​the cell mirrors. For fundamental mode radiation with, for example, a 1 µm wavelength and 1 ps pulse duration, mirrors with a high damage threshold (0.5 Jcm-2) are used. -2 ) and helium as gas filling with the highest ionization threshold (I0= 9.3∙10 13 Wcm -2) this results in an installation length of ~1 m per 15 mJ of pulse energy. If in this example the pulse energy is to be scaled to >> 10 mJ, e.g. 1 J, this quickly results in impractical installation lengths of >> 1 m or 60 m. State of the art An exemplary description of the procedure for nonlinear pulse compression using multi-pass arrangements can be found, for example, in J. Schulte et al., "Nonlinear pulse compression in a multi-pass cell", Opt. Lett. 41, 4511-4514 (2016). DE 102014 007 159 B4 describes a method and an arrangement for the spectral broadening of laser pulses for nonlinear pulse compression with a suitably designed multi-pass arrangement. By appropriately splitting the effect of the nonlinear medium, iethe nonlinear phase, into small amounts per pass and summing over many passes with intermediate sections with negligible self-phase modulation, the pulse power of the laser pulses can be chosen to be greater than the power of the nonlinear medium that is critical for self-focusing. To date, multi-pass arrangements with an optimized geometry with mirrors with the highest damage threshold and helium or argon (< 100 mJ) as the nonlinear medium have usually been used for pulse energies > 10 mJ. To scale the pulse energy, the length of the cell is chosen to be correspondingly large, as described, for example, in A.-L. Viotti et al., "Multi-pass cells for post-compression of ultrashort laser pulses", Optica 9(2), 197-216 (2022). Various modifications have been proposed to shorten the length of an optimized Herriott cell with circulating fundamental mode radiation. For example, L. Omar et al., “Spectral broadening of 2-mJ femtosecond pulses in a compact air-filled convex-concave multi-pass cell”, Opt. Lett. 48, 1458-1461 (2023) a modification in which an arrangement with one convex and one concave mirror is used instead of a gas-filled Herriot cell with concave mirrors. This means that there is no beam waist in the cell and the beam cross-section can be chosen to be larger everywhere than in a concave cell. In CM Heyl et al., “High-energy bow tie multi-pass cells for nonlinear spectral broadening applications”, J. Phys. Photonics 4, 014002 (2022) it is proposed to use a folded bow-tie ring resonator instead of a Herriot cell with two mirrors, which enables a significantly larger beam diameter on all mirrors with the same length. However, the beam waist between the two concave mirrors is significantly smaller, so that a gas would be ionized as a nonlinear medium.Therefore, the cell must be evacuated and a dielectric inserted as a nonlinear medium at a location with a large beam cross-section. However, both modifications lead to a smaller spectral broadening per roundtrip and thus to poorer spectral homogeneity for the respective multipass arrangement. This is because the nonlinear phase per Gouy phase must not be too large if the broadening is to be homogeneous. When propagating between two mirrors, a collimated beam can therefore only collect a smaller nonlinear phase than a focused beam that undergoes almost a complete caustic with a Gouy phase of ^. Another possibility for a more compact multipass arrangement at higher pulse energies is described in M. Kaumanns et al., “Spectral broadening of 112 mJ, 1.3 ps pulses at 5 kHz in a LG. 10multipass cell with compressibility to 37 fs," Opt. Lett. 46, 929-932 (2021). In this case, the geometry of the multipass arrangement was not adjusted, but the Gaussian fundamental mode of the incident laser beam was converted into a Laguerre-Gaussian mode LG using an appropriate phase mask. 01transformed. As a result, with the same geometry of the multipass arrangement, the beam cross-section increases both on the mirrors and in the beam waist, and a correspondingly greater pulse energy can be coupled in. However, during conversion to this donut mode and back through the vortex-shaped phase front, considerable losses occur due to discontinuity in the center of the donut mode, as well as chromatic aberrations on the phase plates used. The object of the present invention is to provide a method for the nonlinear spectral broadening of a pulsed laser beam using a periodic multipass arrangement, as well as an arrangement designed to carry out the method, in which the multipass arrangement can be made more compact at higher pulse energies without the disadvantages described above. Description of the invention The object is achieved with the method and the arrangement according to patent claims 1 and 13.Advantageous embodiments of the method and the arrangement are the subject of the dependent patent claims or can be derived from the following description and the exemplary embodiment. In the proposed method for the nonlinear spectral broadening of a pulsed laser beam, a periodic multipass arrangement is used which comprises a plurality of propagation sections, in each of which a medium or element with nonlinear optical properties suitable for generating self-phase modulation of the laser beam coupled into the multipass arrangement, hereinafter also referred to as a nonlinear medium, is present at one or more positions, and an optical element in the form of a lens or a curved mirror, wherein the laser beam preferably propagates freely between the optical elements of the propagation sections.Free propagation means that the laser beam's propagation is neither influenced by apertures, i.e., its transverse extension, beam distribution, and center of gravity are fixed or restricted, nor is it guided like a waveguide. The proposed method is characterized by the fact that the transverse intensity distribution of the laser beam is transformed into an elliptical quasi-eigenmode of the multipass array before being coupled into the multipass array. This intensity distribution is then coupled into the multipass array.The elliptical quasi-eigenmode exhibits a transverse intensity distribution with two semi-axes whose aspect ratio or ellipticity ^ is constant or at least approximately constant during the propagation of the laser beam in the multi-pass arrangement, whose length differs by at least a factor of 2, and which rotate around the propagation direction during the propagation of the laser beam, thus exhibiting orbital angular momentum. An only approximately constancy of the ellipticity can occur, for example, in the case of astigmatism in the multi-pass arrangement, which is caused by a slight tilt of the curved optics.In a particularly advantageous embodiment of the periodic optical arrangement, there are also several consecutive unit cells with identical optical properties, consisting of one or more propagation sections, so that the transverse intensity distribution and phase front of the elliptical quasi-eigenmode are reproduced, except for the orientation of the principal axes, after passing through each unit cell. In a stable resonator, electromagnetic waves generally have a set of eigensolutions or eigenmodes, e.g., Hermite, Laguerre, or Ince-Gaussian modes. Depending on the hard confinement imposed by apertures, optical elements, or other boundary conditions within the resonator, one of these modes is particularly stable against disturbances—the fundamental mode.In a periodic multipass arrangement, as in a resonator, an eigensolution can be defined which, in the special case of successive identical unit cells, reproduces itself exactly after passing through each unit cell. For a multipass arrangement with a nonlinear medium, an eigensolution exists for a circulating monochromatic field distribution, neglecting time and frequency dependence. This eigensolution is intensity-dependent and stable to small perturbations. For small nonlinearities, for example, the fundamental Gaussian mode is a very good approximation of a stable eigensolution of a multipass arrangement with sufficiently large optical elements and no additional apertures. For spectral broadening, a transverse eigenmode of the multipass arrangement, slightly modified by the Kerr lens of the nonlinear medium, is generally used.The multipass arrangement is ideally chosen so that for this eigenmode, the distance between the ionization threshold in the nonlinear medium and the laser-induced damage threshold (LIDT) of the optics is approximately equal. A Laguerre-Gaussian mode (LG) is used. 01, also called donut mode, is an example of a higher order eigensolution for which the above applies equivalently. Vortex modes with a spiral phase, like the fundamental Gaussian mode, are particularly distinguished by the fact that their shape is, to a first approximation, invariant with respect to the transverse Kerr effect. However, the phase fronts of higher modes such as Hermite or Laguerre-Gaussian have either phase jumps of ^ and / or, like the donut mode, a helical phase front (vortex) with a singularity. This is technically relevant because modes with vortex or phase jumps have field distributions with discontinuities or zeros that cannot be transformed into a Gaussian mode without losses, or can only be done with great effort. In the following, therefore, only modes are considered that can be transformed into modes without vortex and phase jumps using spherical and / or cylindrical optics and / or mirrors alone.Transforming the laser radiation to be spectrally broadened into an eigensolution of the (nonlinear) multipass arrangement using suitable optics or telescopes and propagating this eigensolution through the multipass arrangement is advantageous because the beam profile and beam diameter are approximately reproduced with each revolution, thus maintaining a constant distance from the destruction and ionization thresholds. Furthermore, the propagation (even with a nonlinear element or medium) is particularly stable against small phase and amplitude perturbations. Although a multipass arrangement (with and without nonlinearity) is not a resonator in the strict sense, the restriction to eigensolutions as a condition for stable propagation (against small perturbations) is also useful there.The stable propagation of radiation in a typical multi-pass arrangement with a non-linear medium is, to a first approximation, not determined by limiting apertures, but by the interaction with the non-linear medium, i.e., the phase and amplitude modulation imposed thereby. For stable propagation, a field distribution in a multi-pass arrangement with a non-linear medium does not have to reproduce itself periodically with mathematically exact amplitude and / or phase, but only in such a way that the non-linear interaction and thus the beam cross-sections in the periodic arrangement and within a caustic remain approximately constant. Such a solution is referred to in the present patent application as a quasi-eigensolution or quasi-eigenmode. A transverse beam distribution formed by two principal axes with two different beam diameters orRayleigh lengths (or confocal parameters) is a quasi-eigensolution or quasi-eigenmode if the two beam diameters and their ratio are at least approximately reproduced with each pass through the propagation sections of the multi-pass arrangement. If the multi-pass arrangement has several consecutive unit cells that are identical in their optical effect, the two beam diameters of the quasi-eigensolution or quasi-eigenmode are reproduced with each pass through the unit cell. For a simply astigmatic beam, this is fundamentally not possible for a rotationally symmetric multi-pass arrangement. However, such quasi-eigensolutions do exist for generally astigmatic beams with suitable orbital angular momentum, where the orientation of the principal axes changes with propagation. The quasi-eigensolutions or quasi-eigenmodes used in the proposed methodQuasi-eigenmodes of the multipass arrangement with elliptical beam profile (elliptical quasi-eigenmodes) have five relevant advantages for the design of multipass arrangements for nonlinear spectral broadening or pulse compression: ^ The mode volume and the mode cross-section are increased compared to the fundamental Gaussian mode, depending on the ellipticity ^' of the transverse beam distribution or beam profile. This reduces the intensity on the optics and in the nonlinear medium or element. Conversely, the length of the unit cell or the corresponding propagation section required for a given pulse energy can be reduced at constant intensity. ^ The elliptical quasi-eigenmodes can be transformed back and forth into round, stigmatic beams without orbital angular momentum using only two cylindrical optics.^ In many cases, especially when using Herriott cells as a multipass arrangement, the required beams can be arranged particularly compactly on the mirrors of the multipass arrangement. With beam cross-sections specified by LIDT, the optics of the multipass arrangement can be particularly well utilized and thus selected to be small. ^ The beam profile of the quasi-eigenmodes, in particular ellipticity and orbital angular momentum, is preserved to a first approximation during propagation through a second-order nonlinear medium. ^ The properties mentioned above apply not only to the fundamental quasi-eigenmode, i.e. a generally astigmatic Gaussian mode with suitable orbital angular momentum, but also to elliptic Hermite-Gaussian modes of higher order. The advantages described here can therefore be combined with the potential advantages of exciting higher transverse modes.In the proposed method, the transverse intensity distribution of the input laser beam is transformed into an elliptical quasi-eigenmode of the multi-pass arrangement with the characteristics specified above before being coupled into the multi-pass arrangement. This intensity distribution is then coupled into the multi-pass arrangement. In the case of a generally astigmatic Gaussian beam (according to ISO / DIS 11146-2 (Part 2)) with orbital angular momentum as the lowest-order elliptical quasi-eigenmode, this can be described mathematically as follows. The starting point for this mathematical description is a stigmatic beam whose propagation is determined by a caustic with a Rayleigh length of, for example, rand waist position z0. In the first step, this beam is transformed into a simply astigmatic beam by beam shaping symmetrical in two orthogonal axes, e.g. with an arrangement of cylindrical optics aligned along two orthogonal main axes, such that the following conditions are met: z 01 = z 02 ; e r1 / z r2 > 1 / ( ^2-1) ^ z r1 / z r2 < ( ^2-1) The beam propagation of this elliptical, simply astigmatic beam is determined along two orthogonal symmetry planes by two caustics with a Rayleigh length z r1 or z r2 and a waist position z 01 or z 02 with a waist diameter w 01 or w 02 This allows us to define modes with a Rayleigh length z r and waist position z0, whose intensity distribution at all locations of the beam propagation is elliptical with two beam radii w a(z)and w b (z) that follow a caustic. The principal axes of the ellipse rotate with the propagation around the ray axis. For this purpose, the ray is rotated in the second step at z 01 = z 02 = 0 in the mathematical description by two crossed cylindrical lenses, rotated 45° to the input axes, transformed with the focal lengths f1 and f2: f1= z r / (z r / z r1 -1) = z r ^ / (1-z r / z r2 ) or 1 / f1= 1 / z r1 -1 / z r and f2= -f1with 2 / z r = 1 / z r1 +1 / z r2 on site z 01 = z 02 = z0= 0 Behind the cylindrical lenses, the elliptic quasi-eigenmode with ellipticity ^' and ^ : ^(z r1 / z r2 ) = ^' w a (z) / f b (z)= w 01 / f 02 = ^ The propagation of the elliptic quasi-eigenmode is determined in addition to the beam radii w a and w b described by the orientation angle depending on the location z ^^ the main axes: w a 2 = w 01 2 ^(1+(z / z r ) 2 ) ; w b 2 = w 02 2 ^(1+(z / z r ) 2 ) ^ ^ = arctan(z / z r ) This propagation behavior, ie the description of the elliptic eigenmode by a Rayleigh length z Rand ellipticity ^ or ^' is retained even after any stigmatic beam transformation. In the case of a Herriot cell or other periodic multipass arrangement, in order to excite an elliptical quasi-eigenmode whose beam cross-section is the same at any distance from the cell center for all revolutions in the cell, the following conditions apply for the coupling chosen in the mathematical description in the middle between the two mirrors with radius R and distance L or Gouy parameter ^ (cf. e.g. DE 102014 007 159 B4) and the focal lengths f1 and f2 of the crossed cylindrical lenses rotated by 45° at the coupling location in the cell center: with L = R ^(1-cos( ^ / 2)) ; f2= z r / (z r / z r2 -1) z 01 = z 02 = 0 ; f1= -f2z r1 / z r = a 2 ^(1+ ^(1-1 / a 2 )) ; z r2 / z r = a 2 ^(1- ^(1-1 / a 2 )) Here a = ^(z r1 / z r^ z r2 / z r ) the effective enlargement of the beam cross-section by the elliptical quasi-eigenmode with orbital angular momentum compared to the usual stigmatic eigenmode without orbital angular momentum and the Gouy parameter of the multipass arrangement. Of course, the laser beam can also be coupled into the multipass arrangement at a different location. For the beam transformation required for this, corresponding conditions for the lenses or mirrors used and distances can be set up equivalent to the mathematically compact representation given here. The above representation is only to be understood as an example that can be extended to any multipass arrangement. These only need to have a distinctive eigensolution or eigenmode, which is defined by the quantities z r and z0. The above formula for z ris therefore given only as an example for such an eigensolution of a Herriott cell. In the representation for Gaussian beams with general astigmatism by Arnaud and Kogelnik (J.A. Arnaud, H. Kogelnik, "Gaussian light beams with general astigmatism," Applied optics 8 (8), 1687-1693 (1969)), the following representation of the elliptical quasi-eigenmode results for the sake of completeness: z1 / z r = - ^( ^ ∙ ^(1-1 / a 2 )) ; z2= ​​-z1 In the proposed method, the laser beam, before transformation, preferably has the transverse intensity distribution of any Gaussian beam, in particular a stigmatic, collimated Gaussian beam. In a preferred embodiment, the incoming laser beam to be spectrally broadened is transformed into the elliptical quasi-eigenmode with ellipticity ^ using a (first) optical device which has two cylindrical optics with cylinder axes rotated relative to one another in the plane perpendicular to the propagation direction. The cylindrical optics can be, for example, two cylindrical lenses, two cylindrical mirrors, or a combination of a cylindrical lens and a cylindrical mirror. The (first) optical device is designed such that the cylindrical optics approximately forms a cylindrical telescope with a magnification factor ^ or 1 / ^.The angle at which the two cylinder axes are rotated relative to each other is selected such that the laser beam is transformed into the desired elliptical quasi-eigenmode with orbital angular momentum. A further suitably selected rotationally symmetric optic performs mode matching, in which the size and divergence of the input beam are adapted to the multipass arrangement. The multipass arrangement itself is rotationally symmetric. It is therefore a stigmatic multipass arrangement. In one advantageous embodiment, the nonlinear medium in the multipass arrangement is a gas. For example, the entire multipass arrangement can be enclosed in a housing filled with this gas. In another embodiment, in which the multipass arrangement has a nonlinear element, this element is, for example, a dielectric plate or dielectric lens.The multipass arrangement preferably has a number of N > 5 propagation sections, particularly preferably 10 or more, which in the case of a Herriott cell, which is preferably used as a multipass arrangement, corresponds to the number of passes. In the case of a Herriott cell, the entrance and exit planes that delimit the propagation sections and the unit cell, respectively, correspond to the respective mirror sections. A propagation section or a unit cell contains the propagation path and the nonlinear medium between the mirrors, as well as the reflection at the curved surface of one of the two mirrors of the Herriott cell. For pulse compression, the transverse intensity distribution of the laser beam spectrally broadened in the multipass arrangement should be retransformed into a collimated stigmatic beam after exiting the arrangement.For this purpose, a second optical arrangement with the same optical elements as in the first optical arrangement is preferably used. The laser beam emerging from the multipass arrangement is then finally shortened in its pulse duration, preferably in a suitable compression arrangement to eliminate the chirp. The proposed arrangement for carrying out the method accordingly comprises the multipass arrangement already described in connection with the method as well as an upstream (first) optical device with which a pulsed laser beam to be spectrally broadened or its transverse intensity distribution is transformed into an elliptical quasi-eigenmode of the multipass arrangement and coupled into the multipass arrangement with this intensity distribution.The first optical device preferably has two cylindrical optics and a further rotationally symmetric optic for mode adaptation, as already described in connection with the method. Furthermore, a second optical device is preferably provided, which transforms the transverse intensity distribution of the laser beam emerging from the multipass arrangement back into its original transverse intensity distribution. In advantageous embodiments, the multipass arrangement and first optical device are designed such that they enable the implementation of the various embodiments of the method. By transforming the input beam into an elliptical quasi-eigenmode of the multipass arrangement, the ellipticity of the quasi-eigenmode increases the mode volume or the beam cross-section at every location within the multipass arrangement compared to the fundamental eigenmode.This allows the pulse peak power within the multipass arrangement to be kept constant, while increasing the beam cross-section at higher pulse peak power or pulse energy. Since the ionization of the nonlinear medium or the damage thresholds of the optical elements of the multipass arrangement generally limit the pulse energy that can be spectrally broadened and then compressed in a nonlinear multipass arrangement, this can be reduced by transforming it into an elliptic quasi-eigenmode by a factor of a = 1 / 2∙( ^´), depending on the ellipticity ^´. 2+1) / ^´ can be increased. The transformation and re-transformation into an elliptical quasi-eigenmode can be realized simply and without loss using a stigmatic beam transformation and a cylindrical telescope with mutually rotated cylinder axes or another suitable combination of spherical, elliptical, or cylindrical optics. In addition, a combination with other methods for enlarging the beam cross-section is also possible, in particular the use of higher-order modes. Brief Description of the Drawings The proposed method and the associated arrangement are explained again below using an exemplary embodiment in conjunction with the drawings. Herein: Fig. 1 shows a schematic representation of a Herriot cell as a multi-pass arrangement and the optical devices for coupling and decoupling the laser beam according to the prior art; Fig.2 shows an example of the proposed arrangement with a Herriot cell as a multipass arrangement as in the example of Figure 1; and Figure 3 shows a representation of the rotation of the elliptical beam profile near the beam waist in the arrangement of Figure 2. Ways of Implementing the Invention In the proposed method for spectral broadening of laser pulses for nonlinear pulse compression, the laser pulses are coupled into a stigmatic multipass arrangement in which they propagate through at least one medium with nonlinear optical properties, in which the laser pulses are given a nonlinear phase by self-phase modulation. By means of the inventive beam transformation of the input beam into an elliptical quasi-eigenmode of the multipass arrangement, the beam cross-section can be increased by a factor a everywhere within the multipass arrangement.As a result, the pulse energy to be compressed can also be increased by a factor of a without changing the geometry and length of the multipass arrangement. This is particularly relevant for high pulse energies (>> 10 mJ), where the required size or length of the multipass arrangement is very large (>> 1 m), and thus the size limits the maximum pulse energy. The multipass arrangement used in the proposed method and arrangement consists of several propagation sections or unit cells containing a nonlinear medium or element. The method and arrangement are explained below using a configuration with several consecutive unit cells consisting of one or more propagation sections. However, the explanations can also be applied analogously to configurations with propagation sections that are not completely identical in their optical effect.The number of propagation sections is at least N > 5, particularly preferably 10 or more. Each propagation section has at least one lens or one curved mirror or mirror section, at least one nonlinear element or nonlinear medium, and a preferably free propagation region. In addition, one or more propagation sections form a unit cell, several of which follow one another. The unit cell is delimited in the beam propagation direction by an entrance and exit plane and does not map the entrance plane onto the exit plane. The unit cell represents the smallest possible identical unit of the multipass arrangement. Its optical effect is invariant upon rotation about the propagation axis or optical axis of the unit cell.The elliptical quasi-eigenmode is a beam distribution within the multipass arrangement whose propagation through a unit cell is characterized by the following properties. The field distribution, or intensity distribution, and phase front of the quasi-eigenmode do not reproduce themselves simultaneously after passing through the unit cell. They also do not reproduce themselves when the size and divergence of the input beam are adjusted (mode matching). The transverse intensity distribution of the quasi-eigenmode can be characterized at any location in the multipass arrangement by two principal axes and two beam diameters (or the length of the principal axes). The beam diameters along the two principal axes differ by at least a factor of 2. With the exception of the size and orientation of the principal axes, the transverse intensity distribution of the quasi-eigenmode has the same shape at every propagation location within the multipass arrangement.The intensity distribution and phase front of the elliptical quasi-eigenmode reproduce themselves after passing through the unit cell, with the exception of the orientation of the principal axes. The elliptical quasi-eigenmode possesses an orbital angular momentum, but not necessarily a phase singularity and spiral phase front, vortex, or topological charge (cf. J. Fisser et al., "Orbital angular momentum of general astigmatic modes," Phys. Rev. A 70, 0138091-11 (2004)). The beam distribution, or rather its principal axes, rotate continuously (proportional to the Gouy phase) during the preferably free propagation within the multipass arrangement. However, the intensity distribution is constant except for the orientation of the axes and the size. The beam distribution, or rather its principal axes, rotate by a fixed amount with each pass through the unit cell.The spatially dependent phase and amplitude perturbations caused by the nonlinear medium or element are similar for all passes relative to the local principal axes of the beam. The following figures illustrate the transition from spherical eigenmodes to elliptical quasi-eigenmodes according to the proposed method in a nonlinear multipass arrangement for spectral broadening or pulse compression, using a Herriot cell with two concave mirrors of equal curvature. The required nonlinear medium is not shown in the figures but can be a gas, a dielectric, or even a plasma. Figure 1 shows a Herriot cell consisting of the two concave mirrors 1. The Herriot cell is designed for 25 revolutions of the coupled laser beam 3, thus having 50 propagation sections and 50 unit cells.In this example, the two concave mirrors 1 have a radius of curvature of 1500 mm and are arranged at a distance of 2988 mm (Gouy parameter 24 / 25 ^). The laser beam 3 is coupled into and out of the Herriot cell via rectangular spherical mirrors 2a, 2b, each with a radius of curvature of 2850 mm, which also serve for mode matching. The laser beam 3 is coupled into the Herriot cell via a spherical Galilean telescope with a magnification of 0.2, consisting of the two curved telescope mirrors 4, 5. Beam shaping after the outcoupling is performed via another telescope with the spherical telescope mirrors 6, 7, which are identical to the telescope mirrors 4, 5 of the spherical telescope for the input beam.The beam diameter of the collimated Gaussian beam coupled in this example as laser beam 3 at a wavelength of 1030 nm is 25 mm, the beam diameter in the center of the Herriot cell is 0.35 mm, and on the large concave cell mirrors it is 5.6 mm. With the proposed method and the proposed arrangement, the Herriot cell constructed in the same way, consisting of the concave mirrors 1 and input and output mirrors 2a and 2b, can be used. However, instead of the spherical telescope, a cylindrical telescope with the two rectangular cylindrical mirrors 8, 9 is used for input coupling and beam transformation. The magnification of this cylindrical Galilean telescope is 0.16 in the present example. The two rectangular cylindrical mirrors 8, 9 are slightly rotated relative to each other (1.5°) in order to meet the criteria for generating the elliptical quasi-eigenmode of the multipass arrangement as described above.The beam diameter of the collimated Gaussian beam coupled in as laser beam 3 is also 25 mm at a wavelength of 1030 nm. The beam diameter in the center of the Herriot cell is 0.25 mm x 1.57 mm and on the concave cell mirrors it is 4 mm x 25 mm. Compared to the conventional method according to Figure 1, the proposed method in this example achieves an enlargement of the beam cross-section at every point of propagation within the cell by a factor of 3.2. After exiting the Herriot cell, the spectrally broadened beam 3 is re-transformed into its original transverse intensity distribution using a cylindrical telescope with rectangular cylindrical mirrors 10, 11, which in turn are identical to the cylindrical mirrors 8, 9 of the cylindrical telescope for the input beam.Finally, Figure 3 shows a schematic representation of the rotation of the elliptical beam profile 12 around the propagation axis near the beam waist or the waist of the beam bundle in the unit cell for illustrative purposes. In contrast to all previously known arrangements for pulse compression using a multipass arrangement equivalent to Figure 1, in the proposed method and the proposed arrangement, as shown by way of example in Figure 2, an elliptical mode with orbital angular momentum is excited as a quasi-eigenmode in a rotationally symmetric, in particular spherical, multipass arrangement. This is preferably transformed from or into a stigmatic, collimated input or output beam by a Galilean cylindrical telescope with mutually rotated cylindrical mirrors (alternatively lenses) and another concave, spherical mirror.In the example shown, an ellipticity of ^ = 6.25 increases the beam cross-section of a Gaussian mode at every location within the multipass array by a factor of a = 3.2. The ellipticity was chosen to be consistent with the Gouy parameter chosen in this example. = 24 / 25∙ ^ of the multipass arrangement to make the best possible use of the mirror surface. Any other ellipticities ^ or ^'= ^(z r1 / z r2 ) = a + ^(a 2 -1) and effective magnifications a = 1 / 2∙( ^' 2+1) / ^' of the areas are also possible. With unchanged dimensions of the multipass arrangement and identical nonlinear medium, the elliptical mode, with a pulse energy increased by a factor of a, achieves both the same spectral broadening and the same distance to the damage thresholds of the mirrors and ionization. Without the elliptical mode, the latter would only be possible with an extension of the cell length by a factor of a. The example of a Herriott cell shown can be equivalently transferred to any other periodic multipass arrangement that has stigmatic or spherical periodic eigensolutions. By suitable beam transformation, in particular with a simple cylindrical telescope with mutually rotated cylinder axes, any Gaussian input beam, in particular a collimated, stigmatic round input beam, can be converted into an eigenmode with ellipticities ^ = ^'= a + ^(a 2-1) and effectively increasing the area by a factor of a and back. The elliptical quasi-eigenmode has the same Rayleigh length z r like the fundamental eigenmode of the multipass arrangement, but their principal axes rotate during propagation by the angle If a laser pulse is propagated through the multipass arrangement, which contains a nonlinear medium for spectral broadening, this arrangement behaves (assuming a sufficiently large clear aperture) after transformation of the laser beam into the elliptic quasi-eigenmode and with a pulse energy increased by a factor of a with regard to the effect of the nonlinear medium or nonlinear element, in the same way as without transformation and with a stigmatic fundamental eigenmode. This is particularly true for spectral broadening. Elliptical quasi-eigenmodes of a multipass arrangement in the form of generally astigmatic Gaussian beams can be completely described by ellipticity and Rayleigh length. In addition, other elliptic quasi-eigenmodes exist, e.g. spiral-like elliptic Hermite-Gaussian modes, which are characterized by an atomic number n in addition to ellipticity and Rayleigh length.These elliptic modes have a transverse mode distribution superimposed on the Gaussian mode, which does not change during (free) propagation except for the size and orientation of the principal axes. As in the case of Hermite or Laguere-Gaussian eigenmodes of a resonator or a multipass arrangement, the order number n of the spiral elliptic Hermite-Gaussian modes increases the mode cross-section at all locations by a fixed factor, and the transverse mode distribution exhibits n minima. However, the other propagation properties of the spiral elliptic Hermite-Gaussian modes of nth order correspond to the fundamental elliptic quasi-eigenmode described above, neglecting the nonlinearity.In this respect, higher mode orders and the transformation into an elliptical quasi-eigenmode are independent and combinable possibilities to increase the mode cross-section and thus the possible pulse energy in a nonlinear multipass arrangement for spectral broadening while maintaining unchanged size (propagation length for one revolution in the unit cell).

[0002] List of reference symbols 1 concave cell mirror 2a concave input mirror 2b concave output mirror 3 laser beam 4 concave spherical mirror 5 convex spherical mirror 6 convex spherical mirror 7 concave spherical mirror 8 rectangular concave cylindrical mirror 9 rectangular convex cylindrical mirror 10 rectangular convex cylindrical mirror 11 rectangular concave cylindrical mirror 12 beam profile

Claims

1. A method for the non-linear spectral broadening of a pulsed laser beam in a periodic multi-pass arrangement, - which is formed from a plurality of propagation sections, in each of which a medium or element with non-linear properties suitable for generating a self-phase modulation of the laser beam (3) coupled into the multi-pass arrangement is present at one or more positions, as well as at least one further optical element in the form of a curved mirror or a lens, characterized in that a transverse intensity distribution of the laser beam (3) is transformed into an elliptical quasi-eigenmode of the multi-pass arrangement before being coupled into the multi-pass arrangement and is coupled into the multi-pass arrangement with this intensity distribution, wherein the elliptical quasi-eigenmode has a transverse intensity distribution with two main axes,whose length ratio or ellipticity ^ is constant or at least approximately constant, whose length differs by at least a factor of 2 and which rotate about the propagation direction during the propagation of the laser beam (3).

2. Method according to claim 1, characterized in that the periodic multipass arrangement contains several consecutive unit cells identical in their optical effect, each consisting of one or more propagation sections, wherein the transverse intensity distribution and phase front of the elliptical quasi-eigenmode are reproduced except for the orientation of the principal axes after passing through each unit cell.Method according to claim 1 or 2, characterized in that the transverse intensity distribution of the laser beam (3) is transformed into the elliptical quasi-eigenmode of the multipass arrangement before coupling into the multipass arrangement with the aid of a first optical device which has two cylindrical optics (8, 9) with cylinder axes rotated relative to one another in a plane perpendicular to the propagation direction of the laser beam (3), which are aligned neither parallel nor perpendicular to one another, and optionally one or more further optics (2a).

4. Method according to one of claims 1 to 3, characterized in that the transverse intensity distribution of the laser beam (3) before the transformation is an intensity distribution of any Gaussian beam, in particular a stigmatic Gaussian. Beam.

5. Method according to one of claims 1 to 4, characterized in that the elliptical quasi-eigenmode of the multi-pass arrangement, into which the transverse intensity distribution of the laser beam (3) is transformed, has the shape of a generally astigmatic Gaussian beam.

6. Method according to one of claims 1 to 4, characterized in that the elliptical quasi-eigenmode of the multi-pass arrangement, into which the transverse intensity distribution of the laser beam (3) is transformed, has the shape of a spiral-elliptical Hermite-Gaussian mode.

7. Method according to one of claims 1 to 6, characterized in that the transformation of the transverse intensity distribution of the laser beam (3) is reversed after exiting the multi-pass arrangement by an inverse transformation with a second optical device (10, 11, 2b).Method according to one of claims 1 to 7, characterized in that a Herriott cell is used as the multipass arrangement.

9. Method according to one of claims 1 to 8, characterized in that the medium with nonlinear properties suitable for generating the self-phase modulation is formed by a gas.

10. Method according to one of claims 1 to 8, characterized in that the element with nonlinear properties suitable for generating the self-phase modulation is formed by a dielectric plate or lens.

11. Method according to one of claims 1 to 10, characterized in that the multipass arrangement is designed such that it has more than 5 propagation sections.

12. Method according to one of claims 1 to 11, characterized in that the multipass arrangement is designed such that, at a given pulse energy of the laser beam (3), neither a damage threshold of optical elements of the multipass arrangement nor ionization of the medium with nonlinear properties occurs.Arrangement for the non-linear spectral broadening of a pulsed laser beam, which has a periodic multipass arrangement with an upstream first optical device (8, 9, 2a), wherein- the multipass arrangement is formed from several propagation sections,. in each of which a medium or element with non-linear properties suitable for generating a self-phase modulation of the laser beam (3) coupled into the multi-pass arrangement is present at one or more positions, as well as at least one further optical element in the form of a curved mirror or a lens, and - the first optical device (8, 9, 2a) is designed such that it transforms a transverse intensity distribution of the laser beam (3) before coupling into the multi-pass arrangement into an elliptical quasi-eigenmode of the multi-pass arrangement, wherein the elliptical quasi-eigenmode has a transverse intensity distribution with two principal axes whose length ratio or ellipticity ^ is constant or at least approximately constant, whose length differs by at least a factor of 2, and which rotate about the propagation direction during the propagation of the laser beam (3). 14.Arrangement according to claim 13, characterized in that the periodic multipass arrangement contains several successive unit cells which are identical in their optical effect, each consisting of one or more propagation sections, wherein the transverse intensity distribution and phase front of the elliptical quasi-eigenmode extend up to the orientation of the principal axes. Reproduced by passing through each unit cell.

15. The arrangement according to claim 13 or 14, characterized in that the first optical device (8, 9, 2a) comprises two cylinder optics (8, 9) with cylinder axes rotated relative to one another in a plane perpendicular to the propagation direction of the laser beam (3), which are neither parallel nor perpendicular to one another, and optionally one or more further optics (2a).

16. The arrangement according to one of claims 13 to 15, characterized in that the elliptical quasi-eigenmode of the multipass arrangement, into which the transverse intensity distribution of the laser beam (3) is transformed by the first optical device (8, 9, 2a), has the shape of a generally astigmatic Gaussian beam. 17.Arrangement according to one of claims 13 to 15, characterized in that the elliptical quasi-eigenmode of the multipass arrangement, into which the transverse intensity distribution of the laser beam (3) is transformed by the first optical device (8, 9, 2a), has the shape of a spirally elliptical Hermite-Gaussian mode.

18. Arrangement according to one of claims 13 to 17, characterized in that the arrangement comprises a second optical device. Device (10, 11, 2b) which is designed and arranged such that it reverses the transformation of the transverse intensity distribution of the laser beam (3) by the first optical device (8, 9, 2a) after the laser beam (3) exits the multipass arrangement by means of an inverse transformation.

19. Arrangement according to one of claims 13 to 18, characterized in that the multipass arrangement is a Herriott cell.

20. Arrangement according to one of claims 13 to 19, characterized in that the medium with nonlinear properties suitable for generating the self-phase modulation is a gas.

21. Arrangement according to one of claims 13 to 19, characterized in that the element with nonlinear properties suitable for generating the self-phase modulation is a dielectric plate or lens. 22.Method according to one of claims 13 to 21, characterized in that the multipass arrangement has more than 5 propagation sections.

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