Method and apparatus for nonlinear spectral broadening of a pulsed laser beam

Abstract

The present invention relates to a method and apparatus for nonlinear spectral broadening of a pulsed laser beam 3 in a periodic multipath structure. In this method, the lateral intensity distribution of the incident laser beam 3 is converted into an elliptic quasi-eigenmode of the multipath structure before it is incident on the multipath structure, and the beam is incident on the multipath structure with this intensity distribution. Here, the elliptic quasi-eigenmode has a lateral intensity distribution with two principal axes, the lengths of which differ by at least twice, and the principal axes rotate around the propagation direction during the propagation of the laser beam 3. This method and apparatus makes it possible to configure the multipath structure more compactly even when the pulse energy is high.

Description

[Technical Field]

[0001] The present invention relates to a method for performing nonlinear spectral broadening of a pulsed laser beam in a periodic multipath structure (Multipass-Anordnung). The multipath structure comprises a plurality of propagation segments, each of which contains one or more media or elements having nonlinear properties suitable for generating self-phase modulation of the laser beam incident on the multipath structure, as well as at least one further optical element in the form of a curved mirror or lens. The present invention also relates to an apparatus for carrying out this method. [Background technology]

[0002] Pulsed laser radiation has diverse applications in science, technology, and industry. Here, not only pulse width but also pulse energy and average power play important roles. The pulse width of a laser source is limited by the amplification bandwidth of the laser medium or amplification medium. However, depending on the choice of medium, its material properties also limit other parameters of laser radiation, particularly the average power achievable through thermal management. Therefore, to achieve specific parameters of laser radiation, it is sometimes useful to combine a laser source with a longer pulse width with subsequent pulse compression. This applies, for example, to ultrashort pulses from Yb-based laser sources that enable high average power. In this regard, nonlinear spectral broadening of laser pulses in nonlinear multipath configurations, followed by compression (e.g., from approximately 1 ps to 30 fs), has been proven particularly effective for high average power and pulse energy.

[0003] Exemplary applications requiring short pulses, high average power, and large pulse energies are found particularly in the field of secondary light sources, such as X-ray sources, inverse Compton scattering, electron and proton acceleration, neutron sources, and high-harmonic (HHG) generation, but also in other fields such as large-area material processing.

[0004] A standard method for reducing the pulse width of ultrashort laser pulses is to perform nonlinear spectral broadening of the laser pulse by self-phase modulation in a gas or dielectric, followed by the removal of the resulting spectral chirp. Beyond waveguides such as fibers or capillaries, free propagation in multipath configurations using nonlinear media or elements has been shown to be effective in achieving the required propagation distance and high intensity within the nonlinear medium while maintaining the transverse mode profile. A multipath configuration consists of a periodic optical arrangement, the smallest possible unit of which is represented by a propagation section containing at least one lens or mirror and a nonlinear optical element or nonlinear medium. In a particularly advantageous form of the periodic optical arrangement, there are further consecutive unit cells consisting of one or more propagation sections, defined by an incident and an exit surface, with identical optical action. For such multipath configurations, as in the case of optical resonators, eigenmodes are defined in which the transverse field distribution is accurately reproduced after passing through the unit cells.

[0005] Nonlinear pulse compression in multipath structures is particularly well-suited for high pulse energies and average power because it involves free propagation, i.e., propagation not restricted laterally (e.g., by an aperture), and as a result, is unaffected by variations in Poynting and beam profile. Furthermore, when pulse energies exceed 1 mJ, multipath structures can be made significantly more compact than waveguides or capillaries by folding the beam path. The geometry and minimum external dimensions (i.e., the external length and lateral dimensions of the optical system) of a multipath structure for pulse compression are determined by the intended spectral widening coefficient, and consequently the number of times the beam passes through the nonlinear medium, as well as the damage threshold of the optical components and the ionization threshold of the nonlinear medium. To make the multipath structure as small as possible, the beam cross-section must be made as small as possible, i.e., large enough not to reach the damage threshold and ionization threshold. Therefore, the external dimensions of a particular multipath structure are determined by the pulse energy, and as the energy scales to higher energies, the dimensions of the multipath structure must be increased proportionally.

[0006] For example, considering a gas-filled Herriott cell as a special case of a multipath configuration, this means that, at a given pulse energy, the beam cross-section at the center of the cell's beam waist is defined by the ionization threshold, and the beam cross-section on the mirrors of the Herriott cell is defined by the damage threshold. Therefore, the radius of curvature and spacing of the cell mirrors, i.e., the Gouy parameter of the Herriott cell, are defined. For typical gas, mirrors, pulse width, and wavelength, this is achieved with nearly concentric cells and a practical number of unit cells, i.e., circles (Umlaeufen). In this state with minimum external size, if we want to increase the pulse energy, this can be achieved simply by proportionally increasing the length of the Herriott cell (or unit cell of the multipath configuration) and the area of ​​the cell mirrors. For example, ground mode light with a wavelength of 1 μm and a pulse width of 1 ps, with a high damage threshold (0.5 Jcm). -2A mirror having ) and the best ionization threshold (I0 = 9.3·10) as gas filling. 13 Wcm -2 When using helium with the specified properties, an external length of approximately 1 m can be obtained for every 15 mJ of pulse energy. If we try to scale the pulse energy in this example to a value far exceeding 10 mJ, for example to 1 J, the result will quickly exceed 1 m, i.e., an impractical external length of 60 m.

[0007] A method for nonlinear pulse compression using a multipath configuration is described exemplified in, for example, Non-Patent Document 1. In this regard, Patent Document 1 describes a method and apparatus for spectral widening of a laser pulse for nonlinear pulse compression using a suitably configured multipath configuration. By dividing the effect of the nonlinear medium, i.e., the nonlinear phase, into small amounts corresponding to each pass, adding them over many passes, and inserting sections in between where self-phase modulation can be ignored, the pulse power of the laser pulse can be selected to be greater than the critical power for self-focusing of the nonlinear medium.

[0008] To date, when pulse energies exceed 10 mJ, a multipath configuration with an optimal geometry is typically used, along with a mirror with the highest damage threshold and helium or argon (less than 100 mJ) as a nonlinear medium. To scale the pulse energy, the cell length is selected accordingly, for example, as described in Non-Patent Document 2.

[0009] Various modifications have been proposed to shorten the external length of optimized heliot cells with circulating ground-mode light. For example, Non-Patent Document 3 describes a modification that uses a configuration with convex and concave mirrors instead of a gas-filled heliot cell with concave mirrors. This eliminates beam waist within the cell and allows for a larger beam cross-section across the entire range than in the case of a concave cell. Non-Patent Document 4 proposes using a folded bowtie-type ring resonator instead of a heliot cell with two mirrors, which makes it possible to significantly increase the beam diameter on all mirrors with the same external length. However, in this case, the beam waist between the two concave mirrors becomes significantly smaller, causing the gas, which acts as a nonlinear medium, to ionize. Therefore, the cell must be evacuated and a dielectric material inserted as a nonlinear medium at the position where the beam cross-section is large.

[0010] However, in any of these modifications, the spectral broadening per orbit decreases, and as a result, the spectral uniformity for each multipath construct deteriorates. This is because, in order to achieve uniform broadening, the nonlinear phase per Guy phase must not become too large. When propagating between two mirrors, a collimated beam can only collect a small amount of nonlinear phase compared to a focused beam that passes through nearly perfect caustics (Kaustik) and has a Guy phase of π.

[0011] Further possibilities for more compact multipath configurations at higher pulse energies are described in Non-Patent Document 5. Here, instead of fitting the geometry of the multipath configuration, the Gaussian fundamental mode of the incident laser beam is converted to the Laguerre-Gauss mode LG by the corresponding phase mask. 01This conversion to a donut mode results in the same geometry as the multipath configuration, but the beam cross-section is expanded both on the mirror and at the beam waist, allowing for the coupling of larger pulse energies. However, this conversion to and from the donut mode results in losses due to discontinuities in the center of the donut mode caused by the swirling phase front, and also introduces chromatic aberration in the phase plate used. [Prior art documents] [Patent Documents]

[0012] [Patent Document 1] German patent no. 102014007159

[0013] [Non-Patent Document 1] J. Schulte et al., "Nonlinear plulse compression in a multi-pass cell", Opt. Lett. 41, 4511-4514 (2016) [Non-Patent Document 2] A.-L. Viotti et al., "Multi-pass cells for post-compression of ultrashort laser pulses", Optica 9(2), 197-216 (2022) [Non-Patent Document 3] 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) [Non-Patent Document 4] CM Heyl et al., "High-energy bow tie multi-pass cells for nonlinear spectral broadening applications", J. Phys. Photonics 4, 014002 (2022) [Non-Patent Document 5] M. Kaumanns et al., "Spectral broadening of 112 mJ, 1.3 ps pulses at 5 kHz in a LG10 multipass cell with compressibility to 37 fs," Opt. Lett. 46, 929-932 (2021) [Overview of the project] [Problems that the invention aims to solve]

[0014] The object of the present invention is to provide a method for nonlinear spectral broadening of a pulsed laser beam using a periodic multipath structure, and an apparatus configured to carry out this method, thereby enabling a more compact configuration of the multipath structure even at higher pulse energies, without the aforementioned drawbacks. [Means for solving the problem]

[0015] The above problems are solved by the method described in claim 1 and the apparatus described in claim 13. Advantageous embodiments of the method and apparatus are the subject of the dependent claims or will become apparent from the following description and examples.

[0016] In the proposed method for nonlinear spectral broadening of a pulsed laser beam, a periodic multipath structure is used, the multipath structure comprising multiple propagation sections, each of which contains one or more media or elements (hereinafter also referred to as nonlinear media) having nonlinear optical properties suitable for generating self-phase modulation of the laser beam incident on the multipath structure, as well as optical elements in the form of lenses or curved mirrors. The laser beam preferably freely propagates between the optical elements in the propagation sections. Here, free propagation means that the propagation of the laser beam is not affected by an aperture, that is, its lateral spread, beam distribution and beam center position are not defined or limited, and it is not guided like a waveguide. The proposed method is characterized by converting the lateral intensity distribution of the laser beam into elliptic quasi-eigenmodes of the multipath structure before incidence onto the multipath structure, and then injecting the multipath structure with that intensity distribution. Here, an elliptic quasi-eigenmode has a transverse intensity distribution with two semi-principal axes, where the ratio of the lengths of the semi-principal axes or the ellipticity ε is constant or at least approximately constant during the propagation of the laser beam within the multipath structure, the lengths of the semi-principal axes differ by at least a factor of two, and the semi-principal axes rotate around the propagation direction during the propagation of the laser beam, i.e., they have orbital angular momentum. If the ellipticity is not perfectly constant, astigmatism of the multipath structure may occur, for example, due to a slight tilt of the curved optical element.

[0017] In a particularly advantageous form of the periodic optical structure, there are further multiple consecutive unit cells having the same optical action, each consisting of one or more propagation intervals, and the transverse intensity distribution and phase front of the elliptic quasi-eigenmode are reproduced after passing through each unit cell, except for the orientation of the principal axis.

[0018] In stable resonators, there generally exists an eigensolution or set of eigenmodes for electromagnetic waves, such as Hermitian-Gaussian modes, Laguerre-Gaussian modes, or Ince-Gaussian modes. Depending on the strong constraints imposed by the aperture, optical elements, or other boundary conditions within the resonator, one of these modes is particularly stable to disturbances and is the ground mode. In periodic multipath constructs, as in resonators, an eigensolution can be defined, and especially in the case of consecutive identical unit cells, the eigensolution is accurately reproduced after passing through each unit cell. In multipath constructs with nonlinear media, neglecting time and frequency dependence, there exists an intensity-dependent and small disturbance-stable eigensolution for a circulating monochromatic field distribution. In the case of small nonlinearity, for example, the fundamental Gaussian mode is a very good approximation of the stable eigensolution of a multipath construct with sufficiently large optical elements and no additional apertures. For spectral broadening, the lateral eigenmodes of a multipath construct slightly modified by a Kerr lens in a nonlinear medium are generally used. Ideally, the multipath construct is selected such that the distance between these eigenmodes and the ionization threshold in the nonlinear medium and the laser-induced damage threshold (LIDT) of the optical element are approximately the same.

[0019] Laguerre-Gauss Mode LG 01 The donut mode (also known as the vortex mode) is an example of a higher-order eigenvalue, and the same principles described above apply to it. The vortex mode with a helical phase is particularly characterized by the fact that its shape is invariant under the first approximation to the lateral Kerr effect, similar to the fundamental Gaussian mode. However, the phase fronts of higher-order modes have a phase jump of π, like the Hermitian-Gaussian mode or the Laguerre-Gaussian mode, and / or a helical phase front (vortex) with a singularity, like the donut mode. This is technically important because modes with vortices or phase jumps have field distributions with discontinuities or zeros, and it is technically impossible or extremely difficult to convert them to Gaussian modes without loss.

[0020] Below, we consider only modes that can be converted to modes without vortices and phase jumps using only spherical optical elements and / or cylindrical optical elements and / or mirrors. It is advantageous to convert the laser radiation to be spectrally broadened into the eigensolution of a (nonlinear) multipath construct using a suitable optical system or telescope, and to propagate that eigensolution through the multipath construct. This is because the beam profile and beam diameter are almost reproduced with each orbit, thereby maintaining a constant distance from the damage threshold and ionization threshold. Furthermore, propagation is particularly stable to small phase and amplitude disturbances (even when nonlinear elements or media are involved). Although a multipath construct (with or without nonlinearity) is not a resonator in the strict sense, restricting it to the eigensolution as a condition for stable propagation (to small disturbances) is also useful here. In a typical multipath construct with a nonlinear medium, stable propagation of radiation is determined not by the limiting aperture in the first approximation, but by the interaction with the nonlinear medium, i.e., the phase and amplitude modulation imparted by it. For stable propagation, in a multipath configuration with a nonlinear medium, the field distribution does not need to be mathematically and periodically reproduced with respect to amplitude and / or phase. It is sufficient that the nonlinear interaction, and consequently the beam cross-section within the periodic configuration and caustics, is kept approximately constant. Such solutions are referred to as quasi-eigensolutions or quasi-eigenmodes in this patent application.

[0021] A transverse beam distribution described by two principal axes with two different beam diameters or Rayleigh lengths (or confocal parameters) becomes a quasi-eigensolution or quasi-eigenmode if the two beam diameters and their ratio are at least substantially reproduced each time the beam passes through the propagation section of the multipath structure. If the multipath structure has multiple consecutive unit cells with identical optical properties, the two beam diameters of the quasi-eigensolution or quasi-eigenmode are reproduced each time the beam passes through the unit cell. For simple astigmatic beams, such reproduction is basically impossible in the case of a rotationally symmetric multipath structure. However, for beams with typical astigmatism and favorable orbital angular momentum, such quasi-eigensolutions exist because the orientation of the principal axes changes with propagation.

[0022] The quasi-eigenmodes or quasi-eigenmodes of a multipath configuration having an elliptical beam profile (elliptical quasi-eigenmode) used in the proposed method have five important advantages with respect to the design of multipath configurations for nonlinear spectral widening or pulse compression. • The mode volume and mode cross-section increase compared to the fundamental Gaussian mode, depending on the ellipticity ε' of the transverse beam distribution or beam profile. This reduces the intensity on the optical element and in the nonlinear medium or within the nonlinear element. Conversely, the external length of the unit cell or corresponding propagation section required for a given pulse energy can be shortened if the intensity is constant. • Elliptic quasi-eigenmodes can be converted to or from a circular stigmatic beam without orbital angular momentum by only two cylindrical optical elements. In many cases, especially when using a heliot cell as a multipath configuration, the required beam can be arranged particularly compactly on the mirrors of the multipath configuration. When the beam cross-section is defined by the LIDT, the optical elements of the multipath configuration can be utilized particularly well and therefore selected to be compact. The beam profiles of quasi-eigenmodes, particularly their ellipticity and orbital angular momentum, are, in the first approximation, preserved when propagating through a second-order nonlinear medium. · The above characteristics apply not only to the fundamental quasi-eigenmode, that is, the general astigmatic Gaussian mode with a suitable orbital angular momentum, but also to higher-order elliptical Hermite-Gaussian modes. Therefore, the advantages described herein can be combined with the potential benefits of higher-order transverse mode excitation.

[0023] Therefore, in the proposed method, the transverse intensity distribution of the incident laser beam is converted into the elliptical quasi-eigenmode of the multi-pass arrangement having the above features before incidence into the multi-pass arrangement, and the beam is incident into the multi-pass arrangement with that intensity distribution. When a general astigmatic Gaussian beam (in accordance with ISO / DIS 11146-2 (Part 2)) having orbital angular momentum is used as the lowest-order elliptical quasi-eigenmode, this can be mathematically described as follows.

[0024] The starting point in this mathematical description is a Rayleigh length z r and a stigmatic beam whose propagation is described by a caustic having a waist position z0. In a first step, this beam is converted into a beam with simple astigmatism by beam shaping symmetric with respect to two orthogonal axes, for example by an arrangement of cylindrical optical elements oriented on two orthogonal principal axes, so as to satisfy the following conditions. z 01 =z 02 ;z r1 / z r2 >1 / (√2-1)∨z r1 / z r2 <(√2-1)

[0025] The beam propagation of this elliptical, simple astigmatic beam is along two orthogonal planes of symmetry, described by two caustics having a Rayleigh length z r1 or z r2 and a waist position z 01 or z 02 , as well as a waist diameter w 01 or w 02 From here, the Rayleigh length z rand has a waist position z0, and its intensity distribution is elliptical at all positions during beam propagation, with two beam radii w a (z) and w b A mode can be constructed such that (z) follows caustics. In this case, the principal axis of the ellipse rotates around the beam axis as it propagates. For this reason, in the second step, this beam is described mathematically as z 01 =z 02 The transformation is performed by two intersecting cylindrical lenses rotated 45° with respect to the incident axis, where the focal lengths f1 and f2 are as follows: z 01 =z 02 At the position =z0=0, 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 = -f1 Here, 2 / z r = 1 / z r1 +1 / z r2

[0026] For elliptic quasi-eimodes with ellipticities ε' and ε behind a cylindrical lens, the following applies: √(z r1 / z r2 )=ε' w a (z) / w b (z=w 01 / w 02 =ε

[0027] The propagation of elliptic quasi-eigenmodes occurs at the beam radius w a and w b In addition, the orientation angle φ of the principal axis depends on the position z. ω It is described by. w a 2 =w 01 2 ·(1+(z / zr ) 2 );w b 2 =w 02 2 ·(1+(z / z r ) 2 ) φ ω =arctan(z / z r )

[0028] This propagation behavior, i.e., the elliptic eigenmode, is of Rayleigh length z. R And what is described by the ellipticity ε or ε' is retained even after any stigmatic beam transformation.

[0029] In the case of a heliot cell or other periodic multipath configuration, the following conditions are obtained for incidence at the center between two mirrors with radius R and spacing L, i.e., gui parameter Ψ(psi) (see, for example, Patent Document 1), and for focal lengths f1 and f2 of intersecting cylindrical lenses rotated 45° at the incidence position in the center of the cell, in order to excite an elliptic quasi-eigenmode in which the beam cross section is the same size at any distance from the center of the cell in all orbits within the cell: z r =L / 2·√(2·R / L-1), where L=R·(1-cos(θ / 2)) f1=z r / (z r / z r1 -1); f2=z r / (z r / z r2 -1) z 01 =z 02 =0; f1=-f2 z r1 / z r =a 2 (1 + √(1 - 1 / a) 2 ));z r2 / z r =a 2 ·(1-√(1-1 / a 2 ))

[0030] Here, a = √(zr1 / z r ·z r2 / z r ) is the effective magnification of the beam cross-section by the elliptic quasi-eigenmodes with orbital angular momentum compared to the ordinary stigmatic eigenmodes without orbital angular momentum, and Ψ is the gui parameter of the multipath construct. Naturally, the laser beam can also be incident on the multipath construct at other positions. For the beam transformation required for this, corresponding conditions regarding the lenses or mirrors used and their spacing can be formulated, equivalent to the mathematically concise expression shown here. Here, the above description should be understood as merely an example and is extendable to any multipath construct. These are simply the variable z r It is sufficient to have a specific eigensolution or eigenmode described by z0. Therefore, z r The above equation relating to is merely an illustrative example of such an eigenvalue for a heliot cell.

[0031] In the representation of Gaussian beams with general astigmatism by Arnaud and Kogelnik (JA Arnaud, H. Kogelnik, "Gaussian light beams with general astigmatism," Applied Optics 8 (8), 1687-1693 (1969)), for the sake of completeness, the following representation of elliptic quasi-eigenmodes is obtained. z1 / z r = -√(δ·√(1-1 / a) 2 )); z2 = -z1 p1 / z r = (1 + δ·√(1 - 1 / a) 2 ));p2=p1 tanh(2·α) = 1 - δ / 2·√(a 2 / (a 2 -1)) This holds true in the limit as δ→0.

[0032] In the proposed method, the laser beam preferably has a transverse intensity distribution of any Gaussian beam, particularly a stigmatic collimated Gaussian beam, before conversion. In a preferred embodiment, the incident laser beam to be spectrally broadened is converted by a (first) optical device into an elliptic quasi-eigenmode having ellipticity ε, the optical device comprising two cylindrical optical elements having cylindrical axes twisted relative to each other in a plane perpendicular to the propagation direction. The cylindrical optical elements can be, for example, two cylindrical lenses, two cylindrical mirrors, or a combination of one cylindrical lens and one cylindrical mirror. The (first) optical device is also configured such that the cylindrical optical elements form a substantially cylindrical telescope with a magnification of ε or 1 / ε. The angle at which the two cylindrical axes are twisted relative to each other is selected so that the laser beam is converted into a desired elliptic quasi-eigenmode having orbital angular momentum. Mode-matching is performed by a further rotationally symmetric optical element of a suitably selected configuration so that the magnitude and divergence angle of the incident beam are adapted to the multipath configuration. The multipath structure itself is constructed rotationally symmetrically; that is, it is a stigmatic multipath structure. In an advantageous form, the nonlinear medium within the multipath structure is a gas. Therefore, for example, the entire multipath structure may be surrounded by a housing filled with this gas. In another form in which the multipath structure comprises a nonlinear element, this element is, for example, a dielectric plate or a dielectric lens.

[0033] The multipath configuration preferably comprises a total of N > 5 propagation segments, and particularly preferably 10 or more propagation segments. This corresponds to the number of passes in the case of a heliot cell used as a multipath configuration. In the case of a heliot cell, the incident and exit surfaces defining the propagation segments or unit cells correspond to their respective mirror segments. The propagation segment or unit cell includes the propagation path and nonlinear medium between the mirrors, and the reflection on the curved surface of one of the two mirrors of the heliot cell.

[0034] For pulse compression, the lateral intensity distribution of the spectrally widened laser beam in the multipath configuration should be inversely converted back into a collimated stigmatic beam after emission from this configuration. For this purpose, it is preferable to use a second optical device having the same optical elements as the first optical device. The laser beam emitted from the multipath configuration is then preferably shortened in a suitable compression device for removing chirp in the pulse width.

[0035] The proposed apparatus for carrying out this method comprises, accordingly, a multipath configuration as already described in relation to the above method, and a (first) optical device positioned before it. This optical device converts the pulsed laser beam to be spectrally broadened or its lateral intensity distribution into elliptic quasi-eigenmodes of the multipath configuration and injects the multipath configuration with this intensity distribution. Preferably, the first optical device comprises two cylindrical optical elements and a further rotationally symmetric optical element for mode matching, as already described in relation to the above method. Furthermore, it is preferable to provide a second optical device that inversely converts the lateral intensity distribution of the laser beam emitted from the multipath configuration back into the original lateral intensity distribution. The multipath configuration and the first optical device are also configured in an advantageous form to enable the carrying out of different forms of the method.

[0036] By converting the incident beam into elliptic quasi-eigenmodes of the multipath structure, the mode volume or beam cross-section increases at every position within the multipath structure compared to the fundamental eigenmode, due to the ellipticity of the quasi-eigenmode. This allows the pulse peak power within the multipath structure to remain constant, even at higher pulse peak power or pulse energy, proportional to the increase in the beam cross-section. Generally, the pulse energy that can be spectrally broadened and then compressed in a nonlinear multipath structure is limited by the ionization of the nonlinear medium or the damage threshold of the optical elements of the multipath structure. Therefore, the pulse energy, upon conversion to elliptic quasi-eigenmodes, is given by a coefficient a = 1 / 2·(ε') depending on the ellipticity ε'.2 (+1) / ε' can be increased. The conversion to and inverse conversion from the elliptical quasi-natural mode can be easily realized without loss by stigmatic beam conversion and a cylindrical telescope having cylindrical axes arranged twisted relative to each other, or another suitable combination of spherical, elliptical, or cylindrical optical elements. Furthermore, combinations with other methods for enlarging the beam cross-section are also possible, and in particular the use of higher-order modes is also possible.

[0037] Hereinafter, the proposed method and the associated apparatus will be further described based on embodiments in conjunction with the drawings. Brief Description of the Drawings

[0038] [Figure 1] It is a schematic diagram showing a Herriott cell as a multi-pass structure and an optical device for incidence and emission of a laser beam according to the prior art. [Figure 2] Similar to the example of Figure 1, it is a diagram showing an example of the proposed apparatus using a Herriott cell as a multi-pass structure. [Figure 3] It is a diagram showing the rotation of the elliptical beam profile near the beam waist in the apparatus of Figure 2. Mode for Carrying Out the Invention

[0039] In a proposed method for spectrally broadening a laser pulse for nonlinear pulse compression, the laser pulse is incident on a stigmatic multipath structure and propagated through at least one medium having nonlinear optical properties. Within this medium, the laser pulse acquires a nonlinear phase by self-phase modulation. The beam conversion according to the present invention, which transforms the incident beam into an elliptic quasi-eigenmode of the multipath structure, allows the beam cross-section within the multipath structure to be expanded by a coefficient a over its entire length. This allows the pulse energy to be compressed to be increased by a coefficient a without changing the geometry and external length of the multipath structure. This is particularly important when the required external size or length of the multipath structure is very large (far exceeding 1 m), and consequently, when the pulse energy is large (far exceeding 10 mJ), the external size limits the maximum pulse energy.

[0040] The multipath construct used in the proposed method and apparatus consists of a plurality of propagation sections or unit cells comprising a nonlinear medium or nonlinear element. The method and apparatus will be described below based on a configuration comprising a plurality of consecutive unit cells comprising one or more propagation sections. However, these descriptions also apply to configurations having propagation sections whose optical effects are not entirely identical. The number of propagation sections is at least N > 5, particularly preferably 10 or more. Each propagation section comprises at least one lens or curved mirror or mirror portion, at least one nonlinear element or nonlinear medium, and preferably a free propagation region. Furthermore, one or more propagation sections form a unit cell, and a plurality of these are arranged consecutively. The unit cell is defined by an incident surface and an exit surface in the beam propagation direction, and the incident surface does not image onto the exit surface. Here, the unit cell is the smallest identical unit of the multipath construct. Its optical effect is invariant with respect to rotation of the unit cell about the propagation axis or optical axis.

[0041] An elliptic quasi-eigenmode is a beam distribution within a multipath structure 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 are not simultaneously reproduced after passing through the unit cell. Nor are they reproduced even when the size and divergence angle of the incident beam are matched (mode matching). The transverse intensity distribution of the quasi-eigenmode is characterized by two principal axes and two beam diameters (or principal axis lengths) at each position in the multipath structure. The beam diameters of the two principal axes differ by at least a factor of two. The transverse intensity distribution of the quasi-eigenmode has the same shape at each position of propagation within the multipath structure, except for the size and orientation of the principal axes. The intensity distribution and phase front of the elliptic quasi-eigenmode are reproduced after passing through the unit cell, except for the orientation of the principal axes. Here, elliptic quasi-eigenmodes have orbital angular momentum but do not necessarily have phase singularities and helical phase fronts or vortices or topological charges (see J. Fisser et al., "Orbital angular momentum of general astigmatic modes", Phys. Rev. A 70, 013809 1-11 (2004)). The beam distribution or its principal axis rotates continuously (proportional to the gui phase) during free propagation within the multipath structure, except that the intensity distribution is constant except for the orientation and magnitude of the principal axis. The beam distribution or its principal axis rotates by a fixed amount each time it passes through a unit cell. Position-dependent phase and amplitude fluctuations caused by the nonlinear medium or nonlinear element are similar with respect to the local principal axis of the beam for all passes.

[0042] The following figure illustrates the transition from spherical eigenmodes to elliptic quasi-eigenmodes, according to a proposed method for spectral broadening or pulse compression in a nonlinear multipath configuration, based on a heliot cell having two concave mirrors of the same curvature. The required nonlinear medium is not shown but can be a gas, dielectric, or plasma. Figure 1 shows a heliot cell for this purpose, consisting of two concave mirrors 1. The heliot cell is configured such that the incident laser beam 3 makes 25 orbits, i.e., it has 50 propagation segments and 50 unit cells. In this example, the two concave mirrors 1 have a radius of curvature of 1500 mm and are spaced 2988 mm apart (gui parameter 24 / 25π). The laser beam 3 is incident on and out of the heliot cell via rectangular spherical mirrors 2a and 2b, each with a radius of curvature of 2850 mm, which are also used for mode matching. Here, the laser beam 3 is incident on the heliot cell via a spherical Galilean telescope with a magnification of 0.2, consisting of two spherical telescopic mirrors 4 and 5. Beam shaping after exit is performed via a further telescope having spherical telescopic mirrors 6 and 7, which are configured identically to the telescopic mirrors 4 and 5 of the spherical telescope for the incident beam. In this example, the beam diameter of the collimated Gaussian beam incident as laser beam 3 is 25 mm at a wavelength of 1030 nm, the beam diameter at the center of the heliot cell is 0.35 mm, and on the large concave cell mirror it is 5.6 mm.

[0043] In the proposed method and apparatus, a heliot cell with a similar configuration consisting of a concave mirror 1, an incident mirror 2a, and an exit mirror 2b can be used. However, instead of a spherical telescope, a cylindrical telescope with two rectangular cylindrical mirrors 8 and 9 for incidence and beam conversion is used. The magnification of this cylindrical Galilean telescope is 0.16 in this example. The two rectangular cylindrical mirrors 8 and 9 are positioned slightly (1.5°) twisted relative to each other to satisfy the aforementioned conditions for generating elliptic quasi-eigenmodes of the multipath configuration. The beam diameter of the collimated Gaussian beam incident as laser beam 3 is again 25 mm at a wavelength of 1030 nm. The beam diameter at the center of the heliot cell is 0.25 mm × 1.57 mm, and on the concave cell mirror it is 4 mm × 25 mm. Compared to the conventional method shown in Figure 1, in the proposed method, the beam cross-section at each propagation position within the cell is magnified 3.2 times in this example.

[0044] After exiting the heliot cell, the spectrally widened beam 3 is inversely transformed back to its original lateral intensity distribution by a cylindrical telescope having rectangular cylindrical mirrors 10, 11 configured similarly to the cylindrical mirrors 8, 9 of the cylindrical telescope for the incident beam. Finally, Figure 3 illustrates the rotation of the elliptical beam profile 12 around the propagation axis near the beam waist, i.e., the waist of the beam bundle, within the unit cell.

[0045] In contrast to all known devices that perform pulse compression using a multipath configuration equivalent to that shown in Figure 1, the proposed method and apparatus excite elliptic modes with orbital angular momentum as quasi-eigenmodes within a rotationally symmetric, particularly spherical, multipath configuration, as illustrated in Figure 2. These modes are preferably converted from a stigmatic collimated incident beam to a stigmatic collimated exit beam by a Galilean cylindrical telescope having cylindrical mirrors (or lenses) arranged twisted relative to each other and further concave spherical mirrors. In the illustrated example, the ellipticity ε = 6.25 expands the beam cross-section of the Gaussian modes by a coefficient a = 3.2 at each position within the multipath configuration. The ellipticity was selected in this example to optimally utilize the mirror surfaces with the gui parameter Ψ = 24 / 25·π of the selected multipath configuration, except for any other arbitrary ellipticity ε or ε' = √(z r1 / z r2 ) = a + √(a 2 -1) and the effective area expansion ratio a = 1 / 2·(ε' 2 The same applies to (+1) / ε'. If the dimensions of the multipath structure are not changed and the nonlinear medium is the same, using elliptic modes allows the same spectral broadening to be obtained even when the pulse energy is increased by a coefficient a, while maintaining the same distance from the mirror damage threshold and ionization threshold. The latter is only possible by extending the cell length by a coefficient a when elliptic modes are not used.

[0046] The illustrated example of the heliot cell can be similarly applied to any other periodic multipath construct having stigmatic or spherical periodic eigensolutions. By suitable beam transformation, particularly by using a simple cylindrical telescope with cylindrical axes arranged twisted to one another, any Gaussian incident beam, especially a collimated stigmatic circular incident beam, can be transformed to ellipticity ε=ε'=a+√(a 2 -1) and can be converted to an eigenmode having an effective area expansion factor a, and can also be converted inversely. This elliptic quasi-eigenmode has the same Rayleigh length z as the fundamental eigenmode of the multipath construct.r and its principal axis rotates by an angle φ as it propagates ω . When a laser pulse propagates through a multi-pass arrangement that includes a nonlinear medium for spectral broadening, this apparatus (provided a sufficiently large free aperture is assumed) converts the laser beam into an elliptical quasi-eigenmode, and even after increasing the pulse energy by a factor a, with respect to the action of the nonlinear medium or nonlinear element, it behaves in the same manner as when using a stigmatic fundamental eigenmode without undergoing conversion. This applies in particular to spectral broadening as well.

[0047] The elliptical quasi-eigenmode in the form of a general astigmatic Gaussian beam of a multi-pass arrangement can be completely described by the ellipticity and the Rayleigh length. Furthermore, there are additional elliptical quasi-eigenmodes, for example helical elliptical Hermite-Gaussian modes, which are characterized not only by ellipticity and Rayleigh length, but also by an order n. These elliptical modes have a transverse mode distribution superimposed on the Gaussian mode, and this transverse mode distribution does not change during (free) propagation except for the size and orientation of the principal axis. As in the case of the Hermite-Gaussian or Laguerre-Gaussian eigenmodes of a resonator or multi-pass arrangement, the order n of the helical elliptical Hermite-Gaussian mode causes the mode cross-section to be enlarged by a constant factor at all positions, and the transverse mode distribution exhibits n minima. However, if nonlinearity is neglected, the propagation characteristics of the nth-order helical elliptical Hermite-Gaussian mode correspond to the aforementioned fundamental elliptical quasi-eigenmode. Therefore, a high mode order and conversion to an elliptical quasi-eigenmode are independently combinable means for increasing the mode cross-section, and thus the possible pulse energy for spectral broadening in a nonlinear multi-pass arrangement, without changing the outer dimensions (the propagation length of one round trip within the unit cell). Description of Reference Signs

[0048] 1 Concave cell mirror 2a Concave entrance mirror 2b Concave exit 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 Profiles

Claims

1. A method for performing nonlinear spectral broadening of a pulsed laser beam in a periodic multipath configuration, The multipath structure is formed from a plurality of propagation sections, each of which has one or more media or elements having nonlinear characteristics suitable for generating self-phase modulation of the laser beam (3) incident on the multipath structure, and also has at least one further optical element in the form of a curved mirror or lens. The lateral intensity distribution of the laser beam (3) is converted into an elliptic quasi-eigenmode of the multipath structure before it is incident on the multipath structure, and the laser beam is then incident on the multipath structure with this intensity distribution. A method characterized in that the elliptic quasi-eigenmode has a transverse intensity distribution having two principal axes, the ratio of the lengths of the principal axes or the ellipticity ε is constant or at least substantially constant, the lengths of the principal axes differ by at least twofold, and the principal axes rotate around the propagation direction during the propagation of the laser beam (3).

2. The method according to claim 1, wherein the periodic multipath construct includes a plurality of consecutive unit cells having the same optical action, each consisting of one or more propagation intervals, and the lateral intensity distribution and phase front of the elliptic quasi-eigenmode are reproduced after passing through each unit cell, except for the orientation of the principal axis.

3. The method according to claim 1 or 2, wherein the lateral intensity distribution of the laser beam (3) is converted into the elliptic quasi-eigenmodes of the multipath structure using a first optical device before incidence onto the multipath structure, and the first optical device comprises two cylindrical optical elements (8, 9) that are twisted relative to each other in a plane perpendicular to the propagation direction of the laser beam (3) and have cylindrical axes that are not oriented parallel or perpendicular to each other, and optionally one or more further optical elements (2a).

4. The method according to any one of claims 1 to 3, characterized in that the lateral intensity distribution of the laser beam (3) includes, before the conversion, the intensity distribution of an arbitrary Gaussian beam, in particular a stigmatic Gaussian beam.

5. The method according to any one of claims 1 to 4, characterized in that the elliptic quasi-eigenmode of the multipath construct, on which the lateral intensity distribution of the laser beam (3) is transformed, has the shape of a typical astigmatic Gaussian beam.

6. The method according to any one of claims 1 to 4, characterized in that the elliptic quasi-eigenmode of the multipath construct, on which the lateral intensity distribution of the laser beam (3) is transformed, has the shape of a helical elliptic Hermitian-Gaussian mode.

7. The method according to any one of claims 1 to 6, characterized in that the transformation of the lateral intensity distribution of the laser beam (3) is reversed by an inverse transformation by a second optical device (10, 11, 2b) after emission from the multipath structure.

8. The method according to any one of claims 1 to 7, characterized in that a heliot cell is used as a multipath construct.

9. The method according to any one of claims 1 to 8, characterized in that the medium having nonlinear characteristics suitable for generating the self-phase modulation is formed by a gas.

10. The method according to any one of claims 1 to 8, characterized in that the element having nonlinear characteristics suitable for generating the self-phase modulation is formed by a dielectric plate or a lens.

11. The method according to any one of claims 1 to 10, characterized in that the multipath structure is configured to have more than five propagation intervals.

12. The method according to any one of claims 1 to 11, characterized in that the multipath structure is configured such that, at a given pulse energy of the laser beam (3), the damage threshold of the optical elements of the multipath structure and the ionization of the medium having nonlinear characteristics are not reached.

13. An apparatus for performing nonlinear spectral widening of a pulsed laser beam, the apparatus comprising a periodic multipath configuration having a first optical device (8, 9, 2a) positioned in front of the multipath configuration, the multipath configuration comprising a plurality of propagation sections, each of which has one or more media or elements having nonlinear characteristics suitable for generating self-phase modulation of the laser beam (3) incident on the multipath configuration, and also comprising at least one further optical element in the form of a curved mirror or lens, The first optical device (8, 9, 2a) is configured to convert the lateral intensity distribution of the laser beam (3) into elliptic quasi-eigenmodes of the multipath structure before it is incident on the multipath structure. The apparatus wherein the elliptic quasi-eigenmode has a transverse intensity distribution with two principal axes, the ratio of the lengths of the principal axes or the ellipticity ε is constant or at least approximately constant, the lengths of the principal axes differ by at least twice, and the principal axes rotate around the propagation direction during the propagation of the laser beam (3).

14. The apparatus according to claim 13, wherein the periodic multipath construct includes a plurality of consecutive unit cells having the same optical action, each consisting of one or more propagation intervals, and the transverse intensity distribution and phase front of the elliptic quasi-eigenmode are reproduced after passing through each unit cell, except for the orientation of the principal axis.

15. The apparatus according to claim 13 or 14, characterized in that the first optical apparatus (8, 9, 2a) comprises two cylindrical optical elements (8, 9) that are twisted relative to each other in a plane perpendicular to the propagation direction of the laser beam (3) and have cylindrical axes that are neither parallel nor perpendicular to each other, and one or more additional optical elements (2a) of any choice.

16. The apparatus according to any one of claims 13 to 15, characterized in that the elliptic quasi-eigenmode of the multipath configuration, which is converted by the first optical device (8, 9, 2a) to the lateral intensity distribution of the laser beam (3), has the shape of a typical astigmatic Gaussian beam.

17. The apparatus according to any one of claims 13 to 15, characterized in that the elliptic quasi-eigenmode of the multipath configuration, which is converted by the first optical device (8, 9, 2a) to the lateral intensity distribution of the laser beam (3), has the shape of a helical elliptic Hermitian-Gaussian mode.

18. The apparatus according to any one of claims 13 to 17, wherein the apparatus comprises a second optical device (10, 11, 2b), and the second optical device (10, 11, 2b) is configured and arranged to reverse the conversion of the lateral intensity distribution of the laser beam (3) by the first optical device (8, 9, 2a) after the laser beam (3) has been emitted from the multipath structure.

19. The apparatus according to any one of claims 13 to 18, characterized in that the multipath component is a heliot cell.

20. The apparatus according to any one of claims 13 to 19, characterized in that the medium having nonlinear characteristics suitable for generating the self-phase modulation is a gas.

21. The apparatus according to any one of claims 13 to 19, characterized in that the element having nonlinear characteristics suitable for generating the self-phase modulation is a dielectric plate or a lens.

22. The method according to any one of claims 13 to 21, characterized in that the multipath configuration comprises more than five propagation intervals.

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