Spectral caustics in attosecond science

Many intriguing phenomena in nature—from phase transitions to black holes—occur at singularities. A unique type of singularity common in wave phenomena, known as caustics1,2, links processes observed in many different branches of physics3,4. Here, we investigate the role of caustics in attosecond science and in particular the physical process behind high harmonic generation5. We experimentally demonstrate spectral focusing in high harmonic generation, showing a robust intensity enhancement of an order of magnitude over a spectral width of several harmonics. This new level of control holds promises in both scientific and technological aspects of attosecond science6,7. Moreover, our study provides a deeper insight into the basic mechanism underlying the high harmonic generation process, revealing its quantum nature8 and universal properties. By applying catastrophe theory to high-harmonic generation, researchers identify caustics relating to regions of spectral focusing and greatly enhanced field intensity.

monic generation 5 . We experimentally demonstrate spectral focusing in high harmonic generation, showing a robust intensity enhancement of an order of magnitude over a spectral width of several harmonics. This new level of control holds promises in both scientific and technological aspects of attosecond science 6,7 . Moreover, our study provides a deeper insight into the basic mechanism underlying the high harmonic generation process, revealing its quantum nature 8 and universal properties.
Multiple rays of light can coalesce and form bright focusing features known as caustics. These features are singularities of ray optics in the sense that their intensity, according to the ray approximation, should diverge. Optical caustics are abundant: from the 'cusp' feature inside a coffee cup to the bright spots of focused light on the bottom of a swimming pool 1 . Like many other types of singularities, caustics have universal properties. Their geometry, width, enhancement and diffraction patterns can be described in terms of universal classes, according to the dimensionality of the system. Although originating from optics, caustics can be found in many other fields. Gravitational microlensing enhancements, which are used to detect extrasolar planets 4 , electron flow focusing in 2D electron gases 3 and propagation of radiofrequencies 2 are all described using the language of caustics. Mathematically, caustics are thoroughly described in terms of catastrophe theory, where each caustic can be associated with a catastrophe class 1,2 that dictates its properties.
Here, we show that caustics play a major role in the process of high harmonic generation (HHG) and the production of attosecond pulses. In the HHG process, the interaction of an intense laser field with matter leads to the generation of extreme ultraviolet (XUV) and electron pulses of attosecond duration, opening a broad range of applications from imaging 9 to attosecond time-resolved measurements 10 . Although significant progress has been achieved during the past decade, manipulating the HHG spectrum remains a major challenge. Caustics offer a deeper understanding of the basic mechanism behind HHG in a regime where the simple semiclassical model diverges. Such analysis provides us with a method to accurately 'engineer' the harmonic spectrum, leading to a tunable dramatic enhancement of a narrow spectral band. Experimentally, we demonstrate a unique caustic, known as a 'swallowtail', which is induced by a two-colour field. The observed enhancement can be predicted by catastrophe theory, and cannot be explained by the semiclassical model or by simple quantum calculations 11 . This approach can be integrated with previously demonstrated enhancement methods 6 and put towards the generation of highflux X-ray sources, which are attractive for biological and material processing applications.
A semiclassical picture of HHG depicts the interaction between the intense infrared field and the medium as a three-step process 5 . At first, the strong laser field alters the shape of the Coulomb potential, allowing an electron to tunnel out at an ionization time t ion . Next, the free electron propagates under the influence of the laser field and accumulates high kinetic energy E k (t ion ) before it re-collides with the parent ion. The re-collision and subsequent recombination of the electron with the parent ion leads to the emission of optical radiation in the XUV regime according to v(t ion ) ¼ (E k (t ion ) þ I p )h 21 , where I p is the ionization potential. In this simple classical model, the spectral density I(v) is given by: where for each frequency v the summation is performed over all t ion for which the electron returns with the relevant kinetic energy and E XUV (t ion ) represents the HHG electric field associated with an ionization time t ion . Such a summation is weighted by the Jacobian of the mapping between t ion and v, dt ion /dv. When the interaction is induced by a single colour, v(t ion ) has a single maximum, known as the cut-off energy. This point separates t ion (v) into two branches that span the same spectrum, commonly referred to as 'short' and 'long' trajectories 7 . At the cut-off energy, where the kinetic energy is maximal, dv/dt ion ¼ 0, the Jacobian is singular and I(v) diverges 5 . The quantum-mechanical description of HHG 8 resolves this singularity. In this picture, the process is evaluated by integrating over all possible ionization times: where E XUV (t ion , v) is the amplitude of the quantum trajectory associated with the ionization time t ion and S(t ion , v) is its relative phase. The classical picture can be shown to be a stationary phase approximation (SPA) of the quantum-mechanical picture 8 . For each v, this approximation takes into account only ionization times t st for which (∂S(t ion , v)/∂t ion )| t st = 0, giving rise to the mapping in the semiclassical model between ionization time and v. Such an approximation diverges when the phase is stationary to higher orders, (∂ 2 S(t ion , v)/∂t 2 ion )| t st = 0. At these energies the semiclassical model (equation (1)) diverges 5 , in analogy to the focusing singularities in ray optics.
The catastrophe description of caustics overcomes this singularity and provides an analytical solution near each singular point, while maintaining the simple classical description of the mechanism. In the language of catastrophe theory, the phase S is a generating function, and (∂S(t ion , v)/∂t ion )| t st = 0 defines a gradient map from t ion to v. The variable t ion is identified as a state variable and is being integrated over. Quantities on which the result of the integral explicitly depends, such as v, are termed control parameters.
Department of Complex Systems, Weizmann Institute of Science, 76100 Rehovot, Israel. *e-mail: oren.raz@weizmann.ac.il The singularity in the mapping is structurally stable if by smoothly perturbing the system the singularity shifts or scales while its structure remains unchanged. Examples of stable singularities are the caustic lines at the bottom of a swimming pool. In contrast, the focal point of a perfect lens is not structurally stable, and any small perturbation in the shape of the lens will change the structure at the focus. Here, we consider only structurally stable singularities because in most experiments exact control over the parameters is impractical. Catastrophes, which are structurally stable singularities of gradient mapping, are classified by Thom's theorem according to the number of state and control parameters 1 . Catastrophe theory provides regularization of the divergence near the caustic to a standard diffraction pattern. For one control parameter, as in the HHG spectrum near the cut-off, the only structurally stable catastrophe is the fold catastrophe, which exhibits a diffraction pattern of the form of an Airy function. Indeed, it was recently found 12 that the harmonics near the cut-off behave like an Airy function.
Given the classification of a caustic, catastrophe theory dictates its main characteristics. For example, the relative enhancement in the caustic zone is governed by a simple expression 2 : where I enhanced is the intensity at the caustic, I 0 is the intensity far from the caustic regime, d , 1 is the Arnold index, a number that is determined by the class of catastrophe, and N is a unitless parameter that scales the fast oscillating phase in the SPA. In the case of HHG, N is the harmonic number at the caustic.  (3), the enhancement at the caustic is polynomial in the harmonic number, but exponential in d. Thus, significant enhancement can be achieved by generating caustics with large d. According to catastrophe theory, d can be increased by introducing additional control variables that manipulate the HHG process. Increasing the dimensionality in HHG experiments is commonly achieved by changing the polarization of the fundamental field 14 , adding a second colour field 11,[15][16][17][18][19][20][21][22] or controlling the molecular alignment 23 .
We demonstrate a higher-order caustic in the HHG spectrum using a two-colour configuration by combining the fundamental field with its second harmonic. The total driving field, E tot (t), is given by where v 0 is the fundamental frequency, E IR is the amplitude of the fundamental field, E SH is the amplitude of the second harmonic field and f is their relative phase. This configuration adds two control variables: the field ratio R ¼ E SH /E IR and the two-colour phase f. It has already been noted 16,19 that when R is large enough, there are more than two trajectories that contribute to the HHG spectrum. Multiple trajectories enable higher-order caustics, if more than two trajectories coalesce at a given energy. Catastrophe theory shows that the only structurally stable caustic for the case of three control parameters (v, R, f) and one state variable (t ion ) is the swallowtail catastrophe 1 , where four electron trajectories coalesce. In addition, catastrophe theory directs us to the location in the multidimensional parameter space in which the swallowtail enhancement should appear: at R ¼ 0.44 and f ¼ 0.535 rad. Moreover, it predicts the expected enhancement by equation (3), with d ¼ 3/10 for the swallowtail class. Figure 1a shows the surfaces of the semiclassical model singularities that form the universal swallowtail structure 2 in the (v, R, f ) space. Each point on the surface represents a singularity of the semiclassical model, where at least two derivatives of S with respect to t ion vanish. Figure 1b-e describes returning energy as a function of ionization times according to the semiclassical model. The collection of singular points can be classified in three categories. The first includes a coalescence of two electron trajectories, where the first two derivatives of S vanish, forming a fold caustic given by the yellow and red surfaces as shown in Fig. 1a. These appear at maximal/minimal returning energies (yellow/red circles) in Fig. 1b and c, respectively. The second includes a coalescence of three electron trajectories, where the third derivative of S also vanishes, forming a cusp caustic (blue lines in Fig. 1a). This corresponds to an inflection point of the returning energy, as the blue circle in Fig. 1e. The third category includes a coalescence of four electron trajectories, where all four first derivatives of S vanish, forming a swallowtail caustic (point D in Fig. 1a and the green circle in Fig. 1d). The swallowtail, as with all other caustics, is associated with a standard diffraction pattern. We have measured slices of this diffraction pattern along surfaces of equal R (see Methods for details). Figure 2a,c,e shows the measured HHG spectral intensity (generated from neon atoms) as a function of v and f, for three different values of R: before the swallowtail point (Fig. 2a, R ≈ 0.3, corresponding to point B in Fig. 1a), approximately at the swallowtail point (Fig. 2c, R ≈ 0.45, corresponding to point D in Fig. 1a) and after it (Fig. 2e, R ≈ 0.75, corresponding to points E and C in Fig. 1a). Measurements for additional values of R can be found in the Supplementary Information. In Fig. 2b,d,f, the diffraction patterns of the swallowtail are calculated for the corresponding parameters (see Methods for details).
The geometric structure of the enhanced features, measured in our experiment, agrees with the swallowtail structure. In Fig. 2a, an enhancement by a factor of 3 is observed around the 45th harmonic. This enhancement is narrow along the spectral axis but changes gradually along the f axis. Its maximum is located at a value of f that minimizes the cut-off frequency. A similar response is shown in Fig. 2b. If R is increased, we observe an extremely sharp enhancement along both v and f at the 40th harmonic, as shown in Fig. 2c. The measured enhancement at the 40th harmonic is about an order of magnitude compared to the averaged intensity outside the caustic (low harmonics in Fig. 2a,e), in agreement with equation (3): in this case d ¼ 3/10 and N ¼ 40. This narrow feature is well described by the diffraction pattern in Fig. 2d, which shows a sharp peak at the same spectral and phase location. A further increase in R is described in Fig. 2e and f. In this case, the harmonic number associated with the maximum changes with the phase (Fig. 2e), and the curvature of the enhanced feature has changed its sign with respect to Fig. 2a. A similar response is described in Fig. 2f. Figure 2e demonstrates the spectral controllability induced by the caustic: by changing the relative phase we can scan the dominant harmonics in the spectrum. Moreover, the main features observed in all experiments scale with the total energy, and can therefore be tuned easily. Although the fine structure of the caustic is smeared because of multicycle and spatial averaging in our experiment, the main pattern clearly originates from the expected structure of the caustic. To verify that propagation effects play no significant role in the observed enhancement, we performed a systematic pressure scan (see Supplementary Information).
Caustics offer a simple complementary picture for the semiclassical model that describes HHG. They reveal the quantum nature of the process in a regime where classical analysis fails. We have demonstrated the swallowtail caustic, but this is by no means the only case in which caustics appear in HHG, nor is it the only caustic type that can be expected. Other caustics are expected to occur when additional control parameters are introduced, extending the controllability over the spectrum. Although our work focuses on the HHG process, caustics are at the heart of a broad range of strong field interactions that have analogous semiclassical models, such as above-threshold ionization 24 , electron diffraction 25 or double ionization 26 .

Methods
High harmonics were generated with 30 fs, 1 kHz, 800 nm laser pulses in a neon gas jet operating at 50 Hz. The pulse intensity was in the range of 2 × 10 14 to 4.5 × 10 14 W cm 22 , which was varied according to each specific experiment. The second harmonic field was produced using a 200 mm type-I BaB 2 O 4 crystal, and was orthogonally polarized with respect to the fundamental field. Group-velocity dispersion was compensated using a birefringent crystal (calcite). The relative phase of the second-harmonic field relative to the fundamental field was controlled by shifting two BK7 glass wedges. Parallel polarization was obtained with a l/2 zeroorder waveplate. High harmonics were generated by focusing the two beams into a pulsed gas jet. The harmonic spectrum was measured using an XUV spectrometer. The experimental set-up is described in more detail in ref. 17.
We verified that our observations are related to caustics and not to propagation effects by carrying out a systematic study. Specifically, we scanned over a few focusing distances, laser intensities and gas pressures, performing measurements for both neon and helium atoms. We observed a robust response, independent of the structure of the atomic system or propagation conditions. The diffraction patterns in Fig. 2b,d,f were calculated as follows: a swallowtail diffraction pattern is given by e iS(C 1 ,C 2 ,C 3 ,t) dt, where S is a fifth-order polynomial in the state variable t and with parameters (C 1 , C 2 , C 3 ) as control variables. Hence, we expanded the action S(v, R, f, t ion ) in equation (2) to the fifth order in t ion at the swallowtail point, and used this polynomial to calculate the swallowtail diffraction pattern. b,d,f, Theoretical swallowtail diffraction patterns for the same slices. a, R ≈ 0.3, which corresponds to point B in Fig. 1a. b, The swallowtail standard diffraction pattern for a. c, R ≈ 0.45, which is about the swallowtail point (point D in Fig. 1a). d, The corresponding swallowtail diffraction pattern for c. An enhancement of about an order of magnitude with a few harmonics width around the 40th harmonic can be clearly seen in both the measured data and swallowtail calculation. e,f, Measured (e) versus theoretical (f) data for the case of high conversion (R ≈ 0.75), corresponding to the slice in Fig. 1a containing points E and C. A remarkable qualitative agreement is clearly observed, as the curvature of enhancement changes its direction, facing the lower end of the spectrum in a and the higher end in e. Note that the discrete nature of the harmonics in the measured data is due to the periodicity in time, whereas the fringes in the calculated figures are related to the swallowtail diffraction pattern. The colour code has been kept consistent for all theoretical and experimental figures to allow comparison.