## Abstract

The description of shock waves beyond the shock point is a challenge in nonlinear physics and optics. Finding solutions to the global dynamics of dispersive shock waves is not always possible due to the lack of integrability. Here we propose a new method based on the eigenstates (Gamow vectors) of a reversed harmonic oscillator in a rigged Hilbert space. These vectors allow analytical formulation for the development of undular bores of shock waves in a nonlinear nonlocal medium. Experiments by a photothermal induced nonlinearity confirm theoretical predictions: the undulation period as a function of power and the characteristic quantized decays of Gamow vectors. Our results demonstrate that Gamow vectors are a novel and effective paradigm for describing extreme nonlinear phenomena.

© 2016 Optical Society of America

## 1. Introduction

Shock waves emerge in a wide variety of fields like fluido-dynamics [1, 2], dispersive gas dynamics [3] and plasma physics [4, 5], granular systems [6], Bose-Einstein condensation [7], spatial and temporal nonlinear optics [1,8–12] and polaritons [13]. Shock waves ubiquity arises from the universal properties of hyperbolic systems of partial differential equations, which are present in various contexts. Techniques, like the Whitham approach and very simplified hydrodynamic models, like the Hopf equation, are used in most of the cases [14–20]. These methods allow the description of the wave propagation since the occurrence of the shock point, not providing information on the shock profile for long propagation distances. Issues that are often overlooked include the field decay after the shock, and the long term evolution (far field). In addition, numerical calculations offer ways to compute emerging peculiar features like the oscillation period of the so called undular bores (fast oscillations observed in the wave profile in dispersive systems). One specific open question concerns the appearance of the fast oscillations in the internal part of a Gaussian beam undergoing a shock in a nonlocal medium [21,22]. This is in contrast with the hydrodynamical formulation, that predicts that the bores are expected to appear in the external part, i.e., at the beam edges. During the shock formation the beam displays a characteristic double peaked M-shape that is also a riddle in the field of normal dispersion mode-locked laser [23–26]. No analytical solution is available to describe the phenomenon. Here we propose a new method for describing analytically wave propagation beyond the breaking point.

Recently, unnormalizable wavefunctions named nonlinear Gamow vectors (GVs) proved to be fruitful: shock waves in a nonlocal medium are described by the eigensolutions of the so called reversed harmonic oscillator (RHO) [27,28].

Here we show that this approach can be the key to solve analytically shock wave propagation in the far field. Our analysis allows to describe and analyze the development of the characteristic undular bores during the shock formation, and provides a complete description of the characteristic double-peaked M shpae. We find quantitative agreement with experiments in a optothermal nonlinear medium.

## 2. Shock waves and Gamow vectors

We consider a light beam with amplitude *A* (*I* = |*A*|^{2} is the intensity) and wavelength *λ* propagating in a medium with refractive index *n*_{0}. Letting *Z* and *X* be the propagation and polarization directions respectively, the paraxial propagation equation reads as

*P*= ∫

_{MKS}*IdX*the beam power per unit length in the transverse

*Y*direction and

*k*= 2

*πn*

_{0}

*/λ*the wavenumber. In a nonlocal medium the refractive index variation Δ

*n*can be written as

For optothermal nonlocal nonlinearity Eq. (2) originates from the solution of the Fourier heat equation [19,21]. The function *G* is proportional to the Green function and it is normalized such that *∫ GdX* = 1.

We write Eq. (1) in terms of the normalized variables *x* = *X/W*_{0} and *z* = *Z/Z _{d}* with
${Z}_{d}=k{W}_{0}^{2}$

*ψ|ψ*〉 = 1 denoting the Hilbert space scalar product.

*P*is set as

*P*, with

_{MKS}/P_{REF}*P*=

_{REF}*λ*

^{2}/4

*πn*

_{0}|

*n*

_{2}|.

*n*

_{2}is the nonlinear optical coefficient and

*K*(

*x*) =

*W*

_{0}

*G*(

*xW*

_{0}) is the nonlocality kernel function.

As shown in [27], in a highly nonlocal medium, i.e. when the nonlocality function width is much wider than the field intensity, the highly nonlocal approximation (HNA) is valid *K*(*x*) * ^{*}* |

*ψ*|

^{2}≃

*κ*(

*x*). In this way, beam propagation becomes linear, as for anti-waveguiding GRIN media at fixed power [29]. In the HNA, the solution of Eq. (3) is strongly linked to the generalized eigenstates of a RHO. These states are the so called Gamow vectors, firstly introduced by Gamow in 1920s in nuclear physics in order to describe particle decays and resonances and irreversible quantum mechanics [30]. Indeed, this formalism proved to explain the exponential dynamics of Hamiltonian systems thanks to the excitation of these states. Here we propose it as a solution to the “probability problem” of a classical Hamiltonian system such as a nonlinear shock wave propagation. GVs can be obtained by extending the harmonic oscillator eigenfunctions in the complex plane [31]:

*H*(

_{n}*x*) are the

*n*-order Hermite polynomials. The ${\mathit{f}}_{n}^{\pm}$ are discrete states belonging to a rigged Hilbert space (RHS) ${\mathscr{H}}^{\times}$, which is an extension of the standard Hilbert space $\mathscr{H}$. In ${\mathscr{H}}^{\times}$ the Khalfin theorem [32] does not hold true and exponentially decaying wavefunctions are admitted. Indeed, the eigenvalues of the RHO Hamiltonian

*γ*is the decaying coefficient of the associated classical system. For

*z*< 0 the eigenfunction ${\mathit{f}}_{n}^{+}$ is exponentially increasing while ${\mathit{f}}_{n}^{-}$ is decreasing. For this reason we choose the latter to describe exponential decaying dynamics when

*z*grows. Figure 1 shows the square modulus of ${\mathit{f}}_{n}^{-}$ and their

*tilt*, calculated as the

*x*derivative of the phase

*φ*of ${\mathit{f}}_{n}^{-}$,

*∂*, for even

_{x}φ*n*. Notice the resemblance of these functions with the standard intensity and phase profile observed during numerical simulations and experiments in shock waves [19].

It is worthwhile to notice that we can analyze the beam evolution during wave breaking in a nonlocal nonlinear medium using GVs of RHO. When HNA holds true, writing *ψ* = *e ^{−iPz/}*

^{2}

*with*

^{σ}ϕ*σ*the degree of nonlocality, Eq. (3) reads as (see [27])

The eigenfunctions *φ* can be expanded in terms of Gamow eigenvectors as

*ψ*(

*x*,0) is the initial physical state and Γ

*=*

_{n}*γ*(2

*n*+1) are the GVs quantized decay rates. It can be shown that

We stress here the *γ* dependence on the power *P*, which allows us to discriminate GVs from linear losses that in a GRIN system would make this distinction unfeasible [33].

## 3. Far field

RHO Gamow eigenstates have the peculiar characteristic of being the eigenvectors of Fourier transform operator. Indeed, one can observe that the reversed oscillator eigenvalue equation has the same form as its Fourier transform within a phase factor. Considering the RHO Hamiltonian in the momentum basis $(\widehat{p}\to p\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}\widehat{x}\to i{\partial}_{p})$ we have:

To describe the far field by this formalism, we cannot neglect that GVs have an infinite support, i.e., the *x*-region where the eigenfunction is not null is not finite. Hence, to account for the spatial confinement of the experiment, we introduce the windowed Gamow vectors:

*(*

_{W}*x*) = 0 for |

*x*| >

*W*and rect

*(*

_{W}*x*) = 1 for |

*x*| < W, which is the finite size of the physical system. During the evolution, each Gamow component in Eq. (7) exponential decays with rate Γ

*: the ground state has the lowest decay rate Γ*

_{n}_{0}=

*γ*and higher order Gamow states decay faster than the fundamental one. This allows to consider only the fundamental GV in the far field. We compute the Fourier transform $\mathcal{F}$ of the fundamental state of Eq. (10):

Equation (11) gives an analytical expression of the far field, which is compared below with the experiments (Fig. 2). Equation (11) predicts in closed form the typical double-peaked M shape: the fact that undular bores are internal in the beam profile, and the correct scaling with respect to the power of the undulation period. The period *T* scales with the square root of *γ*, and hence with the forth root of the beam input power.

## 4. Experimental results

We validate this analysis by experiments in a nonlocal optothermal medium. The experimental set-up is illustrated in Fig. 2(a). A continuous wave (CW) laser beam at 532nm wavelength is focused through a cylindrical lens (L1) with focal length *f* = 20cm in order to mimic a nearly one-dimensional propagation. Letting *Z* be the propagation direction, the lens focuses the beam in the *X* direction. The light is collected by a spherical lens (L2) and a Charged Coupled Device (CCD) camera. The spot dimension is 1.0mm in the *Y* direction and 35*μ*m in the *X* direction. These geometrical features make the unidimensional approximation valid and allow to compare experimental results with the theoretical model. The diffraction length in the *X* direction is *L _{dif f}* = 3.0mm.

A solution of Rhodamine B (RhB) and water at 0.1mM acts as a nonlocal optical medium and is placed in a cuvette 1mm thick in the propagation direction. The transverse size of the sample is 10mm. RhB is a dye with a high nonlinear index of refraction *n*_{2} (|*n*_{2}| ≃ 2 × 10^{−12}m^{2}/W), its linear absorption length is *L _{abs}* = 1.0mm [19]. We estimate the degree of nonlocality

*σ*≃ 0.2 [21].

We collect CCD images of the beam in the far field (corresponding to the square modulus of the spatial intensity fourier transform of the beam) for different input powers in a range between 0W and 4W (Fig. 2). For low power (*P _{MKS} ≤* 1W) the elliptical beam profile remains Gaussian along propagation [see inset in Fig. 2(a)]. As the power increases, the transverse

*X*section broadens [Fig. 2(b) and 2(c) with

*P*= 2

*W*and 4W] and the beam develops intensity peaks (double-peaked M shape) on its lateral edges [bottom panels of Fig. 2(b) and 2(c)].

The bottom panels of Fig. 2(b) and 2(c) show the intensity profile at *Y* = 0. These results are in remarkable agreement with Eq. (11) as shown in Fig. 2(d) and 2(e). Indeed, Figure 2(d) and 2(e) are obtained from the square modulus of Eq. (11). Different positions in the *y* direction correspond to different power levels. Any power level furnishes a different value of *γ*, being
$\gamma =\sqrt{P/\sqrt{\pi}{\sigma}^{2}}$, from Eq. (8). The Gaussian beam profile in the *y* direction *P* ∝ *exp*(−*y*^{2}) provides the link between *y* and *P* and *γ*. The bottom panels of Fig 2(d) and 2(e) correspond to *γ* = 12 and *γ* = 40.

We remark that the experimental CCD images display the characteristic undular bores of the shock that appear between the lateral peaks, in the internal part of the Gaussian beam. This is also found in the analytical solution [bottom panel of Fig. 2(d) and 2(e)]. We also observe that the experimental data [bottom of Fig. 2(b) and 2(c)] exhibit a reduction in the central part of the profile. This is mostly caused by the presence of nonlinear losses (*β*_{2} ≃ 10^{−}^{5} m/W *−* not included in the model): the thermal effect induces Rhodamine diffusion out of the highest intensity regions, which, in turn, are hence subject to a reduced absorption [19].

Exponential decays are the major signature of Gamow states [27,31,34]. As discussed above, the elliptical beam has an intensity that varies Gaussianly along *Y*. This implies that, observing a CCD image, intensity profiles at different *Y* correspond to different powers; the link between the *Y* position and the power follows the Gaussian profile
$({P}_{MKS}\propto \mathrm{exp}-{Y}^{2}/{Y}_{0}^{2})$, where *Y*_{0} is the vertical beam waist (*Y*_{0} ≃ 3mm). Correspondingly, the expected exponential trend with respect to the power can be extracted from a single picture by looking at different *Y* positions. Indeed, different *y* means a different *P*, and hence a different *γ*. If the decay rate is quantized, which is predicted by the Gamow theory, this quantization affects the beam intensity along *y*.

Exponential decays are extracted considering the red line in Fig. 2(c); the resulting profile versus power is shown in Fig. 3(a): two exponential trends are clearly evident and the two straight lines corresponding to different decay coefficients are drawn (the conversion from *Y* to *P _{MKS}* correspond to a logarithmic scale in which exponentials are straight lines). The extracted ratio of the two decay coefficients is 5.0

*±*0.4 and hence in agreement with the expected quantized theoretical value [27].

We analyze the undular bores of shock waves [see Fig. 2(b) and 2(c)]. Equation (11) predicts that the field intensity undulation period *T* grows like
$T\propto \sqrt[4]{P}$.

We performed a spectral analysis of the oscillations extracted from CCD images as in Fig. 2(b) and 2(c) collected by removing the second lens (L2) for different injected power *P _{MKS}* [see Fig. 2(c)]. Taking the oscillations in the

*X*-direction at the maximum waist along

*Y*we normalize the intensity oscillations and analyze them by a sinusoidal fit. Figure 3(b) shows the intensity oscillations. The curve at

*P*= 2.0W is symmetric since at that power only few oscillations are present [see Fig. 3(b)]. Data have been shifted on the axes to allow a clearer view of the oscillations. By spectral analysis we extract the period as a function of the input optical power [Fig. 3(c)].

In order to demonstrate unequivocally the period’s
$\sqrt[4]{P}$ trend, we report the period *T* as function of
$\sqrt[4]{P}$ (abscissa axes). As shown in the inset of Fig. 3(c), we obtain a linear behavior which is in agreement with our theory.

## 5. Conclusions

The control of extreme nonlinear phenomena is at the basis of the future developments of nonlinear physics, but requires novel theoretical tools and paradigms.

In this paper, we propose a novel approach to describe the occurrence of undular bores and the M-shape intensity profile during a highly nonlinear evolution ruled by the nonlocal nonlinear Shrödinger equation. The strong nonlinearity produces shock waves, and we provide a description of the wave breaking in the far field by techniques from irreversible quantum mechanics. Our experiments quantitatively confirm the new model proposed. We believe that this approach is not only limited to the spatial case considered here, but has an impact in temporal pulse dynamics (as for example mode-locked lasers in the normal dispersion regime) and also, more in general, in the vast number of fields dealing with shock waves.

## Acknowledgments

We acknowledge support form the ERC project VANGUARD (grant number 664782), the Templeton Foundation (grant number 58277) and the ERC project COMPLEXLIGHT (grant number 201766).

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