Proposal for strong field physics simulation by means of optical waveguide

The theory of bending losses in fiber was developed in the mid 1970s. Here we present only the basic ideas and refer the reader to those early works for further details [30–33]. We start, for simplicity, with a straight step index waveguide, where the waveguide width is L, the refractive index within the waveguide is n₂ and the ambient refractive index is ${n}_{1},$ (${n}_{2}\gt {n}_{1}$) (see figure 2(a)). The electromagnetic waves have an angular velocity ω and obey the Helmholtz wave equation, both inside and outside the waveguide

$$\begin{eqnarray}&&{{\rm{\nabla }}}^{2}E+\displaystyle \frac{{n}_{s}^{2}{\omega }^{2}}{{c}^{2}}E=0.\end{eqnarray} \tag{ 4 }$$

Here s = 1, 2 stands for zone 1 or 2 (outside or inside the waveguide). The solutions in the two zones may be written as: ${E}_{s}=\widetilde{E}{\rm{\exp }}\{{\rm{i}}{\beta }_{{\rm{eff}}}z\pm {{\rm{i}}{q}}_{s}y\}$. Inserting these solutions back into the Helmholtz equation we get:

$$\begin{eqnarray}&&{q}_{s}=\sqrt{{n}_{s}^{2}\displaystyle \frac{{\omega }^{2}}{{c}^{2}}-{\beta }_{{\rm{eff}}}^{2}}.\end{eqnarray} \tag{ 5 }$$

There is a critical wave vector ${\beta }_{{\rm{eff}}}^{c}$, beyond which, q₁ in the ambient zone becomes imaginary, the wave becomes evanescent in the clad and confined to the core. When the waveguide is bent with a radius of curvature R, we may assume that the fiber mode remains the same as the mode of the straight waveguide, provided the radius of curvature is much larger than the waveguide width. To maintain the same pattern of the mode in the bent waveguide we assume that we can write the wave as ${E}_{s}=\widetilde{E}{\rm{\exp }}\{{\rm{i}}(R{\beta }_{{\rm{eff}}})\theta +{{\rm{i}}{q}}_{s}{r}^{\prime }\}$ (see figure 2). Now, the tangential wave vector ${\beta }_{\theta }$ depends on the radius: ${\beta }_{\theta }=(1/r)\partial \phi /\partial \theta =(R/R+{r}^{\prime }){\beta }_{{\rm{eff}}}$. Replacing ${\beta }_{{\rm{eff}}}$ in equation (5) with this tangential local wave vector we get:

$$\begin{eqnarray}&&{q}_{1}({r}^{\prime })=\sqrt{{n}_{1}^{2}\displaystyle \frac{{\omega }^{2}}{{c}^{2}}-{\beta }_{{\rm{eff}}}^{2}{\left(\displaystyle \frac{R}{R+{r}^{\prime }}\right)}^{2}}.\end{eqnarray} \tag{ 6 }$$

We note that there is a critical distance ${r}_{{\rm{cr}}}^{\prime }$ from the waveguide center, beyond which, the transverse wave-number ${q}_{1}({r}^{\prime })$ becomes real again and the waves cease to be evanescent.

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In this section we would like to compare bending losses and tunnel ionization. We start our comparison with the tunneling rate in the adiabatic approximation. There are many ways to calculate the tunneling rate in the adiabatic approximation which give many expressions. However, for exponential accuracy, they are all the same and differ only in the pre-exponential term. Here we give the famous Ammosov–Delone–Krainov [34] (ADK) ionization rate as a reference, though, we are most interested in the exponential term.

The tunneling rate according to the ADK solution is:

$$\begin{eqnarray}&&w{(E)}_{\{{\rm{ADK}}\}}=\displaystyle \frac{{I}_{p}}{{\hslash }}{| {C}_{{n}^{* }}| }^{2}{\left(2\displaystyle \frac{\sqrt{8m}{I}_{p}^{3/2}}{{\hslash }| {eE}| }\right)}^{2{n}^{* }-1}{e}^{-\tfrac{2}{3}\tfrac{\sqrt{8m}{I}_{p}^{3/2}}{{\hslash }| {eE}| }}.\end{eqnarray} \tag{ 7 }$$

Here, $w{(E)}_{\{{\rm{ADK}}\}}$ is the ADK ionization rate as a function of the external electric field, ${n}^{* }=Z\left(\tfrac{13.6\,{\rm{eV}}}{{I}_{p}}\right)$, ${| {C}_{{n}^{* }}| }^{2}\,=\tfrac{{2}^{2n* }}{{n}^{* }{\rm{\Gamma }}({n}^{* }+1){\rm{\Gamma }}({n}^{* })}$, Z is the net resulting charge of the atom and ${\rm{\Gamma }}(x)$ is the gamma function. Our next step is to compare the ADK ionization expression with the expression of the bending losses. As is the case with the tunneling ionization, there are many expressions for bending losses rates, but for exponential accuracy, they are all the same. Here we present the results obtained by Marcuse [32] as a representative expression

$$\begin{eqnarray}&&w{(R)}_{\{{\rm{BL}}\}}=\displaystyle \frac{a{\kappa }^{2}{{\rm{e}}}^{2\gamma a}}{\sqrt{\pi \gamma }{V}^{2}}{\left(\displaystyle \frac{1}{R}\right)}^{\tfrac{1}{2}}{{\rm{e}}}^{-\tfrac{2}{3}\tfrac{{\gamma }^{3}}{{\beta }_{{\rm{eff}}}^{2}}R}.\end{eqnarray} \tag{ 8 }$$

Here, $w{(R)}_{\{{\rm{BL}}\}}$ is the bending loss rate as a function of bending radius R; $2a$ is the waveguide width, $\kappa ={q}_{2},\gamma ={{\rm{i}}{q}}_{1}$ in equation (5), ${V}^{2}=\tfrac{{\omega }^{2}{a}^{2}}{{c}^{2}}({n}_{1}^{2}-{n}_{2}^{2})$. To compare equation (7) with equation (8) we have to relate the radius of curvature R to the external electric field E. For that purpose we can take a closer look at figure 1(a) which describes a potential well in the transformed coordinate $Q(t).$ In this system of coordinates, the potential well follows a curve that is given parametrically by $\{z(t),y(t)\}$ where $z(t)={ct}$ and $y(t)=q(t)$. Therefore, the instantaneous radius of curvature of this 'waveguide' is given by:

$$\begin{eqnarray}&&\widetilde{R}\iff \displaystyle \frac{{({\dot{z}}^{2}+{\dot{y}}^{2})}^{3/2}}{| \dot{z}\ddot{y}-\dot{y}\ddot{z}| }=\displaystyle \frac{{\left[{c}^{2}+\tfrac{{e}^{2}{A}^{2}}{{m}^{2}{c}^{2}}\right]}^{3/2}}{c\tfrac{{eE}}{m}}\approx \displaystyle \frac{{{mc}}^{2}}{{eE}}.\end{eqnarray} \tag{ 9 }$$

Throughout the text, $\{\iff \}$ means equivalent quantities.

Using again the analogy between the Helmholtz equation and the Klein–Gordon equation we get: ${\beta }_{{\rm{eff}}}\iff \tfrac{{mc}}{{\hslash }},\gamma \iff \sqrt{\tfrac{2{{mI}}_{p}}{{{\hslash }}^{2}}}$ (appendix) and we can substitute it back in equation (8) to get: $-\tfrac{2}{3}\tfrac{{\gamma }^{3}}{{\beta }_{{\rm{eff}}}^{2}}\widetilde{R}\iff -\tfrac{2}{3}\left(\tfrac{\sqrt{8m}}{e{\hslash }}\right)\left(\tfrac{{I}_{p}^{3/2}}{E}\right)=-\tfrac{2}{3}\tfrac{2{I}_{p}}{{E}_{{ph}}}{{\rm{\Gamma }}}_{{\rm{k}}}$ which is exactly the exponential term in equation (7) (${{\rm{\Gamma }}}_{{\rm{k}}}$ is the Keldysh parameter). By recognizing that $\tfrac{{n}_{2}-{n}_{1}}{{n}_{1}}\iff \tfrac{{V}_{0}}{{{mc}}^{2}}$ and $\tfrac{{n}_{{\rm{eff}}}-{n}_{1}}{{n}_{1}}\iff \tfrac{{I}_{p}}{{{mc}}^{2}}$ (appendix) we can continue this analogy and calculate the critical radius. From equation (6) we find: ${r}_{{\rm{cr}}}^{\prime }=R\left(\tfrac{{n}_{{\rm{eff}}}-{n}_{1}}{{n}_{1}}\right)\iff \tfrac{{{mc}}^{2}}{{eE}}\left(\tfrac{{I}_{p}}{{{mc}}^{2}}\right)\,=\tfrac{{I}_{p}}{{eE}}$, exactly what one would expect to be the tunneling exit point. It is worth also commenting on the power law of $(1/R)$ or E in the pre-exponential terms in equations (7) and (8). If we take for example the hydrogen atom, n^* is equal to 1 and the pre-exponential term in equation (7) is proportional to ${E}^{-1},$which is different than the ${\left(\tfrac{1}{R}\right)}^{0.5}$ in the pre-exponential term in equation (8). The reason for this difference is the long-range nature of the Coulomb interaction in the hydrogen atom, as opposed to the short range interaction of the step index waveguide [35]. For short range potential, e.g. negative ions, ${n}^{* }\to 0$ and the pre-exponential power law is ranging between ${E}^{-1}$ to ${E}^{+1}$, depends on the detailed structure of the binding potential [13, 35].

The concept of electron trajectories plays an important role in our understanding of strong field processes. Even in the simple-man model, the concept of electron trajectories predicts, with a good accuracy, many aspects of HHG such as: atto-chirp, polarization gating, two color HHG spectroscopy, HHG with helicity and HHG emission from short/long trajectories. Nevertheless, this model sometimes fails to predict the finer details because of the crude assumption about the zero velocity at the tunnel exit and the lack of information about electron motion during tunneling [24, 36, 37]. The Lewenstein model [38, 39] is a more detailed model, which is based on the saddle point method to calculate the most probable trajectories. Briefly, the Lewenstein model assumes a transition from the bound ground state into unbound Volkov-like states. The saddle point approximation yields the following three equations:

$$\begin{eqnarray}&&\displaystyle \frac{{({{\bf{p}}}_{d}+e{\bf{A}}({t}_{i}))}^{2}}{2m}+{I}_{p}=0\end{eqnarray} \tag{ 10a }$$

$$\begin{eqnarray}&&{\int }_{{t}_{i}}^{{t}_{r}}({{\bf{p}}}_{{\rm{d}}}+e{\bf{A}}(\tau )){\rm{d}}\tau =0\end{eqnarray} \tag{ 10b }$$

$$\begin{eqnarray}&&\displaystyle \frac{{({{\bf{p}}}_{{\rm{d}}}+e{\bf{A}}({t}_{r}))}^{2}}{2m}+{I}_{p}=N{\hslash }\omega .\end{eqnarray} \tag{ 10c }$$

Here, t_r is the re-collision time with the parent ion, ${{\bf{p}}}_{d}$ is the electron drift momentum, and N is the harmonic order. The solutions for equation (10) define the electron trajectories, known also as quantum orbits. Equation (10b) requires the electron to return to the parent ion—the prerequisite for recombination. Equation (10c) is a statement about the energy conservation when the electron delivers its energy to the emitted harmonic photon. The solution for equation (10a) describes the electron's trajectories during the tunneling process. Although it gives us some limited knowledge about electron dynamics under the potential barrier, this solution requires us to introduce the concept of complex time and complex momentum, for which the physical meaning is hard to interpret.

If we wish to take the analogy between the electron wave-function and the electromagnetic field, the analogous to the electron trajectories are the optic rays. Optic rays are lines which are anywhere perpendicular to the constant phase planes. In most cases, optic rays are just straight lines, but not for example in an inhomogeneous medium. It is worth noting that the classical electron trajectories, derived from the Hamiltonian (3) without a potential, are just straight lines. Now, If we could fabricate a curved waveguide, as in figure 1(a), and probe the electromagnetic wave anywhere at close proximity to the waveguide, we could find the optical rays even in the 'forbidden zone', thus, finding the 'electron trajectories' under the barrier.

To demonstrate the feasibility of our proposed 'optical simulator', we run a finite element numerical simulation (Comsol Multiphysics). Instead of solving the time dependent Schrödinger equation for the electron, we solve the full Helmholtz equation for the curved optical waveguide. To convert back and forth between the atomic physical problem and the curved waveguide problem, we used the dimensionless time and lengths as mentioned above (see figure 1). Our simulation consists of an electromagnetic wave with a carrier frequency $\nu =c/(1\,\mu {\rm{m}})=2.99\times {10}^{14}\,\mathrm{Hz}$ which propagates in a curved step index waveguide, having a core index of refraction ${n}_{{\rm{co}}}=3$ and an ambient index of refraction ${n}_{{\rm{cl}}}={n}_{{\rm{co}}}-\delta n$. The curved waveguide has a width of $a=34\,\mu {\rm{m}}$ and it follows the curve:

$$\begin{eqnarray}&&q(z)=\displaystyle \frac{A}{4}{e}^{-\tfrac{{Z}_{0}^{2}}{{\rm{\Delta }}{Z}^{2}}}+\displaystyle \frac{A}{2}\left(\displaystyle \frac{1}{2}-\cos (2\pi \,z/{\rm{\Lambda }})\right){e}^{-\tfrac{{(z-{Z}_{0})}^{2}}{{\rm{\Delta }}{Z}^{2}}}.\end{eqnarray} \tag{ 11 }$$

In our simulations we set ${Z}_{0}={\rm{\Delta }}Z=220\,\mathrm{mm},{\rm{\Lambda }}=100\,\mathrm{mm}$, and we varied A and $\delta n$. For example, a simulation with the aforementioned parameters, with $\delta n=1.4\times {10}^{-4}$ and $A=400\,\mu {\rm{m}}$ is equivalent to a particle in a finite square well potential having a width of 4.6 Bohr radius, ionization energy of 20.8 eV, subjected to a laser radiation at 720 nm with intensity of $4.2\times {10}^{14}\tfrac{{\rm{W}}}{{\mathrm{cm}}^{2}}$. Because of limited computer resources, we break the entire medium into small slices, each slice having a length of $500\,\mu {\rm{m}}$. We calculated the wave propagation in one slice and used the wave-function at the end boundary as an initial boundary condition for the next slice and so on. To calculate the 'electron trajectories' (optic rays), we first calculated ${\bf{j}}={\rm{Im}}\{{E}^{* }{\boldsymbol{\nabla }}E\}$ which is the analog to the Klein–Gordon probability current ${j}_{{\rm{KG}}}^{\mu }=\tfrac{1}{2}{\rm{Im}}\{{\psi }^{* }{\partial }^{\mu }\psi \}.$ We next calculated the trajectory direction at each point: $\tan (\theta )={j}_{y}/{j}_{z}$ , and used it to calculate the electron trajectory $y({z}_{i},z)=y({z}_{i})\,+{\int }_{{z}_{i}}^{z}\tan (\theta ){\rm{d}}{z}$.

Figure 3 shows the normalized ${| E| }^{2}$ in a logarithmic false color scale. Here we simulate a $34\,\mu {\rm{m}}$ wide, 300 mm long waveguide with core refractive index ${n}_{{\rm{co}}}=3$ and $\delta n=1.7\times {10}^{-4}$. The oscillation amplitude is $A=400\,\mu {\rm{m}}$. Accordingly, the Keldysh parameter changes from ${{\rm{\Gamma }}}_{{\rm{K}}}=1.2$ ($\tilde{R}=2.3\times {10}^{3}\,\mathrm{mm})$ at $z=50\,\mathrm{mm}$ to ${{\rm{\Gamma }}}_{{\rm{K}}}=0.7$ ($\tilde{R}\,=1.27\times {10}^{3}\,\mathrm{mm}$) at $z=200\,\mathrm{mm}$, i.e., changes from the 'multi-photn ionization' regime into the 'tunneling ionization' regime. In the figure we indicate the curved waveguide with black dash-dot lines. The adiabatic tunneling exit point (TEP), ${r}_{{\rm{cr}}}^{\prime }\iff {I}_{p}/{eE}$ is indicated with the dotted yellow line. Qualitatively, we can see from this figure that each time, near the maximum curvature of the waveguide, radiation is released from the waveguide with an almost point-like radiation source pattern (near the tips of the red arrows). We can see also how these consecutive sources are interfering with each other.

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We want to examine more closely the electron trajectories and compare them to the electron trajectories in the semi-classical model. We focus on ionization points around z = 200 mm where the Keldysh parameter is lower than one. Figure (4) shows long, short and near cut-off trajectories (green, red and yellow solid lines) as well as other representative trajectories (gray solid lines). The dotted pink, red and yellow lines are the corresponding semi-classical trajectories assuming the electron emerges at the adiabatic exit point ${r}_{{\rm{cr}}}^{\prime }={I}_{p}/{eE}$ with zero velocity (figures 4(a) and (c)) or from the waveguide core (figures 4(b) and (d)). The reason for drawing the two cases is because the three-step-model first assumes that the electron emerges into the continuum at the tunneling exit point with zero velocity (equation (10a)), but equation (10b) assumes that the electron motion starts at r = 0. To extract the resulting electron trajectories (optic rays) from our numerical simulation, we first selected a point, which is far enough from the ionization point. Next, we used the above-mentioned procedure to calculate the trajectories from this point back to the ionization region, and forward to where the electron trajectories collide again with the oscillating waveguide (re-collision). From here on we will refer to the trajectories that we extracted from the numerical simulation as the 'numerical trajectories' (NT); to distinguish them from the semi-classical trajectories (SCT). To compare the NT with the SCT, we first recall that in the transformed coordinates (equation (2a)), the electron trajectories away from any potential, are just straight lines. Therefore, we first find the NT slope at the point far away from the ionization region. We then draw a straight line with the same slope and find the position z_i at which this line is tangent to q(t). Finally we shift the line in the $\widehat{y}$ direction, either to start from the waveguide core (figure 4(b)) or from the TEP (figure 4(a)). The dash-dot purple line in figures 4(c)–(d) is proportional to the second derivative of q(z) (proportional to ${\bf{E}}(t)$).

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At first glance, the NT look very similar to the SCT, but if we look more closely we can see some differences. The first difference to note is the ionization times. As it is well known in the three-step-model, the different trajectories emerge at the continuum at different times t_i. These t_i are indicated in figure 4(a) with the red, yellow and pink arrows. The simple-man model gives us no information about the electron trajectories inside the 'forbidden zone' under the barrier. In contrast, from our numerical method we can extract the NT, even in the 'under the barrier zone'. Looking at the NT, we see that all are starting almost at the same point (slanted cyan arrow) and propagating as a bundle until they reach a branching zone (vertical cyan arrow), from which, each trajectory deflects into a different angle. We note also that this branching zone is placed, neither at the TEP, nor at the waveguide edges but placed somewhere in-between.

From figure 4 we can see that, in contrast to the semi-classical model, only the long trajectories are really emerging through the TEP (marked by the dotted cyan line) while the short trajectories never really tunnel out. Figure 5 shows the transition from tunnel ionization to above-the-barrier ionization (ABI). The simulation parameters in this case, $\delta n=2.9\times {10}^{-5}$ and $A=269\,\mu {\rm{m}}$, correspond to an atom with ionization potential ${I}_{p}=4.3\,\mathrm{eV}$ subjected to laser radiation at central wavelength of 720 nm and intensity of $1.9\times {10}^{14}\,{\rm{W}}{\mathrm{cm}}^{-2}$. The Keldysh parameters are ${{\rm{\Gamma }}}_{{\rm{k}}}=0.12$ at $z=55\,\mathrm{mm}$ and ${{\rm{\Gamma }}}_{{\rm{K}}}=0.09$ at $z=104\,\mathrm{mm}$ and ${{\rm{\Gamma }}}_{{\rm{k}}}=0.075$ at $z=150\,\mathrm{mm}$ where the TEP reach the waveguide edges and we are entering into the ABI. We first look at the tunnel ionization, which occurs around $z=50\,\mathrm{mm}$. In terms of matching between the semi-classical trajectories and the actual trajectories, it is not much different than the case in figure 4. The slight difference is that here most of the ionized trajectories start deeper from within the waveguide. This difference is even more pronounced at the ionization event near $z=100\,\mathrm{mm}$ and around $z=150\,\mathrm{mm},$ where we reach the ABI threshold, the waves 'spill' out of the waveguide in almost a straight line.

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