Path integral formulation of light propagation in a static collisionless plasma, and its application to dynamic plasma

Je Hoi Mun; Je Hoi Mun; Cheonha Jeon; Chang-Mo Ryu; Chang-Mo Ryu

doi:10.1364/OE.386404

1. Introduction

Laser-matter interactions have been intensively studied both theoretically and experimentally in the past several decades. When an intense laser impinges on a target, plasma is generated through ionization mechanisms such as multiphoton, tunneling and avalanche ionizations [1–4]. These mechanisms can depend significantly on the initial laser parameters and the target material. Due to the time-and intensity-dependent plasma density, the generated plasma gives rise to a nonlinear refractive index that can distort the phase and amplitude of the laser pulse. Therefore, from the perspectives of fundamental physics issues and applications, the plasma-induced effects have been studied in various branches of optical physics, such as the self-focusing [5], phase matched high-order-harmonic generations (HHG) from gas [6–8] and solid state matter [9]. When the plasma density becomes higher than the critical one, the laser cannot penetrate the plasma and undergoes total reflection. Since the reflection rate increases dramatically at the critical density, the self-induced plasma was employed to rapidly turn off a laser pulse in transmissional plasma optics [10–12] or to achieve a high-contrast relativistic laser pulse in reflectional plasma optics [13]. In the reflectional plasma optics, which is often called the plasma mirror (PM), many studies have been conducted on optimal control of the reflectivity, contrast ratio, and spatial beam profile [13–22]. In Refs. [15,20], for example, to characterize the PM, plasma excitation and hydrodynamic expansion simulations were carried out. In these studies, a dynamically varying plasma density was combined with the solution of light reflection from static plasma to estimate the optical qualities of a reflected laser pulse. The reflection amplitude from static plasma is calculated from a numerical solution for the Helmholtz equation in Ref. [15], and from an approximate analytic equation in Ref. [20]. Therefore, deriving an analytic expression of the reflection and/or transmission amplitude(s) is very important for characterizing and controlling not only the dynamic plasma optics but also various phenomena stemming from the laser-plasma interaction. The approximate analytic equation for the reflection amplitude given in Ref. [19,20] and references therein, after scaling the configuration to a one-dimensional case, can be expressed as

(1)$$R(\omega) = \exp\big[-i\frac{2\omega}{c}\int_{0}^{x_{\textrm{C}}} n(x)dx +i\frac{\pi}{2}\big],$$

where $c$ is the speed of light, $\omega$ is the angular frequency of the light, $n(x)$ is the $x$-dependent complex refractive index, and $x_{\textrm {C}}$ the critical reflection position where $\textrm {Re}[n(x_{\textrm {C}})] = 0$. This equation can be understood as a single optical path (SOP) approximation that considers the most dominant reflection event at the critical plasma density $N_{\textrm {C}}$ (see Fig. 1(a)). As shown in Refs. [19,20] and in the present work, this solution well reproduces both the experimental observations and the numerical results of a particle-in-cell (PIC) simulation when the peak plasma density is higher than the critical one. We note, however, that this reflection amplitude is based on a fundamental assumption that the reflection from underdense plasma and the resonance from near-critical plasma can be ignored. As the Fresnel equation implies, a finite portion of light can be reflected from a vacuum-plasma interface, and light can be absorbed by overdense plasma through nonlinear resonant absorption [23–25], vacuum heating [26–28], and ponderomotive absorption [29].

Fig. 1. Sketch (not to scale) of light reflection from plasma surface. (a) Reflection at point of critical density $N_{\textrm {C}}$. (b) Reflection and transmission amplitudes given by the Fresnel equation near position $x = x_{\textrm {m}}$. (c) Reflection path obtained by applying the Fresnel equations at multiple positions (see Eq. (4) and descriptions in the main text).

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2. Theoretical model

Reflection amplitude of light from an arbitrary inhomogeneous plasma slab can be obtained by solving a Riccati-type integral equation [30], whose analytic solution is given by an iteration. In this work, we derive an intuitive analytic solution that describes both the partial reflection of light from underdense plasma as well as the total reflection from overdense plasma by using a path-integral approach. For simplicity, we confine our attention to a one-dimensional system. In consideration of the nonrelativistic collisionless plasma, the refractive index of plasma is given by $n(x) =\frac {1}{\omega }\sqrt {\omega ^2 - { N_{\textrm {e}}(x)e^{2}}/{\epsilon _{0} m_{\textrm {e}}}}~(\frac {1}{\omega }\sqrt {\omega ^2 - 4\pi N_{\textrm {e}}(x)}$ in atomic units), where $N_{\textrm {e}}(x)$ is the plasma density and $x$ is the spatial coordinate. Atomic units are used unless otherwise stated. $m_{\textrm {e}}$, $e$, and $4\pi \epsilon _{0}$, which represent electron mass, elementary charge, and the inverse Coulomb constant respectively, are all unity in atomic units. We express $n(x)$ in terms of discrete steps, $n(x) \sim n_{m}$, where the integer $m$ represents the corresponding spatial location $x_{m} \equiv m \delta x$. In the discretized layer configuration, the Fresnel equation can be employed for analyzing the reflection and transmission of light [31].

Let us consider a single stepwise variation of the refractive index at $x_{m}$ from $n_{m}$ to $n_{m+1}$ (see Fig. 1(b)). For a plane wave $E_{\textrm {i}} = \exp [i\omega (t - n_{m}x /c)]$, propagating in the positive direction, the transmission and reflection amplitudes at $x=x_{m}$ are determined by the boundary conditions of the electromagnetic fields (Fresnel equation). For propagation of the transmitted and reflected waves considered upto $x = x_{m+1}$ and $x = x_{m-1}$, respectively, the amplitudes are

(2)$$\begin{aligned} E_{\textrm{t}}(x_{m+1}) = &\frac{2n_{m}}{n_{m}+n_{m+1}} \times \exp[-i\omega n_{m+1}\delta x/c ],\\ E_{\textrm{r}}(x_{m-1}) = &\frac{n_{m}-n_{m+1}}{n_{m}+n_{m+1}} \times \exp[-i\omega n_{m}\delta x/c ]. \end{aligned}$$

For the counter-propagating wave $\hat {E}_{\textrm {i}} = \exp [i\omega (t + n_{m+1}x /c)]$, the transmission and reflection amplitudes are

(3)$$\begin{aligned} \hat{E}_{\textrm{t}}(x_{m-1}) = &\frac{2n_{m+1}}{n_{m}+n_{m+1}} \times \exp[-i\omega n_{m}\delta x/c ],\\ \hat{E}_{\textrm{r}}(x_{m+1}) = &\frac{n_{m+1}-n_{m}}{n_{m}+n_{m+1}} \times \exp[-i\omega n_{m+1}\delta x/c]. \end{aligned}$$

By multiplying the above amplitudes, an amplitude of the reflection path where an incident plane wave propagates from vacuum upto $x_{l-1}$, then is reflected at $x_{l}$, and propagates back to the vacuum can be given by

(4)$$\begin{aligned} r(\omega,x_{l}) = \prod_{m = 0}^{l-1}& \frac{2n_{m} }{n_{m}+n_{m+1}}\exp[-i\omega n_{m+1}\delta x /c] \times\\ & \frac{n_{l}-n_{l+1}}{n_{l}+n_{l+1}}\exp[-i\omega n_{l}\delta x /c] \times\\ \prod_{m = 0}^{l-1}& \frac{2n_{m+1}}{n_{m}+n_{m+1}}\exp[-i\omega n_{m}\delta x /c]. \end{aligned}$$

These three terms correspond to the transmission, reflection, and backward transmission amplitudes, respectively (see Fig. 1(c)). Here, $n_{m+1} \sim n_{m} + \frac {dn(x_{m})}{dx}\delta x$ and the product of the first and third lines $\frac {2n_{m} }{n_{m}+n_{m+1}}\times \frac {2n_{m+1}}{n_{m}+n_{m+1}}$ becomes unity by ignoring the term including $\delta x^2$. In the second line, $\frac {n_{l}-n_{l+1}}{n_{l}+n_{l+1}}$ can be written as $\frac {-1}{2n(x_{l})}\frac {dn(x_{l})}{dx}\delta x$. The product of all the exponential terms is given as $\exp \big [-i\frac {2\omega }{c}\sum _{m=0}^{l}n(x_{m}) \delta x\big ]$. By assuming the infinitesimally small $\delta x \sim dx$, the resultant reflection amplitude is given by

(5)$$r(\omega,x_{l}) = -\frac{dx}{2n(x)}\frac{dn}{dx}\vert_{x=x_{l}} \times \exp\big[-i\frac{2\omega}{c}\int^{x_{l}}_{0}n(x) dx\big].$$

We note that Eq. (4) or Eq. (5) is a reflection amplitude derived by considering only one reflection event at $x=x_{l}$. We do not consider the reflection paths with two (even-numbered) reflection events since they do not propagate back to the vacuum in the one dimensional configuration. Contributions from the higher-order reflection paths, i.e., the reflection paths with three, five, and higher numbers of reflection events, are considered later on. By integrating Eq. (5) for $x_{l}$ over the entire $x$ position, we can express the total reflection amplitude as

(6)$$\begin{aligned} R({\omega,x_{\textrm{f}}}) &=\\ &\int^{x_{\textrm{f}}}_{0}dx \frac{-1}{2n(x)}\frac{dn(x)}{dx}\times\exp\big[-i\frac{2\omega}{c} \int^{x}_{{0}}n(x') dx'\big], \end{aligned}$$

where $x = 0$ is the left boundary of the vacuum-plasma surface, and $x_{\textrm {f}}$ is that on the right side. We note that, this analytic result derived from the path-integral approach provides a result identical to equation (3.30) of Ref. [30], equation (36) of Ref. [32], and equation (54) of Ref. [33], which are derived from a nonlinear Riccati-type integral equation in the low-reflectivity condition, i.e., $|R|^2 \ll 1$. In Ref. [30], the more exact analytic solution is expressed by an iteration. In the configuration given in Fig. 1(a), we can set $x_{\textrm {f}}$ at $L_{1}$, because no reflection takes place when the derivative of refractive index $dn(x)/dx$ is equal to zero.

In the summation of the many reflection paths (Eq. (6)), the dominant one has a minimum action $S(x)$ that satisfies $\frac {\partial S(x)}{\partial x} \equiv 0$, and the action term $S(x)$ corresponds to $2\int _{{0}}^{x}n(x')\frac {\omega }{c}dx'$ in Eq. (6). The minimum action condition is satisfied at the critical density, where $n(x)\equiv 0$, being a result qualitatively consistent with the single optical path approximation given in Eq. (1). Using $n(x) = \frac {1}{\omega }\sqrt {\omega ^2 - 4\pi N_{\textrm {e}}(x)}$, Eq. (6) can be expressed as

(7)$$R({\omega,n_{\textrm f}}) = \int^{n_{\textrm f}}_{1}dn \frac{-1}{2n}\times\exp\big[-i\frac{\omega^{3}}{3\pi c a}(1-n^{3})\big].$$

Here, we have assumed a linear slope, i.e., $\frac {dN_{\textrm {e}}(x)}{dx} \equiv a$ . We note that, in deriving Eq. (7) we employed the linear slope instead of a more realistic exponential slope. The analytic solution can also be given for the exponential slope with the left boundary of the integral given by $x=-\infty$. However, in terms of the numerical PIC simulation, we need to provide a well-defined boundary at which the plasma density is exactly zero. By employing the linear slope, we have minimized some possible artifacts arising from boundaries that are not well-defined, and have obtained very good agreement between the PIC results and model calculations as shown in the next sections. Since the underlying physics of light propagation in a static collisionless plasma is universal, the model can be applied to any plasma profile.

We emphasize that Eq. (7) is a reliable solution in the low-reflectivity condition [30,32,33]. In the present work, the other higher-order reflection paths are explicitly introduced. Following the same procedure for the first-order reflection path, a third-order reflection path shown in Fig. 2(a) can be given by

(8)$$\begin{aligned} r_{{\textrm{3rd}}}(\omega,x_{l_{1}},x_{l_{2}},x_{l_{3}}) = \prod_{m_{1} = 0}^{l_{1}-1}& \frac{2n_{m_{1}}}{n_{m_{1}}+n_{m_{1}+1}}\exp(-i\omega n_{m_{1}+1}\delta x /c) \times \frac{n_{l_{1}}-n_{l_{1}+1}}{n_{l_{1}}+n_{l_{1}+1}}\exp(-i\omega n_{l_{1}}\delta x /c)\\ \times \prod_{m_{2} = l_{1}-1}^{l_{2}+1}& \frac{2n_{m_{2}+1}}{n_{m_{2}}+n_{m_{2}+1}}\exp(-i\omega n_{m_{2}}\delta x /c) \times \frac{n_{l_{2}+1}-n_{l_{2}}}{n_{l_{2}}+n_{l_{2}+1}}\exp(-i\omega n_{l_{2}}\delta x /c)\\ \times \prod_{m_{3} = l_{2}+1}^{l_{3}-1}& \frac{2n_{m_{3}}}{n_{m_{3}}+n_{m_{3}+1}}\exp(-i\omega n_{m_{3}+1}\delta x /c) \times \frac{n_{l_{3}}-n_{l_{3}+1}}{n_{l_{3}}+n_{l_{3}+1}}\exp(-i\omega n_{l_{3}}\delta x /c)\\ \times \prod_{m_{4} = 0}^{l_{3}-1}& \frac{2n_{m_{4}+1}}{n_{m_{4}}+n_{m_{4}+1}}\exp(-i\omega n_{m_{4}+1}\delta x /c). \end{aligned}$$

Fig. 2. (a) Third-order and (b) fifth-order reflection paths. In (a), incident light propagating from vacuum is reflected back at $x= x_{l_{1}}$. The reflected light transmits forward at $x= x_{l_{2}}$ and is again reflected back at $x= x_{l_{3}}$. The reflected light propagates to the vacuum. In (b), incident light follows the same reflection path shown in (a) until it is reflected back at $x= x_{l_{3}}$. The light then transmits forward at $x= x_{l_{4}}$ and is reflected back at $x= x_{l_{5}}$. The reflected light propagates to the vacuum.

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Here, the first (i), second (ii) and third (iii) lines of the right-hand side of Eq. (8) represent amplitudes of (i) the transmission from the vacuum to $x=x_{l_{1}-1}$ and the reflection at $x=x_{l_{1}}$, (ii) the back transmission from $x=x_{l_{1}-1}$ to $x=x_{l_{2}+1}$ and the reflection at $x=x_{l_{2}}$, and (iii) the transmission from $x=x_{l_{2}+1}$ to $x=x_{l_{3}}-1$ and the reflection at $x=x_{l_{3}}$. The forth line corresponds to the back transmission amplitude of the light from $x=x_{l_{3}-1}$ to the vacuum. Vanishing $\delta x^{2}$ in Eq. (8) leads us to

(9)$$r_{{\textrm{3rd}}}(\omega,n_{l_{1}},n_{l_{2}},n_{l_{3}}) = \exp\big[-i\frac{\omega^3}{3\pi c a}(1-n^{3}_{l_{1}}+n^{3}_{l_{2}}-n^{3}_{l_{3}}) \big] \times \frac{1}{8n_{l_{1}}n_{l_{2}}n_{l_{3}}}dn_{l_{1}}dn_{l_{2}}dn_{l{3}},$$

where $n_{l}$ is the refractive index at position $x_{l}$, with $dn_{l}$ being $\frac {\partial n_{l}}{\partial x_{l}} dx_{l}$.

We integrate Eq. (9) by $n_{l_{1}}$, $n_{l_{2}}$, and $n_{l_{3}}$ under the geometric conditions given by $0\;<\;x_{l_{2}}\;<\;x_{l_{1}}\;<\;x_{\textrm {f}}$ and $x_{l_{2}}\;<\;x_{l_{3}}\;<\;x_{\textrm {f}}$ ($1\;>\;n_{l_{2}}\;>\;n_{l_{1}}\;>\;n_{\textrm {f}}$ and $n_{l_{2}}\;>\;n_{l_{3}}\;>\;n_{\textrm {f}}$). Here $x_{\textrm {f}}$ is the right boundary of the integrals. Using the constant $X \equiv \frac {\omega ^3}{3\pi c a}$, the resulting equation can be written as

(10)$$\begin{aligned} {R}_{\textrm{3rd}}(\omega,n_{\textrm{f}}) = \frac{1}{8}\exp{\big[ -iX \big]}&\int_{1}^{n_{\textrm{f}}}dn_{l_{1}}\frac{1}{n_{l_{1}}}\exp{\big[ iXn_{l_{1}}^{3} \big]} \\ \times &\int_{1}^{n_{l_{1}}}dn_{l_{2}}\frac{1}{n_{l_{2}}}\exp{\big[ -iXn_{l_{2}}^{3} \big]} \int_{n_{l_{2}}}^{n_{\textrm{f}}}dn_{l_{3}}\frac{1}{n_{l_{3}}}\exp{\big[ iXn_{l_{3}}^{3} \big]}. \end{aligned}$$

The fifth-order reflection path shown in Fig. 2(b) can similarly be expressed as

(11)$$\begin{aligned} r_{\textrm{5th}}(\omega,n_{l_{1}},n_{l_{2}},n_{l_{3}},n_{l_{4}},n_{l_{5}}) = \exp\big[-iX(1-n^{3}_{l_{1}}+n^{3}_{l_{2}}-n^{3}_{l_{3}}+n^{3}_{l_{4}}-n^{3}_{l_{5}}) \big]\\ \times \frac{-1}{32n_{l_{1}}n_{l_{2}}n_{l_{3}}n_{l_{4}}n_{l_{5}}}dn_{l_{1}}dn_{l_{2}}dn_{l_{3}}dn_{l_{4}}dn_{l_{5}}. \end{aligned}$$

We integrate Eq. (11) by $n_{l_{1}}$, $n_{l_{2}}$, $n_{l_{3}}$, $n_{l_{4}}$, and $n_{l_{5}}$ taking into account the geometric conditions given by $0\;<\;x_{l_{2}}\;<\;x_{l_{1}}\;<\;x_{\textrm {f}}, x_{l_{2}}\;<\;x_{l_{3}}\;<\;x_{\textrm {f}}$, $0\;<\;x_{l_{4}}\;<\;x_{l_{3}}\;<\;x_{\textrm {f}}, x_{l_{4}}\;<\;x_{l_{5}}\;<\;x_{\textrm {f}}$ to obtain the final form of Eq. (12) as

(12)$$\begin{aligned} {R}_{\textrm{5th}}(\omega,n_{\textrm{f}}) = -\frac{1}{32}\exp{\big[ -iX \big]}&\int_{1}^{n_{\textrm{f}}}dn_{l_{1}}\frac{1}{n_{l_{1}}}\exp{\big[ iXn_{l_{1}}^{3} \big]} \\ \times &\int_{1}^{n_{l_{1}}}dn_{l_{2}}\frac{1}{n_{l_{2}}}\exp{\big[ -iXn_{l_{2}}^{3} \big]} \int_{n_{l_{2}}}^{n_{\textrm{f}}}dn_{l_{3}}\frac{1}{n_{l_{3}}}\exp{\big[ iXn_{l_{3}}^{3} \big]}\\ \times &\int_{1}^{n_{l_{3}}}dn_{l_{4}}\frac{1}{n_{l_{4}}}\exp{\big[ -iXn_{l_{4}}^{3} \big]} \int_{n_{l_{4}}}^{n_{\textrm{f}}}dn_{l_{5}}\frac{1}{n_{l_{5}}}\exp{\big[ iXn_{l_{5}}^{3} \big]}. \end{aligned}$$

We can generalize these results to an arbitrary-order reflection path. The integral of $(2p+1)$th-order reflection path, where $p\geq 0$ is an integer, $R_{p}(\omega ,n_{\textrm {f}})$ can be given as

(13)$${R}_{p}({\omega,n_{\textrm{f}}}) = \frac{(-1)^{p+1}}{2^{2p+1}}\exp{\big[-iX\big]} \times \int^{n_{\textrm{f}}}_{1}dn' F_{p}(n',n_{\textrm{f}})\times \frac{\exp\big[iX{n'}^{3}\big]}{n'},$$

where

(14)$$F_{p+1}(n,n_{\textrm{f}}) = \int^{n}_{1}dn' \frac{\exp\big[-iX{n'}^{3}\big]}{n'} \times \int^{n_{\textrm{f}}}_{n'}dn'' \frac{\exp\big[iX{n}''^{3}\big]}{n}F_{p}(n'',n_{\textrm{f}}).$$

Therefore, the analytic reflection amplitude is

(15)$$R({\omega,n_{\textrm{f}}}) = \sum_{p=0}^{p_{\textrm{max}}}\hat{R}_{p}({\omega, n_{\textrm{f}}}),$$

where, $(2p_{\textrm {max}} + 1)$ is the highest-order of reflection to be included. As discussed in the next section, the higher-order reflection paths play a role when the peak plasma density is near the critical one.

3. Validity of model

In Sec. 2, we derived an intuitive analytic solution of the reflection amplitude by using a path-integral approach. In this section, our analytic result is reinforced by numerical experiments using a one-dimensional PIC simulation code developed in-house. In developing this code, we benchmarked Ref. [34]. The spatial grid size and the number of particles in a cell are $1/64$ (in units of the wavelength, 800 nm) and 1, respectively. The initial velocities of the particles are set to be zero. In order to give a linear density gradient to a plasma surface, the charge and the mass of each particle are varied depending on the initial position. In the simulation, a nonrelativistic laser with a low intensity ($10^{12}~\textrm {W/cm}^{2}$) is used so that the plasma particles remain in their initial positions. Therefore, the stationary plasma density condition is well satisfied. Pulse widths of the incident laser pulses are fixed at 14.5 fs. The reflected laser pulses are analyzed in the spectral domain. In Fig. 3, open circles show the spectral amplitudes of the central wavelength (800 nm) of the reflected laser pulses as a function of the peak plasma density.

Fig. 3. Absolute amplitude $|R|$ (black solid lines) and phases $\phi _{R}$ (blue solid lines) of path integral for reflection given in Eq. (7). (a), (b), and (c) show results for $a = 0.032, 0.32$, and $3.2$ ($N_{\textrm {C}}/\lambda$), respectively. Black dotted lines are absolute amplitudes $|R|$ including third-order reflection paths (see Eq. (15)). Open circles show results of a one-dimensional PIC simulation (see the main text).

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We start by analyzing the contributions of the first-order reflection paths. In Figs. 3(a), 3(b), and 3(c), the numerical results of the integral given by Eq. (7) are shown for the density gradients of $a = 0.032, 0.32$ and $3.2$ in units of $N_{\textrm {C}}/ \lambda$, respectively, where $N_{\textrm {C}} = \frac {\omega ^{2}}{4\pi }$ and $\lambda = 800$ nm. $R({\omega ,n(L_{1})})$ is plotted by the absolute amplitude $|R({\omega ,n(L_{1})})|$ (black solid lines) and the phase $\phi _{R}(\omega )$ (blue solid lines). Figure 3(a) shows the result of the integral $|R|$ in Eq. (7) with the $a$ set at 0.032 $N_{\textrm {C}}/ \lambda$. $|R|$ becomes significant in the vicinity of the critical density, which comes from two different aspects: on one side, the amplitude of the reflection path $-dn/2n$ becomes very large near the critical density near $n\sim 0$. On the other side, as shown by the blue solid line, the phase of the complex reflection amplitude $\phi _{R}(\omega )$ varies rapidly well before the critical density, which indicates that the interference of many reflection paths is not mutually constructive. As the electron density increases further, the reflection paths can interfere constructively and the $|R|$ increases like a step function, thus supporting the validity of Eq. (1).

When the density slope is larger at $a = 0.32~N_{\textrm {C}}/ \lambda$ (Fig. 3(b)), $\phi _{R}(\omega )$ changes much more slowly. Before $|R|$ becomes significantly large near the critical density, three very small local maxima (illustrated by black arrows and the inset in Fig. 3(b)) of the reflection amplitude are generated owing to the partially phase-matched reflection paths, which agrees with the PIC results. One can notice that well below the critical plasma density, $|R|$ becomes significant (compare Fig. 3(a) and Fig. 3(b)). For the case of $a = 3.2~N_{\textrm {C}}/ \lambda$ (Fig. 3(c)), far below the critical density, the reflection paths are effectively phase matched, and the total reflection amplitude reaches $|R|\sim 0.4$ at $N_{\textrm {e}} = 0.9 ~N_{\textrm {C}}$.

While the PIC simulation and the model calculation Eq. (7) exhibit a qualitative agreement, there are noticeable deviations near the critical density (see Fig. 3(c)), which can be attributed to the inclusion of reflection paths having only one reflection event. One can notice that $|R(\omega ,n(L_{1}))|$ as calculated by Eq. (7) becomes larger than 1 below the critical density in Fig. 3(c). We find that for an even sharper density gradient $a\sim \infty$ (not shown), the total reflection of $|R(\omega ,n(L_{1}))=1|$ is achieved at $n=0.135$, which contradicts the Fresnel equation ($|R(\omega ,n=0)=1|$ for $a=\infty$). We emphasize that near the critical density, we cannot ignore the higher-order reflection paths in our path-integral model. Absolute reflection amplitudes including the third-order reflection paths are plotted as dotted lines in Fig. 3. The PIC results fit better with the dotted lines in the vicinity of the critical density, as shown in Fig. 3(c).

In Fig. 4, the numerical integral of Eq. (15) is shown on a logarithmic scale under the same conditions as in Fig. 3. Higher-order reflection paths are considered by integrating Eq. (15) up to $p_{\textrm {max}}=$0-5, and the resulting reflection amplitudes are well converged in the plotted ranges of $T$=0-4, 0-3.5, and $T$=0-2.5 for (a), (b), and (c), respectively. However, the convergence is not ensured for a larger $T$, where very complex interplay of the higher-order reflection paths is included. In principle, an exact analytic solution is given with $p_{\textrm {max}}=\infty$ for the present collisionless (lossless) plasma case. If the absorption (collisional) effect is considered, the multiple reflection events will lead to energy transfer to electrons, and the multiple reflection paths will vanish before the light propagates back to the vacuum. For example, in a significant absorbing plasma condition where the low-reflectivity condition $|R|^{2}\ll 1$ is satisfied, Eq. (7) becomes the reliable solution even for high-density plasma. Therefore, in the case of collisional plasma, a reliable analytic solution can be obtained by considering up to a finite-order reflection events. In our collisionless plasma model, in contrast, all the contributing infinite number of high-order paths must be included, otherwise the reflection amplitude can be large than 1 at the vicinity of the critical plasma density as shown in Fig. 4.

Fig. 4. Absolute amplitudes of reflection path integral $|R|$ on a logarithmic scale, where $T = \log {(1/n)}$. (a), (b), and (c) show results for the same conditions given in (a), (b), and (c) of Fig. 3. Results with $p_{\textrm {max}}$=0-5, are shown. On each panel, a horizontal green line is drawn at $|R| \equiv 1$. Again, open circles show results of a one-dimensional PIC simulation.

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In our collisionless plasma case, to obtain an approximate reflection amplitude by including up to a finite-order reflection paths, we set the upper limit of the integral $n_{\textrm {f}}$ to be determined by $|R(\omega ,n_{\textrm {f}})|=1$, because $|R|$ cannot be larger than $1$ by its definition of the reflection amplitude. The total reflection $|R|=1$ is obtained at different values of $T$ depending on $p_{\textrm {max}}$. When only the first-order reflection paths are considered ($p_{\textrm {max}}=0$), the total reflections are achieved at $T\sim 4 , 3$, and 2.5 as shown in Figs. 4(a), 4(b), and 4(c), respectively. The reflection amplitude given in Eq. (7), which includes only the first-order reflection paths, with the upper limit of the integral $n_{\textrm {f}}$ defined by $|R(\omega ,n_{\textrm {f}})|=1$ reproduces the PIC simulation results very well in optical qualities of reflected laser pulses, when the plasma density is much higher than the critical one (see Sec. 4). This approximation works well because the other higher-order paths up to $p_{\textrm {max}}=\infty$ mutually interfere destructively. However, this approximation cannot be employed for plasma density slightly lower than the critical density as shown next.

In order to elucidate the higher-order contributions in this collisionless case, we show the results of a PIC simulation in Fig. 5. The laser conditions are the same as in Figs. 3 and 4. The peak plasma density is set at the critical density of the central wavelength of 800 nm. The spectrum of the reflected laser from the near-critical plasma is shown. Figures 5(a), 5(c), and 5(e) are the results for the density gradients $a$ of 0.032, 0.32, and 3.2 $N_{\textrm {C}}/ \lambda$, respectively. According to the model based on Eq. (1), some frequency components of an ultrashort laser pulse can be completely reflected, while the remaining components are transmitted. Thus, owing to this spectral cutoff effect, the spectrum is rapidly truncated at the central frequency of the laser pulse. However, our model indicates that there could be a significant reflection from the underdense plasma, and thus that the spectrum is smoothly truncated near the central frequency of the laser pulse, as shown in Fig. 5. Figure 5(a) shows a more sharply truncated spectrum than those in Figs. 5(c) and 5(e). In Fig. 5(a), result with only the first-order reflection paths well reproduces the spectrum obtained by the PIC simulation.

Fig. 5. Spectra [(a), (c), and (e)] and corresponding phases [(b), (d), and (f)] of reflected laser pulses from plasma surfaces. Peak plasma densities are at critical density of 800-nm laser field. Density gradients are 0.032, 0.32, and 3.2 $N_{\textrm {C}}/\lambda$ for (a), (c), and (e) respectively. Black solid lines are results of a one-dimensional PIC simulation. Solid red, blue, and green lines are results from reflection path integrals with $p_{\textrm {max}}$ = 0, 1, and 5, respectively. Black dashed line shows spectrum of incident laser pulse. Blue dashed line shows spectrum truncated at central frequency of incident laser pulse.

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However, in Fig. 5(c), the cutoff frequency at which near 100 percent reflection occurs is more precisely reproduced by the model that includes the third-order reflection paths, while the result considering only the first-order reflection path shows visible deviation at the cutoff frequency. In Fig. 5(e), where the steep density gradient is assumed ($a = 3.2 N_{\textrm {C}}/\lambda$), neither the result with only the first-order paths nor the one including the third-order paths well reproduces the PIC result, and the result including the paths up to eleventh-order matches better with the PIC result. Inclusion of very higher-order paths will provide better agreement with the PIC results.

The spectra shown in Figs. 5(a) and 5(c), exhibit additional small modulations near the cutoff frequency, because the bulk plasma is resonantly excited by the partially transmitted resonant laser and gives rise to a resonant emission near the plasma frequency, which is not included in the analytic model. In PIC simulations, moving electrons and ions in general coexist so that the local collisions (local Coulomb interaction between electrons and ions) are naturally included, while they are forming a net charge equilibrium. However, in our PIC simulation, since we have assumed a one-dimensional configuration and, at the same time, a weak laser-intensity condition, the vertical and transverse positions of the electrons are invariant as their initial conditions. This means that the local charge equilibrium is satisfied and that the Coulomb interaction is negligible. Therefore, the collisionless plasma condition is well satisfied. However, we note that in this collisionless case, the plasma wave can be excited when the resonant frequency impinges on the plasma. The spectral modulation becomes more significant when the plasma has a lower density gradient (as shown in the comparison between (a) and (c)) because a larger portion of laser can impinges on the bulk plasma. The spectra in Figs. 5(a), 5(c), and 5(e) have the spectral phases shown in Figs. 5(b), 5(d) and 5(f), respectively. The PIC simulation results do not show a complete agreement with our model calculation. This is owing to the resonance effects. However, a smooth phase jump near the cutoff frequency is relatively well reproduced by the inclusion of the third-order reflection paths in Figs. 5(b) and 5(d). On the other hand, in Fig. 5(f), the results with the third- and eleventh-order reflection paths do not match well with the PIC result, indicating that much-higher-order reflection paths are contributing. These results demonstrate that the higher-order paths need to be considered for sharper density gradient plasma. Since the higher-order paths have negligible reflection amplitudes for low density plasma, we can conclude that the maximum order of reflections to be incorporated depends on the density gradient in the vicinity of the critical point.

4. Optical qualities of a reflected laser pulse

In Sec. 2 and Sec. 3, we have derived an analytic reflection amplitude of light from a plasma surface. In this section, we evaluate optical qualities of reflected light from static and dynamic plasmas.

4.1 Static plasma case

In the static plasma case, the laser pulse reflected back from the plasma surface is analyzed with regard to the total reflection rate (energy ratio), delay, and pulse width. In the 1D-PIC simulation, the same initial laser parameters used in Fig. 3 are employed. In analytic models of the single optical path (SOP) solution given in Eq. (1) and the reflection path integral (RPI) including only the first-order paths, the reflected laser pulse $E_{\textrm {ref}}(t)$ is calculated as

(16)$$E_{\textrm{ref}}(t) = \frac{1}{\sqrt{2\pi}}\int_{0}^{\infty}d\omega \tilde{E}_{\textrm{in}}(\omega)R(\omega)\exp{(i\omega t)},$$

where $\tilde {E}_{\textrm {in}}(\omega )$ is the Fourier-transformed incident laser field, and $R(\omega )$ is the analytic reflection amplitude given by either Eq. (1) for the SOP or Eq. (7) for the RPI. In Figs. 6(a), 6(b), and 6(c), results from the PIC simulation and the RPI are indicated by black and blue symbols, respectively. The peak plasma density is set at 0.5 $N_{\textrm {C}}$, where $N_{\textrm {C}}$ is the critical plasma density of the 800-nm laser. Under this low-plasma-density condition, only a small portion of laser can be reflected. While the SOP model does not consider this small reflection amplitude, the small portion of the reflected light can play an important role in laser-matter interaction when the reflected pulse is more tightly focused at the next stage such that the intensity of the laser becomes extremely strong. Therefore, it is of crucial importance to consider the reflection from the underdense plasmas. The results from the RPI model well reproduce those from the PIC simulation.

Fig. 6. Optical qualities of reflected laser pulse as a function of scale length of plasma surface. (a), (b), and (c) show reflection rate, delay and FWHM of reflected laser pulse for a fixed peak plasma density of 0.5 $N_{\textrm {C}}$. (d), (e), and (f) show the same optical qualities for a fixed plasma density of 1.5 $N_{\textrm {C}}$. Black, blue, and red symbols represent results from the PIC, RPI, and SOP, respectively.

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We evaluate the optical qualities of the reflected laser pulse for a peak plasma density of 1.5 $N_{\textrm {C}}$. Interestingly, the RPI model with only the first-order paths, whose upper integral limit is determined by $|R|=1$, reproduces the PIC simulation results very well. This approximative solution works due to the destructive interference of many higher-order reflection paths in the vicinity of the critical density. The SOP model Eq. (1) also provides consistent results. In a very near critical plasma case (slightly lower than the critical density), however, some of the higher-order paths must be included in the analytic model as discussed in section 3. However, since the peak plasma density never stops abruptly at the critical density for the dynamically varying plasma case, we can apply our model with only the first-order paths to the dynamically varying plasma case.

As pointed out earlier, in reality the electrons and ions undergo collisions, giving rise to a nonzero imaginary part of the refractive index, and therefore low reflectivity. To simulate the collisionless static plasma with the PIC method, we have confined our discussions to the low laser intensity ($\leq 10^{12}~\textrm {W/cm}^2$) regime. However, in the relativistic-intensity regime ($>\;10^{18}~\textrm {W/cm}^2$), both the ponderomotive force [35,36] and the relativistic mass variation [37] of electrons become very important sources of nonlinearity. The ponderomotive force, which originates from the $\vec {v}\times \vec {B}$ term of the Lorentz force, can push the electron layer in the direction of the laser propagation. The immobile ion layer driven by the relativistic laser then pulls back the electron layer, thus causing the electron layer to experience a forced oscillation. Furthermore, in such a strong-intensity regime, the effective electron mass changes and the refractive index is given by $n(x) = \frac {1}{\omega }\sqrt {1-4\pi {N_{e}(x)}/{\gamma }}$. Here, $\gamma$ is the relativistic factor given by $\sqrt {1+|p|^2}$, with $|p|$ being the absolute momentum of electron. In the plasma, $\gamma$ is also a function of time and position.

Therefore the refractive index of plasma $n(x)$ is a very dynamic function, which cannot in general be described analytically. In order to explore the extent to which our static collisionless model provides a reliable analytic solution, we obtained results of the PIC simulation with different laser intensity conditions. Figure 7 shows the optical qualities (reflection rate, delay and FHWM) of the reflected light as a function of the peak intensity. In the low plasma density case, the reflection rate (Fig. 7(a)) becomes much higher for the strong laser intensity of $10^{17}~\textrm {W/cm}^2$ as the peak plasma density grows temporally by the ponderomotive effect that forces the plasma surface to be compressed. Furthermore, the delay of the reflected pulse increases at the higher intensity (Fig. 7(b)), which implies that an increased portion of light is reflected from a deep plasma surface. For the high plasma density, the stronger laser intensity provides the lower reflection rate (Fig. 7(d)). This is a consequence of the variation of the dynamic relativistic mass, which results in a lower effective plasma density, and the energy loss of the light due to the collisions that the moving electrons undergo. The delay of laser pulse is not remarkable, suggesting that the reflection occurs in the vicinity of the well-defined turning point. For both low (Fig. 7(c)) and high (Fig. 7(f)) plasma densities, the pulse duration (FWHM) can be compressed for a strong laser intensity limit, which is attributed to the nonlinear spectral broadening stemming from the time-dependent refractive index. We note that an initial condition with a larger number of particles having a thermal energy distribution is desirable for obtaining more realistic simulation results, especially for the higher laser intensity limit. While we have considered only 64 particles per the optical wavelength, it provides a reliable result for the collisionless static plasma model with a big advantage in terms of the computational cost.

Fig. 7. PIC simulation results with several different laser intensities. The vertical dashed lines show results with peak intensity of $10^{12}~\textrm {W/cm}^{2}$. The other laser conditions are the same as the ones used for Fig. 6. The scale length of plasma is set at 1 $\lambda$. For a fixed peak plasma density of 0.5 $N_{\textrm {C}}$, the figure shows (a) reflection rate, (b) delay, and (c) FWHM of reflected laser pulse. The same optical characteristics are shown in (d), (e), and (f) for a fixed plasma density of 1.5 $N_{\textrm {C}}$.

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As can be seen, up to the intensity of $10^{14}\sim 10^{15}~\textrm {W/cm}^2$, the results are virtually the same as those of the low-intensity case. Meanwhile, from intensities of $10^{15}\sim 10^{16}~\textrm {W/cm}^2$ and greater, the result starts to show remarkable deviations both for low (Figs. 7(a), (b) and (c)) and high (Figs. 7(d), (e) and (f)) plasma densities. Therefore, this intensity dependence illustrates that our analytic model is suited for describing the reflection dynamics for laser intensities below $10^{15}~\textrm {W/cm}^2$. Furthermore, using the analytic model derived from the static plasma, we can study reflection of light from a dynamic plasma, as is demonstrated in the following subsection.

4.2 Dynamically varying plasma case

We can combine our analytic reflection amplitude with a dynamic plasma density function to study the dynamic plasma mirror operation [15,20]. In the dynamic case, the reflection amplitude $R(\omega )$ in Eq. (16) is given as a function of time, and the reflected light can be obtained by Eq. (16) in the time domain representation.

Many studies on the plasma mirror have assumed that, as the peak plasma density increases by the incident laser, the reflection probability switches on when the peak plasma density exceeds the critical one. This rapid switching-on mechanism provides a high-contrast relativistic laser pulse. However, there must be a finite amount of reflection from the underdense plasma. We demonstrate an application of our analytic model for understanding the effect of underdense plasma. Figure 8 shows the temporal profile of the reflected light from the dynamic plasma mirror. In this model, the scale length of the linear plasma profile, $L_{1}$ is fixed at a constant, while the peak plasma density $N_{\textrm {e}}(t)$ is a function of time. The peak plasma density is calculated by

(17)$$\frac{dN_{\textrm{e}}}{dt} = [N_{\textrm{SiO}_{2}} - N_{\textrm{e}}(t)] \times \Gamma(|E_{\textrm{in}}(t)|,I_{\textrm{P}}),$$

where $N_{\textrm {SiO}_{2}}$ is the molecular density of fused silica, $2.3\times 10^{22}~\textrm {cm}^{-3}$ [38]. $|E_{\textrm {in}}(t)|$ and $I_{\textrm {P}}$ are the absolute laser field amplitude at time $t$ and band gap energy of the fused silica 9.0 eV, respectively. The ADK model is employed for the ionization rate $\Gamma$ [39]. The initial laser intensity profile is composed of two temporally synchronized Gaussian pulses, a main pulse and a pedestal pulse. Peak intensity of the main pulse is set at $10^{15}~\textrm {W/cm}^{2}$ with the 25 fs of pulse duration by FWHM. The pedestal has the 3% peak intensity of the main pulse and 125-fs pulse duration by FWHM.

Fig. 8. Temporal profile of laser. The initial laser pulse is drawn by a solid black line, and the time-dependent plasma density is shown by a dotted line. Temporal profiles of reflected laser are shown by solid colored lines.

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In Fig. 8 we compare a pair of results from the RPI and SOP models. The scale length of the plasma is fixed at $L_{1} = 0.01 \lambda$. In the RPI model, which considers the effect of underdense plasma, $10^{-10}$ of the peak intensity is reflected at $t = -40$ fs (blue line in Fig. 8), while this reflection is not taken into account in the SOP model (red line in Fig. 8). In the SOP model, the plasma mirror rapidly switches on at approximately $t = -36$ fs, at which the peak plasma density $N_{\textrm {e}}(t)$ becomes equal to the critical density. When the laser reflected by the PM is much tightly focused so that the peak intensity exceeds $\sim 10^{22}~\textrm {W/cm}^{2}$, the tiny fraction reflected from the underdense plasma is not negligible because they can significantly change the initial target condition by generating a plasma before the main pulse arrives on the target material.

Our model can also qualitatively explain a recent experimental result. It has been found that there is an optimal scale length of the preformed plasma for achieving the highest-contrast relativistic laser [18]. As seen in Fig. 2, a step-function-like switching on of the reflection is possible if the plasma surface has a low density gradient. In our RPI model, a rapid turning on of the reflection is observed by setting a scale length at $L_{1} = \lambda$ (green line in Fig. 8). In Ref. [18], a prepulse generates an underdense plasma much earlier than the arrival of the main pulse. Due to the plasma expansion dynamics taking place at the picosecond time scale, the main pulse can interact with a plasma having a long scale length. On the other hand, the long plasma slab can distort the main pulse by optical chirp and absorption. Therefore, an optimal condition for the high-contrast is achieved by striking a balance between these two factors.

5. Summary and outlook

While analytic reflection amplitude of light from an inhomogeneous medium has been solved in a low-reflectivity condition [30,32,33], the result has not been applied to studies on dynamic plasma optics. In many studies, it has been considered that reflection of light occurs at the turning point where the plasma density becomes equal to the critical density. In the present study, we successfully derived an intuitive analytic complex reflection amplitude by integrating the reflection paths from a plasma surface. Our model equation reproduced the PIC simulation results for underdense, overdense, and near-critical plasma density conditions. We demonstrated that infinite-order reflection paths must be included in the case of high-density collisionless plasma. It was also demonstrated that, as an approximative solution, we can use only first-order reflection paths for high-density plasma by defining an upper limit of the reflection integral determined by $|R|=1$ . Our model calculations were well corroborated by the results of a PIC simulation.

Therefore, our model extends the analytic reflection amplitude described by Eq. (1), which is applicable only to overdense plasmas, to general plasma conditions. We expect that, by exploiting this model, a more precise analysis is possible in experimental studies on laser-plasma interactions. For example, we can combine our analytic reflection amplitude with a dynamic plasma density function to study the dynamic PM operation [15,20]. Our model also qualitatively explains how to achieve a high-contrast relativistic laser by using PM optics in the experiments.

For simplicity, in this work, we focused on a case with a low laser intensity and static collisionless plasma. However, our method is not restricted to this simple case. By considering a time-dependent imaginary portion of the refractive index, various phenomena associated with the resonant absorption [23,24] can be studied. Absorption and emission of light can be considered analytically by giving a positive $\textrm {Im}[n](x,t)$ and a negative $\textrm {Im}[n](x,t)$, respectively [31]. For relativistic laser intensity conditions, the plasma density is more dynamic due to the ponderomotive force and the relativistic electron mass variation. We can study the enhanced wave-plasma resonance absorption [40] and high-order harmonic generation [41], by applying the time-dependent reflection amplitude. We can also apply this approach to the oblique incident case by making a Lorentz transformation from the laboratory frame to a moving frame [35,36]. Our approach can be a powerful tool for characterizing and analyzing the laser-plasma interaction.

Funding

Institute for Basic Science (IBS-R012-D1).

Acknowledgements

We are grateful to Prof. Chang Hee Nam for his encouragement and support. JHM is thankful to Dr. Vishwa Bandhu Pathak for his useful advices in development of our PIC simulation code. The authors acknowledge and appreciate the reviewers for their critical review which improved the clarity and quality of the manuscript.

Disclosures

The authors declare no conflicts of interest.

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Path integral formulation of light propagation in a static collisionless plasma, and its application to dynamic plasma

Abstract

1. Introduction

2. Theoretical model

3. Validity of model

4. Optical qualities of a reflected laser pulse

4.1 Static plasma case

4.2 Dynamically varying plasma case

5. Summary and outlook

Funding

Acknowledgements

Disclosures

References

Cited By

Figures (8)

Equations (17)

Optics Express