Frequency-domain modelling of gain in pump-probe experiment by an inhomogeneous medium

For a slab of a single uniform gain medium, the inhomogeneous gain effect originates from two factors, the attenuation of pump beam along the propagation direction and the oscillatory field intensity due to the interference between the forward and backward waves. The former results in non-uniform gain feature in a scale longer than the pump wavelength and the latter in the order of a half of the pump wavelength. Firstly, to ensure the optical pump generates inhomogeneous population inversion without reaching the saturation regime, the pump energy should be comparable to the maximum absorbable energy. Here, the maximum absorbable energy is the photon energy required to excite all electrons in the lower level ($i=0$) to the higher level ($i=3$) during propagation of one wavelength: ${{E}_{{\rm abs}}}=A{{\lambda }_{{\rm a}}}\bar{N}\hbar {{\omega }_{{\rm a}}}/n$, where ${{\lambda }_{{\rm a}}}$ and ${{\omega }_{{\rm a}}}$ are the absorption wavelength and the absorption angular frequency respectively. The pump beam energy is the incident power multiplied by the pulse length: ${{E}_{{\rm pump}}}=n{{c}_{0}}{{\varepsilon }_{0}}AE_{0}^{2}\Delta {{t}_{{\rm pulse}\,{\rm length}}}/2$ where ${{\varepsilon }_{0}}$ is the vacuum permittivity and ${{c}_{0}}$ is the light speed in free space. For the parameters chosen for our pump-probe experiment (see above), the maximum absorbable energy $({{E}_{{\rm abs}}}=0.74\,{\rm J})$ is not negligible compared with the pump energy $({{E}_{{\rm pump}}}=3.44\,{\rm J})$. It implies that the pump loses a considerable amount of its energy by exciting the lower level electrons to the higher states when it enters the slab and the attenuated pump beam excites fewer electrons as it propagates through the slab. In addition, since the electric field at a given point is the sum of forward and backward travelling waves, it will show an oscillatory spatial profile rather than a modest change in amplitude. Therefore, a slab of gain material should be described as an inhomogeneous medium with spatially varying gain parameters as shown in figure 1(b) instead of a uniform effective medium with constant population inversion $\Delta N$ and gain permittivity ${{\varepsilon }_{{\rm g}}}$.

Our four-level system is assumed to have only two optical transitions: absorption (transition $0\leftrightarrow 3$) and stimulated emission (transition $1\leftrightarrow 2$). Three non-radiative decay processes to the right below each level are added to describe temporal dynamics of carrier densities. Time evolution of carrier densities in four-level systems follows these coupled equations [11]:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \frac{\partial {{N}_{3}}}{\partial t}=\frac{1}{\hbar {{\omega }_{{\rm a}}}}\left( \frac{\partial {{\boldsymbol{P}}_{{\bf a}}}}{\partial t}+\frac{\Delta {{\omega }_{{\bf a}}}}{2}{{\boldsymbol{P}}_{{\bf a}}} \right)\cdot {{\boldsymbol{E}}_{{\bf loc}}}-\frac{{{N}_{3}}}{{{\tau }_{32}}},\nonumber \end{align}$$

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \frac{\partial {{N}_{2}}}{\partial t}=\frac{1}{\hbar {{\omega }_{{\rm e}}}}\left( \frac{\partial {{\boldsymbol{P}}_{{\bf e}}}}{\partial t}+\frac{\Delta {{\omega }_{{\bf e}}}}{2}{{\boldsymbol{P}}_{{\bf e}}} \right)\cdot {{\boldsymbol{E}}_{{\bf loc}}}+\frac{{{N}_{3}}}{{{\tau }_{32}}}-\frac{{{N}_{2}}}{{{\tau }_{21}}},\nonumber \end{align}$$

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \frac{\partial {{N}_{1}}}{\partial t}=-\frac{1}{\hbar {{\omega }_{{\rm e}}}}\left( \frac{\partial {{\boldsymbol{P}}_{{\bf e}}}}{\partial t}+\frac{\Delta {{\omega }_{{\bf e}}}}{2}{{\boldsymbol{P}}_{{\bf e}}} \right)\cdot {{\boldsymbol{E}}_{{\bf loc}}}+\frac{{{N}_{2}}}{{{\tau }_{21}}}-\frac{{{N}_{1}}}{{{\tau }_{10}}},\nonumber \end{align}$$

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \frac{\partial {{N}_{0}}}{\partial t}=-\frac{1}{\hbar {{\omega }_{{\rm a}}}}\left( \frac{\partial {{\boldsymbol{P}}_{{\bf a}}}}{\partial t}+\frac{\Delta {{\omega }_{{\bf a}}}}{2}{{\boldsymbol{P}}_{{\bf a}}} \right)\cdot {{\boldsymbol{E}}_{{\bf loc}}}+\frac{{{N}_{1}}}{{{\tau }_{10}}}.\nonumber \end{align} \tag{ 1 }$$

Here, the subscript a and e indicate absorption and emission respectively. ${{\omega }_{{\rm a}}}$ and ${{\omega }_{{\rm e}}}$ are the resonance angular frequencies, $\Delta {{\omega }_{{\rm a}}}$ and $\Delta {{\omega }_{{\rm e}}}$ are the full bandwidths, ${{\boldsymbol{P}}_{{\bf a}}}$ and ${{\boldsymbol{P}}_{{\bf e}}}$ are the polarization densities for absorption and emission respectively. ${{N}_{i}}$ is carrier density in the $i$th level ($i=0,1,2,3$) and the total carrier density $\bar{N}$ is given by $\bar{N}={{N}_{0}}+{{N}_{1}}+{{N}_{2}}+{{N}_{3}}$. ${{\tau }_{ij}}$ is the lifetime of carriers for the non-radiative decay transition from $i$ to $j$. Other non-radiative terms such as successive decays are assumed to be negligible. ${{\boldsymbol{E}}_{{\bf loc}}}$ is a local electric field given by $[\left( \varepsilon +2 \right)/3]\boldsymbol{E}$ from the Lorentz approximation, not the average electric field [16].

In equation (1), the terms that contain the polarization densities ${{\boldsymbol{P}}_{\boldsymbol{i}}}$ and local electric field ${{\boldsymbol{E}}_{{\rm loc}}}$ correspond to excitation of the ground-level electrons and stimulated emission for $i=a$ and $i=e$ respectively while the other terms represent non-radiative relaxation processes. Gain materials such as dyes or semiconductors which are good candidates to be implemented in metamaterials have long ${{\tau }_{21}}$ compared to ${{\tau }_{32}}$ and ${{\tau }_{10}}$[17], so that electrons in the third and first level rapidly decay to the lower level, but stay in the second level. The second level is sometimes called as a metastable level for its relatively long lifetime [12]. In other words, electrons in the first and the third level relax fast whereas they stay in the second level longer compared to the time delay (${{\tau }_{10}},{{\tau }_{32}}\ll \Delta {{t}_{{\rm delay}}}\ll {{\tau }_{21}}$). Therefore, a probe beam experiences the gain materials in which the electrons relaxed from the highest level stay in the metastable level giving rise to a population inversion.

To calculate the absorption and gain effect, population difference of two optical transitions $0\leftrightarrow 3$ and $1\leftrightarrow 2$ should be known. In the remaining part of this section, we will find a formula of the spatial-dependent population difference and then show that effective permittivity for the pump and the probe beam can be expressed in terms of electric field.

For a ground state before the excitation, the initial condition is given by ${{N}_{0}}=\bar{N},{{N}_{i}}=0(i=1,2,3)$. As a pulse of the pump beam propagates in the gain medium, the carrier densities vary in time and then after the pump is turned off and before the probe is turned on, they reach a state which is different from the initial state. Since the carrier densities in all level vary slowly due to the scale mismatch of the lifetime and time delay between the pump and the probe, we approximate this state as a stationary state. In the absence of the probe beam, the carrier density in each level at the stationary state can be obtained by applying $\partial /\partial t=0$ to the second and third equations of (1). Then the carrier densities for $i=1,2,3$ at the stationary state follow

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle {{N}_{1}}:{{N}_{2}}:{{N}_{3}}={{\tau }_{10}}:{{\tau }_{21}}:{{\tau }_{32}}.\nonumber \end{align} \tag{ 2 }$$

Additionally, the time evolution of ${{N}_{3}}$ can be also expressed as a simplified form in terms of stimulated transition probability ${{\Gamma }_{ij}}$ [12].

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \frac{\partial {{N}_{3}}}{\partial t}={{\Gamma }_{03}}{{N}_{0}}-\frac{{{N}_{3}}}{{{\tau }_{32}}}.\nonumber \end{align} \tag{ 3 }$$

We will now derive the expression for the spatial-dependent ${{\Gamma }_{03}}$ starting from a formula ${{\Gamma }_{03}}={{\sigma }_{{\rm abs}}}I/\hbar {{\omega }_{{\rm a}}}$ [14] where $I(z,t)={{c}_{0}}n(z){{\varepsilon }_{0}}{{\left| E(z,t) \right|}^{2}}/2$ is a local intensity and ${{\sigma }_{{\rm abs}}}(z)={{\left( {{\lambda }_{{\rm a}}}/n \right)}^{2}}{{\gamma }_{{\rm rad}}}(z)/2\pi \Delta {{\omega }_{{\rm a}}}$ is the absorption cross section. The Lorentzian term in the absorption cross section in [14] can be neglected by assuming the narrow bandwidth of the pump beam. The radiative decay rate ${{\gamma }_{{\rm rad}}}$ is given by Fermi's golden rule [18]: ${{\gamma }_{{\rm rad}}}(z)={{e}^{2}}d_{{\rm a}}^{2}\omega _{{\rm a}}^{3}/\pi \varepsilon \hbar {{c}^{3}}$. Here, ${{d}_{i}}$ is dipole length when the dipole is oriented parallel to the incident polarization. Note that $\varepsilon$ and $c$ are the permittivity and the speed of light in the medium, both of which depend on the propagation distance $z$. In other words, the absorption cross section and the intensity are both functions of propagation distance. However, ${{\Gamma }_{03}}(z,t)={{e}^{2}}d_{{\rm a}}^{2}{{\left| E(z,t) \right|}^{2}}/{{\hbar }^{2}}\Delta {{\omega }_{{\rm a}}}$ is dependent only on the field amplitude, but not on refractive index as $n(z)$ are cancelled out. Therefore, spatially varying population difference is explicitly determined by the local field amplitude. Note that the population difference is only dependent on the wave propagation direction but uniform with respect to the wavefront plane due to plane symmetry.

We introduce a time-independent variable ${{\Gamma }_{03,{\rm ef\,\!f}}}$ to effectively describe the stimulated transition probability in the frequency domain using slowly varying envelope approximation. The electric field of pump has the Gaussian envelope with an oscillating term of its angular frequency: $\newcommand{\e}{{\rm e}} E\left(z,t \right)=E(z)\sin \left(-{{\omega }_{{\rm c}}}(t-{{t}_{0}}) \right)\exp (-{{\left(t-{{t}_{0}} \right)}^{2}}/2\Delta {{t}^{2}})$ where ${{t}_{0}}$ is center of the Gaussian envelope and $\Delta t=\Delta {{t}_{{\rm pulse}\,{\rm length}}}/2\sqrt{\log 2}$. First, ignoring the fast oscillating term $\sin \left(-{{\omega }_{{\rm c}}}(t-{{t}_{0}}) \right)$, a crude approximation will be taking average of the Gaussian envelope in time, which leads to effective electric field amplitude as ${{E}_{0,{\rm ef\,\!f}}}(z)={{E}_{0}}(z)\sqrt{2\pi }\Delta t/$$\Delta {{t}_{{\rm delay}}}$. Then, we can obtain ${{\Gamma }_{03,{\rm ef\,\!f}}}$ by substituting ${{E}_{0,{\rm ef\,\!f}}}(z)$ instead of $E\left(z,t \right)$ into ${{\Gamma }_{03}}(z,t)$ and multiplying additional $1/2$ to take into account the square term of $\sin \left(-{{\omega }_{{\rm c}}}(t-{{t}_{0}}) \right)$. To sum up, ${{\Gamma }_{03,{\rm ef\,\!f}}}={{\left(\sqrt{\pi }\Delta t/\Delta {{t}_{{\rm delay}}} \right)}^{2}}{{e}^{2}}d_{{\rm a}}^{2}{{\left| E(z) \right|}^{2}}/{{\hbar }^{2}}\Delta {{\omega }_{a}}$ is used instead of ${{\Gamma }_{30}}$ to remove time dependence. Finally, by incorporating equation (2) with (3), we obtained the expression for the population difference of both absorption and emission which are defined as $\Delta {{N}_{{\rm a}}}={{N}_{3}}-{{N}_{0}}$ and $\Delta {{N}_{{\rm e}}}={{N}_{2}}-{{N}_{1}}$:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \Delta N_{{\rm a}}^{{\rm stat}}={{N}_{3}}-{{N}_{0}}=\frac{{{\tau }_{32}}{{\Gamma }_{03,{\rm ef\,\!f}}}-1}{1+\left({{\tau }_{32}}+{{\tau }_{21}}+{{\tau }_{10}} \right){{\Gamma }_{03,{\rm ef\,\!f}}}}\bar{N},\nonumber \end{align} \tag{ 4 }$$

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \Delta N_{{\rm e}}^{{\rm stat}}={{N}_{2}}-{{N}_{1}}=\frac{\left({{\tau }_{21}}-{{\tau }_{10}} \right){{\Gamma }_{03,{\rm ef\,\!f}}}}{1+\left({{\tau }_{32}}+{{\tau }_{21}}+{{\tau }_{10}} \right){{\Gamma }_{03,{\rm ef\,\!f}}}}\bar{N}.\nonumber \end{align} \tag{ 5 }$$

Meanwhile, the population densities are governed by the following equations of motion for $i=a$ and $i=e$ [14].

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \frac{{{\partial }^{2}}{{\boldsymbol{P}}_{\boldsymbol{i}}}}{\partial {{t}^{2}}}+\Delta {{\omega }_{i}}\frac{\partial {{\boldsymbol{P}}_{\boldsymbol{i}}}}{\partial t}+\omega _{i}^{2}{{\boldsymbol{P}}_{\boldsymbol{i}}}=-{{\sigma }_{i}}\Delta {{N}_{i}}{{\boldsymbol{E}}_{{\bf loc}}}.\nonumber \end{align} \tag{ 6 }$$

The coupling constant ${{\sigma }_{i}}$ of ${{\boldsymbol{P}}_{\boldsymbol{i}}}$ to the local electric field is provided by ${{\sigma }_{i}}=2{{\omega }_{i}}{{e}^{2}}d_{i}^{2}/\hbar$. The coupling constant and the population difference determine the strength of the coupling between the induced polarization density and the local electric field and thereby effective gain permittivity. Given that pump beam usually has a picosecond scale pulse length for a narrow bandwidth and the relatively broadband probe is in tens of femtoseconds scale for visible range, both pump and probe beams can be considered as continuous waves for their relatively long pulse lengths compared to the wavelength. Therefore, we can simplify the partial differential equations to get effective permittivity using the time-harmonic assumption.

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle {{\boldsymbol{D}}_{i}}={{\varepsilon }_{0}}\boldsymbol{E}+{{\boldsymbol{P}}_{\boldsymbol{i}}}={{\varepsilon }_{0}}\varepsilon \boldsymbol{E}\nonumber \end{align}$$

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \varepsilon ={{\varepsilon }_{{\rm r}}}+{{\varepsilon }_{{\rm g}}}={{\varepsilon }_{{\rm r}}}+\frac{1}{{{\varepsilon }_{0}}}\frac{{{\sigma }_{i}}\Delta {{N}_{i}}}{{{\omega }^{2}}+{\rm i}\omega \Delta {{\omega }_{i}}-\omega _{i}^{2}}.\nonumber \end{align} \tag{ 7 }$$

The total relative permittivity $\varepsilon$ is a sum of relative permittivity ${{\varepsilon }_{{\rm r}}}$ of the gain material itself and effective gain permittivity ${{\varepsilon }_{{\rm g}}}$ with Lorentzian line shape.

To calculate the changes of the population difference and permittivity in the gain medium, let us consider the propagation of the pump beam which corresponds to absorption process by exciting electrons in the lowest level to the higher levels. Strictly speaking, the excitation of electrons by the pump is a transient process, which requires nonlinear time-domain simulation. In other words, the effective gain permittivity is a function of both time and propagation distance. To remove the time dependence, we used population difference for absorption by averaging the population difference of the initial state ($\Delta {{N}_{{\rm a}}}=-\bar{N}$) and that of the stationary state ($N_{{\rm a}}^{{\rm stat}}$). Then, we can effectively describe the interaction between the pump and the gain slab as a wave propagation in an inhomogeneous medium with the following effective gain permittivity.

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \varepsilon _{{\rm g}}^{{\rm pump}}=\frac{1}{{{\varepsilon }_{0}}}\frac{{{\sigma }_{{\rm a}}}(-\bar{N}+\Delta N_{{\rm a}}^{{\rm stat}})/2}{{{\omega }^{2}}+{\rm i}\omega \Delta {{\omega }_{{\rm a}}}-\omega _{{\rm a}}^{2}}.\nonumber \end{align} \tag{ 8 }$$

Here, the negative sign of the population difference indicates a loss. Note that the contribution of optical transition $2\leftrightarrow 1$ can be disregarded here as the pump beam does not overlap with emission wavelength. On the other hand, the propagation of probe beam is a more complicated process which involves both absorption and stimulated emission. The probe beam usually has broad wavelength range centered at the emission wavelength. In other words, the probe contains waves with absorption wavelength as well as those with emission wavelength. The wave with absorption wavelength works as if it is another pump during the propagation of the gain medium. It excites the electrons in $i=0$ level to $i=3$ level and decays. In contrast, the wave with emission wavelength induces radiative emission of electrons in $i=2$ level to $i=1$ level. Therefore, the probe beam experiences a medium with two Lorentzian permittivities, one is positive (loss from absorption) and the other is negative (gain from stimulated emission).

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \varepsilon _{{\rm g}}^{{\rm probe}}=\frac{1}{{{\varepsilon }_{0}}}\frac{{{\sigma }_{a}}\Delta N_{{\rm a}}^{{\rm stat}}}{{{\omega }^{2}}+{\rm i}\omega \Delta {{\omega }_{{\rm a}}}-\omega _{{\rm a}}^{2}}+\frac{1}{{{\varepsilon }_{0}}}\frac{{{\sigma }_{{\rm e}}}\Delta N_{{\rm e}}^{{\rm stat}}}{{{\omega }^{2}}+{\rm i}\omega \Delta {{\omega }_{{\rm e}}}-\omega _{{\rm e}}^{2}}.\nonumber \end{align} \tag{ 9 }$$

Here, the spatially non-uniform parameters in effective gain permittivity of both pump and probe are population difference $N_{{\rm a}}^{{\rm stat}}$ and $N_{{\rm e}}^{{\rm stat}}$, which can be easily obtained by the local field amplitude as stated earlier. All the other parameters are constant. Therefore, we can get effective permittivity if the field amplitude is known. Importantly, the change of the carrier densities caused by the probe can be neglected as the intensity of the probe beam is several orders of magnitude smaller than that of the pump beam.

So far, a frequency-domain approach has been introduced for an effective modelling of the gain medium in a pump-probe experiment with narrowband laser pump and broadband probe beams. Based on stationary state assumption, time-harmonic assumption and slowly varying envelope approximation, we proposed an approach to calculate effective gain permittivity in frequency-domain. In the next section, we will discuss how to describe the propagation of the pump and probe beams through the spatially varying effective medium.

To calculate the electromagnetic fields in the inhomogeneous medium, we employ the transfer matrix method for a dielectric multilayer film [19, 20]. Wave propagation in a medium with spatially varying permittivity can be readily calculated by taking a limit of thickness to zero in the dielectric multilayer problem. Suppose a dielectric multilayer slab is placed between two semi-infinite media labelled as $i=1$ and $i=M$ (figure 2). We assume that the dielectric media are non-magnetic ($\mu =1$). The slab is composed of $M-2$ layers ($i=2,3,\ldots,M-1$), where $i$th layer has refractive index ${{n}_{i}}$, impedance ${{Z}_{i}}$ and thickness ${\rm d}{{z}_{i}}$, and the total gain slab has thickness $L=\sum\nolimits_{i=2}^{M-1}{{\rm d}{{z}_{i}}}$. Then the electric and magnetic fields at the layer $i-1$ and the layer $i$ are related by transfer matrix ${{T}_{i}}$ according to continuity equation [20]:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \left(\begin{array}{@{}cccccccccccccccccccc@{}} {{E}_{i-1}} \nonumber \\ {{H}_{i-1}} \nonumber \\\end{array}\!\!\! \right)={{T}_{i}}\left(\begin{array}{@{}cccccccccccccccccccc@{}} {{E}_{i}} \nonumber \\ {{H}_{i}} \nonumber \\\end{array} \!\!\!\right)=\left(\begin{array}{@{}cccccccccccccccccccc@{}} \cos {{n}_{i}}{{k}_{0}}{\rm d}{{z}_{i}} & -j{{Z}_{i}}\sin {{n}_{i}}{{k}_{0}}{\rm d}{{z}_{i}} \nonumber \\ -jZ_{i}^{-1}\sin {{n}_{i}}{{k}_{0}}{\rm d}{{z}_{i}} & \cos {{n}_{i}}{{k}_{0}}{\rm d}{{z}_{i}} \nonumber \\\end{array} \!\!\right)\left(\begin{array}{@{}cccccccccccccccccccc@{}} {{E}_{i}} \nonumber \\ {{H}_{i}} \nonumber \\\end{array} \!\!\!\right)\nonumber \end{align} \tag{ 10 }$$

for $i=1,2,\ldots,M$ with ${\rm d}{{z}_{M}}=0$ and the initial condition $\newcommand{\e}{{\rm e}} \left(\begin{array}{@{}cccccccccccccccccccc@{}} {{E}_{M}} \\ {{H}_{M}} \\\end{array} \!\!\!\right)=\left(\begin{array}{@{}cccccccccccccccccccc@{}} 1 \\ 1/{{Z}_{i}} \\\end{array}\!\!\! \right){{E}_{M}}$. Taking a limit of ${\rm d}{{z}_{i}}\to 0$ for $i=2,3,\ldots,M-1$ gives the first order differential equation with zero diagonal terms. It is solvable for a constant or simple form of refractive index. However, since the refractive index of each layer is dependent on the electric field amplitude as described in the previous section, wave propagation in the inhomogeneous gain medium requires a TMM calculation with an infinitesimal thickness. If $\newcommand{\e}{{\rm e}} \left(\begin{array}{@{}cccccccccccccccccccc@{}} {{E}_{i}} \\ {{H}_{i}} \\\end{array}\!\!\! \right)$ is known, ${{n}_{i}}$ can be calculated by the equations (8) and (9) for pump and probe beams respectively. Then $\newcommand{\e}{{\rm e}} \left(\begin{array}{@{}cccccccccccccccccccc@{}} {{E}_{i-1}} \\ {{H}_{i-1}} \\\end{array} \!\!\!\right)$ can be obtained from the equation (10). Consequently, the electromagnetic fields in the layer $i$ is given by

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \left(\begin{array}{@{}cccccccccccccccccccc@{}} {{E}_{i}} \nonumber \\ {{H}_{i}} \nonumber \\\end{array}\!\!\! \right)=\left(\underset{l=i+1}{\overset{M}{\mathop \prod }}\,{{T}_{l}} \right)\left(\begin{array}{@{}cccccccccccccccccccc@{}} {{E}_{M}} \nonumber \\ {{H}_{M}} \nonumber \\\end{array}\!\!\! \right).\nonumber \end{align} \tag{ 11 }$$

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Combined with equations (8) and (9), the equation (11) fully determines the electromagnetic fields at every dielectric layer. Furthermore, we can get transmission and reflection spectra by ${{E}_{M+}}/{{E}_{1+}}$ and ${{E}_{1-}}/{{E}_{1+}}$ respectively.

We performed a numerical pump-probe experiment with a gain slab using (i) nonlinear calculation in time-domain, (ii) homogeneous gain and (iii) inhomogeneous gain approach. Rhodamine 800 of thickness $6.8\,\mu {\rm m}$, which corresponds to ten times of the pump wavelength, is used and modelled as 500 layers with an equal thickness. Since the overall thickness is comparable to the wavelength, the slab acts as a Fabry–Perot resonator with oscillatory transmission spectrum. Therefore, the thickness is determined to have a unity transmission at the emission wavelength to exclude the effect from multiple reflections. We also assume the gain material to have zero imaginary part of the refractive index in the absence of pump to distinguish gain effect from the attenuation from inherent losses.

The simulation parameters are chosen as follows [11]: $\sqrt{{{\varepsilon }_{{\rm r}}}}=1.62,{{\lambda }_{{\rm a}}}=680\,{\rm nm},\,{{\lambda }_{{\rm e}}}=710\ {\rm nm},\Delta {{\omega }_{{\rm a}}}=\Delta {{\omega }_{{\rm e}}}=1\times$${{10}^{14}}\,{{{\rm s}}^{-1}},{{\tau }_{32}}={{\tau }_{10}}=\!1\times {{10}^{-13}}\,{\rm s},{{\tau }_{21}}=\!5\times {{10}^{-10}}\,{\rm s},{{d}_{{\rm a}}}=0.1\,{\rm nm},$${{d}_{{\rm e}}}=0.09\,{\rm nm}$ and $\bar{N}=6\times {{10}^{24}}\,{{{\rm m}}^{-3}}$. The incident pump amplitude is set as $2\times {{10}^{7}}\,{\rm V}\,{{{\rm m}}^{-1}}$. For time-domain calculations, we used Lumerical finite-difference time-domain (FDTD) solution [21] with an additional plugin. In the FDTD simulation, the pulse lengths of pump and probe beams are set to $4\times {{10}^{-12}}\,{\rm s}$ and $1.2\times {{10}^{-14}}\,{\rm s}$ respectively with $2.5\times {{10}^{-11}}\,{\rm s}$ delay between the pump and probe.

Electric fields calculated by the homogeneous gain analysis, inhomogeneous gain approach and the FDTD are shown in figures 3(a)–(c) respectively. In the homogeneous assumption, the field amplitude of pump exhibits standing wave-like amplitude originated from the interference of waves propagating forward (transmitting) and backward (reflecting) direction as shown in figure 3(a). Instead of the oscillatory amplitude, incident field amplitude is used to calculate population inversion, which leads to constant population inversion which is overestimated throughout the slab (figure 3(d)). On the contrary, local field effect included in inhomogeneous gain analysis gives rise to distinct characteristics. The pump attenuates as well as oscillates during the propagation as shown in figure 3(b). Affected by the local field amplitude, the population inversion also decays with oscillations during the propagation which agrees well with the FDTD calculation results (figure 3(f)).

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Due to the modulated population inversion, real and imaginary part of the refractive index ($n+{\rm i}k$) at emission wavelength varies in space for the inhomogeneous gain model and FDTD whereas the refractive index is constant for the homogeneous gain model (figures 4(a)–(c)). Since the amplitude of the incident probe is much smaller than that of the pump, we simulated two cases where the absorption of probe, which corresponds to the first term in the right-hand side of equation (9), is neglected (middle column) and included (right column). In both cases, the imaginary part of the refractive index increases implying the diminished gain effect during the propagation. When the absorption is not included, the imaginary part of the refractive index is negative over the entire slab meaning that the probe beam will experience only the gain effect. Accordingly, the electric field amplifies gradually despite the negative gradient. On the other hand, if the absorption is included, the imaginary part of the refractive index becomes positive after some point and the electric field starts to decay. It means that even at the emission wavelength, absorption occurs and even exceeds the stimulated emission after several wavelengths.

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To check the functionality of our inhomogeneous gain approach, the transmission spectra of the probe beam are calculated with the homogeneous gain analysis, inhomogeneous gain approach and the FDTD for three different slab thickness: $6.8\,\mu {\rm m}$, $1.97\,\mu {\rm m}$ and $1.1\,\mu {\rm m}$ (figure 5). These thicknesses are chosen to give unity transmission without gain, but the scheme is not limited to this condition and applicable for an arbitrary thickness. Transmission spectra obtained from homogeneous gain approach is indicated by lines with asterisk marks with an oscillating feature due to the multiple reflections. Here, the gain effect is overestimated especially when the slab is several orders thicker than the pump wavelength. Although the inhomogeneous gain approach with neglected probe absorption shows reduced gain (dotted lines), which is similar to FDTD calculation (solid lines), it fails to explain low transmission near the absorption wavelength originating from the excitation of electrons by the probe beam. However, if the probe absorption is included (dashed lines), the calculated transmission contains both the gain at emission wavelength and the loss at absorption wavelength. A notable feature here is that the inhomogeneous gain calculated in frequency-domain perfectly reproduces transmission spectra simulated by nonlinear time-domain calculation. A numerical pump-probe experiment which has been regarded to be available at only time-domain can be performed in the frequency-domain either for this simple slab geometry.

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