IZhO 2019, Теория — Задача 3. Laser

Population inversion: two-level system

Consider a medium that is subject to monochromatic pumping light flux $I_0$. The incident light flux leads to transitions of molecules only between two states, i.e. the ground state 0 and the excited state 1. The absorption $\sigma_{01}$ and stimulated emission $\sigma_{10}$ cross sections are equal, i.e. $\sigma_{10} = \sigma_{01} = \sigma$ and the lifetime in the excited state is found to be $\tau$.

3.1 Write down an equation describing the change in the population of the excited state 1 over time, i.e. express the derivative $dn_1/dt$ in terms of $n_1$, $I_0$, $\sigma$ and $\tau$.

3.2 Find the population of the excited state and the difference of the populations between the excited and ground states under steady conditions as functions of the pumping light flux $I_0$. Express your answer in terms of the parameter $I_0\sigma\tau$.

3.3 Is it possible to achieve an amplification of the laser light flux in this case?

Population inversion: three-level system

Let three states of the molecule be involved in possible transitions: the ground state 0 and the two excited states 1, 2. Under the action of the pumping light flux $I_0$, the molecule can be raised from the ground state 0 to the first excited state 1. The absorption cross section of this transition is equal to $\sigma$. As a result of intramolecular relaxation, the molecule that has resided in state 1 almost instantly drops to the lower energy state 2, whose lifetime is $\tau$. In this case, assume that the stimulated emission is completely absent.

3.4 Write down an equation describing the change in the population $n_2$ of the excited state 2 over time.

3.5 Find the population $\bar{n}_2$ of the excited state 2 and the population difference $\left(\bar{n}_2 - \bar{n}_0\right)$ of the excited and the ground states under steady conditions as functions of the pumping light flux $I_0$. Express your answer in terms of the parameter $I_0\sigma\tau$.

3.6 At what minimum value of the parameter $I_0\sigma\tau$ is it possible to amplify the laser light flux with the frequency equal to the frequency of the transition $2 \to 0$?

Population inversion: four-level system

Let four states of the molecule be involved in possible transitions. Under the action of the pumping light flux $I_0$, the molecule can be raised from the ground state 0 to the first excited state 1. The absorption cross section of this transition is equal to $\sigma$. As a result of intramolecular relaxation, the molecule that has resided in state 1 almost instantly drops to the lower energy state 2, whose lifetime is $\tau$. From this state, the molecule undergoes a transition to the intermediate state 3, which results in the emission of a photon. In this case, assume that the stimulated emission is completely absent. As a result of intramolecular relaxation, the molecule that has fallen into state 3 almost instantly drops to the ground energy state 0.

3.7 At what minimum value of the parameter $I_0\sigma\tau$ is it possible to amplify the laser light flux with the frequency equal to the frequency of the transition $2 \to 3$?

Resonator

The four-level system is usually implemented in dye solutions. Dyes consist of complex molecules with multiple energy states. Therefore, the possible energy states are grouped into bands: the ground state band $S_0$ contains an almost continuous spectrum of sub-levels, and the same is true for the first excited state band $S_1$. Thus, there are two bands of possible states. The absorption of the pumping light flux $I_0$ leads to transitions from the sublevels of the ground state band $S_0$ to different sublevels of the excited state band $S_1$. Transitions between the sub-levels of state 1 occur almost instantaneously, so the stimulated emission of the laser light flux $I_1$ occurs at lower frequencies, and the stimulated emission due to the pumping light flux $I_0$ can be neglected. In the approximation described, it is sufficient to know the population of the ground and excited state bands. Rhodamine 6G is used as a dye, for which: the absorption cross section in the transition $S_0 \to S_1$ is $\sigma_A = 3{,}90\cdot 10^{-16}\ \mathrm{cm^2}$; the stimulated emission cross section $S_1 \to S_0$ is $\sigma_E = 2{,}20\cdot 10^{-16}\ \mathrm{cm^2}$; the lifetime of the molecule in the state $S_1$ is $\tau = 4{,}20\cdot 10^{-9}\ \mathrm{s}$.

To generate light, cuvette 1 with the solution of rhodamine 6G is placed between two parallel mirrors 2 and 3, thus forming a resonator. The solution is excited by a uniform pumping light flux 4 of intensity $I_0$, whose frequency strictly corresponds to the maximum absorption of the solution. The pumping light flux 4 is directed perpendicular to the axis of the resonator and fully illuminates the entire cell. The intensity of this flux, of course, decreases as it passes through the solution; however, for carrying out our estimations, consider $I_0$ constant in the entire bulk of the resonator, assuming it to be averaged over the solution volume. The laser light flux 5 generated in the resonator propagates along the resonator axis, and its amplification occurs due to multiple reflections from the resonator mirrors. Consider mirror 2 fully reflective, and the second mirror 3 translucent with the reflectance $\rho$. The absorption of light in the mirrors, the solvent-body, as well as the scattering of light and other losses can be completely ignored. The resonator has the following parameters: the cuvette length is $l = 3{,}00\ \mathrm{cm}$; the rhodamine 6G concentration is $\gamma = 1.30\cdot 10^{16}\ \mathrm{cm^{-3}}$; the reflection coefficient of the translucent mirror is $\rho = 0{,}90$; the refractive index of the solution of rhodamine 6G is $r = 1{,}50$. The speed of light is denoted as $c = 3{,}00\cdot 10^{10}\ \mathrm{cm/s}$.

For a simplified description of the laser light flux propagating along the resonator axis, one can consider the intensity of the light fluxes averaged over the length of the resonator. We denote the average intensity of the laser light flux propagating to the translucent mirror as $I_G$, and the intensity of the laser light flux propagating in the opposite direction as $I'_G$. Since the transmittance of mirror 3 is small, then we can assume that the average intensities of these fluxes be approximately equal $I_G \approx I'_G$.

3.8 Let a laser light flux $I_G$ be created in the resonator. Show that in the absence of absorption and stimulated emission, the change in the intensity of the laser flux over time is described by the equation

where $T$ stands for the so-called photon lifetime in the resonator. Express the parameter $T$ in terms of the parameters of the resonator. Calculate its numerical value.

3.9 Show that in the absence of the photon losses through mirror 3, the laser light flux variation over time obeys the following equation

where $n$ designates the population of the excited state of rhodamine 6G, and $K$ is the resonator gain. Express the resonator gain $K$ in terms of the parameters of the resonator and the stimulated emission cross section $\sigma_E$ of rhodamine 6G. Calculate its numerical value.

Stationary generation mode

In this part, we assume that the pumping light flux is constant and does not depend on time. In the stationary mode, all quantities remain constant: the population of the excited state and the laser light flux. Assume that the population of the excited state is low, i.e. $n \ll 1$.

3.10 Write down a set of equations describing the change in the population $\frac{dn}{dt}$ of the excited state and the laser light flux $\frac{dI_G}{dt}$ in the resonator.

3.11 Obtain the formula and calculate a minimum (threshold) value $n_{th}$ of the population of the excited state at which the amplification (generation) of laser light in the resonator occurs. Express it in terms of the parameters of the resonator $K, T$.

3.12 Derive the formula and calculate a minimum (threshold) value of the pumping light flux $I_{0,th}$ at which the laser light amplification in the resonator starts. Let the wavelength of the pumping light flux be $\lambda = 520\mathrm{nm}$. Calculate the pumping flux in energetic units of $\mathrm{W/cm^2}$.

3.13 Find the laser flux at the output of the resonator as a function of the pumping light flux $I_0$, and express it in terms of the ratio $\eta = I_0 / I_{0,th}$, which is called the threshold overrun, and the characteristics of molecules. Draw a graph of the laser flux at the output of the resonator as a function $\eta$.

3.14 Find the quantum output $f = N_E / N_A$, i.e. the ratio of the number of photons $N_E$ leaving the resonator per unit of time to the number of photons $N_A$ absorbed in the resonator per the same unit of time as a function of the parameter $\eta$.