# Gain medium

The active laser medium (also called gain medium or lasing medium) is the source of optical gain within a laser. The gain results from the stimulated emission of nuclear, electronic or molecular transitions to a lower energy state from a higher energy state previously populated by an electron current, chemical reaction or the pump source.

## Examples of active laser media

In order to lase, the active gain medium must be in a nonthermal energy distribution known as a population inversion. The preparation of this state requires an external energy source and is known as laser pumping. Pumping may be achieved with electrical currents (e.g. semiconductors, or gases via high-voltage discharges) or with light, generated by discharge lamps or by other lasers (semiconductor lasers). More exotic gain media can be pumped by chemical reactions, nuclear fission, or with high-energy electron beams.[7]

## Example of a model of gain medium

Fig.1. Simplified scheme of levels a gain medium.

A universal model valid for all laser types does not exist.[8] The simplest model includes two systems of sub-levels: upper and lower. Within each level, the fast transitions lead to the Boltzmann distribution of excitations among sub-levels (fig.1). The upper level is assumed to be metastable. Also, gain and refractive index are assumed independent of a particular way of excitation.

For good performance of the gain medium, the separation between sub-levels should be larger than working temperature; then, at pump frequency $$~\omega_{\rm p}~$$, the absorption dominates.

In the case of amplification of optical signals, the lasing frequency is called signal frequency. However, the same term is used even in the laser oscillators, when amplified radiation is used to transfer energy rather than information. The model below seems to work well for most optically-pumped solid-state lasers.

### Cross-sections

The simple medium can be characterized with effective cross-sections of absorption and emission at frequencies $$~\omega_{\rm p}~$$ and $$~\omega_{\rm s}~$$.

• Let $$~N~$$ be concentration of active centers in the solid-state lasers.
• Let $$~N_1~$$ be concentration of active centers in the ground state.
• Let $$~N_2~$$ be concentration of excited centers.
• Let $$~N_1+N_2=N~$$.

The relative concentrations can be defined as $$~n_1=N_1/N~$$ and $$~n_2=N_2/N~$$.

The rate of transitions of an active center from ground state to the excited state can be expressed with $$~ W_{\rm u}= \frac{I_{\rm p}\sigma_{\rm ap}}{ \hbar \omega_{\rm p} }+\frac{I_{\rm s}\sigma_{\rm as}}{ \hbar \omega_{\rm s} } ~$$ and The rate of transitions back to the ground state can be expressed with $$~W_{\rm d}=\frac{ I_{\rm p} \sigma_{\rm as}}{ \hbar \omega_{\rm p} }+\frac{I_{\rm s}\sigma_{\rm es}}{ \hbar \omega_{\rm s} } +\frac{1}{\tau}~$$, where $$~\sigma_{\rm as} ~$$ and $$~\sigma_{\rm ap} ~$$ are effective cross-sections of absorption at the frequencies of the pump and the signal.

$$~\sigma_{\rm es} ~$$ and $$~\sigma_{\rm ep} ~$$ are the same for stimulated emission;

$$~\frac{1}{\tau}~$$ is rate of the spontaneous decay of the upper level.

Then, the kinetic equation for relative populations can be written as follows$~ \frac {{\rm d}n_2} {{\rm d}t} = W_{\rm u} n_1 - W_{\rm d} n_2 ~$,

$$~ \frac{{\rm d}n_1}{{\rm d}t}=-W_{\rm u} n_1 + W_{\rm d} n_2 ~$$ However, these equations keep $$~ n_1+n_2=1 ~$$.

The absorption $$~ A ~$$ at the pump frequency and the gain $$~ G ~$$ at the signal frequency can be written as follows$~ A = N_1\sigma_{\rm pa} -N_2\sigma_{\rm pe} ~$, $$~ G = N_2\sigma_{\rm se} -N_1\sigma_{\rm se} ~$$.

In many cases the gain medium works in a continuous-wave or quasi-continuous regime, causing the time derivatives of populations to be negligible.

The steady-state solution can be written$~ n_2=\frac{W_{\rm u}}{W_{\rm u}+W_{\rm d}} ~$ , $$~ n_1=\frac{W_{\rm d}}{W_{\rm u}+W_{\rm d}}.$$

The dynamic saturation intensities can be defined$~ I_{\rm po}=\frac{\hbar \omega_{\rm p}}{(\sigma_{\rm ap}+\sigma_{\rm ep})\tau} ~$, $$~ I_{\rm so}=\frac{\hbar \omega_{\rm s}}{(\sigma_{\rm as}+\sigma_{\rm es})\tau} ~$$.

The absorption at strong signal$~ A_0=\frac{ND}{\sigma_{\rm as}+\sigma_{\rm es}}~$.

The gain at strong pump$~ G_0=\frac{ND}{\sigma_{\rm ap}+\sigma_{\rm ep}}~$, where $$~ D= \sigma_{\rm pa} \sigma_{\rm se} - \sigma_{\rm pe} \sigma_{\rm sa} ~$$ is determinant of cross-section.

Gain never exceeds value $$~G_0~$$, and absorption never exceeds value $$~A_0 U~$$.

At given intensities $$~I_{\rm p}~$$, $$~I_{\rm s}~$$ of pump and signal, the gain and absorption can be expressed as follows$~A=A_0\frac{U+s}{1+p+s}~$, $$~G=G_0\frac{p-V}{1+p+s}~$$,

where $$~p=I_{\rm p}/I_{\rm po}~$$ , $$~s=I_{\rm s}/I_{\rm so}~$$ , $$~U=\frac{(\sigma_{\rm as}+\sigma_{\rm es})\sigma_{\rm ap}}{D}~$$ , $$~V=\frac{(\sigma_{\rm ap}+\sigma_{\rm ep})\sigma_{\rm as}}{D}~$$ .

### Identities

The following identities[9] take place$U-V=1 ~$ , $$~ A/A_0 +G/G_0=1~.\$$

The state of gain medium can be characterized with a single parameter, such as population of the upper level, gain or absorption.

### Efficiency of the gain medium

The efficiency of a gain medium can be defined as $$~ E =\frac{I_{\rm s} G}{I_{\rm p}A}~$$.

Within the same model, the efficiency can be expressed as follows$~E =\frac{\omega_{\rm s}}{\omega_{\rm p}} \frac{1-V/p}{1+U/s}~$.

For the efficient operation both intensities, pump and signal should exceed their saturation intensities; $$~\frac{p}{V}\gg 1~$$, and $$~\frac{s}{U}\gg 1~$$.

The estimates above are valid for a medium uniformly filled with pump and signal light. The spatial hole burning may slightly reduce the efficiency because some regions are pumped well, but the pump is not efficiently withdrawn by the signal in the nodes of the interference of counter-propagating waves.

## Keywords

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## References and notes

1. Hecht, Jeff. The Laser Guidebook: Second Edition. McGraw-Hill, 1992. (Chapter 22)
2. Hecht, Chapter 22
3. Hecht, Chapters 7-15
4. Hecht, Chapters 18-21
5. F. J. Duarte and L. W. Hillman (Eds.), Dye Laser Principles (Academic, New York, 1990).
6. F. P. Schäfer (Ed.), Dye Lasers, 2nd Edition (Springer-Verlag, Berlin, 1990).
7. Encyclopedia of laser physics and technology
8. http://www.uscibooks.com/siegman.htm A.E.Siegman. Lasers. University Science Books (1986) sbn0-935702-11-3
9. http://josab.osa.org/abstract.cfm?id=84730 D.Kouznetsov, J.F.Bisson, K.Takaichi, K.Ueda. Single-mode solid-state laser with short wide unstable cavity JOSAB 22 p.1605–1619 (2005)