Haar measure

2014-02-03T18:34:41Z

2620:101:F000:700:0:0:51:5EDB: /* Preliminaries */

{{Refimprove|date=September 2007}}
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 electronic or molecular transitions to a lower energy state from a higher energy state
previously populated by a [[laser pumping|pump source]].

Examples of active laser media include:
*Certain [[crystal]]s, typically doped with [[rare earth element|rare-earth]] [[ion]]s (e.g. [[neodymium]], [[ytterbium]], or [[erbium]]) or [[transition metal]] ions ([[titanium]] or [[chromium]]); most often [[yttrium aluminium garnet]] (YAG), [[yttrium orthovanadate]] (YVO4), or [[sapphire]] (Al2O3);<ref>Hecht, Jeff. ''The Laser Guidebook: Second Edition.'' McGraw-Hill, 1992. (Chapter 22)</ref>
*[[Glass]]es, e.g. silicate or phosphate glasses, doped with laser-active ions;<ref>Hecht, Chapter 22</ref>
*[[Gas]]es, e.g. mixtures of [[helium]] and [[neon]] (HeNe), [[nitrogen]], [[argon]], [[carbon monoxide]], [[carbon dioxide]], or metal vapors;<ref>Hecht, Chapters 7-15</ref>
*[[Semiconductor]]s, e.g. [[gallium arsenide]] (GaAs), [[indium gallium arsenide]] (InGaAs), or [[gallium nitride]] (GaN).<ref>Hecht, Chapters 18-21</ref>
* Liquids, in the form of dye solutions as used in [[dye lasers]].<ref>[[F. J. Duarte]] and L. W. Hillman (Eds.), ''Dye Laser Principles'' (Academic, New York, 1990).</ref><ref>[[F. P. Schäfer]] (Ed.), ''Dye Lasers'', 2nd Edition (Springer-Verlag, Berlin, 1990).</ref>

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 [[glow discharge|high-voltage discharges]]) or with light, generated by [[discharge lamp]]s or by other lasers ([[semiconductor laser]]s). More exotic gain media can be pumped by [[chemical reactions]], [[nuclear fission]], or with high-energy [[electron beam]]s.<ref name="rp">[http://www.rp-photonics.com/gain_media.html Encyclopedia of laser physics and technology]</ref>

==Example of a model of gain medium==
[[Image:LaserLevels1.png|thumb|right|200px|Fig.1. Simplified scheme of levels a gain medium.]]
A universal model valid for all laser types does not exist.<ref name="siegman">
{{cite book
|url=http://www.uscibooks.com/siegman.htm
|author=A.E.Siegman
|title=Lasers
|year=1986
|publisher=University Science Books
|isbn= 0-935702-11-3
}}
</ref>
The simplest model includes two systems of sub-levels: upper and lower. Within each sub-level system, the fast transitions ensure that thermal equilibrium is reached quickly, leading to the [[Maxwell–Boltzmann statistics]] of excitations among sub-levels in each system ''(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 <math>~\omega_{\rm p}~</math>, the absorption dominates.

In the case of [[Amplifier|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 laser]]s.

===Cross-sections===
The simple medium can be characterized with [[cross section (physics)|effective cross-sections]] of [[Absorption (electromagnetic radiation)|absorption]] and [[Emission (electromagnetic radiation)|emission]] at frequencies <math>~\omega_{\rm p}~</math> and <math>~\omega_{\rm s}~</math>.
*Let <math>~N~</math> be concentration of active centers in the solid-state lasers.
*Let <math>~N_1~</math> be concentration of active centers in the ground state.
*Let <math>~N_2~</math> be concentration of excited centers.
*Let <math>~N_1+N_2=N~</math>.

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

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

<math>~\sigma_{\rm es} ~</math> and <math>~\sigma_{\rm ep} ~</math> are the same for stimulated emission;

<math>~\frac{1}{\tau}~</math> is rate of the spontaneous decay of the upper level.

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

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

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

===Steady-state solution===
In many cases the gain medium works in a continuous-wave or [[quasi-continuous function|quasi-continuous]] regime, causing the time [[derivative]]s of populations to be negligible.

The steady-state solution can be written:
<math>~ n_2=\frac{W_{\rm u}}{W_{\rm u}+W_{\rm d}} ~</math>,
<math>~ n_1=\frac{W_{\rm d}}{W_{\rm u}+W_{\rm d}}.</math>

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

The absorption at strong signal:
<math>~ A_0=\frac{ND}{\sigma_{\rm as}+\sigma_{\rm es}}~</math>.

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

Gain never exceeds value <math>~G_0~</math>, and absorption never exceeds value <math>~A_0 U~</math>.

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

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

===Identities===
The following identities<ref name="uns">{{cite journal
| author=D.Kouznetsov
| coauthors=J.F.Bisson, K.Takaichi, K.Ueda
| title=Single-mode solid-state laser with short wide unstable cavity
|url=http://josab.osa.org/abstract.cfm?id=84730
|journal=[[JOSAB]]|volume=22| issue=8| pages=1605–1619
| year=2005
| doi=10.1364/JOSAB.22.001605
| bibcode=2005JOSAB..22.1605K
}}</ref> take place:
<math>U-V=1 ~ </math>, <math>~ A/A_0 +G/G_0=1~.\ </math>

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
<math>~ E =\frac{I_{\rm s} G}{I_{\rm p}A}~</math>.

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

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

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.

==See also==
*[[Population inversion]]
*[[Laser construction]]
*[[Laser science]]
*[[List of laser articles]]
*[[List of laser types]]

==References and notes==
<references/>

==External links==
*[http://www.rp-photonics.com/gain_media.html Gain media] Encyclopedia of Laser Physics and Technology

{{DEFAULTSORT:Active Laser Medium}}
[[Category:Laser gain media| ]]
[[Category:Laser science]]

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