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| '''Self-pulsation''' takes place at the beginning of [[laser action]].
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| As the pump is switched on, the gain
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| in the active medium rises and exceeds the steady-state value. The number of photons in the cavity increases,
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| depleting the gain below the steady-state value, and so on.
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| The laser pulsates; the output power at the peaks can be orders of magnitude larger than that between pulses. | |
| After several strong peaks,
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| the amplitude of pulsation reduces, and the system behaves as a linear oscillator with damping.
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| Then the pulsation decays; this is the beginning of the [[continuous-wave operation]].
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| Self-pulsation is a transient phenomenon in continuous-wave lasers.
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| ==Equations==
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| The simple model of self-pulsation deals with number <math>X</math> of photons in the laser cavity
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| and number <math>~Y~</math> of excitations in the [[gain medium]]. | |
| The evolution can be described with equations:
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| : <math>~\begin{align}
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| {{\rm d}X}/{{\rm d}t} & = KXY-UX \\
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| {{\rm d}Y}/{{\rm d}t} & = - KXY-VY+W
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| \end{align}
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| </math>
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| where
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| <math>~K = \sigma/(s t_{\rm r})~</math> is coupling constant,<br>
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| <math>~U = \theta L~</math> is rate of relaxation of photons in the [[laser cavity]],<br>
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| <math>~V = 1/\tau~</math> is rate of relaxation of excitation of the [[gain medium]],<br>
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| <math>~W = P_{\rm p}/({\hbar\omega_{\rm p}})~</math> is the [[pumping rate]];<br>
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| <math>~t_{\rm r}~</math> is the round-trip time of light in the [[laser resonator]],<br>
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| <math>~s~</math> is area of the [[pumped region]] (good [[mode matching]] is assumed);<br>
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| <math>~\sigma~</math> is the [[emission cross-section]] at the [[signal frequency]] <math>~\omega_{\rm s}~</math>.<br>
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| <math>~\theta~</math> is the [[transmission coefficient]] of the [[output coupler]].<br>
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| <math>~\tau~</math> is the lifetime of [[excited state|excitation]] of the [[gain medium]].<br>
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| <math>P_{\rm p}</math> is power of pump absorbed in the [[gain medium]] (which is assumed to be constant).
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| Such equations appear in the similar form (with various notations for variables) in [[textbook]]s on [[laser physics]],
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| for example, the monography by A.Siegman.
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| <ref name="siegman">
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| {{cite book
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| |url=http://www.uscibooks.com/siegman.htm
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| |author=A.E.Siegman
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| |title=Lasers
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| |year=1986
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| |publisher=University Science Books
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| |isbn= 0-935702-11-3
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| }}
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| </ref>
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| | |
| ==Steady-state solution==
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| : <math>
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| \begin{align}
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| X_0 & = \frac{W}{U}-\frac{V}{K} \\
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| Y_0 & = \frac{U}{K}
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| \end{align}
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| </math>
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| | |
| ==Weak pulsation==
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| Decay of small pulsation occurs with rate
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| : <math>
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| \begin{align}
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| \Gamma & = KW/(2U) \\
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| \Omega & = \sqrt{w^2-\Gamma^2}
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| \end{align}
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| </math>
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| where
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| <math>
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| w=\sqrt{KW-UV}
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| </math>
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| Practically, this rate can be orders of magnitude smaller than the repetition rate of pulses. In this case, the decay of the self-pulsation in a real lasers is determined by other physical processes, not taken into account with the initial equations above.
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| | |
| ==Strong pulsation==
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| The transient regime can be important for the quasi-continuous lasers that needs to operate in the pulsed regime, for example, to avoid the overheating
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| .<ref name="uns">{{cite journal
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| | author=D.Kouznetsov
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| | coauthors=J.-F.Bisson, K.Takaichi, K.Ueda
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| | title=Single-mode solid-state laser with short wide unstable cavity
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| |url=http://josab.osa.org/abstract.cfm?id=84730
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| |journal=[[JOSAB]]|volume=22| issue=8| pages=1605–1619
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| | year=2005
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| | doi=10.1364/JOSAB.22.001605
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| | bibcode=2005JOSAB..22.1605K
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| }}</ref>
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| | |
| <!-- Deleted image removed: [[Image:SelfPulsed.png|600px|right|thumb|Fig.4. Output power of the pulsed laser versus time measured in milliseconds.
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| Red and green: two oscillograms of the microchip laser.<ref name="kouz07">{{cite journal|url=http://www.iop.org/EJ/abstract/-search=15823442.1/1751-8121/40/9/016| author=D.Kouznetsov|coauthors=J.-F.Bisson, J.Li, K.Ueda|title=Self-pulsing laser as oscillator Toda: Approximation through elementary functions|journal=[[Journal of Physics A]]|volume=40|pages=1–18| year=2007|doi=10.1088/1751-8113/40/9/016|issue=9|bibcode = 2007JPhA...40.2107K }}</ref>
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| Black: scaled prediction from the model with the [[oscillator Toda]].
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| All curves are centered to the maximum of the first spike.]] -->
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| The only numerical solutions were believed to exist for the strong pulsation, '''spiking'''.
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| The strong spiking is possible, when <math>U/V \ll 1</math>, id est,
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| the lifetime of excitations in the active medium is large compared to the lifetime of photons inside the cavity.
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| The spiking is possible at low dumping of self-pulsation, in the corresponding both parameters
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| <math>u</math> and <math>~v^{}~</math>
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| should be small.
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| The intent of realization of the [[oscillator Toda]] at the optical bench is shown in Fig.4.
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| The colored curves are oscillograms of two shouts of the quasi-continuous diode-pumped [[Integrated circuit|microchip]]
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| [[solid-state laser]] on [[Yb:YAG]] ceramics, described by.<ref name="kouz07">{{cite journal|url=http://www.iop.org/EJ/abstract/-search=15823442.1/1751-8121/40/9/016| author=D.Kouznetsov|coauthors=J.-F.Bisson, J.Li, K.Ueda|title=Self-pulsing laser as oscillator Toda: Approximation through elementary functions|journal=[[Journal of Physics A]]|volume=40|pages=1–18| year=2007|doi=10.1088/1751-8113/40/9/016|issue=9|bibcode = 2007JPhA...40.2107K }}</ref>
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| The thick black curve represents the approximation within the simple model with [[oscillator Toda]].
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| Only qualitative agreement takes place.
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| ==Toda Oscillator==
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| Change of variables
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| : <math>
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| \begin{align}
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| X & = X_0 \exp(x) \\
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| Y & = Y_0+X_0 y \\
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| t & = z/w
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| \end{align}
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| </math>
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| lead to the equation for [[Toda oscillator]].
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| <ref name="oppo">{{cite journal|url=http://worldcat.org/issn/0722-3277| author=G.L.Oppo|coauthors=A.Politi|title=Toda potential in laser equations|
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| journal=[[Zeitschrift fur Physik]] B|volume=59|pages=111–115| year=1985|doi=10.1007/BF01325388|bibcode = 1985ZPhyB..59..111O }}</ref>
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| <ref name="kouz07"/>
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| At weak decay of the self-pulsation (even in the case of strong spiking), the solution of corresponding equation can
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| be approximated through elementary function. The error of such approximation of the solution of the initial equations is small compared to the precision of the model.
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| The pulsation of real the output of a real lasers in the transient regime usually show significant deviation from the simple model above, although the model gives good qualitative description of the
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| phenomenon of self-pulsation.
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| ==See also==
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| * [[solid-state laser]]s
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| * [[disk laser]]
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| ==References==
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| <references/>
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| *Koechner, William. ''Solid-state laser engineering'', 2nd ed. Springer-Verlag (1988).
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| ==External links==
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| * http://www.tcd.ie/Physics/Optoelectronics/research/self_pulse.php (self-pulsation in semiconductor lasers)
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| | |
| [[Category:Oscillators]]
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| [[Category:Laser science]]
| |
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