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Self-pulsation is a transient phenomenon in continuous-wave lasers. Self-pulsation takes place at the beginning of laser action. As the pump is switched on, the gain in the active medium rises and exceeds the steady-state value. The number of photons in the cavity increases, depleting the gain below the steady-state value, and so on. The laser pulsates; the output power at the peaks can be orders of magnitude larger than that between pulses. After several strong peaks, the amplitude of pulsation reduces, and the system behaves as a linear oscillator with damping. Then the pulsation decays; this is the beginning of the continuous-wave operation.

Equations

The simple model of self-pulsation deals with number $<img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}">$ of photons in the laser cavity and number $<img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c15c8866731e9630e7b9a6944ed3e43e24ea5878" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:2.935ex; height:2.009ex;" alt="{\displaystyle ~Y~}">$ of excitations in the gain medium. The evolution can be described with equations:

<img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/55222d8439bfaa70e0ba6ea153e832b3ced21f2f" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:30.302ex; height:6.176ex;" alt="{\displaystyle ~{\begin{aligned}{{\rm {d}}X}/{{\rm {d}}t}&amp;=KXY-UX\\{{\rm {d}}Y}/{{\rm {d}}t}&amp;=-KXY-VY+W\end{aligned}}}">

where $<img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38f47171630b6f7196e61b1e7b375cfa7d39e835" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.434ex; height:2.843ex;" alt="{\displaystyle ~K=\sigma /(st_{\rm {r}})~}">$ is coupling constant,
$<img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e98723dcee4e1ef80840fbf6c896e9f931d9b2a0" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.716ex; height:2.176ex;" alt="{\displaystyle ~U=\theta L~}">$ is rate of relaxation of photons in the laser cavity,
$<img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/20072fa7f5b9a49de08c14ee6e0c9ab43330c543" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.574ex; height:2.843ex;" alt="{\displaystyle ~V=1/\tau ~}">$ is rate of relaxation of excitation of the gain medium,
$<img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aa306b77d884a044622b6e0e911b2dda267b2db9" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:16.204ex; height:3.009ex;" alt="{\displaystyle ~W=P_{\rm {p}}/({\hbar \omega _{\rm {p}}})~}">$ is the pumping rate;
$<img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4041c54d64b69feeb5b20830cf454e23756dca72" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.878ex; height:2.343ex;" alt="{\displaystyle ~t_{\rm {r}}~}">$ is the round-trip time of light in the laser resonator,
$<img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4d25cdb814ec5cfae154bd376cad1d7db1600985" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.252ex; height:1.676ex;" alt="{\displaystyle ~s~}">$ is area of the pumped region (good mode matching is assumed);
$<img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cceebc7b4e68237597aa8baeaed1021a9ecbb7ab" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.491ex; height:1.676ex;" alt="{\displaystyle ~\sigma ~}">$ is the emission cross-section at the signal frequency $<img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f87d6ca3d6e04b25ec94053bcfca51e27bfc8526" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.487ex; height:2.009ex;" alt="{\displaystyle ~\omega _{\rm {s}}~}">$ .
$<img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6c656edf3a0b2bca62a5321900d9ebbb3e0a3ce7" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.252ex; height:2.176ex;" alt="{\displaystyle ~\theta ~}">$ is the transmission coefficient of the output coupler.
$<img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a6eaf4f3acc9b2180a3c4413a3da407b1ac67d0a" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.363ex; height:1.676ex;" alt="{\displaystyle ~\tau ~}">$ is the lifetime of excitation of the gain medium.
$<img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d00b4f3d9d63325b248efc068796735182ba2d11" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.638ex; height:2.843ex;" alt="{\displaystyle P_{\rm {p}}}">$ is power of pump absorbed in the gain medium (which is assumed to be constant).

Such equations appear in the similar form (with various notations for variables) in textbooks on laser physics, for example, the monography by A.Siegman.^[1]

Steady-state solution

<img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/02242520e59042d61b1d5beccb028e8fc45ff47f" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -4.589ex; margin-bottom: -0.249ex; width:15.842ex; height:10.843ex;" alt="{\displaystyle {\begin{aligned}X_{0}&amp;={\frac {W}{U}}-{\frac {V}{K}}\\Y_{0}&amp;={\frac {U}{K}}\end{aligned}}}">

Weak pulsation

Decay of small pulsation occurs with rate

<img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e963e1f995ebd878380c03b0d59a660f2bbd1cc2" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:15.946ex; height:6.509ex;" alt="{\displaystyle {\begin{aligned}\Gamma &amp;=KW/(2U)\\\Omega &amp;={\sqrt {w^{2}-\Gamma ^{2}}}\end{aligned}}}">

where

<img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4d766fdabf10bd54975ce97212870a66cf2ce4d9" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.61ex; height:3.009ex;" alt="{\displaystyle w={\sqrt {KW-UV}}}">

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.

Strong pulsation

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.^[2]

The only numerical solutions were believed to exist for the strong pulsation, spiking. The strong spiking is possible, when $<img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2c80cbdc13e2a5dca6fc1f5e393a3c5258ed971a" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.509ex; height:2.843ex;" alt="{\displaystyle U/V\ll 1}">$ , i.e., the lifetime of excitations in the active medium is large compared to the lifetime of photons inside the cavity. The spiking is possible at low dumping of self-pulsation, in the corresponding both parameters $<img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3e6bb763d22c20916ed4f0bb6bd49d7470cffd8" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle u}">$ and $<img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6aeae99d512c48841e31b21ebe22afa42792635e" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.289ex; height:1.676ex;" alt="{\displaystyle ~v^{}~}">$ should be small.

The intent of realization of the oscillator Toda at the optical bench is shown in Fig.4. The colored curves are oscillograms of two shouts of the quasi-continuous diode-pumped microchip solid-state laser on Yb:YAG ceramics, described by.^[3] The thick black curve represents the approximation within the simple model with oscillator Toda. Only qualitative agreement takes place.

Toda Oscillator

Change of variables

<img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6c1247b53ec09f9fd461aab7ea94a39367a8cd64" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:15.887ex; height:9.176ex;" alt="{\displaystyle {\begin{aligned}X&amp;=X_{0}\exp(x)\\Y&amp;=Y_{0}+X_{0}y\\t&amp;=z/w\end{aligned}}}">

lead to the equation for Toda oscillator.^[4]^[3] At weak decay of the self-pulsation (even in the case of strong spiking), the solution of corresponding equation can 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.

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 phenomenon of self-pulsation.

References

^ A.E.Siegman (1986). Lasers. University Science Books. ISBN 978-0-935702-11-8. Archived from the original on 2016-12-06. Retrieved 2007-03-30.
^ D.Kouznetsov; J.-F.Bisson; K.Takaichi; K.Ueda (2005). "Single-mode solid-state laser with short wide unstable cavity". Journal of the Optical Society of America B. 22 (8): 1605–1619. Bibcode:2005JOSAB..22.1605K. doi:10.1364/JOSAB.22.001605.
^ ^a ^b D.Kouznetsov; J.-F.Bisson; J.Li; K.Ueda (2007). "Self-pulsing laser as oscillator Toda: Approximation through elementary functions". Journal of Physics A. 40 (9): 1–18. Bibcode:2007JPhA...40.2107K. CiteSeerX 10.1.1.535.5379. doi:10.1088/1751-8113/40/9/016. S2CID 53330023.
^ G.L.Oppo; A.Politi (1985). "Toda potential in laser equations". Zeitschrift für Physik B. 59 (1): 111–115. Bibcode:1985ZPhyB..59..111O. doi:10.1007/BF01325388. ISSN 0722-3277. S2CID 119657810.

Koechner, William. Solid-state laser engineering, 2nd ed. Springer-Verlag (1988).

External links

https://web.archive.org/web/20070514062058/http://www.tcd.ie/Physics/Optoelectronics/research/self_pulse.php (self-pulsation in semiconductor lasers)

[siegman-1] A.E.Siegman (1986). Lasers. University Science Books. ISBN 978-0-935702-11-8. Archived from the original on 2016-12-06. Retrieved 2007-03-30.

[uns-2] D.Kouznetsov; J.-F.Bisson; K.Takaichi; K.Ueda (2005). "Single-mode solid-state laser with short wide unstable cavity". Journal of the Optical Society of America B. 22 (8): 1605–1619. Bibcode:2005JOSAB..22.1605K. doi:10.1364/JOSAB.22.001605.

[kouz07-3] D.Kouznetsov; J.-F.Bisson; J.Li; K.Ueda (2007). "Self-pulsing laser as oscillator Toda: Approximation through elementary functions". Journal of Physics A. 40 (9): 1–18. Bibcode:2007JPhA...40.2107K. CiteSeerX 10.1.1.535.5379. doi:10.1088/1751-8113/40/9/016. S2CID 53330023.

[oppo-4] G.L.Oppo; A.Politi (1985). "Toda potential in laser equations". Zeitschrift für Physik B. 59 (1): 111–115. Bibcode:1985ZPhyB..59..111O. doi:10.1007/BF01325388. ISSN 0722-3277. S2CID 119657810.

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