### - Art Gallery -

The lasing threshold is the lowest excitation level at which a laser's output is dominated by stimulated emission rather than by spontaneous emission. Below the threshold, the laser's output power rises slowly with increasing excitation. Above threshold, the slope of power vs. excitation is orders of magnitude greater. The linewidth of the laser's emission also becomes orders of magnitude smaller above the threshold than it is below. Above the threshold, the laser is said to be lasing. The term "lasing" is a back formation from "laser," which is an acronym, not an agent noun.
Theory

The lasing threshold is reached when the optical gain of the laser medium is exactly balanced by the sum of all the losses experienced by light in one round trip of the laser's optical cavity. This can be expressed, assuming steady-state operation, as

$$R_1 R_2\exp(2g_\text{threshold}\,l) \exp(-2\alpha l) = 1..$$

Here $$R_{1}$$ and $$R_{2}$$ are the mirror (power) reflectivities, l is the length of the gain medium, $$\exp(2g_\text{threshold}\,l)$$ is the round-trip threshold power gain, and $$\exp(-2\alpha l)$$ is the round trip power loss. Note that $$\alpha >0$$. This equation separates the losses in a laser into localised losses due to the mirrors, over which the experimenter has control, and distributed losses such as absorption and scattering. The experimenter typically has little control over the distributed losses.

The optical loss is nearly constant for any particular laser ( $$\alpha=\alpha_{0}$$), especially close to threshold. Under this assumption the threshold condition can be rearranged as[1]

$$g_\text{threshold} = \alpha_{0} - \frac{1}{2l} \ln (R_1 R_2) .$$

Since $$R_1 R_2 < 1$$, both terms on the right side are positive, hence both terms increase the required threshold gain parameter. This means that minimising the gain parameter $$g_\text{threshold}$$ requires low distributed losses and high reflectivity mirrors. The appearance of l in the denominator suggests that the required threshold gain would be decreased by lengthening the gain medium, but this is not generally the case. The dependence on l is more complicated because $$\alpha_{0}$$ generally increases with l due to diffraction losses.
Measuring the internal losses

The analysis above is predicated on the laser operating in a steady-state at the laser threshold. However, this is not an assumption which can ever be fully satisfied. The problem is that the laser output power varies by orders of magnitude depending on whether the laser is above or below threshold. When very close to threshold, the smallest perturbation is able to cause huge swings in the output laser power. The formalism can, however, be used to obtain good measurements of the internal losses of the laser as follows:[2]

Most types of laser use one mirror that is highly reflecting, and another (called the output coupler) that is partially reflective. Reflectivities greater than 99.5% are routinely achieved in dielectric mirrors. The analysis can be simplified by taking $$R_1 = 1$$. The reflectivity of the output coupler can then be denoted R OC {\displaystyle R_{\text{OC}}} R_\text{OC}. The equation above then simplifies to

$$2g_\text{threshold}\,l = 2\alpha_{0}l - \ln R_\text{OC} .$$

In most cases the pumping power required to achieve lasing threshold will be proportional to the left side of the equation, that is $$P_\text{threshold} \propto 2g_\text{threshold}\,l.$$ (This analysis is equally applicable to considering the threshold energy instead of the threshold power. This is more relevant for pulsed lasers). The equation can be rewritten:

$$P_\text{threshold} = K(\,L - \ln R_\text{OC}\,),$$

where L is defined by $$L = 2\alpha_{0}l$$ and K is a constant. This relationship allows the variable L to be determined experimentally.

In order to use this expression, a series of slope efficiencies have to be obtained from a laser, with each slope obtained using a different output coupler reflectivity. The power threshold in each case is given by the intercept of the slope with the x-axis. The resulting power thresholds are then plotted versus $$-\ln R_\text{OC}$$. The theory above suggests that this graph is a straight line. A line can be fitted to the data and the intercept of the line with the x-axis found. At this point the x value is equal to the round trip loss $$L = 2\alpha_{0}l$$. Quantitative estimates of g threshold {\displaystyle g_{\text{threshold}}} g_\text{threshold} can then be made.

One of the appealing features of this analysis is that all of the measurements are made with the laser operating above the laser threshold. This allows for measurements with low random error, however it does mean that each estimate of $$P_\text{threshold}$$ requires extrapolation.

A good empirical discussion of laser loss quantification is given in the book by W. Koechner.[3]
References

Yariv, Amnon (1989). Quantum Electronics (3rd ed.). Wiley. ISBN 0-4716-0997-8.
Findlay, D.; Clay, R.A. (1966). "The measurement of internal losses in 4-level lasers". Physics Letters. Elsevier BV. 20 (3): 277–278. doi:10.1016/0031-9163(66)90363-5. ISSN 0031-9163.

W. Koechner, Solid-State Laser Engineering, Springer Series in Optical Sciences, Volume 1, Second Edition, Springer-Verlag 1985, ISBN 0-387-18747-2.

vte

Lasers

List of laser articles List of laser types List of laser applications Laser acronyms

Laser physics

Laser optics

Beam expander Beam homogenizer B Integral Chirped pulse amplification Gain-switching Gaussian beam Injection seeder Laser beam profiler M squared Mode-locking Multiple-prism grating laser oscillator Multiphoton intrapulse interference phase scan Optical amplifier Optical cavity Optical isolator Output coupler Q-switching Regenerative amplification

Laser spectroscopy

Cavity ring-down spectroscopy Confocal laser scanning microscopy Laser-based angle-resolved photoemission spectroscopy Laser diffraction analysis Laser-induced breakdown spectroscopy Laser-induced fluorescence Noise-immune cavity-enhanced optical heterodyne molecular spectroscopy Raman spectroscopy Second-harmonic imaging microscopy Terahertz time-domain spectroscopy Tunable diode laser absorption spectroscopy Two-photon excitation microscopy Ultrafast laser spectroscopy

Laser ionization

Above-threshold ionization Atmospheric-pressure laser ionization Matrix-assisted laser desorption/ionization Resonance-enhanced multiphoton ionization Soft laser desorption Surface-assisted laser desorption/ionization Surface-enhanced laser desorption/ionization

Laser fabrication

Laser beam welding Laser bonding Laser converting Laser cutting Laser cutting bridge Laser drilling Laser engraving Laser-hybrid welding Laser peening Multiphoton lithography Pulsed laser deposition Selective laser melting Selective laser sintering

Laser medicine

Computed tomography laser mammography Laser capture microdissection Laser hair removal Laser lithotripsy Laser coagulation Laser surgery Laser thermal keratoplasty LASIK Low-level laser therapy Optical coherence tomography Photorefractive keratectomy Photorejuvenation

Laser fusion

Civil applications

3D laser scanner CD DVD Blu-ray Laser lighting display Laser pointer Laser printer Laser tag

Military applications

Advanced Tactical Laser Boeing Laser Avenger Dazzler (weapon) Electrolaser Laser designator Laser guidance Laser-guided bomb Laser guns Laser rangefinder Laser warning receiver Laser weapon LLM01 Multiple Integrated Laser Engagement System Tactical High Energy Laser Tactical light ZEUS-HLONS (HMMWV Laser Ordnance Neutralization System)

Physics Encyclopedia

World

Index