Lasing threshold

Lasing threshold

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Encyclopedia
The lasing threshold is the lowest excitation level at which a laser's
Laser
A laser is a device that emits light through a process of optical amplification based on the stimulated emission of photons. The term "laser" originated as an acronym for Light Amplification by Stimulated Emission of Radiation...

 output is dominated by stimulated emission
Stimulated emission
In optics, stimulated emission is the process by which an atomic electron interacting with an electromagnetic wave of a certain frequency may drop to a lower energy level, transferring its energy to that field. A photon created in this manner has the same phase, frequency, polarization, and...

 rather than by spontaneous emission
Spontaneous emission
Spontaneous emission is the process by which a light source such as an atom, molecule, nanocrystal or nucleus in an excited state undergoes a transition to a state with a lower energy, e.g., the ground state and emits a photon...

. Below the threshold, the laser's output power rises slowly with increasing excitation
Laser pumping
Laser pumping is the act of energy transfer from an external source into the gain medium of a laser. The energy is absorbed in the medium, producing excited states in its atoms. When the number of particles in one excited state exceeds the number of particles in the ground state or a less-excited...

. Above threshold, the slope of power vs. excitation is orders of magnitude
Order of magnitude
An order of magnitude is the class of scale or magnitude of any amount, where each class contains values of a fixed ratio to the class preceding it. In its most common usage, the amount being scaled is 10 and the scale is the exponent being applied to this amount...

 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
Agent noun
In linguistics, an agent noun is a word that is derived from another word denoting an action, and that identifies an entity that does that action. For example, "driver" is an agent noun formed from the verb "drive". The endings "-er", "-or", and "-ist" are commonly used in English to form agent...

.

Theory


The lasing threshold is reached when the optical gain
Gain
In electronics, gain is a measure of the ability of a circuit to increase the power or amplitude of a signal from the input to the output. It is usually defined as the mean ratio of the signal output of a system to the signal input of the same system. It may also be defined on a logarithmic scale,...

 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
Optical cavity
An optical cavity or optical resonator is an arrangement of mirrors that forms a standing wave cavity resonator for light waves. Optical cavities are a major component of lasers, surrounding the gain medium and providing feedback of the laser light. They are also used in optical parametric...

. This can be expressed, assuming steady-state operation, as
.

Here and are the mirror (power) reflectivities, is the length of the gain medium, is the round-trip threshold power gain, and is the round trip power loss. Note that . 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 (), especially close to threshold. Under this assumption the threshold condition can be rearranged as
.

Since , both terms on the right side are positive, hence both terms increase the required threshold gain parameter. This means that minimising the gain parameter requires low distributed losses and high reflectivity mirrors. The appearance of 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 is more complicated because generally increases with due to diffraction
Diffraction
Diffraction refers to various phenomena which occur when a wave encounters an obstacle. Italian scientist Francesco Maria Grimaldi coined the word "diffraction" and was the first to record accurate observations of the phenomenon in 1665...

 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:

Most types of laser use one mirror that is highly reflecting, and another (called the output coupler
Output coupler
An output coupler is a partially reflective mirror used in lasers to extract a portion of the laser beam from the optical resonator....

) that is partially reflective. Reflectivities greater than 99.5% are routinely achieved in dielectric mirror
Dielectric mirror
A dielectric mirror is a type of a mirror composed of multiple thin layers of dielectric material, typically deposited on a substrate of glass or some other optical material. By careful choice of the type and thickness of the dielectric layers, one can design an optical coating with specified...

s. The analysis can be simplified by taking . The reflectivity of the output coupler can then be denoted . The equation above then simplifies to
.

In most cases the pumping
Laser pumping
Laser pumping is the act of energy transfer from an external source into the gain medium of a laser. The energy is absorbed in the medium, producing excited states in its atoms. When the number of particles in one excited state exceeds the number of particles in the ground state or a less-excited...

 power required to achieve lasing threshold will be proportional to the left side of the equation, that is . (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:
,

where is defined by and is a constant. This relationship allows the variable to be determined experimentally.

In order to use this expression, a series of slope efficiencies
Slope efficiency
The slope efficiency is an important property of a laser. It is obtained by plotting the laser output power against the input pump power. Above the lasing threshold, the resulting curve is usually close to a straight line. The slope efficiency is the slope of this line. Slope efficiency can...

 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 . 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 . Quantitative estimates of 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 requires extrapolation.

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