Laser linewidth is the
spectral linewidth of a
laser
A laser is a device that emits light through a process of optical amplification based on the stimulated emission of electromagnetic radiation. The word "laser" is an acronym for "light amplification by stimulated emission of radiation". The firs ...
beam.
Two of the most distinctive characteristics of laser emission are
spatial coherence
In physics, two wave sources are coherent if their frequency and waveform are identical. Coherence is an ideal property of waves that enables stationary (i.e., temporally or spatially constant) interference. It contains several distinct concepts, ...
and
spectral coherence. While spatial coherence is related to the
beam divergence
In electromagnetics, especially in optics, beam divergence is an angular measure of the increase in beam diameter or radius with distance from the optical aperture or antenna aperture from which the beam emerges. The term is relevant only i ...
of the laser, spectral coherence is evaluated by measuring the linewidth of laser radiation.
Theory
History: First derivation of the laser linewidth
The first human-made
coherent light source was a
maser
A maser (, an acronym for microwave amplification by stimulated emission of radiation) is a device that produces coherent electromagnetic waves through amplification by stimulated emission. The first maser was built by Charles H. Townes, Jam ...
. The acronym MASER stands for "Microwave Amplification by Stimulated Emission of Radiation". More precisely, it was the
ammonia
Ammonia is an inorganic compound of nitrogen and hydrogen with the formula . A stable binary hydride, and the simplest pnictogen hydride, ammonia is a colourless gas with a distinct pungent smell. Biologically, it is a common nitrogeno ...
maser operating at 12.5 mm
wavelength
In physics, the wavelength is the spatial period of a periodic wave—the distance over which the wave's shape repeats.
It is the distance between consecutive corresponding points of the same phase on the wave, such as two adjacent crests, tr ...
that was demonstrated by
Gordon,
Zeiger, and
Townes in 1954.
One year later the same authors derived
theoretically the linewidth of their device by making the reasonable approximations that their ammonia maser
Notably, their derivation was entirely semi-classical,
[ describing the ammonia molecules as quantum emitters and assuming classical ]electromagnetic field
An electromagnetic field (also EM field or EMF) is a classical (i.e. non-quantum) field produced by (stationary or moving) electric charges. It is the field described by classical electrodynamics (a classical field theory) and is the classica ...
s (but no quantized fields or quantum fluctuation
In quantum physics, a quantum fluctuation (also known as a vacuum state fluctuation or vacuum fluctuation) is the temporary random change in the amount of energy in a point in space, as prescribed by Werner Heisenberg's uncertainty principle. ...
s), resulting in the half-width-at-half-maximum (HWHM) maser linewidth[
:
denoted here by an asterisk and converted to the full-width-at-half-maximum (FWHM) linewidth . is the ]Boltzmann constant
The Boltzmann constant ( or ) is the proportionality factor that relates the average relative kinetic energy of particles in a gas with the thermodynamic temperature of the gas. It occurs in the definitions of the kelvin and the gas consta ...
, is the temperature
Temperature is a physical quantity that expresses quantitatively the perceptions of hotness and coldness. Temperature is measured with a thermometer.
Thermometers are calibrated in various temperature scales that historically have relied on ...
, is the output power, and and are the HWHM and FWHM linewidths of the underlying passive microwave resonator, respectively.
In 1958, two years before Maiman demonstrated the laser (initially called an "optical maser"), Schawlow and Townes transferred the maser linewidth to the optical regime by replacing the thermal energy
The term "thermal energy" is used loosely in various contexts in physics and engineering. It can refer to several different well-defined physical concepts. These include the internal energy or enthalpy of a body of matter and radiation; heat, ...
by the photon energy
Photon energy is the energy carried by a single photon. The amount of energy is directly proportional to the photon's electromagnetic frequency and thus, equivalently, is inversely proportional to the wavelength. The higher the photon's frequency, ...
, where is the Planck constant
The Planck constant, or Planck's constant, is a fundamental physical constant of foundational importance in quantum mechanics. The constant gives the relationship between the energy of a photon and its frequency, and by the mass-energy equivalen ...
and is the frequency
Frequency is the number of occurrences of a repeating event per unit of time. It is also occasionally referred to as ''temporal frequency'' for clarity, and is distinct from '' angular frequency''. Frequency is measured in hertz (Hz) which is ...
of laser light, thereby approximating that
: iv. one photon
A photon () is an elementary particle that is a quantum of the electromagnetic field, including electromagnetic radiation such as light and radio waves, and the force carrier for the electromagnetic force. Photons are Massless particle, massless ...
is coupled into the lasing mode by spontaneous emission during the photon-decay time ,
resulting in the original Schawlow–Townes approximation of the laser linewidth:[
:
Also the transfer from the microwave to the optical regime was entirely semi-classical,][ without assuming quantized fields or quantum fluctuations. Consequently, the original Schawlow–Townes equation is entirely based on semi-classical physics][ and is a four-fold approximation of a more general laser linewidth,][ which will be derived in the following.
]
Passive resonator mode: Photon-decay time
We assume a two-mirror Fabry–Pérot resonator of geometrical length , homogeneously filled with an active laser medium of refractive index
In optics, the refractive index (or refraction index) of an optical medium is a dimensionless number that gives the indication of the light bending ability of that medium.
The refractive index determines how much the path of light is bent, o ...
. We define the reference situation, namely the passive resonator mode, for a resonator whose active medium is transparent, i.e., it does not introduce gain
Gain or GAIN may refer to:
Science and technology
* Gain (electronics), an electronics and signal processing term
* Antenna gain
* Gain (laser), the amplification involved in laser emission
* Gain (projection screens)
* Information gain in d ...
or absorption.
The round-trip time of light travelling in the resonator with speed , where is the speed of light
The speed of light in vacuum, commonly denoted , is a universal physical constant that is important in many areas of physics. The speed of light is exactly equal to ). According to the special theory of relativity, is the upper limit fo ...
in vacuum
A vacuum is a space devoid of matter. The word is derived from the Latin adjective ''vacuus'' for "vacant" or " void". An approximation to such vacuum is a region with a gaseous pressure much less than atmospheric pressure. Physicists often di ...
, and the free spectral range are given by[
:
Light in the longitudinal resonator mode of interest oscillates at the ''q''th ]resonance
Resonance describes the phenomenon of increased amplitude that occurs when the frequency of an applied periodic force (or a Fourier component of it) is equal or close to a natural frequency of the system on which it acts. When an oscillat ...
frequency[
:
The exponential outcoupling decay time and the corresponding decay-rate constant are related to the intensity ]reflectance
The reflectance of the surface of a material is its effectiveness in Reflection (physics), reflecting radiant energy. It is the fraction of incident electromagnetic power that is reflected at the boundary. Reflectance is a component of the respon ...
s of the two resonator mirrors
A mirror or looking glass is an object that reflects an image. Light that bounces off a mirror will show an image of whatever is in front of it, when focused through the lens of the eye or a camera. Mirrors reverse the direction of the ima ...
by[
:
The exponential intrinsic loss time and the corresponding decay-rate constant are related to the intrinsic round-trip loss by][
:
The exponential photon-decay time and the corresponding decay-rate constant of the passive resonator are then given by][
:
All three exponential decay times average over the round-trip time ][ In the following, we assume that , , , , and , hence also , , and do not vary significantly over the frequency range of interest.
]
Passive resonator mode: Lorentzian linewidth, ''Q''-factor, coherence time and length
Besides the photon-decay time , the spectral-coherence properties of the passive resonator mode can be equivalently expressed by the following parameters. The FWHM Lorentzian linewidth of the passive resonator mode that appears in the Schawlow–Townes equation is derived from the exponential photon-decay time by Fourier transform
A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed, ...
ation,[
:
The ''Q''-factor is defined as the energy stored in the resonator mode over the energy lost per oscillation cycle,][
:
where is the number of photons in the mode. The coherence time and coherence length of light emitted from the mode are given by][
:
]
Active resonator mode: Gain, photon-decay time, Lorentzian linewidth, ''Q''-factor, coherence time and length
With the population densities and of upper and lower laser level, respectively, and the effective cross sections and of stimulated emission
Stimulated emission is the process by which an incoming photon of a specific frequency can interact with an excited atomic electron (or other excited molecular state), causing it to drop to a lower energy level. The liberated energy transfers to th ...
and absorption at the resonance frequency , respectively, the gain per unit length in the active laser medium at the resonance frequency is given by[
:
A value of induces amplification, whereas induces absorption of light at the resonance frequency , resulting in an elongated or shortened photon-decay time of photons out of the active resonator mode, respectively,][
:
The other four spectral-coherence properties of the active resonator mode are obtained in the same way as for the passive resonator mode. The Lorentzian linewidth is derived by Fourier transformation,][
:
A value of leads to gain narrowing, whereas leads to absorption broadening of the spectral linewidth. The ''Q''-factor is][
:
The coherence time and length are][
:
]
Spectral-coherence factor
The factor by which the photon-decay time is elongated by gain or shortened by absorption is introduced here as the spectral-coherence factor :[
:
All five spectral-coherence parameters then scale by the same spectral-coherence factor :][
:
]
Lasing resonator mode: Fundamental laser linewidth
With the number of photons propagating inside the lasing resonator mode, the stimulated-emission and photon-decay rates are, respectively,[
:
:
The spectral-coherence factor then becomes][
:
The photon-decay time of the lasing resonator mode is][
:
The fundamental laser linewidth is][
:
This fundamental linewidth is valid for lasers with an arbitrary energy-level system, operating below, at, or above threshold, with the gain being smaller, equal, or larger compared to the losses, and in a cw or a transient lasing regime.][
It becomes clear from its derivation that the fundamental laser linewidth is due to the semi-classical effect that the gain elongates the photon-decay time.][
]
Continuous-wave laser: The gain is smaller than the losses
The spontaneous-emission rate into the lasing resonator mode is given by[
:
Notably, is always a positive rate, because one atomic excitation is converted into one photon in the lasing mode.][ It is the source term of laser radiation and must not be misinterpreted as "noise".][ The photon-rate equation for a single lasing mode reads][
:
A CW laser is defined by a temporally constant number of photons in the lasing mode, hence . In a CW laser the stimulated- and spontaneous-emission rates together compensate the photon-decay rate. Consequently,][
:
The stimulated-emission rate is smaller than the photon-decay rate or, colloquially, "the gain is smaller than the losses".][ This fact has been known for decades and exploited to quantify the threshold behavior of semiconductor lasers.][Siegman, A. E. (1986) "Lasers", University Science Books, Mill Valley, California, ch. 13, pp. 510-524.] Even far above laser threshold the gain is still a tiny bit smaller than the losses. It is exactly this small difference that induces the finite linewidth of a CW laser.[
It becomes clear from this derivation that fundamentally the laser is an amplifier of spontaneous emission, and the cw laser linewidth is due to the semi-classical effect that the gain is smaller than the losses.][ Also in the quantum-optical approaches to the laser linewidth,][Sargent III, M.; Scully, M. O.; Lamb, Jr., W. E. (1993) "Laser Physics", 6th edition, Westview Press, Ch. 17.] based on the density-operator master equation, it can be verified that the gain is smaller than the losses.[
]
Schawlow–Townes approximation
As mentioned above, it is clear from its historical derivation that the original Schawlow–Townes equation is a four-fold approximation of the fundamental laser linewidth. Starting from the fundamental laser linewidth derived above, by applying the four approximations i.–iv. one then obtains the original Schawlow–Townes equation.
I.e., by applying the same four approximations i.–iv. to the fundamental laser linewidth that were applied in the first derivation,[ the original Schawlow–Townes equation is obtained.][
Thus, the fundamental laser linewidth is][
:
whereas the original Schawlow–Townes equation is a four-fold approximation of this fundamental laser linewidth and is merely of historical interest.
]
Additional linewidth broadening and narrowing effects
Following its publication in 1958,[ the original Schawlow–Townes equation was extended in various ways. These extended equations often trade under the same name, the "Schawlow–Townes linewidth", thereby creating a veritable confusion in the available literature on the laser linewidth, as it is often unclear which particular extension of the original Schawlow–Townes equation the respective authors refer to.
Several semi-classical extensions intended to remove one or several of the approximations i.–iv. mentioned above, thereby making steps towards the fundamental laser linewidth derived above.
The following extensions may add to the fundamental laser linewidth:
]
Measurement of laser linewidth
One of the first methods used to measure the coherence of a laser was interferometry
Interferometry is a technique which uses the '' interference'' of superimposed waves to extract information. Interferometry typically uses electromagnetic waves and is an important investigative technique in the fields of astronomy, fiber o ...
. A typical method to measure the laser linewidth is self-heterodyne interferometry. An alternative approach is the use of spectrometry.
Continuous lasers
The laser linewidth in a typical single- transverse-mode He–Ne laser (at a wavelength of 632.8 nm), in the absence of intracavity line narrowing optics, can be on the order of 1 GHz. Rare-earth-doped dielectric-based or semiconductor-based distributed-feedback lasers have typical linewidths on the order of 1 kHz. The laser linewidth from stabilized low-power continuous-wave lasers can be very narrow and reach down to less than 1 kHz. Observed linewidths are larger than the fundamental laser linewidth due to technical noise (temporal fluctuations of the optical pump power or pump current, mechanical vibrations, refractive-index and length changes due to temperature fluctuations, etc.).
Pulsed lasers
Laser linewidth from high-power, high-gain pulsed-lasers, in the absence of intracavity line narrowing optics, can be quite broad and in the case of powerful broadband dye lasers it can range from a few nm wide to as broad as 10 nm.
Laser linewidth from high-power high-gain pulsed laser oscillators, comprising line narrowing optics, is a function of the geometrical and dispersive features of the laser cavity.[F. J. Duart]
''Tunable Laser Optics'', 2nd Edition (CRC, New York, 2015)
To a first approximation the laser linewidth, in an optimized cavity, is directly proportional to the beam divergence
In electromagnetics, especially in optics, beam divergence is an angular measure of the increase in beam diameter or radius with distance from the optical aperture or antenna aperture from which the beam emerges. The term is relevant only i ...
of the emission multiplied by the inverse of the ''overall intracavity dispersion''. That is,
:
This is known as the ''cavity linewidth equation'' where is the beam divergence
In electromagnetics, especially in optics, beam divergence is an angular measure of the increase in beam diameter or radius with distance from the optical aperture or antenna aperture from which the beam emerges. The term is relevant only i ...
and the term in parenthesis (elevated to −1) is the overall intracavity dispersion. This equation was originally derived from classical optics. However, in 1992 Duarte derived this equation from quantum interferometric principles, thus linking a quantum expression with the overall intracavity angular dispersion.
An optimized multiple-prism grating laser oscillator can deliver pulse emission in the kW regime at single-longitudinal-mode linewidths of ≈ 350 MHz (equivalent to ≈ 0.0004 nm at a laser wavelength of 590 nm). Since the pulse duration from these oscillators is about 3 ns, the laser linewidth performance is near the limit allowed by the Heisenberg uncertainty principle
In quantum mechanics, the uncertainty principle (also known as Heisenberg's uncertainty principle) is any of a variety of mathematical inequalities asserting a fundamental limit to the accuracy with which the values for certain pairs of physi ...
.
See also
*Laser
A laser is a device that emits light through a process of optical amplification based on the stimulated emission of electromagnetic radiation. The word "laser" is an acronym for "light amplification by stimulated emission of radiation". The firs ...
* Fabry–Perot interferometer
*Beam divergence
In electromagnetics, especially in optics, beam divergence is an angular measure of the increase in beam diameter or radius with distance from the optical aperture or antenna aperture from which the beam emerges. The term is relevant only i ...
* Multiple-prism dispersion theory
* Multiple-prism grating laser oscillator
* N-slit interferometric equation
* Oscillator linewidth
* Solid state dye lasers
References
{{Lasers
Linewidth
A spectral line is a dark or bright line in an otherwise uniform and continuous spectrum, resulting from emission or absorption of light in a narrow frequency range, compared with the nearby frequencies. Spectral lines are often used to ident ...