History: First derivation of the laser linewidth

The first human-made coherent light source was a maser. The acronym MASER stands for "Microwave Amplification by Stimulated Emission of Radiation". More precisely, it was the ammonia maser operating at 12.5 mm wavelength that was demonstrated by Gordon, Zeiger, and Townes in 1954.^[1] One year later the same authors derived^[2] theoretically the linewidth of their device by making the reasonable approximations that their ammonia maser

Notably, their derivation was entirely semi-classical,^[2] describing the ammonia molecules as quantum emitters and assuming classical electromagnetic fields (but no quantized fields or quantum fluctuations), resulting in the half-width-at-half-maximum (HWHM) maser linewidth^[2]

${\displaystyle \Delta \nu _{\rm {M}}^{*}={\frac {4\pi k_{\rm {B}}T(\Delta \nu _{\rm {c}}^{*})^{2}}{P_{\rm {out}}}}\Leftrightarrow \Delta \nu _{\rm {M}}={\frac {2\pi k_{\rm {B}}T(\Delta \nu _{\rm {c}})^{2}}{P_{\rm {out}}}},}$

denoted here by an asterisk and converted to the full-width-at-half-maximum (FWHM) linewidth ${\displaystyle \Delta \nu _{\rm {M}}=2\Delta \nu _{\rm {M}}^{*}}$. ${\displaystyle k_{\rm {B}}}$ is the Boltzmann constant, ${\displaystyle T}$ is the temperature, ${\displaystyle P_{\rm {out}}}$ is the output power, and ${\displaystyle \Delta \nu _{\rm {c}}^{*}}$ and ${\displaystyle \Delta \nu _{\rm {c}}=2\Delta \nu _{\rm {c}}^{*}}$ 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"),^[3] Schawlow and Townes^[4] transferred the maser linewidth to the optical regime by replacing the thermal energy ${\displaystyle k_{\rm {B}}T}$ by the photon energy ${\displaystyle h\nu _{\rm {L}}}$, where ${\displaystyle h}$ is the Planck constant and ${\displaystyle \nu _{\rm {L}}}$ is the frequency of laser light, thereby approximating that

${\displaystyle }$ iv. one photon is coupled into the lasing mode by spontaneous emission during the photon-decay time ${\displaystyle \tau _{\rm {c}}}$,^[5]

resulting in the original Schawlow–Townes approximation of the laser linewidth:^[4]

${\displaystyle \Delta \nu _{\rm {L,ST}}^{*}={\frac {4\pi h\nu _{\rm {L}}(\Delta \nu _{\rm {c}}^{*})^{2}}{P_{\rm {out}}}}\Leftrightarrow \Delta \nu _{\rm {L,ST}}={\frac {2\pi h\nu _{\rm {L}}(\Delta \nu _{\rm {c}})^{2}}{P_{\rm {out}}}}.}$

Again, the transfer from the microwave to the optical regime was entirely semi-classical. Consequently, the original Schawlow–Townes equation is entirely based on semi-classical physics^[2][4] and is a four-fold approximation of a more general laser linewidth,^[5] which will be derived in the following.

Passive resonator mode: Photon-decay time

We assume a two-mirror Fabry–Pérot resonator^[6] of geometrical length ${\displaystyle \ell }$, homogeneously filled with an active laser medium of refractive index ${\displaystyle n}$. 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 or absorption.

The round-trip time ${\displaystyle t_{\rm {RT}}}$ of light travelling in the resonator with speed ${\displaystyle c=c_{0}/n}$, where ${\displaystyle c_{0}}$ is the speed of light in vacuum, and the free spectral range ${\displaystyle \Delta \nu _{\rm {FSR}}}$ are given by^[6][5]

${\displaystyle t_{\rm {RT}}={\frac {1}{\Delta \nu _{\rm {FSR}}}}={\frac {2\ell }{c}}.}$

Light in the longitudinal resonator mode of interest oscillates at the qth resonance frequency^[6][5]

${\displaystyle \nu _{L}={\frac {q}{t_{\rm {RT}}}}=q\Delta \nu _{\rm {FSR}}.}$

The exponential outcoupling decay time ${\displaystyle \tau _{\rm {out}}}$ and the corresponding decay-rate constant ${\displaystyle 1/\tau _{\rm {out}}}$ are related to the intensity reflectances ${\displaystyle R_{i}}$ of the two resonator mirrors ${\displaystyle i=1,2}$ by^[6][5]

${\displaystyle R_{1}R_{2}=e^{-t_{\rm {RT}}/\tau _{\rm {out}}}\Rightarrow {\frac {1}{\tau _{\rm {out}}}}={\frac {-\ln {(R_{1}R_{2})}}{t_{\rm {RT}}}}.}$

The exponential intrinsic loss time ${\displaystyle \tau _{\rm {loss}}}$ and the corresponding decay-rate constant ${\displaystyle 1/\tau _{\rm {loss}}}$ are related to the intrinsic round-trip loss ${\displaystyle L_{\rm {RT}}}$ by^[5]

${\displaystyle 1-L_{\rm {RT}}=e^{-t_{\rm {RT}}/\tau _{\rm {loss}}}\Rightarrow {\frac {1}{\tau _{\rm {loss}}}}={\frac {-\ln {(1-L_{\rm {RT}})}}{t_{\rm {RT}}}}.}$

The exponential photon-decay time ${\displaystyle \tau _{\text{c}}}$ and the corresponding decay-rate constant ${\displaystyle 1/\tau _{\rm {c}}}$ of the passive resonator are then given by^[5]

${\displaystyle {\frac {1}{\tau _{\rm {c}}}}={\frac {1}{\tau _{\rm {out}}}}+{\frac {1}{\tau _{\rm {loss}}}}={\frac {-\ln {[R_{1}R_{2}(1-L_{\rm {RT}})]}}{t_{\rm {RT}}}}.}$

All three exponential decay times average over the round-trip time ${\displaystyle t_{\rm {RT}}.}$^[5] In the following, we assume that ${\displaystyle \ell }$, ${\displaystyle n}$, ${\displaystyle R_{1}}$, ${\displaystyle R_{2}}$, and ${\displaystyle L_{\rm {RT}}}$, hence also ${\displaystyle \tau _{\rm {out}}}$, ${\displaystyle \tau _{\rm {loss}}}$, and ${\displaystyle \tau _{\rm {c}}}$ 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 ${\displaystyle \tau _{\rm {c}}}$, the spectral-coherence properties of the passive resonator mode can be equivalently expressed by the following parameters. The FWHM Lorentzian linewidth ${\displaystyle \Delta \nu _{\rm {c}}}$ of the passive resonator mode that appears in the Schawlow–Townes equation is derived from the exponential photon-decay time ${\displaystyle \tau _{\rm {c}}}$ by Fourier transformation,^[6][5]

${\displaystyle \Delta \nu _{\rm {c}}={\frac {1}{2\pi \tau _{\rm {c}}}}.}$

The Q-factor ${\displaystyle Q_{\rm {c}}}$ is defined as the energy ${\displaystyle W_{\rm {stored}}}$ stored in the resonator mode over the energy ${\displaystyle W_{\rm {lost}}}$ lost per oscillation cycle,^[5]

${\displaystyle Q_{\rm {c}}=2\pi {\frac {W_{\rm {stored}}(t)}{W_{\rm {lost}}(t)}}=2\pi {\frac {\varphi (t)}{-{\frac {1}{\nu _{L}}}{\frac {d}{dt}}\varphi (t)}}=2\pi \nu _{L}\tau _{\rm {c}}={\frac {\nu _{L}}{\Delta \nu _{\rm {c}}}},}$

where ${\displaystyle \varphi =W_{\rm {stored}}/h\nu _{L}}$ is the number of photons in the mode. The coherence time ${\displaystyle \tau _{\rm {c}}^{\rm {coh}}}$ and coherence length ${\displaystyle \ell _{\rm {c}}^{\rm {coh}}}$ of light emitted from the mode are given by^[5]

${\displaystyle \tau _{\rm {c}}^{\rm {coh}}={\frac {1}{c}}\ell _{\rm {c}}^{\rm {coh}}=2\tau _{\rm {c}}.}$

Active resonator mode: Gain, photon-decay time, Lorentzian linewidth, Q-factor, coherence time and length

With the population densities ${\displaystyle N_{2}}$ and ${\displaystyle N_{1}}$ of upper and lower laser level, respectively, and the effective cross sections ${\displaystyle \sigma _{\rm {e}}}$ and ${\displaystyle \sigma _{\rm {a}}}$ of stimulated emission and absorption at the resonance frequency ${\displaystyle \nu _{L}}$, respectively, the gain per unit length in the active laser medium at the resonance frequency ${\displaystyle \nu _{L}}$ is given by^[5]

${\displaystyle g=\sigma _{\rm {e}}N_{2}-\sigma _{\rm {a}}N_{1}.}$

A value of ${\displaystyle g>0}$ induces amplification, whereas ${\displaystyle g<0}$ induces absorption of light at the resonance frequency ${\displaystyle \nu _{L}}$, resulting in an elongated or shortened photon-decay time ${\displaystyle \tau _{\rm {L}}}$ of photons out of the active resonator mode, respectively,^[5]

${\displaystyle {\frac {1}{\tau _{\rm {L}}}}={\frac {1}{\tau _{\rm {c}}}}-cg.}$

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,^[5]

${\displaystyle \Delta \nu _{\rm {L}}={\frac {1}{2\pi \tau _{\rm {L}}}}.}$

A value of ${\displaystyle g>0}$ leads to gain narrowing, whereas ${\displaystyle g<0}$ leads to absorption broadening of the spectral linewidth. The Q-factor is^[5]

${\displaystyle Q_{\rm {L}}=2\pi {\frac {W_{\rm {stored}}(t)}{W_{\rm {lost}}(t)}}=2\pi {\frac {\varphi (t)}{-{\frac {1}{\nu _{L}}}{\frac {d}{dt}}\varphi (t)}}=2\pi \nu _{L}\tau _{\rm {L}}={\frac {\nu _{L}}{\Delta \nu _{\rm {L}}}}.}$

The coherence time and length are^[5]

${\displaystyle \tau _{\rm {L}}^{\rm {coh}}={\frac {1}{c}}\ell _{\rm {L}}^{\rm {coh}}=2\tau _{\rm {L}}.}$

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 ${\displaystyle \Lambda }$:^[5]

${\displaystyle \Lambda$ :={\frac {1}{1-cg\tau _{\rm {c}}}}.}

All five spectral-coherence parameters then scale by the same spectral-coherence factor ${\displaystyle \Lambda }$:^[5]

${\displaystyle {\begin{aligned}\tau _{\rm {L}}&=\Lambda \tau _{\rm {c}},&(\Delta \nu _{\rm {L}})^{-1}&=\Lambda (\Delta \nu _{\rm {c}})^{-1},&Q_{\rm {L}}&=\Lambda Q_{\rm {c}},&\tau _{\rm {L}}^{\rm {coh}}&=\Lambda \tau _{\rm {c}}^{\rm {coh}},&\ell _{\rm {L}}^{\rm {coh}}&=\Lambda \ell _{\rm {c}}^{\rm {coh}}.\end{aligned}}}$

Lasing resonator mode: Fundamental laser linewidth

With the number ${\displaystyle \varphi }$ of photons propagating inside the lasing resonator mode, the stimulated-emission and photon-decay rates are, respectively,^[5]

${\displaystyle R_{\rm {st}}=cg\varphi ,}$

${\displaystyle R_{\rm {decay}}={\frac {1}{\tau _{\rm {c}}}}\varphi .}$

The spectral-coherence factor then becomes^[5]

${\displaystyle \Lambda ={\frac {R_{\rm {decay}}}{R_{\rm {decay}}-R_{\rm {st}}}}.}$

The photon-decay time of the lasing resonator mode is^[5]

${\displaystyle \tau _{\rm {L}}=\Lambda \tau _{\rm {c}}={\frac {R_{\rm {decay}}}{R_{\rm {decay}}-R_{\rm {st}}}}\tau _{\rm {c}}.}$

The fundamental laser linewidth is^[5]

${\displaystyle \Delta \nu _{\rm {L}}={\frac {1}{\Lambda }}\Delta \nu _{\rm {c}}={\frac {R_{\rm {decay}}-R_{\rm {st}}}{R_{\rm {decay}}}}\Delta \nu _{\rm {c}}.}$

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.^[5]

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.^[5]

Continuous-wave laser: The gain is smaller than the losses

The spontaneous-emission rate into the lasing resonator mode is given by^[5]

${\displaystyle R_{\rm {sp}}=c\sigma _{\rm {e}}N_{2}.}$

Notably, ${\displaystyle R_{\rm {sp}}}$ is always a positive rate, because one atomic excitation is converted into one photon in the lasing mode.^[7][5] It is the source term of laser radiation and must not be misinterpreted as "noise".^[5] The photon-rate equation for a single lasing mode reads^[5]

${\displaystyle {\frac {d}{dt}}\varphi =R_{\rm {sp}}+R_{\rm {st}}-R_{\rm {decay}}=c\sigma _{\rm {e}}N_{2}+cg\varphi -{\frac {1}{\tau _{\rm {c}}}}\varphi .}$

A CW laser is defined by a temporally constant number of photons in the lasing mode, hence ${\displaystyle d\varphi /dt=0}$. In a CW laser the stimulated- and spontaneous-emission rates together compensate the photon-decay rate. Consequently,^[5]

${\displaystyle R_{\rm {st}}-R_{\rm {decay}}=-R_{\rm {sp}}<0.}$

The stimulated-emission rate is smaller than the photon-decay rate or, colloquially, "the gain is smaller than the losses".^[5] This fact has been known for decades and exploited to quantify the threshold behavior of semiconductor lasers.^{[8][9][10][11]} 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.^[5]

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.^[5] Also in the quantum-optical approaches to the laser linewidth,^[12] based on the density-operator master equation, it can be verified that the gain is smaller than the losses.^[5]

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 ${\displaystyle \Delta \nu _{\rm {L}}}$ 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 ${\displaystyle \Delta \nu _{\rm {L}}}$ that were applied in the first derivation,^[2][4] the original Schawlow–Townes equation is obtained.^[5]

Thus, the fundamental laser linewidth is^[5]

${\displaystyle \Delta \nu _{\rm {L}}={\frac {1}{\Lambda }}\Delta \nu _{\rm {c}}={\frac {R_{\rm {decay}}-R_{\rm {st}}}{R_{\rm {decay}}}}\Delta \nu _{\rm {c}}=(1-cg\tau _{\rm {c}})\Delta \nu _{\rm {c}}=\Delta \nu _{\rm {c}}-{\frac {cg}{2\pi }},}$

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,^[4] 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: