For many applications, optical frequency combs (OFCs) require a high degree of temporal coherence (narrow linewidth). Commonly, OFCs are generated in nonlinear media from a monochromatic narrow linewidth laser source or from a mode-locked laser pulse, but in all the important mid-infrared (MIR) and terahertz (THz) regions of the spectrum, OFCs can be generated intrinsically by free-running quantum cascade lasers (QCLs) with high efficiency. These combs do not look like conventional OFCs as the phases of each mode are different, and in the temporal domain, OFCs are a seemingly random combination of amplitude- and phase-modulated signals rather than a short pulse. Despite this “pseudo-randomness,” the experimental evidence suggests that the linewidth of a QCL OFC is just as narrow as that of a QCL operating in a single mode. While universally acknowledged, this observation is seemingly not fully understood. In this work, we explicate this fact by deriving the expression for the Schawlow-Townes linewidth of QCL OFCs and offer a transparent physical interpretation based on the orthogonality of laser modes, indicating that despite their very different temporal profiles, MIR and THz QCL OFCs are just as good for most applications as any other OFCs.

Optical frequency combs (OFCs) have been enjoying a healthy increase in interest in the last decade due to their applications in metrology, frequency standards, and spectroscopy.^{1,2} The major difference between an OFC and an arbitrary signal with a discrete periodic spectrum (such as produced by a multi-mode laser) is that in OFCs, the phases of all spectral components are locked, and it assures a narrow linewidth and stability (in the sense that all the frequencies remain equidistant in a long term) which is essential for all the existing and potential OFC applications. Following the original work,^{1,2} the OFCs are routinely generated by mode-locked lasers,^{3–5} which are subsequently spectrally broadened in nonlinear fiber. Nonlinear processes (self-phase modulation in the temporal domain or four-wave mixing (FWM) in the spectral domain) require dispersion compensation, and once, it is attained, a short soliton-like pulse is formed in the time domain. More recently, another strategy has been advancing—using a continuous-wave narrow linewidth laser source to generate OFCs in a nonlinear micro-resonator^{6–8}—once again with a soliton being formed, indicating that all the spectral components are locked with the same phase.^{9–11} Obviously, the linewidth of a micro-resonator or nonlinear fiber-generated OFC can be essentially the same as that of the laser that pumps it since neither scheme has spontaneous emission.

While OFCs have been most successful in the near-IR region of the spectrum, from the point of view of spectroscopy, it is mid-infrared (MIR) and terahertz (THz) regions of the spectra which are more interesting because they contain information about vibrational and rotational structures of many organic and inorganic substances. In the absence of a direct laser pump source, one must revert to using optical parametric oscillators^{9,10,12} or to down converting of near infrared OFCs^{13,14} which greatly increases the complexity, dimensions, and cost of MIR OFC sources.

Most recently, though, an entirely new method of generating MIR^{15} and THz^{16} OFCs based on free running quantum cascade lasers (QCLs)^{17} has been developed, and dual comb spectroscopic measurements have been performed using these unconventional OFCs. What makes these OFCs “unconventional” is the fact that the OFC regime in QCLs does not require any additional intra-cavity phase locking mechanism but is achieved by means of four wave-mixing in the fast (picosecond) saturable gain medium. This fast response time is a salient feature of intersubband transitions in semiconductor quantum wells and is unique to QCLs, and it is the reason that in the time domain, QCL frequency combs^{15,16} look nothing like a short pulse but is a predominately frequency-modulated (FM) signal. As explained in Refs. 18 and 19, the QCL active medium is a fast saturable gain (i.e., negative loss), which is the exact opposite of the fast saturable absorbing medium that enables production of short pulses in a mode-locked laser. Therefore, the effect of the fast saturable gain is just the opposite—it favors a constant intensity output which can be either a single mode lasing or a FM signal. The single mode regime has higher threshold due to burning of spatial and spectral holes, while the multimode FM regime mitigates the spatial and spectral hole burning and is therefore a regular operational regime of free running QCLs in which group velocity dispersion (GVD) has been compensated.^{19,20} More recent measurements have shown that the actual operating regime of THz and MIR QCL OFCs is more complicated than a simple FM^{21,22} and has a significant intensity modulation on it as explained in Refs. 23 and 24. While the signal is obviously periodic with a period of cavity round trip, within this interval, it appears entirely “random” which^{23} is the best way to mitigate spatial hole burning. Despite this, QCL OFCs have been very successfully used in spectroscopy^{25,26} which indicates that all the relevant parameters of QCL OFCs are comparable to those in other OFCs.

One of the most important parameters describing OFCs is the linewidth of each spectral line in the comb. According to the original work of Schawlow and Townes^{27} refined by Lax,^{28} the linewidth of a single mode laser is inversely proportional to the power $Pout$ or to the total number of photons in the cavity $Np$. For a multimode laser operating in N modes, one would then expect that the linewidth of each line would be inversely proportional to the number of photons in that mode or $Pout/N$, i.e., N times wider. However, it is experimentally well known that the linewidth of the mode locked lasers is comparable to the linewidth a single mode laser of equal average power since all the modes are locked into the same phase.^{29,30} The linewidth and phase noise of mode locked lasers have been theoretically explored in a number of works^{31–33} where it was assumed that all the modes are locked into the same phase since until recently that was the only practical way to lock all the phases using a saturable absorber or an active intensity of a phase modulator. However, as QCL OFCs mostly with FM character appeared, it became important to understand what kind of linewidth can be achieved in them. Measurements^{34} have shown that the linewidth is indeed very narrow, comparable to the linewidth of a single-mode QCL operating at the same power. This narrowness has been explained^{34} by introducing a concept of “supermode” but without a detailed derivation of the amount of noise going into that mode and just assuming a Langevin spontaneous emission term. As shown below, this approach is correct but would benefit from a stronger theoretical foundation.

This foundation is provided in this work with a simple derivation of the Schawlow-Townes (ST) linewidth for arbitrary laser OFCs based on the orthogonality of modes and the fact that the spontaneous noise amplitudes in each mode are uncorrelated. We also provide a simple intuitive and physically transparent picture of why only a small fraction of spontaneous noise actually affects the linewidth of the arbitrary OFC. Our conclusion is that despite their seemingly “random” temporary profiles, OFCs produced by the lasers with short gain recovery times (such as QCLs) are every bit as good as OFCs produced by single mode lasers and microresonators.

We consider the situation in which the steady phase locking of the modes has already taken place, and as explained in Ref. 31, following the standard methods in calculating oscillator linewidths,^{35,36} we assume that the noise sources are not strong enough to disrupt the phase relationship between modes, and hence, the total field can be written as

where $\omega n$ is a frequency of the n-th mode, $an(z)\u223c\u2009sin\u2009(\omega nz/v)$ is the normalized shape of the n-th orthogonal mode, $v$ is the phase velocity of light, $\u222bamandz=\delta mn$, and the Fourier amplitudes $fn$ have been normalized as $\u2211n=1N|fn|2=1$. The average power can then be defined as $P\xaf=|A|2$. The normalized distribution of normalized instant electric field $E(z)$ inside the cavity and one of the modes $am(z)$ are shown in Fig. 1(a).

Let us now write the expression for the temporal development of the slow-varying amplitude of the n-th mode^{18,19}

where $12gn(P\xaf)$ is the gain that includes both self-saturation and cross-saturation, $gnlm(P\xaf)\kappa nlm$ is the four-wave mixing (FWM) term (but in the case of active mode locking, it may have a different shape corresponding to the sideband generation), $\tau c$ is the cavity lifetime, $Dn=2n2\pi 2\tau c\u22122\beta 2vg$ is the dispersive term, $vg$ is the velocity, $\beta 2$ is the GVD, and $Sn(t)$ is the noise complex amplitude in the n-th mode. The gain and FWM term dependence on average power becomes important further on [see (8)] because in order to estimate the linewidth, one must take into account the fact that the population inversion is not pinned at the threshold but is slightly less than that. The noise can be caused by the spontaneous emission of photons in the n-th mode, but it can also be due to a far more mundane causes such as cavity length oscillations due to random vibrations. The noise amplitudes of different modes are not correlated, $\u27e8Sn*(t)Sm(t\u2032)\u27e9=S2\delta nm\delta (t\u2212t\u2032)$. Upon substituting (1) into (2), we obtain

The FWM terms under double summation have “amplitude” or in-phase parts and “phase” or quadrature parts. The quadrature terms cause frequency chirp, and once the stable regime has been reached (i.e., once the relative phases and amplitudes of modes $fn$ get locked while the amplitude of the envelope $A(t)$ can still experience a change on a slow scale), they get compensated by the group velocity dispersion terms $Dn$, i.e.,

Multiply now both sides of (3) by $fn*$ and perform summation. The amplitude terms only cause the energy redistribution between the modes, and hence, once the stable operational regime has been reached, they all sum up to zero, i.e.,

The new differential equation then becomes

Obviously, any small deviations from (4) and (5) can be treated as additional sources of noise and simply added to (6). The deviation from (4) caused by the stochastic nature of power $|A(t)|2$ is nothing but additional phase noise due to gain/index coupling which can be taken care of by introducing the linewidth enhancement factor^{37} $\alpha $ later on and which we are not considering here for the sake of simplicity as it is known to be small in the QCLs. Let us now examine the noise term. Since the phases of complex noise amplitudes in each mode are random, one must sum up the noise powers, i.e., the total noise power is $\u27e8|Stot|2\u27e9=\u2211n=1N\u27e8|Sn|2\u27e9|fn|2$.

Under a reasonable assumption that all the noise powers $\u27e8|Sn|2\u27e9$ are identical, we obtain $\u27e8|Stot|2\u27e9=\u27e8|Sn|2\u27e9$, i.e., the total noise added to a given stable phase-locked signal is equal to the noise in any given mode, and we have

where $Stot(t)$ is the noise source whose power is equal to the power of noise in any given mode.

In other words, this equation looks exactly like the equation for a single mode laser and is therefore bound to yield the linewidth that is determined by the total average power $P\xaf=|A|2$ rather than by a power in a given mode. One easy way to interpret this situation is to simply assume that one can take any linear combination of modes in the cavity and just call it an eigenmode of the system.^{34} Of course, as different modes do have different frequencies, their linear combination does not necessarily amount to an eigenmode, which after all is supposed to have a well-defined frequency, but this is still a useful analogy which allows Eq. (7) to be obtained instantly.

One can now obtain the ST linewidth for the case when the only noise source is the spontaneous emission following the framework in original ST work^{27} as elucidated by Siegman^{38} by introducing the photon number in the cavity as $Np=|A|2\tau rt/\u210f\omega $, where $\tau rt$ is the cavity round trip time, so that (7) becomes

where $nsp\u22651$ is the excess noise factor^{38} caused by the finite population of the lower laser level (subband in the case of QCL). In the steady state, the saturated gain is $g(Np)=Np\tau c\u22121/(Np+nsp)$ and the linewidth can be introduced as the effective photon decay rate

where $\Delta \nu 0=\tau c\u22121/2\pi $ is the cold cavity linewidth. Relating the photon density inside the cavity to the output power as $Pout=\u210f\omega \eta out\tau c\u22121Np$, where $\eta out$ is the outcoupling efficiency, the ST linewidth assumes a more familiar shape $\Delta \nu ST=2\pi \eta out\Delta \nu 02\u210f\omega /Pout$

Let us now consider the physical interpretation of the result—how come that despite the fact that each of N modes contributes to the spontaneous emission, only 1/N_{th} of that radiation contributes to both the phase and amplitude noise of the laser comb? For a short (mode-locked with all phases being equal) pulse, the interpretation in the time domain is obvious: the pulse sequence containing N cavity modes has the duty cycle of exactly 1/N—hence, only 1 of each N spontaneously emitted photons will affect the phase (and amplitude) of the pulse—the rest of them will appear as additive noise which can always be filtered out. For a FM frequency comb, a similar time domain interpretation can be made: assume for simplicity that the laser simply jumps from one mode to another—i.e., instant frequency is always equal to one of the cavity frequencies. Obviously, the spontaneous emission into other N-1 modes does not contribute to the multiplicative phase noise but simply presents an additive noise.

To understand the narrow linewidth of the arbitrary comb rather than using the temporal domain, it is helpful to turn instead to the spatial dependence of the signal (1) and consider the change in the phase imposed by the emission of a single photon into the m-th mode. According to (8), each photon is emitted every $\delta tm=\tau c/nsp$ and the power of that photon $\delta Pm=|\delta Em|2=\u210f\omega \tau rt\u22121$. Therefore, the phase change imposed by this photon is [see Fig. 1(a)]

where $\delta EmQ$ is the quadrature component of the spontaneous emission, and we have used the orthogonality of the laser modes in space. Now, in the time interval $\Delta t$, there will be $\Delta Nm=\Delta t/\delta tm=nsp\tau c\u22121\Delta t$ photons emitted into the m-th mode, and according to random walk process theory, the variance of the phase will be

However, according to the theory for the random walk process with the variance $\u27e8\delta \phi 2\u27e9=2\pi C\Delta t$, the power spectral density of the phase noise can be found as $S\phi (\omega )=C/\omega 2$ with the linewidth being $\Delta \nu =C$. Therefore, the linewidth can be found as

Comparing (12) with (9), one can state that the linewidth's contribution from spontaneous emission in the m-th mode is proportional to its relative weight $fm2$, i.e., roughly 1/N. Thus, physical interpretation in the spatial domain is simple*—*since the spontaneous emission into a given resonator mode only weakly overlaps in space with the actual distribution of the laser field inside the cavity, most of this emission ends up as an additive noise and does not contribute to the phase/frequency noise and linewidth. This situation is schematically explained in Fig. 1(b).

The key conclusion reached in this work is that we have confirmed the previously made argument that the linewidth of a given multi-mode phase locked laser emission does not depend on exactly what is the phase relationship between the individual modes as long as this relation exists, i.e., as long as the locking mechanism (which can be a fast saturable absorption as in conventional mode locked lasers or a fast saturable gain in QCLs) is strong enough to overcome the dispersion of group velocity and gain. The OFC signal may indeed be a short pulse,^{1} an FM signal,^{18} or a combined AM/FM signal^{21,22}—from the point of view of most practical OFC applications, the character of combs in the time domain is irrelevant and the measurements with all kinds of combs can be expected to yield the results with roughly the same precision.

Let us compare OFCs obtained directly from the phase-locked multimode laser with the one generated in a micro-resonator pumped by a single mode laser.^{10} For equal average powers, both lasers would have similar linewidths, but the multimode laser will have substantially higher additive noise due to spontaneous emission. This noise can be easily filtered out from the single mode laser. However, the efficiency of the laser OFC generation is typically much higher (precisely due to the presence of laser gain at each mode), and therefore, the micro-resonator comb must be amplified and the spontaneous emission in the amplifier will of course generate significant additive noise which will negate all the purported advantages of micro-resonator combs.

Therefore, an OFC generated by the free-running lasers with fast saturable gain media, such as QCLs, can provide measurements just as good if not superior to the comb produced using nonlinear processes in microresonators given comparable signal to noise ratios combined the simplicity, smaller size, and higher efficiency of the free running laser OFCs.

The authors acknowledge the generous support provided by the DARPA SCOUT Program. Additionally, they would like to thank Professor Jérôme Faist and Matt Singleton at ETH Zurich for many stimulating discussions.