Linear and nonlinear frequency- and time-domain spectroscopy with multiple frequency combs

Mode-locked lasers, widely used for generating ultrashort light pulses, operate by keeping a fixed phase relationship between the modes of the laser, leading to periodic constructive interferences that result in a train of pulses with a well-defined interpulse separation T. In the frequency domain, such an electric field corresponds to a series of frequency spikes with fixed wavenumber separation 1/T, called a frequency comb.^1–5 Their advent has revolutionized high-precision metrology,^5,6 enabled the creation of intense, few-cycle pulses with controlled carrier-envelope phase,^7,8 and shows great promise for novel spectroscopic applications.^2,5,9–15 Frequency combs can be used for fast data acquisition and also offer ultrahigh resolution spectra.

In a standard ultrafast nonlinear spectroscopy set-up, multiple pulses generate several interaction pathways within the matter that may then be separated through various techniques (phase-cycling, phase matching). The laser’s repetition rate is then used to accumulate signals with a satisfactory signal to noise ratio. In the scheme presented here, all the pulses in the comb sequence are used to intrinsically isolate a desired contribution to the signal. A variety of spectroscopic signals have been measured using dual frequency combs (DFCs). Traditionally, experiments have been conducted by mixing the two lasers, with either one or both having travelled through the material, and measuring the interference between the two.^5,16,17 The Fourier transform of this interferogram gives the signal that, at linear order, is the absorption spectrum and higher-order contributions showing Raman and other resonances.^18,19 The use of multiple frequency combs with slightly different frequency spacings induces slow temporal modulations of the various pathways contributing to the signal.²⁰ These down-shifted frequencies can be small enough to allow their detection by electronic means in real time while conserving the high frequency resolution of the comb.

This technique can be alternatively described in the time domain. Since a frequency shift is equivalent to a modulation in the time domain, the advantages of using a frequency comb can also be, in principle, obtained by modulating pulses within a sequence with a varying frequency.^21,22 This technique has been used^23,24 to detect and select various interaction pathways in two-photon fluorescence. The modulation of the pulses is achieved by using an independent acousto-optic modulator (AOM) for each of the four pulses in the interaction scheme. A standard ultrafast laser can be used in this application, and the modulation of the pulses is tailored to match the repetition rate of the laser, releasing the constraint of using a comb in the frequency domain. On the other side, frequency combs offer a frequency shift control that would be difficult to achieve by modulating standard sources in the time domain.

In this manuscript, we develop a unified description for the multi-comb spectroscopies and the phase-modulated AOM detection techniques and compare the respective signals. Both techniques have their merits and limitations. AOM is more straightforward to implement since it requires a simpler ultrafast laser:²⁵ a typical laser oscillator at 100 MHz can be used and the downshifted signals are acquired at 3–13 kHz using a lock-in amplifier. However, phase modulation using AOM still requires varying the delays between the pulses. This requires long acquisition times because of the necessity to scan various delay stages as well as requiring a larger setup. Frequency comb techniques, on the other hand, do not require scanning any delays, and the acquisition can be done at the much higher frequency of the comb repetition rate. The downshifting of the relevant signals is due to the frequency shift between various combs. The signal can then be modulated at very low, few-Hertz frequencies.

We extend the formalism of multidimensional spectroscopy to account for incoming fields composed of multiple combs. This can be used to analyze the aforementioned experiments and to guide the development of these techniques. In Sec. II, we derive general expressions for nonlinear signals using broadband transmission and fluorescence detection. Our derivations make clear that the same ideas of temporal signal modulation can be applied to incoherent detection when the exciting field is an overlap of combs. In Sec. III, we perturbatively expand the signal to linear and third order to demonstrate the effect of using a DFC field. Four-comb spectroscopic techniques for measuring the third order response function are discussed in Sec. IV. Finally, in Sec. V, we compare the phase modulation of ultrafast pulses with the frequency comb techniques.

First, we demonstrate that coherent detection can be described as the rate of material energy change due to the laser combs. We then discuss incoherent fluorescence detection, which has recently been used to measure two-photon absorption of Rubidium vapor.²⁶ We derive expressions for the fluorescence signal, using the excited state population as a proxy quantity for this signal. In both cases, we derive the signal expressions in terms of the material quantities without specifying how they are generated, thus subsuming arbitrary-order interaction and allowing specialization to linear or nonlinear signals in Secs. III and IV.

The time-dependent dissipation of field energy by the interaction with matter can be measured as the nonlinear transmission. We start with a model Hamiltonian given by the sum of a material-only term $Ĥ0$ and a dipolar term coupling the electric field E(t) to the matter

where $V^$ is the dipole operator. We define the signal as the rate of change of material energy

Working in the Schrödinger picture and using the Liouville equation $ρ̇=−i[Ĥ,ρ]$ then gives

where we have used the invariance of the trace operation to cyclic permutations and the fact that only the electric field depends explicitly on time. The total energy change due to the field, given by the time integration,

can thus be found by substituting the Fourier transforms of the electric field and dipole expectation value

to obtain

In the long-time limit (⁠$t→∞$⁠), the time-integration gives a $δ$-function in frequencies, and the total dissipated energy is

where the last equality follows from $E*(ω)=E(−ω)$ and the fact that $ΔH$ is real. This reproduces the standard starting point for arbitrary-order nonlinear spectroscopic signals. To proceed further, we must specify the electric field and some care must be taken when considering one which is temporally unlimited. The simplest example of such a case is a continuous wave E-field, but a continuously applied pulsed laser is also temporally unlimited. In such cases, Eq. (7) diverges, and the material dissipates an infinite total amount of energy. The meaningful observable should then be the rate of this dissipation

or its Fourier transform

which is the rate of energy dissipation as a function of frequency. In the applications given hereafter, we consider infinite combs. We thus use Eqs. (8) and (9) rather than (6) and (7). We then analyze the temporal modulation of the signals for various comb configurations.

So far, we have discussed a coherent detection of the gain (or loss) of material energy due to the field. Alternatively, one can detect the time-dependent population $P^e≡|ee|$ of some selected excited state e, where $P^$ is a projection operator. While multi-frequency comb spectroscopy has traditionally been done in a heterodyne fashion, we demonstrate below that the same ideas can be imported into fluorescence detection schemes. By exciting the system with a field composed of multiple combs, the same phase modulation of the signal occurs in fluorescence as in the heterodyne case. As before, when the field is not temporally confined, the physically relevant signal is the rate-of-change of this population. Fluorescence is a readily identifiable proxy-signal for this quantity. The signal is then proportional to the time-dependent population flux of an emitting excited state

where we have used the Liouville equation and $P^̇e=0$ when $|e$ is an eigenstate of $Ĥ0$⁠. This can be further simplified by making use of the cyclic invariance of the trace and evaluating the resulting commutator

or, in the frequency domain,

These expressions for the incoherent population signal are analogous to Eqs. (8) and (9) for the rate of energy gain/loss and may be similarly expanded order by order. The only difference between the two types of detection is that the material operator of relevance is now the projected dipole product $P^eV^$ rather than the entire dipole $V^$⁠. If we define the projected susceptibilities,

then all results derived formally for the coherent transmission signal will apply immediately to these incoherent signals by the simple substitution $χ→χe$⁠. We note that, since the projection operators satisfy $∑P^e+P^g=1$⁠, we have $∑χe+χg=χ$⁠. However, the $χg$ signal obviously does not contribute to a fluorescence signal since it is the probability of returning to the ground state after the last dipole interaction.

The projected susceptibility can also be written as a Liouville-space superoperator expression

where $1|$ is the trace operator. The subscript “L” indicates action on the left in Hilbert space $ÔL|ρ↔Ôρ$ while the “−” subscript stands for the commutator $Ô−|ρ↔Ôρ−ρÔ$⁠. We could also simplify this by acting with the projection operator to the left on the trace operator $1|P^e,L=ee|$

where we have assumed the equilibrium density matrix to be the ground state. For comparison, the superoperator expression for the full $χ(n)$ is

We now apply the results of Sec. II to frequency combs as shown at the top of Fig. 1. We first consider the linear response of a system to a pair of frequency combs. The time-domain electric field of a pulse train corresponds to a comb in frequency space so that, for the j-th pulse train with T_j, the repetition period of the mode-locked pulses, we have⁴ 

where we have separated the electric field into an envelope function $Ẽ$ and a carrier wave with frequency $ωc$⁠, $Δωj≡2π/Tj$⁠, j = 1, 2 are the laser repetition frequencies, $ϕce,j$ is the pulse-to-pulse carrier envelope phase shift leading to the uniform frequency shift $ωce,j=ϕce,jΔωj$⁠, and $ϕj$ is an overall phase. This definition can also be used for a pulse series of ultrafast lasers. When an AOM with frequency $ϕjn$ is used, an extra $ϕjnTj$ is added to the phase. The ability to measure $ϕce$ for optical frequency combs, by self referencing for example, was instrumental in allowing their use for high-precision metrology since it determines the mapping between optical frequencies $ω$ and RF frequencies $Δω$ and $ωce$⁠.⁵ In dual- and multi-comb spectroscopies, active control of $ϕce$ allows a much higher degree of accuracy,²⁷ and phase-sensitive schemes can be used to separate linear from nonlinear signals.²⁰ While knowledge of $ωce$ is thus necessary to precisely determine the frequency of the comb teeth, it is not essential for understanding the key property of downshifting high frequency material response, where the downshift factor depends on the laser repetition rate difference. As the focus of this paper is the derivation of general expressions for AOM and multi-comb nonlinear spectroscopy, we do not further address these important details in this manuscript and, for simplicity, take $ωce,j$ and $ϕj$ to be zero with the understanding that they can be easily resurrected at any point as needed.

We consider a DFC composed of two frequency combs $E=∑jEj$⁠. We assume that the two combs have the same envelope function and are in phase so that the DFC is given by

In practical experimental implementations,²⁶ the repetition rates are chosen such that

For this reason, we will use the notation $Δω1≡Δω$⁠, $Δω2=Δω+δω2$ which is readily generalizable to $Δωj=Δω+δωj$ where we set $δω1=0$ without loss of generality.

We first consider the linear response in which the dipole expectation value is given by

where $χ(1)$ is the linear susceptibility. Upon substituting Eqs. (18) and (20) into Eq. (8), we find that the time-dependent rate of energy change has four contributions, which we denote by S_jk(t), j, k = 1, 2, depending on whether the material interacts with comb 1 or 2 in each E field. The “diagonal” terms come as

while the cross terms are

with $ΔH21(t)$ following similarly by $1↔2$⁠. The time modulation represented by the exponential factors is critically important. Note that the diagonal terms can come modulated with any multiple of the repetition frequencies while the cross terms come modulated with a multiple of one of the repetition frequencies and a $δω$ shift. The double summation can be simplified by considering the time-dependent signal that would result from a low-pass filtering where the cutoff frequency is less than either $Δωj$⁠. This would eliminate all terms $n≠m$⁠, rendering $ΔHjj$ a static, DC contribution. The cross terms, however, would then come as

where we have set $δω2=δω$ as it is the only such term at linear order. The nth term in this summation carries the information of the material response at $nΔω2$ but is modulated in time by the factor $einδωt$⁠, i.e., at a down-shifted frequency of $δω$⁠. Thus, in a temporal detection of the rate of energy dissipated (absorbed) by the matter, the response at an optical frequency $nΔω2$ will be modulated at a frequency $nδω$⁠. Since the externally controlled ratio between these two factors can reach $δωΔω≈10−6$⁠, this effectively down-shifts the optical frequency to the microwave or even RF regime, where it is easily detected in the time-domain by standard electronics. The material susceptibility at an optical frequency is accompanied by a microwave oscillation.

Equivalently, we can begin with the general expression [Eq. (9)] for the frequency-dependent rate of energy dissipation

Inserting Eq. (18) for the electric field gives

The linear response is then obtained by inserting Eq. (20), which gives

Inserting the form of the E field then results in

where we have used the $δ$-function to simplify the arguments of $χ(1)$⁠. Upon setting j = 1, k = 2, we recover the Fourier transform of Eq. (22). We can also recast this in an alternative form by relabeling $n+m=n′$⁠,

Here the relabeling (and the use of the $δ$-function) has allowed us to put one of the summations purely on the $δ$-function term, into which we have substituted the definition of $Δωj,k$⁠. This makes clear that, for each choice of $n′$⁠, there will be a comb with spikes separated by $δωk−δωj$ corresponding to different values of m, and each of these separate combs will be shifted by $n′Δω$⁠. Finally, we may consider a low-pass filtered signal as before to select the $n′=0$ term and obtain the simple formula

It is then immediately clear that the j = k terms will all contribute a DC component at $ω=0$⁠, while the j = 1, k = 2 term will generate spikes at $ω=mδω2$⁠, the j = 2, k = 1 term generates spikes at $ω=−mδω2$ which overlap since m takes positive and negative integer values. The above frequency-domain approach is clearly equivalent to the time-domain approach of Sec. III A but is easier to visualize because of the explicit appearance of the $δ$-function in the frequency domain. This expression can be simplified further by assuming that only one comb passes through the material while the other is used for the heterodyne detection so that, setting j = 2, k = 1, we obtain

where, for simplicity, we have relabeled $δω2=Δω2−Δω=δω$ since this is the only such quantity in the linear case. Similarly, detecting the reference beam corresponds to setting j = 1, k = 2 to obtain

Extending the previous formalism to any nonlinear response is straightforward. Here, we calculate the third-order signals obtained by the application of 4 combs. Third-order signals measured using 2 or 3 combs are then special cases while the use of higher numbers of combs would simply be a sum over possible 4-sets of the applied combs. A recent implementation²⁰ of a four-wave mixing measurement effectively utilizes 3 combs by starting with 2, splitting one of these, and using an AOM in one path of the apparatus. In that work, a phase cancellation scheme is utilized to separate out the desired signal. While the details of experimental implementation of such techniques are not completely straightforward, the advantages discussed above (short acquisition times at high resolution) justify their use.

Generalizing Eq. (18), we write the electric field as

and the dipole expectation value at this order is

which, upon substitution of Eq. (32), becomes

where each of the $j1,…,j4$ are summed over ${1,…,4}$⁠, signifying the 4 applied combs. To simplify this expression, we first define $n+m+p+q≡n′$ and $ωmpq≡mΔωj2+pΔωj3+qΔωj4$⁠, which gives

where all high-frequency components are at multiples of $n′$⁠. Applying a low-pass filter, as in the linear case, thus finally selects the $n′=0$ term

where the first argument of $χ(3)$ can also be written as $−ωmpq=−(m+p+q)Δωj1−ω$ via the $δ$-function. Setting the detected comb to be the reference comb then gives j₁ = 1, and the signal corresponds to

where we have used the symmetry of $χ$ to eliminate the summations over the j_i [e.g., the transformation $(j2,m)↔(j3,p)$ yields the same expression]. The presence of the integer summations means that the various frequency dimensions of the nonlinear response are essentially “folded” into the single observation axis $ω$⁠, preventing an immediate recovery of the full $χ$ at precise frequencies. By varying $δωj$⁠, we can then produce a 4-dimensional signal $S(ω,δω2,δω3,δω4)$⁠. The summations over mpq can be thought of as tensor contractions (of the tensors represented by the $δ$-function and the product of field envelopes with $χ(3)$⁠) and a generalized tensor equation inversion procedure can then be used to determine $χ(3)$⁠. The essential point remains the same as in the case of linear spectroscopy, i.e., all measurements are conducted at the frequency scale determined by the $δω$⁠, but they are sampling the material response $χ$ at the $Δω$-scale, which may be 5 or 6 orders of magnitude higher in frequency.

The special case of dual-comb, third-order spectroscopy in which all perturbative interactions are with comb 2, and the signal is heterodyne detected with respect to comb 1 gives

It is immediately clear that the third-order DFC technique does not provide enough tunable parameters to allow for a full inversion to obtain $χ(3)$ due to the dimensionality. Another special case is two interactions with each of two combs which becomes

This can be simplified by changing $m↔p$ in the second term and $m↔q$ in the third,