Cavity-enhanced ultrafast two-dimensional spectroscopy using higher-order modes
CCavity-enhanced ultrafast two-dimensionalspectroscopy using higher-order modes
Thomas K. Allison
Stony Brook University, Stony Brook, NY 11794-3400E-mail: [email protected]
August 2016
Abstract.
We describe methods using frequency combs and optical resonators forrecording two-dimensional (2D) ultrafast spectroscopy signals with high sensitivity.By coupling multiple frequency combs to higher-order modes of one or more opticalcavities, background-free, cavity-enhanced 2D spectroscopy signals are naturallygenerated via phase cycling. As in cavity-enhanced ultrafast transient absorptionspectroscopy (CE-TAS), the signal to noise is enhanced by a factor proportional to thecavity finesse squared, so even using cavities of modest finesse, a very high sensitivityis expected, enabling ultrafast 2D spectroscopy experiments in dilute molecular beams.
Keywords : ultrafast spectroscopy, frequency combs, cavity-enhanced spectroscopyInvited submission to
Emerging Leaders issue of
J. Phys. B: At. Mol. Opt. Phys.
1. Introduction
Spectroscopy of gas-phase atoms and molecules was essential in the development ofquantum mechanics and remains essential today for fundamental studies in physicsand chemistry. Particularly impactful for chemical physics have been studies on thedesigner species that can be produced in supersonic expansions, or molecular beams[1, 2, 3]. With molecular beam methods, one can produce cold isolated molecules,specific molecular clusters, radicals, and ions with a high degree of control [4]. Forexample, with gas-phase water clusters (H O) n , one can assemble the liquid “one-molecule at a time” [5], and perform detailed systematic studies of hydrogen bondnetworks. Electro-spray techniques even allow the introduction of very large moleculesand aggregates, with vanishing vapor pressure, into gas-phase experiments.While physicists have demonstrated exquisite control over gas-phase molecularsamples [6], the optical spectroscopy that is performed on these systems is usuallymuch less sophisticated than their solution phase counterparts, due to limitationsimposed by the very small optical densities of dilute gases. For matter in condensedphases, coherent, all-optical, third-order spectroscopies using ultrashort pulses, such as a r X i v : . [ phy s i c s . op ti c s ] A ug avity-enhanced ultrafast two-dimensional spectroscopy using higher-order modes and ultrafast. Using frequency combs and opticalresonators, we performed cavity-enhanced transient absorption (CE-TAS), or simplepump-probe, measurements with a time resolution of 120 fs and a detection limit of∆OD = 2 × − , a nearly four order of magnitude improvement over the previousstate of the art [16]. In this article, we describe how this technology can be applied toperform ultrasensitive 2D spectroscopy. However, instead of simply adapting standardtechniques for recording 2D spectra to cavity-enhancement, we describe here a methoduniquely enabled by the propagation properties of light in optical cavities. We show thatusing higher-order cavity modes, one can naturally record cavity-enhanced 2D signalsby mixing three resonantly-enhanced frequency combs with carrier-envelope offsetfrequencies ( f CEO , , f CEO , , f CEO , ) to generate a fourth resonantly enhanced frequencycomb with carrier-envelope offset frequency f (3)CEO = ± ( f CEO , − f CEO , ) + f CEO , . The2D signal is isolated from background signals via a combination of phase cycling andspatial mode-matching/phase matching. Since the three frequency combs share thesame repetition rate and differ only in their carrier-envelope offset frequencies, theycan be generated using just one mode-locked laser and fixed-frequency acousto-opticmodulators (AOMs).Similar to CE-TAS, the techniques described here are generally applicable to theIR, visible, and UV spectral regions, and while the primary motivation of this workis to record 2D spectroscopy signals from cold gas phase molecules and clusters, themethods also may find application in condensed phase work where higher sensitivity isneeded [17, 18, 19], or a robust, alignment-free instrument is desired. In section 2, wedescribe the critical connections between phase cycling 2D spectroscopy, the nonlinearmixing of frequency combs, and the Gouy phase shifts of cavity modes. In section 3, we avity-enhanced ultrafast two-dimensional spectroscopy using higher-order modes
2. Phase cycling 2D spectroscopy from a frequency comb perspective
For simplicity, we restrict the discussion to the case where all pump and probe pulsesare linearly polarized in the same plane, but the general principles discussed here easilygeneralize to more complicated polarization schemes [20]. Adopting the notation ofHamm and Zanni [20], the nonlinear polarization produced by a sequence of pulses E , E and E (or complex conjugates) arriving at the sample at t , t , and t can beexpressed as [20]: P (3) ( (cid:126)r, t ) ∝ (cid:90) ∞ dt (cid:90) ∞ dt (cid:90) ∞ dt (cid:88) n R n ( t , t , t ) × E ( (cid:126)r, t − t ) E ( (cid:126)r, t − t − t ) E ( (cid:126)r, t − t − t − t ) (1)where R n are the third-order system response terms that encode the molecularinformation of interest, and the sum is over all the double-sided Feynman diagrams thatsurvive the rotating wave approximation, including background signals not explicitlywritten here such as terms proportional to E E ∗ . This nonlinear polarization thenradiates a signal field E (3) , which is optically detected.In commonly employed 2D Fourier-transform spectroscopy methods, the twopump fields E and E correspond to separate ultrashort pulses with adjustable,interferometrically stable, relative delay τ . The pump frequency axis of a 2D spectrumis then generated by scanning τ and performing a Fourier transform. This allows for thesimultaneous combination of high pump frequency resolution and high time resolution,since only short pulses are used [21]. However, there are many terms in the sum ofequation (1), and to collect a background-free 2D spectrum one must isolate the desiredsubsets of this sum. This can be done either via careful arrangement of the wave-vectors( (cid:126)k i ) so that different terms in equation (1) emit phase-matched signals in differentdirections [22], or by selective modulation of the pulses combined with lock-in detection[21]. For the latter, either phase [23, 24, 25] or amplitude modulation [26] on one of thepump pulses may be used to separate the desired signals due to the concerted action of E and E from the undesired transient absorption backround signals due to each pulseacting individually.In the phase modulation approach illustrated in figure 1, commonly called“phase cycling”, the relative phase of two collinear pump pulses is varied and thespectral amplitude of the delayed probe light is detected at the modulation frequency.Mathematically, this works in the following way. The rephasing ( − (cid:126)k + (cid:126)k + (cid:126)k ) andnonrephasing (+ (cid:126)k − (cid:126)k + (cid:126)k ) signals are both emitted in the probe direction (cid:126)k , since (cid:126)k = (cid:126)k . For one pulse sequence, the field emitted from the desired components of thethird order polarization then depends on the carrier-envelope offset phases, φ , φ , φ of avity-enhanced ultrafast two-dimensional spectroscopy using higher-order modes E (3) ∝ R e i ( − φ + φ + φ ) + R e i ( φ − φ + φ ) (2)where we are using the symbol R here to stand in for the sum of all the rephasingterms and R to stand for all the nonrephasing terms for notational simplicity, as hasbeen adopted by other authors [20]. This signal is then self-heterodyned by the probefield in a square-law detector, yielding a signal of the form [20] S ( φ , φ , φ ) ∝ R (cid:2) E ∗ · (cid:0) R e i ( − φ + φ + φ ) + R e i ( φ − φ + φ ) (cid:1) (cid:3) ∝ R (cid:2) (cid:0) R e − iφ + R e iφ (cid:1) (cid:3) (3)where R denotes the real part and φ ≡ φ − φ . By constructing linear combinationsof signals with different phases φ one can recover the rephasing and nonrephasingcomponents of the 2D spectrum or any desired combination [21, 20]. For example,to record purely absorptive 2D spectra, one commonly records signals with phasedifferences φ = 0 and π : S absorptive 2D = S ( φ = 0) − S ( φ = π ) (4)The desired 2D signals add in this construction, while unwanted background signalsdue to the action of one pump pulse alone are subtracted away. One can also usephase differences other than 0 and π to recover the rephasing and non-rephasing signalsseparately [20, 27]. We will return to this point below.Since the excitations probed in 2D spectroscopy typically decohere on picosecondtime scales, the mutual coherence of successive pulse sequences at repetition rate f rep ,usually separated by milliseconds, is of no consequence. The sample has no coherentmemory of the last pulse sequence, so it does not matter whether there is a definite phaserelationship between E pulses n and n + 1 or not. Put another way, it does not matterif E is a phase-coherent frequency comb. The only coherence that matters is that forevery pulse sequence, there is a definite phase relationship between the pulses E and E , separated on the ultrafast time scale. However, to understand how 2D spectroscopysignals can be cavity-enhanced, it is instructive to consider the standard phase-cyclingexperiment described by equation (4) in the case where pulses separated by 1 /f rep are coherent, and E , E and E do constitute frequency combs. Consider the case where φ is incremented by the phase shift ∆ φ every sequence of laser pulses. Since thecarrier envelope offset frequency of a frequency comb, f CEO , is simply given by f CEO = f rep ∆ φ CE π (5)where ∆ φ CE is the pulse-to-pulse carrier envelope phase shift and f rep is the comb’srepetition rate, the combs E and E share the same repetition rate but differ in theircarrier-envelope offset frequency by f CEO , − f CEO , = ∆ φ f rep / π , as illustrated infigure 1b). These two pump combs then mix with the probe comb via the third-orderresponse, with offset frequency f CEO , to produce new combs with offset frequencies f (3)CEO = ∓ f CEO , ± f CEO , + f CEO , = ∓ f rep ∆ φ π + f CEO,3 (6) avity-enhanced ultrafast two-dimensional spectroscopy using higher-order modes k , k k ( ) ωω τT φ φ + π Δ f C E O , / f r e p τT 1/f rep π Δ f C E O , π Δ f C E O , E E (3) rephasing a) b) E (3) non-rephasing E E π Δ f C E O , Detector
Figure 1.
Frequency comb perspective on phase cycling 2Dspectroscopy.
Phase cycling depicted in the time a) and frequency b)domains. Two frequency combs E and E generate a phase cycling excitationwhen their carrier-envelope offset frequencies are detuned by ∆ f CEO , ≡ f CEO , − f CEO , . Four-wave mixing with the probe comb, E , generatesnew frequency combs in the (cid:126)k direction, E (3)rephasing and E (3)non-rephasing . When∆ f CEO , (cid:54) = f rep /
2, the rephasing and non-rephasing signals show up atdistinct frequencies, as shown in b), and can be separated [27, 21]. When∆ f CEO , = f rep /
2, such that the relative phase φ changes by π every 1 /f rep ,then the rephasing and nonrephasing signals appear at degenerate frequenciesand are inseparable, giving a purely absorptive 2D signal. In both cases, asquare law detector (or array) detects the heterodyne beat signal between E and the generated third order fields at modulation frequency ∆ f CEO , . where the upper sign corresponds to the new comb generated via the rephasingcomponents of the third order response and the lower sign to the nopnrephasingcomponents. This is illustrated in the time and frequency domains in figure 1. Thepulses of the probe comb E and the generated combs E (3) are coincident in time,and thus give rise to heterodyne beat signals at the differences between their offsetfrequencies. For the common case where ∆ φ = π , the intensity at the square-law detector is thus modulated at f rep / f rep /
2. Just as the rephasing and nonrephasing signals are added in the conventionalphase cycling method with ∆ φ = π , one can see from figure 1b) that in this case where | f CEO , − f CEO , | = f rep / φ (cid:54) = π , or offset frequency differenceother than f rep /
2, in which case the combs generated via rephasing and norephasingcontributions to the third order response appear at nondegenerate frequencies, and canbe separated [20, 27].Once this connection between phase cycling and wave-mixing of frequency combsis understood, it becomes clear how to cavity-enhance the phase-cycling 2D signal: onetunes the modes of one or more optical cavities such that all four frequency combs, avity-enhanced ultrafast two-dimensional spectroscopy using higher-order modes E , E , E ) and the generated E (3) , are all resonant with modes of optical cavities. As in CE-TAS, the sample has nomemory of prior pump and probe pulses, but the cavity does. Each field is enhancedby a factor proportional to the square root of the cavity finesse ‡ , √F , so in the limitthat the molecular excitation is not saturated the attainable signal to noise scales as F , as in CE-TAS [15]. Thus, even for cavities of modest finesse, very large signalenhancements are possible. There are, in principle, several ways one could achieve thisresonance condition. In this article we focus on using higher-order cavity modes, whichallows the FSR of the cavities employed to remain matched to the frequency comb,providing the optimum enhancement of the intracavity peak power. § In section 3, we describe several possible physical implementations, but here wefirst discuss the basic premise of the idea. Phase cycling using higher-order modesis motivated by the mode structure of optical resonators. In an optical cavity, lightpulses in different spatial modes acquire a round trip differential phase shift due to thedependence of the round trip Gouy phase on the Hermite-Gaussian mode order. Ingeneral, if E is in the TEM l m mode and E is in the TEM l m mode, each round tripthey acquire a phase shift∆ φ | round trip = ( l − l ) ψ tan + ( m − m ) ψ sag (7)with the Gouy phase shifts ψ tan and ψ sag solely determined by the geometry of the cavity,related to the components of the ABCD matrices via ψ = sgn( B ) cos − [( A + D ) / ABCD matrices for the sagittal and tangential planes, respectively. Thesephase shifts are tunable. For example, for a simple resonator with two concave mirrorsof equal curvature, ψ = ψ tan = ψ sag is continuously tunable from near 0 + to + π (nearplanar → confocal) and 0 − to − π (near concentric → confocal). Since the Gouyphase shift depends only on the cavity geometry and is independent of wavelength,it corresponds to a pure carrier-envelope offset frequency shift, viz. f CEO , − f CEO , = f rep π ∆ φ | round trip (8)Thus, by coupling combs to the higher-order modes of an optical cavity they naturallyphase cycle, generating new combs which can also be made resonant. As we discuss inthe next section, mode-matching also provides spatial isolation of the signal analogousto non-collinear phase matching in conventional 2D spectrometers. ‡ The field generated in the cavity, E (3) is actually enhanced ∝ F inside the cavity, but then must bereduced ∝ √F for detection outside the cavity, so that the overall field enhancement for the generated E (3) field (not counting the power enhancements of the driving fields) scales as √F§ One could also, in principle, use an overly long cavity such that f rep of the comb is an integer multipleof the cavity FSR. This would provide extra TEM resonances for coupling multiple, f CEO -shiftedcombs to the same cavity. However, this method would suffer several drawbacks. First, since thereare multiple pulse sequences per round trip circulating in the cavity, the peak power of both the pumpand probe pulses is less, lowering the nonlinear signal size. Second, the cavity linewidth is narrower,increasing the technical difficulty without increasing the signal size. avity-enhanced ultrafast two-dimensional spectroscopy using higher-order modes
3. Implementations
As in conventional 2D spectroscopy setups using mJ-pulsed lasers, there are manyconceivable physical implementations of the resonantly enhanced phase-cycling scenariodiscussed above. In general, since E , E , and E share the same repetition rate and onlydiffer by their carrier envelope offset frequency, they can be generated from one mode-locked laser simply by diffraction from fixed-frequency AOMs, and one does not needthree separate frequency comb lasers. The optimum choice of cavity geometry and modeselection depends on several factors, including system complexity, signal enhancementfactor, signal specificity, ease of alignment, attainable sample length, and signal readout.Design decisions will thus likely be driven by the demands of a particular measurement.In this section, we discuss several possible implementations and their relative strengthsand weaknesses.We restrict the discussion to bow-tie ring cavities for the reason that they allowindependent control of the overall cavity length and the focus size. This allows one toseparately control the peak intensity at the sample and the repetition rate of the system.Ring cavities also allow for the easy introduction of counter-propagating reference beamsfor common-mode noise subtraction, as has been critical for the success of CE-TAS [15].For a bow-tie ring cavity, the sign of the B component of the ABCD matrix is alwaysnegative, such that phase shifts ψ tan and ψ sag are restricted to the range between 0 and − π . Figures 2 and 4 show implementations of cavity-enhanced 2D spectroscopy usingone and two ring cavities, respectively. Both generate signals that are “background-free” in the sense that the signal field is generated in an unoccupied cavity mode. Usingone cavity makes the optical alignment and stabilization of the system very simple, andalso permits the use of an extended slit jet expansion for an increased column density ofmolecules [28], but requires separation of the weak signal field from the intense collinearpump and probe fields. Using two cavities makes the alignment and stabilization morecomplicated but it is easier to isolate the desired 2D signal. We discuss these subtletiesin more detail in the sections below. In the one-cavity scheme illustrated in figure 2, three collinear frequency combs withdifferent f
CEO ’s are coupled to three different Hermite-Gaussian spatial modes of a ringcavity with normalized field amplitudes described mathematically at the beam waist via[29]: u lm ( x, y ) = (cid:18) π (cid:19) / (cid:115) ( l + m ) w x l ! w y m ! H l (cid:32) √ xw x (cid:33) H m (cid:32) √ yw y (cid:33) e − x /w x e − y /w y (9)where l and m are the mode orders in the tangential (x) and sagittal (y) planes,respectively, H l is the l th order Hermite polynomial, and w x and w y are the 1 /e intensity radii of the fundamental TEM mode in the x and y directions. In a ring cavity avity-enhanced ultrafast two-dimensional spectroscopy using higher-order modes w x (cid:54) = w y , which breaks the degeneracybetween horizontal and vertical modes via their different round-trip Gouy phase shifts,described by equation (8). To resonantly enhance a desired 2D signal, the generatedcomb must be resonant with one or more of the cavity’s transverse modes. Analogousto equation 6, mathematically, this means that there exists at least one set of integers l t and m t for the target mode that satisfy l t ψ tan + m t ψ sag = ( ∓ l ± l + l ) ψ tan + ( ∓ m ± m + m ) ψ sag (10)where the upper sign corresponds to the rephasing signal, and the lower sign correspondsto the non-rephasing signal. This can be satisified in simple fashion via l t = ∓ l ± l + l and m t = ∓ m ± m + m , as shown in figure 2, but can also be satisfied in otherways, particularly when either 2 π/ψ tan or 2 π/ψ sag are integers and several modes aredegenerate. For example, with E in the TEM mode, E in the TEM mode, E in the TEM , the rephasing signal is clearly resonant with the TEM mode since − l + l + l = 0 and − m + m + m = 1. In contrast, the simple arithmetic forthe non-rephasing signal gives ( l t , m t ) = (0 , − , − mode, andit appears that this signal is not resonant, as illustrated in figure 2. However, if forexample ψ sag = − π/
2, the the non-rephasing signal is resonant with the TEM mode,which also has the appropriate even-x, odd-y symmetry to accept the signal.Only the spatial component of the generated field that is mode-matched to thetarget cavity mode will be resonantly enhanced. The spatial overlap factor (cid:104) u t | u (3) (cid:105) between the generated E (3) comb, with normalized spatial mode amplitude u (3) , andthe target resonant TEM l t m t mode, with normalized spatial amplitude u t , is given by (cid:104) u t | u (3) (cid:105) = (cid:82) dx (cid:82) dy u ∗ t u l m u l m u l m (cid:82) dx (cid:82) dy u ∗ l m u ∗ l m u ∗ l m u l m u l m u l m (11)Now the generated signal field E (3) is enhanced by a total factor proportional to | (cid:104) u t | u (3) (cid:105) | ( F /π ) . Indeed, this would imply that if one detects the intensity of thegenerated light on its own (homodyne detection), for example with a VIPA spectrometer[30] or spatial-mode division multiplexing [31], then the signal in fact scales as I (3) ∝| E (3) | ∝ | (cid:104) u t | u (3) (cid:105) | ( F /π ) . Each of the three input beams in the driven four-wavemixing process [32] has an intensity enhancement of F /π and the cavity also providesan additional enhancement of | (cid:104) u t | u (3) (cid:105) | F /π for the intensity of the generated light,giving an overall scaling of | (cid:104) u t | u (3) (cid:105) | ( F /π ) . However due to the expected smallabsolute size of the signal, it is still likely advantageous to employ heterodyne detectionof the generated field. Indeed, conventional background-free 2D spectroscopy, isolatedby phase matching, is still in general less sensitive than heterodyne detected signalsrecorded in a pump-probe geometry [33, 34]. In both cases, the fundamental shot-noiselimit on the signal to noise scales only as | (cid:104) u t | u (3) (cid:105) | ( F /π ) , since in the heterodynecase the noise level is determined by the noise of the local local oscillator, but in theintensity (homodyne) measurement, the noise scales as √ I (3) . Also note that althoughthe E (3) field enhancement is only reduced by one power of the mode-matching factor | (cid:104) u t | u (3) (cid:105) | , an additional mode-matching factor less than unity may be encountered in avity-enhanced ultrafast two-dimensional spectroscopy using higher-order modes Signal IsolationSlit supersonic expansion T ω comb ω cavityτ l re = l - l + l m re = m - m + m TEM l m l m l m π f r e p Δ φ π f r e p Δ φ π Δ f C E O , π Δ f C E O , E E E E (3)nr. E (3)re. Frequency Comb
Interferometer with 2 AOMs a)b)
Fast PZT mirror for cavity locking
Curved mirror separation is tuned to achieve desired Δφ Figure 2.
Cavity-enhanced 2D spectroscopy using one cavity. a) Afrequency-comb and an interferometer with two AOMs are used to generatedthree frequency combs with distinct carrier-envelope offset frequencies that arethen resonantly enhanced in a passive optical cavity, generating a resonantly-enhanced 2D spectroscopy signal if the resonance conditions illustrated in b)are met. The one-cavity scheme simplifies the alignment and laser/cavitystabilization and also allows the use of an extended sample (slit expansion), buthas the drawback that the cavity’s transmitted light must be well resolved toseparate the weak signal from the strong pump and probe pulses. By choosingthe symmetry of the excited cavity modes and tuning the resonance frequencies,the spectroscopist can select what signals are resonantly enhanced and suppressbackground. Shown is a case where the rephasing signal is resonantly enhancedbut the non-rephasing signal is not. In this case, the non-rephasing signal canstill be recorded by reversing the time-ordering of E and E . heterodyne detection. For example, if the heterodyne beat between E (3) and E fieldsis detected by simply recording the amplitude modulation on the probe beam in thetwo-cavity scheme (figure 2), the orthogonality of the Hermite-Gaussian modes requiressampling of less than the whole beam to recover a non-zero beat signal.Equation (11) also provides an opportunity to understand the physical origin of thesignal from the perspective of the probe pulse absorption and diffraction. In conventionalthird-order spectroscopy setups using free-space non-collinear beams, one can think ofthe pump pulse(s) generating a spatially dependent excitation pattern that the probelight can be diffracted from. When the pump pulse(s) overfill the volume of the sampleprobed by the probe beam in a pump-probe geometry, only the probe absorption ismodulated, its spatial mode unchanged. However, if the coherent excitation of themedium by the pump pulse(s) is not spatially uniform, a transient excitation grating avity-enhanced ultrafast two-dimensional spectroscopy using higher-order modes E and E act in concert to produce a spatially non-uniform excitation, which causes the probe light to diffract into the higher order modesof the cavity. For resonance, the spatial pattern is modulated (via ∆ f CEO,12 ) such thatthe diffracted probe pulses from successive round trips interfere constructively, and thediffracted probe beam is then resonant with one of the modes of the cavity. Indeed,one can see from equation (11) that spatial inhomogeneity of the pump fields is crucialfor generating a non-zero signal. If the pump modes overfill the probe volume suchthat u l m u l m → const. in equation (11), then | (cid:104) u t | u (3) (cid:105) | is identically zero due to theorthogonality of the Hermite-Gaussian modes! (cid:107) The spatial-mode selectivity of the cavity via equation (11) is analogous to phasematching in conventional 2D spectroscopy setups. Just as one isolates a desired signalin a boxcar geometry by detecting in a certain direction, in CE-2D spectroscopy usinghigher-order modes one can isolate a desired signal by detecting in a certain spatial mode.The generation and resonant enhancement of CE-2D signals using higher order cavitymodes can thus be viewed as selecting a desired third-order response signal througha combination of both phase cycling and spatial discrimination/phase matching. Thiscombination can make CE-2D spectroscopy highly selective, even in the completelycollinear geometry of fig. 2a), since both the cavity and the detection methods[35, 36, 31] can discriminate against undesired signals. As an example, lets again considerthe simple case where E , E , and E are coupled into the TEM , TEM , and TEM modes of one optical cavity. The rephasing signal is resonantly enhanced in the TEM mode with a mode-matching factor | (cid:104) u t | u (3) (cid:105) | = 0 .
65. Without mode degeneracy, thenon-rephasing signal is not resonantly enhanced, and would instead be recorded byreversing the time-ordering of E and E [20]. Many undesired signals, although emittedcollinearly, are suppressed from the target mode via a combination of the spatial andfrequency discrimination. For example the transient absorption signals ∝ | E | E and ∝ | E | E are enhanced in the TEM mode occupied by E but are generated withboth the wrong frequency ( f (3)CEO = f CEO,3 (cid:54) = f CEO,3 − f rep2 π ∆ φ | round trip ) and the wrongspatial symmetry ( (cid:104) u | u (3) (cid:105) = 0) to appear in the target TEM . Similarly, 2 quantumsignals ∝ E E E ∗ are weakly resonant with the TEM mode ( | (cid:104) u | u (3) (cid:105) | = 0 . mode by frequency discrimination. Somefifth-order signals and cascaded third-order signals do satisfy the necessary resonanceand symmetry requirements to be resonantly enhanced in the target mode, but can bedistinguished via power and sample density dependence of the signal, as in conventional2D spectroscopy.Using a cavity where some of the modes are degenerate provides additionalopportunities [37]. This can be done by tuning the curved mirror separation, δ , such (cid:107) For our previous CE-TAS demonstration [15], the modulation frequency was much less than the cavitylinewidth, and thus the target mode is the same as the probe mode ( u t = u ) and then | (cid:104) u t | u (3) (cid:105) | → avity-enhanced ultrafast two-dimensional spectroscopy using higher-order modes π/ψ tan or 2 π/ψ sag (or both) are integers. For cavities with long focal lengthmirrors, as we employed in CE-TAS [15], this does not require particularly precisecontrol of the curved mirror separation. For example, for a four-mirror bow-tie cavitywith FSR = 87 MHz, 75 cm mirror radius of curvature mirrors, and the curved mirrorseperation δ = 90 . ψ x = − π/
2, and dψ x /dδ isonly 0.06 rad/cm. Since a 2 π intracavity phase shift corresponds to one cavity FSR,this corresponds to a frequency shift of only 0.01 FSR/cm. The cavity linewidth is theFSR divided by the finesse, so for a cavity finesse of 1000, a frequency shift of one cavitylinewidth corresponds to a large change in δ of 1 mm. For achieving degeneracy of 5higher-order transvers modes, separated by ∆ l = 4, all within 1/10 of a cavity linewidththus only requires control of δ to the length scale of 2 µ m, which can likely even beachieved passively with careful design. To put this in perspective, to lock the cavity tothe comb, the overall length of the cavity is already being actively stabilized to muchbetter than λ/ F < | f CEO,1 − f CEO,2 | = f rep / ψ x = − π/
2, and the spectroscopist couples E to a superposition of thefirst five degenerate TEM n, and E to a superposition of the first five degenerateTEM n, , so that ∆ φ | round trip = − π and | f CEO,1 − f CEO,2 | = f rep /
2. With E coupledto a superposition of the first five degenerate TEM n, modes, we find via roughnumerical optimization of equation (11) that mode-matching factors | (cid:104) u t | u (3) (cid:105) | > . u t a superposition of the first five TEM n, modes. ¶ The above cases illustrate that selectively cavity-enhancing the desired 2Dspectroscopy signals, either purely absorptive signals or separate components, can bedone well with one cavity in a collinear geometry. The main challenge of using onecavity will be the separation of the miniscule 2D spectrosocpy signal from the muchmore intense intense input beams. E is distinct from the signal field via spatial modeand frequency. E and E are distinct via spatial mode, frequency, and also in the timedomain for waiting times T >
0. Thus, in principle, the signal isolation problem hasalready been solved by the practitioners of mode-division multiplexing [31], in whichsignals in different spatial modes are de-multiplexed at the end of multi-mode fiber, anddirect frequency comb spectroscopy (DFCS) where individual comb-teeth are resolved[38, 35, 36]. However, some simple estimates indicate that a very high degree of isolationwill be necessary. In our demonstration of CE-TAS in a dilute molecular iodine sample, ¶ Note that here we report | (cid:104) u t | u (3) (cid:105) | for the total power enhancement instead of | (cid:104) u t | u (3) (cid:105) | forthe field enhancement because in the superposition state the field enhancement is spatially dependent(in x, y , and z ) and is somewhat meaningless, whereas in the single-mode situation it is reasonablystraightforward in certain cases to achieve mode-matched heterodyne detection, measuring the field ∝ | (cid:104) u t | u (3) (cid:105) | . avity-enhanced ultrafast two-dimensional spectroscopy using higher-order modes ∼ ∼ E , E , and E are coupled to the TEM , TEM , andthe TEM modes, respectively, generating a rephasing signal in the lowest-order TEM mode with | (cid:104) u t | u (3) (cid:105) | = 0 .
65. Light from the cavity is then launched in a single-modegraded index fiber to remove the E , E , and E fields from the beam. Graded indexfiber is used because the modes of graded index fiber are Hermite-Gaussians [39], andthus form the same orthonormal basis as that of the cavity, providing (theoretically)perfect rejection of the unwanted higher order modes. Also coupled to the fiber isan intense local oscillator frequency comb with f CEO ,LO near f (3) CEO . After exiting thefiber, the modulation on the light at frequency f CEO ,LO - f (3)CEO is detected using a lock-inspectrometer, providing discrimination in the frequency domain. A lock-in spectrometercan be achieved trivially using a scanning grating monochromator or Fourier-transformspectrometer with a fast single-element detector, or nontrivially using specialized lock-inarray detectors [40, 41] or time-stretch dispersive Fourier transform techniques [42, 43]for parallel detection. For waiting times T >
0, this scheme additionally discriminatesin the time domain due to the fact that the local-oscillator does not overlap temporallywith E and E . Much of the discussion for one cavity schemes carries over to two-cavity schemes,illustrated in figure 4. Good mode-matching and background suppression can beachieved through appropriate mode selection, and degenerate modes can be used torecord purely absorptive 2D spectroscopy signals. The pump-probe geometry also makesit much easier to use very different frequencies for pump and probe, such as employed in2D electronic-vibrational spectroscopy [44]. But the main advantage of using two cavitiesis that the non-collinear pump and probe allows for much easier isolation of the desired2D signal via simple lock-in detection at the phase cycling frequency | f CEO,1 − f CEO,2 | .This is similar to the amplitude modulation frequency scheme used in our previous CE- avity-enhanced ultrafast two-dimensional spectroscopy using higher-order modes T ω comb ω cavityτ TEM
11 10 E E E E (3)re. Frequency Comb
Interferometer with 3 AOMs
01 00 E LO Ring CavityLock-in spectrometer lock-in single modeGRIN fiber
Local oscillator
Figure 3.
Isolation of the 2D signal.
The generated frequency comb E (3) is different from the three driving fields in time, spatial mode, and frequency.The proposed scheme shown here uses all three to isolate the 2D signal frombackground. First, the TEM beam from the cavity is selected by couplingthe cavity output to a single mode graded index fiber. Second, heterodynedetecting the signal with a local oscillator (LO) frequency comb and a lock-inspectrometer discriminates against the collinear combs E , E , and E in thefrequency domain, since their heterodyne signals with the LO comb appear atthe wrong frequencies. Third, background signals due to E and E are furthersuppressed by adjusting the delay of the LO pulses such that they coincidetemporally with only the E (3) and E combs. TAS demonstration and thus one would expect similar signal to noise considerations,except that now the phase cycling scheme has a distinct advantage: the modulationfrequency is now much larger than the cavity linewidth. This means that the signalappears at a frequency where there is naturally greatly reduced intensity noise on thetransmitted light because the cavity low-pass filters the laser noise at Fourier frequenciesmuch larger than the cavity linewidth [45, 46]! + .The price one pays for simplified detection of the 2D signal is complexity in theoptical layout and its alignment. With two cavities, their foci must be overlapped in themolecular beam, and the required finite crossing angle means a smaller overlap volumecan be achieved in the molecular beam sample. There are also now two cavities thatrequire stabilization and tuning of ψ sag and ψ tan , although if the f CEO frequencies of thetwo cavities can be matched, as in [15], then in principle one can use fewer AOMs. + This can also be exploited in the heterodyne detection scheme with a separate local-oscillator offigure 3 avity-enhanced ultrafast two-dimensional spectroscopy using higher-order modes Lock-in spectrometer τ E ω comb Frequency Comb Interferometer with 1 AOM
TEM l m π f r e p Δ φ Pump Cavity Probe Cavity l m ω cavity Supersonic expansion ω comb ω cavity E (3)nr. a)b) c) Fast PZT mirrors for cavity lockingCurved mirror separationsare fine-tuned to achieve E (3) resonance.Translation Stagefor T-delay
TEM l t m t l m π f r e p Δ φ π Δ f C E O , π Δ f C E O , E E E (3)re. Figure 4.
Cavity-enhanced 2D spectroscopy in a pump-probegeometry using two cavities. a) Light from a frequency comb source issplit into pump and probe beams. An interferometer with a fixed-frequencyacousto-optic modulator (AOM) in one arm generates a pair of pump frequencycombs shifted by the AOM drive frequency and variable delay. A background-free 2D spectroscopy signal is passively amplified in the probe cavity if thenecessary resonance conditions illustrated in b) and c) are met by appropriatetuning of the frequency comb, AOM frequency, and cavity-mirror separations.The signal is recorded via lock-in detection of the heterodyne beating betweenthe generated E (3) comb and the probe light. For completeness, we briefly describe some possibilities for generating multiple collinearfrequency combs with different and tunable f CEO ’s and adjustable delays and in differentspatial modes for efficient mode-matching to the cavity. This can be done in reasonablystraightforward fashion by incorporating AOMs and phase/amplitude masks [31] intoa stabilized “pulse stacker” [47, 48, 49, 23]. For example, Ryf et al. [31] describe amode-multiplexing interferometer incorporating simple phase masks that achieves > f CEO shift with one device is to use a dazzler, as has recently been demonstrated fordual comb spectroscopy [51].Notably absent from this list is the pulse shaper, now commonly employed by many avity-enhanced ultrafast two-dimensional spectroscopy using higher-order modes
4. Conclusions
In conclusion, we have described methods to perform passively amplified 2Dspectroscopy experiments using a frequency comb laser and optical cavities. As we havedemonstrated for transient absorption spectroscopy in both isolated molecules [15] andclusters [52], a large sensitivity improvement of several orders of magnitude is expected,enabling 2D spectroscopy in dilute molecular beams. Additionally, in the one cavityscheme, the pump-probe spatial overlap factors (equation (11)) relating the absolutesize of the signal to strength of the molecule’s nonlinear polarization can be knownprecisely, enabling greater quantification of 2D spectroscopy signals for fundamentalstudies or analytical chemistry applications. As in CE-TAS, the techniques are generallyapplicable to the UV, visible, and infrared regimes - wherever frequency combs withreasonable power can be generated and high reflectivity, low loss, low GDD mirrorscan be fabricated. The necessary optical components and light sources have beendemonstrated in all of these spectral ranges for other purposes, and the simultaneouscoupling of a frequency comb to multiple higher-order modes has been previously usedin the context of intracavity high-order harmonic generation [53, 37].The largest constraints on CE-2D schemes likely come from the simultaneousbandwidth that can be resonantly enhanced. This is not due to the attainable bandwidthof frequency comb sources or the reflectivity bandwidth of high-reflectivity, low losscavity mirrors. Instead, it is set by the dispersion of the cavity mirrors. For a cavitywith a finesse of 1000, to enhance 5 THz of bandwidth the net cavity group delaydispersion (GDD) must be controlled to better than 50 fs . This is quite feasible, butresonantly enhancing 100’s of THz of bandwidth for 2D spectroscopy experiments inthe visible simultaneously is likely not feasible and 2D-electronic spectra will likelyhave to be acquired piecewise, scanning both the pump and probe. However, for manyexperiments in the IR, 5-10 THz (170-330 cm − ) of bandwidth is more than sufficient,and recently there have been major breakthroughs in mid-IR cavity mirror technology[54]. The ability to perform 2D spectroscopy on cold gas-phase molecules allows theultrafast spectroscopist to either (1) study a molecule of interest in a cold, collision freeenvironment, recording the same optical signal as recorded in solution, or (2) study thedynamics of designer molecules that can only be made in cold supersonic expansions.We expect these new capabilities to allow for the study of the dynamics of moleculeswith unprecedented detail and control. Problems of interest include intramolecularvibrational relaxation [55, 56], many-body couplings and dynamics of hydrogen bondnetworks [5], solvation effects on the dynamics of small molecules [57], exciton dynamics avity-enhanced ultrafast two-dimensional spectroscopy using higher-order modes
5. Acknowledgements
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