The superposition principle and cavity ring-down spectroscopy Kevin K. Lehmann Department of Chemistry, Princeton University, Princeton, New Jersey 08544 Daniele Romanini Laboratoire de Spectrome´trie Physique-CNRS URA 08, Universite´ J. Fourier/Grenoble, B.P. 87-38402 Saint Martin d’He`res Cedex, France ͑Received 14 May 1996; accepted 12 September 1996͒ Cavity ring-down is becoming a widely used technique in gas phase spectroscopy. It holds promise for further important extensions, which will lead to even more frequent use. However, we have found widespread confusion in the literature about the nature of coherence effects, especially when the optical cavity constituting the ring-down cell is excited with a short coherence length laser source. In this paper we use the superposition principle of optics to present a general and natural framework for describing the excitation of a ring-down cavity regardless of the relative values of the cavity ring-down time, the input pulse coherence time, or the dephasing time of absorption species inside the cavity. This analysis demonstrates that even in the impulsive limit the radiation inside a high finesse cavity can have frequency components only at the natural resonance frequencies of the cavity modes. As an immediate consequence, a sample absorption line can be detected only if it overlaps at least one of the cavity resonances. Since this point is of particular importance for high resolution applications of the technique, we have derived the same conclusion also in the time domain representation. Finally, we have predicted that it is possible to use this effect to do spectroscopy with a resolution much higher than that of the bandwidth of the excitation laser. In order to aid in the design of such experiments, expressions are derived for the temporal and spatial overlap of a Fourier transform limited input Gaussian beam with the TEMmn modes of the cavity. The expressions we derive for the spatial mode overlap coefficients are of general interest in applications where accurate mode matching to an optical cavity is required. © 1996 American Institute of Physics. ͓S0021-9606͑96͒01847-8͔

I. INTRODUCTION of physical systems having linear response are mathemati- cally equivalent, since they are uniquely mapped into each In the last few years, cavity ring-down spectroscopy other by a Fourier transformation. Given this equivalence, ͑CRDS͒ has been applied with increasing frequency to a one is free to choose the most convenient description for the number of problems, allowing highly sensitive absolute ab- problem at hand, as the final results will not depend on this sorption measurements of weak transitions or rarefied choice. However, the frequency representation has the gen- species.1–8 Very briefly, CRDS uses pulsed laser excitation eral advantage that the spectrum of the ‘‘output’’ of a passive of a stable optical cavity formed by two or more highly device can be simply written as a system response function reflective mirrors. One observes absorption by molecules times the spectrum of the ‘‘input.’’ In contrast, the general contained between the mirrors at the laser wavelength by the time domain system response must be expressed as a convo- decrease it causes in the decay time of photons trapped in the lution integral of the input with a time dependent system cavity. Absorption equivalent noise as low as response function. For an optical cavity, ‘‘input and output’’ Ϫ10 ϳ3•10 /cmͱHz has been demonstrated, and in principle, are the electromagnetic fields arriving from the optical several orders-of-magnitude-further-improvement is pos- source and going towards the detector, respectively. In sible.3,6 CRDS, one may have to deal with input laser pulses that are In several of the recent publications on this subject, we short compared to the light round trip time inside the cavity. have detected a diffuse belief that the behavior of an optical In this limit, it is clear that destructive interference phenom- cavity under impulsive excitation ͑as in CRDS͒ is to be con- ena among the different time components of the injected sidered radically different from that in presence of continu- light should be negligible. More simply, after a pulse is par- ous wave ͑cw͒ radiation. This impression has been confirmed tially transmitted through the cavity input mirror, it does not by direct discussion with different authors. The purpose of spatially overlap or interfere with itself as it ‘‘rings down’’ this paper is to clarify the situation by presenting a rigorous inside the cavity. Therefore, it might appear more convenient analysis of the excitation of an optical cavity that is valid to use the time domain representation to treat CRDS. It turns over the full range of parameters that are likely to be of out that the frequency domain analysis of the light fields importance in CRDS. Further, the framework introduced inside and transmitted by a ring-down cavity is quite simple here may be useful in the design of new experiments. and of general scope. No restrictive assumptions on the pulse The problem can be summed up as follows. Time do- duration and bandwidth, the absorbing transition linewidth, main and frequency domain representations of the dynamics or the cavity quality factor are required. For the case of nar-

J. Chem. Phys. 105 (23), 15 December 1996 0021-9606/96/105(23)/10263/15/$10.00 © 1996 American Institute of Physics 10263

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For complementarity we also give a ͑less gen- CAVITY eral͒ time domain treatment of this important problem, which obviously reaches the same conclusions. This result was pre- Consider a ring-down cavity ͑RDC͒ formed by two mir- 9 viously obtained by Zalicki and Zare, but then recently con- rors with radius of curvature Rc , separated by a distance L 7 12 tested by Scherer et al. ͑which must be Ͻ2Rc to have a stable cavity ͒. We define The only simple but powerful principle to be applied in trϭ2L/c, the round trip time of the cell, where c is the speed the frequency domain analysis is that of linear superposition of light in the medium between the mirrors. For simplicity, of the effects produced by the different frequency compo- we will assume that the mirrors have identical electric field nents in the incoming field.10 We will show that the standard reflectivity ϪR and transmittivity T . The results derived spectral response function of an etalon can be applied regard- below are easily generalized to the case of assymetric cavi- less of the temporal profile of the input field. While we ex- ties and the results are qualitatively the same. If the mirrors pect that many readers will find this result to be so obvious were infinitely thin, continuity of the electric field would as to not require an explicit derivation, the CRDS literature is give the relationship RϩT ϭ1 between these complex 11 rife with statements that contradict this view. Since the quantities. The reflections and transmitions of the multiple comblike structure of this response function reflects the pres- surfaces inside dielectric mirrors can be combined ͑with ence of cavity resonances, the cavity mode structure can be propagation phase and absorption͒ into a single effective fre- considered as a fixed characteristic of the system, and not as quency dependent R and T ͑an application of the superpo- something that builds up only if the excitation coherence sition principle͒, as long as one is outside of the region of the time is sufficiently long compared to the cavity optical round coating. However, this treatment will not yield a continuous trip time, as stated in several recent publications. One must electric field, since this solution is not appropriate inside the be careful not to confuse interference in the time and in the mirror coatings where one is beyond some surfaces and be- frequency domain. Even if the pulse injected into the cavity fore others. The more familiar intensity reflectivity and trans- 2 2 is such that its time components do not overlap and interfere, mittivity are given by Rϭ͉R͉ and Tϭ͉T ͉ . For now, we the multiple reflections from the cavity mirrors still result in will assume that we can treat R and T as constants over the destructive interference for those frequency components of bandwidth of input radiation to the RDC. Let us consider the pulse which do not overlap any cavity resonances and excitation of the RDC by light of arbitrary electric field constructive interference of those that do. Correct predictions Ei(t), as measured at the input mirror of the cavity. We will of the cavity behavior in different situations follows natu- also initially assume that the radiation is mode matched to rally if one thinks in terms of the cavity resonances in fre- the TEM00 mode of the cavity. In these conditions the trans- quency space and the associated spatial mode structure ͑lon- verse beam profile is stationary and we can treat the problem gitudinal and transversal͒ in physical space. as one dimensional along the cavity optical axis z. Below, Contrary to what has been suggested in the CRDS litera- we will consider the effects of excitation of higher order ture, one of the most interesting consequences of our consid- transverse modes. We can calculate the electric field of light erations is that it is possible to turn the high finesse mode leaving the cavity by adding up all paths that lead to output, structure of the ring-down cavity to a tremendous advantage. which make 1, 3, 5, etc., passes through the cell. The light 2 We will show that one can use CRDS with a spectral reso- making one pass has an amplitude of T Ei(tϪtr/2) at time lution much higher than that of the pulsed laser source. In t. Each additional round trip through the cell changes the 2 order to aid in the design of such experiments, expressions amplitude by a factor of R and leads to an additional retar- are derived for the temporal and spatial overlap of a Fourier dation of tr . Summing the contribution of the possibly infi- transform limited input Gaussian beam with the TEMmn nite number of passes leads to the intracavity electric field, modes of the cavity. E(z,t) at position z and the output electric field, Eo(t) ͑mea- A note is warranted about the application of linear re- sured outside the output mirror at zϭL), sponse theory to CRDS. As it has already been noted,8 even ϱ zϩ2nL in presence of intense and short laser pulses, nonlinear ef- 2n E͑z,t͒ϭ͚ TR Ei tϪ fects such as the saturation of molecular transitions are usu- nϭ0 ͩ c ͪ ally negligible in CRDS. This is principally due to the strong 2͑nϩ1͒LϪz attenuation of the input laser pulse upon transmission of the 2nϩ1 ϪTR Ei tϪ , ͑1͒ cavity input mirror. We would like, however, to warn that ͩ c ͪ there may exist special conditions in which nonlinear effects ϱ may become relevant, specifically for very strong transitions 2 2n 1 Eo͑t͒ϭ͚T R Ei͑tϪ͑nϩ2͒tr͒. ͑2͒ and a ring-down cavity employing mirrors of substantial nϭ0

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We can calculate the spectrum ͑in the angular frequency ence time ͑either short or long compared to tr). This is, of ␻) of this field by computing the Fourier transform ͑FT͒. For course, a straightforward consequence of the superposition light transmitted by the cavity we have principle of linear optics, which states that the optical throughput at each frequency can be calculated separately, 1 ˜ and is independent of any other spectral components of the Eo͑␻͒ϭ Eo͑t͒exp͑Ϫi␻t͒dt ͱ2␲ ͵ wave.13 In their recent paper, Scherer et al.7 make the claim ‘‘it 1 ϱ 2 2n 1 seems intuitively more reasonable that the onset of coherent ϭ ͚ T R Ei͑tϪ͑nϩ 2͒tr͒ ͱ2␲͵ nϭ0 effects should explicitly depend upon the coherence length of the light, since if the phase of the overlapping light is not ϫexp͑Ϫi␻t͒dt preserved, a regular interference will not necessarily occur. ϱ These results underscore the complexity associated with pre- 2 2n 1 ˜ ϭ͚ T R exp͑Ϫi͑nϩ 2͒␻tr͒Ei͑␻͒ dicting the behavior of optical resonators which are injected nϭ0 with pulsed laser light, and the subsequent limitation of ap- i ϱ plying cw–based model to the pulsed regime.’’ The deriva- ϭT 2 exp Ϫ ␻t ˜E ͑␻͒ ͓R2 exp͑Ϫi␻t ͔͒n, tion presented above demonstrates that this claim is incon- 2 r i ͚ r ͩ ͪ nϭ0 sistent with the superposition principle of optics. The ͑3͒ i coherence length of the input pulse appears in our analysis as 2 ˜ T exp Ϫ ␻tr part of Ei(␻) and only affects the relative excitation of dif- ˜ ͩ 2 ͪ ˜ Eo͑␻͒ϭ 2 Ei͑␻͒. ferent resonance modes of the cavity. 1ϪR exp͑Ϫi␻tr͒ The following considerations might help to further ˜ clarify pulsed excitation of an etalon. In the case of an im- Above, Ei(␻) is the FT of the input radiation. The output pulsive E (t) shorter than t , it is clear that there is no de- spectral density, Io(␻), of this light is proportional to i r ˜ 2 structive interference of the temporal components of the ͉Eo(␻)͉ , which gives pulse. This is the case if dispersion effects are not suffi- 2 T ciently strong to make the pulse duration become longer than I ͑␻͒ϭ I ͑␻͒. ͑4͒ o 2 2 1 i t before the pulse is completely decayed. However, absorb- ͑1ϪR͒ ϩ4R sin ͑ 2 ␻trϪ␪͒ r ing molecules inside the cavity do not see a single pulse, but In this expression, ␪ϭarg(ϪR), i.e., the phase shift per also its recurrent reflections, with a well defined and constant reflection of the mirrors. Going through the same analysis for period tr . Here, we neglect fluctuations of the cavity length, light inside the cavity, starting from E(z,t) we find which is a good approximation if we consider that a typical optical dephasing time, T , is much shorter than mechanical ͑1ϪͱR͒2ϩ4ͱR sin2͑k͑LϪz͒Ϫ␪͒ 2 I͑␻,z͒ϭT I ͑␻͒, vibrations. It is this strict recurrence that produces coherence 2 2 1 i ͑1ϪR͒ ϩ4R sin ͑ 2 ␻trϪ␪͒ effects if the molecular transition T2 is longer than tr . In the ͑5͒ frequency domain, we have shown that, due to the recur- where kϭ␻/c is the wave vector of the light. This shows rences, the spectrum of the injected light is modulated and that for a fixed ␻ or monochromatic input field, light travel- this modulation is mathematically represented by multiplica- ing in both directions inside the cavity leads to standing tion of the initial spectrum by the comblike transfer function waves with near nodes in the limit that Rϳ1. The time av- of the cavity. eraged total intensity at z is given by ͐I(␻,z)d␻. For exci- If we were to select a single one of the pulses in the cavity ring-down, say by using a fast electro-optic switch, it tation of the cavity with a pulse short compared to tr , whose would have the same spectrum as the input radiation under bandwidth will be larger than ϳ1/tr , this integral will wash out the standing waves except close to the mirrors. the assumptions made above. Of course, such a switch must The above equations are essentially identical to the ex- change its transmission on a time scale less than tr , and thus pression for the transmission of an etalon found in standard will introduce a spectral broadening greater than the spacing optics texts, such as that of Born and Wolf ͑Ref. 13, page between cavity modes of the RDC. The etalon comblike fil- 327͒. The output spectrum is equal to the input spectrum ter convoluted with the spectral broadening produced by the times a transmission function. For a Rϳ1, this intensity switch will just reproduce a highly attenuated version of the transmission function versus frequency consists of a series of input pulse spectrum. Notice that since the optical switch is a narrow peaks with full width at half maximum ͑FWHM͒ time dependent system, it cannot be represented in frequency given by ⌬␯FWϭ(1ϪR)/(ͱR␲tr), separated by the free space by simple multiplication by a spectral response func- spectral range, FSRϭ1/tr , of the etalon. The finesse of the tion, but by a convolution operation. etalon is defined by the ratio of the FSR to the FWHM and is The fact that a series of equally spaced decaying replicas equal to (␲ͱR)/(1ϪR). While Eq. ͑4͒ is usually derived by of the same pulse has a spectrum which is different from that considering an infinite wave of pure frequency, it should be of the single initial generating pulse, is a consequence of the noticed that we have made no assumption as to the shape of properties of the FT. The fact that this frequency spectrum Ei(t), and thus the input can have any pulse length or coher- contains resonances which are closely the harmonics of the

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fundamental 1/tr is borne out by the rigorous application of where we have introduced the ‘‘spectral’’ Green function the superposition principle given above, but may be also un- G˜ (␻) ͑FT of the usual time dependent Green function͒ for derstood in simpler ͑though less general͒ terms as follows. If the frequency response of the cavity. For the spectrum of trӶtd and we neglect the phase shift per reflection, the re- light inside the cavity we have instead peating pulse is well approximated as a product of an expo- nential decaying envelope exp(Ϫt/2td) times a periodic func- I͑␻,z͒ tion f (t) with period tr . It is a well known property of ͓1ϪͱRe␣͑zϪL͔͒2eϪ␣zϩ4ͱReϪ␣L sin2͑k͑LϪz͒Ϫ␪͒ periodic functions that they can be decomposed in a Fourier ϭT ϩϱ 1ϪR 2ϩ4R sin2 kLϪ series as f (t)ϭ͚mϭϪϱfm exp(2␲imt/tr). Therefore, we find ͑ eff͒ eff ͑ ␪͒ again that only harmonics of the base frequency 1/t must be r ϫI ͑␻͒. ͑8͒ present. The effect of the decay is to add a finite Lorentzian i width to these resonances. This is seen in mathematical In order to determine the time dependent intensity of light terms by taking the FT of the product f (t)ϫ exp(Ϫt/2t ) d transmitted by the cavity, which is the quantity measured in and recognizing that this equals the convolution of the indi- CRDS, we must compute the inverse FT of ˜E (␻). Using the vidual FT’s of each operand ͑Faltung theorem͒. The FT of an o convolution or Faltung theorem, this can be expressed in the exponential decay is a Lorentzian, while the FT of the sum following standard form: of circular functions exp(2␲imt/tr) is a sum of delta functions ␦(␻Ϫ2␲m/tr). The convolution then gives a sum of Lorent- zians whose center frequencies are the harmonics of 1/t , r Eo͑t͒ϭ G͑tϪtЈ͒Ei͑tЈ͒dtЈ, ͑9͒ which is a good approximation to the comblike cavity trans- ͵ mission function in Eq. ͑4͒ in the above limit of trӶtd . If one looks at the field in the direction of the reflection where G(tϪtЈ) is the Green’s function, which represents the from the cavity, one will find both the direct reflection of the cavity response to a delta function input. For the ring-down cavity, this can be written from the expression above as fol- input pulse plus a pulse train lasting a time td that comes from intracavity radiation re-transmitted by the input mirror. lows: In computing the spectrum of this pulse, there will be de- 2 Ϫ␣L/2 ϪikL structive interference for those spectral components of the 1 T e e G͑t͒ϭ ei␻͑t͒d␻. ͑10͒ input pulse which are resonant with those of the re- 2␲ ͵ 1ϪR2eϪ␣LeϪik2L transmitted pulse train, leading to ‘‘holes’’ in the spectrum of the reflected light. This is of course a natural consequence In this integral, one should keep in mind that ␣, k, R, and of the need to separately conserve energy for each spectral T are all functions of the integration variable ␻. We wish to component. point out that up to this point in the analysis we have made Let us now relax the assumption that R and T are fre- no assumptions beyond linearity of optical response, which quency independent, and allow for the presence of an absorb- is implied by the use of n(␻) and ␣(␻), and the superposi- ing medium inside the cell with an absorption coefficient tion principle of optics. given by ␣(␻) and index of refraction given by n(␻). In As long as the width of each absorption line is much principle, we could repeat the above calculation directly in greater than the width of the individual resonances ͑i.e., the the time domain, but then we would need to consider the T2 for the optical transition is short compared to td), each time dependent response functions of the mirrors and the resonance of the cavity will still have a Lorentzian shape. re-radiation by the molecules ͑Ref. 14, p. 490͒. However, as For realistic parameters at optical frequencies, the transit long as we remain in the linear response limit, we can exploit time broadening of molecules through the narrow TEM00 the superposition principle to look at the reflection and ab- cavity mode will greatly exceed the width of the resonances sorption of each spectral component separately, which are ͑but it will be much narrower than their separation͒. The just complex multiplicative factors. Adding up the multiple Lorentzian resonance shape implies that the decay of inten- paths for each spectral component, we find the following for sity emitted from the cavity, following impulsive excitation the spectrum of light transmitted by the cavity of a single mode, will be exponential. We point out that this limit is violated in Balle–Flygare-type FT microwave spec- Ϫ␣͑␻͒L Reff͑␻͒ϭR͑␻͒e , troscopy, which is conceptually similar to cavity ring-down, but where typically the T2 due to transit time is long com- n͑␻͒␻ pared to the cavity decay time.15 k͑␻͒ϭ , c In this limit of cavity resonances much narrower than the interval over which the other frequency dependent terms in 2 Ϫ␣L/2 ϪikL T e e the expression for the response function change, we can in- ˜E ͑␻͒ϭ ˜E ͑␻͒ϭͱ2␲G˜͑␻͒˜E ͑␻͒, o 1ϪR2eϪ␣LeϪik2L i i tegrate over each resonance separately. By expanding the ͑6͒ exponential exp(Ϫik2L) to first order around each ␻q where k(␻q)Lϭ␲qϩ␪, we get a sum over Lorentzian amplitude 2 Ϫ␣L T e terms for which the inverse FT can be computed. The result- Io͑␻͒ϭ 2 2 Ii͑␻͒, ͑7͒ ͑1ϪReff͒ ϩ4Reff sin ͑kLϪ␪͒ ing expression is

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͑Ref. 10, p. 302͒, and the d␪/d␻ term which reflects that 1 tϪtr/2 G͑t͒ϭ ͚ A͑␻q͒exp Ϫ different wavelengths have different average penetration into q ͩ 2t ͑␻ ͒ͪ ͱ2␲ d q the optical coating of the mirrors and thus travel different pathlengths. The effective t is given by 1/FSR where FSR ϫexp͓i␻q͑tϪtr/2͔͒⌰͑tϪtr/2͒, ͑11͒ r ͓ is now defined by Eq. ͑15͔͒ for a cavity with dispersion. If where ⌰(x) is the step function ͑ϭ1or0forxϾ0orϽ0, we consider the contribution to n(␻) by the sample absorp- respectively͒ and tion, we see that this will slightly increase the index and thus trReff decrease the mode spacing on the low frequency side of a td͑␻q͒ϭ , ͑12͒ 2͑1ϪReff͒ transition, and the opposite on the high frequency side. Zal- icki and Zare9 suggested that dispersion effects are generally 2 Ϫ␣L/2 T e negligible for the weak absorption strengths investigated by A͑␻q͒ϭͱ2␲ , ͑13͒ tr Reff CRDS. We agree with this if one is observing only the decay of energy in the cavity, as is typically done, since in this case with Reff(␻) and k(␻) as defined above. Note that due to the small but finite propagation delay through the cavity, causal- the interference between different excited modes is filtered out. However, if one attempts to model the time dependent ity requires that G(t) is zero for tϽtr /2 rather than just for tϽ0. shape of the train of peaks leaving the cavity, then the small Equations ͑9͒ and ͑11͒ demonstrate that the requirement shifts in resonance frequency caused by dispersion must be on the input source to observe a cavity ring-down is not that considered along with the frequency dependent absorption. its pulse width be short compared to tr , or even td , but only We have explicitly shown by numerical calculation, using that the falling edge of the input pulse be short compared to Eq. ͑14͒ for the case of a Lorentzian absorption line narrow td(␻q) for the modes q that are significantly excited. Those compared to the input pulse, that the cavity output displays modes will be determined by the spectrum of the input ra- the expected sample free induction decay after the excitation diation, and ͑when we consider the effect of higher trans- pulse, only if these shifts in resonance frequencies are in- verse modes below͒ the spatial properties of the radiation. cluded in the calculation. This is an interesting subject for However, for a typical pulsed laser the total extent in time of future investigation16 since changes in pulse shape may be the input radiation is itself much shorter than the cavity de- useful for extracting information about the sample absorption cay times. In this impulsive limit, we can neglect the expo- spectrum on frequency scales between the linewidth of the nential cavity decay during the input pulse, and if we place laser used to excite the RDC and the spacing between longi- the time origin at the input pulse, we can explicitly evaluate tudinal cavity modes. the integral in Eq. ͑9͒ using Eq. ͑11͒ to give Scherer et al. state in their paper7 that, ‘‘In the case of

imp tϪtr /2 simple exponential decay, the cavity does not act as an eta- Eo ͑t͒ϭ͚ A͑␻q͒exp Ϫ q ͩ 2t ͑␻ ͒ͪ lon, i.e., standing waves are not established in the cavity.’’ d q On the contrary, exponential decay is a natural consequence ˜ ϫexp͑i␻q͑tϪtr /2͒͒Ei͑␻q͒, ͑14͒ of the RDC being a high finesse etalon, if it is excited on a single cavity mode. More typically, we have multiple mode for times greater than the end of the input pulse plus tr /2. The result is a sum of damped exponential decays for each excitation, due both to the bandwidth of the excitation source cavity resonance. Since the electric field has a decay lifetime and due to lack of exact transverse mode matching. As dis- cussed by Zalicki and Zare,9 for multiple mode excitation we of 2td , the intensity in each mode has decay lifetime of td . The decay rate, and thus the spectral width, of those reso- expect in general a multiple exponential decay of the RDC nances which overlap the sample absorption spectrum will unless the excited modes have the same loss Reff . Even in be increased. Sample absorption lines that do not overlap this case, the intensity of the decay will not be exactly expo- excited cavity modes do not contribute to the rate of light nential, but show modulations due to beating between differ- intensity decay and are not detectable in CRDS. ent mode frequencies. Beating among longitudinal modes If we treat n(␻) and ␪(␻) as changing slowly with ␻ ͑same transverse mode numbers͒ will have periods that are ͑thus neglecting dispersion effects due to narrow absorption submultiples of tr. This simply corresponds to the exponen- lines͒, the cavity mode spacing is given by tially decaying series of pulses generated by the initially in- c jected light as it rings down inside the cavity. Thus a simple FSRϭ . ͑15͒ exponential decay of the cavity implies that only one longi- dn d␪ 2L n͑␻͒ϩ␻ Ϫc tudinal mode has been excited, which also requires that a ͩ d␻ͪ d␻ standing wave must be produced inside the cavity, in direct For constant dn/d␻ and d␪/d␻, we have equally spaced conflict with the above statement of Scherer et al. In most resonance modes. If in addition Reff is constant, the pulse in applications of CRDS, however, the difference between tr the cavity travels with no change in shape since the effective and td is large enough that this round-trip beating pattern can loss and group velocity is the same for all frequencies con- be easily filtered electronically without significantly distort- tained in its bandwidth. Dispersion changes the velocity of ing the exponential signal envelope which is the sum of the the wave due to both the ␻ (dn/d␻) term, which is standard decay of each excited cavity mode.

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III. CAVITY RING DOWN WITH A NARROW trix element for the transition ␮21 . Let the molecule be at BANDWIDTH ABSORBER position z0 at tϭ0 and its velocity along the axis of the RDC 12 be vz . Starting with Eq. ͑8.1–6͒ in the text by Yariv, and 9 The paper by Zalicki and Zare considered the effect of then using first order time dependent perturbation theory, we having an intracavity absorption feature with a linewidth nar- find that the coherence between the two states, ␳21 , created rower than the bandwidth of the laser used to excite the by the time dependent electric field is given by cavity. As long as one is exciting many modes of the cavity across the absorption line, this leads to nonexponential cavity i t Ϫi␻0t i␻0tЈ decay. This has been verified by Jongma et al.17 and more ␳21͑t͒ϭe ␮21E͑z0ϩvztЈ,tЈ͒e ប ͵Ϫϱ quantitatively by Hodges et al.18 These effects are closely related to the well known line shape distortions observed in ϫ͑␳11Ϫ␳22͒edtЈ, ͑16͒ conventional absorption spectroscopy when the instrumental resolution is not higher than the width of spectral features. where (␳11Ϫ␳22)e is the equilibrium difference in popula- There, the observed sample absorption ͑integrated over the tions between the lower and upper levels of the optical tran- instrument function͒ does not follow Beer’s law for samples sition. We will assume that the input pulse is shorter than 19 that are optically thick. For optically thin samples, such one round trip time, tr . This is done only for simplicity. In measurements give the correct integrated line strength ͑also fact, we could divide any arbitrary incoming field in portions known as the equivalent width͒, though not the correct peak that are shorter than the round trip time, and the following value due to instrumental broadening of the line. Hodges analysis could be applied to each of these ‘‘time compo- et al.18 have demonstrated that an additional complication nents’’ and the results added. We will use Eq. ͑1͒ for 21 can arise in cavity ring-down due to the spectral structure in E(z,t) for the field inside the RDC. Putting this into the conventional multimode pulsed lasers. This results in inten- above equation, we find for the coherence induced by the sity distortions when the width of an absorption feature is pulse comparable to the width of structures in the spectrum of the int͑t/tr͒ excitation laser. We anticipate that these effects can be re- vz Ϫi␻0t n Ϫi2n␪ ␳21͑t͒ϭe ͚ ␳ϩR e exp i␻0 1ϩ ntr moved by averaging ring-down decays observed as the nϭ0 ͭ ͫ ͩ c ͪ ͬ modes of the excitation laser are swept to produce a smooth average excitation line shape. vz Ϫ␳ Rnϩ1/2eϪi͑2nϩ1͒␪ exp i␻ 1Ϫ ͑nϩ1͒t , Another important consideration discussed by Zalicki Ϫ ͫ 0ͩ c ͪ rͬͮ and Zare is the relationship between the width of the absorp- ͑17͒ tion features and the FSR of the cavity. They report to show that the bandwidth of light admitted into the RDC depends where ␳Ϯ is the coherence produced by a single forward or upon the length of the excitation pulse. We think that this is reverse going pulse in the cavity a consequence of their considering the FT only over a time interval of the input pulse ͑which they took to have a square i z0 ␳ ϭ ␮ ͱT͑␳ Ϫ␳ ͒ exp Ϯi␻ wave amplitude͒ and not of the whole damped series of re- Ϯ ប 21 11 22 e ͩ 0 c ͪ flections the intracavity molecules interact with. Despite this, they recognize that the excitation caused by the coherent sum vz ϫ E*͑t͒exp i␻ 1Ϯ t dt. ͑18͒ of all the cavity reflections will only excite transitions that ͵ i ͩ 0ͩ c ͪ ͪ overlap one of the excited modes of the RDC. Thus they reach the same observable consequence as the present analy- We will get constructive interference ͑and thus net absorp- sis and one may dismiss the differences in our respective tion͒ of the coherence produced on successive round trips of analysis as largely semantic. the cavity only if ␻0 satisfies the following equation with In a frequency domain analysis of the problem, one integer N should integrate ͑in calculating the FT͒ over the whole time of interaction. To consider the spectral content of the radia- vz 2␲ 2␪ ␻ 1Ϯ ϭ Nϩ . ͑19͒ tion field only in a given limited time interval is equivalent to 0ͩ c ͪ t t r r employ a mixed time frequency representation. Time and frequency localized representations are the subject of wave- Thus we will only get net absorption by the sample if either let transform theory, which is becoming quite an active field the forward or backward Doppler shifted Bohr frequency of research and application in recent years.20 ͑the left hand side of the above equation͒ satisfies the same It is often useful to compare descriptions of the same equation as a cavity resonance ͓see Eq. ͑4͔͒. The cavity acts phenomena both in the time and frequency domains, though to produce a multiple-pulse time domain ‘‘Ramsey’’ fringe they must give the same results. The analysis we will present pattern, as previously noted by Zalicki and Zare.9 Including a below is also complimentary in that it focuses on the mol- T2 for the optical transition will allow off resonance absorp- ecules in the cavity, while the earlier treatment focused on tion, but this is just equivalent to considering the resulting the optical properties of the system. Consider excitation of a line broadening in the frequency domain. Thus we recover two level system with optical resonance ␻0 and dipole ma- the condition that CRDS is only sensitive to sample absorp-

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Downloaded¬16¬Feb¬2010¬to¬129.6.144.159.¬Redistribution¬subject¬to¬AIP¬license¬or¬copyright;¬see¬http://jcp.aip.org/jcp/copyright.jsp K. K. Lehmann and D. Romanini: Cavity ring-down spectroscopy 10269 tion that overlaps one of the very narrow cavity resonances using a continuous wave laser locked on a molecular transi- excited by the input radiation, even in the limit of short pulse tion, one could also lock the RDC to a spectroscopic stan- excitation. dard. Since we can repetitively scan the cavity over one FSR The spectral selectivity of sample excitation inside a and observe the time when the reference laser passes through RDC can be shown to be demanded also by the laws of resonance, we can have continuous feedback control of the thermodynamics. Let us suppose, contrary to what we have cavity length. In this case, one can use the time that the demonstrated above from first principles, that the RDC does excitation laser is fired relative to the scanning ramp to con- not act as an etalon for short pulse ͑or short coherence trol the frequency sampled on any laser shot. length͒ excitation, and that the full bandwidth of the pulse Thus rather than being a problem, the frequency selec- enters the cavity and excites all optical transitions that over- tive nature of the cavity resonances opens the possibility for lap its spectrum, as is claimed by Scherer et al.7 It has been a dramatic improvement in the resolution available in pulsed experimentally demonstrated that spontaneous emission is laser experiments. The combination of this technique with turned off in a cavity with no mode resonant with the atomic CRDS should dramatically improve both the sensitivity and transition.22 If black body radiation could enter the cavity resolution that can be realized in sub-Doppler spectroscopy. and excite an atom, but the atom were not able to undergo While this work was being reviewed for publication, we spontaneous emission, an infinite atomic electronic tempera- were informed of an earlier experiment that demonstrated ture would be produced by interaction with a finite tempera- these principles. Teets et al.23 observed a two-photon transi- ture heat bath. This clearly would be a violation of the sec- tion in atomic sodium inside an optical cavity of modest ond law of thermodynamics. (Ϸ16) finesse. By scanning the length of the cavity, they The fact that the RDC acts as a frequency selective filter observed a transition of width 12 MHz, despite a Fourier might lead to the erroneous conclusion that CRDS is not transform limit for the input pulse of about 170 MHz. The suitable for quantitative spectroscopy of high resolution round trip time of the cavity used (13 ns͒ was about twice spectra. This is simply not the case. As Meijer et al.5 have the length of the input pulse. The spectral selectivity ob- shown, the excitation of many transverse modes of the RDC served in this experiment directly invalidates the model of 7 will lead to a near continuous spectrum of cavity resonances, CRDS presented by Scherer et al. eliminating this potential problem. We will return to this The next section will consider the spectrum of longitu- point below. At present, we will show how even with single dinal modes excited by a FT limited pulse having Gaussian mode excitation of the RDC cell, one need not miss absorp- transverse profile. We also need to account for the fact that tion features. without great care, one will excite multiple transverse modes The narrow bandwidth of light admitted into a RDC of- of the cavity, each of which has its own resonance frequency. fers a tremendous opportunity for high resolution spectros- The following section will then consider the excitation of copy. One has to replace the static length cell considered by higher transverse modes. Meijer et al., Zalicki and Zare, and others, by a cell whose length can be varied by at least ␭/2, ͑one-half wavelength͒ IV. FRINGES VERSUS THE EXCITATION PULSE which will shift each mode by one FSR of this ‘‘etalon.’’ LENGTH Since only light with a bandwidth much narrower than the excitation laser enters the cell, it is possible to do spectros- Consider the mode matched excitation of a RDC by a copy with a resolution much greater than that of the laser pulse with center frequency ␻c . For a sufficiently short in- source used to excite the cavity! Consider a commercially put, this excitation will create a traveling pulse inside the available OPO laser with a bandwidth of ϳ125 MHz. By RDC. Neglecting dispersion effects, this pulse will propagate using a cavity of length LϽ75 cm, the mode spacing will be back and forth with no change in intensity profile, except for greater than twice the laser linewidth and if mode matched, a slow attenuation. If dispersion effects are important over we will primarily excite only one mode. We can then scan the bandwidth of the pulse, either from spectral changes in the cavity and observe a spectrum with a full width instru- mirror reflectivity and phase shifts, or from the presence of a mental resolution of ϳ16 kHz for a cavity decay time of narrow absorption feature that is not optically thin, the intra- ϳ10 ␮s. In practice, time of flight of molecules through the cavity pulse shape will change with time. In a frequency laser beam will limit resolution to about 0.1 MHz, 3 orders domain picture, this corresponds to the variable mode spac- of magnitude better than the FT limit for a few nanoseconds ings described by Eq. ͑15͒. laser pulse. Such experiments will require interferometric The most obvious sign that the cell is acting as an etalon control of the length of the RDC, and tracking the excitation would be the observation of ‘‘fringes’’ in the transmitted laser as the cavity is scanned. The latter should be of minor light as the center frequency of the excitation beam is difficulty, since in this situation, high contrast interference scanned. It is the lack of such a modulation when the input fringes will be observed as the cavity and laser are detuned, pulse is short compared to tr that appears to have produced allowing for standard feedback techniques to be used. Main- the belief that a RDC does not act as an etalon in the impul- taining the RDC to interferometric accuracy, say ϳ1 MHz, sive limit. However, it is well known that one does not ob- can, in principle, be achieved in several ways. One can pur- serve interference fringes if one excites an etalon with broad chase a temperature stabilized etalon with such passive sta- bandwidth light. This is just a consequence of the transmis- bility, which could be modified to act as a cell for CRDS. By sion spectrum being periodic with period equal to the FSR.

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Thus if the input pulse spectrum is much wider than one FSR, the expected interference structure is ‘‘washed out,’’ and the transmission coefficient becomes constant with value T2/2(1ϪR), independent of the center frequency of the in- put light pulse. What happens if the input pulse length is closer to the value of the cavity round trip time? Zalicki and Zare consid- ered this question for the case of square wave excitation.9 We presently consider a FT limited Gaussian shaped input pulse with an intensity FWHM equal to ⌬t, since this pro- vides a better model for many real lasers. The frequency spectrum of this input pulse will also be Gaussian, with a FWHM equal to 0.44/⌬t ͑Ref. 24 p. 334͒. In the limit of a high finesse RDC, each resonance will have a nearly Lorent- zian shape with peak transmission equal to 2 ͓T exp(Ϫ␣L/2)/(1ϪReff)͔ , and FWHM equal to 1/2␲td , where tdϭn(␻q)LͱReff/͓c(1ϪReff)͔ is the decay time for the qth resonance mode and R ϭR exp(Ϫ ( )) is the ef- eff ␣ ␻q FIG. 1. Plot of the ratio of the predicted transmission divided by its inco- fective reflectivity for that mode. herent excitation value for the pulse frequency centered on a cavity mode Cavity resonances will be narrow relative to the spectral ͑solid line͒ and for when it is halfway between cavity modes ͑dashed line͒. structure of the input pulse as long as the pulse duration is short compared to t . Therefore, in order to calculate the d 2 total intensity transmitted by the cavity, we can treat the next neighbors is given by exp͓Ϫ(␲⌬t/ͱln 2tr) ͔.When resonances as delta functions. This approximation will break ⌬tϭ0.5tr , this ratio is already only 0.028, while for Ϫ7 down, because of the long tails of the Lorentzian resonances, ⌬tϭtr , this ratio is 6.6 10 . Thus very selective excitation when the width of the input pulse is much narrower than of a transverse mode matched RDC is possible with FT lim- ited pulses with FWHM of ϳ0.5t or longer. Below, we will cavity mode spacing, i.e., when ⌬tӷtr . For each resonance r see that the need for mode matching can be relaxed in case of mode q at frequency ␻q , from Eq. ͑4͒ we find that the en- ergy transmitted by the cavity is a confocal ͑or more generally re-entrant͒ cavity configura- tion. ␲ ⌬t T2 JqϭJ0 V. EFFECT OF HIGHER ORDER TRANSVERSE ͱln 2 t 2͑1ϪR͒ r MODES 2 ␲ ⌬t ͑␻qϪ␻c͒ Our analysis up to this point has been one dimensional, ϫexp Ϫ , ͑20͒ ͫ ͩ ͱln 2 tr FSR ͪ ͬ which is appropriate for light perfectly mode matched into the RDC. In order to consider the excitation of a RDC pro- where J0 is the energy in the input pulse. The total transmit- duced by a beam with an arbitrary spatial distribution, we ted energy is obtained by summing over all the resonance must again use the superposition principle, but now in space modes. In the limit that the input pulse is short compared to rather than time, expanding the input wave in a set of cavity tr the sum over modes can be approximated by an integral modes. Due to the finite transverse extent of an open optical 2 which gives just the factor of T /2(1ϪReff) expected for cavity, its resonances, unlike those of the closed electromag- incoherent excitation of an etalon. In Fig. 1, we plot the ratio netic cavities commonly treated in textbooks, need not be a of the predicted transmission divided by this incoherent ex- complete, orthogonal set ͑Ref. 24, p. 568͒. However, stable citation value for the pulse center frequency exactly resonant cavities with small diffraction losses have resonance modes with a cavity mode and for when it is exactly halfway be- closely approximated by Hermite–Gauss functions up to tween cavity modes. The ratio of the two curves is the ex- relatively high transverse mode orders, which do have this pected fringe modulation as the excitation laser ͑or the RDC attractive property ͑Ref. 24, p. 569͒. We will derive the gen- length͒ is scanned. Significant modulation is observed when eral cavity output, for an arbitrary input pulse, by expansion ⌬t is greater than ϳ0.5tr . This can be rationalized when one in both frequency ͑for the time dependence͒ and cavity remembers both that the Gaussian has a tail and is not a modes ͑for the transverse spatial dependence͒. While the fol- square wave ͑where there would be zero modulation for lowing discussion is for the response of the cavity calculated ⌬tϽtr) and also that interference is given by overlap of the at its output, the same treatment can be applied to the field electric field, which has a FWHM ͱ2⌬t. When one recalls produced inside the cavity and leads to similar results. that for a Gaussian pulse ⌬t⌬␯ϭ0.44, we see that when the In the paraxial approximation ͑small propagation angles FWHM of the pulse is half the cavity round trip time, the and deviations from the optical axis͒, the Maxwell equation spectral width has a FWHM only 0.88 as large as the spacing for wave propagation in a homogeneous dielectric medium between the longitudinal modes. When the laser pulse is cen- ͓refraction index n(␻)] has a set of solutions known as the tered on a mode, the ratio of the mode excitation to that of its transverse electro–magnetic ͑TEM͒ modes,12,24 whose order

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Downloaded¬16¬Feb¬2010¬to¬129.6.144.159.¬Redistribution¬subject¬to¬AIP¬license¬or¬copyright;¬see¬http://jcp.aip.org/jcp/copyright.jsp K. K. Lehmann and D. Romanini: Cavity ring-down spectroscopy 10271 is indexed by two positive integers (m,n). The spatial field stable symmetric cavities LϽ2Rc . While z0 for the TEM set amplitude for these modes is written in terms of Hermite– of cavity modes is frequency independent, note that this is 2 Gauss functions not the case for w0ϭ␭z0 /␲n(␻). We now consider an input field of angular frequency ␻ 1 ͱ2x that has a spatial distribution that just matches the TEMm,n Emn͑x,y,z,␻͒ϭ Hm ͱ2mϩnϪ1␲n!m!w2͑z͒ ͩw͑z͒ͪ mode of the field. We denote the amplitude of this field by ˜ Ei,mn(␻). Similar to the one dimensional case ͓Eq. ͑7͒ and ͱ2y x2ϩy 2 ik͑x2ϩy2͒ ϫH exp Ϫ Ϫ following͔, we can sum over all round trips through the cav- nͩw͑z͒ͪ ͩ w2͑z͒ 2R ͑z͒ ity to arrive at an output electric field, which will still match f the TEMm,n in the perpendicular plane and have an ampli- Ϫikzϩi͑mϩnϩ1͒␩͑z͒ͪ. ͑21͒ tude given by ˜ ˜ ˜ Eo,mn͑␻͒ϭͱ2␲Gmn͑␻͒Ei,mn͑␻͒, ͑26͒ These monochromatic waves have wave vector kϭn(␻)␻/c along z, and time dependence exp(i␻t). They where are characterized by a transverse beam waist of size w(z), by 1 T 2eϪ␣L/2eϪikL a radius of curvature of the phase fronts Rf(z), and by a G˜ ͑␻͒ϭ , ͑27͒ mn 2 Ϫ␣L Ϫi͑kϪ␦kmn͒2L phase shift induced by diffraction ␩(z). For free space ͱ2␲ 1ϪRmne e propagation along the paraxial axis z, these TEM parameters L change as follows: ␦kmnLϭ2͑mϩnϩ1͒arctanͱ . ͑28͒ ͑2RcϪL͒ zϪz 2 w The different transverse amplitude distribution of each TEM w͑z͒ϭw0ͱ1ϩ , ͩ z0 ͪ mode may result in different losses due to spatially varying 2 defects in the mirrors surfaces and to their finite size. We z0 R ͑z͒ϭ͑zϪz ͒ 1ϩ , ͑22͒ account for this by using mode dependent reflectivities f w ͫ ͩ zϪz ͪ ͬ w Rmn(␻). ˜ zϪzw Like in the one dimensional case, Gmn(␻) is a comblike ␩͑z͒ϭarctan , ͩ z ͪ transmission function, and we can identify the longitudinal 0 cavity resonances with the peaks of this function, and their with zw the focal point of the wave, where the beam waist decay rate with their width. The functional dependence of reduces to its minimum w . Once the value of w is given, ˜ 0 0 Gmn(␻) is similar to that for the one dimensional case of Eq. the ‘‘confocal length’’ z0 is also determined ͑7͒, but the position of its resonances has a shift dependent 2 on the TEM order, due to the ␩(z) term in Eq. ͑21͒. The ␲w0n͑␻͒ z ϵ . ͑23͒ frequencies of the longitudinal resonances are found by the 0 ␭ solutions of the following equation ͑obtained numerically by

At fixed frequency ␻, fixed w0 ͑or equivalently z0) and fixed iteration͒ R ͑or z), the set of TEM modes form a complete orthogo- f mn c nal and normalized set of functions ␻ ϭ ␲qϩ␪ϩ2͑mϩnϩ1͒ mnq n͑␻ ͒L ͫ mnq * x,y,z, x,y,z, dxdyϭ , EmЈnЈ͑ ␻͒Emn͑ ␻͒ ␦mЈm␦nЈn L ͵ ϫarctan , ͑29͒ ͑24͒ ͱ͑2R ϪL͒ͬ c

Emn* ͑xЈ,yЈ,z,␻͒Emn͑x,y,z,␻͒ϭ␦͑xЈϪx͒␦͑yЈϪy͒. where ␪ is the phase change upon reflection, introduced ear- ͚m,n lier. In the time domain, each mode will have an exponential In the axially symmetric case, each mode is degenerate with decay with a time constant ͑for power͒ given by respect to polarization. trRmn,eff If we wish to consider light coupled into an optical cav- td,mn͑␻mnq͒ϭ ͑30͒ ity made of two mirrors, then one should select the beam 2͑1ϪRmn,eff͒ parameter so that the radius of the beam just matches the with Rmn,effϭRmn exp(Ϫ␣(␻mnq)L). Note that if Rmn(␻)is radius of curvature of each mirror at its surface. For the not constant due to mode dependent loss, we will have a symmetric cavity considered in here ͑both mirrors with cur- nonexponential decay even without sample absorption. vature Rc , as before͒, this condition implies that the beam We will now consider excitation of a cavity by an arbi- focus is at the center of the cavity and the confocal length trary input pulse whose electric field is given by 12 is Ei(x,y,z,t). Using the superposition principle, we can ex- 1 pand Ei(x,y,z,t) in terms of the set of functions z0ϭ 2ͱ͑2RcϪL͒L. ͑25͒ Emn(x,y,z,␻) for which we have just presented the trans- More general expressions can be found in the text by mission properties. The total output field, Eo(x,y,z,t) can 12 Yariv. The condition that z0 is real defines the range of then be written as

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˜ ˜ i␻t ˜ Eo͑x,y,z,t͒ϭ Gmn͑␻͒Ei,mn͑␻͒Emn͑x,y,z,␻͒e d␻, Eo͑x,y,z,t͒ϭ Amn͑␻mn͒Ei,mn͑␻mnq͒Emn͑x,y,z,␻mnq͒ ͚mn ͵ m͚,n,q ͑31͒ tr tϪtr/2 ϫexp i tϪ ␻ exp Ϫ , ͑34͒ ˜ ͫ ͩ 2 ͪ mnqͬ ͩ 2t ͪ where Gmn(␻) is given by Eq. ͑27͒ and d,mn

˜ 1 Ei,mn͑␻͒ϭ Emn* ͑x,y,z,␻͒ ͱ2␲͵͵͵ where the factor Amn(␻) is given by Eq. ͑13͒ with Reff,mn in place of Reff and td,mn is given by Eq. ͑30͒. For brevity, the Ϫi␻t ϫEi͑x,y,z,t͒e dx dy dt. ͑32͒ dependence on ␻mnq has been left implicit for several of the parameters. The output intensity of the cavity will contain ˜ Ei,mn(␻) is the ‘‘excitation amplitude’’ of the mode with beating between different excited modes. However, if one transverse order mn and frequency ␻. Note that we do not detects the total light intensity crossing the output plane, the have to specify at which value of z the integral is evaluated orthogonality of the mode functions eliminates beats be- at, due to the superposition principle for space propagation. tween modes with different transverse mode numbers. If we assume that the input field has a small fractional The fraction of input pulse energy coupled to a particular bandwidth, which allows one to neglect the frequency depen- TEMmn mode is given by the squared magnitude of the mode 2 dence of the TEM parameters, we can derive another expres- overlap amplitude, ͉Ei,mn͉ , times a spectral density factor, sion for the time dependent cavity output decomposed over which for the case of an input pulse with Gaussian temporal the spatial TEM modes. For each mode, the cavity response profile was given by Eq. ͑20͒, which is easily generalized to is then a convolution of the corresponding Green function the present case. Thus together with the results of Appendix 25 with the time dependent amplitude Ei,mn(t) for that mode B, we have given explicit expressions for the distribution of excitation of different modes of an optical cavity for an input ik͑␻Ј͒z pulse of Gaussian shape in space and time, with alignment Eo͑x,y,z,t͒ϭ Emn͑x,y,z,␻Ј͒e ͚mn restricted by the paraxial conditions and of duration short compared to the cavity decay time. The present results can nz be easily extended, using numerical integration, to treat input ϫ G tϪtЈϪ E ͑tЈ͒dtЈ, ͑33͒ ͵ mnͩ c ͪ i,mn pulses of arbitrary temporal and spatial shape. As seen by Eq. ͑34͒, the output signal in CRDS is only where ␻Ј is the center of the field bandwidth; Gmn(t) is the sensitive to the sample loss at the frequencies of excited ˜ inverse Fourier transform of Gmn(␻) given by Eq. ͑27͒; and modes of the cavity, and thus the spectrum of the cavity is ˜ Ei,mn(t) is the inverse Fourier transform of Ei,mn(␻). important to the design and interpretation of CRDS. We can We leave to Appendix A the discussion of some special see that the longitudinal modes ͑for fixed m,n) are separated cases and examples. An important case is that of a beam by angular frequency 2␲/tr , while transverse modes ͑chang- which is separable as the product of space and time depen- ing mϩn, with q fixed͒ are separated by (4/tr) dent functions. A special separable case is when the spatial ϫ arctanͱL/(2RϪL). The full spectrum of the cavity ͑say component of the beam is Gaussian and astigmatic, with ar- for an incoming plane wave that excites all TEM orders͒ will bitrary spot sizes, focal positions and axis, but only slightly be continuous if the ratio of these spacings is irrational. If it tilted with respect to z as required by the paraxial approxi- is rational with the transverse spacing to the longitudinal mation. Then, recursive analytic expression exist and are de- spacing equal to M/N, the spectrum will consist of a series rived in Appendix B for the amplitude excitation coefficients of resonances with frequencies exactly spaced by 1/(trN), in terms of the beam parameters. These results allow one to i.e., exactly a factor of N less than the separation of the model the coupling of a laser operating on a single TEM00 TEM00 modes. It is easily seen from above that for a confo- mode to an external optical cavity. In addition, the method of cal cavity (LϭRc), the transverse mode separation is exactly the Appendix B can be extended to input beams with higher half of the longitudinal modes, giving a frequency spacing TEM order, and the expressions derived could be applied to between resonances of 1/2tr . However, for other values of the more general case of a beam which is the superposition the spacing we can get other low order rational ratios. For of TEM modes characterized by the same arbitrary param- example, if LϭRc/2 or 3Rc/2, we will get a frequency spac- eters. This would be adequate for modeling the cavity cou- ing of 1/3tr , i.e., just 3 times as dense as the TEM00 modes. pling of a nonseparable beam produced by a laser which is For general multiple mode cavity excitation, the trans- not operating on a single transverse mode. verse profile of the light intensity changes shape on each Let us consider the case when for each mn the longitu- pass of the cell. For a rational ratio as discussed above with ˜ dinal resonances in Gmn(␻) are narrow with respect to other divisor N, the intensity will be exactly periodic ͑except for spectral widths of the problem. In the time domain this im- an overall damping factor͒ after N round trips. This is the plies that the variation of the field is faster than the cavity separation condition for the multipass re-entrant cavity con- decay times. Then, the same result as in Eq. ͑11͒ can be figurations used and discussed by Herriott et al.26 obtained here and Eq. ͑14͒ can be generalized to Meijer et al.5 suggested that one could use a nonconfocal

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Downloaded¬16¬Feb¬2010¬to¬129.6.144.159.¬Redistribution¬subject¬to¬AIP¬license¬or¬copyright;¬see¬http://jcp.aip.org/jcp/copyright.jsp K. K. Lehmann and D. Romanini: Cavity ring-down spectroscopy 10273 cavity to achieve an essentially continuum excitation spec- Hodges, Looney, and van Zee.27 In a series of careful mea- trum and thus eliminate the expected distortions of the cavity surements, they have recently verified that CRDS cells have ring-down spectrum caused by a static cell acting as a fre- the expected etalon transmission properties even when ex- quency filter. They used slightly diverging input radiation cited with laser pulse length a few times the cavity round trip ͑incorrect sign of Rf compared to the mode matched case͒ to time, directly contradicting the claims of Scherer et al. insure that a wide range of transverse modes were excited, For certain applications of cavity ring-down, especially giving a dense spectrum. We would like to point out that one for obtaining spectral resolution higher than the input laser, must be careful not to inadvertently use a mirror separation the mode coupling, or cavity mode structure, or both, will that leads to a rational ratio unless the divisor is large enough have to be carefully controlled. Mode matching requires con- that the resulting ‘‘picket fence’’ spectrum has spacings trol over four experimental variables: ͑1͒ The position of the much smaller than the linewidths in the spectrum one wishes input beam relative to the optic axis of the etalon; ͑2͒ the to observe. relative input angle; ͑3͒ the spot size; and ͑4͒ the radius of We would also like to mention that the clever solution of curvature. Precise adjustment of the first two is a routine part Meijer et al.5 comes at a cost of reduced spatial resolution in of most optical setups and is easily optimized. Control of the CRDS. This is clearly undesirable when the method is used, second two requires the equivalent of a ZOOM telescope as by Zalicki et al.,8 to observe the spatial profile of a spe- lens. In addition, if the laser source is not operating on a cies’ concentration. A perhaps more general problem is that stable TEM00 mode, spatial filtering will be needed to obtain it increases the required size and spatial uniformity of the a Gaussian beam profile that can be matched to the cavity coatings of mirrors used for CRDS. A multiexponential ring- TEM00 mode. Monitoring the size and shape of the intensity down signal will result from excitation of modes with differ- emitted from the cavity, say with a camera,18 appears to pro- ent transverse profile if the different portions of the mirrors vide a direct diagnostic to allow the optimization of the input surfaces have different reflectivity. Such variations in reflec- coupling. The Super Cavity, a high finesse spectrum analyzer tivity may be caused also by dust or dirt. Further, changes in sold by Newport Co., uses a near planar cavity (LӶRc); in the distribution of modes excited as the laser is scanned will our laboratory we have achieved Ͼ90% excitation of the lead to variations in the ‘‘background’’ loss of the cell which TEM00 mode of such a cavity by using a GRIN lens at fixed will translate into a poor zero absorption baseline. It is also distance from the cavity to focus the output of a single mode important to ensure that the entire cross section of the beam fiber.28 One can thus expect high selectivity with careful is collected and detected to prevent transverse mode beating alignment of a near diffraction limited beam, which for most from distorting the cavity ring-down. We wish to emphasize pulsed lasers will require spatial filtering. Likely, it will be that these are surmountable issues in most experiments, but more convenient to use a confocal or another ‘‘degenerate’’ they should be considered in experimental design. mirrors separation. Note, however, that a confocal cavity is The experiments described by Scherer et al.7 have been unstable if one has any finite difference in the curvature of interpreted by them as demonstrating that a RDC does not the two cavity mirrors!12,24 The precision required for the act as an etalon even for input pulses with a FWHM twice as mirror separation will increase linearly with both the number long as the cavity round trip time. They base this in part on of transverse modes excited and with the ultimate resolution the lack of modulation in the observed cavity transmission as required. We note that for exactly coaxial excitation of the a function of the center frequency of their near transform cavity with a symmetric beam, one will only get excitation of limited laser. As is shown in Fig. 1, very strong fringe modu- TEMmn modes with even values of m and n, and thus have lation is expected for a transverse mode matched cavity an effective transverse mode spacing twice what is otherwise when the input pulse length is as long as twice the cavity expected. round trip time. However, these authors make no mention in Thus in applications of CRDS one is often interested in their paper of any attempt to mode match the radiation, nor either deliberately exciting a known range of cavity modes or to carefully adjust the mirror separation for a re-entrant con- in carefully exciting only the lowest order one. It is then figuration. Thus it is almost certain that the reported lack of useful to have expressions for the excitation of different fringe modulation is a consequence of the effects previously modes expected for a given input beam. In Appendix B, we predicted by Meijer et al.5 The interference structure has give an analytic expression for the overlap of a Gaussian been ‘‘washed out’’ due to simultaneous excitation of many input beam with arbitrary parameters and alignment with re- TEMmn cavity modes. This also naturally accounts for the spect to the TEM00 mode of the cavity, and a recursion rela- fact that they observed a methane spectrum with the same tionship that allows the overlap with higher order modes to relative intensities and linewidths when observed with differ- be calculated as well. These general formulae are sufficiently ent length cavities, even though the longitudinal mode spac- complex that it is difficult to directly gain insight by inspec- ing of the shorter cavity was almost twice the observed tion of the results, though they can be easily programmed FWHM of the methane lines. Scherer et al. claim that their and plotted for different ranges of parameters. In Appendix results contradict the analysis of Zalicki and Zare.9 The A limiting examples are considered that will likely be useful present work demonstrates that the experimental results of estimating precision needed for a specific experimental ar- Scherer et al. are completely in accord with the prior predic- rangement. tions of Meijer et al., and in no way contradict the results of We wish to close this section with a brief discussion of Zalicki and Zare. The same conclusion has been reached by diffraction effects in optical cavities used in CRDS. The

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Downloaded¬16¬Feb¬2010¬to¬129.6.144.159.¬Redistribution¬subject¬to¬AIP¬license¬or¬copyright;¬see¬http://jcp.aip.org/jcp/copyright.jsp 10274 K. K. Lehmann and D. Romanini: Cavity ring-down spectroscopy quantitative theory of these effects is rather complicated and culating the transverse mode expansion amplitudes of an ar- we refer the reader to the text by Siegman24 for a good in- bitrary input Gaussian beam that is coupled into an optical troduction to this subject; the material given below is taken cavity. from this source. It is interesting that the simple first order While ring-down cavity spectroscopy is a technique that estimate of mode losses, by calculation of the fraction of a yields good results with a very simple setup, a thorough un- given Gaussian mode that ‘‘spills over’’ the mirrors, substan- derstanding of the properties of resonant cavities is necessary tially overestimates the diffraction losses. The low order for optimal design of the experiment and for this technique resonant modes distort ͑‘‘pull in their skirts’’͒, falling below to be used for quantitative absorption measurements of spec- the Hermite–Gaussian function near the mirror edge, thereby tra with narrow lines. effectively reducing diffraction losses. Diffraction losses are typically characterized by the dimensionless resonator 2 ACKNOWLEDGMENTS Fresnel number, Nfϭa /(L␭), where 2a is the diameter of the resonator limiting aperture ͑usually the mirror coating͒, We are grateful to Dr. Anthony O’Keefe for extensive L is the cavity length, and ␭ the wavelength of resonant correspondence that has clarified the differences in our un- light. Values of Nfϳ100 are common in CRDS experiments. derstanding of cavity ring-down spectroscopy and for en- For a stable cavity not far from a confocal design, the highest couragement that we publish our model. Dr. J. P. Looney order Gaussian modes that will ‘‘fit’’ inside the cavity aper- and Dr. R. D. van Zee are acknowledged for helpful discus- ture will have m,nϷ␲Nf . Given the rapid fall-off of the sions, for sharing their unpublished results, and for bringing Hermite–Gaussian functions beyond their ‘‘classical turning Refs. 29 and 30 to our attention. Professor Richard Saykally points,’’ diffraction losses fall rapidly for modes with m,n is acknowledged for bringing Ref. 23 to our attention. This below this limiting value. For the TEM00 mode of a confocal work was supported by grants from the National Science cavity with circular mirrors, the single pass diffraction power 2 4 Foundation, the National Institute of Standards and Technol- loss is approximately ␲ 2 Nf exp(Ϫ4␲Nf) for NfϾ1. This Ϫ52 ogy, and by a European Community grant under the Human represents a loss of ϳ4ϫ10 even for Nfϭ10! Capital and Mobility program.

APPENDIX A: TRANSVERSE CAVITY MODE VI. CONCLUSION EXPANSION A good approximation if the input radiation derives from We have demonstrated that contrary to widespread belief a laser operating on a single transverse mode, is that the coherence effects are important in cavity ring-down spectros- beam has a separable space and time dependence, copy in presence of narrow absorption lines, independent of E (x,y,z)E (t). Then, if we assume that the TEM param- the physical or coherence length of the laser pulse used to is it eters do not change appreciably over the ͑limited͒ bandwidth excite the cavity. We find that the superposition principle of of the input field, we can deal explicitly with the fast ␻ optics provides a natural and convenient framework for pre- dependent term (exp(Ϫikz)) in the TEM functions and find dicting the effects of cavity excitation, regardless of the spec- that the projection coefficients are also separable tral or spatial characteristics of the light source. Using the frequency domain representation we have shown that for ˜Esep ͑␻͒ϭ˜E ͑␻͒ei͑kϪkЈ͒z E ͑x,y,z͒E* ͑x,y,z,␻Ј͒dx dy mirrors of high reflectivity the radiation field inside the cav- i,mn it ͵ is mn ity will have negligible spectral intensity outside the cavity ˜ i͑kϪkЈ͒z resonances. This gives a simple and general solution to the ϭEis,mnEit͑␻͒e , ͑A1͒ problem of absorption lines narrower than the cavity free zn spectral range, since it indicates that absorption will be ob- Esep ͑t͒ϭE E tϩ , i,mn is,mn it c servable only when the lines overlap the cavity resonances. ͩ ͪ We have shown that the same can also be obtained in the where the Eis,mn are practically constants, and kЈϭk(␻Ј). time domain representation by using first order time depen- Using these results, one can write somewhat simpler expan- dent perturbation theory to account for the total molecular sions in TEM modes for the cavity response excitation produced as an input light pulse bounces back and forth through the cavity. The problem of missing absorption sep Eo ͑x,y,z,t͒ϭ͚ Eis,mnEmn͑x,y,z,␻Ј͒ lines in cavity ring-down measurements has been previously mn considered,5,9 but we have here discussed in more detail the ˜ ˜ iwt advantages and possible limitations of the proposed solution ϫ͵ Gmn͑␻͒Eit͑␻͒e d␻ ͑A2͒ of using high order cavity mode excitation. In addition, we have argued that the frequency selectivity of a ring-down ikЈz ϭ Eis,mnEmn͑x,y,z,␻Ј͒e cavity can be turned in a tremendous advantage, opening the ͚mn possibility of using this technique to observe spectra with a resolution much higher than that of the excitation laser. Fi- ϫ G ͑tϪtЈ͒E ͑tЈ͒dtЈ. ͑A3͒ nally, we have derived general analytic expressions for cal- ͵ mn it

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We will now consider two special cases which we believe plex and appear much more difficult to use than the simple will be helpful in selecting experimental parameters. recursion relationships given below. Further, neither of these First, we would like to consider the case of input radia- previous works dealt with the case of a cavity or input beam tion with the same spot size and curvature as those of the with astigmatism, which is easily treated by the present ap- cavity TEM00 mode, but having its propagation axis mis- proach. aligned by an angle of ␪x with respect to the cavity axis, and The z axis of our coordinate system is defined by the displaced by a distance of x0 in the focal plane of the cavity. optical axis of the cavity. For the case of a symmetric cavity, We assume the angular and displacement misalignments are the origin of the coordinate system is located at its center. in the xz plane. Using the analogy to a harmonic oscillator, We will pick an arbitrary xy plane ͑a fixed value of z)to this is equivalent to initial preparation of a Glauber coherent compute the overlap integrals, where the cavity TEM modes state31,32 with dimensionless phaser, ␣, given by are characterized by the spot size w(z), radius of curvature Rf(z), and phase ␩(z) given in Eqs. ͑22͒ together with Eqs. 1 x0 ␣ϭ Ϫikw sin ␪ . ͑A4͒ ͑23͒ and ͑25͒. These TEMmn modes can be separated as ͩ w 0 x ͪ ͱ2 0 products of two components, Em(x) and En(y) ͓see Eq. ͑21͔͒, with This state will be a superposition of a Poisson distribution of 2 TEM modes, with mean value ␣ and a standard devia- 4 m0 ͉ ͉ ͱ2/␲ ͱ2x tion equal to the square root of the mean, or ͉␣͉. Thus to ͑x͒ϭ H Em m m strongly excite a range of ϳK modes, we either need to ͱ2 m!w ͩ w ͪ displace the input beam ϳK spot sizes off the optic axis, or x2 ikx2 i 1 misalign the input angle by ϳK/kw . For excitation of a 0 ϫexp Ϫ 2 Ϫ Ϫ kzϩi mϩ ␩ . ͑B1͒ large number of modes, this result will be insensitive to ͩ w 2Rf 2 ͩ 2ͪ ͪ small changes in the spot size or radius of curvature away For the input beam, we will assume a generic free space from the mode matched values. elliptic Gaussian beam.12 In the paraxial approximation, the As a second special case, we consider excitation with a axis of this beam will have tilt angles ␪ and ␪ in the x and Gaussian beam aligned with the optic axis of the cavity, but x y y directions. We will take the origin of the beam coordinate with input values for the spot size, w , and radius of curva- x system at the point (x ,y ,0) of the cavity coordinate sys- ture, R , that may not match those, w and R , of the cavity 00 00 x f tem. By writing the overall tilt as a product of a rotation ␪ modes. In this case, we get for the overlap with the TEM x 00 around the y axis and then by ␪ around the rotated x axis, mode of the cavity the amplitude, E ͓see Eq. ͑B6͔͒, y i,00 and using the paraxial approximation of small angles, one 2wwx can write the transformation from the beam coordinates E ϭ . ͑A5͒ i,00 i R ϪR xЈ, yЈ, zЈ to the cavity coordinates like this 2 2 2 2 f x w ϩwxϩ kw wx 2 RxRf xЈӍx cos ␪xϪz sin ␪xϪx00ӍxϪz sin ␪xϪx00 , The only higher order modes that will have nonzero overlap are those with even values for both m,n. The magnitude of yЈӍy cos ␪yϪz sin ␪yϪy 00ӍyϪz sin ␪yϪy 00 , ͑B2͒ these overlaps will decrease approximately exponentially with mϩn ͓see Eq. ͑B14͔͒, with the width of the distribution zЈӍx sin ␪xϩy sin ␪yϩz cos ␪x cos ␪y . being ϳ2/Ei,00 . Applying this transformation to the elliptic Gaussian beam function E(xЈ,yЈ,zЈ) in Eq. ͑6.12–8͒ of Yariv,12 we can then APPENDIX B: GENERAL MODE OVERLAP separate it as a product of two terms, Ex(x) and Ey(y), AMPLITUDES which we normalize for integration on the xy plane. The x We wish to give general expressions for the TEM expan- component is sion amplitudes ˜E (␻) of an arbitrary input Gaussian i,mn 4ͱ2/␲ xϪx ͒2 ik xϪx ͒2 beam that is coupled into an optical cavity. This problem is ͑ 0 ͑ 0 Ex͑x͒ϭ exp Ϫ 2 Ϫ w ͩ w 2Rx mathematically closely related to the calculation of elec- ͱ x x tronic Franck–Condon factors in the harmonic approxima- i i tion. Below we give a derivation that exploits a method one Ϫikx sin ␪xϪ kz cos ␪x cos ␪yϩ ␩x , of the authors developed for that problem.33 Recently, we 2 2 ͪ became aware of two previous publications that reported ͑B3͒ mode ‘‘coupling coefficients.’’ The first was an article by 29 Kogelnik which provided general close form expressions where the beam parameters wx , Rx , and ␩x are functions of for the case of a perfectly aligned cavity, but only presented zЈ as for the TEM modes parameters of Eqs. ͑22͒. However, results for the coupling to the lowest order mode in the case we have to approximate zЈӍz in the argument of these pa- of a misaligned input beam. Bayer–Helms30 presented re- rameters, which is necessary for the evaluation of the overlap sults with arbitrary misalignment as well as input mode mis- integrals below. Taking into account also the position zwx of match. The resulting expressions, however, are rather com- the beam waist in the xz plane, we have therefore

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zϪz 2 wx ˜ 1 w wx͑z͒ϭwx0 1ϩ , E ϭ ␰H ͑␰͒k ͑␰͒d␰ ͱ ͩ z ͪ x,mϩ1 mϪ1 ͱw ͵ m mϩ1 x0 ͱ2 ͑mϩ1͒!␲ x

2 zx0 m R ͑z͒ϭ͑zϪz ͒ 1ϩ , ͑B4͒ Ϫ eϪ2i␩˜E , ͑B7͒ x wx ͫ ͩzϪz ͪ ͬ ͱmϩ1 x,mϪ1 wx where we have introduced for convenience zϪzwx 2 ␩ ͑z͒ϭarctan , x ͑z͒ϭx ϩz sin ␪ , km(␰)ϭexp(Ϫa␰ Ϫb␰Ϫcm), with the simple property x ͩ z ͪ 0 00 x x0 kmϩ1ϭexp(Ϫi␩)km . Using also HmЈ ϭ2mHmϪ1 and integrat- ing by parts, one can show that where x0(z) is the beam displacement in the x direction. The confocal parameter is related to the beam waist w as usual, m x0 ␰H k ͑␰͒d␰ϭ H k ͑␰͒d␰ z ϭ w2 n( )/ . Analogous equations can be written for ͵ m m ͵ mϪ1 m x0 ␲ x0 ␻ ␭ a the Ey(y) function, with different parameters wy0 , zwy which allow for astigmatism of the input beam. Finally, no- b Ϫ Hmkm͑␰͒d␰. ͑B8͒ tice that the wave number k in this equation is the same as 2a ͵ that of the TEM modes we use to evaluate the expansion Substituting this into the previous equation, we find the re- amplitudes, since we are working in the frequency domain, cursion relationship at fixed ␻. Ϫi␩ Since we have chosen the mode functions to be normal- ˜ b e ˜ Ex,mϩ1ϭϪ Ex,m ized with respect to integration over the xy plane, we can a ͱ2͑mϩ1͒ calculate the overlap amplitudes Ei,mn , which give the ex- pansion of the input wave in terms of TEMmn cavity modes, 1 m ϩ Ϫ1 eϪ2i␩˜E . ͑B9͒ by ͩa ͪͱmϩ1 x,mϪ1 ˜ ˜ ˜E ͑␻͒ϭ˜E ˜E , Since this gives the correct Ex,1 if we use Ex,Ϫ1ϭ0, one can i,mn x,m y,n ˜ ˜ calculate all Ex,m starting from the Ex,0 given before. If one correctly propagates along z the parameters both of the input ˜ * Ex,mϭ͵ Em͑x͒Ex͑x͒dx beam and of the cavity modes, one can verify that the Ei,mn(␻) coefficients are invariant in value, as expected. 1 w Using these relationships, it is possible to calculate the ϭ H ͑␰͒exp͑Ϫa␰2Ϫb␰Ϫc ͒d␰, m ͱ m m full set of ˜E overlap coefficients for the coupling of an ͱ2 m!␲ wx ͵ mn arbitrary input pulse having a Gaussian transverse profile. By w2 1 1 ik 1 1 a straightforward extension of the present method, the case aϭ ϩ ϩ Ϫ , ͑B5͒ of an arbitrary input with a Hermite–Gaussian profile ͑and 2 w2 2 2 R R ͫ wx ͩ x fͪͬ therefore any linear combination of such profiles͒ can be handled as well. w 2x0 ikx0 bϭ Ϫ 2 Ϫ ϩik sin ␪x , 2 ͫ w Rx ͬ 1 ͱ x A. O’Keefe and D. A. G. Deacon, Rev. Sci. Instrum. 59, 2544 ͑1988͒. 2 A. O’Keefe, J. J. Scherer, A. L. Cooksy, R. Sheeks, J. Heath, and R. J. 2 2 Saykally, Chem. Phys. Lett. 172, 214 ͑1990͒. x0 ikx0 i c ϭ ϩ Ϫ kz 1Ϫcos cos 3 D. Romanini and K. K. Lehmann, J. Chem. Phys. 99, 6287 ͑1993͒. m 2 ͑ ␪x ␪y͒ 4 wx 2Rx 2 T. Yu and M. C. Lin, J. Am. Chem. Soc. 115, 4371 ͑1993͒. 5 G. Meijer, M. G. H. Boogaarts, R. T. Jongma, D. H. Parker, and A. M. 1 i Wodtke, Chem. Phys. Lett. 217, 112 ͑1994͒. 6 ϩi mϩ ␩Ϫ ␩x , D. Romanini and K. K. Lehmann, J. Chem. Phys. 102, 633 ͑1995͒. ͩ 2ͪ 2 7 J. J. Scherer, D. Voelkel, D. J. Rakestraw, J. B. P. C. P. Collier, R. J. Saykally, and A. O’Keefe, Chem. Phys. Lett. 245, 273 ͑1995͒. ˜ 8 with an analogous equation for the Ey,m terms as integrals P. Zalicki, Y. Ma, R. N. Zare, E. H. Wahl, J. Dadamio, T. G. Owano, and over dy. For the case of ˜E , we are left with an integral C. H. Kruger, Chem. Phys. Lett. 234, 269 ͑1995͒. x,0 9 P. Zalicki and R. N. Zare, J. Chem. Phys. 102, 2708 ͑1995͒. over the exponential term, which is easily solved by comple- 10 J. D. Jackson, Classical Electrodynamics, 2nd ed. ͑Wiley, New York tion of squares to put it in standard form. The result is 1975͒. 11 In this paper we will follow the common convention of using complex w b2 values for the time dependent electric fields, with the implicit understand- ˜ ing that one should add the complex conjugate of the expressions to obtain Ex,0ϭ exp Ϫc0 . ͑B6͒ ͱaw ͩ 4a ͪ the physical field. In the frequency domain, this convention implies that x the value of ˜E(␻) should be replaced by ˜E(␻)ϩ˜E(Ϫ␻). Recursion relationships for the higher overlap terms are 12 A. Yariv, Quantum Electronics, 3rd ed. ͑Wiley, New York, 1989͒. 13 obtained exploiting the properties of the Hermite polynomi- M. Born and E. Wolf, Principles of Optics, 6th ed. ͑Pergamon, Oxford, 1980͒. als ͑Ref. 32, p. 531͒. Using the relationship 14 D. A. McQuarrie, Statistical Mechanics ͑Harper & Row, New York, Hmϩ1(␰)ϭ2␰Hm(␰)Ϫ2mHmϪ1(␰), it is easy to show 1976͒.

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15 ϪiwtЉ T. J. Balle and W. H. Flygare, Rev. Sci. Instrum. 52,33͑1981͒. ͐Gmn(tЉϪtЈ)Ei,mn(tЈ)dtЈe dtЉ. Then, treating explicitly the term 16 K. K. Lehmann, J. Chem. Phys. ͑submitted͒. exp(Ϫikz) present in the TEM functions, we can integrate over ␻ produc- 17 R. T. Jongma, M. G. H. Boogaarts, I. Holleman, and G. Meijer, Rev. Sci. ing a ␦(tЉϪtϩnz/c), which is eliminated by further integration over tЉ. Instrum. 66, 2821 ͑1995͒. 26 18 J. T. Hodges, J. P. Looney, and R. D. v. Zee, Appl. Opt. 35, 4112 ͑1996͒. D. Herriott, H. Kogelnik, and R. Kompfner, Appl. Opt. 3, 523 ͑1964͒. 27 19 M. D. Crisp, Phys. Rev. A 1, 1604 ͑1970͒. J. T. Hodges, J. P. Looney, and R. D. van Zee, J. Chem. Phys. 105, 10278 20 I. Daubechies, Ten Lectures on Wavelets ͑Capital City, Montpelier, 1992͒. ͑1996͒, following paper. 21 As discussed in Ref. 11, we need to use both Eq. ͑1͒ and its complex 28 J. E. Gambogi, M. Becucci, C. J. O’Brien, K. K. Lehmann, and G. Scoles, conjugate. Since only the term oscillating as exp(Ϫi␻t) will cause signifi- Ber. Bunsenges. Phys. Chem. 99, 548 ͑1995͒. cant excitation ͑the rotating wave approximation͒, for positive frequencies 29 H. Kogelnik, in Proceedings of the Symposium on Quasi–Optics, edited ͑by the present phase convention͒, we will retain only the complex con- by J. Fox ͑Polytechnic, Brooklyn, 1964͒. jugate term. 30 F. Bayer-Helms, Appl. Opt. 23 1369 ͑1984͒. 22 R. G. Hulet, E. S. Hilfer, and D. Klemperer, Phys. Rev. Lett. 55, 2137 31 R. J Glauber in Quantum Optics and Electronics, Les Houches Lectures ͑1985͒. 1964, edited by C. De Witt, A. Blandin, and C. Cohen-Tannoudji ͑Gordon 23 R. Teets, J. Eckstein, and T.W. Ha¨nsch, Phys. Rev. Lett. 38, 760 ͑1977͒. 24 A. E. Siegman, Lasers University Science Books, Mill Valley, California, and Breach, New York, 1965͒. ͑ 32 1986͒. C. Cohen-Tannoudji, B. Biu, and F. LaLoe¨, Quantum Mechanics ͑Wiley, 25 We use the Faltung theorem and write the product of FT’s New York, 1975͒. ˜ ˜ 33 K. K. Lehmann ͑unpublished͒. Gmn(␻)ϫEi,mn(␻) as the FT of a convolution (1/2␲)

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