The superposition principle and cavity ring-down spectroscopy

Transcription

1 The superposition priniple and avity ring-down spetrosopy Kevin K. Lehmann Department of Chemistry, Prineton University, Prineton, New Jersey Daniele Romanini Laboratoire de Spetrométrie Physique-CNRS URA 08, Université J. Fourier/Grenoble, B.P Saint Martin d Hères Cedex, Frane Reeived 14 May 1996; aepted 12 September 1996 Cavity ring-down is beoming a widely used tehnique in gas phase spetrosopy. It holds promise for further important extensions, whih will lead to even more frequent use. However, we have found widespread onfusion in the literature about the nature of oherene effets, espeially when the optial avity onstituting the ring-down ell is exited with a short oherene length laser soure. In this paper we use the superposition priniple of optis to present a general and natural framework for desribing the exitation of a ring-down avity regardless of the relative values of the avity ring-down time, the input pulse oherene time, or the dephasing time of absorption speies inside the avity. This analysis demonstrates that even in the impulsive limit the radiation inside a high finesse avity an have frequeny omponents only at the natural resonane frequenies of the avity modes. As an immediate onsequene, a sample absorption line an be deteted only if it overlaps at least one of the avity resonanes. Sine this point is of partiular importane for high resolution appliations of the tehnique, we have derived the same onlusion also in the time domain representation. Finally, we have predited that it is possible to use this effet to do spetrosopy with a resolution muh higher than that of the bandwidth of the exitation laser. In order to aid in the design of suh experiments, expressions are derived for the temporal and spatial overlap of a Fourier transform limited input Gaussian beam with the TEM mn modes of the avity. The expressions we derive for the spatial mode overlap oeffiients are of general interest in appliations where aurate mode mathing to an optial avity is required Amerian Institute of Physis. S I. INTRODUCTION In the last few years, avity ring-down spetrosopy CRDS has been applied with inreasing frequeny to a number of problems, allowing highly sensitive absolute absorption measurements of weak transitions or rarefied speies. 1 8 Very briefly, CRDS uses pulsed laser exitation of a stable optial avity formed by two or more highly refletive mirrors. One observes absorption by moleules ontained between the mirrors at the laser wavelength by the derease it auses in the deay time of photons trapped in the avity. Absorption equivalent noise as low as /mhz has been demonstrated, and in priniple, several orders-of-magnitude-further-improvement is possible. 3,6 In several of the reent publiations on this subjet, we have deteted a diffuse belief that the behavior of an optial avity under impulsive exitation as in CRDS is to be onsidered radially different from that in presene of ontinuous wave w radiation. This impression has been onfirmed by diret disussion with different authors. The purpose of this paper is to larify the situation by presenting a rigorous analysis of the exitation of an optial avity that is valid over the full range of parameters that are likely to be of importane in CRDS. Further, the framework introdued here may be useful in the design of new experiments. The problem an be summed up as follows. Time domain and frequeny domain representations of the dynamis of physial systems having linear response are mathematially equivalent, sine they are uniquely mapped into eah other by a Fourier transformation. Given this equivalene, one is free to hoose the most onvenient desription for the problem at hand, as the final results will not depend on this hoie. However, the frequeny representation has the general advantage that the spetrum of the output of a passive devie an be simply written as a system response funtion times the spetrum of the input. In ontrast, the general time domain system response must be expressed as a onvolution integral of the input with a time dependent system response funtion. For an optial avity, input and output are the eletromagneti fields arriving from the optial soure and going towards the detetor, respetively. In CRDS, one may have to deal with input laser pulses that are short ompared to the light round trip time inside the avity. In this limit, it is lear that destrutive interferene phenomena among the different time omponents of the injeted light should be negligible. More simply, after a pulse is partially transmitted through the avity input mirror, it does not spatially overlap or interfere with itself as it rings down inside the avity. Therefore, it might appear more onvenient to use the time domain representation to treat CRDS. It turns out that the frequeny domain analysis of the light fields inside and transmitted by a ring-down avity is quite simple and of general sope. No restritive assumptions on the pulse duration and bandwidth, the absorbing transition linewidth, or the avity quality fator are required. For the ase of nar- J. Chem. Phys. 105 (23), 15 Deember /96/105(23)/10263/15/$ Amerian Institute of Physis 10263

2 10264 K. K. Lehmann and D. Romanini: Cavity ring-down spetrosopy row absorption lines whih have long dephasing times, the frequeny domain analysis is substantially easier than treating time dependent response funtions and sample reradiation as would be required for a proper time domain treatment. One of the onlusions of this analysis is that no absorption is possible even in the short pulse limit when the frequeny of a narrow absorption feature falls between two avity modes. For omplementarity we also give a less general time domain treatment of this important problem, whih obviously reahes the same onlusions. This result was previously obtained by Zaliki and Zare, 9 but then reently ontested by Sherer et al. 7 The only simple but powerful priniple to be applied in the frequeny domain analysis is that of linear superposition of the effets produed by the different frequeny omponents in the inoming field. 10 We will show that the standard spetral response funtion of an etalon an be applied regardless of the temporal profile of the input field. While we expet that many readers will find this result to be so obvious as to not require an expliit derivation, the CRDS literature is rife with statements that ontradit this view. Sine the omblike struture of this response funtion reflets the presene of avity resonanes, the avity mode struture an be onsidered as a fixed harateristi of the system, and not as something that builds up only if the exitation oherene time is suffiiently long ompared to the avity optial round trip time, as stated in several reent publiations. One must be areful not to onfuse interferene in the time and in the frequeny domain. Even if the pulse injeted into the avity is suh that its time omponents do not overlap and interfere, the multiple refletions from the avity mirrors still result in destrutive interferene for those frequeny omponents of the pulse whih do not overlap any avity resonanes and onstrutive interferene of those that do. Corret preditions of the avity behavior in different situations follows naturally if one thinks in terms of the avity resonanes in frequeny spae and the assoiated spatial mode struture longitudinal and transversal in physial spae. Contrary to what has been suggested in the CRDS literature, one of the most interesting onsequenes of our onsiderations is that it is possible to turn the high finesse mode struture of the ring-down avity to a tremendous advantage. We will show that one an use CRDS with a spetral resolution muh higher than that of the pulsed laser soure. In order to aid in the design of suh experiments, expressions are derived for the temporal and spatial overlap of a Fourier transform limited input Gaussian beam with the TEM mn modes of the avity. A note is warranted about the appliation of linear response theory to CRDS. As it has already been noted, 8 even in presene of intense and short laser pulses, nonlinear effets suh as the saturation of moleular transitions are usually negligible in CRDS. This is prinipally due to the strong attenuation of the input laser pulse upon transmission of the avity input mirror. We would like, however, to warn that there may exist speial onditions in whih nonlinear effets may beome relevant, speifially for very strong transitions and a ring-down avity employing mirrors of substantial transmittivity as in the uv. In suh ases, one must go beyond linear response theory and the superposition priniple should be used with aution. II. THE SPECTRUM OF LIGHT INSIDE AND TRANSMITTED BY A MODE MATCHED RING DOWN CAVITY Consider a ring-down avity RDC formed by two mirrors with radius of urvature R, separated by a distane L whih must be 2R to have a stable avity 12. We define t r 2L/, the round trip time of the ell, where is the speed of light in the medium between the mirrors. For simpliity, we will assume that the mirrors have idential eletri field refletivity R and transmittivity T. The results derived below are easily generalized to the ase of assymetri avities and the results are qualitatively the same. If the mirrors were infinitely thin, ontinuity of the eletri field would give the relationship RT 1 between these omplex quantities. 11 The refletions and transmitions of the multiple surfaes inside dieletri mirrors an be ombined with propagation phase and absorption into a single effetive frequeny dependent R and T an appliation of the superposition priniple, as long as one is outside of the region of the oating. However, this treatment will not yield a ontinuous eletri field, sine this solution is not appropriate inside the mirror oatings where one is beyond some surfaes and before others. The more familiar intensity refletivity and transmittivity are given by RR 2 and TT 2. For now, we will assume that we an treat R and T as onstants over the bandwidth of input radiation to the RDC. Let us onsider exitation of the RDC by light of arbitrary eletri field E i (t), as measured at the input mirror of the avity. We will also initially assume that the radiation is mode mathed to the TEM 00 mode of the avity. In these onditions the transverse beam profile is stationary and we an treat the problem as one dimensional along the avity optial axis z. Below, we will onsider the effets of exitation of higher order transverse modes. We an alulate the eletri field of light leaving the avity by adding up all paths that lead to output, whih make 1, 3, 5, et., passes through the ell. The light making one pass has an amplitude of T 2 E i (tt r /2) at time t. Eah additional round trip through the ell hanges the amplitude by a fator of R 2 and leads to an additional retardation of t r. Summing the ontribution of the possibly infinite number of passes leads to the intraavity eletri field, E(z,t) at position z and the output eletri field, E o (t) measured outside the output mirror at zl), Ez,t n0 E o t n0 TR 2n E it z2nl TR 2n1 E it 2n1Lz, 1 T 2 R 2n E i tn 1 2t r. 2

3 K. K. Lehmann and D. Romanini: Cavity ring-down spetrosopy We an alulate the spetrum in the angular frequeny ) of this field by omputing the Fourier transform FT. For light transmitted by the avity we have Ẽ o 1 2 E o texpitdt 1 T 2 R 2n E i tn 2t 1 r 2 n0 expitdt n0 T 2 R 2n expin 1 2t r Ẽ i T exp 2 i 2 r t Ẽ i R 2 expit r n, n0 3 T exp 2 i 2 r t Ẽ o 1R 2 expit r Ẽ i. Above, Ẽ i () is the FT of the input radiation. The output spetral density, I o (), of this light is proportional to Ẽ o () 2, whih gives T 2 I o 1R 2 4R sin t r I i. 4 In this expression, arg(r), i.e., the phase shift per refletion of the mirrors. Going through the same analysis for light inside the avity, starting from E(z,t) we find I,zT 1R2 4R sin 2 klz I 1R 2 4R sin 2 1 i, 2 t r 5 where k/ is the wave vetor of the light. This shows that for a fixed or monohromati input field, light traveling in both diretions inside the avity leads to standing waves with near nodes in the limit that R1. The time averaged total intensity at z is given by I(,z)d. For exitation of the avity with a pulse short ompared to t r, whose bandwidth will be larger than 1/t r, this integral will wash out the standing waves exept lose to the mirrors. The above equations are essentially idential to the expression for the transmission of an etalon found in standard optis texts, suh as that of Born and Wolf Ref. 13, page 327. The output spetrum is equal to the input spetrum times a transmission funtion. For a R1, this intensity transmission funtion versus frequeny onsists of a series of narrow peaks with full width at half maximum FWHM given by FW (1R)/(Rt r ), separated by the free spetral range, FSR1/t r, of the etalon. The finesse of the etalon is defined by the ratio of the FSR to the FWHM and is equal to (R)/(1R). While Eq. 4 is usually derived by onsidering an infinite wave of pure frequeny, it should be notied that we have made no assumption as to the shape of E i (t), and thus the input an have any pulse length or oherene time either short or long ompared to t r ). This is, of ourse, a straightforward onsequene of the superposition priniple of linear optis, whih states that the optial throughput at eah frequeny an be alulated separately, and is independent of any other spetral omponents of the wave. 13 In their reent paper, Sherer et al. 7 make the laim it seems intuitively more reasonable that the onset of oherent effets should expliitly depend upon the oherene length of the light, sine if the phase of the overlapping light is not preserved, a regular interferene will not neessarily our. These results undersore the omplexity assoiated with prediting the behavior of optial resonators whih are injeted with pulsed laser light, and the subsequent limitation of applying w based model to the pulsed regime. The derivation presented above demonstrates that this laim is inonsistent with the superposition priniple of optis. The oherene length of the input pulse appears in our analysis as part of Ẽ i () and only affets the relative exitation of different resonane modes of the avity. The following onsiderations might help to further larify pulsed exitation of an etalon. In the ase of an impulsive E i (t) shorter than t r, it is lear that there is no destrutive interferene of the temporal omponents of the pulse. This is the ase if dispersion effets are not suffiiently strong to make the pulse duration beome longer than t r before the pulse is ompletely deayed. However, absorbing moleules inside the avity do not see a single pulse, but also its reurrent refletions, with a well defined and onstant period t r. Here, we neglet flutuations of the avity length, whih is a good approximation if we onsider that a typial optial dephasing time, T 2, is muh shorter than mehanial vibrations. It is this strit reurrene that produes oherene effets if the moleular transition T 2 is longer than t r. In the frequeny domain, we have shown that, due to the reurrenes, the spetrum of the injeted light is modulated and this modulation is mathematially represented by multipliation of the initial spetrum by the omblike transfer funtion of the avity. If we were to selet a single one of the pulses in the avity ring-down, say by using a fast eletro-opti swith, it would have the same spetrum as the input radiation under the assumptions made above. Of ourse, suh a swith must hange its transmission on a time sale less than t r, and thus will introdue a spetral broadening greater than the spaing between avity modes of the RDC. The etalon omblike filter onvoluted with the spetral broadening produed by the swith will just reprodue a highly attenuated version of the input pulse spetrum. Notie that sine the optial swith is a time dependent system, it annot be represented in frequeny spae by simple multipliation by a spetral response funtion, but by a onvolution operation. The fat that a series of equally spaed deaying replias of the same pulse has a spetrum whih is different from that of the single initial generating pulse, is a onsequene of the properties of the FT. The fat that this frequeny spetrum ontains resonanes whih are losely the harmonis of the

4 10266 K. K. Lehmann and D. Romanini: Cavity ring-down spetrosopy fundamental 1/t r is borne out by the rigorous appliation of the superposition priniple given above, but may be also understood in simpler though less general terms as follows. If t r t d and we neglet the phase shift per refletion, the repeating pulse is well approximated as a produt of an exponential deaying envelope exp(t/2t d ) times a periodi funtion f (t) with period t r. It is a well known property of periodi funtions that they an be deomposed in a Fourier series as f (t) m f m exp(2imt/t r ). Therefore, we find again that only harmonis of the base frequeny 1/t r must be present. The effet of the deay is to add a finite Lorentzian width to these resonanes. This is seen in mathematial terms by taking the FT of the produt f (t) exp(t/2t d ) and reognizing that this equals the onvolution of the individual FT s of eah operand Faltung theorem. The FT of an exponential deay is a Lorentzian, while the FT of the sum of irular funtions exp(2imt/t r ) is a sum of delta funtions (2m/t r ). The onvolution then gives a sum of Lorentzians whose enter frequenies are the harmonis of 1/t r, whih is a good approximation to the omblike avity transmission funtion in Eq. 4 in the above limit of t r t d. If one looks at the field in the diretion of the refletion from the avity, one will find both the diret refletion of the input pulse plus a pulse train lasting a time t d that omes from intraavity radiation re-transmitted by the input mirror. In omputing the spetrum of this pulse, there will be destrutive interferene for those spetral omponents of the input pulse whih are resonant with those of the retransmitted pulse train, leading to holes in the spetrum of the refleted light. This is of ourse a natural onsequene of the need to separately onserve energy for eah spetral omponent. Let us now relax the assumption that R and T are frequeny independent, and allow for the presene of an absorbing medium inside the ell with an absorption oeffiient given by () and index of refration given by n(). In priniple, we ould repeat the above alulation diretly in the time domain, but then we would need to onsider the time dependent response funtions of the mirrors and the re-radiation by the moleules Ref. 14, p However, as long as we remain in the linear response limit, we an exploit the superposition priniple to look at the refletion and absorption of eah spetral omponent separately, whih are just omplex multipliative fators. Adding up the multiple paths for eah spetral omponent, we find the following for the spetrum of light transmitted by the avity R eff Re L, k n, Ẽ o T 2 e L/2 e ikl 1R 2 e L e ik2l Ẽ i 2G Ẽ i, 6 T 2 e L I o 1R eff 2 4R eff sin 2 kl I i, 7 where we have introdued the spetral Green funtion G () FT of the usual time dependent Green funtion for the frequeny response of the avity. For the spetrum of light inside the avity we have instead I,z T 1RezL 2 e z 4Re L sin 2 klz 1R eff 2 4R eff sin 2 kl I i. In order to determine the time dependent intensity of light transmitted by the avity, whih is the quantity measured in CRDS, we must ompute the inverse FT of Ẽ o (). Using the onvolution or Faltung theorem, this an be expressed in the following standard form: E o t GttE i tdt, where G(tt) is the Green s funtion, whih represents the avity response to a delta funtion input. For the ring-down avity, this an be written from the expression above as follows: Gt 2 1 T 2 e L/2 e ikl 1R 2 e L e ik2l e it d In this integral, one should keep in mind that, k, R, and T are all funtions of the integration variable. We wish to point out that up to this point in the analysis we have made no assumptions beyond linearity of optial response, whih is implied by the use of n() and (), and the superposition priniple of optis. As long as the width of eah absorption line is muh greater than the width of the individual resonanes i.e., the T 2 for the optial transition is short ompared to t d ), eah resonane of the avity will still have a Lorentzian shape. For realisti parameters at optial frequenies, the transit time broadening of moleules through the narrow TEM 00 avity mode will greatly exeed the width of the resonanes but it will be muh narrower than their separation. The Lorentzian resonane shape implies that the deay of intensity emitted from the avity, following impulsive exitation of a single mode, will be exponential. We point out that this limit is violated in Balle Flygare-type FT mirowave spetrosopy, whih is oneptually similar to avity ring-down, but where typially the T 2 due to transit time is long ompared to the avity deay time. 15 In this limit of avity resonanes muh narrower than the interval over whih the other frequeny dependent terms in the expression for the response funtion hange, we an integrate over eah resonane separately. By expanding the exponential exp(ik2l) to first order around eah q where k( q )Lq, we get a sum over Lorentzian amplitude terms for whih the inverse FT an be omputed. The resulting expression is

5 Gt 1 2 q A q exp tt r/2 2t d q expi q tt r /2tt r /2, 11 where (x) is the step funtion 1 or0forx0or0, respetively and t d q t rr eff 21R eff, 12 A q 2 T 2 e L/2, 13 t r R eff with R eff () and k() as defined above. Note that due to the small but finite propagation delay through the avity, ausality requires that G(t) is zero for tt r /2 rather than just for t0. Equations 9 and 11 demonstrate that the requirement on the input soure to observe a avity ring-down is not that its pulse width be short ompared to t r, or even t d, but only that the falling edge of the input pulse be short ompared to t d ( q ) for the modes q that are signifiantly exited. Those modes will be determined by the spetrum of the input radiation, and when we onsider the effet of higher transverse modes below the spatial properties of the radiation. However, for a typial pulsed laser the total extent in time of the input radiation is itself muh shorter than the avity deay times. In this impulsive limit, we an neglet the exponential avity deay during the input pulse, and if we plae the time origin at the input pulse, we an expliitly evaluate the integral in Eq. 9 using Eq. 11 to give E imp o t A q q exp tt r/2 2t d q expi q tt r /2Ẽ i q, 14 for times greater than the end of the input pulse plus t r /2. The result is a sum of damped exponential deays for eah avity resonane. Sine the eletri field has a deay lifetime of 2t d, the intensity in eah mode has deay lifetime of t d. The deay rate, and thus the spetral width, of those resonanes whih overlap the sample absorption spetrum will be inreased. Sample absorption lines that do not overlap exited avity modes do not ontribute to the rate of light intensity deay and are not detetable in CRDS. If we treat n() and () as hanging slowly with thus negleting dispersion effets due to narrow absorption lines, the avity mode spaing is given by FSR 2L n d dn d. 15 d For onstant dn/d and d/d, we have equally spaed resonane modes. If in addition R eff is onstant, the pulse in the avity travels with no hange in shape sine the effetive loss and group veloity is the same for all frequenies ontained in its bandwidth. Dispersion hanges the veloity of the wave due to both the (dn/d) term, whih is standard K. K. Lehmann and D. Romanini: Cavity ring-down spetrosopy Ref. 10, p. 302, and the d/d term whih reflets that different wavelengths have different average penetration into the optial oating of the mirrors and thus travel different pathlengths. The effetive t r is given by 1/FSR where FSR is now defined by Eq. 15 for a avity with dispersion. If we onsider the ontribution to n() by the sample absorption, we see that this will slightly inrease the index and thus derease the mode spaing on the low frequeny side of a transition, and the opposite on the high frequeny side. Zaliki and Zare 9 suggested that dispersion effets are generally negligible for the weak absorption strengths investigated by CRDS. We agree with this if one is observing only the deay of energy in the avity, as is typially done, sine in this ase the interferene between different exited modes is filtered out. However, if one attempts to model the time dependent shape of the train of peaks leaving the avity, then the small shifts in resonane frequeny aused by dispersion must be onsidered along with the frequeny dependent absorption. We have expliitly shown by numerial alulation, using Eq. 14 for the ase of a Lorentzian absorption line narrow ompared to the input pulse, that the avity output displays the expeted sample free indution deay after the exitation pulse, only if these shifts in resonane frequenies are inluded in the alulation. This is an interesting subjet for future investigation 16 sine hanges in pulse shape may be useful for extrating information about the sample absorption spetrum on frequeny sales between the linewidth of the laser used to exite the RDC and the spaing between longitudinal avity modes. Sherer et al. state in their paper 7 that, In the ase of simple exponential deay, the avity does not at as an etalon, i.e., standing waves are not established in the avity. On the ontrary, exponential deay is a natural onsequene of the RDC being a high finesse etalon, if it is exited on a single avity mode. More typially, we have multiple mode exitation, due both to the bandwidth of the exitation soure and due to lak of exat transverse mode mathing. As disussed by Zaliki and Zare, 9 for multiple mode exitation we expet in general a multiple exponential deay of the RDC unless the exited modes have the same loss R eff. Even in this ase, the intensity of the deay will not be exatly exponential, but show modulations due to beating between different mode frequenies. Beating among longitudinal modes same transverse mode numbers will have periods that are submultiples of t r. This simply orresponds to the exponentially deaying series of pulses generated by the initially injeted light as it rings down inside the avity. Thus a simple exponential deay of the avity implies that only one longitudinal mode has been exited, whih also requires that a standing wave must be produed inside the avity, in diret onflit with the above statement of Sherer et al. In most appliations of CRDS, however, the differene between t r and t d is large enough that this round-trip beating pattern an be easily filtered eletronially without signifiantly distorting the exponential signal envelope whih is the sum of the deay of eah exited avity mode.

6 10268 K. K. Lehmann and D. Romanini: Cavity ring-down spetrosopy III. CAVITY RING DOWN WITH A NARROW BANDWIDTH ABSORBER The paper by Zaliki and Zare 9 onsidered the effet of having an intraavity absorption feature with a linewidth narrower than the bandwidth of the laser used to exite the avity. As long as one is exiting many modes of the avity aross the absorption line, this leads to nonexponential avity deay. This has been verified by Jongma et al. 17 and more quantitatively by Hodges et al. 18 These effets are losely related to the well known line shape distortions observed in onventional absorption spetrosopy when the instrumental resolution is not higher than the width of spetral features. There, the observed sample absorption integrated over the instrument funtion does not follow Beer s law for samples that are optially thik. 19 For optially thin samples, suh measurements give the orret integrated line strength also known as the equivalent width, though not the orret peak value due to instrumental broadening of the line. Hodges et al. 18 have demonstrated that an additional ompliation an arise in avity ring-down due to the spetral struture in onventional multimode pulsed lasers. This results in intensity distortions when the width of an absorption feature is omparable to the width of strutures in the spetrum of the exitation laser. We antiipate that these effets an be removed by averaging ring-down deays observed as the modes of the exitation laser are swept to produe a smooth average exitation line shape. Another important onsideration disussed by Zaliki and Zare is the relationship between the width of the absorption features and the FSR of the avity. They report to show that the bandwidth of light admitted into the RDC depends upon the length of the exitation pulse. We think that this is a onsequene of their onsidering the FT only over a time interval of the input pulse whih they took to have a square wave amplitude and not of the whole damped series of refletions the intraavity moleules interat with. Despite this, they reognize that the exitation aused by the oherent sum of all the avity refletions will only exite transitions that overlap one of the exited modes of the RDC. Thus they reah the same observable onsequene as the present analysis and one may dismiss the differenes in our respetive analysis as largely semanti. In a frequeny domain analysis of the problem, one should integrate in alulating the FT over the whole time of interation. To onsider the spetral ontent of the radiation field only in a given limited time interval is equivalent to employ a mixed time frequeny representation. Time and frequeny loalized representations are the subjet of wavelet transform theory, whih is beoming quite an ative field of researh and appliation in reent years. 20 It is often useful to ompare desriptions of the same phenomena both in the time and frequeny domains, though they must give the same results. The analysis we will present below is also omplimentary in that it fouses on the moleules in the avity, while the earlier treatment foused on the optial properties of the system. Consider exitation of a two level system with optial resonane 0 and dipole matrix element for the transition 21. Let the moleule be at position z 0 at t0 and its veloity along the axis of the RDC be v z. Starting with Eq in the text by Yariv, 12 and then using first order time dependent perturbation theory, we find that the oherene between the two states, 21, reated by the time dependent eletri field is given by 21 te i 0 t i e dt, t 21 Ez 0 v z t,te i 0 t 16 where ( ) e is the equilibrium differene in populations between the lower and upper levels of the optial transition. We will assume that the input pulse is shorter than one round trip time, t r. This is done only for simpliity. In fat, we ould divide any arbitrary inoming field in portions that are shorter than the round trip time, and the following analysis ould be applied to eah of these time omponents and the results added. We will use Eq. 1 for E(z,t) for the field inside the RDC. 21 Putting this into the above equation, we find for the oherene indued by the pulse 21 te i 0 t intt/t r n0 R n e i2n exp i 0 1 v z nt r R n1/2 e i2n1 exp i 0 1 v z n1t r, 17 where is the oherene produed by a single forward or reverse going pulse in the avity i 21T e exp i 0 z 0 E i *texp i 0 1 v z t dt. 18 We will get onstrutive interferene and thus net absorption of the oherene produed on suessive round trips of the avity only if 0 satisfies the following equation with integer N 0 1 v z 2 t r N 2 t r. 19 Thus we will only get net absorption by the sample if either the forward or bakward Doppler shifted Bohr frequeny the left hand side of the above equation satisfies the same equation as a avity resonane see Eq. 4. The avity ats to produe a multiple-pulse time domain Ramsey fringe pattern, as previously noted by Zaliki and Zare. 9 Inluding a T 2 for the optial transition will allow off resonane absorption, but this is just equivalent to onsidering the resulting line broadening in the frequeny domain. Thus we reover the ondition that CRDS is only sensitive to sample absorp-

7 K. K. Lehmann and D. Romanini: Cavity ring-down spetrosopy tion that overlaps one of the very narrow avity resonanes exited by the input radiation, even in the limit of short pulse exitation. The spetral seletivity of sample exitation inside a RDC an be shown to be demanded also by the laws of thermodynamis. Let us suppose, ontrary to what we have demonstrated above from first priniples, that the RDC does not at as an etalon for short pulse or short oherene length exitation, and that the full bandwidth of the pulse enters the avity and exites all optial transitions that overlap its spetrum, as is laimed by Sherer et al. 7 It has been experimentally demonstrated that spontaneous emission is turned off in a avity with no mode resonant with the atomi transition. 22 If blak body radiation ould enter the avity and exite an atom, but the atom were not able to undergo spontaneous emission, an infinite atomi eletroni temperature would be produed by interation with a finite temperature heat bath. This learly would be a violation of the seond law of thermodynamis. The fat that the RDC ats as a frequeny seletive filter might lead to the erroneous onlusion that CRDS is not suitable for quantitative spetrosopy of high resolution spetra. This is simply not the ase. As Meijer et al. 5 have shown, the exitation of many transverse modes of the RDC will lead to a near ontinuous spetrum of avity resonanes, eliminating this potential problem. We will return to this point below. At present, we will show how even with single mode exitation of the RDC ell, one need not miss absorption features. The narrow bandwidth of light admitted into a RDC offers a tremendous opportunity for high resolution spetrosopy. One has to replae the stati length ell onsidered by Meijer et al., Zaliki and Zare, and others, by a ell whose length an be varied by at least /2, one-half wavelength whih will shift eah mode by one FSR of this etalon. Sine only light with a bandwidth muh narrower than the exitation laser enters the ell, it is possible to do spetrosopy with a resolution muh greater than that of the laser soure used to exite the avity! Consider a ommerially available OPO laser with a bandwidth of 125 MHz. By using a avity of length L75 m, the mode spaing will be greater than twie the laser linewidth and if mode mathed, we will primarily exite only one mode. We an then san the avity and observe a spetrum with a full width instrumental resolution of 16 khz for a avity deay time of 10 s. In pratie, time of flight of moleules through the laser beam will limit resolution to about 0.1 MHz, 3 orders of magnitude better than the FT limit for a few nanoseonds laser pulse. Suh experiments will require interferometri ontrol of the length of the RDC, and traking the exitation laser as the avity is sanned. The latter should be of minor diffiulty, sine in this situation, high ontrast interferene fringes will be observed as the avity and laser are detuned, allowing for standard feedbak tehniques to be used. Maintaining the RDC to interferometri auray, say 1 MHz, an, in priniple, be ahieved in several ways. One an purhase a temperature stabilized etalon with suh passive stability, whih ould be modified to at as a ell for CRDS. By using a ontinuous wave laser loked on a moleular transition, one ould also lok the RDC to a spetrosopi standard. Sine we an repetitively san the avity over one FSR and observe the time when the referene laser passes through resonane, we an have ontinuous feedbak ontrol of the avity length. In this ase, one an use the time that the exitation laser is fired relative to the sanning ramp to ontrol the frequeny sampled on any laser shot. Thus rather than being a problem, the frequeny seletive nature of the avity resonanes opens the possibility for a dramati improvement in the resolution available in pulsed laser experiments. The ombination of this tehnique with CRDS should dramatially improve both the sensitivity and resolution that an be realized in sub-doppler spetrosopy. While this work was being reviewed for publiation, we were informed of an earlier experiment that demonstrated these priniples. Teets et al. 23 observed a two-photon transition in atomi sodium inside an optial avity of modest (16) finesse. By sanning the length of the avity, they observed a transition of width 12 MHz, despite a Fourier transform limit for the input pulse of about 170 MHz. The round trip time of the avity used (13 ns was about twie the length of the input pulse. The spetral seletivity observed in this experiment diretly invalidates the model of CRDS presented by Sherer et al. 7 The next setion will onsider the spetrum of longitudinal modes exited by a FT limited pulse having Gaussian transverse profile. We also need to aount for the fat that without great are, one will exite multiple transverse modes of the avity, eah of whih has its own resonane frequeny. The following setion will then onsider the exitation of higher transverse modes. IV. FRINGES VERSUS THE EXCITATION PULSE LENGTH Consider the mode mathed exitation of a RDC by a pulse with enter frequeny. For a suffiiently short input, this exitation will reate a traveling pulse inside the RDC. Negleting dispersion effets, this pulse will propagate bak and forth with no hange in intensity profile, exept for a slow attenuation. If dispersion effets are important over the bandwidth of the pulse, either from spetral hanges in mirror refletivity and phase shifts, or from the presene of a narrow absorption feature that is not optially thin, the intraavity pulse shape will hange with time. In a frequeny domain piture, this orresponds to the variable mode spaings desribed by Eq. 15. The most obvious sign that the ell is ating as an etalon would be the observation of fringes in the transmitted light as the enter frequeny of the exitation beam is sanned. It is the lak of suh a modulation when the input pulse is short ompared to t r that appears to have produed the belief that a RDC does not at as an etalon in the impulsive limit. However, it is well known that one does not observe interferene fringes if one exites an etalon with broad bandwidth light. This is just a onsequene of the transmission spetrum being periodi with period equal to the FSR.

8 10270 K. K. Lehmann and D. Romanini: Cavity ring-down spetrosopy Thus if the input pulse spetrum is muh wider than one FSR, the expeted interferene struture is washed out, and the transmission oeffiient beomes onstant with value T 2 /2(1R), independent of the enter frequeny of the input light pulse. What happens if the input pulse length is loser to the value of the avity round trip time? Zaliki and Zare onsidered this question for the ase of square wave exitation. 9 We presently onsider a FT limited Gaussian shaped input pulse with an intensity FWHM equal to t, sine this provides a better model for many real lasers. The frequeny spetrum of this input pulse will also be Gaussian, with a FWHM equal to 0.44/t Ref. 24 p In the limit of a high finesse RDC, eah resonane will have a nearly Lorentzian shape with peak transmission equal to T exp(l/2)/(1r eff ) 2, and FWHM equal to 1/2t d, where t d n( q )LR eff /(1R eff ) is the deay time for the qth resonane mode and R eff R exp(( q )) is the effetive refletivity for that mode. Cavity resonanes will be narrow relative to the spetral struture of the input pulse as long as the pulse duration is short ompared to t d. Therefore, in order to alulate the total intensity transmitted by the avity, we an treat the resonanes as delta funtions. This approximation will break down, beause of the long tails of the Lorentzian resonanes, when the width of the input pulse is muh narrower than avity mode spaing, i.e., when tt r. For eah resonane mode q at frequeny q, from Eq. 4 we find that the energy transmitted by the avity is J q J 0 ln 2 t t r T 2 21R t exp q ln 2 t r FSR 2, 20 where J 0 is the energy in the input pulse. The total transmitted energy is obtained by summing over all the resonane modes. In the limit that the input pulse is short ompared to t r the sum over modes an be approximated by an integral whih gives just the fator of T 2 /2(1R eff ) expeted for inoherent exitation of an etalon. In Fig. 1, we plot the ratio of the predited transmission divided by this inoherent exitation value for the pulse enter frequeny exatly resonant with a avity mode and for when it is exatly halfway between avity modes. The ratio of the two urves is the expeted fringe modulation as the exitation laser or the RDC length is sanned. Signifiant modulation is observed when t is greater than 0.5t r. This an be rationalized when one remembers both that the Gaussian has a tail and is not a square wave where there would be zero modulation for tt r ) and also that interferene is given by overlap of the eletri field, whih has a FWHM 2t. When one realls that for a Gaussian pulse t0.44, we see that when the FWHM of the pulse is half the avity round trip time, the spetral width has a FWHM only 0.88 as large as the spaing between the longitudinal modes. When the laser pulse is entered on a mode, the ratio of the mode exitation to that of its FIG. 1. Plot of the ratio of the predited transmission divided by its inoherent exitation value for the pulse frequeny entered on a avity mode solid line and for when it is halfway between avity modes dashed line. next neighbors is given by exp(t/ln 2t r ) 2.When t0.5t r, this ratio is already only 0.028, while for tt r, this ratio is Thus very seletive exitation of a transverse mode mathed RDC is possible with FT limited pulses with FWHM of 0.5t r or longer. Below, we will see that the need for mode mathing an be relaxed in ase of a onfoal or more generally re-entrant avity onfiguration. V. EFFECT OF HIGHER ORDER TRANSVERSE MODES Our analysis up to this point has been one dimensional, whih is appropriate for light perfetly mode mathed into the RDC. In order to onsider the exitation of a RDC produed by a beam with an arbitrary spatial distribution, we must again use the superposition priniple, but now in spae rather than time, expanding the input wave in a set of avity modes. Due to the finite transverse extent of an open optial avity, its resonanes, unlike those of the losed eletromagneti avities ommonly treated in textbooks, need not be a omplete, orthogonal set Ref. 24, p However, stable avities with small diffration losses have resonane modes losely approximated by Hermite Gauss funtions up to relatively high transverse mode orders, whih do have this attrative property Ref. 24, p We will derive the general avity output, for an arbitrary input pulse, by expansion in both frequeny for the time dependene and avity modes for the transverse spatial dependene. While the following disussion is for the response of the avity alulated at its output, the same treatment an be applied to the field produed inside the avity and leads to similar results. In the paraxial approximation small propagation angles and deviations from the optial axis, the Maxwell equation for wave propagation in a homogeneous dieletri medium refration index n()] has a set of solutions known as the transverse eletro magneti TEM modes, 12,24 whose order

9 K. K. Lehmann and D. Romanini: Cavity ring-down spetrosopy is indexed by two positive integers (m,n). The spatial field amplitude for these modes is written in terms of Hermite Gauss funtions 1 E mn x,y,z, m 2 mn1 n!m!w 2 z H 2x wz H nwzexp 2y x2 y 2 w 2 z ikx2 y 2 2R f z ikzimn1z. 21 These monohromati waves have wave vetor kn()/ along z, and time dependene exp(it). They are haraterized by a transverse beam waist of size w(z), by a radius of urvature of the phase fronts R f (z), and by a phase shift indued by diffration (z). For free spae propagation along the paraxial axis z, these TEM parameters hange as follows: wzw 0 1 zz w z 0 z R f zzz w 1 0, 2, 2 zz w, 22 zartan zz w z 0 with z w the foal point of the wave, where the beam waist redues to its minimum w 0. One the value of w 0 is given, the onfoal length z 0 is also determined z 0 w 0 2 n. 23 At fixed frequeny, fixed w 0 or equivalently z 0 ) and fixed R f or z), the set of TEM mn modes form a omplete orthogonal and normalized set of funtions E m * x,y,z,e n mn x,y,z,dxdy m m n n, 24 E mn * x,y,z,e mn x,y,z,xxyy. m,n In the axially symmetri ase, eah mode is degenerate with respet to polarization. If we wish to onsider light oupled into an optial avity made of two mirrors, then one should selet the beam parameter so that the radius of the beam just mathes the radius of urvature of eah mirror at its surfae. For the symmetri avity onsidered in here both mirrors with urvature R, as before, this ondition implies that the beam fous is at the enter of the avity and the onfoal length is 12 z0 22R 1 LL. 25 More general expressions an be found in the text by Yariv. 12 The ondition that z 0 is real defines the range of stable symmetri avities L2R. While z 0 for the TEM set of avity modes is frequeny independent, note that this is not the ase for w 0 2 z 0 /n(). We now onsider an input field of angular frequeny that has a spatial distribution that just mathes the TEM m,n mode of the field. We denote the amplitude of this field by Ẽ i,mn (). Similar to the one dimensional ase Eq. 7 and following, we an sum over all round trips through the avity to arrive at an output eletri field, whih will still math the TEM m,n in the perpendiular plane and have an amplitude given by Ẽ o,mn 2G mnẽ i,mn, where G mn 1 2 T 2 e L/2 e ikl 1R 2 mn e L e ikk mn 26 2L, 27 k mn L2mn1artan L 2R L. 28 The different transverse amplitude distribution of eah TEM mode may result in different losses due to spatially varying defets in the mirrors surfaes and to their finite size. We aount for this by using mode dependent refletivities R mn (). Like in the one dimensional ase, G mn() is a omblike transmission funtion, and we an identify the longitudinal avity resonanes with the peaks of this funtion, and their deay rate with their width. The funtional dependene of G mn() is similar to that for the one dimensional ase of Eq. 7, but the position of its resonanes has a shift dependent on the TEM order, due to the (z) term in Eq. 21. The frequenies of the longitudinal resonanes are found by the solutions of the following equation obtained numerially by iteration mnq n mnq Lq2mn1 artan L 29 2R L, where is the phase hange upon refletion, introdued earlier. In the time domain, eah mode will have an exponential deay with a time onstant for power given by t d,mn mnq t rr mn,eff 30 21R mn,eff with R mn,eff R mn exp(( mnq )L). Note that if R mn () is not onstant due to mode dependent loss, we will have a nonexponential deay even without sample absorption. We will now onsider exitation of a avity by an arbitrary input pulse whose eletri field is given by E i (x,y,z,t). Using the superposition priniple, we an expand E i (x,y,z,t) in terms of the set of funtions E mn (x,y,z,) for whih we have just presented the transmission properties. The total output field, E o (x,y,z,t) an then be written as

10 10272 K. K. Lehmann and D. Romanini: Cavity ring-down spetrosopy E o x,y,z,t mn G mnẽ i,mn E mn x,y,z,e it d, 31 where G mn() is given by Eq. 27 and Ẽ i,mn 1 E mn * x,y,z, 2 E i x,y,z,te it dx dy dt. 32 Ẽ i,mn () is the exitation amplitude of the mode with transverse order mn and frequeny. Note that we do not have to speify at whih value of z the integral is evaluated at, due to the superposition priniple for spae propagation. If we assume that the input field has a small frational bandwidth, whih allows one to neglet the frequeny dependene of the TEM parameters, we an derive another expression for the time dependent avity output deomposed over the spatial TEM modes. For eah mode, the avity response is then a onvolution of the orresponding Green funtion with the time dependent amplitude E i,mn (t) for that mode 25 E o x,y,z,t E mn x,y,z,e ikz mn G nz mntt E i,mn tdt, 33 where is the enter of the field bandwidth; G mn (t) is the inverse Fourier transform of G mn() given by Eq. 27; and E i,mn (t) is the inverse Fourier transform of Ẽ i,mn (). We leave to Appendix A the disussion of some speial ases and examples. An important ase is that of a beam whih is separable as the produt of spae and time dependent funtions. A speial separable ase is when the spatial omponent of the beam is Gaussian and astigmati, with arbitrary spot sizes, foal positions and axis, but only slightly tilted with respet to z as required by the paraxial approximation. Then, reursive analyti expression exist and are derived in Appendix B for the amplitude exitation oeffiients in terms of the beam parameters. These results allow one to model the oupling of a laser operating on a single TEM 00 mode to an external optial avity. In addition, the method of the Appendix B an be extended to input beams with higher TEM order, and the expressions derived ould be applied to the more general ase of a beam whih is the superposition of TEM modes haraterized by the same arbitrary parameters. This would be adequate for modeling the avity oupling of a nonseparable beam produed by a laser whih is not operating on a single transverse mode. Let us onsider the ase when for eah mn the longitudinal resonanes in G mn() are narrow with respet to other spetral widths of the problem. In the time domain this implies that the variation of the field is faster than the avity deay times. Then, the same result as in Eq. 11 an be obtained here and Eq. 14 an be generalized to E o x,y,z,t m,n,q A mn mn Ẽ i,mn mnq E mn x,y,z, mnq exp i t t r 2 mnqexp tt r/2 34 2t d,mn, where the fator A mn () is given by Eq. 13 with R eff,mn in plae of R eff and t d,mn is given by Eq. 30. For brevity, the dependene on mnq has been left impliit for several of the parameters. The output intensity of the avity will ontain beating between different exited modes. However, if one detets the total light intensity rossing the output plane, the orthogonality of the mode funtions eliminates beats between modes with different transverse mode numbers. The fration of input pulse energy oupled to a partiular TEM mn mode is given by the squared magnitude of the mode overlap amplitude, E i,mn 2, times a spetral density fator, whih for the ase of an input pulse with Gaussian temporal profile was given by Eq. 20, whih is easily generalized to the present ase. Thus together with the results of Appendix B, we have given expliit expressions for the distribution of exitation of different modes of an optial avity for an input pulse of Gaussian shape in spae and time, with alignment restrited by the paraxial onditions and of duration short ompared to the avity deay time. The present results an be easily extended, using numerial integration, to treat input pulses of arbitrary temporal and spatial shape. As seen by Eq. 34, the output signal in CRDS is only sensitive to the sample loss at the frequenies of exited modes of the avity, and thus the spetrum of the avity is important to the design and interpretation of CRDS. We an see that the longitudinal modes for fixed m,n) are separated by angular frequeny 2/t r, while transverse modes hanging mn, with q fixed are separated by (4/t r ) artanl/(2rl). The full spetrum of the avity say for an inoming plane wave that exites all TEM orders will be ontinuous if the ratio of these spaings is irrational. If it is rational with the transverse spaing to the longitudinal spaing equal to M/N, the spetrum will onsist of a series of resonanes with frequenies exatly spaed by 1/(t r N), i.e., exatly a fator of N less than the separation of the TEM 00 modes. It is easily seen from above that for a onfoal avity (LR ), the transverse mode separation is exatly half of the longitudinal modes, giving a frequeny spaing between resonanes of 1/2t r. However, for other values of the spaing we an get other low order rational ratios. For example, if LR /2 or 3R /2, we will get a frequeny spaing of 1/3t r, i.e., just 3 times as dense as the TEM 00 modes. For general multiple mode avity exitation, the transverse profile of the light intensity hanges shape on eah pass of the ell. For a rational ratio as disussed above with divisor N, the intensity will be exatly periodi exept for an overall damping fator after N round trips. This is the separation ondition for the multipass re-entrant avity onfigurations used and disussed by Herriott et al. 26 Meijer et al. 5 suggested that one ould use a nononfoal