N. van Leeuwen and A. Wilson Vol. 21, No. 10 /October 2004 /J. Opt. Soc. Am. B 1713

Measurement of pressure-broadened, ultraweaktransitions with noise-immune

cavity-enhanced optical heterodynemolecular spectroscopy

Nicola J. van Leeuwen and Andrew C. Wilson

Physics Department, University of Otago, P.O. Box 56, Dunedin, New Zealand

Received February 25, 2004; revised manuscript received June 3, 2004; accepted June 18, 2004

We present a theoretical description of the ultrasensitive cavity-enhanced spectroscopic technique called noise-immune cavity-enhanced optical heterodyne molecular spectroscopy (NICE OHMS) for the case of transitionsdescribed by a Voigt line shape. The two levels of modulation used in NICE OHMS are treated with the stan-dard theory for frequency modulation spectroscopy and a Fourier description of wavelength modulation spec-troscopy. We compare predicted line shapes with experimental results for pressure-broadened transitions inmolecular oxygen and show that our description can be used to determine the spectroscopic parameters. Akey aspect of this research is the application of NICE OHMS to broad absorption features across a range ofwavelengths, and etalon effects are shown to limit the detection sensitivity. © 2004 Optical Society ofAmerica

OCIS codes: 020.0020, 020.3690, 300.0300, 300.6380.

1. INTRODUCTIONNoise-immune cavity-enhanced optical heterodyne mo-lecular spectroscopy (NICE OHMS) is an ultrasensitivetechnique developed by Ye et al.1 in 1996. The initial re-search on this technique produced an absorption sensitiv-ity (minimum detectable absorption) of 1.83 10212 cm21, and subsequent efforts by the same groupimproved the sensitivity to 1 3 10214 cm21 (Refs. 2, 3),both with 1-s averaging. This outstanding sensitivitywas achieved because, with NICE OHMS, we can utilizethe features of two methods that are independentlyhighly sensitive. These are frequency modulation spec-troscopy (FMS)4,5 and cavity-enhanced spectroscopy,6,7

and as a result NICE OHMS has also been termed cavity-enhanced FMS.1

FMS is an optical heterodyne technique that involvesfrequency modulation (FM) of laser light to produce side-bands on the laser frequency. In the vicinity of an ab-sorption line the laser frequencies experience absorptionand dispersion independently causing amplitude modula-tion (AM). The AM is detected on a photodetector andsubsequently demodulated to provide the FMS signal,which can be either an absorption or a dispersion-basedsignal. These two signals occur in quadrature, so thechoice of the demodulation phase allows selection of ei-ther signal type. In short, FMS shifts detection to ahigh-frequency, and thus low-noise, region to provide aminimum detectable fractional absorption of around 1027

for 1-s averaging.5

Cavity-enhanced spectroscopy involves use of a high-finesse optical cavity to effectively increase the samplepath length, and hence the amount of absorption. Themost basic method of cavity-enhanced spectroscopy sim-ply involves the measurement of cavity transmission. An

0740-3224/2004/101713-09$15.00 ©

example of this was reported by Nakagawa et al.8 whoused a Nd:YAG laser and a cavity finesse of ;18,000 toproduce a sensitivity of 1028 cm21.

In NICE OHMS, FMS is performed on a sample con-tained within a high-finesse cavity. The laser frequencyis servo locked to a longitudinal mode of the cavity, andthe modulation frequency is set to exactly match the freespectral range (FSR). The light transmitted by the cav-ity is incident on a photodetector, and the output is de-modulated at the modulation frequency to produce theFM signal. Because the sample is inside the cavity (of fi-nesse F), the path length of the light, and hence the FMsignal, is increased by a factor of 2F/p. Therefore, al-though NICE OHMS has the same minimum detectablefractional absorption as FMS, the shot-noise-limited sen-sitivity of NICE OHMS is improved by 2F/p comparedwith FMS.2 However, as in FMS the signal level willfluctuate with changes in the level of the FM imparted tothe laser beam. Ye et al.2 found that, to achieve close toshot-noise-limited sensitivity, a second (low-frequency)modulation was needed to suppress the resultant noiseand baseline changes. This modulation is applied to thecavity length and thus, through locking, the laser fre-quency. The output FM signal is fed to a lock-in ampli-fier and demodulated at this low frequency to produce thefinal NICE OHMS signal.

As with FMS, the form of the NICE OHMS signal de-pends on the FM frequency (FSR), the linewidth of the ab-sorption feature, and the choice of absorption or disper-sion FM signal. The group that developed NICE OHMSused it to investigate saturated absorption of molecularovertone transitions in C2H2 and C2HD.1–3 These satu-rated absorption features are described by a Lorentzianline shape and have a linewidth much smaller than the

2004 Optical Society of America

1714 J. Opt. Soc. Am. B/Vol. 21, No. 10 /October 2004 N. van Leeuwen and A. Wilson

FM frequency. A signal at line center was obtained bymeans of setting the demodulation phase, for the FM, todetect a dispersion-based signal. To describe the effect ofthe low-frequency modulation, the Wahlquist modulationbroadening formalism was used.9 Similar investigationson CH4 and CH3I were carried out by Ishibashi andSasada.10,11 The motivation to investigate such narrowfeatures is to identify and measure transitions that havethe potential to be used as frequency standards.

Gianfrani et al.12 used NICE OHMS in a different re-gime. They investigated transitions in molecular oxygenat pressures up to an eighth of an atmosphere. In thisregime the transitions are predominately Doppler broad-ened, but include some pressure broadening, so the lineshape is best described by a Voigt profile. The demodu-lation phase for the FM signal was set to produce anabsorption-based FM signal, which is simpler to interpretfor the broadened case. However, the line shapes result-ing from the subsequent low-frequency modulation wherenot investigated. Using knowledge of the transition in-tensity and the signal-to-noise ratio for their data, theycalculated a absorption detection limit of 6.93 10211 cm21 Hz21/2.

In this paper we present a theoretical description ofNICE OHMS, including the usual low-frequency modula-tion, for the case of a Voigt profile. We extend the theoryof Ye et al.2,3 and the experimental studies of Gianfraniet al.12 by modeling the NICE OHMS signal for transi-tions that are both pressure and Doppler broadened,which allows us to extract precise values for both the lineintensity and the linewidth. Our motivation is that bothpressure and Doppler broadening are significant for alarge number of practical spectroscopic applications, suchas the monitoring of atmospheric pollutants, and we ex-plore the potential of the NICE OHMS technique for useunder these conditions.

Although schemes for FMS that use multiple frequen-cies have been investigated and line-shape analyses havebeen performed,13,14 to our knowledge there has been noanalysis appropriate for NICE OHMS. We developed atheoretical description for cavity-enhanced FMS with ad-ditional low-frequency modulation for the particular casein which the phase of the FM demodulation is chosen toprovide an absorption-based FM signal. We built on thestandard theory for FMS using a Fourier description ofwavelength modulation spectroscopy (WMS).15 We de-scribe our experimental NICE OHMS setup and presentresults for ultraweak transitions in molecular oxygen.These have a range of absorption intensities and are usedto test the validity of our model. We also identify effectsthat limit the sensitivity of our NICE OHMS apparatus tohighlight practical issues that arise when broad transi-tions are measured at multiple wavelengths.

2. THEORYFor NICE OHMS we apply two stages of FM to the laserlight, which are demodulated sequentially. The firstmodulation is at a high frequency (the cavity FSR) andproduces a FM signal that is enhanced by the presence ofthe cavity. The second modulation is at a low frequency,like in WMS. In this section we briefly summarize the

relevant theories of FMS and WMS and combine them todescribe the NICE OHMS signal. We then describe theparticular case in which the absorption line shape is rep-resented by a Voigt profile.

A. Cavity-Enhanced Frequency ModulationSpectroscopyFirst we consider the high-frequency modulation and theline shape from the cavity-enhanced FM signal. In stan-dard FMS theory5 the electric field of light after FM isgiven by

E~t ! 5 E0 exp~i2pnt !$J0~b! 1 J1~b!@exp~i2pnmt !

2 exp~2i2pnmt !#%, (1)

where b is the modulation index, n is the carrier or centrallaser frequency, nm is the modulation frequency, and Jn isthe Bessel function of order n. Equation (1) is valid inthe limit where b ! 1.

A gas sample, of length L and absorption coefficient a,is described by the complex transmission functionexp(2dn 2 ifn), where dn 5 anL/2 is the amplitude at-tenuation and fn is the optical phase shift. In this nota-tion, n 5 21, 0, 1 corresponds to the frequencies n2 nm , n, and n 1 nm . Upon transmission through anoptically thin sample, where a ! 1, the AM on the light is

I~nm! 5 2I0J0~b!J1~b!@~d21 2 d1!cos ~2pnmt !

1 ~f1 1 f21 2 2f0!sin ~2pnmt !#, (2)

where I0 is the intensity incident on the sample.The transmitted light is incident on a photodetector,

and the output is demodulated to produce the FMS signal.For the case in which the demodulation phase producesan absorption-based FMS signal, this is given by

SFM 5 I0hFMJ0~b!J1~b!~a21L 2 a1L !, (3)

where hFM is an instrumentation factor that accounts forthe response of the photodetector and gain within theFMS detection system.

For NICE OHMS the sample is inside a cavity so theeffective optical path length is increased to Leff 5 2FL/p,where F is the finesse of the cavity and L is the physicalcavity length.12 Because the spectrum of the frequency-modulated light is not altered by the cavity, only by thesample contained within, we can write the cavity-enhanced FM signal as simply

SCE FM 52F

pI0hFMJ0~b!J1~b!~a21L 2 a1L !. (4)

It should be noted that shifts in the frequency of the cav-ity modes will occur as a result of the dispersion of thesample. However, these shifts are too small to cause sig-nificant distortions to the line shape of broadened transi-tions. For example, a transition that causes (at line cen-ter) a large total absorption of 5% upon transmissionthrough the cavity will result in a maximum frequencyshift because of the dispersion of approximately 550 Hzcompared with a typical broadened linewidth of 1.5 GHz.

N. van Leeuwen and A. Wilson Vol. 21, No. 10 /October 2004 /J. Opt. Soc. Am. B 1715

B. Wavelength ModulationAs mentioned above, for NICE OHMS an additionalmodulation is applied at a low frequency f, and the cavity-enhanced FM signal is demodulated at this frequencywith a lock-in amplifier. We treat this modulation usingthe theory for WMS. Kluczynski et al.15 provide a com-prehensive review of recent research into a Fourier for-malism for treating WMS, which we now use. Those fa-miliar with the research of Kluczynski et al. should notethat we use slightly different terminology for the functionparameters, keeping them in natural rather than scaledunits; the WMS and FMS theories are in Subsection 2.C.

The instantaneous laser frequency n(t) of thewavelength-modulated light can be written as

n~t ! 5 nc 1 na cos~2pft !, (5)

where na is the wavelength modulation amplitude and ncis the laser center frequency. Any absorption-based sig-nal derived from this light will therefore be time depen-dent and can be expressed as a Fourier series:

S~x, nc , t ! 5 (n50

`

Sneven~x, nc!cos~2pnft !

1 (n51

`

Snodd~x, nc!sin~2pnft !, (6)

where x represents time-independent parameters (suchas the line center and linewidth of transitions) andSn

even(x, nc) and Snodd(x, nc) are the even and odd compo-

nents of the nth Fourier coefficient. Because the laserfrequency has a cosine dependence, it is the even Fouriercomponent that will carry the WMS signal.15

The signal S(x, nc , t) is fed to a lock-in amplifier fordemodulation, and the nth-harmonic output is propor-tional to a linear combination of Sn

even(x, nc) andSn

odd(x, nc). Adjustment of the detection phase allows se-lection of the even Fourier component to provide theWMS signal:

SWM 5 hLISneven~x, nc!, (7)

where hLI is the gain of the lock-in amplifier.

C. Noise-Immune Cavity-Enhanced Optical HeterodyneMolecular SpectroscopyFor NICE OHMS the absorption-based signal S(x, nc , t),which is fed to the lock-in amplifier, is the cavity-enhanced FM signal SCE FM from Eq. (4). This is time de-pendent because of the modulation (at f ) of the central la-ser frequency as defined in Eq. (5). If we assume that thelight intensity is insensitive to frequency change, thenonly a21 and a1 are time dependent. For resonant detec-tion of a single absorption line these can be written asa21 5 apx̄(x, n 2 nm , t) and a1 5 apx̄(x, n 1 nm , t),where ap is the peak absorption coefficient of the line andx̄(x, nc , t) is a time-dependent peak-normalized line-shape function. Thus we can write the time-dependentcavity-enhanced FM signal as

SCE FM~x, n, t ! 52F

pI0hFMJ0~b!J1~b!apL@ x̄~x, n

2 nm , t ! 2 x̄~x, n 1 nm , t !#, (8)

where x̄(n 2 nm , t) and x̄(n 1 nm , t) contain all thetime-dependent information.

This signal is passed to the lock-in amplifier, and thefirst-harmonic output provides the NICE OHMS signal,so that

SNICE OHMS~x, n! 52F

pI0hFMhLIJ0~b!J1~b!ap

3 L@ x̄1even~x, n 2 nm!

2 x̄1even~x, n 1 nm!#, (9)

where x̄1even(x, nc) is the first even Fourier component of

the time-dependent line-shape function x̄(x, nc , t).A similar expression for the NICE OHMS signal could

be derived for a dispersion-based FM signal, although cal-culation of the Fourier components for the time-dependent dispersion line shapes is more difficult. ForNICE OHMS setups with long cavities, which have theadvantages of a long path length and low-cost electro-optic modulators (EOMs), the broadened transition line-widths will be similar to, or larger than, fFSR . In Eq. (2),which describes a FMS signal before demodulation, wesee that a dispersion-based FM signal contains three com-ponents, centered on the carrier and sideband frequen-cies, compared with the two components of theabsorption-based FM signal. For the broadened transi-tions these overlap, so the dispersion-based signal is morecomplex. Thus the absorption-based signal is a naturalchoice for broadened transitions. For narrow transitionsthe dispersion-based FM spectrum, which has a signalaround the line center, is a good choice. Most of thesetransitions have a Lorentzian line shape, for which theNICE OHMS signal is described by the research of Maet al.3

D. Fourier Components of the Voigt Line ShapeFor absorption by a gas at near-atmospheric pressure weconsider a Voigt line shape. Following Kluczynskiet al.,15 we give an expression for the first Fourier compo-nent of the time-dependent peak-normalized Voigt lineshape.

A Voigt profile with center frequency n0 , LorentzianHWHM DnL , and Gaussian FWHM DnG has a peak-normalized Voigt line shape16 of

x̄V~x, n! 5Re@w~ n̄d8 1 iDLG8!#

Re@w~iDLG8!#, (10)

where n̄d8 5 Aln 2(n 2 n0)/(DnG/2) is a normalized detun-ing, DLG8 5 Aln 2DnL /(DnG/2), and w(x) is the error func-tion of complex argument.17 For the time-dependentpeak-normalized Voigt line shape, n is replaced by the in-stantaneous frequency nc 1 na cos(2pft).

The even component of the first Fourier component ofthe peak-normalized Voigt absorption profile is then

1716 J. Opt. Soc. Am. B/Vol. 21, No. 10 /October 2004 N. van Leeuwen and A. Wilson

x̄V,1even~x, nc! 5

2

t Re@w~iDLG8!#E

0

t

Re@w~ n̄d8

1 n̄a8 cos~2pft ! 1 iDLG8 !#cos~2pft !dt,

(11)

where t is an integration time equal to f 21 and n̄a85 Aln 2na /(DnG/2) is a normalized modulation depth.We are not aware of an analytical solution to this integral;however, it can be easily evaluated by numerical methods.For our fitting program, we used a trapezoidal approxi-mation of the integral. If the Fourier components for theLorentzian or Gaussian functions are required, we referthe reader to Kluczynski et al.15 for the complete analysis.

3. EXPERIMENTAL DETAILSA. Optical ApparatusA schematic of our experimental apparatus is shown inFig. 1. The external-cavity diode laser (ECDL) and high-finesse cavity have been described previously.18 Withcooling, a SANYO (DL-7140-201) laser was tuned to oxy-gen absorption lines near 771 nm (12,970 cm21), and themode-hop-free tuning range of the laser was increased to23 GHz by a feed-forward signal to the injection current.19

The cavity has a FSR of 539.68 MHz and a cavity finesse

of ;11,000 at 771 nm. The vacuum chamber containing

the high-finesse cavity was evacuated to a base pressureof 1027 mbar, then filled with 99.7% pure molecular oxy-gen. The pressure in the cavity was measured with a ca-pacitance manometer (Pfeiffer Vacuum CMR 261) with arange of 1–1100 mbar and a precision of 0.2%.

In the optical path between the ECDL and the cavityare two optical isolators, two EOMs, and a polarization-preserving optical fiber. The optical isolators each pro-vide 35 dB of isolation that protects the ECDL from opti-cal feedback, primarily from the cavity. Thepolarization-preserving optical fiber is used to isolatealignment of the cavity and EOMs from adjustments tothe laser (such as tuning) and to provide a TEM00 modethat is matched to the cavity by the two lenses.

B. Frequency Generation and LockingThere are two servo-locking systems in our NICE OHMSapparatus. The first is used to lock the laser to a cavitymode and the second to lock the FM frequency to fFSR .For this, our setup involves the generatation of threelinked frequencies at approximately 15, 540, and 555MHz. Two voltage-controlled oscillators produce 540 and555 MHz, and these are mixed with a double-balancedmixer and filtered to produce the 15-MHz differencefrequency.20

The laser frequency is locked to a longitudinal mode ofthe cavity by the Pound–Drever–Hall method.21 An

Fig. 1. Setup for our NICE OHMS apparatus. OI, optical isolator; PBS, polarizing beam splitter; PD1, photodiode for laser and fFSRlocking; PD2, photodiode for NICE OHMS detection; DBM, double-balanced mixer.

N. van Leeuwen and A. Wilson Vol. 21, No. 10 /October 2004 /J. Opt. Soc. Am. B 1717

EOM (EOM 1) imprints FM sidebands on the laser fre-quency at 615 MHz. Upon reflection from the cavity, apolarizing beam splitter and a quarter-wave plate directthe light to a fast photodiode (PD1). The photodiode sig-nal is amplified and demodulated at 15 MHz to producethe Pound–Drever–Hall error signal, which is fed to aservo-locking circuit much like that described by Foxet al.22 The servo loop has a bandwidth of ;2 MHz andconsists of two elements. High-speed feedback to the la-ser injection current provides the primary lock, whereas aslower signal to the laser piezoelectric transducer (PZT)provides large gain at low frequency and a wide tuningrange.

The second EOM (EOM 2) frequency modulates thelight at ;540 MHz with a modulation index (b) of 0.43.The noise-immune aspect of NICE OHMS requires a goodmatch between the modulation frequency and fFSR . Forthis we use the locking scheme by DeVoe and Brewer.23

A fraction of the signal from PD1 (reflection from the cav-ity) is split off and demodulated at ;555 MHz (the sum offFSR and the Pound–Drever–Hall locking frequency) toproduce the error signal. A proportional and integrallocking circuit with a 20-kHz bandwidth then adjusts thetuning voltage on the 540-MHz voltage-controlled oscilla-tor to match the FSR frequency.

C. Detection and CalibrationTo obtain the NICE OHMS signal, the transmission of theoptical cavity is monitored by a photodetector (PD2) witha bandwidth of 1.5 GHz. The output of the photodetectoris amplified, high-pass filtered, and then demodulated atfFSR with a double-balanced mixer. After low-pass filter-ing, this gives the cavity-enhanced FM signal. At thispoint, noise and offsets caused by cavity vibrations andfluctuations in the phase modulation at fFSR limit the de-tection sensitivity. To suppress this noise, the cavitylength, and hence the laser frequency, is dithered at f5 25.4 Hz. The cavity-enhanced FM signal is then de-modulated, at this frequency, by a lock-in amplifier to pro-duce the full NICE OHMS signal. Demodulation occurswith a time constant of 30 ms to give a detection band-width of 5.4 Hz.

The modulation at f 5 25.4 Hz, with a depth of ;230MHz, was chosen to produce a large output while still re-taining a good laser lock to the cavity. Although themodulation applied to the cavity PZT has a constant am-plitude, the PZT response varies with dc voltage so thatthe modulation depth varies across a frequency scan. Wemeasured the input amplitude required to produce amodulation of one cavity FSR for various dc voltages.This information was subsequently used, together withthe amplitude of the modulation applied to the PZT, to de-termine the modulation depth across the scan.

To produce our cavity-enhanced FM signal we want thedemodulation to result in a purely absorption-based sig-nal. For the oxygen transitions of interest here, this isnontrivial because the absorption and dispersion-basedsignals are of similar size and shape. To adjust the de-modulation phase we viewed a transition at low pressurewhere the line shape is well described by a Doppler pro-file. Thus the dispersion signal shape is easily calcu-

lated, and we use the methods of North et al.24 to adjustthe phase so that an absorption-based FM signal is pro-duced.

The 25.4-Hz modulation is applied to the cavity length,so the laser locking circuit must transfer this modulationto the laser frequency. However, we want to minimizeany associated residual amplitude modulation (RAM),and this requires careful adjustment of the locking cir-cuit. Clearly, if the laser injection current provides themodulation, the associated AM will be large, so we wantthe PZT to provide the modulation. Therefore our lock-ing circuit provides a large gain to the PZT at 25.4 Hz.

To perform frequency scans and collect our NICEOHMS data we used a National Instruments PCI-MIO-16E-1 data-acquisition board. A ramp from an analogoutput is fed to the cavity PZT to produce a scan of thelaser frequency. Each scan is ;11 GHz (0.35 cm21) wideand takes 30 s. The cavity PZT does not produce a per-fectly linear scan so a frequency scaling measurement isperformed with each data set. This requires us to scanthe cavity, with the laser unlocked, and record the cavitytransmission to obtain well-defined frequency markersseparated by fFSR . A wavemeter (Burleigh WA-1000)provides the wave number at the start of the scan.

The NICE OHMS data presented in this paper are anaverage of nine consecutive scans. The computer simul-taneously records the output of the lock-in amplifier (theNICE OHMS signal) and the output of a photodiode,which monitors the power of the laser before the opticalfiber. To produce a calibrated NICE OHMS measure-ment we need to know the laser power upon cavity trans-mission, the finesse of the cavity, the system gain, and thedepth of both the high- and the low-frequency modulation(FM). Before scanning, we measure the laser powerupon cavity transmission with a powermeter. This isused, together with the photodiode output, to record thelaser power (upon cavity transmission) as it changesacross the scan. The depth of the 25.4-Hz modulationvaries with the dc voltage applied to the cavity PZT and isdetermined for each scan as discussed above. The cavityfinesse slowly changes with frequency and was measuredby cavity ringdown.18 The system gain and depth of thehigh-frequency modulation do not change and were cali-brated prior to the measurements. The total uncertaintyin the size of the NICE OHMS signal as a result of thesecalibrations is 10%.

After the data were collected we used MATLAB to pre-form a fit to the theoretical NICE OHMS line-shape func-tion [Eq. (9)] based on the Voigt line shape. From thetemperature of the sample we calculated the Dopplerwidth, and the fitting gave us the line intensity andLorentzian width.

4. RESULTS AND DISCUSSIONWe present measurements of ultraweak transitions inmolecular oxygen to test the validity of our description ofthe NICE OHMS signal for transitions described by aVoigt profile. We begin by comparing a previously mea-sured (and relatively strong) transition with our fittedline shape. We then consider a weaker transition to de-termine the dominant sources of uncertainty in our mea-

1718 J. Opt. Soc. Am. B/Vol. 21, No. 10 /October 2004 N. van Leeuwen and A. Wilson

surements. Finally, we look at the ability of our methodto determine the line-shape parameters for overlappingtransitions.

The transitions investigated in this paper are from twovibrational bands of the b1 Sg

1 ← X3 Sg2 electronic transi-

tion of molecular oxygen. The first is the v8 5 1 ← v95 1 vibrational band, commonly referred to as a hotband. The second is the v8 5 0 ← v9 5 0 vibrationalband (or A band) for the isotopically substituted oxygen16O 18O. The transitions included in this paper are allweak, and, for most, this is the first time to our knowledgethat the line-shape parameters have been reported.

We first investigate the relatively strong P7P7 hot-band transition at a temperature of 22 °C 6 1 °C andpressure of 397.4 6 0.8 mbar. At this temperature andpressure we expect a Doppler full width of 0.0281 cm21

and a considerable contribution from pressure broaden-ing. Figure 2 shows the NICE OHMS data for this tran-sition with the fitted line shape and the residual to the fit.To facilitate differentiation of the data and the fit, onlyone out of every seven data points is shown; however, allthe data points are retained in the residual. As expected,the measured signal closely matches the fitted NICEOHMS line shape. There is a small distortion in the re-sidual to the fit that we believe is due to RAM at 25.4 Hz.The effect of RAM in WMS was discussed by Kluczynskiet al.15 where similar line-shape distortions were re-ported. In Table 1 we show that our fitted parameters forthis transition are in good agreement with two othersources: the HITRAN database25 and a recent paper byYang et al.26

In Fig. 3 we examine the ultraweak P35Q34 transitionfrom the A band of 16O 18O to investigate the dominantsources of noise and uncertainty. At 22 °C 6 1 °C and396.8 6 0.8 mbars (the conditions of our measurement)the HITRAN database gives a maximum absorption of

Fig. 2. NICE OHMS signal for the P7P7 transition of the hotband at 12,947.31 cm21 (dots) with the fit to the expected lineshape (solid curve) and the corresponding residual to the fit (bot-tom graph).

3.79 3 1029 cm21. There are two sources of uncertaintythat we can identify from Fig. 3: etalon effects and avariation in the background level due to RAM.

Etalon effects, from unwanted reflections between com-ponents, not only produce a variation in the overall inten-sity of the light but also convert some of the FM at 540MHz into AM. This second effect causes the most prob-lems for NICE OHMS because it directly produces a falseabsorption signal. Thus etalon effects are a major issue,particularly across the wide frequency scans needed tomeasure the line shape of pressure and Doppler-broadened transitions. We designed our apparatus withthis in mind. We tilted the vacuum system windows andthe optical components (where possible), particularlythose between the last optical isolator (OI2) and the cav-ity. As suggested by Fox and Hollberg,27 we vibrated themirrors at the cavity input and output to average out anyetalon fringes involving reflections from a cavity mirror.The optical fiber we used has angled ends to minimizebackreflections within the fiber, and the two optical

Fig. 3. NICE OHMS signal for the weak P35Q34 transitionfrom the A band of 16O 18O (dots) with a curve fit to the expectedline shape (solid curve) and the corresponding residual to the fit(bottom graph).

Table 1. Line Parameters at 397.4 mbar and 22 °Cfor the P7P7 Transitiona

Data SourceLine Intensity

(1028 cm22 at 400 mbar)Lorentzian Half-Width

(cm21 at 400 mbar)

Our data 3.1(6) 0.0212(11)HITRAN 3.43 0.0204Yang et al.b 3.61(4) 0.0214

a Estimated last digit total uncertainties are shown in parentheses.b Recorded at 23 °C and intensity corrected for temperature.

N. van Leeuwen and A. Wilson Vol. 21, No. 10 /October 2004 /J. Opt. Soc. Am. B 1719

isolators were used to minimize the etalon effects in ad-dition to stoping feedback to the laser.

The significance of the etalon effects is critically depen-dent on optical alignment, and we found that the effectsvaried considerably as the laser was tuned to differenttransitions. For example, to probe the entire hot bandrequires that the laser be tuned over a 160-cm21 range,for which the optical isolators require adjustment ap-proximately every 6 cm21 and the alignment of the inputto the optical fiber requires adjustment for each new tran-sition. After each wavelength change we spent sometime minimizing the etalon effects. However, with such asensitive spectroscopic method, the remaining etalon ef-fects are still significant. Some of our optical compo-nents, such as the optical isolators, have a laser line (at780 nm) rather than broadband antireflection coatings.Also, as mentioned above, each wavelength changecaused a change in the beam path, which gave rise tofresh etalon fringes. We believe consistent attention tothe tilting of optical surfaces and use of specialized anti-reflection coatings would further decrease the remainingetalon effects.28

The second, and less significant, source of uncertaintyis the RAM at 540 MHz directly on the output of EOM 2and at 25.4 Hz on the laser output as a consequence of thelaser lock to the cavity. For the transitions we studied,this was generally seen as a slow variation in the back-ground level of the NICE OHMS signal and could bemostly removed by background subtraction. The datashown in Fig. 3 have a large RAM at 25.4 Hz to illustratethis effect. In general, the RAM can be at either 540MHz or 25.4 Hz, and we believe we have a contributionfrom both sources. In the case of the 540-MHz modula-tion, the RAM can be suppressed with a servo-locking sys-tem such as that described by Wong and Hall29 and usedin other NICE OHMS experiments. Such a locking sys-tem would also remove the AM from etalon effects thatare caused by reflections prior to the detection point forthe locking error signal. The RAM at 25.4 Hz occurs be-cause we use a particularly simple method of modulation,and it could possibly be reduced by further optimization ofthe laser locking circuit, as discussed above. If RAM at25.4 Hz is still significant after this optimization, a servoloop to produce a constant laser power could be made witha feedback to an acousto-optic modulator.

From the discussion above, it is clear that the dominantsources of uncertainty in our apparatus are not random.Therefore errors were estimated for the line-shape pa-rameters based on visual inspection of the NICE OHMSsignal and the corresponding fit. For the stronger tran-sitions that we consider, the estimated experimental pre-cision is 5% for the maximum absorption and linewidth,and thus 10% for the line intensity. For the weaker lines,with more distortion due to etalon effects and RAM, alarger uncertainty is estimated on a case-by-case basis.The total accuracy of the maximum absorption and theline intensity includes this value for the experimentalprecision plus an additional 10% uncertainty from thecalibration.

It is also useful to compare the sensitivity of this appa-ratus with the shot-noise limit of the NICE OHMS setup.This can be calculated by

~aL !min 5p

2F S 2eB

hP0D 1/2 A2

J0~b!J1~b!, (12)

where e is the electron charge, B is the detection band-width, h is the responsivity of the photodiode (in amperesper watt), and P0 is the power incident on the photodetec-tor in the absence of absorption.2 Here, b 5 0.43, h5 0.4 A/W, F 5 10,800, P0 ; 120 mW, and B 5 5.3 Hz,leading to a shot-noise-limited minimum detectable frac-tion absorption of (aL)min 5 1.9 3 10210. Given the cav-ity length of 27.8 cm, the shot-noise-limited minimum de-tectable absorption is (a)min 5 7.0 3 10212 cm21.However, for broadened transitions measured with NICEOHMS, the absorption components at 6nm do overlap sothe shot-noise limit should be expressed as a minimumdetectable absorption difference of (a21 2 a1)min 5 7.03 10212 cm21.

Using the maximum predicted absorption of 3.793 1029 cm21 for the P35Q34 transition and the size ofthe residuals in Fig. 3, we estimate the minimum detect-able absorption for the apparatus to be (a)min . 83 10210 cm21. Considering the linewidth of the transi-tion, this is equivalent to a minimum detectable absorp-tion difference of (a21 2 a1)min 5 6 3 10210 cm21, ap-proximately a factor of 100 away from the shot-noise limitcalculated for B 5 5.3 Hz. It is therefore clear that a de-crease in the etalon effects and RAM would provide a sig-nificant improvement to the sensitivity of the apparatus.

Finally, we consider the case of a NICE OHMS signalthat includes a contribution from three overlapping tran-sitions. The data were recorded at 22 °C 6 1 °C and397.1 6 0.8 mbar and involve the R15Q16 and R17R17transitions of the 16O2 hot band and a third weaker line.To the best of our knowledge this is the first time thesetransitions have been resolved and quantitatively mea-sured. The third transition is approximately a factor of 8weaker and has its line center between the two othertransitions. As a result the NICE OHMS data appear to

Fig. 4. NICE OHMS signal (dots) for three overlapping lines, ofwhich one is a factor of 8 smaller than the other two. Two the-oretical fits to the data are shown, one containing two transitions(dotted curve) and the second containing three transitions (solidcurve).

1720 J. Opt. Soc. Am. B/Vol. 21, No. 10 /October 2004 N. van Leeuwen and A. Wilson

Table 2. Parameters from the Theoretical Fit of NICE OHMS Data ContainingThree Overlapping Transitions

Our Dataa HITRAN

Transition Lorentz Width Line Intensity Lorentz Width Line Intensity

16O2 R15Q16 0.0184 1.78 3 1028 0.01712 1.96 3 1028

16O2 R17R17 0.0179 1.20 3 1028 0.01668 1.29 3 1028

16O 18O P30P30 0.0126 0.14 3 1028 0.01552 0.20 3 1028

a Lorentz width is in cm21 at 400 mbar and the line intensity is in cm22 at 400 mbar.

involve only two transitions. Although recording thedata at a lower pressure would improve the resolution ofthese transitions and may be preferable for a detailedspectroscopic study, we believed that this pressure pro-vided a better test of the apparatus and fitting.

Figure 4 shows our data for this transition groupingand the results of two theoretical fits. The dotted curveis the fit to two transitions, and the solid curve is the fit toall three transitions. As can be seen, the theoretical fit totwo transitions does not adequately describe the lineshape that we observe, and manual adjustments to thecurve do not improve this, suggesting that an additionalfeature exists within the data. Indeed, we can see in Fig.4 that the introduction of a third transition to the fit al-lows a good match to the data to be found. We have sinceidentified this transition as the P30P30 transition of the16O 18O A band, but it is shifted 0.15 cm21 from the posi-tion predicted in the HITRAN database.

In Table 2 we show the line intensity and linewidth pa-rameters obtained from the three-transition fit and com-pare these with values in the HITRAN database. Theparameters of the two stronger transitions are within10% of their expected values, whereas the parameters forthe smallest transition are within 25% for the linewidthand 50% for the line intensity. Given the level of overlap,we believe that this uncertainty is reasonable.

5. CONCLUSIONSIn summary, we have presented a theoretical descriptionof the NICE OHMS signal for the case of absorption fea-tures described by a Voigt line shape and compared thiswith experimental results for ultraweak transitions inmolecular oxygen. We combined a Fourier description ofWMS and the standard treatment of FMS to model thetwo levels of modulation used in NICE OHMS. For thecase of an absorptive signal, we derived an expression forthe overall line shape, which we then numerically solvedfor fitting to the experimental data. A full description ofour NICE OHMS apparatus was given, and measure-ments of several transitions were presented. These havea range of absorption intensities and were used to test thevalidity of our model. Our results were shown to be con-sistent with values in the HITRAN database and with thesmall number of previous measurements. Although con-siderable effort was made to avoid the AM from etalon ef-fects, for our broad frequency scans made over a widerange of wavelengths, this was shown to significantlylimit our detection sensitivity (compared with that re-ported by Ye et al.1–3). It is hoped that implementation of

a locking scheme for RAM at 540 MHz, like that used byYe et al., will reduce the level of etalon effects to signifi-cantly improve the sensitivity. However, we have shownthat the NICE OHMS technique can be used to obtainquantitative transition parameters for broadened fea-tures that are sufficiently weak as to be out of the reach ofmore traditional spectroscopic techniques.

Further reseach with this apparatus and theory is un-der way and includes making a detailed spectroscopicstudy of molecular oxygen transitions,30 of which a onlysmall sample was presented here. Of particular interestis the capability of the method to determine the linearityof spectroscopic parameters with pressure.

ACKNOWLEDGMENTSThis research is supported by the New Zealand Founda-tion for Research, Science and Technology (New EconomyResearch Fund, contract UOOX0217). We thank RichardW. Fox at the National Institute for Standards and Tech-nology (Boulder, Colorado) and Henrik G. Kjaergaard atthe University of Otago for helpful discussions.

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