Off-Axis ICOS

In 1998, we demonstrated that the effective long sample pathlength which results from the use of highly reflective cavity mirrors results in a strong enhancement in the net time integrated cavity output signal. The use of mirrors with reflectivity = 99.99% and a 0.000001 fractional absorption per pass in the cavity provides an interesting example. In this case, the net integrated absorption becomes 1% which is easily measured using average lab equipment. The theory behind this process is discussed in two recent papers, which also presents pulsed laser experimental results which demonstrate the effect. The extension of this technique to continuous laser application is trivial, and is discussed in our second paper on this topic where scanned cw diode laser spectra of water and carbon dioxide are shown (see below).

In recent years, a highly sensitive absorption based method known as Cavity Ringdown Spectroscopy (CRDS) has been successfully implemented for infrared gas absorption analysis studies^1-7. In CRDS, the intensity decay rate of light trapped in an optical cavity is used to obtain the associated total intra-cavity losses (per pass) as a function of the optical wavelength. When the cavity loss is dominated by cavity mirror scatter and transmission, the frequency resolved "loss" curve maps out the mirror reflectivity function. When a narrow band absorbing species is present, absolute atomic or molecular absorption intensities can be inferred by subtracting the baseline (non-resonant) loss of the cavity, which is determined while the laser is tuned off-resonance with the transitions. The great utility of the CRLAS method lies as much in the extremely high sensitivity as in the simplicity of the technique. Absolute concentrations are easily inferred from the absorption data that CRDS provides. CRDS concentration detection limits for many species have been demonstrated to be in the part-per-billion to part-per-trillion range⁸.

Although the standard Cavity Ringdown technique can provide a significant improvement in the sensitivity of near-infrared and infrared absorption analysis, pulsed IR light sources generally do not possess the high frequency resolution necessary for many IR applications, and often require too much power. In response to this, we have developed the Integrated Cavity Output Spectroscopy (ICOS) method^9,10, which allows narrowband continuous-wave (CW) lasers to be used in conjunction with optical cavities in a simple and effective manner. The absorption signal is obtained through the integration of the total signal transmitted through the optical cavity, in much the same fashion as in conventional absorption measurements. Single pass cavity losses are calculated from the measured cavity output, which is a function of the mirror reflectivity as well as scattering and absorption losses due to the presence of samples located between the mirrors. Previous attempts to use optical cavities as absorption cells often required complex hardware because they attempted to actively control the interaction between the narrowband light and the optical resonance modes of the cavity¹¹. The ICOS approach, in contrast, requires a relatively simple apparatus, because it is based on systematically disrupting the cavity resonances, thereby recovering frequency averaged response of the cavity. It will be shown below that this property is extremely sensitive to the round trip intensity loss experienced by the intracavity light as it circulates through the cavity. Previously, we described a method for disrupting the resonances that involved modulating the cavity length and/or the laser frequency⁹. Here, we describe a more effective method that we have recently developed, which is based on aligning the laser in an off-axis configuration with respect to the cavity. The principles of this technique are discussed below.

The Off-Axis ICOS Approach

Off-axis paths through optical cavities are well understood^12,13, and in effect spatially separate the multiple reflections within the cavity until the "re-entrant" condition is fulfilled, which refers to the time at which the ray begins to retrace its original path through the cavity. The occurrence of this condition is dictated by the specific curvature and spacing of the mirrors forming the cavity. Any stable cavity geometry can produce stable off-axis paths through the cavity, where the stability condition (for a spherical 2-mirror cavity) is defined by the inequality

0 <(1 - d/R1)(1 - d/R2) < 1

where d is the mirror spacing and R1 and R2 are the mirror curvatures. The multiple reflections appear on the mirrors as a series of spots in an elliptical pattern. The per-pass rotation (q) around the ellipse is again determined purely by the geometry of the cavity, and is given by

cosq = 1 - d/R

assuming R=R1=R2. When 2mq = 2pn, where m equals the number of optical round-trip passes and n is an arbitrary integer, the pattern becomes re-entrant. For certain cases, this occurs after only a few passes, however, for others the number of passes can approach infinity. In many respects, the properties of the cavity, including the free spectral range (FSR), become similar to one that is m times longer. For example, a 0.5-m cavity normally has a FSR of 300 MHz, however, when aligned in a 100 pass configuration (50 round-trips) has a FSR of only 6 MHz. The reentrant condition can easily be lengthened to over 1000 passes by using astigmatic mirrors, which results in a Lissajous spot pattern13. For the cases of CRDS and ICOS, since the light is not extracted from the cavity by means of a hole in the mirror as in multi-pass absorption methods, a specific pattern (or even a known pattern) is not required. This fact removes much of the complications associated with using astigmatic mirror cavities.

Once a condition is achieved where the effective cavity FSR is significantly narrower than the laser bandwidth, the "fringe contrast" ratio approaches unity, implying that the energy coupled into the cavity ceases to be a function of the laser wavelength¹⁴. In practice, the more important ratio is between the cavity FSR and the bandwidth of the absorption feature of interest, as once the effective FSR is narrower than the absorption feature the laser frequency can be dithered, or more simply, rapidly scanned linearly through the cavity modes to reduce the fringe contrast while retaining a sufficient number of data points to define the absorption feature.

With these conditions met, all wavelength and electric-field phase information can be neglected, leading to a simplified description of the intracavity optical intensity. In this case, a source term is added to the standard differential equation used to describe the ringdown event15, resulting in the following rate equation describing the change in the intracavity power (traveling in each direction):

where IL is the incident laser power, Cp is a cavity coupling parameter, R and T are the mirror intensity reflection and transmission coefficients, L is the cavity length, and c is the speed of light. The factor of 2 in the loss term accounts for the fact that the light leaves through both mirrors, while only enters through one. Cp has a value between 0 and 1, and generally depends on the mode quality of the light source and the degree of mode matching between the cavity and the laser. For pulsed lasers, this value is often fairly low (~0.1), but it can approach unity for a well matched TEM00 CW laser. Assuming a stable (i.e. time invariant) light source, the solution to Eq. 3 for an initially empty cavity is

When the laser is switched on, a "ring-up" occurs with the same time-constant as the ring-down, given by t = L/(c(1-R)). Steady-state is reached when I = ILCpT/(2(1-R)) in each direction, i.e. the amount transmitting through the rear mirror is »ILCpT/2 (assuming R+T»1). In other words, at steady-state half of the laser power coupling into the cavity leaves through each mirror, as required by energy conservation. In general, this result represents the broadband steady-state transmission of any high finesse etalon, as it also corresponds to the integral of the Airy function over one FSR. Once sufficient laser power is leaving the cavity, the laser can be quickly interrupted to observe the ringdown decay. As the intensity buildup occurs predictably and on a well defined timescale, this can be done with a passive device such as a mechanical chopper.

Effective cavity gain as a function of the gain parameter (G) and intracavity absorption.

With an absorbing medium between the mirrors, R is replaced by R', given

where a(w) represents the absorbance of the medium over the length of the cavity. Thus, the intracavity absorption can be determined by comparing the cavity decay times with and without the absorber present (thereby determining both R and R'), as comparing Eq. (5) with the Beer-Lambert absorption formula for a single pass through the medium (I/Io = e-a(w)) reveals that I/Io=R'/R. Eqs. (4) and (5) also show that absorption information is contained in the steady-state cavity output intensity, which is the basis for ICOS. From these equations, it is easy to show that the change in steady-state output due to the presence of an absorbing species is given by

where A=1 - e-a(w) and G=R/(1-R). For weak absorption (GA << 1), the cavity provides a linear absorption signal gain (Fig. 1). Therefore, G will be referred to as the gain of the cavity. Physically, G equals the number of optical passes occurring within cavity decay time (G » t c/L), and is also simply related to the cavity finesse (G » F/p). For example, a gain of 10⁴ (1-R=100 ppm) would result in a 1% change in cavity output power for a single-pass absorption (A) of 1 ppm. It is also clear from Fig. 1 that as the absorption becomes comparable to the intrinsic cavity loss, the gain "rolls-off" due to a saturation effect.

The above analysis shows that the cavity provides tremendous signal gains even if the resonances are completely suppressed. In fact, the signal gain is only reduced by a factor of two compared with the expected gain for the case of resonantly coupling the laser to a single cavity mode. This factor relates to the relative change between the peak and the area of an individual Lorentzian cavity mode for a given cavity loss figure. The transmitted power level, on the other hand, is reduced by a factor of ~T/2, or 2×104 for of the case of T=100 ppm mirrors. In terms of the ultimately achievable shot-noise limit, however, the difference between the two methods is only the square root of T/2 for a given laser power level, or a factor of 140 in this case.