1. Introduction

Cavity ring-down spectroscopy (CRDS) [1] has proven to be an extremely sensitive method of detecting small absorption levels of gaseous samples [2]. In this method, absorption of a sample is determined by an increase in the decay rate of an optical cavity that contains the sample of interest. Particularly advantageous is excitation with a continuous wave laser that is then rapidly turned off (compared to the decay time of the optical cavity), as this allows for much cleaner excitation of a single mode of the optical cavity. This method is known as CW-CRDS [3, 4, 5]. The sensitivity of CRDS to changes in sample absorption coefficient is equal to the speed of light in the sample times the noise in the determination of the cavity decay rate as determined by repetitive measurement of that decay as either the sample or excitation wavelength is changed. In the near-IR, where mirrors with R ≈ 99.999% and distributed feedback (DFB) diode lasers are available, we can routinely achieve noise levels of about 1 part in 10⁴ in the decay rate in a few seconds of signal averaging, which corresponds to a sensitivity of about 1 part in 10⁹ absorption per pass of the cell.

We have often found, however, that there are times when the sensitivity of the instrument is quite a bit lower due to considerably increased instability of the decay rate at fixed wavelength and quite spectrally sharp variations in the decay rate with the length or inner pressure of the cavity. This paper presents our analysis of the cause of this instabilities and the method we have found that eliminates them. Briefly, we deduce that the noise arises from coupling between the TEM₀₀ mode of the cavity we are exciting and higher order transverse modes which tune into resonance at specific cavity lengths. These modes are coupled by scattering on the mirrors which tends to excite each mode of the cavity. If the modes are not very nearly degenerate, the field amplitude transferred on each pass of the cavity destructively interferes and only a trivial amount of energy is transferred into the higher modes. However, in the case of exact degeneracy, the transfer is coherent on each pass and can add up constructively. Given the high signal to noise ratio of each individual decay (≈10³ : 1), a power transfer ~10^-12 per pass will effect our determined decay rate.

In the rest of this paper we will first describe our experimental setup, then our observations, and finally present an analysis of our results.

2. Experimental setup

Our CW-CRDS setup is similar to that previously described [6] and also that adopted by other groups, so only the essentials will be described. Light is produced by a DFB semiconductor diode laser (NTT Electronics Corporation, NLK1U5EAAA) operating near 1652 nm. The linewidth is measured to be 10MHz with Lorentzian line shape. The laser frequency is stable to 20MHz on seconds due to laser temperature instability and it also drifts about 50MHz per hour. The laser temperature tuning range (0 to 30 °C) corresponds to a change in wavelength of 3 nm. The laser comes mounted on thermoelectric cooler and fiber coupled with a 30 dB optical isolator. The light from the optical fiber is collimated, passed through an 47 dB (measured at 1652 nm) external optical isolator (IsoWave, I-15-UHP-4), and then an acousto-optic modulator (AOM) (IntraAction, ACM-802AA14). The laser frequency of the first order diffraction from the AOM has a positive 80MHz shift to the incident laser frequency in our system. This order passes through a two lens telescope and is mode matched into our CRDS cell.

The cell consists of two 1 inch diameter super mirrors (Research Electro-Optics Inc., with reflectivity R=99.9987% at λ =1652 nm) that are held in mirror mounts with bellows between the fixed and variable plate. This allows for the mirrors to be aδφusted while under vacuum and a seal to be made around the outside of the mirror. The input mirror is flat and the output one has a radius of curvature R_c = 1m and the mirrors are separated by L = 39.5 cm. This gives a free spectral range FSR = c/2L = 379.5MHz and a transverse mode spacing of 82.1MHz for an empty cavity. The TEM₀₀ mode of the cavity has a focus on the flat mirror and a calculated beam waist of ω₀ = 0.507 mm. On the output mirror, the mode has a beam waist of ω₁ = 0.652mm and, of course, a radius of curvature that matches that of the mirror. Both mirrors have the back side wedged by 0.55° to prevent feedback into the cavity which would lead to modulation of the mirror effective reflectivity. The input mirror mount has three piezoelectric transducers (PZT) that allow the cavity length to be scanned over 8 μm.

Light leaving the ring-down cavity is focused onto a 300 μm diameter InGaAs detector using a 4 cm focal length off-axis parabolic mirror. The TEM₀₀ mode of the cavity is calculated to be focused to a spot with beam waist 28.4 μm on the detector. The detector is slightly tilted to prevent the reflection from its front surface from feeding back into the cavity, which could also be a source of instability in the decay rate.

In CW-CRDS, either laser (using current tuning) or cavity (using PZTs) is scanned into resonance. A trigger detector monitors the cavity transmission and when this exceeds a preset threshold, the radio frequency (RF) power to the AOM is turned off. Also, a data acquizition card is triggered, which captures the ring-down event for signal processing. The measured extinction ratio of our AOM is 49 dB, which was achieved by inserting a second RF switch between the RF oscillator and amplifier. Our detector has a transimpedance amplifier with a 3 dB bandwidth of about 1 MHz. The data acquizition digitizes the detector output with a 12 bit A/D and a digitization rate of 1 MHz. The detector noise is measured to be 2.5mV (RMS), which based upon the transimpedance gain of the amplifier (5×10⁶ V/A) and the optical sensitivity (0.9A/W) corresponds to a detector noise of 0.6nW.

By careful alignment of the cavity, we can reduce the transmission of higher order modes to 6% of that of the TEM₀₀ modes. See Fig. 1. This allows us to set the threshold such that we have essentially zero probability of triggering on a higher transverse mode. To maximize stability of the cavity decay rate, and thus CRDS sensitivity, one wants to insure that only decay of TEM₀₀ modes are observed.

The detector signal is digitized for 2 ms after each trigger. We determine the detector offset and noise by fitting the last 0.5 ms signal, by which time the cavity intensity has decayed to well below our detector noise. This baseline is subtracted from the earlier points and a weighted least squares fit [7] is done to the logarithm of the signal to determine the cavity decay rate. This is followed by one or two cycles of a nonlinear least squares fit [7] to the original data, which experience and modeling have shown us to be fully converged. The fitting model is

y(t) is the detector voltage at time t, A is the detector DC offset, B is the amplitude at the beginning of the ring-down decay, and k is the cavity decay rate. We judge the quality of the fit by using the reduced χ² of the fit [7] given by:

where N is the number of data points in the observed decay (typically ~ 2000), A, B, and k are the parameters returned by the least squares fit, and σ is the detector noise voltage. The later should have a mean of unity and standard deviation of $\sqrt{\frac{2}{\left(N-3\right)}}$ if the data fits the single exponential decay model.

 

Fig. 1. Mode structure of the CRDS cavity. The length of the cavity was scanned by PZT at intracavity pressure ~ 30mtorr. Both signals were averages of 128 ramps. The calculated FSR and transverse mode spacing are 379.5MHz and 82.1MHz respectively. The energy coupling efficiency of the three labeled transverse modes are 94.1%, 2.6%, and 3.3% respectively. We believe the three small peaks labeled with * correspond to the unshifted laser frequency. The measured beam waist of the incident laser beam at the flat mirror is 0.517 mm, which is very close to the calculated beam size of TEM₀₀, 0.507 mm. This suggests the resonator is well described by the model.

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The CRDS cavity has a gas inlet/outlet in each mirror mount which allows the gas in the cavity to be flowed, with a rate determined by mass flow controllers. Rates of 20 sccm are typical. The gas flows through a 10 nm filer to remove particles that can be a source of noise in the decay rate. The gas pressure inside the CRDS cell is measured with a capacitance manometer. The gas pressure in the cell is controlled by both the outlet needle valve and gas flow rate.

3. Results

Figure 2 shows a plot of the observed cavity ring-down time (the inverse of the decay rate) as the cavity length is scanned by applying a voltage to the cavity PZTs. Each data point in the plot corresponds to a result of a single cavity decay. The horizontal axis is the recorded PZT voltage, and can be calibrated by observing the PZT voltage required to scan one FSR of the cavity with fixed laser wavelength. A change of 500Volts on the PZTs corresponds to about 10 FSR. During the scan, the laser was subjected to a 10 Hz current modulation to scan the laser by approximately 1.1 FSR of the cavity, insuring that for each cavity length, the laser will be brought into resonance with the cavity. The cavity was either evacuated to a pressure of ~ 30 mtorr or filled with pure Nitrogen or Xenon gases. No absorption feature is expected in these conditions. In this experiment, the cell was filled with 4.43 torr Nitrogen gas. Similar results were observed for an empty cavity (pressure ~ 30mtorr). Yet, as can be seen, there appear to be two well defined “absorption” features corresponding to reduced cavity lifetime. These are in fact not due to absorption of light by the sample. In this scan, the variation of laser frequency is less than the width of Doppler broadened lines of molecules in the gas phase and the dips in the cavity lifetime do not repeat with a period of cavity displacement λ/2 as would be expected if the feature was due to the laser frequency modulation to reach resonance with the cavity. This is an example of the type of noise that we are investigating in this paper. Also shown in Fig. 2 is the reduced χ² for each data point. It is evident that the points associated with reduced cavity lifetime are associated with fit residuals significantly higher than expected based upon the detector noise, dramatically so for the peak of the dip in cavity decay time.

 

Fig. 2. Cavity length scan by PZT. About 50V change of Vpzt corresponds to one FSR. Cavity was filled with 4.43 torr Nitrogen gas but similar results were observed for an empty cavity (pressure ~ 30mtorr).

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Fig. 3. Cavity pressure scan with Xenon gas. About 1.83 torr pressure change corresponds to one FSR. Laser current was modulated at 10 Hz by about 1.1 FSR.

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Figure 3 shows a similar cavity scan, but in this case observed by scanning the pressure of gas, ultra high purity Xenon, inside the cavity. In this case, we can scan considerably further than with the PZTs by increasing the scan range. Ideally, we would be changing the resonance wavelength only because of a change in the index of refraction of the region between the mirrors, i.e. that the intracavity wavelength stays constant while the frequency changes. However, since atmospheric pressure presses on our mirrors and PZTs, changes in the intracavity pressure lead to some change in the cavity length. Based upon pressure change required to scan the cavity one free spectral range (20 torr change corresponds to ~11 FSR), compared to the prediction based upon the refraction index of Xenon, we calculate that 81% of the scan is due to the refraction index of the gas and 19% due to changing mechanical stress on the cavity. The qualitative result is similar to that observed with the pure cavity length modulation.

 

Fig. 4. Cavity pressure scan with Nitrogen gas. About 3.26 torr pressure change corresponds to one FSR. PZTs were modulated at 12 Hz by about 1.1 FSR and the laser wavelength was fixed.

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While the positions of the noise peaks are stable for short time, they drift with time, likely due to thermal drifts in the cavity length. The mirrors mounts are bolted to a conventional steel optical table in a laboratory with poor temperature regulation. Figure 4 shows another cavity pressure scan by using pure Nitrogen gas and modulating PZTs instead of laser current. One feature to note is the significantly longer decay times in Fig. 4. We had recently replaced our cavity mirrors, due to build up of small damage spots over time, and the longer decay time corresponds to scans done with the newer, lower loss mirrors. While the newer mirrors continue to have drops in the decay time, they are smaller in peak size and the magnitude of the reduced χ² at the peak of the lifetime dips is also reduced. This difference is important and will be discussed in the next section. Another feature to point out is that all the dropout peaks in Fig. 2 and 3 can be fitted as single or multiple Gaussian line shapes, while most in Fig. 4 are asymmetric therefore the fitting is not as good. The full widths at half maximum (FWHM) of two peaks in Fig. 2 are 0.8 FSR and 0.44 FSR respectively. The FWHMs of peaks in cavity pressure scans fall in the range from 0.04 FSR to 0.26 FSR. The width of different peaks in the same scan can differ by up to a factor of 4. Some peaks in our results (not shown here) are very different and cannot be fitted as Gaussian line shapes. We also compared the results of these scans at different mode matching conditions, first 94.1% of energy then about 80% of energy coupled into the TEM₀₀ mode. We found that the line shape and width of dropout peaks are insensitive to the mode matching, although the peak pattern of each scan changes with mode matching, caused by the cavity length change in the adjustment. We also tried cavity length and pressure scans with a laser wavelength shifted by 3 nm (the maximum tuning range of our laser) and similar results were observed.

As an example, Fig. 5 shows a ring down decay that has a reduced χ² = 15.54. It is evident that this curve does not fit a single exponential decay, but rather has a distinct modulation on the decay. This is shown clearly in Fig. 6(A), which shows the residuals of the fit to a single exponential decay, and Fig. 6(C) which shows a power spectrum of this residual. In looking at a number of such plots of “bad data”, it was evident that as we scanned into the region of dropouts, there was always a peak in the power spectrum and that the frequency of this modulation reached a minimum near the point corresponding to the peak of the dip. This naturally suggests that the dip is due to a tuning through a resonance of some type. In order to test this model, we decided to do nonlinear least squares fit of the observed decays to a model that includes excitation of two modes that can interfere.

 

Fig. 5. One example of noisy decay signals. The first 10 points of each decay signal are always skipped in the fitting. After fitted to the two-mode beating model (Equation (3)), the reduced χ² is very close to one.

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In this equation, B ₁ and B ₂ are the initial amplitudes of the two modes, k ₁ and k ₂ their power decay rates (inverses of decay time constants), Δv the difference in frequency of the two modes and Δϕ the difference in initial phase. There is quite a bit of parameter correlation in the fits, particularly as the mode difference frequency approaches the width of the modes (k_i/2π). Despite this, the reduced χ² of these fits is always close to one, indicating that this model fits the data as well as can be expected statistically. Figure 6(B) and 6(D) are residuals and noise spectrum of the same example after it is fitted to new model, Eq. (3). It can be seen from them that the two-mode beating model fits the data perfectly. Figure 7 and 8 show a PZT scan and a pressure scan over a lifetime dip, with the fitted values for Δv and the ratio of the amplitudes, B ₂/B ₁. It is evident in both cases, that the dips in the lifetime are due to excitation of two modes that go through resonance and appear to have an avoided crossing of Δv _min ≈ 1 kHz, within the width of the modes.

These results suggest that the mode crossing must be due to excitation of higher order modes. Our cavity length was selected to prevent such a resonance, at least with low order transverse modes. The ratio of the transverse to longitudinal mode spacing is calculated to be 0.216. In order to test this hypothesis, we inserted a 4mm diameter intracavity aperture inside the cavity at a distance of ~ 13.5cm from the flat mirror, where the cavity TEM₀₀ is calculated to have a beam waist of 0.526 mm. We observe that the aperture causes ~ 1% decrease in the cavity ring down time. However, higher transverse modes, TEM_nm have spatial widths that scale as [8] $\sqrt{m+n+1}$ and thus will suffer higher loss, particularly for those with mode index (m+n) larger than about 14. Figure 9 shows both a PZT and pressure scan with this intracavity aperture. We note that the dropouts in the cavity lifetime have been greatly reduced. The PZT scan gives nice straight line. For the pressure scan, we observe a decay of cavity lifetime with increasing pressure. This slope is very sensitive to the alignment of the cavity and can change sign after fine adjustments of mirrors. This rules out weak absorption from possible impurities in the gas or due to Rayleigh scattering. We believe this slope is due to small changes in alignment of the optic axis due to mechanical stress on the mirrors and mounts. In order to obtain the direct proof of our proposed model, we also used an InGaAs area camera from Sensors Unlimited (Model SU320-1.7RT-V) to monitor the spatial structure of light transmitted from the cavity in cavity length scans and pressure scans without the intracavity aperture. The laser current was modulated at 10 Hz by about 1.1 FSR for both types of scans. The camera was put ~ 30 cm behind the output mirror of the cavity and the high-reflective (HR) coating surface of the input mirror (beam waist position) was focused onto the InGaAs array, which is composed by 320×240 pixels. The frame rate of the camera is 30 Hz. In the experiment, the camera was not synchronized with the resonance between the cavity and the laser. By calibration, each pixel of the array corresponds to ~ 1mm transverse distance at the beam waist position. When the cavity is on resonant with the laser, the output power of the TEM₀₀ mode makes a circle area of diameter ~ 6 pixels saturated on the array. In the cavity, the intensity of TEM₀₀ modes is much higher than that of higher order modes excited (see the Discussion part). This saturation acts as a strong background and has to be removed in order to capture images of higher order modes excited. For this purpose one 3mm wide copper foil strip, covered by black Aquadag, was horizontally put right behind the center of the output mirror, which blocked those beams of TEM₀₀, the first higher order and the second higher order modes, reducing the background down to almost zero. The result of this experiment is shown by Fig. 10. At some Vpzt or cavity pressures higher order transverse modes were excited and captured by the camera. The mode in Fig. 10(A) is linear and that in Fig. 10(B) has rectangular cross section. In pressure scans there are several pressures less than 20 torr which can excite higher order modes with different order indices, while in PZT scans typically only one Vpzt less than 500V can generate higher order modes. The modes in Fig. 10(A) and 10(B) have the mode indices n+m ~ 60 and 50 respectively. This experiment clearly shows that higher order transverse modes can be excited at some cavity pressures or PZT voltages.

 

Fig. 6. A is the residuals of the fit to a single exponential decay and C is the noise spectrum of it; B is the residuals of the fit to two-mode beating model and D is the noise spectrum of it. A and B have the same horizontal axis. C and D have the same horizontal axis. The peak ~ 175 kHz in C and D is from the computer system. The time zero point has been shifted because the first 10 points of each decay are always skipped in the fitting.

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Fig. 7. Analysis of one of the noisy peaks in Fig. 2. The cavity was filled with 4.43 torr Nitrogen gas. The slope of Δv changing with Vpzt is ≈120 Hz/V, or 7.215 kHz/μm.

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Fig. 8. Analysis of one of the noisy peaks in Fig. 3. The slope of Δv changing with Xenon pressure is ≈30.7 kHz/torr.

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4. Discussion

Since the experimental results strongly point to crossing of cavity modes as the source of our ring down time dropouts, let us examine the resonance mode frequencies. Resonance modes are characterized by transverse mode numbers n,m and longitudinal mode number q. Within the paraxial ray approximation (which is equivalent optically to neglecting spherical aberration), the resonant modes of an optical cavity formed by a flat mirror and one of radius of curvature R_c, separated by distance L(L<R_c) and filled with a gas of index of refraction n ₀ has resonance frequency nqnm [8], given by

 

Fig. 9. Pressure scan and cavity length scan with 4mm diameter intracavity aperture. Laser current was modulated for the PZT scan and PZTs were modulated for the pressure scan.

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Fig. 10. Images of higher order transverse modes captured by InGaAs camera. Modes like the one in image A were found in both PZT scans and pressure scans and the mode in image B was only found in pressure scans. Each pixel corresponds to ~ 1mm by calibration. The size of the mode in image A is ~ 4mm and that in image B ~ 3.6 mm. When n+m = 60, the size of TEM_nm is ~ 3.9mm at the input mirror of the cavity. All these modes have size less than the limiting aperture size near the output mirror. With a 4mm diameter intracavity aperture, this transverse mode excitation was no longer observed.

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Fig. 11. Image of damaged spots on one of the old supermirror surfaces. The picture represents an area of 146(H)×100(V)μm near the center of the HR surface of the mirror. The three black dots are damaged spots. The size of the spots is ~ 1μm. The nonuniform background is because the objective lens is dirty. This type of damaged spots were not observed for new mirrors.

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where δφ is a constant proportional to the phase shift of the light upon reflection off the mirror. This gives a longitudinal mode spacing FSR = 379.5MHz and a transverse mode spacing of Δv_T = 82.1 MHz. Thus for two modes to be degenerate, we require that ΔqΔv _L + Δ(m+n)Δv_T = 0. The lowest order near resonances calculated by this expression have Δ(n+m) = 37,74 with mode detunings of 1.553, 3.105MHz respectively. The radial size of a TEM_nm mode is given approximately by [8] $\sqrt{n+m+1}$ ω where ω is the beam radius of the Gaussian shaped TEM₀₀ mode (where the field drops to 1/e of its peak value). For our cavity, the limiting aperture has a radius of 7.6mm near the curved mirror where the beam radius is 0.652 mm. Thus, our cavity can support modes with n+m < 135. One thing to note is that a change in n ₀ holding L constant shifts all modes proportional to their frequency and thus does not lead to any mode crossings.

If we consider a change in cavity length, then all the modes will change frequency, but not at the same rates.

For our cavity, the frequency difference of two modes will shift at a rate of 123D(n+m) Hz/μm. Comparing with the data presented in Fig. 7, where a slope of 7.215 kHz/μm is observed, we conclude that for this resonance, the mode crossing the TEM_nm mode must have transverse mode numbers n+m ~ 59. If the cavity length is 39.604 cm, based on Eq. (4), the index of the lowest order mode approaching the TEM₀₀ mode is 60, with a mode detuing of 31 kHz. If we consider the pressure scans, the modes move relative to one another only due to strain induced change in the cavity length. As discussed earlier, we can estimate the magnitude of this effect from the pressure change required for one FSR change in the cavity compared to the calculated index shift. For both gases, the cavity length increases about 0.1 μm per torr, while with Xenon, 81% of the pressure scan is done by refractive index changing and with Nitrogen, the same number is only 59%. Using this factor, we can convert the slope of the mode crossing observed in Fig. 8 to a shift per unit cavity length change, which is 307 kHz/μm. This predicts that for this resonance the transverse mode numbers of the crossing mode are n+m ~ 2500, which is unphysically high.

Comparing with the experimental results in Fig. 10, we can see that the caculation above explains the PZT scan very well but the pressure scan badly. Although the diffraction losses of the modes in Fig. 10 are calculated to be very small (~ 10^-41 per pass), a mode with n+m ~ 2500 is not only too large to exist in this cavity, but also contradictory to the recorded mode images. Results in Fig. 8 are calculated from the data in Fig. 3. Different from Fig. 3, the pressure scan in Fig. 4 was performed by modulating Vpzt at 12 Hz with a range of 1.1 FSR while the laser wavelength was fixed. Slopes of its noise peaks are also on the order of 100 kHz/μm. If the mode crossing in pressure scans was caused by cavity length change, then it would be impossible to do pressure scans by modulating cavity length with a range of 1.1 FSR (~ 0.9 μm). This modulation will also produce a change in frequency of ~ 100D(n+m) Hz according to Eq. (5). This suggests that the pressure scan has a different mechanism of mode crossing. Compared with cavity length scans, those pressure scans produces much denser noise peaks during the same scan range (measured in unit of FSR). These two questions are still not answered. This mode crossing doesn’t occur with a periodicity of one FSR in both PZT and pressure scans; it is determined by the mode structure of the resonator. However, if one modulates the laser wavelength while making the cavity stay at the bad position, one will have noisy decay data with a period of 1 FSR.

An important question to address is why does transfer of intensity to a low loss, high order transverse mode effect the cavity decay? For the standard model of a cavity with mirrors with spatially uniform reflectivity and transmission, as long as the detector is large enough, has no spatial dependence to its quantum efficiency, and there are no other limiting apertures in the beam path after the cell, the transverse mode beating will not be visible on the ring down decay because of the orthogonality between different transverse modes. In order to maximize the Signal to Noise ratio in the detected decay, one does not want to use a detector with too large an area. The capacitance of the detector scales with area, and thus to maintain constant response time, one must scale the feedback resistor of the transimpedance amplifier inversely with this capacitance and thus detector area. The Johnson current noise [9] of this amplifier is inversely proportional to the square root of the feedback resistor and thus is proportional to the detector size. Shot noise in the detector dark current also scales as the detector size. In our instrument, given the focusing into our detector, we estimate that the detector sensitivity will fall for modes with n+m ≥ 27. Another point worth noting is that we know that the mirror loss is spatially dependent. Figure 11 shows a microscope image of one of our old mirrors which clearly shows small imperfections. In such a case, even if we had a uniform detector without limiting aperture, we would expect that in the case of multiple transverse mode excitation, the loss would vary on each pass as the varying spatial distribution sampled mirror regions of different loss. For these two reasons, different eigenmodes of the cavity are no longer exactly orthogonal.

One may assume that we can solve this problem simply by improving the mode matching of our laser light into the cavity to reduce excitation of the higher order modes. Assuming that the input beam is Gaussian, we calculated the excitation intensity of different transverse modes by using the expressions in reference [10]. As expected, for an alignment for which the TEM₀₀ dominates, as in our case, the direct excitation of high transverse modes is completely negligible. Further, we do not find the strength of the dropouts is hypersensitive with alignment as would be expected if it depended upon the overlap of our input beam with a high order transverse mode. Another argument against our directly exciting both 00 and high order modes is that we almost always observe drops in the cavity ringdown time, as shown in the figures above. If we have a simple case of mode beating, the change in the time of a single exponential fit would vary evenly both above and below the decay time of the TEM₀₀ mode as the initial relative phase of the mode excitations changed from 0 to π. This lack of symmetry in the variation in decay times was for a long time a point of confusion.

Figure 7 and 8 also show the ratio of the two mode amplitudes as a function of tuning. We note that the relative amplitudes are strongly dependent upon tuning, peaking when the modes are as nearly degenerate as possible. This result strongly suggests the analogy to the situation of quantum beat between two quantum levels that tune through resonance when there is a coupling between the states [11], and that the excitation of the the higher order modes of the cavity is not due to errors in the mode matching of the input beam, but rather due to some coupling of these modes inside the cavity.

In the standard theory of the resonance modes of an optical cavity, each mode is independent and there is no coupling between modes - it is a linear system like the normal modes of a set of harmonic oscillators. If we consider scattering on each mirror of the cavity, this will create a speckle pattern that will tend to couple the field amplitude of different modes. On each mirror reflection, one will have a matrix that relates the amplitude in different transverse modes before and after reflection, with small off-diagonal scattering elements. These will have a Gaussian random distribution if one assumes that we have a large number of independent isotropic scattering centers, which is speckle scattering [12]. The importance of scattering leading to coupling and mode repulsion in a highly degenerate cavity was discussed by Klaassen et al. [13], who demonstrated that this leads to an apparent width of a transmission peak that was significantly wider than that predicted from the observed cavity decay time.

The assumption that there is a scattering coupling of the modes explains some of the curious features of our observations. Once we excite the TEM₀₀ mode of the cavity, we will have transfer of electric field amplitude from the TEM₀₀ to other modes of the cavity. However, the field transferred on each round trip will change in relative phase by Δv t_r where Δv is the difference in frequency between the modes and t_r = 2L/c is the cavity round trip time. For typical mode spacings, this will lead to destructive interference after a few round trips and we will get very weak excitation of these other modes for small scattering amplitudes. However, if one of the modes, TEM_nm, is very nearly resonant with the excited TEM₀₀ mode, then the field amplitude in it will initially build up linearly with time (intensity of that mode increase quadratically in time) and in fact the power in the TEM₀₀ will decay both due to intrinsic cavity losses and due to this coherent transfer. Thus, we expect that the observed decay will generally no longer be a single exponential function and will decay faster than when the TEM_nm mode is off resonance, if the energy in TEM_nm is not totally collected by the detector. This clearly agrees with our observations. In addition, when these two transverse modes are nearly on resonance, they will mix with each other severely through the coupling, generating two new eigenmodes which are mixtures of the TEM₀₀ and the TEM_nm modes. These two new eigenmodes can be excited together, and decay simultaneously because the spacing between them (~ kHz) is much smaller than the laser linewidth (10 MHz). Thus, beating can happen between them as an analogy of a quantum beat, if the orthogonality between TEM₀₀ and TEM_nm is broken, or the detector has different sensitivity for these modes. Eq. (3) describes one extreme case of this beating, which is, all the energy of the TEM_nm mode is not collected by the detector while no loss for the TEM₀₀ mode. B ₁ and B ₂ are not the initial intensities of these two transverse modes, but those of the two diabatic states. In our system, only modes with the index m+n less than 27 still can be efficiently collected by the detector. The 60th order transverse mode is focused to a size much larger than that of the detector, leading to a calculated energy loss of 54%. Our data shows even with this number Eq. (3) still describes the noisy data very well because the intensity of TEM_nm in the cavity is much smaller than that of the TEM₀₀ mode (see next paragraph). Δv is like that of an avoided crossing with a minimum value given by the scattering coupling. Based upon the quantum analogy, we can predict Δv_min = 2∣S ₁ +S ₂∣/t_r, where S ₁,S ₂ are the scattering amplitudes for each mirror. Based upon our observation that Δv_min ~ 1 KHz, we can estimate S ₁,S ₂ ~ 10^-6, which translates a scattering intensity of only ~ 10^-12 per pass. The point that two new eigenmodes beat with each other is also verified by experimental data. At two wings of each dropout peak, τ₁ (inverse of k ₁) is substantailly larger than τ₂ (inverse of k ₂), and they can be regarded as the lifetime of the TEM₀₀ and TEM_nm modes respectively because the interference between them is not as strong as at the peak. However, when approaching the peak, τ₁ decreases and τ₂ increases to new values which are between τ₁ and τ₂ near dropout peak wings. These two new time constants are very close with each other and can be the same. They are decay time constants of the two mixing eigenmodes. These two mixing modes decay faster (τ decreases) than the TEM₀₀ mode, and slower (τ increases) than the higher order mode. The lifetime τ_nm of the higher order mode is smaller than that of the TEM₀₀ mode (diffraction losses of both are totally negligible). Possible explanations are that higher order modes can have larger scattering losses because of larger mode size, or that they sample different parts of the mirrors, which may not be spatially uniform.

In terms of the two state model we used to fit the decays above, the peak value of the excitation of the TEM_nm mode in the two adiabatic modes, which are mixtures of the TEM₀₀ and TEM_nm modes, even on resonance, will no longer be equal to (1/√2), but Δv_minτ_nm. The maximum decrease in the TEM₀₀ intensity due to coupling to this TEM_nm mode is proportional to (Δv_minτ_nm)2. Without the intracavity aperture, τ_nm is ~50μs for our system. Therefore the peak intensity of the TEM_nm mode is only ~10^-3 of that of the TEM₀₀ mode even on resonance (using the Δv_min ~ 1 kHz). With the intracavity aperture, the TEM_nm mode will have much higher diffraction loss. If we assume the diffraction loss reduces τ_nm down to ~ 100t_r, this maximum intensity loss will be ~ 10^-10. This is well below the signal to noise in the decay and thus will not effect the observed mode decay rate.

We can estimate the size we expect for the scattering coupling. Let the total scatting power loss of each mirror be L_s. Assuming that this mode is isotropic, the fraction of scattering intensity into a given mode should be on the order of L_sΔΩ/2π, where ΔΩ ~ π(λ/πω₀)² = 3∙10^-6 is the solid angle subtended by the TEM₀₀ mode in far field. Thus for a total scattering loss of a few parts per million per reflection (= 1-R-T, T is the transmission of the mirror), we can estimate that the scattering power coupled into other modes will be on the order of ~ 10^-12, of the same order as required to explain the strength of our observed anticrossings between the lowest order and higher order modes. Given that many of the observed defects on the surface of the mirrors are visible under a modest power microscope, the assumption of isotropic scattering is perhaps crude. This estimate supports our model of coupling through surface scattering. Also, as pointed out before, new mirrors give smaller magnitude of depth and reduced χ² at dropout peaks in both cavity pressure scans and PZT scans (not shown here). This means that Δv_min of new mirrors is smaller than that of old mirrors, just as what we observed in the experiments.

Another possible coupling mechanism between TEM₀₀ modes and higher order modes is the reflection from the anti-reflective (AR) coating surface of each wedged mirror. This reflection will generates feedback to the TEM₀₀ mode. For a mirror with HR coating reflectivity of 99.999% and that of 1% for AR coating, the feedback again is on the order of 10^-12. However, the 0.55° wedged angle and the refractive index of the substrate will produce an angle displacement of 29 mrad to this weak feedback compared with the TEM₀₀ mode in the cavity. With this angle the transverse displacement of this feedback beam near the curved mirror is about 11 mm, which is larger than the limiting aperture size (7.6 mm) of the cavity.

In our experiments we also recorded decay signals of TEM₀₁ and TEM₁₀ modes, which can be triggered when their intensities are increased to reach the threshold by misaligning the cavity away from the TEM₀₀ mode. We found they have different decay times from TEM₀₀ modes, likely because they sample different places on the mirrors which are not perfectly homogenous (calculated diffraction losses are completely negligible). We also found those decay signals are no longer single exponential decays but can also be fitted to the modified Eq. (3) quantitatively ($2\sqrt{B_1{B}_2}$ replaced by B ₃). Here B ₁ and B ₂ are initial amplitudes of TEM₁₀ and TEM₀₁ modes. B ₃ is twice of the field overlap integration of these two modes. We found B ₃ does not equal to zero. This suggests these two modes are no longer orthogonal, possibly because the transmission of the mirrors or the detector quantum efficiency are not spatially uniform. Δv equals to 261.3 kHz, which is much larger than that of the cavity length scan and pressure scan situations and can not be explained by the mirror surface scattering coupling. We believe it is due to the lifting of the degeneracy of TEM_nm modes with n+m =constant due to weak astigmatism in the cavity induced by stress on the mirrors, or from the broken cylindrical symmetry of the curvature when the mirrors were made. From Eq. (4) we can estimate the frequency difference between TEM₀₁ and TEM₁₀ modes if the radius of curvature of the mirror in x direction is no longer the same as that in y direction.

For our cavity and n+m = 1, Δv of 261.3 kHz gives ΔR_c ~ 2.6mm according to Eq. (6), which corresponds to a $\frac{\Delta R_c}{R_c}$ change of 0.26%.

5. Conclusion

In conclusion, we have found the time constant of ringdown decays drops to a much lower number at some cavity length or some inner pressure of the cavity, and we deduced the cause of this noise is the coupling between the TEM₀₀ mode and the higher order transverse modes of the cavity. Images of the higher order modes excited at certain cavity conditions were captured by an InGaAs area camera. The noise signals are no longer single exponential decays but can be fitted to the two-mode beating model (Eq. (3)) quantitatively with the reduced χ² very close to one. The noise in both cavity length scans and cavity pressure scans can be removed by inserting a 4mm diameter aperture in the cavity. One possible coupling explanation is scattering from the mirror surfaces because of the coating imperfections of them. This coupling will cause an anti-crossing as transverse modes tune with cavity length. For the cavity length scans, this mechanism explains the experimental data successfully. The strength of this scattering coupling calculated from the experimental data matches with the estimation of it by using the measured scattering loss of the mirrors, which supports our model of surface scattering coupling. We also found decay signals of TEM₀₁ and TEM₁₀ modes are no longer single exponential decays but can also be fitted to the modified Eq. (3) quantitatively. We believe this splitting is due to the lifting of the degeneracy of TEM_nm modes with n+m =constant due to weak astigmatism in the cavity induced by stress on the mirrors, or from the broken cylindrical symmetry of the curvature when the mirrors were made.