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Abstract
The response of an optical cavity to incomplete extinction of nearly resonant incident light was experimentally examined. Measurements were performed using a Pound-Drever-Hall-locked frequency-stabilized cavity ring-down spectrometer (CRDS) that allowed the laser frequency detuning from the cavity resonance center to be controlled at Hz-level resolution. It is shown that an insufficient laser light extinction ratio combined with a phase shift and frequency detuning may lead to non-exponential cavity pumping and decay signals. The experimental results can be explained with a simple analytical model. The non-exponential decay can lead to a systematic shift as high as 0.5% in the ring-down time constants, dependent on the laser frequency detuning from the cavity mode center and on the extinction ratio. This can lead to appreciable systematic errors in the absorption coefficients determined with the CRDS technique.
© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
High quality spectroscopic measurements require very precise and accurate determinations of both the frequency and absorption axes. Whereas the recent developments in optical metrology have tremendously improved the frequency axis determination, the absorption axis is typically based on intensity measurements which are subject to instrumental errors limiting the quality of the spectrum. One of the most accurate spectroscopic techniques is the cavity ring-down spectroscopy (CRDS) [1]. The absorption coefficient of a sample in CRDS is determined from the cavity ring-down decay, which is assumed to be single-exponential [2]. In practice, the relative accuracy of the measurement may be decreased to a few percent by distortion of the ring-down decays due to, for example, insufficiently fast or incomplete extinction of the laser beam [3,4], detuning of laser frequency from the cavity mode center, coupling between the TEM00 mode of the cavity and higher order transverse modes [5] or detection system nonlinearity [6]. On the other hand, spectral parameters with the precision better than 0.1% are required in many practical applications: satellite monitoring of the Earth’s atmosphere [7,8], determination of the Boltzmann constant by means of optical thermometry [9], isotope ratio measurements [10] or the verification of ab initio interaction potentials [11–13]. Although the noise-immune cavity-enhanced optical heterodyne molecular spectroscopy (NICE-OHMS) provides the lowest detection limits [14,15], CRDS allows experimental investigation of spectral line shapes with highest ever achieved signal-to-noise ratios [16–18]. Ultra-precise CRDS measurements [19,20] also provide the most accurate tests of the Standard Model, e.g. based on comparison of theoretical and experimental energies of hydrogen molecule states.
The theoretical description of light interacting with an optical cavity, first given by Kastler [2], is now well established [21] and was recently reviewed by Romanini et al. [22]. Its rich dynamics was so far most clearly observed using frequency sweeps of the probe laser through the cavity resonances [23], but less direct manifestations were also observed. Some deviations from the ordinary exponential decay were observed by Huang and Lehmann [3], who ensured the coincidence between the laser and the cavity by cavity length modulation, as well as by Long et al. [4] who phase-locked the laser to a stabilized cavity. However, both papers focused only on the requirements for high optical extinction to eliminate this problem.
In this work we study the dynamics of light interacting with the optical cavity under well controlled experimental conditions. We achieve that through precise control of the laser light frequency, phase, and power with high temporal resolution. For this purpose we used a Pound-Drever-Hall-locked frequency-stabilized (FS) CRDS spectrometer [24] to investigate the response of optical cavity to rapid switching of the amplitude and phase shift of incident light detuned from the center of a cavity resonance with Hz-level resolution. Our experimental results are compared with a simple analytical model. We tested its predictions by changing the amplitude and phase of the RF signal driving an acousto-optic modulator (AOM) which served as a light switch and found very good agreement. In this study we also show the dependence of the relative accuracy of the experimentally determined cavity ring-down time constant on the optical extinction ratio and the detuning of the laser beam from the cavity mode center. We have verified that even a quite strongly attenuated laser beam (the optical extinction ratio of 58 dB) leads to a 0.5% systematic error on the ring-down time constant when detuned from the cavity resonance by about two mode halfwidths. The case of slightly non-resonant excitation of the cavity is particularly important for the CRDS systems that lack the stabilization of the laser frequency with respect to the cavity mode center. Also, frequency-agile rapid scanning (FARS) CRDS setups [25] ignore the dispersion shift of the cavity modes which also leads to non-resonant excitation. Modeling of cavity pumping with far-detuned lasers is also important in the cavity ring-up spectroscopy (CRUS) [26,27].
2. Theory
The response of an optical cavity to excitation by a laser pulse can be described according to a simple analytical model. The optical electric field Eo(t) transmitted through the cavity at a time t results from the interference of the incident electric field Ei(t) with itself, delayed by multiples of the round-trip time tr = 2nL/c determined by the length of the cavity L, the sample refractive index n, and the speed of light in vacuum c. Thus the field Eo(t) can be written as [21]
where 𝒯 and ℛ are the cavity mirrors transmission and reflection coefficients for electric field, respectively, and α is the sample absorption coefficient.We assume that the AOM acting as the light switch is turned on at time tp, or, in other words, the cavity pumping stage begins at time tp (see Fig. 1). Prior to tp the AOM remains off. Due to the finite optical extinction ratio, expressed in dB as −20 log ζ, a leakage field with amplitude ζℰ is coupled to the cavity. The initial rise of the incident electric field Ei(t) in the cavity pumping stage – after the AOM is switched on – is described by an exponential function with a time constant 2τs, where τs ≈ 50 ns is the typical switching time of an AOM defined for light intensity. After a sufficiently long time a stationary state of electric field amplitude ℰ is reached. The decay stage starts at time td when the AOM is turned off. It is also described by an exponential function with the time constant equal to 2τs. Due to incomplete switching off of the laser beam, a leakage field of amplitude ζℰ overlaps with the decaying signal. We also take into account the possibility that the phase of field Ei(t) can change when the AOM is turned on and off. Therefore, the incident radiation electric field Ei(t) can be described as
where νL = νC + Δν is the probe laser frequency, νC is the cavity resonance center frequency, Δν is the probe laser frequency detuning from the resonance center, Δϕ is the field phase shift during the cavity pumping stage, and θ(t) is a unity step function defined as3. Experiment
In this work we used a Pound-Drever-Hall-locked FS-CRDS instrument similar to one described in [28,29]. Here we will only emphasize the main features of this system. The main part of the experimental setup is a 74 cm-long, high-finesse optical cavity formed by two dual-wavelength-coated spherical mirrors having reflectivity of 99.981% in the spectral range near 690 nm and 96% at 1064 nm, corresponding to the wavelengths of probe and reference lasers, respectively. The free spectral range (FSR) of the cavity is about 204 MHz. One of the cavity mirrors is mounted on a piezo-electric transducer (PZT) used for active stabilization of the cavity length with respect to the reference I2-stabilized Nd:YAG laser working at 1064 nm and having a long-term absolute frequency stability of about 5 kHz. A detailed description of the locking technique based on the frequency modulation of the Nd:YAG laser frequency by the AOM arranged in a double-pass configuration can be found in [30]. The probe laser beam is generated by a continuous-wave external cavity diode laser (cw-ECDL) emitting around 690 nm and split into two beams of orthogonal polarizations. The vertically polarized beam is used to tightly lock the laser frequency νL to a cavity mode by means of the Pound-Drever-Hall (PDH) technique [31] which also narrows the laser linewidth [32] by more than three orders of magnitude compared to the free-running linewidth of about 200 kHz. The horizontally polarized beam, being the actual probe beam, passes through the AOM which shifts the probe beam frequency by one cavity FSR from the laser locking point and enables scanning the cavity mode in small steps in order to measure its shape and determine the mode center frequency. This way the probe beam closely follows any cavity mode frequency shifts (due to e.g. cavity length variations) and the narrow cavity modes can be recorded with high precision [33]. An example of such a mode with half width at half maximum (HWHM) γ = 6.196(3) kHz is presented in Fig. 2. Since the probe and lock beams are separated, the probe laser lock is not interrupted during the acquisition of ring-down events, which ensures the control of probe laser frequency detuning at the Hz-level for the entire measurement.
Fig. 2 Example cavity mode profile with a half width at half maximum γ = 6.196(3) kHz. Color dots indicate the frequency detunings Δν from the mode center at which measurements were performed.
The ring-down switching system is presented in Fig. 3(a). It consists of a trigger source (digital delay generator) and a RF source, whose signals are coupled to the AOM through an electronic RF switching system and an RF amplifier. When the laser power transmitted through the cavity reaches a given threshold, the trigger source generates the transistor-transistor logic (TTL) signal used to rapidly turn off the probe beam by the AOM which initiates the ring-down event. The same TTL signal triggers the data acquisition by a 14-bit analog-to-digital converter operating at a 25 MS/s sampling rate. In this setup we used three different switching systems. The simplest one is a single microwave switch (MiniCircuits, ZYSW-2-50DR) providing RF extinction ratio of (65 ± 1) dB measured using an RF spectrum analyzer (Fig. 3(b)). Its switching time is of the order of several ns. The measured optical extinction ratio of the AOM using this switching system is (58 ± 2) dB. The second system is built with two RF switches connected in series (Fig. 3(c)) which increases the RF extinction ratio to (97 ± 2) dB. In this case the optical extinction ratio is too high to be measured properly. However, the results presented in Section 5 indicate that it is significantly higher than for the single-switch system. The last system is the most versatile and provides the possibility of adjusting the residual RF power level when the AOM is OFF (Fig. 3(d)). It should be also noted that the tunable RF source provides controlled frequency detuning of the incident light from the cavity resonance center. This switching system is similar to those found in [3,4] and consists of two switches, a variable attenuator, and a phase shifter. When the TTL trigger signal is high, output 2 of both switches is ON and output 1 is OFF. The RF signal passes through the upper switch and is subsequently amplified and coupled to the AOM. When the trigger signal is low, the RF signal passes through the variable attenuator, the phase shifter, and the lower switch before it is amplified and coupled to the AOM. Both channels are added together by a combiner. In this case the residual RF power level, and thus the extinction ratio, can be easily adjusted when the AOM is OFF. Moreover, the field phase shift Δϕ can be controlled during the cavity pumping stage, i.e. when the AOM is ON.
Fig. 3 (a) Scheme of the experimental setup with an acousto-optic modulator (AOM) as the light switch and various RF switching systems: (b) single-switch system, (c) double-switch system, and (d) variable attenuation and phase shifting switching system. PBS – polarizing beam splitter, PZT – piezo-electric transducer, DetI – incident cavity light photodetector, DetT – transmitted light photodetector.
4. Measurements and model validation
In order to test the model presented in Sec. 2 we recorded the pumping and decay signals at various experimental conditions. We performed the measurements for five detunings Δν from the cavity mode center as shown in Fig. 2: Δν = 0 kHz (the mode center), Δν = ±6.2 kHz (detuned by about ±γ), and Δν = ±30 kHz (detuned by about ±5γ). All measurements were made with an empty cavity, that is for α = 0 and n = 1. The results obtained for the single-switch system are presented in Fig. 4. The decay signals for all detunings can be reasonably well approximated by the exponential function (see Fig. 6 and discussion in Section 5). On the other hand, only at exact resonance the pumping signal can be described by an exponential function with time constant equal to twice the decay time constant. When the probe beam frequency is slightly detuned from the cavity resonance, there is a small phase difference between the incident field and field injected one round trip earlier. This phase shift accumulates with the number of round trips and results in damped oscillations with period equal to the inverse of the frequency detuning 1/Δν. On the other hand, in the case of zero detuning the incident field interferes constructively with the previously injected field and no oscillations are observed. A detailed theoretical discussion of this phenomenon can be found in [22]. It is shown in Fig. 4 that the signals obtained for detunings of −Δν and +Δν are of the same shape. The simulations were carried out for tp = 50 μs, td = 450 μs, ζ = 0.00126(29) which corresponds to a measured optical extinction ratio of (58 ± 2) dB, and Δϕ = 1.67(13) which is the measured phase shift introduced by the microwave switch. However, given the relatively high optical extinction ratio the results are almost the same regardless of the phase shift. A perfect agreement between the theory and experiment is found.
Fig. 4 Incident cavity light signal (upper panel) and experimental (dots) and simulated (lines) transmitted signal (lower panel) for different frequency detunings Δν from the mode center obtained with the single-switch system which provides an optical extinction ratio of (58 ± 2) dB. The transmitted signals obtained in different experimental conditions are normalized to unity during the stationary state when the AOM in ON. For clarity every 100th experimental point is shown.
Fig. 5 Incident cavity light signals (upper panels) as well as experimental (dots) and simulated (lines) transmitted signals (lower panels) for different frequency detunings Δν from the mode center obtained with the variable attenuation and phase shifting switching system. The data correspond to an optical extinction ratio of 3 dB and a phase shift Δϕ of (a) 0, (b) π/2, (c) π, and (d) 3π/2. The transmitted signals obtained in different experimental conditions are normalized to unity during the stationary state when the AOM in ON. For clarity every 100th experimental point is shown.
Fig. 6 (a) Example ring-down decay signal obtained with the single-switch system. Residuals from fits of an exponential decay function to experimental (dots) and simulated (lines) data for different frequency detunings Δν from the mode center obtained with (b–d) single-switch system and (e–g) double-switch system. For clarity every 20th experimental point is shown.
More complicated behavior can be observed for much lower extinction ratios obtained with the use of the variable attenuation and phase shifting switching system. In Fig. 5 we present, as an example, the results of measurements performed for the optical extinction ratio of only 3 dB (ζ2 = 0.5) and the same frequency detunings as before. Moreover, we chose four values of the phase shift Δϕ: 0, π/2, π, and 3π/2. For Δϕ = 0 the pumping signals for all detunings are similar to these observed with the single-switch system. However, the decay signals no longer even resemble an exponential function (except for Δν = 0 kHz) and their shapes are very similar to the pumping signals. It was already shown by Anderson et al. [34] that the cavity output field has the resonance frequency νC regardless of the original source of the frequency. Therefore damped oscillations observed on the decay signals can be explained by the interference between the output cavity field at resonance frequency νC and the leakage field at frequency νL detuned from νC by Δν. It should be noted that signals obtained for the detunings of −Δν and +Δν have the same shape for Δϕ = 0. However, this is no longer true for Δϕ = π/2. In this case the observed signals cannot be approximated by an exponential function for any detuning. Also, the amplitude of the oscillations seen in these signals is much larger than previously reaching a maximum for Δϕ = π. Here the shapes of the signals obtained for −Δν and +Δν detunings are the same again. For this phase shift the signal for zero detuning shows the biggest departure from the exponential function and even reaches zero. It can be seen that the signal for given Δν and Δϕ = 3π/2 has the same shape as for −Δν and Δϕ = π/2. In all cases where the damped oscillations can be observed before a stationary state is reached, they are characterized by the period equal to inverse of the frequency detuning. Moreover, almost perfect agreement between the theory and experiment was obtained.
5. Application to CRDS
We have investigated the effect of the finite extinction ratio and an off-resonant laser on the ring-down decays in the CRDS spectrometer utilizing the single- and a double-switch system. The residuals of a single exponential function fitted to a signal obtained by averaging 3000 decays are shown in Fig. 6 for different detunings Δν. Clearly, the extinction ratio provided by the single-switch system is too low to avoid non-exponential ring-down signals where damped oscillations, similar to those in Figs. 4 and 5, are visible for non-zero detunings. A completely different situation occurs for the double-switch system where the residuals are very similar for all detunings. The observed slight departure from an ideal exponential decay (identical for all detunings) can be ascribed to nonlinearities in the detection system. In all cases the model agrees well with the experimental results. The simulations were carried out for Δϕ = 1.67(13) and Δϕ = 3.33(25) for the single- and double-switch system, respectively. An optical extinction ratio of 90 dB (ζ = 3.2 × 10−5) was used in the simulations for the double-switch system and good agreement with experimental data was achieved.
In Fig. 7 we present the fractional difference between the ring-down time constants, τ, measured at different frequency detunings and the reference value, τref, for both switching systems. As a reference value we take the ring-down time constant τref = 12.923(4) μs measured with the double-switch system for Δν = 0 kHz. In the case of the single-switch system the ring-down time constants agree with τref for small detunings from the resonance center, but for detunings larger than about one HWHM a significant systematic shift is seen. A difference as large as 0.5% in the ring-down time constant can be observed when the laser frequency detuning from the cavity mode center is equal to about 2γ. The optical extinction ratio of the double-switch system is high enough for the fitted ring-down time constants to be almost equal τref for detunings as large as 6.5γ which corresponds to only about 1% of the maximum cavity transmission. The increase of the uncertainty of τ with increasing frequency detuning is caused by the lower amplitude of transmitted signal and consequently lower signal-to-noise ratio. Here a qualitative agreement between simulation and experiment is found.
Fig. 7 Experimental (symbols) and simulated (lines) fractional differences between ring-down time constants τ, and the reference time constant τref shown for different frequency detunings Δν from the mode center. The presented values correspond to single-switch system (blue) and double-switch system (red).
6. Conclusions
We have examined the dependence of the optical cavity response on the basic properties of the incident light, such as frequency detuning from the cavity resonance center, amplitude variation, and phase shift, which typically affect CRDS experiments. For this purpose we used the PDH-locked FS-CRDS spectrometer. We showed how the cavity pumping and decay signals can differ from single-exponential functions. We experimentally tested the predictions of a simple analytical model and found very good agreement. The effect of finite extinction ratio of almost resonant laser light on the accuracy of the cavity ring-down decay time constant determination was examined. A systematic difference of 0.5% in the ring-down time constant was observed when the laser frequency detuning from the cavity mode center is about twice the mode halfwidth. These effects may play an important role in high fidelity line-shape measurements [9,12,13].
The systematic errors related to off-resonant excitation of the cavity mode can occur in CRDS instruments with tight laser-cavity lock (e.g. with the use of PDH technique) [16,25,35]. The elimination of locking point offset is crucial in such setups and can be done e.g. by using an active servo control to suppress the residual amplitude modulation (RAM) leading to the PDH error signal offset [36,37] or by searching the cavity mode center [29]. Alternatively the optical extinction ratio of at least 90 dB is required in ring-down measurements to suppress systematic distortions below the 10−3 level. On the other hand, in the case of low-bandwidth lock in CRDS setups (e.g. similar to one used in [38,39]) the systematic error related to the probe laser detuning from the cavity mode center should be negligible if the laser frequency fluctuations are random. In this case these systematic errors would contribute to an increased scatter of the fitted time constants.
Funding
Ministry of Science and Higher Education; Polish National Science Centre (DEC-2013/11/D/ST2/02663, 2014/15/D/ST2/05281, 2015/17/B/ST2/02115, 2015/18/E/ST2/00585, 2016/23/B/ST2/00730).
Acknowledgment
National Laboratory FAMO in Toruń, Poland, supported by the subsidy of the Ministry of Science and Higher Education;
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