Measuring finesse and gas absorption with Lorentzian recovery spectroscopy

Si Tan; Paul Berceau; Shailendhar Saraf; John A. Lipa

doi:10.1364/OE.25.007645

1. introduction

Laser frequency stabilization commonly uses two techniques, locking to a Fabry-Perot (FP) cavity resonance or locking to an atomic or molecular transition. Both techniques are widely used in ultra-sensitive measurements, such as gravitational wave interferometers [1–4], precision measurements [5–7], pure metrology experiments such as tests for Lorentz invariance [8,9] or vacuum magnetic birefringence measurements [10,11] and atomic and molecular spectroscopy [12,13].

To characterize the quality of frequency stabilization using these techniques, the spectral linewidth is often a useful metric. The spectral linewidth of FP cavity resonance modes is inversely proportional to their finesse ℱ. The exact expression of ℱ has been represented as an infinite series of Lorentz functions [14], and with certain assumptions, it can be simplified to the well-known Airy equations [15, 16]. One of the methods for finesse measurement is with the cavity ring down spectroscopy (CRDS) technique [17–19], which measures the cavity characteristic time τ. The finesse can be obtained through the relation ℱ = πcτ/L = 2πτ f_FSR, where L is the cavity length, c is the speed of light in the cavity medium and f_FSR is the free spectral range [20].

Another technique for accurate measurement of finesse based on measurement of cavity Lorentzian line shape has been reported [21]. It utilizes the low pass filter feature of the cavity when the laser frequency is locked to the cavity resonance mode. When both intensity and phase modulated sidebands are applied, the resulting cavity transfer function was mathematically demonstrated to be related to the linewidth and free spectral range (FSR) of the cavity. Hence by measurement of this transfer function, finesse can be determined.

In this paper, we present a method called Lorentzian Recovery Spectroscopy (LRS), which can be an alternative for measurement of finesse and weak gas absorption in the cavity-locked regime. It is suitable for measurements where the minimum possible disturbance to the carrier is desired. We start with a theoretical derivation of the method in section 2. The experimental setup for LRS is described in section 3. A discussion of the experimental results is presented in section 4, followed by conclusions in section 5.

2. Theory: analytical solution

The form of LRS we have implemented uses two sets of symmetric phase-modulated sidebands. The first set of sidebands is used to lock the laser to a FP cavity TEM₀₀ mode [20]. The second set is used to sweep across adjacent TEM₀₀ modes that are ± 1 free spectral range (FSR) from the primary resonance. These adjacent modes are probed while maintaining a tight lock of the carrier to the central mode. The derivation below shows that the resulting signal has a Lorentzian line shape, and the finesse can be directly extracted from it.

To illustrate the behavior of the transmitted beam through a FP cavity, we start with the usual expression for the cavity incident field E_i of the carrier with a dual set of sidebands. This can be written as [22]:

E_{i} (t) = E_{0} e^{i (ω t + β_{1} \sin Ω_{1} t + β_{2} \sin Ω_{2} t)} = E_{0} e^{i ω t} \cdot \sum_{N = - \infty}^{+ \infty} J_{N} (β_{1}) e^{i N Ω_{1} t} \cdot \sum_{M = - \infty}^{+ \infty} J_{M} (β_{2}) e^{i M Ω_{2} t}

where E₀ is the amplitude of the incident field at the carrier angular frequency ω, β₁ and β₂ are the modulation indices of the cavity-locking and FSR sidebands at modulation angular frequencies Ω₁ and Ω₂ respectively, and J_N,M are Bessel functions of the first kind of order N, M. To get the transmitted field, we use the FP cavity transmission coefficient t_FP(ω), and for simplicity consider a symmetrical cavity with lossless mirrors [15]:

t_{FP} (ω) = \frac{(1 - R) e^{- i ω L / c}}{1 - R e^{- 2 i ω L / c}}

where L is the cavity length, c is the speed of light and R is the power reflection coefficient of the mirrors. We also define T as the power transmission coefficient. The incident field is multiplied by t_FP(ω) to get the transmitted electrical field E_t, and the corresponding transmitted intensity is given as:

I_{t} = A E_{t} E_{t}^{*} = A \cdot (t_{FP} E_{i}) \cdot {(t_{FP} E_{i})}^{*}

where A = ε₀c/2, and ε₀ is vacuum permittivity.

We note that the components in Eq. (3) with frequencies at ω ± nΩ₁ when n is non-zero, and at ω ± nΩ₁ ± mΩ₂ with non-zero n and m are off-resonance and will not be transmitted by the cavity. There will be other cross terms that will not be seen by a bandwidth limited detector. As an example, when expanding Eq. (3), the lowest order cross terms will be

\begin{array}{l} I_{t} = \dots + J_{0} (β_{1}) J_{0} (β_{2}) \cdot Re [t_{FP} (ω) t_{FP}^{*} (ω + Ω_{2}) - t_{FP}^{*} (ω) t_{FP} (ω - Ω_{2})] \cdot \cos (Ω_{2} t) \\ + J_{0} (β_{1}) J_{0} (β_{2}) \cdot Im [t_{FP} (ω) t_{FP}^{*} (ω + Ω_{2}) + t_{FP}^{*} (ω) t_{FP} (ω - Ω_{2})] \cdot \sin (Ω_{2} t) + \dots \end{array}

With the above considerations, the resulting transmitted intensity given by Eq. (3) can be reduced to:

\begin{array}{l} I_{t} = A E_{0}^{2} T_{0, FP} J_{0}^{2} (β_{1}) [J_{0}^{2} (β_{2}) + \sum_{M = 1}^{+ \infty} \frac{2 J_{M}^{2} (β_{2})}{1 + {(\frac{2 ℱ}{π})}^{2} \sin^{2} (\frac{M Ω_{2} L}{c})}] \\ = T_{0, FP} P_{c} + \sum_{M = 1}^{+ \infty} \frac{2 T_{0, FP} P_{s, M}}{1 + {(\frac{2 ℱ}{π})}^{2} \sin^{2} (\frac{M Ω_{2}}{2 f_{FSR}})} \end{array}

where T_0,FP = |t_FP(ω = 2π f_FSR)|² = T² / (1 − R)² ≈ 1 and

ℱ = π \sqrt{R / (1 - R)}

is the cavity finesse. P_c and P_s,_M are the powers of the carrier and the FSR sidebands respectively defined by:

P_{c} = A E_{0}^{2} J_{0}^{2} (β_{1}) J_{0}^{2} (β_{2}), P_{s, M} = A E_{0}^{2} J_{0}^{2} (β_{1}) J_{M}^{2} (β_{2})

The first term in Eq. (5) is a DC component describing the locked state of the cavity. The second term is a summation of the well-known Airy function of the cavity that can be traced by scanning the frequency Ω₂ while measuring the transmitted power. We note that this summation can be truncated depending on the particular choice of β₂. In our case β₂ = 0.45, truncating terms beyond M = 3 results in a fractional difference of ~ 1.33 × 10⁻⁷, hence M = 3 is used for later calculation of finesse.

In addition to the transmission intensity, the dispersion curve ε [22] from the second set of sidebands can also be obtained for ω ≈ 2pπ f_FSR, and is plotted in Fig. 2, along with the transmitted power profile given by Eq.(5).

3. Experimental procedure

A simplified layout of the experiment setup is shown in Fig. 1. The light source is a 10 mW single-frequency, narrow-linewidth external-cavity diode laser [23] operating near 1563 nm from Redfern Integrated Optics (linewidth ~ 1 kHz at 1 ms). The laser is pre-stabilized on a FP cavity (length ~ 10.0 cm) as described in reference [13]. We then lock the pre-stabilized laser source (PSL) to an ultra-high finesse FP cavity (length ~ 25.6 cm) with the Pound-Drever-Hall (PDH) method [24]. In order to lock the laser to the cavity, a pair of sidebands (f_PDH = Ω₁/2π = 30 MHz) with β₁ ≈ 0.4 is added to the beam using the electro-optic modulator (EOM), and heterodyne detection of the beam reflected by the cavity is performed by the photodiode PD1 (Newfocus, 1611). This produces an error signal that is processed by a custom proportional-integral-differential (PID) stage that drives a fiber-coupled double-pass acousto-optic modulator AOM (Brimrose, IPF-300-100-1550-2FP) to lock the input beam to the cavity. The input power to the cavity is 2 mW and is stabilized using a pickoff photodiode PD2 and a servo loop whose error signal controls the RF power to the AOM. The fractional optical power noise is typically < 10⁻³, using the power control servo shown in Fig. 1. We add a second pair of sidebands at Ω₂/2π = f_FSR ≈ 585 MHz with β₂ ≈ 0.45. The signal generator (Stanford Research Systems, SG382) has an external modulation input that allows us to ramp its frequency over a small range. This is used to trace out the Lorentzian lineshape of the cavity. The light transmitted by the cavity is detected with a 50 MHz bandwidth photodiode PD3 (Thorlabs, PDA10CS), and sent to a high-speed oscilloscope for data acquisition and subsequent analysis.

Fig. 1 Simplified layout of the experimental setup: PSL: pre-stabilized laser source; AOM, acousto- optic modulator; EOM, electro-optic modulator; PBS, polarizing beam splitter; FR, Faraday Rotator; PID, servo control stages; f_PDH, PDH modulation frequency; f_FSR, free spectral range frequency; PD1, reflection photodiode; PD2, input power photodiode; PD3, transmission photodiode; ΔΦ, phase shifter; LPF, low-pass filter; BPF, band- pass filter; [×], double balanced mixer; [+], RF combiner.

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In addition, a second error signal obtained in reflection from the Ω₂ sidebands is used as a cross-check on our FSR scanning technique to verify the location of the line center. We extract this error signal as Ω₂ is swept across the resonances by performing a heterodyne detection in reflection with the photodiode PD1 at the frequency f_FSR − f_PDH = 555 MHz. The detection phase of the error signal is optimized using the phase delay ΔΦ between the signal generator and the band-pass filtered signal from the reflection photodetector. We note that the symmetry of the dispersion error signal and the location of the zero-crossing at line center confirms the validity of the transmission signal, as shown in Fig. 2.

Fig. 2 Optical power transmitted by the high finesse cavity as the FSR sideband frequency Ω₂ is scanned across the resonance. The experimental data (orange circles, left axis) is fitted to Eq. (5) (blue line) yielding the finesse. The dispersion signal obtained in reflection is also plotted with the green dashed line (right axis). At the top of the figure we show the residuals of the fit in grey.

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4. Results

4.1. Finesse measurements

The results of a finesse measurement using the LRS method are shown in Fig. 2. The frequency Ω₂/2π is scanned across the cavity resonance and monitored with a counter locked to an oven-stabilized quartz oscillator. The power of the transmitted light is monitored by photodetector PD3, shown in Fig. 1. By fitting our data to Eq. (5), we obtained for a single scan cavity finesse of ℱ = 165633 ± 58 where the uncertainty is given at the 1σ confidence interval. The residual plot from the fit is shown in grey at the top of Fig. 2 and is < 1% in fractional value, thus showing good agreement between the data and the fit.

In Fig. 2 we also plot the dispersion signal as the dashed green line (right axis) extracted from the demodulation of the cavity reflected light at 555 MHz as indicated above. This signal is odd-symmetric and allows an independent estimate of line center, and its shape provides additional validation of our measurement technique.

We have also used the apparatus to compare the LRS finesse results with those obtained from the CRDS method. The finesse measurements using CRDS are obtained by abruptly switching off the incident beam locked to the cavity and analyzing the exponential decay of the transmitted beam seen by photodiode PD3. In our case, the incident beam is switched off with an ultrafast switch in the RF drive to the AOM used for locking the laser to the cavity. We use the first order diffraction beam, with a switching time of < 10 ns (including the AOM response time and the RF switch transition time). The photon lifetime τ of the resulting decay yields the cavity finesse through the relation ℱ = πcτ/L [20]. For a single ring down curve, we obtained τ = 44.925 ± 0.065 µs, corresponding ℱ = 165352 ± 239, where the uncertainty is again given at the 1σ confidence level.

4.2. Statistical analysis

In order to show the repeatability of LRS, we performed the measurements 100 times, followed by a statistical analysis.

Figure 3 shows the histogram of 100 measured finesse values for LRS, which leads to a finesse of ℱ = 165570 ± 8.5. As a cross check, we performed another 100 measurements using CRDS, which yields a finesse of ℱ = 165558 ± 39.5. Note that the uncertainties for both methods are obtained at 1σ confidence level for a sample size of 100, denoted as σ₁₀₀. As can be seen, the finesse values measured by the two methods are in good agreement within the statistical uncertainties. We note that the σ₁₀₀ of LRS appears to be smaller than the CRDS value. This is not to be taken as any criticism of the method, as our apparatus was not set up with optimization of either technique in mind. The data sets were taken in quick succession in order to minimize introduction of bias from environmental effects like temperature variations during the measurement period. In LRS the cavity was locked to the carrier during the entire measurement while the adjacent resonances were repetitively probed, thus illustrating an advantage of the technique in some situations.

Fig. 3 Histogram with associated Gaussian fit for finesse values measured by LRS.

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We note that the LRS method has more relaxed bandwidth requirements compared to CRDS, primarily due to the time taken to scan the peak. We also note that the two techniques measure different cavity fields: in CRDS we measure the field of the cavity mode to which the laser is locked. However, in LRS multiple neighboring modes are combined and measured while the laser is locked to the central mode. The optical frequency dependence of ℱ is very small in an evacuated cavity, but can be put to use in gas absorption measurements, as described in the next section.

4.3. Carbon monoxide spectroscopy

As a simple illustration of absorption coefficient measurement with LRS, we have measured the coefficient of carbon monoxide ¹²C¹⁶O at the R(14) line. This measurement was performed near 1563.616 nm that is within the tuning range of our laser. The absorption coefficient is well known [25] and our aim here is only to demonstrate the capability of our measurement technique. To apply the method, we set the laser carrier frequency (with a piezo attached to the flat end mirror) such that one resonant sideband is located near the middle of the Doppler-broadened line (450 MHz linewidth at 10 mTorr, our nominal operating pressure). Since the cavity f_FSR = 585 MHz, only the sideband centered on the Doppler line has appreciable interaction with the gas, as shown in Fig. 4.

Fig. 4 Doppler broadened absorption profile for CO molecules at low pressure, as a function of laser relative frequency. In the CRDS technique (blue), the carrier (big arrow, at frequency f_CRDS) is resonant with the gas transition, while in the LRS technique (orange), one of the sidebands (small arrow at frequency f_LRS − f_FSR) is resonant with the gas transition.

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To model the system we consider two types of sidebands: one type interacting with the gas, and the other type, far from the line center. The transmission intensity coefficient of the sideband interacting with gas can be derived, using the field expression [26]:

t_{gas, FP} = \frac{T e^{- i θ} e^{- α_{0} L / 2}}{1 - R e^{- 2 i θ} e^{- α_{0} L}}

where θ = Ω₂nL/c and α₀ is the absorption coefficient in m⁻¹ at a given pressure. In our case n is the refractive index of CO, and is a function of both optical frequency and pressure. For a given pressure, the carrier refractive index n(ω_c) is different from that at line center n(ω₀), it can be calculated by the Kramers-Kronig relations [27], assuming a Gaussian profile absorption line shape (valid under low pressure as collision broadening is insignificant). It is explicitly written as:

n (ω_{c}) = 1 + \frac{c}{π} \int_{0}^{\infty} \frac{α_{0} (P) \exp (- {[(\frac{2 \sqrt{\ln 2}}{Δ ν_{D}}) (ω^{'} - ω_{0})]}^{2})}{ω^{' 2} - ω_{c}^{2}} d ω^{'}

where ω₀ is the absorption center frequency, Δv_D ~ 450 MHz is the FWHM of the absorption line and α₀(P) is the absorption coefficient in m⁻¹ at a given pressure P.

For a worst case analysis, we use the maximum α₀ value (~ 6 × 10⁻⁵ m⁻¹) for our operating pressure range. We numerically evaluate the integral shown in Eq. (8) and find that the variation of refractive index Δn is ~ 2.4 × 10⁻¹² for the carrier (~ 585 MHz away from the particular molecular line center), causing a frequency pulling of ~ 450 Hz. The frequency pulling for all the higher order sidebands will be even smaller, since they are further off the Doppler line, as shown in Fig. 4. This frequency pulling has a negligible effect in the carrier absorption correction calculation shown later.

The transmission intensity coefficient from Eq. (7) is:

{| t_{gas, FP} |}^{2} = \frac{T^{2} e^{- α_{0} L}}{{(1 - R e^{- α_{0} L})}^{2} + 4 R \sin^{2} (θ) e^{- α_{0} L}}

where R is the cavity mirror reflection coefficient of the empty cavity obtained from the measured empty cavity finesse ℱ₀. For the sidebands far from the Doppler line we use an expression similar to that in Eq. (5).

The total power I_t for all transmitted frequencies can then be expressed as:

\begin{array}{l} I_{t} = T_{0, FP} P_{c} + \sum_{M = 2}^{+ \infty} \frac{2 T_{0, FP} P_{s, M}}{1 + {(\frac{2 ℱ_{0}}{π})}^{2} \sin^{2} (\frac{M Ω_{2}}{2 f_{FSR}})} + \frac{T_{0, FP} P_{s, 1}}{1 + {(\frac{2 ℱ_{0}}{π})}^{2} \sin^{2} (\frac{Ω_{2}}{2 f_{FSR}})} \\ + \frac{T_{0, FP} {(1 - R)}^{2} e^{- α_{0} L}}{{(1 - R e^{- α_{0} L})}^{2} + 4 R \sin^{2} (θ) e^{- α_{0} L}} \end{array}

Note that the four terms on the right hand side of Eq. (10) correspond to three components: the first term corresponds to the DC component of the locked carrier. The second and third terms correspond to the non-absorbed sidebands. The fourth term corresponds to the absorbed sideband.

To obtain the absorption coefficient we measure the transmission intensity indicated by Eq. (10), as we scan the frequency Ω₂ across the cavity resonance for 13 pressure settings of CO from 0 to 28.4 mTorr. To avoid complications from the sub-Doppler feature, we offset the mode slightly from the gas line center using a piezo located between the cavity spacer and the end mirror (see ref [13]). Next we subtract the DC component of the intensity and plot the residual in Fig. 5 for these pressures. This data corresponds to the second through fourth terms of Eq. (10). Note that the pressure-induced shifts have been removed by plotting the frequency relative to that of each transmission maximum.

Fig. 5 Relative transmitted power observed in the LRS scanning configuration for several CO pressures between 0 and 28.4 mTorr. Frequencies are plotted relative to the absorption maximum.

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We can see from Fig. 5 that the relative transmission decreases from the empty cavity case as the gas pressure increases. The reduced transmission can be attributed to the reduction of the fourth term in Eq. (10). However, there are two corrections that need to be applied to the results of a fit to this equation: the effects of saturation and the residual absorption of the carrier. We therefore calculate the unadjusted absorption coefficient α_M by directly fitting to Eq. (10), and later apply these correction terms.

The saturation effect was discussed in an earlier paper [13] where we calculated the saturation parameter s at 10 mTorr given by ref. [28] to be $s |_{I_{0}, P_{0}} ~ 0.8$ at the cavity circulation power I₀. Saturation at other powers and pressures can be scaled from this reference value through:

s (P, I) = {(\frac{10 mTorr}{P})}^{2} \cdot \frac{I}{I_{0}} \cdot s |_{I_{0}, P_{0}}

where P is pressure in mTorr, I is the circulating optical power in the cavity.

The saturation effect correction factor η₁ (multiplicative) can be written as:

η_{1} = 1 / \sqrt{1 + s_{1, sb}}

where s_1,sb is the saturation parameter of the first sideband which can be evaluated from Eq. (11).

Due to the rather broad Doppler lineshape of the gas transition there is a small correction needed for the residual absorption at optical frequencies off resonance. In Eq. (10), we used the empty cavity finesse ℱ₀ for evaluation of intensity for the non-absorbing sidebands and carrier, which will lead to an over estimation of the absorption. The total measured absorption includes the main absorbing sideband intensity plus a small contribution from gas molecules resonating with the other first order sideband and carrier away from line center. All the other higher order sidebands will be far from the line center and their contribution will be negligible. Hence considering the contribution of the 1st order non-absorbing sideband and the carrier, a correction factor η₂ (multiplicative) is derived as:

η_{2} = \frac{J_{1}^{2} (β_{2}) / \sqrt{1 + s_{1, sb}}}{ϵ J_{1}^{2} (β_{2}) / \sqrt{1 + s_{1, sb}} + ϵ J_{0}^{2} (β_{2}) / \sqrt{1 + s_{c}} + J_{1}^{2} (β_{2}) / \sqrt{1 + s_{1, sb}}}

where ϵ is the fraction of absorbing molecules relative to the line center 585 MHz away (with an additional few hundred Hz frequency pulling under maximum pressure, which has a very small effect on ϵ). It can be calculated from the Gaussian Doppler broadened line shape, similar to that used in Eq. (8). The quantity s_c is the saturation parameters of the carrier, which can be evaluated from Eq. (11).

The total correction factor η = η₁η₂, and the adjusted absorption coefficient is:

α_{0} = η α_{M} = \frac{J_{1}^{2} (β_{2}) / \sqrt{1 + s_{1, sb}}}{ϵ J_{0}^{2} (β_{2}) \sqrt{\frac{1 + s_{1, sb}}{1 + s_{c}}} + (ϵ + 1) J_{1}^{2} (β_{2})} \cdot α_{M}

We now plot α₀ as a function of the gas pressure in Fig. 6(a), and the absorption per mTorr α^∗ can then be extracted from its linear pressure dependence. For comparison, we also performed CRDS measurements over a similar pressure range.

Fig. 6 Linear absorption coefficient of CO as a function of the pressure. Comparison of the CRDS (grey circles) and the LRS (blue squares) methods. The two linear fits to the data (CRDS: grey dash line, LRS: blue line) give compatible values of the absorption coefficient of CO. The uncertainty is given at 1σ confidence level. (a) Method 1: Direct evaluation from Eq. (10); (b) Method 2: Evaluation from maximum value of scans.

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We can also evaluate α₀ and α^∗ through another simpler approach. This is done by simply calculating the transmission difference from the empty cavity case at the line center. The maximum transmission power of Eq. (10) at its line center (when Ω₂ = 2Nπ f_FSR) with the gas present is given as:

I_{g . \max} = T_{0, FP} [\frac{J_{1}^{2} (β_{2}) {(1 - R)}^{2} e^{- α_{0} L}}{{(1 - R e^{- α_{0} L})}^{2}} + J_{1}^{2} (β_{2}) + \sum_{M = 2}^{\infty} 2 J_{M}^{2} (β_{2})]

Note that the DC term is also taken out for this analysis. For the empty cavity, when evaluated at line center, the maximum transmission (less the DC term) should be:

I_{0, \max} = T_{0, FP} \sum_{M = 1}^{\infty} 2 J_{M}^{2} (β_{2})

If we compare the maximum value of the Lorentzian lines of the gas filled cavity and the empty cavity, we obtain the following:

\frac{V_{0, \max} - V_{g, \max}}{V_{0, \max}} = \frac{I_{0, \max} - I_{g, \max}}{I_{0, \max}}

where V_0,max and V_g,max are the voltages on the transmission PD for the empty cavity and the gas filled cavity. Solving the above equations leads to a simple expression for the linear absorption coefficient:

α_{0} = η \cdot \frac{1 - R}{1 + R} \cdot [{(1 - c_{1} (1 - \frac{V_{g, \max}}{V_{0, \max}}))}^{- \frac{1}{2}} - 1] \cdot \frac{2}{L}

where

c_{1} = \frac{J_{1}^{2} (β_{2})}{1 - J_{0}^{2} (β_{2})}

. The results from this method are shown in Fig. 6 (b).

Comparing the above two approaches and the CRDS measurement, we find that the values are within the statistical uncertainties at the 1σ confidence level:

\begin{array}{l} α_{LRS, method 1}^{*} = 2.19 (07) \times 10^{- 8} / cm \cdot {mTorr}^{- 1} \\ α_{LRS, method 2}^{*} = 2.18 (11) \times 10^{- 8} / cm \cdot {mTorr}^{- 1} \\ α_{CRDS}^{*} = 2.21 (06) \times 10^{- 8} / cm \cdot {mTorr}^{- 1} \end{array}

These values compare well with the HITRAN database value of ~ 2.2 × 10⁻⁸/cm · mTorr⁻¹ [29], as well as others measurements, for example those by Sung et al [30], indicating the reliability of the method. We note that the primary uncertainties in the results come from the pressure measurements in both methods rather than the measurement of α₀ for each pressure. The pressure gauge was a model VG64 thermocouple gauge with a resolution of 1 micron and calibrated against a high precision Baratron absolute pressure gauge using the sample gas. Also, the apparatus was not optimized for spectroscopy so these results can easily be improved on.

5. Conclusion

A direct method for finesse measurement in the locked regime has been demonstrated, in which an auxiliary set of sidebands is swept across the FSR of a FP cavity, while the carrier is continuously locked to a fundamental mode of the cavity. Theoretical analysis has demonstrated that we can accurately extract the Airy function and hence it can be used for finesse measurement. We have also demonstrated the use of the LRS technique in the spectroscopy of molecular CO gas. The resulting value of the linear absorption coefficient is consistent with HITRAN data [29,31] and the value obtained from CRDS within the uncertainties of our measurements. The LRS method has the advantage that the bandwidth of the photodetector is more flexible, since it is only limited by the ramp frequency of the probing sideband, which can be configured for different scanning rates depending on the application. In addition, there is no need for fast data acquisition and the dynamic range requirement for the measurement device is also more relaxed comparing to CRDS. There is no need for optical power switching, and there is no disturbance to the cavity carrier lock during the measurement process. The technique allows in situ finesse measurement in situations where the minimum possible disturbance to an optical setup is required. It also opens up a new way of detecting trace gasses that could be of use in environmental research or commercially exploited.

Funding

NASA Strategic Astrophysics Technology Program (NNX13AC91G).

Acknowledgments

We thank R. Byer and S. Buchman (Stanford University) for supporting the project. We also thank Dr. Shengkai Wang (Stanford University) for helpful discussion.

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Measuring finesse and gas absorption with Lorentzian recovery spectroscopy

Abstract

Corrections

1. introduction

2. Theory: analytical solution

3. Experimental procedure

4. Results

4.1. Finesse measurements

4.2. Statistical analysis

4.3. Carbon monoxide spectroscopy

5. Conclusion

Funding

Acknowledgments

References and links

Cited By

Figures (6)

Equations (19)

Optics Express