Supplementary Material for Quantum-Noise-Limited Cavity Ring-Down Spectroscopy

Supplementary Material for “Quantum-noise-limited cavity ring-down spectroscopy”

D. A. Long, A. J. Fleisher, S. Wójtewicz, J. T. Hodges

Experimental details

The optical cavity had a length of 0.74 m and a finesse of ~20,000. The probe laser was a fiber-coupled external-cavity diode laser (ECDL) with an output power of ~25 mW, a tuning range of 1570 nm to 1630 nm, and a linewidth less than 200 kHz. The output of the ECDL was split into two arms;a locking arm and a spectroscopic probe arm. While in free-space, the locking and probe legs had orthogonal linear polarizations in order to facilitate separation.

The locking arm was phase-modulated atfmod = 13 MHz using a high bandwidth, wave-guide electro-optic phase modulator (EOM) and offset-locked to the cavity using a 2f-modification of the Pound-Drever-Hall (PDH) technique. This locking procedure is described in detail in Ref. 1. The probe arm was then phase-modulated with an EOM at a modulation frequency of fMW = nfFSR ± fmod, where n is an integer, fFSR is the optical cavity’s free-spectral range (FSR), and the sign of the fmodis set based upon the lock phase. With this modulation frequency, it is ensured thatonly a single, selected sideband isin resonance with the optical cavity (see Refs.1,2 for further details on this approach).

Alternating the probe sideband frequency between two cavity modes separated by a multiple of the cavity free spectral range (FSR) creates a heterodyne beat signal between the cavity mode pumping and the cavity ring-down, as shown in Fig. 1 of the main text. For all measurements except those shown in Fig. 3 of the main text, the EOM modulation frequencies were supplied by electronically switching between two radiofrequency signal generators with bandwidths of ~6 GHz. To generate the spectrum shown in Fig. 3, a single arbitrary waveform generator (50 GS/s) was utilized. The probe sideband frequency alternated at a rate of 4.2 kHz. For the high-power experiments conducted using the traditional InGaAs photoreceiver (incident power P0≥400 μW), a fiber amplifier was inserted into the probe arm after phase-modulation. With the aid of optical attenuators and the fiber amplifier, the power incident on either photoreceiver could be varied over nearly four orders of magnitude, from ~100 nW to nearly 1 mW. The fiber amplifier was not neededin order to conduct quantum-noise-limited HD-CRDS using the avalanche photodiode (APD) at those modest incident powers.

Calculation of noise-equivalent absorption coefficients

As seen in equation (1) of the main text, the signal-to-noise ratio (SNR) of each heterodyne waveform (see Fig. 1 of the main text) is related to our achievable noise-equivalent absorption coefficient (NEA). Since all portions of the heterodyne signal contain information about both cavity modes, we fit a 30 μs section around the signal maximum using the model of Ye and Hall3 in order to determine the SNR. Unfortunately, when P0≥100 μW, the traditional InGaAs photoreceiverbegins to exhibit nonlinear behavior and a 9th order polynomial fit to the resulting residuals was included in the fitting routine in order to remove the slow-varying structure in the fit residuals. The expanded uncertainty in NEA plotted in Fig. 2 of the main text is the expanded standard deviation (2σ) of 13 successive heterodyne half-cycle fits.

The NEA reported here using HD-CRDS is limited by up to three different noise sources: detector noise, quantum noise, and the spectrum analyzer dynamic range. We can calculate the relative contributions of these noise sources beginning from the formalism of Ye and Hall.3 The detector current response at the maximum of the heterodyne signal is:

,(S1)

whereη is the detector responsivity (η≈ 1 A/W for both detectors), and P0 is the incident optical power level when a single mode is continuously pumping the cavity. The noise-limited SNRs are defined as SNRnoise=isignal/inoise, where the inoise definitions are:

, and(S2)

, (S3)

for the detector noise and quantum (shot) noise, respectively. We note that B is the acquisition bandwidth (5 MHz), xN is the detector excess noise factor (5 for the avalanche photodiode and 1 for the traditional InGaAsphotoreceiver),e is the electron charge, andNEPdet is the detector’s noise-equivalent power.

At high P0, we find it necessary to attenuate the photoreceiver response before the spectrum analyzer, causing the observable SNR to clamp at ~4000:1 as seen in the lower panel of Fig. 2 of the main text.We can calculate the maximum spectrum analyzer signal-to-noise ratio, SNRsa, as:

,(S4)

whereVmax is the maximum voltage(at 203.076 MHz) which can be recorded on the spectrum analyzer (before overload) and Vnoise is the spectrum analyzer root-mean-square noise. For our spectrum analyzer, we find that Vmax = 0.046 V. Assuming an acquisition bandwidth of 5 MHz, a 50 Ω load and the spectrum analyzer’s specified power noise density of -153 dBm/Hz, we calculate that Vnoise = 1.12×10−5 V. This result leads toa maximum spectrum analyzer SNR of 4110:1. Thus, we can write an analogous expression to those found in S2-S3 for the spectrum analyzer’s noise current asisa = isignal/4110. We note this calculated value for Vnoise is in excellent agreement with our measured value of 1.3(6) ×10−5 V.

The total noise curves plotted as solid red lines in Fig. 2 of the main text require that the individualinoise currents be added in quadrature to produce a total SNR of

.(S5)

This equation can then be inserted into equation (1) of the main text to calculate the total noise curves (shown in red) plotted in Fig. 2. Based upon the measurements found in Fig. 2, the NEPs for the avalanche photodiode and traditional InGaAsphotoreceiver were determined to be ~3 pW Hz-1/2 and ~13 pW Hz-1/2, respectively. These experimental values are in reasonable agreement with the manufacturer’s specifications of ~1.3 pW Hz-1/2 and ~16 pW Hz-1/2, respectively.

Supplementary Material References

1Long, D. A., Truong, G.-W., Van Zee, R. D., Plusquellic, D. F. & Hodges, J. T. Frequency-agile, rapid scanning spectroscopy: Absorption sensitivity of 2×10-12 cm-1 Hz-1/2 with a tunable diode laser. Appl. Phys. B, doi:10.1007/s00340-00013-05548-00345 (2013).

2Truong, G. W. et al. Frequency-agile, rapid scanning spectroscopy. Nat. Photon.7, 532-534 (2013).

3Ye, J. & Hall, J. L. Cavity ringdown heterodyne spectroscopy: High sensitivity with microwatt light power. Phys. Rev. A61 (2000).