the rms shot noise voltage given by

$$v_{\text{shot}} = R_T \sqrt{e(\mathcal{R}\bar{p}_{\text{bkg}} + i_D)/\Delta t}, \qquad (99)$$

and the rms RIN noise voltage given by

$$v_{\text{RIN}} = R_T \mathcal{R} \bar{p}_{\text{bkg}} 10^{\sigma_{\text{RIN}}/20} / \sqrt{2 \Delta t}. \qquad (100)$$

The parameters in these formulae are the detector temperature, T, in Kelvins, the Boltzmann constant, k<sub>B</sub>, the elementary charge, e, the detector dark current, i<sub>D</sub>, and the relative intensity noise, σ<sub>RIN</sub>, centered on mean detection offset frequency and is given in units of dBc Hz<sup>-1</sup>. The power used in the noise calculation, p̄<sub>bkg</sub>, can be either the total signal power or a larger background power, depending on the CRS detection mode. The SNR performance of a particular CRS technique is strongly dependent on the source of the background noise. The signal-to-noise (SNR) ratio will be considered for a range of conditions for each of the CRS spectroscopy techniques. In all cases, we may write the SNR in terms of an SNR coefficient that is independent of the integration time, with the form SNR = κ√Δt.

The noise processes are independent and add in quadrature so that the total rms noise voltage is written as

$$v_t = \sqrt{v_j^2 + v_{\text{shot}}^2 + v_{\text{RIN}}^2}. \qquad (101)$$

For low incident power levels, the noise is dominated by Johnson noise, and is thus independent of the signal power, producing an SNR coefficient in the low-power limit given by

$$\kappa_J = M_{\text{CRS}} \Upsilon_J, \qquad (102)$$

where we have assumed that the signal power is related to the probe pulse power by p̄<sub>sig</sub> = M<sub>CRS</sub> p̄<sub>pr</sub>. We have defined the parameter Υ<sub>J</sub> = p̄<sub>pr</sub>√(R<sub>T</sub>R<sup>2</sup>)/(2k<sub>B</sub>T). This expression shows that in this low power limit, the SNR scales linearly with the average signal power incident on the detector. This limiting case is only relevant for methods that scatter light to new frequencies or spatial positions that can be isolated from the incident light. Such CRS methods include transient grating and single pulse CARS methods.

At intermediate power levels, the noise is dominated by shot noise, leading to an SNR coefficient of

$$\kappa_{\text{shot}} = W_{\text{CRS}} \delta\phi_0^\nu \Upsilon_{\text{pr}}, \qquad (103)$$

In the shot noise limit, the SNR scales with the square root of the average power incident on the detector, which appears in the term Υ<sub>pr</sub> = √R̄p̄<sub>pr</sub>/e. Shot noise is fundamental to light detection and can not be mitigated as in the case of RIN (e.g. balanced detection) or Johnson noise (cooling the detector).

At high enough power levels, when using a single photodetector, the noise will be dominated by intensity fluctuations from the light source. The SNR coefficient in this limiting case is now independent of the power incident on the detector, and reads

$$\kappa_{\text{RIN}} = Y_{\text{CRS}} \delta\phi_0^\nu \Upsilon_{\text{RIN}}. \qquad (104)$$

The parameter Υ<sub>RIN</sub> = √2 10<sup>-σ<sub>RIN</sub>/20</sup> depends entirely on the intensity noise from the laser source. The RIN noise PSD exhibits a strong detector spectral dependence (i.e. detection integration time), and the spectral dependence is generally modelled with an offset frequency scaling of f<sup>-α</sup><sub>off</sub>, where α ≥ 1. Because of the strong spectral dependence of RIN, much lower RIN values can be obtained in the detection of light by using modulation transfer to detect the signal about a narrow frequency range at an offset frequency larger that 1/τ<sub>e</sub>, where τ<sub>e</sub> is the excited state storage lifetime of the laser gain medium. Such a strategy is employed with pump modulation and lock-in detection in many CRS methods.

Due to the linear scaling of RIN rms noise voltage, at high enough power levels, RIN dominates the CRS signal SNR. The power threshold for RIN dominance is computed with

$$p_3 = \frac{2e}{R} 10^{-\sigma_{\text{RIN}}/10}. \qquad (105)$$

Here the dark current can be neglected because it is orders of magnitude smaller than the photocurrent at these power levels. In the RIN limit the SNR scaling is set by Υ<sub>RIN</sub>, and is independent of the average probe pulse power. At very low power levels, Johnson noise will always dominate. Here, the CRS SNR scales with Υ<sub>J</sub> and SNR improvement scales linearly with the average probe pulse power. The low-power limit depends on
<figure><img src="image_23.png" /><figcaption>Figure 12. Typical regions of operation for optical detectors used for CRS showing the ranges of noise-dominated behaviors for a range incident power levels for the detectors with low and high levels of RIN for a photodiode with low (a) and high (b) RIN, a photomultiplier tube (PMT) with low (c) and high (d) RIN, and a cooled (to 200) (k) avalanche photodiode (APD) with low (e) and high (f) RIN. The red line shows the total rms voltage noise, whereas, Johnson noise is in purple, shot noise is in blue, and RIN is orange. The shaded regions highlight the areas of noise domination for these three noise processes.</figcaption></figure>

the value of RIN. For high values of RIN, we will transition directly from a RIN dominated noise to Johnson dominated noise at a threshold power of

$$p_2 = 10^{-\sigma_{\text{RIN}}/20} \sqrt{\frac{4 k_B T}{R^2 R_T}} \quad (106)$$

However, when RIN values are low enough, then below $p_3$ the rms voltage noise is dominated by shot noise and then as the power drops further, the transition to Johnson noise occurs at

$$p_1 = \frac{2 k_B T}{e R R_T} - \frac{i_D}{R} \quad (107)$$

When RIN is low enough, we can directly operate in a shot noise limited regime for an intermediate range of power levels in the range of $p_1 \le \bar{p}_{pr} \le p_3$. In this the shot noise limit, CRS SNR scales with $\Upsilon_{SH,j}$ and SNR improvement scales with the square root of the average probe pulse power. Otherwise, when RIN levels are too high, there is not a shot noise limited regime, and the noise behavior switches from Johnson to RIN at $p_2$. In the RIN-dominated regime RIN, can be suppressed by 20–50 dB with balanced detection, allowing the recovery of shot-noise limited operation even at high power levels. This then allows for an improvement of shot-noise limited SNR up to the saturation power of the detector.

To facilitate identifying favorable detectors for various CRS methods and to ascertain which noise process is dominate for a given operating condition, we consider the typical noise performance of several typical detectors: a photodiode, a photomultiplier tube (PMT), and a cooled avalanche photodiode (APD). For each detector, we consider two levels of RIN, with $\sigma_{\text{RIN}} = -120$ dBc Hz$^{-1}$ for a high value of RIN and $\sigma_{\text{RIN}} = -160$ dBc Hz$^{-1}$ for a low value of RIN.

Figures 12(a) and (b) shows the rms noise voltage as a function of average power on the detector for a typical photodiode with parameters R = 0.5 A W$^{-1}$, $R_T = 50$ Ω, T = 293 K, and $i_D = 0.3$ nA. For this detector, $p_1 = 2$ mW. For low RIN, $p_2 = 3.6$ mW and $p_3 = 6.41$ mW, as we see in figure 12(a). However for high RIN, the shot-noise limited operation vanishes detection noise switches from RIN to Johnson when the average probe power drops below $p_2 = 36$ µW as seen in figure 12(a). For this detector the SNR scales according to $\Upsilon_J = 3.93 \times 10^{10} \bar{p}_{pr}$ in the Johnson noise limit, $\Upsilon_{pr} = 1.77 \times 10^9 \sqrt{\bar{p}_{pr}}$ in the shot limit, $\Upsilon_{\text{RIN}} = 1.41 \times 10^8$ and $\Upsilon_{\text{RIN}} = 4.47 \times 10^5$ in the RIN limit for low and high RIN, respectively.

Figures 12(c) and (d) shows the rms noise voltage as a function of average power on a typical photomultiplier tube with parameters R = 10 000 A W$^{-1}$, $R_T = 50$ Ω, T = 293 K, and $i_D = 0.3$ nA. For this detector, $p_1 = 101$ nW. For low RIN, $p_2 = 180$ nW and $p_3 = 320$ nW, as we see in figure 12(c). However for high RIN, the shot-noise limited operation vanishes detection noise switches from RIN to Johnson when the average probe power drops below $p_2 = 1.80$ µW as seen in figure 12(d). For this detector the SNR scales according to $\Upsilon_J = 3.93 \times 10^{10} \bar{p}_{pr}$ in the Johnson noise limit, $\Upsilon_{pr} = 7.86 \times 10^{14} \sqrt{\bar{p}_{pr}}$ in the shot limit, $\Upsilon_{\text{RIN}} = 2.50 \times 10^{11}$ and $\Upsilon_{\text{RIN}} = 4.47 \times 10^5$ in the RIN limit for low and high RIN, respectively.
Figures 12(e) and (f) show the rms noise voltage as a function of average power on the detector for a typical cooled avalanche photodiode with parameters R = 100 A W⁻¹, RT = 100 kΩ, T = 200 K, and iD = 10⁻¹⁹ A. For this detector, p₁ = 3.45 nW. For low RIN, p₂ = 332 nW and p₃ = 32 µW, as we see in figure 12(e). However for high RIN, the shot-noise limited operation vanishes detection noise switches from RIN to Johnson when the average probe power drops below p₂ = 1.05 nW as seen in figure 12(f). For this detector the SNR scales according to ΥJ = 4.26 × 10¹⁰ p̄pr in the Johnson noise limit, Υpr = 2.50 × 10⁹ √p̄pr in the shot limit, ΥRIN = 1.41 × 10⁸ and ΥRIN = 4.47 × 10⁵ in the RIN limit for low and high RIN, respectively.

ORCID iD

Randy A Bartels https://orcid.org/0000-0003-0530-0435

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