The detectability of a small signal depends on how large it is compared to the noise. This is usually expressed by the signal-to-noise ratio (SNR), defined as the ratio of electrical signal power to electrical noise power. Taking the electrical signal power in the circuits of Fig. 14-2 as that due to the photocurrent i , we have

where Eq. (14-2) has been used. Note that this expression applies only when the detector circuit is well below saturation. Also, in the photoconductive mode the "signal" current is defined as the measured current minus the dark current. The two contributions to the noise were discussed in Section 13-3. For shot noise, Eq. (13-29) gives the electrical noise power as

where the total current consists of both signal current i and dark current i₀. For thermal noise, Eq. (13-33) gives the electrical noise power as

Note that the thermal noise power is independent of R_L, whereas the shot noise power increases linearly with R_L. Therefore, the dominant source of noise tends to be shot noise for large R_L, and thermal noise for small R_L.

Using the above equations, the signal-to-noise ratio can be written as

where the signal current i is related to the incident optical power P_in by i = (P_in/hv) _abse. Since the SNR is a ratio of signal and noise powers, and the power is proportional to the square of voltage or current, we can write

where V_sig = i R_L. It is therefore the square root of the SNR that gives the ratio of signal amplitude to rms noise amplitude.

It is useful to consider the following limiting cases, in which one source of noise dominates.

1. Large signal. When i i₀, and i R_L V_T(recall V_T k_BT/e is the voltage equivalent of temperature), we have

The noise here is dominated by the shot noise from the signal current, and the SNR is independent of load resistance. Since 1/B corresponds to the measurement time, this result says that the SNR is roughly the number of charge carriers produced during this measurement time. Since i P_in, the SNR increases linearly with the incident optical power.

2. Small signal, large R_L. When i i₀and i₀R_L V_T,

In this regime, the SNR is limited by shot noise from the dark current i₀, and is again independent of load resistance. Note that SNR here increases with the square of the incident optical power.

3. Small signal, small R_L. When i i₀and i₀R_l V_T,

In this regime, the SNR is limited by thermal noise from the load resistor R_L. This situation is often encountered in practice, and leads to a trade-off of SNR with detector response time. Increasing R_Limproves the SNR, but degrades the response time due to RC time constant effects. Decreasing R_Limproves the response time, but at a sacrifice in SNR ratio.

When the signal becomes equal to the noise (SNR = 1), it is barely discernible, and this can be considered to be the criterion for signal detectability. The optical power that gives SNR = 1 is known as the noise equivalent power, or NEP, and is a measure of the detector's sensitivity. In the limiting case #2 above, where shot noise from the dark current dominates, the NEP is found by setting SNR = 1 and using Eq. (14-2) with P_in = NEP. This gives

or

The MKS unit for NEP is watts, since it is an optical power. An alternative unit commonly used for optical power is the dBm, defined as the power in dB relative to 1 mW. Thus, for an optical power P measured in mW,

For example, an optical power of-20 dBm is 0.01 mW, whereas an optical power of +20 dBm is 100 mW.

It is useful to separate the NEP for a detector into those factors that are fundamental and those that can be adjusted by the device geometry or detector circuit. The dark current i₀, for example, is not fundamental, since it is proportional to the area of the p-n junction. The fundamental parameter is the dark current density J₀, which depends on the material used and the temperature, but not on the device geometry. Writing i₀ = J₀A, we obtain

This shows that the minimum power that can be detected is proportional to AB. The detector can be made more sensitive by decreasing the junction area A, or by decreasing the detection circuit bandwidth. To obtain a parameter that is independent of B and A, the NEP can be divided by AB. It is conventional to define the reciprical of this as a figure of merit, since it is then a larger number for a better (more sensitive) detector. Designating this figure of merit as D* (pronounced "dee star"), we have

which becomes

where Eq. (14-18) was used in the last step.

The D* parameter provides a good way of comparing the ultimate sensitivity limits for different types of detectors. According to Eq. (14-44), D* varies inversely as the square root of dark current density, so that narrow-band-gap materials (with high J₀) have a smaller D* than wider-band-gap materials. Since a narrower band gap is required to detect light of longer-wavelength, D* is inherently smaller for longer wavelength detectors, all other things being equal. D* also depends on wavelength through the responsivity ( ). Fig. 14-17 shows the wavelength dependence of D* for some common detector materials. Note that since J₀decreases with decreasing temperature, the D* for a given photodetector can generally be improved by lowering the temperature of the semiconductor element.