# Detector Noise

## Photon noise

Photon noise is the noise that originates from the radiation source and is unavoidable. The best a detection system can be is photon noise limited.

Radiation from a laser source is described as a Poisson statistic, while thermal sources (e.g. blackbodies) emit Planck radiation described by Bose-Einstein statistics. However, when the wavelength-temperature product of the emission is much smaller than 14,500 micron-Kelvin, then Bose-Einstein statistics can be approximated by the Poisson statistics.

With a Poisson probability density function, the variance equals the mean, thus the standard deviation (noise) varies as the square root of the average photon flux. Therefore, finding the photon noise is simply a matter of knowing the incident flux.

The input (photon) noise is defined as $$\sigma = \sqrt{\overline{n}_{q}} = \sqrt{E_{q}A_{d}\Delta t}$$ where $E_{q}$ is the pixel photon irradiance, $A_{d}$ is the pixel area, $\Delta t$ is the integration time, and $\eta$ is the quantum efficiency. The input SNR happens to equal the noise for photon noise: $$\left ( \frac{S}{N} \right )_{input} = \frac{\overline{n}_{q}}{\sigma } = \sqrt{\overline{n}_{q}} = \sigma$$ Finally, the output SNR is defined as $$\left ( \frac{S}{N} \right )_{out} = \sqrt{\eta }\left ( \frac{S}{N} \right )_{in}$$

## Johnson Noise

Johnson noise is the fluctuation caused by the thermal motion of the charge carriers in a resistive element, also called Nyquist noise.

Johnson noise is defined as $$v_{j} = \sqrt{4kTR\Delta f}$$ for voltage or $$i_{j} = \sqrt{\frac{4kT\Delta f}{R}}$$ for current where $T$ is the device temperature in Kelvin, $R$ is the device resistance in ohms, and $\Delta f$ is the noise measurement electrical bandwidth in Hz.

Although the detector area does not directly feed into the calculation for Johnson Noise, the Johnson Noise current is in fact proportional to the square root of the detector area. This is because detectors are typically characterized by their R-A product, which enters the equation for Johnson noise as follows: $$i_{j} = \sqrt{\frac{4kT\Delta f}{(RA_{d})}A_{d}}$$

## Shot Noise

Shot noise is associated with dc current flowing through the potential barrier of a diode and follows Poisson statistics. Shot noise is defined as $$i_{s} = q \sqrt{\frac{2 n_{e} \Delta f}{\tau }}$$ where $n_{e}$ is the number of photo-generated carriers in the time inverval (integration time) $\tau$, and $\Delta f$ is the noise measurement electrical bandwidth in Hz. Like most noise, it is proportional to the square root of the detector area because it depends on the total current through the detector.

## Generation-recombination noise

Generation-recombination noise is associarted with statistical fluctuations and variations in the generation and recombination rate of carriers, and carrier lifetimes. G-R noise is defined as $$i_{gr} = 2qG\sqrt{\eta E_{q}A_{d}\Delta f + g_{th}A_{d}\Delta f l_{x}}$$ where $G$ is the photoconductive gain, $E_{q}$ is the photon irradiance, $A_{d}$ is the detector/pixel area, $\Delta f$ is the noise bandwidth in Hz, $\eta$ is the detector quantum efficiency, $g_{th}$ is the rate of thermal carrier generation (carriers/sec), and $l_{x}$ is the detector thickness in optical propogation direction. One of the parameters in G-R noise is the rate of thermally generated carriers $g_{th}$. In cooled infrared detectors this rate is typically set to zero; otherwise you need to know or model the thermal carrier generation rate to determine the g-r noise.

## Temperature noise

Temperature noise the fluctuation of the temperature of the photo-sensitive device, resulting from, among other things, background radiative exchange. This noise is particularly important in cooled bolometers. This calculation shows the mean-square temperature fluctuation spectrum given the stated input. That fluctuation is defined as $$\overline{\Delta T^{2}(f)} = \frac{4kKT^{2}}{K^{2} + (2\pi f)^{2}H^{2}}$$ where $K$ is the thermal conductivity (W/K), $T$ is the kinetic temperature (K), $H$ is the heat capacity (J/K), and $f$ is the electrical frequency in Hz.

## Quantization noise

Quantization noise is the noise resulting from the digitzation of an analog signal. For this calculation, we assume that the probability density function of the measured voltage over the range of one LSB is uniform. This quantization noise is defined as $$\sigma = \frac{2^{-m}v_{max}}{\sqrt{12}}$$ where $m$ is the number of bits in the A/D converter, and $v_{max}$ is the analog voltage or current range of detector output (starvation to saturation).