# Photon Counting SNR Simulator

Signal-to-Noise Ratio (SNR) is a figure of merit that is used to describe how discernable the signal is from the noise. In principle SNR must be at least greater than 1 in order to measure an input.

How to use the simulator:
1. Choose your detector type, up to five detectors can be compared simultaneously (Detector A to E)
2. Input your detector specifications, or simply choose a preset detector from the dropdown menu.
3. Adjust the global parameters such as light input conditions flux $$ph \over second$$ or flux density $$ph \over sec*sq. millimeter$$, and integration time.
4. If you are using a preset detector, the sensitivity will change automatically when you change the wavelength. Please note that adjusting any of the specifications will override the preset and disable the sensitivity vs. wavelength estimating function.

This simulation should be regarded as a reference only, no guarantee of detector performance is implied by the results of this simulation.

Please contact our Applications Engineers by submitting a web inquiry or by calling the Hamamatsu technical support line for a more thorough and in-depth simulation and detector selection.

### Global Settings:

Detector A
Detector B
Detector C
Detector D
Detector E

How to calculate SNR:
SNR can be calculated for photon counting using the following equation. $$SNR = { \Phi_q \eta t \over \sqrt{2(n_d +n_b)t + \Phi_q \eta t } }$$ Where $$\Phi_q$$ is the rate of photons per second which we are trying to measure, $$\eta$$ is the quantum efficiency, $$n_d$$ is the dark count rate, $$n_b$$ is the background photon count rate, and $$t$$ the integration time.
It's interesting to note the absence of excess noise, and readout noise in this equation. These noise terms can be effectively ignored because photon counting is achieved by counting pulses generated from photon and dark events. Thus small fluctuations in pulse height (excess noise), and amplitude (readout noise) do not significantly affect the measurement. This is the primary advantage that enables photon counting to achieve lower limits of detection than analog measurement.

The background photon rate $$n_b$$ are the unwanted photons from our environment originating from various sources such as autofluorescence, stray light, and ambient illumination. The number of background counts is also a function of the quantum efficiency $$\eta$$, and the background photon rate $$\Phi_b$$. $$n_b = { {\Phi_b} \eta }$$ Noise Sources:
In the above SNR formula, the noise of the dark and background light is calculated as twice the contribution of the sum. This means that we considered the case of acquiring the baseline output due to background light or dark other than the light to be measured as a reference, and we assumed that there would be fluctuations at that time as well.
To calculate the true signal from the light being measured, take the difference of the total output obtained from the measurement less the baseline reference value. $$n_{total} = (\Phi_q \eta + n_d + n_b)t$$ $$n_{baseline} = (n_d+n_b) t$$ $$n_{photon} = n_{total} - n_{baseline} = (\Phi_q \eta + n_d + n_b)t - (n_d+n_b) t$$ The total output noise from signal measurement consist of independent noise sources which is true signal light's shot noise, background light's shot noise, and dark count's shot noise. We name total output noise from measurement as $$\sigma_{measure}$$, and noise from reference measurement as $$\sigma_{baseline}$$ . $$\sigma_{measure} = \sqrt{\Phi_q \eta t + (n_d + n_b) t}$$ $$\sigma_{baseline} = \sqrt{(n_d + n_b) t}$$ The measurement including the signal light and the reference measurement are performed independently, and the total noise is given by the root sum of the squares of these noises, considering the difference calculation is performed from these measurement results. $$\sigma_{calculated} = \sqrt{ \sigma_{measure}^2 + \sigma_{baseline}^2 }$$ $$\sigma_{calculated} = \sqrt{ \sqrt{\Phi_q \eta t + (n_d+n_b)t}^2 + \sqrt{(n_d+n_b)t}^2 }$$ $$\sigma_{calculated} = \sqrt{\Phi_q \eta t + 2(n_d +n_b)t}$$ Therefore, we arrive at the final form of the SNR equation below. $$SNR = { (\Phi_q \eta + n_d +n_b)t - (n_d + n_b)t \over \sqrt{ \Phi_q \eta t + 2(n_d + n_b)t} }$$ $$SNR = { \Phi_q \eta t \over \sqrt{ \Phi_q \eta t + 2(n_d +n_b)t } }$$

Count Rate Linearity:
At high rates of photon incidence the SNR will decrease due to saturation. Photon counting circuits have an inherent dead time which limits the minimum arrival time between successive photon detections. Thus at high count rates the detector experiences pulse pile-up effects which results in missed counts. Change the vertical axis to 'Accumulated Counts' or 'Counts per Second' to see how the measured counts are affected, which in turn affects the SNR.

This simulator uses the paralyzable detector equation for approximating the observed counts as a result of non-linearity effects. Where the expected count rate $$N_{expected}$$ is simply the sum of photon counts and dark counts, and $$N_{observed}$$ is the approximate measured counts with non-linearity which factors in the dead time $$\tau$$. $$N_{expected} = [(\Phi_q + \Phi_b) \eta + n_d] t$$ $$N_{observed} = (N_{expected}) e^{ -(N_{expected}) \tau \over t } = [(\Phi_q + \Phi_b) \eta + n_d] t e^{ -[(\Phi_q + \Phi_b) \eta + n_d]\tau}$$