# Analog 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 be 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 noise current and voltage RMS

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.

**Measurement Conditions:**

**Global Settings:**

**Detector A**

**Warnings**

**Cautions**

**Detector B**

**Warnings**

**Cautions**

**Detector C**

**Warnings**

**Cautions**

**Detector D**

**Warnings**

**Cautions**

**Detector E**

**Warnings**

**Cautions**

**How to calculate SNR:**

Signal-to-noise ratio can be estimated for analog measurement using the following formula:
$$ SNR =
{
I_{ph}M
\over
\sqrt{
\sqrt{2qI_{ph}BFM^2}^2+
\sqrt{2q(I_{d}+I_{b})BFM^2}^2+
\sqrt{4k_{B}TB \over R_{f}}^2+
[e_n \sqrt{B} (1+{R_{f} \over Z_{C_{in} \parallel C_{term}}})/R_f]^2 +
[i_n \sqrt{B}]^2
}
}
$$
Here the signal is defined as the cathode current \(I_{ph}={photons \over second} QE \% \) multiplied by the intrinsic gain \(M\) of the detector.

In PMT the resulting cathode current is also multiplied with the collection efficiency \(CE\%\) to account electron collection inefficiency in the dynode chain, primarily between the cathode and the first dynode.

The signal shot noise is easily calculated since the variance of Poisson distributed random photon arrival is simply the mean.
$$ i_s = \sqrt{2qI_{ph}BFM^2} $$
The shot noise of dark current and background light is similarly calculated from their equivalent cathode currents \(I_d\) and \(I_b\).
$$ i_s = \sqrt{2q(I_{d}+I_{b})BFM^2} $$
The term \(F\) is the excess noise factor used to describe the fluctuation of the gain process, this term can be omitted for photodiodes where \(M=1\) and \(F=1\).
The calculation for the actual value of F is dependent on the type of detector.

**Excess Noise Factor for PMT:**

In PMT if the distribution of gain from each dynode is assumed to be Poissonian, then it can be shown that the variance of the first dynode is roughly the variance of the entire PMT. Assuming the gain of the PMT is evenly distributed between each dynode, we can estimate the gain of the first dynode using \(\delta_{dy1}=M^{1 \over {dynodes}}\) then the excess noise factor can be estimated.
$$
F_{pmt} ={ \delta_{dy1} \over {\delta_{dy1} - 1}}
$$

**Excess Noise Factor for APD:**

The probability mass function of APD excess noise was derived by McIntyre for \(m\) number of multiplied electrons resulting from \(n\) initial electrons with an average gain \(M\) and ionization ratio \(k\).
$$
p(m | n) = {{n \Gamma({m \over {1-k}}+1)} \over m(m-n)! \Gamma({{km \over {1-k}}+n+1})} \left[{{1+k(M-1)}\over M}\right]^{n+km/(1-k)}
{\left[{{(1-k)(G-1)}\over G}\right]^{m-n}}
$$
A simplified approximation was derived by Webb et al. However, it should be noted this approximation is not a true probability mass function as it does not sum to one
and it is known that the approximate distribution deviates from McIntyre's exact probability density function for \(m \lt 10 \) output electrons, but nevertheless it is a useful approximation.
$$
p(m|n) =
{
{1 \over {
\sqrt{2 \pi n M^2 F}
(1+ {{m-Mn}\over{nGF/(F-1)}})^{3/2}
}}
\exp{\left[
-{(m-Gn)^2} \over {2 n G^2 F (1+{{m-Gn} \over nGF/(F-1)})}
\right]}
}
$$
Here the Excess Noise Factor is defined as a function of the ionization ratio and gain as \(F=kG+(2-{1\over G})(1-k) \). Further simplification is achieved by measuring F values at varying gain settings.
The formula from empirical approximation is provided below, here the term \(x\) is excess noise figure the measured value provided by the manufacturer.
$$
F_{apd} = M^x
$$

**Excess Noise Factor for SiPM (MPPC):**

A primary source of multiplication noise in SiPM (MPPC) is optical crosstalk, which results from secondary photons emitted during the avalanche migrating to another microcell and triggering a secondary avalanche.
The probability of N microcells fired as a result of exactly one initial microcell fired is shown to have a Borel distribution. The Borel distribution can be written in terms of SiPM parameters as such.
$$
P_{\mu}(N)= {{\exp(-\mu N)(\mu N)^{N-1}} \over N!}
$$
Where \(\mu=-ln(1-P_{ct})\), the expected value of Borel distributed random numbers is \(1 \over {1- \mu}\). Thus we get an approximation for excess noise factor of SiPM (MPPC) as a function of crosstalk probability \(P_{ct}\).
$$
F_{sipm} = {1 \over {1+ln(1-P_{ct})}}
$$

Created by Dino Butron, Hamamatsu Corporation