# 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 for the 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 presence of the \( \sqrt{M^2} \) term in both baseline noise currents implies that the intrinsic gain term \(M\) cancels out when only internal noise of detector is considered.
By simple arithmetic we can transform the SNR equation into a more intuitive form.
$$ SNR =
{
I_{ph}MM^{-1}
\over
M^{-1}
\sqrt{
{2qI_{ph}BFM^2}+
{2q(I_{d}+I_{b})BFM^2}+
{4k_{B}TB \over R_{f}}+
[e_n \sqrt{B} (1+{R_{f} \over Z_{C_{in} \parallel C_{term}}})/R_f]^2 +
[i_n \sqrt{B}]^2
}
}
$$
$$ SNR =
{
I_{ph}
\over
\sqrt{
{2qI_{ph}BF}+
{2q(I_{d}+I_{b})BF}+
{4k_{B}TB \over R_{f}{M^2}}+
[{e_n \sqrt{B} (1+{R_{f} \over Z_{C_{in} \parallel C_{term}}})\over R_f{M}}]^2 +
[{i_n \sqrt{B} \over {M}}]^2
}
}
$$
We can see indeed the advantage of using detectors with intrinsic is to reduce the effect of external noise.
In fact for very high values of \(M\) the external noise is negligible and we can practically calculate the SNR using only the shot noise of the signal photocurrent, background photocurrent, and dark current.
However, intrinsic gain is not without penalty. The multiplication process of photodetectors is shown to have some variance which we factor into the shot noise equation using the excess noise factor \(F\).
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 [1] for \(m\) number of multiplied electrons resulting from \(n\) initial electrons with an average gain \(M\) and ionization ratio \(k\).
$$
p(m | n) = {{n \Gamma({m \over {1-k}}+1)} \over m(m-n)! \Gamma({{km \over {1-k}}+n+1})} \left[{{1+k(M-1)}\over M}\right]^{n+km/(1-k)}
{\left[{{(1-k)(M-1)}\over M}\right]^{m-n}}
$$
A simplified approximation was derived by Webb et al. [2]. 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.
$$
p(m|n) =
{
{1 \over {
\sqrt{2 \pi n M^2 F}
(1+ {{m-Mn}\over{nMF/(F-1)}})^{3/2}
}}
\exp{\left[
-{(m-Mn)^2} \over {2 n M^2 F (1+{{m-Mn} \over nMF/(F-1)})}
\right]}
}
$$
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 by Vinogradov [3]. 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 \(P_{ct}\).
$$
F_{sipm} = {1 \over {1+ln(1-P_{ct})}}
$$

**References**

[1] R. J. McIntyre, "The Distribution of Gains in Uniformly Multiplying Avalanche Photodiodes: Theory,"
IEEE Transactions on Electron Devices, vol. ED-19, pp.703-713, June 1972.

[2] P. P. Webb, R. J. McIntyre, and J. Conradi, "Properties of Avalanche,"
RCA Review, vol. 35, pp. 234-278, June 1974.

[3] S. Vinogradov, "Analytical Models of Probability Distribution and Excess Noise Factor of Solid State Photomultipier Signals with Crosstalk,"
Nuclear Inst. and Methods in Physics Research, A, Volume 695, p. 247-251, Dec. 2012.