# MPPC (SiPM) Linearity

### MPPC (SiPM) Linearity:

The Multi-Pixel Photon Counter is a type of Silicon Photomultiplier which are arrays of Geiger mode APDs called microcells connected in parallel. Due to the discrete photon counting nature of each APD element, the upper limit of MPPC linearity is characterized by the number of microcells and the recovery time of the microcell, in other words how long it takes for the microcell to recovery. The approximation for the number of fired microcells as a function of number of pixels and number of photons has two forms depending on the pulse width of the incident relative to the recovery time:

$$ N_{fired} = \begin{cases} { {N_{pixel} \times PW} \over T_{recovery} }(1-e^{-{{N_{photon} \times PDE \times T_{recovery}}\over{{N_{pixel} \times PW}}}}) & \text{ , if }PW \gt T_{recovery} \cr { {N_{pixel}} }(1-e^{-{{N_{photon} \times PDE}\over{{N_{pixel}}}}}) & \text{ , if }PW \leq T_{recovery} \end{cases} $$The two forms of this equation follows the logic:

1. If the incident pulse width is greater than the recovery time, then each microcell can fire approximately \( PW \over T_{recovery}\) times per incident pulse.

2. If the incident pulse width is equal to or less than the recovery time, then each microcell can fire only once per incident pulse.

*Note: MPPC linearity has a complex dependence on sensitivity, recovery time, and correlated noise. In practice, linearity may be affected by conditions not represented in this approximation.*

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.

**Input Parameters:**

\(N_{pixel} \) \( \text{# of microcells}\)

\(T_{recovery} \) \( seconds\)

PDE \( \%\)

Pulse Width \( \text{seconds}\)

Axis Style