I am a physics PhD student working in optics and I have a bit of a weird problem that I am trying to sort out and I'm hoping you math folks can help me with.
Without boring you with the experimental details, I have a laser, characterized by its pulse energy, which I shine onto a detector which then outputs a voltage in response to the pulse energy. The pulse energy has some coefficient of variation (also known as normalized root-mean-square deviation, percent RMS, or relative standard deviation) and for my purposes this value is usually around $10\%$. Normally this isn't a problem, but for the specific sensor I am working with and interested in using, I want to characterize it's nonlinear response to this pulse energy.
Theoretically, the sensor signal should roughly follow the equation Signal $= E^n$, where $n \in \mathbb{R}$ is greater than $1$, and $E$ is the pulse energy. This value of $n$ is supposed to correspond to the ratio of the bandgap energy of the sensor and the photon energy of the laser being used, but due to some physics which I won't detail here, we believe that for longer wavelengths (smaller values of the photon energy) this relationship will no longer hold and will be begin to decrease.
As a concrete example:
For $\lambda = 2um, n \sim 2$
For $\lambda = 4um, n \sim 4$
For $\lambda = 8um, n \sim 6$
That is to say, the nonlinear scaling begins to break-down at longer wavelengths.
So on to my actual question: Due to the fluctuations in my pulse energy, it is necessary that I average the response from my detector over many pulses, but since the detector has a nonlinear response, will a simple time-average actually give me a representative answer or will it give me garbage?
My intuition is that it will give me garbage because a pulse with energy $10\%$ greater than the average will contribute a signal proportional to $1.1^n$ while a pulse with energy $10\%$ smaller than average will contribute a signal proportional to $0.9^n$.
How should I go about averaging this sort of nonlinear response when I have uncertainty in the input variable as well? Is there a form of "nonlinear averaging" that applies here? Any suggestions for textbooks/papers that would explain how to do this in a relatively straightforward manner?
Thanks,
The Ultrashort Giraffe