# Amplitude Spectral Density from a Fast Fourier Transformation

I want to calculate the Noise-equivalent Power (NEP) of a photodetector. Hereto, I have to measure the Amplitude Spectral Density (ASD) of the detector and divide the result by the Responsivity:

The response of my detector is shown in the figure below: I measure the current as a function of time. Hereby, a pulsed illumination is applied between 0.0 s and 0.5 s.

The Responsivity calculation is straightforward: I integrate the current between 0.0 s and 0.5 s and divide the result by the duration of 0.5 s. I then divide the result by the incoming illumination power to obtain the responsivity in units of [A/W].

Next, I have to determine the Amplitude Spectral Density without any optical input. Hereto, I perform a Fast Fourier Transform on the detector output in the range between -0.7 s and - 0.2 s. The result is shown below. I have already scaled the data to be consistent with Parsval's theorem.

I do not understand how the calculate the Amplitude Spectrum Density in Units of A/sqrt(Hz) that I need for the NEP calculation from this graph.

My plan was to integrate the Amplitude Spectrum over the frequency range and then take the average, but this approach does not end up with units of A/sqrt(Hz). Where is my mistake and where does the sqrt(Hz) term come from?

• A single number in A/rtHz doesnt really make sense unless your noise spectrum is flat. That being said, its common to do so even for non flat noise spectra, you just need to also specify a bandwidth. What is the measurement bandwidth over which you want to calculate NEP?
– Matt
May 28, 2021 at 23:39
• The NEP of a photodetector is given as only one number in the specification photodetectors, e.g. from Thorlabs. As I understand, the bandwidth used to determine the NEP is 1 Hz, which is far lower than the typical measurement bandwidth of a photodiode @Matt May 31, 2021 at 12:25
• Nope. It can be given as a function of measurement bandwidth. See figure 3: thorlabs.com/images/TabImages/… This must be the case since its fairly rare to actually have a flat noise spectrum and the manufacturer cant know how you plan to use the detector.
– Matt
May 31, 2021 at 12:31
• @Matt Yes, you are right. Even though your example is an exception and most of the time, only a single value is quoted in the specifications, which applies to the specified bandwidth: link Back to my original question, I still do not know how to determine the NEP for a bandwidth of e.g. DC to 100 MHz from my amplitude spectrum. May 31, 2021 at 13:36

To calculate NEP in W/Hz1/2 first calculate the total noise power within your measurement bandwidth. (When working with noise, power is the more fundamental quantity you want to use, rather than A or V.) To do this, get the noise power spectrum in units of A2/Hz. In your case, square the current spectrum you are showing and divide by the bandwidth of a single spectral bin. Depending on how you got your spectrum this may simply be equal to the frequency step along your frequency axis, or it may be more complicated if you used windowing functions or measured it directly with a spectrum analyzer.

The total noise power, in A2, in your measurement is the integral of this spectrum over your bandwidth. (You can also get here from the standard deviation of the noise timeseries measurement if it is collected at the same frequency/bandwidth at which you want to calculate NEP. Doing so would automatically account for your measurement bandwidth. There may be an additional conversion I'm forgetting though.)

To get NEP in A2/Hz you take your in-band noise power and divide by the bandwidth you integrated over. Take the square root of this to get A/Hz1/2 and apply your responsivity to get to the desired NEP in W/Hz1/2 for your bandwidth.

Its possible to do the calculation in fewer steps if you work out the math, but I think this breakdown makes it more clear what is going on.

The noise equivalent power (NEP) is a metric which expresses the minimum detectable power per square root bandwidth of a given detector. In other words, NEP is the input signal power that results in a signal-to-noise ratio SNR = 1 per a 1 Hz output bandwidth, the bandwidth here is a signal power bandwidth. The product of NEP and SNR for photonic systems is the total signal power per bandwidth.

To measure PSD of both signal and noise you need not provide "a pulsed illumination" in the measurement setup. The light source is 'on' for all the time of measurement with light, and 'off' for all the time of the noise measurement. Temporal dynamics is the result of both optical noise of the light source (photon noise) and electrical noise of the photodetector and the data acquisition circuit as these noises vary in the time domain. In the frequency domain, you measure the spectral distribution of these noises, and not of the light flashes. You need a fast ADC for these measurements; for measurements in the high-frequency end of your noise spectral range (tens or hundreds MHz) you may require an instrument called electrical spectrum analyzer.

For instructions on building a measurement setup, ask the suppliers of your photodetectors and lab instruments. To begin the process, I recommend you to study the guides of optical instrument manufacturers: NEP – Noise Equivalent Power by Thorlabs and Measuring the Noise Equivalent Power of a photodetector by Koheron.