Signal to noise calculation

[andy]

To my knowledge ignoring attributes like size, cost, flexibility, excitation wavelength, accessories, vendor support, and so on there are three scientific qualities of a raman system:

Signal:Noise Ratio (SNR), Wavelength Range, and Resolution.

According the chemical analysis book I cited elsewhere, the procedure is to directly measure SNR is to:

• Take two snapshots
• Calculate S bar by averaging the peaks in some region of interest
• Calculate sigma y by subtracting the two snapshots (so it is now all just background noise) and then finding the standard deviation in the peak region and dividing that by sqrt(2).
• SNR = (S bar) / (sigma y)

Does that sound right? I've seen SNRs quoted on raman spectrometer websites by themselves, but doesn't it also depend on variables like what chemical they tested it on, exposure time, number of images averaged, and peak region they selected? I also saw this paper for an experimental method of just measuring the limit of detection.

[Luc]

There are two variables that can be used to define the noise of a spectrograph which are often mixed up. Here is a bit of clarity in that mix:

Signal to Noise Ratio (SNR) which is technically how much noise is contained in every pixel of your data. To obtain it you typically take at least 10 spectra (ideally, much more) and perform the ratio between the average of the values (at the peaks apex for example) and their standard deviation.

When properly exposed (i.e. when you are close to sensor saturation), the SNR is directly proportional to the shot noise of the photons which is the square root of the number of photons received by the sensor (and transformed into electrons). You can estimate the maximum SNR of a single pixel using the sensor well size which is typically around 10ke- to 20ke- but can be much more than that (several hundreds of thousands of electrons). A typical camera pixel SNR is therefore around 100:1. In a spectrograph, it can be higher since we average columns of pixels. If all 'N' pixels of a column are illuminated with the same intensity, you increase the SNR by sqrt(N). Same if you average multiple images.

Poorly exposed pixels have SNR mostly dominated by the electronic noise of the camera. But who wants a signal that is poorly exposed in the first place ?

Dynamic Range is a bit like SNR but adds spatiality to the mix. Here you take the ratio between the average max peak height and the standard deviation of the baseline of the spectrum where there is no info. It gives a direct indication on "how much" a peak stands out of the baseline noise because you can assume that the temporal variation of a single pixel of the baseline is equal to the spatial variation of all pixels in the baseline at a given time t. In OpenRAMAN, I measured Dynamic Range > 10,000:1 which means the main peak is ten of thousand times stronger than the fluctuations of the baseline. Camera manufacturers will also give you the dynamic range (at the pixel level) of their sensor, usually expressed in dB. For instance, the camera used in OpenRAMAN has a DR of 69 dB. Again, due to averaging of columns we increase the achievable DR.

In conclusions:

If you would like to know if a given spectrum is readable enough, Dynamic Range is the way to go. On the other hand, if you'd like to quantify a peak height (e.g. concentration of some species), don't forget about the SNR because it'll tell you by how much your peak is fluctuating in time. I think when people talk about SNR in a spectrometer they are actually referring to DR; I did the same in the past but it's technically wrong unless you eventually mention "spatial SNR" in opposition to "temporal SNR".

[andy]

That is very interesting, thanks for clarifying that.

The book mentions that, and I didn't quite understand what it was saying until you broke it down, saying:

Unfortunately, the SNR is often calculated in the literaure as bar S/sigma b, which is a fundamentally different quantity from the SNR defined in Eq (4.1)

Where bar S is average peak high and sigma b is standard deviation of background.

I also noticed that the method ThorLabs uses is the dynamic range method. I'm interested how the relationships are related, if at all. Like you said it makes intuitive sense that SNR is useful for concentration, while DR is useful for untargeted analysis.

The book talks about "figure of merit" which is basically how good a spectrometer is. The equation is:

(Signal to noise) / sqrt((Power Density of laser ) * (Bd, known raman response of an analyte) * (Concentration/Density of analyte) * (measurement time))

Then it talks about how this is theoretically equal to:

sqrt(Ad/AL * Omega * Q * T * K)

Where Ad - Detection area, AL - Laser spot size, Omega "angle of collection", Q Quantum efficiency, T - transmission of optics (to my understanding, optical losses), K "geometric factor".

Which at first was a head scratcher to me, but I think I understand this to mean... if you are comparing two spectrometers, ignoring controllable factors like laser power and time, and uncontrollable factors like the raman response of a given chemical; what you are left with is just comparing much of the laser spot goes through to the spectrometer, optical losses, quantum efficiency, and the factor of geometry and how much of the response you collect (backscattering, etc).