Obtaining photoelectron counts from known radiance

Question

I am attempting to make a rough estimate of the counts seen on a CCD spectrometer assuming I know the spectral radiance of a calibration source. I understand that this is not the same thing as radiometrically calibrating, but I want to make a calculation of the incident light and just come up with a rough estimate of how many photons strike a detector pixel. I know the quantum efficiency, the gain, and the diffraction efficiency, and I have dimensions for the pixel size if that is useful.

For my experimental setup, there is:

1. the calibration source, and integrating sphere that emits light through an aperture. I know the dimensions of this.

2. The spectrometer, with known slight size and field of view, apparent quantum efficiency at all wavelengths, as well as diffraction grating efficiency values. There is no lens or fiber optics cable in my setup.

3. A fixed distance between the spectrometer and radiance source, and I am making the assumption that the optical axes are aligned, and that the field of view of the spectrometer is completely encapsulated by the radiance source.

FIRST - How can I make an estimate of the power incident on the spectrometer slit?

I have some curve that provides radiance as a function of wavelength. It looks like this:

Knowing the area of the slit, and the f/# or NA, I should be able to calculate the throughput (etendue ) of the spectrometer, correct? If T is throughput:

$$ T = FA_{slit}\\ T = 2\pi(1-\cos(\theta))wl\\ $$

Where $\theta$ is the half angle of the slit field of view and $w$ and $l$ are the width and height of the spectrometer slit. If I multiply the radiance $L$ at some wavelength by this throughput, is that the power incident on the lens at a particular wavelength, or am I missing some geometric factor from the source? The units appear to work, so I am confused, $$P = [W/\mu] = TL = [m^2 sr][W/m^2\mu sr]$$. I believe this calculation assumes that the source is an 'extended' source.

From here, I would apply instrument efficiency values (loss due to diffraction, mirror reflectance, quantum efficiency) and integrate the radiance over the width of a pixel (in wavelength units). This would give me the final power incident on the pixel which can be converted to an incident photon rate at a specific wavelength. Finally, applying the Gain of the A/D converter and multiplying by some integration time should yield a rough estimate of the number of photons recorded.

SECOND - Knowing the instrument has some spectral resolution, but the pixels have some wavelength width, what is the best metric to integrate the radiance value over?

Any suggestions or references you may be able to provide are greatly appreciated, thank you.

Answer 1

This is a rather tricky problem. Unfortunately, there is a lot that is unknown about your source, and you are not using the optical system for which your spectrometer likely is designed. Typical grating spectrometers have an optical system designed for light which is diverging from the point of the slit, ideally at the divergence angle to fill the mirrors/grating. Since you are not using a lens or a fiber, you won't have a focus at the slit, nor will you likely be ideally filling the optics. In the "horizontal" direction (in the plane of the spectrometer) you will probably under-fill the optics, but in the "vertical" direction you will probably over-fill them. Normally, this is no big problem, since you could just try it in the lab and see what kind of throughput and resolution you get (resolution could also affect the number of counts per pixel if you have any sharp features in your spectrum). But since you are trying it estimate the counts a priori, all of these inefficiencies become rather complicated to include quantitatively.

But the most important piece here, as you may recognize, is the intensity of the light at the slit. For this, it will be critical to know the geometry of the emitter. Unfortunately, it's not enough to know the amount of radiance; you also need to know how the light is propagating. Naturally, for diverging light, the farther away you are from the emitter, the weaker the intensity at the entrance to your spectrometer. Can your source be estimated as a homogeneous, collimated plane wave? Or is it diverging/converging significantly? Is there a known optical system associated with the source, such as a parabolic/elliptical mirror or lens? Without this information, you will not be able to estimate the detector counts. But even with this information, given the uncertainties of the problem, I wouldn't be confident that you could get more accurate than an order-of-magnitude estimate.

If possible, I highly recommend adding a single bi-convex lens to your optical system (possibly achromatic for broadband use, or you could use an elliptical silver mirror). If you put the source and the spectrometer slit at $2f$ distance from the lens on either side, you will be imaging your source onto your slit, and it will remove a lot of the uncertainty in your estimate.

Good luck!