In our ongoing JASPER VIS-NIR spectrometer project, we have worked through the core components: fixing the geometry, selecting the grating, and deriving the focal lengths for the collimating (L_C) and imaging (L_F) lenses. The final piece of the optical puzzle is determining the optimal entrance slit width (w).

The slit width is critical because it directly controls the amount of light entering the system (the optical throughput) and also dictates the final spectral resolution. We need a slit that is wide enough to capture sufficient light but narrow enough not to degrade the resolution we designed the system for.

To find the optimal slit width, we must first recall the minimum required image size on our detector array.

Step 1: Minimum Resolvable Image Dimension (Δd)

The goal of any spectrometer is to distinctly separate two wavelengths that are very close to each other. This minimum difference in wavelength is our desired spectral resolution, Δλ.

For the spectrometer to register this change, the image of Δλ must be separated by at least two pixels on the sensor array. This means the minimum resolvable image dimension (Δd) must be equal to twice the pixel width.

The relationship between the change in wavelength (Δλ) and the resulting physical separation on the detector (Δd) is governed by the linear dispersion of the system:

Note: The angular dispersion, dβ/dλ, which is the basis for this linear dispersion equation, was derived in detail in Part 6: Angular and Linear Dispersion https://hackaday.io/project/202421-jasper-vis-nir-spectrometer/log/243271-angular-and-linear-dispersion

where:

• m is the diffraction order (usually 1).
• L_F is the focal length of the imaging (focusing) lens.
• d is the grating groove spacing.
• β is the angle of diffraction for the wavelength being resolved (often taken at λmin).

Solving for the smallest resolvable image dimension Δd at the desired spectral resolution Δλ:

For optimal performance, this dimension Δd should be set to match the physical requirement of the detector:

Step 2: Deriving the Optimal Slit Width (w)

In an infinity-corrected optical setup—where the collimating lens (L_C) and the imaging lens (L_F) are used—the slit width (w) is imaged onto the detector plane. The relationship between the object size (w) and the image size (Δd) is simply the ratio of the focal lengths of the two lenses:

We want the image of the slit to be exactly equal to our minimum resolvable image dimension (Δd) to ensure we utilize the maximum optical power without sacrificing resolution.

Now, we solve for the optimal slit width w:

Substituting the expression for Δd from Step 1 into this equation:

Notice that the focal length of the imaging lens,L_F, cancels out, which significantly simplifies the final equation for the optimal slit width:

This final equation elegantly links the physical slit width to the core design parameters: the desired spectral resolution (Δλ), the grating characteristics (m and d), and the focal length of the collimating lens (L_C). By setting the slit width according to this derivation, we achieve a system where the spectral resolution is perfectly matched to the detector's pixel size, thereby optimizing both light throughput and resolution.

In the next part, we will use all these derived equations to plug in our target values and finalize the physical dimensions of the JASPER spectrometer.