At the core of the JASPER VIS-NIR spectrometer is a diffraction grating, a component with thousands of microscopic grooves. When light hits this grating, it is diffracted, with each wavelength bent at a different angle. This is governed by the Grating Equation:

Here, λ is the wavelength of light, $d$ is the spacing between the grating grooves, $m$ is the diffraction order, $\alpha$ is the angle of incidence, and $\beta$ is the diffraction angle. For a given setup, $d$, $m$, and $\alpha$ are constant.

The measure of how well the grating separates different wavelengths is called angular dispersion, defined as the change in diffraction angle (β) for a small change in wavelength (λ). We can find this by differentiating the grating equation with respect to λ:

This simplifies to:

Rearranging for the angular dispersion, we get the corrected equation:

A higher value for dβ/dλ means that two very close wavelengths will exit the grating at noticeably different angles, which is exactly what we want for a high-performance spectrometer

From Angle to Position: The Role of Linear Dispersion

After the light is separated by angle, it passes through a lens that focuses the diffracted rays onto a detector plane, such as a CCD or CMOS sensor. 

This lens transforms the angle (β) into a specific position ($x$) on the detector. The relationship is given by:

where f is the focal length of the focusing lens. This equation is accurate for a flat detector plane normal to the central ray.

To understand how a change in wavelength relates to a change in position on the detector, we need to combine the two concepts. The key metric for a spectrometer is its linear dispersion, dx/dλ, which tells us how many nanometers of wavelength are spread across one millimeter of the detector

Using the chain rule, dλ/dx=(dλ/dβ)⋅(dβ/dx). We can derive the expression for linear dispersion:

This equation is a cornerstone of spectrometer design. A lower value of dλ/ corresponds to a higher resolution, meaning the spectrometer can distinguish between very closely spaced wavelengths. This is achieved by using a grating with a small groove spacing (d), operating at a higher diffraction order (m), or using a lens with a longer focal length (f)