In our previous blogs, we tackled the initial steps of designing a spectrometer: fixing the geometry $\varphi =\alpha +\beta$ and then calculating the angles of incidence (α) and diffraction ($\beta$). We even created a simulator that showed how a varying $\alpha$ could achieve a desired spectral range. Details at 

https://hackaday.io/project/202421-jasper-vis-nir-spectrometer/log/243109-spectrometer-design-choosing-the-right-geometry-for-vis-nir-spectroscopy 

https://hackaday.io/project/202421-jasper-vis-nir-spectrometer/log/243174-spectrometer-design-part-3-deriving-alpha-and-beta-angles
https://hackaday.io/project/202421-jasper-vis-nir-spectrometer/log/243185-simulating-diffraction-grating

While a rotating grating-based monochromator offers supreme flexibility—allowing for variable spectral resolution or even the use of multiple gratings on a turret—it is not the optimal choice when spectral acquisition speed is paramount.

For high-speed spectroscopy, the ideal solution is a spectrometer with a stationary grating and a sensor array detector. This design has no moving parts and relies on a constant angle of incidence ($\alpha$). The entire spectral range is imaged simultaneously onto the array, where each pixel corresponds to a specific wavelength.

Let's explore the key equations that govern this design.

The Grating Equation for a Fixed-Grating System

The fundamental grating equation is: 

Here, m is the diffraction order, λ is the wavelength, and d is the grating groove spacing. For a fixed-grating system, α is constant. We can derive the equation for the diffraction angle β as a function of wavelength:

The total spectral span captured by the detector is the difference in diffraction angles between the maximum and minimum wavelengths of interest, λmax and λmin.

From Angular Dispersion to Physical Dimensions

The spectral span δ is an angular measurement. We must convert this into a physical length that can be imaged onto the detector. This is where the imaging lens and its focal length (L_F)come into play. The total length of the detector (d_D) required to capture the entire spectral span is given by:

This equation highlights the trade-off between the detector's physical size and the focal length of the imaging lens. The span is limited by these two parameters.

To determine the smallest change in wavelength (Δλ) that the spectrometer can resolve, we need to consider the angular dispersion, which describes how the diffraction angle changes with a change in wavelength.

For a very small change, we can approximate this relationship as:

This angular change, Δβ, must be large enough to be detected by the sensor. The smallest resolvable image dimension (Δd) on the detector corresponds to the spectral resolution. This allows us to select a detector with an appropriate pixel size. The linear dispersion is given by:

The physical distance between two adjacent resolvable wavelengths is Δd. To distinctly record two close spectral lines, they must be mapped onto two adjacent pixels.

In practice, a design workflow is typically reversed. We start with a fixed pixel size and sensor length from available detectors and work backward to calculate the achievable spectral resolution. If the result doesn't meet the design goals, we can iterate on other parameters like the angle of incidence (α) or the grating's groove density.

This shows that even with a fixed grating, there are many parameters to tune for an optimal design. In the next blog, we can create a simulator to compute the pixel size required for a given spectral resolution and span, bringing these equations to life