The Fundamental Trade-Off
A spectrometer's entrance slit is a critical gatekeeper. A wider slit allows more light in (high throughput), which is great for detecting faint signals. However, it blurs the resulting spectrum, worsening the ability to distinguish between closely spaced wavelengths (low resolution). Conversely, a narrow slit provides sharp, detailed spectra (high resolution) but might starve the detector of light. The optimal design finds the perfect balance by matching the slit's image on the detector to the physical size of its pixels. This infographic walks through the derivation of that perfect width.
High Throughput
(Wide Slit)
High Resolution
(Narrow Slit)
Step 1: The Goal - Spectral Resolution (螖位)
Our primary design goal is the spectral resolution (螖位), the smallest difference in wavelength the spectrometer can reliably distinguish. To register this separation, the light from two wavelengths, 位 and 位+螖位, must land on different pixels on our detector. The "Two Pixel Rule" is a common design standard: the image of 螖位 should be separated by at least two pixels for a clear distinction.
Minimum Resolvable Image on Detector (螖d):
螖d = 2 脳 (Pixel Size)
Step 2: Linear Dispersion
Linear dispersion describes how the optical system spreads different wavelengths across the detector plane. It connects the desired spectral resolution (螖位, in nanometers) to the required physical separation (螖d, in millimeters) on the detector via the imaging lens focal length (LF) and grating properties.
Step 3: Imaging the Slit
In an infinity-corrected setup, the system images the entrance slit (width 'w') onto the detector. The size of this image is determined by the ratio of the imaging lens (LF) and collimating lens (LC) focal lengths. To optimize performance, we set the slit's image size equal to the minimum resolvable dimension, 螖d.
Step 4: The Final Equation
By setting the equations for 螖d from Step 2 and Step 3 equal to each other, we can solve for the optimal slit width 'w'. Notice that the imaging focal length, LF, conveniently cancels out, simplifying the final design equation.
Interactive Calculator
Use the final derived equation to calculate the optimal slit width for your own design. Adjust the parameters below to see how they impact the result.
Optimal Slit Width (w)
26.5 碌m
Parameter Sensitivity Analysis
How does changing a single parameter affect the required slit width? This chart shows the percentage change in the optimal slit width 'w' when one input is increased by 20%, keeping all others at their default values (螖位=1nm, LC=75mm, 600 gr/mm, 尾=19.5掳). A taller bar indicates a greater impact.