In our last post, we explored the physical layouts of Czerny-Turner and Lens Grating Lens (LGL) spectrometers, which visually represent the two primary conventions for the grating equation. Now, we'll dive deeper into the first convention—where the total deviation angle () is fixed—to mathematically derive the angle of incidence, $\alpha$, and the angle of diffraction, $\beta$.

This derivation is crucial for designing a Czerny-Turner system, as it allows us to precisely calculate the position of the detector array to capture the full spectrum of light. The grating equation is our starting point:

Where m is the diffraction order, $\lambda$ is the wavelength, and is the grating period.

We also have our fixed geometry constraint:

We can use the sum-to-product trigonometric identity to simplify this equation. The derivation is a two-part process depending on whether the incident and diffracted beams are on the same side or opposite sides of the grating normal.

Case 1: Angles of Incidence and Diffraction on the Same Side of the Normal

This is the typical configuration for a Czerny-Turner spectrometer. Both the collimating mirror and focusing mirror are placed on the same side of the grating normal.

The grating equation becomes:

Using the trigonometric identity sinA + sinB= 2sin((A+B)/2) cos((A−B)/2), we can substitute our angles:

Since $\Phi =\alpha +\beta$, we can substitute that in:

Now, we can solve for α. From our geometry constraint, β=Φ−α. Substituting this into the equation:

Rearranging the terms to isolate α, we get:

Finally, we find the equation for α:

Once you have solved for α, you can find β by using the original constraint: β=Φ−α.

Case 2: Angles of Incidence and Diffraction on Opposite Sides of the Normal

While less common for a full Czerny-Turner, this geometry can be relevant in certain off-axis designs. In this case, the angles are on opposite sides of the normal, so the grating equation has a subtraction term:

Using the identity sinA−sinB=2cos((A+B)/2)sin((A−B)/2), we substitute our angles:

Since our constraint is still Φ=α+β, the equation becomes:

Substituting β=Φ−α and rearranging to solve for α:

Which gives us the equation for α:

And again, once α is known, β can be found using β = Φ−α.

Understanding these derivations is a critical part of the optical design process. It provides the mathematical foundation for selecting the correct components and placing them precisely to achieve the desired spectral range and resolution for our JASPER spectrometer.