After struggling to design a spectrometer in Czerny turner configuration using spectrometer design guide https://ibsen.com/resources/spectrometer-resources/spectrometer-design-guide/) and tool provided by Ibsen Photonics (https://ibsen.com/wp-content/uploads/Spectrometer.html), I uncovered a fundamental conflict between common spectrometer geometry and the raw physics of diffraction. This deep dive led to a surprising conclusion: for hobbyists and professionals building Czerny-Turner (CT) spectrometers, trying to calculate the Angle of Incidence (α) based on the fixed deviation angle (φ) is a recipe for failure with common high-resolution gratings.

The core issue? The seemingly simple inverse trigonometric functions, arcsin and arccos, impose severe, hidden limits on your choice of grating and wavelength.

The Grating Equation and the Two Geometries

The physics of diffraction is governed by the Grating Equation:

where:

• G is the groove density (grooves/mm)
• λ is the center wavelength
• α is the Angle of Incidence (AOI)
• β is the Angle of Diffraction (AOD)
• The +/- sign depends on whether alpha and beta are on the same or opposite sides of the grating normal.

The confusion arises from how the total Deviation Angle (φ)—the fixed angle between your collimating and focusing optics—is defined:

1. The Littrow-like Geometry (The Ibsen Tool Approach)

In compact and highly optimized spectrometers (like Ibsen's), the system operates close to the Littrow condition (alpha is approximately beta). The total deviation angle phi is defined as the difference:

When this definition is combined with the grating equation (sin(α) + sin(β) = G*λ), the resulting formula for the Angle of Incidence (α) involves an arcsin function:

The Practical Advantage: For this formula to work, the argument of the arcsin must be <= 1. This sets the limitation: G*m*λ<= 2 * cos(φ/2). Since φ is typically small (30 degrees), cos(φ/2) is close to 1. For φ=30 degrees, this limit is G*λ<= 1.932, which is generous and accommodates almost any commercial grating (e.g., 1200 g/mm at 550 nm gives G*λ=0.66, which works perfectly). This is why the Ibsen tool is so practical for Littrow-like geometry 

2. The Classic Czerny-Turner Geometry (φ = α + β)

In the traditional CT setup, both the input and output rays are on the same side of the Czerny-Turner axis, leading to the simple geometric sum: (φ = α + β)

When this is combined with the same side of normal grating equation (sin(α) + sin(β) = G*λ), the resulting formula for alpha involves an arccos function (derived by exploiting the trigonometric sum-to-product identity, as seen in this derivation log):

The Hidden Czerny-Turner Limitation

This arccos-based formula is where practical design collides with math. For the arccos to return a real angle, its argument must be <= 1. This yields a dramatically tighter limitation:

Let's look at the numbers for a very common fixed deviation angle, φ=30 degrees:

This means that for a φ=30 degrees CT spectrometer using the φ=α+β definition, the product of G and λ MUST NOT exceed 0.5176 mm

Why Your High-Resolution Grating Won't Work (The Real-World Test)

Consider common commercial gratings:

If you attempt to design a φ=30 degrees Czerny-Turner spectrometer (where φ=α+β) using a 1200 g/mm grating to look at green light (550 nm), the required G*λ product is 0.66 mm. Since 0.66 is greater than the 0.5176 limit, the arccos formula fails, and no real angles α and β exist that satisfy both the grating equation and the geometric sum!

The Superior Design Strategy: Fixing α

If you are using a fixed-angle Czerny-Turner setup and high-density gratings, the failure of the inverse trig functions shows that you cannot treat α as the unknown.

The most practical design approach for a CT system is to fix α (Angle of Incidence) and then calculate β (Angle of Diffraction) directly using the simplest form of the grating equation.

 1. Define α: Choose a reasonable AOI, typically 10 to 20 degrees for good performance (or even 0 degrees if possible, though that introduces aberrations).

 2. Calculate β: For your center wavelength (λ_c), calculate the required diffraction angle

 3. Check Geometry (φ): Now, calculate the actual deviation angle required by this combination:

 4. Align Optics: Align collimating and focusing mirrors/lenses to match this φ_required 

By adopting this strategy, you sidestep the fatal G*λ limitations of the arccos and arcsin formulas, allowing you to use high-resolution gratings with typical visible light while guaranteeing a real, physically implementable solution.