You've likely seen spectrometer spans like 400-800 nm or 700-1100 nm, where λ₂ is less than 2λ₁. This might make you wonder why the simplest spectrometers are often limited to a range where λ₂≤ 2λ₁? This phenomenon is a direct result of how a diffraction grating works, and understanding it leads us to a key concept in optics: the Free Spectral Range (FSR).

A diffraction grating is a surface with a series of parallel lines or "slits" that diffract light. The core principle that governs this process is the Grating Equation:

Here, m is the spectral order (an integer), λ is the wavelength of light, d is the grating constant (the distance between the slits), α is the angle of the incident light, and β is the angle of the diffracted light.

For a single wavelength, there are many possible angles that satisfy this equation, corresponding to different integer values of m. These different values of m are called spectral orders. For example, consider two examples: light has an angle α of 15° and a diffracted angle β of 20°. For a grating constant (d) of 1 µm, if the spectral order (m) is 1, the wavelength is 601 nm. If m is 2, the wavelength is 300.5 nm. This is where the problem of spectral overlay arises, as shown in the figure

Deriving the FSR

The Free Spectral Range (Δλ) is the largest range of wavelengths in a given spectral order that does not overlap with the next adjacent order. In other words, it’s the spectral bandwidth that is "free" from overlap.

To derive the equation for FSR, let's consider the point where the spectral orders begin to overlap. This happens when a wavelength λ_m,max in a given order m is diffracted at the same angle as a wavelength λ_m+1,min in the next order, m+1. Since they are diffracted at the same angle, we can set their grating equations equal to each other (for a fixed incident angle):

This simplified form, often written as FSR=λ/m, shows that the FSR is inversely proportional to the spectral order. This means that as you use a higher order (m=2,3,...), the spectral range that is free from overlap gets smaller.

The Role of Order Sorting Filters

Because of this spectral overlap, a spectrometer may require an order sorting filter to ensure that only the desired spectral order is passed through to the detector. These are typically band-pass filters that are carefully selected to block wavelengths from adjacent orders, allowing the spectrometer to provide a clean, unambiguous spectrum within its FSR.

Understanding FSR is crucial for anyone working with spectrometers and is a key factor in designing an optical system that can capture a clean and accurate spectrum.