Introduction

Diffraction gratings offer a significant advantage over prisms for selecting wavelengths in monochromators or spectrometers due to their much higher luminosity for the same resolution. However, unlike prisms, diffraction gratings introduce additional spectral orders that must be eliminated, a challenge particularly severe in the infrared region.

Solutions to Order Sorting

1. Prism Monochromator in Series: A low-resolution prism monochromator, paired with a high-resolution grating monochromator, can handle order sorting efficiently while retaining the grating’s luminosity.
2. Longwave-Pass Thin-Film Filters: Recently, several instruments have employed thin-film filters as a simpler alternative to prisms. These filters effectively block unwanted orders, particularly in the infrared region.

Diffraction Grating Principles

Grating Equation

The angles of incidence (\( \vartheta \)) and diffraction (\( \phi \)) for any wavelength (\( \lambda \)) in the \( m \)-th order are determined by:
\[
\sin \vartheta + \sin \phi = \pm m \frac{\lambda}{\sigma}
\]
where:
– \( \sigma \): Grating constant (groove spacing)
– \( m \): Order number

This relationship reveals the challenge: angles for a wavelength \( \lambda \) in the first order (\( m=1 \)) coincide with those for \( \lambda/2, \lambda/3 \), etc., in higher orders.

Energy Distribution in Orders

The energy distribution across orders is governed by:

1. Grating Design: Grooves are typically “blazed” or tilted to concentrate energy at a specific wavelength (blaze wavelength, \( \lambda_0 \)).
2. Power Distribution:
   \[
   I = I_0 \sin^2 \left[ \pi \nu \sigma \psi (\cos \alpha – \cos \beta) \right]
   \]
   where:
   – \( \alpha, \beta \): Angles of incidence and diffraction relative to the groove normal
   – \( \psi \): Angle between the grating and blaze normal
   – \( \nu \): Frequency
   – \( I_0 \): Incident intensity

Figure 15.11 illustrates power distributions for a typical grating.

Order Sorting Filter Design

Filter Requirements

Order sorting filters must ensure:

1. High rejection in stop bands to block unwanted orders.
2. Sufficient transmittance in pass bands for the desired wavelength range.

Efficiency Considerations

The effectiveness of filters depends on:

• Detector Sensitivity: Assumed flat for simplicity.
• Grating Efficiency (\( \epsilon_\lambda \)): Varies with wavelength.
• Source Output (\( E_\lambda \)): Approximated by Planck’s equation:
  \[
  E_\lambda = \frac{c_1}{\lambda^5 \left( e^{c_2/(\lambda T)} – 1 \right)}
  \]
  where:
  – \( c_1 = 3.74 \times 10^{-16} \, \text{Wm}^2 \)
  – \( c_2 = 1.4388 \times 10^{-2} \, \text{mK} \)

Stray Light Calculation

The stray light \( r_m \) due to the \( m \)-th order as a fraction of first-order energy is:
\[
r_m = \frac{\epsilon_\lambda T_\lambda}{\epsilon_{\lambda/m} T_{\lambda/m}}
\]
where:
– \( T_\lambda \): Filter transmittance

For \( N \) significant orders and total permissible stray light \( S \):
\[
T_{\lambda/m} = \frac{S}{N} \cdot \frac{\epsilon_\lambda}{\epsilon_{\lambda/m}} \cdot \frac{E_\lambda}{E_{\lambda/m}}
\]

Figure 15.12 provides normalized values for \( \lambda E_\lambda / (\lambda/m) E_{\lambda/m} \) to simplify calculations.

Filter Specification

Pass Band Determination

To cover a wavelength range \( \lambda_S \) to \( \lambda_F \):

1. Divide the range into \( n \) filters.
2. For each filter:
   – Transmission starts at \( (1 + \gamma)^i \lambda_F / 2^i \), where \( i = 1, 2, \ldots, n \).
   – Transmission ends at \( \lambda_F / 2 \).

The edge steepness parameter \( \gamma \) is calculated as:
\[
\gamma = \left( \frac{\lambda_S}{\lambda_F} \right)^{1/n} – 1
\]

Filter Tolerances

Allow for sloping edges to ensure manufacturability. Real filters deviate slightly from ideal rectangular transmission curves.

Example: Infrared Region (3–30 μm)

Filter Arrangement

For a grating blazed at \( 5 \, \mu \text{m} \):

1. Use five filters for efficient coverage.
2. Overlap transmission regions slightly to account for source and grating efficiency variations.

Leakage Control

Filter performance is assessed by calculating maximum allowable transmittance in stop bands (Figure 15.13).

General Rules for Filter Design

1. Filters including the first-order blaze wavelength require the tightest specifications.
2. Place blaze wavelengths near the shortwave pass region limit to reduce edge steepness.
3. For filters not covering the blaze wavelength, relaxed rejection criteria can simplify fabrication.

Conclusion

Order sorting filters are essential for achieving high spectral resolution in grating spectrometers. Proper design, considering grating efficiency, detector response, and stray light elimination, ensures optimal performance. Once specifications are defined, fabrication follows the principles discussed before, balancing practical constraints with optical precision.