It is worthwhile considering why interference filters are used in preference to other types of wavelength-selecting devices such as prisms and grating monochromators. The size and mechanical stability of the thin-film filter are powerful arguments in favor of its use.

In cases where space and weight are at a premium, such as in satellite-borne instruments, these advantages are likely sufficient. However, there is an even more compelling reason to adopt thin-film filters: their greatly increased potential energy grasp compared to dispersive systems.

For example, a thin-film filter with the same bandwidth as a grating monochromator can collect several hundred, or even thousands of times, the energy collected by the monochromator, provided the system is correctly designed around the filter. This section compares the interference filter with the diffraction grating, particularly from the perspective of potential energy grasp.

To compare energy-gathering properties, we assume:

1. Each component is used in an ideal system designed to maximize its energy-gathering powers.
2. The bandwidths of the systems are equal.
3. Dispersive components are used well within their limiting resolutions to avoid diffraction effects.
4. The illumination source has equal radiance in all cases.
5. The collecting optics fill the entrance apertures completely.

Under these conditions, energy grasp is computed for each component as a function of its area. Comparisons are then made based on these figures.

Comparison with Diffraction Gratings

Jacquinot analyzed the energy grasp of a diffraction grating, a prism, and a Fabry–Perot interferometer. He demonstrated that diffraction gratings have a clear advantage over prisms, with energy grasp advantages ranging from about 3 to 100 times depending on prism dispersion. This analysis primarily focuses on the interference filter versus the diffraction grating.

Jacquinot also compared the Fabry–Perot interferometer with a diffraction grating, showing that the interferometer offers 3 to 400 times the energy grasp of a grating of the same area. Interference filters, with cavity layers having refractive indices greater than unity (especially in the infrared), provide even greater energy grasp. The following analysis extends Jacquinot’s arguments to include cavities with indices other than unity.

Theoretical Analysis

Jacquinot considered a spectrometer comprising an input slit, a collector, a collimator, a dispersive element (e.g., a grating), and an output imaging element. The resolution is determined by slit width and dispersion. Maximum luminosity is achieved with equal entrance and exit slit widths, resulting in a triangular response function.

Let the source be monochromatic and of uniform radiance \( L \). The energy transmitted by the system is given by:
\[
E = LS\omega T
\]
where \( \omega \) is the solid angle subtended by either slit at the collector, \( S \) is the beam area at the collector, and \( T \) is the monochromator transmittance. For an exit slit of width \( \alpha_2 \) and length \( \beta_2 \):
\[
E = LST\beta_2\alpha_2
\]
The resolving power \( R \) is:
\[
\alpha_2 = \frac{\lambda D_2}{R}
\]
Thus:
\[
E = \frac{LST\beta_2\lambda D_2}{R}
\]

For a grating monochromator, angular dispersion \( D_2 \) is derived from:
\[
\sigma (\sin i_1 + \sin i_2) = m\lambda
\]
where \( \sigma \) is the grating constant, \( m \) is the order number, and \( i_1, i_2 \) are angles of incidence and diffraction. Assuming Littrow mounting (\( i_1 \approx i_2 \)):
\[
D_2 = \frac{\sin i_1 + \sin i_2}{\lambda}
\]
and the energy becomes:
\[
E = \frac{LT\beta_2 A}{R}
\]

For interference filters, the analysis follows Jacquinot’s, incorporating the concept of effective refractive index \( n^* \). For a cone of semiangle \( \Theta \):
\[
W_\Theta = \sqrt{2} W_0
\]
where \( W_0 \) is the bandwidth at normal incidence. From Chapter 8:
\[
\Theta^2 = \frac{2n^{*2}}{R}
\]
and the energy collected is:
\[
E = LAT \left(\frac{\pi}{4}\right)\omega
\]
For interference filters, the relative energy grasp compared to a grating is:
\[
\frac{E_{\text{filter}}}{E_{\text{grating}}} = \frac{\pi n^{*2} \beta^2}{4}
\]
Jacquinot estimated typical \( \beta \) values as 0.01 radians, with \( n^* \) in the visible region exceeding 1.5. The energy ratio ranges from 76 to 760. In the infrared (\( n^* \approx 3.0 \)), the ratio rises to 306–3060.

Interference filters offer significant energy grasp advantages over diffraction gratings. For the DHW filter type, these advantages are even greater due to higher effective transmittance in a cone of illumination.