This article provides a basic overview of the capabilities of transmission gratings optimized for molecular spectroscopy.
These gratings offer near 100%, polarization-insensitive efficiency over a broad wavelength range and are ideal for applications
such as Raman spectroscopy where the available light intensity from the sample is very low. The design, manufacturing, and
performance of the two most commonly used types of transmission gratings — the surface relief transmission grating (SRTG)
and the volume phase hologram (VPH) — are described and examples of the two grating types are discussed.
Most diode array–based diffractive spectrometers for the ultraviolet (UV), visible (vis), and near-infrared (NIR) spectral
ranges use reflection-type diffraction gratings and are generally optimized for high resolution rather than high optical throughput.
During the past 5–10 years, transmission-type gratings have been developed that provide near 100% efficiency for any polarization.
These gratings have found widespread use for Raman spectroscopy and astronomy, where very little light is available, but elsewhere
the awareness of the distinct benefits provided by transmission gratings seems quite limited. The purpose of this article
is to provide a basic overview of the capabilities of transmission gratings optimized for molecular spectroscopy. Specifically,
we will describe the two most commonly used types of gratings: the surface relief transmission grating (SRTG) and the volume
phase hologram (VPH).
Transmission Grating Basics
This section describes the three key performance parameters of a transmission diffraction grating: the angular dispersion,
the diffraction efficiency, and the bandwidth of the grating.
Figure 1: Transmission grating structure. Key grating parameters: Λ = grating period; t = grating thickness; nAVE = average refractive index; and Δn = refractive index modulation.
Angular Dispersion: The Grating Equation
A transmission (or phase) grating consists of a region with a periodic modulation of the refractive index as shown in Figure
1. The period of the index modulation is on the same order of magnitude as the wavelength, and the form of the modulation
can be sinusoidal, rectangular, triangular, or even more complex. From the plane wave ray tracing drawn on Figure 1 it can
be seen that the plane waves add up in phase for exit angles β that fulfill the following criteria:
Equation 1 is the general grating equation that is valid for any grating structure. The term m is the order of diffraction, and in the remainder of this article we will consider first-order diffraction (m = 1) only.
For a certain spectrometer design, the center wavelength as well as the period Λ (or groove density G = 1/Λ) for the grating are often given parameters that cannot be freely chosen. Furthermore, a spectrometer is used in a
range of wavelengths around the center wavelength and equation 1 determines the angular dispersion of this wavelength range.
Although the grating equation determines the angular direction (β) of the diffracted light as a function of wavelength (λ),
it does not say anything about the diffraction efficiency. The calculation of the diffraction efficiency for a transmission
grating geometry generally requires complicated numerical methods such as rigorous coupled wave theory (1) for solving Maxwell's
equations. However, Baldry and colleagues (2) approximated analytical equations based on Kogelnik's original work from 1969
(3) to provide a basic relationship between the physical parameters of the gratings and the diffraction efficiency and bandwidth
of the grating. This analysis is based on a pure sinusoidal index modulation and is surprisingly accurate for a certain range
of gratings. Based on this analysis, it can be shown that a grating's diffraction efficiency, for a fixed grating period and
center wavelength, depends on three factors:
- the average refractive index (n
- the thickness of the grating (t)
- the index modulation (Δn)
Figure 2 illustrates the effect of the average refractive index. The figure shows the diffraction efficiency versus grating
thickness for three different average indices and constant index modulation. First of all, the diffraction efficiency of both
the transverse electric (TE) polarized and the transverse magnetic (TM) polarized wave oscillates between 0% and 100% efficiency
with varying grating thickness. However, while the period of the oscillation of the TE diffraction efficiency is almost independent
of the average index, the oscillation of the TM diffraction efficiency depends strongly on the average index. For certain
values of the average refractive index (for example, 1.285 in Figure 2), the TE and TM can peak at the same grating thickness.
This situation is naturally very desirable because it provides a polarization independent efficiency around the center wavelength.
Figure 2: Diffraction efficiency for TE and TM polarization versus grating thickness for three different average indices in
the grating area. Fixed grating parameters: λc = 840 nm, Λ = 566 nm, and α = β (Littrow configuration).
The effect of the refractive index modulation is illustrated in Figure 3 for the case of a polarization-sensitive design (average
index = 1.200) and a polarization-insensitive design (average index = 1.285). From Figure 3 it can be seen that the oscillation
frequency of the TE and TM waves increases with increasing index modulation but the relative peak position of TE and TM stays
Figure 3: Diffraction efficiency versus grating thickness for average indices of 1.2 (left column) and 1.285 (right column)
and for three different index modulation strengths. Fixed grating parameters: λc = 840 nm, Λ = 566 nm, and α = β (Littrow configuration).
- The average refractive index controls whether polarization independent ~100% diffraction efficiency can be obtained.
- The index modulation controls the ideal thickness of the grating.
As will be demonstrated in the next section, such nearly 100% efficient, polarization-independent gratings can be produced
as both SRTGs and VPHs. The high polarization-independent efficiency is what makes transmission gratings so powerful.
So far, we have only considered the grating efficiency at a single wavelength. However, a spectrometer needs to analyze a
range of wavelengths around the center wavelength. From the article by Baldry and colleagues (2) the full width at half maximum
(FWHM) bandwidth of the grating can be calculated by
It has been assumed that the grating is used in the Littrow configuration where α = β. From equation 2 it can be seen that
the bandwidth of the grating is inversely proportional to the grating thickness t.