Light incident on a transmission grating is often normal to the back surface of the grating (α = 0; see Fig. 1), in which case the familiar grating equation reduces to

mλ = d sinβ

Here m is the (integral) diffraction order (usually |m| = 2), λ is the wavelength, d the groove spacing and β the angle of diffraction (measured from the normal).

The peak efficiency of a blazed (triangular-groove) transmission grating occurs when the refraction of the incident beam though the mini-prism that constitutes a groove lies in the same direction as the diffraction given by the grating equation. Unlike reflection gratings, the groove angle is much larger than the blaze angle for a transmission grating, since the phase retardation doubles upon reflection but is multiplied by n-1 for a transmission grating, where n is the refractive index of the grating medium.

Applying Snell's law to the interface between the groove facet and air,

n sinχ = sin(χ+β)

Combining this relation with the grating equation, yields the relationship between the blaze angle βB and the groove angle χ:

tanχ = sinβ_β / (n - cosβ_β)

For small groove angles and for λ << d, a useful approximation to Eq. (3) relates the blaze wavelength of a reflection grating to that of the corresponding transmission grating. For transmission gratings used in air or vacuum, the ratio of its blaze wavelength (for normal incidence) to that of the equivalent reflection grating (used in Littrow) is

λ_β (trans) / λ_β (ref) ≈ (n - 1) / 2

where  is the blaze wavelength of the corresponding reflection grating. This approximate formula, rarely in error by more than ten percent, simplifies conversion of information in the Grating Catalog from reflection gratings to transmission gratings. Taking n ≈ 1.6, which is true for most transmission gratings, yields

λ_β (trans) / λ_β (ref) ≈ 0.3

A corollary to this approximation is that, for n ≈ 1.6, the grooves of a transmission grating are about 10/3 times as deep as those of the corresponding reflection grating. For small diffraction angles (i.e., diffraction near the normal), this ratio holds for the angle χ as well.

The choice of groove angle for transmission gratings is limited by total internal reflection effects:

χ ≤ arcsin (1/n)

For n ≈ 1.6, this yields about 40° as the upper limit on χ, though in many cases the effective limit is somewhat lower. This means that transmission gratings cannot be used for high-dispersion applications.

All standard Richardson gratings have a part number according to the following format:

AA BBB CC DD - EEE x

• AA indicates the type of grating (e.g., diced, plano, grism).
• BBB indicates the size of the grating substrate.
• CC indicates the substrate material
• DD indicates the type of coating
• EEE indicates the master grating groove parameters
• x indicates the master grating type (e.g., ruled, holographic, echelle)

Please see Diffraction Grating Part Number System for additional information.

Newport is pleased to discuss special and unusual applications that are not addressed by our build to order catalog diffraction gratings.  In some instances, none of the hundreds of master gratings we have in stock meet specifications, so a new master may be required. Please see Custom Diffraction Gratings for additional information on our capabilities.