2.4.1.   Testing of Prisms and Diffraction Rulings

The Twyman–Green interferometer is a very useful instrument for testing prisms. Its
application for testing the accuracy of the 90^o angle between two of the faces of a
right angle (Porro) prism, a roof (Amici) prism, or a cube corner prism is especially
interesting. As explained before, the relative rotation or reversal of the wavefronts
should be corrected, as shown in Figure 2.16, if a gas laser is not used. The
arrangements in Figure 2.17 can be used when a gas laser source is employed.

A very good cube corner prism will give rise to an interferogram like that shown in
Figure 2.18. The fringes are straight throughout the aperture. A cube comer prism
with angular errors produces an interferogram such as that shown in Figure 2.19, in
which the straight fringes abruptly change their direction. Thomas and Wyant (1977)
made a complete study of the testing of cube corner prisms.

Figures 2.20 and 2.21 show similar situations for a right-angle prism of no error
and of some angular error, respectively. If, in addition to angle errors, the surfaces are
not flat or the glass is not homogeneous, an interferogram with curved fringes is
obtained. When a right angle or porro prism is tested in the retroreflective configuration
and the surface flatness as well as the 90^o angle is correct, the fringes look

straight and parallel as in Figure 2.20. If the right angle has an error, the fringes look
like those shown in Figure 2.21 and can be manipulated to look like those in
Figure 2.22. We describe here a brief method for obtaining the angular error in a
right angle prism. If 2L is the width of the face of the prism, π/2 ± ε is the angle of the
prism, d is the distance between two successive fringes, k is the deviation of the fringe
from the straight fringe after bending, n is the refractive index of the prism, and λ is
the wavelength used. As shown in Figure 2.23, the error is given by

where α is the angle between the two exiting wavefronts. For example, for a prism of
100 mm face width and k /d = 0:25, the error ε of the 90^o angle is about 1 s of arc. In
regard to the sign of the error, the hot rod or finger procedure described before can be
used.

Luneburg (1964) showed that the angular error ε in a roof face of a prism is

where n is the refractive index of the material, α is the angle between the two exiting
wavefronts in a single pass through the prism, θ is the angle between de roof edge and
the incident beam, and m is the number of times the lights is reflected on the roof face.
For the arrangements shown in Figs. 2.15 and 2.16 we have the values in Table 2.1.
The angle α is determined from Eq. (2.32), but with the interferometer adjusted in
such a way that all the fringes in one of the faces are eliminated.

A dispersive prism can also be tested as shown in Figure 2.24(a). This arrangement
of smoothly changing inhomogeneities in the glass may be compensated for by
appropriately figuring the faces. An axicon may be tested in a Twyman–Green
interferometer using the method described by Fantone (1981) as well as reflaxicons
(Hayes et al., 1981).

In 1935, Bisacre and Simeon suggested a method whereby a diffraction grating
could be tested by means of a Twyman–Green interferometer. Unfortunately, they
never published their work (Candler, 1951). They used the arrangement shown in
Figure 2.24(b). The interferometer is initially adjusted to obtain horizontal fringes in
the first order. Then the grating is rotated to pass to the third order, in which the
ghosts, if any, are stronger. If there are ghosts and a tilt about an axis along the grating
chromatic dispersion is introduced, the fringes have a sawtooth appearance. When
the spacing between the horizontal fringes is increased by removing the tilt, the teeth
become larger and larger until they form a system of vertical fringes due to the
inteference between the zero order and the ghost wavefront. Using this interferometer,
Jaroszewics (1986) has also tested the spacing error of a plane diffraction
grating.