Previous discussion centered on *isochromatic* fringes,
*i.e.*, fringes caused by the extinction of various
wavelengths or colors when polychromatic light is passed through
birefringent material. If the birefringence is linearly related
to the stresses in the material, these fringes can be used to
calculate the intensity of (s_{1}
- s_{2}), or equivalently the
maximum shear stress, at any point in the field of view. It is,
however, difficult to interpret the colored fringe pattern when
it results from polychromatic light. The overlap of wavelengths
in different orders causes the pattern repetition to be hard to
observe, since the colors do not repeat uniformly. This problem
can be mitigated by the use of monochromatic light, especially
light of the wavelength equal to the so-called *tint-of-passage*.
In this case, sharp narrow fringes can be seen and easily
counted.

However, these isochromatic fringes only give the magnitude of
stress over the field. Stress is a vector, and so it is necessary
to determine the direction of the stress at each point. This
determination requires a different set of fringes, called *isoclinic*
fringes. To understand these fringes, it is necessary to consider
the basic equation for a linear dark-field polariscope:

Here *E* is the monochromatic light wave amplitude
emerging from the linear dark-field polarizer, f is the angle between the orthogonal axes
of the birefringent material and the axes of the
polarizer/analyzer pair, *n*_{1} and *n*_{2}
are the two refractive indicies of the material, *n*_{0}
is the index of the surrounding medium (almost always air), *d*
is the thickness of the birefringent model, and *v* is the
velocity of light in the model. It is assumed that the coordinate
axes of the model are chosen with *z* along the direction
of light flow, and *x* and *y* two orthogonal
directions in the plane of the model. Frequently the *x*-axis
is taken to be along the direction of the applied load.

Looking at the first two sine terms, it is clear that there are two conditions under which light transmission will be cancelled:

- f = mp (the polarizers are aligned with the principal axes of refractive index, or stress)
- (
*n*_{1}-*n*_{2})*d*/*n*_{0}= ml (the relative retardation is an integer multiple of the wavelength)

Cancellation Fringes due to the first cause are called *isoclinics*,
while those resulting from the second cause are *isochromatics*.

It is by using the isoclinics that the direction of the stress can be determined. If the polarizers are rotated slowly, all the points on the image where the birefringent axes are aligned with the polarizer/analyzer axes turn black (the isoclinics). Thus rotating the polarizers and watching the isoclinics will show the direction of the stress components everywhere over the image. Note that the isoclinics are always black and will move over the image when the polarizers are rotated.

It is thus by using both the isoclinics and the isochromatics that the direction and magnitude of the stress at any point can be determined.

Detailed background material at a lower mathematical level
than the course text can be found in Dally and Riley, *Experimental
Stress Analysis*. Cf. Chapter 10: section 6, Chapter 12:
sections 1-6, and Chapter 13: sections 1-4.

Last Modified on April 20, 1997