Interference colors seen under polarized light microscopy and the sign of birefringence

   When birefringent objects are viewed under polarized light microscopy the polarization axis of the light illuminating the object is rotated so that some light passes through the second polarizer (located between the objective and the eyepiece), thus allowing the brightness of the object to appear against the dark background of the field of view. Below is shown an image of a quartz wedge with varying thickness seen under polarized light microscopy.

    Figure 1. Interference colors produced by a quartz wedge with varying thickness under polarized light microscopy with white (polychromatic) light. 

  The colors seen in Fig. 1 are a result of some rotation of the axis of the electric field of the plane linearly polarized light that illuminates the sample. With no birefringent object between crossed polarizers, the background remains dark (as seen in the left edge of the image above). Light is seen only with a birefringent material (like , for instance, quartz, in the image, and dental enamel). Birefringent (BR) is given by:

 BR = ne - no       (1)

Where ne and no are the refractive indexes of the extraordinary and ordinary rays, respectively. These terms were derived some centuries ago when thick birefringent minerals where seen producing double images of an object, one moving as the mineral was rotated (extraordinary image) and the other standing still (ordinary image). Each one has its own refractive index, and their difference yields birefringence. In the case of quartz ne and no are (for wavelength of 550 nm; source: Handbook of optics) 1.55515 and 1.54597, respectively, and BR is 0.00918. As light propagates thought the material, one ray travels at one velocity (related to ne) and another ray travels at a different velocity (related to no), so that one gradually gets ahead of the other. The difference between the distances traveled at a given time interval is called optical retardance (or phase retardance). Retardance (R) is given by:

R = BR x thickness (nm)    (2)

Thus, the thicker the material the higher the retardance. A 0,1 mm (100,000 nm) thick quartz sample has a retardance of 918 nm. Fig. 1 shows a quartz wedge with the same BR but varying thickness (increasing from left to right), resulting in different retardances. The ligth crossing the sample undergoes optical interference, with events of destructive interference playing the central role in the resulting colors seen through the eyepiece. Destructive interference occurs when one wavelength is subtracted. As white light has various wavelengths, the subtraction of one wavelength results in a complementary color (interference color). It is required at least half wavelenght of retardance for any particular wavelength to be subtracted enough to result in the first interference color from white light. For very small retardances, like those presented by biological samples in thin sections (most common situations in biological laboratories), some amounts of all visible wavelengths are transmitted to the second polarizer, and white light is seen. In practice, first interference colors are represented by increasing levels of grey up to white light. Then, when retardance approaches half, and up to equalize half wavelength, of the wavelength of violet, yellow is produced. Subtraction of blue results in orange, and subtraction of green (retardance of 550 nm) results in red.  All interference colors are grouped at 550 nm intervals called orders (usually up to five), separated by red colors:

1) First order: includes interference colors resulted from retardances of up to 550 nm (first red color seen from left to right in Fig. 1). The first order red color is known as "Red I", a term of high practical utility.

2) Second order: include interferences colors resulted from retardances > 550 nm and up to 1100 nm (second order red). A repetition of most of the interference colors (from yellow to red) of the first order are seen;

3) Third order: interference colors resulted from retardances > 1100 nm and up to 1650 nm (third order red). All interference colors of the second order are seen;

4) Fourth order: interference colors resulted from retardances > 1650 nm and up to 2200 nm (fourth order red);

5) Fifth order: inteference colors resulted from retardances > 2200 nm. When retardance is too high, more than one color is subtracted and the interference colors results from more complex subtractions. A successive of pink and green is seen, getting paler as retardance increases.

     Some samples of minerals with certain thickness result in a retardance of 550 nm, resulting in a first order red, and are very useful as filters in polarized ligth microscopy. They are called Red I filter or lambda filter (lambda is the denomination given to retardance of 550 nm), and turn the background from black to red. Some filters with a quarter of 550 nm (quarter of lambda) are also commonly sold as retardance filters. They are particularly very useful for determining the sign of birefringence of dental enamel, as explained ahead. Normal enamel (sound and mature) has an average BR of (-)0.002 under immersion in water. For samples with thickness of 0,1 mm (most common thickness used in labs), retardance is 200 nm. This retardance can be added (750 nm, light blue color) or subtracted (yellow color) from the retardance of a Red I filter. If the prisms' long axis (reference axis in enamel when using polarized light microscopy) is oriented paralell to the higher refractive index axis of the Red I filter (its direction is marked as a line on the body of the filter holder), subtraction occurs. If the prisms's long axis if oriented perpendicular to the higher refractive index axis of the Red I filter, addition occurs. 

    The rule that applies for addition and subtraction of refractive indexes axis (ne and no) is as follows: 

- when filter's higher refractive index axis is aligned with the samples' lower refractive index axis (hence, filter's lower refractive index axis is aligned with the samples' higher refrative index axis) subtraction of retardances occurs;

- when the higher refractive indexes axis (and hence lower refractive indexes axis) of both filter and sample are paralell to each other, addition of retardances occur.

  Normal enamel under water immersion presents the higher refractive index axis perpendicular to the prism's axis. Thus, when prisms are oriented at + 45º (and paralell to the Red I filter's higher refractive index axis) subtraction occurs (yellow color; subtraction diagonal position). At -45º addition occurs (blue color; addition diagonal position).