Polarized light microscopy

Polarized light microscopy (PLM) is transmitted-light microscopy with a polarizer below the specimen and a second polarizer, the analyzer, above the objective, usually crossed at 90° so that the field of view is dark. Isotropic materials such as glass, liquids and cubic crystals stay dark. Birefringent materials split the light into two components that travel at different speeds; they recombine at the analyzer with a phase difference, so the specimen appears bright and, in white light, coloured. The colour measures the retardation, the optical path difference between the two components.

Retardation and intensity

The retardation of a plate of thickness $t$ and birefringence $\Delta n$ is

$$\Gamma = \Delta n\,t.$$

It is usually quoted in nanometres; the phase form is covered under retardance. For quartz, $\Delta n = n_e - n_o = 1.553 - 1.544 = 0.009$, and a 30 µm section gives $\Gamma = 0.009 \times 30\,000$ nm $= 270$ nm.

Between crossed polarizers the transmitted fraction is

$$\frac{I}{I_0} = \sin^2 2\varphi \,\sin^2\!\left(\frac{\pi\Gamma}{\lambda}\right),$$

where $\varphi$ is the angle between a vibration direction of the crystal and the polarizer. The first factor makes a grain go dark (extinct) every 90° of stage rotation and brightest at 45°. The second factor depends on wavelength: for $\Gamma = 270$ nm it is 0.90 at 450 nm, 1.00 at 550 nm and 0.93 at 650 nm, nearly flat across the visible, which is why the grain looks white or grey instead of a saturated colour.

The Michel-Lévy chart

As $\Gamma$ grows, different wavelengths are suppressed and the transmitted mixture passes through a characteristic sequence of interference colours: grey, white, yellow, orange and red in the first order up to about 550 nm, then repeating bands of blue, green, yellow and red in the second and third orders, fading to pale "high-order white" beyond about 1500–2000 nm. The Michel-Lévy chart plots these colours against retardation, with diagonal lines for birefringence, so any two of thickness, birefringence and colour give the third. Calcite, with $\Delta n = 0.172$, reaches 5160 nm in a 30 µm section and sits deep in the high-order region. Quartz, of known birefringence, serves to check section thickness.

Compensators

A compensator is a calibrated retarder inserted at 45° between the specimen and analyzer. The most common is the first-order red plate (also called a full-wave or λ plate), with about 530–550 nm of retardation, which turns the background magenta. When the slow axis of the specimen is parallel to that of the plate the retardations add and the colour rises in the chart; when they are crossed they subtract and it falls. With a 530 nm plate, the quartz grain above moves to 800 nm or down to 260 nm, depending on orientation, which reveals which of its axes is the slow one. Quartz wedges and tilting Berek compensators give quantitative retardation. A related waveplate arrangement gives circularly polarized illumination, which removes the orientation-dependent extinction so all grains appear at once.

Conoscopy

In conoscopy, the specimen is illuminated with a strongly converging cone from a high-numerical-aperture condenser and the back focal plane of the objective is viewed through a Bertrand lens. Each point of that plane corresponds to one propagation direction through the crystal, and the result is an interference figure: a dark cross of isogyres with coloured rings for a uniaxial crystal viewed along its optic axis, and curved isogyres for a biaxial crystal. With a compensator the figure gives the optic sign and, for biaxial crystals, an estimate of the optic angle.

Applications

• Mineralogy and petrology. Minerals in 30 µm thin sections are identified from relief, extinction angle, interference colour, twinning and pleochroism, the orientation-dependent absorption that is a form of dichroism.
• Crystal arthropathy. In synovial fluid, monosodium urate crystals of gout are needle-shaped and strongly negatively birefringent: yellow when their long axis is parallel to the slow axis of the red compensator and blue when perpendicular. Calcium pyrophosphate dihydrate (CPPD) crystals are rhomboid or rod-shaped and weakly positively birefringent, blue when parallel. The sign of elongation distinguishes the two diagnoses.
• Polymers. Spherulites show a Maltese-cross pattern; molecular orientation in fibres and films, and residual stress in moulded parts, appear as retardation.
• Liquid crystals. Nematic and smectic phases are identified from characteristic textures, and phase transitions are followed on a heated stage.

Pitfalls

Strained objectives and condensers add their own retardation and brighten the background; PLM uses strain-free optics. Sections thicker or thinner than 30 µm shift all colours. Strongly dispersive minerals show anomalous colours that do not match the chart. Pleochroic absorption can be mistaken for interference colour; it appears with the analyzer removed.

Common questions

Why is the background dark in polarized light microscopy?

The analyzer is crossed with the polarizer and blocks light that has passed through isotropic regions unchanged. Only light whose polarization the specimen has altered reaches the eyepiece.

How are gout crystals identified under polarized light?

By shape and the sign of birefringence with a first-order red compensator: negatively birefringent needles, yellow when parallel to the compensator slow axis, indicate monosodium urate.

References: E. Hecht, Optics, 5th ed. (Pearson, 2017), Ch. 8; M. Born, E. Wolf, Principles of Optics, 7th ed. (Cambridge University Press, 1999), Ch. 15; W. D. Nesse, Introduction to Optical Mineralogy, 4th ed. (Oxford University Press, 2013).