Cross-polarized crystal of 'Floralife Crystal Clear'

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Post by mtuell »

Sorry, I've been away on vacation and didn't see the excellent replies to this thread!

Interference is a wave phenomenon, not requiring invocation of quantum mechanics at all. Perhaps the confusion arises from the experiment where it was shown that a single particle at a time (electrons or photons), shot at a double slit, still produces an interference pattern, rather than the two discreet clumps that would form in classical mechanics. While this is true, it is evidence of the quantum nature of light, which then goes on to form a wave-based interference pattern.

Next, with the exception of random polarization, all polarized light is elliptical, with circular and linear being special cases, with specific phase shifts between the orthogonal polarization states. It isn't just something interesting to think about, it is the fundamental nature of an E-M wave. If the electric vector is static, we call this linear polarization. If it spirals around as it propagates, we call this elliptical. If the ellipticity is zero (both axes are equal), it is circular polarization.

These non-linear polarization states are obtained by introducing a phase shift and interfering the result. If the two states are in phase, we have linear polarization, but if they are out of phase, we have elliptical or circular (in rare cases). This phase shift is produced by a birefringent material. Birefringence is the difference in index of refraction between the fast and slow (or as noted, ordinary and extraordinary) axes. Light literally travels slower in one axis than the other, creating a phase shift that interferes in the analyzer.

Birefringence (linear) is in terms of ?n, but it is retardance that we see, which is ?n*t, where t is the thickness of the material. So, as the material increases in thickness, the retardance increases. As noted, the retardance is absolute distance (?n is unitless and t is distance), so one must divided by the wavelength of interest to determine how many waves of retardance were experienced. This means a simple crystal is only quarter-wave, for example, at a single wavelength. True zero-order retarders, in which the retardance is literally 1/4 wave, can be difficult to fabricate out of natural crystal, so multiple-order retarders are sometimes found, where it may be 1.25 or 3.75 waves, for example, which will still create the circular polarization (although in this case, one would be right-handed and the other, left-handed). Cementing two of these together, with the fast axis of one aligned with the slow axis of the other causes them to cancel, so one might be 4.25 waves and the other 4.00 waves, making a zero-order retarder, but very thick, which limits the field with "constant" retardance. Polymer waveplates can be made to be true zero-order.

As discussed, the simple waveplate, has different effects for different wavelengths. By a clever stack of polymers, achromatic retarders can be made. Generally in precision optics, details such as what wavelength it is quarter-wave for, would be prominently given, so if they (a reputable firm) don't specify, it could be achromatic. Ideally of course, they would plainly state this, too.

So, in short, Rik's explanation is very close. The edges of crystals may be thin, so have little retardance, altering the phase of all wavelengths a little, resulting in increasingly bright grey. As the shift becomes significant enough to get to a half-wave, the polarization state is rotated 90°, meaning that the color which is retarded such makes it through the analyzer unmolested. Other colors will make it through the analyzer to lesser degrees, resulting in white-light interference fringes.

As the thickness is increased to cause multiple waves of retardance, the colors are all jumbled up, generally making sickly green and magenta fringes. For example, if there is 250 nm retardance, That is half-wave for 500 nm light, so bluish-green makes it through the analyzer, with a drop off in intensity for other wavelengths. At 550 nm, 250 nm is 0.45 waves. If we now say that there is 2500 nm of retardance, or 5 waves at 500 nm, then that light is unchanged, so is blocked. However, 550 nm light is 4.55 waves of retardance. The point is that for thin specimens, This change from 500 to 550 nm is only a difference of 0.05 waves of retardance, but for the thicker specimen, it is 0.45 waves, meaning that the light going through the analyzer changes quickly with varying wavelength, allowing a wider variety of colors to simultaneously be transmitted.

Some examples of this color mash-up can be seen in my post about interference microscopy ... hp?t=33684 and more polarization discussion is on my website in the "Optics Primer" section.

A couple of key points to remember:

If you have crossed polarizers and introduce a birefringent material between them, rotating the material makes it such that the light is extincted every 90° since you are then putting linearly polarized light down one of the principal axes (fast or slow, extraordinary or ordinary) and it is unaffected. As Rik says, it has to be exactly 45° between axes, with exactly a quarter-wave retardance at that wavelength to get circular polarization out. Rik's observation that some parts of his crystal were very dark are likely because that portion has it's axes rotated in a different orientation. Rotating the specimen between crossed polarizers would resolve this question, as those would become brightly colored at different orientations.

This discussion has been about uniaxial crystals. Things get a bit stranger with biaxial crystals.

Optical activity (remember - the term that started this whole thing? :D ) is circular birefringence (and, yes, there is elliptical birefringence, but it refers to something a little different). The discussion about Rik's photos are about linear birefringence.

A useful concept to understand polarization states and how they are modified by birefringence (of all types) is the Poincare' Sphere, where linear polarization is on the equator of the sphere and R/L circular are the poles. Neither a point or a line has any area.... the surface of the sphere are all the elliptical states, which is one way to visualize that all polarized light is elliptical. Birefringence rotates the sphere, causing the polarization state to move to a new location on the sphere. Circular birefringence rotates about the poles, changing longitude, so the latitude cannot be changed (which is why I earlier said that circular birefringence doesn't have an axis.) Linear birefringence pierces through at the equator. A 90° phase shift (quarter-wave), rotates the state up to the pole, if oriented correctly. Spinning the sphere about the point where your polarization state is leaves your state unchanged.

Sorry for the delay and then the length of the reply, but it is a good topic for us all to be familiar with.


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