http://people.csail.mit.edu/jaffer/FreeSnell/coherence.html
	FreeSnell: Optical Coherence
	Refractive Index Spectra. Parametric Data Tabular Data nk spectral database management script All-Dielectric Optics Bandpass Filter Polarizers Anti-Reflection Coatings Dielectric Mirrors Metal Films Thick Reflective Metal Thin Metal Films Matching Measured Data Colors of Zinc Sulfide Granular Metal Films Maxwell Garnett Theory Gold Ruby Glass IRG-II GranFilm The Colors of Silver and Gold Films Optical Coherence Coherence Incoherence Gaussian Filtering Polyethylene Background Emissivity Absorption Lines Thin HDPE Films Thick HDPE Films

This suite is from "coherence.scm", part of the FreeSnell package. "coherence.log" is the text generated by running coherence.scm:

scm -l coherence.scm > coherence.log

The vacillations between transmissions of 0.8 and 1.0 are caused by destructive and constructive interference between the incident and reflective waves at the HDPE surfaces.

But a thin single sheet of plastic also conducts heat well. One way to reduce heat conduction is to use multiple glazings (sheets) separated by air.

Now there is interference across the enclosed air space which, by virtue of its greater thickness, has 50 times more peaks and troughs than the HDPE intererence.

When we add a third glazing the troughs become quite deep, making this analysis suspect; or at least difficult to interpret. The peak and valley wavelengths of the narrow bands are extremely sensitive to changes in the distance between sheets, such as could be caused by temperature changes, vibration, or small changes in incident angle.

Incoherence

But with millimeter layer thicknesses a one degree change can jump several short-wavelength peaks. The reflections are no longer strongly correlated with the incident light; they are incoherent.

Whereas coherent systems are modeled by the electric field intensities whose complex values add and subtract as interference; incoherent light intensities add as quantities of (electrical) power. A calculation analogous to FreeSnell's matrix method will suffice: The 2-by-2 matrices set up are the same as for the coherent case except that the square of the magnitude of each matrix element replaces that matrix element.

The incoherent designation applies to an individual layer. FreeSnell segregates the stack into coherent substacks separated by incoherent layers. Each substack is solved and the resulting matrices are used as the reflection matrices in the incoherent stack.

If we make all layers incoherent, then the graph will lack any ripples. That would be incorrect; each glazing this thin does add ripple. We assign incoherence to only the spaces between the glazings. In the single glazing case there is no difference from the coherent case.

The transmission for two coherent glazings with an incoherent airspace between them is shown to the right. The ripples are twice as deep as for one glazing.

The transmission for three coherent glazings with two incoherent airspaces between them is shown to the right. The ripples are thrice as deep as for one glazing.

Gaussian Filtering

Because of the incoherence effects discussed earlier, and because of the finite apertures of spectrographs, at short wavelengths the wiggles average out. Adding (smooth 0.05e-6) computes the convolution of the spectrum with the Gaussian filter having a sigma of 0.05.um, as was done in the section Matching Measured Data.

Smoothing is much easier to apply in the wavenumber (and eV) domain because the interference ripples are evenly spaced:

Wavenumbers and electron-Volts also have the desirable property that each increment in value corresponds to a proportional increment in photon energy.

The next chapter attempts to deduce the spectral refractive index of polyethylene from published graphs of transmission.