Around 1988, I added phase shift to my optical thin film design program written in Excel 4.0 for the Mac. At the time, this was utterly unique: each spreadsheet row represented a layer and the matrices describing each layer could be calculated right in that row by squashing them down horizontally. The S- and P-polarization matrices could be recorded this way, and the running matrix products similarly maintained. Finally, using a simple one input table, the reflectance of a typically 25-31 layer laser mirror could be calculated. And in less than a second on a 20 MHz 68020 (?) Mac II for about 50 wavelengths. The best part were the graphics which were instantaneous, beautiful, publishable, and pasteable into customer quotations. Semi-technical people could be trained to use the whole thing.

Now about the phase shift. In 1988, atan2 didn’t exist. Anywhere. Not in FORTRAN, Basic, Excel, or a C library. I’m sure phase shift calculators implemented it, each working alone. For us, it was critical. You see we actually cared not about the phase shift, but its second derivative, the group delay dispersion. This was the beginning of the femtosecond laser era, and people needed to check whether these broadband laser pulses would be inadvertently stretched by reflection off or transmission through the mirror coating. So atan2, the QUADRANT-PRESERVING arc tangent, is required for a continuous,differential phase function. An Excel function macro did this, with IF statements correcting the quadrant. And the irony of all this?

I CALLED it atan2.