Do the results of this laser experiment demonstrate nonlinear optical phenomenon?

Question

In my photonics class, we conducted an experiment to observe the Beer-Lambert law and the effects of filters(absorption, transmittance, and reflection). We shined a red(632nm) 5mW Helium-Neon laser through combinations of 3 filters: red, green, and blue. We measured the output beam power using a highly sensitive ThorLabs power meter that can detect nanowatt levels.

Distance between the filters = 3.81 cm Distance between the laser and the 1st filter = 25.4 cm

Here our the surprising results:

What we found is that when we used all 3 filters, the

In table 1: x = thickness of the filter and a = absorption coefficient. Now as you can see from table 2, the actual transmitted power output measured from using combinations of filters is orders of magnitude larger than predicted. For the predicted values the books told us to take the product of the individual filter transmittances.

Also, when using combinations of all 3 filters: the ORDER in which the filters were placed made a big difference. Using the RGB combination(left to right = first to last) got the highest transmittance. To me this suggests some Nonlinear optical phenomenon occurring between the filters.

But what I find curious is how when a blue or green filter was placed in front of the red filter the measured power of the output beam was greater than that of using only the red filter. My guess is there is some back reflection leading to standing wave resonant modes. But also the blue/green filter(s) seem to be interacting with the red filter to increase the red filters transmittance. Is it possible that there is some nonlocality going on here too?

Answer 1

No there is no evidence of nonlinearity. What you are missing is that the transmission spectrum of the filters is not constant and they will overlap somewhat. E.g if the bulk of the work of the green and blue filters is to cut out the red spectrum then used together you won't expect much of a drop in broadband power than when used alone.

The power you are measuring is a sum over all wavelengths. If you wanted to verify nonlinearity you would have to measure at one wavelength. The "nonlinearity" you are seeing is not optical nonlinearity but the nonlinearity inherent in the multiplication of the transmission spectra:

$\sum_i (A_i * B_i) \ne \sum_i(A_i) \sum_i(B_i)$

Answer 2

I don't think there is any nonlinearity, let alone non-locality at this power level.

I cannot be sure the following is relevant to your situation: you don't provide enough details, for example, distances between filters, are unknown, so take this for what it's worth.

Let us assume that there is zero distance between filters when you use more than one of them. The reflection coefficient on the boundary between two filters would be greater if the difference between the refraction coefficients is greater (for the specific wavelength). Thus, the order in which the filters are placed may change the total transmission coefficient. For example, if the refraction coefficients are 1.3, 1.4, and 1.6, I would think the transmission coefficient for the order of filters 1.3, 1.4, 1.6 would be greater than for the order 1.3, 1.6, 1.4.