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Polarized Material. Uh-oh. Where do we find something that's polarized? Luckily enough, there are literally hundreds of everyday objects that can be exploited for their polarized screens.
The easiest to adapt to our purposes is probably a good ol' fashion pair of 3-D glasses – each lens being polarized, just rotated at an angle to each other. (For an example of this approach, check out this YouTube movie.) Your everyday LCD screen has two sets of polarized screens in it, one on each side of the LCD itself. This will have to be removed carefully, but it can be done. Polarized sunglasses, if you can find a clear pair (or a colored pair, depending on what you want) might also work in pinch.
So, ask around if anyone has a damaged LCD screen you can salvage a corner of, or keep an eye out in those thrift stores for a cheap pair of glasses. Make sure that whatever you use is unscratched, and is large enough for your camera.
Cutting utensil. Box cutters work great, especially if you'll be working with hard materials – probably mostly plastic - and will need to make precision cuts.
Sandpaper. Smooth edges are easier to work with, so having sandpaper really helps. Failing that, a good emery board might also work.
Glue. Hot glue is probably a good bet, since you'll probably be working with plastic, though anything that's strong and won't expand too much will work just fine.
Filter Attachment. The details of how to make a DIY filter mounts are explained in this article.
A few notes specific to polarized filters: these filters tend to be the real workhorse of photography filters, so building it in a sturdy fashion – and in such a way that other filters may be used in addition to it – is really a must. You'll want something that can rotate, so simply taping it on to the end of the camera will not suffice. Being able to attach two to your camera at once allows you to do some other neat DIY tricks, such as a variable ND filter. Thus, it is recommended that simply reworking an old, damaged filter is the best solution.
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You'll want to prep your filter mount before anything else, so you know what size to cut the filter to. Methods for making different filter mounts are discussed, as previously linked, in this article.
Whatever material you decided to use for the polarized component, take out and take a good look at it. Are you sure that there are no scratches on it? If there are, is it possible to use a different part of the material? Trace the outline of your lens onto the polarized material, and begin cutting, with a preference to cutting outside of the margins to inside. Once you've got this done, sand the edges smooth. The filter itself is now complete.
After that, attach your future filters to your filter mount of choice and you're done!
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Using The Polarized Filter
From there, the possibilities are endless. Polarized filters can be used to create more dramatic, contrasted skies, to reduce reflections in windows and screens, to darken a bright day... Experiment with your new piece of equipment as something that really can just be kept on the camera for most situations. Many photographers find it to be the single most used (and most useful) piece of equipment in their toolbox.
For more on how to use polarized filters, check out this article.