I have read that it is ok to use a linear polarizer with mirrorless cameras because they do not use a beam splitter to do metering, so the polarizer will not confuse the auto exposure.
However, the OM-D E-M1 that I am getting does have phase detect auto focus, so will that present a problem?
Yes, you can use linear polarizer with mirrorless cameras, including OM-D E-M1.
No, phase detect autofocus in mirrorless cameras is not affected by linear polarizers.
Phase detect autofocus sounds like something that belongs to DSLR, hence the confusion. Indeed, AF might be affected in DSLR cameras (usually it is not), where the light is reflected to a separate AF sensor. Mirrorless cameras use the same main sensor for phase detect AF, without reflections and without polarization issues.
In theory, there is no issue — the potential problem with SLRs is that the half-mirror used to direct some light to the AF sensors also causes polarization. With on-sensor AF (of any type) that isn't an issue.
But, in the real world today, this is an academic distinction because it's hard to find a high-quality filter designed for photography with good transmission, neutral color, and decent coating that isn't a circular type. There's no harm in that, but don't scrimp on those other things just to feel like you are saving money by getting a linear filter.
The polarizing filter is likely the most valuable for the digital photographer. It cuts haze like a UV filter. It darkens blue sky causing the clouds to stand out boldly. It increases saturation without changing the color balance of the vista. It midrates reflections off water and glass and most non-conductive surfaces.
Film photographers used a “linear” polarizing screen however; digital photographers are advised to use a “circular” polarizing screen. This is because; the common polarizing screen can hinder auto focusing and exposure determination. I say, try what you have; it might just work for you. If you are purchasing, get a circular polarizing screen.
The “circular” type is actually two filters sandwiched together. The first one is a “linear” and this does the deed. The second is called a “retarder”. This filter rescinds the polarizing action thus the camera’s apparatuses are not impeded.
As for me, I have the experimenter’s gene, I have used a linear on digital and I have never seen any jeopardy. But, I don't doubt it exists.
