Why can't I view DVDs on my MacBook Air using AirPlay?

I am attempting to view DVDs using an external Apple DVD SuperDrive connected to my 2012 MacBook Air. The DVD plays normally on my computer screen, but when I click on the AirPlay icon in the menu bar, I am able to view my computer screen on my large-screen TV connected to Apple TV. But the DVD no longer shows the movie. Instead it shows a checkerboard pattern on both screens. The audio, however, keeps playing normally. I don't know how to correct this problem. When I turn off AirPlay, the image returns to the computer screen normally.