Target display extremely dark

I'm working from home on my employer-owned Air, trying to use my (personal) 27" 2013 iMac as a monitor for it. I've got a Thunderbolt cable running from the laptop to the iMac, I use command-F2 to put the iMac in target display mode, and I've gotten it to work successfully a few times: I can move windows back and forth between the two screens, everything looks fine on both screens, life is good.

But probably 90% of the times I try this, the iMac display is extremely dark: I can move windows onto the second display and sorta tell that something is moving, and I can sorta tell that one window contains text, but I can't actually read anything on the big monitor. The iMac looks just fine when it's not in target-display mode. I haven't found any consistency in when it works and when it doesn't.

Any suggestions?