I'm trying to make it so that the plane is transparent except for the light from the aqua emission. Or in other words, how can I make it so that the plane's white areas are transparent and the plane's aqua areas are not? I want to achieve this without using the compositor, only with material nodes.
As far as I know this is fundamentally impossible to achieve in Cycles
Cycles being a physically based renderer has to find a solution to the rendering equation this includes calculating all lighting interactions influenced by surface properties. Materials can't possibly depend on the render results; so surfaces properties can't be influenced by the render solution.
As such transparency can never be the result of light intensity which would cause a dependency loop in rendering process. You can't determine surface lighting before you know its transparency, and you can't set its transparency without knowing the light intensity.
So either fake it in the compositor, bake the light into a texture, render in separate layers and separate steps, use a Shadow Catcher material (only available in 2.79+ versions), or change render engines and use something non physically based, or non photorealistic.