How do I add syntax highlighting support for TypeScript string interpolation in an Emacs major mode?

Question

I'm trying to write a custom major mode for TypeScript in Emacs. I would like for my major mode to highlight the following code correctly:

const myString = `The meaning of life is ${40 + 2}`;

Inside of the back ticks, ${40 + 2} should be highlighted as not a string, but a normal TypeScript expression; everything else should be recognized as a string. What's the right way to handle this case? Does the answer have something to do with font-lock or the syntax table for my major mode? I would appreciate any advice or pointers to relevant documentation.