We are replacing old surface mount medicine cabinets with new ones (old ones are literally falling apart). Recessing is not an option in our old lath and plaster home. Old medicine cabinets had integrated lights; cost is prohibitively expensive on new ones. Looking at 4" deep medicine cabinets.

Most vertical bath bars extend or have a depth of 3.75 to the very edges which means much of the light will be shallower than the medicine cabinet. My intuition says the lights should extend beyond the depth of medicine cabinets, but I'm no expert. Thoughts? Any ways anyone has used to 'deepen' the light? Optimal light amount in this type of situation? Ideas welcome. Thank you.