The Lydian mode is an interesting scale which often appears in soundtrack
music. The dreamy, film like quality arises from raising the 4th in the basic
major scale by a semi tone. In the key of D this presents a problem in DADGAD
because the open G string can no longer be used. Some care is needed to work
around this.
The D Lydian key signature matches the A major scale which makes this a
very useful fingering to know in DADGAD. It is not a simple scale to work with,
and it is easy to end up actually playing in A major. It offers an interesting
break, however, from the basic D major scale in DADGAD, one that is not heard
very often.