By using "by using", I believe you are absolved from any dangling in 4, 5, and 6! 1, 2, and 3 are all ambiguous because of the dangling structure. The first is the worst offender because both "they", the detectors, and "the agent" are both mentioned in the sentence and we don't know exactly to whom to ascribe the use of the new method.
New words, one handy idiom, and a 2-minute quiz — delivered to your inbox to keep your streak alive.
By looking through this high-powered microscope, the butterfly seemed even more beautiful.
We saw a man on a hill with a telescope.Yes this sentence is really ambiguous. We can't fix the exact sense without knowing the context.
After (my) having bought an apple tree, the only problem now is where to plant it.Couldn't we be absolved from danglingness even in the presence of the preposition 'after'?