Biomedical Engineering Reference
In-Depth Information
1.5.2 Lagrange invariant
By.combining.various.optical.elements,.light.can.be.manipulated.to.produce.magniied.images..Simple.
two-lens.systems.designed.for.plane.wave.input.and.output.is.shown.in.Figure.1.16..Incoming.light.is.
gathered.with.a.large.aperture.input.lens.and.then.presented.to.the.eye.or.subsequent.camera.system.
with.a.small.aperture.output.lens..An.example.of.an.astronomical.telescope.is.shown.on.the.let.image.
of.Figure.1.16,.although.modern.optical.telescopes.use.mirrors.that.perform.the.same.function.as.the.
lenses. depicted. here.. Incoming. star. light. is. essentially. composed. of. plane. waves. since. the. source. is.
at.a.great.distance.away..A.microscope.optical.system.is.shown.in.the.right.igure..he.microscope.is.
designed.to.image.an.object.that.is.very.close.to.the.objective.lens—a.biological.specimen,.for.example..
he.eyepiece.lens.presents.a.collimated.output.beam.matched.to.the.eye's.pupil.and.the.eye.acts.to.focus.
light.on.the.retina..Again,.the.system.magniies.angles.in.the.sense.that.small.displacements.at.the.object.
plane,.which.give.small.input.ray.angles,.are.magniied.to.larger.angles.in.the.output.beam.presented.
to.the.eye.
he. Lagrange invariant . is. a. statement. about. the. light-gathering. power. of. an. optical. system.. If. the.
optical.system.reduces.the.beam.diameter,.then.angles.are.proportionally.magniied..Conversely,.angle.
magniication.must.be.accompanied.by.a.compression.of.the.beam.size.
Figure.1.17.shows.the.geometry..From.a.wave-optic.perspective,.the.optical.path.length.along.each.ray.
is.what.matters..For.a.normal.plane.wave.beam,.the.optical.system.adds.the.same.optical.path.length.to.
θ
D
θ'
D '
D '
θ'
D
θ
FIGuRE 1.16
Optical.magniication.systems.. Let :.astronomical.telescope.. Right :.microscope.
θ
D
D '
θ'
FIGuRE 1.17
Lagrange.invariant:.compressing.the.beam.diameter.magniies.the.angles.
 
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