Principles of Geometric Optics - Adaptive Optics for Biological Imaging - page 32

2.4 Paraxial Lens equation

It.is.also.useful.to.consider.refraction.from.a.curved.surface,.such.as.the.circular.arc.shown.in.Figure 2.3..
When. the. arc. separates. two. regions. of. diferent. indices. of. refraction,. n 1 . and. n 2 ,. a. ray. of. light. will. be.
refracted.at.the.interface.according.to.Snell's.law..If.we.choose.the.correct.shape.for.the.interface,.we.can.
cause.light.that.originates.from.a.point. O .along.the.axis.of.the.arc.to.pass.through.a.point. I .that.is.also.
along.the.axis..his.is.the.geometry.of.a.lens.and.it.enables.an.object.at.one.point.in.an.optical.system.to.
be.imaged.at.a.diferent.point.in.the.system.
To.ind.the.equation.that.describes.the.action.of.the.lens,.we.consider.the.fact.that.a.light.ray.traveling.
from.position. O .to.position. I .must.take.the.same.amount.of.time.regardless.of.the.path..his.means.that.
the.time.taken.for.light.to.travel.from.point. O .to. P .and.then.from.point. P .to. I .must.equal.the.amount.
of. time. taken. for. light. to. travel. directly. from. point. O . to. I .. Of. course,. the. spatial. distance. OP + PI . is.
greater than.the.distance. OI ,.but.if. n 1 .is.smaller.than. n 2 ,.then.light.will.travel.slower.inside.the.media.
with.index. n 2 .and.thus.will.take.the.same.amount.of.time.along.the.direct.path. OI .as.it.would.along.the.
path. OP .+. PI ..his.is.because.more.of.the.path.of.light.is.inside.the.media.with.higher.index.
To.estimate.these.travel.times,.it.is.useful.to.make.an.approximation.for.light.rays.that.are.close.to.
the.optical.axis.(Feynman.1966)..hese.are.called.paraxial.rays..Consider.the.right.triangle.with.sides.of.
length. s > d > h .shown.in.Figure.2.4.
From.the.Pythagorean.theorem,.we.know.that

s

2

= h

2

+ → = d

2

h

2

s

2

− d

2

= − ( s d s d )( + )

.

To.simplify.this,.we.can.approximate.that. s ≈ d ,.so.that. s + d ≈.2 s .and.substitute.Δ.=.( s .-. d ).to.obtain

h

2

h ( s d s d )( ) 2 s

2

= − + ≈ → ≈ ∆ ∆ (2.4)

2 s

. .

P

S 1 S 2

R

O I

V Q C

n 1 n 2

FIGuRE 2.3 Refraction.at.a.curved.interface..he.light.traveling.from.point. O .(object).to.point. P .is.refracted.at.
the.interface.between.two.media.with.indices.of.refraction. n 1 .and. n 2 ..he.refracted.light.intersects.the.optical.axes.
at.point. I .(image)..he.curved.surface.has.a.radius.of.curvature. R .

Δ

s

h

d

FIGuRE 2.4 he. paraxial. approximation. to. ind. the. diference. in. optical. path. diference. for. two. rays. that. are.
close.to.the.optical.axis.