Biomedical Engineering Reference
In-Depth Information
x
ˆ
ˆ
k
=
k
z
ˆ
FIGuRE 1.5
Electromagnetic.plane.wave.in.a.general.state.of.polarization.
and.
E
x
.=.|
E
x
|.is.a.real.number..δ.represents.the.relative.phase.between.the.
x
.and.
y
.components..he.four.
Stokes.parameters.are.deined:
I
=
E E
*
+
E E
*
,
Q
=
E E
*
−
E E
*
.
x
x
y Y
x
x
y
y
U
=
E E
*
+
E E
*
,
V i E E
=
*
−
E E
*
x
y
y
x
x
y
y
x
he.advantage.of.using.the.Stokes.parameters.is.they.are.physically.measureable.with.a.combination.
of.waveplates,.polarized.ilm,.and.photometers.in.the.laboratory,.and.they.are.a.complete.representation.
from.which.the.helical.picture.of.the.
E
.vector.can.be.reconstructed.
he. polarization. state. represented. in. terms. of. Stoke. parameters. can. be. traced. through. a. series. of.
polarization-dependent.optical.elements.using.the.Mueller.matrix.calculus.[Mueller.1948]..he.polar-
ization.state.expressed.in.terms.of.complex.
E
.components.directly.can.be.traced.through.optical.ele-
ments.using.the.Jones.calculus.[Jones.1941].
Some.light-transmitting.materials.have.the.property.that.the.index.of.refraction.is.diferent.for.the.two.
axes.of.linear.polarization.states..he.property.is.called.
birefringence
..Calcite.crystal.is.a.commonly.used.
material.with.this.property..If.the.wave.polarization.is.aligned.along.the.crystal's.
ordinary
.axis,.it.encoun-
ters.a.relatively.low.index..If.aligned.along.the.
extraordinary
.axis,.it.encounters.a.relatively.high.index..
hese.crystals.are.useful.for.analyzing.or.changing.the.polarization.state,.per.the.following.examples:
.
1.. Wollaston.prism:.An.incoming.wave.at.nonnormal.incidence.to.the.face.of.the.birefringent.crystal.
is.separated.into.its.two.polarization.component.waves.since.the.diferent.indices.impart.diferent.
refraction.angles.
.
2.. Quarter.wave.plate:.An.incident.wave.is.converted.from.linear.to.circular.polarization.or.vice-
versa..One.aligns.the.linear.polarized.input.wave.normal.to.the.crystal.face.but.with.the.polariza-
tion.axis.at.45°.with.respect.to.the.ordinary.axis.of.the.crystal..he.crystal.is.just.thick.enough.
such.that.slow.component,.the.extraordinary.wave,.ends.up.¼.wavelength.retarded.from.the.faster.
ordinary.wave..he.resulting.output.wave.thus.has.the.two.components.shited.π/2.radians.out.of.
phase.and.so.is.circularly.polarized..Similarly,.a.circularly.polarized.input.wave.is.converted.to.
linear.polarized.
.
3.. Half.wave.plate:.he.half.wave.plate.lips.the.polarization.state.of.linear.polarized.waves.by.90°.
and.converts.right-hand.circular.waves.to.let.hand.circular.waves..he.birefringent.plate.is.thick.
enough.that,.at.the.output,.the.extraordinary.wave.lags.½.of.a.wavelength.behind.the.ordinary.
wave..he.resulting.output.wave.has.the.two.components.shited.π.radians.out.of.phase.
1.3.3 optical Path Length, Geometric rays, and Diffraction
Fermat. postulated. that. light. follows. paths. of. least-time,. accounting. for. changes. in. velocity. within.
refractive.media..For.example,.light.appears.to.go.along.in.the.straightest.path.from.point.A.to.point.B.
in.free.space.(veriied.by.the.fact.that.an.opaque.screen.will.block.the.light.only.if.it.is.placed.directly.on.
the.straight.line.path.between.A.and.B),.but.if.the.index.of.refraction.is.not.uniform,.then.the.light.paths.
are.curved.going.from.A.to.B.(veriied.again.by.blocking.the.path.by.an.opaque.screen,.watching.when.
the.light.does.not.arrive.at.point.B)..he.relected.path.of.a.mirror.seems.to.be.a.minimum.distance.path.
