Biomedical Engineering Reference
In-Depth Information
point.on.this.subsequent.surface.reemits.spherical.wavelets.that.propagate.on.to.the.next.surface.and.
so.on..As.we.consider.an.increasing.density.of.these.intermediate.surfaces,.we.realize.that.the.spherical.
emitters.and.their.Huygens-Fresnel.integrals.are.tracing.out.
all possible paths of light
.from.points.on.
A to.points.on.B,.both.straight.and.meandering..Each.path.has.a.complex.phase.factor,.cycling.once.per.
wavelength.of.optical.path,.and.an.amplitude.factor.that.is.inversely.proportional.to.distance.traveled..
Coherent.addition.of.terms.from.all.the.paths.reveals.that.light.energy.prefers.those.paths.conined.to.a.
Fresnel.zone.around.the.minimum.time.path.through.all.the.surfaces,.where.the.total.integrated.phase.
factors.are.nearly.ixed,.so.the.terms.can.add.constructively.
1.4.2 computer Simulation of Wave Propagation
Numerical.evaluation.of.the.Huygens-Fresnel.integral.can.provide.useful.insight.into.the.behavior.of.
an. optical. system,. especially. if. difractive. efects. could. be. important. or. dominant.. To. assess. whether.
difractive.calculations.are.necessary.or.if.a.simpler.ray.trace.will.suice,.one.can.use.the.back-of-the.
envelope. calculation. in.
Section. 1.5
.
. Of. particular. importance. is. to. determine. if. beams. are. going. to.
propagate. a. signiicant. fraction. of. their. Rayleigh. range. (
Section. 1.5.6
)
,. in. which. case. the. ray. tracing.
would.not.predict.the.difraction.rings.that.will.grow.near.the.beam.edge.as.the.beam.propagates..his.
Fresnel ringing
.is.a.result.of.interference.of.the.beam's.plane.wave.with.the.difracted.waves.scattered.of.
the.hard.edges.of.apertures.
Assuming.a.beam.is.propagating.in.a.direction.mostly.normal.to.the.surface.of.Huygens.emitters.(the.
Paraxial
.condition),.we.can.use.the.approximation
2
+
x
L
≅ +
1
2
x
L
2
.
( )
=
ρ
x
L
1
L
ϕ
x x
( )
≅
1
.for.the.obliquity.factor..It.
is. actually. not. necessary. for.
x
.<<.
L
. for. this. approximation. to. work,. as. only. the. cluster. of. paths.
within.the.Fresnel.zone.will.actually.add.substantial.contribution.to.the.integral..Thus,.it.is.only.
necessary.that.
L
in.the.phase.factor,.ρ
x
( )
≅ .in.the.amplitude.factor,.and.
cos
L
,
>> λ ,.that.is.
L
.>>.λ,.to.use.this.approximation..The.Huygens-Fresnel.integral.
L
becomes
2
i
L
x
− ′
x
−
i
π
.
( )
=
(
)
∫∫
u
x
u
x
′
e
d
2
x
′
e
−
ikL
L
λ
λ
S
he.integral.can.be.recognized.as.a.convolution,.and.so.we.can.exploit.Fourier.transform.techniques.
to.evaluate.it.numerically..he.Fourier.transforms.of.the.wave.ield.at.each.surface.are.related.through
4
3
(
)
=
(
)
π
(
)
.
u
k
u
k
′
e
i k L k
2
2
e
−
ikL
Fresnel Method
1
⊥
⊥
⊥
L
λ
where.
k
⊥
. is. the. spatial. frequency. of. the. wave. ield. projected. on. the. planar. Huygens. emitter. surface.
(the.
x
-
y
.plane,.where.
z
.is.the.axis.of.propagation)..he.two-dimensional.Fourier.transform.pair.is.here.
deined.by
(
)
=
1
2
∫∫
( )
u
k
u
x
e
−
i
k x
⋅
d
2
x
⊥
⊥
π
.
S
( )
=
1
2
(
)
∫∫
u
x
u
k
e
i
k x
⋅
d
2
k
⊥
⊥
⊥
π
