Biomedical Engineering Reference
In-Depth Information
where.
k
⊥
,.the.spatial.frequency.of.the.wave.ield.on.the.planar.Huygens.emitter.surface,.which.con-
sists. of. the.
k
x
. and.
k
y
. components. of. the. wavenumber. vector.
k
.. In. a. paraxial. beam,.
k
. and.
z
>>
k
=
k
⊥
⊥
k
z
≅
2π
λ.
An.alternative.approach.is.to.expand.the.square.and.recognize.the.cross.term.as.the.exponent.of.a.
Fourier.transform.kernel:
x
L
2
x
L
′
2
x x
⋅ ′
i
L
−
i
π
−
i
π
i
2
π
.
( )
=
(
)
(
)
∫∫
u
x
e
u
x
′
e
e
d
2
x
′
e
−
ikL
Fresnel Method 2
λ
λ
L
λ
λ
S
he.alternative.method.starts.by.multiplying.the.wave.ield.at.the.starting.surface.by.the.phase.factor.
2
λ
.
For.numerical.accuracy.of.either.method,.it.is.important.that.the.phase.factors.not.vary.too.rapidly.over.
the.sample.grid.of.the.Huygens.surface,.as.stored.in.the.computer..Obviously,.Method.1.works.well.for.
small.
k
⊥
.and.small.
L
.(low.spatial.frequencies.and.short.propagation.distances)..Method.2.works.for.large.
L
.and.
x
.and.
x
′.inside.the.Fresnel.zone,.that.is,.
x
2
λ
,.taking.the.Fourier.transform.of.the.result,.then.multiplying.by.the.phase.factor.
e
x
L
′
x
L
−
i
π
−
i
π
e
< λ.and. ′ <
x L
λ ..he.rule.of.thumb.from..experience.
is.that.Method.1.should.be.used.for.propagation.distances.less.than.10%.of.the.Rayleigh.range.=.
D
2
L
λ.
(see.
Section.1.5.6
)
.and.that.Method.2.should.be.used.for.propagation.distances.larger.than.this.
Figure.1.12.shows.a.slice.through.a.circular.beam.ater.it.has.propagated.20%.of.the.Rayleigh.range.
from.a.hard.aperture..Note.the.ringing.caused.by.difraction.
At.very.long.propagation.distance,.difraction.dominates.and.the.beam.width.forms.a.
far-ield
.pat-
tern.that.expands.linearly.with.distance..At.these.distances,.
x
.<<.
L
.and.
L
,.so.the.phase.factors.are.
x
′ <<
very.close.to.unity..Deining.θ =
x
L
,.the.integral.becomes
i
L
.
( )
=
(
)
(
)
∫∫
u
θ
u
x
′
e
ik
θ
⋅
x
′
d
2
x
′
e
−
ikL
Fraunhofer meth
od
λ
S
he.far.ield.difraction.pattern.of.a.circular.aperture.illuminated.by.a.plane.wave.is.the.
Airy pattern
.
shown.in.
Figure.1.13
.
L = 0
L = 20% of the
Rayleigh Range
FIGuRE 1.12
Results.of.a.numerical.propagation.of.a.circular.beam.using.Fresnel's.approximation.to.Huygens.
integral.
