Biomedical Engineering Reference
In-Depth Information
D
λ / D
L
FIGuRE 1.21
he.Rayleigh.range.is.the.limiting.distance.for.a.geometric.ray.approximation.to.light.propagation.
Recall.from. Section.1.3.1 . that.the.Fresnel.zone.has.a.diameter.of.approximately. λ L ..One.can.think.
of.Fresnel.number.as.the.progress.of.difracted.waves.from.one.edge.of.the.aperture,.opening.at.angle.
λ/ D ,.reaching.the.other.edge.(Figure.1.21)..he.Fresnel.number.is.equal.to.one.at.the.Rayleigh.range.and.
is.ininite.at.the.aperture.
1.6 image Formation and Analysis
In.the.Fraunhofer.analysis,.we.showed.that.the.ield.at.the.focal.plane.is.the.complex.Fourier.transform.
of.the.ield.at.the.aperture..With.this.in.mind,.we.can.describe.the.optical.system.performance.in.terms.
of.linear.system.theory..he.optical.system.is.described.in.terms.of.an.optical.transfer.function.(OTF).
in.the.spatial.frequency.domain,.that.is,.what.it.does.to.the.various.spatial.frequency.components.of.the.
incoming.wavefronts..he.image.is.composed.of.linear.superpositions.of.point.spread.functions.(PSF)..
Assuming.paraxial.beams,.this.is.a.very.accurate.way.to.predict.detailed.properties.of.images.produced.
by.complex.optical.systems.
he. Fourier. transform. relationship. of. aperture. plane. (“spatial. domain”). and. image. plane. (“spatial.
frequency.domain”).is.summarized.in . Figure.1.22 .
he. pupil. function. (a. simple. mask. of. ones. and. zeros). multiplies. the. electric. ield. incident. at. the.
aperture.
An.imaging.system.(as.shown.in. Figure.1.23 ) .gathers.light.waves.from.the.sources.in.the.object.plane.
and.produces.an.image.at.an.image.plane..he.points.in.the.image.plane.correspond.one.to.one.with.
points.in.the.object.plane..However,.since.the.entrance.aperture.is.inite,.the.resolution.of.the.image.is.
limited.by.difraction.
Using.the.fact.that.multiplication.in.the.spatial.domain.maps.to.convolution.in.the.spatial.frequency.
domain,.the.complex.ield.in.the.image.plane.is.the.complex.PSF.(Fourier.transform.of.the.aperture).
convolved.with.the.complex.ield.of.the.object:
I u
=
( ) =
( )
P u O u
( ) ( )
.
(
)
(
)
′ = ( ) ( )
i x
p x
− ′
x o x
d
x
p x
o x
where. O ( u ).is.the.incident.wave.ield.in.front.of.the.aperture.stop,.which.is,.by.Fraunhofer.theory,.the.
Fourier.domain.representation.of.the.object.ield.. I ( u ).is.the.wave.ield.just.beyond.the.aperture.stop,.
which.is,.by.Fraunhofer.theory,.equal.to.the.inverse.Fourier.transform.of.the.image.ield.
he.distribution.of.light.intensity.at.the.image.plane.is.proportional.to.the.squared.modulus.of.the.
complex.ield.incident.on.it:
2
.
( ) =
(
)
(
)
(
)
(
)
∫∫
i x
p x
− ′
x p x
*
− ′′
x
o x o x
*
′′
d d
x
x
In.the.case.of.an.incoherent.distribution.of.sources.in.the.object.plane,.the.time.average.of.cross.cor-
relation.of.point.sources.is.zero..For.a.scene.illuminated.by.incandescent.light.or.sunlight,.the.reemit-
ting.atoms.are.independent.oscillators..For.a.laser.emitter.or.a.scene.illuminated.by.laser.light,.this.is.
 
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