Transcript Diffraction at the back focal plane

Today
• Diffraction from periodic transparencies:
gratings
• Grating dispersion
• Wave optics description of a lens:
quadratic phase delay
• Lens as Fourier transform engine
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Diffraction from periodic array of
holes
incident
plane
wave
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Period: Λ
Spatial frequency: 1/
A spherical wave is generated
at each hole;
we need to figure out how the
periodically-spaced spherical waves
interfere
Diffraction from periodic array of
holes
Period: Λ
incident
plane
wave
Spatial frequency: 1/Λ
Interference is constructive in the
direction pointed by the parallel rays
if the optical path difference
between successive rays
equals an integral multiple of λ
(equivalently, the phase delay
equals an integral multiple of 2π)
Optical path differences
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Diffraction from periodic array of
holes
Period: Λ
Spatial frequency: 1/
From the geometry
we find
Therefore, interference is
constructive iff
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Diffraction from periodic array of
holes
incident
plane
wave
Grating spatial frequency: 1/
Angular separation between diffracted orders:  λ/
2nd diffracted
order
1st diffracted
order
“straight-through”
order (aka DC term)
–1st diffracted
order
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several diffracted plane
waves
“diffraction orders”
Fraunhofer diffraction
from periodic array of holes
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Sinusoidal amplitude grating
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Sinusoidal amplitude grating
incident
plane
wave
Only the 0th and ±1st
orders are visible
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Sinusoidal amplitude grating
one
plane
wave
three
plane
waves
far field
+1st order
three
converging
spherical waves
0th
order
–1st order
diffraction efficiencies
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Dispersion
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Dispersion from a grating
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Dispersion from a grating
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Prism dispersion vs grating
dispersion
Blue light is refracted at
larger angle than red:
Blue light is diffracted at
smaller angle than red:
normal dispersion
anomalous dispersion
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The ideal thin lens
as a Fourier transform engine
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Fresnel diffraction
Reminder
coherent
plane-wave
illumination
The diffracted field is the convolution of the transparency with a spherical wave
Q: how can we “undo” the convolution optically?
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Fraunhofer diffraction
Reminder
The “far-field” (i.e. the diffraction pattern at a
large
longitudinal distance l equals the Fourier
transform
of the original transparency
calculated at spatial frequencies
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Q: is there another
optical element
who
can perform a
Fourier
transformation
without having to
go
too far (to ) ?
The thin lens (geometrical optics)
f (focal length)
object at 
(plane wave)
point object
at finite distance
(spherical wave)
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Ray bending is proportional
to the distance from the axis
The thin lens (wave optics)
incoming wavefront
a(x,y)
outgoing
wavefront
a(x,y) t(x,y)eiφ(x,y)
(thin transparency approximation)
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The thin lens transmission function
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The thin lens transmission function
this constant-phase term can be omitted
where
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is the focal length
Example: plane wave through lens
plane wave: exp{i2πu0x}
angle θ0, sp. freq. u0θ0 /λ
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Example: plane wave through lens
back focal plane
wavefront after lens :
ignore
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spherical
wave,
converging
off–axis
Example: spherical wave through
lens
front focal plane
spherical wave,
diverging
off–axis
spherical wave (has propagated distance ) :
lens transmission function :
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Example: spherical wave through
lens
front focal plane
spherical wave,
diverging
off–axis
plane
wave
at
angle
wavefront after lens
ignore
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Diffraction at the back focal plane
thin
transparency
g(x,y)
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thin
lens
back focal plane
diffraction pattern
gf(x”,y”)
Diffraction at the back focal plane
1D calculation
Field before
lens
Field after lens
Field at back f.p.
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Diffraction at the back focal plane
1D calculation
2D version
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Diffraction at the back focal plane
spherical
wave-front
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Fourier transform
of g(x,y)
Fraunhofer diffraction vis-á-vis a
lens
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Spherical – plane wave duality
point source at (x,y)
amplitude gin(x,y)
plane wave oriented
towards
... of plane
waves
... a superposition ... corresponding to
point sources
in the object
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each output coordinate
(x’,y’) receives ...
Spherical – plane wave duality
produces a spherical wave
a plane wave departing
produces a spherical wave
converging
from the transparency
towards
converging
at angle (θx, θy) has amplitude
equal to the Fourier coefficient
each output coordinate
at frequency (θx/λ, θy /λ) of
(x’,y’) receives amplitude
gin(x,y)
equal
to that of the corresponding
Fourier component
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Conclusions
• When a thin transparency is illuminated coherently by
a monochromatic plane wave and the light passes
through a lens, the field at the focal plane is the
Fourier transform of the transparency times a
spherical wavefront
• The lens produces at its focal plane the Fraunhofer
diffraction pattern of the transparency
• When the transparency is placed exactly one focal
distance behind the lens (i.e., z=f ), the Fourier
transform relationship is exact.
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