1. Fresnel diffraction formulae
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2. spherical wave kernels
Fresnel propagation: Huygens picture
Thin transparency
field at observation plane
superposition of
spherical wave kernels
(Huygens “wavelets”)
weighed by the
incoming wavefront
complicated
incoming
wave-front
field just before
the transparency
field just after
the transparency
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3. Fresnel propagation: Fourier picture
Thin transparency
The field immediately past
the transparency is “decomposed”
into a multitude of plane waves,
weighed by the Fourier transform
optical
axis
complex amplitude
for this component:
complicated
incoming
wave-front
field just before
the transparency
field just after
the transparency
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4. Fresnel propagation: Fourier picture
Thin transparency
The field at the observation
plane is the superposition
of the fields generated by all
these plane waves (i.e., the
“Fourier components” of the
t(x,y)g2(x,y) field);
note the quadratic phase delay
between different components:
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5. Fresnel propagation: Fourier picture
Thin transparency
In the continuous limit,
the plane wave superposition
becomes an inverse Fourier
transform; note that the
quantity being transformed
(the “operand”) is the
product of G2(u,v) times a
quadratic phase representing
the FT of a Huygens “wavelet”
in agreement with the
Convolution theorem
Inverse FT
kernel
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wk11-b-5

6. as a linear, shift-invariant system
Fresnel diffraction
as a linear, shift-invariant system
output
amplitude
impulse response
convolution
Fourier
transform
Fourier
transform
transfer function
(≡plane wave
spectrum)
multiplication
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7. The 4F system object plane Fourier plane Image plane 2.71/2.710 Optics
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8. The 4F system object plane Fourier plane Image plane 2.71/2.710 Optics
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9. The 4F system with FP aperture
object plane
Fourier plane: aperture-limited
Image plane: blurred
(i.e. low-pass filtered)
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10. The 4F system with FP aperture
Transfer function:
circular aperture
Impulse response:
Airy function
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