MIT 2'712'710 Optics

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Today Imaging with coherent light
Coherent image formation space domain
description impulse response spatial
frequency domain description coherent transfer
function
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The 4F system
Fourier transform relationship
Fourier transform relationship
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The 4F system
Theorem
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The 4F system
object plane
Fourier plane
Image plane
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The 4F system
object plane
Fourier plane
Image plane
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The 4F system with FP aperture
object plane
Fourier plane aperture-limited
Image plane blurred i. e. low-pass filtered
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Impulse response transfer function
A point source at the input plane ...
... results not in a point image but in a
diffraction pattern h(x,y)
Point source at the origin ?delta function d(x,y)
h(x,y) is the inpulse response of the
system More commonly, h(x,y) is called
the Coherent Point Spread Function (Coherent PSF)
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Coherent imaging as a linear, shift-invariant
system
Thin transparency
output amplitude
impulse response
convolution
illumi nation
Fourier transform
Fourier transform
transfer function
(plane wave spectrum
multiplication
transfer function H(u,v) akapupil function
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Transfer function impulse response of
rectangular aperture
Impulse response Airy function
Transfer function circular aperture
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Coherent imaging as a linear, shift-invariant
system
Example 4F system with circular aperture _at_
Fourier plane
Thin transparency
output amplitude
Impulse response
convolution
illumi nation
Fourier transform
Fourier transform
transfer function
(plane wave spectrum
multiplication
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Transfer function impulse response of
rectangular aperture
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Coherent imaging as a linear, shift-invariant
system
Example 4F system with circular aperture _at_
Fourier plane
Thin transparency
output amplitude
Impulse response
convolution
illumi nation
Fourier transform
Fourier transform
transfer function
(plane wave spectrum
multiplication
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Aperturelimited spatial filtering
Image plane grating is imaged with
lateral de-magnification
object plane grating generates one spatial
frequency
Fourier plane aperture unlimited
(all orders pass)
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Aperturelimited spatial filtering
Image plane grating is not imaged only 0th
order (DC component) surviving
object plane grating generates one spatial
frequency
Fourier plane aperture limited
(some orders cut off)
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Spatial frequency clipping
field after input transparency
field before filter
field after filter
field at output (image plane)
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Effect of spatial filtering
Fourier plane filter with circ-aperture
Original object (sinusoidal spatial variation,
i.e. grating)
Frequency-filtered image (spatial variation
blurred out, only average survives)
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f120cm ?0.5µm
Spatial frequency clipping
monochromatic coherent on-axis illumination
Fourier plane cire-aperture
object plane Transparency
Fourier filter transitivity
intensity at input plane
Intensity before Fourier Filter (negative
contrast)
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Space-Fourier coordinate transformations
pixel size
frequency resolution
spare domain
Spatial Frequency domain
Nyquist relationships.
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4F coordinate transformations
pixel size
spare domain
Fourier plane
Nyquist relationships.
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Spatial frequency clipping
f120cm ?0.5µm
monochromatic coherent on-axis illumination
Fourier plane cire-aperture
Image plane observed field
object plane transparency
Fourier filter transitivity
intensity at input plane
Intensity before Fourier Filter (negative
contrast)
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Formation of the impulse response
Image plane Fourier transform of aperture, Airy
pattern
object plane pinhole generates spherical wave
Fourier plane circ-aperture limited
(plane wave is clipped)
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Lowpass filtering
field after input transparency
field before filter
field after filter
field at output (image plane)
(Airy pattern)
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Effect of spatial filtering
Fourier plane filter with circ-aperture
Original object (small pinhole ?impulse,
generating spherical wave past the transparency)
Impulse reponse (aka point point-spread
function, original point has blurred to an Airy
pattern, or jinc)
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f120cm ?0.5µm
Lowpass filtering the impulse
monochromatic coherent on-axis illumination
object plane transparency
Fourier plane cire-aperture
intensity at input plane
Fourier filter transitivity
Intensity before Fourier Filter (negative
contrast)
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Spatial frequency clipping
monochromatic coherent on-axis illumination
Fourier plane cire-aperture
object plane transparency
Image plane observed field
intensity at input plane
Intensity after Fourier filter
Intensity at output plane
note pseudo-accentuated sidelobes
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Low-pass filtering with the 4F system
monochromatic coherent on-axis illumination
Fourier plane cire-aperture
object plane transparency
Image plane observed field
field arriving At Fourier plane
Fourier transform
field arriving from Fourier plane
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Spatial filtering with the 4F system
monochromatic coherent on-axis illumination
Fourier plane cire-aperture
object plane transparency
Image plane observed field
field arriving At Fourier plane
Fourier transform
Fourier transform
field arriving from Fourier plane
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Examples the amplitude MIT pattern
Original MIT pattern
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Weak lowpass filtering
Pinhole, radius 2.5mm
Filtered with pinhole, radius 2.5mm
Fourier filter
Intensity _at_ image plane
f120cm ?0.5µm
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Moderate lowpass filtering
(aka blurringblurring)
Pinhole, radius 1mm
Filtered with pinhole, radius 1mm
Fourier filter
Intensity _at_ image plane
f120cm ?0.5µm
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Strong lowpass filtering
Pinhole, radius 0.5mm
Filtered with pinhole, radius 0.5mm
Fourier filter
Intensity _at_ image plane
f120cm ?0.5µm
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Moderate highpass filtering
Reflective disk, radius 0.5mm
Filtered with reflective disk, radius 0.5mm
Fourier filter
Intensity _at_ image plane
f120cm ?0.5µm
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Strong highpass filtering
(aka edge enhancement)
Reflective disk, radius 2.5mm
Filtered with reflective disk, radius 2.5mm
Fourier filter
Intensity _at_ image plane
f120cm ?0.5µm
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1-dimensional blurring
Filtered with horizontal slit, width 2mm
Horizontal slit, width 2mm
Fourier filter
Intensity _at_ image plane
f120cm ?0.5µm
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1-dimensional blurring
vertical slit, width 2mm
Filtered with vertical slit, width 2mm
Fourier filter
Intensity _at_ image plane
f120cm ?0.5µm
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Phase objects
thickness
protruding part phase-shifts coherent
illumination by amount f2p(n-1)t/?
glass plate (transparent)
Often useful in imaging biological objects
(cells, etc.)
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Viewing phase objects
Original 0.1 rad phase MIT pattern (phase)
Original phase MIT pattern (intensity)
Amplitude (need interferometer)
Intensity (object is invisible)
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Zernicke phase-shift mask
phase-shift mask (phase), radii 5mm 1mm (phase)
phase-shift mask (magnitude), radii 5mm 1mm
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Imaging with Zernicke mask
phase-shift mask (phase), radii 5mm 1mm (phase)
phase-shift mask, radii 5mm 1mm
Filtered with,
Fourier filter
Intensity _at_ image plane
f120cm ?0.5µm
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Imaging with Zernicke mask
phase-shift mask (phase), radii 5mm 0.1mm
(phase)
Filtered with,
phase-shift mask, radii 5mm 0.1mm
Fourier filter
Intensity _at_ image plane
f120cm ?0.5µm