MIT 2.71/2.710 Optics 11/08/04 wk10-a-1

Today

Imaging with coherent light

• Coherent image formation

–space domain description: impulse response

–spatial frequency domain description: coherent transfer function

MIT 2.71/2.710 Optics 11/08/04 wk10-a-2

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-limitedImage 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 δ(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

transfer function H(u,v): akapupil function

Thin transparencyoutput

amplitude

impulse response

convolution

Fourier transform

Fourier transform

illumi nation

(≡plane wave spectrum

transfer function

multiplication

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Transfer function & impulse responseof rectangular aperture

Transfer function:circular aperture

Impulse response:Airy function

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Coherent imaging as a linear, shift-invariant system

Example: 4F system with circular aperture @ Fourier plane

Thin transparency

illumination

Impulse response

output amplitude

Fouriertransform

Fouriertransform

transfer function

multiplication

convolution

(≡plane wave spectrum

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Transfer function & impulse responseof rectangular aperture

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Coherent imaging as a linear, shift-invariant system

Example: 4F system with circular aperture @ Fourier plane

Thin transparency

illumination

Impulse response

output amplitude

Fouriertransform

Fouriertransform

transfer function

multiplication

convolution

(≡plane wave spectrum

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Aperture–limited spatial filtering

object plane:grating generates

one spatial frequency

Fourier plane:aperture unlimited

Image plane:grating is imaged

with lateralde-magnification

(all orders pass)

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Aperture–limited spatial filtering

object plane:grating generates

one spatial frequency

Fourier plane:aperture limited

Image plane:grating is not imaged

only 0th order(DC component)

surviving(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 filterwith circ-aperture

Original object(sinusoidal spatial

variation, i.e. grating)

Frequency-filtered image(spatial variation blurred out,

only average survives)

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Spatial frequency clipping f1=20cmλ=0.5μm

monochromatic coherent on-axis

illumination

object planeTransparencyintensity at input plane Intensity before Fourier Filter (negative contrast)

Fourier planecire-aperture

Fourier filter transitivity

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Space-Fourier coordinate transformations

:pixel size :frequencyresolution

sparedomain

SpatialFrequencydomainNyquist

relationships.

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4F coordinate transformations

:pixel size

Nyquistrelationships.

sparedomain

Fourierplane

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Spatial frequency clipping f1=20cmλ=0.5μm

monochromatic coherent on-axis

illumination

object planetransparencyintensity at input plane Intensity before Fourier Filter (negative contrast)

Fourier planecire-aperture

Fourier filter transitivity

Image planeobserved field

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Formation of the impulse response

object plane:pinhole generates

spherical waveFourier plane:

circ-aperture limited

Image plane:Fourier transform

of aperture,Airy pattern

(plane wave is clipped)

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field after input transparency

field before filter

field after filter

field at output(image plane)

Low–pass filtering

(Airy pattern)

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Effect of spatial filtering

Fourier plane filterwith 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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Low–pass filtering the impulse f1=20cmλ=0.5μm

monochromatic coherent on-axis

illumination

object planetransparency

intensity at input plane Intensity before Fourier Filter (negative contrast)

Fourier planecire-aperture

Fourier filter transitivity

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Spatial frequency clipping

monochromatic coherent on-axis

illumination

object planetransparency

intensity at input plane Intensity after Fourier filter

Fourier planecire-aperture

Intensity at output plane

Image planeobserved field

note: pseudo-accentuated sidelobes

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Low-pass filtering with the 4F system

monochromatic coherent on-axis

illumination

object planetransparency

Fourier planecire-aperture

Image planeobserved field

field arrivingAt Fourier plane

field arrivingfrom Fourier plane

Fouriertransform

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monochromatic coherent on-axis

illumination

object planetransparency

Fourier planecire-aperture

Image planeobserved field

Spatial filtering with the 4F system

field arrivingAt Fourier plane

field arrivingfrom Fourier plane

Fouriertransform

Fouriertransform

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Examples: the amplitude MIT pattern Original MIT pattern

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Weak low–pass filtering

Pinhole, radius 2.5mm Filtered with pinhole, radius 2.5mm

Fourier filter Intensity @ image plane f1=20cmλ=0.5μm

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Moderate low–pass filtering(aka blurringblurring)

Fourier filter Intensity @ image plane

Pinhole, radius 1mm Filtered with pinhole, radius 1mm

f1=20cmλ=0.5μm

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Strong low–pass filtering

Pinhole, radius 0.5mm Filtered with pinhole, radius 0.5mm

Fourier filter Intensity @ image plane f1=20cmλ=0.5μm

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Moderate high–pass filtering

Reflective disk, radius 0.5mm Filtered with reflective disk, radius 0.5mm

Fourier filter Intensity @ image plane f1=20cmλ=0.5μm

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Strong high–pass filtering

Reflective disk, radius 2.5mm Filtered with reflective disk, radius 2.5mm

Fourier filter Intensity @ image plane f1=20cmλ=0.5μm

(aka edge enhancement)

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1-dimensional blurring

Horizontal slit, width 2mm Filtered with horizontal slit, width 2mm

Fourier filter Intensity @ image plane f1=20cmλ=0.5μm

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1-dimensional blurring

vertical slit, width 2mm Filtered with vertical slit, width 2mm

Fourier filter Intensity @ image plane f1=20cmλ=0.5μm

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Phase objects

glass plate(transparent)

thickness

protruding partphase-shifts

coherent illuminationby amount φ=2π(n-1)t/λ

Often useful in imaging biological objects (cells, etc.)

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Viewing phase objects

Intensity(object is invisible)

Amplitude(need

interferometer)

Original phase MIT pattern (intensity) Original 0.1 rad phase MIT pattern (phase)

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Zernicke phase-shift mask

phase-shift mask (magnitude), radii 5mm & 1mm phase-shift mask (phase), radii 5mm & 1mm (phase)

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Imaging with Zernicke mask

Fourier filter Intensity @ image plane f1=20cmλ=0.5μm

phase-shift mask (phase), radii 5mm & 1mm (phase) phase-shift mask, radii 5mm & 1mmFiltered with,

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Imaging with Zernicke mask

Fourier filter Intensity @ image plane f1=20cmλ=0.5μm

phase-shift mask (phase), radii 5mm & 0.1mm (phase) phase-shift mask, radii 5mm & 0.1mmFiltered with,