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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

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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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MIT 2.71/2.710 Optics 11/08/04 wk10-a- 3 The 4F system Theorem:

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MIT 2.71/2.710 Optics 11/08/04 wk10-a- 4 The 4F system object plane Fourier plane Image plane

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MIT 2.71/2.710 Optics 11/08/04 wk10-a- 5 The 4F system object plane Fourier plane Image plane

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MIT 2.71/2.710 Optics 11/08/04 wk10-a- 6 The 4F system with FP aperture object planeFourier plane : aperture-limited Image plane: blurred i. e. low-pass filtered

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MIT 2.71/2.710 Optics 11/08/04 wk10-a- 7 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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MIT 2.71/2.710 Optics 11/08/04 wk10-a- 8 Coherent imaging as a linear, shift-invariant system transfer function H(u,v): akapupil function Thin transparency output amplitude impulse response convolution Fourier transform illumi nation (plane wave spectrum transfer function multiplication

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MIT 2.71/2.710 Optics 11/08/04 wk10-a- 9 Transfer function & impulse response of rectangular aperture Transfer function: circular aperture Impulse response: Airy function

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MIT 2.71/2.710 Optics 11/08/04 wk10-a- 10 Coherent imaging as a linear, shift-invariant system Example: 4F system with circular aperture @ Fourier plane Thin transparency illumi nation Impulse response output amplitude Fourier transform Fourier transform transfer function multiplication convolution (plane wave spectrum

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MIT 2.71/2.710 Optics 11/08/04 wk10-a- 11 Transfer function & impulse response of rectangular aperture

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MIT 2.71/2.710 Optics 11/08/04 wk10-a- 12 Coherent imaging as a linear, shift-invariant system Example: 4F system with circular aperture @ Fourier plane Thin transparency illumi nation Impulse response output amplitude Fourier transform Fourier transform transfer function multiplication convolution (plane wave spectrum

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MIT 2.71/2.710 Optics 11/08/04 wk10-a- 13 Aperture–limited spatial filtering object plane: grating generates one spatial frequency Fourier plane: aperture unlimited Image plane: grating is imaged with lateral de-magnification (all orders pass)

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MIT 2.71/2.710 Optics 11/08/04 wk10-a- 14 Aperture–limited spatial filtering object plane: grating generates one spatial frequency Fourier plane: aperture limited Image plane: grating is not imaged only 0 th order (DC component) surviving (some orders cut off)

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MIT 2.71/2.710 Optics 11/08/04 wk10-a- 15 Spatial frequency clipping field after input transparency field before filter field after filter field at output (image plane)

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MIT 2.71/2.710 Optics 11/08/04 wk10-a- 16 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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MIT 2.71/2.710 Optics 11/08/04 wk10-a- 17 Spatial frequency clipping f1=20cm λ =0.5 μ m monochromatic coherent on-axis illumination object plane Transparency intensity at input plane Intensity before Fourier Filter (negative contrast) Fourier plane cire-aperture Fourier filter transitivity

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MIT 2.71/2.710 Optics 11/08/04 wk10-a- 18 Space-Fourier coordinate transformations :pixel size :frequency resolution spare domain Spatial Frequency domain Nyquist relationships.

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MIT 2.71/2.710 Optics 11/08/04 wk10-a- 19 4F coordinate transformations :pixel size Nyquist relationships. spare domain Fourier plane

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MIT 2.71/2.710 Optics 11/08/04 wk10-a- 20 Spatial frequency clipping f 1 =20cm λ =0.5 μ m monochromatic coherent on-axis illumination object plane transparency intensity at input plane Intensity before Fourier Filter (negative contrast) Fourier plane cire-aperture Fourier filter transitivity Image plane observed field

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MIT 2.71/2.710 Optics 11/08/04 wk10-a- 21 Formation of the impulse response object plane: pinhole generates spherical wave Fourier plane: circ-aperture limited Image plane: Fourier transform of aperture, Airy pattern (plane wave is clipped)

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MIT 2.71/2.710 Optics 11/08/04 wk10-a- 22 field after input transparency field before filter field after filter field at output (image plane) Low–pass filtering (Airy pattern)

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MIT 2.71/2.710 Optics 11/08/04 wk10-a- 23 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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MIT 2.71/2.710 Optics 11/08/04 wk10-a- 24 Low–pass filtering the impulse f 1 =20cm λ =0.5 μ m monochromatic coherent on-axis illumination object plane transparency intensity at input plane Intensity before Fourier Filter (negative contrast) Fourier plane cire-aperture Fourier filter transitivity

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MIT 2.71/2.710 Optics 11/08/04 wk10-a- 25 Spatial frequency clipping monochromatic coherent on-axis illumination object plane transparency intensity at input plane Intensity after Fourier filter Fourier plane cire-aperture Intensity at output plane Image plane observed field note: pseudo-accentuated sidelobes

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MIT 2.71/2.710 Optics 11/08/04 wk10-a- 26 Low-pass filtering with the 4F system monochromatic coherent on-axis illumination object plane transparency Fourier plane cire-aperture Image plane observed field field arriving At Fourier plane field arriving from Fourier plane Fourier transform

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MIT 2.71/2.710 Optics 11/08/04 wk10-a- 27 monochromatic coherent on-axis illumination object plane transparency Fourier plane cire-aperture Image plane observed field Spatial filtering with the 4F system field arriving At Fourier plane field arriving from Fourier plane Fourier transform Fourier transform

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MIT 2.71/2.710 Optics 11/08/04 wk10-a- 28 Examples: the amplitude MIT pattern Original MIT pattern

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MIT 2.71/2.710 Optics 11/08/04 wk10-a- 29 Weak low–pass filtering Pinhole, radius 2.5mmFiltered with pinhole, radius 2.5mm Fourier filterIntensity @ image plane f 1 =20cm λ =0.5 μ m

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MIT 2.71/2.710 Optics 11/08/04 wk10-a- 30 Moderate low–pass filtering (aka blurringblurring) Fourier filterIntensity @ image plane Pinhole, radius 1mmFiltered with pinhole, radius 1mm f 1 =20cm λ =0.5 μ m

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MIT 2.71/2.710 Optics 11/08/04 wk10-a- 31 Strong low–pass filtering Pinhole, radius 0.5mmFiltered with pinhole, radius 0.5mm Fourier filterIntensity @ image plane f 1 =20cm λ =0.5 μ m

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MIT 2.71/2.710 Optics 11/08/04 wk10-a- 32 Moderate high–pass filtering Reflective disk, radius 0.5mm Filtered with reflective disk, radius 0.5mm Fourier filterIntensity @ image plane f 1 =20cm λ =0.5 μ m

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MIT 2.71/2.710 Optics 11/08/04 wk10-a- 33 Strong high–pass filtering Reflective disk, radius 2.5mmFiltered with reflective disk, radius 2.5mm Fourier filterIntensity @ image plane f 1 =20cm λ =0.5 μ m (aka edge enhancement)

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MIT 2.71/2.710 Optics 11/08/04 wk10-a- 34 1-dimensional blurring Horizontal slit, width 2mm Filtered with horizontal slit, width 2mm Fourier filterIntensity @ image plane f 1 =20cm λ =0.5 μ m

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MIT 2.71/2.710 Optics 11/08/04 wk10-a- 35 1-dimensional blurring vertical slit, width 2mm Filtered with vertical slit, width 2mm Fourier filterIntensity @ image plane f 1 =20cm λ =0.5 μ m

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MIT 2.71/2.710 Optics 11/08/04 wk10-a- 36 Phase objects glass plate (transparent) thickness protruding part phase-shifts coherent illumination by amount φ =2 π (n-1)t/ λ Often useful in imaging biological objects (cells, etc.)

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MIT 2.71/2.710 Optics 11/08/04 wk10-a- 37 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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MIT 2.71/2.710 Optics 11/08/04 wk10-a- 38 Zernicke phase-shift mask phase-shift mask (magnitude), radii 5mm & 1mm phase-shift mask (phase), radii 5mm & 1mm (phase)

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MIT 2.71/2.710 Optics 11/08/04 wk10-a- 39 Imaging with Zernicke mask Fourier filterIntensity @ image plane f 1 =20cm λ =0.5 μ m phase-shift mask (phase), radii 5mm & 1mm (phase) phase-shift mask, radii 5mm & 1mm Filtered with,

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MIT 2.71/2.710 Optics 11/08/04 wk10-a- 40 Imaging with Zernicke mask Fourier filterIntensity @ image plane f 1 =20cm λ =0.5 μ m phase-shift mask (phase), radii 5mm & 0.1mm (phase) phase-shift mask, radii 5mm & 0.1mm Filtered with,

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