Mirna Lerotic
With Prof. Chris Jacobsen
AIRY PATTERN
Airy pattern is diffraction pattern of circular aperture.
Figure
1.Airy pattern (on log scale)
To get this I made Fourier transform of circular aperture, and plotted absolute value in order to get intensity.
That is in-focus point spread function (PSF).
CONVOLUTION
Image of an object is given by
Convolution can be thought of as shift-and-multiply operation, and is given by
Convolution theorem states that the Fourier transformation of convolution of two functions is equal to the product of the Fourier transforms of the two functions
PROPAGATION
The Fresnel-Kirchoff diffraction integral in the Fraunhofer approximation is
where t is object transmittance.
We can write diffraction integral as
where we recognize two dimensional convolution of the object transmittance and the propagation function h.
So, propagation is described by convolution with a propagator. Propagator function, after some approximations, is given by
.
So to find out-of-focus PSF one has to convolve in-focus PSF with propagator.
Figure
2. - 5. Defocused PSF for monochromatic light
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Defocus
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l=2.5nm,
drn=45nm
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z=-10000nm
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z=-5000nm
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z=0nm
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z=+5000nm
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QUASI-MONOCHROMATIC LIGHT
When doing diffraction with non-monochromatic light, each wavelength will be focused to different point along axis zF. So to get total picture at one point z0 one has to add contributions from all wavelengths at that particular point. So for each wavelength, one has to calculate out-of-focus image at z0 - zF , weighted by Gaussian. And that operation is just convolution with Gaussian.
Therefore, I described quasi-monochromatic light by convolving PSF with Gaussian. Resolution is
where
width of gaussian in coordinate space was equal to twice the depth of focus
for monochromatic light (depth_nm=8200nm).
When treating quasi-monochromatic light there are two limits:
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Defocus
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Resolving
power=1600, l=2.5nm, drn=45nm
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z=-10000nm
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z=-5000nm
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z=0nm
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z=+5000nm
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DEFOCUS IMAGE SERIES OF BARS
Bar width: 33, 71 and 148 nm.
I first calculated image for each defocus image. In order to make picture that depends on defocus I took horizontal section for each defocus image.
Figure 11. Defocus image series of bars for monochromatic light
Figure 12.Defocus image series of bars for quasi-monochromatic light
