Lecture 7: PSF and transfer function
Diffraction at the system aperture
Optical system: works as a low pass filter
truncation of the spherical wave--> a finite angle light cone
In image space: uncomplete constructive interference of partial waves--> spreaded image spreaded
PSF by Huygens Principle
Huygens wavelets--> vectorial field components
phase-->direction
amplitude-->length
constructive and destructive interference
Ideal point spread function
Apodization: variable lengths of arrows
decreasing length of arrows
Aberrations: variable orientation of arrows(Amplitude constant on pupil plane)
real wavefront with aberrations central peak reduced
Fraunhofer Point Spread Function:
Fraunhofer approximation in the far field for large Fresnel number
N_F = r_p^2 / (lambda*z)~1
PSF: Fourier transform of the complex pupil function
Perfect PSF
Circular homogeneous illuminated aperture: intensity distribution
Resolution
transversal better than axial
Axial: sinc scale
R_E = n*lambda / NA^2
in air n=1
transversal: Airy scale:
D_Airy = 1.22*lambda/ NA
Abbe resolution and assumptions
Abbe resolution with scaling to lambda/NA:
A resolution beyond the Abbe limit is only possible with violating of certain assumptions
Perfect Lateral Point Spread Function: Airy distribution
Perfect Axial Point Spread Function
decrease intensity to 80% light (R_E)--> diffraction limited system
Axial distribution of intensity corresponds to defocus
sinc^2
Scale for depth of focus: 4R_E
Rayleigh length R_E = n'lambda / NA^2
PSF with Aberrations
Spherical Aberration
Axial asymmetrical distribution off axis
Peak moves
Gaussian illumination
Axial: lorentzian shape
Strehl Ratio
Definition(for one point): ratio of real peak intensity (with aberrations) referenced on ideal peak intensity
criterion: D_S>0.5 --> good quality
In microscopy: D_S>0.8
Approximation of Marechal
Point resolution
Formula Abbe: delta[x] = k * lambda/NA
Assumption for the validity of the formula
1.no evanescent waves(no near field effects)
2.no non-linear effects(2-photon)
Incoherent resolution
Incoherent 2-Point Resolution: Sparrow criterion
Incoherent 2-Point Resolution: Rayleigh Criterion
Dependence on NA
Incoherent 2-Point resolution: visual
Optical Transfer Function
Normalized OTF in frequency domain
Fourier transform of the PSF-intensity
Absolute value of OTF--> modulation transfer function MTF
Gives the contrast at a special spatial frequency of a sine grating
OTF: Autocorrelation of shifted pupil function
Duffieux-integral interpretation: Interference of 0th and 1st diffraction of the light in the pupil
larger overlapped area --> more interference --> better contrast
The area of the overlap corresponds to the information transfer of the structural details
Frequency limit of resolution:
areas completely seperated
Contrast