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Lecture 8: Further performance criteria (Sine Condition (Lagrange…
Lecture 8: Further performance criteria
Criteria overview
Rayleigh Criterion
in the case of diffraction limited
Marechal Criterion
Strehl Ratio Criterion
Centroid ray
Line of sight
straight line
wave aberrations of odd order in the azimuthal term influence the centroid: tilt and coma-like aberrations
Pupil with apodization
PSF with Coma
PSF with apodization
Deviation of centroid ray from chief ray
Asymmetrical apodization
coincidence in image plane
Coma phase aberration
coincidence in pupil
Depth of Focus
Diffraction Consideration
Normailized aixal intensity for uniform pupil amplitude
I = I_0 * sinc(u)^2
Decrease of intensity onto 80%: R_u/2
Depth of focus: R_u = n'*lambda/(NA)^2
Edge and line imaging
Fresnel Edge Diffraction
Edge Spread Function
Line spread function
Integral over PSF
Realization: narrow slit
Convolution of slit width
Fourier transform of pupil in one dimension
Focus spot size of a lens
Comparison Germetrical Spot
Spot diamater as a function of aberrations
Focal diameter determination
D_foc = beta * (lambda/NA)
Additonal factor beta
no truncation, beamradius w
lambda/NA = lambda*f/w
With truncation at radius a:
lambda/NA = lambda*f/a
Changing the NA/Aperture D
Large NA/D (Diffraction neglectible)
geometrical aberrations dominate
small values of D (Diffraction limited)
diffraction dominates--> Airy formula
Pupil aberration
Spherical aberration of the chief ray/pupil imaging
Exit pupil location depends on the field height
Illumination for decentered pupil
dark zones due to vignetting
Sine Condition
Lagrange invariant for paraxial angles U, U'
-->extension for finite aperture angle u
n
y
sin(u) = n'
y'
sin(u')
corresponds to energy conservation in the system
Constant magnification for all aperture zones
Sin-condition violates--> magnification varies for different points
Pupil shape for finite aperture is a sphere
Tangential and Sagittal Coma
Sagittal coma depends on x_p
describes the asymmetry
Only asymmetry removed with sine condition: sagittal coma vanishes
Tangential coma depends on y_p
corresponds to spherical aberration under skew conditions larger by a factor of 3
Skew spherical aberration
General Aplanatic surface
General approach of Fermat principle
Special case OPD=0:n'
s'=n
s
-->solution: spherical aplanatic surface
--> coma vanishing
Isoplanatism condition of Staeble-Lihotzky
Piecewise isoplanatism
Seidal approach
perturbation of the paraxial ray
intrinsic & induced aberration
Vectorial aberration
Fourier optics