Lecture 5: Geometrical aberrations
Optical Image Formation
Perfect optical image:
All rays coming from one object point intersect in one image point
Real system with aberrations
transverse aberration in the image plane
longitudinal aberrations from the image plane
wave aberration in the exit pupil
perfect spherical wave surface as reference
Representation of Geometrical Aberrations
Longitudinal aberrations delta(s)
Transverse aberrations delta(y)
Angle aberrations delta(u)
Wave aberrations delta(W)
wavefront: function of pupil coordinate
Angle aberrations for a ray bundle: deviation from common direction of the collimated ray bundle
in mrad/°
Representation as a conventional spot diagram
Strong relation to spot diagram
Usually only linear sampling along the x-,y-axis
No information in the quadrant of the aperture
Typical low order polynomial contribution for
Defocus: linear
Coma: quadratic
Spherical: Cubic
Lateral color: offset
Spot diagram
All rays start in one point in the object plane
Entrance pupil is sampled equidistant
In exit pupil, the transferred grid may be distorted
In image plane a spreaded spot diagram is generated.
Symmetry on x comes from sampling on the y-axis.
Polynomial Expansion of the Aberrations
Aberrations occur for larger angle values
Taylor expansion of the deviation:
Transverse aberration: Image height; Pupil height; Pupil azimuth angle
Symmetry invariance
Type of aberration based on field y and aperture r.
Image location: defocus, tilt
Primary aberrations: Spherical, Coma, Astig./Curva., Distortion
Secondary aberrations.
Primary Aberrations
Three types of definition:
1.Only for chief ray
2.small neighborhood around chief ray
3.finite aperture
Coupling relation between Transverse aberration and wave aberration
Transverse Aberrations of Seidel
Surface Contributions
Expansion of the transverse aberration delta(y) on image height y and pupil height r
Abbreviations
Seidel aberrations expression
height ratio of marginal rays
height ratio of chief rays
Surface invariant of Abbe
Abbe invariant of the pupil imaging
Surface contributions of chromatical aberrations
Axial
transverse
total chromatical errors of a system
In height ratio and Abbe invariant
In n', n, R, s', p'
Lens contribution of Seidel
Spherical aberration
On axis, circular symmetry
Real marginal rays, shorter intersection length
Optimal image plane: circle of least rms value
lens bending
spherical aberration and focal spot diameter as a function of the lens bending
Optimal bending for incidence averaged incidence angles
Minimum larger than zero: usually no complete correction possible
Aplanatic surfaces: zero spherical aberration
ns = n's'
condition for aplanatic surface
Virtual image location
Applications:
Microscopic objective lens;
Interferometer objective lens
Aplanatic lenses: combination of one concentric and one aplanatic surface
zero contribution of the whole lens to spherical aberration
A-A: parallel offset
A-C: convergence enhanced
C-A: convergence reduced
C-C: no effect
Astigmatism
Reason:
Chief ray: oblique angle
different refractive power for tangential and sagittal section
Different focal line for tangential and sagittal
tangential rays meet closer
In between: circle of least confusion
Imaging with astigmatism
Tangential and sagittal image shell depending on the azimuth
Field curvature and image shell
Law of Petzval
No effect of bending on curvature, but distribution of lens powers and indices
Sharpe imaged zone changes from centre to margin of the image field
Astigmastism corrected:
remains a curved image shell
Bended field--> Petzval curvature(not an optimal surface with godd imaging resolution)
Coma
Tangential ray fan for coma: caustic
sagittal: groove-like surface
Coma aberration: for oblique bundles and finite aperture due to asymmetry
Relation of spot circles and pupil zones
Distortion
Purely geometrical deviations without any blur
Distortion correspond to spherical aberration of the chief ray
the location of the stop defines the chief ray path
Definition of local magnification changes
D = (y'_real - y'_ideal) / y'_ideal
D<0: barrel, front stop
D>0: pincushion, rear stop
Chromatical aberrations (1st order)
Axial chromatical aberration:
dispersion of marginal ray
different image locations
Transverse chromatical aberration
dispersion of chief ray
different image sizes
higher n -->shorter intersection length
definition of error: change in image location/intersection length
correction by glasses with different dispersion
equal intersection length for outer wavelength(blue, red), residual deviation for middle wavelength.
Secondary spectrum: Residual errors in image location
Apochromat
coincidence of the image location for at least 3 wavelengths
3 glasses necessary, only with anomal partial dispersion
Corresponds to a change of magnification with wavelength
Impression in real images
typical colored fringes/edges
color sequence depends on sigh of CHV ?
color rings isotropic around circle object
color rings uniform over FoV