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Lecture 11: Correction 2 (Petzval Theorem for Field Curvature (Flattening…
Lecture 11: Correction 2
Petzval Theorem for Field Curvature
Formula for surfaces: R^-1 = - n'_m
SIGMA_k( (n'_k-n_k) / (n_k
n'_k
r_k) )
for thin lenses (in air): R^-1=- SIGMA_j ((n_j
f_j)^-1)
no dependence on bending or stop location
Goal: vanish Petzval curvature and get positive total refractive power: f^-1= SIGMA_j(h_j/h_1*f_j^-1)
Positive lenses with:
high refractive index;
large marginal ray height;
large contribution to power;
low weighting in Petzval sum
negative lenses with:
low refractive index;
small marginal ray height;
small negative contribution to power;
high weighting in Petzval sum
Flattening Meniscus Lenses
R^-1= -(n
f)^-1 + ((n-1)/n)^2
d/(r_1*r_2) (2 surfaces of thick lenses)
Hoeghs meniscus: identical radii
Petzval sum =0
positive refractive power: F'=(n-1)^2
d/(n
r^2)
Concentric meniscus (r_2 = r_1 - d)
Petzval sum negative:R^-1 = (n-1)
d/(n
r_1
(r_1-d))
weak negative focal length
refractive power for thickness d: F' = -(n-1)
d/(n
r_1
(r_1 -d))
Thick meniscus without refractive power
r_2 = r_1 - d
(n-1)/n
R^-1 = (n-1)^2
d / (n
r_1
(n
r_1 - d
(n-1)) ) >0
Correcting Petzval Curvature
Triplet Group with +-+ for collimated beam
Lithographic lens
principle: certain number of bulges
Microscope Objective Lens
Goal: reduction of Petzval sum
keeping astigmatism corrected
for one wavelength
single meniscus lenses
two meniscus lenses
symmetrical triplet
for more than two wavelengths
achromatized meniscus len
two achromatized meniscus lenses
modified achromatized triplet
Size reduction by aspheres
Asphere does no favor to the flatterning
Flattening Field Lens
Field lens + CCD: high requirement for cleanness
Field lens:
in or near image planes
Influence on the chief ray-->
Pupil shifted
Lens can be shrinked
Conjugation to image plane
surface errors sharply seen
Field lens in Endoscope
Chromatical aberrations
Axial chromatical aberration
dispersion of marginal ray
different image locations
compensation by appropriate glass choice
cement two different glasses-->achromate
4 combinations
bending: correction of spherical aberration at the full aperture
Aplanatic coma correction
typical
correction for object in infinity
spherical correction at center wavelength with zone
diffraction limited for NA<0.1 (rule of thumb)
only very small field corrected
spherochromatism(Gaussian aberration)
perfect axial color correction (on axis)--> not feasible
Transverse chromatical aberration
dispersion of chief ray
different image sizes
Achromate
Total power: F = F_1 + F_2
Achromatic correction condition
F_1/v_1 + F_2/v_2 =0
Properties
1 positive and 1 negative lens
2 different sequence of plus(crown)/minus(flint)
large difference of v relaxes the bendings
achromatic correction independent from bending
bending corrects spherical Aberration at the margin
aplanatic coma correction for Special glass choices
Further optimization of materials reduces the spherical zonal aberration