Lecture 11: Correction 2

Petzval Theorem for Field Curvature

Formula for surfaces: R^-1 = - n'_m SIGMA_k( (n'_k-n_k) / (n_kn'_kr_k) )
for thin lenses (in air): R^-1=- SIGMA_j ((n_jf_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;

negative lenses with:
low refractive index;
small marginal ray height;

large contribution to power;
low weighting in Petzval sum

small negative contribution to power;
high weighting in Petzval sum

Flattening Meniscus Lenses

R^-1= -(nf)^-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 (nr_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-->

  1. Pupil shifted
  2. 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

Transverse chromatical aberration
dispersion of chief ray
different image sizes

compensation by appropriate glass choice
cement two different glasses-->achromate
4 combinations

spherochromatism(Gaussian aberration)

perfect axial color correction (on axis)--> not feasible

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

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

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