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Prof. Jose Sasian
Chromatic Aberrations
Lens Design OPTI 517
Prof. Jose Sasian
Second-order chromatic aberrations
• Change of image location with λ (axial or longitudinal chromatic aberration)
• Change of magnification with λ (transverse or lateral chromatic aberration
2 2000 200 111 020, cosW H W W H W H W
Prof. Jose Sasian
Chromatic Aberrations
• Variation of lens aberrations as a function of wavelength
• Chromatic change of focus: W020
• Chromatic change of magnification W111
• Fourth-order:W040 and other• Spherochromatism
Prof. Jose Sasian
2020W 111 cosW H
Chromatic Aberrations
Prof. Jose Sasian
Topics• Chromatic coefficients• Optical glass and selection• Index interpolation• Achromats: crown and flint:
different solutions• Achromats: dialyte; single
glass• Mangin lens• Third-order behavior• Spherochromatism• Secondary spectrum• Tertiary spectrum
• Apochromats• Super-apochromats• Buried surface• Monochromatic design: one
task at a time• Lateral color correction as an
odd aberration• Color correction in the
presence of axial color• Field lens to control lateral
color: field lenses in general• Conrady’s D-d sum
Prof. Jose Sasian
Chromatic aberration coefficients
0201
1 /2
j
j j ji
W A n n y
1111
/j
j j ji
W A n n y
For a system of j surfaces
1111
j
i i
W yy
2020
1
12
j
i i
W y
For a system of thin lenses
Prof. Jose Sasian
With stop shift
020 0W
111 0202 yW Wy
Prof. Jose Sasian
Review of paraxial quantitiesy
ru’
s’
1/ nn nn
Prof. Jose Sasian
Chromatic coefficients
rsnn
rsnnyW 111
'1''
2
2
020
nnAy
rsn
rsnyW
2111
'1'
2
2
020
2
0201 1 1 1'
2 'yW n n
s r s r
Prof. Jose Sasian
Stop shifting
Prof. Jose Sasian
Prof. Jose Sasian
Stop shifting
Ey
Ey
Old stop plane at CCNew stop
Marginal ray height at old pupil
New chiefray
Hyyy EEshiftE
HAAH
yy
E
Eshift
New chief ray height at old pupil
Prof. Jose Sasian
Chromatic coefficients
HAAH
yy
E
Eshift
2
2shift shiftA AH H HA A
020 2y nW A
n
111nW A yn
Prof. Jose Sasian
Can show equality using the Lagrange invariant
Prof. Jose Sasian
The ratio E
E
y y Ay y A
Prof. Jose Sasian
The ratio
1 1
2 2
2 1 2 1
2 1 2 1
Ж A y AyЖ A y Ay
A y y y A A
y y A Ay A
/A A
1
2 1
0
cc
cc
Ayy y A
y y A
When the stop is shifted at the cc
Old chief ray
New chief rayMarginal ray
stopParameters are at the surface
Prof. Jose Sasian
The ratio
2 1 2 1q q p py y y yq p
q py yq p
Marginal ray
stop
Old and new chief rays
Does not depend on plane where it is calculated given similar triangles
2 1 cc
cc
yy yy y
pq
Prof. Jose Sasian
For a system of thin lenses
1111
j
i i
W yy
2020
1
12
j
i i
W y
V is the glass V-numberΦ is the optical powery is the marginal ray heighty-bar is the chief ray height
Prof. Jose Sasian
Glass• Schott, Hoya, Ohara glass catalogues (A wealth of
information; must peruse glass catalogue)• Crowns and flints are divided at V=50• Normal glasses: • Soda-lime, silica, lead (older glasses)• Crowns, light flints, flints, dense flints• Barium glasses (~1938)• Lanthanum or rare-earth glasses• Titanium• Fluorites and phosphate• Environmental and health issues in the production of
glass. Lead replaced with Titanium and Zirconium.
Prof. Jose Sasian
Other materials
• For the UV• For the IR• Plastics• Advances come usually with new
materials that extend or have new properties.
• The design is limited by the material
Prof. Jose Sasian
Prof. Jose Sasian
Glass properties
nd-1nF-nC
486.1 587.6 656.3589.8
F (H) d (He) C (H)D (Na)
nd -1 RefractivitynF-nC Mean dispersionnd -nC Partial dispersion
v=(nd-1)/(nF-nC) v-value, reciprocal dispersive power, Abbe numberP=(nd-nC)/(nF-nC) Partial dispersion ratio
λ
n
1
Prof. Jose Sasian
Glass properties
• Homogeneity• Transmission• Stria• Bubbles• Ease of fabrication; soft glasses• Coefficient of thermal expansion• Opto-thermal coefficient• Birefringence
Prof. Jose Sasian
Index of refraction variation
Rate of slope change in the blue makes it more difficult to correct for color
Prof. Jose Sasian
Index interpolation
2 22
2 2 ...b dn ac e
2 2 2 4 6 81 2 4 6 81 ...n A A A A A A
Schott
Sellmeier
HartmannConradyKettler-Drude
Must verify index of refraction
Prof. Jose Sasian
The optical wedge
δα
1n
The deviation is independent of the angle of incidence for small θ(First order approximation)
'1 1
'2 1'2 2
'1 2
1n
n
θ
Prof. Jose Sasian
Wedge
1
1 1
1 1
1
d
F C F C
d C d C
d
F C
d C
F C
n
n n n n
n n n n
nv
n n
n nP
n n
v
Pv
δ Deviation∆ Dispersionε Secondary
dispersion
Prof. Jose Sasian
Achromatic wedge pair
1 21 2
1 2
22 1
1
2 1 21 2 1 1 1 2 1 2
1 1 2
1 1
1 2 1
2 2
1 2 2
1 21 2
0
11
11
1
d
d
v vvv
v v v v vv v v
vv v n
vv v n
P Pv v
Deviation without dispersion
Prof. Jose Sasian
•There is deviation•There is no dispersion•Red and blue rays are parallel•Independent of theta to first order
Achromatic wedge
Schematic drawing
Prof. Jose Sasian
Achromatic doublet(Treated as two wedges)α
2
' '1 2 1
1 2
1 2
1 2
1 2
1 2
; '2
1 11
0
0
d
Y YZ sag Zr r
YZ Z Yr r n
v vY Yv v
Z’
Z
Prof. Jose Sasian
Achromatic doublet1 2
1 2
1 2
1 1
1 2
2 2
1 2
0Y Yv v
vv v
vv v
Independent of conjugateRequires finite difference
1 2v v
Can lead to strong optical powers
Prof. Jose Sasian
Relative sag(for 100 mm focal length)
Zonal spherical aberrationCritical airspace
Prof. Jose Sasian
Achromatic doublet
•Must have opposite power (Glass)•Strong positive and weaker negative lens•Cemented doublet•Crown in front•Flint in front•Corrected for spherical aberration•Degrees of freedom•Large achromats and cementing•Conrady D-d sum•Zonal spherical aberration
Prof. Jose Sasian
Conrady’s D-d sum
• In the presence of sphero-chromatism thebest state of correction is achieved when:
0D d n D
d
Is the difference of optical path between the marginal F and C rays.
Prof. Jose Sasian
Conrady’s D-d sum
_ _ f f c cOptical path difference Dn dn Dn dn D d n
D
d
Minimizes the rms OPD difference by joining the opd curves at the edge ofThe aperture. Valid for fourth order sphero-chromatism.
Prof. Jose Sasian
Cemented doublet solutions
• Correction for chromatic change of focus• Correction for spherical aberration• Degrees of freedom: relative powers for a
set of glasses; shapes• Crown in front: two solutions• Flint in front: two solutions• Note multiple solutions
Prof. Jose Sasian
Crown in front and flint in frontdoublet solutions (BK7 and F2)
Prof. Jose Sasian
Contact options for doublets
Full contact (cemented)Air spaced
Edge contactedCenter contacted
Prof. Jose Sasian
LimitationsSecondary spectrum, spherochromatism and
zonal spherical aberration set limits
F=100 mm, f/4, 0.5 wave scale
Prof. Jose Sasian
Achromatic doublet
20 inch diameter
F/12
BK7
F4
Prof. Jose Sasian
In this lecture
• Chromatic coefficients• Basic glass properties• Achromatic wedge-pair and lens doublets• Examples• D-d method• Achromatic doublet• Diversity of solutions