Chromatic Aberrations
Lens Design OPTI 517
Prof. Jose Sasian
Second-order chromatic
aberrations

 W H ,    W000   W200 H 2   W111 H  cos     W020  2


• Change of image location with λ (axial or
longitudinal chromatic aberration)
• Change of magnification with λ (transverse
or lateral chromatic aberration
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
Chromatic Aberrations
 W020 
Prof. Jose Sasian
2
 W111 H  cos  
Topics
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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
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Prof. Jose Sasian
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
Chromatic aberration coefficients
For a system of j surfaces
 W111   Aj  j  n / n y j
i 1
j
1
Aj  j  n / n y j

2 i 1
Prof. Jose Sasian
 
 W111    yy 
i
i 1 
j
j
 W020 
For a system of thin lenses
 W020
1 j  2 
  y 
2 i 1   i
With stop shift
 W020  0
y
 W111  2
 W020
y
Prof. Jose Sasian
Review of paraxial quantities
y
r
u’
s’
Prof. Jose Sasian
n / n 
n 1
n
Chromatic coefficients
 
y 2   1 1   1 1   
W020        n '     n         
2   s ' r   s r 
 W020
 W020
Prof. Jose Sasian
y2 
 1 1 
 1 1
  n'n'      n  n     
2 
 s r 
 s' r 
y2

2
  1 1
 n 
 1 1  y

n




n

'


     A    

n
 s r  2
  s' r 
Stop shifting
Prof. Jose Sasian
Prof. Jose Sasian
Stop shifting
New chief
ray

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New stop
New chief ray height
at old pupil
Old stop plane at CC
yE

y E  shift

Marginal ray height
at old pupil
Prof. Jose Sasian
yE
 shift

 yE   yE H

yE   A 

H  H
A
yE

Chromatic coefficients
 
y
 n   
 W020        A         
2
n

yE   A 

H  H
A
yE

 
A    A  
  2 H    H H
A
 A
 shift

 shift   shift

2
 n 
 W111  A      y
n
Prof. Jose Sasian
Can show equality using the Lagrange invariant
Prof. Jose Sasian
The ratio
Prof. Jose Sasian
yE   y A


yE
y
A
The ratio
A/ A
Old chief ray
Marginal ray
New chief ray
Parameters are
at the surface
Ж  A1 y  Ay1
Ж  A2 y  Ay2
 A  y2  y1   y  A2  A1 
y2  y1 A2  A1


y
A
Prof. Jose Sasian
stop
When the stop is shifted at the cc
A1  0
y2  y1 ycc A


y
ycc A
The ratio

Marginal ray
y2 q  y1q

stop
q
yq
p
Old and new
chief rays
y2  y1 ycc

y
ycc
q
 
 
y2 p  y1 p
yp
p
q
Does not depend on plane where it is
calculated given similar triangles
Prof. Jose Sasian
p
For a system of thin lenses
 W020
1  2 
  y 
2 i 1   i
j
 
 W111    yy 
i
i 1 
j
Prof. Jose Sasian
V is the glass V-number
Φ is the optical power
y is the marginal ray height
y-bar is the chief ray height
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
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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
n
nF-nC
nd-1
1
λ
nd -1 Refractivity
nF-nC Mean dispersion
nd -nC Partial dispersion
486.1 587.6 656.3
589.8
F (H) d (He) C (H)
D (Na)
v=(nd-1)/(nF-nC) v-value, reciprocal dispersive power, Abbe number
P=(nd-nC)/(nF-nC) Partial dispersion ratio
Prof. Jose Sasian
Glass properties
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Homogeneity
Transmission
Stria
Bubbles
Ease of fabrication; soft glasses
Coefficient of thermal expansion
Opto-thermal coefficient
Birefringence
Prof. Jose Sasian
Index of refraction variation
Prof. Jose Sasian
Rate of slope change in the blue makes it
more difficult to correct for color
Index interpolation
Sellmeier
2
2
b
d


n2  a 

 ...
2
2
c
e
Schott
n 2  1  A  A1 2  A2  2  A4  4  A6  6  A8  8 ...
Hartmann
Conrady
Kettler-Drude
Must verify index of refraction
Prof. Jose Sasian
The optical wedge
α
θ
1
  1
n
'
 2  1  
'
1
  n 2
'
2
    1  
'
2
δ
    n  1 
The deviation is independent of the angle of incidence for small θ
(First order approximation)
Prof. Jose Sasian
Wedge
    nd  1
   nF  1     nC  1     nF  nC   
   nd  1     nC  1     nd  nC   
nd  1



v
  nF  nC 
nd  nC 


P
  nF  nC 



v

 P
v
Prof. Jose Sasian
δ Deviation
∆ Dispersion
ε Secondary
dispersion
Achromatic wedge pair
  1   2 
2  
1  2

0
v1 v2
v2
1
v1
  1   2  1 


v2
1   v1  v2  1    v1  v2  2
v1
v1
v2
 1   v1 
 


1



v
v
n
 1 2   d1 
1
 1   v2 



  v1  v2  nd 2  1 
2
 1 
 
  P1  P2 

 v1  v2 

Prof. Jose Sasian
Deviation without dispersion
Achromatic wedge
•There is deviation
•There is no dispersion
•Red and blue rays are parallel
•Independent of theta to first order
Prof. Jose Sasian
Schematic drawing
α
Achromatic doublet
(Treated as two wedges)
Z
Z’
Y2
Y
; Z'
Z  sag 
2r
r
1 1
Y
'
'
  Z1  Z 2  Y    
1

1
r
r
n
d
 1 2
1  2

0
v1 v2
Y 1 Y 2

0
v1
v2
Prof. Jose Sasian
Achromatic doublet
Y 1 Y 2

0
v1
v2
  1  2
1
v1

 v1  v2
2
v2


v1  v2
Prof. Jose Sasian
Independent of conjugate
Requires finite difference
v1  v2
Can lead to strong
optical powers
Relative sag
(for 100 mm focal length)
Zonal spherical aberration
Critical 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 the
best state of correction is achieved when:
  D  d  n  0
D
d
Is the difference of optical path between
the marginal F and C rays.
Prof. Jose Sasian
Conrady’s D-d sum
D
d

Optical _ path _ difference    Dn f  dn f

 


Dn

dn
   c  c     D  d  n
 

Minimizes the rms OPD difference by joining the opd curves at the edge of
The 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 front
doublet solutions (BK7 and F2)
Prof. Jose Sasian
Contact options for doublets
Full contact (cemented)
Air spaced
Edge contacted
Center contacted
Prof. Jose Sasian
Limitations
Secondary 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
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Chromatic coefficients
Basic glass properties
Achromatic wedge-pair and lens doublets
Examples
D-d method
Achromatic doublet
Diversity of solutions
Prof. Jose Sasian