ECEN 4616/5616
2/4/2013
Dispersion and Achromats
Zemax’s “Glass Map”:
The index of refraction of glasses is traditionally taken at three wavelengths (for use in
the visible). These are the Fraunhofer lines for hydrogen and helium ( a nearby line from
mercury was used before helium became widely available). These were convenient,
because they spanned the visible spectrum and could be duplicated with great precision in
any well-equipped laboratory.
F (hydrogen)  486nm
d (helium)  588 nm
C (hydrogen)  656 nm
pg. 1
ECEN 4616/5616
2/4/2013
In the “glass map”, the index of refraction is the index at the d line (nd), and the ‘Abbe
Number’ (often called the “V-number” by those who aren’t sure how to pronounce
“Abbe”) is:
n 1
Vd  d
nF  nC
The significance of this value follows from the calculation of the chromatic variation of
power for a thin lens.
1
1 
Power at d: K d  nd  1    nd  1C
 R1 R2 
Power at F: nF 1C
Power at C: nC 1C
Consider:
K F  K C nF  1C  nC  1C nF  nC
1



nd  1C
Kd
nd  1 Vd
Hence:
K
K F  KC  d
Vd
From the image equation, we have:
1 1
1 1
  K F , and
  KC
lF lF
lC lC
1 1
K
  K F  KC  d
lF lC
Vd
So, the change in power is proportional to the power divided by the Abbe number.
(It can also be shown that change in magnification with wavelength – transverse
K
chromatic aberration – is also proportional to d .
Vd

Hence, chromatic aberration (longitudinal and transverse) will be corrected (paraxially) if
we use two lenses whos ratios sum to zero:
K1 K 2

0
V1 V2
where the K’s are implicitly calculated at the d wavelength.
For two thin lenses in contact, K  K1  K2 . Solving these two equations, we get:
VK
VK
K1  1
and K 2   2
V1  V2
V1  V2
So, we can create a compound lens without (to first order) chromatic aberration by
combining a positive and negative lens made with glasses that have different V-numbers.
pg. 2
ECEN 4616/5616
2/4/2013
(Note that, since all V-numbers are positive, the first equation can only be satisfied if the
powers are one positive and one negative.)
Equivalent V-Number:
Suppose we design a doublet with a non-zero change in chromatic focal power:
K
K1 K 2
K

 K . Since, from K F  K C  d , we can define V 
, we can say the
K
V1 V2
Vd
K1  K 2  . Hence we can produce a
the effective V-number of our doublet is V 
 K1 K 2 



 V1 V2 
compound lens with an effective V-number that wouldn’t be available from a single
glass. This can prove useful in correcting optical instruments with widely separated
elements such as microscope and telescopes – the eyepiece can be designed to correct the
chromatic aberrations of the objective, for instance, allowing more freedom to design the
objective’s performance.
pg. 3