Page 32
The previous ray-tracing diagrams suggest that chromatic
aberration might be corrected by properly combining a
converging lens with a diverging lens.
Indeed that is the case in the design of the achromatic
doublet lens.
ACHROMATIC
DOUBLET
r11
Consisting of a crown
equiconvex lens
cemented to a negative
flint glass lens
r21
n 1 n2
r12
r22
Let’s assume we want an “achromat doublet” of 15 cm
focal length. Lets call it fdesired= 15 cm. Since, in general,
the focal length depends on the wavelength, fdesired is
conveniently specified as that associated with yellow light
(the Fraunhofer wavelength D= 587.6 nm). Thus,
fdesired = 15 cm,
at  =D
Considering a doublet as two thin lenses, its effective focal
length in terms of its lens components can be expressed as,
1
f doublet

1
1

f1
f2
The focal length of the individual lenses is given by
(1)
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

1
 (n1  1) r1  r1  (n1  1) 1
11
12
f1

(2a)

1
 (n 2  1) r1  r1  (n 2  1) 2
21
22
f2
(2b)
The focal length of any doublet is then given by,
1
 (n1  1) 1  (n 2  1) 2
f doublet
(3)
The value of 1/fdoublet in general will depend on .
n(
Index of
refraction

D
1
D
C
F
C DF
frequency

Dispersion power 
(measured at
minimum deviation
of the D line)

D nF  nC


nD 1
The condition of achromaticity, around the wavelength of
the yellow light D, can be expressed as,
d
( 1 )  0 , at  = D
d f doublet
Or, equivalently, using expression (3),
Page 34
1
dn1
d
D
 2
dn 2
d
0
(4)
D
The derivative of n at =D can be approximated using the
red and blue Fraunhofer wavelengths, C= 656.3 nm and
F= 486.1 nm,
nF  nC
dn

d 
 F  C
D
In addition, using, the reciprocal of the dispersion power 
nD  1
1
known as “Abbe number”,
V  
 nF  nC
we obtain,
1
dn1
d
 D
 1
n1F  n1C
 1
n1F  n1C n1D  1
 1
F  C
 F   C n1D  1
1
1 n1D  1
 F   C V1 1
and using (2a)
1
dn1
d

D
1
1 1
 F   C V1 f1D
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So, the condition for achromaticity, expression (4),
becomes
1 1
1 1
(5)

0
V1 f1D V2 f 2 D
(For two given materials, their corresponding Abbe number
values are tabulated.)
Expression (5), together with the designer’s requirement
that the doublet has a desired focal length of fdesired=15 cm
at  =D, where
1
f desired

1
f1D

1
f 2D
,
(6)
constitute the couple of equations from which f1D and f2D
have to be determined.
The solution to (5) and (6) is:
1
f1D
1
f 2D


V1
1
V2  V1 f desired
V2
1
V2  V1 f desired
and
(7)
Page 36
DESIGN of an ACHROMAT
DOUBLET of 15 cm focal length
r11
DOUBLET
(1) Equiconvex
520/636 crown glass
(2) 617/366 flint glass
Catalog
code
V
1
nD
nD
(1) Borosilicate
crown
(2) Dense flint
n1 n2
r12
nC
nD
nF
656.5 nm
587.6 nm
486.1 nm
520/636
63.59
1.51764
1.52015
1.52582
617/366
36.60
1.61218
1.61715
1.62904
1/f1D
fdesired
15 cm
r22
1
nF  nC
10 V
r21
1/f2D
 V1
V2
 V1
V2  V1
V2  V1
V2  V1
-1.356
0.157 cm-1
-0.0904 cm-1
f1D=6.35 cm
f2D=-11.06 cm
2.356
1
f
desired
 V1
V2  V1
1
f
desired
From expression (2)
1
 (n1D  1) r1  r1  0.52015 r1  r1
11
12
11
12
f1D




r1
Page 37

0.157
1
1
;
cm

0
.
3019
cm
r12
11
0.52015
by choosing a equiconvex lens r12 = -r11<0 gives
1
2
r11  0.3019cm . Thus, r11=6.624 cm

1
Similarly
1
 (n 2D  1) r1  r1  (0.61715) r1  r1
21
22
21
22
f 2D

r1

21
1
r22




 0.0904 1
cm  0.14649 cm1 ;
0.61715
by construction r21 = r12 (cemented lenses), which
gives
r1
r   0.14649 cm . Since r =
obtain  r1 r1   0.14649 cm . Thus,
12

1
1
12
22

1

11
-r11, we
22
r22=-224.21 cm
Observation
d
(1/ fdoublet )  0 , at  = D, is
d
quite appealing to use it as a requirement for
obtaining a achromatic doublet, such a condition
may does not necessarily ensure that 1/f is
independent of the wavelength. The graph below is
an example of what may happen in some cases.
Although the condition
Page 38
That is, 1/f may still be different for different colors,
despite the fact that the derivative of 1/f is zero at 
= D.
1/f
Index of
refraction

C DF
frequency
A stronger condition is to require that, for example,
that the focal length of the doublet for the red (C)
and blue (F) light (Fraunhofer lines) to be equal,
1
f doublet,C

1
f doublet,F
requirement for achromaticity
Using (1) and (2) we obtain,
1
 (n1C  1) 1  (n 2C  1) 2
f doublet , C
1
f doublet , F
 (n1F  1) 1  (n 2 F  1) 2
So, the condition (8) is equivalent to
(n1F  n1C ) 1  (n 2F  n 2C ) 2  0 ,
This expression is nothing but expression (4).
(8)
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Indeed, the resemblance can be more direct if we
divide by the difference F-C.
n  n1C
n  n 2C
1 1F
 ( n 2 F  n 2 C ) 2 2 F
0
F  C
 F  C
where the fractions are the approximate values of
the derivative of n with respect to .
SUMMARY
The procedure above ensures that the achromat
doublet will have equal focal length for at least the
red and blue colors.
Page 40
Ref: S. Inoue aand K. Spring, Video Microscopy, Plenum Press, Second Edition
CLASIFICATION of OBJECTIVE
LENSES
1) Achromats. 2) Fluorites. 3) Apochromats.
Added Plan designation to lenses with wide flat field
(low curvature of field and distortion).
0
Correction for aberrations
Type
Spherical Chromatic
Flatness
correction
Achromat
*
2
No
F-Achromat
*
2
Improved
Neofluar
3
<3
No
Plan Neofluar
3
<3
Yes
Plan Apochromat
*
2
3
4
Yes
4
>4
Corrected for two wavelengths
Corrected for blue and red
Corrected for blue, green and red
Corrected for dark blue, blue, green and red
Fluorites objectives.
 Considerably better corrected than achromats, but
not quite as well corrected as the apochromats.
 Uses fluorite crystals (which have lower dipersion)
in place of some of the glass elements.
 Correct for spherical aberrations in three
wavelengths at considerably lower cost than the
apochromats.
Page 41
Apochromatic objectives.
 Provides the same focal length for three
wavelengths, and free of spherical aberration for
two wavelengths.
 Magnification still vary with wavelength (a
compensating eyepiece is used to cancel the colored
fringes).
60x Plan Apochromat Objective
http://www.microscopyu.com/articles/optics/objectivespecs.html