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2.8.1.
Secondary spectrum and spherochromatism
Fortunately, chromatic aberration can be
significantly reduced with a pair of lenses combined. Defining
"achromatic" as the level of correction where both longitudinal and
lateral chromatism are cancelled for two wavelengths near the opposite
ends of the visual range, a lens pair - or "doublet" - is
achromatic if
the individual lens focal lengths are reciprocals of their Abbe numbers:
-ƒ1/ƒ2
= V2/V1 (42)
with f1,2
being the lens focal lengths (given by
Eq. 1.1.) in e-line, and V1,2
their respective Abbe (dispersion) numbers. The equation implies that
the two lenses need to have opposite powers. For near-contact or contact
pairs, the system focal length ƒ relates to the focal lengths of
individual lenses ƒ1
and ƒ2
as 1/ƒ=(1/ƒ1)+(1/ƒ2).
This determines individual lens focal lengths in terms of the system
focal length and Abbe numbers as:
ƒ1
= (V1-V2)ƒ/V1
and ƒ2
= (V2-V1)ƒ/V2
(43)
If we choose to cancel
chromatism for the F and C lines, then the curvatures of the front and
rear lens need to relate as:
k2/k1
= - (nF1
-nC1)/(nF2-nC2) (44)
with k=(1/R1)-(1/R2),
R1,2
being the front and rear lens radius of curvature, respectively. This
leaves enough room for lens "bending", so that other (monochromatic)
aberrations are also corrected or minimized.
The most successful version of
a doublet achromat is aplanatic (corrected for spherical aberration and
coma) contact doublet created by the Bavarian optician Joseph von Fraunhofer in
the early 18th century. It consists from the positive front crown
element and the negative rear flint element.
In order for a doublet to have identical focal
length for a selected additional third wavelength in the
mid-range (e-line here),
it also needs to satisfy the relation:
k2/k1
= - (nF1
-ne1)/(nF2-ne2)
(45)
which implies the equality
(nF1
-ne1)/(nF1
-nC1)
= (nF2-ne2)/(nF2-nC2).
Such doublet is an apochromat.
Since it is the combination of optical
power and dispersion properties of two glasses that determines degree of
chromatic defocus, it can't be significantly improved by adding more
lens elements of identical or similar glass types. Triplet objective
makes it easier to have better overall correction of aberrations, but
will not have appreciably lower secondary spectrum than a doublet made
of the same type of glass. The
(nF
-ne)/(nF
-nC)
factor is a relative partial
dispersion of glass, hereafter denoted PF,e.
Consequently, the axial separation between the red/blue focus and that of the
e-line - secondary spectrum - is given by:
Df
= (PF1,e1
- PF2,e2)f/(V1-V2)
(46) For the typical e-line optimized
achromat (so called C-e-F correction), using BK7/F3 glasses for the
front and rear lens, respectively, this gives longitudinal chromatic
error (as the separation of the common blue/red focus from the e-line focus) as:
Df
= (3.348732 - 4.5799n1
+ 2.230453n2)ƒ
(47)
with
n1
and n2
being the indici of refraction for the chosen wavelength for the front
and rear lens, respectively, and ƒ the system focal
length. Substituting indici for the e-line (n1=1.51872,
n2=1.61685)
in the above relation gives Dƒ=-0.000546f,
or -ƒ/1832 as the separation between the green (e-line) and red/blue
focus (the minus sign indicating that the latter is farther away from
the objective). Although term secondary spectrum
generally applies to all the wavelengths deviating from the common
red/blue focus, it is most often used to designate the above focal
separation. The limit to
chromatic correction for doublet achromat made with common glasses,
expressed as the red/blue-green focal separation is ~ƒ/2000. Centered on
e-line, defocus increases exponentially toward both ends of the spectrum
(FIG. 30).
FIGURE 30: Secondary spectrum
in a doublet achromat optimized for the green e-line
(C-F correction). Since C- and F-line come to
a common focus, their axial separation from the optimum focus is
near-identical, an so are their respective blurs.
Defocus aberration worsens much more rapidly toward the blue end of spectrum. This is even
more pronounced in C-e corrected achromats, where C- and e-line
are brought to a common focus, optimizing for the d-line, and
reducing relative aberration in the red. However, the blue/violet
end is practically sacrificed, fainting away into much more quickly
expanding chromatic defocus. The downside is not only in losing the
blue/violet end, but also in compromising a portion of the green. Considering eye sensitivity curves for
bright (photopic)
and dim (scotopic)
light, correction
shifted toward the blue end
(d-F) offers best overall visual performance.
Since the sum of the relative partial
dispersions for an apohromatic doublet is, by definition, zero, so is
its secondary spectrum. In reality, there is always some residual
secondary spectrum, but it is entirely negligible with well designed and
made objectives. One last form of chromatism - and the one most
difficult to subdue - may not be quite negligible even in apochromats. It is
chromatic spherical aberration or, for short, spherochromatism.
As mentioned, spherical aberration can't be
made zero in a single lens, but can be cancelled in a doublet. However, it
can be cancelled only for a single wavelength. Others wavelength will be
uncorrected, the farther away from the optimum wavelength, the more so.
From Eq. 7, the wavefront error of spherical aberration
of a thin lens contact doublet at best (diffraction) focus can be expressed as:
W=(S1+S2)(r4-
r2)
(48) with
S1
and S2
being the peak aberration coefficients of the front and rear lens,
respectively, and
r
the ray height in the pupil (aperture) in units of the radius. The peak
aberration coefficients are S1=s1d4
and S2=s2d4,
with s1
and s2
being aberration coefficients for the front and rear lens, respectively,
and d
the pupil (aperture) radius. General expression for the aberration
coefficient of spherical aberration of a thin lens is:
s = - [n3
+ (n+2)q2
+ (3n+2)(n-1)2p2
+ 4(n2-1)pq]/32n(n-1)2ƒ3
(49)
with n being the
refractive index, q=(R2+R1)/(R2-R1)
the lens shape factor, and p=1-2ƒ/i the lens position factor
(FIG. 31), with ƒ
being the lens focal length, and i the image-lens
separation.
FIGURE 31: Position factor p of a thin lens
changes with the lens type (positive/negative) and the light cone
properties with respect to lens' focal point (i.e. resulting image
location), as indicated by the relation
p=1-2ƒ/i.
For the front lens the object is at infinity, so i1gƒ1
and p1=-1,
resulting in:
s1
= - [n3
+ (n+2)q2
+ (3n+2)(n-1)2
- 4(n2-1)q]/32n(n-1)2ƒ3 (50)
For
the rear lens, the final image separation equals doublet's focal length, thus i2gƒd
and p2=1-2ƒ2/ƒd,
with ƒ2
being the rear lens focal length, and ƒd
the doublet focal length. The thin lens's focal length ƒ is given
by
ƒ=R1R2/(n-1)(R2-R1). Calculated values for
p2
and q2
are substituted in Eq. 49 to obtain the rear lens aberration
coefficient s2.
For cancelled spherical aberration, the sum of aberration coefficients
for the front and rear lens must be zero. This can be achieved only for
a single value of the refractive index n, that is, for a single
wavelength. In a doublet achromat with spherical aberration cancelled
for the e-line, the blue end of spectrum will be overcorrected and the
red end under-corrected (FIG. 32). However, the error is generally
low. Chromatic spherical aberration - or spherochromatism - remains
negligible in most doublet achromats.
 FIGURE
32:
Spherochromatism of the red C- and blue F-line in achromats. In an
achromat with spherical aberration corrected for the green
e-line (with the rays from all zones of the lens objective focusing at the same axial distance), the blue
wavelengths are overcorrected (marginal rays focusing farther away than
the paraxial), while the red wavelengths are under-corrected. Properly
designed achromat has its best red and blue foci (those where rays from
the 70% zone
come to focus) coinciding, although it has no practical importance with mid-
and long-focus amateur-sized objectives, due to their
spherochromatism being at negligible levels.
◄
4.8. Chromatic aberration
▐
4.8.2. Measuring chromatic error
►
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