|
telescopeѲptics.net
▪
▪
▪
▪
▪▪▪▪
▪
▪
▪
▪
▪
▪
▪
▪
▪ CONTENTS
4.8.1. Secondary
spectrum, spherochromatism, lateral color
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 the reciprocal of their
Abbe numbers:
with ƒ1,2
being the lens focal lengths (given by
Eq. 1.1.) in e-line, and V1,2
their respective Abbe numbers (dispersion), with the subscripts 1
and 2 referring to the front and rear lens, respectively. 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:
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:
with k=(1/R1)-(1/R2),
R1,2
being the front and rear lens radius of curvature, respectively,
and
nF1,
nC1
and
nF2,
nC2
refractive indici of the selected distant wavelengths for the front and
rear lens, 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 (who also invented German equatorial mount). 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:
which implies the equality
(nF1
-ne1)/(nF1
-nC1)
= (nF2-ne2)/(nF2-nC2).
Such doublet is an apochromat;
bringing two widely separated wavelengths
and a third, mid-wavelength, to a common focus, practically means that
all other wavelength within this range are also nearly parfocal. Thus,
apochromat corrects for both, primary
and secondary color error. That alone still doesn't guarantee
high-level correction of chromatism, since spherochromatism in lens
objectives with strongly curved surfaces can be significant. This is why
Abbe's apochromatic condition requires good correction of spherical
aberration across the wavelength range as well.
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 glass types. The
(nF
-ne)/(nF
-nC)
factor is a relative partial
dispersion of glass, hereafter denoted PF,e.
Consequently, axial separation between the red/blue focus and the
e-line focus - the secondary spectrum - is given by:
 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 green focus from the common blue/red focus,
a standard measure of secondary spectrum) as:
Df
= - (3.348732 - 4.5799n1
+ 2.230453n2)f (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 (since P(F,e)1<P(F,e)2,
secondary spectrum is numerically negative, because it is measured from
the common F/C focus, which is farther away from the objective, to the
right).
Substituting indici for the e-line (n1=1.51872,
n2=1.61685
for BK7/F3 doublet)
in the above relation gives Dƒ=-0.000546ƒ,
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). When measured from d-line focus, which is closer to the
common F/C focus, secondary spectrum is slightly better than ƒ/2000; the
difference is, of course, only numerical, since the level of chromatic
correction doesn't change.
Although term secondary spectrum
generally applies to all 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-to-green focal separation is ~ƒ/2000. Centered on
e-line, defocus increases exponentially toward
either end of the spectrum
(FIG.45).
FIGURE 45: Secondary spectrum
in a doublet achromat optimized for the green e-line
(C-F correction). Since C- and F-line are brought to
a common focus, their axial separation from the optimum focus (focal
shift) 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, fading 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:
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. 46), with ƒ
being the lens focal length, and i the lens-to-image
separation.
FIGURE 46: Position factor p of a thin lens
changes with the lens type (positive/negative) and the properties of
light cone incident to it, with respect to lens' focal point
F and the resulting image
location, as indicated by
p=1-2ƒ/i.
For the front lens the object is at infinity, so i1gƒ1
and p1=-1,
resulting in:
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' 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. 47). However, the error is generally
low. Chromatic spherical aberration - or spherochromatism - remains
negligible in most doublet achromats, the exceptions being larger, fast
achromats, for instance 6" ƒ/5 or 8" ƒ/6.
 FIGURE
47: Spherochromatism of the red C- and blue F-line in achromats.
In an achromat with spherical aberration corrected for the green
e-line (with rays from all zones of the lens objective focusing at the same axial distance
from the objective), 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) nearly coinciding, although it has no practical importance with mid-
and long-focus amateur-sized objectives, due to the level of their sphero-chromatism
being negligible. If, for instance, paraxial blue and red foci are
coinciding, the best red focus is closer, and best blue focus
farther away from the e-line (green) focus, with the greater defocus
error in blue making color correction slightly imbalanced.
In apochromatic objectives,
due to their more strongly curved surfaces, spherochromatism is often
significant, commonly involving both, lower- and higher order spherical
aberration.
Lateral color
Origins of
lateral color are not as obvious as for longitudinal chromatism. In
simplest terms, it is the consequence of unequal refractive compensation
at lens surfaces. Since different wavelengths refract at a different
rate, a single
refractive
surface will always split oblique incident light into a fan-like spread
of wavelengths, diverging at slightly different angles. Unless this
divergence is compensated for at another refracting surface, or
surfaces,
by the time this diverging light hits the image plane, the spread of
wavelengths (i.e. colors) can be significant.
It doesn't take much; for a f/10 lens, two wavelength will arrive at the
image plane separated by the Airy disc diameter when their angle
differential at the exit from the lens nearly equals the angular Airy
disc diameter, or as little as 2.8 arc seconds.
Lateral color also can be generated by fabrication error,
specifically, by wedge-like orientation of refractive surfaces relative
to each other. Following text will limit to lateral color associated
with oblique incidence pencils, but general principles are the same for
both.
Since the reference ray for all aberrations resulting
from obliquity of incident pencils is their chief ray, the chief rays of
optimized wavelengths are unavoidably directed toward image plane at
different angles (FIG. 48). For that reason, lateral color is
sometimes referred to as chief ray chromatism.
FIGURE 48: Cause of lateral color in
a lens. With aperture stop at the lens (A), chief ray of the
incident white light passes near the lens' center (not quite through the
center, as usually depicted for thin lens and shown at left, but the
difference is negligible with respect to lateral color). The white light
chief ray splits into chief rays of different wavelength after the first
surface, but the angle of divergence is very small, resulting in
negligible height differential at the second surface. Due to surface
tangents at the respective refraction points being nearly parallel, this
section of lens acts as plano-parallel plate, with the slight
differential in angular direction of color chief rays compensated for by
their slightly different rate of refraction at the second surface. as a
result, the chief rays travel toward focal plane at nearly identical
angle, nearly equal to the incident angle, staying tightly together.
Note that due to different focal lengths, chromatic difference of
magnification for different colors is present in their respective focal
planes. However, since all chief rays arrive at nearly identical angle,
there is no lateral color in the green light focal plane: the other
colors are merely defocused. When the aperture stop is displaced, either
longitudinally or laterally, the geometry changes (B). The white
chief ray is now directed farther off the lens center, with the tangents
on two lens surfaces at the respective points of refraction being
no longer nearly parallel. As a result, refraction at the second surface
is no longer compensatory, and the chief rays of different colors keep
diverging toward focal plane. Consequently, they reach different heights
in the green light focal plane, producing lateral color error. This
error is now combined with longitudinal defocus, i.e. other colors are
both, defocused and shifted laterally. Obviously, correcting
longitudinal chromatism would only eliminate defocus error - as well as
chromatic difference of magnification due to it - but wouldn't affect
lateral color error, nor chromatic difference of magnification resulting
from it.
As illustrated above, the two main determinants of
lateral color error are stop position and lens shape. If, for instance,
the 2nd surface tangent in (B) was nearly parallel to that at the
1st surface (a weak positive meniscus), the lens would have acted as
plano parallel plate, producing negligible lateral color despite
displaced stop.
Some basic relations for the above simple lens case help
define specific factors of its lateral color. For the white light
incidence angle α, the
height at which it hits first surface is h1=αL,
L being the longitudinal displacement of aperture stop. The angle of
incidence at the first surface β=α+ρ1,
with ρ1=h1/R1
being the angle formed by optical axis and the 1st surface's radius of
curvature R1.
The refracted angle, for small β, is β'=(n/n')β, with n,
n' being the refractive index of incident and refractive medium,
respectively. For lens in air, n=1 and β'=β/n'; since n' varies
with the wavelength, so do their respective angles of refraction.
In
other words, chief ray divergence that would produce lateral color is
initiated at the first surface. In order to cancel this divergence, the tangent at refraction
point on the second lens surface needs to be nearly parallel to that on
the first surface, i.e. ρ1-ρ2~0,
with ρ1=h1/R1
and ρ2=h2/R2,
where h2=h1+β't,
t being the lens thickness (obviously, the curvature radii for
first and second surface, R1
and R2,
respectively, need to be of the same sign).
Consequently, in first approximation, the chief ray angle
of refraction after the second surface is given by β"=n'(ρ1-ρ2).
In other words, for any non-zero value in brackets, the angle of
divergence for any specific wavelengths will equal the product of that
value and the glass refractive index n' corresponding to the
wavelength. For any two wavelengths 1 and 2, the angle of their lateral divergence is:
δ1,2 = β"1-β"2 = (n'1-n'2)(ρ1-ρ2)
(50)
For given aperture, focal length - or focal ratio - of the system is
irrelevant to the magnitude of lateral color error, since the angular
size of this error is constant (i.e. the linear extent of lateral color
error remains constant with respect to the Airy disc size).
With multiple lens systems the calculation is more complex,
but the principle remains the same: cancellation of lateral color
requires the sum of refracted angles at its surfaces to be near zero for
a given range of wavelengths.
As a wavefront aberration, lateral
color error is a consequence of wavefront tilt vs. reference sphere. As
monochromatic aberration, wavefront tilt does not affect point-image
quality, only its location; however, in a wavefront that splits
chromatically through refraction, tilt error varying with the wavelength
does cause spread of energy in the central diffraction maxima (FIG.
49). Unlike secondary spectrum and
spherochromatism, where most of the energy lost to the central maxima
goes to the rings area, lateral color error mainly expands (and deforms)
the central maxima.
FIGURE 49: (A) Simple
geometry of wavefront tilt shows that the P-V error is given by WT=τD,
with the tilt angle τ
=h/ƒ,
where h is the linear shift of point image in the image plane,
ƒ being the focal length (for
object at infinity; image separation for close objects), and D
the aperture diameter. The P-V to RMS wavefront error ratio is 4√32/3.
The angle of tilt τ
is determined by the angular discrepancy between chief ray angle for
specified wavelength, and chief ray angle of the reference wavelength
(in the visual context, usually around green e-line); it equals the
angle of lateral divergence δ, as defined above, with the only
difference being that it expresses the error relative to the primary
wavelength. (B) The
effect of lateral color error on point-image quality and overall
contrast depends on its magnitude and spectral sensitivity of the
detector. Shown is its effect on polychromatic PSF (PPSF, for 0.4-0.7μm
range, photopic eye sensitivity) and MTF, in terms of C/F separation in
units of e-line Airy disc diameter,
l. The system used for
raytracing has negligible other aberrations
(50mm
ƒ/9.56 Maksutov camera, R1=-206,
S1=20.1,
BK7, R2=217.5,
S2=666,
air, R3=-996,
mirror, all mm, stop at the 1st surface), hence the effect is nearly entirely the result of
lateral color error. The C/F separation needs to be half the e-line Airy
disc radius, or less, for the polychromatic Strehl to remain within
diffraction limited range. The primary effect on polychromatic PSF is elongation of
the central maxima in the direction of lateral color shift, wider on the
side of longer wavelengths' shift (top), narrower at the side of shorter
wavelengths' shift (bottom). Asymmetrical expansion of the central
maxima causes largest MTF contrast drop in the high-frequency range,
from the maximum in the orientation coinciding with lateral shift
(tangential) to near-zero in the orientation perpendicular to it
(sagittal).
The tolerance for lateral color error, obviously, depends
on the spectral sensitivity of detector. For photopic eye sensitivity,
diffraction-limited maximum is at the C/F separation nearly equaling the
e-line Airy disc radius; for even sensitivity over the visual range,
polychromatic Strehl drops to 0.80 at only 30% of that separation.
As the PPSF/l graph on FIG. 49B indicates, the effect of lateral color
error is not proportional to its angular magnitude. Similarly to
secondary spectrum, negative effect of lateral color on image quality
changes at a slower rate than its nominal magnitude. For instance, at
C/F separation equaling half the Airy disc radius, the P-V wavefront
error of primary spherical aberration corresponding to the resulting
0.81 Strehl is slightly better than 1/4 wave. Doubling the C/F separation does not
double the corresponding P-V wavefront error, which is 1/2.4 waves P-V
for 0.54 Strehl. Doubling it once more, to twice the Airy disc diameter,
only lowers the polychromatic Strehl to 0.32, with the corresponding
primary spherical aberration error of 1/1.8 wave P-V.
A close empirical approximation for the photopic
polychromatic Strehl resulting from the lateral color error is SP~1-l2/(1+1.2l2),
with l the C/F separation in
units of e-line Airy disc diameter. The difference vs. raytracing values
is within 1% of the nominal PPSF value for l<1,
and doesn't exceed a few percentage points at l~2
(e.g. 0.31 vs. 0.32 by raytrace for l=2).
◄
4.8. Chromatic aberration
▐
4.8.2. Measuring chromatic error
►
Home
| Comments |