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4.7.
Chromatic aberrations
Chromatic aberration, as the name implies
(khroma=color in Greek) is the phenomenon of white light wavefront
splitting into its spectral components' (i.e. individual wavelengths) wavefronts, as light encounters media of different
optical density. It is caused by different wavelengths moving at
different speeds through media denser than air. Denser media slow down
all wavelengths, but shorter wavelengths more than
longer wavelengths, in inverse proportion to the media index of
refraction (the default value for air is 1, with the actual value - 1.00027784
for 550nm wavelength - being
negligibly higher than for vacuum).
FIGURE
62: Refractive
index for the green e-line (546nm, or 0.546μ
Fraunhofer spectral line) of the
visual spectrum is
for most optical glasses in the 1.4-1.9 range; plot to the left
shows variation of refractive index with wavelength for several optical
glasses or materials, and water. The higher refractive index, the
stronger refraction, i.e. power of a refractive surface, or a lens.
As the plot illustrates, refractive
index declines with the increase in wavelength, more rapidly going from
short toward longer wavelengths, nearly flattening out in the infrared
(index change is gradual in 0.2-4μ range, approximately; outside of it
becomes much more volatile, particularly for long waves, about 10
microns and longer). Refractive index also generally
increases with glass density; it is lower for crown and higher for
flint glasses. In the latter, index of refraction increases with their lead
content.
Some other materials used for lens fabrication, such
as calcium fluorite (form of crystal) and plastics like acrylic or
polycarbonate, similarly to glass, refract more strongly shorter
wavelengths, while gradually flattening out from the mid visual range
(approximately) toward red and infrared..
As illustrated on
Fig. 4, points of
wavefront that simultaneously moves through media of different optical
densities, generate wavelength-specific phase difference, due to
different wavelengths now moving at different speeds through the
incident and refractive media. This causes change in
a form of the wavefront (in terms of rays, it results in
refraction).
Obviously, chromatic split of the
wavefront
will take place even if the shape of optical surface coincides with the
shape of wavefront, for instance, when flat wavefront passes through an
optical window at zero inclination angle. However, this effect is
negligibly small; in BK7 glass, the green e-line is only 0.241% faster
than the blue F-line, and 0.29% slower than red C-line. Since the speed
of light in a refractive medium is cn=c/n,
n being the refractive index, the in-glass path differential is
inversely proportional to the index ratio, which is in this case for F
vs. C line given by (1-0.00241)/(1+0.0029)=0.99471, for 0.005295
differential (in terms of the nominal index differential δ it is
δ/n times in-glass path). In, say, 5mm thick
window, axial separation between blue and red wavefront at the rear
surface would
be 5x0.005295=0.0265mm (since the "faster" wavefront gets ahead more
traveling through air while the slower wavefront is still in glass, the
final separation is n times larger, i.e. given by δn/n=δ times
in-glass path).
Much more important is the effect of
change in the wavefront form when the shapes of the wavefront and optical
surface don't coincide - a common scenario with telescope lens
objectives, made to change flat incoming wavefronts into spherical.
Here, the in-glass light speed varying with the wavelength causes shorter
(faster) wavelengths to
form more strongly curved wavefront - or, in terms of rays, refract more strongly - and
focus closer than longer wavelengths (FIG. 63).
The aberration's ray geometry
is governed by the law of refraction, stating
that the
sines of
the
incident (α) and refracted (α') ray angle
to
the
surface normal relate in inverse proportion to the
refractive indici of the incident and refractive media, or
sinα'/sinα
= n/n', with n and n' being the incident and refractive media
refractive index, respectively.
FIGURE 63: The three forms of chromatic
aberration are: (1) chromatic defocus, (2) lateral color and (3)
spherochromatism. Illustrations above show how the the first two
form in a lens,
grossly exaggerated.
LEFT: white-light wavefront
W splits into its component wavelengths wavefronts, as
shorter wavelengths lag behind in a denser media of the
refractive n', forming more strongly curved wavefronts with
shorter foci. This form of chromatism is called primary
color, or primary spectrum: the
longitudinal variation of color foci produced by a single lens,
where the focus falls farther as the wavelength increases. The path of refracted rays (dotted lines) is
orthogonal with respect to the transforming wavefronts.
Note that the axial separation of the
wavefronts themselves is exaggerated; typically, it is negligible in
comparison to the effect of wavefront radii variation (i.e. the
extent of their focal length variation), also grossly exaggerated in
the illustration. In a doublet achromat, primary color is corrected by bringing two
widely separated wavelengths - usually red C and blue F lines - to a
common focus, but other wavelengths still focus differently. This longitudinal focus disparity for
other wavelengths vs. common
focus of two widely separated wavelengths is called
secondary
spectrum (usually measured by the displacement of e-
or d-line focus). Strictly talking, only a single wavelength can come out of a lens
objective with
perfectly spherical wavefront; the rest of them are affected by
spherical aberration. Variation of spherical aberration with the
wavelength is called chromatic spherical aberration, or spherochromatism
(not shown above).
RIGHT: The white-light chief ray (the
one orthogonal to the center of the incident wavefront
at an angle to the optical axis), splits laterally into the component
wavelengths chief rays, setting the stage for
lateral chromatism;
unlike primary and secondary color, and spherochromatism, which are
axial aberrations (meaning that they affect the entire field),
lateral chromatism is off-axis aberration, caused by variation of
refraction power with the wavelength for off axis point.
With the stop at the surface, it passes through the center of the
lens, which acts as a plane-parallel plate, having the chief rays of
different wavelengths exit slightly separated, but at nearly
identical angle. Consequently, lateral chromatism approaches zero,
unless the lens is exceedingly thick. With the aperture stop
displaced from the lens, the chief ray passes through a portion of
lens with significant optical power, resulting in chief rays of
different wavelengths exiting the lens ad different angles, thus
focusing at different heights in the image space. Since lateral
color and secondary spectrum have different origins and magnitude,
correcting one doesn't necessarily cancel the other.
Independently of the longitudinal chromatic error
- either primary or secondary spectrum, which result from defocus error
relative of other wavelengths relative to a specific wavelength - every
refractive objective also generates spherochromatism, as
spherical aberration can be optimally corrected only for a narrow range
of wavelengths. Depending on the specifics of a lens, or lens group,
spherochromatism ranges from entirely negligible to a serious detriment
to optical quality. Both of these chromatic aberrations are axial
aberration, i.e. affect the entire image field. The third main type of
chromatic aberration is lateral chromatism, causing abaxial point
images to be formed at different heights for different wavelengths, thus
smearing the spectral energy radially. When present, it is zero on axis
(assuming properly aligned lenses), increasing with the field
angle.
Longitudinal chromatic aberration This form
of chromatic aberration is traditionally in the focus of amateur
astronomers, because it is the dominant form of chromatic aberration in
achromats - a type of telescope that remained popular among amateurs
long time after it was abandoned by professional astronomy.
The cause of longitudinal chromatism is variation of glass' refractive
index with the wavelength, and with it the inverse to it variation in
lens' focal length (FIG. 64).
FIGURE 64: General scheme of the
change in glass' refractive index with the wavelength, and accompanying
change in lens' focal length. As plots show, refractive index
decreases with the wavelength, while the resulting focal length,
expectedly, follows the opposite dynamics. The pace of change in the
refractive index also decreases as the wavelength increases, and so does
the pace of the resulting change in focal length. While
lens focal length
for any given wavelength is determined by the glass' refractive index
for that particular wavelength, the change in focal length as a function
of change in the wavelength and its corresponding index depends on the
dynamics of index change with the wavelength - or glass
dispersion.
It varies from one glass type to another. As it will be addressed in
more detail ahead, it is this variation in dispersion from one glass
type to another that made possible overcoming the crippling chromatism
of a single lens.
Since both, refractive index and
dispersion are determinants of the chromatic optical properties of a material,
standard glass diagram
groups glasses with respect to their refractive index and dispersion.
Longitudinal chromatism of a single lens
- known as primary spectrum - is overwhelming:
it smears the wavelengths into a rainbow of colors, due to shorter
wavelengths, for which glass has higher refractive indici, refracting
more strongly and focusing closer than longer ones. In terms of lens
power, shorter wavelengths converge more strongly in a positive lens,
and diverge more strongly in a negative lens. Hence red focuses farther
away from blue in both, positive or negative lens (with the focus in the
latter being imaginary, determined by extending refracted rays in the
direction opposite to their travel).
 For object at infinity,
longitudinal chromatism of a single lens, known as primary chromatism,
defined as the separation between red C-line and blue F-line foci, is
proportional to 1/V, V being the glass dispersive constant, or
Abbe
number. Specifically, this means that a lens of the focal
length f will have longitudinal chromatism, as defined above,
equal to f/V. The blue focus, which focuses shorter, is somewhat closer
to the e-line focus than the longer red focus, at approximately 45% of
the F-C separation. General form
of the Abbe number is:
with n being the
glass refractive index for the chosen central wavelength, and
ι
the index differential for the chosen range. The usual choice for n
is either e (λ=0.5461μ) green, or d
(λ=0.5876μ)
yellow line, and for
ι
the
index differential between the blue F-line (λ=0.4861μ) and red C-line
(λ=.6563μ),
nF-nC.
Taking e line
and BK7 glass (ne=1.518722, nF=1.522376
and nC=1.514322),
gives the corresponding Abbe number as:
Thus, the lower Abbe number, the higher glass dispersive power, and vice
versa.
Lens' lateral color is
near-zero with the stop at the surface; for displaced stop, it increases
with stop displacement, in proportion to the field angle, depending on
lens' dispersive and optical powers, as well as its thickness). However,
its longitudinal chromatism is independent of stop location. A single
BK7 lens of the focal length f=1000mm, would have the axial F-C
separation of over 15mm. In other words, it would be all but useless
for observing of
pretty much anything but the chromatism itself.
According to raytrace, a single 100mm diameter crown lens needs to be
about f/340 in order to achieve 0.80
polychromatic Strehl in the visual range (for photopic eye sensitivity).
The required f-ratio scales with the
aperture diameter, thus 50mm lens needs to be "only"
f/170. Obviously, longitudinal
chromatism for given glass changes in proportion to the focal length.
However, the effective error, corresponding to the
polychromatic Strehl,
changes at a somewhat slower rate. For instance, an
f/170 100mm lens will have 0.55
polychromatic Strehl, not ~0.4 that would result from the λ/2 wave P-V
of spherical aberration equivalent (doubled λ/4 wave equivalent at
f/320). Similarly, at
f/640, its polychromatic Strehl is
about 0.93, corresponding to λ/7 wave P-V of spherical aberration, not
nearly 0.95 Strehl that would result from λ/8 wave.
The above numbers imply that the defocus tolerance at F- and C-line for
0.80 poly-Strehl level in a single lens is just over λ/2 P-V.
In order to minimize longitudinal
chromatic defocus in a lens objective, it has to be made of two or more
glass elements of different optical properties. Such compounded refracting objectives
usually consist of two or three lens elements. Depending on the level
of correction of chromatism, most of them belong to one of the two main
groups of lens objectives, achromats and apochromats. In the
former, primary spectrum of a single lens is corrected by
bringing two wavelengths near the opposite ends of the visual spectrum
(usually blue F- and red C-line) to a common focus. However, other
wavelengths still deviate from this focus, resulting in so called
secondary spectrum. Using glass of abnormal dispersion (i.e.
significantly different from that in common-type glasses) for one of the
elements, allows for the correction of secondary spectrum, with the only
potentially significant form of chromatism remaining being
spherochromatism.
Not every abnormal glass has dispersion differential strong enough
to effectively eliminate secondary spectrum. In that case, it is only
partly reduced; such objectives - along with those having secondary
spectrum cancelled, but still retaining strong spherochromatism - belong
to somewhat vaguely defined class of semi-apochromats.
Optical glasses
Obviously, refraction is a more complex phenomenon than reflection.
While the reflected angle is always identical to the incident angle,
relative to the normal to a surface, angle of refraction varies with the optical properties
of a refractive medium. In general, denser glass slows light
propagation more, hence it refracts light more strongly and,
consequently, has higher refractive index.
While the glass' index of refraction and dispersion, as mentioned, are
its two most important optical properties, it is also of significance
how much of the light incident upon glass surface is reflected back
(that applies to both, air-to-glass and glass-to-air surface), and how
much of the light that enters glass medium actually makes it to its rear
surface (FIG. 65)
 FIGURE
65: Two
important properties of the optical glass are reflectance and
transmittance. Both are dependant on the glass' refractive index,
but in a different way.
Plot at left illustrates these dependencies in
terms of the relative index of refraction n*, defined as a ratio of the
index of transmittance medium vs. incidence medium, or n*=nt/ni.
When the incidence medium is air (ni=1),
n* equals the refractive index of glass.
Most optical glasses
fall in the 1.4<n<1.8 refraction index range, with the corresponding
air-to-glass reflectance (R) and transmittance (T) in the
3-8% and 8-3% range, respectively. For glass-to-glass surface, n*
values are most often in the proximity of 1, with near-zero reflectance
and nearly total transmittance.
Obviously, the two curves are symmetrical, i.e. complementary, and
either value can be expressed in terms of the other one: as T=1-R (as a
ratio number; 100-R for T and R in percents) and R=1-T.
As mentioned, the parameter specifying the effect of an optical surface, or material,
on propagation of light is called reflective and refractive index, for
reflective and refractive media, respectively. In the case of
reflection, there is no change in speed of light, only direction, which
is generally opposite to that of incident light. Consequently, for
surface in air (refractive index n=1), the reflective index is n'=-1.
However, after passing refractive surface and entering different medium,
speed of light in it changes, and that also results in the change of
light direction whenever shape of the incident wavefront does not
coincide with the surface shape, as
FIG. 4 illustrates. Since
general direction of light propagation does not change, glass refractive
index has the same sign as the media of incidence (usually air, with
refractive index n=1, or n=-1, depending on the direction), with its nominal value, as mentioned, given as an
inverse of the speed of light in glass vs. that in vacuum (which is
exactly 1, with the refractive index of air,
1.00027784 for 550nm wavelength, being negligibly higher).
The slow-down of light propagation inside glass varies with the
wavelength; in general, the shorter wavelength, the more it is slowed
down. The greater slow-down, the stronger refraction. This variability
of refraction with the wavelength is the origin of chromatic aberration.
The relative strength of refraction for different wavelengths - or glass
dispersion - as mentioned above, is expressed with Abbe number. Due
to their different physical properties, the dispersive properties vary
somewhat from one glass to another. This fortunate phenomenon, expressed
with a parameter called
relative partial dispersion of glass, made
possible to correct one major form of chromatic aberration,
secondary spectrum.
Summing it up, main properties of optical glass are expressed with its
index of refraction, its dispersion (Abbe) number and its relative
partial dispersion. Following graph shows optical glasses of two major
manufacturers grouped according to their refractive index and
dispersion.
FIGURE 66: ne-Vd
glass diagram for Schott and Ohara glasses. Glass
acronims for Schott glasses are: FK-fluorite crown, PK-phosphate
crown, PSK-dense PK ("S" is from German schwer for
heavy; also K from krone for crown), BK-borosilicate
crown, K-crown, BaK-barium crown, SK-dense crown,
KF-crown/flint, BaLF-barium light flint, SSK-very
dense crown, LaK-lanthanum crown, LLF-very light flint, BaF-barium
flint, LF-light flint, F-flint, BaSF-barium dense
flint, FS-dense flint, KzF-special "short" flint.
Additives change glass optical properties. For instance, soda-lime glass in its basic form is made of silica (SiO2), which forms the glass mass (also, greater proportion of silica results in lower thermal expansion), soda (Na2O), which lowers melting temperature (also, increases glass viscosity), and lime, (CaO, stabilizer), in a 74:16:10 proportion. Many other oxides readily dissolve in silica: alumina (Al2O3) increases durability, lead oxide (PbO) increases refractive index and density, barium oxide (BaO) has similar effect as lead oxide, but w/o toxicity, iron oxide (FeO) reduces infrared transmission, while in reduced form (Fe2O3) increases IR transmission; the latter gives to glass grinish tint, as does chromia (Cr2O3), while neodymium oxide (Nd2O3) colors glass purple. Boron oxide (B2O3) is, like silica, glass former, with better chemical resistance, etc.
As the Schott diagram shows, the traditional
crown and
flint
denominations for optical glasses indicates refractive index n≲1.6
and Abbe number V≳55 for the
former, and n≳1.6, V≲55
for the latter. That classification is, however, very general; there are
"transitional" glasses between crown and flint, and some modern crowns
have index of refraction significantly higher than 1.6. The main
ingredient of crown glasses are alkali-lime silicates, with their
optical properties varying with the secondary ingredient, the common
being potassium oxide (crown), boric oxide (borosilicate crown), barium
oxide, zinc oxide, phosphorus pentoxide, fluorite and lanthanum oxide.
Flint glasses had high lead content, which is gradually being lowered
or replaced by less toxic substances.
Common glass, usually called soda-lime float, or plate glass
varies somewhat in its optical properties. Its e-line index of
refraction is about 1.52 and its Abbe number about 60.
In general, the denser glass, the lower its Abbe number and the higher
its refractive index, usually given for e- or d- Fraunhofer line, as
ne
and nd,
respectively. Since all the wavelengths participate in the formation of
a telescopic image, the presence of refractive elements makes it
necessary to determine its effect within entire
spectral range of interest. For the visual spectrum, it encompasses
wavelengths from ~0.4 micron (μ) to ~0.7μ.
The usual way of assessing and presenting the effects of refraction,
i.e. magnitude of chromatism, is
by selecting up to a several selected spectral lines as representatives
of the specific portions of the wavelength range. The most commonly used
spectral lines notation is that established by Fraunhofer. Some of the
Fraunhofer's spectral lines routinely used in analyzing optical
properties of telescopes with refracting elements are listed below.
|
Wavelength |
Fraunhofer line |
Emitting element |
Color |
|
microns (μ) |
nanometers (nm) |
Angstrom
(Å) |
|
0.405 |
404.7 |
4046.6 |
h |
mercury (Hg) |
violet |
|
|
0.436 |
435.8 |
4358.4 |
g |
mercury
(Hg) |
violet |
|
0.480 |
480.0 |
4799.9 |
F' |
cadmium (Cd) |
blue |
|
0.486 |
486.1 |
4861.3 |
F |
hydrogen (H) |
blue |
|
0.546 |
546.1 |
5460.7 |
e |
mercury
(Hg) |
green |
|
0.588 |
587.6 |
5875.6 |
d |
helium (He) |
yellow |
|
0.589 |
589.3 |
5893.0 |
D |
sodium (Na) |
yellow |
|
0.644 |
643.8 |
6438.5 |
C' |
cadmium
(Cd) |
red |
|
0.656 |
656.3 |
6562.7 |
C |
hydrogen
(H) |
red |
|
0.707 |
706.5 |
7065.2 |
r |
helium
(He) |
red |
For assessing the effect of chromatism on the diffraction image
quality, it is also necessary to factor in specific spectral
sensitivity of the detector, i.e. to assign sensitivity weight
to a number of selected lines over the range of interest, for
obtaining the actual intensity distribution over
diffraction PSF. Similarly
to the monochromatic Strehl for
monochromatic light,
polychromatic Strehl indicates the quality of PSF in
polychromatic light, i.e. the amount of energy lost to the rings
area, as well as the average contrast over MTF range of
frequencies.
◄
4.6. Field curvature
▐
4.7.1. Secondary
spectrum and spherochromatism
►
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