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4.7.2. Lateral color
error
Origins of
lateral color error - also known as *transverse chromatic aberration*
- 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. 71**). For that reason, lateral color is
sometimes referred to as *chief ray chromatism*.
**FIGURE 71**: 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 h**1**=αL,
**L** being the longitudinal displacement of aperture stop. The angle of
incidence at the first surface β=α+ρ**1**,
with ρ**1**=h**1**/R**1**
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. ρ**1**-ρ**2**~0,
with ρ**1**=h**1**/R**1**
and ρ**2**=h**2**/R**2**,
where h**2**=h**1**+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'(ρ**1**-ρ**2**).
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:
δ**1,2** = β"**1**-β"**2**** **= (n'**1**-n'**2**)(ρ**1**-ρ**2**)
(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.
72**). 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 72: (A) Simple
geometry of wavefront tilt shows that the P-V error is given by W**T**=τD,
with the tilt angle **τ**
=h/f,
where **h** is the linear shift of point image in the image plane,
**
f** 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, **Δ** (on the graph marked as *l*). The system used for
raytracing has negligible other aberrations
(50mm
f/9.56 Maksutov camera, R**1**=-206,
S**1**=20.1,
BK7, R**2**=217.5,
S**2**=666,
air, R**3**=-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).
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/Δ graph on **FIG. 72**B 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 given by S**P**~1-Δ2/(1+1.2Δ2),
with **Δ** being 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 Δ<1,
and doesn't exceed a few percentage points at Δ~2
(e.g. 0.31 vs. 0.32 by raytrace for Δ=2). The plot has quasi-Gaussian shape, with the drop in the Strehl becoming asymptotical for larger F-to-C separations - a consequence of the rate of wavelength divergence decreasing exponentially toward the central wavelength (the dashed portion of the plot indicates range where the approximation may not be accurate).
◄
4.7.1. Secondary
spectrum and spherochromatism
▐
4.7.3. Measuring chromatic error
►
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