In the above, O is an incoming ray which strikes surface f at point P. If
the surface were not present, the unobstructed ray would cross the axis
at B. Instead, it crosses at B'. C is the center of curvature of the
surface with distance PC equal to the radius r. ZP is the normal to the
surface at P. V is the vertex of the surface from which object distance
VB ( = L) and image distance VB' ( = L') are measured. n and n' are the
indices of refraction in object and image space, respectively.
Angle OBV is the Object Angle ( = U) and angle PB'V is the Image Angle
( = U'). Note that if ray O had struck the surface from an upward angle,
point B would be on the optical axis somewhere left of the vertex V.
If object point B is a very great distance to the left of the surface, as in
the case of an astronomical telescope, then ray O would be parallel to the
axis. In that case angle U = 0, distance L is negative infinity and sin I
= Y / r where Y is the height at which the ray strikes the lens surface.
Snell's Law, expressed in equation [1-1], establishes the relationship
between the angle of incidence OPZ ( = I) and the angle of refraction CPB' ( = I').
The remaining expressions follow from the figure.
sin I' = ( n / n') sin I [1-1]
sin I = ( L - r ) sin U / r [1-2]
U' = U + I - I' [1-3]
L' = r ( 1 + sin I'/ sin U') [1-4]
1.2 Sign and Notation Conventions
The following conventions apply:
1. Primed quantities refer to image space (after refraction or reflection).
Unprimed quantities refer to object space.
2. Lengths and distances to the right of the surface are positive; to the
left they are negative.
3. Curvature radii with centers to the right of the surface are positive.
If the center of curvature is to the left of the surface, the radius
is negative.
4. For angles between the ray and the axis of the system: mentally rotate
the axis into the ray. If the rotation is clockwise, the angle has
positive sign. Otherwise, it is negative.
5. Use n' = - 1 for reflective elements.
1.3 Paraxial Ray-tracing
Ray-trace calculations are performed for rays entering the system at
various distances from the axis of the system. A simplification of the
basic formulae results when rays are traced close to the system axis.
This is because the object and image angles become small. When expressed
in radians, the sine of a very small angle is equal to the angle. For
this reason, equations [1-1] through [1-4] can be rewritten as follows
for small angles:
|
i = u ( l - r ) /
r |
[1-5] |
|
i' = i ( n / n'
) |
[1-6] |
|
u' = u + i - i' |
[1-7] |
|
l' = r ( 1 + i'/ u'
) |
[1-8] |
Lower case letters are used in the formulae for rays passing near the
axis of the optical system. These rays are known as paraxial rays. Rays
which pass through the system elements at the largest distance from the
axis are called marginal rays. It can be shown that l' is independent of
the angles in the formulae. However, it is customary and convenient to
select u for the initial paraxial trace to be equal to the value of
sin U from the initial marginal trace.
2. OPTICAL ABERRATIONS
2.1 Spherical Aberration
A point source of light can be thought of as emitting spherical wavefronts
represented by a series of expanding concentric spheres with the point
source at the center. Those waves which may be intercepted by a converging
optical system will be redirected to create an image of the object point.
If the original object point is on the optical axis of the system, the image
point will also be on-axis. In the absence of spherical aberration and if
the light is monochromatic, all of the rays will converge at a single
image point. Spherical aberration, given the symbol LA' for "longitudinal
aberration,'' is present in an optical system when monochromatic rays
from an object point on the axis of the system cross the axis in image
space at more than a single point. For a lens, rays passing near the
center of the lens cross the axis at a point different from rays which
pass near the edge. The lens in the next figure is said to be undercorrected
( has a ``short'' edge ).
The ray passing near the edge of the lens at height Y has an image
distance VB' equal to L'. The ray passing near the center has a
greater image distance Vb' equal to l'. The measure of spherical
aberration for the system is simply
LA' = L' - l' [2-1]
Generally, L' in the above expression is determined for a marginal ray,
although Zonal Spherical Aberration (LZA') where L' is for an intermediate
zone is also an important design analysis parameter.
2.2 Coma
with the optical axis. Distance g'h' is a measure of coma produced by the system.
Since the magnitude of this aberration varies with field width, another
optical design parameter which provides a measure of a systems inherent
susceptibility to coma is often examined. This parameter is known as the
"offense against sine condition.'' When imaging objects at infinity,
OSC = 1 - u'l'/ L'sin U' [3-2]
The above relation holds true only if we follow the customary ray-
tracing convention of setting u = sin U for the paraxial trace through
the first surface of the system. The expression is also more complicated
if the system contains an aperture stop. A short focus Newtonian
reflecting telescope is the usual example given for exhibiting comatic
star images near the edge of the field, although most systems exhibit
some coma at short focal ratios unless specifically designed for wide
angle work. References [2] and [3] contain additional information on
the use of the parameter OSC.
2.3 Chromatic Aberration
An optical system which is corrected for both spherical aberration and
coma is said to be aplanatic. Chromatic aberration, which is present only
in systems which contain refractive elements, is a result of the fact
that a refractive medium has a slightly different refractive index for
different wavelengths of light. Because white light is made up of a mix
of different wavelengths and each must obey Snell's Law, it is clear that
the amount of bending will vary with the wavelength. The lens behaves
as a prism breaking up the white light into its constituent colors, each
crossing the optical axis at a different point. Longitudinal chromatic
aberration is the calculated difference between the image distances for two
particular wavelengths. The choice of wavelengths depends upon the desired
application. One commonly selected pair is the Cadmium F' and C' lines
(480 and 643.8 nanometers, respectively). For visual instrument design,
the wavelength pair commonly used is one shifted slightly toward the red
end of the spectrum: The Hydrogen F and C lines, 486.1 and 656.3 nm,
respectively. In the figure, distance VC is the image distance for
Hydrogen C-light ( = L'C ) and distance VF is the image distance for
Hydrogen F-light ( = L'F ). Longitudinal Chromatic Aberration is determined
by the following expression:
Chr' = L'F - L'C [2-3]
A lens system which is fully corrected for Longitudinal Chromatic
Aberration will usually still suffer from some residual color known as
the Secondary Spectrum. This residual is caused by the difference
between the now common F/C focus and the slightly different focal lengths
for the other wavelengths of interest. Selection of specialized glasses
can minimize the problem in doublets. A three element lens can bring
three wavelengths to common focus, creating an apochromatic lens.
2.4 Other Aberrations
While it is true that Spherical and Chromatic Aberration and Coma are
often significant sources of image degradation in optical systems, other
aberrations are also of concern. Astigmatism, Distortion, Field Curvature,
and Lateral Color occur in the imaging of extra-axial object points.
The focused image of a round disk object displaced a distance from the
optical axis will appear oval shaped if the lens system suffers from
astigmatism. Distortion results in the familiar "barrel'' or "pincushion''
appearance of a square object such as a window frame. Although distortion
impacts image shape, it does not lower system resolution. Field Curvature
is exactly as implied by the name; the surface of best focus for a flat
object is a curved shell rather than being flat as would be desired when
imaging on a piece of photographic film or a flat projection screen.
Lateral color is the height difference for principal rays of different
wavelengths and results in a colored haze around an image at best focus.
3. EVALUATING OPTICAL SYSTEM PERFORMANCE
The need to predict optical system performance based on a paper design
has resulted in the development of a number of different analysis tools
with corresponding different levels of usefulness.
3.1 Optical Tolerances
Traditional optical tolerances are based on the 1/4-wavelength
criterion established by Lord Rayleigh. That is, that images will not
be badly impaired as long as the longest and shortest optical path
lengths to a selected focus do not exceed 1/4 wavelength of light.
In addition to calculating the achieved value of most aberrations, the
dbOptic software also presents the corresponding tolerance value.
Clearly, the notion of fixed tolerances for performance evaluation
implies the existence of a well-defined boundary between acceptable and
non-acceptable performance. In fact, image degradation or improvement
will be found to be gradual as a design progresses through the tolerance
region.
3.2 Meridional Ray Plots
If several rays in a single plane which includes the optical axis originate
from an off-axis object point, they will be found to intercept the optical
axis in object space at readily determined locations. We can then easily
trace the rays through the optical system using the axis intercepts to define
the object distance for each trace. When this is accomplished, it is also
possible to determine the height H' in image space at which each ray
intercepts a flat plane perpendicular to the optical axis at the paraxial
image distance, l'. When the H' value for each trace is plotted against
(1) Tan U', the final image angle and (2) L, the object distance, we find
two useful graphical indicators of optical system performance.
(1) H' - Tan U' Curves
When final image angle, Tan U', becomes the abscissa then a perfect lens
will plot as a straight horizontal line. The two ends of the line
represent the first and last rays of the fan or beam to pass completely
through the system, called the upper and lower rim rays. The figures below
show a variety of outcomes.
(2) H' - L Curves
If the system includes an aperture stop of diameter D at distance Lo
from the first lens surface, a wide beam of parallel rays at angle ø
to the optical axis are traced as a series of rays from axial object
points. The object distances vary in the range Lo ± D / 2 Tan ø .
If we do not know the best location for the stop, it can be determined
by tracing a number of oblique rays over a range of object distances L
and plotting H' vs L. The stop will allow rays only in the range of
Lo ± D / 2 Tan ø to pass. In the figures below, the upper and lower rim
rays and the principal ray are noted by u, l and p. The principal ray is
defined as the last ray to be cut off as the diameter of the stop is reduced
to zero.
In ray-trace calculations the chief ray, midway between the upper and lower
rim rays, is often used to represent the principal ray. If there is no
vignetting, that is, if the stop is large enough and positioned such that it stops
no incident rays, then the chief and principal rays are the same.
Examining the curvature of the graph in the region of the principal ray
gives an indication of any coma present. If the curve contains an inflection
point, placing a suitable stop at that distance from the lens will minimize
the magnitude of coma. Reference [3] Page 210 contains additional information
on the interpretation of H' - L curves.
3.3 Spot Diagrams
Skew rays are rays from extra-axial object points which enter the first
surface of the optical system outside the meridional plane. Skew rays make
up the majority of rays which enter an optical system from an extra-axial
point source. If the first surface of a lens is divided into a large number
of equal areas and a ray is traced through the center of each area, the ray
intersection pattern on the paraxial image plane is known as a ray spot
diagram. The "tightness'' of the pattern provides an indication of axial as
well as off-axis performance of the system.
The image below shows the results of tracing 2,000 rays through a well-corrected
500mm focal length f/8 telephoto lens, as viewed at the paraxial image plane.
The image is typically examined at two other locations: (1) at the DLC (Disk of Least
Confusion), which is at a point ¼ of the distance from the marginal image plane to
the paraxial image plane and (2) Midpoint, which is half way between the paraxial
image plane and the DLC. For systems that are well-corrected for spherical aberration
there will be little difference between the 3 spot diagrams. The scale shown for the
example below is in millimeters
Spot diagrams are perhaps most useful in determining the quality of off-axis imagery.
In the figure below, the same system is examined with rays entering obliquely, at 5
degrees off-axis.
3. 4 Encircled Energy Plots
Finally, to provide a more quantitative measure of performance using spot diagrams, a useful
tool is the Encircled Energy Plot. This is generated by starting at the center of the spot diagram
(using the center point calculated statistically in the off-axis case) and counting the fraction of
incident rays that are within any radius R. The figure below is an Encircled Energy Plot for the
above lens for the 5 degree off-axis case. In the example, the percentage of incident energy as
shown on the Y-axis does not exceed 80% due the fact that not all rays entering the system at
5 degrees off-axis can reach the image plane.
3.5 Extended Objects
Instead of using a single point as the object for a spot diagram, we could
consider an array of object points at the same finite or non-finite object
distance and create a composite spot diagram at the image plane. The
composite image of this "object map" provides a near-realistic view of
the image that a candidate optical system might generate.
 |
 |
| Original
object from file. |
Image
generated by tracing object array through a well-corrected
76mm f/5 achromat. |
3.6 Modulation Transfer Function
The Modulation Transfer Function (MTF) is a lens evaluation tool designed to
characterize the ability of a system to faithfully deliver the contrast
present in an object to the image. It depends on the spatial frequency
which is simply the number of contrast (light-to-dark) cycles per unit
length in the object. MTF is the ratio of image contrast to object contrast,
determined for a range of object spatial frequencies. That is,
MTF( µ ) = Ci( µ ) / Co
where Ci( µ ) = Image contrast at frequency µ
The usual object intensity function is a sine wave representing a set of
equally spaced lines with the sine wave intensity level cycling between the
dark lines (low intensity) and white spaces (high intensity). If I is the
intensity, then object contrast is given by
Co = ( Imax - Imin ) / ( Imax + Imin )
When a pair of lines in the object are far apart (low spatial frequency) we
could expect that the lens may have little difficulty resolving the lines
and the MTF is nearly one. As the spatial frequency increases, a lens will
begin to lose its resolving ability. For an imperfect lens, this loss will
occur at lower spatial frequencies than for a perfect lens.
MTF is not addressed in the design program, dbOptic. References [4] and
[5] contain additional information about MTF.
Aspheric Surfaces
The spherical ray-tracing equations presented in Section 2 are a special
case of tracing rays through any smooth surface. In order to trace rays
through aspheric surfaces, suitable equations must be developed for the
more general case.
The aspheric surface shown above is in general form x = f(y). That is,
the displacement of the curve measured as a distance x from the y-axis is
a function of some height value y. Note that expressing the surface in
this form represents a slight departure from normal mathematical convention
where the independent variable is usually selected to be x.
If x and y are the ray intercepts at the surface at P, we can write
y / ( L' - x ) = Tan U' [A-1]
If ZP is the normal to the surface at (x,y), then the angle of incidence I
is shown by Angle CPL and the angle of refraction I' is shown by Angle CPL'.
Since U' = U + I - I', then
L' = f(y) + y / Tan ( U + I - I') [A-2]
The relationship between I and I' is determined by Snell's Law:
I' = arcsin ( n sin I / n' ) [A-3]
Noting that the slope of the normal to the surface at (x,y) is simply the
first derivative of x with respect to y and that this value is exactly
equal to Tan ( I + U ); or dx/dy = f '(y) = Tan ( I + U ), then
I = arctan f '(y) - U [A-4]
Equation [A-4] becomes the starting point for the trace through an aspheric
surface. The intercept height y at the surface is determined by simultaneous
solution of the equations of the surface x = f(y) and the ray
y / (L - x) = Tan U. For the paraxial trace, we recall that the sine and
tangent (and arcsin and arctan) of small angles are equal to the angle.
References.
[1] Mackintosh, A., et al (1986). Advanced Telescope Making Techniques,
Vol 1. Willmann-Bell, Inc; Richmond, Virginia.
[2] Conrady, A.E. (1957). Applied Optics and Optical Design, Dover
Publications, Inc; New York (Reprint from 1929 Oxford University Press
edition).
[3] Kingslake, Rudolf (1978). Lens Design Fundamentals, Academic Press;
New York.
[4] Schroeder, Daniel J. (1987). Astronomical Optics, Academic Press;
New York.
[5] Mahajan, Virendra N. (1991). Aberration Theory Made Simple, SPIE Optical
Engineering Press; PO Box 10, Bellingham, Washington 98227.
