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Pupils and Stops:
The system STOP is the aperture that limits the marginal ray from a field point.
For a lens by itself, the stop is the edge of the lens:
Lens with external stop:
Stop and
Entrance
Pupil
Exit Pupil: Image of
Stop from Image
Space
f’
The stop limits the cone of rays that can pass through the lens from any particular object
point. (It may be different for different object points – see wide angle lens examples.)
The Entrance Pupil is the image of the stop seen from object space.
The Exit Pupil is the image of the stop seen from image space.
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Pupils and Ray Tracing:
1. There is no point in tracing a ray that misses the stop, as it will not traverse the
system and contribute to the image.
2. If the stop is behind the lens (or some lenses, in a compound system), then rays
can be aimed into the entrance pupil, which is the image of the stop as seen from
object space. Because of the properties of imaging, a ray aimed into the entrance
pupil is guaranteed to make it through the stop.
Zemax (and most other programs’) conventions:
Many analyses in Zemax will require you to specify both the Field and Pupil
coordinates of a ray. These are normalized coordinates that go from –1 to 1 in X and
Y and have the following meaning:
I)
II)
The field dialog box (you will use in the tutorial) lets you specify points on
the object by a number of measures: angle, object height, etc. The
normalized field coordinates (labeled Hx, Hy in Zemax) go from zero at the
Z-axis to 1.0 at the maximum size specified in the dialog box, in either X or Y
– normalized coordinates are always symmetrical, then don’t have different
scales in X and Y.
The pupil coordinates (labeled Px, Py in Zemax) refer to the X,Y location in
the entrance pupil (and hence, in the stop and exit pupil as well) in the same
way: Px=Py=0 is the center of the pupil, and Px=1,Py=0 is the top edge of the
pupil. (NOTE, that Px=1, Py=1, would fall outside of a circular pupil.)
Zemax does the work of figuring out where the appropriate pupil is, and which way
the ray should be aimed to get from the designated field position to the desired pupil
position. This is usually very convenient.
Caveats about Zemax (and most other programs, as well):
I)
II)
In sequential mode (which is all our version allows) Zemax traces rays from
surface to surface in the order that they are in the Lens Design Editor. If
any of the distances in that editor are negative, then this won’t be the order
that the surfaces are placed in space. This leads to non-physical behavior.
Zemax does not check for this problem – that is up to you. You can simulate
this by deciding to trace rays through a system in a different order than the
elements make from left to right. The equations all still work (if you follow
the sign convention), but the results are nonsense.
Not all analysis methods are meaningful for all systems. Zemax sometimes
will warn you of this.
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Aberrations:
Aberrations are errors in the imaging task – rays that start from the same object point do
not all arrive at the associated image point. We will be covering Aberration Theory later
in the course, but will deal with specific aberrations on a necessary basis.
Some aberrations arise in the paraxial domain: “Chromatic Focal Shift” is an example. It
is simply the change in focal length with wavelength due to materials having different
indices of refraction for different wavelengths. As a paraxial aberration, it can be
corrected paraxially, which we will do in the next several weeks as a prelude to designing
achromatic lenses – lenses that have minimal chromatic focal shift.
Some aberrations of finite-size optics only show up at off-axis fields, some everywhere.
An aberration we will be using Zemax to minimize in the next homework is called
“Spherical Aberration”, which is illustrated in the following drawings.
Spherical aberrations arises for two reasons:
I)
The surface of a sphere diverges from the quadratic (that 1st order optics is
based on) as the radial distance from the Z-axis increases. Ultimately, a
sphere is limited by it’s Radius of Curvature and rays further than that from
the axis will miss it altogether. Long before that happens, however, the angle
of incidence of the ray with the surface is increasing significantly faster with
ray height than for the paraxial approximation.
II)
In addition, Snell’s law using sine functions is also increasingly non-linear as
the angle of incidence increases.
Spherical Aberration:


Lens power changes with ray height
Positive SA: lens power increases with height
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SA is the consequence of the deviation of sin(x) from x outside the paraxial
region:
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Example of Spherical Aberration (SA):
(‘TA’ stands for ‘Transverse Aberration’ – the distance a ray misses the paraxial
focal point on the focal plane.)
Why does reversing the lens above greatly reduce SA?
(Answer: The maximum angle of incidence is much less when both surfaces
participate in refraction.)
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How to do Fourier Optics in a Ray Trace Program?
Given an optical system that is analyzed by tracing rays:
Zemax (and other programs) will give us wave-based data such as:
Wavefront Maps:
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MTF Plots:
Diffraction PSF:
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Wave-based Analyses are calculated from the pupil OPD.
To construct a Optical Path Difference map of the exit pupil, first a ray is traced from the
desired object point through the center of the entrance pupil (and, hence out the center of
the exit pupil). This is called the chief ray.
Then, rays are traced from the same object point through various parts of the entrance
pupil (and hence exit pupil). When the rays arrive at the image plane, the OPD between
each ray and the chief ray is recorded.
This set of OPD values is then assigned to the exit pupil. A perfect optical system will
have the OPD values at the exit pupil describe a spherical wavefront centered on the
image point. The differences between the actual OPD values and the ideal sphere are
what Zemax displays at the “Wavefront Map”.
From this relative (to a sphere) wavefront pupil map, the various wave-based analysis
properties can be calculated as follows:
(Question: What is the geometrical meaning – i.e., in rays -- of these relationships?)
psf  FP
2
OTF  Fpsf 

 F FP

2


 F FP FP
 P  P (autocorre lation the orem)
and

MTF  OTF
What approximations are involved in this method, and when do they hold?
 All diffraction is ignored up to the exit pupil. This approximation assumes that only a
trivial amount of the total light energy is scattered by diffracting from the edges of
lenses, stops and diffraction spreading while propagating through the optical system.
Only the diffraction effects where the light is concentrated in a small area (such as the
PSF) are considered.
 This is a good approximation, as long as all of the structures in the optical
system are much larger than a wavelength of light. When lenses get to be
only a few dozen (or less) wavelengths across, such as pixel lenslets for
detector arrays, the light diffracted by individual elements in the optical
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system will become a significant fraction of the total light and this
approximation breaks down. In these cases, the light propagation through the
optical system must be modeled using wave propagation algorithms.
 Obviously, if dedicated diffractive structures -- such as gratings -- exist in the
optical system, they have to be explicitly accounted for, although it may be
possible to use ray splitting to model the grating if the grating is large w.r.t.
the wavelength and still only consider diffraction from the exit pupil to the
image.
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