Transcript Document

Optics and Optical Design
Richard G. Bingham
Session 1
Rays that may appear odd
Image rotation and parity and an image
Spherical Trigonometry – advanced option
(C) R G Bingham 2005. All rights reserved.
The beginning
Six pages of introductory notes
(C) R G Bingham 2005. All rights reserved.
Ray-tracing software is amazingly accurate and comprehensive, but
practising with the buttons on the toolbar is not enough. Our aims are:
to understand optical systems; to understand what the program is doing;
to gain a feeling for what problems are tractable and what optical
capabilities are provided by existing designs; to assess the performance
of optical designs constructively; to see how complex optics may be
comprised of sub-systems exploiting local aberration correction or
special cases; to express our ideas with the program; to handle the tools
that are available for designing new systems; to see what factors may
stop optical aberrations from being made smaller; to be aware of cost
and weight; to avoid redundant complexity in a design or its
manufacturing specification; and to be practical and economic as
regards the optics that we design and the use of our time. We hope to
achieve all this by properly understanding the designer’s methods, so
that we can learn and keep up to date in our own new designs or others
that we may evaluate.
(C) R G Bingham 2005. All rights reserved.
Format of this text
This material, although created in PowerPoint, was not intended for,
and is not suitable for, projection as slides. PowerPoint was used to
encapsulated successive ideas and topics in a format that ultimately
will be neatly separate screens. A slide is referred to here as a
The material is arranged separate “sessions” of about 25 pages each.
A few sessions may include one or two .doc files as well.
As mentioned above, the material could be viewed as successive
screens by an individual user, but initially, it will be printed (quite
large), again to be read individually. The suggested procedure is to
first to read a session individually and to follow that by discussing it
with others in a group, or if possible, to arrange some kind of tutorial
following the individual reading.
(C) R G Bingham 2005. All rights reserved.
How to handle this
I think that it is impossible fully to practise each topic before moving on. Perhaps
material can be skimmed over if it does not seem helpful. Or perhaps the concepts and
terminology from complex areas may be acquired in an effortless osmosis, later.
Most equations are given here without derivations; the aims of this particular course are
such that we need to press on to the work of the designer. The fundamentals, however,
may be useful for a physicist or engineer to understand. They are written up by Welford
in “Aberrations of Optical Systems”, Hilger 1986. It may be worth considering whether
to work systematically through the whole book, as indeed I did with the previous
edition, for a comprehensive foundation. I refer to the book as ‘Welford’ throughout.
Examples are given in Zemax. If both the .zmx and the .ses files are in the same folder,
graphics windows should open when the Lens Data Editor opens. Also explore any of
the other Zemax features, their local help files and the main manual (.pdf).
Within the ZEMAX examples, there is a window called Title/Notes that appears on a
tab after the Gen button is pressed. As you view one of the examples, it is vital to read
those notes that are within the Zemax data, as I have often put critical information there.
I recommend that you do not personalise the default toolbar buttons in your own copy
of Zemax. If you do, you may find that a different computer is slower to use, and it will
become markedly more difficult to demonstrate things to other users.
(C) R G Bingham 2005. All rights reserved.
How can I get to practise on some real designs?
If you are already working with any optics, a way to practice on
real optics without spending money is to create a ray-tracing
model of your experiment as soon as there is even a minor
modification to explore. So for example, if some existing
instrument needs merely to be differently focused or to have a
thicker colour filter, or some as-made data becomes available,
use that as a reason to set up a ZEMAX lens data file, however
simple, and then use your ZEMAX file to check the modification.
Once you have a few ZEMAX models, you will also be better
placed to solve any further problems arising with your
experiment, and will be able to create realistic graphics for
publication. However, I would say that whilst setting up and
using ZEMAX yourself for even simple purposes is very
instructive, collecting complex data files from other people
would be a waste of time.
(C) R G Bingham 2005. All rights reserved.
• ZEMAX was created by Ken Moore from 1990 to date.
•It is for PCs only.
•We shall use Zemax in this course.
•The ‘hard key’ allows only one person to access it at once. It is
worth the whole value of the software.
•You can open Zemax twice on the same machine (except with
Remote Desktop). The second instance of Zemax is useful for
running long computations whilst getting on with something else, or
for cutting and pasting between systems.
•You need individual, unrestricted, continuous access to the program,
along with technical support and updates, to do much productive
work in optical design, if that is your task. Sharing a system is
frustrating and hinders the design of real lenses.
(C) R G Bingham 2005. All rights reserved.
Example – setting up a ball lens
The example is a glass ball, as available from Melles Griot. It is used as a lens (e.g. for
fibre-optics work). It can be set up by following the details in Ball_lens_data.doc. This
is the main example with such instructions for creating it from absolute scratch, which
will provide a useful exercise if you have not previously set up lenses in ZEMAX. In
any case, please find the further notes that I wrote within that .zmx file. They can be
found on the General / Title/Notes tab ( on the ‘Gen’ button). Such further notes are
held within all my ZEMAX examples in these lectures, so it will be useful to be able to
find them. To check your results, the intended data is in file: Ex00-Ball_Lens.zmx.
If the Ex00-Ball_Lens.SES file is also present, it will bring up relevant graphics as
(C) R G Bingham 2005. All rights reserved.
Coordinate systems
13 pages
(C) R G Bingham 2005. All rights reserved.
x, y, z
Surface of a lens
In Zemax, light must
leave the object surface
in the +z direction. The
thickness of the object
surface must be positive.
‘sag’ z
Positive radius
of curvature
Right-handed axes
Negative sag,
negative radius
of curvature
(C) R G Bingham 2005. All rights reserved.
Lens. Sequential ray tracing
1. The ‘thickness’ or
‘axial thickness’ is
measured positive in
the +z direction
2. The curvature of the front surface is positive here
Example file: Ex01-Lens.ZMX
3. The ‘front’ of a lens
is the face that the light
hits first
4. Lens drawings in most diagrams are
cross-sections that are not shaded
5. This is an ‘optical system’
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Defining surfaces for ray tracing
•Local coordinates. A surface S1 has an
origin (O1 here) that serves to locate
this surface within the optical system.
•The ‘figure’ of surface S1 is defined by
its sag z(x,y) that is thus measured
orthogonal to the x,y plane.
•Each surface also has a following
‘thickness’. For sequential ray tracing,
the origin O2 of the next optical surface
S2 is positioned by this thickness of S1 +x is inwards
along the z axis.
•Each surface can be positioned in this
way with reference to the previous
(C) R G Bingham 2005. All rights reserved.
Thickness after a mirror
The thickness of glass or an air space
entered into ZEMAX’s Lens Data Editor
changes sign at a mirror.
(C) R G Bingham 2005. All rights reserved.
Sequential ray tracing - a Mangin mirror
Example file: Ex02-Mangin.ZMX
The signs of these
This one piece of glass has to
curvatures are all
appear in the data twice for
sequential ray tracing. It has
positive thickness before the
mirror but negative thickness
after the mirror ; in the second
pass, the thickness is measured in
the minus z direction from the
preceding surface, the mirror.
The three rays in this 2D layout are
drawn optionally from a flat surface
numbered 1 that is in the data file. This
surface 1 is a ‘dummy surface’ here. A
dummy surface is one that has the same
refractive index on both sides.
The three rays started from
an object at infinity at
surface number 0.
A backsurface
(C) R G Bingham 2005. All rights reserved.
The manufacturing specification is different
Wrongly made convex because a positive
sign was quoted? A curvature with a positive
sign is not necessarily convex in Zemax!
Lens in the
Lens on machine
Aspheric lens
‘Glass removal’ might correspond to positive z in the
ray-tracing data but negative z on a machine.
There is a necessary intermediate stage where we supply
an engineering drawing or at least a sketch. Zemax will
output a good starting point for a proper drawing.
(C) R G Bingham 2005. All rights reserved.
Local Tilts and Decentres
The program provides for surfaces
that are tilted or moved (‘decentred’).
The local frame of reference can be
tilted or moved, leaving the function
z(x,y) unchanged.
For example, the weird lens can have
a right-hand surface that is tilted
around its local x axis. The tilt shown
is positive around the local +x axis.
Combinations of two or three tilts can
be used. The origin O2 can also be
locally shifted in x and y, but not z –
we have already chosen a z position,
based on the previous surface.
+x is inwards
(C) R G Bingham 2005. All rights reserved.
x, y, z rotations – signs
•We rotate the axes in which subsequent surfaces are then
defined, so the subsequent surfaces move with the rotation.
•A positive rotation in ZEMAX is clockwise as viewed in a
positive axis direction.
Rotations that are consistent with the above:
In the x, y Argand diagram a positive angle is drawn
anticlockwise, but it is clockwise looking along a
+z axis emerging from the paper. A sign like that of
the Argand diagram is also used for the angle of a
ray in some optical diagrams, and for position angle
on the sky, e.g. for the plane of sky polarisation
measured anticlockwise (from north).
International standards for machine tools such as
milling machines also use this sign.
A positive rotation
around the x axis. The
relevant matrix
expression is in the
ZEMAX manual.
(C) R G Bingham 2005. All rights reserved.
x, y, z rotations that are different
OPTICA (Mathematica) is the opposite. It applies a positive rotation looking
towards the origin, that is, in the negative axis direction. The difference arises
from rotating an object the other way in global axes. The global axes in
ZEMAX can also be unaffected, and often one need not be aware of them
anyway. Check how they are defined. In ZEMAX, with a coordinate break,
fresh local axes are obtained for following parts of the system.
In the Code V optical design code (current), a rotation around either the local x
or the local y axis is negative as compared to the right-hand convention used in
ZEMAX, whilst rotation around z is not affected.
In the GRT optical design code (obsolescent university FORTRAN), the x
rotation is reversed as in Code V. However, the y rotation is not reversed. The
z rotation appeared to be reversed when I rotated the lines on a diffraction
Other codes may be different again.
Doubts can be resolved by inspecting 3D graphics that the program will draw!
(C) R G Bingham 2005. All rights reserved.
Sequential ray tracing – a prism
The Lens Data Editor contains a sequence of optical surfaces.
The path of any ray is calculated to intercept the next optical
surface, in the order in which they are listed in the Lens Data
Editor. See next slide for an example in ZEMAX.
(C) R G Bingham 2005. All rights reserved.
Coordinate breaks - a prism
1. The prism angles are defined by ‘coordinate breaks’. A coordinate break is placed at
a dummy optical surface, taking up one line of data in the Lens Data Editor. That line
has boxes for x & y shifts and for x, y & z tilts in degrees. Other boxes on the same line
hold the thickness to the next surface and a flag for reversing the order of the tilts.
Example file: Ex03-Seq_Prism.ZMX
Thickness 4
2. The coordinate breaks are: 2 the tilt of
the front surface; 4 the tilt of the z axis
leading to the required centre of the exit
surface; 5 the tilt of the exit surface; and 7
the tilt of the z axis leading to the required
centre of the image surface.
3. The rays refract passively according to Snell's
law. The rays do not affect the tilts of the axes in
these data.
(C) R G Bingham 2005. All rights reserved.
Coordinate breaks again
1. If we place a coordinate break immediately
before an optical surface, it is sometimes
useful to put in another one with the
opposite sign immediately after that
surface, undoing the tilt etc. Then we can
continue with the previous axis direction.
2. Reversing multiple tilt angles needs be
done in the reverse sequence. Read the
(C) R G Bingham 2005. All rights reserved.
Tilts and shifts again
1. SEQUENTIAL RAY TRACING uses local coordinates:
‘Coordinate breaks’ are inserted in the Lens Data Editor ahead of
optical surfaces that need moving. The values of the angles and
shifts can be varied in optimisation.
Again for sequential ray tracing, individual optical surfaces can be
given an intrinsic tilt or shift on a Surface Properties tab. The
effect is the same as with with coordinate breaks but the data are
not currently optimisable.
For NON-SEQUENTIAL RAY TRACING, a whole object can
have its own tilt and position set up in a line of data for that object
within the Non-Sequential Components Editor. This is essentially
a global reference system, although alternatively, one component
can be referred to another.
(C) R G Bingham 2005. All rights reserved.
Rays that may appear odd
Two pages about geometrical rays in a
ray-tracing program
(C) R G Bingham 2005. All rights reserved.
A geometrical ray is a line to the next surface, but …
•AA is a normal ray path
•In sequential ray tracing, ray B is a valid extrapolation to a virtual image in a
negative space. It has to end the ray trace or to be reversed.
But in a sequential ray trace, rays can fail or“crash”:
•Ray C misses a defined surface aperture
•Ray D, if reflected by TIR, terminates unless the surface is defined as a mirror
•Ray E misses the sphere altogether
•Ray F may intersect the sphere at the wrong point.
(C) R G Bingham 2005. All rights reserved.
Non-sequential ray tracing – a prism
Example: Ex04-NSC_Prism.ZMX
This ‘Rectangular Volume’ object is
defined on a single line of the NonSequential Components Editor. The faces
of this object would form a cuboid by
default, but they can be angled as they are
here, and the whole thing is tilted 30º
around x. Rays shown here intercept 0, 2
or 3 surfaces.
Two sorts of rays appear
that go through this
prism but are not
dispersed in angle.
There are many types of NSC objects.
They can be positioned with respect to a
single origin, or individually with respect to
other defined objects. NSC rays can split,
scatter, etc., but don’t crash.
(C) R G Bingham 2005. All rights reserved.
Image Rotation and Parity and an
image slicer
Ten Pages
(C) R G Bingham 2005. All rights reserved.
Inverted real image in an axially symmetrical
•Inverting the image means rotating it 180 degrees around the axis.
Notice that ZEMAX views the image in the +z direction.
•If there is an intermediate real image, the final image is erect.
• The same applies to mirrors as to lenses.
•Thus a Cassegrain telescope gives an inverted image but a Gregorian
telescope, which has one intermediate image, gives an erect final image.
(C) R G Bingham 2005. All rights reserved.
Image Parity
•Reversal of parity is a three-dimensional effect.
•If the total number of mirrors is odd, the image parity is reversed.
•With an odd number of mirrors in an axially symmetrical system, the
three-dimensional image must be reversed in depth, because neither
lateral direction is special.
•Thus in a 3-mirror camera, if the
object moves one way, the image
moves in the opposite direction to
the object, just as in a single plane
•Using these ideas, in a Cassegrain
telescope, which way does the
focus move between infinity and a
laser guide star?
(C) R G Bingham 2005. All rights reserved.
Image rotators
Field of view
Image 1
Image 2
We have a few
objects around a
field of view that
is fixed in space.
The image
rotator flips their
images across a
diameter A. That
is what it does.
Then if the flipping
diameter rotates by an
angle  to line B, the
field of view rotates
by 2 as compared
with Image 1.
(C) R G Bingham 2005. All rights reserved.
Three-mirror image rotator (K-mirror)
Directions of the z axis and the signs of
the thicknesses. The unusual, reversed
final thickness affects the sign of the
final z rotation in the data file.
Animate with Tools, Slider
(C) R G Bingham 2005. All rights reserved.
Two other devices with flat mirrors
Three pages
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Dihedral mirrors
The direction of the emergent rays is unchanged when tilting a pair of
plane mirrors together. Furthermore, if the tilt is around the line of
intersection of the mirrors, the directions, positions and total path
lengths of the emergent rays are all unchanged. So for example,
mounting a pair of flat mirrors on the same base can provide stability.
The example uses both coordinate breaks and a tilt on a Surface
Properties tab.
(C) R G Bingham 2005. All rights reserved.
Image slicer / assembler
Side view
Two mirrors
simulating TIR
This modified
image slicer cuts
a patch of light
into 5 slices to fit
the slit of a
The slit formed
View into emerging beam
Enlarged View
into beam
(C) R G Bingham 2005. All rights reserved.
Image slicers (illustration)
The bHROS 5-slicers.
Two image
slicers assembled
in one unit,
comprising four
pieces of glass.
Silica base
plate ~55 mm
(C) R G Bingham 2005. All rights reserved.
Spherical Trigonometry - Advanced option
Note. I joined a project that had been running six
months, only to find that no one had been using the
correct 3D geometry in ZEMAX. As a result,
nothing was quite right and no correct 3D layouts
could be drawn. Spherical Trigonometry may seem
a forbidding subject area to step into, but
personally, I don’t expect to solve these problems in
an hour or a morning. Even if it takes a few days to
puzzle out the Spherical Trigonometry, it may be
well worth the effort. Ten pages.
(C) R G Bingham 2005. All rights reserved.
Great and small circles and spherical triangles
•Several ‘great circles’
•One ‘small circle’
•Spherical Triangles. Their angles
add to more than 180 degrees.
The sides of the triangles are great
circles. They are also angles.
Notice the spherical triangles in
which at least one of the angles is
a right angle. These triangles are
often needed and can be solved
by Napier’s rules.
(C) R G Bingham 2005. All rights reserved.
Sketching for calculations in 3D
Mapping a sphere onto a plane
P as
Side view through a sphere,
showing a plane and its pole P
Stereographic projection onto the
equatorial plane from the South Pole.
(C) R G Bingham 2005. All rights reserved.
Good News about the Stereographic
1. We never need to calculate the actual projection. It
serves as a way of sketching diagrams of 3D objects.
2. Angles measured locally on the surface of the sphere
are the same angle in the projection.
3. A circle drawn on the sphere appears as a circle in the
(C) R G Bingham 2005. All rights reserved.
Napier’s rules for a right-angled triangle (C=90º)
C = 90º
Arrange the symbols in five “parts” as shown on the right. Then
sin(middle part) = prod tan adjacent parts
= prod cos opposite parts
e.g. sin(90-c) = cos a cos b = tan(90-A) tan(90-B)
This is the one that we shall use
for the example
(C) R G Bingham 2005. All rights reserved.
Napier’s rules for a quadrantal triangle (c=90º)
c = 90º
sin(middle part) = prod tan adjacent parts
= prod cos opposite parts
(C) R G Bingham 2005. All rights reserved.
Other useful formulae for a right-angled spherical triangle:
sin a / sin A = sin b / sin B = sin c / sin C
cos a = cos b cos c + sin b sin c cos A
cos A = – cos B cos C + sin B sin C cos a
There are others but I have never needed them for
(C) R G Bingham 2005. All rights reserved.
3D example - Positioning a
Use NSC object Triangular Corner – but we want the
hollow side to face the light.
(C) R G Bingham 2005. All rights reserved.
3D example - Positioning a Retroreflector
Turning the Triangular
Corner so that the hollow side
faces the light. The poles of
all the surfaces need to be
equidistant from Z.
1. Plot the poles of the three starting
surfaces at the black dots x, y and z.
2. First, rotate the device through a
negative angle around X. The Y
pole goes to Y'. Z goes to z. Draw the
equator of Y' through z.
(90 - )
(90 - )
3. Second, rotate the device 135
degrees around Y'. The -z pole shifts
to Z'. The -X pole shifts to X'.
Z 4. The poles of the surfaces are now
X', Y' and Z'. They can be all at an
angle 90- from the original Z axis.
5. Now solve for  in the triangle
X' Z z.
(C) R G Bingham 2005. All rights reserved.
Solving the right-angled triangle
Re-draw it:
z  Z
One of Napier’s rules for a rightangled triangle says:
cos c = cos a cos b
(if C = 90 º)
So in this case:
sin  = cos  cos 45º , therefore:
tan  = 1/2
 = 35.2644 degrees
(90- )
( )
90 -  = 54.7356 degrees
(C) R G Bingham 2005. All rights reserved.
Result of positioning the retroreflector using
theTriangular Corner
This example is developed
for the advanced topic on
Spherical Trigonometry.
(C) R G Bingham 2005. All rights reserved.