ECEN 4616/5616
Optoelectronic Design
Class website with past lectures, various files, and assignments:
http://ecee.colorado.edu/ecen4616/Spring2014/
(The first assignment will be posted here on 1/22)
To view video recordings of past lectures, go to:
http://cuengineeringonline.colorado.edu
and select “course login” from the upper right corner of the page.
Right Side Rear-View Mirror
In a previous lecture, the question was asked: What are convex (negative)
mirrors used for?
One very common use is for the right-side rear-view mirror on an automobile.
θ’ R
θ
Driver
The engineering design task is: Given the size constraints of the
mirrors, W, and the geometry of the car layout, L1 , L2 ; Find R
such that θ = θ’. (i.e., make the driver’s field of view equal
on both sides.)
Other Issues with Convex Mirrors
Observer
l
l’
f’
Note that, for a convex mirror, l > l’, but
also M < 1 (i.e., things appear both
nearer and smaller).
Whether objects actually appear closer
or farther depends on if the user
depends mostly on size or binocular
vision (or focus) cues for distance.
C
Or, in other words:
Shape Factor and Aberration
“Bending Lenses”
The shape of a lens is defined by the shape factor, X:
c1  c2
X
c1  c2
The following lenses all have the same focal length, but different shape factors:
X<-1
X=0
X=-1
c1
For the lens above with shape
factor, X=-1, the Zemax LDE is:
X=1
c2
X>1
LDE for shape factor X=1
The “Solve” on surface 2, Radius, is:
This dialog box is accessed by rightclicking on the Radius box in the LDE.
The Solve chosen, “Element Power”
causes the radius to be automatically
calculated so as to maintain a lens
power of 0.05mm-1 or a focal length
of 20 mm.
The first surface of the lens can then be made variable without changing the power of the lens.
Shape Factor and Aberration
The lens’s aberrations will change as its shape factor changes, since
this changes the real angles of incidence of the rays impinging on it.
We can monitor the aberrations using the operands available in the
Merit Function Editor (MFE):
Click on the ‘Help’ menu item in the MFE to bring up the operand list:
The list goes on for several pages. Clicking on an operand brings
up the definition and instructions for use.
For example, clicking on ‘COMA’ in the above list gives this information box:
The terms “Surf” and “Wave” refer to the columns in the MFE (Merit
Function Editor) which will be labeled as such when the COMA operand is
entered.
Other operands for third-order aberrations of interest are:
•SPHA: Spherical Aberration
•FCUR: Field Curvature
•ASTI: Astigmatism
Install these four operands in the MFE, setting the “Surf” value to “0”, so
that the aberration sum for both surfaces is returned:
(Here, the “Hide Column” feature under the “View” menu has been
used to shrink unused columns in the editor.)
We set Zemax up with two field points at ∞, one on axis and one at
5 degrees (since only spherical aberration exists on axis – the
others require an off-axis field):
Likewise, since these are all monochromatic aberrations, we use
only one wavelength:
We can then update the MFE and see what aberrations Zemax reports:
Using our Merit Function which reports 3rd order aberrations, we can write a Zemax
macro (see: “Zemax Programming Language”, chapter 25 in the Manual). Here is a
macro in the “ZPL” language that loops our system through 50 different Shape Factors
and prints out the aberrations:
!ThirdOrd.zpl
!
!Print out 3rd order aberrations vs. shape factor
!Run on Shape.zmx
!Editors LDE and MFE must be open for macro to work correctly
!
!Number of points to print:
N = 50
!
!Surfaces in Shape.zmx
frontsurf = 3
backsurf = 4
!Rows in MFE:
Spha = 1
Coma = 2
Asti = 3
Fcur = 4
!
Note that there are only 20 active
statements in this macro – 20
statements are only comments.
!Get starting curvatures (should already be at one extreme)
c1 = CURV(frontsurf)
c2 = CURV(backsurf)
!
Inc = -(c2+c1)/N
!
!Loop through shape factors:
FOR S1, c1, -c2, Inc
!Set curvature of frontsurf:
SetSurfaceProperty frontsurf, 2, S1
UPDATE all
!Get back surface curvature:
S2 = CURV(backsurf)
!Calculate the shape value:
Shape = (S1+S2)/(S1-S2)
!Extract the aberrations from the MFE:
SA = OPER(Spha,10)
CO = OPER(Coma,10)
AS = OPER(Asti,10)
FC = OPER(Fcur,10)
!
PRINT Shape, " ",SA," ",CO," ",AS," ",FC
NEXT
ZPL is much like early “Basic”, but contains a number of functions that
access Zemax editor values – such as “OPER”, which accesses values in
the MFE – and keywords – like “UPDATE” – which drive Zemax
operations. Macros are run from the “Macros” menu from the top of the
Zemax window, and they must be in the Macro Directory. Not sure
where the Macro Directory is, or you would like to change it? Look under
“File/Preferences/Folders”.
The output of our “ThirdOrd” macro is a text window:
Where the columns are determined
by our PRINT statement:
PRINT Shape, " ",SA," ",CO," ",AS," ",FC
To be Shape factor, Spherical
Aberration, Coma, Astigmatism, and
Field Curvature.
You can save this text to a file and use it in a plotting program (or
you can deal with the primitive plot capabilities of ZPL):
Spherical & Coma vs. Shapefactor
100
Sph Abb
Coma
80
60
Aberration in
Waves
40
20
0
-20
-40
-1.5
-1
-0.5
0
S
h
a
p
e
Factor
Shape Factor
0.5
1
1.5
Expanded View:
Spherical & Coma vs. Shapefactor
Sph Abb
Coma
20
15
10
5
0
-5
-10
-0.4
-0.2
0
0.2
0.4
0.6
Shape Factor
0.8
1
1.2
1.4
As can be seen from the close-up, both S.A. and Coma are
minimized at a Shape Factor of ~0.7
Adding weights to the spherical aberration and Coma lines of our MFA:
Plus making the radius of the first lens surface variable and optimizing,
we get this lens (R1=11.7825, R2=-79.00985, Shape Factor = 0.7405):
This is known as a “Best Form” Singlet.
Example of Aberration Correction
Running our macro on a
negative singlet, we also note
that a negative lens also has a
‘best form’, but the S.A. is
negative whereas the S.A. for
a positive lens is positive.
Let’s try adding a negative lens to
the best form positive lens so as
to maintain positive power, but
with the form(s) adjusted to
(hopefully) elminate S.A.
Aberration in Waves
We’ve seen that Spherical Aberration (S.A.) cannot be eliminated with
a singlet lens – but that, for a given power, the shape of the lens
controls to a great degree how much S.A. the singlet creates.
Spherical Aberration Corrected Doublet
Let’s start with a pair of symmetrical lenses:
•A positive lens with K = 0.02 (f=50 mm)
•A negative lens with K = -0.01 (f = -100mm)
The result will be a combination with f ~ 100 mm. (We can scale the
lens after optimization to get an exact focal length.)
Close up view:
Here is the LDE for the starting system:
And here are the curvature solves on the second surface of each
element:
Here is the 3rd order aberrations plot.
Note that the negative lens (surfaces 4&5) has aberrations of the opposite sign
from the positive lens (surfaces 2&3), and that some aberration correction is
already taking place, even though we haven’t optimized it.
A second off-axis field has been added to allow Zemax to calculate the off-axis aberrations.
No chromatic aberrations are reported, as only one wavelength has been defined.
Here are the TRA plot and the Wavefront Map for the onaxis field:
Here is the Merit Function Editor (MFE)
Notes:
•Only lines 1 & 2 (spherical and coma) are weighted. We are not trying to do
everything at once.
•Lines 10 & 16 give the shape factors of lens 1 and 2 respectively. Many
calculations are possible (with some awkwardness) in the MFE.
•The operand “DMFS” (Default Merit Function Start) at the end makes sure that, if I
add the default merit function (“Design – Sequential Merit Function”), it won’t
overwrite my own operands.
Optimize the doublet shapes by opening the “Local Optimization” dialog box
(“Tools-Design-Local Optimization”, or just the “Opt” tab at the top of the Zemax
window).
Before Optimization
Lens shapes before
optimization:
After Optimization
Lens shapes after
optimization:
The MFE: Spherical and Coma
are now essentially zero:
As also reported by
the Seidel Plot:
While the TRA plot still shows some S.A., note the scale on the
diagram --- 0.1 µm. The wavelength for this system was set at
0.55 µm. A report of 1/5 wave TRA is completely meaningless, as
the ray trace assumptions are not valid at that scale.
This is verified by the MTF plot, which
shows no difference between the
system (on-axis) and a perfect lens:
Alternate Combination of Powers Formula
(How to add negative lenses and still have positive power)
u’1=u2
u1 =0
h1
Assume K1, K2 are thin lenses:
let n1  n1  n2  n2  1.0
h2
u’2
d1
n’1 = n2
n1
K1
n’2
K2
By following a ray entering the system of two lenses parallel to
the axis, we were able to find the standard combination of
powers formula by eliminating all references to ray variables:
K  K1  K2  K1K2d1
We would like to develop an equivalent combination of
powers expression that is given in terms of the parallel input
ray, as this will be useful in designing systems for aberration
correction.
f
u’1=u2
u1 =0
h1
h2
Refract through first element:
d1
u1  u1  h1K1  u2  h1K1
Since:
u1  u2 , and u 1  0
u’2
f
K2
K1
Refract through second element:
h1
u2  u2  h2 K 2 but also u2    h1 K
f
Substituting for u2 , u2 :
We could replace the two lenses with their
equivalent lens, then add a third lens and do
 h1 K   h1 K1  h2 K 2
the same analysis getting:
h2
 K  K1  K 2
h1
And, by induction,
for n lenses:
Ktotal  K 
h3
h
h
K 3  K1  2 K 2  3 K 3
h1
h1
h1
h3
h2
K  K1  K 2  K3 
h1
h1
hn
 Kn
h1
h3
h2
K  K1  K 2  K3 
h1
h1
hn
 Kn
h1
This combination of powers formula gives us some insight into optical
design principles.
For example, in the Cooke Triplet design, the negative element is
positioned such that it subtracts less power from the combination than
it’s intrinsic power:
This allows the negative lens to cancel some of
the aberrations of the positive lenses, without
having too great an effect on the total power.
h1
h2
Remember that the ray heights referred to are those of a ray traced
into the system parallel to the axis – the ray that is used to define
the power of a combination of lenses.
Returning to our corrected doublet:
Suppose we select a range of wavelengths covering the
visible and see how the system behaves:
Looking at a stretched and zoomed section of the layout, we
see that the red, green, and blue wavelengths don’t focus at
the same distance:
This aberration is also shown in the “Chromatic focal Shift” analysis window
(“Analysis – Miscellaneous – Chromatic Focal Shift”), which shows ~ 1.5 mm
change in focal length over the visible.
Here is the Seidel (3rd order) aberration plot. Note that “Axial
Color” (another term for ‘chromatic focal shift’) is the largest
remaining aberration. There are both positive and negative
contributions, however – perhaps to correct for axial color we
need another degree of freedom.
Dispersion and Achromats
Zemax’s “Glass Map” (‘Analysis – Glass and Gradient Index – Glass Map’):
Dispersion
The index of refraction of glasses is traditionally taken at three
wavelengths (for use in the visible). These are the Fraunhofer lines for
hydrogen and helium ( a nearby line from mercury was used before
helium became widely available). These were convenient, because they
spanned the visible spectrum and could be duplicated with great
precision in any well-equipped laboratory.
1. F (hydrogen) ≈ 486nm
2. d (helium) ≈ 588 nm
3. C (hydrogen)≈656 nm
The corresponding indices of refraction
(for a given glass type) at these
wavelengths are tradionally labeled:
1. nF = n(0.486µm) (blue)
2. nd = n(0.588µm) (green)
3. nC = n((0.656µm) (red)
Dispersion
In the “glass map”, the index of refraction is the index at the d line
(nd), and the ‘Abbe Number’ (often called the “V-number” by those
who aren’t sure how to pronounce “Abbe”) is:
nd  1
Vd 
nF  nC
The significance of this value follows from the calculation of the
chromatic variation of power for a thin lens.
Power at d:
Kd   nd  1 c1  c2    nd  1 C
 nF  1 C
Power at C:  nC  1 C
Power at F:
Consider:
K F  KC  nF  1 C   nC  1 C nF  nC
1



Kd
nd  1 Vd
 nd  1 C
Hence:
K F  KC 
Kd
Vd
Dispersion
.
From the image equation, we have:
1 1
1 1
  K F , and
  KC
l F l F
lC lC
Kd
1 1
   K F  KC 
l F lC
Vd
Given that the
Object distances
are all the same:
l F  lC  l d
So, the change in power is proportional to the power divided by the
Abbe number.
(It can also be shown that change in magnification with wavelength
– transverse chromatic aberration – is also proportional to
Hence, chromatic aberration (longitudinal and transverse) will be
corrected (paraxially) if we use two lenses whos ratios sum to zero:
K1 K 2

0
V1 V2
where the K’s are implicitly
calculated at the d wavelength.
Kd
Vd
Achromats
For two thin lenses in contact: K  K1  K2
The condition that the combination have zero
focal shift between the F and C wavelengths is:
K1
K
 2
V1
V2
Solving these two equations simultaneously, we
get the construction equations for an Achromat:
V1K
K1 
V1  V2
and
V2K
K2  
V1  V2
Obviously, this only works if the lenses are made of
glasses with significantly different Abbe Numbers.
The lenses
chromatic focal
shifts cancel each
other.