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.
Lecture #27: 3/17/14
Strehl Ratio
When optimizing an optical design, it is rarely useful to put a lot of work into getting the
geometric spot size much smaller than the diffraction spot size (the Airy Pattern, for
circularly symmetric systems).
When the geometric spot size is similar to the size of the the Airy Pattern, however, the
actual MTF is determined by a complicated interference interaction between the incident
waves at the image plane.
A very useful measure of how an optical system will perform is known as the Strehl
Ratio. This is the ratio of the intensity of the peak of the systems Point Spread Function
compared to the intensity at the peak of the Airy Pattern of a perfect system.
(The measure is named for Karl Strehl, a German mathematician and astronomer, who
proposed it in 1895.)
In Zemax, the Strehl Ratio is returned by the
Merit Operand STRH, which can be used both
in the Merit Function Editor and in the
Universal Plot feature under the ‘Analysis’
menu. It is also returned by the Huygens PSF
window (also in the ‘Analysis’ menu):
Strehl Ratio
The Strehl Ratio is a measure of how well a system actually performs at imaging. It is
often described as the “Simplest, most meaningful way of expressing the effect of
wavefront aberrations on image quality”.
As an example, how does one decide which of these two aberrations will produce
the ‘best’ image?
Going by the geometric spot statistics (RMS radius and Geometric radius), we
would probably choose the right hand one.
Strehl Ratio
The wavefront maps are not much help: How do you compare two different
shapes of distortion?
The Strehl Ratio gives us a simple way to decide which is likely best:
Strehl Ratio
The Strehl Ratio can vary between 0 and 1, with 1 being a perfect system, and 0
meaning “no discernable PSF”.
Approximate Rules for Strehl Ratios:
• Strehl Ratio = 0.8 Most people can’t see any improvement beyond this
point. (A criteria due to Lord Rayleigh, once again!)
• Strehl Ratio = 0.9 A good system – not worth much effort to improve.
• Strehl Ratio = 0.95 A very good system – stop optimizing and build!
The Strehl Ratio is also an approximate measure of how much of the PSF
energy is within the central peak:
Strehl Ratio = 1.0
Strehl Ratio = 0.5
Strehl Ratio
A comparison between MTF’s and Strehl Ratios:
Spherical Aberration:
No Aberration:
TS Diff. Limit
TS 0.0000 (deg)
1.0
0.9
0.9
0.8
0.8
Modulus of the OTF
Modulus of the OTF
TS Diff. Limit
TS 0.0000 (deg)
1.0
0.7
0.6
0.5
0.4
0.3
0.2
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.1
0.0
0.0
0
37
74
111
148
185
222
259
296
333
0
370
36
72
108
180
216
252
288
324
360
Polychromatic Diffraction MTF
Polychromatic Diffraction MTF
3/6/2013
Data for 0.5500 to 0.5500 µm.
Surface: Image
3/6/2013
Data for 0.5500 to 0.5500 µm.
Surface: Image
StrehlRatio.zmx
Configuration 1 of 2
1
StrehlRatio.zmx
Configuration 2 of 2
1
0.9
Relative Irradiance At y = 0.0000 µm
0.9
Relative Irradiance At y = 0.0000 µm
144
Spatial Frequency in cycles per mm
Spatial Frequency in cycles per mm
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.1
0
-12.8
-10.24
-7.68
-5.12
-2.56
0
2.56
5.12
7.68
10.24
12.8
StrehlRatio.zmx
0.00000000E+000 Millimeters Configuration 1 of 2
Strehl Ratio = 1.0
-10.24
-7.68
-5.12
-2.56
0
2.56
5.12
7.68
10.24
12.8
Huygens PSF Cross Section X
Huygens PSF Cross Section X
3/6/2013
0.5500 to 0.5500 µm at 0.0000 (deg).
Image width is 25.60 µm.
Strehl ratio: 1.000
Center coordinates: 0.00000000E+000,
0
-12.8
X-Position (µm)
X-Position (µm)
3/6/2013
0.5500 to 0.5500 µm at 0.0000 (deg).
Image width is 25.60 µm.
Strehl ratio: 0.720
StrehlRatio.zmx
Center coordinates: 0.00000000E+000, 0.00000000E+000 Millimeters Configuration 2 of 2
Strehl Ratio = 0.72
Strehl Ratio
Why the Strehl Ratio is important for optimizing optical designs:
• Optimizing the strehl ratio is a useful way of refining optical
systems, since it reduces the complexities of 2D wavefront errors
(and MTF values) into a single number.
• Since the merit function requires the merit of a trial system to be
a single number, the Strehl ratio is an easy way to get there.
• Other combinations of aberration constraints may over-restrict
the optimizer’s options compared to their importance in the final
system. The more flexibility the optimizer has to traverse the design
space, the more likely it is to find a good solution.
What are the problems with using the Strehl Ratio?
The Strehl Ratio is computationally expensive (hence slow) to calculate. This
is, however, less of an issue each year (as long as Moore’s Law continues).
Strehl Ratio
It is relatively simple to measure the Strehl ratio, if at least part of the object is a
point source. Hence, for astronomical imaging, the Strehl ratio can be used to
detect a “Lucky Imaging” situation.
A rough design for an amateur “Lucky Imaging” telescope system:
Beam Splitter
Gated (switchable)
Light from Objective
Detector at image
“Lucky Image”
Detector
ControlElectronics
The lucky image detector can be as simple as an x-y adjustable pinhole which is centered on a guide
star, with a single detector behind. When the intensity passed through the pinhole exceeds a set
amount, the detector is set to integrate – when the power level falls, the detector is turned off. This
would remove both tip-tilt errors and de-focus errors, at the expense of only integrating for a fraction
of the observing time.
GRIN Lenses
A GRadient INdex lens is one in which the index of refraction changes with
position. The lens in the Human eye is an example of a natural GRIN lens.
A normal lens, but with the index a function of Z, can correct for more aberrations than a
lens with a fixed index. This material is called Gradium, and is available from Light Path
Technologies, Inc. The GRADIUM surface type is included in the Zemax program, and
several stock lenses from Light Path are available in the Zemax lens catalog. This finds
uses where there is not room for a more complex lens, but a higher degree of aberration
correction is required.
A simple planoconvex lens:
With a Z-axis
index gradient:
Can be diffraction limited:
GRIN Lenses
The more usual type of GRIN lens is a cylindrical rod of glass with an index that varies in
a radial direction with distance from the axis of the cylinder. You can see how this could
focus light conceptually by considering a discrete approximation – a cylinder with
layered indices: (θi is the angle inside the central layer, w.r.t. the z-axis.)
n3
n2
n0
Φ0
n1
Θi
n0
n1
n2
n3
Assuming that n0 > n1 > n2 > n3, we can see that the input ray will have decreasing
angles with the axis as it passes each successive interface, until it finally undergoes
Total Internal Reflection and heads back to the axis, since , and θi = 90 – θ0.
Assuming that nk is the index of the last layer (which could be air, or a cladding
layer), then the ray will escape the rod, if n0 sin 0  nk .
GRIN Lenses
n2
ϕ
Z
θ
n1
Since sin(θ) = cos(ϕ) ≡ Z0, the direction cosine of the ray in the layer – and this also
holds true for all other layers (i.e., sin(θn) = Zn), we can re-write Snell’s law for GRIN
media in terms of Z-direction cosines as:
n  r  Z  r   n0Z0
Where Z is the direction cosine w.r.t. the Z-axis (and the projection of a unitlength ray onto the Z-axis).
GRIN Lenses
GRINs as Lenses:
Under what conditions will a rod with radially dependent index act as a lens? When
rays starting from an object point re-collect at an image point, as in the figure below:
This will happen if rays from a launch point follow a periodic path whose period
remains constant over a range of launch angles.
Paraxial Model:
By solving the GRIN Snell’s Law,
n  r  Z  r   n0Z0
Y
for a function, n(r), which causes a ray to follow a sinusoidal path:
X
Z
Z  r   r0 sin  z 
we can find a profile that will achieve periodic imaging.
GRIN Lenses
The text, Mouroulis & Macdonald, does this derivation on pages 168 – 172. In the
paraxial approximation, a quadratic profile does the job:
The index profile:
 2 2 
n  r   n0  1 
r 
2 

will cause rays entering one face of the rod to follow sinusoidal paths of
the form:
r  r0 sin( z ),
where the amplitude, r0 is related to the angle of incidence at the axis of
the rod by tan     r
0
GRIN Lenses
GRIN lenses are characterized by their Pitch Length, P. The pitch length
is the number of cycles that light will make in the given length of the
GRIN rod. The above GRIN has a pitch length of one.
For example: a GRIN lens with a pitch length of ¼ will focus parallel
incoming light to a point on the back of the rod:
(Pitch slightly < ¼)
GRIN Lenses
Our text also develops paraxial ray-tracing equations for GRIN lenses:
u0
Paraxial transfer equation:
h  h0 cos( z ) 
Paraxial refraction equation:
u   h0 sin( z)  u0 cos( z)

sin( z )
The paraxial equations are useful for calculating the Numerical Aperture of
GRIN lenses, for example, by calculating the output ray angles for different
aperture heights. Refer to the text for examples of this.
GRIN Lenses
Finite Ray Trace Model:
For tracing real rays, Zemax approximates a grin lens as a sequence of thin
lenses. Consider a thin slice of a GRIN lens:
decreasing index
Because the index of refraction is less further from the axis, the optical path length is
less there and hence the wavefront edges advance more than the center as it
traverses the slice. This is, to first order, identical to the effect that a thin lens
(shown as dashed lines) would have on the same wavefront. The thin lens has a
constant index of refraction, but the OPD is reduced away from the center, since the
thickness is also reduced further from the axis.
GRIN Lenses
Since the change of index in the GRIN slice corresponds to a change of thickness in
the thin lens, the quadratic GRIN profile is equivalent to the quadratic surface
approximation to a spherical lens.
Hence, real GRIN profiles often diverge from a pure quadratic profile, to allow nonparaxial rays to focus at a point independent of initial ray angle.
Zemax lists nearly a dozen different GRIN profiles that are used in making stock
GRIN lenses.
When Zemax traces rays through a GRIN lens, it divides the lens into a number
of thin slices and treats each as a thin lens. Rays are traced using Snell’s Law
and the local slope (derivative) of the index, dn/dr, at the ray’s intersection
with the surface. (dn/dr corresponds to the slope of a regular len’s surface, and
hence to the angle of incidence.)
GRIN Lenses
There is a parameter in the Lens Data Editor, delta-t, which controls the
thickness of the slices.
By default, Zemax divides a GRIN into 20
slices/pitch. The only way to know if this is
enough is to change the delta-t parameter
and see if anything changes noticeably.
For almost all purposes, the default works
well.
GRIN Lenses
Non-Paraxial GRIN lenses:
The quadratic profile was calculated using paraxial assumptions. Often it is desired
to use GRIN lenses at higher apertures, hence there are many modifications to the
quadratic profile. Zemax lists 10 types of gradient “surface”, which are variations on
the index profile formula. A typical one (for the GRADIENT2 surface) is:
n  n0  n2r 2  n4r 4  n6r 6  n8r 8  n10r 10  n12r 12
Other surface formulas allow index change in the Z direction, and, of course, the
ends of the GRIN rod can also be curved like a normal lens, a modification which
Zemax handles easily.
The reason for so many GRIN surfaces is to model the many stock lenses and
types of GRINs available.
It is easier to design GRIN lenses than to get them built – usually it is better to stay
with commercially available GRINs, unless your budget is very large. Custom GRIN
lenses usually come in (total) lengths of many meters. They can be cut to any
desired pitch length.
GRIN Lenses
Systems that use GRIN lenses:
GRIN lenses are useful in a number of optical systems. Many of these
are due to the fact that a GRIN rod (with P=1) will act as an image relay:
Each integer of pitch acts as a one-to-one, upright image relay:
GRIN Lenses
Because of their inherent image relay ability, GRIN lenses are regularly used in
endoscope relays:
GRIN Lenses
The scanner bar on a common photocopy machine is typically a linear array of
GRIN lenses arranged to produce a one-to-one, upright, image relay, with
images from adjacent lenses overlapping in registration. This works much
better than trying to relay each pixel, as that would result in very small lenses
with poor resolution.
Modeling this phenomena accurately can be somewhat of a
challenge.