ECEN 4616/5616
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Strehl Ratio:
The Strehl Ratio (named after German physicist Karl Strehl, 1864-1940) is a
measure of optical quality for optical systems that are fairly well corrected. It is a
one-number measure of the quality of an optical system.
An (circularly-symmetric) optical system with no aberrations produces a PSF that
is the Airy Pattern:
Huygens PSF
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0.5500 to 0.5500 µm at 0.0000 (deg).
Imageamount
size is 32.of
00 µspherical
m square.
When a small
(or other) aberrations
are present, the energy
Strehl ratio: 1.000
StrehlRatio.zmx
Center coordinates: 0.00000000E+000, 0.00000000E+000 Millimeters Configuration 1 of 2
in the peak of the PSF shifts to the rings:
Huygens PSF
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0.5500 to 0.5500 µm at 0.0000 (deg).
Image size is 32.00 µm square.
Strehl ratio: 0.720
StrehlRatio.zmx
Center coordinates: 0.00000000E+000, 0.00000000E+000 Millimeters Configuration 2 of 2
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The two PSFs (unaberrated and aberrated) look similar, and both even have
reasonable MTFs:
No Aberration:
Sph Aberration:
TS Diff. Limit
TS 0.0000 (deg)
1.0
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)
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
0.0
0
37
74
111
148
185
222
259
296
333
0
370
36
72
Spatial Frequency in cycles per mm
108
144
180
216
252
288
324
360
Spatial Frequency in cycles per mm
Polychromatic Diffraction MTF
Polychromatic Diffraction MTF
However, a profile plot through the PSF shows that the power in the peak of the
Airy pattern is significantly reduced:
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Data for 0.5500 to 0.5500 µm.
Surface: Image
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Data for 0.5500 to 0.5500 µm.
Surface: Image
StrehlRatio.zmx
Configuration 2 of 2
1
0.9
Relative Irradiance At y = 0.0000 µm
1
0.9
0.8
0.7
1
0.6
0.5
0.4
0.9
0.3
0.2
0.1
0
-12.8
-10.24
-7.68
Relative Irradiance At y = 0.0000 µm
Relative Irradiance At y = 0.0000 µm
StrehlRatio.zmx
Configuration 1 of 2
-5.12
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.8
-2.56
0.8
0
2.56
5.12
7.68
10.24
0
-12.8
12.8
-10.24
-5.12
-2.56
0
2.56
5.12
7.68
10.24
12.8
X-Position (µm)
0.7
Huygens PSF Cross Section X
Huygens PSF Cross Section X
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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,
-7.68
X-Position (µm)
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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
The Strehl Ratio is defined
as the ratio of the peak of the actual PSF compared
0.6
to the peak of the ideal (unaberrated) Airy pattern. The Strehl ratio for this
aberrated PSF is reported
(by the Huygen’s PSF analysis window) as 0.720:
0.5
StrehlRatio.zmx
0.00000000E+000 Millimeters Configuration 1 of 2
0.4
0.3
0.2
0.1
0
-12.8
-10.24
-7.68
-5.12
-2.56
0
2.56
5.12
7.68
10.24
12.8
X-Position (µm)
Huygens PSF Cross Section X
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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
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There is an operand (“STRH”) for the Merit Function Editor which returns the
Strehl ratio for a given combination of configurations, wavelengths and field.
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.
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)
Detector at image
Light from Objective
“Lucky Image”
Detector
Control
Electronics
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. A tip-tilt correction mirror would allow an increased
fraction for detection time.
A common way to get increased resolution in small telescopes today is to use a
video camera to take many thousands of pictures, then use computer algorithms
to align and sum the images. It is interesting to imagine how these algorithms
might be improved:
 Before summing an image, “nearby” star images might be scanned to
estimate the Strehl ratio, and only use images with sufficiently high values.
This might require that the PSFs be slightly oversampled, as we want to
find the power in the central peak, and not integrate over the rings as well.
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
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Papers (and research) have discovered that summing images over a
range of focus produces effective PSFs that are invariant over a significant
range of focus and have no zeros in the MTF. Thus, summed images
can, in principle, be filtered to achieve effective diffraction limited images.
(Or even beyond diffraction limited, due to the “Lucky Imaging” effect.)
Both these methods could be simulated (in Matlab and/or Zemax) and their
effectiveness estimated.
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Scheimpflug Principle:
Consider the problem of taking a picture of the top of a table, from an oblique
angle:
Depth of Focus:
camera
Table
Only a portion of the table-top will be in focus. The resulting image would look
something like this:
Is there some way to achieve a focal plane and DOF like this:
camera
DOF:
Table
The problem is solved by a camera type known as a “View Camera”:
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While this simply looks like an antique camera, it is not. (Although there are
antique view cameras, they are also produced today.) The defining characteristic
of a view camera is its ability to move the lens with at least 3 degrees of freedom
(x,y tilt, and z translation):
How does this help? Consider the imaging properties of a lens if the object is
tilted:
Y
Z
X
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When the tilted object is in focus, the focal plane, image plane and lens plane
have the characteristic that they intersect at a line:
Y
Z
X
This is known as the “Scheimpflug Condition”.
For a derivation from paraxial optics, see:
http://en.wikipedia.org/wiki/Scheimpflug_principle
The Scheimpflug principle also works with afocal systems, allowing a constant
magnification system:
Y
X
Z
This has applications in photogrammetry – the use of photographs to determine
the size and shape of objects, and in optical inspection systems.
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