OPTI 517 Image Quality

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OPTI 517
Image Quality
Richard Juergens
Senior Engineering Fellow
Raytheon Missile Systems
520-794-0917
[email protected]
Why is Image Quality Important?
•
Resolution of detail
– Smaller blur sizes allow better reproduction of image details
– Addition of noise can mask important image detail
Original
Blur added
Noise added
OPTI 517
Pixelated
2
Step One - What is Your Image Quality (IQ) Spec?
•
There are many metrics of image quality
– Geometrical based (e.g., spot diagrams, RMS wavefront error)
– Diffraction based (e.g., PSF, MTF)
– Other (F-theta linearity, uniformity of illumination, etc.)
•
It is imperative that you have a specification for image quality when you are
designing an optical system
– Without it, you don't know when you are done designing!
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3
You vs. the Customer
•
Different kinds of image quality metrics are useful to different people
•
Customers usually work with performance-based specifications
– MTF, ensquared energy, distortion, etc.
•
Designers often use IQ metrics that mean little to the customer
– E.g., ray aberration plots and field plots
– These are useful in the design process, but they are not end-product specs
•
In general, you will be working to an end-product specification, but will probably
use other IQ metrics during the design process
– Often the end-product specification is difficult to optimize to or may be time
consuming to compute
•
Some customers do not express their image quality requirements in terms such
as MTF or ensquared energy
– They know what they want the optical system to do
•
It is up to the optical engineer (in conjunction with the system engineer) to
translate the customer's needs into a numerical specification suitable for
optimization and image quality analysis
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4
When to Use Which IQ Metric
•
The choice of appropriate IQ metric usually depends on the application of the
optical system
– Long-range targets where the object is essentially a point source
• Example might be an astronomical telescope
• Ensquared energy or RMS wavefront error might be appropriate
– Ground-based targets where the details of the object are needed to
determine image features
• Example is any kind of image in which you need to see detail
• MTF would be a more appropriate metric
– Laser scanning systems
• A different type of IQ metric such as the variation from F-theta
distortion
•
The type of IQ metric may be part of the lens specification or may be a derived
requirement flowed down to the optical engineer from systems engineering
– Do not be afraid to question these requirements
– Often the systems engineering group doesn't really understand the
relationship between system performance and optical metrics
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5
Wavefront Error
•
Aberrations occur when the converging wavefront is not perfectly spherical
Real Aberrated
Wavefront
Reference sphere
(centered on ideal
image point)
Rays normal to the
reference sphere
form a perfect image
Ideal image
point
Real rays proceed in a
direction normal to the
aberrated wavefront
Optical Path
Difference (OPD)
OPTI 517
Optical path difference (OPD)
and wavefront error (WFE)
are just two different names
for the same error
6
Tilt and Focus of a Wavefront
•
When calculating the wavefront error, the reference sphere is centered on the
"expected" image location (usually the image surface location of the chief ray)
•
It may be that a different reference sphere will fit the actual wavefront better
– If the center of the better reference sphere is at a different axial location
than the expected image location, there is a "focus error" in the wavefront
– If the center of the better reference sphere is at a different lateral location
than the expected image location, there is a "tilt error" in the wavefront
Actual
wavefront
Better fitting
reference
sphere
Center of better
fitting reference
sphere
Tilt
error
Reference
sphere
Focus error
Expected
image location
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7
Optical Path Difference
Peak-to-valley OPD is the difference
between the longest and the shortest
paths leading to a selected focus
Typical
Wavefront
RMS wavefront error is given by:
Specific
OPD
W
n
[W ( x, y )] dxdy
= òò
òò dxdy
n
Wrms = W 2 - W
Peak-To-Valley
OPD
2
For n discrete rays across the pupil
Reference
Sphere
RMS =
S OPD2/n
This wavefront has the same P-V
wavefront error as the example at the
left, but it has a lower RMS
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8
Peak-to-Valley vs. RMS
•
The ratio of P-V to RMS is not a fixed quantity
•
Typical ratios of P-V to RMS (from Shannon's book)
– Defocus
3.5
– 3rd order spherical
13.4
– 5th order spherical
57.1
– 3rd order coma
8.6
– 3rd order astigmatism
5.0
– Smooth random errors
~5
•
In general, for a mixture of lower order aberrations, P-V/RMS ≈ 4.5
•
When generating wavefront error budgets, RMS errors from different sources
can be added in an RSS fashion
– P-V errors cannot be so added
•
In general, Peak-to-Valley wavefront error is a poor choice to use for error
budgeting
– However, Peak-to-Valley surface error or wavefront error is still commonly
used as a surface error specification for individual optical components and
even for complete optical systems
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9
Rayleigh Criterion
•
Lord Rayleigh observed that when the maximum wavefront error across a
wavefront did not exceed l/4 peak-to-valley, the image quality was "not sensibly
degraded"
– This quarter-wave limit is now called the Rayleigh Criterion
•
This is approximately equivalent to the RMS wavefront error being about 0.07
wave or less (using the value for defocus)
•
The Strehl Ratio is a related measure of image quality
– It can be expressed (for RMS wavefront error < 0.1 wave) as
Strehl Ratio = e -(2pF ) » 1 - ( 2pF )2
2
where F is the RMS wavefront error (in waves)
– For F = 0.07 wave, the Strehl Ratio » 0.8
•
Requiring the Strehl Ratio to be 0.80 or greater for acceptable image quality is
often called the Maréchal Criterion
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10
Diffraction-limited Performance
•
Many systems have "diffraction-limited" performance as a specification
– Taken literally, this might mean that no aberrations are allowed
– As a practical matter, it means that diffraction dominates the image and that
the geometric aberrations are small compared to the Airy disk
•
There is a distinction between the best possible performance, as limited by
diffraction, and performance that is below this limit but produces acceptable
image quality (e.g., Strehl Ratio > 80%)
Diffraction spot size
Geometrical spot size
Spot Size
Total spot size
Rule of Thumb:
Total 80% blur = [(Geo 80% blur)2 + (Airy diameter)2]1/2
Amount of Aberration
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11
Image Quality Metrics
•
The most commonly used geometrical-based image quality metrics are
– Ray aberration curves
– Spot diagrams
– Seidel aberrations
– Encircled (or ensquared) energy
– RMS wavefront error
– Modulation transfer function (MTF)
•
The most commonly used diffraction-based image quality metrics are
– Point spread function (PSF)
– Encircled (or ensquared) energy
– MTF
– Strehl Ratio
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12
Ray Aberration Curves
•
These are by far the image quality metric most commonly used by optical
designers during the design process
•
Ray aberration curves trace fans of rays in two orthogonal directions
– They then map the image positions of the rays in each fan relative to the
chief ray vs. the entrance pupil position of the rays
Sagittal
rays
Dy values for
tangential rays
Dx values for
sagittal rays
0.1
Image
position
Tangential rays
1
-y
+y -x
+x
-0.1
Pupil position
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13
Graphical Description of Ray Aberration Curves
•
Ray aberration curves map the image positions of the rays in a fan
– The plot is image plane differences from the chief ray vs. position in the fan
Image plane
differences from
the chief ray
Pupil position
Image plane
•
Ray aberration curves are generally computed for a fan in the YZ plane and a
fan in the XZ plane
– This omits skew rays in the pupil, which is a failing of this IQ metric
Transverse vs. Wavefront Ray Aberration Curves
•
Ray aberration curves can be transverse (linear) aberrations in the image vs.
pupil position or can be OPD across the exit pupil vs. pupil position
– The transverse ray errors are related to the slope of the wavefront curve
ey(xp,yp) = -(R/rp) ¶W(xp,yp)/¶yp, ex(xp,yp) = -(R/rp) ¶W(xp,yp)/¶xp
R/rp = -1/(n'u') » 2 f/#
•
Example curves for pure defocus:
0.001 inch
Transverse
1.0 wave
Wavefront error
More on Ray Aberration Curves
•
The shape of the ray aberration curve can tell what type of aberration is present
in the lens for that field point (transverse curves shown)
Tangential fan
Sagittal fan
0.05
0.05
1
1
-0.05
-0.05
Defocus
Coma
0.05
0.05
1
1
-0.05
-0.05
Third-order spherical
Astigmatism
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16
The Spot Diagram
•
The spot diagram is readily understood by most engineers
•
It is a diagram of how spread out the rays are in the image
– The smaller the spot diagram, the better the image
– This is geometrical only; diffraction is ignored
•
It is useful to show the detector size (and/or the Airy disk diameter)
superimposed on the spot diagram
Different colors
represent different
wavelengths
Detector outline
•
The shape of the spot diagram can often tell what type of aberrations are
present in the image
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17
Main Problem With Spot Diagrams
•
The main problem is that spots in the spot diagram don't convey intensity
– A ray intersection point in the diagram does not tell the intensity at that point
FIELD
POSITION
0.00, 1.00
0.000,14.00 DG
0.00, 0.71
0.000,10.00 DG
The on-axis image appears
spread out in the spot diagram,
but in reality it has a tight core
with some surrounding lowintensity flare
0.00, 0.00
0.000,0.000 DG
.163
DEFOCUSING
MM
0.00000
Double Gauss - U.S. Patent 2,532,751
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18
Diffraction
•
Some optical systems give point images (or near point images) of a point object
when ray traced geometrically (e.g., a parabola on-axis)
•
However, there is in reality a lower limit to the size of a point image
•
This lower limit is caused by diffraction
– The diffraction pattern is usually referred to as the Airy disk
Image intensity
Diffraction pattern of
a perfect image
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19
Size of the Diffraction Image
•
The diffraction pattern of a perfect image has several rings
– The center ring contains ~84% of the energy, and is usually considered to
be the "size" of the diffraction image
d
d
Very important !!!!
•
The diameter of the first ring is given by d » 2.44 l f/#
– This is independent of the focal length; it is only a function of the
wavelength and the f/number
– The angular size of the first ring b = d/F » 2.44 l/D
•
When there are no aberrations and the image of a point object is given by the
diffraction spread, the image is said to be diffraction-limited
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20
Image of a Point Object and a Uniform Background
D1
d
D2
d
Same f/#
•
For both systems, the Airy disk diameter is the same size
– d = 2.44 l f/#
•
For both systems, the irradiance of the background at the image is the same
– EB = LB(p/4f2)
•
The flux forming the image from the larger system is larger by (D2/D1)2
– We get more energy in the image, so the signal-to-noise ratio (SNR) is
increased by (D2/D1)2
– This is important for astronomy and other forms of point imagery
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21
Spot Size vs. the Airy Disk
•
Regime 1 – Diffraction-limited
Airy disk
diameter
Image
intensity
Point image
(geometrically)
Strehl = 1.0
•
Regime 2 – Near diffraction-limited
Non-zero geometric
blur, but smaller
than the Airy disk
Strehl ³ 0.8
•
Regime 3 – Far from diffraction-limited
Airy disk
diameter
Geometric blur
significantly larger
than the Airy disk
OPTI 517
Strehl ~ 0
22
Point Spread Function (PSF)
•
This is the image of a point object including the effects of diffraction and all
aberrations
Intensity peak of the PSF relative to
that of a perfect lens (no wavefront
error) is the Strehl Ratio
Image
intensity
Airy disk (diameter of the first zero)
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23
Diffraction Pattern of Aberrated Images
•
When there is aberration present in the image, two effects occur
– Depending on the aberration, the shape of the diffraction pattern may
become skewed
– There is less energy in the central ring and more in the outer rings
Perfect PSF
Strehl = 1.0
Strehl = 0.80
OPTI 517
25
25
0.002032 mm
0.002032 mm
24
PSF vs. Defocus
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25
PSF vs. Third-order Spherical Aberration
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26
PSF vs. Third-order Coma
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27
PSF vs. Astigmatism
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28
PSF for Strehl = 0.80
Defocus
Balanced 3rd and
5th-order SA
3rd-order SA
Coma
Astigmatism
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29
Encircled or Ensquared Energy
•
Encircled or ensquared energy is the ratio of the energy in the PSF that is
collected by a single circular or square detector to the total amount of energy
that reaches the image plane from that object point
– This is a popular metric for system engineers who, reasonably enough,
want a certain amount of collected energy to fall on a single pixel
– It is commonly used for systems with point images, especially systems
which need high signal-to-noise ratios
•
For %EE specifications of 50-60% this is a reasonably linear criterion
– However, the specification is more often 80%, or even worse 90%, energy
within a near diffraction-limited diameter
– At the 80% and 90% levels, this criterion is highly non-linear and highly
dependent on the aberration content of the image, which makes it a poor
criterion, especially for tolerancing
Ensquared Energy Example
Ensquared energy on a detector of same order of size as the Airy disk
Perfect lens, f/2, 10 micron wavelength, 50 micron detector
Airy disk
(48.9 micron diameter)
Detector
Approximately 85% of the
energy is collected by the
detector
Modulation Transfer Function (MTF)
•
MTF is the Modulation Transfer Function
•
Measures how well the optical system images objects of different sizes
– Size is usually expressed as spatial frequency (1/size)
•
Consider a bar target imaged by a system with an optical blur
– The image of the bar pattern is the geometrical image of the bar pattern
convolved with the optical blur
Convolved with
=
•
MTF is normally computed for sine wave input, and not square bars to get the
response for a pure spatial frequency
•
Note that MTF can be geometrical or diffraction-based
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32
Computing MTF
•
The MTF is the amount of modulation in the image of a sine wave target
– At the spatial frequency where the modulation goes to zero, you can no
longer see details in the object of the size corresponding to that frequency
•
The MTF is plotted as a function of spatial frequency (1/sine wave period)
MTF =
OPTI 517
Max - Min
Max + Min
33
MTF of a Perfect Image
For an aberration-free image and a round pupil, the MTF is given by
2
MTF( f ) = [j - cos j sinj]
p
j = cos -1(f / fco ) = cos -1(
lf
)
2NA
DEFOCUSING 0.00000
1.0
This f is spatial frequency
(lp/mm) and not f/number
0.9
0.8
0.7
MTF
•
0.6
A
0.5
0.4
0.3
Cutoff frequency
fco = 1/(lf/#)
0.2
0.1
50
150
250
350
450
550
650
750
850
950
Spatial frequency (lp/mm)
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34
Abbe’s Construct for Image Formation
•
Abbe developed a useful framework from which to understand the diffractionlimiting spatial frequency and to explain image formation in microscopes
•
If the first-order diffraction angle from the grating exceeds the numerical
aperture (NA = 1/(2f/#)), no light will enter the optical system for object features
with that characteristic spatial period
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35
Example MTF Curve
Direction of
field point
FOV
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36
MTF as an Autocorrelation of the Pupil
•
The MTF is usually computed by lens design programs as the autocorrelation of
the OPD map across the exit pupil
Relative spatial frequency = spacing
between shifted pupils
(cutoff frequency = pupil diameter)
Perfect MTF = overlap area / pupil area
Complex OPD
computed for many
points across the pupil
Overlap
area
MTF is computed as the normalized integral
over the overlap region of the difference
between the OPD map and its shifted
complex conjugate
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37
Typical MTF Curves
Introductory Seminar
f/5.6 Tessar
DIFFRACTION LIMIT
AXIS
DIFFRACTION MTF
T
R
0.7 FIELD ( 14.00 O )
T
R
1.0 FIELD ( 20.00 O )
WAVELENGTH
650.0 NM
550.0 NM
480.0 NM
WEIGHT
1
2
1
MTF is a function
of the spectral
weighting
DEFOCUSING 0.00000
1.0
MTF curves are different
for different points across
the FOV
0.9
0.8
MTF is a function
of the focus
0.7
Diffraction-limited MTF
(as good as it can get)
M
O 0.6
D
U
L 0.5
A
T
I
O 0.4
N
0.3
0.2
0.1
20
40
60
80
100
120
140
160
180
200
SPATIAL FREQUENCY (CYCLES/MM)
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38
Phase Shift of the OTF
•
Since OPD relates to the phase of the ray relative to the reference sphere, the
pupil autocorrelation actually gives the OTF (optical transfer function), which is a
complex quantity
– MTF is the real part (modulus) of the OTF
OTF = Optical Transfer Function
MTF = Modulus of the OTF
PTF = Phase of the OTF
When the OTF goes
negative, the phase
is p radians
OPTI 517
39
What Does OTF < 0 Mean?
•
When the OTF goes negative, it is an example of contrast reversal
OPTI 517
40
Example of Contrast Reversal
1.00
0.75
DEFOCUSING 0.00000
1.0
0.9
0.50
0.8
0.7
0.6
0.25
0.5
0.4
At best focus
0.3
0.00
0.2
-0.1234
-0.0925
-0.0617
-0.0308
0.0000
0.0308
0.0617
0.0925
0.1234
-0.0922
-0.0614
-0.0307
0.0000
0.0307
0.0614
0.0922
0.1229
0.1
1.0
6.0
11.0
16.0
21.0
26.0
31.0
36.0
41.0
46.0
51.0
DISPLACEMENT ON IMAGE SURFACE
(MM)
56.0
1.00
0.75
DEFOCUSING 0.00000
1.0
0.9
0.50
0.8
0.7
0.6
Defocused
0.5
0.25
0.4
0.3
0.00
-0.1229
0.2
0.1
1.0
6.0
11.0
16.0
21.0
26.0
31.0
36.0
41.0
46.0
51.0
DISPLACEMENT ON IMAGE SURFACE
(MM)
56.0
OPTI 517
41
More on Contrast Reversal
Original
Object
OPTI 517
42
Effect of Strehl = 0.80
•
When the Strehl Ratio = 0.80 or higher, the image is considered to be equivalent
in image quality to a diffraction-limited image
•
The MTF in the mid-range spatial frequencies is reduced by the Strehl ratio
DEFOCUSING 0.00000
1.0
0.9
0.8
0.7
M
O 0.6
D
U
L 0.5
A
T
I
O 0.4
N
Diffraction-limited MTF
0.3
1.0
0.2
0.8
0.1
13
91
169
247
325
403
481
559
637
715
793
SPATIAL FREQUENCY (CYCLES/MM)
OPTI 517
43
Aberration Transfer Function
•
Shannon has shown that the MTF can be approximated as a product of the
diffraction-limited MTF (DTF) and an aberration transfer function (ATF)
DTF(n ) =
2é
-1
2ù
cos
n
n
1
n
úû
p êë
2
n = f / fco
(
æW
ö
ATF(n ) = 1 - ç rms ÷ 1 - 4(n - 0.5)2
è 0.18 ø
)
1.0
1.0
0.9
0.9
0.8
0.8
0.7
0.7
0.6
0.5
0.4
0.025 waves rms
0.050 waves rms
0.3
0.075 waves rms
0.2
0.100 waves rms
0.125 waves rms
0.1
0.150 waves rms
0.0
0.00
Diff. Limit
0.025 waves rms
0.050 waves rms
0.075 waves rms
0.100 waves rms
0.6
MTF
ATF
Bob
Shannon
0.20
0.40
0.60
0.80
1.00
Normalized Spatial Frequency
0.125 waves rms
0.150 waves rms
0.5
0.4
0.3
0.2
0.1
0.0
0.00
0.20
0.40
0.60
0.80
1.00
Normalized Spatial Frequency
OPTI 517
44
Demand Contrast Function
•
The eye requires more modulation for smaller objects to be able to resolve them
– The amount of modulation required to resolve an object is called the
demand contrast function
– This and the MTF limits the highest spatial frequency that can be resolved
The limiting resolution is where
the Demand Contrast Function
intersects the MTF
System A will produce a superior
image although it has the same
limiting resolution as System B
System A has a lower limiting
resolution than System B even though
it has higher MTF at lower frequencies
OPTI 517
45
Example of Different MTFs on RIT Target
OPTI 517
46
Central Obscurations
•
In on-axis telescope designs, the obscuration caused by the secondary mirror is
typically 30-50% of the diameter
– Any obscuration above 30% will have a noticeable effect on the Airy disk,
both in terms of dark ring location and in percent energy in a given ring
(energy shifts out of the central disk and into the rings)
•
Contrary perhaps to expectations, as the obscuration increases the diameter of
the first Airy ring decreases (the peak is the same, and the loss of energy to the
outer rings has to come from somewhere)
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47
Central Obscurations
•
Central obscurations, such as in a Cassegrain telescope, have two deleterious
effects on an optical system
– The obscuration causes a loss in energy collected (loss of area)
– The obscuration causes a loss of MTF
A
B
C
D
OPTI 517
So/Sm = 0.00
So/Sm = 0.25
So/Sm = 0.50
So/Sm = 0.75
48
Coherent Illumination
•
Incoherent illumination fills the whole entrance pupil
•
Partially coherent illumination fills only part of the entrance pupil
– Coherent illumination essentially only fills a point in the entrance pupil
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49
MTF of Partially Coherent Illumination
OPTI 517
50
Partial Coherent Image of a 3-Bar Target
GEOMETRICAL SHADOW
RNA (X,Y) FIELD SCAN
INC ( 0.00, 0.00) R
1.50 ( 0.00, 0.00) R
1.00 ( 0.00, 0.00) R
0.50 ( 0.00, 0.00) R
0.00 ( 0.00, 0.00) R
DIFFRACTION INTENSITY PROFILE
PARTIALLY COHERENT ILLUMINATION
WAVELENGTH
500.0 NM
WEIGHT
1
13-Oct-02
1.25
DEFOCUSING 0.00000
RELATIVE
INTENSITY
1.00
0.75
0.50
0.25
0.00
-5.0
-3.8
-2.5
-1.3
0.0
DISPLACEMENT ON IMAGE SURFACE
OPTI 517
1.3
2.5
(MICRONS)
3.8
5.0
51
Example of Elbows Imaged in Partially Coherent Light
0.2145
0.00504 mm
With 1 wave of
spherical aberration
0.1168
0.00504 mm
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52
The Main Aberrations in an Optical System
•
Defocus – the focal plane is not located exactly at the best focus position
•
Chromatic aberration – the axial and lateral shift of focus with wavelength
•
The Seidel aberrations
– Spherical Aberration
– Coma
– Astigmatism
– Distortion
– Curvature of field
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53
Defocus
•
Technically, defocus is not an aberration in that it can be corrected by simply
refocusing the lens
•
However, defocus is an important effect in many optical systems
Spherical reference sphere centered
on defocused point
Ideal focus
point
Defocused
image point
Actual wavefront
When maximum OPD = l/4, you are
at the Rayleigh depth of focus = 2 l f2
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54
Defocus Ray Aberration Curves
Wavefront map
Spot diagram
2.5
0.02
-2.5
-0.02
Wavefront error
Transverse ray aberration
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55
MTF of a Defocused Image
•
As the amount of defocus increases, the MTF drops accordingly
A
B
C
D
E
OPTI 517
OPD = 0
OPD = l/4
OPD = l /2
OPD = 3l /4
OPD = l
56
Sources of Defocus
•
One obvious source of defocus is the location of the object
– For lenses focused at infinity, objects closer than infinity have defocused
images
– There's nothing we can do about this (unless we have a focus knob)
•
Changes in temperature
– As the temperature changes, the elements and mounts change dimensions
and the refractive indices change
– This can cause the lens to go out of focus
– This can be reduced by design (material selection)
•
Another source is the focus procedure
– There are two possible sources of error here
• Inaccuracy in the measurement of the desired focus position
• Resolution in the positioning of the focus (e.g., shims in 0.001 inch
increments)
– The focus measurement procedure and focus position resolution must be
designed to not cause focus errors which can degrade the image quality
beyond the IQ specification
OPTI 517
57
Chromatic Aberration
•
Chromatic aberration is caused by the lens's refractive index changing with
wavelength
1.524
Blue
Green
Red
Refractive Index
1.522
1.520
1.518
1.516
0.800
1.514
480.
520.
560.
600.
640.
680.
Wavelength (nm)
The shorter wavelengths focus
closer to the lens because the
refractive index is higher for the
shorter wavelengths
FOCUS SHIFT (in)
0.400
0.000
480.
520.
560.
600.
640.
680.
-0.400
-0.800
-1.200
OPTI 517
WAVELENGTH (nm)
58
Computing Chromatic Aberration
•
The chromatic aberration of a lens is a function of the dispersion of the glass
– Dispersion is a measure of the change in index with wavelength
•
It is commonly designated by the Abbe V-number for three wavelengths
– For visible glasses, these are F (486.13), d (587.56), C (656.27)
– For infrared glasses they are typically 3, 4, 5 or 8, 10, 12 microns
– V = (nmiddle-1) / (nshort - nlong)
•
For optical glasses, V is typically in the range 35-80
•
For infrared glasses they vary from 50 to 1000
•
The axial (longitudinal) spread of the short wavelength focus to the long
wavelength focus is F/V
– Example 1: N-BK7 glass has a V-value of 64.4. What is the axial chromatic
spread of an N-BK7 lens of 100 mm focal length?
• Answer: 100/64.4 = 1.56 mm
• Note that if the lens were f/2, the diffraction DOF = ±2lf2 = ±0.004 mm
– Example 2: Germanium has a V-value of 942 (for 8 – 12 m). What is the
axial chromatic spread of a germanium lens of 100 mm focal length?
• Answer: 100/942 = 0.11 mm
Note: DOF(f/2) = ±2lf2 = ±0.08 mm
OPTI 517
59
Chromatic Aberration Example - Germanium Singlet
•
We want to use an f/2 germanium singlet over the 8 to 12 micron band
•
Question - What is the longest focal length we can have and not need to color
correct? (assume an asphere to correct any spherical aberration)
•
Answer
– Over the 8-12 micron band, for germanium V = 942
– The longitudinal defocus = F / V = F / 942
– The 1/4 wave depth of focus is ±2lf2
– Equating these and solving gives F = 4*942*l*f2 = 150 mm
waves
0.25
FIELD HEIGHT
( 0.000 O)
DEFOCUSING 0.00000
1.0
0.9
0.8
0.7
Strehl = 0.86
M
O 0.6
D
U
L 0.5
A
T
I
O 0.4
N
0.3
-0.25
0.2
0.1
1.0
6.0
11.0
16.0
21.0
26.0
31.0
36.0
41.0
46.0
51.0
56.0
61.0
SPATIAL FREQUENCY (CYCLES/MM)
OPTI 517
60
Correcting Chromatic Aberration
•
Chromatic aberration is corrected by a combination of two glasses
– The positive lens has low dispersion (high V number) and the negative lens
has high dispersion (low V number)
Red
Blue
Green
Red and blue
focus together
– This will correct primary chromatic aberration
• The red and blue wavelengths focus together
• The green (or middle) wavelength still has a focus error
– This residual chromatic spread is called secondary color
OPTI 517
61
Secondary Color
•
Secondary color is the residual chromatic aberration left when the primary
chromatic aberration is corrected
These two
wavelengths
focus together
0.050
FOCUS SHIFT (mm)
0.040
0.030
0.020
Secondary color
(~ F/2400)
This wavelength
has a focus error
0.010
0.000
480.
520.
560.
600.
640.
680.
-0.010
WAVELENGTH (nm)
•
Secondary color can be reduced by selecting special glasses
– These glasses cost more (naturally)
OPTI 517
62
Lateral Color
•
Lateral color is a change in focal length (or magnification) with wavelength
– This results in a different image size with wavelength
– The effect is often seen as color fringes at the edge of the FOV
– This reduces the MTF for off-axis images
Red
Green
Blue
OPTI 517
63
Higher-order Chromatic Aberrations
•
For broadband systems, the chromatic variation in the third-order aberrations
are often the most challenging aberrations to correct (e.g., spherochromatism,
chromatic variation of coma, chromatic variation of astigmatism)
– These are best studied with ray aberration curves and field plots
OPTI 517
64
The Seidel Aberrations
•
These are the classical aberrations in optical
design
– Spherical aberration
– Coma
– Astigmatism
– Distortion
– Curvature of field
•
These aberrations, along with defocus and
chromatic aberrations, are the main aberrations
in an optical system
OPTI 517
65
The Importance of Third-order Aberrations
•
The ultimate performance of any unconstrained optical design is almost always
limited by a specific aberration that is an intrinsic characteristic of the design
form
•
A familiarity with aberrations and lens forms is an important ingredient in a
successful optimization that makes optimal use of the time available to
accomplish the design
•
A knowledge of the aberrations
– Allows "spotting" lenses that are at the end of the road with respect to
optimization
– Gives guidance in what direction to "kick" a lens that has strayed from the
optimal solution
OPTI 517
66
Orders of Aberrations
OPTI 517
67
Spherical Aberration
•
Spherical aberration is an on-axis aberration
•
Rays at the outer parts of the pupil focus closer to or further from the lens than
the paraxial focus
This is referred to as
undercorrected spherical
aberration (marginal rays
focus closer to the lens
than the paraxial focus)
Ray aberration curve
Paraxial focus
•
The magnitude of the (third-order) spherical aberration goes as the cube of the
aperture (going from f/2 to f/1 increases the SA by a factor of 8)
OPTI 517
68
Third-order SA Ray Aberration Curves
Spot diagram
Wavefront map
Transverse ray
aberration curve
Wavefront error
OPTI 517
69
Spherical Aberration
Minimum
spot size
Marginal
focus
Minimum
RMS WFE
Paraxial
focus
½L
¾L
L
Marginal focus
Minimum spot size
OPTI 517
Minimum RMS WFE
Paraxial focus
70
Scaling Laws for Spherical Aberration
µ q0
µ (f/#)-3
5.0
4.0
f/3
Field Angle (deg)
3.0
f/4
2.0
1.0
0.0
-1.0
-2.0
f/5
-3.0
-4.0
Spot size goes as the
cube of the EPD (or
inverse cube of the f/#)
-5.0
-5.0 -4.0 -3.0 -2.0 -1.0 0.0
1.0
2.0
3.0
4.0
5.0
Field Angle (deg)
Spot size not dependent
on field position
OPTI 517
71
Spherical Aberration vs. Lens Shape
•
The spherical aberration is a function of the lens bending, or shape of the lens
OPTI 517
72
Spherical Aberration vs. Refractive Index
•
Spherical aberration is reduced with higher index materials
– Higher indices allows shallower radii, allowing less variation in incidence
angle across the lens
n = 1.50
Notice the bending
for minimum SA is a
function of the index
n = 1.95
OPTI 517
73
Spherical Aberration vs. Index and Bending
b at K min = r j
3
3
4n2 - n
16(n - 1) (n + 2)
2
n = 1.5
n = 2.0
n = 3.0
n = 4.0
OPTI 517
74
Example - Germanium Singlet
•
We want an f/2 germanium singlet to be used at 10 microns (0.01 mm)
•
Question - What is the longest focal length we can have and not need aspherics
to correct spherical aberration?
•
Answer
– Diffraction Airy disk angular size is b diff = 2.44 l/D
– Spherical aberration angular blur is b sa = 0.00867 / f3
– Equating these gives D = 2.44 l f3 / 0.00867 = 22.5 mm
– For f/2, this gives F = 45 mm
DEFOCUSING 0.00000
1.0
0.9
0.8
Strehl = 0.91
0.7
waves
0.25
M
O 0.6
D
U
L 0.5
A
T
I
O 0.4
N
FIELD HEIGHT
( 0.000 O)
0.3
0.2
0.1
1.0
5.0
9.0
13.0
17.0
21.0
25.0
29.0
33.0
37.0
41.0
SPATIAL FREQUENCY (CYCLES/MM)
-0.25
OPTI 517
75
45.0
49.0
Focus Shift vs. Wavelength (Germanium singlet)
0.1000
FOCUS SHIFT (mm)
0.0500
0.0000
8000.
8500.
9000.
9500.
10000.
10500.
11000.
11500.
12000.
-0.0500
Diffraction-limited
depth of focus
-0.1000
WAVELENGTH (nm)
OPTI 517
76
Spherical Aberration vs. Number of Lenses
•
Spherical aberration can be reduced by splitting the lens into more than one lens
SA = 1
(arbitrary units)
SA = 1/4
(arbitrary units)
SA = 1/9
(arbitrary units)
OPTI 517
77
Spherical Aberration vs. Number of Lenses
OPTI 517
78
Spherical Aberration and Aspherics
•
The spherical aberration can be reduced, or even effectively eliminated, by
making one of the surfaces aspheric
1.0 mm
spherical
0.0001 mm
aspheric
OPTI 517
79
Aspheric Surfaces
•
Aspheric surfaces technically are any surfaces which are not spherical, but
usually refer to a polynomial deformation to a conic
z(r ) =
r2 / R
1 + 1 - (k + 1)(r / R)2
+ A r 4 + B r 6 + C r 8 + D r10 + ...
•
The aspheric coefficients (A, B, C, D, …) can correct 3rd, 5th, 7th, 9th, … order
spherical aberration
•
When used near a pupil, aspherics are used primarily to correct spherical
aberration
•
When used far away (optically) from a pupil, they are primarily used to correct
astigmatism by flattening the field
•
Before using aspherics, be sure that they are necessary and the increased
performance justifies the increased cost
– Never use a higher-order asphere than justified by the ray aberration curves
OPTI 517
80
Optimizing Aspherics
For an asphere far away
(optically) from a pupil, the
ray density need not be
high, but there must be a
sufficient number of
overlapping fields to sample
the surface accurately.
This asphere primarily
corrects field aberrations
(e.g., astigmatism).
For an asphere at (or near) a
pupil, there need to be
enough rays to sample the
pupil sufficiently.
This asphere primarily
corrects spherical aberration.
OPTI 517
81
Asphere Example
• 2 inch diameter, f/2 plano-convex lens
sphere
0.10
asphere
0.00001
Note: Airy disk
diameter is
~ 0.0001 inch
OPTI 517
82
Aspheric Orders
Sag cont relative to base sphere (in)
0.000
-0.002
-0.004
-0.006
Aspheric Sum
4th order
6th order
8th order
10th order
-0.008
-0.010
0.00
0.20
0.40
0.60
Radial position (in)
0.80
1.00
0.0025
Corresponds to ~114
waves of asphericity
Delta Sag
0.0020
0.0015
0.0010
0.0005
0.0000
-1.000E+00
-5.000E-01
0.000E+00
Y Position
OPTI 517
5.000E-01
1.000E+00
83
MTF vs. Aspheric Order
DEFOCUSING
1.0
DEFOCUSING
1.0
0.00000
0.00000
0.9
0.9
0.7
0.7
M
O 0.6
D
U
L 0.5
A
T
I
O 0.4
N
M
O 0.6
D
U
L 0.5
A
T
I
O 0.4
N
0.3
0.3
0.2
0.2
0.1
0.1
84
168
252
336
420
asphere
A term only
0.8
sphere
0.8
504
588
672
756
840
78
SPATIAL FREQUENCY (CYCLES/MM)
156
234
312
390
468
546
624
702
780
SPATIAL FREQUENCY (CYCLES/MM)
DEFOCUSING
1.0
0.00000
DEFOCUSING
1.0
0.00000
0.9
0.9
asphere
A,B terms
0.8
0.7
M
O 0.6
D
U
L 0.5
A
T
I
O 0.4
N
0.7
M
O 0.6
D
U
L 0.5
A
T
I
O 0.4
N
0.3
0.3
0.2
0.2
0.1
0.1
78
156
234
312
390
468
546
asphere
A,B,C terms
0.8
624
702
780
858
SPATIAL FREQUENCY (CYCLES/MM)
78
156
234
312
390
468
546
624
702
780
858
SPATIAL FREQUENCY (CYCLES/MM)
OPTI 517
84
Coma
•
Coma is an off-axis aberration
•
It gets its name from the spot diagram which looks like a comet (coma is Latin
for comet)
•
A comatic image results when the periphery of the lens has a higher or lower
magnification than the portion of the lens containing the chief ray
Chief ray
Spot diagram
•
The magnitude of the (third-order) coma is proportional to the square of the
aperture and the first power of the field angle
OPTI 517
85
Transverse vs. Wavefront 3rd-order Coma
Spot diagram
Wavefront map
5.0
0.001
-0.001
-5.0
Wavefront error
Transverse ray aberration
OPTI 517
86
Scaling Laws for Coma
µ (f/#)-2
f/5
f/4
µ q1
f/3
5.0
4.0
Full Field
0.5 Field
Field Angle (deg)
3.0
2.0
1.0
0.0
-1.0
-2.0
-3.0
-4.0
-5.0
On-axis
-5.0 -4.0 -3.0 -2.0 -1.0 0.0
1.0
2.0
3.0
4.0
5.0
Field Angle (deg)
Spot size goes as the
square of the EPD (or
inverse square of the f/#)
Spot size is linearly
dependent on field height
OPTI 517
87
Coma vs. Lens Bending
•
Both spherical aberration and coma are a function of the lens bending
Coma
Spherical
aberration
OPTI 517
88
Coma vs. Stop Position
•
The size of the coma is also a function of the stop location relative to the lens
Aperture stop
Coma is reduced due to
increased lens symmetry
around the stop
OPTI 517
89
Coma is an Odd Aberration
•
Any completely symmetric optical system (including the stop location) is free of
all orders of odd field symmetry aberrations (coma and distortion)
OPTI 517
90
Astigmatism
•
Astigmatism is caused when the wavefront has a cylindrical component
– The wavefront has different spherical power in one plane (e.g., tangential)
vs. the other plane (e.g., sagittal)
•
The result is different focal positions for tangential and sagittal rays
Rays in YZ
plane focus
here
Rays in XZ
plane focus
here
•
The magnitude of the (third-order) astigmatism goes as the first power of the
aperture and the square of the field angle
OPTI 517
91
Cause of Astigmatism
Radius = RSphere
Radius = RCut
Rcut< RSphere
Non-rotationally
symmetric through
an off-center part
of the surface
Astigmatism
No astigmatism
Rotationally
symmetric through
a centered part of
the surface
OPTI 517
92
Image of a Wagon Wheel With Astigmatism
Wagon
Wheel
Radial lines
Tangential
Focus
Tangential lines
In Focus
Tangential lines
OPTI 517
Sagittal or
Radial
Focus
Radial lines
In Focus
93
Astigmatism vs. Field
FIELD
POSITION
ASTIGMATIC
FIELD CURVES
ANGLE(deg)
T
0.00, 1.00
0.000,5.000 DG
S
5.00
0.00, 0.75
0.000,3.750 DG
3.75
0.00, 0.50
0.000,2.500 DG
2.50
0.00, 0.25
0.000,1.250 DG
1.25
0.00, 0.00
0.000,0.000 DG
-0.10
-0.05
0.0
0.05
0.10
FOCUS (MILLIMETERS)
.715E-01 MM
DEFOCUSING -0.100 -0.090 -0.080 -0.070 -0.060 -0.050 -0.040 -0.030 -0.020 -0.010 -0.000
OPTI 517
94
Scaling Laws for Astigmatism
µ (f/#)-1 q2
5.0
5.0
4.0
4.0
3.0
3.0
Field Angle (deg)
Field Angle (deg)
µ (f/#)-1 q2
2.0
1.0
0.0
-1.0
2.0
1.0
0.0
-1.0
-2.0
-2.0
-3.0
-3.0
-4.0
-4.0
-5.0
-5.0
-5.0 -4.0 -3.0 -2.0 -1.0 0.0
1.0
2.0
3.0
4.0
-5.0 -4.0 -3.0 -2.0 -1.0 0.0
5.0
1.0
2.0
3.0
4.0
5.0
Field Angle (deg)
Field Angle (deg)
Tangential focus
Sagittal focus
OPTI 517
95
Astigmatism Ray Aberration Plots
TANGENTIAL
0.01
1.00 RELATIVE
SAGITTAL
FIELD HEIGHT
0.01
O
( 5.000 )
-0.01
-0.01
TANGENTIAL
0.01
0.01
( 2.500 O)
-0.01
0.01
-0.01
SAGITTAL
0.01
-0.01
FIELD HEIGHT
( 2.500 O)
-0.01
0.00 RELATIVE
0.01
( 5.000 O)
TANGENTIAL
0.01
0.01
( 0.000 O)
-0.01
Tangential focus
0.01
0.01
-0.01
FIELD HEIGHT
( 0.000 O)
-0.01
( 5.000 O)
SAGITTAL
0.01
-0.01
0.01
FIELD HEIGHT
( 2.500 O)
-0.01
0.01
-0.01
0.00 RELATIVE
0.01
0.01
-0.01
-0.01
Medial focus
(best diffraction focus)
Occurs halfway
between sagittal and
tangential foci
OPTI 517
FIELD HEIGHT
0.50 RELATIVE
0.00 RELATIVE
FIELD HEIGHT
1.00 RELATIVE
-0.01
0.50 RELATIVE
FIELD HEIGHT
-0.01
FIELD HEIGHT
-0.01
0.50 RELATIVE
0.01
1.00 RELATIVE
FIELD HEIGHT
( 0.000 O)
0.01
-0.01
Sagittal focus
Note: the sagittal focus does
not always occur at the
paraxial focus
96
Transverse vs. Wavefront Astigmatism
At medial focus
Spot diagram
Wavefront map
0.02
2.0
-0.02
-2.0
Transverse ray aberration
Wavefront error
OPTI 517
97
PSF of Astigmatism vs. Focus Position
Tangential focus
Medial focus
(best diffraction focus)
OPTI 517
Sagittal focus
98
Astigmatism of a Tilted Flat Plate
•
Placing a tilted plane parallel plate into a diverging or converging beam will
introduce astigmatism
t
q
•
The amount of the longitudinal astigmatism (focus shift between the tangential
and sagittal foci) is given by
é n2 cos2q
ù
Ast =
1
ê 2
ú
2
n2 - sin2q ë n - sin q û
t
(
)
- t q 2 n2 - 1
Ast =
n3
Exact
Third-order
OPTI 517
99
Correcting the Astigmatism of a Tilted Flat Plate
•
The astigmatism introduced by a tilted flat plate can be corrected by
– Adding cylindrical lenses
– Adding tilted spherical lenses
– Adding another plate tilted in the orthogonal plane
To correct
for this
Do not do this
(it will double the
astigmatism)
OPTI 517
Do this
100
Reducing the Astigmatism of a Tilted Flat Plate
•
Astigmatism of a flat plate can be reduced
by adding a slight wedge to the plate
Flat plate
0.2
-0.2
0.47° wedge
0.2
-0.2
Transverse ray aberration
OPTI 517
101
Rectilinear Imaging
•
Most optical systems want to image rectilinear objects into rectilinear images
h
Object
q
Image
s'
q
s
h'
•
This requires that m = -s'/s = -h'/h = constant for the entire FOV
•
For infinite conjugate lenses, this requires that h' = F tanq for all field angles
h'
q
F
OPTI 517
102
Distortion
•
If rectilinear imaging is not met, then there is distortion in the lens
•
Effectively, distortion is a change in magnification or focal length over the field of
view
Paraxial image height
Real image height less than paraxial
height implies existence of distortion
Plot of distorted FOV
•
Negative distortion (shown) is often called barrel distortion
•
Positive distortion (not shown) is often called pincushion distortion
OPTI 517
103
More on Distortion
•
Distortion does not result in a blurred image and does not cause a reduction in
any measure of image quality such as MTF
•
Distortion is a measure of the displacement of the image from its corresponding
paraxial reference point
•
Distortion is independent of f/number
•
Linear distortion is proportional to the cube
of the field angle
•
Percent distortion is proportional to the square
of the field angle
DISTORTION
ANGLE(deg)
20.00
15.27
10.31
5.20
-2
-1
0
1
2
% DISTORTION
OPTI 517
104
Implications of Distortion
•
Consider negative distortion
– A rectilinear object is imaged inside the detector
•
This means a rectilinear detector sees a larger-than-rectilinear area in object
space
OPTI 517
105
Curvature of Field
•
In the absence of astigmatism, the focal surface is a curved surface called the
Petzval surface
Petzval Surface
Lens
Flat Object
OPTI 517
106
Third-order Field Curvature
µ (f/#)-1 q2
TANGENTIAL
0.15
1.00 RELATIVE
FIELD HEIGHT
( 15.00 O)
-0.15
SAGITTAL
0.15
-0.15
0.66 RELATIVE
ASTIGMATIC
FIELD CURVES
0.15
FIELD HEIGHT
( 10.00 O)
0.15
ANGLE(deg)
T
15.00
S
-0.15
-0.15
0.33 RELATIVE
0.15
11.36
FIELD HEIGHT
( 5.000 O)
-0.15
0.15
-0.15
7.63
0.00 RELATIVE
0.15
FIELD HEIGHT
( 0.000 O)
0.15
3.83
-0.15
-0.15
Aberrations relative to a flat image surface
-0.02
-0.01
0.0
0.01
0.02
FOCUS (MILLIMETERS)
OPTI 517
107
The Petzval Surface
•
The radius of the Petzval surface is given by
1
RPetzval
æ 1 ö
÷÷
= å çç
i è ni Fi ø
– For a singlet lens, the Petzval radius = n F
•
Obviously, if we have only positive lenses in an optical system, the Petzval
radius will become very short
– We need some negative lenses in the system to help make the Petzval
radius longer (i.e., flatten the field)
•
This, and chromatic aberration correction, is why optical systems need some
negative lenses in addition to all the positive lenses
OPTI 517
108
Field Curvature and Astigmatism
•
As an aberration, field curvature is not very interesting
•
As a design obstacle, it is the basic reason that optical design is still a challenge
•
The astigmatic contribution starts from the Petzval surface
– If the axial distance from the Petzval surface to the sagittal surface is 1
(arbitrary units), then the distance from the Petzval surface to the tangential
surface is 3
Field curvature and
astigmatism can be used
together to help flatten
the image plane and
improve the image quality
3
1
OPTI 517
109
Flattening the Field
•
The contribution of a lens to the focal length is proportional to yF where F is lens
power (1/F)
•
The contribution of a lens to the Petzval sum is proportional to F/n
•
Thus, if we include negative lenses in the system where y is small we can
reduce the Petzval sum and flatten the field while holding the focal length
Y
Y
Lens With Field Flattener
(Petzval Lens)
Cooke Triplet
•
Yet another reason why optical systems have so darn many lenses
Flat-field lithographic lens
Negative lenses in RED
OPTI 517
110
Original Object
OPTI 517
111
Spherical Aberration
Image blur is
constant over
the field
OPTI 517
112
Coma
Image blur
grows linearly
over the field
OPTI 517
113
Astigmatism
Image blurs
more in one
direction over
the field
OPTI 517
114
Distortion
No image
degradation
but image
locations are
shifted
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Curvature of Field
Image blur
grows
quadratically
over the field
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Combined Aberrations – Spot Diagrams
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Balancing of Aberrations
•
Different aberrations can be combined to improve the overall image quality
– Spherical aberration and defocus
– Astigmatism and field curvature
– Third-order and fifth-order spherical aberration
– Longitudinal color and spherochromatism
– Etc.
•
Lens design is the art (or science) of putting together a system so that the
resulting image quality is acceptable over the field of view and range of
wavelengths
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Resolution
•
Resolution is an important aspect of image quality
•
Every image has some resolution associated with it, even if it is the Airy disk
– In this case, the resolution is dependent on the aberrations of the system
•
Resolution is the smallest detail you can resolve in the image
– It determines whether you can resolve two closely spaced objects
25
25
25
Well resolved
0.002016 mm
0.002016 mm
0.002016 mm
Rayleigh criterion
peak of 2nd at
1st zero of first
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Sparrow criterion
overlap at FWHM
119
Resolution vs. P-V Wavefront Error
•
The 1/4 wave rule was empirically developed by astronomers as the greatest
amount of P-V wavefront error that a telescope could have and still resolve two
stars separated by the Rayleigh spacing (peak of one at 1st zero of the other)
Perfect
1/4 wave P-V
25
25
0.002016 mm
0.002016 mm
3/4 wave P-V
1/2 wave P-V
25
25
0.002016 mm
0.002016 mm
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Resolution Examples
•
Angular resolution is given by b » 2.44 l/D
– Limited only by the diameter, not by the focal length or f/number
•
U of A is building 8.4 meter diameter primary mirrors for astronomical telescopes
•
For visible light (~0.5 mm), the Rayleigh spacing corresponds to an angular
separation of (2.44 * 0.5x10-6 / 8.4)/2 = 0.073x10-6 radian (~0.015 arc second)
•
Assume a binary star at a distance of 200 light years (~1.2x1015 miles)
– This would have a resolution of 90 million miles
– Perhaps enough resolution to "split the binary"
•
A typical cell phone camera has an aperture of about 0.070 inch
– This gives a Rayleigh spacing of about 0.34 mrad (for reference, the human
eye has an angular resolution of about 0.3 mrad)
– For an object 10 feet away, this is an object resolution of about 1 mm
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Film Resolution
•
Due to the grain size of film, there is an MTF associated with films
•
A reasonable guide for MTF of a camera lens is the 30-50 rule: 50% at 30 lp/mm
and 30% at 50 lp/mm
•
For excellent performance of a camera lens, use 50% at 50 lp/mm
•
Another criterion for 35 mm camera lenses is 20% at 30 lp/mm over 90% of the
field (at full aperture)
•
As a rough guide for the resolution required in a negative, use 200 lines divided
by the square root of the long dimension in mm
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Detectors
•
All optical systems have some sort of detector
– The most common is the human eye
– Many optical systems use a 2D detector array (e.g., CCD)
•
No matter what the detector is, there is always some small element of the
detector which defines the detector resolution
– This is referred to as a picture element (pixel)
•
The size of the pixel divided by the focal length is called the Instantaneous FOV
(IFOV)
– The IFOV defines the angular limit of resolution in object space
– IFOV is always expressed as a full angle
FOV
Detector
array
IFOV
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Implications of IFOV
•
If the target angular size is smaller than an IFOV, it is not resolved
– It is essentially a point target
– Example is a star
•
If the target annular size is larger than an IFOV it may be resolved
– This does not mean that you can always tell what the object is
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Practical Resolution Considerations
•
Resolution required to photograph written or printed copy:
– Excellent reproduction (serifs, etc.) requires 8 line pairs per lower case e
– Legible reproduction requires 5 line pairs per letter height
– Decipherable (e, c, o partially closed) requires 3 line pairs per height
•
The correlation between resolution in cycles/minimum dimension and certain
functions (often referred to as the Johnson Criteria) is
– Detect
1.0 line pairs per dimension
– Orient
1.4 line pairs per dimension
– Aim
2.5 line pairs per dimension
– Recognize
4.0 line pairs per dimension
– Identify
6-8 line pairs per dimension
– Recognize with 50% accuracy
7.5 line pairs per height
– Recognize with 90% accuracy
12 line pairs per height
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Johnson Resolution Criteria
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126
Examples of the Johnson Criteria
Detect
1 bar pair
Maybe something
of military interest
Recognize
4 bar pairs
Tank
Identify
7 bar pairs
Abrams Tank
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MTF of a Pixel
•
Consider a fixed size pixel scanning across different sized bar targets
When the pixel size equals
the width of a bar pair (light
and dark) there is no more
modulation
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128
MTF of a Pixel
•
If the pixel is of linear width D, the MTF of the pixel is given by
MTF( f ) =
The cutoff frequency (where the MTF goes to zero) is at a spatial frequency 1/D
1.0
0.8
0.6
0.4
MTF
•
sin( pfD )
pfD
0.2
0.0
0.00
0.25
0.50
0.75
1.00
1.25
1.50
1.75
2.00
-0.2
-0.4
Normalized Spatial Frequency
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129
Optical MTF and Pixel MTF
•
The total MTF is the product of the optical MTF and the pixel MTF
1.0
1.0
detector
0.9
1.0
0.9
0.9
0.8
0.8
0.8
0.7
0.7
0.7
0.6
0.6
optics
0.5
0.6
0.5
0.5
0.4
0.4
0.4
product
0.3
0.3
0.3
0.2
0.2
0.2
0.1
0.1
0.1
0.0
0.00
0.0
0.0
0.00
500.00
1000.00
1500.00
2000.00
Case 1 - Optics limited
Best for high resolution
over-sampling
•
2500.00
Case 2 - Optics and
detector are matched
Best for most FLIR-like
mapping systems
500.00
1000.00
1500.00
2000.00
Case 3 - Detector limited
Best for detecting dim point
targets
Of course, there are other MTF contributors to total system MTF
– Electronics, display, jitter, smear, eye, turbulence, etc.
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2500.00
130
Effects of Signal/CCD Alignment on MTF
A sampled
imaging system is
not shift-invariant
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131
MTF of Alignment
•
When performing MTF testing, the user can align the image with respect to the
imager to produce the best image
– In this case, a sampling MTF might not apply
•
A natural scene, however, has no net alignment with respect to the sampling
sites
•
To account for the average alignment of unaligned objects a sampling MTF must
be added
– MTFsampling = sin(pfDx)/(pfDx) where Dx is the sampling interval
– This MTF an ensemble average of individual alignments and hence is
statistical in nature
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132
Aliasing
•
Aliasing is a very common effect but is not well understood by most people
•
Aliasing is an image artifact that occurs when we insufficiently sample a
waveform
– It is evidenced as the imaging of high frequency objects as low frequency
objects
Array of detectors
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133
Sampling of a Sine Wave
1.0
1.0
0.5
0.5
0.0
0.0
11
-0.5
-0.5
-1.0
-1.0
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134
Nyquist Condition
•
If we choose a sampling interval sufficiently fine to locate the peaks and valleys
of a sine wave, then we can reconstruct that frequency from its sample values
•
The Nyquist condition says we need at least two samples per cycle to reproduce
a sine wave
– For a sine wave period x, we need a sampling interval Dx < x/2
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MTF Fold Over
•
The effect of sampling is to replicate the MTF back from the sampling frequency
– This will cause higher frequencies to appear as lower frequencies
Nyquist
frequency
Sampling
frequency
Prefiltered
MTF
•
The solution to this is to prefilter the MTF so it goes to zero at the Nyquist
frequency
– This is often done by blurring
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Final Comments
•
Image quality is essentially a measure of how well an optical system is suited for
the expected application of the system
•
Different image quality metrics are needed for different systems
•
The better the needed image quality, the more complex the optical system will
be (and the harder it will be to design and the higher the cost will be to make it)
•
The measures of image quality used by the optical designer during the design
process are not necessarily the same as the final performance metrics
– It's up to the optical designer to convert the needed system performance
into appropriate image quality metrics as needed
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