OPTI 517 %
`
`Image Quality
`
`Richard Juergens
`
`'
`
`ring Fellow
`
`ile Systems
`
`
`
`--
`
`0917
`
`rcjuergens@raythe0n.com
`
`1
`
`ZEISS 1133
`Zeiss v. Nikon
`
`|PR201 3-00363
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`ZEISS 1133
`Zeiss v. Nikon
`IPR2013-00363
`
`
`
`Image Quality
`
`Richard Juergens
`
`'
`
`ring Fellow
`
`ile Systems
`
`
`
`--
`
`0917
`
`rcjuergens@raythe0n.com
`
`1
`
`ZEISS 1133
`Zeiss v. Nikon
`
`|PR201 3-00363
`
`D
`
`33..
`.5
`
`E I
`
`?‘-
`[U5
`
`no
`
`'.,r
`
`Iii:-_."-'~'35''
`
`1
`
`ZEISS 1133
`Zeiss v. Nikon
`IPR2013-00363
`
`
`
`Why is Image Quality Important?
`
`-
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`Resolution of detail
`
`- Smaller blur sizes allow better reproduction of image details
`— Addition of noise can mask important image detail
`
`
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`Blur added
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`Noise added
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`Pixelated
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`2
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`Sten One - What is Your Image Quality (IQ) Spec?
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`
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`ams, RMS wavefront error)
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`— Other (F-theta linearity, uniformity of illumination. etc.)
`
`it is imgerative that you have a specification for image quality when you are
`‘
`1:;
`J: -
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`' are done designingi
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`3
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`You vs. the Customer
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`Different kinds of image quality metrics are useful to different people
`
`Customers usually work with performancebased 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 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 comunction with the system engineer) to
`translate the customer's needs into a numerical specification suitable for
`optimization and image quality analysis
`
`OPTI 517
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`4
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`When to Use Which I0 Metric
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`=
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`The choice of appropriate IQ metric usually depends on the application of the
`optical system
`
`a oint source
`— Lorig—range targets where the object is essential:
`
`~ Example might be an astrono
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`
`
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`n which you need to see detail
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`ate metric
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`ch as the variation from F—theta
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`
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`- lens specification or may be a derived
`ngineer from systems engineering
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`-
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`relationship between system eformance and ptical mtrics
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`0F’Tl 51?
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`'l.|1
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`5
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`Wavefront Error
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`- Aberrations occur when the converging wavefront is not perfectly spherical
`
`Fleal Aberrated
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`Wavefront
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`image point)
`
`Reference sphere
`(centered on ideal
`
`Rays normal to the
`reference sphere
`form a perfect Image
`
`
`
` Optical Path
`
`Optical path difference (OPD)
`Difference (app)
`and wavefront error (WFE)
`are just two different names
`for the same error
`
`OPTIS17
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`6
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`Ideal image
`point
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`Fleal rays proceed in a
`direction normal to the
`aberrated wavefront
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`6
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`
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`Typical
`Wavefront
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`S%eFc|i:i)‘Ic_~$
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`Optical Path Difference
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`Peak—to—va|ley OPD is the difference
`
`between the longest and the shortest
`
`paths leading to a selected focus
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`HMS wavefront error is given by:
`
`<Wn> = [W()(,y)]"dxdy
`
`fldxdy
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`wm =(w2)—(w)2
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`For n discrete rays across the pupil
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`RMS = ,/ 2 OPD?/n
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`This wavefront has the same P-V
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`wavefront error as the example at the
`left, but it has a lower RMS
`‘
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`517
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`7
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`Peak-To-Valley
`OPD
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`Reference
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`Sphere
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`7
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`Peak-to-Valley vs. HMS
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`The ratio of P-V to RMS is not a fixed quantity
`
`Typical ratios of P—\/ to F-{MS (from Shannon's book)
`Defocus
`V
`3.5
`3rd order spherical
`'
`13.4
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`5th order spherical
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`-
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`3rd order coma
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`3rd order astigmatism
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`Smooth random errors
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`57.1
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`8.6
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`5.0
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`~5
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`in general, for a mixture of lower order aberrations, P—V/HMS = 4.5
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`When generating wavefront error budgets, HMS 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
`OPT] 51?
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`8
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`Rayleigh Criterion
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`
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`Lord Rayleigh observed that when the
`wavefront did not exceed M4 peai<—to
`degraded"
`d thFiieigh Criterion I
`_
`
`
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`MS wavefront error being about 0.0?
`
`S}
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`The Strehl Ratio is a related measure of image quality
`
`— It can be expressed (for Fii\/IS wavefront error < 0.1 wave) as
`
`Strehi Ratio 2 e-fem“ =1—(2n<1>)'~’
`
`where (P is the RMS wavefront error (in waves)
`
`greater for acoeptabie image quality is
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`2 0.8
`
`often called the Maréchal Criterion
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`9
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`
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`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
`—Tota[ spot size
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`Spot
`
`Size Rule of Thumb:
`
`Total 80% blur = [(Geo 80% blur)? + (Airy diameter)'¢‘]"2
`
`Amount of Aberration
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`10
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`10
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`10
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`Image Quality Metrics
`
`-
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`The most commonly used geometricahbased image quaiity metrics are
`
`— Ray aberration curves
`
`
`
`— Modulation transfer function (MTF)
`
`—
`
`The most Commonly used diffraction—based image quality metrics are
`
`
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`11
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`11
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`Ray Aberration Curves I
`
`- These are by far the image quality metric most commonly used by optical
`designers during the design process
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`-
`
`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
`
`sagittai rays
`Tangential rays
`
`Ay values for
`tangential rays
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`Ax values for
`
`4-— Pupil position —>
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`"I517
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`12
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`12
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`12
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`Graphical Description of Ray Aberration Curves
`
`ositions of the rays in a fan
`from the chief ray vs. position in the fan
`
`I
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`. ¢
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`\
`
`
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`",=_.!*=== Te' -we
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`H''‘‘'*----.._ /#,e
`I
`Pupil position
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`_._,'________
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`4
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`*—“
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`|
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`-
`
`
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`\
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`\
`——-X‘:/”
`‘E
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`.
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`//’
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`Image plane
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`Meg 13.3: e
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`in the Y; mus mi -31
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`hich is a failing of this IQ metric
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`
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`13
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`13
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`Transverse vs. Wavefront Ray Aberration Curves
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`-
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`v
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`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 curve is a scaled derivative of the wavefront curve
`
`Example curves for pure defocus:
`
`
`
`Wavefront error
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`14
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`14
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`More on Bay Aberration Curves
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`
`
`an tel! what type of aberration is present
`curves shown)
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`ewe
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`fl.05
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`
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`Defocus
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`Coma
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`%_%fi
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`\\
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`
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`Third—order spherical
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`Astigmatism
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`OPTIS1?
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`15
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`15
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`15
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`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 usetui to show the detector size (and/or the Airy disk diameter)
`superimposed on the spot diagram
`
`. wavelengths Detector outiine <-*-* "I: '
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`'
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`, . Different colors
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`represent different
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`I.
`I I.
`Inn
`,_
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`.
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`.
`I
`_. —u-
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`The shape of the spot diagram can often tell what type of aberrations are
`present in the image
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`CJPTI 51?
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`16
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`16
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`Main Problem With Spot Diagrams
`
`
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`at diagram don't convey intensity
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`m does not tell the intensity at that point
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`L-.l-D.
`. -1,.-
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`.r.
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`The ::m—ax1's image appears
`spread out in the spot diagram,
`but in reality it has a tight core
`with some surrounding low-
`intensity flare
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`‘I?
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`17
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`17
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`Diffraction
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`-
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`~
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`-
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`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
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`
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`Diffraction pattern of
`a perfect image
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`151?
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`1B
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`18
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`18
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`Size of the Diffraction Image
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`-
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`-
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`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
`
`
`
`
`
`The diameter of the first ring is given by d z 2.44 ?l.f/#
`— This is independent of the focal length; it is only a function of the
`wavelength and the finumber
`— The angular size of the first ring 13 = d/F == 2.44 MD
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`Very important !l!!
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`19
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`mage of a point object is given by the
`e ditfraction—limited
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`.17
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`19
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`19
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`Spot Size vs. the Airy Disk
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`-
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`Fiegimei — Ditfraction—|imited
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`Airy disk
`diameter
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`Point image
`(geometrically)
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`-
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`Regime 2 — Near diffraction—|imited
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`
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`Non-zero geometric
`blur, but smaller
`than the Airy disk
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`-
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`Regime 3 — Far from diffraction—limited
`Airy disk
`diameter
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`Geometric blur
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`significantly larger
`than the Airy disk
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`-
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`OPTI 517
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`_ Srel '-
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`20
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`20
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`Point Spread Function (PSF)
`
`
`
`ing the effects of diffraction and alt
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`Image
`intensity
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`Intensity peak of the PSF relative to
`«IT that of a perfect lens (no wavefront
`error) is the Strehl Ratio
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`Airy disk (diameter of the first zero)
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`‘I,
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`lW
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`21
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`21
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`21
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`Diffraction Pattern of Aberrated Images
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`- 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 fess energy in the central ring and more in the outer rings
`
`Perfect PSF
`
`Strehl =- 1 .0
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`
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`22
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`22
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`PSF vs. Defocus
`
`Figure 11.2: Paint. spread functions for different amounts of defocus. (a) 0.125 Wave. (P-
`W; 0.03’? wave rms; 0.85 Strehl. (b) 0.25 wave {P-VJ; D.O74 wave rms; 0.80 Strehl. (cl 0.50
`wave [P-\}'); 0.143 wave nus; 0.39 Strohl- (cl) LOG wave (P-V]; 0.29’? wave mm; 0.00
`Streh].
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`CPTI 517
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`23
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`23
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`23
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`PSF vs. Third-order Spherical Aberration
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`[LE5
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`Figure 11.25 Point spread functions for different amaunts of third-order spherical aber-
`ration. (sz) 0.125 wave (P—V}; 0.040 wave ms; 0.94 Swehl. (b) 0.25 wave {P-V]; 0.080
`wave was; 0.78 Strehl. Kc} 0.50 wave (P-V); 0.159 wave mus; 0.37 Strehl. (d'J 1.00 wave
`(P-V]; 0.313 wave rms; 0.08 Strahl.N0£e: Referenca sphere centered at 0.5LA,,, (midway
`bctwccn marginal and paraxial foci).
`OPT! 517
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`24
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`24
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`24
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`PSF vs. Third-order Coma
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`
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`(‘*1
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`Figure 11.26 Point spread functions for different amounts of third-order coma. (a) 0.125
`wave (P-V); 0.031 wave rms; 0.96 Strehl. {h} 0.25 wave (P-V); 0.061 Wave mm; 0.35
`Strehl. is) 0.50 wave L1’-V): 0.123 wave x-ms; 0.65 Strehl- (cl) 1.00 wave {P-V); 0.25 wave
`rms; 0.18 SI:.reh1.Note: P-V OPD reference sphere centered at 0.25Comag« from chief ray
`intersection point. rms 01-‘D reference sphere centered at 0.22ElComa-,=~ liram chief‘ ray
`intersection paint.
`OPTI 517
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`25
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`25
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`25
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`PSF vs. Astigmatism
`
`0.25
`
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`Figure.-11.27 Point spread functions for different: amounts of astigmatism. (a) 0.125 wave
`[P-V); 0.026 wave rms; 0.97 St:-eh]. (b) 0.25 wave (P-V); 0.052 wave rms; 0.90 Strehl. (c}
`0.50 wave (P-VJ; 0.104 wave mm; 0.65 Strelnl. (d) 1.00 wave. (P-VJ; 0.207 wave rms; 0.18
`Stmhl. Nate: Reference sphere centered midway between sagittal and tangential foci.
`OF"1'l 517
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`25
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`26
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`26
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`PSF for Strehl = 0.80
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`
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`{'3
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`0.25
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`
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`Figum 11.29 Paint spread lilnctions for live different ::.berraLions_. each with a Stnahl
`ratio of 0.80 (the Mamcl1al criterion). In each case the center 01' the reference sphere is
`located to minimize the rms CIPD, which is 0.075 wave For all five abe1'1'al:ioa:'.:..
`(11)
`Defn-:.-us: 0.25 wave ("P-V}. {b} Third-order spherical: 0.235 wave (P-V}. (:1 Balanced third-
`aml fift]1-arder spherical: D221 wave {P-VJ. (d}A3tig'matIsm: 0.359 wave {P-V}. Ie}Cn1na:
`0.305 wave {P-V}.
`
`OPTI 517
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`27
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`27
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`27
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`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—|imited 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 toierancinfi
`5}age
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`28
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`28
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`Ensquared Energy Example
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`Ensquared energy on a detector of same order of size as the Airy disk
`
`Perfect lens, V2, 10 micron wavelength, 50 micron detector
`
`Airy disk
`
`(48.9 micron diameter)
`
`Approximately 85% cf the
`energy is collected by the
`detector
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`29
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`29
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`
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`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
`
`I I I I
`
`Convolvedwith Q
`
`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
`
`OPT! 51?’
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`30
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`30
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`30
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`Comnutina MTF
`
`-
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`-
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`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)
`
`/\
`
`
`
`5PREflD FUNCTION
`
`+ UBJECT[EDGE|'—F
`
`[EDGE] [IMGE
`
`OBJECT BRIGHTNESS
`
`I;
`l
`}
`¥
`J
`‘
`THMGE
` L -»',== ““f=+
`
`Max — Min
`
`Max + Min
`
`I LLUM|N-QTIUN
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`H"-;_
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`'
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`OPTIE
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`31
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`31
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`MTF of a Perfect Image
`
`For an aberration-free image and a round pupil, the MTF is given by
`_
`_ M
`
`MTF(f) = §[(p—cos(p Sincp]
`
`1.0
`
`(P = cos 10‘ Moo): 005 km?)
`This f is spatial frequency
`
`
`
` Cutoff frequency
`
`
`fw = 1/(2.f!#)
`
`32
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`32
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`32
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`
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`Abbe’s Construct for Image Formation
`
`Abbe developed a useful framework from which to understand the diffraction-
`Iimiting spatial frequency and to explain image formation in microscopes
`
`1-‘QC;-_D5c.Op3_
`5:25
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`se grating exceeds the numerical
`er the optical system for object features
`
`
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`33
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`33
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`Example MTF Curve
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`W S00
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`E T
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`MTF depends on target
`orientation
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`:13
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`52
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`‘J1
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`1.13
`IL"?
`20:3"
`211‘:
`2&6
`IIZS
`36th
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`'
`
`OPTI 517
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`34
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`34
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`Direction of
`
`field point
`
`:’.’LlHU-.|3='l'‘-E1C.IOI_’. :.
`
`S = Sagittal
`R = Radial
`
`T = Tangential
`
`FOV
`
`34
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`
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`MTF as an Autocorrelation of the Pupil
`
`«
`
`The MTF is usually computed by lens design programs as the autocorrelation of
`the OPE) map across the exit pupil
`
`Relative spatial frequency 2 spacing
`between shifted pupils
`(cutoff frequency = pupil diameter)
`
`Perfect MTF = overlap area I pupil area
`
`points across the pupil
`
`
`
` IOIIOQIII.
`
`
`
`IUIIIOUOI
`
`MTF is computed as the normalized integral
`over the overlap region of the difference
`between the OPD map and its shifted
`complex conjugate
`
`OPTl 51?
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`35
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`35
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`35
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`
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`Typical MTF Curves
`
`I-1 Ir'.trI3c':L1Ct.:.H.“y Seminar
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`of
`
`M-I-F is a function
`of the focus
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`36
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`
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`\
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` _
`
`MTF curves are different
`for different points across
`the FOV
`
`Diffraction—|imited MTF
`(as good as it can get)
`
`36
`
`36
`
`
`
`
`
`Phase
`
`aftlne
`
`{
`
`-
`
`Since OPD .relates to the phase of the ray relative to ‘the. reference. sphere‘,-the
`.pupilautocorre1atian actually gives -the QTF (optical transfer function), which is a.
`nomplex quantity
`— MTF isthe real part (r'nodu'lus~) of the OTF’
`
`OTF %
`
`'
`
`‘
`
`7
`
`
`
`' OTF = Optical Transfer Fun‘c:'tio'n
`
`MTF = Modulus of-the OTF
`
`PTF a Phase -of the OTF
`
`Whén the OTF gees
`negative, the phase
`.is:s: radians
`
`3?
`
`
`
`37
`
`37
`
`
`
`What Does OTF < 0 Mean?
`
`- When the OTF goes negative, it is an example of contrast reversal
`
`OPT
`
`38
`
`38
`
`
`
`Example of Contrast Reversal
`
`OPT] 51?’
`
`39
`
`39
`
`39
`
`
`
`More on Contrast Reversal
`
`TI 51?
`
`40
`
`40
`
`40
`
`
`
`Effect of Strehl = 0.80
`
`the image is considered to be equivatent
`age
`
`ncies is reduced by the Strehl ratio
`
`
`
`
`Diffractiemlimited MTF
`
`41
`
`41
`
`
`
`Aberration Transfer Function
`
`-
`
`Shannon has shown that the MTF can be approximated as a product of the
`diffraction—|imited MTF (DTF) and an aberration transfer function (ATF)
`
`.
`
`DTF(v) =§[cos‘1 v-—v\i1-1/2]
`
`V = f/foo
`
`9'
`
`Bob
`
`Shannon
`
`
`
`: 0.025 waves rrns
`
`
`—:0.050wavesrm:
`5-5 l\ 1‘
`-—0.075wavesrrns
`0 F,
`-
`
`l\\t\\C—
`*—0.100wavesrn1s
`
`1 ‘
`
`0-4
`03
`
`0-2
`0 1
`'
`o o
`Eon
`
`[L20
`
`mm
`0.40
`Normalized Spatial Frequency
`
`mm
`
`mm
`
`42
`
`
`
`"-
`E 0.5
`
`0.?
`..
`D.5
`
`[La
`
`ATF
`
`'3'"
`0.00
`
`
`
`‘ * 0.025 waves rrns
`-4 * 0.050 waves rrns
`0.3
`"""-
`U 2 — -"0.0?5wavesrms
`‘ - —- 0.100 waves rrns
`0.1
`Z 0.125 waves rms
`—-' 0.1 50 waves rms
`
`0.20
`
`0.60
`0.40
`Nonnallzecl Spatiai Frequency
`
`0.30
`
`1.00
`
`OPTI 517
`
`42
`
`
`
`42
`
`
`
`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
`LIIHTINE
`RESDi.U‘lION
`
`5
`
`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 51?
`
`43
`
`43
`
`
`
`Elnsrgagcgnim
`vL-I&nEt{LATro~
`
`FREEUENCHN -
`
`IUNES PER IIILLIIIEFEHJ
`tat
`
`
`
`LIMIIING
`RESOLUTION
`
`FREQUENCY-h
`lcl
`
`—r-
`
`
`
`MGDIJLATION
`
`
`
`MCIDULATION—0-
`
`MIIDULJITION
`
`43
`
`
`
`Example of Different MTFs on RIT Target
`
`-inn‘!
`g-‘
`
`.
`
`e
`
`W
`
`mm for 1211" Thrgeu
`-_ —————.-..:.--.—-1 .....-»«:l....-7.........=_......_....1__........\.... ............|-.....——-y-------T.----»._.-_.......,_.-_-_-!
`I‘if
`
`
` 3
`
`I
`..._._.I._._.._ _
`3
`I
`PI-h;|rI'v'v-3-u-I-1.i|
`LIJ_'
`1‘.l3I:| Six‘
`
`._l
`m”
`DU]
`
`‘
`I
`i
`lh-nn-i.I-H-1
`11% 5'33 C CD
`Fmqua.
`
`I1
`
`.
`
`II1L. {son He: 1=_=u:n'
`
`
`C}4Ei¢.‘1«’l11m)
`
`44
`
`44
`
`
`
`Central Obscurations
`
`In on-axis teiescope designs, the obscuration caused by the secondary mirror is
`typicalfy 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)
`
`.. ions as the obscuration increases the diameter of
`
`the same, and the loss of energy to the
`
`75 % Linear obs.
`
`CJFTI 51?
`
`45
`
`45
`
`45
`
`
`
`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 sols,“ = 0.00
`B
`s,,/s,,, = 0.25
`c e,/s,,, = 0.50
`
`W ‘"
`08 I
`
`
`
`U
`
`0.1
`
`(12
`
`-0.3
`
`0.4
`
`0.?
`
`-0.3
`
`0'.
`
`1.0
`
`0.5
`0.5
`Vf"I1ro—]|-
`OPT} 517
`
`45
`
`46
`
`46
`
`
`
`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
`JLLUMINATED
`PUF"L
`POINT
`
`O O
`
`g
`
`IJIFFRACTED
`
`
`
`% — ~— — — -
`: E
`/'
`5
`COHEFIENT
`ILLUMINATION
`
`(3)
`
`‘
`MTF
`
`MTF
`
`+
`
`[D]
`
`PUFIL
`\
`
`|1_:_uM|Np.'rEn
`men OF
`FUPIL
`
`l:H_EQUENCY_—_
`M
`
`2 N.“
`’‘
`
`OPTICAL
`SYSTEM
`
`at
`M"
`OBJECT
`/
`
`
`
`SEMI—
`COHERENT
`ILLUMIN.-WON 3
`
`
` GONE OF RAYS
`
`(til
`
`{-9)
`
`FREQUENCY
`(fl
`
`21!:
`3*
`
`Figure 11.20 la—<:) The MTF with coherent illmnination. (d—f) The MTF with semicoher-
`ent illumination (which partially fills the pupil}.
`OPTJ 51?
`
`47
`
`47
`
`
`
`MTF of Partially Coherent Illumination
`
`‘ID036-
`
`
`
`inch-HEHENT
`
`
`
`MTF"
`
`0.8 '
`
`FREQUENCY -—u-
`
`uency for a partially fined pupil (semicoherent illumination).
`Figure 11.21 MTF vs.
`Numbarsaretheratioo i1lumi11ati_ngsystemNAtuo tic-al
`emNA.
`
`
`
`
`
`48
`
`48
`
`
`
`Partial Coherent Image of a 3-Bar Target
`
`DIFFRACTION INTENSITY PRDFILE
`PARTIALLY COHERENT'ILLUMINATION
`
`D01
`ii ”§*%'§”f’?§~”SE?i5i§§T“=a*=
`L50! dam 0
`zunmau
`Dthfiacna
`0D10
`c»:a :59at)ua
`013699°
`
`aa-
`
`WAVELENGTH WEIGHT
`1
`500.0 NM
`
`
`
`RELATIVE
`INTENSITY
`
`DEFOCUSING 0.00000
`
`1.25
`
`1.DD
`
`ELT5
`
`0.50
`
`0.25
`
`0.0
`-1.3
`-2.5
`DISPLACEMENT ON IMAGE SURFACE
`
`1
`
`A
`
`as
`5}
`
`{MICRON
`
`OPTI 517
`
`5.0
`
`49
`
`49
`
`49
`
`
`
`Example of Elbows lmaged in Partially Coherent Light
`
`
`spherical aberration
`
`With 1 wave of
`
`"I 51?
`
`50
`
`50
`
`50
`
`
`
`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
`
`
`
`51
`
`
`
`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
`
`Ideal focus
`
`point
`
`Spherical reference sphere centered
`‘___,on defocused point
`
`
`
`Defooused
`
`
`
`\
`
`A°t“a' WaV9"°'“
`
`image point
`
`
`when maximum OPD = 1/4. you are
`at the Rayleigh depth of focus = 2 1 12
`
`OPTI 517
`
`52
`
`52
`
`
`
`Defocus Flay Aberration Curves
`
`Wavefront map
`
`
`
`Wgggfgggl gggfggr
`
`Spot diagram
`
`
`
`-r;.c:
`
`Transverse ray aberration
`
`53
`
`53
`
`
`
`MTF of a Defocused Image
`
`As the amount of defocus increases. the MTF drops accordingly
`
`A OPD =' 0
`
`
`
`B OPD = N4
`
`C OPD = 1/2
`
`54
`
`54
`
`
`
`Sources of Defocus
`
`- One obvious source of defocus is the location of the object
`
`+ For tenses focused at infinity, objects closer than infinity have defooused
`
`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
`
`the focus (e.g._. shims in 0.001 inch
`
`- of the desired focus position
`
`
`
`and focus position resolution must be
`hich can degrade the image quality
`
`55
`
`55
`
`
`
`Chromatic Aberration
`
`- Chromatic aberration is caused by the Iens's refractive index changing with
`wavelength
`
`
`
`nexracuuInflux
`
`saw.
`
`Inn '
`no
`ilaiuelanuth Inni
`
`an.
`
`inn.
`
`0.30
`SHIFT{in}
`
`The shorter wavelengths focus
`closer to the lens because the
`refractive index is higher for the
`shorter wavelengths
`
`II. !D
`
`FOCUS
`
`I517
`
`NAVELBNGIH {hm}
`
`55
`
`56
`
`56
`
`
`
`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
`
`If is commonly designated by the Abbe \l—number for three wavelengths
`— For visible glasses, these are F (486.13), d (587.56), C (856.27)
`
`— For infrared glasses they are typically 3, 4, 5 or 8, 10, 12 microns
`
`—
`
`V = {”miudie'1) / ("short ' 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
`
`
`
`e diffraction DOF = d:2Xf9 = i0.004 mm
`lue of 942 (for s -12 ii). What is the
`--
`axial chromatic spread of a germanium lens of 100 mm focal length?
`
`' Answer: 100K942 = 0.11 mm
`OPTIS17
`
`Note: DOF(f/2) = i27tf2 = i0.08 mm
`
`5?
`
`57
`
`57
`
`
`
`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 I V = F / 942
`
`— The 1K4 wave depth of focus is 2211‘?
`
`— Equating these and solving gives F = 4"942*?t*l2 = 150 mm
`
`Waves
`
`LL25
`
`FIELD 1-IEIIGHT
`r 0.000 ‘)
`
`‘W I
`'
`
`' Strehl = 0.36
`
`L0
`
`Lfi
`
`ILB
`
`15.3
`
`|1.|1
`25.0
`JLO
`FLU
`21.'.I
`SPATIAL PREQUEMCY {CHCLKEHHH1
`
`|u.l|
`
`§l.|1
`
`El:
`
`OPT] 51 fr‘
`
`58
`
`58
`
`58
`
`
`
`Correcting Chromatic Aberration
`
`(high V number) and the negative lens
`
`
` Red and blue
`
`focus together
`
`— This will correct primary chromatic aberration
`
`- The red and blue wavelengths toous together
`
`- The green (or middle) wavelength still has a focus error
`
`read is called secondary color
`
`59
`
`59
`
`
`
`Secondary Color
`
`-
`
`Secondary color is the residual chromatic aberration iett when the primary
`chromatic aberration is corrected
`
`
`
`This wavelength
`has a focus error
`
`focus together
`
` Secondary color
`
`(~ F/2400)
`
`|'J_|J<1.'
`
`1-1;-1-........u...‘......,
`
`
`F'(')''‘'..‘.‘?.’i!!
`
`'5-i.‘.A'5.
`
`'.«"F'.L E)-iG'1'%J
`
`[ .'i:1}
`
`-
`
`Secondary color can be reduced by selecting special glasses
`..
`gt.
`2-
`
`
`
`60
`
`60
`
`
`
`Lateral Color
`
`or magnification) with wavelength
`
`
`
`ages
`
`" with wavelength
`
`gee at the edge of the FOV
`
`61
`
`
`
`Higher-order Chromatic Aberrations
`
`-
`
`For'broadb-and systems, the chromatic variation in the third-order aberrations.
`are often the most challenging aberrations to correct (e.g., spherochromatism,
`chromatic variation of co ma, chromatic vari-ation of astigmatism)
`- These are best studied with ray aberration curves and field plots
`
`%I'.'.!D'1HL
`EIIIEE.-'.".'.?|L MIKE.
`
`‘n.
` '-I‘
`
`II-Iv
`
`Ifl
`
`II
`
`517
`
`S2
`
`62
`
`62
`
`
`
`The Seridel Aberrations
`
`- These are the ciassical aberrations in optical
`design
`
`— Spherical aberration
`
`- Coma
`
`ations
`
`— Astigmatism
`
`— Distortion
`
`— Curvature of field
`
`- These aberrations, along with defocus and
`'
`'
`'
`
`63
`
`63
`
`
`
`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
`
`
`
`64
`
`64
`
`
`
`Orders of Aberrations
`
`Sphelril
`
`Coma
`
`Astigmatism
`
`I-"Ield Curvature
`
`Tnirdorde-I’
`
`Fiflh-ordgr
`
`-3
`(F#)
`
`u
`, {9}
`
`-2
`(F#)
`
`1
`
`, (6)
`
`-1
`(F#)
`
`2
`_ (6}
`
`-1
`(Fit)
`
`2
`, (6)
`
`-5 '
`(F#)
`,
`
`0
`(8)
`
`-4
`(F#)
`
`1
`
`, (9)
`
`(F#)
`
`-1
`
`-1
`
`4
`, (B)
`
`4
`
`(F#)
`
`, (9)
`
`_
`_
`Distortion
`
`O
`(Fit)
`
`3
`, (9)
`
`0
`
`5
`, (9)
`
`(F#)
`
`65
`
`65
`
`
`
`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 paraxiai focus
`
`Ray aberration curve
`
`This is referred to as
`undero-arrested spherical
`aberration (marginal rays
`focus closer to the lens
`than the paraxial focus)
`
`
`
`\
`
` I
`
`Paraxiai focus
`
`-
`
`The magnitude of the [third-order) spherical aberration goes as the cube of the
`aperture (going from f/21o f/1 increases the SA by a factor of 8)
`
`DF'T| 517
`
`BE
`
`66
`
`66
`
`
`
`%_ Tljjuird-order SA Ray Aberration Curves
`
`.._ -we
`
`
` Wavefrontmap
`
`Spot diagram
`
`
`
`Transverse ray
`aberration curve
`
`6?
`
`_\,/
` =
`
`67
`
`67
`
`
`
`Spherical Aberration
`
`Paraxial
`Minimum
`focus
`RM8 WFE
`/ ____./""'—H'I‘/
`_..—-"""/-
`
`
`
`Marginai
`focus
`
`Mlmimum
`spot SIZE
`
`Minimum FHVIS WFE
`
`Paraxial focus
`
`I
`
`5]?
`
`68
`
`68
`
`68
`
`
`
`Scaling Laws for Spherical Aberration
`
`mmw
`
`
`
`173
`
`1°/4
`
`H5
`
`Spot size goes as the
`
`
`
`9:90
`
`um
`Q: mt IQ: :® 1%!
`Q! gr 1%:
`fig: Q3 Q6 ngi Q @: Q:
`eeeeeeeeeee
`1
`1 lg: lg: 1m: 1m:
`rmt am IQ fig:
`IE
`lg lg lg! I
`l IQ IE1 I®(
`Q‘ 5%‘
`5'5 Q‘ Q *3‘
`131 um Qt lg! fin
`lg: 1?: mt am
`IQ my rm mm
`um: la 1%: Qt
`
`'
`
`IQ mm;
`gt
`rm Q: mg; fit
`
`mm: am: IQ! Q
`1%: lg:
`mgr
`
`am:
`
`Spot size not dependent
`on field position
`
`3'1?
`
`69
`
`69
`
`69
`
`
`
`Spherical Aberration vs. Lens Shape
`
`-
`
`The spherical aberration is a function of the lens bending, or shape of the lens
`
`70
`
`70
`
`
`
`Spherical Aberration vs. Refractive Index
`
`
`n=1.50 F ’ ; A
`
`} Notice the bending
`for minimum SA is a
`
`function of the index
`
`OPTI 5
`
`71
`
`
`
`Spherical Aberration vs. Index and Bending
`K =0
`1;: .1 K=1-2 K=-
`K=—1
`
`2
`4!‘l—l']
`16(n—1)*’(n+2)
`
`-33
`BatKI'I'Iin_r (P
`
`n:2.0
`
`3 7
`
`
`
`1?:AngularBlurDuetoSph:-riualAhemlinn
`
`72
`
`2
`
`72
`
`
`
`Example - Germanium Singlet
`
`
`
`used at 10 microns (0.01 mm)
`
`th we can have and not need aspherics
`
`r-’-‘answer
`
`Diffraction Airy disk angular size is Bum = 2.44 7L/D
`Spherical aberration angular blur is [353 = 0.00857 / f3
`Equaiing these gives D = 2.44 X i3 I 0.00867 = 22.5 mm
`
`For i/2, this gives F = 45 mm
`
`Strehl = 0.91
`
`
`
`OPT1 51?
`
`73
`
`73
`
`73
`
`
`
`Spherical Aberration vs. Number of Lenses
`
`-
`
`Spherical abe'rrati.on can be reduced by splitting the |éns'into more than one lens
`
`¥:
`
`SA = 1
`
`(arbitrary units)
`
`+
`
`(arbitrary units)
`
`
`
`SA = 1/9
`
`(arbitrary units)
`
`74
`
`74
`
`
`
`Spherical Aberration vs. Number of Lenses
`
`
`
`
`
`,4 —o.5v3ri=
`1—=1iu- ‘1J(F~'- 11‘;
`'
`
`
`)2
`{:'\4'+E‘]
`i'-'|
`WHERE i
`= THE mumesa OF E-_EMENTS
`
`_Dl4y5®3
`
`
`
`ix :5,
`ti: : TOTAL POWER :
`,'
`= ELEMENT |\JI‘x1BER
`
`3.
`
`O |'\J -cw
`
`
`
`
`
`QNGLJLERBLURSPOT
`
`|.'\.'DE2'«'.
`
`r3.'—‘
`
`:*EFm:.t:T:3-N
`
`Figure 3.2 The spherical aberration of one, two. three, and four thin positive
`elements, each bent for minimum spherical aberration, plotted as a function
`of the index 0F I‘efI‘£1cti0n_, and showing the reduction in the amount of aber-
`1-atitzm produced by splitting a single elem ent into two or more elements {of
`the same total poweri. Each plot is labeled with i, the number of elements in
`
`75
`
`75
`
`
`
`Spherical Aberration and Aspherics
`
`«
`
`The spherical aberration can be reduced, or even effectively eliminated, by
`making one of the surfaces aspheric
`
`sphefical
`
`76
`
`76
`
`
`
`Aspheric Surfaces
`
`= Aspheric surfaces technically are any surfaces which are not spherioai, but
`usually refer to a polynomial deformation to a conic
`
`-+Ar4+Br6+Cr8+Dr1°+...
`r2/R
`z(r) —
`—1+«_i'1—(k+1)[rXR)2
`
`order
`sed primariiy to correct spherical
`
`-._I can correct 3rd, 5th, 7th, 9th.
`
`
`
`pil, they are primarily used to correct
`
`Before using aspherics, be sure that they are necessary and the increased
`performance justifies the increased cost
`— Never use a higher-order a-sphere than justified by the ray aberration curves
`
`;:
`
`77
`
`77
`
`
`
`Optimizing Aspherics
`
`
`
`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.
`
`
`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
`
`I517
`
`73
`
`(e.g., astigmatism).
`
`78
`
`78
`
`
`
`“R5316
`
`'
`
`' ‘Ta’
`
`E
`
`_E_.M__...%[EZL
`
`79
`
`79
`
`
`
`
`
`
`
`
`
`Aspheric Orders
`
`Aspheric ::‘-um
`41th order
`61:11 order
`
`Bth order
`mm: order
`
`fl .fiflfl T
`—1.naIJ=wa
`-:.nnu¢-n:
`
`. ml:-no
`sition
`
`3’
`
`I
`5.nuLa:.n1
`
`Luna:-0':
`
`OPTI 517
`
`4
`
`so
`
`80
`
`5.50
`Luau tin}
`
`.
`
`'
`
`.00
`
`..__ Gorresponds to -114
`waves of asphericity
`
`~\
`
`n.nn23-
`
`n.w2|r
`
`fl.0fl‘.5'
`
`n.0flL0'
`
`n.0fl05'
`
`%i
`
`i
`52.‘?
`3
`3
`9.
`i
`5-\
`-2
`
`DeltaSag
`
`80
`
`
`
`
`
`MTF vs. Assphefrlc Order
`
`‘
`
`e asphere
`
`"‘f
`
`Aterm only
` asphere
`
`-A_,B terms
`
`II'lIIl|_- 1% -tibia“!
`
`DPTl"E?l T
`
`
`
`81
`
`81
`
`
`
`° Coma is an otf—axis aberration
`
`Coma
`
`It gets its name from the spot diagram which looks like a comet (coma is Latin
`for comet)
`
`A cornatic image results when the periphery of the lens has a higher or tower
`magnification than the portion of the lens containing the chief ray
`
`Chief ray
`
`
`
`Spot di'ag.ram
`
`
`
`The magnitude of the (third—order) coma is proportional to the square of the
`aperture and the first power of the field
`
`a
`
`4»
`
`-
`
`-
`
`82
`
`82
`
`
`
`Transverse vs. Wavefront 3rd-order Coma
`
` Spot diagram
`
`Wavefront map
`
`
`
`
`
`Wavefront error
`
`Transverse ray aberration
`
`OPTI 51?
`
`SS
`
`83
`
`83
`
`
`
`Scaling Laws for Coma
`
`oc (f/#)'3
`
`f/5
`
`f/4
`
`f/3
`
`Full Field
`
`T? V
`
`0.5 Field
`
`V’
`
`3
`
`V
`
`I
`
`I
`
`I
`
`I
`
`c
`
`.
`
`I
`
`
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