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Geometrical OpticsGeometrical Optics for for AOAO
Claire Max
UC Santa Cruz
August 4, 2008
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Some tools for active learningSome tools for active learning
• In-class conceptual questions will aim to engage you inmore active learning and provide me with feedback onwhether concepts are clear
– I will pose a short conceptual question (no calculations)– I will ask you to first formulate your own answer, then discuss
your answer with two other students, finally to report yourconsensus answer to the class
• Some web-sites & books about teaching and learning:– http://ic.ucsc.edu/CTE/teaching/– http://teaching.berkeley.edu/compendium/– How People Learn, Bransford, Brown, and Cocking, Editors;
National Research Council, National Academy Press
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Goals of this lectureGoals of this lecture
• Review of Geometrical Optics– Understand the tools used for optical design of AO
systems
– Understand what wavefront aberrations look like,and how to describe them
– Characterization of the aberrations caused byturbulence in the Earth’s atmosphere
– What the optics of a simple AO system look like
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Keck AO system optical layout: whyKeck AO system optical layout: whydoes it look like this ??does it look like this ??
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Simplest schematic of an AO systemSimplest schematic of an AO system
COLLIMATING LENSOR MIRROR
FOCUSING LENS ORMIRROR
BEAMSPLITTERPUPIL
Optical elements are portrayed as transmitting,for simplicity: they may be lenses or mirrors
WAVEFRONTSENSOR
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What optics concepts are needed for AO?What optics concepts are needed for AO?
• Design of AO system itself:– What determines the size and position of the deformable
mirror? Of the wavefront sensor?
– What does it mean to say that “the deformable mirror isconjugate to the telescope pupil”?
– How do you fit an AO system onto a modest-sized optical bench,if it’s supposed to correct an 8-10m primary mirror?
• What are optical aberrations? How are aberrationsinduced by atmosphere related to those seen in lab?
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Levels of models in opticsLevels of models in optics
Geometric optics - rays, reflection, refraction
Physical optics (Fourier optics) - diffraction, scalar waves
Electromagnetics - vector waves, polarization
Quantum optics - photons, interaction with matter, lasers
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Review of geometrical optics:Review of geometrical optics:lenses, mirrors, and imaginglenses, mirrors, and imaging
• Rays and wavefronts
• Laws of refraction and reflection
• Imaging– Pinhole camera– Lenses– Mirrors
• Resolution and depth of field
Note: Adapted in part from material created by MIT faculty member Prof.George Barbastathis, 2001. Reproduced under MIT’s OpenCourseWarepolicies, http://ocw.mit.edu/OcwWeb/Global/terms-of-use.htm.
© 2001 George Barbastathis.
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Rays andRays and wavefronts wavefronts
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Rays and Rays and wavefrontswavefronts
• In homogeneous media, light propagates in straightlines
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Spherical waves and plane wavesSpherical waves and plane waves
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Refraction at a surface: SnellRefraction at a surface: Snell’’s Laws Law
• Snell’s law:
Medium 1, index of refraction n
Medium 2, index of refraction n′
nsin! = "n sin "!
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Reflection at a surfaceReflection at a surface
• Angle of incidence equals angle of reflection
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HuygensHuygens’’ Principle Principle
• Every point in a wavefrontacts as a little secondarylight source, and emits aspherical wave
• The propagating wave-front is the result ofsuperposing all these littlespherical waves
• Destructive interference inall but the direction ofpropagation
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WhyWhy are imaging systems needed? are imaging systems needed?
• Every point in the objectscatters an incident light into aspherical wave
• The spherical waves from allthe points on the object’ssurface get mixed together asthey propagate toward you
• An imaging system reassigns(focuses) all the rays from asingle point on the object ontoanother point in space (the“focal point”), so you candistinguish details of the object.
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Pinhole camera is simplest imagingPinhole camera is simplest imaginginstrumentinstrument
• Opaque screen with apinhole blocks all but one rayper object point fromreaching the image space.
• An image is formed (upsidedown). Good news.
• BUT most of the light iswasted (it is stopped by theopaque sheet)
• Also, diffraction of light as itpasses through the smallpinhole produces artifacts inthe image.
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Imaging with lenses: doesnImaging with lenses: doesn’’t throw awayt throw awayas much light as pinhole cameraas much light as pinhole camera
Collects allrays that passthrough solid-angle of lens
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““Paraxial approximationParaxial approximation”” or or ““firstfirstorder opticsorder optics”” or or ““Gaussian opticsGaussian optics””
• Angle of rays with respect to optical axis is small
• First-order Taylor expansions:– sin ε ≈ tan ε ≈ ε , cos ε ≈ 1, (1 + ε)1/2 ≈ 1 + ε / 2
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Thin lenses, part 1Thin lenses, part 1
Definition: f-number ≡ f / # ≡ f / D
D = lens diam.
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Thin lenses, part 2Thin lenses, part 2
D = lens diam.
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Ray-tracing with a thin lensRay-tracing with a thin lens
• Image point (focus) is located at intersection of ALLrays passing through the lens from the correspondingobject point
• Easiest way to see this: trace rays passing through thetwo foci, and through the center of the lens (the “chiefray”) and the edges of the lens
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Refraction and the Lens-usersRefraction and the Lens-users EquationEquation
f f
– Any ray that goes through the focal point on its wayto the lens will come out parallel to the optical axis.(ray 1)
ray 1
Credit: J. Holmes, Christian Brothers Univ.
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Refraction and the Lens-usersRefraction and the Lens-users EquationEquation
f f
– Any ray that goes through the focal point on its wayfrom the lens, must go into the lens parallel to theoptical axis. (ray 2)
ray 1
ray 2
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Refraction and the Lens-usersRefraction and the Lens-users EquationEquation
f f
– Any ray that goes through the center of thelens must go essentially undeflected. (ray 3)
ray 1
ray 2
ray 3
object
image
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Refraction and the Lens-usersRefraction and the Lens-users EquationEquation
f f
– Note that a real image is formed.
– Note that the image is up-side-down.
ray 1
ray 2
ray 3
object
image
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Refraction and the Lens-usersRefraction and the Lens-users EquationEquation
f f
• By looking at ray 3 alone, we can see
by similar triangles that M = h’/h = -s’/s
object
images
h s’
h’<0
Note h’ is up-side-downand so is < 0Example: f = 10 cm; s = 40 cm; s’ = 13.3 cm:
M = -13.3/40 = -0.33
ray 3
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Summary of important relationships forSummary of important relationships forlenseslenses
X X
Mx=xi
xo
= !si
so
Ma= !
so
si
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Definition: Field of view (FOV) of anDefinition: Field of view (FOV) of animaging systemimaging system
• Angle that the “chief ray” from an object can subtend,given the pupil (entrance aperture) of the imagingsystem
• Recall that the chief ray propagates through the lensun-deviated
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Optical invariant ( = Lagrange invariant)Optical invariant ( = Lagrange invariant)
y1!1= y
2!2
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Lagrange invariant has importantLagrange invariant has importantconsequences for AO on large telescopesconsequences for AO on large telescopes
From Don Gavel
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Refracting telescopeRefracting telescope
• Main point of telescope: to gather more light than eye.Secondarily, to magnify image of the object
• Magnifying power Mtot = - fObjective / fEyepiece so for highmagnification, make fObjective as large as possible (longtube) and make fEyepiece as short as possible
1
fobj=
1
s0
+1
s1
!1
s1
since s0"#
so s1! fobj
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Lick ObservatoryLick Observatory’’s 36s 36”” Refractor: Refractor:one long telescope!one long telescope!
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Concept QuestionConcept Question
• Give an intuitive explanation for why themagnifying power of a refracting telescope is
Mtot = - fObjective / fEyepiece
Make sketches to illustrate your reasoning
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Imaging with mirrors: spherical andImaging with mirrors: spherical andparabolic mirrorsparabolic mirrors
Spherical surface:in paraxial approx,focuses incoming
parallel rays to(approx) a point
Parabolic surface: perfect focusingfor parallel rays (e.g. satellite dish,
radio telescope)
f = R/2
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Problems with spherical mirrorsProblems with spherical mirrors
• Optical aberrations (mostly spherical aberrationand coma)
– Especially if f-number is small (“fast” focalratio, short telescope, big angles)
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Focal length of mirrorsFocal length of mirrors
• Focal length of sphericalmirror is fsp = − R/2
• Convention: f is positive if itis to the left of the mirror
• Near the optical axis,parabola and sphere arevery similar, so thatfpar = − R/2 as well.
f
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Parabolic mirror: focus in 3DParabolic mirror: focus in 3D
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Mirror equationsMirror equations
• Imaging condition for spherical mirror
• Focal length
• Magnifications
1
s0
+1
s1
= !2
R
f = !R
2
Mtransverse = !s0
s1
Mangle = !s1
s0
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Cassegrain Cassegrain reflecting telescopereflecting telescope
• Hyperbolic secondary mirror: 1) reduces off-axis aberrations,2) shortens physical length of telescope.
• Can build mirrors with much shorter focal lengths than lenses.Example: 10-meter primary mirrors of Keck Telescopes havefocal lengths of 17.5 meters (f/1.75). About same as Lick 36”refractor.
Parabolic primary mirror
Hyperbolicsecondary mirror
Focus
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Heuristic derivation of the diffractionHeuristic derivation of the diffractionlimitlimit
Courtesy of Don Gavel
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Angular resolution and depth of fieldAngular resolution and depth of field
• Diffractive calculation ⇒ light doesn’t focus at a point.“Beam Waist” has angular width λ / D, and length Δz(depth of field)
Diameter D!
"# $%
D
!
"z =8
#
$f 2
D2
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Time for a short breakTime for a short break
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AberrationsAberrations
• In optical systems
• In atmosphere
• Description in terms of Zernike polynomials
• Based on slides by Brian Bauman, LLNL and UCSC, andGary Chanan, UCI
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Third order aberrationsThird order aberrations
• sin θ terms in Snell’s law can be expanded in powerseries
n sin θ = n’ sin θ’
n ( θ - θ3/3! + θ5/5! + …) = n’ ( θ’ - θ’3/3! + θ’5/5! + …)
• Paraxial ray approximation: keep only θ terms (firstorder optics; rays propagate nearly along optical axis)
– Piston, tilt, defocus
• Third order aberrations: result from adding θ3 terms– Spherical aberration, coma, astigmatism, .....
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Different ways to illustrate opticalDifferent ways to illustrate opticalaberrationsaberrations
Side view of a fan of rays
(No aberrations)“Spot diagram”: Image at
different focus positions
Shows “spots” where rays wouldstrike a detector
1 2 3 4 5
1 2 3 4 5
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Spherical aberrationSpherical aberration
Through-focus spot diagramfor spherical aberration
Rays from a sphericallyaberrated wavefront focus
at different planes
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Hubble Space Telescope suffered fromHubble Space Telescope suffered fromSpherical AberrationSpherical Aberration
• In a Cassegrain telescope, the hyperboloid of theprimary mirror must match the specs of the secondarymirror. For HST they didn’t match.
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HST Point Spread FunctionHST Point Spread Function(image of a point source)(image of a point source)
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Spherical aberration as Spherical aberration as ““the mother ofthe mother ofall other aberrationsall other aberrations””
• Coma and astigmatism can be thought of as theaberrations from a de-centered bundle of sphericallyaberrated rays
• Ray bundle on axis shows spherical aberration only
• Ray bundle slightly de-centered shows coma
• Ray bundle more de-centered shows astigmatism
• All generated from subsets of a larger centered bundleof spherically aberrated rays
– (diagrams follow)
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Spherical aberration as the mother ofSpherical aberration as the mother ofcomacoma
Big bundle of sphericallyaberrated rays
De-centered subset ofrays produces coma
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ComaComa
• “Comet”-shapedspot
• Chief ray is at apexof coma pattern
• Centroid is shiftedfrom chief ray!
• Centroid shifts withchange in focus! Wavefront
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ComaComa
Through-focus spotdiagram for coma
Rays from a comaticwavefront
Note that centroid shifts:
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Spherical aberration as the mother ofSpherical aberration as the mother ofastigmatismastigmatism
Big bundle of sphericallyaberrated rays
More-decentered subset of raysproduces astigmatism
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AstigmatismAstigmatism
Through-focus spotdiagram for astigmatism
Side view of rays
Top view of rays
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Wavefront for astigmatismWavefront for astigmatism
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Different view of astigmatismDifferent view of astigmatism
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Where does astigmatism come from?Where does astigmatism come from?
From Ian McLean, UCLA
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Concept QuestionConcept Question
• How do you suppose eyeglasses correct forastigmatism?
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Off-axis object is equivalent to having aOff-axis object is equivalent to having ade-centered ray bundlede-centered ray bundle
Ray bundle from an off-axisobject. How to view thisas a de-centered raybundle?
For any field angle there will be anoptical axis, which is ⊥ to thesurface of the optic and // to theincoming ray bundle. The bundleis de-centered wrt this axis.
Spherical surface
New optical axis
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Zernike Zernike PolynomialsPolynomials
• Convenient basis set for expressing wavefrontaberrations over a circular pupil
• Zernike polynomials are orthogonal to eachother
• A few different ways to normalize – alwayscheck definitions!
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Piston
Tip-tilt
From G. Chanan
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Astigmatism(3rd order)
Defocus
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Trefoil
Coma
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Spherical
“Ashtray”
Astigmatism(5th order)
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Units: Radians of phase / (D / r0)5/6
Reference: Noll
Tip-tilt is single biggest contributor
Focus, astigmatism,coma also big
High-order terms goon and on….
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Seidel polynomials vs. Seidel polynomials vs. ZernikeZernikepolynomialspolynomials
• Seidel polynomials also describe aberrations
• At first glance, Seidel and Zernike aberrations look very similar
• Zernike aberrations are an orthogonal set of functions used todecompose a given wavefront at a given field point into itscomponents
– Zernike modes add to the Seidel aberrations the correct amount oflow-order modes to minimize rms wavefront error
• Seidel aberrations are used in optical design to predict theaberrations in a design and how they will vary over the system’sfield of view
• The Seidel aberrations have an analytic field-dependence that isproportional to some power of field angle
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References for References for Zernike Zernike PolynomialsPolynomials
• Pivotal Paper: Noll, R. J. 1976, “Zernikepolynomials and atmospheric turbulence”,JOSA 66, page 207
• Books:– e.g. Hardy, Adaptive Optics, pages 95-96
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Considerations in the optical design ofConsiderations in the optical design ofAO systems: pupil relaysAO systems: pupil relays
Pupil Pupil Pupil
Deformable mirror and Shack-Hartmann lenslet arrayshould be ““optically conjugate to the telescope pupil.optically conjugate to the telescope pupil.””
What does this mean?mean?
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Define some termsDefine some terms
• “Optically conjugate” = “image of....”
• “Aperture stop” = the aperture that limits the bundle of rays accepted bythe optical system
• “Pupil” = image of aperture stop
optical axisobject space image space
symbol for aperture stopsymbol for aperture stop
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So now we can translate:So now we can translate:
•• ““The deformable mirror should be optically conjugateThe deformable mirror should be optically conjugateto the telescope pupilto the telescope pupil””
meansmeans
•• The surface of the deformable mirror is an image of theThe surface of the deformable mirror is an image of thetelescope pupiltelescope pupil
wherewhere
•• The pupil is an image of the aperture stopThe pupil is an image of the aperture stop
–– In practice, the pupil is usually the primary mirror of theIn practice, the pupil is usually the primary mirror of thetelescopetelescope
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Considerations in the optical design ofConsiderations in the optical design ofAO systems: AO systems: ““pupil relayspupil relays””
Pupil Pupil Pupil
‘PRIMARY MIRROR
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Typical optical design of AO systemTypical optical design of AO system
telescopeprimarymirror
Science camera
Pair of matched off-axis parabola mirrors
Wavefrontsensor(plus
optics)Beamsplitter
Deformablemirror
collimated
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More about off-axis parabolasMore about off-axis parabolas
• Circular cut-out of a parabola, off optical axis
• Frequently used in matched pairs (each cancels out theoff-axis aberrations of the other) to first collimate lightand then refocus it
SORL
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Concept Question: what elementary optical calculationsConcept Question: what elementary optical calculationswould you have to do, to lay out this AO system?would you have to do, to lay out this AO system?(Assume you know telescope parameters, DM size)(Assume you know telescope parameters, DM size)
telescopeprimarymirror
Science camera
Pair of matched off-axis parabola mirrors
Wavefrontsensor(plus
optics)Beamsplitter
Deformablemirror
collimated
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Review of important pointsReview of important points
• Both lenses and mirrors can focus and collimate light
• Equations for system focal lengths, magnifications arequite similar for lenses and for mirrors
• Telescopes are combinations of two or more opticalelements
– Main function: to gather lots of light
• Aberrations occur both due to your local instrument’soptics and to the atmosphere
– Can describe both with Zernike polynomials
• Location of pupils is important to AO system design