Page 1

Geometrical OpticsGeometrical Optics for for AOAO

Claire Max

UC Santa Cruz

August 4, 2008

Page 2

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

Page 3

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

Page 4

Keck AO system optical layout: whyKeck AO system optical layout: whydoes it look like this ??does it look like this ??

Page 5

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

Page 6

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?

Page 7

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

Page 8

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.

Page 9

Rays andRays and wavefronts wavefronts

Page 10

Rays and Rays and wavefrontswavefronts

• In homogeneous media, light propagates in straightlines

Page 11

Spherical waves and plane wavesSpherical waves and plane waves

Page 12

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 "!

Page 13

Reflection at a surfaceReflection at a surface

• Angle of incidence equals angle of reflection

Page 14

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

Page 15

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.

Page 16

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.

Page 17

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

Page 18

““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

Page 19

Thin lenses, part 1Thin lenses, part 1

Definition: f-number ≡ f / # ≡ f / D

D = lens diam.

Page 20

Thin lenses, part 2Thin lenses, part 2

D = lens diam.

Page 21

Page 22

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

Page 23

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.

Page 24

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

Page 25

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

Page 26

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

Page 27

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

Page 28

Summary of important relationships forSummary of important relationships forlenseslenses

X X

Mx=xi

xo

= !si

so

Ma= !

so

si

Page 29

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

Page 30

Optical invariant ( = Lagrange invariant)Optical invariant ( = Lagrange invariant)

y1!1= y

2!2

Page 31

Lagrange invariant has importantLagrange invariant has importantconsequences for AO on large telescopesconsequences for AO on large telescopes

From Don Gavel

Page 32

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

Page 33

Lick ObservatoryLick Observatory’’s 36s 36”” Refractor: Refractor:one long telescope!one long telescope!

Page 34

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

Page 35

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

Page 36

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)

Page 37

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

Page 38

Page 39

Parabolic mirror: focus in 3DParabolic mirror: focus in 3D

Page 40

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

Page 41

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

Page 42

Heuristic derivation of the diffractionHeuristic derivation of the diffractionlimitlimit

Courtesy of Don Gavel

Page 43

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

Page 44

Time for a short breakTime for a short break

Page 45

AberrationsAberrations

• In optical systems

• In atmosphere

• Description in terms of Zernike polynomials

• Based on slides by Brian Bauman, LLNL and UCSC, andGary Chanan, UCI

Page 46

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, .....

Page 47

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

Page 48

Spherical aberrationSpherical aberration

Through-focus spot diagramfor spherical aberration

Rays from a sphericallyaberrated wavefront focus

at different planes

Page 49

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.

Page 50

HST Point Spread FunctionHST Point Spread Function(image of a point source)(image of a point source)

Page 51

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)

Page 52

Spherical aberration as the mother ofSpherical aberration as the mother ofcomacoma

Big bundle of sphericallyaberrated rays

De-centered subset ofrays produces coma

Page 53

ComaComa

• “Comet”-shapedspot

• Chief ray is at apexof coma pattern

• Centroid is shiftedfrom chief ray!

• Centroid shifts withchange in focus! Wavefront

Page 54

ComaComa

Through-focus spotdiagram for coma

Rays from a comaticwavefront

Note that centroid shifts:

Page 55

Spherical aberration as the mother ofSpherical aberration as the mother ofastigmatismastigmatism

Big bundle of sphericallyaberrated rays

More-decentered subset of raysproduces astigmatism

Page 56

AstigmatismAstigmatism

Through-focus spotdiagram for astigmatism

Side view of rays

Top view of rays

Page 57

Wavefront for astigmatismWavefront for astigmatism

Page 58

Different view of astigmatismDifferent view of astigmatism

Page 59

Where does astigmatism come from?Where does astigmatism come from?

From Ian McLean, UCLA

Page 60

Concept QuestionConcept Question

• How do you suppose eyeglasses correct forastigmatism?

Page 61

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

Page 62

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!

Piston

Tip-tilt

From G. Chanan

Astigmatism(3rd order)

Defocus

Trefoil

Coma

Spherical

“Ashtray”

Astigmatism(5th order)

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….

Page 70

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

Page 71

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

Page 72

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?

Page 73

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

Page 74

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

Page 75

Considerations in the optical design ofConsiderations in the optical design ofAO systems: AO systems: ““pupil relayspupil relays””

Pupil Pupil Pupil

‘PRIMARY MIRROR

Page 76

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

Page 77

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

Page 78

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

Page 79

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