Lecture 2: Geometrical Optics 1

Outline

1 Spherical Waves2 From Waves to Rays3 Lenses4 Chromatic Aberrations5 Mirrors

Christoph U. Keller, Utrecht University, C.U.Keller@uu.nl Astronomical Telescopes and Instruments, Lecture 2: Geometrical Optics 1 1

Introduction to Geometrical Optics

Spherical Waves

wave equations for dielectric

∇2~E − µε

c2∂2~E∂t2 = 0

spherical wave solution

~E (r , t) = ~E0Ar

ei(kr−ωt)

~E0: polarizationA: a constantr : radial distance fromcenter/source of wave

mostly part of spherical wave

Christoph U. Keller, Utrecht University, C.U.Keller@uu.nl Astronomical Telescopes and Instruments, Lecture 2: Geometrical Optics 1 2

Light Sources

light source: collection of sources of spherical wavesastronomical sources: almost exclusively incoherentlasers, masers: coherent sourcesspherical wave originating at very large distance can beapproximated by plane wave

Christoph U. Keller, Utrecht University, C.U.Keller@uu.nl Astronomical Telescopes and Instruments, Lecture 2: Geometrical Optics 1 3

Ideal Optics

ideal optics: spherical waves from any point in object space areimaged into points in image spacecorresponding points are called conjugate pointsfocal point: center of converging or diverging spherical wavefrontobject space and image space are reversible

Christoph U. Keller, Utrecht University, C.U.Keller@uu.nl Astronomical Telescopes and Instruments, Lecture 2: Geometrical Optics 1 4

From Waves to Rays: General Optical System

ideal optical system transforms plane wavefront into spherical,converging wavefront

Christoph U. Keller, Utrecht University, C.U.Keller@uu.nl Astronomical Telescopes and Instruments, Lecture 2: Geometrical Optics 1 5

From Waves to Rays: Azimuthal Symmetry

most optical systems are azimuthally symmetricaxis of symmetry is optical axis

Christoph U. Keller, Utrecht University, C.U.Keller@uu.nl Astronomical Telescopes and Instruments, Lecture 2: Geometrical Optics 1 6

From Waves to Rays: Locally Flat Wavefronts

rays are normal to local wave (locations of constant phase)local wave around rays is assumed to be plane wave

Christoph U. Keller, Utrecht University, C.U.Keller@uu.nl Astronomical Telescopes and Instruments, Lecture 2: Geometrical Optics 1 7

From Waves to Rays: Rays

geomtrical optics works with rays onlyrays are reflected and refracted according to Fresnel equationsphase is neglected⇒ incoherent sumrays can carry polarization information

Christoph U. Keller, Utrecht University, C.U.Keller@uu.nl Astronomical Telescopes and Instruments, Lecture 2: Geometrical Optics 1 8

From Waves to Rays: Finite Object Distance

object may also be at finite distancealso in astronomy: reimaging within instruments and telescopes

Christoph U. Keller, Utrecht University, C.U.Keller@uu.nl Astronomical Telescopes and Instruments, Lecture 2: Geometrical Optics 1 9

Geometrical Optics Example: SPEX

Christoph U. Keller, Utrecht University, C.U.Keller@uu.nl Astronomical Telescopes and Instruments, Lecture 2: Geometrical Optics 1 10

Limitations of Geometrical Opticsoptical system cannot collect all parts of spherical wavefront⇒diffractiongeometrical optics neglects diffraction effectsgeometrical optics: λ⇒ 0physical optics λ > 0simplicity of geometrical optics mostly outweighs limitations

Christoph U. Keller, Utrecht University, C.U.Keller@uu.nl Astronomical Telescopes and Instruments, Lecture 2: Geometrical Optics 1 11

Lenses

Definitionslens = refracting device, discontinuity in material propertiesperfect lens for infinite object: makes plane wavefront intospherical wavefront material

Christoph U. Keller, Utrecht University, C.U.Keller@uu.nl Astronomical Telescopes and Instruments, Lecture 2: Geometrical Optics 1 12

Surface Shape of Perfect Lens

lens material has index of refraction no z(r) · n + z(r) f = constantn · z(r) +

√r2 + (f − z(r))2 = constant

solution z(r) is hyperbola with eccentricity e = n > 1

Christoph U. Keller, Utrecht University, C.U.Keller@uu.nl Astronomical Telescopes and Instruments, Lecture 2: Geometrical Optics 1 13

Conic Sections

circle and ellipses: cuts angle < cone angleparabola: angle = cone anglehyperbola: cut along axis

Christoph U. Keller, Utrecht University, C.U.Keller@uu.nl Astronomical Telescopes and Instruments, Lecture 2: Geometrical Optics 1 14

en.wikipedia.org/wiki/Conic_constant

Conic Constant

r2 − 2Rz + (1 + K )z2 = 0 forz(0) = 0

z = r2

R1

1+

r1−(1+K ) r2

R2

R radius of curvatureK conic constantK = −e2, e eccentricityprolate elliptical (K > 0)spherical (K = 0)oblate elliptical (0 > K > −1)parabolic (K = −1)hyperbolic (K < −1)sphere is good approximation toall conic sections close to origin

Christoph U. Keller, Utrecht University, C.U.Keller@uu.nl Astronomical Telescopes and Instruments, Lecture 2: Geometrical Optics 1 15

Foci of Conic Sections

en.wikipedia.org/wiki/File:Eccentricity.svg

sphere has single focusellipse has two fociparabola (ellipse withe = 1) has one focus (andanother one at infinity)hyperbola (e > 1) has twofocal points

Christoph U. Keller, Utrecht University, C.U.Keller@uu.nl Astronomical Telescopes and Instruments, Lecture 2: Geometrical Optics 1 16

Paraxial Optics

assumption 1: Snell’s law for small angles of incidence (sin x ≈ x):n · φ = φ′

assumption 2: ray hight h small so that optics curvature can beneglected (plane optics, (cos x ≈ 1))assumption 3: tanφ ≈ φ = h/f

Christoph U. Keller, Utrecht University, C.U.Keller@uu.nl Astronomical Telescopes and Instruments, Lecture 2: Geometrical Optics 1 17

Spherical Lenses

if two spherical surfaces have same radius, can fit them togethersurface error requirement less than λ/105cm diameter lens, 500 nm wavelength⇒ 1ppm accuracygrinding spherical surfaces is easy⇒ most optical surfaces arespherical

Christoph U. Keller, Utrecht University, C.U.Keller@uu.nl Astronomical Telescopes and Instruments, Lecture 2: Geometrical Optics 1 18

Planoconvex Lenses

Christoph U. Keller, Utrecht University, C.U.Keller@uu.nl Astronomical Telescopes and Instruments, Lecture 2: Geometrical Optics 1 19

Types of Lenses

en.wikipedia.org/wiki/File:Lens2.svg

Christoph U. Keller, Utrecht University, C.U.Keller@uu.nl Astronomical Telescopes and Instruments, Lecture 2: Geometrical Optics 1 20

Positive/Converging Spherical Lens Parameters

commons.wikimedia.org/wiki/File:Lens1.svg

center of curvature and radii with signs: R1 > 0,R2 < 0center thickness: dwith material index of refraction⇒ positive focal length f

Christoph U. Keller, Utrecht University, C.U.Keller@uu.nl Astronomical Telescopes and Instruments, Lecture 2: Geometrical Optics 1 21

Negative/Diverging Spherical Lens Parameters

commons.wikimedia.org/wiki/File:Lens1b.svg

note different signs of radii: R1 < 0,R2 > 0virtual focal pointnegative focal length

Christoph U. Keller, Utrecht University, C.U.Keller@uu.nl Astronomical Telescopes and Instruments, Lecture 2: Geometrical Optics 1 22

General Lens Setup: Real Image

commons.wikimedia.org/wiki/File:Lens3.svg

object distance S1, object height h1

image distance S2, image height h2

axis through two centers of curvature is optical axissurface point on optical axis is the vertexchief ray through center maintains direction

Christoph U. Keller, Utrecht University, C.U.Keller@uu.nl Astronomical Telescopes and Instruments, Lecture 2: Geometrical Optics 1 23

General Lens Setup: Virtual Image

commons.wikimedia.org/wiki/File:Lens3b.svg

note object closer than focal length of lensvirtual image

Christoph U. Keller, Utrecht University, C.U.Keller@uu.nl Astronomical Telescopes and Instruments, Lecture 2: Geometrical Optics 1 24

Thin Lens Approximationthin-lens equation:

1S1

+1

S2= (n − 1)

(1

R1− 1

R2

)Gaussian lens formula:

1S1

+1

S2=

1f

Finite Imagingrarely image point sources, but extended objectobject and image size are proportionalorientation of object and image are inverted(transverse) magnification perpendicular to optical axis:M = h2/h1 = −S2/S1

Christoph U. Keller, Utrecht University, C.U.Keller@uu.nl Astronomical Telescopes and Instruments, Lecture 2: Geometrical Optics 1 25

Thick Lenses

www.newport.com/servicesupport/Tutorials/default.aspx?id=169

basic thick lens equation 1f = (n − 1)

(1

R1− 1

R2+ (n−1)d

nR1R2

)thin means d << R1R2

focal lengths measured from principal planesdistance between vertices and principal planes given by

H1,2 = − f (n − 1)dR2,1n

Christoph U. Keller, Utrecht University, C.U.Keller@uu.nl Astronomical Telescopes and Instruments, Lecture 2: Geometrical Optics 1 26

Chromatic Aberration 1

Christoph U. Keller, Utrecht University, C.U.Keller@uu.nl Astronomical Telescopes and Instruments, Lecture 2: Geometrical Optics 1 27

Chromatic Aberration 2

Christoph U. Keller, Utrecht University, C.U.Keller@uu.nl Astronomical Telescopes and Instruments, Lecture 2: Geometrical Optics 1 28

Chromatic Aberration 3

en.wikipedia.org/wiki/File:Lens6a.svg

due to wavelength dependence of index of refractionhigher index in the blue⇒ shorter focal length in blue

Christoph U. Keller, Utrecht University, C.U.Keller@uu.nl Astronomical Telescopes and Instruments, Lecture 2: Geometrical Optics 1 29

Achromatic Lens

en.wikipedia.org/wiki/File:Lens6b.svg

combination of positive and negative lens made from materialswith differents dispersions

Christoph U. Keller, Utrecht University, C.U.Keller@uu.nl Astronomical Telescopes and Instruments, Lecture 2: Geometrical Optics 1 30

Transmission of Transparent Materials

www.edmundoptics.com/technical-support/technical-library/frequently-asked-questions/

Christoph U. Keller, Utrecht University, C.U.Keller@uu.nl Astronomical Telescopes and Instruments, Lecture 2: Geometrical Optics 1 31

Mirrors

Mirrors vs. Lensesmirrors are totally achromaticreflective over very large wavelength range (UV to radio)can be supported from the backwavefront error is twice that of surface, lens is (n-1) times surfaceonly one surface to ’play’ with

Christoph U. Keller, Utrecht University, C.U.Keller@uu.nl Astronomical Telescopes and Instruments, Lecture 2: Geometrical Optics 1 32

Plane Mirrors: Fold Mirrors and Beamsplitters

Christoph U. Keller, Utrecht University, C.U.Keller@uu.nl Astronomical Telescopes and Instruments, Lecture 2: Geometrical Optics 1 33

Spherical Mirrors

www.tutorvista.com/content/science/science-ii/reflection-light/formation-spherical-mirrors.php

easy to manufacturefocuses light from center of curvature onto itselffocal length is half of curvaturetip-tilt misalignment does not matterhas no optical axisdoes not image light from infinity correctly (spherical aberration)

Christoph U. Keller, Utrecht University, C.U.Keller@uu.nl Astronomical Telescopes and Instruments, Lecture 2: Geometrical Optics 1 34

Parabolic Mirrors

want to make flat wavefront into spherical wavefrontdistance az(r) + z(r)f = const.z(r) = r2/2Rperfect image of objects at infinityhas clear optical axis

Christoph U. Keller, Utrecht University, C.U.Keller@uu.nl Astronomical Telescopes and Instruments, Lecture 2: Geometrical Optics 1 35

Elliptical Mirrors

en.wikipedia.org/wiki/Ellipse

have two foci at finite distancesreimage one focal point into another

Christoph U. Keller, Utrecht University, C.U.Keller@uu.nl Astronomical Telescopes and Instruments, Lecture 2: Geometrical Optics 1 36