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Lecture 2: Geometrical Optics 1
Outline
1 Spherical Waves2 From Waves to Rays3 Lenses4 Chromatic Aberrations5 Mirrors
Christoph U. Keller, Utrecht University, [email protected] 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, [email protected] 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, [email protected] 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, [email protected] 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, [email protected] 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, [email protected] 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, [email protected] 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, [email protected] 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, [email protected] Astronomical Telescopes and Instruments, Lecture 2: Geometrical Optics 1 9
Geometrical Optics Example: SPEX
Christoph U. Keller, Utrecht University, [email protected] 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, [email protected] 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, [email protected] 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, [email protected] 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, [email protected] 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, [email protected] 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, [email protected] 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, [email protected] 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, [email protected] Astronomical Telescopes and Instruments, Lecture 2: Geometrical Optics 1 18
Planoconvex Lenses
Christoph U. Keller, Utrecht University, [email protected] 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, [email protected] 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, [email protected] 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, [email protected] 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, [email protected] 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, [email protected] 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, [email protected] 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, [email protected] Astronomical Telescopes and Instruments, Lecture 2: Geometrical Optics 1 26
Chromatic Aberration 1
Christoph U. Keller, Utrecht University, [email protected] Astronomical Telescopes and Instruments, Lecture 2: Geometrical Optics 1 27
Chromatic Aberration 2
Christoph U. Keller, Utrecht University, [email protected] 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, [email protected] 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, [email protected] 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, [email protected] 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, [email protected] Astronomical Telescopes and Instruments, Lecture 2: Geometrical Optics 1 32
Plane Mirrors: Fold Mirrors and Beamsplitters
Christoph U. Keller, Utrecht University, [email protected] 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, [email protected] 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, [email protected] 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, [email protected] Astronomical Telescopes and Instruments, Lecture 2: Geometrical Optics 1 36