Design and Correction of optical Systems

Transcription

1 Design and Correction of optical Systems Part 3: Components Summer term 0 Herbert Gross

2 Overview. Basics Materials Components Paraxial optics Properties of optical systems Photometry Geometrical aberrations Wave optical aberrations Fourier optical image formation Performance criteria Performance criteria Measurement of system quality Correction of aberrations Optical system classification

3 Part 3: Components 3. Lenses - description and parameters - imaging formulas 3. Mirrors 3.3 Prisms - dispersion prims - reflection prisms - miscellaneous 3.4 Special components - gratings - aspheres - diffractive elements - taper - gardient lenses

4 Optical Imaging Optical image formation: All rays starting in an object point meet in one image point Real image: intersection length positive Virtual image: intersection length negative Region near the optical axis: ideal, paraxial, gaussian linear, aberration-free Object and image: conjugated

5 Single Surface Single surface between two media Radius r, refractive indices n, n Imaging condition, paraxial Abbe invariant alternative representation of the imaging equation ' ' ' ' f r n n s n s n = = = = ' ' s r n s r n Q s

6 Sag of a Spherical Surface Sag z at height y for a spherical surface: z = r r y Paraxial approximation: quadratic term y z p r

7 Cardinal Elements of a Lens Focal points:. incoming parallel ray intersects the axis in F. ray through F is leaves the lens parallel to the axis Principal plane P: location of apparent ray bending y principal plane P' f ' u' s BFL F' focal plane nodal planes s P' u N N' u' Nodal points: Ray through N goes through N and preserves the direction

8 Notations of a Lens P principal point S vertex of the surface O n n n F focal point y s intersection point of a ray with axis u F S P P' N N' S' u' F' f focal length PF y' r radius of surface curvature s s' O' d thickness SS f BFL f f' f' BFL s P s' P' n refrative index a d a'

9 Lens Shapes Different shapes of singlet lenses:. bi-, symmetric. plane convex / concave, one surface plane 3. Meniscus, both surface radii with the same sign Convex: bending outside Concave: hollow surface Principal planes P, P : outside for mesicus shaped lenses P P' P P' P P' P P' P P' P P' bi-convex lens plane-convex lens positive meniscus lens bi-concave lens plane-concave lens negative meniscus lens

10 Bending of a Lens Bending: change of shape for invariant focal length Parameter of bending X = r r + r r X<- meniscus lens r 4 3 X = X = - 0 X = - 0 X = - 7 X = - 5 X = - 4 X = - 3 X = - X = -.5 X=- X=0 planconvex lens planconcave lens biconvex lens biconcave lens 0 X = - - X = X= planconvex lens planconcave lens r X> meniscus lens X = 0

11 Bending of a Lens and Principal Planes Ray path at a lens of constant focal length and different bending The ray angle inside the lens changes The ray incidence angles at the surfaces changes strongly The principal planes move For invariant location of P, P the position of the lens moves P P' F' X = -4 X = - X = 0 X = + X = +4

12 Properties of a Lens Main notations and properties of a lens: - radii of curvature r, r curvatures c sign: r > 0 : center of curvature is located on the right side - thickness d along the axis - diameter D - index of refraction of lens material n Focal length (paraxial) Optical power Back focal length intersection length, measured from the vertex point c = c = r r yf ' f =, f ' = tan u n F = = f s = f + s n' f ' F ' ' H ' y tan u'

13 Formulas of Surface and Lens Imaging Single surface imaging equation Thin lens in air focal length Thin lens in air with one plane surface, focal length Thin symmetrical bi-lens Thick lens in air focal length ' ' ' ' f r n n s n s n = = ( ) = ' r r n f ' = n r f ( ) ' = n r f ( ) ( ) ' r r n d n r r n f + =

14 Thick Lens Definition of thick / non-thin:. geometrical: thickness mach smaller than radius c d <<, c d <<. physical: significant difference of the ray height at front and rear surface Differences in bending point and angle of exit ray y n = d y n c <<

15 Conic Mirror: Paraboloid Equation c : curvature /R s κ : eccentricity ( = - ) z = cy + ( ) y c +κ y y C F z ray R sag vertex circle sagittal circle of curvature tangential circle of curvature R tan vertex circle R s parabolic mirror F y z x R radii of curvature : R tan = s + y R s parabolic mirror R R tan = s + R s y R s f 3

16 Simple Asphere Parabolic Mirror Equation Radius of curvature in vertex: R s Perfect imaging on axis for object at infinity Strong coma aberration for finite field angles Applications:. Astronomical telescopes. Collector in illumination systems z = y R s axis w = 0 field w = field w = 4

17 Conic Mirror: Ellipsoid Equation c: curvature /R z = cy + ( ) y c +κ κ: Eccentricity

18 Simple Asphere Elliptical Mirror Equation Radius of curvature r in vertex, curvature c eccentricity κ Two different shapes: oblate / prolate Perfect imaging on axis for finite object and image loaction Different magnifications depending on used part of the mirror Applications: Illumination systems s z = cy + ( + κ) y c s' F F'

19 Modelling a Mirror Surface Problem in coordinate system based raytracing of mirror systems: right-handed systems becomes left-handed Possible solutions: spherical mirror folded mirror surface. Folding the mirror - light propagation direction changed z-component inverted - tunnel diagram for prism. negative refractive index 3. inversion of the x-axis C F z r f' P=P'

20 Single Plane Dielectric Interface Refraction an single plane interface: intersection length changed n' s' = s n Optical denser medium: length increased Medium Example: View into water, ground seems to be nearer n = n' > s s'

21 Plane Parallel Platte Plane parallel plate: image location changed, Intersection length increased s' = n d n d 3 Reflection prisms works optically as plane parallel plate inside optical systems d Finite numerical aperture: Generation of spherical aberration Application: cover glass in microscopy d s' sph = ( n ) n Non-parallel ray path: Generation of astigmatism Application: Prism positions in collimated beam path preferred s' ast d = 3 ( n ) n 3 sin sin u w u s z

22 Dispersion Prism Parameters for characterization:. length of basis side: b. wedge angle α Angle of ray deviation:. exact ϕ = i α + arcsin[ n sin i sinα cosα sin i ]. approximation for small wedge angles ϕ = (n ) α α I I' ϕ I' I b

23 Dispersion Prism Angle deviation ϕ changes with tilt of prism Minimum value of deviation, approximately realized for a symmetrical ray path The deviations allows to measure the refractive index

24 White light dispersion by a prism wedge Dispersion Prism

25 Dispersion Prism Separation of white light into the spectral components due to dispersion Normal dispersion: blue color stronger bending than red color Application. spectral spreading of the wavelengths in spectrometer dϕ = dn α sin n sin dn α dλ white ϕ ϕ red green blue

26 Prism Magnification A beam with diameter D in changes ist width for a non-symmetrical path through a dispersion prism The magnification of the prism defines the change of diameter (in the main section) as cosi M = cosi' cosi cosi' The prism magnification depends on angle of incidence and refractive index Application: Anamorphotic prism pairs, change of ellipticity of beam cross section in laser beam guiding D in i i' D prism i i' D out

27 Anamorphotic Prism Pair Appropriate combination of two dispersion prisms: Change of cross section ellipticity of a collimated beam without angle deviation prism compressor Application: transform the profile of semiconductor laser beams in a circular cross section D D out in = ( cosθ sinθ tanα ) D in α θ beam cross-section at the entrance D out D in beam cross-section at the exit D out

28 Reflection Prisms Properties of reflection prims: Bending of the beam path, deflection of the axial ray direction Application in instrumental optics and folded ray paths Parallel off-set, lateral displacement of the axial ray Modification of the image orientation with four options:. Invariant image orientation. Reverted image ( side reversal ) 3. Inverted image ( upside down ) 4. Complete image inversion (inverted-reverted image) The number of mirrors is important Every mirror generates a complete inversion, No change for even numbers l/r and u/d separation by roof-edge prisms Off-set of the image position, shift of image position forwards in the propagation direction. Aberrations introduced. Astigmatism. Chromatic aberration 3. Spherical aberration in non-collimated beams

29 Comparison: Mirror vs. Prism Systems Prisms Mirrors Transmission utilizing total internal reflection + Chromatic properties, dispersion + Weight + Centering sensitivity, monolithic components + Complexity, number of mechanical holders + Coatings + Material absorption and inhomogeneities + Aberrations in a non-parallel beam path + Ghost images + Complexity of alignment + Separately adjustable reflecting surfaces +

30 Tunnel Diagram Tunnel diagram: Unfoldung the ray path with invariant sign of the z-component of the optical axis Optical effect of prisms corresponds to plane parallel plates More rigorous model: Exact geometry of various prisms can cause vignetting 3 3

31 Transformation of Image Orientation Modification of the image orientation with four options:. Invariant image orientation. Reverted image ( side reversal ) 3. Inverted image ( upside down ) 4. Complete image inversion y (inverted-reverted image) Image side reversal in the principal plane of one mirror x mirror Inversion for an odd number of reflections y - z- folding plane Special case roof prims: Corresponds to one reflection in the edge plane, Corresponds to two reflections perpendicular to the edge plane mirror y x z y x z

32 Transformation of Image Orientation image reversion in the folding plane (upside down) image unchanged original image reversion perpendicular to the folding plane folding plane image inversion

33 Transform of Image Orientation Rotatable Dove prism: Azimutal angle: image rotates by the double angle object angle of prism rotation Bild angle of image rotation Application: periscopes

34 Roof-Edge Prism Roof edge: - two reflecting surfaces with 90 - change of lateral coordinate in one section Critical in practice: Precision of 90 angle, typical tolerance errors cause image split Coatings critical due to polarization effects roof edge A D s C ϕ β intersection plane with angle of 90 intersection planes with angles of β B

35 Types of Reflection Prisms: 90 Prism Classical 90 prism Version with roof edge Version with arbitrary deviation angle (Amici prism) D α b δ h D 90 β

36 Types of Reflection Prisms: Porro Prism Porrro Prism Incoming ray direction inverted in one section Version with roof-edge: Ray direction inverted in 3D (retro reflector, cats eye reflector) 90 D a v

37 Types of Reflection Prisms: Penta Prism Classical penta prism β Penta prism with roof edge α Penta prism with arbitrary deviation angle D δ b.5 D 90 a D d

38 Types of Reflection Prisms: Bauernfeind Prism Classical Bauernfeind prism D One surface used for entrance and in reflection Prism with roof-edge β δ α D a α/ α

39 Deviation of Light Mechanisms of light deviation and ray bending Refraction Reflection Diffraction according to the grating equation n sinθ = n' sinθ ' θ = θ ' ( sin sin ) g θ θ = m λ o Scattering ( non-deterministic)

40 Reflection Grating Geometry of grating diffraction Generation of diffraction orders incidence y diffracted orders x h z g

41 Grating Diffraction Maximum intensity: constructive interference of the contributions of all periods grating Grating equation ( sin sin ) g θ θ = m λ o grating constant g in-phase +. diffraction order θ θ ο s = λ incident light

42 Grating Equation Intensity of grating diffraction pattern (scalar approximation g >> λ) Product of slit-diffraction and interference function I = N g πug sin λ πug λ Nπug sin λ πug N sin λ Maxima of pattern: coincidence of peaks of both functions: grating equation g ( sinθ sinθ ) = m λ o 0.7 Angle spread of an order decreases with growing number od periods N Oblique phase gradient: - relative shift of both functions - selection of peaks/order - basic principle of blazing u = π/λ sinθ

43 Real Diffraction Grating Real diffraction grating:. Finite number of periods. Finite width of diffraction orders θ λ : spectral width θ finite divergence grating N : finite number of periods +. diffraction orders : finite width incident light 0. -.

44 Spectral Resolution of a Grating Angle dispersion of a grating dθ D = = dλ λ cosθ sinθ m sinθ 0 Separation of two spectral lines λ sinθ m sinθ 0 A = = L = m N λ λ m I(x) Complete setup with all orders: Overlap of spectra possible at higher orders 0 mλ /g θ m(λ+ λ) /g sinθ

45 Diffractive Optics: Local micro-structured surface Location of ray bending : macroscopic carrier surface Direction of ray bending : local grating micro-structure local grating g(x,y) thin layer ϕ bending angle m-th order lens macroscopic surface curvature

46 Diffractive Elements Original lens height profile h(x) Wrapping of the lens profile: h red (x) Reduction on maximal height h π Digitalization of the reduced profile: h q (x)

47 Diffraction Orders Usually all diffraction orders are obtained simultaneously Blazed structure: suppression of perturbing orders Unwanted orders: false light, contrast and efficiency reduced diffractive structure m+3 m+ diffraction orders m+ m m- m- m-3 desired order

48 Fresnel Zone Plate Circular rings at radii r m = m f λ Classical Fresnel zone lens: only rings with same sign of phase have tranmission Modern zone plate (Wood): phase steps of π at the rings, improved power transmission f+5λ f+4λ f+3λ f+λ f+λ r 5 r 4 r 3 r r f F

49 Diffractive Lens Diffractive Fresnel lens Zone rings with radii Blaze in every zone (surface slope) r k = π m k f λ h m λ k = = + n cosθ k ( rk rk ) tanψ k

50 Gradient Lens Types Curved ray path in inhomogeneous media Different types of profiles y n entrance (y) n exit (y) radial gradient rod lens axial gradient rod lens radial and axial gradient rod lens n i n o z radial gradient lens axial gradient lens radial and axial gradient lens n(x,y,z)

51 Selfoc Lens Transverse parabolic profile of refractive index: Rod works as a periodical focussing lens L F' F P P'

52 Gradient Lenses Refocusing in parabolic profile off axis ray bundle axis ray bundle waist points Helical ray path in 3 dimensions y y perspectivic view y' x x' x z view along z

53 Gradient Lenses Types of lenses with parabolic profile y marginal n( r) = n 0 = n = n 0 0 n r ( n r ) r A r 0.5 Pitch Object at infinity y coma Pitch length 0.50 Pitch Object at front surface p = n0 π π = n n r 0.75 Pitch Object at infinity.0 Pitch Object at front surface Pitch

54 Summary of Important Topics 54 Single lens, paraxial imaging Cardinal points: focal point, principal plane, nodal points Simple formulas for focal length of thin and thick lenses Thick lens: lateral changes of ray height inside the lens is not neglectable Important for correction: bending of lenses: shape changed, focal length preserved Mirrors: mainly conic sections are of interest Plane parallel plate: image in z-direction shifted Dispersion prisms: spectral spreading of white light, spectroscopic applications Dispersion prisms: anamorphotic magnification Reflection prisms: use for beam deflection, change of image orientation Of special interest: roof prisms with one-sided image flip Gratings: overlay of diffraction effects of single period and interference function separation of the light into discrete diffraction orders Generalized: diffractive elements, local grating structures, problems with efficiencies and flase light of unwanted orders Gradient lenses: spatiually variant refractive index causes bended ray paths, can be used for imaging or beam profiling

55 Part 4: Paraxial Optics Next lecture: Part 4 Paraxial optics Date: Wednesday, Contents: 4. Imaging - basic notations - paraxial approximation - linear collineation - graphical image construction - lens makers formula 4. Optical system properties - pupil - special rays - special configurations 4.3 Matrix calculus - simple matrices - non-centered systems 4.4 Phase space - basic idea - invariants