Medical Photonics Lecture 1.2 Optical Engineering

Transcription

1 Medical Photonics Lecture 1.2 Optical Engineering Lecture 4: Components Michael Kempe Winter term

2 2 Contents No Subject Ref Detailed Content 1 Introduction Gross Materials, dispersion, ray picture, geometrical approach, paraxial approximation 2 Geometrical optics Gross Ray tracing, matrix approach, aberrations, imaging, Lagrange invariant 3 Diffraction Gross Basic phenomena, wave optics, interference, diffraction calculation, point spread function, transfer function 4 Components Kempe Lenses, micro-optics, mirrors, prisms, gratings, fibers 5 Optical systems Gross Field, aperture, pupil, magnification, infinity cases, lens makers formula, etendue, vignetting 6 Aberrations Gross Introduction, primary aberrations, miscellaneous 7 Image quality Gross Spot, ray aberration curves, PSF and MTF, criteria 8 Instruments I Kempe Human eye, loupe, eyepieces, photographic lenses, zoom lenses, telescopes 9 Instruments II Kempe Microscopic systems, micro objectives, illumination, scanning microscopes, contrasts 10 Instruments III Kempe Medical optical systems, endoscopes, ophthalmic devices, surgical microscopes 11 Optic design Gross Aberration correction, system layouts, optimization, realization aspects 12 Photometry Gross Notations, fundamental laws, Lambert source, radiative transfer, photometry of optical systems, color theory 13 Illumination systems Gross Light sources, basic systems, quality criteria, nonsequential raytrace 14 Metrology Gross Measurement of basic parameters, quality measurements

3 3 Passive Components in Optical Systems 1 from infinity Refraction Lenses F F' real image Beam Shaping Diffraction DOE Reflection Mirrors virtual image a) astigmatism A = 0 b) tangential flat stop F P B T S F' P S B T stop C C Reflection Mirrors virtual object location R R/2 R F F' c) best image flat d) sagittal flat 0.42 R P S B T stop P S B T Wavefront Management Beam Redirection / Splitting Refraction Prisms C R C R stop 0.29 R refractive : one direction diffractive blazed : one direction diffractive binary : all directions Diffraction Gratings +2. order +1. order 0. order -1. order n g g -2. order Beam Guiding Refraction Reflection Graded-Index Fibers Step-Index Fibers

4 4 Passive Components in Optical Systems Refraction Prisms grating +3. diffraction orders +2. Angular Dispersion weiß D incident collimated light rot Diffraction Gratings grün blau g = 1 / s grating constant Wavelength Management Filtering Absorption Absorption Filters Interference Interference and Dielectric Filters Modification Nonlinear Interaction Nonlinear Crystals Angular Dispersion Birefringence Birefringent Prisms etc. Absorption Dichroic Filters Polarisation Management Filtering Reflection Brewster Angle Polarizer Wavefront Retarder / Rotator Modification Birefringence Waveplates

5 virtual object 5 Lenses in Optical Systems Lenses are key elements in optical systems for Optical imaging Optical projection Light focusing (energy concentration) from infinity F F' real image virtual image Virtuell F F' virtual image F F'

6 6 Imaging by Lenses y F F' y' f f s s' y F y' F' s f f s' Lens equation (paraxial approximation, n = n ): 1 s 1 s = 1 f = 1 f Magnification: m = y = s = f s y s f

7 7 Cardinal Elements of a Refractive Lens Focal points: 1. incoming ray parallel to the axis intersects the axis in F 2. ray through F is leaves the lens parallel to the axis The focal lengths are referenced on the principal planes F front focal plane f P P' f ' F' back focal plane principal planes s BFL nodal planes Nodal points: Ray through N goes through N and preserves the direction N N' u' u

8 Notations of a lens P principal point N nodal point O n n n 1 2 y S vertex of the surface u F N N' S' u' F' F focal point S P P' f focal length PF y' O' r radius of surface curvature r > 0 : center of curvature is located on the right side s a f BFL f s P d s' P' f' f' BFL a' s' c curvature (c=1/r) d thickness SS Optical power F-Number n refractive index n f n' f ' F# = f D D: effective diameter

9 Lens shape Different shapes of singlet lenses: 1. bi-convex/concave, symmetric 2. 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 meniscus 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 Spherical Lenses Exhibit Aberrations Example: spherical aberration

11 Aspheres - Geometry Reference: deviation from sphere Deviation Dz along axis Better conditions: normal deviation Dr s y z(y) deviation Dz y height y tangente z(y) deviation Dz along axis z height y sphere perpendicular deviation Dr s aspherical shape spherical surface z aspherical contour

12 Reducing the Number of Lenses with Aspheres Example photographic zoom lens Equivalent performance 9 lenses reduced to 6 lenses Overall length reduced x x Photographic lens f = 53 mm, F# = 6.5 a) all spherical, 9 lenses Dy axis field 22 Dx Dy Dx 436 nm 588 nm 656 nm y p x p y p x p b) 3 aspheres, 6 lenses, shorter, better performance Dy axis field 22 Dx Dy Dx A 1 A 3 A 2 y p x p y p x p Ref: H. Zügge

13 GRIN lenses Gradient index (GRIN) lenses are using a spatially varying index of refraction Example: GRIN lenses with radial parabolic index profile 0.25 Pitch Object at infinity n r n = n 0 n 2 r² n Pitch Object at front surface r 0.75 Pitch Object at infinity 1.0 Pitch Object at front surface Pitch Such lenses are used, e.g., as relay lenses

14 Fresnel Lenses Fresnel lenses are refractive lenses with a surface structure They are used to reduce weight and length of optical systems active surfaces linear Significant aberrations used for illumination

15 Diffractive Optical Elements Diffractive optical elements (DOE s) are based on diffraction to redirect light refractive : one direction diffractive blazed : one direction diffractive binary : all directions +2. order +1. order 0. order -1. order n g g -2. order Different types: Fresnel zone plates (transmission or phase zones) Binary diffractive elements Computer generated diffractive elements Blazed diffractive element

16 Grating Diffraction Maximum intensity: constructive interference of the contributions of all periods grating Grating equation g sin sin m o grating constant g in-phase + 1. diffraction order Ds = incident light

17 DOE s for Chromatic Correction Example: blazed diffractive element For 1st order r k = 2π f λ k sinψ k = sinθ k n sinθ k = λ r k+1 r k One use of DOE s in optical systems is for color correction

18 Micro Lens Array Used for spot array generation or beam homogenization Ref: W. Osten

19 Use of Micro Lens Arrays for Illumination The aperture splitting of the lens array provides a plurality of parallel Köhler illumination systems Optics Express 18(20): September 2010

20 20 Mirrors Mirrors are based on reflection, typically off coated surfaces (dielectric/metal) The reflectivity but not the direction depend on the wavelength and polarization a) s-polarization b) p-polarization reflection E r B r E r reflection transmission B r transmission E t B t i i' B t i i' E t normal to the interface i normal to the interface i incidence E i B i n n' incidence interface B i E i n n' interface Reflectivity of silver In optical systems mirrors are used to redirect light and to control aberrations

21 21 Curved Spherical Mirrors Radius of curvature R Focal length R/2 On-axis imaging with spherical aberration Strong aberrations off-axis dependent on pupil stop position a) astigmatism A = 0 b) tangential flat stop Image P B Tplane S P S B C C R R/2 R c) best image flat d) sagittal flat P S B T P stop

22 22 Aspheric Surface: Conic Sections Explicite surface equation, resolved to z Parameters: curvature c = 1 / R conic parameter z 2 2 cx y c x 1 1 y Influence of on the surface shape z Parameter Surface shape = - 1 paraboloid < - 1 hyperboloid = 0 sphere > 0 oblate ellipsoid (disc) 0 > > - 1 prolate ellipsoid (cigar ) y x

23 Simple Asphere Parabolic Mirror Surface equation z x 2 2 R s y 2 Radius of curvature in vertex: R s Perfect imaging on axis for object at infinity Strong coma aberration for finite field angles Applications: 1. Astronomical telescopes 2. Collector in illumination systems axis w = 0 field w = 2 field w = 4

24 Simple Asphere Elliptical Mirror Equation z 1 Radius of curvature r in vertex, curvature c eccentricity 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 c( x 2 y 2 1 (1 )( x s' ) 2 y 2 ) c 2 F F'

25 Reflection Prisms Right angle prism 90 deflection Bauernfeind prism Beam deviation Penta prism 90 deflection Rhomboid prism Beam offset Dove prism image inversion

26 Properties of Reflection Prisms Functions 1. Bending of the beam path, deflection of the axial ray direction Application in instrumental optics and folded ray paths 2. Parallel off-set, lateral displacement of the axial ray 3. Modification of the image orientation (reversion, inversion) 4. Off-set of the image position, shift of image position forwards in the propagation direction. Aberrations introduced 1. Astigmatism 2. Chromatic aberration 3. Spherical aberration in non-collimated beams

27 Transformation of Image Orientation image reversion in the folding plane (upside down) image unchanged original Modification of the image orientation with four options: 1. Invariant image orientation 2. Reverted image ( side reversal ) 3. Reverted image ( upside down ) 4. Image inversion y image reversion perpendicular to the folding plane folding plane image inversion x mirror 1 y - z- folding plane y mirror 2 y x z x z

28 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

29 Application in Binoculars Double Porro Prism Abbe-König Roof Prism

30 Conical Light Taper Waveguide with conical boundary Lagrange invariant: decrease in diameter causes increase in angle: Aperture transformed D sin u D sin u' Number of reflections: - depends on diameter/length ratio - defines change of aperture angle in out D in / 2 Reflexion No j u' D out / 2 u r i n n L

31 31 Optical Fibers: Step-Index total Totalreflexion r refractive Brechzahlindex cladding Mantel a i core Kern z 1 n 2 n 1 n Index step Critical angle Numerical aperture = n 1 n 2 n 1 = θ c = cos 1 (n 2 /n 1 ) NA = sin θ a = n 1 sin θ c = n 1 ² n 2 ² θ a : acceptance angle

32 32 Optical Fibers: Step-Index V fiber parameter descibes number of guided modes 2 a V NA Number of modes: Requirement for single mode : V E(r) M V² 2 Approximation for fundamental mode: Gaussian beam 1 rule-of-thumb for width: w e a V 0.5 V = 3.0 V = 1.0 V = 2.2 V = r / a

33 Radial Index 33 Fundamental Mode of Step-Index Fibers Scalar solution of mode propagation with step-index boundary condition : Core : r < a E( r,, z) A J ( kr) e m im e iz Cladding : r > a E( r,, z) B K ( kr) e m im Mode matching is requirement for efficient in- and out-coupling A(r) Grundmode fundamental mode J 0 (kr) Anregungsmode J 3 (kr) K 0 (gr) K 3 (gr) r Kern core cladding Mantel Azimutal Index

34 34 Optical Fibers: Graded-Index Continuous index profile in core Rays follow curved trajectories with shorter paths compared to step-index fibers Number of modes: M V² 4 kontinuierliche continuous ray deflection Strahlbiegung r refractive Brechzahlindex cladding Mantel a Kern core z 1 n 2 n 1 n

35 Dispersion Prism Dispersion prism spatially separate light in its colors Blue light is refracted more strongly than red light ( normal dispersion) Application : spectrometer, dispersion control white weiß α For symmetric case D 2 arcsin n sin 2 dφ dλ = rot grün red blau green blue 2sin α/2 1 n 2 sin² α/2 dn dλ

36 Ideal Diffraction Grating Ideal diffraction grating: monochromatic incident collimated beam is decomposed into discrete sharp diffraction orders blue grating red Constructive interference of the contributions of all periodic cells Only two orders for sinusoidal grating constant g blue in-phase + 1. diffraction order sin sin m o / g Ds = incident light

37 Grating Equation Intensity of grating diffraction pattern Product of slit-diffraction and interference function Maxima of pattern: coincidence of peaks of both functions: grating equation g sin sin m 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 o I N s g 2 g 2 us Nug sin sin us ug N sin slit diffraction 2 interference function u = λ g u = λ s u = sin u s λ

38 Spectral Resolution of a Grating Angle dispersion of a grating d D d sin m sin 0 cos Separation of two spectral lines sin m sin 0 A L m N D m I(x) Complete setup with all orders: Overlap of spectra possible at higher orders 0 m /g D m(d /g sin

39 Blaze Grating Blaze grating (echelette): - facets with finite slope - additional phase shifts the slit diffraction function - all orders but one suppressed Blaze condition is only valid for - one wavelength - one incidence angle ψ = θ + θ 0 2 λ = 2g m B sin ψ cos θ 0 ψ slit diffraction working order suppressed orders m B -2 m B -1 m B m B +1 m B +2

40 40 Color Glass Filter Wavelength filtering based on selective absorption of light Insensitive to angle of incidence Heating due to absorption Limited efficiency Neutral density Bandpass Long Pass Short Pass Source: Schott AG

41 41 Dielectric Filters Wavelength filtering based on interference in multilayer systems E R0 E R1 E R2 Sensitive to angle of incidence Low absorption High efficiency n 0 Luft Air Design degrees of freedom increase with number of layers n 1 n 2 d 1 d 2 Layer 1 Layer 2 Example: Anti-reflexion coating For normal incidence Interference Condition n d 1 1 n2d2 / 4 n s Substrate Substrat Amplitude Condition n1 n 2 n n o s E T0

42 42 Interference Filters Wavelength filtering based on multi-path interference Highly sensitive to angle of incidence Low absorption Low efficiency High wavelength selection T R = R = R = R = R = d φ n Semi-reflective surface λ = d 2n cos φ I T 2 (1 R) 2 1 R 2R cos