Physical Optics Lecture 2: Diffraction 2018-04-18 Herbert Gross www.iap.uni-jena.de 2 Physical Optics: Content No Date Subject Ref Detailed Content Complex fields, wave equation, k-vectors, interference, light propagation, 1 11.04. Wave optics G interferometry Slit, grating, diffraction integral, diffraction in optical systems, point spread 2 18.04. Diffraction G function, aberrations Plane wave expansion, resolution, image formation, transfer function, 3 25.04. Fourier optics G phase imaging Quality criteria and Rayleigh and Marechal criteria, Strehl ratio, coherence effects, two-point 4 02.05. G resolution resolution, criteria, contrast, axial resolution, CTF Energy, momentum, time-energy uncertainty, photon statistics, 5 09.05. Photon optics K fluorescence, Jablonski diagram, lifetime, quantum yield, FRET Temporal and spatial coherence, Young setup, propagation of coherence, 6 16.05. Coherence K speckle, OCT-principle Introduction, Jones formalism, Fresnel formulas, birefringence, 7 23.05. Polarization G components Atomic transitions, principle, resonators, modes, laser types, Q-switch, 8 30.05. Laser K pulses, power Basics of nonlinear optics, optical susceptibility, 2nd and 3rd order effects, 9 06.06. Nonlinear optics K CARS microscopy, 2 photon imaging Apodization, superresolution, extended depth of focus, particle trapping, 10 13.06. PSF engineering G confocal PSF Introduction, surface scattering in systems, volume scattering models, 11 20.06. Scattering L calculation schemes, tissue models, Mie Scattering 12 27.06. Gaussian beams G Basic description, propagation through optical systems, aberrations Laguerre-Gaussian beams, phase singularities, Bessel beams, Airy 13 04.07. Generalized beams G beams, applications in superresolution microscopy 14 11.07. Miscellaneous G Coatings, diffractive optics, fibers K = Kempe G = Gross L = Lu 3 Contents . Introduction . Slit diffraction . Grating diffraction . Diffraction integral . Diffraction in optical systems . Point spread function . Aberrations Fraunhofer Farfield Diffraction . Diffraction at a slit: intensity distribution 2 d sin 0. order sin d I() d sin . Angles with side maxima: constructive interference 1st diffraction 1 order sinmax m d 2 order: -3rd -2nd -1st 0th +1st +2nd +3rd . Diffraction pattern Diffraction at a Slit . Equation for the minima : m = asinq . Finite detection with a lens 4 Huygens Principle . Every point on a wave is the origin of a new spherical wave (green) . The envelope of all Huygens wavelets stop forms the overall wave front (red) Huygens- . Light also enters the geometrical shadow Wavelets region . The vectorial superposition principle propagation requires full coherence direction wave fronts geometrical shadow 7 Diffraction at apertures Kirchhoff integral Huygens Principle eik r'r E(r') eit A(r)cos(q ) dF Each point of a wavefront acts as F r'r starting point of spherical waves Their envelope forms a new wavefront Calculation of field at any point r‘ F r F assumption: Field within the aperture equals the field without it, no interaction Ref: B. Böhme 8 Slit and rectangular aperture Slit with infinite length 2 sinx a sin I ~ where x x Rectangular aperture a Square Constructive interference for aperture difference /2 between rims of slit sin( max ) 0,5 2m 1 m a Ref: B. Böhme 9 Sinc-function Sinc-function 1 . sinus cardinalis Slit 4 . „slitfunction“ sin(4x) sin(4x) 2 4x sinx 0 I ~ = [ sinc(x) ]² -1 1 x |1/(4x)| sin(4x) [ 4 x ]² a where x -1 Slit 1 sinc(x) [sinc(x)]² Ref: B. Böhme 10 Diffraction at slit Transmission 1 Variable x @ Object Intensity at screen 0 -2 -1 0 1 1 r sin min d 1 a d Variable @ Screen 0 -2 -1 0 1 2 sin(x) The sinc-function is the cos(x)dx a / 2 2 2 fourier-transform sin(x / 2) I() ~ cos(x)dx ~ ex of the slit function ex dx a / 2 a = width of slit = diffraction angle Ref: B. Böhme 11 Double Slit or slit with infinite length Intensity after double slit with distance g or one slit with width a with screen far behind Constructive interference on axis 2 2 E ~ cos ru cos rd a g Destructive for difference eg. between rim Destructive for path difference /2 and center of aperture /2 min min min 2 min m sin(1 ) sin( ) 2m 1 sin( ) sin( ) 2g m 2g 1 2 a m a Constructive interference for difference Constructive for difference /2 for rims max max 1 sin( m ) m sin( ) g m 2 a Ref: B. Böhme Double Slit Interference . Interference of laser light at a double slit setup Ref: W. Osten 13 Double Slit and grating Grating . Ideal diffraction grating: collimated monochromatic incident beam is decomposed into discrete sharp diffraction orders 0 . Constructive interference as contribution of all periodic cells g www.girlinclouds.worldpress.com . The number of orders depends on grating structure, for sinusoidal structure only two orders Constructive interference for difference Arbitrary incidence angle 0 max max sin( m ) m sin( ) sin( ) m g m 0 g Ref: B. Böhme 14 Double Slit and grating Grating: ~ Number of grating periods N g sin() sin 2 N . Sharpness of orders I() g sin() increases with N sin 2 0 m N . Grating resolution: separation of 2 spectral lines g www.girlinclouds.worldpress.com Number of orders m . depend on grating fine structure . for sinusodial structure only two orders . Blaze/echelette grating has facets with finite slope all orders but one higher suppressed . Complete setup with all orders: Overlap of spectra at higher orders possible -4. +4. -3. +3. -2. -1. +1. +2. 0. Ref: B. Böhme 15 Diffraction at grating – Complex Field 1. Width of slits defines occurrence of orders 2. Number of slits defines fidelity 3. Period of grating defines distance of orders 1-2-3-4 1-2-3-4 5 5 Between the maxima are 5 periods – The number increases with number of periods M Ref: B. Böhme 16 Diffraction at grating – Intensity 1. Width of slits defines occurrence of orders 2. Number of slits defines fidelity 3. Period of grating defines distance of orders 1-2-3-4 1-2-3-4 5 5 Between the maxima are 5 periods – The number increases with number of periods M Ref: B. Böhme 17 Diffraction at circular aperture Transmission 1 Intensity at screen 0 -2 -1 0 1 Variable r @ Object Similar, but 1 about 20% more spreaded distribution non-equidistant zeros DAiry min 1.22 sin1 NA D 0 -2 -1 0 1 2 Variable The Airy-distribution 2 @ Screen 2r follows from the 2 J1 NA 2D-Fourier-transform I(r) NA n'sin u' 2r of the circular aperture NA Ref: B. Böhme 18 Circular aperture a D Airy . Slit: first destructive interference at sin( min ) 1 a . Circular aperture – no separation of coordinates sin( min ) 1,22 1 a . Finite aperture causes finite spot size = Airy-diameter DA More general: DAmin 1,22 mim 1,22 sin(1 ) NA NA = n sin() ~ r / f a = 2r f distance = diameter of first dark rings Ref: B. Böhme 19 Model depth of Light Propagation . Different levels of modelling in optical propagation . Schematical illustration (not to scale) accuracy rigorous waveoptic vectorial waveoptic scalar waveoptic (high NA) paraxial geometrical waveoptic optic (raytrace) paraxial optic calculation effort Ref: A. Herkommer 20 Diffraction Integral . General diffraction integral Follows from wave equation with E(r') G(r,r') Green's theorem E(r) G(r,r') E(r') dF' Green's function G not unique F no no 1 ik rr ' . Choice of G with spherical waves: G(r,r') e 4 rr' . Kirchhoff integral i E(r') E(r) e ikS (r ') r r ' cosq cosq dF' i d 2 F r r' 21 Approximation of the Diffraction Integral i e ik r r ' . Kirchhoff integral E(r) E(r') dF r r' FAP . Fresnel integral: r r' ( x x ')2 ( y y ') 2 z 2 Phase quadratic approximated (x x ')22 ( y y ') (convolution expression) z .... 22zz i ieikz (xx')2 ( y y')2 E(x, y, z) E(x', y',0)e z dx'dy' z x2 xx'''' x 2 y 2 yy y 2 . Fraunhofer integral: r r' z far-field approximation for large 2z z 2 z 2 z z 2 z distances or convergent waves into x'22 y ' xx ' yy' z focal region, x << x' 2zz (Fourier integral) ii2 ieikz x''''22 y xx yy E( x ', y', z ) ezz E ( x , y,0) e dx dy z 22 Fraunhofer Point Spread Function . Rayleigh-Sommerfeld diffraction integral, i eik r r ' Mathematical formulation of the Huygens-principle E (r) E(r') cosq dx'dy' I r r' d . Fraunhofer approximation in the far field r 2 N p 1 for large Fresnel number F z . Optical systems: - pupil amplitude/transmission/illumination T(xp,yp) - wave aberration W(xp,yp) 2iW ( x p ,y p ) - complex pupil function A(xp,yp) A(xp , y p ) T (xp , y p ) e - transition from exit pupil to 2i x x' y y' image plane 2iW x ,y p p E(x', y') T x , y e p p eRAP dx dy p p p p AP . Point spread function (PSF): Fourier transform of the complex exit pupil function 23 Approximation of the Diffraction Integral i e ik r r ' . Kirchhoff integral E(r) E(r') dF r r' FAP . Fresnel integral: - phase quadratic approximated - 1/z is approximated to be constant - corresponds to the paraxial approximation of a parabolic phase of the propagator only - the inital field E(x',y') can have higher order phase contributions: i abberated field can be calculated ieikz (xx')2 ( y y')2 E(x, y, z) E(x', y',0)e z dx'dy' . Fraunhofer integral: z - far-field approximation for large distances in case of plane waves y - quasi far field in case of convergent fields observed near the focal plane all wavelets in phase ii2 ieikz x''''22 y xx yy E( x ', y', z ) ezz E ( x , y,0) e dx dy z final plane: wave in focus front initial plane 24 Fraunhofer Point Spread Function .