Physical Optics Lecture : Diffraction 018-04-18 Herbert Gross www.iap.uni-jena.de
Physical Optics: Content No Date Subject Ref Detailed Content 1 11.04. Wave optics G Complex fields, wave equation, k-vectors, interference, light propagation, interferometry 18.04. Diffraction G Slit, grating, diffraction integral, diffraction in optical systems, point spread function, aberrations 3 5.04. Fourier optics G Plane wave expansion, resolution, image formation, transfer function, phase imaging 4 0.05. Quality criteria and Rayleigh and Marechal criteria, Strehl ratio, coherence effects, two-point G resolution resolution, criteria, contrast, axial resolution, CTF 5 09.05. Photon optics K Energy, momentum, time-energy uncertainty, photon statistics, fluorescence, Jablonski diagram, lifetime, quantum yield, FRET 6 16.05. Coherence K Temporal and spatial coherence, Young setup, propagation of coherence, speckle, OCT-principle 7 3.05. Polarization G Introduction, Jones formalism, Fresnel formulas, birefringence, components 8 30.05. Laser K Atomic transitions, principle, resonators, modes, laser types, Q-switch, pulses, power 9 06.06. Nonlinear optics K Basics of nonlinear optics, optical susceptibility, nd and 3rd order effects, CARS microscopy, photon imaging 10 13.06. PSF engineering G Apodization, superresolution, extended depth of focus, particle trapping, confocal PSF 11 0.06. Scattering L Introduction, surface scattering in systems, volume scattering models, calculation schemes, tissue models, Mie Scattering 1 7.06. Gaussian beams G Basic description, propagation through optical systems, aberrations 13 04.07. Generalized beams G Laguerre-Gaussian beams, phase singularities, Bessel beams, Airy 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 I( ) d sin sin d sin d 0. order Angles with side maxima: constructive interference 1 sin max m d 1st diffraction order order: -3rd -nd -1st 0th +1st +nd +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 forms the overall wave front (red) Light also enters the geometrical shadow region The vectorial superposition principle requires full coherence propagation direction stop Huygens- Wavelets wave fronts geometrical shadow
7 Diffraction at apertures Kirchhoff integral E( r') e it F e A( r)cos( q ) ik r' r r' r Calculation of field at any point df Huygens Principle Each point of a wavefront acts as starting point of spherical waves Their envelope forms a new wavefront r F r F assumption: Field within the aperture equals the field without it, no interaction Ref: B. Böhme
a 8 Slit and rectangular aperture Slit with infinite length I sin ~ x x where x a sin Rectangular aperture Constructive interference for difference / between rims of slit Square aperture sin( max m ) 0,5 m 1 a Ref: B. Böhme
9 Sinc-function Sinc-function sinus cardinalis slitfunction 1 Slit 4 sinx I ~ = [ sinc(x) ]² x where a x sin(4x) 1/(4x) -1 0 1 sin(4x) 4x sin(4x) [ 4x ]² -1 Slit 1 sinc(x) [sinc(x)]² Ref: B. Böhme
10 Diffraction at slit Transmission 1 Intensity at screen 0 - -1 0 1 Variable x @ Object 1 sin min 1 a rd d 0 - -1 0 1 Variable @ Screen The sinc-function is the fourier-transform of the slit function Ref: B. Böhme I( ) ~ a / a / cos( x ) dx sin( x / ) ~ a = width of slit = diffraction angle cos( x ) dx x e dx sin( x ) x e
g a 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 E ~ cos r u cos r d Destructive for path difference / min min sin( 1 ) sin( m ) g g m 1 Constructive interference for difference m max sin( ) m g Destructive for difference eg. between rim and center of aperture / min sin( 1 ) a sin( min m m ) a Constructive for difference / for rims sin( max m 1 ) a Ref: B. Böhme
Double Slit Interference Interference of laser light at a double slit setup Ref: W. Osten
g www.girlinclouds.worldpress.com 13 Double Slit and grating Grating 0 Ideal diffraction grating: collimated monochromatic incident beam is decomposed into discrete sharp diffraction orders Constructive interference as contribution of all periodic cells The number of orders depends on grating structure, for sinusoidal structure only two orders Constructive interference for difference m max sin( ) m g Arbitrary incidence angle 0 max sin( m ) sin( 0) m g Ref: B. Böhme
g www.girlinclouds.worldpress.com 14 Double Slit and grating Grating: ~ 0 Number of grating periods N g sin( ) sin N Sharpness of orders I( ) g sin( ) increases with N sin Grating resolution: separation of spectral lines m N 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. -3. -. -1. 0. +1. +. +3. +4. Ref: B. Böhme
Diffraction at grating Complex Field 15 1. Width of slits defines occurrence of orders. Number of slits defines fidelity 3. Period of grating defines distance of orders Ref: B. Böhme 1--3-4 1--3-4 5 5 Between the maxima are 5 periods The number increases with number of periods M
Diffraction at grating Intensity 16 1. Width of slits defines occurrence of orders. Number of slits defines fidelity 3. Period of grating defines distance of orders 1--3-4 1--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 Similar, but about 0% more spreaded distribution non-equidistant zeros 0 1 - -1 0 1 Variable r @ Object D A sin min 1 1. NA D The Airy-distribution follows from the D-Fourier-transform of the circular aperture 0 - -1 0 1 r J1 NA ( ) I r r NA NA n' sin u' Variable @ Screen Ref: B. Böhme
a = r a 18 Circular aperture D Airy Slit: first destructive interference at sin( Circular aperture no separation of coordinates Finite aperture causes finite spot size = Airy-diameter DA DA Ref: B. Böhme min 1, 1, sin( ) mim 1 NA distance = diameter of first dark rings min 1 ) a sin( min 1 More general: NA = n sin() ~ r / f ) 1, a f
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) geometrical optic (raytrace) paraxial waveoptic Ref: A. Herkommer paraxial optic calculation effort
0 Diffraction Integral General diffraction integral Follows from wave equation with Green's theorem Green's function G not unique Choice of G with spherical waves: Kirchhoff integral E( r ') G( r, r ') E( r) G( r, r ') E( r ') df' n n F 1 G( r, r ') e 4 rr ' i E ( r) F E( r ') e r r ' o ik rr ' ik S ( r ') r r ' cosq cosq o i d df'
1 Approximation of the Diffraction Integral Kirchhoff integral i E( r) F AP ik r r ' e E( r ') r r' df Fresnel integral: Phase quadratic approximated (convolution expression) r r ' ( x x ') ( y y ') z ie E( x, y, z) z ikz ( x x ') ( y y ') z... z z E( x', y',0) e i ( xx') ( y y') z dxdy ' ' Fraunhofer integral: far-field approximation for large distances or convergent waves into focal region, x << x' (Fourier integral) r r ' z x xx ' x ' y yy ' y ' z z z z z z z x ' y ' xx ' yy' z z ikz i ' ' i ie x y xx ' yy ' z z E( x ', y', z ) e E( x, y,0) e dx dy z
Fraunhofer Point Spread Function Rayleigh-Sommerfeld diffraction integral, Mathematical formulation of the Huygens-principle ik r r ' i e EI ( r) E( r') cosq ddx' dy' r r' Fraunhofer approximation in the far field for large Fresnel number N F r p z 1 Optical systems: - pupil amplitude/transmission/illumination T(x p,y p ) - wave aberration W(x p,y p ) - complex pupil function A(x p,y p ) - transition from exit pupil to image plane E( x', y' ) Point spread function (PSF): Fourier transform of the complex exit pupil function AP iw ( x p, y p A( x, y ) T ( x, y ) e T p p p p i x px' y p y' iw x p, y p RAP x, y e e p p ) dx p dy p
3 Approximation of the Diffraction Integral Kirchhoff integral Fresnel integral: z E( x, y, z) E( x', y',0) e - phase quadratic approximated z - 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: abberated field can be calculated Fraunhofer integral: - far-field approximation for large distances in case of plane waves - quasi far field in case of convergent fields observed near the focal plane i E( r) F AP ie e E( r ') r r' ikz ik r r ' df i ( xx') ( y y') dxdy ' ' ikz i ' ' i ie x y xx ' yy ' z z E( x ', y', z ) e E( x, y,0) e dx dy z y all wavelets in phase wave front initial plane final plane: in focus
4 Fraunhofer Point Spread Function Rayleigh-Sommerfeld diffraction integral, Mathematical formulation of the Huygens-principle ik r r ' i e EI ( r) E( r') cosq ddx' dy' r r' Fraunhofer approximation in the far field for large Fresnel number N F r p z 1 Optical systems: numerical aperture NA in image space Pupil amplitude/transmission/illumination T(x p,y p ) Wave aberration W(x p,y p ) complex pupil function A(x p,y p ) Transition from exit pupil to image plane E( x', y' ) Point spread function (PSF): Fourier transform of the complex pupil function iw ( x p, y p A( x, y ) T ( x, y ) e p p AP T p i x px' y p y' iw x p, y p RAP x, y e e p p p ) dx p dy p 1
5 Angular Spectrum Plane wave ik x k y E( x, y, z) A( x, y, z) e Wave number k k x k y k z n k z x y z x z x n/ z Spatial frequence: re-scaling of k k z n x y E( x, y, z) Ae i x y z x y z Fourier transform to get the plane wave spectrum ix x y y z z E x y z E e d d d,,,, x y z x y z
6 Propagation of Plane Waves Phase of a plane wave x z e i e i zncos e i z z e i z n x y h x, y ; z The spectral component is simply multiplied by a phase factor in during propagation i z x, y, 1 x,, y 0, ;,, 0 z x y x y E z E z e h z E z z z 0 1 z the function h is the phase function Back-transforming this into the spatial domain: Propagation corresponds to a convolution with the impulse response function,,, ;,, E x y z H x y z E x y z H 1 0 i xx yy h x, y ; z e d x x, y; z d y Fresnel approximation for propagation: U P 1 i z x z y 1 1... x y x x y i xx yy 1 i 0 0 0 0 x, y; z e Ux0, y 0;0 e z e z dx 0dy iz y 0
7 Propagation by Plane / Spherical Waves Expansion field in simple-to-propagate waves 1. Spherical waves. Plane waves Huygens principle ik r r ' e E( r ') E( r ) d r r ' r x' spectral representation Fourier approach E( r ') Fˆ 1 xy e ik z z Fˆ xy E( r ) x' x x e ikr r e ik z z E(x) E(x) z z
8 Diffraction Ranges Different ranges of edge or slit diffraction as a function of the Fresnel number / distance: 1. Far field Fraunhofer weak structure small N F 1. Fresnel Quadratic approximation of phase ripple structure medium N F > 1 far field Fraunhofer Fresnel regime z [mm] N F = 1 N F = 5 N F = 0 N F = 100 intensity near left edge 3. Near field Behind slit large N F >> 1 N F = 1000 N F = 10000 near field slit coherent plane wave
Typical change of the intensity profile Normalized coordinates Diffraction integral 9 Fresnel Diffraction z geometrical focus f a z stop far zone geometrical phase intensity a r r f a v z f a u ; ; 1 0 0 / 0 ) ( ), ( d e v J e f E ia v u E u i u a f i
30 Geometrical vs Diffraction Ranges Focussed beam with various wavelengths Focal region given by Rayleigh length R u = /NA Geometrical cone outside focal region Diffraction structure in focal range A 1/ = 1 mm =.5 mm = 5 mm = 10 mm geometrical range diffraction focal range
31 Sampling of the Diffraction Integral Fresnel integral: phase quadratic approximated Sampling of the initial field: curvature, W(x,y) ie E( x, y, z) z ikz E( x', y',0) e i ( xx') ( y y') z dxdy ' ' Sampling of the propagator: different distances of points r 1 -r Nearly ideal: compensation of wave sag and distance differences: quasi flattening sag W(x) x P 1 (x) max r 1 x point spread function min r 1 light cone P (x ) wave front initial plane calculation channel final plane
3 Sampling of the Diffraction Integral Oscillating exponent : Fourier transform reduces on period Most critical sampling usually at boundary defines number of sampling points Steep phase gradients define the sampling High order aberrations are a problem 50 40 30 0 phase quadratic phase 10 wrapped phase 0-6 -4-0 4 x smallest sampling intervall
33 Fresnel Edge Diffraction Diffraction at an edge in Fresnel approximation Intensity distribution, Fresnel integrals C(x) and S(x) 1 1 1 I( t) C( t) S( t) scaled argument 1.5 1 0.5 I(t) k t x x z z N F 0-4 - 0 4 6 t Intensity: - at the geometrical shadow edge: 0.5 - shadow region: smooth profile - bright region: oscillations 7
34 Fresnel Diffraction Phase behaviour at edge diffraction Oscillation of phase in the range of the intensity ripples
35 Fresnel Number Fresnel number : critical number for separation of approximation ranges Setup with two stops in distance L N F a1a L Fraunhofer diffraction, farfield, Dominant effect of diffraction: N F < 1 start plane stop a 1 a L sagittal height p final plane stop Fresnel diffraction with considerable influence of diffraction: N F = 1 P spherical wave from axis point P Geometrical-optical range witj neglectable diffraction, near field : N F >> 1
Diffraction at the System Aperture 36 Self luminous points: emission of spherical waves Optical system: only a limited solid angle is propagated, the truncaton of the spherical wave results in a finite angle light cone In the image space: uncomplete constructive interference of partial waves, the image point is spreaded The optical systems works as a low pass filter spherical wave truncated spherical wave point spread function object point x = 1. / NA object plane image plane
37 Hybrid model y y p optical y' p y' system stop image image point rays z real wave front object point ideal reference sphere Object plane Op Entrance pupil EnP Exit plane ExP Image plane Ip
38 Diffraction inside a system Usually neglected: - diffraction inside system with impact on phase and amplitude in exit pupil - mostly extremly good approximation Special cases: - beam cleanup telescope - very large propagation distances in laser systems stop diffraction exit pupil a spreading and intensity redistribution diffraction angle q = / a
39 PSF by Huygens Principle Huygens wavelets correspond to vectorial field components: - represented by a small arrow - the phase is represented by the direction - the amplitude is represented by the length Zeros in the diffraction pattern: destructive interference Ideal point spread function: point spread function zero intensity closed loop single wavelets sum central peak maximum constructive interference wave front pupil stop side lobe peak 1 ½ round trips
40 PSF by Huygens Principle Apodization: variable lengths of arrows r p point spread function homogeneous pupil: same length of all arrows I(x p ) apodization: decreasing length of arrows wave front pupil stop Aberrations: variable orientation of arrows real wave front point spread function real wavefront with aberrations central peak reduced ideal spherical wavefront central peak maximum ideal wave front pupil stop
41 Perfect Point Spread Function Circular homogeneous illuminated aperture: Transverse intensity: Airy distribution v I Dimension: D Airy v normalized lateral coordinate: D Airy 1. NA v = x / NA Axial intensity: sinc-function Dimension: Rayleigh unit R E normalized axial coordinate u = z n / NA J1 0, v I0 sin u / 4 Iu, 0 I u / 4 R E n NA 0 z r image plane optical axis Airy lateral 1,0 0,8 0,6 vertical lateral Rayleigh axial 0,4 0, aperture cone 0,0-5 -0-15 -10-5 0 5 10 15 0 5 u / v
4 Perfect Lateral Point Spread Function: Airy Airy distribution: Gray scale picture Zeros non-equidistant Logarithmic scale Encircled energy 10 10 10 10 10 0-1 - -3-4 log I(r) 10-5 10-6 0 5 10 15 0 5 30 E circ (r) 1 r D Airy 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0. 0.1 3. ring 1.48%. ring.79% 1. ring 7.6% peak 83.8% 0 0 1 3 4 5 1.831.655 3.477 r / r Airy
43 Axial and Lateral Ideal Point Spread Function Comparison of both cross sections Ref: R. Hambach
44 Annular Ring Pupil Generation of Bessel beams r p r z Ref: R. Hambach
45 Spherical Aberration Axial asymmetrical distribution off axis Peak moves Ref: R. Hambach
46 Gaussian Illumination Known profile of gaussian beams Ref: R. Hambach
47 Defocussed Perfect Psf Perfect point spread function with defocus Representation with constant energy: extreme large dynamic changes z = -R E z = -1R E focus z = +1R E z = +R E normalized intensity I max = 5.1% I max = 9.8% I max = 4% constant energy
48 Perfect Axial Point Spread Function Axial distribution of intensity Corresponds to defocus Normalized axial coordinate NA z u z 4 Scale for depth of focus : Rayleigh length I(z) 1 0.9 0.8 0.7 0.6 z sin sin u / 4 I( z) I0 Io z u / 4 R E n' sin u' n' NA 0.5 0.4 0.3 Zero crossing points: equidistant and symmetric, 0. 0.1 Distance zeros around image plane 4R E 0-4 -3 - -1 0 1 3 4 z/ R E z = R E 4R E
Deblurring of PSF and Strehl Ratio 49 reference sphere corresponds to perfect imaging RMS deviation as an integral measure of performance wave aberration pv-value of wave aberration image plane 1 W ( x, y) W ( x, y) dxdy 0 F Strehl ratio SR ExP Approximations SR A( x, y) e I I ( real ) PSF ( ideal ) PSF iw ( x, y) A( x, y) dxdy 0,0 0,0 dxdy phase front exit aperture reference sphere peak reduced Strehl ratio I( x ) 1 ideal, without aberrations Marechal useful SR >0.5 Biquadratic Exponential SR SR M B SR E e 1 4 Wrms W 1 rms 4 W rms distribution broadened real with aberrations r
Zernike Polynomials: Measure of Wave Aberrations 50 describe deviation from reference sphere with orthogonal basis functions circular shape Zernike Polynomials: conventions (amount/sign) due to Gram-Smith-Orthogonalization m W (, ) c nm Z n (, ) Z m n n (, ) ~ R n mn m n sin ( m) for m 0 ( ) cos ( m) for m 0 1 for m 0 Peak-Valey (PV)-values at pupil rim = +1 Fringe Constant RMS-value (Orthonormal) Standard m = + 8 + 7 + 5 + 4 + 3 + Z,7,14 + 1 Z 4,9,16 0 Z 3,8,15-1 - - 3-4 cos SR M 1 4 c nm k n n1 mn Circular obscuration Zernike-Tatian polynomials - 5-6 - 7-8 sin Rectangular pupil Legendre polynomials n = 0 1 3 4 5 6 7 8
Zernike Fringe vs Zernike Standard Polynomials 51 In radial symmetric system for y-field (meridional) sine-terms vanish Fringe coefficients: Z 4,9,16 = n² spherical azimutal order f grows if number increases Standard coefficients - different term numbers Z N = 0.05 RMS = 0.05 SR ~ 1 40 Z N ² = 0.9 Elimination of tilt No Elimination of defocus @ Zemax
Spherical Aberration 49 Single positive lens shorter intersection length for marginal rays undercorrected on axis, no field dependence, circular symmetry Increase of spherical aberration growing axial asymmetry around the nominal image plane paraxial focus perfect symmetry best image plane circle of least RMS, best contrast at special frequency, Z4 = 0,... c 9 = 0 c 9 = 0.3 Example: i c 9 = 0.7 Focus a collimated beam with plano-convex lens n = 1.5 Paraxial focus worse (red) vs. best (green) orientation c 9 = 1 spherical aberration i differs by a factor of 4: i1 sin i i 3 i 6 i sin i 3 i 6 3 i i 4 5
Coma: Spot Construction 50 Term B (+cosf) y asymmetric ray path for non-axial object point construction of spot circles from constant pupil zones: circle radius ~ radius at pupil² 360 @ pupil x360 at image comet shape spot ray @ interval [1 3] aspect ratio :3 circle radius ~ ² angle 60 tangential marginal rays central ray C S ~ C T / 3 Tangential coma C T Saggital coma Cs sagittal coma ~ 1/3 of tangential coma saggital marginal rays Coma PSF correspods to spot 55% of energy in the triangle between tip of spot and saggital coma Separation of peak and the centroid position different image position for center of gravity From the energetic point of view coma induces distortion in the image centroid c 7 = 0.3 c 7 = 0.5 c 7 = 1 53
Astigmatism & Petzval 51 Term (3B3+B4) y² cosf For a single positive lens: chief ray passes surface under oblique angle projection of surface curvatures Tangential ray fan T B S par different powers in tangential and sagittal tangential (blue) and sagittal (red) focal lines Sequence: tangential - circle of least confusion Sagittal ray fan with smallest spots (best) sagittal paraxial Imaging of a circular grid in different planes Tangential focus Circle of least confusion Sagittal focus tangential focus best focus sagittal focus Astigmatism B3 corrected one curved image shell with Petzval curvature B4 Single positive lens image curved toward the system negative Petzval = sign convention In general Petzval image s = ½ ( 3s S -s T ) departs from T, S and Best, is defined @ math 54
55 Psf with Aberrations Psf for some low oder Zernike coefficients The coefficients are changed between c j = 0...0.7 The peak intensities are renormalized trefoil coma 5. order astigmatism 5. order spherical 5. order c = 0.0 c = 0.1 c = 0. c = 0.3 c = 0.4 c = 0.5 c = 0.7 coma astigmatism spherical defocus