IMPRS: Ultrafast Source Technologies
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1 IMPRS: Ultrafast Source Technologies Fran X. Kärtner & Thorsten Uphues, Bldg. 99, O3.097 & Room 6/3 & phone: Lectures: Tuesday Geb 99, Sem Raum IV (1.OG) Thursday Geb 99, Sem Raum IV (1.OG) Start: Course Secretary: Christine Berber O3.095, phone x-6351, Class website: teaching/ imprs_summer_semester_014/lecture_notes 1
2 General Information Prerequisites: Basic course in electrodynamics Text: Class notes will be distributed in class. Grade: None Recommended Texts: Fundamentals of Photonics, B.E.A. Saleh and M.C. Teich, Wiley, Ultrafast Optics, A. M. Weiner, Hoboken, NJ, Wiley 009. Ultrashort Laser Pulse Phenomena, Diels and Rudolph, Elsevier/Academic Press, 006 Optics, Hecht and Zajac, Addison and Wesley Publishing Co., Principles of Lasers, O. Svelto, Plenum Press, NY, Waves and Fields in Optoelectronics, H. A. Haus, Prentice Hall, NJ, Gratings, Mirrors and Slits: Beamline Design for Soft X-Ray Synchrotron Radiation Sources, W. B. Peatman, Gordon and Breach Science Publishers, Soft X-ray and Extreme Ultraviolet Radiation, David Attwood, Cambridge University Press, 1999
3 Schedule Lecturer Topic Time Location Kärtner Classical Optics Tuesday Feb. 18, 9-11am Uphues X-ray Optics Thursday Feb. 0, 9-11am Kärtner Ultrafast Lasers Tuesday Feb. 5, 9-11am Uphues Kärtner Uphues Synchrotron Radiation Electron Sources and Accelerators High Order Harmonic Gen. Thursday Feb. 7, 9-11am Tuesday Mar. 04, 9-11am Thursday Mar. 06, 9-11am Geb. 99 Raum IV, 1.OG Geb. 99 Raum IV, 1.OG Geb. 99 Raum IV, 1.OG Geb. 99 Raum IV, 1.OG Geb. 99 Raum IV, 1.OG Geb. 99 Raum IV, 1.OG 3
4 Classical Optics Classical Optics.1 Maxwell s Equations and Helmholt Equation.1.1 Helmholt Equation.1. Plane-Wave Solutions (TEM-Waves) and Complex Notation.1.3 Poynting Vectors, Energy Density and Intensity. Paraxial Wave Equation.3 Gaussian Beams.4 Ray Propagation.5 Gaussian Beam Propagation.6 Optical Resonators 4
5 . Classical Optics.1 Maxwell s Equations of Isotropic Media Maxwell s Equations: Differential Form Ampere s Law Current due to free charges Faraday s Law Gauss s Law No magnetic charge Free charge density Material Equations: Bring Life into Maxwell s Equations Polariation Magnetiation 5
6 Classical Optics Vector Identity: Vacuum speed of light: 6
7 Classical Optics No free charges, No currents from free charges, Non magnetic Every field can be written as the sum of tansverse and longitudinal fields: Only free charges create a longitudinal electric field: Pure radiation field Simplified wave equation: Wave in vacuum Source term 7
8 .1.1 Helmholt Equation Linear, local medium dielectric susceptibility Medium speed of light: with Refractive Index 8
9 .1. Plane-Wave Solutions (TEM-Waves) and Compl. Notation Real field: Artificial, complex field: Into wave equation (.16): Dispersion relation: Wavelength 9
10 TEM-Waves 10
11 What about the magnetic field? TEM-Waves Faraday s Law: 11
12 Characteristic Impedance Vacuum Impedance: r 1 ( ) Example: EM-Wave polaried along x-axis and propagation along -direction: 1
13 Backwards Traveling Wave Backwards 13
14 .1.3 Poynting Vector, Energy Density and Intensity relative permittivity: r 1 Example: Plane Wave: 14
15 A plane wave is described by: ~ E( x, y, ). Paraxial Wave Equation e x ~ E 0 exp jk x x jk y y jk k 0 kx k y k free space wavenumber Although a plane wave is a useful model, strictly speaking it cannot be realied in practice, because it fills whole space and carries infinite energy. Maxwell s equations are linear, so a sum of solutions is also a solution. An arbitrary beam can be can be formed as a superposition of multiple plane waves: ~ E( x, y, ) ~ E 0 exp jk xx jk y y jk dkxdk y there is no integral over k because once k x and k y are fixed, k is constrained by dispersion relation k 0 kx k y k 15
16 Paraxial Beam Consider a beam which consists of plane waves propagating at small angles with axis such a beam y is called a paraxial beam x k kx k k y k Figure.: Construction of paraxial beam by superimposing many plane waves with a dominant k-component in -direction k can be approximated as (paraxial approximation) 16
17 Paraxial Diffraction Integral The beam ~ E( x, y, ) ~ E 0 ( k x, k y ) exp jk xx jk y y jk dkxdk y can then be expressed as ~ jk 0 E( x, y, ) e ~ E 0 ( k x, k y )exp k j x k k 0 y jk x x jk y y dk x dk y quickly varying carrier wave slowly varying envelope (changing slowly along ) Allows to find field distribution at any point in space. The beam profile is changing as it propagates in free space. This is called diffraction. 17
18 Paraxial Diffraction Integral Finding Field at Arbitrary Given the field at some plane (e.g. =0), how to find the field at any? From the previous slide ~ jk 0 E( x, y, ) At =0 e ~ E( x, y, 0) ~ E 0 ( k x, k y )exp ~ E 0 ( k x k j, k y x k k 0 ) exp y jk x x jk y y dk jk xx jk y y dkxdk y x dk y This is Fourier integral. Using Fourier transforms: ~ E 0 ( k x, k y ) 1 ( ) ~ E( x, y, 0) exp jk x x jk y ydxdy ~ ~ ~ Knowing E( x, y, 0) can find E ( k x, k ) and then E( x, y, ) at any. 0 y 18
19 .3 Gaussian Beam Choose transverse Gaussian distribution of plane wave amplitudes: (Via Fourier transforms, this is equivalent to choosing Gaussian transversal field distribution at =0) Substitution into paraxial diffraction integral gives with called Rayleigh range. Performing the integration (i.e. taking Fourier transforms of a Gaussian) 19
20 0 The quantity can be expressed as ) ( ) ( 1 1 w j R j j R R R ) /( 1 R j ) ( 1 R R ) ( R R w where R() and w() are introduced according to The pre-factor can be expressed as /( R ) j j )) ( exp( ) ( 1 )) ( exp( j w j j j R R R with R / ) ( tan ) ( ) ( ) ( exp ) ( 1 ~ ), ( ~ 0 0 j R r jk w r w r E We get Gaussian Beam
21 1 Finally, the field is normalied, giving ) ( ) ( ) ( exp ) ( 1 4 ), ( ~ j R r jk w r w P Z r E F Field of a Gaussian beam Normaliation means that the total power carried by the beam is P: Final Expression for Gaussian Beam Field
22 Gaussian Beams: Spot Sie and Rayleigh Range We derived ~ 4ZF 0P 1 r r E ( r, ) exp jk0 j ( w( ) w ( ) R( ) 0 ) Field of a Gaussian beam We have Defines transversal intensity distribution w R ( ) R I r r, ) ~ exp w ( ) ( w ( ) R 1 R w() spot sie Spot sie at =0 is min w ( 0) R w R w 0 0 Rayleigh range w ( ) 0 1 w R spot sie as a function of
23 We derived Gaussian Beams: Phase Distribution ~ 4ZF 0P 1 r r E ( r, ) exp jk0 j ( w( ) w ( ) R( ) 0 ) Field of a Gaussian beam We have 1 R( ) R R( ) 1 defines phase distribution R Radius of wavefront curvature Phase fronts (surfaces of constant phase) are parabolic, can be approximated as spherical for small r. R() turns out to be the radius of the sphere. ( ) arctan / R Guoy phase shift Defines extra phase shift as compared to plane wave 3
24 Gaussian Beams: All Equations in One Slide ~ 4ZF 0P 1 r r E ( r, ) exp jk0 j ( w( ) w ( ) R( ) 0 ) Gaussian Beam Field w ( ) 0 1 w R R( ) 1 R Spot sie w 0 min spot sie (=0) Radius of wavefront curvature (.54) (.55) ( ) arctan R Guoy phase shift (.65) R w 0 Rayleigh range (.56) 4
25 Intensity Distribution =0 = R = R Figure.3: The normalied beam intensity I/I₀ as a function of the radial distance r at different axial distances: (a) =0, (b) = R (c) = R. 5
26 Spot Sie as a Function of ( ) 0 1 w R Spot sie w w( R ) = w 0 Waist: minimum radius At = R R Rayleigh range : distance from the waist at which the spot area doubles 4 4 waist w( ) w( ) w0 0 0 w() w 0 w R R 3 R R.R R R 6
27 Divergence Angle and Confocal Parameter For large w( ) w0 1 w0 R R Divergence angle w w( ) 0 R Beam Divergence: Smaller spot sie larger divergence Confocal parameter and depth of focus: Figure.5: Gaussian beam and its characteristics 7
28 Axial Intensity Distribution r=0 Intensity distribution along Z axis (r=0) Figure.4: Normalied beam intensity I(r=0)/I₀ on the beam axis as a function of propagation distance. 8
29 Power Confinement Which fraction of the total power is confined within radius r 0 from the axis? Dependence is exponential 99% of the power is within radius of 1.5 w() from the axis 9
30 Wavefront Wavefronts, i.e. surfaces of constant phase, are parabolic constant Figure.8: Wavefronts of Gaussian beam 30
31 Wavefront and Radius of Curvature Wavefront radius of curvature: R( ) 1 R Figure.7: Radius of curvature R() 31
32 Comparison to Plane and Spherical Waves Figure.9: Wave fronts of (a) a uniform plane wave, (b) a spherical wave; (c) a Gaussian beam. 3
33 Guoy Phase Shift Phase delay of Gaussian Beam, Guoy-Phase Shift: Figure.6: Phase delay of Gaussian beam, Guoy-Phase-Shift 33
34 Gaussian Beams: Summary Solution of wave equation in paraxial approximation Wave confined in space and with finite amount of power Intensity distribution in any cross-section has the same shape (Gaussian), only sie and magnitude is scaled At the waist, the spot sie is minimum and wave fronts are flat Lasers are usually built to generate Gaussian beams Planes of const. Phase Beam Waist / R 34
35 .4 Ray Propagation 1 r 1 Optical r 1 System r r Z Figure.10: Description of optical ray propagation by its distance and inclination from the optical axis. 35
36 Ray Matrix: Transition medium 1 to 1 n 1 n r 1 1 r 1 r r Figure.11: Snell s law for paraxial rays. Z æ M = ç è ö ø 36
37 Ray Matrix: Propagation over Distance L 1 r r 1 r r 1 L Z Figure.1: Free space propagation 37
38 Prop. in Medium with Index n over Distance L 1 r r 1 r r 1 L Z Figure.13: Ray propagation through a medium with refractive index n. 38
39 Thin Plano-convex Lens r 1 r 1 r r 0 R Z for n 1 n Figure.14: Derivation of ABCD-matrix of a thin plano-convex lens. 39
40 Biconvex Lens and Focussing Biconvex Lens æ ç M = ç ç è f 1 ö ø r 1 f Figure.15: Imaging of parallel rays through a lens with focal length f 40
41 Concave Mirror with ROC R r r 1 -R r 1 r 1 r 0 Z Figure.16: Derivation of Ray matrix for concave mirror with Radius R. 41
42 Table of Ray Matrices 4
43 Application: Gauss Lens Formula I II r 1 f r d 1 d Figure.17: Gauss lens formula. 43
44 An Image is formed? 44
45 Gaussian Beam Propagation 8: Gaussian beam transformation by ABCD law, [6], p. 99. this law, we realie that it is true for the case of free space i.e. pure di raction, comparing (.83) with (.53) and (.70). If e that it is additionally true for a thin lens, then we are finished, ABCD matrix (x matrix) can be written as a product of a per triangular matrix (LR-decomposition) like the one for free ation and the thin lens. Note, the action of the lens is identical of free space propagation, Figure.18: but Gaussian in the Fourier-domain. beam transformation In the by ABCD law. in the Gaussian beam parameter is replaced by its inverse (.46) ABCD-marix µ is given by e + 0 ( ) = exp 0 (.84) ( ) ( ) µ = + e 0 ( ) = exp ( ) (.85) 8 CHAPTE where 1 and are the beam parameters at the se q-parameter transforms according to (.83) 1 of the1 optical + = system or component, see Figure 1 1 (.86) 45
46 Telescope d 1 d R1 R Figure.19: Focussing of a Gaussian beam by a lens. 46
47 Gaussian Beam Optics: Physical Optics Ray Optics: B = 0 Gauss lens formula: Physical Optics: Real part of Eq. (.90) = 0 47
48 .6 Optical Resonators w 0 L R 1 R 1 Figure.0: Fabry-Perot Resonator with finite beam sie. 48
49 Curved - Flat Mirror Resonator with Figure.1: Curved-Flat Mirror Resonator 49
50 Curved - Flat Mirror Resonator For given R 1 Figure.: Beam waists of the curved-flat mirror resonator as a function of L/R 1. 50
51 Two Curved Mirror Resonator w 0 L R 1 R 1 Figure.0: Fabry-Perot Resonator with finite beam sie. 51
52 Two Curved Mirror Resonator by symmetry: and: Or: 5
53 Figure.3: Two curved mirror resonator. 53
54 Resonator Stability L 0 R 1 R R 1 +R Figure.75: Stable regions (black) for the two-mirror resonator Or introduce cavity parameters: 54
55 Geometrical Interpretation Figure.4: Stability Criterion 55
56 Stable Unstable R 1 R R1 R R R R 1 R 1 R R Figure.6: Stable and unstable resonators 56
57 Hermite Gaussian Beams Other solutions to the paraxial wave equation: Hermite Polynomials: 57
58 Hermite Gaussian Beams Figure.7: Hermite Gaussians G l (u) 58
59 Hermite Gaussian Beams Figure.8: Intensity profile of TEM lm -beams. by ABCD law. 59
60 Axial Mode Structure: Roundtrip Phase = p : Resonance Frequencies: Special Case: Confocal Resonator: L = R ( ) ( ) 1 60
61 Resonance Frequencies d=l Figure.9: Resonance frequencies of the confocal Fabry-Perot resonator, 61
Course Secretary: Christine Berber O3.095, phone x-6351,
IMPRS: Ultrafast Source Technologies Franz X. Kärtner (Umit Demirbas) & Thorsten Uphues, Bldg. 99, O3.097 & Room 6/3 Email & phone: franz.kaertner@cfel.de, 040 8998 6350 thorsten.uphues@cfel.de, 040 8998
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