Course Secretary: Christine Berber O3.095, phone x-6351,

Transcription

1 IMPRS: Ultrafast Source Technologies Franz X. Kärtner (Umit Demirbas) & Thorsten Uphues, Bldg. 99, O3.097 & Room 6/3 & phone: Course Secretary: Christine Berber O3.095, phone x-6351, Class website: 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 (Demirbas) Uphues Kärtner (Demirbas) Uphues Uphues Uphues Classical Optics Basic Technologies Ultrafast Lasers Ultrafast Electron Sources High Order Harmonic Gen. Synchrotron sources Tuesday Feb. 14, 9:30-11:30 am Thursday Feb. 16 Tuesday Feb. 1 Thursday Feb. 3 Tuesday Feb. 8 Thursday March Bldg. 99 Room IV, 1.OG Bldg. 99 Room IV, 1.OG Bldg. 99 Room IV, 1.OG Bldg. 99 Room IV, 1.OG Bldg. 99 Room IV, 1.OG Bldg. 99 Room IV, 1.OG 3

4 Classical Optics Classical Optics.1 Maxwell s Equations and Helmholtz Equation.1.1 Helmholtz 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 Polarization Magnetization 5

6 Classical Optics Vector Identity: Vacuum speed of light: 6

7 Classical Optics No free charges, No currents from free charges, Nonmagnetic 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 Helmholtz 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: εr = 1 + χω ( ) Example: EM-Wave polarized along x-axis and propagation along z-direction: 1

13 Backwards Traveling Wave Backwards 13

14 .1.3 Poynting Vector, Energy Density and Intensity relative permittivity: εr = 1+ χ Example: Plane Wave: 14

15 A plane wave is described by: ~ E( x, y, z). Paraxial Wave Equation ~ = e E exp x 0 kx + k y kz 0 k + [ jk x jk y jk z] x = free space wavenumber y z Although a plane wave is a useful model, strictly speaking it cannot be realized 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, z) + + ~ = E exp 0 [ jk ] xx jk y y jkz z dkxdk y there is no integral over k z because once k x and k y are fixed, k z is constrained by dispersion relation 0 = kx + k y kz k + 15

16 Paraxial Beam Consider a beam which consists of plane waves propagating at small angles with z axis such a beam y is called a paraxial beam x k k x << k z k << y k z z Figure.: Construction of paraxial beam by superimposing many plane waves with a dominant k-component in z-direction k z can be approximated as (paraxial approximation) 16

17 Paraxial Diffraction Integral The beam ~ E( x, y, z) + + ~ = E ( k, k ) exp 0 x y [ jk ] xx jk y y jkzz dkxdk y can then be expressed as ~ jk 0 z E( x, y, z) = e + + ~ E 0 ( k x, k y )exp kx + k j k0 y z jk x x jk y y dk x dk y quickly varying carrier wave slowly varying envelope (changing slowly along z) 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 z Given the field at some plane (e.g. z=0), how to find the field at any z? From the previous slide ~ jk 0 z E( x, y, z) At z=0 = e ~ E( x, + + ~ E 0 ( k x, k y )exp x kx + k j k0 + + ~ y, z = 0) = E ( k, k ) exp 0 y y z 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 ) ~ = E( x, y, z = 0) exp[ jkxx + jk y y]dxdy (π ) ~ ~ ~ Knowing E( x, y, z = 0) can find E ( k x, k ) and then E( x, y, z) at any z. 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 z=0) Substitution into paraxial diffraction integral gives with called Rayleigh range. Performing the integration (i.e. taking Fourier transforms of a Gaussian) 19

20 Gaussian Beam The quantity 1/( z + jzr ) 1 z + jz 1 R = z R ( z) z + z z z R can be expressed as + jz z R R 1 R( z) j λ π w where R(z) and w(z) are introduced according to The pre-factor j z + jz We get R j /( z + jzr ) exp( jζ ( z)) = z + z ~ E R ( r, z) = ~ ( z) λ z π w + R ( z) z zr can be expressed as λ π z R 1 exp( jζ ( z)) w( z) with 1 r r exp jk0 + jζ ( ) w( z) w ( z) R( z) 0 z tan ζ ( z ) = z / zr 0

21 1 Finally, the field is normalized, giving + = ) ( ) ( ) ( exp ) ( 1 4 ), ( ~ z j z R r jk z w r z w P Z z r E F ζ π Field of a Gaussian beam Normalization means that the total power carried by the beam is P: Final Expression for Gaussian Beam Field

22 Gaussian Beams: Spot Size and Rayleigh Range We derived ~ 4Z F 0P 1 r r E ( r, z) = exp jk0 + jζ ( π w( z) w ( z) R( z) 0 z ) Field of a Gaussian beam We have Defines transversal intensity distribution λ z π w + R ( z) z zr I r r, z) ~ exp w ( z) λ w ( z) = zr 1 + π ( z z R w(z) spot size Spot size at z=0 is min w λ ( 0) = z w π z R = R π w 0 λ 0 Rayleigh range w ( ) 0 1 z z = w + z R spot size as a function of z

23 We derived Gaussian Beams: Phase Distribution ~ 4Z F 0P 1 r r E ( r, z) = exp jk0 + jζ ( π w( z) w ( z) R( z) 0 z ) Field of a Gaussian beam We have 1 ( z R z) z + z R R( z) = z 1 + defines phase distribution z z R Radius of wavefront curvature Phase fronts (surfaces of constant phase) are parabolic, can be approximated as spherical for small r. R(z) turns out to be the radius of the sphere. ζ ( z ) = arctan / ( z ) z R Guoy phase shift Defines extra phase shift as compared to plane wave 3

24 Gaussian Beams: All Equations in One Slide ~ 4Z F 0P 1 r r E ( r, z) = exp jk0 + jζ ( π w( z) w ( z) R( z) 0 z ) Gaussian Beam Field w ( z ) 0 1 z = w + z R R( z) = z 1 + z z R Spot size w 0 min spot size (z=0) Radius of wavefront curvature (.54) (.55) ζ ( z) = arctan z z R Guoy phase shift (.65) z R = π w 0 λ Rayleigh range (.56) 4

25 Intensity Distribution z=0 z=z R z=z R Figure.3: The normalized beam intensity I/I₀ as a function of the radial distance r at different axial distances: (a) z=0, (b) z=z R (c) z=z R. 5

26 Spot Size as a Function of z Spot size w ( ) 0 1 z z = w + z R Waist: minimum radius At z = z R Rayleigh range : distance from the waist at which the spot area doubles 4 4 waist w( z) w( z) w0 0 0 w(z) w 0 w z zr 0z z z 3z 3 R 3z R R z.r R R 6

27 Divergence Angle and Confocal Parameter For large z z z w( z) = w0 1+ w0 z R z R Divergence angle θ ( ) w w z 0 z zr Beam Divergence: Smaller spot size 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: Normalized beam intensity I(r=0)/I₀ on the beam axis as a function of propagation distance z. 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(z) 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( z) = z 1 + z z R Figure.7: Radius of curvature R(z) 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 size and magnitude is scaled At the waist, the spot size is minimum and wave fronts are flat Lasers are usually built to generate Gaussian beams Planes of const. Phase Beam Waist z/z 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 θ 1 r 1 r r θ Figure.11: Snell s law for paraxial rays. Z 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 α r 1 r r α 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 r 1 f z Figure.15: Imaging of parallel rays through a lens with focal length f 40

41 Concave Mirror with ROC R r α r 1 -R α 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 z d 1 d Figure.17: Gauss lens formula. 43

44 An Image is formed? 44

45 Gaussian Beam Propagation Figure.18: Gaussian beam transformation by ABCD law. 45

46 Telescope d 1 d z R1 z 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 z 1 z Figure.0: Fabry-Perot Resonator with finite beam size. 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 z 1 z Figure.0: Fabry-Perot Resonator with finite beam size. 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 Axial Mode Structure: Roundtrip Phase = p π: Resonance Frequencies: Special Case: Confocal Resonator: L = R π ζ( z) ζ( z1) = 59

60 Hermite Gaussian Beams Figure.8: Intensity profile of TEM lm -beams. by ABCD law. 60

61 Resonance Frequencies d=l Figure.9: Resonance frequencies of the confocal Fabry-Perot resonator, 61