Distinguished Visiting Scientist Program. Prof. Michel Piché Université Laval, Québec

Transcription

1 Institute for Optical Sciences University of Toronto Distinguished Visiting Scientist Program Prof. Michel Piché Université Laval, Québec Lecture-1: Vectorial laser beams, hyper resolution and electron acceleration

2 Vectorial laser beams, superresolution and electron acceleration Michel Piché Département de physique, de génie physique et d'optique Centre d'optique, photonique et laser (COPL) Université Laval, Québec Work supported by NSERC, CIPI, FCAR, Femtotech Thanks to Alexandre April and Charles Varin

3 Evolution of laser intensity (from G. Mourou)

4 Evolution of pulse duration (from P. Corkum) Pulse duration (fs) Year

5 What about electron beams? Electrons are fermions, hence they cannot superimpose in the same state. This is contrary to photons. Electrons are subject to electrostatic repulsion, prohibiting extremely intense electron beam intensities. In general, electrons interact more strongly with matter than photons (via resonant and non-resonant interactions). High-energy electron beams are important tools for the study of the fundamental interactions of matter.

6 Short pulses of electrons: present status Conventional accelerators deliver short pulses of electrons (in the picosecond range) Femtosecond electron beams have been generated at the University of Toronto (Siwick, Miller) Recollision electrons produce attosecond optical pulses (Corkum) Laser-plasma accelerators ( wakefield ) have produced 100 MeV s electron pulses Other schemes have been proposed (ponderomotive potential)

7 Outline of presentation 1. Introduction to scalar laser beams 2. Typical laser beams 3. Propagation of paraxial laser beams 4. Characterization of laser beams 5. Gouy phase shift 6. Radially polarized TM 01 laser beam 7. Electron acceleration using the TM 01 laser beam 8. Superresolution using the TM 01 laser beam

8 1. Introduction to scalar laser beams - There are a number of families of laser beams - The most commonly used laser beams are defined by Gauss-Hermite or Gauss-Laguerre field distributions - Other families include Gauss-Ince beams, Bessel beams and Ma thieu beams - There is a growing interest for wave packets where temporal and spatial field distributions are coupled, such as X-waves - In general, the Gaussian (fundamental) beam is preferred - Sometimes, for certain applications, one would prefer a flat-top (or "supergaussian") laser beam - Most often, one assumes that these beams are linearly polarized, ignoring their vector character (e.g. their longitudinal electric field and their magnetic field)

9 - Let us start with the scalar wave equation in free space: 2 u = 1 2 u c 2 t 2, where the field amplitude u is a function of position and time u = u ( r r;t) = u (x, y, z;t) and 2 is the Laplacian operator defined, in Cartesian coordinates, as: 2 2 x y z 2 - For monochromatic laser beams, we i ntroduce the phasor notation: u( r,t) r = Re {%U ( r ) e jω t }.

10 - The phasor U ( r ) is solution of the time-independent partial differential equation: 2 % U ( r r) + ω 2 c 2 % U ( r r) = 0 - Very often one uses conditions where the optical beam travels at a small angl e with respect to the beam axis. One can then used the simplified, paraxial wave equation: 2 %U ( r r) x %U ( r r) y 2 2 jk %U ( z r r) = 0, where k = ω/c. - Throughout this series of seminars we assume that the beam propagates along the z-axis. - The optical intensity (W/m 2) is defined as the square modulus of the phasor % U ( r r) : I( r r) = % U ( r r) 2

11 Remarks - Most analytical expressions used for laser beams are obtained from the paraxial wave equation (or fro m the equivalent Huygens-Fresnel integral equation) - Real lasers may produce modes that do not match any specific solution of the wave equation, due to improper boundary conditions, thermal effects, birefringence, etc. - In fact, they may produce a summation of modes of various form, with time-varying amplitudes and relative phases. - Expressing the beam emitted by a laser in terms of a set of modes is a delicate operation. Very often the phase information is not available. Sometimes it is simpler and more accurate to use a purely numerical approach.

12 2. Typical laser beams Hermite-Gauss beams ψ mn ( r w ) = A mn o w(z) e jm+n+1 ( )ζ(z) 2x H m w(z) H 2y j n w(z) e ( ) I mn ( r 2 2 w ) = (I o ) o 2x mn w 2 H m (z) w(z) H 2y 2 x2 +y 2 n w(z) e w 2 (z) Lagu err e-gauss beams kx ( 2 +y 2 ) 2 %q(z) e jkz ψ l p ( r w ) = A l o j p e ( l 2 p+l +1 )ζ (z) 2r l 2r 2 kr 2 j w(z) w(z) L p w 2 (z) e 2 %q(z) cos(l θ)e jkz I l p ( r 2 w ) = (I o ) o 2r 2 l l p w 2 (z) w 2 (z) l 2r 2 2 L p w 2 (z) 2r2 e w 2 (z) cos 2 (l θ)

13 Intensity profiles of Hermite-Gauss beams

14 Intensity profiles of Laguerre-Gauss beams

15 Intensity profiles of Laguerre-Gauss beams

16 Intensity profiles of Ince-Gauss beams for elliptical cylindrical coordinates (see Bandres et al, Opt. Lett. Vol. 29, 144 (2004))

17 3. Propagation of paraxial laser beams in free space - When a monochromatic paraxial laser beam is propagated in free space over a distance L from plane 1 to plane 2, its input and output field distributions are related by the Huygens-Fresnel integral equation: %f 2 ( x, y)= j λ L e jkl f1 % ( x, y )exp jπ λ L ( x x ) 2 + y y ( ) 2 d xd y - N.B.: The amplitudes f % 1 ( x', y' ) and f % 2 ( x, y) are phasors (complex quantities). - For beams of azimuthal symmetry of order l, the Huygens- Fresnel integral equation is given by: %f 2 r l+1 jkl λ L e f1 % ( r ')exp jπ λ L r2 + r ' 2 2π rr' λ L ()= 2π j where J l J l ( 2πrr ' )r 'dr ' λ L is an ordinary Bessel function of l.

18 - Integral equations are useful to e xtract some fundamental properties of laser beams. However, for s imulation of practical conditions, integral equations are too slow. One then uses the beam propagation method ("BPM" algorithm), which consists of propagating the plane wave spectrum associated to a l aser beam. ( ) = e jkl jπλl ν x +νy T F e %f 2 x, y ( ) T F f1 % x, y { ( )} where the operators T F and T F 1 designate Fourier and inverse Fourier transforms. Using Fast Fourier Transforms (FFT), calculations are much faster than using the integral equation. Caution - Computations using FFT may generate noise and artificial sidebands if not properly done. Be careful.

19 Evolution of the intensity of a Gaussian beam Gaussian decay I(r,z) Lorentzian decay Intensity r z r I(0,z) I o w (z) I o /2 w o z R z z R z

20 Evolution of the phase of a Gaussian beam r R(z) Rayleigh zone w(z) z R w o 2 w o z R z z R min z R 2z R ζ (z) π/2 π/4 z z R Plane wavefront Spherical wavefront π/2

21 4. Characterization of laser beams - Question: How do we define the size of a laser beam? - A common practice was to use the full width at half maximum (3-dB width) of the beam intensity. - The beam size can also be measured in the far field. One can then evaluate the TDL parameter of the beam (TDL = "times diffraction limited") - This approach bears some inherent arbitrariness in the definition of the beam size, particularly with beam profiles having multiple peaks or long tails. - This difficulty can be resolved by using the moments of a laser beam. Moments are of current use in statistics and in quantum mechanics.

22 - The rms width of a beam is defined in 1-D cartesian coordinates as: 2 σ x + + = (x x ) 2 I (x) dx I(x) dx, x = + xi(x) dx + I (x) dx, where x are σ x are, respectively, the moments of order 1 and 2 of the beam intensity I(x). One can define the rms width of t he beam in the spatialfrequency domain (or in the far field): σ νx 2 = + + (νx ν x ) 2 P(ν x ) dν x ν x = ν x P(ν x ) dν x + + P(ν x ) dν x P(ν x ) dν x where ν x et σ νx are the moments of order 1 and 2 of the far-field intensity P(ν x ).

23 - For Gaussian beams with a flat wavefront, one finds that the uncertainty product is given by: σ x σ νx = 1 4π - This number is the minimum of the uncertainty product. For any other laser beam, one finds σ x σ νx 1 4π M2 Parameter ("Beam Quality Factor") - The M2 parameter ("Beam Quality Factor") of an optical beam is defined as: M 2 = 4πσ x,min σ νx i.e. the uncertainty product of a laser beam is normalized to that of a Gaussian beam, with σ x,min the minimum value of the rms width. - A l arger value of M2 means a higher divergence of the beam. If M2 1, then its divergence tends to a minimum for its beam size, and its profile tends toward a Gaussian.

24 The rms width of the beam varies along propagation distance z according to: 2 σ x (z) = σ 2 x,min σ νx + λ 2 2 σ νx (z z0 ) 2 = M 2 /4πσ x,min where z 0 is the position where the rms width is minimum. - The beam profile can also be characterized using an rms "phase" moment (P.A. Bélanger, Opt. Lett. vol. 16, (1991)). Caution - When the beam profile has a discontinuity (due to an hardedge aperture), then the value of M2 diverges. This is a pathological situatin where the use of moments must be exercised with care. One often uses a cut-off for low intensity side lobes.

25 5. Gouy phase shift - The Gouy phase shift for a Gaussian beam is given : z φ G (z) arctan z R - Its derivative with respect to z : dφ G dz = 1 1 z R 1+ z 2 2 z R π /2 π /4 φ G (z) 1 z / z R π /2

26 - We note that the de rivative of the Gouy phase shift is always positive z R φ G (z) 1 z / z R - Would it be possible to have a beam with a Gouy phase shift having a negative slope? φ h (z)? z / z R

27 Some thoughts on the Gouy phase shift - We consider an optical field % f 1 (r 1 ) in plane 1. It produces a field %f 2 (z, r 2 ) in plane 2, loc ated at distance z from plane 1. - Huygens-Fresnel integral, allows to calculate % f 2 (z, r 2 ): 2 2 ( +r2 ) r1 dr 1 %f 2 (z, r 2 ) = j2π 2πr %f 1 (r 1 )J 1 r 2 0 λz λz e j π λz r On the axis (r 2 0 ), the field % f 2 (z, r 2 ) reduces to: %f 2 (z,0)= j2π %f λz 1 (r 1 )e jπr 1 λz r 1 dr We do the following changes of variables: s 1 2 r 1 2, ds = r 1 dr 1, ν s 1 λz 2 - Hence we express the on-axis field as a Fourier transform of f % 1 (s): %f 2 (z,0)= j2πν s %f 1 (s)e j2πν s s ds 0.

28 - Hence the on-axis field % f 2 (z,0) i s the Fourier transform of the transverse field profile f % 1 (r 1 ) at plane z = 0: %f 2 (z,0)= j2πν s T % F { f1 (s)}. - Conversely one can extract the transverse profile f % 1 (r 1 ) (or f % 1 (s)) from an inverse Fourier transform of f % 2 (z,0): 1 1 %f 1 (s) = T F % f2 (z,0) j2πν s = 1 %f 2 (ν s,0) e j2πν ss j2π dνs. ν s - If the on-axis field exhibits a Gouy phase shift with a negative slope along z, it means that its slope with respect to ν s is positive, namely: %f 2 (ν s,0) νs ν 0 A(ν 0 )exp j2π (ν s ν 0 ) a ( ). with a > 0. - When substituted in the inverse Fourier transform, this will produce a nonzero field component at s = r 1 2 = a < 0. - T his is an unphysical situation since the field is only defined for positive values of s. - It can also be shown by the residue theorem that any physical field can be expressed in terms of a ba sis consisting of a Gaussian times a power of the variable s. Each term produces a positive Gouy phase shift.

29 6. Radially polarized TM laser beams Laser beams are vector fields with longitudinal components Transverse magnetic (TM) beams have only an electric field component along the longitudinal axis They exhibit radial polarization if their azimuthal order l is 0 They have an azimuthal magnetic field component Radially polarized transverse magnetic beams B θ E r E z z Configuration of electric and magnetic fields

30 7. Electron acceleration using the TM 01 laser beam This work is the object of the Ph.D. thesis of Charles Varin We benefitted from the expert collaboration of Miguel Porras from Universidad Politécnica de Madrid Our basic approach was to use a time-domain formalism to account for corrections of the paraxial approximation and the Slowly-Varying Envelope Approximation Once the fields were corrected, we simulated their effects on electrons

31 How to produce a relativistic electron beam locked to a laser beam? Challenge : laser fields are transverse to propagation axis But laser beam must obey Maxwell equations (div.e = 0) Hence laser beams have longitudinal E and (or) B fields Practical case: Transverse Magnetic (TM) beams with an axial E field (no axial B field) The TM 01 beam has a ring shape in the transverse direction Its axial E field is a quasi-gaussian with a peak at center Under tight focusing conditions, the axial and transverse E fields have comparable amplitude

32 How to phase match laser and electron beams? Due to its Gouy phase shift, any laser beam has a phase velocity exceeding c The TM 01 laser beam has a Gouy phase shift of 2π from z = - to z = + Hence phase matching is impossible from z = - to z = + However phase matching is possible over a restricted zone Strategy: electrons are accelerated from rest at z = 0 to relativistic speed and remain locked to a single half cycle of the laser field

33 The radially-polarized TM 01 beam (SVEA solution) See C. Varin et al, Phys. Rev. E-71, , pp (2005).

34 Acceleration scheme Axial longitudinal field of the driving pulse 0 Electron of charge q at rest z

35 First series of computations A single electron is considered The electron is initially located on the axis (r = 0) at the beam waist (z = 0) The electron is initially at rest (v z = 0) No effect of transverse fields We made corrections to the temporal form of the wave packet beyond the Slowly-Varying Envelope Approximation

36 Single-electron on-axis trajectories (λ 0 = 800 nm, w 0 = 10 μm)

37 Intensity threshold and phase sensitivity (λ 0 = 800 nm, w 0 = 10 μm)

38 Electron trapping by laser field L ong pu lse ( t p >> T ) E(t ) electron trajectory t S hor t pu lse ( t p ~ T ) E (t ) electron trajectory t

39 Second series of computations Full 3D electromagnetic beam structure was considered An initial cloud of 2000 electrons was at rest The size and position of the electron cloud was changed Pulse duration of 12 fs is assumed for all presented results

40 Two regimes of interaction Low power: acceleration from ponderomotive potential High power: direct acceleration from the field and locking to a single half cycle of the field Analytical criterion to distinguish the two regimes: P > P* P* = (π 5 /2η 0 ) (w 0 /λ 0 ) 4 (m e c 2 /e) 2 Available energy: W[MeV] = (P[TW] 2η 0 /π) 1/2 C. Varin and M. Piché, submitted to Phys. Rev. Lett.

41 Acceleration with P =100 TW and w 0 = 3 μm

42 Effect of carrier phase and initial position Electron energy is very sensitive to absolute phase of laser field Largest energy gain if electrons are located before beam waist

43 Electron pulse duration and divergence Electron pulses of attosecond duration appear possible Milliradian divergence is predicted (no electrostatic repulsion) Large sensitivity to absolute phase of laser pulse

44 Petawatt laser (I = w/cm 2, w 0 = 10 μm)

45 Production of a TM 01 laser beam incident beam (linearly polarized) four half-wave plates (#1,2,3,4) on a quadrant transmitted beam (quasi radially polarized) Figure 2 : Comment produire un mode TM 01 - Thin quartz half-wave plates can be used to rotate beam polarization - Manipulation of polarization must be done after compression

46 Observations and comments Power threshold for electron acceleration: 21 TW for w 0 = 3 μm 2 TW for w 0 = 1 μm Shorter pulses lead to larger pulse energies. Electrons can reach the pulse crest Absolute phase of the pulse affects the energy gain Proper positioning of initial electron cloud

47 Other observations and comments Electron trajectory is stable with respect to transverse field components E r and B φ The effects of E r and B φ cancel each other for relativistic electrons TM 01 beams could be produced by splitting a Gaussian beam into 4 beams disposed on a quadrant TM 01 beams can be focused to narrower spots than Gaussian beams (taking into account longitudinal field)

48 8. Superresolution using the TM 01 laser beam C. Varin et al, J. Opt. Soc. Am. A (in press)

49 Beam profile at focus Corrections up to second order of paraxial beam profile

50 Beam spread and Gouy phase near focus Fast variation of beam size and Gouy phase under extreme temporal and spatial focusing

51 Future avenues Initial electron cloud could come from a nanoparticle or a cluster of C 60 molecules Spreading of electron pulse by electrostatic effects: limitation of electron density and pulse duration Interaction with solid targets: the large axial E field of a TM 01 beam could accelerate the plasma boundary (high harmonics?) Could positive charges (protons) also be accelerated? Is radial polarization necessary? A linearly polarized Laguerre-Gauss donut mode has a doublepeak axial E field It could produce a pair of electron beams

52 Acknowledgements Thanks to Dwayne and to the Institute of Optical Sciences for inviting me Questions are welcome