Part II. Interaction with Single Atoms. Multiphoton Ionization Tunneling Ionization Ionization- Induced Defocusing High Harmonic Generation in Gases

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1 - Part II 27 / 115

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3 Bohr model recap. At the Bohr radius - a B = the electric field strength is: 2 me 2 = cm, E a = e ab 2 (cgs) Vm 1. This leads to the atomic intensity: I a = ce 2 a 8π (cgs) Wcm 2. (4) 29 / 115

4 V(x) V(x) - E ion 0 x E f 0 x a) b) Figure: a) ionization (MPI): Electron with binding energy E ion simultaneously absorbs n photons with energy ω and escapes from atom with minimal kinetic energy. b) Above-threshold ionization (ATI): electron absorbs more photons than necessary for ionization, acquiring momentum. 30 / 115

5 Above-threshold ionization (ATI) - Experiments: distinct peaks in electron spectra beyond the ionization energy E ion, separated by the photon energy ω. Final kinetic energy of electron is given by an extended version of Einstein s formula: E f = (n + s) ω E ion, (5) where n is the number of photons needed for multiphoton ionization; s is the excess absorbed. 31 / 115

6 Above-threshold ionization (ATI): measurements of electron spectra - Source: Yergeau, Petite & Agostini, J. Phys. B (1986) 32 / 115

7 ionization - Keldysh (1965) and Perelomov (1966): introduced a parameter γ separating the multiphoton and tunneling regimes, given by: 2Eion E ion γ = ω L. (6) I L Φ pond where Φ pond = e2 E 2 L 4mω 2 L is the ponderomotive potential of the laser field. γ < 1 tunneling strong fields, long wavelengths γ > 1 MPI (7) 33 / 115

8 Tunnelling: barrier suppression model I V(x) - e ε x 0 x max x Eion Figure: a) Schematic picture of tunneling or barrier-suppression ionization by a strong external electric field. 34 / 115

9 Tunnelling: barrier suppression model II - Coulomb potential modified by a stationary, homogeneous electric field, see Fig. 10: V (x) = Ze2 x eεx. suppressed on RHS of the atom, and for x x max is lower than the binding energy of the electron. If the barrier falls below E ion, the electron will escape spontaneously over-the-barrier (OBT) or barrier suppression (BS) ionization. 35 / 115

10 Barrier suppression model III - Differentiate V(x) to determine the position of the barrier, x max = (Ze/ε), then set V (x max ) = E ion to get the threshold field strength for OTBI: ε c = E 2 ion 4Ze 3. (8) 36 / 115

11 Barrier suppression model IV - Equate critical field to the peak electric field of the laser appearance intensity for ions created with charge Z: or: I app = c 8π ε2 c = ce ion 4 128πZ 2 e 6, (9) ( ) 4 I app Eion Z 2 Wcm 2. (10) ev NB: E ion is the ionization potential of the ion or atom with charge (Z 1). 37 / 115

12 Appearance intensity: Hydrogen example - Hydrogen: Z = 1 E ion = E h = e2 2a B = ev. Making use of Eq. (??), the critical field for hydrogen is: ε c = E 2 h 4e 3 = e 16a 2 B = E a 16, Appearance intensity: I app = I a Wcm 2. (11) 38 / 115

13 Appearance intensities of selected ions according to the BS ionization model Eq. (10). - Ion E ion I app (ev) ( Wcm 2 ) H He He C C N O Ne Ne Ar Xe Xe / 115

14 Experimental appearance intensities - Source: Auguste et al., J. Phys. B (1992) 40 / 115

15 Tunnelling ionization rate - Keldysh formula for H-like ions (stripped down to the last 1s electron): ( ) 5 [ Ei 2 E a α i = 4ω a E h E L (t) exp 2 ( ) 3 ] Ei 2 E a, (12) 3 E h E L (t) where E i and E h, are the ionization potentials of the atom and hydrogen respectively, E a is the atomic electric field, E L is the instantaneous laser field, and is the atomic frequency. ω a = me4 3 = s 1 (13) Ammosov generalization (1986): more complex many-electron atoms & ions 41 / 115

16 Experimental ionization rates - Source: Auguste et al., J. Phys. B (1992) 42 / 115

17 -induced defocussing - Refractive index of plasma created after ionization given by: η(r, t) = ( 1 n ) 1 2 e(r, t), (14) n c where n e (r, t) is the local electron density and n c the critical density for the laser, related to its frequency ω L by: ω 2 L = 4πe 2 n c /m More electrons at beam center η(r) has minimum at centre lens for rest of beam. High gas pressure leads to deflection of beam before it reaches nominal focus. 43 / 115

18 -induced defocussing: ray equation - Trajectory of light ray x(t) in a refractive medium obeys the ray equation (Born & Wolf): ( d η(x) dx ) = η(x), (15) ds ds where ds is an element of length along the ray. Apply paraxial approximation: η/ η λ, and k k Setting x = r + ẑz, and taking ds dz then gives useful form: dr dz dk dz = k k(z), = k 0 η(r, z), (16) where k 0 = ω 0 /c is now the vacuum wave vector of the laser and k(z) = k 0 η(r, z). 44 / 115

19 Beam divergence - Define the divergence as θ = k /k, and assuming for a highly underdense plasma (n e /n c 1), refractive index is approx.: so η(r) 1 1 n e (r), 2 n c ( ) dθ dz 1 ne (r). 2 r For a laser spot size σ L, the total beam deflection scales as: n c θ I 1 σ L ne (0) n c dz, (17) rays bent away from regions of higher electron density 45 / 115

20 Density clamping - Gaussian beam focused in vacuum is diffraction limited : θ D = σ L Z R, (18) where Z R = 2πσL 2 /λ is the Rayleigh length. Find that ionization-induced refraction will dominate (θ I (z R ) > θ D ) when n e n c > λ πz R. Density clamped at value O(λ/πZ R ), because no further focusing can occur. 46 / 115

21 Numerical propagation model - Example: λ = 1 µm τ L = 80 fs, vacuum focal spot size σ L = 4.5 µm and nominal peak intensity of Wcm 2. Initialized with a radial phase modulation corresponding to an f /10 lens; and enters a neutral H 2 gas at different pressures. Figure: a) beam width; b) peak intensity; c) electron density at the 47 / 115

22 High-harmonic generation by atoms - Field-ionized electron may be sent back close to its parent ion, where it can recombine, emitting a single, high-frequency photon. log(intensity) plateau cutoff Harmonic order Cutoff energy U c Krause (1992) given by: U c = I p U p, (19) where I p = E ion and U p are the ionization potential of the atom and the ponderomotive potential (Eq. 7) respectively. 48 / 115

23 Recollision model I - Classical equations of motion for a linearly polarized laser E = ˆxE 0 cos ωt: where v = v os sin ωt + v i, x = v os ω cos ωt + v it + x i, v os ee 0 (20) mω is the electron quiver velocity, and x i, v i are the electron s position and velocity just after ionization. 49 / 115

24 Recollision model II - Now suppose that this occurs at time t = t 0, and let x(t 0 ) = v(t 0 ) = 0: the electron is born with zero velocity close to the ion center. The orbit is then: v(φ) = v os (sin φ sin φ 0 ), x(φ) = v os ω {cos φ 0 cos φ + (φ 0 φ) sin φ 0 )}, (21) where φ = ωt and φ 0 = ωt 0. Look for orbits where the electron returns to x = 0 (the ion center) at some later time t / 115

25 Recollision model III - Electron s K.E. U c = 1 2 mv 2 depends on φ 0, the phase of the laser that the electron is born into. Max. velocity at the recrossing point x = 0 is at v m /v os = ± 3.17/2 Max. U c (x = 0) is 3.17U p for φ 0 = 17 o and 197 o o 197 o o v/v os 0 0 o o 30 o 17 o x/v os 51 / 115