## Photoionization with orbital angular momentum beams

Optics Express, Vol. 18, Issue 4, pp. 3660-3671 (2010)

http://dx.doi.org/10.1364/OE.18.003660

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### Abstract

Intense laser ionization expands Einsteins photoelectric effect rules giving a wealth of phenomena widely studied over the last decades. In all cases, so far, photons were assumed to carry one unit of angular momentum. However it is now clear that photons can possess extra angular momentum, the orbital angular momentum (OAM), related to their spatial profile. We show a complete description of photoionization by OAM photons, including new selection rules involving more than one unit of angular momentum. We explore theoretically the interaction of a single electron atom located at the center of an intense ultraviolet beam bearing OAM, envisaging new scenarios for quantum optics.

© 2010 Optical Society of America

13. P. B. Corkum and F. Krausz, “Attosecond science,” Nature Phys. **3**, 381 (2007). [CrossRef]

14. C. I. Blaga et al., “Strong-field photoionization revisited,” Nature Phys. **5**, 335 (2009). [CrossRef]

15. G. A. Mourou, “Optics in the relativistic regime,” Rev. Mod. Phys. **78**, 309 (2006). [CrossRef]

*h*̄ unit), and the selection rules avoid the possibility of one-photon excitation of atomic transitions with angular momentum variation larger than one

*h*̄. However, one can overcome this limitation by considering multi-photon effects with very intense lasers. In the non-dipole regime, the light-atom interaction is more complex, exciting not only atomic transitions with angular momentum change equal to one unit

*h*̄, but also atomic transitions with larger angular momentum variation. In this work we present for the first time photoionization with light beams carrying OAM which give rise to new selection rules out of both the electric-dipole and the non-dipole regime, opening new perspectives for atomic transition excitations.

2. A. T. O’Neil, I. MacVicar, L. Allen, and M. J. Padgett, “Intrinsic and Extrinsic Nature of the Orbital Angular Momentum of a Light Beam,” Phys. Rev. Lett. **88**, 053601 (2002). [CrossRef] [PubMed]

17. S. M. Barnett, “Optical Angular-Momentum Flux,” J. Opt. B: Quantum Semiclass. Opt. **4** S7 (2002). [CrossRef]

18. S. J. Van Enk, “Selection rules and centre-of-mass motion of ultracold atoms,” Quantum Opt. **6**, 445 (1994). [CrossRef]

19. R. Jáuregui, “Rotational effects of twisted light on atoms beyond the paraxial approximation,” Phys. Rev. A **70**, 033415 (2004). [CrossRef]

20. A. Alexandrescu, D. Cojoc, and E. Di Fabrizio, “Mechanism of Angular Momentum Exchange between Molecules and Laguerre-Gaussian Beams,” Phys. Rev. Lett. **96**, 243001 (2006). [CrossRef] [PubMed]

18. S. J. Van Enk, “Selection rules and centre-of-mass motion of ultracold atoms,” Quantum Opt. **6**, 445 (1994). [CrossRef]

19. R. Jáuregui, “Rotational effects of twisted light on atoms beyond the paraxial approximation,” Phys. Rev. A **70**, 033415 (2004). [CrossRef]

20. A. Alexandrescu, D. Cojoc, and E. Di Fabrizio, “Mechanism of Angular Momentum Exchange between Molecules and Laguerre-Gaussian Beams,” Phys. Rev. Lett. **96**, 243001 (2006). [CrossRef] [PubMed]

19. R. Jáuregui, “Rotational effects of twisted light on atoms beyond the paraxial approximation,” Phys. Rev. A **70**, 033415 (2004). [CrossRef]

## 1. Light-matter interaction scheme

*z*-direction with a temporal envelope wave parameterized by a quadratic sinus (see Fig. 1). This temporal envelope has a frequency

*ω*=

_{e}*π*/

*N*, where

_{cyc}τ*N*and

_{cyc}*τ*are the cycle number and the period of the carrier wave, respectively. A hydrogen atom is assumed to be localized at the origin of the reference system and experiences a vector potential, associated to the pulse, of the form

*θ*is the step function,

*a*the Bohr radius,

_{0}*c*the speed of light,

*ω*the carrier wave frequency, and

**A**

_{0}the amplitude of the wave (it includes the polarization state). The transverse spatial structure of the pulse beam is accounted by the functions

*LG*(

_{ℓ,p}*ρ,ϕ*;

*ω/c*); the Laguerre-Gaussian modes [1

1. L. Allen, M. V. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A **45**, 8185 (1992). [CrossRef] [PubMed]

*k*

_{0}the carrier wave vector,

*w*

_{0}being the width of the mode at

*z*= 0,

*R*(

*z*) =

*z*[1 + (

*z*

_{0}/

*z*)

^{2}] is the phase-front radius,

*z*

_{0}=

*k*

_{0}

*w*

^{2}

_{0}/2 the Rayleigh range,

**Φ**

_{G}(

*z*) = - (2

*p*+ |ℓ| + 1)arctan(

*z*/

*z*

_{0}) the Gouy phase, and

*L*

_{p}^{|ℓ|}(ξ) are the associated Laguerre polynomials

*p*= 0,1,2,… correspond to the winding (or topological charge) and the number of nonaxial radial nodes of the mode. Laguerre-Gaussian modes contain an azimuthal phase

*e*, see Eq. (2), which gives rise to a discrete OAM of ℓ

^{iℓϕ}*h*̄ units per photon along their propagation direction [1

1. L. Allen, M. V. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A **45**, 8185 (1992). [CrossRef] [PubMed]

21. G. F. Calvo, A. Picón, and R. Zambrini, “Measuring the Complete Transverse Spatial Mode Spectrum of a Wave Field,” Phys. Rev. Lett. **100**, 173902 (2008). [CrossRef] [PubMed]

*ψ*(

**r**,

*t*) is the electron quantum state,

*V*̂(

*r*) the Coulomb potential originated by the hydrogen nucleus,

*m*the electron mass,

*q*the electron charge,

**A**̂(

**r**,

*t*) the vector potential, which in our case is given by expression (1), and

**p**̂ = -

*ih̄*∇ the linear momentum operator, satisfying the canonical commutation relation [

**x**̂,

**p**̂] =

*ih*̄.

## 2. Selection rules with OAM

_{0}+ 𝓗̂

_{퓘}+ 𝓗̂

_{𝓘𝓘}, where 𝓗̂

_{0}is the free part, whereas 𝓗̂

_{𝓘}≡ -

*q*(

**p**̂ ∙

**A**̂(

**r**,

*t*) +

**A**̂(

**r**,

*t*) ∙

**p**̂)/2

*m*and 𝓗̂

_{𝓘𝓘}=

*q*

^{2}

**A**̂

^{2}(

**r**,

*t*)/2

*m*refer to the interaction parts. Representing the quantum state in a spherical basis

*ψ*(

**r**) = ∑

_{L,M}

*u*(

_{L,M}*r*)

*Y*(

^{M}_{L}*θ,φ*) (all radial dependence is in the functions

*u*(

_{L,M}*r*), while the angular dependence remains in the spherical harmonic functions

*Y*(

^{M}_{L}*θ, φ*) instead), the first interaction contribution can be written as

*E*and

_{i}*E*are the unperturbed energies of the initial and final states, respectively. Taking into account the vector potential given by (1), and within the dipolar (

_{f}*λ*≫

*a*) and the transverse spatial (

_{0}*w*

_{0}≫

*λ*) approximations, we derive the following set of selection rules for beams carrying any arbitrary ℓ units of OAM (see the Appendix):

*L*| = 1), significant variations of the angular momentum are to be expected. In terms of photons, the selection rules (6) can be conceived as the absorption of photons carrying a total angular momentum

*j*= ℓ +

*s*in the propagation direction, where

*s*indicates the polarization part (or spin momentum,

*s*= ± 1 for right- and left-circular polarization). We would like to remark that these selection rules originate from the transverse profile, despite the dipolar approximation. Moreover, the second contribution for the interaction Hamiltonian 𝓗̂

_{𝓘𝓘}yields, in the case of plane waves, a constant term, producing a ponderomotive force [22

22. Shawn A. Hilbert, Cornelis Uiterwaala, Brett Barwick, Herman Batelaan, and Ahmed H. Zewail, “Temporal lenses for attosecond and femtosecond electron pulses,” Proc. Natl. Acad. Sci. **106**, 10558 (2009). [CrossRef] [PubMed]

## 3. Hydrogen simulations

*N*= 3, ℓ = 1,

_{cyc}*p*= 0, an angular frequency

*ω*= 1 au (atomic units where

*h*̄ =

*m*=

*q*= 1, and

*ω*= 2

*π*× 6.57 ∙ 10

^{15}s

^{-1}, ultraviolet), a period τ = 2

*π*au (152 as), and two possible polarizations: linear (in the

*x*-direction) and left-handed. We choose a beam waist to satisfy the paraxial regime,

*w*

_{0}= 9 ∙ 10

^{4}au (4.79

*μ*m), which is much larger than the characteristic size of the atom (

*w*

_{0}≫

*a*). Our atom is centered at the beam vortex, interacting with its vicinity, where the electric field amplitude is much weaker than the maximum one (reached at a distance

_{0}*w*

_{0}/√2). This imposes the need for very intense lasers; in the proximity of the vortex singularity the electric field amplitude increases linearly (when |ℓ| = 1). For example, an electric field

*E*~

*A*of about 10

_{0}ω^{4}au (10

^{13}V/cm), would give rise to 5 au amplitudes (during the pulse peak) at distances of about 20 au (1 nm) from the singularity. In order to clarify the structure of the electric field, in Fig. 2 we represent the polarization of the electric field in the transverse plane

*z*= 0, when the pulse achieves its maximum value. Figure 2 also plots the pulse beam with respect to time at four different positions in the transverse plane. Notice the variation of the carrier envelope phase (CEP) depending on the azimuthal position. In fact, all the possible CEPs are encompassed in a circle around the singularity, in contrast with standard few cycle pulses technology [23

23. T. Brabec and F. Krausz, “Intense few-cycle laser fields: Frontiers of nonlinear optics,” Rev. Mod. Phys. **72**, 545 (2000). [CrossRef]

*ω*= 1 au (larger than the bound hydrogen energy 0.5 au), ionization is to be expected. In fact, after the first pulse cycle we verify that 52% of the quantum electron state is ionized when the beam is linearly polarized, and 31% when the beam is circularly polarized. Note that Eq. (4) yields a total photoionization probability that depends non-linearly on the light polarization. We can express the electron quantum state at each time as |

*ψ*(

*t*)〉 =

*α*|

*ψ*〉 + |

_{i}*δψ*(

*t*)〉, where

*δψ*is the excited state part. The excited state function can be decomposed in an unbound spherical basis

*δψ*(

**r**) = ∑

_{L,M}

*u*(

_{L,M}*r*)

*Y*(

^{M}_{L}*θ, φ*). Our initial electron state has full spherical symmetry, belonging to the spherical harmonic

*Y*

^{0}

_{0}. However, after the interaction with the pulse, the electron state excites different spherical harmonics. In Fig. 3 the projection of the excited state onto the

*xy*-plane is depicted during the first cycle of the pulse beam, being a superposition of spherical harmonics obeying the selection rules (6). Also, depending on the input polarization, the electron evolution varies noticeably.

*Y*and extract the corresponding probabilities

^{M}_{L}*P*= ∫

_{L,M}*dr*

*r*

^{2}|

*u*(

_{L,M}*r*)|

^{2}. Using this numerical method, the widths of the spherical harmonic superpositions at different times have been derived, showing excellent agreement with the electron state evolution, as represented in Fig. 3(a). Moreover, we have analyzed the lowest spherical harmonic content of the final excited electron state (see Fig. 4) in three scenarios: (i) with a Gaussian pulse beam (in the transverse spatial approximation, it could be considered as a plane wave) linearly polarized in the

*x*-direction (the electron state is ionized about 30%); (ii) the case for a pulse spatially modulated by a Laguerre-Gaussian mode, linearly polarized in the

*x*-direction; (iii) when the Laguerre-Gaussian pulse is left-circularly polarized. The main remark is that the spherical harmonics

*Y*

^{1}

_{1}and

*Y*

^{-1}

_{1}are most efficiently excited by the plane-wave-like pulse, in striking contrast with the Laguerre-Gaussian scenario, where no such excitation exists. If the Laguerre-Gaussian pulse is linearly polarized, then,

*Y*

^{0}

_{2},

*Y*

^{2}

_{2}and

*Y*

^{4}

_{4}(this term is due to two-photon absorption) are the most occupied states, whereas if it is circularly polarized,

*Y*

^{0}

_{0}and

*Y*

_{2}

^{0}are the most relevant. There is a small contribution from

*Y*

_{4}

^{0}(this term is due to two-photon absorption), as the electron is less ionized. We emphasize that these results are in accordance with the derived selection rules (6) and (7).

**L**̂〉 = 〈

**r**̂ × (

**p**̂ -

*q*

**A**̂)〉, during its evolution. Figure 5 shows the time evolution of the orbital angular momentum along the

*z*-direction in the same case of Fig. 3 (a), while a depopulation of the fundamental state occurs. The electron starts in the ground state, with zero OAM. As the pulse begins to interact with the electron, the OAM of the latter in the

*z*-direction oscillates, but notice that at the end of the pulse, the electron quantum state gains a finite amount of OAM: 1.53 au (1.53

*h*̄). There is no OAM contribution in other directions. We expect, as the ground state has no OAM in the absence of a field, that the electron excited states belong to unbound states bearing OAM. On the other hand, when the pulse is left-circularly polarized, as the case of Fig. 3(b), the OAM in the

*z*-direction is negligible, as it is expected. Regarding the excited state position components, mean values are zero except for the

*z*-component, where a small shift of 10

^{-2}au is present, caused by a non-vanishing magnetic field at the origin.

## 4. Discussion

*N*= 14, we notice that the excited electron state remains confined, within a radius of 10 au, due to the ponderomotive force induced by the Laguerre-Gaussian profile. We also simulate the case where the atom is displaced 2 au from the origin in the

_{cyc}*x*and

*y*-directions, with

*N*= 14. In both cases, the atom is ionized much faster since the electric field is more intense now, and the excited electron state remains trapped. From evaluations of the mean values of the position component, we have observed an electron motion around the vortex. Varying the polarization of the beam, in particular for left-circular polarization, an ionized state with a ring structure (Δ

_{cyc}*M*= 0) is predicted, see Figs. 3(b) an 4(c). Furthermore, modifying the initial phase, different ionized state structures can be generated. Thus, by tunning the phase and the polarization, one expects a manipulation of the ionized state. There still remain many open questions, such as the feasibility to achieve high-harmonic generation [25

25. J. Courtial, K. Dholakia, L. Allen, and M. J. Padgett, “Second-harmonic generation and the conservation of orbital angular momentum with high-order Laguerre-Gaussian modes,” Phys. Rev. A **56**, 4193 (1997). [CrossRef]

24. S. Patchkovskii, Z. Zhao, T. Brabec, and D. M. Villeneuve, “High Harmonic Generation and Molecular Orbital Tomography in Multielectron Systems: Beyond the Single Active Electron Approximation,” Phys. Rev. Lett. **97**, 123003 (2006). [CrossRef] [PubMed]

26. K. W. D. Ledingham, P. McKenna, and R. P. Singhal, “Applications for Nuclear Phenomena Generated by Ultra-Intense Lasers,” Science **300**, 1107 (2003). [CrossRef] [PubMed]

27. T. J. Bürvenich, J. Evers, and C. H. Keitel, “Nuclear Quantum Optics with X-Ray Laser Pulses,” Phys. Rev. Lett. **96**, 142501 (2006). [CrossRef] [PubMed]

28. E. Peik and Chr. Tamm, “Nuclear laser spectroscopy of the 3.5 eV transition in Th-229,” Europhys. Lett. **61**, 181 (2003). [CrossRef]

## 5. Appendix

**r**= (

*x,y,z*) =

*r*(sin

*θ*cos

*ϕ*, sin

*θ*sin

*ϕ*, cos

*θ*) and

*ρ*=

*r*sin

*θ*. Hence, the interaction element given by equation (5) is proportional to

*θ*)

^{|ℓ|+1ei(ℓ+1)ϕ}and (sin

*θ*)

^{|ℓ|+1ei(ℓ+1)ϕ}. These two terms can be decomposed in spherical harmonics functions, getting then the selection rules. In order to obtain the spherical harmonic decomposition, we must consider two cases, when ℓ ≥ 0 and ℓ ≤ 0. Let us begin with case ℓ ≥ 0, where the angular terms read as

^{2}

*θ*= 4√5(

*Y*

^{0}

_{0}-

*Y*

_{2}

^{0}/√5)/3 and

*L*| ≤ ℓ + 1, Δ

*L*+ ℓ + 1 is even and Δ

*M*= ℓ + 1. In equation (12) there are two contributions, the first one yields also straightforwardly the selection rule: |Δ

*L*| ≤ ℓ - 1, Δ

*L*+ ℓ - 1 is even and Δ

*M*= ℓ - 1, but the second one is a bit subtle. First of all, we need to decompose the product

*Y*

^{ℓ-1}

_{ℓ-1}

*Y*

_{2}

^{0}into spherical harmonics. Using the following formula;

*l*

_{1},

*l*

_{2};

*m*

_{1},

*m*

_{2}|

*l*

_{3},

*m*

_{3}〉 are the corresponding Clebsch-Gordan coefficients, we know that it is possible to write

*Y*

^{ℓ-1}

_{ℓ-1}

*Y*

_{2}

^{0}=

*aY*

^{ℓ-1}

_{ℓ+1}+

*bY*

^{ℓ-1}

_{ℓ}+

*cY*

^{ℓ-1}

_{ℓ-1}+

*dY*

^{ℓ-1}

_{ℓ-2}+

*eY*

^{ℓ-1}

_{ℓ-3}. We can take

*d*=

*e*= 0, as

*M*≤

*L*must be satisfied in a spherical harmonic function. And

*b*= 0 always, due to 〈ℓ-1,2;0,0|ℓ,0〉 = 0. Concretely,

*L*| ≤ ℓ + 1, Δ

*L*+ ℓ + 1 is even and Δ

*M*= ℓ - 1.

^{2}

*θ*= 4√5(

*Y*

^{0}

_{0}-

*Y*

_{2}

^{0}/√5)/3 and

*L*| ≤ |ℓ| + 1, Δ

*L*+ |ℓ| + 1 is even and Δ

*M*= ℓ - 1. On the other hand, in Eq. (16) there are two contributions, the first one giving rise to: |Δ

*L*| ≤ |ℓ| - 1, Δ

*L*+ |ℓ| - 1 is even and Δ

*M*=ℓ+1. In the second one, the product

*Y*

^{ℓ-1}

_{|ℓ|-1}

*Y*

^{0}

_{2}must be decomposed into spherical harmonics using formula (13). As before, we can write

*Y*

^{ℓ+1}

_{|ℓ|-1}

*Y*

^{0}

_{2}=

*a*́

*Y*

^{ℓ+1}

_{|ℓ|+1}+

*b*́

*Y*

^{ℓ-1}

_{|ℓ|}+

*c*́

*Y*

^{ℓ+1}

_{|ℓ|-1}+

*d*́

*Y*

^{ℓ+1}

_{|ℓ|-2}+

*e*́

*Y*

^{ℓ+1}

_{|ℓ|-3}, where

*d*́ =

*e*́ = 0 owing to |

*M*| ≤

*L*, which must be satisfied in any spherical harmonic function. And

*b*́ = 0 always, due to 〈|ℓ| - 1,2;0,0||ℓ|,0〉 = 0. Therefore,

*L*| = 1), we can expect larger exchange of angular momentum. Of course, playing with the beam polarization (

*α*and

*β*), as we can note in equation (6), we can modify the selection rules. For example, for a right-circular polarization (

*α*= 1 and

*β*=

*i*), the only surviving terms are given by expressions (11) and (16), restricting Δ

*M*= ℓ + 1. Analogously, for a left-circular polarization (

*α*= 1 and

*β*= -

*i*) the only surviving terms are given by expressions (12) and (17), restricting Δ

*M*= ℓ - 1. Selection rules (6), in photon terms, can be thought as the absorption of a photon carrying a total angular momentum in the propagation direction

*m*= ℓ +

*s*, where

*s*indicates the polarization part (spin momentum, for right-circular polarization

*s*= 1 and for left-circular polarization

*s*= -1). We would like to remark that these selection rules are exclusive to the transverse profile. Moreover, the second interaction hamiltonian 𝓗̂

_{𝓘𝓘}, in the case of plane waves, is just a constant term, yielding a ponderomotive force. In our case, it is quite different. Analogously to equation (5), we can write

## Acknowledgments

## References and links

1. | L. Allen, M. V. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A |

2. | A. T. O’Neil, I. MacVicar, L. Allen, and M. J. Padgett, “Intrinsic and Extrinsic Nature of the Orbital Angular Momentum of a Light Beam,” Phys. Rev. Lett. |

3. | M. Bhattacharya and P. Meystre, “Using a Laguerre-Gaussian Beam to Trap and Cool the Rotational Motion of a Mirror,” Phys. Rev. Lett. |

4. | S. Franke-Arnold, L. Allen, and M. Padgett, “Advances in optical angular momentum,” Laser Photonics Rev. |

5. | M. F. Andersen, C. Ryu, P. Cladé, V. Natarajan, A. Vaziri, K. Helmerson, and W. D. Phillips, “Quantized Rotation of Atoms from Photons with Orbital Angular Momentum,” Phys. Rev. Lett. |

6. | A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature |

7. | R. Inoue, N. Kanai, T. Yonehara, Y. Miyamoto, M. Koashi, and M. Kozuma, “Entanglement of orbital angular momentum states between an ensemble of cold atoms and a photon,” Phys. Rev. A |

8. | G. F. Calvo, A. Picón, and A. Bramon, “Measuring two-photon orbital angular momentum entanglement,” Phys. Rev. A |

9. | M. van Veenendaal and I. McNulty, “Prediction of Strong Dichroism Induced by X Rays Carrying Orbital Momentum,” Phys. Rev. Lett. |

10. | I. G. Mariyenko, J. Strohaber, and C. J. G. J. Uiterwaal, “Creation of optical vortices in femtosecond pulses,” Opt. Express |

11. | I. J. Sola, V. Collados, L. Plaja, C. Méndez, J. San Román, C. Ruiz, I. Arias, A. Villamarín, J. Atencia, M. Quin-tanilla, and L. Roso, “High power vortex generation with volume phase holograms and non-linear experiments in gases,” Appl. Phys. B |

12. | A. Einstein, “Concerning an Heuristic Point of View Toward the Emission and Transformation of Light,” Ann. Phys. |

13. | P. B. Corkum and F. Krausz, “Attosecond science,” Nature Phys. |

14. | C. I. Blaga et al., “Strong-field photoionization revisited,” Nature Phys. |

15. | G. A. Mourou, “Optics in the relativistic regime,” Rev. Mod. Phys. |

16. | J. R. Vázquez de Aldana and L. Roso, “Magnetic field effects in strong field ionization of single-electron atoms: Three-dimensional numerical simulations,” Laser and Part. Beams |

17. | S. M. Barnett, “Optical Angular-Momentum Flux,” J. Opt. B: Quantum Semiclass. Opt. |

18. | S. J. Van Enk, “Selection rules and centre-of-mass motion of ultracold atoms,” Quantum Opt. |

19. | R. Jáuregui, “Rotational effects of twisted light on atoms beyond the paraxial approximation,” Phys. Rev. A |

20. | A. Alexandrescu, D. Cojoc, and E. Di Fabrizio, “Mechanism of Angular Momentum Exchange between Molecules and Laguerre-Gaussian Beams,” Phys. Rev. Lett. |

21. | G. F. Calvo, A. Picón, and R. Zambrini, “Measuring the Complete Transverse Spatial Mode Spectrum of a Wave Field,” Phys. Rev. Lett. |

22. | Shawn A. Hilbert, Cornelis Uiterwaala, Brett Barwick, Herman Batelaan, and Ahmed H. Zewail, “Temporal lenses for attosecond and femtosecond electron pulses,” Proc. Natl. Acad. Sci. |

23. | T. Brabec and F. Krausz, “Intense few-cycle laser fields: Frontiers of nonlinear optics,” Rev. Mod. Phys. |

24. | S. Patchkovskii, Z. Zhao, T. Brabec, and D. M. Villeneuve, “High Harmonic Generation and Molecular Orbital Tomography in Multielectron Systems: Beyond the Single Active Electron Approximation,” Phys. Rev. Lett. |

25. | J. Courtial, K. Dholakia, L. Allen, and M. J. Padgett, “Second-harmonic generation and the conservation of orbital angular momentum with high-order Laguerre-Gaussian modes,” Phys. Rev. A |

26. | K. W. D. Ledingham, P. McKenna, and R. P. Singhal, “Applications for Nuclear Phenomena Generated by Ultra-Intense Lasers,” Science |

27. | T. J. Bürvenich, J. Evers, and C. H. Keitel, “Nuclear Quantum Optics with X-Ray Laser Pulses,” Phys. Rev. Lett. |

28. | E. Peik and Chr. Tamm, “Nuclear laser spectroscopy of the 3.5 eV transition in Th-229,” Europhys. Lett. |

**OCIS Codes**

(020.0020) Atomic and molecular physics : Atomic and molecular physics

(260.0260) Physical optics : Physical optics

**ToC Category:**

Atomic and Molecular Physics

**History**

Original Manuscript: November 2, 2009

Revised Manuscript: December 10, 2009

Manuscript Accepted: December 11, 2009

Published: February 5, 2010

**Citation**

A. Picón, J. Mompart, J. R. Vázquez de Aldana, L. Plaja, G. F. Calvo, and L. Roso, "Photoionization with orbital angular momentum beams," Opt. Express **18**, 3660-3671 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-4-3660

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### References

- L. Allen, M. V. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, "Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes," Phys. Rev. A 45, 8185 (1992). [CrossRef] [PubMed]
- A. T. O’Neil, I. MacVicar, L. Allen, and M. J. Padgett, "Intrinsic and Extrinsic Nature of the Orbital Angular Momentum of a Light Beam," Phys. Rev. Lett. 88, 053601 (2002). [CrossRef] [PubMed]
- M. Bhattacharya and P. Meystre, "Using a Laguerre-Gaussian Beam to Trap and Cool the Rotational Motion of a Mirror," Phys. Rev. Lett. 99, 153603 (2007). [CrossRef] [PubMed]
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