## Second harmonic generation in isotropic media: six-wave mixing of optical vortices |

Optics Express, Vol. 21, Issue 10, pp. 12783-12789 (2013)

http://dx.doi.org/10.1364/OE.21.012783

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### Abstract

Optical vortex light can be up-converted into a second harmonic output in an isotropic medium, in which such conversion is normally forbidden, through six-wave mixing. The involvement of orbital angular momentum is tackled for the case of a Laguerre-Gaussian pump comprising *l* = 1 photons. By calculating quantum amplitudes for the emergent radiation states, utilizing a state-sequence method, the analysis identifies the characteristics of the emission and an entangled distribution of conserved orbital momentum. A distinctive conical spread affords a potential means of resolving the individual angular momentum content.

© 2013 OSA

## 1. Introduction

1. D. L. Andrews, “Harmonic-generation in free molecules,” J. Phys. At. Mol. Opt. Phys. **13**(20), 4091–4099 (1980). [CrossRef]

2. K. Dholakia, N. B. Simpson, M. J. Padgett, and L. Allen, “Second-harmonic generation and the orbital angular momentum of light,” Phys. Rev. A **54**(5), R3742–R3745 (1996). [CrossRef] [PubMed]

3. L. C. Dávila Romero, D. L. Andrews, and M. Babiker, “A quantum electrodynamics framework for the nonlinear optics of twisted beams,” J. Opt. B Quantum Semiclassical Opt. **4**(2), S66–S72 (2002). [CrossRef]

*et al*. [4

4. P. Allcock and D. L. Andrews, “Six-wave mixing: secular resonances in a higher-order mechanism for second-harmonic generation,” J. Phys. At. Mol. Opt. Phys. **30**(16), 3731–3742 (1997). [CrossRef]

5. I. D. Hands, S. J. Lin, S. R. Meech, and D. L. Andrews, “A quantum electrodynamical treatment of second harmonic generation through phase conjugate six-wave mixing: Polarization analysis,” J. Chem. Phys. **109**(24), 10580–10586 (1998). [CrossRef]

*et al*. [6

6. K. D. Moll, D. Homoelle, A. L. Gaeta, and R. W. Boyd, “Conical Harmonic Generation in Isotropic Materials,” Phys. Rev. Lett. **88**(15), 153901 (2002). [CrossRef] [PubMed]

## 2. Six-wave mixing theory

4. P. Allcock and D. L. Andrews, “Six-wave mixing: secular resonances in a higher-order mechanism for second-harmonic generation,” J. Phys. At. Mol. Opt. Phys. **30**(16), 3731–3742 (1997). [CrossRef]

*I, F*denote the initial and final system states,

*R*,

*S*,

*T*,

*U*,

*V*, denote virtual intermediate states, and

*E*is the energy of the state defined by the subscript. In the electric dipole approximation the interaction Hamiltonian has as its leading term

**k**, polarization

*η*and circular frequency

*ω*, are converted to two photons, the latter having wave-vectors

**k′**,

**k′′**and polarizations

*η*′,

*η*′′, respectively. By virtue of energy conservation, both of the output photons have the same frequency

*ω*′ = 2

*ω*; however, they are allowed to have different directions of propagation.

*R*,

*S*,

*T*,

*U*,

*V*, it is expedient to use a recently developed method [8

8. R. D. Jenkins, D. L. Andrews, and L. C. Dávila Romero, “A new diagrammatic methodology for non-relativistic quantum electrodynamics,” J. Phys. At. Mol. Opt. Phys. **35**(3), 445–468 (2002). [CrossRef]

*R*,

*S*,

*T*,

*U*,

*V*, thus providing the structure for the fifth rank optical susceptibility tensor

**r**now emerges in a form that is readily cast in terms of the Cartesian components of polarization vectors for the input and output,

**e**and

**e**′ respectively. Implementing the electric dipole approximation using Eq. (1) and the vortex field interaction operator in the mode expansion [10], and using the usual implied summation over repeated tensor indices, we have:This result assumes a basis of plane wave Fock states for the input radiation, quantized in a volume Ω – at this stage without involvement of the orbital angular momentum which is to be our focus in the following. The quantization volume is here defined as the volume that contains the energy of the four input and two output photons. In Eq. (2),

*perm*denotes all permutations of the index set in parentheses. It transpires that the explicitly given term, the first of fifteen that feature within curly braces, delivers the major contribution, corresponding to the lowest pathway through the state-sequences shown in Fig. 1. This term dominates over the other contributions because each of its denominator factors can, in effecting the corresponding summations over electronic states

*r*,

*s*,

*t*,

*u*,

*v*, produce a diminutive result. The other fourteen terms in the curly braces are derived from the other routes through the state-sequence diagram.

6. K. D. Moll, D. Homoelle, A. L. Gaeta, and R. W. Boyd, “Conical Harmonic Generation in Isotropic Materials,” Phys. Rev. Lett. **88**(15), 153901 (2002). [CrossRef] [PubMed]

11. D. L. Andrews and T. Thirunamachandran, “On three-dimensional rotational averages,” J. Chem. Phys. **67**(11), 5026–5033 (1977). [CrossRef]

## 3. Plotting the conical emission

*p*, azimuthal index

*l,*and

*h.c.*denotes Hermitian conjugate. In the following, the subscript 0 represents the LG mode of the four input photons, and subscripts 1 and 2 those of the two, potentially different, emergent photons. Accordingly, the SWM quantum amplitude emerges in a result identical to that in Eq. (2), except for an additional multiplicative factor,where primed variables denote the cylindrical polar coordinates in the rotated frames of reference for the two output photons. It is this part of the matrix element that determines radial variation of the emerging radiation. Here we focus on output that conserves orbital angular momentum; this is a valid approximation as a strict analysis shows the output distribution as dependent on

*et al.*the cone angle for third harmonic generation (determined by arccos of the ratio of refractive indices for the output, relative to the value for the input wavelength) is approximately 10°-12° [6

6. K. D. Moll, D. Homoelle, A. L. Gaeta, and R. W. Boyd, “Conical Harmonic Generation in Isotropic Materials,” Phys. Rev. Lett. **88**(15), 153901 (2002). [CrossRef] [PubMed]

12. D. Flamm, C. Schulze, D. Naidoo, A. Forbes, and M. Duparré, “Mode analysis using the correlation filter method,” Proc. SPIE **8637**, 863717 (2013). [CrossRef]

13. T. Kaiser, D. Flamm, S. Schröter, and M. Duparré, “Complete modal decomposition for optical fibers using CGH-based correlation filters,” Opt. Express **17**(11), 9347–9356 (2009). [CrossRef] [PubMed]

*l*

_{0}

*=*1 input photons, and given

*l*

_{1},

*l*

_{2})

*=*(2,2); (3,1); (4,0). Interestingly, while the output profile displays no structural differences between the unique combinations of emergent pairs, the relative magnitudes have a neat relationship, shown in Table 1. The Pascal’s triangle form, which arises without any assumption of combinatorial weighting, serves as an independent verification of the calculations. Moreover, the pairwise matching of (

*l*

_{1},

*l*

_{2}) values clearly indicates quantum entanglement between the generated optical states, as has recently been observed in other optical vortex studies [14

14. S. Franke-Arnold, S. M. Barnett, M. J. Padgett, and L. Allen, “Two-photon entanglement of orbital angular momentum states,” Phys. Rev. A **65**(3), 033823 (2002). [CrossRef]

16. A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature **412**(6844), 313–316 (2001). [CrossRef] [PubMed]

*p >*0, the beam width,

*w*, increases outwards with a monotonic dependence on

*p*, from the

*p*= 0 counterpart. Hence the Fermi rate, which inherits the

*p*modes.

17. G. Gibson, J. Courtial, M. J. Padgett, M. Vasnetsov, V. Pas’ko, S. M. Barnett, and S. Franke-Arnold, “Free-space information transfer using light beams carrying orbital angular momentum,” Opt. Express **12**(22), 5448–5456 (2004). [CrossRef] [PubMed]

20. H. H. Arnaut and G. A. Barbosa, “Orbital and intrinsic angular momentum of single photons and entangled pairs of photons generated by parametric down-conversion,” Phys. Rev. Lett. **85**(2), 286–289 (2000). [CrossRef] [PubMed]

## 4. Conclusion

21. D. Shwa, E. Shtranvasser, Y. Shalibo, and N. Katz, “Controllable motion of optical vortex arrays using electromagnetically induced transparency,” Opt. Express **20**(22), 24835–24842 (2012). [CrossRef] [PubMed]

22. N. Olivier, D. DéBarre, P. Mahou, and E. Beaurepaire, “Third-harmonic generation microscopy with Bessel beams: a numerical study,” Opt. Express **20**(22), 24886–24902 (2012). [CrossRef] [PubMed]

23. M. T. Cao, L. Han, R. F. Liu, H. Liu, D. Wei, P. Zhang, Y. Zhou, W. G. Guo, S. G. Zhang, H. Gao, and F. L. Li, “Deutsch’s algorithm with topological charges of optical vortices via non-degenerate four-wave mixing,” Opt. Express **20**(22), 24263–24271 (2012). [CrossRef] [PubMed]

24. M. N. O’Sullivan, M. Mirhosseini, M. Malik, and R. W. Boyd, “Near-perfect sorting of orbital angular momentum and angular position states of light,” Opt. Express **20**(22), 24444–24449 (2012). [CrossRef] [PubMed]

26. J. Romero, D. Giovannini, S. Franke-Arnold, S. Barnett, and M. Padgett, “Increasing the dimension in high-dimensional two-photon orbital angular momentum entanglement,” Phys. Rev. A **86**(1), 012334 (2012). [CrossRef]

*l*

_{1},

*l*

_{2}) combinations. Generally, the detection of a photon with a specific OAM

*l*

_{1}in the sorted output can be considered to herald a counterpart with the complementary value

*l*

_{2}= 4

*l*

_{0}–

*l*

_{1}. Thus, at a fundamental level the distribution of orbital angular momentum in the twin harmonic outputs presents new opportunities to explore features in the entanglement of structured light. Such features have become a particular focus [26

26. J. Romero, D. Giovannini, S. Franke-Arnold, S. Barnett, and M. Padgett, “Increasing the dimension in high-dimensional two-photon orbital angular momentum entanglement,” Phys. Rev. A **86**(1), 012334 (2012). [CrossRef]

## Acknowledgment

## References and links

1. | D. L. Andrews, “Harmonic-generation in free molecules,” J. Phys. At. Mol. Opt. Phys. |

2. | K. Dholakia, N. B. Simpson, M. J. Padgett, and L. Allen, “Second-harmonic generation and the orbital angular momentum of light,” Phys. Rev. A |

3. | L. C. Dávila Romero, D. L. Andrews, and M. Babiker, “A quantum electrodynamics framework for the nonlinear optics of twisted beams,” J. Opt. B Quantum Semiclassical Opt. |

4. | P. Allcock and D. L. Andrews, “Six-wave mixing: secular resonances in a higher-order mechanism for second-harmonic generation,” J. Phys. At. Mol. Opt. Phys. |

5. | I. D. Hands, S. J. Lin, S. R. Meech, and D. L. Andrews, “A quantum electrodynamical treatment of second harmonic generation through phase conjugate six-wave mixing: Polarization analysis,” J. Chem. Phys. |

6. | K. D. Moll, D. Homoelle, A. L. Gaeta, and R. W. Boyd, “Conical Harmonic Generation in Isotropic Materials,” Phys. Rev. Lett. |

7. | R. M. Eisberg and R. Resnick, |

8. | R. D. Jenkins, D. L. Andrews, and L. C. Dávila Romero, “A new diagrammatic methodology for non-relativistic quantum electrodynamics,” J. Phys. At. Mol. Opt. Phys. |

9. | D. R. Mazur, |

10. | D. L. Andrews and M. Babiker, eds., |

11. | D. L. Andrews and T. Thirunamachandran, “On three-dimensional rotational averages,” J. Chem. Phys. |

12. | D. Flamm, C. Schulze, D. Naidoo, A. Forbes, and M. Duparré, “Mode analysis using the correlation filter method,” Proc. SPIE |

13. | T. Kaiser, D. Flamm, S. Schröter, and M. Duparré, “Complete modal decomposition for optical fibers using CGH-based correlation filters,” Opt. Express |

14. | S. Franke-Arnold, S. M. Barnett, M. J. Padgett, and L. Allen, “Two-photon entanglement of orbital angular momentum states,” Phys. Rev. A |

15. | J. Romero, D. Giovannini, S. Franke-Arnold, S. M. Barnett, and M. J. Padgett, “Increasing the dimension in high-dimensional two-photon orbital angular momentum entanglement,” Phys. Rev. A |

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

17. | G. Gibson, J. Courtial, M. J. Padgett, M. Vasnetsov, V. Pas’ko, S. M. Barnett, and S. Franke-Arnold, “Free-space information transfer using light beams carrying orbital angular momentum,” Opt. Express |

18. | S. Franke and S. M. Barnett, “Angular momentum in spontaneous emission,” J. Phys. At. Mol. Opt. Phys. |

19. | S. Franke-Arnold, S. M. Barnett, E. Yao, J. Leach, J. Courtial, and M. Padgett, “Uncertainty principle for angular position and angular momentum,” New J. Phys. |

20. | H. H. Arnaut and G. A. Barbosa, “Orbital and intrinsic angular momentum of single photons and entangled pairs of photons generated by parametric down-conversion,” Phys. Rev. Lett. |

21. | D. Shwa, E. Shtranvasser, Y. Shalibo, and N. Katz, “Controllable motion of optical vortex arrays using electromagnetically induced transparency,” Opt. Express |

22. | N. Olivier, D. DéBarre, P. Mahou, and E. Beaurepaire, “Third-harmonic generation microscopy with Bessel beams: a numerical study,” Opt. Express |

23. | M. T. Cao, L. Han, R. F. Liu, H. Liu, D. Wei, P. Zhang, Y. Zhou, W. G. Guo, S. G. Zhang, H. Gao, and F. L. Li, “Deutsch’s algorithm with topological charges of optical vortices via non-degenerate four-wave mixing,” Opt. Express |

24. | M. N. O’Sullivan, M. Mirhosseini, M. Malik, and R. W. Boyd, “Near-perfect sorting of orbital angular momentum and angular position states of light,” Opt. Express |

25. | M. P. J. Lavery, D. Robertson, M. Malik, B. Rodenburg, J. Courtial, R. W. Boyd, and M. J. Padgett, “The efficient sorting of light's orbital angular momentum for optical communications,” Proc. SPIE |

26. | J. Romero, D. Giovannini, S. Franke-Arnold, S. Barnett, and M. Padgett, “Increasing the dimension in high-dimensional two-photon orbital angular momentum entanglement,” Phys. Rev. A |

27. | S. Franke-Arnold, “Orbital angular momentum of photons, atoms, and electrons ,” Proc. SPIE |

**OCIS Codes**

(020.4180) Atomic and molecular physics : Multiphoton processes

(190.2620) Nonlinear optics : Harmonic generation and mixing

(190.4160) Nonlinear optics : Multiharmonic generation

(270.4180) Quantum optics : Multiphoton processes

(270.5580) Quantum optics : Quantum electrodynamics

(050.4865) Diffraction and gratings : Optical vortices

**ToC Category:**

Nonlinear Optics

**History**

Original Manuscript: February 22, 2013

Revised Manuscript: April 10, 2013

Manuscript Accepted: April 10, 2013

Published: May 16, 2013

**Citation**

Matt M. Coles, Mathew D. Williams, and David L. Andrews, "Second harmonic generation in isotropic media: six-wave mixing of optical vortices," Opt. Express **21**, 12783-12789 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-10-12783

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### References

- D. L. Andrews, “Harmonic-generation in free molecules,” J. Phys. At. Mol. Opt. Phys.13(20), 4091–4099 (1980). [CrossRef]
- K. Dholakia, N. B. Simpson, M. J. Padgett, and L. Allen, “Second-harmonic generation and the orbital angular momentum of light,” Phys. Rev. A54(5), R3742–R3745 (1996). [CrossRef] [PubMed]
- L. C. Dávila Romero, D. L. Andrews, and M. Babiker, “A quantum electrodynamics framework for the nonlinear optics of twisted beams,” J. Opt. B Quantum Semiclassical Opt.4(2), S66–S72 (2002). [CrossRef]
- P. Allcock and D. L. Andrews, “Six-wave mixing: secular resonances in a higher-order mechanism for second-harmonic generation,” J. Phys. At. Mol. Opt. Phys.30(16), 3731–3742 (1997). [CrossRef]
- I. D. Hands, S. J. Lin, S. R. Meech, and D. L. Andrews, “A quantum electrodynamical treatment of second harmonic generation through phase conjugate six-wave mixing: Polarization analysis,” J. Chem. Phys.109(24), 10580–10586 (1998). [CrossRef]
- K. D. Moll, D. Homoelle, A. L. Gaeta, and R. W. Boyd, “Conical Harmonic Generation in Isotropic Materials,” Phys. Rev. Lett.88(15), 153901 (2002). [CrossRef] [PubMed]
- R. M. Eisberg and R. Resnick, Quantum Physics Of Atoms, Molecules, Solids, Nuclei and Particles (Wiley, 1985), Appendix K.
- R. D. Jenkins, D. L. Andrews, and L. C. Dávila Romero, “A new diagrammatic methodology for non-relativistic quantum electrodynamics,” J. Phys. At. Mol. Opt. Phys.35(3), 445–468 (2002). [CrossRef]
- D. R. Mazur, Combinatorics: A Guided Tour (Mathematical Association of America, 2010).
- D. L. Andrews and M. Babiker, eds., The Angular Momentum of Light (Cambridge University Press, 2013), Chap. 9.
- D. L. Andrews and T. Thirunamachandran, “On three-dimensional rotational averages,” J. Chem. Phys.67(11), 5026–5033 (1977). [CrossRef]
- D. Flamm, C. Schulze, D. Naidoo, A. Forbes, and M. Duparré, “Mode analysis using the correlation filter method,” Proc. SPIE8637, 863717 (2013). [CrossRef]
- T. Kaiser, D. Flamm, S. Schröter, and M. Duparré, “Complete modal decomposition for optical fibers using CGH-based correlation filters,” Opt. Express17(11), 9347–9356 (2009). [CrossRef] [PubMed]
- S. Franke-Arnold, S. M. Barnett, M. J. Padgett, and L. Allen, “Two-photon entanglement of orbital angular momentum states,” Phys. Rev. A65(3), 033823 (2002). [CrossRef]
- J. Romero, D. Giovannini, S. Franke-Arnold, S. M. Barnett, and M. J. Padgett, “Increasing the dimension in high-dimensional two-photon orbital angular momentum entanglement,” Phys. Rev. A86(1), 012334 (2012). [CrossRef]
- A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature412(6844), 313–316 (2001). [CrossRef] [PubMed]
- G. Gibson, J. Courtial, M. J. Padgett, M. Vasnetsov, V. Pas’ko, S. M. Barnett, and S. Franke-Arnold, “Free-space information transfer using light beams carrying orbital angular momentum,” Opt. Express12(22), 5448–5456 (2004). [CrossRef] [PubMed]
- S. Franke and S. M. Barnett, “Angular momentum in spontaneous emission,” J. Phys. At. Mol. Opt. Phys.29(10), 2141–2150 (1996). [CrossRef]
- S. Franke-Arnold, S. M. Barnett, E. Yao, J. Leach, J. Courtial, and M. Padgett, “Uncertainty principle for angular position and angular momentum,” New J. Phys.6, 103 (2004). [CrossRef]
- H. H. Arnaut and G. A. Barbosa, “Orbital and intrinsic angular momentum of single photons and entangled pairs of photons generated by parametric down-conversion,” Phys. Rev. Lett.85(2), 286–289 (2000). [CrossRef] [PubMed]
- D. Shwa, E. Shtranvasser, Y. Shalibo, and N. Katz, “Controllable motion of optical vortex arrays using electromagnetically induced transparency,” Opt. Express20(22), 24835–24842 (2012). [CrossRef] [PubMed]
- N. Olivier, D. DéBarre, P. Mahou, and E. Beaurepaire, “Third-harmonic generation microscopy with Bessel beams: a numerical study,” Opt. Express20(22), 24886–24902 (2012). [CrossRef] [PubMed]
- M. T. Cao, L. Han, R. F. Liu, H. Liu, D. Wei, P. Zhang, Y. Zhou, W. G. Guo, S. G. Zhang, H. Gao, and F. L. Li, “Deutsch’s algorithm with topological charges of optical vortices via non-degenerate four-wave mixing,” Opt. Express20(22), 24263–24271 (2012). [CrossRef] [PubMed]
- M. N. O’Sullivan, M. Mirhosseini, M. Malik, and R. W. Boyd, “Near-perfect sorting of orbital angular momentum and angular position states of light,” Opt. Express20(22), 24444–24449 (2012). [CrossRef] [PubMed]
- M. P. J. Lavery, D. Robertson, M. Malik, B. Rodenburg, J. Courtial, R. W. Boyd, and M. J. Padgett, “The efficient sorting of light's orbital angular momentum for optical communications,” Proc. SPIE8542, 85421R, (2012). [CrossRef]
- J. Romero, D. Giovannini, S. Franke-Arnold, S. Barnett, and M. Padgett, “Increasing the dimension in high-dimensional two-photon orbital angular momentum entanglement,” Phys. Rev. A86(1), 012334 (2012). [CrossRef]
- S. Franke-Arnold, “Orbital angular momentum of photons, atoms, and electrons,” Proc. SPIE 8637, (in press).

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