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Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 21, Iss. 10 — May. 20, 2013
  • pp: 12783–12789
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Second harmonic generation in isotropic media: six-wave mixing of optical vortices

Matt M. Coles, Mathew D. Williams, and David L. Andrews  »View Author Affiliations


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

When highly intense structured light propagates through an optically nonlinear medium, the output can display a variety of novel features exhibiting physically different attributes of the input. The best known illustration is second harmonic generation (SHG). Like all even harmonics, this process is supported by an optical susceptibility of even order, and therefore generally occurs in a non-centrosymmetric material, being forbidden in isotropic media to all multipole orders [1

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

]. In such a system, the input of a twisted beam, for example, will deliver an output in which each frequency-doubled photon also carries twice the orbital angular momentum of the pump [2

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

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]

]. Wave-vector matching ensures the conservation of linear as well as orbital angular momentum.

However, there are means of generating even order optical harmonics in a centrosymmetric medium. One such process, which engages an optical susceptibility of odd order, corresponds to the concerted interaction of an even number of photons. For example, SHG is allowed by a six-wave process in which four photons of pump input are converted into two photons of harmonic output, the latter equally disposed on the surface of a cone. In each individual conversion event, the two harmonic photons emerge on opposite sides of the cone, and wave-vector matching can be achieved even in the presence of significant optical dispersion by the medium. Following theoretical predictions by Andrews 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

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]

] this type of higher order nonlinear optical phenomenon has subsequently been demonstrated in experiments by Boyd 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]

], showing harmonic signals emerging from sapphire in a tightly confined cone of forward emission.

We now prove that new effects can be anticipated in six-wave mixing, when the input radiation has phase structure associated with orbital angular momentum. When two photons are produced in each harmonic conversion event, it is possible for the combined rules of linear and orbital angular momentum conservation to be satisfied in more than one possible outcome. The conditions over the angular disposition of the harmonic, together with the finely structured radial distributions of each emission mode, can thus generate a distinctive patterning in the output field. The process displays novel characteristics associated with phase singularities and topological charge conservation, and will be of particular interest for the on-going development of quantum state entanglement with heralded photon generation.

2. Six-wave mixing theory

To proceed within a quantum electrodynamic framework it is necessary to sum over all time orderings, or equally over paths through the states, for the process. To this end, and in order to identify the populations of each optical mode in the system states 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]

] based on a Hasse combinatorial diagram [9

9. D. R. Mazur, Combinatorics: A Guided Tour (Mathematical Association of America, 2010).

]. These diagrams are formally nets, in which the edge connections represent ordered pairs; in contrast to Feynman diagrams it is these edges that physically signify photon interactions, and the vertices, states. The corresponding state-sequence diagram is shown on the left in Fig. 1
Fig. 1 In the state sequence diagram (left), each row denotes a state of consecutive photon occupying number, n. Columns denote successive system states. Each vacant column and row in the tabular diagram represents an interaction, signifying a node of the corresponding Feynman diagram (right), for a given permutation. In the middle set of catawampus cells the wave-vector label for the emitted photon provides for a photon in either of the two output modes; both modes are populated in the upper set. Absorption ▬▬▬ (inclination) Emission ▬▬▬ (declination).
: since the two emitted photons have the same frequency, it is unnecessary to distinguish between them in this representation.

It is readily determined that there are fifteen quantum pathways linking the initial radiation state (with four input photons) to the final state (two emitted photons). Each of them corresponds to one of the Feynman graphs, and hence a contributory quantum amplitude, for the six-wave process. One of these fifteen terms – the dominant contribution to the process – corresponds both to the lowest pathway through the state-sequence diagram (as indicated by the bolder connection lines, the sequence traversing the zero-photon state), and the Feynman diagram shown on the right in Fig. 1.

Interpreting the Feynman diagrams in the usual fashion delivers the form of the material states for the intervening states R, S, T, U, V, thus providing the structure for the fifth rank optical susceptibility tensor χ(5). The full result for the quantum amplitude for the six-wave conversion at a position 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

10. D. L. Andrews and M. Babiker, eds., The Angular Momentum of Light (Cambridge University Press, 2013), Chap. 9.

], and using the usual implied summation over repeated tensor indices, we have:
MFI(r)=3(ck2ε0Ω)3e¯ie¯jekelemenχ(ij)(klmn)(5)ei(4kkk).r.
(2)
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), χ(ij)(klmn)(5)signifies a component of the index-symmetrized nonlinear susceptibility tensor, whose exact form is secured by applying the interaction Hamiltonian along each edge (interaction) connecting two nodes (states), for each path through the state sequence diagram:
χ(ij)(klmn)(5)=148r,s,t,u,vk,η,l,p[{μi0vμjvuμkutμltsμmsrμnr0[Er0ck][Es02ck][Et03ck][Eu04ck][Ev02ck]+...}+{perm(i,j)perm(k,l,m,n)}],
(3)
where 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.

The exponential phase factor in Eq. (2) exhibits a position-dependence of the amplitude that leads to the wave-vector matching condition,
4k=k+k,
(4)
responsible for coherent emission. In consequence, a conical form can be anticipated for the emergent harmonic in dispersive media, its apical angle determined by the refractive index of both the input and emergent radiation. It is now expedient to perform isotropic rotational averaging on Eq. (2) to model a random distribution of optical center orientations in the conversion material, as for example would relate to the six-wave mixing (SWM) experiments performed with fused silica glass [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]

]. Furthermore, as a parametric process, the matrix elements associated with different optical centers add constructively, allowing the rotational averaging to be performed on the probability amplitude [11

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

]. This form of second harmonic conversion entails an even number of photons, leading to an even rank rotational average, which delivers a non-zero result. The ensuing Fermi rate emerges as
Γ~|(e¯e¯)(ee)2{23χ(λλ)(μμνν)(5)20χ(λμ)(λμνν)(5)}+(e¯e)(e¯e)(ee){20χ(λλ)(μμνν)(5)+32χ(λμ)(λμνν)(5)}|2,
(5)
where Greek letter indices denote laboratory frame coordinates. As both terms are dependent on (ee), the use of a circularly polarized basis would deliver a vanishing result; the most efficient conversion rate is obtained by the use of plane polarized input radiation. Resolving the output into plane polarized components gives:
Γ~|{3χ(λλ)(μμνν)(5)+12χ(λμ)(λμνν)(5)}|2,
(6)
for polarizations parallel to the input, (e¯e)=(e¯e)=(ee)=(e¯e¯)=1, and
Γ~|{23χ(λλ)(μμνν)(5)20χ(λμ)(λμνν)(5)}|2,
(7)
for polarizations perpendicular to the input, (e¯e)=(e¯e)=0;(ee)=(e¯e¯)=1. For materials in which all tensor components have a similar value and sign, such that χ(λλ)(μμνν)(5)=χ(λμ)(λμνν)(5), the above results suggest the emergent radiation will be primarily polarized parallel to the input.

3. Plotting the conical emission

To address the six-wave mixing process for photons endowed with orbital angular momentum we model the radiation in terms of Laguerre-Gaussian (LG) modes. Calculation of the quantum amplitude proceeds, using the expression for the LG transverse displacement field,
d(r)=ik,η,l,p(ckε02Ω)12{el,p(η)(k)a(η)(k)fl,p(r)eikzilφh.c.},
(8)
where fl,p(r) represents the radial distribution function of the LG mode with radial number 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,
[fl0,p0(r)]4f¯l1,p1(r)f¯l2,p2(r)ei(4l0φl1φl2φ),
(9)
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 sin2(r()/z())/(r()/z())2, centred about a maximum at the angle corresponding to orbital angular momentum (OAM) conversion.

Here the parameters are chosen to simulate a laser at wavelength 800 nm, focused into the conversion material. In work by Boyd 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]

], consistent with the paraxial approximation to within 1%. Assuming an unchanged beam-waist, we calculate the various contributions to the output with different orbital angular momenta, through explicit application of the Fermi rate equation with appropriate initial and final states; this enables us to secure results directly, without mode filtering that would otherwise be required [12

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

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]

].

Plotted in Fig. 2
Fig. 2 Left: Schematic depiction of conical SWM with LG input. Middle: Intensity plots (arbitrary scale) of the emission in the two output beams shown on the left, for OAM combinations (l1, l2), at a distance of 100 wavelengths from the conversion material. Right: Cross-sectional intensity distribution around the (2,2) output delivered by l0 = 1, p = 0 input (centered on the input beam axis: distance as in the middle diagram). Red indicates high intensity; blue and black indicate low and zero intensities, respectively.
is the square modulus of Eq. (9), the part of the rate equation that displays the radial variation. For a process involving four l0 = 1 input photons, and given l1l2, there are three possible pairs of emergent photons: (l1, l2) = (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

Table 1. Relative Magnitudes of the Intensities for OAM Output Photon Pairs

table-icon
View This Table
. 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 (l1, l2) 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

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]

]: we return to this in the Conclusion. For output LG modes with 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 w2p dependence of the radial distribution functions, delivers successively smaller contributions for non-zero p modes.

For simplicity, and also to directly connect with the realm of most practicable experimental measurements, we have considered the form of the output in a region well removed from the conversion material – a distance of around one hundred input or two hundred output wavelengths. Clearly, in a closer region there will be significant overlap of the ring structures across the emission cone, and closer still there will be near-field features further complicating the structure of the emergent fields. However, in such regions where there is greater uncertainty over the resolvable angle of emission, it can be anticipated that there will be manifestations of an increasing angle-angular momentum quantum uncertainty [17

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

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

There is now a rapidly growing interest in the possibilities offered by engaging structured light with nonlinear optical mechanisms [21

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]

], most notably with four-wave parametric processes [22

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

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]

]. The higher-order interaction that we have described offers a fascinating insight into further characteristics deeply connected with the combination of quantum and nonlinear optical character. In practical terms, the angular distribution of the output presents a useful means of diverting beam components towards optical elements that can resolve their orbital angular momentum content, for example using one of the innovative sorting and detection schemes that have recently been reported [24

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

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]

].

One unusual feature of the interaction arises from the fact that it generates two photons of output – in which respect there are some similarities with degenerate down-conversion, despite the intrinsic character of an upconversion process. Prior to measurement of the OAM in either component of the output, the radiation field exists in a superposition of states with different (l1, l2) combinations. Generally, the detection of a photon with a specific OAM l1 in the sorted output can be considered to herald a counterpart with the complementary value l2 = 4l0l1. 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]

], with other interesting recent observations of unexpectedly weighted topological charge distributions in four-wave coupling interactions [27

27. S. Franke-Arnold, “Orbital angular momentum of photons, atoms, and electrons ,” Proc. SPIE 8637, (in press).

].

Acknowledgment

The authors thank EPSRC and the University of East Anglia for funding this research.

References and links

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]

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]

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]

7.

R. M. Eisberg and R. Resnick, Quantum Physics Of Atoms, Molecules, Solids, Nuclei and Particles (Wiley, 1985), Appendix K.

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]

9.

D. R. Mazur, Combinatorics: A Guided Tour (Mathematical Association of America, 2010).

10.

D. L. Andrews and M. Babiker, eds., The Angular Momentum of Light (Cambridge University Press, 2013), Chap. 9.

11.

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

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]

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]

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 86(1), 012334 (2012). [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]

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]

18.

S. Franke and S. M. Barnett, “Angular momentum in spontaneous emission,” J. Phys. At. Mol. Opt. Phys. 29(10), 2141–2150 (1996). [CrossRef]

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. 6, 103 (2004). [CrossRef]

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]

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]

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 8542, 85421R, (2012). [CrossRef]

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]

27.

S. Franke-Arnold, “Orbital angular momentum of photons, atoms, and electrons ,” Proc. SPIE 8637, (in press).

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

  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. A54(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]
  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]
  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]
  7. R. M. Eisberg and R. Resnick, Quantum Physics Of Atoms, Molecules, Solids, Nuclei and Particles (Wiley, 1985), Appendix K.
  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]
  9. D. R. Mazur, Combinatorics: A Guided Tour (Mathematical Association of America, 2010).
  10. D. L. Andrews and M. Babiker, eds., The Angular Momentum of Light (Cambridge University Press, 2013), Chap. 9.
  11. D. L. Andrews and T. Thirunamachandran, “On three-dimensional rotational averages,” J. Chem. Phys.67(11), 5026–5033 (1977). [CrossRef]
  12. D. Flamm, C. Schulze, D. Naidoo, A. Forbes, and M. Duparré, “Mode analysis using the correlation filter method,” Proc. SPIE8637, 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. Express17(11), 9347–9356 (2009). [CrossRef] [PubMed]
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