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

Matt M. Coles; Mathew D. Williams; David L. Andrews

doi:10.1364/OE.21.012783

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]. 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, 3]. 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, 5] this type of higher order nonlinear optical phenomenon has subsequently been demonstrated in experiments by Boyd et al. [6], 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 establish a theoretical foundation for experimental observations of SHG in an isotropic medium, we first derive the Fermi rate [7] for six-wave mixing. It is expedient to formulate theory in terms of the associated quantum amplitude, which, for the process of interest, can be developed by standard methods reported in previous work [4]. The leading term in the quantum amplitude for the considered process is derived from sixth order perturbation theory:

M_{F I}^{(6)} = \sum_{R, S, T, U, V \neq I, F} \frac{〈 F | H_{int} | V 〉 〈 V | H_{int} | U 〉 〈 U | H_{int} | T 〉 〈 T | H_{int} | S 〉 〈 S | H_{int} | R 〉 〈 R | H_{int} | I 〉}{(E_{I} - E_{R}) (E_{I} - E_{S}) (E_{I} - E_{T}) (E_{I} - E_{U}) (E_{I} - E_{V})},

where 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

H_{int} = ε_{0}^{- 1} μ \cdot d^{⊥}

and as usual the electric displacement field has a mode expansion in terms of photon creation and annihilation operators. In the six-wave process of interest, four photons of wave-vector 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 ′ = 2ω; however, they are allowed to have different directions of propagation.

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] based on a Hasse combinatorial diagram [9]. 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: since the two emitted photons have the same frequency, it is unnecessary to distinguish between them in this representation.

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).

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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], and using the usual implied summation over repeated tensor indices, we have:

M_{F I} (r) = \sqrt{3} {(​ \frac{ℏ c k}{\sqrt{2} ε_{0} Ω})}^{3} {\bar{e}}_{i}^{' ​'} {\bar{e}}_{j}^{'} e_{k} e_{l} e_{m} e_{n} χ_{(i j) (k l m n)}^{(5)} e^{i (4 k - k^{'} - k^{″}) . r} .

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_{(i j) (k l m n)}^{(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:

\begin{array}{l} χ_{(i j) (k l m n)}^{(5)} = - \frac{1}{48} \sum_{\begin{array}{l} r, s, t, u, v \\ k, η, l, p \end{array}} [{\frac{μ_{i}^{0 v} μ_{j}^{v u} μ_{k}^{u t} μ_{l}^{t s} μ_{m}^{s r} μ_{n}^{r 0}}{[E_{r 0} - ℏ c k] [E_{s 0} - 2 ℏ c k] [E_{t 0} - 3 ℏ c k] [E_{u 0} - 4 ℏ c k] [E_{v 0} - 2 ℏ c k]} + ...} \\ + {p e r m (i, j) p e r m (k, l, m, n)}], \end{array}

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,

4 k = k^{'} + k^{″},

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]. 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]. 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

\begin{array}{l} Γ ~ | ({\bar{e}}^{'} \cdot {\bar{e}}^{″}) {(e \cdot e)}^{2} {23 χ_{(λ λ) (μ μ ν ν)}^{(5)} - 20 χ_{(λ μ) (λ μ ν ν)}^{(5)}} + \\ {({\bar{e}}^{'} \cdot e) ({\bar{e}}^{″} \cdot e) (e \cdot e) {- 20 χ_{(λ λ) (μ μ ν ν)}^{(5)} + 32 χ_{(λ μ) (λ μ ν ν)}^{(5)}} |}^{2}, \end{array}

where Greek letter indices denote laboratory frame coordinates. As both terms are dependent on

(e \cdot e)

, 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 ~ {| {3 χ_{(λ λ) (μ μ ν ν)}^{(5)} + 12 χ_{(λ μ) (λ μ ν ν)}^{(5)}} |}^{2},

for polarizations parallel to the input,

({\bar{e}}^{'} \cdot e) = ({\bar{e}}^{″} \cdot e) = (e \cdot e) = ({\bar{e}}^{'} \cdot {\bar{e}}^{″}) = 1

, and

Γ ~ {| {23 χ_{(λ λ) (μ μ ν ν)}^{(5)} - 20 χ_{(λ μ) (λ μ ν ν)}^{(5)}} |}^{2},

for polarizations perpendicular to the input,

({\bar{e}}^{'} \cdot e) = ({\bar{e}}^{″} \cdot e) = 0; (e \cdot e) = ({\bar{e}}^{'} \cdot {\bar{e}}^{″}) = 1

. For materials in which all tensor components have a similar value and sign, such that

χ_{(λ λ) (μ μ ν ν)}^{(5)} = χ_{(λ μ) (λ μ ν ν)}^{(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) = i {\sum_{k, η, l, p} (\frac{ℏ c k ε_{0}}{2 Ω})}^{\frac{1}{2}} {e_{l, p}^{(η)} (k) a^{(η)} (k) f_{l, p} (r) e^{i k z - i l φ} - h . c .},

where

f_{l, 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,

{[f_{l_{0}, p_{0}} (r)]}^{4} {\bar{f}}_{l_{1}, p_{1}} (r^{'}) {\bar{f}}_{l_{2}, p_{2}} (r^{″}) e^{- i (4 l_{0} φ - l_{1} φ^{'} - l_{2} φ^{″})},

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

\sin^{2} ({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], 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, 13].

Plotted in Fig. 2 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 $l_{1} \geq l_{2}$ , 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. 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 [14–16]: 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 $w^{- 2 p}$ dependence of the radial distribution functions, delivers successively smaller contributions for non-zero p modes.

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.

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

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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 [17–20].

4. Conclusion

There is now a rapidly growing interest in the possibilities offered by engaging structured light with nonlinear optical mechanisms [21], most notably with four-wave parametric processes [22, 23]. 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 [24–26].

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 = 4l₀ – l₁. 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], with other interesting recent observations of unexpectedly weighted topological charge distributions in four-wave coupling interactions [27].

Acknowledgment

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

References and links

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

Abstract

Corrections

1. Introduction

2. Six-wave mixing theory

3. Plotting the conical emission

4. Conclusion

Acknowledgment

References and links

Cited By

Figures (2)

Tables (1)

Equations (9)

Optics Express

l =	(4,0)	(3,1)	(2,2)
Mag.	1	4	6