1. INTRODUCTION

Second-harmonic generation (SHG) in doped-silica fibers was discovered in the early 1980s and intensively studied in the following years. In 1986 conversion efficiencies as high as 3–5% were reported [1]. A subsequent theoretical study [2] showed that such high efficiencies were orders of magnitude above what could reasonably be explained by interface or bulk multipole contributions to the second-order susceptibility, ${\chi }^{\left(2\right)}$. Since the isotropic nature of amorphous silica forbids the presence of a (bulk) dipole contribution to ${\chi }^{\left(2\right)}$, the origin of the experimentally observed phenomena was initially a puzzle, but was eventually explained by the formation of a bulk dipole ${\chi }^{\left(2\right)}$ through multiphoton processes involving both pump and SHG light [3]. In this model, the multiphoton ${\chi }^{\left(2\right)}$ inscription will naturally lead to the formation of a ${\chi }^{\left(2\right)}$ grating that compensates the phase mismatch arising from waveguide and material dispersion in the fiber. It also explains why the efficient SHG reported in [1] was only found after exposing the fiber to pump radiation for a certain incubation period, which was subsequently found to be shortened by the presence of seed SHG radiation [4]. However, in spite of much effort it has proved impossible to scale up the SHG efficiency beyond the level of a few percent. This can be explained as a self-saturation effect caused by the interference of the SHG light itself with the ${\chi }^{\left(2\right)}$ grating formed in the fiber [3].

Recent experiments on second- and third-harmonic generation in pure-silica nanowires found that a noticeable SHG signal could be observed in a nanowire with an estimated interaction length of only $\sim 100\mu \mathrm{m}$ [5]. The authors attributed this finding to a surface ${\chi }^{\left(2\right)}$ effect. Indeed, the advent of nanowire technology [6] calls for a reexamination of the surface and bulk multipole ${\chi }^{\left(2\right)}$ effects in fibers. This is because the nanowires offer a much larger index contrast than the values assumed in [2], allowing SHG phase matching between the fundamental and second-order modes, and greatly increasing the achievable intensities at the surface. Furthermore, the microstructured optical fibers developed during the last decade offer a possibility of manufacturing nanoscale fiber cores embedded in a fiber with a sufficient outer diameter to allow for practical handling outside of the laboratory. However, the complex geometries of microstructured fibers imply that the analytical expressions valid for circular step-index fibers cannot be used, and must be replaced by numerical simulations.

The purpose of this work is to investigate the prospects for surface and bulk multipole SHG by phase matching between fundamental and second-order modes in circular as well as structured nanowires. While circular nanowires are studied by analytical methods, the structured fibers will be studied numerically using the finite-element method. It will be shown that significant power conversion can in principle be achieved in nanowire lengths of a few cm, but that structural fluctuations along the nanowire are likely to be a severe limitation, at least for silica-based nanowires.

2. THEORY AND NUMERICAL METHODS

In the small-signal limit where pump depletion is negligible the SHG process can be described by the equation

In the case of a circular silica nanowire, analytic solutions for the guided-mode electromagnetic fields are available. Throughout this work, the input (pump) field will be assumed to propagate in the fundamental ${\mathrm{HE}}_{11}$ mode of the fiber, while SHG (signal) field generation in either the ${\mathrm{TM}}_{01}$ or the ${\mathrm{HE}}_{21}$ modes will be considered. Working in cylindrical coordinates, the fields of the ${\mathrm{HE}}_{n1}$ modes inside the silica nanowire can be expressed as [9]

Whereas the case of a nanowire with a circular transverse profile can be treated analytically, numerical solutions are required for the microstructured nanowire. In the present work, guided-mode fields and propagation constants were determined using the finite-element modeling tool COMSOL3.5. The generic fiber structure is depicted in Fig. 1 . The three parameters characterizing the structure are the outer radius of the microstructure R, the bridge width $W_b$, and the curvature radius of the curved core surfaces $r_c$. In the calculations, R was set to $6\mu \mathrm{m}$, and absorbing perfectly-matched-layer boundary conditions were applied beyond that radius, to emulate a fiber with a large ring of massive silica surrounding the microstructure. The parameters $r_c$ and $W_b∕r_c$ were taken as variables. The material dispersion of pure silica was described by a three-term Sellmeier polynomial, with the coefficients given by Okamoto [9].

In the limit of a small SHG signal, and assuming perfect phase matching, $\Delta \beta =0$, the evolution of the SHG intensity is given by:

Away from perfect phase matching, but still assuming an undepleted pump, Eq. (1) can be integrated to

3. NUMERICAL RESULTS

3A. Circular Nanowire

Surface and bulk nonlinear coefficients were obtained for pump radiation in the fundamental ${\mathrm{HE}}_{11}$ mode and SHG radiation in the ${\mathrm{TM}}_{01}$ and ${\mathrm{HE}}_{21}$ modes. The phase-matched SHG wavelengths, ${\lambda }_{\mathit{SHG}}$, for ${\mathrm{TM}}_{01}$ and ${\mathrm{HE}}_{21}$ modes are plotted against the nanowire diameter in Fig. 2 . ${\lambda }_{\mathit{SHG}}$ is seen to be roughly similar to the fiber diameter, which implies that phase matching happens at a V-parameter value of $\sim 1.5$ for the fundamental wavelength, where $V=\pi d\sqrt{n_s^2-1}∕\lambda$ is the usual V-parameter as defined in the theory of step-index fibers. The difference in ${\lambda }_{\mathit{SHG}}$ between ${\mathrm{TM}}_{01}$ and ${\mathrm{HE}}_{21}$ modes is about 10%.

In Fig. 3 , the sum of all SHG nonlinear coefficients, ${\rho }_{\mathit{tot}}$ is plotted together with the individual terms as calculated from Eqs. (20, 21, 22, 23, 24, 25). For the ${\mathrm{TM}}_{01}$ mode, the ${\rho }_s^{\perp }$ term is clearly dominant, whereas for the ${\mathrm{HE}}_{21}$ mode both the ${\rho }_s^{\perp }$ and ${\rho }_s^{\parallel }$ give significant contributions to the total sum. The total SHG coefficient roughly scales with ${\lambda }^{-3}$. This is readily understood from Eq. (2) and Fig. 2. The fiber diameter scales with λ, which implies that the electric field strength for a fixed total power must scale with ${\lambda }^{-1}$. The ${\lambda }^{-1}$ scaling of the ${\omega }_2$ prefactor in Eq. (2) is then cancelled by the λ scaling of the surface area over which the fields are integrated.

3B. Microstructured Nanowire

Replacing the perfectly circular nanowire with the microstructure depicted in Fig. 1 has significant implications for the phase matching conditions. The general modal structure known from the circular nanowire is preserved, and one can find second-order modes with field distributions resembling both the ${\mathrm{TM}}_{01}$ and ${\mathrm{HE}}_{21}$ modes. The symmetry arguments that imply that only two second-order modes are coupled to a given fundamental mode by the surface and bulk SHG terms also hold up because of the mirror symmetry of the fiber in a plane running parallel to any of the three core-supporting struts through its center. However, if the struts are regarded as slab waveguides coupled to the core region of the microstructure, the transverse field of the ${\mathrm{TM}}_{01}$-like core mode matches well onto the TE modes in any of these slab waveguides. The ${\mathrm{HE}}_{21}$ mode, on the other hand, can only match well onto the TE mode in one of the struts, and the ${\mathrm{TE}}_{01}$ mode always matches onto the TM modes of the struts. As a result, the ${\mathrm{TM}}_{01}$-like core mode effectively sees a higher cladding index in the microstructure than the ${\mathrm{HE}}_{21}$ and ${\mathrm{TE}}_{01}$ modes, and therefore ends up having the highest effective index. Furthermore, the effective cladding index increases with increasing frequency, which makes SHG phase matching more problematic, because both the increasing material index and the increasing cladding index must now be compensated by the effective-index difference between fundamental and second-order modes.

In the present work, only values of $W_b∕r_c⩾0.03$ were studied. Since the $r_c$ values where phase matching occurs turn out to be of the order of $1\mu \mathrm{m}$, this corresponds to a strut width of a few tens of nm. It is unclear whether still narrower struts can be fabricated with a reasonable accuracy. With these values of $W_b∕r_c$, it was found that the frequency-doubled ${\mathrm{TM}}_{01}$-like core mode could not be phase matched to the fundamental mode, at least not in a regime where both modes were reasonably well confined in the microstructure. On the other hand, the ${\mathrm{HE}}_{21}$-like mode could be phase-matched in geometries where both the fundamental and second-order modes were reasonably well confined.

In Fig. 4 the phase matching curves and the sum of surface contributions to the SHG coefficients are reported. The bulk contributions were ignored, since the results for the silica nanowires indicated them to be of minor importance. Results are shown as a function of the structural parameter $r_c$ (see Fig. 1) for different values of the strut width relative to $r_c$. For $W_b=0.07r_c$, phase matching to the ${\mathrm{HE}}_{21}$ mode could no longer be found. However, for the lower bridge widths, the variations with $W_b$ are within $\sim 50\mathrm{\%}$. Comparing to the ideal silica nanowire, the nonlinear coefficients for a given SHG wavelength are reduced by a factor of roughly 0.6–0.8 in the microstructures. This can be understood as a consequence of the phase matching appearing in a regime where the guided modes are less well confined because of the above mentioned influence of the struts on the SHG phase matching. Indeed, the effective areas of the phase matched modes in the microstructure are found to be larger than in the nanowire for the same phase matching wavelength. Also the effective index of the modes at phase matching is slightly lower in the microstructure, in spite of the presence of the struts, which should raise the effective cladding index. It must therefore be expected that structures with a larger number of struts, such as the well-known sixfold symmetric microstructures, will be more difficult to phase match, and will have lower values of the surface SHG coefficients.

3C. Impact of Structural Fluctuations

The central question of the present work is whether the nonlinear coefficients calculated in the previous subsections will allow significant power conversion fractions in realistic nanowires. Using Eq. (26) for a quick estimate, the SHG coefficients calculated for the microstructured nanowires lead to power conversion at the 10% level in fiber lengths of a few centimeters at a pump power level of $1\mathrm{kW}$. For nanosecond pulses, such peak power levels have been shown to be feasible in silica nanowires [10]. Also, nanowires with lengths of many centimeters have been demonstrated, with propagation losses of the order of $0.1\mathrm{dB}∕\mathrm{cm}$ or smaller [10]. However, a propagation length of the order of ${10}^4–{10}^5$ wavelengths sets rather severe restrictions on the tolerable phase mismatch $\Delta \beta$. This implies that efficient power conversion to the second-harmonic frequency will only occur over a very narrow bandwidth. In Fig. 5 , the product of $\Delta {\lambda }_{\mathit{FWHM}}$ defined in Eq. (29) with the fiber length L is plotted as a function of the fiber diameter. It is evident that fiber lengths of the order of $10\mathrm{cm}$ will lead to bandwidths of the order of $10–100\mathrm{pm}$. Therefore, the efficiency of the process will be highly sensitive to the uniformity of the nanowire. In this subsection, some simple propagation calculations will serve to illustrate the level of uniformity needed if serious power conversion fractions are to be achieved.

To describe propagation of pulses in a fluctuating fiber structure, Eq. (1) is generalized to

In the numerical simulations, Eq. (30) is discretized in steps much smaller than $L_c$. In each step $\Delta \beta$ is assumed constant, and Eq. (30) is propagated according to

For short propagation distances, fulfilling that

Experimentally, substantial efforts have gone into improving the uniformity of tapered nanowires. Shi et al. reported diameter fluctuations of $\pm 5\mathrm{nm}$ in a $12\mathrm{cm}$ nanowire with a diameter of $900\mathrm{nm}$ [12], which would roughly translate into a $\Delta \lambda ∕L$ figure of $4\cdot {10}^{-8}$. Vukovic et al. studied tapering of microstructured fibers, and were able to maintain an outer-diameter fluctuation of $\sim 1\mathrm{\%}$ over a length of $\sim 1\mathrm{m}$ [13]. Assuming that this level of accuracy carries over to the core dimensions and the phase matching wavelength, this would correspond to a ${\lambda }_{2c}$ fluctuation of $\pm 5\mathrm{nm}$ for the fiber considered here, and thus $\Delta \lambda ∕L\sim 5\cdot {10}^{-9}$. Calculations done at this level of ${\lambda }_{2c}$ fluctuations yielded averaged conversion efficiencies of ${10}^{-3}$ for a fiber length of $10\mathrm{cm}$, and ${10}^{-2}$ for a fiber length of $1\mathrm{m}$, i.e., linear scaling in the incoherent limit. These figures indicate that power conversion factors in excess of 10% would require very long fibers while maintaining a very precise diameter control.

Since the lack of phase matching is due to random fluctuations rather than an inherent phase mismatch in the waveguide structure, the use of quasi-phase matching techniques is of little use to increase the conversion efficiency in this case. Instead, the conversion efficiency may be enhanced either by increasing the peak power of the pump pulses, or by increasing the surface nonlinear coefficient. The conversion efficiency will scale linearly with pulse power until the breakdown threshold is reached. Conversion efficiencies in excess of 10% in fibers shorter than 1 m will therefore require power levels beyond $10\mathrm{kW}$. It is unclear whether such high powers in sub-micrometer cores will be tolerable, even in very short pulses. On the other hand, if the nonlinear coefficient ${\rho }_2$ can be increased, it follows from Eq. (1) that the power conversion efficiency will scale with ${\rho }_2^2$. Thus, increasing the surface nonlinearity by an order of magnitude compared to the values assumed for silica in this work would be enough to provide useful conversion efficiencies in realistic fibers. Such an increase in surface nonlinearity might be achieved by use of soft glasses with a higher polarizability than silica as a base material. Direct drawing of lead silicate microstructured fibers with core dimensions of the order of $0.5\mu \mathrm{m}$ was recently demonstrated experimentally [14]. Another approach would be to keep silica as a base material, but coat the surface of the nanowire with a film of highly polarizable material, as was recently suggested by Xu et al. [15]. The potential of this idea can be appreciated by the following argument: The dominant term in the silica surface ${\chi }^{\left(2\right)}$ tensor has a magnitude of $6\cdot {10}^3{\mathrm{pm}}^2∕\mathrm{V}$. One can think of this as if a surface layer with a thickness of $1\mathrm{nm}$ had a dipole ${\chi }^{\left(2\right)}$ polarizability of $6\mathrm{pm}∕\mathrm{V}$. Using more polarizable materials, this polarizability could be enhanced by at least a factor of 2 [15], and the surface layer could be extended to a thickness of 10 nm or more without drastically changing the guiding properties of the fiber, especially for the longer pump wavelengths. Of course, the uniformity of the polarizable layer would be a crucial issue. Furthermore, an increase of the surface ${\chi }^{\left(2\right)}$ coefficient will in itself not be very useful if it happens at the expense of a similar reduction in the power threshold, a caveat which applies for both the soft-glass and the coated-silica approach.

An alternative application of nanowire SHG may be in sensing applications, since the surface second-order polarizability will be highly sensitive to adsorbed molecules on the silica surface. Indeed, surface SHG on bulk samples has long been used to characterize adsorbates, and a recent paper has suggested a similar application for silica microspheres [16, 17]. Finally, it should be mentioned that a very recent paper demonstrates the use of both second- and third-harmonic generation to characterize the diameter of a nanowire with an accuracy of about 2% [18].

4. CONCLUSION

In conclusion, the surface dipole and bulk multipole contributions to the second-order nonlinear coefficient in silica nanowires have been studied numerically, along with phase matching conditions and bandwidths. It is concluded that the length and loss figures of present-day nanowires allow for significant conversion efficiencies, but that nanowire uniformity is likely to be a severe limitation. Changing the fiber material or applying a polarizable surface coating may have the potential to overcome this limitation by providing an increase in the nonlinear coefficient of an order of magnitude. The use of microstructured fibers instead of circular nanowires is shown to be possible, although the nonlinear coefficients are slightly reduced, and ${\mathrm{TM}}_{01}$-like modes are difficult to phase match.

Table 1. Surface and Bulk ${\chi }^{\left(2\right)}$ Components (in Units of ${\mathrm{pm}}^2∕\mathrm{V}$; from [7])

View Table

 

Fig. 1 Schematic of the threefold symmetric microstructure investigated. The fiber core is characterized by the surface curvature radius $r_c$ and the bridge width $W_b$. The outer radius of the microstructured region was set to $6\mu \mathrm{m}$ in the calculations.

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Fig. 2 Relation between diameter and phase matched SHG wavelength for a circular silica nanowire for the case where the SHG mode is either ${\mathrm{TM}}_{01}$ or ${\mathrm{HE}}_{21}$.

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Fig. 3 Nonlinear coupling parameter ρ versus SHG wavelength for silica nanowires with the phase matched SHG radiation being in either the ${\mathrm{TM}}_{01}$ mode (top panel) or the ${\mathrm{HE}}_{21}$ mode (bottom panel).

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Fig. 4 SHG wavelength ${\lambda }_{\mathit{SHG}}$ versus $r_c$ (top panel) and sum of surface nonlinear coefficients versus ${\lambda }_{\mathit{SHG}}$ (bottom panel) for microstructured nanowires with different relative bridge widths.

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Fig. 5 Product of $\Delta {\lambda }_{\mathit{FWHM}}$ as found from Eq. (30) and the fiber length L for a circular nanowire with phase matching to either the ${\mathrm{TM}}_{01}$ or ${\mathrm{HE}}_{21}$ mode.

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Fig. 6 Typical variation of SHG phase matching wavelength for a particular realization of the random-coefficient Fourier expansion discussed in the text.

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Fig. 7 SHG conversion efficiency versus propagation distance in a $10\mathrm{cm}$ microstructured nanowire. Results averaged over 200 realizations of the random structure are compared with results from the individual realizations giving either maximal or minimal conversion efficiency at the output end.

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Fig. 8 SHG conversion efficiency versus nanowire length for various levels of fluctuations in the SHG phase matching wavelength. The fiber lengths are $10\mathrm{cm}$ (top) and $50\mathrm{cm}$ (bottom). The curves and $\Delta \lambda ∕L$ values are averaged over 200 realizations of the random structure.

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