Abstract

The possibility of second-harmonic generation based on surface dipole and bulk multipole nonlinearities in silica nanowires is investigated numerically. Both circular and microstructured nanowires are considered. Phase matching is provided by propagating the pump field in the fundamental mode, while generating the second harmonic in one of the modes of the LP11 multiplet. This is shown to work in both circular and microstructured nanowires, although only one of the LP11 modes can be phase-matched in the microstructure. The prospect of obtaining large conversion efficiencies in silica-based nanowires is critically discussed, based on simulations of second-harmonic generation in nanowires with a fluctuating phase-matching wavelength. It is concluded that efficient wavelength conversion will require strong improvements in the nanowire uniformity, peak powers well in excess of 10KW, increase of the second-order nonlinearity by an order of magnitude by use of a different base material, or highly polarizable surface coatings.

© 2010 Optical Society of America

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, χ(2). Since the isotropic nature of amorphous silica forbids the presence of a (bulk) dipole contribution to χ(2), the origin of the experimentally observed phenomena was initially a puzzle, but was eventually explained by the formation of a bulk dipole χ(2) through multiphoton processes involving both pump and SHG light [3]. In this model, the multiphoton χ(2) inscription will naturally lead to the formation of a χ(2) 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 χ(2) 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 100μm [5]. The authors attributed this finding to a surface χ(2) effect. Indeed, the advent of nanowire technology [6] calls for a reexamination of the surface and bulk multipole χ(2) 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

dA2dziρ2A12exp(iΔβz)=0,
where A1, A2 are the field amplitudes of the fundamental and second-harmonic signals, respectively; Δβ=2β1β2 is the phase mismatch between the guided waves at the fundamental and second-harmonic frequencies, and ρ2 is the overlap integral
ρ2=ω22A12dre2*P(2)Redr[e2*×h2]z,
where ω2 is the second-harmonic frequency. The fields are normalized to unit power, i.e., the pump and signal powers, P1 and P2, are given by |A1|2 and |A2|2. The electric and magnetic fields of the guided modes are expressed as
E(r,ωj)=Aj(z)ej(r,ωj)exp(i(βjzωjt)),
H(r,ωj)=Aj(z)hj(r,ωj)exp(i(βjzωjt)),
for j=1,2. In the following, the shorthand notation Ej=E(r,ωj) will be used. P(2) is the second-order nonlinear polarization, which will have both dipole contributions from the fiber surface and multipole contributions from the bulk. The bulk contributions can be written as
Pb(2)(r)=ε0γ(E1E1)+ε0δ(E1)E1.
A third term, proportional to E1, vanishes (or can be included in the surface term) for fibers with piecewise constant material profiles, such as the ones considered here. For the surface contributions, it is convenient to express the results in terms of the field and polarization components parallel and perpendicular to the fiber surface. Three distinct terms then contribute to P(2):
Ps(2)(r)=δ(rS)[P(2s)+P(2s)+P(2s)],
P(2s)=ε0χ(2s)E12r̂,
P(2s)=ε0χ(2s)|E1|2r̂,
P(2s)=2ε0χ(2s)E1E1,
where r̂ is the unit vector normal to the surface. The surface χ(2) components appearing in Eqs. (7, 8, 9), as well as the bulk multipole parameters γ and δ have recently been measured by Rodriguez et al. [7]. Two sets of values, differing by about a factor of 1.5 were reported, resulting from the use of two different calibration methods. In the experiments, the contribution from the bulk multipole term proportional to γ in Eq. (5) cannot be separated from the surface contributions in Eqs. (7, 8). However, various indirect arguments support the notion that the bulk susceptibility is predominantly of magnetic dipole character [8], which implies γ=0.5δ. Under this assumption, and taking the lowest estimates put forward in [7], one arrives at the χ(2) components given in Table 1 .

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 HE11 mode of the fiber, while SHG (signal) field generation in either the TM01 or the HE21 modes will be considered. Working in cylindrical coordinates, the fields of the HEn1 modes inside the silica nanowire can be expressed as [9]

Enr(r,θ)=AnβnaunRnr(r)cos(nθ+φ),
Enθ(r,θ)=AnβnaunRnθ(r)sin(nθ+φ),
Enz(r,θ)=iAnJn(unar)cos(nθ+φ),
Rnr(r)=1sn2Jn1(unar)1+sn2Jn+1(unar),
Rnθ(r)=1sn2Jn1(unar)+1+sn2Jn+1(unar),
un=aknns2nn2,wn=aknnn21,sn=n(un2+wn2)Jn(un)unJn(un)+Kn(wn)wnKn(wn),
where Jn are Bessel functions of the first kind, Kn are modified Bessel functions of the second kind, kn is the vacuum wave vector of the light, ns is the silica refractive index, nn=βnkn is the modal effective index, a is the nanowire radius, and φ is a phase that determines the polarization. Two orthogonal polarization states can be obtained by, e.g., setting φ=0,π2. The harmonic dependence on z and t, exp(i(ωtβz)), has been omitted in these formulas. The propagation constant, β can be determined from the implicit equation [9]
[Jn(un)unJn(un)+Kn(wn)wnKn(wn)][Jn(un)unJn(un)+1ns2Kn(wn)wnKn(wn)]=n2(1un2+1wn2)[1un2+1(nswn)2].
From Eqs. (10, 11, 12, 13, 14, 15, 5, 6, 7, 8, 9) it may be shown that pump light in the HE11 mode will only generate SHG signal in the HE21 mode with the same value of φ. Alternatively, it may generate signal in the TM01 mode, whose field distribution can be expressed as [9]
Er(r)=A2β2au2J1(u2ar),
Eθ=0;Ez(r)=iA2J0(u2ar),
and whose propagation constant is determined by the equation [9]
J1(u2)u2J0(u2)=1ns2K1(w2)w2K0(w2).
This mode is often referred to as the “radially polarized” mode because of the strictly radial nature of the transverse electric-field component. It should be noted that because of the large index contrast and small dimensions of silica nanowires, the z components of the fields may be significant, and full-vectorial expressions for the overlap integrals are essential. In the limit of small index contrasts, the TM01 and HE21 modes are both part of the LP11 multiplet, which also comprises the TE01 mode, whose only nonvanishing electric-field component is Eθ. The overlap between this mode and P(2) arising from HE11 pump light can be shown to vanish. The nonvanishing surface contributions to ρ2 then become:
TM01mode:
ρs=πa12a2β12β2a4u12u2R1r2(a)J1(u2)χ(2s),
ρs=πa12a2β2a2u2J1(u2)χ(2s)[(β1au1)2R1θ2(a)J12(u1)],
ρs=2πa12a2β1β2a3u1u2J0(u2)J1(u1)R1r(a)χ(2s);
HE21mode:
ρs=π2a12a2β12β2a4u12u2R1r2(a)R2r(a)χ(2s),
ρs=π2a12a2β2a2u2R2r(a)χ(2s)[(β1au1)2R1θ2(a)+J12(u1)],
ρs=πa12a2β1a2u1R1r(a)χ(2s)[a2β1β2u1u2R2θ(a)R1θ(a)J1(u1)J2(u2)].
Here a1 and a2 are the amplitudes of the fields when normalized to unit power. The expressions for the bulk polarization terms are somewhat more involved and will be omitted here. A discussion of these terms can be found in [2].

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 Wb, and the curvature radius of the curved core surfaces rc. In the calculations, R was set to 6μm, 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 rc and Wbrc 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, Δβ=0, the evolution of the SHG intensity is given by:

P2(z)=[ρ2P1z]2P2P1=(ρ2z)2P1.
Even in the case of perfect phase matching, Eq. (26) will become inaccurate when the SHG power becomes large enough that pump depletion must be taken into account. A complete description of SHG dynamics will not be attempted here, since the undepleted-pump approximation is sufficient to reveal whether significant (e.g. >10%) SHG conversion efficiencies are realistic. As is evident from Eq. (26), this requires (ρ2L)2P10.1 where P1 is the pump power and L the length of the nanowire.

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

A2(λ,L)=2iρ2sinΔβ(λ)L2Δβ(λ).
For a spectrally narrow pump, Δβ(λ) can be found by a first-order expansion around the perfect phase matching wavelength λ2c as
Δβ(λ)dΔβdλ(λλ2c)=4πλ2c2(ng1ng2)(λλ2c),
where ng1, ng2 are the group indices of the fundamental and second-harmonic modes, respectively. From this expression, the spectral FWHM (at the SHG wavelength) can be evaluated to
ΔλFWHM1.39156λ2c22πL(ng1ng2).
The bandwidth thus scales inversely with the interaction length L.

3. NUMERICAL RESULTS

3A. Circular Nanowire

Surface and bulk nonlinear coefficients were obtained for pump radiation in the fundamental HE11 mode and SHG radiation in the TM01 and HE21 modes. The phase-matched SHG wavelengths, λSHG, for TM01 and HE21 modes are plotted against the nanowire diameter in Fig. 2 . λSHG is seen to be roughly similar to the fiber diameter, which implies that phase matching happens at a V-parameter value of 1.5 for the fundamental wavelength, where V=πdns21λ is the usual V-parameter as defined in the theory of step-index fibers. The difference in λSHG between TM01 and HE21 modes is about 10%.

In Fig. 3 , the sum of all SHG nonlinear coefficients, ρtot is plotted together with the individual terms as calculated from Eqs. (20, 21, 22, 23, 24, 25). For the TM01 mode, the ρs term is clearly dominant, whereas for the HE21 mode both the ρs and ρs give significant contributions to the total sum. The total SHG coefficient roughly scales with λ3. 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 λ1. The λ1 scaling of the ω2 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 TM01 and HE21 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 TM01-like core mode matches well onto the TE modes in any of these slab waveguides. The HE21 mode, on the other hand, can only match well onto the TE mode in one of the struts, and the TE01 mode always matches onto the TM modes of the struts. As a result, the TM01-like core mode effectively sees a higher cladding index in the microstructure than the HE21 and TE01 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 Wbrc0.03 were studied. Since the rc values where phase matching occurs turn out to be of the order of 1μm, 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 Wbrc, it was found that the frequency-doubled TM01-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 HE21-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 rc (see Fig. 1) for different values of the strut width relative to rc. For Wb=0.07rc, phase matching to the HE21 mode could no longer be found. However, for the lower bridge widths, the variations with Wb are within 50%. 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 1kW. 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 0.1dBcm or smaller [10]. However, a propagation length of the order of 104105 wavelengths sets rather severe restrictions on the tolerable phase mismatch Δβ. 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 ΔλFWHM 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 10cm will lead to bandwidths of the order of 10100pm. 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

dA2(ωj,z)dz=iρ2kA1(ωk)A1(ωjωk)exp(i0zdzΔβj(z)).
The pulse power is distributed on discrete frequency components ωj, whose spacing is proportional to the inverse of the repetition rate (this is a different notation than the one used in Eqs. (3, 4)). Note that
Δβ=β(ωk)+β(ωjωk)β(ωj)2β(ωj2)β(ωj)Δβj
as long as β1, β2 are expanded to linear order in the frequency arguments. In the numerical simulations, the spectral distribution of the pump pulse is assumed to be an unchirped Gaussian. As a mathematical model for the structural fluctuations, the position-dependent phase matching wavelength λ2c(z) is expressed as a Gaussian Fourier series with random-phase coefficients
λ2c(z)=WλmΦmexp(12km2(Lc2)2)exp(ikmz),
where Φm is a random phase factor. One can then show that
dzλ2c(z)λ2c(z+z)exp[(zLc)2];
that is, Lc can be regarded as a correlation length for the structural fluctuations. A typical realization with Lc=500μm is shown in Fig. 6 . The function Δβ(z) can be found from Eq. (28).

In the numerical simulations, Eq. (30) is discretized in steps much smaller than Lc. In each step Δβ is assumed constant, and Eq. (30) is propagated according to

A2(ωj,zm+1)=A2(ωj,zm)+iρ2ΔzkA1(ωk)A1(ωjωk)sin(ΔβjmΔz2)Δβjm×exp(i0zmdzΔβj(z))exp(iΔβjmΔz2),
where Δβjm is the value of Δβ(z) in the mth discretization segment. The calculations were done for an SHG wavelength of 515nm, with parameters for ρ2 and dΔβdλ appropriate for the microstructured nanowire with Wb=0.05rc. Transform-limited Gaussian pump pulses with a temporal width of 1ns and a peak power of 1kW were used.

For short propagation distances, fulfilling that

0zdzΔβj(z)1,
one has coherent SHG, and the second-harmonic power increases with z2. For longer distances, a transition to an incoherent regime occurs, where the SHG becomes an incoherent sum of contributions from different fiber segments, and the SHG power in this case scales with z [11]. In this limit, there are strong fluctuations in the SHG efficiency between different realizations of the random structure. An example is shown in Fig. 7 . The SHG power as a function of propagation length, averaged over 200 different realizations, is compared to the results for the particular realizations which lead to maximal and minimal SHG power at the output end. In Fig. 8 , curves averaged over 200 realizations are shown for different levels of structural fluctuations, and for fiber lengths of either 10 or 50cm. It is evident that the SHG power initially follows a z2 dependence, which at some point breaks off , and at longer propagation distance becomes a linear dependence. The transition point depends on the magnitude of the structural fluctuations. This magnitude is quantified by ΔλL, where Δλ is the maximal deviation of λ2c(z) from the nominal value, and L is the fiber length. It can be seen that a fiber length of 10cm allows a conversion efficiency of 25% (in the undepleted-pump approximation) for perfect phase matching, but that the conversion efficiency begins to deteriorate for ΔλL2109, corresponding to Δλ=0.2nm. Since the SHG phase matching wavelength was found to scale almost linearly with the nanowire diameter, this corresponds to an atomic-scale roughness, which is unrealistic to maintain over longer fiber segments. For a fiber length of 50cm, a similar conversion efficiency is obtained for Δλ0.4nm, but in this case ΔλL must be lower. This is difficult, because one must expect that in a longer fiber the peak Δλ will also be larger.

Experimentally, substantial efforts have gone into improving the uniformity of tapered nanowires. Shi et al. reported diameter fluctuations of ±5nm in a 12cm nanowire with a diameter of 900nm [12], which would roughly translate into a ΔλL figure of 4108. Vukovic et al. studied tapering of microstructured fibers, and were able to maintain an outer-diameter fluctuation of 1% over a length of 1m [13]. Assuming that this level of accuracy carries over to the core dimensions and the phase matching wavelength, this would correspond to a λ2c fluctuation of ±5nm for the fiber considered here, and thus ΔλL5109. Calculations done at this level of λ2c fluctuations yielded averaged conversion efficiencies of 103 for a fiber length of 10cm, and 102 for a fiber length of 1m, 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 10kW. 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 ρ2 can be increased, it follows from Eq. (1) that the power conversion efficiency will scale with ρ22. 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 0.5μm 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 χ(2) tensor has a magnitude of 6103pm2V. One can think of this as if a surface layer with a thickness of 1nm had a dipole χ(2) polarizability of 6pmV. 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 χ(2) 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 TM01-like modes are difficult to phase match.

Tables Icon

Table 1. Surface and Bulk χ(2) Components (in Units of pm2V; from [7])

 

Fig. 1 Schematic of the threefold symmetric microstructure investigated. The fiber core is characterized by the surface curvature radius rc and the bridge width Wb. The outer radius of the microstructured region was set to 6μm 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 TM01 or HE21.

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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 TM01 mode (top panel) or the HE21 mode (bottom panel).

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Fig. 4 SHG wavelength λSHG versus rc (top panel) and sum of surface nonlinear coefficients versus λSHG (bottom panel) for microstructured nanowires with different relative bridge widths.

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Fig. 5 Product of ΔλFWHM as found from Eq. (30) and the fiber length L for a circular nanowire with phase matching to either the TM01 or HE21 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 10cm 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 10cm (top) and 50cm (bottom). The curves and ΔλL values are averaged over 200 realizations of the random structure.

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16. J. Dominguez-Juarez, G. Kozyreff, and J. Martorell, “Surface nonlinear light generation in microresonators,” CLEO/Europe—EQEC 2009—European Conference on Lasers and Electro-Optics and the European Quantum Electronics Conference (IEEE, 2009), p. 1.

17. G. Kozyreff, J. Dominguez Juarez, J. Martorell, and J. Martorell, “Whispering-gallery-mode phase matching for surface second-order nonlinear optical processes in spherical microresonators,” Phys. Rev. A 77043817 (2008). [CrossRef]  

18. U. Wiedemann, K. Karapetyan, C. Dan, D. Pritzkau, W. Alt, D. Meschede, and S. Irsen, “Measurement of submicrometre diameters of tapered optical fibres using harmonic generation,” Opt. Express 18, 7693–7704 (2010). [CrossRef]   [PubMed]  

References

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    [Crossref]
  6. G. Brambilla, F. Xu, P. Horak, Y. Jung, F. Koizumi, N. Sessions, E. Koukharenko, X. Feng, G. Murugan, J. Wilkinson, and D. Richardson, “Optical fiber nanowires and microwires: fabrication and applications,” Adv. Opt. Photon. 1, 107–161 (2009).
    [Crossref]
  7. F. Rodriguez, F. X. Wang, and M. Kauranen, “Calibration of the second-order nonlinear optical susceptibility of surface and bulk of glass,” Opt. Express 16, 8704–8710 (2008).
    [Crossref] [PubMed]
  8. F. Rodriguez, F. X. Wang, B. Canfield, S. Cattaneo, and M. Kauranen, “Multipolar tensor analysis of second-order nonlinear optical response of surface and bulk of glass,” Opt. Express 15, 8695–8701 (2007).
    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
  12. L. Shi, X. Chen, H. Liu, Y. Chen, Z. Ye, W. Liao, and Y. Xia, “Fabrication of submicron-diameter silica fibers using electric strip heater,” Opt. Express 14, 5055–5060 (2006).
    [Crossref] [PubMed]
  13. N. Vukovic, N. G. R. Broderick, M. Petrovich, and G. Brambilla, “Novel method for the fabrication of long optical fiber tapers,” IEEE Photon. Technol. Lett. 20, 1264–1266 (2008).
    [Crossref]
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  15. Y. Xu, A. Wang, J. Heflin, and Z. Liu, “Proposal and analysis of a silica fiber with large and thermodynamically stable second order nonlinearity,” Appl. Phys. Lett. 90, 211110-1–3 (2007).
    [Crossref]
  16. J. Dominguez-Juarez, G. Kozyreff, and J. Martorell, “Surface nonlinear light generation in microresonators,” CLEO/Europe—EQEC 2009—European Conference on Lasers and Electro-Optics and the European Quantum Electronics Conference (IEEE, 2009), p. 1.
  17. G. Kozyreff, J. Dominguez Juarez, J. Martorell, and J. Martorell, “Whispering-gallery-mode phase matching for surface second-order nonlinear optical processes in spherical microresonators,” Phys. Rev. A 77043817 (2008).
    [Crossref]
  18. U. Wiedemann, K. Karapetyan, C. Dan, D. Pritzkau, W. Alt, D. Meschede, and S. Irsen, “Measurement of submicrometre diameters of tapered optical fibres using harmonic generation,” Opt. Express 18, 7693–7704 (2010).
    [Crossref] [PubMed]

2010 (1)

2009 (2)

2008 (3)

F. Rodriguez, F. X. Wang, and M. Kauranen, “Calibration of the second-order nonlinear optical susceptibility of surface and bulk of glass,” Opt. Express 16, 8704–8710 (2008).
[Crossref] [PubMed]

G. Kozyreff, J. Dominguez Juarez, J. Martorell, and J. Martorell, “Whispering-gallery-mode phase matching for surface second-order nonlinear optical processes in spherical microresonators,” Phys. Rev. A 77043817 (2008).
[Crossref]

N. Vukovic, N. G. R. Broderick, M. Petrovich, and G. Brambilla, “Novel method for the fabrication of long optical fiber tapers,” IEEE Photon. Technol. Lett. 20, 1264–1266 (2008).
[Crossref]

2007 (3)

V. Grubsky and J. Feinberg, “Phase-matched third-harmonic uv generation using low-order modes in a glass micro-fiber,” Opt. Commun. 274, 447–450 (2007).
[Crossref]

Y. Xu, A. Wang, J. Heflin, and Z. Liu, “Proposal and analysis of a silica fiber with large and thermodynamically stable second order nonlinearity,” Appl. Phys. Lett. 90, 211110-1–3 (2007).
[Crossref]

F. Rodriguez, F. X. Wang, B. Canfield, S. Cattaneo, and M. Kauranen, “Multipolar tensor analysis of second-order nonlinear optical response of surface and bulk of glass,” Opt. Express 15, 8695–8701 (2007).
[Crossref] [PubMed]

2006 (2)

X. Vidal and J. Martorell, “Generation of light in media with a random distribution of nonlinear domains,” Phys. Rev. Lett. 97, 013902 (2006).
[Crossref] [PubMed]

L. Shi, X. Chen, H. Liu, Y. Chen, Z. Ye, W. Liao, and Y. Xia, “Fabrication of submicron-diameter silica fibers using electric strip heater,” Opt. Express 14, 5055–5060 (2006).
[Crossref] [PubMed]

2004 (1)

1991 (1)

1987 (2)

1986 (1)

Alt, W.

Anderson, D.

Birks, T.

Brambilla, G.

G. Brambilla, F. Xu, P. Horak, Y. Jung, F. Koizumi, N. Sessions, E. Koukharenko, X. Feng, G. Murugan, J. Wilkinson, and D. Richardson, “Optical fiber nanowires and microwires: fabrication and applications,” Adv. Opt. Photon. 1, 107–161 (2009).
[Crossref]

N. Vukovic, N. G. R. Broderick, M. Petrovich, and G. Brambilla, “Novel method for the fabrication of long optical fiber tapers,” IEEE Photon. Technol. Lett. 20, 1264–1266 (2008).
[Crossref]

Broderick, N. G. R.

N. Vukovic, N. G. R. Broderick, M. Petrovich, and G. Brambilla, “Novel method for the fabrication of long optical fiber tapers,” IEEE Photon. Technol. Lett. 20, 1264–1266 (2008).
[Crossref]

Canfield, B.

Cattaneo, S.

Chen, X.

Chen, Y.

Dan, C.

Dominguez Juarez, J.

G. Kozyreff, J. Dominguez Juarez, J. Martorell, and J. Martorell, “Whispering-gallery-mode phase matching for surface second-order nonlinear optical processes in spherical microresonators,” Phys. Rev. A 77043817 (2008).
[Crossref]

Dominguez-Juarez, J.

J. Dominguez-Juarez, G. Kozyreff, and J. Martorell, “Surface nonlinear light generation in microresonators,” CLEO/Europe—EQEC 2009—European Conference on Lasers and Electro-Optics and the European Quantum Electronics Conference (IEEE, 2009), p. 1.

Ebendorff-Heidepriem, H.

Feinberg, J.

V. Grubsky and J. Feinberg, “Phase-matched third-harmonic uv generation using low-order modes in a glass micro-fiber,” Opt. Commun. 274, 447–450 (2007).
[Crossref]

Feng, X.

Grubsky, V.

V. Grubsky and J. Feinberg, “Phase-matched third-harmonic uv generation using low-order modes in a glass micro-fiber,” Opt. Commun. 274, 447–450 (2007).
[Crossref]

Heflin, J.

Y. Xu, A. Wang, J. Heflin, and Z. Liu, “Proposal and analysis of a silica fiber with large and thermodynamically stable second order nonlinearity,” Appl. Phys. Lett. 90, 211110-1–3 (2007).
[Crossref]

Horak, P.

Irsen, S.

Jung, Y.

Karapetyan, K.

Kauranen, M.

Koizumi, F.

Koukharenko, E.

Kozyreff, G.

G. Kozyreff, J. Dominguez Juarez, J. Martorell, and J. Martorell, “Whispering-gallery-mode phase matching for surface second-order nonlinear optical processes in spherical microresonators,” Phys. Rev. A 77043817 (2008).
[Crossref]

J. Dominguez-Juarez, G. Kozyreff, and J. Martorell, “Surface nonlinear light generation in microresonators,” CLEO/Europe—EQEC 2009—European Conference on Lasers and Electro-Optics and the European Quantum Electronics Conference (IEEE, 2009), p. 1.

Leon-Saval, S.

Liao, W.

Liu, H.

Liu, Z.

Y. Xu, A. Wang, J. Heflin, and Z. Liu, “Proposal and analysis of a silica fiber with large and thermodynamically stable second order nonlinearity,” Appl. Phys. Lett. 90, 211110-1–3 (2007).
[Crossref]

Margulis, W.

Martorell, J.

G. Kozyreff, J. Dominguez Juarez, J. Martorell, and J. Martorell, “Whispering-gallery-mode phase matching for surface second-order nonlinear optical processes in spherical microresonators,” Phys. Rev. A 77043817 (2008).
[Crossref]

G. Kozyreff, J. Dominguez Juarez, J. Martorell, and J. Martorell, “Whispering-gallery-mode phase matching for surface second-order nonlinear optical processes in spherical microresonators,” Phys. Rev. A 77043817 (2008).
[Crossref]

X. Vidal and J. Martorell, “Generation of light in media with a random distribution of nonlinear domains,” Phys. Rev. Lett. 97, 013902 (2006).
[Crossref] [PubMed]

J. Dominguez-Juarez, G. Kozyreff, and J. Martorell, “Surface nonlinear light generation in microresonators,” CLEO/Europe—EQEC 2009—European Conference on Lasers and Electro-Optics and the European Quantum Electronics Conference (IEEE, 2009), p. 1.

Mason, M.

Meschede, D.

Mizrahi, V.

Monro, T. M.

Murugan, G.

Okamoto, K.

K. Okamoto, Fundamentals of Optical Waveguides (Academic, 2000).

Osterberg, U.

Petrovich, M.

N. Vukovic, N. G. R. Broderick, M. Petrovich, and G. Brambilla, “Novel method for the fabrication of long optical fiber tapers,” IEEE Photon. Technol. Lett. 20, 1264–1266 (2008).
[Crossref]

Pritzkau, D.

Richardson, D.

Rodriguez, F.

Russell, P. S. J.

Sessions, N.

Shi, L.

Sipe, J.

Stolen, R.

Terhune, R.

Tom, H.

Vidal, X.

X. Vidal and J. Martorell, “Generation of light in media with a random distribution of nonlinear domains,” Phys. Rev. Lett. 97, 013902 (2006).
[Crossref] [PubMed]

Vukovic, N.

N. Vukovic, N. G. R. Broderick, M. Petrovich, and G. Brambilla, “Novel method for the fabrication of long optical fiber tapers,” IEEE Photon. Technol. Lett. 20, 1264–1266 (2008).
[Crossref]

Wadsworth, W.

Wang, A.

Y. Xu, A. Wang, J. Heflin, and Z. Liu, “Proposal and analysis of a silica fiber with large and thermodynamically stable second order nonlinearity,” Appl. Phys. Lett. 90, 211110-1–3 (2007).
[Crossref]

Wang, F. X.

Warren-Smith, S. C.

Weinberger, D.

Wiedemann, U.

Wilkinson, J.

Xia, Y.

Xu, F.

Xu, Y.

Y. Xu, A. Wang, J. Heflin, and Z. Liu, “Proposal and analysis of a silica fiber with large and thermodynamically stable second order nonlinearity,” Appl. Phys. Lett. 90, 211110-1–3 (2007).
[Crossref]

Ye, Z.

Adv. Opt. Photon. (1)

Appl. Phys. Lett. (1)

Y. Xu, A. Wang, J. Heflin, and Z. Liu, “Proposal and analysis of a silica fiber with large and thermodynamically stable second order nonlinearity,” Appl. Phys. Lett. 90, 211110-1–3 (2007).
[Crossref]

IEEE Photon. Technol. Lett. (1)

N. Vukovic, N. G. R. Broderick, M. Petrovich, and G. Brambilla, “Novel method for the fabrication of long optical fiber tapers,” IEEE Photon. Technol. Lett. 20, 1264–1266 (2008).
[Crossref]

J. Opt. Soc. Am. B (1)

Opt. Commun. (1)

V. Grubsky and J. Feinberg, “Phase-matched third-harmonic uv generation using low-order modes in a glass micro-fiber,” Opt. Commun. 274, 447–450 (2007).
[Crossref]

Opt. Express (6)

Opt. Lett. (3)

Phys. Rev. A (1)

G. Kozyreff, J. Dominguez Juarez, J. Martorell, and J. Martorell, “Whispering-gallery-mode phase matching for surface second-order nonlinear optical processes in spherical microresonators,” Phys. Rev. A 77043817 (2008).
[Crossref]

Phys. Rev. Lett. (1)

X. Vidal and J. Martorell, “Generation of light in media with a random distribution of nonlinear domains,” Phys. Rev. Lett. 97, 013902 (2006).
[Crossref] [PubMed]

Other (2)

J. Dominguez-Juarez, G. Kozyreff, and J. Martorell, “Surface nonlinear light generation in microresonators,” CLEO/Europe—EQEC 2009—European Conference on Lasers and Electro-Optics and the European Quantum Electronics Conference (IEEE, 2009), p. 1.

K. Okamoto, Fundamentals of Optical Waveguides (Academic, 2000).

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Figures (8)

Fig. 1
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 μ m in the calculations.
Fig. 2
Fig. 2 Relation between diameter and phase matched SHG wavelength for a circular silica nanowire for the case where the SHG mode is either TM 01 or HE 21 .
Fig. 3
Fig. 3 Nonlinear coupling parameter ρ versus SHG wavelength for silica nanowires with the phase matched SHG radiation being in either the TM 01 mode (top panel) or the HE 21 mode (bottom panel).
Fig. 4
Fig. 4 SHG wavelength λ SHG versus r c (top panel) and sum of surface nonlinear coefficients versus λ SHG (bottom panel) for microstructured nanowires with different relative bridge widths.
Fig. 5
Fig. 5 Product of Δ λ FWHM as found from Eq. (30) and the fiber length L for a circular nanowire with phase matching to either the TM 01 or HE 21 mode.
Fig. 6
Fig. 6 Typical variation of SHG phase matching wavelength for a particular realization of the random-coefficient Fourier expansion discussed in the text.
Fig. 7
Fig. 7 SHG conversion efficiency versus propagation distance in a 10 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.
Fig. 8
Fig. 8 SHG conversion efficiency versus nanowire length for various levels of fluctuations in the SHG phase matching wavelength. The fiber lengths are 10 cm (top) and 50 cm (bottom). The curves and Δ λ L values are averaged over 200 realizations of the random structure.

Tables (1)

Tables Icon

Table 1 Surface and Bulk χ ( 2 ) Components (in Units of pm 2 V ; from [7])

Equations (37)

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d A 2 d z i ρ 2 A 1 2 exp ( i Δ β z ) = 0 ,
ρ 2 = ω 2 2 A 1 2 d r e 2 * P ( 2 ) Re d r [ e 2 * × h 2 ] z ,
E ( r , ω j ) = A j ( z ) e j ( r , ω j ) exp ( i ( β j z ω j t ) ) ,
H ( r , ω j ) = A j ( z ) h j ( r , ω j ) exp ( i ( β j z ω j t ) ) ,
P b ( 2 ) ( r ) = ε 0 γ ( E 1 E 1 ) + ε 0 δ ( E 1 ) E 1 .
P s ( 2 ) ( r ) = δ ( r S ) [ P ( 2 s ) + P ( 2 s ) + P ( 2 s ) ] ,
P ( 2 s ) = ε 0 χ ( 2 s ) E 1 2 r ̂ ,
P ( 2 s ) = ε 0 χ ( 2 s ) | E 1 | 2 r ̂ ,
P ( 2 s ) = 2 ε 0 χ ( 2 s ) E 1 E 1 ,
E n r ( r , θ ) = A n β n a u n R n r ( r ) cos ( n θ + φ ) ,
E n θ ( r , θ ) = A n β n a u n R n θ ( r ) sin ( n θ + φ ) ,
E n z ( r , θ ) = i A n J n ( u n a r ) cos ( n θ + φ ) ,
R n r ( r ) = 1 s n 2 J n 1 ( u n a r ) 1 + s n 2 J n + 1 ( u n a r ) ,
R n θ ( r ) = 1 s n 2 J n 1 ( u n a r ) + 1 + s n 2 J n + 1 ( u n a r ) ,
u n = a k n n s 2 n n 2 , w n = a k n n n 2 1 , s n = n ( u n 2 + w n 2 ) J n ( u n ) u n J n ( u n ) + K n ( w n ) w n K n ( w n ) ,
[ J n ( u n ) u n J n ( u n ) + K n ( w n ) w n K n ( w n ) ] [ J n ( u n ) u n J n ( u n ) + 1 n s 2 K n ( w n ) w n K n ( w n ) ] = n 2 ( 1 u n 2 + 1 w n 2 ) [ 1 u n 2 + 1 ( n s w n ) 2 ] .
E r ( r ) = A 2 β 2 a u 2 J 1 ( u 2 a r ) ,
E θ = 0 ; E z ( r ) = i A 2 J 0 ( u 2 a r ) ,
J 1 ( u 2 ) u 2 J 0 ( u 2 ) = 1 n s 2 K 1 ( w 2 ) w 2 K 0 ( w 2 ) .
TM 01 mode :
ρ s = π a 1 2 a 2 β 1 2 β 2 a 4 u 1 2 u 2 R 1 r 2 ( a ) J 1 ( u 2 ) χ ( 2 s ) ,
ρ s = π a 1 2 a 2 β 2 a 2 u 2 J 1 ( u 2 ) χ ( 2 s ) [ ( β 1 a u 1 ) 2 R 1 θ 2 ( a ) J 1 2 ( u 1 ) ] ,
ρ s = 2 π a 1 2 a 2 β 1 β 2 a 3 u 1 u 2 J 0 ( u 2 ) J 1 ( u 1 ) R 1 r ( a ) χ ( 2 s ) ;
HE 21 mode :
ρ s = π 2 a 1 2 a 2 β 1 2 β 2 a 4 u 1 2 u 2 R 1 r 2 ( a ) R 2 r ( a ) χ ( 2 s ) ,
ρ s = π 2 a 1 2 a 2 β 2 a 2 u 2 R 2 r ( a ) χ ( 2 s ) [ ( β 1 a u 1 ) 2 R 1 θ 2 ( a ) + J 1 2 ( u 1 ) ] ,
ρ s = π a 1 2 a 2 β 1 a 2 u 1 R 1 r ( a ) χ ( 2 s ) [ a 2 β 1 β 2 u 1 u 2 R 2 θ ( a ) R 1 θ ( a ) J 1 ( u 1 ) J 2 ( u 2 ) ] .
P 2 ( z ) = [ ρ 2 P 1 z ] 2 P 2 P 1 = ( ρ 2 z ) 2 P 1 .
A 2 ( λ , L ) = 2 i ρ 2 sin Δ β ( λ ) L 2 Δ β ( λ ) .
Δ β ( λ ) d Δ β d λ ( λ λ 2 c ) = 4 π λ 2 c 2 ( n g 1 n g 2 ) ( λ λ 2 c ) ,
Δ λ FWHM 1.39156 λ 2 c 2 2 π L ( n g 1 n g 2 ) .
d A 2 ( ω j , z ) d z = i ρ 2 k A 1 ( ω k ) A 1 ( ω j ω k ) exp ( i 0 z d z Δ β j ( z ) ) .
Δ β = β ( ω k ) + β ( ω j ω k ) β ( ω j ) 2 β ( ω j 2 ) β ( ω j ) Δ β j
λ 2 c ( z ) = W λ m Φ m exp ( 1 2 k m 2 ( L c 2 ) 2 ) exp ( i k m z ) ,
d z λ 2 c ( z ) λ 2 c ( z + z ) exp [ ( z L c ) 2 ] ;
A 2 ( ω j , z m + 1 ) = A 2 ( ω j , z m ) + i ρ 2 Δ z k A 1 ( ω k ) A 1 ( ω j ω k ) sin ( Δ β j m Δ z 2 ) Δ β j m × exp ( i 0 z m d z Δ β j ( z ) ) exp ( i Δ β j m Δ z 2 ) ,
0 z d z Δ β j ( z ) 1 ,

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