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Abstract
Optical nanofibers have recently emerged as attractive nanophotonic platforms for many applications ranging from quantum technologies to nonlinear optics, due to both their tight optical confinement and their wide evanescent field. Herein we examine theoretically the optical Kerr effect induced by the evanescent field of a silica nanofiber surrounded by different nonlinear liquids such as water, ethanol and acetone and we further compare them with air cladding. Our results show that the evanescent Kerr effect significantly dominates the usual Kerr effect inside the silica core for sub-wavelength diameters below $560$ nm, using acetone. We further report the observation of the evanescent Kerr effect through surrogate measurements of stimulated Raman-Kerr scattering (SRKS) in an acetone-immersed silica nanofiber. Our findings open the way towards potential applications of optical nanofibers to ultra-sensitive liquid sensing or to enhancing the nonlinear effects through the evanescent field.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Optical nanofibers are ultra-thin optical fibers manufactured by heating and stretching standard silica-based optical fibers down to submicron scale over a few centimeters [1]. These ultra-thin fibers have many optical properties that make them interesting for a number of applications including plasmonics, quantum and atom optics, optical sensing, and nonlinear photonics. Among the most important are the strong light confinement in the waist region, the wide evanescent field in the sub-wavelength limit, and the tailorable group-velocity dispersion [2–5]. These remarkable optical properties have been exploited in many key experimental demonstrations such as octave-spanning supercontinuum generation over short propagation distance [6], atom trapping in the evanescent field of the nanofiber [7,8], ultra-sensitive chemical or biological sensors [9], high-Q micro-resonator coupling [10], and nanofiber-based interferometers and resonators [1]. They also possess exceptional mechanical and elastic properties that make them attractive for surface acoustic wave generation and Brillouin light scattering, as recently reported in [11,12]. Among third-order nonlinear effects, stimulated Raman scattering has recently been evidenced in the evanescent field of silica nanofibers dipped in liquid such as ethanol and toluene [13]. Although the evanescent field has already been exploited for enhancing nonlinear effects using metals and plasmons [14,15], to the best of our knowledge, no investigation about the optical Kerr effect in the evanescent field of an optical nanofiber has been reported yet.
In this paper, we report on a numerical investigation of the optical Kerr effect in silica nanofiber immersed in several nonlinear liquids such as ethanol, acetone and water and we further compare them with air cladding. We provide formula of the effective nonlinear coefficients including the contribution of the evanescent field for varying nanofiber diameter. It is shown for instance that the evanescent field contribution to the total Kerr effect using acetone liquid is greater than that of the silica core for a nanofiber diameter smaller than $560$ nm. Furthermore, the experimental evidence of the evanescent Kerr effect (EKE) is reported through the observation of the stimulated Raman-Kerr scattering (SRKS) using an acetone-immersed optical nanofiber.
2. Theoretical model
Figure 1(a) schematically shows the concept of the evanescent Kerr effect in a liquid-immersed optical nanofibers with cylindrical geometry. The conditions for enhancing the optical Kerr effect using the evanescent field are twofolds. Firstly the nonlinear liquid must have a lower refractive index than silica for mode guidance. Secondly, the silica core must be as small as possible to get the widest evanescent field that can nonlinearly interact with the Kerr liquid. Then the objective is to demonstrate that the evanescent contribution is greater than the core contribution. As a starting point, let us recall that the optical Kerr effect in a nonlinear optical waveguide is usually defined using the following nonlinear parameter,
where $n_2$ is the nonlinear Kerr index of the nonlinear medium, $A_{eff}$ is the effective area of the guided mode, and $\lambda$ is the wavelength of the field propagating in the waveguide. For a tapered fiber with a waist smaller than the optical wavelength, the evanescent field of the fundamental mode which interacts with the external environment becomes more important than that the electric field guided into the core. To include the evanescent field into the Kerr effect, we can write the effective nonlinear parameter $\gamma _{eff}$ for a mode as [16],
Fig. 1. (a) Scheme of a subwavelength-diameter optical nanofiber immersed in a nonlinear liquid for observing the evanescent Kerr effect. (b) Numerical simulations of the normalized Poynting vector $S_z$ component of the fundamental mode for a nanofiber with a core diameter of $560$ nm and immersed in acetone. The dashed line corresponds to the core surface.
Table 1 gathers the linear and nonlinear refractive indices, $n_0$ and the $n_2$, of the different media used in our model, and at wavelength of $\lambda ~=~532$ nm [16,17]. Although acetone seems to be the best candidate for increasing the Kerr nonlinearity, we have investigated different liquids for the sake of comparison. Indeed, ethanol and acetone have nearly the same refractive index but different Kerr coefficients. The comparison will allow us for estimating the sensitivity of the nonlinear parameter to the Kerr coefficient of the liquids. On the other hand, water and ethanol (or acetone) will be used to compare two liquids with different refractive indices and nonlinear Kerr coefficients. The results of this paper will therefore be helpful in estimating the effective Kerr parameter $\gamma _{eff}$ for other nonlinear liquids. To get the effective Kerr coefficient, we first computed the electric and magnetic fields of the fundamental mode using a finite element method (FEM, COMSOL Multiphysics). As an example, Fig. 1(b) shows the Poynting vector component $S_z$ for a nanofiber of $560$ nm of diameter immersed in acetone. A perfectly matched layer (PML) with cylindrical symmetry has also been included in the numerical method to absorb light at open boundaries. From the computed fields, we then calculated a number of parameters including the effective Kerr coefficient $\gamma _{eff}$, the effective area of the fundamental mode $A_{eff}$, the fraction of intensity in the evanescent field $f~=~\frac {\iint _{cl}S_zdxdy}{\iint ^{+\infty }_{-\infty }S_zdxdy}$, the contributions to the Kerr parameter of the silica core $\gamma _{co}$ and of the cladding $\gamma _{cl}$, respectively. The ratio $\gamma _{cl}/\gamma _{eff}$ was further derived to quantify the cladding contribution in the effective Kerr parameter.
3. Numerical results and discussion
Figures 2(a)–2(b) show the effective area $A_{eff}$ of the fundamental mode and the evanescent field intensity fraction $f$ (in percentage) for an increasing diameter $d$ from $0.3$ to $0.8~\mu$m and using different claddings (air, water, ethanol, acetone).
Fig. 2. (a) Effective mode area as a function of the nanofiber diameter for air cladding and different nonlinear liquids, (b) Fraction of power density in the evanescent field for an optical wavelength $\lambda =532$ nm.
The evolution of the effective area as a function of the nanofiber diameter shown in Fig. 2(a) is well-known (see [18,19]). To sum up, the fundamental mode confinement increases with the index step $\Delta n$, and decreases when the taper diameter becomes identical or lower than the wavelength, meaning that the mode field extends spatially by increasing its evanescent part outside the silica core [3,4]. From Fig. 2(a), we find that the fiber diameter for the smallest effective area is $d=350$ nm, $525$ nm, $580$ nm, and $590$ nm for cladding of air, water, acetone and ethanol, respectively. Figure 2(b) further shows that an evanescent field fraction of $50\%$ is reached for diameter smaller than $d=421$ nm, $467$ nm and $472$ nm using as a cladding water, acetone and ethanol, respectively, while this threshold can never be reached for the air-clad nanofiber (Fig. 2(b)), due to the larger index step $\Delta n$ in air compared to the other liquids.
In Figs. 3(a) to 3(d), we plotted the nonlinear parameter contribution of the silica core $\gamma _{co}$ and of the cladding $\gamma _{cl}$, the effective nonlinear coefficient $\gamma _{eff}$ and the evanescent Kerr fraction, with respect to the nanofiber diameter. Concerning the silica core contribution $\gamma _{co}$, we can see that this parameter dramatically changes when using liquids as cladding compared to air. It first increases until reaching a maximum value for diameters close to those of the minimal effective area (See Fig. 2(a)). This is due to the fact that if the effective mode area decreases, the field intensity and therefore the Kerr effect in the silica core increase. Consequently, the ascending order of the $\gamma _{co}$ follows the descending order of the effective area or of the refractive index step for the different external media. Then, it is clear that $\gamma _{co}$ is much greater for nanofiber in air than in liquids, in agreement with Fig. 3(a). For small diameters, the core contribution drastically decreases and tends to zero since the fundamental mode spreads out spatially with increasing mode effective area. Concerning the cladding contribution (Fig. 3(b)), the evolution is similar. Since the intensity fraction in the evanescent field increases (Fig. 2(b)) with decreasing nanofiber diameter, $\gamma _{cl}$ increases up to reach a maximum and to decrease thereafter. This peak value corresponds to an optimal balance between the increase of the evanescent field that gives rise to the Kerr effect enhancement and to the decrease of the field intensity. This peak value is achieved for diameter lower that those for minimal effective area. For a given nanofiber diameter, the dependence of $\gamma _{cl}$ on the cladding materials follows the order of their Kerr indices : the higher the Kerr index is, the stronger the gamma parameter is. However, the ratio between two $\gamma _{cl,ac}$ for two different media depends not only on the ratio of the respective Kerr coefficients $n_2$ of the media but also on the fraction of the evanescent field intensity, i.e., on the refractive indices ratio between the liquids. Therefore, for acetone and ethanol, the $\gamma _{cl,ac}/\gamma _{cl,eth}$ ratio is almost equivalent to their $n_2$ ratio as their refractive indices are very close. This is however not the case for acetone and water. Figure 3(c) shows the effective Kerr parameter $\gamma _{eff}=\gamma _{co}+\gamma _{cl}$. First of all, it is clear that the Kerr parameter $\gamma _{eff}$ remains maximum for a nanofiber in air, even if the contribution of the evanescent field to the Kerr effect is null, and even compared to a nanofiber immersed in acetone which presents the highest Kerr coefficient $n_2$. This is due to the fact that the effective area of the fundamental mode for a nanofiber in air remains the smallest, leading to high field intensity and Kerr effect. The parameter $\gamma _{eff}$, for the core contribution $\gamma _{co}$, has a maximum value of $2.3~W^{-1}.$m$^{-1}$ for a small diameter $d=375$ nm.
Fig. 3. Evolution in function of the nanofiber diameter and for different surrounding media of (a) the nonlinear parameter contribution of the silica core $\gamma _{co}$ (b) the nonlinear parameter contribution of the cladding $\gamma _{cl}$, (c) the whole effective nonlinear parameter $\gamma _{eff}$, and (d) the $\gamma _{cl}/\gamma _{eff}$ ratio. The optical wavelength is $\lambda =532$ nm.
Although $\gamma _{eff}$ is lower for a nanofiber immersed in a liquid, the contribution of the evanescent field to the Kerr effect may be comparable or even higher than that from the silica core. This is shown when comparing Fig. 3(a) to Fig. 3(b). For acetone, $\gamma _{eff}$ has a maximal value of $1.54~$W$^{-1}.$m$^{-1}$ for a core diameter of $d=410$ nm. For that diameter, $\gamma _{cl}=1.22~$W$^{-1}.$m$^{-1}$ and $\gamma _{co}=0.32~$W$^{-1}.$m$^{-1}$. Therefore, almost $80\%$ of the Kerr parameter is in the evanescent field. We note from this analysis that this large evanescent fraction is no more correlated to the effective area of the mode which is minimum for a nanofiber diameter of $d=580$ nm.
If we now compare the role of ethanol and acetone that have the same effective mode area (Fig. 2(b)), the contribution $\gamma _{co}$ is therefore equivalent for both liquids (Fig. 3(b)). In addition, as the Kerr coefficient $n_2$ of acetone is higher than that of ethanol, the contribution $\gamma _{cl}$ is clearly greater for acetone. Thus, the effective Kerr coefficient is about 2 times larger for acetone than ethanol. The same analysis can be performed when comparing water and ethanol. As the refractive index of water is the lowest compared to ethanol or acetone, a nanofiber in water will have a higher core field and a lower evanescent field than in other liquids (See Fig. 2(b)), leading to similar amounts for nonlinear parameter contributions. Furthermore, the Kerr coefficient of ethanol is higher than that of water which balances the low field intensity in the evanescent part and gives rise to similar nonlinear parameters $\gamma _{eff}$ for both liquids (See Fig. 3(c)).
A further comparison of Fig. 3(d) and Fig. 2(b) shows that the evolution of the contribution of the evanescent field to the Kerr effect is indeed greater than that of the outer fraction of the field intensity, due to the Kerr coefficient of the nonlinear liquid. To quantify this difference, we plot in Fig. 4 the 2D spatial profile of the $\gamma _{eff}$ parameter for a nanofiber with a core diameter of $560$ nm immersed in acetone, before the 2D spatial integration in Eq. 4. Therefore, it corresponds to $\frac {\partial ^2 \gamma _{eff}}{\partial x \partial y}$ called thereafter $\partial _{xy}\gamma _{eff}$, and written as follows
Fig. 4. Normalized 2D spatial profile of the $\gamma _{eff}$ parameter before 2D spatial integration in Eq. 4 (called $\partial _{xy}\gamma _{eff}$ - read text and Eq. 5) for a nanofiber with a core diameter of $560$ nm and immersed in acetone. Right window: normalized Y-cut spatial profiles of the Poynting vector (blue dotted curve) of Fig. 1(b), and of $\partial _{xy}\gamma _{eff}$ (orange curve).
It is noteworthy that the Kerr effect studied here has both electronic and molecular contributions, according to the considered medium, with different time scales for the nonlinear response in function of the origin of the nonlinear change in the refractive index. Thus, the electronic polarization of liquids and silica is associated to an instantaneous response, whereas the liquid molecule reorientations lead to several non instantaneous responses with picosecond time scales [20].
4. Experimental observation
Experimentally, the evanescent Kerr effect can be observed using different ways. The first one is the self-phase modulation (SPM) which gives rise to a spectral broadening of the pump laser pulse scaling with to the $\gamma _{eff}$ parameter [21]. The theory predicts that the total Kerr effect, and therefore the SPM-induced spectral broadening, will decrease in a liquid-clad nanofiber compared to an air-clad nanofiber (see Fig. 3(c)). However, this decrease of SPM can also be due to other effects such as scattering by impurities or absorption. It seems thus difficult to experimentally demonstrate the evanescent Kerr effect by comparing the SPM-induced spectral broadening in liquids and air. This quantitative approach is delicate to implement, and anyway, it does not prove the presence of EKE as this latter is mixed with the core contribution to the Kerr effect.
Another experimental approach that would allow for the observation of the evanescent Kerr effect is based on using stimulated Rayleigh-Kerr scattering (SKS) and stimulated Raman-Kerr scattering (SRKS). This inelastic light scattering comes from the inertial nature of reorientation motion of anisotropic molecules of many liquids, as those used in this study, and it gives rise to a broad asymmetric broadening of the pump spectrum (SKS) or of the Raman Stokes spectrum (SRKS) towards the red [22–26]. It can be understood as follows: unlike the case of gaseous medium, molecules in liquid phase cannot rotate freely due to the viscous damping of the medium. To make a re-orientational motion within a liquid, a molecule has to get more energy to overcome the viscosity, leading to inelastic light scattering over a broad Stokes frequency range of the pump spectrum (SKS) or beyond the initial Raman frequency shift (SRKS). A broadening on the anti-Stokes side due to the thermal collisions between molecules that promote this re-orientational motion can also be observed [22], however with a lower extent than for the Stokes side as we will explain thereafter. Therefore, if we could observe SRKS on the Stokes and anti-Stokes sides of the Raman lines of liquids, this would mean that the evanescent nonlinearities as Raman and inertial Kerr effects are simultaneously present in the liquids. Without Kerr effect in the liquid, no asymmetric broadening would be observed and the Raman lines would remain perfectly symmetric.
The experimental setup is drawn in Fig. 5(a). In our experiment, we made a silica nanofiber using the heat-brush technique described in [12]. It has a uniform diameter of $580$ nm over a $4$ cm length and the two transition tapers are $8$ cm long. As a laser source, we used a frequency-doubled Nd:YAG laser at $532$ nm with a repetition rate of $21$ kHz and delivering Gaussian pulses of $500$ ps of duration (FWHM). A detailed scheme of the nanofiber immersed in acetone is shown in Fig. 5(b). The pump evanescent field is drawn around the nanofiber, such as the Raman spectral component generated along the nanofiber in the acetone by the evanescent part of the pump field.
Fig. 5. (a) Experimental setup: MO microscope objective $\times$10, OSA optical spectrum analyser. (b) Detailed scheme of the nanofiber, with the spectral component generated with the pump evanescent field by stimulated Raman scattering in acetone.
Figure 6(a) shows the experimental spectra at the fiber output of the nanofiber when increasing the input mean power from $5$ to $15$ mW. The spectral components other than that of the pump at $532$ nm correspond to the Raman lines with the highest gain due to symmetrical C$H_3$ stretching mode with the a Raman shift of $2925$ cm$^{-1}$ [27]. The first Raman order is generated around $630$ nm. When the input power increases, the Raman cascading process occurs and the second Raman order appears at a wavelength of $772.5$ nm. Figure 6(b) compares the spectra of the first Raman order for at $5$ mW and for at $15$ mW pumping. For an input mean pump power of $5$ mW, the first Raman order shown in red has a spectrum perfectly symmetric such as the pump. However, for higher power, an asymmetric spectral broadening of the first Raman peak appears (blue spectrum) and increases with the input pump power up to 500 cm$^{-1}$ ($\Delta \nu =15$ THz) and 200 cm$^{-1}$ in the Stokes and anti-Stokes sides, respectively, for the highest pump power of $15$ mW. Furthermore, it is important to note that the pump spectrum does not present such a strong broadening in the same range of input powers.
Fig. 6. Experimental measurements for an acetone-immersed silica nanofiber: (a) output spectra as a function of input mean pump power. Spectra are separated by 30 dB for clarity. (b) First Raman order spectra for 5 mW and 15 mW input pump power, showing the stimulated Raman-Kerr scattering induced by the evanescent field. The red dotted staight line and the black dotted curve corresponds to the theoretical model of SRKS broadening in the Stokes and anti-Stokes side, respectively.
Let us examine now several physical processes, which could be the origin of this phenomenon. As previously discussed, SPM process leads to a symmetrical broadening with periodic oscillation structure, which is not our case. Moreover, the pulse duration of $600$ ps is too long to generate such large spectral broadening. As the dispersion is highly normal, no modulation instability or four-wave mixing processes can occur and generate spectral components around powerful spectral peaks. Besides, the spectral change of scattered light by a liquid due to molecular dynamics is well-known ([28,29] and references therein). From a general point of view, anisotropic molecules scatter incident light with a change in frequency when their orientations are changing with time. The motion of the individual molecules in a fluid is therefore influenced by thermal collisions with other molecules and by the viscous damping of the medium during Kerr-induced reorientation for example [28]. The spectrum profile of this elementary scattering process can be described by a Lorentzian function, with a linewidth that depends on the orientational relaxation time of molecules [28]. For instance, spontaneous Rayleigh wing scattering is due to fluctuations of molecular orientation via thermal collisions and leads to symmetric broadening of the pump spectrum with a line shape modeled by a Lorentzian profile. The broadening can be large, as observed in CS$_2$ capillary fiber with a broadening of up to 160 cm$^{-1}$ [28]. In the same way, stimulated Rayleigh wings scattering (SRWS) corresponds to fluctuations of dielectric constant of the medium from reorientation of molecules induced by intense light. Theory of SRWS predicts a positive gain distribution in the Stokes side with a peak at a frequency shift inversely proportional to the orientational relaxation time [30]. Therefore the maximal gain frequency should be shifted by few cm$^{-1}$ from the main spectral line for the pump and Raman orders spectra, which does not explain the broadening of several hundred of cm$^{-1}$ and only for the first Raman order. SRWS cascading process could occur, but it would lead to oscillations on the Stokes side. Moreover, SRWS also presents a negative gain distribution on the anti-Stokes side, and cannot generate broadening in the blue side. Therefore, spontaneous and stimulated Rayleigh wing scattering processes cannot be the main origins of the observed spectral broadening because they should occur for all powerful spectral components including the pump and Raman orders with symmetric broadening or absorption in blue side. The last process corresponds to SRKS previously described. The theoretical model predicts an asymmetric broadening with a wide range in the Stokes side and a lower extent in the anti-Stokes side. For that process, the molecules have to be anisotropic, which is the case of acetone [31]. Moreover, the broadening of the powerful spectral components can reach several hundred of cm$^{-1}$, as already observed [24–26]. The fact that the pump does not present a broadening induced by SKS is because the molecular vibration during Raman process leads to the increase of the anisotropy of the molecule, such as benzene or toluene for example [24,25]. The profiles of the spectral components broadened by SRKS have been already theoretically modeled in [22,23]. As explain in [22], the final spectrum for the Raman components corresponds to the overlap of all scattering spectra shifted from the main Raman shift by random spectral shifts corresponding to different and random molecular orientation angle change. Consequently, in the Stokes side, due to the large possible reorientation angles, the resulting theoretical gain presents a shape which can be approximated to an exponential behavior with $\Delta \nu$ [22]. Therefore, a linear behavior in logarithm scale should be obtained for large spectral shift, as experimentally observed in this work and shown in Fig. 6(b) (dotted red straight line in Stokes side). On the anti-Stokes side, the spectrum also corresponds to the superposition of shifted spectra but for only scattering associated to a very small induced reorientation angle change, i.e., $\Delta \nu \approx 0$. Therefore, He et al. [22,23] predicts a gain described by a Lorentzian profile, as experimentally observed in the Fig. 6(b) (black dotted curve in anti-Stokes side, Lorentzian function in dB scale). For all these reasons, we think that SRKS is responsible of the large broadening on both the Stokes and anti-Stokes sides.
To sum up, as the broadening in the Stokes and anti-Stokes sides due to SRKS is clearly visible when the input pump power increases, the Kerr effect then occurs in the liquid which clearly demonstrates the presence of evanescent Kerr field. Moreover, as SRKS process has a power threshold, its observation for a weak propagation distance indicates that Kerr effect in acetone is strong. Let us recall that without the Kerr effect the Raman lines should present a perfectly symmetric and narrow spectrum in sub-nanosecond regime. It is noteworthy that this is the first time to our knowledge that SRKS phenomenon is observed in acetone and in the evanescent field, and also that SRKS occurs over a so weak interaction length, compared to few meters of capillaries filled with liquids [22–26]. Note that the Raman effect in acetone appears clearly before the Raman effect in silica, showing that evanescent Raman scattering is very efficient in comparison with Raman effect in silica core due to the gap between their Raman gain coefficients.
5. Conclusion
In summary, we theoretically investigated the optical Kerr effect in the evanescent field of a silica nanofiber surrounded by nonlinear liquids such as water, ethanol and acetone. We have shown that the effective Kerr coefficient contains two contributions: the contribution of the silica core which depends on the diameter of the nanofiber and follows the inverse of the effective area, and the contribution of the cladding via the evanescent field which depends on the diameter of the nanofiber and the nonlinear properties of the external environment. It has been further shown that the contribution to the optical Kerr effect in the evanescent field of the fundamental mode overpasses that of the core when the nanofiber is immersed in a highly nonlinear liquid such as acetone for diameter below $560$ nm. The presence of a strong evanescent Kerr effect has been experimentally confirmed in silica nanofiber immersed in acetone by the observation of the stimulated Raman-Kerr scattering effect leading to the asymmetric spectral broadening of the first Raman order generated by the evanescent Raman effect. The evanescent Kerr and Raman effects shown in here may find potential applications to ultra-sensitive liquid sensing and Raman spectroscopy, as the optical mode propagating in the optical nanofiber essentially interacts with the outer environment without any major contribution from the nanofiber.
Funding
Agence Nationale de la Recherche (ANR-15-IDEX-0003, ANR-16-CE24-0010, ANR-17-EURE-0002); H2020 Marie Skłodowska-Curie Actions (722380); Conseil régional de Bourgogne-Franche-Comté.
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