1. Introduction

Optofluidics [1–6] and optomechanics [7–9] have recently emerged as two important areas of research. Optofluidics aims to synergize microfluidics and optics to achieve novel functionalities including reconfigurable optical systems [1, 2], integrated optics [3, 4], lasers [5], and sensing [6]. Optomechanics, on the other hand, involves the dynamic interplay between optical field and mechanical motion [7–9]. To date, most optomechanics-related research utilizes solid resonators [10–12]. Only a limited number of studies investigated the mechanical interaction between optical field and fluids [13–18]. The earliest example is perhaps the classic work in [13], where the authors used a focused high power laser beam to create a bulge over a flat air-liquid interface. Similarly, in [15, 16], the authors used a high power laser beam focused onto a liquid droplet to distort its interface. More recently, optical radiation pressure was used in [17] to distort a flat liquid interface and form a tunable lens. By using a liquid mixture with extremely small interfacial tension, which can occur near the critical temperature of phase transition, it is possible to significantly reduce the optical power required for large interface distortion [17–19]. Additionally, stimulated Brillouin scattering in a hollow capillary tube filled with liquid has been reported recently [20]. The existence of high-Q whispering gallery mode (WGM) in an all-liquid droplet has also been experimentally confirmed [21, 22]. The focus of this work, however, is distinct from existing studies. Specifically, our goal is to demonstrate that under appropriate conditions, the dynamic interplay between optical force and liquid interface can lead to processes that are very similar to classical third-order nonlinear processes such as Kerr effects and optical solitons. As will be made clear in this paper, a defining feature of these nonlinear optofluidic processes is that the nonlinearity arises from the distortion of liquid interfaced induced by optical radiation pressure.

To analyze nonlinear optofluidic processes, it is necessary to solve the optical equations (e.g., Maxwell’s equations) and the fluidic equations (e.g., Navier-Stokes equations) in a self-consistent manner, where optical radiation pressure must be balanced by fluidic forces such as surface tension and buoyancy. Consequently, by tuning fluid parameters such as surface tension and density, one should be able to modulate and control the effective strength of optofluidic nonlinearity. This is in sharp contrast with traditional nonlinear optics, where nonlinear susceptibility (e.g., ${\text{χ}}^{\left(2\right)}$ or ${\text{χ}}^{\left(3\right)}$) is an intrinsic material property and cannot be easily tuned. Furthermore, by using liquids with low surface tension, it should be possible to achieve large nonlinearity with low optical power. In fact, in this paper, we find that by reducing surface tension to a low but experimentally attainable level, the effective strength of nonlinear optofluidics can be several orders of magnitude stronger than the traditional Kerr effects. To the best of our knowledge, this possibility has never been discussed in existing literature.

To quantitatively analyze nonlinear optofluidic effects, we consider two examples in this paper. Figure 1(a) shows a liquid structure which contains a high index liquid as the waveguiding core, air as the top cladding, and a lower index liquid as the bottom cladding. Assuming low interfacial tension for the bottom liquid-liquid interface, the presence of optical field should deform the interface and create a bulge. At the same time, this bulge can enhance optical confinement along the transverse direction. Under certain conditions, we find that the propagation of optical field within the liquid bulge can be described by the nonlinear Schrödinger equation, the solutions to which are in the familiar form of optical solitons. This example illustrates a key feature of nonlinear optofluidics, namely that the presence of the optical field distorts the liquid interface; simultaneously, the change in the liquid interface shape also modifies the optical field. As a result, analyses of nonlinear optofluidics often require solving the coupled system of optical waves and fluids. Specifically, the optical field must satisfy the Maxwell’s equations plus the boundary conditions imposed by the deformable liquid interface. Meanwhile, the liquid system is governed by the relevant fluid equations, where the impact of optical radiation pressure must also be included. Clearly, the most important feature of nonlinear optofluidics is the coupling between the optical field and the liquids, where the coupling strength is effectively determined by the radiation pressure and the interfacial tension of the liquid interface.

 

Fig. 1 (a) An optofluidic soliton formed from a self-guided optical wave that is confined within a liquid bulge formed through radiation pressure. (b) A liquid droplet that contains a high-Q WGM circulating along the equator. The radiation pressure of the WGM forms the bulge, which in turn shifts the WGM resonance frequency.

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The optofluidic soliton depicted in Fig. 1(a) may not be easy to implement experimentally. A more practical example is the system shown in Fig. 1(b), which is based on a liquid droplet that supports high-Q WGMs. In this example, the optical field of the WGM exerts radiation pressure on the droplet surface. Since the direction of radiation pressure always points from the high index core to the low index cladding [16], a bulge should form along the droplet equator. As a result, given sufficiently high optical power, both the effective optical path of the resonator and the corresponding resonance frequency should change. This phenomenon is very similar to the Kerr effect, where high optical power also shifts the effective cavity length by changing the refractive index of the liquid. In this paper, we derive a closed-form formula that can provide an order of magnitude estimate for this nonlinear optofluidic process. Using common liquid parameters, we find that this nonlinear optofluidic effect can be significantly stronger than the Kerr effect. In fact, for liquids with low but experimentally achievable surface tension, it may even be possible to produce measurable change in WGM resonance frequency at single photon energy level. Such a possibility may ultimately enable us to demonstrate nonlinear interaction between two single photons, which is obviously important for quantum information technology.

2. Optofluidic soliton

The optofluidic solitons can exist in the asymmetric liquid waveguide shown in Fig. 2(a). The refractive indices of the three layers are represented by$n_1$, $n_2$, and $n_3$, while their fluid densities are given by ${\rho }_1$, ${\rho }_2$, and ${\rho }_3$, respectively. For simplicity, we assume layer 3 is air and$n_3=1$. In the absence of optical signals, the waveguide core is the planar middle layer with thickness $h_0$ and has the highest refractive index (i.e., $n_2>n_1$ and $n_2>n_3$). By using the water-in-oil microemulsion system described in [17], it is possible to ensure that the core liquid layer (layer 2) possesses a higher refractive index but a lower mass density than the cladding liquid layer (layer 1), i.e., $n_2>n_1$ and ${\rho }_2<{\rho }_1$. Thus the configuration in Fig. 2 should be hydrodynamically stable for small deformation.

 

Fig. 2 (a) Illustration of an optofluidic soliton. The structure contains two liquids (refractive indices$n_1$ and $n_2$) and air ($n_3=1$). The thickness of the waveguide is $h_0$ in the absence of optical field. The radiation pressure of the guided optical signal produces the bulge shown in the figure. The thickness of the bulge that serves as the waveguide core is denoted as $h\left(x\right)=h_0+\Delta h\left(x\right)$. (b) The effective index of the asymmetric dielectric waveguide defined in (a) as a function of the core layer thickness $h\left(x\right)$. The waveguide parameters are $n_1=1.5$, $n_2=1.5043$, and $n_3=1$. The operation wavelength is 1$\mu m$. The exact solution (blue line) is calculated using standard waveguide theory. The dashed black line represents linear fitting of the exact solution in the range of $2.5 \mu m<h<4 \mu m$. The slope gives $\partial \text{n}/\partial h=1\times {10}^{-3}\mu m^{-1}$.

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After coupling light into layer 2, the presence of the optical field generates radiation pressure on the dielectric boundaries. We assume that the interfacial tension between layer 1 and 2 is much smaller than that of layer 2 and 3. As a result, we can ignore the deformation of the air-liquid interface (the interface between layer 2 and 3) and focus on the liquid-liquid interface instead. We note that the direction of radiation pressure always points from the high refractive index medium towards the low refractive index medium [16]. Consequently, the presence of optical field should generate a liquid bulge as illustrated in Fig. 2(a). In turn, the geometrical deformation can lead to better optical confinement within the bulge. Qualitatively, such a nonlinear optofluidic process is very similar to the self-focusing of spatial solitons. In the following analysis, we establish an analytical framework and confirm that under certain conditions, the aforementioned optofluidic process can indeed be described by the nonlinear Schrödinger equation.

Our first step is to apply the effective index theory [23], a widely used theory that allows us to reduce the original three-dimensional (3D) problem into a two-dimensional (2D) one. In our case, similar to the treatment of structures such as ridge waveguides, we assume that the optical field depends only on $x$ and $z$ and use an effective index $n_{eff}\left(x,z\right)$ to account for the field’s variation along the y direction. For the specific example in Fig. 2(a), we assume that the optical field’s distribution along the y axis is simply given by the fundamental y-polarized mode in a planar waveguide. Then, we approximate the $x$ and $z$ dependence of the optical field as:

Given the effective index in Eq. (2), the electric field in Eq. (1) should satisfy:

In order to determine $\Delta h\left(x,z\right)$, we must first derive its governing equation. For the electric field polarized along the y axis, the optical radiation pressure at the layer 1 and 2 interface is given by $p_{opt}={\epsilon }_0\left(n_2^2-n_1^2\right)n_2^2{\left|A\left(x,z\right)\right|}^2/(2n_1^2)$ [16]. Since we look for a solution that is self-guided and propagates along the $x$ direction, its intensity, i.e., ${\left|A\left(x,z\right)\right|}^2$, should not depend on $x$. Therefore, the shape of the bulge $\Delta h\left(x,z\right)$ should not depend on $x$ either. Then, in equilibrium, interfacial tension on a deformed interface is balanced with buoyancy and radiation pressure, according to the Young-Laplace equation. When the bulge has a small slope ($\partial \Delta h/\partial z\text{<<}1$), this balance leads to:

Our assumptions in deriving the form of optofluidic solitons are either the standard approach or can be justified using carefully controlled experimental conditions. For example, the slow varying envelope assumption and the effective index theory are commonly used in nonlinear optics and integrated optics. Our assumptions on liquid properties such as density, refractive index, and surface tension, however, warrant additional discussion. For example, two key requirements are: 1) in comparison with the cladding medium (layer 3), the core medium (layer 2) should possess lower density and higher refractive index; 2) the surface tension between layer 1 and 2 should be small. Both requirements can be satisfied by using a water-in-oil microemulsion system that contains a mixture of water, sodium dodecyl sulfate, toluene, and n-butanol-1 [17]. At a temperature slightly higher than the critical temperature ($T_c$ = 35 °C) for phase transition, the mixture phase separates into two different micellar phases, with interfacial tension given by $\sigma ={\sigma }_0{\left(T/T_c-1\right)}^{2v}$, where ${\sigma }_0=1.04\times {10}^{-4} N/m$, and $v=0.63$. The two micellar phases possess different densities and refractive indices, with the density difference given by $\Delta \rho ={\left(\Delta \rho \right)}_0{\left(T/T_c-1\right)}^{\beta }$, and the index difference given by $\Delta n={\left(\partial n/\partial \rho \right)}_T\left({\rho }_1-{\rho }_2\right)$, where ${\left(\Delta \rho \right)}_0=285 kg/m^3$,$\beta =0.325$ and ${\left(\partial n/\partial \rho \right)}_T=-1.22\times {10}^{-4} m^3/kg$. Given these conditions, at the temperature of $T=35.5℃$, the interfacial tension is $\sigma =3.18\times {10}^{-8}N /m$, the density and refractive index differences are $\Delta \rho =35kg/m^3$, and $\Delta n=-0.0043$. The relative magnitude of buoyancy to interfacial tension force is defined as the Bond number ($B=\Delta \rho g\Delta h/(\sigma \Delta h/\Delta z^2)$). Assuming a soliton solution with a bulge size of $\Delta z=20 \mu m$ and height $\Delta h=1 \mu m$, the Bond number is in the order of O(10). Such a large Bond number indicates that the interfacial tension effect is negligible compared to buoyancy effect. When the Bond number is small, the interfacial tension effect becomes comparable or even stronger than gravity, then we must solve the two coupled equations (Eqs. (4) and (5)) without any approximations. For such cases, analytical solutions would unfortunately be unattainable and numerical methods must be applied.

An attractive feature of the optofluidic soliton is that its nonlinear behaviors can be tuned by varying the parameters of the liquid systems. In particular, an interface with weak interfacial tension should possess high nonlinear efficiency. As an example, consider the waveguide structure analyzed in Fig. 2(b). From linear fitting of effective index for $2.5 \mu m<h<4 \mu m$, we find $\partial n/\partial h=1\times {10}^{-3}\mu m^{-1}$. Using this value and liquid parameters defined in the previous paragraph, the effective nonlinear coefficient defined in Eq. (6b) is approximately ${\chi }^{eff}=7.44\times {10}^{-13} m^2/V^2$. Comparing Eq. (6a) to the equation that describes spatial self-focusing, e.g., Eq. (7).1.18) of [24], it is clear that ${\chi }^{eff}$ essentially represents the third order nonlinear susceptibility of the optofluidic system. For typical optical materials, the third order nonlinear susceptibility ${\chi }^{\left(3\right)}$ varies in the range of ${10}^{-22} m^2/V^2$ to ${10}^{-18} m^2/V^2$ [24]. Therefore, by using liquid systems with weak interfacial tension, we can effectively increase the third order nonlinearity by several orders of magnitude.

We can use Eq. (7) to estimate the optical power required to form the aforementioned optofluidic soliton. For this calculation, we assume$z_0=10 \mu m$, $\lambda =1 \mu m$, $n_0\approx 1.5$, and use ${\chi }^{eff}=7.44\times {10}^{-13} m^2/V^2$ given in the paragraph above. Then, based on Eq. (7b), the required electric field strength is $A_0=2.61\times {10}^4 V/m$. If we assume that the optical beam is confined in a cross-section of width $W=10 \mu m$ (i.e., soliton width) and height $H=3 \mu m$ (i.e., waveguide thickness), the optical power of such a beam can be simply estimated as $n_{0}c{\epsilon }_0{\left|A_0\right|}^{2}WH/2$. Using parameters given above, we find that the optical power of this optofluidic soliton is only $41 \mu W$. We can also use the value for $A_0$ to estimate the bulge height. Applying the discussion immediately above Eq. (6), we can relate the magnitude of bulge height to the optical field amplitude as $\Delta h\approx {\epsilon }_0\left(n_2^2-n_1^2\right)n_2^2{\left|A_0\right|}^2/\left(2n_1^2\Delta \rho g\right)$. Using waveguide parameters given in the caption of Fig. 2, $g=9.8m/s^2$, and assuming $\Delta \rho =35kg/m^3$ (refer to the discussion on the water-in-oil microemulsion system), we find the bulge height is approximately $\Delta h\approx 114 nm$. The relative deformation ($\Delta h/h$) is approximately 4%.

The discussion in this section is mainly theoretical, where our primary aim is to show that radiation pressure induced interface deformation can lead to nonlinear processes very similar to traditional third order nonlinear processes such as optical solitons. In terms of experiment, it is perhaps easier to investigate nonlinear optofluidic processes using a high-Q optical resonator based on a liquid droplet, as will be discussed in the next section.

3. WGM induced droplet deformation

Figure 3(a) illustrates the second example of radiation pressure induced nonlinear optofluidic processes, where the key element is the high-Q WGM circulating along the equator of a liquid droplet in air. Since the direction of the radiation pressure always points from the high refractive index material to the low index material, the presence of a high-Q WGM would push the spherical droplet surface outwards and consequently enlarge the circumference of the droplet’s equator. Since radiation pressure is proportional to the optical power carried by the WGM, under the limit of small droplet deformation, we expect that the equator circumference should increase linearly as a function of WGM power and shift the WGM resonance frequency as a direct result. This phenomenon is very similar to the Kerr effect. In the following analysis, we establish an analytical framework and provide an order of magnitude estimate for the frequency shift associated with this nonlinear optofluidic process. Our analysis suggests that the effective strength of the aforementioned nonlinear optofluidic process can be several orders of magnitude larger than the traditional Kerr effect.

 

Fig. 3 (a) A liquid droplet with a high-Q WGM circulating near its equator. The radiation pressure of the WGM deforms the original spherical droplet (the blue circle) and generates the bulge (the solid black line), which is approximated as an oblate spheroid (the dashed black line). The normalized equator radius $x_e$ is defined as the ratio of the spheroid radius at the equator $(a+\Delta R)$ and the radius of the original sphere $a$. (b) The integral $F(x_e)=-{\displaystyle {\int }_0^{\pi }a\overline{\kappa }\left(x_e,\theta \right)Y_{20}\left(\theta \right)\sin \theta d\theta }$ (blue circles) as a function of the normalized equator radius $x_e$. $\overline{\kappa }\left(x_e,\theta \right)$ is given in Eq. (12). The dimensionless constant ${\Gamma }_{\sigma }$ is extracted using Eq. (14) and least square fitting (dashed red line).

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The starting point of our analysis is the Young-Laplace equation. We assume a small Bond number, i.e., buoyant force is much smaller than surface-tension force and is ignored. As a result, we find:

Numerically solving Eq. (9) to (11) can give us the form of droplet deformation. However, the mathematical process is complex and can obscure the physics. In this paper, we emphasize the physical reality and utilize several assumptions to obtain an intuitive estimate of droplet deformation. First, we assume that the droplet deformation can be approximated by an oblate spheroid, as shown in Fig. 3(a). This assumption simplifies our analysis considerably. For example, assuming a spheroid shape, the mean curvature of the droplet interface is simply:

The next step is to expand both sides of Eq. (8) using spherical harmonic function $Y_{LM}\left(\theta ,\varphi \right)$. From symmetry arguments, we can conclude that most of the spherical harmonic expansion coefficients are zero. First, $\Delta p$ is constant over the droplet interface, whereas both $p_{opt}$ and $\overline{\kappa }$ are independent of $\varphi$. Therefore, any $M\ne 0$ term must be zero. Additionally, both $p_{opt}$ and $\overline{\kappa }$ are symmetric with respect to $\theta =\pi /2$. Hence, any term with odd $L$ number must also be zero. The first two non-zero terms of spherical harmonic expansion involve overlap integrals with $Y_{00}$ and $Y_{20}$, and are given by:

To further simplify Eq. (13b), we note that the right hand side of Eq. (13b) is a function of $x_e$ only. Multiplying the integral by droplet radius $a$, we can define a dimensionless parameter $F(x_e)=-{\displaystyle {\int }_0^{\pi }a\overline{\kappa }\left(x_e,\theta \right)Y_{20}\left(\theta \right)\sin \theta d\theta }$ and numerically evaluate it as a function of $x_e$. The result, which is shown in Fig. 3(b), suggests that this integral is almost a perfect linear function of $x_e$. Thus we introduce a dimensionless parameter ${\Gamma }_{\sigma }$ as:

To obtain a closed form formula, we express ${\left|{\overrightarrow{E}}_{surf}\right|}^2$ in Eq. (11) as:

Equation (18) relates droplet deformation to the peak electric field intensity on the droplet interface and is the key result of this paper. In the next section, we consider several examples of liquid droplets and calculate their deformation as a function of optical power associated with the circulating WGM.

4. Estimate of nonlinear optofluidic effects based on high-Q WGMs

Using Eq. (18), we can calculate the radiation pressure induced WGM frequency shift and quantitatively compare nonlinear optofluidics with the traditional Kerr effect. We choose the following parameters in our calculations. The core index is $n_{co}=1.33$, the cladding index is $n_{cl}=\mathrm{1.0.}$ The droplet radius varies from $10\mu m$ to $400\mu m$. All WGMs are the fundamental TE mode $|ll〉$, and their angular momenta $l$ are chosen to ensure that the WGM wavelengths $\lambda$ are in the vicinity of $1.56 \mu m$. Table 1 lists the $l$ and $\lambda$ of the TE $|ll〉$ mode in such droplets with different radii $a$. The resonance frequencies are obtained by matching transverse field components across the droplet interface [27], which is assumed to be a perfect sphere.

Table 1. The Angular Mode Number $l$ and Resonance Wavelength $\lambda$ of WGMs in Liquid Droplets.

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Equation (18) links droplet deformation with peak surface field intensity, which is unfortunately not directly measurable. Therefore, we need to further relate the peak field intensity in Eq. (18) with the optical power or energy of the circulating WGM. To accomplish this, we take advantage of the fact that the deformation is linearly proportional to ${\left|{\overline{E}}_{surf}^{peak}\right|}^2$. Specifically, we choose an electric field amplitude such that $\left|{\overline{E}}_{surf}^{peak}\right|=1$ and calculate the corresponding field distribution, WGM power flux and total energy. Figure 4 shows the field distribution and the $\varphi$ component of the Poynting vector of a WGM ($\text{l}=257$) in a droplet with radius $a=50\mu m$. By integrating the Poynting vector across the $\varphi =0$ plane, we can calculate the total optical power carried by the WGM. Similarly, through volume integration of energy density, we can obtain the total energy stored within the WGM. Then, we multiply the electric field intensity with an appropriate normalization factor such that the total power or the energy of the WGM becomes the desired value. This normalization factor is then used to calculate ${\left|{\overline{E}}_{surf}^{peak}\right|}^2$ associated with the desired WGM power or energy. Once ${\left|{\overline{E}}_{surf}^{peak}\right|}^2$ is obtained, the only unknown parameter in Eq. (18) is the dimensionless factor ${\Gamma }_{\theta }^{lm}$, which can be obtained through numerical integration of Eq. (17) and is shown in Fig. 4(d).

 

Fig. 4 (a) The radial dependence of ${\left|\overrightarrow{E}\right|}^2$ of a fundamental TE mode ($l=$257) in a spherical droplet with radius $a=50\mu m$. We assume $\left|{\overrightarrow{E}}_{surf}^{peak}\right|=1$ and take $\theta =\pi /2$ and $\varphi =0$. (b) The angular dependence of ${\left|\overrightarrow{E}\right|}^2$ for the WGM in (a). The value of ${\left|\overrightarrow{E}\right|}^2$ is evaluated over the droplet surface, with $\varphi =0$. Due to our normalization scheme, the curve is also $f_{lm}\left(\theta \right)$. (c) The power flux of the WGM in (a) within the $\varphi =0$ plane. Only the ${\overrightarrow{e}}_{\varphi }$ component of the Poynting vector is shown. The white circle represents the droplet surface. (d) The value of ${\text{Γ}}_{\theta }^{lm}$ for the fundamental TE mode $|ll〉$ in droplets with different radius. The sphere radii and WGM parameters are listed in Table 1. The ${\Gamma }_{\theta }^{lm}$ values (blue circles) are obtained numerically using Eq. (17) and simply connected together using the dashed line.

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Using the procedure described above, we set the power associated with the circulating WGM to be 1 W and use Eq. (18) to calculate the radiation induced interface deformation for different droplet radius. The results are shown in Fig. 5(a). The red circles give the normalized droplet deformation ($\Delta R/a$) induced by the WGM radiation pressure, where we assume the surface tension is that of water ($\sigma =72mN/m$). The blue and black diamonds represent the estimated Kerr effect in water (blue) and CS₂ (black), produced by the same peak surface field ${\left|{\overrightarrow{E}}_{surf}^{peak}\right|}^2$. These estimates are simply obtained by using $\Delta n\approx {\chi }^{\left(3\right)}{\left|{\overrightarrow{E}}_{surf}^{peak}\right|}^2$, where ${\chi }^{\left(3\right)}$ is the third order nonlinear susceptibility of water ($2.5\times {10}^{-22}{m}^2/V^2$) or CS₂ ($3.1\times {10}^{-20}{m}^2/V^2$). Since both $\Delta R/a$ and $\Delta n$ represent the same physical effects, i.e., the relative change in the optical path of the high-Q resonator, they can be shown in the same figure for direct comparison. Clearly, the nonlinear optofluidic effect is three to five orders of magnitude stronger than the Kerr effect.

 

Fig. 5 (a) Radiation pressure induced droplet deformation ($\Delta R/a$) in droplets with different radii (red circles). The total power of the WGM that circulates along the droplet equator is fixed at 1 W. For comparison, the changes in refractive index ($\Delta n$) due to the Kerr effect are shown in the same figure, which are estimated using $\Delta n\approx {\chi }^{\left(3\right)}{\left|{\overrightarrow{E}}_{surf}^{peak}\right|}^2$. The Kerr effects in water and in CS₂ are represented as the blue and black diamonds, respectively. All points are connected by dashed lines. (b) The relative shift in WGM frequency induced by the radiation pressure of a single photon. Two different values are used for surface tension: $\sigma =72mN/m$ (blue circles), and $\sigma =1mN/m$ (red crosses). All points are connected by dashed lines.

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Since the nonlinear optofluidic process can be significantly stronger than the Kerr effect, it is worth asking the question: Is it possible to use only a few photons to generate measurable WGM frequency shift? To answer this question, we first calculate the value of ${\left|{\overrightarrow{E}}_{surf}^{peak}\right|}^2$ such that the total WGM energy equals to single photon energy $\hslash \text{ω}$. Then, we substitute this value into Eq. (18) and calculate the normalized radius change $\Delta R/a$ induced by the presence of a single photon. Finally, using the perturbation theory results in [26], the WGM frequency shift associated with the deformed spheroid is:

Combining Eqs. (18) and (19), we can now calculate the radiation pressure induced frequency shift. The results are shown in Fig. 5(b). One set of data (blue circles) is obtained using $\sigma =72mN/m$, which is a typical value for air-liquid interface. The other set of data (red crosses) assumes a low but experimentally achievable surface tension $\sigma =1mN/m$. Given the fact that WGMs with Q factors as high as 2.3 × 10⁶ have been observed using a liquid resonator [22], Fig. 5(b) suggests that one should be able to use only a few photons to produce experimentally observable resonance shift. If one can reduce the surface tension further down to the level of $\sigma =0.1mN/m$, then the presence of even a single photon can significantly change the characteristics of the liquid droplet resonator.

It is worth remarking that liquid systems with ultralow surface tensions have been experimentally attained using surfactants [28–30]. For example, by introducing a bimolecular layer of preformed ferric stearate, one can reduce the surface tension of water down to $1mN/m$ [28]. In emulsion or microemulsion systems containing water (a polar fluid) and oil (a nonpolar fluid), surfactants can lower surface tension as small as $1\mu N/m$ at an optimal concentration [29]. Therefore, for these liquid systems with ultralow surface tension, nonlinear optics at single photon energy level should be experimentally feasible.

In our WGM analyses, the effect of gravity is ignored. To justify this choice, we can estimate the Bond number of the liquid droplet. For a sphere with radius $a$, the gravitational and the surface tension effects can be estimated as $\Delta \rho g\Delta R$ and $\sigma \Delta R/a^2$, respectively. Consequently, the Bond number is simply $B=\Delta \rho g\Delta R/(\sigma \Delta R/a^2)$. For an order of magnitude estimate, we can assume $\Delta \rho ~{10}^{3}kg/m^3$, $g~10m/s^2$, and $a=100 \mu m$. With these values, even if the surface tension is as low as $\sigma =1mN/m$, the Bond number remains small ($B~0.1$) and we can safely neglect the gravitational effect in Eq. (8).

It is also of interest to briefly consider other types of liquid resonators such as an oil droplet ($n_{co}=1.45$) immersed in water ($n_{co}=1.33$). Such a resonator is similar to the one investigated in [21]. As a specific example, we estimate the deformation induced by 1W WGM power circulating within an oil-in-water droplet with radius $a=70 \mu m$ and interfacial tension $\sigma =72 mN/m$. This resonator supports an TE-polarized WGM with $l=397$ and resonance wavelength $\lambda =1.5610 \mu m$. Using the procedure described above, we find that 1 W WGM power should induce a relative radius change of $\Delta R/a=2.94\times {10}^{-4}$. In comparison with the results shown in Fig. 5(a), for an water-in-air droplet with identical radius, interfacial tension and WGM power, the deformation is $\Delta R/a=2.70\times {10}^{-4}$. This result suggests that the conclusions we obtained using water-in-air droplets are broadly applicable to other types of liquid resonators.

Equation (18) suggests that droplet deformation depends on interfacial tension but not on liquid viscosity. This is to be expected, since our discussion is based on static analysis, i.e., the Young-Laplace equation in Eq. (8), which does not involve viscosity. We do, however, expect that viscosity should play a role in the dynamics of droplet deformation.

Finally, we point out that both the spherical WGM system analyzed here and the planar soliton case discussed in section 2 can be connected to liquid-based spheroid resonators. On the one hand, the spherical droplet analyzed in section 3 is a special case of the liquid spheroid resonators. On the other hand, a prolate spheroid resonator with an extremely large major axis can essentially be regarded as an infinitely long dielectric cylinder, which can also support WGM that circulates along the perimeter of the cylinder. And if we “unwrap” the infinitely long cylinder, the WGM traveling along the cylinder perimeter becomes similar to the self-guided soliton waves analyzed in section 2.

5. Summary

In this paper we analyze the possibility of using all liquid systems to achieve nonlinear optical effects with extremely low power threshold. The defining feature of the proposed nonlinear optofluidic processes is the interface deformation due to optical radiation pressure. In particular, we find that through the formation of a bulged interface, optical waves could self-focus within a liquid slab waveguide. Under appropriate conditions, these self-guided optical waves are governed by the nonlinear Schrödinger equation and can be described by the familiar soliton solutions. We then consider a spherical liquid droplet that supports high-Q WGMs. With sufficiently high power, the optical force associated with the circulating WGM could deform the liquid droplet and induce a frequency shift of the WGM resonance. By applying spherical harmonic expansion, we estimate that the radiation pressure induced nonlinearity is several orders of magnitude stronger than the traditional Kerr effect. In fact, we find that by using a liquid system with low but experimentally achievable surface tension, it is possible to produce measurable frequency shift at the energy level of a few photons. Such effect may ultimately lead to nonlinear interactions between two single photons.