Laser induced surface stress on water droplets

Neng Wang; Zhifang Lin; Jack Ng

doi:10.1364/OE.22.023770

1. Introduction

Since its first demonstration in 1970 [1], optical trapping - the confinement of small particles by light induced forces - has developed into a very useful tool for small particle manipulation [2–5]. Later, the technique is also applied to rotate [6–8] or stretch [9–14] a particle. Compared with stretching or deforming a particle, the theoretical modeling of optical force and torque are relatively simple. Consider a particle in air for example, to compute its optical force and torque, one only needs to perform a surface integral in air, therefore only the stress tensor in air is needed. In contrast, in order to calculate the optical stress, one has to know the stress tensor inside particle, which can be difficult.

We employ the Mie theory [15–18] to compute the light induced stress. For nano to micro scale spherical water droplets are considered, where the stress depends strongly on size. When the droplets’ whispering gallery mode [19–26] is excited, the stress is enhanced by two orders of magnitudes and can be comparable with the surface tension stress.

2. Computation of the optical surface stress

We consider the light induced stress acting on water droplets (with relative dielectric constant ε_w = 1.33²) in air under cw laser (wavelength λ = 532nm) illumination. For incompressible fluid like water, when it is in mechanical equilibrium, we model the light induced stress by the following stress tensor [27]

{\bar{T}}_{water} = \frac{1}{2} ε_{w} ε_{0} E E^{*} + \frac{1}{2} μ_{0} H H^{*} - \frac{1}{4} [ε_{w} ε_{0} E \cdot E^{*} + μ_{0} H \cdot H^{*}] \bar{I}

As a consequence of mechanical equilibrium, the internal force density, given by f = ∇ · T̄_water, vanishes, but the surface stress does not. The light induced surface stress of a spherical water droplet under intense laser illumination is given by

σ = [{\bar{T}}_{air} (at outer boundary) - {\bar{T}}_{water} (at inner boundary)] \cdot e_{r}

By applying the boundary conditions for the electromagnetic field, Eq. (2) can be written as

σ = \frac{ε_{0}}{4} (ε_{w} - 1) ({| E_{t} |}^{2} + \frac{{| E_{r} |}^{2}}{ε_{w}}) e_{r}

where E_t and E_r are the tangential and radial component of the electric field outside the droplet. Accordingly, one only needs to solve for the boundary electromagnetic field. For a spherical particle, we adopted the accurate semi-analytically Mie theory to accomplish this task. According to Eq. (3), the surface stress is always radially outward and positive definite.

3. The droplets with different sizes

We consider droplets with size ranging from sub-wavelength scale to several microns. The former can be treated by dipole approximation while the later has to be treated by Mie theory.

Consider a dipolar particle illuminated by a circularly polarized plane wave propagating in the z-direction of the form

E_{i} = (e_{x} + i e_{y}) E_{0} e^{i k z} / \sqrt{2}

If not otherwise stated, the incident plane wave has an intensity of 1.3 × 10⁵W/cm² (E₀ = 10⁶V/m). The dipole moment will then be given by p = αE, where α = i6πε₀a₁/k³ is the polarizability [18]. The scattered field near the particle surface can be obtained by

E_{s} = \frac{1}{4 π ε_{0}} [3 e_{r} (e_{r} \cdot p) - p] \frac{1}{a^{3}}

where a is the droplet radius. With Eq. (4) and (5), the boundary field is known, and making use of Eq. (3), one arrives at

σ = \frac{ε_{0}}{8} (ε_{w} - 1) [(1 + {cos}^{2} θ) {| 1 - \frac{α}{4 π ε_{0} a^{3}} |}^{2} + \frac{1}{ε_{w}} {sin}^{2} θ {| 1 + \frac{α}{2 π ε_{0} a^{3}} |}^{2}] E_{0}^{2} e_{r}

where θ is the polar angle. Eq. (6) is the analytical expression for the surface stress of a dipolar spherical water droplet under laser illumination.

The polar plots of the surface stress for spherical droplet of various sizes are shown in Fig. 1(a). For a small sub-wavelength droplet with radius a = 0.02λ, the stress computed by Mie theory and the analytical formula Eq. (6) agree perfectly. The strength of the stress increases with particle radius. At an intensity of 1.3 × 10⁵W/cm², the a = 0.02λ droplet has a stress of ∼ 1N/m², while the stress could be over 10N/m² for a droplet with size comparable to a wavelength. Moreover, the stress distributions are also very different among droplets of different sizes. For a sub-wavelength droplet, the stress is slightly stronger near 90° and 270°. As the droplet size increases, the stress in the forward direction is significantly enhanced, due to the focusing effect of the spherical droplet. As shown by the green line on Fig. 1(a), the forward (0°) stress is more than 13 times larger than the stress in other directions (90°, 180°, 270°). Accordingly, for these particles, the forward optical force is dominated by the stress on the forward direction. Figure 1(b) is the polar plots of a droplet (a = 0.5λ) illuminated by a x-polarized plane wave at different azimuthal angles. The forward stress is the strongest, and the distribution of the stress near the forward direction is the widest for ϕ = 0°, wider for ϕ = 45°, and narrowest for ϕ = 90°.

Fig. 1 Light induced surface stress on water droplets. (a) Droplet of different sizes illuminated by an circularly polarized incident plane wave. (b) Droplet with a radius a = 0.5λ illuminated by a linearly polarized (E on the ϕ = 0° plane) plane wave. Different color curves correspond to different azimuthal angles.

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4. The whispering gallery mode

The whispering gallery modes (WGMs) are high quality factor (Q) resonances that exhibit extremely large boundary electromagnetic field. The large surface tension of water ensures a smooth surface even at the nano-metric scale, and this enables a high-Q resonance. One then expects a very large stress. The optical forces associated with WGMs have been considered in the literature [19–26], but not the associated stress. Figure 2 shows the radiation pressure as functions of the radii of the droplets being illuminated by a circularly polarized plane wave. There are two sharp peaks, each with Q ∼ 10⁴. They are precisely the WGMs labeled by $b_{44}^{1}$ and $a_{44}^{1}$ . Here b and a denote TE and TM modes, respectively. The superscript 1 denotes a first order resonance and the 44 denotes its angular mode number. The polar plots of the stress are shown in Fig. 3. The WGMs are plotted by red ( $b_{44}^{1}$ ) and blue ( $b_{44}^{1}$ ) lines on the left panel, and the off resonance case is plotted by the green line on the right panel for comparison. By comparing Figs. 3(a) and 3(b), it is clear that the stress is greatly enhanced at the WGMs. Since at resonance, a single mode is dominated, the profiles of the resonance stress is similar to the corresponding vector spherical wave functions ${| M_{1, 44}^{(3)} (k, r) |}^{2}$ and ${| N_{1, 44}^{(3)} (k, r) |}^{2}$ . In contrast, the off resonance stress is not symmetric, and is stronger in the forward direction due to the focusing effect of the spherical droplet. For $b_{44}^{1}$ mode, the maximum stress occurs in the forward and backward directions (∼ 1.5×10³N/m²). For $a_{44}^{1}$ mode, the maximum stress occurs at ∼2° and ∼ 178°, reaching over 600N/m². The off resonance stress is ∼ 10N/m², so there is a two orders of magnitude enhancement in stress here.

Fig. 2 Radiation pressure acting on a spherical droplet illuminated by a circularly polarized plane wave. The radii of the two sharp peaks, denoted by $b_{44}^{1}$ and $a_{44}^{1}$ , respectively, are 5.93895λ and 6.00526λ.

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Fig. 3 Light induced stress for (a) whispering gallery modes and (b) off resonance case where a = 6λ. The incident wave is a circular polarized plane wave.

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The stress for $a_{44}^{1}$ under a 1 Watt Gaussian beam illumination is also plotted in Fig. 4. The vector generalization of Debye’s integral [8, 28] is used to model the focused Gaussian beam. In Fig. 4(a), the mode is excited by a Gaussian beam (N.A.=0.9) that is focused to the droplet center. The excitation of the optical mode also depends on factors such as the beam waist and the filling factor. Nevertheless, the single most important factor is to have the correct frequency of the incident light. The induced stress is not very strong. This is expected, as WGMs are most easily excited by light illumination at the particle’s edge rather than its center. As we move the particle by 1 wavelength in a direction perpendicular to the beam axis (i.e. along 90°), the stress increases tremendously, as shown in Fig. 4(b). At 270°, the direction opposite to particle displacement, the stress is over 2 × 10⁴N/m², which is comparable with the surface tension induced stress for a water droplet with a radius of 3μm, typically 10⁴N/m². It is then clear that the light induced stress can result in significant deformation on the droplet.

Fig. 4 Light induced surface stress for a droplet excited at the $a_{44}^{1}$ mode. (a) A Gaussian beam is focused on the droplet center and (b) the droplet is displaced perpendicularly to the beam axis, along 90°. The incident circularly polarized Gaussian beam has a power of 1 W and a numerical aperture of 0.9.

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5. Conclusion

To summarize, the light induced stress acting on a spherical water droplet is studied. Sub-wavelength droplet shows an asymmetry stress distribution as the stress is weaker along the light propagation direction. On the other hand, larger particle shows significant enhanced forward stress, which is due to the focusing effect of spherical droplets. For WGMs with Q ∼ 10⁴, the stress is enhanced by two orders of magnitude. The stress distribution approximately follows that of the corresponding vector spherical wave function. For ∼ 3 micron radius particles with a ∼ 10⁴ illuminated by a 1 Watt Gaussian beam, the stress for the whispering gallery mode can be comparable with the stress due to surface tension.

The light induced stress is in general small compared with the surface tension of water. An exception is when the high-Q WGMs are excited. In reality, the water droplets are likely not as spherical as we assume. However, the strength of the stress is closely tied to Q, and water droplets with high-Q do exist, albeit it may not be perfectly spherical. We remark that the strong electromagnetic fields associated with the optical whispering gallery modes may deform the droplet, modifying the resonance frequency and thus driving the system out of the optical resonance. However, as soon as the droplet is out of resonance, the strength of the fields drops dramatically, and the droplet will restore to its initial position. Depending on conditions, such optomechanics coupling can lead to a series of very interesting phenomena [29].

Acknowledgments

The supports from HK RGC through grants HKBU 604011, HKBU 603312, and HKBU 209913 are gratefully acknowledged. NW and ZL were also supported by the China 973 Project (Nos. 2011CB922004 and 2013CB632701) and China National Natural Science Foundation (No. 11174059). Computational resources were supported by the Information Technology Office of HKBU.

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Laser induced surface stress on water droplets

Abstract

1. Introduction

2. Computation of the optical surface stress

3. The droplets with different sizes

4. The whispering gallery mode

5. Conclusion

Acknowledgments

References and links

Cited By

Figures (4)

Equations (6)

Optics Express