Localized optical manipulation in optical ring resonators

Haotian Wang; Xiang Wu; Deyuan Shen

doi:10.1364/OE.23.027650

1. Introduction

At present, optical tweezers have been a powerful non-invasive tool for the manipulation of microscale dielectric particles like cells, bacteria, and dielectric spheres etc. in biology and medicine science [1–5]. However, the manipulation for nanoscale particles (<100 nm) is very difficult to achieved by traditional free space optical trapping techniques due to the fundamental physical limitations such as the diffraction limit [6]. To address this issue, alternative near-field optical trapping techniques have been recently developed [7–18]. Since the evanescent fields surrounding the photonic structures such as planar waveguides, optical resonators and plasmonics decay rapidly, the extremely high optical gradients can provide greatly enhanced optical forces to manipulate nanoscale particles. Among these photonic architectures, optical ring resonators supporting whispering-gallery modes (WGMs) become one of the most promising optical near-field trapping schemes. The first systematic study on manipulating the small particles by using microsphere resonators was demonstrated by Arnold et al. in 2009 [11]. In this case, a tapered fiber is utilized as the bus waveguide to couple the light effectively into a dielectric microsphere for the excitation of WGMs. When the light is set at a resonance wavelength, the incoming light in the fiber is in phase with the light which is circulating in the microsphere and these two parts light will constructively interferes with each other. Consequently, the light intensity in the microsphere is much stronger than that in the bus waveguide. Due to the strong light intensity and the high optical gradients of the evanescent fields near the microsphere surface, the manipulation for the dielectric nanoparticles as small as 280 nm has been achieved. Afterwards, the detection and trapping of nanoparticles in the WGM resonators with ultrahigh Q factor are demonstrated [19, 20]. Similar optical trapping schemes based on on-chip ring resonators have been demonstrated by several research groups [12, 13, 21–24]. Compared to the microsphere resonators, these on-chip optical trapping systems are more compatible with other photonics devices and can be integrated conveniently into a promising optofluidic platform for the optical manipulation of high-throughput nanofluidics.

In all of the above trapping cases, the particles are attached onto the surface of the optical ring resonator, where the azimuthal momentum flux of the circulating light propels the particles to move in a circular orbit due to the optical scattering force. Therefore, it is impossible to “stop” the particles at the expected location. To solve this “non-localization” problem, the optical trapping technique based on photonic crystal (PC) resonators, which can confine the light within an extremely small volume, have been proposed in recent works [25–27]. In these PC resonators, the light is highly compressed within a microcavity smaller than Rayleigh limit, which generates great enough optical gradient forces to achieve the controlled nanoscale trapping. However, the fabrication process of PC resonators is complex and expensive. Once the PC resonators have been fabricated, the locations of the trapped particles will be immutable.

In this work, we propose a modified localized optical trapping system based on on-chip optical ring resonators. In the trapping scheme, two bus waveguides are utilized to couple two coherent beams into a ring resonator in opposite directions, respectively. When light is set at resonance wavelengths, two sets of excited WGMs circulate in opposite directions in the ring resonator and constructively interfere with each other to form standing waves. The particles could be trapped within the volume near the nodes of standing waves for the balance of the optical scattering forces, which means the optical trapping scheme is “localization”. Moreover, the localizations of the trapped particles are tunable when the phase or wavelengths of the input beam are selected appropriately. The field distributions of excited WGMs in the localized optical trapping systems based on different optical ring resonators including microring, microdisk and ring-shaped slot structure as well as the corresponding optical forces on the nanopartices are calculated numerically. The location-tunable trapping of very small particles are achieved in the designed trapping systems which possess the potential in several promising applications, such as trapping single molecules, optical chromatography and directed nanoassembly.

2. Basic structure and theoretical analysis

Figure 1 schematically shows the on-chip system of the localized optical trapping. The SiN planar ring resonator with two paralleled planar waveguides are placed on the SiO₂ substrate and immersed in water which is served as the particles solution. Figure 1 (b) and (c) show the top-view of the ring resonator and its transmission spectra, respectively. The dip of the transmission spectra is corresponding to the resonant wavelength of the WGM excited in the ring resonator. The mode field E(r, φ, z) of the WGM can be separated as E = E_r(r)•E_φ(φ)•E_z(z) in cylindrical coordinate. The azimuthal distribution E_φ(φ) is:

E_{φ} (φ) = \exp (i m φ)

and the radial distribution E_r(r) can be expressed as [28]:

E_{r} = {\begin{matrix} A J_{m} (k_{φ}^{(m, l)} n_{1} r), r \leq R_{1} \\ B J_{m} (k_{φ}^{(m, l)} n_{2} r) + C H_{m}^{(1)} (k_{φ}^{(m, l)} n_{2} r) R_{1} < r \leq R_{2} \\ D H_{m}^{(1)} (k_{φ}^{(m, l)} n_{1} r) r > R_{2} \end{matrix}

where J_m and H_m⁽¹⁾ are the mth Bessel function and the first kind mth Hankel function, respectively. k_φ(m,l) is the amplitude of the resonant wave vector component in φ direction labeled by the azimuthal index m and the radial index l. n₁ and n₂ are the refractive index of ring resonator and the water, respectively. R₁ and R₂ are the inner and external radius of the ring resonator.

Fig. 1 (a) Schematic of the on-chip localized optical trapping system. (b) The top-view of the ring resonator with the two bus waveguides. (c) The transmission spectra of the trapping system based on the microring resonator.

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As shown in Fig. 1, the input beam is split equally into two paralleled bus waveguides to make the dual-port inputs. When the two input beams are at the resonant wavelength, they will be coupled into the ring resonator more effectively and excite two sets of WGMs circulating in opposite directions, respectively. If we only consider the azimuthal distribution, the mode field distributions of the two sets of WGMs E₁(φ) and E₂(φ) in the ring optical resonator can be expressed as:

\begin{array}{l} E_{1} (φ) = A \cos (ω t - \frac{2 π n_{1} R φ}{λ} + ϕ_{1}) \\ E_{2} (φ) = A \cos (ω t + \frac{2 π n_{1} R φ}{λ} + ϕ_{2}) \end{array}

Where

ϕ_{1}

and

ϕ_{2}

are the two input beam phases at the coupling gaps, respectively. R is the radius of the ring resonator. The two sets of WGMs interfere with each other in the ring resonator leading to the formation of standing wave which can be expressed as:

E (φ) = 2 A \cos (ω t + \frac{ϕ_{1} + ϕ_{2}}{2}) \cos (\frac{2 π n_{1} R φ}{λ} + \frac{ϕ_{2} - ϕ_{1}}{2})

From Eq. (4), the azimuthal orientations of the standing wave nodes

φ_{n o d e}

are derived as:

φ_{n o d e} = (n π - \frac{ϕ_{2} - ϕ_{1}}{2}) \cdot \frac{λ}{2 π n_{1} R}, n = 0, 1, 2, \dots

The optical gradient force and scattering force are proportional to the 3rd and 6th power of particle radius, respectively [6]. In generally, for the nanoscale particles the optical scattering force is negligible. Therefore, we will focus on the gradient force on the nanoparticles in this paper. If the particle scale is much smaller than the light wavelength, the optical gradient force will be written as [29]:

F_{g r a d} = π ε_{0} \nabla {| E |}^{2} R_{p}^{3} (\frac{ε_{m} - ε}{ε_{m} + 2 ε})

where

| E |

is the electric field amplitude and R_p is particle radius. ε_m and ε are the permittivities of the particle and water, respectively. It is clear that the optical gradient force is proportional to the gradient of the electric field intensity. Therefore, the particle will be trapped within a small volume near one of the standing wave nodes where the gradient of the electric field intensity is much higher. It should be pointed out that the particle trapped in the resonator is not only influenced by the optical field but also be affected by the other forces, such as the waveguide surface adhesion force, electrical double layer (EDL) repulsive force and liquid drag force. However, compared to the optical force for our designed system in this paper, it is reasonable to neglect the above-mentioned forces.

3. Trapping the nanoparticles in optical ring resonator

To investigate the trapping capability of our trapping system, we first study the amplification of light intensity in the ring resonator. In our simulation, the width and height of the planar waveguides are 500 and 650 nm, respectively. The gap between the bus waveguide and the ring resonator are 500 nm and the ring resonator radius R is set as 9 μm. Here, we assume the waveguide mode is in the TM polarization that has a stronger optical field gradient near the waveguide top surface than TE polarization. Using the axisymmetric mode developed by Oxborrow [30], we calculate the normalized $| E_{z} |$ distribution profile at the cross section of the ring resonator, which is shown in Fig. 2 (a). A considerable portion of TM mode field is above the top surface of the planar waveguide. The 2-dimensional (2D) model is adopted to calculate the optical field distribution profile at the top surfaces of the ring resonator and two bus waveguides. To replace the practical 3-dimensional (3D) model, in our 2D model we set the effective refractive index of SiN waveguides as 1.83 calculated by Effective Index Method.

Fig. 2 (a) The normalized $| E_{z} |$ distribution profile in x-z plane. (b) Normalized $| E |$ distribution profile in x-y plane. (c), (d) and (e) are the optical gradient force on a particle of 100 nm in diameter along the azimuthal, radial and vertical directions, respectively. (f)-(h) are the corresponding potential energy of the trapped particle.

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Figure 2 (b) shows the normalized $| E |$ distribution profile at the resonance wavelength (~1002.8 nm) which is corresponding to the dip in the transmission spectra in Fig. 1 (c). A typical mode field distribution of the standing wave featuring the high field confinement in the ring resonator is presented. To evaluate the amplification of the optical field in the ring resonator, we estimate the Q factor of the ring resonator according to the transmission spectra. The Q factor is inferred as approximately 5 × 10⁴ and the field intensity in the ring resonator is 500 times as strong as that in the bus waveguides. For the nanoscale particles trapped in our designed WGM resonator with Q factor of ~10⁴, the influence of the particle on the WGM is negligible due to the extremely tiny changing of resonance wavelength. However, when the nanoparticle is trapped in the ultrahigh-Q factor resonator, mode splitting will occur. The counter-propagating WGMs induced by the trapped particle will be excited in the resonator, therefore it is necessary to take the particle-perturbation into account [20, 31, 32]. Figure 2 (c) and (d) demonstrate the optical gradient force on a particle of 100 nm in diameter along the azimuthal and radial directions indicated by the dash lines in the inset of (b), respectively. Figure 2 (e) shows the corresponding vertical optical gradient force along the vertical dash line over the top surface of the ring resonator in (a). The corresponding potential energy of the trapped particles are shown in Fig. 2 (f)-(h). Here, we define equilibrium position of the trapped particles in the resonator as potential energy zero point. The optical gradient force along the vertical direction is a magnitude higher than the azimuthal and radial directions because the high optical gradient of evanescent field outside the ring resonator.

Under a fixed input power, the optical gradient force will decrease drastically as reducing the particle size. Since the trapped particle needs enough optical force to resist the Brownian diffusion, the realizable trapping size has a lower limit. Thus, to access the performance of our trapping scheme, it is necessary to analyze the stability of the trapped particle which is under the Brownian motion. As suggested in Ref [33], the stability number is defined as:

S = \frac{W_{t r a p}}{k_{B} T}

where W_trap is the necessary work to release the particle from trapping region, k_B is the Boltzmann constant and T is the temperature in Kelvin unit. In previous ring-resonator-based trapping systems, the trapped particles circulate in the ring resonator without the confinement in azimuthal direction. For our localized trapping system, since we “stop” the particle within a small volume it is essential to analyze the stability in all directions. On the condition that the temperature and the light power before the beam splitter are set as 300 K and 10 mW, respectively, the relationship between the stability number and the particle diameter is given in Fig. 3. The horizontal dash line represents S = 10 which means the trapping edge for the particles under Brownian motion [34]. Based on this, the smallest diameter of the particles could be trapped stably in azimuthal, radial and vertical directions are 57, 61.2 and 33.6 nm, respectively. Practically, for the 10 mW input power the reasonable smallest trapping size is the maximum of them, i.e. 61.2 nm. The relationship between the input power and the minimal particle size which could be stably trapped in the resonator is shown in the inset.

Fig. 3 The relationship between the stability number and the particle diameter. The relationship between the input power and the minimal trapped particle size is shown in the inset.

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4. Tunable locations of the trapped particles

According to Eq. (6), the locations of the standing wave nodes are dependent on the phase difference $ϕ_{2} - ϕ_{1}$ . Commonly, we can control the effective index of the waveguide mode to change the beam phase by some mechanical means, such as applying a stress on the bus waveguide. Referring to Fig. 1, a particle is trapped near a standing wave node indicated by the red ball when the beam phase in the upper bus waveguide is $ϕ_{1}$ . If we change the beam phase from $ϕ_{1}$ into $ϕ_{1}^{'}$ , the standing wave nodes will have a azimuthal displacement which drives the trapped particle to a new location indicated by the blue ball.

Figure 4 gives the electric field amplitude distribution near the coupling gap corresponding to Fig. 2 (b) when phase difference are at different values. Every nodes move to their neighbouring nodes along the azimuthal direction when the phase difference is changed from 0 to 2π.

Fig. 4 The electric field amplitude distribution near the coupling gap when phase differences between the two input beams are (a) 0 (b) 0.5π, (c) 1π and (d) 1.5π.

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Because the width of the ring resonator is smaller than the wavelength, there are only fundamental modes in radial direction. Therefore the location of the trapped particle in radial direction is fixed, which implies we could not enable a tunable radial trapping. To push the limitation of the ring width, we employ the microdisk instead of the microring as the resonator, as shown in Fig. 5 (a). Differ from the microring structure, microdisk resonator could support high order WGMs in radial direction. We set the gap between the bus waveguides and the microdisk and the radius of the microdisk as 500 nm and 9 μm, respectively. Figure 5 (c) and (d) show electric field amplitude at two different resonance wavelengths (996.2 nm and 990.4 nm) corresponding to mode 1 and mode 2 of the transmission spectra in Fig. 5 (b). A example of radial trapping of nanoparticles is shown in Fig. 6. When the input light is at resonant wavelength 996.2 nm, the WGM with only fundamental modes is excited in the microdisk resonator. The particle is trapped near one standing wave node and its location is indicated by circle A in Fig. 6 (a). Then we change the input wavelength into 990.4 nm to excite the high order radial modes. In this case, the excited mode filed has two radial nodes implying the particle will be trapped at two possible locations, i.e. circle B and C in Fig. 6 (b). Figure 6 (c) and (d) are the radial optical gradient force along the vertical dash lines in (a) and (b). The radial locations of the trapped particles are the zero potential energy points shown by the points A, B and C in Fig. 6 (e) and (f). It should be pointed out that being different from the azimuthal trapping we only could enable the radial trapping for finite locations.

Fig. 5 (a) The top-view of the microdisk resonator with two bus waveguides. (b) The transmission spectra of the trapping system based on the microdisk resonator. (c) and (d) are the normalized electric field amplitude distribution in the microdisk resonators at the resonance wavelengths 996.2 nm 990.4 nm, respectively. The corresponding zoom-in-view images near the coupling gap are shown in the insets.

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Fig. 6 (a) and (b) are the zoom-in-view images of the insets of Fig. 6 (c) and (d), respectively. (c) and (d) are the optical gradient forces along vertical dash line in (a) and (b), respectively. (e) and (f) are the corresponding potential energy of the trapped particle.

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5. Manipulation of the smaller nanoparticles in a slot ring resonator

The trapping configurations based on planar waveguides are less effective because they depend on the interaction between the particle and the evanescent field outside of the waveguides that is a small part of the whole optical power. A slot waveguide is composed by two parallel solid strips which are placed closely to make a narrow slot in the middle of them. The narrow slot region is filled with a low refractive index medium, commonly the liquids, where most of the optical field is confined. The particles can be dissolved in the liquids and interact directly with the optical field inside the waveguide rather than the evanescent field. Using the slot waveguides, some researchers have successfully realized the manipulation of the nanoparticles [8, 9]. Here, we will adopt the slot structure to construct the ring resonator for trapping extreme small particles. In our design, the ring slot waveguide is 700 nm in width with a 50 nm-wide slot. Figure 7 (a) and (b), respectively, give the cross section of the normalized $| E_{x} |$ distribution and the $| E |$ profile in x-y plane at 325 nm above the SiO₂ substrate in the slot ring resonator. The most part of electric field intensity is compressed within the narrow slot. Figure 7 (c) and (d) are the azimuthal and vertical optical gradient forces on a 40 nm particle along the black dash lines in the inset of (b) and the vertical dash line inside the slot region in (a), respectively, (e) and (f) are the corresponding potential energy of the trapped particle. The radial trapping is replaced by the solid waveguides on both sides. As same as the stability analysis in the microring resonator, the size of the particle that could be trapped stably is minimized to 29 nm when the input power is also set as 10 mW.

Fig. 7 (a) The normalized $| E_{x} |$ distribution profile in x-z plane. (b) Normalized $| E |$ distribution profile in x-y plane. (c) and (d) are the optical gradient force along the azimuthal and vertical directions in the slot ring resonator, respectively. (e) and (f) are the corresponding potential energy of the trapped particle.

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6. Conclusion

In summary, a tunable localized trapping system of the nanoparticles based on generating standing waves in ring resonators is proposed. The input beams from two paralleled bus waveguides are coupled into the ring resonator in opposite directions to generate the standing wave. The mode filed distributions of standing waves and the optical gradient forces are calculated numerically. The particles will be trapped within a small volume near the nodes of the standing wave and their azimuthal orientations could be controlled by the phases of the input beams. Meanwhile, in the trapping system based on the microdisk resonator, the radial locations are tunable due to the excited high order modes in radial direction. The stability analysis shows our localized trapping system has a relatively high stability number. The result indicates the smallest size of the particle could be stably trapped is 61.2 nm when the input power before the beam splitter is set as 10 mW in the microring resonator. To improve the trapping ability for much smaller particles, we apply the slot ring structure to increase the usage ratio of the optical field and the achieved smallest diameter of the stably-trapped particles could be minimized to 29 nm under the same input power. The trapping system presented in this work permits to trap extremely small particle at a selectable location, which possess enormous potential in precisely manipulation of nanoparticles, such as directed nanoassembly and manipulating single molecules.

Acknowledgements

This work is supported in part by National Natural Science Foundation of China (grant # 61378080, 61327008, 60907011, 61177045)and the open project of State Key Laboratory of Modern Optical Instrumentation, Zhejiang university, China.

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Localized optical manipulation in optical ring resonators

Abstract

1. Introduction

2. Basic structure and theoretical analysis

3. Trapping the nanoparticles in optical ring resonator

4. Tunable locations of the trapped particles

5. Manipulation of the smaller nanoparticles in a slot ring resonator

6. Conclusion

Acknowledgements

References and links

Cited By

Figures (7)

Equations (7)

Optics Express