Systematic analysis of optical gradient force in photonic crystal nanobeam cavities

Shoubao Han; Yaocheng Shi

doi:10.1364/OE.24.000452

1. Introduction

To stably trap nanoparticles with lower input power, there are various near-field trapping configurations to increase the field amplitude inside the device or/and create a large spatial field gradient, such as slot waveguide [1,2], microring resonator [3] and photonic crystal (PhC) cavities [4–8]. Among them, PhC cavities have been investigated for the cavity-enhanced optomechanical coupling due to their high quality factors (Q) and small mode volumes (V). Especially, 1D PhC nanobeam cavities receive extensive attention due to its compact size, ease of fabrication and comparable performances to their 2D counterparts [9–12]. Furthermore, it is very difficult to trap particles with sizes beyond 200 nm by using the conventional optical tweezers due to the diffraction limitation in free space while PhC resonators can trap and release nanoparticles of a few tens of nanometers in size [13,14].

Combining with microfluidic channels, PhC cavities are widely used for the label-free detection of single particle [15], sorting [16] and subsequent analysis using sensitive cavity-enhanced spectroscopic techniques such as Raman spectroscopy [17]. Lin et al. combined a photonic crystal cavity waveguide cavity with a nanoslot structure to get a ~1300 times enhancement of optical trapping force compared with a conventional waveguide- trapping devices [4]. X. Serey et al. compared several PhC cavities and shows that particles as small as 7.7 nm diameter can be trapped with the input power of 10 mW [5]. Usually, the higher the optical power the larger the optical force. However, trapping biomolecules using PhC resonators is limited by the heat arising from the optical absorption of aqueous solution as in the case of silicon devices [18]. The temperature increase can affect or even damage biomolecules in biophysical studies. Thus, a low input power is of a prime importance to avoid the influence of temperature. Most of the reported works are focusing on how to increase the optical trapping force, but they don’t investigate the relationship between the optical trapping force and the figure of merits of the cavity: quality factor (Q), mode volume (V) and transmission (T).

In this work, we provide systematic analysis of the optical force for a particle trapped in the waveguide/cavity/waveguide system. To the best of our knowledge, this is the first time to show the relationship between the optical force and the quality factor, mode volume and the transmission of the cavity. The analysis shows that the optical trapping force is proportional to QT^1/2/V at a fixed position. We provide numerical proof for the proposed principles and systematically optimize the design recipe to investigate optical tapping force of the PhC nanobeam cavities. The numerical results are in accordance with the theoretical analysis and show that the maximum optical trapping force occurs when the transmission of cavity is near 0.25 and the maxima are almost the same for different PhC cavities. We believe that our study is beneficial to the design and optimization of a high-performance cavity-based optical tweezers [14, 16]. With the proposed method, it is much easier to achieve a maximized optical force by optimizing the geometry of the cavity systematically.

2. Theory and analysis

Figure 1 shows the schematic diagram of a waveguide/cavity/waveguide system. For such a resonant cavity, the optical trapping force for a particle around the cavity can be expressed as [8,20]

Fig. 1 Schematic diagram of a waveguide/cavity/waveguide system.

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F = \oint_{S} \vec{T} \cdot d \vec{A} .

where $\vec{A}$ is the outgoing vector of surface area A with the direction normal to the surface and $\vec{T}$ is the time-averaged Maxwell Stress Tensor. The integration performs on the outer surface of the particle.

Usually, the optical trapping force is calculated according to Eq. (1) and this has already been verified by experiments. However, the relationship between the optical trapping force and the parameters for the cavity is still not clear.

In this work, we theoretically analyze the optical trapping force for the cavity from the point view of the energy. The input waveguide #1 is identical to the output waveguide #2, according to the coupled mode theory [21], the amount of energy (U) stored in the cavity can be written as:

U = \frac{2 Q_{w}}{ω_{0}} {(\frac{Q_{r}}{Q_{w} + Q_{r}})}^{2} P_{0} .

In Eq. (1), P₀ is the launched power to excite the cavity mode, Q_w denotes the quality factor related to the field energy leakage from the cavity via the dielectric waveguide and Q_r is the scattering loss. Thus the total quality factor Q can be written as:

\frac{1}{Q} = \frac{1}{Q_{w}} + \frac{1}{Q_{r}} .

The transmission of the cavity T is:

T = {(\frac{Q}{Q_{w}})}^{2} .

By combing Eqs. (2)-(4), the energy (U) of the cavity can be expressed as:

U = \frac{2}{ω_{0}} Q T^{1 / 2} P_{0} .

The energy (U) stored in the cavity can also be calculated by integrating the energy density over the entire volume of the cavity,

U = \frac{1}{2} ∭ ε {| E |}^{2} d V .

The mode volume (V) of the cavity can be expressed as:

V = \frac{∭ ε {| E |}^{2} d V}{\max (ε {| E |}^{2})} .

By combing Eqs. (5)-(7), the energy stored in the particle U_p is:

U_{p} = \frac{1}{2} ∭_{p} ε {| E |}^{2} d V = \frac{∭_{p} ε {| E |}^{2} d V}{\max (ε {| E |}^{2})} \frac{Q T^{1 / 2}}{V} \frac{2}{ω_{0}} P_{0} .

Thus, the polarized energy of the particle can be expressed as:

W_{p} = s U_{p} .

Here, s is the ratio between the energy difference of the cavity with/without the particle and the energy of the cavity with particle. The optical trapping force is given by the change of the polarized energy of the particle with the particle’s coordinates. By assuming the z direction is normal to the direction of light propagation, the electric field can be shown by E = E₀ exp(-kz), where k is the the penetration depth of the cavity mode. Then we find the optical trapping force along z direction can be written as:

F_{t r a p} = - \frac{d W_{p}}{d z} = \frac{∭_{p} ε {| E |}^{2} d V}{\max (ε {| E |}^{2})} \frac{Q T^{1 / 2}}{V} \frac{4 k s}{ω_{0}} P_{0} .

Equation (7) shows the relation between optical trapping force and the performance of the cavity and it is more intuitive than Eq. (1). For a particle at a certain location, the term $∭_{p} ε {| E |}^{2} d V / \max (ε {| E |}^{2})$ related to the electric filed distribution is nearly a constant, so the normalized optical trapping force (F_trap/P₀) is directly proportional to QT^1/2/V of the cavity.

3. Numerical results and discussions

In this work, we take PhC nanobeam cavity as a numerical example to validate our theoretical analysis. Silicon-on-Insulator (SOI) platform with a 220 nm thick Si device layer and a 2 μm thick buffer layer is considered in our work. The refractive index of the silicon core, silica insulator layer are 3.476 and 1.444, respectively. Since particles are usually dispersed in DI water in the practical application, the top cladding is assumed to be water (n = 1.311) in our numerical simulation [4,15,16]. The schematic diagram of the PhC nanobeam cavity is shown in Fig. 2(a). It consists of an array of air-holes etched into silicon nanowire waveguide in decreasing radii. Here we take polystyrene sphere as the trapped particle (n = 1.59, radius r = 100 nm).

Fig. 2 (a) Schematic illustrations of the PhC nanobeam cavity for optical trapping; (b) The electric field distribution (top view) of PhC nanobeam cavity taken at the center of the silicon layer; (c) The red and blue curves are the band diagrams for the hole radius of r = 86 nm (central hole) and r = 33 nm (side hole), respectively (period a = 330 nm).

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In order to achieve PhC nanobeam cavities with high Q and high transmission, the cavity is formed by keeping the period while changing the hole sizes from the center to the side of the input/output feeding waveguides. The holes size of the cavity is designed as following [20]: (1) The dielectric band-edge of the PhC with radius of the central hole is located near our target resonant frequency; (2) The target resonant frequency is in the middle of the bandgap of the PhCs with radius of the side holes. Figure 2(c) shows the band diagrams for the PhCs with the different hole radii at a period a = 330 nm. The red and blue curves are the band diagrams for the hole radius of r = 86 nm (central hole) and r = 33 nm (side hole), respectively. Here, we keep the dielectric band-edge of the first mirror hole located at 195 THz which deviates our desired resonance frequency 193.4 THz, as a result of frequency perturbation [19].

Three dimensional finite-difference-time-domain method (3D-FDTD) with commercially available software from Lumerical Solutions [22] is utilized for the simulation. Figure 2(b) shows the electric field distribution of the cavity. From Fig. 2(b), we can find that the maximum of the electric field occurs at the center of the cavity and the particle will be stably trapped on this place due to the huge electric field gradient force of the cavity. In our simulation, the particle is placed on the center of the cavity and 10 nm off the waveguide surface. The optical trapping force is calculated by integrating Maxwell Stress Tensor on the outer surface of the particle.

PhC nanobeam cavities with different tapering profiles are investigated in our work. In order to investigate the influence of the Q, V, and T, we change the period of the cavity to obtain different transmissions. Figure 3 shows how the mirror holes tapering profiles influence the performance of the cavity and optical trapping force. From Fig. 3, it is noted that PhC cavities with different mirror holes tapering profile all have their maximum optical trapping force at the transmission around 0.25 rather than which has the highest value of Q/V. It can be easily understood by using the analysis given in Section 2. For PhC nanobeam cavities with the same tapering profile and the small variations in period, the difference in mode volume can be neglected compared to the large changes of quality factor, thus, the mode volume V can be regarded as a constant. From Eq. (10) and Eq. (6), we know that the optical trapping force is proportional to the energy of the cavity U. Thus, the maxima optical force can be obtained in the case of a cavity with highest U. From Eq. (3), the energy stored in the cavity achieves its maxima when Q_w = Q_r for a given Q_r. It follows that T = 0.25 from Eq. (5). Thus, the result shows that the maximum optical force occurs at the transmission of the cavity near 0.25, which agrees well with our theoretical analysis.

Fig. 3 (a) The ratio of quality factor to mode volume Q/V and transmission T as a function of period for PhC cavities with different tapering profile; (b) The optical trapping force normalized to the input light power F_trap/P₀ vary with different period for PhC cavities with different tapering profile.

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From Eq. (10), we conclude that the normalized optical trapping force is directly proportional to the QT^1/2/V of the cavity. Thus, we investigate how optical trapping force varies with QT^1/2/V, as shown in Fig. 4. One can find that it is in accordance with our theoretical analysis.

Fig. 4 Normalized optical force (F_trap/P₀) for PhC nanobeam cavited with different QT^1/2/V.

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To further verify the above conclusion, we calculate the optical force for PhC cavites with different periods and different mirror numbers. Figure 5 shows the transmission of the cavities and the normalized optical trapping force vary with the number of mirrors. In Fig. 5, we can see that the achieved maximum optical trapping forces are almost the same for PhC cavities with different mirror numbers. Furthermore, all the maximum values are obtained at the transmission around 0.25, which again verifies our theoretical analysis.

Fig. 5 (a) The transmission T as a function of mirror number N for the PhC cavities with different periods; (b) The optical trapping force normalized to the input light power F_trap/P₀ vary with mirror number N for the PhC cavities with different periods.

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4. Conclusion

In summary, we provide systematic analysis of the optical force for a particle trapped in the waveguide/cavity/waveguide system. The theoretical analysis shows that the optical trapping force is proportional to QT^1/2/V for a fixed position of the particle. The numerical results are in accordance with theoretical analysis and shows that the maximum optical trapping force for a particle occurs at the transmission of cavity near 0.25 and the maxima are almost the same for PhC nanobeam cavities with different parameters. Our method provide an easy way to achieve a maximized optical force by optimizing the geometry of the cavity systematically. We believe that the proposed method will greatly ease the optimization processes of optical trapping/manipulation, and enable both fundamental studies in strong light-matter interactions, and practical applications in optical tweezers. Although we use PhC nanobeam cavity as an example, it can also be used for the design of other resonant cavities.

Acknowledgments

This work was partially supported by the National Nature Science Foundation of China (61377023), Zhejiang Provincial Natural Science Foundation of China (LY13F050002), and the Program of Zhejiang Leading Team of Science and Technology Innovation.

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Systematic analysis of optical gradient force in photonic crystal nanobeam cavities

Abstract

1. Introduction

2. Theory and analysis

3. Numerical results and discussions

4. Conclusion

Acknowledgments

References and links

Cited By

Figures (5)

Equations (10)

Optics Express