1. Introduction

Since A. Ashkin’s demonstration of micron-sized particle trapping using tightly focused laser beams [1, 2], micro-manipulation using optical force has witnessed a glorious development [3–5]. Now, optical tweezer has become an indispensable tool in multiple disciplines including atomic physics [6], biological science [7], chemistry [8], and quantum physics [9]. Now, optical force tailoring has been extended to more sophisticated systems including Fano resonance structure [10], chiral material [11], metamaterial [12], and even graded index structures [13, 14]. In optical tweezers, the key point is to generate a large intensity gradient using a high numerical aperture objective lens which focuses the incident beam into a tight spot, where the optical trapping occurs. This mechanism is high efficiency in trapping object around a fixed point while it is lack of flexibility for long distance transportation.

For the purpose of long range transportation, two different mechanisms have been proposed. The first one is utilizing the evanescent wave occurred in total internal reflection on a dielectric interface [15–31]. In this case, the evanescent wave generates a large intensity gradient (exponentially decaying law of evanescent mode) force along the normal direction of the interface, which pulls the object to the interface. At the same time, the scattering force parallel the interface can transport the object over long distance, as shown in Fig. 1(a). In a similar manner, one can also use a waveguide structure (such as a micro-fiber) to generate the evanescent mode. In this case, the guiding mode extends outside of the physical boundary of the waveguide, and can trap and manipulate objects over long distance. Then second more interesting mechanism is the optical tractor beams [32–44], which can transport object backwardly, reverse to the beam propagation direction. Unfortunately, it is a great challenge to achieve an optical tractor beam in practice, where special optical beams [32–34], or special operation objects with exotic optical parameters [36, 45], or a rather complex background configurations are needed [43].

 

Fig. 1 Schematic illustration of electromagnetic force in evanescent modes. (a) Traditional configuration where the gradient force and scattering force are perpendicular to each other. (b) Our new configuration with collinear gradient force and scattering force, which can be regarded as the combination of traditional evanescent mode manipulation and optical tractor beam.

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In this paper, we propose a distinct optical manipulation method using the attenuation propagation mode, as illustrated in Fig. 1(b), which can be regarded as the combination of evanescent wave trapping and optical tractor beam. In this case, the transportation over relatively long distance is achieved by the attenuation mode, which provides both backward gradient force and forward scattering force. The direction of the total force is determined by their relative amplitudes, which can be tuned by the oscillating frequency and/or the damping rate of the mode. Thus, both the optical pushing force and the tractor-beam-like pulling force can be realized. We also create a solid model in sub-wavelength photonic crystal (PC) waveguides, and obtain the forward and backward forces quantitatively. Both the pushing and pulling forces can overcome the Brownian motion within a critical distance, thus can be used to transport and sort particles. More importantly, due to the physical boundary of the channel, the transverse size can be much smaller than the wavelength, which is advantage for nano-sized object manipulation along straight and bend trajectories. This optical-straw-like manipulation may find potential applications in multidisciplinary fields.

2. Principle, results and discussion

2.1. Optical force in attenuation fields

For an attenuated propagation mode along x direction, the propagation constant $\beta$ is a complex number of $\beta =\alpha -i\gamma$ with $\alpha$ and $\gamma$ being real and positive numbers. Then, the dominant electric field$E_z$of the mode can be expressed as

Since the attenuation parameter $\gamma$ is positive, $F_{\text{grad},x}$is a negative force reverse to the propagation direction. However, we still cannot judge the motion direction by this force only since there is also a forward radiation pressure. For the case of Rayleigh particle, we can also derive the scattering force analytically, which is [46]

From Eq. (4), we can see that $F_{\text{scat},x}$ is always positive. Finally, the competition between the negative and positive forces gives the total force of

For the mode with $\gamma =0$, one can see that $F_{\text{grad,}x}=0$ from Eq. (3). Then the total optical force is determined by the positive scattering force only. For a purely attenuation mode without propagation (i.e., $\alpha =0$), one can get that $F_{\text{scat,}x}=0$, and the total force is contributed only by the intrinsic gradient force. For the intermediate cases, i.e., attenuation propagation modes, both the scattering and gradient forces are nonzero, and the total force is determined by their relative amplitude.

In order to get a tractor-beam-like negative force, from Eq. (5) we can see that the propagation constant $\beta =\alpha -i\gamma$ also plays an important role, apart from the relative refractive index m and radius r of the scatterer. For the usual optical manipulations operated in boundless spaces (such as water or air), $\beta =n_2{k}_0$ can only be tuned by replacing the background medium (for the same source). Here, we propose to tune the scattering force by introducing optical channel, in which the effective mode refractive n₂ can be tuned by selecting different modes. In the extreme case of $\alpha \text{=0}$, the forward scattering force is vanished, and then a tractor-beam-like negative optical force will appear.

2.2. Optical manipulation in sub-wavelength photonic crystal waveguide

Now, we quantitatively consider the damped propagating mode discussed above in a solid structure of two dimensional (2D) photonic crystal (PC) waveguide. As shown in Fig. 2(a), the square lattice PC is formed by silicon rods in water (refractive indexes of $n_a=3.56$and $n_b=1.33$, respectively) with the lattice constant of a. A W1 guiding channel is formed by removing a line of silicon rods along the ΓX direction. Then a circular object with a refractive index of $n_o$ and radius of $r_o$ is launched into the channel. For the TM mode (H_x,_y and E_z are nonzero), the band structure is calculated using the plane wave expansion method [47], and the results are shown in Fig. 2(b). The shaded region between the guiding mode (upper left curve, blue dashed) and band edge mode (lower right curve, solid red) is the expected attenuation propagation band with complex propagation constants. The upper right inset shows the field pattern of $\left|E_z\left(x,y\right)\right|$ near the upper band edge of ${\omega }_1=0.226\left(2\pi c/a\right)$. It can be seen that the damping rate is small, and $\left|E_z\right|$ still keeps significant at a distance of 50a (more than 11 times of the wavelength in vacuum λ₀). The lower left inset shows the field pattern of ${\omega }_{\text{2}}$ near the lower edge of the damping band. We can see that the damping rate is relatively large, and $\left|E_z\left(x,y\right)\right|$ tends to zero within a distance of 10a (about 2λ₀). One also can find that the modes are well confined on the transverse direction, of which the dimension is determined by the geometry size of the channel, which is at the sub-wavelength scale. This feature is advantage for the nano-sized object manipulation.

 

Fig. 2 Structure and the evanescent mode. (a) Schematic of the photonic crystal (PC) waveguide channel, which supports attenuation propagating mode. The parameters are n_a = 3.56 and n_b = 1.33. (b) Band structure of TM mode (nonzero component of H_x,y and E_z) of the PC channel. The blue dashed curve (upper left curve) shows the guiding mode of the waveguide, and the red solid (lower right) curve is the band edge of the PC. The shaded region between those two curves is the band of attenuation propagation modes with complex propagation constant. The upper right inset shows the field pattern of |E_z| at the frequency ${\omega }_{\text{1}}$ with a smaller damping rate, and the lower left inset shows the mode of ${\omega }_{\text{2}}$ with a larger damping rate.

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We calculate the optical force using the integration of Maxwell stress tensor $\overleftrightarrow{T}$along an enclosed surface (it is a closed curve in 2D case), i.e.

Here $〈\cdot 〉$ means the time average operation over an oscillation period, $\overleftrightarrow{I}$is the unit tensor, and $\otimes$is the dyad operator. $\mathrm{Re}(A)$ and $A^{*}$ represent the real part and the complex conjunction of A (with A an arbitrary vector). All the involved electric and magnetic components of D, E, B and H are numerically calculated using the finite-difference in time-domain method. Since the object investigated here is embedded in water but not in vacuum, there is some controversy on the proper stress tensor to use. According to our results reported in Ref [48], Minkowski stress tensor can always predict the accurate time averaged optical force even in a non-vacuum background, such as in the inhomogeneous background of interfacial tractor beam case [48]. The Chu and Nelson stress tensors, however, lead to pushing force for the interfacial tractor beam, which is conflict with the experimental observations [48]. Therefore, the Minkowski stress tensor is used here.

For the scatterer with a radius of $r_o=0.2a$ and a moderate refractive index of $n_o=1.5$, we calculate the optical force F_x on the object when it locates at different positions along the x-axis, and the results are shown in Fig. 3(a). Both positive pushing and negative pulling forces are obtained when different damping rate of the mode is selected by setting different incident frequency, which agrees well with the analysis above. For the frequencies of 0.2259 and 0.2260 (normalized by 2πc/a), the decaying rate $\gamma$is small, which is overcome by the forward scattering force, and a net pushing force is obtained. On the other hand, for the frequencies of 0.2257 and 0.2258, the decaying rate $\gamma$ is large enough to generate a strong backward gradient force to overcome the forward scattering force, and then a tractor-beam-like like net pulling force is obtained. This shows that the force can be reversed easily by slight tuning the mode damping rate, or equivalently the incident frequency.

 

Fig. 3 (a) Optical forces change with x for different modes. The radius and refractive index of the object are $r=0.2a$ and $n_o=1.5$. (b) Optical force changes with the refractive index and size of the object at the fixed frequency of $\omega =0.2258$. The regions I (III) means F_x is always pulling (pushing) at any positions of x, while the middle region II means F_x maybe pushing or pulling dependent on the position.

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When the incident frequency is fixed, we calculate the optical forces $F_x$for different objects that are characterized by the parameters of $n_o\in [1.36,1.77]$ and $r_o\in [0.14a,0.36a]$, and the results are shown in Fig. 3(b). One can see that the entire domain is divided into three regions. For the lower left region I with smaller radius and smaller relative refractive index, F_x is always negative wherever the scatterer is in the channel. For the upper right region III, however, F_x is positive for arbitrary value of x, which means that the pushing force is always larger than the gradient force. The middle region II is the transitional region where the sign of F_x is dependent on the position. Those results show that the optical force can be utilized to sort objects with different size, and/or different refractive index.

Comparing with other tractor beams and optical tweezers investigated in boundless spaces, one advantage of current manipulation is the ability of transporting objects along curved trajectories. In boundless space, self-acceleration beams have to be used for optical manipulations along curved paths, such as the Airy beam that is proposed as an “optical snowblower” [49], or helical profile beam both in phase and amplitude [50, 51]. In current situation, however, the curve manipulation path can be achieved easily by design the channel geometries, rather than by a complex self-acceleration light beam or helical beam, which makes this scheme more flexible in practice.

In order to illustrate this feature, we create different bend channels, and consider the optical forces in them. The bending channels are shown by the gray open circles in Fig. 4, where the angular and radial distances between the units are both a, the lattice constant of the photonic crystal structure, while the whole structure is bend with a radius of R. In Fig. 4(a), the red color shows the field pattern of the mode. One can see clearly that the mode can bend smoothly according to the channel, and the field pattern is almost the same as that in straight. The thick blue curve with arrows shows the motion trajectory of a dipole object pulled by the optical force, which is calculated by Eq. (5). Figure 4(b) is the same as Fig. 4(a) except for the bend radius is smaller of R = 12a. For the case of pushing force, it is much easier to achieve using the mode with smaller damped factor, and not shown here. From those results, one can see that the manipulation along bend trajectories is an intrinsic feature of our mechanism free from complicated self-acceleration beams.

 

Fig. 4 Optical manipulation through a bend channel with bending radius of (a) R = 16a, and (b) R = 12a. The gray open circles show the position of the photonic crystal and red color shows the field pattern of the mode. The blue thick curves with arrows show the motion trajectory of a dipole object driven by the optical force.

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2.3. Effect of Brownian motion on the manipulation

When Brownian motion is considered, the manipulation property will be modified slightly. Results show that the most significant influence of Brown motion is the decrease of the effective operation range. In order to measure the influence of Brownian motion, we consider the trapping parameter of $C\left(x\right)={\displaystyle {\int }_x^{\infty }{F}_x\left(x\text{'}\right)dx\text{'}}/k_{B}T$. Here, $k_B$ is the Boltzmann constant, and T is the absolute temperature in Kelvin. Stable trapping and transportation is excepted only when C is larger than certain critical value C₀, such as $\left|C\left(x\right)\right|\ge 10=C_0$ [2]. When $F_x<0$ and $C<C_0$, the object cannot escape from the potential formed by the negative pulling force, and the object will be pulled back to the source direction. Figure 5 shows the profiles of $C\left(x\right)$ for different incident frequencies at the incident level of 10 mW, and at the temperature of 300K. We define the effective length $L_{eff}$for the optical manipulation parameter by the criteria of $\left|C\left(L_{eff}\right)\right|=C_0$. We can see that, although the amplitude of force is decaying exponentially with distance, the operation distance can extend as long as several times of wavelength. For the case of tractor-beam-like pulling force, the effective length is about $\text{4}{\lambda }_0$ with ${\lambda }_{\text{0}}$ the optical wavelength in vacuum. For the object within this distance, the object can be pulled back to the source. For the case of optical pushing force, the effective length is about $\text{5}{\lambda }_0$. An object within this distance can be pushed away. What’s more, the effective pushing distance can be increased very easily by tuning the damping rate $\gamma$ through the incident frequency. It is noted that the effective length (both the pushing length and pulling length) can be tuned by changing the incident power.

 

Fig. 5 The manipulation factor C(x) changes with position at the incident level of 10mW.

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3. Discussion and conclusion

From those results and discussions presented above, one can find that when the channel mode works as an optical tractor beam like operation, our method possesses several advantages. In the usually investigated optical tractor beam, the incident field firstly exerts a pushing force that is closely related to the momentum per photon of $\hslash k$ (with $k$being the wavevector, $\hslash$ the reduced Plank’s constant), and then the negative pulling force is possible only when the forward scattering is enhanced enough to generate a larger backward recoil force. Therefore, an intense endeavor is spent to engineer the source and/or object to increase the forward scattering. In our case, the incident pushing force can be greatly reduced due to the present of the waveguide channel, which makes the real part of the propagation constant $\beta$ much smaller (or even vanish is some extreme cases) than that in homogeneous backgrounds. Therefore, only a moderate damping rate can result in a backward gradient force that overcomes the undesired forward radiation pressure. Also, the waveguide channel provides a physical boundary for the manipulation, which performs like an optical analog of drinking straw for micro particle in the bidirectional transportation process. The physical boundary is also beneficial for the stability on the transverse direction in manipulation, and provides the manipulation ability along curved trajectories (channels). Our scheme releases almost all the other tight restrictions adopted in tractor beams before, such as nonparaxial Bessel beam, gain medium, or negative refractive materials. Those properties are beneficial for practical application.

In summary, we have proposed an optical manipulation method using the attenuation propagation mode in a guiding channel, rather than in a boundless space. The operation can be regarded as a mixing of the optical tweezer and the optical tractor beam with tunable direction of total force. Due to the mode engineering ability and physical boundary of the channel, an object could be bi-directionally transported along straight and bended trajectories. Using the finite-difference in time-domain method, we quantitatively investigated the optical channel manipulation properties in different photonic crystal waveguide channels. This optical pulling force can be extended straightly to other kinds of channels which support attenuation and propagation modes. Therefore, it provides a flexible and universal method for optical manipulations, which can be used in optical sorting and extraction.