Quasi 3-dimensional optical trapping by two counter-propagating beams in nano-fiber

Lihua Zhao; Yudong Li; Jiwei Qi; Jingjun Xu; Qian Sun

doi:10.1364/OE.18.005724

1. Introduction

Recently, there has been an increased focus in optical manipulation of micro and nanoparticles [1–3]. For instance, as the traditional excellent tools for manipulating particles ranging in size from several micrometers to a few hundred nanometers, the optical tweezer [4] has been used to manipulate and trap micro-scale objects [5], liquid droplets [6], and even some submicron objects such as viruses [7] and silver nanoparticles [8]. Besides, there is also indirectly using of optically induced microfluidic effects [9,10]. The precision with which particles can be manipulated with these techniques makes them particularly useful for applications ranging from flow cytometry [11,12] to self-assembly [13].

Alternatively, evanescent field is another way to realize optical manipulation. Almeida et al. [14] developed a slot waveguide, composing of a nanoscale slot sandwiched between two materials of much higher refractive index. This structure provides an excellent method for not only possible long distance transportation of the particles along the slot but also trapping of particles in 2-dimension, which is inside and along the slot. Very recently, Yang et al. [15] demonstrated the trapping and transport of polystyrene nanoparticles and DNA molecules using this kind of slotted silicon waveguides. It also has been shown that the nano-fiber [16] can be used for the trapping or guiding atom [17–19], where the atom can be confined along two straight lines parallel to the fiber axis, or further efficient tools as the nano-probes for atoms [20, 21]. Besides, the scattering of particles in proximity of a nano-fiber surface has also been investigated [22].

Inspired by the previous work, herein we investigate the optical force on a nanoparticle around a dielectric nano-fiber which is illuminated by two counter-propagating beams. It should be noted that due to the application of the nano-fiber, there is no need to consider the collimation of the two beams. In this work, the scattering and the gradient force on the nanoparticle are mainly concerned. The results show that steady trapping is achieved both in the longitudinal and azimuth directions. The steady trapping position can be shifted by altering the polarization direction or the incident intensity ratio of the two incident beams.

2. Model and theoretical investigation

Figure 1 shows the schematic of the model. A dielectric nano-fiber, with the length of L and the diameter of D, is illuminated by two coherent counter-propagating beams, beam I and II, which propagate along +z and –z directions, respectively. The incident beams are assumed to be linear polarized along Y-axis with the same uniform amplitude of 10⁵V/m. The incident wavelength is taken as 550nm. The nano-fiber only includes the high refractive index core and is surrounded by infinite water cladding (n=1.33). The core is taken as fused silica, of which the refractive index is taken as 1.46 from the dispersion formula [23].

Fig. 1 The schematic diagram of calculation model. Two counter-propagating beams, beam I and II, are incident upon the nano-fiber end. The rectangular coordinates for each beam are also given. L presents the length of the nano-fiber.

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E₁ and E₂ present the electric field of beam I and II, respectively. For the convenience, we used two cylindrical coordinates (r, φ, z) and (r’, φ’, z’) to calculate E₁ and E₂. The left and right end faces of the nano-fiber were set to be z=0 and z’=0, respectively. So it can be obtained that z=L-z’. And the unit vectors of the two cylindrical coordinates follow the relations: ${\vec{r}}_{0} = {\vec{r}}_{0}^{'}$ , ${\vec{φ}}_{0} = - {\vec{φ}}_{0}^{'}$ , ${\vec{z}}_{0} = - {\vec{z}}_{0}^{'}$ .

It is assumed that the nano-fiber can only support fundamental guiding mode, HE₁₁ mode [24]. With the assumption of monochromatic incidence, the electric field distribution of the nano-fiber is described as:

\vec{E} = {\vec{E}}_{1} + {\vec{E}}_{2} = [(E_{r} {\vec{r}}_{0} + E_{φ} {\vec{φ}}_{0} + E_{z} {\vec{z}}_{0}) e^{i β z} e^{- γ z / 2} + (E_{r}^{'} {\vec{r}}_{0}^{'} + E_{φ}^{'} {\vec{φ}}_{0}^{'} + E_{z}^{'} {\vec{z}}_{0}^{'}) e^{i β z^{'}} e^{- γ z^{'} / 2}] e^{- i ω t}

where (

E_{r}

,

E_{φ}

,

E_{z}

) and (

E_{r}^{'}

,

E_{φ}^{'}

,

E_{z}^{'}

) are the electric field components along the unit vector (

{\vec{r}}_{0}

,

{\vec{φ}}_{0}

,

{\vec{z}}_{0}

) and (

{\vec{r}}_{0}^{'}

,

{\vec{φ}}_{0}^{'}

,

{\vec{z}}_{0}^{'}

) in the two cylindrical coordinates, respectively. γ presents the absorption coefficient of the fiber core. According to the guiding mode theory [24],

E_{r}

,

E_{φ}

and

E_{z}

are described as:

E_{r} = \frac{- U}{W} \frac{a_{1} K_{0} (W R) - a_{2} K_{2} (W R)}{K_{1} (W)} f_{1} (φ), E_{φ} = \frac{- U}{W} \frac{a_{1} K_{0} (W R) + a_{2} K_{2} (W R)}{K_{1} (W)} g_{1} (φ), E_{z} = \frac{- i U}{a β} \frac{K_{1} (W R)}{K_{1} (W)} f_{1} (φ)

Here

f_{1} (φ) = \cos (φ), g_{1} (φ) = - \sin (φ)

. The initial phase of beams I and II are all assumed to be 0. β is the transfer constant, determined by its eigen-value equation [24]. Considering the relationship of (

{\vec{r}}_{0}

,

{\vec{φ}}_{0}

,

{\vec{z}}_{0}

) and (

{\vec{r}}_{0}^{'}

,

{\vec{φ}}_{0}^{'}

,

{\vec{z}}_{0}^{'}

), it can be obtained that

E_{r}^{'} = E_{r}, E_{φ}^{'} = - E_{φ}, E_{z}^{'} = E_{z}

.So the components of the electric field along

{\vec{r}}_{0}

,

{\vec{φ}}_{0}

and

{\vec{z}}_{0}

outside the nano-fiber can be described as:

E_{a} = E_{r} e^{i β z} e^{- γ z / 2} + E_{r}^{'} e^{i β z'} e^{- γ z' / 2} = E_{r} e^{i β z} e^{- γ z / 2} + E_{r} e^{i β (L - z)} e^{- γ (L - z) / 2}

E_{b} = E_{φ} e^{i β z} e^{- γ z / 2} - E_{φ}^{'} e^{i β z'} e^{- γ z' / 2} = E_{φ} e^{i β z} e^{- γ z / 2} + E_{φ} e^{i β (L - z)} e^{- γ (L - z) / 2}

E_{c} = E_{z} e^{i β z} e^{- γ z / 2} - E_{z}^{'} e^{i β z'} e^{- γ z' / 2} = E_{z} e^{i β z} e^{- γ z / 2} - E_{z} e^{i β (L - z)} e^{- γ (L - z) / 2}

Hence the intensity of light outside the nano-fiber can be written as:

I = \frac{1}{2} n_{2} ε_{0} c \vec{E} \cdot {\vec{E}}^{*} = \frac{1}{2} n_{2} ε_{0} c ({| E_{a} |}^{2} + {| E_{b} |}^{2} + {| E_{c} |}^{2})

According to the Rayleigh mode of optical trapping, the nanoparticle undergoes two kinds of force, scattering force (F_scat) and gradient force (F_grad), which are described as [25]:

{\vec{F}}_{s c a t} = \frac{C_{p r} < \vec{S} (r, t) >_{T}}{c / n_{2}}

F_{g r a d} = \frac{2 π n_{2} a^{3}}{c} (\frac{n^{2} - 1}{n^{2} + 2}) \nabla I

\vec{S} (r, t) = {\vec{S}}_{1} (r, t) + {\vec{S}}_{2} (r, t)

.

{\vec{S}}_{1}

and

{\vec{S}}_{2}

are the Poynting vectors of beam I and II, respectively, where

S_{n} (r, t) = ε E_{n}^{2} (r, t)

, n=1, 2. Here

< \vec{S} (r, t) >_{T}

is the time average of the Poynting vector.

C_{p r} = \frac{8}{3} π {(k a)}^{4} a^{2} {(\frac{n^{2} - 1}{n^{2} + 2})}^{2}

is the scattering cross-section. c and k present the light velocity and the wave vector in the vacuum. a is the particle diameter, and I is the intensity. n=n_p/n₂ is the relative refractive index, while n_p=1.592 and n₂=1.33 present the refractive index of the particle and the surrounding, respectively. From the Eq. (8), it can be known that the direction of the gradient force always points to the position where the intensity I reaches the maximum.

In this work, trapping target is limited to the nanoparticle with diameter of tens of nanometers due to the Rayleigh mode. Besides, if the diameter of particle reached hundreds of nanometers, the electric field outside the nano-fiber would be influenced by the coupling between the fiber and the particles and therefore the formulas above are no longer available.

3. Results and discussions

Firstly, we consider the trapping along the longitudinal axis, i.e. z-axis in Fig. 1. Different from the situation that particles are driven to move along the nano-fiber or the nano-waveguide, upon which only one beam is incident, in this work, steady trapping is achieved.

In our model, beams I and II illuminate the nano-fiber simultaneously. When the nano-fiber is non-absorptive, the total Poynting vector $\vec{S}$ keeps zero along z-axis and the scattering force is zero. The gradient force variation is of uniform sinusoidal profile. So there is no steady trapping. If absorption coefficient γ ≠ 0, $\vec{S}$ is no longer uniform. $< {\vec{S}}_{1} >_{T}$ > $< {\vec{S}}_{2} >_{T}$ at positions z<5.0μm, therefore $\vec{F}$ points to the +z direction, while at position z>5.0μm, $< {\vec{S}}_{1} >_{T}$ < $< {\vec{S}}_{2} >_{T}$ and hence $\vec{F}$ points to the -z direction. At position z=5.0μm, $\vec{F}$ =0 because $< {\vec{S}}_{1} >_{T}$ = $< {\vec{S}}_{2} >_{T}$ . Therefore, particles can be steadily trapped at position z=5.0μm. According to our calculation, the scattering force is several orders of magnitude larger than the gradient force. So in the longitudinal axis the scattering force plays the key role for the trapping.

Figure 2(a) shows the scattering force on a nanoparticle with diameter of 20nm versus z, while in the transverse plane, the center of the nanoparticle stays at r=0.21μm and φ=0. The diameter of the nano-fiber was taken as 400nm and the length of the nano-fiber L=10.0μm. The γ of the nano-fiber is taken as 0.01μm⁻¹, 0.1μm⁻¹, 0.2μm⁻¹ and 0.5μm⁻¹ respectively. The results prove that position z=5.0μm is the equilibrium position in the longitudinal direction. It also shows that scattering force reaches its maximum at each end of the nano-fiber, due to the propagation length difference of the two beams.

Fig. 2 The scattering forces versus z on a particle with diameter of 20nm (a) and the normalized stiffness versus the absorption coefficient γ (b). Here D=400nm, L=10.0μm, r=0.21μm and φ=0. In (a), the four lines present the cases that γ=0.01/μm, 0.1/μm, 0.2/μm and 0.5/μm, respectively. In (b), z=5.0μm. The normalized condition is $C_{p r} < {\vec{S}}_{1} >_{T} / (2 c / n_{2}) = 1$ .

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One can see that absorption coefficient causes the change in the scattering force curve profiles. In order to investigate the influence of γ on the trapping, herein we employ the parameter stiffness, defined as k_f=∂F/∂z|_F₌₀, which represents the trapping ability at the equilibrium position. k_f has the similar meaning to the stiffness in an oscillating spring system. In this work, $k_{f} = C_{p r} < {\vec{S}}_{1} >_{T} γ (- e^{- γ z} - e^{- γ (L - z)}) / (2 c / n_{2})$ . Figure 2(b) shows the normalized stiffness versus the absorption coefficient γ with z=5.0μm, L=10.0μm, in which the normalized condition is $C_{p r} < {\vec{S}}_{1} >_{T} / (2 c / n_{2}) = 1$ . That the stiffness keeps negative means particles deviating from the equilibrium position always undergo the force which drives them back to the equilibrium position. So the trapping at z=5.0μm is stable trapping.

It is also worthy to mention that, in this work, the absorption coefficient γ we take is several orders larger than the one for the telecommunication fiber. From Fig. 2(a), one can see that for a large absorption, the scattering force is larger than the cases that the absorption is small, e.g. γ=0.01μm⁻¹. Furthermore, according to Fig. 2(b), a large absorption means a better trapping stability. Under our calculation conditions, when γ=0.2μm⁻¹, k_f reaches its minimum and therefore most steady trapping is achieved.

Now we consider the trapping in the transverse plane of the nano-fiber, i.e. along the azimuth direction ${\vec{φ}}_{0}$ and the radius direction ${\vec{r}}_{0}$ . In transverse plane, $< \vec{S} (r, t) >_{T}$ =0 and hence scattering force is zero. Gradient force becomes the source of trapping.

Figures 3(a) and (b) shows the gradient force variation versus φ and r respectively. It can be seen that in azimuth direction there are two steady trapping positions, which are φ=0 and φ=π. It is parallel to the incident polarization direction. Steady position in azimuth direction can be shifted by changing the polarization state of the incident beam. In our calculation, the polarization of the two beams is assumed to parallel with Y-axis. When the polarization is rotated to X-axis, the electric field and hence the steady trapping position in azimuth direction will also rotate to π/2 and 3π/2. In the radius direction, seeing Fig. 3 (b), negative gradient force indicates the direction of the gradient force is –r-axis and the particle is always pushed towards the nano-fiber. If the placement of the nano-fiber is altered so that the gradient force in the r-axis can be used to cancellation the particle gravity, the particle will stay at a distance from the nano-fiber. However, it is unstable trapping. In contrast, the gradient force is more equal to transport particles in the radius direction. So in this work, it is called quasi 3-dimensioal trapping.

Fig. 3 The variation of gradient force acting on a particle versus φ (a) and r (b) Here D=400nm, z=5.0μm and γ=0.2/μm. In (a) r=0.21μm. In (b), φ=0.

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In the discussion above, the intensity of Beam I and II are assumed to be equal. Therefore, the equilibrium position of the particles along the longitudinal direction is in the middle of the nano-fiber. It is interesting that the equilibrium position can be altered if the intensities of the two beams are different. In this situation, the scattering force on one particle can be described as:

F = C_{p r} (< {\vec{S}}_{1} >_{T} e^{- γ z} - < {\vec{S}}_{2} >_{T} e^{- γ (L - z)}) / (2 c / n_{2})

Figure 4 shows the scattering forces versus z when the amplitude of Beam I and II are different. While E₁<E₂ represents E₁=10⁵V/m, E₂=2.227 × 10⁵V/m; E₁=E₂ represents E₁=E₂=10⁵V/m and E₁>E₂ represents E₁=10⁵V/m, E₂=4.49×10⁴V/m. The equilibrium position for the three situations are z=3.0μm (E₁<E₂), 5.0μm (E₁=E₂) and 7.0μm (E₁>E₂), respectively. This also infers that the manipulation of the particles is very easy in our design, which is just altering the intensity of one incident beam.

Fig. 4 The scattering force versus position z on a particle with diameter of 20nm at r = 0.21μm and φ = 0. The steady position along the z-axis for the three situation: z=3.0μm (E₁<E₂), 5.0μm (E₁=E₂) and 7.0μm (E₁>E₂), where D=400nm, γ=0.2/μm, L=10.0μm.

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4. Conclusion

In conclusion, we have shown optical forces on a nanoparticle by using the evanescent field of two counter-propagating beams in an absorptive nano-fiber. Results show that the particles can be trapped steadily along the longitudinal direction and in the azimuth direction of the nano-fiber. By changing the intensity ratio or the polarization direction of two incident beams, optical manipulation along longitudinal or azimuth axis can be realized. In the radius direction, the particle is always pushed towards the nano-fiber. The unstable trapping can be achieved if the particle gravity is considered. The gradient force in the radius direction is more equal to transport particles. Therefore, it is called quasi 3-dimensional trapping.

In our analysis, due to the complex theoretical calculation of the energy coupling between the particles and the nano-fiber, we did not considered particles with large diameter, but chose them such that we could highlight the advantages of such a method where there is no restriction on the uniformity of the electromagnetic field over the particles. This type of analysis could be used in the future for the design of a more sophisticated system incorporating complex photonic elements.

Acknowledgments

The authors thank Dr. Yanfeng Zhang for helpful discussion. This work is supported by Natural Science Foundation of Tianjin (06TXTJJC13500), National Natural Science Foundation of China (60978020), Foundation of “973” Project (2007CB307002) and Key International S&T Cooperation Project (2005DFA10170).

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Quasi 3-dimensional optical trapping by two counter-propagating beams in nano-fiber

Abstract

1. Introduction

2. Model and theoretical investigation

3. Results and discussions

4. Conclusion

Acknowledgments

References and links

Cited By

Figures (4)

Equations (9)

Optics Express