Analytical prediction of stable optical trapping in optical vortices created by three TE or TM plane waves

Tomasz M. Grzegorczyk; Jin Au Kong

doi:10.1364/OE.15.008010

1. Introduction

Laguerre-Gaussian beams have been shown to carry and to transfer orbital momentum to matter [1]. Such beams are characterized by a helical phase structure and a zero intensity at their center due to the destructive interference close to the vortex phase singularity [2, 3, 4]. These properties have allowed for the recent creation of microfluidic devices such as sorting devices, mixers, and pumps [5, 6], and have been able to trap low index particles [7], as an extension of the more traditional optical tweezer based on Gaussian beams [8]. In practice, Laguerre-Gaussian beams are typically realized using diffractive optical elements or computer-generated holograms [7, 9] from which single and multiple-particle traps have been realized, as already shown experimentally [10]. Independently, multiple particle trapping has also been demonstrated using three interfering beams which create an hexagonal pattern of high intensity regions [11, 12]. This simple configuration has been shown to trap hundreds of particles, typically in the high field regions.

In this paper we show that, independently of the polarization, the simple three-beam interference pattern exhibits a phase singularity which can be used to trap properly designed particles in a spiral-like fashion when the curvature of the optical pseudo-potential is close to zero (we denote by pseudo-potential the potential-like function along the x̂ direction obtained from the force). In order to show this, we first derive a closed-form analytical expression of the force on an infinite lossless dielectric cylinder subject to multiple TE (magnetic field parallel to the axis of the cylinder) plane waves. This derivation completes our previous work [13] where the general framework of the computation was laid out but only carried through for the TM polarization. We show here that the forces due to TE or TM incidences are written in a similar way, in which the polarization dependent coefficients do not affect the qualitative form of the force. With this conclusion, we proceed by analytically computing the curvature of the pseudo-potential and showing that close to its zero, the optical landscape creates a force field that induces spiral-like motions of particles, centered at the points of undetermined phases also called optical vortices [14]. By analyzing the forces, we establish the threshold in particle size between an outward spiraling motion and an inward spiraling motion, the latter yielding a stable trap at the optical vortex.

Fig. 1. Configuration of the problem: an infinitely long cylinder of permittivity ε_p is embedded in a background of permittivity ε and is subject to multiple in-plane incidences.

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2. Analytical expression of the force

The framework of the problem under consideration is represented in Fig. 1: multiple plane waves of similar amplitude and incident angles (θ = π/2,ϕ) impinge on an infinite lossless dielectric cylinder of radius a and permittivity ε_p, in a background medium of permittivity ε. The interference of the multiple in-plane incidences creates an optical landscape which in turn induces a force on the cylindrical particle. This force has so far been computed either numerically using techniques such as the discrete-dipole approximation [15, 16], or semi-numerically using an analytical expansion of the fields but a numerical evaluation of the integral in the Maxwell stress tensor formulation [17, 18, 19, 20]. Although not immediately straightforward, however, a series of algebraic manipulations can be performed on the force expression obtained from the Maxwell stress tensor, using an expansion of the electromagnetic fields in the cylindrical coordinate system. An overview of the method along with some mathematical details have been presented in [13]: a generic formula for the force has been obtained for both TE and TM incidences, and further simplifications were performed to reduce the TM formula to a closed-form expression from which physical conclusions were drawn. The first purpose of this paper is to generalize the closed-form formula to the TE case as well. In order to do this, we rewrite without further details the generic formula of the force due to a single plane wave first (the generalization to multiple waves follows), as obtained in [13, Eq. (16a)]:

F^{TE} = \hat{x} \frac{aπε}{2} ℜ (\sum_{n} (\frac{n (n + 1)}{{(ka)}^{2}} - 1) H_{z_{n}} H_{z_{n + 1}}^{*} - E_{ϕ_{n}} E_{ϕ_{n + 1}}^{*})

where

E_{ϕ_{n}} = k [a_{n}^{(M) s} H_{n}^{(1)'} (k ρ) + a_{n}^{(M)} J_{n}^{'} (k ρ)]

H_{z_{n}} = k [a_{n}^{(M) s} H_{n}^{(1)} (k ρ) + a_{n}^{(M)} J_{n} (k ρ)]

a_{n}^{(M)} = \frac{i^{n + 1} e^{- in ϕ_{i}}}{k} E_{hi},

E_hi is the amplitude of the incident wave of wave vector k and incident angle ϕ_i, H _n ⁽¹⁾ and J _n are the Hankel and Bessel functions, respectively, and the prime denotes a derivative with respect to the argument. The coefficients a _n ^(M)s are related to a _n ^(M) by the T _n ^M coefficients of a cylinder [13, Eq. (5b)].

Although (1) provides an exact relation to compute the force on a lossless dielectric cylinder under TE illumination, it does not provide an intuitive physical understanding of momentum transfer to particles. We therefore simplify (1) further by replacing the expression of a _n ^(M)s into Eqs. (2) and using the Wronskian to obtain

E_{ϕ_{n}} = \frac{2 k}{iπa} \frac{J_{n}^{'} (k_{p} a)}{D_{n}^{TE}}, H_{z_{n}} = \frac{2 k_{p}}{iπa} \frac{J_{n} (k_{p} a)}{D_{n}^{TE}},

where k_p is the wave vector inside the particle and D^TE _n is given by

D_{n}^{TE} = k H_{n}^{(1)} (ka) J_{n}^{'} (k_{p} a) - k_{p} H_{n}^{(1)'} (ka) J_{n} (k_{p} a) .

Inserting Eqs. (3) into (1) and extracting the x̂ component of the force, we write F_x^TE = (πε/2)ℜ(A+B) where, after some algebra,

A = - {(\frac{2}{πa})}^{2} \frac{i}{k^{2}} {∣ E_{hi} ∣}^{2} \sum_{n} \frac{e^{i ϕ_{i}}}{D_{n}^{TE} D_{n + 1}^{T E^{*}}} \times

B = - {(\frac{2}{πa})}^{2} \frac{i}{k^{2}} {∣ E_{hi} ∣}^{2} \sum_{n} [e^{i ϕ_{i}} \frac{k_{p} n J_{n}^{2} (k_{p} a)}{D_{n}^{TE} D_{n + 1}^{T E^{*}}}

The force is obtained by adding Eqs. (5a) to (5b) and taking the real part. Details of the algebraic manipulations are provided in the appendix, after which the force is simplified into the expression:

F^{TE} = {\hat{k}}_{i} \frac{4 ε}{πa} \frac{k_{p}^{2} - k^{2}}{k^{2}} {∣ E_{hi} ∣}^{2} \sum_{0}^{+ \infty} ℑ (D_{n}^{T E^{*}} D_{n + 1}^{TE}) \times

for a single plane wave (k̂_i = k _i/|k _i|), and

F_{ij}^{TE} = (\hat{x} \cos Φ_{ij} + \hat{y} \sin Φ_{ij}) \frac{4 ε}{πa} \frac{k_{p}^{2} - k^{2}}{k^{2}} {∣ E_{hi} ∣}^{2}

for the interaction between plane wave #i and #j, where Φ_ij = (k _i - k _j) ∙ ρ + (ϕ_i + ϕ_j)/2. Naturally, (7) reduces to (6) when i = j in a straightforward manner.

A comparison between (7) for the TE polarization and [13, Eq. (27)] for the TM polarization reveals that the two formulae are fundamentally similar, with only a few polarization dependent coefficients. Unifying the two expressions, we write the total force for a given polarization (p) as

F^{(p)} = \sum_{i, j = 1}^{M} F_{ij}^{(p)},

F_{ij}^{(p)} = e^{i Φ_{ij}} K^{(p)} \sum_{n = 0}^{+ \infty} Λ^{(p)} ℑ (γ_{n}^{(p)} β_{n}),

where

K^{(p)} = \frac{4 ε}{πa} \frac{k_{p}^{2} - k^{2}}{k^{2}} {∣ E^{(p)} ∣}^{2},

Φ_{ij} = (k_{i} - k_{j}) ∙ ρ + \frac{ϕ_{i} + ϕ_{j}}{2},

β_{n} = e^{- i (n + 0.5) (ϕ_{i} - ϕ_{j})},

γ_{n}^{(p)} = D_{n}^{{(p)}^{*}} D_{n + 1}^{(p)},

Λ^{TM} = \frac{J_{n} (K_{p} a) J_{n + 1} (k_{p} a)}{{∣ D_{n}^{TM} ∣}^{2} {∣ D_{n + 1}^{TM} ∣}^{2}},

Λ^{TE} = \frac{\frac{n (n + 1)}{{(ka)}^{2}} J_{n} (k_{p} a) J_{n + 1} (k_{p} a) + J_{n}^{'} (k_{p} a) J_{n + 1}^{'} (k_{p} a)}{{∣ D_{n}^{TE} ∣}^{2} {∣ D_{n + 1}^{TE} ∣}^{2}},

D_{n}^{TE} = k H_{n}^{(1)'} (ka) J_{n} (k_{p} a) - k_{p} H_{n}^{(1)} (ka) J_{n}^{'} (k_{p} a),

D_{n}^{TE} = k H_{n}^{(1)} (ka) J_{n}^{'} (k_{p} a) - k_{p} H_{n}^{(1)'} (ka) J_{n} (k_{p} a) .

The symbols in Eqs. (9) are defined as follows: ρ is the position of the particle in the (xy) plane, (p) =TE, TM indicates the polarization, E ^(p) is the amplitude of the incident polarization, while (k _i, ϕ_i) and (k _j, ϕ_j) are the two wavevectors and incident angles of the two plane waves with |k _i,| = |k _j| = k = k ₀√ε, k_p = k ₀√ε_p, k ₀ = 2π/λ. Under the notation of (8), the x and y components of the force are given by F_{x_ij} ^(p) = ℜ (F_ij ^(p)) and F_{y_ij} ^(p) = ℑ (F_ij ^(p)), where ℜ (∙) and ℑ (∙) denote the real and imaginary part operators, respectively.

It is important to emphasize that (8) is a fundamental relation which is straightforward to evaluate and which gives the force on a cylinder with no approximation. In addition, it immediately demonstrates a series of known results, previously established either experimentally or analytically using some approximations (typically a Rayleigh approximation when particles are very small [21] or a ray optics approximation when they are large compared to the wavelength [22]): in the case of a single incidence (F ^(p) = F_ii ^(p)), the force is in the same direction as the incident wave, the force vanishes if there is no permittivity contrast, it is always positive and pushing the particle; in the case of a Gaussian beam (written as a superposition of weighted plane waves), the operating mode of an optical tweezer can be demonstrated and more generally, the longitudinal and transverse forces can be shown to be separately controlled by the size and the permittivity contrast, respectively [13]. In addition, the polarization dependent coefficients in (8) can be assimilated to simple constants: they do not influence the qualitative dependence of the force on the various incident waves or on the position in space ρ. The subsequent study on the pseudo-potential is therefore in essence identical for both TE or TM polarizations.

3. Pseudo-potential and zero curvature points

In addition to providing a closed-form expression of the force, (8) provides a way to study the pseudo-potential (along the x̂ direction) created by the interference of three plane waves in an analytical manner. The plane waves are supposed to have identical amplitudes and their wavevectors are separated by an angle of 2π/3, yielding an hexagonal interference pattern of high and low intensity regions. These regions can either attract or repel particles, as function of their size and permittivity contrast compared to the background medium. The attraction or repulsion can be quantitatively studied by analyzing the curvature of the pseudo-potential, obtained by a simple derivative of (8) with respect to x, which can be here obtained analytically. This process is simplified by combining the symmetric force terms as

F_{ij}^{(p)} + F_{ji}^{(p)} = 2 K^{(p)} e^{\frac{i (ϕ_{i} + ϕ_{j})}{2}}

The curvature of the pseudo-potential can be obtained from

\frac{\partial^{2} U^{(p)}}{\partial x^{2}} = - \frac{\partial F^{(p)}}{\partial x},

which yields:

\frac{\partial^{2} U^{(p)}}{\partial x^{2}} = - 2 i k_{i j_{x}} K^{(p)} e^{\frac{i (ϕ_{i} + ϕ_{j})}{2}} \sum_{i = 1}^{2} \sum_{j = i + 1}^{3} \sum_{n = 0}^{+ \infty} Λ^{(p)} .

where k_{ij_x} = (k _i -k _j) ∙ x̂. A positive (negative) curvature yields a stable (unstable) trap due to a pseudo-potential well (mountain). Various curvatures as function of particle size are shown in Fig. 2 for both TE and TM polarizations, at two highly symmetric locations in the lattice: ρ ₀ = (0,0) and ρ ₁ = (2λ/(3√3ε), 0), the latter point being denoted by the right-most cross in Fig. 3 and corresponding to a phase singularity as we shall illustrate hereafter. Although Fig. 2 only shows the curvature along the x direction, which is not sufficient to conclude on the efficiency of the two-dimensional trapping at either ρ ₀ or ρ ₁, it is straightforward to extend the study to other directions and unambiguously characterize the trapping properties in two dimensions. It can be seen from Fig. 2 that at the origin, the curvature is positive for small particles under a TM illumination. This is a direct illustration of the fact that small dielectric particles are well trapped in high intensity regions due to the gradient force, as it has already been demonstrated multiple times using the Rayleigh approximation. Interestingly, the same particles are not stably trapped in the high intensity regions for a TE polarization (note that the optical landscape refers to the magnetic field in this case). A dual behavior occurs at ρ ₁ for both polarizations.

The oscillating nature of the curve indicates that the stable and unstable equilibria alternate as the particle size increases, with the first crossing points located at a ≃ 0.2264λ for the TM incidence and a ≃ 0.2105λ for the TE incidence. The locations of these crossing points can be approximated by realizing that for small particles, the sole n = 0 term in (8) suffices to obtain a good approximation of the force [13]. A zero curvature coincides in this case with the first zero of J ₀(k_pa) for the TM polarization (k_pa ≃ 2.405), and the first zero of J́₁[(k_pa) for the TE polarization (k_pa ≃ 1.841). Using the numerical values given in the caption of Fig. 2, the two zeros yield a = 0.239λ for TM and a = 0.183λ for TE. The disagreement with the exact values stems from the fact that such sizes of particles are not very small compared to the wavelength so that the approximation of using only the n = 0 term is less accurate.

Using (8), it can be easily demonstrated that the crossing points (and in fact the whole curve at ρ ₀ presented in Fig. 2) are independent of frequency if the electrical size of the particle remains unchanged, i. e. k ₀ a = constant: at the origin the sine term vanishes and the cosine term reduces to unity while the other dependencies are K ^(p) ~ k ₀, Λ^(p) ~ 1/k₀ ², and γ _n ^(p) ~ k ₀ ². With the additional k_{ij_x} term, ∂ ² U ^(p)/∂x ² becomes independent of frequency at ρ ₀. This property allows us to make a connection with the results presented in [18, Fig. 3], where the attraction and repulsion of the high intensity regions was clearly illustrated for a TM incidence and particle sizes of a = 0.15λ and a = 0.3λ, respectively, although the results were obtained at a different wavelengths. The previous argument can be generalized to the location ρ ₁ to show that in this case, only the crossing points in Fig. 2 are independent of frequency.

fig.curv also provides the basis to study the trapping efficiency of a given particle in the optical lattice, which is often a concern in experiments. The best trapping efficiency at ρ ₀ is obtained when the curvature of the corresponding pseudo-potential well is positive and maximum, which corresponds to the positive maxima of the solid (dashed) curves for the TM (TE) polarization. The first such sizes are a ~ 0.142λ and a ~ 0.293λ for a TM and TE incidence, respectively. Not only these points correspond to the steepest well at ρ₀ but they also correspond to the steepest mountain at ρ ₁, thus combining the best attraction by the high intensity regions to the best repulsion by the low intensity regions. Correspondingly, it is also possible to trap in the low intensity regions by selecting the negative extrema at ρ ₀ or the positive extrema at ρ ₁. In two dimensions, however, this trapping is not symmetric because of the asymmetry of the three incident waves so that the pseudo-potential well is steeper in some directions than in others (data not shown for brevity).

Finally, we emphasize the importance of the points of zero curvature in Fig. 2, which correspond to a flat pseudo-potential at both ρ ₀ and ρ ₁. The force distribution for a TM polarization at a = 0.2264λ (the first crossing point) is illustrated by the arrows on the right side of Fig. 3, while the background represent the incident intensity. It is clearly seen that neither the high intensity regions nor the low intensity regions are attractive or repulsive, but that optical vortices are formed around ρ ₁. The left side of Fig. 3 shows the phase of the field, where singularities are clearly seen at ρ ₁ symmetry points, yielding optical vortices. These vortices, however, are only effective at trapping particles if the curvature of the pseudo-potential is close to zero, as we show subsequently.

Fig. 2. Curvature of the pseudo-potential along the x̂ axis at ρ ₀ = (0,0) and ρ ₁ = (2λ/(3√3ε), 0) a (ρ ₀ corresponds to the point of maximum field intensity while ρ ₁, shown by the right-most ‘+’ sign in Fig. 3, corresponds to the point of minimum field intensity). The incident field is obtained from three plane waves of equal amplitude and directions separated by 2π/3, either TE or TM polarized. The inset depicts a zoomed version of the black rectangle and shows the identical crossing points for the respective polarizations. Other parameters: λ = 1064 nm, ε= 1.69ε ₀, ε_p = 2.56ε ₀.

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A vortex trapping occurs at ρ ₁ when the phase singularity points induce an attractive force on the particle. This can be quantified by looking at a circle centered at ρ ₁ and requiring the force in the normal direction to be pointing inward (toward ρ ₁). Such a circle is represented in Fig. 3, with a radius arbitrarily chosen of 140 nm, along which we compute the force and the normal component as F̂^TM ∙ n̂ where n̂ is the outward pointing normal vector to the circle. A positive (negative) value of the dot product indicates a normal force pointing away from (toward) ρ ₁, the negative value yielding a spiral-like attractor. The results are presented in Fig. 4 for two sizes of particles. The first one corresponds to the zero curvature of the pseudo-potential, i.e. a = 0.2264λ, and shows that various locations around the circle experience different directions of normal force. Such configuration is therefore not adequate for an inward spiraling motion, and a larger size of particle needs to be chosen (a smaller size would yield a less attractive ρ ₁ point). The minimum value obtained for which the dot product is always negative all around the circle is a = 0.243λ, which is the second curve shown in Fig. 4. The motion corresponding to these two sizes of particles is illustrated in Fig. 3. The first trajectory (marked ‘1’) corresponds to a = 0.2264λ and is seen to spiral away from the singular phase points. Eventually, the particle takes a triangular path between the three closest high intensity points, exactly following the force flow shown on the right side of Fig. 3. Although theoretically the particle can be trapped at ρ ₀, Brownian motion will prevent such a trapping to occur in experiments. The second trajectory illustrated in Fig. 3 (marked ‘2’) corresponds to a = 0.243λ. As predicted in Fig. 4, the normal force is here always attractive resulting in an inward spiral-like motion toward the point of low intensity (a symmetric point to ρ ₁), resulting in a stable trap. The final position exactly corresponds to the phase singular point, as predicted.

Fig. 3. Background pattern left: phase of the incident electric field (in degrees). Background pattern right: incident TM intensity distribution (in V ²/m ²). Arrows (symmetric in x, only x > 0 is shown for clarity): force on a particle with parameters a = 0.2264λ, λ = 1064 nm, ε = 1.69ε ₀, ε_p = 2.56ε ₀. The ‘+’ signs indicate the minimum field intensity points corresponding to possible optical vortices (the right-most sign corresponds to ρ ₁). Trajectory ‘1’ corresponds to a particle with a = 0.243λ starting at (-200 nm, 380 nm) (denoted by the ‘x’ sign), and shows that the particle is not trapped by the landscape. Trajectory ‘2’ corresponds to a particle with a = 0.2264λ starting at (-200 nm, -380 nm) (denoted by the ‘x’ sign), and shows that the particle is trapped by the landscape in a spiral-like manner. Dimensions along the horizontal axes are given in μm and the two axes are equal.

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Fig. 4. Normalized normal component of the force along the white circle (centered around ρ ₁) for two sizes of particles. The always negative values for a = 0.243λ indicate that the normal force is always attracting the particle toward ρ ₁, yielding a spiral attractor.

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4. Conclusion

The main contribution of this paper is to provide an analytical framework for the study of the pseudo-potential due to multiple plane waves incident on a lossless cylindrical dielectric particle. In this regard, we have generalized the work proposed in [13] in two main aspects. The first has been to provide an analytic expression of the force for a TE incidence, in addition to the TM incidence already known. The second aspect has been to study the curvature of the pseudo-potential generated by three plane waves, which has allowed us to provide a deeper analysis of the trapping results obtained in earlier works. As a result, we have predicted the formation of periodic force vortices in the optical landscape, which induce spiral-like trajectories of particles whose parameters yield points close to the zeros of the pseudo-potential. As function of their parameters (size and index contrast with the background medium), the particles following these trajectories have been shown to be either trapped at the location of the optical vortices, or to be permanently moving between three intensity maxima. These phenomena are shown here to occur at particle sizes of about 0.2λ, but quantitatively similar results are obtained at sizes corresponding to the other zeros of the pseudo-potential in Fig. 2, at about 0.4λ, 0.6λ, etc. Since these crossing points correspond to increasing particle size, they yield configurations that are less and less sensitive to Brownian motion and more appropriate for experimental verifications.

This work is sponsored by NASA-USRA under Contracts No. NAS5-03110 and No. 07605-003-055, by the Department of the Air Force under Air Force Contract No. FA8721-05-C-0002.

Appendix

We provide here a few algebraic details to simplify Eqs. (5a) and (5b) into (7). The first step is to generalize Eqs. (5a) and (5b) to take multiple incidences into account. This is simply done by the transformation

e^{i ϕ_{i}} \to e^{- in ϕ_{i}} e^{i (n + 1) ϕ_{j}},

where ϕ_i, and ϕ_j are the angles of incidence of two plane waves, the total force being obtained by adding all (i, j) contributions. In addition, the space dependency exp(ik _i, ∙ ρ) is also added to each a _n ^(M) coefficient. The equations become

A = {(\frac{2}{πa})}^{2} \frac{2 a}{k^{2}} {∣ E_{hi} ∣}^{2} e^{\frac{(ϕ_{i} + ϕ_{j})}{2}} \sum_{n = 0}^{+ \infty} \frac{1}{{∣ D_{n}^{TE} ∣}^{2} {D_{n + 1}^{TE} ∣}^{2}} \times

B = - {(\frac{2}{πa})}^{2} \frac{2 i k_{p}}{k^{2}} {∣ E_{hi} ∣}^{2} \cos (\frac{ϕ_{i} + ϕ_{j}}{2}) \times

where it is understood that the argument of the Bessel functions is k_pa (i.e. J _n = J _n(k_pa)). Further simplifications require the conversion of the sums to have the same limits (a sum from ∑_n = ∑_n=-∞ ^+∞ is converted to ∑_n=0 ^+∞ by adding the symmetric terms (n,-n-1) and conversely), the proper combination of terms from (14a) and (14b), and the use of identities on the derivative of the Bessel functions. We finally obtain (for a single incidence for example):

A = - {(\frac{2}{πa})}^{2} \frac{i a}{k^{2}} {∣ E_{hi} ∣}^{2} e^{i ϕ_{i}} \times

B = - {(\frac{2}{πa})}^{2} \frac{2 i k_{p}}{k^{2}} {∣ E_{hi} ∣}^{2} \cos ϕ_{i} \sum_{n} \frac{n J_{n} J_{n + 1}^{'}}{D_{n}^{TE} D_{n + 1}^{T E^{*}}} .

It is then straightforward to further simplify the summation of (15a) and (15b) and to convert the sums to ∑_n=0 ^+∞. Upon doing so, the force given by (7) is directly obtained.

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Analytical prediction of stable optical trapping in optical vortices created by three TE or TM plane waves

Abstract

1. Introduction

2. Analytical expression of the force

3. Pseudo-potential and zero curvature points

4. Conclusion

Appendix

References and links

Cited By

Figures (4)

Equations (44)

Optics Express