Toward Waveguide-Based Optical Chromatography

1 Introduction

Since Ashkin’s pioneering paper [1], Optical Tweezers (OT) have opened up the possibility to perform micro- and nanoscale non–contact mechanical manipulation and measurements on particles in the nano-micron scale. Single and multiple optical beams can hold, stretch, and compress particles, filaments, vesicles in a suspension, enabling novel applications such as optical force spectroscopy of microspheres [2, 3]. The room temperature force sensitivity of OT based techniques ranges from 20 fN [4] to 200 pN [5], making this tool ideal to handle particles in the biological micro–environment. Single–beam optical tweezers have allowed the development of several applications in life science, and use optical forces created by a tightly focused laser beam not only to manipulate micron to nanometer–sized structures, but also to measure their mechanical properties [6–8]. The single–beam optical tweezers require gradient forces that exceed scattering forces, a condition usually met with in the case of small differences between medium/particle refractive indexes. This situation is commonly found in aqueous solutions, but not in air or vacuum. The Brownian motion sets the limit for the size of the smallest particle that can be trapped, although at low temperatures (not compatible with live cells), it is possible to even optically trap atomic gases [9]. The success of the single–beam optical tweezers has encouraged the development of new, complementary optical trapping configurations, including optical chromatography, a method that combines the use of optical forces to manipulate and size–select particles in microfluidic systems [6].

Optical chromatography was initially proposed by Imasaka et al. in 1995 [10]. Their approach consisted of mildly focusing a laser beam into a solution containing particles that are counter–flowing coaxially with a capillary. When in equilibrium, there is a balance between the laser scattering force and the drag force from the fluid medium. A challenge in this configuration is to keep the laser beam aligned at the center of the of capillary. Since these pioneering experiments, optical chromatography has led to a range of different applications, for example: to visualize immunological reactions by flowing a sample of antigens past optically–trapped beads that are coated with antibodies. Selective binding changes the effective size of the particles an thus their retention distance. This allowed to determine antigen concentrations in the range from nanomolar to nanograms per milliliter [11]. Optical chromatography can also be used to separate same–sized particles of different refractive indices [12], and be combined with microfluidic channels with well–defined velocity flow patterns [13]. This approach was used to measure low concentrations of anthrax bacteria, by accumulating them from a flow in an optical trap [14]. Further improvements of the technique include hydrodynamic focusing, enabling the separation of equal–sized microsphere differing only 0.1 in refractive index [15]. Further optical chromatography and other optical sorting strategies were summarized in a recent review [6].

Optical chromatography requires a good understanding of the forces acting on the analyzed particles. Initial models were based on geometrical–optics models, and identified analogies with separation parameters for conventional chromatography such as: retention distance, selectivity, dynamic range, theoretical plate number, and resolution [16]. Using a focused Gaussian beam, the retention distance can be related to the Rayleigh length, and can be shown to scale as the square root of particle size. The selectivity in chromatography is directly related to the slope of the retention distance vs. particle size, which can be tuned by carefully changing the laser power and the flow velocity. The dynamic range, defined as the ratio between the maximum and minimum particle size that can be analyzed, is ultimately limited by the size-range of particles that can be trapped. The theoretical plate number, another chromatography parameter, determines the separation efficiency in optical chromatography. In the case of optical traps, this is typically limited by Brownian fluctuations and instrumental noise. Finally, the resolution of particle separation can be determined from the theoretical plate number and selectivity. In theory, particles can be resolved if they differ from at least 0.4% in size [16].

A more recent approach to optical separation uses hollow–core photonic crystal fibers with core diameters between 10 and 30 μm, to optically trap and guide particles along a length of fiber [17]. Advantages are the presence of well–defined waveguide modes, that are well-approximated by the modes of cylindrical waveguides [18], as well as the long interaction length. Optical forces provided by the fundamental fiber mode can robustly align particles in the center of the channel, resulting in well–defined optical- and fluidic forces that can be used to determine the refractive index and size of spherical microspheres by monitoring their speed by Doppler–velocimetry [19–21]. Besides the long distance, stability, and robustness of such waveguide-based chromatography, it also allows to operate in either solutions, air, or vacuum, and at a wide range of pressures and temperatures, as long as the integrity of the waveguide is not compromised. Moreover, the waveguide enclosure around the particles protects the operators and guarantees a perturbation–free environment. Considering the nature of the particles, protection can be an issue in cases such as aerosols loaded with viruses, or radioactive and toxic particles.

The further development of fiber–based optical chromatography techniques requires a better understanding of the optical forces exerted on the particles by the waveguide modes, whose description strongly depends on particles size. The dimensionless size parameter used in scattering is defined as $x=ka=2\pi n_{1}a/\lambda$, where a is the particle’s radius, λ is the wavelength and M = n₁/n₀ is the relative refraction index between the particle and the medium. In the case of Rayleigh particles with radius $x\ll 1$, radial forces are provided by the gradient force which is proportional to x³. If a particle is accelerated in a fiber loop, its maximum velocity will depend on the centripetal force required to keep the particle in its circular trajectory. As both the gradient force and centripetal force scale with x³, the maximum particle velocity will be independent of particle size. The axial forces, on the other hand, are provided by the scattering force and therefore proportional to x⁶. This results in a particle acceleration that is proportional to x³. In the laminar flow regime, the terminal velocity off Rayleigh particles will depend on the ratio between the optical force (x⁶) and the Stokes drag force (x), and will therefore be proportional to x⁵, offering excellent conditions for size–sorting the particles through their velocity. For microparticles, on the other hand, particle resonances and their coupling to the waveguide modes must be taken into account to obtain expressions for the size–dependent optical forces. This requires an integrated scattering theory that combines a Mie description of the particles with analytical waveguide modes, which has not been demonstrated to date.

In this paper, we will develop analytical equations for optical forces, including the scattering forces, on particles within waveguides and will discuss their implications to sorting of particles based on particle size, refraction index, absorption, and mass density. We will start from the full electromagnetic force theory to obtain expressions for the forces acting on waveguide–embedded particles spanning the Rayleigh and Mie regime. Analytical results provide more than just fast calculations compared to numerical methods. They provide powerful insights of how to analyze the role of each parameter, allowing to optimize the design of the system for a given task. The main challenge of analytical optical force calculations for spherical particles lies in the expansion of the fields into vector spherical wave functions, which, except for the plane wave case developed by Mie in 1908 [22], could only be obtained numerically. However, in 2010 we discovered how to perform this expansion for any type of electromagnetic field in the Fourier space and provided analytical integrals for the Beam Shape Coefficients for general waveguides. Analytical results from the integrals were found in the cases of cylindrical and rectangular metallic waveguides [23, 24].

Our current analysis focuses on cylindrical metallic waveguides, whose modes are largely similar to those in readily available hollow-core fibers waveguides. However, the work can be easily extended to rectangular waveguides, using recently develop analytical expressions for their modes [24], which can be of interest for typical on-chip microfluidic geometries. Recently, Ref. [25] presented an analytical solution for the scattered intensities of a sphere on the axis of an infinite metallic cylinder with a plane wave incident on one end of the cylinder. While the combination of the spherical and cylindrical symmetries is interesting, they did not calculate the BSCs for off-axis positions, nor the optical forces on the sphere.

In our analysis, the particle size is assumed to be much smaller than the waveguide size, resulting in a scattering intensity much smaller than the incident light. This implies that we can neglect reflections from the waveguide walls back to the particle, as well as interference between different particles. Furthermore, the waveguides are assumed to be lossless. For low–loss waveguides, losses to be taken into account by considering a slow variation in the electromagnetic field intensity with axial position.

Our manuscript is organized as follows: Section 2 presents the electromagnetic fields for any propagation mode of the cylindrical waveguide. Section 3 defines the Mie Coefficients as well is the Beam Shape Coefficients (BSCs), which are obtained for any waveguide mode and any particle position. These results are valid for any particle size smaller than the waveguide size and any waveguide mode. In Section 5 the BSCs will be used to obtain a complete theory for the optical forces, for any waveguide mode, and any particle size and position within the waveguide. At this point the theory is complete.

We will then discuss the implication of the theory for three specific applications. First we focus on nanoparticles that are well in the Rayleigh regime. We will obtain their optical forces using a small–value series expansion of Mie Coefficients, and will demonstrate that radial forces are proportional to x³, and axial forces proportional to x⁶ for non–absorbing particles. Secondly, we discuss the microparticles, for which the full Mie calculation is required. Section 6, studies the optical force as a function of particle size and refractive index in case of an on-axis particle position. For larger particles, strong Mie resonances appear that can potentially be used be used to perform in highly–sensitive particle sorting experiments. Third, in Section 7, we will study how the effect of particle absorption on axial force depends on particle size. Finally, we will summarize illustrative results for special cases, applications, and experimental implementation.

2 Waveguides Electromagnetic Fields

The general formalism for the TM and TE modes of a hollow waveguide of arbitrary cross–section starts with the scalar solution $g\left({\mathbf{r}}^{\prime }\right)=g_{\gamma }\left({\mathit{\rho }}^{\prime }\right)e^{\text{i}{k}_z{z}^{\prime }}$ of the Helmholtz equation in three dimensions

which leads to a two–dimensional solution $g_{\gamma }\left(\mathit{\rho }\right)$ of the transverse Helmholtz equation [26].

satisfying the boundary conditions at the waveguide surface S. The TM modes satisfy $g_{\gamma }\left(\mathit{\rho }\right){|}_S=0$ and the TE modes satisfy ${\partial }_n{g}_{\gamma }\left(\mathit{\rho }\right){|}_S=0$ with ${\partial }_n$ the normal derivative, k_z is the wavevector in the z–direction, $k^2=k_z^2+{\gamma }^2$, γ is the transverse wavevector, $\text{d}/\text{d}\mathit{\rho }=\widehat{x}{\partial }_x+\widehat{y}{\partial }_y$ is the transverse gradient operator, and $\text{d}/\text{d}\mathbf{r}=\text{d}/\text{d}\mathit{\rho }+\widehat{z}{\partial }_z$ is the three–dimensional gradient operator, $k^2=n_o{\omega }^2/c^2$, n_o is the waveguide’s interior refractive index. It is useful to introduce the cylindrical coordinate system in the r–space and k–space. In the r–space we have $\mathit{\rho }=x\widehat{x}+y\widehat{y}$, ${\rho }^2=\mathit{\rho }\cdot \mathit{\rho }$ and $\widehat{\mathit{\rho }}=\mathit{\rho }/\rho$. We have also $\varphi =-y\widehat{x}+x\widehat{y}$ and $\widehat{\varphi }=\varphi /\rho$. In the same way, for spherical coordinates in k–space, we change $\left(r,\theta ,\varphi \right)\to \left(k,\xi ,\zeta \right)$ and for cylindrical coordinates, we change $\left(\rho ,\varphi ,z\right)\to \left(\gamma ,\zeta ,k_z\right)$, obtaining equivalent systems of coordinates. The boundary conditions define possible values for γ based on the geometry alone. The cutoff frequency of a given mode happens when $k^2={\gamma }^2$. Below this frequency, k_z < 0, the mode becomes evanescent. Given the scalar solution, the TM and TE fields are given by:

For a general waveguide, the task is to find g(r) satisfying boundary conditions. For a cylindrical waveguide, the scalar wavefunction is given by $g_{\gamma }\left(\mathit{\rho }\right)=J_m\left({\gamma }_{mn}\rho \right)e^{\pm \text{i}m\varphi }$, where $J_m(.)$ are Bessel functions. The transverse vectors are given by ${\gamma }_{mn}^{\text{TE}}={\chi }_{mn}^{\prime }$ where ${\chi }_{mn}^{\prime }$ is the nth–root of $J_m(.)$ derivative, that is $J_m^{\prime }\left({\chi }_{mn}^{\prime }\right)=0$ for the TE mode, and ${\gamma }_{mn}^{\text{TM}}={\chi }_{mn}$ where ${\chi }_{mn}$ is the n^th–root of $J_m(.)$, that is $J_m\left({\chi }_{mn}\right)=0$, for the TM mode. The cutoff frequencies are ${\omega }_{cmn}^{\text{TE}}=\left(c/n_o\right)\left({\chi }_{mn}^{\prime }/R\right)$ for TE modes, and ${\omega }_{cmn}^{\text{TM}}=\left(c/n_o\right)\left({\chi }_{nm}/R\right)$ for TM mode. The cutoff wavelengths are given by ${\lambda }_{cnm}=2\pi R/{\chi }_{nm}$. The first ten modes, from highest to lowest ${\lambda }_{nm}$ are ${\text{TE}}_{11}\left[{\chi }_{11}^{\prime }\right]=1.8412$, ${\text{TM}}_{10}\left[{\chi }_{10}\right]=2.4048$, ${\text{TE}}_{21}\left[{\chi }_{21}^{\prime }\right]=3.0542$, ${\text{TE}}_{01}\left[{\chi }_{01}^{\prime }\right]=3.8317$, ${\text{TM}}_{11}\left[{\chi }_{11}\right]=3.8317$, ${\text{TE}}_{31}\left[{\chi }_{31}^{\prime }\right]=4.2012$, ${\text{TM}}_{12}\left[{\chi }_{12}\right]=5.1356$, ${\text{TE}}_{41}\left[{\chi }_{41}^{\prime }\right]=5.3175$, ${\text{TE}}_{12}\left[{\chi }_{21}^{\prime }\right]=5.3314$, ${\text{TM}}_{20}\left[{\chi }_{20}\right]=5.5201$. Even for $R=2.5\mu \text{m}$ and n_o = 1 there are 16 modes for wavelengths shorter than 2,000 nm and 45 modes for wavelengths shorter than 1,000 nm. Therefore, in the visible–NIR region, the waveguide will be essentially multimode. We are mostly interested in the range 400 to 2,000 nm to avoid direct IR and NIR absorption. Assuming a wavelength of 500 nm and a $10\mu \text{m}$ waveguide diameter, the first mode’s propagation constant becomes $k_z/k=0.9995\approx 1$ and $\gamma /k=\sqrt{1-{\left(k_z/k\right)}^2}=0.032$. Linear combinations of $g^{\pm }\left(\mathit{\rho }\right)$ can be used to build the cosine and sine modes, as well.

The analytical expression for the fields of TE, TM, ±, cos and sin are condensed in the expressions:

where the form TE/TM and 3 × 1 array represents the (±, cos, sin) forms and $Z=\sqrt{\mu /\mathit{ϵ}}$ is the waveguide medium impedance. It is interesting to note the interchange between electric and magnetic fields between TM and TE modes does not change the result. This shows that the Poynting vector expression will be the same for both the TE and TM modes. The TE mode is obtained by: ${\mathbf{E}}^{\text{TE}}\to -Z{\mathbf{H}}^{\text{TM}}$ and $Z{\mathbf{H}}^{\text{TE}}\to {\mathbf{E}}^{\text{TM}}$. The first modes have m = 1 and for a very large waveguide ($R\to \infty$, tending to free space), $\gamma \to 0$ and $k_z\to k$, we can use $\gamma \rho \to 0$ to obtain $J_m\left(x\right)/x\approx x^m/\left(2^{m}m!\right)$, $J_m^{\prime }\left(x\right)\approx m{x}^{m-1}/2^{m}m!$ and so $J_1\left(x\right)/x=J_1^{\prime }\left(x\right)=1/2$, which allows us to obtain:

This is clearly the plane wave limit, from which we can notice the association of circularly polarized light for the mode ±, x-polarized beam to cos mode and y-polarized to sin mode. One can couple to ± modes using circular polarized light while a polarized beam will couple to cos and sin modes.

We also have the explicit form of the electromagnetic fields written as

and it is important to calculate the gradients, given by

and

The ϕ component of the gradient is given by

and

3 Vector Spherical Wave Function Expansion

To calculate the electromagnetic fields inside and scattered by a spherical particle located at an arbitrary position r inside a cylindrical waveguide, one has to expand the incident fields in the Hansen multipoles. For the force calculation, only the incident and scattered fields are necessary. The VSWF expansion is given by

where E₀ is an electric field dimension constant, $k{\mathbf{N}}_{pq}^{\text{in}}\left(\mathbf{r}\right)=\text{i}\nabla \times {\mathbf{M}}_{pq}^{\text{in}}\left(\mathbf{r}\right)$, and ${\mathbf{M}}_{pq}^{\text{in}}\left(\mathbf{r}\right)=j_p\left(kr\right){\mathbf{X}}_{pq}\left(\widehat{r}\right)$ are the Hansen multipoles for the incident beam, and $k{\mathbf{N}}_{pq}^{\text{sc}}\left(\mathbf{r}\right)=\text{i}\nabla \times {\mathbf{M}}_{pq}^{\text{sc}}\left(\mathbf{r}\right)$, and ${\mathbf{M}}_{pq}^{\text{sc}}\left(\mathbf{r}\right)=h_p^{\left(1\right)}\left(kr\right){\mathbf{X}}_{pq}\left(\widehat{r}\right)$ the Hansen multipoles for the scattered beam, j_p(kr) are spherical Bessel functions, y_p(kr) are spherical Neumann functions and $h_p^{\left(1\right)}\left(kr\right)=j_p\left(kr\right)+\text{i}{y}_p\left(kr\right)$${\mathbf{X}}_{pq}\left(\widehat{r}\right)=\mathbf{L}{Y}_{pq}\left(\widehat{r}\right)/\sqrt{p\left(p+1\right)}$ are the vector spherical harmonics, $Y_{pq}\left(\widehat{r}\right)$ are the scalar spherical harmonics, $Z=\sqrt{\mu /\epsilon }$, $k=\omega \sqrt{\mu \epsilon }$, and $\mathbf{L}=-i\mathbf{r}\times \text{d}/\text{d}\mathbf{r}$ is the angular momentum operator in direct space.

The coefficients $G_{pq}^{\text{TE}/\text{TM}}$ are known as Beam Shape Coefficients (BSC) [27]. It is important to note that, as the VSWF expansion is performed with the origin of the axis at the center of a sphere, we need to apply boundary conditions that are independent of the particular position of the beam. The a_p and b_p coefficients, called Mie coefficients, come from the boundary conditions over the surface of a sphere, and are given by

Here $x=ka=2\pi n_{1}a/\lambda$ is the size parameter, where a is the particle’s radius and λ is the wavelength and M = n₁/n₀ is the relative refraction index. The functions $\psi \left(x\right)=x{j}_p\left(x\right)$ and $\xi \left(x\right)=x{h}_p^{\left(1\right)}\left(x\right)$ are known as Riccati-Bessel functions. The BSCs depend on the incident field only and all further calculations rely on them. The position of the center of the sphere with respect to the axis of the incident field in cylindrical coordinates is defined as $\mathbf{r}=\left({\rho }_o,{\varphi }_o,z_o\right)$. It is clear, then, that BSCs are functions of this position, $G_{pq}^{\text{TE},\text{TM}}=G_{pq}^{\text{TE},\text{TM}}\left(\mathbf{r}\right)$. Gustav Mie solved this problem in 1908 [22] for the incident plane wave using the Rayleigh expansion of a plane wave in terms of spherical Bessel functions. For the plane wave, the position of the sphere with respect to the beam is not important. For an arbitrary beam, the usual procedure is to take the dot product with ${\mathbf{X}}_{pq}\left(\widehat{r}\right)$ to obtain:

which is not complete because there is a spherical Bessel function embedded in the right–hand side that did not cancel explicitly the one on the left side. We solved this problem for an arbitrary electromagnetic field in 2010 [23, 24], by taking the Fourier transform, explicitly canceling the spherical Bessel functions, and obtaining the final result:

where

are Fourier transforms of each other and

are the Fourier transforms in a sphere, requiring that $|\mathbf{k}\prime |=k$. By using the translation theorem of Fourier transform it was easy to calculate the BSC for any position respect to the incident beam. The geometry of the problem is shown in Figure 1.

FIGURE 1. Cylindrical waveguide coordinate system (CWG). The natural coordinate system of the waveguide is rʺ and a point r at which we perform a partial wave expansion (PWE) in vector spherical wave functions (VSWF). In the new system, the expansion is done in the rʹ coordinate system, the fields are described by the Hansen multipoles and the G^TE/TM coefficients, that are constant in the rʹ frame, but varies depending on the position of the expansion r.

4 Cylindrical Waveguide BSC’s

Using Eqs 24, 25 and 32 of Ref. [24] the explicit expression for the BSCs for any mode of the waveguide is given by:

Here, our notation is that superscript TE/TM means the spherical Transverse Electric/Magnetic in the sphere, while (TE)/(TM) means the waveguide TE/TM modes, $P_p^q(.)$ is the associated Legendre function. To shortening the equations we will omit the arguments with the convention that all Legendre functions $P_p^q=P_p^q\left(k_z/k\right)$, and all Bessel functions $J_m=J_m\left(\gamma {\rho }_o\right)$. The form factor is:

5 Optical Forces

In a recent review, it was demonstrated how optical forces can be derived from the Maxwell stress tensor given the BSCs in the Minkowski formalism [28]. Explicitly, in Eq. 93 in Ref. [28] provides the expression for the transverse forces, while the force in the z–direction is given by Eq. 94 in Ref. [28]:

where $A_p=a_p+a_{p+1}^{\text{∗}}-2a_p^{\text{∗}}{a}_{p+1}$, $B_p=b_p+b_{p+1}^{\text{∗}}-2b_p^{\text{∗}}{b}_{p+1}$ and $C_p=a_p+b_p^{\text{∗}}-2a_p^{\text{∗}}{a}_p^{\text{∗}}$. The definition of the literature, including the a recent review [28], is $A_p^{lit}=A_p^{\text{*}}$ which is counter-intuitive, because the lowest order Mie coefficient appears conjugated $a_p^{\text{*}}$. We therefore prefer the current definition. The formulas must thus be corrected by changing $A_p^{lit}\to A_p^{\text{*}}$ and $B_p^{lit}\to B_p^{\text{*}}$ and vice-versa, while keeping the same C_p.

From the expression of z-component of the force we substitute the BSCs expressions into the force equation, and using the normalization for the force $\mathbf{C}=\mathbf{F}/\left(\pi \epsilon |E_o/k{|}^2\right)$ we obtain

We display the equation in this way to demonstrate that all particle properties, such as size and refractive index, are embedded in the Mie factors A_p, B_p and C_p. Breaking the q sum into q = 0, q > 0 and q < 0, and using $\left(p+q\right)!P_p^{-q}={\left(-1\right)}^q\left(p-q\right)!P_p^q$, we can simplify this expression to:

The [radial, azimuthal] forces can be written in the same way:

While these equations could be written in a more compact way, we want to highlight the dependence of the real and imaginary parts of the Mie factors A_p, B_p and C_p, because these factors define the Mie resonances, as well as their scaling with particle size and relative refraction index, which are important to the sorting process. The dependence of the position of the particle inside the waveguide is embedded in the form factors $g_{m,q}^{\text{form}}\left({\rho }_o,{\varphi }_o\right)$.

It becomes immediately clear that F_z depends on the real part of A_p, B_p and C_p only. In principle, $F_{\rho }$ depends on the real and imaginary parts of A_p, B_p and C_p, but the form factors will cancel the real part. It is also interesting to see that dependence on z disappeared, as expected from the axial translation symmetry of the waveguide (assumed lossless). Furthermore, there is no azimuthal dependence for the ± modes, corresponding to circular polarization, due to their rotational symmetry. For the $\text{cos}\varphi$ and $\text{sin}\varphi$ forms, the result does depend on ϕ, as the symmetry is broken by the incident beam polarization.

These expressions are general and can be used to obtain the force for any waveguide mode and particle size. Special care must be taken in the choice of highest p to truncate the series. The Wiscombe criterion was to truncate the sum when $|a_p{|}^2+|b_p{|}^2<5{10}^{-14}$ [29], leading to the known truncation of the infinite series of $p_{\text{max}}=\text{int}\left\{x+4.05x^{1/2}+2\right\}$. Meanwhile, Neves and Pisignano included the relative error, allowing faster computation within an fixed error bound, as $p_{\text{max}}=\text{int}\left\{x+0.76{\mathit{ϵ}}^{2/3}{x}^{1/3}-4.1\right\}$, where ϵ is related to a relative error of ${10}^{-\mathit{ϵ}}$ [30].

In the following sections we will discuss two special cases. The first is related to use of the optical chromatography (OC) for nanoparticles that are well in the Rayleigh limit $x\ll 1$. In this limit, we need the Mie coefficients at $x\to 0$. Our results will show that for any waveguide mode in any position inside the guide, the radial forces are proportional to x³ and the gradient of the field intensity $F_{\rho ,\varphi }\propto x^3\nabla |\mathbf{E}{|}^2$. On the other hand, for non–absorbing particle, the axial force is proportional to x⁶ and the field intensity $F_z\propto x^6|\mathbf{E}{|}^2$. This shows that radial forces are gradient forces while z forces are scattering forces. The axial force result is obvious because there is no intensity gradient in a lossless waveguide.

The second case is waveguide-based OC for microparticles, such as dielectric spheres or virus–loaded droplets, with sizes in the Mie regime $x\approx 1$. In the Mie–limit, the radial force is assumed to keep the particle on the z-axis, $\rho =0$, resulting in a significant simplification of the expressions. The analysis can be done for any particle size and any waveguide mode, allowing us to observe the role played by the Mie resonances in the optical sorting process.

6 Mie Coefficients in Rayleigh Regime

We can use the series expansion of spherical Bessel and Hankel function keeping the lowest orders of the imaginary and real parts to obtain [31]:

For the lowest terms we obtain

Reports on the scattering of metallic nanoparticles usually call the lowest terms approximations as modified long–wavelength approximation [32–35]. We shall not neglect the second term because they are real while the first are imaginary. Except for TE[m = 0] modes, we will keep only terms with $a_1$ meaning that transverse forces scale with x³ while the axial force scales with x⁶. However, for particles with absorption, the refractive index becomes complex in the form $M=M^{\prime }+\text{i}{M}^{\prime \prime }$. In this case the lowest order terms are:

Without absorption, the leading term comes only from a₁, and only A₁ = a₁ and C₁ = a₁ shall be considered. The only $P_p^q$ to take into account are given by $P_2^2=3{\gamma }^2/k^2$, $P_2^1=-3\gamma k_z/k^2$ and $P_2^0=1/2\left[\left(3k_z^2/k^2\right)-1\right]$ for the q =2 family whose derivative are $P_2^2\prime =-6\left(kz/k\right)$, $P_2^1\prime =3\left(k_z^2/k\gamma -\gamma /k\right)$, $P_2^0\prime =3\left(k_z/k\right)$, and for q = 1 $P_1^1=-\gamma /k$, $P_1^1\prime =k_z/\gamma$, $P_1^0=k_z/k$, $P_1^0\prime =1$.

In the expressions for the force, we get