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Self-induced backaction in optical waveguides

Mohammad Ali Abbassi and Khashayar Mehrany

Open Access Open Access

Abstract

In this paper, we study the backaction effect on the force exerted upon Rayleigh particles in guided structures. We show that the backaction becomes stronger as the group velocity of the guided modes is decreased, which is not unexpected since the fall of group velocity increases the interaction time between the particle and the electromagnetic field. Interestingly, the sign of the group velocity affects the pushing and pulling nature of the exerted electromagnetic force. We specifically investigate the case of a single mode optical waveguide both in the propagating and evanescent regimes, and show that the backaction enables us to enhance the ratio of the potential depth to the trapping intensity, and thereby can be a beneficial tool for nondestructive trapping of small nanoparticles. We further show that backaction can induce some resonances in the optical force in the evanescent regime. These resonances can be employed for sorting of nanoparticles.

© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

1. Introduction

Light can exert force upon nanoparticles by exchanging momentum with them [1–3]. The force exerted by light can be used to trap [4–7], rotate [8–10], or transport nanoparticles [11,12]. The exerted force can be calculated by integrating Maxwell stress tensor over the outside surface of the particle [2,13]. In the Rayleigh regime when the size of the particle is much smaller than the electromagnetic wavelength, the force can be approximated by using the dipole approximation method [14,15]. In this regime, the scattered field by the particle is approximated by an equivalent electric dipole [14,16]. Then, the force exerted upon the particle is obtained by calculating the force exerted upon the equivalent electric dipole [14,15]. It has been shown that the time-averaged force can be decomposed to gradient force, radiation pressure, and spin curl force [15]. The gradient force, is a conservative force that is proportional to the gradient of the intensity, and can be used to trap nanoparticles [4,14]. Radiation pressure and spin curl force are non-conservative forces originating from the radiation reaction of the particle, and sometimes are referred to as the scattering forces. Radiation pressure is proportional to the Poynting vector, and spin curl force is proportional to the curl of the spin angular momentum of the light field [15].

When other scattering objects are present in the vicinity of the particle, the radiation of the dipole will be scattered back from the surrounding objects. Then, the back-scattered field modifies the polarizability of the equivalent dipole, and exerts a force upon the particle which is referred to as the backaction effect [17]. Thus far, the backaction effect has been extensively studied in resonant structures [17,18]. In a high-Q cavity, the backaction is restricted to the shift of the cavity’s resonance frequency due to the presence of the particle [18]. It has been shown that the backaction can reshape the trapping potential, and reduce the local intensity seen by the particle [18]. Thus, the backaction can be advantageous in nondestructive trapping of small nanoparticles by reducing the intensity seen by the particle that was a challenge for many years. The backaction effect is not limited to the high-Q cavities [17]. It has also been reported in plasmonic nanoapertures employed for trapping of small dielectric nanoparticles [19–22]. In these structures, the transmission of the aperture strongly depends on the position of the particle, and the particle plays an active role in enhancing the restoring force exerted upon itself which is sometimes called the self-induced backaction (SIBA) [19]. Furthermore, the backaction has also been observed in photonic crystal structures [23,24]. In [17], a general formulation based on the scattering Green’s function of the structure is presented to study the backaction effect. This formulation has also been extended to study the dynamical backaction effect, such as cooling and heating of the particle’s motion [25].

In this paper, we study the backaction effect in guided structures. We use the modal expansion of the Green’s function to develop a general formulation for studying the backaction effect in optical waveguides. We show that the backaction relates with the group velocity of the guided modes. By reducing the group velocity of the guided modes, stronger backaction can be achieved in the optical waveguides. We further investigate the case of a single mode waveguide, and show that backaction can help us to reduce the local intensity seen by the particle. Moreover, we show that the force exerted upon the particle can resonate in the evanescent regime due to the backaction effect, which can be advantageous for sorting of nanoparticles.

In the following, we present a general formulation for studying the backaction effect in guided structures in Sec. 2. Then, we investigate the case of a single mode optical waveguide in detail in Sec. 3. In that section, we study the cases of propagating and evanescent modes, separately. Finally, we make conclusions in Sec. 4.

2. Backaction formulation in guided structures

Consider a spherical nanoparticle with radius $R_p$, and relative permittivity $\epsilon _p$, which lies within a guided structure extended along the z direction. We assume that the radius of the particle is much smaller than the electromagnetic wavelength. In such a case, the particle can be modeled by an electric dipole. As shown in Appendix B in detail, the equivalent dipole moment of the particle is given by

$$\boldsymbol{p}=\overset{\leftrightarrow}{\boldsymbol{\alpha}}\cdot \boldsymbol{E}_0,$$
where
$$\overset{\leftrightarrow}{\boldsymbol{\alpha}}=\alpha_0\left[\overset{\leftrightarrow}{\boldsymbol{I}}-\alpha_0 \langle\overset{\leftrightarrow}{\boldsymbol{G}}_\perp(\boldsymbol{r}_p,\boldsymbol{r}_p)\rangle\right]^{{-}1},$$
is the polarizability of the particle, and $\boldsymbol {E}_0$ is the illuminating electric field in the absence of the particle within the waveguide. Here, $\alpha _0=3\epsilon _0V_p (\epsilon _p-1)/(\epsilon _p+2)$ is the static polarizability of the particle in the free space, where $V_p$ is the volume of the particle and $\epsilon _0$ is the permittivity of free space. Furthermore, $\langle \overset {\leftrightarrow }{\boldsymbol {G}}_\perp (\boldsymbol {r}_p,\boldsymbol {r}_p)\rangle =V_p^{-1}\int _{V_p}\overset {\leftrightarrow }{\boldsymbol {G}}_\perp (\boldsymbol {r}_p,\boldsymbol {r}')d^{3}r$ is the spatial averaged of the transverse Green’s function inside the particle. As shown in Appendix A, the transverse Green’s function can be decomposed to $\overset {\leftrightarrow }{\boldsymbol {G}}_\perp =\overset {\leftrightarrow }{\boldsymbol {G}}_{0\perp }+\overset {\leftrightarrow }{\boldsymbol {G}}_s$, where $\overset {\leftrightarrow }{\boldsymbol {G}}_{0\perp }$ is the transverse component of the free space Green’s function, and $\overset {\leftrightarrow }{\boldsymbol {G}}_s$ represents the scattering Green’s function. Hence, the averaged of $\overset {\leftrightarrow }{\boldsymbol {G}}_\perp$ which can be written as:
$$\begin{aligned}\langle\overset{\leftrightarrow}{\boldsymbol{G}}_\perp(\boldsymbol{r}_p,\boldsymbol{r}_p)\rangle=&\langle\overset{\leftrightarrow}{\boldsymbol{G}}_{0\perp}(\boldsymbol{r}_p,\boldsymbol{r}_p)\rangle+\langle\overset{\leftrightarrow}{\boldsymbol{G}}_s(\boldsymbol{r}_p,\boldsymbol{r}_p)\rangle\\ =&\frac{ik_0^{3}}{6\pi\epsilon_0}\overset{\leftrightarrow}{\boldsymbol{I}}+\frac{1}{V_p}\int_{V_p}\overset{\leftrightarrow}{\boldsymbol{G}}_s(\boldsymbol{r}_p,\boldsymbol{r}') d^{3}r', \end{aligned}$$
contains both the radiation reaction and the backaction effects [17].

Now, the time-averaged force exerted upon the particle can be obtained from

$$\boldsymbol{F}=\frac{1}{2}\mathrm{Re}\left[\sum_i p_i \boldsymbol{\nabla}{E_{i}^{{\ast}}}\right]_{\boldsymbol{r}=\boldsymbol{r}_p},$$
in which
$$\boldsymbol{E}(\boldsymbol{r},\boldsymbol{r}_p)=\boldsymbol{E}_0(\boldsymbol{r})+\overset{\leftrightarrow}{\boldsymbol{G}}_s(\boldsymbol{r},\boldsymbol{r}_p)\cdot \boldsymbol{p}(\boldsymbol{r}_p),$$
is the electric field seen by the particle that comprises both the incident and back-scattered fields. It can be easily seen that $\boldsymbol {E}$ is a function of both the observation point $\boldsymbol {r}$, and the particle’s position $\boldsymbol {r}_p$. It should be noted that $\boldsymbol {\nabla }$ is the gradient operator with respect to $\boldsymbol {r}$. Then, the time-averaged total force exerted upon the particle can be separated to two parts:
$$\boldsymbol{F}=\boldsymbol{F}_0+\boldsymbol{F}_{\mathrm{bs}}.$$
Here, $\boldsymbol {F}_0$ represents the force that $\boldsymbol {E}_0$ exerts upon the particle, and is given by
$$\boldsymbol{F}_0=\frac{1}{2}\mathrm{Re}\left[\sum_i p_i \boldsymbol{\nabla}{E_{0_i}^{{\ast}}}\right]_{\boldsymbol{r}=\boldsymbol{r}_p}.$$
The other term, $\boldsymbol {F}_{\mathrm {bs}}$, represents the force exerted upon the particle by the back-scattered field, and is given by
$$\boldsymbol{F}_{\mathrm{bs}}=\frac{1}{2}\mathrm{Re}\left[\sum_{i,j} p_i p_j^{{\ast}} \langle\boldsymbol{\nabla}G_{s_{ij}}^{{\ast}}\rangle\right]_{\boldsymbol{r}=\boldsymbol{r}_p},$$
which can also be written as
$$\boldsymbol{F}_{\mathrm{bs}}=\frac{1}{2}\mathrm{Re}\left[\sum_{i,j} p_i p_j^{{\ast}} \langle\boldsymbol{\nabla}G_{\perp_{ij}}^{{\ast}}\rangle\right]_{\boldsymbol{r}=\boldsymbol{r}_p},$$
since $\langle \boldsymbol {\nabla }\overset {\leftrightarrow }{\boldsymbol {G}}_{0\perp }\rangle$ vanishes due to the symmetry of $\overset {\leftrightarrow }{\boldsymbol {G}}_0$.

As shown in Appendix C, the average of $\overset {\leftrightarrow }{\boldsymbol {G}}_{\perp }$ and its derivatives can be obtained from:

$$\langle\overset{\leftrightarrow}{\boldsymbol{G}}_{{\perp}}\rangle|_{z=z'}\simeq \sum_n \frac{i\omega_L}{2\epsilon_0 A_n v_{g_n}} \bigg[\boldsymbol{\phi}_{n_t}(x,y)\boldsymbol{\phi}_{n_t}(x',y')+\phi_{n_z}(x,y)\phi_{n_z}^{{\ast}}(x',y')\hat{\boldsymbol{z}}\hat{\boldsymbol{z}}\bigg],$$
$$\langle\partial_z\overset{\leftrightarrow}{\boldsymbol{G}}_{{\perp}}\rangle|_{z=z'}\simeq \sum_n \frac{-\omega_L\beta_n}{2\epsilon_0A_n v_{g_n}}\bigg[\boldsymbol{\phi}_{n_t}(x,y)\hat{\boldsymbol{z}}\phi_{n_z}^{{\ast}}(x',y')+\phi_{n_z}(x,y)\hat{\boldsymbol{z}}\boldsymbol{\phi}_{n_t}(x',y')\bigg],$$
using the modal expansion. Here, $\omega _L$ is the frequency of the illuminating field. $\boldsymbol {\phi }_{n_t}$ and $\phi _{n_z}$ respectively represent the transverse and longitudinal components of the n-th mode of the waveguide which has the propagation constant of $\beta _n$, the group velocity of $v_{g_n}$, and surface mode of $A_n$. It should be noted that the sum is taken over the propagating modes and near cut-off evanescent modes in the above expressions.

Now by inserting Eq. (10) in Eq. (2), it can be shown that the polarizability of the particle in a guided structure can be written as:

$$\overset{\leftrightarrow}{\boldsymbol{\alpha}}=\alpha_0 \left[\hat{\boldsymbol{z}}\times\frac{-\overset{\leftrightarrow}{\boldsymbol{I}}+i\sum_n \gamma_n \boldsymbol{\phi}_{n_t} \boldsymbol{\phi}_{n_t} }{1-i\sum_n \gamma_n |\boldsymbol{\phi}_{n_t}|^{2}}\times \hat{\boldsymbol{z}} +\frac{\hat{\boldsymbol{z}}\hat{\boldsymbol{z}}}{1-i\sum_n \gamma_n |\phi_{n_z}|^{2}}\right]_{\boldsymbol{r}=\boldsymbol{r}'=\boldsymbol{r}_p},$$
where $\gamma _n=\omega _L \alpha _0/(2\epsilon _0 A_n v_{g_n})$. In the following, we will show that the backaction effect will be stronger as the magnitude of $\gamma _n$ becomes greater. Hence, we refer to $\gamma _n$ as the backaction parameter of the n-th mode. It is clear that $\gamma _n$ is inversely proportional to $v_{g_n}$. Hence, the backaction can be reinforced by reducing the group velocity of the guided modes. One other interesting point to observe is that while $\gamma _n$ is real for a lossless propagating mode, it becomes imaginary for an evanescent one. The consequences of such a difference are further investigated below.

3. Single mode waveguide

As seen in the previous section, when the particle is located inside a single mode optical waveguide, its polarizability can be approximated by

$$\overset{\leftrightarrow}{\boldsymbol{\alpha}}=\alpha_0 \left[\hat{\boldsymbol{z}}\times\frac{-\overset{\leftrightarrow}{\boldsymbol{I}}+i \gamma \boldsymbol{\phi}_{t} \boldsymbol{\phi}_{t} }{1-i \gamma |\boldsymbol{\phi}_{t}|^{2}}\times \hat{\boldsymbol{z}} +\frac{\hat{\boldsymbol{z}}\hat{\boldsymbol{z}}}{1-i \gamma |\phi_{z}|^{2}}\right],$$
where $\gamma =\omega _L\alpha _0/(2\epsilon _0 A v_g)$ is the backaction parameter, and $\boldsymbol {\phi }_t$ and $\phi _z$ are the transverse and longitudinal components of the normalized mode profile. Furthermore, $\boldsymbol {E}_0$ can also be written as:
$$\boldsymbol{E}_0(\boldsymbol{r})=E_0 \left(\boldsymbol{\phi}_t+\hat{\boldsymbol{z}} \phi_z \right)e^{i\beta z},$$
where, $E_0$ and $\beta$ are the amplitude and the propagation constant of the guided mode. Hence, the equivalent electric dipole moment of the particle is given by:
$$\boldsymbol{p}=\alpha_0 E_0 \bigg[\frac{\boldsymbol{\phi}_t}{1-i\gamma |\boldsymbol{\phi}_t|^{2}}+\frac{\phi_z \hat{\boldsymbol{z}}}{1-i\gamma |\phi_z|^{2}}\bigg]e^{i\beta z_p}.$$
Substitution of the equivalent electric dipole moment in the force terms in Eqs. (7) and (9) leads to
$$\boldsymbol{F}_0=\frac{ E_0^{2}}{4}\mathrm{Re} \bigg[\alpha_0\frac{\left(\boldsymbol{\nabla}_t-2i\beta^{{\ast}} \hat{\boldsymbol{z}}\right) |\boldsymbol{\phi}_t|^{2}}{1-i\gamma |\boldsymbol{\phi}_t|^{2}}+\alpha_0\frac{\left(\boldsymbol{\nabla}_t-2i\beta^{{\ast}} \hat{\boldsymbol{z}}\right) |\phi_z|^{2}}{1-i\gamma |\phi_z|^{2}}\bigg]e^{i(\beta-\beta^{{\ast}})z_p},$$
and
$$\begin{aligned} \boldsymbol{F}_{\mathrm{bs}}&=\frac{ E_0^{2} }{8}\mathrm{Im}[\alpha_0\gamma^{{\ast}}] \bigg[\frac{\boldsymbol{\nabla}_t |\boldsymbol{\phi}_t|^{4}}{\left|1+i\gamma |\boldsymbol{\phi}_t|^{2}\right|^{2}}+\frac{\boldsymbol{\nabla}_t |\phi_z|^{4}}{\left|1+i\gamma |\phi_z|^{2}\right|^{2}}\bigg]e^{i(\beta-\beta^{{\ast}})z_p}\\ &-E_0^{2}\hat{\boldsymbol{z}} \mathrm{Re}\left[\alpha_0\beta^{{\ast}}\gamma^{{\ast}}\right] \mathrm{Re}\bigg[\frac{|\boldsymbol{\phi}_t|^{2}|\phi_z|^{2}}{\left(1-i\gamma |\boldsymbol{\phi}_t|^{2}\right)\left(1+i\gamma^{{\ast}} |\phi_z|^{2}\right)}\bigg]e^{i(\beta-\beta^{{\ast}})z_p}.\end{aligned}$$
The total time-averaged force can then be obtained from the summation of $F_0$ and $F_{\mathrm {bs}}$.

In the following we will investigate in detail the cases of propagating and evanescent modes.

3.1 Propagating mode

Here, we want to investigate the case of a lossless propagating mode where $\beta$ and $\gamma$ are real. In such a case, $F_0$ and $F_{\mathrm {bs}}$ can be further simplified to

$$\boldsymbol{F}_0=\frac{\alpha_0 E_0^{2}}{4} \bigg[\frac{\boldsymbol{\nabla}_t |\boldsymbol{\phi}_t|^{2}}{1+\gamma^{2} |\boldsymbol{\phi}_t|^{4}}+\frac{\boldsymbol{\nabla}_t |\phi_z|^{2}}{1+\gamma^{2} |\phi_z|^{4}}\bigg]+\frac{\alpha_0 E_0^{2} \beta \gamma}{2}\hat{\boldsymbol{z}}\bigg[\frac{|\boldsymbol{\phi}_{t}|^{4}}{1+\gamma^{2} |\boldsymbol{\phi}_{t}|^{4}}+\frac{|\phi_z|^{4}}{1+\gamma^{2} |\phi_z|^{4}}\bigg],$$
and
$$\boldsymbol{F}_{\mathrm{bs}}={-}\alpha_0E_0^{2} \beta\gamma\hat{\boldsymbol{z}}|\boldsymbol{\phi}_t|^{2}|\phi_z|^{2} \frac{1+\gamma^{2} |\boldsymbol{\phi}_t|^{2} |\phi_z|^{2}}{\left(1+\gamma^{2} |\boldsymbol{\phi}_t|^{4}\right)\left(1+\gamma^{2} |\phi_z|^{4}\right)},$$
respectively. Therefore, the time-averaged total force exerted upon the particle will be given by
$$\boldsymbol{F}=\frac{\alpha_0 E_0^{2}}{4} \bigg[\frac{\boldsymbol{\nabla}_t |\boldsymbol{\phi}_t|^{2}}{1+\gamma^{2} |\boldsymbol{\phi}_t|^{4}}+\frac{\boldsymbol{\nabla}_t |\phi_z|^{2}}{1+\gamma^{2} |\phi_z|^{4}}\bigg]+\frac{\alpha_0 E_0^{2}}{2}\beta\gamma\hat{\boldsymbol{z}} \frac{\left(|\boldsymbol{\phi}_t|^{2}-|\phi_z|^{2}\right)^{2}}{\left(1+\gamma^{2} |\boldsymbol{\phi}_t|^{4}\right)\left(1+\gamma^{2} |\phi_z|^{4}\right)}.$$
From the above expression, it can be found out that $F_z$ has the same sign as $\gamma$. Hence, when $v_g$ is positive, $\gamma$ and $F_z$ will also be positive and the force acted upon the particle will be pushing. But, when $v_g$ is negative, $\gamma$ and $F_z$ will also be negative and the force acted upon the particle will be pulling.

The total force exerted upon the particle can be further simplified when $|\gamma |\ll 1$ which is hereafter referred to as the weak backaction regime:

$$\boldsymbol{F} \simeq \frac{\alpha_0 E_0^{2}}{4} \boldsymbol{\nabla}_t \left(|\boldsymbol{\phi}_t|^{2}+|\phi_z|^{2}\right)+\frac{\alpha_0 E_0^{2} \beta \gamma}{2}\hat{\boldsymbol{z}} \left(|\boldsymbol{\phi}_t|^{2}-|\phi_z|^{2}\right)^{2}.$$
It is obvious from this expression that the transverse force has been simplified to the well-known gradient force in the free space. But, the longitudinal force component, $F_z$, differs from the well-known scattering forces in the free space, given by
$$F_z^{\mathrm{(fs)}}=\frac{\beta E_0^{2}}{2}\mathrm{Im}[\alpha^{\mathrm{(fs)}}],$$
where $\alpha ^{\mathrm {(fs)}}=\alpha _0[1-ik_0^{3}\alpha _0/(6\pi \epsilon _0)]^{-1}$ is the polarizability of the particle in the free space.

Now we present some numerical examples to illustrate this difference. For the first example, assume a silica nanosphere with relative permittivity of 2.1, is located inside a rectangular waveguide which can be formed by PEC boundary conditions (Fig. 1) or a metallic cladding (Fig. 2) which is more realistic in the optical frequencies. The width and height of the waveguide are 750[nm] and 375[nm], respectively. The frequency of the incident field is 300[THz] at which the waveguide is single mode. The particle’s radius is small enough to ensure that we are in the Rayleigh regime. In Fig. 1(c), $F_z$ at the center of the waveguide is plotted as a function of the particle’s radius. For the case of the metallic waveguide, as seen in Fig. 2(c) the particle traps away from the center of the waveguide. Hence, we plot $F_z$ for a particle located near the cladding in Fig. 2(d). We have used three different methods for calculating $F_z$. The first is based on the proposed formulation that includes the backaction effect given in Eq. (21), which is referred to as the dipole approximation in non-free space. The second method is the dipole approximation formulation in the free-space that neglects the backaction effect. Finally, the last method is calculating the Maxwell stress tensor numerically by COMSOL Multiphysics software. As it can be easily seen in this figure, the non-free space dipole approximation perfectly coincides with the COMSOL results, while the results from the free space dipole approximation significantly differ from them.

 figure: Fig. 1.

Fig. 1. The scattering force exerted upon a silica nanosphere inside a rectangular waveguide with PEC boundary condition. (a) Schematic of the structure. (b) The electric field magnitude of the mode. (c) The longitudinal force component $F_z$ at the center of the waveguide is plotted as a function of the particle’s radius. The incident frequency is 300[THz], and the width and height of the waveguide are 750[nm] and 375[nm], respectively.

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 figure: Fig. 2.

Fig. 2. The scattering force exerted upon a silica nanosphere inside a rectangular waveguide with metallic cladding. (a) Schematic of the structure. (b) The electric field magnitude of the mode. (c) The transverse force component $F_y$ along the y-axis for a 20[nm] radius particle. (d) The longitudinal force component $F_z$ is plotted as a function of the particle’s radius. The incident frequency is 300[THz], and the width and height of the waveguide are 750[nm] and 375[nm], respectively.

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As another example, $F_z$ is plotted as a function of the particle’s radius inside a photonic crystal waveguide in Fig. 3. The waveguide is formed by missing a single rod in a $\mathrm {9\times 9}$ lattice of silicon rods ($\epsilon =12.7$). The radius of the rods is $0.2a$ where $a=\mathrm {380[nm]}$ is the lattice constant. The frequency of the incident field is 300[THz] at which the waveguide supports a single TM mode. According to the figure, the results obtained from the dipole approximation in non-free space coincides with the Maxwell stress tensor results while the results obtained by applying dipole approximation at free space do not. Hence, it can be concluded that the backaction effect cannot be neglected when calculating the scattering forces even at the weak regime ($|\gamma |\ll 1$).

 figure: Fig. 3.

Fig. 3. The scattering force exerted upon a silica nanosphere inside a photonic crystal waveguide. (a) Schematic of the structure. (b) The electric field magnitude of the mode. (c) The longitudinal force component $F_z$ at the center of the waveguide is plotted as a function of the particle’s radius. The lattice constant $a$ is 380[nm], and the radius of silicon rods are $0.2a$.

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Although the longitudinal force obtained in Eq. (21) differs from the scattering force in the free space, it still relates with the rate at which the energy is scattered by the particle. In Appendix D, we show that when the mode is TE and the waveguide is surrounded by PEC boundary conditions, the longitudinal force component can be written as

$$F_z=\frac{v_g}{c^{2}}W_s,$$
where $c$ is the speed of light and $W_s$ is the amount of power scattered by the particle, given by:
$$W_s=\frac{\alpha_0\gamma\beta c^{2}}{v_g}E_0^{2}|\boldsymbol{\phi_t}|^{4}.$$

Now, we want to study the transverse trapping of the particle beyond the weak regime in more detail. According to the transverse force component given in Eq. (20), we can define a trapping potential for the transverse motion of the particle as:

$$U(x_p,y_p)={-}\frac{\alpha_0 E_0^{2}}{4 \gamma}\Big[\mathrm{tan}^{{-}1}(\gamma |\boldsymbol{\phi}_t|^{2})+\mathrm{tan}^{{-}1}(\gamma |\phi_z|^{2})\Big].$$
The local intensity seen by the particle is also given by
$$I=\frac{E_0^{2}}{2\eta_0} \bigg[\frac{|\boldsymbol{\phi}_t|^{2}}{1+\gamma^{2} |\boldsymbol{\phi}_t|^{4}}+\frac{|\phi_z|^{2}}{1+\gamma^{2} |\phi_z|^{4}}\bigg].$$
As shown in Appendix E, the ratio of $\Delta U/I_{\mathrm {trap}}$ is an important quantity in investigating the stability of the trap, where $\Delta U$ is the potential depth, and $I_{\mathrm {trap}}$ is the local intensity at the trapping position. In general, the trapping position and also the ratio of $\Delta U/I_{\mathrm {trap}}$ depend on the mode profile. However, when the guided mode is TE ($\phi _z=0$), the particle will trap where $|\boldsymbol {\phi }_t|^{2}=1$, and the ratio of $\Delta U /I_{\mathrm {trap}}$ will be independent of the mode profile, given by
$$\frac{\Delta U}{I_{\mathrm{trap}}}=\frac{\alpha_0\eta_0}{2}\left(\frac{1}{\gamma}+\gamma\right)\mathrm{tan}^{{-}1}(\gamma).$$

In Fig. 4(a), the ratio of $\Delta U /I_{\mathrm {trap}}$ is plotted as a function of $\gamma$. It is clear that the ratio of $\Delta U /I_{\mathrm {trap}}$ is raised by increasing $\gamma$. This means that as backaction becomes stronger, the required $\Delta U$ for stable trapping (usually $10k_BT$) can be realized by applying lower local intensity. Hence, backaction can enable us to trap small nanoparticles without incurring thermal damage. Furthermore, the functionality of $\gamma$ is plotted in Fig. 4(b) as a function of the mode’s group velocity and the particle’s radius. It is clear that as $v_g$ is reduced, higher values of $\gamma$ can be reached, and thereby the backaction effect becomes stronger.

 figure: Fig. 4.

Fig. 4. (a) The ratio of $\Delta U /I_{\mathrm {trap}}$ as a function of $\gamma$ for a TE mode. (b) The backaction parameter ($\gamma$) versus $v_g$ and $R_p/\lambda$.

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In Fig. 5, the profile of the trapping potential is plotted for different values of $\gamma$ caused by a single TE mode inside the waveguide. According to the figure, in the weak backaction regime where $\gamma \ll 1$, the trapping potential has a linear function of $|\boldsymbol {\phi }_t|^{2}$. However, the shape of the trapping potential modifies by increasing $\gamma$ and approaches to a rectangular potential well in the strong backaction limit.

 figure: Fig. 5.

Fig. 5. The normalized profile of the trapping potential due to a single TE mode

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3.2 Evanescent mode

When the guided mode is evanescent, $\beta$ and $\gamma$ will be pure imaginary. Then, the expressions for $\boldsymbol {F}_0$ and $\boldsymbol {F}_{\mathrm {bs}}$ given in Eqs. (16) and (17) can be further simplified to

$$\begin{aligned}\boldsymbol{F}_0&=\frac{\alpha_0 E_0^{2}}{4}e^{{-}2\beta_iz_p} \bigg[\frac{\boldsymbol{\nabla}_t |\boldsymbol{\phi}_t|^{2}}{1+\gamma_i |\boldsymbol{\phi}_t|^{2}}+\frac{\boldsymbol{\nabla}_t |\phi_z|^{2}}{1+\gamma_i |\phi_z|^{2}}\bigg]\\ &-\frac{\alpha_0 E_0^{2} \beta_i}{2}e^{{-}2\beta_iz_p}\hat{\boldsymbol{z}}\bigg[\frac{|\boldsymbol{\phi}_{t}|^{2}}{1+\gamma_i |\boldsymbol{\phi}_{t}|^{2}}+\frac{|\phi_z|^{2}}{1+\gamma_i |\phi_z|^{2}}\bigg], \end{aligned}$$
and
$$\begin{aligned}\boldsymbol{F}_{\mathrm{bs}}&={-}\frac{\alpha_0 E_0^{2} \gamma_i}{8} e^{{-}2\beta_iz_p}\bigg[\frac{\boldsymbol{\nabla}_t |\boldsymbol{\phi}_t|^{4}}{(1+\gamma_i |\boldsymbol{\phi}_t|^{2})^{2}}+\frac{\boldsymbol{\nabla}_t |\phi_z|^{4}}{(1+\gamma_i |\phi_z|^{2})^{2}}\bigg]\\ &+\alpha_0E_0^{2} e^{{-}2\beta_iz_p} \hat{\boldsymbol{z}} \frac{\beta_i\gamma_i|\boldsymbol{\phi}_t|^{2}|\phi_z|^{2}}{\left(1+\gamma_i |\boldsymbol{\phi}_t|^{2}\right)\left(1+\gamma_i |\phi_z|^{2}\right)}. \end{aligned}$$
Here, $\gamma _i$ and $\beta _i$ are the imaginary part of $\gamma$ and $\beta$, respectively. It is worth noting that for the conventional guided modes $\gamma _i$ is negative. Then, the time-averaged total force exerted upon the particle can be obtained from the sum of $\boldsymbol {F}_0$ and $\boldsymbol {F}_{\mathrm {bs}}$:
$$\begin{aligned} \boldsymbol{F}&=\frac{\alpha_0 E_0^{2}}{4}e^{{-}2\beta_iz_p} \bigg[\frac{\boldsymbol{\nabla}_t|\boldsymbol{\phi}_t|^{2}}{(1+\gamma_i |\boldsymbol{\phi}_t|^{2})^{2}}+\frac{\boldsymbol{\nabla}_t|\phi_z|^{2}}{(1+\gamma_i|\phi_z|^{2})^{2}}\bigg]\\ &-\frac{\alpha_0 E_0^{2}}{2}\beta_i e^{{-}2\beta_iz_p} \hat{\boldsymbol{z}} \frac{|\boldsymbol{\phi}_t|^{2}+|\phi_z|^{2}}{\left(1+\gamma_i |\boldsymbol{\phi}_t|^{2}\right)\left(1+\gamma_i |\phi_z|^{2}\right)}.\end{aligned}$$
Now, according to the transverse component of the force, we can define a trapping potential for the transverse motion of the particle as:
$$U(x_p,y_p)=\frac{\alpha_0 E_0^{2}}{4\gamma_i}e^{{-}2\beta_iz_p}\bigg[\frac{1}{1+\gamma_i |\boldsymbol{\phi}_t|^{2}}+\frac{1}{1+\gamma_i|\phi_z|^{2}}\bigg].$$
The local intensity seen by the particle is also given by
$$I=\frac{E_0^{2}}{2\eta_0}e^{{-}2\beta_i z_p} \bigg[\frac{|\boldsymbol{\phi}_t|^{2}}{\left(1+\gamma_i |\boldsymbol{\phi}_t|^{2}\right)^{2}}+\frac{|\phi_z|^{2}}{\left(1+\gamma_i |\phi_z|^{2}\right)^{2}}\bigg].$$

According to Eq. (30), the denominator of the force becomes zero when $|\boldsymbol {\phi }_t|^{2}=-1/\gamma _i$ or $|\phi _z|^{2}=-1/\gamma _i$. In such a case, a strong optical force will be exerted upon the particle which stems from the resonance in the polarizability of the particle. Now, we want to investigate the case of the TE-mode ($\phi _z=0$) in more detail. Assume that $f_c$ is the cut-off frequency of the guided mode, and $f_r$ is the frequency at which $\gamma _i=-1$. In other words, $f_r$ is the lowest frequency at which the resonance in the polarizability and the force takes place. When the frequency is between $f_r$ and $f_c$, the potential depth tends to infinity. Nevertheless, the local intensity at the trap, $I_{\mathrm {trap}}=E_0^{2}e^{-2\beta _i z_p}/\left [2\eta _0(1+\gamma _i)^{2}\right ]$, will be finite. Hence, the ratio of $\Delta U / I_{\mathrm {trap}}$ also tends to infinity which will be advantageous for nondestructive trapping of small nanoparticles. The maximum stiffness can also be obtained near the frequency of $f_r$ where the resonance takes place at $|\boldsymbol {\phi }_t|^{2}=1$.

Now, we want to investigate the impact of the particle’s material loss on these resonances. If the particle has material loss the polarizability $\alpha _0$ becomes complex. Hereafter, $\alpha _0^{\prime }$ and $\alpha _0^{\prime \prime }$ represent the real and imaginary parts of $\alpha _0$, respectively. Since the backaction parameter $\gamma$ is proportional with $\alpha _0$, it also becomes complex as a result of the particle’s material loss. In such a case, the transverse and longitudinal components of the force for a TE mode will be given by

$$\boldsymbol{F}_t=\frac{\alpha_0^{\prime} E_0^{2}}{4}e^{{-}2\beta_iz_p}\frac{\boldsymbol{\nabla}_t|\boldsymbol{\phi}_t|^{2}}{\left(1+\gamma_i|\boldsymbol{\phi}_t|^{2}\right)^{2}+\gamma_r^{2}|\boldsymbol{\phi}_t|^{4}},$$
and
$$F_z={-}\frac{\beta_i E_0^{2}}{2}e^{{-}2\beta_iz_p}|\boldsymbol{\phi}_t|^{2} \frac{\alpha_0^{\prime}\left(1+\gamma_i|\boldsymbol{\phi}_t|^{2}\right)-\alpha_0^{\prime\prime}\gamma_r|\boldsymbol{\phi}_t|^{2}}{\left(1+\gamma_i|\boldsymbol{\phi}_t|^{2}\right)^{2}+\gamma_r^{2}|\boldsymbol{\phi}_t|^{4}},$$
respectively. Here, $\gamma _r$ represents the real part of $\gamma$. According to the above expressions, the force resonances will be damped as a result of the particle’s material loss.

Fig. 6 shows the transverse force component, $F_x$, inside a rectangular waveguide for three different frequencies below the cut-off frequency of the dominant mode. In Fig. 6(d), the frequency is less than $f_r$ in which no resonances can be seen in the force. In Fig. 6(e), the frequency is set at $f_r$. In such a case, $F_x$ resonates at $x=a/2$. Eventually, Fig. 6(f) shows the case where the frequency is higher than $f_r$. In such a case, two resonances can be seen in $F_x$. It is worth noting that the place of resonances moves away from the center of the waveguide as the frequency becomes closer to $f_c$. To discuss the origin of the force resonances further, we can use the transmission line model shown in Fig. 6(c). In the evanescent regime, the waveguide’s mode can be modeled by a transmission line with purely imaginary characteristic impedance $Z_0=i\chi _0$. The transmission line will be loaded by an impedance $Z_p=R_p+i\chi _p$ due to the presence of the particle. If the particle has no material loss, $R_p$ will be zero. When $2\chi _p=-\chi _0$, the reflection coefficient resonates and the energy scattered from the particle blows up that causes the resonance of the polarizability and the force exerted upon the particle.

 figure: Fig. 6.

Fig. 6. The transverse force exerted upon a nanosphere inside a PEC rectangular waveguide in the evanescent regime. (a) Schematic of the structure. (b) The electric field magnitude of the mode. (c) The equivalent transmission line model. (d-f): The transverse force component $F_x$ along the x-axis at three different frequencies. Here, we consider $R_p=100[nm]$, $\epsilon _p=2.25$, $a=750[nm]$ and $E_0e^{-\beta _i z_p}=10^{6} [V/m]$.

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 figure: Fig. 7.

Fig. 7. The longitudinal force exerted upon a nanosphere at the center of a PEC rectangular waveguide in the evanescent regime. (a) Schematic of the structure. (b) $F_z$ as a function of the particle’s radius. The incident frequency is 199.8[THz] at which the dominant mode is evanescent. The width and height of the waveguides are 750[nm] and 375[nm], respectively. Other parameters are $\epsilon _p=2.25$ and $E_0e^{-\beta _i z_p}=10^{6} [V/m]$

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Now, we want to investigate the z-component of the force exerted upon the particle at the trap. According to Eq. (34), $F_z$ can resonate at the trapping position when $\gamma _i=-1$. This resonance condition will be met for a specific radius of the particle that hereafter is referred to as $R_r$. The sign of $F_z$ differs for particles smaller than $R_r$ from the particles bigger than $R_r$, as seen in Fig. 7. For particles with radius bigger than $R_r$, $F_z$ is positive and thereby a pushing force will be exerted upon the particle. However, for particles smaller than $R_r$, $F_z$ is negative and thereby a pulling force will be exerted upon the particle. This feature can be used for sorting the particles which are smaller and bigger than $R_r$.

4. Conclusion

In conclusion, we have presented a general formulation for calculating the time-averaged force in optical waveguides, and thereby studied the backaction effects in optical waveguides at different regimes. We have shown that the backaction inversely relates to the group velocity of the guided modes, and thereby it becomes stronger by reducing the group velocity. We have further discussed the case of a single mode waveguide in detail, and shown that backaction can enable us to stably trap small nanoparticles without thermal damage by enhancing the ratio of $\Delta U / I_{\mathrm {trap}}$. We have also shown that the backaction can cause resonances in the polarizability of the particle at the evanescent regime thanks to which the particle can experience much stronger electromagnetic force.

Appendix A: Longitudinal and transverse Green’s function

In general, the dyadic Green’s function can be decomposed to a transverse $\overset {\leftrightarrow }{\boldsymbol {G}}_\perp$ and a longitudinal component $\overset {\leftrightarrow }{\boldsymbol {G}}_\parallel$

$$\overset{\leftrightarrow}{\boldsymbol{G}}=\overset{\leftrightarrow}{\boldsymbol{G}}_\parallel{+}\overset{\leftrightarrow}{\boldsymbol{G}}_\perp,$$
where $\boldsymbol {\nabla }\times \overset {\leftrightarrow }{\boldsymbol {G}}_\parallel =0$ and $\boldsymbol {\nabla }\cdot \overset {\leftrightarrow }{\boldsymbol {G}}_\perp =0$ [26–28]. It is worth nothing that $\overset {\leftrightarrow }{\boldsymbol {G}}_\perp$ and $\overset {\leftrightarrow }{\boldsymbol {G}}_\parallel$ are sometimes called the solenoidal and irrotational components of the Green’s function, respectively [28].

As another decomposition, the dyadic Green’s function can also be written as:

$$\overset{\leftrightarrow}{\boldsymbol{G}}=\overset{\leftrightarrow}{\boldsymbol{G}}_0+\overset{\leftrightarrow}{\boldsymbol{G}}_s.$$
The first term $\overset {\leftrightarrow }{\boldsymbol {G}}_0$ is the free space Green’s function, given by:
$$\overset{\leftrightarrow}{\boldsymbol{G}}_0=\Big[k_0^{2}\overset{\leftrightarrow}{\boldsymbol{I}}+\boldsymbol{\nabla}\boldsymbol{\nabla}\Big]g,$$
where
$$g(\boldsymbol{r},\boldsymbol{r}')=\frac{e^{ik_0|\boldsymbol{r}-\boldsymbol{r}'|}}{4\pi\epsilon_0|\boldsymbol{r}-\boldsymbol{r}'|}.$$
The second term $\overset {\leftrightarrow }{\boldsymbol {G}}_s$ represents the scattering part of the Green’s function [13]. Since $\boldsymbol {\nabla }\cdot \overset {\leftrightarrow }{\boldsymbol {G}}=\boldsymbol {\nabla }\cdot \overset {\leftrightarrow }{\boldsymbol {G}}_0=-\boldsymbol {\nabla }\delta (\boldsymbol {r}-\boldsymbol {r}')/\epsilon _0$, it can be concluded that $\overset {\leftrightarrow }{\boldsymbol {G}}_s$ is always transverse, i.e., $\boldsymbol {\nabla }\cdot \overset {\leftrightarrow }{\boldsymbol {G}}_s=0$. However, $\overset {\leftrightarrow }{\boldsymbol {G}}_0$ contains both transverse and longitudinal components. According to Eq. (37), $\overset {\leftrightarrow }{\boldsymbol {G}}_0$ can also be written as
$$\overset{\leftrightarrow}{\boldsymbol{G}}_0=\boldsymbol{\nabla}\boldsymbol{\nabla}g_0+\Big[k_0^{2}\overset{\leftrightarrow}{\boldsymbol{I}}g+\boldsymbol{\nabla}\boldsymbol{\nabla}(g-g_0)\Big],$$
where $g_0(\boldsymbol {r},\boldsymbol {r}')=|\boldsymbol {r}-\boldsymbol {r}'|^{-1}/4\pi \epsilon _0$ is the static scalar Green’s function in the free space. The first term in the above expression represents the longitudinal component of $\overset {\leftrightarrow }{\boldsymbol {G}}_0$:
$$\overset{\leftrightarrow}{\boldsymbol{G}}_{0\parallel}=\boldsymbol{\nabla}\boldsymbol{\nabla}g_0,$$
which is clearly curl free since $\boldsymbol {\nabla }\times \boldsymbol {\nabla }=0$. Then, the second term in Eq. (39) represents the transverse component of $\overset {\leftrightarrow }{\boldsymbol {G}}_0$:
$$\overset{\leftrightarrow}{\boldsymbol{G}}_{0\perp}=k_0^{2}\overset{\leftrightarrow}{\boldsymbol{I}}g+\boldsymbol{\nabla}\boldsymbol{\nabla}(g-g_0),$$
which can be shown that is divergence free as follows:
$$\begin{aligned}\boldsymbol{\nabla}\cdot\overset{\leftrightarrow}{\boldsymbol{G}}_{0\perp}&=\boldsymbol{\nabla}\Big(k_0^{2}g+\boldsymbol{\nabla}^{2}g-\boldsymbol{\nabla}^{2}g_0\Big)\\ &=\boldsymbol{\nabla}\Big(k_0^{2}g+\boldsymbol{\nabla}^{2}g+\frac{1}{\epsilon_0}\delta(\boldsymbol{r}-\boldsymbol{r}')\Big)=0, \end{aligned}$$
using $\boldsymbol {\nabla }^{2} g_0=-\delta (\boldsymbol {r}-\boldsymbol {r}')/\epsilon _0$ and the Helmholtz equation for $g$.

Hence, the longitudinal and transverse components of the Green’s function are given by

$$\overset{\leftrightarrow}{\boldsymbol{G}}_{{\parallel}}=\overset{\leftrightarrow}{\boldsymbol{G}}_{0\parallel}=\boldsymbol{\nabla}\boldsymbol{\nabla}g_0,$$
and
$$\begin{aligned}\overset{\leftrightarrow}{\boldsymbol{G}}_{{\perp}}&=\overset{\leftrightarrow}{\boldsymbol{G}}_{0\perp}+\overset{\leftrightarrow}{\boldsymbol{G}}_s\\ &=k_0^{2}\overset{\leftrightarrow}{\boldsymbol{I}}g+\boldsymbol{\nabla}\boldsymbol{\nabla}(g-g_0)+\overset{\leftrightarrow}{\boldsymbol{G}}_s, \end{aligned}$$
respectively. It is worth nothing that longitudinal component of the Green’s function equals the quasi-static part of $\overset {\leftrightarrow }{\boldsymbol {G}}_0$.

Appendix B: Dipole approximation

Assume a spherical particle with relative permittivity $\epsilon _p$ is illuminated by an electromagnetic field. The polarization field inside the particle can be written as the following integral equation:

$$\boldsymbol{P}(\boldsymbol{r})=\epsilon_0(\epsilon_p-1)\bigg[\boldsymbol{E}_0(\boldsymbol{r})+\boldsymbol{\nabla}\boldsymbol{\nabla}\cdot\int g_0(\boldsymbol{r},\boldsymbol{r}')\boldsymbol{P}(\boldsymbol{r}')d^{3}r'+\int \overset{\leftrightarrow}{\boldsymbol{G}}_{{\perp}}(\boldsymbol{r},\boldsymbol{r}')\cdot \boldsymbol{P}(\boldsymbol{r}')d^{3}r'\bigg].$$
Here, $\boldsymbol {E}_0$ is the electric field in the absence of the particle, and the second and third terms represent the longitudinal and transverse parts of the electric field scattered by the particle.

In the Rayleigh regime, where the radius of the particle is much smaller than the electromagnetic wavelength, the particle can be solely modeled by an equivalent electric dipole. In such a case, the distribution of $\boldsymbol {P}(\boldsymbol {r})$ will be almost uniform inside the spherical particle which can be obtained from:

$$\boldsymbol{P}(\boldsymbol{r}_p)=\epsilon_0(\epsilon_p-1)\bigg[\boldsymbol{E}_0(\boldsymbol{r}_p)-\frac{1}{3\epsilon_0}\boldsymbol{P}(\boldsymbol{r}_p)+\int \overset{\leftrightarrow}{\boldsymbol{G}}_{{\perp}}(\boldsymbol{r}_p,\boldsymbol{r}')d^{3}r' \boldsymbol{P}(\boldsymbol{r}_p) \bigg],$$
since
$$\boldsymbol{\nabla}\boldsymbol{\nabla}\int g_0(\boldsymbol{r},\boldsymbol{r}')d^{3}r'=\frac{-1}{3\epsilon_0}\overset{\leftrightarrow}{\boldsymbol{I}},$$
for a spherical particle [17]. Then, the equivalent electric dipole moment can be obtained from:
$$\boldsymbol{p}=V_p\boldsymbol{P}(\boldsymbol{r}_p)=\alpha_0 \bigg[\overset{\leftrightarrow}{\boldsymbol{I}}-\frac{1}{V_p}\int \overset{\leftrightarrow}{\boldsymbol{G}}_{{\perp}}(\boldsymbol{r}_p,\boldsymbol{r}')d^{3}r' \bigg]^{{-}1}\cdot \boldsymbol{E_0}(\boldsymbol{r}_p),$$
where $\alpha _0=3\epsilon _0 V_p (\epsilon _p-1)/(\epsilon _p+2)$ is the static polarizability of a spherical particle in the free space.

Appendix C: Modal expansion

Inside a uniform waveguide, the transverse Green’s function can be written as:

$$\overset{\leftrightarrow}{\boldsymbol{G}}_\perp(\boldsymbol{r},\boldsymbol{r}')= \omega_L^{2} \mu_0\sum_n \int \frac{c^{2}}{A_n(\beta)} \frac{\boldsymbol{\phi}_n(x,y,\beta)\boldsymbol{\phi}_n^{{\ast}}(x',y',\beta)}{\omega_n^{2}(\beta)-\omega_L^{2}}e^{i\beta (z-z')}\frac{d\beta}{2\pi},$$
by using the modal expansion [13,29]. Here, $\omega _L$ is the frequency of the illuminating field, $\boldsymbol {\phi }_n$ is the profile of the n-th mode of the waveguide, which obeys
$$\frac{1}{\epsilon(x,y)} \boldsymbol{\nabla}\times \boldsymbol{\nabla}\times \boldsymbol{\phi}_n(x,y,\beta)e^{i\beta z}= \frac{\omega_n^{2}(\beta)}{c^{2}} \boldsymbol{\phi}_n(x,y,\beta)e^{i\beta z},$$
and its amplitude is normalized in accordance with
$$\mathrm{max}\left[\epsilon(x,y) |\boldsymbol{\phi}_n(x,y,\beta)|^{2}\right]=1.$$
Furthermore, $\omega _n(\beta )$ is the angular frequency of the n-th mode whose propagation constant along z is $\beta$, and
$$A_n(\beta)=\int_A \epsilon(x,y)|\boldsymbol{\phi}_n(x,y,\beta)|^{2} dxdy,$$
is the mode surface of the n-th mode. The integral in Eq. (49) can be further simplified by applying the residue theorem:
$$\begin{aligned} \overset{\leftrightarrow}{\boldsymbol{G}}_\perp(\boldsymbol{r},\boldsymbol{r}')&=\frac{i\omega_L}{2\epsilon_0}\sum_n \frac{\boldsymbol{\phi}_n(x,y,\beta_n)\boldsymbol{\phi}_n^{{\ast}}(x',y',\beta_n)}{A_n(\beta_n)v_{g_n}}e^{i\beta_n (z-z')}\mathcal{U}(z-z')\\ &+\frac{i\omega_L}{2\epsilon_0}\sum_n \frac{\boldsymbol{\phi}_n(x,y,-\beta_n)\boldsymbol{\phi}_n^{{\ast}}(x',y',-\beta_n)}{A_n(\beta_n) v_{g_n}}e^{{-}i\beta_n (z-z')}\mathcal{U}(z'-z).\end{aligned}$$
Here, $\beta _n$ is the propagating constant of the n-th mode that satisfies $\omega _n(\beta _n)=\omega _L$, and $v_{g_n}=\frac {\partial \omega }{\partial \beta }|_{\beta _n}$ is its group velocity. It is also noteworthy that according to the parity symmetry, the transverse and longitudinal components of the guided modes obey the following relations:
$$\boldsymbol{\phi}_t(x,y,-\beta_n)=\boldsymbol{\phi}_t(x,y,\beta_n),$$
$$\phi_z(x,y,-\beta_n)={-}\phi_z(x,y,\beta_n).$$
It should be noted that $\overset {\leftrightarrow }{\boldsymbol {G}}_\perp$ is singular at $\boldsymbol {r}=\boldsymbol {r}'$ since the sum in Eq. (53) does not converge [27]. However, the averaged of $\overset {\leftrightarrow }{\boldsymbol {G}}_\perp$ and its derivatives inside the particle are not singular, and can be approximated by:
$$\langle\overset{\leftrightarrow}{\boldsymbol{G}}_{{\perp}}\rangle|_{z=z'}\simeq \sum_n \frac{i\omega_L}{2\epsilon_0 A_n v_{g_n}} \bigg[\boldsymbol{\phi}_{n_t}(x,y)\boldsymbol{\phi}_{n_t}(x',y')+\phi_{n_z}(x,y)\phi_{n_z}^{{\ast}}(x',y')\hat{\boldsymbol{z}}\hat{\boldsymbol{z}}\bigg],$$
$$\langle\partial_z\overset{\leftrightarrow}{\boldsymbol{G}}_{{\perp}}\rangle|_{z=z'}\simeq \sum_n \frac{-\omega_L\beta_n}{2\epsilon_0A_n v_{g_n}}\bigg[\boldsymbol{\phi}_{n_t}(x,y)\hat{\boldsymbol{z}}\phi_{n_z}^{{\ast}}(x',y')+\phi_{n_z}(x,y)\hat{\boldsymbol{z}}\boldsymbol{\phi}_{n_t}(x',y')\bigg],$$
where the sum is taken over the propagating modes and evanescent ones which are near cut-off.

Appendix D: Power scattered by the particle

If the waveguide supports a single propagating TE mode, the electric field scattered by the particle can be approximated by

$$\boldsymbol{E}_s=\overset{\leftrightarrow}{\boldsymbol{G}}_s\cdot \boldsymbol{p}\simeq \begin{cases}i\gamma E_0|\boldsymbol{\phi}_t(x_p,y_p)|^{2} \boldsymbol{\phi}_t(x,y) e^{i\beta z} & z>z_p \\ i\gamma E_0|\boldsymbol{\phi}_t(x_p,y_p)|^{2} \boldsymbol{\phi}_t(x,y) e^{i\beta (2z_p-z)} & z<z_p\end{cases},$$
in the weak backaction regime ($|\gamma |\ll 1$). Hence, the power scattered by the particle can be written as:
$$W_s=2\gamma^{2} |\boldsymbol{\phi}_t(x_p,y_p)|^{4} W_0,$$
in which $W_0$ is the incident power, and is given by
$$W_0=\iint \frac{1}{2} \mathrm{Re}\left[\boldsymbol{E}_0 \times \boldsymbol{H}_0^{{\ast}} \right] \cdot \hat{\boldsymbol{z}}dxdy= \frac{\beta}{2\omega_L \mu_0}E_0^{2} \iint |\boldsymbol{\phi}_t(x,y)|^{2} dxdy.$$
If the waveguide is hollow core surrounded by PEC boundary conditions, the surface integral of $\iint |\boldsymbol {\phi }_t(x,y)|^{2} dxdy$ equals the mode surface $A$. In such a case, the power scattered by the particle can be obtained from:
$$W_s=\frac{\gamma^{2}\beta A}{\omega_L\mu_0}E_0^{2} |\boldsymbol{\phi}_t(x_p,y_p)|^{4} =\frac{c^{2}}{2v_g}\alpha_0\beta\gamma E_0^{2} |\boldsymbol{\phi}_t(x_p,y_p)|^{4}.$$

Appendix E: Stability of the trap

Assume a Brownian particle is trapped inside a potential well with depth $\Delta U$. Then, the escape rate of the particle from the well under the overdamping conditions is proportional to $\exp (-\Delta U/k_B T)$ where $k_B$ is the Boltzmann constant and $T$ is the temperature of the ambient, and is referred to as the Kramer’s escape rate [30]. Hence, the tapping time will be increased by increasing $\Delta U$, and a standard of $\Delta U \ge 10k_BT$ is usually considered as a stable trap [4].

In the free space, the trapping force exerted upon a Rayleigh particle is given by

$$\boldsymbol{F}=\frac{1}{4}\alpha_0\boldsymbol{\nabla}|\boldsymbol{E}_0|^{2},$$
that leads to a trapping potential as:
$$U(\boldsymbol{r}_p)={-}\frac{1}{2}\alpha_0\eta_0I(\boldsymbol{r}_p),$$
where $I=|\boldsymbol {E}_0^{2}|/2\eta _0$ is the intensity of the incident field, and $\eta _0$ is the characteristic impedance of the free space. Hence, the ratio of the potential depth $\Delta U$ to the local intensity at the trapping position $I_{\mathrm {trap}}$ equals
$$\frac{\Delta U}{I_{\mathrm{trap}}}=\frac{1}{2}\alpha_0 \eta_0,$$
which is independent of the incident field’s profile. Since $\alpha _0$ is proportional to $R_p^{3}$, the local intensity required to generate $10k_BT$ potential depth grows rapidly by reducing the size of the particle. As shown in this paper, the backaction can help us to increase the ratio of $\Delta U/I_{\mathrm {trap}}$ and thereby reduces the local intensity required for the generation of $10k_BT$ potential depth.

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Figures (7)
Fig. 1.
Fig. 1. The scattering force exerted upon a silica nanosphere inside a rectangular waveguide with PEC boundary condition. (a) Schematic of the structure. (b) The electric field magnitude of the mode. (c) The longitudinal force component $F_z$ at the center of the waveguide is plotted as a function of the particle’s radius. The incident frequency is 300[THz], and the width and height of the waveguide are 750[nm] and 375[nm], respectively.

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Fig. 2.
Fig. 2. The scattering force exerted upon a silica nanosphere inside a rectangular waveguide with metallic cladding. (a) Schematic of the structure. (b) The electric field magnitude of the mode. (c) The transverse force component $F_y$ along the y-axis for a 20[nm] radius particle. (d) The longitudinal force component $F_z$ is plotted as a function of the particle’s radius. The incident frequency is 300[THz], and the width and height of the waveguide are 750[nm] and 375[nm], respectively.

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Fig. 3.
Fig. 3. The scattering force exerted upon a silica nanosphere inside a photonic crystal waveguide. (a) Schematic of the structure. (b) The electric field magnitude of the mode. (c) The longitudinal force component $F_z$ at the center of the waveguide is plotted as a function of the particle’s radius. The lattice constant $a$ is 380[nm], and the radius of silicon rods are $0.2a$.

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Fig. 4.
Fig. 4. (a) The ratio of $\Delta U /I_{\mathrm {trap}}$ as a function of $\gamma$ for a TE mode. (b) The backaction parameter ($\gamma$) versus $v_g$ and $R_p/\lambda$.

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Fig. 5.
Fig. 5. The normalized profile of the trapping potential due to a single TE mode

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Fig. 6.
Fig. 6. The transverse force exerted upon a nanosphere inside a PEC rectangular waveguide in the evanescent regime. (a) Schematic of the structure. (b) The electric field magnitude of the mode. (c) The equivalent transmission line model. (d-f): The transverse force component $F_x$ along the x-axis at three different frequencies. Here, we consider $R_p=100[nm]$, $\epsilon _p=2.25$, $a=750[nm]$ and $E_0e^{-\beta _i z_p}=10^{6} [V/m]$.

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Fig. 7.
Fig. 7. The longitudinal force exerted upon a nanosphere at the center of a PEC rectangular waveguide in the evanescent regime. (a) Schematic of the structure. (b) $F_z$ as a function of the particle’s radius. The incident frequency is 199.8[THz] at which the dominant mode is evanescent. The width and height of the waveguides are 750[nm] and 375[nm], respectively. Other parameters are $\epsilon _p=2.25$ and $E_0e^{-\beta _i z_p}=10^{6} [V/m]$

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Equations (64)

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p = α ↔ ⋅ E 0 ,
α ↔ = α 0 [ I ↔ − α 0 ⟨ G ↔ ⊥ ( r p , r p ) ⟩ ] − 1 ,
⟨ G ↔ ⊥ ( r p , r p ) ⟩ = ⟨ G ↔ 0 ⊥ ( r p , r p ) ⟩ + ⟨ G ↔ s ( r p , r p ) ⟩ = i k 0 3 6 π ϵ 0 I ↔ + 1 V p ∫ V p G ↔ s ( r p , r ′ ) d 3 r ′ ,
F = 1 2 R e [ ∑ i p i ∇ E i ∗ ] r = r p ,
E ( r , r p ) = E 0 ( r ) + G ↔ s ( r , r p ) ⋅ p ( r p ) ,
F = F 0 + F b s .
F 0 = 1 2 R e [ ∑ i p i ∇ E 0 i ∗ ] r = r p .
F b s = 1 2 R e [ ∑ i , j p i p j ∗ ⟨ ∇ G s i j ∗ ⟩ ] r = r p ,
F b s = 1 2 R e [ ∑ i , j p i p j ∗ ⟨ ∇ G ⊥ i j ∗ ⟩ ] r = r p ,
⟨ G ↔ ⊥ ⟩ | z = z ′ ≃ ∑ n i ω L 2 ϵ 0 A n v g n [ ϕ n t ( x , y ) ϕ n t ( x ′ , y ′ ) + ϕ n z ( x , y ) ϕ n z ∗ ( x ′ , y ′ ) z ^ z ^ ] ,
⟨ ∂ z G ↔ ⊥ ⟩ | z = z ′ ≃ ∑ n − ω L β n 2 ϵ 0 A n v g n [ ϕ n t ( x , y ) z ^ ϕ n z ∗ ( x ′ , y ′ ) + ϕ n z ( x , y ) z ^ ϕ n t ( x ′ , y ′ ) ] ,
α ↔ = α 0 [ z ^ × − I ↔ + i ∑ n γ n ϕ n t ϕ n t 1 − i ∑ n γ n | ϕ n t | 2 × z ^ + z ^ z ^ 1 − i ∑ n γ n | ϕ n z | 2 ] r = r ′ = r p ,
α ↔ = α 0 [ z ^ × − I ↔ + i γ ϕ t ϕ t 1 − i γ | ϕ t | 2 × z ^ + z ^ z ^ 1 − i γ | ϕ z | 2 ] ,
E 0 ( r ) = E 0 ( ϕ t + z ^ ϕ z ) e i β z ,
p = α 0 E 0 [ ϕ t 1 − i γ | ϕ t | 2 + ϕ z z ^ 1 − i γ | ϕ z | 2 ] e i β z p .
F 0 = E 0 2 4 R e [ α 0 ( ∇ t − 2 i β ∗ z ^ ) | ϕ t | 2 1 − i γ | ϕ t | 2 + α 0 ( ∇ t − 2 i β ∗ z ^ ) | ϕ z | 2 1 − i γ | ϕ z | 2 ] e i ( β − β ∗ ) z p ,
F b s = E 0 2 8 I m [ α 0 γ ∗ ] [ ∇ t | ϕ t | 4 | 1 + i γ | ϕ t | 2 | 2 + ∇ t | ϕ z | 4 | 1 + i γ | ϕ z | 2 | 2 ] e i ( β − β ∗ ) z p − E 0 2 z ^ R e [ α 0 β ∗ γ ∗ ] R e [ | ϕ t | 2 | ϕ z | 2 ( 1 − i γ | ϕ t | 2 ) ( 1 + i γ ∗ | ϕ z | 2 ) ] e i ( β − β ∗ ) z p .
F 0 = α 0 E 0 2 4 [ ∇ t | ϕ t | 2 1 + γ 2 | ϕ t | 4 + ∇ t | ϕ z | 2 1 + γ 2 | ϕ z | 4 ] + α 0 E 0 2 β γ 2 z ^ [ | ϕ t | 4 1 + γ 2 | ϕ t | 4 + | ϕ z | 4 1 + γ 2 | ϕ z | 4 ] ,
F b s = − α 0 E 0 2 β γ z ^ | ϕ t | 2 | ϕ z | 2 1 + γ 2 | ϕ t | 2 | ϕ z | 2 ( 1 + γ 2 | ϕ t | 4 ) ( 1 + γ 2 | ϕ z | 4 ) ,
F = α 0 E 0 2 4 [ ∇ t | ϕ t | 2 1 + γ 2 | ϕ t | 4 + ∇ t | ϕ z | 2 1 + γ 2 | ϕ z | 4 ] + α 0 E 0 2 2 β γ z ^ ( | ϕ t | 2 − | ϕ z | 2 ) 2 ( 1 + γ 2 | ϕ t | 4 ) ( 1 + γ 2 | ϕ z | 4 ) .
F ≃ α 0 E 0 2 4 ∇ t ( | ϕ t | 2 + | ϕ z | 2 ) + α 0 E 0 2 β γ 2 z ^ ( | ϕ t | 2 − | ϕ z | 2 ) 2 .
F z ( f s ) = β E 0 2 2 I m [ α ( f s ) ] ,
F z = v g c 2 W s ,
W s = α 0 γ β c 2 v g E 0 2 | ϕ t | 4 .
U ( x p , y p ) = − α 0 E 0 2 4 γ [ t a n − 1 ( γ | ϕ t | 2 ) + t a n − 1 ( γ | ϕ z | 2 ) ] .
I = E 0 2 2 η 0 [ | ϕ t | 2 1 + γ 2 | ϕ t | 4 + | ϕ z | 2 1 + γ 2 | ϕ z | 4 ] .
Δ U I t r a p = α 0 η 0 2 ( 1 γ + γ ) t a n − 1 ( γ ) .
F 0 = α 0 E 0 2 4 e − 2 β i z p [ ∇ t | ϕ t | 2 1 + γ i | ϕ t | 2 + ∇ t | ϕ z | 2 1 + γ i | ϕ z | 2 ] − α 0 E 0 2 β i 2 e − 2 β i z p z ^ [ | ϕ t | 2 1 + γ i | ϕ t | 2 + | ϕ z | 2 1 + γ i | ϕ z | 2 ] ,
F b s = − α 0 E 0 2 γ i 8 e − 2 β i z p [ ∇ t | ϕ t | 4 ( 1 + γ i | ϕ t | 2 ) 2 + ∇ t | ϕ z | 4 ( 1 + γ i | ϕ z | 2 ) 2 ] + α 0 E 0 2 e − 2 β i z p z ^ β i γ i | ϕ t | 2 | ϕ z | 2 ( 1 + γ i | ϕ t | 2 ) ( 1 + γ i | ϕ z | 2 ) .
F = α 0 E 0 2 4 e − 2 β i z p [ ∇ t | ϕ t | 2 ( 1 + γ i | ϕ t | 2 ) 2 + ∇ t | ϕ z | 2 ( 1 + γ i | ϕ z | 2 ) 2 ] − α 0 E 0 2 2 β i e − 2 β i z p z ^ | ϕ t | 2 + | ϕ z | 2 ( 1 + γ i | ϕ t | 2 ) ( 1 + γ i | ϕ z | 2 ) .
U ( x p , y p ) = α 0 E 0 2 4 γ i e − 2 β i z p [ 1 1 + γ i | ϕ t | 2 + 1 1 + γ i | ϕ z | 2 ] .
I = E 0 2 2 η 0 e − 2 β i z p [ | ϕ t | 2 ( 1 + γ i | ϕ t | 2 ) 2 + | ϕ z | 2 ( 1 + γ i | ϕ z | 2 ) 2 ] .
F t = α 0 ′ E 0 2 4 e − 2 β i z p ∇ t | ϕ t | 2 ( 1 + γ i | ϕ t | 2 ) 2 + γ r 2 | ϕ t | 4 ,
F z = − β i E 0 2 2 e − 2 β i z p | ϕ t | 2 α 0 ′ ( 1 + γ i | ϕ t | 2 ) − α 0 ′ ′ γ r | ϕ t | 2 ( 1 + γ i | ϕ t | 2 ) 2 + γ r 2 | ϕ t | 4 ,
G ↔ = G ↔ ∥ + G ↔ ⊥ ,
G ↔ = G ↔ 0 + G ↔ s .
G ↔ 0 = [ k 0 2 I ↔ + ∇ ∇ ] g ,
g ( r , r ′ ) = e i k 0 | r − r ′ | 4 π ϵ 0 | r − r ′ | .
G ↔ 0 = ∇ ∇ g 0 + [ k 0 2 I ↔ g + ∇ ∇ ( g − g 0 ) ] ,
G ↔ 0 ∥ = ∇ ∇ g 0 ,
G ↔ 0 ⊥ = k 0 2 I ↔ g + ∇ ∇ ( g − g 0 ) ,
∇ ⋅ G ↔ 0 ⊥ = ∇ ( k 0 2 g + ∇ 2 g − ∇ 2 g 0 ) = ∇ ( k 0 2 g + ∇ 2 g + 1 ϵ 0 δ ( r − r ′ ) ) = 0 ,
G ↔ ∥ = G ↔ 0 ∥ = ∇ ∇ g 0 ,
G ↔ ⊥ = G ↔ 0 ⊥ + G ↔ s = k 0 2 I ↔ g + ∇ ∇ ( g − g 0 ) + G ↔ s ,
P ( r ) = ϵ 0 ( ϵ p − 1 ) [ E 0 ( r ) + ∇ ∇ ⋅ ∫ g 0 ( r , r ′ ) P ( r ′ ) d 3 r ′ + ∫ G ↔ ⊥ ( r , r ′ ) ⋅ P ( r ′ ) d 3 r ′ ] .
P ( r p ) = ϵ 0 ( ϵ p − 1 ) [ E 0 ( r p ) − 1 3 ϵ 0 P ( r p ) + ∫ G ↔ ⊥ ( r p , r ′ ) d 3 r ′ P ( r p ) ] ,
∇ ∇ ∫ g 0 ( r , r ′ ) d 3 r ′ = − 1 3 ϵ 0 I ↔ ,
p = V p P ( r p ) = α 0 [ I ↔ − 1 V p ∫ G ↔ ⊥ ( r p , r ′ ) d 3 r ′ ] − 1 ⋅ E 0 ( r p ) ,
G ↔ ⊥ ( r , r ′ ) = ω L 2 μ 0 ∑ n ∫ c 2 A n ( β ) ϕ n ( x , y , β ) ϕ n ∗ ( x ′ , y ′ , β ) ω n 2 ( β ) − ω L 2 e i β ( z − z ′ ) d β 2 π ,
1 ϵ ( x , y ) ∇ × ∇ × ϕ n ( x , y , β ) e i β z = ω n 2 ( β ) c 2 ϕ n ( x , y , β ) e i β z ,
m a x [ ϵ ( x , y ) | ϕ n ( x , y , β ) | 2 ] = 1.
A n ( β ) = ∫ A ϵ ( x , y ) | ϕ n ( x , y , β ) | 2 d x d y ,
G ↔ ⊥ ( r , r ′ ) = i ω L 2 ϵ 0 ∑ n ϕ n ( x , y , β n ) ϕ n ∗ ( x ′ , y ′ , β n ) A n ( β n ) v g n e i β n ( z − z ′ ) U ( z − z ′ ) + i ω L 2 ϵ 0 ∑ n ϕ n ( x , y , − β n ) ϕ n ∗ ( x ′ , y ′ , − β n ) A n ( β n ) v g n e − i β n ( z − z ′ ) U ( z ′ − z ) .
ϕ t ( x , y , − β n ) = ϕ t ( x , y , β n ) ,
ϕ z ( x , y , − β n ) = − ϕ z ( x , y , β n ) .
⟨ G ↔ ⊥ ⟩ | z = z ′ ≃ ∑ n i ω L 2 ϵ 0 A n v g n [ ϕ n t ( x , y ) ϕ n t ( x ′ , y ′ ) + ϕ n z ( x , y ) ϕ n z ∗ ( x ′ , y ′ ) z ^ z ^ ] ,
⟨ ∂ z G ↔ ⊥ ⟩ | z = z ′ ≃ ∑ n − ω L β n 2 ϵ 0 A n v g n [ ϕ n t ( x , y ) z ^ ϕ n z ∗ ( x ′ , y ′ ) + ϕ n z ( x , y ) z ^ ϕ n t ( x ′ , y ′ ) ] ,
E s = G ↔ s ⋅ p ≃ { i γ E 0 | ϕ t ( x p , y p ) | 2 ϕ t ( x , y ) e i β z z > z p i γ E 0 | ϕ t ( x p , y p ) | 2 ϕ t ( x , y ) e i β ( 2 z p − z ) z < z p ,
W s = 2 γ 2 | ϕ t ( x p , y p ) | 4 W 0 ,
W 0 = ∬ 1 2 R e [ E 0 × H 0 ∗ ] ⋅ z ^ d x d y = β 2 ω L μ 0 E 0 2 ∬ | ϕ t ( x , y ) | 2 d x d y .
W s = γ 2 β A ω L μ 0 E 0 2 | ϕ t ( x p , y p ) | 4 = c 2 2 v g α 0 β γ E 0 2 | ϕ t ( x p , y p ) | 4 .
F = 1 4 α 0 ∇ | E 0 | 2 ,
U ( r p ) = − 1 2 α 0 η 0 I ( r p ) ,
Δ U I t r a p = 1 2 α 0 η 0 ,
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