Dynamics of angular momentum-torque conversion in silicon waveguides

Wenjia Li; Jianlong Liu; Yang Gao; Keya Zhou; Shutian Liu

doi:10.1364/OE.27.010208

1. Introduction

The interaction between light and matter brings about the transfer of angular momenta (AM) of light which induces an optical torque (OT) exerted on an illuminated object. The OT can be generated by light beams with the AM [1–5] or propagating in anisotropic media [6–9]. The OT has been applied to optical tweezers [10,11], integrated light devices [12–18] and microscopic machines [19–21]. Recently, there have been growing interests in the relationship between the AM and OT [22,23] which is beneficial to understand the essence of the interaction between light and matter. The AM for a light beam about its axis or propagation direction can be divided into the spin angular momentum (SAM) and the orbital angular momentum (OAM) [24], which is associated with the polarization states and the spatial structure of light field, respectively. The spin-orbit interaction (SOI) plays an important role in modern optics that is inevitable in inhomogeneous or anisotropic media [25]. The SOI results in the conversion between the SAM and OAM. To date, various methods have been presented to achieve the SOI such as uniaxial birefringent crystals [26,27], biaxial birefringent crystals [28], subwavelength metallic structures [29–31], strongly focused circularly polarized beams [32,33] and so on.

In the last decade, the optical AM conservation has attracted considerable attention in free-space [24,34], crystals [26–28], dipoles [35], waveguides [22,36] and so on. For the pure free-space light propagation, the local conservation of the SAM and OAM was derived in [34] and their theory is gauge invariant but not Lorentz covariant. On this basis, the electromagnetic spin conservation law in a lossy material was derived in [23], which shows the spin torque corresponding to the photonic spin absorption. It has been reported that the AM is conserved along the optical axis in uniaxial crystals [26,27] and biaxial crystals [28]. The SAM and OAM contributions to the OT are indistinguishable in the biaxial crystals [28], which indicate that the OT is correspond to the change of TAM. The respective conservation laws of the SAM and OAM for the scattering of arbitrary wave fields by a small particle was reported in [35], which hints the different parts of the torque from SAM and OAM exerted by the field on the particle. The OT exerted by rotating waveguide eigenmodes on a guiding structure was investigated in [36] that contains a dielectric cylinder bounded by a perfectly electric conducting waveguide. The torque was understood in terms of the AM flux flowing in the waveguide. However, to the best of our knowledge, relevant works have not yet been reported much about the relationship between the OT and the optical AM in the dielectric waveguide. In [22], the relationship of the OT and SAM in a birefringent dielectric waveguide was studied in theory and experiment but they neglected the OAM. Here, we give a refined analysis on the AM-OT conversion where the SAM and OAM coexist and both contribute to the OT.

In this paper, we present a general description of the relationship between OT and TAM in a birefringent dielectric waveguide through theory and simulation. We choose a freestanding silicon waveguide which is suspended from a substrate [37–39] for our analysis. Compared with the waveguide adhered on a substrate, the freestanding silicon waveguide has more degree of freedom in mechanical effects, e.g. torsion or vibration. In order to better understand the optical the AM-induced OT effect, we investigate the evolution rules of OT, TAM, SAM and OAM along the propagation direction in the waveguide by analyzing the waveguide modes with the vector angular spectrum [40,41] of the electromagnetic field. Our results show that the OT, SAM and OAM coexist and vary synchronously in the interaction between light and dielectric waveguide. In addition, we demonstrate that the ratio between the OAM and TAM in the propagation is highly dependent on the incident wavelength and the size of the waveguide. In integrated photonic devices, the photon AM-induced mechanical effects are ubiquitous due to the interaction between light and matter. As a result, we propose a three-layer waveguide structure to convert light chirality and generate a higher OT. Such structure might be used as a torque generator in integrated photonic circuits.

2. Theory and structure

We consider a birefringent silicon waveguide shown in Fig. 1(a) which supports two different orthogonal modes, transverse electric (TE) and transverse magnetic (TM) modes. The width w and height h of the waveguide can be chosen so that only two fundamental propagation modes are supported. The effective refractive indices of the TE-like and TM-like modes are n₁ and n₂, respectively. The difference of the refractive indices of two orthogonal modes is $Δ n$ = n₂ $-$ n₁. Two orthogonal mode sources with $π / 2$ of phase difference are normally incident on the silicon waveguide that is equivalent to a circularly polarized light corresponding to the SAM. We assume the incident light to be left circularly polarized that propagates along z direction. The TAM of the light field is the sum of the SAM and OAM and the change of TAM leads to the OT exerted on the waveguide. The periodic variation of the TAM leads to the periodic variation of the OT in the birefringent silicon waveguide, which is indicated in Fig. 1(b).

Fig. 1 (a) The three-dimensional schematic of the birefringent silicon waveguide. The width and height of the waveguide are w and h, respectively. The red rotatory arrows represent SAM and the green belts represent OAM. (b) Longitudinal AM densities and longitudinal torque density vary periodically along the waveguide. TAM (black line) is the sum of OAM (blue line) and SAM (red line). The torque density is equal to the negative derivative of AM flux density.

Download Full Size | PDF

The interaction between light and waveguide causes the transfer of TAM between light and waveguide that generate an OT. The conservation law for the TAM in the whole system consisting of light and waveguide can be expressed as [24,34,35]

\frac{\partial J}{\partial t} + \nabla \cdot \hat{σ} = - Γ,

where J is the AM density of the light field, the tensor

\hat{σ}

is the AM flux density of the light field and

Γ

is the OT density exerted on the waveguide. In the case of a silicon waveguide, the longitudinal components of the AM and OT are significant. For an incident monochromatic wave, we consider the longitudinal component of the integration of Eq. (1) over an x-y surface and take the time average,

〈 Γ_{z} 〉 = - \frac{\partial}{\partial z} 〈 σ_{z} 〉,

where

Γ_{z}

is the z component of the OT density and

σ_{z}

is the longitudinal component of the AM flux density indicating the longitudinal component of the AM crossing the x-y surface in unit time. The time-averaged longitudinal AM flux density (

σ_{z}

) is defined as

〈 σ_{z} 〉 = \frac{P}{ℏ ω} 〈 j_{z} 〉,

where P is the electromagnetic field energy flux,

ℏ ω

is single photon energy and

j_{z}

is the longitudinal TAM of a single photon. By considering the longitudinal TAM density (

J_{z}

) that is the integration of

j_{z}

over the surface in x-y plane along the z direction,

〈 σ_{z} 〉

can be written as

〈 σ_{z} 〉 = \frac{P}{W} 〈 J_{z} 〉 = v_{g} 〈 J_{z} 〉,

where W is the time-averaged electromagnetic field energy of the surface. P/W can be understood as the energy velocity

v_{g}

. Thus the relationship of the OT and the TAM crossing each x-y surface in the silicon waveguide can be expressed as

〈 Γ_{z} 〉 = - v_{g} \frac{\partial}{\partial z} 〈 J_{z} 〉 .

It indicates that the torque density is equal to the negative derivative of AM flux density.

The longitudinal OAM is generated at the moment of the circularly polarized light contacting the waveguide due to the SOI, whose existence is independent of the shape of the waveguide. The longitudinal TAM density is the sum of the longitudinal SAM and OAM densities. According to the definition of the SAM and OAM, the SAM density and the OAM density crossing a surface in x-y plane along the z direction can be expressed as [23,42]

S = \frac{ε_{0}}{2 i ω} \iint (E^{*} \times E) d x d y + \frac{μ_{0}}{2 i ω} \iint (H^{*} \times H) d x d y,

L = \frac{ε_{0}}{2 i ω} \sum_{σ = x, y, z} \iint E_{σ}^{*} (r \times \nabla) E_{σ} d x d y + \frac{μ_{0}}{2 i ω} \sum_{σ = x, y, z} \iint H_{σ}^{*} (r \times \nabla) H_{σ} d x d y .

The transverse components of the electric and magnetic fields ( $E {(H)}_{x (y)}^{TE(M)}$ ) can be real and the longitudinal components of the electric and magnetic fields ( $E {(H)}_{z}^{TE(M)}$ ) can be imaginary by correctly choosing the phase factor of the eigenmodes in the silicon waveguide. Consider the TE-like and TM-like mode components of the electric field and magnetic field, the time-averaged longitudinal TAM, SAM and OAM densities crossing a surface in x-y plane derived from the vector angular spectrum [40,41] can be expressed as

〈 J_{z} 〉 = 〈 S_{z} 〉 + 〈 L_{z} 〉,

\begin{matrix} 〈 S_{z} 〉 = - \frac{ε_{0}}{2 ω} \cos (k_{0} Δ n z) \iint [E_{x}^{TM} (x, y) E_{y}^{TE} (x, y) - E_{x}^{TE} (x, y) E_{y}^{TM} (x, y)] d x d y \\ - \frac{μ_{0}}{2 ω} \cos (k_{0} Δ n z) \iint [H_{x}^{TM} (x, y) H_{y}^{TE} (x, y) - H_{x}^{TE} (x, y) H_{y}^{TM} (x, y)] d x d y, \end{matrix}

\begin{array}{l} 〈 L_{z} 〉 = \cos (k_{0} Δ n z) [\frac{ε_{0}}{4 ω} (ζ_{e t} + ζ_{e l}) + \frac{μ_{0}}{4 ω} (ζ_{m t} + ζ_{m l})], \\ with \\ ζ_{e t} = \sum_{σ = x, y} \iint [E_{σ}^{TE} (x, y) (x \frac{\partial}{\partial y} - y \frac{\partial}{\partial x}) E_{σ}^{TM} (x, y) - E_{σ}^{TM} (x, y) (x \frac{\partial}{\partial y} - y \frac{\partial}{\partial x}) E_{σ}^{TE} (x, y)] d x d y, \\ ζ_{e l} = \iint Re [E_{z}^{TE}^{*} (x, y) (x \frac{\partial}{\partial y} - y \frac{\partial}{\partial x}) E_{z}^{TM} (x, y) - E_{z}^{TM}^{*} (x, y) (x \frac{\partial}{\partial y} - y \frac{\partial}{\partial x}) E_{z}^{TE} (x, y)] d x d y, \\ ζ_{m t} = \sum_{σ = x, y} \iint [H_{σ}^{TE} (x, y) (x \frac{\partial}{\partial y} - y \frac{\partial}{\partial x}) H_{σ}^{TM} (x, y) - H_{σ}^{TM} (x, y) (x \frac{\partial}{\partial y} - y \frac{\partial}{\partial x}) H_{σ}^{TE} (x, y)] d x d y, \\ ζ_{m l} = \iint Re [H_{z}^{TE}^{*} (x, y) (x \frac{\partial}{\partial y} - y \frac{\partial}{\partial x}) H_{z}^{TM} (x, y) - H_{z}^{TM}^{*} (x, y) (x \frac{\partial}{\partial y} - y \frac{\partial}{\partial x}) H_{z}^{TE} (x, y)] d x d y, \end{array}

where the integral parts are real constants and independent of the propagation distance z. From Eqs. (9) and (10), we can see that the longitudinal SAM density and longitudinal OAM density coexist and vary with z according to a cosine function during the propagation in the birefringent silicon waveguide. The longitudinal SAM and OAM densities become the opposite states after it propagates a distance of L₀ =

λ / 2 | Δ n |

. However, the longitudinal SAM and OAM densities coexist in a square silicon waveguide but without a periodic evolution, for the fact that the refractive indices of the two orthogonal modes are identical.

The TAM of the whole system consisting of light and waveguide is conserved but the TAM of the light field is not. The variation of TAM of the light field results in the OT exerted on the waveguide. The OT density crossing a surface in x-y plane inside a homogeneous, linear, isotropic medium can be written as [22,43]

Γ = \frac{1}{2} \iint {r \times [(P \cdot \nabla) E + (M \cdot \nabla) H + \frac{d P}{d t} \times B - \frac{d M}{d t} \times D] + M \times H + P \times E} d x d y,

where P is the electric susceptibility and M is the magnetic susceptibility. For the birefringent silicon waveguide, Eq. (11) can be simplified and the time-averaged longitudinal OT density for each x-y surface inside the volume of the silicon waveguide is expressed as

\begin{array}{l} 〈 Γ_{z}^{v} 〉 = - \frac{1}{4} ε_{0} [χ_{e} \sin (k_{0} Δ n z) (ζ_{t} + ζ_{l})], \\ with \\ ζ_{t} = \sum_{σ = x, y} \iint [E_{σ}^{TE} (x, y) (x \frac{\partial}{\partial y} - y \frac{\partial}{\partial x}) E_{σ}^{TM} (x, y) - E_{σ}^{TM} (x, y) (x \frac{\partial}{\partial y} - y \frac{\partial}{\partial x}) E_{σ}^{TE} (x, y)] d x d y, \\ ζ_{l} = \iint Re [E_{z}^{TM}^{*} (x, y) (x \frac{\partial}{\partial y} - y \frac{\partial}{\partial x}) E_{z}^{TE} (x, y) - E_{z}^{TE}^{*} (x, y) (x \frac{\partial}{\partial y} - y \frac{\partial}{\partial x}) E_{z}^{TM} (x, y)] d x d y, \end{array}

where

χ_{e}

is the real part of P and the integral parts are all real constants that are independent of the propagation distance z. We consider air-waveguide boundaries at x = x₊ line, x = x_- line, y = y₊ line and y = y_- line which are the waveguide boundaries in x-y surface. The associated time-averaged longitudinal torques exerted on the x-y surface of the silicon waveguide are written as [22]

〈 Γ_{z}^{x_{+}} 〉 = \frac{1}{2} ε_{0} χ_{e}^{2} \sin (k_{0} Δ n z) \int y E_{x, mat}^{TE} (x_{+}) E_{x, mat}^{TM} (x_{+}) d y,

〈 Γ_{z}^{x_{-}} 〉 = - \frac{1}{2} ε_{0} χ_{e}^{2} \sin (k_{0} Δ n z) \int y E_{x, mat}^{TE} (x_{-}) E_{x, mat}^{TM} (x_{-}) d y,

〈 Γ_{z}^{y_{+}} 〉 = \frac{1}{2} ε_{0} χ_{e}^{2} \sin (k_{0} Δ n z) \int x E_{y, mat}^{TE} (y_{+}) E_{y, mat}^{TM} (y_{+}) d x,

〈 Γ_{z}^{y_{-}} 〉 = - \frac{1}{2} ε_{0} χ_{e}^{2} \sin (k_{0} Δ n z) \int x E_{y, mat}^{TE} (y_{-}) E_{y, mat}^{TM} (y_{-}) d x,

where

E_{x, mat}

and

E_{y, mat}

are the transverse components of the electric field inside the waveguide, and the integral parts are also real constants that are independent of the propagation distance z. Therefore, the total time-averaged longitudinal OT density as the sum of

〈 Γ_{z}^{v} 〉

,

〈 Γ_{z}^{x_{+}} 〉

,

〈 Γ_{z}^{x_{-}} 〉

,

〈 Γ_{z}^{y_{+}} 〉

and

〈 Γ_{z}^{y_{-}} 〉

for the silicon waveguide is given by

〈 Γ_{z} 〉 = 〈 Γ_{z}^{v} 〉 + 〈 Γ_{z}^{x_{+}} 〉 + 〈 Γ_{z}^{x_{-}} 〉 + 〈 Γ_{z}^{y_{+}} 〉 + 〈 Γ_{z}^{y_{-}} 〉 .

From Eqs. (12)-(17), we can see that the total longitudinal OT varies with z according to a sine function during the propagation in the birefringent silicon waveguide. It can be deduced that the longitudinal OT and the longitudinal TAM have periodic evolutions in the propagation process and the longitudinal OT is related to the negative derivative of the longitudinal TAM as Eq. (5).

3. Simulations

We use the Finite-Difference Time-Domain (FDTD) method to do the simulation and check the analysis. First, we explore the distributions of the longitudinal OT, TAM, SAM and OAM in the x-y surface of a birefringent silicon waveguide. The width and height of the birefringent silicon waveguide are set as w = 0.4 μm and h = 0.26 μm, respectively. The incident light is two orthogonal mode sources with a $π / 2$ phase difference, which is equivalent to a right circularly polarized light. The incident wavelength is 1.45 μm. There are only two orthogonal fundamental modes in the waveguide. Figure 2 indicates the longitudinal torque density and longitudinal AM densities in x-y surface at z = 3 μm. Figures 2(a)-2(d) depict the simulated longitudinal OT, TAM, SAM and OAM density distributions, respectively. It can be seen that longitudinal OT and TAM are concentrated at the silicon waveguide where the longitudinal OT is mainly located at the boundaries, the longitudinal SAM is located at the middle and the longitudinal OAM is located at the boundaries.

Fig. 2 The simulated longitudinal OT density distribution (a), the longitudinal TAM density distribution (b), the longitudinal SAM density distribution (c) and the longitudinal OAM density distribution (d) in the x-y surface at z = 3 μm of the birefringent silicon waveguide with 0.4 μm-width and 0.26 μm-height, respectively.

Download Full Size | PDF

To confirm the theoretical analysis above, we give some concrete examples for different sizes of the waveguide. The incident light is the same as described above. Figure 3(a) depict the simulated longitudinal OT density and the derivative of longitudinal AM flux density with the longitudinal propagation distance z in eight different waveguides whose (w, h) are (0.4 μm, 0.26 μm), (0.4 μm, 0.3 μm), (0.4 μm, 0.34 μm), (0.35 μm, 0.35 μm), (0.5 μm, 0.3 μm), (0.6 μm, 0.3 μm), (0.7 μm, 0.3 μm) and (0.8 μm, 0.3 μm), denoted by ${WG}_{m} (m = 1, 2, ..., 8)$ , respectively. The dashed line represents the longitudinal torque density and the solid line the negative derivative of longitudinal TAM flux density. The curves demonstrated that the longitudinal OT density is almost equal to the negative derivative of the longitudinal AM flux density. The results verify well our theory. Figure 3(b) depicts the simulated longitudinal OT density, the negative derivative of longitudinal OAM flux density, the negative derivative of longitudinal SAM flux density and the negative derivative of longitudinal TAM flux density versus z in the same birefringent silicon waveguide with w = 0.4 μm and h = 0.34 μm as [22]. The OAM’s contribution to the AM flux (denoted by the red solid line) was neglected in [22]. According to the results shown in Fig. 3(b), we can see that the existence of the longitudinal OAM flux density and its contribution to the OT is inevitable due to the SOI.

Fig. 3 (a) The longitudinal OT density and the negative derivative of longitudinal TAM flux density versus the propagation distance z in the different silicon waveguides with different sizes denoted by WG_m (m = 1,2,...,8), the size parameters are listed in the above context. The dashed line represents the longitudinal torque density and the solid line the negative derivative of longitudinal TAM flux density. (b) The longitudinal OT density, the negative derivative of longitudinal OAM flux density, the negative derivative of longitudinal SAM flux density and the negative derivative of longitudinal TAM flux density versus the distance z in the silicon waveguide with 0.4 μm-width and 0.34 μm-height.

Download Full Size | PDF

4. OAM ratio

The SAM-to-OAM conversion can occur strongly under some specific conditions such as light-matter interaction in anisotropic [44], inhomogeneous [45], or structured media [46], focusing, scattering, and imaging systems [47]. In the silicon waveguide case, the SAM-to-OAM conversion occurs at the moment of light-waveguide contact. We define that the OAM ratio is the ratio of the peak of the longitudinal OAM to the peak of the longitudinal TAM for the average per photon in the x-y surface in the propagation. According to Fig. 2, it can be discovered that different AM has different magnitude distribution in the x-y surface. In the following, the calculations of the longitudinal AM adopt the method of the average per photon in x-y surface for convenience. The method can obtains the longitudinal AM of the average per photon in x-y surface. Figures 4(a) and 4(b) show the simulated electric field distributions of the TE-like and TM-like modes in the waveguide with a height of 0.26 μm and width of 0.4 μm at the wavelength of 1.45 μm, respectively. Within the incident wavelength interval between 1.25 μm and 1.75 μm, the relationship of the propagation constant (β) and the incident wavelength (λ) for the TE-like and TM-like modes are shown in Fig. 4(c). We can see that the difference of the propagation constants between two orthogonal modes has a maximum at the wavelength of 1.7 μm. Figure 4(d) depicts the changes in the longitudinal OAM ratio, longitudinal TAM density and longitudinal OAM density of average per photon in x-y surface with incident wavelength. It is easy to observe that increasing the incident wavelength can bring down the longitudinal TAM of average per photon in x-y surface and can enhance the longitudinal OAM of average per photon in x-y surface, specifically the wavelength up to 1.65 μm at the maximum value. Therefore, increasing the incident wavelength can enhance the longitudinal OAM ratio and the maximum value is almost at the wavelength of 1.7 μm. It suggests that the OAM ratio is inversely proportional to the difference of the propagation constants of two orthogonal modes in a fixed size of the waveguide.

Fig. 4 The electric field distributions of the TM-like mode (a) and the TE-like mode (b) supported by a 0.4 μm-width and 0.26 μm-height silicon waveguide at the wavelength of 1.45 μm. (c) The dispersion characteristics of the TE-like and TM-like modes. The results show the relationship between the propagation constant β and incident wavelength λ. (d) The changes in the longitudinal OAM ratio, the longitudinal TAM density and the longitudinal OAM density per photon with the incident wavelength.

Download Full Size | PDF

To understand the underlying mechanism of OAM ratio with the size of waveguide, we illustrate the changes in the OAM ratio with the height of waveguide in a silicon waveguide with 0.4 μm-width at wavelength of 1.45 μm in Fig. 5. Figures 5(a) and 5(c) are dispersion relations of the TE-like and TM-like modes in different height range. It is clearly found that the difference of propagation constants decrease gradually with the height approaching to the width, where the birefringence disappears. Figures 5(b) and 5(d) depict the changes in the longitudinal OAM ratio, the longitudinal TAM density and the longitudinal OAM density of average per photon in the x-y surface with the height of waveguide. It is easy to identify that the OAM ratio decrease gradually with the height of the waveguide in the case, which is irrelevant to the difference of propagation constants of the two orthogonal modes. As stated above, we can achieve different OAM ratio through changing the size of the waveguide and the incident wavelength.

Fig. 5 (a) and (c) The dispersion characteristics of the TE-like and TM-like modes supported by a 0.4 μm-width waveguide with a varied height $h$ at the wavelength of 1.45 μm. It shows the relationship between the propagation constant (β) and the height of waveguide (h). (b) and (d) show the changes in the longitudinal OAM ratio, longitudinal TAM density and longitudinal OAM density per photon with the height of the waveguide.

Download Full Size | PDF

5. Chirality conversion and torque generator

We can apply the above theory to integrated photonic devices for achieving a chirality convertor and a torque generator. We propose a three-layer waveguide structure with the functions of the chirality conversion and the torque generation as shown in Fig. 6(a). It cascades a rectangular-shaped silicon waveguide in center with two square-shaped silicon-waveguides on both sides. The converter is the middle waveguide which can convert the incident AM state into the opposite one. The widths of the rectangular-shaped silicon waveguide and the two square-shaped silicon waveguides are h₁ and h₂, respectively. The heights of the rectangular-shaped silicon waveguide and the two square-shaped silicon waveguides are all h₂. The length of the rectangular-shaped silicon waveguide L can be designed so that we can get a maximum AM conversion. According to Eqs. (8)-(10), a maximum AM conversion indicates that $\cos (k_{0} | Δ n | z)$ turns from $\pm 1$ to $\mp 1$ , i.e., the propagation distance $z$ satisfies $z_{p} = (2 p + 1) λ / 2 | Δ n |$ , where $p$ is an integer. We take the designed length of the converter as $L = z_{0} = λ / 2 | Δ n |$ , i.e., the minimum length of $z_{p}$ . Obviously, $L$ depends on the incident wavelength $λ$ and the difference of the refractive indices of the orthogonal modes $| Δ n |$ controlled by the width h₁ and height h₂ of the waveguide. Two square-shaped silicon waveguides are stretched to infinity in our theoretical analysis.

Fig. 6 (a) The three-dimensional schematic of the ultra-compact waveguide structure. The width and height of the rectangular-shaped silicon waveguide in center are h₁ and h₂, respectively. The width and height of the two square-shaped silicon waveguides on both sides are all h₂. The length of the rectangular-shaped silicon waveguide is L which corresponds to the minimum propagation distance for an opposite AM state conversion. The two square-shaped silicon waveguides are stretched to infinity. (b) The simulated longitudinal TAM, SAM and OAM of average per photon in the x-y surface versus the propagation distance z in the waveguide, where h₁ = 0.26 μm, h₂ = 0.4 μm, L = 1.26 μm and λ = 1.45 μm.

Download Full Size | PDF

We give a simulation example here for a waveguide with the width and height of h₁ = 0.26 μm and h₂ = 0.4 μm at the incident wavelength of 1.45 μm. There are two orthogonal fundamental modes in the middle waveguide whose difference of the refractive indices $Δ n$ is 0.57549. As a result, the optimized length of the rectangular-shaped silicon waveguide is $L = λ / 2 | Δ n | = 1.26 μm$ . Figure 5(b) shows the simulations for the longitudinal TAM, SAM and OAM of average per photon in the x-y surface versus the propagation distance z in the waveguide. It is worth noticing that there is an opposite AM state conversion in the position of the middle waveguide with a high OT.

Figure 7 shows the OT exerted on the middle waveguide and the length L versus the width h₁ of the middle waveguide and the incident wavelength. Figure 7(a) shows the changes of the simulated OT per unit incident intensity (1 mW/μm²) and L versus the width h₁ of the middle waveguide with 0.4 μm-width at wavelength of 1.45 μm. The OT and L all increase gradually with the width of the middle waveguide. It means that the OT and L are positively correlated. Figure 7(b) depicts the changes of the simulated OT and L with the incident wavelength in a 0.26 μm-width and 0.4 μm-height middle silicon waveguide. It is easily observed that the OT increase gradually with the incident wavelength, specifically the wavelength up almost to 1.5 μm at the maximum value of the OT and then decrease gradually with the incident wavelength. It can be seen that increasing the incident wavelength can bring down the L and the minimum value is almost at the wavelength of 1.65 μm. We can also change the incident polarization state to control the direction of the OT. It is an effective way to achieve and control OT by changing the incident source and the size of the waveguide.

Fig. 7 (a) The simulated OT and L versus the width h₁ in a 0.4 μm-height middle silicon waveguide at the wavelength of 1.45 μm. (b) The simulated OT and L versus the incident wavelength in a 0.26 μm-width and 0.4 μm-height middle silicon waveguide.

Download Full Size | PDF

6. Conclusion

In summary, we gave a detailed analysis on the relationship between the optical TAM and the OT in a birefringent silicon waveguide in theory and simulation which lay the foundation for understanding the optical AM-induced mechanical effects. We further demonstrated the evolution rules of the OT, TAM, SAM and OAM by using the vector angular spectrum. The evolution rule of the OAM is the same with the SAM in the waveguide. Meanwhile, we found that the optical OAM exists in different waveguide with a circularly polarized incident light incidence, which is a refinement of the previous work of He et al. in [22]. By changing the size of the waveguide and the incident wavelength, we found the OAM ratio varies, which gives an effective way to obtain a considerable OAM ratio. In addition, we designed an ultracompact structure with three waveguides in series as a chirality convertor and a torque generator. It is found that the OT can be controlled with the width of the middle waveguide and the incident wavelength. We believe that the new scheme might provide a meaningful torque generator in integrated circuits and is promising for related technologies of optically driven micro-machines.

Funding

National Basic Research Program of China (2013CBA01702); National Natural Science Foundation of China (NSFC) (61575055, 11874132, 61405056, 61307072, 61308017, 61377016).

References

1. L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992). [CrossRef] [PubMed]

2. M. Babiker, W. L. Power, and L. Allen, “Light-induced torque on moving atoms,” Phys. Rev. Lett. 73(9), 1239–1242 (1994). [CrossRef] [PubMed]

3. J. E. Curtis and D. G. Grier, “Structure of optical vortices,” Phys. Rev. Lett. 90(13), 133901 (2003). [CrossRef] [PubMed]

4. S. Albaladejo, M. I. Marqués, F. Scheffold, and J. J. Sáenz, “Giant enhanced diffusion of gold nanoparticles in optical vortex fields,” Nano Lett. 9(10), 3527–3531 (2009). [CrossRef] [PubMed]

5. J. Ng, Z. Lin, and C. T. Chan, “Theory of optical trapping by an optical vortex beam,” Phys. Rev. Lett. 104(10), 103601 (2010). [CrossRef] [PubMed]

6. M. Khan, A. K. Sood, F. L. Deepak, and C. N. R. Rao, “Nanorotors using asymmetric inorganic nanorods in an optical trap,” Nanotechnology 17(11), S287–S290 (2006). [CrossRef]

7. K. Ding, J. Ng, L. Zhou, and C. T. Chan, “Realization of optical pulling forces using chirality,” Phys. Rev. A 89(6), 063825 (2014). [CrossRef]

8. D. Hakobyan and E. Brasselet, “Left-handed optical radiation torque,” Nat. Photonics 8(8), 610–614 (2014). [CrossRef]

9. S. B. Wang and C. T. Chan, “Lateral optical force on chiral particles near a surface,” Nat. Commun. 5(1), 3307 (2014). [CrossRef] [PubMed]

10. V. L. Y. Loke, T. Asavei, A. B. Stilgoe, T. A. Nieminen, and H. Rubinsztein-Dunlop, “Driving corrugated donut rotors with laguerre-gauss beams,” Opt. Express 22(16), 19692–19706 (2014). [CrossRef] [PubMed]

11. H. Ukita and H. Kawashima, “Optical rotor capable of controlling clockwise and counterclockwise rotation in optical tweezers by displacing the trapping position,” Appl. Opt. 49(10), 1991–1996 (2010). [CrossRef] [PubMed]

12. M. Liu, T. Zentgraf, Y. Liu, G. Bartal, and X. Zhang, “Light-driven nanoscale plasmonic motors,” Nat. Nanotechnol. 5(8), 570–573 (2010). [CrossRef] [PubMed]

13. T. Harada and K. Yoshikawa, “Mode switching of an optical motor,” Appl. Phys. Lett. 81(25), 4850–4852 (2002). [CrossRef]

14. A. La Porta and M. D. Wang, “Optical torque wrench: Angular trapping, rotation, and torque detection of quartz microparticles,” Phys. Rev. Lett. 92(19), 190801 (2004). [CrossRef] [PubMed]

15. K. Kim, X. Xu, J. Guo, and D. L. Fan, “Ultrahigh-speed rotating nanoelectromechanical system devices assembled from nanoscale building blocks,” Nat. Commun. 5(1), 3632 (2014). [CrossRef] [PubMed]

16. S. Lin and K. B. Crozier, “An integrated microparticle sorting system based on near-field optical forces and a structural perturbation,” Opt. Express 20(4), 3367–3374 (2012). [CrossRef] [PubMed]

17. S. Lin, W. Zhu, Y. Jin, and K. B. Crozier, “Surface-enhanced Raman scattering with Ag nanoparticles optically trapped by a photonic crystal cavity,” Nano Lett. 13(2), 559–563 (2013). [CrossRef] [PubMed]

18. S. Lin and K. B. Crozier, “Trapping-assisted sensing of particles and proteins using on-chip optical microcavities,” ACS Nano 7(2), 1725–1730 (2013). [CrossRef] [PubMed]

19. S. Kuhn, A. Kosloff, B. A. Stickler, F. Patolsky, K. Hornberger, M. Arndt, and J. Millen, “Full rotational control of levitated silicon nanorods,” Optica 4(3), 356–360 (2017). [CrossRef]

20. S. H. Simpson and S. Hanna, “Holographic optical trapping of microrods and nanowires,” J. Opt. Soc. Am. A 27(6), 1255–1264 (2010). [CrossRef] [PubMed]

21. W. A. Shelton, K. D. Bonin, and T. G. Walker, “Nonlinear motion of optically torqued nanorods,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 71(3 Pt 2A), 036204 (2005). [CrossRef] [PubMed]

22. L. He, H. Li, and M. Li, “Optomechanical measurement of photon spin angular momentum and optical torque in integrated photonic devices,” Sci. Adv. 2(9), e1600485 (2016). [CrossRef] [PubMed]

23. M. Durach and N. Noginova, “Spin angular momentum transfer and plasmogalvanic phenomena,” Phys. Rev. B 96(19), 195411 (2017). [CrossRef]

24. S. M. Barnett, “Optical angular-momentum flux,” J. Opt. B Quantum Semiclass. Opt. 4(2), S7–S16 (2002).

25. K. Y. Bliokh, F. J. Rodríguez-Fortuño, F. Nori, and A. V. Zayats, “Spin–orbit interactions of light,” Nat. Photonics 9(12), 796–808 (2015). [CrossRef]

26. A. Ciattoni, G. Cincotti, and C. Palma, “Angular momentum dynamics of a paraxial beam in a uniaxial crystal,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 67(3 Pt 2), 036618 (2003). [CrossRef] [PubMed]

27. E. Brasselet, Y. Izdebskaya, V. Shvedov, A. S. Desyatnikov, W. Krolikowski, and Y. S. Kivshar, “Dynamics of optical spin-orbit coupling in uniaxial crystals,” Opt. Lett. 34(7), 1021–1023 (2009). [CrossRef] [PubMed]

28. X. Lu and L. Chen, “Spin-orbit interactions of a Gaussian light propagating in biaxial crystals,” Opt. Express 20(11), 11753–11766 (2012). [CrossRef] [PubMed]

29. L. T. Vuong, A. J. L. Adam, J. M. Brok, P. C. M. Planken, and H. P. Urbach, “Electromagnetic spin-orbit interactions via scattering of subwavelength apertures,” Phys. Rev. Lett. 104(8), 083903 (2010). [CrossRef] [PubMed]

30. M. Kang, J. Chen, B. Gu, Y. Li, L. T. Vuong, and H.-T. Wang, “Spatial splitting of spin states in subwavelength metallic microstructures via partial conversion of spin-to-orbital angular momentum,” Phys. Rev. A 85(3), 035801 (2012). [CrossRef]

31. J. Chen, S. Wang, X. Li, and J. Ng, “Mechanical effect of photonic spin-orbit interaction for a metallic nanohelix,” Opt. Express 26(21), 27694–27704 (2018). [CrossRef] [PubMed]

32. Y. Zhao, J. S. Edgar, G. D. Jeffries, D. McGloin, and D. T. Chiu, “Spin-to-orbital angular momentum conversion in a strongly focused optical beam,” Phys. Rev. Lett. 99(7), 073901 (2007). [CrossRef] [PubMed]

33. H. Adachi, S. Akahoshi, and K. Miyakawa, “Orbital motion of spherical microparticles trapped in diffraction patterns of circularly polarized light,” Phys. Rev. A 75(6), 063409 (2007). [CrossRef]

34. K. Y. Bliokh, J. Dressel, and F. Nori, “Conservation of the spin and orbital angular momenta in electromagnetism,” New J. Phys. 16(9), 093037 (2014). [CrossRef]

35. M. Nieto-Vesperinas, “Optical torque: electromagnetic spin and orbital-angular-momentum conservation laws and their significance,” Phys. Rev. A 92(4), 043843 (2015). [CrossRef]

36. A. Mizrahi, M. Horowitz, and L. Schächter, “Torque and longitudinal force exerted by eigenmodes on circular waveguides,” Phys. Rev. A 78(2), 23802 (2008). [CrossRef]

37. P. Y. Yang, G. Z. Mashanovich, I. Gomez-Morilla, W. R. Headley, G. T. Reed, E. J. Teo, D. J. Blackwood, M. B. H. Breese, and A. A. Bettiol, “Freestanding waveguides in silicon,” Appl. Phys. Lett. 90(24), 241109 (2007). [CrossRef]

38. P. Sun and R. M. Reano, “Submilliwatt thermo-optic switches using free-standing silicon-on-insulator strip waveguides,” Opt. Express 18(8), 8406–8411 (2010). [CrossRef] [PubMed]

39. Y. Akihama and K. Hane, “Single and multiple optical switches that use freestanding silicon nanowire waveguide couplers,” Light Sci. Appl. 1(6), e16 (2012). [CrossRef]

40. H. Guo, J. Chen, and S. Zhuang, “Vector plane wave spectrum of an arbitrary polarized electromagnetic wave,” Opt. Express 14(6), 2095–2100 (2006). [CrossRef] [PubMed]

41. R. P. Chen and G. Li, “The evanescent wavefield part of a cylindrical vector beam,” Opt. Express 21(19), 22246–22254 (2013). [CrossRef] [PubMed]

42. K. Y. Bliokh, A. Y. Bekshaev, and F. Nori, “Dual electromagnetism: helicity, spin, momentum and angular momentum,” New J. Phys. 15(3), 033026 (2013). [CrossRef]

43. M. Mansuripur, “Electromagnetic force and torque in ponderable media,” Opt. Express 16(19), 14821–14835 (2008). [CrossRef] [PubMed]

44. L. Marrucci, C. Manzo, and D. Paparo, “Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media,” Phys. Rev. Lett. 96(16), 163905 (2006). [CrossRef] [PubMed]

45. V. S. Liberman and B. Y. Zel’dovich, “Spin-orbit interaction of a photon in an inhomogeneous medium,” Phys. Rev. A 46(8), 5199–5207 (1992). [CrossRef] [PubMed]

46. N. Shitrit, I. Yulevich, E. Maguid, D. Ozeri, D. Veksler, V. Kleiner, and E. Hasman, “Spin-optical metamaterial route to spin-controlled photonics,” Science 340(6133), 724–726 (2013). [CrossRef] [PubMed]

47. K. Y. Bliokh, E. A. Ostrovskaya, M. A. Alonso, O. G. Rodríguez-Herrera, D. Lara, and C. Dainty, “Spin-to-orbital angular momentum conversion in focusing, scattering, and imaging systems,” Opt. Express 19(27), 26132–26149 (2011). [CrossRef] [PubMed]

Dynamics of angular momentum-torque conversion in silicon waveguides

Abstract

1. Introduction

2. Theory and structure

3. Simulations

4. OAM ratio

5. Chirality conversion and torque generator

6. Conclusion

Funding

References

Cited By

Figures (7)

Equations (17)

Optics Express