1. Introduction

The longitudinal electric field component (E_z) of a propagating electromagnetic wave has recently been the subject of increasing interest largely as a result of two unique attributes. First, it can be shown to focus tighter than the diffraction limit [1–4], which lends itself to applications such as lithography [5], near-field microscopy [6, 7], and optical data storage [8]. Second, because of its directionality, applications have been proposed to take advantage of this unique polarization, such as particle acceleration [9], absorption dipole moment probing [10], multiple quantum well heterostructure excitation [11], optical tweezing [12], grating couplers in high-index-contrast slab waveguides [13], as well as nonlinear optical response probing [14]. The longitudinal field originates from the spatial derivative of the transverse fields [15]; hence by means of strong optical confinement a large E_z component can be generated. For a Gaussian beam, the generation of a large E_z is nontrivial as it can be shown to be small for the paraxial case [16]. A common approach to creating large longitudinal fields, as used in many of the above applications, is through radially or azimuthally polarized beams [17, 18].

We propose an alternative, integrated approach to the generation of large longitudinal electric fields, especially desirable due to the recent trend of miniaturizing photonic devices and the growing interest in lab-on-chip micro-systems. Since E_z originates from the spatial gradient of the transverse fields, it is reasonable to expect strong optical confinement associated with silicon (Si) photonic wire waveguides (Si nanowires) to yield a large E_z. Si nanowires based on the silicon-on-insulator (SOI) platform have been heavily investigated recently due to their high potential for nanoscale photonic devices and photonic integrated circuits (PICs). The ultratight optical confinement of Si nanowires has led to the realization of unique optical properties. As a recent example, it was shown that the dispersion in these guides is dominated by the cross-section geometry as opposed to material properties, allowing waveguide dispersive properties to be engineered [19, 20]. The high electric field intensity has also led to the exploration of nonlinear optical effects in silicon [21]. In fact, a recent work proposed an optical isolator based on nonreciprocal Raman gain resulting from the longitudinal electric field in Si waveguides [22].

In this paper, we show numerically that an ultralarge longitudinal electric field component of a guided mode may be obtained in Si nanowires excited at a wavelength of 1.55 μm. Specifically, we show that the magnitude of the E_z component can be comparable to the transverse field and optimized through careful adjustment of waveguide geometry. We demonstrate that waveguide nonlinearity is strongly affected by the longitudinal electric field for the case of Si nanowires and explore a dual waveguide design, which can enhance the longitudinal electric field in free space and provide control over modal symmetry.

2. Basic theoretical considerations

The origin of the longitudinal electric field within a waveguide can be understood through a consideration of Maxwell’s equations. In a waveguide, the electric field vector may be expressed as E(r) =E ₀(r)exp(- jβz) where β is the propagation constant and z the direction of propagation [15]. Because of waveguide translational symmetry, E ₀(r) =E ₀(r _T), where r _T is the transverse position vector. Assuming an invariant dielectric profile in the z direction, through Gauss’s law, the longitudinal electric field can be expressed as ∂E_z/d_z = - (1/n ²)∇ _T · (n ² E _T)[23]. In piecewise homogeneous regions, one can show that E_z = ∇ _T · E _T/jβ, where E _T denotes the transverse electric field complex amplitudes (E_x,E_y), and ∇ _T the transverse gradient. Equiva-lently,

where N _eff is the effective index experienced by the mode and λ ₀ the free space wavelength. For quasi-TE and -TM modes, ∇ _T · E _T ≈ ∂E_x/d_xand ∇ _T · E _T ≈ ∂E_y/d_y respectively. For the following discussion, we consider quasi-TE mode propagation. Examination of Eq. (1) shows that the complex amplitude of the longitudinal field is purely imaginary (by convention that the transverse fields are purely real) meaning that it is in quadrature (π/2) phase with respect to the transverse fields. Similarly, E_z is in quadrature phase with the transverse magnetic fields and examination of the complex Poynting vector, $\mathbf{S}=\frac{1}{2}\mathbf{E}\times {\mathbf{H}}^{*},$, reveals no real contribution via the longitudinal components. Therefore E_z does not contribute to net energy transport and carries no net momentum; however it does contribute to the total energy density, meaning that the longitudinal component acts locally as an energy reservoir, which can be tapped.

Equation (1) is useful within the usual limits of the effective index approximation. We do note exact solutions for the electric field have been discussed for the high-index-contrast cylindrical waveguide [24] including the effect of varying radial extent, while no exact solutions exist for the channel case leaving numerical analysis as the best approach for investigating these wire waveguides; the difference between the cylindrical and channel cross-sectional waveguides will be discussed in more detail below. Nevertheless, Eq. (1) does serve as a useful approximation for conveying the impact of tight confinement on increasing the longitudinal field.

3. Calculation and optimization of the longitudinal electric field

Calculations of the full vectorial modal components supported by a Si nanowire were accomplished using the finite element method (FEM), carried out through the software package Fem-SIM^TM by RSoft. Figure 1 shows the E_x and E_y components of the fundamental quasi-TE and -TM modes at wavelength λ = 1.55 μm supported by a Si nanowire with 260 × 400 nm² (height, h × width, w) dimensions surrounded by silicon-dioxide (SiO₂). A grid size of w/45 and h/45 in the x and y directions respectively was used, and for all simulations in this paper we assume an index of refraction of 3.5, 1.5 and 1 for Si, SiO₂, and air respectively. In the figure, all fields are normalized to the peak dominant electric field component, i.e., the E_x(E_y) component for the quasi-TE(-TM) mode. With this geometry, the maximum E_z amplitude is found to be ~ 63%(78%) that of the maximum x-component(y-component) amplitude. Since ∣E_z∣ is maximized where the transverse spatial derivative is maximized, the largest E_z amplitude occurs near the waveguide sidewalls for quasi-TE modes and the top/bottom waveguide boundaries for quasi-TM modes.

 

Fig. 1. The E_x(E_y) and E_z components of the fundamental quasi-TE(-TM) mode supported by a 260 × 400² nm Si nanowire waveguide surrounded by SiO₂ cladding. Graded yellow and blue colors indicate a π-phase difference.

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The magnitude of the E_z amplitude is highly dependent on the waveguide geometry since the transverse profile approximately assumes the shape of the core for these highly confining Si nanowires. E_z can therefore be modified significantly by variation of the height or width of the waveguide. Figure 2 shows a contour plot of the normalized maximum E_z amplitude relative to the peak amplitudes of the corresponding quasi-TE and -TM mode fields. Notice that for quasi-TE(-TM) modes, the near vertical(horizontal) contour lines indicate that waveguide width(height) dominates the strength of E_z. Note E_z is additionally related to the dimensionally dependent N_eff as shown in Eq. (1), however our numerical analysis has shown that the longitudinal electric field strength is predominantly determined by the spatial derivative, a consequence of confinement. As the width is decreased to ~ 300 nm, the magnitude of E_z increases due to the increasing spatial derivative and further width reduction results in loss of optical confinement, and a sharp decrease in E_z.

As an example, a 330 × 320 nm² Si nanowire on an SiO₂ substrate with air cladding yields ∣E _z(max)/E _T(max)∣= 97%(89%) for the quasi-TE(-TM) mode. This result shows that it is possible to design a single waveguide such that its E_z component is comparable to that of the transverse field for both quasi-TE and -TM modes. This is in stark contrast to standard fiber optic cable where E_z is only a few percent the transverse field and is often ignored in analysis [25]. Comparing Figs. 2(a) and 2(b) shows that a slightly smaller E_z exists for the quasi-TM mode compared with the quasi-TE mode for the air/SiO₂ cladding system and arises from a larger SiO₂ area being sampled by the field for the quasi-TM mode. Similarly, E_z is smaller for the full SiO₂ cladding system [Figs. 2(c)–2(d)] compared to the air/SiO2 system [Figs. 2(a–2(b)].

To provide a comparison with our results for a channel waveguide, we investigated computationally the longitudinal field of the fundamental HE₁₁ mode for cylindrical-waveguides. Note that these computations agree with analytic results of Tong et al [24], thus providing a validating case for the computations. In addition, the results show that the increased confinement factor associated with a cylindrical Si nanowire allows for a larger E_z normalized to the transverse field [Fig. 3]. In general, this comparison shows clearly that for comparable size the longitudinal field in a channel guide is less than that in a cylindrical waveguide. For example, we find that a 400 nm diameter cylindrical silicon waveguide with SiO₂ cladding supports an E_z component ~ 0.69 times that of the transverse field [Fig. 3], while a square Si nanowire waveguide (400 × 400 nm²) with SiO₂ cladding supports an E_z component ~ 0.58 times that of the transverse field for both the quasi-TE and -TM modes [Figs. 2(c)–2(d)]. This difference makes it important to provide specific calculations for the channel guide. Numerical analysis of the channel structure also allows for the study of unsymmetric cladding and rectangular cross-sections which are not well approximated by circular cross-sections. Finally, we note that rectangular channel guides have the practical geometry now being used in virtually all SOI PICs today.

 

Fig. 2. ∣E_z(max)/E _T(max)∣ for the quasi-TE and -TM modes of Si nanowires with variable dimensions, SiO₂ substrate, and air (a,b) or SiO₂ (c,d) cladding.

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Fig. 3. ∣E_z(max)/E _T(max)∣ for the HE11 mode in a cylindrical Si nanowire of variable diameter, and air (top blue line) or SiO₂ (bottom red line) cladding.

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4. Harnessing and controlling the longitudinal electric field

To harness the axial electric field component from a Si nanowire waveguide for practical applications, one first needs to analyze the fields existing at the edge of a terminated waveguide. We explore this problem using the finite-difference time-domain (FDTD) method, via Full-WAVE ^TM by RSoft, to simulate the propagation of the fundamental quasi-TE mode supported by a 260 × 500 nm² Si nanowire surrounded by SiO₂ cladding and terminated into air. A grid size of 10 nm, 26 nm, and 10 nm was used in the x, y, and z directions respectively, with a step time (cT) of 0.006 μm. Here we consider prorogation in a semi-infinite wire. Note that for a real finite-length wire, the length of the wire itself would affect the electric field intensity in the waveguide and consequently the electric field strength of all modal components due to Fabry-Perot resonances; however even in the finite wire case the normalized strengths as presented in this paper would remain the same. This result was confirmed by varying the wire length over one λ and finding the ratio E_z/E _T.

Figures 4(a) and 4(b) show the E_x and E_z components respectively for a single wire, where both have been normalized to the maximum E_x component. Note that a jump in the E_z amplitude occurs due to the continuity of the normal electric displacement D [Fig. 4(b)]. The transverse components of the electric field, on the other hand, are continuous across the interface by virtue of tangential electric field continuity [Fig. 4(a)], and numerical calculations are in accord with the boundary conditions. An examination of the fields beyond the waveguide’s endface clearly show a substantial presence of E_z, and Fig. 4(c) shows a contour plot of ∣E_z∣² 40 nm from the endface normalized to ∣E _x(max)∣² in the wire.

 

Fig. 4. The (a) E_x and (b) E_z electric field components of a mode propagating in a 260×500 nm² Si nanowire with SiO₂ cladding. (c) ∣E_z/E _T(max)∣² 40 nm from the endface.

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The relationship between the longitudinal and transverse modal components allows control of E_z, in principle, by modifying the transverse profile of the beam. In order to achieve this capability, one may introduce a second identical waveguide at close proximity [Fig. 5(c)] carrying an identical but phase-delayed propagating mode. This may be accomplished, for example, by integrating two waveguides as part of a Mach-Zehnder interferometer with one phase-controllable arm. The degree of interference between the transverse modes, and consequently E_z and the modal symmetry, can be controlled via the relative phase difference between the propagating modes. To demonstrate, we employ dual 260 × 500 nm² waveguides excited at λ = 1.55μm. Through a FEM mode solver it can be seen that such a structure supports two fundamental system modes with the symmetric system mode yielding a highly confined transverse field in the gap region corresponding to the heavily analyzed slot waveguide [26, 27], and the antisymmetric system mode where the transverse fields in the two arms have a π phase difference and remain in the Si regions [Fig. 5(a)]. For the antisymmetric mode, the longitudinal component of the evanescent tails are in phase in the gap and constructively interfere [Fig. 5(b)] while also constructively interfering at the terminated output [Fig. 6]. In fact, for this mode the gap region contains a purely longitudinal electric field. Similar analysis has been reported from the perspective of optical forces induced by evanescently coupled waveguides [28].

 

Fig. 5. The (a) transverse (E_x) and (b) longitudinal (E_z) field components of the fundamental antisymmetric system mode supported by (c) dual waveguides where each waveguide has cross-section dimensions 260 × 500 nm² separated by a 50 nm gap.

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Fig. 6. The(a)E_z and(b)E_x electric field components at the endface of dual 260 × 500 nm² Si nanowire waveguides with SiO₂ cladding, 50 nm gap, terminated into air, and excited by the fundamental antisymmetric system mode. (c) ∣E_z∣² near the edge of the dual waveguide structure. Contour plot (d) and line scan (e) of ∣E_z∣² 40 nm from the edge of the waveguide.

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For this antisymmetric waveguide design, the mode propagation of the E_x and E_z components into air was found through FDTD [Figs. 6(a)–6(b)]. At the output, the longitudinal field dominates that of the transverse field and is further enhanced in free space due to constructive interference [Fig. 6(c)]. A plot of the spatial contours of ∣E_z∣² 40 nm from the endface [Figs. 6(d)–6(e)] reveals a very narrow peak; in fact it is beyond the diffraction limit and has a 320 nm (~ λ/5) full-width half-maximum (FWHM) along the x-direction. Furthermore, at this distance, ∣E_z∣² is ~ 1.4 times that of ∣E_x∣² in the waveguide. It is important to reiterate the significant difference between the slot waveguide structures [26, 27] and the design discussed here; the slot waveguide is optimized to enhance the transverse fields in the gap, while here we are interested in enhancing the longitudinal electric fields at the terminated output; thus different geometry and phase relations must be considered.

5. The effect of the longitudinal field on waveguide nonlinearity

The presence of a large longitudinal electric field can also be shown to be important in devices based on optical nonlinearities. Our preliminary calculations show that the longitudinal electric field may have an adverse effect on the effective nonlinear parameter (γ). These preliminary calculations clearly show the importance of considering E_z in predicting γ As an example, following the method presented in Ref. [19] to calculate γ, we find the effective nonlinear parameter for a 260 × 500 nm² SiO₂ clad Si nanowire to be ~ 245 W^-1 m^-1 when the longitudinal field is included in the calculation and ~ 345 W^-1m^-1 when the longitudinal field is excluded. As the waveguide’s confinement factor increases via size reduction, the total transverse electric field intensity increases however so does the longitudinal electric field; these results have an opposite effect on γ and is in clear contrast to similar calculations in fiber optics where the longitudinal field is ignored under the assumption its magnitude is negligible [25]. Such an approximation is valid for low-index-contrast fiber optic cables, however the longitudinal field plays a critical role in the case of high-index-contrast Si nanowires. Currently a more extensive investigation of this phenomenon is underway in our laboratory and will be reported elsewhere.

6. Conclusions

In this paper we have shown numerically that the longitudinal electric-field component may become comparable to the amplitude of the transverse field in high-index-contrast Si nanowire waveguides. E_z may be engineered via waveguide geometry and optimized to be ~ 97% that of the transverse field, which allows Si nanowire supported E_z fields to have potential uses in novel integrated waveguide applications such as optical isolation. Using the antisymmetric system mode of a dual waveguide design, a purely longitudinal electric field exists in the gap and the longitudinal field becomes enhanced and sharply peaked (~ λ/5) at the output, which serves as an important new route to ultrahigh-resolution applications such as subwavelength optical microscopy. We further show that E_z has a large effect when calculating the nonlinear parameter of a Si nanowire and, in contrast to low-index-contrast platforms such as standard fiber optic cables, cannot be ignored in analysis.