1. Introduction

In the standard thin-element approximation (TEA), the optical field is propagated through a diffraction grating or other scattering structure with the aid of rays that travel along parallel, straight lines [1, 2]. Further propagation beyond the modulated region is then accomplished by means of wave optics. While the limitations of TEA are well-established, its simplicity attracts one to search for extensions that would retain some of its intuitiveness while improving its accuracy. In general, one can consider using non-paraxial geometrical optics together with Fresnel’s transmission and reflection coefficients to propagate the rays inside the scatterer. This leads to the concepts of local plane-interface and spherical-interface approximations [3–5]. In particular the local plane-interface approximation (LPIA) has been applied to a variety of different structures, most notably to dielectric gratings with different types of triangular profiles [6–10]. Comparisons with rigorous electromagnetic theory have shown that LPIA can provide heuristic understanding of some striking resonant features of grating diffraction, especially if multiple refractions and reflections inside the modulated region are taken into account [7, 9, 10]. This is true even if the feature sizes of the structure are in the wavelength scale, in which case the validity of LPIA would appear to be seriously in doubt. Furthermore, LPIA allows one to improve the results of TEA substantially at large angles of incidence even if applied only to correct the phase delay introduced by the structure [11, 12].

In the previous studies on LPIA all the plane waves propagating inside the modulated region were assumed to be homogeneous. However, when total internal reflection (TIR) occurs at any facet of the interface, an evanescent field is generated; it is well known that such fields can play an important role in the diffraction process. Here we study the extension of LPIA to such cases. In particular, we associate a ray with an evanescent wave propagating along the interface where TIR occurs. This ray is perpendicular to the surfaces of constant phase of the evanescent wave. The decay of the field amplitude in the direction perpendicular to the ray is also included in the analysis. We chose an example structure, which resembles a scanning near-field microscope, the motivation being that LPIA in its conventional form would then predict no transmission of light through the structure. We then show that intuitive information on the diffraction process can be obtained by adding the evanescent field in the LPIA analysis to allow tunneling of light through the structure.

2. Theoretical models

Figure 1 illustrates the geometrical configuration to be studied. A homogeneous plane wave with wavelength λ is incident from the half-space z < 0, filled with a material of refractive index n, at an angle of incidence θ that exceeds the critical angle at the interface z = 0. Thus an evanescent field propagating in the positive x direction is generated above the plane z = 0. This field is probed at height h with a dielectric wedge of refractive index n₁. The geometrical shape of the wedge is defined by the parameters H and α. In the half-space above the wedge (z > h + H) we finally have a planar waveguide with core index n₁, cladding index n₂, and core half-width a = H tanα. Our goal is to study the coupling of light into the waveguide modes.

 

Fig. 1 The geometry involving the generation of an evanescent field on top of the substrate and detection with a wedge-shaped core of a waveguide.

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The LPIA model of the structure illustrated in Fig. 1 is shown in Fig. 2. The evanescent field in the region z > h is first refracted inside the wedge through its left-hand-side facet. We assume that α is sufficiently large for total internal reflection to occur at the right-hand-side facet. Ignoring diffraction effects in the spirit of LPIA, we then obtain a projection of the incident evanescent field tail in the input plane z = h + H of the waveguide. This projection is a non-uniform plane wave, which vanishes when x < x₀, decays exponentially in x direction when x₀ < x < a, and propagates in the direction defined by the angle ϕ. Once the amplitude and phase of this field have been determined, the coupling efficiencies into different waveguide modes can be evaluated using the standard overlap integral method. Hence, as usual in LPIA models, we move from a geometrical-optics picture to a wave-optical picture at the upper boundary z = h + H. Instead of free-space propagation beyond this boundary as in, e.g., the geometries considered in [7, 9, 10], we now propagate the field onwards as a superposition of waveguide modes.

 

Fig. 2 Geometrical model of the propagation of an evanescent wave into and inside the wedge, and coupling into the waveguide along a typical ray path ABCDE.

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Let us assume that the incident plane wave is TM polarized (the TE case can be treated similarly). If the half-space above the interface at z = 0 were empty, the only non-vanishing component of the magnetic field of the evanescent wave in the region z > 0 would be

It is not difficult to show, using geometrical optics, that the projection of the field in Eq. (2) into the entrance plane z = h + H of the waveguide is of the form (see the Appendix for some details of the calculation)

If we denote the magnetic field of an arbitrary waveguide mode by H₂(x), the coupling efficiency from an input field H₁(x) into this particular mode is generally defined as [13]

The predictions provided by the LPIA approach outlined above are compared to rigorous results obtained using the Fourier Modal Method (FMM) with the S-matrix algorithm to evaluate the coupling efficiencies into the waveguide modes [15]. As illustrated in Fig. 3, the wedge region is divided into J layers thin enough to model the continuous facets with sufficient accuracy, and the computational period was chosen large enough to ensure that the non-periodic structure is modeled correctly. Typically we chose J = 256, d ∼ 25λ, and included ∼ 100 Floquet–Bloch modes in the calculations.

 

Fig. 3 Computational box showing the quantization of the wedge region into J layers and the computational period d.

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The spatial dependence of magnetic-field amplitude |H_y(x, z)| in the structure being considered, as predicted by FMM, is shown in Fig. 4. Rich-shaped diffraction patterns within the wavelength-scale structure are seen, as expected. However, these rigorously calculated results also justify the use of LPIA to study phenomena related to evanescent fields. A beam-like field emerges from the wedge and is coupled smoothly into the (guided and radiation) modes of the waveguide. Although the spatial profile differs from that predicted by LPIA because of diffraction, the beam propagation direction changes with the wedge angle α in qualitative agreement with Eq. (8). Media 1 illustrates the temporal behavior of the (harmonic) real field ℜ{H_y(x, z)}. The in-coupled beam gains wavefront curvature as it propagates upwards, but in the entrance plane of the waveguide the wavefronts are still essentially planar as assumed in the LPIA model. Also evident from Media 1 is the generation of a cylindrical edge diffraction wave at the tip of the wedge, which interferes with the ‘transmitted’ evanescent field outside the wedge in the region x > 0. Normally such a wave needs to be considered since it can contribute significantly to the field at the output plane z = H of the diffracting object. This would be the case, e.g., if the output medium in the region z > H in Fig. 1 would be homogeneous. However, in our case the edge diffraction wave propagates towards the cladding of the waveguide and does not contribute to the field within the core. Moreover, as can be seen from Media 1, the wave fronts in the cladding region are strongly curved. Hence the overlap integral of the edge diffracted wave and the tail of any waveguide mode in the cladding has an insignificant value compared to the contribution from Eq. (8), and can be safely ignored.

 

Fig. 4 Distributions of magnetic field amplitude inside the structure for several combinations of the wedge angle α and core width 2a. Media 1 illustrates the temporal evolution of the real field within the structure. Here λ = 633 nm, n = 1.4569, θ = 44.91°, n₁ = 1.52, and n₂ = 1.49.

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3. Coupling efficiency from the geometrical and rigorous models

Let us first compare the coupling efficiencies predicted by FMM and LPIA as a function of the wedge angle α. In all examples λ = 633 nm, n = 1.4569, and θ = 44.91°. We keep the core width 2a of the waveguide constant, and hence the wedge height varies as H = a/tanα. Figure 5 illustrates the results for a single-mode waveguide with parameters n₁ = 1.52, n₂ = 1.49, and 2a = 1000 nm, the coupling efficiency curves evaluated by FMM and LPIA are normalized to their respective maximum values. Although the two curves are not identical, their general behavior is similar. We obtain maximum coupling at a certain optimum wedge angle α, which is nearly the same according to FMM and LPIA. The upward-pointing arrow in Fig. 5 corresponds to the value of α for which the angle ϕ in Fig. 2 matches the geometrical propagation angle θ₀ of the fundamental waveguide mode [14].

 

Fig. 5 Normalized coupling efficiency into the fundamental mode of a single-mode waveguide with 2a = 1000 nm as a function of the wedge angle α. Solid black: FMM calculation. Solid blue: overlap integral method based on LPIA. Arrow: optimum angle given by the phase matching condition.

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Figure 6 illustrates similar results for a waveguide with 2a = 1600 nm, which supports two guided modes. Again we find fairly good agreement between the FMM and LPIA calculations around the optimum angle, which is now different for the fundamental mode m = 0 and the antisymmetric mode m = 1. The arrows indicate the values of α that satisfy the phase matching conditions ϕ = θ₀ and ϕ = θ₁. The total coupling efficiency is, of course, the sum of the coupling efficiencies for the two individual modes. The maximum coupling efficiency into the antisymmetric mode, compared to that into the fundamental mode, is ∼ 88% according to FMM and ∼ 59% according to LPIA.

 

Fig. 6 Same as Fig. 5, but for a two-mode waveguide with 2a = 1600 nm. (a) Fundamental mode m = 0. (b) First antisymmetric mode m = 1.

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Finally, Fig. 7 illustrates the results for a waveguide with 2a = 2200 nm, which supports three guided modes. The agreement between FMM and LPIA is again reasonably good for the fundamental mode m = 0 and the antisymmetric mode m = 1, but not for the second symmetric mode m = 2, for which the geometrical propagation angle θ₂ = 11.33° is already rather close to the cut-off θ_c = 11.40°. As the propagation angle approaches the cut-off the field extends much into the cladding region resulting in poor confinement in the core as most of the energy propagates in the cladding region. This results in disagreement between LPIA and FMM. The maximum coupling efficiency into mode m = 0, compared to that into mode m = 1, is now ∼ 90% according to FMM and ∼ 55% of the mode m = 1, compared to that into mode m = 0 according to LPIA. The corresponding ratios for the m = 2 mode are ∼ 11% of mode m = 1 rigorously and ∼ 76% of mode m = 0 according to LPIA, i.e., the rigorously evaluated coupling into this mode is relatively weak. As evident from Media 1, it is clear that with increase in diameter of the waveguide, the in-coupled planar wavefront is slowly modulated to the spherical wavefront due to diffraction, unlike in LPIA where the primary assumption at the waveguide interface was planar wavefront. Likewise with increase in diameter in Figs. 5, 6, and 7, we can see that the LPIA and FMM results seems to diverge slowly. Nevertheless, while the agreement between the two disparate approaches is not perfect, there is considerable similarity even though diffraction has not been included inside the triangular wedge for LPIA.

 

Fig. 7 Same as Fig. 5, but for a three-mode waveguide with 2a = 2200 nm. (a) Fundamental mode m = 0. (b) First antisymmetric mode m = 1. (c) Second symmetric mode m = 2.

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4. Observation of evanescent-wave interference patterns

As a further illustration of the LPIA model we proceed to consider the case of two incident plane waves arriving from the half-space z < 0 at symmetrical angles of incidence ±θ. When θ exceeds the critical angle, a standing interference pattern of two counter-propagating evanescent waves is generated in the region z > 0 if there is no structure above the interface. The magnetic-field intensity of this pattern is

The results of the analysis are shown in Fig. 8. Here we assume the two-mode waveguide considered in Fig. 6 with fixed 2a. We compare the intensity distribution given by Eq. (12) with the normalized fundamental-mode and antisymmetric-mode coupling efficiencies obtained by FMM and LPIA. The two models give indistinguishable results, which also agree with Eq. (12) for the fundamental mode. However, since the planar waveguide is symmetric about the plane x = 0 as considered in Fig. 2, the mode solutions are either even or odd in x. For odd modes H_y(x, z) = −H_y(−x, z) is a solution, which results in half a period phase shift between coupling efficiency curves and the undisturbed interference pattern as shown in Fig. 8(b).

 

Fig. 8 Observation of evanescent-wave interference patterns. Red circles: magnetic-field intensity given by Eq. (12). Normalized coupling efficiencies given by FMM (black crosses) and LPIA (solid blue) into (a) the fundamental mode m = 0 with α = 40° and (b) the antisymmetric mode m = 1 with α = 42°.

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5. Conclusions

We have applied a geometrical-optics-based local-plane-interface approach to model near-field phenomena involving evanescent waves in wavelength-scale structures. Fair agreement between this model and rigorous Fourier Modal Method analysis was found. The results on coupling efficiencies do not match perfectly since the geometrical model ignores diffraction effects within the wedge region, but the agreement is in our opinion better than one might have expected.

6. Appendix

In this Appendix we provide some details of the derivation of the LPIA model, Eqs. (3)–(9). Figure 9 illustrates the propagation of some representative rays through the wedge. The red ray is a typical one, originating at height z and hitting the entrance plane of the waveguide at position x. The black path illustrates the marginal ray incident at the tip of the wedge, z = h, which enters the waveguide boundary at position x = x₀. Finally, the green path illustrates the other marginal ray, which hits the core-cladding boundary at x = a. Only the part of the evanescent field within the range z₀ < z < z₁ contributes significantly to the coupling efficiency since, with the chosen parameters, the part z > z₁ couples primarily to radiation modes.

 

Fig. 9 Ray propagation model inside the wedge.

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Since, according to the geometry of Fig. 9, a = H tanα and

(13)$$b=(H+h-z)\text{tan}\alpha ,$$

we find that

(14)$$f=(z-h)\text{sec}\alpha .$$

Furthermore, β = π/2 + γ − 2α and the propagation direction of the output rays is given by

(15)$$\varphi =3\alpha -\gamma -\pi /2.$$

We also find that the x-coordinate of the ray at z = h + H is related to the z-coordinate of the incident ray through

(16)$$x=c\text{cos}(\alpha -\gamma )-d\text{cos}(3\alpha -\gamma )-(z-h)\text{tan}\alpha$$

and that

(17)$$f\text{sin}(2\alpha )=c\text{cos}(2\alpha -\gamma ),$$

(18)$$H=(z-h)+c\text{sin}(\alpha -\gamma )+d\text{sin}(3\alpha -\gamma ).$$

Inserting Eq. (14) in Eq. (17) yields

(19)$$c=(z-h)\frac{\text{sin}(2\alpha )}{\text{cos}\alpha \text{cos}(2\alpha -\gamma )}.$$

Thus a straightforward calculation using Eq. (18) gives

(20)$$d=\frac{H-(z-h)\text{cos}\gamma \text{sec}(2\alpha -\gamma )}{\text{sin}(3\alpha -\gamma )}.$$

Inserting from Eqs. (13) and (19) in Eq. (16) yields

(21)$$x=(z-h)\frac{\text{cos}\gamma }{\text{cos}\alpha \text{sin}(3\alpha -\gamma )}-H\text{cot}(3\alpha -\gamma )$$

and hence x₀ = −H cot(3α − γ). Expressing the z − h coordinate in terms of x we get

(22)$$(z-h)=\frac{\text{cos}\alpha }{\text{cos}\gamma }\left[x\text{sin}(3\alpha -\gamma )+H\text{cos}(3\alpha -\gamma )\right].$$

The total path length inside the tip is then

(23)$$c+d=(z-h)\frac{\text{sin}(2\alpha )}{\text{cos}\alpha \text{cos}(2\alpha -\gamma )}+\frac{H-(z-h)\text{cos}\gamma \text{sec}(2\alpha -\gamma )}{\text{sin}(3\alpha -\gamma )}.$$

After some further steps we arrive at the final expression of the (linear) phase at the waveguide boundary:

(24)$$\phi (x)=-x{k}_0{n}_1\text{cos}(3\alpha -\gamma )=x{k}_0{n}_1\text{sin}\varphi .$$

Acknowledgments

This work was supported by the Academy of Finland, project 252910.

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