1. Introduction

Modeling of scattering from finite periodic gratings have become important in the field of micro-optics for optical switching and multiplexing/demultiplexing applications. The accurate simulation of scattered near-field from finite gratings can clearly show the phase interactions in the near field. This in turn opens a rich opportunity for optical near-field manipulation of nano and micro-optical device.

On first glance, one can view the problem of scattering from finite number of arbitrary grooves as a superposition of scattering from single (isolated) groove with an appropriate accounting of phase differences for the locations of various grooves (superposition approach). Treating the mutual coupling between the grooves as insignificant will make the formulation straight forward and easy to implement. In order to study the importance of such coupling, here we extended the formulation of the scattering from a single arbitrary-shaped groove [1] to two grooves problem. The importance and effect of the mutual coupling will be studied based on the calculation of the scattered near and far-fields using the superposition and exact analysis for the two grooves models for different separation period between the two grooves and different angle of incident.

2. Problem formulation

Figure 1 depicts the geometry of the two dimensional scattering problem from two arbitrary shaped-grooves in a perfectly conducting plane. For simplicity, the formulation assumes an identical general-shaped grooves with groove width of 2a and the period between the two groove is T. The problem space is divided into two regions: upper half space (region I) and the two general-shaped grooves region (region II). In region (I) (z > 0), a uniform plane electromagnetic wave polarized along the grooves axis (y-axis), E^inc_y(x,z), is assumed to be incident on the grooves with an incident angle θ_inc. The wave numbers in region (I) and (II) are $k_0=\omega \sqrt{{\mu }_0{\epsilon }_0}\text{and}k=\omega \sqrt{\mathrm{\mu \epsilon }},$ respectively. Throughout the paper, the assumed exp (jωt) time-harmonic factor is suppressed.

 

Fig. 1. Schematic diagram of plane wave scattering from two general-shaped grooves

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The total electric field in the upper-half space (region I) is considered a summation of incident E_y^inc, reflected E_y^r, and scattered E_y^s fields

where, $k_x=k_0\sin {\theta }_{\mathrm{inc}},k_z=k_0\cos {\theta }_{\mathrm{inc}},\mathrm{and}{\kappa }_0=\pm \sqrt{k_0^2-{\zeta }^2}.$ In region II, the field representations inside both groove are identical with different unknown harmonics coefficients. In earlier work [1], each general-shaped groove is divided into L number of layers, with individual layer thickness of d and total groove depth D = Ld. The width of an arbitrary inter-layer l is 2w_l = a_l + b_l. The first layer in each groove (l = 1) has a common interface with the upper-half space. Fields in the inter-layers of the general-shaped grooves, through a stair-case approximation [2,3], must satisfy Maxwell’s equations subject to the boundary conditions at the perfectly conducting walls of the grooves. The tangential electric field component in the first layer(l = 1) for both grooves, can be written as:

Here, A_n^P,l and B^p,l_n are the nth unknown harmonic coefficients of the first layer (l=1) where p =0 or 1 for the first and second grooves, respectively. Throughout the paper, subscript“n” is the harmonic index and superscripts “p” and “l” are the groove and layer numbers, respectively. β_n ¹ = nπ/2a and ζ_n ¹ = (k ² - β_n ¹)^1/2 are the nth harmonic field variation along x and z-axis in the first layer (l = 1), respectively. For the identical grooves case considered here, β_n ¹ and ζ_n ¹ are identical for each groove (hence superscript p is redundant). Application of the boundary conditions between tangential electric and magnetic fields in general-shaped groove interlayers result in a linear set of equations by combining all unknown space harmonics coefficients of the general-shaped groove. Details of the derivation can be found in reference [3].

Field matching at the region I and II interface (z = 0)

The tangential electric and magnetic-fields continuity at the interface z = 0 along x-axis gives

where

and

This integration can be determined by residue calculus [4–6]. More details can be found in the appendix. Rewriting (5) in a matrix form

where,

Here, C^q(q,p), CA^q,p(q,p), and CB^q,p(q,p) are elements of the matrices C ^q, CA ^q,p, CB ^q,p, respectively, and q,p = 0,1. CA ^q,P and CB ^q,p are square matrices of dimensions equal to number of harmonics (N ₁) in the first layer (l = 1) of all grooves.

Solving (20) combined with the equations from boundary conditions between grooves inter-layers [1] for the harmonic coefficients A ^0,1, A ^1,1, B ^0,1 and A ^1,1 of the electric fields in the first layers of both grooves which are the scattered near-field at the grooves interface. Once the harmonic coefficients are calculated, the far-field can be determined the scattered field formula from (3) using the saddle-point method [7]. In the superposition method, the amplitudes of the e-field are identical in both grooves, while there are a phase difference between the fields of the two grooves equal to k_xTsinθ_inc.

3. Numerical simulation results

we conducted rigorous simulations of scattering from two isosceles right triangle (IRT) grooves using the superposition analysis and formulation from [1] and the exact formulation presented in section 2 in this paper. The two grooves are assumed to be identical with each side 2a = 1.2λ and the period between the two grooves is T as shown in Fig. 1, and λ being the operating wavelength is assumed to be equal to μm. The incident electric field is polarized along the grooves axis (y-axis in Fig. 1) with an incident angle θ_inc. The scattered near-field (magnitude and phase) at each groove opening and far-field are calculated for different separation distances T between the two grooves for different incident angles while keeping all other parameters unchanged. The separation distance T between the grooves is varied from a small fraction of (λ) to tens of λ. The simulations are performed at θ_inc = 0°, 45°, and 60°.

Figure 2 and 3show plots of the scattered near-field (magnitude and phase) at the groove interface for nine different cases by changing the incident angle and the groove periods while Fig. 4 shows the scattered far-field for the same cases. The two extreme cases are when T = 1.3 λ and θ_inc = 60° (strong coupling effects) and T = 13λ and θ_inc = 0° (negligible coupling effects). The near-field ,magnitude and phase, and far-fields for the first case are significantly different. These difference in the simulation results decreases when increasing the groove separation distance and decreasing the angle of incidence. In the second extreme case no noticeable differences in the scattered near-field between the superposition and exact analysis for the two grooves simulations, and little difference in the far-field plots were observed. As expected, the differences are accounted for by the significant coupling between the two grooves which is missing in the superposition simulations. The simulation results indicate significant dependency of the coupling on the incident angle and the groove separation distance. Even for large separation distance, small difference in the simulation results between the superposition and exact simulations are noticed for larger angle of incident in the far-field plot in Fig 4(c). For large value of T and the special case of normal incidence, coupling between the two grooves starts to be insignificant and superposition analysis with appropriate care of the phase difference between the fields in the grooves is sufficient.

Most of the finite grating have a period T that is comparable to λ. In addition, the incident angle is usually oblique for almost all applications. Hence the coupling between grooves is significant and need to be included in the calculations. The inter-groove couplings are more significant in the case of non-identical grooves, where individual scattering fields at each groove is no longer identical. Another noteworthy point is that the coupling between grooves depends on the relative location of the groove itself. Coupling strengths differ if the groove is in the middle of other grooves or at the edges (tapering effects).

The above numerical experiment quantitatively shows the importance of coupling between grooves in the scattering problem under consideration here. Thus we were motivated to continue developing the rigorous formulation of scattering from finite number of arbitrary-shaped grooves. We realized that the extension from a single groove to finite number of grooves is not straight forward, as the superposition approach do not provide sufficient accuracy required for near-field designs. We were able to produce an efficient and flexible numerical implementation for practical problems that was not obtainable intuitively from superposition approach to the problem. Based on this numerical study, we recommend that in multi-groove scattering formulations, when the angle of incidence exceeds 30 degrees from broadside and the inter-groove separation is less than a wavelength or so the inter-groove coupling should not be neglected.

4. Conclusions

A quantitative study of the importance of coupling between grooves was proved to be significant in formulating near and far-field em scattering problems from finite array of grooves. The study results show a significant contribution of the grooves separation distance and the incident angle of the electromagnetic waves on the scattered near and far-fields. This effect is minimized for normal angle of incident and large groove period. Significant effects were observed for sub-wavelength separation distance and large angle on incidence. It is noted that, in the near field design considerations, often accurate knowledge of the location of near-null fields is very important. Our results indicate that significant error would arise if the coupling effects were not taken into account properly.

This study proves for the importance of extending the current method to finite number of non-identical grooves, where fully numerical simulations are required to solve this type of problems.

5. Appendix A

The integration in 6 will be determined by the residue calculus and contour integrations methods. The integration to be to be evaluated is:

(12)$$R_{m,n}^{q,p}\left(k_0\right)={\int }_{-\infty }^{\infty }{a}^2{G}_n\left(\mathrm{\zeta a}\right)G_m\left(-\mathrm{\zeta a}\right)e^{-j\left(p-q\right)\mathrm{T\zeta }}{\kappa }_o^2\mathrm{d\zeta }.$$

where s = q - p and

(13)$$f\left(\zeta \right)=\frac{\left({\left(-1\right)}^{m+n}+1\right)e^{\mathrm{jsT\zeta }}-{\left(-1\right)}^m{e}^{j\left(\mathrm{sT}+2a\right)\zeta }-{\left(-1\right)}^n{e}^{j\left(\mathrm{sT}-2a\right)\zeta }}{\left({\zeta }^2-{\beta }_n^2\right)\left({\zeta }^2-{\beta }_m^2\right)}{\kappa }_o^2$$

The contour integration ∮f(ζ)dζ will be performed along Γ contours as shown in Fig. 5 where the required integration is R^S_m,n(k ₀) = ∫Γ₁ f(ζ)dζ. The special case when s = 0,R^S_m,n(k_O) will be the same as R_m,n(k_O) for a single groove. Details of the calculation can be found in [5]. In general

(14)$$R_{m,n}^s\left(k_0\right)=\frac{2\mathrm{\pi a}}{{\beta }_m^2}\beta \sqrt{k_0^2-{\beta }_m^2}{\delta }_{\mathrm{mn}}{\delta }_{\mathrm{qp}}-2I$$

Where I is the integration along the branch cut. Note that the integration along the branch cuts Γ₅ and Γ₇ are equal. The first term of the integration is the reside value when m = n and q = p. If s ≠ 0 and m + n : odd, then

(15)$$f\left(\zeta \right)={\left(-1\right)}^{m+1}\frac{e^{j\left(\mathrm{sT}+2a\right)\zeta }-e^j(\mathrm{sT}-2a)\zeta }{\left({\zeta }^2-{\beta }_n^2\right)\left({\zeta }^2-{\beta }_m^2\right)}{\kappa }_o^2$$

f(ζ) has four simple poles at ±β_n and ±β_m and no second order poles since m ≠ n. The residues at these four simple poles are Res_ζ±βm f(ζ) = 0 and Res_ζ=±βn f(ζ) = 0. If s > 0, then the term (sT -2a) is always positive and the branch cut integration I will be performed in the upper half space of the complex plane. let ζ = k ₀ + jk ₀ v and dζ, = jk ₀ dv where v = 0 → ∞. The branch cut integration is

(16)$$I=\frac{j{\left(-1\right)}^{m+1}}{k_0^2}{\int }_0^{\infty }\frac{e^{\left({\mathrm{jk}}_0-{\mathrm{vk}}_0\right)\left(\mathrm{sT}+2a\right)}-e^{\left({\mathrm{jk}}_0-{\mathrm{vk}}_0\right)\left(\mathrm{sT}-2a\right)\zeta }}{\left({\left(1+\mathrm{jv}\right)}^2-{\beta }^2\right)\left({\left(1+\mathrm{jv}\right)}^2-{\alpha }^2\right)}\sqrt{v\left(-j2+v\right)}\mathrm{dv}={-R}_{m,n}^s\left(k_0\right)$$

If s < 0, then the term (sT + 2a) is always negative and the branch cut integration I will be performed in the lower half space of the complex plane. let s → -s, then

(17)$$f\left(\zeta \right)={\left(-1\right)}^m\frac{e^{-j\left(\mathrm{sT}+2a\right)\zeta }-e^{-j\left(\mathrm{sT}-2a\right)\zeta }}{\left({\zeta }^2-{\beta }_n^2\right)\left({\zeta }^2-{\beta }_m^2\right)}{\kappa }_o^2$$

let ζ = - (k ₀ + jk ₀ v) and dζ = -jk ₀ dv where v = 0 → ∞. The branch cut integration is

(18)$$I=\frac{j{\left(-1\right)}^{m+1}}{k_0^2}{\int }_0^{\infty }\frac{e^{\left({\mathrm{jk}}_0-{\mathrm{vk}}_0\right)\left(\mathrm{sT}+2a\right)}-e^{\left({\mathrm{jk}}_0-{\mathrm{vk}}_0\right)\left(\mathrm{sT}-2a\right)\zeta }}{\left({\left(1+\mathrm{jv}\right)}^2-{\beta }^2\right)\left({\left(1+\mathrm{jv}\right)}^2-{\alpha }^2\right)}\sqrt{v\left(-j2+v\right)}\mathrm{dv}={-R}_{m,n}^{-s}\left(k_0\right)$$

In general for m + n: odd and any s, R^-S_m,n = R^S_m,n. Similarly we can prove the same case when m + n is even that R^-S_m,n = R^S_m,n ,Hence for any m, n, and s R^S_m,n can be written as

(19)$$R_{m,n}^s={\int }_{-\infty }^{\infty }\frac{\left({\left(-1\right)}^{m+n}+1\right)e^{j\mid \mathrm{sT}\mid \zeta }+{\left(-1\right)}^{m+1}{e}^{j\mid \mathrm{sT}\mid +2a\mid \zeta }+{\left(-1\right)}^{n+1}{e}^{j\mid \mathrm{sT}-2a\mid \zeta }}{\left({\zeta }^2-{\beta }_n^2\right)\left({\zeta }^2-{\beta }_m^2\right)}{\kappa }_o^2\mathrm{dv}$$

And the branch cut integration is

(20)$$I=\left(1+{\left(-1\right)}^{m+n}\right)I_o\left(\mathrm{sT}\right)+{\left(-1\right)}^{m+1})I_o\left(\mathrm{sT}+2a\right)+{\left(-1\right)}^{n+1})I_o\left(\mathrm{sT}-2a\right)$$

where α ² =β_n ²/k ₀ ²,β ² = β_m ²/k ₀ ², and

(21)$$I_0\left(x\right)={\int }_0^{\infty }\frac{{\mathrm{je}}^{j{k}_0\mid x\mid }{e}^{-k_0\mid x\mid v}\sqrt{v\left(-j2+v\right)}}{k_0^2\left({\left(1+\mathrm{jv}\right)}^2-{\beta }^2\right)\left({\left(1+\mathrm{jv}\right)}^2-{\alpha }^2\right)}\mathrm{dv}$$

This integration will be determined by numerical method. The integration will decay very fast due the term e ^-k₀~x~v. The integrand is a well behaved function and numerical integration will be efficient. In the other hand, I ₀(x) can also be expressed in terms of asymptotic series approximation [5].

 

Fig. 2. Comparison between the scattered near-field using the superposition and the exact formulations of two grooves for different incident angle θ_inc and period T where (a) T = 1.3λ, (b) T = 5λ, and (c) T = 13λ.

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Fig. 3. Comparison between phases of scattered near-field using the superposition and the exact formulations of two grooves for different incident angle θ_inc and period, where T (a) T = 1.3λ, (b) T = 5λ, and (c) T = 13λ.

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Fig. 4. Comparison between the scattered far-field using the superposition and the exact formulations of two grooves for different incident angle θ_inc and period T, where (a) T = 1.3λ, (b) T = 5λ, and (c) T = 13λ.

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Fig. 5. Contour path in the complex Γ-plane [5]

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References and links

1. M. A. Basha, S. Chaudhuri, S. Safavi-Naeini, and H. J. Eom, “Rigorous formulation for electromagnetic plane wave scattering from a general shaped-groove in a perfectly conducting plane,” J. Opt. Soc. Am. A. 24, 1647–1655 (2007). [CrossRef]  

2. Z. Ma and E. Yamashita, “Modal analysis of open groove guide with arbitrary groove profile,rdquo; IEEE Microwave Guided Wave Lett. 2, 364–366 (2007). [CrossRef]  

3. M. A. Basha, “Optical MEMS Switches: Theory, Design, and Fabrication of a New Architecture,” PhD Thesis, Electrical and Computer Engineering, University of Waterloo (2007), (http://uwspace.uwaterloo.ca/handle/10012/3116).

4. H.J. Eom, “Electromagnetic Wave Theory for Boundary-Value Problems,” (chapter 7),(Springer-Verlag, 2004).

5. H.J. Eom, “Wave Scattering Theory: A series Approach Baesd on The Fourier Transform,” (chapter 1) (Springer-Verlag, 2001).

6. P. M. Morse and H. Feshbach. newblock Method of theoretical Physics, volume I. McGraw-Hill Book Company (1953).

7. J. A. Kong, “Electromagnetic Wave Theory,” (New York: John Wiley & Sons, c1990).