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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 18, Iss. 20 — Sep. 27, 2010
  • pp: 21083–21089
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Coupling interaction of electromagnetic wave in a groove doublet configuration

Lan Ding, Jinsong Liu, Dong Wang, and Kejia Wang  »View Author Affiliations


Optics Express, Vol. 18, Issue 20, pp. 21083-21089 (2010)
http://dx.doi.org/10.1364/OE.18.021083


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Abstract

Based on the waveguide mode (WGM) method, coupling interaction of electromagnetic wave in a groove doublet configuration is studied. The formulation obtained by WGM method for a single groove [Prog. Electromagn. Res. 18, 1–17 (1998)] is extended to two grooves. By exploring the total scattered field of the configuration, coupling interaction ratios are defined to describe the interaction between grooves quantitatively. Since each groove in this groove doublet configuration is regarded as the basic unit, the effects of coupling interaction on the scattered fields of each groove can be investigated respectively. Numerical results show that an oscillatory behavior of coupling interaction is damped with increasing groove spacing. The incident and scattering angle dependence of coupling interaction is symmetrical when the two grooves are the same. For the case of two subwavelength grooves, the coupling interaction is not sensitive to the incident angle and scattering angle. Although the case of two grooves is discussed for simplicity, the formulation developed in this article can be generalized to arbitrary number of grooves. Moreover, our study offers a simple alternative to investigate and design metallic gratings, compact directional antennas, couplers, and other devices especially in low frequency regime such as THz and microwave domain.

© 2010 OSA

1. Introduction

The scattering of electromagnetic wave on a finite collection of grooves has recently sparked a wealth of experimental and theoretical works, which are aiming at understanding the underlying physics and at developing new applications of wave manipulation. In the past twenty years, a lot of methods, such as impedance boundary condition approach [1

1. K. Barkeshli and J. L. Volakis, “TE scattering by a two-dimensional groove in a ground plane using higher order boundary conditions,” IEEE Trans. Antenn. Propag. 38(9), 1421–1428 (1990). [CrossRef]

,2

2. K. W. Whites, E. Michielssen, and R. Mittra, “Approximating the scattering by a material-filled 2-D trough in an infinite plane using the impedance boundary condition,” IEEE Trans. Antenn. Propag. 41(2), 146–153 (1993). [CrossRef]

] and rigorous coupled-wave analysis [3

3. M. G. Moharam, E. B. Grann, D. A. Pommet, and T. K. Gaylord,, “Formulation for stable and efficient implementation of the rigorous coupled-wave analysis of binary gratings,” J. Opt. Soc. Am. A 12(5), 1068–1076 (1995). [CrossRef]

], were established to study this scattering problem. Among these methods, the waveguide mode (WGM) method [4

4. M. A. Morgan and F. K. Schwering, “Mode expansion solution for scattering by a material filled rectangular groove,” Prog. Electromagn. Res. PIER 18, 1–17 (1998). [CrossRef]

], which was formulated for the case of an isolated groove, required neither matrix inversions nor special function evaluations, thus it is efficient in computation. For a finite collection of grooves, treating the coupling interaction between the grooves as insignificant will make the formulation of WGM straight forward and easy to implement. However, although the importance of mutual coupling between grooves was demonstrated in [5

5. M. A. Basha, S. Chaudhuri, and S. Safavi-Naeini, “A study of coupling interactions in finite arbitrarily-shaped grooves in electromagnetic scattering problem,” Opt. Express 18(3), 2743–2752 (2010). [CrossRef] [PubMed]

], the formulation of WGM including the coupling interaction has not been developed for a finite collection of grooves.

In this paper, based on the WGM method, coupling interaction of electromagnetic wave in a groove doublet configuration is studied. The formulation obtained by WGM method for a single groove [4

4. M. A. Morgan and F. K. Schwering, “Mode expansion solution for scattering by a material filled rectangular groove,” Prog. Electromagn. Res. PIER 18, 1–17 (1998). [CrossRef]

] is extended to two grooves. By exploring the total scattered field of the groove doublet configuration, coupling interaction ratios are defined to describe the interaction between grooves quantitatively. Since each of the two grooves is regarded as the basic unit here, one can even investigate the effects of coupling interaction on the scattered fields of each groove respectively. Numerical results show that an oscillatory behavior of coupling interaction is damped with increasing groove spacing. The incident and scattering angle dependence of coupling interaction is symmetrical when the two grooves are the same. For the case of two subwavelength grooves, the coupling interaction is not sensitive to the incident angle and scattering angle. It should be noted that the results obtained here can be generalized to a finite collection of grooves. Moreover, our study offers a simple alternative to investigate and design scattering devices, such as metallic gratings, compact directional antennas, and couplers, etc.

2. Problem formulation

In WGM method, the incident field is replaced by an impressed surface current in the aperture, and fields in and above the cavity are represented by the use of waveguide modes [4

4. M. A. Morgan and F. K. Schwering, “Mode expansion solution for scattering by a material filled rectangular groove,” Prog. Electromagn. Res. PIER 18, 1–17 (1998). [CrossRef]

]. Based on this simplified approach, coupling interaction of electromagnetic wave in a groove doublet configuration is studied here. For simplicity, the configuration is assumed as two rectangular grooves in a perfect electrical conductor (PEC) plane, as shown in Fig. 1
Fig. 1 Schematic of the groove doublet configuration, including the geometrical parameters W 1, W 2, d 1, d 2, and Λ. A THz plane TM wave is incident upon the configuration, with the angle of incidence θi. The material in both regions are the same, where εr I = εr II = εr, and μr I = μr II = μr.
. A transverse magnetic (TM) plane wave ( Ei=Eixx^+Eizz^ , and Hi=Hiyy^ ) is obliquely incident at an arbitrary angle of incidence θi, which is between the incident wave vector k 0 and the z direction, upon the configuration.

In this groove doublet configuration, each groove can be regarded as the basic unit. The compound incident field for each groove consists of the TM plane wave and the field scattered by the other groove. For example, the compound incident wave H2icom for groove 2 can be represented as Hi+H1S , where H1S is the scattered field of groove 1 obtained by ignoring the effect of groove 2. In other words, the compound scattered field of groove 2 can be expressed as H2Scom=H2S+H2S , in which H2S is the scattered field generated directly by H i and H2S is the additional scattered field generated by H1S , as illustrated in Fig. 1. Since the additional scattered field has been taken into account, the coupling interaction of electromagnetic wave between grooves is included here. In this section, we first explore the compound scattered field of each groove. Then, the total scattered field of the groove doublet configuration can be obtained as HStotal=H1Scom+H2Scom .

Next, we will deduce the additional scattered field H2S , which is generated by H1S . For simplicity, we only consider the scattered field propagating in the + x direction from groove 1. Substituting ρ1=x and θ1=π/2 into Eq. (1), we can obtain
H1S(x)H0W1[n1=0N1c1n1L1n1(π2)]ik02πxeik0xy^=H2eik0xxy^,
(4)
in which H2 represents

H2=H0W1[n1=0N1c1n1L1n1(π2)]ik02π.
(5)

Fourier cosine series are used to represent the additional impressed surface current J2S for the scattered field H1S generated by groove 1. These series can be written as
J2S=2H2eik0xxx^H2n2=0N2a2n2cos[n2πW2(xΛ)]x^,
(6)
where Fourier cosine series integration yield

a2n2=2W2ΛΛ+W2eik0xxcos[n2πW2(xΛ)]dx.
(7)

The cavity modes for groove 2 are formed from pairs of ±z directed parallel plate waveguide modes [6

6. R. E. Collin, Field Theory of Guided Waves (IEEE Press, New York, 1991).

] to satisfy E2||(x,d2)=0 on the groove bottom. Field components parallel to the aperture are given for d2z0 by
H2y(x,z)H2n2=0N2b2n2cos[n2πW2(xΛ)]cosh[γ2n2(z+d2)],
(8a)
E2x(x,z)iE2k0εr0n2=0N2b2n2γ2n2cos[n2πW2(xΛ)]sinh[γ2n2(z+d2)],
(8b)
in which γ2n2=(n2π/W2)2k02εrμr .

Above groove 2, the fields are expanded for +z directed waveguide modes as
H2y(x,z)H2n2=0N2c2n2cos[n2πW2(xΛ)]exp(u2n2z),
(9a)
E2x(x,z)iE2k0εr0n2=0N2c2n2u2n2cos[n2πW2(xΛ)]exp(u2n2z),
(9b)
where u2n2=γ2n2=(n2π/W2)2k02εrμr .

By enforcing continuity conditions at the aperture boundary, field expansion coefficients are found as

c2n2=sinh(γ2n2d2)b2n2=[exp(2u2n2d2)1]a2n2.
(10)

As mentioned above, the compound scattered field of groove 2 consists of the scattered field generated directly by the TM plane wave and the additional field generated by H1S , where H1S is given by Eq. (1). Thus, the compound far-zone scattered field of groove 2 can be represented as
H2Scom(ρ2,θ2)=H2S+H2S=H0W2ik02πρ2eik0ρ2n2=0N2(c2n2+Fn212)L2n2(θ2)y^,
(12)
where Fn212 describes the electromagnetic mode effect of groove 1 on groove 2. Comparing Eq. (12) with Eq. (1) and (5), we can obtain

Fn212=W1[n1=0N1c1n1L1n1(π2)]ik02πc2n2.
(13)

By using the same method discussed above, the additional scattered field H1S and the compound scattered field of groove 1 can also be obtained
H1Scom(ρ1,θ1)=H1S+H1S=H0W1ik02πρ1eik0ρ1n1=0N1(c1n1+Fn121)L1n1(θ1)y^,
(14)
in which Fn121 describes the effect of groove 2 on groove 1, and has the form

Fn121=W2[n2=0N2c2n2L2n2(π2)]ik02πc1n1.
(15)

In Eq. (15), the field expansion coefficients are decided by c1n1=[exp(2u1n1d1)1]a1n1 , where Fourier cosine series integration yield

a1n1=2W10W1eik0(Λx)Λxcos(n1πW1x)dx.
(16)

It should be noted that the polar coordinate origin of Eq. (12) is set at (x = Λ, z = 0), while that of Eq. (14) is set at (x = 0, z = 0). The compound scattered fields of groove 1 and groove 2 are given by Eq. (12) and Eq. (14), respectively. We can analyze the effect of coupling interaction on each of the grooves through the two equations. Then, by using of coordinate transformation and vector addition, the total scattered field of the groove doublet configuration can be expressed as

HStotal(ρ1,θ1)=H1Scom(ρ1,θ1)+H2Scom(ρ2(ρ1,θ1),θ2(ρ1,θ1)).
(17)

In Eq. (17), the process of coordinate transformation has the form

{ρ2(ρ1,θ1)=ρ12+Λ22Λρ1sinθ1)θ2(ρ1,θ1)=arctan(tanθ1Λρ1cosθ1).
(18)

According to Eq. (12) to (18), coupling interaction has been included in the compound field of each groove and the total scattered field of the groove doublet configuration. Equation (17) is the main result of this paper.

Although the case of two grooves is discussed for simplicity in this article, the formulation developed for the groove doublet configuration can be generalized to arbitrary number of grooves. We take a configuration consisted of m grooves (m>2) for an example. The total scattered field HStotal can be represented as l=1mHlScom(ρl,θl) , in which (ρl,θl) can be expressed by (ρ1,θ1) through coordinate transformation. It means that Eq. (17) and Eq. (18) can be generalized to the case of m grooves. It should be noted that the compound scattered field HlScom for 1<l<m contains two additional scattered fields generated by the two adjacent grooves respectively.

3. Numerical results

To investigate the effects of coupling interaction on the total scattered field, we pay our attention on the scattering width σ (also referred to as radar cross section per unit length), which represents the scattering ability of a scatterer and can be evaluated using Green’s function integration [4

4. M. A. Morgan and F. K. Schwering, “Mode expansion solution for scattering by a material filled rectangular groove,” Prog. Electromagn. Res. PIER 18, 1–17 (1998). [CrossRef]

]. Based on the concept of scattering width, we propose a coupling interaction ratio (CIR) to describe the interaction between grooves quantitatively, which can be defined as
CIR=σtotalσσ=limρ12πρ1(|HStotal|2|HS|2)/|HS|2,
(19)
where H S can be obtained by H 1 S + H 2 S, σtotal is the scattering width corresponding to the total scattered field with coupling interaction, and σ is the one ignoring the coupling interaction. It is clear that the coupling interaction gets weaker when CIR approaches 0 while it gets stronger when CIR departs from 0. In order to establish a concept of magnitude, we carried out the calculation of Eq. (19) for different Λ, θi, and θ 1 with given εr , μr , W 1, W 2, d 1 and d 2. To make the numerical results clear, all size parameters of the grooves are compared with the incident wavelength λ 0, thus the two grooves are either subwavelength or non-subwavelength. The groove doublet configurations can be categorized into three types as the insets to Fig. 2
Fig. 2 The CIR versus the groove spacing for (a) D-S, (b) D-NS, and (c) S-NS, with θi = θ 1 = 0, εr = 1, and μr = 1.
, which are named as double-subwavelength (D-S), double-non-subwavelength (D-NS), and subwavelength-non-subwavelength (S-NS), respectively.

Figure 2 shows the groove spacing (Λ) dependence of CIR. It is clear that CIR approaches 0 with increasing Λ, which means that coupling interaction between the two grooves decreases when the spacing between them is increasing. Meanwhile, total scattered field of the configuration has been strengthened/weakened by considering the coupling interaction when CIR is more/less than zero. Moreover, a damped oscillatory behavior with period of λ 0, which is similar to slit-groove interaction [7

7. L. Chen, J. T. Robinson, and M. Lipson, “Role of radiation and surface plasmon polaritons in the optical interactions between a nano-slit and a nano-groove on a metal surface,” Opt. Express 14(26), 12629–12636 (2006). [CrossRef] [PubMed]

], can be observed in Fig. 2. These oscillations come from interference between the compound scattered fields of groove 1 and 2.

The incident angle (θi) and scattering angle (θ 1) dependence of CIR for D-S, D-NS, and S-DS is displayed in Fig. 3
Fig. 3 The CIR versus the incident angle and scattering angle for (a) D-S with Λ = 2.1λ 0, (b) D-NS with Λ = 5λ 0, and (c) S-NS with Λ = 3.7λ 0, when εr = 1, and μr = 1.
. When the two grooves of the configuration are the same such as the cases of D-S and D-NS, the angle dependence of CIR is symmetrical, as illustrated in Fig. 3(a) and 3(b). Furthermore, one can see from Fig. 3 that the CIR surface changes smooth for D-S, and those for D-NS and S-DS change sharp. We have also calculated the angle dependence of CIR for different sizes of the three types of configurations (not displayed here), and the similar results have been obtained. It means that the coupling interaction is not sensitive to θi and θ 1 when both of the two grooves are subwavelength.

These numerical examples indicate that the introducing of CIR is reasonable to study coupling interaction between grooves. Based on the results obtained above, the coupling interaction in the groove doublet configuration could be conveniently analyzed and designed by the formulation developed here.

Since our formulation is based on WGM, the numerical simulation is rapid, and has improved accuracy with increasing electrical size [4

4. M. A. Morgan and F. K. Schwering, “Mode expansion solution for scattering by a material filled rectangular groove,” Prog. Electromagn. Res. PIER 18, 1–17 (1998). [CrossRef]

]. Moreover, as mentioned in part 2, since the compound scattered fields of groove 1 and 2 are deduced respectively, our approach would be more convenient when the two grooves are filled with different materials. However, the material of the groove doublet configuration is assumed as perfect electrical conductor (PEC), thus our results are applicable for metallic configurations in low frequency regime such as THz and microwave domain. Surface plasmon polaritons (SPPs) may be stimulated by optical waves on metallic groove configurations under some conditions. Then the coupling interaction between grooves will be different, thus the results obtained here will not be applicable any more.

4. Conclusion

In this article, coupling interaction of electromagnetic wave in a groove doublet configuration is studied based on WGM method. The formulation of total scattered field including coupling interaction between two grooves has been developed. To investigate the coupling interaction quantitatively, CIR is introduced and defined. Numerical results show that an oscillatory behavior of coupling interaction is damped with increasing groove spacing. The incident and scattering angle dependence of coupling interaction is symmetrical when the two grooves are the same. For the case of two subwavelength grooves, the coupling interaction is not sensitive to the incident angle and scattering angle. Although the case of two grooves is discussed for simplicity, the formulation developed in this article can be generalized to arbitrary number of grooves.

Moreover, our study offers a simple alternative to investigate and design metallic gratings, compact directional antennas, couplers, and other devices especially in low frequency regime such as THz and microwave domain.

Acknowledgments

The National Natural Science Foundation of China under grant No. 10974063, and the Research Foundation of Wuhan National Laboratory under Grant No. P080008 have supported this research.

References and links

1.

K. Barkeshli and J. L. Volakis, “TE scattering by a two-dimensional groove in a ground plane using higher order boundary conditions,” IEEE Trans. Antenn. Propag. 38(9), 1421–1428 (1990). [CrossRef]

2.

K. W. Whites, E. Michielssen, and R. Mittra, “Approximating the scattering by a material-filled 2-D trough in an infinite plane using the impedance boundary condition,” IEEE Trans. Antenn. Propag. 41(2), 146–153 (1993). [CrossRef]

3.

M. G. Moharam, E. B. Grann, D. A. Pommet, and T. K. Gaylord,, “Formulation for stable and efficient implementation of the rigorous coupled-wave analysis of binary gratings,” J. Opt. Soc. Am. A 12(5), 1068–1076 (1995). [CrossRef]

4.

M. A. Morgan and F. K. Schwering, “Mode expansion solution for scattering by a material filled rectangular groove,” Prog. Electromagn. Res. PIER 18, 1–17 (1998). [CrossRef]

5.

M. A. Basha, S. Chaudhuri, and S. Safavi-Naeini, “A study of coupling interactions in finite arbitrarily-shaped grooves in electromagnetic scattering problem,” Opt. Express 18(3), 2743–2752 (2010). [CrossRef] [PubMed]

6.

R. E. Collin, Field Theory of Guided Waves (IEEE Press, New York, 1991).

7.

L. Chen, J. T. Robinson, and M. Lipson, “Role of radiation and surface plasmon polaritons in the optical interactions between a nano-slit and a nano-groove on a metal surface,” Opt. Express 14(26), 12629–12636 (2006). [CrossRef] [PubMed]

OCIS Codes
(050.0050) Diffraction and gratings : Diffraction and gratings
(050.2770) Diffraction and gratings : Gratings
(290.5880) Scattering : Scattering, rough surfaces
(040.2235) Detectors : Far infrared or terahertz

ToC Category:
Scattering

History
Original Manuscript: July 21, 2010
Revised Manuscript: September 8, 2010
Manuscript Accepted: September 9, 2010
Published: September 21, 2010

Citation
Lan Ding, Jinsong Liu, Dong Wang, and Kejia Wang, "Coupling interaction of electromagnetic wave in a groove doublet configuration," Opt. Express 18, 21083-21089 (2010)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-20-21083


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References

  1. K. Barkeshli and J. L. Volakis, “TE scattering by a two-dimensional groove in a ground plane using higher order boundary conditions,” IEEE Trans. Antenn. Propag. 38(9), 1421–1428 (1990). [CrossRef]
  2. K. W. Whites, E. Michielssen, and R. Mittra, “Approximating the scattering by a material-filled 2-D trough in an infinite plane using the impedance boundary condition,” IEEE Trans. Antenn. Propag. 41(2), 146–153 (1993). [CrossRef]
  3. M. G. Moharam, E. B. Grann, D. A. Pommet, and T. K. Gaylord, “Formulation for stable and efficient implementation of the rigorous coupled-wave analysis of binary gratings,” J. Opt. Soc. Am. A 12(5), 1068–1076 (1995). [CrossRef]
  4. M. A. Morgan and F. K. Schwering, “Mode expansion solution for scattering by a material filled rectangular groove,” Prog. Electromagn. Res. PIER 18, 1–17 (1998). [CrossRef]
  5. M. A. Basha, S. Chaudhuri, and S. Safavi-Naeini, “A study of coupling interactions in finite arbitrarily-shaped grooves in electromagnetic scattering problem,” Opt. Express 18(3), 2743–2752 (2010). [CrossRef] [PubMed]
  6. R. E. Collin, Field Theory of Guided Waves (IEEE Press, New York, 1991).
  7. L. Chen, J. T. Robinson, and M. Lipson, “Role of radiation and surface plasmon polaritons in the optical interactions between a nano-slit and a nano-groove on a metal surface,” Opt. Express 14(26), 12629–12636 (2006). [CrossRef] [PubMed]

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