## Coupling interaction of electromagnetic wave in a groove doublet configuration |

Optics Express, Vol. 18, Issue 20, pp. 21083-21089 (2010)

http://dx.doi.org/10.1364/OE.18.021083

Acrobat PDF (1341 KB)

### 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

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]

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]

## 2. Problem formulation

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]

*θ*, which is between the incident wave vector

_{i}

*k*_{0}and the

*z*direction, upon the configuration.

**H**

*and*

_{i}**PIER 18**, 1–17 (1998). [CrossRef]

**H**

*can be expressed as where*

_{i}*j =*1, 2 represent groove 1 and groove 2, respectively,

*θ*is the scattering angle, and

_{j}**PIER 18**, 1–17 (1998). [CrossRef]

*x*direction from groove 1. Substituting

*θ*

_{2}is the scattering angle, and

*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

*m*grooves (

*m*>2) for an example. The total scattered field

*m*grooves. It should be noted that the compound scattered field

*l*<

*m*contains two additional scattered fields generated by the two adjacent grooves respectively.

## 3. Numerical results

*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

**PIER 18**, 1–17 (1998). [CrossRef]

*coupling interaction ratio*(

*CIR*) to describe the interaction between grooves quantitatively, which can be defined as where

**H**

*can be obtained by*

_{S}**H**

_{1}

*+*

_{S}**H**

_{2}

*,*

_{S}*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,

*θ*, and

_{i}*θ*

_{1}with given

*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 , which are named as double-subwavelength (D-S), double-non-subwavelength (D-NS), and subwavelength-non-subwavelength (S-NS), respectively.

*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]

*θ*) and scattering angle (

_{i}*θ*

_{1}) dependence of

*CIR*for D-S, D-NS, and S-DS is displayed in Fig. 3 . 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

*θ*and

_{i}*θ*

_{1}when both of the two grooves are subwavelength.

*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.

**PIER 18**, 1–17 (1998). [CrossRef]

## 4. Conclusion

*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.

## Acknowledgments

## 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. |

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. |

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 |

4. | M. A. Morgan and F. K. Schwering, “Mode expansion solution for scattering by a material filled rectangular groove,” Prog. Electromagn. Res. |

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 |

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 |

**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

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