## A note on plane wave diffraction by a perfectly conducting strip in a homogeneous bi-isotropic medium

Optics Express, Vol. 16, Issue 17, pp. 13203-13217 (2008)

http://dx.doi.org/10.1364/OE.16.013203

Acrobat PDF (215 KB)

### Abstract

We studied the problem of diffraction of an electromagnetic plane wave by a perfectly conducting finite strip in a homogeneous bi-isotropic medium and obtained some improved results which were presented both mathematically and graphically. The problem was solved by using the Wiener-Hopf technique and Fourier transform. The scattered field in the far zone was determined by the method of steepest decent. The significance of present analysis was that it recovered the results when a strip was widened into a half plane.

© 2008 Optical Society of America

## 1. Introduction

4. C. F. Bohren, “Light scattering by an optically active sphere,” Chem. Phys. Lett. **29**, 458 (1974). [CrossRef]

5. A. Lakhtakia, *Beltrami Fields in Chiral Media*, (World Scientific, Singapore, 1994). [CrossRef]

6. A. Lakhtakia, “Time-dependent scalar Beltrami-Hertz potentials in free space,” Int. J. Infrared Millim. Waves **15**, 369–394 (1994). [CrossRef]

7. A. Lakhtakia and W. S. Weiglhofer, “Covariances and invariances of the Beltrami-Maxwell postulates,” IEE Proc. Sci. Meas. Technol. **142**, 262–26 (1995). [CrossRef]

*l*→∞ whereas this is not possible in [31]. It is found that the two edges of the strip give rise to two diffracted fields (one from each edge) and an interaction field (double diffraction of two edges).

## 2. Formulation of the problem

*z*=0, -

*l*≤

*x*≤0. In the Drude-Born-Fedorov representation [5

5. A. Lakhtakia, *Beltrami Fields in Chiral Media*, (World Scientific, Singapore, 1994). [CrossRef]

*ε*and

*µ*are the permittivity and the permeability scalars, respectively, while

*α*and

*β*are the bi-isotropy scalars.

**D**is the electric displacement,

**H**is the magnetic field strength,

**B**is the magnetic induction, and

**E**is the electric field strength. The bi-isotropic medium with

*α*=

*β*is reciprocal and is then called a chiral medium. Recently, it has been proved [26

26. A. Lakhtakia and W. S. Weiglhofer, “Constraint on linear, homogeneous constitutive relations,” Phys. Rev. E , **50**, 5017–5019 (1994). [CrossRef]

33. Mackay and Lakhtakia, “Electromagnetic fields in linear bianisotropic mediums, Prog. Opt. **51**, 121–209 (2008). [CrossRef]

*α*≠

*β*may also be considered for generality.

*iωt*), where

*ω*is the angular frequency. The source free Maxwell curl postulates in the bi-isotropic medium can be set up as

*γ*

_{1}and

*γ*

_{2}are given by

**E**and the magnetic field

**H**, as given in [27], are :

**Q**

_{1}is the left-handed Beltrami field and

**Q**

_{2}is the right-handed Beltrami field. In Eqs. (7) and (8), the two impedances

*η*

_{1}and

*η*

_{2}are given by

**Q**

_{1t}and

**Q**

_{2t}lie in the

*xz*-plane and

**j**is a unit vector along the

*y*-axis such that

**j**.

**Q**

_{1t}=0 and

**j**.

**Q**

_{2t}=0. Now, the Eq. (3) can be written as:

*ik*) dependence on the variable

_{y}y*y*and comparing

*x*and

*z*components on both sides of the above equation, we obtain

*Q*

_{1y}and

*Q*

_{2y}because the other components of

**Q**

_{1}and

**Q**

_{2}can then be completely determined by using Eqs. (15–19).

**E**=

*iω*

**B**-

**K**and ∇×

**H**=-

*iω*

**D**+

**J**may be written as:

**S**

_{1}and

**S**

_{2}are the corresponding source densities and are given by

**Q**

_{1}is

**E**like and

**Q**

_{2}is

**H**like. Similarly

**S**

_{1}is

**K**like and

**S**

_{2}is

**J**like where

**J**and

**K**are the electric and magnetic source current densities, respctively. The boundary condition which is necessary is that the tangential component of the electric field must vanish on perfectly conducting finite plane. This implies that

*E*=

_{x}*E*=0, for

_{y}*z*=0, -

*l*≤

*x*≤0. Using this fact in Eqs. (7) and (8), the boundary conditions on the finite plane take the form

*Q*

_{1y}and

*Q*

_{2y}satisfy the boundary conditions (26

*a*) and (27). Now, eliminating

*Q*

_{2y}from Eqs. (26

*a*) and (27), we obtain

*x*<-

*l*,

*x*>0,

*z*=0. Therefore the continuity conditions are given by

*Q*

_{2y}also satisfies Eqs. (28–30). Finally, the scattered field must satisfy the radiation conditions in the limit (

*x*

^{2}+

*z*

^{2})

^{1/2}→∞. We must also observe at this juncture that, in effect, we need to consider the diffraction of only one scalar field, that is either

*Q*

_{1y}or

*Q*

_{2y}, at a time, but the presence of the other scalar field is reflected in the complicated nature of the boundary condition (28). If we set the incident field to be a plane wave, then

*Q*

^{sca}_{1y}satisfies the following homogeneous Helmholtz equation

*b*), we introduce the Fourier transform

*w.r.t*variable

*x*as:

*z*) is regular for

*Im*υ<

*Imk*

_{1xz}, and

_{+}(υ,

*z*) is regular for

*Im*υ>-

*Imk*

_{1xz}and

_{1}(υ,

*z*) is analytic in the common region-

*Imk*

_{1xz}<

*Im*υ<

*Imk*

_{1xz}. The Fourier transform of Eq. (32

*a*) in the region -

*l*≤

*x*≤0,

*z*=0 gives

*b*), respectively, yields

*b*) from Eq. (44

*a*) and Eq. (44

*d*) from Eq. (44

*c*) and then by adding and subtracting the resultant equations, we obtain

_{1}(υ,0

^{+}) from Eqs. (49

*a*) and (45) and

_{1}(υ,0

^{-}) from Eqs. (49

*b*) and (46) and then by adding the resultant equations, we get

_{1}(υ,0

^{+}) from Eqs. (49

*a*) and (40),

_{1}(υ,0

^{-}) from Eqs. (49

*b*) and (41), and then subtracting the resulting equations, we get

## 3. Solution of theWiener-Hopf equations

*L*

_{+}(υ) and κ

_{+}(υ) are regular for

*Im*υ>-

*Im k*

_{1xz}, i.e., for upper half plane and

*L*_(υ) and κ

_{-}(υ) are regular for

*Im*υ<

*Imk*

_{1xz}, i.e., lower half plane. The factorization expression (52

*a*) has been accomplished by Asghar et al [30

30. S. Asghar, T. Hayat, and B. Asghar, “Cylindrical wave diffraction by a perfectly conducting strip in a homogeneous bi-isotropic medium,” J. Mod. Opt. , **3**, 515–528 (1998). [CrossRef]

*J*

_{1}(υ,0) and

*J*′

_{1}(υ,0) from Eqs. (50) and (51) into Eqs. (45) and (46), we get

*O*(

*δ*

_{1}) are neglected while in the present analysis the

*δ*

_{1}parameter is taken up to order one so that the results due to semi infinite barrier [10] can be recovered by taking an appropriate limit. To accomplish this, we have to solve both theWiener- Hopf equations to find the values of unknown functions

*A*(υ) and

*C*(υ). For this we use Eqs. (52a) and (52b) in Eqs.(50) and (51), which gives

*S*

_{+}(υ) and

*S*

_{-}(υ) are regular in upper and lower half plane, respectively. Equations of types (55) and (56) have been considered by Noble [29] and a similar analysis may be employed to obtain an approximate solution for large

*p*=-

*i*(

*k*

_{1xz}+υ)

*l*and

*W*is known as a Whittaker function. Now, making use of Eqs. (58–61) in Eqs. (53) and (54), we get

_{m,n}*A*(

*ν*) corresponds to

*z*>0 and

*C*(

*ν*) corresponds to

*z*<0. We can see that the second term in the above equation was altogether missing in Eq. (70) of [31]. This term includes the effect of

*δ*

_{1}parameter in it which can be seen from the solution also. Now,

*Q*

^{sca}_{1y}(

*x, z*) can be obtained by taking the inverse Fourier transform of Eq. (43). Thus

*A*(υ) and

*C*(υ) are given by Eq. (75). Substituting the value of

*A*(υ) and

*C*(υ) from Eq. (75) into Eq. (76) and using the approximations (63–70), one can break up the field Ψ(

*x, z*) into two parts

^{sep}(

*x,z*) consists of two parts each representing the diffracted field produced by the edges at

*x*=0 and

*x*=-

*l*, respectively, although the other edge were absent while Ψ

*(*

^{int}*x,z*) gives the interaction of one edge upon the other.

## 4. Far field solution

*x*=

*r*cos

*ϑ*, |

*z*|=

*r*sin

*ϑ*and deform the contour by the transformation υ=-

*k*

_{1xz}cos(

*ϑ*+

*iξ*), (0<

*ϑ*<

*π*, -∞<ξ<∞). Hence, for large

*k*

_{1xz}

*r*, Eqs. (76), (78) and (79) become

*δ*

_{1}parameter, which were altogether missing in the analysis of [31].

## 5. Remarks

*k*

_{1xz}has positive imaginary part and using the L Hospital rule successively, the value of

*E*

_{-1}, reduces to

*T*(υ),

*R*

_{1,2}(υ),

*G′*

_{2}(υ),

*C′*

_{1}and

*G*

_{2}(υ) in zero. The third term in Eqs. (63), (64) and (67) also becomes zero as

*l*→∞. The Eq. (75), after these eliminations reduces to

*L*(υ)=

*L*

_{+}(υ)

*L*

_{-}(υ),

_{+}(υ)κ

_{-}(υ).

*k*

_{1x}, the above result reduces to Eq. (26

*a*) of the Half Plane [10]. Subsequently, Eq. (82), i.e., the interacted field vanishes by adopting the same procedure as in case of Eq. (75), while the separated field results into the diffracted field [10] as the strip is widened to half plane.

## 6. Graphical results

*δ*

_{1}are taken from 0.2 to 0.4. The following situations are considered:

- When the source is fixed in one position (for all values of
*δ*_{1}) relative to the finite barrier, (*θ*_{0}=45°,*l*and*θ*are allowed to vary). - When the source is fixed in one position, relative to the infinite barrier (
*θ*_{0}=45°,*l*and*θ*are allowed to vary).

*θ*

_{0}=45

*°*, the graphs (1), (2), (3), (4) and (5) show that the field, in the region 0<

*θ*≤

*π*, is most affected by the changes in

*δ*

_{1},

*l*and

*k*

_{1xz}. The main features of the graphical results, some of which can be seen in graphs (1), (2), (3), (4) and (5) are as follows:

- In graphs (1), (2) and (3) by increasing the value of strip length
*l*and*δ*_{1}, the number of oscillations increases and the amplitude of the separated field decreases, respectively. - The graphs of the diffracted field corresponding to the half plane is given in fig. (5). It is observed that the figs. (1)–(4) are in comparison with fig. (5) for various values of the different parameters.

## 7. Conclusion

*δ*parameter was not taken into account which ends up in an equation from which one cannot deduce the results for semi infinite barrier [10]. This has been proved mathematically as well as numerically which can be considered as check of the validity of the analysis in this paper. Thus, the new solution can be regarded as a correct solution for a perfectly conducting barrier.

## Acknowledgments

## References and links

1. | E. Beltrami, “Considerazioni idrodinamiche,” Rend. Inst. Lombardo Acad. Sci. Lett.22,122–131(1889). An English translation is available: Beltrami, E. “Considerations on hydrodynamics,” Int. J. Fusion Energy 3(3),53–57(1985). V. Trkal, “Paznámka hydrodynamice vazkych tekutin,” C2d8 asopis pro Pe2d8 stování Mathematiky a Fysiky 48,302–311 1919. An English translation is available: V. Trkal, “A note on the hydrodynamics of viscous fluids,” Czech J. Phys. 44,97–106 1994. |

2. | S. Chandrasekhar, “Axisymmetric Magnetic Fields and Fluid Motions,” Asrtophys. J ,. |

3. | A. Lakhtakia, “Viktor Trkal, Beltrami fields, and Trkalian flows,” Czech. J. Phys. |

4. | C. F. Bohren, “Light scattering by an optically active sphere,” Chem. Phys. Lett. |

5. | A. Lakhtakia, |

6. | A. Lakhtakia, “Time-dependent scalar Beltrami-Hertz potentials in free space,” Int. J. Infrared Millim. Waves |

7. | A. Lakhtakia and W. S. Weiglhofer, “Covariances and invariances of the Beltrami-Maxwell postulates,” IEE Proc. Sci. Meas. Technol. |

8. | V. V. Fisanov, “Distinctive features of edge fields in a chiral medium,” Sov. J. Commun. Technol. Electronics |

9. | S. Przezdziecki, “Field of a Point Source Within Perfectly Conducting Parallel-Plates in a Homogeneous Biisotropic Medium,” Acta Physica Polonica |

10. | S. Asghar and A. Lakhtakia, “Plane-wave diffraction by a perfectly conducting half-plane in a homogeneous bi-isotropic medium,” Int. J. Appl. Electromagn. Mater. |

11. | J. P. McKelvey, “The case of the curious curl,” Amer. J. Phys ,. |

12. | H. Zaghloul and O. Barajas, “Force- free magnetic fields,” Amer. J. Phys ,. |

13. | V. K. Varadan, A. Lakhtakia, and V. V. Varadan, “A comment on the solutions of the equation ∇× |

14. | A. Lakhtakia, V. K. Varadan, and V. V. Varadan, |

15. | A. Lakhtakia, “Recent contributions to classical electromagnetic theory of chiral media,” Speculat. Sci. Technol. . |

16. | B. D. H. Tellegen, Phillips Res. Rep.3, 81 (1948).; errata: M. E. Van Valkenburg, ed., Circuit Theory: Foundations and Classical Contributions (Stroudsberg, PA: Dowden, Hutchinson and Ross, 1974). |

17. | L. I. G. Chambers, “Propagation in a gyrational medium,” Quart. J. Mech. Appl. Math,. 9, 360 (1956).; addendum: Quart. J. Mech. Appl. Math, 11, 253–255, (1958). [CrossRef] |

18. | J. C. Monzon, “Radiation and scattering in homogeneous general biisotropic regions,” IEEE Trans. Antennas Propagat ,. |

19. | A. H. Sihvola and I. V. Lindell, “Theory of nonreciprocal and nonsymmetric uniform transmission lines,” Microwave Opt. Technol. Lett. |

20. | A. Lakhtakia and J. R. Diamond, “Reciprocity and the concept of the Brewster wavenumber,” Int. J. Infrared Millim. Waves , |

21. | A. Lakhtakia, “Plane wave scattering response of a unidirectionally conducting screen immersed in a biisotropic medium,” Microwave Opt. Technol. Lett , |

22. | A. Lakhtakia and T. G. Mackay, “Infinite phase velocity as the boundary between positive and negative phase velocities,” Microwave Opt Technol Lett , |

23. | A. Lakhtakia, M. W. McCall, and W. S. Weiglhofer, |

24. | T. G. Mackay, “Plane waves with negative phase velocity in isotropic chiral mediums,” Microwave Opt. Technol. Lett. , |

25. | T. G. Mackay and A. Lakhtakia, “Plane waves with negative phase velocity in Faraday chiral mediums,” Phys. Rev. E , |

26. | A. Lakhtakia and W. S. Weiglhofer, “Constraint on linear, homogeneous constitutive relations,” Phys. Rev. E , |

27. | A. Lakhtakia and B. Shanker, “Beltrami fields within continuous source regions, volume integral equations, scattering algorithms, and the extended Maxwell-Garnett model,” Int. J. Appl. Electromagn. Mater. |

28. | W. S. Weiglhofer, “Isotropie chiral media and scalar Hertz potential,” |

29. | B. Noble, |

30. | S. Asghar, T. Hayat, and B. Asghar, “Cylindrical wave diffraction by a perfectly conducting strip in a homogeneous bi-isotropic medium,” J. Mod. Opt. , |

31. | S. Asghar and T. Hayat, “Plane wave diffraction by a perfectly conducting strip in a homogeneous biisotropic medium,” Int. J. Appl. Electromagn. Mechanics , |

32. | E. T. Copson, |

33. | Mackay and Lakhtakia, “Electromagnetic fields in linear bianisotropic mediums, Prog. Opt. |

34. | F. I. Fedorov, |

**OCIS Codes**

(050.1940) Diffraction and gratings : Diffraction

(290.5838) Scattering : Scattering, in-field

**ToC Category:**

Diffraction and Gratings

**History**

Original Manuscript: June 10, 2008

Revised Manuscript: July 4, 2008

Manuscript Accepted: July 18, 2008

Published: August 13, 2008

**Citation**

M. Ayub, M. Ramzan, and A. B. Mann, "A note on plane wave diffraction by a perfectly conducting strip in a homogeneous bi-isotropic medium," Opt. Express **16**, 13203-13217 (2008)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-17-13203

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### References

- E. Beltrami, "Considerazioni idrodinamiche," Rend. Inst. Lombardo Acad. Sci. Lett.22,122 -131(1889). An English translation is available: Beltrami, E. "Considerations on hydrodynamics," Int. J. Fusion Energy 3,53- 57(1985). V. Trkal, "Paznámka hydrodynamice vazkych tekutin," C2d8 asopis pro Pe2d8 stování Mathematiky a Fysiky 48, 302 -311 1919. An English translation is available: V. Trkal, "A note on the hydrodynamics of viscous fluids," Czech J. Phys. 44, 97-1061994.
- S. Chandrasekhar, "Axisymmetric Magnetic Fields and Fluid Motions," Asrtophys. J, 124, 232 (1956). [CrossRef]
- A. Lakhtakia, "Viktor Trkal, Beltrami fields, and Trkalian flows," Czech. J. Phys. 44, 89 (1994). [CrossRef]
- C. F. Bohren, "Light scattering by an optically active sphere," Chem. Phys. Lett. 29, 458 (1974). [CrossRef]
- A. Lakhtakia, Beltrami Fields in Chiral Media (World Scientific, Singapore, 1994). [CrossRef]
- A. Lakhtakia, "Time-dependent scalar Beltrami-Hertz potentials in free space," Int. J. Infrared Millim. Waves 15, 369-394 (1994). [CrossRef]
- A. Lakhtakia andW. S. Weiglhofer, "Covariances and invariances of the Beltrami-Maxwell postulates," IEE Proc. Sci. Meas. Technol. 142, 262-26 (1995). [CrossRef]
- V. V. Fisanov, "Distinctive features of edge fields in a chiral medium," Sov. J. Commun. Technol. Electronics 37, 93 (1992).
- S. Przezdziecki, "Field of a Point Source Within Perfectly Conducting Parallel-Plates in a Homogeneous Biisotropic Medium," Acta Physica Polonica A 83, 739 (1993).
- S. Asghar and A. Lakhtakia, "Plane-wave diffraction by a perfectly conducting half-plane in a homogeneous bi-isotropic medium," Int. J. Appl. Electromagn. Mater. 5, 181-188, (1994).
- J. P. McKelvey, "The case of the curious curl," Amer. J. Phys. 58, 306 (1990). [CrossRef]
- H. Zaghloul and O. Barajas, "Force- free magnetic fields," Amer. J. Phys. 58, 783 (1990). [CrossRef]
- V. K. Varadan, A. Lakhtakia, and V. V. Varadan, "A comment on the solutions of the equation.�?a = ka," J. Phys. A: Math. Gen. 20, 2649 (1987). [CrossRef]
- A. Lakhtakia, V. K. Varadan, and V. V. Varadan, Time-Harmonic Electromagnetic Fields in Chiral Media (Springer, Heidelberg, 1989).
- A. Lakhtakia, "Recent contributions to classical electromagnetic theory of chiral media," Speculat. Sci. Technol. 14, 2-17 (1991).
- B. D. H. Tellegen, Phillips Res. Rep. 3, 81 (1948).; errata: M. E. Van Valkenburg, ed., Circuit Theory: Foundations and Classical Contributions (Stroudsberg, PA: Dowden, Hutchinson and Ross, 1974).
- L. I. G. Chambers, "Propagation in a gyrational medium," Quart. J. Mech. Appl. Math. 9, 360 (1956), addendum: Quart. J. Mech. Appl. Math 11, 253-255 (1958). [CrossRef]
- J. C. Monzon, "Radiation and scattering in homogeneous general biisotropic regions," IEEE Trans. Antennas Propagat, 38, 227 (1990). [CrossRef]
- A. H. Sihvola and I. V. Lindell, "Theory of nonreciprocal and nonsymmetric uniform transmission lines," Microwave Opt. Technol. Lett. 4, 292 (1991).
- A. Lakhtakia and J. R. Diamond, "Reciprocity and the concept of the Brewster wavenumber," Int. J. Infrared Millim. Waves 12, 1167-1174 (1991). [CrossRef]
- A. Lakhtakia, "Plane wave scattering response of a unidirectionally conducting screen immersed in a biisotropic medium," Microwave Opt. Technol. Lett, 5, 163 (1992). [CrossRef]
- A. Lakhtakia and T. G. Mackay, "Infinite phase velocity as the boundary between positive and negative phase velocities," Microwave Opt Technol. Lett. 20, 165-166 (2004). [CrossRef]
- A. Lakhtakia, M. W. McCall, and W. S. Weiglhofer, "Negative phase velocity mediums," W. S. Weiglhofer and A. Lakhtakia (Eds.), Introduction to complex mediums for electromagnetics and optics, (SPIE Press. Bellingham, W. A, 2003).
- T. G. Mackay, "Plane waves with negative phase velocity in isotropic chiral mediums," Microwave Opt. Technol. Lett. 45, 120-121 (2005). [CrossRef]
- T. G. Mackay and A. Lakhtakia, "Plane waves with negative phase velocity in Faraday chiral mediums," Phys. Rev. E 69, 026602 (2004). [CrossRef]
- A. Lakhtakia and W. S. Weiglhofer, "Constraint on linear, homogeneous constitutive relations," Phys. Rev. E 50, 5017-5019 (1994). [CrossRef]
- A. Lakhtakia and B. Shanker, "Beltrami fields within continuous source regions, volume integral equations, scattering algorithms, and the extended Maxwell-Garnett model," Int. J. Appl. Electromagn. Mater. 4, 65-82 (1993).
- W. S. Weiglhofer, "Isotropie chiral media and scalar Hertz potential," J. Phys. A 21, 2249 (1988).
- B. Noble, Methods Based on the Wiener-Hopf Technique (Pergamon, London, 1958).
- S. Asghar, T. Hayat, and B. Asghar, "Cylindrical wave diffraction by a perfectly conducting strip in a homogeneous bi-isotropic medium," J. Mod. Opt. 3, 515-528 (1998). [CrossRef]
- S. Asghar and T. Hayat, "Plane wave diffraction by a perfectly conducting strip in a homogeneous biisotropic medium," Int. J. Appl. Electromagn. Mechanics 9, 39-51 (1998).
- E. T. Copson, Asymptotic Expansions (Cambridge University Press, 1967).
- Mackay and Lakhtakia, "Electromagnetic fields in linear bianisotropic mediums, Prog. Opt. 51, 121-209 (2008). [CrossRef]
- F. I. Fedorov, Theory of Gyrotropy (Minsk: Nauka i Tehnika), 1976).

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