1958 J. Opt. Soc. Am. A/Vol. 6, No. 12/December 1989

Nontrivial grating that possesses only specularcharacteristics: normal incidence: reply to comment

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

Department of Engineering Science and Mechanics, Pennsylvania State University, University Park,Pennsylvania 16802

Received April 3, 1989; accepted June 8, 1989

An error made in Eq. (9) of a previous paper [J. Opt. Soc. Am. A 3, 1788 (1986)] was found as a result of commentsmade by Petit and Tayeb [J. Opt. Soc. Am. A 6, 1955 (1989)]. This error had propagated into the final equation[Eq. (12c)] and gave rise to inaccurate numerical results. The corrected versions are given here.

Comments made by Petit and Tayebl are greatly appreciat-ed, since they have resulted in the identification of an errorin the mathematical analysis of Ref. 2. At the outset, it is tobe observed that the effects of diffraction by the knife edgeswere neglected in Ref. 2, an assumption that was statedclearly in that paper and is also implicit in the analysis ofPetit and Tayeb. Also, only normal plane-wave incidencewas considered in Ref. 2.

As a result of the comments provided by Petit and Tayeb,the mathematical analysis of Ref. 2 was rechecked, and anerror was identified. This error was made in the conversionof the simultaneous algebraic equations arising from theimposition of the boundary conditions (7) of Ref. 2 into thematrix Eq. (9). The correct version of Eq. (12a), which is areexpression of Eq. (9), reads as

I2

2~~~2

-, 2a -00

3perfectly conducting

plates

Fig. 1. Schematic of the two-dimensional grating considered inRefs. 1 and 2 and here.

Table 1. Comparison of Data Computed from the Correct Version of Eq. (12c) with the Data by Petit and Tayeba

Specularly Reflected PowerConditions Petit and Tayeb Corrected Eq. (12c)

Number k1(2a) d/a Incident Polarization (Read from Graphs) (Computed)

1 0.6ir 0.55555 TE 0.7412 0.7372342 0.63r 0.55555 TM 0.1466 0.1479293 0.6ir 1.33333 TE 0.9634 0.9531644 0.67r 1.33333 TM 0.0555 0.0565875 1.47r 2.85714 TM 0.0000 6.6 X 10-116 1.4ir 14.2857 TM 0.0000 6.8 10-

a See Fig. 1 for the geometry used; only normal incidence can be considered. Also, I = 2 = /13 90, = 3 = E0, 2 = 2.25eo.

[J 1 1[DjjIAj = [L1 2][D2 1{BI,

in which the supermatrices

[D1] = diag([D 1]; [D1*]), (Di)m, = exp(-i2dfjin)6mn,

[D2 J = diag([D2]; [D2*]), (D2 )mn = exp(-i2d# 2"EM)5mn

With this change, the correct version of Eq. (12c) is given by

{C} = [J3 J-'[L32 ] [D2]-l[Ll2]1l[Jl] [D1]fAI

and holds, within the stated assumptions, even when theisotropic media in zones 1, 2, and 3 are all different.

Since the computer program was based on the incorrectversion of Eq. (12c), it follows that the numerical resultspresented in Ref. 2 are incorrect, as are the conclusions

drawn from them. In particular, nonspecular modes canalso carry energy away from the grating, contrary to theconclusion obtained in Ref. 2. Shown in Table 1 are com-parisons made with the data of Petit and Tayeb, whichsuggest that the use of the correct version of Eq. (12c) yieldscomparable results. In conclusion, the regrettable error inRef. 2 has been corrected here.

REFERENCES

1. R. Petit and G. Tayeb, "Nontrivial grating that possesses onlyspecular characteristics: normal incidence: comment," J. Opt.Soc. Am. A 6, 1955-1957 (1989).

2. A. Lakhtakia, V. V. Varadan, and V. K. Varadan, "Nontrivialgrating that possesses only specular characteristics: normal in-cidence," J. Opt. Soc. Am. A 3, 1788-1793 (1986).

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