[31] The effect of the wavelength and angle of incidence and the grating's characteristics on the transmission and reflection coefficients, given by (37) and (38), is demonstrated by the numerical results established hereafter. For each set of parameters, the required number of expansion functions in the Fourier series (25) is determined by applying the convergence check, discussed in section 6.1.
6.1. Convergence and Numerical Efficiency
[32] The method's convergence and numerical efficiency are tested against two examples, investigated previously in the literature. The first example concerns the structure with n_{0} = n_{1} = n_{2} = 1, n_{3}^{2} = 3, d = 0.2λ, w = 2d, Λ = 0.6λ, q = 1, a_{1} = 0, s_{1} = 0.5Λ, proposed in the work of Pai and Awada [1991], and the second one concerns the “metallic” grating structure with n_{0} = n_{1} = n_{2} = 1, n_{3} = 1.8 + 7.12i, d = 0.05, w = 2d, Λ = 1, q = 1, a_{1} = 0, s_{1} = 0.1Λ, θ_{inc} = 85°, proposed in the work of Morf [1995]. The 0order coefficients convergence patterns for the above two examples are shown in Figures 4a and 5a. The curves of Figures 4a and 5a, for which convergence is achieved by the present method, are in excellent agreement with the corresponding ones of Figures 4 and 5 in Pai and Awada [1991] and Figure 15 in Morf [1995]. In addition, Figures 4b and 5b depict the variation of ∣r_{0}∣^{2} and ∣t_{0}∣^{2} near the resonances, as obtained by the present method and by those of Pai and Awada [1991] and Morf [1995], respectively. High accuracy of the present method is achieved by considering only 2N + 1 = 7 expansion functions in expression (25) for the first example and 2N + 1 = 5 for the second one. These numbers of required coefficients are by far smaller than the respective ones (NT = 31, N = 21) of the multiple reflection series method [Pai and Awada, 1991] and (N = 59 modes) of the method of eigenfunction expansion in polynomial basis functions [Morf, 1995]. Moreover, as it is shown in Figure 5a, the resonant wavelength λ = 0.9132 in the metallic grating is calculated accurately by using only 2N+ 1 = 3 expansion functions, while the method of Morf [1995] requires 59 Legendre basis functions to achieve the respective convergence. This high numerical efficiency of the present method is justified by the facts that the unknown electric field factor and the entire domain expansion functions satisfy the same physical laws (not satisfied in the considerations of Morf [1995]) and that the Green's function, utilized herein, satisfies inherently the boundary conditions in the nongrating structure.
[33] Also, the present semi analytical method is very efficient in terms of CPU time, due to the fact that all the computations and integrations are analytically carried out. For the calculation of a specific order's reflectivity or transmittivity 0.1 seconds are sufficient (Pentium IV, 2.80 GHz with 1 GByte of RAM) per wavelength or incident angle sampling point.
[34] In the numerical results of sections 6.2–6.4, for the determination of the required number N of expansion functions in (25) the following two criteria, in consistency with literature, are used: (a) an energy conservation condition [Bertoni et al., 1989] and (b) convergence to the solution with increasing N for all the grating and the incident wave parameters [Moharam et al., 1995]. Criterion (a) is implemented by demanding the reflected and transmitted fields to conserve power within 1 part in 10^{8}; for example in the case of the 0order propagating ∣r_{0}∣^{2} + ∣t_{0}∣^{2} differs from unity by less than 1 × 10^{−8}. In all the following results the choice N = 7 is sufficient in order for both criteria to be satisfied, a fact that validates the method's numerical efficiency.
6.2. Wood's Anomalies
[35] Recent applications of resonant integrated optics frequencyselective devices such as dielectric filters, spectrally selective polarized laser mirrors, and beam samplers [Wang and Magnusson, 1994] have revived the interest in the investigation of anomalous resonance effects, commonly known as Wood's anomalies (first observed by Wood [1902] in metallic gratings). Such effects occur when under special illumination conditions (with respect to the wavelength or angle of incidence) the intensity of the grating diffracted spectral orders drops from maximum to minimum (approximate ratio 10 to 1) within a very narrow range [Nevière et al., 1995; Tamir and Zhang, 1997]. Fano [1941] was the first suggesting that these anomalies could be associated with the excitation of a surface wave along the grating waveguide. This explanation has been fully confirmed by Hessel and Oliner [1965], who demonstrated that the stronger anomalies express a forced resonance associated with leaky waves of the grating structure. Furthermore, Nevière et al. [1995] have asserted that Wood's anomalies can be predicted and studied from the complex poles and zeros of the scattering matrix.
[36] In order to investigate the Wood's anomalies with the developed integral equation methodology and compare its results with previously established techniques, we study the proposed by Tamir and Zhang [1997] grating configuration with constant parameters n_{0} = 1, n_{1} = 2, n_{2}^{2} = 2.31, n_{3}^{2} = 3.61, d = 0.1 μm, w = 0.15 μm, Λ = 0.39 μm, q = 1, a_{1} = 0.3Λ, s_{1} = 0.4Λ. The analysis concerns the rapid changes in the reflection and transmission coefficients, due to the interaction between the incident field and the grating structure. These changes occur for wavelengths λ or incident angles θ_{inc} lying in regions where the diffracted orders of the incident wave are phase matched to leaky wave space harmonics. The amplitudes of the 0 and −1orders of the reflected and transmitted fields as functions of λ for fixed θ_{inc} = 52° are depicted in Figures 6a and 6b. Furthermore, the respective amplitudes as functions of θ_{inc} for fixed λ = 0.47736 μm are shown in Figures 7a and 7b. Figures 6a and 7a also indicate the variations of the term ∣t_{0} − T∣ (i.e. the 0order transmitted field without the contribution of the respective field in the nongrating structure; see equation (38)).
[37] The curves of Figures 6a and 7a and Figures 6b and 7b for ∣r_{0}∣, ∣t_{0}∣ and ∣r_{−1}∣, ∣t_{−1}∣ essentially coincide with the respective ones of Figures 4a and 10a and Figures 9a and 11a in the work of Tamir and Zhang [1997], where the modal method is utilized. Abrupt changes (namely the Wood's anomalies mentioned above) are observed in the 0order reflectivity and transmittivity. On the other hand, the −1order diffraction curves are smoother than the 0order ones. Besides, narrow peaks appear in the curves of Figures 6 and 7 centered at λ_{1} = 0.4686 μm, λ_{2} = 0.4743 μm and θ_{inc,1} = 50.7°, θ_{inc,2} = 53.8°. The locations of these resonant wavelengths and angles are independent of the order p. The width of the narrow peaks is smaller for p = 0 than for p = −1, due to the rapid changes of the coefficients amplitude for p = 0.
[38] Also, it is worth to emphasize the Lorentzian behavior (in the terminology of Norton et al. [1997]) of the curve of ∣t_{0} − T∣. As reported in the statements of Figure 8a of Tamir and Zhang [1997], an almost identical Lorentzian behavior of the 0order reflection and transmission coefficients is due to the ignorance of a complex zero in the fractional expressions of the coefficients. Here, by the statements of Figures 6a and 7a we conclude that this Lorentzian behavior can also be justified by the subtraction of the field component T due to the nongrating structure from the 0order transmitted field. This conclusion follows by the use of the integral equation methodology, which separates in the analysis the fields in the nongrating structure (function Ψ_{0} in the mathematical formulation) from the fields due to the presence of the grating. On the contrary, the methodology of Tamir and Zhang [1997] involves uniform field expansions in the different layers, without making the above mentioned separation. Moreover, the −1order exhibits a similar type of Lorentzian behavior. This order is not affected by the field on the nongrating structure, because this field contributes only in the 0order.
6.3. Array of Dielectric Rectangular Cylinders
[39] Plane wave diffraction by an array of dielectric rectangular cylinders, which has been analyzed separately in the literature [see, e.g., Zunoubi and Kalhor, 2006], is treated here as a special case of the developed method. Specifically, in this section by selecting n_{0} = n_{2} = 1, w = 2d, q = 1 and a_{1} = 0, we study the structure of an infinite Λperiodic array composed of two adjacent rectangular cylinders per unit cell with indices n_{3} and n_{1} and lengths s_{1} and Λ − s_{1} (see Figure 8).
[40] Figures 9a and 9b depict the dependence of the amplitude of the 0reflection order with respect to the normalized frequency Λ/λ for n_{3}^{2} = 2.44 and n_{3}^{2} = 5, corresponding to three different values of s_{1} with n_{1} = 1, Λ = 2d, θ_{inc} = 90°. For n_{3}^{2} = 2.44 occur two wavelengths of total reflection for each value of s_{1}. These reflection peaks become wider as s_{1} increases. The reflection curves become smoother as s_{1} approaches Λ (mainly demonstrated by the behavior for s_{1} = 0.7Λ); i.e. as the array approaches the homogeneous dielectric slab. On the other hand, for n_{3}^{2} = 5 occur two wavelengths of total reflection for s_{1} = 0.1Λ and 0.3Λ and four for s_{1} = 0.5Λ. For both values of n_{3} one and two narrow total transmission wavelengths occur for s_{1} = 0.1Λ and s_{1} = 0.5Λ. Also, for fixed s_{1}, the reflection peaks are wider for n_{3}^{2} = 5 than n_{3}^{2} = 2.44.
[41] Furthermore, Figures 10a and 10b show the variations of ∣r_{0}∣ and ∣r_{0}∣^{2} with respect to the normalized frequency 2k_{0}d for Λ = 2d/1.713 and Λ = 2d/2.037, with n_{1}^{2} = 2.56, n_{3}^{2} = 1.44, θ_{inc} = 45°. In each figure, three curves are plotted corresponding to s_{1} = 0.1Λ, 0.3Λ and 0.5Λ. We have plotted up to the value 2k_{0}d = 6.3 for Figure 10a and 2k_{0}d = 7.5 for Figure 10b, since at these points p_{r}^{−} = −1 and hence the reflected −1order switches from cutoff to propagation in the air region along the xaxis. Note the agreement between the curves of Figures 10a and 10b for s_{1} = 0.5Λ and the corresponding ones in Figures 3 and 8 of Bertoni et al. [1989]. For Λ = 2d/1.713 occur five total transmission and three total reflection frequencies for lengths s_{1} = 0.1Λ, 0.3Λ and four total transmission and two total reflection frequencies for s_{1} = 0.5Λ. On the other hand, for Λ = 2d/2.037 occur two total transmission frequencies for each value of s_{1} and two and three total reflection frequencies for s_{1} = 0.1Λ and s_{1} = 0.3Λ, 0.5Λ. The physical mechanism generating these total reflection and transmission resonances is strongly connected to the excitation of the waveguide modes and is discussed thoroughly in the work of Bertoni et al. [1989]. For example, in the case of Λ = 2d/1.713 the first two total transmission frequencies for each s_{1} occur when the 0order mode propagates inside the periodic layer. For higher frequencies, when the −1order mode also propagates inside the layer, total reflection occurs at three frequencies for s_{1} = 0.1Λ, 0.3Λ and two for s_{1} = 0.5Λ, in the vicinity of which exist the same number of frequencies with zero reflection (total transmission).
[42] Furthermore, we define the fractional bandwidth of the above described resonant frequencies as the ratio of the difference between the frequencies at which ∣r_{0}∣^{2} = 0.9 (reflection) and ∣r_{0}∣^{2} = 0.1 (transmission) over the total reflection and transmission frequency respectively. This bandwidth constitutes a measure of the narrowness of a reflection or transmission peak. The values of the total reflection and transmission frequencies and the respective fractional bandwidths for the cases of Figures 10a and 10b are summarized in Table 1. For Λ = 2d/1.713 the total reflection and transmission frequencies and the fractional bandwidths increase with s_{1}. It is worth to emphasize the very narrow reflection peaks at 2k_{0}d = 4.899 and 5.085 for s_{1} = 0.1Λ and 0.3Λ with bandwidths 0.002 and 0.016%. On the other hand, the reflection peak at 2k_{0}d = 5.311 of the structure proposed by Bertoni et al. [1989], corresponding also to that of the curve of Figure 10a with s_{1} = 0.5Λ, is wider (bandwidth 0.043%) than the above mentioned for s_{1} = 0.1Λ and 0.3Λ. Moreover, for Λ = 2d/2.037 the bandwidth of the reflection and transmission peaks increases with the normalized frequency 2k_{0}d and the length s_{1}. Very narrow resonances occur for s_{1} = 0.3Λ at 2k_{0}d = 5.9941 and 5.9944 with bandwidths 0.003 and 0.008% and for s_{1} = 0.5Λ at 2k_{0}d = 6.2315 and 6.2316 both with bandwidths 0.002%.
Table 1. Normalized Frequencies 2k_{0}d and Bandwidths for Total 0Order Reflection and Transmission, Corresponding to Three Different Values of s_{1} and Λ = 2d/1.713 and Λ = 2d/2.037^{a}  Total Reflection Resonance  Total Transmission Resonance 

2k_{0}d  Bandwidth, %  2k_{0}d  Bandwidth, % 


Λ = 2d/1.713 
s_{1} = 0.1Λ    2.250  29.95 
  4.495  14.91 
4.899  0.002  4.899  0.041 
5.366  0.045  5.361  0.093 
6.060  0.074  6.071  0.313 
s_{1} = 0.3Λ    2.388  33.42 
  4.733  14.98 
5.085  0.016  5.088  0.511 
5.577  0.380  5.534  1.066 
6.213  0.542  6.282  0.637 
s_{1} = 0.5Λ    2.556  38.38 
  4.995  14.37 
5.311  0.043  5.317  5.492 
5.830  0.952  5.730  4.589 
   
Λ = 2d/2.037 
s_{1} = 0.1Λ  6.2141  0.031  6.2081  0.203 
6.8842  0.044  6.7604  6.745 
s_{1} = 0.3Λ  5.9941  0.003  5.9944  0.008 
6.4640  0.254  6.4211  1.055 
7.1167  0.327   
s_{1} = 0.5Λ  6.2315  0.002  6.2316  0.002 
6.7681  0.587  6.6788  1.613 
7.4154  0.859   
6.4. Effect of Modulation Inside the Grating's Unit Cell
[43] In this last section, we investigate the dependence of the grating waveguide's reflectance spectrum from the characteristics a_{i}, s_{i} and q inside each grating's unit cell. So far in the literature has been investigated only the particular case of binary gratings (q = 1). The developed integral equation method admits the arbitrary selection of the number q and the geometrical characteristics a_{i}, s_{i} of the rectangles, hence offering additional degrees of freedom to the designing of grating waveguides with the desirable operational standards.
[44] Two representative frequency selective structures, incorporating a grating waveguide and operating in the microwave region, are analyzed. The grating periods and the angles of incidence are chosen so that both grating structures admit only the 0order propagating in the substrate and cover regions. First, we investigate a reflection filter under normal incidence (θ_{inc} = 90°) with air in the substrate and cover (n_{0} = n_{2} = 1) and grating waveguide characteristics n_{1}^{2} = 2.59 (Plexiglas), n_{3}^{2} = 2.05 (Teflon), w = 2d, Λ = 1.65 cm. Figures 11a and 11b depict the filter's spectral response ∣r_{0}∣^{2} for 2d = 0.5 cm and 2d = 0.7 cm. In each figure are plotted three curves corresponding to the sets of parameters (α) q = 1, a_{1} = 0, s_{1} = 0.5Λ, (β) q = 2, a_{1} = 0, a_{2} = 0.5Λ, s_{1} = 0.1Λ, s_{2} = 0.3Λ and (γ) q = 3, a_{1} = 0, a_{2} = 0.3Λ, a_{3} = 0.6Λ, s_{1} = 0.2Λ, s_{2} = 0.1Λ, s_{3} = 0.2Λ. The curves of Figure 11 for the set (α) are the same with that of Figure 2 in the work of Tibuleac et al. [2000], where the rigorous coupled wave analysis is utilized. The slab thickness 2d = 0.7 cm results in symmetric reflectance curves with small sideband reflection for the three sets of parameters, justified also by the fact that this thickness satisfies the halfwavelength condition (analyzed in Wang and Magnusson [1994]). On the other hand, the thickness 2d = 0.5 cm yields asymmetric reflectance curves with larger sideband reflection compared to 2d = 0.7 cm. For set (α) one reflection resonance occurs in the frequency range 13–15 GHz for both values of 2d. However, for sets (β) and (γ) two reflection resonances are generated in the same frequency range. Furthermore, the bandwidth of the resonant reflection frequencies (defined in section 6.3) decreases with increasing q. More precisely, for 2d = 0.5 cm the total reflection bandwidths are: 0.17% for set (α), 0.012% and 0.04% for (β) and 0.0007% and 0.0013% for (γ), while for 2d = 0.7 cm: 0.15% for (α), 0.011% and 0.032% for (β) and 0.0022% and 0.0014% for (γ). The slab's thickness does not significantly influence the distances between the two resonant frequencies, which are approximately 600 MHz for set (β) and 500 MHz for (γ) for both values of 2d.
[45] Second, we investigate the reflection filter with n_{0} = n_{2} = 1, n_{1}^{2} = 6.13(Eglass), n_{3}^{2} = 3.7(silica), w = 2d = 4.37 mm, Λ = 11.28 mm. Figures 12a and 12b depict the filter's spectral response ∣r_{0}∣^{2} corresponding to (a) the sets of parameters (α), (β), (γ) of Figure 11 with θ_{inc} = 90° and (b) θ_{inc} = 75°, 80° and 85° with the parameters of set (γ). We also observe in Figure 12a (as in Figure 11) that an additional reflection resonance is generated for sets (β) and (γ) and the bandwidth of the resonant frequencies decreases with increasing q. In addition, Figure 12b indicates that for set (γ) the bandwidth of the two resonant frequencies remains very small independently of θ_{inc}. However, the distance between the two resonant frequencies decreases with increasing θ_{inc}, making hence the incident angle a control mechanism of the resonances distance. For frequencies different from the resonant ones, the reflectance has almost the same values for all θ_{inc}.
[46] Besides, it is interesting to investigate the convergence for q > 1 of the present method, utilizing essentially the Fourier series expansion (25) with respect to z. Figure 13 depicts the 0order convergence patterns near the resonances with respect to the number N of expansion functions for the sets of parameters of Figure 11b with (a) q = 1, (b) and (c) q = 2 and (d) and (e) q = 3. The sufficient order N for high accuracy of the present method increases with the number q of rectangles per grating unit cell; namely N = 2 for q = 1, N = 6 for q = 2 and N = 10 for q = 3. However, even for q>1 these numbers of required coefficients in the present method are by far smaller than the numbers of the multiple reflection series terms (NT = 31, N = 21) and the polynomial basis functions (N = 59), needed in the methods of Pai and Awada [1991] and Morf [1995], corresponding to the simple case of q = 1. Therefore, the basis functions (25) utilized herein exhibit superior convergence for q > 1. In addition, we note that for ∣r_{0}∣^{2} < 0.8 (i.e. relatively far from the resonance) only a small number of terms is required for convergence, and moreover the convergence is independent of q. More precisely, for the specific parameters of Figure 13 simulations have indicated that 2N + 1 = 5 terms are sufficient for all q.
[47] We conclude this section by noting that the modulation inside each grating's unit cell provides flexibility in the design of reflection grating filters and leads to very narrow resonances for specific choices of the parameters a_{i}, s_{i}, q. This offers an alternative way of generating narrow resonances to those proposed so far in the literature [Tibuleac et al., 2000; Coves et al., 2004], concerning mainly the addition of extra dielectric layers in the grating structure.