The paper addresses the leaky stop bands associated with resonant photonic crystal slabs and periodic waveguides. We apply a semianalytical model pertinent to the second band to compute the dispersion curves describing the leaky stop band and verify its correctness by rigorous band computations. This approximate model provides clear insights into the physical properties of the leaky stop band in terms of explicit analytical expressions found. In particular, it enables comparison of the structure of the bands computed in complex propagation constant, implying spatially decaying leaky modes, with the bands computed in complex frequency, implying temporally decaying modes. It is shown that coexisting Bragg-coupling and energy-leakage mechanisms perturb the bands in complex propagation constant whereas these mechanisms are decoupled in complex frequency. As a result, the bands in complex frequency are well defined exhibiting a clear gap. These conclusions are verified by numerical diffraction computations for both weak and strong grating modulations where the resonance peaks induced by external illumination are shown to closely track the band profile computed in complex frequency. Thus, in general, phase matching to a resonant leaky mode occurs via real propagation constant that is found by dispersion computations employing complex frequency.
© 2007 Optical Society of America
Light propagation in periodic waveguides, diffractive elements, and photonic crystals is of practical interest in many areas of physics and engineering. Knowledge of the frequency stop bands associated with these structures is important in numerous applications as widely reported [1–3]. Consequently, the band structure of photonic lattices in one, two, and three dimensions has received considerable attention recently, in part motivated by improved ability to fabricate functional periodic elements with nanoscale feature control.
The objective of this paper is to formulate and explain the leaky stop bands characteristic of planar periodic waveguides, also referred to as waveguide gratings or photonic crystal slabs. These bands differ from the usual Bragg-type stop bands in that there exists an out-of-plane radiative energy-coupling channel; namely, the leaky mode resides above the light line in the first zone of the Brillouin diagram. An incident plane wave can induce a leaky waveguide mode on a waveguide-grating structure thereby generating a guided-mode resonance (GMR) field response in the spectrum. The resonance effect leads to the redistribution of diffracted energy and manifests as reflection and transmission peaks arising out of the background provided by the effective-medium thin-film characteristics of the structure. Nanophotonic devices operating under GMR conditions provide versatile functionalities and can be used for both narrowband and broadband filtering and polarization control . Moreover, GMR elements may possess high Q factors and corresponding narrow linewidths thus being suitable for high sensitivity bio- and chemical sensors [5, 6]. The strong surface-localized electromagnetic fields associated with high-Q resonance also provide interesting means to study the nonlinear properties of biological and polymeric materials as well as for radiation generation in surface-localized active molecular species .
In recent papers, we discussed the application of leaky-mode resonance layers in photonic device design [4, 8]. Key to achieving functional devices with versatile spectral properties is placement and manipulation of the resonance peaks in the second stop band. Although not specifically addressing guided-mode resonance effects, the semi-analytical perturbation-theory formalism provided by Kazarinov and Henry  is well suited for explaining resonance behavior at the second band. Rosenblatt et al. applied the Kazarinov-Henry (KH) model in their studies of resonant grating behavior by including an external excitation wave as necessary for actual guided-mode resonance generation . For gratings with symmetric profiles, the KH model shows that the radiated fields generated by the two counterpropagating leaky modes can be in phase or out of phase at the edges of the second stop band. At one edge, there is zero phase difference and radiation is enhanced whereas at the other edge there is π phase difference quenching the radiation. In resonance elements with asymmetric profiles, there appears a resonance at each side of the second band gap [11, 8, 4]; for clarity, in this paper, we will only address the leaky-mode stop bands associated with symmetric profiles.
Vincent and Neviere presented dispersion curves for the second-order stop bands of periodic waveguides . They showed results in real frequency k0 versus complex propagation coefficient β=βR+jβI that included the null in the imaginary part at the nonleaky edge. Noting that the k0-βR plot exhibited a forbidden gap that was not well defined, they turned to the complex frequency plane and demonstrated a clear band by numerical computations . Leaky stop bands exhibiting distorted forbidden regions in k0-βR have since been reported in several papers [12, 13, 4]. In particular, large grating modulation has the effect of essentially obliterating the stop bands [4, 13] for the various supported leaky modes. In general, Brillouin diagrams computed in real frequency and complex propagation constant are valuable in analyzing resonance behavior in the various stop band regions. Use of complex frequency, however, results in clear stop bands [14, 15] as first observed by Vincent and Neviere . One of the results of the present paper shows why the distortion of the real frequency bands occurs and why it does not occur in the complex frequency band.
In this paper, we compute dispersion curves for one-dimensional single-layer periodic waveguides. We are particularly interested in the dispersion behavior around the second stop band. Results generated with a semi-analytical coupled-mode model [9, 10] are verified with rigorous computations. Shown in Fig. 1 is the model periodic waveguide, or photonic crystal slab, sandwiched by a cover region and a substrate. Because the effective refractive index of the periodic layer is higher than the cover and substrate indices, it can function as a guiding layer. The grating has binary modulation and the modulation profile can, in principle, be either symmetric or asymmetric ; however, only the symmetric profile as presented in Fig. 1 will be treated in this paper. The structures under study will have differing high and low refractive indices (nh and nl); here, the effective index of the guiding layer (navg) is kept constant to highlight the effect of changes in modulation level. We use a parameter Δε=nh2-nl2 to represent the level of modulation. The diffraction problem is solved using a rigorous coupled-wave analysis (RCWA) formulation [16, 17] whereas rigorous dispersion results are computed with the method introduced by Peng et al.  and Tamir et al. . The frequency and propagation constant are presented in normalized form as k0/K (=ωΛ/2πc where ω is the frequency and c is the speed of light in vacuum) and β/K, respectively, with K=2π/Λ and Λ being the grating period. We assume that between k0/K and β/K, one quantity is complex and the other real. Which one of these is complex is implied by the usage of subscripts R and/or I.
2. Dispersion characteristics of periodic waveguides
Dispersion plots for a periodic waveguide are shown in Fig. 2 for the fundamental TE mode (TE0). Figure 2(a) shows the first Brillouin zone where we display the first three stop bands. The solid line denotes the leaky mode and the dashed curves provide the light lines corresponding to the substrate (k0ns/K) and cover (k0nc/K). At a particular frequency beyond the first stop band, a leaky wave with βI≠0 appears as shown in Fig. 2(b). When the frequency increases, a new leaky wave emerges and radiates into the cover or substrate when the Rayleigh conditions, expressed in the first zone,
are satisfied. There, k0 is the wave number in free space, m is the index of the diffracted order, K=2π/Λ, and β is the propagation constant. These conditions are satisfied at the intersections of the folded light lines (dashed lines) and the βR curve (solid line) as shown in Fig. 2(a).
Based on the Rayleigh points of Eq. (1), the dispersion diagram can be separated into regimes. In the non-leaky regime, the periodic waveguide supports purely bounded modes and no waves leak into the cover or substrate. As the refractive index of the substrate is higher than that of the cover in this example, the folded light line of the substrate intersects the dispersion curve at the first Rayleigh point. Starting from this point, which is beyond the first stop band, a radiation wave leaks into the substrate as illustrated in Fig. 2(c). The purely bounded mode thereby changes into a leaky mode and the propagation constant becomes complex as indicted in Fig. 2(b). Subsequently, the second Rayleigh point will be reached and a radiation wave also leaks into the cover. Additional waves emerge from the grazing angle as the frequency increases. The appearance of a new leaky wave may generate a perturbation on the βI curve as shown in Fig. 2(b).
The stop bands form at the Bragg condition βR=qK/2, where q is an integer denoting the band. The stop band can be identified by its significantly high value of βI as shown in Fig. 2(b). As the figure shows, the coupling of two counter-propagating waveguide modes within a band is generally considerably stronger than the coupling between the waveguide mode and the radiating wave at moderate modulation levels. In the present example, the first stop band (q=1) occurs around normalized frequency k0/K=0.315; the second stop band (q=2) is near frequency 0.571; and the third stop band (q=3) is at frequency 0.816. Details of these stop bands are presented in Fig. 3. As the first band is located in the non-leaky regime, it is a non-leaky stop band. Higher bands, such as the second and third stop bands, are thus leaky stop bands. In the first stop band, βR is a constant across the band.
Dispersion plots can be presented in two different ways. One approach uses real frequency and complex propagation constant as in Figs. 2 and 3. The second method employs complex frequency and real propagation constant. Even though both are solutions to the same homogeneous system of equations, the two sets of dispersion results offer different, albeit complementary, perspectives on the leaky mode. In the first type of representation, the imaginary part of the propagation constant accounts for spatial attenuation. As the mode propagates along the periodic waveguide, its amplitude decays due to Bragg reflection and energy leakage by radiation. In the second type of representation, the imaginary part of the frequency defines a temporal decay rate. The difference between these points of view will be clarified with the coupled-mode model.
3. Second-band coupled-mode model
For small modulation Δε, the dispersion properties at the second stop band can be approximated as 
or, expressed in a different form, as
where Δβ is the deviation from the Bragg condition (Δβ=β-K) at the second stop band and Δk is the deviation of the wave number from the stop band center kcenter whence Δk=k0-kcenter. As presented in Fig. 4, the coefficient h1 represents the coupling between a guided wave and a leaked wave. This coupling is first-order coupling that is assisted by the grating vector K⃗. The coefficient h2 represents the second-order coupling between the two counter-propagating waveguide modes and h3 is a coefficient related to the group velocity of the unperturbed waveguide mode and can be treated as a group index [9, 10]. Note that Δβ=Δkh3 when there is no modulation (homogenous waveguide). We refer to this theoretical approach as the Kazarinov-Henry (KH) coupled-mode model.
where γ0, γ1 and γ2 are the 0th, 1st and 2nd Fourier coefficients of the relative permittivity modulation profile, respectively. For a binary profile, γ0, γ1 and γ2 are constant throughout (i. e. not x-dependent) the grating layer. G(x,x́) is the Green’s function for the diffracted field  and ϕ(x) denotes the mode profile. All the coefficients are wavelength/frequency dependent. However, in this paper, for convenience, the coefficients used in the coupled mode model are assumed to be wavelength independent with numerical values evaluated at normalized frequency 0.571, the second band center (estimated from the dispersion properties of the equivalent homogenous waveguide). Under this treatment, h3 is constant, while h1 and h2 are proportional to Δε2 and Δε, respectively. These assumptions are appropriate for small modulation with the frequency under consideration varying across a limited range. At high modulation, these assumptions do not work well because of two factors. First, the stop-band width generally expands as modulation increases. Second, the band may shift when higher-order coupling becomes significant . Nevertheless, as shown in the paper, the KH model, as expressed in the form of Eqs. (2) and (3), can provide valuable information even under relatively strong modulation.
4. Physical properties of the leaky stop band
By forcing either frequency or propagation constant to be real, two sets of dispersion results can be obtained from the coupled-mode approach expressed by Eqs. (2) and (3). One set applies real frequency and complex propagation constant; the other uses real propagation constant and complex frequency. We will first discuss the versions of Eqs. (2) and (3) relevant to the nonleaky stop band. In this case, h1=0 and Eqs. (2) and (3) become
Equation (5) can be used to generate dispersion curves with real frequency. As shown in Fig. 5(a), the propagation constant has an imaginary part for a range of frequencies satisfying |Δkh3|<|h2|. Figure 5(b) is computed with Eq. (6) presenting the dispersion characteristics with real propagation constant; note that there exists no solution for frequencies Re(Δkh3)<|h2|. Outside the bands in Fig. 5, the imaginary part of either propagation constant or frequency is zero as there is no radiation leakage in this case. In addition, the real parts of the curves in Figs. 5(a) and 5(b) are exactly the same.
4.1. Band structure in complex propagation constant
Figure 6 presents leaky-band dispersion results computed with real frequency and complex propagation constant. In this case, the real part of the propagation constant is not constant in the band and the imaginary part is nonzero both inside and outside the band. As illustrated in Fig. 2, outside the band, the imaginary part originates in energy leaked into the substrate and cover. Inside the band, both leaked energy and Bragg coupling contribute to the imaginary part of the propagation constant. In this example with modulation Δε=0.34, the results obtained from Eqs. (2) or (3) agree with rigorous computations verifying that the coupled-mode model holds well around the leaky band.
For additional insights, we examine special points in frequency. At frequency Δk = h2/h3, we get Δβ = 0 from Eq. 2, which defines the upper band edge in Fig. 6. At this edge, h1 has no effect on the propagation constant as βI = 0 and the wave appears nonleaky. The lower edge is not as obvious. If we use the criterion that the band edge should appear at minimum ∂k/∂βR (which is analogous to group velocity), it can be shown that the lower edge locates at approximately Δk = -h 2/h 3 in the case of small modulation. At this edge, by Eq. (2), we have (note that h1 is generally a complex number as indicated in Eq. 4). Thus, both the real and imaginary parts of the propagation constant are nonzero and proportional to both h1 and h2. At frequencies far from the band edges with |Δk| ≫ h 2/h 3, one has . Therefore, the imaginary part of the propagation constant converges to Re(h1) away from the band as shown in Fig. 6. This indicates that the real part of h1 represents the actual amount of energy leaked out of the waveguide as discussed by Rosenblatt et al. .
4.2. Band structure in complex frequency
Figure 7 presents the dispersion characteristics of the same leaky band using a real propagation constant. Unlike the previous rendition in Fig. 6, an unambiguous stop band is observed. Outside the band, the frequency has an imaginary part. Again, the characteristics of the stop band are well captured by the KH coupled-mode model with simplified analytical expressions available in special cases. At Δβ = 0, the model gives two solutions Δk = h 2/h 3 and Δk = -(h 2 + 2jh 1)/h 3. The former represents the upper band edge as shown in Fig. 7. At this edge, Δk is real agreeing with the results from analysis obtained with real frequency; since k0,I=0, no leakage occurs. The latter expression represents the lower band edge as shown in the figure where Δk has an imaginary part equal to-2Re(h 1)/h 3 inducing energy leakage. At frequencies far from the band edges, one has Δk = (±Δβ - jh 1)/h 3 with the imaginary part of Δk converging to -Re(h 1)/h 3 as shown in Fig. 7.
4.3 Summary of analytical results
These results are summarized in Table I. At the nonleaky edge, the two representations give the same results. Away from the band, the results are comparable since Δβ=Δkh3. In the stop band and at the leaky edge, the results in real and complex frequency differ significantly in their relations to the coexisting coupling mechanisms (Bragg coupling and leaky wave coupling). For example, the representation with complex β keeps track of the spatial decay associated with each mechanism whereas the representation in complex k lumps the effects of these mechanisms together with the mode decay represented as temporal decay or radiation lifetime given by Im(k)-1. In particular, as seen in the table, both ΔβR and ΔβI in the band center and at the lower (leaky) edge, depend on both coefficients h1 (leak) and h2 (Bragg). In contrast, for complex k, ΔkR at the lower edge depends only on h2 whereas ΔkI depends only on h1. Thus, for the band in complex frequency, the two mechanisms are decoupled accounting for the well-defined band gap observed. Moreover, there are no solutions found in the band center consistent with no states existing there. Figure 8 shows a comparison of the band structure for these two cases computed with RCWA.
The location of the nonleaky bandedge relates to the sign of h2. For the example structure treated (Fig. 1) whose results are presented in Figs. 6–8, the coefficient h2 is real and positive since the modulation profile is symmetric. The fill factor F<0.5 and thus Δβ is zero at the upper edge in this particular example [13, 8]. Both the real and imaginary parts of Δβ are proportional to Δε1.5 since h1∝Δε2 and h2∝Δε. Again, the difference between the real parts of β in the two representations originates in the leakage of the energy from the waveguide grating. At moderate modulation, the real parts of the two sets converge quickly when the frequency is away from the band as shown in Fig. 8.
4.4 Extension to strong modulation
Because the leakage of the energy is defined by h1∝Δε2, the difference between the real and complex frequency dispersion formulations will be amplified under strong modulation. This is illustrated in Fig. 9. From Figs. 9(a) and 9(b), we see that the dispersion plots in real frequency do not give a clearly defined band gap while the set in complex frequency does. The KH model provides qualitatively correct results even for this high level of grating modulation. Again, both representations obtain the same nonleaky edge located at approximately k0/K=0.577 for this example. It is interesting to note that the nonleaky edge has shifted to the lower side of the band (compare with Figs. 6 and 7). This will be explained in a future publication; we can show that the transition point is at h2=Re(h1).
5. Guided-mode resonance excitation on periodic waveguides
Thus far, we have treated the leaky-mode resonance eigenvalue problem obtaining the dispersion properties pertaining to periodic waveguides or photonic crystal slabs. Since we are additionally interested in comparing these results with diffraction efficiency maps in the ω-β plane, we turn our attention to the resonance response of these elements on excitation with external illumination. Figures 10(a) and 10(b) show resonance interactions between the periodic waveguide and an incident plane wave. At short wavelength (high frequency), as the grating has low spatial frequency with respect to the incident wave, there exist propagating orders other than the zero orders. As the wavelength increases, the propagation angles of the higher-order waves increase. Eventually, all the higher-order waves will become evanescent, or cut off, and the zero-order regime prevails. In multi-wave and in zero-order regimes, the higher-order evanescent waves can induce a resonance by coupling to a leaky mode as illustrated in Figs. 10(a) and 10(b). Because of the involvement of a waveguide mode, the physical picture associated with the resonance is clearly communicated by the term “guided-mode resonance.” Since the resonance involves a particular diffraction order coupled to a particular mode, we label the GMRs as TEm,ν; for example, TE2,1 is the resonance formed by the coupling between the second diffracted order and the TE1 mode.
The GMRs manifest as rapid spectral variations of the diffraction efficiencies of the propagating waves. In Fig. 11, GMRs TE1,0, TE2,0 and TE2,1 can be observed at normalized frequencies (k0/K) 0.57, 1.06 and 1.23 respectively. Because GMR TE1,0 resides in the zero-order diffraction regime, there is no energy lost to propagating higher-order diffracted waves and the resonance peak has 100% reflectance theoretically. The higher-order diffraction regime begins approximately at frequency 0.67, at which the ±1 diffraction orders emerge from the waveguide as propagating waves on the substrate side. Additional propagating orders will appear as the frequency increases.
We remark that these computed results are obtained assuming plane-wave incidence. Thus, in this paper, we do not address effects associated with the coupling of optical beams with finite transverse cross section and the attendant non-planar phase fronts. For discussion of grating coupling involving finite beams, we refer the reader to references [21, 22].
6. Association of dispersion and resonance
The resonance effects induced on photonic-crystal waveguides are closely related to their dispersion properties. Specifically, the locations of the GMRs and Rayleigh transition points can be found from the dispersion plots as discussed above. For an obliquely incident excitation wave, as shown in Figs. 12(a) and 12(b), GMRs appear when the dispersion curve of the incident wave (k0ncsinθ) crosses the βR curve because the cross point is a solution of the phase matching condition. Note that these GMRs are not near any of the stop bands. The graph reveals the order-mode pair contributing to the resonance; for example, TE +2,0 is identified as the corresponding cross point on the βR curve of the TE0 mode that is folded backwards twice. A new diffracted order emerges when the dispersion curve of the incident wave intersects the folded light lines seen by comparing Figs. 12(a) and 12(c). Thus the higher diffraction orders whose diffraction efficiency η is plotted in Fig 12(c) emerge when
is satisfied. In Eq. (7), a minus sign is used when the dispersion curve is folded odd number of times while the positive sign corresponds to even number of foldings. In Fig. 10(b), it is seen that the resonance efficiency can be substantially high even in a multiwave regime.
Figures 13(a)–13(c) provide rigorously computed diffraction and dispersion results near the second stop band for the same structure. Figure 13(a) presents the GMR reflection efficiency calculated as function of frequency and angle near normal incidence. This reflectance map in ω-β contains information on both the resonance location and resonance linewidth. The reflectance map shows that the lower edge is the leaky edge indicated by the continuity of the reflectance to the θ=0 point; the upper edge shows no resonance at θ=0 since the profile is symmetric . Figure 13(b) shows a dispersion plot computed with complex frequency with GMR locations noted also. Figure 13(c) provides similar results but with the dispersion computed in real frequency. Comparison of Figs. 13(a)–13(c) shows that the dispersion curves computed with real propagation constant and complex frequency accurately predict the locations at frequencies near the lower band edge.
To excite a resonance with a plane wave incident at angle θ, the phase matching condition
is generally used. The results presented in Sec. 4 show that different values of Re(β) pertain to the real ω and complex ω dispersion pictures. The difference between these solutions increases with the modulation Δε. The width of the stop band is ~2h2/h3 (see Table I), in both cases, is set primarily by h2 which is proportional to the value of the second Fourier grating harmonic with amplitude γ2. At the resonant (leaky) edge, the difference in the β-values is due to the leaked wave as demonstrated in Sec. 4. The representation in complex frequency delivers the real propagation constant directly and matches the results from the diffraction computations as demonstrated in Fig 13. Figure 13(c) indicates phase matching to the deviated part of the curve. It implies that, as an example, for incident wave with θ=0.2°, a resonance should be found at the point marked by ‘*’ in Fig. 13(c) based on Eq. (8). However, there are no states in the band and this picture is not correct. The phase matching relation that pertains to the correct dispersion plot in Fig. 13(b) is
where βν is real and found by dispersion computations employing complex frequency.
A possible interpretation of these results is as follows: The incident wave can phase-match only to the leaky mode and not to the leaked wave since the leaky mode generates the leaked wave. In Sec 4.3, it is shown that the propagation constant computed in real frequency in the band center and at the leaky edge depends on both coefficients h1 (leak) and h2 (Bragg). Thus, the leaked wave perturbs the real part of β on which phase-matching proceeds. Assignment of a complex propagation constant to the problem requires spatial Bragg- and radiation decay via Im(βν). In contrast, by casting the dispersion computation into the complex frequency domain, the spatial decay mechanisms associated with Bragg reflection and radiation leakage are converted into a single time decay coefficient as described in Sec. 4.3. This renders an unperturbed, real βν for input-wave phase matching, avoiding explicit dependence on separate spatial decay mechanisms, and provides consistent results as shown in Fig. 13.
With increased modulation, the amplitude of the leaked wave increases substantially such that dispersion curves computed in real frequency may fail to reveal the band structure. Figure 14 shows a Brillouin diagram for a waveguide grating with heavy modulation computed in real and complex frequency whose dispersion characteristics are given in Fig. 9. As in Fig. 13, a clear band gap is shown in complex frequency. Additionally, estimated GMR peak locations are shown agreeing well with the dispersion curve. In Fig. 14, the resonance locations estimated from diffraction computations do not fall precisely on the upper band. This is because accurate resonance locations are difficult to define when the Q factor is very low. Actually, it is infeasible to find the resonance locations based on the diffraction efficiency in this example. Figure 15 shows the zero-order reflectance at normal incidence. Solely based on reflectance, one would expect the resonance to locate at frequency 0.64. Thus, we turn to the locations of the peak amplitudes of the leaky mode excited by the ±1 evanescent diffraction orders (Fig. 15); these appear at frequency 0.605, which is consistent with the dispersion results as shown in Fig. 14. At oblique incidence, the resonance locations at the upper band for certain angles are still difficult to identify even using the near field peaks. This is because the resonance in the lower band, which has narrower linewidth but stronger excited modes, affects the mode amplitude of the resonance at the upper band . With increasing angle, the two resonances separate. Indeed, the upper resonance becomes clearly detectable again for higher incident angles (i.e. θ>5°).
In this paper, we have endeavored to formulate and explain the leaky stop bands characteristic of planar periodic waveguides or photonic crystal slabs. These bands differ from Bragg-type stop bands as there exists an out-of-plane radiative energy-coupling channel. We apply a semianalytical model due to Kazarinov and Henry  to compute the dispersion curves describing the leaky stop band (second band) and verify its correctness by rigorous band computations. This approximate model provides clear insights into the properties of the leaky stop band on account of analytical expressions obtained that relate band features to the operative physical processes. In particular, it enables comparison of the structure of the bands computed in complex propagation constant, implying spatially decaying leaky modes, with the bands computed in complex frequency, implying temporally decaying modes. It is explicitly shown that coexisting Bragg-coupling and energy-leakage mechanisms perturb the bands in complex propagation constant whereas these mechanisms are decoupled in complex frequency. As a result, the bands in complex frequency are well defined and clear band gap emerges. These conclusions are verified by numerical diffraction computations for both weak and strong grating modulations.
Formulations in terms of complex propagation constant are often used in analysis of the spectral response of periodic waveguides . Band computations in complex propagation constant and complex frequency give similar results when a single mechanism is at work. For example, in the nonleaky stop band, illustrated in Fig. 5, only Bragg reflection is at work and both formulations give equivalent results. Another example is GMR occurrence away from the band. At sufficiently large angle of incidence, there is only out-of-plane radiation induced by the resonant leaky-mode and the solutions again agree as shown in Fig. 13. For very strong modulation, the results of the KH model become inaccurate while retaining qualitative agreement with numerical results. For such cases, as demonstrated in Fig. 14, phase matching to a leaky mode resonance, in or out of band, will accurately proceed via real propagation constant βν that is found by dispersion computations employing complex frequency.
This material is based, in part, upon work supported by the National Science Foundation under Grant No. ECS-0524383.
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