[41] From the relationship struck in section 4 between the positivity and negativity of the reflection phase (for TM^{z} polarized incidence in the *ϕ*_{inc}= 0 plane perpendicular to the corrugations) with the surface-wave passband and stopband (in the O X dispersion diagram), respectively, it may be anticipated that the analytic expression of the surface-wave propagation constant of(21b)can be used to derive a likewise explicit closed-form formula for the reflection-phase in terms of the various corrugation parameters. The motivation behind this idea stems from the herein-discovered phenomenon that at least the behavioral connection (in the cyclical sense) between the propagation constant and the reflection-phase, when both are plotted against the frequency, has already been established, and thus the*k*_{x} expression of (21b) as a function of all parameters (except the incident theta angle, which will be taken care of later) offers an excellent starting platform to build on.

[42] The thought process of the derivation shall now be described. First, concentrate on the graphs of the reflection-phase and surface-wave propagation constant versus frequency (simply, the reflection-phase and dispersion diagrams which we have been seeing in the preceding figures, but with the axes of the latter diagram swapped, i.e., the propagation constant is now represented by the vertical axis). It is observed that the rise of each modal propagation constant with frequency gets increasing steep as the surface-wave enters deeper into the slow-wave region. This is corresponded with the increasing steep fall of the reflection-phase with frequency from 180°. As the slope of the propagation-constant versus frequency graph approaches infinity (entry into a surface-wave stopband), the counterpart situation in the reflection-phase diagram is a zero-crossing, i.e., AMC or HIS condition. Although no linkage between the two graphs can be detected for the frequency regions of falling propagation-constant values in one and increasingly negative reflection-phase values in the other, it is of no necessity finding any connection since it can be observed from all preceding reflection-phase diagrams that the traces for positive and negative phases are vertically and horizontally flip-symmetric. As such, it suffices to consider just positive values of the reflection-phase. All these foregoing observations constitute the foundational concept on which the subsequent derivation stages rest.

[43] With the above laid out, it would not be difficult to deduce that the reflection phase may assume a reciprocal relation with the propagation constant according to:

noticing how the equality of the reflection-phase to 180° whenever*k*_{x}equals the free-space wave number, i.e., when the dispersion curve grazes the light-line, has been enforced as required. However, from our computational experiments, it was found that the traces of the reflection-phase diagram produced by this(23)just decay too slowly from 180° initially with frequency (i.e., initial entry into surface-wave passbands) due to the strong quasi-asymptote of the dispersion trace with the light-line pertaining to initial gentle entry of the surface-wave into the slow-wave region. Nonetheless, as the surface wave moves deeper into the slow-wave region, the rate of increase of its propagation constant with frequency gradually matches up fairly well with the rate of fall of the reflection-phase, thereby suggesting that the reciprocal relationship of(23)can be viable over those frequency ranges (nearer toward the upper edge of each surface-wave passband).

[44] Now, if we take the liberty to break away from the correction term *g/d*_{x} ≤ 1 of (21b) and let it exceed unity to become a new quantity, say *ζg/d*_{x}, it has been observed from our computational experiments that the larger this *ζ*is, the higher will be the rate of increase of the propagation constant with frequency as the surface-wave initially moves into the slow-wave region. In other words, as each mode preliminarily moves from the fast-wave region into the slow-wave region, the modified trace produced by using the abovementioned new quantity no longer creeps as slowly along the light-line with rising frequency as before the*ζ*> 1 term was implemented, i.e., reduced lingering on the light-line.

[45] With these above-described aspects, we may write the modified term as

with associated

where *a*_{1} > 1 and *a*_{2}are coefficients to be specified later. In this way, when the surface-wave number*k*_{x}equals the free-space wave number (dispersion trace touches the light-line, i.e., entry into surface-wave passband, with corresponding flipping of the reflection-phase diagram curve from −180° to +180° and subsequent falling toward zero-phase from 180°), the*ζ* term assumes the value of 1, being the maximum amplitude of the scaling coefficient where it is needed to ‘pull’ the dispersion trace off the light-line at a faster rate (with frequency) in order for an associated higher fall-rate of the resultant ∠Γ^{modif}of (26) with frequency. Whereas as *k*_{x}rises toward infinity (with associated zero-crossing of the reflection-phase),*ζ*drops to a value closer to unity in accordance with the increasingly matching rate of change between the reflection-phase and dispersion diagrams as mentioned at the end of the paragraph preceding the previous.

#### 6.1. Dependency on Incident Theta Angle

[46] However, the dependency on the incident theta angle of the impingent plane wave (as required by reflection-phase studies) onto the corrugated surface has not yet been considered, i.e., the term*θ*_{inc} must be incorporated into the functional expression for the reflection phase, the necessity of which being obvious from Figure 6. This variable is absent from the original formula of (21b). Intuitively, the terms *a*_{1} and *a*_{2} of (24) are picked out to be assigned as functions of *θ*_{inc}, i.e., *a*_{1}(*θ*_{inc}) and *a*_{2}(*θ*_{inc}). Subsequently, the final form of the reflection-phase is anticipated to look like:

whereby *a*_{0} is a new quantity also dependent on *θ*_{inc}, and the dependency of *k*_{x}^{modif} from (25) on *ζ* of (24) and in turn on *a*_{1}(*θ*_{inc}) and *a*_{2}(*θ*_{inc}) are explicitly shown. The dependency on the rest of the parameters are implicit within *k*_{x}^{modif}.

[47] Next, a parametric study of *θ*_{inc}in terms of the reflection-phase diagram generated using the full-wave rigorous moment-method code is performed, for an arbitrary set of controlled (fixed) values of all the other parameters. The results of this study shall then constitute the reference data needed to determine the functional forms of*a*_{0}(*θ*_{inc}), *a*_{1}(*θ*_{inc}) and *a*_{2}(*θ*_{inc}). To do so, the first step would be to compute ∠Γ^{final} of (27) for the abovementioned controlled values of all parameters (except *θ*_{inc}) over ranges of values of *a*_{0}, *a*_{1} and *a*_{2}; specifically, from 0 to *π*/6 for *a*_{0} and 1.725 to 2.6 for both *a*_{1} and *a*_{2}, each of them in some number of step-divisions. The plotted graph of every resultant ∠Γ^{final} versus frequency is then checked up with a certain *θ*_{inc} case from the set of reference graphs of ∠Γ^{MOM}against frequency (where the superscript MOM denotes moment-method), from which the most matching set of (*a*_{0}, *a*_{1}, *a*_{2}) values is selected to be assigned to that *θ*_{inc} case. Repeating this for all *θ*_{inc} cases in the reference parametric data set reveals the *a*_{0}, *a*_{1} and *a*_{2} as discrete functions of *θ*_{inc}. For the present example of eighteen *θ*_{inc} cases in the parametric set, the discretized *a*_{0}, *a*_{1} and *a*_{2} are constructed as discrete functions of *θ*_{inc}. These 18-element vectors of numerical data for*a*_{0}, *a*_{1} and *a*_{2}may then be curve-fitted into polynomial functions of*θ*_{inc} by standard techniques. Upon doing so with a polynomial degree of 5 for decent modeling of the discrete functions, the functional forms of these coefficients are explicitly stated as follow.

where the coefficients of the polynomial functions are placed in 6-element vectors*P*_{0}, *P*_{1}, and *P*_{2}:

[48] With these established, the efficacy of the formula for ∠Γ^{final}in producing the reflection-phase diagrams had been demonstrated in terms of parametric studies; however these results are not presented in this paper due to space constraints. Five parameters for characterizing the corrugations and its impingent excitation plane wave had been considered. They are (a) the period*d*_{x}, (b) the relative permittivity of the groove-filling:*ε*_{rel,cav} = *ε*_{cav}/*ε*_{0}, (c) the corrugation-depth to period ratio:*h*/*d*_{x}, (d) the ratio between the width of the cavity (or groove) and the period: *g*/*d*_{x}, and (e) the elevation theta angle of incidence: *θ*_{inc}. Let us henceforth refer to these five parameters as *p*_{1}, *p*_{2}, *p*_{3}, *p*_{4}, and *p*_{5}, respectively, for convenience and brevity. Table 1 provides the numerical values of these investigated five parameters, being 6 of them per parameter.

Table 1. Range of Values of the Five Parameters*p*_{1} (mm) | *p*_{2} | *p*_{3} | *p*_{4} | *p*_{5} (rad) |
---|

1 | 2 | 2 | 0.45 | 0.001745 |

1.3 | 3 | 2.4 | 0.55 | 0.30194 |

1.6 | 4 | 2.8 | 0.65 | 0.6021 |

1.9 | 5 | 3.2 | 0.75 | 0.9023 |

2.2 | 6 | 3.6 | 0.85 | 1.2025 |

2.5 | 7 | 4 | 0.95 | 1.503 |

#### 6.2. Inclusion of Incident Phi Angle Variation

[49] In order to include the dependency of the reflection-phase for TM^{z} polarization on the azimuthal incident angle *ϕ*_{inc}, a parametric study of the reflection-phase over those aforementioned five parameters (defined as*p*_{1} to *p*_{5}) is conducted (again using the full-wave moment method), but now including*ϕ*_{inc}as the sixth parameter. From the computed reflection-phase diagrams (not presented here due to the voluminous size), comparison by observation of the reflection-phase diagram between the*ϕ*_{inc}= 0 case (being the situation thus-far considered, i.e., propagation in the plane perpendicular to the corrugations) and that of a nonzero*ϕ*_{inc} case for a certain common set of the other (five) parameters (period, groove permittivity, depth, width, and *θ*_{inc}) reveals frequency shifting between the two traces, the amount of deviation being a function of frequency (increases with it). Repeating such a comparison for another *θ*_{inc} case, it is found that this frequency shift also grows with *θ*_{inc}, but slowly for initially small *θ*_{inc} values (near broadside incidence), becoming increasingly appreciable as the incident plane wave gets closer to grazing. Repeating these comparisons yet for another *ϕ*_{inc} case shows that the frequency shift also varies with the azimuthal incidence angle. Hence, this frequency shift is a function of three parameters: (a) frequency, (b) *θ*_{inc}, and (c) *ϕ*_{inc}. Interestingly, this shift does not depend on the other four physical properties of the corrugations. Again in a similar way to the modeling of the coefficients *a*_{0}, *a*_{1} and *a*_{2} earlier on as polynomial functions, the frequency shift can also be modeled as an analytic polynomial function of those three parameters using the numerical data generated by the parametric study. The considered ranges of the three parameters are as follow: (a) five values of *ϕ*_{inc}: 0.1°, 22.55°, 45°, 67.45°, and 89.9°; (b) eight values of *θ*_{inc}: 0.1°, 10.85°, 21.6°, 32.35°, 43.1°, 53.85°, 64.6°, and 75.35°; and (c) nine values of frequency *f*: 5 GHz, 10 GHz, 15 GHz, 20 GHz, 25 GHz, 30 GHz, 35 GHz, 40 GHz, and 45 GHz. For convenience, we shall henceforth use serial indices to denote any of these values, e.g., *ϕ*_{inc}#3 = 45°, *θ*_{inc}#5 = 43.1°, *f*#7 = 35 GHz, etc.

[50] Performing the polynomial curve-fitting for a degree of 2, i.e., a quadratic-equation modeling, the frequency shift of the reflection-phase diagram trace for any nonzero*ϕ*_{inc} case (but sharing the same set of the other parameters of the *ϕ*_{inc} = 0 case) from that of the *ϕ*_{inc} = 0 case may be expressed as

where any is in turn a polynomial (quadratic) function of , i.e.

for all combination-pairs of (*u*, *v*) in (30a), and whereby

where *μ* and *σ* are the mean and standard deviation of the parameter denoted by their respective subscripts. For our range of parametric values stated above, these are: *μ*_{ϕ} = 0.7854, *σ*_{ϕ} = 0.61953; *μ*_{θ} = 0.6584, *σ*_{θ} = 0.45958; *μ*_{f} = 2.5 × 10^{10}, *σ*_{f} = 1.3693 × 10^{10}. The numerical values of the coefficients of (30b), where *u*, *v*, and *w* may denote 2, 1 or 0, are tabulated in Figure 8.

[51] Therefore, the *ultimate*explicit formula for the reflection phase as an entirely closed-form analytic function of*all parameters* of the corrugations, incident plane wave (particularly now with even *ϕ*_{inc} included), and the frequency is stated as:

from (27), in which *a*_{0}, *a*_{1} and *a*_{2} those of (28a), (28b) and (28c), and with

from (25), where

from (24), in which *a*_{1} and *a*_{2} are those of (28b) and (28c), and with

from (21b), and of course finally, with the shifted frequency:

where *f*_{shift} is from (30a) and is a function of *ϕ*_{inc}, *θ*_{inc} and *f.*

[52] The ultimate results for the demonstration of the efficacy of equations (32a)–(32e) are now presented in Figure 9for five cases of entirely arbitrary parameters. Each graph within this figure contains the reflection-phase diagram generated byequations (32a)–(32e)and the rigorous full-wave moment method formulated insection 2. Clearly, the agreement between the two approaches is seen to be outstanding. The CPU times for both were also computed and the derived formula is found to be around 40 times faster than the moment method.