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Takashige Omatsu, Editor-in-Chief
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Characterization of extraordinary transmission through conducting screens pierced by arrays of rectangular flared holes for quantitative comparison among various taper profiles and study of metal losses
Malcolm Ng Mou Kehn and Wei-Yu Lai
Malcolm Ng Mou Kehn1,2,* and Wei-Yu Lai1,2
1College of Electrical and Computer Engineering, National Chiao Tung University, Taiwan
2Center for mmWave Smart Radar Systems and Technologies, National Chiao Tung University, Taiwan
*Corresponding author: malcolm.ng@ieee.org
Malcolm Ng Mou Kehn https://orcid.org/0000-0002-2196-8112
M Ng Mou Kehn
W Lai
Vol. 2,
•https://doi.org/10.1364/OSAC.2.002728
Malcolm Ng Mou Kehn and Wei-Yu Lai, "Characterization of extraordinary transmission through conducting screens pierced by arrays of rectangular flared holes for quantitative comparison among various taper profiles and study of metal losses," OSA Continuum 2, 2728-2754 (2019)
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Modal analysis of metallic screen with finite conductivity perforated by array of subwavelength rectangular flared holes (OE)
Effective medium and equivalent circuit analysis of extraordinary transmission through metallic grating in the infrared range (OSAC)
Parametric study of optical transmission through plasmonic hole arrays modulated by the phase transition of vanadium dioxide (OSAC)
Holography, Gratings, and Diffraction
Extraordinary optical transmission
Fluorescence correlation spectroscopy
Fluorescence spectroscopy
Near field microscopy
Transmission coefficient
Original Manuscript: April 22, 2019
Revised Manuscript: July 2, 2019
Manuscript Accepted: July 6, 2019
Angular coordinates defining plane-wave incidence and validation of the tool
Characterization of extraordinary transmission
Various flare profiles
Conductor losses and sub-efficiencies
Oblique incidence
Experimental results from measurements of manufactured prototypes
Equations (17)
When illuminated by incident plane waves, conducting screens pierced by an array of holes may permit strong diffusions through them even at frequencies where the size of each hole is considerably less than the wavelength, amounting to extraordinary transmission (ET). This well-known phenomenon is herein further investigated beyond conventional topologies by considering square holes with tapered profiles, which are thereupon used as the vehicle for characterizing the degree of ET in a quantifiable manner, achieved by relating the transmissivity of an actual screen with that predicted by classical Bethe's theory. In terms of such quantified ET, the ordinary subwavelength square hole array is compared with a screen of equal thickness and perforated by holes of the same size as the former at the input side but get larger as they tunnel through to the exit apertures. It is found that the latter flared holes provide higher levels of ET than their unflared counterparts, the advantage being more pronounced for larger ratios of exit-to-entry hole sizes. Investigations of several hole-flare profiles also reveal that the flanged-type small input diaphragm directly peering (abruptly flared) into a single large square hole provides the greatest degree of ET. Oblique angles of incidence and conductor losses are investigated as well. Prototypes of perforated screens were also manufactured and measured with success in corroborating with theoretical predictions.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
By virtue of the light confinement that is offered, extraordinary transmission (ET) through metallic films pierced by arrays of subwavelength holes [1] lends itself to a wide range of applications such as sensing and spectroscopy, filters and polarizers, near-field microscopy, wavelength-tunable filters, subwavelength photolithography, optical modulators, nonlinear and quantum optics, subwavelength imaging [2–4], tracking and detection of single molecule fluorescence in biology and chemistry, ultrafast photodetectors for optoelectronics, vibrational spectroscopy of molecular monolayers, as well as fluorescence correlation spectroscopy [5–8].
There had been studies reported on arrays of slits with tapered profiles [9–14]. Those works, however, involved grille structures with just one dimensional (1D) periodicities, unlike [15] which treats two-dimensional (2D) lattices of flared-open rectangular holes that penetrate a conducting screen by a modal approach, the validity of which has been rigorously established by commercial simulation software in that paper and on which the findings of this paper are based. The papers of [9–11] looked into the prospects of enhancing the transmission through the gratings by linearly tapering each slit into a V-shape such that it is narrower at the output side, among which [10] further investigated the effects of higher order slit resonances and diffraction orders. A study of the same phenomenon was carried out in [12] but this time with the use of a transmission line model to explain the mechanism. Endeavors of a similar nature can be found in [13], whereby a circuit model was used. Breaking away from linear tapering, efforts were put into [14] for exploring nonlinearly tapered elliptically-profiled walls of the grooves.
Two dimensional (2D) periodic arrays of tapered holes were also explored in [16–18]. None of them, however, offered any theoretical solution, as opposed to the full-wave method based on modal analysis presented in [15] by which the herein results are computed.
Other works on 1D slit arrays include [19] and [20] while more of those on 2D hole arrays are found in [21–24]. Reports on the mode-matching of tapered holes take on the likes of [25] while the treatment of losses in the modeling of such structures has been addressed in [26]. General topics related to extraordinary transmission through holes array can also be looked up in [27–31].
In the entire vast literature on this subject, the notion of ET through subwavelength hole arrays stems universally from the anomalous phenomenon of diffused power which far exceeds that expected by the classical theory of Bethe [32]. Besides merely mentioning this basic concept textually, or at best going only as far as to just state that the transmitted power falls as the fourth power of the aperture size normalized to the wavelength, no prior work has gone further to relate its studies with Bethe's theory. Instead, most if not all papers have simply presented spikes in transmission spectra and identified them as being exceptional just because they occur at wavelengths that exceed the hole sizes. A more proper characterization could thus be in order to quantify just how exceptional any one scenario of ET through a certain perforated screen is as compared to that of another. It is among the objectives of this paper to attempt this endeavor for the first time, after which one may have a figure-of-merit to tell if a certain hole array is a stronger performer than another in terms of its ability in achieving ET. The aforementioned array of flared rectangular holes as modally treated in [15] shall be used as the vehicle for this study, of which that quantitative index will be ascertained for various taper profiles, thereby shedding insights into how the different flare topologies compare with one another in terms of wave diffusivity. Through such investigations and with the quantitative measure of ET, not only are tapered holes reaffirmed to provide stronger ET than ordinary straight ones that bore through the same screen thickness, as is already known, a specific attribute of the flare profile that further enhances this phenomenon is discovered, the precise details of which shall be revealed later. The portrayal of this latter constitutes the other main purpose of this paper.
Figure 1 illustrates a periodic 2D array of generally rectangular (although shown as square ones) flared holes pierced through a metallic sheet of thickness d, with periods along x and y denoted by dx and dy respectively. Each pyramidal tunnel is modeled as a cascaded stepped series of cavity sections, in connection with the modal analysis of [15] from which the herein computed results are obtained. The width a0 and height h0 along x and y of the input smallest section are as indicated, while those of the output largest one are given by aZ and hZ, respectively. Two perspectives showing the front and back of the screen are displayed. Several hole-flare profiles that are special cases of this general schematic shall be studied, particularly in the context of how they fare in producing the phenomenon of ET. Experiments on manufactured prototypes shall also be reported in the section that precedes the final one which summarizes the work.
Fig. 1. Two-dimensional periodic array of generally rectangular (although shown as square ones) flared holes pierced through a metallic sheet of thickness d, with periods along x and y denoted by dx and dy respectively as annotated. Each pyramidal tunnel modeled as a cascaded stepped series of cavity sections, in connection with the modal analysis of [15] from which the herein computed results are obtained. The width a0 and height h0 along x and y of the input smallest section are as indicated, while those of the output largest one are given by aZ and hZ, respectively. Two perspectives showing front and back of the screen are displayed.
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2. Angular coordinates defining plane-wave incidence and validation of the tool
As annotated in the diagram, Fig. 2 depicts the propagation direction of an incident plane wave that illuminates a planar conducting film of thickness d perforated by a 2D array of rectangular holes with periods dx and dy along x and y. A reference coordinate system is also drawn, the origin of which is located at an arbitrary distance of z0 perpendicularly from the surface of the grating. The path of irradiation in relation with the structure is represented by the angles symbolized as θinc and ϕinc subtended respectively from the z and x axes, the latter phi angle defining the plane of incidence in the usual way and as portrayed by the tilted shaded sheet in the schematic.
Fig. 2. Rectangular hole array illuminated by incident plane wave that propagates along direction defined by θinc measured from the z axis and contained within azimuth plane of incidence defined by ϕinc measured from the x axis.
As a brief single case of validation of the computer code based on the theoretical formulation of [15] all developed by the first author, Fig. 3 presents the validation with CST of the modal formulation of [15] for the zeroth-order transmission coefficient versus frequency for plane wave incidence that illuminates an array of square holes, each divided into two sections (single-step flare), with unit cell period of dx = dy = 5 mm, input square hole sizes: a0 = a1 = h0 = h1 = 2.5 mm, output square hole sizes: aZ = hZ = 4.5 mm, thicknesses of the input and output sections being 0.5 mm and 0.75 mm, respectively, a total screen thickness of d = 1.25 mm, and for TMz polarized incidence, θinc = 15°, ϕinc = 0. As can be seen, the agreement between both solution approaches is excellent.
Fig. 3. Validation with CST of modal formulation of [15] for zeroth-order transmission coefficient versus frequency for ϕinc = 0 azimuth plane of incidence of arriving plane wave that illuminates array of square holes each divided into two sections (single-step flare), with unit cell period: dx = dy = 5 mm, input square hole sizes: a0 = a1 = h0 = h1 = 2.5 mm, output square hole sizes: aZ = hZ = 4.5 mm, thicknesses of input and output sections = 0.5 mm and 0.75 mm, respectively, total screen thickness d = 1.25 mm: for TMz polarized incidence, θinc = 15°.
3. Characterization of extraordinary transmission
3.1. Screen porosity and transmission efficiency
To facilitate the upcoming developments, amongst the foremost quantities to establish is the screen porosity, ℘, defined as the ratio of the area of the rectangular aperture on the input side to that of the unit cell, which in the present context is
(1)$$\wp = {a_0}{h_0}/{d_x}{d_y},$$
noting that the perceived penetrability of the screen is defined by the hole size on the entrance (input) side of the screen at z = z0, and is limited by this size even if the exit aperture on the output side (at z = z0 + d = zZ) may be larger. Hence, the screen is regarded to be only as porous as its least penetrable side, either the input or the output side.
In the most physically intuitive sense, the usual zeroth-order transmission coefficient, T, defined as the fraction of the incident power that is transmitted through the perforated screen by the dominant Floquet harmonic mode, is then equal to the porosity ℘, since it is indeed just this proportion of the screen that receives (or 'feels') the impingent radiation and is thus permeable. If T is less than ℘, then it is still a conceivable outcome since there is certainly the logical possibility of imperfections (such as reflective or dissipative losses) such that some of the incident power fails to get through the screen. However, if T exceeds ℘, more output power than the available input amount collected by the holes penetrates the screen, resulting in an unusual phenomenon; an extraordinary transmission. Therefore, a so-called transmission efficiency, ɛT, may be defined as the ratio of T to ℘, i.e.,
(2)$${\varepsilon _\textrm{T}} = \textrm{T}/\wp ,$$
In this way, when T equals ℘, the latter serving as an apparent upper limit, a 100% efficiency arises, amounting to nothing more than what is expected of a lossless perfect screen; what goes into the holes gets out of them on the other side. If T falls short of ℘, the notion of losses sets in, falling in line with an ɛT that is understandably less than unity. But if T is larger than ℘, an extraordinary efficiency greater than 100% unexpectedly shows up.
3.2. Bethe hole theory
According to Bethe's theory of diffraction by small holes [32], for an incident plane wave of wavelength λ with power density Sinc (W/m2) that is obliquely impingent on an electric conducting screen having a total area of Atotal and pierced by an array of circular holes, each of diameter dhole, and arriving at an angle θinc measured from the surface normal, the hypothesized total power of the wave transmitted across the screen may be stated for the two principal polarizations as follows:
(3)$$P_{trans}^{T\zeta } = \left\{ \begin{array}{ll} P_{holes}^{\theta = 0}\cos {\theta_{inc}}; & {d_{hole}} > {\Upsilon ^{ - 1/4}}\lambda \\ \Upsilon {{({{{{d_{hole}}} \mathord{\left/ {\vphantom {{{d_{hole}}} \lambda }} \right.} \lambda }} )}^4}P_{holes}^{\theta = 0}\cos {\theta_{inc}}; & {d_{hole}} \le {\Upsilon ^{ - 1/4}}\lambda \end{array} \right.$$
where $\Upsilon = {{64{\pi ^2}\Lambda } \mathord{\left/ {\vphantom {{64{\pi^2}\Lambda } {27}}} \right.} {27}}$,
(4)$$\Lambda = \left\{ {\begin{array}{ll} \cos {\theta_{inc}}; & \zeta \equiv E\\ {({4 + {{\sin }^2}{\theta_{inc}}} )} \mathord{\left/ {\vphantom {{({4 + {{\sin }^2}{\theta_{inc}}} )} {({4\cos {\theta_{inc}}} );\ \zeta \equiv M}}} \right.} ({4\cos {\theta_{inc}}} ); & \zeta \equiv M \end{array}} \right.$$
(5)$$P_{holes}^{\theta = 0} = {S_{inc}}A_{holes}^{all};\textrm{ with }A_{holes}^{all} = \wp {A_{total}}$$
this latter being the total power received by all the holes of the array from that same illuminating wave had it been normally incident instead, such that
(6)$${P_{holes}} = P_{holes}^{\theta = 0}\cos {\theta _{inc}} = \wp \overbrace{{{P_{wholescreen}}}}^{{{S_{inc}}{A_{total}}\cos {\theta _{inc}}}}$$
is the total power collected by all holes from the obliquely incident wave, with ℘ being the screen porosity as defined earlier, and where Pwholescreen symbolizes the total power of the tilted irradiation that is presented to the entire screen (holes included). It is noted that under oblique incidence, the porosity remains unchanged, since both the hole and cell areas as 'felt' by the tilted propagating waves are effectively reduced by the same factor. Substituting this (6) into (3), and with
(7)$${\rm T}_{Bethe}^{T\zeta } = {{P_{trans}^{T\zeta }} \mathord{\left/ {\vphantom {{P_{trans}^{T\zeta }} {{P_{wholescreen}}}}} \right.} {{P_{wholescreen}}}},\ \zeta \equiv E\textrm{ or }M$$
being the usual zeroth-order transmission coefficient, (3) is rewritten as
(8)$$\overbrace{{{{{\rm T}_{Bethe}^{T\zeta }} \mathord{\left/ {\vphantom {{{\rm T}_{Bethe}^{T\zeta }} \wp }} \right.} \wp }}}^{{\varepsilon _{{{\rm T}_{_{T\zeta }}}}^{Bethe}}} = \left\{ {\begin{array}{ll} \cos {\theta_{inc}}; & {d_{hole}} > {\Upsilon ^{ - 1/4}}\lambda \\ \Upsilon {{({{{{d_{hole}}} \mathord{\left/ {\vphantom {{{d_{hole}}} \lambda }} \right.} \lambda }} )}^4}\cos {\theta_{inc}}; & {d_{hole}} \le {\Upsilon ^{ - 1/4}}\lambda \end{array}} \right.$$
The indication with braces as $\varepsilon _{{{\rm T}_{T\zeta }}}^{Bethe}$ on the left-hand side is by virtue of (2), as ${{{\rm T}_{Bethe}^{T\zeta }} \mathord{\left/ {\vphantom {{{\rm T}_{Bethe}^{T\zeta }} \wp }} \right.} \wp }$ bears the quantity of the transmission efficiency, ɛT. Scripts 'Bethe' attached to T and ɛT are indicative of the fact that these quantities are merely those expected of the Bethe hole theory, thereby being distinguished from the actual ${\rm T}_{actual}^{T\zeta }$ and $\varepsilon _{{{\rm T}_{T\zeta }}}^{actual} = {{{\rm T}_{actual}^{T\zeta }} \mathord{\left/ {\vphantom {{{\rm T}_{actual}^{T\zeta }} \wp }} \right.} \wp }$ that are obtained (computed) for any actual scenario. This same formula of (8) but only for normal incidence (θinc = 0) has also appeared in [5] although it was not derived there with the same details as here. Notice from (4) that Λ equals unity when θinc = 0 for both cases of ζ. Hence, the same transmitted power and transmission efficiency of (3) and (8) apply under normal incidence regardless of which one of the two principal polarizations, which is as required.
Therefore, at adequately high frequencies such that the hole diameter approaches half the wavelength, diffusion of normally-incident irradiation through the screen is permitted according to the Bethe hole theory. However, when the wavelengths associated with lower frequencies exceed twice the hole diameter, $\varepsilon _{{{\rm T}_{T\zeta }}}^{Bethe}$ decays as the fourth power of the hole diameter normalized to the wavelength. This means virtually none of the impingent radiation is expected to pass through the screen when the hole is electrically small.
3.3. "Extraordinarity" in transmission
With only the porosity being factored in (2), this original simple formula for the transmission efficiency alone offers only a partial manifestation of how truly extraordinary (if at all) is the transmission through hole arrays. For a more complete characterization, the hole-size relative to the wavelength should also come into play, as elucidated by the preceding subsection, thereby factoring in the effects of the frequency. With this established, the overall degree of extraordinary transmission, or herein termed as, "extraordinarity", may be conveyed via a figure-of-merit, symbolized here by Ψ and defined as the ratio of the actual transmission efficiency, $\varepsilon _{{{\rm T}_{T\zeta }}}^{actual}$, to that predicted by the Bethe hole theory, $\varepsilon _{{{\rm T}_{T\zeta }}}^{Bethe}$, i.e.,
(9)$$\Psi = {{\varepsilon _{{{\rm T}_{T\zeta }}}^{actual}} \mathord{\left/ {\vphantom {{\varepsilon_{{{\rm T}_{T\zeta }}}^{actual}} {\varepsilon_{{{\rm T}_{T\zeta }}}^{Bethe}}}} \right.} {\varepsilon _{{{\rm T}_{T\zeta }}}^{Bethe}}} = {{{\rm T}_{actual}^{T\zeta }} \mathord{\left/ {\vphantom {{{\rm T}_{actual}^{T\zeta }} {{\rm T}_{Bethe}^{T\zeta }}}} \right.} {{\rm T}_{Bethe}^{T\zeta }}}$$
(10)$$\varepsilon _{{{\rm T}_{T\zeta }}}^{actual} = {{{\rm T}_{T\zeta }^{actual}} \mathord{\left/ {\vphantom {{{\rm T}_{T\zeta }^{actual}} \wp }} \right.} \wp }$$
is the actual transmission efficiency obtained via the transmission coefficient ${\rm T}_{T\zeta }^{actual}$ of any actual scenario under Tζ polarization. This indicator Ψ thus serves to signal the true presence of ET when it exceeds unity (or its dB value is above zero), the degree of which is quantified by how much greater this ratio is than one (or how positive its dB value is).
4. Various flare profiles
This section shall be concerned with the investigations into four flared square-hole configurations, all with a universal square unit cell size of dxy = 5 mm and a common screen thickness of d = 1.25 mm; namely (A) linearly-flared, (B) quadratically-flared, (C) a so-called reverse-quadratically flared case, and (D) a case of input diaphragm opening with its surrounding thin metallic flange having a non-zero thickness (5% of the 1.25 mm film thickness), peering into a single large square cavity-hole (4.52 mm2) that tunnels all the way to the exit aperture of the same size at the output end. This latter input iris is thus represented by a short cavity-hole section having a length that equals its associated flange thickness. A normally incident (θinc = 0) TMz polarized plane wave (with ϕinc = 90° plane of incidence; Fig. 2) is also assumed to illuminate the perforated screens of all cases. Before proceeding, it is declared at this juncture that, with the exception of Section 6 later, the metal considered in the computations of all the upcoming results is perfectly electric conducting (PEC).
4.1. Linearly flared
For a certain entrance aperture size a0 × h0, several exit hole sizes aZ × hZ that are at least as large as that input mouth are considered, thus ranging from a0 × h0 tantamount to an unflared straight hole to an output hole size close to that of the unit cell (near dx × dy). Ten cavity-hole sections of uniform thickness d/10 = 0.125 mm along z are used to model each flare from the input to the output openings in a linear progressive fashion.
With these laid out, Fig. 4 contains six subplots of the zeroth power transmission coefficient against frequency for the linearly-flared square hole array, computed by the numerical code based on the modal technique reported and validated in [15], for input square hole sizes of (a) 1 mm2, (b) 1.52 mm2, (c) 22 mm2, (d) 2.52 mm2, (e) 32 mm2, and (f) 3.52 mm2, within each are several traces pertaining to various exit hole sizes as annotated. The corresponding variation of ${\rm T}_{Bethe}^{TM}({\theta _{inc}} = 0)$ as predicted by Bethe theory according to (7) is also graphed in each subplot as a single trace curve which pertains only to the common input hole porosity constituting the limiting factor (the 'weakest link'), being independent of the exit hole size. Evidently, for any one entry hole size (a subplot), the penetration of the impingent wave through the screen, for all frequencies, gets enhanced as the exit aperture is enlarged. The general similarity in trend between the actual computed results and the Bethe prediction is also discernable, whereby the transmission escalates initially with increasing frequency within lower bands, but tapering off in its climb beyond the frequency at which the hole diameter exceeds approximately half the wavelength of the irradiation, in correspondence with the straight horizontal-line portion of the Bethe curve. However, the concurrences in terms of the transmission levels are better for larger output holes. The reason for this is that Bethe's theory strictly applies to screens of ideally zero thickness, whereas finitely thick ones are considered here. Bigger exit apertures as compared to the entry portals mean wider flare angles. This creates a thinning effect of the metallic flanges surrounding the input holes which 'feel' them. As such, the metallic flanges as 'sensed' by the input holes are thinner for widely-flared open holes than those of gently-flared ones, thus more closely resembling the idealized infinitesimally-thin screen. Better agreement with Bethe's theory is thus achieved for such cases of widely-flared holes. In the extreme case of unflared ordinary holes, the full thickness d of the screen is sensed, leading to the poorest agreement.
Fig. 4. Variation of zeroth-modal power transmission coefficient for linearly-flared holes (modeled by 10 cavity sections with numerical tool of [15]) with frequency, for various output hole sizes (at least the size of the input hole) as annotated in each subplot, for input hole size: (a) 1 mm2, (b) 1.52 mm2, (c) 22 mm2, (d) 2.52 mm2, (e) 32 mm2, and (f) 3.52 mm2. All for TMz polarized incidence, ϕinc = 90°, θinc = 0, dx = dy = 5 mm, d = 1.25 mm.
The three peaks at 60 GHz, 85 GHz, and 120 GHz in the transmission spectrum of Fig. 4(a) are now explained. These spikes are due to the intersection of the respective k0 circles (at those frequencies) in the Floquet kx-ky diagram with spectral nodes, as had been described in [33]. It is the onset of surface waves that leads to surges in transmission.
For the present case of normal incidence and lattice constant of dx = dy = dx&y = 5 mm, consider first the Floquet diagram of Fig. 5(a) with the dominant (m = 0, n = 0) Floquet node located at the origin (due to the normal incidence). Anchored about this latter nodal point is a two-dimensional (2D) lattice of diffraction nodes with inter-nodal spacing of 2π/dx and 2π/dy (both equal to 2π/dx&y = 1256.6 rad/m) along the two axes: kx and ky, which is as shown. In this graph, the k0-circle has a radius of 2π/dx&y, i.e.,
(11)$${k_0}^{\# 1} = 2\pi /{d_{\textrm{x}\&\textrm{y}}} = 1256.6\textrm{ rad/m},$$
thereby intersecting all four nearest-in nodes (m = ±1, n = 0), (m = 0, n = ±1) along the two axes. With c = 3 × 108 m/s being the speed of light in free space, then upon applying (11), the frequency f #1 = k0#1c/(2π) = c/dx&y of this scenario (at which the wavelength equals the period) is readily seen to be indeed 60 GHz, thus accounting for the first spike (superscript #1).
Fig. 5. Grating lobe diagram for square cell (dx&y = 5 mm) and with dominant Floquet node at origin (normal incidence); (a) k0-circle intersects the four nodes along the axes, and (b) k0-circle intersects the four diagonal nodes.
Proceeding to the second spike at 85 GHz, refer to the spectral plane of Fig. 5(b), in which the k0-circle now intersects the four diagonal Floquet nodes at (m = ±1, n = 1) and (m = ±1, n = –1), thus having a radius of
(12)$${k_0}^{\# 2} = (\sqrt 2 )(2\pi /{d_{x\& y}}) = 1777\textrm{ rad/m},$$
Subsequently, the frequency f #2 = k0#2c/(2π) of this case is (√2)(c/dx&y), thus being 1.414 times the 60 GHz of the previous case, yielding 85 GHz indeed, thereby explaining the second peak (superscript #2).
As for the third and last spike at 120 GHz, being twice the 60 GHz of the first one, there is no need for an illustration of a third Floquet lattice diagram since the reason for this peak can be directly deduced to be due to the intersection of the k0-circle with the nodes of (m = ±2, n = 0), (m = 0, n = ±2) along the two axes. And when the radius of this circle doubles from that of the first spike case, so would the associated frequency.
The corresponding frequency responses of the extraordinarity, Ψ, are plotted in Fig. 6. The higher this factor rises above 0 dB, the more extraordinary is the transmission. Focusing on the topmost trace in each subplot pertaining to the largest output hole providing the strongest diffusion, it is observed that the peak Ψ occurring near and slightly below 60 GHz is stronger for smaller input holes, although at the expense of bandwidth, which gets narrower about higher summits. It is stressed at this point that it is only through the present quantitative characterization of ET that this finding could have been possible, without which that outcome cannot be seen by just looking at the transmission spectra of Fig. 4 alone.
Fig. 6. Variation of extraordinariness Ψ of linearly-flared square holes (modeled by 10 cavity sections with numerical tool of [15]) with frequency for various output square hole sizes (at least the size of the inputs square hole) as annotated in each subplot, for input square hole size: (a) 1 mm2, (b) 1.52 mm2, (c) 22 mm2, (d) 2.52 mm2, (e) 32 mm2, and (f) 3.52 mm2. All for TMz polarized incidence, ϕinc = 90°, θinc = 0, dx = dy = 5 mm, d = 1.25 mm.
This study thus puts forth the notion that, for a common output hole size, the smaller the input hole is as compared to the fixed output hole, the stronger will be the degree of ET, thereby demonstrating the benefit of steeply flaring open the holes for attaining ET.
4.2. Quadratically flared
A schematic of the quadratic flare is given in Fig. 7(a), which is described by the function: f(z) = (aZ − a0)z2/(2d2), as appreciated by Fig. 7(b). This profile is flared in both the x and y directions and fifteen cavity-hole sections are used to model this hole profile in the computations by the tool of [15], the results of which are to follow.
Fig. 7. (a) Schematic, and (b) functional representation of quadratic flare profile
As was for linearly-flared holes, the same earlier observation about smaller input holes in tandem with larger output ones having the potential for the greatest Ψ is again prevalent in Fig. 8. Comparing the graphs in this figure with those of Fig. 6, it is found that the ability of quadratically-flared holes to achieve high Ψ is not distinctly superior to that of linearly-flared ones; instead, they even suffer from narrower bandwidths about the peaks at which Ψ is locally maximum.
Fig. 8. Variation of extraordinarity Ψ of quadratically-flared square holes (modeled by 15 cavity sections with numerical tool of [15]) with frequency for various output square hole sizes (larger than the input square hole) as annotated in each subplot, for input square hole size: (a) 1 mm2, (b) 1.52 mm2, (c) 22 mm2, and (d) 2.52 mm2. All for TMz polarized incidence, ϕinc = 90°, θinc = 0, dx = dy = 5 mm, d = 1.25 mm.
4.3. Reverse-quadratically flared
In the foregoing study of quadratic flare profiles, the growth rate of the hole is initially low from the entry port but speeds up progressively as it tunnels towards the output end. The inverse scenario may also be investigated, and that is, the hole starts off at the input side already with a maximum expansion pace but slows down as it approaches the exit aperture. This is depicted by Fig. 9(a) and its mathematical description is provided by the functional relation in Fig. 9(b), expressed as: f(z) = [(aZ − a0)/2]√(z/d), representative of the reverse case of the preceding quadratic profile. As before, this hole profile is flared in both the x and y directions and fifteen cavity-hole sections are used in the modal analysis of [15] to model it.
Fig. 9. (a) Schematic, and (b) functional representation of reverse- quadratic flare profile
In the same fashion as the preceding two flare-profiles, the corresponding graphs of Ψ versus frequency are presented in Fig. 10 for the reverse-quadratic case. With some scrutiny of these plots, it is observed that the performance of this reverse-quadratic flare-profile where achieving high Ψ is concerned is much better than that of the preceding quadratically-flared case, and even surpasses the linearly-flared one earlier, both in terms of the maximum achievable degree of extraordinary transmission and the bandwidth about this peak. A more precise comparison shall be made later for a fixed output hole size of 4.52 mm2 out of a 52 mm2 square cell size.
Fig. 10. Variation of extraordinarity Ψ of reverse-quadratically-flared holes (modeled by 15 cavity sections with numerical tool of [15]) with frequency for various output hole sizes (larger than the input hole) as annotated in each subplot, for input hole size: (a) 1 mm2, (b) 1.52 mm2, (c) 22 mm2, and (d) 2.52 mm2. All for TMz polarized incidence, ϕinc = 90°, θinc = 0, dx = dy = 5 mm, d = 1.25 mm.
As a closing remark of this section, it is noted that the peaks in extraordinarity at 60 GHz in Figs. 6, 8, and 10 coincide with the period of the structure, which is 5 mm. This is in agreement with the theory of extraordinary transmission.
4.4. Flanged input iris peering into a single large cavity
The investigations of the foregoing two hole-flare topologies spark the suggestion of a certain notion: that the earlier the stage of the tunneling at which the metallic flange gets thinner, the stronger will be the attainable Ψ (the reverse-quadratic case surpasses the linear one, which in turn is better than the quadratic-flared case). This inspires the consideration in this subsection of an extreme configuration comprising input diaphragm openings with surrounding metallic flanges each having a non-zero but small thickness (5% of the 1.25 mm screen thickness), all peering into a single large square cavity-hole (4.52 mm2) near the size of the square unit cell (52 mm2) and which tunnels all the way (through the remaining distance of 95% of 1.25 mm) to the exit aperture of the same size (as the cavity) at the output end, as depicted in Fig. 11. In this way, an extremely abrupt flare essentially of infinite slope (90° flare angle) is in place. A perspective view looking from the output side is also provided.
Fig. 11. Input iris with metallic flange of thickness 5% of 1.25 mm, peering into a single large square cavity-hole (4.52 mm2) near the size of the square unit cell (52 mm2). A perspective view is also furnished.
Each trace in the graph of Fig. 12 presents the variation of Ψ with frequency for a certain input iris size as annotated and listed here as: (a) 1 mm2, (b) 1.52 mm2, (c) 22 mm2, (d) 2.52 mm2, (e) 32 mm2, and (f) 3.52 mm2. It is observed that the best achievers of Ψ at the same resonance near 60 GHz are those with the smallest input holes; although the input iris with 1.52 mm2 size appears to offer the greatest Ψ, it is the one with 1 mm2 size that actually attains the highest level but just that its bandwidth is so small that its true peak value gets missed from the computed set of discrete frequencies. For all other sizes larger than this, the maximal attainable Ψ falls off as the entry diaphragm enlarges and occurs at a frequency that is increasingly displaced below 60 GHz.
Fig. 12. Variation of extaordinarity Ψ with frequency for flanged-type input iris peering into a single 4.52 mm2 cavity-hole for various input diaphragm sizes as annotated.
4.5. Field distributions within cavities
The cross-sectional field variations within the cavities are now presented. Because of the similarities among them all, only those of a couple of various flare profiles are showcased. Figure 13(a) displays the Ey and Hx field distributions within an a = 4.5 mm by h = 4.5 mm cavity section of the quadratic-flared hole profile whereas Fig. 13(b) offers those of its reverse-quadratic counterpart, both at 60 GHz. The well-known TE10 modal field variation is evident, nothing less than which would be expected of the cavity Green's functions that satisfy the boundary conditions.
Fig. 13. Modal field distributions at 60 GHz within an a = 4.5 mm by h = 4.5 mm cavity section of the (a) quadratic, and (b) reverse-quadratic flared hole profile; left plots: |Ey| field, and right plots: |Hx|-field profiles.
It has been learnt from the foregoing section that the larger the exit aperture relative to its entry port, the greater will be the transmission and thus the attainable extraordinarity. As such, a universal output square hole size of 4.52 mm2 shall now be considered in this section, one that almost occupies the entire square unit cell of 52 mm2, this latter likewise applying throughout. Then for a of 1 mm2 input aperture size and a common screen thickness d = 1.25 mm, the performances of Ψ (plotted versus frequency) among the various hole-flare profiles are superposed onto the same graph in Fig. 14, namely, the linearly-flared of Section 4.1, the quadratically-flared and its reverse counterpart of Sections 4.2 and 4.3 respectively, as well as the flanged input iris peering into a single large 4.52 mm2 cavity as of Section 4.4. Two further traces are included in the plot: one pertaining to a uniform (ordinary untapered type) square hole bearing the same 1 mm2 size as the input port and which tunnels through a metallic screen with thickness d = 1.25 mm (serving as the base reference unflared case - the top item in the legend), and the other associated with a likewise regular straight square hole also of that same 1 mm2 input-hole size, but this time pierced through a screen which is only as thin (0.05 × 1.25 mm) as the flange of the input iris considered in Section 4.4 (constituting another reference scenario - bottom item in the legend).
Fig. 14. Comparison of extraordinarity spectra amongst various topologies as identified in legend, all for fixed input 1 mm2 hole and screen thickness d = 1.25 mm except one case (last item in legend) where it is 0.05 × 1.25 mm2 instead. Output size of 4.52 mm2 except two uniform unflared hole reference cases (first and last items in legend).
In terms of attaining high Ψ, it is obvious that all configurations surpass the uniform-hole reference by tremendous amounts (in the orders of 20 dB) throughout the entire considered frequency range, thus again exemplifying the effects of flaring open the exit hole. Focusing next on the resonance around 60 GHz, it is seen that the flanged-type input diaphragm peering into just a single 4.52 mm2 cavity achieves the greatest Ψ amongst all its contending hole-flare topologies, surpassing even the case of just ordinary unflared holes pierced through a metallic sheet that is of the same thickness as the aforementioned flange, being 5% of the total screen thickness d = 1.25 mm. In other words, even if the square metallic fence defining (bordering) the 4.52 mm2 cavity were to be broken off (see perspective view of Fig. 11), the resultant flat perforated thin screen (of thickness 0.05 × 1.25 mm) would still offer a lower maximum Ψ at around 60 GHz than that afforded by its original unbroken form. This demonstration addresses any concerns about the possible redundancy of that latter fence creating the 4.52 mm2 cavity - it is not superfluous for this case. On passing this juncture, it is worth mentioning that, for the same screen porosity, thinner sheets are known to offer stronger transmission than thicker ones. Thus, the just-described superiority over the thin perforated screen with broken-off fences cannot be taken for granted; in fact, it is not expected.
6. Conductor losses and sub-efficiencies
With flared-open holes comes the risk of greater conductor losses due to larger exposed metallic surfaces, if imperfect metals with finite conductivities are considered. It is the intent of this section to look into this matter.
The transmission coefficient dealt with so far, being the ratio of the transmitted (Ptrans) to the incident (Pinc) power per unit cell of the hole array, may also be perceived as a form of efficiency, symbolized as ɛtrans = Ptrans/Pinc. Consequently, by defining the conductivity efficiency as: ɛcond = (Ptrans − Pcond)/Ptrans where Pcond is the power loss per unit cell due to finite metal conductivity, the total efficiency, ɛtot is stated as the product of the latter two subefficiencies, i.e. ɛtot = ɛtransɛcond, indeed being (Ptrans − Pcond)/Pinc as required. The conductivity power lost to the imperfect metal walls is given by:
(13)$${P_{cond}} = \frac{{{R_s}}}{2}\int\!\!\!\int\limits_{{S_{metal}}} {{{|{{{\vec{H}}_t}} |}^2}ds} ;\textrm{ where }{R_s} = \sqrt {{{\pi f{\mu _{cond}}} \mathord{\left/ {\vphantom {{\pi f{\mu_{cond}}} \sigma }} \right.} \sigma }}, $$
the latter being the surface resistance of the metal, in which μcond and σ are its permeability and conductivity, respectively, and where Smetal denotes all conducting surfaces inside that cell, comprising the interior walls of the flared hole as well as the exterior surfaces of the metallic flanges on both the input (incidence) and output (transmission) sides of the screen, with Ht being the tangential magnetic field component over those aforementioned metal surfaces made available by the full-wave modal analysis presented in [15].
For normal incidence, the same unit cell size (dx = dy = 5 mm) and total screen thickness (d = 1.25 mm) as before, and with σ = 5.8 × 107 S/m and μcond = μ0 (copper assumed as the lossy metal), the variation of these aforesaid efficiencies with frequency is given in Fig. 15 for three hole-flare profiles, namely (a) linear, (b) quadratic, and (c) flanged input iris peering into a single cavity. For all of them, the input and output hole sizes are 1.5 mm2 and 4.5 mm2, respectively. For the last case, the thickness of the metallic flange surrounding the 1.52 mm2 input iris is 5% of d while the depth of the 4.5 mm2 hole is 95% of d as before. As can be seen, the conductivity efficiency is observed to commensurate with the transmission efficiency for all hole topologies; as in, when ɛtrans rises with frequency, ɛcond climbs as well (albeit not in the same fashion), both of them peaking at the same frequency, and when the former encounters a sharp dip at a certain frequency, so does the latter. This is an important finding since extraordinary transmissions pertaining to summits in ɛtrans are thus in concert with the peaks in ɛcond as well, being a highly favorable outcome. Another significant aspect learned is that, despite the flaring open of the holes resulting in increased metallic surfaces, the conductivity losses are not found to be severe. In fact, the metal losses are actually very low at most frequencies in the considered band.
Fig. 15. Graphs of transmission, conductivity, and total efficiencies versus frequency, for three hole-flare topologies: (a) linear, (b) quadratic, and (c) flanged input iris peering into a single cavity. Copper assumed as the lossy metal, with σ = 5.8 × 107 S/m and μcond = μ0.
7. Oblique incidence
This section investigates hole arrays illuminated by plane waves with directions of incidence that are not perpendicular to the surfaces of the screen, i.e. θinc ≠ 0. With that same universal lattice size of dx = dy = 5 mm as before, two types of holes shall be considered: the input square iris with flange thickness 5% of 1.25 mm and peering into a 4.52 mm2 cavity of depth 95% of 1.25 mm, and the ordinary unflared square hole pierced through a screen of the same thickness as the aforementioned metal flange. Across the board, TEz polarized plane-waves are henceforth assumed to arrive along directions of propagation contained within a fixed azimuth plane of incidence ϕinc = 0 (see Fig. 2). For the flanged input iris topology, Fig. 16 presents through its right-side plots of Figs. 16(aii) and 16(bii) the transmission spectra for two entrance diaphragm sizes: 1 mm2 and 1.52 mm2, respectively, in each of which any one trace pertaining to a certain θinc, as annotated by the shared legends located in the middle. The corresponding results as expected by Bethe theory of (7) are conveyed by the left-side plots of Figs. 16(ai) and 16(bi). Good concurrence between the hypothetical prediction and the rigorous modal analysis of [15] is evident.
Fig. 16. Transmission spectra for various θinc as annotated, for two input square hole sizes: upper plots (a) 1 mm2, and lower plots (b) 1.52 mm2; left-side plots of (ai) and (bi) predicted by Bethe theory as of (8), right-side plots of (aii) and (bii) computed by present modal analysis for input square iris with flange thickness 5% of 1.25 mm and peering into a 4.52 mm2 cavity of depth 95% of 1.25 mm.
Closer observations of the computed actual cases of Figs. 16(aii) and 16(bii) reveal the presence of kinks between 30 and 50 GHz, which are just abrupt spikes. Let us explain each one in turn, the reasons for all of which being the same as those of Figs. 5(a) and 5(b) earlier, whereby the incipience of surface waves gives rise to transmission resonances. Before that, for a 5 mm period along both x and y, the inter-nodal spacing of 2π/dx&y = 1256.6 rad/m along the kx and ky axes of the Floquet diagram is first laid out. Then starting with the resonance at 48 GHz (with corresponding k048G = 1005 rad/m) for the case of θinc = 15°, then with its associated ϕinc = 0, the dominant Floquet node in the spectral diagram is at
(14)$$({{k_x}_{00},\;{k_y}_{00}} )= ({k_0}^{48\textrm{G}}\sin {\theta _{inc}}\cos {\phi _{inc}} = {k_0}^{48\textrm{G}}\sin {15^{\circ}} = 260.2,\;{k_0}^{48\textrm{G}}\sin {\theta _{inc}}\sin {\phi _{inc}} = 0),$$
This results in a (m = –1, n = 0) node to be located at (kx,(−1,0) = 260.2–1256.6 = −996.4 rad/m, ky,(−1,0) = 0), as portrayed in Fig. 17(a). When the k0 circle intersects this node, meaning that k048G equals 996.4 rad/m, the associated frequency is f = k048Gc/(2π) = 47.6 ≈ 48 GHz indeed!
Fig. 17. Grating lobe diagram for square cell (dx&y = 5 mm) and with dominant Floquet node at: (a) (260.2, 0) pertaining to (θinc = 15°, ϕinc = 0), with k0-circle radius of 996.4 rad/m corresponding to 48 GHz, and (b) (419, 0) pertaining to (θinc = 30°, ϕinc = 0), with k0-circle radius of 838 rad/m corresponding to 40 GHz.
Proceeding to the second kink at 40 GHz (with k040G = 838 rad/m) for the case of θinc = 30° (again with ϕinc = 0), the dominant node is now at
(15)$$({{k_x}_{00},\;{k_y}_{00}} )= ({k_0}^{40\textrm{G}}\sin {\theta _{inc}}\cos {\phi _{inc}} = {k_0}^{40\textrm{G}}\sin {30^{\circ}} = 419,\;{k_0}^{40\textrm{G}}\sin {\theta _{inc}}\sin {\phi _{inc}} = 0),$$
The consequent (m = −1, n = 0) node is then located at (kx,(−1,0) = 419–1256.6 = –838 rad/m, ky,(−1,0) = 0), as portrayed in Fig. 17(b). Expectedly, the frequency pertaining to the k0-circle that intersects this node, i.e., k040G = 838 rad/m, is indeed f = k040G c/(2π) = 40 GHz.
Moving on to kink at 35 GHz (with k035G = 733 rad/m) for θinc = 45°, deductively without any more lattice diagrams, the following relations are directly written in the same analogous way:
(16)$$({{k_x}_{00},{k_y}_{00}} ) = ({k_0}^{30G}\sin {\theta _{inc}}\cos {\phi _{inc}} = {k_0}^{30G}\sin 45^\circ = 518,{k_0}^{30\textrm{G}}\sin {\theta _{inc}}\sin {\phi _{inc}} = 0),$$
(17)$${k_0} = |{518 - 1256.6} | = 738\textrm{ rad/m} \Rightarrow f = {k_0}c/(2\pi ) = 35.2 \approx 35\textrm{ GHz indeed}.$$
The same form of portrayal for the ordinary straight square holes pierced through a metallic sheet with identical thickness as the metal flange of the preceding configuration is given by Fig. 18. Once again, fine agreement between the formula of (8) and the present modal technique is achieved. It is noted that the Bethe-generated left-side plots of Figs. 18(ai) and 18(bi) are the same as those of Figs. 16(ai) and 16(bi), but are repeated just for ease of comparison.
Fig. 18. Transmission spectra for various θinc as annotated, for two input square hole sizes: upper plots (a) 1 mm2, and lower plots (b) 1.52 mm2; left-side plots of (ai) and (bi) predicted by Bethe theory as of (8), right-side plots of (aii) and (bii) computed by present modal analysis for ordinary unflared square holes pierced through a screen with thickness 5% of 1.25 mm.
For both Figs. 16 and 18, a universal phenomenon is obvious, and that is, as the obliquity of the impingent ray increases, the diffusivity of the perforated screen drops.
To ascertain if the finding made earlier for normally impingent irradiation about the flanged input iris peeking into a single large cavity as being the topology that yields the greatest extraordinarity at the resonance about 60 GHz holds also for oblique incidences, Fig. 19 conveys the same kind of graphs as Fig. 14, but this time each panel pertaining to one of four angles of incidence, namely θinc being (a) 15°, (b) 30°, (c) 45°, and (d) 60°, in this same order of the subplots of Figs. 19(a)–(d), all for a universal input 1 mm2 hole (and 4.52 mm2 output except two reference cases of unflared holes). As observed, for all four oblique directions of plane wave arrival, that aforementioned flanged type of input diaphragm retains its top spot amongst the various contending configurations in achieving the highest attainable Ψ at resonances. Not only that, it also preserves its superiority in terms of bandwidth about the peaks in its Ψ spectra. Importantly, as before, it surpasses the reference case of just a single (unflared) cavity-hole section tunneling through a metallic screen of a thickness that equals that of its thin metallic flange (of thickness 0.05 × 1.25 mm). This demonstrates that the square fences defining the large cavity are not redundant; instead they are needed for the flanged-type input iris to outperform the otherwise identical case without them.
Fig. 19. Comparison in each panel of extraordinarity spectra amongst various topologies as identified in legend, all for fixed input 1 mm2 and output 4.52 mm2 holes except two uniform 1 mm2 hole reference cases in each subplot, and common screen thickness d = 1.25 mm except one case (where it is 0.05 × 1.25 mm instead) as specified in each legend; for various θinc: (a) 15°, (b) 30°, (c) 45°, and (d) 60°.
8. Experimental results from measurements of manufactured prototypes
Three prototypes of aluminum sheets perforated by arrays of square holes were manufactured, the photographs of which are shown in Fig. 20. The periods along both directions of the square unit cells for all of them are equal to 13 mm. Sheets 1 and 2 share the same thickness of 1 mm, differing only in their hole sizes, being respectively 8 × 8 mm2 and 10 × 10 mm2. Like the first sheet, each aperture size of Sheet 3 is also 8 × 8 mm2, but with a thickness of 2 mm. A fourth configuration is created by piling the first two sheets together to form a screen of cascaded stepped holes with combined thickness of 2 mm, thereby constituting a coarsely discretized form of flared holes each composed of just two hole sizes, as pictured in Fig. 21.
Fig. 20. Photograph of the three manufactured perforated metallic screens.
Fig. 21. Photograph of Sheets 1 and 2 piled on top of each other to create an array of stepped square holes, representing a coarsely discretized form of flared holes made of only two sections.
The setup for the measurement of reflection from and transmission through the perforated screens is photographed in Figs. 22(a) and 22(b) for normal and oblique incidences. Two horn antennas serving as the transmitter and receiver are connected to the two ports of an Anritsu MS4644B vector network analyzer and the perforated screen is placed in between both of them as shown. This latter is also surrounded by absorbers to shield off unwanted ambient microwave fields. Normal incidence is assumed throughout the experiments.
Fig. 22. Photographs of the experimental setup comprising two mutually facing horns both connected to a network analyzer (not shown) and between which the perforated screen is placed, for (a) normal incidence, and (b) θinc = 30°
As the experiment was not carried out in an anechoic chamber, the measured results are not for the far field. Several issues affecting the measurements may then arise. Firstly, the incidence is not a true plane wave and only a certain number of holes get illuminated in the intended way. Secondly, there may also be standing-waves between the transmitter and the metallic sheet as well as between the latter and the receiver. The measurements may thus depart from numerical calculations, as observed by the green curves later in Fig. 24.
Each comprising two subplots, the upcoming Figs. 23 and 24 present respectively the graphs of the reflection and transmission coefficients against frequency. The upper (a) plot of each one of them is obtained from computations by the program code based on the analysis of [15] whereas the lower (b) graph is from measurements. Every subplot contains four traces, each pertaining to one of the four configurations described at the start of this section and as indicated in each legend. For all four hole-array topologies, good agreements between the computed and measured reflection spectra are observable from Figs. 23(a) and 23(b) respectively. The same can be said about the frequency responses of the transmission through all four types of pierced screens, as the congruence between computed and measured results of Figs. 24(a) and 24(b) exemplifies.
Fig. 23. Graphs of reflection coefficient versus frequency for various hole-array topologies as indicated in the legends, generated by (a) computations of program code based on analysis in [15], and (b) measurements.
Fig. 24. Graphs of transmission coefficient of normal incidence versus frequency for various hole-array topologies as indicated in the legends, generated by (a) computations of program code based on analysis in [15], and (b) measurements.
The best performer in terms of being able to transmit incident waves with the least reflection is seen to be the one with the largest square hole-size of 102 mm2 as well as having the thinnest screen of 1 mm thickness (the cross markers in both Figs. 23 and 24). Whereas the least porous screen is the one with the smallest hole size of 82 mm2 made worse by its largest sheet thickness of 2 mm (the plus markers in both figures). These observations cohere perfectly with the findings acquired in Section 4.
Further inspection of the graphs (their two middle traces) shows that, despite being thicker overall (2 mm) and thus expected to fare worse in wave diffusivity (than one with a smaller thickness of 1 mm), the cascaded screen with stepped holes (8 × 8 mm2 and 10 × 10 mm2) deemed only as porous as the smaller of its two composite aperture sizes (8 × 8 mm2), is actually comparable (in its ability to allow waves to penetrate) to the thinner sheet with a smaller 1 mm thickness and of the same porosity of 8 × 8 mm2 (divided by 132 mm2). This elucidates the wondrous effect of the stepped, coarsely-flared holes in making up for the larger screen thickness. And importantly, this phenomenon is reproduced in the experiments, as the measurement results of Figs. 23(b) and 24(b) show.
Finally, as importantly, the piled-up sheets with staircase-type flared holes 'beats' the unflared hole-array with the same 8 × 8 mm2 aperture size and having the same 2 mm thickness, as predicted by computations and again verified by experiments. This unequivocally confirms the key message of this research – that flared holes, even as crudely as by the use of only two steps, can and will permit stronger penetration of waves through them, thereby lending themselves to greater potentials for achieving extraordinary transmission.
By the reasoning of Fig. 5, a transmission resonance is expected at the frequency where the period equals the wavelength, which for the present prototype with cell size of 13 mm, is 23.06 GHz. The computed graph of Fig. 25(a), being of the same kind as Fig. 24(a) but just zoomed into a narrow band about this 23 GHz, indeed demonstrates a very abrupt and narrow-banded spike at precisely that 23.06 GHz for every one of the four cases of perforated screens. The corresponding experimental results are given in Fig. 25(b), in which peaks for all cases can be observed at around 22.98 GHz, extremely close to the predicted frequency.
Fig. 25. Same type as Fig. 24 but zoomed into range about 23 GHz, for (a) computations, and (b) measurements.
A case of oblique incidence was also measured for TMz polarization, with θinc = 30° and ϕinc = 90°, the experimental setup of which is photographed in Fig. 22(b). The experimental results are compared with computations as given in Figs. 26(a) and 26(b). As seen, the measured transmission coefficients cohere well with theoretical predictions in terms of their relative levels among the four hole-array topologies; specifically, the 102 mm2 hole array of 1 mm thickness is the highest while the 82 mm2 hole array of 2 mm thickness being the lowest, with the other two cases approximately of the same levels. The peaking at 15 GHz in the computations for all cases is captured experimentally at around 14.5 GHz whereas the common theoretical dip at 15.4 GHz has been detected at about 16.5 GHz in the measurements. These slight shifts in frequency are due to many factors such as the gratings not being infinitely large, the impingent waves from the horns are not true plane waves as mentioned earlier, and of course the fact that the measurement scenario is nowhere near being in an ideal unbounded free space environment. The kinks at around 13 GHz in the experimental data may be attributed to resonances within the two layers separating both the transmitter as well as the receiver from the gratings.
Fig. 26. Transmission coefficient versus frequency for various hole-array topologies as indicated in the legends, for TMz polarized oblique (ϕinc = 0, θinc = 30°) incidence generated by (a) computations of program code based on analysis in [15], and (b) measurements.
With the numerical tool developed based on the modal formulation in [15], computed results in the forms of transmission spectra (graphs of transmission versus frequency) can be generated. Enhanced energy diffusion occurring even at frequencies where each hole is considerably less than the wavelength of the irradiation has been demonstrated, being a phenomenon well known as extraordinary transmission (ET) as it violates classical aperture theory which predicts the inability of electrically small holes in allowing electromagnetic fields and waves to pass through them.
The degree of ET has herein been characterized by relating the computed transmission property of the actual perforated screen with the corresponding one predicted by Bethe's classical hole theory, thus serving as a figure-of-merit that allows one to gauge just how exceptional the ET of one screen is relative to another. This means for quantifying ET is then applied to the comparison of ordinary straight holes with flared ones of the same sizes on the input side as the former and drilled through metal boards of equal thicknesses, thus leading to the discovery that flared holes provide higher levels of ET than their unflared but otherwise equivalent counterparts; the extent of which being greater for larger output-to-input aperture sizes. Investigations of a variety of hole-flare profiles have shown that, for the same input and output hole sizes, holes which flare open the most rapidly right from the start of their tunneling journeys at the input side but slow down as they approach the output ports provide higher degrees of ET than those which begin sluggishly but speed up as they reach the exit apertures. This notion is confirmed by the verification of the fact that, amongst all its contending hole-flare topologies with the same input and output hole sizes, the metal-flanged-type small input-diaphragm directly peering (abruptly flared open) into a single large cavity-hole the size of the common exit portal affords the greatest enhancement of the degree of ET from that of the unflared counterpart. It was also found to possess the widest bandwidth of this attribute as well. Importantly, this flanged-type input iris has proven to be superior even to the case of ordinary unflared holes with the same size on the input side but pierced into a metallic sheet that is only as thin as its flange. This points to the non-redundancy of the large cavity, whose fences are thus not to be broken off. Besides, as thinner screens are known to provide higher transmission, this superiority is not expected, the demonstration of which is therefore not trivial.
Comparisons of rigorously-computed transmission spectra of actual scenarios with those predicted by Bethe theory, applicable strictly to screens of zero thickness, have shown that thinner screens provide better agreements. Related to this finding, for equal input and output aperture sizes, flared holes that start growing abruptly right from the entrance window but slowing down as they pierce through the sheet also concur better with Bethe's predictions than those which enlarge slowly initially but expand more rapidly as they approach the exit apertures. This can be appreciated by the fact that the surrounding metal flange 'felt' by each hole gets thinner more 'quickly' in the former case than the latter, thus attaining conditions similar to thinner screens earlier in the drill-stage.
Oblique incidences with consideration of the polarization of the arriving plane wave can also be managed by the modal approach presented in [15]. It is found that those same main phenomena for normal incidences generally reapply for oblique ones, but just with reduced transmission and levels of ET. Particularly, the benefits of flared holes are preserved and the flanged-input iris peering into a single large cavity retains its edge over all other contending hole-profiles in terms of the ability to attain high degrees of ET and wide bandwidths about this property.
Conductor losses to imperfect metal surfaces of the hole array have also been studied, which are quantified by a so-called conductivity efficiency. Computed results have shown that the occurrences of lowest metal losses coincide with those of surges in wave penetration through the screen, which is an extremely favorable behavior and a significant finding.
Experiments on manufactured hole-array prototypes of various forms in terms of hole sizes and screen thicknesses were also successfully conducted, as the measured results corroborate the theoretical findings established in the earlier parts of the paper in resounding fashions.
It is hoped that the materials presented in this work could trigger the attention on the new notion of characterizing ET in a quantitative fashion and also raise awareness of the capabilities which flared holes have in attaining ET that are even more exceptional than those achievable by their unflared but otherwise equivalent counterparts, particularly now for 2D periodic arrays as opposed to just 1D arrays of slits or periodic gratings which have been formerly reported. This may then crystallize new ideas for continued research in this exciting and growing subject that lends itself to a vast range of applications in forefront topics of physical sciences as well as electrical engineering and optical technologies.
Ministry of Science and Technology, Taiwan (MOST) (MOST 107-2221-E-009 -051 -MY2, MOST 107-3017-F-009-001).
This work was partially supported by the "Center for mmWave Smart Radar Systems and Technologies" under the Featured Areas Research Center Program within the framework of the Higher Education Sprout Project by the Ministry of Education (MOE) in Taiwan, and partially funded by the Ministry of Science and Technology (MOST) of Taiwan under Grant numbers MOST 107-3017-F-009-001 and MOST 107-2221-E-009 -051 -MY2.
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10. T. Sondergaard, S. I. Bozhevolnyi, J. Beermann, S. M. Novikov, E. Devaux, and T. W. Ebbesen, "Extraordinary optical transmission with tapered slits: effect of higher diffraction and slit resonance orders," J. Opt. Soc. Am. B 29(1), 130–137 (2012). [CrossRef]
11. J. Beermann, T. Sondergaard, S. M. Novikov, S. I. Bozhevolnyi, E. Devaux, and T. W. Ebbesen, "Field enhancement and extraordinary optical transmission by tapered periodic slits in gold films," New J. Phys. 13(6), 063029 (2011). [CrossRef]
12. H. Shen and B. Maes, "Enhanced optical transmission through tapered metallic gratings," Appl. Phys. Lett. 100(24), 241104 (2012). [CrossRef]
13. F. Medina, R. Rodriguez-Berral, and F. Mesa, "Circuit model for metallic gratings with tapered and stepped slits," Proc. 42nd European Microwave Conference, Oct. 2012.
14. Y. Liang, W. Peng, R. Hu, and H. Zou, "Extraordinary optical transmission based on subwavelength metallic grating with ellipse walls," Opt. Express 21(5), 6139–6152 (2013). [CrossRef]
15. M. N. M. Kehn, "Modal analysis of metallic screen with finite conductivity perforated by an array of subwavelength rectangular flared holes," Opt. Express 26(25), 32981–33004 (2018). [CrossRef]
16. J.-C. Yang, H. Gao, J. Y. Suh, W. Zhou, M. H. Lee, and T. W. Odom, "Enhanced optical transmission mediated by localized Plasmons in anisotropic, 3D nanohole arrays," Nano Lett. 10(8), 3173–3178 (2010). [CrossRef]
17. R. J. Moerland, J. E. Koskela, A. Kravchenko, M. Simberg, S. van der Vegte, M. Kaivola, A. Priimagio, and R. H. A. Ras, "Large-area arrays of three-dimensional plasmonic subwavelength-sized structures from azopolymer surface-relief gratings," Mater. Horiz. 1(1), 74–80 (2014). [CrossRef]
18. H. L. Chen, S. Y. Chuang, W. H. Lee, S. S. Kuo, W. F. Su, S. L. Ku, and Y. F. Chou, "Extraordinary transmittance in three-dimensional crater, pyramid, and hole-array structures prepared through reversal imprinting of metal films," Opt. Express 17(3), 1636–1645 (2009). [CrossRef]
19. Q. Wang, Y. Zhai, S. Wu, Z. Qi, L. Wang, and X. Li, "A modified transmission line model for extraordinary optical transmission through sub-wavelength slits," Plasmonics 10(6), 1545–1549 (2015). [CrossRef]
20. C. Molero, R. Rodriquez-Berral, and F. Mesa, "Wideband analytical equivalent circuit for coupled asymmetrical nonaligned slit arrays," Phys. Rev. E 95(2), 023303 (2017). [CrossRef]
21. M. Beruete, M. Navarro-Cia, and M. S. Ayza, "Understanding anomalous extraordinary transmission from equivalent circuit and grounded slab concepts," IEEE Trans. Microwave Theory Tech. 59(9), 2180–2188 (2011). [CrossRef]
22. P. Rodriquez-Ulibarri, M. Navarro-Cia, R. Rodriquez-Berral, F. Medsa, F. Medina, and M. Beruete, "Annular apertures in metallic screens as extraordinary transmission and frequency selective surface structures," IEEE Trans. Microwave Theory Tech. 65(12), 4933–4946 (2017). [CrossRef]
23. M. Navarro-Cia, V. Pacheco-Pena, S. A. Kuznetsov, and M. Beruete, "Extraordinary THz transmission with a small beam spot: the leaky wave mechanism," Adv. Opt. Mater. 6(8), 1701312 (2018). [CrossRef]
24. M. Camacho, R. R. Boix, and F. Medina, "Computationally efficient analysis of extraordinary optical transmission through infinite and truncated subwavelength hole arrays," Phys. Rev. E 93(6), 063312 (2016). [CrossRef]
25. L. Novotny and C. Hafner, "Light propagation in a cylindrical waveguide with a complex, metallic, dielectric function," Phys. Rev. E 50(5), 4094–4106 (1994). [CrossRef]
26. F. Mesa, R. Rodriquez-Berral, and F. Medina, "Unlocking complexity using the ECA," IEEE Microw. Mag. 19(4), 44–65 (2018). [CrossRef]
27. L. Wang, B. Ai, H. Mohwald, Y. Yu, and G. Zhang, "Invertible nanocup array supporting hybrid plasmonic resonances," Adv. Opt. Mater. 4(6), 906–916 (2016). [CrossRef]
28. B. Ai, Y. Yu, H. Mohwald, L. Wang, and G. Zhang, "Resonant optical transmission through topologically continuous films," ACS Nano 8(2), 1566–1575 (2014). [CrossRef]
29. A. Peer and R. Biswas, "Extraordinary optical transmission in nanopatterned ultrathin metal films without holes," Nanoscale 8(8), 4657–4666 (2016). [CrossRef]
30. Z. Chen, P. Li, S. Zhang, Y. Chen, P. Liu, and H. Duan, "Enhanced extraordinary optical transmission and refractive-index sensing sensitivity in tapered plasmonic nanohole arrays," Nanotechnology 30(33), 335201 (2019). [CrossRef]
31. A. Peer, Z. Hu, A. Singh, J. A. Hollingsworth, R. Biswas, and H. Htoon, "Photoluminescence enhancement of CuInS2 quantum dots in solution coupled to plasmonic gold nanocup array," Small 13(33), 1700660 (2017). [CrossRef]
32. H. A. Bethe, "Theory of diffraction by small holes," Phys. Rev. 66(7-8), 163–182 (1944). [CrossRef]
33. M. N. M. Kehn, "Extraordinary transmission through subwavelength hole arrays for general oblique incidence - mechanism as related to surface wave dispersion and Floquet lattice diagrams," Prog. Electromag. Res. (PIER) 82, 49–71 (2018). [CrossRef]
Article Order
T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, "Extraordinary optical transmission through sub-wavelength hole arrays," Nature (London) 391(6668), 667–669 (1998).
[Crossref]
J. R. Sambles, "More than transparent," Nature (London) 391(6668), 641–642 (1998).
T. Thio, H. J. Lezec, and T. W. Ebbesen, "Strongly enhanced optical transmission through subwavelength holes in metal films," Phys. B (Amsterdam, Neth.) 279(1-3), 90–93 (2000).
R. Gordon, A. G. Brolo, D. Sinton, and K. L. Kavanagh, "Resonant optical transmission through hole-arrays in metal films: physics and applications," Laser Photonics Rev. 4(2), 311–335 (2010).
C. Genet and T. W. Ebbesen, "Light in tiny holes," Nature 445(7123), 39–46 (2007).
J. V. Coe, J. M. Heer, S. Teeters-Kennedy, H. Tian, and K. R. Rodriguez, "Extraordinary transmission of metal films with arrays of subwavelength holes," Annu. Rev. Phys. Chem. 59(1), 179–202 (2008).
M. E. Stewart, C. R. Anderton, L. B. Thompson, J. Maria, S. K. Gray, J. A. Rogers, and R. G. Nuzzo, "Nanostructured plasmonic sensors," Chem. Rev. 108(2), 494–521 (2008).
R. Gordon, D. Sinton, K. L. Kavanagh, and A. G. Brolo, "A new generation of sensors based on extraordinary optical transmission," Acc. Chem. Res. 41(8), 1049–1057 (2008).
T. Sondergaard, S. I. Bozhevolnyi, S. M. Novikov, J. Beermann, E. Devaux, and T. W. Ebbesen, "Extraordinary optical transmission enhanced by nanofocusing," Nano Lett. 10(8), 3123–3128 (2010).
T. Sondergaard, S. I. Bozhevolnyi, J. Beermann, S. M. Novikov, E. Devaux, and T. W. Ebbesen, "Extraordinary optical transmission with tapered slits: effect of higher diffraction and slit resonance orders," J. Opt. Soc. Am. B 29(1), 130–137 (2012).
J. Beermann, T. Sondergaard, S. M. Novikov, S. I. Bozhevolnyi, E. Devaux, and T. W. Ebbesen, "Field enhancement and extraordinary optical transmission by tapered periodic slits in gold films," New J. Phys. 13(6), 063029 (2011).
H. Shen and B. Maes, "Enhanced optical transmission through tapered metallic gratings," Appl. Phys. Lett. 100(24), 241104 (2012).
F. Medina, R. Rodriguez-Berral, and F. Mesa, "Circuit model for metallic gratings with tapered and stepped slits," Proc. 42nd European Microwave Conference, Oct. 2012.
Y. Liang, W. Peng, R. Hu, and H. Zou, "Extraordinary optical transmission based on subwavelength metallic grating with ellipse walls," Opt. Express 21(5), 6139–6152 (2013).
M. N. M. Kehn, "Modal analysis of metallic screen with finite conductivity perforated by an array of subwavelength rectangular flared holes," Opt. Express 26(25), 32981–33004 (2018).
J.-C. Yang, H. Gao, J. Y. Suh, W. Zhou, M. H. Lee, and T. W. Odom, "Enhanced optical transmission mediated by localized Plasmons in anisotropic, 3D nanohole arrays," Nano Lett. 10(8), 3173–3178 (2010).
R. J. Moerland, J. E. Koskela, A. Kravchenko, M. Simberg, S. van der Vegte, M. Kaivola, A. Priimagio, and R. H. A. Ras, "Large-area arrays of three-dimensional plasmonic subwavelength-sized structures from azopolymer surface-relief gratings," Mater. Horiz. 1(1), 74–80 (2014).
H. L. Chen, S. Y. Chuang, W. H. Lee, S. S. Kuo, W. F. Su, S. L. Ku, and Y. F. Chou, "Extraordinary transmittance in three-dimensional crater, pyramid, and hole-array structures prepared through reversal imprinting of metal films," Opt. Express 17(3), 1636–1645 (2009).
Q. Wang, Y. Zhai, S. Wu, Z. Qi, L. Wang, and X. Li, "A modified transmission line model for extraordinary optical transmission through sub-wavelength slits," Plasmonics 10(6), 1545–1549 (2015).
C. Molero, R. Rodriquez-Berral, and F. Mesa, "Wideband analytical equivalent circuit for coupled asymmetrical nonaligned slit arrays," Phys. Rev. E 95(2), 023303 (2017).
M. Beruete, M. Navarro-Cia, and M. S. Ayza, "Understanding anomalous extraordinary transmission from equivalent circuit and grounded slab concepts," IEEE Trans. Microwave Theory Tech. 59(9), 2180–2188 (2011).
P. Rodriquez-Ulibarri, M. Navarro-Cia, R. Rodriquez-Berral, F. Medsa, F. Medina, and M. Beruete, "Annular apertures in metallic screens as extraordinary transmission and frequency selective surface structures," IEEE Trans. Microwave Theory Tech. 65(12), 4933–4946 (2017).
M. Navarro-Cia, V. Pacheco-Pena, S. A. Kuznetsov, and M. Beruete, "Extraordinary THz transmission with a small beam spot: the leaky wave mechanism," Adv. Opt. Mater. 6(8), 1701312 (2018).
M. Camacho, R. R. Boix, and F. Medina, "Computationally efficient analysis of extraordinary optical transmission through infinite and truncated subwavelength hole arrays," Phys. Rev. E 93(6), 063312 (2016).
L. Novotny and C. Hafner, "Light propagation in a cylindrical waveguide with a complex, metallic, dielectric function," Phys. Rev. E 50(5), 4094–4106 (1994).
F. Mesa, R. Rodriquez-Berral, and F. Medina, "Unlocking complexity using the ECA," IEEE Microw. Mag. 19(4), 44–65 (2018).
L. Wang, B. Ai, H. Mohwald, Y. Yu, and G. Zhang, "Invertible nanocup array supporting hybrid plasmonic resonances," Adv. Opt. Mater. 4(6), 906–916 (2016).
B. Ai, Y. Yu, H. Mohwald, L. Wang, and G. Zhang, "Resonant optical transmission through topologically continuous films," ACS Nano 8(2), 1566–1575 (2014).
A. Peer and R. Biswas, "Extraordinary optical transmission in nanopatterned ultrathin metal films without holes," Nanoscale 8(8), 4657–4666 (2016).
Z. Chen, P. Li, S. Zhang, Y. Chen, P. Liu, and H. Duan, "Enhanced extraordinary optical transmission and refractive-index sensing sensitivity in tapered plasmonic nanohole arrays," Nanotechnology 30(33), 335201 (2019).
A. Peer, Z. Hu, A. Singh, J. A. Hollingsworth, R. Biswas, and H. Htoon, "Photoluminescence enhancement of CuInS2 quantum dots in solution coupled to plasmonic gold nanocup array," Small 13(33), 1700660 (2017).
H. A. Bethe, "Theory of diffraction by small holes," Phys. Rev. 66(7-8), 163–182 (1944).
M. N. M. Kehn, "Extraordinary transmission through subwavelength hole arrays for general oblique incidence - mechanism as related to surface wave dispersion and Floquet lattice diagrams," Prog. Electromag. Res. (PIER) 82, 49–71 (2018).
Ai, B.
Anderton, C. R.
Ayza, M. S.
Beermann, J.
Beruete, M.
Bethe, H. A.
Biswas, R.
Boix, R. R.
Bozhevolnyi, S. I.
Brolo, A. G.
Camacho, M.
Chen, H. L.
Chen, Y.
Chou, Y. F.
Chuang, S. Y.
Coe, J. V.
Devaux, E.
Duan, H.
Ebbesen, T. W.
Gao, H.
Genet, C.
Ghaemi, H. F.
Gordon, R.
Gray, S. K.
Hafner, C.
Heer, J. M.
Hollingsworth, J. A.
Htoon, H.
Hu, Z.
Kaivola, M.
Kavanagh, K. L.
Kehn, M. N. M.
Koskela, J. E.
Kravchenko, A.
Ku, S. L.
Kuo, S. S.
Kuznetsov, S. A.
Lee, M. H.
Lee, W. H.
Lezec, H. J.
Li, P.
Li, X.
Liang, Y.
Liu, P.
Maes, B.
Maria, J.
Medina, F.
Medsa, F.
Mesa, F.
Moerland, R. J.
Mohwald, H.
Molero, C.
Navarro-Cia, M.
Novikov, S. M.
Novotny, L.
Nuzzo, R. G.
Odom, T. W.
Pacheco-Pena, V.
Peer, A.
Peng, W.
Priimagio, A.
Qi, Z.
Ras, R. H. A.
Rodriguez, K. R.
Rodriguez-Berral, R.
Rodriquez-Berral, R.
Rodriquez-Ulibarri, P.
Rogers, J. A.
Sambles, J. R.
Shen, H.
Simberg, M.
Singh, A.
Sinton, D.
Sondergaard, T.
Stewart, M. E.
Su, W. F.
Suh, J. Y.
Teeters-Kennedy, S.
Thio, T.
Thompson, L. B.
Tian, H.
van der Vegte, S.
Wang, L.
Wang, Q.
Wolff, P. A.
Wu, S.
Yang, J.-C.
Yu, Y.
Zhai, Y.
Zhang, G.
Zhang, S.
Zhou, W.
Zou, H.
Acc. Chem. Res. (1)
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Appl. Phys. Lett. (1)
Chem. Rev. (1)
IEEE Microw. Mag. (1)
IEEE Trans. Microwave Theory Tech. (2)
J. Opt. Soc. Am. B (1)
Laser Photonics Rev. (1)
Mater. Horiz. (1)
Nano Lett. (2)
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(1) ℘ = a 0 h 0 / d x d y ,
(2) ε T = T / ℘ ,
(3) P t r a n s T ζ = { P h o l e s θ = 0 cos ⁡ θ i n c ; d h o l e > Υ − 1 / 4 λ Υ ( d h o l e / d h o l e λ λ ) 4 P h o l e s θ = 0 cos ⁡ θ i n c ; d h o l e ≤ Υ − 1 / 4 λ
(4) Λ = { cos ⁡ θ i n c ; ζ ≡ E ( 4 + sin 2 θ i n c ) / ( 4 + sin 2 θ i n c ) ( 4 cos ⁡ θ i n c ) ; ζ ≡ M ( 4 cos ⁡ θ i n c ) ; ζ ≡ M
(5) P h o l e s θ = 0 = S i n c A h o l e s a l l ; with A h o l e s a l l = ℘ A t o t a l
(6) P h o l e s = P h o l e s θ = 0 cos ⁡ θ i n c = ℘ P w h o l e s c r e e n ⏞ S i n c A t o t a l cos ⁡ θ i n c
(7) T B e t h e T ζ = P t r a n s T ζ / P t r a n s T ζ P w h o l e s c r e e n P w h o l e s c r e e n , ζ ≡ E or M
(8) T B e t h e T ζ / T B e t h e T ζ ℘ ℘ ⏞ ε T T ζ B e t h e = { cos ⁡ θ i n c ; d h o l e > Υ − 1 / 4 λ Υ ( d h o l e / d h o l e λ λ ) 4 cos ⁡ θ i n c ; d h o l e ≤ Υ − 1 / 4 λ
(9) Ψ = ε T T ζ a c t u a l / ε T T ζ a c t u a l ε T T ζ B e t h e ε T T ζ B e t h e = T a c t u a l T ζ / T a c t u a l T ζ T B e t h e T ζ T B e t h e T ζ
(10) ε T T ζ a c t u a l = T T ζ a c t u a l / T T ζ a c t u a l ℘ ℘
(11) k 0 # 1 = 2 π / d x & y = 1256.6 rad/m ,
(12) k 0 # 2 = ( 2 ) ( 2 π / d x & y ) = 1777 rad/m ,
(13) P c o n d = R s 2 ∫ ∫ S m e t a l | H → t | 2 d s ; where R s = π f μ c o n d / π f μ c o n d σ σ ,
(14) ( k x 00 , k y 00 ) = ( k 0 48 G sin ⁡ θ i n c cos ⁡ ϕ i n c = k 0 48 G sin ⁡ 15 ∘ = 260.2 , k 0 48 G sin ⁡ θ i n c sin ⁡ ϕ i n c = 0 ) ,
(15) ( k x 00 , k y 00 ) = ( k 0 40 G sin ⁡ θ i n c cos ⁡ ϕ i n c = k 0 40 G sin ⁡ 30 ∘ = 419 , k 0 40 G sin ⁡ θ i n c sin ⁡ ϕ i n c = 0 ) ,
(17) k 0 = | 518 − 1256.6 | = 738 rad/m ⇒ f = k 0 c / ( 2 π ) = 35.2 ≈ 35 GHz indeed .