IEEE MICROWAVE AND WIRELESS COMPONENTS LETTERS, VOL. 14, NO. 7, JULY 2004
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Enhanced Transmission Through Two-Period Arrays of Subwavelength Holes Vitaliy Lomakin, Nan-Wei Chen, Shuqing Li, Senior Member, IEEE, and Eric Michielssen, Fellow, IEEE
Abstract—A two-period array of subwavelength holes in a perfect electrically conducting plate permits enhanced transmission of incident plane waves with wavenumbers near those of leaky waves supported by the array. A simple periodic impedance sheet model explains this phenomenon and provides an approximate expression for the transmission peak location. Transmission through thin perforated plates is studied using a full-wave solver to verify modeltheoretic observations. Transmission coefficient peak splitting and narrowing phenomena, associated with thick perforated plates, are explored. Index Terms—Enhanced transmission, leaky-wave (LW), perfect electrically conducting (PEC), subwavelength holes, surface plasmon polaritons (SPPs), surface waves (SWs).
I. INTRODUCTION
E
VER since Bethe’s influential paper [1], small holes and weak transmission are often mentioned in the same breath. Recently, however, several interesting mechanisms leading to enhanced transmission of electromagnetic energy through small holes have been identified. In the optical realm, observations of unusually high transmission through small holes caused by surface plasmon polaritons (SPPs) on metal interfaces [2]–[4] were elucidated using leaky-wave (LW) SPP theory [5], [6]. In the RF regime, measurements of enhanced transmission through a single small hole were attributed to hole-antenna interactions [7]. Likewise, enhanced transmission through arrayed holes was ascribed to the excitation of surface waves (SWs) on perforated plates [8] and/or waveguide modes on supporting dielectric slabs [9], [10]. This paper extends the work in [8]. First, a configuration that allows for enhanced transmission through arrayed small holes in an infinitesimally thin perfect electrically conducting (PEC) plate is described. Next, conditions leading to enhanced transmission are formulated and the nature of the phenomenon is elucidated using a LW model. Finally, effects of plate thickness are investigated. Although our focus is on a specific geometry, viz. the two-period perforated plate (TPPP) (Fig. 1), the phenomena discussed occur on a large family of related structures. Throughout this letter, , , and represent unit vectors along the -, -, and -directions. In addition, , , , , and denote the frequency, free-space speed of light, wavelength, wavenumber, and wave impedance, respectively. The time dependence is suppressed. Manuscript received December 15, 2003; revised March 15, 2004. This research was supported by the Defense Advanced Research Projects Agency VET Program under Contract F49620-01-1-0228. The review of this letter was arranged by Associate Editor A. Weisshaar. The authors are with the Center for Computational Electromagnetics, University of Illinois at Urbana-Champaign, Urbana, IL 61801 USA (e-mail:
[email protected]). Digital Object Identifier 10.1109/LMWC.2004.829280
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Fig. 1. TPPP configuration and T ( ) for thin TPPP with h = 0, s = 2:2 mm, d = d = 4:5 mm, M = 5 and D = 45 mm. Total transmission = 0:331 50 and 0.305 25 for f = 5 GHz (=s is obtained at 27) and f = 5:1 GHz, respectively.
II. TWO-PERIOD PERFORATED PLATES Consider a PEC plate of thickness that is loaded with a two-period array of subwavelength circular holes of radius (Fig. 1). The plate bottom face resides in the plane. Along the -direction holes are spaced uniformly with period . Along the -direction they are arranged in groups of with small and large periods (spacing between hole centers) and (extent of a group of holes plus the adjacent “conducting strip”), respectively. In other words, along the -direction the plate comprises of adjacent perforated and conducting strips of approximate width and , respectively. The plate is excited by a plane wave with magnetic field with and being the angle of incidence. Zeroth-order reflected and transmitted magnetic fields are denoted and , respectively. Associated electric fields are denoted , , and . The transmission coefficient is . III. PERIODIC IMPEDANCE SHEET (PIS) MODEL FOR THIN TPPPS A TPPP with and can be modeled as an impedance sheet. That is, incident, reflected, and transmitted electric and magnetic fields approximately satisfy the boundary condition
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(1)
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where the Ch. 12.5]
IEEE MICROWAVE AND WIRELESS COMPONENTS LETTERS, VOL. 14, NO. 7, JULY 2004
-dependent impedance is given by [11; Ch. 1.8 and (2)
when . The latter condition will be observed below. SWs, Due to its inductive nature, the plate can support which manifest themselves as poles in the transmission coefficient resulting from (1) and (2). but can It follows that the TPPP with be modeled as a periodic impedance sheet (PIS) comprised of -oriented alternating impedance and conducting strips of and , respectively. Previously, a approximate width Floquet-based analysis of impenetrable periodic impedance surfaces was carried out by Hessel and Oliner [12] and fast variations of the reflection coefficient were attributed to resonant-type Wood anomalies. Recently, their scheme was extended to analyze penetrable PISs with square holes [8] relying on ( , )-independent estimates for the sheet impedance. The latter shortcoming and other deficiencies in [8] are remedied here by considering PISs with circular holes and the impedance model (2). Higher order Floquet modes with electric/magnetic reflected , and , and normalized and transmitted fields wavenumbers , at satisfy (3) where is the Kronecker symbol and is given by (2) with replaced by . Then, by expanding the scattered field in Floquet modes and by using (3) leads to a matrix equation for the transmission coefficients, see [8, (1)]. Assuming that only the zeroth-order Floquet mode propagates and considering cou1st-order modes yields (see pling only between the 0th- and derivation of [8, (2)])
(4) Here, is the characteristic impedance for mode , , accounts for coupling between the 0th- and 1st-order modes, and is the period-averaged sheet impedance. Total transmission occurs when the numerator of the second term on the right-hand side in (4) vanishes, i.e., when with (5) where
is the positive root of the eigenvalue equation
and hence is the wavenumber of the SWs existing on the sheet of averaged nonmodulated impedance . It is noted that has prominent resonant behavior originating from a complex pole, viz. the zero of the denominator of the second term on the right-hand side of (4), or (6)
with the appropriate sign dictated by (5). These poles reside near since, referring to (2), the difference is small . It follows that approximately predicts when the transmission peak location. In addition, the bandwidth of . From (6), it the resonance can be defined as BW Im follows that the BW is very small when or whereas the maximal value is obtained around the value (chosen in all numerical simulations that follow). and The complex poles reside in the visible spectrum therefore give rise to LWs. Even though the LWs cannot be directly excited from, nor couple into, plane waves, the LW poles are seen to play a crucial role in shaping the transmission coefficient. The transmission enhancement through PIS/TPPPs has a clear physical interpretation. With a mild abuse of language, one can say that the PIS/TPPPs perforated strips support SWs that, through the action of the PIS/TPPPs conducting strips, are transformed into LWs. The latter, while propagating along the structure, convert their power into the transmitted plane wave (Localized PIS/TPPP excitations that directly excite LWs will be described elsewhere). IV. NUMERICAL STUDY OF THIN TPPP , mm, Plane wave scattering from a TPPP with mm, , and mm was analyzed via a full-wave Floquet modal approach [13]. In Fig. 1, plots of the magnitude of the transmission coefficient versus reveal transmission maxima for and 0.305 25 at GHz and 5.1 GHz , respectively. Peak locations predicted via (2) are and 0.306 03 and agree, thereby validating the PIS model. Note that, , grows as the fredue to the dispersive character of quency decreases. Fig. 2 describes the dispersion of the complex poles depicting their real and imaginary parts versus (normalized) frequency for the above studied TPPP. The poles were found numerically by tracking the complex zeros of the determinant of the system of the equations governing the full-wave solution , there [13]. For low frequencies, when exist only real poles in the invisible spectrum corresponding to purely bound SWs. As the frequency increases, when , a stopband appears in which wavenumbers are purely imaginary. For yet larger frequencies the pole remains complex while its real part splits into two branches. The right branch [Fig. 2(b)] resides in the invisible spectrum and represents slow waves. The left branch enters the visible spectrum and there represents LWs. Visible and invisible spectra are separated by the light line, which resides slightly to the left of the slow wave pole trajectory. For the frequencies GHz and 5.1 GHz the poles are located at and , respectively, and their real part agrees well with the corresponding peak locations in Fig. 1. Note that (6), while providing a good approximation for the real part of the poles, gives a noticeable discrepancy for the imaginary part. This is likely because (2) ignores the effect of the transition between the hole and PEC regions in the TPPPs. A more sophisticated model for TPPPs is under development and is not presented here. Note that enhanced transmission can be achieved even for structures with a single period along both and direction
LOMAKIN et al.: ENHANCED TRANSMISSION THROUGH TWO-PERIOD ARRAYS
Fig. 2. Dispersion relation for thin TPPP of Fig. 1. For k D = > 1:001, LW poles appear. For f = 5 GHz and 5.1 GHz, the LW poles are located at = 0:331 54+j 3:713 10 and 0:305 32+j 3:827 10 , respectively. The pole real part agrees well with the transmission maxima in Fig. 1.
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single transmission peak, the magnitude of which decreases exponentially with . The peak narrowing for but thin plates is due to the decrease of the hole polarizability/equivalent impedance. The peak splitting was also observed for SPP enhanced transmission [3] and dielectric slab guided waves [10]. It occurs due to resonantly excited and hole-coupled LWs on two plate faces. When the plate is thin, a combined resonance leads to a single peak. The coupling changes, however, for thicker plates leading to peak splitting. The faces of very thick plates are almost uncoupled, hence only a single peak is observed. It is found that this transition occurs when approximately equals the hole diameter. The peak splitting in the transmission through perforated plates residing between dielectric slabs is also detailed in [10]. VI. CONCLUSION A TPPP with subwavelength holes is transparent to an incident plane wave when its wavenumber is near that of a TPPP LW. PIS models, LW pole dispersion curves, and studies on plate thickness provide quantitative/qualitative data and insights that permit tuning of the TPPP characteristics. Applications of TPPP-like structures in near-field imaging and target identification are being studied. TPPPs use is anticipated in the construction of novel antennas, tunable filters, and mode converters. REFERENCES
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Fig. 3. T ( ) for several thick TPPP of Fig. 1. An additional peak appears for h 1:5 mm. It gradually approach the first peak as h grows (h = 2:5 mm) and merge it for h = 4:5 mm. For all thicker plates (h = 6 mm) a single transmission peak decreases exponentially with h.
(i.e., ). However, the bandwidth of the resulting enhanced transmission regime is very small; moreover, such a structure permits less independent control over the peak location and bandwidth. It should be also noted that two-period perforations can be added along -direction, leading to additional transmission bands. V. NUMERICAL STUDY OF THICK TPPP Thick TPPPs also exhibit enhanced transmission that once again can be traced to LW poles. However, finite plate thickness also causes additional phenomena that are described next. Fig. 3 transmission coefficient versus of the predepicts the viously studied TPPP but now for (different values of) at GHz, calculated using an extension of the approach in [13]. When is small compared to , e.g., mm, the transmission peak has essentially the same shape as when , though it narrows. For thicker plates, e.g., mm, a weak and narrow additional peak appears. As the thickness further increases, e.g., mm, these two peaks approach each other. For sufficiently thick plates, e.g., mm, the peaks merge. Even thicker plates, e.g. mm, exhibit a
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