Using the Bloch modes of a periodic, semi-infinite array of slits in a metallic host, we study the transmission of obliquely incident plane-waves through sub-wavelength slits. Matching the tangential E- and H-fields at the entrance facet of the periodic structure yields the complex amplitudes of the various Bloch modes, which exist and propagate within the slit array independently of each other. The computational scheme is robust, convergence is rapid, and a good match at the boundaries is obtained in every case. The regions examined in some detail include the vicinity of the Wood anomaly (where new diffraction orders appear/disappear on the horizon), the neighborhood of a point where surface plasmon polaritons (SPPs) are excited, and an ordinary situation in which the incidence angle is far from the angles that invoke Wood’s anomaly or cause the excitation of SPPs. Field distributions and energy flow diagrams in and around the slits reveal the existence of transmission minima (and reflection maxima) at incidence angles associated with the excitation of SPPs.
©2006 Optical Society of America
This paper is a follow-up to our previous publication , where we employed the Bloch modes of a slit array in a semi-infinite metallic host to investigate the transmission of a normally incident plane-wave through sub-wavelength slits. Toward the end of that paper, we briefly discussed the case of oblique incidence at small angles (i.e., close to normal incidence), thus demonstrating the utility of the Bloch mode analysis for interpreting the results of experiments involving oblique-incidence on metallic gratings, such as those conducted by R. W. Wood early in the 20th century [2, 3]. In the present paper we expand upon the earlier analysis and explore the transmission properties of a metallic slit array having a fixed period and a fixed slit width under plane-wave illumination at angles of incidence ranging from 0° to 90°.
The setup for our calculations is shown in Fig. 1. Because the system is invariant along the x-axis, its optical behavior can be studied separately for the two cases of transverse electric (TE) polarization (involving the field components Ex , Hy , Hz ) and transverse magnetic (TM) polarization (involving Hx , Ey , Ez ). As in the previous paper, the focus of attention will be the case of TM polarization, as illumination with TE-polarized light does not excite any guided modes within subwavelength slits.
Section 2 describes the propagation constants and field profiles of the various electromagnetic modes of a semi-infinite slit array in a metallic host. For any given angle of incidence θ, the modes satisfy the corresponding Bloch condition , and are therefore referred to as Bloch modes. Matching the tangential components of the E- and H-fields at the entrance facet of the array is discussed in Section 3, where we demonstrate the convergence of the Bloch mode series for three representative angles of incidence. The three skew angles selected for discussion represent the cases of ordinary behavior (i.e., incidence angles that are sufficiently far from anomalous angles), Wood’s anomaly, observed when a new diffraction order appears/disappears on the horizon , and the excitation of surface plasmon polaritons (SPPs) at the entrance facet of the array [5–7]. In Section 4 we analyze the behavior of a slit array throughout the entire range of skew angles (θ= 0: 90°), and examine the aforementioned three types of behavior for specific incidence angles. The anomalous angles revealed by the Bloch mode analysis will be shown to satisfy simple formulas involving the incident light’s wavelength, the period of the array, and the SPP’s characteristic refractive index. Also presented and discussed in Section 4 are the field profiles in and around the slits.
As in the previous paper , we use Raether’s definition of the SPP , a localized electromagnetic wave confined to the vicinity of a dielectric-metal interface, consisting of a single evanescent plane-wave on the dielectric side of the interface, and a single inhomogeneous plane-wave  on the metallic side. At λ o= 1.0μm, at the interface between silver (dielectric constant εm = - 48.8 + 2.99i; see Ref. ) and free-space (εd = 1.0), the real part of the effective refractive index of the SPP, , is ~ 1.01.
2. Propagation constants and Bloch mode profiles
With reference to Fig. 1, we express the E- and H-fields inside the slits - which are either empty or filled with a transparent material of dielectric constant εs - and also the fields in the metallic regions between adjacent slits as superpositions of two (generally inhomogeneous) plane-waves bouncing back and forth between the vertical walls. The H-field transmitted below the surface at z = 0 is thus written
Here n is the mode index (1, 2, 3, …), k o=2π/λo is the vacuum wave-number, is the n th mode’s propagation constant along the z-axis, the subscript s denotes the slit region, and the subscript m denotes the metallic host medium (i.e., cladding for the slit waveguides). Inside the slits ( )2 + ( )2 = εs , where εs is the relative permittivity of the filling material, while in the metal ( )2 + ( )2 = εm , where εm is the relative permittivity of the host material. Maxwell’s equations relate the E-field to the H-field, and, subsequently, the continuity of the tangential components of E and H at the (vertical) slit walls, in conjunction with the Bloch condition for a given incidence angle θ, enable one to determine the unknown parameters of each and every Bloch mode .
Compared to the case of normal incidence discussed in Ref. , for a given period p, oblique incidence involves twice as many Bloch modes within a given region of the complex plane in which the propagation constant resides. The reason is that normal incidence invokes only the even modes of the array, whereas oblique incidence breaks the left-right symmetry, thus admitting additional modes. (The designations even and odd are relative to an axis of symmetry of the array, say, the x-axis passing through the center of a slit). As a matter of fact, anti-symmetric modes are also found in the case of normal incidence, but their coefficients turn out to be zero when the boundary conditions at the entrance facet are matched.
We searched for the propagation constants in a rectangular region of the complex plane. The region was sampled with a small grid size, fine enough to guarantee that no roots of the characteristic equation were missed . We scanned the grid for local minima, and searched for the roots in the vicinity of each such minimum. For the slit array of Fig. 1 at θ= 30°, Fig. 2 shows the complex-plane location of some of the roots, σz , as well as the corresponding σys and σym . The modes are numbered according to the strength of the imaginary part of σz , which means that the lower-order modes propagate deeper into the slits. Except for the first mode - the sole guided mode in the present example - all other modes have a large imaginary component in their σz , thus residing mainly at the entrance facet of the slit array. Most of these higher-order modes, however, have a negligible loss along the y-axis (see the values of σym in Fig. 2); consequently, they propagate parallel to the upper surface of the slit array, helping to establish the continuity of the tangential E- and H-fields at this facet.
Profiles of the first twelve modes of the slit array of Fig. 1 at θ= 30° are shown in Fig. 3. Here the Hx profiles appear on the left-hand side, while those of Ey appear on the right. In each case, the field’s magnitude and phase profiles are shown in the top and bottom frames, respectively. The mode number in each frame ranges from 1 to 12, top to bottom, as shown. Some modes are symmetric-like (e.g., modes 6, 8), while others are anti-symmetric-like (e.g., modes 5, 7). The phase profile of the incident plane-wave is preserved through the Bloch condition embedded in the structure of each and every Bloch mode, thus guaranteeing, for instance, that the 0th-order beam that exits the slit array will propagate in the same direction as the incident beam.
From the phase plots of Fig. 3, the first mode is seen to be symmetric; this may also be inferred from the computed amplitudes of the corresponding plane-waves that reside within the slits, namely, =; see Eq. (1). The second mode in Fig. 3 is anti-symmetric; again this may be inferred from the computed amplitudes of the mode’s constituent plane-waves, namely, = -. The first two modes in this example are essentially the same for all angles of incidence θ, the reason being that their large absorption coefficients along y in the metallic region - see Im in caption to Fig. 2 - prevent the modal fields of adjacent slits from interacting with each other. As for the higher-order modes, although the magnitudes tend to be similar for different values of θ, the modal phase profiles have a certain well-defined dependence on the incidence angle.
3. Matching the boundary conditions at the entrance facet of the slit array
For the system depicted in Fig. 1 with λ o = 1.0 μm, θ = 30°, p = 1.2 μm, w = 0.1 μm, silver host, we computed the coupling coefficients by matching the tangential fields, Ey and Hx , at the entrance facet using a total of N= 120 modes in each space (i.e., 120 modes in the free space of incidence and 120 modes in the slit array). Our method of minimizing the mismatch at the entrance facet is described in Ref . The matched field magnitudes on both sides of the z = 0 interface are plotted in Fig. 4. The match is excellent, and the difference between the incident optical power and the combined reflectance and transmittance of the array, R + T, was found to be less than 0.1%. The sharp peaks of the Ey -field at the slit edges, y = ±0.05 μm, represent a significant accumulation of electrical charge at these sharp corners.
Next, we examine the convergence of the Bloch mode series expansion. With reference to Fig. 5(a), the magnitudes of the first five modes of the slit array, C 1, C 2, … C 5, were computed using values of N ranging from 2 to 120, N being the total number of modes included in the calculations. For the first five modes, we have plotted (versus N) the difference between the intermediate values of each coefficient (computed with N< 120) and the final, steady-state value (obtained with N = 120). Shown in Fig. 5(a) are plots of the mode-magnitude error, |ΔCn |, versus the number N of modes used to match the boundary conditions. It is seen that, with increasing N, the mode coefficients rapidly converge to their steady-state values. In general, we found N~50 to be sufficient for obtaining the coupling coefficients with less than 0.1% error. (This is roughly twice the number of modes needed in the case of normal incidence , where only “even modes” are excited, while odd modes are ignored at the outset.)
Figure 5(b) shows the magnitudes of the first 20 modes of the slit array (computed with N= 120) for three cases of interest, labeled as Ordinary, Wood, and SPP. The “Ordinary” case (blue symbols) corresponds to θ= 30°, representing a typical situation that is free from anomalies (more about anomalies in Section 4). In the case of the Wood anomaly at θ= 9.6° (green), convergence behavior is not too different from that of the Ordinary case. In the case of SPP resonance at θ= 10.2° (red), the first mode, which is the guided mode of the slit, is fairly weak, modes 3, 4 and 5 have relatively large amplitudes, and the high-order modes (beyond n = 5) are weak again. As was the case at normal incidence , the SPP excitation tends to suppress the guided mode as well as all high-order modes of the slit array. A fraction of the incident light is then strongly absorbed within the skin-depth of the metallic host at the entrance facet, while the remaining light returns (in the form of specularly reflected or diffracted plane-waves) to the incidence space.
4. Wood’s anomaly and SPP excitation at oblique incidence
Using an expansion of the transmitted fields into the Bloch mode series, we computed, for angles of incidence θ ranging from 0° to 89°, the reflectance R, guided mode’s transmittance T1 (i.e., fraction of the incident Poynting vector component Sz entering the slits), and total transmittance T (including losses at the entrance facet); the results are shown in Fig. 6. For each value of θ, a total of N = 120 modes (in each space) were included in the calculations; this number did not have to be adjusted as θ rose from small to large, and the quality of the match at the entrance facet was high for all incidence angles.
The values of R, T, and T 1 in Fig. 6 were computed in the xy-plane at the entrance facet of the slit array by integrating the Poynting vector component Sz (y, z = 0) over one period p of the array for the following cases:
- incident plane-wave at z = 0- (used for normalization purposes);
- superposition of all reflected modes in the incidence space at z = 0- (used to compute R);
- superposition of all the Bloch modes of the slit array at z = 0+ (used to compute T);
- guided mode entering the slits, namely, the first Bloch mode of the array at z = 0+.
In Fig. 6, the difference between the green curve (total transmittance T) and the red curve (guided mode’s transmittance T 1) is the fraction of the optical power absorbed within the skin-depth at the entrance facet. The two sharp dips in T and T 1 (coinciding with the spikes in R) at θ= 10.2° and θ= 41° correspond to SPP anomalies. The small, highly localized peaks in T and T 1 adjacent to SPP anomalies (coinciding with the tiny dips in R) are manifestations of the Wood anomaly at θ= 9.6° and θ= 41.8°. Transmissivity reaches a maximum of T ~ 60% at θ= 82.5°, before diminishing at the grazing incidence, θ→90°, where R approaches unity.
The anomalous incidence angles may be readily estimated from simple formulas [1, 4]. At Wood’s anomaly, the Bragg condition should indicate the appearance, on the horizon, of a new diffraction order m. The SPP anomaly occurs when a diffracted order m acquires the SPP wavelength of λ o/Re[nspp ], where . Thus θ Wood, θspp may be found from:
For p = 1.2μm, λ o = 1.0μm, m = 1, Re [nspp ] = 1.01, we find θ Wood= 9.6°, θspp = 10.2°; these correspond to the first set of anomalies in Fig. 6. The second set, θ Wood = 41.8°, θspp = 41.0°, corresponds to m=-2. Carrying out the Bloch mode calculations in steps of Δθ= 0.1°, we found close agreement with these predicted anomalous angles.
The field profiles of Fig. 7 show in detail how the slit array modifies the field distributions in the incidence space near the slits. A non-zero Poynting vector component Sy (parallel to the surface) is, of course, expected at skew incidence. In the ordinary case (θ= 30°) shown in the top row of Fig. 7, the fields do not differ all that much from those at normal incidence , but the charge accumulation at the slit’s left edge (evident in the |Ey | plot) and the one-sided flow of energy into the slit are noteworthy. At the Wood anomaly (θ= 9.6°, middle row), the incident fields are seen to be coupled into the guided mode of the slit, there is charge accumulation on the left edge, and the energy flow behavior is fairly complex. In the case of the SPP anomaly at θ= 10.2° (bottom row), the incident optical power strikes the metallic surface on the left side of the slit, bounces back, and lands again on the metal on the right-hand side. The incident power flux thus misses the slits altogether, resulting in negligible transmission through the array. Note the left-right symmetry of the surface charge and surface current profiles on the top metallic surface in the case of SPP excitation [surface current density along y = Hx (y, z= 0-); surface charge density ≈Dz (y, z= 0-)].
Shown in Fig. 8 are the field profiles at θ= 82.5°, the point in Fig. 6 where R reaches a minimum (and T a maximum). As in the “ordinary” case of θ=30° discussed in conjunction with Fig. 7, a hot spot of accumulated charge appears on the slit’s left edge, but no strong tendency is observed on the part of the Poynting vector S to turn around and head for the slits. The peaking of T and T 1 around θ= 82.5° is thus a consequence of the reduced incident Sz at large skew angles rather than an enhancement of the transmitted Sz .
5. Concluding remarks
The Bloch mode expansion of the optical field in a metallic slit array is a viable calculation scheme, which is readily applicable to the case of incidence at arbitrary skew angles. Each Bloch mode, being a natural mode of the structure, propagates within the array independently of all the other modes. The strength of each mode is determined at the entrance facet by matching the incident E and H fields to the collective profile of the reflected and transmitted fields. One advantage of the Bloch mode method over the traditional Rigorous Coupled Wave Analysis (RCWA) [11, 12] is its rapid convergence, although restriction to one-dimensional structures is a serious drawback. In this paper we employed the Bloch mode scheme to show that the excitation of surface plasmon polaritons at the entrance facet of a semi-infinite slit array leads to a substantial reduction in the strength of the guided mode through the slits. This behavior is quite distinct from that of the Wood anomaly, where the transition of a diffracted order from just below to just above the horizon produces a tiny peak in the plot of T versus θ, but does not diminish the strength of the guided mode through the slits.
The authors are grateful to Moysey Brio, John Weiner, Krishna Gundy, Philippe Lalanne, and Hongbo Li for many helpful discussions. JVM acknowledges support from the Alexander von Humboldt Foundation. This work has been supported by the AFOSR contracts F49620-03-1-0194, FA9550-04-1-0213, FA9550-04-1-0355 awarded by the Joint Technology Office.
References and links
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