Grating-coupled excitation of multiple surface plasmon-polariton waves
aa r X i v : . [ phy s i c s . op ti c s ] M a y Grating-coupled excitation of multiple surface plasmon-polaritonwaves
Muhammad Faryad and Akhlesh Lakhtakia ∗† Nanoengineered Metamaterials Group (NanoMM),Department of Engineering Science and Mechanics,Pennsylvania State University, University Park, PA 16802-6812, USA
Abstract
The excitation of multiple surface-plasmon-polariton (SPP) waves of different linear polarizationstates and phase speeds by a surface-relief grating formed by a metal and a rugate filter, both offinite thickness, was studied theoretically, using rigorous coupled-wave-analysis. The incident planewave can be either p or s polarized. The excitation of SPP waves is indicated by the presence ofthose peaks in the plots of absorbance vs. the incidence angle that are independent of the thicknessof the rugate filter. The absorbance peaks representing the excitation of s -polarized SPP waves arenarrower than those representing p -polarized SPP waves. Two incident plane waves propagating indifferent directions may excite the same SPP wave. A line source could excite several SPP wavessimultaneously. PACS numbers: ∗ Corresponding author † Electronic address: [email protected] . INTRODUCTION Surface plasmon-polariton (SPP) waves are surface waves guided by a planar interfaceof a metal and a dielectric material. SPP waves find applications for sensing, imaging andcommunication [1, 2]. If the dielectric partnering material is isotropic and homogeneous,only one SPP wave—that too, of the p -polarization state—can be guided by the metal-dielectric interface at a given frequency [3, 4]. If a periodic nonhomogeneity normal tothe wave-guiding interface is introduced in the dielectric partnering material, multiple SPPwaves with different polarization states, phase speeds, and spatial profiles can be guidedby the metal-dielectric interface. This has recently been shown both theoretically [5–7] andexperimentally [8–10]. In all of these studies, the dielectric partnering material is also locallyorthorhombic.Very recently, we have solved a canonical boundary-value problem [11] to show thatmultiple SPP waves can be guided even if the dielectric partnering material is isotropic—provided that material is also periodically nonhomogeneous normal to the interface. Thisis a very attractive result, because both partnering materials are isotropic and because thedielectric partnering material can be fabricated as a rugate filter [12–16].The canonical boundary-value problem does not possess direct practical significance,because both partnering materials are assumed to be semi-infinite normal to the planarinterface. Therefore, we set out to investigate the excitation of multiple SPP waves bythe periodically corrugated interface of a metal and a rugate filter. This grating-coupledconfiguration [2, pp. 35–41] is popular, when the dielectric partnering material is homoge-neous, because it allows the excitation of an SPP wave by a nonspecular Floquet harmonic.The interplay of the periodic nonhomogeneity of the dielectric partnering material and aperiodically corrugated interface is phenomenologically rich [17, 18], and should lead to theexcitation of multiple SPP waves as different Floquet harmonics.The relevant boundary-value problem was formulated using the rigorous coupled-waveanalysis (RCWA) [19, 20]. In this numerical technique, the constitutive parameters are ex-panded in terms of Fourier series with known expansion coefficients, and the electromagneticfield phasors are expanded in terms of Floquet harmonics whose coefficients are determinedby substitution in the frequency-domain Maxwell curl postulates. The accuracy of solutionis conventionally held to depend only on the number of Floquet harmonics actually used2n the computations [21]. The RCWA has been used to solve for scattering by a variety ofsurface-relief gratings [20–23], generally with both partnering materials being homogeneous.The theoretical formulation of the boundary-value problem is provided in Sec. II and thenumerical results are discussed in Sec. III. Concluding remarks are presented in Sec. IV.An exp( − iωt ) time-dependence is implicit, with ω denoting the angular frequency. Thefree-space wavenumber, the free-space wavelength, and the intrinsic impedance of free spaceare denoted by k = ω √ ǫ µ , λ = 2 π/k , and η = p µ /ǫ , respectively, with µ and ǫ being the permeability and permittivity of free space. Vectors are in boldface, columnvectors are in boldface and enclosed within square brackets, and matrixes are underlinedtwice and square-bracketed. The Cartesian unit vectors are identified as ˆ u x , ˆ u y , and ˆ u z .The superscript T denotes the transpose. II. BOUNDARY-VALUE PROBLEMA. Description
Let us consider the schematic of the boundary-value problem shown in Fig. 1. Theregions z < z > d are vacuous, the region 0 ≤ z ≤ d is occupied by the dielectricpartnering material with relative permittivity ǫ d ( z ), and the region d ≤ z ≤ d by themetallic partnering material with spatially uniform relative permittivity ǫ m . The region d < z < d contains a surface-relief grating of period L along the x axis. The relativepermittivity ǫ g ( x, z ) = ǫ g ( x ± L, z ) in this region is taken to be as ǫ g ( x, z ) = (1 / ǫ m + ǫ d ( z )] − [ ǫ m − ǫ d ( z )] × (cid:8) U [ d − z − g ( x )] − (cid:9) , x ∈ (0 , L ) ,ǫ d ( z ) , x ∈ ( L , L ) , (1)for z ∈ ( d , d ), with g ( x ) = ( d − d ) sin (cid:18) πxL (cid:19) , L ∈ (0 , L ) , (2)and U ( ζ ) = , ζ ≥ , , ζ < . (3)3he depth of the surface-relief grating defined by Eq. (2) is d − d . This particular gratingshape is chosen for the ease of fabrication; however, the theoretical formulation given in theremainder of this section is independent of the shape of the surface-relief grating.
2Ω Surface wavesDielectricMetal Reflected lightIncident light z x z = 0z = d z = d z = d Transmitted light
FIG. 1: Schematic of the boundary-value problem solved using the RCWA.
In the vacuous half-space z ≤
0, let a plane wave propagating in the xz plane at an angle θ to the z axis, be incident on the structure. Hence, the incident, reflected, and transmittedfield phasors may be written in terms of Floquet harmonics as follows: E inc ( r ) = X n ∈ Z (cid:0) s n a ( n ) s + p + n a ( n ) p (cid:1) exp (cid:2) i (cid:0) k ( n ) x x + k ( n ) z z (cid:1)(cid:3) , z ≤ , (4) H inc ( r ) = η − X n ∈ Z (cid:0) p + n a ( n ) s − s n a ( n ) p (cid:1) exp (cid:2) i (cid:0) k ( n ) x x + k ( n ) z z (cid:1)(cid:3) , z ≤ , (5) E ref ( r ) = X n ∈ Z (cid:0) s n r ( n ) s + p − n r ( n ) p (cid:1) exp (cid:2) i (cid:0) k ( n ) x x − k ( n ) z z (cid:1)(cid:3) , z ≤ , (6) H ref ( r ) = η − X n ∈ Z (cid:0) p − n r ( n ) s − s n r ( n ) p (cid:1) exp (cid:2) i (cid:0) k ( n ) x x − k ( n ) z z (cid:1)(cid:3) , z ≤ , (7) E tr ( r ) = X n ∈ Z (cid:0) s n t ( n ) s + p + n t ( n ) p (cid:1) exp (cid:8) i (cid:2) k ( n ) x x + k ( n ) z ( z − d ) (cid:3)(cid:9) , z ≥ d , (8) H tr ( r ) = η − X n ∈ Z (cid:0) p + n t ( n ) s − s n t ( n ) p (cid:1) exp (cid:8) i (cid:2) k ( n ) x x + k ( n ) z ( z − d ) (cid:3)(cid:9) , z ≥ d , (9)4here k ( n ) x = k sin θ + nκ x , κ x = 2 π/L , and k ( n ) z = + q k − ( k ( n ) x ) , k > ( k ( n ) x ) + i q ( k ( n ) x ) − k , k < ( k ( n ) x ) . (10)The unit vectors s n = ˆ u y (11)and p ± n = ∓ k ( n ) z k ˆ u x + k ( n ) x k ˆ u z (12)represent the s - and p -polarization states, respectively. B. Coupled ordinary differential equations
The relative permittivity in the region 0 ≤ z ≤ d can be expanded as a Fourier serieswith respect to x , viz., ǫ ( x, z ) = X n ∈ Z ǫ ( n ) ( z ) exp( inκ x x ) , z ∈ [0 , d ] , (13)where κ x = 2 π/L , ǫ (0) ( z ) = ǫ d ( z ) , z ∈ [0 , d ] , L R L ǫ g ( x, z ) dx , z ∈ ( d , d ) ,ǫ m , z ∈ [ d , d ] , (14)and ǫ ( n ) ( z ) = L R L ǫ g ( x, z ) exp( − inκ x x ) dx , z ∈ [ d , d ]0 , otherwise ; ∀ n = 0 . (15)The field phasors may be written in the region 0 ≤ z ≤ d in terms of Floquet harmonics as E ( r ) = X n ∈ Z (cid:2) E ( n ) x ( z )ˆ u x + E ( n ) y ( z )ˆ u y + E ( n ) z ( z )ˆ u z (cid:3) exp( ik ( n ) x x ) H ( r ) = X n ∈ Z (cid:2) H ( n ) x ( z )ˆ u x + H ( n ) y ( z )ˆ u y + H ( n ) z ( z )ˆ u z (cid:3) exp( ik ( n ) x x ) , z ∈ [0 , d ] , (16)with unknown functions E ( n ) x,y,z ( z ) and H ( n ) x,y,z ( z ).5ubstitution of Eqs. (13) and (16) in the frequency-domain Maxwell curl postulates resultsin a system of four ordinary differential equations and two algebraic equations as follows: ddz E ( n ) x ( z ) − ik ( n ) x E ( n ) z ( z ) = ik η H ( n ) y ( z ) , (17) ddz E ( n ) y ( z ) = − ik η H ( n ) x ( z ) , (18) k ( n ) x E ( n ) y ( z ) = k η H ( n ) z ( z ) , (19) ddz H ( n ) x ( z ) − ik ( n ) x H ( n ) z ( z ) = − ik η X m ∈ Z ǫ ( n − m ) ( z ) E ( m ) y ( z ) , (20) ddz H ( n ) y ( z ) = ik η X m ∈ Z ǫ ( n − m ) ( z ) E ( m ) x ( z ) , (21) k ( n ) x H ( n ) y ( z ) = − k η X m ∈ Z ǫ ( n − m ) ( z ) E ( m ) z ( z ) . (22)Equations (17)–(22) hold ∀ z ∈ (0 , d ) and ∀ n ∈ Z . These equations can be reformulatedinto an infinite system of coupled first-order ordinary differential equations. This systemcan not be implemented on a digital computer. Therefore, we restrict | n | ≤ N t and thendefine the column (2 N t + 1)-vectors[ X σ ( z )] = [ X ( − N t ) σ ( z ) , X ( − N t ) σ ( z ) , ..., X (0) σ ( z ) , ..., X ( N t − σ ( z ) , X ( N t ) σ ( z )] T , (23)for X ∈ { E, H } and σ ∈ { x, y, z } . Similarly, we define (2 N t + 1) × (2 N t + 1)-matrixes[ K x ] = diag[ k ( n ) x ] , [ ǫ ( z )] = (cid:2) ǫ ( n − m ) ( z ) (cid:3) , (24)where diag[ k ( n ) x ] is a diagonal matrix.Substitution of Eqs. (19) and (22) into (17), (18), (20) and (21), to eliminate E ( n ) z and H ( n ) z ∀ n ∈ Z , gives the matrix ordinary differential equation ddz [ f ( z )] = i (cid:2) P ( z ) (cid:3) · [ f ( z )] , z ∈ (0 , d ) , (25)where the column vector [ f ( z )] with 4(2 N t + 1) components is defined as[ f ( z )] = h [ E x ( z )] T , [ E y ( z )] T , η [ H x ( z )] T , η [ H y ( z )] T i T (26)and the 4(2 N t + 1) × N t + 1)-matrix (cid:2) P ( z ) (cid:3) is given by (cid:2) P ( z ) (cid:3) = (cid:2) (cid:3) (cid:2) (cid:3) (cid:2) (cid:3) h P ( z ) i(cid:2) (cid:3) (cid:2) (cid:3) − k (cid:2) I (cid:3) (cid:2) (cid:3)(cid:2) (cid:3) h P ( z ) i (cid:2) (cid:3) (cid:2) (cid:3)h P ( z ) i (cid:2) (cid:3) (cid:2) (cid:3) (cid:2) (cid:3) . (27)6hereas (cid:2) (cid:3) is the (2 N t + 1) × (2 N t + 1) null matrix and (cid:2) I (cid:3) is the (2 N t + 1) × (2 N t + 1)identity matrix, the three non-null submatrixes on the right side of Eq. (27) are as follows: h P ( z ) i = k (cid:2) I (cid:3) − k h K x i · (cid:2) ǫ ( z ) (cid:3) − · h K x i , (28) h P ( z ) i = 1 k h K x i − k (cid:2) ǫ ( z ) (cid:3) , (29) h P ( z ) i = k (cid:2) ǫ ( z ) (cid:3) . (30) C. Solution algorithm
The column vectors [ f (0)] and [ f ( d )] can be written using Eqs. (4)–(9) as[ f (0)] = h Y + e i h Y − e ih Y + h i h Y − h i · [ A ][ R ] , [ f ( d )] = h Y + e ih Y + h i · [ T ] , (31)where [ A ] = (cid:2) a ( − N t ) s , a ( − N t +1) s , ..., a (0) s , ..., a ( N t − s , a ( N t ) s ,a ( − N t ) p , a ( − N t +1) p , ..., a (0) p , ..., a ( N t − p , a ( N t ) p (cid:3) T , (32)[ R ] = (cid:2) r ( − N t ) s , r ( − N t +1) s , ..., r (0) s , ..., r ( N t − s , r ( N t ) s ,r ( − N t ) p , r ( − N t +1) p , ..., r (0) p , ..., r ( N t − p , r ( N t ) p (cid:3) T , (33)[ T ] = (cid:2) t ( − N t ) s , t ( − N t +1) s , ..., t (0) s , ..., t ( N t − s , t ( N t ) s ,t ( − N t ) p , t ( − N t +1) p , ..., t (0) p , ..., t ( N t − p , t ( N t ) p (cid:3) T , (34)and the non-zero entries of (4 N t + 2) × (4 N t + 2)-matrixes h Y ± e,h i are as follows: (cid:0) Y ± e (cid:1) nm = 1 , n = m + 2 N t + 1 , (35) (cid:0) Y ± e (cid:1) nm = ∓ k ( n ) z k , n = m − N t − , (36) (cid:0) Y ± h (cid:1) nm = ∓ k ( n ) z k , n = m ∈ [1 , N t + 1] , (37) (cid:0) Y ± h (cid:1) nm = − , n = m ∈ [2 N t + 2 , N t + 2] . (38)In order to devise a stable algorithm [20, 21, 23, 24], the region 0 ≤ z ≤ d is dividedinto N d slices and the region d < z < d into N g slices, but the region d ≤ z ≤ d is keptas just one slice. So, there are N d + N g + 1 slices and N d + N g + 2 interfaces. In the j thslice, j ∈ [1 , N s + N g + 1], bounded by the planes z = z j − and z = z j , we approximate (cid:2) P ( z ) (cid:3) = (cid:2) P (cid:3) j = (cid:20) P (cid:18) z j + z j − (cid:19)(cid:21) , z ∈ ( z j , z j − ) , (39)7o that Eq. (25) yields[ f ( z j − )] = (cid:2) G (cid:3) j · exp n − i ∆ j (cid:2) D (cid:3) j o · (cid:2) G (cid:3) − j · [ f ( z j )] , (40)where ∆ j = z j − z j − , (cid:2) G (cid:3) j is a square matrix comprising the eigenvectors of (cid:2) P (cid:3) j as itscolumns, and the diagonal matrix (cid:2) D (cid:3) j contains the eigenvalues of (cid:2) P (cid:3) j in the same order.Let us define auxiliary column vectors [ T ] j and transmission matrixes (cid:2) Z (cid:3) j by the relation[23] [ f ( d j )] = (cid:2) Z (cid:3) j · [ T ] j , j ∈ [0 , N d + N g + 1] , (41)where d = 0, [ T ] N d + N g +1 = [ T ] , (cid:2) Z (cid:3) N d + N g +1 = h Y + e ih Y + h i . (42)To find [ T ] j and (cid:2) Z (cid:3) j for j ∈ [0 , N d + N g ], we substitute Eq. (41) in (40), which results inthe relation (cid:2) Z (cid:3) j − · [ T ] j − = (cid:2) G (cid:3) j · e − i ∆ j [ D ] uj e − i ∆ j [ D ] lj · (cid:2) G (cid:3) − j · (cid:2) Z (cid:3) j · [ T ] j ,j ∈ [1 , N d + N g + 1] , (43)where (cid:2) D (cid:3) uj and (cid:2) D (cid:3) lj are the upper and lower diagonal submatrixes of (cid:2) D (cid:3) j , respectively,when the eigenvalues are arranged in decreasing order of the imaginary part.Since [ T ] j and (cid:2) Z (cid:3) j cannot be determined simultaneously from Eq. (43), let us define [23][ T ] j − = exp n − i ∆ j (cid:2) D (cid:3) uj o · (cid:2) W (cid:3) uj · [ T ] j , (44)where the square matrix (cid:2) W (cid:3) uj and its counterpart (cid:2) W (cid:3) lj are defined via (cid:2) W (cid:3) uj (cid:2) W (cid:3) lj = (cid:2) G (cid:3) − j · (cid:2) Z (cid:3) j . (45)Substitution of Eq. (44) in (43) results in the relation (cid:2) Z (cid:3) j − = (cid:2) G (cid:3) j · (cid:2) I (cid:3) exp (cid:8) − i ∆ j [ D ] lj (cid:9) · (cid:2) W (cid:3) lj · n(cid:2) W (cid:3) uj o − · exp (cid:8) i ∆ j [ D ] uj (cid:9) ,j ∈ [1 , N d + N g + 1] . (46)8rom Eqs. (45) and (46), we find (cid:2) Z (cid:3) in terms of (cid:2) Z (cid:3) N d + N g +1 . After partitioning (cid:2) Z (cid:3) = (cid:2) Z (cid:3) u (cid:2) Z (cid:3) l , (47)and using Eqs. (31) and (41), [ R ] and [ T ] are found as follows: [ T ] [ R ] = (cid:2) Z (cid:3) u − h Y − e i(cid:2) Z (cid:3) l − h Y − h i − · h Y + e ih Y + h i · [ A ] . (48)Equation (48) is obtained by enforcing the usual boundary conditions across the plane z = 0. After [ T ] is known, [ T ] = [ T ] N d + N g +1 is found by reversing the sense of iterations inEq. (44). III. NUMERICAL RESULTS AND DISCUSSIONA. Homogeneous dielectric partnering material
Let us begin with the dielectric partnering material being homogeneous, i.e., ǫ d ( z ) isindependent of z . This case has been numerically illustrated by Homola [2, p. 38] and weadopted the same parameters: λ = 800 nm, ǫ d = 1 .
766 (water), ǫ m = −
25 + 1 . i (gold),and L = 672 nm. The incident plane wave is p polarized ( a ( n ) p = δ n and a ( n ) s ≡ ∀ n ∈ Z )and the quantity of importance is the absorbance A p = 1 − N t X n = − N t (cid:16)(cid:12)(cid:12) r ( n ) s (cid:12)(cid:12) + (cid:12)(cid:12) r ( n ) p (cid:12)(cid:12) + (cid:12)(cid:12) t ( n ) s (cid:12)(cid:12) + (cid:12)(cid:12) t ( n ) p (cid:12)(cid:12) (cid:17) Re (cid:2) k ( n ) z /k (0) z (cid:3) , (49)which simplifies to A p = 1 − N t X n = − N t (cid:16)(cid:12)(cid:12) r ( n ) p (cid:12)(cid:12) + (cid:12)(cid:12) t ( n ) p (cid:12)(cid:12) (cid:17) Re (cid:2) k ( n ) z /k (0) z (cid:3) , (50)because all materials are isotropic.Figure 2 shows the variation of A p versus the incidence angle θ for a sinusoidal surface-relief grating defined by [2] g ( x ) = 12 ( d − d ) (cid:20) (cid:18) πxL (cid:19)(cid:21) (51)9 A p (deg) FIG. 2: Absorbance A p as a function of the incidence angle θ for a sinusoidal surface-relief gratingdefined by Eq. (51). Black squares represent d = 1500 nm, red circles d = 1000 nm, and bluetriangles d = 800 nm. The grating depth ( d − d = 50 nm) and the thickness of the metalliclayer ( d − d = 30 nm) are the same for all three cases. The arrow identifies an SPP wave. A p (deg) FIG. 3: Same as Fig. 2 except that the surface-relief grating is defined by Eq. (2) with L = 0 . L . and Fig. 3 shows the same for the surface-relief grating defined by Eq. (2) with L = 0 . L .For computational purposes, we set N d = 1, N g = 50, and N t = 10, after ascertaining thatthe reflectances (cid:12)(cid:12)(cid:12) r ( n ) p (cid:12)(cid:12)(cid:12) Re h k ( n ) z /k (0) z i and the transmittances (cid:12)(cid:12)(cid:12) t ( n ) p (cid:12)(cid:12)(cid:12) Re h k ( n ) z /k (0) z i convergedfor all n ∈ [ − N t , N t ].Each figure shows plots of A p vs. θ for three different values of the thickness d , in order10o distinguish [25] between(i) surface waves [11], which must be independent of d for sufficiently large values of thatparameter, and(ii) waveguide modes [26, 27], which must depend on d as has been shown elsewhere [28, 29]. In both figures, an absorbance peak at θ ≃ ◦ for allthree values of d indicates the excitation of an SPP wave.The relative wavenumbers k ( n ) x /k of a few Floquet harmonics at θ = 12 ◦ are given inTable I. The solution of the canonical boundary-value problem (when both partneringmaterials are semi-infinite along the z axis and their interface is planar) [2] shows that therelative wavenumber κ/k of the SPP wave that can be guided by the planar gold-waterinterface is κ/k = p ǫ d ǫ m / ( ǫ d + ǫ m ) = 1 . . i . (52)A comparison of Table I and Eq. (52) confirms that an SPP wave is excited at θ = 12 ◦ as theFloquet harmonic of order n = +1. We note that the absorbance peak in Fig. 3 is not onlywider than in Fig. 2, but also of lower magnitude, which points out the critical importanceof the shape function g ( x ) of the surface-relief grating. The incidence angle determined byHomola [2, p. 38] is approximately 11 ◦ , the small difference between his and our results being(i) due to the different methods of computation and (ii) the fact that, while Homola hadsemi-infinite dielectric and metallic partnering materials, we have the two of finite thickness. TABLE I: Relative wavenumbers k ( n ) x /k of Floquet harmonics for a gold-water grating when θ = 12 ◦ . Boldface entries signify SPP waves. n = − n = − n = 0 n = 1 n = 2 θ = 12 ◦ − . − . . . . B. Periodically nonhomogeneous dielectric partnering material
Now let us move on to the excitation of multiple
SPP waves by a surface-relief gratingwhere the dielectric partnering material has a periodic nonhomogeneity normal to the mean11lane of the surface-relief grating: ǫ d ( z ) = (cid:20)(cid:18) n b + n a (cid:19) + (cid:18) n b − n a (cid:19) sin (cid:18) π d − z Ω (cid:19)(cid:21) , z > , (53)where 2Ω is the period. We chose n a = 1 .
45 and n b = 2 .
32 from an example provided byBaumeister [30, Sec. 5.3.3.2 ]. For all calculations reported in the remainder of this paper,we chose the metal to be bulk aluminum ( ǫ met = −
56 + i
21) and the free-space wavelength λ = 633 nm. The surface-relief grating is defined by Eq. (2) with L = 0 . L . We fixed N t = 8 after ascertaining that the absorbances for N t = 8 converged to within < N t = 9. The grating depth d − d = 50 nm and thethickness d − d = 30 nm were also fixed, as their variations would not qualitatively affectthe excitation of multiple SPP waves. Numerical results for Ω = λ and Ω = 1 . λ are nowpresented. Ω = λ Let us commence with Ω = λ . The solution of the corresponding canonical boundary-value problem results in five p -polarized and one s -polarized SPP waves [11], the relativewavenmbers κ/k being provided in Table II. To analyze the excitation of s -polarized SPPwaves in the grating-coupled configuration, we calculated the absorbance A s = 1 − n = N t X n = − N t (cid:16)(cid:12)(cid:12) r ( n ) s (cid:12)(cid:12) + (cid:12)(cid:12) t ( n ) s (cid:12)(cid:12) (cid:17) Re[ k ( n ) z /k (0) z ] (54)for a ( n ) s = δ n and a ( n ) p ≡ ∀ n ∈ Z . Both A p and A s were calculated as functions of θ for d ∈ { λ , λ , λ } , with N g and N d selected to have slices of thickness 2 nm in the region0 ≤ z ≤ d but 1 nm in the region d < z < d . Figures 4, 5(a), 5(b), and 5(c) present theabsorbances as functions of θ for L = λ , 0 . λ , 0 . λ , and 0 . λ , respectively.For all three values of d , a peak is present at θ = 37 . ◦ in the plots of A p vs. θ in Fig. 4.The relative wavenumbers k ( n ) x /k of several Floquet harmonics at this incidence angle aregiven in Table III. At θ = 37 . ◦ , k (1) x /k = 1 . κ/k ] = 1 . κ/k is the relative wavenumber of an SPP wave in the canonical boundary-value problem [11]as provided in Table II; likewise, k ( − x /k = 1 . κ/k ] = 1 . A p -peak couldrepresent the grating-coupled excitation of either one or two p -polarized SPP waves. To12 ABLE II: Relative wavenumbers κ/k of possible SPP waves obtained by the solution of thecanonical boundary-value problem [11] for Ω = λ . Other parameters are provided in the beginningof Sec. III B. s -pol . . ip -pol . . i . . i . . i . . i . . i
20 25 30 35 40 450.20.40.60.8 20 25 30 35 40 450.00.20.40.60.81.0 A p A s (deg) FIG. 4: Absorbances A p and A s as functions of the incidence angle θ for a surface-relief gratingdefined by Eq. (2) with L = 0 . L , when λ = 633 nm, Ω = λ , and L = λ . Black squaresare for d = 6 λ , red circles for d = 5 λ , and blue triangles for d = 4 λ . The grating depth( d − d = 50 nm) and the thickness of the metallic layer ( d − d = 30 nm) are the same for allplots. Each arrow identifies an SPP wave. resolve this issue, we computed A p vs. θ for L = 1 . λ (not shown). For the new gratingperiod, the A p -peak shifted to a higher value of θ . This shift showed that the SPP wave atthe A p -peak in Fig. 4 is excited as a Floquet harmonic with a positive index n and not theone with a negative index. So the A p -peak represents the excitation of a p -polarized SPPwave by a Floquet harmonic of order n = 1 with k (1) x /k = 1 . ABLE III: Relative wavenumbers k ( n ) x /k of Floquet harmonics when L = λ . Boldface entriessignify SPP waves. n = − n = − n = 0 n = 1 n = 2 θ = 28 ◦ − . − . . . . θ = 37 . ◦ − . − . . . . A peak is also present at θ = 28 ◦ in the plots of A s vs. θ , for all three values of d inFig. 4. The solution of the canonical boundary-value problem [11] indicates the excitationof an s -polarized SPP wave when κ/k = 1 . . k (1) x /k = 1 . θ = 28 ◦ ; see Table III. So the A s -peak representsthe grating-coupled excitation of an s -polarized SPP wave. TABLE IV: Same as Table III except for L = 0 . λ . n = − n = − n = 0 n = 1 n = 2 θ = 32 . ◦ − . − . . . . θ = 51 ◦ − . − . . . . θ = 64 ◦ − . − . . . . Since not all possible p -polarized SPP waves (predicted after the solution of the canonicalboundary-value problem) can be excited with period L = λ of the surface-relief grating,the grating period needs to be changed to excite the remaining SPP waves. The plots of A p vs. θ for L = 0 . λ are presented in Fig. 5(a), again for d ∈ { λ , λ , λ } . Thefigure shows three A p -peaks at θ = 32 . ◦ , 51 ◦ , and 64 ◦ that are present for all three chosenvalues of d . The relative wavenumbers of several Floquet harmonics at these values of theincidence angle are given in Table IV. The A p -peak at θ = 32 . ◦ represents the excitationof a p -polarized SPP wave as a Floquet harmonic of order n = 1 because k (1) x /k = 1 . . . i ] in Table II. Remarkably, the A p -peak at θ = 51 ◦ representsthe excitation of the same SPP wave as a Floquet harmonic of order n = −
2. Finally, the A p -peak at θ = 64 ◦ represents the excitation of another p -polarized SPP wave as a Floquetharmonic of order n = 1. 14 A p A p (c)(b) A p (deg) (a) FIG. 5: Absorbance A p as a function of the incidence angle θ for a surface-relief grating defined byEq. (2) with L = 0 . L , when λ = 633 nm, Ω = λ ; and (a) L = 0 . λ , (b) L = 0 . λ , and (c) L = 0 . λ . Black squares are for d = 6 λ , red circles for d = 5 λ , and blue triangles for d = 4 λ .The grating depth ( d − d = 50 nm) and the thickness of the metallic layer ( d − d = 30 nm) arethe same for all plots. Each arrow identifies an SPP wave. Two A p -peaks, at θ = 27 ◦ and 38 ◦ , are present for all values of d in Fig. 5(b) for L = 0 . λ . The relative wavenumbers of several Floquet harmonics at these two values ofthe incidence angle are given in Table V. The A p -peak at θ = 27 ◦ represents the excitationof a p -polarized SPP wave because k (1) x /k = 1 . κ/k ] = 1 . p -polarized SPP wave predicted by the canonical problem.15 ABLE V: Same as Table III except for L = 0 . λ . n = − n = − n = 0 n = 1 n = 2 θ = 27 ◦ − . − . . . . θ = 38 ◦ − . − . . . . The A p -peak at θ = 38 ◦ represents another p -polarized SPP wave because k (1) x /k = 2 . κ/k ] = 2 . TABLE VI: Same as Table III except for L = 0 . λ . n = − n = − n = 0 n = 1 n = 2 θ = 25 ◦ − . − . . . . All the SPP waves that can be guided by the planar interface of the rugate filter (forthe chosen value of Ω) and aluminum have been shown to be excited by the grating-coupledconfiguration except the one with κ/k = 1 . . i (Table II). This p -polarizedSPP wave was found to be excited with a surface-relief grating of period L = 0 . λ .The variation of A p vs. θ for this case is shown in Fig. 5(c). The A p -peak at θ = 25 ◦ is independent of the thickness d of the rugate filter. The relative wavenumbers k ( n ) x /k of several Floquet harmonics at this value of the incidence angle are given in Table VI.Comparing Tables II and VI, we see that k ( − x /k is close to Re [1 . . i ] and k (1) x /k is close to Re [2 . . i ]. This suggests that either of the two or both SPPwaves are excited at θ = 25 ◦ . To resolve this issue, we computed A p vs. θ for L = 0 . λ (not shown) and the A p -peak shifted to a smaller value of the incidence angle. This shiftindicated that the absorbance peak at θ = 25 ◦ represents an SPP wave due to a Floquetharmonic of negative order. So the A p -peak at θ = 25 ◦ represents the excitation of the SPPwave as a Floquet harmonic of order n = − Ω = 1 . λ The relative wavenumbers of possible SPP waves that can be guided by the planar in-terface of the chosen rugate filter and the metal are given in Table VII for Ω = 1 . λ .16n this case, the solution of the canonical boundary-value problem indicated that multiple s -polarized SPP waves can also be guided in addition to multiple p -polarized SPP waves.Absorbances A p and A s , calculated for d ∈ { λ , . λ , λ } and θ ∈ [0 ◦ , ◦ ), arepresented in Figs. 6, 7, 8(a), and 8(b) for L = 0 . λ , 0 . λ , 0 . λ , and 0 . λ , respectively.For the computations, the region d < z < d was again divided into 1-nm-thick slices;however, the region 0 ≤ z ≤ d was divided into 3-nm-thick slices to reduce the computationtime, after ascertaining that the accuracy of the computed reflectances and transmittanceshad not been adversely affected. TABLE VII: Same as Table II except for Ω = 1 . λ . s -pol . . . . ip -pol . . i . . i . . i . . i . . i . . i TABLE VIII: Same as Table III except for L = 0 . λ . n = − n = − n = 0 n = 1 n = 2 θ = 17 ◦ − . − . . . . θ = 21 ◦ − . − . . . . θ = 27 . ◦ − . − . . . . θ = 32 ◦ − . − . . . . θ = 40 ◦ − . − . . . . θ = 44 ◦ − . − . . . . In the plots of A p vs. θ in Fig. 6, the excitation of p -polarized SPP waves is indicatedat four values of the incidence angle: θ = 17 ◦ , 21 ◦ , 27 . ◦ , and 40 ◦ . The relative wavenum-bers k ( n ) x /k of a few Floquet harmonics at these values of the incidence angle are given inTable VIII. The A p -peak at θ = 17 ◦ represents the excitation of a p -polarized SPP wave, be-cause k (1) x /k = 1 . κ/k ] = 1 . κ/k is a solution ofthe canonical boundary-value problem. The A p -peak at θ = 21 ◦ also represents a p -polarizedSPP wave because k ( − x /k = 2 . κ/k ] = 2 . A p A s (deg) FIG. 6: Absorbances A p and A s as functions of the incidence angle θ for a surface-relief gratingdefined by Eq. (2) with L = 0 . L , when λ = 633 nm, Ω = 1 . λ , and L = 0 . λ . Black squaresare for d = 9 λ , red circles for d = 7 . λ , and blue triangles for d = 6 λ . The grating depth( d − d = 50 nm) and the width of the metallic layer ( d − d = 30 nm) are the same for all theplots. Each arrow indicates an SPP wave. another solution of the canonical problem. The relative wavenumber k (1) x /k = 1 . θ = 21 ◦ was ruled out, after examining the plots for L = 0 . λ (not shown). Similarly, thepeak at θ = 27 . ◦ is due to the excitation of another p -polarized SPP wave as a Floquetharmonic of order n = 1 and not of the order n = −
2. At θ = 40 ◦ , both k (1) x /k = 1 . k ( − x /k = 1 . . . i ] (Table VII). However, the plotsof A p vs. θ for L = 0 . λ (not shown) made us conclude that the SPP wave is excited asthe Floquet harmonic of order n = 1.In the plots of A s vs. θ in Fig. 6, two peaks at θ = 32 ◦ and 44 ◦ are present for allvalues of d . The A s -peak at θ = 32 ◦ represents the excitation of an s -polarized SPP wavebecause k (1) x /k = 1 . κ/k ] = 1 . s -polarized SPP wave. The other A s -peak at θ = 44 ◦ represents the excitation of the same s -polarized SPP wave as a Floquet harmonic18f order n = − A p A s (deg) FIG. 7: Same as Fig. 6 except for L = 0 . λ .TABLE IX: Same as Table III except for L = 0 . λ . n = − n = − n = 0 n = 1 n = 2 θ = 10 ◦ − . − . . . . θ = 16 ◦ − . − . . . . θ = 27 . ◦ − . − . . . . For L = 0 . λ , the absorbances A p and A s are presented as functions of θ in Fig. 7. Therelative wavenumbers k ( n ) x /k of several Floquet harmonics at those values of the incidenceangles where peaks are present independent of the value of d are given in Table IX. In theplots of A p vs. θ , the peak at θ = 16 ◦ represents the excitation of a p -polarized SPP wavebecause k (1) x /k = 1 . κ/k ] = 1 . A p -peakat θ = 27 . ◦ represents the excitation of a p -polarized SPP wave, as k (1) x /k = 1 . κ/k ] = 1 . A s vs. θ , the peak at θ = 10 ◦ representsthe excitation of an s -polarized SPP wave because k (1) x /k = 1 . κ/k ] = 1 .
40 45 50 55 600.20.40.6 15 20 25 30 35 40 45 500.00.20.40.60.81.0 A p (b) A p (deg) (a) FIG. 8: Absorbance A p as a function of the incidence angle θ for a surface-relief grating definedby Eq. (2) with L = 0 . L , when λ = 633 nm, Ω = 1 . λ ; and (a) L = 0 . λ and (b) L = 0 . λ .Black squares are for d = 9 λ , red circles for d = 7 . λ , and blue triangles for d = 6 λ . Thegrating depth ( d − d = 50 nm) and the width of the metallic layer ( d − d = 30 nm) are thesame for all the plots. Each arrow indicates an SPP wave.TABLE X: Same as Table III except for L = 0 . λ . n = − n = − n = 0 n = 1 n = 2 θ = 43 ◦ − . − . . . . θ = 51 . ◦ − . − . . . . θ = 55 ◦ − . − . . . . In Fig. 8(a), three peaks are present for all values of d at θ = 43 ◦ , 51 . ◦ , and 55 ◦ . Therelative wavenumbers of several Floquet harmonics at these values of the incidence angle aregiven in Table X. The comparison of Tables VII and X shows that each A p -peak representsthe excitation of a p -polarized SPP wave. Similarly, each of the three A p -peaks at θ = 22 . ◦ ,20 ABLE XI: Same as Table III except for L = 0 . λ . n = − n = − n = 0 n = 1 n = 2 θ = 22 . ◦ − . − . . . . θ = 33 . ◦ − . − . . . . θ = 46 ◦ − . − . . . . . ◦ , and 46 ◦ in Fig. 8(b) represents the excitation of a p -polarized SPP wave as a Floquetharmonic of order n = 1. The relative wavenumbers of a few Floquet harmonics at thesevalues of the incidence angle are given in Table XI. At each angle, k (1) x /k is close to the realpart of one of the κ/k of the solutions of the canonical boundary-value problem problemprovided in Table VII. C. General conclusions
In the last two subsections, we have deciphered a host of numerical results and identifiedthose absorbance peaks that indicate the excitation of SPP waves in the grating-coupled con-figuration, when the dielectric partnering material is periodically nonhomogeneous normalto the mean plane of the surface-relief grating. We found that(i) the periodic nonhomogeneity of the dielectric partnering material enables the excita-tion of multiple SPP waves of both p - and s -polarization states;(ii) fewer s -polarized SPP waves are excited than p -polarized SPP waves;(iii) for a given period of the surface-relief grating, it is possible for two plane waves withdifferent angles of incidence to excite the same SPP wave (Figs. 5(a) and 6);(iv) not all SPP waves predicted by the solution of the canonical problem may be excitedin the grating-coupled configuration for a given period;(v) the absorbance peaks representing the excitation of p -polarized SPP waves are gener-ally wider than those representing s -polarized SPP waves; and(vi) the absorbance peak is narrower for an SPP wave of higher phase speed (i.e. smallerRe [ κ ]). 21et us note that some other combination of the periodic functions ǫ d ( z ) and g ( x ) may allowall solutions of the canonical boundary-value problem to be excited in the grating-coupledconfiguration.The solution of the canonical boundary-value problem [11] indicates that the period 2Ωof the rugate filter needs to be greater than a certain value in order for more than one SPPwaves to be excited, and the excitation of s -polarized SPP waves to exist requires an evenlarger period. However, the number of possible SPP waves increases as the period increasesup to a certain value. We chose Ω = λ and Ω = 1 . λ to allow the excitation of multiple p -polarized SPP waves for both values, and multiple s -polarized SPP waves for the secondvalue. Our numerical results confirm that the conclusions on the number of SPP wavesdrawn in the predecessor paper [11] also hold for the grating-coupled configuration. IV. CONCLUDING REMARKS
The excitation of multiple surface-plasmon-polariton (SPP) waves by a surface-relief grat-ing formed by a metal and a dielectric material, both of finite thickness, was studied theoret-ically using the rigorous coupled-wave-analysis technique for the practically implementablesetup. The presence of an SPP wave was inferred by a peak in the plots of absorbance vs.the angle of incidence θ , provided that the θ -location of the peak tuned out to be indepen-dent of the thickness of the partnering dielectric material. If that material is homogeneous,only one p -polarized SPP wave, that too of p -polarization state, is excited. However, the pe-riodic nonhomogeneity of the partnering dielectric material normal to the mean plane of thesurface-relief grating results in the excitation of multiple SPP waves of different polarizationstates and phase speeds. In general, the absorbance peak is narrower for an s -polarized SPPwave than for of a p -polarized SPP wave, and the absorbance peak is narrower for an SPPwave of higher phase speed.Since the electromagnetic field radiated by a line source can be considered as a spectrumof plane waves propagating at all angles [31, Sec. 2.2], the grating-coupled configurationdiscussed in this paper can be used to excite multiple SPP waves simultaneously by a linesource. The excitation of multiple SPP waves may be significant for practical applications—for example, to increase the absorption of light in solar cells due to the increased possibilityof excitation of SPP waves [32]. This application is currently under investigation by the22uthors. Acknowledgments
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