Propagation of surface waves and waveguide modes guided by a

PHYSICAL REVIEW A 83, 013814 (2011)

Propagation of surface waves and waveguide modes guided by a dielectric slab inserted in a sculptured nematic thin film Muhammad Faryad* and Akhlesh Lakhtakia Nanoengineered Metamaterials Group (NanoMM), Department of Engineering Science and Mechanics, Pennsylvania State University, University Park, Pennsylvania 16802-6812, USA (Received 7 September 2010; published 14 January 2011) Wave propagation guided by a dielectric slab inserted in a sculptured nematic thin film (SNTF) was studied theoretically. Two types of guided waves can be identified: (i) surface (Dyakonov–Tamm) waves guided by one or both of the two planar interfaces of the dielectric slab and the SNTF, and (ii) waveguide modes in the dielectric waveguide formed by the slab with the SNTF as the cladding. As the thickness of the dielectric slab is increased, the number of waveguide modes increases. If the slab thickness is less than twice the e-folding distance into the dielectric slab, the Dyakonov–Tamm waves propagate coupled to both interfaces; the coupling decreases and eventually vanishes as the slab thickness increases, so that Dyakonov–Tamm waves are guided by the individual dielectric-SNTF interfaces independently. The chosen structure supports the propagation of Dyakonov–Tamm waves in all directions, in contrast to the restricted range of propagation supported by a single SNTF-dielectric interface. Propagation of both Dyakonov–Tamm waves and waveguide modes should occur in practice with negligible attenuation, in contrast to that of surface-plasmon-polariton waves that are guided when the dielectric slab is replaced by a metal slab. DOI: 10.1103/PhysRevA.83.013814

PACS number(s): 42.25.−p, 42.70.−a

I. INTRODUCTION

One hundred years ago, the first all-dielectric waveguide was studied theoretically by Hondros and Debye [1]. Since then, dielectric waveguides have been investigated intensively [2]. Optical waveguides not only form the backbone of the global communication network today, but are also widely used in devices such as endoscopes, laser systems, and optical sensors [3]. In a traditional step-index optical dielectric waveguide, the electromagnetic energy propagates in a central core housed inside a cladding, both the core and the cladding being made of homogeneous, isotropic, dielectric materials. In a graded-index optical waveguide, the refractive index in the transverse plane decreases away from the central axis, thereby concentrating the electromagnetic energy along that axis [2,4]. Most importantly, the cladding is always made of a homogeneous material. The last two decades have opened new technoscientific avenues, including the capability to design materials with nanoscale morphology [5–7]. Since solid-state technology is mostly planar, the need for two distinct modalities of guided-wave propagation, among others, has arisen. First, efficient planar optical waveguides in the cladding-corecladding configuration are needed to transport optical signals [8]; second, surface-plasmon-polariton (SPP) waves guided by planar metal/dielectric interfaces are attractive as they are faster than purely electronic waves traveling in good conductors [9,10]. SPP waves have also generated enormous research interest due to potential and actual applications in sensing and imaging [9–12]. A major challenge to the adoption of SPP-based communication is the high attenuation of SPP waves [13], as becomes evident from the solutions of a relevant canonical boundary-value problem [14], primarily due to ohmic losses in

*

Corresponding author: [email protected]

1050-2947/2011/83(1)/013814(9)

the metal. Though several mitigation strategies are being tried out [15], an obvious one is to replace the metal by a dielectric material. That is the attraction of Dyakonov waves [16–18], which are surface waves guided by the planar interface of two homogeneous dielectric materials of which at least one must be anisotropic [19,20]. Solutions of canonical boundary-value problems [21] indicate the possibility of lossless propagation, provided that dissipation in both partnering dielectric materials is negligible, leading to potential applications in integrated optics, optical sensing and waveguiding [22]. However, very restrictive conditions need to be satisfied in order for Dyakonov waves to exist [23,24]. Therefore, the directions of propagation in the interface plane are confined to a tiny angular sector. The introduction [25] of periodic nonhomogeneity normal to the interface into the partnering anisotropic dielectric material resulted in surface waves of a new type. These are called Dyakonov–Tamm waves [21], because they share certain features of Tamm states [26] from solid-state physics. Dyakonov–Tamm waves can propagate over a wider angular sector compared to Dyakonov waves [27,28], and have similar spatial-localization characteristics as SPP waves but negligible attenuation. Because of continued interest in conventional dielectric waveguides and optical surface waves, we decided to find a structure which can transport electromagnetic energy via waveguide modes and Dyakonov–Tamm waves. The interface of a homogeneous, isotropic dielectric material and a periodically nonhomogeneous sculptured nematic thin film (SNTF) [29] can support Dyakonov–Tamm waves [27]. Therefore, we set out to investigate wave propagation guided by a homogeneous, isotropic dielectric slab inserted in an SNTF, the dielectric slab being the core and the SNTF being the cladding. The structure thus formed can be considered to be a canonical structure that allows (i) interaction between waves localized to different parallel surfaces, as well as (ii) waveguide modes with energy largely confined to the region between two parallel surfaces.

013814-1

©2011 American Physical Society

MUHAMMAD FARYAD AND AKHLESH LAKHTAKIA


An SNTF is a special type of sculptured thin film (STF). STFs are grown by directing a collimated vapor flux toward a suitably rotating and/or rocking substrate in vacuum [30,31]. Macroscopically, an STF is a material continuum that is periodically nonhomogeneous in a particular direction. At the microscopic length scale, an STF is an assembly of parallel columns of nanoscale cross-sectional diameter, all columns having the same shape. The columnar morphology of an SNTF is described by a curve embedded in two-dimensional space, ([31], Chap. 8) as the substrate is only rocked about a tangential axis while the film is growing. The plane in which the curve lies is the morphologically significant plane of the SNTF. The relevant canonical boundary-value problem is formulated in Sec. II and numerical results are discussed in Sec. III. Concluding remarks are given in Sec. IV. An exp(−iωt) timedependence is implicit,√ with ω denoting the angular frequency, t the time, and i = −1. The free-space wave number, the free-space wavelength, and the intrinsic impedance of √ free space are denoted by k0 = ω ε0 µ0 , λ0 = 2π/k0 , and √ η0 = µ0 /ε0 , respectively, with µ0 and ε0 being the permeability and permittivity of free space. Vectors are in boldface, dyadics are underlined twice, column vectors are in boldface and enclosed within square brackets, and matrices are underlined twice and square-bracketed. The dyadics have been treated as 3 × 3 matrices in this paper [32,33]. The superscript T denotes the transpose, the asterisk denotes the complex conjugate, and the Cartesian unit vectors are identified as uˆ x , uˆ y , and uˆ z . II. THEORETICAL FORMULATION

Suppose that the region L− z L+ is occupied by an isotropic and homogeneous dielectric material with relative permittivity scalar εs , as shown in Fig. 1. The thickness of the dielectric slab is denoted by Ls = L+ − L− . The regions zL± are occupied by the chosen SNTF with periodically nonhomogeneous permittivity dyadic [29] ε

SNTF

(z) = ε0 S (γ ± ) · S (z) · ε o (z) · S −1 (z) · S −1 (γ ± ), z

ref

y

y

z

zL± ,

= εa (z) uˆ z uˆ z + εb (z) uˆ x uˆ x + εc (z) uˆ y uˆ y z Ls

y

+ (uˆ z uˆ x − uˆ x uˆ z ) sin[χ (z)] + uˆ y uˆ y

x, y dielectric slab

FIG. 1. (Color online) Schematic illustration of the geometry of the boundary-value problem for γ + = γ − .

(3)

describes the nematicity. Both the relative permittivity scalars εa,b,c (z) and the tilt angle χ (z) are supposed to have been nanoengineered by a periodic variation of the direction of the vapor flux during fabrication by physical vapor deposition [29,31]. This periodic variation is captured by the vapor incidence angle [29] π (z − L± ) χv (z) = χ˜ v ± δv sin , zL± , (4) that varies sinusoidally with z, and describes the shape of the columns of the SNTF in the morphologically significant plane. The third dyadic in Eq. (1) was chosen as S (γ ± ) = (uˆ x uˆ x + uˆ y uˆ y ) cos γ ± z

+ (uˆ y uˆ x − uˆ x uˆ y ) sin γ ± + uˆ z uˆ z ,

(5)

so that plane formed by the unit vectors uˆ z and uˆ x cos γ ± + uˆ y sin γ ± is the morphologically significant plane for zL± . Thus, there is sufficient flexibility in the formulation with respect to the twist γ + -γ − of the two morphologically significant planes. Without loss of generality, let us choose the direction of guided-wave propagation (encompassing both surface waves and waveguide modes) in the xy plane to be parallel to the x axis. Accordingly, we set E(r) = e(z) exp(iκx) , (6) H(r) = h(z) exp(iκx) where κ is a real-valued scalar. When the twist γ + -γ − is fixed, the range of γ + for guided-wave propagation is effectively the range of propagation directions in the xy plane. The axial field components ez (z) and hz (z) can be expressed in terms of the column vector [f(z)] = [ex (z) ey (z) hx (z) hy (z)]T via

(2)

2Ω SNTF

S (z) = (uˆ x uˆ x + uˆ z uˆ z ) cos[χ (z)]

(1)

where the locally orthorhombic symmetry is expressed through the diagonal dyadic ε o (z) ref

and the local tilt dyadic

(7)

⎡

⎤ ez (z)

[ A± (z)] · [f(z)] , ⎢ 0 ⎥ = κ⎣ ⎦ hz (z) [ As (z)] · [f(z)], 0

zL± , z ∈ (L− ,L+ ) ,

(8)

where the 4×4 matrices ⎡ εd (z) ⎤ κ2 0 0 0 − ωε 0 εa (z)εb (z) ⎢0 0 0 ⎥ 0 ⎢ ⎥ [ A± (z)] = ⎢ ⎥ 2 ⎣0 κ ⎦ 0 0 ωµ0 0 0 0 0 εd (z) [εa (z) − εb (z)] +κ sin [χ (z)] cos [χ (z)] εa (z)εb (z) ⎡ ⎤ cos γ ± sin γ ± 0 0 ⎢ 0 ⎥ 0 0 0 ⎢ ⎥ ×⎢ (9) ±⎥ ⎣ 0 0 0 − sin γ ⎦ 0 0 0 cos γ ±

013814-2

PROPAGATION OF SURFACE WAVES AND WAVEGUIDE . . .

and

⎡

0 ⎢0 ⎢ [ As (z)] = ⎢ ⎣0 0

0 0

0 0

κ2 ωµ0

0 0

0

2 − ωεκ0 εs

0 0 0


and

⎤ ⎥ ⎥ ⎥; ⎦

(10)

here and hereafter, εd (z) = εa (z)εb (z)/{εa (z) cos [χ (z)] + εb (z) sin [χ (z)]}. 2

2

(11)

The column vector [f(z)] satisfies the matrix differential equations d [f(z)] = i [ P s (z)] · [f(z)] , z ∈ (L− ,L+ ) , dz

d [f(z)] = i [ P ± (z)] · [f(z)] , zL± , dz where the 4×4 matrices ⎡ ⎤ 0 0 0 µ0 ⎢ 0 0 −µ0 0 ⎥ ⎢ ⎥ [ P s (z)] = [ As (z)] + ω ⎢ ⎥, ⎣ 0 0 0⎦ −ε0 εs

Equation (12) can be solved analytically to yield (16)

Equation (13) can be solved numerically by the piecewise uniform approximation technique [27]. The optical response of one period of the SNTF on either side of the dielectric slab can be expressed using the matrices [Q± ] that appear in the relations (17)

+

˜ ] such that Im[α + ] > 0, where αn± = the eigenvalues of [Q 1,2 ∓(i/2 )ln σn± , and then set [28] + A1 , A+ 2

0 0

0 −µ0 2 ± 2 ± −ε0 [εc (z) cos γ + εd (z) sin γ ] 0 −ε0 [εc (z) − εd (z)] cos γ ± sin γ ± 0

⎤ µ0 0⎥ ⎥ ⎥. 0⎦ 0

(15)

Enforcing standard boundary conditions at the interfaces z = L± using Eqs. (16), (18), and (19), we obtain the matrix equation + A1 + (1) + (2) [[t ] [t ] ] · A+ 2 − A s − (1) − (2) = exp{i[ P ](L+ − L− )} · [[t ] [t ] ] · 1− , (20) A2 which may be rearranged as ⎡

⎡ ⎤ 0 ⎢ +⎥ ⎢ ⎥ ⎢A2 ⎥ ⎢0⎥ ⎥ ⎢ ⎥ [ M(κ)] · ⎢ ⎢A− ⎥ = ⎢0⎥ . ⎣ 1⎦ ⎣ ⎦ 0 A− 2 A+ 1

The electromagnetic fields of the guided waves must diminish in magnitude as z → ±∞. Let [t± ](n) , n ∈ [1,4], be the eigenvector corresponding to the nth eigenvalue σn± of [Q± ]. Therefore, in the half-space z > L+ , we first label

[f(L+ )] = [[t+ ](1) [t+ ](2) ] ·

0

(12)

0 ⎢ 0 ⎢ [ P ± (z)] = [ A± (z)] + ω ⎢ ⎣ ε0 [εc (z) − εd (z)] cos γ ± sin γ ± ε0 [εc (z) sin2 γ ± + εd (z) cos2 γ ± ]

[f(L± ± 2 )] = [ Q± ] · [f(L± )].

0

(14)

and

⎡

[f(L+ )] = exp{i[ P s ](L+ − L− )} · [f(L− )].

0

ε0 εs

(13)

⎤

(21)

The dispersion equation for guided-wave propagation is then as follows: det [ M (κ)] = 0.

(22)

(18)

where A+ 1,2 are unknown scalars. A similar argument for the − half-space z < L− requires us to ensure that Im[α1,2 ] < 0 and then to set − A (19) [f(L− )] = [[t− ](1) [t− ](2) ] · 1− , A2 where A− 1,2 are unknown scalars. Parenthetically, although an SNTF of finite thickness can be expected to function as bulk dielectric waveguide [2,34], both SNTFs here are semiinfinitely thick and can only sustain guided-wave propagation if the fields decay as z → ±∞. This requirement has been satisfied by the representation chosen for [f (L± )].

III. NUMERICAL RESULTS AND DISCUSSION

A MATHEMATICATM program was written and implemented to solve (22) to obtain κ for specific values of γ + and γ − . The dispersion equation (22) was solved using the Newton– Raphson technique [35]. The free-space wavelength was fixed at λ0 = 633 nm for all results presented here. The relative permittivity of dielectric slab was set at εs = n2s = (1.8)2 . The SNTF was chosen to be made of titanium oxide, with [29] ⎫ εa (z) = [1.0443 + 2.7394v(z) − 1.3697v 2 (z)]2 ⎪ ⎪ ⎪ εb (z) = [1.6765 + 1.5649v(z) − 0.7825v 2 (z)]2 ⎬ , (23) εc (z) = [1.3586 + 2.1109v(z) − 1.0554v 2 (z)]2 ⎪ ⎪ ⎪ ⎭ χ (z) = tan−1 [2.8818 tan χv (z)]

013814-3



where v(z) = 2χv (z)/π . Following Agarwal et al. [27], we fixed δv = 16.2◦ , χ˜ v = 19.1◦ , and = 197 nm.

A. Dyakonov–Tamm waves guided by a single dielectric-SNTF interface

Before we discuss the solutions of the dispersion equation for guided-wave propagation by the SNTF-dielectric-SNTF system described in the previous section, let us discuss the solutions of the dispersion equation for a single dielectricSNTF interface. The waves localized to this interface are Dyakonov–Tamm waves [27]. Effectively, we set L+ = 0 and revised the formulation in the limit L− → ∞. The relative wave number κ/k0 , the relative phase speed v¯r = ns k0 /κ, the e-folding distance

= 1/Im[+ (k0 ns )2 − κ 2 ] into the homogeneous dielectric material perpendicular to the interface, and the decay constants + exp(u1,2 ) = exp(2 Im[α1,2 t]) are presented as functions of + γ in Fig. 2. Either none, one, or two Dyakonov–Tamm waves can be guided by this interface, depending on the angle γ + between the direction of propagation and the morphologically significant plane of the SNTF. This situation has been analyzed in detail by Agarwal et al. [27] for various values of εs , χ˜ v , and δv . However, for the specific case chosen, the shorter branch of the solutions given in Fig. 2 was missed in Ref. [27]. The e-folding distance varies between 0.8 and 2 . The decay constants exp(u1,2 ) represent the decay of the Dyakonov–Tamm waves after one structural period (i.e., 2 ) into the SNTF and perpendicular to the interface. If both of its decay constants are close to zero, a Dyakonov–Tamm wave is strongly localized. If either one or both of its decay constants are close to unity, a Dyakonov–Tamm wave is loosely bound to the interface on the SNTF side.

FIG. 2. (Color online) Relative wave number κ/k0 , relative phase speed v¯r , the e-folding distance , and the decay constants + ]) as functions of γ + for Dyakonov– exp(u1,2 ) = exp(2 Im[α1,2 Tamm waves guided by the single interface of the chosen dielectric material and the SNTF. The black symbols (square) identify the solutions also found by Agarwal et al. [27], but the solutions identified by the red symbols (circular) were missed in that work.

B. SNTF-dielectric-SNTF system

Returning to the boundary-value problem (with finite L± ) actually tackled for this paper, we solved the dispersion equation (22) for two different values of the twist between the morphologically significant planes on either side of the dielectric slab. We chose (i) γ − = γ + , and (ii) γ − = γ + + 90◦ , while γ + was kept as a variable. For each choice, the boundaries of the dielectric slab were taken to be L± = ± , ±1.5 , ±3 , or ±4 . These selections were made so as to have Ls to be less than and greater than twice the e-folding distance within the dielectric slab along the z axis, in order to study the effect of coupling of the two interfaces. 1. γ − = γ +

Let us begin with the case when the morphologically significant planes of the SNTF on both sides of the dielectric slab are aligned with each other and make an angle γ − = γ + with respect to the direction of wave propagation in the xy plane. By virtue of symmetry, the solutions for 180◦ + γ + and 360◦ − γ + are the same as for γ + . The solutions of the dispersion equation for various values of slab thickness—given in Fig. 3 as functions of γ + ∈ [0◦ ,90◦ ]—can be grouped into two categories,based on the value of κ. Within the dielectric slab, kz = + (k0 ns )2 − κ 2 is the wave number along the z axis. If κ/k0 > ns , kz /k0 must be a complex number, signifying that these solutions do not represent waveguide modes, but Dyakonov–Tamm waves localized to the interfaces between the dielectric slab and the SNTF. However, the solution of the dispersion equation with κ/k0 < ns represent waveguide modes as kz would be a real number. This categorization is supported by a comparative analysis of the solutions presented in Figs. 2 and 3 as follows. The solutions given in Fig. 3(a) for L± = ± may be organized in three branches with κ/k0 > ns and several branches with κ/k0 < ns . As the slab thickness Ls increases to 8 in Fig. 3(d), the three branches with κ/k0 > ns merge into two branches, but the number of branches for κ/k0 < ns goes on increasing. The decrease in the number of solutions representing Dyakonov–Tamm waves is due to the uncoupling of the two interfaces z = L− and z = L+ as the thickness L+ − L− increases. For L± = ± , the slab thickness Ls is smaller than twice the e-folding distance = 1/Im (kz ). Hence, due to the coupling of the two interfaces we have more solutions representing Dyakonov–Tamm waves. But Ls > 2 when L± = ±4 (because / ∈ [0.8,2]), and the two dielectricSNTF interfaces are uncoupled from each other so that we have exactly the same solutions with κ/k0 > ns , as given in Fig. 2 for a single dielectric-SNTF interface. The increase in the number of waveguide modes with the increase in the thickness Ls of the dielectric slab is in accordance with the general behavior of waveguiding structures [2,4]: the number of waveguide modes increases with the increase in the cross-sectional dimensions of a waveguide.

013814-4



The foregoing analysis is buttressed by the spatial profiles of the time-averaged Poynting vector P(x,z) =

1 2

Re[E(x,z) × H∗ (x,z)];

(24)

as κ is real, P(x,z) ≡ P(z). Representative plots of P(z) against z are presented in Figs. 4 and 5 for four values of κ at γ + = 10◦ and two values of κ at γ + = 80◦ . These spatial −1 and deterprofiles were calculated by setting A+ 1 = 1 Vm mining the remaining coefficients using Eq. (21). The shaded region in each plot represents the region occupied by the dielectric slab. Each of Figs. 4(a), 4(b), and 4(c) represents the spatial profile of the power density of a Dyakonov–Tamm wave, as the relative wave number κ/k0 > ns and the power density is strong at or close to the interfaces. In contrast, Figs. 4(d), 4(e), and 4(f) contain waveguide modes because κ/k0 < ns . The distinction between Dyakonov–Tamm waves and the waveguide modes is more evident in the spatial profiles presented in Fig. 5 for L± = ±4 . The spatial profiles given in Figs. 5(a) and 5(b) represent Dyakonov–Tamm waves, whereas Figs. 5(c), 5(d), 5(e), and 5(f) contain the spatial profiles of waveguide modes. Some of the waveguide modes do have maximums in their profiles at or near the two dielectric SNTF/interfaces, but some other waveguide modes contain most of the power near the central axis. 2. γ − = γ + + 90◦

The solutions of the dispersion equation for γ − = γ + + 90 —the morphologically significant planes of the SNTF on both sides are perpendicular to each other—are given in Fig. 6 as functions of γ + ∈ [0◦ ,90◦ ] for various values of slab thickness. Due to the symmetry of the problem, the solutions for 90◦ ± γ + , 180◦ ± γ + , 270◦ ± γ + , and 360◦ − γ + are the same as for γ + . The solutions for this case can also be divided into two categories based on the value of κ/k0 . The solutions ◦

0.004

(a)

(b)

0.003 0.002

P x,y,z

0.007 0.006 0.005 0.004 0.003 0.002 0.001 0.000

P x,y,z

P x,y,z

FIG. 3. (Color online) Variation of relative wave number κ/k0 with γ + , when γ − = γ + . (a) L± = ±1 , (b) L± = ±1.5 , (c) L± = ±3 , and (d) L± = ±4 . Solutions in the shaded regions represent Dyakonov–Tamm waves, but those in the unshaded regions represent waveguide modes, the boundary between the two regions being delineated for the chosen parameters by κ/k0 = ns .

0.001 0.000

4

2

0

2

4

6

4

2

z

4

6

0.4

0.10 0.05

0.2 0.0

0.00 2

0 z

4

2

2

4

4

2

0 z

0

2

4

6

0

2

4

6

z

P x,y,z

P x,y,z

0.6

4

6

(e)

0.15

0.8 P x, y,z

2

(c)

z

(d)

1.0

0

0.010 0.008 0.006 0.004 0.002 0.000

2

4

0.025 0.020 0.015 0.010 0.005 0.000

(f)

6

4

2 z

FIG. 4. (Color online) Variation of the Cartesian components of P(z) with z for γ − = γ + and L± = ± . The x-, y-, and z-directed components are represented by solid red, dashed blue, and chain-dashed black lines. The orange-shaded region represents the dielectric slab. γ + = 10◦ (a–d) and 80◦ (e–f). κ/k0 = (a) 1.856 08, (b) 1.831 54, (c) 1.822 69, (d) 1.594 88, (e) 1.770 07, and (f) 1.693 57. 013814-5

(a)


(b)

0.006 0.004

0.0

0.000 5

0

5

10

15

10

5

0

z

5

10

10

15

5

0

0.8

0.10 0.05

(e)

0.020

0.6 0.4 0.2

0.00

10

5

10

(f)

0.015 P x, y,z

(d)

5

z

z

P x,y,z

P x,y,z

0.15

1.0 0.5

0.002

10

(c)

1.5 P x,y,z

0.006 0.005 0.004 0.003 0.002 0.001 0.000

P x, y,z

P x, y,z


0.010 0.005 0.000

0.0 10

5

0

5

10

10

5

0 z

z

5

10

10

5

0 z

FIG. 5. (Color online) Same as Fig. 4 except for L± = ±4 and κ/k0 = (a) 1.844 89, (b) 1.819 41, (c) 1.604 96, (d) 1.511 1, (e) 1.793 81, and (f) 1.707 05.

with κ/k0 > ns represent Dyakonov–Tamm waves that strongly couple to either one or both of the dielectricSNTF interfaces, while the solutions with κ/k0 < ns represent waveguide modes. For the thinnest dielectric slab, there are three branches representing Dyakonov–Tamm waves that are coupled to both interfaces. As the slab thickness increases up to 8 , the coupling diminishes and the solutions

FIG. 6. (Color online) Same as Fig. 3 except for γ − = γ + + 90◦ .

representing Dyakonov–Tamm waves eventually become identical to the solutions of two independent dielectric-SNTF interfaces with the morphologically significant plane located either at γ + = 0◦ or at γ + = 90◦ . The Dyakonov–Tamm waves propagate independently guided by individual interfaces for Ls = 8 because the slab thickness is then much greater than the e-folding distance for all γ + . The number of solution branches representing waveguide modes increases as the thickness of the dielectric slab increases from 2 to 8 . Representative plots of P(z) vs. z for L± = ±1 are given in Fig. 7 for three values of κ each at γ + = 10◦ and 40◦ . The spatial profiles given in Figs. 7(a), 7(b), and 7(d) represent Dyakonov–Tamm waves while those in Figs. 7(c), 7(e), and 7(f) represent waveguide modes. Figure 8 contains representative spatial profiles of the time-averaged Poynting vector for L± = ±4 for three values of κ at γ + = 6◦ , two values of κ at γ + = 30◦ , and one value of κ at γ + = 20◦ . The spatial profiles given in Figs. 8(a) and 8(d) represent Dyakonov–Tamm waves localized to one interface while the profiles given in Figs. 8(b), 8(c), 8(e), and 8(f) represent waveguide modes. However, in Figs. 8(e) and 8(f) the decay rate of the Dyakonov–Tamm wave in the SNTF z > L+ is very low. The asymmetry in the power profiles in Figs. 7 and 8 is due to the asymmetry in the arrangement of the morphologically significant planes of the SNTF on either side of the dielectric slab. Before continuing, let us note that Dyakonov–Tamm waves alone would propagate if the dielectric slab were to be absent (i.e., Ls = 0), as has been shown elsewhere [36]. Analysis of Figs. 2, 3, and 6 reveals a major advantage of using the SNTF-dielectric-SNTF system for the propagation of Dyakonov–Tamm waves as compared to the single SNTF-dielectric interface: the angular range of propagation and the possible number of Dyakonov–Tamm waves can be increased if the thickness of the dielectric slab is sufficiently small. Guided by a single SNTF-dielectric interface, two Dyakonov–Tamm waves can propagate for γ + ∈ [0,10◦ ], one for γ + ∈ (10,66◦ ], and none for γ + ∈ (66,90◦ ]. However, the

013814-6

(a)


(b)

0.008 0.006

P x,y,z

0.005 0.004 0.003 0.002 0.001 0.000

P x, y,z

P x,y,z


0.004 0.002 0.000

4

2

0

2

4

4

2

0

z 0.08

8

15

(e)

0.02

0.000

0.00 2

5

0

2

4

6

0

5

(f)

0.06

0.04

0.005

4

10 z

P x,y,z

P x,y,z

P x, y,z

6

0.06

0.010

6

4

(c)

z

(d)

0.015

2

12 10 8 6 4 2 0

0.04 0.02 0.00

6

4

2

z

0

2

4

6

6

4

2

z

0

2

4

6

z

FIG. 7. (Color online) Same as Fig. 4 except for γ − = γ + + 90◦ . γ + = 10◦ (a–c), and 40◦ (d–f). κ/k0 = (a) 1.838 52, (b) 1.817 81, (c) 1.580 16, (d) 1.901 88, (e) 1.724 64, and (f) 1.557 86.

presented data indicate that an SNTF-dielectric-SNTF system with a dielectric slab of thickness 2 or less can support the propagation of

as in the predecessor papers [37,38]. The waves propagating in the SNTF-metal-SNTF system are classified as SPP waves which may be coupled to either one or both of the metal-SNTF interfaces, depending on the thickness of metal slab. Some attributes of wave propagation in the SNTF-metalSNTF system are similar to those of guided-wave propagation in the SNTF-dielectric-SNTF system, but others are not. The following differences were noted: (i) The wave number κ is a complex number for the SNTFmetal-SNTF system so that P(x,z) = P(z) exp [−2Im(κ)x], but real for the SNTF-dielectric-SNTF system so that P(x,z) = P(z). Consequently, whereas attenuation must occur in the direction of propagation in the SNTF-metal-SNTF system, the propagation is lossless in the SNTF-dielectric-SNTF system. (ii) The dielectric core can support the propagation of waveguide modes whereas the metallic core cannot. Therefore, both surface waves (Dyakonov–Tamm waves) and waveguide

(i) three Dyakonov–Tamm waves for γ + ∈ [0,10◦ ], two for γ + ∈ (10,48◦ ], and one for γ + ∈ (48,90◦ ], when γ − = γ +, (ii) two Dyakonov–Tamm waves for γ + ∈ [0,10◦ ] ∪ [22◦ ,68◦ ] ∪ [80◦ ,90◦ ] and one for γ + ∈ (10◦ ,22◦ ) ∪ (68◦ ,80◦ ), when γ − = γ + + 90◦ . C. Comparison with SNTF-metal-SNTF system

Wave propagation guided by an aluminum slab inserted in an SNTF was studied in detail in Refs. [37] and [38] for = 200 nm, χ˜ v = 45◦ , and δv = 30◦ . However, we performed similar calculations with = 197 nm, χ˜ v = 19.1◦ , and δv = 16.2◦ , for direct comparison with the data presented in Figs. 3–8, and the results showed same characteristics

P x,y,z

P x,y,z

0.006 0.004 0.002 0.000 10

5

0

5

0.4

(b)

0.05 0.04 0.03 0.02 0.01 0.00

0.0 5

0

(d)

0.001 0.000 0 z

0

5

10

5

10

(f)

0.04

0.010 0.005

0.03 0.02 0.01 0.00

0.000 5

5

0.05

(e) P x,y,z

P x,y,z

0.002

10

10

10

z

0.015

0.003 P x,y,z

5

z

z 0.004

0.2 0.1

10

10

(c)

0.3 P x, y,z

(a)

0.008

10

5

0

5 z

10

15

20

10

5

0

5

10

15

20

z

FIG. 8. (Color online) Same as Fig. 4 except for γ − = γ + + 90◦ , and L± = ±4 . γ + = 6◦ (a–c), 30◦ (d, e), and 20◦ (f). κ/k0 = (a) 1.839 87, (b) 1.696 41, (c) 1.666 92, (d) 1.872 42, (e) 1.665 33, and (f) 1.787 69. 013814-7



modes can propagate in the SNTF-dielectric-SNTF system, but only surface waves (SPP waves) can propagate in the SNTF-metal-SNTF system. (iii) Coupling between the two interfaces exists only for very < 60 nm), but for much thicker dielectric thin metal slabs (Ls ∼ < 1000 nm). This is because the imaginary parts slabs (Ls ∼ of the relative permittivities of metals are very high in the optical regime, but those of commonplace dielectric materials are vanishingly small. (iv) When both the morphologically significant planes of the SNTF on either side of the dielectric slab are aligned parallel to each other, the allowable range of the direction of propagation for the Dyakonov–Tamm waves is restricted, as can be seen from Fig. 3. However, SPP waves can propagate in any direction in the interface plane ( [38], Fig. 2), when the dielectric slab is replaced by a metal slab. If we compare only surface-wave propagation in the two systems, three similarities must be noted: (i) The wave number κ for Dyakonov–Tamm waves guided by the dielectric slab and Re (κ) for SPP waves guided by the metallic slab are greater than k0 , so both SPP and Dyakonov– Tamm waves have smaller phase speeds than the speed of light in free space. (ii) The surface waves are strongly coupled to both interfaces when the slab thickness is less than twice the e-folding distance into the slab material. As the slab gets thicker, the coupling decreases. (iii) The coupling of two interfaces increases the number of possible surface waves that can be guided by the dielectric or metallic slab, although the effect is more pronounced with a metallic slab. IV. CONCLUDING REMARKS

The canonical boundary-value problem of a dielectric slab inserted in a sculptured nematic thin film was formulated and solved numerically to analyze guided-wave propagation in the SNTF-dielectric-SNTF system. The problem was solved for two cases: (i) when the morphologically significant planes of the SNTF on either sides are aligned with each other and (ii) when they are perpendicular to each other. For both cases, multiple surface (Dyakonov–Tamm) waves and waveguide

[1] D. Hondros and P. Debye, Ann. Phys. (Leipzig) 32, 465 (1910). [2] N. S. Kapany and J. J. Burke, Optical Waveguides (Academic Press, New York, 1972). [3] Handbook of Optical Fibre Sensing Technology, edited by J. M. L´opez-Higuera (Wiley, Chicester, UK, 2002). [4] D. Marcuse, Theory of Dielectric Optical Waveguides (Academic Press, San Diego, 1991). [5] National Research Council, Condensed-Matter and Material Physics (National Academy Press, Washington, DC, 1999). [6] G. Cao, Nanostructures and Nanomaterials: Synthesis, Properties and Applications (Imperial College Press, London, 2004). [7] R. J. Mart´ın-Palma and A. Lakhtakia, Nanotechnology: A Crash Course (SPIE Press, Bellingham, WA, 2010), Chap. 6.

modes—with different phase speeds and spatial profiles— propagate. As the thickness of the dielectric slab increases, the number of waveguide modes increases. However, with the increase of the slab thickness, the two dielectric-SNTF interfaces begin to uncouple and ultimately, each interface guides multiple Dyakonov–Tamm waves all by itself. If the dielectric slab is replaced by a metal slab, coupling of the two metal-SNTF interfaces will also occur when the slab is very thin. Multiple surface (SPP) waves guided by one or both of the metal-SNTF interfaces can propagate, but the SNTFmetal-SNTF does not exhibit waveguide modes. One slight disadvantage of the SNTF-dielectric-SNTF system is the shorter angular sector—when both the interfaces are uncoupled—for the propagation directions of Dyakonov–Tamm waves when compared to SPP waves. However, propagation guided by the SNTF-dielectric-SNTF system is lossless (or almost lossless with actual materials), but not by the SNTF-metal-SNTF system. Let us also note that the replacement of the SNTF in an SNTF-dielectric-SNTF system by a homogeneous, isotropic dielectric material will eliminate Dyakonov–Tamm waves, and only waveguide modes will propagate in the dielectric slab [2]. The capability for lossless propagation of surface waves in any direction could be useful for sensing and communication applications that are currently restricted by the attenuative propagation of SPP waves [10–12,15,39]. Multiple Dyakonov–Tamm waves can lead to enhanced confidence in sensing measurements and may increase the capabilities of multianalyte sensors. Since an SNTF is a porous material, an infiltrating fluid will change the dielectric properties of the SNTF, thereby, changing the wave numbers and spatial profiles of both the Dyakonov–Tamm waves and waveguide modes. So the coexistence of the Dyakonov–Tamm waves and the waveguide modes could lead to enhanced sensing modalities. The presence of multiple Dyakonov–Tamm waves and waveguide modes may also be useful for multichannel communication at a specific frequency. ACKNOWLEDGMENTS

Both authors are grateful to Charles Godfrey Binder Endowment at the Pennsylvania State University for partial support of this work.

[8] J. I. Mackenzie, IEEE J. Sel. Top. Quantum Electron. 13, 626 (2007). [9] S. A. Maier, Plasmonics: Fundamentals and Applications (Springer, New York, 2007). [10] M. Dragoman and D. Dragoman, Prog. Quantum Electron. 32, 1 (2008). [11] Surface Plasmon Resonance Based Sensors, edited by J. Homola (Springer, Heidelberg, 2006). [12] I. Abdulhalim, M. Zourob, and A. Lakhtakia, Electromagnetics 28, 214 (2008). [13] M. Takabayashi, M. Haraguchi, and M. Fukui, J. Mod. Opt. 44, 119 (1997). [14] A. V. Zayats, I. I. Smolyaninov, and A. A. Maradudin, Phys. Rep. 408, 131 (2005).

013814-8



[15] P. Berini, Adv. Opt. Photon. 1, 484 (2009). [16] F. N. Marchevski˘ı, V. L. Strizhevski˘ı, and S. V. Strizhevski˘ı, Sov. Phys. Solid State 26, 911 (1984) [Fiz. Tverd. Tela 26, 1501 (1984)]. [17] M. I. D’yakonov, Sov. Phys. JETP 67, 714 (1988) [Zh. Eksp. Teor. Fiz. 94, 119 (1988)]. [18] O. Takayama, L. Crasovan, D. Artigas, and L. Torner, Phys. Rev. Lett. 102, 043903 (2009). [19] D. B. Walker, E. N. Glytsis, and T. K. Gaylord, J. Opt. Soc. Am. A 15, 248 (1998). [20] V. M. Galynsky, A. N. Furs, and L. M. Barkovsky, J. Phys. A: Math. Gen. 37, 5083 (2004). [21] J. A. Polo Jr. and A. Lakhtakia, Laser Photon. Rev., doi:10.1002/lpor.200900050. [22] L.-C. Crasovan, D. Artigas, D. Mihalache, and L. Torner, Opt. Lett. 30, 3075 (2005) [23] O. Takayama, L.-C. Crasovan, S. K. Johansen, D. Mihalache, D. Artigas, and L. Torner, Electromagnetics 28, 126 (2008). [24] S. R. Nelatury, J. A. Polo Jr., and A. Lakhtakia, Microwave Opt. Technol. Lett. 50, 2360 (2008). [25] A. Lakhtakia and J. A. Polo Jr., J. Eur. Opt. Soc. Rapid Publ. 2, 07021 (2007). [26] H. Ohno, E. E. Mendez, J. A. Brum, J. M. Hong, F. Agull´o– Rueda, L. L. Chang, and L. Esaki, Phys. Rev. Lett. 64, 2555 (1990).

[27] K. Agarwal, J. A. Polo Jr., and A. Lakhtakia, J. Opt. A: Pure Appl. Opt. 11, 074003 (2009). [28] J. Gao, A. Lakhtakia, and M. Lei, Phys. Rev. A 81, 013801 (2010). [29] M. A. Motyka and A. Lakhtakia, J. Nanophoton. 2, 021910 (2008). [30] R. Messier and A. Lakhtakia, Mater. Res. Innovations 2, 217 (1999). [31] A. Lakhtakia and R. Messier, Sculptured Thin Films: Nanoengineered Morphology and Optics (SPIE Press, Bellingham, WA, 2005). [32] H. C. Chen, Theory of Electromagnetic Waves: A Coordinatefree Approach (McGraw-Hill, New York, 1983). [33] T. G. Mackay and A. Lakhtakia, Electromagnetic Anisotropy and Bianisotropy (World Scientific, Singapore, 2010). [34] E. Ertekin and A. Lakhtakia, Proc. R. Soc. London A 457, 817 (2001). [35] Y. Jaluria, Computer Methods for Engineering (Taylor & Francis, Washington, DC, 1996). [36] M. Faryad and A. Lakhtakia, Opt. Commun. 284, 160 (2011). [37] M. Faryad and A. Lakhtakia, Phys. Stat. Sol. RRL 4, 265 (2010). [38] M. Faryad and A. Lakhtakia, J. Opt. 12, 085102 (2010). [39] S. A. Maier, M. L. Brongersma, P. G. Kik, S. Meltzer, A. A. G. Requicha, and H. A. Atwater, Adv. Mater. 13, 1501 (2001).

013814-9