Theory of Dyakonov-Tamm waves at the planar interface of a sculptured nematic thin film and an isotropic dielectric material
aa r X i v : . [ phy s i c s . op ti c s ] F e b Theory of Dyakonov–Tamm waves at the planarinterface of a sculptured nematic thin film and anisotropic dielectric material
Kartiek Agarwal, , John A. Polo, Jr., and AkhleshLakhtakia , ‡ Department of Electrical Engineering, Indian Institute of Technology Kanpur,Kanpur 208016, India NanoMM—Nanoenginered Metamaterials Group, Department of EngineeringScience and Mechanics, Pennsylvania State University, University Park, PA 16802,USA Department of Physics and Technology, Edinboro University of Pennsylvania,Edinboro, PA 16444, USA Department of Physics, Indian Institute of Technology Kanpur, Kanpur 208016,India
Abstract.
In order to ascertain conditions for surface–wave propagation guided bythe planar interface of an isotropic dielectric material and a sculptured nematic thinfilm (SNTF) with periodic nonhomogeneity, we formulated a boundary–value problem,obtained a dispersion equation therefrom, and numerically solved it. The surface wavesobtained are Dyakonov–Tamm waves. The angular domain formed by the directionsof propagation of the Dyakonov–Tamm waves can be very wide (even as wide as toallow propagation in every direction in the interface plane), because of the periodicnonhomogeneity of the SNTF. A search for Dyakonov–Tamm waves is, at the presenttime, the most promising route to take for experimental verification of surface–wavepropagation guided by the interface of two dielectric materials, at least one of which isanisotropic. That would also assist in realizing the potential of such surface waves foroptical sensing of various types of analytes infiltrating one or both of the two dielectricmaterials.
Keywords:
Dyakonov wave, optical sensing, sculptured nematic thin film, surfacewave, Tamm state, titanium oxide ‡ Corresponding author; e-mail:[email protected] yakonov–Tamm waves at nematic thin film/isotropic dielectric interface
1. Introduction
In 1988, Dyakonov [1] theoretically predicted the propagation of electromagnetic wavesguided by the planar interface of two homogeneous dielectric materials, one of whichis isotropic and the other uniaxial with its optic axis aligned parallel to the interface.Since then, the existence of Dyakonov surface waves has been theoretically proved formany sets of dissimilar dielectric partnering materials, at least one of which is anisotropic[2, 3]. The possibility of the anisotropic partnering material being artificially engineered,either as a photonic crystal with a short period in comparison to the wavelength[4] or as a columnar thin film (CTF) [5], has also emerged. Just like that of manysurface phenomena [6], the significance of Dyakonov surface waves for optical sensingapplications is obvious: the disturbance of the constitutive properties of one or both ofthe two partnering materials—due to, say, infiltration by any analyte—could measurablychange the characteristics of the chosen Dyakonov surface wave.However, the directions of propagation of Dyakonov surface waves parallel to anysuitable planar interface are confined to a very narrow angular domain, typically of theorder of a degree or less [2, 7]. Not surprisingly, Dyakonov surface waves still remainto be experimentally observed. Clearly then, the potential of Dyakonov surface wavescannot be realized if they cannot be demonstrably excited; a wider angular domain isneeded.Recently, Lakhtakia and Polo [8] examined surface–wave propagation guided bythe planar interface of an isotropic dielectric material and a chiral sculptured thin film[9] with its direction of periodic nonhomogeneity normal to the planar interface. Theauthors named the surfaces waves after Dyakonov and Tamm, the latter being theperson who indicated the possibility of finding surface states of electrons at exposedplanes of crystals and other periodic materials [10, 11]. The angular domain formedby the directions of propagation of the Dyakonov–Tamm waves turned out to be verywide, as much as 98 ◦ in one case for which realistic constitutive parameters of the twopartnering materials were employed [8]. This implies that Dyakonov–Tamm waves couldbe detected much more easily than Dyakonov surface waves.Chiral sculptured thin films (STFs) are structurally chiral materials: § the § An object is said to be chiral if it cannot be made to coincide with its mirror-image by translationsand/or rotations. There are two types of chiral materials: (i) microscopically or molecularly chiralmaterials, and (ii) structurally chiral materials. The first type of chiral materials either have chiralmolecules or are composite materials made by embedding, e.g., electrically small helixes in a hostmaterial. Materials comprising chiral molecules have been known for about two hundred years, as aperusal of an anthology of milestone papers [12] will show to the interested reader. These materialsare generally (optically) isotropic [13]. Composite materials comprising electrically small [14, 15] chiralinclusions were first reported in 1898 [16], and these materials can be either isotropic [17, 18, 19]or anisotropic [20]. The first type of chiral materials can be considered as either homogeneous ornonhomogeneous continuums at sufficiently low frequencies. In contrast, the second type of chiralmaterials can only be nonhomogeneous and anisotropic continuums at the length–scales of interest,their constitutive parameters varying periodically in a chiral manner about a fixed axis. Take awaythe nonhomogeneity of a structurally chiral material, and its (macroscopic) chirality will also vanish yakonov–Tamm waves at nematic thin film/isotropic dielectric interface Figure 1.
Schematic of a collimated vapor flux responsible for the growth of tiltedstraight nanowires growing at an angle χ ≥ χ v . permittivity dyadic rotates at a uniform rate along the direction of nonhomogeneity.As a result, chiral STFs are periodically nonhomogeneous. Thus, in a chiral STF,structural chirality and periodic nonhomogeneity are inseparable. Is the huge angularexistence domain of Dyakonov–Tamm waves for the case investigated by Lakhtakia andPolo [8] due to the periodicity or due to the structural chirality of the chiral STF? Inorder to answer this question, we devised a surface–wave–propagation problem whereinthe chiral STF is replaced by a periodically nonhomogeneous and anisotropic materialthat is not structurally chiral. Specifically, we replaced the chiral STF with a sculpturednematic thin film (SNTF) [9].CTFs, chiral STFs, and SNTFs are all fabricated by physical vapor deposition[9, 22, 23]. In the simplest implementation of this technique, a boat containing a certainmaterial, say titanium oxide, is placed in an evacuated chamber. Under appropriateconditions, the material evaporates towards a substrate such that the vapor flux ishighly collimated. The collimated vapor flux coalesces on the substrate as an ensembleof more or less identical and parallel nanowires, due to self–shadowing. If the substrateis held stationary during deposition, a CTF grows wherein the nanowires are straightand aligned at an angle χ ≥ χ v with respect to the substrate plane, where χ v is the anglebetween the collimated vapor flux and the substrate plane, as shown in Fig. 1. If thesubstrate is rotated about an axis passing normally through it, a chiral STF comprisinghelical nanowires grows. Finally, if the substrate is rocked about an axis tangential tothe substrate plane, χ v and χ are not constant and an SNTF grows. Scanning-electron-microscope images of a CTF, an SNTF, and a chiral STF are presented in Fig. 2.This paper is organized as follows. Section 2 presents the boundary–value problemand the dispersion equation for the propagation of Dyakonov–Tamm waves guidedby the interface of a homogenous, isotropic dielectric material and a periodicallynonhomogeneous SNTF. Section 3 contains numerical results when the SNTF is chosento be made of titanium oxide [5, 24]. An exp( − iωt ) time–dependence is implicit, with ω [9, 21]. yakonov–Tamm waves at nematic thin film/isotropic dielectric interface Figure 2.
Scanning-electron-microscope images of sculptured thin films. Top left:columnar thin film; top right: sculptured nematic thin film; bottom: chiral sculpturedthin film. Courtesy: Russell Messier and Mark Horn. denoting the angular frequency. The free–space wavenumber, the free–space wavelength,and the intrinsic impedance of free space are denoted by k o = ω √ ǫ o µ o , λ o = 2 π/k o , and η o = p µ o /ǫ o , respectively, with µ o and ǫ o being the permeability and permittivity of freespace. Vectors are underlined, dyadics underlined twice; column vectors are underlinedand enclosed within square brackets, while matrixes are underlined twice and similarlybracketed. Cartesian unit vectors are identified as u x , u y and u z . The dyadics employedin the following sections can be treated as 3 ×
2. Formulation
Let the half–space z ≤ n s . The region z ≥ ǫ ( z ) = ǫ o S z ( γ ) · S y ( z ) · ǫ ◦ ref ( z ) · S Ty ( z ) · S Tz ( γ ) , z ≥ . (1)The dyadic function S z ( γ ) = (cid:0) u x u x + u y u y (cid:1) cos γ + (cid:0) u y u x − u x u y (cid:1) sin γ + u z u z (2)contains γ as an angular offset, and the superscript T denotes the transpose. The dyadics S y ( z ) = ( u x u x + u z u z ) cos [ χ ( z )] + ( u z u x − u x u z ) sin [ χ ( z )] + u y u y (3)and ǫ ◦ ref ( z ) = ǫ a ( z ) u z u z + ǫ b ( z ) u x u x + ǫ c ( z ) u y u y (4) yakonov–Tamm waves at nematic thin film/isotropic dielectric interface χ v ( z ) = ˜ χ v + δ v sin (cid:16) πz Ω (cid:17) (5)that varies sinusoidally as a function of z with period 2Ω. Whereas ˜ χ v ≥
0, we have toensure that χ v ( z ) ∈ (0 , π/ ǫ a,b,c ( z ) and χ ( z ) to χ v ( z ) have been reportedfor SNTFs ( δ v = 0), we decided to use available data for CTFs ( δ v = 0). Opticalcharacterization experiments on CTFs of titanium oxide at λ o = 633 nm [24] lead tothe following expressions for our present purpose: ǫ a ( z ) = [1 . . v ( z ) − . v ( z )] ǫ b ( z ) = [1 . . v ( z ) − . v ( z )] ǫ c ( z ) = [1 . . v ( z ) − . v ( z )] χ ( z ) = tan − [2 . χ v ( z )] v ( z ) = 2 χ v ( z ) /π . (6)Let us note that the foregoing expressions—with v ( z ) independent of z —are applicableto CTFs produced by one particular experimental apparatus, but may have to bemodified for CTFs produced by other researchers on different apparatuses.Without loss of generality, we take the Dyakonov–Tamm wave to propagate parallelto the x axis in the plane z = 0. There is no dependence on the y coordinate, whereasthe Dyakonov–Tamm wave must attenuate as z → ±∞ . In the region z ≤
0, the wave vector may be written as k s = κ u x − α s u z , (7)where κ + α s = k o n s , (8) κ is positive and real–valued for unattenuated propagation along the x axis, andIm [ α s ] > z → −∞ . Accordingly, the field phasors in the region z ≤ E ( r ) = (cid:20) A s u y + A p (cid:18) α s k o u x + κk o u z (cid:19)(cid:21) exp( ik s · r ) , z ≤ , (9)and H ( r ) = η − o (cid:20) A s (cid:18) α s k o u x + κk o u z (cid:19) − A p n s u y (cid:21) exp( ik s · r ) , z ≤ , (10)where A s and A p are unknown scalars representing the amplitudes of s – and p –polarizedcomponents. yakonov–Tamm waves at nematic thin film/isotropic dielectric interface z ≥ E ( r ) = e ( z ) exp( iκx ) H ( r ) = h ( z ) exp( iκx ) ) , z ≥ , (11)and create the column vector (cid:2) f ( z ) (cid:3) = [ e x ( z ) e y ( z ) h x ( z ) h y ( z )] T . (12)This column vector satisfies the matrix differential equation [9] ddz (cid:2) f ( z ) (cid:3) = i (cid:2) P ( z, κ ) (cid:3) · (cid:2) f ( z ) (cid:3) , z > , (13)where the 4 × P ( z, κ )] = ω µ o − µ o ǫ o [ ǫ c ( z ) − ǫ d ( z )] cos γ sin γ − ǫ o (cid:2) ǫ c ( z ) cos γ + ǫ d ( z ) sin γ (cid:3) ǫ o (cid:2) ǫ c ( z ) sin γ + ǫ d ( z ) cos γ (cid:3) − ǫ o [ ǫ c ( z ) − ǫ d ( z )] cos γ sin γ + κ ǫ d ( z ) [ ǫ a ( z ) − ǫ b ( z )] ǫ a ( z ) ǫ b ( z ) sin χ ( z ) cos χ ( z ) cos γ sin γ − sin γ γ + − κ ωǫ o ǫ d ( z ) ǫ a ( z ) ǫ b ( z ) κ ωµ o (14)and ǫ d ( z ) = ǫ a ( z ) ǫ b ( z ) ǫ a ( z ) cos χ ( z ) + ǫ b ( z ) sin χ ( z ) . (15)For sin γ = 0, (13) splits into two autonomous matrix differential equations, eachemploying a 2 × N ] which appears in the relation[ f (2Ω)] = [ N ] · [ f (0+)] (16)to characterize the optical response of one period of the chosen SNTF. By virtue of theFloquet–Lyapunov theorem [27], we can define a matrix [ Q ] such that[ N ] = exp n i Q ] o . (17) yakonov–Tamm waves at nematic thin film/isotropic dielectric interface N ] and [ Q ] share the same eigenvectors, and their eigenvalues are also related.Let [ t ] ( n ) , ( n = 1 , , , n th eigenvalue σ n of[ N ]; then, the corresponding eigenvalue α n of [ Q ] is given by α n = − i ln σ n . (18) For the Dyakonov–Tamm wave to propagate along the x axis, we must ensure thatIm[ α , ] >
0, and set[ f (0+)] = (cid:2) [ t ] (1) [ t ] (2) (cid:3) · " B B , (19)where B and B are unknown scalars; the other two eigenvalues of [ Q ] describe wavesthat amplify as z → ∞ and cannot therefore contribute to the Dyakonov–Tamm wave.At the same time,[ f (0 − )] = α s k o α s k o η − o − n s η − o · " A s A p , (20)by virtue of (9) and (10). Continuity of the tangential components of the electric andmagnetic field phasors across the plane z = 0 requires that[ f (0 − )] = [ f (0+)] , (21)which may be rearranged as[ M ] · A s A p B B = . (22)For a nontrivial solution, the 4 × M ] must be singular, so thatdet [ M ] = 0 (23)is the dispersion equation for the Dyakonov–Tamm wave. The value of κ satisfying (23)was obtained by employing the Newton–Raphson method [28].
3. Numerical Results and Discussion
We set λ o = 633 nm in accordance with (6), since calculations were performed onlyfor CTFs or SNTFs composed of titanium oxide. All calculations for the periodicallynonhomogeneous SNTFs were performed for ˜ χ v = 19 . ◦ and Ω = 197 nm withtwo oscillation amplitudes: δ v = 7 . ◦ and 16 . ◦ . For comparison, calculations were yakonov–Tamm waves at nematic thin film/isotropic dielectric interface χ v ( z ) ≡ . ◦ and 19 . ◦ ∀ z ≥
0. The former is the approximatelower limit of χ v obtainable with current STF technology. For a chosen set of values of n s , ˜ χ v , and δ v , surface–wave propagation was found to occur in four separate γ -ranges: γ ∈ [ ± γ m − ∆ γ/ , ± γ m + ∆ γ/
2] and γ ∈ [ ± γ m + 180 ◦ − ∆ γ/ , ± γ m + 180 ◦ + ∆ γ/ γ m ∈ (0 ◦ , ◦ ) is used to describe the mid–point of a γ -range while ∆ γ ≤ ◦ is the extent of that range. Since the surface–wave characteristics are the same in allfour γ -ranges, results are displayed only for γ ∈ [ γ m − ∆ γ/ , γ m + ∆ γ/ v = v DT n s √ ǫ o µ o , (24)where v DT = ω/κ is the phase speed of the Dyakonov–Tamm wave.Let us begin the presentation and discussion of results with the characteristics ofDyakonov waves guided by the interface of a CTF ( δ v = 0 ◦ ) and an isotropic dielectricmaterial [5]. Figure 3 shows plots of v versus γ with: χ v = 7 . ◦ for n s = 1 .
57, 1.59,1.61, 1.63, 1.65, 1.67, 1.69, 1.71, and 1.73; and χ v = 19 . ◦ for n s = 1 .
80, 1.82, and 1.84.Three characteristics of Dyakonov waves can be garnered from this figure:(i) the nearly vertical curves demonstrate the extremely small width ∆ γ of the γ -range, leading to very narrow angular existence domains as we remarked upon inSection 1;(ii) higher values of n s result in higher values of γ m ; and(iii) lower values of χ v also result in higher values of γ m .Figure 4 shows v as a function of γ when ˜ χ v = 19 . ◦ for three groups of parameters: • δ v = 0 ◦ for n s = 1 .
80, 1.82, and 1.84; • δ v = 7 . ◦ for n s = 1 .
73, 1.77, 1.82, 1.88, 1.92, and 1.96; and • δ v = 16 . ◦ for n s = 1 .
80 and 1 . dramatic increase in ∆ γ brought about by the introductionof sinusoidal oscillation in the vapor incidence angle. With ∆ γ on the order of tens ofdegrees when an SNTF is the anisotropic partnering material, the width of the γ -range isorders of magnitude greater than that obtained with a CTF as the anisotropic partneringmaterial. As n s increases, the γ -range supporting surface–wave propagation widens andthe mid-point γ m shifts to higher values. When δ v = 16 . ◦ and n s is increased to 1.84,∆ γ actually increases to 90 ◦ , which means that the four separate γ -ranges merge intoone, thereby allowing surface–wave propagation to occur for any value of γ .Figure 4 also indicates that the average value of v over the γ -range rises as n s increases, for both δ v = 7 . ◦ and 16 . ◦ . For a given value of γ , v takes on similar valuesfor CTFs and SNTFs. yakonov–Tamm waves at nematic thin film/isotropic dielectric interface Figure 3.
Relative phase speed v as a function of γ when δ v = 0 ◦ . The values of ˜ χ v are as follows: (1)-(9) 7 . ◦ , (10)-(12) 19 . ◦ . The values of n s are as follows: (1) 1.57,(2) 1.59, (3) 1.61, (4) 1.63, (5) 1.65, (6) 1.67, (7) 1.69, (8) 1.71, (9) 1.73, (10) 1.80,(11) 1.82, (12) 1.84. Figure 4.
Relative phase speed v as a function of γ when ˜ χ v = 19 . ◦ . The values of δ v are as follows: (1)-(3) 0 ◦ , (4)-(9) 7 . ◦ , (10) and (11) 16 . ◦ . The values of n s are asfollows: (1) 1.80, (2) 1.82, (3) 1.84, (4) 1.73, (5) 1.77, (6) 1.82 (7) 1.88, (8) 1.92, (9)1.96, (10) 1.80, (11) 1.84. yakonov–Tamm waves at nematic thin film/isotropic dielectric interface Figure 5.
Normalized decay constants Im[ α ] /k o and Im[ α ] /k o as functions of γ when δ v = 0 ◦ . The values of ˜ χ v are as follows: (1)-(9) 7 . ◦ , (10)-(12) 19 . ◦ . Thevalues of n s are as follows: (1) 1.57, (2) 1.59, (3) 1.61, (4) 1.63, (5) 1.65, (6) 1.67, (7)1.69, (8) 1.71, (9) 1.73, (10) 1.80, (11) 1.82, (12) 1.84. The localization of the surface wave about the bimaterial interface is described bythe decay constants Im[ α ] and Im[ α ] in the anisotropic partnering material and byIm[ α s ] in the isotropic partnering material. Figure 5 displays the two decay constantsin the CTF normalized to the free-space wavenumber, Im[ α ] /k o and Im[ α ] /k o , asfunctions of γ for χ v = 7 . ◦ and 19 . ◦ for the same values of n s as in Figure 3. Thevalues of Im[ α ] /k o show considerable variation over the γ -range for a given set of χ v and n s even though the γ -range is very narrow, as shown by the nearly vertical curvesin Figure 5a. Unlike Im[ α ] /k o , Im[ α ] /k o shows little variation over the γ -range and,for most values of n s , appears as a single point in Figure 5b. Furthermore, Im[ α ] /k o isroughly a linear function of γ with a positive slope, for a given value of χ v . The slopeincreases as χ v increases.Figures 6 and 7) show the decay constants in the anisotropic partnering materialas functions of γ for for ˜ χ v = 19 . ◦ , the remaining parameters being the same as inFigure 4. The values of Im[ α ] /k o in Figure 6a are larger in the SNTFs than they are inthe CTFs for the same values of n s and γ . The values of Im[ α ] /k o appear to decreasetowards zero at the upper range of γ for each value of n s , except for those cases wherethe range of γ extends to 90 ◦ . In the two cases where the γ -range extends to 90 ◦ ,Im[ α ] /k o shows a minimum at 90 ◦ . It should be kept in mind that neither 0 ◦ nor 90 ◦ truly represent the limits of a γ -range for surface–wave propagation, since curves whichseem to end at these points are joined to curves of one of the other three, in general,separate γ -ranges, thereby resulting in only two distinct γ -ranges.Values of Im[ α ] /k o in the SNTF when δ v = 7 . ◦ are nearly identical to those inthe CTF at a given value of γ , as shown in Figure 6b, for ˜ χ v = 19 . ◦ . The curvesof Im[ α ] /k o versus γ for the various values of n s nearly join to form a single smoothcontinuous curve, except near γ = 0 ◦ where the curves for n s = 1 .
73, 1.77 and 1.82bifurcate. The value of Im[ α ] /k o increases as n s and γ increase. yakonov–Tamm waves at nematic thin film/isotropic dielectric interface Figure 6.
Normalized decay constants Im[ α ] /k o and Im[ α ] /k o as functions of γ when ˜ χ v = 19 . ◦ . The values of δ v are as follows: (1)-(3) 0 ◦ , (5)-(9) 7 . ◦ , (10) and(11) 16 . ◦ . The values of n s are as follows: (1) 1.80, (2) 1.82, (3) 1.84, (4) 1.73, (5)1.77, (6) 1.82, (7) 1.88, (8) 1.92, (9) 1.96, (10) 1.80, (11) 1.84. Figure 7.
Normalized decay constant Im( α ) /k o as a function of γ when ˜ χ v = 19 . ◦ and δ v = 16 . ◦ . The values of n s are (1) 1.80, (2) 1.84. The behaviour of Im[ α ] /k o for higher modulation amplitude, δ v = 16 . ◦ , ispresented in Figure 7 for n s = 1 .
80 and 1.84. At low values of γ , Im[ α ] /k o is largerfor δ v = 16 . ◦ than for δ v = 7 . ◦ . As γ approaches 90 ◦ , however, the value of Im[ α ] /k o for δ v = 16 . ◦ , for both n s = 1 .
80 and 1 .
84, approaches a value only slightly less thatobserved when δ v = 7 . ◦ and n s = 1 . δ v = 0 ◦ ). Figure 8 shows the ratio of the s -polarization amplitude to thethe p -polarization amplitude for n s = 1 .
57, 1.59, 1.61, 1.63, 1.65, 1.67, 1.69, 1.71, and yakonov–Tamm waves at nematic thin film/isotropic dielectric interface Figure 8.
Ratio | A s | / | A p | as a function of γ when δ v = 0 ◦ . The values of ˜ χ v areas follows: (1)-(9) 7 . ◦ , (10)-(12) 19 . ◦ . The values of n s are as follows: (1) 1.57, (2)1.59, (3) 1.61, (4) 1.63, (5) 1.65, (6) 1.67, (7) 1.69, (8) 1.71, (9) 1.73, (10) 1.80, (11)1.82, (12) 1.84. χ v = 7 . ◦ ; and for n s = 1 .
80, 1.82 and 1.84 when ˜ χ v = 19 . ◦ . When thevapor incidence angle is low (7 . ◦ ∀ z ≥ s -polarized with 4 < | A s | / | A p | <
5. When the vapor incidenceangle is increased to 19 . ◦ ∀ z ≥
0, the surface wave is only mildly polarized with1 < | A s | / | A p | <
2, and | A s | / | A p | shows a positive slope as a function of γ .Figure 9 shows the ratio | A s | / | A p | versus γ , when ˜ χ v = 19 . ◦ , for CTFs ( n s = 1 . δ v = 7 . ◦ ( n s = 1 .
73, 1.77, 1.82, 1.88, 1.92, 1.96). At thesame values of γ , the values of | A s | / | A p | are slightly smaller for the SNTF than the CTF.As with the curves describing Im[ α ] /k o in Figure 6b, the curves describing | A s | / | A p | for the SNTFs nearly join to form a single, continuous curve. At γ = 0 ◦ , the surfacewave is entirely p -polarized in the isotropic partnering material. The s -polarizationstate intensifies rapidly but then levels off at larger values of γ . The value of | A s | / | A p | is nearly constant at ∼ . γ > ◦ . For δ v = 19 . ◦ , the behaviour of | A s | / | A p | issimilar, as shown in Figure 10, but plateaus at a value of about 1.3 at large values of γ .
4. Concluding Remarks
To conclude, we examined the phenomenon of surface–wave propagation at the planarinterface of an isotropic dielectric material and a sculptured nematic thin film withperiodic nonhomogeneity. The boundary–value problem was formulated by marrying theusual formalism for the Dyakonov wave at the planar interface of an isotropic dielectricmaterial and a columnar thin film with the methodology for Tamm states in solid–statephysics. The solution of the boundary–value problem led us to predict the existence of yakonov–Tamm waves at nematic thin film/isotropic dielectric interface Figure 9.
Ratio | A s | / | A p | as a function of γ when ˜ χ v = 19 . ◦ . The values of δ v areas follows: (1)-(3) 0 ◦ , (4)-(9) 7 . ◦ . The values of n s are as follows: (1) 1.80, (2) 1.82,(3) 1.84, (4) 1.73, (5) 1.77, (6) 1.82, (7) 1.88, (8) 1.92, (9) 1.96. Figure 10.
Ratio | A s | / | A p | as a function of γ when ˜ χ v = 19 . ◦ and δ v = 16 . ◦ . Thevalues of n s are (1) 1.80 and (2) 1.84. Dyakonov–Tamm waves.Dyakonov surface waves may propagate guided by the bimaterial interface of anisotropic dielectric material and a columnar thin film (or any other biaxial dielectricmaterial). The angular domain of their existence is very narrow, of the order of a degree.Although several techniques have been suggested [2, 7], the widening of that domainhas not been impressive. By periodically distorting the CTF in two ways, either as achiral STF [8] or now as a periodically nonhomogeneous SNTF, the angular existence yakonov–Tamm waves at nematic thin film/isotropic dielectric interface
Acknowledgment.
This work was supported in part by the Charles Godfrey BinderEndowment at Penn State. KA thanks the Department of Engineering Science andMechanics, Penn State, and AL is grateful to the Department of Physics, IIT Kanpur,for hospitality.
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