Sculptured-thin-film plasmonic-polaritonics
aa r X i v : . [ phy s i c s . op ti c s ] J a n Sculptured–thin–film Plasmonic–Polaritonics
A. Lakhtakia, J. A. Polo Jr., and M. A. Motyka
Abstract —The solution of a boundary–value problemformulated for the Kretschmann configuration showsthat the phase speed of a surface–plasmon–polariton(SPP) wave guided by the planar interface of a suffi-ciently thin metal film and a sculptured thin film (STF)depends on the vapor incidence angle used while fabri-cating the STF by physical vapor deposition. Further-more, it may be possible to engineer the phase speedby periodically varying the vapor incidence angle. Thephase speed of the SPP wave can be set by selectinghigher mean value and/or the modulation amplitude ofthe vapor incidence angle.
I. Introduction
A resonance phenomenon arises from the interactionof light with free electrons at a planar metal–dielectricinterface [1], [2]. Under certain conditions, the energycarried by photons in the dielectric medium is trans-ferred to collective excitations of free electrons in themetal. Because the free electrons in the metal are cou-pled to the photons in the dielectric medium, the quan-tum is called a surface plasmon polariton , often short-ened to surface plasmon . A classical understanding ofthe surface plasmon polariton (SPP) is in terms of aelectromagnetic surface wave that propagates along theinterface and decays exponentially with distance normalto the interface.Research on electromagnetic surface waves has a longhistory, dating back about a hundred years. Zenneck[3] proposed in 1907 a mode of electromagnetic wavepropagation localized at the Earth–atmosphere inter-face. The wave, since named the Zenneck wave, propa-gates parallel to the interface with an amplitude whichdecays exponentially with distance from the interface.Credit is also given to Sommerfeld for a clean analysisof the phenomenon published in 1909 [4]. Basically thesame kind of wave at optical frequencies, the SPP wavebecame the subject of intense investigation[5], [6], [7]from the middle part of the 20th century. It was soonrealized that, because SPP waves are highly sensitive toconditions at the interface, they might be employed todetect various chemical species. SPP–wave–detectiontechniques are currently used for a wide range of sensingapplications especially in the detection of biomolecules[8], [9], [10], [11].
Akhlesh Lakhtakia and Michael A. Motyka are with the De-partment of Engineering Science and Mechanics, Pennsylva-nia State University, University Park, PA 16802, USA, E-mail:[email protected] A. Polo Jr. is with the Department of Physics and Tech-nology, Edinboro University of Pennsylvania, 235 Scotland Rd.,Edinboro, PA 16444, USA, E-mail: [email protected]
Much of the SPP literature is restricted to the pla-nar interface of a good conductor and an isotropic di-electric medium, though there are noticeable exceptionswherein the dielectric medium could be anisotropic [12],[13] and be additionally endowed with magnetic prop-erties [14], [15], [16]. The SPP wave is excited throughone of several different types of couplers [17].
Fig. 1. Scanning electron micrography of a sculptured nematicthin film. The columnar morphology is essentially two–dimensional. Courtesy: R. Messier.
In the Kretschmann [18] configuration, the bulkmetal is in the form of a sufficiently thin film of uniformthickness, bounded by dielectric mediums on both sides,one medium being optically denser than the other. Aplane wave is launched in the optically denser dielec-tric medium towards the metal film, in order to excite asurface–plasmon wave at the interface of the metal withthe optically rarer dielectric medium. The plane wavemust be p –polarized. The telltale sign is a sharp peak inabsorbance (i.e., a sharp trough in reflectance withouta compensatory peak in transmittance) as the angle ofincidence (with respect to the thickness direction) of thelaunched plane wave is changed. The absorbance peakoccurs in the vicinity of the critical angle (of incidence)that would exist if the metal film were absent.Generally, the optically rarer medium is homoge-neous normal to its planar interface with the metal film,at least within the range of the SPP field. In this pa-per, however, we take the optically rarer medium tobe continuously nonhomogeneous in the thickness di-ection. Specifically, this medium is a sculptured thinfilm (STF).STFs are nanostructured materials with unidirection-ally varying continuum properties that can be designedand realized in a controllable manner using physicalvapor deposition [19]. The ability to virtually instan-taneously change the growth direction of their colum-nar morphology, through simple variations in the di-rection of the incident vapor flux, leads to a wide va-riety of ∼ Fig. 2. Scanning electron micrography of a chiral sculptured thinfilm, which has a three–dimensional columnar morphology.Courtesy: R. Messier.
This paper is organized as follows. Section II be-gins with a description of the Kretschmann configura-tion to launch a SPP wave at the planar interface ofa metal and a STF. The relative permittivity dyadicof the STF is described in that section, along with theboundary–value problem to determine planewave ab-sorbance. Section III contains numerical results to elu-cidate the effects of the nonhomogeneity of the STFon the SPP wave. An exp( − iωt ) time–dependence isimplicit, with ω denoting the angular frequency. Thefree–space wavenumber, the free–space wavelength, andthe intrinsic impedance of free space are denoted by k = ω √ ǫ µ , λ = 2 π/k , and η = p µ /ǫ , respec-tively, with µ and ǫ being the permeability and per-mittivity of free space. Vectors are in boldface, dyadicsunderlined twice; column vectors are in boldface andenclosed within square brackets, while matrixes are un-derlined twice and similarly bracketed. Cartesian unitvectors are identified as ˆ u x , ˆ u y and ˆ u z . II. Theory
In conformance with the Kretschmann configuration[18], [20] for launching SPP waves, the half–space z ≤ ǫ ℓ . Dissipa-tion in this material is considered to be negligible andits refractive index n ℓ = √ ǫ ℓ is real–valued and positive.The laminar region 0 ≤ z ≤ L met is occupied by a bulk metal with relative permittivity scalar ǫ met . The region L met ≤ z ≤ L met + L stf is occupied by a STF. Withoutsignificant loss of generality in the present context, thehalf–space z ≥ L met + L stf is taken to be occupied bythe same material as fills the half–space z ≤ A. Sculptured Thin Film
The relative permittivity dyadic ǫ stf ( z ) of the STFis factorable as ǫ stf ( z ) = S z ( ζ ) • S y ( χ ) • ǫ refstf • S Ty ( χ ) • S Tz ( ζ ) ,L met ≤ z ≤ L met + L stf , (1)wherein the reference relative permittivity dyadic ǫ refstf = ǫ a ˆ u z ˆ u z + ǫ b ˆ u x ˆ u x + ǫ c ˆ u y ˆ u y (2)captures the locally orthorhombic character of STFs,the dyadic function S z ( ζ ) = (ˆ u x ˆ u x + ˆ u y ˆ u y ) cos ζ + (ˆ u y ˆ u x − ˆ u x ˆ u y ) sin ζ + ˆ u z ˆ u z (3)denotes rotation about the z axis, the dyadic function S y ( χ ) = (ˆ u x ˆ u x + ˆ u z ˆ u z ) cos χ + (ˆ u z ˆ u x − ˆ u x ˆ u z ) sin χ + ˆ u y ˆ u y (4)involves the angle χ ∈ [0 , π/ T denotes the transpose. The quantities ζ , χ , ǫ a , ǫ b , and ǫ c can all be functions of z .Although STFs have been made by evaporating awide variety of materials [19], [21], the constitutive pa-rameters of STFs have not been extensively measured.However, the constitutive parameters of certain colum-nar thin films (CTFs) are known. Both CTFs and STFsare fabricated by physical vapor deposition. The basicprocedure to deposit CTFs has been known for morethan a century. At low enough temperature and pres-sure, a solid material confined in a boat evaporates to-wards a stationary substrate. The vapor flux is col-limated into a well–defined beam, and its average di-rection is quantified by the angle χ v ∈ (0 , π/
2] withrespect to the substrate plane, as illustrated in Fig. 3.Provided the adatom mobility is low, the resulting filmturns out to be an assembly of parallel and nominallyidentical columns. The columns have elliptical cross–sections and are tilted at an angle χ ≥ χ v with respectto the substrate plane. The parameters χ , ǫ a , ǫ b , and ǫ c have to be functions of χ v , at the very least because thenanoscale porosity of a CTF depends on the directionof the vapor flux.A series of optical characterization experiments oncertain CTFs were carried out some years ago [22], [23].After ignoring the effects of dispersion and dissipation, ubstrate χ χ v Growing columnsVapor flux
Fig. 3. Schematic of the growth of a columnar thin film. Thevapor flux is directed at an angle χ v , whereas columns growat an angle χ ≥ χ v . at least in some narrow range of frequencies, the re-sults can be put in the following form for our presentpurposes: ǫ a = (cid:0) n a + n a v + n a v (cid:1) ǫ b = (cid:0) n b + n b v + n b v (cid:1) ǫ c = (cid:0) n c + n c v + n c v (cid:1) χ = tan − ( m tan χ v ) . (5)Here, v = χ v / ( π/
2) is the vapor incidence angle ex-pressed as a fraction of a right angle. The quantities m and n a , etc., in Eq. 5 depend on the evaporant materialas well as the deposition conditions.When the substrate is rotated about either the y or the z axes, parallel columns of specific shape growinstead of straight columns, and a STF is depositedinstead of a CTF. Although the substrate is nonsta-tionary, the functional relationships connecting χ , ǫ a , ǫ b , and ǫ c to χ v for CTFs would substantially applyfor STFs, provided the substrate rotation is sufficientlyslow. Thus, we need only to specify the z –dependencesof ζ and χ v .For our present purposes, we chose ζ ( z ) = h π Ω ( z − L met ) χ v ( z ) = ˜ χ v + δ v sin (cid:2) π Ω ( z − L met ) (cid:3) ) . (6)Here, Ω is a characteristic length along the z axis,whereas the angles ˜ χ v ∈ (0 , π/
2] and δ v ∈ [0 , ˜ χ v ].The structural–handedness parameter h = 1 for right–handedness, h = − h = 0for no handedness. For theoretical investigations, wedecided to focus on the following three types of STFs: (i) columnar thin films ( h = 0, δ v = 0); (ii) sculptured nematic thin films with periodicallyvarying tilt angle ( h = 0, δ v > (iii) chiral sculptured thin films ( h = ± δ v = 0). B. Boundary–Value Problem
Suppose that a p –polarized plane wave, propagatingin the half–space z ≤ θ ∈ [0 , π/
2) to the z axis in the xz plane, is incident on the metal–coatedSTF in the Kretschmann configuration. The electricfield phasor associated with the incident plane wave is E inc ( r ) = p + e iκx e ik n ℓ z cos θ , z ≤ . (7)The reflected electric field phasor is expressed as E ref ( r ) = ( r s s + r p p − ) e iκx × e − ik n ℓ z cos θ , z ≤ , (8)and the transmitted electric field phasor as E tr ( r ) = ( t s s + t p p + ) e iκx × e ik n ℓ ( z − L met − L stf ) cos θ ,z ≥ L met + L stf . (9)Here, κ = k n ℓ sin θ s = ˆ u y p ± = ∓ ˆ u x cos θ + ˆ u z sin θ , (10)where ω/κ is the phase speed parallel to the interfacialplane z = L met of interest, and the unit vectors s and p ± denote the s – and the p –polarization states of theelectric field phasors.The reflection amplitudes r s and r p , as well as thetransmission amplitudes t s and t p , have to be deter-mined by the solution of a boundary–value problem.The required procedure is standard [19]. It suffices tostate here that the following set of four algebraic equa-tions emerges (in matrix notation): t s t p = [ K ] − · [ M stf ] · exp (cid:16) i [ P met ] L met (cid:17) · [ K ] · r s r p . (11)The procedure to obtain the 4 × M stf ] canbe gleaned from two predecessor papers [24], [25]. Theremaining two 4 × K ] = − cos θ θ − (cid:16) n ℓ η (cid:17) cos θ (cid:16) n ℓ η (cid:17) cos θ − n ℓ η − n ℓ η , (12)[ P met ] = ωµ − ωµ − ωǫ ǫ met ωǫ ǫ met − κ ωǫ ǫ met κ ωµ . (13)Equation 11 can be solved for r s , r p , t s , and t p usingstandard algebraic techniques. The quantity of interestfor establishing the existence of the SPP wave is theabsorbance A p = 1 − (cid:0) | r s | + | r p | + | t s | + | t p | (cid:1) (14)as a function of θ . III. Numerical Results and Discussion
Calculations of A p against θ are reported in thispaper for the following parameters of the STF: Ω =200 nm, L stf = 4Ω, and ˜ χ v = 30 ◦ . Calculations haveshown that the selection of higher values of L stf doesnot impact the SPP wave significantly, whereas the ef-fect of the nanostructured periodicity of SNTFs and chi-ral STFs is not appreciable for lower values of L stf . Thefree–space wavelength λ = 633 nm is the same at whichthe parameters in Eqs. 5 were measured. The STF wastaken to be made of titanium oxide: n a = 1 . n a = 2 . n a = − . n b = 1 . n b =1 . n b = − . n c = 1 . n c = 2 . n c = − . m = 2 . ǫ met = −
56 + i
21 [26] and L met = 15 nm. The metal film is thus thin enough thatit allows sufficient penetration of the evanescent wave toexcite the SPP at the metal–STF interface; at the sametime, the metal film is thus thick enough to preventtunneling of photons across it from the medium of in-cidence and reflection to the STF. The two half–spaceswere taken to be filled with zinc selenide ( n ℓ = 2 . h = 0 and δ v = 0. Figure 4 shows A p as a functionof θ . The SPP wave is excited at θ = 55 . ◦ , with theabsorbance in excess of 0 .
988 denoting a very efficientconversion of the energy of the incident plane wave intothe SPP wave.Let θ spp denote the value of θ at which the SPP waveis excited. A study of θ spp versus χ v reveals that θ spp for a CTF fabricated with a specific evaporant materialincreases as χ v increases [27]. Therefore, the wavenum-ber of the SPP wave, given by κ spp = k n ℓ sin θ spp , (15)is a monotonically increasing function of χ v , whichmeans that the phase speed v spp = ω/κ spp (16) θ (deg) Α p Fig. 4. Absorbance A p as a function of θ when the STF is acolumnar thin film ( h = 0 and δ v = 0); see the text for otherparameters. The SPP wave is excited at θ = 55 . ◦ . θ (deg) Α p Fig. 5. Absorbance A p as a function of θ when the STF isa sculptured nematic thin film ( h = 0 and δ v = 20 ◦ ); seethe text for other parameters. The SPP wave is excited at θ = 62 . ◦ . The sharp changes for two values of θ < ◦ donot indicate the excitation of SPP waves. of the SPP wave is a monotonically decreasing functionof χ v .The effect of nonhomogeneity in the dielectricmedium on the SPP wave becomes evident when weconsider the metal–backed sculptured nematic thin film: h = 0 and δ v >
0. Figure 5 shows A p as a function of θ when δ v = 20 ◦ . The SPP wave is excited at θ = 62 . ◦ ,with the absorbance in excess of 0 . χ v is periodically modulated with a significant amplitude,but the high efficiency of conversion of the energy of theincident plane wave into the SPP wave is not affectedthereby.Nonhomogeneity is also introduced in the dielectricmedium when the vapor deposition angle is held fixedduring fabrication but the substrate is rotated aboutthe z axis. Figure 6 shows A p as a function of θ when δ v = 0 and h = ±
1; i.e., it is drawn for a metal–backed chiral STF. Now, θ spp = 55 . ◦ , which is onlymarginally higher than in Fig. 4. IV. Concluding Remarks
We conclude that the solution of a boundary–valueproblem formulated for the Kretschmann configuration θ (deg) Α p Fig. 6. Absorbance A p as a function of θ when the STF is achiral sculptured thin film ( h = ± δ v = 0); see the textfor other parameters. The SPP wave is excited at θ = 55 . ◦ . shows that the phase speed of a SPP wave guided bythe planar interface of a sufficiently thin metal film anda sculptured thin film depends on the vapor incidenceangle used while fabricating the STF by physical vapordeposition. Therefore, it may be possible to engineerthe phase speed quite simply by selecting an appro-priate value of the vapor incidence angle (in additionto the metal and the evaporant species). Furthermore,by periodically varying the vapor incidence angle, thephase speed of the SPP wave can be reduced. Ade-quate selection of the phase speed should be importantfor controlled data flow in plasmonic circuits.The high degree of porosity [19] of STFs may pro-vide certain advantages in the application of SPP wavesin detectors. A properly chosen STF could preventparticulates from reaching its interface with the metalfilm while allowing molecular species through for de-tection. The detection of biomolecules typically relieson recognition molecules, which are bound to the inter-face, binding with the analyte molecules. The poros-ity and surface roughness of the STF may offer someadvantage for adherence of the recognition molecule.STFs can now be patterned using standard photolitho-graphic techniques [28]. Partitioning of the STF intomany sectors could permit the use of multiple speciesof recognition molecules on a single chip for the detec-tion of many different types of analyte molecules. Re-cently, the activity of living cells has been monitoredusing SPP detection techniques [29]. Cells have beenshown to grow well on the rough STF surface [21], [30],which thus may offer an advantage. Finally, titaniumoxide is a very well–known photocatalyst. Although thephotocatalytic properties of titanium oxide STFs [31],[32] have been investigated, the combination of SPPdetection and this catalyst may allow for other usefulapplications to emerge. References [1] A. V. Zayats, I. I. Smolyaninov, and A. A. Maradudin,“Nano–optics of surface plasmon polaritons,”
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