On determining gas concentrations using dielectric thin-film helicoidal bianisotropic medium bilayers

Sensors and Actuators B 52 (1998) 243 – 250

On determining gas concentrations using dielectric thin-film helicoidal bianisotropic medium bilayers Akhlesh Lakhtakia * Department of Engineering Science and Mechanics, CATMAS—Computational and Theoretical Materials Sciences Group, Pennsyl6ania State Uni6ersity, Uni6ersity Park, PA 16802 -1401, USA Received 10 March 1998; received in revised form 14 July 1998

Abstract Bilayers comprising two matched layers of dielectric thin-film helicoidal bianisotropic mediums (TFHBMs) with opposite structural handedness reflect arbitrarily polarized, normally incident, plane waves almost perfectly in a certain frequency range. As TFHBMs are porous, a model is devised to show that the infiltration of a TFHBM bilayer by a gas can shift as well as reduce the span of this range. Accordingly, a suitably designed TFHBM bilayer can serve as the active element of an optical device for sensing gas concentration. © 1998 Elsevier Science S.A. All rights reserved. Keywords: Gas concentrations; Dielectric thin film; Helicoidal bianisotropic medium bilayers

1. Introduction The morphology of the so-called columnar thin films (CTFs) resembles a mass of parallel match sticks or microcolumns [1–3]. A CTF can be highly porous or not, depending on the conditions prevailing during its deposition on a substrate, and can therefore be considered as a two-phase composite material. When probed with electromagnetic waves whose wavelengths are considerable larger than the microcolumnar diameter, a CTF responds like a homogeneous uniaxial continuum [4]. During the past 5 years, controlled motion of substrates has allowed the engineering of the shapes of the parallel microcolumns into bedsprings, corkscrews, C’s, S’s and other combinations [5]. This development catalyzed the concept of sculptured thin films (STFs), which comprises two canonical types of thin films with engineered nanostructure [6]. One canonical type has a nematic morphology [7], the other has a cholesteric morphology [8,9]. Many applications of STFs are possible [5] and experimental [10] as well as theoretical results [11,12] on the optical responses of STFs initiated device—oriented research last year at Penn State. * Tel.: +1 814 8634319; fax: [email protected]

+1 814 8637969; e-mail:

As a first step in that direction, the possibility of bilayers of dielectric thin-film helicoidal bianisotropic mediums (TFHBMs)—which are STFs with cholesteric morphology—as laser mirrors and notch filters was established [13]. The dynamic tunability of these thinfilm devices as well as their design flexibility offers definite advantages over comparable liquid crystal devices [14]. The tunability of the notch-filter response of a TFHBM bilayer may be advantageously harnessed for optical sensing of gas concentrations, possibly toxic. Provided the shaped microcolumns of the bilayer are made of a material chemically inert to the gas being sensed, immersion of the bilayer in the gaseous environment would cause gas molecules to diffuse in the intermicrocolumnar void regions. As a result, the permittivity dyadic would change and the notch in the transmission spectrum would shift—which could be used to estimate the ambient gas concentration. This exciting possibility motivated the present communication, wherein a nanoscopic-to-continuum model is devised to predict the electromagnetic response of a TFHBM bilayer. Briefly, the microcolumns are supposed to be made of needles of an isotropic material with bulk relative permittivity es (v), v being the angular frequency of the electromagnetic probe. The gas to be sensed has a molar refractivity Amg and it completely

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Fig. 1. Reflection and transmission of a normally incident plane wave by a TFHBM bilayer. The letters L and R respectively identify the structurally left- and right-handed layers.

occupies the inter-microcolumnar void region. This composite medium can be locally homogenized into a uniaxial dielectric medium, using the Bruggeman formalism [15]. The relative permittivity dyadic of the homogenized medium may closely approximate the socalled reference relative permittivity dyadic e ref(v) of the TFHBM bilayer [6,13], particularly if the value of the ratio u of the needle’s length to its cross-sectional radius is chosen after calibrating the model with the results of carefully selected experiments. The plan of this paper is as follows: Equations for the described model are developed in Section 2, while numerical results are presented and discussed in Section 3. The last section contains notes for device development. A note on notation: vectors are underlined once in this paper, while dyadics [17] are underlined twice.

2. Model development

2.1. Constituti6e relations Shown in Fig. 1 is a dielectric TFHBM bilayer consisting of two layers of thickness D each. The

two layers are matched, being identical in their electromagnetic response properties except that the lower layer is structurally right-handed and the upper one is left-handed. The half-spaces zB 0 and z \ 2D are vacuous. The electromagnetic constitutive relations for time harmonic fields in the chosen TFHBM bilayer may be set down as [13] D(r, v)=e(z, v) · E(r, v)

?

; 0B zB 2D

(1)

B(r, v)= m0 H(r, v) where the z- and the v-dependences of the permittivity dyadic e (z, v) can be factorized as follows: e(z, v)=e0 S z (z) · e ref(v) · S Tz (z)

(2)

Here and hereafter, an exp (− ivt) time-dependence is implicit; e0 and m0 are the permittivity and the permeability of free space (i.e., vacuum), respectively; I= u6 x u6 x + u6 y u6 y + u6 z u6 z is the identity dyadic [17], with u6 x, u6 y and u6 z as the three cartesian unit vectors; while the superscript T denotes the transpose. The rotation dyadic S z (z) is specified in a piecewise manner for the TFHBM bilayer [13]; thus,

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Á pz (u x u x +u y u y ) cos + à V à pz +u z u z ; 05 z5 D (u y u x −u x u y ) sin à V S z (z) = Í p(z− D) à (u x u x +u y u y ) cos − V à Ã(u y u x −u x u y ) sin p(z− D) +u z u z ; D5 z5 2D V Ä





 n

n

This specification reflects the fact that the z axis is the axis of rotational nonhomogeneity with 2V as the pitch in both layers, while the two layers are of opposite structural handedness. We take D  V so that the Bragg phenomenon manifests itself unambiguously [11]. Finally, the reference relative permittivity dyadic in (2) must display the locally uniaxial nature of the TFHBM bilayer. Thus, e ref(v)= ea (v)I+ [eb (v) − ea (v)]u t u t

(4)

wherein the unit vector u t = u x cos x +u z sin x

(5)

is inclined above the xy plane by an angle x. This is the angle of rise of the parallel helical microcolumns that a TFHBM is composed of [9,6] and the vapor deposition procedure for STF fabrication entails that x \ 0° [7].

2.2. Localized homogenization Eq. (1) and the following ones in the previous subsection incorporate the assumption of the TFHBM bilayer as a unidirectionally nonhomogeneous continuum. This is certainly valid in a macroscopic sense, i.e., when the length scale of the film morphology is considerably smaller than the electromagnetic probe wavelength [4]. This assumption is valid for optical and infra-red applications, and even for high-frequency millimeter-wave applications, because TFHBMs with appropriate morphological length scales can be fabricated [9,18]. As the TFHBM bilayer is definitely a two-phase composite material, e(z, v) must emerge from its microstructural details. The mathematical process describing this transition from the nanoscopic to the continuum length scales is called homogenization. It is very commonly implemented in various fashions for random distributions of electrically small particles in a homogeneous host medium [4]. But, as the particles are randomly distributed therein, the constitutive properties after homogenization are independent of position. In contrast, a TFHBM is a nonhomogeneous continuum. Therefore, we have to localize the homogenization process [19]. Accordingly, we concentrate on e ref(v) instead of e(z, v). Let the chosen thin film be made of an

(3)

isotropic material whose relative permittivity in the bulk state is denoted by es (v). In a thin slice of the film, this material is distributed in the form of identical parallel-oriented needles whose length is u times their cross-sectional radius. The film is porous, the void regions being occupied by a gas of molar refractivity Amg (v) and molar density rmg [20]. Whereas the molar refractivity is largely independent of pressure [20], the molar density—being a measure of gas concentration—is not. Temperature changes will affect both Amg (v) and rmg [21], but we assume that the internal temperature of the film is temporally constant and spatially uniform. The Mossotti–Clausius formalism predicts [20] eg (v)=

1+ 2rmgAmg (v) 1− rmgAmg (v)

(6)

as the relative permittivity of the gas. This equation here represents homogenization at the nanoscopic length scales. In using (6) to estimate eg (v) and in specifying es (v), we have ignored the electromagnetic interactions between the two types of materials. Treatment of this interaction with the Bruggeman formalism leads to the dyadic equation [15] −1 1 fg [e ref(v)− eg (v)I] · {I− W(v) · [I− eg (v)e − ref (v)]}

+ (1− fg )[e ref(v)− es (v)I] −1 1 =0 · {I− W(v) · [I− es (v)e − ref (v)]}

(7)

where the shape factor u of the needles occurs in the dyadic W(v)=

 ! 

n

1 e (v) 1+ 4u − 2 b 2 ea (v) + 1−

− 1/2

I

n "

3 e (v) 1+ 4u − 2 b ea (v) 2

− 1/2

ut ut

(8)

The volumetric proportion of the thin film occupied by the gas is denoted by fg, (0 5fg 5 1), which is also the porosity of the TFHBM bilayer. Eq. (7) has to be solved numerically to estimate the components ea (v) and eb (v) of e ref(v)—see [15] for further details—and

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the solution represents homogenization at the microscopic length scales.

2.3. Model inputs The final transition to a continuum is represented by (2). Thus, the presented model requires a knowledge of V and the porosity fg of the TFHBM bilayer, along with the layer thickness D. These quantities can be easily measured [10]. The model also requires that Amg (v) and eg (v) be known. The remaining input is the shape factor u, which must be \ 1 for needles. Its value can estimated by running the model with various values for u and then comparing against the experimentally determined planewave response of the TFHBM bilayer when the gas phase is replaced simply by air (i.e., eg (v)# 1.00058 at standard temperature and pressure and at optical frequencies). A program of experimental research in this direction has recently commenced at Penn State. Parenthetically, if u \10, then setting u = 10 may be justified by the results of several homogenization exercises [15,16].

2.4. Boundary 6alue problem The molar density rmg of the gas in the TFHBM bilayer remains unknown, being the quantity to be sensed. Let the TFHBM bilayer be illuminated by, say, a normally incident Gaussian pulse whose bandwidth is appropriately wide. The spectrum of the transmitted pulse depends critically on e ref(v) and can therefore function as a predictor of rmg. In order to demonstrate this possibility, let us now consider the planewave response of a TFHBM bilayer. An arbitrarily polarized plane wave is normally incident on the TFHBM bilayer from the lower half-space z 5 0, as shown in Fig. 1. The electric field phasor E inc(r, v)= [aL (v)u + +aR (v)u − ] exp (ik0z);

z5 0, (9)

delineates the incident plane wave. Here, k0 =2p/l0 = v(e0 m0)1/2 is the free-space wavenumber and l0 is the free-space wavelength; aL (v) and aR (v) are the amplitudes of the left- and right-circularly polarized components; and u6 9 = (u6 x 9iu6 y )/ 2. The electric field phasors associated with the reflected and the transmitted plane waves are similarly specified as follows: E ref(r, v)= {[rLL (v)aL (v) + rLR (v)aR (v)]u − + [rRL (v)aL (v) + rRR (v)aR (v)]u + } × exp (− ik0z);

z5 0

E tr (r, v)= {[tLL (v)aL (v) + tLR (v)aR (v)]u + +[tRL (v)aL (v) + tRR (v)aR (v)]u − }

(10)

× exp [ik0(z− 2D)];

z ]2D.

(11)

As expressions for the electromagnetic fields induced inside a TFHBM layer are given elsewhere in detail [11], they are not repeated here for the sake of economy. The calculation of the unknown reflection coefficients (rLL (v), etc.) and transmission coefficients (tLL (v), etc.) then requires the solution of a simple boundary value problem. As the standard 4 × 4 matrix procedure described elsewhere [22,11] for TFHBMs was employed to solve the boundary value problem, there is no need to provide details here.

3. Numerical results and discussion The presented model and the solution procedure for the boundary value problem were implemented using Mathematica 3.0. Based on experience gained from earlier work [13], the half-pitch was fixed at V=190 nm, the layer thickness at D=60V and the angle of rise at x=30°, all realistic values. Calculations of the reflection and transmission coefficients were confined to the wavelength regime 4005l0 5 700 nm and the satisfaction of the principle of conservation of energy was verified at all l0. Furthermore, in order to provide simple illustrative results, dissipation as well as dispersion in the gaseous and the solid material phases were ignored over the selected wavelength regime and the following values were used: Amg (v) Amg =3.72× 10 − 3 m3 mol − 1 and es (v) es = 5. Fig. 2 shows the computed values of the co-polarized reflectivities rLL (v) 2 and rRR (v) 2 as functions of l0 and rmg, when fg = 0.6. Both plots in Fig. 2 display a very prominent ridge-like feature, due to the Bragg phenomenon in the so-called Zone II of operation [11,23]. The feature is marked by very high co-polarized reflectivities which can be virtually perfect in some portions of the ridge. The undulations in the ridge are matched by those of the corresponding cross-polarized reflectivities so that rRL (v) 2 : 1− rLL (v) 2 and rLR (v) 2 : 1− rRR (v) 2. Therefore, both the co- and the cross-polarized transmissivities are virtually zero when the Bragg phenomenon occurs [11], which is why TFHBM bilayers may be used as notch filters [13]. Fig. 2 also demonstrates the independence of the shape of the ridge from the polarization state of the incident plane wave. The wavelength-span as well as the central wavelength of the ridge at any fixed rmg are essential to the possible use of TFHBM bilayers as gas concentration sensors. This becomes clearer from Fig. 3, wherein rLL (v) 2 is plotted as a function of l0 at rmg = 0 mol m − 3 and rmg = 50 mol m − 3. Clearly from the two figures, the central wavelength of the ridge shifts towards higher values as rmg increases. Simultaneously, the wavelength-span of the ridge also decreases.

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Fig. 2. Computed values of rLL (v) 2 and rRR (v)2 versus the molar density rmg of the gas and the free-space wavelength l0. The other parameters are as follows: V =190 nm, D = 60V, x =30°, fg = 0.6, while Amg (v) =3.72 ×10 − 3 m3 mol − 1 and es (v) =5 are assumed independent of v in the 4005l0 5700 nm range. Note the ridge-like features in both plots.

At any rmg, the central wavelength may be estimated as [13]

< D

eb /ea e cos2 x + b sin2 x ea and the wavelength-span as

lII : V ea 1+

F< D

=

(12)

=F

eb /ea (13) eb 2 2 cos x + sin x ea for the plots of Figs. 2 and 3. Both lII and DlII are plotted in Fig. 4 as functions of rmg and fg. Two major inferences can be drawn from Fig. 4. First, lII is affected by both the porosity of the thin film DlII : V ea 1−

and the molar density of the gas. When fg = 0, the film is totally solid and lII is obviously unaffected by rmg. On the other extreme (i.e., fg = 1), the film becomes totally gaseous and lII is almost a linear function of rmg, because eg : 1+3rmgAmg so that lII : 2V(1 +(3/ 2) rmg Amg ) for the usual case of rmgAmg  1. Clearly then, the variation of lII with respect to rmg is fairly monotonic, regardless of the value of fg. Second, in the mathematical context, the conditions fg \ 0 and fg B 1 are needed because the Bragg phenomenon does otherwise not appear. The optimal values of fg must lie in between the two extremes. In Fig. 4, fg : 0.6 for maximum DlII, which is a reasonable result. After all, DlII 0 for fg = 0 as well as for fg =1. Optimization for use as a sensor may necessitate a value for fg that makes the shift of lII proportional to rmg, a feature that should not be difficult to realize

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A. Lakhtakia / Sensors and Actuators B 52 (1998) 243–250

Fig. 3. Computed values of rLL (v)2 versus the free-space wavelength l0, when rmg =0 mol m − 3 and rmg =50 mol m − 3. See Fig. 2 for other details.

practically as the ridges in Fig. 2 do not seem to be curved. The use of a TFHBM bilayer as a gas concentration sensor requires that the bilayer be excited by a normally incident pulse of wavelength-span larger than DlII and the transmission spectrum be measured. A zero-transmission notch of span DlII and centered about lII will appear in this spectrum. As both quantities depend on rmg, either may be used to determine the molar density. Errors may be reduced if measured values of both quantities are fed into an artificial neural network (ANN) which fuses the inputs [24] for robust and reliable estimation of rmg. The use of ANNs may also reduce the performance variability in nominally identical TFHBM bilayers produced in one batch as well as in different batches.

4. Concluding remarks Several comments on the presented model and its proposed optical application are now in order. First, the interaction between the gas molecules and the solid material is taken into account somewhat imperfectly. This is because we assume an aciculate morphology in (7), whereas the solid material has helicoidal morphology. As of now, this imperfection does not appear to be avoidable in any reasonably tractable model for deriving the structure-property relations of STFs. While the incorporation of correction factors arising from careful experimentation may improve device design, we surmise that the use of ANNs may render fine scale corrections unnecessary.

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Fig. 4. Computed values of lII /V and DlII /V versus the molar density rmg of the gas and the porosity fg of the TFHBM bilayer. See Fig. 2 for other details.

Second, although the presented model does take the frequency-dependences of Amg (v) and es (v) into account, we assume both parameters to be independent of frequency for the numerical results presented here. Furthermore, neither of the two materials is dissipative in the calculations of Figs. 2 – 4, which issue is related to that of frequency-dependence [11]. The simple calculations presented shall have to be modified, once the frequency-dependence of e ref(v) is experimentally ascertained in relation to the frequency-dependences of Amg (v) and es (v). Third, actual TFHBMs may not have the rigorous structural periodicity required by the presented model through (3). This is likely to contribute to the broadening of the zero-transmission notch, but improved fabrication methods will eventually minimize the broadening effect. Fourth, temperature variations inside the TFHBM bilayer may cause the dielectric response periodicity to degenerate slightly, because Amg (v) does depend on temperature [21] and es (v) is also likely to

exhibit similar but muted behavior. Again, the use of ANNs is expected to successfully deal with both issues. The gas sensor proposed here may not necessarily be superior to existing ones for sensing specific gases; certainly, extensive experimental studies need to be carried out for efficient fabrication as well as optimized use. However, TFHBM bilayers may provide a sensing modality for gases whose sensing is currently infeasible, costly or otherwise problematic. For instance, TFHBM bilayers are likely to be used for gas sensing in toxic environments—e.g., in the decontamination chambers of hospitals, wherein textiles and apparatuses are routinely sterilized with highly toxic gases—because they may be optically interrogated from remote locations. Additionally, optical interrogation of embedded TFHBM bilayers may provide real-time monitoring capabilities for sensing toxic as well as non-toxic gases. Patches of TFHBM bilayers—of different materials and/or different pitches—may be compactly mounted on the same platform for sensing different gases, either

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serially or synchronously. Integration with electronic circuitry remains an exciting possibility to be explored. Finally, sensors based on TFHBM bilayers are expected to be reusable, a key component of sustainable development strategies [25,26]. At this stage, one can safely state that these and other possibilities require serious experimental research.

[14]

[15]

[16]

Acknowledgements [17]

The author thanks Professor K. Vedam of the Pennsylvania State University for assistance with relevant literature and Dr U.B. Unrau of Technische Universita¨t Braunschweig for discussions on practical implementation of the TFHBM bilayer as a gas sensor. This paper is dedicated to the hope for a truly comprehensive and global nuclear disarmament treaty.

[18]

[19]

[20] [21]

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Biographies Akhlesh Lakhtakia was born July 1, 1957, in Lucknow, India. Educated in Lucknow, New Delhi, Varanasi and Salt Lake City, he is currently a Professor in the Department of Engineering Science and Mechanics, Pennsylvania State University. He is a Fellow of the Optical Society of America, SPIE—The International Society for Optical Engineering and the Institute of Physics (UK). He has published more than 340 papers in peer-reviewed journals and has authored, co-authored, edited or co-edited seven research books. Please see his homepage for information on his current research activities: http://www.esm.psu.edu/HTMLs/Faculty/Lakhtakia/ALakhtakia.html.