High precision position sensor based on CPA in a composite multi-layered system

Sanjeeb Dey; Suneel Singh; Desai Narayana Rao

doi:10.1364/OE.26.010079

1. Introduction

In recent years many proposals related to highly sensitive position measurements in nanometers regime have emerged. While a few of these utilize all optical techinques of spectral interferometry and Vernier shift [1], spectral shifts in a white-light interferometer [2] others are based on piezoelectric transducer mirror design [3–5] for measuring nano displacements. In this context it would also be of interest to explore the feasibility of utilizing coherent perfect absorption (CPA) -another optical interference based phenomenon-for the purpose of high sensitivity nano displacement measurements.

As the name indicates, CPA is the process [6–10] in which incident coherent electromagnetic radiations interfere destructively under right conditions of amplitudes and phases and are fully absorbed in a medium. CPA is of tremendous interest in recent years and has been studied in meta-materials [11–15], meta-surfaces [16–18], and graphene [19], etc. using various geometries for active device applications. Metal dielectric composites [20–22] are also found particularly suitable for studying CPA as its absorption properties can be tuned by varying the composition of the constituents. In such metal-dielectric composite materials where the metal and dielectric components are arranged randomly, localized plasmon resonances can be excited in the nanosize metal inclusions that are segregated by dielectric material. By varying the amount of metal inclusions it is possible to control the overall absorption and broadening and shift of the (localized) plasmon resonances. For such disordered distribution of the metal and dielectric components, generalized analytic approaches exist, that using the mean-field approximation estimate the average electromagnetic response of composite materials in terms effective dielectric function ϵ_eff of the macroscopically uniform medium [23, 24]. This effective dielectric function is given in terms of the permittivities of the individual components as well as their respective volume fractions. Earlier studies of CPA in a single metal-dielectric thin structure using plane-wave approximation [20] and the more realistic Gaussian beams [21, 22] of finite widths have demonstrated similar behaviour in both the cases.

In this article, we study CPA in a layered medium consisting of three layers of metal-dielectric composite media (CM) separated by two spacer (air) layers. Here, the outermost (CM) layers are fixed while the middle (CM) layer can be moved between the adjacent spacer layers causing asymmetry in the layered structure. The organization of the article is as follows: Section 2 deals with the presentation of the geometry of the layered medium and formulation of the problem. In this section, we will briefly outline the effective medium theory (EMT), scattering intensity (SI) and the CPA, the expression and details of which are presented in the Appendix. Section 3 contains numerical results and discussion of CPA in symmetric and asymmetric layered structures. Also in this section we propose a design of high precision position sensing device based on the theoretical formalism developed on CPA with multilayer (metal-dielectric composite and space) structure. Conclusions are presented in the final Section 4.

2. Geometry and formulation

Consider a five layer structure made of three thin slabs of metal-dielectric composite material (CM) and two spacer (air) layers as shown in the Fig. 1. The permittivities of the CM and spacer layers are denoted respectively, by ϵ_CM and ϵ_air. The widths of the CM layers are d₁, d₃ and d₅ whereas the widths of the remaining spacer (air) layers are d₂ and d₄. Note that the widths d_j, j = 1 – 5 of the layers can in general be arbitrary. We however begin with a particular situation in which the outermost CM layers have the same width (d₁ = d₅, d₃). The middle layer d₃ is free to move whereas the outermost layers d₁ and d₅ are fixed in their position. Initially the layered structure is symmetric when the middle layer d₃ is exactly in the center such that the width of the spacer layers is equal, d_sp = d₂ = d₄. Asymmetry is caused in the layered system if the d₃ layer is moved left towards d₁ by a distance δ, in which case the spacer (air) layer widths change by d₂ = d_sp − δ and d₄ = d_sp + δ. Similarly the spacer (air) layer widths will change by d₂ = d_sp + δ and d₄ = d_sp − δ if the d₃ layer is moved right towards d₅ by a distance δ. Obviously a left (right) shift (of CM layer d₃) by a distance δ decreases (increases) the width of spacer layer d₂ accompanied with an equal and opposite effect in the width of the spacer layer d₄. Hence it suffices to mention the shift in the width of just one of the spacer layers (say d₂). However as the outermost layers d₁ and d₅ are fixed in their position and the thickness of the solid (CM) layers d₁, d₃ and d₅ is also constant, the total spacer thickness is also constant despite translation of the middle layer towards left (or right) direction. In other words the optical path length of the structure remains fixed.

We now determine the permittivity of the CM layers. Considering CM as non magnetic and a homogeneous mixture of Au (metal) and SiO₂ (dielectric), its effective permittivity is calculated from the following expression for the Bruggeman effective medium theory [23, 24]

ϵ_{C M} = \frac{1}{4} {(3 f_{1} - 1) ϵ_{1} + (3 f_{2} - 1) ϵ_{2} \pm \sqrt{{[(3 f_{1} - 1) ϵ_{1} + (3 f_{2} - 1) ϵ_{2}]}^{2} + 8 ϵ_{1} ϵ_{2}}},

where, ϵ₁, {f₁} and ϵ₂, {f₂ = (1 − f₁)} are permittivity {volume fraction} of metal and dielectric respectively. The sign of the term under square root (in above equation) is to be chosen in such a way that the imaginary part of the effective permittivity is positive, for the medium to be a dissipative (absorbing) medium. The permittivity of metal (Au) was obtained for all optical wavelength from Johnson and Christy’s [25] experimental data through numerical interpolation [as depicted in the Figs. 2(a) and 2(b)]. Dielectric’s permittivity [Figs. 2(c) and 2(d)] of SiO₂ was computed through interpolation of Palik’s [26] data in the visible wavelengths region.

Now two similar s-polarized (or TE) light are incident on the outermost CM layers (i. e. d₁ and d₅) from opposite directions at equal angles of incidence θ_i and −θ_i (with the normal to the surfaces) at z = 0 and z = d respectively as shown in the Fig. 1. We use the label L in subscript for the wave incident from left on z=0 interface. Thus r_L and t_L respectively, are coefficients of the reflected (at z=0) and transmitted (at z=d) waves due to the wave incident from left on z=0 interface. Similarly r_R and t_R respectively are coefficients of the reflected (at z=d) and transmitted (at z=0) waves due to the wave incident from right on z=d interface. The amplitude of scattering at the interface z = 0 (z = d) is given by r_L + t_R (r_R + t_L), square of the absolute value of which, yields the SI. For ease of computation of the reflection and transmission coefficients at the two outermost interfaces of the layered medium, we use a modified T matrix approach (see appendix) that differs slightly from the standard approaches [27, 28]. Thus for the illumination geometry of Fig. 1, the superposition of the reflected and transmitted amplitudes of the two incident fields (with identical incident amplitudes α_i₊ = α_f₋) yields the scattering amplitude from LHS (RHS) interface at z=0 (z=d) as S = r_L + t_R (r_R + t_L).

Fig. 1 Schematic view of the layered medium. Two identical light waves focus (or incident) on opposite interfaces of the system on CM and air layered medium. Here, r_L (t_L) and r_R (t_R) are reflection (transmission) coefficient at the left hand side (LHS) and right hand side (RHS) of the interfaces respectively. Various layers thickness (are not scaled) and the coordinate systems are shown in the figure.

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Fig. 2 First and second rows of the figure depict the real (a, b and e) and imaginary (b, d and f) values of permittivity of respective medium. First, second and third columns show the permittivity of gold (Au) (a and b), silica (SiO₂) (c and d) and composite medium (CM) (e and f with volume fraction f_m = 0.00641) in the optical region respectively.

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Fig. 3 (a) Plot of Log₁₀|SI| as a function of volume fraction (f_m) for a fixed wavelength, λ = 590.03 nm. (b) Absolute values of the amplitude of reflection (|r_L|) (solid line) and transmission (|t_R|) (dash line) coefficients and (c) phase difference, ∆ϕ (units of π) between the right transmitted $(Φ_{t_{R}})$ and left reflected $(Φ_{t_{L}})$ plane waves.

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3. Numerical results and discussion

In this section we present numerical results for the scattered intensity SI (= |r_L + t_R|²) at interface, z = 0 (LHS interface), using the modified characteristic matrix method described in the preceding section and appendix. Results for the reflection (r_L), transmission t_R coefficients and phase difference between them are also shown to understand the origin of CPA. The various parameters used in numerical calculation are as follows. The permittivity of initial, final and spacer layers is the same ϵ_i = ϵ_f = ϵ_air = 1, because these are space (or air) medium. Initially when the layered system is symmetric, i.e., the middle layer d₃ is exactly in the center of the structure, the width of both the spacer layers is identical, d₂ = d₄ and equal to 1180 nm. The fixed widths of the CM layers are d₁ = d₅ = 3000 nm and d₃ = 4000 nm. As described in the preceding section, layers d₁, d₃ and d₅ are composite materials made of gold (Au) and silica (SiO₂) and their permittivities are calculated using BEMT Eq. (1). The angle of incidence of incident light waves is θ_i = 45° in all simulations. Previous studies of CPA in single layer metal-dielectric CM [20–22] have explored in detail, the dependence of the wavelength λ at which CPA occurs, on the layer thickness d and metal volume fraction f_m. They reveal that by fixing any two among these parameters, the remaining third parameter can be optimized by studying the CPA characteristics of the single CM layer. For instance, one may choose a layer thickness d and CPA wavelength λ and determine the required metal volume fraction f_m by studying the CPA characteristics as a function of the volume fraction.

Figure 3 shows such a scheme for determining f_m of CM layers for the present layered structure. Figure 3(a) shows the absorption of incident light waves calculated by evaluating Log₁₀|SI| as a function of volume fraction (f_m) for different positions of d₃ layer. Solid red line depicts the symmetric case i.e., when CM layer d₃ is exactly in the middle of the structure so that d₂ = d₄ = 1180 nm. Dashed blue depicts (left shifted) asymmetric structure case where the d₃ layer was shifted (from the middle of the structure) by a small distance δ = 59 nm towards the left side so that the widths of spacer layers d₂ and d₄ respectively, were modified to 1121 nm and 1239 nm. Similarly the dotted green lines are for right shift of the d₃ layer (from the middle of the structure) by a small distance δ = 59 nm such that the new widths of spacer layers d₂ and d₄ respectively, were 1239 nm and 1121 nm. It is observed in Fig. 3(a) that the scattered intensity (SI) from the symmetric structure, i.e., when d₃ is exactly in the middle (depicted by red line curve) is as low as 10⁻⁸ [see also Fig. 5] at metal volume fraction value, f_m =0.00641 for a fixed incident light wavelength, λ = 590.03 nm. In other words CPA occurs for the symmetric structure case but not for the other two asymmetric cases. This can be understood by examining the magnitude of the reflection and transmission coefficients and the relative phase between them as depicted respectively, in Fig. 3(b) and Fig. 3(c) as a function of f_m. It is clear from these figures that whereas for symmetric case the magnitude of the amplitudes of the reflection and transmission coefficients are equal and they differ in phase by π radians, this is not true for the asymmetric cases at wavelength 590.03 nm.

Now a question arises: is symmetric position the only position at which CPA can occur for these particular values of wavelength (λ = 590.03 nm) and metal volume fraction (f_m =0.00641)? However, before addressing this issue we need to first determine the effective permittivity of the CM layers. The permittivity of Au at this particular wavelength 590.03 nm, is found to be ϵ_Au = −8.3421 + 1.6348i. Using this value of permittivity and volume fraction, f_m = 0.00641 (for Au) in BEMT (Eq. (1)) the effective permittivity of the CM layers of this structure was calculated as ϵ_CM = 2.3667 + 0.0371i. To answer the above question let us look at Fig. 4, where Log₁₀|SI| is plotted as a function of layer spacing d₂ varied by shifting the position of CM layer d₃ (relative to CM layer d₁). Only values of d₂ spacing are shown because d₂+d₄=2360 nm. That is, as the spacing d₂ increases continually from 0 nm to 2360 nm, simultaneously the spacing d₄ decreases from 2360 nm to 0 nm. The solid line curve is the result for a 5 layered structure in which the outermost CM layers have identical widths (d₁ = d₅) that differ from the width d₃ of the middle layer. The dash line curve in Fig. 4 show result for a 5 layered structure in which all the CM layers have different width (d₁ ≠ d₃ ≠ d₅). It is observed that apart from a slight reduction and shift in the position of CPA dips, other features remain essentially the same as those seen for structure with matched outer layer thickness. In this figure the SI for both the cases is observed to be below 10⁻⁵ (corresponding to Log₁₀|SI| = −5) i.e. there is more than 99.999 % absorption. Therefore, at a fixed value of wavelength 590.03 nm and metal volume fraction 0.00641 the layered structure exhibits CPA even at several distinct asymmetric positions of the middle d₃ layer.

Another possibility that is essential to verify is: for a fixed volume fraction and under the same translation conditions, does the structure exhibit CPA at other wavelengths?

Figure 5(a) shows the scattering Log₁₀|SI| at LHS interface as a function of wavelength (λ) for a fixed value of the volume fraction f_m = 0.00641. Also shown in Fig. 5(b) are the reflection coefficient (|r_L|)(solid line) and transmission coefficient |t_R| (dotted line). Figure 5(c) shows the phase difference (∆Φ) between them as a function of wavelengths (λ). In all these figures the volume fraction is fixed at f_m = 0.00641 for all the CM layers. Now, the middle CM layer d₃ layer was again translated by a small distance δ = 59.0 nm to the left side into the adjacent spacer layer with respect to the symmetric (middle) position of the system, so that d₂ and d₄ were changed to 1121.0 nm and 1239.0 nm respectively. Similarly for right shift of the middle layer, the spacing d₂ and d₄ were 1239.0 and 1121.0 nm respectively. Here we find that symmetric position of d₃ layer also diplays some absorption; (around ≤ 10⁻²). The symmetric structure shows CPA at two different wavelengths (590.0 nm and 610.5 nm). They are similar to the single slab result shown in [6, 20, 22]. These results also indicate the possibility of attaining CPA at many more distinct wavelengths for asymmetric structures under oblique angle of illumination. From Fig. 5(b) and 5(c) we observe that CPA dips occur due to destructive interference, that is when the reflection (|r_L|)(solid line) and transmission |t_R||(dotted line) coefficients are equal in magnitude but differ in phase by π radians.

Figure 6(a) shows a three dimensional plot of the scattering (Log₁₀|SI|) as a function of both wavelength (λ) and air spacing d₂ created by changing the position of the middle layer d₃ relative to the first layer d₁. The volume fraction (f_m = 0.00641) was constant for all CM layers. This figure shows the occurrence of multiple CPA dips as a function of d₂ at many values of wavelengths for a particular value of volume fraction. To better understand this behaviour we plot Log₁₀|SI| in Fig. 6(b) as a function of the position of middle layer (d₂) for a fixed value of volume fraction f_m = 0.00641. It is seen from this figure that for the same f_m (=0.00641) CPA is observed not at a single but four distinct wavelengths labeled [in Fig. 6(a)] as λ_G = 587.4 nm, λ_Y = 590.0 nm, λ_O = 610.5 nm and λ_R = 614.0 nm. Thus for a given wavelength say λ_Y = 590.0 represented by blue line in Fig. 6(b), the five CPA dips observed as a function of d₂ correspond to five CPA dips labeled Y₁ to Y₅ in Fig. 6(a) and so on.

To further explore the sensitivity of the layered structure to the thickness of CM layers, we show in Fig. 7, CPA for very thick and thin CM layers for same values of wavelength (λ = 590 nm) and metal volume fraction (f_m = 0.00641) as in previous graphs. Figure 7 shows results for a 5 layered structure - in which width of all the CM layers is the same-for two different CM slab thicknesses d₁ = d₃ = d₅ = 300 nm and d₁ = d₃ = d₅ = 3000 nm. It is obvious from above results that the thickness of the CM layers does not cause any significant deviation and all the layered structures display similar features. However the half-width at half maximum (HWHM) of the CPA resonance does depend upon the thickness of the CM layer and is found to be higher for larger thickness. To verify this fact we show results for much thicker CM slabs [see Fig. 8] where the HWHM is found to be around 10 nm. Similar feature was observed in the studies of Chong et al [6] where the width of the CPA resonance was found to increase with increasing thickness of the CPA film. This similarity arises as here we are operating in the regime of very low metal content such that the specific dispersion of the material resembles that of the dielectric. On the other hand opposite behavior is seen if the metal content is very high, as then the specific dispersion of a thin film resembles that of the metal which can lead to occurrence of broadband CPA [10].

Figure 9 demonstrates the effect of angle θ_i on CPA for both normal and oblique incidence of the light waves. In normal incidence case, higher absorption and number of CPA dips can be seen compared with oblique incidence under same condition of d₃ translations. These results indicate that a potential application of CPA could be in position sensitivity measurements. Although several possibilities exist, we propose one such precise position sensor device [see Fig. 10] capable of detecting the changes in position to very high degree of accuracy. The reflected beams are detected by two photo diodes PD₁ and PD₂, output of which can be given either to a scope or a PC. The two outermost layers d₁ and d₅ are fixed to the end of the CM tube while the movable middle CM layer d₃ is connected to the external device whose position needs to be controlled and also to a piezoelectric transducer (PZT) which monitors the position through an external voltage. The voltage can be very accurately measured, the displacement of the middle layer can easily be tracked. The voltage across the PZT can thus be a very accurate measure of the displacement and accuracies better than a nanometer are achievable with this type of set ups depending upon the thickness of the CM layers.

In the present case however this precision is limited by HWHM of around 4nm (of CPA resonance) and is found to be governed by the thickness of the CM layers. Thus by appropriate choice of CM layer thickness it is possible to attain the desired accuracy. For example using thicker CM slabs (d₁ = d₃ = d₅= 0.1 mm) [see Fig. 8] would limit the sensor accuracy to around 10 nm HWHM. On the other hand compared with the complicated designs of earlier proposals [1–5], the present scheme apparently is more rugged and simpler from fabrication viewpoint as it involves a multilayer stack.

Fig. 4 Variation of Log₁₀ |SI| as a function of d₂ (width of spacer layer) for fixed values of wavelength, λ (= 590.03 nm) and metal volume fraction, f_m (=0.00641). Note that width d₂ of spacer layer is actually the position of CM layer d₃ from the first CM layer d₁. Also shown is the case (red dashed line) when widths of all three CM layers are unequal, i.e., d₁ ≠ d₃ ≠ d₅.

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Fig. 5 (a) Plot of log₁₀|SI| as a function of wavelength (λ) for a fixed value of volume fraction f_m = 0.00641. (b) Absolute values of reflected (|r_L|) (solid line) and transmitted (|t_R|) (dash line) amplitudes and (c) phase difference, ∆Φ (scale of π) between the right transmitted $(Φ_{t_{R}})$ and left reflected $(Φ_{t_{L}})$ plane waves.

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Fig. 6 Scattering Log₁₀|SI| plotted as functions of (a) wavelength (λ) and spacing d₂ caused by translation of middle layer d₃ relative to first (d₁) layer. (b) Log₁₀|SI| plotted as a functions of function of the position of middle layer (d₂). Here, volume fraction f_m = 0.00641, angle of incidence θ_i = 45° and CPA observed at λ_G = 587.4 nm, λ_Y = 590.0 nm, λ_O = 610.5 nm and λ_R = 614.0 nm.

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Fig. 7 Log₁₀|SI| plotted as a function of position of the middle d₃ layer for 45° angles of incidence.

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Fig. 8 Variation of scattering intensity ((Log₁₀|SI|) as a function of (a) wavelength λ and (b) position of middle layer relative to d₁. Width of CM layers is d₁ = d₃ = d₅ = 0.1 mm and total spacer width is d₂ + d₄ = 0.02 mm. The angle of incidence θ_i = 45°, λ = 644 nm and f_m = 0.001164. Here HWHM of CPA resonances is around 10 nm.

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Fig. 9 log₁₀|SI| is plotted as a function of position of the middle layer for normal incidence (dash line, λ = 589.1 nm and f_m = 0.00552 and oblique incidence (solid line, λ = 590 nm and f_m = 0.00641)).

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Fig. 10 Top view of proposed setup for high precision position sensing device based on CPA in multilayered structure. BS₁, BS₂ and BS₃ are 50% - 50% beam splitters, M is a mirror, PD₁ and PD₂ are photodetectors. One end of the piezoelectric transducer (PZT) is rigidly fixed to the device wall while the other movable end is in contact with both the device whose position is to be monitored and the central movable CM slab d₃ mounted on a ring. A narrow slots is cut along the length of the wall of the tube so that the shaft connected through it to the movable central CM slab can move freely along the length of the tube between the other two CM slabs d₁ and d₅ fixed to both ends of the CM tube.

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4. Summary and conclusions

We propose here, a scheme for pecision position sensing based on the phenomenon of CPA in a five layer medium, under symmetric and asymmetric structure conditions. A detailed analysis of the CPA as a function of volume fraction, incident field wavelength and position of the middle layer is developed. Translation of middle layer leads to multiple exteremely narrow (HWHM = 4 nm) CPA resonances. These can be exploited for the purpose of position sensing of an external device. Moreover the detection accuracy (sensitivity) can be altered by changing the thickness of the composite medium layers. An experimental set up based on this idea is proposed.

Appendix

For a TE polarized light beam propagating in j^th layer, the spatial part of the the positive frequency component of the electric field can be expressed as $E_{j y} = α_{j +} e^{i k_{j z} z} + α_{j -} e^{- i k_{j z} z}$ where α_j₊ denotes the amplitude of the forward moving wave (in positive z direction) and α_j₋ is that for the the wave moving in backward (negative z) direction. Here $k_{j z} = \sqrt{k_{j}^{2} - k_{j x}^{2}} = \sqrt{k_{0}^{2} ϵ_{j} - k_{0}^{2} \sin^{2} θ}$ , ϵ_j is permittivity of j^th layer, $k_{0} = \frac{ω}{c}$ is the wave vector in vacuum, and θ is the angle of incidence. Using the relation for the corresponding magnetic field, $i k_{0} H_{j x} = - \frac{\partial E_{j y}}{\partial z}$ , the propagation equation for j^th layer can be expressed in matrix form as $[\begin{array}{l} E_{j y} \\ H_{j x} \end{array}] = D_{j} (z) [\begin{array}{l} α_{j +} \\ α_{j -} \end{array}]$ where

D_{j} (z) = [\begin{matrix} e^{i k_{j z} z} & - e^{i k_{j z} z} \\ - \frac{k_{j z}}{k_{0}} e^{i k_{j z} z} & - \frac{k_{j z}}{k_{0}} e^{- i k_{j z} z} \end{matrix}]

is the dynamical matrix for j^th layer. Thus for a TE field incident (at an angle θ) from left on z = 0 interface between the i (air) and first (CM) layers, α_i₊ and α_i₋ correspond respectively, to the amplitudes of the incident and reflected fields. However the field transmitted through the interface at z = d from CM (layer-5) into air (layer-f) has only forward amplitude, α_f₊ i.e., α_f₋ = 0. For a given incident (from left) field amplitude α_i₊ we can determine the reflected field amplitude α_i₋ and transmitted field amplitude α_f₊ through the following relation obtained by matching the field amplitudes at the boundaries of various interfaces

[\begin{array}{l} α_{i +} \\ α_{i -} \end{array}] = [D_{i}^{- 1} (0) D_{1} (0)] [D_{1}^{- 1} (d_{1}) D_{2} (d_{1})] \dots [D_{5}^{- 1} (d) D_{f} (d)] [\begin{array}{l} α_{f +} \\ 0 \end{array}] = [\begin{array}{l} M_{11} \\ M_{21} \end{array}] α_{f +}

From the above equation we can find the transmission and reflection coefficients as $t_{L} = \frac{α_{f +}}{α i +} = \frac{1}{M_{11}}$ and $r_{L} = \frac{α_{i -}}{α_{i +}} = \frac{M_{21}}{M_{11}}$ respectively, for a TE electromagnetic field incident from left on air-CM interface at z = 0. In a similar manner for a TE field incident from right on z = d interface between air (layer-f) and the last CM (layer-5), α_f₋ and α_f₊ correspond respectively, to the amplitudes of the incident and reflected fields. Here too the field transmitted through the interface at z = 0 from CM (layer-1) into air (layer-i) has only single (backward) amplitude, α_i₋ i.e., α_i₊ = 0. Proceeding in the same manner as outlined above, for a TE electromagnetic field incident from right on air-CM interface at z = d, we can determine the transmission and reflection coefficients respectively, as $t_{R} = \frac{α_{i -}}{α_{f -}} = \frac{1}{{M^{'}}_{22}}$ and $r_{R} = \frac{α_{f +}}{α_{f -}} = \frac{{M^{'}}_{12}}{{M^{'}}_{22}}$ . Here ${M^{'}}_{12}$ and ${M^{'}}_{22}$ are obtained from the following relation attained through matching of field amplitudes at boundaries of various interfaces

[\begin{array}{l} α_{f +} \\ α_{f -} \end{array}] = [D_{f}^{- 1} (d) D_{5} (d)] [D_{5}^{- 1} (d - d_{5}) D_{4} (d - d_{5})] \dots [D_{1}^{- 1} (0) D_{i} (0)] [\begin{matrix} 0 \\ α_{i -} \end{matrix}] = [\begin{matrix} {M^{'}}_{12} \\ {M^{'}}_{22} \end{matrix}] α_{i -}

However when both the (LHS and RHS) beams are incident simultaneously as shown in Fig. 1, the scattering amplitude at any outermost layer can be calculated using the superposition principal, that is, (r_L + t_R) at the first (z=0) interface and (r_R + t_L) on the last (z=d) interface.

Acknowledgments

Financial support from DST-PURSE program of University of Hyderabad, Hyderabad, India is acknowledged. DNR acknowledges DAE for Raja Ramanna fellowship.

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High precision position sensor based on CPA in a composite multi-layered system

Abstract

1. Introduction

2. Geometry and formulation

3. Numerical results and discussion

4. Summary and conclusions

Appendix

Acknowledgments

References and links

Cited By

Figures (10)

Equations (4)

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