Four-Layer Slab Waveguide Sensors Supported with

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SENSOR LETTERS Vol. 9, 1–7, 2011

Four-Layer Slab Waveguide Sensors Supported with Left Handed Materials M. M. Abadla1 ∗ , S. A. Taya2 , and M. M. Shabat2 Physics Department, Alaqsa University, Gaza Strip, Palestinian Authority Physics Department, Islamic University, Gaza, P.O.Box 108, Gaza Strip, Palestinian Authority 1

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(Received: 9 February 2011. Accepted: 8 April 2011) The present paper deals with a multilayer slab waveguide structure for sensor applications. We assume a four layer slab waveguide structure with one of these layers made of a left-handed material (LHM) of negative permittivity () and permeability (). Using the Fresnel reflection coefficients, the reflectance of the structure is studied in details. The sensitivity of the effective refractive index to variations in the refractive index of a measurand homogeneously distributed in the cladding layer is also presented.

Keywords: Slab Waveguides, Left-Handed Materials, Reflectance, Sensitivity.

1. INTRODUCTION

∗

Corresponding author; E-mail: [email protected]

Sensor Lett. 2011, Vol. 9, No. 5

1546-198X/2011/9/001/007

doi:10.1166/sl.2011.1729

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In general, electric permittivity () and magnetic permeability () determine how the material interacts with electromagnetic radiation. Almost all materials encountered in optics, such as glass and water, have positive values for permittivity and permeability and hence the positive square √ root is chosen to give a positive value of the index of refraction n. When a beam of light enters a material at an angle, it is bent by an amount determined by the positive refractive index of the material. In a newly discovered type of artificial materials called left-handed materials (LHMs) with negative index of refraction,1–4 a beam of light will be bent in the opposite way. Due to the negative index, metamaterials have been subjected to research interest in the field of optoelectronics because they are promising for a variety of optical and microwave applications, such as new types of modulators, band pass filters, lenses, and microwave couplers. One of the first applications of the LHM was reported by Pendry,5 who demonstrated that a slab of a lossless left-handed material can provide a perfect image of a point source. Grbic et al.6 verified by simulation the enhancement of evanescent waves in a transmission-line network by using a negative refractive index material. Qing and Chen7 showed that LHMs can enhance the evanescent field in planar slab waveguides. Recently, LHMs

have been proposed as a mechanism of building cloaking devices.8 In this mechanism, the object to be cloaked is surrounded by a shell of LHM that affects the passage of light near it. LHMs have been intensely studied in the field of antenna arrays.9 As superstrates, LHMs showed a remarkable improvement in microwave and millimeter wave antenna arrays. In a recent application of LHMs, Taya et al. showed by simulation that using a thin layer of LHM between the guiding layer and the cladding layer can dramatically enhance the sensitivity of slab waveguide sensors.10 Slab waveguides have been studied extensively as optical sensors11–14 in which the effective refractive index is used to probe the variations in the refractive index of an analyte. This recent application of LHMs in the filed of optical waveguide sensing deserves more interest. In Ref. [10], the thin layer of the LHM was inserted between the guiding layer and the cladding layer and the argument was restricted to TE modes. In the present work, we assume a similar structure to that considered in Ref. [10] with the LHM layer inserted between the guiding layer and the substrate layer. The advantage of the present work compared to Taya’s structure is that we here discuss both types of optical waveguide sensing. In the first, the angular shift of the reflectance dip is treated as the probe for detecting the changes in the refractive index of an analyte. In the second type, the change in the effective refractive index is considered as the probe. As another advantage of the present work, we treat both TE and TM polarizations. The sensitivity

Four-Layer Slab Waveguide Sensors Supported with Left Handed Materials

enhancement in this configuration is explained in scope of the evanescent field in the cladding layer due to the generation of surface polaritons at the boundary between the metamaterial and the dielectric layers. The presented work studies theoretically four-layer slab waveguide structure in which one of the layers is assumed to be a lossy LHM with both the and being simultaneously negative and complex. We assume that the multilayer waveguide is operated as an optical sensor in reflected mode. The reflectance of the sensor is derived and studied in details with different parameters of the structure. The sensitivity of the proposed sensor is also presented and plotted with the parameters of the structure.

2. THEORY

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In Figure 1, a schematic diagram of the planar waveguide structure under consideration is presented. The waveguide basically comprises four layers: substrate (usually glass); a thin layer of lossy LHM; waveguiding film (usually lossless dielectric) and a cover medium whose RI is to be measured. The proposed sensor may be operated in reflection mode, in which the waveguide is illuminated from below while the reflected intensity is studied as a function of the angle of incidence (, or instead as a function of the effective refractive index of the structure. The field F in the sensor layers is given by: F C = AC e

ikc z−dM −dF ix−t

e

(1)

FF = AF eikF z−dM + BF e−ikF z−dM eix−t

(2)

FM = AM eikM z + BM e−ikM z eix−t

(3)

FS = BS e

−ikS z ix−t

e

(4)

where F represents the electric field for TE modes and the magnetic field for TM modes, is the propagation constant in x-direction, ki = ± ko2 n2i − 2 , ni is the refractive index of layer i, where i = S, M, F, C to denote substrate, LHM, film and clad; respectively, and ko is the free space wave number. The effective index (N ) can be written in terms of as N = /ko .

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Matching the fields and their derivatives at the interfaces and solving for AM and BM we get: At the substrate-LHM interface (z = 0) B (5) AM = S kM − M 1− kS 2kM S B BM = S kM + M 1− kS (6) 2kM S where denotes the relative permeability of the LHM and i denotes the relative permittivity of layer i. is called the polarization parameter and is given by = 0 for TE modes and = 1 for TM modes. The film, the cladding, and the substrate are assumed to be nonmagnetic materials. The film, the cladding, and the substrate are assumed to be nonmagnetic materials. At the LHM-film interface (z = dM 1 AM eikM dM F AF = kF + k 2kF M 1− M B e−ikM dM 1 F kF − (7) + M k 2kF M 1− M

F 1 kF − k M 1− M B eikM dM 1 F + M kF + k 2kF M 1− M

A eikM dM BF = M 2kF

rMF + rFC e2ikF dF 1 + rMF rFC e2ikF dF

(12)

rSM =

S 1− kM − M kS S 1− kM + M kS

(13)

rMF =

M 1− kF − F kM M 1− kF + F kM

(14)

rMFC =

Guiding layer (film)

dF

Left-handed material

dM

Substrate

(8)

At the film-clad interface (z = dM + dF A AF = C kF + F (9) kC e−ikF dF 2kF C A kC eikF dF (10) BF = C kF − F 2kF C If we use Fresnel’s reflection coefficients of the fourlayer structure shown in Figure 1, the total reflectance can be found as rSM + rMFC e2ikM dM 2 2 (11) R = rSMFC = 1 + rSM rMFC e2ikM dM where

Cladding

C kF − F kC (15) C kF + F kC The four-layer mode equation can be obtained using the above set of equations as rFC =

Fig. 1. A schematic diagram of the multilayer slab waveguide sensor under consideration.

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FC + FMS + 2kF dF = 2m

where

FC = −2i tan

FMS = FS +

−1

F C

kC kF

2 2irMS 1 − rFM ·k d rFM + rMS 1 + rFM rMS M M

FS = −2 tan

−1

F S

kS kF

(16) (17)

(18)

(19)

where FC , FMS , and FS are the phase shifts at the film-cover interface, at the film-LHM interface, and at LHM-substrate interface. The sensitivity of effective refractive index to changes in the refractive index of an analyte uniformly distributed in the cladding can be calculated as S = − / nC / / N ,15 where denotes the dispersion relation. Carrying out these differentials, we get:

2k2 n k F =G o C F (20) nC kC C

2ko2 N (21) T j + dF + H M =− N kF J =S C where

G = kF2 + F C TJ =

1 + 21− M /S 2 − 21− M /S 2 kS2 1 + F /S 2 − 2 kF − F /S 2 kS2 2 kM

−1 2 N 2 kC 2 −1 nC

kF2 + kJ2 1 F kJ J kF2 + F /J 2 kJ2

(22)

(23)

(24)

Following the definition, the sensitivity of the proposed sensor can be expressed in the general form as:

−1 kF2 nC F S=G T J + dF + H M (25) kC N C S C

3. NUMERICAL RESULTS AND DISCUSSION A computer code was developed to solve the dispersion relation given by Eq. (16) and evaluate various quantities according to some given parameters. The effective index ranging between ns and nc (N = ns sins was used to evaluate the appropriate range of wave-guiding film thickness that must be maintained for the sensor to show these Sensor Letters 9, 1–7, 2011

Fig. 2. Fresnel reflectance R versus the effective refractive index N for dF = 3 m (leftmost), dF = 35 m (middle), and dF = 4 m (rightmost) for TE (solid lines) and TM (dotted) modes. M = −4+001i, M = −1+ 001i, dM = 07 m, ns = 15, nf = 16, nc = 133, and = 6300 nm.

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H = 1 + 2kF2

values of N . The calculations showed that the appropriate wave-guiding film thickness starts from nearly 1 m. Lossless LHM only represents an ideal case, which cannot be realized in the present designs. However, LHMs with minimal absorption coefficient were recently proposed.16 LHMs with minimal absorption coefficient have shown interesting features. It was verified that LHMs with low loss can focus light onto an area smaller than a square wavelength in near fields.17 18 This super-resolution is attributed to an important feature of LHMs which is amplification of evanescent waves.19 For these reasons, in the present work we treat LHMs with and having the form M = r + 001i and M = r + 001i with r and r are both negative. Another reason that supports this assumption is the requirements that rFC and rFMS must be unity in magnitude. Figure 2 shows the reflectance versus the effective refractive index for different waveguiding film thicknesses for TE (solid lines) and TM (dotted lines) modes. The figure reveals many interesting features. The reflectance has a dip at a specific value of the effective refractive index. This dip shifts towards lower values of the reflectance at higher values of the effective refractive index with increasing the wave-guiding film thickness. The reflectance at these dips drops to minimum values but not zero as if using a metal instead of the LHM layer.20 The width of the dip gets narrower with increasing wave-guiding film thickness. In surface plasmon resonance (SPR) slab waveguide sensors, the angular shift of the reflectance dip is adopted as the probe for measuring the change in the refractive index of the measurand (analyte). The narrow reflectance dip is preferable in such sensors.20

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The reflectance versus the effective refractive index for different thicknesses of the LHM layer for TE (solid lines) and TM (dotted lines) modes is shown in Figure 3. Almost the same features observed in Figure 2 can be seen in Figure 3 with an essential difference. As the thickness of the LHM layer increases, the depth of the dip increases and in contrary to its behavior with the waveguiding film thickness, it shifts towards lower values of the effective refractive index. Also the LHM layer thickness doesn’t have a large impact on the dip width. As can be seen from the figure, increasing the thickness of the LHM layer from 0.4 m to 1 m is not associated with a considerable change in the dip width. Moreover, Figure 3 also reveals another important issue, namely the difference in dip depth between TE and TM modes. In TM modes, the change in dip depth with increasing the thickness of the LHM layer is barely detectable whereas it is considerable in TE modes. It is very important to study the effect of the negative parameters of the LHM layer on the reflectance of the proposed structure. The variation of the reflectance with the effective refractive index and the negative permittivity of the LHM layer is depicted in Figure 4. It is apparent that for TE modes the positions of reflectance dips are shifted left due to increasing the absolute value of the real part of the permittivity. Moreover, these dips get deeper and narrower with increasing the absolute value of the real part of the permittivity. In TM case, the situation is completely reversed. The reflectance dips get less deep and wider with increasing the absolute value of the real part of the permittivity. Thus, if the proposed structure is used as an optical

Fig. 3. Fresnel reflectance R versus the effective refractive index N for dM = 1 m (leftmost), dM = 07 m (middle), and dM = 04 m (rightmost) for TE (solid lines) and TM (dotted) modes. M = −4 + 001i, M = −1 + 001i, dF = 3 m, ns = 15, nf = 16, nc = 133, and = 6300 nm.

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Fig. 4. Fresnel reflectance R versus the effective refractive index N for M = −14 + 001i (leftmost), M = −9 + 001i (middle), and M = −4+001i (rightmost) for TE (solid lines) and TM (dotted) modes. M = −1 + 001i, dF = 4 m, dM = 07 m, ns = 15, nf = 16, nc = 133, and = 6300 nm.

sensor (with the angular shift of the reflectance dip as the probe for detection of the variations in the refractive index of an analyte uniformly distributed in the cladding layer), higher TE modes with higher (absolute) real parts of permittivity is recommended. This contradicts the conventional three-layer slab waveguide optical sensor in which TM0 mode is usually recommended.15 The effect of the negative permeability of the LHM layer on the reflectance is also shown in Figure 5. The dependence of the reflectance on the permeability is slightly different from its dependence on the permittivity. As can be seen from the figure, the positions of the reflectance dips are shifted left due to increasing the absolute value of the real part of the permeability. These dips get deeper and narrower with decreasing the absolute value of the real part of the permeability. The behavior of the reflectance for TM modes is similar to that for TE modes. However, these dips are less deep and wider for TM modes. Thus we again recommend TE modes for applications depending on measuring the location of the dip such as optical sensing. The sensitivity S of the proposed structure is also calculated using Eq. (25) and plotted versus the effective refractive index for different parameters of the LHM layer. These plots are shown in Figures 6–8. In Figure 6, the sensitivity shows a considerable decay with the effective index of the structure. This behavior is attributed to the fraction of power flowing in each layer. When effective index value approaches that of the cladding, most of the power is flowing in the analyte distributed in the cladding and the sensitivity is high. On the other hand, when the Sensor Letters 9, 1–7, 2011

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Fig. 5. Fresnel reflectance R versus the effective refractive index N for M = −2 + 001i (leftmost), M = −15 + 001i (middle), and M = −1 + 001i (rightmost) for TE (solid lines) and TM (dotted) modes. M = −6 + 001i, dF = 4 m, dM = 07 m, ns = 15, nf = 16, nc = 133, and = 6300 nm.

Fig. 6. The sensitivity S versus the effective refractive index N for dM = 05 m (uppermost), dM = 1 m (middle), and dM = 15 m (lowermost) for TE (solid lines) and TM (dotted) modes. M = −3 + 001i, M = −1 + 001i, dF = 3 m, ns = 15, nf = 16, nc = 133, and = 6300 nm.


in Figure 7. As can be seen from the two figures, the sensitivity of both modes drops in Figure 7 compared to Figure 6. The effect of decreasing the real part of M from −3 (Fig. 6) to −5 (Fig. 7) reduces the sensitivity of TM modes by a slight amount and that of TE modes by a considerable amount. To study the effect of the permeability of the LHM layer, Figure 8 is plotted at the same conditions of Figure 7 with one difference, namely M = −1 + 001i in Figure 7 and M = −35 + 001i in Figure 8. A considerable decrease in the sensitivity of both modes is observed in Figure 8 due to changing the real part of M from −1 (Fig. 7) to −3.5 (Fig. 8). The impact of this change on TM modes is much greater than on TE’s. The effect of the LHM layer thickness on the sensitivity is shown in Figures 6–8. Inspection of these two figures shows clearly that the lower the thickness of the LHM layer is the higher the sensitivity of the proposed structure to changes in the index of the cladding. The explanation of this behavior may be based on the evanescent field extended from the guiding layer to the surrounding layer. The evanescent field extended to the cladding layer is responsible for the sensing operation. This part of the field can be enhanced by decreasing the thickness of the LHM layer. It is significant to discuss the preference of a mode over the other in optical sensing applications. Slab waveguides can be used for optical sensing in two configurations. In the first configuration, the reflectance dip shift is used as an indicator of changes in the refractive index of the cladding. TE modes exhibit a sharp and deep reflectance dip, therefore it is recommended for such a configuration. This behavior for the reflectance for TE modes was 5

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effective index approaches that of the waveguiding layer or that of the substrate, a few percent of the total power flows in the cladding and the sensitivity is low. Figure 7 is plotted at the same conditions of Figure 6 with one difference, namely M = −3 + 001i in Figure 6 and M = −5 + 001i




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reported by Nina et al. for a four-layer structure comprising a metal layer in the place of the LHM in our structure. In the second configuration, the change in the effective refractive index of the mode is used to detect the changes in the index of the cladding. As can be seen form Figures 6–8, the preference of a mode over the other depends on the parameter of the structure. In Figure 6, the two modes exhibit almost the same sensitivity with TE modes having a little bit smaller values. But changing the real part of M from −3 (Fig. 6) to −5 (Fig. 7) reduces the sensitivity of TE modes by a considerable amount giving a preference for TM modes. On the other hand, changing the real part of M from −1 (Fig. 7) to −3.5 (Fig. 8) reverses the situation and shows a preference for TE modes. Thus, in the second configuration of optical sensing, the preference of one of the modes is critically dependent of the chosen parameters. Finally, it is worth to compare our results for the sensitivity with the four-layer structure proposed in Ref. [20] in which a layer of metal is used in the place of the LHM layer of the present structure. In fact, studying the effect of a thin LHM layer (especially the negative permittivity) is expected to give some characteristics similar to the so called metal clad sensors20 (using thin metallic layers on substrates). For the sake of comparison, we plotted sensitivity against the effective refractive index for two sensors. The result is shown in Figure 9. As can be seen from the figure, the sensitivity of the proposed sensor is much higher than that of the metal layer. This is again due to the surface polaritons that enhance the evanescent waves caused by the LHM layer.7 10 6

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Fig. 9. The sensitivity S versus the effective refractive index N for a structure with a LHM layer (solid lines) and a structure with a metal layer (dotted). dM = 05 m, M = −3 + 001i, M = −1 + 001i, dF = 3 m, ns = 15, nf = 16, nc = 133, and = 6300 nm.

4. CONCLUSION Four-layer slab waveguide structure in which one of the layers is made of a LHM has been investigated in details. The study has been devoted to effect of the negative parameters on the reflectance of the structure and the sensitivity of the effective index to changes in the refractive index of the cladding. The simulation results have shown many interesting features in the reflectance curves and the sensitivity values as a result of the negative parameters of the LHM layer. These features include: dependence of the shape and position of the reflectance dip on the negative parameters of the LHM layer, the obvious difference between the reflectance curves between TE and TM modes, the preference of TE modes over TM modes for sensing applications depending of the reflectance dip shift, and the high sensitivity of the proposed structure. It is clear that LHMs can be used for promising generations of optical waveguide sensors.

References and Notes 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.

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11. M. M. Shabat, H. Khalil, S. A. Taya, and M. M. Abadla, Int. J. Optomechatronics 1, 284 (2007). 12. H. M. Khalil, M. M. Shabat, S. A. Taya, and M. M. Abadla, Int. J. Modern Phys. B 21, 5075 (2007). 13. S. A. Taya, M. M. Shabat, H. Khalil, and Dieter S. Jäger, Sen. and Act. A 147, 137 (2008). 14. S. A. Taya, M. M. Shabat, and H. Khalil, Int. J. Light and Elect. Optics, Optik 121, 860 (2010).

15. K. Tiefenthaler and W. Lukosz, J. Opt. Soc. Am. B 6, 209 (1989). 16. J. Kästel stel, M. Fleischhauer, S. F. Yelin, and R. L. Walsworth, Phys. Rev. Lett. 99, 073602 (2007). 17. V. A. Podolskiy and E. E. Narimanov, Opt. Lett. 30, 75 (2005). 18. L. Zhao and T. J. Cui, Appl. Phys. Lett. 89, 141904 (2006). 19. C. Yan, Q. Wang, and Y. Cui, Optik 121, 63 (2010). 20. N. Skivesen, R. Horvath, and H. C. Pedersen, Sens. Actuators B 106, 668 (2005).

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