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Enhancement of sensitivity in optical waveguide sensors using left-handed materials
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Optik
Optics
Optik 120 (2009) 504–508 www.elsevier.de/ijleo
SHORT NOTE
Enhancement of sensitivity in optical waveguide sensors using left-handed materials S.A. Tayaa, M.M. Shabata,b,, H.M. Khalilc a
Physics Department, Islamic University, P.O. Box 108, Gaza, Gaza Strip, Palestinian Authority Max-Planck-Institut fu¨r Physik komplexer Systeme, No¨thnitzer Street, 3801187 Dresden, Germany c College for Girls, Ain Shams University, Cairo, Egypt b
Received 1 October 2007; accepted 23 December 2007
Abstract We show analytically that the sensitivity of an optical waveguide sensor can be dramatically enhanced by using a metamaterial with negative permittivity and permeability. The variation of the sensitivity of the proposed waveguide sensor with different parameters of the waveguide is studied. It is found that the sensitivity of the sensor increases with the increasing thickness of the metamaterial due to the surface polariton generation. r 2008 Elsevier GmbH. All rights reserved. Keywords: Left-handed materials; Optical waveguide sensors; Sensitivity analysis
1. Introduction Optical evanescent wave sensors have been widely used for various purposes such as humidity sensing [1], chemical sensing [2], biochemical sensing [3], and biosensing [4]. Optical waveguide sensors have shown many attractive features such as the immunity to electromagnetic interference, the use in aggressive environments, and, in general, a high sensitivity [5]. The sensing of the waveguide sensors is performed by the evanescent tail of the modal field in the cover medium. The guided electromagnetic field of the waveguide mode extends as an evanescent field into the cladding and substrate media and senses an effective refractive index of the waveguide. The effective refracCorresponding author at: Max-Planck-Institut fur Physik kom¨ plexer Systeme, No¨thnitzer Street, 3801187 Dresden, Germany. Tel.: +35 18712115; fax: +35 18711999. E-mail addresses: [email protected] (S.A. Taya), [email protected] (M.M. Shabat).
0030-4026/$ - see front matter r 2008 Elsevier GmbH. All rights reserved. doi:10.1016/j.ijleo.2007.12.001
tive index of the propagating mode depends on the structure parameters, e.g., the guiding layer thickness and dielectric permittivity and magnetic permeability of the media constituting the waveguide. As a result, any change in the refractive index of the covering medium results in a change in the effective refractive index of the guiding mode. The basic sensing principle of the planar waveguide sensor is to measure the changes in the effective refractive index due to changes in the refractive index of the covering medium. Parriaux and Velduis [6] presented an extensive theoretical analysis for the design of evanescent linear waveguide sensors and derived the conditions for the maximum achievable sensitivity for both TE and TM polarizations. Horvath et al. [7] demonstrated the design and implementation of a waveguide sensor configuration called reverse symmetry in which the refractive index of the aqueous cladding is higher than that of the substrate material. The reverse symmetry waveguide has been tested for bacterial and cell detection and it showed a considerably high sensitivity compared with the
ARTICLE IN PRESS S.A. Taya et al. / Optik 120 (2009) 504–508
conventional waveguide sensor. We have investigated the sensitivity of a nonlinear waveguide structure where one of the layers of the waveguide structure is considered to be nonlinear medium [8]. We showed that the sensitivity can be enhanced when the covering medium has an intensity-dependent refractive index. In recent years, much attention is drawn to metamaterials with simultaneously negative permittivity and permeability that are often referred to as left-handed materials. The history of these materials begins with the paper of Veselago [9], who proposed a hypothetical material with simultaneously negative dielectric permittivity and magnetic permeability and studied the propagation of electromagnetic waves in such a medium. In his article in 1968, Vesegalo predicted a number of interesting features of waves in metamaterials, including negative refraction. These materials are not found in nature, and need to be artificially constructed. So far the left-handed materials are demonstrated only in the microwave frequency range [10]. One of the first applications of the left-handed materials was reported by Pendry [11], who demonstrated that a slab of a lossless left-handed material can provide a perfect image of a point source. Gribe and Eleftheriades [12] verified by a simulation the enhancement of evanescent waves in a transmission-line network by using a negative refractive index material. In [13], it was shown that left-handed materials can enhance the evanescent field in slab waveguides. The main aim of the present paper is to show that the sensitivity of the conventional optical sensors can be dramatically enhanced by inserting a layer of lefthanded material between the cladding and the guiding layer and to study the variation of the sensitivity of the proposed sensor with the parameters, i.e., thickness, permittivity, and permeability, of the left-handed material.
2. Structure analysis Fig. 1 shows a schematic structure of the slab waveguide under consideration. A guiding layer with permittivity ef, permeability mf, and thickness d1 is sandwiched between a semi-infinite substrate with permittivity es and permeability ms and a semi-infinite cladding with permittivity ec and permeability mc. In the working region, there is a layer of metamaterial with negative permittivity em, negative permeability mm, and thickness d2 between the cladding and the guiding layer. Here we assume that all the materials are lossless, i.e., both e’s and m’s are real. We also consider the TE waves in which the electric field E is polarized along the y-axis. The well-known Helmholtz equations for TE modes are
505
z Non working region
workingregion
Non working region
Cladding Metamaterial
d2 z=0
x d1
Guiding layer
Substrate
Fig. 1. Schematic structure of integrated waveguide sensor with a metamaterial layer.
described by the following scalar equation: q2 E y ðzÞ þ ðo2 ðzÞmðzÞ b2 ÞE y ðzÞ ¼ 0, (1) qz2 where o is the angular frequency of the field and b is the propagation constant in x-direction, which can be written as b ¼ ko N, where ko is the free space wave number and N is the modal effective index. In the working region, the solution of Eq. (1) in each layer has the forms E y1 ðzÞ ¼ F egc ðzd 2 Þ ;
z4d 2 ,
E y2 ðzÞ ¼ Cegm z þ Degm z ;
(2)
0ozod 2 ,
E y3 ðzÞ ¼ B1 cosðgf zÞ þ B2 sinðgf zÞ;
d 1 ozo0,
(3) (4)
and (5) E y4 ðzÞ ¼ Aegs ðzþd 1 Þ ; zo d 1 , qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where gc ¼ b2 o c mc o2 , gm ¼ b2 o m mm o2 , qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi gf ¼ o f mf o2 b2 , and gs ¼ b2 o s ms o2 . Using Eqs. (2)–(5), we can calculate Hx in different media using H x ðzÞ ¼ ði=omðzÞÞðqE y ðzÞ qzÞ. As a result of matching Ey and Hx at z ¼ d1, 0, and d2, we obtain the following dispersion relations: gs m f g m ðg m þ gc mm Þ þ arctan m f m c gf d 1 ¼ arctan gf ms gf mm ðgm mc þ gc mm Þ ðgm mc gc mm Þe2gm d 2 þ mp, (6) þðgm mc gc mm Þe2gm d 2 where m ¼ 0, 1, 2, y is the mode order. As d2 approaches zero, i.e., no metamaterial is available and considering all the layers are nonmagnetic materials (mc ¼ mf ¼ ms ¼ mo), Eq. (6) reduces to the well-known three-layer linear waveguide dispersion relations given by g g (7) gf d 1 ¼ arctan s þ arctan c þ mp. gf gf
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For the sake of simplicity in the evaluation of the sensitivity, we assume the cladding, the film, and the substrate are nonmagnetic materials and the permeability of the metamaterial is given by mm ¼ nmo, where n is a negative number. We also assume that gc ¼ ko qc , gm ¼ ko qm , gf ¼ ko qf and gs ¼ ko qs , where qc ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi N 2 c , qm ¼ N 2 nm , qf ¼ f N 2 , and pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qs ¼ N 2 s . To obtain the sensitivity of the proposed sensor in a condensed form, we define three normalized effective indices Xs, Xc, and Xm and three asymmetry parameters as, ac, and am as q q q s X s ¼ s ; X c ¼ c ; X m ¼ m ; as ¼ , qf qf qf f c m (8) ac ¼ ; and am ¼ . f f In the light of these assumptions, Eq. (6) can be written as X m b1 þ mp, (9) ko qf d 1 ¼ arctanðX s Þ þ arctan n b2 where b1 ¼ ðX m þ nX c Þ ðX m nX c Þe2ko X m qf d 2 and b2 ¼ ðX m þ nX c Þ þ ðX m nX c Þe2ko X m qf d 2 . In the case of homogenous sensing, the sensitivity S is defined as the rate of change of the modal effective index N under an index change of the cover nc. The sensitivity S 2 ¼ qN=qnc of the proposed sensor shown in Fig. 1 is calculated by differentiating Eq. (9) with respect to N. After some algebraic manipulations the sensitivity can be written as
3. Numerical simulation In the analysis below, we will assume the guiding layer to be Si3N4 (ef ¼ 4), the free space wavelength to have the value 1550 nm, and m ¼ 0 which corresponds to the fundamental mode. We will also assume gc mm ¼ gm mc , which corresponds to the surface polariton conditions at the boundary between the metamaterial and the dielectric cladding [13,14]. In Fig. 2 the sensitivity of the proposed sensor is shown as a function of the guiding layer thickness d1. When the guiding layer thickness approaches the cutoff thickness, the effective refractive index approaches the substrate refractive index, the penetration depth of the evanescent field into the substrate medium becomes infinite, and the total power of the mode flows mainly in the substrate. In this case the sensitivity of the sensor approaches zero. For thick waveguides, the sensitivity decreases to zero again because the power of the guided mode flows mainly in the guiding layer itself. The effective refractive index approaches the guiding layer refractive index. The figure shows that the sensitivities have their maxima between these two limits at waveguide thickness somewhat higher than the cutoff thickness of the guided mode considered. Moreover, Fig. 2 shows a comparison between the sensitivity of the proposed sensor with the left-handed medium and the sensitivity of conventional three-layer waveguide sensors. As can be seen, the presence of the
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ X 2c X m ½b2 b1 þ ef ðb1 þ b2 Þ S 2 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi , X c ac þ X 2c AmTE þ ð1=X s Þ b22 þ ðX 2m b21 n2 Þ þ G 1 þ G 2 þ ðb1 b2 C 1 =nX m Þ pffiffiffiffiffi ac
where f ¼ 2ko X m qf d 2 , G 1 ¼ ðb2 b1 Þ Cn1 þ CX2 Xc m , G 2 ¼ h i ef ðb1 þb2 Þ ðX m nX c Þf C 1 þ C 2XXcm n , AmTE ¼ arctanðX s Þþ n Xm arctan Xnm bb12 þ mp, C 1 ¼ 1 þ X 2m , and C 2 ¼ 1 þ X 2c . In a similar manner, we differentiate Eq. (7) to obtain the sensitivity S1 of the three-layer waveguide conventional sensor without the left-handed material. As a result, we obtain
pffiffiffiffiffi ac
S1 ¼ Xc
left-handed material can considerably enhance the sensitivity. The variation of the sensitivity S2 with the thickness, the negative dielectric permittivity, and the negative magnetic permeability of the metamaterial is shown in Figs. 3–5, respectively. Sensitivity increases with the increasing thickness of the metamaterial, with the decreasing absolute value of am ¼ em/ef, and with the increasing absolute value of mm. Sensitivity enhancement
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi . ac þ X 2c 1 þ X 2c arctanðX s Þ þ arctanðX c Þ þ mp þ ð1=X c Þ þ ð1=X s Þ
To evaluate the enhancement effect due to the lefthanded material, we define the sensitivity enhancement factor Fen as F en ¼ S2 =S1 .
(10)
(11)
with the thickness of the metamaterial can be interpreted by surface polariton effects: the evanescent wave generated at the metamaterial–guiding layer interface
ARTICLE IN PRESS S.A. Taya et al. / Optik 120 (2009) 504–508
507
0.6
0.22 0.5
0.2
0.3
0.18
0.2
0.16
0.1
S2
S2 , S1
0.4
0.14
0 100
200
300
400
500
600
700
800
-2
-1.6
-1.2 n
d1(nm)
Fig. 2. Sensitivity versus the guiding layer thickness d1 for as ¼ 0.62, ac ¼ 0.6, am ¼ 0.5, n ¼ 0.6, and d2 ¼ 80 nm for the proposed sensor with the metamaterial (solid line) and a conventional three-layer waveguide sensor without the metamaterial (dotted line).
-0.4
-0.8
Fig. 5. Sensitivity S2 with n where mm ¼ nmo for ac ¼ 0.7, as ¼ 0.72, am ¼ 0.5, d2 ¼ 80 nm, and d1 ¼ 400 nm.
m = -1.5 o
14 12 10
ac = 0.3
m = -1.0 o
Fen
0.6 0.5
8 6
ac = 0.45
0.3
ac = 0.6
2
0.2
50
0.1 0
20
40
60
80
100
120
d2(nm)
Fig. 3. Sensitivity S2 versus the thickness of the metamaterial d2 for different values of ac, as ¼ 0.62, am ¼ 0.5, and n ¼ 0.6. 0.5 0.45
ac = 0.7
0.35
S2
0.4
0.3 ac = 0.5
-0.7
m = -0.5 o
4
S2
0.4
-0.6
-0.5
-0.4
-0.3
0.25 0.2 -0.2
100
150 d2(nm)
200
250
300
Fig. 6. The sensitivity enhancement factor as a function of the thickness of the metamaterial for different values of n, ac ¼ 0.6, as ¼ 0.62, and d1 ¼ 400 nm.
conditions of Maxwell equations are satisfied for the whole waveguide structure. For s-polarized waves, a greater mm will lead to a larger power fraction in the cladding and a larger value of the sensitivity. Fig. 6 shows the sensitivity enhancement factor versus the thickness of the left-handed material for different values of mm. As can seen, the sensitivity enhancement factor increases with the thickness of the metamaterial due to the generation of the surface polaritons as we discussed above. Also for a given metamaterial thick ness, the sensitivity enhancement factor increases as mm increases. To obtain a given sensitivity enhancement factor, a smaller thickness of the metamaterial is required as mm increases.
am
Fig. 4. Sensitivity S2 versus am for different values of ac, as ¼ 0.72, n ¼ 0.6, d2 ¼ 80 nm, and d1 ¼ 400 nm.
excites a surface wave at the metamaterial–cladding interface. The field intensities in both the metamaterial and the cladding keep on building until the boundary
4. Conclusion We have analytically proved that the sensitivity of waveguide optical sensors can be enhanced when there is a layer of left-handed material between the cladding layer and the guiding layer. We believe that metamaterials with simultaneously negative dielectric permittivity
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and magnetic permeability could be used to improve the performance of waveguide chemical and biochemical sensors.
References [1] K. Tiefenthaler, W. Lukosz, Sensitivity of grating couplers as integrated-optical chemical sensors, J. Opt. Soc. Am. B 6 (1989) 209–220. [2] D. Qing, I. Yamaguchi, Analysis of the sensitivity of optical waveguide chemical sensors for TM modes by the group-index method, J. Opt. Soc. Am. B 16 (1999) 1359–1369. [3] W. Lukosz, Integrated optical chemical and direct biochemical sensors, Sensors Actuators B 29 (1995) 37–50. [4] R. Horvath, H. Pederson, N. Larsen, Demonstration of reverse symmetry waveguide sensing in aqueous solutions, Appl. Phys. Lett. 81 (2002) 2166–2168. [5] F. Prieto, A. Liobera, D. Jimenez, C. Domengues, A. Calla, L. Lechuga, Design and analysis of silicon antiresonant reflecting optical waveguide for evanescent field sensor, J. Lightwave Technol. 18 (2000) 966–972.
[6] O. Parriaux, G. Velduis, Normailzed analysis for the sensitivity optimization of integrated optics evanescentwave sensors, J. Lightwave Technol. 16 (1998) 573–582. [7] R. Horvath, H. Pederson, N. Skivesen, Optical waveguide sensor for on-line monitoring of bacteria, Opt. Lett. 28 (2003) 1233–1235. [8] M. Shabat, H. Khalil, S. Taya, M. Abadla, Analysis of the sensitivity of self-focused nonlinear optical evanescent waveguide sensors, Int. J. Optomechatronics 1 (2007) 284–296. [9] V. Veselago, The electrodynamics of substance with simultaneously negative values of e and m, Sov. Phys. Usp. 10 (1968) 509–514. [10] R. Shelby, Experimental verification of a negative index of refraction, Science 292 (2001) 77–79. [11] J. Pendry, Negative refraction makes a perfect lens, Phys. Rev. Lett. 85 (2000) 3966–3969. [12] A. Gribe, G. Eleftheriades, Growing evanescent waves in negative-refractive index, Appl. Phys. Lett. 82 (2003) 1815–1817. [13] D. Qing, G. Chen, Enhancement of evanescent waves in waveguides using metamaterials of negative permittivity and permeability, Appl. Phys. Lett. 21 (2003) 669–671. [14] R. Ruppin, Surface polaritons of a left-handed medium, Phys. Lett. A 227 (2000) 61–64.
Optik
Optics
Optik 120 (2009) 504–508 www.elsevier.de/ijleo
SHORT NOTE
Enhancement of sensitivity in optical waveguide sensors using left-handed materials S.A. Tayaa, M.M. Shabata,b,, H.M. Khalilc a
Physics Department, Islamic University, P.O. Box 108, Gaza, Gaza Strip, Palestinian Authority Max-Planck-Institut fu¨r Physik komplexer Systeme, No¨thnitzer Street, 3801187 Dresden, Germany c College for Girls, Ain Shams University, Cairo, Egypt b
Received 1 October 2007; accepted 23 December 2007
Abstract We show analytically that the sensitivity of an optical waveguide sensor can be dramatically enhanced by using a metamaterial with negative permittivity and permeability. The variation of the sensitivity of the proposed waveguide sensor with different parameters of the waveguide is studied. It is found that the sensitivity of the sensor increases with the increasing thickness of the metamaterial due to the surface polariton generation. r 2008 Elsevier GmbH. All rights reserved. Keywords: Left-handed materials; Optical waveguide sensors; Sensitivity analysis
1. Introduction Optical evanescent wave sensors have been widely used for various purposes such as humidity sensing [1], chemical sensing [2], biochemical sensing [3], and biosensing [4]. Optical waveguide sensors have shown many attractive features such as the immunity to electromagnetic interference, the use in aggressive environments, and, in general, a high sensitivity [5]. The sensing of the waveguide sensors is performed by the evanescent tail of the modal field in the cover medium. The guided electromagnetic field of the waveguide mode extends as an evanescent field into the cladding and substrate media and senses an effective refractive index of the waveguide. The effective refracCorresponding author at: Max-Planck-Institut fur Physik kom¨ plexer Systeme, No¨thnitzer Street, 3801187 Dresden, Germany. Tel.: +35 18712115; fax: +35 18711999. E-mail addresses: [email protected] (S.A. Taya), [email protected] (M.M. Shabat).
0030-4026/$ - see front matter r 2008 Elsevier GmbH. All rights reserved. doi:10.1016/j.ijleo.2007.12.001
tive index of the propagating mode depends on the structure parameters, e.g., the guiding layer thickness and dielectric permittivity and magnetic permeability of the media constituting the waveguide. As a result, any change in the refractive index of the covering medium results in a change in the effective refractive index of the guiding mode. The basic sensing principle of the planar waveguide sensor is to measure the changes in the effective refractive index due to changes in the refractive index of the covering medium. Parriaux and Velduis [6] presented an extensive theoretical analysis for the design of evanescent linear waveguide sensors and derived the conditions for the maximum achievable sensitivity for both TE and TM polarizations. Horvath et al. [7] demonstrated the design and implementation of a waveguide sensor configuration called reverse symmetry in which the refractive index of the aqueous cladding is higher than that of the substrate material. The reverse symmetry waveguide has been tested for bacterial and cell detection and it showed a considerably high sensitivity compared with the
ARTICLE IN PRESS S.A. Taya et al. / Optik 120 (2009) 504–508
conventional waveguide sensor. We have investigated the sensitivity of a nonlinear waveguide structure where one of the layers of the waveguide structure is considered to be nonlinear medium [8]. We showed that the sensitivity can be enhanced when the covering medium has an intensity-dependent refractive index. In recent years, much attention is drawn to metamaterials with simultaneously negative permittivity and permeability that are often referred to as left-handed materials. The history of these materials begins with the paper of Veselago [9], who proposed a hypothetical material with simultaneously negative dielectric permittivity and magnetic permeability and studied the propagation of electromagnetic waves in such a medium. In his article in 1968, Vesegalo predicted a number of interesting features of waves in metamaterials, including negative refraction. These materials are not found in nature, and need to be artificially constructed. So far the left-handed materials are demonstrated only in the microwave frequency range [10]. One of the first applications of the left-handed materials was reported by Pendry [11], who demonstrated that a slab of a lossless left-handed material can provide a perfect image of a point source. Gribe and Eleftheriades [12] verified by a simulation the enhancement of evanescent waves in a transmission-line network by using a negative refractive index material. In [13], it was shown that left-handed materials can enhance the evanescent field in slab waveguides. The main aim of the present paper is to show that the sensitivity of the conventional optical sensors can be dramatically enhanced by inserting a layer of lefthanded material between the cladding and the guiding layer and to study the variation of the sensitivity of the proposed sensor with the parameters, i.e., thickness, permittivity, and permeability, of the left-handed material.
2. Structure analysis Fig. 1 shows a schematic structure of the slab waveguide under consideration. A guiding layer with permittivity ef, permeability mf, and thickness d1 is sandwiched between a semi-infinite substrate with permittivity es and permeability ms and a semi-infinite cladding with permittivity ec and permeability mc. In the working region, there is a layer of metamaterial with negative permittivity em, negative permeability mm, and thickness d2 between the cladding and the guiding layer. Here we assume that all the materials are lossless, i.e., both e’s and m’s are real. We also consider the TE waves in which the electric field E is polarized along the y-axis. The well-known Helmholtz equations for TE modes are
505
z Non working region
workingregion
Non working region
Cladding Metamaterial
d2 z=0
x d1
Guiding layer
Substrate
Fig. 1. Schematic structure of integrated waveguide sensor with a metamaterial layer.
described by the following scalar equation: q2 E y ðzÞ þ ðo2 ðzÞmðzÞ b2 ÞE y ðzÞ ¼ 0, (1) qz2 where o is the angular frequency of the field and b is the propagation constant in x-direction, which can be written as b ¼ ko N, where ko is the free space wave number and N is the modal effective index. In the working region, the solution of Eq. (1) in each layer has the forms E y1 ðzÞ ¼ F egc ðzd 2 Þ ;
z4d 2 ,
E y2 ðzÞ ¼ Cegm z þ Degm z ;
(2)
0ozod 2 ,
E y3 ðzÞ ¼ B1 cosðgf zÞ þ B2 sinðgf zÞ;
d 1 ozo0,
(3) (4)
and (5) E y4 ðzÞ ¼ Aegs ðzþd 1 Þ ; zo d 1 , qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where gc ¼ b2 o c mc o2 , gm ¼ b2 o m mm o2 , qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi gf ¼ o f mf o2 b2 , and gs ¼ b2 o s ms o2 . Using Eqs. (2)–(5), we can calculate Hx in different media using H x ðzÞ ¼ ði=omðzÞÞðqE y ðzÞ qzÞ. As a result of matching Ey and Hx at z ¼ d1, 0, and d2, we obtain the following dispersion relations: gs m f g m ðg m þ gc mm Þ þ arctan m f m c gf d 1 ¼ arctan gf ms gf mm ðgm mc þ gc mm Þ ðgm mc gc mm Þe2gm d 2 þ mp, (6) þðgm mc gc mm Þe2gm d 2 where m ¼ 0, 1, 2, y is the mode order. As d2 approaches zero, i.e., no metamaterial is available and considering all the layers are nonmagnetic materials (mc ¼ mf ¼ ms ¼ mo), Eq. (6) reduces to the well-known three-layer linear waveguide dispersion relations given by g g (7) gf d 1 ¼ arctan s þ arctan c þ mp. gf gf
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For the sake of simplicity in the evaluation of the sensitivity, we assume the cladding, the film, and the substrate are nonmagnetic materials and the permeability of the metamaterial is given by mm ¼ nmo, where n is a negative number. We also assume that gc ¼ ko qc , gm ¼ ko qm , gf ¼ ko qf and gs ¼ ko qs , where qc ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi N 2 c , qm ¼ N 2 nm , qf ¼ f N 2 , and pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qs ¼ N 2 s . To obtain the sensitivity of the proposed sensor in a condensed form, we define three normalized effective indices Xs, Xc, and Xm and three asymmetry parameters as, ac, and am as q q q s X s ¼ s ; X c ¼ c ; X m ¼ m ; as ¼ , qf qf qf f c m (8) ac ¼ ; and am ¼ . f f In the light of these assumptions, Eq. (6) can be written as X m b1 þ mp, (9) ko qf d 1 ¼ arctanðX s Þ þ arctan n b2 where b1 ¼ ðX m þ nX c Þ ðX m nX c Þe2ko X m qf d 2 and b2 ¼ ðX m þ nX c Þ þ ðX m nX c Þe2ko X m qf d 2 . In the case of homogenous sensing, the sensitivity S is defined as the rate of change of the modal effective index N under an index change of the cover nc. The sensitivity S 2 ¼ qN=qnc of the proposed sensor shown in Fig. 1 is calculated by differentiating Eq. (9) with respect to N. After some algebraic manipulations the sensitivity can be written as
3. Numerical simulation In the analysis below, we will assume the guiding layer to be Si3N4 (ef ¼ 4), the free space wavelength to have the value 1550 nm, and m ¼ 0 which corresponds to the fundamental mode. We will also assume gc mm ¼ gm mc , which corresponds to the surface polariton conditions at the boundary between the metamaterial and the dielectric cladding [13,14]. In Fig. 2 the sensitivity of the proposed sensor is shown as a function of the guiding layer thickness d1. When the guiding layer thickness approaches the cutoff thickness, the effective refractive index approaches the substrate refractive index, the penetration depth of the evanescent field into the substrate medium becomes infinite, and the total power of the mode flows mainly in the substrate. In this case the sensitivity of the sensor approaches zero. For thick waveguides, the sensitivity decreases to zero again because the power of the guided mode flows mainly in the guiding layer itself. The effective refractive index approaches the guiding layer refractive index. The figure shows that the sensitivities have their maxima between these two limits at waveguide thickness somewhat higher than the cutoff thickness of the guided mode considered. Moreover, Fig. 2 shows a comparison between the sensitivity of the proposed sensor with the left-handed medium and the sensitivity of conventional three-layer waveguide sensors. As can be seen, the presence of the
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ X 2c X m ½b2 b1 þ ef ðb1 þ b2 Þ S 2 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi , X c ac þ X 2c AmTE þ ð1=X s Þ b22 þ ðX 2m b21 n2 Þ þ G 1 þ G 2 þ ðb1 b2 C 1 =nX m Þ pffiffiffiffiffi ac
where f ¼ 2ko X m qf d 2 , G 1 ¼ ðb2 b1 Þ Cn1 þ CX2 Xc m , G 2 ¼ h i ef ðb1 þb2 Þ ðX m nX c Þf C 1 þ C 2XXcm n , AmTE ¼ arctanðX s Þþ n Xm arctan Xnm bb12 þ mp, C 1 ¼ 1 þ X 2m , and C 2 ¼ 1 þ X 2c . In a similar manner, we differentiate Eq. (7) to obtain the sensitivity S1 of the three-layer waveguide conventional sensor without the left-handed material. As a result, we obtain
pffiffiffiffiffi ac
S1 ¼ Xc
left-handed material can considerably enhance the sensitivity. The variation of the sensitivity S2 with the thickness, the negative dielectric permittivity, and the negative magnetic permeability of the metamaterial is shown in Figs. 3–5, respectively. Sensitivity increases with the increasing thickness of the metamaterial, with the decreasing absolute value of am ¼ em/ef, and with the increasing absolute value of mm. Sensitivity enhancement
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi . ac þ X 2c 1 þ X 2c arctanðX s Þ þ arctanðX c Þ þ mp þ ð1=X c Þ þ ð1=X s Þ
To evaluate the enhancement effect due to the lefthanded material, we define the sensitivity enhancement factor Fen as F en ¼ S2 =S1 .
(10)
(11)
with the thickness of the metamaterial can be interpreted by surface polariton effects: the evanescent wave generated at the metamaterial–guiding layer interface
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507
0.6
0.22 0.5
0.2
0.3
0.18
0.2
0.16
0.1
S2
S2 , S1
0.4
0.14
0 100
200
300
400
500
600
700
800
-2
-1.6
-1.2 n
d1(nm)
Fig. 2. Sensitivity versus the guiding layer thickness d1 for as ¼ 0.62, ac ¼ 0.6, am ¼ 0.5, n ¼ 0.6, and d2 ¼ 80 nm for the proposed sensor with the metamaterial (solid line) and a conventional three-layer waveguide sensor without the metamaterial (dotted line).
-0.4
-0.8
Fig. 5. Sensitivity S2 with n where mm ¼ nmo for ac ¼ 0.7, as ¼ 0.72, am ¼ 0.5, d2 ¼ 80 nm, and d1 ¼ 400 nm.
m = -1.5 o
14 12 10
ac = 0.3
m = -1.0 o
Fen
0.6 0.5
8 6
ac = 0.45
0.3
ac = 0.6
2
0.2
50
0.1 0
20
40
60
80
100
120
d2(nm)
Fig. 3. Sensitivity S2 versus the thickness of the metamaterial d2 for different values of ac, as ¼ 0.62, am ¼ 0.5, and n ¼ 0.6. 0.5 0.45
ac = 0.7
0.35
S2
0.4
0.3 ac = 0.5
-0.7
m = -0.5 o
4
S2
0.4
-0.6
-0.5
-0.4
-0.3
0.25 0.2 -0.2
100
150 d2(nm)
200
250
300
Fig. 6. The sensitivity enhancement factor as a function of the thickness of the metamaterial for different values of n, ac ¼ 0.6, as ¼ 0.62, and d1 ¼ 400 nm.
conditions of Maxwell equations are satisfied for the whole waveguide structure. For s-polarized waves, a greater mm will lead to a larger power fraction in the cladding and a larger value of the sensitivity. Fig. 6 shows the sensitivity enhancement factor versus the thickness of the left-handed material for different values of mm. As can seen, the sensitivity enhancement factor increases with the thickness of the metamaterial due to the generation of the surface polaritons as we discussed above. Also for a given metamaterial thick ness, the sensitivity enhancement factor increases as mm increases. To obtain a given sensitivity enhancement factor, a smaller thickness of the metamaterial is required as mm increases.
am
Fig. 4. Sensitivity S2 versus am for different values of ac, as ¼ 0.72, n ¼ 0.6, d2 ¼ 80 nm, and d1 ¼ 400 nm.
excites a surface wave at the metamaterial–cladding interface. The field intensities in both the metamaterial and the cladding keep on building until the boundary
4. Conclusion We have analytically proved that the sensitivity of waveguide optical sensors can be enhanced when there is a layer of left-handed material between the cladding layer and the guiding layer. We believe that metamaterials with simultaneously negative dielectric permittivity
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and magnetic permeability could be used to improve the performance of waveguide chemical and biochemical sensors.
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