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Nonlinear planar asymmetrical optical waveguides for sensing applications
ARTICLE IN PRESS
Optik
Optics
Optik 121 (2010) 860–865 www.elsevier.de/ijleo
Nonlinear planar asymmetrical optical waveguides for sensing applications Sofyan A. Tayaa, Mohammed M. Shabata,b,, Hala M. Khalilc a
Physics Department, Islamic University, Gaza, P.O. Box 108, Gaza Strip, Palestinian Authority Max Planck Institute for the Physics of complex Systems, No¨thnitzer Straße 38, 01187 Dresden, Germany c College for Girls, Ain Shams University, Cairo, Egypt b
Received 8 June 2008; accepted 28 September 2008
Abstract This paper provides a theoretical analysis for TE polarized waves guided by a linear film surrounded by two asymmetrical nonlinear media for sensing applications. The sensitivity of the proposed sensor is derived. The conditions required for the sensor to exhibit its maximum sensitivity are presented. This analysis covers the case when the measurand is homogeneously distributed in the covering medium. We show that nonlinear sensors have sensitivities higher than those of linear sensors. The authors believe that optical sensing can be improved by introducing nonlinear waveguides. r 2009 Elsevier GmbH. All rights reserved. Keywords: Nonlinear planar waveguides; Optical sensors; Sensitivity analysis
1. Introduction Optical evanescent wave sensors have been widely used for various purposes such as humidity sensing [1,2], chemical sensing [3,4], biochemical sensing [5,6] and biosensing [7]. The sensors are based on the detection of changes in refractive index occurring close to the guiding layer surface. Optical sensors have shown many attractive features such as the immunity to electromagnetic interference, the use in aggressive environment and, in general, a high sensitivity. Moreover, optical sensors based on integrated optics add some other advantages as a better control of the light path by the
Corresponding author at: Physics Department, Islamic University, Gaza, P.O. Box 108, Gaza Strip, Palestinian Authority. E-mail addresses: [email protected] (S.A. Taya), [email protected], [email protected] (M.M. Shabat).
0030-4026/$ - see front matter r 2009 Elsevier GmbH. All rights reserved. doi:10.1016/j.ijleo.2008.09.036
use of optical waveguides, a higher mechanical stability and a reduced size [8]. In optical waveguide sensors, although light travels confined within the optical waveguide, there is a part of the guided mode called the evanescent field that extends to the surrounding media. The evanescent field detects any refractive index variation at the clad–film surface in surface sensing and in the refractive index of the covering medium in homogeneous sensing. A modification of the optical properties of the guided mode is induced giving an indication of the concentration, thickness or refractive index of the analyte (the material to be measured). Parriaux et al. [9,10] 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. [7,11,12] demonstrated the design and implementation of a waveguide sensor configuration called reverse-symmetry in which the refractive
ARTICLE IN PRESS S.A. Taya et al. / Optik 121 (2010) 860–865
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 conventional waveguide sensor. We have investigated the sensitivity of nonlinear waveguide structure where one of the layers of the waveguide structure is considered to be nonlinear medium [13,14]. In the present paper we demonstrate a new nonlinear asymmetrical waveguide sensor structure. The waveguide under consideration consists of a linear film embedded between two Kerr-type nonlinear media. The characteristics of the sensor are presented. The fraction of the total power flowing in the cladding is derived due to the close connection between the sensitivity of the sensor and this fraction.
861
n2i co n2i [17], where c is the speed of light in vacuum and eo is free space permittivity. Continuity of Ey and qE y =qz gives the dispersion relation tanðk0 qf hÞ ¼
qc qf tanh C c þ qs qf tanh C s , q2f qs qc tanh C s tanh C c
(6)
where ko is the free space wave number, Cc ¼ k0qc(hzc), and Cs ¼ k0qszs which are called the cladding film interface nonlinearity and the substrate film interface nonlinearity respectively. To simplify calculations in the following steps [10], we will define two asymmetry parameters as and ac and two normalized variables Xs and Xc as as ¼
s ; f
ac ¼
c ; f
Xs ¼
qs ; qf
Xc ¼
qc . qf
(7)
Xs and Xc are linked by
2. Theory The nonlinear waveguide sensor under consideration consists of a linear thin film of thickness h and dielectric constant ef is supported by a substrate of nonlinear permittivity e2 and is covered by a cladding of nonlinear permittivity e1, where e1 and e2 are given by [15–17]: 1 ¼ c þ ac jE y1 j2 ,
(1)
2 ¼ s þ as jE y3 j2 ,
(2)
where ac and as are the nonlinearity coefficients of cladding and substrate, respectively, ec and es are the linear parts of the permittivities and Ey1 and Ey3 are the TE fields in cladding and substrate, respectively. The electric field in each layer has the following solution for TE modes for ac40 and as40: sffiffiffiffiffi 2 q sec hðk0 qc ðz zc ÞÞ; z4h, E y1 ¼ (3) ac c E y2 ¼ A cosðk0 qf zÞ þ B sinðk0 qf zÞ;
0ozoh,
(4)
sffiffiffiffi 2 q sec hðk0 qs ðzs zÞÞ; zo0, E y3 ¼ (5) as s qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where qc ¼ N 2 c , qf ¼ f N 2 , qs ¼ N 2 s , N is the effective refractive index, constants A and B represent the amplitude of the waves, and zc and zs are constants related to the field distribution in the covering medium and the substrate, respectively. In nonlinear optics, the nonlinear coefficient a is usually written in terms of the intensity-dependent refractive index n ¼ ni+n2iS, where S is the local pffiffiffiffi intensity (W/m2), ni ¼ i is the low power refractive index and i is either c for the cladding or s for the substrate. Simple algebra in the limit nibn2iS gives ai ¼
ð1 ac Þð1 þ X 2s Þ 1. ð1 as Þ
X 2c ¼
(8)
The effective refractive index can be written in terms of as and Xs as sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffi as þ X 2s N ¼ f . (9) 1 þ X 2s In terms of Xs and Xc Eq. (6) can be written as k0 qf h arctanðX c tanh C c Þ arctanðX s tanh C s Þ mp ¼ 0,
(10)
where m ¼ 0,1y is the mode order. From the dispersion relation given by Eq. (10), we derive the sensor sensitivity Sh, i.e., the changes in the effective refractive index N for changes in the refractive index nc of the cover medium. Differentiating Eq. (10) with respect to N and calculating Sh as ðqnc =qNÞ1 ) we obtain qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffi ac 1 þ X 2c ðH c þ tanh C c Þ , S h ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X c ac þ X 2c ð1 þ X 2c tanh2 CÞðATE þ G s þ G c Þ (11) where pffiffiffiffi H c ¼ k0 ðh zc ÞX c f
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ac ð1 tanh2 C c Þ, 1 þ X 2c
(12)
Gc ¼
H c þ tanh C c ð1 þ X 2c Þ , X c ð1 þ X 2c tanh2 C c Þ
(13)
Gs ¼
H s þ tanh C s ð1 þ X 2s Þ , X s ð1 þ X 2s tanh2 C s Þ
(14)
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sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffi 1 as H s ¼ k 0 zs X s f 1 þ X 2s
tanh C s bs ss þ 3 tanh2 C s X 2s ð1 tanh2 C s Þ,
(15) d TE ¼
X s ss bs þ tanh C s X 2c sc bc þ wX s tanh C c þ , js X c jc
ATE ¼ arctanðX s tanh C s Þ þ arctanðX c tanh C c Þ þ mp. (16) The sensitivity of the proposed nonlinear sensor is given by Eq. (11) which strongly depends on the waveguide structure. To maximize the sensitivity on the quantity to be measured, we cancel the derivative of the sensitivity Sh with respect to guiding layer thickness h. To simplify the expression of the maximum sensitivity we consider M1 ¼ ATE+Gs+Gc, M2 ¼ Hc+tanh Cc, qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R1 ¼ ac þ X 2c , R2 ¼ 1 þ X 2c , and R3 ¼ ðwX s =X c þ 2sc bc X 3c tanh C c þ 3wX s X c tanh2 C c Þ. After some algebraic manipulations and using the fact that qS h =qh ¼ ðqS h =qX s ÞðqX s =qhÞ and qX s =qha0, the maximum sensing sensitivity condition can be written as pffiffiffiffiffi pffiffiffiffiffi wX s X c jc R1 M 1 ac 2bc sc R2 ð1 C c tanh C c Þ þ M 2 a c R2 M 2 R2 j wX s X c ¼ 0, X c R1 jc ðd TE þ gs þ gc Þ þ M 1 R1 R3 þ c R1
(17) where
bc ¼
(22) sc ¼ 1 tanh2 C c ;
ss ¼ 1 tanh2 C s ,
jc ¼ 1 þ X 2c tanh2 C c ; 1 ac . w¼ 1 as
js ¼ 1 þ X 2s tanh2 C s ,
One of the most important quantities when speaking about optical sensing is the power flow in different layers of the slab waveguide. The sensitivity of the sensor is critically dependent on the fraction of total power propagating in the cladding medium. The energy flux per unit length is given by Z 1 E H dz ¼ Ps þ Pf þ Pc . (24) P¼ 1
We consider j1 ¼ 1tanh Cc, j2 ¼ 1tanh Cs, j3 ¼ 1 cos(2k0qfh), xþ ¼ 1 þ X 2s tanh2 C s , and x ¼ 1 X 2s tanh2 C s . The fraction of total power flowing in the nonlinear cladding is
X s X c X s ss bs X c tanh C s X s w tanh C c pffiffiffiffi X s X c Cw 2 þ þ þ þ k 0 zc f js js jc Xc 1 þ X 2s
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 as 1 þ X 2s
X 2 sc 1 c jc
,
Xc j Pc ac 1 ¼ . Ptotal X c X 2 sec h2 C s 1 Xs j1 þ s ko qf hxþ þ sinð2k0 qf hÞx þ X s tanh C s j 3 þ j 2 ac 2as as 2
pffiffiffiffi bs ¼ k 0 z s f
gc ¼
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 as 1 , 2 1 þ X s 1 þ X 2s
X c jc r1 r2 r3 X 2c j2c
(19)
(20)
with r1 ¼ bc sc ð1 2C c tanh C c Þ þ 2wX s tanh C c þ bc ð1 þ X 2c Þsc , r2 ¼ H c þ tanh C c ð1 þ X 2c Þ, r3 ¼ wðX s = X c Þ þ 2X 3c tanh C c bc sc þ 3 tanh2 C c wX s X c gs ¼
X s j s r4 r5 r6 X 2s j2s
(21)
with r4 ¼ bs ss ð1 2C s tanh C s Þ þ 2X s tanh C s þ bs ð1 þ X 2s Þss , r5 ¼ H s þ tanh C s ð1 þ X 2s Þ, r6 ¼ 1 þ 2X 3s
(23)
(18)
(25)
3. Representation and discussion In our calculations below, we will assume the guiding layer to be Si3N4 (nf ¼ 2), the free space wavelength to have the value 1550 nm, tanh Cc ¼ 0.6 and tanh Cs ¼ 0.7. Only the fundamental mode (m ¼ 0) will be considered since it has the highest sensitivity [1,10]. In Figs. 1 and 2 the sensitivity of the proposed nonlinear sensor is shown as a function of the guiding layer thickness h. The sensitivity versus the waveguide thickness behaves quite differently in the case when the refractive index of the cover is lower (ncons) and higher (nc4ns) than the substrate index. The case (ncons) is called the conventional waveguide symmetry and the
ARTICLE IN PRESS S.A. Taya et al. / Optik 121 (2010) 860–865 0.16
Sh
0.12
0.08
0.04
0
200
300
400 500 h(nm)
600
700
800
Fig. 1. Sensitivity versus the waveguide thickness h for as ¼ 0.65 and ac ¼ 0.45 for the proposed nonlinear sensor (solid line) and a linear sensor (dotted line).
0.7
ac = 0.7
863
sensor is similar to the conventional waveguide discussed above. Moreover, Fig. 1 shows a comparison between linear and nonlinear sensors. We see that nonlinear sensors have higher sensitivities than linear sensors. We believe that optical sensing can be improved by using nonlinear sensors. The variation of the sensitivity with the terms tanh Cc and tanh Cs, the clad–film interface nonlinearity and the substrate–film interface nonlinearity, respectively, which arise from the nonlinearity of the cladding and the nonlinearity of the substrate respectively, is plotted in Figs. 3 and 4. As the terms tanh Cc and tanh Cs go to unity in Eq. (10), we obtain the well known characteristic equations for linear waveguides [9,10]. Fig. 3 shows the sensitivity to have the minimum value as tanh Cc goes to one, the linear cladding, and Fig. 4 shows that the sensitivity increases as tanh Cs increases to one. Thus we conclude, the nonlinearity of the cladding is essential to enhance the sensitivity of the proposed waveguide sensor.
0.6
Sh
0.5
ac = 0.6
0.12
0.4 0.3
ac = 0.5
0.1 Sh
0.2 0.1 0
ac = 0.5
0.1
ac = 0.4
0.08 0.06 0.04
100 200 300 400 500 600 700 800 h(nm)
Fig. 2. Sensitivity versus the thickness for a waveguide with normal symmetry (as4ac) and a waveguide with reverse symmetry (asoac). as ¼ 0.55 for all cases.
0.2
0.4
0.6 tanhCc
0.8
1
Fig. 3. Sensitivity against tanh Cc for different values of ac, as ¼ 0.7, and h ¼ 0.2 mm.
0.08 0.07
ac = 0.6
0.06 Sh
case (nc4ns) is called the reverse symmetry waveguide. For the conventional symmetry (ncons), the two figures show that the sensitivities have their maxima at waveguide thickness somewhat higher than the cut-off thickness of the guided mode considered. For thick waveguides, the sensitivities decrease to zero because the power of the guided mode flows mainly in the guiding layer itself. For the case of reverse symmetry (nc4ns), Fig. 2 shows that the sensitivities decrease with increasing the guiding layer thickness, starting from the maximum value near cut-off thickness. This behavior is attributed to the power considerations. When the guiding layer thickness approaches the cutoff thickness, the effective refractive index (N) approaches the cover index (nc), the penetration depth of the evanescent field into the cover medium becomes infinite, and the total power of the mode flows mainly in the cover. As the guiding layer thickness of reverse symmetry waveguide increases, the behavior of the
ac = 0.3
0.02
0.05
ac = 0.45
0.04
ac = 0.3
0.03 0.02 0.2
0.4
0.6 tanhCs
0.8
1
Fig. 4. Sensitivity with tanh Cs for different values of ac, as ¼ 0.65, and h ¼ 0.65mm.
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0.6 0.5 0.3
0.6
0.2
0.5
0.1 0
0.4 0.7 0.6 0.3 0.5 0.4 0.3 0.2 as 0.2 0.1
ac
Xs
0.4
be improved by using nonlinear materials in the cladding and substrate. The optimum structure can be obtained by using a nonlinear material in the cladding layer and a linear material in the substrate. It has been shown that the thickness of the guiding layer is a critical parameter for the sensitivity of the optical sensor with the optimum thickness is equal to the cut-off thickness in case of reverse symmetry waveguides and just above the cut-off thickness in case of normal symmetry waveguides.
Fig. 5. Normalized effective refractive index Xs versus the asymmetry parameters as and ac ensuring maximum sensitivity for homogeneous sensing.
Eq. (17) contains the condition required for the sensor to exhibit its maximum sensing sensitivity. If we consider Eq. (8), the solution of Eq. (17) is one value of the normalized refractive index Xs for each pair of the asymmetry parameters (as, ac). The effective refractive index N to which an Xs solution corresponds is given by Eq. (9). The thickness of the waveguide ensuring maximum sensitivity can be easily found by substituting the optimized value of Xs obtained from the solution of Eq. (17) into Eq. (10). Eq. (17) is plotted in Fig. 5 and surface Xs (as, ac) is obtained. All the necessary information to construct the optimized sensor is found in this surface and the above set of equations. A typical way of proceeding is as follows: usually the sensor usage imposes the cover material, thus the refractive index nc is determined. The choice of the substrate (consequently ns) is controlled by temperature, mechanical stability and cost criteria. The optical and chemical stability are the criteria considered in the choice of the guiding layer material. Thus the optogeometrical parameters as and ac are determined. The designer will look at the chart representing Xs as a function of as and ac (Fig. 5) to find the value of Xs which provides the highest sensitivity. Introducing the solution found for Xs into Eqs. (8)–(10) gives optimum normalized parameter Xc, effective index N and the guiding layer thickness h, respectively. Substituting these values into Eq. (11), the maximum sensitivity achievable is calculated.
4. Conclusion In this work a nonlinear three-layer slab waveguide optical sensor is analyzed. The case of homogeneous sensing is considered. The sensitivity of the effective refractive index to variations of the cladding index can
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] L. Xu, J. Fanguy, K. Soni, S. Tao, Optical fiber humidity sensor based on evanescent-wave scattering, Opt. Lett. 29 (2004) 1191–1193. [3] 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. [4] K. Remley, A. Weisshaar, Design and analysis of a silicon-based antiresonant reflecting optical waveguide chemical sensors, Opt. Lett. 21 (1996) 1241–1243. [5] W. Lukosz, Integrated optical chemical and direct biochemical sensors, Sen. Actuators B 29 (1995) 37–50. [6] R. Kunz, Miniature integrated optical modules for chemical and biochemical sensing, Sen. Actuators B 38 (1997) 13–28. [7] R. Horvath, H. Pederson, N. Larsen, Demonstration of reverse symmetry waveguide sensing in aqueous solutions, App. Phys. Lett. 81 (2002) 2166–2168. [8] 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. [9] O. Parriaux, P. Dierauer, Normalized expressions for the optical sensitivity of evanescent wave sensors, Opt. Lett. 19 (1994) 508–510. [10] O. Parriaux, G. Velduis, Normailzed analysis for the sensitivity optimization of integrated optics evanescentwave sensors, J. Lightwave Technol. 16 (1998) 573–582. [11] R. Horvath, N. Skivesen, H. Pederson, Measurement of guided light-mode intensity: an alternative waveguide sensing principle, App. Phys. Lett. 84 (2004) 4044–4046. [12] R. Horvath, H. Pederson, N. Skivesen, Optical waveguide sensor for on-line monitoring of bacteria, Opt. Lett. 28 (2003) 1233–1235. [13] 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. [14] H. Khalil, M. Shabat, S. Taya, M. Abadla, Nonlinear optical waveguide structure for sensor application: TM case, Int. J. Mod. Phys. B 21 (2007) 5075–5089.
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[15] C. Xuelong, H. Zhaoming, Z. Yowei, Spatial bistability in nonlinear optical waveguides, Chin. Phys.Lasers 15 (1988) 381–383. [16] C. Chelkowski, J. Chrostowski, Scaling rules for slab waveguides with nonlinear substrate, Appl. Opt. 26 (1987) 3681–3688.
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[17] C. Seaton, J. Valera, R. Shoemaker, Calculations of nonlinear TE waves guided by thin dielectric films bounded by nonlinear media, IEEE J. Quantum Electron. QE-21 (1985) 774–782.
Optik
Optics
Optik 121 (2010) 860–865 www.elsevier.de/ijleo
Nonlinear planar asymmetrical optical waveguides for sensing applications Sofyan A. Tayaa, Mohammed M. Shabata,b,, Hala M. Khalilc a
Physics Department, Islamic University, Gaza, P.O. Box 108, Gaza Strip, Palestinian Authority Max Planck Institute for the Physics of complex Systems, No¨thnitzer Straße 38, 01187 Dresden, Germany c College for Girls, Ain Shams University, Cairo, Egypt b
Received 8 June 2008; accepted 28 September 2008
Abstract This paper provides a theoretical analysis for TE polarized waves guided by a linear film surrounded by two asymmetrical nonlinear media for sensing applications. The sensitivity of the proposed sensor is derived. The conditions required for the sensor to exhibit its maximum sensitivity are presented. This analysis covers the case when the measurand is homogeneously distributed in the covering medium. We show that nonlinear sensors have sensitivities higher than those of linear sensors. The authors believe that optical sensing can be improved by introducing nonlinear waveguides. r 2009 Elsevier GmbH. All rights reserved. Keywords: Nonlinear planar waveguides; Optical sensors; Sensitivity analysis
1. Introduction Optical evanescent wave sensors have been widely used for various purposes such as humidity sensing [1,2], chemical sensing [3,4], biochemical sensing [5,6] and biosensing [7]. The sensors are based on the detection of changes in refractive index occurring close to the guiding layer surface. Optical sensors have shown many attractive features such as the immunity to electromagnetic interference, the use in aggressive environment and, in general, a high sensitivity. Moreover, optical sensors based on integrated optics add some other advantages as a better control of the light path by the
Corresponding author at: Physics Department, Islamic University, Gaza, P.O. Box 108, Gaza Strip, Palestinian Authority. E-mail addresses: [email protected] (S.A. Taya), [email protected], [email protected] (M.M. Shabat).
0030-4026/$ - see front matter r 2009 Elsevier GmbH. All rights reserved. doi:10.1016/j.ijleo.2008.09.036
use of optical waveguides, a higher mechanical stability and a reduced size [8]. In optical waveguide sensors, although light travels confined within the optical waveguide, there is a part of the guided mode called the evanescent field that extends to the surrounding media. The evanescent field detects any refractive index variation at the clad–film surface in surface sensing and in the refractive index of the covering medium in homogeneous sensing. A modification of the optical properties of the guided mode is induced giving an indication of the concentration, thickness or refractive index of the analyte (the material to be measured). Parriaux et al. [9,10] 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. [7,11,12] demonstrated the design and implementation of a waveguide sensor configuration called reverse-symmetry in which the refractive
ARTICLE IN PRESS S.A. Taya et al. / Optik 121 (2010) 860–865
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 conventional waveguide sensor. We have investigated the sensitivity of nonlinear waveguide structure where one of the layers of the waveguide structure is considered to be nonlinear medium [13,14]. In the present paper we demonstrate a new nonlinear asymmetrical waveguide sensor structure. The waveguide under consideration consists of a linear film embedded between two Kerr-type nonlinear media. The characteristics of the sensor are presented. The fraction of the total power flowing in the cladding is derived due to the close connection between the sensitivity of the sensor and this fraction.
861
n2i co n2i [17], where c is the speed of light in vacuum and eo is free space permittivity. Continuity of Ey and qE y =qz gives the dispersion relation tanðk0 qf hÞ ¼
qc qf tanh C c þ qs qf tanh C s , q2f qs qc tanh C s tanh C c
(6)
where ko is the free space wave number, Cc ¼ k0qc(hzc), and Cs ¼ k0qszs which are called the cladding film interface nonlinearity and the substrate film interface nonlinearity respectively. To simplify calculations in the following steps [10], we will define two asymmetry parameters as and ac and two normalized variables Xs and Xc as as ¼
s ; f
ac ¼
c ; f
Xs ¼
qs ; qf
Xc ¼
qc . qf
(7)
Xs and Xc are linked by
2. Theory The nonlinear waveguide sensor under consideration consists of a linear thin film of thickness h and dielectric constant ef is supported by a substrate of nonlinear permittivity e2 and is covered by a cladding of nonlinear permittivity e1, where e1 and e2 are given by [15–17]: 1 ¼ c þ ac jE y1 j2 ,
(1)
2 ¼ s þ as jE y3 j2 ,
(2)
where ac and as are the nonlinearity coefficients of cladding and substrate, respectively, ec and es are the linear parts of the permittivities and Ey1 and Ey3 are the TE fields in cladding and substrate, respectively. The electric field in each layer has the following solution for TE modes for ac40 and as40: sffiffiffiffiffi 2 q sec hðk0 qc ðz zc ÞÞ; z4h, E y1 ¼ (3) ac c E y2 ¼ A cosðk0 qf zÞ þ B sinðk0 qf zÞ;
0ozoh,
(4)
sffiffiffiffi 2 q sec hðk0 qs ðzs zÞÞ; zo0, E y3 ¼ (5) as s qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where qc ¼ N 2 c , qf ¼ f N 2 , qs ¼ N 2 s , N is the effective refractive index, constants A and B represent the amplitude of the waves, and zc and zs are constants related to the field distribution in the covering medium and the substrate, respectively. In nonlinear optics, the nonlinear coefficient a is usually written in terms of the intensity-dependent refractive index n ¼ ni+n2iS, where S is the local pffiffiffiffi intensity (W/m2), ni ¼ i is the low power refractive index and i is either c for the cladding or s for the substrate. Simple algebra in the limit nibn2iS gives ai ¼
ð1 ac Þð1 þ X 2s Þ 1. ð1 as Þ
X 2c ¼
(8)
The effective refractive index can be written in terms of as and Xs as sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffi as þ X 2s N ¼ f . (9) 1 þ X 2s In terms of Xs and Xc Eq. (6) can be written as k0 qf h arctanðX c tanh C c Þ arctanðX s tanh C s Þ mp ¼ 0,
(10)
where m ¼ 0,1y is the mode order. From the dispersion relation given by Eq. (10), we derive the sensor sensitivity Sh, i.e., the changes in the effective refractive index N for changes in the refractive index nc of the cover medium. Differentiating Eq. (10) with respect to N and calculating Sh as ðqnc =qNÞ1 ) we obtain qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffi ac 1 þ X 2c ðH c þ tanh C c Þ , S h ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X c ac þ X 2c ð1 þ X 2c tanh2 CÞðATE þ G s þ G c Þ (11) where pffiffiffiffi H c ¼ k0 ðh zc ÞX c f
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ac ð1 tanh2 C c Þ, 1 þ X 2c
(12)
Gc ¼
H c þ tanh C c ð1 þ X 2c Þ , X c ð1 þ X 2c tanh2 C c Þ
(13)
Gs ¼
H s þ tanh C s ð1 þ X 2s Þ , X s ð1 þ X 2s tanh2 C s Þ
(14)
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sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffi 1 as H s ¼ k 0 zs X s f 1 þ X 2s
tanh C s bs ss þ 3 tanh2 C s X 2s ð1 tanh2 C s Þ,
(15) d TE ¼
X s ss bs þ tanh C s X 2c sc bc þ wX s tanh C c þ , js X c jc
ATE ¼ arctanðX s tanh C s Þ þ arctanðX c tanh C c Þ þ mp. (16) The sensitivity of the proposed nonlinear sensor is given by Eq. (11) which strongly depends on the waveguide structure. To maximize the sensitivity on the quantity to be measured, we cancel the derivative of the sensitivity Sh with respect to guiding layer thickness h. To simplify the expression of the maximum sensitivity we consider M1 ¼ ATE+Gs+Gc, M2 ¼ Hc+tanh Cc, qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R1 ¼ ac þ X 2c , R2 ¼ 1 þ X 2c , and R3 ¼ ðwX s =X c þ 2sc bc X 3c tanh C c þ 3wX s X c tanh2 C c Þ. After some algebraic manipulations and using the fact that qS h =qh ¼ ðqS h =qX s ÞðqX s =qhÞ and qX s =qha0, the maximum sensing sensitivity condition can be written as pffiffiffiffiffi pffiffiffiffiffi wX s X c jc R1 M 1 ac 2bc sc R2 ð1 C c tanh C c Þ þ M 2 a c R2 M 2 R2 j wX s X c ¼ 0, X c R1 jc ðd TE þ gs þ gc Þ þ M 1 R1 R3 þ c R1
(17) where
bc ¼
(22) sc ¼ 1 tanh2 C c ;
ss ¼ 1 tanh2 C s ,
jc ¼ 1 þ X 2c tanh2 C c ; 1 ac . w¼ 1 as
js ¼ 1 þ X 2s tanh2 C s ,
One of the most important quantities when speaking about optical sensing is the power flow in different layers of the slab waveguide. The sensitivity of the sensor is critically dependent on the fraction of total power propagating in the cladding medium. The energy flux per unit length is given by Z 1 E H dz ¼ Ps þ Pf þ Pc . (24) P¼ 1
We consider j1 ¼ 1tanh Cc, j2 ¼ 1tanh Cs, j3 ¼ 1 cos(2k0qfh), xþ ¼ 1 þ X 2s tanh2 C s , and x ¼ 1 X 2s tanh2 C s . The fraction of total power flowing in the nonlinear cladding is
X s X c X s ss bs X c tanh C s X s w tanh C c pffiffiffiffi X s X c Cw 2 þ þ þ þ k 0 zc f js js jc Xc 1 þ X 2s
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 as 1 þ X 2s
X 2 sc 1 c jc
,
Xc j Pc ac 1 ¼ . Ptotal X c X 2 sec h2 C s 1 Xs j1 þ s ko qf hxþ þ sinð2k0 qf hÞx þ X s tanh C s j 3 þ j 2 ac 2as as 2
pffiffiffiffi bs ¼ k 0 z s f
gc ¼
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 as 1 , 2 1 þ X s 1 þ X 2s
X c jc r1 r2 r3 X 2c j2c
(19)
(20)
with r1 ¼ bc sc ð1 2C c tanh C c Þ þ 2wX s tanh C c þ bc ð1 þ X 2c Þsc , r2 ¼ H c þ tanh C c ð1 þ X 2c Þ, r3 ¼ wðX s = X c Þ þ 2X 3c tanh C c bc sc þ 3 tanh2 C c wX s X c gs ¼
X s j s r4 r5 r6 X 2s j2s
(21)
with r4 ¼ bs ss ð1 2C s tanh C s Þ þ 2X s tanh C s þ bs ð1 þ X 2s Þss , r5 ¼ H s þ tanh C s ð1 þ X 2s Þ, r6 ¼ 1 þ 2X 3s
(23)
(18)
(25)
3. Representation and discussion In our calculations below, we will assume the guiding layer to be Si3N4 (nf ¼ 2), the free space wavelength to have the value 1550 nm, tanh Cc ¼ 0.6 and tanh Cs ¼ 0.7. Only the fundamental mode (m ¼ 0) will be considered since it has the highest sensitivity [1,10]. In Figs. 1 and 2 the sensitivity of the proposed nonlinear sensor is shown as a function of the guiding layer thickness h. The sensitivity versus the waveguide thickness behaves quite differently in the case when the refractive index of the cover is lower (ncons) and higher (nc4ns) than the substrate index. The case (ncons) is called the conventional waveguide symmetry and the
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Sh
0.12
0.08
0.04
0
200
300
400 500 h(nm)
600
700
800
Fig. 1. Sensitivity versus the waveguide thickness h for as ¼ 0.65 and ac ¼ 0.45 for the proposed nonlinear sensor (solid line) and a linear sensor (dotted line).
0.7
ac = 0.7
863
sensor is similar to the conventional waveguide discussed above. Moreover, Fig. 1 shows a comparison between linear and nonlinear sensors. We see that nonlinear sensors have higher sensitivities than linear sensors. We believe that optical sensing can be improved by using nonlinear sensors. The variation of the sensitivity with the terms tanh Cc and tanh Cs, the clad–film interface nonlinearity and the substrate–film interface nonlinearity, respectively, which arise from the nonlinearity of the cladding and the nonlinearity of the substrate respectively, is plotted in Figs. 3 and 4. As the terms tanh Cc and tanh Cs go to unity in Eq. (10), we obtain the well known characteristic equations for linear waveguides [9,10]. Fig. 3 shows the sensitivity to have the minimum value as tanh Cc goes to one, the linear cladding, and Fig. 4 shows that the sensitivity increases as tanh Cs increases to one. Thus we conclude, the nonlinearity of the cladding is essential to enhance the sensitivity of the proposed waveguide sensor.
0.6
Sh
0.5
ac = 0.6
0.12
0.4 0.3
ac = 0.5
0.1 Sh
0.2 0.1 0
ac = 0.5
0.1
ac = 0.4
0.08 0.06 0.04
100 200 300 400 500 600 700 800 h(nm)
Fig. 2. Sensitivity versus the thickness for a waveguide with normal symmetry (as4ac) and a waveguide with reverse symmetry (asoac). as ¼ 0.55 for all cases.
0.2
0.4
0.6 tanhCc
0.8
1
Fig. 3. Sensitivity against tanh Cc for different values of ac, as ¼ 0.7, and h ¼ 0.2 mm.
0.08 0.07
ac = 0.6
0.06 Sh
case (nc4ns) is called the reverse symmetry waveguide. For the conventional symmetry (ncons), the two figures show that the sensitivities have their maxima at waveguide thickness somewhat higher than the cut-off thickness of the guided mode considered. For thick waveguides, the sensitivities decrease to zero because the power of the guided mode flows mainly in the guiding layer itself. For the case of reverse symmetry (nc4ns), Fig. 2 shows that the sensitivities decrease with increasing the guiding layer thickness, starting from the maximum value near cut-off thickness. This behavior is attributed to the power considerations. When the guiding layer thickness approaches the cutoff thickness, the effective refractive index (N) approaches the cover index (nc), the penetration depth of the evanescent field into the cover medium becomes infinite, and the total power of the mode flows mainly in the cover. As the guiding layer thickness of reverse symmetry waveguide increases, the behavior of the
ac = 0.3
0.02
0.05
ac = 0.45
0.04
ac = 0.3
0.03 0.02 0.2
0.4
0.6 tanhCs
0.8
1
Fig. 4. Sensitivity with tanh Cs for different values of ac, as ¼ 0.65, and h ¼ 0.65mm.
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0.6 0.5 0.3
0.6
0.2
0.5
0.1 0
0.4 0.7 0.6 0.3 0.5 0.4 0.3 0.2 as 0.2 0.1
ac
Xs
0.4
be improved by using nonlinear materials in the cladding and substrate. The optimum structure can be obtained by using a nonlinear material in the cladding layer and a linear material in the substrate. It has been shown that the thickness of the guiding layer is a critical parameter for the sensitivity of the optical sensor with the optimum thickness is equal to the cut-off thickness in case of reverse symmetry waveguides and just above the cut-off thickness in case of normal symmetry waveguides.
Fig. 5. Normalized effective refractive index Xs versus the asymmetry parameters as and ac ensuring maximum sensitivity for homogeneous sensing.
Eq. (17) contains the condition required for the sensor to exhibit its maximum sensing sensitivity. If we consider Eq. (8), the solution of Eq. (17) is one value of the normalized refractive index Xs for each pair of the asymmetry parameters (as, ac). The effective refractive index N to which an Xs solution corresponds is given by Eq. (9). The thickness of the waveguide ensuring maximum sensitivity can be easily found by substituting the optimized value of Xs obtained from the solution of Eq. (17) into Eq. (10). Eq. (17) is plotted in Fig. 5 and surface Xs (as, ac) is obtained. All the necessary information to construct the optimized sensor is found in this surface and the above set of equations. A typical way of proceeding is as follows: usually the sensor usage imposes the cover material, thus the refractive index nc is determined. The choice of the substrate (consequently ns) is controlled by temperature, mechanical stability and cost criteria. The optical and chemical stability are the criteria considered in the choice of the guiding layer material. Thus the optogeometrical parameters as and ac are determined. The designer will look at the chart representing Xs as a function of as and ac (Fig. 5) to find the value of Xs which provides the highest sensitivity. Introducing the solution found for Xs into Eqs. (8)–(10) gives optimum normalized parameter Xc, effective index N and the guiding layer thickness h, respectively. Substituting these values into Eq. (11), the maximum sensitivity achievable is calculated.
4. Conclusion In this work a nonlinear three-layer slab waveguide optical sensor is analyzed. The case of homogeneous sensing is considered. The sensitivity of the effective refractive index to variations of the cladding index can
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