Magneto-optical phase modulation in integrated Mach–Zehnder interferometric sensors

Sensors and Actuators A 134 (2007) 339–347

Magneto-optical phase modulation in integrated Mach–Zehnder interferometric sensors B. Sep´ulveda ∗ , G. Armelles, L.M. Lechuga Instituto de Microelectr´onica de Madrid, IMM (CNM-CSIC), Isaac Newton 8 (PTM), Tres Cantos, Madrid 28760, Spain Received 5 January 2006; received in revised form 24 May 2006; accepted 29 May 2006 Available online 24 July 2006

Abstract The integrated Mach–Zehnder interferometric biosensors are one of the most promising optical biosensors due to their extreme sensitivity and possibility of integration in a lab-on-a-chip. However, the periodic response of the interferometric devices complicates the interpretation of experimental results due to the ambiguity and fading of the signal. To overcome these problems we present a phase modulation system based on the introduction of a periodic phase shift in the reference arm of the interferometers and the Fourier analysis of the output signal. This system allows the direct and unambiguous detection of the phase changes induced by the biosensing measurement. As phase modulation mechanism, we propose and theoretically analyze two different magneto-optic (MO) methods compatible with the standard microelectronic processes. The phase modulators are based on silicon-on-insulator (SOI) waveguides and yttrium iron garnets (YIG) as magnetic material. The first one exploits the non-reciprocal phase shifts induced in the TM guided modes when the orientation of the magnetization of an YIG layer is rotated within the plane of the layer. On the other hand, the MO phase modulation can also rely on the birefringence induced in magnetic liquids under an external magnetic field. We demonstrate that the MO phase modulation is compatible with SOI waveguides showing very high surface sensitivity in the biosensing measurements. In addition, the MO phase modulation can be achieved by using MO interaction lengths of only a few millimetres, facilitating the integration within the interferometric biosensors. © 2006 Elsevier B.V. All rights reserved. Keywords: Integrated optics; Mach–Zehnder interferometer; Biosensors; Phase modulation; Magneto-optic effects

1. Introduction The progress of integrated optics has allowed the fabrication of compact biosensing devices with high sensitivity, fast response time and able to monitor real-time interactions. Additional advantages, as compared to the conventional optical biosensing systems, are the possibility of miniaturization, robustness, reliability, potential for mass production with consequent reduction of production costs, low energy consumption and simplicity in the alignment of the individual optical elements. The integrated optical biosensors share two characteristics: the propagation of light within a waveguide, and the evanescent field detection. Among the evanescent field sensors, the integrated Mach–Zehnder interferometric (MZI) biosensor is one of

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the most interesting due to their high sensitivity and the possibility of optoelectronic integration in lab-on-a-chip microsystems [1]. The combination of these features can open the way to the development of compact and portable biosensing devices which will allow the real time detection of very low concentrations of small molecules, like some infection agents, SNPs in DNA strands or low weight environmental pollutants, without labeling [2]. In an integrated MZI, the light traveling in a rib waveguide is split into two arms, the sensing arm and the reference arm, by means of an Y-divisor. The two beams are recombined again, after a certain distance, producing their interference. A schematic of this sensor is illustrated in Fig. 1. For the biosensing application, the integrated MZI is covered with a protective layer except the sensor area of the sensor arm, thus the evanescent field of the guided mode can probe the external medium. The local changes of refractive index produced inside of the evanescent field in the sensor area, as the ones generated by a biomolecular interaction at the surface, will change

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Fig. 1. Integrated MZI biosensor scheme.

the effective propagation index of the guided mode, inducing a phase difference between the light beams traveling in both arms. Such phase difference will be translated as a change in the interference signal of the interferometric device. As a consequence, the output signal of the MZI is given by the following expression:  IT = IS + IR + 2 IS IR cos(ϕS (t)) (1) where IS and IR are the intensities of the light in the sensor and reference arms, respectively. The term ϕS is the phase difference between the light beams traveling in both arms, and is represented by: ϕS (t) =

2π (NS (t) − NR )L λ

(2)

where NS and NR are the effective propagation indexes of the guided modes in the sensor and reference arms, respectively, λ the wavelength of the light and L is the length of the sensor area. As can be deduced from Eq. (1), the output intensity of the interferometric device is periodic with respect to the phase changes induced in the sensor area. The periodicity of the output signal is an important drawback for the interpretation of the MZI sensor response. There are mainly three problems associated with this type of signal: (a) Ambiguity: it is not possible to deduce the direction of the phase changes when the phase difference between the two arms of the MZI is an integer multiple of π (see Fig. 2). (b) Intensity variations produced by fluctuations of the light source, misalignments of the optical components or changes in the absorption of the environment may be misinterpreted as phase changes of the interferometer. (c) Signal fading: the sensitivity depends on the initial phase difference between the interferometric arms. If the interferometer is tuned close to one of the extreme values of the transmission curve, small phase changes will generate low intensity variations (see Fig. 2). However, the sensitivity of the device reaches its maximum value if the interferometer is tuned close to the quadrature condition, which corresponds

Fig. 2. Output signal of a MZI showing the fading at extreme values of the transmission curve.

to a phase difference between the arms of π/2 plus an integer multiple of π. The disadvantages derived from the periodicity of the output signal can be solved by tuning the phase difference between the arms of the interferometer by means of a phase modulation system integrated in the reference arm of the MZI [1,3–5]. Several methods to obtain the phase modulation have been proposed as, for example, the electro-optic [1], thermo-optic [6,7] or optomechanical [8,9] modulations. The electro-optic modulation requires the use of materials with high electro-optic coefficients, like LiNbO3 or KD2 PO4 , widely employed for the fabrication of modulators in the telecommunications field. However, the growth of these materials is not compatible with standard microelectronic processes. The compatibility can be achieved using ZnO, as Ref. [1] describes, or electro-optic polymers. Another negative aspect of the electro-optic modulation is the requirement of the introduction of electric contacts in the sensor, making more complex the fabrication of the device, as the contacts must be isolated to work in aqueous environments. On the other hand, in the thermo-optic modulation it is difficult to control the phase shifts due to the thermal coupling between the arms of the interferometer. In addition, it is necessary to introduce metal films to generate heat by Joule effect, which results in high power consumptions. Finally, an opto-mechanical modulator can be developed by fabricating a diaphragm in the sensor arm of the interferometer, which involves complicated micromechanical fabrication. To overcome these technological problems we propose and theoretically study in this paper a novel magneto-optical (MO) phase modulation method by placing a MO medium in the reference arm of the integrated MZI, and driven with a low external magnetic field. The proposed MO phase modulation is compatible with the standard microelectronic processes, which can help to the inexpensive and rapid development of the integrated MZI biosensors.

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where Jn is the Bessel function of order n. Therefore, in this modulation scheme, the DC term in is represented by:  (6) IDC = IS + IR + 2 IS IR cos(ϕS (t))J0 (μM ) while the first harmonic and second harmonics are, respectively:  (7) I1ω = 4 IS IR sin(ϕS (t))J1 (μM )  I2ω = 4 IS IR cos(ϕS (t))J2 (μM ) (8) If we choose a modulation amplitude, μM , that satisfies: J1 (μM ) = J2 (μM )

(9)

using the rate between the first and the second harmonics, we can find that   I1ω ϕS (t) = arctan (10) I2ω Fig. 3. Output signal of an MZI with a sinusoidal phase modulation system, assuming that IS = IR = 1/4 and μM = 0.8371π. The output signal of the interferometer when the biosensing measurement induces a phase change ϕS = π/3 is also represented (dashed line).

The outline of this paper is as follows: Section 2 presents the modulation principle that allows the direct measurement of the phase changes produced in the sensor area of the MZI. Section 3 describes the fundamentals of the MO phase modulation and analyzes the characteristics of the MO materials required for this application. Section 4 shows the results of the theoretical simulations and presents the optimal designs of the MO phase modulator. Finally, the conclusions are summarized in Section 5. 2. Phase modulation principle The proposed phase modulation system is based on the introduction of a periodical phase change in the reference arm of the interferometer and the subsequent Fourier analysis of the output signal. Firstly, we consider an arbitrary phase modulation function, f(ωM ), and we assume that the period of the modulation is much shorter than the response time of the biosensing reactions, which is typically in the order of seconds or minutes. The introduction of the phase modulation system in the reference arm of the interferometer transforms Eq. (1) into:  IT = IS + IR + 2 IS IR cos[f (ωM ) + ϕS (t)] (3) If we consider that the phase modulation system produces a periodic phase shift with μM amplitude and ωM frequency, the output signal of the interferometer, shown in Fig. 3, can be expressed as:  IT = IS + IR + 2 IS IR cos[μM sin(ωM t) + ϕS (t)] (4)

Eq. (10) directly provides ϕs (t), which is the quantity to be determined in the biosensing measurements. The modulation amplitude that satisfies relation (9) is 0.8371π. This kind of phase modulation scheme solves the problems of the ambiguity and fading of the output signal of the interferometer. In addition, the detection of the phase changes is immune to the intensity changes produced by the absorption of the environment, fluctuations of the light source or misalignments of the optical components. The modulation frequency can be in the order of tens of hertz, since the changes caused by the biosensing interactions will be, generally, in the order of seconds or minutes. 3. Magneto-optic modulation The phase modulation of the light traveling within an optical waveguide can be induced using materials with magnetooptical activity. The optical properties of a magnetic medium are described by its dielectric tensor. If the magnetization of the medium is in an arbitrary direction and the material has cubic symmetry or is polycrystalline, the dielectric tensor is given by: ⎛ ⎞ εxx εxy εxz ⎜ ⎟ ε = ⎝ −εxy εxx −εyz ⎠ (11) −εxz εyz εxx The magneto-optic non-diagonal elements are responsible of the non-reciprocal coupling between the different components of the electromagnetic fields. In general, the transversal electric (TE) and transversal magnetic (TM) polarizations are coupled in the MO waveguides, and the MO effects depend on the direction of the magnetization of the magnetic materials with respect to the direction of propagation of the light. If we assume that the guided light propagates in the x direction, the transversal configuration is found when the

Eq. (4) can be expanded in a Fourier series, resulting:     ∞  ∞    IT = IS + IR + 2 IS IR cos(ϕS (t)) J0 (μM ) + 2 J2n cos(2nωM t) + sin(ϕS (t)) 2 J2n+1 cos((2n + 1)ωM t) n=1

n=0

(5)

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magnetization is perpendicular to the propagation plane of the waveguide (the XZ plane in Fig. 1). In this configuration, the dielectric tensor (11) is transformed into: ⎛ ⎞ εxx 0 εxz ⎜ ⎟ εxx 0 ⎠ ε=⎝ 0 (12) −εxz 0 εxx The MO non-diagonal elements couple the x and z components of the electric field, maintaining the independence of the TE and TM polarizations in Maxwell equations. In this configuration, the MO effect can induce a non-reciprocal propagation wavevector for the TM guided modes, i.e. the forward and backward traveling modes exhibit different effective propagation indexes (N): N + = −N −

(13)

where the (+) and (−) symbols represent the forward and backward directions of propagation, respectively. Such existence of a non-reciprocal effective propagation constant has been proposed and used to develop integrated non-reciprocal optical isolators [10]. In addition, in this direction of the magnetization the following relation between the forward and backward modes is satisfied [11]:  = −N − (−M)  N + (M)

(14)

where M is the magnetization of the magnetic materials. Considering Eqs. (13) and (14), it is concluded that  = N + (−M)  N + (M)

(15)

Therefore, the inversion of the magnetization can produce a nonreciprocal phase shift in the guided light, given by: ϕM =

2π  − N(−M))L  (N(M) M λ

(16)

where LM represents the length of the MO waveguide, and the  − N(−M),  is a linear function of the magdifference N(M) netization. Such a non-reciprocal phase shift can be used to generate the phase modulation in the MZI sensor by placing a magnetic material in the reference arm of the interferometer, and changing its magnetization state with an external magnetic field. Although the non-reciprocal phase modulation depends linearly on the magnetization, the relation between the magnetization of the ferromagnetic material and the external magnetic field is not, generally, linear. As an example, if the ferromagnetic material exhibits a square magnetization-reversing loop in the transversal configuration, the change or inversion of the external magnetic field only induces two different magnetization states. As a consequence, this kind of materials only permits a square-shape modulation, which will not solve the problems of the periodicity of the output signal of the MZI. In these cases it will be possible to achieve the sinusoidal phase modulation by rotating the magnetization within the plane of the magnetic film (XY plane). However, a magnetization parallel to the direction of propagation of the light (longitudinal configuration) produces a rotation of the polarization plane of the guided light,

which will generate as well an undesirable intensity modulation. Such effect is very small in the waveguide configurations required for biosensing applications, as we will show, and can be neglected. Among the different magnetic materials which can be employed to develop a MO phase modulator, the yttrium iron garnets (YIG) are the most promising and most widely used in integrated optics, since they present an excellent transparency at communication wavelengths, and show high MO effects. In addition, these materials are compatible with the microelectronic technology and can be integrated in SOI wafers [12]. The YIG films used in integrated optics present in-plane magnetization and square magnetization loops with low coercitivity. Therefore, the employment of these materials in the development of integrated phase modulation systems will require the use of a sinusoidal phase modulation scheme with a rotating magnetization. Another interesting MO media to perform a phase modulation are the magnetic liquids which contain acicular magnetic nanoparticles. If the nanoparticles do not interact between them, the magnetic liquid can be treated as a superparamagnetic system. In the absence of a magnetic field, the orientation of these nanoparticles will be random, and the magnetic liquid will be optically isotropic. The application of an external magnetic field will tend to align the long axis of the nanoparticles parallel to the magnetic field, inducing an optical birefringence in the liquid. As a result, two different MO effects can be observed in a magnetic liquid: the pure MO effect due to the non-diagonal elements of the dielectric tensor of the nanoparticles, and the birefringence caused by their alignment. The optic and magneto-optic effects of the magnetic liquids can be theoretically studied using a self-consistent effective medium formalism [13], providing the nanoparticles are much smaller than the light wavelength. In this formalism, the optical properties of the composite medium are described by an effective dielectric tensor that depends on the optical and magneto-optical constants of the matrix material (solvent) and the nanoparticles, as well as their concentration, shape, and orientation. For simplicity, it is assumed that the nanoparticles are ellipsoidal with revolution symmetry in the long axis, and present an aspect ratio, AR, defined as c/a, where a and c are the lengths of the short and long axis of the nanoparticles, respectively. Thus, when long axis of the nanoparticles is oriented in the x direction, the dielectric tensor of the magnetic liquid can be expressed as [20]: ⎛

εe11

⎜ ε = ⎝ −εe12 0

εe12 εe22 0

0

⎞

⎟ 0 ⎠

(17)

εe22

where εeij are the effective dielectric constants of the magnetic liquid. In this tensor εe11 and εe22 are different due to the birefringence induced by the shape and orientation of the nanoparticles, and εe12 is owed to the magneto-optic activity of the nanoparticle. The change of the orientation of the nanoparticles to the

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y-axis will transform Eq. (17) into: ⎛ ⎞ εe22 0 εe12 ⎜ ⎟ εe11 0 ⎠ ε=⎝ 0 −εe12 0 εe22

343

(18)

If the magnetic liquid is placed over the core of the waveguide, the change of the orientation of the magnetic nanoparticles will modify the effective propagation index of the TE and TM guided modes, allowing its use to generate the phase modulation in the integrated MZI: ϕM =

2π (N(Mx ) − N(My ))LM λ

(19)

Being N(Mx ) and N(My ) the effective propagation index of the guided mode when the nanoparticles are oriented in the x and y directions, respectively. However, if the nanoparticles show optical absorption, their alignment will produce as well an intensity modulation that can complicate the detection of the phase changes generated by the biosensing interactions. In consequence, the magnetic nanoparticles must be composed of materials with low optical absorption, like YIG. 4. Theoretical calculations 4.1. Structure of the optical waveguides for biosensing The waveguides of an integrated MZI for biosensing applications must satisfy two main conditions: high surface sensitivity and monomode behavior. The monomode behavior is necessary to prevent the superposition of the interference signals of the different independent guided modes. On the other hand, a high surface sensitivity is required since the detection of the biomolecular interactions is produced at surface of the core of the waveguide. The highest surface sensitivity is obtained for core layers with high refractive index [14,21]. For that reason and, in order to achieve the compatibility with the microelectronic technology, we study SOI waveguides. In these waveguides, calculations based on effective index method [15] show that the monomode behavior is achieved for core thickness below 350 nm in both the TE and TM polarizations and for rib depths of 3 nm when the rib width is 3 ␮m. For a waveguide-based biosensor, the surface sensitivity can be defined as the rate of change of the effective propagation index of the guided modes as the thickness of a biological layer (db ) varies: η=

∂N ∂db

Fig. 4. Thickness dependence of the surface sensitivity of the SOI waveguides.

thickness of 210 nm for the TM polarization and 60 nm for the TE. 4.2. Structure of the magneto-optic waveguide Two different MO media are analyzed to perform the phase modulation in SOI waveguides. In the first one, the MO medium is a continuous YIG film located over the silicon core layer, while in the second structure the MO medium is a magnetic liquid with YIG acicular magnetic nanoparticles, placed as well over the Si core. The general structure under study is depicted in Fig. 5. In such guiding structure, the MO waveguide with the continuous or nanostructured magnetic medium is situated between two SOI waveguides. 4.2.1. YIG film phase modulator The MO phase modulation with the YIG film requires the rotation of the magnetization within the plane of the layer and TM guided modes. When the magnetization is in the transversal orientation, an incident TM mode coming from the first SOI waveguide mode can only excite the TM mode in the MO waveguide and, therefore, the transmitted and reflected guided modes through the MO waveguide will be TM. In contrast, when the magnetization is parallel to the x-axis, the guided modes of the MO waveguide (M1 and M2 in Fig. 5) are not pure TE or TM, having the six components of the elec-

(20)

In Fig. 4 we represent such sensitivity as a function of the core thickness. In these calculations we have assumed that the refractive index of the silicon core layer is nc = 3.45 at a wavelength of 1.5 ␮m, the index of the biological layer is nb = 1.45 and the external medium is water (ne = 1.33). As can be seen in Fig. 4, the highest surface sensitivity is obtained for a core

Fig. 5. Structure of the layers of the integrated MO phase modulator in SOI waveguides. The arrows denote the incident, transmitted and reflected modes in the guiding structure.

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tromagnetic fields. In this case, an incident TM mode can excite both modes of the MO waveguide inducing transmitted and reflected TE and TM modes in the SOI waveguides. Such MO effect can be considered as a rotation of the polarization plane of the transmitted and reflected modes. The TE output modes are not phase modulated by the magnetization and produce an additional intensity modulation. Such intensity modulation can mix the different terms of the Fourier analysis of the output signal hindering the direct extraction of the biosensing phase changes. The theoretical treatment of the guiding structure of Fig. 5 can be performed using a theoretical formalism that takes into account the non-reciprocity of the MO waveguide and the continuity of the electromagnetic field components (Ey , Ez , Hy and Hz ) parallel to the discontinuities between the SOI and the magneto-optic waveguides [16,17]. This formalism requires, firstly, the calculation of N and the field distribution of the guided modes in the SOI and MO waveguides, which can be determined by using a 4 × 4 matrix formalism [17,18]. Once N and the field distribution of the modes are obtained, the amplitude of the transmitted and reflected TE and TM modes in the SOI waveguides can be evaluated. We firstly analyze the core thickness dependence of the nonreciprocal shift of the effective propagation index of the TM mode when the magnetization is reversed in the MO waveguide, N(φ = 0◦ ) − N(φ = 180◦ ). In these calculations we assume that the dielectric and magneto-optic constants of YIG at a wavelength of 1.5 ␮m are, respectively, εii = 4.84 and εij = 0.005i [19]. As can be deduced from Fig. 6, the strongest MO effect is achieved for a 200 nm core thickness. Therefore, it will be possible to obtain simultaneously high MO effects and high surface sensitivity for the biosensing measurements using the SOI waveguides. This figure also shows the length of the MO waveguide necessary to obtain a 2π phase modulation, which is calculated through the following

Fig. 7. Non-reciprocal phase shift of the TM mode and amplitude of the excited TE mode in the second SOI waveguide as a function of the angle (φ) of the magnetization (φ = 0◦ and 90◦ represent the transversal and longitudinal magnetizations, respectively). We assume a Si core thickness of 210 for the SOI waveguide.

expression: LM (2π) =

λ N(φ = 0◦ ) − N(φ = 180◦ )

(21)

As can be observed, the MO phase modulation can be achieved with a MO waveguide length of only 1.5 mm. Now we analyze the MO effects created by the rotating magnetization. For such purpose we calculate in Fig. 7 the amplitude of the excited TE mode in the second SOI waveguide with respect to the amplitude of an incident TM mode in the first SOI waveguide (TTE ), as a function of the orientation angle (φ) of the magnetization using the formalism developed in Ref. [16]. Such effect is compared to the induced phase change when the magnetization is rotated, taking as a reference the longitudinal configuration, that is: ϕM = ϕ(φ) − ϕ(φ = 90◦ )

(22)

As Fig. 7 shows, when the magnetization is in the transversal configuration (0◦ and 180◦ ), the amplitude of the excited TE mode is zero, as a result of the decoupling of TE and TM polarizations. In contrast, the maximum amplitude of the TE mode is obtained when the magnetization is in the longitudinal configuration (φ = 90◦ ). However, such amplitude is five orders of magnitude lower than the amplitude of the excited TM mode, which is around 0.8 in this configuration, and can be neglected in practical applications. From these calculations we can also deduce that 20% of the light is lost by radiation at the discontinuities, due to the different refractive index between YIG and SiO2 .

Fig. 6. Thickness dependence of the non-reciprocal change of the effective propagation index of the TM guided mode of the MO waveguide with a continuous YIG layer; and length of the MO waveguide required to obtain a 2π phase shift when the magnetization is reversed.

4.2.2. Magnetic liquid phase modulator The phase modulation based on magnetic liquids can be a simple alternative to avoid the integration of YIG films into the SOI waveguides, since the magnetic liquids can be flowed and kept within microfluidic channels over the reference arm of the integrated MZI devices. In addition, the change of the orientation

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Fig. 8. MO effect produced by the rotation of the applied magnetic field from the transversal to the polar configuration for spherical YIG nanoparticles and for ellipsoidal nanoparticles whose aspect ratio is 1.1. In this figure, 0◦ represents the magnetization in the transversal configuration and 90◦ the longitudinal configuration. It is also represented the MO effect of the ellipsoidal nanoparticles when their MO activity is not considered in the calculations.

of the nanoparticles also allows the phase modulation in the TE modes of the waveguide. In order to separate the MO effects produced by the nondiagonal MO elements of the dielectric tensor of the nanoparticles, from the effects caused by the change of the orientation of the nanoparticles, we first analyze a magnetic fluid composed by spherical nanoparticles. We consider YIG nanoparticles showing the same optical constants as the continuous film, dispersed in water (n = 1.33), with a volume concentration (f) of 1%. If the nanoparticles are spherical, the change in the orientation of the applied magnetic field will not induce birefringence, i.e. εe11 = εe22 in Eq. (20), and the MO effects will be due to the MO activity of the nanoparticles (εe12 component). Fig. 8 represents the non-reciprocal shift of N in the TM mode induced by the inversion of the magnetization of the nanoparticles. Such effect is compared to the variation of N in the TM mode when acicular nanoparticles with 1.1 aspect ratio are rotated from the transversal configuration (φ = 0◦ ) to the longitudinal configuration (φ = 90◦ ): N = N(φ) − N(φ = 0)

(23)

The non-reciprocal effect induced by the MO constants of the nanoparticles is very weak. Fig. 8 shows that the variation of N in the ellipsoidal nanoparticles is one order of magnitude higher when the AR is only 1.1. In this figure we also represent the variation of N when the MO constants of the ellipsoidal nanoparticles are not considered. As can be observed, the difference in the calculation is small and can be neglected for AR bigger than 1.1. In the case of the magnetic liquids, the MO effect induced by the change of orientation of the particles can be observed in the TE and TM guided modes. In Fig. 9 we analyze such effect for both polarizations and different aspect ratios of the

Fig. 9. Variation of the effective propagation index of the TM and TE modes, when the nanoparticles are rotated from the transversal (φ = 0◦ ) to the longitudinal configuration (φ = 90◦ ), for different aspect ratios, as a function of the volume concentration of particles. The results are normalized by the volume concentration.

ellipsoidal nanoparticles. The guiding structures are chosen to obtain the highest surface sensitivity in each polarization (i.e. Si core thickness of 60 or 210 nm for the TE or the TM polarizations, respectively). Fig. 9 shows the change of N induced by the rotation of the nanoparticles from the transversal to the longitudinal configuration, for different AR of the nanoparticles and as a function of the volume concentration (f). The results are normalized by f for the comparison. As Fig. 9 evidences, the MO effect is stronger as the aspect ratio of the nanoparticles increases. The effect is higher for the TE mode, since the highest MO effect for the TM polarization is found when the nanoparticles rotate from the longitudinal to the polar configuration (nanoparticles pointing to the z direction). Finally, we test the viability of the orientational effect of the magnetic nanoparticles to achieve a phase modulation in the integrated MZI. For such purpose we calculate in Fig. 10 the length of the MO waveguide necessary to obtain a 2π phase modulation. In this particular case such length is given by:

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ity in the biosensing measurements. In this modulator the MO medium can be a YIG film or a magnetic liquid formed by nanometric and ellipsoidal YIG nanoparticles dispersed in a fluid. The results of the theoretical calculations have revealed that is possible to achieve the 2π non-reciprocal phase modulation for the TM modes by integrating an YIG film with a length as short as 1.5 mm. In the case of the magnetic fluids, the modulation can be induced in the TE and the TM modes with MO waveguides ranging from 1.5 cm to 1 mm, depending on the aspect ratio of the nanoparticles, using very low concentrations of nanoparticles. These MO effects using YIG and SOI waveguides could also find important applications within the telecommunications field to develop, for instance, integrated isolators, modulators or switches. Fig. 10. Length of the MO waveguide necessary to obtain a 2π phase change when the orientation of the nanoparticles is rotated from the transversal (φ = 0◦ ) to the longitudinal (φ = 90◦ ) configuration as a function of the aspect ratio of the particles. We assume a volume concentration of 1%.

λ LM (2π) = N(φ = 0◦ ) − N(φ = 90◦ )

(24)

In these calculations we have assumed a 1% volume concentration of nanoparticles. For both polarizations LM (2π) ranges from 1.5 cm to a few millimeters, showing the compatibility with the integrated MZI biosensors. 5. Conclusions In this paper we have proposed a phase modulation system to solve the problems derived from the periodic output signal of the Mach–Zehnder interferometric biosensors. The phase modulation allows the direct detection of the phase changes induced by the biosensing measurement through the introduction of a periodic phase shift in the reference arm of the interferometer and the Fourier analysis of the output signal. The phase modulation system solves the problems of ambiguity and fading of the periodic output signal of the MZI. In addition, the detection of the phase changes is immune to the fluctuations of the light source or the misalignment of the optical components. As a consequence, the introduction of the phase modulation system will increase the signal-to-noise ratio of the biosensing measurements, which will allow the reduction of the limits of detection of these biosensors. We have analyzed integrated magneto-optic phase modulators based on two different effects. In the first one the phase modulation is obtained by the non-reciprocal phase change induced in the TM guided modes when the magnetization of the MO medium is inverted in the transversal configuration. On the other hand, the phase modulation can be produced changing the orientation of ellipsoidal magnetic nanoparticles dispersed in liquids. In both cases the MO materials must show a very good transparency, like the YIG in the near infrared, which offers, in addition, the possibility to be integrated in CMOS compatible integrated MZI biosensors. We have shown that the MO phase modulator can be achieved with SOI waveguides presenting a very high surface sensitiv-

Acknowledgements This work was supported by the Ministerio de Ciencia y Tecnolog´ıa, Project MAT2002-04484-C03-01. B. Sep´ulveda acknowledges the I3P program of the Consejo Superior de Investigaciones Cient´ıficas (CSIC) and PHOREMOST European Excellence Network for financial support. References [1] R.G. Heideman, P.V. Lambeck, Remote opto-chemical sensing with extreme sensitivity: design, fabrication and performance of a pigtailed integrated optical phase-modulated Mach–Zehnder interferometer system, Sens. Actuators B 61 (1999) 100–127. [2] L.M. Lechuga, F. Prieto, B. Sep´ulveda, Interferometric biosensors for environmental pollution detection Optical Sensors for Industrial, vol. E, Springer, 2003 (Springer Series on Chemical Sensors and Biosensors). [3] G.W. Johnson, D.C. Leiner, D.T. Moore, Phase-locked interferometry, Proc. SPIE 126 (1977) 152–160. [4] D.A. Jackson, A. Dandridge, S.K. Sheem, Measurement of small phase shifts using a single-mode optical fiber interferometer, Opt. Lett. 5 (1980) 139–141. [5] A.T.M. Lenferink, E.F. Schipper, R.P.H. Kooyman, Improved detection method for evanescent wave interferometric chemical sensing, Rev. Sci. Instrum. 68 (1997) 1582–1586. [6] G. Cocorullo, F.G.D. Corte, M. Iodice, I. Rendina, P.M. Sarro, A temperature all silicon micro-sensor based on the thermo-optic effect, IEEE Trans. Electron Dev. 44 (1997) 766–774. [7] K. Benaissa, Y. Lu, A. Nathan, Design and fabrication of ARROW thermooptic modulators, Proc. SPIE 2641 (1995) 28–31. [8] A. Vadekar, A. Nathan, W.P. Huang, Analysis and design of integrated silicon ARROW Mach–Zehnder micromechanical interferometer, J. Lightwave Technol. 12 (1994) 157–162. [9] K. Benaissa, A. Nathan, IC compatible optomechanical pressure sensors using Mach–Zehnder interferometer, IEEE Trans. Electron Dev. 43 (1996) 1571–1582. [10] J. Fujita, M. Levy, R.M. Osgood, L. Wilkens, H. Dotsch, Waveguide optical isolator based on Mach–Zehnder interferometer, Appl. Phys. Lett. 76 (2000) 2158. [11] P.R. McIsaac, Mode orthogonality in reciprocal and non-reciprocal waveguides, IEEE Trans. Microw. Theory Tech. 38 (1991) 1808. [12] R.L. Espinola, T. Izuhara, M.C. Tsai, R.M. Osgood, Magneto-optical nonreciprocal phase shift in garnet/silicon-on-insulator waveguides, Opt. Lett. 29 (2004) 941–943. [13] D. Stroud, Generalized effective-medium approach to conductivity of an inhomogeneous material, Phys. Rev. B 12 (1975) 3368–3373.

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Biographies ´ Dr. Borja Sepulveda received his PhD degree in physics from the Complutense University of Madrid in 2005. His post-graduate research was carried out at the Microelectronics Institute of Madrid (CNM-CSIC) and was related to the “Magneto-optic effects in evanescent field optical sensors”. Currently he has a postdoctoral position at the Nanoplasmonic Group of Chalmers University (Sweden).

Dr. Lechuga received his PhD degree in chemistry by the Universidad Complutense de Madrid. She is the Head of the Biosensors Group at the National Centre for Microelectronics (CSIC) in Madrid. The Biosensors Group develops plasmonics, integrated nanophotonics and nanomechanical biosensors for lab-on-a-chip devices. The biosensors microsystems are applied in the environmental control, early clinical diagnostics and in proteomics.