The interaction frequencies between longitudinal acoustic waves and fiber Bragg grating are numerically and experimentally assessed. Since the grating modulation depends on the acoustic drive, the combined analysis provides a more efficient operation. In this paper, 3-D finite element and transfer matrix methods allow investigating the electrical, mechanical and optical resonances of an acousto-optical device. The frequency response allows locating the resonances and characterizing their mechanical displacements. Measurements of the grating response under resonant excitation are compared to simulated results. A smaller than <1.5% average difference between simulated-measured resonances indicates that the method is useful for the design and characterization of optical modulators.
©2013 Optical Society of America
Acousto-optical modulators (AOM) have been widely used as drivers in the control of the modal properties of optical fibers [1–4]. When the device is excited by an electrical signal, it induces mechanical waves in the fiber, which modulate the characteristics of the guided light. In particular, longitudinal acoustic waves allow modulating the properties of fiber Bragg gratings (FBGs), intrinsic light reflectors which are of interest for applications such as, dynamic modulators , grating writing , and fiber lasers . Since the grating reflectivity and wavelength can be controlled by the amplitude and frequency of the acoustic wave, the grating strain sensibility to axial deformations allows obtaining the fast switching of wavelengths in fibers . Previous analyses, in which only the horn and the fiber of the modulator are considered, show that the excitation of longitudinal modes occurs mainly at higher frequencies, and depends on isotropic material and geometric silica properties [6, 7]. However, these works do not consider the electrical properties of the modulator, which depend on the piezoelectric transducer (PZT) used as a driver. The horn-fiber setup is considered to be excited by perfect parallel axial forces which are uniformity distributed over the horn basis. In this way, the PZT modes are supposed to have the same displacement behavior over all excitation frequencies, situation that can diverge from the real PZT behavior. Although the presence of uniform forces on the horn basis allows estimating the FBG spectrum under the acousto-optical effect, the ideal axial excitation does not provide the real resonances and the mechanical behavior of the longitudinal modes.
Piezoceramics usually do not respond to any excitation frequency, but only to specific resonances in which the vibration modes manifest larger and particular mechanical displacements. Each resonance possesses a specific factor of electromechanical coupling that results in a different displacement for each vibration mode . Since the amplitude modulation of the FBG spectrum is proportional to the acoustically induced strain, such behavior can cause reflectivity variations when the device is tuned from one resonance to other. Moreover, the non-uniform axial PZT deformation forces applied to the horn basis can induce transversal bends in fiber causing modal coupling or undesired optical instabilities . The transducer excitation at non-resonant frequencies also implies in higher electric power consumption and reduction of the device lifetime. Therefore, the combined analysis of the electrical, mechanical and optical properties required for the resonant acousto-optical interaction, considering the frequency response, anisotropic properties, electric polarization and geometric dimensions is useful for the proper operation of acousto-optical devices.
In this paper, the multiphysics analyses of the modulator response and displacement behavior of longitudinal acoustic waves are performed, and the experimental characterization of the resultant acousto-optical effect in a Bragg grating is assessed. The device is numerically modeled using 3-D finite element method (FEM) [8–10] and the transfer matrix method (TMM)  to obtain the piezo spectral response, fiber displacements and modulated FBG spectrum. PZT anisotropy is included in the simulations and takes into account the displacement change of the PZT when excited at different frequencies. The PZT is electrically excited and its deformations are automatically coupled to the horn-fiber setup. The method is based on the frequency response of the entire modulator which allows estimating the modes that are useful in the real device. Simulation results are compared to experimental measurements of the wavelength shift, as the grating is under the excitation of the resonant acoustic modes. The proposed approach allows investigating the main intrinsic properties of an acousto-optical driver when is excited. The combined characterization is essential to the design, efficiency control and operation of acousto-optical devices.
2. Operation principle of the modulator
Figure 1 illustrates the main electrical, mechanical and optical properties of a coaxial AOM device composed by a PZT disc with electrodes deposited on its polarization surfaces, an acoustic horn and a FBG segment. Due to the high dielectric material constant and the electrode polarization, the PZT can be compared to a capacitor and represented by the equivalent circuit illustrated in Fig. 1(a) . The circuit is composed of a RLC series within a parallel circuit, where the resistor, R, the inductor, L, and the capacitor, C, are related to the damping, mass and elastic constant, respectively. Co is the electric capacitance between the electrodes. The transducer frequency response ω = 2πf is obtained solving the RLC circuit in terms of the impedance, written as, Z = R + j(ωL–j(ωC)−1). Figure 1(a) illustrates the PZT response in terms of the impedance magnitude |Z| and phase. The PZT resonance frequency, fs, and anti-resonance frequency, fp, correspond to the condition of minimum and maximum impedance, respectively. As the electrical current flowing across the transducer is related to the mechanical deformation, the reduction in impedance |Z| causes a maximum current flow I, which induces the maximum PZT deflection at fs. The resultant axial strain S is amplified by the horn and propagates harmonically into the fiber (Fig. 1(b)). When an optical mode with effective index neff propagates in the non-perturbed grating, whose period is Λ, and satisfies the Bragg condition (λB = 2neffΛ), it is reflected on the grating planes and produces a narrow wavelength band centered at λB (Fig. 1(c)). However, when a longitudinal acoustic wave is excited along the fiber, it produces a periodic deformation that compress and stretches the grating planes. The axial strain modulates both the modal effective index neff and the grating pitch Λ, causing reflection bands to appear on both sides of the Bragg wavelength, λB. The distance Δλ between the main λB and the side lobe obeys the relationship Δλ = faλB2/(2neffva), where fa and va are the acoustic frequency and velocity, respectively .
3. Numerical modeling
The piezoelectric material associated to the AOM device is modeled using coupled electromechanical anisotropy. In this way, it is possible to electrically excite the vibration modes that oscillate in each resonance. The mechanical behavior of the deformation manifested by the modes also depends on the direction in which the electric field is polarized, dimensions and geometric shape. When the transducer is exposed to an electric field Ε, it is mechanically deformed by a stress σ or strain S. However, if a mechanical deformation is applied to the material, an electric field is induced as an electric displacement D. For the piezoceramic used in simulations, the strain-charge relationship is written as ,Fig. 2(b) . The PZT is polarized in z direction and excited by a sinusoidal electric field Ε. The resulting longitudinal wave causes the periodic strain S(z) = S0cos(2πfz) into the fiber core, where a uniform Bragg grating supports a propagating and a counter-propagating optical mode with electric field amplitudes A(z) and B(z), respectively. The interaction between incident-reflected modes is described by coupled-mode equations ,6],Eq. (8) into a second order matrix (FB(2x2)), and applying appropriate boundary conditions, the amplitude and power reflection coefficients are obtained. The PZT is modeled using the coupled electromechanical relations and material constants included in Eqs. (3) and (4), which are solved by the 3-D FEM Comsol Multiphysics v3.5a tool. Figure 2 shows the 3-D design of the acoustic-optic modulator, which is composed by a PZ26 model disc, a silica horn and a standard single mode fiber, which is modeled as a solid rod. Details of the lateral and frontal profiles are shown in Figs. 2(a) and 2(c), respectively. The PZT has silver electrodes that are deposited on its polarization surfaces as illustrated in the detail of Fig. 2(a). The mechanical parameters of the electrodes, acoustic horn and optical fiber are modeled using isotropic elastic properties (density ρ, Young modulus Y and Poisson ration ν). The piezo anisotropic properties are obtained from the fabricant catalogue . Further material and geometrics properties are included in Fig. 2. In order to calculate the frequency response and the displacements of the natural resonant modes, the transducer is initially set free of loads and simulated alone. The transducer is excited by a 500 mV harmonic voltage in the 1-1200 kHz range, with 0.5 kHz steps, and the correspondent impedance-phase values are solved for each frequency. Later, the PZT is coupled to the horn-fiber setup and the modulator end faces are fixed for the calculation of the fiber acoustic resonances. The device is excited by a 10 V voltage in a 600 – 1200 kHz range, with 1 kHz steps, and the fiber axial (z direction) and the transversal displacements (xy directions) are solved for each frequency. After locating the longitudinal resonances (acoustic modes with larger displacements in z direction), 1000 strain samples (M = 1000 sections) are obtained in the fiber core to simulate the FBG spectrum using the TMM. To accurately describe the real device properties, a finite element mesh of approximately 3 million tetrahedral elements is utilized. The simulations are performed using a computer with 8 x 2.8 GHz Intel® processor cores and 96 GB RAM.
4. Experimental setup
The modulator is assembled considering the same components and boundary conditions illustrated in Fig. 2. The device includes a 5 cm long FBG, in which the reflectivity is 25 dB and the Bragg wavelength, is set at λΒ = 1538.42 nm. The FBG was inscribed by the direct UV exposure of the fiber through a 5 cm long phase mask using a 248nm KrF laser. Figure 3 illustrates the experimental setup used for the characterization of the reflectivity spectrum when the resonant longitudinal acoustic waves are excited in the grating. The FBG is illuminated by a superluminescent Amonics LED (SLED) with central wavelength of λ = 1475.7 nm and full width at half maximum FWHM = 37.6 nm. The reflection spectrum is obtained through a circulator and an optical spectrum analyzer (OSA) Agilent 86142B, with a 60 pm wavelength resolution and −90 dBm optical sensitivity. The fiber was carefully adjusted and aligned with help of XYZ positioning stages and microscopes to avoid the presence of stress when no acoustic wave is applied. Bragg wavelength shifts were also monitored during the alignment. Since a stress applied in the fiber or horn can induce block forces on the PZT, the presence of initial stress can cause a change of resonances of the PZT, since that PZT, horn and fiber work as a resonant cavity. Considering that a longitudinal fiber stress can induce an initial strain in the grating and consequently a Bragg wavelength shift, the “DC” strain can also induce a side lobe shift. The PZT is excited by a 10 V maximum sinusoidal signal from an arbitrary signal generator (SG) Tektronix, in the 600 – 1200 kHz frequency range. By fixing the PZT base and the fiber tip, the modulator works as a resonant acoustic cavity that allows exciting standing acoustic waves at certain resonant frequencies. The grating spectrum is obtained for the resonances in which the acousto-optical effect is observed. In order to verify the natural PZT frequency response, the PZT is set free of load and excited by a 500 mV harmonic voltage in the 1-1200 kHz range using an arbitrary impedance analyzer.
Figure 4(a) shows the measured-simulated PZT responses in terms of the impedance magnitude and phase as a function of the excitation frequency. The FEM response presents larger impedance-phase amplitudes compared to experimental values. This difference is related to distinct losses that each mode presents in the resonance. Since the anisotropic constants used in simulations do not include the piezoelectric transducer losses, the simulated amplitudes become higher and sharper in the resonances. The losses reduce the impedance-phase amplitude, which affects the localization of some resonances by the condition of minimum impedance and null phase. A better distinction of the resonances is observed in low frequencies, since much attenuated and closer resonances in higher frequencies result in narrower bandwidths. The PZT response in terms of displacements indicates also the localization of the modes by the maximum values in Fig. 4(b). However, a good agreement between simulated-experimental resonances is obtained, mainly in the f = 1-400 kHz and f = 650-1200 kHz ranges. It has been observed from previous simulation and experimental works, that only the second frequency band is useful to acoustically excite side lobes in FBGs [6, 7]. In the lower frequency range, the PZT presents radial modes that excite mainly flexural acoustic waves in the fiber [2,8]. Therefore, in this paper only the resonances in the second range are analyzed. The transducer responses in Figs. 4(a) and 4(b) are used to locate the vibration modes, considering the correspondent measured-simulated resonances.
The displacement vibration behavior of the resonant modes is shown in Fig. 5 and illustrates the lateral yz and frontal xy planes for each resonance. The PZT analysis of these profiles allows verifying the longitudinal and radial deformations, respectively. The displacements of the transducer are magnified to provide better observation of the maximum (red) and minimum (blue) displacements. It is observed that all modes possess points of longitudinal displacements at their polarization surfaces, which is required for the excitation of axial waves in the fiber. However, due to particular and different electromechanical properties that each mode presents in resonance, the deformation shows that the displacements are not perfectly uniform and axial. More uniform z displacement is observed only at 958, 1049, 1058.5 kHz resonances, in which the maximum deflection is located at the disc center. It is observed in Figs. 5(h) and 5(i) that the 1049 and 1058 kHz resonances are strongly coupled and present a similar deformation behavior.
Figure 6(a) shows the modulator response (PZT is coupled to the horn-fiber) in terms of fiber displacements, which is decomposed into axial (z direction) and transversal displacements (xy directions). Although all analyzed acoustic modes present components of both xyz displacements, the longitudinal acoustic waves are characterized by the larger displacements in z direction. The transversal displacements are composed by lower amplitude flexural oscillations which are polarized in xy plane transversally to the fiber axis. The proximity between the axial-transversal displacements observed in some resonances (for example in the narrow frequency band around 1150 kHz) can produce complex acoustic waves that are not useful for grating modulation. An increase in the number of resonances is also observed if compared to PZT ones in Fig. 4(a). The modulator response shows that the device only works at certain frequencies, which limits the wavelength shift step of the Bragg side lobe in grating spectrum. Figure 6(b) shows the ratio between axial and transversal displacements (z/x and z/y displacement ratios), which allows improved distinction of the longitudinal acoustic resonances. The agreement between fiber-PZT resonances allows verifying the strong dependence of the fiber deformations originated by the PZT excitation. Because of the different deflection that each PZT mode shows at resonance, amplitude variations in fiber displacements are also observed. Since the acousto-optical efficiency is proportional to the strain amplitude, theses discrepancies can induce optical variations in a FBG spectrum. It is important to note in Fig. 6(b), that although the main longitudinal modes present a well-defined frequency, the fiber resonances are composed by narrow frequency bands in which other modes oscillate with lower intensity. Consequently, variations in the geometry or material parameters, or alterations in device boundary conditions can suppress, reinforce or even couple energy from one mode to other. However, the analysis allows locating the fiber resonances and identifying the longitudinal modes that control the optical grating spectrum.
Figures 7(a) and 7(b) show the 3-D modulator displacement behavior with detail of the xy plane for the 688 kHz and 1013 kHz longitudinal resonances, respectively. These resonances are excited by the 683 kHz and 1014 kHz PZT modes, respectively, previously shown in Figs. 5(a) and 5(g). The differences in deflection behavior and resonances are related to alterations in piezo boundary conditions, since one transducer face is fixed, and the other is coupled to the horn-fiber setup. However, the mode in Fig. 7(a) presents stronger and more uniform axial displacement if compared to the one in Fig. 7(b). Although both modes are induced to axially deform the fiber, the complex deformation in the 1014 kHz PZT mode still affects the horn-fiber displacement. The modulator mode shown in Fig. 7 (b) is also affected by transversal acoustic displacements. Figures 7(c)-7(d) show the yz section of the horn displacement field (red arrow) and the fiber strain (for one fiber acoustic wavelength) for the assessed modes, respectively. Figure 7(c) shows that PZT axial deflections are strongly coupled and amplified by the horn, which causes the maximum strain in the fiber (red color) in Fig. 7(d). On the other hand, the transversal deflections in Fig. 7(e) induce acoustic losses and reflections in the horn reducing the strain in the fiber as shown in Fig. 7(f).
Figure 8(a) shows the measured FBG spectrum when the grating is excited with the 679 and 1019 kHz resonances. The measured spectra are compared to the modeled TMM spectra obtained from previously assessed FEM simulations shown in Fig. 7. The differences in reflectivity and wavelength bandwidth between measured-simulated results is due to grating variations originating in the grating inscription process and the OSA resolution that cannot distinguish the lowest side lobes. However, an experimental side lobe reflectivity of about 40% higher with the 679 kHz resonance is observed, which corresponds to the strongest longitudinal mode in Fig. 7(a). The comparison is also made considering other resonances and the measured-simulated side lobe wavelength shift Δλ is compared to theoretical curve and plotted in Fig. 8(b). The results show that the acousto-optic effect occurs only at specific frequencies that depend on the PZT-horn-fiber resonances. The response is approximately linear and the frequency tuning resolution is discrete and non-uniform. The modulator is adjusted to the 1072 kHz longitudinal resonance and the measured amplitude of the side lobe reflectivity is also investigated for a 10 V maximum voltage variation. Figures 8(c) and 8(d) show the FBG modulated reflection spectrum and side lobe reflectivity variation, respectively. Although it is not demonstrated, this behavior has also been observed for other acoustic resonances and the linear response allows the modulator to be used as a tuned optical power modulation at specific resonances.
The reflectivity of the side lobe is proportional to the FBG length  and the voltage applied to the PZT. As the voltage has been limited by the signal generator, the use of a long FBG was useful to obtain a better distinction of side lobes, which facilitates the location of the side lobe wavelength. The use of different grating lengths would change the reflectivity, but would not change the side lobe shift, since it does not depend on the FBG length, but on the excitation frequency. However, variations in the fiber length change the modulator resonances.
Table 1 summarizes the FBG resonances measured experimentally in comparison to the resonances obtained in simulations with the PZT and the modulator (shown in Figs. 4(b) and 8(b), respectively). The relative maximum differences, rd, between FBG-PZT and FBG-fiber values are 4.09% and 2.53%, respectively. However, the average differences are 1.45% and 1.07%, respectively. The better agreement between FBG-fiber results shows that the decomposition of the fiber displacements is useful to accurately locate the longitudinal acoustic modes that induce axial strain in the grating. Since the PZT modes can also induce transversal deflections in the horn-fiber setup, the analysis of the entire AOM device is necessary for this purpose. The results indicate that the non-uniform axial PZT deformations coupled to the horn basis induce transversal displacements in the horn-fiber setup, which can cause acoustic loss, reflections and reduction of the strain in grating.
Differences between experimental-simulated values are caused by geometric variations in the modulator components due to fabrication process or experimental setup building, piezoelectric constants tolerances, element size used in the FEM mesh and step resolution in frequency response used in the simulations. A quantitative analysis of the PZT-fiber deflections, amplitude strain and FBG reflectivity can be obtained by considering the imaginary part of the piezoelectric constants, which were not available from the manufacturer.
The interaction analysis of the longitudinal acousto-optical resonances of a fiber Bragg modulator is numerically and experimentally investigated. Electromechanical anisotropy, 3-D finite element and transfer matrix methods are used to simulate the modulator response and locate the resonant modes. The simulated impedance-phase response can be used to localize the transducer vibration modes and characterize its mechanical displacements. The decomposition of the fiber displacements allows accurately locating the longitudinal acoustic resonances, and estimating the axial core strain, which is required for FBG TMM spectrum simulations. It allows identifying the electric voltage frequencies in which the driver should be excited to obtain better operation efficiency. Experimental measures of the grating spectrum at resonances are compared to simulations. The agreement between measured-simulated values shows that the acousto-optic effect occurs only at specific frequencies that depend on the entire modulator. The grating spectral response is approximately linear, but the frequency tuning resolution is discrete and non-uniform. A better accuracy between measured-simulated values can be obtained by considering the imaginary part of the piezo constants, by refining the FEM mesh and reducing the frequency response step, which result in more computational cost and time processing. However, the results indicate the proposed approach is useful to investigate the intrinsic properties of the acousto-optical modulator as it is electrically excited. The numerical method is useful to characterize and estimate possible operation instabilities and to improve the device efficiency. In addition, it assists the design of novel acousto-optical drives, in which acoustic waves can be controlled to produce specific reflectivity or spectral modulation of fiber gratings.
This work was supported in part by the Cordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES), Ministério da Defesa - Projeto Pró-Defesa, and CNPq/FAPESP – INCT (FOTONICOM).
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