Sensitivity and phase response of FBG based acousto-optic sensors for real-time MRI applications

Fiber Bragg grating (FBG) based sensors have recently been introduced to the field of magnetic resonance imaging (MRI). Real-time MRI applications demand highly amplitude and phase sensitive MRI compatible sensors. Thus, a model and detailed analysis of FBG based ultrasound detection are required for designing better performing sensors. A hybrid FBG model incorporating numerical and FEA methods was developed and used for sensitivity and linearity analysis. The transfer matrix method was used for the modeling of optical modulation whereas FEA was used for pressure field calculations within the grating. The model was verified through reflection spectrum and acoustic pressure sensitivity testing of two π -phase shifted FBGs in a side slope read-out configuration. The sensitivity curves with respect to the operation point on the side slope was characterized in terms of amplitude and phase, and nonlinearity of the phase response has been quantified. Lastly, the impact of phase linearity of the FBG based acousto-optic sensor was tested under MRI when the sensor was used as a position marker and an analog phase shifter based solution was demonstrated.


Introduction
Fiber Bragg gratings (FBGs) have been widely used for strain and temperature measurements over the last few decades in nondestructive structural health monitoring [1][2][3][4][5]. FBG sensors offer high sensitivity, small size and immunity to electromagnetic field; similar to other fiber optic based sensors. Moreover, multiple gratings can be embedded onto a single optical fiber enabling multiplexed sensor topologies [6][7][8]. In addition to intrinsic advantages of fiber optic sensors, FBG sensors offer amplitude/intensity fluctuation independent results due to spectrum based detection mechanisms [9,10], which ensures reproducible measurements despite the optical losses.
FBGs are constructed by introducing periodic variations in the refractive index of the fiber core, which generates a wavelength-specific dielectric mirror ( Fig. 1(a)). The wavelength of the reflected light depends on the geometry of the grating as well as the refractive index modulation. Due to the strain and temperature dependence of the refractive index and grating geometry, the wavelength of the reflected light will change as function of temperature and strain. Strain sensitivity is proportional to the slope of the FBG spectrum and high spectral slope requires a narrowband response of the FBG. Bandwidth of FBG spectrum is controlled by the refraction index modulation of the grating and the grating length as well as grating profile [11,12]. The most obvious way to achieve a narrowband response is using weak refractive index modulations over a long grating length. However, long grating length limits the minimum sensor size and spatial resolution. One way to achieve a narrowband response without extending the grating length is employing a phase shifted grating profile in which an abrupt discontinuity is introduced in the center of the grating [13][14][15], shown in ( Fig. 1(a)). A narrow notch is formed within the reflection lobe of the FBG spectrum due to the phase shift in the refractive index modulation ( Fig. 1(b)).
The most common read out scheme for FBG sensors employs a spectrometer and generates output based on the spectrum shift induced by applied strain or change in temperature. Spectrometer based approaches are well suited for multiplexed measurements where the inspection of the full response of the FBG is required [16,17]. However, this approach is limited by the speed of the spectrometer, which limits the use of FBG sensors in higher frequency applications. In ultrasound detection and acousto-optic modulation based applications frequencies are in MHz range whereas strain levels induced by the acoustic waves are in μ-strain range or less [18]. Since the spectrum shift of the FBG is fast and amplitudes are small, the most sensitive region in the spectrum of the FBG is monitored instead of the whole spectrum [10,19]. A narrow linewidth laser is used as input light source and wavelength of the laser is fixed in the middle section of the slope of the FBG spectrum, shown in Fig. 1 Although both normal and phase shifted FBGs have been used as acousto-optic sensors with the side slope detection method, large portion of the slope of the spectrum has been assumed linear in the previous studies [12,15,18,20]. This assumption is valid when the sensitivity of the FBG is relatively low and acousto-optic modulation is very small. However, when the sensitivity and strain levels are high, effect of non-linearity becomes prevalent in the form of sensitivity variations to amplitude and phase. Several groups [21,22] have studied the linearity of Fabry-Perot based optical fiber sensors, which incorporate the side slope detection method. On the other hand, linearity analysis for FBG sensors are limited in the literature.
A particular case where linearity is crucial is when fiber optic sensors are used in magnetic resonance imaging (MRI) [23][24][25][26]. Electromagnetic field interference during MRI makes sensing in that environment challenging, thus fiber optic based sensors are well suited for MRI applications due to intrinsic immunity to electromagnetic field. Along these lines, we recently introduced a FBG based acousto-optic catheter tracking sensor for interventional MRI procedures [27,28]. Since image formation in MRI is achieved by frequency and phase encoding, the linearity of the FBGs is very crucial in such applications, especially phase. In terms of phase sensitivity, miliradian range phase stability is needed for acceptable image quality [29]. Each phase-encoding step usually takes an entire MR excitation step and takes up significantly longer times than frequency encoding leading to build up of any nonlinearity in phase. Moreover, high levels of acoustic noise and temperature changes under MRI further accentuates the problem as FBGs are inherently sensitive to all strain and temperature changes. In order to characterize FBG response in such dynamically challenging environments and highly sensitive applications requiring a detailed analysis of linearity and sensitivity of FBG sensors with FBG side slope read-out are required.
In this paper, we present linearity and sensitivity analysis of pi-phase shifted FBGs for acousto-optic modulation for position sensing application in MRI. Two pi-phase shifted FBGs with different bandwidths are analyzed. A model for FBG is developed using both numerical methods and finite element analysis (FEA) tool. In order to validate the model, experiments are carried out with piezoelectric transducers as ultrasound sources. Sensitivity and linearity of the FBGs are measured and compared with the simulation results. Lastly, the FBGs were tested under MRI for phase sensitivity and a phase correction method for improved position sensing is demonstrated. This paper is organized as follows: first, the FBG sensor is modelled based on coupled mode theory using transfer matrix method for completeness and background in section 2, then sensitivity and noise of the FBG sensor is measured in Section 3, nonlinear response of the FBG sensor is analyzed and compared with experimental results in Section 4, finally, several conclusions are provided in Section 5.

Modeling of FBG
A detailed model of the FBG sensor is required for the sensitivity analysis. Numerical methods are well suited for analyzing optical characteristics of the FBG whereas complex mechanical models utilize finite element analysis (FEA) tools. Thus, a composite model of the FBG utilizing both numerical and FEA methods is developed in order to investigate the linearity and sensitivity of the FBG as an ultrasound sensor. Numerical methods, namely coupled mode theory using transfer matrix method, is used for optical simulations. This method allows random grating profile and refractive index modulation. Mechanical response of the FBG to acoustic waves are simulated using a FEA tool. Two models are coupled through the pressure field within the fiber in the grating region.
FBG is manufactured by laterally exposing the core of a single-mode fiber to a periodic pattern of ultraviolet light, which alters the refractive index of the core according to the periodic pattern ( Fig. 1(a)). In essence, FBG is a narrowband dielectric mirror with peak reflection occurring at the Bragg wavelength λ Bragg = 2n eff Λ (1) where n eff is the effective refractive index of the fiber core and Λ is the grating period. πphase shifted FBG (πFBG) has a jump in the periodicity in the middle of the grating. In essence, this discontinuity separates the grating into two highly reflective mirror resulting in a high quality factor Fabry Perot like cavity. This introduces a sharp notch in the middle of the reflection spectrum FBG (at λ Bragg ), shown in Fig. 1(b), enabling higher sensitivity.
Reflection spectrum of the πFBG can be calculated using a coupled mode theory and transfer matrix method [30], [31]. πFBG with a total length of L is divided into smaller uniform sections with length of ( Fig. 1(a)). Relations between forward-going modes, A i , and backward-going modes, B i , before and after each section can be represented by a 2 × 2 transfer matrix T i .
where T i is defined as In Eq. (3) and (4), σ ⌢ is dc self-coupling coefficient, κ is ac coupling coefficient and Coupling coefficients FBG is given as follows σ ⌢ = δ + 2πn(z)/λ (7) κ = πn(z)/λ (8) where δ = 2πn eff (1/λ − 1/λ D ) is detuning parameter, λ wavelength and λ D = 2n eff Λ(z)is design wavelength and n is local refractive index. A single transfer matrix T(λ) can be defined for the overall grating by multiplying individual transfer matrices T i at each wavelength. Phase shift in the grating can be introduced into the model by Hence the system matrix T will become where A (z=0) and B (z=0) are complex modes at the start of the grating, A (z=L) and B (z=L) are complex modes at the end of the grating and M = L/Δz. Lastly, by applying boundary conditions A (z=0) = 1 and B (z=0) = 0 are in Eq. (10), amplitude and power reflection coefficients can be calculated as Γ(λ) = −T 21 /T 22 and r = |Γ(λ)| 2 respectively.
When the πFBG is impinged by an acoustic wave, refractive index n and grating period Λ will change due to elasto-optic effect. Normalized refractive index Δn and relative grating period changes ΔΛ by an acoustic wave with pressure field (z,) can be expressed as follow (1 − 2v) 2P 12 + P 11 where P 11 and P 12 are elements of the strain-optic tensor, E is Young's modulus and v is Poisson's ratio. Since coupling coefficients σ ⌢ and κ depend on refractive index n and grating period Λ, new reflection spectrum of the grating can be calculated by plugging Eq. (11) and (12) into (7) and (8). When choosing Δz, acoustic wavelength and grating length should be taken into consideration such that Λ ≪ Δz ≪ λ acoustic where λ acoustic is acoustic wavelength inside the fiber in order to satisfy uniformity assumption.
FEA is used for pressure calculations within the optical fiber. Optical fiber is modelled as a silica cylinder in COMSOL Multiphysics platform since the mechanical properties of both core and cladding of the fiber are the same, shown in Fig. 2. A uniform acoustic wave is radially applied in the grating region of the fiber. In order to minimize the acoustic wave reflections at the fiber ends, a low reflecting boundary condition is applied on both fiber ends. The pressure field in the core of the fiber induced by the external acoustic wave is calculated and inserted as P(z,t) into (11) and (12). Any arbitrary pressure field distribution can be used as an input for the model as long as Δz is adequately smaller than smallest acoustic wavelength.

Model validation
We first validated the model by comparing the reflection spectrums of the fabricated gratings with simulations. In this study, custom made πFBGs with two bandwidths operating at 1550nm (Teraxion Inc., Quebec, Canada) were used. Overall grating length for both πFBGs are 8 mm with 2 pm (π-FBG-1) and 0.4 pm (π-FBG-2) notch bandwidth at FWHM. The πFBGs were fabricated on a polarization maintaining (PM) fiber so that there are two Bragg wavelengths due to slightly different refractive index in the slow axis and fast axis of PM fiber. In this work, only the slow axis mode was excited with a laser source polarized in slow axis. Details of the gratings and the simulation parameters were given in Table 1. In order to achieve different notch bandwidths for π-FBG-1 and π-FBG-2, refractive index modulation of 3.7×10 −6 and 4.85×10 −6 were used, respectively.
We used the same side slope optical read-out scheme as we have reported in [27] (Fig. 3). In its simplest form, a narrow linewidth continuous wave (CW) tunable laser source (NKT Photonics, Denmark) was coupled to the πFBG through an optical circulator. Reflected light from the πFBG was captured by a high trans-impedance gain InGaAs photo detector with 125MHz bandwidth (New Focus Model 1811, CA, USA). Output of the photodetector was also used for tuning the wavelength of the laser to the side-slope of the πFBG. Figure 4, shows the reflection spectrum around the center notch. Bragg wavelength of each πFBG was tracked by adjusting the temperature of the laser. Then, spectrum of the πFBGs were measured by sweeping the laser source using the piezo-electric wavelength tuning mode of the laser around the Bragg wavelength. Note that, measured reflection spectrums are normally not centered around 1550nm due to fabrication tolerances. Thus, the measured reflection spectrums were shifted to 1550nm in order to compare the spectrum shape with the simulations. The spectrum shape is more critical than the absolute Bragg wavelength since readout is carried on the side slope of the center notch.
The reflection spectrum of the π-FBG-1 is in good agreement with the simulation results around the Bragg wavelength. Similarly, the reflection spectrum of the π-FBG-2 close to simulation results except the notch shape is a little distorted. It should be noted that fabrication of πFBGs with a notch bandwidth of 0.4pm is very challenging, and ideal behavior is not expected.

Sensitivity measurements
We characterized the sensitivity of the gratings to acoustic pressure (and resulting strain) by exciting them using a piezoelectric transducer immersed in water. Water immersion ensures uniform pressure distribution over the FBG region of the fiber. Figure 5(a) shows the components of the experimental set-up used for the sensitivity testing of the acousto-optic sensor. The πFBG is immersed in water at the focal region of a piezoelectric transducer (Panametrics Model A306S) with a center frequency of 2.25 MHz. In water, acoustic wavelength at 2.25MHz is 666 μm, which is more than twice the diameter of the fiber (125 μm), thus a uniform ultrasound generated strain over the radial thickness of the grating is achieved. This is further validated via FEA simulation; a fiber model was placed into water and excited with a point pressure source away from the fiber and the radial pressure distribution was investigated. The simulation results showed a continuous and uniform pressure field with 85% of the surrounding pressure field amplitude within the optical fiber ( Fig. 5(b)).
The acoustic pressure produced by the transducer at the same location was measured using a calibrated hydrophone (ONDA Model HGL0200). An input signal of 10 cycle tone burst with 10Vpp amplitude was used for acoustic wave generation which resulted in 107kPa peak pressure around the fiber, corresponding to 91kPa peak pressure at the fiber core.
Laser intensity was adjusted such that the photodetector output is 10V when maximum reflection is achieved from the gratings. For the same pressure field measured by the hydrophone, π-FBG-1 generated 1.1V output whereas π-FBG-2 generated 7.3V output (Fig.  6). Therefore, the pressure sensitivity of the sensors can be calculated as 80 mV/kPa for π-FBG-2 and 12 mV/kPa for π-FBG-1.

Linearity analysis
We investigated the linearity of the FBG by inspecting amplitude and phase sensitivity with respect to reflectivity level on the side slope. These experiments were carried out at 23.4 MHz, the Larmor frequency of the prototype 0.55T MRI system (Siemens Healthineers Erlangen) in which the subsequent imaging experiments were conducted. The strain field on the FBG was generated using a custom lithium niobate (LiNbO3) piezoelectric transducer (Boston Piezo-Optics) with 23MHz thickness mode resonance. The transducer was used in a configuration for MRI position tracking as described earlier [27,28]. To record the sensitivity variation, the bias point on the side slope was slowly changed from minimum reflection to maximum reflection while using a continuous wave acoustic excitation. This configuration was also simulated where the piezoelectric transducer in contact with the optical fiber was modelled using the FEA tool. The resulting pressure distribution in the core of the fiber was used for time domain simulations in the optical model described above.
Amplitude sensitivity of the FBG sensors were calculated by taking the first derivative of the output signal with respect to bias as measured in Fig. 4. Moreover, the model was used for time domain simulations. A time domain pressure signal at 23 MHz with an amplitude of 1kPa was used as the input to the model and the corresponding optical modulation is calculated at different bias points. Simulation results were normalized to the experimental results for better comparison and amplitudes were presented in arbitrary units (AU) as the absolute amplitude values were not of interest for this study. Moreover, pressure output from the piezoelectric transducer was kept such that the optical modulation does not exceed 1% in order to ensure linearity across the spectrum.
As seen in Fig. 7, simulation results are in good agreement with the experimental results. Maximum sensitivity is observed around 25% bias on the side slope rather than 50% as reported earlier by [12]. In π-FBG-2, the experimental data shows that the maximum sensitivity is observed around 35% rather than 25% due to the fabrication tolerances. Maximum sensitivity of FBG-2 is 5.7 times greater than π-FBG-1, which is comparable to the hydrophone measurements reported in the previous section; sensitivity of π-FBG-2 was 6.6 times larger than π-FBG-1. Moreover, amplitude linearity is observed only at a limited range around the 25% bias rather than 20-80% as assumed by [18]. Sensitivity is within 1% of its maximum value from 21% to 29% bias. Thus, unwanted bias change on the side slope should be kept under 8% for a maximum amplitude variation of 1%. Note that the response of the FBG sensors becomes unreliable under 5% and above 80% bias as the reflectivity curve becomes noisy and signal levels are significantly reduced.
Phase response of the FBG sensors were tested using the same set-up. A continuous wave excitation at 23 MHz was used as the input and the optical signal and the input signal was recorded at different bias points. Then the phase difference between the input signal and the optical signal was calculated. Similarly, simulations were performed at 23 MHz with an 1kPa pressure input then the phase difference between the pressure input and the modulated optical signal was calculated for different bias points. A linear relation between the bias point and the phase was observed between 5% and 80% reflectivity (Fig. 8). Simulation and experimental results are in good agreement except for π-FBG-2. There is a less then 4° deviation from the simulations up to 50% bias, then the difference becomes more pronounced. As discussed above, this discrepancy can be attributed to the non-ideal fabrication tolerances. The phase changes 0.165° per 1% bias change for π-FBG-1, whereas phase changes 0.24° per 1% bias change for π-FBG-2 up to 50% reflectivity. Highly phase sensitive MRI applications requires the unwanted phase due to bias change to be strictly monitored and corrected in order to minimize image distortion.

Demonstration of FBG phase correction for position tracking application under realtime MRI
The impact of phase nonlinearity of FBG sensor for position tracking during real-time MRI is observed when the bias point is changed due to finite response time of the wavelength controller [27,28]. Since π-FBG-2 has higher nonlinearity, to demonstrate and test a phase nonlinearity correction method, the π-FBG-2 based sensor was tested under MRI.
The sensor was tested under a 0.55T prototype MRI system using the experimental set-up shown in Fig. 9 and detailed description of the overall system can be found in [27]. A 30 AWG insulated copper wire wound into a tight-pitched solenoid coil with 2.4 mm diameter and 8 turns was used as an active MRI marker. Since the electrical length of the solenoid coil is electrically short, additional temperature variation on the FBG due to RF induced heating will be minimized. The acousto-optic sensor was inserted into an MRI phantom, which is a saline solution mimicking electromagnetic properties of human tissue [32]. The laser output was adjusted such that the maximum reflection corresponds to 10V voltage at the photodetector output. Bias was set to 4.5V for maximum amplitude sensitivity where the phase sensitivity of the π-FBG-2 is 24°/V. An analog phase shifter (SigaTek Model SF50A2) is used for phase correction. The DC voltage output from photodetector is used for controlling the phase shifter. Since the phase shifter introduces 18° phase shift per Volt of control voltage input, a gain of 0.75 was applied to the DC voltage output of photodetector for proper phase correction.
A balanced steady-state free precession (bSSFP) sequence with following parameters was used: Flip Angle, 90°; TR, 471 ms; TE, 1.7 ms; slice thickness, 10mm; bandwidth, 500 Hz/ pixel and matrix size, 128 [notdef] 128. Figure 10(a) shows the MRI image of the acoustooptic sensor without the phase correction has a size of 5.6 mm with severe image distortion at the corners. Moreover, some lines of the coil image were missing which indicates the phase was distorted during the acquisition of MR signal associated with that particular region. On the other hand, as seen in Fig. 10(b), the image of the coil was uniform with a size of 3.2 mm when phase correction was applied. These results indicate the importance of phase linearity for FBG based position sensing in MRI as predicted by the careful modeling and characterization results, and demonstrate that these artifacts can be corrected in realtime using proper phase shifting electronics.

Conclusion
A hybrid FBG model incorporating numerical and FEA methods was developed and used for sensitivity and linearity analysis for ultrasonic applications. Transfer matrix method was used for the modeling of optical modulation whereas FEA based model was used for pressure field calculations within the grating. Two πFBGs with notch bandwidths of 2 pm and 0.4 pm were investigated. The reflection spectrums of both FBGs measured and compared with the model. Simulation results based on the model were found in good agreement with the experimental results. Pressure sensitivities were measured at 80mV/kPa and 12mV/kPa for FBGs with bandwidths of 2pm and 0.4pm respectively. Maximum sensitivity was observed and calculated at 25% bias on the side slope of the πFBG notch, which is lower than the widely accepted optimum bias at 50% reflectivity. Moreover, the sensitivity dropped 1% within ±4° of the optimum bias point which indicates a much smaller linear range on the side-slope than previously reported. Phase dependency on the bias point was investigated and found that phase changes 0.165° and 0.44° per 1% bias change for FBGs with bandwidths of 2pm and 0.4pm respectively. Lastly, a FBG based sensor was tested in a 0.55T prototype MRI system as an active position marker. Image distortion was observed due to phase distortion caused by the high acoustic noise of MRI system. Using the FBG model and measurement results, an analog phase shifter based phase correction scheme was implemented and demonstrated successful image recovery.  a) Experimental set-up for pressure sensitivity. The hydrophone is placed at the FBG location for pressure calibration b) FEA simulation showing pressure distribution inside the FBG and the surrounding water in the axial direction. Note that the pressure field inside the fiber is 85% of the surrounding pressure field. Pressure field captured by π-FBG-1 (a) and π-FBG-2 (b) with the hydrophone measurement. The time delay between the FBG and hydrophone signals is due to the ~4 mm distance between them as shown in Fig. 5(a). Image of acousto-optic sensor without phase correction (a) and with phase correction (b) under MRI. Simulation parameters forπ-FBG-1 andπ-FBG-2.