Nonlinear Distortion of the Fiber Optic Microphone

Analytical expressions are obtained for predicting the harmonic and intermodulation performance of the fiber optic microphone. These expressions are in terms of the ordinary Bessel functions with arguments dependent on the amplitudes of the acoustical exciting signal.


INTRODUCTION
At present, there is a growing interest in designing fiber optic microphones using different principles [1][2][3][4][5][6]. The fiber optic microphone offers a simple and cheap solution for a digital telephone line based entirely on optical fiber support. Moreover, the use of acoustic vibrations to modulate the light signal, without an electrical intermediate, reduces the power consumption [6]. However, the large signal performance of this microphone has not yet been investigated. While it is conjectured that a zone of linearity exists in the static characteristic of the fiber optic microphone reported in Ref. [6], and shown in Figure 1, no analytical expressions have been obtained for the harmonic and intermodulation performance of the microphone under large signal conditions. The major intention of this paper is, therefore, to present analytical expressions for predicting the harmonic and intermodulation 2 M.T. ABUELMA'ATTI Membrane FIGURE Fiber optic microphone of Ref. [6]. Pi: incident acoustic power, Pe: incident optical power, Pr: reflected optical power.
performance of the fiber optic microphone of Figure when excited by a multisinusoidal acoustical signal. Using these expressions, it is possible to select the parameters of the microphone for a predetermined distortion performance.

ANALYSIS
According to Malki et al. [6], the coupling coefficient of the fiber optic microphone can be expressed as where Pr is the optical power reflected by the membrane, Pe is the radiated power provided by the fiber, x z/zo, z is the membrane-fiber distance, Zo a/2NA, a is the core radius of the optical fiber and NA is the numerical aperture of the fiber. Here we propose to approximate the characteristic of Figure 2 by the Fourier-series of Eq. (2). x Z/Zo The coefficients Bk can be obtained using the discrete Fouriertransform (DFT) technique. This technique invariably demands a well developed software. Moreover, to reduce the number of multiplications and additions involved in the DFT of a large number .of equally-spaced data points, it is essential to organize the problem so that the number of data points can be easily factored, particularly into powers of two [7]. Futhermore, in the DFT technique the number of terms of a Fourier-series function must be less than or equal to the number of data points available. Thus with a limited number of data points, as may be the case, the desired accuracy, in approximating the nonlinear term of Eq. (1) by a Fourier-series, may not be attained.
Alternatively, first we make the characteristic of Figure 2 periodic by removing the offset at x 0 and using the resulting curve in mirror image to generate a complete period of the periodic function f(x) K(x)-1 as shown in Figure 3. Secondly, we choose a number of data points and connect them using straight line segments joined end to end as shown in Figure 3. The x-values of the segment joins are termed knots. The number of knots and their positions must generally  Figure 2 from which it is obvious that the proposed Fourier-series accurately represents the nonlinear term of Eq. (1) with RRMS error 0.0017.

HARMONIC AND INTERMODULATION PRODUCTS
One of the potential applications of the proposed approximation of (2) is in the prediction of the amplitudes of the harmonic and 6 M.T. ABUELMA'ATTI intermodulation products resulting from multisinusoidal acoustical excitation of the fiber optic microphone. Thus, assuming that the normalized membrane displacement, resulting from exciting the membrane by a multisinusoidal acoustical signal, can be expressed as x Xo + Z Xn sin Wnt, Xo + Xn _ Xmax (4) n=l n=l then combining (2) and (4) (7)-(11) the relative second-order and third-order harmonic and intermodulation products can be calculated for any number of input signals. Figure 4 shows the results obtained for an exciting signal resulting in a membrane displacement of the form (sinolt + sin o2t)) (12) x=X 1+ Equation (12) implies that the amplitude of the membrane displacement at frequencies Wl and a)2 is equal to the bias. From Figure 4 it can be seen that the second-order intermodulation is dominant. Figure 5 shows the results obtained for an exciting signal resulting in a membrane displacement of the form x Xmax(1 "+-a(sinwlt + sin w2t)), for different values of a. Equation (13) implies that the amplitude of the membrane displacements at frequencies 1 and 2 is a fraction of the fixed bias.
From Figures 4 and 5 it can be seen that the amplitudes of the third-order harmonic and intermodulation products are smaller than the amplitudes of the second-order harmonic and intermodulation products. Moreover, the second-order intermodulation component of the form wr-(,dq is dominant.

CONCLUSION
By approximating the coupling coefficient of Eq. (1), using a Fourierseries, analytical expressions can be obtained for the harmonic and intermodulation performance owing to an acoustical signal resulting in multisinusoidal membrane displacement of a fiber optic microphone of Figure 1. The Fourier-series coefficients can be evaluated using simple calculations without recourse to numerical integration or DFT techniques. The analytical expressions obtained for the harmonic and intermodulation products are in terms of the ordinary Bessel functions and can be easily evaluated using programmable hand calculators.
The results show that the second-order intermodulation component is the dominant nonlinear distortion component. The results obtained can be used for selecting the parameters of the fiber optic microphone to meet a prespecified large signal performance under multisinusoidal acoustical excitation.
It is worth mentioning here that the accuracy of the results predicted using the present analysis depends on the accuracy of Eq. (1). It is known, however, that from the physical point of view, Eq. (1) is an approximation based upon several assumptions, for example uniform modal distribution. Thus, while in principal, the analysis presented here is correct, the discripancies between theoretical and analytical resuits is due mainly due the accuracy by which Eq. (1) represents the operation of the fiber optic microphone of Figure 1.