1. Introduction

In fiber sensor circuits, particularly when operating at a wavelength of 633 nm or 830 nm, a silicon avalanche photo diode (APD) is often used as a detector in the receiver. At higher wavelengths Ge and InGaAs type of APD's with moderate gain are used for detection purposes [1-3].

In analog optical communication links, in interferometric optical fiber sensors, and in analog radio-over-fiber systems, the performance at the receiving end is limited by second order nonlinearity, which is induced by the photodiode acting as the photodetector [4-13]. That is, the degree of nonlinearity of the photodiode is determined by the total harmonic distortion appearing at the output power which in particular includes the second and third order harmonic components [13]. Works have been reported to improve the linearity of photodiodes for the frequency range of 1-2 GHz. [14, 15].

In another attempt, a method was used to improve higher frequency performance of photodiodes by using Mach-Zehnder modulator (MZM) at quadrature bias with balanced detection to nullify the second order nonlinearity of photodiode at high frequencies [16].

In a recent report, yet another method used dual parallel MZM to cancel out the photodiode-induced second harmonic distortion [17]. It was shown that the second harmonic generated by this method was 180 deg. out of phase of that of the photodiode nonlinearity. Recently, the nonlinearity of a commercial photodiode was measured, using three setups of a one-tone heterodyne, two-tone heterodyne and three-tone MZM designs. Mathematically developed data on multiple devices are compared to find under which conditions the measurements by three setups are comparable [18]. A new method is reported theoretically with experimental results to cancel even-order distortion induced by photodiode in microwave photonic links. A single Mach-Zehnder modulator, biased slightly away from the quadrature point, is shown to suppress photodiode second-order intermodulation distortion in excess of 40 dB without affecting the fundamental power [19].

In an another report, a measurement of the nonlinearity responsivity of two commercial photodiodes of types p-n Ge and p-i-n InGaAs, used in optical fiber power measurements, was presented. The photodiodes in measurements were under high irradiance levels. It was shown that the photodiodes nonlinearities were of the saturation type, which depended on the beam diameter of the radiation source [20]. In newly reported work, a model of simulating photodiode microwave nonlinearities is proposed, which includes the effects of non-uniform absorption in three dimensions and self-heating of the photodiode. The saturated output power and third order output intercept points of two different waveguide photodiodes are simulated, with excellent agreement between measurement and theory [21].

In an early work, a simple method was used to obtain harmonic distortion levels of a pin photodiode at different frequencies from the microwave reflection coefficient of the photodiode under dc illumination for different incident optical powers [22].

The gain of the APD is a function of the actual bias voltage across the diode, and if the output voltage across the load resistance changes corresponding to a change in the received optical power, the bias voltage of the APD gets modulated leading to a non-linear response of the overall APD receiver. However, when receiving weak optical signals, this non-linearity will be acceptably small [23].

In this paper, a simple analysis on the nonlinear behavior of a practical silicon APD photodetector circuitry is presented, by considering the gain characteristics of the APD photodiode. The analysis of the total harmonic distortion (THD), which is an indication of degree of nonlinearity in an APD photodetector, includes the effects of received input power and the load resistance at the receiving output. The behavior of the THD is formulated and graphically illustrated in terms of the input power and load resistance.

The analysis is given in an easy-to-understand manner that can be readily applied to a practical system and would be useful when dealing with the optical fiber sensor systems and analog optical fiber communication systems.

2. Distortion in APD Receiver

Figure 1 shows a practical circuit diagram of an APD receiver using a silicon APD. The resistor is being used to protect the avalanche photodiode from an accidental break-down, if any, by limiting the maximum possible current flow. The capacitors at the cathode of the APD keeps the voltage at the cathode at ac ground. Figure 2 shows the plot of the APD gain as a function of the bias voltage, obtained experimentally. Around the break-down voltage of the APD, the gain of the receiver increases rapidly.

Figure 1. Circuit diagram of the APD receiver

Figure 2. Characteristic curve of the APD gain at 5 kHz as a function of APD bias voltage for a peak ac optical input power

The APD receiver, theoretically, is nonlinear if it is operated at high values of APD gain. The source of nonlinearity arises due to the finite value of the load resistor and the ac output voltage For a given dc bias voltage at the cathode of the APD (in Fig. 1), the actual voltage across APD changes with the output voltage as where is the dc voltage across At a given average optical power level, and will remain constant but will change with the modulation in the optical power. Since itself is a function of the actual APD bias voltage, distortion results when the output voltage modulates the APD gain. This effect will be worse whenever the load resistance and the optical signal level (and hence the electrical output signal voltage) are high.

The APD gain characteristic curve of Fig. 2 can be assumed to be linear over a small range of the APD bias voltage and accordingly can be expressed by a straight line equation. This assumption is valid when receiving a small optical signal and helps in simplifying the distortion analysis. For example, the curve AB of Fig. 2 can be expressed as:

(1)

where is the gain of the APD, is the instantaneous bias voltage across the APD, is a constant equal to the slope of curve AB, and is the voltage as indicated in Fig. 2, obtained by extending the line passing through points A and B. Equation (1) is valid only for points along the curve AB. The value of for Fig. 2 was calculated to be about 11.197 and was equal to 145 .

An expression for the current multiplication factor has been given as [24]:

(2)

where we have:

(3)

(4)

where is the instantaneous reverse bias voltage, is the reverse voltage at which avalanche multiplication becomes significant, and is the reverse breakdown voltage of the APD. In this case the slope, of the characteristic curve is given by:

(5)

Then, from Eq. (2) we can write:

(6)

Figure 3 shows a segment of the characteristic curve of For a given bias voltage of the value of can be evaluated from Eq. (6) along a linear segment PQ shown on the curve in Fig. 3. The value of on the -axis can then be evaluated by:

(7)

Figure 3. A general characteristics curve of APD for evaluation of and from the equation of

The empirical expression Eq. (2) is valid for values of up to 0.8 [24]. More accurate expression for , if any, can be used instead of Eq. (2) without any loss of generality.

3. Derivation of Harmonic Distortion in APD

To derive the expression for harmonic distortion in the APD circuit, we will consider the linearized characteristic curve (AB in Fig. 2) over a small range around the operating point of the APD voltage. The expression for the instantaneous ac current generated by the APD receiver corresponding to an ac input optical power can be written, referring to Eq. (1), as:

(8)

where is the responsivity of the photodiode (with gain equal to unity). After rearranging Eq. (8) we obtain:

(9)

Let us assume a sinewave for the input optical power at an angular frequency expressed as:

(10)

where is the peak amplitude of the ac optical power. The optical power does not contain any dc component and is equal to the instantaneous optical power minus the dc component of the optical power . Substituting for P_ac(t) in Eq. (9), we get:

(11)

Or by rearranging, can be alternatively shown as:

(12)

On multiplying current by the load resistance the instantaneous ac output voltage can be obtained as:

(13)

(14)

Where we have:

(15)

3.1. Fourier Series Expansion of the ac Output Voltage of the APD

In general, the Fourier transform of the periodic signal (Eq. 14) in interval can be expressed as:

(16)

Since is an even function, the coefficient ; then Eq. (16) simplifies to:

(17)

where the constant is the average value of given by:

(18)

and the coefficient is given as:

(19)

By solving the integral in Eq. (18), we obtain:

(20)

For one obtains

To determine the value of we proceed as follows. From Eq. (19), we can write:

(21)

The value of first term in Eq. (21) in the interval to is zero, then we have:

(22)

(23)

Assuming we can expand the term in the bracket in Eq. (23) as:

(24)

In the given range of to the first term in Eq. (24) is zero. Therefore, can be reduced to:

(25)

Or, in a compact form, we can show Eq. (25) as follows:

(26)

In Eq. (26), with no loss of generality, we can interchange the position of the summation and the integration operators. Then,

(27)

Since the factor tends to zero for high values of by assuming the expression for can be approximated by limiting the maximum value of to 5.

The total harmonic distortion (THD), which indicates the degree of nonlinearity of an APD receiver, can be characterized by an analysis of the output spectral components with an input driven by a pure sinewave. Therefore, the THD in dB and percentage (%) can be defined, respectively, as [25-27]:

(28)

where and are total signal powers and fundamental signal power, respectively.

Table 1 lists the approximate expressions for the different Fourier coefficients in terms of fundamental, second- and third harmonics components. From these expressions, second- (HD)₂ and third harmonic distortion (HD)₃ in dB, with respect to the fundamental, can be calculated, respectively, as:

(29)

(30)

Table 1. Fourier components of the output voltage in the APD receiver circuit

The value of can be determined from Eq. (15) for a given values of and The harmonic distortion will be low, if the value of is high. In other words, the distortion value will be low, when either the APD receiver's load resistance or the peak optical power is small.

4. Results of the Distortion Analysis

The harmonic distortion values were computed for load resistance of for different peak values of ac optical power . The plot is shown in Fig. 4, at for which and . The second HD was about -71 dB (below fundamental) at which increased to -31 dB at for . The corresponding theoretical value of third HD was -142 dB and -62 dB, respectively. The corresponding THD values are -214 dB for and -93 dB for

Figure 4. Theoretical harmonic distortion of the APD receiver versus peak optical input power for load resistance

Figure 5 shows the HD as a function of the APD receiver's load resistance for two fixed power levels of and At , the second HD increases from a value of about -51 dB for to a value of about -11 dB for

Figure 5. Theoretical harmonic distortion of the APD receiver versus load resistance for peak optical input power and

In Table 2, typical values of second- and third harmonic distortions are listed for two optical powers of and The results indicate that when the load resistance is lower, the THD is also lower for a specified received optical power.

Table 2. Typical values of second- and third harmonic distortion

The theoretical curves of Figs. 4 and 5 are valid for the typical APD circuit shown in Fig. 1, for a case when and . For low load resistance, when received power has increased 10 times, the THD only increased to 9 dB, i.e. 4.7% degradation, whereas for high load resistance, the same power increase would cause an increase of 53 dB in the THD, which corresponds to 61% degradation of the THD. That is to say for a high load resistance, a 10 order of magnitude increase in received power, would result in 13 order of magnitude degradation in the THD.

For a general APD characteristic given by Eq. (2), the value of and at different APD bias voltages can be theoretically evaluated, and from Table 1, the harmonic distortion values can be accordingly calculated.

For sake of comparison, to the best of our knowledge a newer than the Ref. [28] was not traced to have reported an experimental replica of the present analysis. The then experimental results showed that for an optical input power of 10 the second order (HD) and third order harmonic distortions were found to be -60 dB and -118 dB, respectively. The biasing voltage was 155 V, the gain was 76, and the load resistance was 75

In the present analysis, for a gain of 42, biasing voltage of 145 V, and input power of 10 the second order HD and third order HD are found to be -55 dB and -110 dB, respectively, for the same load resistance at 75 In Table 2, it shown that the percentage differences between 2^nd HDs of the present analysis and Ref [28] is 9% and for 3^rd HDs is nil.

5. Conclusions

In this paper, a simple model is proposed to compute the total harmonic distortion in an APD receiver circuit. The model is explained, by using the gain characteristic curve of an experimental circuit. For making the approach more general, a procedure is given so that the whole analysis could be based on an analytical equation of the gain Direct expressions, for the second and third harmonic distortions in dB (with reference to the fundamental), are derived.

It is shown that when input power increases, the 2^nd HD and 3^rd HD would increase, and at a given input power, the 2^nd HD is higher than 3^rd HD. Typically, for the given circuit, the THD was found to be -191 dB, for and If a high impedance preamplifier is used with the THD will become worse to a value of -34 dB for The results of the analysis indicates that for low THD, the load resistance and the received optical power at the output end should be at low level. In a given comparison shows that the obtained results almost tally with a reported experimental results.

The analysis presented here will be useful for analog optical fiber transmission systems and optical sensor applications.