## Abstract

The third-order intermodulation product intercept (IP3) has become an important figure of interest for photodiodes in high-performance, externally modulated, analog links. With the desire for highly linear photodiodes comes the need to be able to accurately measure the device without additional variables. The purpose of this work is to redesign the current IP3 setup to accurately measure the third order intermodulation distortions of the photodiode and to also be able to measure the second order intercept point (IP2) using the same setup. The new setup will isolate equipment contributed nonlinearities by maintaining a small RF drive voltage amplitude on the modulators and by optically changing the photodiode input modulation depth while maintaining a constant input optical power to the photodiode. The goal is to measure a specific knee point where different slopes will indicate the change to photodiode dominated nonlinearity.

©2008 Optical Society of America

## 1. Introduction

The spurious free dynamic range of analog fiber-optic links is closely tied to the linearity of the transmitter and the photodiode. As power increases, the linearity performance of the photodiode progressively becomes a limitation. When two RF tones coming from two modulated optical signals are input into the detector many spurious signals arise. Particularly third-order intermodulation distortions (IMD3) are important due to the fact that they appear very close to the fundamental signals and cannot be easily removed by filters [1]. The measurement of IMD3 and the third-order intercept point (IP3) are essential to understanding the nonlinearities caused by photodiodes. IP3 is the intersection of the extrapolation of IMD3 and output fundamental RF power, P_{f}. Additionally the mixing of the two tones will create second order intermodulation (IMD2) that is also of interest. The second order intercept (IP2) limits broad-band dynamic range in the same way IP3 limits narrow-band dynamic range [2]. As the need for detectors with higher linearity increases with the desire for larger spurious-free dynamic range, the need for more accurate and larger measurement range is essential to meeting this goal. In order to accurately perform the measurement all other nonlinearities contributed by equipment must be minimized or calibrated out.

Many methods of measuring IP3 have been employed including a three-tone distortion less setup [3-4] and various two-tone setups [1, 5]; however with the increasing desire for highly linear detectors, problems arise from the limitations and additional nonlinearities at both ends of the measurement spectrum. The three-tone measurement relaxes some of the demands for eliminating harmonics when working with a highly linear diode, but is nonetheless not as ideal as the two-tone measurement [4]. In one of the two-tone setups, Jiang *et al*. used two Mach-Zehnder modulators (MZM) to modulate two optical beams respectively; the resulting modulated signals are detected. The modulators have inherent nonlinearities, in particular, the second order harmonics (HD2) from the first laser beam, that can mix in the detector with the fundamental RF tone detected from the second signal and contribute to IMD3. The second method employed in Scott *et al*. uses a four-laser heterodyning technique in which two pairs of two lasers with slightly offset frequencies are used to generate the two RF tones at the detector. This scheme does not depend on the linearity of any RF source, but does require the lasers to operate with high wavelength stability up to high power.

In this work we describe a new technique for the measurement of third order and second order distortions specifically in the interest of highly linear photodiodes where nonlinearities contributed by measurement components cannot be ignored. The setup will no longer rely on changing the RF input power to the MZM, but will optically attenuate the amplitude modulated beams and use a DC laser to compensate. We demonstrate that nonlinearities arising from the MZM have been successfully calibrated out of the measurement and the setup is valid for a large dynamic range.

## 2. Three laser two-tone setup

The previous two laser setup employed a MZM to modulate the optical beam [1]. The setup typically biases the MZM at the quadrature point to minimize HD2 which contributes to the IMD3 when mixed inside the detector. Although biasing at the quadrature reduces the HD2 signal, we cannot be sure the MZM is not contributing significantly to the IMD3 that is measured. The quadrature point of the MZM also changes as a function of time and temperature due to DC bias drift [6]. To accurately measure the IP3 we will need to show that the IMD3 due to the MZM as a result of the second order mixing at the detector (we denote it IMD3_{MZM}) can be isolated.

The setup we present incorporates the technique used in the previous two laser setup [1], as depicted in Fig. 1. Two external cavity lasers (ECL) at wavelengths of 1541nm and 1549nm are input into their respective MZM. A polarization rotator is inserted in front of the MZM to properly rotate the beam for minimum loss. Two synthesizers shown in Fig. 1 along with their respective driver are used to provide the input RF power (RF_{in}) to their respective MZM. A bandpass filter is placed just before the MZM to ensure no second order signals are contributed by the synthesizers. The output of the MZM is amplified by an erbium doped fiber amplifier (EDFA) shown in yellow in Fig. 1. Following the EDFAs are variable optical attenuators (VOA) that will be used to control the input optical modulation depth to the
photodiode (MD_{in}). Since both DC power and MD_{in} will be attenuated with the VOA, a third ECL is needed to maintain the constant input optical power to the photodiode. The MD_{in} corresponds to the modulation depth of the total fundamental output power (P_{f}) seen at the signal analyzer (ESA) from the photodiode. The two modulated lasers are combined using a polarizing beam combiner (PBC) and appropriate polarization rotators to ensure the beams are orthogonal. The purpose of the PBC is to ensure there will be no optical mixing before combining in the photodiode. The third ECL also uses an EDFA and VOA for amplification and control of the DC optical power. The combined first two lasers are coupled with the third using a 50/50 coupler, D, as shown in Fig. 1.

## 3. Experiment I: second and third order intercept measurement

#### 3.1 Experiment I: measurement setup

Initially the measurement was designed to include a fair amount of modulator nonlinearity in order to distinguish the different IMD3 tones. To do this we used a large value for RF_{in} at the synthesizers. For a given value of RF_{in} the RF component of the intensity modulated output of the MZM in log scale is:

where x denotes the first or second input frequency. In this set up the main contributions to IMD3 tones measured at the ESA, are the IMD3 of the detector and the IMD3_{MZM} mentioned earlier, which arises from the second order mixing of a second harmonic (e.g. 2f_{1} from the first modulator) and a fundamental signal (e.g. f_{2} from the second modulator) at the detector. The nonlinearities due to the RF sources and the ESA are assumed negligible in comparison. When the VOA is used to attenuate the MD_{in} by a factor of Δ the value of each coefficient (A, B, C…) will decrease at the same factor. The HD2 due to the MZM of f_{1} will mix with the fundamental of f_{2} inside the photodiode to generate IMD3_{MZM}. The change in IMD3_{MZM} in log scale will be:

Equation (2) shows that with a decrease in fundamental RF power by the amount of Δ, the contribution of IMD3_{MZM} should change with a slope of two with respect to fundamental RF power, P_{f}, in the log scale.

As the measurement is made the results should contain both contributions. Hayes *et al*. determined the slope of IMD3 due to the photodiode should be three [7]. For instance, for the IMD3 term 2∙f_{1}-f_{2} due to nonlinear mixing of f_{1} and f_{2} tones at the photodiode, as each tone is reduced by 1dB, the IMD3 will be reduced by 3dB (2dB from f_{1} tone and 1 dB from f_{2} tone). In the beginning, when the measured RF fundamental tone, P_{f}, is small, the IMD3_{MZM} (slope = 2) dominates; as P_{f} increases, the IMD3 from the photodiode (IMD3_{PD}) will dominate because it is increasing at a faster rate than IMD3_{MZM}. Therefore, the IMD3 curve should indicate a knee point at which IMD3 will change from a slope of two to a slope of three as P_{f} is increased. After the two regimes have been distinguished, the IMD3_{PD}, with a slope of three, can be extrapolated with P_{f} to find the intercept and determine the IP3 as in previous measurements [1]. The output IP3 (OIP3) which is the value of output power at the intercept can be calculated as according to [8]:

Additionally, the second order intermodulation (IMD2) of the photodiode can be measured at f_{1}+f_{2}, where we will expect a slope of two. The IMD2 will not contain contributions from the modulators since there is no mixing of the signals prior to the photodiode and only f_{1} and f_{2} are of concern. The IMD2 can then be extrapolated with the fundamental to give a value for the output second order intercept point (OIP2) and is calculated according to:

#### 3.2 Experiment I: results

A PIN InGaAs/InP waveguide photodiode (PD) was used for the measurement test. The PD was a 5μm width by 55μm length device with a 0.32μm intrinsic region. The PD has a maximum responsivity around 0.9A/W but was measured at about half the maximum responsivity for ease of measurement and to ensure stable current throughout the measurement. The measurement was made at frequencies 1.1GHz and 1.2GHz for f_{1} and f_{2} respectively, with 10dBm RF_{in}, 11.5dBm total input optical power, with bias voltage (V_{b}) of 4V, and 5.8mA photocurrent.

The results in Fig. 2 indicate a knee point in the IMD3 data that corresponds to the change in slope from two to three; a slope of two is measured for IMD2. From Fig. 2 we can see that as predicted the slope at lower input RF values is due to the IMD3_{MZM}, but that as P_{f} increases, the slope shifts to three for IMD3 as the photodiode’s IMD3 begins to dominate. On the upper end, the IMD3 measurement is limited by a number of factors that will be discussed below.

## 4. Experiment II: investigating the IMD3_{MZM} nonlinearity

#### 4.1 Experiment II: setup

The second experiment will investigate the IMD3_{MZM} and the sensitivity of the setup to bias drift over time. Looking at the intrinsic properties of a single modulator and starting with a single-tone input to the MZM as in [9]:

the transfer function can be expanded into a Taylor series:

where T^{n} is the n^{th} derivative of T at V=V_{b} and V_{RF} is the input RF voltage, (V-V_{b}), that can be related to RF_{in}. The IMD3_{MZM} in equation (2) is proportional to the product of the second harmonic modulation term of the first modulator which is a function of V_{RF1} and the fundamental in the second modulator, which is a function of V_{RF2}:

Equation (7) shows that the IMD3_{MZM} contribution to IMD3 increases as the V_{RF} increases. Using the same set up, we will measure IMD3 as a function of P_{f} for different values of RF_{in} to determine where the MZM must be operated so that the IMD3_{MZM} is suppressed below the noise floor of the ESA (NF_{ESA}). In addition, IMD3 will be measured at various MZM biasing points, quadrature and off quadrature, to assess the effects of the MZM bias stability on the IMD3 measurement.

#### 4.2 Experiment II: results

As described in section 4.1 the nonlinearities due to IMD3_{MZM} were investigated in the system. The first experiment was to vary RF_{in} to the MZM by changing the output at the synthesizer. The RF_{in} to the MZM corresponds to 16dBm, 10dBm and 7dBm for test 1, test 2 and test 3 respectively. The results can be seen in Fig. 3 where the blue data corresponds to the maximum RF_{in} and the red to the lowest RF_{in}. Figure 3 shows that as RF_{in} is decreased the resulting IMD3_{MZM} is reduced. As the IMD3_{MZM} is reduced the IMD3 due to the device remains along the same trend line as expected. The data allows us to conclude that the RF_{in} to the MZM has no effect on the IMD3_{PD}, but does affect the ability to measure lower values for the device by contributing to the IMD3_{MZM} signal. In other words, in previous measurement setups [1] without knowing the exact contribution of IMD3_{MZM} the results may have been affected by this nonlinearity as RF power is raised even when biasing the MZM at the quadrature.

The second investigation performed in experiment II explored the effects of stability of the modulator bias. In Jiang *et al*., the MZMs were biased at the quadrature point to maximize the fundamental tone [1]. The first test was performed with the MZM biased at the quadrature point. Fig. 4 shows the result in red with the contributions to IMD3 from both the PD and IMD3_{MZM} as indicated on the graph. The IMD3_{MZM} limited knee point corresponds to ~ - 115dBm. As the biasing of the MZM drifts off the quadrature (blue points), the knee point shifts to the right as shown in Fig. 4. Since IMD3_{MZM} is only increased by a few dB, the bias drift has only a small effect on the IMD3_{MZM} and the knee point (which is slightly shifted to the right). Also, the IMD3_{PD} (from the f_{1} and f_{2} tones) measured remains the same as the bias drifts off quadrature.

## 5. Theoretical discussion of measurement limit

Now that the limitations of the setup due to the IMD3_{MZM} have been determined and it can be optimized for dynamic range, it is useful to project the maximum IP3 measureable with the setup. First we define the limited IMD3 in log scale as:

where NF_{ESA} (in units of dB) is the noise floor of the ESA, with assumed negligible nonlinearity from the ESA. We will assume the first term dominates provided that the MZM has an appropriate RF_{in}. Since both tones need to operate at the same RF power, we will calculate P_{f} using one-tone for simplicity. Using the photocurrent stated in [9], the one-tone fundamental output power at the load R_{L} can be defined as:

where I_{ac} is the peak AC part of the photocurrent, P_{IN} is the optical power into the modulator, γ is the gain of the EDFA, T’(V_{b}) is the first derivative of the transfer function at V=V_{b}, η_{D} is the photodiode responsivity, η_{loss} is the loss in the fiber, R_{L} is the load resistance and V_{RF} is the input RF voltage to the modulator. Since the IMD3 is limited by the RF_{in} to the MZM the term V_{RF} cannot be used to boost P_{f}. The important terms are γ, R, and η_{loss}. From equation (8) and (9) we can calculate the maximum OIP3 measurable by our setup using equation (3). With the current MZMs, EDFAs and photodiodes we have calculated a maximum IP3 of 80dBm by measuring the maximum fundamental power (9dBm; EDFA limited) achieved with our setup and NF_{ESA} (-140dBm) and assumed the IMD3_{MZM} would be suppressed sufficiently.

## 6. Conclusions

We successfully demonstrated a technique for simultaneous measurement of IP3 and IP2 that calibrates out nonlinearities contributed by setups previously using amplitude modulators, ensuring accurate measurement in the range of the noise floor which is necessary for ultra-linear photodiodes. In addition we have investigated nonlinearities due to optical components in the setup and analyzed the effects IMD3_{MZM} has on the measurement of IP3. Analysis of these investigations show that reducing the RF input of the MZM until the IMD3_{MZM} is just below the noise floor and biasing close to the quadrature will optimize the measurement setup for minimum contributed IMD3_{MZM}. The setup implies a maximum measurable IP3 of ~ 80dBm with room for improvements.

## Acknowledgment

The authors would like to acknowledge Dr. Steve Pappert at DARPA and Dr. Don Kimball at Calit2, UCSD for helpful discussion and Calit2 for the loan of some of the equipment for the measurement. This work was supported by DARPA under the ULTRA and STTR programs.

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