Analysis of magnitude and relative phase of photodiode IMD2 using amplitude matched MZM-distortion cancellation technique

Meredith N. Hutchinson; Sharon R. Harmon; Vincent J. Urick; Keith J. Williams

doi:10.1364/OE.22.000962

1. Introduction

High fidelity multi-octave analog optical links have long suffered from reduced or limited dynamic range due to photodiode (PD) power and linearity limitations. Recently a technique was presented that utilized the second order intermodulation distortion (IMD2) of the Mach Zehnder modulator (MZM) in an intensity modulated direct detection link (IMDD) to cancel the PD IMD2 [1]. The advantage of the technique over other linearization techniques is that no additional components are required, as previously with balanced photodiodes or photodiode arrays [2–4]. In addition to providing significant increase in dynamic range and distortion cancellation, the theory postulates that the magnitude and the relative sign of $a_{2}$ , which represents the Taylor expansion coefficient of PD IMD2, can be calculated purely from the bias phase point of the modulator. The use of this linearization technique as a method to characterize PD IMD2 has been briefly introduced [5]. The purpose of this paper is to gain a greater understanding of photodiodes by expanding upon the initial analysis of this system as a method to gain more insight into the PD distortion amplitude and phase.

The structure of this paper will be as follows. A brief background of the measurement setup and method will be discussed along with the corresponding theoretical analysis as it pertains to $a_{2}$ . A sample device will be analyzed over a range of parameters such as voltage, photocurrent and frequency to study their effect on the magnitude and relative phase of $a_{2}$ . For the purposes of this paper, the analysis will focus on the technique of accurately measuring the PD distortion and the information that can be gained via this technique over the traditional two-tone OIP2 measurement setup.

2. Background and theory

As demonstrated in [1], the second order photodiode distortion can be cancelled using MZM generated distortion in an IMDD link. Additionally, the setup has been shown to be useful for measuring the relative sign of photodiode second order distortion. The analysis of this cancellation technique for the purposes of linearization was previously covered [1]. We will cover the terms relevant to the photodiode distortion. The IMDD link is shown in Fig. 1. We assume an ideal push-pull MZM with the transfer function:

[\begin{matrix} E_{1} (t) \\ E_{2} (t) \end{matrix}] = \frac{1}{2} [\begin{matrix} 1 & i \\ i & 1 \end{matrix}] [\begin{matrix} e^{i ϕ (t) / 2} & 0 \\ 0 & e^{- i ϕ (t) / 2} \end{matrix}] [\begin{matrix} 1 & i \\ i & 1 \end{matrix}] [\begin{matrix} E_{i n} (t) \\ 0 \end{matrix}],

where

E_{1}

and

E_{2}

are the fields corresponding to the two MZM outputs,

ϕ (t)

is the phase shift induced by the applied voltage and

E_{i n}

is the field at the MZM input. The frequency-dependent MZM half-wave voltage is

V_{π} (Ω)

. The input field is written as

E_{i n} = κ \sqrt{2 P_{o}} e^{i ω t}

, where

P_{o}

is the average optical power at angular frequency

ω

and

κ

is a constant such that

P_{o} = E^{*} E / (2 κ^{2})

. The input to the MZM comprises a DC bias voltage

V_{d c}

and a two-tone RF signal of the form

a_{2 -}

, where Ω are the angular frequencies [1]. It is important to ensure that the PD distortion is not corrupted by some other non-linearity in the system, such as from the electrical spectrum analyzer (ESA) or the RF mixer which include their own second order distortion at

Ω_{1} \pm Ω_{2}

. The phase shift assuming a pure two-tone signal can be described as:

ϕ (t) = ϕ_{d c} + ϕ_{1} \sin (Ω_{1} t) + ϕ_{2} \sin (Ω_{2} t),

where

ϕ_{d c} = π V_{d c} / V_{π}

and

ϕ_{1, 2} = π V_{1, 2} / V_{π}

. The fundamental MZM photocurrent at

Ω_{1, 2}

and the even order MZM contributed photocurrent that occurs at

Ω_{1} \pm Ω_{2}

can be calculated by expanding the Bessel functions of the first kind using Jacobi series. Assuming small-signal with equal amplitude two-tone input, the Bessel function can be written as

J_{n} (ϕ) \approx ϕ^{n} / (2^{n} n!)

, this gives us [6]:

\begin{array}{l} I_{f u n d, M Z M} = ϕ I_{d c, q} \sin (ϕ_{d c}) [\sin (Ω_{1} t) + \sin (Ω_{2} t)] \\ I_{e v e n, M Z M} = 2 \cos (ϕ_{d c}) I_{d c, q} \end{array}

\times {\frac{1}{4} ϕ^{2} \cos [(Ω_{2} - Ω_{1}) t] - \frac{1}{4} ϕ^{2} \cos [(Ω_{2} + Ω_{1}) t]},

where

I_{d c, q}

is the photocurrent at quadrature. The quadrature condition is given by

ϕ_{d c} = (2 k + 1) π / 2

where

k

is an integer. The small-signal IMD2 photocurrent in (3b) can then be simplified to:

Fig. 1 Intensity-modulation direct-detection link employing an external MZM.

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I_{Ω_{2} \mp Ω_{1}, M Z M} = \pm \frac{ϕ^{2} I_{d c, q} \cos (ϕ_{d c})}{2} \cos [(Ω_{2} \mp Ω_{1}) t] .

The photodiode distortion can be represented as follows. Assuming the fundamentals from the MZM drive the photodiode, a Taylor series expansion is applied:

I_{P D} = a_{0} + a_{1} (I_{i n} - I_{d c}) + a_{2} {(I_{i n} - I_{d c})}^{2} + \cdot \cdot \cdot,

where

I_{p d}

is the output current of the photodiode with an injection current of

I_{i n}

and an average photocurrent

I_{d c}

. The Taylor coefficients are defined as:

a {}_{m}= \frac{1}{m!} \frac{d^{m} I_{p d}}{d I_{i n}^{m}} .

Assuming

I_{i n} = I_{d c} + I_{f u n d, M Z M}

as given by Eq. (3a), then we have:

I_{P D} = (a_{0} + a_{2} I^{2}) + a_{1} I \sin (Ω_{1} t) + a_{1} I \sin (Ω_{2} t) - \frac{a_{2} I^{2}}{2} \cos (2 Ω_{1} t)

- \frac{a_{2} I^{2}}{2} \cos (2 Ω_{2} t) + a_{2} I^{2} \cos [(Ω_{1} - Ω_{2}) t] - a_{2} I^{2} \cos [(Ω_{1} + Ω_{2}) t] + \cdot \cdot \cdot,

where

I = ϕ I_{d c, q} \sin (ϕ_{d c})

. Separating the photocurrent into IMD2 terms we have:

I_{i m d 2, P D} = \pm a_{2} ϕ^{2} I_{d c, q}^{2} \sin^{2} (ϕ_{d c}) \cos [(Ω_{2} \mp Ω_{1}) t] .

The OIP2 can be determined by:

O I P 2_{P D} = \frac{P_{f u n d, P D}^{2}}{P_{i m d 2, P D}^{2}} = \frac{a_{1}^{4} R}{2 a_{2}^{2}} .

As shown previously the peak MZM- and photodiode-generated distortion can be combined to derive a cancellation condition by adding Eqs. (4) and (8) [1]:

I_{i m d 2, p e a k} = \pm ϕ^{2} I_{d c, q} [\frac{\cos (ϕ_{d c})}{2} + a_{2} I_{d c, q} \sin^{2} (ϕ_{d c})],

where the “+” and “−“ signs correspond to the terms at

(Ω_{2} - Ω_{1})

and

(Ω_{2} + Ω_{1})

respectively. Setting Eq. (10) to zero and solving for

a_{2}

gives us the relative sign and amplitude of the photodiode IMD2:

a_{2} = - \frac{\cos (ϕ_{d c})}{2 I_{d c, q} \sin^{2} (ϕ_{d c})} .

The above equation allows us to analyze the photodiode second order distortion as a function of phase as well as amplitude. Previously the nulls in IMD2 observed in certain photodiodes and microwave devices have been attributed to competing nonlinearities that are out of phase [7–11]. The method presented here will allow us to investigate the presence of multiple nonlinearities that are out of phase, as well as the behavior of IMD2 over a variety of operating conditions. It should be noted that the equation above only allows us to evaluate the strongest distortion signal present, as the cancellation will only be apparent if the strongest signal is 180 degrees out of phase with the MZM. Additionally the null will only be as deep as the maximum of any additional distortion that is in phase with the MZM at the location of the null. For example, if the strongest distortion is negative in sign, but there is additional distortion from the PD that is of a smaller magnitude but positive in sign, the null measurement will be equal to or greater than this.

We should also point out that the treatment of PD distortion with a power series analysis does not account for differences in distortion as a function of the frequency at which the distortion occurs. Previously it has been shown that treatments of nonlinearity would require the use of Volterra series analysis to capture the different values of distortion at different harmonic frequencies [12]. For the purposes of this paper we will use the power series analysis which sufficiently relates the photodiode IMD2 to MZM bias phase without accounting for the potential difference in IMD2 at $f_{1} \pm f_{2}$ .

3. Measurement setup

The setup shown in Fig. 1 uses the architecture of an intensity-modulation direct-detection link employing a MZM. The setup consists of a CW laser input to a MZM with two-tone RF input and the output to a photodiode, which is biased by a voltage source. The output is measured on an electrical spectrum analyzer (ESA). A dual-output modulator was used to monitor the power level of the arm not in use such that the measurements made were accurate. The accuracy relies on the precise measurement of MZM bias. As the measurements are made, quadrature bias drift will occur, thus using the second arm to monitor optical power is essential to the measuring of bias phase, $ϕ_{d c}$ , at the cancellation condition.

Once the off-quadrature bias phase has been measured where cancellation is observed, Eq. (11) is employed to determine the measured IMD2 and relative sign. Since the sign of the distortion is extracted by Eq. (11), the setup is limited however to our benefit to only measuring 180° phase shift in PD IMD2. In other words, if there are for a given set of operating conditions two PD-generated IMD2 mechanisms that are out of phase, the setup in Fig. 1 can in general only measure one distortion, namely the dominant. Additionally, if the PD is operating in an area of extreme nonlinearity (i.e. at very low bias) a null may not be observed even if we move far off quadrature. In this case the PD nonlinearity could potentially be greater than the maximum possible MZM generated distortion. Therefore, as we move off quadrature we will see cancellation of distortion, but no minimum once we reach the bias phase point that is some integer multiple of π, the point at which the MZM distortion is maximized. Despite this limitation, we can gain valuable insight into the areas of operation where there may be competing nonlinearities, which can sometimes result in distortion cancellation where the device will operate with a much higher OIP2. Cases of believed competing distortions have been reported before [7]. Until now the ability to study only one of those distortions at a time was not possible. This ability to isolate one particular distortion at a time by measurement would be very useful for model validation and development. For the purposes of this paper we focused on a commercially available diode (DSC50S), which has a responsivity of 0.75A/W and bandwidth of 12 GHz, to investigate the second order distortion over many variables.

4. Measurement and analysis

To begin, we wanted to establish that the setup was indeed measuring photodiode IMD2 by cancellation. The first measurement checked that the OIP2 measured over a range of small-signal inputs (<0dBm) would result in roughly the same OIP2. The input frequencies were set to $f_{1} = 2.789 G H z$ and $f_{2} = 3.124 G H z$ to be well within the test PD (DSC50S) bandwidth of 12GHz and non-integers to avoid any extraneous signals present in the environment that might mix and add distortion. At 5 V bias and $I_{d c, q} = 6 m A$ we calculated OIP2 using Eqs. (9) and (11) for input RF powers −10 dBm to 0 dBm in 1 dB increments. The result was OIP2 varied ± 1.64 dB and ± 0.55 dB for $O I P 2_{+}$ and $O I P 2_{-}$ where the distortion occurs at $f_{1} + f_{2}$ and $f_{2} - f_{1}$ respectively indicating our relative tolerance due to accurately measuring the null. Since the measurement is reliant upon an accurate bias phase measurement, we deemed these measurements to be within reason for the analyses we will conduct.

First we analyze $a_{2}$ as a function of voltage. The modulator was characterized where $V_{π} (Ω = 3 G H z) = 5.4 V$ . The input frequencies were as mentioned previously with a DC photocurrent at quadrature of 6mA. From here on out, $a_{2 +}$ will refer to the distortion occurring at $f_{1} + f_{2}$ and $a_{2 -}$ will refer to the distortion occurring at $f_{2} - f_{1}$ . As mentioned previously, the theoretical treatment used does not account for different distortion at $f_{1} + f_{2}$ , however we will analyze the measured $a_{2 +}$ and $a_{2 -}$ separately as the data clearly shows that in general the two are not equal. In Fig. 2, $a_{2 +}$ and $a_{2 -}$ show similar behavior, where both are near zero at 0 V and significantly increase at 0.5 V, then shift phase by 180°. In the inset, it is apparent that another 180° phase shift occurs at ~2.5 V to positive and finally another phase shift of 180° at ~4.5 V where both flatten out from 5 to 7 V. Similar behavior was previously detailed in [5]. As seen previously, the increase in distortion from 0 to 2 V is commonly seen with space charge being the dominant source of nonlinearity at these low bias voltages [13]. Thus the data show the presence of multiple nonlinearities with opposite phase dominating in various operating regions with bias. It should be noted that when the phase of $a_{2}$ suddenly shifts 180° as from 0.5 V to 1 V most likely the dominant linearity has switched and the setup can for the most part resolve only the nonlinearity with the greatest magnitude. At low bias voltage we notice that the magnitude of $a_{2 +}$ and $a_{2 -}$ greatly differ. We suspect nonlinear mechanisms may be dependent on the frequency at which the distortion occurs, which has been seen before [14]. As mentioned in Section 2, the theory does not account for this frequency dependence.

Fig. 2 Second order intermodulation distortion coefficient $(a_{2})$ as a function of voltage for input RF tones, $f_{1} = 2.789 G H z$ and $f_{2} = 3.124 G H z$ at 6 mA photocurrent.

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The second analysis focused on $a_{2}$ as a function of photocurrent. In Fig. 2 at 5 V the device is in a stable operating regime. In Fig. 3, the PD was biased at 5 V and the quadrature photocurrent was swept from 0.5 to 7mA. Since the bias was sufficient to create a high electric field, space-charge screening was not expected to be an issue. As can be seen in Fig. 3 the magnitude of both $a_{2 +}$ and $a_{2 -}$ do not change much, however there is a 180° phase shift around 2.5 mA for both and another at 6.5 mA for $a_{2 +}$ . Checking the values at 6 mA, we find they are similar to those measured at the same conditions in Fig. 2, indicating we generally have good measurement agreement using the setup.

Fig. 3 Second order intermodulation distortion coefficient $(a_{2})$ as a function of photocurrent for input RF tones, $f_{1} = 2.789 G H z$ and $f_{2} = 3.124 G H z$ at 5 V bias voltage.

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A second more interesting bias voltage to sweep the photocurrent at is 0.5 V. From Fig. 2 we remember at this value there is a positive spike for both $a_{2 +}$ and $a_{2 -}$ . At this operating point space charge is most likely a dominant nonlinearity; however it would be useful to see the behavior over photocurrent. The results for biasing at 0.5 V and sweeping photocurrent from 0.5 to 7mA are shown in Fig. 4. As seen previously in Fig. 2, $a_{2 +}$ has a much larger magnitude than $a_{2 -}$ at low bias voltage, most likely due to frequency dependent nonlinearity that may strongly impact $a_{2 +}$ which occurs at a higher frequency. Both $a_{2 +}$ and $a_{2 -}$ tend to decrease in Fig. 4, as was the case in Fig. 3, however $a_{2 -}$ is slightly positive at 0.5 mA but becomes negative at 3 mA. Both also shift phase by 180° ~5 mA. Interestingly we were able to observe two nulls on either side of quadrature for $a_{2 +}$ from 5.5 to 6 mA. Two nulls on either side of quadrature that are roughly the same magnitude would indicate the presence of two competing nonlinearities which would cancel and create a spike in OIP2. Recent simulation work incorporating multiple sources of physical nonlinearities detailed the peaks resulting from multiple nonlinearities interacting [10]. In Hu’s case, impact ionization at higher voltages results in an increase in harmonic power, where if this were not present the second order harmonic distortion would decrease and ultimately OIP2 will increase where only one nonlinearity is present [10]. Since the observation of two nulls occurred at low voltage and high photocurrent, most likely one source is space charge and the other is nonlinear capacitance. Checking again the values at 0.5 V and 6 mA with the measurement from Fig. 2, we find good agreement, although only by making sure to observe both nulls which captures the $a_{2 +}$ with positive phase. For $a_{2 -}$ the inability to see two nulls in the same region may be due to the magnitude of the nonlinearity and the measurement setup limitations.

Fig. 4 Second order intermodulation distortion coefficient $(a_{2})$ as a function of photocurrent for input RF tones, $f_{1} = 2.789 G H z$ and $f_{2} = 3.124 G H z$ at 0.5 V. The measurement for $a_{2 +}$ observed two nulls between 5.5 mA and 6 mA, one on either side of quadrature. The data for $a_{2 +}$ with negative relative phase is graphed with blue squares and for $a_{2 +}$ with positive relative phase is graphed with blue diamonds.

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Finally, we set the operating parameter to 6 mA for quadrature photocurrent and sweep the frequency from 1 to 12 GHz at 5 V and 1-15 GHz at 2 V. We will keep the frequency separation constant at 335 MHz as before, such that the frequency for $f_{2} - f_{1}$ is constant. Thus the fundamental frequencies are swept and the distortion at $f_{1} + f_{2}$ increases as twice the fundamental. The modulator $V_{π} (Ω)$ was previously characterized and varies by less than 2 V over the measurement range. Modulator $V_{π} (Ω)$ should not affect the measurement as we ensure that we are operating in the small-signal regime. The results are shown in Fig. 5. For $a_{2 -}$ at 5 V and 2 V there is very little change over frequency and the distortion is slightly less at 2 V than 5 V. Previously, similar behavior was observed where the distortion was seen to be a function of the output frequency where it was concluded that there is some frequency dependence for where the distortion is centered [14]. At 5 V $a_{2 -}$ is negative and quite consistent up to 12 GHz which is the bandwidth of the device. At 2 V $a_{2 -}$ is also consistently negative in phase until 12 GHz where the distortion shifts in phase 180°. In contrast, $a_{2 +}$ behaves more conventionally where the magnitude increases with frequency. However the behavior differs between 2 V and 5 V, where at 2 V the distortion increases in magnitude and is negative in phase until 9 GHz where $a_{2 +}$ decreases in magnitude and finally shifts in phase by 180° at 12 GHz. Measuring up to 15 GHz we see that the magnitude of $a_{2 +}$ continues to increase, as the distortion is expected to increase at higher frequency. At 5 V the behavior is much more conventional where the magnitude of distortion does not change significantly but experiences a 180° shift in phase at 6 GHz and as the 3-dB bandwidth is approached, the magnitude of $a_{2 +}$ increases at 10 GHz. Again checking the values at 3 GHz for consistency with earlier measurements in Fig. 2, we find at 2 V both $a_{2 -}$ and $a_{2 +}$ are negative in phase, where $a_{2 +}$ has a larger magnitude. At 5 V again both phases are negative and consistent with Fig. 2, $a_{2 -}$ has a larger magnitude than $a_{2 +}$ .

Fig. 5 Second order intermodulation distortion coefficient $(a_{2})$ as a function of frequency for input RF tones at 6mA photocurrent and 2V (red) and 5V (blue) bias voltage.

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The study of $a_{2}$ as a function of voltage, photocurrent and frequency has yielded many insights into the complicated behavior of second order PD intermodulation distortion. We have been able to understand the phase of the dominant nonlinearity in various scenarios and to see whether $a_{2 +}$ and $a_{2 -}$ are in or out of phase with each other. This technique also allowed us to observe that multiple nonlinearities are occurring out of phase as determined at 0.5 V for $a_{2 +}$ , confirming earlier predictions. The ability to differentiate nonlinearities will be useful for future device design. Ideally if the physical mechanism causing nonlinearity cannot be mitigated, we desire to tailor nonlinearities that are out of phase to cancel each other and thus increase OIP2. The setup has limitations in that it cannot differentiate nonlinearities that are in phase and generally can only measure the strongest nonlinearity. Finally, the study of $a_{2}$ as a function of frequency confirmed earlier measurements where there appeared to be a dependence on the frequency at which the distortion occurs. By studying the relative phase and magnitude of $a_{2}$ we can begin to build more complete and comprehensive models for similar device structures.

5. Summary

We presented an IMDD link for use as a method to characterize photodiode IMD2, both the magnitude and relative phase. The theory predicts that the MZM IMD2 can be used to cancel PD IMD2 and therefore indicate the approximate magnitude and phase of $a_{2}$ . We used the setup to characterize a test device over a range of parameters: voltage, photocurrent and frequency. The analysis confirmed previous predictions as well gave us insight into the more detailed behavior of $a_{2}$ , including the difference between $a_{2 +}$ and $a_{2 -}$ over the parameter set. We plan to continue to use this method to characterize and build models for current and future devices with the goal of better controlling nonlinearity mechanisms.

References and links

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Analysis of magnitude and relative phase of photodiode IMD2 using amplitude matched MZM-distortion cancellation technique

Abstract

1. Introduction

2. Background and theory

3. Measurement setup

4. Measurement and analysis

5. Summary

References and links

Cited By

Figures (5)

Equations (13)

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