Suppression of even-order photodiode distortions via predistortion linearization with a bias-shifted Mach-Zehnder modulator

Vincent J. Urick; Meredith N. Hutchinson; Joseph M. Singley; Jason D. McKinney; Keith J. Williams

doi:10.1364/OE.21.014368

1. Introduction

High-linearity photodiodes are actively researched in the field of microwave photonics, with applications in the academic, industrial and military sectors. A recent survey collects reported state-of-the-art results from the component level [1]. The concentration of high-linearity photodiode work is largely in terms of single-octave third-order-limited intermodulation distortion as quantified by a third-order output intercept point (OIP3). One of the inherent advantages of photonic solutions is the wide bandwidth available in the optical domain, making analog optical links attractive for multi-octave applications. However, even-order distortion generated by photodiodes can be inhibiting in such implementations [1]. Previous works have described the photodiode requirements in high-linearity photonic links for single- and multi-octave applications in terms of OIP3 and second-order output intercept point (OIP2), respectively [1,2]. Oftentimes the present photodiode technology falls short of the system requirements, particularly in multi-octave applications. Architectural techniques have been devised to mitigate the component limitations. For example, photodiode arrays have been shown to achieve better linearity than the individual photodiodes are capable of alone. Two- [3,4] and four-photodiode [5,6] arrays have been demonstrated. This simple but quite effective technique is based on dividing the input signal between numerous non-linear devices and then linearly combining their outputs. The “array gain” scales with the number of elements for both even- and odd-order distortion, assuming that each element exhibits the same nonlinearity. Balanced photodiode arrays have been demonstrated that improve the OIP3 by the array gain but suppress photodiode-generated even-order distortion through the balanced detection process [2,7]. This technique is attractive for multi-octave applications but requires two phase-matched fibers for the transmission span when implemented with a Mach-Zehnder modulator (MZM).

In this work we demonstrate cancellation of photodiode even-order distortion via predisortion linearization with a MZM biased slightly away from quadrature. This technique employs a single fiber run and a single photodiode. Improvements in carrier-to-intermodulation ratio (CIR) upwards of 40 dB are reported. A theoretical analysis of the technique is provided in Section 2, where closed-form expressions are used to describe the MZM nonlinearities and a Taylor-series expansion is applied to the photodiode. Experimental data supporting the theory are presented in Section 3. Implications of this work are discussed in Section 4.

2. Theory

A calculation demonstrating cancellation of photodiode even-order distortion with MZM-generated distortion is conducted assuming the architecture in Fig. 1. One of the two available MZM outputs is connected to a photodiode. The bias of the MZM is adjusted to match the amplitude of the photodiode second-order distortion and under certain conditions these two terms are 180 degrees out of phase. The following analysis is conducted in three steps. First, the response for the intensity-modulation direct-detection (IMDD) link in Fig. 1 is reviewed and cast in well-known closed-form expressions assuming an ideal linear photodiode. A Taylor series analysis is then employed to model the photodiode nonlinearity. Finally, these two sets of equations are combined to predict the composite response.

Fig. 1 Intensity-modulation direct-detection link employing an external Mach-Zehnder Modulator (MZM).

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2.1 Link with ideal photodiode

The response for an IMDD link employing a MZM is well-known [8]. Here, the terms relevant to the cancellation technique are highlighted. We assume an ideal push-pull MZM with the following 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_{in} (t) \\ 0 \end{matrix}],

where

E_{1}

and

E_{2}

are the fields corresponding to the two MZM outputs,

ϕ

is the phase shift induced by the applied voltage, and

E_{in}

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_{in} = 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_{dc}

and a two-tone RF signal of the form

V_{1} \sin (Ω_{1} t) + V_{2} \sin (Ω_{2} t)

, where

Ω

are the angular frequencies. With these input voltages the phase shift

ϕ (t) = ϕ_{dc} + ϕ_{1} \sin (Ω_{1} t) + ϕ_{2} \sin (Ω_{2} t)

results, where

ϕ_{dc} = π V_{dc} / V_{π}

and

ϕ_{1,2} = π V_{1, 2} / V_{π}

. Assuming an ideal photodiode with responsivity

ℜ

, the total photocurrent due to

E_{1}

can be calculated and separated into three components [1]:

I_{dc,mzm} = I_{dc,q} - I_{dc,q} J_{0} (ϕ_{1}) J_{0} (ϕ_{2}) \cos (ϕ_{dc})

\begin{array}{l} I_{odd,mzm} = 2 \sin (ϕ_{dc}) I_{dc,q} \\ \times {J_{0} (ϕ_{2}) \sum_{j = 0}^{\infty} J_{2 j + 1} (ϕ_{1}) \sin [(2 j + 1) Ω_{1} t] \\ + J_{0} (ϕ_{1}) \sum_{k = 0}^{\infty} J_{2 k + 1} (ϕ_{2}) \sin [(2 k + 1) Ω_{2} t] \\ - \sum_{j = 0}^{\infty} \sum_{m = 1}^{\infty} J_{2 j + 1} (ϕ_{1}) J_{2 m} (ϕ_{2}) \sin [(2 m Ω_{2} - (2 j + 1) Ω_{1}) t] \\ - \sum_{k = 0}^{\infty} \sum_{h = 1}^{\infty} J_{2 k + 1} (ϕ_{2}) J_{2 h} (ϕ_{1}) \sin [(2 h Ω_{1} - (2 k + 1) Ω_{2}) t] \\ + \sum_{j = 0}^{\infty} \sum_{m = 1}^{\infty} J_{2 j + 1} (ϕ_{1}) J_{2 m} (ϕ_{2}) \sin [(2 m Ω_{2} + (2 j + 1) Ω_{1}) t] \\ + \sum_{k = 0}^{\infty} \sum_{h = 1}^{\infty} J_{2 k + 1} (ϕ_{2}) J_{2 h} (ϕ_{1}) \sin [(2 h Ω_{1} + (2 k + 1) Ω_{2}) t]} \end{array}

\begin{array}{l} I_{even,mzm} = 2 \cos (ϕ_{dc}) I_{dc,q} \\ \times {- J_{0} (ϕ_{2}) \sum_{k = 1}^{\infty} J_{2 k} (ϕ_{1}) \cos (2 k Ω_{1} t) \\ - J_{0} (ϕ_{1}) \sum_{m = 1}^{\infty} J_{2 m} (ϕ_{2}) \cos (2 m Ω_{2} t) \\ + \sum_{n = 0}^{\infty} \sum_{p = 0}^{\infty} J_{2 n + 1} (ϕ_{1}) J_{2 p + 1} (ϕ_{2}) \cos [((2 p + 1) Ω_{2} - (2 n + 1) Ω_{1}) t] \\ - \sum_{n = 0}^{\infty} \sum_{p = 0}^{\infty} J_{2 n + 1} (ϕ_{1}) J_{2 p + 1} (ϕ_{2}) \cos [((2 p + 1) Ω_{2} + (2 n + 1) Ω_{1}) t] \\ - \sum_{k = 1}^{\infty} \sum_{m = 1}^{\infty} J_{2 k} (ϕ_{1}) J_{2 m} (ϕ_{2}) \cos [2 (m Ω_{2} - k Ω_{1}) t] \\ - \sum_{k = 1}^{\infty} \sum_{m = 1}^{\infty} J_{2 k} (ϕ_{1}) J_{2 m} (ϕ_{2}) \cos [2 (m Ω_{2} + k Ω_{1}) t]} \end{array}

where

I_{dc,q}

is the photocurrent at quadrature and J is a Bessel function of the first kind. The quadrature condition is given by

ϕ_{dc} = (2 k + 1) π / 2

where k is an integer. Equation (2a) is the average (DC) current, Eq. (2b) are the odd-order RF terms, and Eq. (2c) are the even-order RF terms. Thus, the total photocurrent for this output is

I_{1} (t) = I_{dc,mzm} + I_{odd,mzm} + I_{even,mzm}

. The photocurrent associated with

E_{2}

is

I_{2} (t) = 2 I_{dc,q} - I_{1} (t)

.

The treatment here will assume a small-signal two-tone stimulus with equal amplitude tones, thus $ϕ_{1} = ϕ_{2} = ϕ < < 1$ . A small-signal approximation allows for the Bessel functions to be written as $J_{n} (ϕ) \approx ϕ^{n} / (2^{n} n!)$ . These conditions can be applied to Eq. (2) to yield the fundamental photocurrents

I_{fund,mzm} = ϕ I_{dc,q} \sin (ϕ_{dc}) [\sin (Ω_{1} t) + \sin (Ω_{2} t)] .

Assuming all of the current is delivered to a load with resistance R, the average output power for both of the fundamentals is $P_{fund,mzm} = ϕ^{2} I_{dc,q}^{2} \sin^{2} (ϕ_{dc}) R / 2$ . The work here concentrates on even-order distortion. The largest small-signal distortion in Eq. (2) is second-order intermodulation distortion (IMD2) at frequencies $| f_{1} \pm f_{2} |$ given by the first two double summations in Eq. (2c) with $n = p = 0$ . The small-signal photocurrent for these two terms is

I_{imd2,mzm} = \pm \frac{ϕ^{2} I_{dc,q} \cos (ϕ_{dc})}{2} \cos [(Ω_{2} \mp Ω_{1}) t] .

The average power associated with Eq. (4) is

P_{imd2,mzm} = ϕ^{4} I_{dc,q}^{2} \cos^{2} (ϕ_{dc}) R / 8

. Finally, the OIP2 due to MZM-generated IMD2 is

OIP 2_{mzm} = \frac{2 \sin^{4} (ϕ_{dc})}{\cos^{2} (ϕ_{dc})} I_{dc,q}^{2} R .

As given by Eq. (5) and detailed previously [1], small deviations from quadrature bias can significantly degrade the OIP2. In fact, the tolerance on MZM bias can be quite stringent to maintain third-order-limited performance in multi-octave links.

2.2 Photodiode distortion

Numerous models have been developed to describe photodiode distortions in microwave photonics applications [9,10]. Here, we assume that the fundamentals from the MZM drive the photodiode, which can be described by a Taylor series expansion. We apply the following definition for a Taylor series expansion

I_{pd} = a_{0} + a_{1} (I_{in} - I_{dc}) + a_{2} {(I_{in} - I_{dc})}^{2} + \dots

where

I_{pd}

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

I_{in}

and an average current

I_{dc}

. The Taylor coefficients are defined as usual,

a_{m} = \frac{1}{m!} \cdot {\frac{d^{m} I_{pd}}{d I_{in}^{m}} |}_{I_{in} = I_{dc}} .

Now, if we assume that

I_{in} = I_{dc} + I_{fund,mzm}

as given by Eqs. (2a) and (3), that is, the ideal IMDD link provides the injection current to a nonlinear photodiode described by Eq. (6), then

\begin{array}{l} I_{pd} = (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] + \dots \end{array}

where

I = ϕ I_{dc,q} \sin (ϕ_{dc})

and the expansion has been carried out to terms of second order. The currents for the IMD2 terms in Eq. (8) are

I_{imd2,pd} = \pm a_{2} ϕ^{2} I_{dc,q}^{2} \sin^{2} (ϕ_{dc}) \cos [(Ω_{2} \mp Ω_{1}) t] .

The OIP2 for the photodiode can be determined by the expression

OIP 2_{pd} = P_{fund,pd}^{2} / P_{imd2,pd}

, where

P_{fund,pd} = a_{1}^{2} I^{2} R / 2

and

P_{imd2,pd} = a_{2}^{2} I^{4} R / 2

are the average powers for the fundamental and IMD2, respectively. Thus,

OIP 2_{pd} = \frac{a_{1}^{4} R}{2 a_{2}^{2}} .

2.3 Combined response

Our proposition is that the MZM bias can be adjusted to generate even-order distortion matching the amplitude of that arising from the photodiode. The forms of Eqs. (4) and (9) predict that two sources of distortion can be out of phase. The treatments of MZM- and photodiode-generated distortion above can be combined to derive a cancellation condition. The peak current at both IMD2 terms is obtained by addition of Eqs. (4) and (9):

I_{imd2,peak} = \pm ϕ^{2} I_{dc,q} [\frac{\cos (ϕ_{dc})}{2} + a_{2} I_{dc,q} \sin^{2} (ϕ_{dc})],

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

(Ω_{2} - Ω_{1})

and

(Ω_{2} + Ω_{1})

, respectively. Setting Eq. (11) to zero yields the cancellation condition as

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

The analysis above also predicts that second-harmonic distortion will cancel as well; expanding the Taylor series to higher orders shows that all even-order distortion is suppressed with this technique. The parameters in Eq. (12) are readily determined. The bias phase and photocurrent at quadrature are easily measured. The small-signal gain of the link will allow for the magnitude of

a_{1}

to be calculated. With this information, a measurement of the photodiode OIP2 will give the magnitude for

a_{2}

by way of Eq. (10). Cancellation of the IMD2 is then predicted by Eq. (12) to be cyclic as a function of

ϕ_{dc}

.

3. Experiments

The experimental investigation involves two apparatuses, a single-MZM link (Fig. 1) to demonstrate the cancellation and a two-MZM setup to isolate photodiode IMD2. The two-MZM setup is shown in Fig. 2. Two lasers at different wavelengths are modulated via two quadrature-biased MZMs each connected to separate signal generators. The signal generator power is adjusted to yield the same modulation depth on each laser. Variable optical attenuators are then employed to equalize the average optical power from each MZM output before coupling the two channels onto the photodiode being evaluated. If the two lasers are spaced at a frequency difference much larger than the modulation frequencies, then the IMD2 from the apparatus should be dominated by the photodiode nonlinearity. As compared to three-tone architectures employing three or more MZMs for odd-order distortion measurements [11,12], this relatively simple two-tone setup is adequate for IMD2.

Fig. 2 Apparatus for characterization of photodiode linearity, where two lasers are both intensity modulated via an external Mach-Zehnder modulator (MZM). Variable optical attenuators (VOA) are employed to balance the output power between MZMs.

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The structure shown in Fig. 2 was constructed using two 100-mW semiconductor lasers (EM4, Inc.) at 1548 nm and 1560 nm. The two MZMs (EOSPACE) exhibited 20 GHz of analog bandwidth and had nearly equal $V_{π} (Ω)$ . The two signal generators were set at $f_{1}$ = 0.9 GHz and $f_{2}$ = 1.1 GHz. With the variable optical attenuators set to output the same average optical power, the signal generators were adjusted to establish the same modulation depth on each laser. The photodiode being examined was an Applied Optoelectronics PD3000 with 3-dB bandwidth of about 3-GHz. The photodiode was reversed biased with 1 V and the OIP2 was measured by sweeping the input power at $f_{1}$ and $f_{2}$ . The measured OIP2 = 13.5 dBm due to IMD2 at 2.0 GHz for an average photocurrent of $I_{dc}$ = 3.0 mA as shown in Fig. 3. Also measured were OIP2 = 13.8 and 13.5 dBm at $I_{dc}$ = 2.5 and 3.5 mA, respectively.

Fig. 3 Measured OIP2 due to intermodulation distortion for the photodiode at 3 mA average photocurrent. Shown are the measured fundamentals (circles), the measured IMD2 (squares), and the first- and second-order fits with slopes m = 1 and m = 2, respectively.

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The same photodiode was used in a link such as shown in Fig. 1. An 80-mW semiconductor laser near 1550 nm (EM4, Inc.) was used as the source at the input to a dual-output 20-GHz MZM (EOSPACE) with $V_{π}$ = 4.5 V at 1 GHz. One output of the MZM was fed to an optical power meter to monitor the bias, while the other was attenuated and connected to the photodiode. The quadrature photocurrent was set at $I_{dc,q}$ = 3.0 mA. A network analyzer was employed to measure the single-tone small-signal gain with an input power of −20 dBm. The response was 4-dB less than that predicted by Eq. (3) with R = 50 Ω at 1 GHz. With this result, Eq. (8) allows for $| a_{1} |$ = 0.631 to be calculated. Equation (10) can then be solved for $| a_{2} | = 13.3 A^{- 1}$ .

A two-tone test was applied to the link with frequencies $f_{1}$ = 0.9 GHz and $f_{2}$ = 1.1 GHz, the results of which are shown in Fig. 4. The fundamentals both exhibit the same gain. The IMD2 at 2.0 GHz resulted in an OIP2 = 12.5 dBm when the MZM was biased at quadrature. This level is very close to that measured for the photodiode alone with the setup in Fig. 2, indicating that the photodiode is limiting the IMD2. The MZM bias was adjusted to determine the minimum IMD2, which was observed well away from quadrature at an average current of 2.5 mA. The IMD2 at this cancellation point is also plotted in Fig. 4(a). Very strong suppression was measured at input powers below −15 dBm; the suppression is good but much less at higher input powers. A second-order function is fit to the data at higher inputpowers resulting in OIP2 = 30 dBm. This limiting OIP2 is due to the measurement apparatus and attributed to reflections in the apparatus but is still 16.5 dB better than the OIP2 of the photodiode. As opposed to other linearization techniques, the fundamental is not affected by this method. The carrier-to-intermodulation ratio (CIR) is also plotted in Fig. 4(b) to demonstrate this point. At high input powers, the CIR is 17.5 dB higher at 2.5 mA than that at 3.0 mA (MZM quadrature). In the strong cancellation region, the CIR difference is upwards of 40 dB.

Fig. 4 (a) Measured OIP2s for the link at quadrature and at the cancellation point. Shown are the measured fundamentals (circles), the measured IMD2 at quadrature (squares), the measured IMD2 at the cancellation condition (triangles), and the first and second order fits with slopes m = 1 and m = 2, respectively. (b) The CIR at quadrature (squares) and at the cancellation condition (triangles).

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To confirm the cancellation condition, equations from the previous section were plotted against measured data as a function of $ϕ_{dc}$ . As shown in Fig. 5, the calculated fundamental power [Eq. (8)], the IMD2 power [Eq. (11)] and the DC photocurrent [Eq. (2a)] follow the experimental results. The measured and calculated DC photocurrents agree precisely. The calculated and experimental fundamental powers agree well at quadrature ( $I_{dc,q}$ = 3.0 mA). However, the measured values diverge above the calculation below 3.0 mA and go below the calculation at photocurrents above 3.0 mA. The IMD2 curves follow the same trend with a larger divergence but do agree quite well at the cancellation condition. The reason for the divergences is attributed to photodiode compression, which is worse at higher photocurrents and at higher frequencies. As described previously, the three measured OIP2 values using the setup in Fig. 2 also decreased at higher photocurrents supporting this claim. With this caveat, the measured results very much support the theory in Section 2 in predicting the cancellation of photodiode IMD2 with MZM-generated IMD2.

Fig. 5 Measured fundamental output power (open circles), measured IMD2 (triangles) and measured DC photocurrent (gray circles) as a function of MZM bias for the link at −20 dBm input power to the fundamentals. The solid lines show the calculated fundamental power, IMD2 power and average photocurrent.

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4. Summary and conclusions

A new technique to suppress photodiode even-order distortion has been demonstrated. The method employs a single-output MZM with a single photodiode, where the bias of the MZM is adjusted to achieve cancellation of photodiode distortion. The amplitude of the MZM-induced even order distortion is symmetric about quadrature but the phase of the even-order distortion shifts by 180 degrees. However, the MZM-generated fundamental phase is constant on either side of quadrature. Therefore, if the fundamental is assumed to drive the photodiode from which the photodiode-induced even-order distortion originates, then the phase and amplitude of the MZM-generated even-order distortion can be adjusted to cancel that from the photodiode. The cancellation bias phase depends on the magnitude of the photodiode distortion and the average photocurrent at quadrature bias.

This technique has numerous applications in microwave photonics. Photodiode even-order distortion can be inhibiting in multi-octave photonic links. This is especially true as the modulation frequency increases, where smaller photodiodes will exhibit reduced OIP2s. The technique presented here provides a mitigation technique for such links. The equations and experimental procedure described in this work suggest a path to determine the amplitude and relative phase of the photodiode even-order distortion terms. This has implications in photodiode research for microwave photonics. The bias condition to achieve cancellation is as sensitive as locking precisely at quadrature in an ideal MZM link. On the other hand, Mach-Zehnder modulator bias techniques based on minimizing the second harmonic of a calibration signal in a feedback loop may not provide the optimal bias for a multi-octave link. That is, the MZM-generated second harmonic may cancel those arising from photodiodes in the bias circuit near true quadrature. The resulting bias condition may not be optimal for the photodiodes on the receive end. Feedback from the back to the front end of the link is not practical in long point-to-point links. Therefore, some engineering challenges exist to implement this technique in deployed systems.

Other analog optical modulation formats should be capable of suppressing photodiode-generated even-order distortion using methods similar, if not identical, to those presented here. For example, an analog optical phase modulated link employing an asymmetric Mach-Zehnder interferometer (MZI) and a single photodiode exhibits a transfer function much like that of the MZM. Likewise, polarization modulation with a polarization beam splitter used to convert the signal to intensity modulation can utilize the technique. These architectures can be implemented with the bias circuits for the MZI and alignment to the beam splitter at the receive end of the link. Therefore, the operating point to achieve cancellation can be locked with the co-located photodiode. Future work on the concepts in this paper includes investigation of these propositions, applying the results to photodiode characterization and implementing the method in millimeter-wave links.

References and links

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6. Y. Fu, H. Pan, Z. Li, and J. Campbell, “High linearity photodiode array with monolithically integrated Wilkinson power combiner,” in 2010IEEE International Meeting on Microwave Photonics Digest, pp. 111–113. [CrossRef]

7. A. S. Hastings, V. J. Urick, C. Sunderman, J. F. Diehl, J. D. McKinney, D. A. Tulchinsky, P. S. Devgan, and K. J. Williams, “Suppression of even-order photodiode nonlinearities in multioctave photonic links,” J. Lightwave Technol. 26(15), 2557–2562 (2008). [CrossRef]

8. B. H. Kolner and D. W. Dolfi, “Intermodulation distortion and compression in an integrated electrooptic modulator,” Appl. Opt. 26(17), 3676–3680 (1987). [CrossRef] [PubMed]

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Suppression of even-order photodiode distortions via predistortion linearization with a bias-shifted Mach-Zehnder modulator

Abstract

1. Introduction

2. Theory

2.1 Link with ideal photodiode

2.2 Photodiode distortion

2.3 Combined response

3. Experiments

4. Summary and conclusions

References and links

Cited By

Figures (5)

Equations (14)

Optics Express