## Abstract

We provide the first experimental demonstration of the impact of bias-frequency on second-order distortion in sampled analog optical links. We show proper selection of bias frequency yields >48 dB improvement in second-order distortion performance. In addition, we demonstrate that measurement of the average frequency of the optical comb may be used to determine the optimum bias frequency – without the need for involved radio-frequency distortion measurements.

© 2013 OSA

## 1. Introduction

Optical comb sources and sampled analog links are of current interest for a number of applications including long-haul and sub-sampled optical links [1, 2]. As with conventional intensity-modulated direct detection (IMDD) links – i.e. those employing a continuous-wave laser – much emphasis is placed on the third-order distortion performance of the link. For conventional links, third-order limited performance is predicated on the link operating at quadrature bias where even-order distortion terms in the radio-frequency (RF) photocurrent are analytically zero [3] and that any even-order distortion contributed by the photodiode is significantly smaller than the third-order distortion arising from the optical modulation process. In externally modulated links employing interferometric optical modulators, quadrature bias is set by adjusting the static optical path length between two interfering optical signals (those traversing the two arms of an interferometer or those propagating in different waveguide modes) electro-optically such that the relative phase difference between these two signals at the optical carrier frequency is *ϕ _{b}* =

*ω*Δ

_{o}L*n/c*= (2

*m*+ 1)

*π*/2 where

*ω*is the carrier frequency,

_{o}*L*is the interferometer arm length,

*c*is the speed of light, Δ

*n*is the difference in effective refractive index between the two signals, and

*m*is an integer. For a sampled analog link employing a broadband pulsed optical carrier (an optical frequency comb) this phase condition is clearly only satisfied at one particular frequency within the source bandwidth. If we consider each comb line individually then, it is clear that its contribution to the total photocurrent depends intimately on its bias phase as well as its amplitude. Therefore, minimization of the composite even-order distortion requires balancing the distortion components arising from each element (optical carrier) in the comb. Here, we illustrate the dramatic impact of bias frequency on second-order distortion in a sampled analog link and show judicious choice of bias frequency can make the difference between sub-octave and wideband operational utility. In addition, we show that the average frequency of the optical comb – determined from a simple measurement of the comb spectrum – provides sufficient bias-frequency accuracy to enable wideband operation even for high-performance optical links (dynamic ranges > 110 dB).

## 2. Theory

In this work, we consider optical links which exhibit negligible dispersion (that is, they are either very short or dispersion-managed) and that the input RF bandwidth is limited to one-fourth of the combline spacing *f*_{rep} = *ω*_{rep}/2*π*. In this case, adjacent comblines do not interact to produce any RF signals within the first Nyquist band (DC − *f*_{rep}/2) and a full optical field treatment is not required [1]. Rather, we may begin with the optical power spectrum of the *N*-element optical comb

*f*=

_{o}*ω*/2

_{o}*π*is the source laser frequency,

*P*is the power of the

_{n}*n*-th optical combline, and * denotes convolution. Each combline acts as an optical carrier thus, the total photocurrent in the link may is given by the sum of the photocurrent contributed from each of the

*N*modulated carriers Here, the photocurrent is related to the modulated optical power by the relation where

*α*

_{pd}is the DC responsivity of the photodiode (A/W) and

*h*

_{pd}(

*t*) is the photodiode impulse response normalized such that

*H*

_{pd}(

*ω*) [4].

From previous analyses it may be readily shown that – for a link excited with two equal-amplitude RF tones (at angular frequencies *ω*_{1} and *ω*_{2}) in the small-signal regime – the magnitudes of the fundamental (*ω*_{1,2}), second-harmonic (2*ω*_{1,2}), third-harmonic (3*ω*_{1,2}), and third-order intermodulation distortion (IMD, 2*ω*_{1,2} ± *ω*_{2,1}) currents arising from the n-th combline are given by [1, 3, 5]

*V*is the applied RF voltage amplitude,

_{o}*V*is the half-wave voltage of the optical modulator, and

_{π}*ϕ*is the DC bias phase of the modulator. In a conventional link, for wideband operation one chooses a bias phase of

_{b}*ϕ*= (2

_{b}*m*+ 1)

*π*/2 where

*m*is an integer such that, according to Eq. (6), the second-order distortion is ideally zero [3] and the limiting distortion is the in-band IMD component given by Eq. (8). Here, we include the third-harmonic term [Eq. (7)] because we perform our measurements with a single-tone excitation and extract the two-tone performance from the measured third harmonic. Because this quadrature condition is frequency-dependent in interferometric modulators, we must consider the case where the bias phase varies across the comb bandwidth. If we consider a narrow bandwidth Δ

*f*=

*f*−

*f*around the bias frequency

_{b}*f*such that the change in bias phase

_{b}*ϕ*= (2

_{b}*m*+ 1)

*π*/2. The resulting photocurrent expressions are then

*ϕ*

_{b,n}**|**. Also, given the fractional bandwidths of interest are quite small

*dϕ*/

_{b}*df*is taken to be constant in the frequency band of interest (the designation that it is to be evaluated at the bias frequency will be suppressed from this point on). The total fundamental and third-order currents are found by inserting Eqs. (10), (12) and (13) into Eq. (2), respectively. The resulting expressions for the fundamental and third-order current magnitudes are where the total average photocurrent is given by Inserting Eq. (11) into Eq. (2) and then substituting Eq. (17) into the result and rearranging yields the expression for the magnitude of the total second-harmonic current

In order to characterize the dynamic range of the optical comb-based link we first calculate the RF power in the fundamental, second- and third-harmonic, and third-order IMD and calculate the corresponding output intercept points [8]. Note, the latter is a useful – though not physically realizable – output power at which the RF power in the fundamental and a given distortion power are equal. A general expression for this point given a *k*-th order nonlinearity is (in terms of the fundamental and *k*-th order distortion output powers, *P*_{fund} and *P _{k}*)

*R*(taken to be 50 Ω) under the assumption that the photodiode has an equal matching resistance (

_{o}*P*= 1/4 ×

*i*

^{2}

*R*/2, the 1/4 accounts for maximum power transfer to a matched load) yields the power in each component of the photocurrent

_{o}_{2h}), third-harmonic (OIP

_{3h}), and third-order IMD (OIP

_{3imd}) intercepts are given by [after inserting Eqs. (21)–(24) into Eq. (20)]

While the output intercept point of the link is a useful ”component” metric for comparing the link linearity to other microwave devices, from a system perspective the spurious-free dynamic range is a more useful quantity as it takes into account the effect of noise in the system [8]. For a given output noise power spectral density (PSD) of *N _{o}* (W/Hz) and a receiver electrical bandwidth of

*B*the second- and third-order-limited spurious-free dynamic ranges (in a bandwidth of

_{e}*B*) are then calculated by inserting the appropriate intercept point into the relation (

_{e}*k*denotes the distortion order)

*f*=

*f*−

_{b}*f*

_{avg}) subject to the condition SFDR

_{2}> SFDR

_{3imd}yields the result

## 3. Experiment

We utilize the optical comb-based link shown schematically in Fig. 1(a) to verify our analysis and demonstrate the impact of choice of bias frequency on the link second-harmonic distortion performance. The optical carrier in the link is an optical comb with 20 GHz line spacing generated via cascaded phase- and intensity modulation of a 100 mW distributed feedback laser (EM4) [9, 10]. The RF input signal – a single tone at *f*_{1} = 500.0 MHz with an amplitude of *V _{o}* = 0.85 V (input power ∼ 8.6 dBm) – is impressed on the optical comb via a GaAs polarization mode converter-based intensity modulator [11] (Versawave Technologies,

*V*∼ 3.3 V). The modulated comb is then filtered using a liquid-crystal-on-silicon multi-port programmable optical filter (Finisar, WaveShaper 4000S). This filter architecture allows us to send the bias frequency to one port of the device and the remainder of the comb (sans the bias frequency component) to a separate port. The RF output of the link corresponding to each of these signals is recovered via direct detection with two internally RF-matched ∼20 GHz photodiodes (Discovery Semiconductors, DSC30S) and subsequently measured on two electrical spectrum analyzers (ESAs, Agilent Technologies, 8563EC). This architecture allows us to set the link bias at quadrature for a particular frequency by adjusting the DC bias voltage to minimize the second-harmonic signal (1000 MHz) at the bias port while simultaneously measuring the composite second-harmonic distortion of the comb-based link. This allows us to measure the bias frequency-dependence of the composite second-harmonic distortion. Note, the absence of the bias frequency line in the composite measurement has negligible impact on the measured total distortion as the contribution of the bias frequency component is ideally zero [see Eq. (22)] and the photocurrent due to any single line is < 1/20-th of the total photocurrent. A symmetric example comb is shown in Fig. 1(b). Here the comb (gray) exhibits ∼ 22 lines within a 10 dB power variation. The red, black, and blue curves illustrate the filtered bias-frequency signals corresponding to the −10, 0 (source laser) and +10 order comblines obtained from the optical filter. We note, comblines adjacent to the desired bias frequency are not perfectly extinguished in the filtering operation; however, these lines are suppressed by 25 dB (∼ 300×) or more which renders their contribution to the measured RF photocurrent inconsequential. The link components and sample rate allow alias-free operation up to a fundamental frequency approaching 10 GHz – the choice of

_{π}*f*= 500 MHz was dictated by the availability of harmonically-related RF filters used in our measurements.

_{o}To illustrate the bias frequency dependence of the second-harmonic distortion (generally, all even-order distortion), we measure the RF power of the composite fundamental, second-harmonic, and third-harmonic as the bias frequency is varied discretely across the symmetric comb. For this measurement, an individual combline is selected as the bias frequency and its second-harmonic is minimized by adjusting the DC bias voltage of the modulator. The measured fundamental, second- and third-harmonic RF powers are shown in Fig. 2(a) versus the bias frequency offset from the laser center frequency. Here, the measured powers show excellent agreement with the theoretical values calculated from Eqs. (21)–(23) for an average photocurrent of *I*_{avg} = 0.58 mA (shown by the solid green, black, and gray curves). Using the measured average photocurrent the derivative of the bias phase with respect to frequency is determined to be *dϕ _{b}*/

*df*= 0.63 mrad/GHz and the bias frequency corresponding to minimum second-harmonic distortion is found to be

*f*

_{b}−

*f*≈ 5.02 GHz from a numerical fit of Eq. (22) to the measured second-harmonic data. The dashed black line shows the minimum measured second-harmonic power achieved through fine adjustment of the bias voltage – this value is attributed to photodiode-induced second-harmonic distortion and agrees well with the value predicted by the measured photodiode nonlinearity (to be discussed further below). From the data it is clearly seen that the second-harmonic power varies dramatically (Δ

_{o}*P*

_{2h}> 48 dB) as the bias frequency is varied across the comb. In contrast, the fundamental and third-harmonic powers remain fixed at their “quadrature” values indicating the total variation in bias phase is quite small (the small-angle approximation to the change in bias phase is justified).

In terms of the output intercept points, the measured variation in second-harmonic power translates directly to an improvement in the second-harmonic output intercept point OIP_{2h} (dB-for-dB). This is shown in Fig. 2(b). The third-harmonic output intercept (blue circles) remains fixed at OIP_{3h} ∼ −14 dBm as determined from the average photocurrent and the fixed fundamental and third-harmonic powers as shown in Fig. 2(a). The second-harmonic OIP (red circles), however, varies by more than 48 dB (2.17 < OIP_{2h} < 50.6 dBm) as the bias frequency is varied. Once again, the maximum intercept point of OIP_{2h} = 50.6 dBm (dashed black line) is believed to be set by the nonlinearity of the photodiode used in our measurement. This is supported by a laser heterodyne measurement of the photodiode distortion using a pair of phase-locked Nd:YAG lasers [12] in which the second-harmonic intercept was measured to be OIP_{2h,pd} = 49 dBm at 1310 nm. The solid black, solid gray, and dashed gray curves illustrate the second-harmonic, third-harmonic, and third-order IMD intercept points calculated from Eqs. (25)–(27). Note, the second-harmonic intercept is maximized (nonlinearity is minimized) near the laser center frequency given that the comb is nearly symmetric.

As the optical comb deviates from symmetry about the laser center frequency, for example due to a temporal misalignment of the amplitude and phase modulation in the comb generator or due to an asymmetric modelocked laser spectrum, the second-order linearity will again be optimized at the average frequency of the optical comb. To illustrate this, the comb generator is intentionally misaligned to produced combs with average frequencies lower and higher than that of the symmetric comb and the RF powers are once again measured as a function of bias frequency. Figure 3 shows the second-harmonic power (top row) normalized to the maximum value achieved either at high- or low-bias (*ϕ _{b}* = 0,

*π*) and the corresponding optical comb (bottom row) as the comb is tuned to have a (a) low-frequency weighting, (b) near optimum symmetry, and (c) a high-frequency weighting. Here, the measured second-harmonic data are shown by the red circles and the calculated powers are shown by the solid black lines. The dashed red lines show the measured average comb frequency, calculated from the measured comb spectra via Eq. (19). In each case, the second-harmonic power is (optimally) minimized to the level determined by the photodiode near the comb average frequency. The frequency corresponding to minimum second-harmonic distortion is determined again from a numerical fit of Eq. (11). Comparisons of the calculated minimum distortion frequency (

*f*

_{b}−

*f*) determined from the measured second-harmonic data and the corresponding average frequency (

_{o}*f*

_{avg}−

*f*) determined from the optical comb are given in Table 1. The error is defined as Error = 100 × |

_{o}*f*

_{avg}−

*f*

_{b}|/Δ

*f*

_{rms}, where Δ

*f*

_{rms}is the full root-mean-square bandwidth of the optical comb [13]. We see excellent agreement between the ideal bias frequency calculated from the measured second-harmonic distortion data and the comb average frequency, with errors less than 0.5% in all cases.

To determine if measurement of the optical comb average frequency provides sufficient accuracy relative to the ideal bias frequency, we must consider the link dynamic range [Eq. (28)], as opposed to solely the output intercept points, because of the difference in rate of growth of second- versus third-order distortion. In particular, we must consider the second-harmonic (*k* = 2) limited dynamic range relative to the in-band third-order IMD limited dynamic range (*k* = 3) with the case of SFDR_{2} > SFDR_{3imd} corresponding to wideband operation. To illustrate the required bias frequency accuracy, we calculate the dynamic range of an intrinsic analog optical link (no optical amplification) as a function of average current and receiver bandwidth (see Appendix). Subsequently, we calculate the magnitude of the maximum allowable frequency error |Δ*f*| = |*f*_{b} − *f*_{avg}| required to maintain wideband operation from Eq. (29). The results are shown in Fig. 4, where we have used the worst-case phase slope corresponding to the symmetric comb (*dϕ _{b}*/

*df*∼ 0.63 mrad/GHz) for calculation purposes and the dashed black line marks the average photocurrent used in our experiments. Here, we see that bias errors of |

*f*

_{b}−

*f*

_{avg}| < 4 GHz enable wideband operation of the sampled link. Comparing the ideal bias frequencies determined from the measured second-harmonic distortion and the optical combs we see the error is well below 4 GHz for all combs utilized here as detailed in Table 1. Therefore, the ideal bias frequency may be readily predicted from measurement of the optical comb spectrum without the need for precise RF distortion measurements. Once the optimum bias frequency is determined one of several bias-control techniques (power-balancing or pilot tone-based) may be used to stabilize the bias point for applications.

## 4. Conclusion

As the use of sampled analog optical links continues to increase, understanding and minimization of RF distortion mechanisms unique to these architectures will become increasingly important. For wideband links (>octave bandwidth) minimizing second-order distortions is of key importance. Here we have shown that bias-dependent distortion can severely impact link performance, but that it is readily alleviated by proper choice of bias frequency. Proper selection of bias frequency has been demonstrated to improve second-order distortion performance by over 48 dB. Additionally, we establish that measurement of the optical spectrum of the sampling or optical comb source provides sufficient information to determine the bias frequency – without the need for involved RF distortion measurements.

## 5. Appendix

We note that the RF metrics for conventional [5] and sampled links [1, 4] have been well-documented in the literature. The current work is geared to describe the bias-dependent second-order distortion mechanism in the sampled link architecture. To accomplish this we do not need to re-derive expressions for the RF gain, noise figure and noise floor. Calculation of the spurious-free dynamic range [Eq. (28), Fig. 4], however, does require the output noise power spectral density, which in turn, requires the RF gain. Therefore, we will simply state the RF gain and the constituents of the output noise PSD. The interested reader may consult the references for derivations of these quantities.

For an intrinsic link, i.e. one which does not employ optical amplification, constructed with a shot-noise limited laser (for the operating photocurrent) the noise PSD at the output of the link is given by

Here,*N*

_{th}=

*kT*is the thermal noise PSD where

*k*is Boltzmann’s constant and

*T*is the temperature in Kelvin. The shot noise contribution is given by (the additional factor of 1/4 assumes a resistively-matched photodiode) where

*q*is the magnitude of the electronic charge. The link gain

*G*is given by where

*R*is the input resistance of the modulator and the additional factor of 1/4 again arises due to photodiode resistance matching. For the calculation shown in Fig. 4 we take

_{i}*T*= 290 K and

*R*= 50Ω.

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