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 ϕb = ωoLΔn/c = (2m + 1)π/2 where ωo is the carrier frequency, L is the interferometer arm length, c is the speed of light, Δn 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 frep = ωrep/2π. In this case, adjacent comblines do not interact to produce any RF signals within the first Nyquist band (DC − frep/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

P(ω)=δ(ωωo)*n=0N1Pnδ(ωnωrep)
where fo = ωo/2π is the source laser frequency, Pn is the power of the 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
itotal(t)=n=0N1in(t).
Here, the photocurrent is related to the modulated optical power by the relation
i(t)=p(t)*αpdhpd(t)
where αpd is the DC responsivity of the photodiode (A/W) and hpd(t) is the photodiode impulse response normalized such that
dthpd(t)=αpd.
Note, in the remainder of our analysis we take the photodiode frequency response as unity – to include a non-uniform frequency response, the photocurrent expressions later in this section should be multiplied by the diode frequency response Hpd(ω) [4].

From previous analyses it may be readily shown that – for a link excited with two equal-amplitude RF tones (at angular frequencies ω1 and ω2) in the small-signal regime – the magnitudes of the fundamental (ω1,2), second-harmonic (2ω1,2), third-harmonic (3ω1,2), and third-order intermodulation distortion (IMD, 2ω1,2 ± ω2,1) currents arising from the n-th combline are given by [1, 3, 5]

ifund,n=αpdPn(πVoVπ)|sinϕb|,
i2h,n=αpdPn14(πVoVπ)2|cosϕb|,
i3h,n=αpdPn124(πVoVπ)3|sinϕb|
i3imd,n=αpdPn18(πVoVπ)3|sinϕb|.
In the above expressions Vo is the applied RF voltage amplitude, Vπ is the half-wave voltage of the optical modulator, and ϕb is the DC bias phase of the modulator. In a conventional link, for wideband operation one chooses a bias phase of ϕb = (2m + 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 Δf = ffb around the bias frequency fb such that the change in bias phase
Δϕb,n=(fnfb)dϕbdf|f=fb.
is small, we may expand the trigonometric functions in Eqs. (5)(8) about ϕb = (2m + 1)π/2. The resulting photocurrent expressions are then
ifund,n=αpdPn(πVoVπ),
i2h,n=αpdPn14(πVoVπ)2|Δϕb,n|,
i3h,n=αpdPn124(πVoVπ)3
i3imd,n=αpdPn18(πVoVπ)3.
Note, we are not considering cancellation effects (e.g., between modulator- and photodiode-induced distortion [6]). Therefore, we are only concerned with the magnitude of the bias error |Δϕb,n|. Also, given the fractional bandwidths of interest are quite small 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
ifund=Iavg(πVoVπ),
i3h=Iavg124(πVoVπ)3
i3imd=Iavg18(πVoVπ)3,
where the total average photocurrent is given by
Iavg=αpdn=0N1Pn.
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
i2h=Iavg14(πVoVπ)2|(fbfavg)dϕbdf|,
where the average frequency of the comb is given by [7]
favg=n=0N1fnPnn=0N1Pn.
Here, we see the magnitude of the modulator-induced bias dependent distortion grows linearly as the bias frequency deviates from the average frequency of the optical comb and that the rate of growth is determined by the slope of the bias phase versus frequency at the bias point. While this quantity may be calculated based on a particular modulator design (to be discussed further in the following section), for the purposes of this work it is treated as an experimentally measured quantity. Clearly, devices exhibiting smaller variations in bias phase as a function of optical frequency are desirable from a second-order distortion perspective.

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, Pfund and Pk)

OIPk=(PfundkPk)1/(k1).
Calculating the RF power delivered to a load resistance Ro (taken to be 50 Ω) under the assumption that the photodiode has an equal matching resistance (P = 1/4 × i2Ro/2, the 1/4 accounts for maximum power transfer to a matched load) yields the power in each component of the photocurrent
Pfund=18Iavg2Ro(πVoVπ)2,
P2h=1128Iavg2Ro(πVoVπ)4|(fbfavg)dϕbdf|2,
P3h=14608Iavg2Ro(πVoVπ)6
P3imd=1512Iavg2Ro(πVoVπ)6.
The corresponding second-harmonic (OIP2h), third-harmonic (OIP3h), and third-order IMD (OIP3imd) intercepts are given by [after inserting Eqs. (21)(24) into Eq. (20)]
OIP2h=2Iavg2Ro|(fbfavg)dϕbdf|2,
OIP3h=3Iavg2Ro,
and
OIP3imd=Iavg2Ro.
It should be noted that the second-harmonic intercept point given in Eq. (25) corresponds to that derived previously for modulator-induced second-harmonic distortion [6] for the specific case of small optical-frequency dependent variations in bias phase.

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 No (W/Hz) and a receiver electrical bandwidth of Be the second- and third-order-limited spurious-free dynamic ranges (in a bandwidth of Be) are then calculated by inserting the appropriate intercept point into the relation (k denotes the distortion order)

SFDR=(OIPkNo×Be)(k1)/k.
In order to maintain the third-order limited dynamic range imposed by the intrinsic nonlinearity of the optical modulator, the second-harmonic limited dynamic range must exceed the third-order IMD limited dynamic range. Solving for the magnitude of the maximum allowable bias frequency offset from the mean frequency of the optical comb (Δf = fbfavg) subject to the condition SFDR2 > SFDR3imd yields the result
|Δf|<2(SFDR3imd)1/4×|dϕbdf|1.
This result is intuitively satisfying in that it clearly shows that the constraints on the choice of bias frequency (a) become more stringent as the desired dynamic range increases and (b) are relaxed as the change in modulator bias phase as a function of frequency decreases.

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 f1 = 500.0 MHz with an amplitude of Vo = 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, Vπ ∼ 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 fo = 500 MHz was dictated by the availability of harmonically-related RF filters used in our measurements.

 

Fig. 1 (a) Schematic of the optical comb-based analog link (PMCIM: polarization mode converter-based intensity modulator; ESA: electrical spectrum analyzer). (b) Example optical comb (gray) and filtered bias frequency lines corresponding to the −10, 0, and +10 order comblines (red, black, and blue, respectively).

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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 Iavg = 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 b/df = 0.63 mrad/GHz and the bias frequency corresponding to minimum second-harmonic distortion is found to be fbfo ≈ 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 (ΔP2h > 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).

 

Fig. 2 (a) Measured RF power in the fundamental (f1 = 500 MHz, black), second-harmonic (2f1 = 1000 MHz, red) and third-harmonic (3f1 = 1500 MHz) as the bias frequency is varied across a symmetric optical comb [bottom, Fig. 3 (b)]. The dashed black line shows the minimum second-harmonic power obtained by optimizing the bias voltage (∼ −112 dBm) and the solid green, black, and gray lines show the theoretical values calculated from Eqs. (21)(23). (b) Output intercept points (OIP2h: red, OIP3h: blue) corresponding to the measured RF powers in (a). The solid black, solid gray, and dashed gray curves show the theoretical values calculated from Eqs. (25)(27). The dashed black line shows the maximum OIP2h = 50.6 dBm achieved by optimizing the bias voltage.

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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 OIP2h (dB-for-dB). This is shown in Fig. 2(b). The third-harmonic output intercept (blue circles) remains fixed at OIP3h ∼ −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 < OIP2h < 50.6 dBm) as the bias frequency is varied. Once again, the maximum intercept point of OIP2h = 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 OIP2h,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 (ϕb = 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 (fbfo) determined from the measured second-harmonic data and the corresponding average frequency (favgfo) determined from the optical comb are given in Table 1. The error is defined as Error = 100 × |favgfb|/Δfrms, where Δfrms 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.

 

Fig. 3 Top row: Normalized second-harmonic power as a function of bias frequency. The measured data are shown in red, calculated values [Eq. (11)] are shown by the black lines. Bottom row: Optical combs corresponding to data in the top row. Here, the average frequency favg is shown by the dashed red line. (a) Asymmetric comb weighted toward low offset frequencies (favgfo ≈ −108.64 GHz). (b) Symmetric comb (favgfo ≈ +5.16 GHz). Asymmetric comb weighted toward high offset frequencies (favgfo ≈ +111.57 GHz)

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Tables Icon

Table 1. Comparison of minimum second-harmonic distortion frequencies (relative to the laser center frequency) determined from the measured second-harmonic powers (bias frequency) and the measured optical combs (average frequency) shown in Fig. 3.

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 SFDR2 > SFDR3imd 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 |Δf| = |fbfavg| 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 (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 |fbfavg| < 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.

 

Fig. 4 Contours of fixed dynamic range (labeled in blue) for an intrinsic link as a function average current and receiver bandwidth. The maximum allowable bias frequency offset to maintain a given dynamic range is shown in red. The dashed black line marks the operating current for this work (Iavg = 0.58 mA).

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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

No=Nsh+(1+G)Nth.
Here, Nth = 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)
Nsh=14×2qIavgRo
where q is the magnitude of the electronic charge. The link gain G is given by
G=14×(πIavgVπ)2RoRi,
where Ri 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 T = 290 K and Ri = 50Ω.

References and links

1. J. D. McKinney, V. J. Urick, and J. Briguglio, “Optical comb sources for high dynamic range single-span long-haul analog optical links,” IEEE Trans. Microwave Theory Tech. 59, 3249–3257 (2011). [CrossRef]  

2. B. C. Pile and G. W. Taylor, “Performance of subsampled analog optical links,” J. Lightwave Technol. 30, 1299–1305 (2012). [CrossRef]  

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

4. J. D. McKinney and K. J. Williams, “Sampled analog optical links,” IEEE Trans. Microwave Theory Tech. 57, 2093–2099 (2009). [CrossRef]  

5. V. J. Urick, F. Bucholtz, J. D. McKinney, P. S. Devgan, A. L. Campillo, J. L. Dexter, and K. J. Williams, “Long-haul analog photonics,” J. Lightwave Technol. 29, 1182–1205 (2011). [CrossRef]  

6. V. J. Urick, M. N. Hutchinson, J. M. Singley, J. D. McKinney, and K. J. Williams, “Suppression of even-order photodiode distortions via predistortion linearization with a bias-shifted mach-zehnder modulator,” Opt. Express 21, 14368–14376 (2013). [CrossRef]   [PubMed]  

7. A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, 1991), 3rd ed.

8. D. M. Pozar, Microwave Engineering (John Wiley and Sons, Inc., 2005), 3rd ed.

9. H. Murata, A. Morimoto, T. Kobayashi, and S. Yamamoto, “Optical pulse generation by electrooptic-modulation method and its application to integrated ultrashort pulse generators,” IEEE J. Select. Topics Quantum Electron. 6, 1325–1331 (2000). [CrossRef]  

10. R. Wu, V. R. Supradeepa, C. M. Long, D. E. Leaird, and A. M. Weiner, “Generation of very flat optical frequency combs from continuous-wave lasers using cascaded intensity and phase modulators driven by tailored radio frequency waveforms,” Opt. Lett. 35, 3234–3236 (2010). [CrossRef]   [PubMed]  

11. F. Rahmatian, N. A. F. Jaeger, R. James, and E. Berolo, “An ultrahigh-speed AlGaAs–GaAs polarization converter using slow-wave coplanar electrodes,” IEEE Photon. Technol. Lett. 10, 675–677 (1998). [CrossRef]  

12. K. J. Williams, L. Goldberg, R. D. Esman, M. Dagenais, and J. F. Weller, “6–34 GHz offset phase-locking of Nd:YAG 1319 nm nonplanar ring lasers,” Electron. Lett. 25, 1242–1243 (1989). [CrossRef]  

13. E. Sorokin, G. Tempea, and T. Brabec, “Measurement of the root-mean-square width and the root-mean-square chirp in ultrafast optics,” J. Opt. Soc. Am. B 17, 146–150 (2000). [CrossRef]  

References

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  1. J. D. McKinney, V. J. Urick, and J. Briguglio, “Optical comb sources for high dynamic range single-span long-haul analog optical links,” IEEE Trans. Microwave Theory Tech.59, 3249–3257 (2011).
    [CrossRef]
  2. B. C. Pile and G. W. Taylor, “Performance of subsampled analog optical links,” J. Lightwave Technol.30, 1299–1305 (2012).
    [CrossRef]
  3. B. H. Kolner and D. W. Dolfi, “Intermodulation distortion and compression in an integrated electrooptic modulator,” Appl. Opt.26, 3676–3680 (1987).
    [CrossRef] [PubMed]
  4. J. D. McKinney and K. J. Williams, “Sampled analog optical links,” IEEE Trans. Microwave Theory Tech.57, 2093–2099 (2009).
    [CrossRef]
  5. V. J. Urick, F. Bucholtz, J. D. McKinney, P. S. Devgan, A. L. Campillo, J. L. Dexter, and K. J. Williams, “Long-haul analog photonics,” J. Lightwave Technol.29, 1182–1205 (2011).
    [CrossRef]
  6. V. J. Urick, M. N. Hutchinson, J. M. Singley, J. D. McKinney, and K. J. Williams, “Suppression of even-order photodiode distortions via predistortion linearization with a bias-shifted mach-zehnder modulator,” Opt. Express21, 14368–14376 (2013).
    [CrossRef] [PubMed]
  7. A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, 1991), 3rd ed.
  8. D. M. Pozar, Microwave Engineering (John Wiley and Sons, Inc., 2005), 3rd ed.
  9. H. Murata, A. Morimoto, T. Kobayashi, and S. Yamamoto, “Optical pulse generation by electrooptic-modulation method and its application to integrated ultrashort pulse generators,” IEEE J. Select. Topics Quantum Electron.6, 1325–1331 (2000).
    [CrossRef]
  10. R. Wu, V. R. Supradeepa, C. M. Long, D. E. Leaird, and A. M. Weiner, “Generation of very flat optical frequency combs from continuous-wave lasers using cascaded intensity and phase modulators driven by tailored radio frequency waveforms,” Opt. Lett.35, 3234–3236 (2010).
    [CrossRef] [PubMed]
  11. F. Rahmatian, N. A. F. Jaeger, R. James, and E. Berolo, “An ultrahigh-speed AlGaAs–GaAs polarization converter using slow-wave coplanar electrodes,” IEEE Photon. Technol. Lett.10, 675–677 (1998).
    [CrossRef]
  12. K. J. Williams, L. Goldberg, R. D. Esman, M. Dagenais, and J. F. Weller, “6–34 GHz offset phase-locking of Nd:YAG 1319 nm nonplanar ring lasers,” Electron. Lett.25, 1242–1243 (1989).
    [CrossRef]
  13. E. Sorokin, G. Tempea, and T. Brabec, “Measurement of the root-mean-square width and the root-mean-square chirp in ultrafast optics,” J. Opt. Soc. Am. B17, 146–150 (2000).
    [CrossRef]

2013 (1)

2012 (1)

2011 (2)

J. D. McKinney, V. J. Urick, and J. Briguglio, “Optical comb sources for high dynamic range single-span long-haul analog optical links,” IEEE Trans. Microwave Theory Tech.59, 3249–3257 (2011).
[CrossRef]

V. J. Urick, F. Bucholtz, J. D. McKinney, P. S. Devgan, A. L. Campillo, J. L. Dexter, and K. J. Williams, “Long-haul analog photonics,” J. Lightwave Technol.29, 1182–1205 (2011).
[CrossRef]

2010 (1)

2009 (1)

J. D. McKinney and K. J. Williams, “Sampled analog optical links,” IEEE Trans. Microwave Theory Tech.57, 2093–2099 (2009).
[CrossRef]

2000 (2)

H. Murata, A. Morimoto, T. Kobayashi, and S. Yamamoto, “Optical pulse generation by electrooptic-modulation method and its application to integrated ultrashort pulse generators,” IEEE J. Select. Topics Quantum Electron.6, 1325–1331 (2000).
[CrossRef]

E. Sorokin, G. Tempea, and T. Brabec, “Measurement of the root-mean-square width and the root-mean-square chirp in ultrafast optics,” J. Opt. Soc. Am. B17, 146–150 (2000).
[CrossRef]

1998 (1)

F. Rahmatian, N. A. F. Jaeger, R. James, and E. Berolo, “An ultrahigh-speed AlGaAs–GaAs polarization converter using slow-wave coplanar electrodes,” IEEE Photon. Technol. Lett.10, 675–677 (1998).
[CrossRef]

1989 (1)

K. J. Williams, L. Goldberg, R. D. Esman, M. Dagenais, and J. F. Weller, “6–34 GHz offset phase-locking of Nd:YAG 1319 nm nonplanar ring lasers,” Electron. Lett.25, 1242–1243 (1989).
[CrossRef]

1987 (1)

Berolo, E.

F. Rahmatian, N. A. F. Jaeger, R. James, and E. Berolo, “An ultrahigh-speed AlGaAs–GaAs polarization converter using slow-wave coplanar electrodes,” IEEE Photon. Technol. Lett.10, 675–677 (1998).
[CrossRef]

Brabec, T.

Briguglio, J.

J. D. McKinney, V. J. Urick, and J. Briguglio, “Optical comb sources for high dynamic range single-span long-haul analog optical links,” IEEE Trans. Microwave Theory Tech.59, 3249–3257 (2011).
[CrossRef]

Bucholtz, F.

Campillo, A. L.

Dagenais, M.

K. J. Williams, L. Goldberg, R. D. Esman, M. Dagenais, and J. F. Weller, “6–34 GHz offset phase-locking of Nd:YAG 1319 nm nonplanar ring lasers,” Electron. Lett.25, 1242–1243 (1989).
[CrossRef]

Devgan, P. S.

Dexter, J. L.

Dolfi, D. W.

Esman, R. D.

K. J. Williams, L. Goldberg, R. D. Esman, M. Dagenais, and J. F. Weller, “6–34 GHz offset phase-locking of Nd:YAG 1319 nm nonplanar ring lasers,” Electron. Lett.25, 1242–1243 (1989).
[CrossRef]

Goldberg, L.

K. J. Williams, L. Goldberg, R. D. Esman, M. Dagenais, and J. F. Weller, “6–34 GHz offset phase-locking of Nd:YAG 1319 nm nonplanar ring lasers,” Electron. Lett.25, 1242–1243 (1989).
[CrossRef]

Hutchinson, M. N.

Jaeger, N. A. F.

F. Rahmatian, N. A. F. Jaeger, R. James, and E. Berolo, “An ultrahigh-speed AlGaAs–GaAs polarization converter using slow-wave coplanar electrodes,” IEEE Photon. Technol. Lett.10, 675–677 (1998).
[CrossRef]

James, R.

F. Rahmatian, N. A. F. Jaeger, R. James, and E. Berolo, “An ultrahigh-speed AlGaAs–GaAs polarization converter using slow-wave coplanar electrodes,” IEEE Photon. Technol. Lett.10, 675–677 (1998).
[CrossRef]

Kobayashi, T.

H. Murata, A. Morimoto, T. Kobayashi, and S. Yamamoto, “Optical pulse generation by electrooptic-modulation method and its application to integrated ultrashort pulse generators,” IEEE J. Select. Topics Quantum Electron.6, 1325–1331 (2000).
[CrossRef]

Kolner, B. H.

Leaird, D. E.

Long, C. M.

McKinney, J. D.

V. J. Urick, M. N. Hutchinson, J. M. Singley, J. D. McKinney, and K. J. Williams, “Suppression of even-order photodiode distortions via predistortion linearization with a bias-shifted mach-zehnder modulator,” Opt. Express21, 14368–14376 (2013).
[CrossRef] [PubMed]

J. D. McKinney, V. J. Urick, and J. Briguglio, “Optical comb sources for high dynamic range single-span long-haul analog optical links,” IEEE Trans. Microwave Theory Tech.59, 3249–3257 (2011).
[CrossRef]

V. J. Urick, F. Bucholtz, J. D. McKinney, P. S. Devgan, A. L. Campillo, J. L. Dexter, and K. J. Williams, “Long-haul analog photonics,” J. Lightwave Technol.29, 1182–1205 (2011).
[CrossRef]

J. D. McKinney and K. J. Williams, “Sampled analog optical links,” IEEE Trans. Microwave Theory Tech.57, 2093–2099 (2009).
[CrossRef]

Morimoto, A.

H. Murata, A. Morimoto, T. Kobayashi, and S. Yamamoto, “Optical pulse generation by electrooptic-modulation method and its application to integrated ultrashort pulse generators,” IEEE J. Select. Topics Quantum Electron.6, 1325–1331 (2000).
[CrossRef]

Murata, H.

H. Murata, A. Morimoto, T. Kobayashi, and S. Yamamoto, “Optical pulse generation by electrooptic-modulation method and its application to integrated ultrashort pulse generators,” IEEE J. Select. Topics Quantum Electron.6, 1325–1331 (2000).
[CrossRef]

Papoulis, A.

A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, 1991), 3rd ed.

Pile, B. C.

Pozar, D. M.

D. M. Pozar, Microwave Engineering (John Wiley and Sons, Inc., 2005), 3rd ed.

Rahmatian, F.

F. Rahmatian, N. A. F. Jaeger, R. James, and E. Berolo, “An ultrahigh-speed AlGaAs–GaAs polarization converter using slow-wave coplanar electrodes,” IEEE Photon. Technol. Lett.10, 675–677 (1998).
[CrossRef]

Singley, J. M.

Sorokin, E.

Supradeepa, V. R.

Taylor, G. W.

Tempea, G.

Urick, V. J.

Weiner, A. M.

Weller, J. F.

K. J. Williams, L. Goldberg, R. D. Esman, M. Dagenais, and J. F. Weller, “6–34 GHz offset phase-locking of Nd:YAG 1319 nm nonplanar ring lasers,” Electron. Lett.25, 1242–1243 (1989).
[CrossRef]

Williams, K. J.

Wu, R.

Yamamoto, S.

H. Murata, A. Morimoto, T. Kobayashi, and S. Yamamoto, “Optical pulse generation by electrooptic-modulation method and its application to integrated ultrashort pulse generators,” IEEE J. Select. Topics Quantum Electron.6, 1325–1331 (2000).
[CrossRef]

Appl. Opt. (1)

Electron. Lett. (1)

K. J. Williams, L. Goldberg, R. D. Esman, M. Dagenais, and J. F. Weller, “6–34 GHz offset phase-locking of Nd:YAG 1319 nm nonplanar ring lasers,” Electron. Lett.25, 1242–1243 (1989).
[CrossRef]

IEEE J. Select. Topics Quantum Electron. (1)

H. Murata, A. Morimoto, T. Kobayashi, and S. Yamamoto, “Optical pulse generation by electrooptic-modulation method and its application to integrated ultrashort pulse generators,” IEEE J. Select. Topics Quantum Electron.6, 1325–1331 (2000).
[CrossRef]

IEEE Photon. Technol. Lett. (1)

F. Rahmatian, N. A. F. Jaeger, R. James, and E. Berolo, “An ultrahigh-speed AlGaAs–GaAs polarization converter using slow-wave coplanar electrodes,” IEEE Photon. Technol. Lett.10, 675–677 (1998).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (2)

J. D. McKinney, V. J. Urick, and J. Briguglio, “Optical comb sources for high dynamic range single-span long-haul analog optical links,” IEEE Trans. Microwave Theory Tech.59, 3249–3257 (2011).
[CrossRef]

J. D. McKinney and K. J. Williams, “Sampled analog optical links,” IEEE Trans. Microwave Theory Tech.57, 2093–2099 (2009).
[CrossRef]

J. Lightwave Technol. (2)

J. Opt. Soc. Am. B (1)

Opt. Express (1)

Opt. Lett. (1)

Other (2)

A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, 1991), 3rd ed.

D. M. Pozar, Microwave Engineering (John Wiley and Sons, Inc., 2005), 3rd ed.

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Figures (4)

Fig. 1
Fig. 1

(a) Schematic of the optical comb-based analog link (PMCIM: polarization mode converter-based intensity modulator; ESA: electrical spectrum analyzer). (b) Example optical comb (gray) and filtered bias frequency lines corresponding to the −10, 0, and +10 order comblines (red, black, and blue, respectively).

Fig. 2
Fig. 2

(a) Measured RF power in the fundamental (f1 = 500 MHz, black), second-harmonic (2f1 = 1000 MHz, red) and third-harmonic (3f1 = 1500 MHz) as the bias frequency is varied across a symmetric optical comb [bottom, Fig. 3 (b)]. The dashed black line shows the minimum second-harmonic power obtained by optimizing the bias voltage (∼ −112 dBm) and the solid green, black, and gray lines show the theoretical values calculated from Eqs. (21)(23). (b) Output intercept points (OIP2h: red, OIP3h: blue) corresponding to the measured RF powers in (a). The solid black, solid gray, and dashed gray curves show the theoretical values calculated from Eqs. (25)(27). The dashed black line shows the maximum OIP2h = 50.6 dBm achieved by optimizing the bias voltage.

Fig. 3
Fig. 3

Top row: Normalized second-harmonic power as a function of bias frequency. The measured data are shown in red, calculated values [Eq. (11)] are shown by the black lines. Bottom row: Optical combs corresponding to data in the top row. Here, the average frequency favg is shown by the dashed red line. (a) Asymmetric comb weighted toward low offset frequencies (favgfo ≈ −108.64 GHz). (b) Symmetric comb (favgfo ≈ +5.16 GHz). Asymmetric comb weighted toward high offset frequencies (favgfo ≈ +111.57 GHz)

Fig. 4
Fig. 4

Contours of fixed dynamic range (labeled in blue) for an intrinsic link as a function average current and receiver bandwidth. The maximum allowable bias frequency offset to maintain a given dynamic range is shown in red. The dashed black line marks the operating current for this work (Iavg = 0.58 mA).

Tables (1)

Tables Icon

Table 1 Comparison of minimum second-harmonic distortion frequencies (relative to the laser center frequency) determined from the measured second-harmonic powers (bias frequency) and the measured optical combs (average frequency) shown in Fig. 3.

Equations (32)

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P ( ω ) = δ ( ω ω o ) * n = 0 N 1 P n δ ( ω n ω rep )
i total ( t ) = n = 0 N 1 i n ( t ) .
i ( t ) = p ( t ) * α pd h pd ( t )
d t h pd ( t ) = α pd .
i fund , n = α pd P n ( π V o V π ) | sin ϕ b | ,
i 2 h , n = α pd P n 1 4 ( π V o V π ) 2 | cos ϕ b | ,
i 3 h , n = α pd P n 1 24 ( π V o V π ) 3 | sin ϕ b |
i 3 imd , n = α pd P n 1 8 ( π V o V π ) 3 | sin ϕ b | .
Δ ϕ b , n = ( f n f b ) d ϕ b d f | f = f b .
i fund , n = α pd P n ( π V o V π ) ,
i 2 h , n = α pd P n 1 4 ( π V o V π ) 2 | Δ ϕ b , n | ,
i 3 h , n = α pd P n 1 24 ( π V o V π ) 3
i 3 imd , n = α pd P n 1 8 ( π V o V π ) 3 .
i fund = I avg ( π V o V π ) ,
i 3 h = I avg 1 24 ( π V o V π ) 3
i 3 imd = I avg 1 8 ( π V o V π ) 3 ,
I avg = α pd n = 0 N 1 P n .
i 2 h = I avg 1 4 ( π V o V π ) 2 | ( f b f avg ) d ϕ b d f | ,
f avg = n = 0 N 1 f n P n n = 0 N 1 P n .
OIP k = ( P fund k P k ) 1 / ( k 1 ) .
P fund = 1 8 I avg 2 R o ( π V o V π ) 2 ,
P 2 h = 1 128 I avg 2 R o ( π V o V π ) 4 | ( f b f avg ) d ϕ b d f | 2 ,
P 3 h = 1 4608 I avg 2 R o ( π V o V π ) 6
P 3 imd = 1 512 I avg 2 R o ( π V o V π ) 6 .
OIP 2 h = 2 I avg 2 R o | ( f b f avg ) d ϕ b d f | 2 ,
OIP 3 h = 3 I avg 2 R o ,
OIP 3 imd = I avg 2 R o .
SFDR = ( OIP k N o × B e ) ( k 1 ) / k .
| Δ f | < 2 ( SFDR 3 imd ) 1 / 4 × | d ϕ b d f | 1 .
N o = N sh + ( 1 + G ) N th .
N sh = 1 4 × 2 q I avg R o
G = 1 4 × ( π I avg V π ) 2 R o R i ,

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