## Bias-dependent distortion in optical comb-based analog optical links |

Optics Express, Vol. 21, Issue 20, pp. 23695-23705 (2013)

http://dx.doi.org/10.1364/OE.21.023695

Acrobat PDF (1320 KB)

### 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

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]

*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

*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

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]

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]

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]

*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

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]

*f*=

*f*−

*f*around the bias frequency

_{b}*f*such that the change in bias phase is small, we may expand the trigonometric functions in Eqs. (5)–(8) about

_{b}*ϕ*= (2

_{b}*m*+ 1)

*π*/2. The resulting photocurrent expressions are then Note, we are not considering cancellation effects (e.g., between modulator- and photodiode-induced distortion [6

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]

*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 where the average frequency of the comb is given by [7] 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.

*k*-th order nonlinearity is (in terms of the fundamental and

*k*-th order distortion output powers,

*P*

_{fund}and

*P*) Calculating the RF power delivered to a load resistance

_{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 The corresponding second-harmonic (OIP

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

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

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

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]

*N*(W/Hz) and a receiver electrical bandwidth of

_{o}*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) 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 (Δ

*f*=

*f*−

_{b}*f*

_{avg}) subject to the condition SFDR

_{2}> SFDR

_{3imd}yields the result 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

*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).

_{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

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]

_{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.

*ϕ*= 0,

_{b}*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

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]

*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

## 5. Appendix

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]

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]

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

*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.

_{i}## 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. |

2. | B. C. Pile and G. W. Taylor, “Performance of subsampled analog optical links,” J. Lightwave Technol. |

3. | B. H. Kolner and D. W. Dolfi, “Intermodulation distortion and compression in an integrated electrooptic modulator,” Appl. Opt. |

4. | J. D. McKinney and K. J. Williams, “Sampled analog optical links,” IEEE Trans. Microwave Theory Tech. |

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. |

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 |

7. | A. Papoulis, |

8. | D. M. Pozar, |

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. |

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. |

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. |

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. |

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 |

**OCIS Codes**

(060.2360) Fiber optics and optical communications : Fiber optics links and subsystems

(350.4010) Other areas of optics : Microwaves

**ToC Category:**

Fiber Optics and Optical Communications

**History**

Original Manuscript: August 12, 2013

Revised Manuscript: September 9, 2013

Manuscript Accepted: September 9, 2013

Published: September 27, 2013

**Citation**

Jason D. McKinney, Vincent J. Urick, and Alexander S. Hastings, "Bias-dependent distortion in optical comb-based analog optical links," Opt. Express **21**, 23695-23705 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-20-23695

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### References

- 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]
- B. C. Pile and G. W. Taylor, “Performance of subsampled analog optical links,” J. Lightwave Technol.30, 1299–1305 (2012). [CrossRef]
- B. H. Kolner and D. W. Dolfi, “Intermodulation distortion and compression in an integrated electrooptic modulator,” Appl. Opt.26, 3676–3680 (1987). [CrossRef] [PubMed]
- J. D. McKinney and K. J. Williams, “Sampled analog optical links,” IEEE Trans. Microwave Theory Tech.57, 2093–2099 (2009). [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]
- 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]
- 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.
- 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]
- 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]
- 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]
- 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]
- 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]

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