## Dynamic range enhancement of a novel phase-locked coherent optical phase demodulator

Optics Express, Vol. 15, Issue 1, pp. 33-44 (2007)

http://dx.doi.org/10.1364/OE.15.000033

Acrobat PDF (296 KB)

### Abstract

We report on a novel cancellation technique, for reducing the nonlinearity associated with the tracking phase-modulator in recently proposed phase-locked coherent demodulator for phase modulated analog optical links. The proposed cancellation technique is input RF signal power and frequency independent leading to a significant increase in dynamic range of the coherent demodulator. Furthermore, this technique demonstrates that large values of the signal-to-intermodulation ratio of the demodulated signal can be obtained even though the tracking phase modulator is fairly nonlinear, and thereby relaxing the linearity requirements for the tracking phase modulator. A new model is developed and the calculated results are in good agreement with measurements.

© 2007 Optical Society of America

## 1. Introduction

1. C. H. Cox, E. I. Ackerman, G. E. Bets, and J. L. Prince, “Limits on performance of RF-over-fibre links and their impact on device design,” IEEE Trans. on Microwave Theory Tech. **54**, Part 2,906–920 (2006) [CrossRef]

2. Alwyn J. Seeds, “Microwave photonics,” IEEE Trans. on Microwave Theory Tech. **50**,877–887 (2002) [CrossRef]

1. C. H. Cox, E. I. Ackerman, G. E. Bets, and J. L. Prince, “Limits on performance of RF-over-fibre links and their impact on device design,” IEEE Trans. on Microwave Theory Tech. **54**, Part 2,906–920 (2006) [CrossRef]

3. R.F. Kalman, J.C. Fan, and L.G. Kazovsky, “Dynamic range of coherent analog fiber-optic links,” IEEE J. Light-wave Technol. **12**,1263–1277 (1994) [CrossRef]

4. H. F. Chou, A. Ramaswamy, D. Zibar, L.A. Johansson, L. Coldren, and J. Bowers, “SFDR Improvement of a Coherent Receiver Using Feedback,” in *Optical Amplifiers and Their Applications/Coherent Optical Technologies and Applications, Technical Digest (CD) (Optical Society of America, 2006)*, paper CFA3.

7. D. Zibar, L. A. Johansson, H. F. Chou, A. Ramaswamy, and J. E. Bowers, “Time Domain Analysis of a Novel Phase-Locked Coherent Optical Demodulator,” in *Optical Amplifiers and Their Applications/Coherent Optical Technologies and Applications, Technical Digest (CD) (Optical Society of America, 2006)*, paper JWB11.

4. H. F. Chou, A. Ramaswamy, D. Zibar, L.A. Johansson, L. Coldren, and J. Bowers, “SFDR Improvement of a Coherent Receiver Using Feedback,” in *Optical Amplifiers and Their Applications/Coherent Optical Technologies and Applications, Technical Digest (CD) (Optical Society of America, 2006)*, paper CFA3.

9. M. N. Sysak, L. A. Johannson, J. Klamkin, L. A. Coldren, and J. E. Bowers, “Characterization of Distortion in In-GaAsP Optical Phase Modulators Monolithically Integrated with Balanced UTC Photodetector”, *in Proceedings of IEEE Lasers and Electro-Optics Society (LEOS) 19th Annual Meeting 2006*, Montreal, Canada, paper TuU2, (2006)

1. C. H. Cox, E. I. Ackerman, G. E. Bets, and J. L. Prince, “Limits on performance of RF-over-fibre links and their impact on device design,” IEEE Trans. on Microwave Theory Tech. **54**, Part 2,906–920 (2006) [CrossRef]

## 2. Novelty of the work

8. C. Cox, *Analog optical links*, (Cambridge, U.K. Cambidge Univ. Press, 2004) [CrossRef]

8. C. Cox, *Analog optical links*, (Cambridge, U.K. Cambidge Univ. Press, 2004) [CrossRef]

## 3. Model set-up

*V*(

_{in}*t*) is used to directly modulate an optical phase modulator at the remote antenna unit. The corresponding optical signal

*E*(

_{in}*t*), see Fig. 1, is then written in complex notation as 1:

*ω*

_{0}is the optical frequency and

*P*is the power of the optical field. Taking into consideration the nonlinearities associated with the (input) phase-modulator located at the remote antenna unit the phase of the optical signal,

_{in}*φ*(

_{in}*t*), is expressed as:

*V*

_{π,in}is the voltage of the input phase-modulator, in order to obtain

*π*phase shift and

*a*

_{1},

*a*

_{2}and

*a*

_{3}represent the terms of the polynomial expansion of the input modulator nonlinear phase response. In order to characterize dynamic range of the demodulator, the input RF signal

*V*(

_{in}*t*) is assumed to consist of relatively closely spaced tones [10]:

*V*

_{1}and

*V*

_{2}are the amplitudes of the input RF signals and

*ω*

_{1}and

*ω*

_{2}are the input RF signal frequencies. The optical signal

*E*(

_{in}*t*) is then transported to the receiver unit where its phase, is compared to the phase of the local optical signal

*E*(

_{LO}*t*), using the balanced detector pair with load resistance

*R*. A single optical source is used for both the remote antenna and the receiver unit. The optical LO signal,

_{L}*E*(

_{LO}*t*), is thereby expressed as:

*ϕ*(

_{LO}*t*) is reference phase (signal) and is function of the feedback loop parameters, see Fig. 1 and

*P*is the power of the optical field2. Following Fig. 1 after the 3-dB coupler, we have in one arm:

_{LO}*I*

_{1}(

*t*) and

*I*

_{2}(

*t*), containing the phase difference between

*ϕ*(

_{in}*t*) and

*ϕ*(

_{LO}*t*) are then expressed as:

*R*is the responsivity of the photodetectors and is assumed equal for both photodetectors.

_{pd}*b*represent the terms of polynomial expansion of the non-linear response of the photodiodes and

_{n}*n*is an integer. In practise, it is only necessary to consider

*n*=1..3. The output signal from the balanced photodetector pair with load resistance

*R*contains the phase difference between

_{L}*ϕ*(

_{in}*t*) and

*ϕ*(

_{LO}*t*) is expressed as:

*V*(

_{pd}*t*) is then used to control the feedback loop. (After the loop has acquired lock, the phase difference

*ϕ*(

_{in}*t*)-

*ϕ*(

_{LO}*t*) will approach zero.) The signal

*V*(

_{pd}*t*) is then passed through the loop filter (low pass) and amplified:

*τ*=1/2

_{LF}*πf*is inversely proportional to the bandwidth of the loop filter and

_{LF}*A*is the gain of the loop filter.

*V*(

_{out}*t*) is the output of the loop filter and the desired demodulated RF signal.

*V*(

_{out}*t*) is then applied to the tracking LO phase modulator. The phase vs. voltage characteristic of the LO phase modulator is nonlinear. In practice, for the semiconductor phase modulators the quadratic and cubic nonlinearity terms will dominate over the higher order terms, and the phase-voltage relation can thereby be expressed as:

*c*

_{1},

*c*

_{2}and

*c*

_{3}represent the terms of the polynomial expansion of the LO phase modulator response. In addition to nonlinearity associated with phase vs. voltage characteristic of the LO phase-modulator, any residual amplitude modulation, as would be expected in practice, may affect the performance of the demodulator in an adverse way. The normalized E-field amplitude of the (optical) LO signal can therefore be expressed as:

*A*

_{0}if the E-field amplitude of the LO signal in the absence of amplitude modulation. In order to determine the overall dynamical response of the loop, the total phase error is defined as:

*V*(

_{out}*t*), is obtained characterizing the overall nonlinear response of the loop. Eq. (14) and (10) are first order non-linear differential equations and their solutions can be obtained numerically.

*V*(

_{out}*t*) is then characterized by the Signal-to-Intermodulation Ratio (SIR) which is the ratio between the power of the demodulated signal (

*ω*

_{1}or

*ω*

_{2}) and 3rd order mixing product (2

*ω*

_{1}-

*ω*

_{2},2

*ω*

_{2}-

*ω*

_{1}). Loop gain is defined as:

## 4. Linearity analysis based on perturbation theory

*V*is assumed equal for the input and tracking LO phase modulator.

_{π}*V*(

_{ref}*t*) denote the signal incident at the tracking LO phase modulator when the loop is open:

*c*′

_{2}=

*c*

_{2}/(

*c*

_{1}

*V*

_{π,LO}),

*c*′

_{3}=

*c*

_{3}/(

*c*

_{1}

*V*

^{2}

_{π,LO}) and

*ϕ*

_{0}is a constant.

*V*(

_{ref}*t*) is a nonlinear function), the output signal of the demodulator,

*V*(

_{out}*t*), after locking the loop can be approximated as [10]:

*A*

_{1},

*A*

_{2}and

*A*

_{3}are constants. We assume that 3rd mixing product

*A*

_{3}

*V*

^{3}

*(*

_{in}*t*) will have larger impact on the SFDR than the second order mixing product

*A*

_{2}

*V*

^{2}

_{in}(

*t*) and we chose therefore not to consider the second order mixing product. Furthermore, the input signal

*V*(

_{in}*t*) consists of closely spaced tones:

*V*(

_{in}*t*)=

*V*

_{1}sin[

*ω*

_{1}

*t*]+

*V*

_{1}sin[

*ω*

_{2}

*t*]. In order to find

*A*

_{1}and

*A*

_{3}, we lock the loop

*V*(

_{ref}*t*)=

*V*(

_{out}*t*) and insert Eq. (16) in (15). Using the method of harmonic balance the coefficients

*A*

_{1}and

*A*

_{3}are found. The demodulated signal,

*V*(

_{out}*t*) is then expressed as:

*c*′

_{2}by adjusting the loop parameters, the 3rd order mixing product of the demodulated signal can be (theoretically) brought to zero, i.e.

*A*

_{3}=0. In other words the feedback circuit in combination with second order nonlinearity associated with tracking LO phase modulator response, results into cancellation of 3rd order mixing product of the demodulated signal.

*V*(

_{ref}*t*)=

*V*(

_{out}*t*), the output signal can be expressed as:

## 5. Experimental results

4. H. F. Chou, A. Ramaswamy, D. Zibar, L.A. Johansson, L. Coldren, and J. Bowers, “SFDR Improvement of a Coherent Receiver Using Feedback,” in

*f*

_{1}=150 kHz and the loop filter bandwidth is 1.1 MHz. The amplitude of the signal after balanced photodetection,

*V*(

_{pd}*t*), is plotted as a function of the loop gain,

*K*. Experimental and simulation results show that as the loop gain is increased, the amplitude of

*V*(

_{pd}*t*) is reduced, i.e. the linearity of the demodulator is improved. Good agreement between the experimental and simulation results is obtained for one tone measurement. In Fig. 2 (b), results of the two tone measurement are shown together with the simulation results. The SIR is plotted as a function of the modulation depth,

*M*=(

_{in}*π*/

*V*

_{π,in})

*V*

_{1}, of the input RF signal.

*V*

_{1}is the amplitude of the input RF signal and is assumed equal for both tones. The input RF signal frequencies are:

*f*

_{1}=150 kHz and

*f*

_{2}=170 kHz. As expected, the SIR decreases as

*M*is increased. Once again good agreement between the model and experimental results is observed.

_{in}## 6. Effects of loop gain and LO phase-modulator nonlinearities

*π*/2. As mentioned in the introduction the modulator distortion, especially of the tracking LO phase modulator in the considered case, will usually dominate over the photodiode distortion. We therefore assume that the tracking LO phase modulator is much more nonlinear than the photodiodes, i.e.

*c*

_{2}/

*b*

_{2}>>1 and

*c*

_{3}/

*b*

_{3}>>1. Furthermore, electronics nonlinearities can be suppressed by the feedback loop and only need to be lower than the nonlinearities of the tracking LO phase modulator response. In contrast, the tracking LO phase modulator nonlinearities are not suppressed and must therefore be carefully considered. However, a linear input phase modulator is considered.

*f*/

_{LF}*f*

_{1}), is varied. Input RF signal

*V*(

_{in}*t*) includes two closely spaced frequencies:

*ω*

_{1}/2

*π*=

*f*

_{1}and

*ω*

_{2}/2

*π*=

*f*

_{2}, as shown in Eq. (3). Linear tracking LO phase-modulator is assumed in Fig. 3 (a). The intermodulation is the magnitude of the mixing terms (2

*f*

_{1}-

*f*

_{2},2

*f*

_{2}-

*f*

_{1}).

*K*is increased, the performance of the phase-locked demodulator improves in terms of SIR, i.e. the SIR of the demodulated signal increases. As the ratio, (

*f*/

_{LF}*f*

_{1}), is significantly increased, the SIR converges. Furthermore, the slope of the SIR line is approximately 3. As observed in Fig. 3 (a) relatively large values of the SIR can be obtained provided large loop gain and linear tracking LO phase-modulator. Using Eq. (17) and setting

*c*

_{2},

*c*

_{3}=0, the expression for the SIR can be obtained:

*c*

_{1}) and cubic term (

*c*

_{3}) of the LO phase-modulator response for selected values of the loop gain. In general, the SIR decreases as the ratio

*c*

_{1}/

*c*

_{3}decreases. The values of

*c*

_{1}/

*c*

_{3}for which SIR starts to decrease are loop gain dependent since the nonlinearities of the LO phase-modulator become more enhanced as the loop gain is increased. This is also in accordance with Eq. (17), i.e. as the loop gain is increased 3rd order mixing product increases as well. For the loop gain of

*K*=10∙10

^{7}rad/s, the ratio

*c*

_{1}/

*c*

_{3}needs to be > 200, to maintain the SIR of 90 dB. Very recently, the the ratio between cubic term and linear term

*c*

_{1}/

*c*

_{3}of the semiconductor phase modulator has been measured to be 26 [9

9. M. N. Sysak, L. A. Johannson, J. Klamkin, L. A. Coldren, and J. E. Bowers, “Characterization of Distortion in In-GaAsP Optical Phase Modulators Monolithically Integrated with Balanced UTC Photodetector”, *in Proceedings of IEEE Lasers and Electro-Optics Society (LEOS) 19th Annual Meeting 2006*, Montreal, Canada, paper TuU2, (2006)

*c*

_{1}/

*c*

_{3}=200.

*c*

_{2}of the nonlinear response of the phase modulator can be used to cancel out 3rd order mixing product of the demodulated signal.

*c*

_{1}/

*c*

_{3}and

*c*

_{1}/

*c*

_{2}in order to obtain SIR peaking, so we need to investigate what happens if we are slightly off. In Fig. 4 (b), the SIR is computed as a function of loop gain when the ratio

*c*

_{1}/

*c*

_{3}is varied from the exact value of

*c*

_{1}/

*c*

_{3}for which the SIR peaking is obtained, i.e.

*c*

_{1}/

*c*

_{3}=20+Δ

_{offset}. The ratio

*c*

_{1}/

*c*

_{2}=40 is held constant. Fig. 4 (b) shows that the SIR peaking is dependent on the loop gain and it occurs in a relatively wide band of the loop gain. It is also noticed that as the ratio

*c*

_{1}/

*c*

_{3}is varied, the resonant peak of the SIR moves as well. So, by adjusting the loop gain resonant peaking of the SIR can be re-obtained. Another, thing which should be addressed is frequency dependence of

*c*

_{3}if the demodulator is operated over wide frequency range. Frequency dependence will cause the

*c*

_{1}/

*c*

_{3}to vary, and Fig. 4 (b) can be used to observe the effect of varying

*c*

_{1}/

*c*

_{3}. If the ratio

*c*

_{1}/

*c*

_{3}varies with frequency for a specific loop gain, we will move away from the resonant peaking of the SIR. One solution could be to re-adjust the loop gain or to design wide band tracking LO phase modulator. Furthermore, the demodulator could be designed to operate in narrow frequency band.

## 7. Effects of residual amplitude modulation

*D*where

_{x}*x*=1,2 and 3. We assume that terms

*D*where

_{x}*x*> 3 are negligible, as it would be expected in practice. Fig. 5 shows that as

*D*

_{1}(

*D*=0,

_{x}*x*=2,3) is increased beyond 5∙10

^{-3}1/V, the SIR starts to decrease. When

*D*

_{2}is varied (

*D*=0,

_{x}*x*=1,3) resonant behavior of the SIR, similar to Fig. 4(b), is observed, i.e. there exist a value of

*D*

_{2}for which the 3rd order mixing product is minimized. Furthermore, we observe, Fig. 5 (a), that the effect of

*D*

_{1}is more deteriorating than that of

*D*

_{2}. For the case when

*D*

_{3}is varied (

*D*=0,

_{x}*x*=1,2), the SIR is not affected and this is in accordance with Eq. (18). Usually, for the semiconductor phase modulator, the quadratic term of the amplitude modulation is more difficult to minimize than the linear term. We therefore need to concentrate on the quadratic term of amplitude modulation.

## 8. Combined effects of nonlinearities and their cancellation

*D*

_{2}, of the residual amplitude modulation since it is more difficult to reduce (in practice) for semiconductor phase modulators, compared to the linear term,

*D*

_{1}.

*c*

_{1}/

*c*

_{3}when

*D*

_{2}is varied from 10

^{-3}1/V to 10

^{-1}1/V. The ratio

*c*

_{1}/

*c*

_{2}is set to 40 and

*D*

_{1}=0.03 1/V. It is observed from Fig. 5 (b) that for low values of cubic nonlinearity (

*c*

_{3}) of the phase modulator response, the SIR is fully limited by the quadratic term of residual amplitude modulation,

*D*

_{2}. However, Fig. 5(b) also shows that there exists a combination of

*c*

_{3},

*c*

_{2},

*D*

_{1}and

*D*

_{2}for which the 3rd order mixing product is minimized, i.e. peaking (resonance) of the SIR. The resonant peak moves towards lower values of

*c*

_{1}/

*c*

_{3}ratio as

*D*

_{2}is increased. Fig. 5 (b) is in accordance with Eq. (18) which states that 3rd order mixing product can be reduced by proper combination of the loop gain and tracking LO phase modulator nonlinearities. The cascaded sources of nonlinearities associated with balanced detector, phase and amplitude modulation of the tracking LO phase modulator cancel each other. Fig. 5 (b) illustrates that even though the ratio

*c*

_{1}/

*c*

_{3}is low for the measured semiconductor phase modulator (

*c*

_{1}/

*c*

_{3}=26[9]), high values of the SIR are still obtainable.

*f*

_{1}) and amplitude of the 3rd order intermodulation product (2

*f*

_{1}-

*f*

_{2}) as a function of input signal voltage,

*V*. The amplitude of the fundamental and 3rd order intermodulation product (IM

_{in}_{3}curve) of the demodulated signal increases with the input signal voltage, as expected. It should be observed that there are no dips in the IM

_{3}curve as the input signal voltage is varied. This means that the cancellation of the nonlienarities associated with the balanced receiver and phase modulator occurs over broad range of input RF signal powers. In Fig. 6 (b), the SIR is computed as a function of input RF signal frequency as the input signal modulation depth,

*M*takes values from

_{in}*π*to

*π*/7. Fig. 6 (b) shows that the SIR remains constant as the input RF signal frequency is increased. As the input RF signal modulation depth decreases, the SIR increases as expected.

## 9. Conclusion

## Acknowledgments

## Footnotes

The scalar notation is used for both E(_{in}t) and E(_{LO}t) by assuming that the two fields are identically polarized. | |

Due to the residual amplitude modulation of the tracking LO phase modulator P will be time dependent. This is explained in more details later in the text_{LO} |

## References and links

1. | C. H. Cox, E. I. Ackerman, G. E. Bets, and J. L. Prince, “Limits on performance of RF-over-fibre links and their impact on device design,” IEEE Trans. on Microwave Theory Tech. |

2. | Alwyn J. Seeds, “Microwave photonics,” IEEE Trans. on Microwave Theory Tech. |

3. | R.F. Kalman, J.C. Fan, and L.G. Kazovsky, “Dynamic range of coherent analog fiber-optic links,” IEEE J. Light-wave Technol. |

4. | H. F. Chou, A. Ramaswamy, D. Zibar, L.A. Johansson, L. Coldren, and J. Bowers, “SFDR Improvement of a Coherent Receiver Using Feedback,” in |

5. | H. F. Chou, A. Ramaswamy, D. Zibar, L.A. Johansson, J. E. Bowers, M. Rodwell, and L. Coldren, “Highly-linear coherent receiver with feeback,” submitted to IEEE Photon. Technol. Lett. |

6. | H. F Chou, L.A. Johansson, Darko Zibar, A. Ramaswamy, M. Rodwell, and J.E. Bowers, “All-Optical Coherent Receiver with Feedback and Sampling,” |

7. | D. Zibar, L. A. Johansson, H. F. Chou, A. Ramaswamy, and J. E. Bowers, “Time Domain Analysis of a Novel Phase-Locked Coherent Optical Demodulator,” in |

8. | C. Cox, |

9. | M. N. Sysak, L. A. Johannson, J. Klamkin, L. A. Coldren, and J. E. Bowers, “Characterization of Distortion in In-GaAsP Optical Phase Modulators Monolithically Integrated with Balanced UTC Photodetector”, |

10. | David M. Pozar, |

**OCIS Codes**

(060.1660) Fiber optics and optical communications : Coherent communications

(120.5060) Instrumentation, measurement, and metrology : Phase modulation

**ToC Category:**

Fiber Optics and Optical Communications

**History**

Original Manuscript: September 21, 2006

Revised Manuscript: November 15, 2006

Manuscript Accepted: December 16, 2006

Published: January 8, 2007

**Citation**

Darko Zibar, Leif A. Johansson, Hsu-Feng Chou, Anand Ramaswamy, and John E. Bowers, "Dynamic range enhancement of a novel phase-locked coherent optical phase demodulator," Opt. Express **15**, 33-44 (2007)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-1-33

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### References

- C. H. Cox, E. I. Ackerman, G. E. Bets and J. L. Prince,"Limits on performance of RF-over-fibre links and their impact on device design," IEEE Trans. on Microwave Theory Tech. 54, Part 2, 906-920 (2006) [CrossRef]
- AlwynJ. Seeds,"Microwave photonics," IEEE Trans. on Microwave Theory Tech. 50, 877-887 (2002) [CrossRef]
- R.F. Kalman, J.C. Fan and L.G. Kazovsky, "Dynamic range of coherent analog fiber-optic links," IEEE J. Lightwave Technol. 12, 1263-1277 (1994) [CrossRef]
- H. F. Chou, A. Ramaswamy, D. Zibar, L.A. Johansson, L. Coldren and J. Bowers, "SFDR Improvement of a Coherent Receiver Using Feedback," in Optical Amplifiers and Their Applications/Coherent Optical Technologies and Applications, Technical Digest (CD) (Optical Society of America, 2006), paper CFA3.
- <jrn>H. F. Chou, A. Ramaswamy, D. Zibar, L.A. Johansson, J. E. Bowers, M. Rodwell and L. Coldren,"Highly-linear coherent receiver with feeback," submitted to IEEE Photon. Technol. Lett.</jrn>
- H. F Chou, L.A. Johansson, Darko Zibar, A. Ramaswamy, M. Rodwell and J.E. Bowers, "All-Optical Coherent Receiver with Feedback and Sampling," in proceedings of IEEE International Topical Meeting on Microwave Photonics (MWP) 2006, Grenoble France, paper W3.2, (2006)
- D. Zibar, L. A. Johansson, H. F. Chou, A. Ramaswamy and J. E. Bowers, "Time Domain Analysis of a Novel Phase-Locked Coherent Optical Demodulator," in Optical Amplifiers and Their Applications/Coherent Optical Technologies and Applications, Technical Digest (CD) (Optical Society of America, 2006), paper JWB11.
- C. Cox, Analog optical links, (Cambridge, U.K. Cambidge Univ. Press, 2004) [CrossRef]
- M. N. Sysak, L. A. Johannson, J. Klamkin, L. A. Coldren, J. E. Bowers,"Characterization of Distortion in In- GaAsP Optical Phase Modulators Monolithically Integrated with Balanced UTC Photodetector", in Proceedings of IEEE Lasers and Electro-Optics Society (LEOS) 19th Annual Meeting 2006, Montreal, Canada, paper TuU2, (2006)
- David M. Pozar, Microwave engineering, 2nd edition (John Wiley and sons, USA, 1998)

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