## Abstract

The stability of single-sideband (SSB) modulator based recirculating frequency shifter (RFS) is analyzed theoretically. The optimum radio frequency (RF) drive peak-to-peak voltage used to drive the modulator is studied with considering the amplified spontaneous emission (ASE) noise of optical amplifier and crosstalk so as to obtain a maximum overall effective optical signal to noise ratio (OSNR) which is defined to quantify the quality of generated tones. Small desired tones number and lower RF peak-to-peak voltage can reduce the crosstalk effectively. While the trade-off should be considered since the larger desired tones number it is, the higher optimum drive voltage should be used when the SSB-based RFS reached the maximum OSNR. The theoretical results show that the optimum operation condition is helpful to improve the performance of RFS which can be a good application for the T-bit/s optical transmission in practice.

©2010 Optical Society of America

## 1. **Introduction**

In order to support the huge and rapid increase of the capacity in data communication and the increasing demands of rich content such as images and videos, the high-speed large capacity optical communication transmission experiments toward to T-bit/s [1–10] have been demonstrated in recent years. These demonstrations are most implemented by using multi-carrier modulation or dense-wavelength-division-multiplexing (DWDM). Therefore, these experiments need multi-wavelength or multi-subcarrier to carry the large capacity of information. In order to generate multi-wavelength continuous-wave, the most way used in optical communication field is supercontinuum [11] or optical frequency comb [12,13]. However, both of them are difficult to achieve a flat and free-controllable output spectrum in practice. In multi-carrier transmission, it is desired to have frequency-locked carriers in order to take the advantage of orthogonal frequency-division multiplexing (OFDM) for high spectral efficiency. Recently, a novel way using the SSB modulator based on the RFS has been shown to generate 24-tone and 36-tone in [3,4] and [7,8] respectively, which both obtained a T-bit/s high data transmission rate. Even though they do really perfect work on experiments and theories, such as a perfect experiment result in [3,4], and theoretical analysis of principle and noise in [8], the instability of SSB-based RFS due to the higher harmonics originated from the modulator nonlinearity that can affect the output performance seriously has not been reported. Therefore, to achieve a steady output of the SSB modulator is still a challenging task.

In this paper, we mainly theoretically analyze the stability of the SSB-based RFS as a function of modulator drive voltage, number of desired generated tones, input laser power, and optical amplifier’s output power. A concept of effective OSNR is used to quantify the performance of the generated carriers. The results show that the modulator nonlinearity induced crosstalk is an important factor that affects the stability of the modulator, and the optimal operation conditions for different tones number can be obtained for a given set of system parameters.

## 2. Theoretical analysis of the stability

The schematic of SSB modulator based on RFS is shown in Fig. 1 . The configuration is composed of a closed fiber loop, which consists of a 50:50 coupler, an I/Q modulator, a tunable optical band-pass filter which is used to control the number of desired tones, and two Erbium doped fiber amplifier (EDFA) which are used to compensate the modulation losses in the loop. The I/Q modulator is driven with two equal-amplitude but π/2 phase shifted RF clock signals through I and Q ports, to induce a positive (or negative) frequency shifting to the input signal which acts as a seed signal.

The I/Q modulator consists of two Mach-Zehnder modulators (MZMs) placed parallel in two arms, and a *π/2* phase shifter in one arm. Considering the input signal as *E _{in}*(

*t*) =

*A*exp(

*j*2

*πf*

_{0}

*t*), and RF drive signals as

*f*(t) =

_{I}*V*sin(2

_{pp}*πf*),

_{s}t*f*(t) =

_{Q}*V*cos(2

_{pp}*πf*). Assuming the total number of desired tones number is

_{s}t*N*+ 1, the generated frequencies of the SSB-based RFS can be denoted as

*f*

_{0},

*f*

_{1}, …,

*f*

_{N}. The output electronic field of I/Q modulator can be expressed as follows [14]

*δ*= (

_{m}*πV*)/(2

_{pp}*V*) denotes for the phase modulation depth,

_{π}*J*

_{2}

_{k-}_{1}(

*δ*) are the first kind odd-order Bessel functions, and

_{m}*ϕ*is the phase delay per round trip (RT). In the most cases, the third order harmonic is dominant among all the harmonics and ultimately affects the output properties of the RFS. Ignoring all the high order harmonics beyond the third order, then the output of SSB-based RFS after the first round can be obtained by Eq. (2)

_{RT}*b*= −

*J*

_{3}/

*J*

_{1}, stands for the crosstalk coefficient which depends on the modulator drive voltage

*V*

_{pp}, which can affect the output of RFS. In order to decrease the crosstalk, we must apply proper

*V*

_{pp}to make |

*b*|<<1. In addition, the required gain of the amplifier for a ideal case must satisfy

*G*= = 1/(

_{R}*J*

_{1})

^{2}to compensate the modulation loss of the I/Q modulator, which can be expressed as follows (in dB)

And then we can obtain a same power of the input seed signal *f*
_{0} with that of the desired tone signal *f*
_{1} = *f*
_{0} + *f _{s}*. By denoting the transfer function of each frequency-shifting as

*n*(

*n*= 1,2,…,

*N*) RTs are as follows

*b*|<<1, we can make an approximation of the output after the

*N*-th RT by keeping up to the first order terms of

*b*as

*j*2

*πnf*) corresponds to the desired frequency components at [0,1,2…

_{s}t*N*]

*f*, which we refer to as the “signal components”. The term exp[

_{s}*j*2

*π*(

*n*− 4)

*f*] corresponds to the “unwanted” crosstalk components at [−4, −3, −2…

_{s}t*N*−4]

*f*, which we refer to as the “crosstalk components”. The physical intuition is that at the (

_{s}*N*−4)-th frequency, the crosstalk can come from any one of the

*N*shifting process that is 4 RTs in advance.

Assuming that the built-in optical filter inside the RFS blocks any frequency less than or equal to the seed frequency *f _{0}*, we have

*N*+ 1)-th RT, we have

*Nf*, and with the first order approximation, Eq. (9) can be rewritten as

_{s}*N*−3)

*b*exp[

*j*2

*π*(

*N*−3)

*f*] exp[

_{s}t*j*(

*N*+ 1)

*ϕ*] by

_{RT}*F*, Let the round trip continues next

_{r}*n*times, then the outputs after the (

*N*+

*n*)-th RT can be shown in Tab.1,

where *A _{r}* = 4

*b*exp(

*jϕ*), which can be ignored as |

_{RT}*b*|<<1. Therefore, we can conclude that the “signal components” and “crosstalk components” have reached the “steady states” when finish another 4 RTs after the

*N*-th RT from Tab.1, i.e., we can just consider the recirculating number up to (

*N*+ 4) as the final output of the SSB-based RFS. This is a new finding that has not been reported in previous analysis [8]. So we have the final expression as follows after the (

*N*+ 4)-th RT

*c*

_{n}is the normalized crosstalk components according to the signal components at their corresponding frequencies shown as follows

*n*, reaches the maximum value of |(

*N*−3)

*b*| at the (

*N*−3)-th frequency, and remains at this maximum value for the last four frequencies. This maximum value is defined as the worst-case crosstalk

*C*

_{n}_{max}and it varies with

*N*and

*V*

_{pp}. The dependence of the crosstalk amplitude on the number of tones to be generated (N) is also a new finding that was not reported in the previous analysis [8]. Note that due to thermal fluctuation-induced path length changes in the RFS,

*ϕ*can be considered as a random variable within the range of [0, 2π].

_{RT}Assuming a 20-carrier SSB-based RFS with the crosstalk coefficient |*b*| = −15dB (which is a relative good value), we have *N* = 19, *b* = 0.032, and therefore |*C _{n}*

_{max}| = 0.5. We can define the worst-case crosstalk-induced power reduction (in dB) as

*β*

_{max}= 6dB, which is unacceptably large.

However, above analysis has not taken into account the ASE noise of optical amplifier. Actually, the optical ASE noise is also an important aspect to affect the output of the modulator. Assuming the normalized saturation amplified gain and output power of the EDFA is *G* and *P*
_{out} (in dBm) respectively, and then can define the effective OSNR from the EDFA as follows

*P*

_{in_tone},

*P*

_{out_tone}and

*P*

_{ASE_total}stand for the input and saturation output power per tone and total ASE noise power after the (

*N +*4)-th RT respectively. Assuming the total loss is

*L*

_{total}which contains the losses of coupler, filter, I/Q modulator insertion and the modulation (described in Eq. (4)), then we have

*P*

_{ASE_RT}is the ASE noise power per RT,

*F*

_{n}is the noise figure (

*N*F is denoted as in dB),

*hν*is the photon energy and

*B*

_{r}is the reference bandwidth of optical band-pass filter. Combining the above equations, we obtain the

*OSNR*

_{EDFA}as follows (in dB)

On the other hand, we may assume all the *J*
_{3} terms add up quadratically when *ϕ*
_{RT} considered as a random variable within the range of [0, 2π], so the E-field sum of all the *J*
_{3} terms will be scaled as $\sqrt{N-3}$. Therefore, we can define an effective OSNR from 3rd-order harmonics after EDFA using Eq. (12) as follows

*P*

_{3}stands for the power of 3rd-order harmonics. Equation (19) can be rewritten in dB as follows

From above equations, we can see that the *OSNR*
_{EDFA} and *OSNR*
_{3rd-harmonic} essentially vary with *V*
_{pp} and desired tones number *N* respectively. Furthermore, the OSNR degradation due to EDFA goes severer (than that due to 3rd-order terms) when *N* increases. So obviously, the larger *N*, the smaller OSNR. Therefore, we can define an effective OSNR for each generated tone as the ratio of the tone power, and the sum of the crosstalk and ASE noise powers, to quantify the overall quality of the generated tones. Assuming the ASE noise and crosstalk induced by 3rd-order harmonics independently each other, we have

Evidently, we can improve the overall *OSNR*
_{eff} by applying an optimum peak-to-peak RF voltage when the desired tones number is given. Moreover, when given the required OSNR for a Tb/s multi-carrier signal, the effective OSNR penalty for each tone can be calculated by using the *OSNR*
_{eff}. The model of calculating the OSNR penalty is shown as Fig. 2
.

In Fig. 2, *OSNR*
_{req} is the required OSNR under the basic bit-error-ratio (BER). We can get the OSNR penalty when *OSNR*
_{eff} changes in the SSB-Modulator. Assuming that, (i) the *OSNR*
_{req} is a fix value which obtained in the experiment with the back-to-back configuration, so *OSNR*
_{req} is equal to *OSNR*
_{eff1}, i.e., satisfies the condition *OSNR*
_{req} = *OSNR*
_{eff1}; (ii) when *OSNR*
_{eff} decreases, we should increase OSNR penalty to keep the same *OSNR*
_{req} in the receiver, so the condition changes to *OSNR*
_{req} = *OSNR*
_{eff} + *OSNR* penalty. Therefore, the expression of *OSNR* penalty can be obtained by using the above analysis and is shown as follows

*OSNR*

_{eff}must keep larger than

*OSNR*

_{req}. Otherwise the

*OSNR*penalty will become too larger to apply for the optical transmission.

## 3. Numerical results

Under the theoretical analysis, the numerical simulation results have obtained by using following parameters: *P*
_{in} = 0dBm, *P*
_{out} = 25dBm (the saturation power of EDFA), *L*
_{Coupler} = 3dB, *L*
_{I/Q} = 13dB and *L*
_{Fliter} = 3dB.

#### 3.1 The basic properties of SSB-based RFS

Figure 3
shows the variation of the 1st-order signal (∝ *J*
^{2}
_{1}(*δ _{m}*)) and 3rd order terms (∝

*J*

^{2}

_{3}(

*δ*)) represented by

_{m}*P*

_{1}and

*P*

_{3}with different peak-to-peak RF drive voltage. With increasing of the

*V*

_{pp},

*P*

_{1}initially increases at a maximum (signed as

*P*

_{1 max}in the figure) around 1.17

*V*

_{π}, and then begins to decrease even lower than the

*P*

_{3}. This means that we can obtain a larger

*P*

_{1}with larger

*V*

_{pp}as long as it does not exceed 1.17

*V*

_{π}.

Figure 4
shows the relationship between the crosstalk coefficient |*b*| and *V*
_{pp}. It shows that we can get a less crosstalk (< −15dB) and corresponding crosstalk power (< −30dB) when *V*
_{pp} takes the value lower than 0.54*V*
_{π}. This has a good agreement with experiment results in [8]. And the result will be helpful to be applied in practice. At point A, |*b*| comes to zero, which means that the 3rd harmonic power *P*
_{3} begins to exceed the desired signal. Therefore, we should avoid the case to come into this undesired operation.

#### 3.2 The influence on SSB-based RFS dues to the Crosstalk induced by the 3rd-order harmonics

Figure 5
shows the worst-case crosstalk *C _{n}*

_{max}induced by the 3rd order harmonics, as a function of

*V*

_{pp}and desired tones number

*N*. We can see that the worst-case crosstalk increases with increased desired tones number

*N*and RF drive voltage

*V*

_{pp}. Under an acceptable worst-case crosstalk, the parameters we can apply will be just a small zone from the figure. Assuming the acceptable maximum power reduction induced by 3rd-order harmonic

*β*

_{max}= 3dB, the worst-case crosstalk

*C*

_{n}_{max}will be just around −1.5dB. So the parameters that we can apply are just within white color zone for this example, while the other zones cannot be considered. Taking the example of Ref [3,4]. and [7,8], we need 24 tones to implement a 1.2-Tb/s multi-carrier signal or 36 tones to expand the bandwidth of Tb/s uncorrelated multi-band OFDM signal. So we show the worst-case crosstalk for these two cases in Fig. 6 . Obviously, as marked with arrow A and B,

*V*

_{pp}should be lower than 0.54

*V*

_{π}and 0.43

*V*

_{π}to keep the crosstalk lower than −1.5dB respectively.

To illustrate the influence of crosstalk on the output of SSB-based RFS, the output spectrum at certain *V*
_{pp} should be studied. Figure 7
shows the normalized steady output spectrum when *V*
_{pp} is 0.54*V*
_{π} (this value is similar to that used in [8]) for *N* = 23 and 36. From the two figures, we can see that the flatness of the generated tone spectrum is worse as *N* increases. Influenced by the crosstalk, the normalized power of the last generated tone is ~2.3dB and ~4dB lower than the first tone for *N* = 23 and 36, respectively. These results are in good agreement with the experiment results of [3,4] and [8].

#### 3.3 The optimum operation condition of SSB-based RFS

It can be seen that smaller *V*
_{pp} is desired to reduce the crosstalk. However, to obtain a relative larger output signal power and lower ASE noise from EDFAs, *V*
_{pp} is desired to be as large as possible. There is thus a trade-off between crosstalk and optical noise, and an optimum *V*
_{pp} that gives the best overall tone quality. Deduced from Eq. (18), (20) and (23), the *OSNR*
_{EDFA}, *OSNR*
_{3rd-harmonics} and overall *OSNR*
_{eff} are shown in Fig. 8
−10 respectively.

We can find that the *OSNR*
_{EDFA} at a given desired tones number has a peak when the RF driven voltage increasing to around 1.17*V*
_{π}. This result due to the signal power has a larger value around 1.17*V*
_{π} from Fig. 3 and has a good in agreement with Eq. (18). Subsequently, the larger desired tones number is, the smaller *OSNR*
_{EDFA} will be.

As show in Fig. 9
, the *OSNR*
_{3rd-harmonic} has obtained with different desired tones number *N* and *V*
_{pp}. Obviously, the *OSNR*
_{3rd- harmonic} decreases sharply with increasing the *V*
_{pp} at a given desired tones number. Moreover, the *OSNR*
_{3rd- harmonic} will be larger when the *V*
_{pp} and *N* are smaller.

Indeed, as seen from Fig. 8 and 9, the *OSNR*
_{EDFA} has a maximum value around 1.17*V*
_{π}, while the *OSNR*
_{3rd-harmonic} has relative larger value at small *V*
_{pp}. Therefore, the trade-off between signal power and crosstalk and ASE noise can be obtained.

Figure 10
shows the overall *OSNR*
_{eff} as the function of *N* and *V*
_{pp}. The overall *OSNR*
_{eff} curve appears to have a maximum as we expected. We can then obtain the optimum *V*
_{pp} (denotes by *V*
_{opt}) and maximum overall *OSNR*
_{eff} (denotes by *OSNR*
_{eff}_max) through these peaks as showing in the figure. The *V*
_{opt} and *OSNR*
_{eff}_max for different desired tones number *N* are shown in Fig. 11
. The *V*
_{opt} is around 0.30*V*
_{π} for *N* within the range of [4,14], and about 0.365*V*
_{π} for *N* within the range of [15, 50]. The *V*
_{opt} is a significant parameter to obtain the larger overall *OSNR*
_{eff}. Take *N* = 23 in [3,4] and 36 in [8] for example, the optimum RF drive voltage is same around 0.365*V*
_{π}.

Furthermore, the effective OSNR penalty for Tb/s multi-carrier signal can be obtained by using Eq. (24). Taking the experiment result of [4] as an example, the required OSNR is ~26dB at a BER of 1 × 10^{−3} in the back-to-back configuration. The relationship between *N* and the effective OSNR penalty for this example is calculated and shown in Fig. 12
. We can see that the effective OSNR penalty is ~0.9dB for the 1.2-Tb/s 24-carrier signal in the back-to-back configuration. This again is in good agreement with the experimental results reported in [4]. On the other hand, generating more than 25 tones (as marked using the arrow A in Fig. 12), the effective OSNR penalty would be larger than 1dB at BER = 1 × 10^{−3}. When the desired tones number *N* is up to 35 (as marked using the arrow B in Fig. 12), the OSNR penalty will exceed 2dB.

## 4. Conclusions

The influence of the crosstalk induced by high order harmonics originated from the modulator nonlinearity on the output of SSB-based RFS has been theoretically analyzed. And it is found to be dependent on the number of desired tones number. An effective OSNR has been defined to quantify the overall quality of the generated tones in the presence of the crosstalk and optical noise. By using the analytical model, the optimum modulator drive voltage is obtained for various RFS configurations (different desired tones number and RF drive voltage), and obtained results are in good agreement with the previous experimental results. This theoretical model may provide a useful guide for optimizing the performance of the SSB-based RFS modulator for Tb/s multi-carrier transmission in practice.

## Acknowledgement

This work is supported by National High Technology Research and Development Program of China (Grant No. 2009AA01Z224) and National Natural Science Foundation of China (Grant No. 60977049).

## References and Links

**1. **S. Chandrasekhar and D. Xiang Liu, “Kilper, C. R. Doerr, A. H. Gnauck, E. C. Burrows, and L. L. Buhl. “Terabit Transmission at 42.7-Gb/s on 50-GHz Grid Using Hybrid RZ-DQPSK and NRZ-DBPSK Formats Over 1680 km SSMF Spans and 4 Bandwidth-Managed ROADMs,” J. Lightwave Technol. **26**, 85–89 (2008). [CrossRef]

**2. **Xiang Liu, Gill, D.M., Chandrasekhar, S., Buhl, L.L., Earnshaw M., Cappuzzo M.A., Gomez L.T., Chen Y., Klemens F.P., Burrows E.C., Chen, Y.-K., Tkach R.W.. “Compact and broadband coherent receiver front-end for complete demodulation of a 1.12-terabit/s multi-carrier PDM-QPSK signal,” ECOC. **Paper 10.3.2,** (2009).

**3. **Liu Xiang, Chandrasekhar S., Zhu Benyuan, Peckham David W. “Efficient Digital Coherent Detection of A 1.2-Tb/s 24-Carrier No-Guard-Interval CO-OFDM Signal by Simultaneously Detecting Multiple Carriers Per Sampling,” OFC. **OWO2**, (2010).

**4. **S. Chandrasekhar, Xiang Liu, B. Zhu, and D. W. Peckham, “Transmission of a 1.2-Tb/s 24-Carrier No-Guard-Interval Coherent OFDM Superchannel over 7200-km of Ultra-Large-Area Fiber,” ECOC. **PD2.6,** (2009).

**5. **Roman Dischler, Fred Buchali. “Transmission of 1.2 Tb/s Continuous Waveband PDM-OFDM-FDM signal with Spectral Efficiency of 3.3 bit/s/Hz over 400 km of SSMF,” OFC. **PDPC2**, (2009).

**6. **William Shieh, “High Spectral Efficiency Coherent Optical OFDM for 1 Tb/s Ethernet Transport,” OFC. **OWW1**, (2009).

**7. **Y. Ma, Q. Yang, Y. Tang, S. Chen, and W. Shieh, “1-Tb/s single-channel coherent optical OFDM transmission over 600-km SSMF fiber with subwavelength bandwidth access,” Opt. Express **17**(11), 9421–9427 (2009). [CrossRef] [PubMed]

**8. **Y. Ma, Q. Yang, Y. Tang, S. Chen, and W. Shieh, “1-Tb/s Single-Channel Coherent Optical OFDM Transmission with Orthogonal-band Multiplexing and Subwavelength Bandwidth Access,” J. Lightwave Technol. **28**(4), 308–315 (2010). [CrossRef]

**9. **Xiang Zhou, Jianjun Yu, Mei Du, and Guodong Zhang. “2Tb/s (20´107 Gb/s) RZ-DQPSK straight-line transmission over 1005 km of standard single mode fiber (SSMF) without Raman amplification,” OFC. **OMQ3**, (2008).

**10. **Yu Jianjun, Zhou Xiang, “32Tb/s DWDM Transmission System,” ACP. **TuEE1**, (2009).

**11. **Toshiaki Kuri, Hiroyuki Toda, Jose Vegas Olmos Juan, and Kitayama Ken-ichi. “Reconfigurable Dense Wavelength Division Multiplexing Millimeter-Wave-Band Radio-over-Fiber Access System Technologies,” J. Lightwave Technol. **28****,** (2010 accepted). [CrossRef]

**12. **T. Sakamoto, T. Yamamoto, K. Kurokawa, and S. Tomita, “DWDM transmission in O-band over 24 km PCF using optical frequency comb based multicarrier source,” Electron. Lett. **45**(16), 850–851 (2009). [CrossRef]

**13. **Sheng Liu, Trina T. Ng, David J. Richardson, Periklis Petropoulos. “An Optical Frequency Comb Generator as a Broadband Pulse Source,” OFC. **OThG7**, (2009).

**14. **McGhan D., O'Sullivan M., Sotoodeh M., Savchenko A., Bontu C., Belanger M., Roberts K. “Electronic Dispersion Compensation,” OFC. **OWK1**, (2006).