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
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  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 , 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 Ein(t) = Aexp(j2πf 0 t), and RF drive signals as fI (t) = Vppsin(2πfst), fQ (t) = Vppcos(2πfst). Assuming the total number of desired tones number is 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 Eq. (1). can be expressed asEq. (2)
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 + fs. By denoting the transfer function of each frequency-shifting as
Assuming that the built-in optical filter inside the RFS blocks any frequency less than or equal to the seed frequency f0, we haveEq. (9) can be rewritten asEq. (10) represents the “signal components”, which have reached a “steady state”. This is true by comparing Eq. (10) with Eq. (8). The second and third term represent the “crosstalk components”, which have not yet reached a “steady state”. To simplify the analysis, we denote (N−3)b exp[j2π(N−3)fst] exp[j(N + 1)ϕRT] by Fr, Let the round trip continues next n times, then the outputs after the (N + n)-th RT can be shown in Tab.1,
where Ar = 4b exp(jϕRT), which can be ignored as |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 . So we have the final expression as follows after the (N + 4)-th RT8]. Note that due to thermal fluctuation-induced path length changes in the RFS, ϕRT can be considered as a random variable within the range of [0, 2π].
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 |Cn max| = 0.5. We can define the worst-case crosstalk-induced power reduction (in dB) as
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 followsEq. (14), 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
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 . Therefore, we can define an effective OSNR from 3rd-order harmonics after EDFA using Eq. (12) as followsEquation (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 followsEq. (24), we can deduce that the 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(δm)) and 3rd order terms (∝ J 2 3(δm)) represented by 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 1.17V 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 1.17V 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 0.54V π. This has a good agreement with experiment results in . 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 Cn 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 Cn 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 0.54V π and 0.43V 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 0.54V π (this value is similar to that used in ) 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 .
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 1.17V π. This result due to the signal power has a larger value around 1.17V 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 1.17V 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 0.30V π for N within the range of [4,14], and about 0.365V 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  for example, the optimum RF drive voltage is same around 0.365V π.
Furthermore, the effective OSNR penalty for Tb/s multi-carrier signal can be obtained by using Eq. (24). Taking the experiment result of  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 . 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.
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.
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).
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