Analysis of the stability and optimizing operation of the single-side-band modulator based on re-circulating frequency shifter used for the T-bit/s optical communication transmission

Jianping Li; Xuan Li; Xiaoguang Zhang; Feng Tian; Lixia Xi

doi:10.1364/OE.18.017597

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.

Fig. 1 Schematic of SSB modulator based on RFS

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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) = Aexp(j2πf ₀ t), and RF drive signals as f_I (t) = V_ppsin(2πf_st), f_Q (t) = V_ppcos(2πf_st). 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 [14]

E_{o u t} (t) = \frac{E_{i n} (t)}{2} [j \sin (\frac{π}{2} \frac{f_{I} (t)}{V_{π}}) + \sin (\frac{π}{2} \frac{f_{Q} (t)}{V_{π}})]

when using the Jacobi-Anger expansion, Eq. (1). can be expressed as

\begin{matrix} E_{o u t} (t) = \frac{E_{i n} (t)}{2} \cdot [j \sin (δ_{m} \sin (2 π f_{s} t)) + \sin (δ_{m} \cos (2 π f_{s} t))] \exp (j ϕ_{R T}) \\ = E_{i n} (t) \cdot j \sum_{k = 1}^{\infty} J_{2 k - 1} (δ_{m}) \sin [2 π (2 k - 1) f_{s} t] \exp (j ϕ_{R T}) + \\ E_{i n} (t) \cdot \sum_{k = 1}^{\infty} j^{2 k - 2} J_{2 k - 1} (δ_{m}) \cos [2 π (2 k - 1) f_{s} t] \exp (j ϕ_{R T}) \\ = E_{i n} (t) \cdot [J_{1} (δ_{m}) \exp (j 2 π f_{s} t) - J_{3} (δ_{m}) \exp (- j 6 π f_{s} t)] \exp (j ϕ_{R T}) + \\ E_{i n} (t) \cdot [J_{5} (δ_{m}) \exp (j 10 π f_{s} t) - \dots] \exp (j ϕ_{R T}) \end{matrix}

where δ_m = (πV_pp)/(2V_π) denotes for the phase modulation depth, J ₂ _k- ₁(δ_m) are the first kind odd-order Bessel functions, and ϕ_RT 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)

E_{1} (t) = E_{i n} (t) + E_{i n} (t) \cdot J_{1} (δ_{m}) [\exp (j 2 π f_{s} t) + b \exp (- j 6 π f_{s} t)] \exp (j ϕ_{R T})

where 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_R = = 1/(J ₁)² to compensate the modulation loss of the I/Q modulator, which can be expressed as follows (in dB)

L_{M o d} = G_{R} = - 20 \lg (| J_{1} |)

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

T = [\exp (j 2 π f_{s} t) + b \exp (- j 6 π f_{s} t)] \exp (j ϕ_{R T})

And assuming that the loss can be compensated exactly by the amplifier, the normalized outputs after passing n (n = 1,2,…,N) RTs are as follows

\begin{array}{l} E_{1} (t) = E_{i n} (t) + T \cdot E_{i n} (t) = E_{i n} (t) \cdot (1 + T) \\ E_{2} (t) = E_{i n} (t) + T \cdot E_{1} (t) = E_{i n} (t) \cdot (1 + T + T^{2}) \\ ... \\ E_{N} (t) = E_{i n} (t) + T \cdot E_{N - 1} (t) = E_{i n} (t) \cdot (1 + T + T^{2} + ... + T^{N}) \end{array}

Under the less crosstalk condition |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

\begin{array}{l} E_{N} (t) = E_{i n} (t) \cdot \sum_{n = 0}^{N} T^{n} \\ \approx E_{i n} (t) \cdot \sum_{n = 0}^{N} {\exp (j 2 π n f_{s} t) + n b \exp [j 2 π (n - 4) f_{s} t)]} \cdot \exp (j n ϕ_{R T}) \end{array}

In the above equation, the term exp(j2πnf_st) corresponds to the desired frequency components at [0,1,2…N]f_s, which we refer to as the “signal components”. The term exp[j2π(n − 4) f_st] corresponds to the “unwanted” crosstalk components at [−4, −3, −2…N−4]f_s, which we refer to as the “crosstalk components”. The physical intuition is that at the (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 f₀, we have

\begin{array}{l} E_{N} (t) \approx E_{i n} (t) \cdot \sum_{n = 0}^{N} \exp (j 2 π n f_{s} t) \exp (j n ϕ_{R T}) \\ + E_{i n} (t) \cdot \sum_{n = 1}^{N - 4} n b \exp (j 2 π n f_{s} t) \cdot \exp [j (n + 4) ϕ_{R T}] \end{array}

Then after the (N + 1)-th RT, we have

\begin{matrix} E_{N + 1} (t) = E_{i n} (t) \cdot \sum_{n = 0}^{N + 1} T^{n} = E_{i n} (t) \cdot \sum_{n = 0}^{N} T^{n} + E_{i n} (t) \cdot T^{N + 1} \\ = E_{i n} (t) \cdot \sum_{n = 0}^{N} {[\exp (j 2 π f_{s} t) + b \exp (- j 6 π f_{s} t)]}^{n} \\ + E_{i n} (t) \cdot [\exp (j 2 π f_{s} t) + b \exp (- j 6 π f_{s} t)]^{N + 1} \exp [j (N + 1) ϕ_{R T}] \end{matrix}

Assuming that the optical filter inside the RSF also blocks any frequency components higher than Nf_s, and with the first order approximation, Eq. (9) can be rewritten as

\begin{matrix} E_{N + 1} (t) = E_{i n} (t) \cdot \sum_{n = 0}^{N} \exp (j 2 π n f_{s} t) \exp (j n ϕ_{R T}) \\ + E_{i n} (t) \cdot [\exp (j 2 π f_{s} t) + b \exp (- j 6 π f_{s} t)]^{N + 1} \exp [j (N + 1) ϕ_{R T}] \\ \approx E_{i n} (t) \cdot \sum_{n = 0}^{N} \exp (j 2 π n f_{s} t) \exp (j n ϕ_{R T}) \\ + E_{i n} (t) \cdot \sum_{n = 1}^{N - 4} n b \exp (j 2 π n f_{s} t) \cdot \exp [j (n + 4) ϕ_{R T}] \\ + E_{i n} (t) \cdot (N - 3) b \exp [j 2 π (N - 3) f_{s} t] \cdot \exp [j (N + 1) ϕ_{R T}] \end{matrix}

Evidently, the first term at right-hand side of Eq. (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)f_st] exp[j(N + 1)ϕ_RT] by F_r, Let the round trip continues next n times, then the outputs after the (N + n)-th RT can be shown in Tab.1,

Table 1. The output of RFS after n RTs of N

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where A_r = 4b exp(jϕ_RT), 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 [8]. So we have the final expression as follows after the (N + 4)-th RT

\begin{matrix} E_{N + 4} (t) = E_{N} (t) + E_{i n} (t) \cdot \sum_{n = 0}^{3} T^{n} F_{r} \\ \approx E_{i n} (t) \cdot \sum_{n = 0}^{N} \exp (j 2 π n f_{s} t) \exp (j n ϕ_{R T}) \\ + E_{i n} (t) \cdot \sum_{n = 1}^{N} c_{n} \exp (j 2 π n f_{s} t) \cdot \exp (j n ϕ_{R T}) \end{matrix}

where c _n is the normalized crosstalk components according to the signal components at their corresponding frequencies shown as follows

\begin{array}{l} c_{n} = n b \exp (j 4 ϕ_{R T}), n \leq (N - 3) \\ = (N - 3) b \exp (j 4 ϕ_{R T}), n > (N - 3) \end{array}

Evidently, the magnitude of the normalized crosstalk initially increases with 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_RT 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 |C_n _max| = 0.5. We can define the worst-case crosstalk-induced power reduction (in dB) as

β_{\max} = - 20 \lg (1 - | c_{n \max} |)

For this example, we have 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

O S N R_{E D F A} = \frac{P_{o u t_t o n e}}{P_{A S E_t o t a l}} = \frac{G P_{i n_t o n e}}{P_{A S E_t o t a l}}

In Eq. (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

L_{t o t a l} (d B) = L_{c o u p l e r} + L_{f i l t e r} + L_{I / Q} + L_{M o d}

P_{o u t_t o n e} (d B) = 10 \lg (G P_{i n_t o n e}) = 10 \lg G + P_{o u t} (d B m) - 10 \lg N - L_{t o t a l} (d B)

P_{A S E_t o t a l} (d B) = 10 \lg [(N + 4) P_{A S E_R T}] = 10 \lg [(N + 4) F_{n} (G - 1) h ν B_{r}]

where P _{ASE_RT} is the ASE noise power per RT, F _n is the noise figure (NF 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)

\begin{matrix} O S N R_{E D F A} (d B) = P_{o u t_t o n e} (d B m) - P_{A S E_t o t a l} (d B m) \\ \approx 58 + (P_{o u t} (d B m) - N F (d B) - L_{t o t a l} (d B)) - 20 \lg N \end{matrix}

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

O S N R_{3 r d - h a r m o n i c} = \frac{P_{o u t_t o n e}}{(N - 3) G P_{3}} = \frac{P_{i n_t o n e}}{(N - 3) P_{3}} = \frac{1}{(N - 3) | b |^{2}}

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

O S N R_{3 r d - h a r m o n i c} (d B) = - 20 \lg (| b |) - 10 \lg (N - 3)

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

P_{A S E_t o t a l} = P_{i n_t o n e} 10^{- \frac{O S N R_{E D F A}}{10}}

P_{3} = P_{i n_t o n e} 10^{- \frac{O S N R_{3 r d - h a r m o n i c}}{10}}

And therefore, the overall effective OSNR can be obtained as

O S N R_{e f f} = 10 \lg \frac{P_{o u t_t o n e}}{P_{A S E_t o t a l} + P_{3}} = - 10 \lg (10^{- \frac{O S N R_{E D F A}}{10}} + 10^{- \frac{O S N R_{3 r d - h a r m o n i c}}{10}})

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 .

Fig. 2 The model of calculating OSNR penalty

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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

O S N R_{p e n a l t y} (d B) = - 10 \lg (10^{- \frac{O S N R_{r e q}}{10}} - 10^{- \frac{O S N R_{e f f}}{10}}) - O S N R_{r e q} (d B)

From Eq. (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(δ_m)) and 3rd order terms (∝ J ² ₃(δ_m)) 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.

Fig. 3 The relationship between normalized power and peak-to-peak RF Drive Voltage

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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 [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.

Fig. 4 The relationship between crosstalk coefficient and peak-to-peak RF Drive Voltage

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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 0.54V _π and 0.43V _π to keep the crosstalk lower than −1.5dB respectively.

Fig. 5 . The worst-case crosstalk at with different desired tones number and RF drive voltage

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Fig. 6 The worst-case crosstalk particular for N = 23 (green line) and N = 36 (blue line) respectively.

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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 [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].

Fig. 7 The output spectrum for (a) N = 23 and (b) N = 36.

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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.

Fig. 8 The OSNR from EDFA with different desired tones number at different V _pp

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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.

Fig. 9 The OSNR _3rd-harmonic with different desired tones number at different V _pp

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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 [8] for example, the optimum RF drive voltage is same around 0.365V _π.

Fig. 10 The overall OSNR _eff with different desired tones number at different V _pp

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Fig. 11 The optimum V _pp and maximum overall OSNR for different desired tones number

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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.

Fig. 12 The effective OSNR penalty with different desired tones number N

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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).

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The number of RT	The Output
k	$E_{k} (t)$
N	$E_{N} (t)$
N + 1	$E_{N + 1} (t) = E_{N} (t) + E_{i n} (t) \cdot F_{r}$
N + 2	$E_{N + 2} (t) = E_{N} (t) + E_{i n} (t) \cdot (F_{r} + T F_{r})$
N + 3	$E_{N + 3} (t) = E_{N} (t) + E_{i n} (t) \cdot \sum_{n = 0}^{2} T^{n} F_{r}$
N + 4	$E_{N + 4} (t) = E_{N} (t) + E_{i n} (t) \cdot \sum_{n = 0}^{3} T^{n} F_{r}$
N + 5	$E_{N + 5} (t) = E_{N} (t) + E_{i n} (t) \cdot (1 + A_{r}) F_{r} + E_{i n} (t) \cdot \sum_{n = 1}^{3} T^{n} F_{r}$
N + 6	$E_{N + 6} (t) = E_{N} (t) + E_{i n} (t) \cdot (1 + A_{r}) \sum_{n = 0}^{1} T^{n} F_{r} + E_{i n} (t) \cdot \sum_{n = 2}^{3} T^{n} F_{r}$
N + 7	$E_{N + 7} (t) = E_{N} (t) + E_{i n} (t) \cdot (1 + A_{r}) \sum_{n = 0}^{2} T^{n} F_{r} + E_{i n} (t) \cdot F^{3} F_{r}$
N + 8	$E_{N + 8} (t) = E_{N} (t) + E_{i n} (t) \cdot (1 + A_{r}) \sum_{n = 0}^{3} T^{n} F_{r}$
⋮	⋮
N + 12	$E_{N + 12} (t) = E_{N} (t) + E_{i n} (t) \cdot (1 + \sum_{n = 1}^{2} A_{r}^{n}) \sum_{n = 0}^{3} T^{n} F_{r}$
⋮	⋮
N + 4n	$E_{N + 4 n} (t) = E_{N} (t) + E_{i n} (t) \cdot (1 + \sum_{n = 1}^{\infty} A_{r}^{n}) \sum_{n = 0}^{3} T^{n} F_{r}$

Analysis of the stability and optimizing operation of the single-side-band modulator based on re-circulating frequency shifter used for the T-bit/s optical communication transmission

Abstract

1. Introduction

2. Theoretical analysis of the stability

3. Numerical results

3.1 The basic properties of SSB-based RFS

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

3.3 The optimum operation condition of SSB-based RFS

4. Conclusions

Acknowledgement

References and Links

Cited By

Figures (12)

Tables (1)

Equations (24)

Optics Express