Abstract

Polarization-insensitive conversion of return-to-zero (RZ) ON-OFF keying (RZ-OOK) to RZ binary phase-shift keying (RZ-BPSK) has been achieved by cross-phase modulation (XPM) in a nonlinear birefringent fiber. This work presents a theoretical analysis of the dependence of format conversion on pump-probe detuning, and the pump state-of-polarization (SOP) that can fluctuate unpredictably in a realistic system. An investigation of the impact of pump polarization fluctuation on receiver sensitivity and receiver optimal threshold for the converted RZ-BPSK probe is also carried out. It was found that although the desired XPM-induced n phase shift can be achieved by launching both the RZ-OOK pump and the probe along the same birefringent axis of the fiber, the phase shift degrades to π/3 if the SOP of the RZ-OOK pump unpredictably switches to the other axis of the fiber, resulting in a large receiver sensitivity penalty fluctuation of 14 dB. By contrast, launching the probe at 45° relative to the birefringent axes can reduce the polarization-dependent receiver sensitivity penalty fluctuation to about 2 dB as the SOP of the RZ-OOK pump is swept over the Poincaré sphere. These conclusions are in good agreement with recently published experimental results.

©2009 Optical Society of America

1. Introduction

Compared with traditional RZ-OOK, RZ differential phase-shift keying (RZ-DPSK) requires about 3-dB lower optical signal-to-noise ratio (OSNR) to achieve the same bit-error rate (BER), but a more complex transmitter and receiver [1]. The RZ-DPSK and RZ-OOK formats can fit different networks [2], and mixed RZ-OOK/RZ-DPSK transmission in wavelength division multiplexed (WDM) systems could happen through optical cross-connects (OXC) interconnecting different networks. However, RZ-OOK can seriously distort RZ-DPSK by cross-phase modulation (XPM) [3], but this impairment may be mitigated by implementing RZ-OOK-to-RZ-BPSK format conversion at the OXC level. All-optical RZ-OOK-to-RZ-BPSK format conversion has been carried out by the XPM effect in nonlinear media, such as semiconductor optical amplifiers (SOAs) [2, 4–6] and highly nonlinear fibers (HNLFs) [7]. Although the ultra-fast χ (3) effect in an HNLF can achieve high speed conversion unlike a SOA limited by a slow response time, the χ (3) effect is polarization-sensitive and the strength of the XPM effect induced by arbitrarily polarized RZ-OOK signals would fluctuate unpredictably. Recently, the XPM effect in a linearly birefringent photonic crystal fiber (PCF) has been proposed to achieve polarization-insensitive RZ-OOK-to-BPSK conversion [8]. In this scheme, the local probe beam is launched equally to both birefringent axes of the fiber to reduce polarization sensitivity, obviating the necessity of sophisticated and costly polarization tracking of the RZ-OOK signal. To further study the polarization sensitivity, a theoretical analysis on the format conversion in a linearly birefringent HNLF is presented here for the first time. The penalty induced by insufficient XPM-induced phase shift is also analytically investigated.

2. The XPM effect in a birefringent fiber

As shown in Fig. 1, while a space in an RZ-OOK signal induces negligible phase shift (close to zero), a mark with specific power is supposed to generate a π phase shift to realize all-optical RZ-OOK-to-BPSK conversion. In the conversion process, the pulse width of the RZ-OOK signal is normally required to be broader than that of the probe beam to reduce unnecessary chirping. Moreover, to avoid erroneous reception of the RZ-BPSK signal in a direct-detection DPSK receiver, an electronic decoder would be required in the receiver as noted by Mishina et al. [2].

 

Fig. 1. RZ-OOK-to-RZ-BPSK format conversion in a birefringent fiber.

Download Full Size | PPT Slide | PDF

To investigate the polarization-dependent XPM effect, it is assumed that χ xxyy (3) = χxyxy (3) = χxyyx (3) = χxxxx (3)/3, so that the third order nonlinear effect in a linearly birefringent HNLF is described by the nonlinear Schrödinger Eqs. (NLSE) [9, 10],

dAmpdz=(Amp2+2Anp2+23Amq2+23Anq2)Amp+jγ3Amp*Amq2ej2Δβmpqz
+j2γ3Anp*AnqAmqej(Δβmpq+Δβnpq)z+j2γ3AnpAnq*Amqej(ΔβmpqΔβnpq)z,

where the subscripts m ∊ {1,2} and n ∊ {1,2} such that mn refer to either the RZ-OOK pump (m or n = 1), or the RZ pulse probe (m or n = 2); the subscripts p ∊ {x, y} and q ∊ {x, y} such that pq represent two birefringent axes of the fiber; Amp is the amplitude of the electrical field; γ is the nonlinear coefficient, and ∆βmpq = βmp - βmq is the wave-vector mismatch due to the linear birefringence of the fiber. In Eq. (1), the retarded time frame has been adopted, and the dispersion and loss have been neglected assuming that the fiber length L is such that LLeff = (1 - e -αL)/α and LLD = T 2 0/ |β″|, where Leff , LD, α , T 0 and β″ respectively are the effective length, the dispersion length, the fiber attenuation, the input pulse width and the group-velocity dispersion parameter. In Eq. (1), the principal axes are assumed to coincide with the birefringent axes of the fiber. Because the probe beam is a RZ pulse train, its launch power has to be sufficiently low to reduce the undesirable self-phase modulation effect. Besides, since most high (10-5 - 10-3) linear-birefringence fibers can satisfy the condition of L |∆βmpq| ≫ 2π , the second and third terms in the RHS of Eq. (1) frequently change signs so that their contributions can be neglected [9]. Hence, the NLSEs for the pump and probe fields reduce to the following,

dA1pdz=(A1p2+23A1q2)A1p,
dA2pdz=(2A1p2+23A1q2)A2p+j2γ3A1A1q*A2qej(Δβ1pqΔβ2pq)z.

If the peak power of the pump beam P 1, its field can be represented as A 1(z = √P 1 cosψ 1 e 1x + √P 1 sinψ 1 e 1y ŷ ,where ψ 1 (∊[0,π/2] and θ 1 collectively determine the SOP of the launched pump beam. The corresponding solutions to Eq. (2) are,

A1x(z)=A1x(0)exp[jγP13(2+cos2ψ1)z],
A1y(z)=A1y(0)exp[jγP13(2+sin2ψ1)z].

After using Eqs. (4) and (5), setting  2x = A 2x exp(jκz) and  2y = A 2y exp(-jκz) with κ = ∆β 1xy - ∆β 2xy - γ P 1 cos(2ψ 1)/3 in Eq. (3), Eq.(3) is then differentiated into a second-order homogenous ordinary differential equation that may be subsequently solved for  2x and  2y . If the input probe is represented as A 2(z = 0) = √P 2 cosψ 2 e 2x + √P 2 sin ψ 2 e 2y ŷ , the output probe becomes,

A2x(L)=A2x(0)[cos(kL2)+jμxsin(kL2)]exp[j(3γP12jγP13sin2ψ1jΔK2)L],
A2y(L)=A2y(0)[cos(kL2)+jμysin(kL2)]exp[j(3γP12jγP13cos2ψ1+jΔK2)L],

where

k=49γ2P12sin2(2ψ1)+[γP1cos(2ψ1)+ΔK]2,
μx=1k[γP1cos(2ψ1)+ΔK+2γP13tanψ2sin(2ψ1)ejΔθ],
μy=1k[γP1cos(2ψ1)ΔK+2γP13cosψ2sin(2ψ1)ejΔθ],

θ = θ 1x -θ 1y -θ 2x +θ 2y is the differential phase that derives from the launch SOPs of the pump and the probe, and ∆K = ∆β 1xy - ∆β is the differential wave-vector mismatch, which is dependent on the fiber phase-index birefringence and pump-probe detuning (PPD, ∆λ = λ 1 - λ 2) [8]. Although |A 2p(L)|2 may not be equal to |A 2(0)|2 due to the last term of Eq. (3) being complex, the total power is unchanged, i.e. |A 2(L)|2 = P 2. Furthermore, because the nonlinear phase shifts induced in A 2x(L) and A 2y(L) are not the same, an effective phase shift is needed to investigate polarization-dependent effects. Considering differential balanced direct-detection, the detected signal will be proportional to |E(t)E *(t-T)|cos(∆ϕ+ξ 0) , where E(t) is the optical field; T is the bit period, and ∆ϕ is the phase difference between E(t) and E(t - T). Additional phase-shift (ξ 0) in the receiver Mach-Zehnder interferometer may favorably influence the performance of the received signal, but for the ensuing analysis, it is set to zero as it would be for a standard RZ-DPSK signal expected by the receiver (thus, the converted probe would be analyzed under the same conditions as the baseline signal.) When the phase shift induced by the space of the RZ-OOK pump is zero, the effective XPM-induced phase shift of Eqs. (6) and (7) can be evaluated as,

ϕeff=cos1(A2x(0)A2x(L)cosϕx+A2y(0)A2y(L)cosϕyA2x(0)2+A2y(0)2),

where ϕx = arg{A 2x(L)/A 2x(0)} and ϕeff = arg{A 2y(L)/A 2y(0)} are the nonlinear phase shifts of the probe field components. In general, the effective phase shift ϕeff is a function of ψ 1, ψ 2 , ∆θ , and ∆K . When the RZ-OOK pump is arbitrarily polarized, ψ 1 and ∆θ become random. To focus on optimizing the controllable SOP of the probe, L|∆K|≫2π is assumed. Therefore, the last term of Eq. (3) can be neglected, and the effective phase shift becomes,

ϕeff=cos1(cos2ψ2cos(Φ(1+2cos2ψ1)3)+sin2ψ2cos(Φ(1+2sin2ψ1)3)),

where Φ (= 2γ P 1 L) is termed the nonlinear phase shift due to XPM. Equations (11) and (12) reveal a complex relationship between the nonlinear phase shift (Φ), and the actual effective phase-shift (ϕeff) acquired by a probe pulse in the interaction with the pump in the fiber. It can be seen that Eq. (12) degenerates to the nonlinear phase shift if the pump and the probe are both launched along either birefringent axes.

Figure 2(a) shows an animation of ϕeff as a function of ψ 2 (Eq. (12)), under the constraints of L|∆K|≫2π and Φ = π, where multiple curves correspond to all possible pump SOPs. Since ϕeff in Eq. (12) is independent of ∆θ , pump SOPs lying on identical S 1 planes on the Poincaré sphere induce the same ϕeff in Fig. 2(a). Although launching the probe along a birefringent axis (ψ 2 = 0) can achieve the desired ϕeff = π at one pump SOP (ψ 1 = 0), as the pump SOP is varied, this probe orientation results in the largest fluctuation (~2π/3) in ϕeff , i.e. with a minimum ϕeff of π/3 and a maximum ϕeff of π. By contrast, although launching the probe at π/4 relative to a principal axes sacrifices some nonlinear phase shift, as the pump SOP is varied, this probe orientation leads to the smallest fluctuation (< π/10) in ϕeff, with a minimum of ~0.58π, and a maximum of ~0.67π. In fact, it can be observed from Fig. 2(a) that a probe launch of π/4 yields the global optimum for ϕeff fluctuations. In order to analyze the PPD-dependence of the fluctuations of ϕeff for variable pump SOP, ψ 2 = π/4 is now defined as the best scenario, while ψ 2 = 0 (or π/2), the worst scenario. An animation of ϕeff for the best scenario (with Φ = π) is plotted as a function of L|∆K| in Fig. 2(b), where the maximum and minimum effective phase shifts of the worst scenario are also shown for comparison. At L|∆K|=2 (N, a positive integer), the minimum phase fluctuations are achieved; because the last term of Eq. (3) contributes least. In fact, for L|∆K|=2 , the minimum phase shift remains about the same, while the phase fluctuation changes by no more than 0.02 radians. Hence, polarization-insensitive operation can be achieved by launching the probe at π/4 and requiring L|∆K|≥2π. Using experimental data from [8], a 30-m commercial PCF (NL-1550-NEG-1 fabricated by Crystal Fibre A/S) has Leff of 29.2 m (≅ L for α = 8×10-3 dB/m) and LD of 1 km (≫L for T 0 = 1.13 ps, β″ = 1.27 ps2/km at λ = 1550 nm), all of which fit the assumptions made for Eq. (1). These parameters result in a differential wave-vector mismatch (obtained from the measured group-index birefringence of the PCF [8]) given by

 

Fig. 2. With Φ = π , (a) (Media 1)the effective phase shift with L|∆K|≫2π as a function of ψ 2 , and (b) (Media 2) the effective phase shift for the best scenario (ψ 2 = π/4) as a function of L|∆K|.

Download Full Size | PPT Slide | PDF

ΔK=1.02×106×(1λ11λ2)+7.04×102×Inλ1λ2,

where ∆K is in 1/m and λ is in nm. With both pump and probe wavelengths within the ITU-T C-band (1530-1565 nm) Eq. (13) approximates to 0.03×∆λ, so that a PPD greater than about 7 nm is required to attain polarization-insensitive operation. However, the PPD can be significantly reduced with a longer or a more birefringent PCF.

Figure 3 plots Eq. (12) for the maximum and minimum effective phase shifts versus Φ for several different values of L|∆K| (as previously noted, the effective phase shift ϕeff fluctuates between two extremes termed the “maximum” and the “minimum” that generally depend on launch conditions.) Once again, ψ 2 = π/4 is termed the best scenario, while ψ 2 = 0 (or π/2), the worst scenario. Figure 3 reveals that the phase shifts for the best scenario with L|∆K| = 2π and L|∆K|≫2π are about the same. Fig. 3 also demonstrates that a polarization-independent phase shift can be attained by making ψ 2 = π/4 , Φ = 3π/4 and L |∆K|≫ 2π , although the resultant effective phase shift is only 0.5π (which can alsobe derived using Eq. (12)). However, to optimize the minimum phase shift to ϕeff ~ 0.59π for the best scenario, Φ = 1.1π will be required. Additionally, to achieve the largest minimum phase shift for the worst scenario, Φ = 1.5π will be required and that phase shift is only 0.5π. Moreover, at Φ = 1.5π, the effective phase shift expressed by Eq. (12) reduces to π-π cos(2ψ 1)| /2 , which is independent of ψ 2.

 

Fig. 3. Maximum and minimum effective phase shifts as functions of Φ (Eq. (12)).

Download Full Size | PPT Slide | PDF

3. Penalty induced by insufficient phase shift

After understanding the polarization-dependent phase shift, it is necessary to investigate how the receiver performance of the converted RZ-BPSK signal is affected by insufficient XPM-induced nonlinear phase shift (< π). To simplify the discussion, the induced nonlinear phase shift over the entire pulse is set constant assuming that the pump pulses are significantly broader than the probe pulses: for instance, the pump pulse was almost three-times larger than the probe pulse in an earlier work [8]. Additionally, if an optical matched filter is used preceding the receiver, the pulse shape of the RZ-BPSK signal will not affect the discussion. Further, the RZ-BPSK probe E(t) = Ex(t) + Ey(t)ŷ is equivalent to either A 2(L) or A 2(0), depending on whether the RZ-OOK pump beam is a mark or a space. Considering a system limited by amplified spontaneous emission (ASE) noise, the received RZ-BPSK signal can be represented as E r(t) = (Ex(t) + nx(t)) + (Ey(t) + ny(t))ŷ [11]. The ASE noise components nx and ny are independent, identically distributed (i.i.d.) complex zero-mean circular Gaussian random variables (GRVs), and the variances of the real and imaginary parts of the noise components are identically σn 2 , i.e. 〈|nx|2〉 = 〈|ny|2 = 2σn 2 where 〈·〉 indicates the expectation value. At the output of the one-bit-delay interferometer, in which the RZ-BPSK signal interferes with its delayed replica, the output duobinary (DB) and alternate-mark inversion (AMI) signals respectively are (E r(t) + E r(t-T))/2 and (E r(t)-E r(t-T))/2.

Setting the detector responsivity to unity, the DB and AMI photocurrents of the detectors are represented as,

iDB(t)=p=x,yEp(t)+Ep(tT)2+np(t)+np(tT)22,
iAMI(t)=p=x,yEp(t)Ep(tT)2+np(t)np(tT)22.

Because the real and imaginary parts of all [np(tnp(t-T)] /2 combinations are still i.i.d. GRVs with variances of σn 2/2 , normalizing the photocurrents by σn 2/2 results in χ 2 distributions with four degrees of freedom and a non-centrality parameter of [12],

ξ=p=x,yEp(t)±Ep(tT)22σn2.

Accordingly, the probability density functions (p.d.f.) of the normalized photocurrents v = i/|A 2|2 are [12],

pv(v)=4ρsρsvξe2ρsvξ2I1(2ξρsv)

where ρs=|A 2|2/(2σn 2) is the signal-to-noise ratio (SNR), and I 1(x) is the first-order modified Bessel function of the first kind. For the in-phase case (denoted by Vin), both of E(t) and E(t - T) are equal to either A 2(0) or A 2(L), and the non-centrality parameters become ξ = 4ρs for the DB signal and ξ = 0 for the AMI signal. Therefore, the corresponding p.d.f.s are,

pVDBin(v)=2ρsve2ρs(1+v)I1(4ρsv),
pVAMIin(v)=4ρs2ve2ρsv,

Furthermore, if either one of E(t) and E(t-T) is A 2(0) and the other is A 2(L) for the out-of-phase case (denoted by Vout), the non-centrality parameters become ξ = 4ρscos2 (ϕeff/2) for the DB signal and ξ =4ρssin2 (ϕeff/2) for the AMI signal, and the p.d.f.s of the normalized photocurrents become

pVDBout(v)=2ρsvsec(ϕeff2)e2ρs[cos2(ϕeff2)+v]I1(4ρsvcos(ϕeff2)),
pVAMIout(v)=2ρsvcsc(ϕeff2)e2ρs[sin2(ϕeff2)+v]I1(4ρsvsin(ϕeff2)),

where the definition of ϕeff is given by Eq. (11). Because Eqs. (18)(21) are mutually independent, and iDB(t)-iAMI(t) is used to determine the logic of the received RZ-BPSK signal, the error probabilities are pein = 〈VDBin - VAMIin < h〉 and pDBout = 〈VDBout - VAMIout > h〉, where h is the normalized receiver threshold, and h = 100% indicates the threshold is equal to |A 2|2. Note that as ϕeff < π only affects VDBout and VAMIout h ≥ 0 is assumed here. Following Stein [13] and Ho [14], the analytical forms of error probabilities can be derived as

Pein=1Q1(2ρs,2ρsh)+12eρs+2ρshQ1(2ρs,22ρsh)
18e2ρs2ρshhI1(4ρsh)+18e2ρs2ρshn=1n(2h)nIn(4ρsh),
Peout=eρs(1+2h)n=0(an4an1an2+8m=0n1am)×cotn1(ϕeff2)In1(ρssin(ϕeff2)),
=eρs[1+cos2(ϕeff2)+2h]m=0(a˜n+4m=0n1(2nm1)a˜mm=1n1m=0m1a˜m)
×(cos(ϕeff2)2h)n1In1(4ρscos(ϕeff2)),

where

am=18δ0,m+18l=1m(1)nll!(m1l1)(4ρsh)l,
a¯m=18δ0,m+18l=1m1l!(m1l1)(ρssin2(ϕeff2))l,

Q 1(α, β) is the first-order Marcum Q-function; nk is the binomial coefficient, and δm,n is the Kronecker delta function. Equations (22) and (23) are equivalent, but they are numerically efficient for respectively low and high thresholds. If the in-phase and out-of-phase cases are equally probable, the total error probability is then pe = (pein + peout)/2. Recall that the fixed threshold of zero for an ideal DPSK signal is one of its advantages, compared with the SNR-dependent threshold of an RZ-OOK signal. For the converted RZ-BPSK signal however, the optimal threshold is not zero for ϕeff < π , and can be obtained by setting (dpe/dh)|h=hopt = 0.

Furthermore, hopt could be approximated as opt = (〈VDBin - VAMIout)-〈VDBout - VAMIout〉)/2 = cos2(ϕeff/2) assuming that the in-phase and out-of-phase distributions are symmetric.

The exact and approximated optimal thresholds are depicted in Fig. 4(a). In fact, hopt is SNR-dependent, and those plotted in Fig. 4(a) correspond to sensitivities (SNR required to reach the error probability 10-9), which are plotted in Fig. 4(b). The maximum sensitivity difference between using hopt and opt is only about 0.7 dB indicating that the approximation is acceptable. The case of zero threshold is also shown for comparison, for which the RZ-BPSK signal cannot achieve error free reception for ϕeff > π/2. The optimal thresholds and the penalties in receiver sensitivity (relative to ideal or baseline RZ-BPSK) as functions of Ф are plotted in Fig. 5 according to Figs. 3 and 4, where both cases of the maximum and the minimum ϕeff are shown to illustrate the fluctuations in threshold and sensitivity penalty as the pump SOP is varied. For the worst scenario with L|∆K|≫2π , and Φ = π (which yields a ϕeff fluctuation of π - π/3 from Fig. 3 as the pump SOP is swept over the Poincaré sphere), the resultant threshold and sensitivity penalty of the RZ-BPSK probe respectively fluctuate as 0 - 76.2%, and 0 - 14 dB; but those fluctuations are respectively mitigated to 0 - 53.8% and 0 -7.8 dB by setting Φ = 1.5π . By contrast, for the best scenario with L|∆K|≫ 2π and Φ = π (which yields a ϕeff fluctuation of ~ 0.67π - 0.58π from Fig. 3 as the pump SOP is swept over the Poincaré sphere), the respective threshold and sensitivity penalty fluctuations reduce to 31.5 - 42.8% and 3.7 - 5.6 dB. Thus, launching the probe at π/4 with L|∆K|≫2π and a fixed Φ can decrease the fluctuations of both threshold and sensitivity caused by an arbitrarily polarized RZ-OOK pump, but the penalty is never zero. Moreover, for perfect polarization-independent operation (ψ 2 = π/4 , Φ = 3π/4 and L|∆K|≫2π in Fig. 3), the phase shift is always only π/2, and the sensitivity penalty is 7.8 dB (relative to ideal RZ-BPSK) which agrees well with the experimental value of ~ 7 dB [8]. For this case, it implies that RZ-BPSK requires higher SNR to achieve an error probability compared with ideal RZ-OOK due to different statistics, even though their symbol distances on the constellation diagram are the same (since the phase shift for RZ-BPSK is only π/2 for polarization-independent operation).

 

Fig. 4. (a) The optimal thresholds, and (b) the sensitivities as functions of ϕeff.

Download Full Size | PPT Slide | PDF

 

Fig. 5. (a) The optimal thresholds, and (b) the sensitivity penalty as functions of Φ.

Download Full Size | PPT Slide | PDF

4. Conclusion

This work theoretically analyzes the XPM effect for RZ-OOK-to-BPSK format conversion in a highly birefringent HNLF with an arbitrary polarized RZ-OOK pump beam. The nonlinear phase-shift-dependent receiver sensitivity and threshold of the resultant RZ-BPSK signal are also derived analytically for the first time. When the XPM-induced nonlinear phase shift ϕeff is set to π at large (L|∆K|ɫ2π) PPD, the probe’s effective phase shift ϕeff will fluctuate between π and π/3 once the pump SOP begins to sweep the Poincaré sphere. For this scenario, the corresponding sensitivity (SNR required to reach the error probability 10-9) penalty and threshold (normalized to |A 2|2) of the RZ-BPSK probe fluctuate as 13.4 - 27.5 dB and 0 - 76.2% respectively. By contrast, although launching the probe at π/4 relative to the birefringent axes with sufficient PPD (L|∆K|≥ 2π) and Φ = π may not achieve the desired π effective phase shift, it can reduce the respective fluctuations in sensitivity penalty and threshold to 3.7 - 5.6 dB and 31.5 - 42.8%.

References and links

1. A. H. Gnauck and P. J. Winzer, “Optical phase-shift-keyed transmission,” J. Lightwave Technol . 23, 115–130 (2005). [CrossRef]  

2. K. Mishina, A. Maruta, S. Mitani, T. Miyahara, K. Ishida, K. Shimizu, T. Hatta, K. Motoshima, and K.-I. Kitayama, “NRZ-OOK-to-RZ-BPSK modulation-format conversion using SOA-MZI wavelength converter,” J. Lightwave Technol . 24, 3751–3758 (2006). [CrossRef]  

3. W. Astar, A. S. Lenihan, and G. M. Carter, “Performance of DBPSK in a 5 × 10 Gb/s mixed modulation format Raman/EDFA WDM system,” IEEE Photon. Technol. Lett . 17, 2766–2768 (2005). [CrossRef]  

4. W. Astar and G. M. Carter, “10 Gbit/s RZ-OOK to RZ-BPSK format conversion using SOA and synchronous pulse carver,” Electron. Lett . 44, 369–370 (2008). [CrossRef]  

5. Cishuo Yan, Yikai Su, Lilin Yi, Lufeng Leng, Xiangqing Tian, Xinyu Vu, and Yue Tian, “All-optical format conversion from NRZ to BPSK using a single saturated SOA,” IEEE Photon. Technol. Lett . 18, 2368–2370 (2006). [CrossRef]  

6. H. Jiang, He Wen, Liuyan Han, Yili Guo, and Hanyi Zhang, “All-optical NRZ-OOK to BPSK format conversion in an SOA-based nonlinear polarization switch,” IEEE Photon. Technol. Lett . 19, 1985–1987 (2007). [CrossRef]  

7. K. Mishina, S. Kitagawa, and A. Maruta, “All-optical modulation format conversion from on-off-keying to multiple-level phase-shift-keying based on nonlinearity in optical fiber,” Opt. Express 15, 8444–8453 (2007), http://www.opticsexpress.org/abstract.cfm?uri=OE-15-13-8444. [CrossRef]   [PubMed]  

8. W. Astar, C.-C. Wei, Y.-J. Chen, J. Chen, and G. M. Carter, “Polarization-insensitive, 40 Gb/s wavelength and RZ-OOK-to-RZ-BPSK modulation format conversion by XPM in a highly nonlinear PCF,” Opt. Express 16, 12039–12049 (2008), ttp://www.opticsinfobase.org/abstract.cfm?URI=oe-16-16-12039. [CrossRef]   [PubMed]  

9. S. Kumar, A. Selvarajan, and G. Anand, “Nonlinear propagation of two optical pulses of two different frequencies in birefringent fibers,” J. Opt. Soc. Am . B 11, 810–817 (1994). [CrossRef]  

10. G. P. Agrawal, Nonlinear Fiber Optics (Academic Press, San Diego, CA, 2001), Chap. 7.

11. D. Marcuse, “Derivation of analytical expressions for the bit-error probability in lightwave systems with optical amplifiers,” J. Lightwave Technol . 8, 1819–1826 (1990). [CrossRef]  

12. E. W. Weisstein, “Noncentral Chi-Squared Distribution.” From MathWorld--A Wolfram Web Resource . http://mathworld.wolfram.com/NoncentralChi-SquaredDistribution.html.

13. S. Stein, “Unified analysis of certain coherent and noncoherent binary communications systems,” IEEE. Trans. Info. Theory 10, 43–51 (1964). [CrossRef]  

14. K.-P. Ho, Phase-Modulated Optical Communication System (Springer, 2005), Appendix 3.A.

References

  • View by:
  • |
  • |
  • |

  1. A. H. Gnauck and P. J. Winzer, “Optical phase-shift-keyed transmission,” J. Lightwave Technol.  23, 115–130 (2005).
    [Crossref]
  2. K. Mishina, A. Maruta, S. Mitani, T. Miyahara, K. Ishida, K. Shimizu, T. Hatta, K. Motoshima, and K.-I. Kitayama, “NRZ-OOK-to-RZ-BPSK modulation-format conversion using SOA-MZI wavelength converter,” J. Lightwave Technol.  24, 3751–3758 (2006).
    [Crossref]
  3. W. Astar, A. S. Lenihan, and G. M. Carter, “Performance of DBPSK in a 5 × 10 Gb/s mixed modulation format Raman/EDFA WDM system,” IEEE Photon. Technol. Lett.  17, 2766–2768 (2005).
    [Crossref]
  4. W. Astar and G. M. Carter, “10 Gbit/s RZ-OOK to RZ-BPSK format conversion using SOA and synchronous pulse carver,” Electron. Lett.  44, 369–370 (2008).
    [Crossref]
  5. Cishuo Yan, Yikai Su, Lilin Yi, Lufeng Leng, Xiangqing Tian, Xinyu Vu, and Yue Tian, “All-optical format conversion from NRZ to BPSK using a single saturated SOA,” IEEE Photon. Technol. Lett.  18, 2368–2370 (2006).
    [Crossref]
  6. H. Jiang, He Wen, Liuyan Han, Yili Guo, and Hanyi Zhang, “All-optical NRZ-OOK to BPSK format conversion in an SOA-based nonlinear polarization switch,” IEEE Photon. Technol. Lett.  19, 1985–1987 (2007).
    [Crossref]
  7. K. Mishina, S. Kitagawa, and A. Maruta, “All-optical modulation format conversion from on-off-keying to multiple-level phase-shift-keying based on nonlinearity in optical fiber,” Opt. Express 15, 8444–8453 (2007), http://www.opticsexpress.org/abstract.cfm?uri=OE-15-13-8444.
    [Crossref] [PubMed]
  8. W. Astar, C.-C. Wei, Y.-J. Chen, J. Chen, and G. M. Carter, “Polarization-insensitive, 40 Gb/s wavelength and RZ-OOK-to-RZ-BPSK modulation format conversion by XPM in a highly nonlinear PCF,” Opt. Express 16, 12039–12049 (2008), ttp://www.opticsinfobase.org/abstract.cfm?URI=oe-16-16-12039.
    [Crossref] [PubMed]
  9. S. Kumar, A. Selvarajan, and G. Anand, “Nonlinear propagation of two optical pulses of two different frequencies in birefringent fibers,” J. Opt. Soc. Am. B  11, 810–817 (1994).
    [Crossref]
  10. G. P. Agrawal, Nonlinear Fiber Optics (Academic Press, San Diego, CA, 2001), Chap. 7.
  11. D. Marcuse, “Derivation of analytical expressions for the bit-error probability in lightwave systems with optical amplifiers,” J. Lightwave Technol.  8, 1819–1826 (1990).
    [Crossref]
  12. E. W. Weisstein, “Noncentral Chi-Squared Distribution.” From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/NoncentralChi-SquaredDistribution.html.
  13. S. Stein, “Unified analysis of certain coherent and noncoherent binary communications systems,” IEEE. Trans. Info. Theory 10, 43–51 (1964).
    [Crossref]
  14. K.-P. Ho, Phase-Modulated Optical Communication System (Springer, 2005), Appendix 3.A.

2008 (2)

2007 (2)

H. Jiang, He Wen, Liuyan Han, Yili Guo, and Hanyi Zhang, “All-optical NRZ-OOK to BPSK format conversion in an SOA-based nonlinear polarization switch,” IEEE Photon. Technol. Lett.  19, 1985–1987 (2007).
[Crossref]

K. Mishina, S. Kitagawa, and A. Maruta, “All-optical modulation format conversion from on-off-keying to multiple-level phase-shift-keying based on nonlinearity in optical fiber,” Opt. Express 15, 8444–8453 (2007), http://www.opticsexpress.org/abstract.cfm?uri=OE-15-13-8444.
[Crossref] [PubMed]

2006 (2)

Cishuo Yan, Yikai Su, Lilin Yi, Lufeng Leng, Xiangqing Tian, Xinyu Vu, and Yue Tian, “All-optical format conversion from NRZ to BPSK using a single saturated SOA,” IEEE Photon. Technol. Lett.  18, 2368–2370 (2006).
[Crossref]

K. Mishina, A. Maruta, S. Mitani, T. Miyahara, K. Ishida, K. Shimizu, T. Hatta, K. Motoshima, and K.-I. Kitayama, “NRZ-OOK-to-RZ-BPSK modulation-format conversion using SOA-MZI wavelength converter,” J. Lightwave Technol.  24, 3751–3758 (2006).
[Crossref]

2005 (2)

W. Astar, A. S. Lenihan, and G. M. Carter, “Performance of DBPSK in a 5 × 10 Gb/s mixed modulation format Raman/EDFA WDM system,” IEEE Photon. Technol. Lett.  17, 2766–2768 (2005).
[Crossref]

A. H. Gnauck and P. J. Winzer, “Optical phase-shift-keyed transmission,” J. Lightwave Technol.  23, 115–130 (2005).
[Crossref]

1994 (1)

S. Kumar, A. Selvarajan, and G. Anand, “Nonlinear propagation of two optical pulses of two different frequencies in birefringent fibers,” J. Opt. Soc. Am. B  11, 810–817 (1994).
[Crossref]

1990 (1)

D. Marcuse, “Derivation of analytical expressions for the bit-error probability in lightwave systems with optical amplifiers,” J. Lightwave Technol.  8, 1819–1826 (1990).
[Crossref]

1964 (1)

S. Stein, “Unified analysis of certain coherent and noncoherent binary communications systems,” IEEE. Trans. Info. Theory 10, 43–51 (1964).
[Crossref]

Agrawal, G. P.

G. P. Agrawal, Nonlinear Fiber Optics (Academic Press, San Diego, CA, 2001), Chap. 7.

Anand, G.

S. Kumar, A. Selvarajan, and G. Anand, “Nonlinear propagation of two optical pulses of two different frequencies in birefringent fibers,” J. Opt. Soc. Am. B  11, 810–817 (1994).
[Crossref]

Astar, W.

W. Astar and G. M. Carter, “10 Gbit/s RZ-OOK to RZ-BPSK format conversion using SOA and synchronous pulse carver,” Electron. Lett.  44, 369–370 (2008).
[Crossref]

W. Astar, C.-C. Wei, Y.-J. Chen, J. Chen, and G. M. Carter, “Polarization-insensitive, 40 Gb/s wavelength and RZ-OOK-to-RZ-BPSK modulation format conversion by XPM in a highly nonlinear PCF,” Opt. Express 16, 12039–12049 (2008), ttp://www.opticsinfobase.org/abstract.cfm?URI=oe-16-16-12039.
[Crossref] [PubMed]

W. Astar, A. S. Lenihan, and G. M. Carter, “Performance of DBPSK in a 5 × 10 Gb/s mixed modulation format Raman/EDFA WDM system,” IEEE Photon. Technol. Lett.  17, 2766–2768 (2005).
[Crossref]

Carter, G. M.

W. Astar and G. M. Carter, “10 Gbit/s RZ-OOK to RZ-BPSK format conversion using SOA and synchronous pulse carver,” Electron. Lett.  44, 369–370 (2008).
[Crossref]

W. Astar, C.-C. Wei, Y.-J. Chen, J. Chen, and G. M. Carter, “Polarization-insensitive, 40 Gb/s wavelength and RZ-OOK-to-RZ-BPSK modulation format conversion by XPM in a highly nonlinear PCF,” Opt. Express 16, 12039–12049 (2008), ttp://www.opticsinfobase.org/abstract.cfm?URI=oe-16-16-12039.
[Crossref] [PubMed]

W. Astar, A. S. Lenihan, and G. M. Carter, “Performance of DBPSK in a 5 × 10 Gb/s mixed modulation format Raman/EDFA WDM system,” IEEE Photon. Technol. Lett.  17, 2766–2768 (2005).
[Crossref]

Chen, J.

Chen, Y.-J.

Gnauck, A. H.

A. H. Gnauck and P. J. Winzer, “Optical phase-shift-keyed transmission,” J. Lightwave Technol.  23, 115–130 (2005).
[Crossref]

Guo, Yili

H. Jiang, He Wen, Liuyan Han, Yili Guo, and Hanyi Zhang, “All-optical NRZ-OOK to BPSK format conversion in an SOA-based nonlinear polarization switch,” IEEE Photon. Technol. Lett.  19, 1985–1987 (2007).
[Crossref]

Han, Liuyan

H. Jiang, He Wen, Liuyan Han, Yili Guo, and Hanyi Zhang, “All-optical NRZ-OOK to BPSK format conversion in an SOA-based nonlinear polarization switch,” IEEE Photon. Technol. Lett.  19, 1985–1987 (2007).
[Crossref]

Hatta, T.

K. Mishina, A. Maruta, S. Mitani, T. Miyahara, K. Ishida, K. Shimizu, T. Hatta, K. Motoshima, and K.-I. Kitayama, “NRZ-OOK-to-RZ-BPSK modulation-format conversion using SOA-MZI wavelength converter,” J. Lightwave Technol.  24, 3751–3758 (2006).
[Crossref]

Ho, K.-P.

K.-P. Ho, Phase-Modulated Optical Communication System (Springer, 2005), Appendix 3.A.

Ishida, K.

K. Mishina, A. Maruta, S. Mitani, T. Miyahara, K. Ishida, K. Shimizu, T. Hatta, K. Motoshima, and K.-I. Kitayama, “NRZ-OOK-to-RZ-BPSK modulation-format conversion using SOA-MZI wavelength converter,” J. Lightwave Technol.  24, 3751–3758 (2006).
[Crossref]

Jiang, H.

H. Jiang, He Wen, Liuyan Han, Yili Guo, and Hanyi Zhang, “All-optical NRZ-OOK to BPSK format conversion in an SOA-based nonlinear polarization switch,” IEEE Photon. Technol. Lett.  19, 1985–1987 (2007).
[Crossref]

Kitagawa, S.

Kitayama, K.-I.

K. Mishina, A. Maruta, S. Mitani, T. Miyahara, K. Ishida, K. Shimizu, T. Hatta, K. Motoshima, and K.-I. Kitayama, “NRZ-OOK-to-RZ-BPSK modulation-format conversion using SOA-MZI wavelength converter,” J. Lightwave Technol.  24, 3751–3758 (2006).
[Crossref]

Kumar, S.

S. Kumar, A. Selvarajan, and G. Anand, “Nonlinear propagation of two optical pulses of two different frequencies in birefringent fibers,” J. Opt. Soc. Am. B  11, 810–817 (1994).
[Crossref]

Leng, Lufeng

Cishuo Yan, Yikai Su, Lilin Yi, Lufeng Leng, Xiangqing Tian, Xinyu Vu, and Yue Tian, “All-optical format conversion from NRZ to BPSK using a single saturated SOA,” IEEE Photon. Technol. Lett.  18, 2368–2370 (2006).
[Crossref]

Lenihan, A. S.

W. Astar, A. S. Lenihan, and G. M. Carter, “Performance of DBPSK in a 5 × 10 Gb/s mixed modulation format Raman/EDFA WDM system,” IEEE Photon. Technol. Lett.  17, 2766–2768 (2005).
[Crossref]

Marcuse, D.

D. Marcuse, “Derivation of analytical expressions for the bit-error probability in lightwave systems with optical amplifiers,” J. Lightwave Technol.  8, 1819–1826 (1990).
[Crossref]

Maruta, A.

K. Mishina, S. Kitagawa, and A. Maruta, “All-optical modulation format conversion from on-off-keying to multiple-level phase-shift-keying based on nonlinearity in optical fiber,” Opt. Express 15, 8444–8453 (2007), http://www.opticsexpress.org/abstract.cfm?uri=OE-15-13-8444.
[Crossref] [PubMed]

K. Mishina, A. Maruta, S. Mitani, T. Miyahara, K. Ishida, K. Shimizu, T. Hatta, K. Motoshima, and K.-I. Kitayama, “NRZ-OOK-to-RZ-BPSK modulation-format conversion using SOA-MZI wavelength converter,” J. Lightwave Technol.  24, 3751–3758 (2006).
[Crossref]

Mishina, K.

K. Mishina, S. Kitagawa, and A. Maruta, “All-optical modulation format conversion from on-off-keying to multiple-level phase-shift-keying based on nonlinearity in optical fiber,” Opt. Express 15, 8444–8453 (2007), http://www.opticsexpress.org/abstract.cfm?uri=OE-15-13-8444.
[Crossref] [PubMed]

K. Mishina, A. Maruta, S. Mitani, T. Miyahara, K. Ishida, K. Shimizu, T. Hatta, K. Motoshima, and K.-I. Kitayama, “NRZ-OOK-to-RZ-BPSK modulation-format conversion using SOA-MZI wavelength converter,” J. Lightwave Technol.  24, 3751–3758 (2006).
[Crossref]

Mitani, S.

K. Mishina, A. Maruta, S. Mitani, T. Miyahara, K. Ishida, K. Shimizu, T. Hatta, K. Motoshima, and K.-I. Kitayama, “NRZ-OOK-to-RZ-BPSK modulation-format conversion using SOA-MZI wavelength converter,” J. Lightwave Technol.  24, 3751–3758 (2006).
[Crossref]

Miyahara, T.

K. Mishina, A. Maruta, S. Mitani, T. Miyahara, K. Ishida, K. Shimizu, T. Hatta, K. Motoshima, and K.-I. Kitayama, “NRZ-OOK-to-RZ-BPSK modulation-format conversion using SOA-MZI wavelength converter,” J. Lightwave Technol.  24, 3751–3758 (2006).
[Crossref]

Motoshima, K.

K. Mishina, A. Maruta, S. Mitani, T. Miyahara, K. Ishida, K. Shimizu, T. Hatta, K. Motoshima, and K.-I. Kitayama, “NRZ-OOK-to-RZ-BPSK modulation-format conversion using SOA-MZI wavelength converter,” J. Lightwave Technol.  24, 3751–3758 (2006).
[Crossref]

Selvarajan, A.

S. Kumar, A. Selvarajan, and G. Anand, “Nonlinear propagation of two optical pulses of two different frequencies in birefringent fibers,” J. Opt. Soc. Am. B  11, 810–817 (1994).
[Crossref]

Shimizu, K.

K. Mishina, A. Maruta, S. Mitani, T. Miyahara, K. Ishida, K. Shimizu, T. Hatta, K. Motoshima, and K.-I. Kitayama, “NRZ-OOK-to-RZ-BPSK modulation-format conversion using SOA-MZI wavelength converter,” J. Lightwave Technol.  24, 3751–3758 (2006).
[Crossref]

Stein, S.

S. Stein, “Unified analysis of certain coherent and noncoherent binary communications systems,” IEEE. Trans. Info. Theory 10, 43–51 (1964).
[Crossref]

Su, Yikai

Cishuo Yan, Yikai Su, Lilin Yi, Lufeng Leng, Xiangqing Tian, Xinyu Vu, and Yue Tian, “All-optical format conversion from NRZ to BPSK using a single saturated SOA,” IEEE Photon. Technol. Lett.  18, 2368–2370 (2006).
[Crossref]

Tian, Xiangqing

Cishuo Yan, Yikai Su, Lilin Yi, Lufeng Leng, Xiangqing Tian, Xinyu Vu, and Yue Tian, “All-optical format conversion from NRZ to BPSK using a single saturated SOA,” IEEE Photon. Technol. Lett.  18, 2368–2370 (2006).
[Crossref]

Tian, Yue

Cishuo Yan, Yikai Su, Lilin Yi, Lufeng Leng, Xiangqing Tian, Xinyu Vu, and Yue Tian, “All-optical format conversion from NRZ to BPSK using a single saturated SOA,” IEEE Photon. Technol. Lett.  18, 2368–2370 (2006).
[Crossref]

Vu, Xinyu

Cishuo Yan, Yikai Su, Lilin Yi, Lufeng Leng, Xiangqing Tian, Xinyu Vu, and Yue Tian, “All-optical format conversion from NRZ to BPSK using a single saturated SOA,” IEEE Photon. Technol. Lett.  18, 2368–2370 (2006).
[Crossref]

Wei, C.-C.

Weisstein, E. W.

E. W. Weisstein, “Noncentral Chi-Squared Distribution.” From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/NoncentralChi-SquaredDistribution.html.

Wen, He

H. Jiang, He Wen, Liuyan Han, Yili Guo, and Hanyi Zhang, “All-optical NRZ-OOK to BPSK format conversion in an SOA-based nonlinear polarization switch,” IEEE Photon. Technol. Lett.  19, 1985–1987 (2007).
[Crossref]

Winzer, P. J.

A. H. Gnauck and P. J. Winzer, “Optical phase-shift-keyed transmission,” J. Lightwave Technol.  23, 115–130 (2005).
[Crossref]

Yan, Cishuo

Cishuo Yan, Yikai Su, Lilin Yi, Lufeng Leng, Xiangqing Tian, Xinyu Vu, and Yue Tian, “All-optical format conversion from NRZ to BPSK using a single saturated SOA,” IEEE Photon. Technol. Lett.  18, 2368–2370 (2006).
[Crossref]

Yi, Lilin

Cishuo Yan, Yikai Su, Lilin Yi, Lufeng Leng, Xiangqing Tian, Xinyu Vu, and Yue Tian, “All-optical format conversion from NRZ to BPSK using a single saturated SOA,” IEEE Photon. Technol. Lett.  18, 2368–2370 (2006).
[Crossref]

Zhang, Hanyi

H. Jiang, He Wen, Liuyan Han, Yili Guo, and Hanyi Zhang, “All-optical NRZ-OOK to BPSK format conversion in an SOA-based nonlinear polarization switch,” IEEE Photon. Technol. Lett.  19, 1985–1987 (2007).
[Crossref]

Electron. Lett (1)

W. Astar and G. M. Carter, “10 Gbit/s RZ-OOK to RZ-BPSK format conversion using SOA and synchronous pulse carver,” Electron. Lett.  44, 369–370 (2008).
[Crossref]

From MathWorld--A Wolfram Web Resource (1)

E. W. Weisstein, “Noncentral Chi-Squared Distribution.” From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/NoncentralChi-SquaredDistribution.html.

IEEE Photon. Technol. Lett (3)

W. Astar, A. S. Lenihan, and G. M. Carter, “Performance of DBPSK in a 5 × 10 Gb/s mixed modulation format Raman/EDFA WDM system,” IEEE Photon. Technol. Lett.  17, 2766–2768 (2005).
[Crossref]

Cishuo Yan, Yikai Su, Lilin Yi, Lufeng Leng, Xiangqing Tian, Xinyu Vu, and Yue Tian, “All-optical format conversion from NRZ to BPSK using a single saturated SOA,” IEEE Photon. Technol. Lett.  18, 2368–2370 (2006).
[Crossref]

H. Jiang, He Wen, Liuyan Han, Yili Guo, and Hanyi Zhang, “All-optical NRZ-OOK to BPSK format conversion in an SOA-based nonlinear polarization switch,” IEEE Photon. Technol. Lett.  19, 1985–1987 (2007).
[Crossref]

IEEE. Trans. Info. Theory (1)

S. Stein, “Unified analysis of certain coherent and noncoherent binary communications systems,” IEEE. Trans. Info. Theory 10, 43–51 (1964).
[Crossref]

J. Lightwave Technol (3)

D. Marcuse, “Derivation of analytical expressions for the bit-error probability in lightwave systems with optical amplifiers,” J. Lightwave Technol.  8, 1819–1826 (1990).
[Crossref]

A. H. Gnauck and P. J. Winzer, “Optical phase-shift-keyed transmission,” J. Lightwave Technol.  23, 115–130 (2005).
[Crossref]

K. Mishina, A. Maruta, S. Mitani, T. Miyahara, K. Ishida, K. Shimizu, T. Hatta, K. Motoshima, and K.-I. Kitayama, “NRZ-OOK-to-RZ-BPSK modulation-format conversion using SOA-MZI wavelength converter,” J. Lightwave Technol.  24, 3751–3758 (2006).
[Crossref]

J. Opt. Soc. Am (1)

S. Kumar, A. Selvarajan, and G. Anand, “Nonlinear propagation of two optical pulses of two different frequencies in birefringent fibers,” J. Opt. Soc. Am. B  11, 810–817 (1994).
[Crossref]

Opt. Express (2)

Other (2)

G. P. Agrawal, Nonlinear Fiber Optics (Academic Press, San Diego, CA, 2001), Chap. 7.

K.-P. Ho, Phase-Modulated Optical Communication System (Springer, 2005), Appendix 3.A.

Supplementary Material (2)

» Media 1: AVI (704 KB)     
» Media 2: AVI (1726 KB)     

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (5)

Fig. 1.
Fig. 1. RZ-OOK-to-RZ-BPSK format conversion in a birefringent fiber.
Fig. 2.
Fig. 2. With Φ = π , (a) (Media 1)the effective phase shift with L|∆K|≫2π as a function of ψ 2 , and (b) (Media 2) the effective phase shift for the best scenario (ψ 2 = π/4) as a function of L|∆K|.
Fig. 3.
Fig. 3. Maximum and minimum effective phase shifts as functions of Φ (Eq. (12)).
Fig. 4.
Fig. 4. (a) The optimal thresholds, and (b) the sensitivities as functions of ϕeff .
Fig. 5.
Fig. 5. (a) The optimal thresholds, and (b) the sensitivity penalty as functions of Φ.

Equations (29)

Equations on this page are rendered with MathJax. Learn more.

d A mp dz = ( A mp 2 + 2 A np 2 + 2 3 A mq 2 + 2 3 A nq 2 ) A mp + j γ 3 A mp * A mq 2 e j 2 Δ β mpq z
+ j 2 γ 3 A np * A nq A mq e j ( Δ β mpq + Δ β npq ) z + j 2 γ 3 A np A nq * A mq e j ( Δ β mpq Δ β npq ) z ,
d A 1 p dz = ( A 1 p 2 + 2 3 A 1 q 2 ) A 1 p ,
d A 2 p dz = ( 2 A 1 p 2 + 2 3 A 1 q 2 ) A 2 p + j 2 γ 3 A 1 A 1 q * A 2 q e j ( Δ β 1 pq Δ β 2 pq ) z .
A 1 x ( z ) = A 1 x ( 0 ) exp [ j γ P 1 3 ( 2 + cos 2 ψ 1 ) z ] ,
A 1 y ( z ) = A 1 y ( 0 ) exp [ j γ P 1 3 ( 2 + sin 2 ψ 1 ) z ] .
A 2 x ( L ) = A 2 x ( 0 ) [ cos ( kL 2 ) + j μ x sin ( kL 2 ) ] exp [ j ( 3 γ P 1 2 j γ P 1 3 sin 2 ψ 1 j ΔK 2 ) L ] ,
A 2 y ( L ) = A 2 y ( 0 ) [ cos ( kL 2 ) + j μ y sin ( kL 2 ) ] exp [ j ( 3 γ P 1 2 j γ P 1 3 cos 2 ψ 1 + j ΔK 2 ) L ] ,
k = 4 9 γ 2 P 1 2 sin 2 ( 2 ψ 1 ) + [ γ P 1 cos ( 2 ψ 1 ) + ΔK ] 2 ,
μ x = 1 k [ γ P 1 cos ( 2 ψ 1 ) + ΔK + 2 γ P 1 3 tan ψ 2 sin ( 2 ψ 1 ) e j Δ θ ] ,
μ y = 1 k [ γ P 1 cos ( 2 ψ 1 ) ΔK + 2 γ P 1 3 cos ψ 2 sin ( 2 ψ 1 ) e j Δ θ ] ,
ϕ eff = cos 1 ( A 2 x ( 0 ) A 2 x ( L ) cos ϕ x + A 2 y ( 0 ) A 2 y ( L ) cos ϕ y A 2 x ( 0 ) 2 + A 2 y ( 0 ) 2 ) ,
ϕ eff = cos 1 ( cos 2 ψ 2 cos ( Φ ( 1 + 2 cos 2 ψ 1 ) 3 ) + sin 2 ψ 2 cos ( Φ ( 1 + 2 sin 2 ψ 1 ) 3 ) ) ,
ΔK = 1.02 × 10 6 × ( 1 λ 1 1 λ 2 ) + 7.04 × 10 2 × In λ 1 λ 2 ,
i DB ( t ) = p = x , y E p ( t ) + E p ( t T ) 2 + n p ( t ) + n p ( t T ) 2 2 ,
i AMI ( t ) = p = x , y E p ( t ) E p ( t T ) 2 + n p ( t ) n p ( t T ) 2 2 .
ξ = p = x , y E p ( t ) ± E p ( t T ) 2 2 σ n 2 .
p v ( v ) = 4 ρ s ρ s v ξ e 2 ρ s v ξ 2 I 1 ( 2 ξ ρ s v )
p V DB in ( v ) = 2 ρ s v e 2 ρ s ( 1 + v ) I 1 ( 4 ρ s v ) ,
p V AMI in ( v ) = 4 ρ s 2 v e 2 ρ s v ,
p V DB out ( v ) = 2 ρ s v sec ( ϕ eff 2 ) e 2 ρ s [ cos 2 ( ϕ eff 2 ) + v ] I 1 ( 4 ρ s v cos ( ϕ eff 2 ) ) ,
p V AMI out ( v ) = 2 ρ s v csc ( ϕ eff 2 ) e 2 ρ s [ sin 2 ( ϕ eff 2 ) + v ] I 1 ( 4 ρ s v sin ( ϕ eff 2 ) ) ,
P e in = 1 Q 1 ( 2 ρ s , 2 ρ s h ) + 1 2 e ρ s + 2 ρ s h Q 1 ( 2 ρ s , 2 2 ρ s h )
1 8 e 2 ρ s 2 ρ s h h I 1 ( 4 ρ s h ) + 1 8 e 2 ρ s 2 ρ s h n = 1 n ( 2 h ) n I n ( 4 ρ s h ) ,
P e out = e ρ s ( 1 + 2 h ) n = 0 ( a n 4 a n 1 a n 2 + 8 m = 0 n 1 a m ) × cot n 1 ( ϕ eff 2 ) I n 1 ( ρ s sin ( ϕ eff 2 ) ) ,
= e ρ s [ 1 + cos 2 ( ϕ eff 2 ) + 2 h ] m = 0 ( a ˜ n + 4 m = 0 n 1 ( 2 n m 1 ) a ˜ m m = 1 n 1 m = 0 m 1 a ˜ m )
× ( cos ( ϕ eff 2 ) 2 h ) n 1 I n 1 ( 4 ρ s cos ( ϕ eff 2 ) ) ,
a m = 1 8 δ 0 , m + 1 8 l = 1 m ( 1 ) n l l ! ( m 1 l 1 ) ( 4 ρ s h ) l ,
a ¯ m = 1 8 δ 0 , m + 1 8 l = 1 m 1 l ! ( m 1 l 1 ) ( ρ s sin 2 ( ϕ eff 2 ) ) l ,

Metrics