## Accurate BER evaluation for lumped DPSK and OOK systems with PMD and PDL

Optics Express, Vol. 15, Issue 15, pp. 9418-9433 (2007)

http://dx.doi.org/10.1364/OE.15.009418

Acrobat PDF (264 KB)

### Abstract

New forms using Dirac bra-ket notations and their transformations to express electrically filtered currents are presented for optical systems using either binary differential phase-shift keying (2-DPSK) or ON-OFF keying (OOK) with lumped first-order PMD and PDL, arbitrary optical and electrical filtering and pulse shaping. Based on these forms, the moment generating functions (MGFs) and bit-error-ratios (BERs) for different systems are obtained. Our results show that, for a given BER, 2-DPSK requires ~5dB lower input signal-to-noise ratio than OOK. By comparing BERs for different polarization systems, we also show that the PDL-induced partially polarized noise can significantly improve system performance and reduce BER variation caused by the random couplings between signal polarization, PDL and PMD vectors.

© 2007 Optical Society of America

## 1. Introduction

1. B. Huttner, C. Geiser, and N. Gisin, “Polarization-induced distortion in optical fiber networks with polarization-mode dispersion and polarization-dependent losses,” IEEE J. Select. Topics Quantum Electron. **6**, 317–329 (2000). [CrossRef]

2. I. T. Lima, A. O. Lima, Y. Sun, H. Jiao, J. Zweck, C. R. Menyuk, and G. M. Carter, “A receiver model for optical fiber communication systems with arbitrarily polarized noise,” J. Lightwave Technol. **23**, 1478–1490 (2005). [CrossRef]

3. A. Mecozzi and M. Shtaif, “Signal-to-noise-ratio degradation caused by polarization-dependent loss and the effect of dynamic gain equalization,” J. Lightwave Technol. **22**, 1856–1871 (2004). [CrossRef]

4. M. Shtaif and O. Rosenberg, “Polarization-dependent loss as a waveform-distorting mechanism and its effect on fiber-optical systems,” J. Lightwave Technol. **23**, 923–930 (2005). [CrossRef]

5. L. Chen, Z. Zhang, and X. Bao, “Combined PMD-PDL effects on BERs in simplified optical systems: an analytical approach,” Opt. Express **15**, 2106–2119 (2007). [CrossRef] [PubMed]

6. P. J. Winzer, S. Chandrasekhar, and H. Kim, “Impact of filtering on RZ-DPSK reception,” IEEE Photon. Technol. Lett. **15**, 840–842 (2003). [CrossRef]

^{-3}, typical BERs for optical systems are in the range of 10

^{-9}–10

^{-15}or below, where the usual Gaussian fitting Q-factor approximation does not work well. In this case, the BER can be obtained by calculating the characteristic function, or equivalently the moment-generating function (MGF) [7

7. D. Marcuse, “Derivation of analytical expressions for the bit-error probability in lightwave systems with optical amplifiers,” J. Lightwave Technol. **8**, 1816–1823 (1990). [CrossRef]

8. P. A. Humblet and M. Azizoglu, “On the bit error rate of lightwave systems with optical amplifiers,” J. Lightwave Technol. **9**, 1576–1582 (1991). [CrossRef]

9. E. Forestieri, “Evaluating the error probability in lightwave systems with chromatic dispersion, arbitrary pulse shape and pre-and postdetection filtering,” J. Lightwave Technol. **18**, 1493–1503 (2000). [CrossRef]

10. J. L. Rebola and A. V. T. Cartaxo, “Performance evaluation of optically preamplified receivers with partially polarized noise and arbitrary optical filtering: a rigorous approach,” IEE Proc. Optoelectron. **152**, 251–262 (2005). [CrossRef]

9. E. Forestieri, “Evaluating the error probability in lightwave systems with chromatic dispersion, arbitrary pulse shape and pre-and postdetection filtering,” J. Lightwave Technol. **18**, 1493–1503 (2000). [CrossRef]

10. J. L. Rebola and A. V. T. Cartaxo, “Performance evaluation of optically preamplified receivers with partially polarized noise and arbitrary optical filtering: a rigorous approach,” IEE Proc. Optoelectron. **152**, 251–262 (2005). [CrossRef]

5. L. Chen, Z. Zhang, and X. Bao, “Combined PMD-PDL effects on BERs in simplified optical systems: an analytical approach,” Opt. Express **15**, 2106–2119 (2007). [CrossRef] [PubMed]

9. E. Forestieri, “Evaluating the error probability in lightwave systems with chromatic dispersion, arbitrary pulse shape and pre-and postdetection filtering,” J. Lightwave Technol. **18**, 1493–1503 (2000). [CrossRef]

**18**, 1493–1503 (2000). [CrossRef]

10. J. L. Rebola and A. V. T. Cartaxo, “Performance evaluation of optically preamplified receivers with partially polarized noise and arbitrary optical filtering: a rigorous approach,” IEE Proc. Optoelectron. **152**, 251–262 (2005). [CrossRef]

## 2. System modeling and BER calculation

*s⃗*is launched into the fiber system with lumped chromatic dispersion (CD) and lumped PMD-PDL1 (first-order) [5

_{in}(t)5. L. Chen, Z. Zhang, and X. Bao, “Combined PMD-PDL effects on BERs in simplified optical systems: an analytical approach,” Opt. Express **15**, 2106–2119 (2007). [CrossRef] [PubMed]

*G*. The normalized ASE noise added at “4” in Fig. 1 is considered as additive white Gaussian noise

*n⃗*with two-sided power spectral density

_{in}(t)*n*≥1 is the spontaneous-emission or population-inversion parameter and

_{sp}*hn*is the photon energy. We assume that

*G*≫1 so that

*N0≈n*[9

_{sp}hnv**18**, 1493–1503 (2000). [CrossRef]

11. J. Wang and J. M. Kahn, “Impact of chromatic and polarization-mode dispersions on DPSK systems using interferometric demodulation and direct detection,” J. Lightwave Technol. **22**, 362–371 (2004). [CrossRef]

*t*. The balanced direct-detection with responstivity

_{k}*R*=1 yields the current at “7” in Fig. 1 which is given by [11

11. J. Wang and J. M. Kahn, “Impact of chromatic and polarization-mode dispersions on DPSK systems using interferometric demodulation and direct detection,” J. Lightwave Technol. **22**, 362–371 (2004). [CrossRef]

12. L. Xie, L. Chen, S. Hadjifaradji, and X. Bao, “WDM high speed chirped DPSK fiber optical system transmission modeling in presence of PMD, PDL, and CD,” Opt. Fiber Technol. **12**, 276–281 (2006). [CrossRef]

*T*is the bit-time interval,

_{b}*c.c*. or […]* stands for complex conjugation and

*s⃗o(t)*[n⃗o(t)] is the signal (noise) at the output of the optical filter (“6” in Fig. 1).

**18**, 1493–1503 (2000). [CrossRef]

*s⃗*in Fig. 1 can be assumed to be a periodic repetition of signal

_{in}(t)*NT*, i.e.,

_{b}*p(t)*determines the elementary input pulse shape and

*a*determines the logic value of the ith bit. In this paper, the input pulse shape is assumed to be

_{i}*E*is the optical energy per transmitted bit [11

_{b}11. J. Wang and J. M. Kahn, “Impact of chromatic and polarization-mode dispersions on DPSK systems using interferometric demodulation and direct detection,” J. Lightwave Technol. **22**, 362–371 (2004). [CrossRef]

*a*} for the OOK model is assumed to be a 2

_{i}^{5}-bit de Bruijn sequence [9

**18**, 1493–1503 (2000). [CrossRef]

**22**, 362–371 (2004). [CrossRef]

*a*(∈{

_{i}*e*}) is determined by requiring the received codes at sampling instants

^{j0},ejπ*t*-1), normalized as “0” or “1” with no signal distortion, form a de Bruijn sequence (see inset of Fig. 2). Therefore, the input

_{k}(t_{k}=t_{0}+kT_{b},k=0, …,N*s⃗*can be expanded in Fourier series,

_{in}(t)*p*⃗

*=[*

_{s}i*x,y*]

*in 2D Jones space.*

^{T}*n⃗*added at “4” in Fig. 1 can be decomposed along two orthonormal Jones vectors (e.g., |

_{in}(t)*e*⃗

*x*〉 and |

*e*⃗

*〉) and can be expressed in a Fourier series using a Karhunen-Loève expansion [9*

_{y}**18**, 1493–1503 (2000). [CrossRef]

*B*) and electrical (bandwidth

_{o}*B*) filters is

_{r}*i(t*in (1) is caused by the noise input within the time interval (

_{k})*t*) [9

_{k}-T_{o}, t_{k}**18**, 1493–1503 (2000). [CrossRef]

**152**, 251–262 (2005). [CrossRef]

*(N*are treated as complex independent and identically distributed (i.i.d.) random variables (r.v.) with Gaussian pdfs of zero mean and variance

_{in})_{i,m}(i=x,y)*σ*[9

^{2}=N_{0}/(2T_{o})**18**, 1493–1503 (2000). [CrossRef]

**152**, 251–262 (2005). [CrossRef]

### 2.1. BER in a 2-DPSK system with PMD, PDL and unpolarized noise

*y(t*given by (35), (40) and (42) in Appendix B, its MGF can be obtained by averaging over the “canonical” noise at “4” in Fig. 1. By using the formula

_{k})=y_{ss}+y_{nn}+y_{ns}*(Z*in (40) have zero mean and variance σ

_{i})m (i=α0,α⊥;m=-M, …,M)^{2}[10

**152**, 251–262 (2005). [CrossRef]

*y(t*can be written as

_{k})*y*is given by (35) and |b̃

_{ss}*D*|2 by (44). In this work the noise average is denoted as 〈.〉 and the Hermitian inner product as 〈.|.〉. Mean and variance of the detected current are given, respectively, by

_{m}**18**, 1493–1503 (2000). [CrossRef]

*y*is the detection threshold, + and

_{th}*C*+ correspond for

*y*<

_{ss}*y*, while - and

_{th}*C*_ for

*y*>

_{ss}*y*[9

_{th}**18**, 1493–1503 (2000). [CrossRef]

*t*)

_{k}=t_{0}+kT_{b}*D*(

^{i}*i=ss,nn,ns*) introduced in (27) to be unity, we can get the reduced

*y*from (35), |

_{ss}*b*̃

*|*

_{m}^{2}from (44),

*λ*from (23) and then the MGF from (5) for the 2D OOK system.

_{m}### 2.2. MGF of a 2-DPSK system with PMD, PDL and partially polarized noise

**152**, 251–262 (2005). [CrossRef]

*K*is Hermitian, it can be diagonalized with its two real eigenvalues

**152**, 251–262 (2005). [CrossRef]

**152**, 251–262 (2005). [CrossRef]

**152**, 251–262 (2005). [CrossRef]

*k*≥1, matrix

_{0}*K*contains both PDL and polarization-dependent gain (PDG), whereas

*T*has pure PDL effect.

_{PDL2}*β*=

_{m}*2σ*)

^{2}/λ^{D}_{m}*T*(10) needs to be replaced by matrix

_{PDL2}*K*(9). In this case the signal-signal beating

*y*in (35) still hold, except

_{ss}*)*

_{k}*′=*

_{ll}*k*

^{2}

_{0}Θ

*′. Also (45) and (46) now become*

_{ll}**152**, 251–262 (2005). [CrossRef]

*|(b̃*in MGF (12) now reduces to

^{D}_{i})m|^{2}*(i=k*

_{0}:+;i=k⊥:-)*b*|

_{m}^{2}near (24)] and the signal-signal beating term (34) now yields

*R*given by (22). Substituting (13) and (14) into (12) and replacing

_{ss}*λ*with

^{D}_{m}*λ*for the OOK system, we obtain (

_{m}*β*=2

_{m}*σ*)

^{2}λ_{m}**152**, 251–262 (2005). [CrossRef]

*p⃗*equals to -cos(2

_{s}·k⃗_{0}*θ*) of Ref. [10

**152**, 251–262 (2005). [CrossRef]

*q*is the polar angle between signal polarization and the assumed

*x*direction in the 2

*D*Jones space. In fact, if the assumed

*y*axis in Ref. [10

**152**, 251–262 (2005). [CrossRef]

*k⃗*〉, then we have

_{0}*p⃗*=-cos(2θ). (See formulas between (10) and (11) in Ref. [5

_{s}·k⃗_{0}**15**, 2106–2119 (2007). [CrossRef] [PubMed]

*k⃗*is the mapping of its Jones vector |

_{0}*k⃗*〉.)

_{0}## 3. Results and discussion

**22**, 362–371 (2004). [CrossRef]

### 3.1. Special cases of OOK and 2-DPSK systems

*H*=1/(1+

_{o}(f)*). The transfer function of the electrical filter in Fig. 1 is a fifth-order Bessel type, i.e.,*

_{j}2f/B_{o}*H*with F=2.43f/B

_{r}(f)=945/(jF^{5}+15F^{4}-j105F3-420F^{2}+ j945F+945)_{r}[11

**22**, 362–371 (2004). [CrossRef]

*y*and initial detection time

_{th}*t*are adjusted to get the minimal BER (8). For a system using RZ-2-DPSK format, its optimum threshold is fixed at

_{0}=t_{k}-kT_{b}(k=0, …,N-1)*y*≈0. This is because, for the cases discussed in this work, the filtered signal-signal beating is always symmetrically “balanced” with respect to

_{th}*y*≈0 [see the dashed curve in inset of Fig.2 (a)]. Therefore it is reasonable to minimize the BER at

_{th}*y*≈0. For an OOK system, its optimum threshold is always changed because the currents of marks and spaces are directly affected by any system variation, such as

_{th}*E*(i.e., OSNR [9

_{b}/N_{0}**18**, 1493–1503 (2000). [CrossRef]

*τ/T*[the normalized differential group delay (DGD)], etc.

_{b}^{-9}BER. They are almost the same as those given in Ref. [11

**22**, 362–371 (2004). [CrossRef]

### 3.2. Effect of polarized noise on the optical performance

*DOP*for these two systems using RZ-2-DPSK format with negligible PMD (τ=0,

*θ*=0). For the first system where the noise is unpolarized,

_{ατ}*BER*(assuming |

^{pa}_{PDL1}*p*⃗

*s*〉‖|α⃗0〉) and

*BER*(assuming |

^{or}_{PDL1}*p*⃗

*s*〉⊥|α⃗

_{0}〉), obtained from (5), (7) and (8) with PDL1-induced

*BER*(thin dashed) keeps unchanged, because the signal is polarized in the zero attenuation direction |α⃗

^{pa}_{PDL1}_{0}〉 and the two orthogonal components of the ASE noise are not affected by PDL1. On the other hand, when the signal is polarized in the maximum attenuation direction (|

*p*⃗

*s*〉 ⊥ |α⃗

_{0}〉), the signal will be severely distorted when the PDL1-induced

*DOP*becomes large. Therefore we get the thick dashed curve in Fig. 3(a). For the second system where the noise is partially polarized, we get

*BER*(thin solid) and

^{pa}_{PDL2}*BER*(thick solid) versus PDL2-induced

^{or}_{PDL2}*DOP*curves, obtained from (7), (8) and (11). As shown in Fig. 3(a),

*BER*(

^{pa}_{PDL2}*BER*) is lower than

^{or}_{PDL2}*BER*(

^{pa}_{PDL1}*BER*), respectively. This is because, no matter how the signal is polarized (|

^{or}_{PDL1}*p*⃗

*〉‖|*

_{s}*k*⃗

_{0}〉 or |

*p*⃗

*〉 ⊥ |*

_{s}*k*⃗

_{0}〉), the overall noise in the second system (no PDL1) is always smaller than that in the first system (no PDL2). Moreover

*BER*(thin solid) shows that although in the zero attenuation direction (|

^{pa}_{PDL2}*k*⃗

_{0}〉) both noise and signal are not changed, the noise attenuation in the orthogonal direction (|

*k*⊥0〉) can also improve the optical performance. Besides, at

*DOP*=0.28 (

*α*=

*α*≈2.5dB), the ratio

_{k}*BER*≈75 is much smaller than the ratio

^{or}_{PDL2}/BER^{pa}_{PDL2}*BER*≈600.

^{or}_{PDL1}/BER^{pa}_{PDL1}*p*⃗

*〉 and |α⃗*

_{s}_{0}〉 (dashed) and the coupling between |

*p*⃗

*〉 and |*

_{s}*k*⃗

_{0}〉 (solid), obtained by setting

*DOP*=0.28 and randomly changing parameters

*φ*) (weighting factor

_{ατ}-φ_{sτ}^{5}random realiztions. (To show clearly the two pdf curves in this inset, the dashed pdf curve for the unpolarized noise case is shifted up by 0.3.) Each pdf can be approximated as a rectangular pulse. This means, BERs with random directional couplings determined by

*θ*and (

_{i}(i=sτ,ατ)*φ*) are homogeneously distributed between thick and thin curves. As shown in this inset, the BER variation range and its average value for the case of unpolarized noise (dashed) are about one order of magnitude larger than those for the partially polarized noise case (solid). Therefore the PDL effect on the ASE noise can improve the system performance and reduce the BER variance caused by the directional coupling between the signal polarization and the PDL vector.

_{ατ}-φ_{sτ}^{-7}, calculated by using the 1D balanced DPSK model discussed in Appendix A, is used to verify

*BER*. Actually

^{pa}_{PDL2}*BER*approaches closely to this horizontal line when

^{pa}_{PDL2}*DOP*>0.6. We also consider the PDG effect by replacing (11) with (12) in the BER calculation. The two curves thus obtained (

*BER*and

^{or}_{K}*BER*, not shown) coincide almost exactly with the two solid curves (

^{pa}_{K}*BER*and

^{or}_{PDL2}*BER*) in Fig. 3(a), respectively. Mathematically one can prove that, for a 2-DPSK format with

^{pa}_{PDL2}*y*≈0, by scaling transformation

_{th}*s*→

*sk*[cf. the discussion between (11) and (12)], the BER obtained using (11) and (7) is the same as the BER using (12) and (7). Physically,

^{2}_{0}*K*is equivalent to

*PDL2*followed by an ideal amplifier with

*G*=

*k*>1, which does not cause further signal and noise distortion and therefore no additional effect on the BER.

_{0}*τ/T*=0.3 are depicted in Fig. 3(b). As shown, the PMD increases the BER and further reduces the BER fluctuation caused by the PMD-PDL-induced directional coupling (

_{b}*BER*≈66 and

^{or}_{PDL1}/BER^{pa}_{PDL1}*BER*≈12 with

^{or}_{PDL2}/BER^{pa}_{PDL2}*DOP*=0.28). In the inset of Fig. 3(b), we show the pdf versus BER with

*DOP*=0.28 for the cases of unpolarized and partially polarized noise. Due to the PMD-PDL interaction, each pdf can be approximated as a “smoothed” rectangular pulse. Note that, in this inset, the BER range of each pdf curve is different from the corresponding gap between

*BER*and

^{or}_{i}*BER*(

^{pa}_{i}*i=PDL1,PDL2*) in Fig. 3(b). As an example, for a system with partially polarized noise, the BER variation range of the solid curve in the inset of Fig. 3(b) is determined by BERs obtained by setting

*θ*=0 with

_{ατ}*θ*=π (~6×10

_{sτ}^{-5}) and

*θ*=0 (~8×10

_{sτ}^{-7}), whereas the thick (thin) solid curve in Fig. 3(b), obtained using

*θ*=

_{ατ}*θ*=

_{sτ}*π*/2 with (

*φ*)=

_{ατ}-φ_{sτ}*π*[(

*φ*)=0], yields

_{ατ}-φ_{sτ}*BER*≈3×10

^{or}_{PDL2}^{-5}(

*BER*≈1.8×10

^{pa}_{PDL2}^{-6}) at

*DOP*=0.28.

## 4. Conclusion

## Appendix A: Expressions of the filtered current: from 1D OOK to 1D DPSK

*h*], only those components with frequencies within the filter bandwidth

_{o}(t) [or H_{o}(f)*B*need to be considered. Because of this the Dirac bra-ket notations for input signal and noise at time

_{o}*t*can be introduced as

_{k}*h*and m must be determined iteratively [10

**152**, 251–262 (2005). [CrossRef]

**18**, 1493–1503 (2000). [CrossRef]

**22**, 362–371 (2004). [CrossRef]

_{2}=-λ

^{2}

*D*(λ)/(

*2πc*). For a nondispersion shifted fiber at

*λ*=1550nm, the dispersion parameter

*D*≈17ps/(km·nm). As usual, the Hermitian conjugate of a matrix

*A*is defined as (

*A*

^{†})

*=*

_{ij}*A**. The photoelectric current prior to the electrical filter is

_{ji}*i*(

_{1D}*t*)=〈

*s*(

^{o}*t*)+

*n*(

^{o}*t*)|

*s*(

^{o}*t*)+

*n*(

^{o}*t*)〉 (

*R*=1). Due to the response function of the electrical filter,

*h*(

_{r}*t*) [or

*H*(

_{r}*f*)], the filtered current at sampling time

*t*consists of three parts, i.e., the signal-signal beating, noise-noise beating and noise-signal beating

_{k}*m*=-

*M*, …,

*M, l=-L, …,L*. In the

*y*term given by (21), because

_{nn}*R*is a Hermitian matrix satisfying

_{nn}*R*is also Hermitian and it can be diagonalized by a orthogonal and unitary transformation

^{†}_{nn}=R_{nn}, O†_{nn}R_{nn}O_{nn}*U*

*Λmm′=δ*and

_{m,m}′l_{m}*U*being composed of eigenvectors |

*λ*〉 of

_{m}*λ*. Thus we have

_{m}*i*

^{D}

_{1 D}(

*t*)=[〈

*s*(

^{o}*t*+

*T*)+

_{b}*n*(

^{o}*t*+

*T*)|

_{b}*s*(

^{o}*t*)+

*n*(

^{o}*t*)

*i*+

*c.c*.]/2 [11

**22**, 362–371 (2004). [CrossRef]

12. L. Xie, L. Chen, S. Hadjifaradji, and X. Bao, “WDM high speed chirped DPSK fiber optical system transmission modeling in presence of PMD, PDL, and CD,” Opt. Fiber Technol. **12**, 276–281 (2006). [CrossRef]

*≡*

^{D}*U*

^{†}

*O*

^{†}

_{nn}

*R*with (Λ

^{D}_{nn}O_{nn}U^{D})

_{mm}′=

*δ*and

_{m,m}′λ^{D}_{m}*b*|

_{m}^{2}discussed above.

## Appendix B: Filtered current in DPSK system with unpolarized ASE noise

*ω*=2

_{l}*πl*/(

*NT*), σ⃗ is the standard physics notation for Pauli spin matrices satisfying σ⃗×σ⃗=2

_{b}*jσ*⃗ [5

**15**, 2106–2119 (2007). [CrossRef] [PubMed]

_{0}=τ⃗/τ are assumed to be frequency independent. Matrix (28) has two orthonormal eigenvectors (|τ⃗

_{0}〉 and |τ⃗⊥〉) in 2D Jones space, indicating the directions of slow (|τ⃗

_{0}〉) and fast (|τ⃗⊥〉) principal state of polarizations (PSPs) with matrix elements [5

**15**, 2106–2119 (2007). [CrossRef] [PubMed]

_{0}〉 is the Jones vector corresponding to Stokes vector τ⃗

_{0}[5

**15**, 2106–2119 (2007). [CrossRef] [PubMed]

*τ*⃗ points in the direction of the slow PSP, which is the usual convention. Note that, although the PMD Jones matrix in Ref. [5

**15**, 2106–2119 (2007). [CrossRef] [PubMed]

**15**, 2106–2119 (2007). [CrossRef] [PubMed]

*T*=exp(-α/2)exp(

_{PDL1}*α*⃗·σ⃗/2) connecting the input and output of PDL1, its two orthonormal eigenvectors (|α⃗

*0i*and |α⃗⊥〉) in 2D Jones space represent the minimum (zero) and maximum attenuation states which yield [5

**15**, 2106–2119 (2007). [CrossRef] [PubMed]

*p*⃗

*s*〉 is the unit Jones vector used to represent the input signal polarization. Because of this, each element of |

*s*⃗

*(*

_{in}*t*)〉 is a 2D vector. Assuming the optical filter has no effect on the signal polarization, we express the signal at “6” in Fig. 1 as

_{k}*P*(τ⃗,α⃗)ll′=

*δ*. (The possible PDL effect of the optical filter on signal and noise can be included into PDL2, which is the case discussed in subsection 2.2 and Appendix C.)With the help of (29), (30) and the completeness relations |

_{l,l}′TPDLTPMD*k*⃗

_{0}〉〈

*k*⃗

_{0}|+|

*k*⃗⊥〉〈

*k*⃗⊥|=1(

*k*⃗=τ⃗,α⃗) in the 2D Jones space, we have

*P*(τ⃗,α⃗)ll′ projects the input polarization |

*p*⃗

*〉 into two orthogonal directions, i.e.,*

_{s}*s*(

^{o}*t*)〉 given by (19). Because each of the three (2

_{k}*L*+1)-dimensional matrices in (34) is diagonal, it can be commutated with any of other two matrices in (34). The electrically filtered signal-signal beating in (20) can be obtained easily by replacing |

*s*(

^{o}*t*)i in the first part of (25) by |

_{k}*s*⃗

^{o}(

*t*)〉 and using (34), which yields

_{k}*θ*and φ

_{ατ}(θ_{sτ})_{ατ}(φ

_{sτ}) are polar and azimuthal angles between α⃗ (

*p*⃗

*) and τ⃗ in 3D Stokes space. For a system with given*

_{s}*τ*and

*α*, the PMD-PDL-induced factor

_{0},α⊥) is determined by the relative directional relations between the input signal polarization

*p*⃗

*, the PMD vector τ⃗*

_{s}_{0}and the PDL vector α⃗

_{0}. In this context, we called it the PMD-PDL-induced directional coupling factor. For a system with α=

*τ*=0 (and

*θ*=0), (38) yields e-αΘll′ (τ⃗,α⃗p⃗

_{ατ}*)=1.*

_{s}*U*

^{†}is the Hermitian matrix of

*U*, which is used to diagonalize

*O†*. Thus the filtered current caused by noise-noise beating yields

_{nn}R^{D}_{n n}O_{nn}*and*

^{D}*λ*are given between (25) and (26).

^{D}_{m}*b*〉=

^{D}*B*|so(

^{D}*t*)i in (25) now becomes

_{k}*B*given by (26) and

^{D}*P*. So the electrically filtered noise-signal beating can be written as

_{i}|p⃗_{s}′ (i=α_{0},α⊥) by (33)*cl*given by (33). Thus we obtain

_{i}*p*⃗

*) given by (36) and*

_{s}_{0}〉, |α⃗⊥〉). For example, when

*t*=0, we can take τ⃗

_{0}‖α⃗

_{0}(i.e., θατ=0) and get

*p*⃗

*and the PDL vector α⃗0.*

_{s}*s*(

^{o}*t*)〉 is the optical field at the output of the optical filter. But in Appendix B it is not the magnitude of |

_{k}*s*⃗

_{o}(

*t*)〉. In fact it is the magnitude of the optically filtered field without the influence of PMD and PDL. The effect of PMD and PDL has been included in the directional factors Θill′ (i=α

_{k}_{0},α⊥) given by (37), since the matrices

*P*(τ⃗,α⃗) and

*O*in (34) and (41) can commutate with each other.

_{ss}## Appendix C: Filtered current in DPSK system with partially polarized noise

*α*is assumed to be frequency independent,

_{k}*P*is a constant diagonal matrix, which means

_{PDL2}*U*†

*P*. Due to the effect of the partially polarized noise, the noise-signal beating in (42) now becomes

_{PDL2}U=_{PPDL2}*α*now becomes

*α*.

_{k}## Acknowledgment

## References and links

1. | B. Huttner, C. Geiser, and N. Gisin, “Polarization-induced distortion in optical fiber networks with polarization-mode dispersion and polarization-dependent losses,” IEEE J. Select. Topics Quantum Electron. |

2. | I. T. Lima, A. O. Lima, Y. Sun, H. Jiao, J. Zweck, C. R. Menyuk, and G. M. Carter, “A receiver model for optical fiber communication systems with arbitrarily polarized noise,” J. Lightwave Technol. |

3. | A. Mecozzi and M. Shtaif, “Signal-to-noise-ratio degradation caused by polarization-dependent loss and the effect of dynamic gain equalization,” J. Lightwave Technol. |

4. | M. Shtaif and O. Rosenberg, “Polarization-dependent loss as a waveform-distorting mechanism and its effect on fiber-optical systems,” J. Lightwave Technol. |

5. | L. Chen, Z. Zhang, and X. Bao, “Combined PMD-PDL effects on BERs in simplified optical systems: an analytical approach,” Opt. Express |

6. | P. J. Winzer, S. Chandrasekhar, and H. Kim, “Impact of filtering on RZ-DPSK reception,” IEEE Photon. Technol. Lett. |

7. | D. Marcuse, “Derivation of analytical expressions for the bit-error probability in lightwave systems with optical amplifiers,” J. Lightwave Technol. |

8. | P. A. Humblet and M. Azizoglu, “On the bit error rate of lightwave systems with optical amplifiers,” J. Lightwave Technol. |

9. | E. Forestieri, “Evaluating the error probability in lightwave systems with chromatic dispersion, arbitrary pulse shape and pre-and postdetection filtering,” J. Lightwave Technol. |

10. | J. L. Rebola and A. V. T. Cartaxo, “Performance evaluation of optically preamplified receivers with partially polarized noise and arbitrary optical filtering: a rigorous approach,” IEE Proc. Optoelectron. |

11. | J. Wang and J. M. Kahn, “Impact of chromatic and polarization-mode dispersions on DPSK systems using interferometric demodulation and direct detection,” J. Lightwave Technol. |

12. | L. Xie, L. Chen, S. Hadjifaradji, and X. Bao, “WDM high speed chirped DPSK fiber optical system transmission modeling in presence of PMD, PDL, and CD,” Opt. Fiber Technol. |

13. | P. Lu, L. Chen, and X. Bao, “Polarization mode dispersion and polarization dependent loss for a pulse in single-mode fiber,” J. Lightwave Technol. |

14. | H. Kogelnik, L. E. Nelson, and J. P. Gordon, “Emulation and inversion of polarization-mode dispersion,” J. Lightwave Technol. |

**OCIS Codes**

(060.2310) Fiber optics and optical communications : Fiber optics

(060.2330) Fiber optics and optical communications : Fiber optics communications

(260.2030) Physical optics : Dispersion

(260.5430) Physical optics : Polarization

**ToC Category:**

Fiber Optics and Optical Communications

**History**

Original Manuscript: April 13, 2007

Revised Manuscript: June 29, 2007

Manuscript Accepted: July 3, 2007

Published: July 16, 2007

**Citation**

Zhongxi Zhang, Liang Chen, and Xiaoyi Bao, "Accurate BER evaluation for lumped DPSK and OOK systems with PMD and
PDL," Opt. Express **15**, 9418-9433 (2007)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-15-9418

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### References

- B. Huttner, C. Geiser, and N. Gisin, "Polarization-induced distortion in optical fiber networks with polarizationmode dispersion and polarization-dependent losses," IEEE J. Select. Topics Quantum Electron. 6, 317-329 (2000). [CrossRef]
- I. T. Lima, A. O. Lima, Y. Sun, H. Jiao, J. Zweck, C. R. Menyuk, and G. M. Carter, "A receiver model for optical fiber communication systems with arbitrarily polarized noise," J. Lightwave Technol. 23, 1478-1490 (2005). [CrossRef]
- A. Mecozzi and M. Shtaif, "Signal-to-noise-ratio degradation caused by polarization-dependent loss and the effect of dynamic gain equalization," J. Lightwave Technol. 22, 1856-1871 (2004). [CrossRef]
- M. Shtaif and O. Rosenberg, "Polarization-dependent loss as a waveform-distorting mechanism and its effect on fiber-optical systems," J. Lightwave Technol. 23, 923-930 (2005). [CrossRef]
- L. Chen, Z. Zhang, and X. Bao, "Combined PMD-PDL effects on BERs in simplified optical systems: an analytical approach," Opt. Express 15, 2106-2119 (2007). [CrossRef] [PubMed]
- P. J. Winzer, S. Chandrasekhar, and H. Kim, "Impact of filtering on RZ-DPSK reception," IEEE Photon. Technol. Lett. 15, 840-842 (2003). [CrossRef]
- D. Marcuse, "Derivation of analytical expressions for the bit-error probability in lightwave systems with optical amplifiers," J. Lightwave Technol. 8, 1816-1823 (1990). [CrossRef]
- P. A. Humblet and M. Azizoglu, "On the bit error rate of lightwave systems with optical amplifiers," J. Lightwave Technol. 9, 1576-1582 (1991). [CrossRef]
- E. Forestieri, "Evaluating the error probability in lightwave systems with chromatic dispersion, arbitrary pulse shape and pre-and postdetection filtering," J. Lightwave Technol. 18, 1493-1503 (2000). [CrossRef]
- J. L. Rebola and A. V. T. Cartaxo, "Performance evaluation of optically preamplified receivers with partially polarized noise and arbitrary optical filtering: a rigorous approach," IEE Proc. Optoelectron. 152, 251-262 (2005). [CrossRef]
- J. Wang and J. M. Kahn, "Impact of chromatic and polarization-mode dispersions on DPSK systems using interferometric demodulation and direct detection," J. Lightwave Technol. 22, 362-371 (2004). [CrossRef]
- L. Xie, L. Chen, S. Hadjifaradji, and X. Bao, "WDM high speed chirped DPSK fiber optical system transmission modeling in presence of PMD, PDL, and CD," Opt. Fiber Technol. 12, 276-281 (2006). [CrossRef]
- P. Lu, L. Chen, and X. Bao, "Polarization mode dispersion and polarization dependent loss for a pulse in singlemode fiber," J. Lightwave Technol. 19, 856-859 (2001). [CrossRef]
- H. Kogelnik, L. E. Nelson, and J. P. Gordon, "Emulation and inversion of polarization-mode dispersion," J. Lightwave Technol. 21, 482-495 (2003). [CrossRef]

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