Pulse Propagation in Presence of Polarization Mode Dispersion and Chromatic Dispersion in Single Mode Fibers

Abstract

The presence of (first and second orders) polarization mode dispersion (PMD), chromatic dispersion, and initial chirp makes effects on the propagated pulses in single mode fiber. Nowadays, there is not an accurate mathematical formula that describes the pulse shape in the presence of these effects. In this work, a theoretical study is introduced to derive a generalized formula. This formula is exactly approached to mathematical relations used in their special cases. The presence of second-order PMD (SOPMD) will not affect the orthogonality property between the principal states of polarization. The simulation results explain that the interaction of the SOPMD components with the conventional effects (chromatic dispersion and chirp) will cause a broadening/narrowing and shape distortion. This changes depend on the specified values of SOPMD components as well as the present conventional parameters.

1. Introduction

As a pulse propagates through a light-wave transmission system with a polarization mode dispersion (PMD), the pulse is spilt into a fast and slow one and therefore becomes broadened. This kind of PMD is commonly known as first-order PMD (FOPMD). Under FOPMD, a pulse at the input of a fiber can be decomposed into two pulses with orthogonal states of polarization (SOP). Both pulses will arrive at the output of the fiber undistorted and polarized along different SOPs, the output SOPs being orthogonal [1]. Both the PSPs and the differential group delay (DGD) are assumed to be frequency independent when only first-order PMD is being considered [2].

PMD also has higher-order components. Second-order PMD (SOPMD) is formed due to dependence of DGD on wavelength, because of that PMD is random variable. Statistical nature of PMD makes compensation very difficult [3]. Statistical properties of first- and second-order PMD have been reported in [4–6]. In these approaches, mean DGD plays a central role. Further, analytical calculations of pulse broadening and PMD compensation have also been reported [7, 8]. However, these analytical results make use of autocorrelation of the PMD vector [4].

Also, it is shown that FOPMD and SOPMD vectors are statistically dependent and tend to be mutually perpendicular. It is important to calculate the probability that a certain value of SOPMD happens, when the value of DGD is known in a transmission line [8, 9]. The frequency dependence of the DGD causes polarization-dependent chromatic dispersion (PCD), resulting in polarization-dependent pulse compression and broadening. While PCD is a simple mechanism, it is a relatively minor component of the SOPMD vector since the frequency dependence of PMD vector has a statistical tendency to point away from the PSP. The dominant impairment mechanism is due to the depolarization, which causes distortion of the pulse shape. Statistical analyses and numerical simulations indicate a strong interaction between CD and the SOPMD in general [10–13].

Chirp in the transmitted signal has a significant impact on second-order impairments. Impairments were found to be dependent on chirp and CD and to be highly dependent on launch SOP. Impairments due to SOPMD and higher-order PMD occur for larger signal bandwidth, particularly when these PMD components combine with chromatic dispersion or signal chirp [1].

The chromatic dispersion and SOPMD can have a significant influence on pulse shape. Whereas FOPMD deals with two pulses delayed relative to each other, pulse distortions due to SOPMD can be visualized in terms of the interplay of six different pulse replicas. These mechanisms result in pulse overshoots and the generation of satellite pulses [2, 14].

Section 1 presents introduction of CD, FOPMD, SOPMD, and chirp effects. Section 2 discusses theoretical treatments prevalent in the area. Section 3 describes the suggested theory to find the reconstructed pulses under the effects of these effects. Section 4 gives special results, while Section 5 presents the results and discussion. Finally in Section 6, conclusions of the work done have been drawn.

2. Statement of the Problem

Consider a generic input field 𝐴_in(𝑇)=𝐴_in(𝑇)|𝑠⟩, where 𝐴_in(𝑇) represents the field complex amplitude and |𝑠⟩ represents the input Jones polarization vector, which depends on the azimuth 𝜃 and the ellipticity 𝜂 of the input SOP. Mathematically, the SOP is defined as [15] |𝑠⟩=𝑎𝑏=cos𝜃cos𝜂−𝑖sin𝜃sin𝜂cos𝜃cos𝜂+𝑖cos𝜃sin𝜂. (1)

The SOP and the pulse shape of the propagated pulses are affected by the FOPMD. The SOPMD induces other changes at the pulse properties. However, the SOPMD contains the PCD that may be taken as an additional CD. The origin CD is influenced by the chirp. These effects will be studied simultaneously in this paper to explain an accurate expressions of the reconstructed pulses.

The frequency response of a single mode fiber with CD and PMD has the form [2, 14] 𝐻(𝑤)=𝑇_PMD(𝑤)exp𝑖𝛽₂𝐿𝑤²2, (2) where 𝛽₂ is the fiber’s GVD parameter, 𝐿 is the fiber length, and 𝛽₂ is the carrier frequency. The matrix 𝑇_PMD(𝑤) is responsible for PMD effects, which is a 2×2 unitary matrix that is defined as [16] 𝑇_PMD=exp−𝑖𝑤2𝜏(𝑤)⋅𝜎, (3) where 𝜏(𝑤) is the PMD vector, |𝜏(𝑤)| is the DGD between the PSPs, and 𝜎=(𝜎₁,𝜎₂,𝜎₃) is the vector of Pauli spin matrices which can be defined as [1] 𝜎₀=1001,𝜎₁=,𝜎100−1₂=0110,𝜎₃=.0−𝑖𝑖0 (4) Here 𝜏⋅𝜎=𝜏₁𝜎₁+𝜏₂𝜎₂+𝜏₃𝜎₃ is a 2×2 matrix. Now, the output field is given by 𝐴_out(𝑤)=exp𝑖𝛽₂𝐿𝑤²2𝐴_in(𝑤)𝑇_PMD(𝑤)|𝑠⟩=𝐴_out(𝑤)|𝑡(𝑤)⟩, (5) where 𝐴_in(𝑤) is the Fourier transform of 𝐴_in(𝑇) and |𝑡(𝑤)⟩=𝑇_PMD(𝑤)|𝑠⟩ is the output SOP. Note that all the fiber birefringence effects are included in the frequency-dependent vector |𝑡(𝑤)⟩, while the CD effects and pulse characteristics are accounted for inside the output field 𝐴_out(𝑤).

In the three-dimensional Poincare representation, PMD is described by the dispersion vector 𝜏=Δ𝜏𝜏, where Δ𝜏=|𝜏| and 𝜏 is the direction of one of the two orthogonal PSP’s of 𝑇_PMD(𝑤). In a first-order approximation both Δ𝜏 and 𝜏 are frequency independent; that is, 𝜏=Δ𝜏₀𝜏₀, where Δ𝜏₀ is the DGD between two input signals polarized along the two PSP’s of the fiber. The Taylor series of 𝜏 vector up to second order is defined as [12, 13] 𝜏=Δ𝜏₀𝜏₀+Δ𝜏^′𝜏₀+Δ𝜏₀𝜏_𝑤𝑤, (6) where 𝜏₀=𝜏|_𝑤=𝑤0, Δ𝜏₀=Δ𝜏|_𝑤=𝑤0, Δ𝜏^′=𝜕Δ𝜏/𝜕𝑤|_𝑤=𝑤0, 𝜏_𝑤=𝜕𝜏/𝜕𝑤|_𝑤=𝑤0, 𝑤₀ is the central frequency, and 𝑤 is the respective deviation to the central frequency 𝑤₀.

It is important to note that 𝜏_𝑤 is not unit vector and the vectors 𝜏₀ and 𝜏_𝑤 are orthogonal in Stokes space. The value of Δ𝜏^′ is responsible for PCD, while the factor 𝜏_𝑤 represents the depolarization.

3. Theoretical Representation

In this section, we will explain the effects of PMD, chromatic dispersion, and initial chirp on the degree of polarization, state of polarization, and pulse shape. Substituting (6) into (3) and using the notation 𝜏_𝑤=2𝑘𝑚, one may obtain 𝑇_PMD−𝑖=exp2𝜑₁𝜏₀+𝜑₂𝑚⋅𝜎, (7) where 𝜑₁=𝑤(Δ𝜏₀+Δ𝜏^′𝑤) and 𝜑₂=𝑤²Δ𝜏₀2𝑘. Under the assumption that the combined effect of PMD may be represented as a concatenation of FOPMD and SOPMD processes, (7) may be rewritten as follows: 𝑇_PMD=𝐶₁𝜎₀−𝑖𝜏₀⋅𝜎𝑆₁𝐶₂𝜎₀−𝑖𝑚⋅𝜎𝑆₂, (8) where we have used the abbreviations 𝐶₁, 𝐶₂, 𝑆₁, and 𝑆₂ to represent cos(𝜑₁/2), cos(𝜑₂/2), sin(𝜑₁/2), and sin(𝜑₂/2), respectively. Using the property 𝜏₀⟂𝑚 and the identity (𝑎⋅𝜎)(𝑏̂⋅𝜎)=𝑎⋅𝑏̂+𝑖𝑎×𝑏̂⋅𝜎, (8) will be 𝑇_PMD=𝐶₁𝐶₂𝜎₀−𝑖𝑆₁𝐶₂𝜏₀+𝑆₂𝐶₁𝑚+𝑆₁𝑆₂𝑛⋅𝜎, (9) where 𝑛=𝜏₀×𝑚. Note that the vectors 𝜏₀, 𝑚, and 𝑛 constituted the orthogonal basis of Stokes space. However, to advance analysis, let 𝑆𝑄=¹𝐶₂𝜏₁+𝑆₂𝐶₁𝑚₁+𝑆₁𝑆₂𝑛₁𝑆₁𝐶₂𝜏₂+𝑆₂𝐶₁𝑚₂+𝑆₁𝑆₂𝑛₂𝑆₁𝐶₂𝜏₃+𝑆₂𝐶₁𝑚₃+𝑆₁𝑆₂𝑛₃, (10) such that (9) will be 𝑇_PMD=𝐶₁𝐶₂𝜎₀−𝑖𝑄⋅𝜎. (11) Using (11), it may be found that 𝑇^†_PMD𝑇_PMD=𝐶²₁𝐶²₂𝜎₀+𝑄²𝜎₀. (12) Also, using (10), it is easy to get 𝑄²=𝑆²₁𝐶²₂+𝑆²₂𝐶²₁+𝑆²₁𝑆²₂. (13) Substituting (13) into (12) will make 𝑇^†_PMD𝑇_PMD=𝜎₀. The eigenvalues of the matrix 𝑇_PMD are 𝜆_1,2=𝐶₁𝐶₂∓𝐶²₁𝐶²₂−1. (14) In presence of PMD, we have 𝐶²₁𝐶²₂<1, such that the eigenvalues are different and complex and their product is 𝜆₁𝜆₂=1. Using these facts and the analysis presented in [16], we concluded a very important conclusion, which is the principle states of polarization (PSPs) vectors stay orthogonal but the orientation of antiparallel PSP’s vector may be changed. That is, the orthogonality property will not be affected by SOPMD.

Return back to (11), which can be written as 𝑇_PMD=𝐶₁𝐶₂−𝑖𝑞₁−𝑖𝑞₂−𝑖𝑞₃−𝑖𝑞₂+𝑖𝑞₃𝐶₁𝐶₂+𝑖𝑞₁. (15) Using (11) and the definition of |𝑠⟩ from (1), the output SOP may be |𝑡(𝑤)⟩=𝑇_PMD|𝑠⟩=𝑎𝐶₁𝐶₂−𝑖𝑞₁−𝑖𝑏𝑞₂−𝑖𝑞₃−𝑖𝑎𝑞₂+𝑖𝑞₃+𝑎𝐶₁𝐶₂+𝑖𝑞₁. (16) Using (10) into (16) yields |𝑡(𝑤)⟩=𝐶₁𝐶₂𝑎𝑏+𝑆₁𝐶₂𝜃₁𝜃₄+𝑆₂𝐶₁𝜃₂𝜃₅+𝑆₁𝑆₂𝜃₃𝜃₆, (17) where 𝜃₁𝜃₄=−𝑖𝜏₀⋅𝜎|𝑠⟩,𝜃₂𝜃₅𝜎|𝑠⟩,𝜃=−𝑖𝑚⋅₃𝜃₆=−𝑖𝑛⋅𝜎|𝑠⟩. (18)

In order to explain the effects of PMD on the signal DOP, it is interesting to rewrite (16) as follows: |𝑡(𝑤=𝑡)⟩^𝑥𝑡_𝑦=𝑢₁𝑎+𝑢₂𝑏−𝑢^∗₂𝑎+𝑢^∗₁𝑏, (19) where 𝑢₁=𝐶₁𝐶₂−𝑖𝑞₁ and 𝑢₂=−𝑖(𝑞₂−𝑖𝑞₃).

The corresponding Stokes vector is ⃗𝑡𝑡=𝑡¹₂𝑡₃=𝑡_𝑥²−𝑡_𝑦²𝑡_𝑥𝑡^∗_𝑦+𝑡_𝑦𝑡^∗_𝑥𝑖𝑡_𝑥𝑡^∗_𝑦−𝑡_𝑦𝑡^∗_𝑥. (20) Since different frequency components of the input broadband signal will undergo different polarizations evaluation when they propagate through the fiber, the signal DOP will degrade. Consequently, the weighted average of the Stokes vector is necessary, which is calculated by multiplication the normalized spectral intensity function 𝐼=|𝐴_in(𝑤)|² and integrating over 𝑤. The DOP will be [14] DOP=∫^∞_−∞𝑡₁𝐼𝑑𝑤²+∫^∞_−∞𝑡₂𝐼𝑑𝑤²+∫^∞_−∞𝑡₃𝐼𝑑𝑤²∫^∞_−∞.𝐼𝑑𝑤 (21) Note that the parameters 𝑡₁, 𝑡₂, and 𝑡₃ are functions of Δ𝜏₀, Δ𝜏^′, and 𝑘. That is, the signal DOP is a function of FOPMD and SOPMD. The integrations in (21) are very hard to solve analytically because the parameters 𝑡₁, 𝑡₂, and 𝑡₃ are complicated functions of frequency. However, determining the analytic form of the signal DOP is beyond this paper.

Now, the trigonometric functions in (17) will be simplified as follows:𝐶₁𝐶₂=14𝜓₁+𝜓^∗₁+𝜓₂+𝜓^∗₂,𝑆₁𝐶₂=−𝑖4𝜓₁−𝜓^∗₁+𝜓₂−𝜓^∗₂,𝑆₂𝐶₁=−𝑖4𝜓₁−𝜓^∗₁−𝜓₂+𝜓^∗₂,𝑆₁𝑆₂=14−𝜓₁−𝜓^∗₁+𝜓₂+𝜓^∗₂, (22a)(22b)(22c)(22d)where 𝜓_𝑞𝑖=exp2Δ𝜏₀𝑤+𝛼_𝑞𝑤²,𝑞=1,2, (23)𝛼₁=Δ𝜏^′+2𝑘Δ𝜏₀, and 𝛼₂=Δ𝜏^′−2𝑘Δ𝜏₀. Substituting (18) and (22a)–(22d) into (17) will get |𝑡(𝑤=1)⟩4𝜓₁𝑠₁+𝜓^∗₁𝑠₂+𝜓₂𝑠₃+𝜓^∗₂𝑠₄, (24) where𝑠₁=[(𝑖(𝑚⋅𝜎𝑠⟩,𝑠𝐼+𝑛−𝜏+))]|₂=[(𝑖(𝑚⋅𝜎𝑠⟩,𝑠𝐼+𝑛+𝜏+))]|₃=[((𝑚⋅𝜎𝑠⟩,𝑠𝐼+−𝑖𝑛−𝜏−))]|₄=[𝐼+(−𝑖𝑛+(𝜏−𝑚))⋅𝜎]|𝑠⟩. (25a)(25b)(25c)(25d)Assuming that the input pulse is a normalized Gaussian pulse, this is defined as [1] 𝐴(𝑇)=exp−1+𝑖𝐶2𝑇²𝑇²₀, (26) where 𝐶 and 𝑇₀ are the initial chirp and pulse width. Substituting (24) and the Fourier transform of (26) into (5) yields 𝐴_out()=1𝐿,𝑤4𝐹⁺₁𝑠₁+𝐹⁻₁𝑠₂+𝐹⁺₂𝑠₃+𝐹⁻₂𝑠₄, (27) where 𝐹^±_𝑞=2𝜋𝑇²₀1+𝑖𝐶exp±𝑖𝑤Δ𝜏₀2−𝑇²₀+𝛼^±_𝑞𝐶−𝑖𝛼^±_𝑞𝑤²2().1+𝑖𝐶 (28) Also𝛼₁=𝛽₂𝐿+𝛼₁=𝛽₂𝐿+Δ𝜏^′+2𝑘Δ𝜏₀,𝛼₂=𝛽₂𝐿−𝛼₁=𝛽₂𝐿−Δ𝜏^′+2𝑘Δ𝜏₀,𝛼₃=𝛽₂𝐿+𝛼₂=𝛽₂𝐿+Δ𝜏^′−2𝑘Δ𝜏₀,𝛼₄=𝛽₂𝐿−𝛼₂=𝛽₂𝐿−Δ𝜏^′−2𝑘Δ𝜏₀. (29a)(29b)(29c)(29d)The inverse Fourier transform on (27) will obtain 𝐴_out()=1𝐿,𝑇4𝐴₁𝑠₁+𝐴₂𝑠₂+𝐴₃𝑠₃+𝐴₄𝑠₄, (30) where𝐴_𝑞=𝑀_𝑞exp𝑖𝜑_𝑞−()exp𝑇+−1^𝑞+1Δ𝜏₀/2²2𝑇²_𝑞,𝑇_𝑞=𝑇₀21+𝛼_𝑞𝐶𝑇²₀+𝛼²_𝑞𝐶²+1𝑇⁴₀,𝜑_𝑞=12tan⁻¹𝛼_𝑞𝑇²₀+𝛼_𝑞𝐶,𝑀−𝛽_𝑞=𝑇₀𝑇_𝑞(),𝑞=1,2,3,4,𝛽=𝑇+−1^𝑞+1Δ𝜏₀/2²𝐶𝑇²₀+𝛼_𝑞𝐶²+1𝑇²₀+𝛼_𝑞𝐶²+𝛼²_𝑞. (31a)(31b)(31c)(31d)(31e)Here, 𝑇_𝑞, 𝜑_𝑞, and 𝑀_𝑞 represent the reconstructed pulse width, generated phase, and amplitude, respectively. Using the vectors in (25a)–(25d) into (30) and conducting a routine simplification, the output power equation takes the following formula: 𝑃_out()=1𝐿,𝑇4𝐴₁²+𝐴₂²+𝐴₃²+𝐴₄²+𝜏₀⋅𝑠̂2Re𝐴₂𝐴^∗₄−𝐴₁𝐴^∗₃+𝑚⋅𝑠̂4𝐴₂²+𝐴₃²−𝐴₁²−𝐴₄²−𝑛⋅𝑠̂2Im𝐴₂𝐴^∗₄−𝐴₁𝐴^∗₃, (32) where Re and Im represent real and imaginary parts, respectively. Equation (32) represents the most important achievement in this paper as it contains the effects of CD, chirp, FOPMD, and SOPMD. It is clear from this equation that the second term results from the FOPMD effects and the third term results because of the SOPMD effects, while the fourth term results from the interaction between FOPMD and SOPMD. The first term has nothing to do with the SOP or PSPs, so it represents the CD effects in the absence of other effects. By the adoption of Table 1, (32) will be reduced to special formulas according to the following effects that accompany the propagated pulse.

Table 1 

Parameters specifications for different effects.

4. Simplified Analytic Forms

In this section, we will address the mathematical analysis for special cases resulting from (32) to highlight theoretically the individual effects of the phenomena under study as well as to check the validity of the evidence contained in the previous section.

(a) CD Only. In this simple case, (32) will be reduced to 𝑃_out()=𝑇𝐿,𝑇⁰𝑇₁exp−𝑇²𝑇²₁, (33) where 𝑇₁ is computed from (31b) and Table 1. Here there is no PSPs because the CD has nothing to do with SOP, but this case is causing broadening or compressing of the output pulse depending on 𝛽₂𝐶 sign. Note that this formula is well known in the scientific researches.

(b) FOPMD and CD. In this case, (32) will be 𝑃_out()=𝑇𝐿,𝑇⁰𝑇₁𝑅₋exp−𝑇²₊𝑇²₁+𝑅₊exp−𝑇²₋𝑇²₁, (34) where 𝑅_∓=(1∓𝜏₀⋅𝑠)̂/2, 𝑇₁ is as in the previous case, and 𝑇_±=𝑇±Δ𝜏₀/2. We note here that the PSP is exactly ±𝜏₀, and the CD will be broadened or compressed each of the two orthogonal components. Neglecting the CD effects will make the orthogonal components not suffer from any change expect the shifting to right or left by Δ𝜏₀/2. However, the resulted pulse will be broadened by Δ𝜏₀.

(c) SOPMD and CD. Assuming that the SOPMD will be compensated, then 𝑃_out(𝐿,𝑇)=𝑅₋𝐴₁²+𝐴₄²+𝑅₊𝐴₂²+𝐴₃². (35) Here 𝑅_∓=(1∓𝑚⋅𝑠)̂/2. In this case, the PSPs are ±𝑚, where there are two orthogonal components and each one contains a sum of two pulses that are shifted to right and left.

(d) Orthogonal SOPMD and CD. Suppose that Δ𝜏^′=0; thus 𝑃_out(𝐿,𝑇)=𝑇₀𝑅₋exp−𝑇²₊𝑇²₁+exp−𝑇²₋𝑇²₁/𝑇₁+𝑅₊exp−𝑇²₊𝑇²₂+exp−𝑇²₋𝑇²₂/𝑇₂, (36) where 𝑅_∓ is as in the previous case and 𝑇_1,2 are computed from (31b) and Table 1. The PSPs are to be ±𝑚 here, where there are two orthogonal components and each one has different amplitudes and widths. This means that the orthogonal SOPMD will create the new PSPs, which are ±𝑚.

(e) Parallel SOPMD and CD. Suppose that 2𝑘=0; thus 𝑃_out()=𝑇𝐿,𝑇⁰21𝑇₁exp−𝑇²₊𝑇²₁+1𝑇₂exp−𝑇²₋𝑇²₂, (37) where 𝑇_1,2 are computed from (31b) and Table 1. There is no relationship here with the SOP or the PSPs. This confirms that the changes will be to the pulse width, where each component will be broadened or compressed, and the result is equivalent to a polarization-dependent CD. This result is physically similar to that of case (a).

(f) FOPMD and SOPMD. This case involved general where all terms of (32) will be used in creating the resulting pulse. In this case, PSPs will not be ±𝜏₀, but they are created from both ±𝜏₀ and ±𝑚; that is, the presence of SOPMD will cause a change in the fiber optic PSPs and this is what is called depolarization.

5. Results and Discussion

Equation (32) has been used for having simulated the behavior of propagated pulse that the input pulse has been considered as a normalized Gaussian one (one can see (23)). The orthogonal vectors 𝜏₀, 𝑚, and 𝑛 are randomly generated. The direction of 𝑠̂ vector for these orthogonal ones is important such that it will determine the output pulse shape. If 𝑠̂ is parallel to 𝜏₀, the effects of SOPMD will not appear, while the case that 𝑠̂ is parallel to 𝑚 will prevent the effects of FOPMD. But as 𝑠̂ is parallel to 𝑛, the individual effects of FOPMD and SOPMD will be discarded save the last term of (32) which represents the interaction between two kinds of PMD. Since these vectors are random ones, the pulse shape differ from simulation process to another one. To determine the actual behavior, it should calculate the statistical average. However, this leads to not clarifying individual effects for parameters in research. So, the pulse’s behavior is proposed by using a specific simulation to point out the effects of the parameters 𝛽₂, 𝐶, Δ𝜏_𝑤, and 𝑘. We have not focused on the investigation of Δ𝜏₀ as FOPMD is a well-covered in the scientific literatures. Some parameters are considered constant within a simulation process as 𝐿=50km, 𝑇₀=10ps, and Δ𝜏₀=5ps, and the others parameters 𝛽₂, Δ𝜏_𝑤, 𝐶, and 𝑘 are changed within simulation. All figures are labeled by colors: blue, red, and green stand for the cases 𝑘=0,5,10ps, respectively.

Figure 1 illustrates the output pulse at Δ𝜏_𝑤=0 and 𝛽₂=0 with various values of 𝑘 and 𝐶. If 𝐶=0, the effects are only concerned to attenuation that increases by increasing 𝑘. This is in a good agreement with applied physics because 𝑘 stands for normal component of SOPMD. The effects of 𝑘 are shown in the form of broadening/compression and distortion in output pulses by increasing 𝐶. The calculated variables in Figure 2 are the same as those in Figure 1 except that 𝛽₂=1ps²/km. As is seen that pulses are tended to broaden because the chromatic dispersion. The amount of broadening will raise by increasing the chirp. At higher value of 𝑘, the SOPMD tends to balance chromatic dispersion effect. So, the output pulses are shown less broadening at a higher value of 𝑘. As the pulse with high values of 𝑘 is affected by the parallel component of SOPMD and chromatic dispersion, then it tends to be distorted.

Figure 1 

The resulted pulses with Δ𝑡𝑤=0ps2 and 𝛽2=0ps2/km for different values of chirp and 𝑘.

Figure 2 

The resulted pulses with Δ𝑡𝑤=0ps2 and 𝛽2=1ps2/km for different values of chirp and 𝑘.

The calculated variables in Figure 3 are the same as those in Figure 1 except that Δ𝜏_𝑤=30ps². As is seen the action of the two components of SOPMD is given to rise broadening/compression and distortion which increase their rates with increasing 𝐶. For high values of 𝑘 the pulses tend for broadening and vice versa. This is corresponding to applications as long the components of SOPMD are orthogonal. The broadening is attributed to the interaction among 𝐶, Δ𝜏_𝑤, and 𝑘. The calculated variables in Figure 4 are the same as those in Figure 3 except that 𝛽₂=1ps²/km. Note that the effects become in contrast to these in Figure 3 because the presence of chromatic dispersion is about to rebalance among those parameters mentioned above so as to be projected with existing high values of 𝐶.

Figure 3 

The resulted pulses with Δ𝑡𝑤=30ps2 and 𝛽2=0ps2/km for different values of chirp and 𝑘.

Figure 4 

The resulted pulses with Δ𝑡𝑤=30ps2 and 𝛽2=1ps2/km for different values of chirp and 𝑘.

The results are implicitly similar with those in Figure 1 in [12], where the power pulses are exceeded “1” as PSPs are rotated at a high value of 𝐶. Overshoot in power may happen by increasing the SOPMD effects, especially, with higher values of 𝐶.

6. Conclusions

The above equations and results illustrated the following conclusions. The short pulses are more affected by the initial chirp and vice versa. The SOPMD will add a new complexity that is presented by PCD and depolarization. The interaction between the FOPMD and SOPMD will cause a more degradation on the resulted pulses. The compensation of FOPMD will make the pulse under the affection of SOPMD only, which may destroy the pulse because of the presence of Δ𝜏_𝑤 and 𝑘. That is, the omitting or compensation of FOPMD may set serious variations. The increasing amount of 𝑘 will make a more distortion on the propagated signal, while the variation of Δ𝜏_𝑤 presents a new type of CD.

The resulted pulses are affected, too, by 𝛽₂𝐶, 𝛽₂Δ𝜏^′, and 𝐶Δ𝜏^′. The presence of CD only broadens/narrows the resulted pulses. The mixing of CD and PMD pictures will alter the known behavior, where the values of Δ𝜏_𝑤 and 𝑘 are very important which enhanced/complicated the certain problem. However, the problem is very small for 𝐶=0, while the variations of pulse shape will be large for 𝐶≠0.