Stochastic interference in a dispersive fiber excited by a partially coherent source

Jing Shao; Shiva Kumar

doi:10.1364/OE.23.029163

1. Introduction

The theory of diffraction of a partially coherent stationary electromagnetic field is well-known [1–5]. In [6], the diffraction of a non-stationary electromagnetic field is studied. In this paper, we consider the temporal propagation of partally coherent non-stationary optical pulses in a dispersive fiber. As an optical pulse propagates down the fiber, it broadens due to chromatic dispersion [7, 8]. If the dispersion is not compensated in optical/electrical domain, it could lead to performance degradations. Marcuse developed an expression for the pulse broadening in optical fiber taking into account the partial coherence properties of the source and chromatic dispersion, and showed that the amount of pulse broadening increases significantly if the source spectral width is much larger the signal bandwidth [8]. The results are modified by taking into account the influence of the asymmetry in the spectral shape of the source [9]. Later, this result was extended to include the chirped Gaussian pulse [10]. In [11], the concept of partially coherent Gaussian Schell-Model pulse is introduced in time domain. In this paper, we generalize the results of Marcuse [8] to multiple optical pulses with the introduction of correlation functions to describe the stochastic interference of pulses.

We first consider the problem of pulse propagation and interference in a dispersive optical fiber when two pulses are launched to the fiber. We take into account the partial coherence of the source. If the source has a very small spectral width, the pulses broaden due to dispersion and interference fringes (in time domain) occur over a very long distance. However, as the coherence time of the source reduces, visibility of fringes drops after a short distance. In the context of interferometers, this distance is called the coherence length which is simply the product of speed of light times the coherence time [12]. However, in our case, coherence length does not adequately describe the length scale of fringe visibility since the interference properties depend on both coherence time and chromatic dispersion. We have developed an analytic expression for the correlation function $Γ$ that describes the stochastic interference of the pulses, assuming that the pulse shape and spectral distribution of the source are both Gaussian. Even for the case of fully coherent source, $| Γ |$ broadens over distance. For the case of a partial coherent source with finite coherence time, there is an additional amount of broadening which depends on accumulated dispersion, pulse width and coherence time. Finally, we extend the analysis to treat the problem of interference of a large number of temporally separated pulses and develop an analytical expression for the correlation function $Γ_{m, n}$ that describes the interference between the pulses centered at $m T_{b}$ and $n T_{b}$ ,where $T_{b}$ is the bit period.

In this paper, we have ignored the nonlinear effects. When the nonlinear effects are weak, it may be expected that the statistics of the random process is still Gaussian and higher order moments can be factorized into products of second-order moments. Accordingly, different forms of kinetic equations can be derived that accurately describe the evolution of the correlation function of the partially coherent wave propagating through the nonlinear medium [13]. Although the assumption of the ergodicity does not hold after the amplitude modulator, it may be reestablished by fiber nonlinear effects.

2. Stochastic interference

Suppose the source of the fiber optic communication system is not an ideal laser, but a partially coherent source. The electric field intensity of the source is

ψ_{i n} (t) = A (t) e^{- i ω_{0} t},

where

ω_{0}

is the mean angular frequency and

A (t)

is a stationary random process with the correlation function

R (τ) = 〈 A (t) A^{*} (t + τ) 〉 .

After passing through the amplitude modulator, the field at the fiber input is

ψ (t, 0) = q (t, 0) e^{- i ω_{0} t},

q (t, 0) = A (t) \sum_{n} a_{n} p (t - n T_{b}),

where q(t,0) is the input field envelope, p(t) is the pulse shape function,

{a_{n}}

is the random bit pattern, and

T_{b}

is the bit interval.

2.1. Case i: Two pulses

Let us first consider the case of two pulses launched to an optical fiber. The field envelope can be written as

q (t, 0) = A (t) [a_{1} p (t + \frac{T_{b}}{2}) + a_{2} p (t - \frac{T_{b}}{2})] .

The optical signal propagates through a dispersive fiber with transfer function [7,14]

H (ω) = \exp [(i β_{1} ω + \frac{i}{2} β_{2} ω^{2}) z],

where z is the length of the fiber,

β_{2}

is the dispersion coefficient and

β_{1}

is the inverse group speed. The field spectrum at a distance L is

\tilde{q} (ω, L) = \tilde{q} (ω, 0) H (ω),

where

\tilde{q} (ω, z) = F [q (t, z)]

, and F denotes the Fourier transform. Inverse Fourier transforming Eq. (7), we find

q (t, L) = \frac{1}{\sqrt{- i 2 π S}} \int e^{- i {(t - β_{1} L - t_{1})}^{2} / 2 S} q (t_{1}, 0) d t_{1},

where

S = β_{2} L

is the accumulated dispersion. The mean optical power at L is

P (t, L) = < {| q (t, L) |}^{2} >,

where <> denotes the ensemble average. Due to the presence of the optical modulator, the envelope function becomes time dependent. Although the source complex amplitude A(t) is an ergodic process, due to the time dependence of the envelope function, the assumption of ergodicity breaks down. Substituting Eq. (5) in Eq. (8) and then in Eq. (9), we find

\begin{array}{l} P (t, L) = \frac{1}{2 π | S |} \iint e^{\frac{- i [{(T - t_{1})}^{2} - {(T - t_{1} - τ)}^{2}]}{2 S}} 〈 q (t_{1}, 0) q^{*} (t_{1} + τ, 0) 〉 d t_{1} d τ \\ = \frac{1}{2 π | S |} \iint e^{\frac{- 2 i (T - t_{1}) τ + i τ^{2}}{2 S}} R (τ) \times {{| a_{1} |}^{2} p (t_{1} + \frac{T_{b}}{2}) p^{*} (t_{1} + τ + \frac{T_{b}}{2}) + {| a_{2} |}^{2} p (t_{1} - \frac{T_{b}}{2}) p^{*} (t_{1} + τ - \frac{T_{b}}{2}) \\ + a_{1} a_{2}^{*} p (t_{1} + \frac{T_{b}}{2}) p^{*} (t_{1} + τ - \frac{T_{b}}{2}) + a_{1}^{*} a_{2} p^{*} (t_{1} + τ + \frac{T_{b}}{2}) p (t_{1} - \frac{T_{b}}{2})} d t_{1} d τ, \end{array}

where

T = t - β_{1} L

. Let

P_{0} (t) = \frac{1}{2 π | S |} \iint e^{\frac{- 2 i (T - t_{1}) τ + i τ^{2}}{2 S}} R (τ) p (t_{1}) p^{*} (t_{1} + τ) d t_{1} d τ,

and

Γ (t, T_{b}) = \frac{1}{2 π | S |} \iint e^{\frac{- 2 i (T - t_{1}) τ + i τ^{2}}{2 S}} R (τ) p (t_{1} + \frac{T_{b}}{2}) p^{*} (t_{1} + τ - \frac{T_{b}}{2}) d t_{1} d τ .

Now, Eq. (10) may be rewritten as

P (t, L) = {| a_{1} |}^{2} P_{0} (t + \frac{T_{b}}{2}) + {| a_{2} |}^{2} P_{0} (t - \frac{T_{b}}{2}) + a_{1} a_{2}^{*} Γ (t, T_{b}) + a_{1}^{*} a_{2} Γ^{*} (t, T_{b}) .

The function

Γ (t, T_{b})

is similar to the self-coherence function [1, 12], yet it is quite different. The self-coherence function is independent of time due to the stationarity of the random process. However,

Γ (t, T_{b})

depends explicitly on time due to the time-varying output of the optical modulator. When

Γ

is negligible, the ouput power is simply the addition of powers due to individual pulses, and hence,

Γ

accounts for the stochastic interference of the fields due to temporally separated pulses. If the fiber is not dispersive, temporally separated rectangular pulses would not interfere. However, in the presence of dispersion, pulses broaden and interfere. The degree of interference depends on the temporal coherence of the source and accumulated dispersion.

To facilitate the derivation of analytical expressions, we assume that the pulse shape is Gaussian,

p (t) = \exp (- t^{2} / 2 T_{0}^{2}),

and the source has a Gaussian spectral distribution with the spectral width W or equivalently, the auto-correlation function of the source is

R (τ) = P_{i n} \exp (- τ^{2} / τ_{0}^{2}),

where

τ_{0} = 2 / W .

Under these conditions, Marcuse has derived an analytical expression for the stochastic broadening for the case of a single pulse as [8]

P_{0} (t) = \frac{P_{i n}^{}}{η} \exp [- T^{2} / {(η T_{0})}_{}^{2}],

where

η = {[1 + {(\frac{S}{T_{0}^{2}})}^{2} + {(\frac{2 S}{T_{0} τ_{0}^{}})}^{2}]}^{1 / 2} .

For the case of two temporally separated pulses, we substitute Eqs. (14) and (15) in Eq. (12), and find (See Appendix)

{| Γ (t, T_{b}) |}^{} = \frac{P_{i n}^{}}{η} e^{- \frac{1}{{(η T_{0})}_{}^{2}} [T^{2} + {(\frac{T_{b}}{2} λ)}^{2}]},

where

λ = \sqrt{1 + \frac{4 S^{2}}{τ_{0}^{2} T_{0}^{2}}} .

From Eq. (18), we see that

| Γ |

has a peak at T = 0. The full width at half-maximum (FWHM) of

| Γ |

is

T_{F W H M} = 1.665 η T_{0} .

From Eqs. (17) and (18), we see that

| Γ |

broadens due to chromatic dispersion even when the source is fully coherent (

τ_{0} \to \infty

). When

τ_{0} \to \infty

, let

η_{\infty} = {[1 + {(\frac{S}{T_{0}^{2}})}^{2}]}^{1 / 2} .

We define the broadening factor due to partially coherent source in the presence of dispersion as

F ≜ {(\frac{η_{}}{η_{\infty}})}^{2} = 1 + \frac{{(\frac{2 S}{T_{0} τ_{0}^{}})}^{2}}{1 + {(\frac{S}{T_{0}^{2}})}^{2}} .

Using Eq. (22), Eq. (18) may be rewritten as

{| Γ (t, T_{b}) |}^{} = \frac{P_{i n}^{}}{η_{\infty}^{} \sqrt{F}} e^{- \frac{1}{F {(η_{\infty} T_{0})}^{2}} [T^{2} + {(\frac{T_{b}}{2} λ)}^{2}]}

As the temporal coherence (

\propto τ_{0}

) decreases, F increases and the peak of

| Γ |

decreases. For a fully incoherent source, F becomes infinity,

| Γ |

becomes zero and hence, the ouput power is the addition of power of individual pulses. From Eq. (22), we see that as

S \to \infty

and

τ_{0}

is non-zero and finite,

F \to 1 + 4 T_{0}^{2} / τ_{0}^{2}

and if

T_{0}^{} < < τ_{0}^{}

broadening due to partial coherence of the source is negligible. As

S \to 0

,

F \to 1

, and

η \to 1

. If

τ_{0} \neq 0

, it implies that dispersion is necessary for the broadening of

| Γ |

.

2.2. Case ii: Multiple pulses

Now we will consider a more general case and suppose that the fiber input consists of multiple pulses. The input filed envelope can be written as

q (t, 0) = A (t) \sum_{m} a_{m} p [t - (m T_{b} + \frac{T_{b}}{2})]

Using Eqs. (24) and (8) in Eq. (9), the mean power at distance L is

P (t, L) = \sum_{m} \sum_{n} a_{m} a_{n}^{*} Γ_{m, n},

where

Γ_{m, n} = \frac{1}{2 π | S |} \iint e^{\frac{- 2 i (T - t_{1}) τ + i τ^{2}}{2 S}} R (τ) p^{*} (t_{1} - m T_{b} - \frac{T_{b}}{2}) p^{*} (t_{1} + τ - n T_{b} - \frac{T_{b}}{2}) d t_{1} d τ .

Here,

Γ_{m, n}

represents the correlation between the pulses centered at

m T_{b} + \frac{T_{b}}{2}

and

n T_{b} + \frac{T_{b}}{2}

. When the pulse shape is Gaussian and the source has a Gaussian spectral distribution, an analytical expression for

Γ_{m, n}

is derived as (See Appendix)

Γ_{m, n} (t, T_{b}) = \frac{P_{i n}^{}}{η} e^{- \frac{T_{0}^{2} T^{2} + α T + β}{{(η T_{0}^{2})}_{}^{2}}},

where

α = - (m + n + 1) T_{b} (T {}_{0}^{2}+ i S) + 2 i T_{b} S (n + 1 / 2),

β = - \frac{1}{2} i (m + n + 1) (n - m) T_{b}^{2} S + {(n - m)}^{2} \frac{T_{b}^{2} S^{2}}{τ_{0}^{2}} + \frac{1}{2} (m^{2} + n^{2} + m + n + \frac{1}{2}) T_{b}^{2} T_{0}^{2} .

3. Results and discussion

In this section, we provide numerical examples in order to illustrate the implications of the theoretical results for fiber optic systems. The following parameters are used throughout this section unless otherwise specified: fiber input power $P_{i n} = 2$ dBm, initial pulse width $T_{0} = 7.15$ ps, bit period $T_{b} = 35.7$ ps, transmission distance L = 80 km, dispersion coefficient of the transmission fiber $β_{2} = - 21 {ps}^{2} / km$ , and the transmission fiber is assumed to be linear and fiber loss is ignored. Figure 1 shows $| Γ |$ (given by Eq. (23)) as a function of time and transmission distance. As can be seen, $| Γ |$ broadens even when the temporal coherence is very large (Fig. 1(a)). When the temporal coherence is small (Fig. 1(b)), the amount of broadening is larger.

Fig. 1 $| Γ |$ as a function of time and distance. (a) $τ_{0} = 250 T_{0}$ , (b) $τ_{0} = 0.1 T_{0}$ .

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Figure 2 shows the relationship between $| Γ |$ (evaluated at T = 0) and the spectral width W for different S. As can be seen from Fig. 2, with the increase in the spectral width of the light source, the correlation between two neighboring pulses decreases. We note that a certain amount of dispersion is necessary to alter the correlation function. If $β_{2} = 0 {ps}^{2} / km$ , from Eqs. (16) and (18), we see that $λ = 1$ and $η = 1$ , and $| Γ |$ does not change as a function of the spectral width W. For the given W, as $| S |$ increases (from a non-zero value), $| Γ |$ decreases monotomically. As dispersion increases, the amount of phase shifts acquired by the frequency components increases which leads to “dephasing” of the sinusoids or equivalently the loss of degree of correlation.

Fig. 2 Relationship between $| Γ |$ and $W$ .

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Next, let us consider the case of multiple pulses. Eight pulses (with $a_{n} = 1$ ) are launched to a fiber and the mean output power at various distances are shown in Figs. 3 and 4. In Fig. 3, the spectral width of light source is quite small (W = 1.12 GHz). As can be seen, there is a strong interference among pulses leading to peaks and troughs, whose locations change with distance. The role of dispersion can be understood as follows. Each pulse can be considered as a superposition of sinusoids of different frequencies. Due to chromatic dispersion, each sinusoidal component acquires a frequency-dependent phase shift. If the source is fully coherent, at certain times, these frequency components add up constructively, we get peaks which could occur at times that may be different from the locations of peaks at $z = 0$ . In Fig. 4, the spectral width of light source is so large that the interference between pulses disappear at the distance of 0.4 km, and thereafter the optical power is nearly constant over a long time period. This can be explained as follows: Correlation functions $Γ_{m, n}$ , $m \neq n$ in Eq. (25) become quite small due to large spectral width at relatively shorter distance ( $\approx$ 0.3 km). When the accumulated dispersion is large, the width of $Γ_{m, n}$ far exceeds the bit period $T_{b}$ and their superposition is roughly constant over a long time period.

Fig. 3 Signals propagate in fibers. $τ_{0} = 250 T_{0}$ = 1.79ns,

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Fig. 4 Signals propagate in fibers. $τ_{0} = 0.1 T_{0}$ = 0.72 ps,

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4. Conclusion

The stochastic interference of temporally separated pulses in an optical fiber excited by a partialy coherent source due to chromatic dispersion is investigated. An analytical expression for the correlation between temporally separated pulses which describes their stochastic interference is derived under the assumption that the spectral distribution of the source and the pulse shape are both Gaussian. The broadening of the correlation function due to finite coherence time and chromatic dispersion is discussed. For the given spectral width of the source, as the accumulated dispersion becomes very large, correlation function becomes negligibly small and the total mean power is simply the sum of power of individual pulses. Finally, the analysis is extened to treat the problem of stochastic interference of a large number of temporally separated pulses. The results show that the interference fringes in time domain disappear after a characteristic distance that depends on the spectral width of the source and chromatic dispersion.

Appendix

Substituting Eqs. (14) and (15) in Eq. (26), we find

\begin{array}{l} Γ_{m, n} = \frac{P_{i n}}{2 π | S |} \iint e^{\frac{- 2 i (T - t_{1}) τ + i τ^{2}}{2 S}} e^{- \frac{τ^{2}}{τ_{0}^{2}}} e^{\frac{- {(t_{1} - m T_{b} - \frac{T_{b}}{2})}^{2} - {(t_{1} + τ - n T_{b} - \frac{T_{b}}{2})}^{2}}{2 T_{0}^{2}}} d t_{1} d τ, \\ = \frac{P_{i n}}{2 π | S |} \iint e^{- τ^{2} A - 2 τ B} e^{\frac{- {(t_{1} - m T_{b} - \frac{T_{b}}{2})}^{2} - {(t_{1} + τ - n T_{b} - \frac{T_{b}}{2})}^{2}}{2 T_{0}^{2}}} d t_{1} d τ, \end{array}

where

A = - \frac{i}{2 S} + \frac{1}{τ_{0}^{2}} + \frac{1}{2 T_{0}^{2}},

and

B = \frac{i (T - t_{1})}{2 S} + \frac{(t_{1} - n T_{b} - \frac{T_{b}}{2})}{2 T_{0}^{2}} .

Since

\int e^{- σ τ^{2}} d τ = \sqrt{\frac{π}{σ}},

Equation (30) is simplified as

Γ_{m, n} = \frac{P_{i n}}{2 π | S |} \sqrt{\frac{π}{A}} \cdot e^{- \frac{[{(m + \frac{1}{2})}^{2} + {(n + \frac{1}{2})}^{2}] T_{b}^{2}}{2 T_{0}^{2}}} \int e^{y} d t_{1},

where

y = \frac{B^{2}}{A} - \frac{t_{1}^{2}}{T_{0}^{2}} + \frac{2 (m + n + 1) T_{b} t_{1}}{2 T_{0}^{2}} .

Next, we want to simplify the exponential term of Eq. (34), by using Eqs. (31) and (32)

y = \frac{{[\frac{i (T - t_{1})}{2 S} + \frac{(t_{1} - n T_{b} - \frac{T_{b}}{2})}{2 T_{0}^{2}}]}^{2}}{- \frac{i}{2 S} + \frac{1}{τ_{0}^{2}} + \frac{1}{2 T_{0}^{2}}} - \frac{t_{1}^{2}}{T_{0}^{2}} + \frac{2 (m + n + 1) T_{b} t_{1}}{2 T_{0}^{2}},

= - t_{1}^{2} A' - 2 t_{1} B' - \frac{{[T T_{0}^{2} + i (n + \frac{1}{2}) T_{b} S]}^{2}}{4 A S^{2} T_{0}^{4}},

where

A' = \frac{{(i S + T_{0}^{2})}^{2}}{4 A S^{2} T_{0}^{4}} + \frac{1}{T_{0}^{2}},

and

B' = \frac{(T_{0}^{2} + i S) [T T_{0}^{2} + i (n + \frac{1}{2}) T_{b} S]}{- 4 A S^{2} T_{0}^{4}} - \frac{(m + n + 1) T_{b}}{2 T_{0}^{2}}

Substituting Eq. (36) to Eq. (34), we obtain

\begin{array}{l} Γ_{m, n} = \frac{P_{i n}}{2 π | S |} \sqrt{\frac{π}{A}} \cdot e^{- \frac{[{(m + \frac{1}{2})}^{2} + {(n + \frac{1}{2})}^{2}] T_{b}^{2}}{2 T_{0}^{2}}} \cdot e^{- \frac{{[T T_{0}^{2} + i (n + \frac{1}{2}) T_{b} S]}^{2}}{4 A S^{2} T_{0}^{4}}} \sqrt{\frac{π}{A'}} e^{\frac{B'^{2}}{A'}}, \\ = \frac{P_{i n}^{}}{η} e^{- \frac{T_{0}^{2} T^{2} + α T + β}{{(η T_{0}^{2})}_{}^{2}}}, \end{array}

where

α = - (m + n + 1) T_{b} (T {}_{0}^{2}+ i S) + 2 i T_{b} S (n + 1 / 2),

β = - \frac{1}{2} i (m + n + 1) (n - m) T_{b}^{2} S + {(n - m)}^{2} \frac{T_{b}^{2} S^{2}}{τ_{0}^{2}} + \frac{1}{2} (m^{2} + n^{2} + m + n + \frac{1}{2}) T_{b}^{2} T_{0}^{2} .

When

m = - 1

and

n = 0

, we obtain Eq. (18).

References and links

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2. M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999).

3. E. Wolf, Introduction to the Theory of Coherence and Polarisation of Light (Cambridge University, 2007).

4. A. Burvall, A. Smith, and C. Dainty, “Elementary functions: propagation of partially coherent light,” J. Opt. Soc. Am. A 26(7), 1721–1729 (2009). [CrossRef] [PubMed]

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7. G. P. Agrawal, Nonlinear Fiber Optics, 4th ed. (Academic, 2007).

8. D. Marcuse, “Pulse distortion in single-mode fibers,” Appl. Opt. 19(10), 1653–1660 (1980). [CrossRef] [PubMed]

9. D. Marcuse, “Pulse distortion in single-mode fibers. Part 2,” Appl. Opt. 20(17), 2969–2974 (1981). [CrossRef] [PubMed]

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13. A. Picozzi, J. Garnier, T. Hansson, P. Suret, S. Randoux, G. Millot, and D. N. Christodoulides, “Optical wave turbulence: towards a unified nonequilibrium thermodynamic formulation of statistical nonlinear optics,” Phys. Rep. 542(1), 1–132 (2014). [CrossRef]

14. S. Kumar and M. J. Deen, Fiber Optic Communications: Fundamentals and Applications (Wiley, 2014).

Stochastic interference in a dispersive fiber excited by a partially coherent source

Abstract

1. Introduction

2. Stochastic interference

2.1. Case i: Two pulses

2.2. Case ii: Multiple pulses

3. Results and discussion

4. Conclusion

Appendix

References and links

Cited By

Figures (4)

Equations (42)

Optics Express