2.1 Self-phase modulation

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Chapter 2 Nonlinear interactions in optical fibers

When an intense electromagnetic (e.m.) field runs through a dielectric medium, the response of the medium becomes nonlinear. The origin of nonlinear effects is related to the motion of boundary electrons due to the applied field. As a result, under intense e.m. fields the induced polarization from the electric dipoles is no longer approximable with a linear function of the electric field E , but satisfies the more general relation

P

(

⁽¹⁾ ⁽²⁾ ⁽³⁾

)

P= ε χ ⋅Ε + χ ⋅Ε Ε + χ ⋅Ε Ε Ε + ⋅⋅⋅₀

where ε0 is the vacuum permittivity, χ⁽ⁱ⁾ (i=1,2,…) is the i^th order susceptibility and it is a 3×3ⁱ matrix [5]. The operator “⋅” denotes the matrix product and “” is the Kronecker product (or matrix tensorial product): given A = (aij) and B = (bij), with dimensions na×ma and nb×mb, is:

a

a a

11 1m

n 1 n m

a B a B

A B =

a B a B

⎛ ⎞

⎜ ⎟

⎝ ⎠

…

a

whose dimensions are nanb×mamb.

Since is a 3×3 matrix, then E EE is a 9×1 and E E E is 27×1.

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refractive index n and in the attenuation coefficient α. The second order susceptibility χ⁽²⁾is nonzero only for asymmetric molecules, so it vanishes for silica glasses because of the symmetry of SiO2 molecules. As a result, optical fibers usually don’t exhibit second-order nonlinear effects. The lowest-order nonlinear effects in optical fibers are related to the third order susceptibility χ⁽³⁾, responsible for phenomena such as third-harmonic generation, four- wave mixing and nonlinear refraction. Most of the nonlinear effects in optical fibers originates from nonlinear refraction, a phenomenon related to the intensity dependence of the refractive index resulting from the contribution of χ⁽³⁾. The refractive index of the fiber becomes

2 2

n( | E | ) n(ω, = ω) +n | E |₂

where n(ω) is the linear part, |E|² is the optical intensity inside the fiber and n2 is the nonlinear-index coefficient related to χ⁽³⁾ by the relation

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n 3 Re(

8n _xxxx)

= χ

where the optical field is assumed to be linearly polarized so that only one component χ⁽³⁾_xxxxof the fourth-rank tensor contributes to the refractive index [2]. In the next paragraphs we will focus on two of the most widely studied nonlinear effects, self-phase modulation (SPM) and cross-phase modulation (XPM), which will be then exploited in NOLMs described in next chapters.

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2.1 Self-phase modulation

Self-phase modulation refers to the self-induced phase shift experienced by an optical field during its propagation along the fiber. The phase of an optical field changes by

2

0 2

nk L (n n | E | )k L

φ = = + ₀

where k0 = 2π/λ and L is the fiber length.

The intensity-dependent nonlinear phase-shift φNL = n2k0L|E|² is due to SPM. The nonlinear coefficient γ is defined by

2 n2

A_eff γ = π

λ

where Aeff is the effective core area. The nonlinear coefficient γ mainly determines the behaviour of nonlinear optical devices based on optical fibers. The smaller effective core area is preferable to enhance γ. Typical values of γ for SiO2-based fibers are 1.8[W^-1km^-1] for standard Single-mode Fiber (SMF) and 2.6[W^-1km^-1] for Dispersion-Shifted Fiber because Aeff is smaller than SMF’s one and the Ge doped in the core enhances n2.

Let us consider a pulsed signal A0(z,t) propagating along a fiber span of length L and fiber attenuation per unit of length α = 0. It can be demonstrated that the total SPM contribution on the signal is given by:

tot 2

SPM(t) | A (0, t) | L0 P (0, t)L0

φ = γ = γ

The unavoidable presence of fiber losses reduces the SPM efficiency because the signal power is reduced along the fiber as:

P(z) P(0)e= ^−αz

In order to consider this effect the effective fiber length can be introduced:

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L_eff = α

Leff plays the role of the effective distance interested by self-phase modulation and it is smaller than L because of the fiber loss.

As a result, when α ≠ 0 the total SPM contribution can be described by:

tot 2

SPM(t) | A (0, t) | L0 _eff P (0, t)L0 _eff

φ = γ = γ

SPM induces spectral broadening as a consequence of the time dependence of φNL(z,T). A temporally varying phase implies that the instantaneous optical frequency differs across the signal from its central value f0. The difference δf is given by:

(

²

)

NL

1 1 L

f (T) | U(0,T) |

2 T 2 T L

δφ δ

δ = − = −

π δ π δ

eff

where U(z,T) is the normalized amplitude and LNL = (γP0)^-1 is the nonlinear length. The time dependence of δf can be viewed as a frequency chirp induced by SPM that increases in magnitude with the propagated distance.

Fig.2.1 Temporal variation of the phase shift and the frequency chirp induced by SPM for the cases of a Gaussian (dashed curve) and a super-Gaussian (solid curve) pulse.

As a result, new frequency components are continuously generated as the signal propagates down the fiber. These SPM-generated frequency components broaden the spectrum over its

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2.2 Cross-phase modulation

Cross-phase modulation refers to the nonlinear phase shift of an optical field induced by another field propagating in the same fiber. Let us consider two optical fields E1 and E2 at frequencies f1 and f2, polarized along the same axis, copropagating simultaneously inside the fiber. The nonlinear phase shift experienced by the signal at f1 is given by:

2 2

NL n k L(| E |2 0 1 2 | E | )2

φ = + .

The two terms on the right side are due to SPM and XPM and it is noticeable that for equally intense optical fields the contribution of XPM is twice compared with SPM’s one [4]. The total XPM contribution introduced by a fiber span of length L and attenuation α ≠ 0 on the signal at f1 is:

tot 2

XPM(t) 2 | A (0, t) | L1 _eff 2 P (0, t)L1 _eff

φ = γ = γ

It is important to notice that the fiber Group-Velocity Dispersion (GVD) reduces the XPM interaction length. In fact as a consequence of chromatic dispersion, pulses at different wavelength propagate at different speeds inside the fiber because of the group-velocity mismatch. This leads to a walk-off effect involving two overlapping optical pulses, so the nonlinear interactions vanish when the faster moving pulse overtakes the slower moving one [3]. If the two signals and are counterpropagating, the XPM appears as well, but its value is due to the mean power of the signal inducing the effect, instead of its instantaneous power. Moreover, in this condition it can be f

E1 E2

1 = f2.