Figure 2.

MI gain spectra. +++ result of small-signal analysis. –––– result of perturbation approach. The parameters are P0 = 10 dBm, β<sup>2</sup> = 15 ps<sup>2</sup> /km, λ = 1550 nm, a = 0.21 dB/km, γ = 0.015W�<sup>1</sup> /m, and z = 0 m.

ΔP zð Þ þ dz;ω 2h i P zð Þ φð Þ z þ dz;ω

transfer function cos <sup>1</sup>

1

cos 1 2 β2ω<sup>2</sup> dz � �

Nonlinear Optics ‐ Novel Results in Theory and Applications

0

BBB@

ΔP zð Þ¼ þ dz;ω 2h i P zð Þ e

gMIð Þ¼ <sup>z</sup>;<sup>ω</sup> j j <sup>Δ</sup>P zð Þ� <sup>þ</sup> dz;<sup>ω</sup> <sup>Δ</sup>Pðz;ω<sup>Þ</sup>

h i P zð Þ dz

2 β2dzω<sup>2</sup> � �

�adz=2�iωβ1dz sin <sup>1</sup>

Even for large modulation index <sup>1</sup>

¼ 2e

sin <sup>1</sup> 2 β2ω<sup>2</sup> dz � �

tion ΔP zð Þ þ dz;ω due to FM-IM conversion is given as

CA <sup>¼</sup> <sup>e</sup>�adz=2�iωβ1dzei<sup>γ</sup> h iþ P zð Þ <sup>2</sup> <sup>P</sup><sup>0</sup> ½ � h i ð Þ<sup>z</sup> dz

� sin <sup>1</sup> 2 β2ω<sup>2</sup> dz � � 1

0

B@

ΔP zð Þ ;ω 2h i P zð Þ φð Þ z;ω

1

(9)

.

dz

(11)

CA

φð Þ z þ dz;ω (10)

CCCA

<sup>2</sup> <sup>β</sup>2ω2dz � � is obtained. The 3 dB cutoff frequency corresponds

<sup>2</sup> <sup>β</sup>2ω<sup>2</sup>dz <sup>¼</sup> <sup>π</sup>=2, the difference is within 0.5 dB.

P zð Þþ ;ω 2P<sup>0</sup> ½ � ð Þ z;ω dz � �.

/m, and z = 0 m.

2 β2dzω<sup>2</sup> � �

cos 1 2 β2ω<sup>2</sup> dz � �

When only intensity modulation is present and no phase modulation exists, the

<sup>2</sup> <sup>β</sup>2ω2dz <sup>¼</sup> <sup>π</sup>=4 in [22, 23]. This treatment is also adaptable to the case that only the nonlinear phase (frequency) modulation is present; then, the intensity modula-

�adz=2�iωβ1dz sin <sup>1</sup>

This is in very good agreement with [24] for small-phase modulation index.

The corresponding MI gain gMI in the side bands of ω<sup>0</sup> (the frequency of signal) is

γ ð<sup>z</sup>þdz z

MI gain spectra. +++ result of small-signal analysis. –––– result of perturbation approach. The parameters are

/km, λ = 1550 nm, a = 0.21 dB/km, γ = 0.015W�<sup>1</sup>

Eq. (10) does not include a Bessel function, so it is simpler than that in [24]. Obviously, the above process can be used to treat NLSE with higher-order dispersion (β3, β4) [25]. Similarly, the result in Eq. (10) will include ω<sup>3</sup> and ω<sup>4</sup>

0

B@

to <sup>1</sup>

given by

Figure 2.

18

P0 = 10 dBm, β<sup>2</sup> = 15 ps<sup>2</sup>

Figure 2 shows a comparison of the gain spectra between Eq. (11) and [6] for the case h i P zð Þ = P<sup>0</sup> h i ð Þz ¼ 1. The maximum frequency modulation index caused by dispersion corresponds to <sup>1</sup> <sup>2</sup> <sup>β</sup>2ω2dz <sup>¼</sup> <sup>π</sup> [22, 23], and the maximum value of the sideband is <sup>ω</sup><sup>c</sup> <sup>¼</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>4</sup>γh i P zð Þ <sup>=</sup> <sup>β</sup><sup>2</sup> j j <sup>p</sup> , so the choice of dz satisfies 1 <sup>2</sup> <sup>β</sup>2ω2dz <sup>¼</sup> <sup>π</sup>, which makes Eq. (11) have the same frequency regime as [26]. In Figure 2, the curves are different but have the same maximum value of gMI. In practice, researchers generally utilize the maximum value of gMI to estimate the amplified noises and SNR [3]. The result of small-signal analysis in Figure 2 has a phase delay of around ω0. Compared with the experiment result of [27], the reason is taking the fiber loss into account, the gain spectrum exhibits a phase delay close to ω0, and the curve descends a little [27]. Fiber loss results in the difference of gMI between the small-signal analysis method and the perturbation approach.
