With the development of nanophotonic technologies, it is available to engineer waveguides with the complex dispersion profiles. This is not adequately described by a simple second-order term in the expansion of the wave number with frequency [1]. Similarly, the nonlinear Schrodinger equation cannot describe fast modulations adequately because it is based on a Taylor expansion in the frequency domain and only several dispersion items are taken into account [2]. Therefore, the influence of the modulation of optical fiber dispersion on soliton propagation has attracted a lot of attentions recently.

More dispersion items have been included [2]. In [3], authors studied the effects of the entire modal dispersion curve and the frequency dependence of the nonlinear coefficients on the formation of modulation instabilities by using the harmonic analysis. New regions of modulation for the dispersion-flattened fibers were found and characterized by this approach [4]. In the strongly dispersion fiber, the chirped Gaussian pulse, as in a linear superposition of the Hermite-Gaussian harmonics and as the zeroth harmonic, presented a periodically varying modulation width [5]. The multipeak modulation instability spectrum was observed in the dispersion oscillating optical fibers [6].

In the fiber with a periodic dispersion, by the variation approach, the stochastic decay of pulses was predicted [7]. Also, the finite-energy Airy waves may endlessly retain their shapes and follow the pre-engineered temporal trajectories [8].

With an appropriately designed dispersion profile, the chirped pulses can be retrieved through the chirp reversal [9] [10]. In this case, of the nonlinear Schrödinger equation with designed group velocity dispersion, variable nonlinearity, and gain or loss, had been demonstrated that the chirp reversal is crucial for the pulse reproduction [9]. A wave suffered the strong modulation of dispersion and then it resulted in a significant chirp development. The periodic modulation of dispersion led to the radiation damping [11] [12], the existence of vibrating solitons [13] and splitting of solitons [14].

Pulse evolution is highly sensitive to the input pulse profile [15]. It was found that light pulses with Gaussian input profile produced less radiation in the fiber system than that hyperbolic-secant or raised-cosine pulses could do.

In this paper, based on our established theory on the field in PCFs [16] which contained both the second order differential to the transmission distance ( ∂ 2 A ∂ z 2 ) and the higher-order dispersion items, we will re-discuss the dispersion effect on pulse. These discussions include the various pulses with profiles of Gauss, chirped Gauss, super-Gauss and hyperbolic functions (the common features of pulse profile). It also presents a Darboux state description for the field in a dispersion medium [17].

In Micro-fibers (the PCFs in Figure 1), the transmission equation of field should include the second-order differential to transmission distance. It is [16]:

∂ 2 A ∂ z 2 + 2 i β ∂ A ∂ z = − ( Δ β 2 − 2 i β Δ β ) A (1)

Generally, for the discussion of field in fibers, people adopt the nonlinear Schordinger equation which ignores the item ∂ 2 A ∂ z 2 because of the suppose of the slow-varying envelope. But in PCFs, the envelope of field varies very fast. Thus, the field equation should be (1).

And the dispersion item β should include many orders of its Taylor expansion or be the entire β coefficient curve.

Then only taking the dispersion effect into account ( Δ β = 0 ), we get:

∂ 2 A ∂ z 2 + 2 i β ∂ A ∂ z = 0 (2)

The field is:

A ( z , ω ) = A ( 0 , ω ) exp [ − 2 i z ( β 0 + β 1 ω + β 2 / 2 ω 2 + ⋅ ⋅ ⋅ ) ]                               / [ − 2 i ( β 0 + β 1 ω + β 2 / 2 ω 2 + ⋅ ⋅ ⋅ ) ] (3)

From the view of the quantum theory, in the phase-amplitude representation, the system’s Hamiltonian operator is [17]:

H ^ S = ω a ′ + a ′ − i β tanh ( β β ) (4)

In [18], we have derived the (4). In the Darboux space, the lax pair is:

β 2 = β 0 2 + 2 ∂ 2 ( ln A ) ∂ z 2 (5)

A = ∂ A ( β 2 ) ∂ z − ∂ A ( β 0 ) / ∂ z A ( β 0 ) A ( β ) (6)

The simulation results based on (6) are the same as those on (3).

In this section, we will compare the simulations of this theory with those of [19] [20] [21] [22] to prove the correction of our theory. The simulation procedure is shown in Figure 2.

For the Gauss pulse, there is:

A ( 0 , T ) = exp [ − ( 1 + C ) T 2 / 2 / T 0 2 ] (7)

A ( 0 , ω ) = 1 2 π ∫ − ∞ + ∞ A ( 0 , T ) e − i ω T d T (8)

A ( z , T ) = ∫ − ∞ + ∞ d ω { A ( 0 , ω ) exp [ − 2 i z ( β 0 + β 1 ω + β 2 / 2 ω 2 + ⋅ ⋅ ⋅ ) ]                             / [ − 2 z i ( β 0 + β 1 ω + β 2 / 2 ω 2 + ⋅ ⋅ ⋅ ) ] exp [ i ω T ] } (9)

We simulate the Gauss pulse evolution in Figure 3 to prove the corrections of (3) and (6).

Although the form of (3) is completely different from the equations in [19] [20], the pulse evolution is the same in the following points:

1) The time axis values minus z β 1 . It is T = t − z β 1 . The whole pulse envelope is moved at the velocity of 1 / β 1 ;

2) The second order dispersion β 2 causes the pulse broaden;

3) The compressed chirped pulse (when C = 0.5 and Z = 0.1L_D, L_D is the dispersion length [23] ) is shown in subfigure (c).

And differences as follow:

1) The second order dispersion β 2 causes the symmetrical fluctuations at the pulse edges (high frequency brims);

2) When the chirped pulse is compressed, as subfigure (c), the edge fluctuations turn severer;

3) The item 2 z i ( β 0 + β 1 ω + β 2 / 2 ω 2 + ⋅ ⋅ ⋅ ) in phase will cause the fluctuations in the envelopes of pulse.

These differences occur in our simulation, are not apparently shown in [19] [20].

Figure 4 is the demonstration of hyperbolic Secant pulse,

A ( 0 , T ) = sech ( T T 0 ) exp ( − i C T 2 2 T 0 ) .

By a transmission, this pulse is broadened due to the second-order dispersion according to the Formula (3). The fluctuations are more evident than those in Figure 3, and even the pulse splittings occur. The simulations based on our theory exhibit more severe pulse expandings and splittings (e)-(g) than those in [19] [21]. This is consistent with the experiment result that in PCFs [2], and we have observed strong super-continuum spectrum which is not induced in the common fibers.

The same results are obtained from Figure 5 for the super-Gauss pulse. A ( 0 , T ) = exp [ − 1 + i C 2 ( T T ) 2 m ] . There are fluctuations in the pulse edges and the pulses will split with increase of the distance [19] [22]. These phenomena are more evident than those occurred in a Gauss-pulse.

By resolving the transmission equation of field in PCFs, the dispersion effect is re-discussed. Although the second order differential to the transmission distance ( ∂ 2 A / ∂ z 2 ) is taken into account, the first-order (β₁) and second-order dispersion (β₂) effects have the same influences as those of the nonlinear Schordinger equation, which cause the envelope of pulse move and pulse broadened, respectively. Additionally, the second-order dispersion also brings the fluctuations in the pulse edges, and with the increase of distance, even results in the pulse splittings.

But by our calculation, the third-order dispersion (β₃) will also induce pulse compression while the second-order dispersion is assumed as zero. The dispersion coefficient β ( ω ) alters with the distance.

The authors declare no conflict of interests.

Gui, M.X. and Huang, J. (2022) The Dispersion Effect in Photonic Crystal Fibers. Journal of Modern Physics, 13, 509-517. https://doi.org/10.4236/jmp.2022.134033