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In this paper, a flatly broadened and highly coherent supercontinuum generation by induced modulation instability in ANDi-PCF is proposed. The numerical results show that the bandwidth of supercontinuun can be increased by increasing the modulation depth and the coherence property of supercontinuum can be improved with the optimal modulation frequency. A highly coherent supercontinuum with the 10 dB bandwidth of 1305 nm and the flatness of less than 10 dB can be obtained when seeding at the modulation depth of 0.3 and modulation frequency of 24 THz.

Supercontinuum (SC) is a physical phenomenon in which a large intensity ultrashort optical pulse passes through a nonlinear medium, and the spectrum is greatly broadened due to the interaction of nonlinear effect and dispersion effect. SC spectrum is wide, relatively stable, simple and inexpensive, so it has been widely used in spectral analysis, optical coherence photography and optical sensing [

In contrast to the normal dispersion region, the anomalous dispersion region can produce a wider SC spectrum, but its coherence properties are very low and it is difficult to be used in many practical applications [

In this study, pulse propagation in the PCF is simulated using the Runge-Kutta algorithm to solve the generalized nonlinear Schrodinger equation [

∂ A ∂ z = ∑ m ≥ 2 i m + 1 β m m ! ∂ m A ∂ τ m − α 2 A + i γ ( 1 + i ω 0 ∂ ∂ τ ) [ A ( z , τ ) ∫ − ∞ + ∞ d τ ′ R ( τ − τ ′ ) | A ( z , τ ′ ) | 2 ] (1)

where A ( z , τ ) is the complex field envelope at a propagation distance z in a retarded reference time frame t traveling at the envelope group velocity. Since only short pieces of the PCF will be considered, loss in the fiber is neglected ( α = 0 ). The dispersion coefficients β m are associated with the Taylor expansion of the propagation constant β ( w ) around the center frequency w 0 , and g is the frequency dependent nonlinear coefficient. Dispersion effects are described by the first term on the right hand side of Equation (1); nonlinear effects correspond to the second one. The nonlinear response function R ( t ) = ( 1 − f R ) σ ( t ) + f R h R ( t ) with f R = 0.18 contains both instantaneous and delayed Raman contributions, the response h R which we use is the same as the analytic approximation in [

As Dudley has pointed out the coherence property of supercontinuum can be improved by using shorter pulses [^{−1}∙km^{−1})，and the up to 8-th order dispersion coefficients are β 2 = 6.9276 × 10 − 2 ps^{2}/km, β 3 = − 3.59804 × 10 − 3 ps^{3}/km, β 4 = 6.57756 × 10 − 4 ps^{4}/km, β 5 = 2.82896 × 10 − 6 ps^{5}/km, β 6 = 2.82896 × 10 − 10 ps^{6}/km, β 7 = 1.45745 × 10 − 12 ps^{7}/km, β 8 = 2.91289 × 10 − 14 ps^{8}/km, the effect of more than 8-th order dispersion can be neglected due to the weak dispersion effect of ANDi-PCF.

The modulated Gaussian input pulse envelope is assumed to have the following form

A ( z = 0 , τ ) = ( P 0 + d 0 P 0 e i 2 π f mod τ ) exp ( − τ 2 / ( 2 τ p 2 ) ) exp [ i δ ϕ ( τ ) ] (2)

where P 0 = 20 k w is the input peak power and we define the power P s e e d = d 0 2 P 0 as the power of the seed, d 0 means the modulation depth, f mod is the modulation frequency, τ p is the temporal width, which is related to FWHM pulse duration by T F W H M ≈ 1.665 τ p , δ ϕ ( τ ) is a small random phase fluctuation which is caused by Gaussian white noise [

We take the Raman response function and high-order dispersion up to β 8 into account to obtain the MI gain g ( Ω ) , it can be written as [

g ( Ω ) = Im { Δ k 0 ± ( Δ k e + 2 γ P 0 R ˜ ( Ω ) ) Δ k e } (3)

where Ω is the modulation frequency, R ˜ ( Ω ) is Raman response for silica, Δ k e and Δ k o are sums over odd and even order derivatives of propagation constant β, respectively [

Δ k 0 = ∑ m = 1 ∞ β 2 m + 1 ( 2 m + 1 ) Ω 2 m + 1 ， Δ k e = ∑ m = 1 ∞ β 2 m 2 m ! Ω 2 m , (4)

The coherence properties of SC spectra can be characterized by calculating the wavelength dependence of the first-order coherence as following equation [

| g 12 ( 1 ) ( λ , t 1 − t 2 ) | = | < A 1 ∗ ( λ , t 1 ) A 2 ( λ , t 2 ) > < | A 1 ( λ , t 1 ) | 2 > < | A ( λ , t 2 ) 2 | 2 > | , (5)

where the term in the angular brackets indicates an ensemble average over independently generated pairs of SC spectra through 20-times simulation results. The value of first-order coherence lies in the interval [0, 1], and a value of 1 denotes perfect coherence.

In order to obtain the coherent SC, we simulate the nonlinear propagation of an ultrashort pulse and SC generation in the ANDi-PCF with nearly zero flattened dispersion at 1550 nm by solving the nonlinear Schrödinger equation using the Runge-Kutta algorithm.

spectra are symmetrically broadened due to SPM as shown in

of short wavelength side, for example, when the modulation frequency is 14 THz near the peak of gain spectrum as shown in

formed the flat and highly coherent SC spectrum as shown in

In conclusion, the nonlinear propagation and SC generation is numerically investigated. The results show that the flatness and coherence of SC can be improved by seeding with a proper modulation frequency and the bandwidth of SC can be increased by increasing the modulation depth. By selecting the optimal modulation parameters, a highly coherent supercontinuum with the 10 dB bandwidth of 1305 nm and the flatness of less than 10 dB can be obtained.

This work was supported by the National Nature Science Foundation of China (Grant Nos. 61475044), the International Science & Technology Cooperation Program of China (No. 2016YFE0124300), the Doctoral Scientific Research Foundation of Hubei University of Technology (No. BSQD2016047) and the Open Foundation of Hubei Collaborative Innovation Center for High-efficiency Utilization of Solar Energy (No. HBSKFZD2017003).

Cheng, C.F., Zeng, Y., Ou, Y.W., Yang, Z.Y. and Chen, Z.H. (2018) Flatly Broadened and Highly Coherent Supercontinuum Generation by Induced Modulation Instability in ANDi-PCF. Journal of Applied Mathematics and Physics, 6, 640-646. https://doi.org/10.4236/jamp.2018.64056