Method for efficient four-wave mixing generation and short pulse generation equipment using the method

A method for generating four-wave mixing in an optical fiber, in which probe light fs and pumping light fp, both set on a frequency axis on condition of fp≠fs, are launched into the optical fiber to generate the four-wave mixing in the optical fiber,

• wherein
• length z of said optical fiber is adapted to satisfy Lmin≦z≦Lmax, the Lmin and Lmax being determined by the following A and B;
• wherein,
• A:
• nonlinear ordinary differential equations for describing the four-wave mixing in an optical fiber are to be given by
• (dEp/dz)+(½)αEp=iγ[(|Ep|2+2|Es|2+2|Ec|2)Ep+2E*pEsEcexp(iΔβz)] (2.1) (dEs/dz)+(½)αEs=iγ[(|Es|2+2|Ec|2+2|Ep|2)Es+E*cEp2exp(−iΔβz)] (2.2) (dEc/dz)+(½)αEc=iγ[(|Ec|2+2|Ep|2+2|Es|2)Es+E*sEp2exp(−iΔβz)] (2.3),
• where E denotes an electric field,
• subscripts p, s, and c denote the pumping light, the signal light (the probe light), and wavelength converted light (idler light),
• α denotes loss of the optical fiber per unit distance,
• γ denotes a non-linear coefficient,
• the γ satisfying a relationship among a pumping light wavelength λp, nonlinear refractive index n2, and effective area Aeff,
• γ=(2π/λp)·(n2/Aeff) (3),
• Δβ being phase mismatching of a propagation constant and satisfying a phase matching condition in terms of frequency given by
• 2ωp=ωs+ωc (4)
• (where ω is an angular frequency, having a relationship of ω=2πf with frequency f), and Δβ also satisfying the following equation,
• Δβ(ωs)=(λp2/2πc)D(λp)(ωs−ωp)2 (5),
• where D is a chromatic dispersion coefficient of the optical fiber, the coefficient being normally expressed in ps/nm/km unit, and c is the speed of light in vacuum,
• wherein
• said differential equations are integrated over the entire length of the optical fiber L with respect to Es and Ec with accuracy of calculation of an error equal to or less than 0.1% along the coordinate z in the longitudinal direction of the fiber, and then a resulting absolute value is squared to determine the corresponding optical power Ps(z) and Pc(z),
• using the solutions, conversion efficiency Gc of the idler light is calculated by a ratio of idler light intensity Pc(L) measured at an output end of the fiber to intensity Ps(0) of the probe light at an input edge of the fiber,
• moreover, probe light gains Gs are calculated by a ratio of probe light intensity Ps(L) measured at the emitting edge of the fiber to probe light intensity Ps(0) at the input edge of the fiber,
• the Ps and Pc being the probe light intensity and the idler light intensity, respectively, which are expressed by functions with an distance z from the input end of the fiber,
• L being an entire length of the fiber in question,
• when this calculation is performed, calculated result can be plotted with the horizontal axis representing distance z and the vertical axis representing the probe light gain or idler light conversion efficiency,
• allowing z of a target quantity for calculation to take on a sufficiently large finite value would cause variations in the longitudinal direction of the conversion efficiency of the idler light to oscillate periodically as a general property of the solutions, and
• values of z, by which the solutions take on maximal values due to periodic behavior, are discrete and have a finite number of solutions,
• letting z=Lm for the smallest value of z among the values of z,
• in addition, letting z Lmax for a distance satisfying a condition of z>Lm and longer 10% than Lm,
• B:
• next, as for aforementioned equations (2.1)-(2.3), two different approximate solutions are known under appropriate conditions,
• in this application, for the sake of convenience, after the initials of the reserchers, the solutions are referred to as SB solution (Stolen and Bjorkholm) [8] and HJKM solution (Hill, Johnson, Kawasaki, and MacDonald) [24],
• idler light conversion efficiency Gc is given by a ratio of idler light intensity Pc(L) measured at the emitting edge of the fiber to probe light intensity Ps(0) at the input edge of the fiber,
• moreover, probe light gains Gs is given by a ratio of probe light intensity Ps(L) measured at the output end of the fiber to probe light intensity Ps(0) at the input end of the fiber,
• the Ps and Pc being the probe light intensity and the idler light intensity, respectively, which are expressed by functions with an argument of distance from the input edge in the fiber,
• L being an entire length of the fiber in question,
• with only results being shown, signal light gain Gs and idler light generating efficiency Gc, which are obtained by respective approximate solutions, are expressed as follows,
• (a) SB solution;
• (a-1) for 4γPp>−Δβ,
• Gs=1+γ2Pp2(0)L2[(sinh(gnL)/gnL]2 (6.1.1.a), Gc=γ2Pp2(0)L2[(sin(gaL)/gnL]2 (6.1.1.b),
• where
• ga=(½)[−Δβ(Δβ+4γPp)]1/2 (6.1.2),
• next,
• (a-2) for 4γPp<−Δβ,
• Gs=1+γ2Pp2(0)L2[(sinh(gbL)/gbL]2 (6.2.1.a) Gc=γ2Pp2(0)L2[(sinh(gbL)/gbL]2 (6.2.1.b),
• where
• gb=(½)[Δβ(Δβ+4γPp)]1/2 (6.2.2),
• (b) HJKM solution;
• Gs=Ps(L)/Ps(0)=exp(−αL) (7.1) Gc=Pc(L)/Ps(0)=γ2Pp2(0)e−αL{(1−e−αL)2+e−αL sin2(ΔβL/2)}/(α2+Δβ2) (7.2),
• when the calculated results are plotted regarding the fiber length L as z in the equations above, as equations (2.1)-(2.3) are solved, such a graph can be plotted as provides a plot with the horizontal axis representing distance z and the vertical axis representing the idler light conversion efficiency Gc, where, as a general property of the solutions, at a value of z less than the aforementioned Lmax and greater than zero, any of the solutions obtained by solving numerically equations (2.1)-(2.3), the equations (6.1.1.a) and (6.1.1.b), the equations (6.2.1.a) and (6.2.1.b) (SB solutions), and the equations (7.1) and (7.2) (HJKM solutions) produces differences not less than 1% which are greater than the calculation error, and
• values of z producing the differences take on two different values at maximum, letting z=Lmin for the greatest value z of the values.