Simulation and Discussion

Numerical simulations are carried out to explain the experimental observations.

There are numerous numerical simulation studies on supercontinuum generation in tapered fibers, and typically supercontinuum generation in a biconical tapered fiber is theoretically investigated using nonlinear Schrödinger equation as numerical model [13]-[19]. The nonlinear Schrödinger equation can be expressed as

( ) ( ) ( )

where A is the amplitude of the pulse envelope, α is the absorption coefficient, t is the time, and z is the propagation axis along the fiber. The βn are nth-order dispersion coefficients, and ω₀ is the carrier frequency of the pulse. The nonlinear parameter is defined as γ = n2ω0/cAeff, where n2 is nonlinear-index coefficient and Aeff is the effective mode area. The left-hand side of Eq. 1 represents the linear terms and the

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right-hand side represents nonlinear terms. The first moment of the nonlinear response function is defined as

( ) (

) ( )

( )

R t = − f δ t + f h t (4.2)

where f_R, that be estimated to be about 0.18, represents the fractional contribution of the delayed Raman response to nonlinear polarization PNL, δ(t) represents the delta

Fig. 4-4. (a) Effective mode area and (b) nonlinear parameter versus the waist of the tapered fiber at 830 nm.

0 20 40 60 80 100 120 140

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We used symmetrized split-step Fourier algorithm to solve this nonlinear system numerically. The waist region of the tapered fiber is 10-mm-long with a uniform diameter of 1-µm. The transition region is 45 mm at each side. The fiber parameters adopt Corning SMF-28 specification. The effective core area Aeff is given by

( ( ) )

where F(r) is the amplitude of the electric field for the fundamental mode at a radius r from the axis of the fiber. In our simulation, Aeff and nonlinear parameter γ is individually plotted against the diameter of the tapered fiber at the wavelength of 830 nm, as shown in Figs. 4-4(a) and (b). The Aeff is getting smaller as the cladding being thinner when the fiber diameter is larger than 35.8 µm; at this region the power is confined by the fiber core. However, the tapered fiber being thinner than 35.8 µm, the core leaves a large amount of light guided outside the core and the A_eff begins to nonlinear parameter than untapered fiber. The nonlinear parameter increases rapidly as the fiber diameter thinner than 13.4 µm. Finally, the 1 µm-diameter waist area, the nonlinear parameter γ is 54.86 W^-1km^-1 which is approximately 25 times larger than that at the untapered region, thus exhibits large nonlinear effects to contribute to intense nonlinear processes.

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The refractive index of the standard single-mode fiber is obtained using Sellmeier formula, and the propagation constant β is derived by solving the eigen-value equation. Expand β in a Taylor series about the pump frequency that obtain the n-order dispersion coefficients. The zero-, first, second, and third order dispersion coefficients β_j are plotted in Figs. 4-5(a)-(d). At the waist region, negative second dispersion coefficient (β2<0) denote anomalous dispersion, which favors soliton formation and also benefits generation of octave-spanning white light.

However, the β2 is positive when the fiber diameter is larger than 2.6 µm, and becomes negative with the zero dispersion cross point located at the fiber diameter being 2.6 µm, as can be seen from Fig. 4-5(c). The β2 has maximum value when the fiber diameter is 11.5 µm. This dispersion variation indicates pulse propagation behavior can be greatly influenced by the sign of the dispersion in the transition region of the tapered fiber, which will be discussed in the next section.

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(a) (b)

(c) (d)

Fig. 4-5. (a)-(d) the dispersion coefficients versus the diameter of the tapered fiber at 830 nm.

Figures 4-6 show the simulation results of supercontinuum spectra for light propagates in the tapered fiber under different average powers using the hyperbolic secant pulses with a center wavelength of 830nm and repetition rate of 82 MHz.

Owing to the inaccuracy measured the geometric parameters of the tapered fiber and the coupling efficiency of the entire tapered fiber, there are slightly different between the simulation results and experiment results. However, the generated spectra broadens when the launch power increases, and the long wavelength grows faster than the short wavelength region, which agrees well with the experimental observation as shown in Fig. 4-6. The spectral bandwidth ∆λ defined at the detuning wavelength

Fiber diameter (µm) 49000 20 40 60 80 100 120

4950

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with intensity of -20 dB to that at peak wavelength is around 180 nm at 400 mW input power in the simulation result, which agrees well with the experimental results in Fig.

4-6.

Fig 4-6. Simulation results of generated spectra at different pump powers using 1-µm-diameter fiber tapers with different launched pump powers.

The evolutions of the spectral intensity and the time domain pulse shape along the whole region of the tapered fiber are plotted in Fig. 4-7 for investigating the processes of the supercontinuum generation. The color from red to blue represents the intensity from strong to weak and the vertical white lines denote the boundary of the waist region. During the input transition region, the spectrum is slightly broadening but the time domain pulse shape is broadening obviously consequent on the interaction of the linear dispersion. The spectrum spread quickly in the waist region where the nonlinearity is 25 times larger than that at the untapered region and the stimulated Raman scattering play significant roles therefore its spectrum is asymmetrical. In the time domain, the pulse width gradually narrows in the waist region because of anomalous dispersion; however in the end of the waist region the pulse is splitting from the nonlinearity and anomalous dispersion interaction. In the

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subsequent output transition region, the pulse split into several group when the transition region is in anomalous dispersion regime; however, those sub-pulse cross the zero dispersion point and reach normal dispersion regime, they broaden with propagation direction.

(a)

(b) Fig. 4-7. Simulation results of (a) the spectrum evolved along the tapered fiber and (b) the time domain pulse evolved along the tapered fiber, that the input average power is 400 mW.

Many previous studies assume that the effect of nonlinearity in taper transition region is much smaller than that in the waist region, thus it can be neglected when the numerical simulation is performed. However, according to our result of numerical simulation, the nonlinear effect in taper transition region apparently influences the spectral broadening so that it cannot be ignore. The red line in Fig. 4-8(a) accounts

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both the linear and nonlinear effect in the transition of the taper fiber, while the blue line represents the numerical result considering only the nonlinear effect in the taper transition region. According to Fig. 4-8, the effect of nonlinearity narrows the spectral broadening range and suppresses the blue-shift. To analyze the importance of the nonlinear effect in taper transition region in another viewpoint, the induced frequency chirp when the pulse propagates through the first taper transition region is calculated, as shown in Fig. 4-8(b). Due to the normal-dispersion regime in the taper transition region, the frequency chirp induced by the nonlinear effect and the linear dispersion cannot cancel each other, on the contrary, spectral broadening range degrades as a result of the frequency chirp enhancement due to the nonlinear effect. The pulse shape in time domain is displayed in Fig. 4-8(c). The nonlinear effect in taper transition region broadens the pulse width since the individual frequency chirp induced by the normal-dispersion and the nonlinear effect has the same sign. The broaden pulse width at the exit of the first taper transition lowers the capacity of the supercontinuum generation when the pulse propagates along the following waist region.

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(a)

(b)

(c) Fig. 4-8. (a)Spectral broadening, (b)frequency chirp when pulse launching into the fiber waist, (c)time domain pulse shape when pulse launching into the fiber waist for comparison between the nonlinear effect being neglected and considered.

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investigate the influence on supercontinuum generation. Assume the shape of the fiber taper is linear tapered and exponential tapered, and the calculated broadening spectra are respectively shown in Fig. 4-9(a). The blue line presents the linear tapered shape, while the red line shown the exponential tapered shape. It can be shown in the Figure that the spectral broadening for linear tapered shape is narrower than that of the exponential tapered shape. The corresponding frequency chirp and pulse width in time-domain when the pulse propagates through these two different shapes of taper transition region are shown in Figs. 4-9(b) and 4-9(c). Owning to different shapes of taper fiber, the amount and ratio of normal-dispersion, anomalous-dispersion, and nonlinear effect slightly vary when pulse propagates in taper transition region. The exponential tapered shape contains the larger pulse width and lower peak power than that of the linear tapered shape, therefore, the supercontinuum generation efficiency decreases when propagating in the following waist region.

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(a)

(b)

(c) Fig. 4-9. (a)Spectral broadening, (b)frequency chirp when pulse launching into the fiber waist under different shapes of the fiber taper.

From above discussion, the amount of frequency chirp of the pulse when it launches into the fiber waist can greatly change the supercontinuum generation spectrum. Therefore, the normalized optical field associated with incident pulse takes

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the form

( )

0 0

0, sec exp

2

T iCT

U T h

T T

   

=   − 

   , (4.5)

The dispersion effect in fiber taper is too complicated to totally compensate by simply adopting conventional dispersion compensation methods. However, the adjustment on the value C can partially compensate the pulse broadening result due to the pulse propagation in taper transition. Figure 4-10(a) presents the supercontinuum generation result using a chirped pulse with C=-1 launched into the fiber taper. The red line in Fig. 4-10(a) is the result for launching an unchirped pulse, and the blue line is the chirped pulse. The spectral broadening range is wider and the efficiency of supercontinuum generation if higher for launching an unchirped pulse rather than launching a chirped pulse. Figures 4-10(b) and 4-10(c) individually plot the frequency chirp and the pulse width in time domain corresponding to a chirp and an unchirp pulse propagation the input of the fiber waist. Because the pulse propagates in the first taper transition experiences mainly normal-dispersion (β₂>0), the initially pulse-width narrowing stage in time domain can be observed for the propagation of the chirped pulse with C=-1.

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(a)

(b)

(c) Fig. 4-10. (a)Spectral broadening, (b)frequency chirp when pulse launching into the fiber waist (c)time domain pulse shape when pulse launching into the fiber waist for chirped and unchirped pulse.