At a given pump wavelength, in contrast to in which the input power is set to generate relatively strong CR, we lower the input power (by up to 50%) toward a threshold where the CR is minimally observable. For the same three fibers (i.e., the 71-cm NL-1.7-770 before the cut-back, the 40-cm NL-1.7-790, and the 40-cm NL-1.8-845), the threshold input powers, the corresponding CR wavelengths, and CR spectra as functions of the pump wavelength are shown in , respectively. The threshold input power at a given pump wavelength and the tuning range of the CR differs considerably among the fibers []. The CR, however, has a Gaussian-like spectrum with a nearly constant bandwidth (FWHM) of ~10 nm [], regardless of the fiber type and the pump wavelength. All these results can be understood from the dispersion profiles of the fibers.

The cross-sectional images of the fibers () can be used to determine Λ and

*d*/Λ, which permit the dispersion profile of the fibers to be calculated from a multipole method [

32]. This profile has little dependence on the number of rings of holes surrounding the guided core (10 for the three PCFs shown in ) if this number is larger than 3, and therefore the actual calculation is always performed by assigning an integer of 4 to this number. Although the cross-sectional image itself can only yield highly approximate values of Λ and

*d*/Λ, the corresponding dispersion profile must reproduce the manufacturer-specified ZDW (presumably measured by the time-of-flight method from long fiber) and the measured ZDW

_{I} (). These two constraints allow the two values to be accurately refined for each of the three fibers (), and the corresponding dispersion profile can be determined [].

The dispersion profile can in turn be used to predict the CR wavelength, using the phase-matching condition involving the source soliton and its resonant CR [

21]. If the nonlinear phase of the soliton is ignored, the condition can be written as [

19],

where

*β*_{n} is the

*n*-th order derivative of the propagation constant of the fiber calculated at the central frequency of the soliton

*ω*_{S}, while

*ω*_{CR} is the central frequency of the CR. In the numerical calculation of

*ω*_{CR},

*ω*_{S} is taken as the pump frequency

*ω*_{P} while

*β*_{n} up to the 9-th order are derived from the known profile of the dispersion coefficient

*β*_{2} []. The calculation produces the function between the CR wavelength and the pump wavelength for each of the three fibers []. Although

Eq. (1) overestimates the CR wavelength in comparison to the observed values, it satisfactorily predicts the overall CR tuning ranges of the three fibers. Fine adjustments of Λ and

*d*/Λ may slightly minimize the overestimation, but at the cost of generating large discrepancy between the predicted and measured ZDW and ZDW

_{I} values. Since the absolute value of the slope of this function approximates unity [], the CR spectrum should approximate the pump spectrum, which has a Gaussian-like spectrum of ~10 nm bandwidth (FWHM) across the tuning range of 820–1020 nm. This explains why the observed CR from each of the three fibers also has a Gaussian-like spectrum with a nearly constant bandwidth (FWHM) of ~10 nm [].

Improved prediction of CR wavelength must take into account the nonlinear phase of the soliton, which was previously estimated in a rather inconsistent manner [

7,

8,

18,

19,

28,

31,

33]. Following ref.

28, we write the condition as

where

*γ* is the nonlinear coefficient of the fiber (), and

*P*_{S} is the peak power of the source soliton (not that of the initial input pulse). This applies to the situation where an input power is large enough to excite higher-order solitons because the initial multi-solitonic state of the pump pulse can be approximated by a single fundamental soliton [

28]. For a higher-order soliton of order

*N* less than or equal to 5, the input energy is primarily coupled into the first and the strongest constituent soliton of the higher-order soliton [

34], and therefore

*P*_{S} can be taken as the peak intensity of this constituent soliton. This leads to [

1]

where

*P*_{in} is the peak power of the pump pulse which can be calculated from the known (average) input power

*P*_{0}, pulse width

*T*_{0}, laser repetition rate (80 MHz) and free-space-to-fiber power coupling efficiency

*η* (50%). The soliton frequency

*ω*_{S} is taken as the pump frequency

*ω*_{P}. Despite the large difference among the threshold input powers of the three fibers at a given pump wavelength,

Eq. (4) calculates that the measured threshold input power corresponds to

*N* = 3 at the blue wavelength end of the tuning range and

*N* = 5 at the red wavelength end of the tuning range []. The large threshold input power of NL-1.7-770 is offset by the large |

*β*_{2}| of NL-1.7-770 while the small threshold input power of NL-1.8-845 is offset by the small |

*β*_{2}| of NL-1.8-845 []. In other words, the threshold for CR generation depends on the order of the excited high-order soliton

*N* rather than the absolute value of the input power.

Despite the uncertainty of the above model, the CR wavelengths predicted by

Eq. (2) are in good agreement with the observed CR wavelengths []. It should be emphasized that the nonlinear phase of the soliton can never be neglected for accurate prediction even though the threshold input power is as small as a few milliwatts. Because the threshold input power does not excite very high-order solitons (

*N* > 6), the spectral recoil effect that deviates

*ω*_{S} from the pump frequency is small [

28], justifying the equalization of the two in the model. We note that two previous studies attempted to correlate the observed pump wavelength-dependent CR wavelength with the phase-matching condition [

18,

19]. However, no definitive quantitative agreement was obtained due to the uncertainty of the dispersion profile of the fiber [

18] or the limited number of experimental data [

19].

Due to the coexistence of ZDW and ZDW

_{I} in the fibers there are two solutions for the CR wavelength; one in the visible and another one in the infrared normal dispersion region [

31]. To predict the wavelength of the infrared CR for NL-1.7-770, we assign 1343 nm to

*ω*_{S} [] in

Eq. (1). The CR wavelength is predicted at 1539 nm, in good agreement with the observed value of 1564 nm []. The relatively small discrepancy can be attributed to the nonlinear phase of the soliton. The similar agreement can be obtained for the other two fibers with results summarized in . It is interesting to note that the onsets of the visible CR and the infrared CR can both be attributed to the first soliton constituting the excited higher-order soliton. The visible CR is generated over a narrow range of propagation distances within the first few centimeter of the fiber where the high-order soliton fission begins and the first soliton emerges [

19,

28], while the infrared CR is generated further down the fiber where the first soliton approaches the ZDW

_{I} due to the Raman self-frequency shift.