We demonstrate supercontinuum generation in a highly nonlinear photonic crystal fiber with two closely lying zero dispersion wavelengths. The special dispersion of the fiber has a profound influence on the supercontinuum which is generated through self-phase modulation and phasematched four-wave mixing and not soliton fission as in the initial photonic crystal fibers. The supercontinuum has high spectral density and is extremely independent of the input pulse over a wide range of input pulse parameters. Simulations show that the supercontinuum can be compressed to ultrashort pulses.
©2004 Optical Society of America
The field of nonlinear optics has recently undergone a minor revolution with the invention of highly nonlinear photonic crystal fibers . In particular, the ability to do nonlinear frequency conversion and generate octave spanning supercontinua  using only low power pulses from standard femtosecond lasers has led to significant breakthroughs in several areas [3, 4, 5, 6, 7].
However, the anomalous dispersion needed to generate broad spectra through soliton fission  makes the supercontinuum generation very susceptible to noise in the input pulse as noise is amplified through modulation instabilities . The interplay between the various nonlinear effects gives rise to temporal and spectral structure, and compression of the spectrally broad fiber output to a few-cycle pulse is thus extremely difficult. Moreover, small changes in the input pulse parameters can lead to very different output spectra, making the selection of an optimum fiber for a given application difficult.
In this paper we demonstrate, experimentally and theoretically, that supercontinuum generation in a highly nonlinear photonic crystal fiber with two closely lying zero dispersion wavelengths solves the problems discussed above. In particular, such a fiber generates stable, compressible spectra with high spectral density and the spectra are only weakly dependent on the input pulse over a wide range of pulse parameters.
2. Photonic crystal fiber with two zero dispersion wavelengths
The photonic crystal fiber designed for this work is made of pure silica using the stack and draw technique. The pitch of the fiber is 0.98 µm and it has a relative holesize of 0.54. The fiber core diameter is 1.5 µm with a corresponding effective area of ~1.3 µm2 and a high nonlinear coefficent γ  of 0.15 W-1m-1. The numerical aperture of the fiber is 0.49 at 800 nm and the fiber is single mode but not polarization maintaining. A scanning electron micrograph image of the fiber cross section is shown in Fig. 1.
The dispersion of the fiber has been measured using white light interferometry . The dispersion parameter β 2(ω) is defined as the second derivative of the mode propagation constant β (ω)=n(ω)ω/c where n is the effective refractive index of the mode 
The result of the dispersion measurement is shown in Fig. 2(a). The dispersion properties of the fiber are highly unusual with two closely lying zero dispersion wavelengths at 780 nm and 945 nm and anomalous dispersion in the narrow range between the two zero dispersion wavelengths. The dispersion profile should be compared to the dispersion of standard highly nonlinear photonic crystal fibers which have only one zero dispersion wavelength in the visible/near-infrared range and anomalous dispersion at wavelengths above the zero dispersion wavelength. We note that a number of photonic crystal fibers exhibit a second but widely separated zero dispersion wavelength at wavelengths typically above 1400 nm.
The dispersion of the fiber is of utmost importance as it governs the phasematching of nonlinear processes. Here we concentrate on degenerate four-wave mixing where two photons from a pump beam are converted into a signal photon and an idler photon. Under the constraint of energy conservation
the phasematching condition is given by 
Where ΔkNL is the nonlinear contribution to phasematching originating in self-phase and cross-phase modulation. By integration of the measured dispersion we can solve the above equations to find the phasematching conditions as shown in Fig. 2(b). The result (full line) is that phase-matching can be obtained for all pump wavelengths in the range in-between the two zero dispersion wavelengths. This is in sharp contrast to the results for photonic crystal fibers with only one zero dispersion wavelength where phasematching is only possible for pump wavelengths below the zero dispersion wavelength . In fibers with one zero dispersion wavelength, phase-matched four-wave mixing for pump wavelengths above the zero dispersion wavelength can be obtained by inclusion of a nonlinear phaseshift  but in our fiber phasematching is present at vanishing intensity. When self-phase and cross-phase modulation is included in our calculations the phasematching curve splits into two sets of phasematched wavelengths as shown by the dashed curve in Fig. 2(b). The power induced splitting has the benefit of providing phase-matched wavelengths in immediate vicinity of the pump wavelength. For increased intensity the phasematching curves shrink and for intensities higher than approximately 1 kW phase-matching is no longer possible.
Experimentally, light from a standard Ti:sapphire femtosecond laser is coupled into a 50 mm long piece of the photonic crystal fiber with a standard 60×microscope objective. Typically an input coupling efficiency above 50% is obtained. The input intensity to the fiber is adjusted with a λ/2 plate and a polarizer. The light emanating from the photonic crystal fiber is butt-coupled to a standard fiber connected to an spectrum analyzer where spectra are recorded as the input pulse parameters are varied.
Our major experimental findings are shown in Fig. 3(a) and Fig. 4(a) with all spectra plotted on a linear scale. In the first experiment a 40 fs pulse centered at 790 nm is launched into the fiber and the spectral evolution is followed as the input pulse energy is increased. At low pulse energies we observe a fast broadening of the spectrum but as the pulse energy is increased above ~100 pJ the spectral broadening enters a saturation regime where the output spectra are remarkably similar as the pulse energy is increased. Indeed, the supercontinuum is characterized by two peaks with fixed, sharp inner edges at ~740 nm and ~950 nm and two outer edges which slowly move outward with increasing pulse energy. Above the threshold we measure an almost complete depletion of power in the region between 740 nm and 950 nm and more than 99% of the light emanating from the fiber is contained in the two spectral peaks. In Fig. 4(a) we show the spectral output for 3 different femtosecond pulses where we have varied the central wavelength and pulse chirp while keeping the pulse energy constant. Irrespective of the input pulse wavelength and chirp we obtain almost identical spectra with depletion of power in the region between ~740 nm and ~950 nm. For all pump wavelengths and durations we observe identical behaviour with increasing pulse energy i.e. above the threshold the output consists of two peaks with fixed, sharp inner edges and slowly outward moving outer edges.
4. Pulse evolution in the fiber
To obtain a thorough understanding of the parameters governing the formation of the supercontinuum, we model the pulse evolution in the fiber with a generalized nonlinear Schrödinger equation
where A(z, t) is the slowly varying envelope of the electric field and γ is again the nonlinear coefficient of the fiber. The measured dispersion of the fiber enters the equation through the operator D̂. The action of D̂ is evaluated in the frequency domain as D̂A(z,ω)=i(β (ω)-ω/Vg)A(z,ω) where Vg is the group velocity of the pulse. The nonlinear effects of Raman scattering, self-steepening and self-phase modulation are described by the second term on the right hand side, where R(t′) is the Raman response function  and the time derivative is responsible for the self-steepening. Numerically Eq. (4) has been solved using the split step method with the fourth order Runge-Kutta method for the nonlinear term . Unless stated otherwise the following numerical results are based on Eq. (4).
Intuitive images of the pulse evolution are obtained by plotting spectrograms as the pulse propagates through the fiber. The spectrograms are calculated as
with α=100 fs.
Figure 5(a)-(f) show spectrograms of a 40 fs pulse centered at 790 nm as it propagates through the fiber with a pulse energy of 700 pJ. Once launched in the fiber, the pulse spectrum begins to broaden due to self-phase modulation. After a few mm of propagation the spectrum breaks into two major peaks. Upon further propagation the two major peaks move outward from the original pump wavelength while a steadily increasing number of sub-peaks develop in-between the two major peaks. After approximately 2 cm of propagation the outer edges of the spectrum reach their final values and the number of subpeaks increases no further. Upon further propagation we find that the intensity is redistributed within the spectrum. In particular the region between 770 nm and 980 nm is depleted in excellent agreement with the experimental observation. During the propagation the major peaks are influenced by the dispersion of the fiber as seen from the bending of the major peaks in the spectrogram. The simulated spectra extend further into the infrared region than the experimental spectra. This discrepancy is explained by leakage losses of the present fiber at wavelengths above 1100 nm.
These supercontinuum results are in marked contrast to what is observed in standard photonic crystal fibers with only one zero dispersion wavelength or two widely separated zero dispersion wavelengths [14, 15]. In such fibers the launched pulse evolves towards a higher-order soliton. However, because of higher-order dispersion and Raman scattering such solitons are not stable and break up into fundamental solitons while emitting blue-shifted nonsolitonic radiation. The fission of solitons is known to create excess noise in the supercontinuum due to modulation instability gain which amplifies noise in the input pulse .
In the present fiber, self-phase modulation broadens the spectrum and hereby provides seed wavelengths for degenerate and non-degenerate four-wave mixing. Initially the intensity is too high to provide phasematching but as soon the the intensity is low enough four-wave mixing proceeds effectively. Because the four-wave mixing process can be phasematched for low intensities, it becomes the dominant mechanism whereas soliton dynamics is arrested and plays only a minor role in the formation of the supercontinuum.
In the absence of soliton fission we expect the supercontinuum to contain less noise than supercontinua from conventional PCFs. Experimentally the amplitude noise across the optical spectrum has been measured. Using a grating and a slit the supercontinuum was spectrally divided into 2 nm slots and then detected by a fast Si photo diode. The resulting photo diode voltage was Fourier-analysed by a frequency analyser in the range 0-80 MHz. For each 2 nm slot the signal power at 76 MHz was compared to the maximum noise feature in the spectrum. The spectra were however seen to be flat and without noise over the entire frequency range. In conclusion only an upper limit of the noise could be established. The optical spectrum in the range 600 nm to 700 nm was analysed in this manner and the noise level was found to be at least 60 dB below the signal power at all wavelengths. Consequently the supercontinuum from this fiber contains less noise than previously reported supercontinua in which significant white noise was present . It should be noted that in the present experiment and in the experiments reported in  it was not possible to measure the noise on the laser, only an upper limit was established.
Both experimentally and theoretically we find that the spectral depletion extends outside the region for phasematched degenerate four-wave mixing given by the two zero dispersion wavelengths, Fig. 2. The additional depletion is caused by non-degenerate four-wave mixing being phasematched for a broader range of wavelengths. The reversed non-degenerate fourwave mixing process moving energy back into the depleted region is suppressed, due to group velocity walk-off.
This understanding of the supercontinuum generation process in the fiber is confirmed by simulations where the dispersion of the fiber has been shifted β 2(ω)→β 2(ω)+β′, so the fiber has normal dispersion for all wavelengths. The shift of β 2(ω) removes the possibility of phasematched degenerate four-wave mixing, and with this channel closed we find only a minor depletion of the intensity between 770 nm and 980 nm. We can therefore conclude that phase-matched four-wave mixing is the main cause of the intensity depletion between 770 nm and 980 nm in our fiber. In our simulations with the shifted dispersion, we still observe formation of subpeaks in the 770 nm to 980 nm region but the substructure is removed if β 2(ω) is shifted even further into the normal dispersion regime. The shifting reduces the influence of higher-order dispersion and we therefore conclude that the interplay between self-phase modulation and higher-order dispersion is the cause of the sub-structure.
We have also performed simulations using a standard nonlinear Schrödinger equation, where higher-order nonlinear effects, i.e., Raman scattering and self-steepening have been omitted in Eq. (4) and we find negligible differences in the output. This result originates in the fact that the threshold pump power for Raman scattering is higher than the threshold for four-wave mixing when phasematching is achieved . The fact that four-wave mixing supersedes Raman scattering results in a higher output power of the field as no energy is lost to phonons in the fiber. Spectral and temporal effects originating in self-steepening are effectively superseded by phasematched four-wave mixing. From the simulations we can thus conclude that the influence of higher-order nonlinear effects are suppressed due to the phasematching of four-wave mixing.
Figure 3(b) shows a simulation of the power dependence of the spectral evolution using the standard nonlinear Schrödinger equation with input pulse parameters identical to what is used in Fig. 3(a) that is a 40 fs pulse centered at 790 nm. The experiment and simulation are in almost complete agreement and the overall shape is extremely well reproduced. Leakage losses at wavelengths above 1100 nm again cause the simulated spectra to extend further into the infrared region than the experimental spectra. The mean photon energy is conserved after 5.0 cm of propagation within the numerical accuracy of the simulations. The simulated threshold for depletion of the 770 nm to 980 nm region is at ~50 pJ pulseenergy which is lower than the experimentally observed value of ~100 pJ. We attribute this discrepancy to the lack of polarization maintenance in the fiber.
In Fig. 4(b)-(d) we show the fiber output for a variety of pulses. The simulations allow us to use a greater span of pulses than experimentally available. In Fig. 4(b) the central wavelength of the 40 fs input pulse is varied from 700 nm to 1000 nm. The generated spectra consist of two major peaks located outside the depleted region, which extends from 770 nm to 980 nm regardless of the pump wavelength. Using a blueshifted pump wavelength the energy content of the blue peak is enhanced and vice versa. The double peaked spectrum is obtained with pump wavelengths spanning from below 600 nm to above 1200 nm. Figure 4(c) shows the outcome when the pulse duration is changed from 20 fs to 160 fs while keeping the peak intensity fixed. Again, we observe very similar spectra consisting of two peaks with a fixed, strongly depleted region in between the peaks. For pulse wavelengths above ~500 fs the formation of the double peaked spectrum stops. This change is due to insufficient self-phase modulation for long pulse durations i.e. the spectrum does not broaden enough to provide seed wavelengths for the four-wave mixing. Finally, in Fig. 4(d) we show simulations where the 40 fs input pulse has been up-or down-chirped to a pulse duration of 80 fs. As is evident from the figure the supercontinuum is only weakly dependent on the chirp of the input pulses.
In a fiber with a single zero dispersion wavelength phase matched four-wave mixing can assist in generating a depleted region in the output spectrum. In that case the different phase-matching condition is responsible for a depletion around the zero dispersion wavelength, and the fiber has to be pumped near the zero dispersion wavelength for the conversion to work [17, 18]. In contrast the present fiber can be pumped anywhere in between the two zero dispersion wavelengths with high conversion efficiency and it is the region between the zero dispersion wavelengths that is depleted.
Figure 6 shows a plot of the spectral intensity and phase after 5 cm of propagation. The amplitude and phase of the output are smooth as expected from the spectrograms. Due to the smooth phase of the single pulse spectrum, the experimentally realized bandwidth of more than 75 THz in the visible peak centered at 640 nm is therefore compressible to a sub-10 fs pulse. Due to the leakage loss the experimentally realized bandwidth is only ~30 THz in the near-infrared peak, which can sustain a 15 fs pulse.
Computer simulations show that the spectral position of the major peaks can be tuned by tuning of the two zero dispersion wavelengths. Using a set of carefully designed photonic crystal fibers short pulse generation throughout most of the visible and near-infrared region is therefore possible. Such tunable pulses are applicable in e.g. two color high repetition rate time-resolved pump-probe spectroscopy.
The possibility to generate a stable, compressible supercontinuum with high spectral density leads to a number of improvements in the applications of supercontinua. Frequency metrology  has recently been greatly improved by octave spanning frequency combs generated in photonic crystal fibers. The beat signal from frequencies ω and 2ω generated by a modelocked laser allows for a direct measurement of optical frequencies. By using a fiber with two properly chosen zero dispersion wavelengths the spectral density at ω and 2ω can be optimized. Moreover, the smaller noise present in a supercontinuum generated through phasematched four-wave mixing can increase the signal to noise ratio in experiments, e.g., in fiber based CARS microscopy . Similarly the supercontinuum provides easy access to new wavelength regions for two- and three-photon microscopy . Optical coherence tomography based on supercontinuum sources has recently been demonstrated . Similar to the above examples this application will also benefit from a stable supercontinuum.
In summary, we have demonstrated a new way of generating a stable and intense supercontinuum through self-phase modulation and phasematched four-wave mixing in a newly designed photonic crystal fiber with two closely lying zero dispersion wavelengths. Standard supercontinuum generation in photonic crystal fibers relies on anomalous dispersion and the supercontinuum generated through soliton fission becomes very susceptible to the characteristics of the input pulse as noise is amplified through modulation instabilities. However, in the newly designed photonic crystal fiber we observe that both the soliton fission mechanism and the noise amplification though modulation instabilities are suppressed. Consequently, both the amplitude and the phase of the supercontinuum are well-behaved, and extremely unsusceptible to the character of the input pulse. By optimizing the dispersion profile of the double zero dispersion wavelength fibers the generated stable supercontinuum can be tailored for the many applications envisaged, in particular compression to ultrashort pulses, metrology, pump-probe spectroscopy and non-linear microscopy.
The authors would like to thank Britt H. Larsen for providing us with the scanning electron micrograph image of the fiber. K. M. Hilligsøe, T. V. Andersen, C. K. Nielsen, and H. N. Paulsen would like to thank NKT Academy for financial support and the authors would like to thank Jan Thøgersen for fruitful discussions. The project was supported by the Danish Center for Scientific Computing.
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