We study theoretically, numerically and experimentally the effect of self-phase modulation of ultrashort pulses with spectrally narrow phase features. We show that spectral enhancement and depletion is caused by changing the relative phase between the initial field and the nonlinearly generated components. Our theoretical results explain observations of supercontinuum enhancement by fiber Bragg gratings, and predict similar enhancements for spectrally shaped pulses in uniform fiber. As proof of principle, we demonstrate this effect in the laboratory using a femtosecond pulse shaper.
©2006 Optical Society of America
Supercontinuum (SC) generation  in optical fibres is a striking effect with applications in imaging , frequency metrology , and interferometry . The large spectral broadening, often spanning more than an octave, results from the interaction of an intense laser pulse with the fiber’s dispersion and cubic nonlinearity . Attention has recently turned to tailoring and improving SC for specific applications. The pump wavelength, duration, energy, and fiber non-linearity and dispersion can be chosen to gain a limited degree of control, whilst more complex methods include pump chirp control , pump spectral shaping [7, 8], UV postprocessing , cascading different fibres  and tapering . Another such scheme involves writing a Bragg grating in the fiber, resulting in a spectrally narrowenhancement of up to 15 dB around the long-wavelength side of the Bragg resonance . This wavelength selective amplification is useful in applications that use only a small part of the spectrum, such as frequency-comb stabilization, where Bragg gratings were shown to improve the signal-to-noise ratio by 24 dB .
This paper generalizes and strengthens grating-enhanced SC generation in two ways. First, although semi-analytic results and modeling agree qualitatively with experiments , an intuitive physical explanation has not been presented. Whilst the enhancement has been linked to the magnitude of dispersion around the grating bandgap , experimental and calculated spectra exhibit complex and unexplained features such as depletion on the short wavelength side of the bandgap and fine spectral fringes with varying period. To realize the scheme’s full potential these effects must be understood. Providing a simple explanation involving interference between initial and nonlinearly generated components is our main goal here. Additionally, we generalize this notion to arbitrarily shaped narrowband spectral phase features. We show that the phase features can be imparted on the field either continuously throughout the fiber, as is the case with Bragg gratings, or as an initial condition before nonlinear pulse propagation commences, as schematically shown in Fig. 1. We implement the latter configuration using a femtosecond pulse shaper, and as proof of principle demonstrate a tunable spectral enhancement after nonlinear propagation in a uniform photonic crystal fibre. This scheme could potentially address one practical disadvantage of Bragg gratings, namely, their limited tunability.
2. Nonlinear pulse propagation in Bragg grating
Supercontinuum generation in a fibre Bragg grating has recently been well described by Westbrook and Nicholson using a perturbative form of the nonlinear Schrödinger equation (NLSE) . Here, we focus on the dynamics of the enhancement mechanism for a single pulse rather than full supercontinuum description . Our primary aim is an intuitive, quantitative description of the enhancement mechanism rather than full broadband agreement with the NLSE. The key technical difference between this work and Ref.  is that we consider the nonlinearity as a perturbation to the linear evolution in the presence of the grating, whereas Ref.  treats the grating as a perturbation to the full nonlinear solution for a uniform fibre.
We initially consider a fibre Bragg grating, and a single transform-limited incident pulse, rather than the complicated fields associated with SC generation. This is justified since the use of scaling laws , in combination with experimental parameters [12, 13] indicate that the gratings were written after the soliton fission point, so the SC field consists of a train of discrete pulses. Since the grating is apodized, incident pulses are coupled into the forward-propagating Bloch mode , permitting the use of the NLSE rather than coupled mode equations. The narrowband grating dispersion cannot be represented over the full pulse bandwidth by the usual Taylor expansion of dispersion coefficients, unlike previous studies of nonlinear propagation in Bragg gratings  where the pulse bandwidth was small or comparable to the bandgap. We therefore work in the frequency domain, and have
where A(ω′, z) is the Fourier transform of the field envelope, β T(ω) the transformed wavenumber, γ=80W-1 km-1 is the fiber’s nonlinear coefficient and B(ω′, z)=FT[|A(T, z)|2 A(T, z)] is the Fourier transform of the usual self-phase modulation term, where T is retarded time. Bandgap reflection is ignored for reasons discussed below. In this frame, ω′=ω-ω 0 is the frequency relative to the carrier ω 0. The transformed wavenumber is
where β(ω) and β F(ω) are the wavenumbers with and without the grating. Equation (2) transforms β(ω) from the laboratory frame to a comoving frame with the phase and group velocity of the uniform fibre at ω 0 - we do not wish to “transform away” the grating dispersion. The grating is modeled as a 1-D photonic bandgap , with Bragg wavelength λ B. In the chosen frame, β T=0 away from the bandgap, β T<0 for λ<λ B, since β<β F, and β T>0 for λ>λ B since β>β F. We ignore background fiber dispersion because of the short propagation lengths under consideration.
Though our analysis is perfectly general, for our modelling we choose the experimental parameters of Li et al. : grating Bragg wavelength λ B=950 nm and coupling strength κ=600 m-1, and pulse duration T FWHM=150 fs and centre wavelength λ 0=945 nm. The pulse bandwidth (6.3 nm) considerably exceeds that of the grating (0.7 nm), so only a small fraction of the pulse energy is affected, justifying the absence of the bandgap reflection in Eq. (1). At the bandgap edges the transformed wavenumber is ±κ, so the typical grating length scale L G=1/κ=1.67 mm is much smaller than the nonlinear length L NL=1/(γP 0)=25 mm where P 0=0.5 kW is the pulse peak power.
Neglecting the nonlinearity, Eq. (1) the field evolves linearly as
and the effect of the grating is thus to rotate the field’s spectral phase in opposite directions on either side of the gap. To treat the nonlinearity, we make two simplifications. First, all our calculations are truncated to first order in z/L NL; equivalently we assume z≪L NL. Whilst this may seem overly restrictive, we show that much of the essential physics can be explained by this simple model. We then write
with A NL(ω′, z) the nonlinear contribution correct to first order in z/L NL. Without the grating, the nonlinearity causes self-phase modulation (SPM). The equivalent frequency domain description of SPM is intrapulse four-wave mixing (FWM) between all energy conserving frequency combinations in the pulse spectrum A L(ω′, z). Hence the second simplification: since the FWM contribution at frequency ω′ results from the many mixing processes just described, the spectrally narrow effect of the grating on A NL(ω′, z) is small. Alternatively the spectrally narrow feature is broad and weak in time, lacking nonlinear effect. By ignoring the grating’s effect on the FWM, the nonlinear term B(ω, z), which drives A NL(ω′, z), can be evaluated at z=0.
Formally, we proceed by substituting Eq. (4) into Eq. (1). By the first assumption, only A L(ω′, z) terms remain in the nonlinear driving term, which by the second assumption can be evaluated at z=0. We obtain
which can be solved using integrating factors to yield
This forms the starting point to explain a key feature of the reported spectra: depletion for wavelengths below the bandgap, accompanying the enhancement on the long wavelength side. For now, we consider A NL(ω′, z) in the initial stages of propagation i.e. z≪L G; equivalently we ignore grating dispersion of the nonlinear terms. This additional assumption is not intrinsic to our formalism and is only required for the simple argument which follows. In this limit, Eq. (6) reduces to
The argument proceeds by considering the relative phases of the two terms in Eq. (4) for various cases. Initially, the pulse is transform limited at T=0, so ∠A(ω′, 0)=∠B(ω′, 0)=0. Thus ∠A L(ω′, z)=β T(ω)z and from Eq. (7), ∠A NL(ω′, z)=π/2, leading to the simple vector construction in Fig. 2. For λ<λ B, β T(ω)<0 and the resultant vector is shortened, corresponding to a depletion of the spectrum, whereas for λ>λ B, β T(ω)>0 leading to an enhancement. By rotating the spectral phase for a narrow wavelength range without changing the nonlinear processes, therefore, the grating alters the phase angle between the original field at a frequency, and the nonlinearly generated field at that same frequency, thereby changing the vector sum.
As noted, Westbrook and Nicholson  presented approximate formulae for supercontinuum generation in a Bragg grating. We now compare their results with the argument just presented here. Since we assume z≪L G, the analogous result in Ref.  is Eq. (4), which shows an interference whose sign is affected by the sign of the wavenumber, but is otherwise very difficult to interpret. Note that, like Westbrook and Nicolson , we use the basic assumption that the grating’s influence on nonlinear processes is negligible. Our argument requires the additional assumptions of a weak nonlinearity, so that z≪L NL, and transform limited pulses. In return, however, we provide the simple physical interpretation described above, and show that the sign of the interference depends only on the sign of the transformed wavenumber.
for transform-limited pulses. The factor of 1/2 in the sine argument in Eq. (8) arises because the nonlinear contribution is generated along the entire length, so that the contribution generated near the end of the fibre is not affected by the grating. We now point out the salient features of Eq. (8) plotted as the solid red curve in Fig. 3(b). For reference, the transformed wavenumber βT outside the bandgap is shown in Fig. 3(a). The sign of the β T(ω) term in the numerator determines whether Eq. (8) corresponds to enhancement or depletion. We shall “approach” the bandgap from the long wavelength side, where β T(ω)>0 and enhancement occurs. As expected, well away from the bandgap there is no spectral modulation owing to the rapid falloff of the transformed wavenumber. Moving closer, β T(ω) begins to increase monotically and hence the sin2 argument in Eq. (8) increases. For small β T(ω)z (e.g. at 950.5 nm), an enhancement proportional to z 2 is observed. Closer to the bandgap, however, β T(ω)z>1 and the oscillatory behaviour of the sine term in Eq. (8) becomes important. The enhancement goes through a maximum where β T(ω)z≈π. (The relation is not exact because of the β T(ω) term in the denominator of Eq. (8)). The enhancement peak therefore moves to longer wavelengths with increasing z. Even closer to the bandedge, fine spectral fringes appear since β T(ω)z>2π. However, because of the large β T(ω) term in the numerator of Eq. (8), the amplitude of these fringes is very small. Published experimental spectra have suggested the presence of fine fringes [12,14], but limited resolution prevents quantitative comparison with Eq. (8). As shown in Fig. 4, Eq. (8) agrees well with the exact numerical solution over a typical experimental propagation length of L=30 mm. The approximation begins to break down for z>L NL as second-order nonlinear effects begin to dominate.
Our Eq. (8) is essentially a special case of Eq. (3) in Ref. , employing our additional assumptions of a weak nonlinearity and a transform-limited pulse. In return for this loss of generality we can predict numerous qualitative features of the spectrum, as discussed in the paragraph above. Indeed, the NLSE results given in Fig. 1(b) of Ref.  show spectral fringes whose period and visibility decrease approaching the bandgap, consistent with Eq. (8). Our work, in offering physical insight and simple analytic results, is therefore complementary to Ref. , which is useful for rapid numerical approximation of grating-enhanced supercontinuum spectra.
The grating phase profile thus changes femtosecond pulse propagation, particularly the relative phase between the incident pulse and nonlinearly generated components, causing asymmetric spectral modulation around the Bragg wavelength. Our interpretation applies to any Bragg grating enhanced supercontinuum experiment where the grating is written after the soliton fission point, where the field incident on the grating consists of discrete solitons. We now discuss in Sec. 3 a generalization of the concept to a wider class of experimental geometries. Section 4 is concerned with establishing proof of principle of this generalization.
3. Nonlinear pulse propagation with arbitrary narrow phase feature
The simplifications in the theoretical analysis in Sect. 2 remain valid if the phase is applied as a single narrowband phase discontinuity of Δϕ(ω)=β T(ω)L at z=0 mm. This means that the two roles played by the grating in the process, namely the inscription of a phase feature, and providing the medium for nonlinear propagation, can be separated. In other words, the process is equivalent to inscribing a spectral feature, by whatever means, followed by self-phase modulation in a fiber. Since this insight allows the two elements to be selected and optimized independently, it broadens the possible application of this process.
We now treat the more general situation in which the application of the spectral feature and the nonlinear propagation are separated. Instead of solving Eq. (1) for a transform limited pulse with a narrowband dispersive term, as in the previous section, we use the initial condition
where A(ω′, 0-) is the transform limited pulse before the phase discontinuity. Since we are ignoring fibre dispersion and the phase feature has been incorporated into the initial condition, we solve Eq. (1) with β T(ω)=0. As with the previous section, we work to first order in z/L NL. However, whereas before we ignored the grating’s influence on the nonlinear contribution, here we ignore the effect of the phase filter instead. This enables us to evaluate the nonlinear driving term at z=0- i.e. with a transform limited pulse, and we obtain
The relative enhancement is then given by
Equation Eq. (11) is shown as the dashed blue curve in Fig. 3 for L=30 mm. Compared with Eq. (8), the fringes are larger close to the bandgap because Eq. (11) does not have the large phase term in the denominator. Also, the sign of the effect (enhancement or depletion) is now determined by the sign of sin Δϕ(ω) rather than simply β T(ω) as in Eq. (8). Physically, both of these differences correspond to the entire grating phase β T(ω)L being pre-applied at z=0 mm, before any nonlinear components are generated, rather than continuously throughout the propagation. Besides this, the two curves are qualitatively similar.
This generalization may now be applied to other, narrow phase perturbations of the pulse. We therefore posit a scenario in which a narrowband phase feature is imprinted on a pulse prior to propagation through a nonlinear medium. The experiment described in the following section tests such a scenario. We note that the application of spectrally narrow amplitude features, followed by self-phase modulation was earlier considered by Präkelt et al. , though not in the context of super continuum generation. Moreover their treatment is qualitative and does give rise to quantitative predictions.
In our experiment we show that our theory correctly describes spectral enhancement of femtosecond pulses which have narrowband phase features applied before nonlinear propagation in photonic crystal fibre. Because this technique limits the spectral enhancement to be within the pump spectrum, its purpose here is to demonstrate proof of principle of the ideas developed in the previous section. A more complicated geometry which permits a wider choice of enhancement wavelengths is described in section 5.
We applied spectrally narrow phase features to 80 fs pulses with centre wavelength λ 0=778 nm from a modelocked Ti:Sapph laser using a 4-f femtosecond pulse shaper  as in Fig. 5. A ruled reflection grating (1800 rules/mm) disperses the spectrum which is then imaged by a lens (D=50 mm, f=200 mm) onto the Fourier plane. A transparent phase mask in the Fourier plane imparts a phase profile Δϕ(ω), then a second lens and grating pair recollimate the beam which is coupled into 4.5 m of small core PCF (Crystal Fibre NL-1.8-750) with measured zero dispersion wavelength λ ZD=760 nm and manufacturer specified nonlinearity γ=99 W-1km-1. The output is spectrally resolved on an optical spectrum analyser. Our phase mask was a d=92 µm thick piece of polished BK7 glass, which was rotated to adjust the effective thickness and hence phase delay within a range equivalent to 74–80 wavelengths at a wavelength λ D=778 nm.
Figure 6(a) shows the enhancement and depletion achievable with a pulse peak power of approximately 14W, giving a nonlinear length L NL=0.7 m.Without the phase mask, SPM has only a small effect. With a phase delay such that sin Δϕ=1, a 5 dB enhancement is observed as predicted by Eq. (11), whilst a corresponding depletion is seen for sin Δϕ=-1. Across the width of the feature Δλ=1.7 nm, the enhancement is not uniform; because of the wavelength dependent phase delay (ϕ∝2π/λ) the relatively large thickness produces a perceptible change in phase over the spectral feature. Material dispersion also plays a small role in the asymmetry. Figure 6(b) shows the spectral intensity at λ D from a series of spectra taken with varying delay. As predicted in Eq. (11), sinusoidal dependence on Δϕ with period 2π is observed.
It is clear from simulation, theory, and our proof of principle experiment, that the sort of enhancements first reported by Li et al.  may be generalized to other situations in which a narrowband phase feature is imposed on a pulse, with the pulse then subject to propagation in a Kerr-like nonlinear medium. The nonlinear propagation has the effect of imprinting a pre-conditioned phase feature on the spectral density of the pulse. While it is not easy to separate the phase filtering and propagation when using a Bragg grating, our experiment clearly shows the distinct roles of phase filtering and propagation. This insight follows immediately from our quantitative, yet intuitive theoretical description of the process, which differs from that of West-brook and Nicholson  which provides less physical insight, and that of Präkelt et al. , which is qualitative.
In our geometry, the enhancement is limited to within the pump spectrum. Its main purpose is to verify the theoretical development in this paper. To obtain spectral peaks at other frequencies, a high power could be propagated through enough fibre to generate a supercontinuum encompassing all required enhancement wavelengths. The resulting field, consisting of various pulses distributed over the spectrum, would have phase features applied using the pulse shaper and would then be re-injected into a second fibre for subsequent spectral enhancement and further broadening. This scheme is similar to that of Li et al.  with the Bragg grating substituted by a pulse shaper. In addition, replacing the BK7 glass phase mask with a liquid-crystal spatial light modulator  would allow far greater control of the applied phase and attenuation. We note that the experiments presented here are reminiscent of those of Präkelt et al. However, their work, in which the self-phase modulation occurs in a water jet, was not conducted in the context of fiber supercontinuum generation.
In conclusion, we have shown that Bragg gratings cause nonlinear spectral enhancement by inducing relative phase rotation between linear and nonlinear components. Our experiment confirms that any narrow phase feature followed by nonlinear propagation exhibits similar effects. Though the enhancement was restricted to within the pump spectrum, it nonetheless demonstrates that that our theoretical argument is sound. The analytic result and simple explanation will, we believe, enable spectral tailoring of supercontinuum light sources in novel ways.
TGB acknowledges support through the Denison Distinguished Visitor scheme. This work was supported by an Australian Research Council Discovery Grant.
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