## Abstract

Compression of linearly chirped picosecond pulses in hollow-core photonic bandgap fibers is investigated numerically. The modal properties of the fibers are modeled using the finite-element technique, whereas nonlinear propagation is described by a generalized nonlinear Schrödinger equation, which accounts both for the composite nature of the nonlinearity and the strong mode profile dispersion. Power limits for compression with more than 90% of the pulse energy in the main peak of the compressed pulse are investigated as a function of fiber design, and the temporal and spectral widths of the input pulse. The validity of approximate scaling rules is investigated, and figures of merit for fiber design are discussed.

© 2008 Optical Society of America

## 1. Introduction

The fabrication of fully fiber-integrated, high-power femtosecond (fs) laser systems is currently an active research topic, since this technology holds the promise of greatly reduced production and maintenance costs compared to existing fs lasers. This in turn would vastly expand the potential for use of fs lasers in e.g. micromachining and various clinical applications. A typical design feature of these laser systems is a low-power picosecond (ps) or fs seed oscillator, followed by several amplifier stages, and in some cases dispersive pulse-stretching elements. In the final stage, the amplified pulses are compressed dispersively to fs duration, which in most current systems is done using a pair of bulk Bragg gratings. For a truly fiber-integrated system, hollow-core photonic bandgap (HC-PBG) fibers constitute a key enabling technology, because they offer anomalous dispersion, suitable for compression of pulses chirped by self-phase modulation (SPM), in combination with very low nonlinear coefficients. This enables linear or near-linear dispersive compression, similar to what may be obtained using bulk diffraction gratings, in an all-fiber system at pulse energies where standard fibers would be strongly nonlinear. While the practical feasibility of this idea has been demonstrated in a number of recent experimental works [1, 2, 3], the limits to power scaling in the HC-PBG compressor are still an open question. The main limitations are expected to be the dielectric breakdown thresholds of air and silica, and the nonlinear effects in the HC-PBG fiber which eventually set in as the pulse power is increased. Several investigators have reported evidence of nonlinear phenomena in HC-PBG fibers. [4, 5, 6, 7] Of particular interest for the present work is the study by Ouzounov *et al* demonstrating compression of 120 fs input pulses to 50 fs pulses with MW peak power by the soliton effect in a Xe-filled HC-PBG fiber. This clearly demonstrates that a regime of strongly nonlinear pulse propagation exists below the material breakdown thresholds. For linear dispersive compression, it is therefore of interest to investigate the limits to power scaling imposed by fiber nonlinearities, and how HC-PBG fibers should be designed for optimal performance.

The purpose of this work is to address these questions through detailed numerical modeling of both the modal properties of HC-PBG fibers and the pulse compression process in the presence of optical nonlinearities. Since there is, in principle, an infinity of both HC-PBG designs and input pulse shapes to consider, we must make some simplifying restrictions to provide clear answers. First of all, we will assume that a high quality of the compressed pulse is desired. Secondly, for the input pulses we will exclusively consider the class of linearly chirped parabolic pulses, which allow us to study trends with pulse duration and bandwidth in a simple way. This pulseform constitutes an asymptotic solution to nonlinear pulse propagation in a fiber with gain and normal dispersion, and can be made experimentally by launching fs seed pulses into fiber amplifiers with the proper combination of gain, effective area and dispersion [8]. Finally, we will consider three HC-PBG structures, chosen to be representative of present-day state-of-the-art designs, taking only the overall scale of the fiber microstructure as an optimization parameter.

The rest of this paper is organized as follows: in section 2, the equation describing nonlinear propagation in HC-PBG fibers is derived, and some approximate scaling relations are discussed. In section 3, the modeling of HC-PBG dispersion and mode profiles is presented. Section 4 contains the central results on pulse compression, along with a short discussion of possible roads to further power scaling. Section 5 summarizes our conclusions.

## 2. Nonlinear propagation equations

#### 2.1. General formulation

We consider the Maxwell equations in the presence of a nonlinear polarization term:

Here *ε* is the relative dielectric constant of the medium, assumed *z*-independent for a straight waveguide, and **P**
* _{NL}* is the nonlinear part of the induced polarization. The

**E**and

**H**fields are expanded into modal fields:

Each modal field in itself fulfills Eq. (1) and Eq. (2) without the nonlinear term. Note that this assumption, together with Eqs. (1), (2), implies a neglect of the *z*-derivatives of *G* in comparison with *β*, that is, a slowly-varying envelope approximation. The modal fields satisfy the orthogonality relation:

It was shown in Ref. [9] that these assumptions leads to the 1+1D propagation equation:

The nonlinear polarization, **P**
* _{NL}* is given by:

In isotropic materials, the independent tensor components are *χ*
^{(3)}
* _{xxxx}*,

*χ*

^{(3)}

*,*

_{xxyy}*χ*

^{(3)}

*,*

_{xyyx}*χ*

^{(3)}

*. At any particular point, we can express the tensor in a local coordinate system whose*

_{xyxy}*x*-axis is aligned with the electric field. Since

*χ*

^{(3)}

*is the same in all coordinate systems, this implies that*

_{xxxx}where *R*(**r**,*t*-*t*′) is the Raman response and *χ*
^{(3)}(**r**
_{⊥})=*χ*
^{(3)}
* _{xxxx}*(

**r**

_{⊥}). This neglects asymmetric contributions to

*χ*

^{(3)}near surfaces. It has also been assumed that

*χ*

^{(2)}processes, even those at interfaces, are unimportant due to a lack of phase-matched guided modes at second-harmonic frequencies.

Using the field expansion (4), **P**
* _{NL}* becomes:

$${\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\mathbf{e}}_{n}(\mathbf{r},t;{\omega}_{1})\left[{\mathbf{e}}_{p}(\mathbf{r},t\prime ;{\omega}_{2})\phantom{\rule{.2em}{0ex}}\xb7\phantom{\rule{.2em}{0ex}}{\mathbf{e}}_{q}^{*}(\mathbf{r},t\prime ;{\omega}_{3})\right]{\chi}^{\left(3\right)}\left(\mathbf{r}\right)R(\mathbf{r},t-t\prime )$$

Typically, the fiber cross section can be divided into sections of constant *χ*
^{(3)}, e.g. silica and air regions. Let us suppose that the fiber is made up of *N* distinct materials. Combining Eqs. (6) and (10) we obtain

$$\int d{\omega}_{1-2}{\tilde{G}}_{n}(z,{\omega}_{1}){\tilde{G}}_{p}(z,{\omega}_{2}){\tilde{G}}_{q}^{*}\left(z,{\omega}_{1}+{\omega}_{2}-\omega \right){R}_{\nu}\left(\omega -{\omega}_{1}\right)\times $$

$${\int}_{v}d{\mathbf{r}}_{\perp}\left[{\mathbf{e}}_{m}^{*}({\mathbf{r}}_{\perp},\omega )\xb7{\mathbf{e}}_{n}({\mathbf{r}}_{\perp},{\omega}_{1})\right]\left[{\mathbf{e}}_{p}({\mathbf{r}}_{\perp},{\omega}_{2})\xb7{\mathbf{e}}_{q}^{*}\left({\mathbf{r}}_{\perp},{\omega}_{1}+{\omega}_{2}-\omega \right)\right],$$

where

So far, the time-domain electric and magnetic fields have been assumed real, which implies that the frequency-domain expansions must run over both positive and negative frequencies, with **e**
* _{m}*(

**r**

_{⊥},-

*ω*)=

**e***

*(*

_{m}**r**

_{⊥},

*ω*), (and a similar relation for

**h**

*),*

_{m}*G*(

*z*,-

*ω*)=

*G*(

*z*,

*ω*)* and

*β*(-

_{m}*ω*)=-

*β*(

_{m}*ω*), ∀

*m*. Since the negative-frequency components are fully determined by their positive-frequency counterparts, it is a commonly used trick to formulate the equations in terms of only the positive-frequency components, assuming that positive- and negative-frequency components are well separated, also in the nonlinear term. This requires that the spectral width of the pulse is smaller than ~

*ω*

_{0}/3, where

*ω*

_{0}is some suitably chosen base frequency [10]. In the present case, this assumption is automatically fulfilled due to the finite width of the photonic bandgap. Performing the frequency separation in the usual way, we obtain:

$$\frac{1}{{\left(2\pi \right)}^{2}}\int d{\omega}_{1-2}{\tilde{G}}_{n}(z,{\omega}_{1}){\tilde{G}}_{p}(z,{\omega}_{2}){\tilde{G}}_{q}^{*}\left(z,{\omega}_{1}+{\omega}_{2}-\omega \right){R}_{v}\left(\omega -{\omega}_{1}\right)\times $$

$${\int}_{v}d{\mathbf{r}}_{\perp}\left[{\mathbf{e}}_{m}^{*}({\mathbf{r}}_{\perp},\omega )\xb7{\mathbf{e}}_{n}({\mathbf{r}}_{\perp},{\omega}_{1})\right][{\mathbf{e}}_{p}({\mathbf{r}}_{\perp},{\omega}_{2})\xb7{\mathbf{e}}_{q}^{*}\left({\mathbf{r}}_{\perp},{\omega}_{1}+{\omega}_{2}-\omega \right)]$$

where it is now understood that the frequency integrals only extend over positive frequencies. Note also that we have now fixed the normalization of the modal fields to *N _{m}*=1. In the following, only a single field state will be considered, and the indices on

*G*and

**e**are suppressed.

Due to the frequency dependence of the modal fields, the frequency integrals in Eq. (13) cannot be evaluated by a series of convolutions, as would be the case if the fields were constant in frequency. Since the evaluation of the full two-dimensional frequency integral at all values of *ω* will scale with *N*
^{3}
* _{w}*,

*N*being the number of points in the frequency grid, whereas convolutions can be done in

_{w}*O*(

*N*log

_{w}*N*) operations by the fast Fourier transform method, this is a very serious drawback. In a previous study of solid-core photonic bandgap fibers [11], it was found that the modal overlap integral could be well approximated by:

_{w}$${\left[C\left(\omega \right)C\left({\omega}_{1}\right)C\left({\omega}_{2}\right)C\left({\omega}_{1}+{\omega}_{2}-\omega \right)\right]}^{-\frac{1}{4}}$$

Although the previous study did not concern fibers with several nonlinear materials, this approximation will also be adopted here, as it is favoured by two general arguments: Firstly, it is correct to linear order in the modal field frequency variations, and secondly, it preserves photon number conservation [11]. It is customary to express the nonlinearities in terms of an *n*
_{2} coefficient defined as:

where *n _{ν}* is the linear refractive index of material

*ν*. Introducing

*n*

^{(ν)}

_{2}instead of

*χ*

^{(3)}

_{ν}and using Eq. (15), we arrive at the final propagation equation:

$$\frac{1}{{\left(2\pi \right)}^{2}}\int d{\omega}_{1-2}{\hat{G}}_{\left(v\right)}(z,{\omega}_{1}){\hat{G}}_{\left(v\right)}(z,{\omega}_{2}){\hat{G}}_{\left(v\right)}^{*}\left(z,{\omega}_{1}+{\omega}_{2}-\omega \right){R}_{v}\left(\omega -{\omega}_{1}\right)$$

where in the last equality, Eq. (5) expressing the normalization has been explicitly used. This definition of the effective area is equivalent to that used in earlier investigations of silica/air PBG fibers [12, 13].

In the numerical calculations, the
${n}_{2}^{{\mathrm{SiO}}_{2}}$
coefficient was set to 2.66·10^{-20} W/m^{2}, and the Raman response function was parametrized as suggested by Agrawal [14]. For air, the coefficient of the instantaneous response (the delta-function part of *R _{air}*), was set to 2.9·10

^{-23}W/m

^{2}[15], and the delayed response was parametrized using the simple form given by Mlejnek

*et al*[16].

#### 2.2. Scaling laws and figures of merit

Despite the complexity of nonlinear propagation in the HC-PBG fibers, it is useful to consider the approximate scaling relations that follow from a more simplistic model. To this end, we reformulate the nonlinear propagation equation in terms of *G̃*(*z*,*ω*), given by Eq. (12) to display the dispersive term explicitly. If third- and higher-order dispersion, the effects of self-steepening and mode profile dispersion are neglected, and the delayed (Raman) nonlinear response is approximated as an instantaneous (Kerr) response, we arrive at the traditional nonlinear Schrödinger equation, which can be written in the dimensionless form [14]

where *t*
_{0} is some characteristic time scale of the pulse, *ω*
_{0} is the base carrier frequency of the pulse, and the second frequency derivative of the propagation constant,
${\beta}_{2}=\frac{{d}^{2}\beta}{d{\omega}^{2}}|{\omega}_{0}$
, was assumed negative. The *A _{eff}* parameter is here understood as a combination of the silica and air effective areas according to:

*A _{eff}* can be considered as the effective area in a solid-core silica fiber which would give the same level of nonlinear effects as the air-filled HC-PBG fiber. A value

*n*

^{air}_{2}=5.7·10

^{-23}W/m

^{2}was used to reflect the magnitude of the combined Kerr and Raman effects.[15, 16] For an evacuated HC-PBG fiber, ${A}_{\mathrm{eff}}={A}_{\mathrm{eff}}^{Si{O}_{2}}$ should be used in this analysis. Eq. (19) implies two important scaling rules for the pulse power: if

*t*

^{2}

_{0}is scaled down, or |

*β*

_{2}|

*A*is scaled up,

_{eff}*P*

_{0}and hence the peak power of the compressed pulse, may be scaled up correspondingly to leave the evolution equation (expressed in reduced coordinates

*τ*and

*z*) unchanged. This establishes

_{c}*F*≡|

*β*

_{2}|

*A*as a figure of merit for a HC-PBG compressor. It should be noted that for a fixed shape and chirp of the pulse in dimensionless units, and a fixed value of

_{eff}*P*

_{0}

*t*

^{2}

_{0}, the spectral width of the input pulse,

*W*, and the pulse energy both scale with

*t*

^{-1}

_{0}.

The figure of merit *F* can be changed by shifting the pulse wavelength relative to the bandgap edges. In this work, a fixed wavelength is considered, so this shift is effected by scaling the fiber pitch. As is characteristic of core mode dispersion within a cladding bandgap, an increase in |*β*
_{2}| comes at the expense of an increase in the dispersion slope. The leading correction to the dispersive part of Eq. 19 is found to be:

which shows that this correction scales with *W*∝*t*
^{-1}
_{0}, and the relative dispersion slope (RDS) *β*
_{3}/|*β*
_{2}|. A second correction comes from the dispersion in the effective area. This correction also becomes more important with increasing spectral width of the pulse. In the simulations, the RDS was found to be the most important correction. Mode profile dispersion, and higher-order dispersion terms shift the quantitative results somewhat, but does not influence the qualitative conclusions.

## 3. Fiber dispersion properties

The mode effective indices and field distributions were obtained using the commercial finite element mode solver JCMwave [17]. An adaptive mesh and quadratic elements were employed to achieve convergence of the mode dispersion, and to sufficiently fine-sample the mode field spatial distributions for an accurate determination of the mode effective areas.

The three example HC-PBG fiber structures to be studied are shown in Fig. 1. The structure labeled HC1 is intended as a realistic model of a fiber which includes structural fluctuations characteristic of the fabrication process, whereas the HC2 and HC3 structures are idealized geometries. All three fibers have a cladding air-filling fraction of 92%, and the cladding hole shapes were chosen to be characteristic of fabricated fibres. HC1 has an antiresonant core surround with four silica nodules incorporated within the inner core wall [18, 19]. These features serve to reduce the overlap between light and silica, but most importantly to make the structure birefringent and polarization-maintaining. The HC2 fiber has core of a size similar to that of HC1, but with a thinned core wall without antiresonant features. This fiber does not provide polarization maintenance but on the other hand has a broader transmission window with a reduced dispersion slope [20]. The HC3 fiber has an enlarged core, where 19 elementary cells of the periodic cladding structure have been removed, compared to 7 for the HC1 and HC2 fibers. This serves to lower the loss and nonlinear coefficients, but also implies lower values of the dispersion coefficient, and a larger number of higher-order core modes. While the present study assumes that the pulse is in a single-mode single-polarization state throughout the compression process, it must be kept in mind that scattering of light into other modes may be a serious practical problem and that the highly multi-moded nature of the HC3 fiber could thus be a significant drawback in comparison with HC1 and HC2.

The dispersion and RDS curves for the different fiber structures are shown in Fig. 2, and in Fig. 3, the silica- and air-related effective area curves, as defined by Eq. (18), are shown. The figures do not show the full HC-PBG transmission windows, since only the wavelength range where strong anomalous dispersion occurs is of interest for compression. In these plots, the pitch, Λ_{0}, which is the lattice constant of the periodic cladding structure, was set to 2.53 *µ*m for HC1, 2.6 *µ*m for HC2 and 2.5 *µ*m for HC3. In the investigations of compressor performance, Λ≡*S*Λ_{0} will be taken as an optimization variable. Since effects of material dispersion are insignificant for the HC-PBG fibers, the wavelengths and RDS values in Figs. 2 and 3 scale with *S*, *D* scales with *S*
^{-1} while the effective areas scale with *S*
^{2}. All rescalings to be considered are within ~10 % of the Λ_{0} values cited above. Therefore, the most significant effect of changing Λ is the shift of the wavelength axis. In Figs. 2 and 3, the points on the curves corresponding to a wavelength of 1064 nm in the rescaled structures have been indicated by red dots. Note that all sampled points lie on the anomalous part of the core mode dispersion curve. The minimal RDS/*S* values over the range of probed *λ*/*S* values are 19.3 fs for the HC1 fiber, 8.5 fs for HC2 and ~14.4 fs for HC3. At the highest *F*-values, the RDS is ~3 times higher than its minimum value for HC2 and HC3, whereas for HC1 the ratio is ~4.

For the 7-cell fibers, the HC1 design has higher dispersion, but lower *A _{eff}* in the air-filled case compared to HC2. The lower

*A*is mostly due to the smaller Λ

_{eff}_{0}value for HC1 and to a confining effect of the core surround geometry. The

*F*-values calculated from dispersion and area curves are shown in Fig. 4. In the air-filled HC1 and HC2 fibers, the

*F*-values turn out to be comparable over a quite broad range of

*λ*/

*S*. In the evacuated case, the two designs have comparable

*A*-values near

_{eff}*λ*/

*S*=1.05

*µ*m despite the smaller geometric mode area of HC1, because the core wall in this fiber is a little more effective in expelling the optical power from the silica regions. The

*F*-values of HC1 are in this case ~1.5-2 times higher than those of HC2. In the air-filled HC3 fiber, the improvement in

*A*over the 7-cell designs roughly reflects the increase in the geometrical core area by a factor of 2.7. The price is somewhat reduced dispersion values, but the HC3 fiber still comes out with

_{eff}*F*-values 2-3 times higher than the HC1 and HC2 designs. On evacuation, the gain is around a factor of 10, showing that the enlarged core is an effective way of reducing the modal overlap with silica. In all three fiber structures it is noted that the variation of

*F*is smaller in the evacuated case, due to the larger slope of

*A*which partially cancels the increase in

_{eff}*D*when the bandgap edge is approached. Note also that for HC2 and HC3, the RDS increases with

*F*for the scalings investigated, whereas for HC1, the RDS is minimized at

*F*≈1.2 ps

^{2}nm (air-filled) and

*F*≈7.2 ps

^{2}nm (evacuated).

## 4. Modeling of pulse compression

#### 4.1. The compression process

We consider compression of parabolic input pulses of the mathematical form

A slight rounding of the pulses near |*t*|=*t*
_{0} was applied to facilitate the numerics. The chirp parameter *C* allows one to vary the temporal (2*t*
_{0}) and spectral width of the pulse independently. We quantify the spectral width (in nm) by the parameter *W*, defining *C* by:

In all calculations, the center frequency *ω*
_{0} corresponds to a wavelength of 1.064 *µ*m. Since no frequency-dependent material parameters (e.g. material dispersion) are considered, the conclusions are not crucially dependent upon this choice of wavelength.

In Fig. 5 the shape of pulses with input width *t*
_{0}=6 ps compressed in the HC2 fiber are shown for various values of the pulse energy. The pulses are scaled to *P*
_{0}, the peak power of the input pulse, so that the compression ratio can be read off directly. In the left panel, pulses having *W*=5 nm are shown, whereas in the right panel *W*=10 nm pulses are shown. In all cases, the length of the fiber was chosen to maximize the peak power of the compressed pulse. Due to the onset of nonlinear effects, this length depends somewhat on the pulse energy. For the 5 nm pulses, the optimum length was 17.4 m in the low-power limit, decreasing to 12.7 m for the pulse with *P*
_{0}=14 kW. For the 10 nm pulses, the corresponding numbers were 8.9 m and 7.66 m, respectively.

The significance of nonlinear effects already at pulse energies below 100 nJ is evident from the power dependence of the pulse shapes in Fig. 5. Two facts are particularly interesting: firstly, the maximal compression ratio for the 5 nm pulses is more than 3 times greater than the compression ratio in the linear (*P*
_{0}=100 W) limit, and for the 10 nm pulses the difference is about a factor of 2. This shows that the optimal compression ratio is reached at power levels where bandwidth generation in the compression process is significant. Secondly, the difference between the maximal compression ratio for 5 nm and 10 nm pulses is only about 20%, although for linear compression the difference is a factor of two. This rough equality in the nonlinear regime may be explained by a rough cancellation effect. Since *t*
_{0} in Eq. (22) is fixed at 6 ps, the chirp of the 5 nm pulses is 2.4 ps/nm, compared to 1.2 ps/nm for the 10 nm pulses. Thus, in the absence of nonlinear effects, the 10 nm pulses compress to double the peak power over half the length when compared to the 5 nm pulses. Since the accumulated SPM scales with the product of power and length, it will to a first approximation be similar in the two cases.

In the near-linear case (*P*
_{0}=100 W), the 5 nm pulses are found to compress to a nearly symmetric shape, whereas for 10 nm pulses, a slight asymmetry is present. This asymmetry is due to third-order dispersion. In the dispersive compression process, the long wavelengths at the leading edge of the input pulse are caught by the short wavelengths at the trailing edge due to second-order dispersion. However, due to the third-order dispersion present in the PBG fiber both the longest and shortest wavelengths move a little too slow, and therefore end up trailing the compressed pulse. As the power is increased and the spectral width broadens, this asymmetry becomes more pronounced, and a substantial part of the pulse energy ends up trailing the main peak. The issue of pulse quality is a very important aspect of pulse compression, and in this work will be quantified by the quality parameter Q defined as the fraction of pulse energy present in the main peak, i.e. between the two power minima surrounding the point of maximum power. This definition was adequate for the present study, since all the pulses encountered in the simulations had the generic shape seen in Fig. 5 at maximal compression.

#### 4.2. Scaling the fiber pitch

In this and the following subsection, a more comprehensive investigation of pulse compression will be undertaken. The focus is to obtain limits on pulse energy, or peak power, for a high quality of the compressed pulses. For definiteness, it will be assumed that *Q*≥0.9 at optimal compression is desired. Also, here optimal compression is defined as the point of maximal peak power. The available optimization parameters are the pitch of the HC-PBG fiber in use, and the temporal and spectral widths of the input pulses. A key issue is to determine the accuracy of the approximate scaling relations discussed in section 2.2, since such rules are highly useful as design guidelines.

This subsection investigates the scaling of pulse power with the parameter *F*=|*β*
_{2}|*A _{eff}*, which may be changed by scaling the fiber pitch, as seen from Fig. 4. For each scaled fiber design, the quality factor Q at maximal compression was calculated as a function of

*P*

_{0}, and the pulse energy,

*E*, peak power,

^{c}_{p}*P*, and main peak FWHM,

_{max}*t*, where Q=0.9 were determined by interpolation. This was done at a fixed value of

_{c}*t*

_{0}=3 ps, and input pulse spectral widths of 5, 10 and 20 nm, for both air-filled and evacuated fibers. In Fig. 6, results for the fibers with 7-cell cores, HC1 and HC2, are shown, while Fig. 7 shows the corresponding data for the HC3 fiber.

Several of the observed trends are readily understandable from the scaling discussion in section 2.2. In the air-filled HC2 fiber, effects of high-order dispersion are weak for input pulses with *W*=5 and 10 nm. For *W*=5 nm, one has the expected linear scaling of *E ^{c}_{p}* and

*P*with

_{max}*F*, whereas for

*W*=10 nm, some curvature can be observed. For

*W*=20 nm, effects of third-order dispersion are significant, and the linear scaling with

*F*breaks down as compression close to the bandgap edge is disfavoured due to the increased RDS. Similar trends are seen in the HC3 fiber, where the linear scaling with

*F*breaks down sooner due to the higher RDS values in this fiber. For the HC1 fiber, the behaviour is more complex, since higher-order dispersion is significant in most cases, and the RDS is minimal around

*F*~7 ps

^{2}nm (

*S*=1). For

*W*=5 nm, compression close to the bandgap edge is still favourable, but already at

*W*=10 nm, the point of maximal

*E*or

^{c}_{p}*P*

_{max}more or less coincides with the RDS minimum. Concerning

*t*, one notes that this is approximately constant when

_{c}*E*and

^{c}_{p}*P*

_{max}scale linearly with

*F*, which is also to be expected from Eq. 19 because

*t*

_{0}is fixed in these calculations. When the scaling breaks down, the shortest

*t*values are almost invariably found where the RDS is minimal.

_{c}Comparing results for air-filled and evacuated fibers, the improvement in *E ^{c}_{p}* and

*P*

_{max}on evacuation roughly corresponds to the increase in

*F*. A notable difference between the two cases is that the highest values of

*E*and

^{c}_{p}*P*occur at lower

_{max}*F*-values than in the air-filled case. This occurs because the increase in

*F*with decreasing

*S*is significantly smaller in the evacuated case, as seen from Fig. 4, whereas the increase in RDS remains unaffected by evacuation. The net result is that operation close to the bandgap edge is less favorable in the evacuated fibers than in the air-filled ones, i.e. the curvature in the

*E*and

^{c}_{p}*P*

_{max}curves is magnified.

The compression factor, or ratio between peak power at maximal compression, *P _{max}*, and input peak power,

*P*

_{0}, for Q=0.9 can be found by comparing the first and second rows in Figs. 6 and 7, noting that

*P*

_{0}is related to

*E*by the relation

_{p}*P*

_{0}=3

*E*/(4

_{p}*t*

_{0}). The highest compression factors come out for the HC2 fiber, which has the lowest RDS value. In Fig. 8, the data from all calculations with pulses having

*t*

_{0}=3 ps launched into the HC2 fiber have been combined in scatter plots to show how the compression factor and

*P*vary with pulse quality. The relation between Q and

_{max}*P*strongly depends on the fiber pitch (i.e. the scaling factor

_{max}*S*), which leads to a strong scatter in these plots. The compression factors, on the other hand, fall on a single line for the narrow-band input pulses where

*P*to a good approximation scales linearly with

_{max}*F*. For

*W*=20 nm and high-power pulses with

*W*=10 nm, the final pulse bandwidth is large enough that higher-order dispersion is important, and a significant scatter develops in the compression factor plots as well. It is noteworthy that the compression factors appear to saturate with decreasing Q. While the compression factor is proportional to

*W*(for fixed

*t*

_{0}) in the linear regime, the values at saturation differ only slightly, as discussed in the previous subsection.

An interesting difference between air-filled and evacuated fibers is that the compression factors in the evacuated case are somewhat higher, and correspondingly the temporal widths of the compressed pulses are shorter in the evacuated case, as seen from Figs. 6 and 7. This would imply that the SPM-induced spectral broadenings in the evacuated fibers at a particular value of Q are stronger (note that the decrease in nonlinearity by evacuation is offset by the larger pulse energies and peak powers in the evacuated fibers). For example, *W*=20 nm pulses in the HC2 fiber have a minimal *t _{c}* of 125 fs in the air-filled case, compared to 109 fs for the evacuated fiber, and

*W*=10 nm pulses in the HC1 fiber have a minimal

*t*of 197 fs air-filled and 171 fs evacuated. Similarly, in the HC3 fiber

_{c}*W*=10 nm pulses have a shortest

*t*of 192 fs with air, and 153 fs without air. It is important to point out that these are not ‘ultimate’ limits on the pulse duration, but are specific for the compression process and quality requirements adopted in the present study. The trend of shorter pulses in the evacuated fibers is also seen for the

_{c}*W*=5 nm input pulses, indicating that the effect is not due to effective-area dispersion. In fact, a closer analysis revealed the difference in the Raman responses of silica and air as the cause of the phenomenon. This was found by comparing the full compression calculations to results where the delayed (Raman) response was approximated by instantaneous (Kerr) response. In Fig. 9, an example of output spectra is displayed. Compression of

*P*

_{0}=20 kW pulses with and without Raman scattering in an air-filled HC2 fiber is compared to compression of

*P*

_{0}=106 kW pulses with and without Raman scattering in an evacuated HC2 fiber. In these cases, higher-order dispersion and effective-area variations are of little importance. All pulses were propagated over a distance of 3.74 m, approximately the optimal length for maximum compression. The initial pulse width was 10 nm, and the scaling factor

*S*=1. The ratio between the

*P*

_{0}values in the airfilled and evacuated fibers corresponds to the ratio between their

*F*-values, i.e. the calculations should be equivalent if Eq. (19) is valid. In the figure, the spectra of the 106kW pulses have been rescaled to facilitate comparison. The results show that Raman scattering is insignificant in the evacuated fiber, and more strikingly, that the spectral broadening in the air-filled fiber closely matches that in the evacuated fiber at the same value of

*P*

_{0}/

*F*if Raman scattering is neglected. Including the Raman effects in air, however, leads to a pronounced asymmetry in the spectrum, where a substantial part of the spectral weight is shifted from short to long wavelengths. This effectively reduces the bandwidth, leading to a longer main peak of the compressed pulse, and also the pulse quality is found to be slightly degraded.

The differences between the Raman responses of silica and air are firstly that the delayed response in air contributes about half of the total nonlinear response [15, 16], whereas for silica, delayed response accounts for only ~20 % of the total response. Secondly, the Raman gain peak in air, using the parametrization of Mlejnek *et al* [16], occurs at a frequency shift of about 2.6 THz, compared to 13.2 THz for silica [14]. For the pulses considered here, intrapulse Raman scattering is therefore strong in air, but not in silica. Still further analysis showed that it is in fact this difference in the frequency shift which is responsible for the spectral distortion seen in Fig. 9. If the Raman shift of air was artificially changed to the value of silica, leaving all other parameters constant, the difference between spectra of air-filled and evacuated fibers vanished.

Although the *F* values of the air-filled HC3 exceeds those of HC1 and HC2 by a factor 2–3, the improvement in *E _{p}* over the HC2 fiber is only around 50% due to the higher RDS of the 19-cell core. In the case of an evacuated fiber, the gain in

*F*is roughly a factor of 5, and the maximal

*E*values are up by a factor of 4 compared to the best 7-cell designs. This reflects the fact that the use of a 19-cell core strongly reduces the overlap of the guided mode with silica.

^{c}_{p}#### 4.3. Scaling the pulse shape

In section 2.2, an approximate scaling of the output pulse power with *t*
^{-2}
_{0} was suggested, provided that
$\sqrt{{P}_{0}}$
and *W* were scaled with *t*
^{-1}
_{0}. Thus, peak power levels can be scaled up by decreasing *t*
_{0}, but it must be noted that the peak power of the input pulse is scaled correspondingly. This may set practical limitations in a fully fiber-integrated system, since interfacing the medium-sized hollow cores studied here to large-core amplifier fibers may not be straightforward.

In Fig. 10, the Q=0.9 peak power levels for different *t*
_{0} values are shown for the HC1 and HC2 fiber designs. The input pulses are scaled so that both *P*
_{0}
*t*
^{2}
_{0} and *Wt*
_{0} are constant. As expected, the peak power level increases with decreasing *t*
_{0}, but the *t*
^{-2}
_{0} scaling predicted from Eq. (19) is somewhat suppressed. The suppression is strongest for short (broadband) pulses, and for fibers with high RDS, i.e. the HC1 fiber, and the HC2 fiber at large *F*-values. In line with the findings of the previous subsection, the optimal position of the input pulse in the transmission window shifts away from the bandgap edge as *W* increases. The low RDS of the HC2 fiber is seen to be a major advantage when using this route to power scaling.

As *t*
_{0} is scaled down, so is the duration of the compressed pulses, but the scaling is sublinear. As an example, in the HC2 fiber an input pulse having *t*
_{0}=1.5 ps, *W*=20 nm gives a *t _{c}* of 97 fs (air-filled) and 81 fs (evacuated), whereas for

*t*

_{0}=1 ps,

*W*=30 nm, the lowest

*t*values were 70 and 64 fs respectively. This should be compared to values of 170 fs and 138 fs for

_{c}*t*

_{0}=3 ps and

*W*=10 nm.

#### 4.4. Discussion

The numerical analysis presented above indicates that dispersive compression of linearly chirped few-ps pulses is limited to peak power levels ranging from 1–3 MW (7-cell air-filled core) to a few tens of MW (19-cell evacuated core). Pulse durations at the highest powers ranged from ~110 fs (HC2, evacuated) to ~250 fs (HC3 air-filled) for a *t*
_{0}=3 ps input pulse (note that the full width of the parabolic pulse is 2*t*
_{0}). Due to the breakdown of the approximate scaling relations at large bandwidths, it is difficult to extrapolate the results to the case of very long (e.g. hundreds of ps) input pulses with finite bandwidths. However, it must be expected that the limits are more stringent in this case, and trial simulations seemed to confirm this. It is important to reiterate that these bounds will move significantly upwards if the Q=0.9 requirement is relaxed. In the case where high pulse quality is desired, various strategies for further power scaling may be considered:

-Shaping of pulse chirp. If the chirp of the input pulse can be shaped appropriately, the higher-order dispersion terms of the PBG fibers may be fully or partially compensated, which will increase the compressed quality of broadband pulses. It must be noted that accurate phase control of a high-power pulse in a fully fiber-integrated setup may not be a straightforward matter.

-Soliton formation and spectral filtering. In the simulation runs it has been observed that one or more solitons form beyond the point of optimal compression. In some cases, a single soliton may carry more than 50% of the total pulse energy. Since the soliton redshifts by the Raman effect, whereas the remaining dispersive waves do not, the latter may be removed if spectral filtering after the fiber output is possible. [21] A recent experiment has actually demonstrated this process experimentally (without filtering) in an air-filled hollow-core fiber at *µ*J energy levels [22].

-Use of tapered HC-PBG fibers. Adiabatic soliton compression using a tapered HC-PBG fiber has already been demonstrated [7]. One can further envision a combination of initial dispersive compression, soliton formation as described above, and a final tapering to further compress the soliton and perhaps filter the dispersive waves. A drawback of this scheme is that the use of a tapered fiber adds complexity to the manufacturing process, and may significantly increase the production costs of the laser system.

## 5. Conclusions

Dispersive compression of linearly chirped ps pulses in hollow-core PBG fibers has been investigated numerically. The mode profiles and dispersion properties of three different fiber designs were modeled by the finite-element method, and compression of few-ps linearly chirped parabolic pulses was simulated by solving a generalized nonlinear Schrödinger equation accounting for the hybrid (silica/air) nonlinearity of hollow-core fibers. Thresholds for pulse energies and peak powers were found under the requirement that at least 90% of the total pulse energy should be present in the main peak of the compressed pulse. The quantity *F*=|*β*
_{2}|*A _{eff}* was found to be a useful figure of merit for a fiber compressor at a low bandwidth of the input pulse, whereas for higher bandwidths, minimization of the relative dispersion slope RDS=

*β*

_{3}/|

*β*

_{2}| is equally important. Fibers with a thinned core wall and a broad transmission window were found to yield superior power scaling and shorter pulse durations than fibers with antiresonant features on the core surround. The use of an enlarged (19-cell) core allowed for higher pulse powers, but at somewhat increased pulse durations due to higher RDS values. The use of evacuated fibers raised the peak powers significantly, in some cases by more than an order of magnitude, and also decreased the pulse durations. The latter effect was found to be due to a detrimental influence of intrapulse Raman scattering in air on the duration and quality of the compressed pulses.

## Acknowledgments

This work was financially supported by the Danish High Technology Foundation.

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