## Dispersive pulse compression in hollow-core photonic bandgap fibers

Optics Express, Vol. 16, Issue 13, pp. 9628-9644 (2008)

http://dx.doi.org/10.1364/OE.16.009628

Acrobat PDF (1845 KB)

### 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

1. C. J. S. De Matos, J. R. Taylor, T. P. Hansen, K. P. Hansen, and J. Broeng, “All-fiber chirped pulse amplification using highly-dispersive air-core photonic bandgap fiber,” Opt. Express **11**, 2832–2837 (2003). [CrossRef] [PubMed]

2. J. Limpert, T. Schreiber, S. Nolte, H. Zellmer, and A. Tunnermann, “All fiber chirped-pulse amplification system based on compression in air-guiding photonic bandgap fiber,” Opt. Express **11**, 3332–3337 (2003). [CrossRef] [PubMed]

3. C. K. Nielsen, K. G. Jespersen, and S. R. Keiding, “A 158 fs 5.3 nJ fiber-laser system at 1 *µ*m using photonic bandgap fibers for dispersion control and pulse compression,” Opt. Express **14**, 239–244 (2006). [CrossRef]

4. D. G. Ouzounov, F. R. Ahmad, D. Muller, N. Venkataraman, M. T. Gallagher, M. G. Thomas, J. Silcox, K. W. Koch, and A. L. Gaeta, “Generation of megawatt optical solitons in hollow-core photonic band-gap fibers,” Science **301**, 1702–1704 (2003). [CrossRef] [PubMed]

5. D. G. Ouzounov, C. J. Hensley, A. L. Gaeta, N. Venkateraman, M. T. Gallagher, and K. W. Koch, “Soliton pulse compression in photonic band-gap fibers,” Opt. Express **13**, 6153–6159 (2005). [CrossRef] [PubMed]

6. C. J. Hensley, D. G. Ouzounov, A. L. Gaeta, N. Venkataraman, M. T. Gallagher, and K. W. Koch, “Silica-glass contribution to the effective nonlinearity of hollow-core photonic band-gap fibers,” Opt. Express **15**, 3507–3512 (2007). [CrossRef] [PubMed]

7. F. Gerome, K. Cook, A. K. George, W. J. Wadsworth, and J. C. Knight, “Delivery of sub-100fs pulses through 8m of hollow-core fiber using soliton compression,” Opt. Express **15**, 7126–7131 (2007). [CrossRef] [PubMed]

*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.

8. M. E. Fermann, V. I. Kruglov, B. C. Thomsen, J. M. Dudley, and J. D. Harvey, “Self-similar propagation and amplification of parabolic pulses in optical fibers,” Phys. Rev. Lett. **84**, 6010–6013 (2000). [CrossRef] [PubMed]

## 2. Nonlinear propagation equations

### 2.1. General formulation

*ε*is the relative dielectric constant of the medium, assumed

*z*-independent for a straight waveguide, and

**P**

*is the nonlinear part of the induced polarization. The*

_{NL}**E**and

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

*z*-derivatives of

*G*in comparison with

*β*, that is, a slowly-varying envelope approximation. The modal fields satisfy the orthogonality relation:

9. M. Kolesik, E. M. Wright, and J. V. Moloney, “Simulation of femtosecond pulse propagation in sub-micron diameter tapered fibers.” Appl. Phys. B: Lasers Opt. **79**, 293–300 (2004). [CrossRef]

**P**

*is given by:*

_{NL}*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}*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.

*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

**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

10. K. J. Blow and D. Wood, “Theoretical description of transient stimulated Raman scattering in optical fibers,” IEEE J. Quantum Electron. **25**(12), 2665–2673 (1989). [CrossRef]

*N*=1. In the following, only a single field state will be considered, and the indices on

_{m}*G*and

**e**are suppressed.

*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

_{w}11. J. Lægsgaard, “Mode profile dispersion in the generalised nonlinear Schrödinger equation,” Opt. Express **15**(24), 16,110–16,123 (2007). [CrossRef]

11. J. Lægsgaard, “Mode profile dispersion in the generalised nonlinear Schrödinger equation,” Opt. Express **15**(24), 16,110–16,123 (2007). [CrossRef]

*n*

_{2}coefficient defined as:

*n*is the linear refractive index of material

_{ν}*ν*. Introducing

*n*

^{(ν)}

_{2}instead of

*χ*

^{(3)}

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

12. J. Lægsgaard, N. A. Mortensen, and A. Bjarklev, “Mode areas and field energy distribution in honeycomb photonic bandgap fibers,” J. Opt. Soc. Am. B **20**, 2037–2045 (2003). [CrossRef]

13. J. Lægsgaard, N. A. Mortensen, J. Riishede, and A. Bjarklev, “Material effects in airguiding photonic bandgap fibers,” J. Opt. Soc. Am. B **20**, 2046–2051 (2003). [CrossRef]

^{-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*), was set to 2.9·10

_{air}^{-23}W/m

^{2}[15

15. E. T. J. Nibbering, G. Grillon, M. A. Franco, B. S. Prade, and A. Mysyrowicz, “Determination of the inertial contribution to the nonlinear refractive index of air, N2, and O2 by use of unfocused high-intensity femtosecond laser pulses,” J. Opt. Soc. Am. B (Optical Physics) **14**, 650–660 (1997). [CrossRef]

*et al*[16

16. M. Mlejnek, E. M. Wright, and J. V. Moloney, “Dynamic spatial replenishment of femtosecond pulses propagating in air,” Opt. Lett. **23**(5), 382–384 (1998). [CrossRef]

### 2.2. Scaling laws and figures of merit

*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]

*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,

*A*parameter is here understood as a combination of the silica and air effective areas according to:

_{eff}*A*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

_{eff}*n*

^{air}_{2}=5.7·10

^{-23}W/m

^{2}was used to reflect the magnitude of the combined Kerr and Raman effects.[15

15. E. T. J. Nibbering, G. Grillon, M. A. Franco, B. S. Prade, and A. Mysyrowicz, “Determination of the inertial contribution to the nonlinear refractive index of air, N2, and O2 by use of unfocused high-intensity femtosecond laser pulses,” J. Opt. Soc. Am. B (Optical Physics) **14**, 650–660 (1997). [CrossRef]

16. M. Mlejnek, E. M. Wright, and J. V. Moloney, “Dynamic spatial replenishment of femtosecond pulses propagating in air,” Opt. Lett. **23**(5), 382–384 (1998). [CrossRef]

*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}.

*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:

*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

17. “JCMwave GmbH, www.jcmwave.com,”.

18. P. J. Roberts, D. P. Williams, H. Sabert, B. J. Mangan, D. M. Bird, T. A. Birks, J. C. Knight, and P. S. J. Russell, “Design of low-loss and highly birefringent hollow-core photonic crystal fiber,” Opt. Express **14**(16), 7329–7341 (2006). [CrossRef] [PubMed]

19. P. J. Roberts, “Birefringent hollow core fibers,” Proc.SPIE **6782**, 67821R (2007). [CrossRef]

20. R. Amezcua-Correa, N. Broderick, M. Petrovich, F. Poletti, and D. Richardson, “Design of 7 and 19 cells core air-guiding photonic crystal fibers for low-loss, wide bandwidth and dispersion controlled operation,” Opt. Express **15**, 17577–17586 (2007). [CrossRef] [PubMed]

*A*in the air-filled case compared to HC2. The lower

_{eff}*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

*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:

*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.

*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.

*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.

*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

*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.

*F*=|

*β*

_{2}|

*A*, 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

_{eff}*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.

*W*=5 and 10 nm. For

*W*=5 nm, one has the expected linear scaling of

*E*and

^{c}_{p}*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}*E*and

^{c}_{p}*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.

*W*=20 nm pulses in the HC2 fiber have a minimal

*t*of 125 fs in the air-filled case, compared to 109 fs for the evacuated fiber, and

_{c}*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.

15. E. T. J. Nibbering, G. Grillon, M. A. Franco, B. S. Prade, and A. Mysyrowicz, “Determination of the inertial contribution to the nonlinear refractive index of air, N2, and O2 by use of unfocused high-intensity femtosecond laser pulses,” J. Opt. Soc. Am. B (Optical Physics) **14**, 650–660 (1997). [CrossRef]

16. M. Mlejnek, E. M. Wright, and J. V. Moloney, “Dynamic spatial replenishment of femtosecond pulses propagating in air,” Opt. Lett. **23**(5), 382–384 (1998). [CrossRef]

*et al*[16

**23**(5), 382–384 (1998). [CrossRef]

*F*values of the air-filled HC3 exceeds those of HC1 and HC2 by a factor 2–3, the improvement in

*E*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

_{p}*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

*t*

^{-2}

_{0}was suggested, provided that

*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.

*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.

*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*of 97 fs (air-filled) and 81 fs (evacuated), whereas for

_{c}*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

*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:

21. G. P. Agrawal, “Effect of intrapulse stimulated Raman scattering on soliton-effect pulse compression in optical fibers,” Opt. Letters **15**, 224–226 (1990). [CrossRef]

*µ*J energy levels [22

22. F. Gérôme, J. Dupriez, J. C. Knight, and W. J Wadsworth, “High power tunable femtosecond soliton source using hollow-core photonic bandgap fiber, and its use for frequency doubling,” Opt. Express **16**, 2381–2386 (2008). [CrossRef] [PubMed]

7. F. Gerome, K. Cook, A. K. George, W. J. Wadsworth, and J. C. Knight, “Delivery of sub-100fs pulses through 8m of hollow-core fiber using soliton compression,” Opt. Express **15**, 7126–7131 (2007). [CrossRef] [PubMed]

## 5. Conclusions

*F*=|

*β*

_{2}|

*A*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=

_{eff}*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

## References and links

1. | C. J. S. De Matos, J. R. Taylor, T. P. Hansen, K. P. Hansen, and J. Broeng, “All-fiber chirped pulse amplification using highly-dispersive air-core photonic bandgap fiber,” Opt. Express |

2. | J. Limpert, T. Schreiber, S. Nolte, H. Zellmer, and A. Tunnermann, “All fiber chirped-pulse amplification system based on compression in air-guiding photonic bandgap fiber,” Opt. Express |

3. | C. K. Nielsen, K. G. Jespersen, and S. R. Keiding, “A 158 fs 5.3 nJ fiber-laser system at 1 |

4. | D. G. Ouzounov, F. R. Ahmad, D. Muller, N. Venkataraman, M. T. Gallagher, M. G. Thomas, J. Silcox, K. W. Koch, and A. L. Gaeta, “Generation of megawatt optical solitons in hollow-core photonic band-gap fibers,” Science |

5. | D. G. Ouzounov, C. J. Hensley, A. L. Gaeta, N. Venkateraman, M. T. Gallagher, and K. W. Koch, “Soliton pulse compression in photonic band-gap fibers,” Opt. Express |

6. | C. J. Hensley, D. G. Ouzounov, A. L. Gaeta, N. Venkataraman, M. T. Gallagher, and K. W. Koch, “Silica-glass contribution to the effective nonlinearity of hollow-core photonic band-gap fibers,” Opt. Express |

7. | F. Gerome, K. Cook, A. K. George, W. J. Wadsworth, and J. C. Knight, “Delivery of sub-100fs pulses through 8m of hollow-core fiber using soliton compression,” Opt. Express |

8. | M. E. Fermann, V. I. Kruglov, B. C. Thomsen, J. M. Dudley, and J. D. Harvey, “Self-similar propagation and amplification of parabolic pulses in optical fibers,” Phys. Rev. Lett. |

9. | M. Kolesik, E. M. Wright, and J. V. Moloney, “Simulation of femtosecond pulse propagation in sub-micron diameter tapered fibers.” Appl. Phys. B: Lasers Opt. |

10. | K. J. Blow and D. Wood, “Theoretical description of transient stimulated Raman scattering in optical fibers,” IEEE J. Quantum Electron. |

11. | J. Lægsgaard, “Mode profile dispersion in the generalised nonlinear Schrödinger equation,” Opt. Express |

12. | J. Lægsgaard, N. A. Mortensen, and A. Bjarklev, “Mode areas and field energy distribution in honeycomb photonic bandgap fibers,” J. Opt. Soc. Am. B |

13. | J. Lægsgaard, N. A. Mortensen, J. Riishede, and A. Bjarklev, “Material effects in airguiding photonic bandgap fibers,” J. Opt. Soc. Am. B |

14. | G. P. Agrawal, |

15. | E. T. J. Nibbering, G. Grillon, M. A. Franco, B. S. Prade, and A. Mysyrowicz, “Determination of the inertial contribution to the nonlinear refractive index of air, N2, and O2 by use of unfocused high-intensity femtosecond laser pulses,” J. Opt. Soc. Am. B (Optical Physics) |

16. | M. Mlejnek, E. M. Wright, and J. V. Moloney, “Dynamic spatial replenishment of femtosecond pulses propagating in air,” Opt. Lett. |

17. | “JCMwave GmbH, www.jcmwave.com,”. |

18. | P. J. Roberts, D. P. Williams, H. Sabert, B. J. Mangan, D. M. Bird, T. A. Birks, J. C. Knight, and P. S. J. Russell, “Design of low-loss and highly birefringent hollow-core photonic crystal fiber,” Opt. Express |

19. | P. J. Roberts, “Birefringent hollow core fibers,” Proc.SPIE |

20. | R. Amezcua-Correa, N. Broderick, M. Petrovich, F. Poletti, and D. Richardson, “Design of 7 and 19 cells core air-guiding photonic crystal fibers for low-loss, wide bandwidth and dispersion controlled operation,” Opt. Express |

21. | G. P. Agrawal, “Effect of intrapulse stimulated Raman scattering on soliton-effect pulse compression in optical fibers,” Opt. Letters |

22. | F. Gérôme, J. Dupriez, J. C. Knight, and W. J Wadsworth, “High power tunable femtosecond soliton source using hollow-core photonic bandgap fiber, and its use for frequency doubling,” Opt. Express |

**OCIS Codes**

(140.3510) Lasers and laser optics : Lasers, fiber

(190.4370) Nonlinear optics : Nonlinear optics, fibers

(060.4005) Fiber optics and optical communications : Microstructured fibers

**ToC Category:**

Nonlinear Optics

**History**

Original Manuscript: March 18, 2008

Revised Manuscript: June 4, 2008

Manuscript Accepted: June 4, 2008

Published: June 16, 2008

**Citation**

J. Laegsgaard and P. J. Roberts, "Dispersive pulse compression in hollow-core photonic bandgap fibers," Opt. Express **16**, 9628-9644 (2008)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-13-9628

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### References

- C. J. S. De Matos, J. R. Taylor, T. P. Hansen, K. P. Hansen, and J. Broeng, "All-fiber chirped pulse amplification using highly-dispersive air-core photonic bandgap fiber," Opt. Express 11, 2832-2837 (2003). [CrossRef] [PubMed]
- J. Limpert, T. Schreiber, S. Nolte, H. Zellmer, and A. Tunnermann, "All fiber chirped-pulse amplification system based on compression in air-guiding photonic bandgap fiber," Opt. Express 11, 3332-3337 (2003). [CrossRef] [PubMed]
- C. K. Nielsen, K. G. Jespersen, and S. R. Keiding, "A 158 fs 5.3 nJ fiber-laser system at 1 ???m using photonic bandgap fibers for dispersion control and pulse compression," Opt. Express 14, 239-44 (2006). [CrossRef]
- D. G. Ouzounov, F. R. Ahmad, D. Muller, N. Venkataraman, M. T. Gallagher, M. G. Thomas, J. Silcox, K. W. Koch, and A. L. Gaeta, "Generation of megawatt optical solitons in hollow-core photonic band-gap fibers," Science 301, 1702-1704 (2003). [CrossRef] [PubMed]
- D. G. Ouzounov, C. J. Hensley, A. L. Gaeta, N. Venkateraman, M. T. Gallagher, and K. W. Koch, "Soliton pulse compression in photonic band-gap fibers," Opt. Express 13, 6153-6159 (2005). [CrossRef] [PubMed]
- C. J. Hensley, D. G. Ouzounov, A. L. Gaeta, N. Venkataraman, M. T. Gallagher, and K. W. Koch, "Silica-glass contribution to the effective nonlinearity of hollow-core photonic band-gap fibers," Opt. Express 15, 3507-3512 (2007). [CrossRef] [PubMed]
- F. Gerome, K. Cook, A. K. George, W. J. Wadsworth, and J. C. Knight, "Delivery of sub-100fs pulses through 8m of hollow-core fiber using soliton compression," Opt. Express 15, 7126-7131 (2007). [CrossRef] [PubMed]
- M. E. Fermann, V. I. Kruglov, B. C. Thomsen, J. M. Dudley, and J. D. Harvey, "Self-similar propagation and amplification of parabolic pulses in optical fibers," Phys. Rev. Lett. 84, 6010-13 (2000). [CrossRef] [PubMed]
- M. Kolesik, E. M. Wright, and J. V. Moloney, "Simulation of femtosecond pulse propagation in sub-micron diameter tapered fibers." Appl. Phys. B: Lasers Opt. 79, 293-300 (2004). [CrossRef]
- K. J. Blow and D. Wood, "Theoretical description of transient stimulated Raman scattering in optical fibers," IEEE J. Quantum Electron. 25, 2665-2673 (1989). [CrossRef]
- J. Lægsgaard, "Mode profile dispersion in the generalised nonlinear Schr¨odinger equation," Opt. Express 15(24), 16,110-123 (2007). [CrossRef]
- J. Lægsgaard, N. A. Mortensen, and A. Bjarklev, "Mode areas and field energy distribution in honeycomb photonic bandgap fibers," J. Opt. Soc. Am. B 20, 2037-45 (2003). [CrossRef]
- J. Lægsgaard, N. A. Mortensen, J. Riishede, and A. Bjarklev, "Material effects in airguiding photonic bandgap fibers," J. Opt. Soc. Am. B 20, 2046-51 (2003). [CrossRef]
- G. P. Agrawal, Nonlinear Fiber Optics (Academic Press, San Diego, 2001).
- E. T. J. Nibbering, G. Grillon, M. A. Franco, B. S. Prade, and A. Mysyrowicz, "Determination of the inertial contribution to the nonlinear refractive index of air, N2, and O2 by use of unfocused high-intensity femtosecond laser pulses," J. Opt. Soc. Am. B(Optical Physics) 14, 650-60 (1997). [CrossRef]
- M. Mlejnek, E. M. Wright, and J. V. Moloney, "Dynamic spatial replenishment of femtosecond pulses propagating in air," Opt. Lett. 23, 382-384 (1998). [CrossRef]
- "JCMwave GmbH, www.jcmwave.com,".
- P. J. Roberts, D. P. Williams, H. Sabert, B. J. Mangan, D. M. Bird, T. A. Birks, J. C. Knight, and P. S. J. Russell, "Design of low-loss and highly birefringent hollow-core photonic crystal fiber," Opt. Express 14, 7329-7341 (2006). [CrossRef] [PubMed]
- P. J. Roberts, "Birefringent hollow core fibers," Proc. SPIE 6782, 67821R (2007). [CrossRef]
- R. Amezcua-Correa, N. Broderick, M. Petrovich, F. Poletti, and D. Richardson, "Design of 7 and 19 cells core air-guiding photonic crystal fibers for low-loss, wide bandwidth and dispersion controlled operation," Opt. Express 15, 17577-17586 (2007). [CrossRef] [PubMed]
- G. P. Agrawal, "Effect of intrapulse stimulated Raman scattering on soliton-effect pulse compression in optical fibers," Opt. Letters 15, 224-6 (1990). [CrossRef]
- F. Gérôme, J. Dupriez, J. C. Knight, and W. J. Wadsworth, "High power tunable femtosecond soliton source using hollow-core photonic bandgap fiber, and its use for frequency doubling," Opt. Express 16, 2381-2386 (2008). [CrossRef] [PubMed]

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