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Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 16, Iss. 13 — Jun. 23, 2008
  • pp: 9628–9644
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Dispersive pulse compression in hollow-core photonic bandgap fibers

J. Lægsgaard and P. J. Roberts  »View Author Affiliations


Optics Express, Vol. 16, Issue 13, pp. 9628-9644 (2008)
http://dx.doi.org/10.1364/OE.16.009628


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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

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

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

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

], 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

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

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

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

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]

] 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 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:

×E=μ0Ht
(1)
×H=ε0ε(r)Et+PNLt
(2)

H(r,t)=12πmdωGm(z,ω)hm(r,ω)exp(i(βm(ω)zωt))
(3)
E(r,t)=12πmdωGm(z,ω)em(r,ω)exp(i(βm(ω)zωt))
(4)

dr[em×hn*+en*×hm]·ẑ=Nmδmn
(5)

It was shown in Ref. [9

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]

] that these assumptions leads to the 1+1D propagation equation:

NmGm(z,ω)z=iωdrdtem*(r,t;ω)·PNL(r,t),
(6)
em(r,t;ω)=em(r,ω)exp(i(βm(ω)zωt)).
(7)

The nonlinear polarization, P NL is given by:

PNL(r,t)=ε0dtR(r,tt)χ(3)(r)E(r,t)E(r,t)E(r,t)
(8)

In isotropic materials, the independent tensor components are χ (3) xxxx, χ (3) xxyy, χ (3) xyyx, χ (3) xyxy. At any particular point, we can express the tensor in a local coordinate system whose x-axis is aligned with the electric field. Since χ (3) xxxx is the same in all coordinate systems, this implies that

PNL(r,t)=ε0χ(3)(r)E(r,t)dtR(r,tt)E(r,t)2
(9)

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:

PNL(r,t)=ε0(2π)3npqdω13dtGn(z,ω1)Gp(z,ω2)Gq*(z,ω3)×
en(r,t;ω1)[ep(r,t;ω2)·eq*(r,t;ω3)]χ(3)(r)R(r,tt)
(10)

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

NmGm(z,ω)z=iωε0(2π)2exp(iβm(ω)z)npqv=1Nχν(3)
dω12G˜n(z,ω1)G˜p(z,ω2)G˜q*(z,ω1+ω2ω)Rν(ωω1)×
vdr[em*(r,ω)·en(r,ω1)][ep(r,ω2)·eq*(r,ω1+ω2ω)],
(11)

where

G˜m(z,ω)=Gm(z,ω)exp(iβm(ω)z)
(12)

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 βm(-ω)=-βm(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]

]. 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:

Gm(z,ω)z=iωε0exp(iβm(ω)z)npqv=1N3χv(3)
1(2π)2dω12G˜n(z,ω1)G˜p(z,ω2)G˜q*(z,ω1+ω2ω)Rv(ωω1)×
vdr[em*(r,ω)·en(r,ω1)][ep(r,ω2)·eq*(r,ω1+ω2ω)]
(13)

dr[e*(r,ω)·e(r,ω1)][e(r,ω2)·e*(r,ω1+ω2ω)]
[C(ω)C(ω1)C(ω2)C(ω1+ω2ω)]14
(14)
C(ω)=[dre*(r,ω)4]1
(15)

n2(v)=3χv(3)4nv2ε0c
(16)

where nν 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:

G(z,ω)z=iωcexp(iβ(ω)z)v=1Nn2(v)[Aeff(v)(ω)]14×
1(2π)2dω12Ĝ(v)(z,ω1)Ĝ(v)(z,ω2)Ĝ(v)*(z,ω1+ω2ω)Rv(ωω1)
(17)
Ĝ(v)(z,ω)=G˜(z,ω)[Aeff(v)(ω)]14,Aeff(v)=C(v)4c2nv2ε02=μ0[Redre×h*·ẑ]2ε0nv2vdre(r)4
(18)

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

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

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]

].

In the numerical calculations, the n2SiO2 coefficient was set to 2.66·10-20 W/m2, and the Raman response function was parametrized as suggested by Agrawal [14

14. G. P. Agrawal, Nonlinear Fiber Optics (Academic Press, San Diego, 2001).

]. For air, the coefficient of the instantaneous response (the delta-function part of Rair), was set to 2.9·10-23 W/m2 [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]

], and the delayed response was parametrized using the simple form given by Mlejnek 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

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 (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]

G˜(zc,τ)zc=i22G˜(zc,τ)τ2+in2SiO2ω0P0t02cβ2AeffG˜(zc,τ)G˜(zc,τ)2,τ=tt0,zc=zβ2t02
(19)

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, β2=d2βdω2|ω0 , was assumed negative. The Aeff parameter is here understood as a combination of the silica and air effective areas according to:

1Aeff=n2airn2SiO2Aeffair+1AeffSiO2
(20)

Aeff 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 nair 2=5.7·10-23 W/m2 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

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]

] For an evacuated HC-PBG fiber, Aeff=AeffSiO2 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|Aeff is scaled up, 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 zc) unchanged. This establishes F≡|β 2|Aeff 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 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:

β36t0β23G˜(zc,τ)τ3,β3=d3βdω3ω0
(21)

which shows that this correction scales with Wt -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

17. “JCMwave GmbH, www.jcmwave.com,”.

]. 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

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

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

]. 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

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]

]. 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.

Fig. 1. Hollow-core PBG fiber structures investigated in the present work. More cladding layers than shown were included in the simulations to ensure confinement loss within the probed band gap region does not affect dispersion.
Fig. 2. Dispersion curves for the HC-PBG fibers investigated in the present work. The scaling factor S=Λ/Λ0, where Λ0=2.53 µm for HC, 2.6 µm for HC2, and 2.5 µm for HC3. Red dots denote the position of the input center wavelength (1064 nm) on the dispersion curves of the fibers with scaled pitches investigated here.

Fig. 3. Left panel: Total effective area at atmospheric pressure of the three HC-PBG designs studied. Right panel: Silica effective area of the HC-PBG fibers, i.e. effective area at zero air pressure. Red dots indicate position of the input wavelength in the scaled fibers. The areas of the fiber with a 19-cell core have been reduced by factors of 3 (left panel) and 10 (right panel) to facilitate comparison.

Fig. 4. Values of the figure of merit F=|β 2|Aeff for the different fiber designs. The results for the evacuated HC3 fiber have been scaled down by a factor of 3 to facilitate comparison. Note that F scales with S 3, so for S≠1 the values in the figure should be scaled to compare with figures in the following subsections.

4. Modeling of pulse compression

4.1. The compression process

We consider compression of parabolic input pulses of the mathematical form

G(t)=P0[1(tt0)2]exp(i(Ct2+ω0t)),t<t0;G(t)=0,t>t0
(22)

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 (2t 0) and spectral width of the pulse independently. We quantify the spectral width (in nm) by the parameter W, defining C by:

C=ω02W8πct0
(23)

In all calculations, the center frequency ω 0 corresponds to a wavelength of 1.064 µm. 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.

Fig. 5. Normalized power of compressed pulses for different values of P 0. Input pulse parameters are t 0=6 ps, W=5 nm (left) and 10 nm (right).

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.

Fig. 6. Pulse energies (top row), peak powers (middle row) and temporal FWHM (bottom row), at maximal compression for Q=0.9 in the HC1 and HC2 fiber structures as a function of the figure of merit |β 2|Aeff. Left and right columns give data for air-filled and evacuated fibers, respectively. All input pulses have t 0=3 ps.

This subsection investigates the scaling of pulse power with the parameter F=|β 2|Aeff, 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, Ecp, peak power, Pmax, and main peak FWHM, tc, where Q=0.9 were determined by interpolation. This was done at a fixed value of 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.

Comparing results for air-filled and evacuated fibers, the improvement in Ecp 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 Ecp and Pmax occur at lower 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 Ecp and P max curves is magnified.

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 tc 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 tc of 197 fs air-filled and 171 fs evacuated. Similarly, in the HC3 fiber W=10 nm pulses have a shortest tc 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 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.

Fig. 7. Same as Fig. 6 for the HC3 fiber.
Fig. 8. Compression factors (upper row) and maximal peak power (lower row) versus Q for compression of t 0=3 ps pulses in the HC2 fiber. Left column shows results for air-filled fibers, right column for evacuated fibers. Data points from all fiber scalings investigated have been included in the plots.
Fig. 9. Spectral densities of pulses compressed in the HC2 fiber, calculated with or without Raman scattering. The P 0 value in the air-filled and evacuated examples were chosen so that P 0/F was the same in the two cases.

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

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

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]

], 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

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]

], occurs at a frequency shift of about 2.6 THz, compared to 13.2 THz for silica [14

14. G. P. Agrawal, Nonlinear Fiber Optics (Academic Press, San Diego, 2001).

]. 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 Ep 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 Ecp 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.

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 P0 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.

Fig. 10. Maximal peak powers for Q=0.9 compression in the HC1 (upper row) and HC2 (lower row) fibers for different input pulse durations scaled so that W t 0 and P 0 t 2 0 are constants. Left column shows results for air-filled fibers, right column for evacuated fibers. Legends indicate W and t 0. Note the logarithmic scale in the lower right panel.

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 tc of 97 fs (air-filled) and 81 fs (evacuated), whereas for t 0=1 ps, W=30 nm, the lowest tc values were 70 and 64 fs respectively. This should be compared to values of 170 fs and 138 fs for 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 2t 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

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]

] A recent experiment has actually demonstrated this process experimentally (without filtering) in an air-filled hollow-core fiber at µJ 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]

].

-Use of tapered HC-PBG fibers. Adiabatic soliton compression using a tapered HC-PBG fiber has already been demonstrated [7

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]

]. 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|Aeff 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.

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 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 µm 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]

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]

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]

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]

11.

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

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]

14.

G. P. Agrawal, Nonlinear Fiber Optics (Academic Press, San Diego, 2001).

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]

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]

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]

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]

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

  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 ???m using photonic bandgap fibers for dispersion control and pulse compression," Opt. Express 14, 239-44 (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]
  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, 6010-13 (2000). [CrossRef] [PubMed]
  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]
  10. 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]
  11. J. Lægsgaard, "Mode profile dispersion in the generalised nonlinear Schr¨odinger equation," Opt. Express 15(24), 16,110-123 (2007). [CrossRef]
  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, 2037-45 (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, 2046-51 (2003). [CrossRef]
  14. G. P. Agrawal, Nonlinear Fiber Optics (Academic Press, San Diego, 2001).
  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, 650-60 (1997). [CrossRef]
  16. 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]
  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, 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]
  21. G. P. Agrawal, "Effect of intrapulse stimulated Raman scattering on soliton-effect pulse compression in optical fibers," Opt. Letters 15, 224-6 (1990). [CrossRef]
  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]

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