Ultrafast nonlinear optics in gas-filled hollow-core photonic crystal fibers
                [Invited]

doi:10.1364/JOSAB.28.000A11

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Ultrafast nonlinear optics in gas-filled hollow-core photonic crystal fibers [Invited]

John C. Travers, Wonkeun Chang, Johannes Nold, Nicolas Y. Joly, and Philip St. J. Russell »View Author Affiliations

JOSA B, Vol. 28, Issue 12, pp. A11-A26 (2011)
http://dx.doi.org/10.1364/JOSAB.28.000A11

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Abstract

We review the use of hollow-core photonic crystal fibers (PCFs) in the field of ultrafast gas-based nonlinear optics, including recent experiments, numerical modeling, and a discussion of future prospects. Concentrating on broadband guiding kagomé-style hollow-core PCF, we describe its potential for moving conventional nonlinear fiber optics both into extreme regimes—such as few-cycle pulse compression and efficient deep ultraviolet wavelength generation—and into regimes hitherto inaccessible, such as single-mode guidance in a photoionized plasma and high-harmonic generation in fiber.

1. INTRODUCTION

Although 2011 is the fiftieth anniversary of nonlinear optics, it is perhaps worth noting that 2012 is both the fortieth anniversary of nonlinear fiber optics [1

R. H. Stolen, “Raman oscillation in glass optical waveguide,” Appl. Phys. Lett. 20, 62–63 (1972). [CrossRef] [CrossRef]

, 2

E. P. Ippen, “Stimulated Brillouin scattering in optical fibers,” Appl. Phys. Lett. 21, 539–540 (1972). [CrossRef]

] and the tenth anniversary of the first time hollow-core photonic crystal fiber (HC-PCF) was used to achieve ultralow-threshold stimulated Raman scattering in hydrogen [3

F. Benabid, J. C. Knight, G. Antonopoulos, and P. St. J. Russell, “Stimulated Raman scattering in hydrogen-filled hollow-core photonic crystal fiber,” Science 298, 399–402 (2002). [CrossRef] [PubMed]

]. In 2002, Downer, writing about this last result in the journal Science [4

M. C. Downer, “A new low for nonlinear optics,” Science 298, 373–375 (2002). [CrossRef] [PubMed]

], expressed the view that “a new era in the nonlinear optics of gases, and maybe even plasmas, is about to begin”—a prophetic statement, because this year the first reports of controlled plasma generation in gas-filled hollow-core PCFs have emerged [5

P. Hölzer, W. Chang, J. C. Travers, A. Nazarkin, J. Nold, N. Y. Joly, M. Saleh, F. Biancalana, and P. St. J. Russell, “Femtosecond nonlinear fiber optics in the ionization regime,” Phys. Rev. Lett. 107, 203901 (2011). [CrossRef] [PubMed]

The remarkable success of solid-core fiber (conventional or photonic crystal) as a vehicle for discovering, observing, and exploring a wide range of different nonlinear optical effects [6

E. M. Dianov, P. V. Mamyshev, and A. M. Prokhorov, “Nonlinear fiber optics,” Sov. J. Quantum Electron. 18, 1–15 (1988). [CrossRef]

, 7

G. Agrawal, Nonlinear Fiber Optics , 4th ed. (Academic, 2006).

, 8

J. M. Dudley and J. R. Taylor, “Ten years of nonlinear optics in photonic crystal fibre,” Nat. Photon. 3, 85–90 (2009). [CrossRef]

], is due to ultralong effective path lengths, a consistent and predictable dispersion, configurable birefringence, and a nonlinear figure of merit (the balance between nonlinearity and loss) that, for silica fiber, is the highest of any medium. These characteristics led to the observation of optical solitons [9

A. Hasegawa, “Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. I. anomalous dispersion,” Appl. Phys. Lett. 23, 142–144 (1973). [CrossRef]

, 10

L. F. Mollenauer, R. H. Stolen, and J. P. Gordon, “Experimental observation of picosecond pulse narrowing and solitons in optical fibers,” Phys. Rev. Lett. 45, 1095–1098 (1980). [CrossRef]

], the discovery of the soliton self-frequency shift [11

E. M. Dianov, A. Y. Karasik, P. V. Mamyshev, A. M. Prokhorov, V. N. Serkin, M. F. Stel’Makh, and A. A. Fomichev, “Stimulated- Raman conversion of multisoliton pulses in quartz optical fibers,” JETP Lett. 41, 242–244 (1985).

, 12

F. M. Mitschke and L. F. Mollenauer, “Discovery of the soliton self-frequency shift,” Opt. Lett. 11, 659–661 (1986). [CrossRef] [PubMed]

], the development of ultrafast all-optical switches [13

N. J. Doran and D. Wood, “Nonlinear-optical loop mirror,” Opt. Lett. 13, 56–58 (1988). [CrossRef] [PubMed]

], and the generation of octave-spanning supercontinua [14

C. Lin and R. H. Stolen, “New nanosecond continuum for excited-state spectroscopy,” Appl. Phys. Lett. 28, 216–218 (1976). [CrossRef]

, 15

J. M. Dudley, G. Genty, and S. Coen, “Supercontinuum generation in photonic crystal fiber,” Rev. Mod. Phys. 78, 1135–1184 (2006). [CrossRef]

] and even continuous-wave pumped supercontinua [16

J. C. Travers, “Continuous wave supercontinuum generation,” in Supercontinuum Generation in Optical Fibers , J. M. Dudley and J. R. Taylor, eds. (Cambridge Univ., 2010), pp. 142–177. [CrossRef]

], as well as advancing our fundamental understanding of four-wave mixing [17

R. Stolen, “Phase-matched-stimulated four-photon mixing in silica-fiber waveguides,” IEEE J. Quantum Electron. 11, 100–103 (1975). [CrossRef]

], modulation instability (MI) [18

A. Hasegawa and W. Brinkman, “Tunable coherent IR and FIR sources utilizing modulational instability,” IEEE J. Quantum Electron. 16, 694–697 (1980). [CrossRef]

], self-phase-modulation [19

E. P. Ippen, “Self-phase modulation of picosecond pulses in optical fibers,” Appl. Phys. Lett. 24, 190–192 (1974). [CrossRef]

], Raman [1

R. H. Stolen, “Raman oscillation in glass optical waveguide,” Appl. Phys. Lett. 20, 62–63 (1972). [CrossRef] [CrossRef]

] and Brillouin [2

E. P. Ippen, “Stimulated Brillouin scattering in optical fibers,” Appl. Phys. Lett. 21, 539–540 (1972). [CrossRef]

] scattering, and self-organized second-harmonic generation [20

R. H. Stolen and H. W. K. Tom, “Self-organized phase-matched harmonic generation in optical fibers,” Opt. Lett. 12, 585–587 (1987). [CrossRef] [PubMed]

Extending nonlinear fiber optics beyond interactions with a solid-core material, to cores filled with gases and plasmas, cannot be achieved using conventional fiber designs based on total internal reflection. This is because the core refractive index must be higher than that of the cladding, but it is close to unity for gases—lower than any dielectric material at optical frequencies. In cases where use of a gas is essential, for example to avoid optical damage when broadening the spectrum of millijoule-level femtosecond-duration

800 nm

pulses, the glass capillary fiber provides a partial solution. Although such capillaries do not support bound modes, the loss of the fundamental

{HE}_{11}

leaky mode scales as the inverse cube of the bore diameter, and it can be as low as

1.2 dB / m

for diameters of

\sim 200 μm

at the

800 nm

wavelength. Even though many higher order leaky modes are supported in bores of this size, it is possible to excite only the

{HE}_{11}

mode if careful attention is paid to the launching optics. Such large-bore capillary fibers have been used with great success in the compression of very high energy pulses [21

M. Nisoli, S. De Silvestri, and O. Svelto, “Generation of high energy $10 fs$ pulses by a new pulse compression technique,” Appl. Phys. Lett. 68, 2793–2795 (1996). [CrossRef]

, 22

M. Nisoli, S. De Silvestri, O. Svelto, R. Szipöcs, K. Ferencz, C. Spielmann, S. Sartania, and F. Krausz, “Compression of high-energy laser pulses below $5 fs$ ,” Opt. Lett. 22, 522–524 (1997). [CrossRef] [PubMed]

, 23

A. Suda, M. Hatayama, K. Nagasaka, and K. Midorikawa, “Generation of sub- $10 fs$ , $5 mJ$ -optical pulses using a hollow fiber with a pressure gradient,” Appl. Phys. Lett. 86, 111116 (2005). [CrossRef]

], and the generation of both low- [24

C. G. Durfee III, S. Backus, M. M. Murnane, and H. C. Kapteyn, “Ultrabroadband phase-matched optical parametric generation in the ultraviolet by use of guided waves,” Opt. Lett. 22, 1565–1567 (1997). [CrossRef]

] and high-order harmonics [25

A. Rundquist, C. G. Durfee, Z. Chang, C. Herne, S. Backus, M. M. Murnane, and H. C. Kapteyn, “Phase-matched generation of coherent soft x-rays,” Science 280, 1412–1415 (1998). [CrossRef] [PubMed]

An additional attractive feature of a gaseous core, impossible in solid-core fiber, is that the nonlinearity and the group velocity dispersion (GVD) can be varied by changing the gas pressure. The GVD of the

{HE}_{11}

mode in an evacuated large-bore capillary is very weakly anomalous, whereas the GVD of noble gases is normal at optical frequencies. It turns out that the zero-dispersion points in a gas-filled capillary are shifted far into the infrared (IR) at useful gas pressures, and the overall GVD is strongly normal in the ultraviolet (UV) to near-IR (NIR) spectral region, greatly restricting the range of accessible ultrafast nonlinear dynamics.

HC-PCFs [26

R. F. Cregan, B. J. Mangan, J. C. Knight, T. A. Birks, P. St. J. Russell, P. J. Roberts, and D. C. Allan, “Single-mode photonic band gap guidance of light in air,” Science 285, 1537–1539 (1999). [CrossRef] [PubMed]

, 27

F. Couny, F. Benabid, and P. S. Light, “Large-pitch kagome- structured hollow-core photonic crystal fiber,” Opt. Lett. 31, 3574–3576 (2006). [CrossRef] [PubMed]

, 28

P. St. J. Russell, “Photonic-crystal fibers,” J. Lightwave Technol. 24, 4729–4749 (2006). [CrossRef]

, 29

P. St. J. Russell, “Photonic crystal fibers,” Science 299, 358–362 (2003). [CrossRef] [PubMed]

, 30

J. C. Knight, “Photonic crystal fibres,” Nature 424, 847–851 (2003). [CrossRef] [PubMed]

] overcome these limitations, offering ultralong single-mode interaction lengths (

\sim 10

or even

\sim 1000 m

), high intensities per watt (due to the small core diameter), a high optical damage threshold, and exquisite control of the GVD. These features have been utilized in a wide range of experiments [31

A. R. Bhagwat and A. L. Gaeta, “Nonlinear optics in hollow-core photonic bandgap fibers,” Opt. Express 16, 5035–5047 (2008). [CrossRef] [PubMed]

, 32

F. Benabid and P. J. Roberts, “Linear and nonlinear optical properties of hollow core photonic crystal fiber,” J. Mod. Opt. 58, 87–124 (2011). [CrossRef]

], including clean demonstrations of solitary waves and self-similarity in stimulated Raman scattering [33

A. Abdolvand, A. Nazarkin, A. V. Chugreev, C. F. Kaminski, and P. St. J. Russell, “Solitary pulse generation by backward Raman scattering in $H_{2}$ -filled photonic crystal fibers,” Phys. Rev. Lett. 103, 183902 (2009). [CrossRef] [PubMed]

, 34

A. Nazarkin, A. Abdolvand, A. V. Chugreev, and P. St. J. Russell, “Direct observation of self-similarity in evolution of transient stimulated Raman scattering in gas-filled photonic crystal fibers,” Phys. Rev. Lett. 105, 173902 (2010). [CrossRef]

], well-controlled coherent atomic physics experiments [35

P. Londero, V. Venkataraman, A. R. Bhagwat, A. D. Slepkov, and A. L. Gaeta, “Ultralow-power four-wave mixing with Rb in a hollow-core photonic band-gap fiber,” Phys. Rev. Lett. 103, 043602 (2009). [CrossRef] [PubMed]

, 36

S. Ghosh, J. E. Sharping, D. G. Ouzounov, and A. L. Gaeta, “Resonant optical interactions with molecules confined in photonic band-gap fibers,” Phys. Rev. Lett. 94, 093902 (2005). [CrossRef] [PubMed]

], delivery of intense, high-energy optical solitons [37

D. G. Ouzounov, F. R. Ahmad, D. Müller, 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]

], and particle guidance [38

F. Benabid, J. C. Knight, and P. St. J. Russell, “Particle levitation and guidance in hollow-core photonic crystal fiber,” Opt. Express 10, 1195–1203 (2002). [PubMed]

], to name just a few.

In this paper we discuss ultrafast nonlinear dynamics in HC-PCF. We briefly introduce the linear properties of the two main types of HC-PCF in Section 2, and then we concentrate on the advantageous properties of kagomé PCF for ultrafast applications. In Section 3 we present a generalized analysis, based on the soliton order and zero-dispersion wavelength (ZDW), which simplifies the description of the highly tunable properties of gas-filled kagomé PCF. This is followed by an overview of techniques to compress pulses to a few optical cycles with kagomé PCF (Section 4). A recently reported, highly efficient, compact, and tunable deep-UV source is discussed in Section 5. It has also recently been shown that soliton self-compression can lead to intensities above the ionization threshold of the filling gas in a kagomé PCF, allowing, for the first time, controllable and extended single-mode laser-plasma interactions in fiber. We review this new area of nonlinear fiber optics in Section 6. Finally, in Section 7 we offer a perspective on the use of kagomé PCF for high-harmonic generation (HHG).

2. PROPERTIES OF HC-PCF

HC-PCF comes in two main varieties [26

, 27

F. Couny, F. Benabid, and P. S. Light, “Large-pitch kagome- structured hollow-core photonic crystal fiber,” Opt. Lett. 31, 3574–3576 (2006). [CrossRef] [PubMed]

, 28

P. St. J. Russell, “Photonic-crystal fibers,” J. Lightwave Technol. 24, 4729–4749 (2006). [CrossRef]

, 29

P. St. J. Russell, “Photonic crystal fibers,” Science 299, 358–362 (2003). [CrossRef] [PubMed]

, 30

J. C. Knight, “Photonic crystal fibres,” Nature 424, 847–851 (2003). [CrossRef] [PubMed]

]. The first, reported in 1999 [26

], confines light by means of a full two- dimensional photonic bandgap (we call it PBG-guiding HC-PCF). The most common structure is formed by a hexagonal lattice of air holes, as shown in Fig. 1a. It can provide extremely low transmission loss (

< 1 dB / km

1550 nm

in the best case) [39

P. Roberts, F. Couny, H. Sabert, B. Mangan, D. Williams, L. Farr, M. Mason, A. Tomlinson, T. Birks, J. Knight, and P. St. J. Russell, “Ultimate low loss of hollow-core photonic crystal fibres,” Opt. Express 13, 236–244 (2005). [CrossRef] [PubMed]

], guiding a tightly confined single mode over a restricted spectral range. This feature makes it useful for spectral filtering, for example, to suppress unwanted Stokes bands in Raman scattering [40

F. Benabid, G. Bouwmans, J. C. Knight, P. St. J. Russell, and F. Couny, “Ultrahigh efficiency laser wavelength conversion in a gas-filled hollow core photonic crystal fiber by pure stimulated rotational Raman scattering in molecular hydrogen,” Phys. Rev. Lett. 93, 123903 (2004). [CrossRef] [PubMed]

]. The GVD of PBG-guiding HC-PCF has a steep slope, passing through zero inside the transmission window and attaining very large values at its edges. To illustrate this, Fig. 1b shows the results of finite-element modeling (FEM) calculations [41

JCMwave, http://www.jcmwave.com (JCMwave GmbH, n.d.).

] for an ideal structure designed to operate at

800 nm

. The calculated loss around

800 nm

0.01 dB / m

. The ZDW is at

768 nm

, and the dispersion changes considerably across the narrow transmission band, with a dispersion slope greater than

2000 {fs}^{3} / cm

at the zero- dispersion point. The very low loss makes these fibers useful for a wide range of applications, but the limited transmission windows and extreme dispersion slopes prevent application to extreme ultrafast pulse experiments.

The second type of HC-PCF, first reported in 2002 [3

], has a kagomé-lattice cladding, characterized by a star-of-David pattern of glass webs, as shown in Fig. 1c. It provides ultrabroadband (several hundred nanometers) guidance at loss levels of

\sim 1 dB / m

, and it displays weak anomalous GVD (

| β_{2} | < 15 {fs}^{2} / cm

, when evacuated) over the entire transmission window, with a low dispersion slope. FEM calculations, for one example structure (

30 μm

core diameter, designed for operation in the UV and around

800 nm

), are shown in Fig. 1d, illustrating the broadband guidance windows (

200 - 350 nm

and

600 - 850 nm < 2 dB / m

), with relatively small all-anomalous dispersion magnitude (below

15 {fs}^{2} / cm

) across the whole band, excluding the anticrossing with a strong cladding resonance at

\sim 380 nm

. Such cladding resonances disrupt the transmission window but are usually quite narrow and so have little influence on the global dispersion properties. Also, they can be tuned away from the wavelength bands required in specific applications by varying the web thickness. As we shall see, unlike for capillary fibers, the normal GVD of a noble gas can be balanced against the anomalous GVD of the kagomé PCF, allowing the ZDW to be tuned across the UV, visible, and NIR spectral regions, simply by varying the pressure. This makes the system ideal for ultrafast applications [42

S.-J. Im, A. Husakou, and J. Herrmann, “Guiding properties and dispersion control of kagome lattice hollow-core photonic crystal fibers,” Opt. Express 17, 13050–13058 (2009). [CrossRef] [PubMed]

]. We now take a closer look at kagomé PCF.

2A. Dispersion and Loss of Gas-Filled Kagomé HC-PCF

To a good approximation, the modal refractive index of kagomé HC-PCF closely follows that of a capillary fiber [42

, 43

J. Nold, P. Hölzer, N. Y. Joly, G. K. L. Wong, A. Nazarkin, A. Podlipensky, M. Scharrer, and P. St. J. Russell, “Pressure- controlled phase matching to third harmonic in Ar-filled hollow-core photonic crystal fiber,” Opt. Lett. 35, 2922–2924 (2010). [CrossRef] [PubMed]

, 44

E. Marcatili and R. Schmeltzer, “Hollow metallic and dielectric waveguides for long distance optical transmission and lasers (long distance optical transmission in hollow dielectric and metal circular waveguides, examining normal mode propagation),” Bell Syst. Tech. J. 43, 1783–1809 (1964).

, 45

W. Chang, A. Nazarkin, J. C. Travers, J. Nold, P. Hölzer, N. Y. Joly, and P. St. J. Russell, “Influence of ionization on ultrafast gas-based nonlinear fiber optics,” Opt. Express 19, 21018–21027 (2011). [CrossRef] [PubMed]

n_{m n} (λ, p, T) = \sqrt{n_{gas}^{2} (λ, p, T) - \frac{u_{m n}^{2}}{k^{2} a^{2}}} ≃ 1 + δ (λ) \frac{p}{2 p_{0}} \frac{T_{0}}{T} - \frac{u_{m n}^{2}}{2 k^{2} a^{2}},

(1)

where k is the vacuum wave vector,

n_{gas}

is the refractive index of the filling gas,

δ (λ)

is a Sellmeier expansion for

n_{gas}^{2}

[46

A. Börzsönyi, Z. Heiner, M. P. Kalashnikov, A. P. Kovács, and K. Osvay, “Dispersion measurement of inert gases and gas mixtures at $800 nm$ ,” Appl. Opt. 47, 4856–4863 (2008). [CrossRef] [PubMed]

], p is the pressure,

p_{0}

is the atmospheric pressure, T is the temperature,

T_{0} = 273.15 K

, and

u_{m n}

is the nth zero of the mth-order Bessel function of the first kind, where

m = n = 1

corresponds to the

{HE}_{11}

mode of the fiber. Several core radii, a, can be defined for the circle used to approximate the kagomé fiber core in Eq. (1), including one that preserves core area (

a_{A}

) and one based simply on the flat-to-flat distance of the hexagonal core (

a_{f}

), as shown in the inset of Fig. 2a. Comparison with full finite-element simulations shows that Eq. (1) is highly accurate [Figs. 2a, 2b] [42

, 45

a_{A}

providing the best agreement when comparing effective mode index and

a_{f}

providing the best agreement with the GVD calculations. The validity of Eq. (1) has been further confirmed by phase-matched third-harmonic generation in a higher order guided mode [43

Given that kagomé PCF has dispersion properties so similar to those of a simple capillary, why bother to use it at all? The reason is that it has much lower loss than is theoretically possible in a capillary fiber [Fig. 3a]. For core diameters of

\sim 30 μm

(required for anomalous dispersion at reasonable levels of nonlinearity—see below), the loss in a capillary exceeds several

100 dB / m

, whereas that of a kagomé PCF can be below

1 dB / m

Apart from broadband guidance with sufficiently low loss, another key feature that makes kagomé PCF particularly useful in ultrafast applications is its pressure-tunable dispersion. Figure 3c shows how the ZDW of an Ar-filled kagomé PCF (

30 μm

core diameter) can be tuned across the whole UV- visible spectral region, allowing, for example, solitons to form in the deep UV. It is interesting to compare this behavior with that of solid-core PCF. Although known for the wide flexibility of its dispersion, the strong material dispersion of the glass limits the shortest attainable ZDW to

\sim 500 nm

[Fig. 3b]. Whereas solid-core PCF permitted soliton propagation at

800 nm

and into the visible spectral region [47

J. C. Knight, J. Arriaga, T. A. Birks, A. Ortigosa-Blanch, W. J. Wadsworth, and P. St. J. Russell, “Anomalous dispersion in photonic crystal fiber,” IEEE Photon. Technol. Lett. 12, 807–809 (2000). [CrossRef]

, 48

J. K. Ranka, R. S. Windeler, and A. J. Stentz, “Visible continuum generation in air-silica microstructure optical fibers with anomalous dispersion at $800 nm$ ,” Opt. Lett. 25, 25–27 (2000). [CrossRef]

], kagomé PCF brings anomalous dispersion to the UV for the first time. Additionally, the dispersion magnitude is much smaller in kagomé PCF, which means that ultrafast pulses broaden much less quickly, and pulses at discrete frequencies walk off less rapidly, increasing the nonlinear interaction length compared to solid-core PCF. The relatively smaller dispersion slope also means that perturbations to soliton propagation are reduced, leading, for example, to the generation of shorter self-compressed pulses (see Subsection 4B).

Figure 4 summarizes how the ZDW can be extensively tuned by varying the gas type, pressure, and core diameter. Not only can it be pushed deep into the UV, but it also can be tuned into the IR beyond

1000 nm

, allowing normal dispersion at

800 nm

—useful for applications such as the fiber-grating/mirror compressors discussed below.

2B. Nonlinearity of Gas-Filled Kagomé PCF

Assuming a modal intensity distribution in the form

J_{0}^{2} (u_{11} r / a)

for the fundamental

{HE}_{11}

mode, with r the radial coordinate, the conventional definition of effective mode area [7

G. Agrawal, Nonlinear Fiber Optics , 4th ed. (Academic, 2006).

] yields

A_{eff} \sim 1.5 a^{2}

; i.e., a kagomé PCF with a

30 μm

core diameter will have

A_{eff} = 338 {μm}^{2}

. Experiments and numerical modeling show that the contribution of the glass to the nonlinearity of kagomé PCF is usually negligible because the fraction of light in glass is

< 0.01 %

away from the cladding resonances [Fig. 5a] [49

N. Y. Joly, J. Nold, W. Chang, P. Hölzer, A. Nazarkin, G. K. L. Wong, F. Biancalana, and P. St. J. Russell, “Bright spatially coherent wavelength-tunable deep-UV laser source using an ar-filled photonic crystal fiber,” Phys. Rev. Lett. 106, 203901 (2011). [CrossRef] [PubMed]

, 50

F. Couny, F. Benabid, P. J. Roberts, P. S. Light, and M. G. Raymer, “Generation and photonic guidance of multi-octave optical-frequency combs,” Science 318, 1118–1121 (2007). [CrossRef] [PubMed]

Gas nonlinearity scales linearly with gas density, or with pressure at constant temperature [51

H. J. Lehmeier, W. Leupacher, and A. Penzkofer, “Nonresonant third order hyperpolarizability of rare gases and $N_{2}$ determined by third harmonic generation,” Opt. Commun. 56, 67–72 (1985). [CrossRef] [CrossRef]

]. For a kagomé PCF with a

30 μm

core diameter, the nonlinear coefficient γ is (at

10 bar

1.5 \times 10^{- 7} m^{- 1} W^{- 1}

(Ne),

2 \times 10^{- 6} m^{- 1} W^{- 1}

(Ar),

5.3 \times 10^{- 6} m^{- 1} W^{- 1}

(Kr), and

1.5 \times 10^{- 5} m^{- 1} W^{- 1}

(Xe). These are over 4 orders of magnitude smaller than in a solid-core PCF (

0.24 m^{- 1} W^{- 1}

for a core diameter of

1 μm

) but much higher than in a gas-filled capillary with a bore diameter of

200 μm

(

\sim 5 \times 10^{- 8} m^{- 1} W^{- 1}

10 bar

Ar).

Raman scattering is absent in monatomic gases such as Ar, allowing for the study of soliton dynamics and pulse compression in the absence of the soliton self-frequency shift. However, by choosing molecular gases such as

N_{2}

{SF}_{6}

, Raman scattering can also be studied in kagomé PCF. For ultrafast applications, one intriguing possibility is to mix gases with differing Raman contributions and hence add to our pressure-tuned system the ability to continuously adjust the fractional Raman contribution.

Apart from perturbative bound electron nonlinearities, the intensities reached in kagomé PCF at few-microjoule energy levels can be sufficient to ionize the gas and produce a partial plasma. Although the field-plasma interaction itself is usually a linear process at the free-electron densities and optical intensities encountered in kagomé PCF, the electron generation process is both ultrafast and highly nonlinear, leading to new nonlinear effects such as a soliton self-frequency blueshift, which we discuss in Section 6.

2C. Energy Handling of Kagomé HC-PCF

Finite-element calculations of the power-in-glass fraction, shown in Fig. 5a, show that, at around

800 nm

, over 99.99% of the light can be guided in the hollow regions of a kagomé PCF, meaning that optical damage thresholds are much higher than in glass-core fibers. The power-in-glass fraction significantly increases at cladding resonances (there is one at

\sim 380 nm

in Fig. 5), where light couples into the glass webs [compare Figs. 5b, 5c]. Optical damage is far more likely at these resonances; thus, it is important to locate them away from operating wavelengths by tuning the web thickness. Additionally optimizing launch efficiency is important because slight input coupling misalignment excites cladding modes, also leading to optical damage.

In capillary fibers with core diameters of

45 - 70 μm

, guided intensities over

10^{16} W / {cm}^{2}

have been demonstrated [52

F. Dorchies, J. R. Marquès, B. Cros, G. Matthieussent, C. Courtois, T. Vélikoroussov, P. Audebert, J. P. Geindre, S. Rebibo, G. Hamoniaux, and F. Amiranoff, “Monomode guiding of $1016 W / {cm}^{2}$ laser pulses over 100 Rayleigh lengths in hollow capillary dielectric tubes,” Phys. Rev. Lett. 82, 4655–4658 (1999). [CrossRef]

]; guidance of similar intensities in kagomé PCF can be expected. Our group has coupled ultrafast pulses (

65 fs

) with over

12 μJ

into a kagomé PCF with a

26 μm

core diameter, with efficiencies up to 80%. After self-compression (Subsection 4B), we have reached intensities of over

10^{14} W / {cm}^{2}

]. By careful mode matching, another group reported 98% launch efficiency of

40 fs

pulses into a PBG-guiding HC-PCF (core diameter

6.8 μm

), reaching a transmitted energy of

\sim 2 μJ

[53

A. A. Ishaaya, C. J. Hensley, B. Shim, S. Schrauth, K. W. Koch, and A. L. Gaeta, “Highly-efficient coupling of linearly- and radially-polarized femtosecond pulses in hollow-core photonic band-gap fibers,” Opt. Express 17, 18630–18637 (2009). [CrossRef]

3. ULTRAFAST NONLINEAR DYNAMICS IN KAGOMÉ PCF

The experimental setup typically used for ultrafast pulse propagation experiments in kagomé PCF filled with gas is illustrated in Fig. 6. It consists of two gas cells, one at each fiber end, both equipped with a high-quality window for launching and extracting light. The fiber is first evacuated, purged several times, and then filled with the gas to the desired pressure. Pressure gradients can be easily achieved, and gas filling times are

< 1 \min

for the pressures and core sizes used [54

J. Henningsen and J. Hald, “Dynamics of gas flow in hollow core photonic bandgap fibers,” Appl. Opt. 47, 2790–2797 (2008). [CrossRef] [PubMed]

]. The simplicity of this setup belies its versatility, as we shall see in the next sections.

All of the numerical results presented in this paper are calculated using a unidirectional field propagation equation [45

], including the full dispersion curve based on Eq. (1), the Kerr effect, photoionization, and the influence of free electrons created through photoionization (Appendix A).

3A. Interplay of Dispersion and Nonlinearity

Ultrafast nonlinear dynamics depend on a complicated interplay between the dispersion and nonlinearity. A combined analysis is therefore necessary. Here we use a simplified envelope equation to facilitate discussion of the physics underlying ultrafast processes in gas-filled kagomé PCF. Taking the guided mode to be linearly polarized (sufficient for comparing the global properties of both kagomé PCF and capillary fiber), the generalized nonlinear Schrödinger equation (GNLSE) takes the normalized form [7

G. Agrawal, Nonlinear Fiber Optics , 4th ed. (Academic, 2006).

\partial_{z} U = - i \frac{sgn (β_{2})}{2 L_{D}} \partial_{τ}^{2} U + \frac{sgn (β_{3})}{6 L_{D}^{'}} \partial_{τ}^{3} U + \frac{i e^{- α z}}{L_{NL}} (U | U |^{2} + i s \partial_{τ} (U | U |^{2})),

(2)

where

U (τ, z)

is the normalized optical field envelope and α is the power loss coefficient. Note that the Raman effect is absent in the monatomic gases considered here, and although dispersion orders

β_{4}

and higher can play an important role in the dynamics, they do not alter the results in this section. Defining

τ_{0}

as the pump pulse duration,

ω_{0}

its central angular frequency, and

P_{0}

its peak power, the following normalized parameters fully define the propagation dynamics:

L_{D} = \frac{τ_{0}^{2}}{| β_{2} |}, L_{D}^{'} = \frac{τ_{0}^{3}}{| β_{3} |}, β_{n} = \partial_{ω}^{n} β |_{ω_{0}}, L_{NL} = \frac{1}{γ P_{0}}, s = \frac{1}{ω_{0} τ_{0}} .

(3)

Without loss of generality, we assume a pump pulse propagating at

800 nm

with a duration of

30 fs

full-width at half- maximum (FWHM) intensity. The length-scales in Eq. (3) can then be reduced to just two parameters—the ZDW

λ_{0}

and the soliton order N:

(L_{D}, L_{D}^{'}) \to λ_{0} (L_{D}, L_{NL}) \to N = \sqrt{\frac{γ P_{0} τ_{0}^{2}}{| β_{2} |}} .

(4)

Note that although bright solitons only occur in regions of anomalous dispersion, this definition of the soliton order can also be used in the normal dispersion regime, providing a convenient means of comparing the relative nonlinearity in all cases. The reduction of the full parameter space to (

λ_{0}

, N) is useful for extracting the global dynamics and understanding the scaling behavior of a given system. For example, if it is known that

λ_{0} = 600 nm

and

N = 9

leads to UV generation, then any combination of core diameter, gas species, filling pressure, and pulse energy that yields the same values of (

λ_{0}

, N) can also be expected to generate UV, even though the physical system parameters may be drastically different. Note that it makes no sense to compare

λ_{0}

or N on their own. For example, although a nearly evacuated capillary fiber will exhibit anomalous dispersion well below

200 nm

, it will have close to zero nonlinear coefficient.

3B. Energy Scaling of Solitons: Dependence on the Dispersion Landscape

The extraordinary scalability of the kagomé PCF system can be fully appreciated by considering again Fig. 4. For a given choice of (

λ_{0}

, N), we can select a gas species and, for a wide choice of kagomé PCF diameters, find the correct filling pressure to fix

λ_{0}

. We then simply choose the correct pulse energy to obtain the given soliton order N. Figures 7a, 7b, 7c show the range of energies required for given (

λ_{0}

, N) for some selected gas species and kagomé PCF diameters, illustrating that, for example, the dynamics of

λ_{0} = 600 nm

and

N = 9

can be accessed with pulse energies ranging from

\sim 100 nJ

\sim 10 μJ

For soliton orders

2 \leq N \leq 15

, while pumping in the anomalous dispersion region, the propagation dynamics are usually dominated by soliton fission, which approximately occurs at a characteristic length scale [6

E. M. Dianov, P. V. Mamyshev, and A. M. Prokhorov, “Nonlinear fiber optics,” Sov. J. Quantum Electron. 18, 1–15 (1988). [CrossRef]

, 15

J. M. Dudley, G. Genty, and S. Coen, “Supercontinuum generation in photonic crystal fiber,” Rev. Mod. Phys. 78, 1135–1184 (2006). [CrossRef]

L_{fiss} = \frac{L_{D}}{N} .

(5)

Nonlinear propagation will be limited if power losses are too high over this length scale (here we take a guideline value of

1 dB

over one soliton fission length). For capillary fibers, nonlinear interaction would be prohibited over the entire param eter space in all three cases presented in Figs. 7a, 7b, 7c; although the loss in a capillary decreases with larger core diameter, the concomitant increase in nonlinear interaction length more than cancels this out (due to lower dispersion and lower nonlinearity). This means that none of the dynamics we discuss below are accessible in capillary fibers. Conversely, most of the parameter space in Figs. 7a, 7b, 7c is accessible to kagomé PCF.

From the discussion in Subsection 2C (on the energy- handling capabilities of kagomé PCF) it is clear that, for some of the parameter sets in Fig. 7, the energies required may be too high. Even so, all of the dynamics can be accessed by at least one system without exceeding the damage threshold. For example,

\sim 1 mJ

of energy is required to create an eighth- order soliton in a kagomé PCF with a core diameter of

50 μm

filled with He so as to yield a ZDW of

250 nm

. Such a high pulse energy may be beyond what such a kagomé PCF can handle; however, the same dynamics would result if the core diameter is reduced to

10 μm

, He is replaced with Xe, and the pulse energy is lowered to

10 μJ

—within the capabilities of kagomé PCF.

A more difficult question is whether self-focusing due to the Kerr effect, or defocusing due to free-electron creation can significantly disturb the guided mode. In the case of large-bore capillary fibers, it was found that spatial perturbations to the leaky modes occurs at power levels similar to those required for critical self-focusing in bulk gas [55

G. Fibich and A. L. Gaeta, “Critical power for self-focusing in bulk media and in hollow waveguides,” Opt. Lett. 25, 335–337 (2000). [CrossRef]

, 56

G. Tempea and T. Brabec, “Theory of self-focusing in a hollow waveguide,” Opt. Lett. 23, 762–764 (1998). [CrossRef]

]. Although the launched peak pulse power remains below this limit in every case in Fig. 7, the pulses can undergo strong temporal compression as they propagate (see Subsection 4B). To quantify this, Fig. 7d plots the ratio of the peak power after self- compression, to the critical self-focusing power at the relevant gas pressure. This ratio reaches a maximum value of

\sim 0.2

for

λ_{0} = 600 nm

and

N = 9

, which are approximately the parameters used in [49

], where no degradation in modal quality was observed. This is probably because the peak self- compressed intensity is maintained over too short a distance for self-focusing to have any effect.

In the following sections we discuss the ultrafast dynamics in a number of exemplary systems, characterized by the (

λ_{0}

, N) values summarized in Table 1 and indicated with colored dots and labeled (i) to (iii) in Figs. 7, 15.

4. PULSE COMPRESSION

4A. Fiber-Grating/Mirror Compression

A common pulse-compression technique is a two-stage process: in the first stage nonlinear spectral broadening is produced by self-phase modulation (SPM), which, in the presence of correctly tuned normal dispersion, produces a parabolic temporal phase, i.e., a linear chirp; in the second stage the parabolic phase is compensated for in an anomalously dispersive external compressor, such as a bulk grating or a chirped mirror. This is known as fiber-grating or fiber-mirror compression, depending on which dispersive compression element is used. It has been applied successfully to the generation of few-cycle pulses both in solid-core fiber [57

C. V. Shank, “Compression of femtosecond optical pulses,” Appl. Phys. Lett. 40, 761–763 (1982). [CrossRef]

, 58

W. J. Tomlinson, R. H. Stolen, and C. V. Shank, “Compression of optical pulses chirped by self-phase modulation in fibers,” J. Opt. Soc. Am. B 1, 139–149 (1984). [CrossRef]

] and at very high pulse energies in gas-filled capillary fibers [21

M. Nisoli, S. De Silvestri, and O. Svelto, “Generation of high energy $10 fs$ pulses by a new pulse compression technique,” Appl. Phys. Lett. 68, 2793–2795 (1996). [CrossRef]

, 22

, 23

PBG-guiding HC-PCF can be designed to have very low nonlinearity and strong anomalous dispersion. These properties have been used to replace the grating/mirror in the compression stage of an all-fiber pulse-compression system [59

C. J. S. de Matos, J. R. Taylor, T. Hansen, K. 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]

, 60

C. J. S. de Matos, S. V. Popov, A. B. Rulkov, J. R. Taylor, J. Broeng, T. P. Hansen, and V. P. Gapontsev, “All-fiber format compression of frequency chirped pulses in air-guiding photonic crystal fibers,” Phys. Rev. Lett. 93, 103901 (2004). [CrossRef] [PubMed]

, 61

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

] at peak powers of

\sim 1 MW

, well above what is possible in solid-core fibers because of nonlinear effects during compression and the onset of optical damage. However, the restricted guidance bandwidth and large dispersion slope limit the use of PBG HC-PCF in the nonlinear spectral broadening stage.

Kagomé PCF offers the opportunity of performing both stages of this pulse-compression technique in fiber because it has a broad guidance band, tunable GVD, and, most importantly, a small GVD slope. Additionally, the nonlinearity of kagomé PCF is intermediate between solid-core and hollow capillary fibers (Subsection 2B), uniquely enabling its use in the nonlinear spectral broadening stage at pulse energies (

0.1 - 100 μJ

) much lower than possible with capillary fibers, but significantly exceeding what solid-core fiber can handle. Results on spectral broadening in Xe-filled kagomé PCF, and external compression, have been reported [62

O. H. Heckl, C. J. Saraceno, C. R. E. Baer, T. Südmeyer, Y. Y. Wang, Y. Cheng, F. Benabid, and U. Keller, “Temporal pulse compression in a xenon-filled kagome-type hollow-core photonic crystal fiber at high average power,” Opt. Express 19, 19142–19149 (2011). [CrossRef] [PubMed]

] at high average power, demonstrating a compression factor of

\sim 4

for

\sim 1 ps

pulses. However in that system, the dispersion was weakly anomalous, which is suboptimal for the production of clean parabolic pulses for optimal external compression to a few optical cycles.

4A1. Illustrative Design

Figure 8 shows numerical results for one potential system design using a kagomé PCF for the spectral broadening stage, and a chirped mirror as the linear compressor. Figure 8a shows propagation of a

10 μJ

30 fs

pulse in a kagomé PCF with a core diameter of

70 μm

filled with

43 bar

of Ar. This causes the ZDW to shift to

1300 nm

so that the pulse (at

800 nm

) is propagating in the normal dispersion region. Both temporal and spectral broadening are observed, as expected for a combination of normal dispersion and SPM. Figure 8b shows the broadened temporal shape of the intensity and the smooth parabolic temporal phase after

8 cm

of propagation. This phase profile can be compensated for with a linear compressor such as a chirped mirror. In Fig. 8c we plot the duration that would be achieved by linearly compressing the spectrally broadened output pulses from the given length of kagomé PCF. After

8 cm

of propagation, the pulse can be compressed to

\sim 3 fs

after the equivalent of approximately two bounces from commercially available chirped mirrors. The resulting compressed pulse shape is shown in Fig. 8d, showing a very clean few-cycle electric field, with a corresponding increase of peak power from 0.3 to

3 GW

. Note that the final peak power does not occur inside the fiber—where the pulse is still temporally broadened—but after a free-space compression stage, allowing much higher peak powers than can exist inside the gas-filled kagomé PCF. The quality of the fiber- grating compression scheme is quantified using a quality factor defined as the ratio of the energy within 1 FWHM intensity of the compressed pulse to that contained within 1 FWHM of the uncompressed pulse. In the example in Fig. 8d, this factor is over 80%, and in fact from Fig. 8c (right-hand axis) we see that it never falls below 70% in this particular system design.

Further energy scaling may require the use of positive gas gradients, as demonstrated in capillary systems [23

]. In this way, the impact of ionization is reduced at the input of the fiber, where the peak intensity is highest, due to low gas pressure. As the pulse temporally broadens on propagation, the intensity decreases [Fig. 8a], but with a positive pressure gradient, consistent SPM can be achieved. Alternatively, downtapered kagomé PCF may be used to reduce the intensity at the input end of the fiber, while preserving SPM broadening and normal dispersion throughout propagation.

4B. Soliton-Effect Compression

A different scheme involves soliton-effect self-compression, as commonly implemented in solid-core fiber [63

L. F. Mollenauer, R. H. Stolen, J. P. Gordon, and W. J. Tomlinson, “Extreme picosecond pulse narrowing by means of soliton effect in single-mode optical fibers,” Opt. Lett. 8, 289–291 (1983). [CrossRef] [PubMed]

, 64

K. Tai and A. Tomita, “ $1100 \times$ optical fiber pulse compression using grating pair and soliton effect at $1.319 μm$ ,” Appl. Phys. Lett. 48, 1033–1035 (1986). [CrossRef]

, 65

A. S. Gouveia-Neto, A. S. L. Gomes, and J. R. Taylor, “Generation of $33 fsec$ pulses at $1.32 μm$ through a high-order soliton effect in a single-mode optical fiber,” Opt. Lett. 12, 395–397 (1987). [CrossRef] [PubMed]

] but ideally suited to kagomé PCF. In this process, the interplay of anomalous dispersion and SPM leads to pulse self-compression—the initial stages of higher order soliton propagation. The use of PBG-guiding HC-PCF for this process has been successfully demonstrated [66

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

, 67

F. Gérôme, P. Dupriez, J. Clowes, 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]

, 68

P. J. Mosley, W. C. Huang, M. G. Welch, B. J. Mangan, W. J. Wadsworth, and J. C. Knight, “Ultrashort pulse compression and delivery in a hollow-core photonic crystal fiber at $540 nm$ wavelength,” Opt. Lett. 35, 3589–3591 (2010). [CrossRef] [PubMed]

], allowing the generation of pulses with megawatt peak powers, beyond the damage limits of fused silica. Kagomé PCF is even more advantageous because of its inherently weaker higher order dispersion—often the limiting factor in extreme pulse compression—and its broader guidance band.

While exploring the efficient generation of deep-UV light in an Ar-filled kagomé PCF, Joly et al. [49

] experimentally demonstrated soliton-effect self-compression of

\sim 30 fs

pulses to shorter than

10 fs

. In these experiments, the PCF was much longer than the optimum compression length. Numerical simulations show that pulses as short as

\sim 4 fs

must have occurred at the position of maximum compression, coinciding with the onset of UV emission (see Section 5).

4B1. Compression Ratio and Quality Factor Scaling

Detailed analysis of a large number of numerical simulations [69

E. M. Dianov, Z. S. Nikonova, A. M. Prokhorov, and A. A. Podshivalov, “Optimal compression of multi-soliton pulses,” Sov. Tech. Phys. Lett. 12, 311–313 (1986).

, 70

C. M. Chen and P. L. Kelley, “Nonlinear pulse compression in optical fibers: scaling laws and numerical analysis,” J. Opt. Soc. Am. B 19, 1961–1967 (2002). [CrossRef]

] suggests the following scaling rules for compression ratio and quality factor in terms of soliton order:

F_{c} \sim 4.6 N, Q_{c} \sim 3.7 / N,

(6)

implying that a

30 fs

pulse at

800 nm

can be compressed to less than a single optical cycle for

N \sim 3

. Such extreme compression is inevitably limited by higher order effects, for example, by higher order dispersion [71

A. A. Voronin and A. M. Zheltikov, “Soliton-number analysis of soliton-effect pulse compression to single-cycle pulse widths,” Phys. Rev. A 78, 063834 (2008). [CrossRef]

]. Figure 9 shows numerical results for propagation through a kagomé PCF (

30 μm

core diameter) filled with Ar. Figure 9a shows the shortest self- compressed pulse duration as a function of (

λ_{0}

, N). As N increases, the compressed pulse gets shorter until it saturates between

N = 4

and 6. The duration of the compressed pulse depends weakly on

λ_{0}

; as

λ_{0}

moves closer to the pump wavelength (

800 nm

), higher order dispersion starts to perturb the self-compression, causing premature pulse fission. As

λ_{0}

moves further away from

800 nm

, which requires lower gas pressures and hence higher pump intensities, ionization of the gas starts to occur, decreasing the effectiveness of the self- compression process. Figure 9b shows the compressed pulse quality factor as a function of (

λ_{0}

, N). As stated in Eq. (6), this degrades significantly with increasing N and also degrades as

λ_{0}

shifts far from

800 nm

. There is a trade-off between the compressed pulse duration and quality factor. A good choice is

λ_{0} = 500 nm

N = 3.5

[corresponding to point (iii) in Figs. 7, 15]; the spectral and temporal evolution of a

30 fs

pulse for these parameters is shown in Fig. 9c. In contrast to propagation in the normal dispersion regime [Fig. 8a], the temporal profile is seen to dramatically sharpen upon propagation. At the point of optimum compression, a

\sim 2 fs

pulse is produced [Fig. 9c]. For higher values of N (

\sim 9

), the self-compressed pulses can reach subcycle durations, though with a significant loss in quality, and the pulse spectrum extends into the UV. This is the initiation of the UV dispersive-wave emission discussed in Section 5.

4B2. Effect of Longer Pulses

What happens to the compression if we increase the pump pulse duration? From Eq. (6) we can write the compressed pulse duration

τ_{c}

in the following form:

τ_{c} = \frac{τ_{0}}{F_{c}} \propto \frac{τ_{0}}{N} \propto \sqrt{\frac{| β_{2} |}{γ P_{0}}},

(7)

which shows that

τ_{c}

remains the same if the peak power of the pump pulse is maintained; i.e., the soliton order is increased in proportion to the pulse duration

τ_{0}

. Equation (6) shows, however, that the quality factor significantly decreases with increasing soliton order, leading to worse compression as

τ_{0}

gets longer.

4C. Adiabatic Pulse Compression

Adiabatic soliton compression [72

H. H. Kuehl, “Solitons on an axially nonuniform optical fiber,” J. Opt. Soc. Am. B 5, 709–713 (1988). [CrossRef]

, 73

S. V. Chernikov and P. V. Mamyshev, “Femtosecond soliton propagation in fibers with slowly decreasing dispersion,” J. Opt. Soc. Am. B 8, 1633–1641 (1991). [CrossRef]

, 74

J. C. Travers, J. M. Stone, A. B. Rulkov, B. A. Cumberland, A. K. George, S. V. Popov, J. C. Knight, and J. R. Taylor, “Optical pulse compression in dispersion decreasing photonic crystal fiber,” Opt. Express 15, 13203–13211 (2007). [CrossRef] [PubMed]

] has also been demonstrated in tapered PBG-guiding HC-PCF [75

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

]. Numerical analysis suggests that the shortest pulse durations can be reached if a pressure gradient is applied so as to cause the GVD to decrease with propagation distance [76

J. Lægsgaard and P. J. Roberts, “Theory of adiabatic pressure-gradient soliton compression in hollow-core photonic bandgap fibers,” Opt. Lett. 34, 3710–3712 (2009). [CrossRef] [PubMed]

], thus avoiding the need to taper the fiber structure itself. Simulations also show that the combination of decreasing dispersion and soliton-effect compression allows for higher compression ratios without impairment to the quality factor.

Kagomé PCF can be used for adiabatic compression, and a positive pressure gradient would lead to both increasing γ and decreasing

β_{2}

, perfect for this process. However, the fiber length must be many times the dispersion length if the compression is to remain adiabatic, so that this technique would be restricted to only the shortest input pulse durations because best-case losses of

\sim 0.3 dB / m

limit the usable fiber length to

\sim 10 m

5. DISPERSIVE-WAVE GENERATION IN THE UV

Recently, highly efficient deep-UV generation was reported using Ar-filled kagomé PCF [49

]. The emitted UV wavelength was tunable by changing the gas pressure and hence the dispersion. To illustrate these results, Fig. 10a shows a series of experimentally achieved UV spectra obtained with an Ar-filled kagomé PCF with a

27 μm

core diameter, pumped with

45 fs

pulses at

800 nm

[77

K. F. Mak, J. C. Travers, W. Chang, P. Hölzer, J. Nold, N. Y. Joly, and P. St. J. Russell are preparing a manuscript to be called “Efficiency and tunability of UV-visible dispersive wave emission in gas-filled kagome photonic crystal fiber.”

]. Each spectrum was obtained by tuning the pulse energy (

0.5 - 3 μJ

) and filling pressure (

4 - 12 bar

) to demonstrate the range of clean UV output tunable from

200 - 300 nm

. The spectra have relative widths of

\sim 0.03

and can support ultrafast pulses. Figure 10b shows the experimentally achieved UV conversion efficiencies from [49

], where 6% to 8% conversion was measured over a wide power tuning range, indicating output energies in the UV of around

50 - 80 nJ

for a kagomé PCF with

λ_{0} \sim 600 nm

N \sim 9

[corresponding to point (i) in Figs. 7, 15].

5A. Phase Matching to Dispersive Waves

The generation of UV light results from extreme soliton-effect pulse compression of the input pulse, resulting in a spectral expansion that overlaps with resonant dispersive-wave frequencies, which are consequently excited [78

P. K. A. Wai, C. R. Menyuk, Y. C. Lee, and H. H. Chen, “Nonlinear pulse propagation in the neighborhood of the zero-dispersion wavelength of monomode optical fibers,” Opt. Lett. 11, 464–466 (1986). [CrossRef] [PubMed]

, 79

P. K. A. Wai, H. H. Chen, and Y. C. Lee, “Radiations by ‘solitons’ at the zero group-dispersion wavelength of single-mode optical fibers,” Phys. Rev. A 41, 426–439 (1990). [CrossRef] [PubMed]

, 80

V. I. Karpman, “Radiation by solitons due to higher-order dispersion,” Phys. Rev. E 47, 2073–2082 (1993). [CrossRef]

, 81

N. Akhmediev and M. Karlsson, “Cherenkov radiation emitted by solitons in optical fibers,” Phys. Rev. A 51, 2602–2607 (1995). [CrossRef] [PubMed]

]. Solitons are stable close to their central frequency, because they propagate with a higher phase velocity than dispersive waves with the same frequency [9

A. Hasegawa, “Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. I. anomalous dispersion,” Appl. Phys. Lett. 23, 142–144 (1973). [CrossRef]

, 10

L. F. Mollenauer, R. H. Stolen, and J. P. Gordon, “Experimental observation of picosecond pulse narrowing and solitons in optical fibers,” Phys. Rev. Lett. 45, 1095–1098 (1980). [CrossRef]

]. However, in the presence of higher order dispersion, they can phase match to dispersive waves at other frequencies [79

, 80

V. I. Karpman, “Radiation by solitons due to higher-order dispersion,” Phys. Rev. E 47, 2073–2082 (1993). [CrossRef]

, 81

N. Akhmediev and M. Karlsson, “Cherenkov radiation emitted by solitons in optical fibers,” Phys. Rev. A 51, 2602–2607 (1995). [CrossRef] [PubMed]

]; i.e.,

β (ω) = β_{sol} (ω)

, where

β (ω) = n_{m n} k

is the linear mode propagation constant and

β_{sol} (ω) = β (ω_{sol}) + β_{1} (ω_{sol}) [ω - ω_{sol}] + \frac{γ P_{c}}{2},

(8)

where

P_{c}

is the self-compressed pulse peak power and

ω_{sol}

is the soliton frequency. In Fig. 11a we plot the propagation constant mismatch [

β (ω) - β (ω_{sol})

] for a range of Ar pressures in a

30 μm

diameter kagomé PCF. We see that the phase-matched points occur in the UV spectral region [82

S.-J. Im, A. Husakou, and J. Herrmann, “High-power soliton- induced supercontinuum generation and tunable sub- $10 fs$ VUV pulses from kagome-lattice HC-PCFs,” Opt. Express 18, 5367–5374 (2010). [CrossRef] [PubMed]

]. Using the results of Subsection 4B on pulse compression,

P_{c} \approx P_{0} F_{c}

[70

C. M. Chen and P. L. Kelley, “Nonlinear pulse compression in optical fibers: scaling laws and numerical analysis,” J. Opt. Soc. Am. B 19, 1961–1967 (2002). [CrossRef]

], and expanding

β (ω)

around

ω_{sol}

, the phase-matching condition reduces to

\sum_{n \geq 2} \frac{(ω - ω_{sol})^{n}}{n!} β_{n} (ω_{sol}) = \frac{γ P_{c}}{2} \sim 2.3 γ P_{0} N .

(9)

In Figs. 11b, 11c we plot this phase-matching condition for two distinct systems (a

10 μm

diameter Xe-filled and a

50 μm

diameter He-filled kagomé PCF) as a function of (

λ_{0}

, N). For an

800 nm

pump pulse, phase-matched solutions can be found at least from 150 to

500 nm

. While the dependence on

λ_{0}

(and hence gas pressure) is strong, the dependence on N is relatively weaker in the UV region, although it becomes more significant as

λ_{0}

approaches

800 nm

and the nonlinear contribution to Eq. (9) becomes more dominant. The similarity between these two plots shows the utility of the scaling analysis presented in Section 3. Noticeable disagreement is only found for extreme phase matching to around

150 nm

, where higher order dispersion becomes important.

Further evidence of this scaling can be seen in Figs. 12a, 12b, which show that both

50 nJ

and

10 μJ

pulses can experience essentially identical spectral evolution along the fiber, including clear UV dispersive-wave emission at

230 nm

, even though they have very different physical parameters.

To illustrate the (as-yet unexplored) degree to which this system may be extended in both wavelength and UV energy, Figs. 12c, 12d show the temporal and spectral evolution of a

10 fs

pulse with

20 μJ

energy propagating in a

50 μm

kagomé PCF filled with

10 bar

He (corresponding to

λ_{0} = 380 nm

N = 3

). After the characteristic pulse self-compression at around

5 cm

, a UV dispersive wave at

150 nm

is emitted, containing over

3.5 μJ

(17.5% of pump energy). Such high energies in the vacuum-UV region, coupled with the inherent wavelength tunability of this system, should lead to numerous applications (Subsection 5C).

5B. Efficiency of Deep-UV Generation

The efficiency of dispersive-wave generation depends strongly on the spectral power density of the compressed pulse at the phase-matched frequency. Tight temporal self-compression is therefore critical (see Subsection 4B) if the initial pump spectrum is to broaden far into the UV. In [49

] it was found that in the kagomé PCF system, self-steepening and optical shock formation [83

D. Anderson and M. Lisak, “Nonlinear asymmetric self-phase modulation and self-steepening of pulses in long optical waveguides,” Phys. Rev. A 27, 1393–1398 (1983). [CrossRef]

], which asymmetrically enhance the blue edge of an SPM broadened spectrum, were the keys to enhancing this process.

What effect does the input pulse duration have? In Fig. 13a we see that

> 15 %

conversion efficiencies to the UV (fraction of total power at wavelengths shorter than

350 nm

) are possible regardless of pulse duration and for a wide range of pump energies. We quantify the quality of the UV emission with the factor

Q_{UV} = P_{FWHM} / P_{UV}

, where

P_{FWHM}

is the spectral power within the FWHM of the strongest UV peak and

P_{UV}

is the total spectral power in the UV region. Figure 13b shows that

Q_{UV}

is strongly dependent on the pump pulse duration. For

15 fs

, over 90% of the UV power is in the main UV peak for a wide range of pump parameters, compared to less than 30% for a

120 fs

pulse. This is clearly evident in Fig. 13c, which shows the spectral evolution of 15, 30, 60, and

90 fs

pulses along the fiber for a normalized soliton order of

S = N / τ_{FWHM} = 0.26

(

N = 7.8

for

30 fs

pulse duration). For the

15 fs

pulse, a high-quality UV band emerges approximately at the soliton fission length [Eq. (6)]. For the

30 fs

pump, the UV band is still of relatively high quality, but at

60 fs

, it degrades considerably, and at

120 fs

, evidence of MI can be observed—the UV band developing a considerably fine structure, as is visible in the spectral slices in Fig. 13d. The reason for this quality degradation is the reduced quality factor of the pulse self-compression for high values of N. Pulse durations of

\sim 30 fs

or shorter are necessary for obtaining high-quality UV spectra at high conversion efficiencies. High-quality conversion is possible with much longer pulses if we reduce the peak power but with significantly lower efficiency. A viable approach for the efficient generation of high-quality UV light from longer pump pulses would be to make use of the fiber-grating/mirror compression system proposed in Subsection 4A, with the UV system as a third stage.

The emitted UV light can be a few optical cycles at the point of generation, but it propagates in a region of normal dispersion, resulting in group velocity broadening. The relatively flat and simple dispersion properties of the kagomé PCF should, however, make it straightforward to externally compress the chirped UV pulses at the output of the fiber. Compression may even be achievable by propagation through an evacuated kagomé PCF, which has anomalous dispersion at all wavelengths.

5C. Applicability of UV Source

These results suggest that an ultrafast coherent light source could be constructed that tunes continuously from 500 to

150 nm

. Whether a further extension deeper into the UV is possible is an open question. Phase matching could be achieved to well below

150 nm

if the pump wavelength is shifted to

400 nm

. But nonlinear absorption and two-photon resonances will disrupt the process at certain wavelengths that depend on the filling gas (in the vacuum-UV for noble gases such as Ar and He). Furthermore, ionization becomes increasingly disruptive to phase-matching at shorter wavelengths [45

The fact that the same tunable UV emission dynamics can be achieved with both

\sim 100 nJ

and

\sim 10 μJ

pulses, with similar efficiencies, promises wide applicability of this technique. Energies of

> 100 nJ

are available from high-repetition-rate chirped oscillators [84

S. Naumov, A. Fernandez, R. Graf, P. Dombi, F. Krausz, and A. Apolonski, “Approaching the microjoule frontier with femtosecond laser oscillators,” New J. Phys. 7, 216 (2005). [CrossRef]

] and ultrafast fiber lasers [85

J. Limpert, S. Hädrich, J. Rothhardt, M. Krebs, T. Eidam, T. Schreiber, and A. Tünnermann, “Ultrafast fiber lasers for strong‐field physics experiments,” Laser Photon. Rev. 5, 634–646 (2011). [CrossRef]

], enabling highly compact UV sources operating at repetition rates suitable for creating deep-UV frequency combs—with numerous applications in spectroscopy. Alternatively, the use of the

\sim 10 μJ

pumped system should enable the generation of deep- UV pulses with energies in excess of

1 μJ

. Such a simple and compact source of spatially coherent, ultrafast deep-UV light has a wide range of potential applications, including femtochemistry [86

A. H. Zewail, “Laser femtochemistry,” Science 242, 1645–1653 (1988). [CrossRef] [PubMed]

] and UV-resonant Raman spectroscopy [87

S. A. Asher, “UV resonance Raman spectroscopy for analytical, physical, and biophysical chemistry. part 1,” Anal. Chem. 65, 59A–66A (1993). [CrossRef]

Another promising application is the seeding of free- electron lasers (FELs) [88

N. Y. Joly, P. Hölzer, J. Nold, W. Chang, J. C. Travers, M. Labat, M.-E. Couprie, and P. St. J. Russell, “New tunable DUV light source for seeding free-electron lasers,” presented at the 33rd Free Electron Laser Conference (FEL 2011), Shanghai, China August 22–26 2011.

], which have the major advantage over traditional laser systems of providing gain over the entire electromagnetic spectrum [89

P. G. O’Shea and H. P. Freund, “Free-electron lasers: status and applications,” Science 292, 1853–1858 (2001). [CrossRef] [PubMed]

]. When seeded, the temporal coherence of the light emitted by an FEL is greatly improved and pulse-to-pulse energy fluctuations are reduced. Conventional lasers have been used to seed FELs in the visible and near IR, but practical seed lasers are unavailable in the UV. Instead, UV seed light has been generated in nonlinear crystals or by HHG in noble gases (see Section 7) [90

G. Lambert, T. Hara, D. Garzella, T. Tanikawa, M. Labat, B. Carre, H. Kitamura, T. Shintake, M. Bougeard, S. Inoue, Y. Tanaka, P. Salieres, H. Merdji, O. Chubar, O. Gobert, K. Tahara, and M.-E. Couprie, “Injection of harmonics generated in gas in a free-electron laser providing intense and coherent extreme- ultraviolet light,” Nat. Phys. 4, 296–300 (2008). [CrossRef]

]. Recent numerical simulations, using the kagomé-based UV source described above as the seed in a single-pass FEL, show that the energy available at

300 nm

is sufficient to achieve saturation (i.e., exhaust the exponential FEL gain) [88

5D. UV Supercontinuum Generation

Raising the soliton order to

N \sim 245

by launching, e.g.,

10 μJ

600 fs

pulses moves the system well into the regime of MI, where smooth and broad (although temporally incoherent) supercontinuum generation can be expected [15

J. M. Dudley, G. Genty, and S. Coen, “Supercontinuum generation in photonic crystal fiber,” Rev. Mod. Phys. 78, 1135–1184 (2006). [CrossRef]

, 91

J. C. Travers, “Blue extension of optical fibre supercontinuum generation,” J. Opt. 12, 113001 (2010). [CrossRef]

]. To illustrate this, Fig. 14 shows propagation of such a long pulse in a kagomé PCF with

λ_{0} = 750 nm

(

25 bar

Ar,

30 μm

core diam eter). Figure 14a shows the temporal evolution of one shot, and Fig. 14b shows the spectral evolution of an ensemble average of 30 simulations (in the MI regime, each laser shot produces a different spectrum, modeled by including quantum noise and averaging over multiple shots). In the temporal picture, the splitting of the input pulse into a large number of ultrashort solitons, that subsequently undergo multiple collisions, can be clearly observed. In the spectral domain, this leads to a smooth and flat, high-energy supercontinuum spanning the range from 350 to

1500 nm

. Note that in the absence of Raman scattering, the MI supercontinuum shows a blue- enhanced asymmetry, in contrast to what is seen in glass-core fibers.

The unique opportunity to tune the ZDW in kagomé PCF into the UV enables us to shift these dynamics to shorter wavelengths. For example, if we pump with the same param eters as above, but at

400 nm

in a fiber designed for zero dispersion at

350 nm

(

8.8 bar

Ar,

10 μm

core diameter), we obtain a very flat and smooth, MI-based, high-energy supercontinuum extending from

140 - 1000 nm

[Figs. 14c, 14d]; experimental realization of such a system would provide a unique source for metrology and spectroscopy.

6. PLASMA-INFLUENCED NONLINEAR FIBER OPTICS

The peak intensities attainable after self-compression of few-microjoule ultrafast pulses in a kagomé PCF can be sufficient to ionize the gas, thus creating a partial plasma inside the fiber. This allows, for the first time, controlled light–plasma interactions in an anomalously dispersive waveguide geometry. In recent experiments, estimated peak intensities of

\sim 2 \times 10^{14} W / {cm}^{2}

were reached, causing plasma generation and a soliton blueshift [5

]. This experiment demonstrated that high-field effects can be observed in kagomé PCF at relatively low pump energies (

< 10 μJ

), compared to other systems.

6A. Comparison with the Kerr Effect

The influence of a partial plasma on pulse propagation is determined by the evolution of the free-electron density, which depends on the time-dependent interaction of the strong electric field with the atoms in the fiber core. Some insight into the strength of the plasma effects can be gained by considering the peak free-electron density throughout pulse propagation. Figure 15a shows the calculated peak free-electron density

N_{e}

in a

30 μm

diameter kagomé PCF as a function of ZDW and soliton order (see Appendix A). Under the conditions of deep-UV generation [marked with (i), Section 5], or of optimum soliton-effect compression [marked with (iii), Subsection 4B], the free-electron densities are below

10^{22} m^{- 3}

. But when the ZDW is shifted to shorter wavelengths, which requires a reduction in gas pressure, the intensity required for soliton formation increases, so that, after self-compression, solitons of order 4 to 10 experience free-electron densities of

\sim 10^{23}

> 10^{24} m^{- 3}

, indicated by the vertical dashed arrow in Fig. 15a.

What is the effect of these free-electron densities? The strong negative polarizability of the free electrons causes a time-dependent reduction in the local refractive index, opposing the increase due to the optical Kerr effect of the noble gas in the visible-IR spectral region. The relative importance of the plasma and Kerr effects can be estimated by considering the ratio between the refractive index changes caused by the two processes [5

, 92

P. Sprangle, J. R. Peñano, and B. Hafizi, “Propagation of intense short laser pulses in the atmosphere,” Phys. Rev. E 66, 046418 (2002). [CrossRef]

R = | \frac{Δ n_{Kerr}}{Δ n_{plasma}} | = n_{2} I \frac{2 n_{0} ω_{0}^{2}}{ω_{p}^{2}},

(10)

where I is the optical intensity;

n_{0}

is the linear refractive index;

ω_{0}

is the pump frequency; and

ω_{p}

is the plasma frequency

ω_{p} = [N_{e} e^{2} / (m_{e} ε_{o})]^{0.5}

, where

ε_{0}

is the vacuum permittivity, e is the electronic charge, and

m_{e}

is the electron mass.

Figure 15b shows R for the same parameters as in Fig. 15a. Over most of the parameter range, the Kerr effect dominates; for example, at free-electron densities of

\sim 10^{22} m^{- 3}

the Kerr effect is over 10 times stronger almost independently of the ZDW (or gas pressure). However at

N_{e} \sim 10^{23} m^{- 3}

, the effects become comparable, and the ratio

R \sim 1

[indicated by the black dashed curve in Fig. 15b]. The experiments by Hölzer et al. [5

] operated in this regime [marked by the vertical dashed arrow in Fig. 15a].

As shown in Fig. 16a, the propagation of

65 fs

pulses in a

26 μm

diameter kagomé PCF filled with

1.7 bar

Ar (corresponding to a ZDW at

380 nm

) leads to the generation of blueshifted sidebands. The first one occurs at a pump energy of

\sim 2 μJ

. Detailed investigations have shown that extended, and more intricate, light–plasma interactions occur in this system [5

, 45

], for example, multiple compression points and the subsequent emission of additional blueshifted sidebands in Fig. 16a. Comparison with full propagation simulations [Fig. 16b] shows excellent agreement only when the ionization terms are included, and analysis of transmission losses also show excellent agreement only when photoionization- induced losses are accounted for [Fig. 16c].

6B. Soliton Blueshift

Because the creation of free electrons is highly nonlinear with pulse intensity, a fast transient change in the refractive index is created by few-cycle pulses, which does not recover on ultrafast time scales due to the slow rate of electron recombination (Appendix A). This creates a steep positive phase-shock across the optical pulse and induces a blueshift, as has been thoroughly investigated in bulk geometries [93

S. C. Rae and K. Burnett, “Detailed simulations of plasma- induced spectral blueshifting,” Phys. Rev. A 46, 1084–1090 (1992). [CrossRef] [PubMed]

, 94

S. P. Le Blanc, R. Sauerbrey, S. C. Rae, and K. Burnett, “Spectral blue shifting of a femtosecond laser pulse propagating through a high-pressure gas,” J. Opt. Soc. Am. B 10, 1801–1809 (1993). [CrossRef]

, 95

W. M. Wood, C. W. Siders, and M. C. Downer, “Femtosecond growth dynamics of an underdense ionization front measured by spectral blueshifting,” IEEE Trans. Plasma Sci. 21, 20–33 (1993). [CrossRef]

]. Results by Saleh et al. [96

M. Saleh, W. Chang, P. Hölzer, A. Nazarkin, J. C. Travers, N. Joly, P. St. J. Russell, and F. Biancalana, “Theory of photoionization-induced blueshift of ultrashort solitons in gas-filled hollow-core photonic crystal fibers,” Phys. Rev. Lett. 107, 203902 (2011). [CrossRef] [PubMed]

] show that solitons propagating in a kagomé PCF undergo a self-frequency blueshift under these conditions, opposite in sign to the well-known redshift caused by intrapulse Raman scattering and that when combined, the two shifts can cancel out. To illustrate this, Saleh et al. derived an equation which extends the conventional GNLSE to account for free-electron effects on the propagation of the complex field envelope

ψ (z, t)

[96

\partial_{z} ψ = i [\sum_{m \geq 2} \frac{β_{m} (i \partial_{t})^{m}}{m!} + γ R (t) \otimes | ψ (t) |^{2} - \frac{ω_{p}^{2}}{2 ω_{0} c} + i \frac{A_{eff} I_{p} \partial_{t} N_{e}}{2 | ψ |^{2}}] ψ,

(11)

where z is the longitudinal coordinate along the fiber, t is the time in a reference frame moving at the group velocity,

R (t)

is the normalized Kerr and Raman response function of the gas,

I_{p}

is the ionization energy of the gas, and

N_{e} (t)

is the temporally varying free-electron density that is modeled with an additional coupled equation, Eq. (A1) (Appendix A).

The first two terms on the right-hand side are the conventional dispersion and nonlinear operators, well known in nonlinear fiber optics [7

G. Agrawal, Nonlinear Fiber Optics , 4th ed. (Academic, 2006).

]. The last two terms are novel in the context of fiber optics (although well known in other communities), and represent phase modulation caused by free electrons and photoionization-induced losses. It is worth noting that this is the first major addition to the GNLSE for over two decades, introducing significant new dynamics into the already rich field of nonlinear fiber optics. In a noble gas-based system without Raman scattering, Eq. (11) simplifies further, allowing the dynamics to be considered as weakly perturbed soliton solutions at certain peak power levels, allowing the derivation of analytical solutions and consequently improving physical insight [96

One such analytical solution describes the continuous frequency blueshift of a soliton propagating with just sufficient peak intensity to ionize the medium. Figure 17 shows numerical simulations of an

N = 4

soliton propagating in a

30 μm

diameter Ar-filled kagomé PCF with a ZDW at

\sim 380 nm

, corresponding to the parameters of the experimental results shown in Fig. 16. A clear blueshift is seen in the spectrum after

\sim 12 cm

of propagation, confirming the analysis of Saleh et al. and allowing us to attribute the blueshifted sidebands in Fig. 16 to the presence of blueshifting solitons.

6C. Prospects for Plasma Generation in Fiber

Although the effects of partial plasmas have been studied in many other systems, the dispersion and guiding properties of kagomé PCF introduce unique possibilities. In capillary fibers, the lack of anomalous dispersion at reasonable pressures prevents the observation of soliton effects (see Section 2). In glass-core systems, ionization is synonymous with material damage. In bandgap fibers, although small plasma blueshifts (a few nanometers) have been observed [97

E. E. Serebryannikov and A. M. Zheltikov, “Ionization-induced effects in the soliton dynamics of high-peak-power femtosecond pulses in hollow photonic-crystal fibers,” Phys. Rev. A 76, 013820 (2007). [CrossRef]

, 98

A. B. Fedotov, E. E. Serebryannikov, and A. M. Zheltikov, “Ionization-induced blueshift of high-peak-power guided-wave ultrashort laser pulses in hollow-core photonic-crystal fibers,” Phys. Rev. A 76, 053811 (2007). [CrossRef]

], the restricted guidance band and strong dispersion limit the soliton effects. Finally, in filaments in free space, stable self-guidance requires the free-electron density to be fixed relative to the Kerr nonlinearity [92

P. Sprangle, J. R. Peñano, and B. Hafizi, “Propagation of intense short laser pulses in the atmosphere,” Phys. Rev. E 66, 046418 (2002). [CrossRef]

, 99

L. Bergé, S. Skupin, R. Nuter, J. Kasparian, and J.-P. Wolf, “Ultrashort filaments of light in weakly ionized, optically transparent media,” Rep. Prog. Phys. 70, 1633–1713 (2007). [CrossRef]

], again limiting the range of possible dynamics. In contrast, kagomé PCF offers an elegant and simple means to study light–plasma interactions in a well-controlled transverse mode and over extended length scales.

It is likely that the results described above are merely the beginning of a new chapter on nonlinear single-mode fiber optics in the presence of free electrons. A number of directions are possible at such high in-fiber intensities. It would be worthwhile to study ionization in kagomé PCF filled with different gas types. For example, the rate of ionization in a gas with closely spaced ionization energies, such as

{SF}_{6}

, appears to modify the propagation dynamics [45

]. Also, the possibility of a third-order nonlinearity resulting from the free electrons was recently proposed [100

C. Rodríguez, Z. Sun, Z. Wang, and W. Rudolph, “Characterization of laser-induced air plasmas by third harmonic generation,” Opt. Express 19, 16115–16125 (2011). [CrossRef] [PubMed]

], and it could be studied as a means of enhancing third-harmonic generation in fiber. Alternatively, Raman-like interactions with plasmas [101

V. M. Malkin, G. Shvets, and N. J. Fisch, “Fast compression of laser beams to highly overcritical powers,” Phys. Rev. Lett. 82, 4448–4451 (1999). [CrossRef]

], used for high-pulse-energy amplification and compression, could be transferred to a fiber context. A further degree of freedom in plasma-influenced propagation in kagomé PCF would result if the plasma was generated with an external excitation source, such as electrodes [102

X. Shi, X. B. Wang, W. Jin, and M. S. Demokan, “Investigation of glow discharge of gas in hollow-core fibers,” Appl. Phys. B 91, 377–380 (2008). [CrossRef]

] or direct microwave excitation [103

B. Debord, R. Jamier, F. Gérôme, C. Boisse-Laporte, P. Leprince, O. Leroy, J.-M. Blondy, and F. Benabid, “UV light generation induced by microwave microplasma in hollow-core optical waveguides,” in CLEO:2011—Laser Science to Photonic Applications (Optical Society of America, 2011), paper CThD5.

], effectively removing the need for very high optical intensities in the gas. This would make possible the study of longer pulses or even CW light in a plasma-filled kagomé PCF, when ponderomotive effects could become important.

7. HIGH-HARMONIC GENERATION

A key driver in the development of few-cycle pulsed lasers has been HHG [104

A. McPherson, G. Gibson, H. Jara, U. Johann, T. S. Luk, I. A. McIntyre, K. Boyer, and C. K. Rhodes, “Studies of multiphoton production of vacuum-ultraviolet radiation in the rare gases,” J. Opt. Soc. Am. B 4, 595–601 (1987). [CrossRef]

, 105

P. B. Corkum, “Plasma perspective on strong field multiphoton ionization,” Phys. Rev. Lett. 71, 1994–1997 (1993). [CrossRef] [PubMed]

, 106

T. Popmintchev, M.-C. Chen, P. Arpin, M. M. Murnane, and H. C. Kapteyn, “The attosecond nonlinear optics of bright coherent x-ray generation,” Nat. Photon. 4, 822–832 (2010). [CrossRef]

], which offers a means of producing coherent extreme-UV and x-ray radiation in the laboratory and is the foundation technology of attosecond science [107

P. B. Corkum, N. H. Burnett, and M. Y. Ivanov, “Subfemtosecond pulses,” Opt. Lett. 19, 1870–1872 (1994). [CrossRef] [PubMed]

, 108

G. Sansone, E. Benedetti, F. Calegari, C. Vozzi, L. Avaldi, R. Flammini, L. Poletto, P. Villoresi, C. Altucci, R. Velotta, S. Stagira, S. De Silvestri, and M. Nisoli, “Isolated single-cycle attosecond pulses,” Science 314, 443–446 (2006). [CrossRef] [PubMed]

, 109

E. Goulielmakis, M. Schultze, M. Hofstetter, V. S. Yakovlev, J. Gagnon, M. Uiberacker, A. L. Aquila, E. M. Gullikson, D. T. Attwood, R. Kienberger, F. Krausz, and U. Kleineberg, “Single-cycle nonlinear optics,” Science 320, 1614–1617 (2008). [CrossRef] [PubMed]

]. Given the discussions above on compression to few optical cycles and the generation of free electrons, it is natural to ask what the prospects are for HHG in kagomé PCF.

7A. HHG in HC-PCF

One preliminary demonstration has been reported using a kagomé PCF (core diameter

15 μm

) filled with

30 mbar

of Xe [110

O. H. Heckl, C. R. E. Baer, C. Kränkel, S. V. Marchese, F. Schapper, M. Holler, T. Südmeyer, J. S. Robinson, J. W. G. Tisch, F. Couny, P. Light, F. Benabid, and U. Keller, “High harmonic generation in a gas-filled hollow-core photonic crystal fiber,” Appl. Phys. B 97, 369–373 (2009). [CrossRef]

]. A conversion efficiency of

10^{- 9}

up to the thirteenth harmonic was observed, which is several orders of magnitude below the non-phase-matched conversion efficiencies observed in free space. The threshold energy, however, was only

200 nJ

, whereas millijoule pulses are usually used with gas jets—a result of the intensity and interaction length enhancement possible in kagomé PCF. Although this experiment is useful as a proof of principle, higher efficiencies are desirable. Additionally, no use was made of pump pulse self-compression and nonlinear phase matching—uniquely offered by kagomé PCF—to enhance the process.

It is well established that the conversion efficiency to high harmonics can be significantly enhanced by phase-matching techniques [106

T. Popmintchev, M.-C. Chen, P. Arpin, M. M. Murnane, and H. C. Kapteyn, “The attosecond nonlinear optics of bright coherent x-ray generation,” Nat. Photon. 4, 822–832 (2010). [CrossRef]

]; for example efficiencies of up to

\sim 10^{- 5}

have been obtained to individual harmonics with photon energies of

\sim 45 eV

(

\sim 28 nm

) [25

]. The use of capillary waveguides, rather than gas jets, adds an extra degree of tunability to achieve phase matching [106

T. Popmintchev, M.-C. Chen, P. Arpin, M. M. Murnane, and H. C. Kapteyn, “The attosecond nonlinear optics of bright coherent x-ray generation,” Nat. Photon. 4, 822–832 (2010). [CrossRef]

] and naturally transfers to HC-PCF; indeed, several techniques have been proposed for phase-matched HHG in PBG-guiding HC-PCF [111

E. E. Serebryannikov, D. von der Linde, and A. M. Zheltikov, “Phase-matching solutions for high-order harmonic generation in hollow-core photonic-crystal fibers,” Phys. Rev. E 70, 066619 (2004). [CrossRef]

, 112

E. E. Serebryannikov, D. von der Linde, and A. M. Zheltikov, “Broadband dynamic phase matching of high-order harmonic generation by a high-peak-power soliton pump field in a gas-filled hollow photonic-crystal fiber,” Opt. Lett. 33, 977–979 (2008). [CrossRef] [PubMed]

]. However, the principal difficulty of phase matching to high harmonics in any waveguide geometry arises from the strong free-electron dispersion, which is unavoidable at the intensities required for very high order harmonics.

7B. Phase Matching with a Counterpropagating Wave

One elegant solution to this problem is the use of quasi-phase-matching with a counterpropagating laser beam [113

S. L. Voronov, I. Kohl, J. B. Madsen, J. Simmons, N. Terry, J. Titensor, Q. Wang, and J. Peatross, “Control of laser high- harmonic generation with counterpropagating light,” Phys. Rev. Lett. 87, 133902 (2001). [CrossRef] [PubMed]

]. In this approach the counterpropagating beam modulates the driving laser intensity. As a result, the phase acquired by the freed electron—which is determined by its trajectory under the influence of the pump laser field before it recombines with the atom—is also modulated, leading to phase-modulation of the HHG emission. Tuning the frequency and intensity of the counterpropagating beam therefore provides a simple means of controlling the phase matching. In capillary waveguides, for phase matching to very high photon energies, this technique would, however, require an intense laser operating at wavelengths in the far-IR. A theoretical analysis of trans ferring this technique to HC-PCF was reported in [114

H. Ren, A. Nazarkin, J. Nold, and P. St. J. Russell, “Quasi-phase-matched high harmonic generation in hollow core photonic crystal fibers,” Opt. Express 16, 17052–17059 (2008). [CrossRef] [PubMed]

]. It suggested that using a correctly tuned counterpropagating laser at

1.6 μm

(an easily attainable laser wavelength), one could phase match to the

\sim 99

th harmonic using a driving laser at

800 nm

with only a few microjoules of pulse energy.

7C. Pumping with Ultrashort Mid-IR Pulses

There is ongoing work to create compound-glass-based kagomé PCFs, which should allow for low-loss optical guidance in the mid-IR spectral region [115

X. Jiang, T. G. Euser, A. Abdolvand, F. Babic, F. Tani, N. Y. Joly, J. C. Travers, and P. St. J. Russell, “Single-mode hollow-core photonic crystal fiber made from soft glass,” Opt. Express 19, 15438–15444 (2011). [CrossRef] [PubMed]

]. This region is well known as being advantageous for extending the range of high harmonics that can be produced to the kiloelectron volt region, because the longer optical half-cycle means that the released electron can be accelerated to higher energy before it recombines with the atom. Although single-atom HHG efficiencies are lower when pumped at longer wavelengths, the combination of reduced ionization due to relaxed intensity requirements (and hence easier phase-matching) and higher gas transparency for high-energy electrons, means that overall efficiencies can be enhanced [106

T. Popmintchev, M.-C. Chen, P. Arpin, M. M. Murnane, and H. C. Kapteyn, “The attosecond nonlinear optics of bright coherent x-ray generation,” Nat. Photon. 4, 822–832 (2010). [CrossRef]

]. The possibility of transferring the advantages of kagomé PCF to HHG driven by mid-IR pulses is very attractive. This might allow few-cycle pulses created through self-compression to be used for phase-matched HHG in kagomé PCF, thus providing an all-fiber table-top x-ray (and possibly attosecond) light source.

8. CONCLUSIONS

HC-PCFs allow one to do single-mode nonlinear fiber optics in cores made from gases, vapors, and plasmas. Gas-filled kagomé PCF, in particular, has almost perfect dispersion and loss properties for ultrafast nonlinear fiber optics along with the ability to guide high intensities without glass damage. This opens the door to the combination of high-field physics with more conventional nonlinear fiber processes. Varying the combination of filling gas, pressure, and PCF design provides flexibility in both the operation wavelength and the energy scale of the desired system. Additionally, kagomé PCF extends the region of fiber-optic operation (both low-loss and zero-dispersion points) deep into the UV.

Few-cycle pulse compression is possible using either a fiber-grating/chirped-mirror scheme, or by soliton self- compression, with energies covering at least the

0.1 - 30 μJ

range. With the correct dispersion profile, self-compression can lead to a highly efficient and compact, tunable deep-UV source, tuning between

200 - 350 nm

having been demonstrated, and

150 - 500 nm

being realistically feasible. UV energies

> 1 μJ

, together with a threshold energy low enough to allow direct pumping from a high-repetition-rate oscillator, will enable the creation of an all-fiber deep-UV frequency comb. Many applications seem possible in spectroscopy and metrology or even in the seeding of a free-electron laser.

At few-microjoule pump energies, ultrafast pulses can self-compress to intensities over the ionization threshold of the filling gas, allowing, for the first time, controllable, extended, single-mode laser-plasma interactions in an anomalously dispersive fiber. This breaks new ground in nonlinear fiber optics, dramatically increasing the range of phenomena possible in fibers beyond those possible in solid-core fibers, including, for example, a soliton self-frequency blueshift. Finally, phase-matched HHG in kagomé PCF may lead to the creation of an all-fiber table-top x-ray light source.

Appendices

APPENDIX A

All of the numerical results presented in this paper are calculated using a unidirectional field equation [45

], including the full dispersion curve based on Eq. (1), the Kerr effect, and the influence of free electrons created through photoionization.

In these calculations, the free electron density is given by [92

P. Sprangle, J. R. Peñano, and B. Hafizi, “Propagation of intense short laser pulses in the atmosphere,” Phys. Rev. E 66, 046418 (2002). [CrossRef]

]

\frac{\partial N_{e}}{\partial t} = W (t) N_{a} - η N_{e} - β_{r} N_{e}^{2},

(A1)

where

W (t)

is the ionization rate of the atoms in the presence of the time varying electric field

E (z, t)

N_{a}

is the density of neutral atoms, and η and

β_{r}

are the electron attachment and recombination rates, respectively, both of which are negligible for durations

< \sim 1 ps

[92

P. Sprangle, J. R. Peñano, and B. Hafizi, “Propagation of intense short laser pulses in the atmosphere,” Phys. Rev. E 66, 046418 (2002). [CrossRef]

]. The calculation of the free- electron density

N_{e} (t)

requires the use of a model for the ionization rate

W (t)

. In this work, we used the Yudin–Ivanov modification of the Perelomov–Popov–Terent’ev technique [116

G. L. Yudin and M. Y. Ivanov, “Nonadiabatic tunnel ionization: looking inside a laser cycle,” Phys. Rev. A 64, 013409 (2001). [CrossRef]

, 117

V. S. Popov, “Tunnel and multiphoton ionization of atoms and ions in a strong laser field (Keldysh theory),” Phys. Usp. 47, 855–885 (2004). [CrossRef]

], which accounts for both tunnel- and multiphoton-based ionization; although it should be noted that simulations using the formulation in [45

] have accurately recreated experimental results discussed in this review when using the Ammosov–Delone–Krainov (ADK) rate [118

M. Ammosov, N. Delone, and V. Krainov, “Tunnel ionization of complex atoms and of atomic ions in an alternating electromagnetic field,” Sov. Phys. JETP 64, 1191–1194 (1986).

], indicating that the parameters chosen where within the tunnel ionization regime [5

, 45

The full free-electron density calculations using Eq. (A1) and the associated models for

W (t)

are not straightforwardly amenable to analytical manipulation. Therefore, Saleh et al. [96

] used a simplified model to derive their results, based on a linear approximation to the ionization rate, valid in a restricted range of intensities above the ionization threshold. This works surprisingly well due to the self-limiting of pulse peak intensities resulting from photoionization-induced optical loss.

ACKNOWLEDGMENTS

The authors thank K. F. Mak and P. Hölzer for providing some of the materials for the figures and for useful discussions and suggestions.

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Table 1 Examples of Parameters Describing Certain Sets of Propagation Dynamics (as Described in the Text) Scaled across Different Core Diameters (d), Filling Gases and Gas Pressures (P), and Energies (E)

System^{^a}	$λ_{0} / nm$	N	He			Ar			Xe
System^{^a}	$λ_{0} / nm$	N	$d / μm$	$P / bar$	$E / μJ$	$d / μm$	$P / bar$	$E / μJ$	$d / μm$	$P / bar$	$E / μJ$
Efficient UV emission (i)	600	9	50	69	6.6	30	10	2	10	18	0.14
Plasma blueshift (ii)	380	4	50	10	12.6	30	1.4	3.8	10	2.4	0.28
Fiber-grating/mirror compression	1300	14^{^b}	500^{^c}	16^{^c}	612^{^c}	70	42	10	40	27	1.9
Soliton-effect compression (iii)	500	3.5	50	33	1.7	30	6.8	0.36	10	12.4	0.025

^a The roman numerals in the system column correspond to the labeled dots in Figs. 7, 15.

^b Bright solitons only occur in regions of anomalous dispersion, but we use the soliton order in this case for scaling relative nonlinearity to normal dispersion in different systems.

^c These parameters are for capillary fiber rather than kagomé PCF and are shown merely for comparison.

Fig. 1 Scanning electron micrographs (SEMs) and FEM of the two main types of HC-PCF: (a) SEM of PBG-guiding HC-PCF, (b) GVD and loss calculated using FEM of an idealized PBG-guiding HC-PCF structure, designed for operation around

800 nm

, with

11 μm

core diameter and

2.1 μm

pitch (Λ), (c) SEM of a kagomé PCF designed for operation in the UV and around

800 nm

, and (d) GVD and loss calculated using the FEM of an idealized kagomé PCF structure with

30 μm

core diameter,

15 μm

pitch (Λ), and

0.23 μm

web thickness.

Fig. 2 Comparison between the FEM and the capillary model: (a) dispersion of the effective mode index and (b) the GVD. (a) Inset, definitions of the two core radii: flat-to-flat radius (

a_{f}

) and area preserving radius (

a_{A}

); A is the hexagonal core area. The kink at

\sim 380 nm

in the FEM results is caused by an anticrossing with a cladding state, and it strongly affects the dispersion.

Fig. 3 (a) Comparison between the lowest theoretical loss possible in a capillary fiber and the measured loss of a kagomé PCF with a core diameter of

30 μm

, (b) dispersion curves for silica-core PCFs (silica strands) with the shortest ZDWs, with core diam eters of 0.47 to

2.86 μm

(

0.2 μm

steps), and (c) the dispersion of a kagomé PCF with a

30 μm

core diameter, filled with 0 to

20 bar

Ar (

2 bar

steps)—note the change of scale compared to (b).

Fig. 4 ZDW scaling for kagomé PCFs with diameters from 10 to

70 μm

(steps of

10 μm

), filled with 0 to

60 bar

of (a) Xe, (b) Kr, (c) Ar, and (d) He.

Fig. 5 FEM calculations for an idealized kagomé PCF with

30 μm

core diameter,

15 μm

pitch, and

0.23 μm

web thickness—designed for operation in the UV and around

800 nm

: (a) wavelength dependence of the fraction of power in glass, (b) axial component of the Poynting vector of the guided mode at the wavelength of a cladding resonance (

380 nm

), and (c) the same as (b) at

800 nm

Fig. 6 Experimental setup typically used for ultrafast nonlinear experiments in gas-filled kagomé PCF.

Fig. 7 (a)–(c) Contour plots of the pump energy (

μJ

) required for formation of

30 fs

solitons at

800 nm

, plotted against soliton order and ZDW, the ZDW being adjusted by varying the gas pressure: (a)

10 μm

diameter filled with Xe, (b)

30 μm

diameter filled with Ar, and (c)

50 μm

diameter filled with He. The entire parameter spaces would be severely loss-limited in a capillary fiber. The blue region (lower, shaded region) is where the estimated loss of the kagomé PCF would limit nonlinear interaction. (d) Contour plots of the ratio of the peak power to critical self-focusing power for pulses propagating in a kagomé PCF (

30 μm

core diameter, filled with Ar), plotted against soliton order and ZDW, including pulse self- compression effects (discussed in Subsection 4B). The labeled dots correspond to the systems defined in Table 1.

Fig. 8 Fiber-grating/mirror pulse compression with kagomé PCF; (a) density plots of the temporal and spectral evolution of a

10 μJ

30 fs

pulse at

800 nm

through a kagomé PCF with

70 μm

core diameter filled with

43 bar

of Ar, placing the ZDW at

1300 nm

; (b) the broadened temporal shape of the intensity, and the smooth parabolic temporal phase after

8 cm

of propagation in the kagomé PCF [dashed line in (a)]; (c) pulse duration after linear compression against the kagomé PCF length used for spectral broadening (left axis), and compression quality factor (right axis); (d) pulse after linear compression of (b).

Fig. 9 Soliton-effect self-compression at

800 nm

in a kagomé PCF with a

30 μm

diameter core filled with Ar: (a) dependence of the compressed pulse duration on

λ_{0}

(horizontal axis) and N (numbered curves), (b) corresponding quality factor dependence, (c) temporal and spectral evolution of self-compression of a pulse for

λ_{0} = 500 nm

N = 3.5

, and (d) field of the compressed output pulse at the point of optimum compression in (c).

Fig. 10 Deep-UV dispersive-wave generation; (a) a series of experimentally achieved tunable output spectra [77

] from an Ar-filled,

27 μm

core diameter kagomé PCF (

45 fs

pulses at

800 nm

with

0.5 - 3 μJ

pulse energy and

4 - 12 bar

filling pressure) and (b) reported UV conversion efficiencies [49

Fig. 11 (a) Propagation constant mismatch between solitons (

N = 9

800 nm

) and dispersive waves at a given wavelength for a

30 μm

kagomé PCF filled with 2 to

20 bar

Ar (

2 bar

steps); (b), (c) phase-matching wavelengths for (b) a

10 μm

diameter Xe-filled fiber, and (c) for a

50 μm

diameter He-filled fiber, assuming pressure- tunable ZDW

λ_{0}

Fig. 12 (a), (b) Spectral evolution of two systems, both with

λ_{0} = 563 nm

and

N = 7.5

, over two fission lengths [Eq. (5)]: (a) a

50 nJ

30 fs

pulse through

0.68 cm

6 μm

diameter kagomé PCF filled with

38 bar

Xe and (b)

10 μJ

30 fs

pulse through

68 cm

60 μm

diameter fiber filled with

36 bar

He. (c), (d) Temporal and spectral evolution of a

10 fs

pulse with

20 μJ

energy propagating in a

50 μm

kagomé PCF with

10 bar

He (corresponding to

λ_{0} = 380 nm

N = 3

Fig. 13 Dependence of UV generation on pulse duration (15, 30, 60, and

120 fs

) and soliton order for a fiber with

λ_{0} = 600 nm

(kagomé PCF with

30 μm

core filled with

9.8 bar

Ar): (a) UV conversion efficiency (to wavelengths shorter than

350 nm

) as a function of normalized soliton order

S = N / τ_{FWHM}

, corresponding to

N = 3

to 9 for a

30 fs

pulse; (b) quality factor of UV conversion as defined in the text for the same parameters as (a); (c) spectral evolution along the fiber for each of the pulse durations at

S = 0.26

; and (d) spectral slices at the position of optimum UV conversion for each of the pulse durations.

Fig. 14 (a), (b) Propagation of an

N \sim 245

(

600 fs

10 μJ

) pulse at

800 nm

in a kagomé PCF with

λ_{0} = 750 nm

(

30 μm

core diameter filled with

25 bar

Ar): (a) temporal evolution of one shot and (b) spectral evolution of an ensemble average of 30 simulations. (c), (d) Propagation of an

N \sim 429

(

600 fs

10 μJ

) pulse at

400 nm

in a kagomé PCF with

λ_{0} = 350 nm

(

10 μm

core diameter filled with

8.8 bar

Ar): (c) temporal evolution of one shot and (d) spectral evolution of an ensemble average of 30 simulations.

Fig. 15 Contour plot of (a) the calculated free- electron density (

m^{- 3}

) in a

30 μm

diameter kagomé PCF filled with Ar, as a function of ZDW and soliton order, taking into account the self-compression of the solitons upon propagation; the vertical dashed arrow indicates the operation region of Hölzer et al. [5

] and (b) the nonlinear refractive index ratio R (defined in the text); the black dashed line indicates a ratio of 1.

Fig. 16 Results of Hölzer et al. [5

]: (a) experimental output spectra from a

26 μm

diameter kagomé PCF filled with

1.7 bar

Ar as a function of input pulse energy, (b) corresponding numerical simulations, and (c) comparison of the transmission between the experiment and the numerical simulations with and without the ionization terms included.

Fig. 17 Numerical simulation of the propagation of an

N = 4

soliton through a

30 μm

diameter kagomé PCF with

1.7 bar

Ar filling pressure (ZDW at

380 nm

OCIS Codes
(190.4370) Nonlinear optics : Nonlinear optics, fibers
(190.7110) Nonlinear optics : Ultrafast nonlinear optics
(260.5210) Physical optics : Photoionization
(060.5295) Fiber optics and optical communications : Photonic crystal fibers

History
Original Manuscript: October 4, 2011
Manuscript Accepted: October 11, 2011
Published: November 30, 2011

Virtual Issues
(2011) Advances in Optics and Photonics
February 24, 2012 Spotlight on Optics

Citation
John C. Travers, Wonkeun Chang, Johannes Nold, Nicolas Y. Joly, and Philip St. J. Russell, "Ultrafast nonlinear optics in gas-filled hollow-core photonic crystal fibers [Invited]," J. Opt. Soc. Am. B 28, A11-A26 (2011)
http://www.opticsinfobase.org/josab/abstract.cfm?URI=josab-28-12-A11

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Journal of the Optical Society of America B

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Ultrafast nonlinear optics in gas-filled hollow-core photonic crystal fibers [Invited]

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Abstract

1. INTRODUCTION

2. PROPERTIES OF HC-PCF

2A. Dispersion and Loss of Gas-Filled Kagomé HC-PCF

2B. Nonlinearity of Gas-Filled Kagomé PCF

2C. Energy Handling of Kagomé HC-PCF

3. ULTRAFAST NONLINEAR DYNAMICS IN KAGOMÉ PCF

3A. Interplay of Dispersion and Nonlinearity

3B. Energy Scaling of Solitons: Dependence on the Dispersion Landscape

4. PULSE COMPRESSION

4A. Fiber-Grating/Mirror Compression

4A1. Illustrative Design

4B. Soliton-Effect Compression

4B1. Compression Ratio and Quality Factor Scaling

4B2. Effect of Longer Pulses

4C. Adiabatic Pulse Compression

5. DISPERSIVE-WAVE GENERATION IN THE UV

5A. Phase Matching to Dispersive Waves

5B. Efficiency of Deep-UV Generation

5C. Applicability of UV Source

5D. UV Supercontinuum Generation

6. PLASMA-INFLUENCED NONLINEAR FIBER OPTICS

6A. Comparison with the Kerr Effect

6B. Soliton Blueshift

6C. Prospects for Plasma Generation in Fiber

7. HIGH-HARMONIC GENERATION

7A. HHG in HC-PCF

7B. Phase Matching with a Counterpropagating Wave

7C. Pumping with Ultrashort Mid-IR Pulses

8. CONCLUSIONS

Appendices

APPENDIX A

ACKNOWLEDGMENTS

References and links

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