## Abstract

The potential of hollow core photonic crystal fiber as a nonlinear gas cell for efficient high harmonic generation is discussed. The feasibility of phase-matching this process by modulating the phase of ionization electrons using a counter-propagating laser field is shown. In this way, harmonics with energies of several hundreds of eV can be produced using fs-laser pump pulses of *µ*J energy.

©2008 Optical Society of America

## 1. Introduction

High-order harmonic generation (HHG) is a unique source of extreme ultraviolet and soft x-ray radiation that can be highly spatially and temporally coherent [1]. It has led to many new applications, such as attosecond pulse generation, lithography, high-resolution imaging, site- and element-specific spectroscopy and bio-microscopy [2,3]. In HHG, the laser field first ionizes the atom or molecule of a gas, then accelerates the liberated electron, finally generating high-order harmonic photons when the field reverses and the electron recollides with its parent ion. The highest photon energy (the cutoff energy) which can be produced by this process is given by *E*
* _{c}*=

*I*

*+3.17*

_{p}*U*

*, where*

_{p}*I*

*is the ionization potential of the atom and*

_{p}*U*

*is the ponderomotive energy of the electron in the laser field [4]. Because the HHG process is mediated by ionization electrons, the gas of free electrons produced by a laser field introduces a phase mismatch which changes the phases of the pump and harmonic field, leading to low (10*

_{p}^{-8}) conversion efficiencies. Since higher harmonic orders are generated at higher intensities (after a significant part of the gas is ionized) this phase mismatch limits the highest achievable photon energy. Much effort has been devoted to improving the efficiency of HHG. Considerable progress was made several years ago with the implementation of gas-filled capillaries. In capillaries, phase-matching is achieved by properly balancing the contribution of the neutral atoms and free electrons to the propagation constant. This is possible if the ionization fraction is below a critical level. As a result, the highest harmonic energy reported so far has been limited to an energy of 130eV, with a conversion efficiency into a single harmonic peak of about 10

^{-5}[5]. Although implementation of quasi-phase-matching by periodically modulating the capillary bore diameter can break this limitation and extend the photon energy into the water window, the efficiency of quasi-phase-matched HHG is still quite low [6] due to high leakage loss of light from the capillary.

In recent years, there has been much progress in the development of novel optical guiding systems – photonic crystal fibers (PCF). Hollow core (HC-PCF) filled with gas has attracted particular interest and has led to a number of breakthroughs in nonlinear optics [7]. In HC-PCF, light is guided in a small hollow core region due to high reflectivity within the photonic bandgaps of the surrounding photonic crystal cladding within photonic bandgaps. Because of the low leakage loss of HC-PCF, ultra-long interaction lengths in the regime of tightly confined high-intensity laser light have become possible, thus offering completely new possibilities for strong field nonlinear optics [8–12]. The aim of this paper is to discuss the specific features of HHG in gas-filled HC-PCF and explore the potential of these novel guiding systems for efficient conversion of laser radiation to the extreme UV and soft x-ray regions. We show that gas-filled HC-PCFs offer interesting possibilities for phase-matched HHG by fs pulse energies several orders of magnitude lower than that typically used in capillaries. The feasibility of quasi-phase-matching this process in HC-PCF by modulating the phase of the ionization electrons with a counter-propagating laser field (as recently reported in [20]) is examining. We show that harmonics with photon energies of several hundreds of eV could be produced in this way from fs-laser sources of µJ energy. It is suggested that, by exploiting the longer interaction lengths of a low loss fiber and using low gas pressures to minimize the influence of space charge effects, the efficiency of this process can be increased. This is possible because in the soft x-ray region, the relation between the refractive index of the core and the cladding of a PCF is such that x-rays can be guided by total internal reflection, though the loss level attainable in practice will depend on the surface roughness. Phase matching of a fundamental mode of the pump field and higher order guided modes of its harmonics has been analysed and shown to be feasible for special regimes of HHG in HC-PCF [26].

## 2. Phase mismatch

The wavevector mismatch in HHG is given by Δ*k*=*β*
_{m}(*λ*
_{q})-_{q}
*β*
_{n}(*λ*
_{0}), where *β*
_{n}(*λ*
_{0}) and *β*
_{m}(*λ*
_{q}) are the propagation constants of the driving laser field and the harmonic field in HCPCF. By analogy with HHG in gas-filled capillary [5], one can derive the following expression for the wavevector mismatch in gas-filled HC-PCF:

$$q{k}_{0}{N}_{\text{a}}\left[\delta \left({\lambda}_{0}\right)-\delta \left({\lambda}_{\text{q}}\right)\right]\u2044{N}_{0}-q{k}_{0}\left(1-\eta \right){n}_{2}I$$

Here the first term (Δ*k*
* _{w}*=

_{q}*k*

_{0}[

*n*(λ

_{q})-

*n*

_{eff}]=

_{q}*k*

_{0}Δ

*n*, where

*n*

_{eff}=β

_{n}(λ

_{0})/

*k*

_{0}), describes the contribution of the HC-PCF dispersion, the second and third terms result from the dispersion of plasma Δ

*k*

_{p}and neutral gas Δ

*k*

_{g}, while the last term Δ

*k*

_{n}comes from the nonlinear refractive index of the gas. The wavelength of the driving laser is λ

_{0},

*k*

_{0}=2

*π*/λ

_{0}is the wavevector,

*q*is the harmonic order, λ

_{q}=λ

_{0}/

*q*is the

*q*-th order harmonic wavelength,

*N*

_{e}and

*N*

_{a}are the number densities of free electrons and neutral atoms, respectively,

*η*is the ionization fraction,

*N*

_{0}is Avogadro’s constant,

*I*is the driving laser intensity and

*n*

_{2}the nonlinear refractive index of the gas. Also,

*r*

_{e}=

*e*

^{2}/(

*m*

_{e}

*c*

^{2}) is the classical radius of an electron, and

*δ*is the dispersion of the neutral gas as found in [13, 14]. Note that, neglecting spatial diffraction, in Eq.(1) the wavevectors of the harmonics are assumed to be approximately equal to their values in the gas, and

*n*(λ

_{q}) ≈1. To include diffraction effects, the analysis can be extended as in [31].

The phase-matching condition Δ*k*=0 can be fulfilled by adjusting different parameters: the dispersion of the HC-PCF, the driving laser wavelength, the gas pressure, and the mode of propagation. Since the signs of Δ*k*
* _{g}* and Δ

*k*

*are opposite to those of Δ*

_{n}*k*

_{w}and Δ

*k*

_{p}, it is possible to choose them so that their sum is zero. It should be emphasized that the main difference in phase-matching conditions between a capillary and a HC-PCF is that the waveguide dispersion of a HC-PCF is much larger than the dispersion of a capillary. In a capillary, the phase mismatch due to waveguide dispersion is Δ

*k*

_{w}=

*qu*

^{2}

_{11}λ

_{0}/4

*πa*

^{2}where

*u*

_{11}=2.405 is the first root of the Bessel function

*J*

_{0}and

*a*is the bore radius of the capillary. Since the damping length of the fundamental mode in a capillary scales as

*a*

^{3}[15], a large bore radius must be used to reduce the leakage loss. The dispersion slope in HC-PCF, with a core radius typically 10 to 30 times smaller, is much larger than that of capillary [25], and the value of Δ

*n*lies in the range 10

^{-2}~10

^{-1}. In order to achieve phase-matching of HHG in a HC-PCF in the same manner as in a capillary, e.g. by tuning the gas pressure, the pressure should be as high as several tens of bar. This would lead to strong absorption of the generated harmonics and the development of space-charge effects which would decrease the probability of electron recombination with the parent ions after the electrons are freed by the laser field. Thus the phase-matching technique commonly used in capillaries, based on the adjustment of gas pressure, can hardly be applied to HHG in a HC-PCF. One more limitation comes from the fact that phase-matching condition can only be fulfilled below the critical ionization fraction

*η*

*=1/+1+*

_{c}*N*

_{0}

*r*

_{e}

*λ*

^{2}

_{0}/2

*π*Δ

*δ*) (for example, for HHG in Ne this is about 1%).This requirement limits the maximum laser intensity to the level of ~5×10

^{14}W/cm

^{2}even for relatively short, < 20fs, pump pulses [16]. Because the cutoff energy of HHG radiation is determined by the driving field intensity, the upper limit for the energy of HHG photons is dictated by the phase-matching conditions (for Ne it corresponds to 110eV).

Recently, a scheme of quasi-phase-matching of HHG in capillaries was proposed based on the use of a weak quasi-CW counter-propagating field [20, 27]. This field introduces a periodic modulation of the phase of the generated harmonic via modulation of the laser intensity. The phase modulation results from the extra phase acquired by an ionization electron after the electron is freed from the atom and is proportional to the laser intensity [28, 29]. This approach is more flexible than the conventional quasi-phase-matching technique [6, 17–19] because the modulation period and the modulation depth can be changed directly by varying the wave-length and the intensity of the counter-propagating laser field, and, more importantly, the technique has been shown to work under conditions of full gas ionization. Whereas implementing this technique in capillaries would require intense (*I* > 10^{10}W/cm^{2}) far infrared radiation at a wavelength λ ~100*µ*m, in the case of a gas-filled HC-PCF, commercial laser systems with λ ~1*µ*m could be used to achieve synchronous HHG with photon energies of several hundreds of eV. Because in HC-PCF light is guided with extremely low loss (~0.01dB/m), this makes possible the use of much longer interaction lengths. Using the definition of the figure of merit [9] for a bore radius of 5*µ*m, we find that a HC-PCF with a loss of 0.3dB/m is almost 10,000 times better than a capillary. Note that at sufficiently high gas pressures (> 1bar), the space-charge effect may disturb the movement of the electron in the laser field and, as a result, decrease the probability of recombination with the parent ion. The total number of atoms in the interaction volume is proportional to *S*(*P*·*L*) where *S* is the effective area of the fundamental mode of a HC-PCF and *P* is the gas pressure. For a constant number of atoms interacting with the laser field in a HC-PCF, the working pressure scales with the length as *P* ∝ 1/*L*. Thus, the efficiency of HHG could be further increased by using longer interactions lengths and lower gas pressures (< 1mbar).

## 3. Numerical results and discussion

To demonstrate the feasibility of phase-matching HHG in HC-PCF using the backward seeding scheme, we consider propagation of intense light at 800nm in a HC-PCF filled with Ne. We assume that the pitch (inter-hole spacing in the cladding) is 2.2*µ*m and the core diameter is 6.8*µ*m. Compared to HHG in a capillary, the pump pulse energy required to generate the same harmonic order in a gas-filled HC-PCF is 1000 times smaller because of the very small mode diameter of the field. The attenuation achievable at 800nm in such fibers is of the order of 0.25dB/m, i.e., 94% of the light is transmitted through 1m of fiber. Because the dispersion is anomalous throughout most of the transmission window of HC-PCF, and varies continously with optical frequency, it is possible to choose appropriate conditions for soliton propagation and thus prevent dispersive pulse broadening [24, 30]. In the discussion below we assume that the change of the pump pulse intensity along the fiber is insignificant and focus on the analysis of phase matching of HHG in a HC-PCF.

To determine the phase mismatch under these conditions, one has to calculate the ionization fraction produced during the pulse interaction with the gas, which is given by:

Here *w*(*t*
^{′}) is the ionization rate, which is calculated using theory [21]:

where

In these formulae, *Z* is the charge of the atomic ion, *I*
_{ph} is the ionization potential of the hydrogen atom, *I*
_{p} is the ionization potential of the atoms in the gas (which is 21.564eV for Ne), *e* and *m*
_{e} are the electron charge and the mass, respectively, and *E*
_{1}(*t*) is the electric field of the laser pulse. The results of calculations of the ionization fraction are shown in Fig. 1. The driving laser radiation has a wavelength λ_{0}=800nm, pulse duration *τ*=50fs and a peak intensity *I*
_{0}=1×10^{15}W/cm^{2}. The ionization fraction at the peak laser intensity is seen to be 0.63, finally reaching 0.86. At such an ionization level, the phase-matching condition cannot be fulfilled using the conventional approach. To quasi-phase-match HHG in a HC-PCF filled with Ne, we first need to calculate the coherence length L_{c}=*π*/Δ*k*, which is related to the modulation period Λ by

The calculated coherence length as a function of harmonic order for a gas pressure of 15mbar is shown in Fig. 2. It can be seen that the phase mismatch due to the dispersion of the HC-PCF Δ*k*
_{w} is many times larger than the contribution of the neutral gas Δ*k*
_{g} and ionization electrons Δ*k*
_{p}, which are pressure dependent.

The coherence length is seen to change very slightly with a gas pressure because Δ*k*
_{g} and Δ*k*
_{p} are much smaller than Δ*k*
_{w}. The coherence length corresponding to a harmonic order near *q*=99 is shown in inset of Fig. 3. The fact that it lies in the *µ*m and sub-*µ*m range is especially important for implementation of quasi-phase-matching in HC-PCF, because, as we will see below, it allows one to use available laser sources to modulate the phase of the harmonics. Note that for the same parameters (e.g., gas pressure, intensity, pulse duration) the modulation period in a capillary would be of the order of one mm. To implement this method, the condition

has to be met, where λ_{2} is the wavelength of the counter-propagating light. For example, if we use a counter-propagating mode at a wavelength 1.6*µ*m, the modulation period is Λ=0.8*µ*m, and harmonics around harmonic order *q*=99 will be quasi-phase-matched. Since the phase mismatch is mainly due to the contribution of waveguide dispersion, it can be approximated by Δ*k*=Δ*k*
* _{w}*=

_{q}*k*

_{0}[

*n*(λ

*)-*

_{q}*n*

_{eff}]. The order of quasi-phase-matched harmonic then can be written as

This shows that the quasi-phase-matched harmonic order depends, through the value of *n*
_{eff}, on the parameters of the HC-PCF (pitch, hole size, core size etc.), and can be varied over a broad range by altering the parameters of the guiding structure. If desired, quasi-phase-matched HHG could even be achieved in a suitably designed HC-PCF using the same wavelength for both the pump and the counter-propagating light. In addition, the quasi-phase-matching can be finely tuned around *q*
_{qpm} (see Eq.(6)) by changing the gas pressure. For example, to quasi-phasematch harmonic order *q*=99 the gas pressure should be chosen equal to 20mbar. By setting the gas pressure equal to 120mbar, the harmonic order *q*=95 will be quasi-phase-matched, and so on. When the HHG process is modulated by a counter-propagating laser field at a wavelength λ_{2}, the phase of the emitted harmonic can be expressed as

where *z* is the propagation distance along the fiber, the first term is the phase mismatch due to the gas-filled HC-PCF and the second term describes the phase modulation (amplitude *A* and periodicity λ_{2}/2) induced by the counter-propagating wave. The amplitude *A* is a function of the field ratio parameter *r* which is defined as the ratio of the counter-propagating laser electric field amplitude to the driving electric field amplitude. The optimized amplitude of the counter-propagating field is a function of many parameters, such as the harmonic order *q*, the driving field intensity *I*
_{0} and wavelength λ_{0}, the ionization potential of the gas atom *I*
_{p} and the wavelength of the counter-propagating field λ_{2}. Using a simple one-dimensional model, for the generated HHG field amplitude, we obtain

where *E*
^{0}
_{HHG} is the harmonic field generated over the interval d*z*. When the condition Λ=λ_{2}/2 is fulfilled, the integration leads to the expression:

where a *J*
_{1} is first order Bessel function of the first kind. We do not consider here quasi-phasematching to a higher order because the first order process is relatively easy to implement and more effective.

Let us summarize the results. In order to achieve quasi-phase-matching to the *q*th order harmonic in HC-PCF, one first needs to calculate the modulation period Λ and select the modulation laser source so that the condition Λ=λ_{2}/2 is satisfied, or, given the modulation laser source wavelength is fixed, select a HC-PCF with a proper *n*
_{eff}. Second, *A*(*r*) is calculated and the optimum counter-propagating light intensities required to maximize *J*
_{1}(*A*) are determined (the method of calculation of A(r) used here is described in detail in [20]). The value of *J*
_{1}(*A*) is maximized when *A*(*r*)=1.84, and the maximum value is 0.58. Using the above mentioned parameters, we calculate the ratio parameter *r* by solving the equation *A*(*r*)=1.84 for different harmonic orders. The results are shown in Fig. 3. The field ratio *r* is smaller for the long trajectory, in which the electron spends more time in the continuum and interacts longer with the counter-propagating wave. For the short trajectory, the time interval between ionization and recombination of the electron with the parent ion is shorter. Therefore, to achieve the same modulation amplitude *A*(*r*)=1.84, more intense counter-propagating laser light should be used. For example, to phase-match harmonic order *q*=99, the corresponding field ratio parameter should be *r*=3×10^{-3} for a long trajectory. The intensity of the counter-propagating modulation wave is *I*
_{2}=*I*
_{0}
*r*
^{2} ≈ 9.13×10^{9}W/cm^{2}. If we choose to select a short trajectory, the field ratio for *q*=99 is *r*=1.25×10^{-2} and the corresponding modulation light intensity is *I*
_{2}=1.57×10^{11}W/cm^{2}.

In summary, we have discussed the potential of gas-filled hollow core photonic crystal fibers for HHG. The feasibility of quasi-phase-matching this process by modulating the phase of the ionization electrons with a counter-propagating laser field has been shown. The results suggest that harmonics with photon energies of several hundreds of eV can be produced using available fs-laser sources of *µ*J energy. It is expected that the efficiency of HHG will be increased through the long interaction lengths offered by low loss HC-PCF.

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