1. Introduction

It is well known that lasers generate strongly directed coherent light with the highest possible brightness. However, a typical laser is a quasi-monochromatic source emitting light of only one color. Many applications require light sources which share with a laser its unidirectional and coherent properties but span the whole spectral range of a rainbow like the sun or an electric bulb. The latter sources, however, are unable to emit coherent, unidirectional and bright radiation.

A milestone on the way towards a coherent white-light high-brightness source (supercontinuum, SC) was achieved by the application of photonic crystal fibres (PCF) [1, 2]. When a femtosecond pulse with only nJ energy from a laser oscillator is focused into such fiber, a dramatic conversion from narrow band light to two-octaves-broad spectra was observed [3]. The discovery of SC generation in PCFs encouraged extensive research activities (see e.g. [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14]) and is now considered to be one of the leading-edge research areas in photonics. This coherent white-light source is applied in physics, chemistry, biology, telecommunication and medicine using numerous methods ranging from the frequency comb method [15, 16], optical coherence tomography [17], absorption spectroscopy[18], and flow cytometry to WDM in telecommunication and others. There are now a few commercial suppliers for these purposes (see e.g. [19]).

Supercontinuum generation in PCFs is based on an effect for spectral broadening [4] which differ from the main previously known mechanism for spectral broadening by self-phase modulation. It is connected with the soliton dynamics in the anomalous dispersion region of the PCFs. The high refractive-index contrast in PCFs leads to a shift of the zero dispersion wavelength to the visible and anomalous dispersion at the wavelength of typical lasers, such as a Ti:sapphire laser. Therefore the input pulse splits into several fundamental solitons with different amplitudes and distinct red-shifted input frequencies. Per definition, a fundamental soliton does not change its energy, however if it is perturbed by third- and higher-order dispersion of the fibre it can emit blue-shifted nonsolitonic radiation (also known as dispersive radiation) at a wavelength determined by phase matching [20]. Since multiple solitons with different central frequencies emit radiation at distinct frequency intervals, a very large spectral range is covered. Subsequent experimental studies [5, 6, 7] by many groups provided evidence for this soliton-induced mechanism for SC generation.

Supercontinuum spectra from PCFs are 10⁶ times brighter than sunlight (~10³ W/cm²/sterad) and have the same bandwidth. Unfortunately, the small radii in PCFs and material damage limit severely the maximum peak power densities to tens of W/nm. In this Letter we propose and study a novel approach for supercontinuum generation in metallic hollow waveguides coated with a dielectric. The key idea in this approach is to use a hollow-core waveguide filled with a noble gas instead of the solid-core microstructure fibers to increase the diameter to a range between 20 to 80 µm and the damage threshold. The introduction of such waveguides has the aim to fulfill the conditions of small loss and anomalous dispersion at optical frequencies which are the basic requirements for the soliton-induced mechanism of supercontiunuum generation, similar to the situation in microstructure fibers [4]. In recent years the application of standard dielectric hollow waveguides has led to impressive progress in ultrafast optics such as the generation of few-cycle [21] and attosecond pulses [22] and high-order harmonics [23]. However, these waveguides can not be used for the purpose studied here, because anomalous dispersion at optical frequencies and high pressures can only be achieved for small diameters in the range of 20–80 µm, for which the loss is too high. Dielectric-coated metallic hollow waveguides can be produced by chemical vapor deposition [24, 25].

For many applications the noise and coherence properties of this octave-spanning white light is of crucial importance. In particular, for pulse compression a shaper is needed to adjust the phases of the different spectral components with respect to each other. Therefore the pulse-to-pulse phase noise of the spectral components is very detrimental for pulse compression. Low-noise supercontinua are also essential for optical frequency metrology, optical coherence tomography, and other applications. Several experimental studies and theoretical simulations have shown that in general the SC coherence is very sensitive to both the fundamental quantum noise and the shot-to-shot pump intensity fluctuations (technical noise) [7–14] These studies have shown that a sufficient level of coherence of the SCs can only be obtained using pulses with a relatively small intensity and durations of about 50 fs or less [12, 9, 11, 14], or for input wavelengths tuned deeper into the anomalous dispersion region[10].

In this paper we predict that in such waveguides two-octave broad SC with up to five orders of magnitude higher spectral peak power densities than in PCFs can be generated. We also predict the generation of UV/VUV supercontinua in the range from 160 to 540 nm from such type of waveguide, a spectral range which can not be achieved by the previously used methods but which is of particular importance in many applications. We also show that, besides its very high spectral peak power density, soliton-induced SCs in metallic hollow waveguides coated with a dielectic has the additional advantage of suppressed coherence degradation due to the contribution of the plasma.

2. Theoretical fundamentals

In Fig. 1, the geometry of the studied metal-dielectric hollow waveguide is presented. The hollow core of the fibre with diameter D is surrounded by a metallic cladding (blue) coated by a dielectric material such as silica (yellow) with thickness a on the inner side.

For the numerical simulations we use a generalized model based on the propagation equation for forward-going waves [4] with inclusion of higher-order transverse modes and the effects of ionization and plasma formation in the waveguide. The Fourier transform E⃗(z,x,y,ω) of the electromagnetic field E⃗(z,x,y, t) in the waveguide can be represented by E⃗(z,x,y,ω)=∑_j E_j(z,ω)F_j(r,ω), where F_j(r,ω) is the transverse mode profiles of the j-th mode and E_j(z,ω) describes the evolution of the field with propagation. The inclusion of higher-order transverse modes takes into account a possible energy transfer to higher-order modes by the nonlinear polarization. We consider only linearly-polarized input fields which couple only to those EH1 j modes which have the same direction of the transverse electric field. The z components of the modes scale with the ratio of the wavelength to the radius, which is below 0.05 in our case. Therefore the influence of z components can be neglected both in Kerr response and in the plasma contribution. The following first-order differential equation can be written for E_j(z,ω) in the EH_1j mode [4]:

where β_j(ω) and α_j(ω) are the wavenumber and the loss. This equation neglects backward-propagating components and contains only the first-order derivative over the propagation coordinate, but in difference to the nonlinear Schrödinger equation (NSE) it does not rely on the slowly-varying envelope approximation and refer to the field components E_j(z,ω) and not to the amplitudes of the field. This approach allows to model the evolution of fs pulses with extremly broad spectra. The quantities β_j(ω), α_j(ω) and F_j(r,ω) are calculated by the transfermatrix approach assuming a circular waveguide structure as shown in Fig. 1, including the bulk dispersion of argon.

 

Fig. 1. Scheme of the hollow waveguide for supercontinuum generation.

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Additional scattering loss due to the roughness of the inner surface is calculated by the following formula [26, 27]

which assumes that roughness can be represented by many uncorrelated pointlike scatterers [26] with surface density ρ_s and volume V_S at the interface between the two layers of the waveguide. In Eq. (2), Δε is the difference of the dielectric constants of the materials on the two sides of the interface, < I >_s is the intensity of the mode profile near the interface averaged over the scatterer volume, I ₀ is the intensity of the mode profile in the center of the waveguide, and a_M is a dimensionless mode factor which is equal to 0.269 for the fundamental EH ₁₁ mode. In the calculation, we have assumed hemispherical scatterers with V_S=(1/12)πσ ³, ρ_s=σ ^-2 where σ is the mean scatterer size. In the case of coated waveguides, experimental data suggests that the scatterers are present on both interfaces between the layers, and we assumed that on both surfaces they have the same size and density. We note that in the Eq. 2 the modification of the density of states near the metal interface is not included. However, in the case of coated fibers the field at the metal-dielectric interface is quite weak, and the corresponding enhancement factor is estimated to be in the range from 0.3 to 2. Besides, the plasmonic resonance, the local field enhancement near the metal scatterers, and the possibility of long-range correlations are also not included in the model, but magnitude and wavelength dependence of α_r can be assessed in this way.

The Fourier transform of the jth component of the nonlinear polarization P ^(j) _NL (z,ω) is given by

Here P_NL(z,r,t) includes the Kerr nonlinearity, the plasma-induced refraction index change and absorption of light energy due to ionization. A theoretical description of light propagation in gases in the presense of photoionization and plasma can be found in numerous papers (see e.g. [28]). Taking these effects into account we can write for P_NL(z,r,t):

where χ3=(4/3)cε ₀ n ₂ is the third-order polarizability of the gas filling and Ĩ(z, r, t) is the intensity averaged over few optical periods. In this equation, the first term is the Kerr polarization, the second one is the dipole moment created by the free elelctrons, and the third one describes the loss of the electric field due to the energy necessary to ionize the argon atoms[29]. We neglect the nonlinearity of the fiber walls, since the field in the wall region is by orders of magnitude lower than in the center of the hollow core.

For argon we have n ₂=1×10^-19 cm²/W at 1 atm, E_g=15.95 eV is the ionization potential, ρ(z,r,t) is the electron density and d(z,r,t) is the mean free-electron displacement in the electric field. The evolution of the two latter quantities is described by

where Γ(Ĩ(z,r,t)) is the photoionization rate calculated from the Faisal-Reiss-Keldysh model[30] which describes both the multiphoton and tunnelling regime and N ₀ is the initial argon density. The latter of these equation is just the second Newton’s law for the free electrons. In the following we study the coherence degradation of the supercontinua due to the influence of the fundamental quantum noise. In the Wigner quasi-probability representation the evolution equations for quatum field operators are mapped onto the classical equations [31] where the effect of quatum noise is included by adding to the input field E(t) the quantum shot noise ΔE(t) which corresponds to a half photon per mode) defined by <ΔE(t ₁)ΔE(t ₂)>=δ(t ₁-t ₂)h̄ω ₀ τ ₀/(2ΔtE_tot) where ω ₀ is the input frequency, E_tot is the total energy of the pulse, τ ₀ is the pulse duration, Δt is the step of the time grid, and the average <…> is taken over the noise realisations (see [31, 13] for details). The quantity which characterizes the coherence of the SC is given by the first order coherence function g(λ) which directly corresponds to the visibility measured in interference experiments and is defined as

where E_a(λ) is the spectrum in the fundamental mode after the waveguide and the indices a,b denote the realization. Equation (1) is solved by the split-step Fourier method with Runge-Kutta nonlinear steps, while Eq. (4) and (5) are solved by the second-order Runge-Kutta method without further approximations at each propagation step.

3. High-power SC generation in the visible

First we study the loss and dispersion properties of a silver waveguide with a diameter of 80 µm coated from the inner side with a 45-nm fused-silica layer. In Fig. 2(a) can be seen that the loss remains relatively low in the range of 10^-2-10^-1 dB/m for wavelengths from 0.4 µm to 1.4 µm. This means that at most 10% of the energy is lost during a propagation over 1 m, while in a conventional dielectric hollow waveguide with the same diameter, roughly 80% of the energy is lost. The physical reason for this difference is that the reflection coefficient for grazing incidence of light is higher for fused-silica-coated silver than for a layer of fused silica. The increased transmission of dielectric-coated metallic waveguides allow a reduction of the diameter of the waveguides while keeping the losses acceptable. This, in turn, leads to significant modification of the dispersion properties and an extended range of anomalous dispersion[27]. The group velocity dispersion, illustrated in Fig. 2(a) by the green curve, is anomalous for λ>570 nm at 1 atm of the argon filling.

The optical properties of the studied waveguide as illustrated in Fig. 2(a) allows the conjecture that soliton-induced supercontinuum generation can be generated in such waveguide. To study this question we consider the evolution of a 100-fs input pulse at 830 nm with a peak intensity of 50 TW/cm² propagating a maximum distance of 2 m in the above-described fiber. We assume that the input field is completely in the EH₁₁ mode, but during propagation roughly 20% of energy is transferred to higher-order modes. It can be seen in Fig. 2(b) that the generated radiation cover the spectral range from 250 nm to 1200 nm, with a total width corresponding to more than two octaves. This supercontinuum is generated already after 50 cm of propagation, as shown in Fig. 2(d). In Fig. 2(b) the first-order coherence is shown by the green curve demonstrating an average value $\stackrel{}{g\left(\lambda \right)}$ of 0.13. The wavelength-averaged peak power spectral density is about 10⁶ W/nm with a peak intensity at the output of about 10¹⁴ W/cm². The obtained spectral power density is 10⁵ times higher than in standard PCFs. For comparison, the Sun at its surface has a brightness of 10³ W/cm²/sterad, which is roughly 10¹¹ times lower than the results predicted here. The temporal shape presented in Fig. 2(c) shows that, after intial compression, the pulse is split into many solitons and phase-matched background radiation which move with different velocities. These solitons are produced from the input pulse, which corresponds to a higher-order soliton; this higher-order soliton splits into many fundamental solitons owing to the perturbing effects. The supercontinuum is explained by the same mechanism as in photonic crystal fibres due to the emission of nonsolitonic radiation by several frequency-shifted solitons[4].

 

Fig. 2. High-power supercontinuum generation. Waveguide characteristics (a), generated output supercontinuum (red) and first- order coherence (green) (b), and evolution of temporal shape (c) and spectrum (d) in a D=80 µm silver waveguide coated with a 45-nm layer of fused silica and filled with argon at 1 atm. The input 100-fs pulse at 800 nm has the intensity of 50 TW/cm². In (a), loss (red) and group velocity dispersion (green) are presented. The roughness size σ equals 100 nm. The spectrum in (b) is presented after 50-cm propagation.

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4. VUV supercontinuum generation

Supercontinua from PCFs have the shortest wavelengths in the ultraviolet at about 300 nm. There exists a strong interest in molecular physics and material science in extending the achievable spectrum still further towards the vacuum ultraviolet (VUV). Here we show that dielectric-coated metallic hollow waveguides can also be used for this purpose and predict a high-power VUV supercontinuum source, which is based on an appropriately designed fused-silica-coated aluminium waveguides filled with argon.

We consider an optimized aluminum waveguide with the diameter of only 20 µm coated by a 10-nm layer of fused-silica. The linear optical properties of this waveguide are shown in Fig. 3(a), demonstrating moderate loss and anomalous dispersion for λ>260 nm at 1 atm. In Fig. 3(b)–3(d) the generation of an ultraviolet/vacuum ultraviolet supercontinuum from a third harmonic pulse of a Ti:sapphire amplifier laser system with input pulses at 266 nm and intensity of 40 TW/cm² is illustrated. The evolution of spectra (d) and the temporal shape (c) demonstrates an initial generation of two side peaks by four-wave mixing followed by fission into several solitons and soliton emission. The output spectrum reaches from 160 to 540 nm, as shown by the red curve in Fig. 3(b).

 

Fig. 3. UV/VUV supercontinuum generation. Waveguide characteristics (a), generated output supercontinuum (b), and evolution of the temporal shape (c) and spectrum (d) in a 20-µm aluminum waveguide with a 10-nm coating and filled with argon at 0.5 atm. The input 50-fs pulse at 266 nm has the input intensity of 40 TW/cm². In (a), loss (red) and group velocity dispersion (green) are presented.

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5. High coherence of the SC in hollow waveguides

We now study the evolution of a 100-fs input pulse at 830 nm with a peak intensity of 100 TW/cm² propagating a distance of 50 cm in the above-described fiber. It can be seen in Fig. 4(a) and 4(c) that the generated radiation after 50 cm of propagation covers the spectral range from 200 nm to 1400 nm, with a total width corresponding to more than two octaves. The interpretation of the predicted spectra and temporal profiles is similar to that for the Fig. 2. In Fig. 4(c) the first-order incoherence is shown by the green curve demonstrating a surprisingly high coherence which deviates from unity by no more than 10^-5. This high coherence of soliton-induced SC generation in hollow waveguides is in contrast to SC generation in MFs based on the same broadening mechanism.

To understand the suppression of coherence degradation for soliton-induced SC generation in hollow waveguides we study the role of the plasma contribution in the SC formation process. In Figs. 5 and 6 we present the evolution of the spectrum and incoherence function 1-g(λ) for models without ( Fig. 5) and with (Fig. 6) plasma contributions. It can be seen that without plasma, at the propagation length of 15 and 20 cm seed spectral components are formed [Fig. 5(a) and 5(b)] within spectral wings between 550 nm and 750 nm as well as around 1200 nm. They arise by the amplification of noise owing to the gain of the modulation instability. The coherence of the side maxima decreases with propagation from 15 to 30 cm, with an incoherence 1-g(λ) on the level of 10^-2 to 0.1. With further propagation after fission of the pulse into several solitons, the spectrum is dramatically broadened as shown by the red curve in Fig. 5(c), but the coherence is also significantly degraded (green curve) with a low average coherence of the output spectrum $\overline{g\left(\lambda \right)}=0.49$. In contrast, in the case when the plasma contribution is included in the numerical model (Fig. 6), the seed components on the short-wavelength side of the spectrum form a smooth spectral wing from 450 to 700 nm in Fig. 6(b). The mechanism of this smooth short-wavelength wing formation is related to the abrupt rise of the plasma density near the peak of the pulse leading to a strong phase modulation due to time-dependent plasma contribution in the refractive index, as shown in Fig. 6(d). We note that this wing is highly coherent in our case with 1-g(λ)~10^-7 [green curve in Fig. 6(b)], as should be expected since its spectral phase is determined by the phases of the input pulse, similar to the case of self-phase modulation. Now in contrast with Fig. 5(c) further propagation and spectral broadening of the pulse up to 50 cm do not lead to a dramatic coherence degradation but to the preservation of the high coherence of the SC. The resulting spectrum [Fig. 6(c) and Fig. 4(c)] is characterised by the average coherence of $\overline{g\left(\lambda \right)}=1-{3\text{x}10}^{-7}$.

 

Fig. 4. Evolution of the spectrum (a) and temporal shape (b) for the propagation of a 100-TW/cm², 100-fs pulse in a fused-silica-coated silver waveguide. In (c) the output spectrum (red) and incoherence 1-g(λ) (green) are shown and in (d) the ouput temporal shape are presented. The propagation length is 50 cm, input wavelength is 800 nm, the waveguide parameters are D=80 µm, a=45 nm.

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Fig. 5. Spectrum and coherence for a propagation model without plasma contribution. The spectrum (red curve) and coherence (green curve) are presented at the distances of 15 cm (a), 20 cm (b) and 50 cm (c). The input pulse and waveguide parameters are the same as in Fig. 2.

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Fig. 6. Spectrum and coherence for a propagation model with plasma contribution. The spectrum (red curve) and coherence (green curve) are presented at the distances of 15 cm (a), 20 cm (b) and 50 cm (c). In (d), the contribution of the free electrons to the refractive index is depicted at the beginning of the propagation. The input pulse and waveguide parameters are the same as in Fig. 2.

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Fig. 7. Schematic presentation of the supercontinuum generation at the initial stage. Black dotted curve indicates the input spectrum, the red curve is the spectrum at the position of the seed component formation, green dashed curve is the incoherence 1-g(λ). In (a), arrows indicate the four-wave-mixing gain bands; in (b), the arrow indicates the spectral broadening induced by plasma.

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The mechanism of the suppression of coherence degradation in soliton-induced SC generation is schematically presented in Fig. 7(a) and 7(b). In the case of a relatively long and intense pulse owing to gain by modulation instability seed spectral components arise from noise, as represented in Fig. 7(a) by the black arrows. These seed components are further amplified by the gain produced by the fundamental solitons after fission owing to third-order dispersion leading to the large spectral broadening but accompanied by coherence degradation. In contrast, in the case of the seed components formation due to plasma, as schematically presented in Fig. 7(b) the short-wavelength wing arise from the pulse itself. It is smooth and highly coherent, and after amplification by the gain which leads to the emission of non-solitonic radiation the coherence remains high.

For comparison we calculated the spectrum and coherence function for a shorter input pulse, as illustrated in Fig. 8. The spectrum in this case is not significantly broader although slighly smoother, and the coherence increases to values in the range of 1-10^-7. In this case the input spectrum is already quite broad, which facilitates the formation of the coherent sidebands from the broad spectrum during the soliton compression and leads to high final coherence.

 

Fig. 8. Output spectrum (red) and coherence (green) for the propagation of a 100-TW/cm², 10-fs pulse in a fused-silica-coated silver waveguide. The propagation length is 50 cm, input wavelength is 800 nm, the waveguide parameters are D=80 µm, a=45 nm.

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Fig. 9. Evolution of the spectrum (a) and temporal shape (b) for the propagation of a 100-TW/cm², 100-fs pulse in a fused-silica-coated silver waveguide. In (c) the output spectrum (red) and the coherence (green) are shown and in (d) the ouput temporal shape are presented. The propagation length is 50 cm, input wavelength is 800 nm, the waveguide parameters are D=200 µm, a=45 nm.

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Finally, we have also performed simulations for a larger waveguide diameter of 200 µm with smaller waveguide contribution to dispersion and normal GVD at the input wavelength. The results are presented in Fig. 9. In the temporal shape shown in Fig. 9(b) and 9(d) a modulation is formed but without sharp edge, and the spectrum shown in Fig. 9(a) and 9(c) remains relatively narrow. This indicates that the anomalous dispersion and the soliton mechanism are essential for the predicted two-octave spectral broadening in the considered hollow waveguide, which can not be achieved by self-phase modulation and plasma-related broadening. The coherence shown by the green curve in Fig. 9(c) is very high, typical for spectral broadening by SPM.

6. Conclusion

In conclusion, we studied the generation of high-power supercontinua in specially designed metallic dielectric coated waveguides. We predicted that supercontinua with two-octave width, spectral power densities in the range of MW/nm, five orders of magnitude higher than in microstructured fibers, and high coherence, can be achieved in such waveguides. Additionally, we predicted the generation of UV/VUV supercontinuum in an aluminum hollow waveguide. We have shown that the coherence degradation in soliton-induced SC generation as typically observed in MFs can be suppressed by the plasma contributions. The coherence preservation is explained by the formation of seed components of the SC generation due to sharp raise of the plasma density. In this way highly coherent supercontinua with two-octave broad spectra and spectral power densities in the range of MW/nm can be achieved in such waveguides.

The findings that we report could have applications in a wide range of fields. Let us remark just a few. In combination with a multichannel frequency filter the high- power SC source can replace many lasers at separated frequencies including wavelengths where lasers are not available (as e.g. in the UV/VUV). The predicted VUV supercontinuum could lead to advances in VUV frequency comb spectroscopy [32]. Further on, direct difference frequency mixing of two portions of the SC [33] eliminates the carrier-offset frequency and its noise which can be used in frequency metrology. The detection sensitivity of nonlinear spectroscopic methods such as CARS microscopy [34] can be significantly increased by an increase of the CS intensity.