1. Introduction

In the last decade impressive progress has been achieved in the field of ultrafast optics. Examples of breakthrough concepts and findings in this area include pulse compression to a duration of about one optical cycle and the generation of attosecond pulses.

The application of hollow waveguides [1] has been instrumental for this progress. Combination of self-phase modulation in noble-gas-filled dielectric hollow waveguides and pulse compression by negative group delay [2] performed outstandingly in the generation of few-cycle mJ pulses recently approaching a duration of almost one optical cycle in the near-infrared region [3]. Such few-cycle intense pulses have become an efficient pump source for high-order harmonic generation and isolated attosecond pulse generation [4]. An alternative method for the compression of isolated pulses in hollow waveguides is the use of Raman molecular modulations in a pump-probe regime [5, 6] enabling also the compression of UV and VUV pulses [7, 8]. High-order harmonic generation [9] and four-wave mixing (FWM) for UV [10] and VUV [11] fs pulse generation are other examples for the application of hollow waveguides in ultrafast nonlinear optics. Unfortunately, dielectric hollow waveguides provide tolerable levels of losses only for radii larger than 50 µm. For these parameters the waveguide contribution to dispersion is relatively small. However, controlling the group velocity dispersion (GVD) by the fiber and gas parameters is a key factor for the efficiency of nonlinear optical processes, because it determines the phase-matching between different spectral components.

Hollow-core photonic bandgap fibers [12] and omniguide fibers [13] are other interesting waveguide types in which light is guided in the core due to photonic bandgaps in the two-dimensional periodic cladding. Low-loss guiding in such fibers with radii of 5-25 µm has been demonstrated, and anomalous dispersion in the optical range at 1 atm enables the generation of megawatt optical solitons [14]. However, their intrinsically narrow-band transmission seriously restricts the usable range of frequencies and hinders their use for broadband nonlinear optical processes. Note that other waveguides using a kagome lattice have been developed [15, 16] which do guide light over a broad wavelength range.

Supercontinuum generation in microstructure fibers [17] is another phenomenon in broad-band nonlinear optics which has recently attracted much attention and has found many applications. Spectral broadening with the output width of two octaves or more is not caused by self-phase modulation but is based on the soliton dynamics in fibers with anomalous dispersion at the input frequency [18]. However, the low output spectral power density cannot be easily scaled up, since low fiber core radii are required for anomalous dispersion, and increasing input intensity leads to damage of the solid core.

The purpose of the present paper is to find an alternative waveguide type which has low loss in a broad optical, UV/VUV and near-IR spectral range for waveguide diameters which simultaneously yields anomalous dispersion in a desired frequency range even at a relatively high gas pressure. As will be shown, dielectric-coated metallic hollow waveguides represent a suitable option for the realization of this aim. The dielectric coating of the metallic wall reduce the waveguide loss and also the scattering loss in comparison to purely metallic or dielectric waveguides. Using a matrix method [19] we study the dependence of the loss (including scattering loss) and group velocity dispersion on the waveguide diameter. We predict tolerable loss in the optical and near-IR region even for small waveguide radii in the range of 10–25 µm with a broad range of anomalous GVD in the visible even for high pressures of the gas filling.

These findings can have interesting applications in ultrafast nonlinear optics and supercontinuum generation. To mention a few examples, the anomalous dispersion combined with the higher nonlinearity due to a higher pressure can enable high-order soliton self-compression of mJ pulses without technically demanding chirp compensation by external anomalous-GVD elements. Second, such waveguides are promising for the generation of soliton-induced supercontinua with more than one octave broad spectra of unprecedentedly high power. Finally, anomalous dispersion allows phase-matching for FWM at much higher pressures and therefore higher efficiencies of frequency conversion, which can significantly increase the accessible pulse energies in UV and VUV femtosecond pulse generation in hollow waveguides.

Note that dielectric-coated metallic waveguides have been known already for a long time and have been fabricated by a variety of techniques, such as wet chemistry, chemical vapor deposition and sputtering [20, 21, 22]. Up to now they have been successfully applied in IR laser-power delivery for medical and industrial applications involving the delivery of CO ₂ laser radiation. A theoretical study and an optimization of the guiding properties of such waveguides in the IR has been presented in Ref. [23]. Up to our knowledge there exist no applications of such waveguides in ultrafast laser physics.

2. Theoretical approach

We consider a straight waveguide with circular cross-section, which consists of a gas-filled hollow core with a refractive index around unity, surrounded by an metallic cladding coated by a layer of dielectric from the inner side. The geometry of such a waveguide is presented in Fig. 1. The guiding properties of this system are best described by the transfer matrix formalism developed in [19]. For each mode, the dependence of all field components on time t and longitudinal coordinate z is given by the same factor exp(-iωt)exp(iβ(ω)z), where ω is the frequency and β(ω) is the (generally complex-valued) wavenumber of the mode. In this case, the transverse components E_r, E_ϕ, H_r, H_ϕ of the electric and magnetics fields E⃗ and H⃗ can be expressed in terms of the longitudinal components E_z and H_z. The latter are the eigenfunctions of the operator Δ_⊥+ω ² εμ/c ², where Δ_⊥≡∂²/∂x ²+∂²/∂y ², ε and μ are the dielectric and magnetic constants of the material. Therefore they can be expressed as a linear combination of Bessel functions of the radial coordinate r within each of the regions, core (j = 1), coating (j = 2), and cladding (j = 3):

E_z = [A_jJ_l(k_jr)+B_jY_l(k_jr)]exp(iϕl)

 

Fig. 1. The scheme of a dielectric-coated metallic hollow waveguide.

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Here k_j = [ω ² ε_jε/c ²-β(ω)²]^1/2, J_l and Y_l are Bessel functions of first and second kind, A_j, B_j, C_j, D_j are complex-valued coefficients, l is the azimuthal mode number, and ϕ is the azimuthal angle. The boundary conditions at the interface j between two regions j+1 and j can be expressed in terms of a transfer matrix T_j which relates the coefficients A, B, C, D in one region with those in another region. The explicit expression [19] for this matrix is

where

Here r _{j, in} is the inner radius of the region j. The matrix M _{j, out} is written in the same way but using the outer radius of the region j. In the multilayer waveguide as considered here, the matrix connecting the coefficients in the core and the outermost region is obtained as a product of matrices on the boundaries. Finally, the characteristic equation for the eigenvalue β (ω) can be obtained requiring finiteness of the field in the core (B ₁=D ₁ = 0) and absence of the incoming waves in the outermost layer (A ₃-iB ₃=C ₃-iD ₃ = 0). The resulting equation [24] has the form F(β(ω)) = 0 where F is a complex analytical function of β(ω). The solution of this equation yields β(ω), which can be complex even in a waveguide made from lossless materials, with an imaginary part accounting for the mode loss α = (λ/π)Im(β).

Additional scattering loss due to the roughness of the inner surface is calculated by the following formula

which assumes that roughness can be represented by many uncorrelated pointlike scatterers [25] with surface density ρ_s and volume V_S at the interface between the two layers of the waveguide. In Eq. (4), 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 as illustrated in Fig. 1, and we assumed that on both surfaces they have the same size and density. We note that in the Eq. 4 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.

3. Loss and group-velocity dispersion of metallic hollow waveguides

First we consider a waveguide with a cladding made from silver and a SiO ₂ inner coating. For an lossless metal cladding, the waveguide loss would be exactly equal to zero. Silver has a relatively low loss in the visible range and is characterized by a dielectric function ε = 1 - ω ² _p/[ω(ω + iν)] with ω_p = 12.0 fs⁻¹ and ν = 0.076 fs⁻¹[26].

In Fig. 2 we present the calculated loss and the GVD of the fundamental EH₁₁ mode for waveguides with inner radius R of 125 µm, 40µm, and 10 µm. Roughness loss for a mean scatterer size σ of 100 nm is included. In Figs. 2(a),(c), and (e) the loss in a waveguide with cladding made of silver without coating is presented by the red solid curve. This loss is by more than one order of magnitude lower than that of hollow waveguide made of fused silica (black dotted curve). For a 40-µm-radius waveguides the predicted loss for a metallic waveguide is at the level of 1 dB/m, while for a 10-µm radius it is in the range of 10-100 dB/m and too high for applications in nonlinear optics. A dielectric-coated metallic surface has a higher reflection coefficient for grazing incidence than an uncoated metallic surface. Therefore by introducing a coating the loss can be reduced to an acceptable level. For fused-silica coating with a thickness a of 60 nm, the results are presented by the green long-dashed curves. Even for the 10-µm-radius coated waveguides the loss remains in the range of 10 dB/m over a relatively wide range of wavelengths, which suggests that they still can be used in nonlinear optics with an effective length around 30 cm. However, a peak of loss appears at short wavelengths for all radii. This peak moves to longer wavelengths for larger values of coating thickness a. This peak is caused by a resonance of the mode localized in the core of the waveguide with the mode localized in the coating. The latter is the mode of a three-layer almost-planar waveguide formed by the core, coating, and cladding.

Here and in the rest of the paper we consider only the fundamental transverse EH₁₁ mode. Higher-order transverse modes are characterised by higher waveguide contribution to dispersion and also by higher loss. Roughness leads to mode coupling and energy transfer to the higher-order modes, with efficiency related to the roughness loss which is quite low for the considered parameters [see Figs. 3(a),(b)]

In Figs. 2(b),(d), and (f) the GVD of the corresponding waveguides is presented for argon filling at 1 atm (thin red solid curves) and at 10 atm (thick red solid curves). For a 125-µm waveguide which is typical for applications, for both pressures the GVD is normal in the visible range, which limits the possibilities to achieve phase-matching for nonlinear processes. For a waveguide radius of 40 µm, the GVD is anomalous in the visible for the lower pressure of 1 atm but not for 10 atm. Finally, a 10-µm radius allows to achieve anomalous GVD in the visible range even for high pressure, which enables advantageous phase-matching conditions for many processes of nonlinear optics. This favorable combination of a still acceptable loss [Fig. 2(e)] and anomalous GVD even at high pressures is made possible by using a coated metal as the waveguide material instead of fused silica. The presence of the cladding introduces a peak of the GVD near the resonance, otherwise its influence is not significant [dashed green curve in Figs. 2(b),(d),(f)].

 

Fig. 2. Loss (a),(c),(e) and GVD (b),(d),(f) of argon-filled hollow waveguides with radii of 125 µm (a),(b), 40 µm (c),(d) and 10 µm (e),(f). The silver cladding is coated by a fused-silica layer with thickness a of 0 nm (red solid curves) and 60 nm (green long-dashed curves). The loss of the fused-silica hollow waveguide of the corresponding radius is shown in (a),(c), and (e) by the black dotted curve. The GVD is presented in (b),(d), and (f) for the uncoated silver waveguide for the pressures of 1 atm (thin red solid curves), 10 atm (thick red solid curves), and for 1 atm with coating thickness of a = 60 µm (dashed green curves).

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In Fig. 3 we analyze the scattering part α_r of the waveguide loss (dashed curves) and the relative intensity at the inner wall of the waveguide 〈I〉_s/I ₀ (red solid curve) for a waveguide with optimized parameters described later in Fig. 5. It can be seen that for σ = 100 nm α_r is below the level of 1 dB/m in most of the relevant spectral region, constituting roughly 20% of the total loss. However, the scattering loss sensitively depends on the scatterer size σ: α_r~σ ⁴ which varies significantly depending on the manufacturing. In Fig. 3(b) the scattering loss is presented for σ = 200 nm, in this case α_r is much larger and constitutes the main part of the total waveguide loss. Dielectric coating reduces not only the waveguiding loss but also the scattering loss (cf. the blue short-dashed and green long-dashed curves in Fig. 3). Note that far from the resonance, the scattering loss of the coated waveguide is by roughly one order of magnitude lower than the loss of the uncoated waveguide. The reason is that dominant scattering which occurs due to the roughness of the metal is not so effective for the coated waveguide, since the coating reduces the intensity near the surface of the metal. Surprisingly, the total variation of the scattering loss over the spectral range far from the planar-waveguide resonance is below one order of magnitude. In Eq. (4) three factors dominate the wavelength dependence of α_r: the ω ⁴ factor, the average relative intensity at the inner wall 〈I〉_s/I ₀, and the squared metal dielectric constant |Δε|²~1/ω ⁴. While the first factor decreases with increasing wavelength, the second and the third factors increase for longer wavelength, thus compensating each other. Finally, in the vicinity of the resonance scattering loss has a pronounced and strong peak, just as the total loss. The reason can be easily understood analyzing the wavelength dependence of the relative intensity at the surface, presented in Fig. 3(c) by the red solid curve. The relative intensity has a strong peak at the resonance, and then increases with wavelength as expected. Note that for the spectral range from 600 nm to 1200 nm the value of the relative intensity remains below 10⁻³. This ensures that even for high peak intensities of the pulse in the range of 100 TW/cm² near the photoionization threshold of argon, the intensity at the inner surface will remain significantly below the damage threshold of fused silica or other dielectric coatings.

 

Fig. 3. Scattering loss (a),(b) for a silver waveguide with R = 15 µm and a = 84 nm (green long-dashed curves) and a = 0 (blue short-dashed curves) and relative intensity at the inner surface of the waveguide (c) (red solid curve). The average scatterer size σ is 100 nm in (a) and 200 nm in (b).

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Let us now consider a hollow waveguide which can be used in the UV/VUV range. This range is of particular importance for spectroscopic and other applications. However, silver has a too large loss in the UV/VUV and cannot serve as a material for an efficient waveguide. Fortunately other metals such as aluminum (characterized by ω_p = 22.5 fs⁻¹ and ν = 0.88 fs⁻¹ [26]) can be used for this purpose, as illustrated in Fig. 4. The loss remains moderate in the spectral range from 150 to 600 nm, and anomalous GVD in the UV for wavelengths larger than 340 nm can be reached even at relatively high pressure of 0.3 atm. For a 40-µmwaveguide, the zero-GVD wavelength is 430 nm.

 

Fig. 4. Loss (a) and GVD (b) of a coated aluminum hollow waveguide with radius of 25 µm (red solid curve) and 40 µm (green dashed curves). The thickness of fused-silica coating a is 15 nm. The GVD is presented for argon pressure of 0.3 atm.

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4. Design optimization for specific nonlinear processes

The large number of free waveguide and material parameters of metal-dielectric waveguides requires a significant numerical effort to find the optimum design for specific nonlinear processes. Therefore simplifying criteria for the optimization are desirable. First, one has to define the wavelength range necessary for the specific process which determines the choice of a metal for the cladding. The guiding properties of the waveguide are characterized by the loss $\overline{\alpha }$ averaged over the corresponding wavelength range. For fixed parameters of the metallic cladding and the dielectric coating $\overline{\alpha }$(R, a) is a function of the waveguide radius R and the coating thickness a. Next one has to define the useful waveguide length L limited by practical reasons or by the effective length in the specific process as well as the admissible transmission T. Then, the acceptable design parameters R and a lie in the region defined by exp(-L$\overline{\alpha }$(R, a)) > T. From this region, we choose the point with the lowest waveguide radius, thus the highest waveguide contribution to GVD allowing us to achieve phase matching at higher pressures and higher efficiencies of the corresponding nonlinear process.

Let us discuss two examples for such design optimization. First we consider nonlinear processes in the optical-IR range from 600 nm to 1200 nm, for example soliton-induced supercontinuum generation or high-order soliton compression without external chirp compensation. We assume L = 1 m and T = 1/e, which yields $\overline{\alpha }$(R, a) < 4.34 dB/m. In Fig. 5(a), the dependence of the average loss in the range from 600 nm to 1200 nm on the waveguide radius and the coating thickness is illustrated. The loss reduces with increasing radius as expected, and there is an optimum value a = 84 nm which is almost independent on the radius. The green line in Fig. 5(a) connects the parameters which yield the average loss $\overline{\alpha }$ of 4.343 dB/m and thus the effective length 1/$\overline{\alpha }$ of 1 m. The optimum waveguide in terms of the above design criteria has the radius of 15 µm and a coating thickness of 84 nm. Note that this radius is significantly lower than that typically used currently, which is made available by the better guiding properties of the metal cladding in comparison to a dielectric one. In Figs. 5(b) and (c), the GVD and the loss for the optimum waveguide are presented as functions of the wavelengths. The group velocity dispersion for the argon gas filling at 1 atm is anomalous for wavelengths larger than 520 nm and up to far IR. Even if the pressure is increased to 25 atm, the zero-GVD wavelength does not move to far IR as in a large-radius waveguide, but remains in the visible at 760 nm. A zero-GVD wavelength in the visible is especially important for soliton-related effects such as soliton-induced supercontinuum generation. A high gas pressure leads to a high nonlinearity therefore to a very efficient spectral broadening. The loss exhibits a resonance peak at about 450 nm, a minimum at 663 nm and then increases with the wavelength.

 

Fig. 5. Optimization map for supercontinuum generation (a), and GVD (b) and loss (c) of a coated silver waveguide with optimum parameters R = 15 µm and a = 84 nm. In (a), the average loss of a silver waveguide in the wavelength range from 600 nm to 1200 nm as a function of the waveguide radius R and the coating thickness a is presented. In (b), the gas filling is argon at 1 atm (green dashed curve) and 25 atm (red solid curve).

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As a second example we discuss four-wave mixing in hollow waveguides. To be specific, we consider VUV pulse generation by four-wave mixing using the fundamental (800 nm) and the third harmonic (266 nm) of Ti:sapphire laser as idler and pump source, respectively [11]. Then the signal frequency of the 3ω+3ω-ω→5ω process is at the fifth harmonic (160 nm). Therefore the average loss at these wavelengths has to be used for optimization. In Fig. 6(a), the corresponding optimization map is presented for the generation of the fifth harmonic by the four-wave mixing process. Aluminum is chosen instead of silver as the waveguide material. Coating thickness a around 25 nm, 50 nm, and 100 nm lead to resonances close to one of the above wavelengths, and low loss is achieved far from these values. From the green optimization curve we obtain the optimum waveguide parameters R = 19.5 µm and a = 75 nm in this case. In Figs. 6(b), (c), and (d) the GVD, loss, and the wavevector mismatch of such waveguide are illustrated. The group velocity dispersion [Fig.6(b)] exhibits jumps at the positions of the resonances, but is anomalous in the wavelength range above 400 nm and low at the relevant frequencies. The loss of the waveguide [Fig. 6(c)] is especially low at the wavelength of fifth and third harmonic. Finally, in Fig. 6(d) the phase mismatch for this waveguide is illustrated by the red solid curve as a function of argon pressure, showing phase-matching at 0.226 atm. In comparison, using fused-silica waveguide with the same average loss (R = 45.5 µm), the phase-matching pressure would be only 0.036 atm, as illustrated by the green dashed curve in Fig. 6(d). Thus the use of the fused-silica-coated aluminum waveguide allows to increase the efficiency, which scales as the second power of pressure, by a factor of roughly 35.

 

Fig. 6. Optimization map for 4-wave-mixing (a), and GVD (b), loss (c), and wavevector mismatch (d) of a coated aluminum waveguide with optimum parameters R = 19.5 µm and a = 75 nm. In (b), the gas filling is argon at 0.226 atm. In (a), the average loss at the wavelengths of 160 nm, 266 nm, and 800 nm is presented as a function of R and a. The green curves connect the points with average loss of 4.34 dB/m. In (d), the phase mis-match for the 5th-harmonic generation by four-wave mixing is presented for the optimum waveguide (red solid curve) and for a fused-silica waveguide with the same average loss (green dashed curve).

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5. Conclusion

In conclusion, we predict that hollow waveguides with a metal cladding and a dielectric coating at the inner have a promising potential for various ultrabroadband nonlinear processes. It has both low loss in a wide spectral range and a strong waveguide contribution to dispersion which allows to achieve anomalous GVD in the visible even at high pressures and improved phase matching for nonlinear processes, as well as good mode localization. Dielectric coating contributes to the reduction of the waveguide and scattering loss, allowing relatively small waveguide radii in the range of 10–40 µm.