1. Introduction

Supercontinuum (SC) generation in optical fibers has been widely applied in diverse research fields such as telecommunication [1], spectroscopy [2], microscopy [3], and optical coherence tomography [4]. Most of these applications are found in the visible and near-infrared wavelength region due to the availability of suitable pump sources and the design flexibility offered by silica photonic crystal fibers (PCFs) at these wavelengths [5].

However, there is an emerging requirement especially in spectroscopy and fluorescence microscopy for light sources in the ultraviolet (UV), since many photo-induced processes are excited in this wavelength region [6]. Therefore, recent studies have tried to extend the bandwidth of the generated SC on the short-wavelength edge [7–16]. These studies rely on dispersive wave generation accompanied by soliton-related effects in PCFs with one or two zero dispersion wavelengths (ZDWs).

Usually, the group velocity dispersion (GVD) of PCFs is carefully chosen in relation to the pump wavelength. However, efficient generation of a dispersive wave at UV wavelengths requires phase matching not only with the original soliton arising at the pump wavelength, but also with the self-frequency-shifted solitons in every state of propagation. This can be achieved by a careful design of the PCF dispersion characteristics, which have to change during pulse propagation in relation to the local soliton wavelength.

This concept was initially demonstrated by combining multiple photonic crystal fibers with sequentially decreasing zero-dispersion wavelengths [11] and was later improved by using a 5 m long fiber taper with continuously varying dispersion properties [12,13]. Following these approaches, significant spectral power densities were generated down to a wavelength of 330 nm. Shorter wavelengths seem to be inaccessible by this approach due to the fact that phase matching is difficult to achieve for these short wavelengths. In addition, relatively long fibers are required, so that the material losses become significant, which rise sharply for wavelengths in this region.

Very recently, taper transitions with a length in the mm-region were successfully used for supercontinuum generation down to 280 nm wavelength [16]. Therefore, a modified approach compared to the one described above was pursued. These short wavelengths are generated by the soliton fission process itself. This requires an ultra-intense spike of electromagnetic radiation whose breakup is triggered within the taper transition at a very small core diameter.

Furthermore, taper transitions with a length in the cm-region [14,15] and, likewise, the homogeneous waist of optical fiber tapers [9,10] were investigated with respect to their suitability for UV wavelength generation. In both cases the shortest spectral components were situated at wavelengths around 400 nm.

In the present paper we suggest another approach for SC generation in the UV wavelength range, based on all-normal dispersion (ANDi) optical fibers. The main advantage of this approach is the generation of highly coherent SC spectra and the preservation of a single temporal pulse featuring plain phase distribution. These unique properties are ideal for the application of the SC in broadband time-resolved spectroscopy or the generation of frequency combs for ultra-high resolution spectroscopy [17]. In contrast to conventional fibers, ANDi fibers do not exhibit any ZDWs across the wavelength region of interest, and pumping occurs with femtosecond pulse sources close to the maximum of the convex dispersion curve in the normal dispersion region. Consequently, noise-sensitive soliton dynamics are suppressed, and spectral broadening is dominated by self-phase modulation (SPM) followed by optical wave breaking and four-wave mixing (FWM), which are highly reproducible processes and therefore conserve the temporal coherence of the pump pulse [18]. Optimization of the ANDi fiber design allows the generation of more-than-octave-spanning SC spectra, which are generally highly coherent over the entire bandwidth and do not exhibit significant noise-seeded fluctuations or fine structures [19]. The well-behaved temporal profile of the continuum in combination with the excellent coherence properties allows recompression to Fourier-limited single-cycle pulses [20,21]. Therefore, this type of SC offers improved stability and attractive temporal properties compared to continua generated by the soliton fission process examined in previous studies.

Within the diversity of ANDi optical fibers [22], optical nanofibers with a dispersion maximum around 500 nm were recently proposed for deep UV SC generation due to a promising combination of low dispersion values, short pump wavelength and short propagation distance [23].

In this paper we report on a comprehensive numerical study about such nanoscale ANDi optical fibers and their applicability for deep UV wavelength generation. We discuss the possibilities of nanofibers as well as suspended-core fibers (SCFs) and compare them to each other. At first, we briefly explain the dispersion properties as well as the numerical model employed for nonlinear pulse propagation. Then we describe detailed investigations into the influence of important parameters such as pump wavelength, peak power, fiber geometry, and propagation loss.

Taper transitions are often used for enhancing the efficiency of light input coupling into nanoscale optical fibers. Their influence on nonlinear pulse propagation cannot be neglected, and we therefore discuss their impact on SC generation as well. We report on extensive investigations concerning the shape and length of potential taper transitions and their suitability for transferring the free-space pulse parameters into nanofibers as well as nanoscale SCFs.

2. Dispersion properties

ANDi optical fibers exhibit a convex dispersion profile, which solely covers the normal dispersion region. Dispersion values closest to zero can be found at the maximum of the dispersion curve. The corresponding wavelength is the maximum dispersion wavelength (MDW).

The possibilities of creating optical fibers with ANDi behavior by exploiting various fiber geometries were investigated recently in [22]. It was shown that cylindrically symmetric strands of silica surrounded by air, termed nanofibers in the present paper, exhibit the shortest MDW. An all-normal dispersion profile can be found for nanofiber diameters below 470 nm with an MDW of 490 nm. The MDW decreases with decreasing fiber diameter, while the absolute dispersion value increases simultaneously. This behavior is indicated in Fig. 1(a) . SCFs with a nanoscale core can also exhibit an all-normal GVD with MDWs between 500 nm and 600 nm. The GVD profiles of different SCFs with varying incircle core diameter supported by four walls with a thickness of 50 nm are shown in Fig. 1(b). In this paper, both fiber geometries are considered for SC generation at deep UV wavelengths.

 

Fig. 1 ANDi profiles a) of nanofibers for different diameters and b) of tetragonal suspended-core fibers for different incircle diameters with 50 nm wall thickness. The inset of a) is a side view of a nanofiber, and b) shows an exemplary cross-section of a tetragonal suspended-core fiber.

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3. Numerical model

Pulse propagation in the nanofiber is simulated according to the generalized nonlinear Schrödinger equation (GNLSE) and implemented by means of the Runge-Kutta in-the-interaction-picture method for integration [24], while the conservation quantity error method is used for adaptive stepping [25]. For a detailed discussion of the numerical model we refer to our previous explanations in [18,19]. A non-chirped Gaussian input pulse intensity profile is assumed.

Although it has been argued that the scalar GNLSE is inadequate for describing ultra- short pulse propagation in waveguides with wavelength-scale structures and high index contrast [26], good agreement between simulated and experimental spectral bandwidth and shape was obtained in tapered suspended core fibers with core diameters similar to the ones discussed here [27]. Therefore, the model is considered sufficient.

In the case of ANDi PCFs [19], and also in the case of tapered suspended-core fibers [27], it was found that the model with a constant nonlinear parameter γ(ω) = γ(ω₀), where ω₀ is the pump frequency, delivered better agreement with experimental results than the model with a varying nonlinear parameter. Therefore, a constant γ is used in this paper, too. Concerning the subject of UV SC generation, this is a conservative approach, since a rigorous treatment of γ results in the generation of even shorter wavelengths.

Note that the constancy of the nonlinear parameter refers to the wavelength only. The nonlinear parameter is still a function of the fiber diameter. For spectral evolution along the taper transitions, the nonlinear parameter and the dispersion were changed in steps corresponding to the current position on the taper transition.

Propagation loss was not allowed for in the calculation, except in the section discussing the influence of propagation loss.

4. Supercontinuum generation

4.1 Influence of pump wavelength

The choice of the pump wavelength is an important aspect. Should it be as short as possible to push the whole spectral range towards the desired direction, or should it be close to the dispersion maximum to achieve maximum spectral broadening? Using a nanofiber with an MDW around 500 nm, the first case is conveniently satisfied by using a frequency-doubled Ti:sapphire laser at 400 nm as the pump, and the second one by using a frequency-doubled Yb laser at 515 nm.

Figure 2 shows the spectra generated in a 10 mm long piece of a nanofiber of 440 nm diameter as a function of the difference between pump wavelength and MDW. No additional spectral broadening occurs after this propagation distance. The pump wavelength is varied by 300 nm around the MDW located at 468 nm.

 

Fig. 2 Influence of the pump wavelength λ_pump on the generated SC for a nanofiber 440 nm in diameter. The MDW λ_MDW is 468 nm. The pulse duration is 50 fs, and the pulse energy is 1 nJ. The white line indicates the pump wavelength.

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Figure 2 reveals a positive correlation between the shift of the pump wavelength and the shift of the spectral boundaries within the calculated parameter range. Shifting the pump wavelength to shorter values shifts the short wavelength edge to shorter wavelengths as well. Furthermore, a mismatch of up to 50 nm between MDW and pump wavelength has only a minor influence on spectral width and spectral homogeneity. Beginning at a mismatch of about ± 100 nm, spectral flatness drops due to the formation of a low intensity shoulder.

Comparable behavior was found for simulations in the near infrared wavelength range and for very small dispersion values at the MDW. Due to the large parameter range covered, this behavior is expected to be generally applicable to ANDi fibers. Hence, the shorter the pump wavelength, the shorter the generated wavelength components will be. For pushing the SC towards the UV, then, a pump wavelength of 400 nm is preferable to one of 515 nm. All subsequent simulations are performed at 400 nm pump wavelength.

4.2 Influence of peak power

In ANDi optical fibers, the SC spectral width is governed by the peak power of the input pulse and not affected by specific pulse duration or pulse energy values [18]. Figure 3(a) shows several spectra as a function of input pulse peak power for a nanofiber with a diameter of 450 nm. The dispersion maximum of −65 ps/nm/km is located at 476 nm. Figure 3(a) illustrates that already at moderate peak power levels of 5 kW, spectral components around 320 nm (−20 dB) can be expected, which is a shorter wavelength than experimentally obtained for ANDi fibers so far. With higher peak power, the spectrum broadens at both the short and long wavelength ends. Thereby the short wavelength edge gradually approaches values near 200 nm wavelength.

 

Fig. 3 Generated SC as a function of peak power within a) a nanofiber with 450 nm diameter and b) an SCF with 500 nm incircle core diameter and four supporting walls 50 nm in thickness. The pulse duration is 50 fs, and the propagation length is 20 mm in all cases.

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Figure 3(b) shows generated spectra as a function of input pulse peak power for a nanoscale SCF with 500 nm incircle diameter and 50 nm wall width. The dispersion maximum of −47 ps/nm/km is located at 565 nm. The overall appearance and behavior is similar to that of the nanofiber. However, due to the red-shifted dispersion, all spectra generated are red-shifted as well, and less UV wavelengths are generated. Moderate peak power levels around 5 kW result in a UV wavelength edge around 340 nm, whereas peak power levels around 20 kW and above are required to generate wavelength components below 300 nm.

Cylindrically symmetric nanofibers perform slightly better than SCFs due to their blue-shifted GVD. However, the SCF's reduced spectral extension into the UV is accompanied by definite practical advantages like increased mechanical stability and inherent shielding against environmental influences. The GVD of the SCF can be pushed towards the nanofiber GVD by increasing the number of supporting walls [22]. This modification is counteracted by the wall width. Increasing the number of walls from four to six at the current wall width of 50 nm does not reveal any visible advantages with regard to the UV wavelength edge. A greater number of walls is only beneficial if a wall thickness of 20 nm and below is used.

The limit regarding the shortest UV wavelengths generated arises from the maximum amount of peak power that can be coupled into the fiber experimentally. There are no limits resulting from the nonlinear optics point of view, as no phase matching or similar conditions need to be fulfilled. Previous SC experiments in the visible wavelength range using an ANDi SCF with an incircle core diameter of 540 nm were performed at a peak power of 25 kW [27]. This peak power level would lead to wavelengths around 270 nm in case of the nanofiber, or to wavelengths around 290 nm in case of the SCF. Pump peak power levels in the order of 50 kW, as used in the experiments in [16], would generate wavelengths down to 240 nm in case of the nanofiber, and down to 270 nm in case of the SCF.

4.3 Influence of fiber diameter

Besides pump wavelength and peak power, the appropriate fiber size is of major interest as well. Are small dispersion values preferable for an extended spectral width, or should the maximum dispersion wavelength be as short as possible? These are opposite properties as indicated in Fig. 1(a). The influence of the nanofiber diameter or SCF incircle diameter on the generated SC is shown in Fig. 4(a) and 4(b), respectively.

 

Fig. 4 Influence of fiber diameter on the generated SC for a) a nanofiber and b) a suspended-core fiber. With decreasing fiber diameter, both edges of the spectra shift to shorter wavelengths. The inset shows an increased illustration of the UV edge. The pulse duration is 50 fs, the peak power is 20 kW, and the fiber length is 20 mm.

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In both cases, the broadest spectrum is generated in the largest diameter fiber due to its smallest absolute dispersion value. With decreasing diameter, an obvious reduction of the long wavelength edge by several tens of nm occurs. As explained in detail in [19], the extreme wavelength components ω_FWM in ANDi fibers are generated by wave-breaking-induced FWM processes according to

Dispersion in the UV range is only marginally affected by the fiber diameter. Nonetheless, the short wavelength can be extended by a few nm using smaller nanofibers. The slightly larger dispersion at UV wavelengths in smaller nanofibers is counterbalanced by an increased nonlinearity, which enhances both SPM broadening and wave breaking effects.

In summary, a smaller fiber enhances the UV edge of the generated SC by a few nanometers. But the comparatively small benefit must be balanced against potential practical disadvantages, e.g. with regard to input coupling efficiency.

4.4 Influence of propagation loss

All the above simulations were performed with propagation losses left out of consideration. However, propagation losses can affect SC generation especially in the UV wavelength range, where losses tend to increase in comparison to the low propagation loss regions of fused silica in the visible and near infrared wavelength ranges. As an exemplary loss behavior, the measured propagation loss of a tetragonal SCF was chosen as shown in Fig. 5 (black line). These loss values were taken into account, and pulse propagation in a nanofiber with 440 nm diameter with and without these losses was simulated. A pulse width of 50 fs and a pulse energy of 1 nJ were assumed.

 

Fig. 5 Influence of propagation loss on SC generation. The SCs generated after a nanofiber of 50 mm in length and 440 nm in diameter with and without loss are shown in blue and red, respectively. The propagation losses considered (measurement) of a tetragonal SCF are shown in black. For wavelengths below 350 nm, the linear slope of the attenuation curve was extrapolated.

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Figure 5 compares the final SC of both simulations and shows that only minor deviations can be found. The inclusion of propagation loss, especially, does not affect the UV wavelength edge. This is due to the overall very short propagation distance required in ANDi optical fibers for the nonlinear effects to evolve. For the SC displayed in Fig. 5, the propagation distance was set to 50 mm for differences to become visible. At the minimum required propagation distance of 5 mm, where SC generation is mostly completed, both displayed SCs are virtually undistinguishable. Thus, consideration of propagation loss does not produce any relevant effect on the UV edge of the generated SC.

Note that recently two-photon absorption (TPA) has been suspected to contribute to the propagation losses in silica in the order of 100 dB/mm for wavelengths below 250 nm [16]. These nonlinear losses are not included in our simulations, as no reliable data is yet available. Should these values be confirmed, the TPA cut-off wavelength would constitute a hard barrier for any UV SC generation process in silica and naturally would also limit the achievable short wavelength edge in ANDi fibers.

5. Taper transitions for input coupling purposes

Taper transitions are often used to enhance the input coupling efficiency for nanoscale fibers. However, they cannot be used without affecting pulse propagation, which is why their influence has to be considered. At a pump wavelength of 400 nm, in particular, the high dispersion of silica and the high nonlinearities in the nanoscale sections of the taper transition are expected to distort the input pulse significantly before the actual ANDi taper waist is reached. In this section we investigate different taper profiles for their potential to minimize pulse distortions in the transition region.

5.1 Adiabaticity and shape

All aforementioned analysis and simulations rely on the properties of the fundamental mode. Therefore, all the power launched into a taper transition must be maintained within the fundamental mode, and energy transfer to higher-order modes due to mode coupling must not occur. This requirement is governed by the adiabatic criterion [28]. The equations in [28] are based on the core radius and core taper angle and were rearranged in terms of cladding radius and cladding taper angle by a linear coordinate transformation.

Based on the difference Δβ = |β₁-β₂| of the local propagation constants β₁ of the fundamental mode and β₂ of the competitive higher-order mode, it determines a delineation taper angle Ω that separates adiabatic and hence, lossless, from non-adiabatic propagation:

Regarding the transition shape, three selected profiles are compared to each other. These are the shortest exponential profile, the shortest linear profile and the overall shortest profile that satisfy the adiabatic criterion. For simplification they are termed exponential, linear, and adiabatic profile, respectively. A theoretical procedure of how to fabricate these profiles is outlined in [29].

Subsequent considerations on nanofibers are based upon the commercially available fiber S405-HP from Nufern. This fiber has a core diameter of 2.5 µm, a numerical aperture of 0.115, a pure silica core with very low loss, and is single-mode at the pump wavelength of 400 nm. The light is coupled into the fiber core and is initially guided by the core-cladding interface. When the original core gets small enough somewhere along the taper transition, core-cladding guidance changes into cladding-air guidance.

Figure 6(a) demonstrates the relation between the adiabatic criterion and the taper angle Ω for all three profiles. The taper angle variation of the adiabatic profile exactly equals the adiabatic criterion. Steeper taper angles at any point of the taper transition would violate the adiabatic criterion and cause energy transfer from the fundamental mode HE₁₁ to the higher-order mode HE₁₂. The exponential and linear profiles show notably smaller taper angles. In both cases the adiabatic criterion is exhausted at a single point, and every contraction of these profiles would violate Eq. (2) at this point.

 

Fig. 6 Properties of the taper profiles discussed. Taper angles Ω as a function of fiber radius r are shown in a). Resulting profiles r(z) are shown in b).

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Based on the local taper angles, the overall profile r(z) for the transition from 62.5 µm to 0.2 µm in radius is calculated as displayed in Fig. 6(b). Most notably, the overall transition length varies to a large extent. With 31.6 mm, the exponential profile is more than seven times longer than the adiabatic profile (4.3 mm).

5.2 Nonlinear pulse propagation

Nonlinear pulse propagation within these profiles is illustrated in Fig. 7 . The input pulse has a peak power of 25 kW and a pulse width of 100 fs. The subsequent statements are not restricted to these specific parameters but are valid for similar values as well.

 

Fig. 7 Nonlinear pulse propagation within the taper transitions and the nanofiber. Spectral evolution is shown along a transition of a) adiabatic, b) linear and c) exponential profile for an input pulse of 25 kW peak power and 100 fs pulse duration. Spectral evolution along these transitions and a subsequent nanofiber is shown in d), e), and f), respectively. The vertical white line marks the end of the taper transition.

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Figures 7(a) to 7(c) show the spectral evolution along the adiabatic, linear and exponential taper profile, respectively. The adiabatic profile leads to only minor spectral changes along the taper transition (Fig. 7(a)). Therefore, this profile is excellently suitable to convey the free-space properties of the pump pulse into the nanofiber. The linear profile features spectral broadening right from the start of pulse propagation (Fig. 7(b)). SPM broadens the spectrum as long as the fundamental mode is confined within the original core. As the fiber diameter is further decreased along the taper transition, the fundamental mode extends well beyond the original core, and guiding by the silica-air interface begins. SPM broadening stops at this point due to the accompanied increase in mode field diameter and the drop in nonlinearity. The nanoscale part of the linear profile with high nonlinearity is sufficiently short to prevent further spectral broadening. The exponential profile exhibits additional spectral broadening in the nanoscale part of the transition (Fig. 7(c)), since it is much longer compared to the previous ones. The result is a strongly modulated spectrum. Therefore, the exponential profile is inappropriate as a pulse-conserving input coupling geometry.

The transition geometry has a considerable impact on temporal pulse properties as well. For instance, starting at 47 kW, the peak power after the adiabatic, linear, and exponential profiles is decreased to 35 kW, 8.3 kW, and 6.3 kW, respectively. Since the peak power defines the spectral bandwidth achievable in the subsequent SC generation process, much narrower spectra are generated for the latter two profiles.

The impact of the pulse distortions occurring within the taper transition on the subsequent spectral evolution and the resulting spectrum is shown in Fig. 7(d) to 7(f). Figure 7(d) reveals that nonlinear processes start immediately after the adiabatic transition, and SPM and FWM proceed as expected in ANDi optical fibers. By contrast, significant distortions are obvious for the linear (Fig. 7(e)) and the exponential profile (Fig. 7(f)). Most importantly, the UV wavelength edge is significantly reduced from 270 nm for the adiabatic profile to 310 nm and 320 nm for the linear and exponential profiles, respectively. Thus, the adiabatic profile is to be preferred for deep UV wavelength generation.

For the profiles discussed, the adiabatic criterion was evaluated concerning a potential coupling between the fundamental mode HE₁₁ and the next higher-order mode of equal azimuthal symmetry HE₁₂. This condition is valid only if the rotational symmetry of the fiber is maintained during taper fabrication. This cannot be taken for granted, since in most cases a one-sided heat source is used. For deformed geometries without rotational symmetry, the adiabatic criterion has to be evaluated concerning the higher-order mode that is closest to HE₁₁, e.g. TE₀₁. This amplifies all discussed distortions of the injected pulse. For instance, the peak power after the transitions drops down to 30 kW, 5.1 kW, and 3.8 kW for the adiabatic, linear and exponential profile, respectively.

The aforementioned results and conclusions apply not only to the specifically discussed fiber S405-HP, but to all optical fibers with a solid design. In every case the linear profile will lead to SPM in the original fiber core due to its moderate diameter reduction in this range. Likewise, the exponential profile will always lead to additional nonlinear effects when the transition diameter approaches the final nanoscale value. Both critical diameter ranges have to be overcome as fast as possible. These demands perfectly coincide with the adiabatic profile, making it the best choice.

Suspended-core fibers require a much smaller diameter reduction from the original core size to the tapered size, because the transition from core to cladding guidance, which occurs in all-solid fiber designs, is avoided. The initial core diameter may be in the order of a few micrometers, causing a large difference of the involved propagation constants of fundamental and higher-order modes for the entire transition profile. Thus, larger taper angles as before can be applied for the linear and the exponential transition profile without violating adiabaticity as demonstrated in Fig. 8(a) . This results in the very short transition lengths seen in Fig. 8(b).

 

Fig. 8 Typical taper transition properties in case of suspended-core fibers. Taper angle as a function of radius is shown in a), and resulting profiles are shown in b). Spectral evolutions along the adiabatic, linear, and exponential profile and additional 2.5 mm of the nanoscale fiber are shown in c), d), and e), respectively. The transitions start at a radius of 2.5 µm and end at 200 nm. The vertical white line marks the end of the taper transition.

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The numerical estimation is based on the S405-HP and not on a particular SCF geometry. Nevertheless, the results are generally representative for all ANDi nanoscale fibers. This is due to the fact that the transition properties are defined by the difference of the propagation constants involved, which essentially depends on core size.

With an untapered core radius of e.g. 2.5 µm to start from, the shortest possible lengths of the taper transitions while maintaining adiabaticity are 0.04 mm, 0.07 mm and 0.19 mm for the adiabatic, the linear and the exponential profile, respectively. The spectral evolution along these transitions including an additional 2.5 mm length along the nanoscale fiber with a constant radius of 200 nm is displayed in Figs. 8(c) to 8(e), with initial pulse properties the same as before. As can be seen, the spectral evolution is almost identical for all three profiles. Due to the overall very short transition length, all profiles are equally suited for conserving the input pulse properties. Thus the utilization of taper transitions for input coupling purposes favors the use of suspended-core fibers.

6. Summary

We examined in detail the possibilities of nanoscale all-normal dispersion optical fibers, represented by nanofibers and suspended-core fibers, for deep ultraviolet coherent supercontinuum generation. The influences of important parameters such as pump wavelength, pump peak power, fiber diameter and propagation losses were discussed. As a result, wavelengths below 300 nm seem to be accessible by these geometries. Without preceding taper transitions, cylindrically symmetric nanofibers perform slightly better than suspended-core fibers. This is due to the blue-shifted group velocity dispersion of the nanofibers.

Furthermore, the influence of adiabatic taper transitions for input coupling purposes was investigated. In case of the nanofiber, only the shortest adiabatic profile was shown to be well suitable with regard to the conservation of input pulse properties. Longer profiles cannot sufficiently prevent nonlinear processes within the transitions and therefore significantly reduce the generated components at the UV wavelength edge. In contrast, all discussed adiabatic transition profiles of suspended-core fibers are equally well feasible for input coupling purposes. Due to their short overall length they have no significant impact on the UV wavelength edge generated, and are considered the best choice for extending current supercontinuum limits in ANDi optical fibers.