Generation of infrared supercontinuum radiation: spatial mode dispersion and higher-order mode propagation in ZBLAN step-index fibers

: Using femtosecond upconversion we investigate the time and wavelength structure of infrared supercontinuum generation. It is shown that radiation is scattered into higher order spatial modes (HOMs) when generating a supercontinuum using ﬁbers that are not single-moded, such as a step-index ZBLAN ﬁber. As a consequence of intermodal scattering and the difference in group velocity for the modes, the supercontinuum splits up spatially and temporally. Experimental results indicate that a signiﬁcant part of the radiation propagates in HOMs. Conventional simulations of super-continuum generation do not include scattering into HOMs, and including this provides an extra degree of freedom for tailoring supercontinuum sources.

*keiding@chem.au.dk Abstract: Using femtosecond upconversion we investigate the time and wavelength structure of infrared supercontinuum generation. It is shown that radiation is scattered into higher order spatial modes (HOMs) when generating a supercontinuum using fibers that are not single-moded, such as a step-index ZBLAN fiber. As a consequence of intermodal scattering and the difference in group velocity for the modes, the supercontinuum splits up spatially and temporally. Experimental results indicate that a significant part of the radiation propagates in HOMs. Conventional simulations of supercontinuum generation do not include scattering into HOMs, and including this provides an extra degree of freedom for tailoring supercontinuum sources.

Introduction
In recent years, supercontinuum generation has been a field of great interest, and the physical processes involved in the generation have been studied vigorously (for a review, see [1]). Numerous applications have emerged since the first supercontinuum in bulk glass was reported by Alfano et al. [2], in such diverse fields as spectroscopy [3,4] and optical metrology [5,6]. Recently, supercontinua have been pushed into the mid-infrared, making room for new applications in spectroscopy and microspectroscopy [7][8][9][10], as the molecular fingerprint region can be reached. Novel soft glass fibers such as ZBLAN or chalcogenide are often used to be able to reach into the mid-infrared. Often microstructured fibers are used to tailor the dispersion profile to a specific pump wavelength and supercontinuum profile. However, ZBLAN fibers are not easily microstructured and to develop new fibers optimized for supercontinua generation, it is important to consider the generation process correctly. It is often assumed that the radiation is only propagating single-moded, and simulations are therefore carried out within this premise, i.e. the excitation of higher order spatial modes is not considered. The normalized frequency parameter V determines the number of modes that can propagate through the fiber at a given wavelength, and for step-index fibers, V is defined as [11] where a is the fiber core radius, and NA is the numerical aperture. When V < 2.405, only the fundamental mode is supported by the fiber, and in this case of a 10 μm step-index fiber with NA = 0.2, the cut-off wavelength is 2.6 μm. However, since the generation of infrared wavelengths depends on nonlinear mixing processes across the bandwidth of the supercontinuum, the influence of HOMs on the generation process must be considered.
Here, experiments are carried out to get a closer look at the temporal and spectral distribution of a supercontinuum generated in a step-index ZBLAN fiber. It is shown that even though the fiber is pumped above the cut-off wavelength of specific higher order spatial modes, these modes are still excited as the supercontinuum is generated. This leads to significant delays between the radiation propagating in different modes, as a consequence of their different group velocities. This must therefore be considered, when modelling supercontinuum generation in multimoded fibers, or in pulse compression applications. The multimode propagation is also important if diffraction-limited focussing is desired.
The approach for investigating the temporal and spectral distribution of the supercontinuum is based on a crosscorrelation using sum frequency generation (SFG) in a nonlinear crystal [12]. By upconverting the entire supercontinuum with a femtosecond pulse, spectrograms are recorded that can reveal the propagation in higher-order modes. A schematic overview of the setup is shown in Fig. 1. A 1 kHz Ti:sapphire system emitting 100 fs pulses with a pulse energy of 0.5 mJ is used to pump an OPA. The idler from the OPA is used to pump a 28 cm Fiberlabs ZBLAN fiber with a core diameter of 10.0 μm, and a numerical aperture of NA = 0.2. The fundamental mode has a zero dispersion wavelength close to 1.65 μm. In this setup, the fiber pumping power is adjusted using neutral density filters, to achieve typical pulse-energies of 100 nJ before coupling into the fiber. The autocorrelation of the idler from the OPA shows a temporal width of 75 fs. The resulting supercontinuum is passed through a 1 mm LiIO 3 crystal. The crystal is mounted on a rotation stage to be able to control the phase matching condition for sum frequency generation. Another arm with 800 nm fs pulses from the Ti:sapphire source is sent through a linear delay stage, and then passed through the LiIO 3 crystal as well. The signal is measured by a Si-diode. Since each wavelength of the supercontinuum is phase matched at a particular crystal angle, this angle can be used to measure the wavelength uniquely, thus no monochromator is needed. This measure of wavelength is calibrated by using a spectrometer to make sure the crystal angles correspond to the correct supercontinuum wavelengths. The spectral resolution is given by the phase matching bandwidth of the crystal, and differs over the spectral range covered by the supercontinuum. At 4 μm the phasematching bandwidth is approximately 0.3 μm, whereas it is only approximately 0.03 μm at 2 μm. The resolution on the time axis of the spectrograms is limited by the temporal width of the 800 nm upconversion pulse.

Results
Supercontinua are commonly described using the generalized nonlinear Schrödinger equation (GNLSE), accounting for the important processes that act together to constitute the generation process [1,11]. The dispersion profile of the fiber is of major importance, and pumping at wavelengths in the anomalous dispersion regime will allow soliton dynamics to be a governing where γ is the nonlinear parameter, P 0 is the peak power, T 0 is the temporal width, and β 2 is the group velocity dispersion parameter. The soliton number becomes larger when pumping near the ZDW in the anomalous dispersion regime. A high soliton number and pumping near the ZDW facilitates radiation into the normal dispersion regime through emission of dispersive waves [11,13]. The dispersive wave generation relies on a soliton fission causing emission of these waves, which extend into the region of normal dispersion. When pumping close to the ZDW the phase matched wavelengths will be spectrally closer to the soliton spectrum giving better overlap. In addition, solitons created through soliton fission close to the ZDW are temporally shorter and thus spectrally broader and with higher peak power, providing higher gain for the dispersive waves. Therefore, the supercontinuum is expected to be spectrally broad when the soliton number is high. In this case, the fiber is pumped by free space coupling with pulse energies of approximately 30 nJ (including coupling losses), centered at 1.8 μm.  measured using a monochromator, and is seen in Fig. 2. This spectrum is broad as expected, and correspond well to [10], Agger et. al.
The corresponding spectrogram from the crosscorrelation of this supercontinuum is seen in Fig. 3(a). It is seen that the spectral features in the spectrogram correspond qualitatively well to the measured spectrum, spanning from 0.8 μm to 3.5 μm. The leading edge of the pulse is seen at 0 ps. The supercontinuum is split up temporally, especially in the short wavelength region of the spectrum. This clearly indicates that HOMs have been excited, and since these modes have different group velocity dispersion, temporal splitting occurs. The impact of input polarization on the spectrograms has been investigated. No changes in the spectrogram were observed when turning the input end of the fiber without changing the polarization of the pump laser. A halfwave plate was furthermore introduced to the supercontinuum to turn the polarization. This did not change the spectrograms in any way, indicating that the supercontinuum is unpolarized, since the SFG crystal acts as a polarizer. In Fig. 3  Here it is evident that the supercontinuum is not broadened as efficiently, as expected since the pump is further away from the ZDW, giving a lower soliton number of approximately 8. When pumping at 1.8 μm, a range of strong echoes trailing the main pulse are seen. When the pump wavelength is moved to 2.2 μm, these echoes are absent. When pumping at 2.6 μm (not shown) the echoes appear again. This is identified as free induction decay in the atmosphere [15,16], as water has a strong absorption line at around 1.8 μm, and the pump propagates approximately 3 m in free space. This effect is absent at 2.2 μm, as the absorption in the atmosphere is low at this wavelength, and is seen again (not shown) when pumping at 2.6 μm, due to a combination of H 2 O and CO 2 absorption lines.

Discussion
A discussion of the temporal and spectral distribution of the supercontinuum is provided in this section. It is important to consider how the radiation is scattered into the higher-order modes to understand the resulting spectrograms. The propagation of a supercontinuum in higher-order modes has earlier been discussed in detail in the context of simulations [17,18]. Simulations are carried out using the multi-mode variant of the GNLSE (MM-GNLSE) in [18]. This approach requires complex computer simulations, and restrictions due to computation time apply, hence numerical approximations are necessary. Here, in order to get further insight into the mode scattering process, the expected arrival time for each wavelength component of the supercontinuum is calculated using the group velocity of the particular mode that the radiation propagates within. The arrival times are expected to follow this group delay, reported earlier for propagation in the fundamental mode in [12]. Therefore, this qualitative and simple analysis for identification of the excited modes is useful, based on a model, where the pulse is fully broadened into its spectral components instantaneously at the input end of the fiber, and then propagates in the first 6 modes with their group velocity along the entire fiber length. This is a good assumption, as the soliton fission distance L fiss [1] is only 3-4 cm, and the change in the dispersion of the arrival times is comparable to the resolution of the experiment. The resulting calculations are shown along with the experimental data in Fig. 4(a). The ZBLAN index profile is calculated on the basis of the Sellmeier equations provided by Gan et. al. [14] combined with the waveguide dispersion.  From Eq. (1), the fiber is single-moded for wavelengths longer than 2.6 μm. It is clearly seen that above this wavelength, the supercontinuum is only propagating in the fundamental mode. Therefore this mode is identified as the LP 01 mode, and the zero dispersion wavelength for the fundamental mode is found to be around 1.65 μm as expected. Furthermore, integration across the spectrogram shows that approximately 40 % of the energy is propagating in other modes than the fundamental mode. It is evident from Fig. 4 that the excited modes are LP 01 , LP 11 and LP 21 . These modes are excited exclusively, and excitation of the LP 02 and LP 03 modes was not observed. In order to exclude propagation in HOMs due to excitation caused by direct coupling of pump radiation into these modes, the fiber is pumped at 2.2 μm. The LP 21 mode is excited even when pumping above the cut-off wavelength of this mode, as seen in Fig. 4(b). Therefore, the excitation of the LP 21 mode is due to an intermodal scattering process in the fiber, as it is not possible to couple into this mode at 2.2 μm. As also observed in [18,19], the excitation of HOMs is inevitable when dealing with high-power supercontinuum generation in fibers where HOMs are supported.
Several situations are considered in [18], pumping in various modes and allowing different mode coupling mechanisms to be present. When pumping in the fundamental mode of the fiber, four-wave mixing (FWM) into modes of the same symmetry class is found to be dominant. In particular, when pumping in the anomalous dispersion regime, radiation on the short-wavelength side of the ZDW is emitted by the solitons created from the soliton fission process. This phenomenon is known as emission of dispersive waves. Emission of intermodal dispersive waves has previously been reported experimentally [20]. The transfer of radiation into higher-order modes happens along the fiber, and is governed by a group velocity matching condition (or soliton trapping, [11]) for this process. This makes the dispersive wave from a given soliton travel at the same speed as the soliton. In this way, the time structure is welldefined to each mode and follows the group velocity, even though the intermodal scattering happens along the fiber, as the supercontinuum is broadening spectrally. The probability of the intermodal scattering process by emission of dispersive waves is dependent on the electric field overlap integral, and certain combinations of intermodal scattering are not allowed, depending on the angular symmetry of the modes. To scatter into the LP 11 mode from the LP 01 mode, the overlap integral f 01,01,01,11 (for four-wave mixing between 3 LP 01 and one LP 11 photon) can be written as [11,17,20] f 01,01,01, 11 where F kl is the spatial distribution of fiber mode LP kl , and the angle brackets denote integration over the transverse coordinates. As LP 01 and the LP 11 modes belong to different symmetry classes, the spatial overlap integral f 01,01,01,11 is zero for a perfect cylindrical fiber symmetry. This strict selection rule for intermodal scattering by dispersive waves could be relaxed by imperfections in the fiber. However, in the fiber used in our experiments, an argument against this process is made, as experiments with fiber bending did not change the spectrogram in any way, even though this changes the overlap integral [20]. A situation where the two modes LP 01 and LP 11 are both excited when coupling into the fiber is now considered. In relation to the simulations in [18], pumping different symmetry classes adds complexity to the system, allowing FWM and cross-phase modulation (XPM) to transfer energy between all existing modes, regardless of the previously mentioned symmetry considerations. XPM was also shown in [18] to be important in the broadening of the spectrum in the HOMs, when radiation is present in more than one mode. This situation cannot be ruled out completely, though thorough experimental efforts to couple into the fundamental mode only have been made.
Considering a final mechanism, Raman scattering could potentially scatter into higher order modes, as previously reported in [21] for a hydrogen-filled hollow-core photonic crystal fiber. This requires spatial intensity overlap. The overlap integral f 01,11 for Raman scattering from LP 01 into LP 11 can be written as [11] where F kl is the transverse distribution of fiber mode LP kl . The angle brackets denote integration over the transverse coordinates. Calculations of the effective area (which is the inverse overlap integral for Raman scattering [11]) between the fundamental and the HOMs considered in this paper are shown in Fig. 5  It is seen that intramode scattering has the best overlap (Raman scattering from the fundamental mode into itself), but intermodal scattering is indeed possible. The overlap integral between the LP 01 and the two modes that are actually excited (LP 11 and LP 21 ) is slightly smaller than that of the LP 02 , LP 03 and LP 12 modes, though it is noted that since the excited modes have the highest cut-off wavelengths of the HOMs, they are the first modes available for scattering into as the pump spectrum broadens early in the fiber. The effective areas of the HOMs are comparable to that of the fundamental mode and one must also consider nonlinear broadening effects in the HOMs themselves. However, as evident from Fig. 4, the slope of the calculated group delays in the HOMs are all negative, corresponding to normal dispersion in the HOMs. This will restrict the non-linear broadening mechanisms to Raman scattering and self-phase modulation.
It is clear that multi-mode supercontinuum generation is very complex. When looking at the intensities in the spectrogram, it can be argued that maximum intensities correspond to solitons in the anomalous dispersion regime of the fundamental mode, which are in turn matched in group velocity with maximum intensities in the normal dispersion regime, corresponding to a soliton-trapped dispersive wave at delays of around 3 ps. The wavelengths of the dispersive wave can be calculated [11], and when pumping at 1.8 μm and 2.2 μm, the corresponding dispersive waves are generated at 1.24 μm and 0.98 μm in the LP 01 mode, respectively. This corresponds well to the observed maximum intensities in the fundamental mode, seen in the spectrograms in Fig. 3 for both pump wavelengths. The overlap integral for intermodal scattering from LP 01 to LP 11 is zero, and therefore initial Raman scattering into this mode is likely, transferring energy into both symmetry groups. This is followed by complex scattering mechanisms such as FWM and XPM between the excited modes, as the symmetry restrictions are removed.
To conclude, femtosecond upconversion of an infrared supercontinuum generated in a stepindex ZBLAN fiber is used to obtain detailed spectrograms. The upconversion technique may be used for measuring the group velocity dispersion in soft-glass fibers. It has been shown that radiation can be scattered into higher order modes as the supercontinuum broadens, even when pumping above the cut-off wavelength of a given mode. The modes are identified by calculating arrival times at the fiber end according to the group velocity. These arrival times are in good agreement with the crosscorrelation measurements. A discussion of the possible intermodal scattering mechanisms reveals that supercontinuum scattering into HOMs is very complex, and a combination of effects including Raman scattering, FWM and XPM are likely to scatter into the HOMs. The LP 11 and LP 21 modes are excited exclusively, though other modes such as the LP 02 and LP 03 modes exist in the wavelength region of interest. The supercontinuum breaks up into multiple pulses, and a temporal separation occurs due to the significant difference in group velocity dispersion of the excited modes. For many applications, such as time resolved experiments, or for diffraction limited focussing, it is very important that these dynamics take place, and they must therefore be considered. Approximately 40 % of the supercontinuum radiation is propagating in HOMs. As fiber drawing and tailoring improves, this may be used as an extra degree of freedom for optimization of supercontinuum sources.