Diario de campo

We obtain the GVD and the effective area of the fiber’s fundamental mode by applying the empirical formulae presented in Ref. [28]. We can achieve sufficiently high nonlinearity required for the spectral broadening by filling the fiber with heavy gas, such as xenon at high pressure [45]. This compensates for the loss of intensity due to having a relatively large mode area that is necessary for a low-loss guidance in mid-infrared. Figure4.9presentsGVDand the nonlinear parameterγ when the fiber is filled with xenon at35 barpressure. At this pressure, there is a

ZDWat around2.1µmwith the normal dispersion regime on the short-wavelength side and the anomalous dispersion regime on the other side.

Figure 4.9 GVD(red solid-line) and nonlinear parameter (γ, blue dashed-line) in the negative-

curvature fiber filled with xenon at35 bar. The inset shows the intensity profile of the fundamental

mode. ZD andλ0denote the zero dispersion and pump wavelength, respectively.

4.2.4.

Nonlinear pulse propagation and supercontinuum genera-

tion

The propagation of femtosecond pulses in a nonlinear dispersive medium, such as the xenon-filled negative-curvature fiber, can be modeled using the generalized nonlinear Schrödinger equation [57] given in Eq. (1.24). For an atomic gas such as xenon, we can drop the Raman effect in the nonlinear term. Therefore, Eq. (1.24) has the following form [38]:

∂A ∂z = X k≥₂ ik+1 k! βk ∂k_A ∂Tk +iγ 1+ i ω0 ∂ ∂t ! |_A|2 _A, (4.3)

whereA(z,T)is the complex envelope of the optical field at propagation distance z;T = t−β_1zis time in the frame moving with the group velocity of the pulse

vg=1/β1,βkis thekth order Taylor series expansion coefficient of the wavevector

β(ω) evaluated at the carrier frequency ω0; and γ0 is the nonlinear parameter

at pump wavelength which is determined by the filling gas species and their density. The time derivative in the nonlinear operator represents the effect of

4.2 Normal dispersion regime pumping 65 self-steepening characterized on a time scale1/ω0. Note that xenon may be subject

to photoionization at high optical intensities. However, this has a negligible effect in our examples, and Eq. (4.3) captures all the major effects of the femtosecond pulse propagation studied below.

We consider the propagation of a15µJpump pulse with30 fsduration in the near- infrared at1.06µmwavelength. Such a pulse can be obtained by compressing the output of a high-power fiber laser system that is now made available commercially [109]. The pump wavelength in this example is deep in the normal dispersion regime of the xenon-filled fiber with its ZDW (2.1µm) almost an octave away from the pump as shown in Fig.4.9. One major advantage of pumping in the normal dispersion regime is that the output supercontinuum spectrum is relatively insensitive to the fluctuations of the pump [107]. Figures4.10(a) and (b) show the temporal and spectral evolutions of the pump propagating along a15 cmlength of the negative-curvature fiber. Notice the exceptional spectral broadening that covers a wide spectrum from0.5to4.2µm. In particular, the strong radiations are observed in mid-infrared in the later stage of the propagation.

Soon after the launch of the pump, the combined effect of the self-phase modulation and the normal dispersion causes the low-/high-frequency components generated in the central part of the pulse to advance/lag towards the leading/trailing edges of the pulse, respectively. This results in the pulse shape becoming rectangular with steep edges on the sides. Then the self-steepening effect becomes significant, which further enhances the steep edge in its tail. As a result, the pulse becomes highly asymmetric in time leaning towards its trailing edge. Further steepening leads to breaking of the trailing edge atz∼0.40 cm, with an oscillation forming at the back

of the pulse. We can describe the same phenomena also in the frequency domain. The self-steepening introduces an asymmetric spectral broadening that enhances the short-wavelength side. In Fig.4.10(b), we can see that the asymmetric extent of the spectrum significantly enhances the development of a separate sideband radiation known as dispersive shock wave (DSW) [110] at560 nmatz∼0.55 cm.

This coincides with the appearance of the time domain shock indicated by the arrow in Fig.4.10(a). The pulse spectrum atz=1 cmin Fig.4.10(c) clearly shows

Figure 4.10 (a) Temporal and (b) spectral evolutions of a30 fspump propagating in the xenon-

filled negative-curvature fiber. The pump wavelength is at1.06µmand has an energy of15µJ. The

vertical dotted line in (b) indicatesZDW, where A and N denote anomalous and normal dispersion regimes, respectively. (c) Normalized spectral intensity,Iωatz=1 cm. The spectral band formed as

a result ofDSWis marked with a vertical arrow. (d) The total energy in the anomalous dispersion regime,EA, as a function of the propagation length.

the formation ofDSWat∼560 nm. As we shall see, a phase-matched four-wave

mixing process is responsible for theDSWformation.

One important observation in Fig.4.10(b) is that a substantial portion of the pump energy enters the anomalous dispersion regime during the early spectral broaden- ing process. This, in fact, leads to further broadening of the spectrum deeper into the mid-infrared range in the later stage of the propagation. Figure4.10(d) shows that∼ 2.7%of the pump energy has entered the anomalous dispersion regime

after the pump having propagated7 cmof the xenon-filled fiber. We stress that this happens despite havingZDWfar away – by almost an octave span – from the pump wavelength. Therefore, a simple explanation of the pump tunneling into the

4.2 Normal dispersion regime pumping 67 anomalous dispersion region through self-phase modulation-induced broadening, as has been suggested in a number of previous reports where the pump was near

ZDW[111,112], cannot fully clarify the case presented here.

The phase mismatch∆βplotted in Fig.4.11(a) reveals how the conversion of the pump to two frequencies, one in the form ofDSWand the other deep into the anomalous dispersion regime, occur. The phase mismatch is calculated as below [93], which is slightly different from Eq. (1.34):

∆β=β(ω)−β(ω

0)+β1(ω0) [ω−ω0]+γ0P0[ω/ω0] , (4.4)

whereωis the angular frequency with ω0 denoting that of the pump. P0 is the

peak power of the pump pulse, andβ(ω)is the wavevector. The strong effect of the self-steepening means that for an accurate determination of the phase-matching wavelengths, we need to account for the nonlinear parameter,γ0by multiplying

the term byω/ω0as given in Eq. (4.4) [93].

In Fig. 4.11 (a), the phase mismatch becomes zero at two wavelengths, one at 560 nmand the other at2.55µm, which correspond also to the wavelengths where the radiation bands develop as demonstrated in Fig.4.11(b). Due to its proximity to the pump wavelength as well as the blue-enhanced spectral broadening as a result of the self-steepening effect, the DSW band at 560 nm forms in the very earlier stage (z<1 cm) of the pulse propagation. On the other hand, the2.55µm spectral band appears much later (z > 7 cm), due to the fact that this phase- matching wavelength is further out from the pump, and hence it requires a longer propagation distance for the low-intensity tail in the pump to reach this point through self-phase modulation.

Once the seed photons reach the phase-matching wavelength at2.55µm, the con- version from the pump into the anomalous dispersion regime becomes highly efficient. This is clearly noticeable in Fig.4.10 (d), where the energy in the an- omalous dispersion regime increases rapidly in the propagation length5 cmto 8 cm. In fact, atz=8.10 cm, the energy in the2.55µmspectral band, shown with the green shaded area in Fig.4.11(b), is sufficient to form a higher-order optical

Figure 4.11 (a) Phase mismatch ∆βas a function of wavelength. (b) normalized spectral intensity,

Iωatz=8.10 cm. The inset shows the normalized time-domain intensity profile,Itof the green-

shaded part in the spectrum.

soliton on its own. This spectral band translates into a pulse with17.2 fsfull-width at half-maximum duration and 25.55 MW peak power in the time domain, as presented in the inset of Fig.4.11(b). Taking into account theGVDand nonlinear parameters of the xenon-filled fiber at2.55µmwavelength, this corresponds to a soliton withN=1.90. By increasing/decreasing the gas pressure, we can shift the

ZDWup/down, respectively. As a result, the phase-matched wavelength can be shifted up/down, respectively. Here note that, the choice of 35 bar xenon is not an ultimate target. It is just an example that energy conversion at the long wavelength becomes possible even in the case when theZDWis more than an octave wide. This possibility has never been studied before in the literature.

4.2 Normal dispersion regime pumping 69

Figure 4.12 (a) Spectrogram atz=8.2 cm. (b) Phase-matching diagram of the degenerate four-

wave mixing process as a function of the pump wavelength. The figure represents the signal/idler photon pair that satisfies the phase-matching conditions in Eqs. (4.5) and (4.6). The peak power of

25.55 MWwas used in the calculations. ωp,ωs, andωiare the angular frequencies of the pump,

signal and idler photons involved in degenerate four-wave mixing process. (c) Spectrogram at

z=9.0 cm. (d) Phase mismatch (∆β) as a function of the wavelength for pump at2.55µm.

Further spectral broadening takes place in the later stage of the propagation. In Fig.4.10(b), the initial accumulation of the light at2.55µmleads to the formation of another spectral band at around 3.71µm. This is also clearly evident in the spectrogram of the pulse atz=8.2 cmpresented in Fig.4.12(a). We can see that the energy is moving towards longer wavelength and initiating to generate another radiation in mid-infrared. In fact, this occurs while depleting the energy in the 2.55µm spectral band. We find that this arises due to a degenerate four-wave

mixing process. It satisfies the conservation of photon energy and momentum [57]:

2ωp = ωs+ωi, (4.5)

2βp = βs+βi+ ΓNL, (4.6)

whereωp, ωs andωi are the angular frequencies of the pump, signal and idler

photons involved in degenerate four-wave mixing process, andβp,βs andβi are

the wavevectors at the idler, pump and signal, respectively. ΓNL = 2γpPis the

nonlinear correction term, whereγpis the nonlinear coefficient atωp. A fixed pulse

peak power ofP=25.55 MWis used which is shown in the inset of Fig.4.11(b). In degenerate four-wave mixing process, two pump photons are annihilated to create an signal/idler photon pair. Figure4.12(b) is the phase-matching diagram as a function of the pump wavelength. It shows that for2.55µmpump, the signal and the idler photon pair is generated at1.94µmand3.71µm, respectively. In this study, the pump(2.55µm)and the signal(1.94µm)are presented at the early stage of the degenerated four-wave mixing process. Consequently, an idler is generated at3.71µmshown in Fig.4.12(a). Note that, the photons involved in this degenerate four-wave mixing process overlap at−0.1 psdelay. This means that the process will

occur over an extended length thus enhancing the new spectral band formation in the3.71µmspectral region.

In the presence of higher-order dispersion, the annihilating soliton centered at 2.55µminvolves in another new radiation generation process. Atz=8.7 cm, a dis- persive wave is generated in the normal dispersion regime at∼1.1µm[68]. This is

clearly evident in the spectrogram taken atz=9.0 cmpresented in Fig.4.12(c). We can see that the phase-matching condition given in Eq.(4.4) is met at the wavelength of dispersive wave as shown in Fig.4.12(d). It should be mentioned that the energy in dispersive wave is weak due to the small spectral overlap between the soliton and dispersive wave, and therefore, it cannot be seen easily in Fig.4.10(b). Moreover, since the soliton at2.55µmand its dispersive wave have different group velocities,

4.2 Normal dispersion regime pumping 71

Figure 4.13 (a) Group delay as a function of the wavelength.∆tis the initial time delay between

the soliton and dispersive wave. (b) Normalized spectral intensity,Iωatz=15 cm. The inset shows

the normalized time-domain intensity profile,Itof the green-shaded part in the spectrum.

they do not interact upon further propagation [113]. However, as we shall see, dispersive wave contributes to another spectral process that takes place in the later stage.

After its formation, the spectral band at 3.71µm undergoes a further red-shift, reaching3.85µmatz=15 cmas shown in Fig.4.10(b). We notice from the group delay,β1 =1/vgplotted in Fig.4.13(a) that the soliton-like radiation at3.71µmand

the generated dispersive wave at1.1µmhave similar group velocities. Moreover, when the 3.71µmband forms atz = 8.40 cm, it marginally leads the dispersive wave by approximately60 fswith a slightly smaller group velocity. Consequently,

Figure 4.14 The normalized spectral intensity profileIωfor different amount of the pump energy

fluctuation. Solid and dotted lines indicate percentage increment and decrement in the pump energy, respectively, except blue solid line. The coherence of the generated supercontinuum based on the first order correlation function (orange colored curve) is shown by the arrow.

they undergo a collision, which induces the cross-phase modulation. In this case the dispersive wave and the soliton-like radiation are temporally locked and during this time they co-propagate along the fiber [114]. This results in the energy of the dispersive wave to be converted to the mid-infrared spectral band. The collision also causes both pulses to decelerate [114], red-shifting the frequency of the mid- infrared band, while shifting the dispersive wave towards the blue as shown in Fig.4.13(a). Figure4.13(b) indicates that a big chunk of the pump energy is in the3.85µmspectral band at z=15 cm. The green-shaded area translates to aN =1 soliton with75.2 fsfull-width at half-maximum duration and4.96 MWpeak power in the time domain.

We have also investigated the influence of the pump energy fluctuation on the out- put spectrum atz=15 cm, as shown in Fig.4.14. This study gives an indication of the experimental tolerance, i.e. the system’s sensitivity to the pump noise. Notice that the output spectrum change is insignificant up to±_5%pump energy fluctu-

ation, which clearly shows that the system is highly robust against the presence of noise in the pump. In addition, Fig.4.14presents further clarification regarding the system’s sensitivity to the noise by providing a coherence curve based on first order correlation function. A bright and spectrally coherent supercontinuum can

4.2 Normal dispersion regime pumping 73 be achieved in the spectral region where coherence is 1. This curve provides clear evidence that pumping in the normal dispersion regime is beneficial and less sensitive to noise.

4.2.5.

Summary

In summary, we found numerically a promising pathway to generate supercon- tinuum in a nonlinear hollow-core fiber using a high-power femtosecond pump source at1.06µmwavelength. We exploited the method of exciting a higher-order soliton in the mid-infrared by pumping in the normal dispersion regime, which fur- ther enhances the spectral broadening in the mid-infrared region while preserving the output coherence. We showed that a number of nonlinear optical effects is responsible for a wide spectral output of supercontinuum generation spanning over three octaves, including the dispersive shock wave, phase-matched four-wave mixing and soliton effects.

CHAPTER 5

Conclusions and future scope

For many years, optical fibers have been an integral building block in science and technology. They are the backbone and central piece of high-speed data communication systems. Optical fibers are also crucial components in lasers generating high-quality light sources that are finding applications in various sectors including steel production industry, modern military equipment, remote sensing devices, biology, and medicine. Presently, the cutting edge fiber technology is specialty hollow-core fibers that guide light in non-traditional ways. They have gained a lot of attention in the past decade, thanks to their versatile optical guiding properties. They have enabled, for example, single-mode guidance over a very wide spectrum, the generation of exceptionally broadband light and low-loss guidance. Yet, the possibilities have not been fully explored, as there are a large set of fiber design parameters that can be varied to obtain completely different optical characteristics.

In this thesis, a detailed study have focused on the negative curvature based spe- cialty hollow-core fibers and on investigating their linear and nonlinear properties. The linear properties are analysed by using finite-element method. The nonlinear optical properties and short pulse propagation in the antiresonent guiding hollow- core fibers are modeled by using a generalized nonlinear Schrödinger equation or unidirectional field propagation equation.

75 Chapter 2 proposed a silica-based negative curvature fiber design with an elliptical element nested in the antiresonent tubes. The study have investigated the guiding properties of the positive and negative curvature nested hollow-core fiber in the near infrared spectral region. It is shown that the single mode property and the confinement loss depend on the ellipticity of the elliptical element. The numerical study also demonstrated that the ellipticity of the elliptical element has a significant effect on bending loss.

Chapter 3 presented empirical formulae for antiresonant guiding hollow-core fibers. These formulae provide the guiding properties without the help of numerical simulations. From this chapter investigation, an important model has been obtained for accurate group-velocity dispersion calculation over a wider spectral range. Moreover, it introduced an empirical formula for effective mode area that can be retrieved from the fiber structure parameters.

Chapter 4 presented nonlinear application of antiresonant guiding hollow-core fiber. A particular attention was paid to generating supercontinuum in silica based hollow-core fibers. Guiding the light in the hollow region allows the silica to be used as a base material for transmission of light in the critical spectral regions such as in deep-ultraviolet and mid-infrared. This would not be possible in the conventional fibers. The necessary nonlinearity can be achieved by filling the hollow core with the noble gas. In addition, this gas based system provides extra degree of freedom to easily control the optical properties by changing the filling gas pressure or gas species. This chapter considered the gas based system and provided numerical investigations of supercontinuum generation that covers mid-infrared