NONLINEAR WAVE PROPAGATION IN PHOTONIC CRYSTAL FIBERS

Wave propagation is a fundamental phenomenon occurring in several physical systems. The spectra have been used by others to develop optical frequency standards. The process can potentially be used for frequency conversion in fiber optic network. In this system the dispersive properties can be controlled by the optical lattice making it possible to achieve phase-matched four wave mixing, like look the process taking place in the photonic crystal fibers (PCFs). In this paper will focus on two such systems the propagation nonlinear wave in photonic crystal fibers and the propagation of matter waves in optical lattices.

geometry [31,32], while a strong light-field confinement is achieved due to the high refractiveindex step between the core and the microstructure cladding [33]. Controlled dispersion of PCFs is the key to new solutions in optical telecommunications and ultrafast photonics. The high degree of light-field confinement, on the other hand, radically enhances the whole catalogue of nonlinearoptical processes and allows observation of new nonlinear-optical phenomena. Fig. 1, a, b display the cross-section views of PCFs with a high refractive-index step from the fibre core to the fibre cladding, controlled by the air-filling fraction of the microstructure cladding. The fibres of this type can strongly confine the electromagnetic field in the fibre core, providing high optical nonlinearities, thus radically enhancing nonlinearoptical interactions of light fields. Highly efficient fibre-format frequency converters of ultrashort light pulses [29] and PCF supercontinuum sources [34,35] based on highly nonlinear PCFs ( Fig. 1, a, b) are at the heart of advanced systems used in optical metrology [36,37], ultrafast optical science [38,39], laser biomedicine [40], nonlinear spectroscopy [41,42], and nonlinear microscopy [43,44]. The possibility of dispersion tailoring makes PCFs valuable components for dispersion balance and dispersion compensation in fibre-optic laser oscillators intended to generate ultrashort light pulses with a high quality of temporal envelope. Lim et al. [45] have demonstrated an ytterbium-fibre laser source of 100-fs pulses with an energy of about 1 nJ with dispersion compensation based on a PCF instead of free-space diffraction gratings. A highly birefringent hollow-core PCF [46] provides a robust polarization-maintaining generation of 70-fs laser pulses with an energy of about 1 nJ in a fibre laser system [47]. Isomäki and Okhotnikov [48] have achieved dispersion balance in an ytterbium femtosecond fibre laser using an all-solid PBG fibre [26]. In contrast to silica-air index-guiding microstructure fibres, including silica and air holes, an all-solid PBG fibre guides light along a silica core surrounded with a two-dimensional periodic lattice of high-index glass inclusions. Dispersion tailoring and a high nonlinearity of small-core PCFs, on the other hand, allow efficient optical parametric oscillation and amplification due to the third-order optical nonlinearity of the fibre material [49,50]. Optical parametric oscillators based on PCFs can serve as efficient sources of correlated photon pairs [51,52]. The maximum laser fluence in an optical system is limited by the laser damage of material of optical components. An increase in a fibre cross section is a standard strategy for increasing the energy of laser pulses delivered by fibre lasers. Standard largecore-area fibres are, however, multimode, making it difficult to achieve a high quality of the transverse beam profile. This difficulty can be resolved by using PCFs with small-diameter air holes in the cladding, which filter out high-order waveguide modes [30,53]. This strategy can provide single-mode waveguiding even for largecore-area fibres [54,55] (Fig. 1, c). A dual-clad PCF design helps to confine the pump field in the microstructured cladding and to optimize a spatial overlap between the pump field and laser radiation. In this type of PCFs, the microstructured part of the fibre is isolated from the cladding by an array of large-diameter air holes (Fig. 1, c). Large-modearea ytterbium-doped PCFs [56,57] are employed for the creation of high-power lasers [55,58,59]. Large-mode-area silica PCFs are also used for the compression of high-power subpicosecond laser pulses [60] and the generation of supercontinuum with an energy in excess of 1 μJ [61,62]. Photonic-crystal fibre design presented in (Fig. 1, d) is of special interest also for the development of novel fibre-optic sensors [63,64]. In sensors of this type, excitation radiation is delivered to an object along the fibre core. The inner part of the microstructured cladding features micrometerdiameter air holes and serves for a high-numerical-aperture collection of the scattered or fluorescent signal from the object, as well as for the fibre delivery of this signal to a detector. With such a scheme of sensing, a detector can be placed next to a radiation source [65,64]. This fibre design is advantageous for sensing chemical and biological samples by means of one-and two-photon luminescence. A microstructured cladding of PCF can be also conveniently filled with a liquid-phase analyte. Radiation propagating along the fibre core will then induce luminescence of the analyte, allowing the detection of specific types of molecules from the minimal amount of analyte [64]. Such fibre sensors can be integrated into chemical and biological data libraries and data analyzers, including biochips, suggesting an attractive format for the readout and processing of the data stored in such devices. The energy of laser pulses in fibre-optic devices can be radically increased through the use of hollow-core fibres. For standard, capillary-type fibres, however, the loss rapidly grows (as  a−3) with a decrease in the core radius a [27,28]. Because of this problem, such fibres cannot provide single-mode guiding or help to achieve high intensities for pulses with moderate peak powers. The loss of coreguided modes in hollow fibres can be radically reduced if the fibre has a two dimensionally periodic (photonic crystal) cladding [24,22,25] (Fig. 2, a, b). A strong coupling of incident and reflected waves, occurring within a limited frequency range, called a photonic band-gap, leads to a high reflectivity of a periodically structured cladding, allowing low-loss guiding of light in a hollow fibre core. Hollow PCF compressors in fibre-laser systems [66,67] allow the generation of output light pulses with a pulse width on the order of 100 fs in the megawatt range of peak powers. Thus, PCFs play the key role in the development of novel fibrelaser sources of ultrashort light pulses and creation of fibre-format components for the control of such pulses. In what follows, we examine the physical mechanisms behind supercontinuum generation in such fibres, analyze various scenarios of spectral broadening and wavelength conversion, and discuss applications of PCF white-light sources and frequency converters in nonlinear spectroscopy and microscopy, as well as in optical metrology.

Dispersion properties of (PCFs)
In a homogeneous medium the dispersion relation between wave vector k and frequency ω of the propagating light is given through the refractive index of the material ω=c|k|/n. In a PCF it is the combined effect of the material dispersion and the band structure arising from the 2D photonic crystal that determines the dispersion characteristics of the fiber. For propagation in fibers it is the dispersion for the wave vector component along the z-direction kz that is the interest- where ω fund . denotes the frequencies of the lowest lying mode in the fiber. The higher derivatives of the propagation constant are given as ( ) ,

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and the second order dispersion A cross-section of an index guiding PCF is shown in Fig. 3, a calculation of the dispersion properties and effective area of this fiber will be sketched. The dispersion given by β 2 (λ) is shown in Fig. 3 and the fiber has λ ZD = 721 nm, whereas the zero dispersion wavelength for bulk silica is found around 1300 nm. The zero dispersion wavelength for this fiber has consequently been shifted into the visible regime due to the micro-structuring. This widely tunable group velocity dispersion is an extremely valuable property of the PCFs. The dispersion can be tuned by a proper choice of the size of the air holes, the distance between the holes (pitch) and the size of the central defect. A general tendency is that the zero dispersion wavelength is found at a shorter wavelength when the fraction of air filling is increased and the central defect is decreased [68]. It is possible to manufacture fibers with zero dispersion wavelengths between 500 and 1500 nm. Another general trend is that decreasing either the pitch or the hole-size leads to a higher curvature of the dispersion profile, eventually leading to two closely lying zero dispersion wavelengths. The fibers can be made with cores down to 1μm in diameter. Due to the small core areas huge intensities can be obtained in the cores of the fibers. Consequently, such fibers will exhibit a highly nonlinear response. Another very useful property of the fibers is that they can be made endlessly single mode. Only one mode should have a propagation constant between the effective propagation constants for the cladding and the core i.e. n core k > β > n clad k, where k is the free space propagation constant. The restriction corresponds where ρ is the core radius. For the fiber to be single mode Veff should be below 2.405. As λ decreases, the effective index of the cladding n clad increases, because more intensity of the light will be confined to the silica part of the cladding. Consequently, Veff can be kept below 2.405 for a wide range of wavelengths and the fiber is said to be endlessly single mode. In this way fibers, even with a very large core, can be made endlessly single mode [70]. As the mode area of the fiber increases the relative intensity in the core will decrease. Hence the fibers can be used for linear propagation, where a lot of power can be delivered without going into a nonlinear propagation regime.

Attenuation properties of (PCFs)
If the transverse scale of a (PCFs) changes without otherwise changing the fibre's structure, the wavelength λ c of minimum attenuation must scale in proportion [71]. Without recourse to the approximations of the previous section, the mean square amplitude of the roughness component that couples light into modes with effective indices between n and n+δn is: where γ -the surface tension; k B -Boltzmann's constant; T -the temperature.
The attenuation to these modes is proportional to u 2 [72] but the only other independent length scale it can vary with is λ c . As attenuation has units of inverse length, it must therefore by dimensional analysis be inversely proportional to the cube of λ c . If this is true for every set of destination modes, it must be true for the net attenuation α to all destination modes, so: This equation [71], predicts the attenuation of a given fibre drawn to operate at different wavelengths. The result differs from the familiar 1/ λ 4 dependence of Rayleigh scattering in bulk media [73], and importantly applies to inhomogeneities at all length scales not just those small compared to λ. The fibres had 7-cell cores but were drawn to different scales, giving them different λ c but otherwise comparable properties [71]. The minimum attenuation is plotted in Fig. 4 against λ c on a log-log scale. A straight-line fit is shown and has a slope of 3.07, supporting the predicted inverse cubic dependence in Eq. (5). The minimum optical attenuation of ~0.15 dB/km in conventional fibres is determined by fundamental scattering and absorption processes in the high-purity glass [73], leaving little prospect of much improvement. Over 99% of the light in (PCFs) can propagate in air [71] and avoid these loss mechanisms, making (PCFs) promising candidates as future ultra-low loss telecommunication fibres. The lowest loss reported in photonic crystal fibres is 1.7 dB/km [71], though we have since reduced this to 1.2dB/km. Since only a small fraction of the light propagates in silica, the effect ofmaterial nonlinearities is insignificant and the fibers do not suffer from the same limitations on loss as conventional fibers made from solid material alone.

Maxwells equations
To get the dispersion characteristics (ω versus β) of the fiber structure Maxwell's equations have to be solved: Decoupling Maxwells equations with no free charges and currents, assuming linear response of the medium and no losses leads to a wave equation for the Hω(r) field where ε is the dielectric function. Here the fields have been expanded into a set of harmonic modes Hω(r, t) = Re (Hω(r)e −iωt ) with frequency ω. This can be done without further loss of generality since Maxwell's equations have already been assumed linear [74,75]. Because of translational symmetry along the z-axis the dielectric function only depends on (x, y), consequently the harmonic modes can be expressed on the following form: where m denotes the m th eigenmode with transverse part hm(x, y) and propagation constant β (m) (ω). After expanding in a plane wave basis the matrix eigenvalue problem is solved leading to the (fully vectorial) eigenmodes. The method is described in [74,76]. Johnson and Joannopoulos have developed a freely available code, to solve Maxwell's equations [77]. With this code and a dielectric function based on the image of  The effective propagation constant β of the fundamental mode can subsequently be found from Eq. (1) by inserting neff. The refractive index of silica n material has been calculated from the Sellmeier formula where λ j is an atomic resonance in the fused silica. For the calculations the parameters given in [79] have been used: . The contribution to the effective refractive index from the micro-structuring n eff,bandstructure, has been calculated by assuming a frequency independent refractive index of silica (n constant =1.45) in the dielectric function ε(x, y) in Eq. (6). By solving the equation the propagation constant of the fundamental mode β (1) is found, giving n eff , bandstructure = cβ (1) /ω. To include the contribution to n eff from silica only once, the constant offset n constant in Eq. (8) is introduced. In fact this constant term will have no impact on the simulations in the following chapters, since a frame of reference moving with the group velocity of the propagating pulse is chosen. Based on a SEM-picture of the fiber end face shown in Fig. 5 the propagation constant of the shown 1.7μm core diameter PCF has been calculated using the method sketched above. The group velocity dispersion β2, calculated from the propagation constant, is shown on Fig. 3. A mode corresponding to the other polarization state exists, but in the calculations presented in the following chapters only propagation in one polarization mode will be considered, even though for example the fiber on Fig.(5) is not polarization maintaining. The frequency dependency of the refractive index of silica can also be taken into account initially through a frequency dependent dielectric function ε(x,y,ω). Eq. (6) then has to be solved self consistently. When comparing the two methods no major differences appear. An effective area of a mode in a fiber can be defined as [78,79] where |hn(x, y)| 2 is proportional to the intensity distribution in the fiber. Fig. 6 shows the effective area of the fundamental mode for the 1.7 μm core diameter PCF. It is the high index contrast between silica and air that makes the relatively low effective areas in PCFs possible [78].

The nonlinear Schrödinger equation
The simplest form of the nonlinear Schrödinger equation is given by On the way to the equation above a more general version of the nonlinear Schrödinger equation will be found. The first term describes the second order dispersion determined by the material and the geometrical structure of the fiber as described in the previous chapter. The second term is the nonlinearity, which depends upon the polarizability of the material through χ(3) and scales with the third power of the electric field. The nonlinear Schrödinger equation has been applied in fiber optics since the beginning of the eighties, where it was used to describe Mollenauer's first experimental observations of solitons in optical fibers [80]. Solitons emerge as fundamental solutions to the nonlinear Schrödinger equation because the dispersion term can balance the nonlinear term. In quantum optics the Gross-Pitaevskii equation is used to describe the evolution of the Bose-Einstein condensate ground state wave function.
The propagation constant can be achieved either from calculations or experimental investigations of the fiber and is often expressed in terms of a Taylor expansion

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(2015), «EUREKA: Physical Sciences and Engineering» Number 1 Any dispersion profile can be fitted with a Taylor polynomial, the question is only how many terms are needed to make a good fit over the width of the spectrum.

Calculation of the propagation
The linear part is and Eq. (6) both originate from Maxwell's linear equations. By considering the magnetic field Hω(r) as given by Eq. (7) and taking the second derivative with respect to z the following equation arises The magnetic and electric fields are related by where with translational symmetry ε(x, y) is independent of z. Consequently, Eω(r) also fulfills Eq. (14) and β(ω) in this and the previous chapter is the same.

Nonlinear effects
As mentioned, Eq. (17) can be implemented directly as it is including both full frequency dependency of the propagation constant and the effective area as well as self-steepening and Raman effects.

Raman response
For the Raman response function the expression g(t) = (1− fR)δ(t) + f R g R (t) has been used, where the delta function term originates from the electronic response i.e. the Kerr interaction and the last term takes the Raman scattering into account. The function g R (t) can be chosen on the form as given by [81]. Raman scattering can be explained as scattering of light on the optical phonons and 1/τ1 gives the optical phonon frequency. 1/τ2 gives the bandwidth of the Lorentzian line (see Fig.7) . The same values as in [79] have been applied for the constants: τ1 = 12.2fs, τ2 = 32fs, f R = 0.18.

Kerr nonlinearity
The Kerr effect is the effect of an instantaneously occurring nonlinear response, which can be described as modifying the refractive index. In particular, the refractive index for the high intensity light beam itself is modified according to 2   n n (20) with the nonlinear index n 2 and the optical intensity I. The n 2 value of a medium can be measured e.g. with the z-scan technique. Note that in addition to the Kerr effect, electrostriction can significantly contribute to the value of the nonlinear index [82,83]. A Kerr nonlinearity can be assumed by ignoring the Raman response in the fibers corresponding to setting g(t) = δ(t). If the nonlinearity factor is assumed constant γ = γ0 the following equation arises If all terms are transformed to the time domain and only up to second order dispersion is taken into account the following equation appears This is exactly the simple form of the nonlinear Schrödinger equation (11) .

Nonlinearity factor γ
For the nonlinearity factor the convention suggested in [79] has been followed The frequency dependent nonlinearity factor γ(ω) in Eq. (23) . Since the effective area A eff (ω) often does not vary too drastically with frequency as seen on Fig. 6 a valuable approximation is to assume the effective area to be constant A eff ,0. With this approximation the nonlinearity factor can be written as

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(2015), «EUREKA: Physical Sciences and Engineering» Number 1 where ω 0 = ω − ω1 is the frequency of the input pulse and In the time domain the nonlinearity factor above is given by 0 , where the time derivative takes self-steepening and shock formation into account. Consequently, for very long pulses this time derivative can be omitted corresponding to assuming a constant nonlinearity factor 0 ( )     .
If the computational grid is centered at a frequency ωc different from the central frequency of the pulse ω 0 the nonlinearity factor has to be changed accordingly

Conclusion
The transverse micro-structuring makes the dispersion of the fibers highly tunable and together with the high index contrast it leads to the small effective area, cade of nonlinear effects can take place in the fibers. The interplay between the special dispersion of the fibers and these nonlinear effects makes the phenomenon of supercontinuum generation possible. The linear Maxwell's equations have been solved for the transverse structure of the fibers. Starting with Maxwell's equations it has been sketched how a nonlinear Schrödinger equation for wave propagation in the PCFs can be achieved. The full frequency dependency of the propagation constant as well as the effective transverse area serve as input for the model and these parameters can either be calculated as measured. The model includes the instantaneous nonlinear response of silica. Additionally, the effects of Raman scattering, self steepening and shock formation can be included. In the following chapter most of the simulations shown will be based on Eq. (25) and the influence of the nonlinear effects will be investigated by comparison with the simpler version Eq. (21).