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We show that a genetic algorithm can be used to optimize the layer thicknesses of a hollow waveguide with a multilayer dielectric cladding. It is shown that in such “chirped” hollow waveguide low loss over a wavelength range with more than one octave width can be achieved. We show that dispersion control is possible in such waveguides.
©2009 Optical Society of America
1. Introduction
The guiding of light within a hollow core has attracted great interest because of its capability for applications in several scientific and technological fields, especially for transmission and nonlinear optical transformations of high-power laser pulses. Standard hollow fibers with a solid dielectric cladding guide light by reflection from a higher-index interface and are thus inherently leaky[1]. Recently the development of two new types of hollow waveguides has provided an attractive alternative to reduce loss. In hollow-core photonic bandgap fibers (HC PBF) the cladding consists of a periodic array of air holes in silica glass [2, 3] and light is guided with very low loss in the range of 1 dB/km by photonic bandgaps. On the other hand, omniguide-type fibers [4, 5, 6] or Bragg fibers[7] have a cladding of periodic layers of two materials alternating in the radial direction and also allow low-loss guiding by photonic bandgaps. Other approaches include the kagome-lattice fibers which are able to provide a low loss over a broad spectral range without using bandgap guiding[8, 9]. The outstanding guiding properties of omniguide fibers and photonic bandgap fibers are useful for high-power delivery, especially for CO2 laser light for medical and material- processing applications. However, the intrinsically narrowband transmission due to bandgaps seriously restricts the usable range of frequencies and limits their applications in several emerging fields in broadband nonlinear optics such as femtosecond pulse delivery [10, 11], pulse compression [12] and low-intense nonlinear optics [9]. In most of these applications only a moderate level of loss in the range of 1 to 10 dB/m is really required. Besides that, the desired broad transmission range the control of the group velocity dispersion (GVD) is an additional key factor for important interesting applications in ultrafast nonlinear optics and supercontinuum generation. Neither HC PBGs or omniguide fibers nor standard hollow fibers with a solid dielectric cladding can up to now resolve this trade-off in the fiber characteristics in a satisfactory way. Standard hollow waveguides have important applications in ultrafast nonlinear optics, such as the generation of few-cycle mJ pulses[13, 14], high-order harmonic generation [15, 16] and four-wave mixing (FWM) for UV [17] and VUV [18] femtosecond pulse generation. Unfortunately they provide a tolerable level of loss only for diameters larger than 100 μm. For these parameters the waveguide contribution to dispersion is relatively small and anomalous dispersion at optical frequencies can only be achieved for very small pressures of the gas filling. Recently, we have shown that dielectric-coated metallic hollow waveguides provide a viable option for the combination of broad-band transmission and dispersion control[19]. We predicted a sufficiently small loss in a broad transmission range and anomalous dispersion at optical frequencies for high gas pressures for waveguide diameters from 20 μm to 80 μm.
In this paper, we discuss an alternative concept and study the question whether the design of omniguide fibers can be modified in such a way that its transmission range is significantly broadened while keeping the loss at a sufficiently low level. The spectral position of the guiding range in standard omniguide fibers is determined by the bandgap which depends on the thickness of the periodic layers of the two materials. For a waveguide with varying layer thickness, i.e. in a chirped multilayer waveguide, the condition of maximum reflexivity will be fulfilled at different positions in the cladding for different frequencies. This should lead to a stretching of the transmission range over a broader wavelength range, at the cost of increasing loss. The key problem is to find an optimum for the free parameters of the variable thickness of the layers. For this aim we introduce a genetic algorithm with a fitness function defined by the averaged loss over a desired spectral range. The transfer matrix theory is used to calculate the losses and the dispersion parameters. In this way we show that it is possible to design a waveguide with a broad transmission range (more than one octave) with moderate losses in the range of 3 dB/m, and an oscillating GVD curve with low maximum values of about 50 fs2/cm. Note that the concept of chirping the structural parameters has already been successfully applied for chirped mirrors [20], chirped photonic crystal waveguides [21] and chirped photonic crystal fibers [22]. Genetic algorithm [23] has been used for coherent control by chirped femtosecond pulses for various application such as e.g. optimization of high-order harmonic generation [24].
2. Model and numerical approach
We consider a chirped hollow multilayer waveguide consisting of a hollow core with circular cross-section and a cladding of a multilayer dielectric structure consisting of alternating layers of two different dielectrics with refractive indices of n 1 and n 2 < n 1. The waveguide is surrounded by a medium with the refractive index n 2. The geometry of such a waveguide is presented in Fig. 1. The guiding properties of this system are described by the transfer matrix formalism developed in [7]. In this method, the wavenumber β(ω) is determined by a characteristic eigenvalue equation obtained by the requirement of finiteness of the field in the core and absence of incoming waves in the outermost layer. The resulting equation 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). For details see Ref. [7 , 5, 19]. Additional scattering loss due to the roughness of the inner surface is neglected since the roughness size is quite low for dielectric surfaces obtained by the standard stack-and-draw technique.
For a given waveguide diameter, the waveguide geometry shown in Fig. 1 has many free parameters in form of the thicknesses d 1i and d 2i of the layers with the refractive indices n 1 and n 2. They have to be optimized with the aim to achieve a low loss in a broad spectral range. For an approximate solution of this optimization problem we use a genetic algorithm defined by a fitness function in which a set of candidate solutions (population) evolves towards better solutions. Here the fitness function f is given by the average loss ã, f=α̃ over a spectral range from ωmin to ωmax
The optimization of the thicknesses of 20 layers is performed using a genetic algorithm with a population of 100 realizations. At each step, the fitness function is calculated for all realizations. The next generation comprises then the best realizations from the previous generation, as well as realizations obtained from them by the processes of mutations, cross-over and mixing. Mutations here are obtained by randomly changing the thickness of each layer with a low probability (0.04 in our case). In the process of crossovers the thickness of each layer is taken randomly from one of the two “parents” from the previous generation. We have additionally introduced a process of mixing, in which each thickness in the next generation is the average of the corresponding layer thicknesses of two “parents”. A few hundred generations were necessary for the convergence to an optimum.
3. Numerical results and discussion
First we consider a waveguide with an inner diameter of 40 μm and a spectral range with the boundaries ωmin and ωmax corresponding to the wavelengths of 1383 and 592 nm, respectively. This spectral range (called target domain in the following) is more than one octave wide and the central frequency of typical solid-state lasers (e.g. Ti:sapphire) is located within these limits. For the refractive indices of the alternating layers we choose n 1 = 1.6 and n 2 = 4.6 which are typically used for omniguide fibers[5]. These values maximize the air/dielectric and dielectric/dielectric refractive index contrast and therefore allow to achieve a lower loss.
In Fig. 2(a), the fitness function of the best realization is shown dependent on the generation number Ngen. It can be seen that the fitness function (red curve) drops quickly during the first few tens of generations, and reaches a plateau after 300 generation when the minimum is found. The achieved average loss is 3 dB/m, which is sufficiently small for the use of such waveguide in ultrafast nonlinear optics with typical waveguide lengths in the range of 1 m or less, such as frequency conversion and supercontinuum generation. The optimized parameters for the layer thicknesses are in the range from 50 to 200 nm as presented in the Fig. 2(b). In Fig. 2(a), by the horizontal lines the losses of the two intuitive designs are indicated. The first one is an omniguide-type with a constant layer thicknesses (d 1i = 0.225 μm and d 2i = 0.04 μm for the low- and high-index layers, respectively) with bandgap in the middle of the target wavelength domain (green line). The second is a waveguide with thickness of the low-index layer linearly varying with radius (d 1i = 0.135+0.018i μm), so that the ‘local’ bandgap position shifts from the lower to the higher limit of the target domain (blue line). It can be seen that the genetic optimization provides a more than one order of magnitude reduction of the loss compared with the both intuitive designs.
Fig. 2. Evolution of the fitness function (red curve) α̃ with the generation number (a) and the optimum layer thicknesses (b). The fitness function of the intuitive design with constant layer thicknesses and a bandgap in the middle of the target domain is indicated by the green horizontal line. For a design with a low-index layer thickness linearly varying with radius (d 1i = 0.135+0.018i), the same is shown by the blue line. The inner waveguide diameter D is 40 μm.
In Fig. 3(a), the loss of an optimized chirped multilayer waveguide is shown as a function of the wavelength by the thick red curve. It can be seen that in the target domain from 580 to 1380 nm (indicated by the black bracket) the loss is significantly lower than in the spectral range outside of this domain. In the target domain the average loss is roughly two orders of magnitude lower than that in hollow dielectric waveguide (green dashed curve) with a cladding refractive index of 1.6 or 4.6. The loss peaks arise due to the coupling to the modes localized in the cladding. By the blue short-dashed curve, the loss of an omniguide-type fiber which has a constant thicknesses of the layers and a band in the middle of the target range is shown. It can be seen that the loss of the omniguide-type waveguide is lower in the middle of its transmission range at the cost of a narrower transmission range. In contrast, in the optimized chirped multilayer hollow waveguide the transmission range exceeds one octave. The average loss does not reach the very low values in the range of few dB/km possible in omniguide fibers, but for typical applications in ultrafast optics moderate values predicted here are sufficient.
In Fig. 3(b) the group-velocity dispersion of the designed fiber is shown by the red curve. One can see that coupling to the cladding-localized modes visible as loss peaks in Fig. 3(a) manifests itself in a strong GVD at these wavelengths. Far from these resonances, the GVD parameter has moderate values in the range of tens of fs2/cm. However, the inhomogeneities of the waveguide in the longitudinal direction will cause a shift of the bandgap position and smoothing of the GVD and loss curve. In Fig. 3(b) the green long-dashed curve shows the result of such smoothing by a convolution with a 40-nm-wide Gaussian profile which corresponds to 5% modification of the parameters. It can also be seen from the smoothed GVD curve that the dispersion oscillates from positive to negative values with low extrema of about 50 fs2/cm. The GVD is anomalous for all wavelengths above 1 μm.
In Fig. 4 the loss curves of other modes, such as TE01, TM01, and HE21, are presented. The corresponding average loss over the target range of the TM01 mode is roughly 20% lower than for the fundamental HE11 mode; the average loss of the TE01 and the HE21 modes are roughly 4 times higher. It should be noted, however, that although TM01 mode has a lower loss, its excitation requires a special setup to shape the input beam into the “onut” form. This can be a challenge to implement experimentally. On the other hand, standard focusing of a beam by a lens at the fiber entrance couples the major fraction of the beam to the HE11 mode, and the energy transfer to higher-order modes during propagation is usually weak. Therefore this mode is the main object of our study.
Fig. 3. Loss of an optimized waveguide with D = 40 μm (red cruve) with layer thicknesses indicated in Fig. 2. The loss of the waveguide with constant layer thicknesses is indicated by the blue curve, and the green curve designates the loss of a hollow waveguide with homogeneous dielectric cladding with a refractive index of 1.6. The black bracket indicates the target domain. In (b), the GVD of the optimized waveguide is shown by the red solid curve and the smoothed GVD is shown by the green long-dashed curve.
Fig. 4. Loss of the TE01 (long-dashed green curve), TM01 (short-dashed blue curve), and HE21 (red solid curve) modes in the optimized waveguide.
Fig. 5. Loss (a) and GVD (b) of an optimized waveguide (red curve) with larger diameter D = 80 μm. The layer thicknesses are close to that shown in Fig. 1. In (b), GVD (red solid curve) and smoothed GVD (green long-dashed curve) are shown.
Let us now study the optimization and the waveguide loss of a chirped multilayer waveguide with a larger diameter D=80 μm and the same target domain as before. As illustrated in Fig. 5, the loss parameter α(ω) is about ten times smaller than for a waveguide diameter D=40 μm with an average loss in the target domain of 0.4 dB/m. However the position of resonances and the optimized layer thicknesses show only a small deviation in both cases. This can be explained by the fact that for grazing incidence of light the position of the bandgaps depends on the incidence angles on the surfaces between the low-index and high-index layers which is determined by the refractive indices of layers. The dependence of the bandgaps on the incidence angle at the core-cladding interface, which is a function of the core radius, is much less sensitive. Therefore changing the radius reduces the loss but leads only to a marginal change of the optimized layer thicknesses and the position of the resonances.
For specific applications in nonlinear optics the predicted resonances within the working wavelength range could be impedimental. However, in difference to Fig. 3 and Fig. 5 the loss function α(ω) can be made smoother if we apply an another fitness function which is more sensitive to the high values of the loss at the resonances and therefore yields a waveguide configuration without resonances in the target domain. As an example we consider the fitness function f
Fig. 6. Loss of an optimized waveguide (red curve) with D = 40 μm and a fitness function given by Eq. (2). The green curve designates the loss of the hollow waveguide with homogeneous dielectric cladding with a refractive index of 1.6. The black bracket indicates the target domain.
The loss of a waveguide optimized by using this fitness function is shown in Fig. 6. It can be seen that this waveguide is characterized by a loss curve without resonances within the target domain, but at the cost of increased average loss. This example shows the possibility to obtain chirped multilayer waveguides with smooth linear characteristics in a broad wavelength range, as well as the versatility of the genetic algorithm to design waveguides with desired characteristics.
4. Supercontinuum generation as an example of a nonlinear process in the optimized fiber
Let us finally consider an example of the application of an optimized chirped multilayer fiber with a design as shown in Figs. 2 and 3. We study the nonlinear propagation of a 100-fs pulse with a peak intensity of 50 TW/cm2 through the hollow core of a fiber with D = 40 μm filled with argon at 1 atm. The pulse propagation is simulated by the model without slowly varying envelope approximation which includes the effects of the Kerr nonlinearity, dispersion to all orders, as described in Ref. [25, 26], with addition of plasma contribution. Since the losses of the higher-order HEn1 modes are significantly higher than the loss of the HE11 mode, only the latter was considered in this example. The input wavelength at 870 nm is chosen in the middle between two resonances, in the range of anomalous GVD. The output spectrum presented in Fig. 7 with parameters given in the caption demonstrates the generation of a supercontinuum with a spectrum covering almost the whole spectral solution range from 620 to 1150 nm. The presence of resonances within this range is detrimental to the smoothness of the generated supercontinuum but does not hinders the spectral broadening by soliton fission and emission[25]. This mechanism of supercontinuum generation is confirmed by the presence of the peaks in the temporal shape shown in Fig. 7(b) which can be identified as fundamental solitons.
Fig. 7. Spectrum (a) and temporal shape (b) of a 100-fs, 50-TW/cm2 pulse after the propagation of 1 m in an optimized fiber with D = 40 μm with geometry given in Fig. 2 filled with argon at 1 atm. The input wavelength is 870 nm.
5. Conclusion
In conclusion, we used a genetic algorithm for optimization of chirped multilayer hollow waveguides. As an example, we design a waveguide with a more than one octave broad transmission range, a moderate level of losses in the range of 3 dB/m, an oscillating GVD curve and anomalous dispersion for λ > 1 μm. These fibers offer interesting applications, especially for pulse compression and supercontinuum generation of μJ femtosecond pulses. This first application of a genetic algorithm for optimization of a photonic fiber certainly did not utilized the full potential of this method, but offers various extensions depending of the specific application. This method is also expected to provide optimized design for other types of photonic fibers, such as different microstructure fibers and hollow core photonic bandgap fibers.
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