An ultra-flattened chromatic dispersion in circular C 6 H 6 -infiltrated photonic crystal fibers

Introduction : In this study, we report a new design of a circular lattice photonic crystal fiber and investigate its optical properties numerically. Methods : Near-zero ultra-flattened chromatic dispersion is achieved by allowing benzene (C 6 H 6 ) to infiltrate the hollow core and induce a difference in the air hole size in the cladding. By solving Maxwell's wave equations using the full-vector finite-difference eigenmode, the electromagnetic field modes are analyzed. Results : The results show that the variation in dispersion over the broad wavelength range of 527 nm is (cid:6) 0.753 ps/nm.km. The nonlinearity coefficient is significantly improved with a value of several thousands W (cid:0) 1 .km (cid:0) 1 ; which is favorable for supercontinuum generation with low input power even though the confinement loss has not yet reached its desired value. Conclusion : Two PCFs with optimal lattice parameters and suitable characteristic quantities are selected for supercontinuum generation orientation compatible with application fields such as spectroscopy, temperature sensing, and telecommunication.


INTRODUCTION
Compared with conventional optical fibers, photonic crystal fibers (PCFs) are preferred in many areas of application due to their special optical properties, such as wideband single-mode operation, high birefringence, great tailorable chromatic dispersion, and higher nonlinearity [1][2][3][4][5][6] . Ultrashort and intense laser pulses propagating in a nonlinear medium such as PCF lead to the appearance of various nonlinear effects. As a result, the output pulse is continuously broadened. Supercontinuum generation (SCG) performance is strongly influenced by the dispersion profile of the PCF. When the input pulse is pumped in the anomalous dispersion regime, stimulated Raman scattering 7 , including soliton fission (SF) and soliton self-frequency shift (SSFS), is the key nonlinear effect responsible for the quality of the SCG spectrum. However, the presence of solitons causes the spectrum to be noisy. This phenomenon will be eliminated immediately if the PCF is pumped in the normal or allnormal dispersion region. Then, effects such as selfphase modulation (SPM) followed by optical wave breaking (OWB), cross-phase modulation, and fourwave mixing 8 dominate the spectrum expansion. In this case, the SCG spectrum is smoother, with less noise, but the expansion efficiency is not high.
With design flexibility, PCFs that can achieve flat, near-zero dispersion and higher nonlinearity can satisfy two important elements of SCG quality: spectral width and flatness over broadband wavelengths 9 . Flat, small, and even ultra-flattened dispersion in the broadband wavelength for better SCG has been achieved in PCFs with lattice holes of different sizes, as presented in works [10][11][12][13][14][15][16] . However, the low nonlinear coefficients and large effective mode area are unfavorable for efficient SCG processes, which are still reported [12][13][14][15] . In recent years, liquid core PCFs with highly nonlinear coefficients have become a useful solution that is being widely studied both numerically and experimentally to replace traditional glass core PCFs. Dispersion has been improved markedly, and the quantities characterizing the nonlinear properties of PCF have suitable values, helping to increase SCG efficiency. Diverse dispersion profiles can be easily obtained, including flat all-normal and anomalous dispersions, as expected in liquid-infiltrated PCFs such as toluene 17,18 , carbon tetrachloride 19,20 , chloroform 21 , benzene 22,23 , nitrobenzene 24,25 , and carbon disulfide 26,27 . These publications also demonstrate the ability to generate a broader SCG spectrum together with a significant improvement in the peak power of the input pulse. It is even possible to achieve a broad SCG spectrum up to 2900 nm in benzene in-filtrated PCF 22,23 . Even so, an ultra-flattened dispersion with low values and near zero has not yet been found in these PCFs. Many researchers have been striving to develop PCFs with ultra-flattened nearzero dispersion to further improve the quality of SCG. By combining the modification of the structural parameters and infiltration of C 6 H 6 into the hollow core of the PCF, we designed circular lattice PCFs with different air hole diameters and lattice constants in the cladding. Then, we simulate light propagation in these PCFs to achieve both all-normal and anomalous dispersion configurations. In particular, ultraflattened and near-zero anomalous dispersion, with a low value of ±0.753 ps/nm km in the wavelength region of 527 nm, was obtained. Furthermore, quantities that characterize nonlinear properties such as nonlinear coefficients, effective mode area, and attenuation also obtain values suitable for SCG applications.

MATERIALS-METHODS
Lumerical Mode Solution (LMS) software was used to model the structure of the PCF circular lattice. First, the Sellmeier coefficient of the refractive index's real parts of SiO 2 is calculated according to Equation (1) 28 and declared in the system data. Then, the circular lattice is selected from the available data of symmetric geometries. The core center coordinates, core diameter values D c , and diameter d of air holes are computed and matched to create a hollow core fiber with 8 rings of parallel air holes along the axis center. Finally, benzene is filled into the hollow core of the PCF by determining and entering the refractive index coefficients from Cauchy's equation (2) of C 6 H 6 29 .
The survey wavelength was varied from 0.5 to 2.0 µm. (1) (2) Figure 1a shows the cross-sectional geometry of the PCF. The crystal lattice is circular in shape. The cladding has 8 rings of air holes that are evenly spaced parallel to the axis of the hollow core, which is filled with C 6 H 6 . The design diversity of PCFs with different lattice types or air hole sizes makes it easier to control the dispersion characteristics 30 . Furthermore, work 10,31 also demonstrates that the dispersion properties, including the dispersion configuration and wavelength shift at which the dispersion is zero (ZDW), are strongly dictated by the air hole size in the innermost ring. The other rings are responsible for the loss of PCFs in the basic mode or some cases in higher order. Many results testify that it is not easy to optimize the optical properties of PCF at the same time, so dispersion optimization is still considered the most necessary in SCG research. From these ideas, we designed the diameter (d 1 ) of the air hole of the innermost ring to be smaller than the others (d 2 ). The distance from the axis of the PCFs to the air holes in the innermost ring is Λ 1 , which is different from the distance between the two adjacent air holes Λ, with Λ 1 = 1.09Λ. The filling factor d 2 /Λ was kept constant at 0.95. The lattice constant Λ was chosen as follows: Λ = 0.9 µm, 1.0 µm, 1.5 µm and 2.0 µm. The optical properties of the structures were investigated according to the variation of d 1 /Λ, from 0.3 to 0.65 for each 0.05. During the design of PCFs, the core size of the PCFs is also a factor that should be considered. Neither too large nor too small a core size is necessary because the electromagnetic field modes will easily leak into the coating if the core size is too large. In contrast, a core that is too small makes the actual fabrication process more difficult. In our study, the core size is calculated by the formula D c = 2Λ -1.2d 1 . The cross-section of an optical fiber is divided into hundreds of thousands of small rectangles, known as "Yee meshes" 32 . A minimum mesh step of 10 −6 µm and 300 mesh cells without override regions are set for the PCFs. The dispersion and nonlinear properties of the C 6 H 6infiltrated PCFs were simulated by solving Maxwell's wave equation with the full-vector finite-difference eigenmode method (FDE). For accurate simulation, FDE divides the fiber cross-section into hundreds of thousands of very small rectangles with the assumption that the loss of C 6 H 6 is negligible. The selected boundary conditions are perfectly matched layers with strong absorption of the waves coming from the calculated region. This helps the LMS to solve the wave equation without any reflection, obtaining the basic-mode field intensity profile with minimal loss and calculating the propagation β with high accuracy. Figure 1b verifies the fine confinement of the electromagnetic field modes in the core of the PCFs. The linear refractive index (n) of SiO 2 and C 6 H 6 decreases with increasing investigated wavelength, as displayed in Figure 1c. Although C 6 H 6 has a higher linear refractive index value than SiO 2 , the difference is not too large. This makes the process of coupling standard and liquid-core fibers in the fabrication of PCFs more convenient. Furthermore, C 6 H 6 has much higher nonlinearity than SiO 2 (n 2 = 169.75×10 −20 m 2 . W −1 ) 33 is approximately 60 times that of SiO 2 (n 2 = 2.74×10 −20 m 2 . W −1 ) 33 . This will result in high nonlinearity of the PCF and improve dispersion, including flatness and value, which is beneficial for SC processes. In addition, the transmittance of C 6 H 6 is in a very broad wavelength range from 1.28 µm to 25 µm 33 , and the strongest absorption peak is at approximately 15 µm, with some lower peaks at 2.6 µm, 7.5 µm, and 10 µm. In the investigated wavelength region of 0.5 µm to 2.5 µm (smaller figure), the absorption peaks are very small at approximately 1.67 µm wavelength corresponding to a k value (imaginary part of the refractive index) of 4×10 −5 . Then, the attenuation A factor is calculated according to the formula 34 , which has a corresponding value of 23.3×10 −3 dB/cm. In our numerical model, optical fiber losses are mainly attributed to the material loss of C 6 H 6, as shown in Figure 1d, but these fibers have very low losses when λ < 1.67 µm. When λ is larger, the fiber loss is assumed to be 23.3×10 −3 dB/cm. The LMS with the full-vector finite-difference eigenmode (FDE) method has been used to create a C 6 H 6 -filled PCF and achieve the field intensity profile of the fundamental mode of the PCF. The data on wavelength-dependent characteristic quantities of all PCFs, including the effective refractive index, dispersion, nonlinear coefficient, effective mode area, and loss, are obtained. Different frequency components of the input pulse propagate at different velocities in nonlinear media, causing dispersion. This dispersion is affected by the size and location of the air holes in the cladding. Thus, achieving preferred values of dispersion, including value, number of ZDWs, and flatness, often depends on the structural design and choice of fiber material. A low dispersion value with a flat profile having one or more ZDWs is the main requirement for achieving coherent SC. Furthermore, the nonlinear effects occurring during the SCG process are related to the occurrence of solitons or nonsolitons depending on the all-normal or anomalous dispersion properties of PCFs. The total fiber dispersion includes material dispersion D m and waveguide dispersion D w , which is determined by Equation 35 .
Here, Re[n e f f ] is the real part of n e f f , and c is the speed of light in a vacuum. The nonlinear coefficient is an important parameter that directly affects the spectral extension and input pulse peak power. Normally, PCFs with high nonlinearities are preferred in SCG. In addition, the higher the nonlinearity of the PCF is, the lower the input pulse peak power required for the SCG process. The nonlinear coefficient (γ) characterizes the nonlinearity of the medium, which is determined by Equation (4) below 35 where n 2 is the nonlinear refractive index of SiO 2 and the effective mode area A e f f is inversely proportional to the nonlinear coefficient. The effective mode area is also a vital factor of an optical fiber because it determines how tightly the light is confined to the core and is related to the nonlinear effect in the fiber. A e f f is computed through the horizontal electric field across the cross-section of the PCF as Equation (5) 35 below : where E is the electric field amplitude. As light propagates in the PCF, the power gradually decreases with the propagation distance and is characterized by the confinement loss L c . It is calculated through the wavelength and imaginary part of the effective refractive index according to Equation (6) 35 below:

RESULTS
The refractive index difference between the cladding and core is the main cause for the light restriction in the core of PCFs. Therefore, it is necessary to investigate the effective refractive index first and then determine the other characteristic parameters. The propagation constant (β ) is related to the effective index of the fiber and can be computed using β = n e f f K 0 36 , with K 0 denoting the free space wavenum- 36 . The real part of the effective refractive index (Re[n e f f ]) of the basic mode in the study wavelength region from 0.7 µm to 2.5 µm has been found and quoted in Figure 2. In all cases, the further toward the long wavelength, the more monoton-  (Table 2), which helps us to consider the selection of pump wavelengths suitable for commercial laser sources in practice. We obtain four all-normal dispersion profiles with d 1 /Λ = 0.5 − 0.65, where the dispersion curve with d 1 /Λ = 0.5 is flattest and closest to zero. When Λ is increased to 1.0 µm, this dispersion curve shifts very close to zero dispersion, becoming ultra-flattened. The flatness covers a broadband wavelength of 527 nm with a low dispersion value of ±0.753 ps/nm.km. With ethanol infiltration into the hollow core of SiO 2 -based PCFs, work 38 achieved a symmetric ultraflat value of 1.04 ps/nm.km normal dispersion in the 500 nm range. An ultra-flattened chromatic dispersion of ±0.947 ps/nm.km was observed with flatness above 500 nm wavelength in publication 18 when using toluene to fill the hollow core of SiO 2 -based PCFs. Thus, it can be seen that the dispersion profile with Λ = 1.0 µm and d 1 /Λ = 0.5 has a lower value and the wavelength region has a broader flat dispersion than the works 18,38 . Compared with the #CBF 1 fiber (circular lattice) in reference 23 , the #F 2 fiber in this study also has allnormal dispersion, but it is flatter, and the dispersion value at 1.55 µm is much smaller. This can be beneficial for SC generation with a very broad SC spectrum with nonsoliton dynamics.      PCFs with near-zero ultra-flattened chromatic dispersions have shown the ability to broaden the SCG spectrum with a rather complex structure but can still be fabricated [40][41][42] . Furthermore, the publication 18 also reported the ability to achieve near-zero ultraflattened chromatic dispersion with a small value of ±0.947 ps/nm.km over a broadband of 500 nm of toluene-infiltrated PCFs, with Λ 1 = 1.095Λ. In this work, to achieve near-zero ultra-flattened chromatic dispersion in PCF infiltrated with C 6 H 6 , we only changed the lattice parameters of the innermost ring near the core, including the air hole size (d 1 ) and lattice constant (Λ 1 ). To clearly see the difference in flatness of dispersion, we compare two dispersion curves with parameters Λ 1 = Λ, d 1 /Λ = 0.5 (green curve) and Λ 1 = 1.09Λ, d 1 /Λ = 0.5 (orange curve) with Λ = 1.0 µm, illustrated in Figure 4. The green curve is not flat and has a rather high dispersion value. While the orange curve is flatter, ultra-flattened chromatic dispersion covers from 1343 nm to 1870 nm with a dispersion value of ±0.753 ps/nm.km.

DISCUSSION
In SCG, the various nonlinear effects that govern the characteristics of the spectrum relate mainly to the dispersion profiles of PCFs. To obtain smooth, low noise, flat peaks, and high coherence spectra that result from SPM effects followed by OWB, PCF is used with an all-normal dispersion regime. An ultrabroad SC spectrum in the PCF is generated due to the interaction between several nonlinear effects, including SF, SSFS, and Raman scattering, in the anomalous dispersion region. Although the spectrum achieved is much wider than that of the SCG using PCF with all-normal dispersion, the coherence is lower and the noise is more. This results in a larger input pulse power. However, SCG with different spectral properties in each case has its own application. Therefore, optimization to achieve diverse dispersion profiles is essential. The optimization criteria for generating supercontinuum at the chosen 1.55 µm wavelength were flatness, the indication of dispersion properties, and the difference between ZDW and pump wavelength. Based on the above dispersion analysis, we propose two PCFs with optimal dispersion suitable for SCG to discuss more characteristic quantities, such as dispersion chromatic, nonlinear coefficient, effective mode area, and confinement loss (Figure 4). The first PCF (#F 1 ) with parameters Λ = 0.9 µm and d 1 /Λ = 0.45 emits a highly coherent supercontinuum in the anomalous regime at a pump wavelength of 1.55 µm (Figure 4a). There are three reasons for this selection. First, this fiber has the flattest anomalous dispersion and the closest to zero among the studied anomalous dispersion curves. Second, it has the ZDW at 1.531 µm, which is the closest wavelength compared to the one pumped. Third, #F 1 has a low dispersion value of 0.883 ps/nm.km at a 1.55 µm pump wavelength. Those are in favor of spectral broadening, effective suppression, and dynamics of noise generation. Second fiber #F 2 (Λ = 1.0 µm, d 1 /Λ = 0.5) will produce a very broad SCG spectrum, flat top, higher coherence, and lower noise with low input peak power. This fiber has near-zero ultra-flattened chromatic dispersion spanning a broadband wavelength. There is no ZDW for PCF with all-normal dispersion, so the fiber is pumped at a wavelength close to the local maximum of the dispersion curve. The low dispersion value at a 1.55 µm pump wavelength is -1.447 ps/nm.km (Figure 4a).
The nonlinear properties of these two fiber structures are studied in detail. Figure 4 (b, c, d) depicts the dependence of the nonlinear coefficient, effective mode area, and confinement loss on the wavelength. As the wavelength increases, the nonlinear coefficient decreases. At wavelengths greater than 1.67 µm, the nonlinear coefficients of the two fibers are quite close, but they are separate in the remaining wavelength region. At a 1.55 µm pump wavelength, the γ value of the #F 1 fiber is 2726.561 W −1 .km −1, which is higher than that of #F 2 (2621.599 W −1 .km −1 ) (Figure 4b).
Because of the inverse relationship between the effective mode area and the nonlinear coefficient, the A e f f of the #F 1 fiber (2.531 µm 2 ) is smaller than that of the #F 2 fiber (2.632 µm 2 ) (Figure 4c). PCFs with larger core sizes typically exhibit poorer light confinement because light modes easily leak from the core to the cladding or between air holes in the cladding. The confinement loss increases with wavelength and increases rapidly in the long wavelength region (λ > 1.67 µm), which is consistent with the above analysis of the transmission spectrum of C 6 H 6 . The L c values of #F 1 and #F 2 at a 1.55 µm pump wavelength are 77.804 and 70.833 dB/m, respectively. The #F 1 fiber has higher loss but lower dispersion and a higher nonlinear coefficient at the pump wavelength, which will improve the SCG efficiency. Table 4 describes the structural parameters at the pump wavelength of the two optimal fibers and compares them with some previous work. The two optimal PCFs #F 1 and #F 2 have flat dispersion and much lower dispersion values than works 21,[44][45][46] . In particular, a near-zero ultraflattened dispersion was not found in these works. With a combination of skillful modification of structural parameters and selection of C 6 H 6 to fill the hollow core, the two proposed PCFs have higher nonlinear coefficients than 21,44,46 , which suggests that when using these PCFs for the SCG process, the peak power may be lower than that of PCFs previously infiltrated with other fluids. Although the confinement loss of the two proposed PCFs is higher than that of previous publications 44,46 (Table 4), the flat dispersion and low dispersion value will help the SCG spectrum to be extended. It can be seen that it is very difficult to obtain flat dispersion and low value while simultaneously limiting confinement loss. Depending on the different application purposes of the SCG, a PCF with suitable optical characteristics will be preferred.

CONCLUSIONS
We numerically simulate the chromatic dispersion and nonlinear properties of the C 6 H 6 -filled PCF by using the FDE. Numerical calculations and simulations reveal that the PCFs exhibit diverse dispersion properties, including all-normal and anomalous dispersion, which strongly depend on the change in lattice constant Λ and the filling factor d 1 /Λ. Further- more, a near-zero ultra-flattened dispersion has been achieved with an all-normal dispersion profile, with dispersion variation as low as ±0.753 ps/nm.km covering the 527 nm wavelength region. Therefore, it can be used to generate a broadened and smooth supercontinuum spectrum. Two optimal structures, #F 1 and #F 2, with flat dispersion and a high nonlinear coefficient are proposed and analyzed in detail for SCG application orientation. In the case of a C 6 H 6filled core, PCFs can produce a reasonably broad spectrum when injected at 1.55 µm. The nonlinear coefficients of the two fibers as high as several thousand W −1 .km −1 show that liquid-core PCF could be an interesting approach for powerful compact all-fiber systems for SCG with low-power femtosecond pumping systems in the fields of spectroscopy, temperature sensing, and telecommunication, … There have been many publications on the study of SCG using other liquids-infiltrated PCFs by experiment 19,44,47 . Therefore, the two optimal structures #F 1 and #F 2 completely meet the requirements of investigating the SCG process both by simulation and experiment. Carbon tetrachloride-infiltrated PCFs with low attenuation 19,44 are more favorable for broadband SCG allfiber sources, but the propagation length may be lower than that of toluene-infiltrated fibers 47 . PCFs with benzene can also experience the same phenomenon.