Simulation study on supercontinuum generation at normal dispersion regime of a carbon disulfide-core photonic crystal fiber

 Communications in Physics, Vol.31, No. 2 (2021), pp. 169-178
 DOI:10.15625/0868-3166/15564
 SIMULATION STUDY ON SUPERCONTINUUM GENERATION AT NORMAL
 DISPERSION REGIME OF A CARBON DISULFIDE-CORE PHOTONIC
 CRYSTAL FIBER
 BIEN CHU VAN1, DINH QUANG HO2, LE THI HA1, VAN CAO LONG3, VU VAN HUNG4
 AND HIEU LE VAN1,†
1Faculty of Natural Sciences, Hong Duc University, 565 Quang Trung Street, Thanh Hoa City,
 Vietnam
2School of Chemistry, Biology and Environment, Vinh University, 182 Le Duan Street, Vinh City,
 Nghe An Province, Vietnam
3Institute of Physics, University of Zielona Gora, Prof. Szafrana 4a, 65-516 Zielona Gora, Poland
4Office of Thanh Hoa People’s Committee, 35 Le Loi Street, Thanh Hoa City, Thanh Hoa
 Province, Vietnam
 E-mail: †levanhieu@hdu.edu.vn
 Received 7 October 2020
 Accepted for publication 17 November 2020
 Published 15 April 2021
 Abstract. A photonic crystal fiber with a hollow core filled with carbon disulfide (CS2) is proposed
 as a new source of supercontinuum light. We numerically study guiding properties of modeled
 fibers including the dispersion and the effective mode area of the fundamental mode. As a result,
 octave spanning of the SC spectrum was achieved in the wavelength range of near-IR from 1.25
 µm to 2.3 µm with 90 fs pulse and energy of 1.5 nJ at a pump wavelength of 1.55 µm. The
 proposed fibers are fully compatible with all-silica fiber systems, in particular, could be used for
 all-fiber SC sources and new low-cost all-fiber optical systems.
 Keywords: nonlinear optics, photonic crystal fiber, liquid, supercontinuum generation.
 Classification numbers: 42.65.Jx; 42.55.Tv; 77.84.Nh; 88.60.np.
 I. INTRODUCTION
 Photonic crystal fibers (PCFs), also known as micro-structured fibers or holey fibers have
 been a considerably attractive topic for optical community all over the world for the past decades.
 ©2021 Vietnam Academy of Science and Technology
170 SIMULATION STUDY ON SUPERCONTINUUM GENERATION AT NORMAL DISPERSION REGIME ...
PCFs can be used in a variety of research fields and practical applications. Here it is worth men-
tioning about fiber light sources, supercontinuum generation devices, fiber optic sensors or non-
linear devices [1, 2], endless single-mode properties [3], single-polarization single-mode opera-
tion [4], high birefringence [5], tailorable dispersion profiles and flat or ultra-flat dispersion [6,7].
Among others, the generation of supercontinuum (SC) is one of the most important applications.
Because of its unique properties, SC generation has been exploited in various prospects including
optical coherence tomography, optical metrology, multimodal bio-photonic imaging, and high-
speed optical communication [8–11].
 The SC generation occurs when the ultra-short optical pulses are pumped into a highly
nonlinear medium [12]. It is a complex process of spectral broadening of ultra-short optical pulses
such as femtosecond or picosecond, which undergo a number of nonlinear interactions in/with
the optical nonlinear medium, such as modulation instability, self-phase-modulation, four-wave-
mixing [13–15] in the normal dispersion region or stimulated Raman scattering, self-steepening
and soliton fission [16, 17] in the anomalous dispersion region.
 In order to get an efficient broadband SC generation, PCFs with flat dispersion characteristic
and highly nonlinear glass is usually utilized. In this manner, the PCFs are usually made of silica
or highly nonlinear glasses [11, 18, 19]. Silica fibers might efficiently create SC spectra in the
visible to the near-infrared (NIR) range [11]. Meanwhile, PCFs made of highly nonlinear glasses
offer a higher nonlinear refractive index in comparison with silica as well as have a broadband
transmission until the mid-IR range. Thus, it allows the SC generation in the mid-IR range [18,19].
However, the SC generation sources of PCFs from non-silica with highly nonlinear solid core
suffer from high costs and a complex fabrication process.
 Recently, liquid-core PCFs have been practically demonstrated as a considerable useful
replacement for obtaining efficient spectral broadening. In this way, several nonlinear liquids,
named a few, carbon disulfide (CS2), ethanol (C2H5OH), carbon tetrachloride (CCl4), chloroform
(CHCl3), toluene (C7H8) and nitrobenzene (C6H5NO2) have been used for this approach [20–25].
The results indicated that the generated SC spectra can be controlled by changing temperature,
pressure or applying an electric field. Moreover, the SC spectrum can be achieved in both the
normal and anomalous dispersion regimes with high coherence [26]. It can be explained that the
liquids have higher nonlinear refractive indices than those for solids. In detail, this leads to the
appearance of interesting nonlinear phenomena, especially, generated SC having a lower peak
power than for solid fiber [21]. Although the SC generation in liquid-infiltrated PCFs has been
widely studied, their perfor ...  1. The Sellmeier’s coefficients of fused silica and CS2 [31, 32].
 Sellmeier’s coefficients Values
 Fused silica
 B1 0.6694226
 B2 0.4345839
 B3 0.8716947
 −3 2
 C1 4.4801 x 10 µm
 −2 2
 C2 1.3285 x 10 µm
 2
 C3 95.341482 µm
 Carbon disulfide
 B1 1.50387
 −2 2
 C1 3.049 × 10 µm
 The material dispersion is directly included in the calculations through the three-term Sell-
meier equation. The chromatic dispersion D (λ) of a PCF is easily calculated from the neff values
172 SIMULATION STUDY ON SUPERCONTINUUM GENERATION AT NORMAL DISPERSION REGIME ...
versus the wavelength using the following formula [33]:
 2  
 λ d Re neff
 D(λ) = − , (2)
 c dλ 2
where Re[neff] is the real part of the refractive index, λ is the operating wavelength, and c is the
velocity of light in a vacuum.
 The effective area Aeff is a qualitative measurement of the cross section area covered by
guided mode of the fiber and is calculated as follows [33]:
  2
 |E|2 dxdy
 s
 Ae f f = , (3)
 |E|4 dxdy
 s
where E is the electric field in the medium obtained by solving an eigenvalue problem derived
from Maxwell’s equations.
 Meanwhile, based on the effective mode area we can obtain another important parameter
for optical communication which is the nonlinearity of the fiber. The nonlinear coefficient of the
PCF can be defined as [33]:
 2πn
 γ(λ) = 2 , (4)
 λAeff
where n2 is the nonlinear coefficient.
III. OPTIMIZATION OF ALL NORMAL DISPERSION PROPERTIES FOR OPTICAL
 FIBER WITH VARIOUS SIZES
 Table 2. The CS2-core diameter of the designed PCF.
 Λ (µm)
 d/Λ
 1.0 1.5 2.0 2.5
 0.20 0.88 1.32 1.76 2.20
 0.25 0.85 1.28 1.70 2.13
 0.30 0.82 1.23 1.64 2.05
 0.35 0.79 1.19 1.58 1.98
 0.40 0.76 1.14 1.52 1.90
 0.45 0.73 1.10 1.46 1.83
 0.50 0.70 1.05 1.40 1.75
 0.55 0.67 1.01 1.34 1.68
 0.60 0.64 0.96 1.28 1.60
 0.65 0.61 0.91 1.22 1.53
 0.70 0.58 0.87 1.16 1.45
 0.75 0.55 0.82 1.10 1.38
 0.80 0.52 0.78 1.04 1.30
 BIEN CHU VAN et al. 173
 a) Ʌ = 1.0 µm b) Ʌ = 1.5 µm
 c) Ʌ = 2.0 µm d) Ʌ = 2.5 µm
 Fig. 2. Characteristics of the PCF mode dispersion for filling factor f values in the range from
 0.20 to 0.80 and lattice constants (a) 1.0 µm, (b) 1.5 µm, (c) 2.0 µm, and (d) 2.5 µm.
 The optimization criteria aimed at the generation of SC in all normal dispersion regimes
with pumping at 1.55 µm, followed the flatness, sign of the dispersion charactericties and distance
from ZDW if located in the analyzed wavelength range. Looking for the optimal structure of a
PCF, we consider the structure with the lattice constants changing from 1.0 µm to 2.5 µm with
step of 0.5 and the linear filling factors varied from 0.20 to 0.80 with step of 0.05. The smallest
CS2-core diameter was 0.53 µm for Λ = 1.0 µm and f = 0.8. The biggest CS2-core diameter
diameter was 2.20 µm for Λ = 2.5 µm and f = 0.2. For more details of the CS2-core diameters
are illustrated in Table 2.
 Normally, the first ring of air-holes surrounding the core strongly affects the dispersion
properties of PCF as well as the zero-dispersion wavelength (ZDW), while the outer rings play
the role in the mode attenuation, especially for higher modes [26]. In the simulation, we used a
constant filling factor for all rings of PCF to simplify the development of this fiber in the future.
174 SIMULATION STUDY ON SUPERCONTINUUM GENERATION AT NORMAL DISPERSION REGIME ...
On the other hand, we do not also check if the modelled fiber is a single-mode fiber or multi-mode
fiber. This step is only performed for optimized fibers.
 Figure 2 shows that the dispersion characteristics of CS2-core PCF can be tuned by chang-
ing the linear filling factor and lattice constant. We observe that all CS2-core PCFs have flat
dispersion characteristics in the considered range.
 For a given Λ (Λ = 1.0 µm),
the dispersion characteristics are all-
normal dispersion in the full range
wavelength. In the case of Λ =
1.5 µm, the dispersion characteris-
tics exist only in the normal dis-
persion regime or a part exist both
regimes. On the contrary, the maxi-
mum dispersion is greater than zero,
which means there is both normal
and anomalous dispersion.
 Moreover, for a given f value,
the dispersion characteristics and
ZDWs are shifted toward longer
waves and flattened with increas- Fig. 3. Numerical calculations of dispersion
ing Λ. Meanwhile, for a given Λ characteristics in CS2-filled core optimal fiber
value, ZDW is usually shifted toward structure.
longer waves with decreasing f .
 On the basis of initial numer-
ical investigations, we choose the
PCF structure with following param-
eters: Λ = 1.5 µm and f = 0.30.
The numerical calculations of dis-
persion characteristics of the opti-
mal structure are presented in Fig. 3.
This fiber has optimum dispersion
characteristics since this fiber has all
normal dispersion and the dispersion
curve has achieved flatness. In ad- 
dition, the dispersion at the pump
wavelength equal – 8.1 ps/nm/km, Fig. 4. Calculations of effective mode area
which is the closest to the zero for all and the nonlinear coefficient of the PCF infil-
lattice constants which can obtained trated with CS2.
all normal dispersion region.
 Figure 4 presents the effective mode area and nonlinear coefficients of the optimal fiber. In
this case, because the core diameter is relatively small, it leads also to smallness in the effective
mode area of the optimal structure. In addition, the modal area of the fundamental mode increases
linearly with the wavelength. For the wavelength of 0.5 µm the modal area equals 2.60 µm2, while
 BIEN CHU VAN et al. 175
for the wavelength of 2.0 µm, the modal area equals 4.11µm2. Thus, the mode area is not changed
much within the wavelength range.
IV. SUPERCONTINUUM GENERATION IN THE OPTIMAL STRUCTURE
 The SC generation of the optimized PCF structure was simulated by numerically solving
of the generalized nonlinear Schrodinger¨ equation (GNLSE) when the split-step Fourier method
is used [33]:
 ∂A α in+1 ∂ n
 = − A + β A
 ∂z 2 ∑ n n! ∂T n
 n≥2 (5)
   Z ∞ 
 1 ∂ 2 2
 + iγ 1 + (1 − fR)|A| A + fRA hR(t)|A(z,T −t)| dt ,
 ω0 ∂T 0
where, A = A(z,t) is the complex amplitude of the optical field, α is the total loss in the PCF, βn
are the dispersion coefficients associated with the Taylor series expansion, γ is the nonlinear coef-
ficient, λc is the central wavelength, fR is the Raman fraction response to nonlinear polarization,
hR(t) represents the Raman response function which is given by [33]:
 2 2 −1 −2
 hR(t) = (τ1 + τ2 )τ1 τ2 exp(−t/τ2)sin(−t/τ1).
 In simulations, the following parameters were used: the fiber length 20 cm, the Gaussian-
shape pulse of duration 90 fs and the Raman fraction fR = 0.89, τ1 = 1.68 ps, τ2 = 0.14 ps [27],
 −19 2 −1
the nonlinear refractive index of CS2 n2 = 3.2 × 10 m W [27] and the pump wavelength of
1.55 µm.
 Fig. 5. Spectral intensity of PCF with various energies.
 Fig. 5(a) presents the evolution of the broadened spectra as a function of input energy at the
length 20 cm. In the case of input pulse energy are smaller than 0.5 nJ, the initial widening of the
spectrum is mainly from phase self-modulation (SPM). The optical wave breaking (OWB) begins
176 SIMULATION STUDY ON SUPERCONTINUUM GENERATION AT NORMAL DISPERSION REGIME ...
to show up when the input pulse energy is higher than 0.5 nJ. The spectral width will increase with
the increase of the input pulse energy.
 For input pulse energy of 1.5 nJ, the SC generation is expected with bandwidth of 1040 nm
around the pumping wavelength in the range 1252 - 2292 nm of wavelengths after propagating
about 20 cm inside the PCF as shown in Fig. 5(b).
 Fig. 6. Numerical calculations of the spectral (a) and temporal evolution (b) - (c) of the
 pulse along the fiber in CS2 - filled core PCF.
 Meanwhile, Fig. 6 depicts the spectral and temporal evolution of the pulse along the prop-
agation distance with input pulse energy 1.5 nJ. In this case, the location of pump wavelength
is in the normal dispersion region, after the initial widening of the spectrum due to phase self-
modulation, which is characterized by temporal spectrogram in S-shape as Fig. 6 (b). Next, we
observe a further widening of the spectrum in the short wavelength range due to OWB. As shown
in Fig. 6 (a), OWB firstly occurs in the trailing edge of the pulse at 2.0 cm of propagation and
generates the new wavelength band around 1.20 µm. After around 2.0 cm of propagation, the
 BIEN CHU VAN et al. 177
broad spectrum is asymmetric with a larger broadening on the long wavelength side as the propa-
gation distance increase. On the leading edge, OWB occurs only after 14 cm of propagation and
creates a new wavelength band around 2.0 µm. For further propagation, the energy of the pulse
is redistributed from the central area to the edges resulting in the further flat on the wings of the
output spectrum. It is clear that because different frequency components usually have different
velocities, time delay between different frequencies becomes larger with a longer propagation as
shown in Fig. 6 (c).
 Table 3. The comparison between the properties of proposed PCFs and some other CS2-
 core PCFs
Type of fibers Pump wavelength Regime Pulse length Pulse Energy SC range (nm) Refs.
CS2-core PCF 1.55 µm Normal 90 fs 1.5 nJ 1252-2292 This work
CS2-core PCF 1.56 µm Normal 180 fs 50 mW 1460-2100 [20]
CS2-core PCF 1.55 µm Normal 200 fs 0.4 nJ 1000-2000 [34]
CS2-core PCF 1.55 µm Normal 500 fs 1.0 nJ 1355-2110 [35]
CS2-core PCF 1.55 µm Normal 450 fs 45 mW 1000-2200 [36]
 Table 3 shows the comparison between the properties of the proposed PCF and those ob-
tained in some previous works. Here the lasers emitting wavelength of around 1.55 µm were used
as pump sources. It can be seen that we obtained a SC spectra range with a similar bandwidth as
those obtained in previous works but higher coherence and lower noise.
V. CONCLUSION
 In this paper, we have presented a numerical simulation on optimum struture of a PCF made
of fused silica with CS2-filled-core for obtaining all normal dispersion charactericties. The large
scope optimization process of the PCF structure, due to the modifications of their micro-structured
geometries, has been carried out in order to achieve the flat dispersion, and the generation of SC
in the whole normal dispersion region with pumping at 1.55 µm. According to the conducted
simulations, optimized fibre with the lattice constant Λ = µm, filling factor f = 0.3 exhibited an
all-normal dispersion and its peak equals −8.1 ps/nm/km at 1.55 µm.
 Our numerically simulated results demonstrated that in CS2 filled-core optimal PCF struc-
ture, the SC with a broadened spectral bandwidth of 1252 nm to 2292 nm was generated by a pump
pulse with a central wavelength of 1.55 µm, 90 fs duration and energy of 1.5 nJ. Further increase
in the spectral width can be expected if we increase input pulse energy. Due to the higher nonlin-
earity of CS2 than that of fused silica, lower power of input pulses is required than in the case of
silica PCFs [23]. Those fibers would be good candidates for all-fiber SC sources as cost-effective
alternatives to glass core fibers.
ACKNOWLEDGEMENT
 This work was supported by the project 796/2019/ HD- KHCN-D- TKHCN.
178 SIMULATION STUDY ON SUPERCONTINUUM GENERATION AT NORMAL DISPERSION REGIME ...
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