I. INTRODUCTION
Optical properties in PCF can be easily tailored to obtain desire results. It can be achieved by designing a particular pattern or arrangement of holes and also by changing the dimension of the hole. Moreover pitch factor of hole and the number of hole in the cladding region can also result desired changes. Photonic crystal fiber due to their extraordinary properties compared to standard optical fibers has drawn the attention of many researchers[^{1}]-[^{4}]. These properties include chromatic dispersion [^{5}], birefringence [^{6}], ultra low loss [^{7}], endless single mode propagation [^{8}], large nonlinear co-efficient [^{9}] and effective mode area [^{10}]. In conventional fiber, achieved birefringences are very low due to very low refractive index contrast. Different to this, PCF are having large index contrast and design flexibility as well. Hence very high value of birefringence can be easily achieved in PCF [^{11}]. Researchers have reported high birefringence by using squeezed crystal lattice. In this structure number of holes are different along two orthogonal axes [^{12}]. Researchers have also used elliptical holes to obtain high birefringence [^{13}]. Lyngso et al [^{14}] reported polarization maintaining index guiding PCF. Cho et al shown large birefringence in plastic PCF [^{15}]. Highly birefringent PCFs can be obtained by adopting two ways. First way is to break the symmetry in the cladding region. And other way is to introduce asymmetry in a porous core. Highly birefringent PCFs have wide application in optic sensor, Précised optical instruments, and high transmission speed, optical communication system. Control over chromatic dispersion in PCF is vital for their partial application. Again there is a tradeoff between high birefringence and dispersion with low losses. Large index contrast sometimes result excessive chromatic dispersion. Hence to make a control over dispersion with high birefringence has always been a challenge for the researchers. Several PCF structures with zero dispersion and high birefringence have been achieved by the researchers [^{16}]-[^{18}]. To achieve more significant result PCF with arrangement like square [^{19}], rectangular [^{20}], octagonal [^{21}], decagonal [^{22}], and do-decagonal [^{23}] have been investigated. Moreover, various PCF structures with nanometric holes have been reported [^{24}]-[^{28}]. PCF structures with such holes have been selectively filled or fully filled with liquids, gases and other materials [^{29}]-[^{32}]. Very tight confinement of optical mode in core results large non linearity. It can be achieved by reducing the core area. It can also be obtained by increasing the index contrast between core and cladding. The contrast can be increased by use of soft glass material. Supercontinuum generation is an inherent property of nonlinear optics. It has wide application in pulse compression, coherence tomography and spectroscopy meteorology [^{33}]-[^{34}]. Basically, supercontinuum generation, in nonlinear process produces broadband light. Interaction of short and intense pulses data output of nano band sources 81elps in achieving this. Zero ultra flattened dispersion is the primary need of achieving a broadband supercontinuum generation. Ranka et al was the first to demonstrated supercontinuum generation [^{35}]. Supercontinuum spectrum are generated by pumping ultra short pulses in fiber whose wavelength lie in dispersion regime close to zero dispersion wavelength. Razak et al achieved large nonlinearity with octagonal PCF [^{36}]. Moreover Camerlengo et al introduced a W-type index profile PCF with nonlinearity 0.82 W^{-1}m^{-1} at zero dispersion wavelengths [^{37}].
In this work, do-octagonal PCF structures doped with material of alcoholic group have been studied. Initially, a hexagonal PCF has been designed. Later a do-octagonal shaped core has been made by introducing defects at the centre. This defect is made by removing holes at the centre of fiber. After that new holes, smaller in dimension has been arranged in a rhombic fashion. The structure has a rhombic arrangement of holes fill with materials of alcoholic group and air. Alcoholic material like butanol, ethanol, methanol and propanol has been used. The structure report ultra flattened zero dispersion at visible range and at infrared region. Birefringence shown by the structure is much higher and is of the order of 10^{-1}. PCF structure doped with methanol reported highest birefringence. Moreover this structure reported ultra low confinement loss and very large nonlinear coefficient. Supercontinuum generation has also been generated at a wavelength of 1250 nm.
II. DESIGN OF DO-OCTAGONAL PCF STRUCTURE
Six ring hexagonal PCF structures have been studied. The PCF structure has holes doped with different material and air arranged in a rhombic fashion at the centre. These holes have much smaller dimension than the holes arranged in the external six rings. Material used for doping are butanol, ethanol, methanol and propanol. The schematic diagram of the investigated structure is shown in Fig. 1(a). Fig. 1(b) shows the three dimensional view of the same investigated structure. Table I shows the description of all five PCF structure simulated in this work.
PCF Structures | Descriptions |
---|---|
Silica_Air | This structure has six rings of holes of circular dimension filled with air. At centre it has smaller circular holes arranged in rhombic fashion filled with air. Smaller circular holes have diameter of d2=0.2*˄, where ˄=1μm. External six rings have air holes of diameter d1=0.6*˄. |
Silica_Butanol | This structure has six rings of holes of circular dimension filled with air. At centre it has smaller circular holes arranged in rhombic fashion filled with butanol. Smaller circular holes have diameter of d2=0.2*˄, where ˄=1μm. External six rings have air holes of diameter d1=0.6*˄. |
Silica_Ethanol | This structure has six rings of holes of circular dimension filled with air. At centre it has smaller circular holes arranged in rhombic fashion filled with ethanol. Smaller circular holes have diameter of d2=0.2*˄, where ˄=1μm. External six rings have air holes of diameter d1=0.6*˄.. |
Silica_Methanol | This structure has six rings of holes of circular dimension filled with air. At centre it has smaller circular holes arranged in rhombic fashion filled with methanol. Smaller circular holes have diameter of d2=0.2*˄, where ˄=1μm. External six rings have air holes of diameter d1=0.6*˄. |
Silica_Propanol | This structure has six rings of holes of circular dimension filled with air. At centre it has smaller circular holes arranged in rhombic fashion filled with propanol. Smaller circular holes have diameter of d2=0.2*˄, where ˄=1μm. External six rings have air holes of diameter d1=0.6*˄. |
Full vector finite element method (FEM) numerical technique have been adopted to analyse the different propagation characteristics of PCF. First FEM processes complex structured PCFs into homogeneous subspaces. Secondly, it is computed with Maxwell's vector equation as given by equation (1)) [^{38}].
Where [s] is perfectly matched layer matrix of dimension 3X3. It includes even parameters S_{x} and S_{y}. Electric vector is denoted by ‘E’. Wave number in vacuum is obtained as:
A circularly perfectly matched layer, boundary condition having thickness 10% of the fiber radius have been used to absorb the scattered light towards the surface of the fiber. Propagation constant (β) and effective refractive index at different wavelength (λ) are provided by the simulating software COMSOL Multiphysics 5.2. These are represented as
Modal birefringence defined as the difference between two polarized modes, is considered to play important role in determining PCFs to work as sensors. Mathmatically, it can be expressed as [^{30}]:
Again, obtained effective refractive index of the guided fundamental modes at different wavelengths helps in calculating chromatic dispersion parameters of PCF by using the following mathematical expressions [^{39}]:
The cross sectional area of mode field, over which the field gets confined along the fiber during its propagation is termed as effective area (A_{eff}). High optical intensities are the results of smaller effective area. Hence, PCF with small effective area results large nonlinear coefficient (γ) [^{40}].
Where |E| is the transverse electric field. n_{2} = 3.0×10^{–20}m^{2}W^{–}^{1}, is the nonlinear refractive index of silica used. n_{2} decides the degree to which non linear effects occur when light with high intensity propagate into the fiber.
Modes having leaky nature, together with non-perfect structure of PCFs result confinement loss. It is an extra form of loss which occurs in PCFs, usually made up of silica. These modes get leaky due to the finite lattice structure of PCFs. Confinement loss is calculated by considering imaginary part of the refractive index. It is calculated using formula [^{40}]:
Investigation of supercontinuum generation in the design fiber has been made. Modified nonlinear Schrodinger equation describes the mathematical model of supercontinuum generation. Schrodinger equation has different linear and nonlinear effects. This equation is solved using Split step Fourier method [^{41}-^{43}].
Where A (Z, T) represents slow varying envelope of the electric field of the optical pulse. The pulses move in a frame of reference along the z- direction at the pump frequency of the group velocity. α represent the attenuation constant of the fiber. β_{n} is the n^{th} order propagation constant at the center frequency ω_{o}. R (T) shows the nonlinear response function. It includes Raman contribution. It is defined as
f_{r} represent fractional contribution of the delay Raman response. Its value is taken to be 0.18. h_{r} represent form of Raman response function and it is calculated as
Where τ_{1}=12.2 fs and τ_{2}=32 fs use for silica. Possibility of cross phase modulation between pulses of two different polarizations are minimized as the assume input pulse with either posses vertical or horizontal polarization during launching into the fiber.
III. SIMULATION AND RESULTS
Simulated PCF structures have circular holes arranged in rhombic fashion. External six rings are arranged in hexagonal arrangements. Dimension of six rings with hexagonal arrangements have diameter d1, where d1=0.6*˄. Here ˄ is the pitch factor and its value taken is 1.0 μm. However holels arranged in rohmbic fashion have smaller diameter d2, where d2=0.2*˄. Designed PCFs have been simulated by finite element method (FEM) method of COMSOL Multiphysics 5.2 software.
Investigated PCF structures report highly birefringent behavior. Do-octagonal structure with air hole and do-octagonal PCF structure dopped with methanol report highest birefringence in comparison to other PCF structures studied in this work. Achieved birefringence for these two PCF structure is of the order of 10^{-1}. Microstructured core are feasible to obtain enhanced birefringence. Further increasing or decreasing the diameter of air hole in the microstructure core and reducing the size of spacing between two consecutive holes (pitch factor) results much higher birefringence. Methanol is an active, volatile, toxic and flammable raw material for the production of synthetic resin and plastic. Advantage of photonic crystal slabs doped with methanol solve the problem of accuracy, sensitivity and security [^{44}], [^{45}]. Moreover researchers have reported a highly birefringence spiral photonic crystal fiber for gas sensing application [^{46}] and Shengsi et al have reported a new two dimensional photonic slab with methanol doped photonic crystal slab [^{47}]. Fig. 2 shows birefringence behavior at diffrent wavelengths. Fig. 3 shows linearily polarized mode of the simulated structure. Due to rhombic arrangement of smaller holes, achieved birefringence at higher wavelengths is much higher.
Do-octagonal PCF structure has shown ultra flattened zero dispersion. From visible range to far infrared region, all the five PCF structure have shown flattened zero dispersion. Group velocity dispersion in standard fiber is dominated by the dispersion of bulk silica. However in photonic crystal fiber design parameters like hole diameter, center to center spacing and geometry of holes enable flexibility in designing PCF to control over dispersion [^{48}], [^{49}].
Zero dispersion photonic crystal fiber with very small core and efficient at generating supercontinuum light have been reported [^{50}], [^{51}]. By optimizing d and Λ it become possible to obtain zero dispersion with ±1 ps/(nm.km) center approximately around 1.52 μm, (d=0.73 μm and Λ=3.02 μm) [^{17}]. Similarly, a dispersion of D = ±0.5ps / (nm.km) is achieved with d=0.63 μm and Λ=2.64 μm [^{52}]. This makes the fiber applicable for high data transmission. Also such fibers can be used supercontinuum generations. Dispersion behavior of the studied structures is shown in Fig. 4.
Confinement loss at different wavelength have been plotted in Fig. 5. Do-octagonal structure with air hole has shown the highest loss in comparison to the doped do-octagonal PCF structure. However it is to be noted that the loss observed for all the structure is ultra low. The confinement loss obtained is of the order of 10^{-14}.
Do-octagonal PCF structure doped with methanol has the highest effective mode area in comparison to the other structures. Ethanol doped do-octagonal has the lowest effective mod e area in comparison to other structures. Figure 6 shows the effective mode area at different wavelength. Nonlinearity of a fiber is inversely proportional to the effective mode area. Nonlinearity of designed fiber is plotted in Fig. 7.
Supercontinuum generation has been explained with a PCF of length 15 cm. In order to get desire characteristic of supercontinuum generation, zero dispersion wavelength of dispersion profile are important. Desire characteristics include width and flatness. Zero dispersion behavior has been obtained over a wide band of wavelength. Pump pulses at a wavelength of 1250 nm has been used for simulation. PCF has shown zero dispersion at this wavelength other values of different high order term of dispersion at 1250 nm are
Ultra short laser pulses with hyperbolic secant shape are introduced as the input to the PCF. These pulses equation can be characterize as:
Where T_{0} represents the temporal width of input pulse.
Higher order soliton become a dominant phenomena in the case of ultra short pulses pumped into zero dispersion regime. Soliton number determines these high order soliton. It can be calculated as:
Here L_{D} represent dispersion length and it is equal to L_{D} = T_{0}^{2} /| β_{2} |. Similarly L_{NL} = 1/ γΡ_{0} where L_{NL} is the nonlinear length. P_{0} is the peak power used. β_{2} is the group velocity dispersion coefficient. In Fig. 8, soliton numbers for the generation of supercontinuum are calculated against the variation of input pulse duration. It reveal that decrease in the duration of the input pulse leads to the generation of more number of high order soliton. Calculated soliton number are very large at the given peak power.
Such a large generation of soliton may cause intra soliton interaction [^{53}], [^{54}]. Again dispersion length calculated is found to be 55 cm. It shows much larger fiber than the actual fiber length of 15 cm. Dispersion effect degrade in comparison to the nonlinear terms. Nonlinear effect determine the pulse evolution and result spectral broadening of these pulses.
Effect of pulse compression are affected by soliton fission length. These length are calculated using L_{fiss} = L_{D} / N. The calculated value have been plotted in Fig. 9. Small duration of pulses cause in early pulse compression. It results smaller fission length and hence result wide spectrum.
Optical pulse of duration 50 fs at 1250 nm has been used in pumping pulse for supercontinuum generation. Numerical simulation for supercontinuum generation has been done at four different peak power levels 1 kW, 2 kW, 5 kW and 10 kW for air filled PCF structure. However, it can also be done for the liquid filled fibres, as also done previously by many researchers [^{55}]-[^{58}]. Liquid evaporation is an important limiting issue which prevents the realization of fiber for long term application in practical. Many application have been enabled by the feasibility of inserting gas, liquids, polymers, and colloids into the microstructure core fully or partially. It result and efficient interaction of the guided light and thus modifies the waveguide characteristics [^{59}], [^{60}]. However in some cases contact of the liquid with the external environment may degrade its properties. It is expected that very low fiber loss will not affect supercontinuum generation. First of all a numerical simulation for supercontinuum generation for fiber having different length have been perform by using pulse of 1kW power. Fig. 10 (a) represent the spectral and temporal evolution of the input pulse of five different length of fiber respectively. A density plot has been used for getting more information about spectra broadening dynamic. Both spectral and temporal intensity have been plot using logarithmic density scale shorter at −40 dB relative to maximum value. Fig. 10 (b) displays density plot. This plot is utilized for studying the low amplitude spectral and temporal components.
Initialy spectral broadening is dominated by modulation instability. Later it gets dominated by nonlinear phenomena like four wave mixing, dispersion wave generation, Raman scattering and self phase modulation. The spectrum obtained at the end of 15 cm fiber is 600 nm wide. Similarly more simulation has been made for supercontinuum generation by increasing the peak power to 2 kW, 5 kW and 10 kW respectively. The mechanism remain same except getting the wider broadening than in previous case. The spectral and temporal dynamic have been shown in Fig. 11 (a), Fig. 11 (b), Fig. 12 (a), Fig. 12 (b), Fig. 13 (a) and Fig. 13 (b) respectively.
The supercontinuum spectra obtained for peak power 1 kW, 2 kW, 5 kW and 10 kW are 600 nm, 950 nm, 1450 nm and 2050 nm respectively.
Results obtained in this paper are compared with the previously reported results of PCF structures at the wavelength 1.33 μm. This comparison is tabulated in Table II.
References | Birefringence | Confinement loss in dB/Km | Dispersion in ps/nm-km | Effective mode area in (Micrometer^{2}) | Nonlinear coefficient in W^{– 1}Km^{-1} |
---|---|---|---|---|---|
Ferhat et al. [61] | -- | -- | 30 | 4.3 | 50 |
Sharma et al. [62] | -- | -- | 50 | 0.7 | 300 |
Kirishna et al. [63] | -- | -- | 100 | 4.3 | 50 |
Ademgil et al. [64] | 0.0019 | 2.1*10^{-2} | -- | -- | -- |
Ademgil et al. [65] | 0.00061 | 2.01*10^{-1} | -- | -- | -- |
Arif et al. [66] | 0.002825 | 2.07*10^{-6} | -- | -- | -- |
[Present Work] | 0.14 | 3.2*10^{14} | Zero | 10.5 | 3430 |
IV. CONCLUSION
Do-octagonal PCF structure with holes arranged in rhombic fashion is investigated. These holes of circular dimension have been doped with different materials of alcoholic groups (-OH group). In one of the investigated structure holes are filled with air. Results of doped PCF structures are compared with PCF structure having air holes. Ultra flattened zero dispersion with much high birefringence has been achieved. Apart from this result, ultra low confinement loss with large nonlinearity has also been obtained. Number of solitons generated is large. Numerical simulation for supercontinuum generation has been performed. The supercontinuum spectra obtained for peak power 10kW is 2050nm.