Realization of Attenuator and Amplifier Using Photonic Crystal Fiber

The present paper realises the attenuator and amplifier of electromagnetic signals with silicon-based square-type photonic crystal fiber. In this research, the input signal varies from 30 eV to 30,000 eV, which lies within the x-ray regime. The operational mechanism deals with the electric field distribution in the structure, which is carried out using the plane wave expansion method. The numerical outcomes affirm that the increasing and decreasing amplitudes of incident signals occur at the output end of the fiber. The intrinsic mechanism of such an interesting result is due to the nonlinear properties of field distribution at a different value of lattice spacing and the diameter of the proposed fiber’s air holes. Finally, the immediate upshot’s outcome resembles the amplifier and attenuator about the optical system.


Introduction
Research on the 2-D photonic structure is a hot arena nowadays due to many exciting applications that have been realised about the current research scenario in optical technology. The best example of two dimensional photonic crystal is photonic crystal fiber. Photonic crystal fibers (PCFs) have recently gained distinction in research and have been implemented in different noteworthy applications due to the special characteristics of these photonic Fibers. Though research on photonic crystal fiber has been started before two decades, many of such works have been found with the help of square, triangular and honeycomb photonic crystal fiber. However, recently, various research works related to biomedical sensing, computer networking, environment, and communication have been proposed [1][2][3][4][5][6][7][8][9][10]. For example, Ref. [1] designs a photonic crystal fiber for the measurement of liquid with high sensitivity and high birefringence. Similarly in the ref. [2], authors presented the design and simulation for the investigation of different blood components with the help of honeycomb photonic crystal fiber at THz regime. Likewise ref. [3] determines the early detection of blood cancer using twin core photonic crystal fiber using refractive index sensor. Ref. [4] deals with high sensitive D-shaped photonic crystal fiber sensor with Vgroove analyte channel where Ref. [5] measures potassium chloride concentration using a square type photonic crystal fiber. Similarly, low dispersion photonic fiber application can be realised through the Ref. [6]. Further photonic crystal fiber-based UV light torch is proposed in the Ref. [7], which could solve the problem related to the contamination of germs in the water. Thereafter Ref. [8] and Ref. [9] deals with the optical MUX/ DEMUX and dual channel detection application. Ultraviolet laser torch has been thoroughly discussed in the ref. [10] to solve out environmental predicament. Nevertheless, the above said research deals with different noteworthy forms; the present work bestows a new type of claim with the help of square type photonic crystal fiber. Besides, photonic crystal fiber has certain advantages over conventional optical fiber with respect to the efficiency, scalability, beam quality, high quality, operating cost and lower attenuation. Apart from this, photonic crystal fiber works under flexible conditions pertaining to the variation of refractive indices of core and cladding. For example; the propagation of the signal in the PCF is possible if the refractive index of core is either greater or less than the refractive index of the cladding. However conventional optical fiber works with total internal reflection principle only where the refractive index of the core is greater than cladding. When X-ray signal is incident to the photonic crystal fiber, the output signals will have either less or more energy than the input energy. So we can deduce that if the energy of the output signal will be less than the input signal, it acts as an attenuator (fiber is said to be attenuator). On the other hand, suppose the output signal will have more energy than the input signal then it simply function like an amplifier (fiber is said to be an amplifier). To understand the same in better ways, we shall discuss the working principle in section 2. The general mathematical formulation is mentioned in section 3, wherein section 4 brings forth the result and discussion. Finally, the conclusion part of the work is presented in section 5.

Working Principle
This research aims to increase or decrease the amplitude of energy related to the input signal. Which indicates that, the value of output energy would be either decreased or increased by the proposed optical fiber. For example, the input energy of the x-ray regime is incident to the proposed structure, and then the emerging signal from the fiber would have less or more energy, which refers to the attenuating or amplifying of the signal as compared to the input energy. The proposed optical fiber structure is shown in Fig. 1.
In Fig.1(a), photonic crystal fiber with a square type structure is seen. Further, in Fig. 1(b), there are 24 numbers of air holes have been arranged regularly in such a way that the central region is realized through a defect because of air holes not appearing in the middle. Moreover, the background material of the same is considered as silicon. The reason for choosing such material is that a silicon-based optical circuit will be compatible with the optoelectronic circuit. Apart from this, silicon is suitable for high frequency applications, which is stable and does not exhibit the hygroscopic nature. It can also be drawn in the fiber easily at high temperatures. Silicon also exhibits a nonlinear behavior at high frequencies. The size of the structure varies as per the desired output (whether output energy is more or less than input), which is discussed in the result and interpretation section, however the length of the fiber is kept 1 μm. Further focusing on the operational mechanism, high energy of 30 eV to 30,000 eV is applied to the aforementioned photonic crystal fiber, and the fiber is designed in such a way that it will either alleviate or aggravate to the input counterpart. As far as the intrinsic mechanism of this research is concerned, the configuration of the proposed photonic fiber, such as lattice constant of structure and radius of air holes play a vital role because the output results differ from different structure parameters, which is clearly realized in the result and interpretation section. In this research the radius of the air hole is 0.4 times of the lattice spacing. As far as the internal mechanism is concerned, the transportation of electric field distribution in the photonic crystal fiber determines the emerging signal from the proposed fiber that appeared at the output end. To realize the same in a better way, let us focus on the following diagram, Fig. 1(c), which shows the complete aforementioned information. Here the energy of 30 eV to 30,000 eV (its corresponding the wavelength of the signal (λ = hc/eV) incidents to the proposed silicon-based photonic crystal fiber, then the structure is designed in such a way that it will either enhance or reduce its energy, and this increasing or decreasing of energy shall appear at the output end of the fiber. Such impressive result is possible due to the nonlinear properties of the proposed fiber pertaining to the electric field distribution. Moreover the physics of such interesting result is due to the nonlinear properties of the structure which depends on the effective area. The effective area can be defined as the ratio of total area of air holes to the surface area of the fiber structure. Here the lattice spacing of structures increases leading to the increase of the radius of the air holes. As a result of which the effective area increases leading to the allowance of more filed distribution due to the nonlinear properties of the photonic fiber. Further, as far as the amplification is concerned, the amplification of a weak signal takes place in a nonlinear medium such as a non-Centro symmetric nonlinear medium silicon optical fiber via the Kerr effect. In addition, the attenuation of the signal with respect to the small effective area is due to a finite amount of loss takes place for silicon background material where effective area is less. Before going to analyzes the mathematical formulation relating to the generation of the filed distribution in the fiber, let us concentrate on the coupling efficiency between source to the fiber, which can be written as [11].
Where η g , η r and η a are coupling efficiency between source and the fiber with respect to the geometry, reflection, and angular factor respectively. As far as geometrical efficiency (η g ) of the fiber is concerned, it depends on the size of the source and photonic crystal fiber. Since the size of the source is less than photonic crystal fiber, the value of η g is '1', therefore whatever amount of signal coming from the source shall reach at the fiber without any loss.
Similarly, the efficiency pertaining to the Fresnel's reflection at the interface between source and fiber (η r ) depends on the following expression as In the right hand side, RI substrate and RI air hole be the refractive indices of substrate and air hole respectively.
Putting the refractive indices of substrate and air holes, the coupling efficiency (ηr) pertaining to the Fresnel's reflection is 90%.
Further, the angular coupling efficiency can be written as Since the angle of incident from source is normal to the surface of the fiber, the value of θ is 90 0 . So the value of η a would be '1'.
So putting all the values of the different coupling efficiency, the resultant coupling efficiency will be around 90%.

Mathematical Formulation for Electric Field in Photonic Crystal Fiber
The general mathematics to find out the electric field distribution in the fiber is derived from the Maxwell's electromagnetic differential equations and the same dot and curl equation can be written as [12].
Here E x; ! t À Á , H x; ! t À Á are the electric and magnetic fields, and B x; ! t À Á and D x; ! t À Á are the magnetic induction fields and displacement.
Considering free from current and charges, the above said equations can be written as where ω > 0 is the angular frequency. Further combining these equations, we can write The solution of electric field equation can be written as From Maxwell's equations, the full-vector wave equation is found out for electric field where k is the wave vector, and ε(r) refers to the dielectric constant of the structure. The proposed structure is signified as a periodic cell, which includes a crystal structure and its defects. Since the structure is periodic in nature, we can convey E k as a sum of plane-waves based on Bloch theorem, where G k is a reciprocal lattice vector at k th state. The dielectric constant ε(r) is characterized by a Fourier expansion method.
Furthermore the energy coming from the photonic fiber in terms of electric field can be expressed as Further, photon particles' interaction with structure material leads to a nonlinear characteristic due to the periodic nature of the crystal. The induced polarization P O of nonlinear optical medium is originated by the susceptibility coefficients as [13,14].
Where χ (1) , χ (2) and χ (3) ) are the 1st, 2nd and 3rd order susceptibility coefficients correspondingly. Here the structures do not have Y (2) which is seen in noncentrosymmetric photonic crystals. The 3rd order for both centrosymmetric and non-centrosymmetric media of the proposed structures show the susceptibility which provides to Kerr nonlinearity.
The induced polarization of the medium can be written for monochromatic electric field, E = E t cosωt is Further taking all into consideration of the following susceptibility as a sum of linear YL and nonlinear YNL terms, the index of refraction are often written in terms of the electric field as The above said expression is the resultant refractive index of the structure which depends on the field distribution in the fiber. In the right side, the first term is due to χL and second term is due to χNL. The field distribution relies on the input wavelengths. So it concludes that the refractive index is a function of the wavelength for nonlinear phenomena in this research.

Result and Discussion
The plane wave expansion method is used to find out the electric field distribution inside the silicon-based photonic crystal fiber with the help of Eqs. 4 and 5. Before going to realize the result, let us focus on the structure parameter, including the nature of the material and configuration of the same. In this case, the lattice spacing of the photonic structure is taken of lattice spacing of 1 nm, 10 nm, 20 nm, 30 nm, 40 nm, 49 nm, 50 nm, 60 nm, 100 nm, 150 nm 160 nm, 170 nm, 180 nm, 190 nm, 200 nm, 210 nm, 220 nm, and 350 nm to examine both optical attenuator and amplifier. Further the radius of the air holes is taken of 0.4 times of the lattice spacing. Aside these, the nature of the structure are chosen as silicon material with fiber length of 1 μm. Furthermore, considering an operational mechanism, it is understood that signal with energy from 30 eV to 30,000 eV incidents to the structure. For example; we consider 30 eV,70 eV,100 eV 300 eV,700 eV,1000 eV,3000 eV,7000 eV,10,000 eV,30,000 eV as input signal for the sake of simulation purposes. So, with the help of Eqs. 4 and 5 and applying the PWE technique, electric field distribution at the output end of the fiber has been computed with the MATLAB, R2018a. Even though simulation has been conducted for a different configuration, we have shown for the lattice spacing of 10 nm and diameter of holes 8 nm corresponding to the input signal of 30 eV, which is shown in Fig. 2. Fig. 2 shows the electric field's variation at the output end of the fiber, which represents the output energy of 30 eV for lattice spacing of 10 nm and diameter air holes of 8 nm. In this figure, length and breadth are taken along x and y, respectively, where the electric field distribution of output (V/μm) is taken along the z-axis. The peak electric field is shown in the figure (red colour), which signifies that the signal is emerging from for the said photonic structure. The peak electric field at a particular position of fiber (inset in the figure) is shown clearly. Similarly, the peak electric field corresponding to other modes of propagation is also estimated. Finally, the amount of energy in terms of eV is computed using Eq. 8. As far as the configuration of structure is concerned, the lattice spacing of the structure and diameter of air holes play a vital role. For example, different lattice spacing and the air holes' diameter bestows different outcomes pertaining to the same input.
To summarize the above statement, it is found that the energy for the lattice spacing of '10 nm' and diameter (radius is 0.4 times of lattice spacing) of air hole of '8 nm' is found as 4.622 eV, which is less compared to the input energy of 30 eV. This implies that the input energy (30 eV) is attenuated because the output energy is lower (14.383 eV). Similarly, we .640 eV respectively pertaining to 30 eV as input signal. After analyzing the above said numerical result, it is understood that the output energy value is less than 30 eV for lattice spacing up to 30 nm, and output energy is more than 30 eV for lattice spacing more than 30 nm. As far as the mode of the propagation in the proposed photonic crystal fiber is concerned, the self-consistent of certain electric field distribution during the propagation is called as modes. Basically, self-consistent depends on different situations. In this case, the waveguide is spatially inhomogeneous in structure which can guide the waves. For light propagating in a waveguide, the self-consistency condition for a mode is more restricted which relies on the propagation constant. A waveguide has only a finite number of guided propagation modes, the intensity distributions of the same have a finite extent around the waveguide core. The number of guided modes, their transverse amplitude profiles and their propagation constants depend on the details of the waveguide structure and on the input frequency. Here the principle of mode of propagation depends on the photonic bandgap which is governed by the phenomenon of both interference and diffraction. Therefore the field distribution is not seen at the cladding region rather in the core or central region.
So it is realised that the proposed structure behaves as an attenuator for the value of lattice spacing up to 30 nm and amplifier for lattice spacing more than 30 nm. The above-said explanation was for 30 eV only, but we have made a simulation for input energy, which varies from 30 eV to 30,000 eV. Similarly, with the help of the same modus operandi, the simulations for all input energy have been made. The entire output result for realising both attenuator and amplifier are plotted in Fig. 3(a) to (j) corresponding to the lattice spacing of 1 nm, 10 nm, 20 nm, 30 nm, 40 nm, 49 nm, 50 nm, 60 nm, 100 nm, 150 nm, 350 nm, respectively.
In the Fig. 3, energy in eV is taken along the vertical axis, where lattice spacing in nm is taken in the horizontal axis. In these graphs, two types of colour are chosen, such as input (green colour) energy to the structure and output (red colour) energy emerging from the structure. It is found that input energy remains constant pertaining to the lattice spacing with respect to each graph. Further output energy increases with the increasing of lattice spacing because of size of the entire structure increases as the increase of lattice spacing. It also indicates that larger air holes structure allows more signal as compared to the smaller one. Moreover, the variation of the output energy is found nonlinear (Eqs. 6-9) as the principle of generation of the electric field is realised nonlinear at the output end of fiber. It is also envisaged that the output energy is less or more than the input signal, which is decided by the structure's configuration and nature. For example, output energy is less than input for certain values of 'a', referred to as attenuation of the input signal. Similarly, it is more at certain values of 'a', referred to as an amplification of input signal. For example; the proposed structure could be realised as attenuator for lattice spacing less than 30 nm, 50 nm,60 nm,100 nm,160 nm, 180 nm,200 nm,350 nm for the input of 30 eV,70 eV,100 eV 300 eV,700 eV,1000 eV,3000 eV,7000 eV,10,000 eV, 30,000 eV respectively. Similarly amplification of the signal could be found for lattice spacing more than 30 nm, 50 nm, 60 nm,100 nm,160 nm,180 nm,200 nm,350 nm for the input of 30 eV,70 eV,100 eV 300 eV,700 eV,1000 eV,3000 eV, 7000 eV,10,000 eV,30,000 eV respectively. Fig. 3 Comparison of output and input energy with respect to lattice spacing at incident of a 30 eV, b 70 eV, c 100 eV, d 300 eV, e 700 eV, f 1000 eV, g 3000 eV, h 7000 eV, i 10,000 eV, and j 30,000 eV

Conclusion
In this paper, the attenuation and amplification of the input signal can be realized through the silicon-based photonic crystal fiber. The operational principle deals with the transportation of the signal through numerical computation, which in turn is carried out with the help of the plane wave expansion method. The numerical investigation indicates that the configuration of the proposed photonic structure and nature of the material plays a pivotal role to decide the application of attenuation or amplification. The research outcomes affirm that silicon-based photonic crystal fiber could be suitable for both attenuator and amplifier.