In a beam propagation method, the wave equation is solved with respect to z, which is the general direction of propagation- i.e., the length of time it takes for a beam to travel from one place to another. Various assumptions can be made based on the device's complexity. Maxwell's equations can be used to obtain the vectorial wave equation. Through this equation, all components of the electric field are connected. The vector wave equation is solved using a full vectorial technique. It attempts simultaneous evaluation of all field components as well as mixed derivatives of the polarisation coupling parameters. As a result, it has a lot of time and memory requirements. In devices that are sensitive to polarisation, such as polarisation splitters and converters, full vectorial approaches are required.
The transverse components of 3D structures are frequently related to each other, and the axial component of axially changing structures is also coupled to the other components. To get reliable results, a comprehensive vectorial analysis is required. Full vector approaches are especially important for hybrid modes in 3D nonlinear optical waveguides. Full vectorial approaches produce the most precise results, but because they are computationally intensive, they are rarely used, and simpler methods with lower computational requirements are preferred. Two groups of approaches have been developed in this direction: semi-vectorial and scalar methods. The coupling between the multiple polarizations is believed to be weak in a semi-vectorial technique, therefore the polarisation coupling terms are ignored. However, the refractive index gradient is not overlooked. As a result, a semi-vectorial technique is commonly utilised in devices with a high index contrast and weak polarisation coupling or when polarisation is irrelevant. The interaction between the multiple polarizations is weak in most practical devices, therefore a semi-vectorial technique would produce reasonably accurate results. Fields become discontinuous in devices where the index contrast is great or the index shift is sudden, such as waveguides produced in semiconductors and polymers, and semi-vectorial approaches are required. The semi-vectorial equation can be reduced even more by ignoring.
Scalar methods are frequently used and are appropriate for devices with a small change in refractive index across a single wavelength. In such instances, all field components must fulfil the scalar wave equation and be continuous in all directions. The scalar wave equation offers rather accurate answers in the case of weakly steering structures. A Fresnel or paraxial approximation, in addition to the above, is frequently utilised. The paraxial approximation is only valid for beams that make a modest angle with the simulation axis and are usually unidirectional. When the beam creates a large angle with the propagation axis, paraxial methods produce considerable inaccuracies, necessitating the use of nonparaxial methods. Several approaches for wide-angle beam propagation through guided-wave devices have been proposed. Most wide-angle beam propagation algorithms take an iterative approach, requiring a numerical effort equivalent to solving the paraxial equation many times. Most of these methods solve the one-way wave equation without taking into account the backward propagating components. The square root of the propagation operator in the wave equation is approximated in several ways in these approaches. The Padé approximants are a widely used approach for approximating the square root of the propagation operator. An approach based on symmetrized splitting of the propagation operator was recently devised for nonparaxial beam propagation. The formulation eliminates the need for the slowly variable envelope and one-way propagation approximations in order to solve the second-order scalar wave equation. The approach has been proven to be both numerically efficient and very accurate. It is non-iterative and takes less processing work than most Padé approximant-based approaches. The method can also be applied to bidirectional propagation. All of the aforementioned characteristics make it a very appealing strategy.
A new method for analysing scalar beam propagation in three dimensions has been described. The method is based on the non-paraxial finite-difference split-step method. By appropriately modifying the matrix representation of the field and the operators, both the transverse and transverse derivative operators are made to work simultaneously. The propagation matrix is evaluated totally analytically. Even with substantial discretization, the approach provides acceptable accuracy and remains stable when the reference refractive index varies. The approach can also be used to determine a waveguide's modes.
We give an example to demonstrate the method's applicability and efficiency. We investigate an FBG in which the refractive index inside the optical fibre core is disturbed. In the computations of reflection and transmission, the approach has been found to be quite efficient. So the method is capable of computing reflections and transmission.