The purpose of this paper is to develop highly efficient decomposition finite difference methods for computing solutions of highly oscillatory beam propagation partial differential equations.
Highly oscillatory optical wave equations, such as the multidimensional paraxial Helmholtz equation, have been used extensively in modelling propagation of the light from lens to the focal region in various engineering applications. Numerical approximations of solutions of such equations contain crucial light information in focal regions even when the f-number is small. However, it has been difficult to acquire highly oscillatory numerical solutions efficiently. This paper proposes two correlated eikonal decomposition strategies for fast computations of the oscillatory solutions. Structures of the numerical methods are designed via an eikonal, or exponential, transformation. The approach converts successfully the oscillatory problems to non-oscillatory subproblems. Therefore, the underlying beam simulation equations can be solved readily with great accuracy and stability.
It is found that the two correlated eikonal transformation based decomposition methods effectively remove the highly oscillatory features of the wave equations. The coupled non-oscillatory subproblems resulted are easier to solve. Discretization steps in computations can be chosen to be relatively large and this ensures the efficiency of computations. The decomposed finite difference schemes are simple to use in different optical applications.
The computational approach provides a valuable tool to practical applications, such as those in the defence industry.
Although the eikonal transformation has been used in the theory of nonlinear optics, this is the first time it has been utilized for effective engineering computations.
The authors would like to thank their colleagues for the many discussions which helped to shape their investigations and computational experiments. This study is supported in part by a research grant (No. F-5400-04-06-SC01-00) from the Air Force Research Laboratory and General Dynamics Information Technology. The authors wish to thank the AFRL and GDIT for their generous support and encouragement. Last, but not least, the authors would like to thank the referees for their valuable suggestions which helped to improve the content and presentation of this paper.
Sheng, Q., Guha, S. and Gonzalez, L. (2012), "Eikonal decomposition methods for fast computations of beam propagations", Engineering Computations, Vol. 29 No. 1, pp. 4-18. https://doi.org/10.1108/02644401211190537Download as .RIS
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