Document Type : Original Research Paper


1 Optical Communications Research Group (NCRLab), Northumbria University, NE1 8ST, UK

2 Microwave Imaging Research Group, Northumbria University, NE1 8ST, UK


Studying Gaussian beam is a method to investigate laser beam propagation and ABCD matrix is a fast and simple method to simulate Gaussian beam propagation in different mediums. Of the ABCD matrices studied so far, reflection and refraction matrices at various surfaces have attracted a lot of researches. However in previous work the incident beam and the principle axis of surface are in parallel. As an extension to those investigations, a general scheme that the incident beam is oblique is discussed here and the full analysis of the reflection and refraction of a Gaussian beam at the surface of a tilted concave/convex elliptic paraboloid surface is addressed. Based on the optical phase matching, analytic mathematical equations are derived for the spot size and the wavefront radius of a beam. Expressions are converted into the ABCD matrices, which are more convenient and practical to use. Finally, a practical case is analyzed by applying the obtained formulas. This analysis is important since paraboloid surfaces in optics or terahertz waves are used as mirrors or lenses.


[1] H. Kogelnik and T. Li, "Laser beams and resonators," Appl. Opt., vol. 5, pp. 1550-1567, 1966.
[2] M. Shabani and A. A. Shishegar, "Vectorial Gaussian beam expansion for high-frequency wave propagation," IET Microwaves, Antennas & Propagation, vol. 4, pp. 2014-2023, 2010.
[3] C. Qi, X. Shi, and G. Wang, "High-order circuit-level thermal model of vertical-cavity surface-emitting lasers," IET Optoelectronics, vol. 5, pp. 19-27, 2011.
[4] Z. Zhao, K. Duan, and B. Lü, "Non-equiphaseHermite–Gaussian model of diodelaserbeams," Optik - International Journal for Light and Electron Optics, vol. 119, pp. 167-170, 2008.
[5] J. H. Song, "Fibre coupling tolerance modelling of uniform grating coupler on silicon on insulator," Electronics Letters, vol. 47, pp. 1290-1292, 2011.
[6] A. Chabory, J. r. m. Sokoloff, S. Bolioli, and P. F. o. Combes, "Computation of electromagnetic scattering by multilayer dielectric objects using Gaussian beam based techniques," ComptesRendus Physique, vol. 6, pp. 654-662, 2005/8/ 2005. [7] A. Chabory, J. Sokoloff, and S. Bolioli, "Physically based expansion on conformal Gaussian beams for the radiation of
curved aperture in dimension 2," IET Microwaves, Antennas & Propagation, vol. 2, pp. 152-157, 2008.
[8] J. S. Gardner, "Approximate expansion of a narrow Gaussian beam in spherical vector wave functions," Antennas and Propagation, IEEE Transactions on, vol. 55, pp. 3172-3177, 2007.
[9] I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, "Free space optical system performance for a Gaussian beam propagating through non-Kolmogorov weak turbulence," Antennas and Propagation, IEEE Transactions on, vol. 57, pp. 1783-1788, 2009.
[10] H. Mao and D. Zhao, "Intensity distribution and coherence property for the broadband Gaussian Schell-model array beams in free space," Optics Communications, vol. 284, pp. 3795-3801, 2011.
[11] R. Shavit, J. Sangiolo, and T. Monk, "Scattering analysis of arbitrarily shaped cylinders in a focused beam system-oblique incidence case," IEE Proceedings - Microwaves, Antennas and Propagation, vol. 148, pp. 73-78, 2001.
[12] W. Zhen-Sen, L. Zheng-Jun, L. Huan, Y. Qiong-Kun, and L. HaiYing, "Off-axis Gaussian beam scattering by an anisotropic coated sphere," Antennas and Propagation, IEEE Transactions on, vol. 59, pp. 4740-4748, 2011.
[13] C. Rieckmann, M. R. Rayner, and C. G. Parini, "Diffracted Gaussian beam analysis of quasi-optical multi-reflector systems," Electronics Letters, vol. 36, pp. 1600-1601, 2000.
[14] H. T. Chou and P. H. Pathak, "Fast Gaussian beam based synthesis of shaped reflector antennas for contoured beam applications," Microwaves, Antennas and Propagation, IEE Proceedings, vol. 151, pp. 13-20, 2004.
[15] D. Lugara, D. Lugara, A. Boag, and C. Letrou, "Gaussian beam tracking through a curved interface: comparison with a method of moments," IEE Proceedings - Microwaves, Antennas and Propagation, vol. 150, pp. 49-55, 2003.
[16] H. Liu, L. Liu, R. Xu, and Z. Luan, "ABCD matrix for reflection and refraction of Gaussian beams at the surface of a parabola of revolution," Appl. Opt., vol. 44, pp. 4809-4813, 2005.
[17] Y. Yu and W. Dou, "ABCD matrix for reflection and refraction of Gaussian beams on the interface of an elliptic paraboloid," Journal of Infrared, Millimeter, and Terahertz Waves, vol. 31, pp. 1304-1311, 2010.
[18] G. A. Massey and A. E. Siegman, "Reflection and refraction of Gaussian light beams at tilted ellipsoidal surfaces," Appl. Opt., vol. 8, pp. 975-978, 1969.
[19] S. Gangopadhyay and S. Sarkar, "ABCD matrix for reflection and refraction of Gaussian light beams at surfaces of hyperboloid of revolution and efficiency computation for laser diode to single-mode fiber coupling by way of a hyperbolic lens on the ϐiber tip," Appl. Opt., vol. 36, pp. 8582-8586, 1997.
[20] T. J. Finn, N. Trappe, J. A. Murphy, and S. Withington, "The Gaussian beam mode analysis of off-axis aberrations in long wavelength optical systems," Infrared Physics Technology, vol. 51, pp. 351-359, 2008.
[21] A. W. M. Lee, Q. Qin, S. Kumar, B. S. Williams, Q. Hu, and J. L. Reno, "Real-time terahertz imaging over a standoff distance (> 25 meters)," Applied Physics Letters, vol. 89, pp. 141125-3, 2006.
[22] X. Wang, Y. Cui, D. Hu, W. Sun, J. Ye, and Y. Zhang, "Terahertz quasi-near-field real-time imaging," Optics Communications, vol. 282, pp. 4683-4687, 2009.
[23] R. Yano, H. Gotoh, Y. Hirayama, T. Hattori, and S. Miyashita, "Synthesis of terahertz electromagnetic wave pulses using amplitude-and-phase masks," Chemical Physics, vol. 326, pp. 577-582, 2006.
[24] B. E. A. Saleh and M. C. Teich, Fundamentals of photonics. New York, Chichester: Wiley, 1991.
[25] D. E. Goldberg, Genetic algorithms in search, optimization, and machine learning. Reading, Mass: Addison-Wesley, 1989.
[26] J. F. Bonnans, Numerical optimization: theoretical and practical aspects, 2nd ed. ed. Berlin, New York: Springer, 2006.
[27] B. Stephen and V. Lieven, Convex Optimization: Cambridge University Press, 2004. 


Journal of Electrical and Computer Engineering Innovations (JECEI) welcomes letters to the editor for the post-publication discussions and corrections which allows debate post publication on its site, through the Letters to Editor. Letters pertaining to manuscript published in JECEI should be sent to the editorial office of JECEI within three months of either online publication or before printed publication, except for critiques of original research. Following points are to be considering before sending the letters (comments) to the editor.

[1] Letters that include statements of statistics, facts, research, or theories should include appropriate references, although more than three are discouraged.

[2] Letters that are personal attacks on an author rather than thoughtful criticism of the author’s ideas will not be considered for publication.

[3] Letters can be no more than 300 words in length.

[4] Letter writers should include a statement at the beginning of the letter stating that it is being submitted either for publication or not.

[5] Anonymous letters will not be considered.

[6] Letter writers must include their city and state of residence or work.

[7] Letters will be edited for clarity and length.