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.