Mathieu functions

The Mathieu functions are solutions of the Ordinary and the Modified Mathieu differential equations

Notes on the theory and numerical analysis of the Mathieu functions
Theory and numerical analysis of the Mathieu functions. v.2.11 [pdf (921 kB)]

Mathieu functions, a visual approach
Julio C. Gutiérrez-Vega, R. M. Rodríguez-Dagnino, M. A. Meneses-Nava, and S. Chávez-Cerda
Am. J. Phys. 71 (3), 233-242, Mar. 2003.

Biography of Emile Mathieu
http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Mathieu_Emile.html

Mathieu beam

Mathieu beams

Mathieu beams are orthogonal eigensolutions of the three-dimensional wave equation in elliptic coordinates.
Their applications are in optics, acoustics, and related wave phenomena.

Related papers on Mathieu beams and Mathieu functions

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If you know about other paper related to Mathieu beams that is not included in this list, let me know.

43	Generation of a vectorial elliptic hollow beam by an elliptic hollow fiber Huirong Li and Jianping Yin Opt. Lett., 36, (4) 457-459 (2011)	[Journal]
42	Numerical study for selective excitation of Mathieu-Gauss modes in end-pumped solid-state laser systems Shu-Chun Chu and Ko-Fan Tsai Opt. Express, 19, (4) 3236-3250 (2011)	[Journal]
41	Experimental generation of Mathieu–Gauss beams with a phase-only spatial light modulator R. J. Hernández-Hernández, R. A. Terborg, I. Ricardez-Vargas and K. Volke-Sepúlveda Appl. Opt., 49, (36) 6903-6909 (2010)	[Journal]
40	Propagation of generalized Mathieu–Gauss beams through paraxial misaligned optical systems A. Chafiq, Z. Hricha, A. Belafhal Opt. Commun., 282, (19) 3934-393 (2009)	[Journal]
39	Parametric characterization of Mathieu-Gauss beams A. Chafiq, Z. Hricha, A. Belafhal Opt. Commun., 282, (13) 2590-2594 (2009)	[Journal]
38	Dynamical constants for electromagnetic fields with elliptic-cylindrical symmetry B. M. Rodríguez-Lara and R. Jáuregui Phys. Rev. A, 78, 033813 (2008)	[Journal]
37	Elliptical beams M. A. Bandres and Julio C. Gutiérrez-Vega Opt. Express, 16 (25), 21087–21092, 2008	[pdf] [Journal]
36	Stable solitons in elliptical photonic lattices Adrian Ruelas, Servando López-Aguayo, and Julio C. Gutiérrez-Vega Opt. Lett. 33 (23), 2785-2787 (2008)	[pdf] [Journal]
35	Generation of Mathieu-Gauss modes with an axicon-based laser resonator Martha B. Alvarez-Elizondo, Rodolfo Rodríguez-Masegosa, and Julio C. Gutiérrez-Vega Opt. Express, 16 23, 18770-18775, 2008	[pdf] [Journal]
34	Propagation aspects of Mathieu-Gaussian beams in turbulence H. T. Eyyuboğlu Appl. Phys. B, 91 (3), 629-637, 2008	[Journal]
33	A study of source plane Mathieu beams H. T. Eyyuboğlu Appl. Phys. B, 91 (3), 611-619, 2008	[Journal]
32	Focal shift in vector Mathieu-Gauss beams Raul I. Hernandez-Aranda and Julio C. Gutiérrez-Vega Opt. Express, 16 (8), 5838-5848, 2008	[pdf] [Journal]

31	Rytov theory for Helmholtz-Gauss beams in turbulent atmosphere Rodrigo J. Noriega-Manez and Julio C. Gutiérrez-Vega Opt. Express, 15 (25), 16328-16341, 2007	[pdf] [Journal]
30	Propagation properties of vector Mathieu–Gauss beams A. Chafiq, Z. Hricha, A. Belafhal Opt. Commun. 275, 165-169 (2007).	[Journal]
29	Helical Mathieu and Parabolic localized pulses Josué Davila-Rodriguez and Julio C. Gutiérrez-Vega J. Opt. Soc. Am. A, 24 (11), 3449-3455, 2007	[pdf] [Journal]
28	Self-trapped modes in highly nonlocal nonlinear media Servando López-Aguayo and J. C. Gutiérrez-Vega Phys. Rev. A, 76, 023814, 2007	[pdf] [Journal]
27	Normalization of the Mathieu-Gauss optical beams J. C. Gutiérrez-Vega and Miguel A. Bandrés, J. Opt. Soc. Am. A, 24(1), 215-220, 2007	[pdf] [Journal]
26	The generation of nondiffracting beams using unexpensive computer-generated holograms Carlos López-Mariscal and Julio C. Gutiérrez-Vega , Am. J. Phys, 75(1), 36-42, 2007	[pdf] [Journal]

25	Propagation of generalized vector Helmholtz-Gauss beams through paraxial optical systems Raul I. Hernandez-Aranda, J. C. Gutiérrez-Vega, Manuel Guizar-Sicairos, and Miguel A. Bandres Opt. Express, 14 (20), 8974-8988, 2 Oct. 2006	[pdf] [Journal]
24	Generalized Helmholtz-Gauss beams and its transformation by paraxial optical systems Manuel Guizar-Sicairos and J. C. Gutiérrez-Vega, Opt. Lett. 31 (19), 2912-2915, 1 Oct. 2006	[pdf] [Journal]
23	General construction and connections of vector propagation invariant optical fields: TE and TM modes and polarization states K. Volke-Sepulveda and E. Ley-Koo J. Opt. A: Pure Appl. Opt. 8, 867-877 (2006)	[Journal]
22	A detailed study of Mathieu–Gauss beams propagation through an apertured ABCD optical system A. Chafiq, Z. Hricha, A. Belafhal Opt. Commun. 265, 594-602 (2006).	[Journal]
21	Propagation of Helmholtz-Gauss beams in absorbing and gain media Manuel Guizar-Sicairos and J. C. Gutiérrez-Vega J. Opt. Soc. Am. A, 23 (8), 1994-2001, Aug. 2006	[pdf] [Journal]
20	Observation of the experimental propagation properties of Helmholtz-Gauss beams Carlos López-Mariscal, Miguel A. Bandrés, and J. C. Gutiérrez-Vega Opt. Eng. 45 (6), 068001, June 2006	[pdf] [Journal]
19	Shaping soliton properties in Mathieu lattices Y. V. Kartashov, A. A. Egorov, V. A. Vysloukh, and L. Torner Opt. Lett. 31, 238–240 (2006)	[Journal]
18	Orbital angular momentum transfer in helical Mathieu beams Carlos López-Mariscal, Julio C. Gutiérrez-Vega, Graham Milne and Kishan Dholakia Opt. Express, 14 (9), 4182-4187, May 2006	[pdf] [Journal]

17	Rows of optical vortices from elliptically perturbing a high-order beam Marc. R. Dennis Opt. Lett., 31(9), 1325-1327 (2006)	[Journal]
16	Paraxial propagation of Mathieu beams through an apertured ABCD optical system A. Chafiq, Z. Hricha, A. Belafhal Opt. Commun. 253, 223-230 (2005).	[Journal]
15	Vector Helmholtz-Gauss and vector Laplace-Gauss beams Miguel A. Bandres and J. C. Gutiérrez-Vega Opt. Lett., 30 (16), 2155-2157, 15 Aug. 2005	[pdf] [Journal]
14	Helmholtz-Gauss waves Julio C. Gutiérrez-Vega and Miguel A. Bandres J. Opt. Soc. Am. A, 22(2), 289-298, Feb. 2005	[pdf] [Journal]

13	Properties of a localized Mathieu pulse C. A. Dartora and H. E. Hernández-Figueroa J. Opt. Soc. Am. A 21, 662-667 (2004).	[Journal]
12	Nondiffracting optical beams: physical properties, experiments, and applications Z. Bouchal Czech. J Phys., 53, 537-578 (2003).	[Journal]
11	Focusing evolution of Generalized Propagation Invariant Optical Fields Julio C. Gutiérrez-Vega, Rodolfo Rodríguez-Masegosa, and S. Chávez-Cerda J. Opt. A: Pure Appl. Opt. 5, 276-282, May 2003.	[pdf] [Journal]
10	General formulation for the analysis of scalar diffraction-free beams using angular modulation: Mathieu and Bessel beams C.A. Dartora, M. Zamboni-Racheda, K.Z. Nobrega, E. Recami, H.E. Hernandez- Figueroa Opt. Commun. 222, 75–80 (2003)	[Journal]
09	Mathieu functions, a visual approach Julio C. Gutiérrez-Vega, R. M. Rodríguez-Dagnino, M. A. Meneses-Nava, and S. Chávez-Cerda Am. J. Phys. 71 (3), 233-242, Mar. 2003.	[pdf] [Journal]

	08	Holographic generation and orbital angular momentum of high-order Mathieu beams S. Chávez-Cerda, M.J. Padgett, I. Allison, G.H.C. New, Julio C. Gutiérrez-Vega, A.T. O’Neil, I. MacVicar, and J. Courtial J. Opt. B: Quantum Semiclass. Opt. 4, S52–S57, Apr. 2002.	[pdf] [Journal]
	07	Attenuation Characteristics in Confocal Annular Elliptic Waveguides and Resonators Julio C. Gutiérrez-Vega, R. M. Rodríguez-Dagnino, and S. Chávez-Cerda IEEE Trans. Microwave Theory Tech., 50 (4), 1095-1100, April 2002. Additional material and extended version of the paper	[pdf] [Journal]

06	Elliptic vortices of electromagnetic wavefields S. Chávez-Cerda, Julio C. Gutiérrez-Vega, and G.H.C. New Opt. Lett., 26 (22), 1803-1805,15-Nov. 2001.	[pdf] [Journal]
05	Probability distributions in classical and quantum elliptic billiards Julio C. Gutiérrez-Vega, S. Chávez-Cerda y R. M. Rodríguez-Dagnino Rev. Mex. Fis., 47 (5), 480-488, Oct. 2001.	[pdf] [Journal]
04	Experimental demonstration of optical Mathieu beams Julio C. Gutiérrez-Vega, M. D. Iturbe-Castillo, G. A. Ramírez, E. Tepichín, R. M. Rodríguez-Dagnino, S. Chávez-Cerda, and G.H.C. New Opt. Comm., 195 (1-4), 35-40, 1 Aug. 2001.	[pdf] [Journal]

03	New Member in the Family of Propagation Invariant Optical Fields: Mathieu beams Julio C. Gutiérrez-Vega, M. D. Iturbe-Castillo, E. Tepichin, G. Ramírez, R. M. Rodríguez-Dagnino, and S. Chávez-Cerda Opt. and Phot. News, 11 (12), 35-36, Dec. 2000	[pdf] [Journal]
02	Alternative formulation for invariant optical fields: Mathieu beams Julio C. Gutiérrez-Vega, M. D.Iturbe-Castillo, and S.Chávez-Cerda Opt. Lett., 25 (20), 1493-1495, Oct. 15, 2000.	[pdf] [Journal]
01	Free oscillations in an elliptic membrane Julio C. Gutiérrez-Vega, S. Chávez-Cerda and R. M. Rodríguez-Dagnino Rev. Mex. Fis., 45 (6), 613-622, Dec. 1999.	[pdf] [Journal]