This paper presents an improved approach for the propagation of electromagnetic (EM) fields in the case of the exible hollow waveguide that consists of two bendings in the same direction. In this case, the objective is to develop a mode model for infrared (IR) wave propagation along the exible hollow waveguide, in order to provide a numerical tool for the calculation of the output fields, output power density and output power transmission. The main steps of the method for the two bendings will introduced in the derivation, in detail, for small values of step angles. The derivation for the first section and the second section of the waveguide with the two bendings is based on Maxwell's equations. The separation of variables is obtained by using the orthogonal-relations. The longitudinal components of the fields are developed into the Fourier-Bessel series. The transverse components of the fields are expressed as functions of the longitudinal components in the Laplace plane and are obtained by using the inverse Laplace transform by the residue method. This model can be a useful tool in all the cases of the hollow toroidal waveguides, e.g., in medical and industrial regimes.
2. Harrington, J. A., "A review of IR transmitting, hollow waveguides," Fiber and Integrated Optics, Vol. 19, 211-228, 2000.
3. Marcatili, E. A. J. and R. A. Schmeltzer, "Hollow metallic and dielectric waveguides for long distance optical transmission and lasers," Bell Syst. Tech. J., Vol. 43, 1783-1809, 1964.
4. Marhic, M. E., "Mode-coupling analysis of bending losses in IR metallic waveguides," Appl. Opt., Vol. 20, 3436-3441, 1981.
5. Miyagi, M., K. Harada, and S. Kawakami, "Wave propagation and attenuation in the general class of circular hollow waveguides with uniform curvature," IEEE Trans. Microwave Theory Tech., Vol. 32, 513-521, 1984.
6. Croitoru, N., E. Goldenberg, D. Mendlovic, S. Ruschin, and N. Shamir, "Infrared chalcogenide tube waveguides," SPIE, Vol. 618, 140-145, 1986.
7. Melloni, A., F. Carniel, R. Costa, and M. Martinelli, "Determination of bend mode characteristics in dielectric waveguides ," J. Lightwave Technol., Vol. 19, 571-577, 2001.
8. Bienstman, P., M. Roelens, M. Vanwolleghem, and R. Baets, "Calculation of bending losses in dielectric waveguides using eigenmode expansion and perfectly matched layers ," IEEE Photon. Technol. Lett., Vol. 14, 164-166, 2002.
9. Mendlovic, D., E. Goldenberg, S. Ruschin, J. Dror, and N. Croitoru, "Ray model for transmission of metallic-dielectric hollow bent cylindrical waveguides," Appl. Opt., Vol. 28, 708-712, 1989.
10. Morhaim, O., D. Mendlovic, I. Gannot, J. Dror, and N. Croitoru, "Ray model for transmission of infrared radiation through multibent cylindrical waveguides," Opt. Eng., Vol. 30, 1886-1891, 1991.
11. Kark, K. W., "Perturbation analysis of electromagnetic eigenmodes in toroidal waveguides," IEEE Trans. Microwave Theory Tech., Vol. 39, 631-637, 1991.
12. Lewin, L., D. C. Chang, and E. F. Kuester, Electromagnetic Waves and Curved Structures, Ch. 6, 58-68, Peter Peregrinus Ltd., London, 1977.
13. Menachem, Z., "Wave propagation in a curved waveguide with arbitrary dielectric transverse profiles," Progress In Electromagnetics Research, Vol. 42, 173-192, 2003.
14. Menachem, Z., N. Croitoru, and J. Aboudi, "Improved mode model for infrared wave propagation in a toroidal dielectric waveguide and applications," Opt. Eng., Vol. 41, 2169-2180, 2002.
15. Menachem, Z. and M. Mond, "Infrared wave propagation in a helical waveguide with inhomogeneous cross section and applications," Progress In Electromagnetics Research, Vol. 61, 159-192, 2006.
16. Menachem, Z. and M. Haridim, "Propagation in a helical waveguide with inhomogeneous dielectric profiles in rectangular cross section," Progress In Electromagnetics Research B, Vol. 16, 155-188, 2009.
17. Collin, R. E., Foundation for Microwave Engineering, McGraw-Hill, New York, 1996.
18. Yariv, A., "Optical Electronics," Holt-Saunders Int. Editions, 1985.
19. Baden Fuller, A. J., Microwaves, Ch. 5, 118-120, A. Wheaton and Co. Ltd, Pergamon Press, Oxford, 1969.
20. Olver, F. W. J., "Royal Society Mathematical Tables, Zeros and Associated Values," 2-30, University Press Cambridge, 1960.
21. Jahnke, E. and F. Emde, "Tables of Functions with Formulae and Curves," Ch. 8, 166, Dover publications, New York, 1945.
22., The Numerical Algorithms Group (NAG) Ltd., Wilkinson House, Oxford, UK.
23. Croitoru, N., A. Inberg, M. Oksman, M. Ben-David, "Hollow silica, metal and plastic waveguides for hard tissue medical applications," SPIE, Vol. 2977, 30-35, 1997.