volume | PIER Journals

Abstract

In this paper, Maslov's method has been used to obtain the high frequency field refracted by a hyperboloidal focusing lens. High frequency problem which contains caustic region is transformed into caustic free problem by transforming the situation into mixed domain. The high-frequency solution that includes the caustic region is obtained from geometrical optics. The defect in high frequency solution due to geometrical optics is overcomed by Maslov's method. Numerical computations are made for the field pattern around the caustic. The results are found in good agreement withobtained using Debye-Wolf focusing integral.

1. Hansen, R. C. (ed.), Geometrical Theory of Diffraction, IEEE Press, 1988.

2. Felson, L. B., Hybrid Formulation of Wave Propagation and Scattering, Nato ASI Series, Martinus Nijhoff, Dordrecht, 1984.

3. Maslov, V. P., "Perturbation theory and asymptotic method," Moskov., Gos. Univ., Moscow,1965(in Russian translated into Japanese by Ouchi et al., Iwanami, Tokyo, 1976.). Google Scholar

4. Kravtsov, Y. A., "Two new methods in the theory of wave propagation in inhomogeneous media (review) ," Sov. Phys. Acoust., Vol. 14, No. 1, 1-17, 1968. Google Scholar

5. Kravtsov, Y. A. and Y. I. Orlov, Caustics, Catastrophes, Wave Fields, 2nd Ed., 1997; 306, Springer-Verlag, 1993.

6. Kravtsov, Y. A. and Y. I. Orlov, Geometrical Optics in Inhomogeneous Media, 312, Springer-Verlag, 1990.

7. Kravtsov, Y. A. and Y. I. Orlov, "Geometrical optics in engineering physics," Alpha Science, International Ltd., UK, 2005, 1990. Google Scholar

8. Gorman, A. D., S. P. Anderson, and R. B. Mohindra, "On caustic related to several common indices of refraction," Radio Sci., Vol. 21, 434-436, 1986.
doi:10.1029/RS021i003p00434 Google Scholar

9. Gorman, A. D., "Vector field near caustics," J. Math. Phys., Vol. 26, 1404-1407, 1985.
doi:10.1063/1.526954 Google Scholar

10. Ziolkowski, R. W. and G. A. Deschamps, "Asymptotic evaluation of high frequency fields near a caustic, an introduction to Maslov's method," Radio Sci., Vol. 19, No. 4, 1001-1025, 1984.
doi:10.1029/RS019i004p01001 Google Scholar

11. Hongo, K., Y. Ji, and E. Nakajimi, "High-frequency expression for the field in the caustic region of a reflector using Maslov's method," Radio Sci., Vol. 21, No. 6, 911-919, 1986.
doi:10.1029/RS021i006p00911 Google Scholar

12. Hongo, K. and Y. Ji, "High-frequency expression for the field in the caustic region of a cylindrical reflector using Maslov's method," Radio Sci., Vol. 22, No. 3, 357-366, 1987.
doi:10.1029/RS022i003p00357 Google Scholar

13. Hongo, K. and Y. Ji, "Study of the field around the focal region of spherical reflector antenna," IEEE Trans. Antennas Propagat., Vol. 36, 592-598, 1988.
doi:10.1109/8.192134 Google Scholar

14. Ji, Y. and K. Hongo, "Field in the focal region of a dielectric spherical lens," J. Opt. Soc. Am. A, Vol. 8, 1721-1728, 1991.
doi:10.1364/JOSAA.8.001721 Google Scholar

15. Ji, Y. and K. Hongo, "Analysis of electromagnetic waves refracted by a spherical dielectric interface," J. Opt. Soc. Am. A, Vol. 8, 541-548, 1991.
doi:10.1364/JOSAA.8.000541 Google Scholar

16. Hongo, K. and H. Kobayashi, "Radiation characteristics of a plano convex lens antenna," Radio Sci., Vol. 31, No. 5, 1025-1035, 1987.
doi:10.1029/96RS01330 Google Scholar

17. Aziz, A., Q. A. Naqvi, and K. Hongo, "Analysis of the fields in two dimensional Cassegrain system," Progress In Electromagnetics Research, PIER 71, 227-241, 2007. Google Scholar

18. Aziz, A., A. Ghaffar, Q. A. Naqvi, and K. Hongo, "Analysis of the fields in two dimensional Gregorian system," Journal of Electromagnetic Waves and Applications, Vol. 22, No. 1, 85-97, 2008.
doi:10.1163/156939308783122733 Google Scholar

19. Ghaffar, A., Q. A. Naqvi, and K. Hongo, "Analysis of the fields in three dimensional Cassegrain system," Progress In Electromagnetics Research, PIER 72, 215-240, 2007.
doi:10.2528/PIER07031602 Google Scholar

20. Ghaffar, A., A. Hussain, Q. A. Naqvi, and K. Hongo, "Radiation characteristics of an inhomogeneous slab," Journal of Electromagnetic Waves and Applications, Vol. 22, No. 2, 301-312, 2008.
doi:10.1163/156939308784160695 Google Scholar

21. Ghaffar, A., Q. A. Naqvi, and K. Hongo, "Study of focusing of field refracted by a cylindrical plano-convex lens into a uniaxial crystal by using Maslov's method," Journal of Electromagnetic Waves and Applications, Vol. 22, 665-679, 2008.
doi:10.1163/156939308784159499 Google Scholar

22. Ghaffar, A., Q. A. Naqvi, and K. Hongo, "Focal region fields of three dimensional Gregorian system," Optics Communications, Vol. 281, 1343-1353, 2008.
doi:10.1016/j.optcom.2007.11.011 Google Scholar

23. Ghaffar, A. and Q. A. Naqvi, "Focusing of electromagnetic plane wave into uniaxial crystalby athre dimentional plano convex lens ," Progress In Electromagnetics Research, PIER 83, 25-42, 2008. Google Scholar

24. Ghaffar, A. and Q. A. Naqvi, "Study of focusing of field refracted by an inhomogeneous slab into a uniaxial crystal by using Maslov's method," Journal of Modern Optics, Vol. 55, No. 21, 3513-3528, 2008.
doi:10.1080/09500340802308300 Google Scholar

25. Fiaz, M. A., A. Ghaffar, and Q. A. Naqvi, "High-frequency expressions for the field in the caustic region of a PEMC cylinderical reflector using Maslov's method," Journal of Electromagnetic Waves and Applications, Vol. 22, 385-397, 2008.
doi:10.1163/156939308784160794 Google Scholar

26. Fiaz, M. A., A. Ghaffar, and Q. A. Naqvi, "High-frequency expressions for the field in the caustic region of a PEMC gregorian system using Maslov's method," Progress In Electromagnetics Research, PIER 81, 135-148, 2008. Google Scholar

27. Ashraf, M. R., A. Ghaffar, and Q. A. Naqvi, "Fields in the focal space of symmetrical hyperbolic focusing lens," Journal of Electromagnetic Waves and Applications, Vol. 22, 815-828, 2008.
doi:10.1163/156939308784159480 Google Scholar

28. Ghaffar, A. and Q. A. Naqvi, "Study of focusing of field refracted by a Wood lens into a uniaxial crystal by using Maslov's method," Cent. Eur. J. Phys..
doi:10.2478/s11534-008-0134-x Google Scholar

29. Ghaffar, A., M. Sajjad, and Q. A. Naqvi, "Study of focusing of field refracted by a quartic Wood lens into a uniaxial crystal by using Maslov's method ," Progress In Electromagnetics Research, PIER 86, 321-339, 2008. Google Scholar

30. Hussain, A., Q. A. Naqvi, and K. Hongo, "Radiation characteristics of the Wood lens using Maslov's method," Progress In Electromagnetics Research, PIER 73, 107-129, 2007. Google Scholar

31. Faryad, M. and Q. A. Naqvi, "Highfrequency expressions for the field in the caustic region of a cylindrical reflector placed in chiral medium," Progress In Electromagnetic Research, PIER 76, 153-182, 2007. Google Scholar

32. Faryad, M. and Q. A. Naqvi, "Highfrequency expressions for the field in the caustic region of a cylindrical reflector placed in chiral medium," Progress In Electromagnetic Research, PIER 76, 153-182, 2007. Google Scholar

33. Faryad, M. and Q. A. Naqvi, "Cylindrical reflector in chiral medium supporting simultaneously positive phase velocity and negative phase velocity ," Journal of Electromagnetic Waves and Applications, Vol. 22, 563-572, 2008.
doi:10.1163/156939308784150344 Google Scholar

34. Dou, W. B., Z. L. Sun, and X. Q. Tan, "Fields in the focal space of symmetrical hyperbolic focusing lens," Progress In Electromagnetics Research, PIER 20, 213-226, 1998. Google Scholar

35. Dou, W. B., Z. L. Sun, and X. Q. Tan, "Diffraction fields in the focal space of symmetrical hyperbolic focusing lens," International Journal of Infrared and Millimeter Waves, Vol. 20, No. 10, 1843-1854, 1999.
doi:10.1023/A:1021721917087 Google Scholar

36. Wolf, E., "Electromagnetic diffraction in optical systems I. An integral representation of the image field," Proc. Roy. Soc. London A, Vol. 253, 349-357, 1959. Google Scholar

37. Richards, B. and E. Wolf, "Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system," Proc. Roy. Soc. London A, Vol. 253, 358-379, 1959. Google Scholar

38. Sherif, S. S. and P. Török, "Eigenfunction representation of the integrals of the Debye-Wolf diffraction formula," J. Mod. Opt., Vol. 52, 857-876, 2005.
doi:10.1080/09500340512331309084 Google Scholar

39. Sherif, S. S., M. R. Foreman, and P. Török, "Eigenfunction representation of the electric field s in focal region of a high numerical aperture focusing system," Opics Express, Vol. 16, 3397-3307, 2008.
doi:10.1364/OE.16.003397 Google Scholar

40. Foreman, M. R., S. S. Sherif, P. R. T. Munro, and P. Török, "Inversion of the Debye-Wolf diffraction integral using an eigenfunction representation of the electric fields in the focal region," Opics Express, Vol. 16, 3397-3307, 2008. Google Scholar

41. Török, P., P. R. T. Munro, and Em. E. Kriezis, "Highn umerical aperture vectorial imaging in coherent optical microscopes," Opics Express, Vol. 21, 507-523, 2008. Google Scholar

42. Leutenegger, M., R. Rao, R. A. Leitgeb, and T. Lasser, "Fast focus field calculations," Opics Express, Vol. 14, No. 23, 11277-11291, 2006.
doi:10.1364/OE.14.011277 Google Scholar

Dr. Abdul Ghaffar

Professor Azhar Abbas Rizvi

Dr. Qaisar Abbas Naqvi