Vector and scalar potential formulation is valid from quantum theory to classical electromagnetics. The rapid development in quantum optics calls for electromagnetic solutions that straddle quantum physics as well as classical physics. The vector potential formulation is a good candidate to bridge these two regimes. Hence, there is a need to generalize this formulation to inhomogeneous media. A generalized gauge is suggested for solving electromagnetic problems in inhomogenous media which can be extended to the anistropic case. The advantages of the resulting equations are their absence of low-frequency catastrophe. Hence, usual differentialequation solvers can be used to solve them over multi-scale and broad bandwidth. It is shown that the interface boundary conditions from the resulting equations reduce to those of classical Maxwell's equations. Also, classical Green's theorem can be extended to such a formulation, resulting in similar extinction theorem, and surface integral equation formulation for surface scatterers. The integral equations also do not exhibit low-frequency catastrophe as well as frequency imbalance as observed in the classical formulation using E-H fields. The matrix representation of the integral equation for a PEC scatterer is given.
2. Heaviside, O., Electromagnetic Theory, Vol. 3, Cosimo, Inc., 2008.
3. Aharonov, Y. and D. Bohm, "Signifiance of electromagnetic potentials in the quantum theory," Physical Review, Vol. 115, No. 3, 485, 1959.
4. Gasiorowicz, S., Quantum Physics, John Wiley & Sons, 2007.
5. Cohen-Tannoudji, C., J. Dupont-Roc, and G. Grynberg, Atom-photon Interactions: Basic Processes and Applications, Wiley, New York, 1992.
6. Mandel, L. and E. Wolf, Optical Coherence and Quantum Optics, Cambridge University Press, 1995.
7. Scully, M. O. and M. S. Zubairy, Quantum Optics, Cambridge University Press, 1997.
8. Loudon, R., The Quantum Theory of Light, Oxford University Press, 2000.
9. Gerry, C. and P. Knight, Introductory Quantum Optics, Cambridge University Press, 2005.
10. Fox, M., Quantum Optics: An Introduction, Vol. 15, Oxford University Press, 2006.
11. Garrison, J. and R. Chiao, Quantum Optics, Oxford University Press, USA, 2014.
12. Manges, J. B. and Z. J. Cendes, "A generalized tree-cotree gauge for magnetic field computation," IEEE Transactions on Magnetics, Vol. 31, No. 3, 1342-1347, 1995.
13. Lee, S.-H. and J.-M. Jin, "Application of the tree-cotree splitting for improving matrix conditioning in the full-wave finite-element analysis of high-speed circuits," Microwave and Optical Technology Letters,, Vol. 50, No. 6, 1476-1481, 2008.
14. Wilton, D. R. and A. W. Glisson, "On improving the electric field integral equation at low frequencies," Proc. URSI Radio Sci. Meet. Dig., Vol. 24, 1981.
15. Vecchi, G., "Loop-star decomposition of basis functions in the discretization of the EFIE," IEEE Transactions on Antennas and Propagation, Vol. 47, No. 2, 339-346, 1999.
16. Zhao, J.-S. and W. C. Chew, "Integral equation solution of Maxwell's equations from zero frequency to microwave frequencies," IEEE Transactions on Antennas and Propagation, Vol. 48, No. 10, 1635-1645, 2000.
17. Nisbet, A., "Electromagnetic potentials in a heterogeneous non-conducting medium," Proc. R. Soc. Lond. A, Vol. 240, No. 1222, 375-381, Jun. 11, 1957.
18. Chawla, B. R., S. S. Rao, and H. Unz, "Potential equations for anisotropic inhomogeneous media," Proceedings of the IEEE, Vol. 55, No. 3, 421-422, 1967.
19. Geselowitz, D. B., "On the magnetic field generated outside an inhomogeneous volume conductor by internal current sources," IEEE Transactions on Magnetics, Vol. 6, No. 2, 346-347, 1970.
20. Demerdash, N. A., F. A. Fouad, T. W. Nehl, and O. A. Mohammed, "Three dimensional finite element vector potential formulation of magnetic fields in electrical apparatus," IEEE Transactions on Power Apparatus and Systems, Vol. 8, 4104-4111, 1981.
21. Biro, O. and K. Preis, "On the use of the magnetic vector potential in the finite-element analysis of three-dimensional eddy currents," IEEE Transactions on Magnetics, Vol. 25, No. 4, 3145-3159, 1989.
22. MacNeal, B. E., J. R. Brauer, and R. N. Coppolino, "A general finite element vector potential formulation of electromagnetics using a time-integrated electric scalar potential," IEEE Transactions on Magnetics, Vol. 26, No. 5, 1768-1770, 1990.
23. Dyczij-Edlinger, R. and O. Biro, "A joint vector and scalar potential formulation for driven high frequency problems using hybrid edge and nodal finite elements," IEEE Transactions on Microwave Theory and Techniques, Vol. 44, No. 1, 15-23, 1996.
24. Dyczij-Edlinger, R., G. Peng, and J.-F. Lee, "A fast vector-potential method using tangentially continuous vector finite elements," IEEE Transactions on Microwave Theory and Techniques, Vol. 46, No. 6, 863-868, 1998.
25. De Flaviis, F., M. G. Noro, R. E. Diaz, G. Franceschetti, and N. G. Alexopoulos, "A time-domain vector potential formulation for the solution of electromagnetic problems," IEEE Microwave Guided Wave Lett., Vol. 8, No. 9, 310-312, 1998.
26. Biro, O., "Edge element formulations of eddy current problems," Computer Methods in Applied Mechanics and Engineering, Vol. 169, No. 3, 391-405, 1999.
27. De Doncker, P., "A volume/surface potential formulation of the method of moments applied to electromagnetic scattering," Engineering Analysis with Boundary Elements, Vol. 27, No. 4, 325-331, 2003.
28. Dular, P., J. Gyselinck, C. Geuzaine, N. Sadowski, and J. P. A. Bastos, "A 3-D magnetic vector potential formulation taking eddy currents in lamination stacks into account," IEEE Transactions on Magnetics, Vol. 39, No. 3, 1424-1427, 2003.
29. Zhu, , Y. and A. C. Cangellaris, Multigrid Finite Element Methods for Electromagnetic Field Modeling, Vol. 28, John Wiley & Sons, 2006.
30. He, Y., J. Shen, and S. He, "Consistent formalism for the momentum of electromagnetic waves in lossless dispersive metamaterials and the conservation of momentum," Progress In Electromagnetics Research, Vol. 116, 81-106, 2011.
31. Rodriguez, A. W., F. Capasso, and S. G. Johnson, "The Casimir effect in microstructured geometries," Nature Photonics, Vol. 5, No. 4, 211-221, 2011.
32. Atkins, P. R., Q. I. Dai, W. E. I. Sha, and W. C. Chew, "Casimir force for arbitrary objects using the argument principle and boundary element methods," Progress In Electromagnetics Research, Vol. 142, 615-624, 2013.
33. Jackson, J. D., Classical Electrodynamics, 3rd Ed., Wiley-VCH, Jul. 1998.
34. Harrington, R. F., Time-harmonic Electromagnetic Fields, 224, 1961.
35. Kong, J. A., Theory of Electromagnetic Waves, 348-1, Wiley-Interscience, New York, 1975.
36. Balanis, C. A., Advanced Engineering Electromagnetics, Vol. 111, John Wiley & Sons, 2012.
37. Greengard, L. and V. Rokhlin, "A fast algorithm for particle simulations," Journal of Computational Physics, Vol. 73, 325-348, 1987.
38. Coifman, R., V. Rokhlin, and S. Wandzura, "The fast multipole method for the wave equation: A pedestrian prescription," IEEE Ant. Propag. Mag., Vol. 35, No. 3, 7-12, Jun. 1993.
39. Chew, W. C., J. M. Jin, E. Michielssen, and J. M. Song Eds., Fast and Efficient Algorithms in Computational Electromagnetics, Artech House, Boston, MA, 2001.
40. Chew, W. C., M.-S. Tong, and B. Hu, Integral Equation Methods for Electromagnetic and Elastic Waves, Morgan & Claypool Publishers, 2008.
41. Yang, K.-H., "The physics of gauge transformations," American Journal of Physics, Vol. 73, No. 8, 742-751, 2005.
42. Lee, S.-C., M. N. Vouvakis, and J.-F. Lee, "A nonoverlapping domain decomposition method with nonmatching grids for modeling large finite antenna arrays," J. Comp. Phys., Vol. 203, 1-21, Feb. 2005.
43. Chew, W. C., Waves and Fields in Inhomogeneous Media, Vol. 522, IEEE Press, New York, 1995 (First Published in 1990 by Van Nostrand Reinhold).
44. Ishimaru, A., Electromagnetic Wave Propagation, Radiation, and Scattering, Prentice Hall, Englewood Cliffs, NJ, 1991.
45. Sun, L., "An enhanced volume integral equation method and augmented equivalence principle algorithm for low frequency problems,", Ph.D. Thesis, University of Illinois at Urbana-Champaign, 2010.
46. Ma, Z. H., "Fast methods for low frequency and static EM problems,", Ph.D. Thesis, The University of Hong Kong, 2013.
47. Atkins, P. R., "A study on computational electromagnetics problems with applications to Casimir force calculations,", Ph.D. Thesis, University of Illinois at Urbana-Champaign, 2013.
48. Galerkin, B. G., "Series solution of some problems of elastic equilibrium of rods and plates," Vestn. Inzh. Tekh., Vol. 19, 897-908, 1915.
49. Kravchuk, M. F., "Application of the method of moments to the solution of linear differential and integral equations," Ukrain. Akad, Nauk, Kiev, 1932.
50. Harringotn, R. F., Field Computation by Moment Method, Macmillan, NY, 1968.
51. Andriulli, F. P., K. Cools, H. Bagci, F. Olyslager, A. Buffa, S. Christiansen, and E. Michielssen, "A multiplicative Calderon preconditioner for the electric field integral equation," IEEE Transactions on Antennas and Propagation, Vol. 56, No. 8, 2398-2412, 2008.
52. Dai, Q. I., W. C. Chew, Y. H. Lo, and L. J. Jiang, "Differential forms motivated discretizations of differential and integral equations," IEEE Antennas Wireless Propag. Lett., Vol. 13, 1223-1226, 2014.
53. Zhang, Y., T. J. Cui, W. C. Chew, and J.-S. Zhao, "Magnetic field integral equation at very low frequencies," IEEE Transactions on Antennas and Propagation, Vol. 51, No. 8, 1864-1871, 2003.
54. Qian, Z.-G. and W. C. Chew, "Fast full-wave surface integral equation solver for multiscale structure modeling," IEEE Transactions on Antennas and Propagation, Vol. 57, No. 11, 3594-3601, 2009.
55. aghjian, A. D., "Augmented electric and magnetic field integral equations," Radio Science, Vol. 16, No. 6, 987-1001, 1981.
56. Vico, F., L. Greengard, M. Ferrando, and Z. Gimbutas, "The decoupled potential integral equation for time-harmonic electromagnetic scattering," Mathematical Physics, arXiv: 1404.0749, 2014.
57. Dai, Q. I., Y. H. Lo, W. C. Chew, Y. G. Liu, and L. J. Jiang, "Generalized modal expansion and reduced modal representation of 3-D electromagnetic fields," IEEE Transactions on Antennas and Propagation, Vol. 62, No. 2, 783-793, 2014.