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By Avijit Lahiri
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Extra info for Basic Optics I-II
For the spherical polar co-ordinate system with co-ordinates r, θ, φ, for instance, one has h1 = 1, h2 = r, h3 = r sin θ, while for the cylindrical co-ordinate system made up of coordinates ρ, φ, z, the scale factors are h1 = 1, h2 = ρ, h3 = 1. In this book, a differential expression such as, say, dx will often be used loosely to express a small increment that may alternatively expressed as δx. Strictly speaking, expressions like dx are meaningful only under integral signs. When used in an expression in the sense of 37 a small increment, it will be implied that terms of higher degree in the small increment are not relevant in the context under consideration.
Once again, there remains an arbitrariness in the definition of the energy flux, though the above expression is acceptable on the ground that it is a consistent one. 39) Assuming that the temporal variation of the field vectors is a harmonic one, and making use of the complex representation of vectors as explained in sec. 40) where Ẽ and stand for the complex amplitudes corresponding to the respective real time dependent vectors (appearing in eq. 39)) at the point under consideration. 41) where the angular brackets indicating the time average has been omitted for the sake of brevity and ŝ denotes the unit vector along S .
This is not to convey the impression that one cannot acquire working knowledge in ray optics or diffraction theory without a thorough grounding in electromagnetic theory. In this book, however, my approach will be to trace the origins of the working rules of these approximation schemes to the principles of electromagnetic theory. In working out solutions to the Maxwell equations, it is often found convenient to look at regions of space where there are no free charge or current sources as distinct from those containing the sources.