Complex Variables
Analytic functions of a complex variable. Cauchy-Riemann conditions. Power series. Cauchy’s integral theorem. Conformal mapping. Singularities: poles, essential singularities. Residue theorem. Contour integration and examples. Analytic continuation. Multiple-valued functions, branch points and branch cut integration.

Partial Differential Equations :
Partial differential equations in Physics: Laplace, Poisson and Helmholtz equations; diffusion and wave equations. Applications.

Integral transforms
Laplace transforms and Fourier transforms. Parsevall’s theorem. Convolution theorem. Applications.
Calculus of Variations
Functionals. Natural boundary conditions. Lagrange multipliers. Rayleigh-Ritz method.

Group theory : Elements of group theory. Discrete groups with examples. Contionuos groups (Lie groups) [rotation group in 2 and 3 dimensions, U(1) and SU(2)]. Generators. Representations, Character tables for some point groups and the orthogonality theorem.

Text Books:

1. G. Arfken and H.J. Weber, Mathematical Methods for Physicists, Academic Press, 6th Edition, Indian Edition, (2005).

2. P. Dennerey and A. Kryzwicki, Mathematics for Physicists, Dover (Indian Edition), (2005).

3. K.F. Riley, M.P. Hobson and S.J. Bence, Mathematical Methods for Physics and Engineering, Cambridge University Press (Cambridge Low-priced Edition) (1999).

1. Schaum’s outline series, McGraw Hill (1964): (i) Complex Variables, (ii) Laplace Transforms, (iii) Group Theory.

2. M. Boas, Mathematical Methods in Physical Sciences, 2nd Edition, Wiley International Edition, (1983).

3. E. Kreyszig, Advanced Engineering Mathematics, Wiley Eastern, 5th Edition, (1991).

4. L.A. Pipes and L.R. Harwell, Applied Mathematics for Engineers and Physicists, McGraw-Hill, (1995).

5. M.Artin, Algebra, Prentice-Hall India, (2002).

6. I.N. Sneddon, The Use of Integral Transforms, Tata McGraw Hill, (1985).

7. D.H. Sattinger and O.L. Weaver, Lie Groups and Algebras with Applications to Physics, Geometry and Mechanics, Springer, (1986).

8. M. Tinkham, Group Theory and Quantum Mechanics, Dover (2003).

Description: This course will cover the basic principles o Optics and Photonics in depth and provide introduction to the upcoming frontiers.

Course Content: Fourier Optics: Diffraction integral; fourier transformation in beam propagation – Fresnel and Fraunhoffer approximations; Fourier filtering, Image processing; Abbe’s principle of image formation; principle of phase contrast microscope; holography – principles of recording and reconstruction. Optics of periodic media: multilayer dielectric interference coatings and their applications photonic crystals, Bragg reflectors. Lasers: optical amplification and lasers; characteristics of laser radiation; spatial and temporal coherence, optics of Gaussian beams. Fibre and Integrated Optics: Guided modes; attenuation and dispersion in optical fibres; application in sensors and communication. Photonic devices based on acousto-optics, electro-optics and magneto-optics: Intensity, phase and frequency modulation; frequency shifters; optical diode and isolator; directional coupler; spatial light modulators. Introductory treatment of: nano-photonics, negative refraction and meta-materials, nonlinear optical processes, slowing of light and other contemporary topics.

Text Books:

1. E. Hecht and A R Ganesan, Optics, 4th Edi., Pearson Education (2008) or earlier editions: E Hecht and A Zajac.

2. B E A Saleh and M C Teich, Fundamentals of Photonics, 2nd Ed. Wiley (2007)

3. A K Ghatak and K. Thyagarajan, Optical Electronics, Cambridge University Press (1989)

Reference Books:

1. M. Born and E. Wold, Principles of Optics, Pergamon Press (1985)

2. J W Goodman, Introduction to Fourier Optics, McGraw Hill (1996)

3. K. Izuka, Engineering Optics, Springer Verlag (2008)

4. J D Joannopoulos, S G Johnson and J N Winn, Photonic crystals: molding the flow of light, Princeton University Press (2008)

5. L. Novotny and B. Hecht, Principles of Nano Optics, Cambridge University Press, UK (2006)

6. P N Prasad, Nanophotonics, Wiley Interscience (2004)

7. S A Maier, Plassmonics: Fundamentals and Applications, Springer (2007)