# ECE 403A/OPTI 403A

## Mathematical Methods for Optics & Photonics

### Course Level

### Units

### Instructor(s)

### Prerequisite(s)

### Course Texts

**The following books are recommended but not required:**

- R. Bracewell, “Fourier Transform and its Applications,” McGraw-Hill, 1999.
- G. Arfkin, H. Weber, “Mathematical Methods for Physicists,” Academic Press, 2000.
- M. Mansuripur, “Introduction to Information Theory,” Prentice-Hall, New Jersey, 1987.

### Schedule

### Course Description

**Specific Course Information:
2021-2022 Catalog Data:** This course covers the basic mathematics needed for an in-depth understanding of the science and technology of fiber-optical communication systems. Every mathematical tool/technique developed in this course will first be motivated by the relevant application. The students are not expected to have a broad-based prior knowledge of the topics covered in this course, but they should generally be familiar with the basics of algebra, Euclidean geometry, trigonometry, integral and differential calculus, simple differential equations, and the rudiments of complex number analysis. The course will cover Complex Analysis, Fourier transform theory, and method of stationary phase (in the context of optical diffraction), vector algebra, linear algebra, ordinary and partial differential equations (e.g., Maxwell's electrodynamics, wave equation, diffusion equation), special functions (e.g., Bessel functions needed to study the guided modes of optical fibers), and probability theory (needed for understanding various sources of noise in communication systems, photodetection theory, digital communication via noisy channels, Information theory, etc.).

### Course Topics

**Brief list of topics to be covered:**

- Elementary calculus, exponential and logarithmic functions, Taylor series expansion
- Approximation methods
- Complex number theory, complex integration and differentiation, simple functions in the complex domain
- Special functions: Rectangular and triangular functions, delta-function and its derivatives, sinc function, etc.
- The convolution operation
- Linear, shift-invariant systems
- Fourier transform theory, theorems, useful Fourier transform pairs
- The method of stationary phase
- Applications of Fourier theory to optical diffraction
- Linear algebra, operations with matrices, matrix inversion
- Eigen-values and eigen-vectors, matrix diagonalization
- Vector algebra, vector identities
- Divergence, curl, gradient, and Laplacian operators
- Ordinary differential equations; elementary methods of solution
- Partial differential equations, method of separation of variables
- The diffusion equation
- Maxwell’s equations; the wave equation
- Solutions of the wave equation in Cartesian, cylindrical, and spherical coordinate systems
- Special functions: Bessel functions of the 1st, 2nd, and 3rd kind; modes of an optical fiber
- Probability theory
- Statistical properties of thermal noise, shot noise, and modal noise in fiber optics systems
- Introduction to Information Theory and Coding
- Communication via noisy channels; Shannon’s noisy channel capacity
- Compression codes, error-correction codes, modulation coding

### Relationship to Student Outcomes

ECE/OPTI 403A contributes directly to the following specific electrical and computer engineering student outcomes of the ECE department:

1. An ability to identify, formulate, and solve complex engineering problems by applying principles of engineering, science, and mathematics.

7. An ability to acquire and apply new knowledge as needed, using appropriate learning strategies.