REAL AND IMAGINARY NUMBERS AND THEIR
APPLICABILITY TO LIGHTWAVE PROPAGATION PROBLEMS

AN ON-SITE COURSE
One day duration

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Course Description:

This unique interactive course is designed to introduce real and imaginary numbers, and to show how their use has for over 100 years been the standard way of describing the subject of electromagnetic wave propagation. No prior knowledge of complex numbers is assumed. This course is intended to give participants a working familiarity with the convenience of applying complex notation to practical wave propagation situations.

The course starts at the very beginning, dealing first with complex numbers, and with their intercombination through addition, subtraction, multiplication, and division.   We see how to find the real or the imaginary parts of complex numbers, and we manipulate usages of the complex conjugation notation.  Next, we introduce trigonometric functions of complex variables (concentrating on sine and cosine), and the exponential functions of complex arguments. We also learn how to rapidly recall formulas for sines or cosines of sums, differences, or multiples. While concentrating on usages relating to electromagnetic wave propagation, we show how phase shifts and attenuation are treated through the complex embodiments of both the propagation constant and the refractive index. In the second half of this course we will identify the interrelationships between the complex index of refraction, the complex dielectric function, the complex propagation vector, the complex susceptibility, and the real absorption coefficient. Throughout the course, examples are shown which demonstrate the convenience of solving simple wave propagation problems in the complex representation.

During this course, much personal attention will be given to each participant in the class. There will be several short homework sessions during the course, where participants will be able to practice some of the procedures and to review some of the concepts that they had learned. These homework activities will be immediately "self-graded" so that each attendee will be most readily able to assess his/her progress, and further, will be able to identify any special needs. This is intended to be an interactive course, encouraging participation to the fullest extent.

Each idea is explained in clear and simple terms with an emphasis on underlying principles.  Pertinent references and review material are identified. The course notes contain copies of each vuegraph used during the day so that participants can spend a maximum amount of time listening and understanding.

Benefits:
This course will enable you to:

• Find the real and the imaginary part of a complex number.
• Add, subtract, multiply, and divide complex numbers.
• Factor sums into products of complex numbers.
• Understand exponentiation of a purely imaginary number.
• Appreciate the relationships between circular functions and exponential functions.
• Appreciate the extension of waves in linear systems to their complex counterparts.
• Understand what it means when the index of refraction is a complex quantity.
• Understand the different roles of the real and imaginary parts of the index of refraction.
Objectives:
Upon completion of this course, participants:
• Find the real and imaginary part of a complex number
• Add, subtract, multiply andc divide complex numbers
• Factor sums and differences into products of complex numbers
• Understand exponentiation of a purely imaginary number.
• Appreciate the relationships bbetween circular functions and exponential functions
• Appreciate the now 100 year long tradition of extension of waves in linear systems to their complex counterparts.
• Understand what it means when the index of refraction is a complex quantity
• Understand the different roles of the real and imaginary parts of the index of refraction

Intended Audience
Engineers, technicians, and managers who need a fundamental understanding of real and imaginary numbers, complex numbers, complex functions, and persons needing to deal with wave propagation issues.

Continuing Education Units (CEU's) available upon request.

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