A
Complex Phase Notation, Engineer's vs. Physicist's

A.1 Sinusoidal Waves

This book presents mathaematical models for the propagation of light through various apertures, optical elements, and linear and nonlinear media. Light is an electromagnetic wave where both the electric and magnetic fields are sinusoidal in terms of both position and time. The simplest mathematical representation of sinusoidal waves is using the trigonometric functions sin and cos. Choosing the cosine function, the general expression for a sinusoidal wave can be written as

where the phase of the wave is everything inside the trigonometric function argument, or [k(z−vt)+φ].

Equation A.1 represents a sinusoidal wave traveling in the +z-direction with an amplitude of A_o, a wavenumber of k, and travelling with a speed of v. The phase constant, φ, determines the value of the oscillating function at the origin at a time of t = o. For example, if we wanted our oscillation wave to look like a cosine function (where the value of the function is a maximum at the origin at t = 0) then we would set our phase constant equal to zero. If we wanted our oscillating wave to look like a sine function (where the value of the function at the origin is zero with a positive slope at t = 0) then we would set our phase constant to be equal to −π/2 or +3π/2. In other words, if we delay a cosine function by a factor of +3π/2 we have created a sine function. It is due to this reason that φ is also known as the “phase delay.”

Equation A.1 can also be written as

where we have multiplied the wavenumber into the expression (z−vt) and used the relationship between the wavenumber, wave speed, and angular frequency of the oscillation, ω,

We should also note here that the values of k and ω are assumed to be positive. We have also previously stated that Eqs. A.2 and A.2 represent a wave traveling in the +z-direction. Without this being explicitely stated, the direction of propagation of the wave can be determined from the signs of the position and time terms of the phase. In the case of Eq. A.2 that is the kz and ωt terms:

• if kz and ωt have opposite signs the wave is traveling in the positive z direction, and
• if kz and ωt have the same sign, the wave is traveling in the negative z direction.

A.2 Complex Notation Using Euler's Formulas

“Complex” notation simply means that we are using complex numbers, functions, or phases. A complex number is defined to be a number that can be represented as a sum of two numbers, a and b, in the form

where i is the fundamental imaginary number

a is the real part of z, and b is the imaginary part of z. The real and imaginary parts of z can also be written as

and

Similarly, complex functions or phases are simply functions of complex numbers and/or complex variables.

Euler’s formulas allow the user to express oscillatory trigonometric functions as an exponent of the natural number e as either

or

Euler’s equations can also be rewritten in the form where the trigonometric fuctions are expressed as complex exponentials:

and

A simpler way of extracting the trigonometric functions from complex exponentials is to apply Eqs. A.6 and A.7 to Eq. A.8, or

and

Finally, the traveling wave described by Eq. A.2 can now be expressed as

When expressing traveling waves using Euler’s form of complex exponentials the function for the wave is more conveniently written simply as

where it is understood that the actual wave is only the real part of Eq. A.15.

Now, many students when they first encounter Euler’s formulas and begin representing traveling waves as complex exponentials have the reaction, “This just seems like a lot of extra work! Why would you even want to go through all of the work to take a simple trig funtion, convert it into a complex function, put it into an exponent, and then have to find the real part of this complex exponential just to get back to where we started? That doesn’t make any sense to me.” This is quite the valid question.

First, if we start the problem using complex notations and stay in complex notation throughout the entire problem we do not have to deal with converting the waves back and forth between trigonometric functions and complex exponentials. However, the major advantage of using complex exponetial notation for waves is that performing mathematical functions on the waves (i.e., adding waves, multiplying waves, finding the phase, etc.) is much, much easier than dealing with sines and cosines and vast tables of trigonometric identites.¹

Let us assume that we do not want to have to write “+φ” in the phase of the complex exponential every time we write the function for the wave. Using complex notation, it is fairly straightforward to absorb this part of the phase into the amplitude due to the fact that a sum of phase terms in an exponent is mathematically the same as a product of exponential functions. In other words, separating out the phase constant is achieved by

or

where A is now the complex amplitude

which contains both the amplitude of the actual wave and the initial phase delay of the wave.

A.3 Engineer's vs. Physicist's Notation

When comparing the complex exponential functions representing sinusoidal waves typically used by engineers to those typically used by physicists, there are two subtle differences.

First, the fundamental imaginary number, $\sqrt{-1}$, is typically found to be represented in physics textbooks as an i. In engineering textbooks, the fundamental imaginary number can be found to be represented by a j.

The second, and slightly more subtle, difference between engineering notation and physicist notation comes from a choice of representing the time-dependence of the oscillation:

• Engineer’s choice for the time-dependence → e^iωt
• Physicist’s choice for the time-dependence → e^−iωt.

Recall that it is the relationship between the position-dependent and time-dependent terms in the phase that determine the direction of propagation. For a wave traveling in the +z-direction, the two phase terms need to have opposite signs.

Using the engineer’s choice for the time-dependence, a function representing a wave traveling in the +z-direction can be written as a product of complex exponetials having opposite signs for the position and time-dependence, or

which can be re-written as

Similarly, using the physicist’s choice for the time dependence, the function for the same wave takes the form

TABLE A.1
Similarities and differences between engineering complex notation and physicist complex notation for various mathematical functions.

which can be re-written as

Even though Eqs. A.20 and A.22 are mathematically different, they do represent the same physical wave by analyzing Eqs. A.8 and A.9 and utilizing the fact that the actual wave is only the real part of the complex exponential. The actual waves for each notation can be written as

and

Recalling the fact that cos(−θ) = cosθ, we see that the two notation styles do represent the same physical wave.

Expanding beyond the simple traveling wave described here, Table A.1 illustrates how the similarities and differences between the two notations manifest themselves for various mathematical functions.

TABLE A.2
Complex notation styles used within this book according to topic, section, or chapter.

A.4 Use of Engineer's and Physicist's Complex Notation in This Book

In this book we use both engineering and physicist complex notation. However, within each topic, or beam propagation model, we are consistent with using only a single complex notation style. Whether engineer’s or physicist’s complex notation is used for a particular topic of the book depends upon the style of notation used in other books, or references, from which the model in this book is derived or closely resembles. Table A.2 clarifies which topics, or sections, of this book use which style of complex notation.

A.5 Some Commonly Used Electrodynamics and Optics Books

• I. These commonly used, or cited, books use engineer's complex notation:
• • G. Bekefi, "Diffraction of electromagnetic waves by an aperture in a large Screen," J. App. Phys. 24, 1123-1130 (1953).²
• • D. Marcuse, Light Transmission Optics, Second Edition (Van Nostrand Reinhold Company, New York, 1982.)
• • A. Siegman, Lasers (University Science Books, Mill Valley, CA 1986.)
• • W. T. Sifvast, Laser Fundamentals, Second Edition (Cambridge University Press, New York, NY, 2004.)
• • A. Yariv, Optical Electronics, Fourth Edition (Saunders College Publishing, Chicago, IL, 1991.)
• • A. Yariv, Quantum Electronics, Third Edition (John Wiley & Sons, New York, 1989.)
• II. These commonly used, or cited, books use physicist’s complex notation:
• • M. Born and E. Wolf, Principles of Optics, Seventh Edition (Cambridge University Press, Cambridge, UK, 1999.)
• • R. W. Boyd, Nonlinear Optics, Second Edition (Academic Press, New York, 2003.)
• • J. W. Goodman, Introduction to Fourier Optics, Third Edition (Roberts & Company Publishers, Englewood, CO, 2005.)
• • D. J. Griffiths, Introduction to Electrodynamics, Third Edition (Prentice Hall, Inc., Upper Saddle River, NJ, 1999.)
• • E. Hecht, Optics, Fourth Edition (Addison Wesley, New York, 2002.)
• • J.D.Jackson, Classical Electrodynamics, Third Edition (John Wiley & Sons, New York, 1999.)
• • R. K. Luneburg, Mathematical Theory of Optics (University of California Press, Berkely, CA, 1964.)
• • F. L. Pedrotti, L. S. Pedrotti, and L. M. Pedrotti, Introduction to Optics, (Pearson Prentice Hall, Upper Saddle River, NJ, 2007.)
• • J. A. Stratton, Electromagnetic Theory (McGraw-Hill Book Company, Inc., New York, 1941.)