Mathematical conventions

In this work, a forward-propagating wave is implicitly taken to be the real part of a signal of the form:

\[ p(x,t) = e^{i(\omega t - kx)} \]

This matches the convention that is typically used in signal processing, where the argument that contains the angular frequency \(\omega\) is positive. It is also the convention in most of the optics and communication texts cited here, as well as several others that include, notably, Haus (1984), Siegman (1986), Kolner (1994a), Cohen (1995), Fletcher and Rossing (1998), Kinsler et al. (1999), Blinchikoff and Zverev (2001), New (2011), Couch II (2013) and Kuttruff (2017). Therefore, formulas that were taken from Morse and Ingard (1968), Jackson (1999), Whitham (1999), Born et al. (2003) and Goodman (2017) sometimes had to be adopted by changing the sign as they originally appeared in the form \(p(x,t) = e^{i(kx- \omega t)}\).

The Fourier transform that is used in the text conforms to

\[ F(\omega) = \int_{-\infty}^\infty f(t) e^{-i\omega t} dt \]

whereas the inverse Fourier transform is

\[ f(t) = \frac{1}{2\pi}\int_{-\infty}^\infty F(\omega) e^{i\omega t} d\omega \]

Thus, throughout this work, lowercase signal functions are time domain and their spectral domain counterparts are capitalized.

References

Born, Max, Wolf, Emil, Bhatia, A. B., Clemmow, P. C., Gabor, D., Stokes, A. R., Taylor, A. M., Wayman, P. A., and Wilcock, W. L. Principles of Optics. Cambridge University Press, 7th (expanded) edition, 2003.

Cohen, Leon. Time-Frequency Analysis. Prentice Hall PTR, Upper Saddle River, NJ, 1995.

Couch II, Leon W. Digital and Analog Communication Systems. Pearson Education Inc., Upper Saddle River, NJ, 8th edition, 2013.

Fletcher, Neville H and Rossing, Thomas D. The physics of musical instruments. Springer Science+Business Media New York, 2nd edition, 1998.

Goodman, Joseph W. Introduction to Fourier Optics. W. H. Freeman and Company, New York, NY, 4th edition, 2017.

Haus, Hermann A. Waves and Fields in Optoelectronics. Prentice-Hall Inc., Englewood Cliffs, NJ, 1984.

Jackson, John David. Classical Electrodynamics. John Wiley & Sons, Inc., Hoboken, NJ, 3rd edition, 1999.

Kinsler, Lawrence E, Frey, Austin R, Coppens, Alan B, and Sanders, James V. Fundamentals of Acoustics. John Wiley & Sons, Inc., New York, NY, 4th edition, 1999.

Kolner, Brian H. Space-time duality and the theory of temporal imaging. IEEE Journal of Quantum Electronics, 30 (8): 1951–1963, 1994a.

Kuttruff, Heinrich. Room acoustics. CRC Group, Taylor & Francis, Boca Raton, FL, 6th edition, 2017.

Morse, Philip M. and Ingard, K. Uno. Theoretical Acoustics. Princeton University Press, Princeton, NJ, 1968.

New, Geoffrey. Introduction to Nonlinear Optics. Cambridge University Press, 2011.

Siegman, Anthony E. Lasers. University Science Books, Mill Valley, CA, 1986.

Whitham, G. B. Linear and Nonlinear Waves. John Wiley & Sons, Inc., New York, NY, 1999. First published in 1974.

Blinchikoff, Herman J. and Zverev, Anatol I. Filtering in the time and frequency domains. Scitech Publishing Inc., Rayleigh, NC, 2001.