Photonics and Lasers

Appendix B - Fourier Synthesis and the Uncertainty Principle

Appendix B

 

Fourier Synthesis and the Uncertainty Principle

At several points in this book, we encounter the so-called uncertainty relation, which relates the minimum uncertainties in time and frequency, or in position and wavelength. A complete description of this involves the Fourier transform, the mathematical treatment of which is beyond the scope of this book. However, we can obtain an intuitive understanding of this concept by considering qualitatively what happens when we add together sinusoidal waves of different frequencies. We take this approach here, and also obtain an exact expression relating the two uncertainties for one important special case.

FOURIER SYNTHESIS

 

The fundamental idea of Fourier synthesis is that any arbitrary waveform can be constructed by adding together an infinite number of pure sinusoidal waves. This concept can be applied to either the time dependence or the position dependence of the wave, but to be concrete we will emphasize here the time dependence. When the waveform is periodic in time, it can be written as a Fourier series,

where v0 = 1/T, T is the repetition time of the waveform, and n is an integer ranging from 0 to . In general, there are also cos(n2 v0t) terms, but for this discussion we can neglect them. The sum is over a set of discrete frequencies, including the fundamental at v0 and higher harmonics at nv0. For example, a square wave can be constructed by choosing coefficients An = 1/n for all odd values of n, and An = 0 for all even values. Figure B-1 shows the calculated y(t) using the four lowest-frequency terms, along with the component waves that are added.

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