By Francis Graham, Sir Smith, J. H. Thomson, Francis Graham-Smith

Optics as a subject matter has developed dramatically lately, with many purposes all through technology and expertise.

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Optics as a topic has advanced dramatically in recent times, with many purposes all through technology and know-how.

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Another useful transformation is that of the Gaussian function 40 Fourier theory and wave groups Chap. 3 Evaluation by the same process, using the identity gives the transform The transform of a Gaussian function is therefore another Gaussian function, whose width l / r u is inversely proportional to the original width a. The two functions chosen for these examples are symmetrical about the origin. In this case the transform is always real. Any asymmetrical part of f (t) would result in a complex transform, in which any component of F(v) must be specified in terms of a phase as well as an amplitude.

We define Now take the Fourier transform of this and manipulate = 1[I + A (t 1 r)AS(t)dt exp (2rirv)dr This is the power spectrum. 9 WAVE GROUPS Modulated waves, and in particular wave groups such as those of Fig. 4(c), are of great importance in many branches of physics. 5. We start with two wave components only. The addition of two waves with equal amplitude a but different angular frequencies w f A d 2 and wave numbers k k Ak/2 is expressed as: Fourier theory and wave groups 44 a x[ = tz exp,i(ot = 2o sin ( i Chap.

2). An example of this summation of a cosine and sine is given in Fig. 1. Looked at in this way it is clear that if there were further disturbances to be added to that already at P 1 , this could be done by adding all the A , and all the As terms, and replacing A, and As in Eq. 4) with CAc and C A s . A, and As thus behave like components of a two-dimensional vector, and the result of the superposition of a number of disturbances at a point may be obtained by means of a vector diagram. To save confusion with threedimensional vectors like E and H, A, and As are said to be the components of a phasor A, and the diagram on which they are added is called a phasor diagram.

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