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I am currently studying Optics, fifth edition, by Hecht. I am presented with the plane wave in Cartesian coordinates as follows:

$$\psi(x, y, z, t) = Ae^{i[k(\alpha x + \beta y + \gamma z) \mp \omega t]}$$

I am then told that $\alpha^2 + \beta^2 + \gamma^2 = 1$. Can someone please explain why $\alpha^2 + \beta^2 + \gamma^2 = 1$?

Thank you.

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1 Answer 1

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Here $\alpha$, $\beta$, and $\gamma$ are the Cartesian components of the unit wavenumber vector. The wavenumber vector is $\mathbf k=(k_x,k_y,k_z)=k(\alpha,\beta,\gamma)$. The squares of the components of any unit vector sum to $1$.

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