= Modal amplitude norm versus wavefront RMS
For a real <wavefront error> $h=\sum_k a_k s_k$, its squared spatial <root mean square> is $\langle h^2\rangle=\sum_{k,l}a_ka_l\langle s_ks_l\rangle$. Orthogonal unit-peak <sine> modes have mean square $1/2$, so $h_{\rm rms}^2=\frac12\sum_k a_k^2$, whereas the coefficient norm is $\sqrt{\sum_k a_k^2}$. On a finite <optical pupil> <orthogonality> needs checking; independent uniform random <wave phases> give the same $1/2$ diagonal after <wave phase> averaging. This normalization matters when a <speckle contrast of a sinusoidal wavefront error> is specified by peak <wave amplitude>.
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