Wave amplitude ($A$)

For a scalar complex wave amplitude $E$, the squared-amplitude scalar wave intensity convention is $I=E{E}^{\ast }$. It is nonnegative and unchanged by multiplying $E$ by a unit-modulus wave phase factor. This convention is useful in paraxial propagation and phase screen models. Conversion to physical energy flux also depends on the field variable and medium; an arbitrary superposition need not have its full flux determined by this scalar alone. For a random field, $⟨I⟩$ is the mean wave intensity, whereas $|⟨E⟩{|}^2$ is the squared coherent field; these generally differ.

A complex number can encode both the wave amplitude and phase of a harmonic disturbance: $a(x,t)=\mathrm{Re}\{\hat{a}{e}^{i(kx-\omega t)}\}$. Its modulus is the real oscillation amplitude and its complex argument fixes the wave phase.

Reference wave phases label complex amplitudes without changing measured powers. If $\mathbf{b}=S\mathbf{a}$, choose diagonal matrices $D_{\mathrm{in}},D_{\mathrm{out}}$ whose entries have modulus one, and set ${\mathbf{a}}^{\prime }=D_{\mathrm{in}}\mathbf{a}$, ${\mathbf{b}}^{\prime }=D_{\mathrm{out}}\mathbf{b}$. Then $S^{\prime }=D_{\mathrm{out}}S{D}_{\mathrm{in}}^{-1}$ and each component has unchanged squared modulus. For a beam splitter scattering matrix, transforming $S$ without transforming input complex amplitudes changes the stated physical input.

A phasor is a complex amplitude representing a sinusoid at one specified angular frequency: $a(t)=\mathrm{Re}(\hat{A}{e}^{\mathrm{i}\omega t})$. Its modulus gives the peak amplitude and its complex argument gives the wave phase in this convention. Both may instead use root mean square magnitudes if that normalization is stated.