The function is a beam-like solution to the scalar wave equation (in the paraxial approximation) where The

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The function

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is a beam-like solution to the scalar wave equation (in the paraxial approximation) where

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The Gouy phase, α(z), arises from the fact that the beam has a finite size in the transverse direction. To see this, use the fact that k= k2x + k2+ k2to motivate the definition of an effective propagation “constant”

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In this formula, the averages are defined over the distribution of transverse wave vectors that make up the beam. That is,

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where

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Show that kz + α(z) = ∫0dz k̅z . Confirm that α(z) = 0 if there is no localization in the transverse direction.

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