Variational Principle for Optical Phase

TL;DR

This work develops a variational framework to maximize laser beam concentration in the far field by optimizing the near-field phase $\psi$ under a fixed amplitude. It formulates a beam propagation functional $T[\psi]$ with explicit Cartesian and cylindrical forms derived from paraxial scalar diffraction, demonstrating that $T[\psi]$ is real and positively defined. The first variation $\delta T[\psi]/\delta \psi=0$ admits uniform phases $\psi(\vec r)=\text{const}$ as extremals, and the second variation satisfies $\delta^2 T[\psi]/\delta \psi^2<0$, indicating a local maximum for the transmitted flux through the aperture. The results yield necessary and sufficient conditions for the maximum, apply to phase-locked laser arrays and Shack–Hartmann type wavefront control, and provide exact solutions for optimal wavefront shapes under fixed near-field amplitudes.

Abstract

{The problem of laser beam concentration in a focal spot via wavefront variations is formulated as a maximization of the $beam$ $propagation$ $functional$ defined as the light power passing through aperture of an arbitrary shape located in the far field. Variational principle provides the necessary and sufficient conditions for at least the $local$ $maximum$ of the $beam$ $propagation$ $functional$. The wavefront shape is obtained as an exact solution of nonlinear integral equation. }

Figures (3)

• Figure 1: (Color online) Different variational principles for different situations and geometries. a) Fermat principle for refraction of rays at the boundary. b) Hamilton principle for massive particle $m$ in Newtonian potential and Lagrangian $L=T-U$. c) Variational principle for the relativistic massive $m$ charged $\pm e$ particles moving from point $\vec{r}_1$ to point $\vec{r}_2$ in magnetic field $\vec{B}=rot \vec{A}$ encircled by vector potential $\vec{A}$. d) Feynman path integral variational principle for particle moving from point $\vec{r}_1$ to point $\vec{r}_2$ whose classical trajectory appears as a result of constructive interference of adjacent paths (green). The origin of classical trajectory may be considered as a result of constructive interference of partial $linear$ waves emitted by Huygens-Frenel wavefronts. The fractal curve connecting points $\vec{r}_1$ and $\vec{r}_2$ is a trajectory of particle that appears under attempts of continuous measurements. The evaluation of Hausdorf fractal dimension $D_H$ in this case gives $D_H=2$Abbot:1981 . e) Variational principle for the spatial soliton Malomed:2002. The particle-like localized excitation Okulov:1988 moves in a nonlinear medium with gain and losses from point $\vec{r}_1$ to point $\vec{r}_2$. The trajectory of soliton is a result of constructive interference of partial $nonlinear$ waves emitted by Huygens-Frenel wavefronts.
• Figure 2: (Color online) Composition of $target$$functional$. a) Far field of $spatial$$soliton$Okulov:2020 or a Gaussian fundamental cavity mode. b) Far field of $laser$$array$ (\ref{['TFT from solutions ']}) Mourou:2013. c) $Rectangular$ diaphragm in far field $D({\vec{r}})$. d) $Circular$ diaphragm in far field $D({r})$. e) Near field of $spatial$$soliton$Okulov:1988 or a Gaussian fundamental cavity mode. f) Near field of thin disk $laser$$array$Okulov:1990 or fiber laser coherent network Mourou:2013.
• Figure 3: (Color online) Phase-locked LMA (large mode area) chirped pulse laser network Mourou:2013. $MO$- master oscillator emitting 30-$fs$ transform-limited pulses, the $stretcher$ elongates pulses to 1 $ns$, and $compressor$ compresses amplified chirped pulses to initial 30-$fs$, each with a pair of diffraction gratings $G_1, G_2$ and $G_3, G_4$. Binary trees of beamsplitters $BS$ and mirrors $M$ provide beam multiplexing and beam combination. Array of large mode area $LMA$ fiber amplifiers and wavefront control $adaptive$$phase$$mask$ with $\lambda/(10 \div 100)$ accuracy form the diffraction-limited focal region with $\lambda^3$ volume.