Questions A diffraction-limited light field of wavelength, λo propagates in air as shown in the following figure. It starts with a flat phase front (R₁ → ∞0) at plane 1 and as such has a minimum waist there of w₁ = σ. It diffracts as it propagates to plane 2 over a distance, d. Its q-parameters at plane 1 and plane 2 are q₁ and q2 respectively. Plane 1 01 = σ R₁ → ∞o d n = 1 (air) wavelength, 10 Plane 2 W2 R₂ 1. Using equations 2 and 3, derive the following equation for the q-parameter, q2 at plane 2: = 92 ino²/o+d [2] Background A light field (a wave-function generally) with a Gaussian spatial amplitude, U; at plane i, may be written as (scaled to unity): Where: U(x) = exp(-inx²/qiλo) eq. 1 1/q₁ = 1/Â¡ — iλo/πw - eq. 2 • Ŕ is the 'reduced' (index-modified) radius of curvature (ROC) defined as R = R₁/n; where R₁ is ROC as measured within a medium at plane i; • 10 is the wavelength for the field where in vacuum defined as λ = n;λ; where λ; is the wavelength as measured within a medium at plane i; • The waist size, w; is defined as the value of the transverse coordinate, x, where the field, U falls to a value 1/e. This is also where the intensity falls to a value 1/e² where intensity I = U₁U}; • Note that n=1.000 for air and so for air we approximate R = R₂ and λ = λi. We refer to the reduced q-parameter, generally referred to as the q-parameter. It can be shown (Siegman ch 20) that the paraxial transfer of the light field of equation 1 between any two planes 1 and 2, is given by: 92 = Aq1+B Cq1+D eq. 3

Questions A diffraction-limited light field of wavelength, λo propagates in air as shown in the following figure. It starts with a flat phase front (R₁ → ∞0) at plane 1 and as such has a minimum waist there of w₁ = σ. It diffracts as it propagates to plane 2 over a distance, d. Its q-parameters at plane 1 and plane 2 are q₁ and q2 respectively. Plane 1 01 = σ R₁ → ∞o d n = 1 (air) wavelength, 10 Plane 2 W2 R₂ 1. Using equations 2 and 3, derive the following equation for the q-parameter, q2 at plane 2: = 92 ino²/o+d [2] Background A light field (a wave-function generally) with a Gaussian spatial amplitude, U; at plane i, may be written as (scaled to unity): Where: U(x) = exp(-inx²/qiλo) eq. 1 1/q₁ = 1/Â¡ — iλo/πw - eq. 2 • Ŕ is the 'reduced' (index-modified) radius of curvature (ROC) defined as R = R₁/n; where R₁ is ROC as measured within a medium at plane i; • 10 is the wavelength for the field where in vacuum defined as λ = n;λ; where λ; is the wavelength as measured within a medium at plane i; • The waist size, w; is defined as the value of the transverse coordinate, x, where the field, U falls to a value 1/e. This is also where the intensity falls to a value 1/e² where intensity I = U₁U}; • Note that n=1.000 for air and so for air we approximate R = R₂ and λ = λi. We refer to the reduced q-parameter, generally referred to as the q-parameter. It can be shown (Siegman ch 20) that the paraxial transfer of the light field of equation 1 between any two planes 1 and 2, is given by: 92 = Aq1+B Cq1+D eq. 3