# British Infantry Colours by Dino Lemonofides By Dino Lemonofides

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The phase φ is a function of time and should not be confused with ψ, the angle of 30 GAUSSIAN BEAMS AND POLARIZATION polarization. 46) φ = φy − φx . 4a. If the ratio of Ex to Ey changes, the polarization angle ψ changes. 4b), and when Ex = Ey , the ellipse reverts to a circle for circular polarization. For linear polarization, the angle ψ, measured from direction x, is such that the two orthogonal polarizations in the transverse plane are φ = φy − φx = 0 for linear polarizations from 0 ≤ ψ < π/2 and φ = π for linear polarizations from π/2 ≤ ψ < π.

3) x where kx is the wave number in the x direction in radians per meter and x is distance in meters. Spatial frequency fx = 2πkx has units of cycles per meter (or lines per meter). Physical space has more than one dimension; consequently, higher dimension transforms are of interest. In considering beams or propagation of images between planes, we will consider 2D spatial Fourier transforms. 4) y where spatial frequencies kx and ky are in radians per meter in x and y directions, respectively. Note that the input wave pattern is decomposed into a set of plane waves with different propagation directions (kx , ky ) and amplitudes.

By tracking these two parameters, we can analyze propagation of Gaussian beams through optical elements and turbulence. 1) where we ignored polarization effects for simplicity. 4). As A(r, z) exp{−jkz} is a product of two functions in z, the second derivative with respect to z gives four terms. 7) provides ∂A/∂z, which describes how the envelope A propagates in z. 8) where ρ2 = x2 + y2 . 7) can be verified by substitution and appropriate differentiation . 10), fully describes the propagation of a Gaussian beam at z: its real part specifying the radius of curvature R(z) of the phase front and the imaginary part specifying the spot size W(z), which is the radius of the spot at 1/e of the peak.