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ls1_unit_3

# ls1_unit_3 - ON CLASSICAL ELECTROMAGNETIC FIELDS PAGE 22...

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ON CLASSICAL ELECTROMAGNETIC FIELDS PAGE 22 R. Victor Jones, February 7, 2000 III. THE PARAXIAL WAVE EQUATION -- PROPAGATION OF GAUSSIAN BEAMS IN UNIFORM MEDIA D ERIVATION OF P ARAXIAL W AVE E QUATION : In point-to-point communication, we may think of the electromagnetic field as propagating in a kind of "searchlight" mode -- i.e. a beam of finite width that propagates in some particular direction. In analyzing this mode of wave propagation, we make use of an important solution to the so call paraxial approximation of the electromagnetic wave equation (or, more precisely, the paraxial approximation of the Helmholz equation). To that end, we first derive the paraxial approximation and then examine the free-space Gaussian Beam solution(s). We start with the homogeneous Helmholz equation for the vector potential in the form -- see Equation [ I-13a ] 2 r A r r , ω ( ) + ω 2 μ 0 ε ω ( ) r A r r , ω ( ) = 2 r A r r , ω ( ) + k 2 r A r r , ω ( ) = 0 [ III-1 ] We are looking for a wave propagating in, say, the z-direction, so we write a particular component of the potential in the form A α r r , ω ( ) = Ψ r r , ω ( ) exp ikz ( ) [ III-2 ] The function Ψ r r , ω ( ) represents a spatial modulation or "masking" of a plane wave propagating in the z-direction. The z-direction is obviously special and it is useful to appropriately parse the differential operators. For the grad operator we may write grad { } = r { } = r t { } + ˆ z z { } [ III-3 ] where, for example,

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ON CLASSICAL ELECTROMAGNETIC FIELDS PAGE 23 R. Victor Jones, February 7, 2000 r t { } = ˆ x x { } + ˆ y y { } . [ III-4 ] so that r A α r r , ω ( ) = r t Ψ r r , ω ( ) + ˆ z z Ψ r r , ω ( ) ik ˆ z Ψ r r , ω ( ) Ρ Σ ΢ Τ Φ Υ exp ik z ( ) [ III-5 ] For the Laplacian operator we may write 2 A α r r , ω ( ) = t 2 Ψ r r , ω ( ) exp ik z ( ) + z z Ψ r r , ω ( ) ik Ψ r r , ω ( ) Ρ Σ ΢ Τ Φ Υ exp ikz ( ) Χ ψ Ω Ϋ ά έ [ III-6 ] where, for example, t 2 { } = 2 x 2 { } + 2 y 2 { } [ III-7 ] Therefore, 2 A α r r , ω ( ) = t 2 Ψ r r , ω ( ) + 2 z 2 Ψ r r , ω ( ) 2 ik z Ψ r r , ω ( ) k 2 Ψ r r , ω ( ) Ρ Σ ΢ Τ Φ Υ exp ikz ( ) [ III-8 ] and the parsed Helmholz equation ( without approximation ) becomes t 2 Ψ r r , ω ( ) 2 ik z Ψ r r , ω ( ) + 2 z 2 Ψ r r , ω ( ) = 0 [ III-9 ] The paraxial approximation is precisely defined by the condition 2 ik z Ψ r r , ω ( ) >> 2 z 2 Ψ r r , ω ( ) [ III-10 ]
ON CLASSICAL ELECTROMAGNETIC FIELDS PAGE 24 R. Victor Jones, February 7, 2000 which means that the longitudinal variation in the modulation function,

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ls1_unit_3 - ON CLASSICAL ELECTROMAGNETIC FIELDS PAGE 22...

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