Gaussian beam component vectors

In order to make these qualitative statements more precise, we will derive all the electromagnetic field components of a fundamental Gaussian beam from two vector functions written in cylindrical coordinates as... 

In order to proceed, we will accept that the transverse components of the electromagnetic field are the only ones that are relevant in the problem on the basis of the exact calculation that we have performed for the fundamental Gaussian beam. Instead, we will use trial functions for u that will lead to self-consistent expressions for the transverse components of Gaussian beams of arbitrary order when substituted into the vector Helmholtz equation. The derivation is clearest for the fundamental. We will redrive the transverse field components of the fundamental Gaussian beam here. The deviation of higher order modes is outlined in the Appendix. 

We have used scalar diffraction theory in this calculation, which is an approximation in two parts. The first part consists of approximating the electromagnetic field as a transverse field. We have derived the conditions under which it is permissible to do so. In the Appendix, we discuss the conditions under which it is possible to replace the vector Helmholtz equation by the scalar Helmholtz equation for transverse fields. In a sense, we have reduced the problem to a solution of the scalar Helmholtz equation. The second part of the approximation consists of exploiting the reduction of the vector Helmholtz equation to a scalar Helmholtz equation. Scalar diffraction theory is based on the scalar Helmholtz equation. Hence, when it is permissible to neglect the longitudinal and cross-polarized components of the Gaussian beam, we may use solutions of the scalar Helmholtz equation for transverse fields and may take over the results of scalar diffraction theory with confidence for this special case. 

The second term on the right-hand side of Eq. (93) may be expanded in terms of the Gaussian beam modes discussed in the Appendbc. The vector d in Eq. (93) represents a displacement of a fundamental Gaussian beam along the

beam radius w [cf. Eq. (20)] and radius of curvature R [cf. Eq. (21)] at the output of the PTR are nearly identical for the two components of the output beam because the path difference A.5 beam waist, R and so we neglect a phase correction in Eq. (93) that is proportional to ik/2R. We include the phase correction in the subsequent analysis for completeness, although its effect is small. 

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