gaussian beam derive

Gaussian beams play such an important role in optical lasers as well as in longer wavelength
systems that they have been extensively analyzed, starting with some of the classic treatments
mentioned in Chapter 1. Almost every text on optical systems discusses Gaussian
beam propagation in some detail, and several comprehensive review articles are available.
However, for millimeter and submillimeter wavelength systems there are naturally certain
aspects that deserve special attention, and we emphasize aspects ofquasioptical propagation
that have proven to be of greatest importance at these relatively long wavelengths.
In the following sections we first give a derivation of Gaussian beam formulas based
on the paraxial wave equation, in cylindrical and in rectangular coordinates. We discuss
normalization, beam truncation, and interpretation of the Gaussian beam propagation formulas.
We next cover higher order modes in different coordinate systems and consider the
effective size of Gaussian beam modes. We then present inverse formulas for Gaussian
beam propagation, which are of considerable use in system design. Finally, we consider
the paraxial approximation in more detail and present an alternative derivation of Gaussian
beam propagation based on diffraction integrals.
2.1 DERIVATION OF BASIC GAUSSIAN BEAM
PROPAGATION
2.1.1 The Paraxial Wave Equation
Only in very special cases does the propagation of an electromagnetic wave result in
a distribution of field amplitudes that is independent of position: the most familiar example
is a plane wave. If we restrict the region over which there is initially a nonzero field, wave
propagation becomes a problem of diffraction, which in its most general form is an extremely
complex vector problem. We treat here a simplified problem encountered when a beam of
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10 Chapter 2 • Gaussian Beam Propagation
radiation that is largely collimated; that is, it has a well-defined direction of propagation but
has also some transverse variation (unlike in a plane wave). We thus develop the paraxial
wave equation, which forms the basis for Gaussian beam propagation. Thus, a Gaussian
beam does have limited transverse variation compared to a plane wave. It is different from
a beam originating from a source in geometrical optics in that it originates from a region of
finite extent, rather than from an infinitesimal point source.
A single component, l/J, ofan electromagnetic wave propagating in a uniform medium
satisfies the Helmholtz (wave) equation
(2.1)
where 1/1 represents any component of E or H. We have assumed a time variation at angular
frequency W of the form exp(jwt). The wave number k is equal to 21l IA, so that
k = W(ErJ-Lr )0.5 [c, where Er and J-Lr are the relative permittivity and permeability of the
medium, respectively. For a plane wave, the amplitudes of the electric and magnetic fields
are constant; and their directions are mutually perpendicular, and perpendicular to the propagation
vector. For a beam of radiation that is similar to a plane wave but for which we
will allow some variation perpendicular to the axis of propagation, we can still assume
that the electric and magnetic fields are (mutually perpendicular and) perpendicular to the
direction of propagation. Letting the direction of propagation be in the positive z direction,
we can write the distribution for any component of the electric field (suppressing the time
dependence) as
E(x,y,z) =u(x,y,z) exp(-jkz), (2.2)
(2.3)
(2.4)
where u is a complex scalar function that defines the non-plane wave part of the beam. In
rectangular coordinates, the Helmholtz equation is
a2E a2E a2E - + - + - + k2 ax2 ay E =