The Propagation Constant
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The propagation of a pure Gaussian beam can be fully specified by either
its beam waist diameter or its far-field divergence. So, in principle, full
characterization of a beam can be made by simply measuring the waist
diameter (2w0), or by measuring
the diameter (2wz) at a known and
specified distance (z) from the beam waist,
using the equations |
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and |
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where l is the wavelength of the laser radiation,
and w(z) and
R(z) are the
beam radius and wavefront radius, respectively, at distance
z from the beam waist. In practice, however, this approach is fraught with problems - it is extremely difficult, in many instances, to locate the beam waist; relying on a single-point measurement is inherently inaccurate; and, most importantly, pure Gaussian laser beams do not exist in the real world. The beam from a well-controlled helium neon laser comes very close, as does the beam from a few other gas lasers. However, for most lasers (even those specifying a fundamental TEM00 mode) the output contains some component of higher order modes that do not propagate according to the formula shown above. The problems are even worse for lasers operating in high-order modes. The need for a figure of merit for laser beams that can be used to determine the propagation characteristics of the beam has long been recognized. Specifying the mode is inadequate, because, for example, the output of a laser can contain up to 50% higher order modes and still be considered TEM00. The concept of a dimensionless beam propagation parameter was developed in the early 1970s to meet this need, based on the fact that, for any given laser beam (even those not operating in the TEM00 mode) the product of the beam waist radius (w0) and the far-field divergence (q) is constant as the beam propagates through an optical system, and the ratio |
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 , |
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where w0R and q
R are the beam waist and far-field divergence
of the real beam, respectively, is an accurate indication of the
propagation characteristics of the beam. For a true Gaussian beam,
M2 = 1 |
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Embedded Gaussian |
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The concept of an "embedded Gaussian," shown in the figure below, is useful
as a construct to assist with both theoretical modeling and laboratory
measurements. |
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The embedded Gaussian |
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A mixed-mode beam that has a waist M (not
M2) times larger than the embedded
Gaussian will propagate with a divergence M times greater than the
embedded Gaussian. Consequently the beam diameter of the mixed-mode beam will
always be M times the beam diameter of the embedded Gaussian, but
will have the same radius of curvature and the same Rayleigh range
(z=R). |
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| Optics Guide Copyright 2002 Melles Griot Inc. |