Diode Laser Collimation & Beam Shaping
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resonator consisting of one straight and
one circular section can readily be analyzed this way by using one constant and
one circular phase function, respectively (see Fig. 3). In a more general case, the
phase function can be given by a look-up
table that is a simple-enough function of
the coordinate vector. In addition, by superimposing fields, two or more directions of propagation can be handled by
supplying multiple sets of phase functions.
Applications in nonlinear optics
Nonlinear optical effects are often very
weak and occur over long interaction
lengths. Here, the beam-envelope method
is useful. Self-focusing is one such non-
linear phenomenon—important from a
laser engineering perspective—where the
modification of the beam must be incor-
porated into the design. The effect may be
seen, for example, in laser rods or glass
components placed at a focal point. If the
threshold for self-focusing is exceeded,
Self-focusing occurs in dielectrics, like
optical glasses and laser rod materials,
such as Nd:YAG.
Other nonlinear effects for which the
method is applicable include second-har-monic generation, sum- and difference-fre-
quency generation, parametric generation and amplification, and self-phase
The beam-envelope method extends the use of
to previously unattainable model
sizes. It fills a gap
tionally heavy, but
and fast-to-compute ray-tracing methods.
Successes within nonlinear optics show the
method’s applicability for real-world design tasks. In the future, we will most likely see this method being combined with
traditional full-wave and ray-tracing meth-
ods to reach new frontiers in computational optics.
Bjorn Sjodin is vice president of product management at COMSOL, Burlington, MA; e-mail:
FIGURE 3. A ring-resonator analysis is done at a 1.559 µm
wavelength. The left view shows the finite-element mesh, the center
view shows the physical fast-varying field, and the right view shows
the slowly varying field envelope, which is the actual unknown field
solved for in the beam-envelope method.