Section 2:
Gaussian Beam Propagation
2.1
2.2 2.3
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1
0.8
0.6
0.4
0.2
0.135
w
w
O
Relative Intensity
Radial Position
Figure 2.1: The waist of a Gaussian beam is defi ned as the location
where the irradiance is 1/e2 (13.5%) of its maximum value
w(z)
z
wo
wr(zr)
zr
Intensity Intensity Intensity
Radial Position Radial Position Radial Position
Figure 2.2: Gaussian beams are defi ned by their beam waist (w0),
Rayleigh range (zR), and divergence angle (θ)
In many laser optics applications, the laser beam is assumed to be Gaussian
with an irradiance profi le that follows an ideal Gaussian distribution.
All actual laser beams will have some deviation from ideal Gaussian
behavior. The M2 factor, also known as the beam quality factor, compares
the performance of a real laser beam with that of a diff ractionlimited
Gaussian beam (see Section 4 pages 13-15).1 Gaussian irradiance
profi les are symmetric around the center of the beam and decrease as
the distance from the center of the beam perpendicular to the direction
of propagation increases (Figure 2.1). This distribution is described by
Equation 2.12:
In Equation 2.1, I0 is the peak irradiance at the center of the beam, r is the
radial distance away from the axis, w(z) is the radius of the laser beam
where the irradiance is 1/e2 (13.5%) of I0, z is the distance propagated
from the plane where the wavefront is fl at, and P is the total power of
the beam.
However, this irradiance profi le does not stay constant as the beam
propagates through space, hence the dependence of w(z) on z. Due to
diff raction, a Gaussian beam will converge and diverge from an area
called the beam waist (w0), which is where the beam diameter reaches
a minimum value. The beam converges and diverges equally on both
sides of the beam waist by the divergence angle θ (Figure 2.2). The beam
waist and divergence angle are both measured from the axis and their
relationship can be seen in Equation 2.2 and Equation 2.31:
In the above equations, λ is the wavelength of the laser and θ is a far
fi eld approximation. Therefore, θ does not accurately represent the divergence
of the beam near the beam waist, but it becomes more accurate
as the distance away from the beam waist increases. As seen in
Equation 2.3, a small beam waist results in a larger divergence angle,
while a large beam waist results in a smaller divergence angle (or a more
collimated beam). This explains why beam expanders can reduce beam
divergence by increasing beam diameter.
Variation of the beam diameter in the beam waist region is defi ned by:
2.4
2.5
The Rayleigh range of a Gaussian beam is defi ned as the value of z
where the cross-sectional area of the beam is doubled. This occurs