Optical Coatings

Even if a Gaussian TEM 00 laser-beam wavefront were made
perfectly flat at some plane, with all elements moving in precisely
parallel directions, it would quickly acquire curvature and begin
spreading in accordance with
⎡
2
⎛ 2
p w ⎞ ⎤
0
R(z) = z
⎢
1 +
⎥
⎢ ⎜ ⎟
⎝ lz
⎥
⎠
⎣⎢
⎦⎥
and
⎡
lz
w(z) = w 0 + ⎛ 2
⎞ ⎤
⎢1
2
⎝ ⎜ ⎥
⎢ ⎟
p w0
⎠ ⎥
⎣
⎦
12 /
where z is the distance propagated from the plane where the wavefront
is flat, l is the wavelength of light, w 0 is the radius of
the 1/e 2 irradiance contour at the plane where the wavefront is flat, w(z)
is the radius of the 1/e 2 contour after the wave has propagated a
distance z, and R(z) is the wavefront radius of curvature after
propagating a distance z. R(z) is infinite at z = 0, passes through
a minimum at some finite z, and rises again toward infinity as
z is further increased, asymptotically approaching the value of z itself.
The plane z = 0 marks the location of a Gaussian waist, or a place
where the wavefront is flat, and w 0 is called the beam waist radius.
A waist occurs naturally at the midplane of a symmetric confocal
cavity. Another waist occurs at the surface of the planar mirror
of the quasi-hemispherical cavity used in many Melles Griot lasers.
The irradiance distribution of the Gaussian TEM 00 beam,
namely,
I (r) = I e =
2P
w e 2
p
42r
2 / w
2 42r
2 / w
2
0
where w = w(z) and P is the total power in the beam, is the same
at all cross sections of the beam. The invariance of the form of the
distribution is a special consequence of the presumed Gaussian
distribution at z = 0. If a uniform irradiance distribution had been
presumed at z = 0, the pattern at z = ∞ would have been the familiar
Airy disc pattern given by a Bessel function, while the pattern at
intermediate z values would have been enormously complicated. (See
Born and Wolf, Principles of Optics, 2d ed, Pergamon/ Macmillan).
Simultaneously, as R(z) asymptotically approaches z for large
z, w(z) asymptotically approaches the value
w(z)
lz
≅
p
w 0
where z is presumed to be much larger than pw 0 /l so that the 1/e 2
irradiance contours asymptotically approach a cone of angular
radius
v
= w(z)
z
l
= .
pw 0
,
(2.1)
(2.2)
(2.3)
(2.4)
(2.5)
This value is the far-field angular radius of the Gaussian TEM 00
beam. The vertex of the cone lies at the center of the waist (see
figure 2.2).
It is important to note that, for a given value of l, variations of
beam diameter and divergence with distance z are functions of a
single parameter. This is often chosen to be w 0 , or the beam waist
radius.
The direct relationship between beam waist and divergence
(v ∝ 1/w 0 ) must always be considered when focusing a TEM 00 laser
beam. Because of this relationship, the spectrally selective coating
of the spherical output mirror of a Melles Griot laser is actually supported
on the concave inner surface of a weak meniscus lens. In
this paraxial, high f-number configuration, the lens introduces no
significant aberration. A new beam waist, larger than the intracavity
beam waist, is formed by this lens near its output pupil. The
transformed beam has greatly reduced divergence, which is
advantageous for most applications. Note that it is the 1/e 2 beam
diameter of this extracavity waist that is published in this catalog.
As an example to illustrate the relationship between beam waist
and divergence, let us consider the real case of a Melles Griot red
5-mW HeNe laser, 05 LHR 151, with a specified beam diameter of
0.8 mm (i.e., w 0 = 0.4 mm). In the far-field region,
l
v =
pw = 632.8 ×
( p)
(0.4)
Using the asymptotic approximation, at a distance of z = 100 m,
w
w 0
w 0
0
w(z) = zv
5 4
= (10 )( 5.04 × 10
4 )
= 50.4 mm
1
e 2
6
10 5 54
irradiance surface
v
= 5.04 × 10 rad.
which is approximately 126 times larger than w 0 .
asymptotic cone
z
w 0
Figure 2.2 Growth in 1/e 2 contour radius with distance
propagated away from Gaussian waist
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