Optical Coatings

Fundamental Optics
Material Properties Optical Specifications Gaussian Beam Optics
Suppose instead that we decide to reduce the divergence
by directing the laser into a beam expander (reversed telescope)
of angular magnification m = 10, such as Melles Griot model
09 LBM 013 (figure 2.3). Consider the case in which the expander
is focused to form a waist of radius w 0 = 4.0 mm at the expander
output lens. Since v ∝ 1/w 0 , by definition, v is reduced by a factor
of 10; therefore, for z = 100 m,
For the expanded beam, the ratio w(z)/w 0 is only a factor of 12.6
for a distance of 100 m, but it is a factor of 126 for the same distance
when the laser is used alone.
OPTIMUM COLLIMATION
Typically, one has a fixed value for w 0 and uses the previously given
expression to calculate w(z) for an input value of z. However, one can
also utilize this equation to see how final beam radius varies with starting
beam radius at a fixed distance, z. Figure 2.4 shows the Gaussian
beam propagation equation plotted as a function of w 0 , with the
particular values of l = 632.8 nm and z = 100 m.
The beam radius at 100 m reaches a minimum value for a starting
beam radius of about 4.5 mm. Therefore, if we wanted to achieve
the best combination of minimum beam diameter and minimum
beam spread (or best collimation) over a distance of 100 m, our
optimum starting beam radius would be 4.5 mm. Any other starting
value would result in a larger beam at z = 100 m.
We can find the general expression for the optimum starting
beam radius for a given distance, z. Doing so yields
w (optimum) =
0
5 5 4
w(z) = (10 )( 504 . × 10 )
= 5.04 mm.
10
1/2
⎛ lz
⎞
⎜
⎝ p
⎟ . (2.6)
⎠
Using this optimum value of w 0 will provide the best combination
of minimum starting beam diameter and minimum beam
spread (ratio of w(z)/w 0 ) over the distance z. The previous example
of z = 100 and l=632.8 nm gives w 0 (optimum) = 4.48 mm, shown
graphically in figure 2.4. If we put this value for w 0 (optimum) back
into the expression for w(z), w(z) = √}}2 w 0 . Thus, for this example,
w(100) = √}}2 (4.48) = 6.3 mm.
By turning this previous equation around, we can define a
distance, called the Rayleigh range (z R ), over which the beam radius
spreads by a factor of √}}2 as
z R =
with
If we use beam-expanding optics (such as the 09 LBC, 09 LBX,
09 LBZ, or 09 LCM series), which allow us to adjust the position
of the beam waist, we can actually double the distance over which
beam divergence is minimized. Figure 2.5 illustrates this situation,
in which the beam starts off at a value of w(z R ) = (2lz/p) 1/2 , goes
through a minimum value of w 0 = w(z R )/√}}2 , and then returns to
w(z R ). By focusing the beam-expanding optics to place the beam
waist at the midpoint, we can restrict beam spread to a factor of √}}2
over a distance of 2z R , as opposed to just z R .
This result can now be used in the problem of finding the starting
beam radius that yields the minimum beam diameter and beam
spread over 100 m. Using 2z R = 100, or z R = 50, and l = 632.8 nm,
we get a value of w(z R ) = (2lz/p) 1/2 = 4.5 mm, and w 0 = 3.2 mm.
Thus, the optimum starting beam radius is the same as previously
calculated. However, by focusing the expander we achieve a final
beam radius that is no larger than our starting beam radius, while
still maintaining the √}}2 factor in overall variation.
Alternately, if we started off with a beam radius of 6.3 mm
(√}}2w 0 ), we could focus the expander to provide a beam waist of
w 0 = 4.5 mm at 100 m, and a final beam radius of 6.3 mm at 200 m.
FINAL BEAM RADIUS (mm)
100
80
60
40
20
pw
l
2
0
w(z ) = 2w
.
R 0
0
0 1 2 3 4 5 6 7 8 9 10
STARTING BEAM RADIUS w 0 (mm)
(2.7)
Optical Coatings
Figure 2.3
telescope)
Laser beam expander 09 LBM 013 (reversed
Figure 2.4 Beam radius at 100 m as a function of starting
beam radius for a HeNe laser at 632.8 nm
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