## Beam Expanders C-BE ## Beamexpander theory

As the name suggests, a beamexpander’s purpose is to enlarge the diameter of a laser beam. This is done
for two possible reasons:
1) When focused, a larger diameter beam will give a smaller spot. Thus a beam is expanded to produce
higher energy densities on the target.
2) A large beam also has a smaller divergence and does not change its diameter as much when
propagating over many meters (unless deliberately defocused). In a flying-optics type system, this helps
to keep the energy density at the focus constant as the optics vary their distance from the laser and also
reduces the focus shift due to the wavefront curvature variation.
The above two points can be expressed mathematically by the equations:

S = 4M²λF/πD        (1)

Where S = Focused spot diameter, F = Lens focal length, D = Beam diameter at the lens, M2 = Beam Msquared
factor (not the magnification). Note that the focusing lens must be diffraction limited.

D0θ= 4M²λ/π         (2)

Where D0 = Beam waist diameter, θ = Beam far-field divergence (full angle). The right-hand side of this
equation is a constant for any particular beam, so as D0 increases, θ will decrease.

If a beamexpander’s magnification is ‘m’, then the input and output beams are related by the following
equations:

DOut = mDIn                                 (3)

1/ROut = 1/m²RIn = 1/G            (4)

θOut = θIn/m                                (5)

Where D are the beam diameters and R the wavefront radii (see Fig 1). ‘G’ is the geometric focus setting of
the beamexpander. On more expensive models than the CO2MPACT, this will be the value engraved on the
focus ring. Normally, however, G is set by experimentation. Note that if G is set to infinity (beamexpander
has zero optical power), the output wavefront radius is not equal to the input wavefront radius, unless a beam
waist is located at the input \begin{equation}R_{in}=\infty\end{equation} ## CO2MPACT beamexpanders

CO2MPACT beamexpanders are optically identical to ULO Optics’s BSL12 series (BSLs now being phased
out). The difference is that the CO2MPACT version has a slightly longer body and is modified to take a thread
insert at the output end. The input end has a built-in thread. The slide-and-lock focus mechanism allows
beam wavefront radius to be adjusted. The central position being factory set for infinity focus. However, the
actual setting needed, will be determined by experimentation.
The magnifications available are listed the Table 1 below along with input clear apertures. Note that the
lowest and highest magnifications possible in the housing are 1.3 and 7.0 respectively. Although a longer
housing is possible to cover magnifications 7 to 12. The output lens is common to all magnifications and together with the mechanical length of the slide-and-lock
slot, determines the focusing range of +1 metre through infinity to –1 metre. Thus the beamexpander can be
used as a ‘long working distance’ focusing lens. When used as such, however, the focal length is equal to
the working distance (=G) divided by the magnification, m, i.e. F = G/m
E.g. when a x7 beamexpander is adjusted to focus over 1400mm, the (effective) focal length is
1400 / 7 = 200mm, not 1400mm. You can use F = 200mm in the equation above for spot size, S, provided D
is the input beam diameter (alternatively, you could use 1400mm for ‘F’ and the output beam diameter for ‘D’:
both are larger by the magnification factor).

### Beamreducers

Occasionally, someone may wish to use a beamexpander in reverse, as a beamreducer. In principle there is
no reason why this cannot be done, however, there are a few cautions to bear in mind:
1) The power density at the output will increase with the square of magnification, m (‘m’ being the
magnification of the beamexpander).
2) The far-field divergence angle of the output beam and any angular misalignment errors of the beam,
increase by the magnification, m. Centring the beam to the beamexpander will therefore be more critical
and it may grow more rapidly than you expect.
3) The focusing becomes much more critical. The range will be approximately +1000/m2 through infinity to
-1000/m2 millimeters. Therefore some short focusing distances are possible. It is advisable when setting
up the beamreducer, not to have any other optics further down the beam until the focusing is set
correctly. This will prevent possible damage from having a focused beam accidentally directed onto
another optical component.