Aberration Corrected 1.1″ Optics

Introduction
In laser systems where it is desirable to achieve a small focused spot size and high energy density
the performance can be limited by aberrations if a single element focusing lens is used. In 1986, in
response to this limitation which occurs at short focal lengths, ULO Optics introduced a range of
aberration-corrected lenses and beamexpanders. Because the items in the range are mutually
compatible and intended for use together where necessary, the range was originally known as
“modular focusing optics”. The extent of any aberrations can be calculated from geometrical optics
considerations, whereas the behaviour of laser beams is dealt with by Gaussian optics.
These approaches are incompatible and so we have developed a series of approximations to deal with
the interaction of real laser beams and aberrations which are sufficiently accurate to cope with dayto-
day laser-optics problems. In this document we hope to show how focused spot sizes can be
calculated. An appreciation of this topic will give the laser user a technique for estimating whether
or not aberrations are limiting his system performance and whether the use of aberration-corrected
optics could have significant benefits.

Terminology
Focal length : The effective focal length of a lens, F (mm).
Beam diameter : The (1/e²) diameter of the laser beam, usually considered at the lens,
D or Dlens (mm).
F/No : The focal ratio at which the lens in used, F/No = F/D.
Q-factor : A measure of the relative quality of the laser beam. See below.
Divergence : The angle at which the laser beam eventually diverges at an infinite
distance. Expressed as “V” milliradians

Q-factor
Laser beams from stable optical cavities generally consist of a superimposition of what are termed
‘lateral modes’. The narrowest and least divergent lateral mode has a single central peak and is
termed the TEM00 mode. Stable laser cavities form, at some point in space (possibly inside the
cavity), a beam waist. For simplicity consider a laser with a flat output coupler. In this case the
‘waist’ is at the output coupler, and the ongoing beam increases in size, slowly at first, as it travels
through space. At an indefinately large distance the beam will diverge at an angle ‘V’ milliradians
– the far field divergence. If the beam waist diameter ‘D’ mm is measured, and the value ‘V’
milliradians is also measured, then the two numbers can be multiplied together. For the TEM00 mode,
the product DV is equal to 13.5 (mm x milliradians at 10.6μm). For real lasers which do not emit
a pure TEM00 beam, we can obtain a measure of the beam quality ‘Q’:
Q = DV/13.5         for a CO2 laser at 10.6μm.
Small waveguide lasers often achieve a Q-factor of around 1.1 or 1.2. Medium sized industrial lasers
of 500W to 2KW power output typically emit beams with Q-factor in the range 1.5 to 3.0.
A true measure of the far-field divergence can be obtained using a calibrated beamexpander. Some
laser manufacturers use a technique of measuring the output beam diameter near the laser and several
metres away from the laser. This latter technique can give results for far-field divergence which are
severely understated. The presentation below assumes that true values of ‘V’ and ‘Q’ have been
obtained.
Lens ‘shape’
The effects of spherical aberration, the on-axis aberration which is of concern when focusing laser
beams, can be minimized for a single element lens with spherical surfaces. The ideal shape of the
lens, bi-convex, plano-convex or meniscus, depends upon the refractive index of the lens raw
material. For ZnSe lenses the ideal shape is of meniscus form. Plano-convex lenses give rise to about
50% greater aberration and will not be dealt with here. All comparisons below are between
aberration-corrected lenses and ‘optimum’ meniscus lenses.
Blurring due to spherical aberration
First we can consider the effect of the spherical aberration contribution to the focused spot size.
We can denote the extent of the lateral geometrical blur due to spherical aberration as ‘X’, where:
X=0.018D3/F2            for a ZnSe meniscus lens.
X/F = 0.018/(F/No.)3
(This is not the size of the circle-of-least-confusion). It is useful to show how ‘X’ increases with
reduction of F/No, but since the actual spot size increases with F, the focal length, we plot X/F (in
microns) against F/No.
To obtain the actual value of ‘X’, multiply the factor X/F by the focal length ‘F’ in mm. For example,
at F/2 and 50mm focal length the spherical aberration contribution is 50 x 5.0mm = 250mm.

Aberration-free spot size
If an aberration-free lens of focal length F mm is
placed sensibly near to the beam waist, the lens will
focus the beam to a new, small waist of diameter ‘Y’
mm, where:
Y = FV
or
Y/F = V
(Y in microns if V in milliradians).
To obtain a comparison with Fig. 20.01 we plot the
same factor “Microns of spot size per mm of focal
length” against F/No. for various laser F/Q values.
The “microns of spot size (etc)” factor is simply
equal to V which is given by,

Y/F = 4l(F/No)/p(F/Q)

Again, to obtain the actual spot size contribution
‘Y’, multiply by the focal length of the
lens in question. Comparison of figures 20.01
and 20.02 shows that the aberration contribution
is predominant below F/3 (for normal
focal lengths and Q no.) and the divergence
contribution is predominant above F/4.
Total spot size
The total spot size, adding together the aberration
and divergence contributions to spot size
is given by:
Total spot size ‘Z’ = (X² + Y²)½
A graphical plot of Z against F/No. and overlaid
onto Figure 20.02 would, in principle,
indicate the conditions under which aberration corrected optics are useful. Such a plot, for various
F/Q values, would contain too many curves to be anything other than confusing. A summary of the
relative improvement that can be obtained using an aberration-corrected lens compared with a
meniscus lens can be demonstrated in tabular form.
In Table 20.03 we show the ratio of spot sizes for a meniscus to an aberration-free lens for various
values of F/No. and divergence ‘V’ (milliradians). In the table we use an expression Ve – effective
divergence (see above) . The spot size ratios can be squared to give the relative improvement in
energy density. If the lens is not at a waist, then a better effective value for ‘V’ divergence can be found
if the Q-factor is known, and the diameter Dlens of the beam at the lens is known. In the Table (and

for spot size calculations), instead of “far-field divergence ‘V'” use:
Ve = 13.5Q/Dlens           the “effective” divergence at the lens.
If it is considered that a 5% decrease in spot size (10% energy density increase) is the lower limit of
significant usefulness, then the dotted line indicates the region below which it is not worthwhile
using aberration-corrected focusing lenses. For the mathematically minded, the ratios given in Table
20.03 were calculated from a general formula which can be applied to a comparison of aberrationfree-
to-meniscus ZnSe lenses wherever the effective divergence and the F/No. are known:
Z/Y = {