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1.1″ Aberration corrected optical assemblies

**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 = {

A laser beam may be increased in diameter by means of a beamexpander. Where a beam is increased

in diameter by a factor ‘M’, the far-field divergence is reduced by the same factor. This is an

extremely useful technique for further reducing focused spot sizes when using aberration-corrected

focusing lenses. However, care must be taken when using a beamexpander with a meniscus focusing

lens, as the introduction of beam expansion will reduce the divergence contribution to spot size but

increase the aberration contribution (by a factor M2).

Focusing lenses and fixed focus beamexpanders, zoom and extended working distance optics, are

described in Technical Data Sections 20.1 to 20.6. Typically of 1″ (25.4mm) clear aperture; larger

aberration-corrected optics are dealt with in Technical Data Section 21.0.

The behaviour of laser beams, and far-field divergence are dealt with in technical clarification

document 1. CO2 laser beamexpanders are dealt with in detail in Technical Clarification Document

4. Beamexpanders for control of the laser beam in large or moving-optics systems are detailed in

Technical Data Section 67.0. A comparison of our multi-element technique with competitors

diamond machining approach to the production of aberration-free optics is given in Technical

Clarification Document 15.