ISO/TR 11146-3:2004

Lasers and laser-related equipment — Test methods for laser beam widths, divergence angles and beam propagation ratios — Part 3: Intrinsic and geometrical laser beam classification, propagation and details of test methods

ISO/TR 11146-3:2004 specifies methods for measuring beam widths (diameter), divergence angles and beam propagation ratios of laser beams in support of ISO 11146-1. It provides the theoretical description of laser beam characterization based on the second-order moments of the Wigner distribution, including geometrical and intrinsic beam characterization, and offers important details for proper background subtraction methods recommendable for matrix detectors such as CCD cameras. It also presents alternative methods for the characterization of stigmatic or simple astigmatic beams that are applicable where matrix detectors are unavailable or deliver unsatisfying results.

Lasers et équipements associés aux lasers — Méthodes d'essai des largeurs du faisceau, des angles de divergence et des facteurs de limite de diffraction — Partie 3: Classification intrinsèque et géométrique du faisceau laser, propagation et détails des méthodes d'essai

General Information

• Status: Published
• Publication Date: 21-Jan-2004
• ICS: 31.260 - Optoelectronics. Laser equipment
• Technical Committee: ISO/TC 172/SC 9 - Laser and electro-optical systems
• Drafting Committee: ISO/TC 172/SC 9/WG 1 - Terminology and test methods for electro-optical systems
• Current Stage: 9093 - International Standard confirmed
• Start Date: 14-Dec-2007
• Completion Date: 19-Apr-2025

Standards Content (Sample)

TECHNICAL ISO/TR
REPORT 11146-3
First edition
2004-02-01
Lasers and laser-related equipment —
Test methods for laser beam widths,
divergence angles and beam propagation
ratios —
Part 3:
Intrinsic and geometrical laser beam
classification, propagation and details of
test methods
Lasers et équipements associés aux lasers — Méthodes d'essai des
largeurs du faisceau, des angles de divergence et des facteurs de
propagation du faisceau —
Partie 3: Classification intrinsèque et géométrique du faisceau laser,
propagation et détails des méthodes d'essai

Reference number
©
ISO 2004
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ii © ISO 2004 – All rights reserved

Contents Page
Foreword. iv
Introduction . v
1 Scope. 1
2 Second-order laser beam characterization .1
2.1 General. 1
2.2 Wigner distribution . 1
2.3 First- and second-order moments of Wigner distribution. 2
2.4 Beam matrix. 3
2.5 Propagation though aberration-free optical systems . 4
2.6 Relation between second-order moments and physical beam quantities. 4
2.7 Propagation invariants . 8
2.8 Geometrical classification. 9
2.9 Intrinsic classification . 9
3 Background and offset correction . 10
3.1 General. 10
3.2 Coarse correction by background map subtraction . 10
3.3 Coarse correction by average background subtraction. 11
3.4 Fine correction of baseline offset . 11
4 Alternative methods for beam width measurements . 13
4.1 General. 13
4.2 Variable aperture method. 14
4.3 Moving knife-edge method. 16
4.4 Moving slit method . 17
Annex A (informative) Optical system matrices. 20
Bibliography . 22

Foreword
ISO (the International Organization for Standardization) is a worldwide federation of national standards bodies
(ISO member bodies). The work of preparing International Standards is normally carried out through ISO
technical committees. Each member body interested in a subject for which a technical committee has been
established has the right to be represented on that committee. International organizations, governmental and
non-governmental, in liaison with ISO, also take part in the work. ISO collaborates closely with the
International Electrotechnical Commission (IEC) on all matters of electrotechnical standardization.
International Standards are drafted in accordance with the rules given in the ISO/IEC Directives, Part 2.
The main task of technical committees is to prepare International Standards. Draft International Standards
adopted by the technical committees are circulated to the member bodies for voting. Publication as an
International Standard requires approval by at least 75 % of the member bodies casting a vote.
In exceptional circumstances, when a technical committee has collected data of a different kind from that
which is normally published as an International Standard (“state of the art”, for example), it may decide by a
simple majority vote of its participating members to publish a Technical Report. A Technical Report is entirely
informative in nature and does not have to be reviewed until the data it provides are considered to be no
longer valid or useful.
Attention is drawn to the possibility that some of the elements of this document may be the subject of patent
rights. ISO shall not be held responsible for identifying any or all such patent rights.
ISO/TR 11146-3 was prepared by Technical Committee ISO/TC 172, Optics and photonics, Subcommittee
SC 9, Electro-optical systems.
This first edition of ISO/TR 11146-3, together with ISO 11146-1, cancels and replaces ISO 11146:1999, which
has been technically revised.
ISO 11146 consists of the following parts, under the general title Lasers and laser-related equipment — Test
methods for laser beam widths, divergence angles and beam propagation ratios:
 Part 1: Stigmatic and simple astigmatic beams
 Part 2: General astigmatic beams
 Part 3: Intrinsic and geometrical laser beam classification, propagation and details of test methods
(Technical Report)
iv © ISO 2004 – All rights reserved

Introduction
The propagation properties of every laser beam can be characterized within the method of second-order
moments by ten independent parameters. However, most laser beams of practical interest need less
parameters for a complete description due to their higher symmetry. These beams are stigmatic or simple
astigmatic, e.g. due to the used resonator design.
The theoretical description of beam characterization and propagation as well as the classification of laser
beams based on the second-order moments of the Wigner distribution is given in this part of ISO 11146.
The measurement procedures introduced in ISO 11146-1 and ISO 11146-2 are essentially based on (but not
restricted to) the acquisition of power (energy) density distributions by means of matrix detectors, as for
example CCD cameras. The accuracy of results based on these data depends strongly on proper data
pre-processing, namely background subtraction and offset correction. The details of these procedures are
given here.
In some situations accuracy obtainable with matrix detectors might not be satisfying or matrix detectors might
simply be unavailable. In such cases, other, indirect methods for the determination of beam diameters or
beam width are viable alternatives, as long as comparable results are achieved. Some alternative
measurement methods are presented here.

TECHNICAL REPORT ISO/TR 11146-3:2004(E)

Lasers and laser-related equipment — Test methods for
laser beam widths, divergence angles and beam
propagation ratios —
Part 3:
Intrinsic and geometrical laser beam classification, propagation
and details of test methods
1 Scope
This part of ISO 11146 specifies methods for measuring beam widths (diameter), divergence angles and
beam propagation ratios of laser beams in support of ISO 11146-1. It provides the theoretical description of
laser beam characterization based on the second-order moments of the Wigner distribution, including
geometrical and intrinsic beam characterization, and offers important details for proper background
subtraction methods recommendable for matrix detectors such as CCD cameras. It also presents alternative
methods for the characterization of stigmatic or simple astigmatic beams that are applicable where matrix
detectors are unavailable or deliver unsatisfying results.
2 Second-order laser beam characterization
2.1 General
Almost any coherent or partially coherent laser beam can be characterized by a maximum of ten independent
parameters, the so-called second-order moments of the Wigner distribution. Laser beams showing some kind
of symmetry, stigmatism or simple astigmatism, need even fewer parameters. The knowledge of these
parameters allows the prediction of beam properties behind arbitrary aberration-free optical systems.
Here and throughout this document the term “power density distribution E(x,y,z)” refers to continuous wave
sources. It might be replaced by “energy density distribution H(x,y,z)” in the case of pulsed sources.
Furthermore, a coordinate system is assumed where the z axis is almost parallel to the direction of beam
propagation and the x and y axes are horizontal and vertical, respectively.
2.2 Wigner distribution
The Wigner distribution h(x,y,Θ ,Θ ;z) is a general and complete description of narrow-band coherent and
x y
partially coherent laser beams in a measurement plane. Generally speaking, it gives the amount of beam
power of a beam passing the measurement plane at the lateral position (x,y) with a horizontal paraxial angle of
Θ and a vertical paraxial angle of Θ to the z axis, as shown in Figure 1.
x y
NOTE The Wigner distribution is a function of the axial location z, i.e. the Wigner distribution of the same beam is
different at different z locations. Hence, quantities derived from the Wigner distribution are in general also functions of z.
Throughout this document this z dependence will be dropped. The Wigner distribution then refers to an arbitrarily chosen
location z, the measurement plane.
x,y spatial coordinates
Θ , Θ corresponding angular coordinates
x y
Figure 1 — Coordinates of Wigner distribution
The power density distribution E(x,y) in a measurement plane is related to the Wigner distribution by
∞
Ex,,y = h xy,Θ,ΘΘddΘ (1)
()
()
x yx y
∫
−∞
NOTE The integration limits in the equation above are finite, representing the maximum angles of the rays contained
in the beam, in paraxial; they are conventionally extended to infinity.
2.3 First- and second-order moments of Wigner distribution
The first-order moments of the Wigner distribution are defined as
xh= x,,y Θ ,ΘΘxdxdyd dΘ
(2)
()
xyxy
∫
P
yh= x,,y Θ ,ΘΘydxdyd dΘ (3)
()
xyxy
∫
P
Θ = hx,,yΘΘ, Θ dxdydΘ dΘ (4)
x ()xy x x y
∫
P
Θ = hx,,yΘΘ, Θ dxdydΘ dΘ (5)
()
yxyy xy
∫
P
where P is the beam power given by
Ph= x,,y Θ ,ΘΘdxdyd dΘ (6)
()
xyxy
∫
or, using Equation (1),
PE= x,dy xdy (7)
( )
∫
2 © ISO 2004 – All rights reserved

The spatial moments 〈x〉 and 〈y〉 give the lateral position of the beam centroid in the measurement plane. The
angular moments 〈Θ 〉 and 〈Θ 〉 specify the direction of propagation of the beam centroid.
x y
The (centred) second-order moments are given by
∞
n
1 kmA
kmA n
xy ΘΘΘ=−h x,,y ,Θ x x y− yΘ−ΘΘ−Θ dxdydΘ dΘ (8)
()()()()
xyxy xx()yy xy
∫
P
−∞
where km,A, and n are non-negative integers and km+A++n= 2 . Therefore, there are ten different second-
order moments.
The three spatial second-order moments xy, and xy are related to the lateral extent of the power
density distribution in the measurement plane, the three angular momentsΘΘ, and ΘΘ to the
x yxy
beam divergence, and the four mixed moments xxΘΘ, ,yΘ and yΘ to the phase properties in
x yx y
the measurement plane. More details on the relation between the ten second-order moments and the physical
beam properties are discussed below.
The spatial first- and second-order moments can be directly obtained from the power density distribution E(x,y).
From Equation (1) it follows:
x = E x,dy xxyd (9)
()
∫
P
y = Ex,dy y xdy (10)
()
∫
P
and
xE=−x,dyxx xdy
() (11)
()
...