Lasers and laser-related equipment - Test methods for laser beam widths, divergence angles and beam propagation ratios - Part 2: General astigmatic beams (ISO 11146-2:2021)

This document specifies methods for measuring beam widths (diameter), divergence angles and beam
propagation ratios of laser beams. This document is applicable to general astigmatic beams or unknown
types of beams. For stigmatic and simple astigmatic beams, ISO 11146-1 is applicable.
Within this document, the description of laser beams is accomplished by means of the second order
moments of the Wigner distribution rather than physical quantities such as beam widths and divergence
angles. However, these physical quantities are closely related to the second order moments of the
Wigner distribution. In ISO/TR 11146-3, formulae are given to calculate all relevant physical quantities
from the measured second order moments.

Laser und Laseranlagen - Prüfverfahren für Laserstrahlabmessungen, Divergenzwinkel und Beugungsmaßzahlen - Teil 2: Allgemein astigmatische Strahlen (ISO 11146-2:2021)

Dieses Dokument legt Verfahren zum Messen der Strahlabmessungen (Strahldurchmesser), Divergenzwinkel und Beugungsmaßzahlen von Laserstrahlen fest. Dieses Dokument ist für allgemein astigmatische Strahlen oder Strahlen unbekannter Art anzuwenden. Bei stigmatischen und einfach astigmatischen Strahlen kann ISO 11146-1 angewendet werden.
In diesem Dokument wird die Beschreibung von Laserstrahlen durch die Momente zweiter Ordnung der Wigner-Verteilung gegenüber der Beschreibung durch physikalische Größen wie Strahlabmessungen und Divergenzwinkel bevorzugt. Diese physikalischen Größen stehen jedoch in engem Zusammenhang mit den Momenten zweiter Ordnung der Wigner-Verteilung. In ISO/TR 11146-3 sind Gleichungen zur Berechnung aller relevanten physikalischen Größen aus den Momenten zweiter Ordnung angegeben.

Lasers et équipements associés aux lasers - Méthodes d'essai des largeurs du faisceau, angles de divergence et facteurs de limite de diffraction - Partie 2: Faisceaux astigmatiques généraux (ISO 11146-2:2021)

Le présent document spécifie les méthodes pour le mesurage des largeurs (diamètres) du faisceau, angles de divergence et  facteurs de limite de diffraction. Le présent document s'applique aux faisceaux astigmatiques généraux ou si le type de faisceau est inconnu. Pour les faisceaux stigmatiques et astigmatiques simples, l'ISO 11146-1 s'applique.
Au sein du présent document, la description des faisceaux laser est réalisée au moyen des moments de second ordre de la distribution de Wigner, plutôt que par des grandeurs physiques telles que les largeurs de faisceau et les angles de divergence. Toutefois, ces grandeurs physiques sont étroitement liées aux moments de second ordre de la distribution de Wigner. Dans l'ISO/TR 11146‑3, des formules sont données pour calculer toutes les grandeurs physiques concernées à partir des moments de second ordre mesurés.

Laserji in laserska oprema - Preskusne metode za širine laserskega žarka, kota divergence in faktorja širjenja žarkov - 2. del: Osnovni astigmatični žarki (ISO 11146-2:2021)

General Information

Status
Published
Public Enquiry End Date
19-Jul-2020
Publication Date
22-Aug-2021
Technical Committee
Current Stage
6060 - National Implementation/Publication (Adopted Project)
Start Date
28-Jul-2021
Due Date
02-Oct-2021
Completion Date
23-Aug-2021

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SLOVENSKI STANDARD
SIST EN ISO 11146-2:2021
01-september-2021
Nadomešča:
SIST EN ISO 11146-2:2005
Laserji in laserska oprema - Preskusne metode za širine laserskega žarka, kota
divergence in faktorja širjenja žarkov - 2. del: Osnovni astigmatični žarki (ISO
11146-2:2021)
Lasers and laser-related equipment - Test methods for laser beam widths, divergence
angles and beam propagation ratios - Part 2: General astigmatic beams (ISO 11146-
2:2021)
Laser und Laseranlagen - Prüfverfahren für Laserstrahlabmessungen, Divergenzwinkel
und Beugungsmaßzahlen - Teil 2: Allgemein astigmatische Strahlen (ISO 11146-2:2021)
Lasers et équipements associés aux lasers - Méthodes d'essai des largeurs du faisceau,
angles de divergence et facteurs de limite de diffraction - Partie 2: Faisceaux
astigmatiques généraux (ISO 11146-2:2021)
Ta slovenski standard je istoveten z: EN ISO 11146-2:2021
ICS:
31.260 Optoelektronika, laserska Optoelectronics. Laser
oprema equipment
SIST EN ISO 11146-2:2021 en
2003-01.Slovenski inštitut za standardizacijo. Razmnoževanje celote ali delov tega standarda ni dovoljeno.

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SIST EN ISO 11146-2:2021

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SIST EN ISO 11146-2:2021


EN ISO 11146-2
EUROPEAN STANDARD

NORME EUROPÉENNE

July 2021
EUROPÄISCHE NORM
ICS 31.260 Supersedes EN ISO 11146-2:2005
English Version

Lasers and laser-related equipment - Test methods for
laser beam widths, divergence angles and beam
propagation ratios - Part 2: General astigmatic beams (ISO
11146-2:2021)
Lasers et équipements associés aux lasers - Méthodes Laser und Laseranlagen - Prüfverfahren für
d'essai des largeurs du faisceau, angles de divergence Laserstrahlabmessungen, Divergenzwinkel und
et facteurs de limite de diffraction - Partie 2: Faisceaux Beugungsmaßzahlen - Teil 2: Allgemein astigmatische
astigmatiques généraux (ISO 11146-2:2021) Strahlen (ISO 11146-2:2021)
This European Standard was approved by CEN on 4 July 2021.

CEN members are bound to comply with the CEN/CENELEC Internal Regulations which stipulate the conditions for giving this
European Standard the status of a national standard without any alteration. Up-to-date lists and bibliographical references
concerning such national standards may be obtained on application to the CEN-CENELEC Management Centre or to any CEN
member.

This European Standard exists in three official versions (English, French, German). A version in any other language made by
translation under the responsibility of a CEN member into its own language and notified to the CEN-CENELEC Management
Centre has the same status as the official versions.

CEN members are the national standards bodies of Austria, Belgium, Bulgaria, Croatia, Cyprus, Czech Republic, Denmark, Estonia,
Finland, France, Germany, Greece, Hungary, Iceland, Ireland, Italy, Latvia, Lithuania, Luxembourg, Malta, Netherlands, Norway,
Poland, Portugal, Republic of North Macedonia, Romania, Serbia, Slovakia, Slovenia, Spain, Sweden, Switzerland, Turkey and
United Kingdom.





EUROPEAN COMMITTEE FOR STANDARDIZATION
COMITÉ EUROPÉEN DE NORMALISATION

EUROPÄISCHES KOMITEE FÜR NORMUNG

CEN-CENELEC Management Centre: Rue de la Science 23, B-1040 Brussels
© 2021 CEN All rights of exploitation in any form and by any means reserved Ref. No. EN ISO 11146-2:2021 E
worldwide for CEN national Members.

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SIST EN ISO 11146-2:2021
EN ISO 11146-2:2021 (E)
Contents Page
European foreword . 3

2

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SIST EN ISO 11146-2:2021
EN ISO 11146-2:2021 (E)
European foreword
This document (EN ISO 11146-2:2021) has been prepared by Technical Committee ISO/TC 172 "Optics
and photonics" in collaboration with Technical Committee CEN/TC 123 “Lasers and photonics” the
secretariat of which is held by DIN.
This European Standard shall be given the status of a national standard, either by publication of an
identical text or by endorsement, at the latest by January 2022, and conflicting national standards shall
be withdrawn at the latest by January 2022.
Attention is drawn to the possibility that some of the elements of this document may be the subject of
patent rights. CEN shall not be held responsible for identifying any or all such patent rights.
This document supersedes EN ISO 11146-2:2005.
Any feedback and questions on this document should be directed to the users’ national standards
body/national committee. A complete listing of these bodies can be found on the CEN websites.
According to the CEN-CENELEC Internal Regulations, the national standards organizations of the
following countries are bound to implement this European Standard: Austria, Belgium, Bulgaria,
Croatia, Cyprus, Czech Republic, Denmark, Estonia, Finland, France, Germany, Greece, Hungary, Iceland,
Ireland, Italy, Latvia, Lithuania, Luxembourg, Malta, Netherlands, Norway, Poland, Portugal, Republic of
North Macedonia, Romania, Serbia, Slovakia, Slovenia, Spain, Sweden, Switzerland, Turkey and the
United Kingdom.
Endorsement notice
The text of ISO 11146-2:2021 has been approved by CEN as EN ISO 11146-2:2021 without any
modification.

3

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SIST EN ISO 11146-2:2021

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SIST EN ISO 11146-2:2021
INTERNATIONAL ISO
STANDARD 11146-2
Second edition
2021-07
Lasers and laser-related equipment —
Test methods for laser beam
widths, divergence angles and beam
propagation ratios —
Part 2:
General astigmatic beams
Lasers et équipements associés aux lasers — Méthodes d'essai des
largeurs du faisceau, angles de divergence et facteurs de limite de
diffraction —
Partie 2: Faisceaux astigmatiques généraux
Reference number
ISO 11146-2:2021(E)
©
ISO 2021

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SIST EN ISO 11146-2:2021
ISO 11146-2:2021(E)

COPYRIGHT PROTECTED DOCUMENT
© ISO 2021
All rights reserved. Unless otherwise specified, or required in the context of its implementation, no part of this publication may
be reproduced or utilized otherwise in any form or by any means, electronic or mechanical, including photocopying, or posting
on the internet or an intranet, without prior written permission. Permission can be requested from either ISO at the address
below or ISO’s member body in the country of the requester.
ISO copyright office
CP 401 • Ch. de Blandonnet 8
CH-1214 Vernier, Geneva
Phone: +41 22 749 01 11
Email: copyright@iso.org
Website: www.iso.org
Published in Switzerland
ii © ISO 2021 – All rights reserved

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SIST EN ISO 11146-2:2021
ISO 11146-2:2021(E)

Contents Page
Foreword .iv
Introduction .v
1 Scope . 1
2 Normative references . 1
3 Terms and definitions . 1
4 Coordinate system . 6
5 Test principles . 6
5.1 General . 6
5.2 Spatial second order moments of the Wigner distribution . 6
5.3 Second order moments of the Wigner distribution . 6
5.4 Derived quantities . 6
6 Measurement arrangement and test equipment . 6
6.1 General . 6
6.2 Preparation . 6
6.3 Control of environment . 7
6.4 Detector system . 7
6.5 Beam-forming optics and optical attenuators . 8
7 Measurement of the second order moments . 8
7.1 General . 8
7.2 Measurement of the second order moments of power density distributions . 8
7.3 Measurement of all second order moments of the Wigner distribution .10
8 Determination of effective beam propagation ratio .12
9 Determination of intrinsic astigmatism.12
10 Determination of the twist parameter .13
11 Test report .13
Bibliography .16
© ISO 2021 – All rights reserved iii

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SIST EN ISO 11146-2:2021
ISO 11146-2:2021(E)

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.
The procedures used to develop this document and those intended for its further maintenance are
described in the ISO/IEC Directives, Part 1. In particular, the different approval criteria needed for the
different types of ISO documents should be noted. This document was drafted in accordance with the
editorial rules of the ISO/IEC Directives, Part 2 (see www .iso .org/ directives).
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. Details of
any patent rights identified during the development of the document will be in the Introduction and/or
on the ISO list of patent declarations received (see www .iso .org/ patents).
Any trade name used in this document is information given for the convenience of users and does not
constitute an endorsement.
For an explanation of the voluntary nature of standards, the meaning of ISO specific terms and
expressions related to conformity assessment, as well as information about ISO's adherence to the
World Trade Organization (WTO) principles in the Technical Barriers to Trade (TBT), see www .iso .org/
iso/ foreword .html.
This document was prepared by Technical Committee ISO/TC 172, Optics and photonics, Subcommittee
SC 9, Laser and electro-optical systems, in collaboration with the European Committee for Standardization
(CEN) Technical Committee CEN/TC 123, Lasers and photonics, in accordance with the Agreement on
technical cooperation between ISO and CEN (Vienna Agreement).
This second edition cancels and replaces the first edition (ISO 11146-2:2005), which has been
technically revised. The main changes compared to the previous edition are as follows:
— The terms and definitions were harmonized with the new ISO 11145.
— The "principal axes" were defined more thoroughly and named as x' and y'. Quantities related to the
principal axes coordinate system refer to this definition and use x' and y' in their indices.
— The requirements for the integration range for the determination of the second order moments
have been relaxed.
A list of all parts in the ISO 11146 series can be found on the ISO website.
Any feedback or questions on this document should be directed to the user’s national standards body. A
complete listing of these bodies can be found at www .iso .org/ members .html.
iv © ISO 2021 – All rights reserved

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SIST EN ISO 11146-2:2021
ISO 11146-2:2021(E)

Introduction
The propagation properties of laser beams can be characterized by ten independent parameters when
applying the method of second order moments (see ISO/TR 11146-3). Most laser beams need few
parameters for a complete description due to their higher symmetry. Lasers emit beams which are
stigmatic or simple astigmatic due to their resonator design.
ISO 11146-1 describes the measurement methods for stigmatic and simple astigmatic beams while
this document deals with the measurement procedures for general astigmatic beams. This document
is applicable to beams of unknown type. Beam characterization, based on the method of second
order moments as described in ISO 11146-1 and this document, is only valid within the paraxial
approximation.
The theoretical description of beam characterization and propagation as well as the classification of
laser beams is given in ISO/TR 11146-3, which is a Technical Report. The procedures for background
subtraction and offset correction are also given in ISO/TR 11146-3.
In ISO 11146, the second order moments of the power (energy) density distribution function are used
for the determination of beam widths. If problems are experienced in the direct measurements of these
quantities, other indirect methods of measurement of second order moments may be used as long as
comparable results are achievable.
In ISO/TR 11146-3, three alternative methods for beam width measurement and their correlation with
the method used in this document are described. These methods are:
— variable aperture method;
— moving knife-edge method;
— moving slit method.
© ISO 2021 – All rights reserved v

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SIST EN ISO 11146-2:2021

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SIST EN ISO 11146-2:2021
INTERNATIONAL STANDARD ISO 11146-2:2021(E)
Lasers and laser-related equipment — Test methods
for laser beam widths, divergence angles and beam
propagation ratios —
Part 2:
General astigmatic beams
1 Scope
This document specifies methods for measuring beam widths (diameter), divergence angles and beam
propagation ratios of laser beams. This document is applicable to general astigmatic beams or unknown
types of beams. For stigmatic and simple astigmatic beams, ISO 11146-1 is applicable.
Within this document, the description of laser beams is accomplished by means of the second order
moments of the Wigner distribution rather than physical quantities such as beam widths and divergence
angles. However, these physical quantities are closely related to the second order moments of the
Wigner distribution. In ISO/TR 11146-3, formulae are given to calculate all relevant physical quantities
from the measured second order moments.
2 Normative references
The following documents are referred to in the text in such a way that some or all of their content
constitutes requirements of this document. For dated references, only the edition cited applies. For
undated references, the latest edition of the referenced document (including any amendments) applies.
ISO 11145, Optics and photonics — Lasers and laser-related equipment — Vocabulary and symbols
ISO 11146-1, Lasers and laser-related equipment — Test methods for laser beam widths, divergence angles
and beam propagation ratios — Part 1: Stigmatic and simple astigmatic beams
EN 61040:1992, Power and energy measuring detectors, instruments and equipment for laser radiation
3 Terms and definitions
For the purposes of this document, the terms and definitions given in ISO 11145, ISO 11146-1, EN 61040
and the following apply.
ISO and IEC maintain terminological databases for use in standardization at the following addresses:
— ISO Online browsing platform: available at https:// www .iso .org/ obp
— IEC Electropedia: available at https:// www .electropedia .org/
Note The x−, y− and z-axes in the following definitions refer to the laboratory system (as described in
Clause 4). 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 case of pulsed sources.
© ISO 2021 – All rights reserved 1

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SIST EN ISO 11146-2:2021
ISO 11146-2:2021(E)

3.1
generalized beam diameter
d
g
measure of the extent of the power density distribution of a beam in a cross-section at an axial location
z, derived from the second order moments by
22
dx=+22 y (1)
g
Note 1 to entry: This definition is similar to the beam diameter defined in ISO 11145 or ISO 11146-1. But in this
context the definition is not restricted to circular power density distributions.
3.2
generalized beam waist location
z
0,g
position where the generalized beam diameter (3.1) reaches its minimum value along the axis of
propagation
3.3
generalized Rayleigh length
z
R,g
distance along the beam axis from the generalized beam waist where the generalized beam diameter is
a factor of 2 larger than the generalized beam waist diameter
3.4
Wigner distribution
phase space distribution representing a laser beam in a transverse plane at location z
Note 1 to entry: The Wigner distribution is a function of two spatial and two angular coordinates, giving the
amount of beam power propagating through the point (x,y) in the direction (Θ , Θ ).
x y
3.5
spatial first order moments of the Wigner distribution
xy,
subset of the first order moments, which can be directly obtained from measured power density
distribution by
∞ ∞
Ε ()xy,, z xxy d d
∫ ∫
−∞−∞
xz() = (2)
∞ ∞
Ε ()xy,, zx d dy
∫ ∫
−∞−∞
and
∞ ∞
Ε ()xy,, zy dxy d
∫ ∫
−∞−∞
yz() = (3)
∞ ∞
Ε ()xy,, zx d dy
∫ ∫
−∞−∞
where E(x,y,z) is the power density distribution at the specific plane z = constant
2 © ISO 2021 – All rights reserved

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SIST EN ISO 11146-2:2021
ISO 11146-2:2021(E)

3.6
second order moments of the Wigner distribution
2 22 2
xy,,,,xy ΘΘ,, ΘΘ ,,xxΘΘ yyΘΘ,
xy xy xy xy
ten second order moments of the Wigner distribution (3.4) of the beam at location z
Note 1 to entry: The ten second order moments contain information on the following physical beam properties:
beam size and orientation, divergence angles and their orientation, radii of curvature of the phase paraboloid
and their orientation and the twist parameter. Details on these relations are given in ISO/TR 11146-3.
2 2 2
Note 2 to entry: In ISO 11146-1, the three spatial second order moments are defined as σ , σ and σ . In this
x y xy
document and ISO/TR 11146-3, the angular brackets are used to emphasize the coordinates of the moments. This
22 22 2
means that σ = x , σ = y and σ = xy .
x y xy
2 2
Note 3 to entry: Three angular moments 〈Θ 〉, 〈Θ 〉 and 〈Θ Θ 〉 are independent of z. The other seven second
x y x y
order moments are, in general, functions of z.
3.7
spatial second order moments of the Wigner distribution
22
xy,, xy
subset of the second order moments, which can be directly obtained from measured power density
distribution by
∞ ∞
2
Ε ()xy,, zx d()− xx dy
∫ ∫
−∞−∞
2
xz () = (4)
∞ ∞
Ε ()xy,, zxd dy
∫ ∫
−−∞−∞
∞ ∞
2
Ε ()xy,, zy d()− yx dy
∫ ∫
−∞−∞
2
yz () = (5)
∞ ∞
Ε ()xy,, zxd dy
∫ ∫
−−∞−∞
and
∞ ∞
Ε ()xy,, zx()−−xy()yxd dy
∫ ∫
−∞−∞
xy ()z = (6)
∞ ∞
Ε ()xy,, zxd dy
∫ ∫
−∞−∞∞
3.8
beam matrix
P
symmetric and positive definite 4×4 matrix containing all ten second order moments of the Wigner
distribution (3.6) and its elements and given by
2
 
xxyxΘΘx
xy
 
 
2
xy yyΘΘy
xy
 
P= (7)
 
2
xyΘΘ ΘΘ Θ
 
xx xx y
 
2
 
xyΘΘ ΘΘ Θ
yy xy y
 
© ISO 2021 – All rights reserved 3

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SIST EN ISO 11146-2:2021
ISO 11146-2:2021(E)

3.9
effective beam propagation ratio
2
M
eff
invariant quantity related to the focusability of a general astigmatic beam, defined as:
1

2
M = []det()P 4 (8)
eff
λ
where det (P) is the determinant of matrix P
2
Note 1 to entry: The effective beam propagation ratio M is an invariant related to the volume that the beam
eff
occupies in the four-dimensional phase space (two lateral spatial and two lateral angular dimensions) and thus a
measure for the focusability of the beam.
Note 2 to entry: For simple astigmatic beams, the effective beam propagation ratio is the geometric mean of the
22 2 22
beam propagation ratios of the principal axes of the beam: MM=×M . For stigmatic beams MM= .
xy
eff eff
3.10
intrinsic astigmatism
a
degree of how close to a stigmatic beam the general astigmatic beam can be transformed by using
lenses and free space propagation
2
2
2

2
22 22 2
 
ax=−ΘΘxy+−ΘΘyx+−2 yxΘΘ ΘΘyM− ≥0
() ()
()
xx ( yy ) xy yx efff
2  
 
λ
(9)
Note 1 to entry: Beams are classified according to their intrinsic astigmatism, a, which is an invariant quantity. A
beam with a = 0 is called intrinsic stigmatic, a beam with a > 0 is called intrinsic astigmatic. For simple astigmatic
2
22
beams aM=()12 −M . More details are given in ISO/TR 11146-3.
()
xy
3.11
twist parameter
t
parameter related to the rotational properties of the phase front of a beam, and also to the orbital
angular momentum carried by the beam
tx=−ΘΘy (10)
yx
Note 1 to entry: The twist parameter is invariant under propagation through free space and spherical lenses. It
might be altered under propagation through cylindrical lenses.
3.12
principal axes
x’, y’
axes of the maximum and minimum beam extent based on the second
order moments of the power density distribution in a cross-section of the beam
4 © ISO 2021 – All rights reserved

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SIST EN ISO 11146-2:2021
ISO 11146-2:2021(E)

Figure 1 — Beam profile with the laboratory and principle axes coordinate systems
Note 1 to entry: The axes of maximum and minimum extent are always perpendicular to each other.
Note 2 to entry: Unless otherwise stated, in this document x’ is the principal axis which is closer to the x-axis
of the laboratory coordinate system, and y’ is the principal axis which is closer to the y-axis of the laboratory
coordinate system.
Note 3 to entry: If the principal axes make the angle π/4 with the x- and y-axes of the laboratory coordinate
system, then the x-axis is by convention the direction of maximum extent.
Note 4 to entry: See Figure 1.
[SOURCE: ISO 11146-1:2021, 3.3]
3.13
azimuthal orientation
φ
azimuthal angle between the x-axis of the laboratory system and the
principal axis x’
[SOURCE: ISO 11146-1:2021, 3.4]
3.14
beam widths
dz() , dz()
σσx' y'
extent of a power density distribution in a cross-section of the beam at an axial location z along the
principal axes x’ and y’, respectively, based on the second order moments of the power density
distribution
Note 1 to entry: This definition differs from that given in ISO 11145:2018, 3.5.2, where the beam widths are
defined only in the laboratory system, whereas for the purposes of this document the beam widths are defined in
the principal axes (3.12) system of the beam.
© ISO 2021 – All rights reserved 5

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SIST EN ISO 11146-2:2021
ISO 11146-2:2021(E)

Note 2 to entry: Formulae for calculation of the beam widths from the three second order moments are given in
7.2.
[SOURCE: ISO 11146-1:2021, 3.5]
4 Coordinate system
The x−, y− and z−axes define the orthogonal space directions in the laboratory axes system and shall
be specified by the user. The z−axis shall approximately coincide with the direction of the beam. The x−
and y− axes are transverse axes, usually horizontal and vertical, respectively. The origin of the z-axis is
in a reference x–y plane defined by the manufacturer, e.g. the front of the laser enclosure.
5 Test principles
5.1 General
The following test principles are valid for general astigmatic beams. For stigmatic and simple astigmatic
beams ISO 11146-1 may be applied.
5.2 Spatial second order moments of the Wigner distribution
Spatial second order moments are obtained by acquisition of power density distributions by means
of spatially resolving detectors, correcting the measured profiles and calculating the first and second
order moments.
5.3 Second order moments of the Wigner distribution
For the determination of all ten second order moments two different measurement set-ups are required.
Eight of the ten second order moments and the sum xyΘΘ+ are obtained by acquisition of
()
yx
power density distributions along the propagation axis z in different planes near the generalized waist
location, calculating the three spatial second order moments of each measured power density profile
and fitting three independent parabolas to them.
The difference xyΘΘ− is obtained from the spatial moments of a power density distribution
()
yx
acquired behind a cylindrical lens, see 7.3.
5.4 Derived quantities
2
The effective beam propagation ratio M , the intrinsic astigmatism, a, and the twist parameter, t, are
eff
obtained from the second order moments of the Wigner distribution according to the Formulae (8) to
(10).
6 Measurement arrangement and test equipment
6.1 General
The test is based on the measurement of the cross-sectional power density distribution at several axial
locations along the beam propagation axis.
6.2 Preparation
The optical axis of the measuring system should be coaxial with the laser beam to be measured. Suitable
optical alignment devices are available for this purpose (e.g. aligning lasers or steering mirrors).
6 © ISO 2021 – All rights reserved

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SIST EN ISO 11146-2:2021
ISO 11146-2:2021(E)

The aperture of the optical system should accommodate the entire cross-section of the laser beam.
Clipping shall be smaller than 1 % of the total beam power or energy.
The attenuators or beam-forming optics should be mounted so that the optical axis runs through the
geometrical centres. Care shall be taken to avoid systematic errors. Reflections, interference effects,
external ambient light, thermal radiation or air draughts are all potential sources of error.
An evaluation shall be made to determine if the entire laser beam reaches the detector surface. For
testing this, apertures of different widths can be introduced into the beam path in front of each optical
component. The aperture that reduces the output signal by 5 % should have a diameter less than 0,8
times the aperture of the optical component.
6.3 Control of environment
Suitable measures such as mechanical and acoustical isolation
...

SLOVENSKI STANDARD
oSIST prEN ISO 11146-2:2020
01-julij-2020
Laserji in laserska oprema - Preskusne metode za širine laserskega žarka, kota
divergence in faktorja širjenja žarkov - 2. del: Osnovni astigmatični žarki (ISO/DIS
11146-2:2020)
Lasers and laser-related equipment - Test methods for laser beam widths, divergence
angles and beam propagation ratios - Part 2: General astigmatic beams (ISO/DIS 11146-
2:2020)
Laser und Laseranlagen - Prüfverfahren für Laserstrahlabmessungen, Divergenzwinkel
und Beugungsmaßzahlen - Teil 2: Allgemein astigmatische Strahlen (ISO/DIS 11146-
2:2020)
Lasers et équipements associés aux lasers - Méthodes d'essai des largeurs du faisceau,
angles de divergence et facteurs de limite de diffraction - Partie 2: Faisceaux
astigmatiques générau (ISO/DIS 11146-2:2020)
Ta slovenski standard je istoveten z: prEN ISO 11146-2
ICS:
31.260 Optoelektronika, laserska Optoelectronics. Laser
oprema equipment
oSIST prEN ISO 11146-2:2020 en
2003-01.Slovenski inštitut za standardizacijo. Razmnoževanje celote ali delov tega standarda ni dovoljeno.

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oSIST prEN ISO 11146-2:2020

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oSIST prEN ISO 11146-2:2020
DRAFT INTERNATIONAL STANDARD
ISO/DIS 11146-2
ISO/TC 172/SC 9 Secretariat: DIN
Voting begins on: Voting terminates on:
2020-04-21 2020-07-14
Lasers and laser-related equipment — Test methods
for laser beam widths, divergence angles and beam
propagation ratios —
Part 2:
General astigmatic beams
Lasers et équipements associés aux lasers — Méthodes d'essai des largeurs du faisceau, angles de
divergence et facteurs de limite de diffraction —
Partie 2: Faisceaux astigmatiques généraux
ICS: 31.260
THIS DOCUMENT IS A DRAFT CIRCULATED
This document is circulated as received from the committee secretariat.
FOR COMMENT AND APPROVAL. IT IS
THEREFORE SUBJECT TO CHANGE AND MAY
NOT BE REFERRED TO AS AN INTERNATIONAL
STANDARD UNTIL PUBLISHED AS SUCH.
IN ADDITION TO THEIR EVALUATION AS
ISO/CEN PARALLEL PROCESSING
BEING ACCEPTABLE FOR INDUSTRIAL,
TECHNOLOGICAL, COMMERCIAL AND
USER PURPOSES, DRAFT INTERNATIONAL
STANDARDS MAY ON OCCASION HAVE TO
BE CONSIDERED IN THE LIGHT OF THEIR
POTENTIAL TO BECOME STANDARDS TO
WHICH REFERENCE MAY BE MADE IN
Reference number
NATIONAL REGULATIONS.
ISO/DIS 11146-2:2020(E)
RECIPIENTS OF THIS DRAFT ARE INVITED
TO SUBMIT, WITH THEIR COMMENTS,
NOTIFICATION OF ANY RELEVANT PATENT
RIGHTS OF WHICH THEY ARE AWARE AND TO
©
PROVIDE SUPPORTING DOCUMENTATION. ISO 2020

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oSIST prEN ISO 11146-2:2020
ISO/DIS 11146-2:2020(E)

COPYRIGHT PROTECTED DOCUMENT
© ISO 2020
All rights reserved. Unless otherwise specified, or required in the context of its implementation, no part of this publication may
be reproduced or utilized otherwise in any form or by any means, electronic or mechanical, including photocopying, or posting
on the internet or an intranet, without prior written permission. Permission can be requested from either ISO at the address
below or ISO’s member body in the country of the requester.
ISO copyright office
CP 401 • Ch. de Blandonnet 8
CH-1214 Vernier, Geneva
Phone: +41 22 749 01 11
Fax: +41 22 749 09 47
Email: copyright@iso.org
Website: www.iso.org
Published in Switzerland
ii © ISO 2020 – All rights reserved

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oSIST prEN ISO 11146-2:2020
ISO/DIS 11146-2:2020(E)

Contents Page
Foreword .iv
Introduction .v
1 Scope . 1
2 Normative references . 1
3 Terms and definitions . 1
4 Coordinate system . 5
5 Test principles . 5
5.1 General . 5
5.2 Spatial second order moments of the Wigner distribution . 5
5.3 Second order moments of the Wigner distribution . 5
5.4 Derived quantities . 6
6 Measurement arrangement and test equipment . 6
6.1 General . 6
6.2 Preparation . 6
6.3 Control of environment . 6
6.4 Detector system . 6
6.5 Beam-forming optics and optical attenuators . 7
7 Measurement of the second order moments . 7
7.1 General . 7
7.2 Measurement of the second order moments of power density distributions . 7
7.3 Measurement of all second order moments of the Wigner distribution . 9
8 Determination of effective beam propagation ratio .11
9 Determination of intrinsic astigmatism.11
10 Determination of the twist parameter .12
11 Test report .12
Bibliography .15
© ISO 2020 – All rights reserved iii

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oSIST prEN ISO 11146-2:2020
ISO/DIS 11146-2:2020(E)

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.
The procedures used to develop this document and those intended for its further maintenance are
described in the ISO/IEC Directives, Part 1. In particular, the different approval criteria needed for the
different types of ISO documents should be noted. This document was drafted in accordance with the
editorial rules of the ISO/IEC Directives, Part 2 (see www .iso .org/ directives).
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. Details of
any patent rights identified during the development of the document will be in the Introduction and/or
on the ISO list of patent declarations received (see www .iso .org/ patents).
Any trade name used in this document is information given for the convenience of users and does not
constitute an endorsement.
For an explanation of the voluntary nature of standards, the meaning of ISO specific terms and
expressions related to conformity assessment, as well as information about ISO's adherence to the
World Trade Organization (WTO) principles in the Technical Barriers to Trade (TBT), see www .iso .org/
iso/ foreword .html.
This document was prepared by Technical Committee ISO/TC 172, Optics and photonics, Subcommittee
SC 9, Laser and electro-optical systems.
This second edition cancels and replaces the first edition (ISO 11146-2:2005), which has been
technically revised.
The main changes compared to the previous edition are as follows:
— The terms and definitions were harmonized with the new ISO 11145.
A list of all parts in the ISO 11146 series can be found on the ISO website.
Any feedback or questions on this document should be directed to the user’s national standards body. A
complete listing of these bodies can be found at www .iso .org/ members .html.
iv © ISO 2020 – All rights reserved

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oSIST prEN ISO 11146-2:2020
ISO/DIS 11146-2:2020(E)

Introduction
The propagation properties of laser beams can be characterized by ten independent parameters when
applying the method of second order moments (see ISO/TR 11146-3). Most laser beams need few
parameters for a complete description due to their higher symmetry. Lasers emit beams which are
stigmatic or simple astigmatic due to their resonator design.
Part 1 of ISO 11146 describes the measurement methods for stigmatic and simple astigmatic beams
while this document deals with the measurement procedures for general astigmatic beams. This
document is applicable to beams of unknown type. Beam characterization, based on the method
of second order moments as described in Part 1 and this document, is only valid within the paraxial
approximation.
The theoretical description of beam characterization and propagation as well as the classification of
laser beams is given in ISO/TR 11146-3, which is an informative Technical Report. The procedures for
background subtraction and offset correction are also given in ISO/TR 11146-3.
In ISO 11146, the second order moments of the power (energy) density distribution function are used
for the determination of beam widths. If problems are experienced in the direct measurements of these
quantities, other indirect methods of measurement of second order moments may be used as long as
comparable results are achievable.
In ISO/TR 11146-3, three alternative methods for beam width measurement and their correlation with
the method used in this document are described. These methods are:
— variable aperture method;
— moving knife-edge method;
— moving slit method.
The problem of the dependence of the measuring result on the truncation limits of the integration area
was investigated and evaluated by an international interlaboratory experiment carried out in 1997. The
results of this interlaboratory testing were taken into consideration in this document.
© ISO 2020 – All rights reserved v

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oSIST prEN ISO 11146-2:2020

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oSIST prEN ISO 11146-2:2020
DRAFT INTERNATIONAL STANDARD ISO/DIS 11146-2:2020(E)
Lasers and laser-related equipment — Test methods
for laser beam widths, divergence angles and beam
propagation ratios —
Part 2:
General astigmatic beams
1 Scope
This document specifies methods for measuring beam widths (diameter), divergence angles and beam
propagation ratios of laser beams. This document is applicable to general astigmatic beams or unknown
types of beams. For stigmatic and simple astigmatic beams, ISO 11146-1 is applicable.
Within this document, the description of laser beams is accomplished by means of the second
order moments of the Wigner distribution rather than physical quantities such as beam widths and
divergence angles. However these physical quantities are closely related to the second order moments
of the Wigner distribution. In ISO/TR 11146-3, formulae are given to calculate all relevant physical
quantities from the measured second order moments.
2 Normative references
The following documents are referred to in the text in such a way that some or all of their content
constitutes requirements of this document. For dated references, only the edition cited applies. For
undated references, the latest edition of the referenced document (including any amendments) applies.
ISO 11145, Optics and photonics — Lasers and laser-related equipment — Vocabulary and symbols
ISO/DIS 11146-1:2020, Lasers and laser-related equipment — Test methods for laser beam widths,
divergence angles and beam propagation ratios — Part 1: Stigmatic and simple astigmatic beams
EN 61040:1992, Power and energy measuring detectors, instruments and equipment for laser radiation
3 Terms and definitions
For the purposes of this document, the terms and definitions given in ISO 11145, ISO 11146-1,
EN 61040:1992 and the following apply.
ISO and IEC maintain terminological databases for use in standardization at the following addresses:
— ISO Online browsing platform: available at https:// www .iso .org/ obp
— IEC Electropedia: available at http:// www .electropedia .org/
Note 1 to entry The x−, y− and z-axes in the following definitions refer to the laboratory system (as described in
Clause 4). Here and throughout this document the term “power density distribution” refers to continuous wave
sources. It might be replaced by “energy density distribution” in case of pulsed sources.
© ISO 2020 – All rights reserved 1

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oSIST prEN ISO 11146-2:2020
ISO/DIS 11146-2:2020(E)

3.1
generalized beam diameter
d
g
measure of the extent of the power density distribution of a beam in a cross-section at an axial location
z, derived from the centred second order moments by
22
dx=+22 y (1)
g
Note 1 to entry: Note 1to entry: This definition is similar to the beam diameter defined in ISO 11145 or
ISO 11146-1. But in this context the definition is not restricted to circular power density distributions.
3.2
generalized beam waist location
z
0,g
position where the generalized beam diameter reaches its minimum value along the axis of propagation
3.3
generalized Rayleigh length
z
R,g
distance along the beam axis from the generalized beam waist where the generalized beam diameter is
a factor of 2 larger than the generalized beam waist diameter
3.4
Wigner distribution
phase space distribution representing a laser beam in a transverse plane at location z
Note 1 to entry: The Wigner distribution is a function of two spatial and two angular coordinates, giving the
amount of beam power propagating through the point (x,y) in the direction (Θ , Θ ).
x y
3.5
spatial first order moments of the Wigner distribution
xy,
subset of the first order moments, which can be directly obtained from measured power density
distribution by
∞ ∞
Ex(,yz,)xxydd
∫ ∫
−∞ −∞
xz()= (2)
∞ ∞
Ex(,yz,)ddxy
∫ ∫
−∞ −∞
and
∞ ∞
Ex(,yz,)yxddy
∫ ∫
−∞ −∞
yz()= (3)
∞ ∞
Ex(,yz,)ddxy
∫ ∫
−∞ −∞
where E(x,y,z) is the power density distribution at the specific plane z = constant
2 © ISO 2020 – All rights reserved

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oSIST prEN ISO 11146-2:2020
ISO/DIS 11146-2:2020(E)

3.6
second order moments of the Wigner distribution
2 22 2
xy,,, xy ,,,ΘΘ ΘΘ ,,xxΘΘ yyΘΘ,
xy xy xy xy
ten second order moments of the Wigner distribution of the beam at location z
Note 1 to entry: The ten second order moments contain information on the following physical beam properties:
beam size and orientation, divergence angles and their orientation, radii of curvature of the phase paraboloid
and their orientation and the twist parameter. Details on these relations are given in ISO/TR 11146-3.
2 2 2
Note 2 to entry: In ISO 11146-1, the three spatial second order moments are defined as σ , σ and σ . In this
x y xy
document and ISO/TR 11146-3, the angular brackets are used to emphasize the coordinates of the moments. This
22 22 2
means that σ = x , σ = y and σ = xy .
x y xy
2 2
Note 3 to entry: Three angular moments 〈Θ 〉, 〈Θ 〉 and 〈Θ Θ 〉 are independent of z. The other seven second
x y x y
order moments are, in general, functions of z.
3.7
spatial second order moments of the Wigner distribution
22
xy,, xy
subset of the second order moments, which can be directly obtained from measured power density
distribution by
∞ ∞
2
Ex(,yz,)()xx− ddxy
∫ ∫
−∞ −∞
2
xz()= (4)
∞ ∞
Ex(,yz,)ddxy
∫ ∫
−∞ −∞
∞ ∞
2
Ex(,yz,)()yy− ddxy
∫ ∫
−∞ −∞
2
yz()= (5)
∞ ∞
Ex(,yz,)ddxy
∫ ∫
−∞ −∞
and
∞ ∞
Ex(,yz,)()xx−−()yy ddxy
∫ ∫
−∞ −∞
xy ()z = (6)
∞ ∞
Ex(,yz,)ddxy
∫ ∫
−∞ −∞
3.8
beam matrix
P
symmetric and positive definite 4×4 matrix containing all ten second order moments of the Wigner
distribution and its elements and given by
2
 
xxyxΘΘx
xy
 
 
2
xy yyΘΘy
 
xy
 
P = (7)
2
 
xyΘΘ ΘΘ Θ
xx xx y
 
 
2
xyΘΘ ΘΘ Θ
 
yy xy y
 
© ISO 2020 – All rights reserved 3

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oSIST prEN ISO 11146-2:2020
ISO/DIS 11146-2:2020(E)

3.9
effective beam propagation ratio
2
M
eff
invariant quantity related to the focusability of a general astigmatic beam, defined as:
1

2
4
 
M = det P (8)
()
eff
 
λ
where det (P) is the determinant of matrix P
2
Note 1 to entry: The effective beam propagation ratio M is an invariant related to the volume that the beam
eff
occupies in the four-dimensional phase space (two lateral spatial and two lateral angular dimensions) and thus a
measure for the focusability of the beam.
Note 2 to entry: For simple astigmatic beams, the effective beam propagation ratio is the geometric mean of the
22 2 22
beam propagation ratios of the principal axes of the beam: MM=×M . For stigmatic beams MM= .
eff xy eff
3.10
intrinsic astigmatism
a
degree of how close to a stigmatic beam the general astigmatic beam can be transformed by using
lenses and free space propagation
2
2
2
8π  
2  
22 22 2
ax=−ΘΘxy+−ΘΘyx+−2 yxΘΘ ΘΘy − M ≥ 0
 xx  yy  ()xy yx  ()eff
()
2
 
λ  
(9)
Note 1 to entry: Beams are classified according to their intrinsic astigmatism a, which is an invariant quantity. A
beam with a = 0 is called int
...

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