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)

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.

---------------------- Page: 1 ----------------------
SIST EN ISO 11146-2:2021
---------------------- Page: 2 ----------------------
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

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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.
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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

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
dx=+22 y (1)

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
0,g

position where the generalized beam diameter (3.1) reaches its minimum value along the axis of

propagation
3.3
generalized Rayleigh length
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
xy,, xy

subset of the second order moments, which can be directly obtained from measured power density

distribution by
∞ ∞
Ε ()xy,, zx d()− xx dy
∫ ∫
−∞−∞
xz () = (4)
∞ ∞
Ε ()xy,, zxd dy
∫ ∫
−−∞−∞
∞ ∞
Ε ()xy,, zy d()− yx dy
∫ ∫
−∞−∞
yz () = (5)
∞ ∞
Ε ()xy,, zxd dy
∫ ∫
−−∞−∞
and
∞ ∞
Ε ()xy,, zx()−−xy()yxd dy
∫ ∫
−∞−∞
xy ()z = (6)
∞ ∞
Ε ()xy,, zxd dy
∫ ∫
−∞−∞∞
3.8
beam matrix

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
 
xxyxΘΘx
 
 
xy yyΘΘy
 
P= (7)
 
xyΘΘ ΘΘ Θ
 
xx xx y
 
 
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
eff

invariant quantity related to the focusability of a general astigmatic beam, defined as:

M = []det()P 4 (8)
eff
where det (P) is the determinant of matrix P

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 eff
3.10
intrinsic astigmatism

degree of how close to a stigmatic beam the general astigmatic beam can be transformed by using

lenses and free space propagation
22 22 2
 
ax=−ΘΘxy+−ΘΘyx+−2 yxΘΘ ΘΘyM− ≥0
() ()
xx ( yy ) xy yx efff
2  
 
(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

beams aM=()12 −M . More details are given in ISO/TR 11146-3.
3.11
twist parameter

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)

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

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

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

acquired behind a cylindrical lens, see 7.3.
5.4 Derived quantities

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
---------------------- Page: 18 ----------------------
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.

---------------------- Page: 1 ----------------------
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
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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
---------------------- Page: 3 ----------------------
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

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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

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
dx=+22 y (1)

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
0,g

position where the generalized beam diameter reaches its minimum value along the axis of propagation

3.3
generalized Rayleigh length
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
xy,, xy

subset of the second order moments, which can be directly obtained from measured power density

distribution by
∞ ∞
Ex(,yz,)()xx− ddxy
∫ ∫
−∞ −∞
xz()= (4)
∞ ∞
Ex(,yz,)ddxy
∫ ∫
−∞ −∞
∞ ∞
Ex(,yz,)()yy− ddxy
∫ ∫
−∞ −∞
yz()= (5)
∞ ∞
Ex(,yz,)ddxy
∫ ∫
−∞ −∞
and
∞ ∞
Ex(,yz,)()xx−−()yy ddxy
∫ ∫
−∞ −∞
xy ()z = (6)
∞ ∞
Ex(,yz,)ddxy
∫ ∫
−∞ −∞
3.8
beam matrix

symmetric and positive definite 4×4 matrix containing all ten second order moments of the Wigner

distribution and its elements and given by
 
xxyxΘΘx
 
 
xy yyΘΘy
 
 
P = (7)
 
xyΘΘ ΘΘ Θ
xx xx y
 
 
xyΘΘ ΘΘ Θ
 
yy xy y
 
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oSIST prEN ISO 11146-2:2020
ISO/DIS 11146-2:2020(E)
3.9
effective beam propagation ratio
eff

invariant quantity related to the focusability of a general astigmatic beam, defined as:

 
M = det P (8)
eff
 
where det (P) is the determinant of matrix P

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

degree of how close to a stigmatic beam the general astigmatic beam can be transformed by using

lenses and free space propagation
8π  
2  
22 22 2
ax=−ΘΘxy+−ΘΘyx+−2 yxΘΘ ΘΘy − M ≥ 0
 xx  yy  ()xy yx  ()eff
 
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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