ISO/TR 11146-3:2004
(Main)Lasers and laser-related equipment — Test methods for laser beam widths, divergence angles and beam propagation ratios — Part 3: Intrinsic and geometrical laser beam classification, propagation and details of test methods
Lasers and laser-related equipment — Test methods for laser beam widths, divergence angles and beam propagation ratios — Part 3: Intrinsic and geometrical laser beam classification, propagation and details of test methods
ISO/TR 11146-3:2004 specifies methods for measuring beam widths (diameter), divergence angles and beam propagation ratios of laser beams in support of ISO 11146-1. It provides the theoretical description of laser beam characterization based on the second-order moments of the Wigner distribution, including geometrical and intrinsic beam characterization, and offers important details for proper background subtraction methods recommendable for matrix detectors such as CCD cameras. It also presents alternative methods for the characterization of stigmatic or simple astigmatic beams that are applicable where matrix detectors are unavailable or deliver unsatisfying results.
Lasers et équipements associés aux lasers — Méthodes d'essai des largeurs du faisceau, des angles de divergence et des facteurs de limite de diffraction — Partie 3: Classification intrinsèque et géométrique du faisceau laser, propagation et détails des méthodes d'essai
General Information
Standards Content (Sample)
TECHNICAL ISO/TR
REPORT 11146-3
First edition
2004-02-01
Lasers and laser-related equipment —
Test methods for laser beam widths,
divergence angles and beam propagation
ratios —
Part 3:
Intrinsic and geometrical laser beam
classification, propagation and details of
test methods
Lasers et équipements associés aux lasers — Méthodes d'essai des
largeurs du faisceau, des angles de divergence et des facteurs de
propagation du faisceau —
Partie 3: Classification intrinsèque et géométrique du faisceau laser,
propagation et détails des méthodes d'essai
Reference number
ISO/TR 11146-3:2004(E)
©
ISO 2004
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ISO/TR 11146-3:2004(E)
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ISO/TR 11146-3:2004(E)
Contents Page
Foreword. iv
Introduction . v
1 Scope. 1
2 Second-order laser beam characterization .1
2.1 General. 1
2.2 Wigner distribution . 1
2.3 First- and second-order moments of Wigner distribution. 2
2.4 Beam matrix. 3
2.5 Propagation though aberration-free optical systems . 4
2.6 Relation between second-order moments and physical beam quantities. 4
2.7 Propagation invariants . 8
2.8 Geometrical classification. 9
2.9 Intrinsic classification . 9
3 Background and offset correction . 10
3.1 General. 10
3.2 Coarse correction by background map subtraction . 10
3.3 Coarse correction by average background subtraction. 11
3.4 Fine correction of baseline offset . 11
4 Alternative methods for beam width measurements . 13
4.1 General. 13
4.2 Variable aperture method. 14
4.3 Moving knife-edge method. 16
4.4 Moving slit method . 17
Annex A (informative) Optical system matrices. 20
Bibliography . 22
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ISO/TR 11146-3:2004(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.
International Standards are drafted in accordance with the rules given in the ISO/IEC Directives, Part 2.
The main task of technical committees is to prepare International Standards. Draft International Standards
adopted by the technical committees are circulated to the member bodies for voting. Publication as an
International Standard requires approval by at least 75 % of the member bodies casting a vote.
In exceptional circumstances, when a technical committee has collected data of a different kind from that
which is normally published as an International Standard (“state of the art”, for example), it may decide by a
simple majority vote of its participating members to publish a Technical Report. A Technical Report is entirely
informative in nature and does not have to be reviewed until the data it provides are considered to be no
longer valid or useful.
Attention is drawn to the possibility that some of the elements of this document may be the subject of patent
rights. ISO shall not be held responsible for identifying any or all such patent rights.
ISO/TR 11146-3 was prepared by Technical Committee ISO/TC 172, Optics and photonics, Subcommittee
SC 9, Electro-optical systems.
This first edition of ISO/TR 11146-3, together with ISO 11146-1, cancels and replaces ISO 11146:1999, which
has been technically revised.
ISO 11146 consists of the following parts, under the general title Lasers and laser-related equipment — Test
methods for laser beam widths, divergence angles and beam propagation ratios:
Part 1: Stigmatic and simple astigmatic beams
Part 2: General astigmatic beams
Part 3: Intrinsic and geometrical laser beam classification, propagation and details of test methods
(Technical Report)
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ISO/TR 11146-3:2004(E)
Introduction
The propagation properties of every laser beam can be characterized within the method of second-order
moments by ten independent parameters. However, most laser beams of practical interest need less
parameters for a complete description due to their higher symmetry. These beams are stigmatic or simple
astigmatic, e.g. due to the used resonator design.
The theoretical description of beam characterization and propagation as well as the classification of laser
beams based on the second-order moments of the Wigner distribution is given in this part of ISO 11146.
The measurement procedures introduced in ISO 11146-1 and ISO 11146-2 are essentially based on (but not
restricted to) the acquisition of power (energy) density distributions by means of matrix detectors, as for
example CCD cameras. The accuracy of results based on these data depends strongly on proper data
pre-processing, namely background subtraction and offset correction. The details of these procedures are
given here.
In some situations accuracy obtainable with matrix detectors might not be satisfying or matrix detectors might
simply be unavailable. In such cases, other, indirect methods for the determination of beam diameters or
beam width are viable alternatives, as long as comparable results are achieved. Some alternative
measurement methods are presented here.
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TECHNICAL REPORT ISO/TR 11146-3:2004(E)
Lasers and laser-related equipment — Test methods for
laser beam widths, divergence angles and beam
propagation ratios —
Part 3:
Intrinsic and geometrical laser beam classification, propagation
and details of test methods
1 Scope
This part of ISO 11146 specifies methods for measuring beam widths (diameter), divergence angles and
beam propagation ratios of laser beams in support of ISO 11146-1. It provides the theoretical description of
laser beam characterization based on the second-order moments of the Wigner distribution, including
geometrical and intrinsic beam characterization, and offers important details for proper background
subtraction methods recommendable for matrix detectors such as CCD cameras. It also presents alternative
methods for the characterization of stigmatic or simple astigmatic beams that are applicable where matrix
detectors are unavailable or deliver unsatisfying results.
2 Second-order laser beam characterization
2.1 General
Almost any coherent or partially coherent laser beam can be characterized by a maximum of ten independent
parameters, the so-called second-order moments of the Wigner distribution. Laser beams showing some kind
of symmetry, stigmatism or simple astigmatism, need even fewer parameters. The knowledge of these
parameters allows the prediction of beam properties behind arbitrary aberration-free optical systems.
Here and throughout this document the term “power density distribution E(x,y,z)” refers to continuous wave
sources. It might be replaced by “energy density distribution H(x,y,z)” in the case of pulsed sources.
Furthermore, a coordinate system is assumed where the z axis is almost parallel to the direction of beam
propagation and the x and y axes are horizontal and vertical, respectively.
2.2 Wigner distribution
The Wigner distribution h(x,y,Θ ,Θ ;z) is a general and complete description of narrow-band coherent and
x y
partially coherent laser beams in a measurement plane. Generally speaking, it gives the amount of beam
power of a beam passing the measurement plane at the lateral position (x,y) with a horizontal paraxial angle of
Θ and a vertical paraxial angle of Θ to the z axis, as shown in Figure 1.
x y
NOTE The Wigner distribution is a function of the axial location z, i.e. the Wigner distribution of the same beam is
different at different z locations. Hence, quantities derived from the Wigner distribution are in general also functions of z.
Throughout this document this z dependence will be dropped. The Wigner distribution then refers to an arbitrarily chosen
location z, the measurement plane.
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ISO/TR 11146-3:2004(E)
x,y spatial coordinates
Θ , Θ corresponding angular coordinates
x y
Figure 1 — Coordinates of Wigner distribution
The power density distribution E(x,y) in a measurement plane is related to the Wigner distribution by
∞
Ex,,y = h xy,Θ,ΘΘddΘ (1)
()
()
x yx y
∫
−∞
NOTE The integration limits in the equation above are finite, representing the maximum angles of the rays contained
in the beam, in paraxial; they are conventionally extended to infinity.
2.3 First- and second-order moments of Wigner distribution
The first-order moments of the Wigner distribution are defined as
1
xh= x,,y Θ ,ΘΘxdxdyd dΘ
(2)
()
xyxy
∫
P
1
yh= x,,y Θ ,ΘΘydxdyd dΘ (3)
()
xyxy
∫
P
1
Θ = hx,,yΘΘ, Θ dxdydΘ dΘ (4)
x ()xy x x y
∫
P
1
Θ = hx,,yΘΘ, Θ dxdydΘ dΘ (5)
()
yxyy xy
∫
P
where P is the beam power given by
Ph= x,,y Θ ,ΘΘdxdyd dΘ (6)
()
xyxy
∫
or, using Equation (1),
PE= x,dy xdy (7)
( )
∫
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The spatial moments 〈x〉 and 〈y〉 give the lateral position of the beam centroid in the measurement plane. The
angular moments 〈Θ 〉 and 〈Θ 〉 specify the direction of propagation of the beam centroid.
x y
The (centred) second-order moments are given by
∞
n
1 kmA
kmA n
xy ΘΘΘ=−h x,,y ,Θ x x y− yΘ−ΘΘ−Θ dxdydΘ dΘ (8)
()()()()
xyxy xx()yy xy
∫
P
−∞
where km,A, and n are non-negative integers and km+A++n= 2 . Therefore, there are ten different second-
order moments.
22
The three spatial second-order moments xy, and xy are related to the lateral extent of the power
22
density distribution in the measurement plane, the three angular momentsΘΘ, and ΘΘ to the
x yxy
beam divergence, and the four mixed moments xxΘΘ, ,yΘ and yΘ to the phase properties in
x yx y
the measurement plane. More details on the relation between the ten second-order moments and the physical
beam properties are discussed below.
The spatial first- and second-order moments can be directly obtained from the power density distribution E(x,y).
From Equation (1) it follows:
1
x = E x,dy xxyd (9)
()
∫
P
1
y = Ex,dy y xdy (10)
()
∫
P
and
2
1
2
xE=−x,dyxx xdy
() (11)
()
∫
P
1
xy=−E()x,dy x x y− y xdy (12)
()()
∫
P
1 2
2
y=−Ex,dy y y xdy (13)
()()
∫
P
The other second-order moments are obtained by measuring the spatial moments in other planes and using
the propagation law of the second-order moments (see below).
NOTE The details of measuring all ten second-order moments are given in ISO 11146-2.
2.4 Beam matrix
The ten second-order moments are collected into the symmetric 4 × 4 beam matrix
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ISO/TR 11146-3:2004(E)
2
xxy xΘΘx
xy
2
xy y yΘΘy
WM xy
P== (14)
T
2
MU
xyΘΘ Θ ΘΘ
xxx xy
2
xyΘΘ ΘΘ Θ
yy xy x
with the symmetric submatrix of the spatial moments
2
xxy
(15)
W =
2
xy y
the symmetric submatrix of the angular moments
2
ΘΘΘ
xyy
U = (16)
2
ΘΘ Θ
yy y
and the submatrix of the mixed moments
xxΘΘ
xy
M = (17)
yyΘΘ
xy
2.5 Propagation though aberration-free optical systems
Aberration-free optical systems are represented by 4 × 4 system matrices S known from geometrical optics.
The propagation of the second-order moments through such a system is given by
T
PS=⋅P ⋅S (18)
out in
where P and P are the beam matrices in entry and exit plane of the optical system, respectively.
in out
Examples for system matrices are given in Annex A.
2.6 Relation between second-order moments and physical beam quantities
The ten second-order moments are closely related to well known physical quantities of a beam.
The three spatial moments describe the lateral extent of the power density distribution of the beam in the
measurement plane. The directions of minimum and maximum extent, called principal axes, are always
orthogonal to each other. Any power density distribution is characterized by the extents along its principal
axes and the orientation of those axes. The beam width along the direction of the principal axis that is closer
to the x-axis of the laboratory system is given by
1
1
2
2 2
2
22 2 2
dx=+22 y+ γx−y+4xy (19)
()
σ x
( ) ( )
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and the beam width along the direction of that principal axis, which is closer to the y-axis by
1
1
2
2 2
2
22 2 2
dx=+22 y− γx−y+4xy (20)
()
σ y ( ) ( )
where
22
xy−
22
γ=−sgnxy= (21)
( )
22
xy−
2 2
If the principal axes make the angle + or − π/4 with x- or y-axis, when 〈x 〉 = 〈y 〉, then d is by convention the
σx
larger of the two beam widths, and
1
2
22
dx=+22 y+2xy (22)
σ x
{}( )
1
2
22
dx=+22 y−2xy (23)
σ y
{}( )
The azimuthal angle between that principal axis, which is closer to the x-axis, and the x-axis is obtained by
1
22
ϕ=−arctan 2 xy x y (24)
( )
2
22 22
valid for xy≠ ; for xy= , ϕ(z) is obtained as
π
ϕ()zx=⋅sgny (25)
()
4
where
xy
sgn xy = (26)
()
xy
See Figure 2.
Figure 2 — Azimuthal angle and beam widths along principal axes of power density distribution
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Very similar, the three angular moments describe the beam divergence characterized by the orthogonal
directions of its maximum and minimum extent. These directions are called the principal axes of the beam
divergence and may not coincide with the principal axes of the power density distribution in the measurement
plane. The beam divergence along the direction of that principal axis, which is closer to the x-axis of the
laboratory system is given by
1
1
2
2 2
2
22 2 2
ΘΘ=+22 Θ+τΘ−Θ+4ΘΘ (27)
()
σxx( y ) (x y ) xy
and the beam divergence along the direction of that principal axis, which is closer to the y-axis by
1
1
2
2 2
2
22 2 2
(28)
ΘΘ=+22 Θ−τΘ−Θ+4ΘΘ
σyx y ) x y ) ()xy
( (
where
22
ΘΘ−
x y
22
τΘ=−sgn Θ= (29)
(xy )
22
ΘΘ−
x y
If the principal axes of the beam divergence make the angle + or − π/4 with x- or y-axis, when 〈Θ 〉 = 〈Θ 〉,
x y
then Θ is by convention the larger of the two beam divergences, and
σx
1
2
22
ΘΘ=+22 Θ+2ΘΘ (30)
σxx( y ) xy
{}
1
2
22
ΘΘ=+22 Θ−2ΘΘ (31)
σyx{}( y ) xy
The divergence azimuthal angle between that principal axis, which is closer to the x-axis, and the x-axis is
obtained by
1
22
ϕΘ=−arctan 2ΘΘΘ (32)
Θ xy ( x y )
2
22 22
valid for ΘΘ≠ ; for ΘΘ= , ϕ (z) is obtained as
x y x y Θ
π
ϕΘz=⋅sgnΘ (33)
()
()
Θ xy
4
where
ΘΘ
x y
sgn ΘΘ = (34)
()xy
ΘΘ
x y
The four mixed moments are related to the average phase properties of the beam in the measurement plane.
The best-fitting phase paraboloid is characterized by the orthogonal directions of maximum and minimum
curvature. These curvatures can take also negative or zero values, independently, along the principal axes.
The directions of maximum and minimum curvature, called the principal axes of the phase paraboloid, may
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ISO/TR 11146-3:2004(E)
not coincide with the principal axes of the power density distribution in the measurement plane nor with the
principal axes of the beam divergence. To retrieve the azimuthal angle of the these principal axes and the
radii of curvature of the average phase front along them, the curvature matrix C has to be calculated by
CC −1
xx xy
TT
CA==−A⋅B−B⋅A−B (35)
()( )
CC
xy yy
where
01
−1
(36)
AW=⋅
−10
and
−1
BM=⋅W (37)
The radius of curvature of the average phase front along the direction of that principal axis, which is closer to
the x-axis of the laboratory system, is given by
2
R =− (38)
x
2
2
CC++ µ C−C + 4C
()( )
xxyy xx yy xy
and, similarly, the radius of curvature of the average phase front along the direction of that principal axis,
which is closer to the y-axis by
2
R =− (39)
y
2
2
CC+− µ C−C + 4C
()( )
xxyy xx yy xy
where
CC−
xxyy
µ=−sgnCC= (40)
()
xx yy
CC−
xxyy
If the principal axes of the average phase curvature make the angle + or − π/4 with the x- or y-axis, i.e. when
CC= , then R is by convention the larger of the two principal radii (including the sign), and
xx yy x
2
(41)
R =−
x
CC++ 2C
()
xxyy xy
2
R =− (42)
y
CC+− 2C
()
xxyy xy
The phase curvature azimuthal angle between that principal axis, which is closer to the x-axis, and the x-axis
is obtained by
1
ϕ=−arctan 2CC C (43)
()
P xy xx yy
2
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ISO/TR 11146-3:2004(E)
valid for CC≠ ; for CC= the phase curvature azimuthal angle is obtained by
xx yy xx yy
π
ϕ=⋅sgn C (44)
()
Pxy
4
where
C
xy
sgn C = (45)
()
xy
C
xy
Another physical beam parameter related to the phase properties is the twist parameter t defined as
tx=−Θ yΘ
yx
The twist parameter is proportional to the orbital angular momentum of the beam and is invariant under
propagation through stigmatic optical systems.
2.7 Propagation invariants
From the propagation law of the beam matrix the invariance of the following two independent quantities results.
The effective beam propagation ratio is defined as
1
4π
2
4
M = det P W1 (46)
()
eff
λ
and the intrinsic astigmatism as
2
2
8π
2
22 22 22
ax=−ΘΘx+y Θ−yΘ+2(xyΘΘ−xΘyΘ−M) W0
xxy y xy y x )
() (
eff
2
λ
(47)
The effective beam propagation ratio is related to the focusability of a beam.
NOTE More general beam propagation ratio is a measure for the overall beam spread, or overall near- and far-field
localization.
For a stigmatic beam (see below) the effective beam propagation ratio equals the beam propagation ratio:
22
M = M (48)
eff
22
For simple astigmatic beams (see below) having M ≠ M , the effective beam propagation ratio equals the
xy
geometric mean of the beam propagation ratios along both principal axes of the beam:
222
MM= M (49)
eff x y
The intrinsic astigmatism is related to the visible and hidden astigmatism of a beam. For a stigmatic beam the
intrinsic astigmatism a vanishes. For a simple astigmatic beam the intrinsic astigmatism a is given by
2
1
22
aM=−M (50)
()xy
2
These quantities are invariant under propagation in lossless and aberration-free optical systems only. In other
systems they may vary. Any combinations of these invariants are also invariant.
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2.8 Geometrical classification
Beams may be classified according to their propagation behaviour in stigmatic or simple astigmatic optical
systems. A stigmatic system is an optical system that can be realized using ideal spherical lenses only. A
simple astigmatic system is an optical system that can be realized using ideal cylindrical lenses all having the
same orientation.
Geometrical beam classification refers to the symmetry of the power density distributions a beam takes on
under propagation. The term “symmetry” is meant in the sense of the second-order moments. A power density
distribution is classified as circular if the ratio of the minimum beam width to the maximum beam width, both
measured along the principal axes, is greater than 0,87. Otherwise it is classified as elliptical.
NOTE In this sense even flat top power density distribution with a square shaped footprint is classified as circular.
Elliptical power density distributions are characterized by their orientation, specified by the azimuthal angle ϕ.
Under free space propagation the power density distributions of a beam may all be circular, all be elliptical or
some may be circular and some elliptical.
A beam is called stigmatic if all power density distributions under free propagation are circular and, in addition,
if the beam were to pass through a cylindrical lens of arbitrary orientation, all of the elliptical power density
distributions behind the lens have the same or orthogonal orientation as the axis parallel to the cylindrical
surface of the lens.
A beam is called simple astigmatic if all elliptical power density distributions have the same (or orthogonal)
azimuthal orientation under free space propagation and, in addition, were the beam to pass through a
cylindrical lens of the same orientation as the beam, all of the elliptical power density distributions behind the
lens would have the same or orthogonal orientation as the cylindrical lens.
All other beams are classified as general astigmatic.
Geometrical beam classification can be derived from the beam matrix. If all three submatrices W, M, and U
are approximately proportional to the identity matrix, the beam is stigmatic. A beam is simple astigmatic and
aligned with the axes x and y if all its submatrices W, M, and U are diagonal. A beam is simple astigmatic, but
rotated with an azimuthal angle ϕ, if the submatrix M is approximately symmetric, i.e. the twist parameter t
vanishes, and the principal axes of the power density distribution in the measurement plane, the principal axes
of the beam divergences and those of the phase paraboloid approximately do coincide:
ϕ≈≈ϕϕ (51)
Θ P
All other beams are geometrically classified as general astigmatic.
2.9 Intrinsic classification
The geometrical classification of a beam may change after passing an optical system. But it can be shown
that not any beam can be transformed into a stigmatic one. The possibility of a beam being transformed into a
stigmatic beam is called intrinsic stigmatism. Hence, a simple astigmatic or even general astigmatic beam
may be classified as intrinsic stigmatic.
A beam is classified as intrinsic stigmatic if
a
< 0,039
2
2
M
()eff
(52)
NOTE The threshold of 0,039 is in coincidence with the threshold value of 0,87 for the circularity of power density
distributions. It ensures that a geometrically stigmatic beam is also intrinsic stigmatic.
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ISO/TR 11146-3:2004(E)
3 Background and offset correction
3.1 General
Signals recorded as the measured power density distribution E (x,y) or H (x,y) can be divided into two
meas meas
parts: the “true” power density distribution E(x,y) or H(x,y), generated by the beam under test, and a possibly
inhomogeneous background map E (x,y), generated by other sources such as external or ambient radiation or
B
by the sensor device itself (noise):
E (,xy)=+E(x,y) E (x,y)
meas B
(53)
where the background signals can be further divided into a homogeneous part E (Baseline offset), an
B,offset
inhomogeneous part E (x,y) (for example baseline tilt) and the high-frequency noise components
B,inh
E (x,y).
B,noise
E (,xy)=+E (x,y) E (,x y)+E (x,y)
B B,offset B,inh B,noise
(54)
NOTE Usually, neither can the high-frequency noise components be corrected nor is it necessary to do so. Due to
the integrations involved in calculating beam parameters the high-frequency noise components determine the intrinsic
statistical errors and therefore affect only the reproducibility of the measurements, whereas the other background signals
cause systematic errors.
The background distribution can be characterized by its mean value (E ) and its standard deviation (E ).
B,offset B,σ
If the variations of the background signal across the detector, which can be characterized by the differences of
local mean values to the overall mean value, are smaller than the standard deviation E the detector
B,σ
background can be considered as homogeneous (cf. 3.4).
Before evaluating the beam parameters, background correction procedures have to be applied to prevent
background signals in the wings of the distribution from dominating the integrals involved. In a first step a
coarse correction has to be carried out by subtracting either a background map or an average background
from the measured power density distribution. For detection systems having a constant background level
across the full area of the sensor, average background level subtraction correction can be used. In all other
cases the subtraction of the complete background map is necessary.
The subtraction of a background map or an average background determined from a background map (see 3.2
and 3.3) may provide offset errors of less than 0,1 digits in most cases, but do not always result in a baseline
offset of zero. Due to the statistical nature of the background noise (the baseline offset is defined as the
average of all non-illuminated pixels), to fluctuations of ambient radiation sources, or to scattered light or other
non-coherent light emissions caused by the laser (for example fluorescence and/or residual pump light), the
baseline offset can only be determined exactly from the measured power density distribution. Even small
baseline offsets can create large errors in the evaluation of parameters characterizing the measured power
density distribution. Therefore, especially for small beams (beam widths of less th
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Questions, Comments and Discussion
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