SIST EN ISO 15367-2:2005

Lasers and laser-related equipment - Test methods for determination of the shape of a laser beam wavefront - Part 2: Shack-Hartmann sensors (ISO 15367-2:2005)

This International Standard specifies methods for measurement and evaluation of the wavefront distribution function in a transverse plane of a laser beam utilizing Hartmann or Hartmann-Shack wavefront sensors. The standard applies to fully coherent, partially coherent and general astigmatic laser beams, both for pulsed and continuous operation.
Furthermore, reliable numerical methods for both zonal and modal reconstruction of the two dimensional wavefront distribution together with their uncertainty are described. The knowledge of the wavefront distribution enables the determination of several wavefront parameters which are defined in ISO 15367-1.

Laser und Laseranlagen - Prüfverfahren für die Bestimmung der Wellenfrontform von Laserstrahlen - Teil 2: Shack-Hartmann-Sensoren (ISO 15367-2:2005)

Dieser Teil der ISO 15367 legt die Verfahren für die Messung und Auswertung der Wellenfrontverteilungsfunktion in einer transversalen Ebene eines Laserstrahls unter Verwendung von Hartmann- oder Shack-Hartmann-Wellenfrontsensoren fest. Dieser Teil der ISO 15367 ist für vollständig kohärente, teilweise kohärente und allgemein astigmatische Laserstrahlen sowohl bei Dauerstrichbetrieb als auch bei gepulstem Betrieb anwendbar.
Des Weiteren werden sowohl für die zonale als auch die modale Rekonstruktion der zweidimensionalen Wellenfrontverteilung zuverlässige numerische Verfahren zusammen mit deren Messunsicherheit beschrieben. Die Kenntnis der Wellenfrontverteilung ermöglicht die Bestimmung einer Reihe von Wellenfrontparametern, die in ISO 15367-1 definiert sind.

Lasers et équipements associés aux lasers - Méthodes d'essai pour la détermination de la forme du front d'onde du faisceau laser - Partie 2: Senseurs Shack-Hartmann (ISO 15367-2:2005)

L'ISO 15367-2:2005 spécifie les méthodes pour le mesurage et l'évaluation de la fonction de distribution du front d'onde, dans un plan transversal, d'un faisceau laser par l'utilisation de senseurs de front d'onde Hartmann ou Shack-Hartmann. Elle s'applique aux lasers totalement cohérents, partiellement cohérents et astigmatiques généraux, aussi bien impulsionnels que continus.
En outre, sont décrites des méthodes numériques raccordables pour une reconstruction par mode ou par zone de la distribution bidimensionnelle du front d'onde, en même temps que leur incertitude. La connaissance de la distribution du front d'onde permet la détermination de plusieurs paramètres de front d'onde définis dans l'ISO 15367-1.

Laserji in laserska oprema – Preskusne metode za ugotavljanje oblike valovne fronte laserskega žarka – 2. del: Shack-Hartmannovi senzorji (ISO 15367-2:2005)

General Information

• Status: Published
• Publication Date: 31-May-2005
• ICS: 31.260 - Optoelectronics. Laser equipment
• Technical Committee: VAZ - Healthcare
• Current Stage: 6060 - National Implementation/Publication (Adopted Project)
• Start Date: 01-Jun-2005
• Due Date: 01-Jun-2005
• Completion Date: 01-Jun-2005
• Ref Project: EN ISO 15367-2:2005 - Lasers and laser-related equipment - Test methods for determination of the shape of a laser beam wavefront - Part 2: Shack-Hartmann sensors (ISO 15367-2:2005)

Standards Content (Sample)

SLOVENSKI STANDARD
SIST EN ISO 15367-2:2005
01-junij-2005
Laserji in laserska oprema – Preskusne metode za ugotavljanje oblike valovne
fronte laserskega žarka – 2. del: Shack-Hartmannovi senzorji (ISO 15367-2:2005)
Lasers and laser-related equipment - Test methods for determination of the shape of a
laser beam wavefront - Part 2: Shack-Hartmann sensors (ISO 15367-2:2005)
Laser und Laseranlagen - Prüfverfahren für die Bestimmung der Wellenfrontform von
Laserstrahlen - Teil 2: Shack-Hartmann-Sensoren (ISO 15367-2:2005)
Lasers et équipements associés aux lasers - Méthodes d'essai pour la détermination de
la forme du front d'onde du faisceau laser - Partie 2: Senseurs Shack-Hartmann (ISO
15367-2:2005)
Ta slovenski standard je istoveten z: EN ISO 15367-2:2005
ICS:
31.260 Optoelektronika, laserska Optoelectronics. Laser
oprema equipment
SIST EN ISO 15367-2:2005 en
2003-01.Slovenski inštitut za standardizacijo. Razmnoževanje celote ali delov tega standarda ni dovoljeno.

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SIST EN ISO 15367-2:2005

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SIST EN ISO 15367-2:2005
EUROPEAN STANDARD
EN ISO 15367-2
NORME EUROPÉENNE
EUROPÄISCHE NORM
March 2005
ICS 31.260
English version
Lasers and laser-related equipment - Test methods for
determination of the shape of a laser beam wavefront - Part 2:
Shack-Hartmann sensors (ISO 15367-2:2005)
Lasers et équipements associés aux lasers - Méthodes Laser und Laseranlagen - Prüfverfahren für die
d'essai pour la détermination de la forme du front d'onde du Bestimmung der Wellenfrontform von Laserstrahlen - Teil
faisceau laser - Partie 2: Senseurs Shack-Hartmann (ISO 2: Shack-Hartmann-Sensoren (ISO 15367-2:2005)
15367-2:2005)
This European Standard was approved by CEN on 21 February 2005.
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 Central Secretariat 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 Central Secretariat has the same status as the official
versions.
CEN members are the national standards bodies of Austria, Belgium, Cyprus, Czech Republic, Denmark, Estonia, Finland, France,
Germany, Greece, Hungary, Iceland, Ireland, Italy, Latvia, Lithuania, Luxembourg, Malta, Netherlands, Norway, Poland, Portugal, Slovakia,
Slovenia, Spain, Sweden, Switzerland and United Kingdom.
EUROPEAN COMMITTEE FOR STANDARDIZATION
COMITÉ EUROPÉEN DE NORMALISATION
EUROPÄISCHES KOMITEE FÜR NORMUNG
Management Centre: rue de Stassart, 36  B-1050 Brussels
© 2005 CEN All rights of exploitation in any form and by any means reserved Ref. No. EN ISO 15367-2:2005: E
worldwide for CEN national Members.

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SIST EN ISO 15367-2:2005

EN ISO 15367-2:2005 (E)





Foreword


This document (EN ISO 15367-2:2005) has been prepared by Technical Committee ISO/TC 172
"Optics and optical instruments" in collaboration with Technical Committee CEN/TC 123 "Lasers
and laser-related equipment", 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 September 2005, and conflicting national
standards shall be withdrawn at the latest by September 2005.

According to the CEN/CENELEC Internal Regulations, the national standards organizations of
the following countries are bound to implement this European Standard: Austria, Belgium,
Cyprus, Czech Republic, Denmark, Estonia, Finland, France, Germany, Greece, Hungary,
Iceland, Ireland, Italy, Latvia, Lithuania, Luxembourg, Malta, Netherlands, Norway, Poland,
Portugal, Slovakia, Slovenia, Spain, Sweden, Switzerland and United Kingdom.


Endorsement notice

The text of ISO 15367-2:2005 has been approved by CEN as EN ISO 15367-2:2005 without any
modifications.

2

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SIST EN ISO 15367-2:2005


INTERNATIONAL ISO
STANDARD 15367-2
First edition
2005-03-15


Lasers and laser-related equipment —
Test methods for determination of the
shape of a laser beam wavefront —
Part 2:
Shack-Hartmann sensors
Lasers et équipements associés aux lasers — Méthodes d'essai pour la
détermination de la forme du front d'onde du faisceau laser —
Partie 2: Senseurs Shack-Hartmann




Reference number
ISO 15367-2:2005(E)
©
ISO 2005

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SIST EN ISO 15367-2:2005
ISO 15367-2:2005(E)
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ii © ISO 2005 – All rights reserved

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SIST EN ISO 15367-2:2005
ISO 15367-2:2005(E)
Contents Page
Foreword. iv
Introduction . v
1 Scope. 1
2 Normative references . 1
3 Terms and definitions. 1
4 Symbols and units . 3
5 Test principle of Hartmann and Shack-Hartmann wavefront sensors . 4
6 Measurement arrangement and test procedure. 4
6.1 General. 4
6.2 Detector system . 4
6.3 Measurement . 7
6.4 Calibration. 8
7 Evaluation of wavefront gradients . 9
7.1 Background subtraction. 9
7.2 Evaluation . 9
8 Wavefront reconstruction . 9
8.1 General. 9
8.2 Direct numerical integration (zonal method). 10
8.3 Modal wavefront reconstruction . 10
9 Wavefront representation. 11
10 Uncertainty. 11
10.1 General. 11
10.2 Statistical measurement errors . 11
10.3 Environmental effects. 12
10.4 Deficiencies in data acquisition . 12
10.5 Uncertainties due to geometrical misalignment. 13
11 Test report. 13
Annex A (informative) Wavefront reconstruction. 17
Annex B (informative) Zernike polynomials for representation of wavefronts. 19
Bibliography . 20

© ISO 2005 – All rights reserved iii

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SIST EN ISO 15367-2:2005
ISO 15367-2:2005(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.
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 15367-2 was prepared by Technical Committee ISO/TC 172, Optics and photonics, Subcommittee SC 9,
Electro-optical systems.
ISO 15367 consists of the following parts, under the general title Lasers and laser-related equipment — Test
methods for determination of the shape of a laser beam wavefront:
 Part 1: Terminology and fundamental aspects
 Part 2: Shack-Hartmann sensors
iv © ISO 2005 – All rights reserved

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SIST EN ISO 15367-2:2005
ISO 15367-2:2005(E)
Introduction
Characterization of the beam propagation behaviour is necessary in many areas of both laser system
development and industrial laser applications. For example, the design of resonator or beam delivery optics
strongly relies on detailed and quantitative information over the directional distribution of the emitted radiation.
On-line recording of the laser beam wavefront can also accomplish an optimization of the beam focusability in
combination with adaptive optics. Other relevant areas are the monitoring and possible reduction of thermal
lensing effects, on-line resonator adjustment, laser safety considerations, or “at wavelength” testing of optics
including Zernike analysis.
There are four sets of parameters that are relevant for the laser beam propagation:
 power (energy) density distribution (ISO 13694);
 beam widths, divergence angles and beam propagation ratios (ISO 11146-1 and ISO 11146-2);
 wavefront (phase) distribution (ISO 15367-1 and this part of ISO 15367);
 spatial beam coherence (no current standard available).
In general, a complete characterization requires the knowledge of the mutual coherence function or spectral
density function, at least in one transverse plane. Although the determination of those distributions is possible,
the experimental effort is large and commercial instruments capable of measuring these quantities are still not
available. Hence, the scope of this standard does not extend to such a universal beam description but is
limited to the measurement of the wavefront, which is equivalent to the phase distribution in case of spatially
coherent beams. As a consequence, an exact prediction of beam propagation is achievable only in the limiting
case of high lateral coherence.
A number of phase or wavefront gradient measuring instruments are capable of determining the wavefront or
phase distribution. These include, but are not limited to, the lateral shearing interferometer, the Hartmann and
Shack-Hartmann wavefront sensor, and the Moiré deflectometer. In these instruments, the gradients of either
wavefront or phase are measured, from which the two-dimensional phase distribution can be reconstructed.
In this document, only Hartmann and Shack-Hartmann wavefront sensors are considered in detail, as they are
able to measure the wavefront of both fully coherent and partially coherent beams. A considerable number of
such instruments are commercially available.
The main advantages of the Hartmann technique are
 wide dynamic range,
 high optical efficiency,
 suitability for partially coherent beams,
 no requirement of spectral purity,
 no ambiguity with respect to 2π increment in phase angle,
 wavefronts can be acquired/analysed in a single measurement.
Instruments which are capable of direct phase or wavefront measurement, as, e.g. self-referencing
interferometers, are outside the scope of this part of ISO 15367.
© ISO 2005 – All rights reserved v

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SIST EN ISO 15367-2:2005

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SIST EN ISO 15367-2:2005
INTERNATIONAL STANDARD ISO 15367-2:2005(E)

Lasers and laser-related equipment — Test methods for
determination of the shape of a laser beam wavefront —
Part 2:
Shack-Hartmann sensors
1 Scope
This part of ISO 15367 specifies methods for measurement and evaluation of the wavefront distribution
function in a transverse plane of a laser beam utilizing Hartmann or Shack-Hartmann wavefront sensors. This
part of ISO 15367 is applicable to fully coherent, partially coherent and general astigmatic laser beams, both
for pulsed and continuous operation.
Furthermore, reliable numerical methods for both zonal and modal reconstruction of the two-dimensional
wavefront distribution together with their uncertainty are described. The knowledge of the wavefront
distribution enables the determination of several wavefront parameters that are defined in ISO 15367-1.
2 Normative references
The following referenced documents are indispensable for the application 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 optical instruments — Lasers and laser-related equipment — Vocabulary and symbols
ISO 13694, Optics and optical instruments — Lasers and laser-related equipment — Test methods for laser
beam power (energy) density distribution
ISO 15367-1:2003, Lasers and laser-related equipment — Test methods for determination of the shape of a
laser beam wavefront — Part 1: Terminology and fundamental aspects
3 Terms and definitions
For the purposes of this document, the terms and definitions given in ISO 11145 and ISO 15367-1 as well as
the following apply.
3.1
array element spacing
d , d
x y
distance between the centres of adjacent pinholes or lenslets in x and y direction
3.2
sub-aperture screen to detector spacing
L
H
spacing of the sub-aperture screen (lenslet array or Hartmann screen) to the detector array
NOTE For Shack-Hartmann sensors this is often set to the lenslet focal length.
© ISO 2005 – All rights reserved 1

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SIST EN ISO 15367-2:2005
ISO 15367-2:2005(E)
3.3
lenslet focal length
f
focal length of the lenslets for a Shack-Hartmann sensor
3.4
sub-aperture width
d
s
aperture width of the pinholes of a Hartmann screen or lenslets of a Shack-Hartmann array, respectively
3.5
angular dynamic range
β
max
maximum usable angular range of Hartmann or Shack-Hartmann sensors
NOTE For square apertures, the angular dynamic range is given by
d λ
x
β =−
max
2L d
H x
3.6
wavefront measurement repeatability
w
r,rms
root-mean-square (r.m.s.) difference between single subsequent measurements w (x, y) of the same
n
wavefront and the average wavefront w (x, y)
2
2

  
E()x,,y w(xy)−−w()x,y E()x,y w()x,y w()xy,
nn nn
∑∑ ∑∑
  
k

1
xy xy

w=−
r,rms
∑
k Ex,,yE xy
() ()
∑∑nn∑∑
n = 1

xy xy

where
n is the number of the measurement;
k is the number of samples taken;
k
E (,xy)×w (x,y)
∑nn
n = 1
wx( ,y) =
k
Ex(,y)
∑ n
n = 1
3.7
wavefront measurement accuracy
w
a,rms
average of the r.m.s. difference between a reference wavefront w and the tilt-corrected wavefront w after
r tc,n
various amounts of tilt θ have been applied to the reference wavefront
n
2
 
Ex(,y)w (x,y) −w (,xy)
nntc, r
∑∑  
k
1 xy
w =
a,rms
∑
k Ex(,y)
∑∑ n
n = 1
xy
2 © ISO 2005 – All rights reserved

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SIST EN ISO 15367-2:2005
ISO 15367-2:2005(E)
where
n is the nth measurement of the wavefront with tilt θ and θ applied;
x,n y,n
k is the number of samples taken;
w is the tilt-corrected wavefront as follows:
tc,n
wx(,y) = wx(,y)−−θθx y
nx,,n yn
tc,n
NOTE See also ISO 15367-1:2003, 3.4.7.
4 Symbols and units
Table 1 — Symbols and units
Symbol Parameter Units Defined in
2 2
E(x, y), H(x, y) power (energy) density distribution W/cm , J/cm ISO 13694
x, y, z mechanical axes (Cartesian coordinates) mm ISO 15367-1:2003, 3.1.5
z beam axis mm ISO 15367-1:2003, 3.1.5
λ wavelength nm
z location of measurement plane mm ISO 15367-1:2003, 3.1.4
m
w(x, y) average wavefront shape nm ISO 15367-1:2003, 3.1.1
ISO 15367-1:2003, 3.1.1,
Φ(x, y) phase distribution rad
Note 1
w (x, y) corrected wavefront nm ISO 15367-1:2003, 3.4.2
c
s(x, y) approximating spherical surface — ISO 15367-1:2003, 3.4.3
R defocus or radius of best sphere mm ISO 15367-1:2003, 3.4.5
ss
w (x, y) wavefront aberration function nm ISO 15367-1:2003, 3.4.6
AF
w wavefront irregularity nm
PV
w weighted r.m.s. deformation nm ISO 15367-1:2003, 3.4.7
rms
d , d array element spacing mm 3.1
x y
L sub-aperture screen to detector spacing mm 3.2
H
f lenslet focal length mm 3.3
d spot size µm
p
d sub-aperture width µm 3.4
s
β angular dynamic range mrad 3.5
max
beam centroid coordinates in sub-aperture ij
(x , y ) i.e. the first order moments of the power mm ISO 11146-1
c c
ij
density distribution in sub-aperture ij
(x , y ) reference beam coordinates in sub-aperture ijmm
r r
ij
(β β ) local wavefront gradient components (tilt, tip) — ISO 15367-1:2003, 3.5.1, 3.5.3
x, y ij
w wavefront measurement repeatability nm 3.6
r,rms
w wavefront measurement accuracy nm 3.7
a,rms
geometry matrix in wavefront reconstruction
B —
algorithms
C covariance matrix —
© ISO 2005 – All rights reserved 3

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SIST EN ISO 15367-2:2005
ISO 15367-2:2005(E)
5 Test principle of Hartmann and Shack-Hartmann wavefront sensors
The Hartmann principle is based on a subdivision of the beam into a number of beamlets. This is either
accomplished by an opaque screen with pinholes placed on a regular grid (Hartmann sensor), or by a lenslet
or micro-lens array (Shack-Hartmann sensor), resulting in an average wavefront gradient sampling (see
Figure 1) and a better radiation collection efficiency. The power (energy) density distribution behind the array
is recorded by a position sensitive detector, most commonly a CCD sensor or an array of quadrant detectors
(quadcells). The detector signals can be accumulated by a computerized data acquisition and analysis system.

Key
1 laser
2 attenuator
3 lenslet array
4 position sensitive detector
5 data acquisition and analysis system
Figure 1 — Experimental arrangement for wavefront measurement using Shack-Hartmann technique
The position of the beamlet centroids shall be determined within each sub-aperture, both for the beam under
test and a reference source, preferably a collimated laser beam. The displacements of the centroids with
respect to the reference represent the local wavefront gradients, from which the wavefront w(x, y) is
reconstructed by direct integration or modal fitting techniques (see Clause 8).
The type, manufacturer and model identifier of the instrument used for Hartmann or Shack-Hartmann
wavefront measurement, as well as the array size and the lens/hole spacing, shall be recorded in the test
report.
6 Measurement arrangement and test procedure
6.1 General
Questions concerning different laser types, laser safety, test environment, beam modification (including
sampling/attenuation and beam manipulating optics) as well as general requirements on detectors to be
employed for phase gradient measurements are treated in ISO 15367-1.
All details on the beam sampling and attenuating optics shall be recorded in the test report.
6.2 Detector system
The detector system used for Hartmann and Shack-Hartmann wavefront measurements shall consist of two
elements:
a) a device for segmentation of the beam under test into ray bundles (sub-aperture screen), for example an
array of (refractive or diffractive) lenslets (Shack-Hartmann) or a pinhole array (Hartmann).
4 © ISO 2005 – All rights reserved

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SIST EN ISO 15367-2:2005
ISO 15367-2:2005(E)
b) a position sensitive detector (e.g. a CCD camera) positioned at a distance L behind the segmenting
H
array (L may be set to f in case of Shack-Hartmann detector, or an appropriate correction may be
H
applied).
The detector area shall be partitioned into sub-apertures corresponding to the segmenting array used for
subdivision of the beam. Most commonly, an orthogonal array of lenslets/pinholes with a fixed spacing d , d
x y
(in x-, y-direction, respectively) is employed. In this case the detector array shall be partitioned into N × M
rectangular sub-apertures with a spacing d , d and indexed (ij).
x y
The angular dynamic range of the wavefront sensor with respect to the wavefront variation is directly related
to the ratio of the size of the spots generated on the detector to the size of the sub-apertures. To avoid
overlapping, the spot size shall be smaller than the sub-aperture size. According to the local wavefront
gradient, the spot of a sub-aperture moves towards the border of its assigned region on the detector. If the
spot crosses the border, its position may not be correctly obtained anymore. This effect limits the angular
dynamic range of the sensor.
For Shack-Hartmann sensors, the spot size d is approximately given by
p
λ f
d = 2 (1)
p
d
s
where
f is the focal length of the lenslets;
d is the width of the square lenslet apertures;
s
and where it is assumed that the sub-aperture screen to detector spacing equals the focal length. The
displacement ∆x of a spot due to a horizontal local wavefront gradient β at its corresponding sub-aperture is
x
given by
∆=xfβ × (2)
x
The maximum allowed displacement ∆x to prevent the spot from crossing its assigned region is
max
1
∆=xd()−d (3)
max x p
2
and the according maximum horizontal wavefront gradient
d
λ
x
β =− (4)
x,max
2f d
s
If the size of the lenslet aperture d equals the array element spacing d , the maximum horizontal wavefront
s x
gradient yields
d
λ
x
β =− (5)
x,max
2f d
x
2
Thus, to avoid spot overlap, the focal length of the lenslets is required to be less than d /2λ . To achieve a
x
useful dynamic range and minimize cross talk, the focal length shall be less than 2d /5λ . A smaller focal
x
length will result in a greater angular dynamic range, but may also result in greater measurement uncertainty.
For the vertical direction a similar expression holds.
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SIST EN ISO 15367-2:2005
ISO 15367-2:2005(E)
In the case of round lenslet apertures of diameter d , the maximum wavefront gradient is given by
s
d λ
x
β =−1,22 (6)
x,max
2f d
s
If the size of the lenslet aperture d equals the array element spacing d , the maximum horizontal wavefront
s x
gradient yields
d λ
x
β =−1,22 (7)
x,max
2 f d
x
2
and hence, to achieve a useful dynamic range, the focal length shall be less than d / 2λ .
x
For Hartmann sensors the spot size d is approximately given by
p
λL
H
d = 2 (8)
p
d
s
where
d is the width of the square screen apertures;
s
L is the sub-aperture screen to detector spacing.
H
2
This approximation is only valid for L  d /λ. The displacement ∆x of a spot due to a horizontal local
H x
wavefront gradient β at its corresponding sub-aperture is given by
x
∆=xLβ × (9)
x H
The according maximum horizontal wavefront gradient is
d λ
x
β =− (10)
x,max
2L d
Hs
Thus, to avoid spot overlap the ratio, L / d is required to be less than d / 2λ . To achieve a useful dynamic
Hs x
range and minimize cross talk, the ratio L / d shall be less than 2d /5λ . A smaller ratio will result in a
Hs x
greater angular dynamic range, but may also result in greater measurement uncertainty. For the vertical
direction, a similar expression holds.
In the case of round screen apertures of diameter d , the maximum wavefront gradient is given by
s
d
λ
x
β =−1,22 (11)
x,max
2L d
Hs
and hence, to achieve a useful dynamic range, the ratio L / d shall be less than d / 2λ .
Hs x
NOTE The dynamic range can be extended from this definition by a number of software algorithms. These algorithms
can result from scaling of the sub-aperture grid mapping or other image processing algorithm.
The uncertainty of the measurement is related to the signal-to-noise ratio of the detector and to the number of
detector elements covered by the spots. The uncertainty depends upon the characteristics of the detector
(detector element size and signal-to-noise ratio) and the geometric screen parameters (distance to the
detector, array element spacing, size of sub-apertures and, for Shack-Hartmann sensors, focal length). For
accurate measurement, it is necessary that the lenslet/pinhole spots illuminate at least two detector elements
in each direction.
6 © ISO 2005 – All rights reserved

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SIST EN ISO 15367-2:2005
ISO 15367-2:2005(E)
Since the uncertainty in the measurements is directly related to the signal-to-noise ratio, the dynamic range of
the detector with respect to power (energy) density shall be at least 100:1.
For a proper evaluation of the spot positions, the spatial resolution of the detector shall be at least two times
greater than the spacing of the lenslet or pinhole array d , d .
x y
6.3 Measurement
6.3.1 Alignment
The laser beam to be analysed and the optics employed for beam manipulation shall be adjusted coaxial to
the phase measuring instrument, which is positioned in the measurement plane z .
m
6.3.2 Setting of sub-apertures
While monitoring the spot distribution produced by the lenslet or pinhole array with the help of the
two-dimensional detector array, the spots shall be properly centered with respect to the detector grid. In
particular, each detector sub-area shall contain only one single spot (see Figure 2). Centering of the spot
distributio
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