ISO 11421:2025

International
Standard
ISO 11421
Second edition
Optics and photonics — Uncertainty
2025-11
of optical transfer function (OTF)
measurement
Optique et photonique — Incertitude de mesurage de la fonction
de transfert optique (OTF)
Reference number
© ISO 2025
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ii
Contents Page
Foreword .v
Introduction .vi
1 Scope . 1
2 Normative references . 1
3 Terms, definitions and symbols . 1
3.1 Terms and definitions .1
3.2 Symbols .2
4 Sources of uncertainty in measuring equipment. 3
4.1 General .3
4.2 Geometry of optical bench system .4
4.2.1 General .4
4.2.2 Finite object and image distance .4
4.2.3 Infinite object distance and finite image distance .5
4.2.4 Infinite object and image distance .6
4.2.5 Suppression of image distance errors by refocusing .6
4.2.6 Mounting of test piece .6
4.3 Azimuth changing .7
4.3.1 General .7
4.3.2 Finite object and image distance .7
4.3.3 Infinite object distance and finite image distance .7
4.3.4 Infinite object and image distance .7
4.3.5 Suppression of image distance errors by refocusing .7
4.4 Alignment (orientation) of TTU and image analyser .7
4.5 Correction factors .8
4.5.1 General .8
4.5.2 Slit width errors .9
4.5.3 Correction for MTF of incoherently coupled relay lenses .9
4.5.4 Spatial frequency correction for field angle .9
4.5.5 Off-axis magnification errors due to image distortion using grating objects .10
4.6 Image distance error .10
4.7 Spatial frequency errors .11
4.8 Residual aberrations in relay optics . .11
4.9 Spectral characteristics.11
4.10 Extent of test target and/or scan and/or camera detector .11
4.11 Angular response characteristics of image analyser . 12
4.12 Polar luminance/radiation characteristics of object generator . 12
4.13 Signal and data processing . 12
4.14 Stray radiation . 12
4.15 Coherent radiation . 12
4.16 Baseline error . 12
4.17 Linearity of camera detector . 13
5 Methods of assessing measurement errors .13
5.1 General . 13
5.2 Geometry of optical bench system . 13
5.2.1 Straightness of slideways . 13
5.2.2 Parallelism of surfaces and/or perpendicularity to reference axes .16
5.2.3 Errors of rotation angles .17
5.3 Collimation error (departure from infinite object distance) .18
5.4 Image distance setting . 20
5.5 Spectral characteristics.21
5.6 Extent of target and/or scan and/or camera detector . 22
5.7 Signal and data processing . 22
5.8 Polar response to image analyser. 22

iii
6 Calculation of overall uncertainty of a measurement .23
7 Specifying a general equipment uncertainty .24
7.1 General .24
7.2 Nominal uncertainty value (NUV) .24
7.3 Standard-lens measurements (SLM) . 25
7.4 Audit-lens measurements (ALM) . 25
7.5 Slit aperture test (SAT) . 26
8 Routine performance evaluation .26
Annex A (normative) Uncertainty of PTF measurement.27
Annex B (informative) Determination of rate of change of MTF with various parameters .29
Annex C (informative) Example calculation of NUV .32
Bibliography .44

iv
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 document 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).
ISO draws attention to the possibility that the implementation of this document may involve the use of (a)
patent(s). ISO takes no position concerning the evidence, validity or applicability of any claimed patent
rights in respect thereof. As of the date of publication of this document, ISO had not received notice of (a)
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www.iso.org/patents. ISO shall not be held responsible for identifying any or all such patent rights.
Any trade name used in this document is information given for the convenience of users and does not
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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 1,
Fundamental standards.
This second edition cancels and replaces the first edition (ISO 11421:1997), which has been technically
revised.
The main changes are as follows:
— sagittal and tangential OTF were defined;
— symbols, formulae and nomenclature have been revised
— off-axis magnification errors due to image distortion using grating objects has been newly added;
— the document has been revised to be in agreement with the terms and definitions of ISO/IEC Guide 98
(GUM) and ISO/IEC Guide 99 (VIM) regarding the expression of measurement uncertainties;
— Explanations for the calculation of measurement uncertainties have been added;
— Annex C was revised.
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.

v
Introduction
The optical transfer function (OTF) is one of the main criteria used for objectively evaluating the image-
forming capability of optical, electro-optical and photographic systems.
The terms used in the measurement of OTF are defined in ISO 9334, whilst ISO 9335 covers the actual
principles and procedures of measurement. A further International Standard, ISO 9336 (all parts), deals
with specific applications in various optical and electro-optical fields and is in several parts, each dealing
with a particular application.
Although ISO 9335 lists the main factors which influence the uncertainty of OTF measurement and describes
procedures which are aimed at achieving accurate and repeatable results, it does not cover in detail the
techniques and procedures for evaluating the uncertainty of OTF measuring equipment and for estimating
the uncertainty in measurements made on specific imaging systems.
The present document lists the main sources of uncertainty in OTF measuring equipment and provides
guidance on how these can be assessed and how the results of these assessments can be used in estimating
the uncertainty in any measurement of OTF. One of the aims in preparing this document is to encourage the
setting of more realistic uncertainty levels for the results of OTF measurements. Another is to encourage
the use of methods of expressing the uncertainty of OTF test equipment which recognize the fact that the
uncertainty of a particular measurement is a function of both the equipment and the test piece.

vi
International Standard ISO 11421:2025(en)
Optics and photonics — Uncertainty of optical transfer
function (OTF) measurement
1 Scope
This document gives general guidance on evaluating the sources of error in optical transfer function
(OTF) equipment and in using this information to estimate errors in a measurement of OTF. It also gives
guidance on assessing and specifying a general uncertainty for a specific measuring equipment, as well as
recommending methods of routine assessment.
The main body of this document deals exclusively with the modulation transfer function (MTF) part of the
OTF. The phase transfer function (PTF) is dealt with relatively briefly in Annex A.
2 Normative references
There are no normative references in this document.
3 Terms, definitions and symbols
For the purposes of this document, the following terms and definitions apply.
ISO and IEC maintain terminology 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/
3.1 Terms and definitions
3.1.1
standard lens
single- or multi-element lens which has been constructed with a level of uncertainty which is sufficient
to ensure that for precisely specified conditions of measurement the MTF is equal to that predicted from
theoretical calculations to an uncertainty of better than 0,05 (MTF units)
Note 1 to entry: In order to achieve this uncertainty, standard lenses are usually of simple construction and therefore
of limited performance. An example of a widely used lens is the 50 mm focal length piano-convex lens described
in Reference [3]. This and several other standard test lenses (including afocal systems and lenses operating in the
infrared wavelength bands) are available commercially.
3.1.2
audit lens
single- or multi-element lens of stable construction whose uncertainty of construction is not sufficient to
enable the MTF to be predicted by calculation from design data (usually as a result of the complexity of
the lens), but whose “accepted” values for the MTF under precisely defined measuring conditions have been
obtained by measurements done by a reputable authority (preferably a national standards laboratory, if
such a service is available)
3.1.3
sagittal OTF
test pattern which is constant (shows no periodic variation) in the direction parallel to a line through the
centre of the image circle
Note 1 to entry: A fixed and defined azimuthal orientation of the sample under test is required for the terms sagittal
OTF and tangential OTF to be unambiguous.
[6]
Note 2 to entry: See also ISO 9334 and Figure 1. An illustration can also be found in Reference [9].
Note 3 to entry: In literature, the term “sagittal” is used synonymous with “radial”.
3.1.4
tangential OTF
test pattern which is constant (shows no periodic variation) in the direction parallel to a tangent of the
image circle
Note 1 to entry: A fixed and defined azimuthal orientation of the sample under test is required for the terms sagittal
OTF and tangential OTF to be unambiguous.
[6]
Note 2 to entry: See also ISO 9334 and Figure 1. An illustration can also be found in Reference [9].
Note 3 to entry: In literature, the term tangential is employed synonymous with “meridional.”
Key
1 sagittal to image circle
2 tangential to image circle
3 image pattern vector
4 center of image field
5 image circle
Figure 1 — Sagittal and tangential OTF, excerpt from ISO 9334:2012, Figure 1
3.2 Symbols
Symbol Meaning Unit
h object height mm, mrad, degree
h′ image height mm, mrad, degree
error in image height mm, mrad, degree
Δh′
l object distance mm
l′ image distance mm
Δl error in object distance, is indexed to distinguish error sources mm
NOTE The notation m(r,l,h), m′(r,l′,h′), p′(r,l′,h′) etc. denotes that these parameters are functions of both spatial frequency
r, image or object distance l′ and l, and image or object height h′ or h (i.e. the value of the parameter is different for different
frequencies, distance and different image heights).

Symbol Meaning Unit
Δl′ error in image distance, is indexed to distinguish error sources mm
P best focus point mm
f
Δa angular departure of object slide from perpendicularity to refer- rad
ence axis
Δa′ angular departure of image slide from perpendicularity to refer- rad
ence axis
Auxiliary distance describing intermediate object distance or mm
ii,
ll′
intermediate image distance.
M magnification dimensionless
−1 −1 −1
r spatial frequency mm , mrad , degree
−1 −1 −1
Δr
error in spatial frequency mm , mrad , degree
−1
m(r,l,h) rate of change of MTF with object distance mm
−1
m′(r,l′,h′) rate of change of MTF with image distance mm
−1
p(r,l,h) rate of change of MTF with object height mm
−1 −1 −1
p′(r,l′,h′) or rate of change of MTF with image height mm , mrad , degree
p′(r,l′,ω)
ω field angle mrad, degree
Δω
error in field angle mrad, degree
focal length of collimator mm
f
c
f focal length of sample under test mm
s
ψ azimuth angle degree
Δψ error in azimuth angle between slits degree
R (test lens focal length)/(collimator focal length) or (decollimator dimensionless
focal length)/(collimator focal length)
g′ width of slit referred to image plane mm
L′ length of shorter slit referred to image plane mm
MTF MTF of relay lens dimensionless
C
−1 −1 −1
r spatial frequency for zero field angle mm , mrad , degree
n′(r,l′,h′) rate of change of MTF with spatial frequency mm, mrad, degree
Δ MTF(r) error in MTF, is indexed to distinguish error sources dimensionless
Δ MTF (r) MTF error of the relay lens dimensionless
C
Δ MTF (r) MTF errors resulting from aberrations of relay lens error dimensionless
rl
NOTE The notation m(r,l,h), m′(r,l′,h′), p′(r,l′,h′) etc. denotes that these parameters are functions of both spatial frequency
r, image or object distance l′ and l, and image or object height h′ or h (i.e. the value of the parameter is different for different
frequencies, distance and different image heights).
4 Sources of uncertainty in measuring equipment
4.1 General
In this clause the main sources of uncertainty in OTF measuring equipment are listed and the effects on a
measurement of MTF are described (brief comments on the measurement of PTF is found in Annex A). This
[4]
clause does not establish a GUM (Guide to the Expression of Uncertainty in Measurement, ISO/IEC Guide 98)
conform uncertainty budget, but is treating common measurement errors. To establish a measurement
uncertainty, the probability density functions of the respective measurement errors need to be identified.
The measurement uncertainty is then established based on the probability density function. Usually, an
expectation value of the measurand with an assigned standard uncertainty of the measurand is reported. In
case of an expanded uncertainty, also a coverage interval is defined (expressed by the coverage factor k
p
[5]
see VIM ISO/IEC Guide 99). The examples of common measurement errors reported in this section form
the basis for a detailed analysis in terms of measurement uncertainty.
NOTE In the current revision of this document the term “accuracy” has been replaced by “uncertainty” or “error”
in many sections of the text to use ISO/IEC Guide 98 (GUM) and SO/IEC Guide 99 (VIM) conform vocabulary.
4.2 Geometry of optical bench system
4.2.1 General
The function of the optical bench is to provide a means for supporting the “test target unit”, the “test
specimen” and the “image analyser” in the correct geometrical relationship (i.e. that defined by the
[6]
chosen I-state, in accordance with ISO 9334 ). To achieve this one normally relies on such things as the
straightness of slideways, their parallelism to each other and/or to the surface to which the test specimen
is referenced, the accuracy of angle scales etc. Departures from the assumed geometry result in deviations
from the ideal I-state and therefore errors in the measured OTF. The important bench parameters depend
on the test arrangement being used (note that for bench arrangements such as “nodal slide benches” which
are not covered by this document, the user shall make his own assessment of errors). For the arrangements
recommended in ISO 9335, the main sources of uncertainty and the resulting MTF errors are as follows.
4.2.2 Finite object and image distance
Both the test target unit (TTU) and image analyser slideways shall be straight and perpendicular to the
“reference axis”.
Departures from straightness and perpendicularity produce departures from the ideal object and image
distance, as given by Formulae (1) and (2):
ΔΔlh()= lh()+ha·Δ (1)
straxis
for the TTU and
′′ ′′ ′′
ΔΔlh()= lh()+ha·Δ (2)
straxis
for the image analyser, where h and h′ are object and image heights, Δl and Δl′ are departures from
axis axis
straightness of the object and image slideways and Δa and Δa′ angular (radian) departures from
perpendicularity to the reference axis, for the TTU and image analyser slideways respectively.
The combined effect as image distance error is given by Formula (3):
h′
2  
′′ ′ ′
ΔΔlh = lh +⋅MlΔ (3)
() ()
 
strstr
M 
′
h
where M = is the magnification.
h
If m′(r,l′,h′) is the rate of change of MTF(r) with image distance, then the error in MTF is given by Formula (4):
′′ ′ ′′
ΔΔMTF()rm= ()rl,,hl⋅ ()h (4)
Further possible sources of error are in the error with which the image height h′ is set and the error with
which the object height h is set. The error in MTF is in this case given by Formula (5) (assuming image
height and object height are the parameters set):
′′ ′ ′
ΔΔMTF()rp= ()rl,,hh⋅+pr(),,lh ⋅Δh (5)
where Δh′ and Δh are the errors in image height and object height respectively and p′ and p are the
corresponding rates of change in MTF. Usually, p′ and p are small in comparison to the rate of change with
image and object distance m′ and m and this source of error may be ignored.
4.2.3 Infinite object distance and finite image distance
Similar considerations as for 4.2.2 apply except that there is only a single slideway for the image analyser.
Departures from the ideal object and image distance are given in this instance by Formulae (6) and (7):
Δlh()=0 (6)
str
ΔΔlh′′ = lh′ + ′′·Δa (7)
()
straxis
and the corresponding error in MTF is given once again by Formula (4):
ΔΔMTF rm= ′′rl,,hl′ ⋅ ′′h
() () ()
Errors may also arise from errors in setting image height or field angle (whichever is used in defining the
I-state). These give MTF errors as previously, i.e. as per Formula (8):
′′ ′′
ΔΔMTF rp= rl,,hh⋅ (8)
() ()
or, if field angle rather than image height is specified in Formula (9):
′ ′
ΔΔMTF()rp= ()rl,,ω ⋅ ω (9)
An error may also result from collimation errors as given by Formula (10) (see 5.3 for details):
′′ ′
ΔΔMTF()rm= ()rl,,hl⋅ ()h (10)
ce
In the above equations h , h′, Δl ,Δl′, p and p′ are as defined in 4.2.2, ω is the field angle and Δω is the error in
the field angle, mr ,,lh′′ is the rate of change of the MTF with image distance. The value of Δl′ shall be
()
ce
determined from the known error Δl in object distance. The relevant equation is copied from 5.3 and
ce
given by Formula (11):
f
s
′
ΔΔl ≈ l (11)
ce ce
f
c
where f is the focal length of the collimator and f is the focal length of the sample under test.
c s
Usually, errors in MTF from these latter sources are small and may be ignored except where, instead of
using a collimator, a very long object distance is used on the assumption that it provides a sufficiently close
approximation to an infinite distance.

4.2.4 Infinite object and image distance
With the recommended bench arrangement for this type of measurement (see ISO 9335) the separation
between image analyser and decollimator should not change as the image angle varies. There is therefore no
MTF error resulting from a change in focus setting with image angle (or field angle).
If bench arrangements are used where this error can occur, or, if as a result of mechanical flexing of the focal
slide which supports the decollimator and image analyser their separation may change, then an error in the
MTF may result, given by Formula (4):
ΔΔMTF rm= ′′rl,,hl′ ⋅ ′′h
() () ()
′′ ′′ ′
Where the error in image distance Δlh() is caused by this mechanical error, and mr(),,lh is the rate of
change of MTF with image distance.
Errors may also arise from errors in setting image height or field angle (whichever is used in defining the
I-state). These give MTF errors as previously, i.e. as per Formula (12):
′′ ′′
ΔΔMTF()rp= ()rl,,hh⋅ (12)
or, if field angle rather than image height is specified in Formula (13):
′′
ΔΔMTF()rp= ()rl,,ω ⋅ ω (13)
An error may also result from collimation errors (see. 5.3 for details) as described in 4.2.3 for a pair of
collimator and sample under test. In the case of infinite object and image distance it needs to be considered,
that the first pair of collimator and sample is followed by a second pair of a collimating and decollimating optic.
4.2.5 Suppression of image distance errors by refocusing
For certain types of systems (e.g. image intensifiers) an accepted test procedure involves refocusing of the
sample under test for each position h′ in the image field. This eliminates the image distance error Δlh′′
()
caused by misalignments as discussed in 4.2.2, 4.2.3 and 4.2.4.
′
MTF errors still arise from errors in the object height Δh or errors in the image height Δh as described in
4.2.2 in Formula (5).
ΔΔMTF rp= ′′rl,,hh′′⋅+pr,,lh ⋅Δh
() () ()
′ ′
where Δh and Δh are the errors in image height and object height respectively and p and p are the
corresponding rates of change in MTF.
4.2.6 Mounting of test piece
The test piece may not always locate exactly as intended on the mount to which it is attached on the
equipment. This introduces some variability in the results of a sequence of measurements where the test
piece has been removed from and remounted on the equipment between each measurement. The main effect
is likely to be a small tilt of the image plane. The effect on the measured MTF, which can be very significant,
is given by the same equations as for angular errors in the slideways (see 4.2.2).

4.3 Azimuth changing
4.3.1 General
With most OTF equipment a change in measurement azimuth is achieved by rotating the TTU and the image
analyser. This rotation can result in a movement of the TTU and or the image analyser along the direction of
the axis of rotation (which is the optical axis). This produces an object distance and/or image distance change:
ΔΔllψψand ′
() ()
for the TTU and image analyser respectively, where ψ is the azimuth angle. The MTF error resulting from
this distance change is given in 4.3.2 to 4.3.4 for each of the bench configurations.
4.3.2 Finite object and image distance
The MTF error for finite object and image distance is expressed by Formula (14):
 
ΔΔMTF rm= ′′rl,,hl′ ′ ψψ+⋅MlΔ (14)
() () () ()
 
4.3.3 Infinite object distance and finite image distance
The MTF error for infinite object and finite image distance is expressed by Formula (15):
 
′′ ′ ′
ΔΔMTF()rm= ()rl,,hl ()ψψ+⋅RlΔ () (15)
 
f
s
whereiRRstheratio = , with f test lens focal length and f collimator focal length.
s c
f
c
Usually, R will be small and the second term in the brackets may be ignored.
4.3.4 Infinite object and image distance
The MTF error for infinite object and infinite image distance is expressed by Formula (16):
 
′′ ′ ′
ΔΔMTF()rm= ()rl,,hl ()ψψ+⋅()MR Δl() (16)
T
 
f
dc
where R is the ratio , with f decollimator focal length, f collimator focal length, and M is the
dc c T
f
c
f
s
magnification of the test specimen (see ISO 9335:2025, Figure 4), with f test lens focal length and
s
f
cs
f test lens collimator focal length.
cs
4.3.5 Suppression of image distance errors by refocusing
If a test procedure is used where the test specimen is refocused for every test azimuth ψ (see also 4.2.5)
then no image distance errors will result from a change in azimuth.
4.4 Alignment (orientation) of TTU and image analyser
If both the TTU and the image analyser use mask patterns which are not circularly symmetric, then their
relative orientation is important. Usually one or both of the masks is in the form of a slit perpendicular to

the scan direction. The effect of any angular misalignment Δψ between the two (see Figure 1) results in an
effective increase in width of the slit, given by Formula (17):
′′
ΔΔgL=⋅ ψ (17)
where L′ is the length of the shorter of the two slits, referred to the image plane. The error in MTF resulting
from this is given by Formula (18):
 11 
′
ΔΔMTFM()rr=⋅π ⋅⋅Lrψ ⋅ TF()⋅ − (18)
 
ππ⋅⋅rg′ tan ⋅⋅rg′
() ()
 
where g′ is the assumed width of the slit, referred to the image plane.
It is important to note that in some types of equipment a combination of a slit and a grating is used to
generate a periodic target whose spatial frequency can be altered by changing the orientation of the grating
with respect to the slit. Spatial frequency errors usually result from any errors in the relative orientation of
the slit and grating. The user shall make his own assessment of the effect of such errors (see 4.6 for the effect
of spatial frequency errors on MTF).
′ ′
Wg= +L sinΔψ
Key
1 LSF line spread function
2 image analyser slit
3 TTU mask/slit
L′ length of slit
g′ width of slit
Δψ angular misalignment
W effective slit width
Figure 2 — Errors from alignment of analysing slit with respect to object pattern
4.5 Correction factors
4.5.1 General
Correction factors are applied to MTF measurements to allow for the effect of equipment constants such as
the finite width of target and/or analyser slits, the MTF of incoherently coupled relay lenses and the effect
of off-axis measurement geometry on spatial frequency (see ISO 9335). Errors in MTF occur either if these
factors are not applied, or if there is an error in the value of the applied correction factor. Only the most
common correction factors are considered here. However, these may be taken as examples of how to deal
with other types of correction factors.

4.5.2 Slit width errors
Errors or uncertainties in the widths of slits introduce errors in the measured MTF, given by Formula (19):
 11 
′
ΔΔMTFM()rr=⋅π ⋅ TF()r ⋅ − ⋅ g (19)
 
ππ⋅⋅rg′′tan ⋅⋅rg
() ()
 
where g′ is the width of the slit, referred to the image plane and Δg′ is the error in its value.
4.5.3 Correction for MTF of incoherently coupled relay lenses
Incoherently coupled relay lenses are frequently used in equipment for measuring the MTF of electro-
optical devices and systems such as image intensifier tubes. The reciprocal of the MTF of these relay lenses
is applied to the measured MTF as a correction factor. Therefore, any errors in the value of MTF of such relay
lenses introduce errors into the final MTF value for the system under test. If the error in the MTF of such a
relay lens is ΔMTF (r) and the actual value of its MTF is MTF (r), then the error in the MTF of the test system
c c
is given by Formula (20):
ΔMTF r
()
c
ΔMTFM()rr= TF()⋅ (20)
MTF ()r
c
where MTF(r) is the MTF of the system.
The MTF has values in the interval [0,1], when the MTF value approaches MTF ()r →0 the signal to noise
c
ratio decreases and the result of the above formula is dominated by noise. Therefore, the noise level and the
required uncertainty need to be considered when checking if MTF ()r is big enough.
c
The above relation remains valid and useful if and only if the MTF of the relay lens MTF r 0 for the
()
c
whole range of frequencies 0≤≤rr , over which the measurement is made. The condition for MTF r
()
max c
being great enough depends on the noise level and requires custom consideration.
Similar considerations apply to other measurement situations where a correction is applied for the MTF of a
device in the measurement train.
4.5.4 Spatial frequency correction for field angle
In making off-axis measurements with a grating test pattern positioned on the axis and in the focal plane of
a collimator, a correction of the frequency scale should be performed (this also applies whenever frequency
is measured in this plane). The corrected frequency is given by:
rr= ·cos ()ω for the tangential azimuth and
rr= ·cos()ω for the sagittal azimuth, where r is the on-axis frequency.
Errors in the value of r are produced by errors in the value of ω the field angle. The values of these errors
are given by Formula (21):
ΔΔrr=⋅2 ⋅sinc()ωω⋅ os()⋅ ω (21)
and Formula (22):
ΔΔrr=⋅2 ⋅sin ωω⋅ (22)
()
The effect of such errors on the MTF can be calculated in the manner indicated in 4.7.

4.5.5 Off-axis magnification errors due to image distortion using grating objects
Image distortion, introduced by the sample under test, causes a dependence of the magnification from the
image height Mh′ . This may cause considerable differences in the magnification in sagittal Mh′ and
() ()
S
′
tangential Mh() direction which needs to be considered in the scaling of the grating period for off-axis
T
measurements.
Consider a grating of period p in the object plane, which is imaged to the image plane by the optical system
g
consisting of the collimator and the sample under test. The magnification in dependence of the image height
is defined as Mh′ . In case of a distortion free imaging system with constant magnification M the spatial
()
frequency exited by the grating is given by Formula (23):

r = (23)
pM⋅
g 0
Considering distortion, the excited frequency becomes dependent on the image height as shown in
Formula (24):
′
rh() = (24)
pM⋅ h′
()
g
and is
′
rh() = for the sagittal,
pM⋅ h′
()
gS
and
′
rh() = for the tangential case.
′
pM⋅ ()h
gT
Additional to the influence of distortion on the magnification as mentioned above, the field angle corrections
as described in 4.5.4 have to be applied to the spatial frequency in making off-axis measurements with a
grating test pattern placed in the focal plane of a collimator.
For intermediate orientation of a grating with the meridional plane by an azimuth angle ψ, its image in a
system with distortion gives rise to a grating in the image space with a different angle since the imagery is
anamorphotic in the presence of distortion. Therefore, in general, the image space value of the orientation of
the grating, should be taken as the azimuth angle for the specific MTF measurement. Also, it should be noted
that, in this case, this azimuth angle is dependent on the value of h.
4.6 Image distance error
An error or uncertainty in image distance of Δl′ (referred to the image plane) results in an MTF error or
uncertainty given by Formula (25):
′′ ′′′ ′
ΔΔMTF()rl,,hm= ()rl,,hl⋅ (25)
where m′(r, l′, h′) is the rate of change of MTF with image distance for a spatial frequency r and an image
height h′.
′
The value of Δl depends on several factors. The most important of these are: the sensitivity of the focus
control, the technique of focusing used, the spatial frequency at which the MTF is maximized (low frequency
generally results in a low focusing accuracy), the numerical aperture (NA) of the test lens, the MTF of the
test lens, the signal/noise ratio associated with the particular equipment and test configuration.

Uncertainties in the focus position normally only lead to small errors in MTF at the field position where the
lens is focused (usually on-axis). However large errors may result at other field positions, particularly where
astigmatism and/or field curvature are present.
4.7 Spatial frequency errors
An error r in the spatial frequency produces an MTF error given by Formula (26):
′′ ′′ ′
ΔΔMTF()rl,,hn= ()rl,,h ⋅ r (26)
where n′(r,l′,h′) is the rate of change of MTF with spatial frequency. Some sources of spatial frequency errors
are: calibration errors, non-linearity and/or zero offset in transducers or mechanisms generating the spatial
frequency reading. Note that the relationship between spatial frequency in image and object space may
change with image height in the presence of distortion.
4.8 Residual aberrations in relay optics
Any optical system in the MTF measurement chain which is coherently coupled to the system under test (e.g.
collimators and image relay lenses) should be aberration-free, since corrections cannot be applied for their
effect on the measured MTF.
Accurate assessment of the errors resulting from known residual wavefront aberrations in relay lenses
require the aberrations of the test system to also be known. Moreover, complex calculations are required for
its determination.
If information is available about the MTF errors ΔMTF (r) which would result from the aberrations of the
rl
relay system when testing a diffraction-limited lens with the same NA and aperture diameter as the test
system, then this represents the largest error which is introduced into the measurements from this source.
The value of ΔMTF (r) can either be measured directly or can be computed from the measured aberrations
rl
of the relay lens.
Unfortunately, this approach overestimates the errors when the system under test is poorly corrected.
4.9 Spectral characteristics
The mismatch between the actual and desired spectral response characteristics of the measurement
equipment introduces errors in the measured MTF. The magnitude of the errors depends on the sensitivity
of the MTF of the system under test to the
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