ISO/TR 16743:2013

TECHNICAL ISO/TR
REPORT 16743
First edition
2013-03-15
Optics and photonics — Wavefront
sensors for characterising optical
systems and optical components
Optique et photonique — Capteurs de front d’onde pour
caractérisation des systèmes optiques et des composants optiques
Reference number
©
ISO 2013
© ISO 2013
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ii © ISO 2013 – All rights reserved

Contents Page
Foreword .iv
1 Scope . 1
2 Introduction to wavefront sensing techniques . 1
3 Foucault knife-edge test . 2
3.1 The knife-edge test . 2
3.2 Variations on the knife-edge test . 3
3.3 Application of knife-edge test to diode lasers . 3
3.4 The pyramid sensor . 3
4 Screen testing . 4
4.1 General . 4
4.2 Hartmann test . 4
4.3 The development of automated wavefront sensing . 5
4.4 Shack-Hartmann test . 6
4.5 Measurements with a Shack-Hartmann sensor . 7
5 Wavefront curvature sensors . 8
5.1 General . 8
5.2 Wavefront curvature sensing and phase diversity techniques . 8
5.3 Phase diversity wavefront sensor with diffraction grating . 9
6 Wavefront sensing by interferometry .10
6.1 General .10
6.2 Self-referencing interferometry .11
6.3 Electronic detection and phase measurement .12
6.4 Shearing interferometry .12
6.5 Point-diffraction interferometers with error-free reference wavefronts .17
6.6 Lateral shearing and the Ronchi test .20
6.7 Lateral shearing with a double frequency grating .21
7 Summary of wavefront sensing methods .22
Bibliography .25
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
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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 16743 was prepared by Technical Committee ISO/TC 172, Optics and photonics, Subcommittee
SC 1, Fundamental standards.
iv © ISO 2013 – All rights reserved

TECHNICAL REPORT ISO/TR 16743:2013(E)
Optics and photonics — Wavefront sensors for
characterising optical systems and optical components
1 Scope
This Technical Report gives terms and definitions and describes techniques for the characterization of
wavefronts influenced by optical systems and optical components. It describes basic configurations for
a variety of wavefront sensing systems and discusses the usefulness of tests in different situations.
The aim is to cover practical instruments and techniques for measuring the wavefronts produced by
optical systems and optical components. This Technical Report includes various implementations of
the Hartmann method, the curvature sensor and applications of the knife-edge method. The use of
interferometers is discussed. This Technical Report also includes techniques such as phase diversity
and pyramid sensors, currently used in astronomy and being developed for other areas.
NOTE More information on interferometry can be found in ISO/TR 14999-1, ISO/TR 14999−2 and
ISO/TR 14999−3.
This Technical Report explains briefly how these techniques work and includes diagrams illustrating
the use of this type of equipment for making the measurements required for ISO 10110-5, ISO 10110-8,
ISO 10110-12 (slope requirements) and ISO 10110-14.
2 Introduction to wavefront sensing techniques
Interferometry is a well-established technique for comparing a test wavefront with a reference
wavefront, usually spherical or planar, and requires a degree of coherence between the two wavefronts
to produce an interference pattern. Some interferometers for wavefront characterization are self-
referencing, such as shearing interferometers. These reveal the slope of the wavefront at various points
with values deduced from the interferogram and integrated to calculate the phase profile.
More recently non-interferometric techniques have been developed, partly driven by the needs of
adaptive optics, and it is possible to apply these to wavefronts with limited coherence. The majority of
these techniques are based on measuring the wavefront slope values.
Many of the non-interferometric techniques can be categorized as screen tests. A screen test is a general
term for the test of a beam with an opaque plate placed or moved in the focusing beam and the irradiance
pattern transmitted by the opaque plate analysed. The screen may have one or more holes, slits or edges
to transmit part of the beam while blocking with the opaque part.
Non-interferometric techniques include focused waist (image of source) measurements and wavefront
sampling which gives slope measurements. The knife-edge is a simple test that isolates regions of the
wavefront to reveal aberrations. The Hartmann test uses a perforated screen to isolate bundles of rays
and the direction of these bundles is measured to calculate the wavefront slopes. The Shack-Hartmann
test uses an array of small lenses to sample the wavefront. The wavefront slopes are deduced from the
positions of the focal spots generated by the lens array and the slope values are integrated to calculate
the phase profile.
Wavefront curvature sensing and phase diversity techniques are a class of wavefront retrieval
mechanisms that infer the wavefront from measurements of the intensity of the light as the beam
propagates. Typically this involves the measurement of two images along the beam path, from which
the intensity gradient is derived. Two standard approaches are to measure the intensity either side of
a focus or either side of a pupil plane in an optical system. Phase diversity techniques use calculation
algorithms for the retrieval of wavefront phase. Once the intensity data are collected, a processing step
is required to calculate the wavefront. This can be achieved using the intensity transport equation,
[11]
and solved by direct integration or iteratively using Fourier transform techniques. Phase diversity
methods are finding more use with faster phase retrieval methods as computation speeds increase.
3 Foucault knife-edge test
3.1 The knife-edge test
a) Test set-up
b) typical intensity pattern seen by observer
Key
1 mirror
2 knife-edge
3 source
Figure 1 — Knife-edge test for wavefront from a concave mirror
The knife-edge test is one of a family of techniques in which elements of a deformed wavefront are detected
by blocking with an opaque mask that has an edge that clearly defines the boundary between the opaque
and transmitting regions of the mask. The mask is placed in the plane of best focus of the wavefront.
The knife-edge test was reported by Foucault in 1858 as a method for examining the form of a concave
[12]
mirror surface. The mirror is used to form an off-axis image of a pinhole source placed in a plane
containing the centre of curvature of the mirror and a blade with a knife-edge as shown in Figure 1a).
The mirror is observed from the vicinity of the image and the knife-edge is moved across the line of
sight until a shadow is seen to cross the mirror. The direction of movement of the shadow will depend on
whether the image is formed in front of or behind the knife-edge. If the centre of curvature lies exactly in
the plane of the pinhole and the knife-edge, the whole aperture of the mirror will darken simultaneously
revealing imperfections in the image. When the knife-edge is scanned across, imperfections in the image
will be revealed as dark and bright regions, simulated in Figure 1b).
Knife-edge scanning is a relatively simple method for measuring spot sizes but inaccuracies can occur
due to uncertainties in the location of the knife-edge. The method has been applied to the measurement
of spot diameters to a precision of better than 50 nm using interferometry to monitor the knife-edge
2 © ISO 2013 – All rights reserved

[13]
position. In applications such as optical data recording, the size and structure of the smallest possible
focused spot is important. These parameters are described by the point-spread function, a measure of
[14-16]
the quality of the optical system.
3.2 Variations on the knife-edge test
The Foucault knife-edge test can be used to make precise measurements of zonal errors in near-spherical
[17]
wavefronts and is useful as a null test. It is not, however, so useful for testing aspherical wavefronts.
The asymmetry of the single knife-edge allows systematic errors to accrue. A better arrangement is to
use two knife-edges that block the light simultaneously in opposite directions. Either a thin wire or a
thin slit will achieve this. A thin wire blocks out a very narrow region of the wavefront and the shadow
patterns consist of thin dark contours. Platzeck and Gaviola applied the thin-wire method to the testing
[18]
of parabolic mirrors. They measured the caustic, which is the line of centres of curvature of the
surface elements.
3.3 Application of knife-edge test to diode lasers
The characteristics of diode laser sources are dependent on the choice of semiconductor materials and the
geometry of the junction structure. The active layer of a typical laser is about one tenth of a wavelength
thick while the waveguide dimensions are typically 1 μm by 3 μm in the planes perpendicular and
parallel to the junction respectively. Mode confinement is achieved either by gain guiding or by index
guiding. A laser beam is emitted from the facet and diverges strongly due to the small size of the source.
Diffraction causes the beam to spread over large angles, typically ranging from 25° to 60° in the
perpendicular direction and 7° to 35° in the junction plane. The radiation pattern is asymmetrical.
Depending on the type of diode the mechanisms of constraining the beam in the plane of the active
laser and in the plane normal to it are quite different and the wavefront becomes strongly astigmatic.
The window in the canister containing the diode source is usually a thin plate and this can introduce
[19,20]
spherical aberration. Anamorphic lenses are sometimes used to collimate the elliptical beam.
[21]
Near-field and far-field measurements are often required to fully characterize a laser diode.
Knife-edge scanning is often recommended for the measurement of the astigmatic distance of wavefronts
from diode lasers. The diode source is imaged at the plane of the knife edge with a relay lens such as a
microscope objective. The total optical power passing the knife edge is measured with a photodiode and
meter. This is plotted against knife edge position and an integrated power profile drawn. Differentiation
of this curve gives the actual power profile from which the beam diameter, defined as full width at half
maximum height, can be determined.
A beam diameter profile is produced by measuring the beam diameter at different points along the axis,
achieved by displacing the source. The beam diameter profile is produced for movement of the knife edge
in the direction perpendicular to the junction plane of the laser diode and the other for travel parallel
to the junction plane. The astigmatic distance is defined as the displacement of the source between the
minima of these plots.
An advantage is that it is a simple test with no demands on coherence properties of source. It is sensitive
to small deviations in slope of the sampled wavefront or to slope deviations that are changing slightly in
magnitude or direction.
3.4 The pyramid sensor
The pyramid sensor may be considered as an extension of the knife-edge test and was primarily
[22]
developed as a wavefront gradient sensor for astronomy. In that application it uses a shallow glass
pyramid with four faces, placed in the focal plane of a telescope and with the tip in line with the optical
axis. An incident wavefront is collected by the telescope and focused to the pyramid which generates
four beams, as shown in Figure 2. Another lens collects this light to reimage the telescope aperture
and form four images on a detector array. The relative intensities of the light in these four images are
compared and the wavefront gradients in two orthogonal directions are calculated. By modulating the
position of the pyramid a linear gradient signal may be obtained. The accuracy of the pyramid sensor has
been investigated and the linearity considered from both the geometrical optical model and diffraction
[23,24]
calculations.
The test makes efficient use of the light and is able to sense two orthogonal directions. The main use of
the pyramid sensor is with adaptive optics in astronomical and ophthalmic applications but its use is
gradually being extended to more general wavefront sensing.
Key
1 wavefront
2 lens
3 pyramid prism
4 field lens
5 four images of entrance pupil on detector array
Figure 2 — Pyramid prism sensor
4 Screen testing
4.1 General
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 as in the Hartmann sensor or by
a lenslet or microlens array as in the Shack-Hartmann sensor. The use of these techniques to determine
the shape of a laser beam wavefront is described in ISO 15367-2. The techniques are also suitable for the
measurement of non-laser wavefronts with reduced spatial coherence.
4.2 Hartmann test
The Hartmann test is based on a geometrical optics approach and uses a perforated screen or mask with
[25]
multiple holes to sample the wavefront at a number of locations in a predetermined fashion. The
[26]
Hartmann test is often used for testing optical components such as telescope mirrors.
The principle is that a portion of the wavefront, when tilted relative to an ideal wavefront in that region,
causes light to come to a focus at a place other than the intended focus, or to intercept a chosen plane at
4 © ISO 2013 – All rights reserved

a location other than the one which would be obtained with light coming from the ideal wavefront and
from that region.
The converse can be used to determine the tilt error in a portion of a wavefront, by determining where the
light from that region intercepts a chosen plane, and what difference there is between that intersection
and the one to be expected from a perfect wavefront.
With interferometry, deviations of the wavefront are found using the interference of light from two
different regions. With screen testing, the light from each of the various regions of the wavefront is
recorded on a photosensitive material, such as a photographic plate, which is measured after processing.
The beam deviations are measured to obtain a surface slope error at the sampled points.
One of the major difficulties in screen testing is the introduction of errors through the method used to
reduce the data in order to obtain the surface deviations. The main assumption is that the wavefront
changes between samples are gradual rather than abrupt. Abrupt changes can be readily detected by
other means such as the Foucault knife-edge test. The screen test is better suited for smooth wavefronts.
Several screens have been designed for testing including the Hartmann radial pattern. The holes in the
screen have to be accurately placed and small but not so small that their diffraction images overlap at
the recording stage.
The advantages of this test include the low number of components and low requirements for source coherence.
Disadvantages include:
a) the Hartmann test samples the wavefront at a discrete number of points;
b) the highest accuracy is achieved for the smoothest wavefronts;
c) considerable time is required to record, process and measure the images on the photographic plate.
The main advantages of the Hartmann technique are:
1) wide dynamic range;
2) high optical efficiency;
3) suitability for partially coherent beams;
4) no requirement of spectral purity;
5) no ambiguity with respect to increment in phase angle.
Kingslake considered using the Hartmann test to measure spherical aberration in microscope
objectives. A relatively small screen was needed and he realized that interference effects between the
light transmitted by adjacent holes in the screen would be a problem. He therefore devised a method
[27]
that used a single hole, traversed across the aperture to isolate a series of light beams in succession.
4.3 The development of automated wavefront sensing
In the Hartmann test a spot diagram is generated by the intersection of a defined plane and the ray
bundles generated by a mask with an array of apertures. An automated device for achieving this was
[28]
described by Baker and Whyte in 1964. They used a rotating polygon mirror to scan the pupil of the
optical element under test. In a modified version a spiral disc scanner was used instead of the polygon
[29]
mirror. The position at which each ray intersected the image plane was measured with a position-
sensitive photodiode and the results displayed in real time on a cathode ray tube. A silicon quadrant
detector was also used to measure the ray intersection coordinates. In a further development the system
[30]
was able to measure wavefront aberration. In 1972 Williams described a system that measured both
[31]
spot diagrams and wavefront aberrations using an image dissector tube as detector. The subsequent
availability of large detector arrays and powerful portable computers has led to the development of
equipment that operates on these principles and offers a serious alternative to the interferometers more
commonly found in optical workshops. One particularly successful configuration has been that due to
[32]
Shack and Platt and described below.
4.4 Shack-Hartmann test
[33]
The Shack-Hartmann sensor uses an array of lenslets or microlenses to sample the wavefront.
Compared with the Hartmann screen test this gives a better optical efficiency and in comparison with
the automated system described above, the static lenslet array obviates the need to scan a pencil of light
over the pupil of the test piece to sample the wavefront.
12 3
Key
1 wavefront
2 lenslet array
3 detector array
Figure 3 — Array of lenslets (microlenses) used to sample an incident wavefront
23 4
Key
1 source of wavefront
2 beam expansion or compression optics
3 lenslet array
4 detector array
Figure 4 — Experimental arrangement for wavefront measurement using Shack-Hartmann
technique
6 © ISO 2013 – All rights reserved

The lenslet array divides the incident wavefront into a number of small areas, each of which is focused
to a spot as in Figure 3. If a plane reference wavefront is used at normal incidence the spots are formed
on the lens axes. If the test wavefront is non-planar the spots are formed away from the axes. A charge-
coupled device (CCD) is commonly used to capture the images of the multiple spots allowing the positions
of the centroids of the spots to be determined. The transverse positions of the spots are related to the
local slopes of the wavefront in the regions of the lens apertures and the form of the wavefront can be
calculated using a small computer and dedicated software. In practice it is usually necessary to expand
or contract the dimensions of the wavefront to match the dimensions of the lenslet and CCD arrays, as
shown in the schematic layout in Figure 4.
The average slope of each area of the wavefront that is sampled is determined from the focal length of
each lens and the displacement of each image on the array relative to its position when a perfect test
wavefront is used. The wavefront is computed from the slope data.
The method yields values for the local wavefront slopes. Two approaches, the zonal method and the
modal method, can be used to fit the data to compute the wavefront. The choice depends on whether the
estimate is a phase value in a local zone or a coefficient of an aperture function. In either case, the least
[34]
squares method may be used for the phase reconstruction.
The Shack-Hartmann test can achieve a high sensitivity and is commonly used with adaptive optic
[35]
systems where a deformable mirror is adjusted to give an optimum shape wavefront. The test has
been applied to the measurement of wavefronts from diode lasers and to industrial infrared lasers.
[36]
Michau et al. describe testing a collimated beam from a laser diode for space communication. One
advantage of the Shack-Hartmann test is that the system is relatively simple with few components and
can be used with low coherence wavefronts. Another advantage is the speed of the measurement. By use
of the CCD and software to analyse the spot positions, the wavefront error is determined immediately.
This measurement can then be used in the feedback controls of adaptive optics.
Disadvantages include the fact that the local wavefront is determined from the slope averaged over
the lenslet aperture. The lateral resolution of the wavefront is limited by the pitch of the lenslets in
the array. Wavefront measurement accuracy is limited by the accuracy with which the centroids of the
focused spots can be determined.
4.5 Measurements with a Shack-Hartmann sensor
4.5.1 General
There are several Shack-Hartmann systems that are commercially available and each will come with its
own instructions for use. Some common aspects include those described in 4.5.2 to 4.5.5.
4.5.2 Wavefront dynamic range
The wavefront shape must lie within the dynamic range of the sensor. The dynamic range depends
primarily on the geometry of the lens array, for example lenses of longer focal length produce greater
lateral displacement of the spots for a given angular tilt of wavefront. This limits the angular range that
can be detected because the spot from one lens translates to a region of the detector assigned to an
adjacent lens. However manufacturers have evolved different techniques for identifying and tracking
the spots and thus extending the dynamic range.
4.5.3 Alignment
To make the most of the dynamic range the sensor must be carefully aligned. It is usually best to fit the
sensor head to a precision adjustable tip/tilt table. The sensor should be aligned to be normal to the
incident beam. One manufacturer recommends placing an adjustable aperture in the beam to define
a small area and placing a small plane mirror on the sensor. The sensor is then tilted until the spot
transmitted by the aperture is reflected back on itself. If this procedure is not possible and it is necessary
to use the sensor in an off-normal position then the normal incidence calibration may not apply and a
new reference measurement should be taken.
4.5.4 Illumination levels
The level of illumination reaching the sensor must be adjusted to within the limits of the sensor. Modern
sensors have software that shows the signal levels along with minimum, maximum and average values.
The irradiance level can be adjusted by inserting neutral density filters in front of the camera but if
the filters are not of high quality they may introduce additional aberrations to the wavefront. In that
case a reference measurement should be made with the filter in place and the measured aberrations
subtracted from the test wavefront results.
Another way of ensuring the light levels are within limits is to adjust the camera gain and exposure
time. Typically, a value of 80 % of the maximum enables accurate measurements to be made without the
danger of saturating the sensor. Changing the camera gain may change the signal to noise ratio.
4.5.5 Calibration
4.5.5.1 General
The calibration of the system should be checked before making a measurement. Some manufacturers will
supply a data file from a calibration check made beforehand. Alternatively a live check may be performed
by illuminating with a wavefront of known form such as a spherical or plane wavefront to provide an
absolute calibration. Where it is only required to monitor the change in a wavefront, a measurement may
be made before and after the change without necessarily carrying out an absolute calibration.
4.5.5.2 Calibrating sphericity component
The performance of the sensor in measuring spherical wavefronts of different radii can be assessed by
monitoring a wavefront emitted by a point source such as an optical fibre and making measurements
[37]
at different distances from the source. It is desirable to use a very long precision slideway for this.
5 Wavefront curvature sensors
5.1 General
Wavefront curvature sensing is a technique which uses local tilt values to calculate wavefront
curvature and commonly finds applications with adaptive optics. It measures the local wavefront
curvature, together with the wavefront tilts at the aperture edge in a direction perpendicular to the
edge. A curvature sensor shown in Figure 5 consists of two image detectors which detect the irradiance
[38]
distributions either side of the wavefront focus produced by the action of lens L on the wavefront W.
P2
L
P1
W
F
Figure 5 — Curvature sensing by measuring irradiance distributions either side of the focus
5.2 Wavefront curvature sensing and phase diversity techniques
Phase diversity techniques are a class of wavefront retrieval mechanisms that infer the wavefront
from measurements of the intensity of the light as the beam propagates. Typically this involves the
measurement of two images along the beam path, from which the intensity gradient is derived. Two
8 © ISO 2013 – All rights reserved

standard methods involve measuring the intensity either side of a focus as shown in Figure 5 or either
side of a pupil plane in an optical system. There are several techniques for imaging these planes onto a
detector, including the use of a number of detectors, a dynamic mirror that changes focal length over
time which is then synchronised to measurement on a detector, or a diffraction grating with optical
power that images both measurement planes onto the same detector. The diffraction grating resembles
the off-axis structure of a Fresnel zone plate and its use is described later in this Technical Report.
The curvature sensor shown in Figure 5 consists of two detectors which detect the irradiance
distributions in planes P1 and P2, either side of the wavefront focus F. The difference divided by the sum
of the intensities of the two images at each position of the beam provides the curvature at that position.
The sign of the calculation at the edge of the beam yields the slope of the wavefront compared to the
reference spherical wavefront.
In Figure 6 plane B may represent an input pupil plane in an optical system or simply a plane designated
for wavefront characterization. The difference between the intensities in the images recorded when
focused on plane A and plane C, divided by the sum of those intensities, provides an estimation of the rate
of change of the intensity as the wavefront propagates (the axial intensity derivative). This derivative
can be used to estimate the local wavefront curvature, indicated in Figure 6 by the curved line in plane
B, using a differential equation known as the Intensity Transport Equation, or ITE. Shrinkages of the
illuminated region at the boundaries provide estimates of the wavefront slope around the boundary,
which give the boundary conditions to use when solving the ITE.
12 3
Key
1 plane A
2 plane B
3 plane C
Figure 6 — Curvature sensing by measuring irradiance in successive planes
5.3 Phase diversity wavefront sensor with diffraction grating
A recent development in wavefront sensing uses a special diffraction grating that resembles the off-axis
structure of a Fresnel zone plate. The grating images planes on either side of an input pupil plane and,
because the images are laterally separated, they can be recorded simultaneously with a single camera
[39]
as shown in Figure 7 Phase diversity algorithms are used to retrieve the wavefront phase from
two images symmetrically placed about the wavefront to be reconstructed, and normal to the axis of
[40-43]
propagation. From the wavefront phase of the two images, the wavefront at plane B is calculated.
+1
−1
12 3
Key
1 plane A
2 plane B
3 plane C
Figure 7 — Wavefront sampling with a special diffraction grating
If the images are captured about the system input pupil, this is essentially the same as a wavefront
curvature algorithm where the measured intensity is directly related to the wavefront shape. By
determining the wavefront shape, the location, direction and magnitude of the wavefront error can be
calculated. This information can then be used to drive a corrective element in an adaptive optics system.
Hall et al. report a comparison of measurements made with a prototype instrument that used a cross-
distorted diffraction grating. The wavefront measuring capability of the instrument was used to derive
[44]
the beam propagation factor, M2, which is an important parameter for characterizing optical beams.
6 Wavefront sensing by interferometry
6.1 General
Interferometry is a widely used technique for analysing the shape of optical wavefronts and deducing
information about the source of a wavefront and any optical components the wavefront may have traversed.
ISO/TR 14999-1 describes interferometric techniques commonly used for the purpose of characterizing
the quality of optical components. The techniques covered include Newton, Fizeau, Haidinger, Mach-
Zehnder and Twyman-Green interferometers. In these interferometers the wavefront returning from
the item under test is compared to a reference wavefront generated by reflection at a reference surface,
usually a partially reflecting plane or spherical surface manufactured to a high degree of precision
and placed between the source and the surface under test. A range of methods for combining the test
and reference wavefronts have been evolved for different types of interferometers and these include,
for example, scatterplates which split wavefronts by optical scattering, gratings which operate by
diffraction and birefringence prisms which operate by double refraction.
The techniques described above are used where it is possible to intercept the wavefront before it
illuminates the surface under test and generate a reference wavefront. In another technique known as
shearing inter ferometry the reference wavefront is a replica of, and can be generated from, the wavefront
under test. The two wavefronts are given a relative displacement either in a transverse or radial sense,
to generate an interference pattern from which the shape of the wavefront under test is calculated. In
some designs of interferometer the reference wavefront is formed by expanding such a small portion of
the test wavefront that the reference wavefront may be considered to be purely spherical.
10 © ISO 2013 – All rights reserved

The analysis of the phase profile of an optical wavefront is a task that is routinely encountered in
metrology of optical components. The component under test is placed in the path of a test wavefront
and the phase variations that are introduced are examined by interferometry. Some interferometric
techniques assume that a coherent reference wavefront of simple form is readily available, others
require a reference wavefront to be generated from the test wavefront.
Two self-referencing techniques are used to generate a reference wavefront from the test wavefront:
a) a small portion of the test wavefront is selected and expanded to give a spherical reference wave;
b) a replica of the entire wavefront is generated and compared to the test wavefront by shearing
interferometry.
The first method treats the selected portion as being relatively unaberrated. This is expanded to compare
with the aberrated test wavefront. The second method means that the aberrated wave is compared with
an identical reference wave. By shearing the wavefronts, path differences are generated that enable the
wave front shapes to be determined. The wavefronts can be inverted, rotated and displaced in a variety
of ways for comparison, depending on the information that is required to be extracted.
6.2 Self-referencing interferometry
This subclause considers the analysis of an isolated wavefront and assumes that a separate reference
beam is not available. Self-referencing interferometers can be classified according to:
a) the shearing geometry (the relation between the test and reference beams);
b) the design layout (optical routes or beam paths);
c) the method of beamsplitting.
The shearing geometry describes the generation and orientation of the reference beam and includes:
1) Lateral shear. This is obtained by displacing a copy of the wavefront laterally.
2) Reverse shear. Obtained by reversing a copy of the wavefront and superposing.
3) Inverted shear. Obtained by folding one part of the wavefront onto the other.
4) Radial shear. This is obtained by expanding or contracting a copy of the wavefront.
5) Large radial (exploded) shear. The copy wavefront is greatly expanded.
6) Point diffraction (error-free reference).
Beamsplitting can be achieved as follows:
— wavefront division - by division of area;
— amplitude division - by partial reflection;
— amplitude division - by diffraction.
The reference and test wavefronts may follow a common path on-axis or follow separate paths, as for
example in the Mach-Zehnder design.
6.3 Electronic detection and phase measurement
Electronic phase measurement techniques enable interference patterns to be processed to a high degree
of accuracy, ignoring noise generated by intensity variations that are not directly associated with the
main interference pattern. Two general categories of phase measurement exist.
a) Heterodyne or AC interferometry. The phase difference between two interfering beams is changed
at a constant rate by producing a frequency difference between the two beams.
b) Phase shifting or phase stepping. The phase difference between interfering beams is changed either
by stepping a piezoelectric translator (PZT) pushing a mirror or by ramping the PZT to push the
mirror as the detector array is integrating. The former is known as phase stepping and the latter as
the integrating-bucket technique. The latter is faster.
Some of the interferometers described in this Technical Report are more readily adapted to electronic
phase measurement than others. In many designs it is the detector that limits the range of wavelengths
over which the interferometer can be used. The spatial geometry of the interference pattern must
be known accurately and it is usual to use a charge coupled array (CCD) to detect the pattern. The
wavelength range is limited by the spectral sensitivity of the CCD.
6.4 Shearing interferometry
6.4.1 General outline
[45]
The advantages of wavefront shearing interferometry were pointed out by Bates in 1946. He reported
that the asphericity of an optical wavefront could be measured by testing it against itself with lateral
displacement or shear. There was no need for an error-free wavefront to act as a reference standard.
Different categories of shear include radial shear, rotational shear and reversal shear, the geometries being
[46]
evident from the category name. Saunders introduced the concept of the inverting interferometer.
In the days before laser sources were invented, path lengths in interferometers had to be closely
matched to retain coherence between the reference and test wavefronts. These often followed long
and separate paths, rendering the test sensitive to environmental disturbances. The development of
shearing interferometry reduced these restrictions and increased the ease of testing components such
[47,48]
as large spherical and non-spherical mirrors.
When two wavefronts containing identical asphericities are exactly overlapped, a single bright fringe
will cover the field of view. If a tilt is applied between the wavefronts, straight fringes parallel to
the intersection of the wavefronts will result. When two wavefronts are sheared, an interferogram
representing the difference between the two wavefronts will be obtained in the region of overlap and
the interferogram reveals directly the slope of the wavefront. This is particularly true for small values
of shear but becomes less exact for larger shears. Because the sensitivity of the test decreases as the
magnitude of the shear decreases, the latter has to be carefully chosen and a compromise made between
sensitivity and ease of interpretation. When the aberrated wavefront does not have circular symmetry
the direction of shear is also important.
The asphericity revealed in the shearing interferogram will not be the same as that in the original
wavefront but will depend upon it in a precise way. This is shown by expressing the wavefront under
test by the polynomial expression.
2 3
22 22 22
wx,yA=+xy ++Ax yA++xy +.
()
() () ()
2 4 6
where x and y represent the coordinates of points on the wavefront. The aberrated wavefront is
divided and sheared to form two interfering wavefronts, centred at C and C where C − C = Δx. Their
1 2 1 2
combination centred at C can be written:
2 2
   
ΔΔx x
   
2 2
   
Wx,yA=+x + yA−−x + y
()
c 2   2  
     
   
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2 2
2 2
   
ΔΔx x
   
2 2
   
++Ax + yA−−x + y
4 4
   
 22 
    
   
3 3
2 2
   
 ΔΔ
...