Pseudo Shear Interferometry
Peter B. Keenan
Spectra Physics, Inc., Optics Division
1250 W. Middlefied Road, Mountain View, California 94042
Abstract
Pseudo Shear Interferometry (PSI) is a technique for obtaining enhanced, absolute accuracy in interferometric measurements.
Whereas conventional interferometry yields measure
ments limited in accuracy by the optical system and by the reference surface, Pseudo ShearInterferometry provides the capability for measurement accuracy which substantially surpas
ses it.
The technique consists of a regimen for data taking and a mathematical procedure
for analyzing the resulting data.
Introduction
A digital, phase measuring interferometer provides sets of measurements on equal spaced
array of points.
Between measurements, a translation is made of the optic being measured.The translation is adjusted to correspond to the sampling interval of the phase measuring
interferometer.
Differences between the measurement sets are then taken numerically in a
computer.
These differences comprise a pseudo shear interferogram.Further processing of
these difference sets is then required to separate the deterministic variation, the syste
matic error, and the random error.The analysis of the data sets ultimately yields the following:
1.
Absolute surface measurements,
2.
Absolute interferometer calibration, and
3.
An estimate of the random errors.
The accuracy requirements for interferometric testing have become increasingly severe in
the past few years.
There are two primary reasons for this trend.First, fringe scanninginterferometers have become commercially available.
These interferometers provide impres
sive levels of measurement resolution and accuracy.Second, there are increasing numbers of
complex optical systems being attempted which contain large numbers of surfaces, systems
which require a transmitted wavefront error of a fraction of a wavelength.Thus, the re
quirement for single surface measuring accuracy has continued to pressure the state of the
art.
Although digital interferometers have provided the user with a measurement resolution on
the order of 1 /1000 wave, there remains the unavoidable fact that all interferometric measurements are comparisons.Each value is a path difference.Some method is required to
separate the component of the path difference due to the optic under test from the component of the path difference which is due to surface variation in other elements of the sys
tem.
The procedure described here has somewhat the flavor of lateral shear interferometry.The processing includes obtaining the differences between successive measurements with a
lateral shift of the optic.
The term pseudo is used because the comparison of the twowavefronts is affected in a computer rather than between the wavefronts themselves.
The pseudo shear interferogram may be processed to yield the actual profile of the opticand the systematic errors of the instrument.
This process does not require the use of a
reference standard of known accuracy.Elimination of the requirement for a reference stan
dard simplifies the testing and reduces the errors introduced by a reference which has lost
its figure due to thermal or mechanical stress.It may be that the reference is not certified to the level of accuracy required for the test.Two position PSI
Conceptually, the method requires placing the test piece on a fixture that may be tran
slated laterally.
After taking a first measurement, the part is translated and a second
measurement is made.
An analysis is performed on the results of the two measurements to
separate the component which moved from the component which remained fixed.The measurement geometry is shown in Figure 1.At each position, the measurement yieldsthe distance between the surfaces.The measurements, Mi, are made at equal spaced positions.
2
PseudoShear Interferometry
Peter
B
KeenanSpectraPhysics,
Inc.,
Optics Division
1250
W
Middlefied
Road,
Mountain
View,
California
94042Abstract
PseudoShear Interferometry
PSI)
is
a
technique
for
obtaining
enhanced,
absolute
ac
curacy
in
interferometric measurements. Whereas conventional interferometry
yields measure ments limited
in
accuracy
by the
optical
system
and
by
the
reference
surface,
PseudoShear Interferometry provides
the
capability
for
measurement accuracy which substantially
surpas
ses
it
The
technique consists
of
a
regimen
for
data taking
and
a
mathematical procedure
for
analyzing
the
resulting
data.
Introduction
A
digital,
phasemeasuring interferometer provides
sets
of
measurements
on
equalspaced
array
of
points.
Between measurements,
a
translation
is
made
of
the
optic being
measured.
The
translation
is
adjusted
to
correspond
to
the
sampling interval
of
the
phasemeasuring
interferometer.
Differences between
the
measurement
sets
are
then taken
numerically
in
a
computer.
These differences comprise
a
pseudoshear
interferogram.
Further processing
of
these
difference
sets
is
then
required
to
separate
the
deterministic
variation,
the
syste matic
error, and
the
random
error.The
analysis
of the
data sets
ultimately
yields
the
following:
1
Absolute surface measurements,
2
Absolute interferometer
calibration,
and
3
An
estimate
of
the
random
errors.The
accuracy requirements
for
interferometric testing
have
become increasingly
severe
in
the past
few
years.
There
are
two
primary
reasons
for
this
trend. First,
fringe
scanning interferometers
have
become commercially
available. These
interferometers provide
impres sive levels
of
measurement resolution
and
accuracy. Second,
there
are
increasing
numbers
of
complex optical
systems
being
attempted which contain
large
numbers
of
surfaces, systems which
require
a
transmitted wavefront
error
of
a
fraction
of
a
wavelength.
Thus, the
re
quirement
for
single surface measuring accuracy
has
continued
to
pressure
the
state
of
the
art.
Although digital interferometers
have
provided
the
user with
a
measurement resolution
on
the
order
of
1/1000
wave,
there remains
the
unavoidable
fact that all
interferometric
mea surements
are
comparisons. Each value
is
a
path difference. Some
method
is
required
to
separate the
component
of
the
path
difference
due
to the
optic under
test
from
the
compo
nent
of
the
path difference which
is
due
to
surface
variation
in
other elements
of
the sys
tem.
The
procedure described
here
has
somewhat
the
flavor
of
lateral shear interferometry.
The
processing includes obtaining
the
differences between successive measurements
with
a
lateral shift
of
the
optic.
The
term pseudo
is
used because
the
comparison
of
the two
wavefronts
is
affected
in
a
computer
rather than
between
the
wavefronts
themselves.The
pseudoshear interferogram may
be
processed
to
yield
the actual
profile
of
the
optic
and the
systematic
errors
of the
instrument. This
process
does
not
require
the
use
of
a
reference
standard
of
known
accuracy.
Elimination
of
the
requirement
for
a
reference
stan dard
simplifies
the
testing
and
reduces
the
errors
introduced
by
a
reference which
has lost
its
figure due
to
thermal
or
mechanical
stress.
It
may
be
that the
reference
is
not
cer
tified
to
the
level
of
accuracy required
for the test.
Two
position
PSI
Conceptually, the
method requires placing
the test
piece
on
a
fixture
that
may
be
tran slated laterally.
After taking
a
first
measurement,
the
part
is
translated
and
a
second
measurement
is
made.
An
analysis
is
performed
on
the
results
of
the two
measurements
to
separate
the
component which moved
from
the
component
which remained
fixed.The
measurement geometry
is
shown
in
Figure
1
At
each position,
the
measurement
yields
the
distance between
the
surfaces. The
measurements,
Mi
are
made
at
equalspaced
positions.
Downloaded From: http://proceedings.spiedigitallibrary.org/ on 09/04/2014 Terms of Use: http://spiedl.org/terms
Also shown in Figure 1 is a mathematical axis for describing the two surfaces.
The surface
of the test optic is described by its distance, Ti, from the axis.Likewise, the distancefrom the axis to the reference surface is given by Ri.Each measurement is simply a difference, Ti  Ri.In conventional interferometry it isonly this difference which can be determined and not the individual values of Ri and Ti.Consider in more detail this two measurement scheme.The expected values that would be
obtained are shown in Figure 2.
In the left hand column are the measurement values foreach position of the initial configuration.The right hand column shows the expected measurement values after the test optic has been translated.Note that the reference values,Ri, stay fixed with respect to the measurement positions.
The difference between the items in the two columns is referred to as a Pseudo Shear In
terferogram.
In Figure 3, the group of equations that result is shown.This set is readilysolved for the values of Ti after choosing an arbitrary value for T one.
After obtaining the
values for each Ti, thé equations of Figure 2 may be solved for each Ri.This two position technique is simple and readily produces a set of values for both Ti,
and Ri.
However, there is no provision for including additional data nor is there provision
for error estimation.
These deficiencies are alleviated in the following multiposition
analysis.
Multiposition PSI
Combining the results of three or more measurements with lateral shifts makes it possible
to extend the previous analysis and simultaneously include a least squares error reduction.
This provides a means to estimate the measurement error as well as the best estimate of theprofile of the test optic and the reference surface.The analysis presented will be for the
case of three measurements made with equal spaced translations as represented in Figure 4.
Each column corresponds to one of the measurement sets; each being a translation of one
unit from the next.
Also appearing in these equations is an error term, E.
It is assumed
that there is an independent, random error associated with each value.There are two independent sets of differences which may be obtained from these; they are
shown in Figure 5.
Note that D12 and D23 are both estimates of the same quantity and com
parable pairs comprise the lists.
If additional measurements are made, they can be easily
included at this point.
Because the number of unknowns, including the error terms, exceeds the number of equations, this set is solved in the least square sense.Figure 6 shows the derivation of thenormal equations from the conditions for minimizing the sum of the squares of the errors.Substitution and evaluation of the partial derivatives gives the normal equations.
The nor 
mal equations may be displayed in somewhat more compact form by the use of matrices.
Addi
tionally, display of the matrix for a specific size array shows the starting and ending
values explicitly.
Figure 7 shows the matrix equation for the case of five positions on the
test optic.
The matrix on the left is a symetric band matrix.All diagonal elements are four except
the last.
This same form prevails for any matrix size greater than two.The column vectoron the right maintains the pattern shown for arrays of any size.This matrix may be readily inverted to solve for the profile of the test optic, Ti.
Ob
taining the values for Ti might be considered the end of the problem.However, for most accurate work, we are required to determine an error estimate for the measurements.Returning to Figure 4 for a moment, it will be noted the differences might be arranged toeliminate Ti and thus be used to determine Ri.The resulting differences are shown in Figure
8.
These are shifted differences and yield PSI data for the interferometer.
Application of
the least square methodology to this set of equations produces the matrix equation for determination of the reference as shown in Figure 9.Again, the patterns of the components of this equation is shown for dimension five by
five.
Extension to matrices of other sizes is readily accomplished.The solution for the
values of Ri is straight forward.
The solutions of these two matrix equations have given values for Ri and Ti.
Substitu
tion of these values into the equations of Figure 4 permit determination of the errors as
sociated with each value.
From the enumerated values, error statistics can be developed inorder to place confidence bounds on the absolute measurements.
3
Also shown
in
Figure
1
is
a
mathematical
axis
for
describing
the
two
surfaces. The surface
of
the
test
optic
is
described
by
its
distance,
Ti,
from the
axis.
Likewise,
the
distance from
the axis
to
the
reference
surface
is
given
by
Ri.
Each
measurement
is
simply
a
difference,
Ti

Ri.
In
conventional interferometry
it is
only this
difference which
can
be
determined
and
not
the
individual values
of
Ri
and
Ti.
Consider
in
more
detail this two
measurement
scheme. The
expected values
that
would
be
obtained
are
shown
in
Figure
2.
In
the
left
hand column
are the
measurement
values
for
each
position
of
the
initial configuration.
The
right hand column shows
the
expected
mea surement values after
the test
optic
has
been translated. Note
that the
reference
values,
Ri
stay
fixed
with
respect
to
the
measurement
positions.
The
difference between
the
items
in
the two
columns
is
referred
to as
a
PseudoShear
In
terf erogram.
In
Figure
3
the group
of
equations
that
result
is
shown.
This
set
is
readily solved
for the
values
of Ti
after
choosing
an
arbitrary
value
for
T one.
After
obtaining
the
values
for
each
Ti,
the
equations
of
Figure
2
may
be
solved
for
each
Ri.
This
two
position technique
is
simple
and
readily
produces
a
set of
values
for
both
Ti
and
Ri.
However, there
is
no
provision
for
including additional
data nor
is
there
provision
for
error estimation. These
deficiencies
are
alleviated
in
the
following multiposition
analysis.
Multiposition
PSI
Combining
the results
of
three
or
more measurements
with lateral shifts
makes
it
possible
to
extend
the
previous analysis
and
simultaneously
include
a
leastsquares
error reduction. This
provides
a
means
to
estimate
the
measurement
error
as
well
as
the best
estimate
of
the
profile
of
the
test
optic
and
the
reference
surface. The
analysis presented
will
be
for
the case
of
three
measurements made with equalspaced translations
as
represented
in
Figure
4.
Each column
corresponds
to
one
of
the
measurement
sets;
each
being
a
translation
of
one unit
from
the
next.
Also
appearing
in
these
equations
is
an
error
term,
E. It
is
assumed
that
there
is
an
independent,
random
error
associated with
each
value.
There
are
two
independent sets
of
differences which may
be
obtained
from
these;
they
are
shown
in
Figure
5.
Note
that
D12
and
D23
are
both
estimates
of
the
same
quantity
and
com
parable pairs
comprise
the
lists.
If
additional
measurements
are made,
they
can
be
easily included
at
this
point.
Because
the
number
of
unknowns,
including
the
error
terms,
exceeds the number
of
equa
tions,
this
set
is
solved
in
the least
square
sense.
Figure
6
shows the
derivation
of
the
normal equations from the
conditions
for
minimizing
the
sum
of
the squares
of
the
errors.
Substitution
and
evaluation
of
the partial
derivatives
gives the normal equations.
The nor mal
equations may
be
displayed
in
somewhat more compact form
by the
use
of
matrices. Addi tionally,
display
of
the
matrix
for
a
specific
size array shows
the
starting
and
ending values explicitly.
Figure
7
shows
the
matrix
equation
for
the case
of
five
positions
on
the
test optic.
The
matrix
on
the
left
is
a
symetric
band matrix.
All
diagonal
elements
are
four except
the
last.
This same form
prevails
for
any
matrix
size greater than
two.
The column vector
on
the right
maintains
the
pattern
shown
for
arrays
of
any
size.
This
matrix
may
be
readily
inverted
to
solve
for
the
profile
of
the
test optic,
Ti.
Ob
taining
the
values
for
Ti
might
be
considered
the
end
of
the
problem. However,
for
most
ac
curate
work,
we are
required
to
determine
an
error
estimate
for
the
measurements.Returning
to
Figure
4
for
a
moment,
it
will
be
noted the
differences might
be
arranged
to
eliminate
Ti
and
thus
be
used
to
determine
Ri.
The
resulting differences
are
shown
in
Figure
8.
These
are
shifted differences
and
yield
PSI
data
for
the
interferometer. Application
of
the
leastsquare methodology
to
this
set of
equations
produces
the
matrix equation
for de
termination
of
the
reference
as
shown
in
Figure
9.
Again,
the
patterns
of
the
components
of
this
equation
is
shown
for
dimension
five
by
five.
Extension
to
matrices
of
other sizes
is
readily accomplished. The
solution
for
the
values
of
Ri is
straight forward.
The
solutions
of
these two
matrix
equations have given
values
for
Ri
and
Ti.
Substitu
tion
of
these values into the
equations
of
Figure
4
permit
determination
of
the errors
as
sociated with each
value.
From
the
enumerated
values,
error
statistics
can be
developed
in
order
to
place
confidence bounds
on
the
absolute measurements.
Downloaded From: http://proceedings.spiedigitallibrary.org/ on 09/04/2014 Terms of Use: http://spiedl.org/terms
Summary
A technique for processing interferometric data has been described which allows absolute
measurements to be obtained without reference to a standard.
The Pseudo Shear Interferogram
is obtained by digital processing of a sequence of measurements with lateral shifts between
them.
The algorithm has provision for including additional data and for error analysis.Surface profile of Test optic
M1
M2
M3
M4M5
Surface profile of Reference
Figure 1.
Measurement geometry.
At each position, themeasurement yields the distance between the surfaces.The measurements, Mi, are made at equally spaced positions.MEASUREMENT
(Initial)
M11= Tl R1
1
MEASUREMENT 2
(Translated)M12=R2 R2M22=
T1 R2
M13=T3 R3M23=
T2 R3
M14=T4 R4M24=
T3 R4Figure 2.
Measurements before and after
translation of test optic.
4
Summary
A
technique
for
processing interferometric
data
has
been
described which
allows absolute
measurements
to
be
obtained
without reference
to
standard. The
PseudoShear
Interferogram
is
obtained
by
digital
processing
of
sequence
of
measurements with
lateral shifts between
them.
The
algorithm
has
provision
for
including additional data
and for
error
analysis.
Surface profile
of
Test
optic
Ml M2 M3 M4
M5
Surface profile
of
Reference
Figure
1
Measurement geometry.
At
each position,
the
measurement
yields
the
distance between
the
surfaces.
The
measurements,
Mi,
are
made
at
equally spaced positions.
MEASUREMENT
1
Initial)
MEASUREMENT
2
Translated)Mll=
Tl
Rl
M12=
R2
R2
M13=
T3
R3
M14=
T4
R4
M22=
Tl
R2
M23=
T2
R3
M24=
T3
R4
Figure
Measurements
before
and
after
translation
of
test optic.
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M12  M22
=
T2  T1M13  M23
=
T3  T2M14  M24
=
T4  T3M15  M25
=
T5  T4
Figure 3.
Measurement differences.
This set of equations is readily
solved for the values of Ti.
MEASUREMENT 1
(Initial)
M11= T1 R1+ EllM12= T2 R2+ El2M13= T3 R3+ E13M14= T4 R4+ El4M15= T5 R5+ E15
MEASUREMENT 2
MEASUREMENT 3
(1st translation)(2nd translation)M22= T1 R2+ E22M23= T2 R3+ E23M24= T3 R4+ E24M25= T4 R5+ E25
M33= Ti R3+ E33M34= T2 R4+ E34M35= T3 R5+ F35
Figure 4.
Measurements before and after each translation.
Error terms are included.
FIRST DIFFERENCESD12 = M12  M22 = T2  Ti + E112D13 = M13  M23 = T3  T2 + E113D14 = M14  M24 = T4  T3 + E114SECOND DIFFERENCESD23 = M23  M33 = T2  Tl + E223D24 = M24  M34 = T3  T2 + E224D25 = M25  M35 = T4  T3 + E225
Figure 5.
Measurement differences.
These arerepeated estimates of the same quantities.
5
M12 M22
=
T2

Tl
M13 M23
=
T3

T2
M14 M24
=
T4 T3
M15 M25
=
T5 T4
Figure
3
Measurement
differences. This
set
of
equations
is
readily solved
for the
values
of
Ti.
ME SUREMENT
1
ME SUREMENT
2
ME SUREMENT
3
(Initial)
(1st
translation)
(2nd
translation
Mll=
Tl R1+ Ell
M12=
T2 R2+ E12
M22=
Tl R2+ E22
M13=
T3
R3
+
E13 M23 T2 R3+ E23
M33=
Tl
R3
+
E33
M14=
T4 R4+ E14 M24= T3 R4+ E24
M34=
T2 R4+ E34
M15=
T5 R5+ E15
M25=
T4 R5+ E25
M35=
T3 R5+ E35
Figure
4.
Measurements
before
and
after each translation.
rror
terms
are included.
FIRST
DIFFERENCES
D12
=
M12 M22
=
T2 Tl
+
E112
D13
=
M13 M23
=
T3
T2'+ E113
D14
=
M14

M24
=
T4

T3
+
E114
SECOND DIFFERENCES
D23
=
M23

M33
=
T2

Tl
£223
D24
=
M24 M34
=
T3 T2
+
E224D25
=
M25

M35

T4
T3
+
E225
Figure
5
Measurement
differences.
These
are
repeated estimates
of
the same quantities.
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