# US 7,612,893 B2 · Scanning interferometric methods and apparatus for measuring aspheric surfaces and wavefronts

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- Number: US 7,612,893 B2
- Document ID: US7612893B2
- Status: Granted
- Kind: B2
- Application number: 11/901,010
- Filed: 2007-09-14
- Published: 2008-03-20
- Granted: 2009-11-03
- Earliest claimed: 2006-09-19
- Assignees: Zygo Corporation
- Inventors: Michael Küchel
- Web page: https://patentgrep.com/patent/US7612893B2

## CPC

- G01B 11/2441: {using interferometry}
- G01M 11/025: {by determining the shape of the object to be tested (measuring contours or curvatures by optical means G01B11/24)}
- G01M 11/0271: {by using interferometric methods}
- G01B: MEASURING LENGTH, THICKNESS OR SIMILAR LINEAR DIMENSIONS; MEASURING ANGLES; MEASURING AREAS; MEASURING IRREGULARITIES OF SURFACES OR CONTOURS
- G01M: TESTING STATIC OR DYNAMIC BALANCE OF MACHINES OR STRUCTURES; TESTING OF STRUCTURES OR APPARATUS, NOT OTHERWISE PROVIDED FOR

## Abstract

Interferometric scanning method(s) and apparatus for measuring test optics having aspherical surfaces including those with large departures from spherical. A reference wavefront is generated from a known origin along a scanning axis. A test optic is aligned on the scanning axis and selectively moved along it relative to the known origin so that the reference wavefront intersects the test optic at the apex of the aspherical surface and at one or more radial positions where the reference wavefront and the aspheric surface intersect at points of common tangency (“zones”) to generate interferograms containing phase information about the differences in optical path length between the center of the test optic and the one or more radial positions. The interferograms are imaged onto a detector to provide an electronic signal carrying the phase information. The axial distance, ν, by which the test optic is moved with respect to the origin is interferometrically measured, and the detector pixel height corresponding to where the reference wavefront and test surface slopes match for each scan position is determined. The angles, α, of the actual normal to the surface of points Q at each “zone” are determined against the scan or z-axis. Using the angles, α, the coordinates z and h of the aspheric surface are determined at common points of tangency and at their vicinity with αmin≦α≦αmax, where αmin and αmax correspond to detector pixels heights where the fringe density in the interferogram is still low. The results can be reported as a departure from the design or in absolute terms.

## Claims

### Claim 1

An interferometric scanning method for measuring test optics having aspheric surfaces including those with large departures from spherical, said method comprising the steps of:
generating a reference wavefront from a known origin along a scanning axis;
aligning a test optic on the scanning axis and selectively moving it along said scanning axis relative to said known origin so that said reference wavefront intersects the test optic at the apex of the aspheric surface and at one or more radial positions where the reference wavefront and the aspheric surface intersect at points of common tangency in circular zones around said scanning axis to generate interferograms containing phase information about the differences in optical path length between the center of the test optic and the one or more radial positions;
imaging the interferograms onto a detector to provide an electronic signal carrying the phase information;
measuring the axial distance, ν, by which the test optic is moved with respect to said origin;
determining the detector pixel height corresponding to where the reference wavefront and test surface slopes match for each scan position;
determining the angles, α, of the actual normal to the surface of points at each zone as a function of the distance along the scanning axis; and
using the angles, α, determining the coordinates z and h of the aspheric surface at the common points of tangency.

### Claim 2 (depends on 1)

The interferometric scanning method of claim 1 wherein the angles α are calculated by finding the positions in the image of the surface at scan positions w where:
ⅆ
q
ⅆ
h
pix
=
zero
for every pixel, where q is the difference between reference and test surfaces near the point where their slopes match minus the difference of the test and reference surfaces at the apex and then calculating the corresponding angles α by:
arccos
⁡
(
1
-
ⅆ
q
ⅆ
w
)
.

### Claim 3 (depends on 1)

The interferometric scanning method of claim 1 further including the step of determining the coordinates z and h of the aspheric surface at said common points of tangency and at their vicinity with αmin≦α≦αmax, where αmin and αmax correspond to detector pixels heights for a scan position ν where the fringe density in said interferogram is still low.

### Claim 4 (depends on 1)

The interferometric scanning method of claim 1 further including reporting the shape of said aspheric surface as the difference between its measured shape and its design shape.

### Claim 5 (depends on 1)

The interferometric scanning method of claim 1 wherein said reference wavefront is at least a partial spherical wavefront generated from said known origin along said scanning axis through the use of a spherical reference surface along said scanning axis upstream of said known origin.

### Claim 6 (depends on 1)

The interferometric scanning method of claim 1 wherein the angles, α, are calculated using the optical properties of the system used to image the interferograms.

### Claim 7 (depends on 6)

The interferometric scanning method of claim 6 wherein said optical properties used to calculate said angles α comprise a mathematical function that relates pixel height, hpix, in image space to said angle α in object space.

### Claim 8 (depends on 7)

The interferometric scanning method of claim 7 wherein said function is in form a polynomial.

### Claim 9 (depends on 8)

The interferometric scanning method of claim 8 wherein said polynomial function is of form:
M
⁡
(
h
pix
)
=
h
pix
sin
⁢
⁢
α
=
a
0
+
a
2
·
h
pix
2
+
a
4
·
h
pix
4
+
a
6
·
h
pix
6
+
…
M
⁡
(
sin
⁢
⁢
α
)
=
h
pix
sin
⁢
⁢
α
=
b
0
+
b
2
·
sin
2
⁢
α
+
b
4
·
sin
4
⁢
α
+
b
6
·
sin
6
⁢
α
+
…
where the coefficients an and bn are determined experimentally or by ray tracing the interferometer.

### Claim 10 (depends on 9)

The interferometric scanning method of claim 9 wherein said coefficients an and bn are determined experimentally prior to making any measurements of an aspheric surface.

### Claim 11 (depends on 6)

The interferometric scanning method of claim 6 wherein said optical properties are determined while measuring an aspheric test surface.

### Claim 12 (depends on 11)

The interferometric scanning method of claim 11 wherein said optical properties are determined using an optimization routine.

### Claim 13 (depends on 10)

The interferometric scanning method of claim 10 wherein said coefficients an and bn are determined with an artifact of known shape.

### Claim 14 (depends on 9)

The interferometric scanning method of claim 9 wherein sin α can be substituted by:
sin
⁢
⁢
α
=
h
pix
M
⁡
(
h
pix
)
⁢
⁢
⁢
or by
⁢
⁢
sin
⁢
⁢
α
=
h
pix
M
⁡
(
sin
⁢
⁢
α
)
.

### Claim 15 (depends on 9)

The interferometric scanning method of claim 9 wherein said coordinates h and z are given by:
(
h
z
)
=
(
sin
⁢
⁢
α
0
-
cos
⁢
⁢
α
1
)
·
(
R
s
R
v
)
=
(
sin
⁢
⁢
α
0
-
1
-
sin
2
⁢
α
1
)
·
(
R
s
R
v
)
(
h
z
)
=
(
h
pix
M
⁡
(
h
pix
)
0
-
1
-
(
h
pix
M
⁡
(
h
pix
)
)
2
1
)
·
(
R
s
⁡
(
h
pix
)
R
v
⁡
(
h
pix
=
0
)
)
(
h
z
)
=
(
h
pix
M
⁡
(
h
pix
)
0
-
1
-
(
h
pix
M
⁡
(
h
pix
)
)
2
1
)
·
(
R
2
-
d
zone
⁡
(
h
pix
)
R
2
-
d
apex
⁡
(
h
pix
=
0
)
)
.

### Claim 16

Interferometric scanning apparatus for measuring test optics having aspheric surfaces including those with large departures from spherical, said apparatus having a scanning axis and comprising:
a radiation source and optical arrangement for generating a reference wavefront from a known origin along said scanning axis;
a precision manipulator for aligning a test optic on said scanning axis and selectively moving it along said scanning axis relative to said known origin so that said reference wavefront intersects the test optic at the apex of the aspheric surface and at one or more radial positions where the reference wavefront and the aspheric surface intersect at points of common tangency in circular zones around said scanning axis to generate interferograms containing phase information about the differences in optical path length between the center of the test optic and the one or more radial positions;
a two-dimensional detector;
optics for imaging the interferograms onto said detector to provide an electronic signal carrying the phase information;
a distance measuring device for measuring the axial distance, ν, by which the test optic is moved with respect to said origin; and
a programmable device for:
determining the detector pixel height corresponding to where the reference wavefront and test surface slopes match for each scan position;
determining the angles, α, of the actual normal to the surface of points at each zone as a function of the distance along the scanning axis; and
using the angles, α, determining the coordinates z and h of the aspheric surface at the common points of tangency.

### Claim 17 (depends on 16)

The interferometric scanning apparatus of claim 16 wherein the angles α are calculated by finding the positions in the image of the surface at scan positions w where:
ⅆ
q
ⅆ
h
pix
=
zero
for every pixel, where q is the difference between reference and test surfaces near the point where their slopes match minus the difference of the test and reference surfaces at the apex and then calculating the corresponding angles α by:
arccos
⁡
(
1
-
ⅆ
q
ⅆ
w
)
.

### Claim 18 (depends on 16)

The interferometric scanning apparatus of claim 16 wherein said programmable device is further configured to determine the coordinates z and h of the aspheric surface at said common points of tangency and at their vicinity with αmin≦α≦αmax, where αmin and αmax correspond to detector pixels heights where the fringe density in said interferogram is still low.

### Claim 19 (depends on 16)

The interferometric scanning apparatus of claim 16 wherein said programmable device is further configured to report the shape of said aspheric surface as the difference between its measured shape and its design shape.

### Claim 20 (depends on 16)

The interferometric scanning apparatus of claim 16 wherein optical arrangement includes a spherical reference surface upstream of said known origin and said radiation source and optical arrangement are configured and arranged with respect to one another to provide said reference wavefront as at least a partial spherical wavefront generated from said known origin along said scanning axis.

### Claim 21 (depends on 16)

The interferometric scanning apparatus of claim 16 wherein the optical properties of the system used to image the interferograms is used to calculate said angles α and comprises a mathematical function that relates pixel height, hpix, in image space to said angle α in object space.

### Claim 22 (depends on 21)

The interferometric scanning apparatus of claim 21 wherein said optical properties are determined while measuring an aspheric test surface.

### Claim 23 (depends on 22)

The interferometric scanning method of claim 22 wherein said optical properties are determined using an optimization routine.

### Claim 24 (depends on 21)

The interferometric scanning apparatus of claim 21 wherein said function is in form a polynomial.

### Claim 25 (depends on 16)

The interferometric scanning apparatus of claim 16 wherein said interferometric scanning apparatus has the general form of a Fizeau.