Scanning interferometer for aspheric surfaces and wavefronts

• US 6,879,402 B2
• Filed: 11/15/2002
• Issued: 04/12/2005
• Est. Priority Date: 11/16/2001
• Status: Active Grant

• Current Assignee: Zygo Corporation (Ametek Incorporated)
• Original Assignee: Zygo Corporation (Ametek Incorporated)
• Inventors: Küchel, Michael
• Primary Examiner(s): Turner, Samuel A.
• Assistant Examiner(s): Lyons, Michael A.
• Application Number:
  US10/295,479
• Publication Number:
  US 20030103215A1
• Time in Patent Office: 879 Days
• Field of Search: 356/457, 356/458, 356/489, 356/495, 356/511, 356/512, 356/513, 356/515
• US Class Current: 356/513
• CPC Class Codes: G01B 11/2441 using interferometry
• CPC Class Codes: G01B 2290/65 Spatial scanning object beam
• CPC Class Codes: G01B 9/02039 by matching the wavefront with a particular object surface shape
• CPC Class Codes: G01B 9/02057 by using common path configuration, i.e. reference and object path almost entirely overlapping
• CPC Class Codes: G01B 9/02072 by calibration or testing of interferometer
• CPC Class Codes: G01J 9/02 by interferometric methods using interferometers for measuring optically the linear dimensions of objects G01B9/02
• CPC Class Codes: G01M 11/0242 by measuring geometrical properties or aberrations
• CPC Class Codes: G01M 11/025 by determining the shape of the object to be tested measuring contours or curvatures by optical means G01B11/24
• CPC Class Codes: G01M 11/0271 by using interferometric methods

Abstract

Interferometric scanning method(s) and apparatus for measuring rotationally and non-rotationally symmetric test optics either having aspherical surfaces or that produce aspherical wavefronts. A spherical or partial spherical wavefront is generated from a known origin along an optical axis. The test optic is aligned with respect the optical axis and selectively moved along it relative to the known origin so that the spherical wavefront intersects the test optic at the apex of the aspherical surface and at radial positions where the spherical wavefront and the aspheric surface intersect at points of common tangency. An axial distance, ν, and optical path length, p, are interferometrically measured as the test optic is axially scanned by the spherical wavefront where ν is the distance by which the test optic is moved with respect to the origin and p is the optical path length difference between the apex of an aspherical surface associated with the test optic and the apex of the circles of curvature that intersect the aspherical surface at the common points of tangency. Coordinates of the aspherical surface are calculated wherever the circles of curvature have intersected the aspherical surface and in correspondence with the interferometrically measured distances, ν and p. Afterwards, the shape of the aspheric surface is calculated. Where the test optic comprises a refracting optic a known spherical reflecting surface is provided upstream of the refracting optic for movement along the optical axis and a known wavefront is made to transit the refracting optic, reflects from the known spherical surface, again transits the refracting optic traveling towards the known origin after which the interferogram is formed. In another aspect of the invention, a spherical reference surface is provided to form a Fizeau that is used to generate phase information for measuring spheres, mild aspheres, and multiple mild aspheres.

Claims

1. A scanning method for measuring rotationally and non-rotationally symmetric test optics having aspherical surfaces, said method comprising the steps of:

generating at least a partial spherical wavefront from a known origin along a scanning axis through the use of a spherical reference surface positioned along said scanning axis upstream of said known origin;

aligning a test optic with respect to said scanning axis and selectively moving said test optic along said scanning axis relative to said known origin so that said spherical wavefront intersects the test optic at the apex of the aspherical surface and at one or more radial positions where the spherical wavefront and the aspheric surface intersect at points of common tangency 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 said interferogram onto a detector to provide an electronic signal carrying said phase information;

interferometrically measuring the axial distance, ν

, by which said test optic is moved with respect to said origin and calculating the optical path length differences, p, between the center of test optic and the one or more radial positions based on said phase differences contained in said electronic signal, said axial distance ν

being given by;

$v=R_2-R_0-\frac{1}{2}<br><br>O<br><br>\text{\hspace{1em}<br><br>}<br><br>P<br><br>\text{\hspace{1em}<br><br>}<br><br>D_{{\stackrel{\_}{U}}_1<br><br>{\stackrel{\_}{U}}_2}<br><br>\left({\mathrm{\delta <br><br>}}_c=0\right),$

where R₂ is the radius of said spherical reference surface, R₀ is the radius of the apex sphere of the aspherical surface, and OPD<sub>Ū

1Ū

2</sub> (δ

_c=0) is the optical path difference between the spherical reference surface and the aspherical surface measured in the vicinity of the apex where δ

is the angular deviation from the apex to the aspherical surface as seen from said known origin, and p is given by;

$p=\frac{1}{2}<br><br>\left(O<br><br>\text{\hspace{1em}<br><br>}<br><br>P<br><br>\text{\hspace{1em}<br><br>}<br><br>D_{{\stackrel{\_}{P}}_1<br><br>{\stackrel{\_}{P}}_2}<br><br>\left(\mathrm{\delta <br><br>}\right)-O<br><br>\text{\hspace{1em}<br><br>}<br><br>P<br><br>\text{\hspace{1em}<br><br>}<br><br>D_{{\stackrel{\_}{U}}_1<br><br>{\stackrel{\_}{U}}_2}<br><br>\left({\mathrm{\delta <br><br>}}_c=0\right)\right),$

where OPD_{{overscore (P)}1{overscore (P)}2} (δ

) is the optical path difference between said spherical reference surface and the aspheric surface at said radial positions;

calculating the coordinates, z and h, of the aspherical surface wherever said circles of curvature have intersected the aspherical surface at common points of tangency and in correspondence with the interferometrically measured distance, ν and

calculated optical path lengths, p; and

determining the shape of said aspheric surface based on said coordinate values and said optical path length differences.

2. The scanning method of claim 1 wherein the radius of the best fitting spheres to the aspherical surface at said radial positions is given by:

$R_e=\frac{\left(2-p^{\mathrm{\prime <br><br>}}\right)<br><br>p^{\mathrm{\prime <br><br>}}}{p^{\mathrm{″<br><br>}}}+\left(R_0+v-p\right)$

with their center locations being given by;

$h_e=-\frac{\sqrt{{\left(p^{\mathrm{\prime <br><br>}}<br><br>\left(2-p^{\mathrm{\prime <br><br>}}\right)\right)}^3}}{p^{\mathrm{″<br><br>}}}<br><br>\text{\hspace{1em}<br><br>}<br><br>\mathrm{and}<br><br>\text{\hspace{1em}<br><br>}<br><br>z_e=\frac{\left(2-3<br><br>p^{\mathrm{\prime <br><br>}}+p^{\mathrm{\prime <br><br>}<br><br>\text{\hspace{1em}<br><br>}<br><br>2}\right)<br><br>p^{\mathrm{\prime <br><br>}}}{p^{\mathrm{″<br><br>}}}+R_0+v.$

3. The scanning method of claim 1 wherein said phase information contained in said electronic signal is extracted to determine said optical path length differences through the step of phase shifting analysis.

4. The scanning method of claim 3 wherein said step of phase shifting analysis is effected by modulating the wavelength of the source for generating said at least partial spherical wavefront.

5. The scanning method of claim 1 wherein said detector comprises a two-dimensional CCD camera.

6. The scanning method of claim 5 wherein the measurable departure of said aspherical surfaces from said spherical reference sphere vary in accordance with the ability of said CCD camera to resolve resultant fringes with meaningful spatial detail.

7. The scanning method of claim 6 wherein said aspherical surfaces depart from said spherical reference surface by up to on the order of 20 micrometers, more or less.

8. The scanning method of claim 6 wherein said CCD camera is moved axially during a scan of a test aspherical surface to maintain optimal imaging conditions.

9. The scanning method of claim 1 wherein phase distribution of a trace along any azimuthal angle at the apex of the aspherical surface and at the one or more radial positions can, to first order, be approximated by parabolas as given by:

Δ

_centre(r)=φ

_centre+ar² 

 

Eq. A

for the apex part of the trace, and like another parabola given by;

$\begin{array}{cc}{\mathrm{\Delta <br><br>}}_z<br><br>\left(r\right)={\mathrm{\phi <br><br>}}_{\mathrm{centre}}+2<br><br>p<br><br>\frac{2<br><br>\mathrm{\pi <br><br>}}{\mathrm{\lambda <br><br>}}+{b<br><br>\left(r-r_{\mathrm{int}}\right)}^2& \mathrm{Eq}.\text{\hspace{1em}<br><br>}<br><br>B\end{array}$

for a radial position, where a is a constant, which depends on R₀ and v, and b is a constant which depends on the radius of curvature at the point (h, z) of the aspherical surface as well as on R₀ and v, a and b normally having different signs;

the only quantity to be evaluated being 2p, which is included in the difference of the phases between the center and the zone.

10. The scanning method of claim 9 wherein, to determine the difference in phase, the minimum of Δ

_centre(r) and Δ

_z(r) are evaluated and then the difference of both phase values at this minimum are taken.

11. The scanning method of claim 10 further including the step of using the measured phase-values in the neighborhood of the minimum.

12. The scanning method of claim 10 wherein, from the difference in phase from Equations (A) and (B), p is evaluated as a function of the scan-length v, and the values of h and z.

13. The scanning method of claim 12 wherein $z=p+\left(R_0+v-p\right)$

ⅆ

pⅆ

v$a<br><br>\text{\hspace{1em}<br><br>}<br><br>n<br><br>\text{\hspace{1em}<br><br>}<br><br>d$$h=\left(R_0+v-p\right)<br><br>\sqrt{\frac{d<br><br>p}{d<br><br>v}<br><br>\left(2-\frac{d<br><br>p}{d<br><br>v}\right)}$

where R=R₀+ν

−

p, with R₀ equal to the reciprocal apical curvature of the aspherical shape and $\frac{d<br><br>p}{d<br><br>v}$

is measured.

14. Scanning apparatus for measuring rotationally and non-rotationally symmetric test optics having aspherical surfaces, said apparatus comprising:

means for generating at least a partial spherical wavefront from a known origin along a scanning axis, said generating means including a spherical reference surface positioned along said scanning axis upstream of said known origin;

means for aligning a test optic with respect to said scanning axis and selectively moving said test optic along said scanning axis relative to said known origin so that said spherical wavefront intersects the test optic at the apex of the aspherical surface and at one or more radial positions where the spherical wavefront and the aspheric surface intersect at points of common tangency 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 areal detector;

means for imaging said interferogram onto said areal detector to provide an electronic signal carrying said phase information;

means for interferometrically measuring the axial distance, ν

, by which said test optic is moved with respect to said origin and calculating the optical path length differences, p, between the center of test optic and the one or more radial positions based on said phase differences contained in said electronic signal, said axial distance ν

being given by;

$v=R_2-R_0-\frac{1}{2}<br><br>O<br><br>\text{\hspace{1em}<br><br>}<br><br>P<br><br>\text{\hspace{1em}<br><br>}<br><br>D_{{\stackrel{\_}{U}}_1<br><br>{\stackrel{\_}{U}}_2}<br><br>\left({\mathrm{\delta <br><br>}}_c=0\right),$

where R₂ is the radius of said spherical reference surface, R₀ is the radius of the apex sphere of the aspherical surface, and OPD<sub>Ū

1Ū

2</sub>(δ

_c=0) is the optical path difference between the spherical reference surface and the aspherical surface measured in the vicinity of the apex where δ

is the angular deviation from the apex to the aspherical surface as seen from said known origin, and p is given by;

$p=\frac{1}{2}<br><br>\left(O<br><br>\text{\hspace{1em}<br><br>}<br><br>P<br><br>\text{\hspace{1em}<br><br>}<br><br>D_{{\stackrel{\_}{P}}_1<br><br>{\stackrel{\_}{P}}_2}<br><br>\left(\mathrm{\delta <br><br>}\right)-O<br><br>\text{\hspace{1em}<br><br>}<br><br>P<br><br>\text{\hspace{1em}<br><br>}<br><br>D_{{\stackrel{\_}{U}}_1<br><br>{\stackrel{\_}{U}}_2}<br><br>\left({\mathrm{\delta <br><br>}}_c=0\right)\right),$

where OPD_{{overscore (P)}1{overscore (P)}2}(δ

) is the optical path difference between said spherical reference surface and the aspherical surface at said radial positions; and

means for calculating the coordinates, z and h, of the aspherical surface wherever said circles of curvature have intersected the aspherical surface at common points of tangency and in correspondence with the interferometrically measured distance, ν and

calculated optical path lengths, p; and

determining the shape of said aspheric surface based on said coordinate values and said optical path length differences.

15. The scanning apparatus of claim 14 wherein the radius of the best fitting spheres to the aspherical surface at said radial positions is given by:

$R_e=\frac{\left(2-p^{\mathrm{\prime <br><br>}}\right)<br><br>p^{\mathrm{\prime <br><br>}}}{p^{\mathrm{″<br><br>}}}+\left(R_0+v-p\right)$

with their center locations being given by;

$h_e=-\frac{\sqrt{{\left(p^{\mathrm{\prime <br><br>}}<br><br>\left(2-p^{\mathrm{\prime <br><br>}}\right)\right)}^3}}{p^{\mathrm{″<br><br>}}}<br><br>\mathrm{and}<br><br>\text{\hspace{1em}<br><br>}<br><br>z_e=\frac{\left(2-3<br><br>p^{\mathrm{\prime <br><br>}}+p^{\mathrm{\prime <br><br>2}}\right)<br><br>p^{\mathrm{\prime <br><br>}}}{p^{\mathrm{″<br><br>}}}+R_0+v.$

16. The scanning apparatus of claim 14 wherein said means for calculating and determining is structured to extract said phase information contained in said electronic signal to determine said optical path length differences through by phase shifting analysis.

17. The scanning apparatus of claim 16 further including means for modulating the wavelength of the source for generating said at least partial spherical wavefront to effect said phase shifting analysis.

18. The scanning apparatus of claim 14 wherein said areal detector comprises a two-dimensional CCD camera.

19. The scanning apparatus of claim 18 wherein the measurable departure of said aspherical surfaces from said spherical reference sphere vary in accordance with the ability of said CCD camera to resolve resultant fringes with meaningful spatial detail.

20. The scanning apparatus of claim 19 wherein said aspherical surfaces depart from said spherical reference surface by up to on the order of 20 micrometers, more or less.

21. The scanning apparatus of claim 19 further including means for axially moving said CCD camera during a scan of a test aspherical surface to maintain optimal imaging conditions.

22. The scanning apparatus of claim 14 wherein phase distribution of a trace along any azimuthal angle at the apex of the aspherical surface and at the one or more radial positions can, to first order, be approximated by parabolas as given by:

Δ

_centre(r)=φ

_centre+ar² 

 

Eq. A

for the apex part of the trace, and like another parabola given by;

$\begin{array}{cc}{\mathrm{\Delta <br><br>}}_z<br><br>\left(r\right)={\mathrm{\phi <br><br>}}_{\mathrm{centre}}+2<br><br>p<br><br>\frac{2<br><br>\mathrm{\pi <br><br>}}{\mathrm{\lambda <br><br>}}+{b<br><br>\left(r-r_{\mathrm{int}}\right)}^2& \mathrm{Eq}.\text{\hspace{1em}<br><br>}<br><br>B\end{array}$

for a radial position, where a is a constant, which depends on R₀ and v, and b is a constant which depends on the radius of curvature at the point (h, z) of the aspherical surface as well as on R₀ and v, a and b normally having different signs;

the only quantity to be evaluated being 2ρ

, which is included in the difference of the phases between the center and the zone.

23. The scanning apparatus of claim 22 wherein said calculating and determining means determines the difference in phase by evaluating the minimum of Δ

_centre(r) and Δ

_Z(r) and then the difference of both phase values at this minimum are taken.

24. The scanning apparatus of claim 23 wherein said calculating and determining means is further is further operative to use the measured phase-values in the neighborhood of the minimum.

25. The scanning apparauts of claim 23 wherein, from the difference in phase from Equations (A) and (B), ρ

is evaluated as a function of the scan-length v, and the values of h and z.

26. The scanning apparatus of claim 14 wherein $z=p+\left(R_0+v-p\right)$

ⅆ

pⅆ

v$a<br><br>\text{\hspace{1em}<br><br>}<br><br>n<br><br>\text{\hspace{1em}<br><br>}<br><br>d$$h=\left(R_0+v-p\right)<br><br>\sqrt{\frac{d<br><br>p}{d<br><br>v}<br><br>\left(2-\frac{d<br><br>p}{d<br><br>v}\right)}$

where R=R₀+ν

−

p, with R₀ equal to the reciprocal apical curvature of the aspherical shape and $\frac{d<br><br>p}{d<br><br>v}$

is measured.