Error compensation in phase shifting interferometry
In certain aspects, disclosed methods include combining reference light reflected from a reference surface with test light reflected from a test surface to form combined light, the test and reference light being derived from a common source, sinusoidally varying a phase between the test light and reference light, where the sinusoidal phase variation has an amplitude u, recording at least one interference signal related to changes in an intensity of the combined light in response to the sinusoidal variation of the phase, determining information related to the phase using a phase shifting algorithm that has a sensitivity that varies as a function of the sinusoidal phase shift amplitude, where the sensitivity of the algorithm at 2 u is 10% or less of the sensitivity of the algorithm at u.
1. A method, comprising:
combining reference light reflected from a reference surface with test light reflected from a test surface to form combined light, the test and reference light being derived from a common source;
sinusoidally varying a phase between the test light and reference light, where the sinusoidal phase variation has an amplitude u;
recording at least one interference signal related to changes in an intensity of the combined light in response to the sinusoidal variation of the phase;
determining information related to the phase using a phase shifting algorithm that has a sensitivity that varies as a function of the sinusoidal phase shift amplitude, where the sensitivity of the algorithm at 2 u is 10% or less of the sensitivity of the algorithm at u.
2. The method of claim 1 , wherein the sensitivity characterizes how the phase shifting algorithm responds to the interference signal as a function of the amplitude u.
3. The method of claim 1 , wherein the sensitivity of the algorithm at 2 u is 5% or less of the sensitivity of the algorithm at u.
4. The method of claim 1 , wherein the sensitivity of the algorithm at 2 u is 2% or less of the sensitivity of the algorithm at u.
5. The method of claim 1 , wherein the sensitivity of the algorithm at 2 u is 1% or less of the sensitivity of the algorithm at u.
6. The method of claim 1 , wherein the sensitivity of the algorithm at N u is 10% or less of the sensitivity of the algorithm at U, where N is an integer greater than 2.
7. The method of claim 1 , wherein the sensitivity of the algorithm at N u is 5% or less of the sensitivity of the algorithm at U, where N is an integer greater than 2.
8. The method of claim 1 , wherein the sensitivity of the algorithm at N u is 2% or less of the sensitivity of the algorithm at u, where N is an integer greater than 2.
9. The method of claim 1 , wherein the sensitivity of the algorithm at N u is 1% or less of the sensitivity of the algorithm at U, where N is an integer greater than 2.
10. The method of claim 1 , wherein the sensitivity of the algorithm has a minimum close to 2 u and in a range from 2 u±0.2 u, the sensitivity is 10% or less of the sensitivity of the algorithm at u.
11. The method of claim 1 , wherein the phase shifting algorithm has a sensitivity to contributions to the combined light from test light that makes a double pass between the test and reference surfaces that is 10% or less of the sensitivity of the algorithm to contributions to the combined light from test light that makes a single pass between the test and reference surfaces.
12. The method of claim 1 , wherein the phase shifting algorithm is a function of a plurality of values of the interference signal and coefficients h odd and h even , where h odd and h even are sensitive to the odd and even harmonics of the sinusoidal phase variation frequency.
13. The method of claim 12 , wherein the phase shifting algorithm is a function of a ratio of the coefficients h odd to the coefficients h even .
14. The method of claim 12 , wherein the phase shifting algorithm determines a phase value, θ, of the interference signal according to the equation
tan
(
θ
)
=
∑
j
(
h
odd
)
j
I
j
∑
j
(
h
even
)
j
I
j
,
where j=0, 1, . . . , P−1 and I j are intensity samples of the combined light composing the interference signal acquired during P corresponding phase shifts.
15. The method of claim 14 , wherein the coefficients h odd and h even are selected subject to the constraints
∑
j
(
h
odd
)
j
(
h
even
)
j
=
0
,
∑
j
(
h
odd
)
j
cos
[
v
(
α
j
+
φ
)
]
=
0
for
v
=
2
,
4
,
6
…
and
∑
j
(
h
even
)
j
cos
[
v
(
α
j
+
φ
)
]
=
0
for
v
=
1
,
3
,
5
…
.
wherein the sinusoidal phase variation is of the form
φ j =u cos(α j ),
where α j =jΔα+Δα/2 and Δα=2π/P.
16. The method of claim 12 , wherein the sensitivity of the phase shifting algorithm is determined based on the coefficients h odd and h even .
17. The method of claim 12 , wherein the sensitivity of the phase shifting algorithm is determined based on a filter function for the odd harmonics, F odd (u), and a filter function for the even harmonics, F even (u).
18. The method of claim 12 , wherein the sensitivity is expressed as a sensitivity function, sens(u), which is given by
sens
(
u
)
=
F
odd
(
u
)
2
+
F
even
(
u
)
2
2
Γ
where Γ is a constant.
19. The method of claim 18 , wherein the filter function F odd (u) is a function of odd-ordered Bessel functions and the filter function F even (u) is a function of even-ordered Bessel functions.
20. The method of claim 1 , wherein u is greater than π/2 radians.
21. The method of claim 1 , wherein the recording comprises recording a plurality of interference signals, each corresponding to a different location of the test surface.
22. The method of claim 1 , wherein the recording comprises detecting an interference pattern formed by the combined light using a multi-element detector.
23. The method of claim 1 , wherein the test and reference light is combined using a Fizeau interferometer.
24. The method of claim 1 , wherein the phase is sinusoidally varied by varying an optical path length difference between the test and reference surfaces.
25. The method of claim 1 , wherein the phase is sinusoidally varied by varying a wavelength of the test and reference light.
26. The method of claim 1 , wherein the information comprises a phase of the interference signal.
27. The method of claim 1 , wherein the information comprises a profile of the test surface.
28. The method of claim 1 , further comprising modifying the test surface based on the information.
29. The method of claim 28 , wherein modifying the test surface comprises modifying a shape of the test surface.
30. The method of claim 1 , wherein the test surface is a surface of an optical element.
31. The method of claim 30 , wherein the optical element comprises a lens, a mirror, or an optical flat.
32. A method, comprising:
combining reference light reflected from a reference surface with test light reflected from a test surface to form combined light, the test and reference light being derived from a common source;
sinusoidally varying a phase between the test light and reference light;
recording at least one interference signal based on a modulation of the combined light in response to varying the phase;
determining information related to the phase using a phase shifting algorithm that has a sensitivity to contributions to the combined light from test light that makes a double pass between the test and reference surfaces that is 10% or less of the sensitivity of the algorithm to contributions to the combined light from test light that makes a single pass between the test and reference surfaces.
33. A system, comprising:
an interferometer which during operation combines test light from a test surface with reference light from a reference surface to form combined light, where the test and reference light are derived from a common source;
a phase shifting component which during operation introduces a sinusoidal phase shift between the test light and reference light, where the sinusoidal phase variation has an amplitude u;
a detector positioned to detect the combined light and provide at least one interference signal based on the modulation of the combined light beam in response to varying the phase; and
an electronic controller coupled to the phase shifting component and the detector, wherein the electronic controller is configured to determine information related to the phase during operation of the system, where the electronic controller determined the information using a phase shifting algorithm that has a sensitivity that varies as a function of the sinusoidal phase shift amplitude, where the sensitivity of the algorithm at 2 u is 10% or less of the sensitivity of the algorithm at u.
34. The system of claim 33 , wherein the interferometer is a Fizeau interferometer.
35. The system of claim 33 , wherein the phase shifting component is the common source, the common source being a wavelength tunable light source.
36. The system of claim 33 , wherein the phase shifting component is a transducer coupled to the test surface or the reference surface.
37. A method, comprising:
combining reference light reflected from a reference surface with test light reflected from a test surface to form combined light, the test and reference light being derived from a common source;
sinusoidally varying a phase between the test light and reference light, where the sinusoidal phase variation has an amplitude u;
recording at least one interference signal related to changes in an intensity of the combined light in response to the sinusoidal variation of the phase;
determining information related to the phase using a phase shifting algorithm that has a sensitivity that varies as a function of the sinusoidal phase shift amplitude, wherein the sensitivity of the algorithm has a minimum close to 2 u and in a range from 2 u+−0.2 u, the sensitivity is 10% or less of the sensitivity of the algorithm at u to suppress calibration errors in u.