SINUSOIDAL PHASE SHIFTING INTERFEROMETRY

• US 20080180679A1
• Filed: 12/17/2007
• Published: 07/31/2008
• Est. Priority Date: 12/18/2006
• Status: Active Grant

• Current Assignee: Zygo Corporation (Ametek Incorporated)
• Original Assignee: Zygo Corporation (Ametek Incorporated)
• Inventors: de Groot, Peter
• Granted Patent: US 7,933,025 B2
• Time in Patent Office: Days
• US Class Current: 356/450
• CPC Class Codes: G01B 2290/35 Mechanical variable delay line
• CPC Class Codes: G01B 9/02004 using frequency scans
• CPC Class Codes: G01B 9/02007 Two or more frequencies or sources used for interferometric measurement using only beat G01B9/02003
• CPC Class Codes: G01B 9/0201 using temporal phase variation
• CPC Class Codes: G01B 9/02057 by using common path configuration, i.e. reference and object path almost entirely overlapping
• CPC Class Codes: G01B 9/02083 characterised by particular signal processing and presentation

Abstract

Disclosed is a method that includes combining a first light beam and at least a second light beam to form a combined light beam, introducing a sinusoidal phase shift with a frequency f between a phase of the first light beam and a phase of the second light beam, recording at least one interference signal based on a modulation of the combined light beam in response to the sinusoidal phase shift, where the interference signal includes at least three different frequency components, and outputting the information. For each interference signal, information related to the difference in optical path lengths of the first and second light beam is determined by comparing the intensity of the at least three different frequency components of the interference signal.

Claims

1. A method comprising:

combining a first light beam and at least a second light beam to form a combined light beam;

introducing a sinusoidal phase shift with a frequency f between a phase of the first light beam and a phase of the second light beam;

recording at least one interference signal based on a modulation of the combined light beam in response to the sinusoidal phase shift, the interference signal comprising at least three different frequency components;

for each interference signal, determining information related to the difference in optical path lengths of the first and second light beam by comparing the intensity of the at least three different frequency components of the interference signal; and

outputting the information.

2. The method of claim 1, wherein the comparing comprises:

assigning a respective weight to the intensity of each of the at least three different frequency components to provide a corresponding weighted intensity; and

comparing the weighted intensities.

3. The method of claim 2, wherein each of the at least three different frequency components has a frequency which is an integer multiple of f.

4. The method of claim 3, wherein the comparing further comprises:

comparing a sum of the weighted intensities corresponding to the at least three different frequency components at even multiples of f to a sum of the weighted intensities corresponding to the at least three different frequency components at odd multiples of f.

5. The method of claim 4, wherein the respective weights are selected so that the effect of an error on the intensity of a first of the at least three different frequency components is compensated by the effect of the error on the intensity of a second of the at least three different frequency components.

6. The method of claim 5 wherein the frequencies of the first and second frequency components are same-parity integer multiples of f.

7. The method of claim 2, wherein the at least three different frequency components comprise at least one frequency component with frequency greater than twice f.

8. The method of claim 3 wherein each of the at least three different frequency components has a frequency greater than three times f.

9. The method of claim 3, wherein the respective weights are selected to compensate an error.

10. The method of claim 9, wherein the respective weights are selected so that the effect of the error on the weighted intensity corresponding to a first frequency component is compensated by the effect of the error on the weighted intensity corresponding to a second frequency component.

11. The method of claim 9, wherein the error comprises a variation in the excursion of the sinusoidal phase shift from a nominal value.

12. The method of claim 9, wherein the error comprises additive random noise.

13. The method of claim 9, wherein the error comprises additive synchronous noise.

14. The method of claim 13, wherein the additive synchronous noise comprises noise at frequency ν

″

, and the at least three different frequency components do not comprise a component with frequency ν

″

.

15. The method of claim 9, wherein the error comprises multiplicative synchronous noise.

16. The method of claim 9, wherein the error comprises synchronous vibration noise.

17. The method of claim 16, wherein the synchronous vibration noise comprises noise at low frequencies, and the at least three different frequency components have frequencies greater than the low frequencies.

18. The method of claim 9, wherein the error comprises phase shift nonlinearity.

19. The method of claim 18, wherein the nonlinearity comprises a quadratic nonlinearity, and the at least three frequency components do not comprise a frequency component with a frequency equal to 2f.

20. The method of claim 9, wherein the error comprises phase shift calibration error.

21. The method of claim 9, wherein the error comprises phase shift timing offset error.

22. The method of claim 1, wherein the recording comprises sampling the interference signal at a sample rate.

23. The method of claim 22, wherein the Nyquist frequency corresponding to the sample rate is greater than the frequency of each of the at least three different frequency components.

24. The method of claim 22, wherein the Nyquist frequency corresponding to the sample rate is greater than three times f.

25. The method of claim 22, the Nyquist frequency corresponding to the sample rate is greater than seven times f.

26. The method of claim 1, wherein the sinusoidal phase shift φ

(t) is of the form

φ

(t)=u cos [α

(t)+φ

]

where u is the excursion of the sinusoidal phase shift, φ

is a timing offset, and

α

(t)=2π

ft

is the scaled time dependence with f equal to the frequency of the sinusoidal phase shift.

27. The method of claim 26, wherein the recording comprises, during a cycle of the sinusoidal phase shift, acquiring intensity data g_j for N successive sample positions each corresponding to a time t_j, where j=0, 1, 2, . . . N−

1.

28. The method of claim 27, further comprising arranging the sample positions symmetrically about the midpoint of a cycle of the sinusoidal phase shift such that

cos(α

(t_j)+φ

)=cos(α

(t_N-1-j)+φ

)

for j=0, 1, 2, . . . (N−

1)/2.

29. The method of claim 26, comprising providing a sinusoidal phase shift excursion u large enough that the interference signal recorded in response to the phase shift comprises frequency components with frequencies at three distinct integer multiples of f.

30. The method of claim 26, comprising providing a sinusoidal phase shift excursion u great enough that the interference signal recorded in response to the phase shift comprises frequency components at the first six integer multiples of f.

31. The method of claim 26 where u>

π

/2 radians.

32. The method of claim 27, wherein the determining comprises:

assigning a first respective weight w_j⁽¹⁾ to each of the intensity data g_j to provide a corresponding first weighted intensity;

assigning a second respective weight w_j⁽²⁾ to each of the intensity data g_j to provide a corresponding second weighted intensity;

calculating the ratio of the sum of first weighted intensities to the sum of the second weighted intensities; and

determining information related to the difference in optical path lengths based on the ratio.

33. The method of claim 32, comprising selecting the first and second respective weights to compensate an error.

34. The method of claim 33, wherein the timing offset is set to a nominal value φ

=0.

35. The method of claim 34, wherein the excursion u is set to a nominal value.

36. The method of claim 35, wherein

w_j⁽¹⁾=Γ

_even×

(h_odd)_j

and

w_j⁽²⁾=Γ

_odd×

(h_even)_j

where (h_odd)_j is the j^th element of a sampling vector h_odd, (h_even)_j is the j_th element of a sampling vector h_even and Γ

_even and Γ

_odd are normalization coefficients based on a model of the interference signal.

37. The method of claim 36, wherein the sampling vectors h_odd, h_even are selected subject to the constraints

$\sum _{}$

j 

( h odd ) j 

( h even ) j = 0. $\underset{j}{\sum <br><br>}<br><br>{\left(h_{\mathrm{odd}}\right)}_j<br><br>\cos <br><br>\left[v<br><br>\left({\mathrm{\alpha <br><br>}}_j+\mathrm{\varphi <br><br>}\right)\right]=0<br><br><br><br>\mathrm{for}<br><br><br><br>v=2,4,6<br><br><br><br>\mathrm{\dots <br><br>}$ $\mathrm{and}$ $\underset{j}{\sum <br><br>}<br><br>{\left(h_{\mathrm{even}}\right)}_j<br><br>\cos <br><br>\left[v<br><br>\left({\mathrm{\alpha <br><br>}}_j+\mathrm{\varphi <br><br>}\right)\right]=0<br><br><br><br>\mathrm{for}<br><br><br><br>v=1,3,5<br><br><br><br>\mathrm{\dots <br><br>}<br><br>.$

38. The method of claim 37, wherein the error comprises a variation in the excursion of the sinusoidal phase shift from the nominal value.

39. The method of claim 38, wherein the sampling vectors h_odd, h_even are selected such that the ratio of the normalization coefficients remains stable in response to the variation of the excursion from the nominal value.

40. The method of claim 37, wherein the error comprises additive random noise.

41. The method of claim 40, wherein the additive random noise comprises mean noise.

42. The method of claim 41, wherein the sampling vectors h_odd, h_even are selected subject to the constraint

$\frac{{\Gamma }_{}}{}$

odd p odd = Γ

even p even $\mathrm{where}$ $p_{\mathrm{odd}}=\sqrt{\underset{j}{\sum <br><br>}<br><br>{\left(h_{\mathrm{odd}}\right)}_j^2},<br><br>p_{\mathrm{even}}=\sqrt{\underset{j}{\sum <br><br>}<br><br>{\left(h_{\mathrm{even}}\right)}_j^2}.$

43. The method of claim 40, wherein the additive random noise comprises root mean square noise.

44. The method of claim 43, wherein the sampling vectors h_odd, h_even are selected subject to the constraint that the magnitude of the quantities Γ

_odd/p_odd and Γ

_even/p_even be maximized where

$p_{\mathrm{odd}}=\sqrt{\underset{j}{\sum <br><br>}<br><br>{\left(h_{\mathrm{odd}}\right)}_j^2},<br><br>p_{\mathrm{even}}=\sqrt{\underset{j}{\sum <br><br>}<br><br>{\left(h_{\mathrm{even}}\right)}_j^2}.$

45. The method of claim 37, wherein the error comprises additive synchronous noise.

46. The method of claim 45, wherein the additive synchronous noise comprises noise at frequency ν

″

, and the sampling vectors h_odd, h_even are selected subject to the constraint that the magnitude of the quantities

$\underset{j}{\sum <br><br>}<br><br>{\left(h_{\mathrm{odd}}\right)}_j<br><br>\cos <br><br>\left[v^{\mathrm{″<br><br>}}<br><br>\left({\mathrm{\alpha <br><br>}}_j+\mathrm{\varphi <br><br>}\right)\right]$ $\mathrm{and}$ $\underset{j}{\sum <br><br>}<br><br>{\left(h_{\mathrm{even}}\right)}_j<br><br>\cos <br><br>\left[v^{\mathrm{″<br><br>}}<br><br>\left({\mathrm{\alpha <br><br>}}_j+\mathrm{\varphi <br><br>}\right)\right]$

be minimized.

47. The method of claim 37, wherein the error comprises multiplicative synchronous noise.

48. The method of claim 47, wherein the multiplicative synchronous noise comprises noise at frequency ν

″

, and the sampling vectors h_odd, h_even are selected to minimize a predicted sensitivity of the determined information to the noise at frequency ν

″

, based on the model of the interference signal.

49. The method of claim 47, wherein the multiplicative synchronous noise comprises a sinusoid with frequency f oscillating in phase with the sinusoidal phase shift;

and

the sampling vectors h_odd, h_even are selected subject to the constraint that the magnitude of the quantities

$\underset{j}{\sum <br><br>}<br><br>{\left(h_{\mathrm{odd}}\right)}_j<br><br>\cos <br><br>\left[\left({\mathrm{\alpha <br><br>}}_j+\mathrm{\varphi <br><br>}\right)\right]$

and

$\underset{j}{\sum <br><br>}<br><br>{\left(h_{\mathrm{even}}\right)}_j<br><br>\cos <br><br>\left[\left({\mathrm{\alpha <br><br>}}_j+\mathrm{\varphi <br><br>}\right)\right]$

be minimized.

50. The method of claim 49, wherein:

the common source comprises a laser diode;

the providing a sinusoidal phase shift comprises sinusoidally varying the wavelength of a diode laser light source; and

the multiplicative synchronous noise is diode laser intensity noise.

51. The method of claim 37, wherein the error comprises synchronous vibration noise.

52. The method of claim 51, wherein the synchronous vibration noise comprises noise at frequency ν

″

, and sampling vectors h_odd, h_even are selected to minimize a predicted sensitivity of the determined information to the noise at frequency ν

″

, based on the model of the interference signal.

53. The method of claim 37, wherein the error comprises nonlinearity of the sinusoidal phase shift.

54. The method of claim 53, wherein the nonlinearity is a quadratic nonlinearity, and the sampling vectors h_odd, h_even are selected subject to the constraint that the magnitude of the quantities

$\sum _{}$

j 

( h odd ) j 

cos 

[ 2 

( α

j + ϕ

) ]

and

$\underset{j}{\sum <br><br>}<br><br>{\left(h_{\mathrm{even}}\right)}_j<br><br>\cos <br><br>\left[2<br><br>\left({\mathrm{\alpha <br><br>}}_j+\mathrm{\varphi <br><br>}\right)\right]$

be minimized.

55. The method of claim 37, wherein the error comprises phase shift timing offset error.

56. The method of claim 55, wherein sampling vectors h_odd, h_even are selected subject to the constraint that the magnitude of the quantity

$\frac{{\Gamma }_{}}{}$

even 

( ϕ

+ δ

ϕ

) Γ

odd 

( ϕ

+ δ

ϕ

) 

Γ

odd 

( ϕ

) Γ

even 

( ϕ

) - 1

be minimized, where φ

is a nominal value for the timing offset and δ

φ

is the deviation from the nominal value.

57. The method of claim 32, wherein the determining comprises calculating the inverse tangent of the ratio.

58. The method of claim 32, wherein:

the recording comprises acquiring intensity data g_j for N=8 successive measurement frames each corresponding to a time t_j such that α

(t_j)=jπ

/4+π

/8 for j=0, 1, 2, . . . 7; and

the determining comprises calculating a value for the phase difference θ

between the phase of first light beam and the phase of the second light beam based on the expression;

$\mathrm{\theta <br><br>}={\tan }^{-1}<br><br>\left(\frac{1.6647<br><br>\left(g_1-g_2\right)}{-g_0+g_1+g_2-g_3}\right),$

where g_i= g_i+ g_7-i, for i=0, 1, 2, 3.

59. The method of claim 58, where in the sinusoidal phase shift excursion u is set to a nominal value of 2.93 radians and the timing offset φ

is set to a nominal value of 0.

60. The method of claim 32, wherein:

the recording comprises acquiring intensity data g_i for N=16 successive measurement frames each corresponding to a time t_j such that α

(t_j)=jπ

/8+π

/16 for j=0, 1, 2, . . . 7; and

the determining comprises calculating a value for the phase difference θ

between the phase of first light beam and the phase of the second light beam based on the expression;

$\mathrm{\theta <br><br>}={\tan }^{-1}<br><br>\left(\frac{\begin{array}{c}2.646<br><br>\left(-g_0+g_7\right)+7.248<br><br>\left(g_1-g_6\right)+\\ 2.507<br><br>\left(-g_2+g_5\right)+6.758<br><br>\left(-g_3+g_4\right)\end{array}}{\begin{array}{c}1.375<br><br>\left(g_0+g_7\right)+1.410<br><br>\left(g_1+g_6\right)+\\ 8.099<br><br>\left(-g_2-g_5\right)+5.314<br><br>\left(g_3+g_4\right)\end{array}}\right),$

where g_i= g_i+ g_15-i, for i=0, 1, . . . 7.

61. The method of claim 60, where in the sinusoidal phase shift excursion u is set to a nominal value of 5.9 radians and the timing offset φ

is set to a nominal value of 0.

62. The method of claim 1, wherein the comparing comprises:

calculating a frequency transform of the interference signal at each of at least three frequencies; and

comparing the magnitudes of the calculated frequency transforms to determine information related to the difference in optical path lengths of the first and second light beam.

63. The method of claim 62 wherein the at least three frequencies are integer multiples of the sinusoidal phase shift frequency.

64. The method of claim 63, further comprising:

extracting the phases of one or more of the calculated frequency transforms, and determining additional information based on the extracted phases.

65. The method of claim 64, wherein the additional information is a value of the excursion of the sinusoidal phase shift.

66. The method of claim 64, wherein the additional information is a value of a timing offset.

67. The method of claim 62, wherein the frequency transforms are Fourier Transforms.

68. The method of claim 62, wherein the frequency transforms are Fast Fourier Transforms.

69. The method of claim 62, wherein the frequency transforms are Discrete Cosine Transforms.

70. The method of claim 68, wherein the Nyquist frequencies of the Fast Fourier Transforms are greater than three times f.

71. The method of claim 69, wherein the Nyquist frequencies of the Discrete Cosine Transforms are greater than three times f.

72. The method of claim 1, wherein the combining comprises

directing the first light beam to a first surface, directing the second light beam to a second surface, and forming an optical interference image from the combined light beam;

wherein the at least one interference signals each correspond to different location on the interference image.

73. The method of claim 72, wherein the information comprises a surface profile of one of the surfaces.

74. A system comprising:

an interferometer which during operation combines a first light beam and a second light beam derived from a common source to form combined light beam;

a phase shifting component which during operation introduces a sinusoidal phase shift between a phase of the first light beam and a phase of the second light beam;

a photo detector positioned to detect the combined light beam and provide at least one interference signal based on the modulation of the combined light beam in response to the phase shift; and

an electronic controller coupled to the phase shifting component and the photo detector, wherein the controller is configured to;

determine information related to the difference in optical path lengths of the first and second light beam by comparing the intensity of at least three frequency components of the interference signal.

75. The system of claim 74, wherein the interferometer is a Fizeau interferometer.

76. The system of claim 74, wherein the interferometer is an unequal path interferometer, and the phase-shifting component is configured to vary the wavelength at least one of the light beams.

77. The system of claim 76, wherein the phase shifting component is a wavelength tunable diode laser.

78. The system of claim 74, wherein the first light beam is directed to a surface and the phase-shifting component is a transducer coupled to the surface.

79. The system of claim 74, wherein the phase-shifting component is an acousto-optic modulator

80. The system of claim 74, wherein the phase-shifting component is an electro-optic modulator.

81. The system of claim 74, wherein:

the interferometer during operation directs the first light beam to a first surface, directs the second light beam to a second surface and forms an optical interference image from the combined light beam; and

the at least one interference signals each correspond to different location on the interference image.

82. The system of claim 81, wherein the information comprises a surface profile of one of the surfaces.

83-105. -105. (canceled)