Cyclic error compensation in interferometry systems

• US 20050166118A1
• Filed: 02/17/2005
• Published: 07/28/2005
• Est. Priority Date: 07/08/2002
• Status: Active Grant

• Current Assignee: Zygo Corporation (Ametek Incorporated)
• Original Assignee: Zygo Corporation (Ametek Incorporated)
• Inventors: Hill, Henry A., Demarest, Frank C.
• Granted Patent: US 7,428,685 B2
• Time in Patent Office: Days
• US Class Current: 714/746
• 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 9/02059 Reducing effect of parasitic reflections, e.g. cyclic errors
• CPC Class Codes: G01B 9/0207 Error reduction by correction of the measurement signal based on independently determined error sources, e.g. using a reference interferometer
• CPC Class Codes: G01B 9/02084 Processing in the Fourier or frequency domain when not imaged in the frequency domain
• CPC Class Codes: G03F 7/70775 Position control, e.g. interferometers or encoders for determining the stage position

Abstract

The invention features a method including: (i) providing an interference signal S(t) from two beams directed along different paths, wherein the signal S(t) is indicative of changes in an optical path difference n{tilde over (L)}(t) between the different paths, where n is an average refractive index along the different paths, {tilde over (L)}(t) is a total physical path difference between the different paths, and t is time; (ii) providing one or more coefficients representative of one or more errors that cause the signal S(t) to deviate from an ideal expression of the form A₁ cos(ω_Rt+φ(t)+ζ₁), where A₁ and ζ₁ are constants, ω_R is an angular frequency difference between the two beams before being directed along the different paths, and φ(t)=nk{tilde over (L)}(t), with k=2π/λ and λ equal to a wavelength for the beams; (iii) calculating a linear combination of values of the signal S(t); and (iv) reducing the effect of the deviation of S(t) from the ideal expression on an estimate of {tilde over (L)}(t) using an error signal generated from the coefficients and one or more error basis functions derived at least in part from the linear combination of values of the signal S(t).

Claims

1. A method comprising:

providing an interference signal S(t) from two beams directed along different paths, wherein the signal S(t) is indicative of changes in an optical path difference n{tilde over (L)}(t) between the different paths, where n is an average refractive index along the different paths, {tilde over (L)}(t) is a total physical path difference between the different paths, and t is time;

providing one or more coefficients representative of one or more errors that cause the signal S(t) to deviate from an ideal expression of the form A₁ cos(ω

_Rt+φ

(t)+ζ

₁), where A₁ and ζ

₁ are constants, ω

_R is an angular frequency difference between the two beams before being directed along the different paths, and φ

(t)=nk{tilde over (L)}(t), with k=2π

/λ and

λ

equal to a wavelength for the beams;

calculating a linear combination of values of the signal S(t); and

reducing the effect of the deviation of S(t) from the ideal expression on an estimate of {tilde over (L)}(t) using an error signal generated from the coefficients and one or more error basis functions derived at least in part from the linear combination of values of the signal S(t).

2. The method of claim 1, wherein the linear combination comprises a quadrature signal {tilde over (S)}(t) based on values of the signal S(t).

3. The method of claim 2, wherein the quadrature signal {tilde over (S)}(t) is based on two values of the signal S(t).

4. The method of claim 2, wherein the quadrature signal {tilde over (S)}(t) is updated at a rate at which the signal S(t) is sampled.

5. The method of claim 1, wherein the linear combination comprises a transform signal D_q(t₁) based on values of the signal S(t), for a frequency value determined by q, updated at a time t₁.

6. The method of claim 5, wherein the transform signal D_q(t₁) comprises a sample of a discrete Fourier transform of the signal S(t).

7. The method of claim 6, wherein

$D_q$

( t 1 ) = ∑

n = 0 N - 1 ⁢

 

⁢

W n ⁢

S ⁢

 

⁢

( n ⁢

 

⁢

τ

+ t 1 ) ⁢

exp ⁢

{ - j ⁡

( 2 ⁢

π

⁡

( n - N - 1 2 ) ⁢

q / N ) } ,

where τ

=1/ƒ

_S, W_n window function centered at

$n=\frac{N-1}{2}.$

8. The method of claim 7, further comprising selecting q as an integer approximately equal to Nƒ

_M/ƒ

_S and storing the transform signal D_q(t₁) under conditions for which ƒ

_M can be estimated as ƒ

_M≈

(ω

_R+dφ

(t)/dt)/2π

.

9. The method of claim 6, wherein the transform signal D_q(t₁) is updated at a rate lower than a rate ƒ

_S at which the signal S(t) is sampled.

10. The method of claim 9, wherein the transform signal D_q(t₁) is updated at a rate ƒ

_R/m based on a time-shifted signal S(t+lm/ƒ

_R) to yield an updated transform signal, where l and m are positive integers, and q is an integer approximately equal to Nƒ

_M/ƒ

_S.

11. The method of claim 10, further comprising storing the transform signal D_q(t₁) under conditions for which ƒ

_M can be estimated as ƒ

_M≈

(ω

_R+dφ

(t)/dt)/2π

.

12. The method of claim 6, wherein the discrete Fourier transform includes a window function.

13. The method of claim 6, wherein the discrete Fourier transform is folded.

14. The method of claim 2, wherein the error signal comprises a time-domain signal S_ψ

(t).

15. The method of claim 14, wherein reducing the effect of the deviation of S(t) from the ideal expression on an estimate of {tilde over (L)}(t) comprises subtracting the error signal from S(t).

16. The method of claim 1, wherein the error signal comprises a signal D_ψ

(t₁) updated at a time t₁.

17. The method of claim 16, wherein reducing the effect of the deviation of S(t) from the ideal expression on an estimate of {tilde over (L)}(t) comprises subtracting the error signal from a discrete Fourier transform of values of S(t).

18. The method of claim 1, further comprising directing the two beams along the different paths and measuring the interference signal S(t).

19. The method of claim 18, wherein at least one of the beams is directed to reflect from a movable measurement object before producing the interference signal S(t).

20. The method of claim 19, wherein the beam directed to contact the movable measurement object reflects from the measurement object multiple times before producing the interference signal S(t).

21. The method of claim 19, wherein the beams are directed to reflect from different locations of the movable measurement object before producing the interference signal S(t).

22. The method of claim 1, wherein the errors correspond to spurious beam paths.

23. The method of claim 1, wherein the deviation can be expressed as

$\sum _{}$

m , p ⁢

 

⁢

A m , p ⁢

cos ⁡

( ω

R ⁢

t + m p ⁢

φ

⁡

( t ) + ζ

m , p ) ,

where p=1, 2, 3 . . . , and m is any integer not equal to p, and where the provided coefficients comprise information corresponding to at least some of A_m,p and ζ

_m,p.

24. The method of claim 2, wherein ω

_R is non-zero.

25. The method of claim 24, further comprising:

providing a reference signal S_R(t)=A_R cos(ω

_Rt+ζ

_R), where A_R and ζ

_R are constants; and

calculating a quadrature reference signal {tilde over (S)}_R(t) based on the signal S_R(t),

wherein the error basis functions are derived from the signals S(t), {tilde over (S)}(t), S_R(t), and {tilde over (S)}_R(t).

26. The method of claim 25, further comprising measuring the reference signal S_R(t) based on an output from a common source of the two beams.

27. The method of claim 25, further comprising generating reference signals S_R(t) and {tilde over (S)}_R(t) from a lookup table indexed synchronously with sampling of S(t).

28. The method of claim 2, wherein calculating the quadrature signal {tilde over (S)}(t) comprises calculating the quadrature signal {tilde over (S)}(t) based on the expression

$\tilde{S}$

( t ) = ( cot ⁢

 

⁢

ω

M ⁢

τ

) ⁢

S ⁡

( t - 2 ⁢

τ

) - cos ⁢

 

⁢

2 ⁢

 

⁢

ω

M ⁢

τ

sin ⁢

 

⁢

ω

M ⁢

τ

⁢

S ⁡

( t - τ

) ,

where τ

>

0 and ω

_M is an instantaneous rate of change of a phase of the interference signal S(t).

29. The method of claim 28, wherein calculating the quadrature signal {tilde over (S)}(t) further comprises approximating ω

_M according to ω

_M≈

ω

_R+dφ

(t)/dt, where φ

(t) in the expression for ω

_M is determined from the interference signal S(t) assuming the deviation of S(t) from the ideal expression is negligible.

30. The method of claim 28, wherein calculating the quadrature signal {tilde over (S)}(t) further comprises approximating ω

_M according to ω

_M≈

ω

_R.

31. The method of claim 30, wherein calculating the quadrature signal {tilde over (S)}(t) comprises calculating the quadrature signal {tilde over (S)}(t) according to the simplified expression

$\tilde{S}$

( t ) = 1 3 ⁡

[ S ⁡

( t - τ

) + S ⁡

( t - 2 ⁢

τ

) ] ⁢

 

⁢

for ⁢

 

⁢

τ

= ( π

+ 6 ⁢

π

⁢

 

⁢

N ) / 3 ⁢

 

⁢

ω

R ,

where N is a non-negative integer.

32. The method of claim 25, wherein calculating the quadrature reference signal {tilde over (S)}(t) comprises calculating the quadrature reference signal {tilde over (S)}_R(t) based on the expression

${\tilde{S}}_R$

( t ) = ( cot ⁢

 

⁢

ω

R ⁢

τ

) ⁢

S R ⁡

( t - 2 ⁢

τ

) - cos ⁢

 

⁢

2 ⁢

 

⁢

ω

R ⁢

τ

sin ⁢

 

⁢

ω

R ⁢

τ

⁢

S R ⁡

( t - τ

) ,

where τ

>

0.

33. The method of claim 32, wherein calculating the quadrature reference signal {tilde over (S)}_R(t) comprises calculating the quadrature reference signal {tilde over (S)}_R(t) according to the simplified expression

${\tilde{S}}_R$

( t ) = 1 3 ⁡

[ S R ⁡

( t - τ

) + S R ⁡

( t - 2 ⁢

τ

) ] ⁢

 

⁢

for ⁢

 

⁢

τ

= ( π

+ 6 ⁢

π

⁢

 

⁢

N ) / 3 ⁢

 

⁢

ω

R ,

where N is a non-negative integer.

34. The method of claim 1, wherein the interference signal S(t) is provided at a data rate that is an integer multiple of ω

_R/2π

.

35. The method of claim 1, wherein each of the error basis functions corresponds to a function that includes one or more leading sinusoidal terms having a time-varying argument that corresponds to a time-varying argument of an error term that represents a portion of the deviation of S(t) from the ideal expression.

36. The method of claim 35, wherein the error basis functions correspond to one or more pairs of sine and cosine functions having an argument whose time-varying component has the form ω

_Rt+(m/p)φ

(t), where p is a positive integer and m is an integer not equal to p.

37. The method of claim 36, wherein a leading term for a function corresponding to one of the error basis functions has the same values of m and p as a time-varying argument of one of the error terms.

38. The method of claim 37, wherein each of the error basis functions is proportional to the function that includes one or more leading sinusoidal terms.

39. The method of claim 35, wherein the error basis functions correspond to one or more pairs of sine and cosine functions having an argument whose time-varying component has the form (m/p)φ

(t), where p is a positive integer and m is an integer not equal to p.

40. The method of claim 39, wherein a leading term for a function corresponding to one of the error basis functions has the same values of m and p as a time-varying argument of one of the error terms.

41. The method of claim 40, wherein each of the error basis functions is proportional to a linear combination of samples of the function that includes one or more leading sinusoidal terms.

42. The method of claim 39, wherein each of the pairs of sine and cosine functions corresponds to imaginary and real parts, respectively, of a complex signal.

43. The method of claim 36, wherein the error basis functions correspond to multiple pairs of the sine and cosine functions.

44. The method of claim 43, wherein the error basis functions comprise multiple pairs of the sine and cosine functions from a family of the sine and cosine functions with {(p=1, m=−

1), (p=1, m=0), (p=1, m=2), (p=1, m=3), and (p=2, m=1)}.

45. The method of claim 2, further comprising generating the error basis functions from the signals S(t) and {tilde over (S)}(t).

46. The method of claim 25, further comprising generating the error basis functions from the signals S(t), {tilde over (S)}(t), S_R(t), and {tilde over (S)}_R(t).

47. The method of claim 46, wherein the error basis functions are generated from algebraic combinations of the signals S(t), {tilde over (S)}(t), S_R(t), and {tilde over (S)}_R(t).

48. The method of claim 14, further comprising generating the error signal S_ψ

(t).

49. The method of claim 48, the error signal S_ψ

(t) is generated from a superposition of the error basis functions weighted by the coefficients representative of the errors.

50. The method of claim 1, further comprising determining a value for the optical path difference n{tilde over (L)}(t) from the interference signal S(t) after the effect of its deviations are reduced.

51. The method of claim 2, wherein the quadrature signal {tilde over (S)}(t) is calculated from the interference signal S(t) based on prior values of S(t) according to the approximation S(t)≈

S(t−

2π

N/ω

_R), where N is a positive integer.

52. The method of claim 14, wherein the error basis functions used to generate the error signal S_ψ

(t) are derived from prior values of the signals S(t) and {tilde over (S)}(t) according to the approximations S(t)≈

S(t−

2π

N/ω

_R) and {tilde over (S)}(t)≈

{tilde over (S)}(t−

2π

M/ω

_R), where N and M are positive integers.

53. The method of claim 1, wherein ω

_R>

100·

dφ

(t)/dt.

54. The method of claim 1, wherein ω

_R>

500·

dφ

(t)/dt.

55. An apparatus comprising a computer readable medium which during operation causes a processor to perform the method of claim 1.

61. A lithography method for use in fabricating integrated circuits on a wafer, the method comprising:

supporting the wafer on a moveable stage;

imaging spatially patterned radiation onto the wafer;

adjusting the position of the stage; and

monitoring the position of the stage using the method of claim 1.

62. A lithography method for use in the fabrication of integrated circuits comprising:

directing input radiation through a mask to produce spatially patterned radiation;

positioning the mask relative to the input radiation;

monitoring the position of the mask relative to the input radiation using the method of claim 1;

and

imaging the spatially patterned radiation onto a wafer.

63. A lithography method for fabricating integrated circuits on a wafer comprising:

positioning a first component of a lithography system relative to a second component of a lithography system to expose the wafer to spatially patterned radiation; and

monitoring the position of the first component relative to the second component using the method of claim 1.

64. A method for fabricating integrated circuits, the method comprising the lithography method of claim 61.

65. A method for fabricating integrated circuits, the method comprising the lithography method of claim 62.

66. A method for fabricating integrated circuits, the method comprising the lithography method of claim 63.

69. A method for fabricating a lithography mask, the method comprising:

directing a write beam to a substrate to pattern the substrate;

positioning the substrate relative to the write beam; and

monitoring the position of the substrate relative to the write beam using the interferometry method of claim 1.

56. An apparatus comprising:

an interferometry system which during operation directs two beams along different paths and provides an interference signal S(t) from the two beams, wherein the signal S(t) is indicative of changes in an optical path difference n{tilde over (L)}(t) between the different paths, where n is an average refractive index along the different paths, {tilde over (L)}(t) is a total physical path difference between the different paths, and t is time,

wherein imperfections in the interferometry system produce one or more errors that cause the signal S(t) to deviate from an ideal expression of the form A₁ cos(ω

_Rt+φ

(t)+ζ

₁), where A₁ and ζ

₁ are constants, ω

_R is an angular frequency difference between the two beams before being directed along the different paths, and φ

(t)=nk{tilde over (L)}(t), with k=2π

/λ and

λ

equal to a wavelength for the beams; and

an electronic processor which during operation receives the interference signal S(t) from the interferometry system, receives one or more coefficients representative of the one or more errors, calculates a linear combination of values of the signal S(t), and reduces the effect of the deviation of S(t) from the ideal expression on an estimate of {tilde over (L)}(t) using an error signal S_ψ

(t) generated from the coefficients and one or more error basis functions derived at least in part from the linear combination of values of the signal S(t).

57. A lithography system for use in fabricating integrated circuits on a wafer, the system comprising:

a stage for supporting the wafer;

an illumination system for imaging spatially patterned radiation onto the wafer;

a positioning system for adjusting the position of the stage relative to the imaged radiation; and

the apparatus of claim 56 for monitoring the position of the wafer relative to the imaged radiation.

58. A lithography system for use in fabricating integrated circuits on a wafer, the system comprising:

a stage for supporting the wafer; and

an illumination system including a radiation source, a mask, a positioning system, a lens assembly, and the apparatus of claim 56,

wherein during operation the source directs radiation through the mask to produce spatially patterned radiation, the positioning system adjusts the position of the mask relative to the radiation from the source, the lens assembly images the spatially patterned radiation onto the wafer, and the apparatus monitors the position of the mask relative to the radiation from the source.

59. A beam writing system for use in fabricating a lithography mask, the system comprising:

a source providing a write beam to pattern a substrate;

a stage supporting the substrate;

a beam directing assembly for delivering the write beam to the substrate;

a positioning system for positioning the stage and beam directing assembly relative one another; and

the apparatus of claim 56 for monitoring the position of the stage relative to the beam directing assembly.

67. A method for fabricating integrated circuits, the method comprising using the lithography system of claim 57.

68. A method for fabricating integrated circuits, the method comprising using the lithography system of claim 58.

60. An apparatus comprising:

an interferometry system which during operation directs two beams along different paths and provides an interference signal S(t) from the two beams, wherein the signal S(t) is indicative of changes in an optical path difference n{tilde over (L)}(t) between the different paths, where n is an average refractive index along the different paths, {tilde over (L)}(t) is a total physical path difference between the different paths, and t is time,

wherein imperfections in the interferometry system produce one or more errors that cause the signal S(t) to deviate from an ideal expression of the form A₁ cos(ω

_Rt+φ

(t)+ζ

₁), where A₁ and ζ

₁ are constants, ω

_R is an angular frequency difference between the two beams before being directed along the different paths, and φ

(t)=nk{tilde over (L)}(t), with k=2π

/λ and

λ

equal to a wavelength for the beams; and

an electronic processor which during operation receives the interference signal S(t) from the interferometry system, calculates a linear combination of values of the signal S(t), and calculates an estimate for one or more coefficients representative of the one or more errors based on filtering a signal derived at least in part from the linear combination of values of the signal S(t).