Method and apparatus for determining best focus using dark-field imaging

1. A method of determining a best focus position of an object relative to a reference position in an optical imaging system, comprising the steps of:

a) forming a dark-field image of the object at different focus positions, each said dark-field image having a corresponding image intensity distribution with an average intensity and a variance of intensity;

b) forming a set of contrast values by calculating a contrast value for each said dark-field image based on said variance and said average intensity; and

c) determining the best focus position by fitting a Lorentzian function to said set of contrast values plotted as a function of said different focus positions;

wherein said step b) further includes the steps of:

d) digitizing each said dark-field image such that said image intensity distribution for each said dark-field image is a digitized image intensity distribution comprising discrete gray-scale intensity levels I n corresponding to a discrete plurality of n pixels; and

e) arranging each said digitized image intensity distribution into a histogram, H(I n ), of an amount of said pixels having a given said gray-scale intensity level, I n .

2. A method according to claim 1 , further including the steps of:

f) calculating said average intensity for each said discrete image intensity distribution via the equation

〈

I

〉

=

∑

I

n

⁢

H

⁡

(

I

n

)

⁢

I

n

∑

I

n

⁢

H

⁡

(

I

n

)

;

g) calculating said variance for each said discrete image intensity distribution via the equation

σ

I

2

=

∑

I

n

⁢

H

⁡

(

I

n

)

⁢

(

I

n

-

〈

I

〉

)

2

∑

I

n

⁢

H

⁡

(

I

n

)

;

and

h) calculating said contrast value, C, for each said discrete image intensity distribution via the equation

C

=

σ

I

〈

I

〉

.

3. A method according to claim 2 , wherein in said step c) of determining the best focus position includes performing a curve fit to the equation

C

=

a

1

+

a

2

(

z

-

a

3

)

2

+

a

4

,

wherein a 1 , a 2 , a 3 and a 4 are Lorentzian curve-fitting parameters, z indicates a distance along a focus direction, and said parameter a 3 corresponds to the best focus position along said focus direction.

4. A method according to claim 1 , wherein the object comprises a region of interest on a substantially reflective substrate.

5. A method according to claim 4 , wherein said region of interest includes a pattern formed on said substrate.

6. A method according to claim 5 , wherein said pattern is a predetermined structure capable of scattering light.

7. A method according to claim 5 , wherein said pattern is surface roughness.

8. A method according to claim 1 , wherein said reference position is a coordinate system of an apparatus that forms said plurality of dark-field images.

9. A method according to claim 1 , wherein the optical imaging system has a depth of field, and an effective focusing range of up to 10 times said depth of field.

10. In a computer system, a method of determining the best focus position of an object relative to a reference position, based on a plurality of dark-field images of the object, comprising the steps of:

a) storing, in a computer readable medium, data corresponding to the plurality of dark-field images, each said dark-field image being associated with a different focus position and having an associated image intensity distribution with an average and a variance;

b) forming, in said computer system, a set of contrast values by calculating a contrast for each said dark-field image, based on said variance and said average; and

c) determining, in said computer system, the best focus position by fitting a Lorentzian function to said set of contrast values slotted as a function of said different focus positions;

wherein said step b) further includes the steps of:

d) digitizing each said dark-field image with said image intensity distribution for each said dark-field image having a digitized image intensity distribution comprising discrete gray-scale intensity levels, I n corresponding to a discrete plurality of n pixels; and

e) arranging each said digitized image intensity distribution into a histogram, H(I n ), of an amount of said pixels having a given said gray-scale intensity level, I n .

11. A method in a computer system according to claim 10 , further including the steps of:

f) calculating said average intensity for each said discrete image intensity distribution via the equation

〈

I

〉

=

∑

I

n

⁢

H

⁡

(

I

n

)

⁢

I

n

∑

I

n

⁢

H

⁡

(

I

n

)

;

g) calculating said variance for each said discrete image intensity distribution via the equation

σ

I

2

=

∑

I

n

⁢

H

⁡

(

I

n

)

⁢

(

I

n

-

〈

I

〉

)

2

∑

I

n

⁢

H

⁡

(

I

n

)

;

and

h) calculating said contrast value, C, for each said discrete image intensity distribution via the equation

C

=

σ

I

〈

I

〉

.

12. A method in a computer system according to claim 11 , wherein said step c) of determining the best focus position includes performing a curve fit to the equation

C

=

a

1

+

a

2

(

z

-

a

3

)

2

+

a

4

,

wherein a 1 , a 2 , a 3 and a 4 are Lorentzian curve-fitting parameters, z indicates a distance along a focus direction, and said parameter a 3 corresponds to the best focus position along said focus direction.

13. A computer readable medium capable of holding data corresponding to at least four dark-field image intensity distributions measured at different focus positions, and holding instructions for determining a best focus position based on said data, by performing the steps of:

a) calculating an average intensity and a variance for each of said dark-field image intensity distributions;

b) forming a set of contrast values by calculating a contrast value for each image intensity distribution based on said variance and said average intensity; and

c) determining the best focus position by fitting a Lorentzian function to said set of contrast values clotted as a function of said different focus positions

wherein said step b) further includes the steps of:

d) digitizing each said dark-field image such that said image intensity distribution for each said dark-field image is a digitized image intensity distribution comprising discrete gray-scale intensity levels, I n , corresponding to a discrete plurality of n pixels; and

e) arranging each said digitized image intensity distribution into a histogram, H(I n ), of an amount of said pixels having a given said gray-scale intensity level, I n .

14. A computer readable medium according to claim 13 , further including the steps of:

f) calculating said average intensity for each said discrete image intensity distribution via the equation

〈

I

〉

=

∑

I

n

⁢

H

⁡

(

I

n

)

⁢

I

n

∑

I

n

⁢

H

⁡

(

I

n

)

;

g) calculating said variance for each said discrete image intensity distribution via the equation

σ

I

2

=

∑

I

n

⁢

H

⁡

(

I

n

)

⁢

(

I

n

-

〈

I

〉

)

2

∑

I

n

⁢

H

⁡

(

I

n

)

;

and

h) calculating said contrast value, C, for each said discrete image intensity distribution via the equation

C

=

σ

I

〈

I

〉

.

15. A computer readable medium according to claim 14 , wherein said step c) of determining the best focus position includes performing a curve fit to the equation

C

=

a

1

+

a

2

(

z

-

a

3

)

2

+

a

4

,

wherein a 1 , a 2 , a 3 and a 4 are Lorentzian curve-fitting parameters, z indicates a distance along a focus direction, and said parameter a 3 corresponds to the best focus position along said focus direction.