Method and apparatus for alias free measurement of optical transfer function
First Claim
1. A method of alias free measurement of an optical transfer function of an optical system with a test pattern wherein the test pattern comprises a tilt angle that is relative to a predetermined axis, a period, a duty cycle and a bar with a bar center line, the method comprising the steps of:
- (a) imaging the test pattern with the optical system to generate an image of the test pattern wherein the image has a sampled signal representation;
(b) measuring the tilt angle from the sampled signal representation;
(c) measuring the period from the sampled signal representation;
(d) measuring the duty cycle from the sampled signal representation;
(e) measuring the bar center line from the sampled signal representation;
(f) calculating a plurality of one dimensional Fourier transforms of the sampled signal representation to generate a plurality of frequency representations; and
(g) determining the optical transfer function based on the plurality of frequency representations, the tilt angle, period, duty cycle, and bar center line.
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Abstract
Measurement of the optical transfer function of an optical system imaging a bar pattern. A one dimensional fast Fourier transform processes sampled image data from the bar pattern. A model of the system utilizes robust measurements of the period, duty cycle and center of each stripe in the pattern. A signal alias free optical transfer function is estimated from the plurality of one dimensional frequency representation of the signal. An idealized bar pattern synthesized from the measured parameters of period, duty cycle and the center of each stripe in the pattern generates an ideal optical transfer function. The noise reduced optical transfer function, OTF, is estimated from OTF of the signal and OTF of the synthesized bar pattern.
25 Citations
45 Claims
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1. A method of alias free measurement of an optical transfer function of an optical system with a test pattern wherein the test pattern comprises a tilt angle that is relative to a predetermined axis, a period, a duty cycle and a bar with a bar center line, the method comprising the steps of:
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(a) imaging the test pattern with the optical system to generate an image of the test pattern wherein the image has a sampled signal representation; (b) measuring the tilt angle from the sampled signal representation; (c) measuring the period from the sampled signal representation; (d) measuring the duty cycle from the sampled signal representation; (e) measuring the bar center line from the sampled signal representation; (f) calculating a plurality of one dimensional Fourier transforms of the sampled signal representation to generate a plurality of frequency representations; and (g) determining the optical transfer function based on the plurality of frequency representations, the tilt angle, period, duty cycle, and bar center line. - View Dependent Claims (2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23)
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14. The method of claim 1 wherein the step of measuring the duty cycle of the test pattern comprises the steps of:
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(a) estimating centers of dark and bright pulses in the sampled signal representation; (b) calculating bright pixel values and dark pixel values in the neighborhood of the centers; (c) computing a period of the test pattern from the centers; (d) calculating an average pixel value for a given period; and (e) computing the duty cycle from the average pixel value and the bright pixel values and dark pixel values.
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15. The method of claim 1 wherein the step of measuring the bar center line further comprises the steps of:
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(a) estimating centers of a plurality of pulses in the sampled signal representation representing a bar of the bar pattern; and (b) estimating a center line of the bar pattern from the plurality of centers.
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16. The method of claim 15 wherein the step of estimating a center of a pulse in the plurality of pulses in a sampled signal representation representing a bar of the test pattern, where a center line of a single pulse, X.sup.(n)k, center of kth single pulse and nth iteration of procedure, for fixed y, are determined by ##EQU39## and where ψ
- (x)=ψ
(-x) is a symmetric windowing function.
- (x)=ψ
- 17. The method of claim 16 wherein an initial position
- space="preserve" listing-type="equation">X.sub.k.sup.(0)
is chosen by satisfying ##EQU40## where
space="preserve" listing-type="equation">q(X.sub.k.sup.(0))=∫
xψ
(x-X.sub.k.sup.(0))s.sub.0 (x,y)dx/∫
ψ
(x-X.sub.k.sup.(0))s.sub.0 (x,y)dx.
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18. The method of claim 1 wherein a data representation s0)(x,y) represents the sampled signal representation, further comprising the steps of:
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(a) multiplying s0 (x,y) by a windowing function w(x) which results in a windowed sampled signal representation s(x,y)=w(x-τ
y)s0 (x,y) where τ
is tan(θ
) and θ
is rotated angle;(b) taking a one dimensional Fourier transform in a x-dimension of s(x,y) which results in a frequency representation of a windowed signal Sx (fx,y) where fx is a frequency corresponding to the x-dimension; (c) multiplying Sx (fx,y) by exp (j2π
fx X0), where X0 is a center of a center line, which results in a frequency representation of a shifted windowed signal Sx (fx,y)(d) measuring an alias free frequency representation, B(fx), from Sx (fx,y); (e) extract nth harmonic components, α
n, from Sx (fx,y) and(f) the optical transfer function, given α
n, for frequency nf0 is computed by;
##EQU41## and f0 =1/T where T is the period of the bar pattern.
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19. The method of claim 18 where the alias free frequency representation, B(fx), can be estimated from a frequency representation of the shifted window signal Sx (fx,y) by ##EQU42## for i=1, . . . , N where x =1/x where x is the sampling dimension in the x direction and where y is the y position in the ith row and where θ
- is the tilt angle.
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20. The method of claim 18 further comprising the steps of estimating a frequency component of B(fx) by minimizing a cost function.
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21. The method of claim 19, wherein the cost function to estimate B(fx) is represented by the following function:
- ##EQU43## where r(i) is a weight value for the cost function for the ith row where x =1/X where X is the sampling dimension in the x direction and where yi is the y position in the ith row and where θ
is the tilt angle.
- ##EQU43## where r(i) is a weight value for the cost function for the ith row where x =1/X where X is the sampling dimension in the x direction and where yi is the y position in the ith row and where θ
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22. The method of claim 21 wherein the cost function is minimized by satisfying the following linear equation for each fx :
- ##EQU44## where k=. . . , -2, -1, 0, 1, 2, . . . .
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23. The method of claim 18 wherein α
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n is determined by minimizing the following cost function;
##EQU45## where N is some interval containing nf0 and W is the Fourier transform of the windowing function w(x), and where n is the nth harmonic of the sampled signal representation.
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n is determined by minimizing the following cost function;
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24. A method of noise reduced alias free measurement of an optical transfer function of an optical system with a test pattern wherein the test pattern comprises a tilt angle that is relative to a predetermined axis, a period, a duty cycle and a bar with a bar center line, the method comprising the steps of:
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(a) imaging the test pattern with the optical system to generate a sampled signal representation of the test pattern; (b) calculating a first optical transfer function for the optical system, the period and tilt angle; (c) synthesizing a synthesized test pattern from the tilt angle, period and optical transfer function; (d) calculating a second optical transfer function for the synthesized test pattern test pattern; and (e) calculating a noise reduced optical transfer function from the first optical transfer function and the second optical transfer function. - View Dependent Claims (25, 26, 27, 28, 29)
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30. An apparatus for alias free measurement of an optical transfer function of an optical system with a test pattern wherein the test pattern comprises a tilt angle that is relative to a predetermined axis, a period, a duty cycle and a bar with a bar center line, the apparatus comprising:
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(a) means for imaging the test pattern with the optical system to generate an image of the test pattern wherein the image has a sampled signal representation, wherein the means for imaging the test pattern has a sampled signal representation output; (b) means for measuring the tilt angle, connected to the sampled signal representation output, having a tilt angle output; (c) means for measuring the period, connected to the sampled signal representation output, having a period output; (d) means for measuring the duty cycle, connected to the sampled signal representation output, having a duty cycle output; (e) means for measuring the bar center line, connected to the sampled signal representation output, having a bar center line output; (f) means for calculating a plurality of one dimensional Fourier transforms, connected to the sampled signal representation output, having a plurality of frequency representations output; and (g) means for determining the optical transfer function connected to the plurality of frequency representations output, the tilt angle output, period output, duty cycle output, and bar center line output, wherein the means for determining the optical transfer function has an optical transfer function output. - View Dependent Claims (31, 32)
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33. An apparatus for noise reduced alias free measurement of an optical transfer function of an optical system with a test pattern wherein the test pattern comprises a tilt angle that is relative to a predetermined axis, a period, a duty cycle and a bar with a bar center line, the apparatus comprising:
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(a) means for imaging the test pattern with the optical system having a sampled signal representation of the test pattern output; (b) means for calculating an optical transfer function for the optical system, connected to the sampled signal representation of the test pattern output, having a first optical transfer function output, a tilt angle output and period output; (c) means for synthesizing a synthesized test pattern, connected to the first optical transfer function output, tilt angle output and period output, having a synthesized test pattern output; (d) means for calculating a second optical transfer function for the synthesized test pattern, connected to the test pattern output, having a second optical transfer function output; and (e) means for calculating a noise reduced optical transfer function, connected to the first optical transfer function output and second optical transfer function output, having a noise reduced optical transfer function output.
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34. A method of alias free measurement of an optical transfer function of an optical system with a test pattern wherein the test pattern comprises a tilt angle that is relative to a predetermined axis, a period, a duty cycle and a bar with a bar center line, the method comprising the steps of:
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(a) imaging the test pattern with the optical system to generate an image of the test pattern wherein the image has a sampled signal representation; (b) measuring the tilt angle from the sampled signal representation; (c) measuring the period from the sampled signal representation; (d) measuring the duty cycle from the sampled signal representation; (e) measuring the bar center line from the sampled signal representation; (f) calculating a plurality of one dimensional Fourier transforms of the sampled signal representation to generate a plurality of frequency representations; (g) determining the optical transfer function based on the plurality of frequency representations, the tilt angle, period, duty cycle, and bar center line; (h) synthesizing an ideal test pattern from the tilt angle, period, duty cycle and center line; (i) calculating an ideal optical transfer function for the ideal test pattern; (j) calculating a noise reduced optical transfer function from the optical transfer function and the ideal optical transfer function; and (k) describing the ideal test pattern by;
##EQU46## wherein a bar pattern function Bp comprises ##EQU47## wherein a sampling function comprises ##EQU48## wherein a bar function comprises ##EQU49## wherein a pixel function comprises ##EQU50## and wherein;
X,Y are spacial sampling dimensions,T is the period, X0 is the centerline, r is the duty cycle, Yc is the center of the sampled image in the y dimension, τ
is the tan(θ
) and θ
is the tilt angle, and* is the two dimensional convolution operator.
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35. A method of alias free measurement of an optical transfer function of an optical system with a test pattern wherein the test pattern comprises a tilt angle that is relative to a predetermined axis, a period, a duty cycle and a bar with a bar center line, the method comprising the steps of:
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(a) imaging the test pattern with the optical system to generate an image of the test pattern wherein the image has a sampled signal representation; (b) measuring the period from the sampled signal representation; (c) measuring the duty cycle from the sampled signal representation; (d) measuring the bar center line from the sampled signal representation; (e) calculating a plurality of one dimensional Fourier transforms of the sampled signal representation to generate a plurality of frequency representations; (f) determining a cost function by the equation;
##EQU51## wherein;
s0 =the sampled signal representation,τ
=tanθ
where θ
represents the tilt angle,i=ith row of the sampled signal representation, N=number of rows of the sampled signal representation, and yi =the position in the image corresponding to the ith row of the sampled signal representation. (g) determining the tilt angle by a minimization of the cost function; and (h) determining the optical transfer function based on the plurality of frequency representations, the tilt angle, period, duty cycle, and bar center line.
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36. A method of alias free measurement of an optical transfer function of an optical system with a test pattern wherein the test pattern comprises a tilt angle that is relative to a predetermined axis, a period, a duty cycle and a bar with a bar center line, the method comprising the steps of:
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(a) imaging the test pattern with the optical system to generate an image of the test pattern wherein the image has a sampled signal representation; (b) measuring the tilt angle from the sampled signal representation; (c) measuring the period from the sampled signal representation; (d) measuring the duty cycle from the sampled signal representation; (e) measuring the bar center line from the sampled signal representation; (f) calculating a plurality of one dimensional Fourier transforms of the sampled signal representation to generate a plurality of frequency representations; (g) determining the optical transfer function based on the plurality of frequency representations, the tilt angle, period, duty cycle, and bar center line; and (h) measuring the period of the test pattern by, (i) estimating centers of dark and bright pulses in the sampled signal representation representing a bar of the test pattern, and (ii) computing the period from the centers; and (i) estimating a center of a pulse in a sampled signal representing a bar of the test pattern wherein a center line of a single pulse, X.sup.(n)k, center of kth single pulse and nth iteration of procedure, for fixed y, by ##EQU52## and where ψ
(x)=ψ
(-x) is a symmetric windowing function. - View Dependent Claims (37)
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38. A method of alias free measurement of an optical transfer function of an optical system with a test pattern wherein the test pattern comprises a tilt angle that is relative to a predetermined axis, a period, a duty cycle and a bar with a bar center line, the method comprising the steps of:
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(a) imaging the test pattern with the optical system to generate an image of the test pattern wherein the image has a sampled signal representation; (b) measuring the tilt angle from the sampled signal representation; (c) measuring the period from the sampled signal representation; (d) measuring the duty cycle from the sampled signal representation; (e) measuring the bar center line from the sampled signal representation; (f) calculating a plurality of one dimensional Fourier transforms of the sampled signal representation to generate a plurality of frequency representations; (g) determining the optical transfer function based on the plurality of frequency representations, the tilt angle, period, duty cycle, and bar center line; and (h) measuring the bar center line by; (i) estimating centers of a plurality of pulses in the sampled signal representation representing a bar of the bar pattern, (ii) estimating a center line of the bar pattern from the plurality of centers; (i) estimating a center of a pulse in the plurality of pulses in a sampled signal representation representing a bar of the test pattern, where a center line of a single pulse, X.sup.(n)k, center of kth single pulse and nth iteration of procedure, for fixed y, by ##EQU54## and where ψ
(x)=ψ
(-x) is a symmetric windowing function. - View Dependent Claims (39)
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40. A method of alias free measurement of an optical transfer function of an optical system with a test pattern wherein the test pattern comprises a tilt angle that is relative to a predetermined axis, a period, a duty cycle and a bar with a bar center line, the method comprising the steps of:
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(a) imaging the test pattern with the optical system to generate an image of the test pattern wherein the image has a sampled signal representation; (b) measuring the tilt angle from the sampled signal representation; (c) measuring the period from the sampled signal representation; (d) measuring the duty cycle from the sampled signal representation; (e) measuring the bar center line from the sampled signal representation; (f) calculating a plurality of one dimensional Fourier transforms of the sampled signal representation to generate a plurality of frequency representations; (g) determining the optical transfer function based on the plurality of frequency representations, the tilt angle, period, duty cycle, and bar center line; and (h) determining a data representation s0 (x,y) to represent the sampled signal representation, by; (i) multiplying s0 (x,y) by a windowing function w(x) which results in a windowed sampled signal representation s(x,y)=w(x-τ
y)s0 (x,y) where τ
is tan(θ
) and θ
is rotated angle;(ii) taking a one dimensional Fourier transform in a x-dimension of s(x,y) which results in a frequency representation of a windowed signal Sx (fx,y) where fx is a frequency corresponding to the x-dimension; (iii) multiplying Sx (fx,y) by exp (j2π
fx X0), where X0 is a center of a center line, which results in a frequency representation of a shifted windowed signal Sx (fx,y);(iv) measuring an alias free frequency representation, B(fx), from Sx (fx,y); (v) extract and harmonic components, α
n, from Sx (fx,y); and(vi) the optical transfer function, given α
n, for frequency nf0 is computed by;
##EQU56## and f0 =1/T where T is the period of the bar pattern. - View Dependent Claims (41, 42, 43, 44, 45)
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Specification