Optics: measuring and testing – By light interference – For dimensional measurement
Reexamination Certificate
2001-09-04
2003-03-11
Turner, Samuel A. (Department: 2877)
Optics: measuring and testing
By light interference
For dimensional measurement
Reexamination Certificate
active
06532073
ABSTRACT:
RELATED APPLICATIONS
This application claims the priorities of Japanese Patent Application No. 2000-277444 filed on Sep. 13, 2000 and Japanese Patent Application No. 2001-022633 filed on Jan. 31, 2001, which are incorporated herein by reference.
BACKGROUND OF THE INVENTION
1. Field of the Invention
The present invention relates to a fringe analysis error detection method and fringe analysis error correction method using Fourier transform method when analyzing fringe images by using phase shift methods; and, in particular, to a fringe analysis error detection method and fringe analysis error correction method in which PZTs (piezoelectric elements) are used for shifting the phase, and Fourier transform method is utilized when analyzing thus obtained image data with fringe patterns such as interference fringes, whereby the analyzed value can be made more accurate.
2. Description of the Prior Art
While light wave interferometry, for example, has conventionally been knows as an important technique concerning precise measurement of a wavefront of an object, there have recently been urgently demanded for developing an interferometry technique (sub-fringe interferometry) which can read out information from a fraction of a single interference fringe (one fringe) or less due to the necessity of measuring a surface or wavefront aberration at an accuracy of {fraction (1/10)} wavelength or higher.
Known as a typical technique widely used in practice as such a sub-fringe interferometry technique is the phase shift fringe analyzing method (also known as fringe scanning method or phase scanning method) disclosed in “Phase-Measurement Interferometry Techniques,” Progress in Optics, Vol. XXVI (1988), pp. 349-393.
In the phase shift method, one or more phase shift element such as PZTs (piezoelectric element), for example, are used for phase-shifting the relative displacement between an object to be observed and the reference, interference fringe images are captured each time when a predetermined phase amount is shifted, the interference fringe intensity at each point on the surface to be inspected is measured, and the phase of each point on the surface is determined by using the result of measurement.
For example, when carrying out a four-step phase shift method, respective interference fringe intensities I
1
, I
2
, I
3
, I
4
at the individual phase shift steps are expressed as follows:
I
1
(
x
,
y
)
=
I
0
(
x
,
y
)
[
1
+
γ
(
x
,
y
)
]
cos
[
φ
(
x
,
y
)
]
I
2
(
x
,
y
)
=
I
0
(
x
,
y
)
[
1
+
γ
(
x
,
y
)
]
cos
[
φ
(
x
,
y
)
+
π
/
2
]
I
3
(
x
,
y
)
=
I
0
(
x
,
y
)
[
1
+
γ
(
x
,
y
)
]
cos
[
φ
(
x
,
y
)
+
π
]
I
4
(
x
,
y
)
=
I
0
(
x
,
y
)
[
1
+
γ
(
x
,
y
)
]
cos
[
φ
(
x
,
y
)
+
3
π
/
2
]
(
1
)
where
x and y are coordinates;
&phgr;(x, y) is the phase;
I
0
(x, y) is the average optical intensity at each point; and
&ggr;(x, y) is the modulation of interference fringes.
From these expressions, the phase &phgr;(x, y) can be determined and expressed as:
φ
(
x
,
y
)
=
tan
-
1
I
4
(
x
,
y
)
-
I
2
(
x
,
y
)
I
1
(
x
,
y
)
-
I
3
(
x
,
y
)
(
2
)
Though the phase shift methods enable measurement with a very high accuracy if the predetermined step amount can be shifted correctly, it may be problematic in that errors in measurement occur due to errors in the step amount and in that it is likely to be influenced by the disturbance during measurement since it necessitates a plurality of interference fringe image data items.
For sub-fringe interferometry other than the phase shift method, attention has been paid to techniques using the Fourier transform method as described in “Basics of Sub-fringe Interferometry,” Kogaku, Vol. 13, No. 1 (February, 1984), pp. 55 to 65, for example.
The Fourier transform fringe analysis method is a technique in which a carrier frequency (caused by a relative inclination between an object surface to be observed and a reference surface) is introduced, so as to make it possible to determine the phase of the object with a high accuracy from a single fringe image. When the carrier frequency is introduced, without consideration of the initial phase of the object, the interference fringe intensity i(x, y) is represented by the following expression (3):
i
(
x,y
)=
a
(
x,y
)+
b
(
x,y
)cos[2&pgr;
f
x
x+
2&pgr;
f
y
y+
&PHgr;(
x,y
)] (3)
where
a(x, y) is the background of interference fringes;
b(x, y) is the visibility of fringes;
&phgr;(x, y) is the phase of the object to be observed; and
f
x
and f
y
are carrier frequencies in the x and y directions respectively expressed by:
f
x
=
2
·
tan
θ
x
λ
,
f
y
=
2
·
tan
θ
y
λ
where &lgr; is the wavelength of light, and &thgr;
x
and &thgr;
y
are the respective inclinations of the object in the x and y directions.
The above-mentioned expression (3) can be rewritten as the following expression (4):
i
(
x,y
)=
a
(
x,y
)+
c
(
x,y
)
exp[i
(2&pgr;
f
x
x+
2&pgr;
f
y
y
)]+
c
*(
x,y
)
exp[i
(2&pgr;
f
x
x+
2&pgr;
f
y
y
)] (4)
where c(x, y) is the complex amplitude of the interference fringes, and c*(x, y) is the complex conjugate of c(x, y).
Here, c(x, y) is represented as the following expression (5):
c
(
x
,
y
)
=
b
(
x
,
y
)
exp
[
ⅈ
φ
(
x
,
y
)
]
2
(
5
)
The Fourier transform of expression (4) gives:
I
(&eegr;,&zgr;)=
A
(&eegr;,&zgr;)+
C
(&eegr;−
f
x
,&zgr;−f
y
)+
C
*(&eegr;−
f
x
,&zgr;−f
y
) (6)
where A(&eegr;, &zgr;) is the Fourier transform of a(x, y), and C(&eegr;−f
x
, &zgr;−f
y
) and C*(&eegr;−f
x
, &zgr;−f
y
) are the Fourier transforms of c(x, y) and c*(x, y), respectively.
Subsequently, C(&eegr;−f
x
, &zgr;−f
y
) is taken out by filtering, the peak of the spectrum positioned at coordinates (f
x
, f
y
) is transferred to the origin of a Fourier frequency coordinate system (also referred to as Fourier spectra plane coordinate system; see FIG.
6
), and the carrier frequencies are eliminated. Then, inverse Fourier transform is carried out, so as to determine c(x, y), and the wrapped measured phase &phgr;(x, y) can be obtained by the following expression (7):
φ
(
x
,
y
)
=
tan
-
1
Im
[
c
(
x
,
y
)
]
Re
[
c
(
x
,
y
)
]
(
7
)
where Im[c(x, y)] is the imaginary part of c(x, y), whereas Re[c(x, y)] is the real part of c(x, y).
Finally, unwrapping processing is carried out, so as to determine the phase &PHgr;(x, y) of the object to be measured.
In the Fourier transform fringe analyzing method explained in the foregoing, the fringe image data modulated by carrier frequencies is subjected to a Fourier transform method as mentioned above.
As mentioned above, the phase shift method captures and analyzes the brightness of images while applying a phase difference between the object light of an interferometer and the reference light by a phase angle obtained when 2&pgr; is divided by an integer in general, and thus can theoretically realize highly accurate phase analysis.
For securing highly accurate phase analysis, however, it is necessary to shift the relative displacement between the sample and the reference with a high accuracy by predetermined phase amounts. When carrying out the phase shift method by physically moving the reference surface or the similar by using phase shift elements, e.g., PZTs (piezoelectric elements), it is necessary to control the amount of displacement of PZTs (piezoelectric elements) with a high accuracy. However, errors in displacement of the phase shift elements or errors in inclination of the reference surf
Fuji Photo Optical Co., Ltd.
Lyons Michael A.
Snider Ronald R.
Snider & Associates
Turner Samuel A.
LandOfFree
Fringe analysis error detection method and fringe analysis... does not yet have a rating. At this time, there are no reviews or comments for this patent.
If you have personal experience with Fringe analysis error detection method and fringe analysis..., we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Fringe analysis error detection method and fringe analysis... will most certainly appreciate the feedback.
Profile ID: LFUS-PAI-O-3054862