Optics: measuring and testing – By light interference – For dimensional measurement
Reexamination Certificate
2002-01-22
2004-08-17
Turner, Samuel A. (Department: 2877)
Optics: measuring and testing
By light interference
For dimensional measurement
C356S508000, C356S512000
Reexamination Certificate
active
06778281
ABSTRACT:
RELATED APPLICATIONS
This application claims the priorities of Japanese Patent Application No. 2001-23200 filed on Jan. 31, 2001 and Japanese Patent Application No. 2001-399179 filed on Dec. 28, 2001, which are incorporated herein by reference.
BACKGROUND OF THE INVENTION
1. Field of the Invention
The present invention relates to a phase shift fringe analysis method using Fourier transform when analyzing a fringe image by using a phase shift method, and an apparatus using the same. In particular, the present invention relates to a phase shift fringe analysis method comprising the steps of obtaining image information of interference fringes and the like while shifting a phase by using a phase shift device such as PZT (piezoelectric device), and analyzing thus obtained plurality of image data items having a fringe pattern of interference fringes and the like, thereby attaining highly accurate phase information of an object to be observed; and an apparatus using the same.
2. Description of the Prior Art
While light-wave interference method, for example, has conventionally been known as important means concerning precise measurement of object surfaces, there have recently been urgent needs for developing an interferometry technique (sub-fringe interferometry) for reading out information from a fraction of a single interference fringe (one fringe) or less from the necessity to measure a surface or wavefront aberration of {fraction (1/10)} wavelength or higher.
An example of typical techniques widely in practice as such sub-fringe interferometry is a phase shift fringe analysis 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, a phase shift device such as a PZT (piezoelectric) device is used for phase-shifting the relative relationship between an object to be observed and a reference, interference fringe data is captured each time a predetermined step amount shifts, so as to measure the interference fringe intensity of each point of the object surface, and the phase of each point of the surface is determined by using the results of measurement.
In the case of a 4-step phase shift method, for example, respective interference fringe intensities I
1
, I
2
, I
3
, and I
4
at individual phase shift steps are expressed as follows:
I
1
(
x, y
)=
I
0
(
x, y
){1+&ggr;(
x, y
)cos[&phgr;(
x, y
)]}
I
2
(
x, y
)=
I
0
(
x, y
){1+&ggr;(
x, y
)cos[&phgr;(
x, y
)+&pgr;/2]} (2)
I
3
(
x, y
)=
I
0
(
x, y
){1+&ggr;(
x, y
)cos[&phgr;(
x, y
)+&pgr;]}
I
4
(
x, y
)=
I
0
(
x, y
){1+&ggr;(
x, y
)cos[&phgr;(
x, y
)+3&pgr;/2]}
where
x and y are coordinates;
&phgr; (x, y) is a 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 as
φ
(
x
,
y
)
=
tan
-
1
I
4
(
x
,
y
)
-
I
2
(
x
,
y
)
I
1
(
x
,
y
)
-
I
3
(
x
,
y
)
.
(
3
)
Though the phase shift method enables measurement with a very high accuracy if a predetermined step amount can shift accurately, it has been problematic in terms of measurement errors occurring due to errors in step amount and in that it is likely to be affected by disturbances during measurement since it necessitates a plurality of interference fringe image data items.
As a sub-fringe interferometry technique other than the phase shift method, attention has been focused on one using a Fourier transform method, for example, as disclosed in “Basics of Sub-fringe Interferometry,”
Kogaku
, Vol. 13, No. 1 (February 1984), pp. 55 to 65.
The Fourier transform fringe analysis method is a technique in which a carrier frequency (caused by a relative tilt between an object surface to be observed and a reference surface) is introduced, whereby the phase of the object can be determined with a high accuracy from a single fringe image. When a carrier frequency is introduced without the initial phase of the object being taken into consideration, the interference fringe intensity i(x, y) is represented by the following expression (4):
i
(
x, y
)=
a
(
x, y
)+
b
(
x, y
)cos[2&pgr;ƒ
x
x+
2&pgr;ƒ
y
y
+&phgr;(
x, y
)] (4)
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; and
f
x
and f
y
are respective carrier frequencies in x and y directions 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 respective inclinations of the object surface in x and y directions.
The above-mentioned expression (4) can be deformed as the following expression (5):
i
(
x, y
)=
a
(
x, y
)+
c
(
x, y
)exp[
i
(2&pgr;ƒ
x
+2&pgr;ƒ
y
)]+
c
*
(
x, y
)exp[−
i
(2&pgr;ƒ
x
+2&pgr;ƒ
y
)] (5)
where c(x, y) is the complex amplitude of interference fringes, and c
*
(x, y) is the complex conjugate of c(x, y).
Here, c(x, y) is represented as the following expression (6):
c
(
x
,
y
)
=
b
(
x
,
y
)
exp
[
ⅈ
φ
(
x
,
y
)
]
2
.
(
6
)
When the above-mentioned expression (5) is Fourier-transformed, the following expression (7) is obtained:
I
(&eegr;,&zgr;)=
A
(&eegr;,&zgr;)+
C
(&eegr;−ƒ
x
,&zgr;−ƒ
y
)+
C
*
(&eegr;+ƒ
x
,&zgr;+ƒ
y
) (7)
where
A(&eegr;, &zgr;) is the Fourier transform of a(x, y);
C(&eegr;−f
x
, &zgr;−f
y
) is the Fourier transform of c(x,y)exp[i(2&pgr;ƒ
x
+2&pgr;ƒ
y
)]; and
C
*
(&eegr;+f
x
,&zgr;+f
y
) is the Fourier transform of c
*
(x,y)exp[−i(2&pgr;ƒ
x
+2&pgr;ƒ
y
)].
Subsequently, C(&eegr;−f
x
, &zgr;−f
y
) is taken out by filtering, and the peak of a spectrum positioned at coordinates (f
x
, f
y
) is moved to the origin of a frequency coordinate system (also known as a Fourier spectrum coordinate system; see FIG.
8
), so as to eliminate the carrier frequency. Then, inverse Fourier transform is carried out, so as to determine c(x, y), and the wrapped phase &phgr; (x, y) is determined by the following expression (8):
φ
(
x
,
y
)
=
tan
-
1
Im
[
c
(
x
,
y
)
]
Re
[
c
(
x
,
y
)
]
(
8
)
where Im[c(x, y)] is the imaginary part of c(x, y), and Re[c(x, y)] is the real part of c(x, y).
Finally, unwrapping is carried out, so as to determine the phase &PHgr;(x, y) of the object to be measured.
In the Fourier transform analysis method explained in the foregoing, the fringe image data modulated by the carrier frequency is subjected to Fourier transform as stated above.
In general, as mentioned above, the phase shift method captures the brightness of an image while imparting a phase difference between object light of an interferometer and reference light by a phase angle which is an integral fraction 2&pgr; and analyzes thus captured brightness, thereby theoretically enabling a highly accurate phase analysis.
For securing highly accurate phase analysis, however, it is necessary that the relative relationship between a sample and a reference be displaced at a high accuracy by a predetermined phase amount (very short distance). When the phase shift method is carried out by physically moving a reference surface or the like by using a PZT (piezoelectric device), for example, it is necessary that the amount of displacement of the PZT (piezoelectric) device be controlled highly accurately. However, the displacement error of the phase shift device or the tilt error of the object surface is hard to eliminate completely, whereby control
Fuji Photo Optical Co., Ltd.
Lyons Michael A.
Snider Ronald R.
Snider & Associates
Turner Samuel A.
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