Measurement of the surface shape and optical thickness variation of a polishing crystal wafer by wavelength tuning interferometer

Abstract

Interferometric surface measurement of parallel plates presents considerable technical difficulties owing to multiple beam interference. To apply the phase-shifting technique, it is necessary to use an optical-path-difference-dependent technique such as wavelength tuning that can separate interference signals in the frequency domain. In this research, the surface shape and optical thickness variation of a lithium niobate wafer for a solid Fabry-Perot etalon during the polishing process were measured simultaneously using a wavelength-tuning Fizeau interferometer with a novel phase shifting algorithm. The novel algorithm suppresses the multiple beam interference noise and has sidelobes with amplitudes of only 1% of that of the main peak. The wafer, which was in contact with a supporting glass parallel plate, generated six different interference fringes that overlapped on the detector. Wavelength-tuning interferometry was employed to separate the specific interference signals associated with the target different optical paths in the frequency domain. Experimental results indicated that the optical thickness variation of a circular crystal wafer 74 mm in diameter and 5-mm thick was measured with an uncertainty of 10 nm PV.

Full text

Measurement of the surface shape and optical thickness variation of a
polishing crystal wafer by wavelength tuning interferometer
Yangjin Kim*a, Kenichi Hibinob, Ryohei Hanayamac, Naohiko Sugitaa, Mamoru Mitsuishia
aUniversity of Tokyo, Bunkyo-ku, Tokyo 113-8656, Japan; bNational Institute of Advanced
Industrial Science and Technology (AIST), Tsukuba 305-8564, Japan; cGraduate School for the
Creation of New Photonics Industries, Hamamatsu 431-1202, Japan.
ABSTRACT
Interferometric surface measurement of parallel plates presents considerable technical difficulties owing to multiple
beam interference. To apply the phase-shifting technique, it is necessary to use an optical-path-difference-dependent
technique such as wavelength tuning that can separate interference signals in the frequency domain. In this research, the
surface shape and optical thickness variation of a lithium niobate wafer for a solid Fabry-Perot etalon during the
polishing process were measured simultaneously using a wavelength-tuning Fizeau interferometer with a novel phase
shifting algorithm. The novel algorithm suppresses the multiple beam interference noise and has sidelobes with
amplitudes of only 1% of that of the main peak. The wafer, which was in contact with a supporting glass parallel plate,
generated six different interference fringes that overlapped on the detector. Wavelength-tuning interferometry was
employed to separate the specific interference signals associated with the target different optical paths in the frequency
domain. Experimental results indicated that the optical thickness variation of a circular crystal wafer 74 mm in diameter
and 5-mm thick was measured with an uncertainty of 10 nm PV.
Keywords: Phase measurement, wavelength tuning interferometer, surface shape, optical thickness, optical parallel
1. INTRODUCTION
The semiconductor industry requires the mass production of fine chips. Lithography has been used to satisfy the
requirements of mass production and miniaturization of semiconductor chips. Conventional lithography uses an optical
mask, but a laser with very short wavelength, such as an EUV laser with a wavelength of 13.5 nm, should be used to
achieve further miniaturization. The requirements of an EUV mask are more severe than those of the optical mask
because the EUV mask is a reflection-type mask. Therefore, to satisfy the requirements of the semiconductor industry, it
is necessary to precisely measure the surface shape, optical thickness distribution, and refractive index of the optical
devices.
Wavelength-tuning interferometry allows simultaneous measurements of the surface shape and the optical thickness
variation of a transparent parallel plate1–13. In wavelength-tuning interferometry, the optical path difference between the
two reflecting surfaces is proportional to the frequency of their interference signal. Simultaneous measurements of the
surface shape and the refractive index variation of a parallel plate were first demonstrated using a wavelength-tuning
Michelson interferometer2. The wavelength-tuning Fizeau interferometer is more insensitive to air turbulence and
mechanical vibration than the Michelson interferometer and also has the capability to dilate to accommodate a large-
diameter measurement sample. However, it suffers from multiple beam interference noise because the sample surfaces
and the reference surface are parallel. For more precise measurement, it is important to remove this interference noise in
the frequency domain.
For a simple parallel plate, it was shown that all signal and noise frequencies can be reduced approximately to harmonic
frequencies and can be detected using Fourier analysis if the air gap distance between the top surface and the reference
surface and the optical thickness of the plate are adjusted to differ by an integral factor. However, nonlinearity in the
wavelength tuning is common in the tunable laser diode, which can shift these signal and noise frequencies from the
exact harmonic frequencies and cause phase-shifting error. Error-compensating phase-shifting algorithms that suppress
the effect of phase-shift error have been reported by several authors5–7,10,12–23. However, no error-compensating algorithm
has been reported for a parallel plate consisting of several reflecting surfaces.
Optical Micro- and Nanometrology V, edited by Christophe Gorecki, Anand Krishna Asundi,
Wolfgang Osten, Proc. of SPIE Vol. 9132, 91320V · © 2014 SPIE
CCC code: 0277-786X/14/$18 · doi: 10.1117/12.2051193
Proc. of SPIE Vol. 9132 91320V-1

CCD
Sample
PBS
QWP
Collimator Lens
Reference
Surface L
Grating
Diode
Mirror
Polarizer
HWP
Screen Imaging Lens
Microscope Objective
Microscope
Objective
T1
T2
Figure 1. Wavelength-tuning Fizeau interferometer optical system. PBS, polarization beam splitter; HWP, half-wave plate;
QWP, quarter-wave plate.
Conventional Fourier analysis utilizes the window function to reduce phase-shift error caused by multiple beam
interference noise10,11. However, most conventional window functions do not compensate for phase-shift miscalibration
and are sensitive to the frequency detuning of the interference signal. For the simultaneous phase measurement of
multiple interference components, the signal frequency is not always an integral harmonic of the fundamental signal, and
it has a fractional detuning error.
In this research, the surface shape and the optical thickness variation of a lithium niobate wafer for a solid Fabry-Perot
etalon during the polishing process were measured simultaneously using a wavelength-tuning Fizeau interferometer with
a novel 3N - 2 phase-shifting algorithm. By aligning the air gap distance between the reference surface and the top
surface of the wafer, we separated the phase distribution related to the surface shape and the optical thickness variation.
2. MEASUREMENT PRINCIPLE
2.1 Wavelength-tuning interferometer
Lithium niobate (LN) is a transparent, uniaxial crystal that is yellow in color. It is widely used in optical modulators such
as light guides and second harmonic generators because its refractive index can be changed substantially by changing the
superimposed voltage and temperature. For a high finesse number, it is necessary to raise the accuracy of measurement
of both the homogeneity of the surface shape and the optical thickness. Although it is necessary to measure the optical
thickness and surface shape of LN crystal wafers during polishing, this can potentially damage the wafer surface.
Therefore, it is desirable to perform the measurement with the crystal adhered to the substrate.
Figure 1 shows the setup for measuring the surface shape of LN crystal wafers using a Fizeau interferometer. The source
is a tunable diode laser with a Littman external cavity (New Focus TLB-6900) comprising a grating and a cavity
Proc. of SPIE Vol. 9132 91320V-2

Table 1. Relative frequencies of interference fringes.
Frequency
Relative frequency (

m/

1)
Optical path difference
(p, q, r)

1
1
1 = n1T1
(0, 1, 0)

2
2
2 = n2T2
(0, 0 ,1)
2 = 2n1T1
(0, 2, 0)

3
3
3 = n1T1 + n2T2
(0, 1, 1)

4
5
5 = L
(1, 0, 0)

5

6
6 = L + n1T1
(1, 1, 0)

6

7
7 = L + n2T2
(1, 0, 1)

7

8
8 = L + n1T1 + n2T2
(1, 1, 1)
mirror24. The source wavelength is scanned linearly from 632.8 nm to approximately 0.23 nm using a piezo-electric
transducer (PZT). The focused output beam from the fiber is reflected by a polarization beam splitter. The linearly
polarized beam is then transmitted to a quarter-wave plate to become a circularly polarized beam. This beam is
collimated to illuminate the reference surface of the sample. The reflections from the sample surfaces and the reference
surface travel back along the same path, and then they are transmitted through the quarter-wave plate again to attain an
orthogonal linear polarization. The resulting beams pass through the polarization beam splitter and combine to generate
interference fringes on the CCD detector with resolution of 640 × 480 pixels.
2.2 Multiple reflection noise in optical parallel plate
When changing a light source wavelength from

0 at the rate of d

/dt, the phase modulation frequency

is given by the
following equation using the optical path difference D:
0
2
0
dd
1dd
Dn
nt




 



(1)
Here, n[1 - (

0 / n)(dn/d

)] is called the group refractive index ng, and it is present in the multiple beam interference.
When an LN crystal wafer is compared with the flat reference surface in a wavelength-tuning Fizeau interferometer as
shown in Figure 1, the irradiance signal observed by the CCD detector is formed by multiple beam interference between
the reflection beams from the reference surface and wafer surfaces. The irradiance signal of a function of time t is given
by
   
01
, , cos ,
m m m
m
I x y t A A t x y



  



(2)
Here, m is the number of interference components, Am and

m are the amplitude and frequency, respectively, of the mth-
order component, and A0 is the DC component. If we denote the optical path difference D of the two interfering beams in
terms of the air gap distance L, the thickness Ti, and the refractive index ni of the transparent parallel plate, the
modulation frequency

m of the interference component is the time derivative of the phase difference and is calculated as
follows:
 
00
1 1 1 2 2 2
2
0 1 2
dd
4π
, , 1 1
dd
nT n n T n
L
p q r p q r
L n L n

  

   
    

   
   

. (3)
Here, p, q, and r are integers, and

0 is the central wavelength. Although the air gap distance L can be set arbitrarily,
because the thicknesses of a crystal and a substrate cannot be chosen freely, it is generally impossible to make the
modulation frequencies of all the ingredients correspond to a simple integer ratio. However, it is possible to set the air
Proc. of SPIE Vol. 9132 91320V-3

1.0-
0.5 -
-0.5 -
-1.0-
- iF1F2
4 8 12 16 20 24 28
Relative Frequency
32
1.0 -
0.5 -
-0.5 -
-1.0-
4 8 12 16 20 24 28 32
Relative Frequency
Figure 2. Sampling functions of synchronous algorithm (N = 32, m = 1, 3).
gap distance such that the specific signal frequency does not overlap with the frequency of other main signals. For
example, if the air gap distance is set as 61.2 mm when the thickness of the LN crystal is 5 mm, the ratio of the optical
thickness of the air gap distance to that of the LN crystal to that of the fused silica substrate is set to approximately 5:1:2.
Six types of interference fringes are formed by the direct reflection beam from the three surfaces of the sample and the
reference surface. Table 1 shows the relative frequencies of the interference fringes.
2.3 Novel phase-shifting algorithm and sampling function
The phases of the frequency components of the interference signal in Eq. (2) are measured using a phase-shifting
algorithm. The interference intensity I(x, y, t) is sampled M times with an equal phase interval

= 2/N. The phase
shifting algorithm can described as
 
 
1
1
,,
arctan ,,
M
rr
r
mM
rr
r
b I x y t
a I x y t





. (4)
Here, ar and br are the sampling amplitudes. Eq. (4) can be transformed to the frequency domain using Fourier transform
and Parseval’s formula, as follows:
   
   
1
2
d
arctan d
m
FJ
FJ
  
  






. (5)
Here, J(

) is the Fourier transform of the interference intensity I(x, y, t) given by Eq. (2), and F1(

) and F2(

) are the
sampling functions, which reveal the frequency characteristics of the algorithm. Using these functions, the specifications
of the phase-shifting algorithm are presented graphically. The sampling functions F1 and F2 are related to sine and cosine
components, respectively, of the extracted phase. The sampling amplitudes ar and br can always be chosen to become
symmetric and antisymmetric19. Accordingly, the sampling functions F1 and F2 become an odd and an even function,
respectively, and purely imaginary and real, respectively. In the following, we consider the sampling functions iF1 and
F2. The sampling amplitudes of the N-sample synchronous algorithm for detecting the mth harmonic signal are defined
by the following equations25:
Proc. of SPIE Vol. 9132 91320V-4

1.0
0.5
0.0
-0.5
iFl
F2
-1.0
036912 15
Relative Frequency
18 21
1.0 -
0.5 -
0.0
iFlF2
-0.5 -
-1.0
36912 15
Relative Frequency
18 21
Figure 3. Sampling functions of novel 3N - 2 algorithm (N = 21, m = 1, 3).
22π1
cos 2
22π1
sin 2
r
r
mN
ar
NN
mN
br
NN
  






  






. (6)
Figure 2 shows the sampling functions of a 32-sample Fourier algorithm for first- and third-harmonic detection. The two
functions iF1 and F2 have equal amplitude at the detection frequency and zero amplitude at the other harmonic
frequencies. The synchronous algorithm is very sensitive to the detuning of the signal frequency from the detection
frequency. It is also observed that the amplitudes of the sidelobes around the main peak are approximately 20 to 30% of
that of the main peak, and they do not decrease very quickly.
In this research, a novel 3N - 2 algorithm was derived using a synchronous algorithm and a novel polynomial window
function. The sampling amplitudes of the novel 3N - 2 algorithm are determined using the following equations:
   
       
    
2
2
2
22π31
cos 2
22π31
sin 2
11 11
2
11 1 2 1 1 2 2
2
11
3 1 3 2 1 3 2
2
rr
rr
r
r
r
mN
a w r
NN
mN
b w r
NN
w r r r N
N
w N N r N N r N r N
N
w N r N r N r N
N
  






  







   



         



       


(7)
Figure 3 shows the sampling functions of the novel 3N - 2 algorithm for first- and third-harmonic detection. The two
functions iF1 and F2 have equal amplitude at the detection frequency and zero amplitude at the other harmonic
frequencies. The new 3N - 2 algorithm is insensitive to the detuning of the signal frequency from the detection
frequency. It is also observed that the amplitudes of the sidelobes around the main peak are approximately 1% of that of
the main peak.
Proc. of SPIE Vol. 9132 91320V-5

L)
(a) (b)
Figure 4. (a) Lithium niobate wafer; (b) raw interferogram (

= 632.8 nm).
(a) (b) 0
2
Figure 5. (a) Phase of surface shape; (b) phase of optical thickness.
0
472.629 nm
(a) 0
804.797 nm
(b)
Figure 6. (a) Surface shape of lithium niobate (LN) wafer; (b) optical thickness of LN wafer.
Proc. of SPIE Vol. 9132 91320V-6

3. EXPERIMENT
The test sample comprises a circular lithium niobate wafer 5 mm thick and 74 mm in diameter as well as a fused silica
substrate 16.5 mm thick and 80 mm in diameter. Therefore, we have four reflecting surfaces, including the reference
surface. Figure 4 (a) shows the measurement sample, which has several parallel surfaces, and (b) shows the raw
interferogram at a wavelength 632.8 nm. The air gap distance was set to 61.2 mm to separate the phase distribution of
each surface in the frequency domain. With this parameter, the ratio of the modulation frequencies in the harmonic
spectrum comprises near-integer values, as shown in Table 1.
Figure 5 (a) shows the phase distribution of the surface shape of an LN wafer, and (b) shows the phase distribution of the
optical thickness. Figure 6 (a) shows the surface shape of an LN wafer, and (b) shows the optical thickness. The
repeatability of the phase measurement, that is, the rms of the difference between two measurements taken successively,
was approximately 2 nm.
4. CONCLUSION
The surface shape and the optical thickness variation of lithium niobate wafers were measured using a novel 3N - 2
algorithm and a wavelength-tuning Fizeau interferometer. The novel algorithm comprises a polynomial window function
and a discrete Fourier transform term. It suppresses the multiple beam interference noise and yields sidelobes with
amplitudes of only 1% of that of the main peak. Using the novel 3N - 2 algorithm, the phase distributions of the surface
shape and the optical thickness were separated in the frequency domain. The repeatability of the phase measurement was
approximately 2 nm.
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