A sinusoidal wavelength-scanning interferometer for measuring thickness and surfaces profiles of a thin film has been proposed in which a superluminescent laser diode and an acousto-optic tunable filter are used. The interference signal contains an amplitude Zb of a time-varying phase and a constant phase α. Two values of an optical path difference (OPD) obtained from Zb and α, respectively, are combined to measure an OPD longer than a wavelength. The values of Zb and α are estimated by minimizing the difference between the detected signals and theoretical ones. From the estimated values, thickness and surface of a silicon dioxide film coated on an IC wafer with different thicknesses of 1 μm and 4 μm are measured with an error less than 5 nm.
©2005 Optical Society of America
It is important to measure positions of the surfaces of a thin film in three dimensions with a high accuracy of a few nanometers. For example, it is required in the manufacturing process of liquid crystals displays and semiconductors that three-dimensional profiles of transparent conductive films of ITO (Indium Tin Oxide) and silicon dioxide films coated on an IC wafer are measured. Many instruments for measuring thickness of a film on one measuring point are available, but they can not measure the surface profiles and need a long time to obtain two-dimensional distribution of thickness of the film. To achieve the three-dimensional measurement of thickness and surface profiles of a thin film, white light interferometers and wavelength-scanning interferometers have been developed. In white light interferometers, the positions of the reflecting surfaces are determined by finding positions where the amplitude of the interference signal has a peak by scanning the optical path difference (OPD) . In wavelength-scanning interferometers, spectral phase of the interference signal, which varies according to the scanning of the wavelength instead of the scanning of the OPD, is utilized for thickness measurement. In the case of linear wavelength-scanning, the positions of the reflecting surfaces are determined by the peaks of the frequency spectrum of the interference signal . When thickness of a film is very thin, the distance between the two peaks of the amplitude of the interference signal in white light interferometers or the distance between the two peaks of the frequency spectrum of the interference signal in wavelength-scanning interferometers, which are caused by the two reflecting surfaces, become too short to distinguish the positions of the two peaks. Therefore, these conventional methods of finding the peaks are not suitable to measure the positions of the two reflecting surfaces in a very thin film. In reference , a spectral phase function of an interference signal was detected around a position where OPD is almost zero in a white light interferometer. An error function was defined by the difference between the detected spectral phase function and the theoretical one. By minimizing the error function, the surface profiles and the thickness of the film were estimated. In this case, the measurement accuracy strongly depends on the mechanical scanning of the OPD by use of a piezoelectric transducer. On the other hand, in references [4, 5] linear wavelength-scanning interferometers are proposed in which an acoustic-optic tunable filter (AOTF) was used to obtain the scanning width of about 100 nm. In this case also, an error function about the spectral phase function of the interference signal was minimized to estimate the surface profile and the thickness whose range was within a few microns.
In this paper we propose a different method using a sinusoidal wavelength-scanning interferometer compared with the methods in references [4,5]. Signal components caused by interference between the lights reflected from a film and a reference light are completely selected from a detected interference signal by the use of a sinusoidal phase modulation produced by a vibrating reference mirror. The double sinusoidal modulation of the wavelength and the phase leads to an error function for the signal estimation which is defined not for the spectral phase of the interference signal, but for the signals derived from the detected interference signal. This error function allows a good estimation of the positions of the two reflecting surfaces of a film even when the wavelength-scanning width is small. The detected interference signal contains a time-varying phase produced by sinusoidal wavelength-scanning and a constant phase α. The amplitude of the time-varying phase is called modulation amplitude Zb which is proportional to the OPD and the wavelength scanning width. Since a rough value and a fine value of the OPD are obtained from Zb and α respectively, the OPD longer than a wavelength can be measured with a high accuracy of few nanometers [6–8]. The important unknowns in the error function are the values of Zb and a for the two reflecting surfaces, and initial values of Zb are obtained with double sinusoidal phase-modulating interferometry . Combination of the estimated values of Zb and α provide the positions of the two reflecting surfaces of a film with a high accuracy of a few nanometers. In the experiment, a superluminescent laser diode (SLD) and an AOTF are used to achieve a wavelength-scanning width of 47 nm. The thickness distribution and surfaces profiles of a silicon dioxide film on an IC wafer with different thicknesses of 1 μm and 4 μm are measured with an error less than 5nm.
Figure 1 shows an interferometer for measuring thickness and surface profiles of a thin film. The output light of the SLD is collimated by lens L1 and incident on the AOTF. The wavelength of the first-order diffracted light from the AOTF is proportional to the frequency of the applied signal. Modulating the frequency of the applied signal sinusoidally, the wavelength of the light from the AOTF is scanned as follows:
where λ0 is the central wavelength. The intensity of the light source is also changed, and it is denoted by M(t). The light is divided into an object light and a reference light by a beam splitter (BS). The reference light is sinusoidally phase modulated with a vibrating mirror M1 whose movement is a waveform of acos(ωct+θ).
The object is a silicon dioxide film coated on IC wafer as shown in Fig. 2, and the refractive index of air, silicon dioxide and IC wafer are denoted by n1, n2 and n3, respectively. The film has two surfaces A and B, and multiple-reflection light from the two surfaces is defined by Ui (i=1, 2, 3, …). The amplitudes of the interference signals caused by reference light and object light Ui are denoted by ai (i=1, 2, 3, …). The constant ratios of ai to a1 are defined by Ki=ai/a1 (i=1, 2, 3, …), and it is calculated with the refractive index of n1, n2 and n3. Since n1=1.00, n2=1.46 and n3=3.70, the constant ratio K2, K3 and K4 are K2=2.24, K3=0.18 and K4=0.01, respectively. Since the amplitude of a4<<a1, the interference signal caused by U4 and higher reflection light can be neglected. Lights Ui (i=1, 2, 3, …) interfere with each other to cause an interference signal. Since the reference light is sinusoidally phase modulated, this interference signal and intensity components of each light can be filtered out through Fourier transform of the interference signals detected with the CCD . The positions of two surfaces A and B are expressed by OPDs L1 and L2. Among the detected interference signals the interference signals caused by interference between Ui (i=1, 2, 3) and the reference light are expressed as
Putting Φi=Zbicos(ωbt)+αi, the interference signal S(t) is rewritten as
The intensity modulation M(t) is obtained by detecting the intensity of the reference light. The Fourier transform of S(t)/M(t) is denoted by F(ω). If the following conditions are satisfied,
where ℑ[y] is the Fourier transformation of y, the frequency components of F(ω) in the regions of ωc/2 < ω < 3ωc/2 and 3ωc/2 < ω < 5ωc/2 are designated by F1(ω) and F2(ω), respectively. Then we have
where Jn(Zc) is the nth-order Bessel function . The values of Zc and θ are measured by sinusoidal phase-modulation interferometry beforehand. Taking the inverse Fourier transform of -F1(ω-ωc)/J1(Zc)exp(jθ) and -F2(ω-2ωc)/J2(Zc)exp(j2θ), we obtain
When the absolute value of Zbi increases, the frequency distributions of F1(ω) and F2(ω) have a wider band around ωc and 2ωc, respectively, due to the terms of Zbicos(ωbt). Since the conditions given by Eq. (6) must be satisfied, maximum detectable value of |Zbi| depends on the ratio of the ωc/ωb. In contrast, when the absolute value of Zbi decrease, the magnitude of the spectra in F1(ω) and F2(ω) becomes so small that they can not be distinguished from noise. Therefore the absolute value of Zbi must be between 1 rad and 12 rad at ωc=32ωb.
The detected values of As(tm) and Ac(tm) are obtained from the detected interference signals at intervals of Δt, where tm=mΔt and m is an integer. Using the detected values of As(tm), Ac(tm), and known values of Ki (i=2, 3), we define an error function
where Âs(tm) and Âc(tm) are the estimated signals which contain unknowns of a1, Zbi, and αi. The values of unknowns of a1, Zbi, and αi are searched to minimize H by multidimensional nonlinear least-squares algorithm.
We obtain values of Li from the values of Zbi that is denoted by Lzi, and also obtain other values of Li from the values of αi that is denoted by Lαi. Since the measurement range of αi=2πLαi/λ0 is limited -π to π, a value of Lαi is limited to the range from -λ0/2 to λ0/2. On the other hand, a value of Zbi=2πbLzi/ provides a rough value Lzi of Li. To combine Lzi and Lαi, the following equation is used:
If the measurement error εLzi in Lzi is smaller than λ0/2, a fringe order mi is obtained by rounding off mci. The suffixes of i=1 and 2 in the Lzi, Lαi, mci, and mi correspond to surface A and B, respectively. Then an OPD Li longer than a wavelength is given by
Since the measurement accuracy of Lαi is a few nanometers, an OPD over several ten micrometers can be measured with a high accuracy.
The positions P1 and P2 of the front and rear surfaces, respectively, are obtained from the estimated values as follows:
where m=m2-m1. The thickness d is given by P2-P1. Thus we can measure the thickness and the two surface profiles of the object.
3. Determination of initial values
While searching for the real values of the unknowns, the existences of numerous local minima was recognized. The conditions of the initial values were examined by computer simulations. The initial values move to the real values almost certainly when differences between the initial values and the real values are within the following values: about 2rad for Zb1 and Zb2, about 1.5rad for α1 and α2, and about 50% accuracy for a1. However if one of these condition for the differences is not satisfied, the initial values do not always reach the global minimum. Good initial values are required to reach the global minimum in a short time.
First we consider how to determine a better initial value of a1. We adjust the position of the object so that L1 ≃ 0 or Zb1 ≃ 0. In this case Eqs. (8) is reduced to
where C1=a1sinα1 and C2=a1cosα1 are constant with time. The constant ratio K3=a3/a1 is almost 10 times smaller than K2. Therefore, these third terms of Eq. (13) can be neglected when rough values of a1, Zb1 and Zb2 are sought as the initial values. The position of L1=0 can be found by checking whether the signals of As(t) and Ac(t) are changing from K2a1 to -K2a1 with a constant amplitude of K2a1. Since K2 is a known value, a value of a1 is obtained from the amplitude of K2a1. Next we consider how to determine the initial values of Zb1 and Zb2. Equation (13) is the same as the equations appeared in the double sinusoidal phase-modulating interferometry  since the first and third terms can be eliminated from Eq. (13). Using the signal processing of double sinusoidal phase modulating interferometry for Eq. (13), the value of Zb2 can be calculated. After that the value of Zb1 is changed from 0 rad to some value by moving the position of the object with a micrometer so that the absolute values of Zb1 and Zb2 are between 1rad and 12rad according to the condition described in Section 2. Then the initial values of Zb1 and Zb2 can be obtained knowing rough values of the thickness and the refractive index of the object. On the other hand the initial values of α1 and α2 can not be determined from the detected signals. Therefore the initial value of α1 is given at intervals of 1.0 rad in the range from -π to π rad for the initial value of α2=0. When a global minimum can not be obtained, the initial value of α2 is changed by 1.0 rad and the search is repeated again. Considering all combinations of α1 and α2, the search becomes successful at most after 36 repetitions. The values estimated first at one measuring point are used as the initial values of the adjacent measuring points, because the difference in real values of α1 and α2 between the adjacent measuring points is within π/2 to detect the interference signal with a sufficient amplitude.
4. Experimental result
We constructed the interferometer shown in Fig. 1 and tried to measure the front and rear surface positions of a silicon dioxide film coated on an IC wafer whose configuration is shown in Fig. 3. The central wavelength and spectral bandwidth of the SLD was 830 nm and 46 nm, respectively. The central wavelength λ0 of the first-order diffracted light from the AOTF was 837.1 nm, and its spectral bandwidth was about 4 nm. The wavelength scanning frequency of ωb/2π was 15.8 Hz and the wavelength-scanning width 2b was 47.3 nm. The phase modulating frequency of ωc/2π was 32(ωb/2π)=506 Hz. A two-dimensional CCD image sensor was used to detect the interference signals. Lenses L2 and L3 formed an image of the object on the CCD image sensor with magnification of 2/3. Number of the measuring point was 60 × 30 in a region of 1.8 mm×0.9 mm on the object surfaces along the x and y axes, respectively. Positions of the pixels of the CCD image sensor are denoted by Ix and Iy, respectively. Intervals of the measuring points were Δx=30 μm and Δy=30 μm.
The object has two thicknesses of dL ≃ 1 μm and dR ≃ 4 μm as shown in Fig. 3. First, we estimated values of unknowns Zb1, Zb2, α1 and α2 at the two points of Ix=1, Iy=1 and Ix=60, Iy=1 by minimizing the error function given by Eq. (9). The estimated values at the point of Ix=1, Iy=1 were used as initial values of the adjacent measuring points in the region of 1μm thickness, and the estimated values at the point of Ix=60, Iy=1 were also used as initial values in the region of 4μm thickness. Values of Zb1, Zb2, α1, and α2 were estimated on all of the measuring points. Figure 4 shows the OPD Lzi (i=1, 2) calculated from Zbi with Eq. (3). Figure 5 also shows the OPD Lαi (i=1, 2) calculated from αi with Eq. (3). Exact measured values could not be obtained in the region of Ix=25-30 because light was strongly diffracted on the boundary of the two different thickness part of the object. By combining Lzi and Lαi with Eq. (10), the fringe order m1 of the front surface and the fringe order m2 of rear surface were obtained. Fringe order mi in the region of Ix=1-24 is denoted by miL, and mi in the region of Ix=31-60 is denoted by miR, as shown in Fig. 3. Figure 6(a) shows that fringe order m1L was almost 20, while there were two different values for the fringe order m1R. A value of m1R=12 appeared in 33% of the measuring points and m1R=13 appeared in 63% of the measuring points. Figure 6(b) shows that fringe order m2L and m2R were almost 24 and 27, respectively. Considering that the estimated OPDs Lαi of the front and rear surface changed smoothly as shown in Fig. 5, it was clear that fringe order miL and miR were constant values on each of the measuring regions. It is certainly decided that fringe order m1L m2L and m2R were 20, 24 and 27, respectively. Figure 7 shows the two different positions of P2 calculated in the cases of m1R=12 and m1R=13 along Ix at Iy=15 with Eq. (12). Considering that position P2 is the position of the IC wafer surface which does not contain a discontinuous part, fringe order m1R can be determined to be 13. Figure 8 shows the measured positions P1 and P2 of the front and rear surfaces of the object with m1L=20, m1R=13, m2L=24, and m2R=27. Figure 9 shows the thickness distribution calculated from P2-P1. Table 1 shows the measured values of Lzi, Lαi, mci, Pi and d along Ix at Iy = 15. In the region of dL the average value of the thickness was 1071 nm, and in the region of dR the average value of the thickness was 4114 nm. It was made clear by repeating the measurement three times that the measurement repeatability was less than 5 nm. The object was also measured with a commercially available white light interferometer to examine the measurement accuracy. The average values of dL and dR measured with the white light interferometer were 1074 nm and 4113 nm. This measurement result indicated that the measurement accuracy of the proposed interferometer was in the rage of a few nanometers.
A sinusoidal wavelength-scanning interferometer for measuring thickness and surface profiles of a thin film has been proposed in which the SLD and the AOTF were used. The interference signal contains the amplitude of the phase modulation Zb and the constant phase α related to the thickness and the surfaces profiles. Values of Zb and α were estimated by reducing the difference between the detected signal and the estimated signal. By combining the two estimated values of Zb and α, the positions of the front and rear surfaces of the silicon dioxide film coated on an IC wafer were measured with an error less than 5 nm.
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