3-D profiles of discontinuous surfaces patterned with high step structures are measured using four wavelengths generated by phase-locking to the frequency comb of an Er-doped fiber femtosecond laser stabilized to the Rb atomic clock. This frequency-comb-referenced method of multi-wavelength interferometry permits extending the phase non-ambiguity range by a factor of 64,500 while maintaining the sub-wavelength measurement precision of single-wavelength interferometry. Experimental results show a repeatability of 3.13 nm (one-sigma) in measuring step heights of 1800, 500, and 70 μm. The proposed method is accurate enough for the standard calibration of gauge blocks and also fast to be suited for the industrial inspection of microelectronics products.
© 2013 OSA
Fast, precise measurement of large surface profiles patterned with high steps is of importance for quality assurance of microelectronic products such as semiconductors, flat panel displays, and photovoltaic cells. Step heights to be dealt with for the purpose usually fall in the so-called meso-scale that ranges from tens to hundreds of micrometers, which cannot be readily covered by conventional surface profilers relying on a monochromatic light source. The reason is the ambiguity encountered in determining the absolute phase of interferometric fringes produced from steps higher than half of the wavelength [1,2]. In order to overcome the phase ambiguity problem, quite a few methods of extended non-ambiguity ranges have been made available, which include two-wavelength interferometry [3,4], multi-wavelength interferometry [5,6], wavelength sweeping interferometry [7,8], and white light interferometry [9–14]. These methods require multiple wavelengths to be provided individually in sequence or mixed from a broad-band source.
In this paper, we revisit the principle of multi-wavelength interferometry in which step heights are determined over a wide non-ambiguity range (NAR) extended by introducing additional wavelengths. The NAR is found sensitive to the wavelength uncertainty, thus the measurement precision is guaranteed only when the used wavelengths are precisely calibrated to the well-defined absorption lines of atoms or molecules . The recent advent of femtosecond lasers allows the optical wavelength calibration to be performed conveniently with direct traceability to the atomic clock through the frequency comb [15–17]. In addition, the frequency comb enables generation of accurate, stable optical wavelengths in active ways as demanded in various applications including length/distance metrology [18–23]. Motivated by the breakthrough, frequency-comb-referenced multi-wavelength interferometry was first demonstrated in calibration of the absolute height of a 25 mm gauge block with an uncertainty of 15 nm . The frequency comb was also applied to the measurement of long absolute distances [24–26] based on the principles of variable synthetic wavelength interferometry , dispersive interferometry , multi-wavelength interferometry [29,30], dual-comb interferometry , and time-of-flight measurement [32,33].
It is noted that most of the previous frequency-comb-referenced interferometers have been investigated so far mainly for length/distance measurements, no much work being done on surface metrology. Here, we present a multi-wavelength interferometer that enables profile measurement of large step heights over an extensive NAR without phase ambiguity by generating four different wavelengths with reference to the frequency comb. This frequency-comb-referenced surface profiling method is capable of maintaining sub-wavelength precision with direct traceability to the Rb atomic clock and fast enough to be well suited for not only standard metrology but also industrial uses.
2. Multi-wavelength interferometry
In monochromatic light interferometry using the phase shifting technique, the interferometric intensity I(x,y) is given asEq. (1) limits the measurement range to the half wavelength, which causes the 2π phase ambiguity in measuring large-stepped discontinuous surfaces.
Multi-wavelength interferometry to overcome the 2π phase ambiguity is implemented by taking additional phase maps with different wavelengths [5,6,34]. For a given wavelength (λ), the target height (h) is expressed as h = λ(m + e)/2, where m is a positive integer and e denotes an excess fraction (where 1>e≥0). Combining all the N wavelengths used in the measurement, the height h can be expressed in the form of a set of simultaneous inequalities:Eq. (2) within a confined boundary, which is defined as the NAR, because the number of unknowns (m1, m2, …, mN, and h) is always one larger than that of given equations. Therefore, the extension of the NAR is an important task in realizing the principle of multi-wavelength interferometry for practical uses dealing with high step heights.
The NAR is affected by the number of the wavelengths in use and also by their individual precision. The utmost precision in generating multiple wavelengths can be achieved by resorting to atomic/molecular absorption lines, but this cannot be easily accomplished for practical applications requiring in situ wavelength calibrations. Instead, the frequency comb can effectively be utilized, because it is capable of supplying numerous well-defined optical modes simultaneously with reference to the time standard [19–23]. In our implementation of multi-wavelength interferometry, four wavelengths of 765.266, 765.452, 777.173, and 777.371 nm are selected with the aim of extending the NAR over 50 mm to be sufficient for industrial meso-scale step-height measurements. Environmental and systematic error contributions are minimized so that the repeatability of phase measurement is suppressed to be less than 2.00 × 10−3; the fractional error bound α in Eq. (2) is set to be 7.78 × 10−3 with the confidence level of 99.99% in a normal distribution.
3. Measurement system design
Figure 1 illustrates the system layout configured for our measurements. The overall system is comprised of three main blocks: a multi-wavelength generation unit, a second-harmonic generation unit, and a step-height measuring interferometer. The multi-wavelength generation unit produces four precise wavelengths with reference to the Rb atomic clock of time standard. Then, the generated four wavelengths are sequentially selected and transferred through a high-speed optical switch to the Twyman-Green interferometer where the target specimen is located. A charge coupled device (CCD) camera is used to capture interferograms, while phase shifting is provided by use of a piezoelectric actuator (PZT) attached to the reference mirror.
3.1. Generation of multiple wavelengths referenced to the frequency comb
A custom-built Er-doped fiber femtosecond laser is used to generate a train of ultrashort pulses at a 100 MHz repetition rate with an average power of ~40 mW. The resulting frequency comb yields a 50 nm bandwidth centered at 1550 nm, which provides several millions of evenly spaced optical modes. The frequency comb is stabilized by controlling two independent parameters; the repetition rate (fr) and the carrier offset frequency (fo). Since both fr and fo fall in the radio-frequency regime, they can be directly phase-locked to the Rb atomic clock; fr by use of a PZT actuator actuating the fiber cavity length and fo by adjusting the injection current to the pump laser diode [22,23].
In the multi-wavelength generation unit, the stabilized frequency comb is sorted out into four channels with different wavelengths through an array waveguide grating (AWG) comprised of multiple fiber Bragg grating filters (FBG; DWDM101CXX, AC Photonics), each being supplemented with a 100 GHz band-pass filter. Each channel is equipped with a distributed feedback laser (DFB; FOL15DCWB-A81, FITEL) and a photodetector to monitor the beat frequency (fb) between the DFB laser and the filtered frequency comb. The signal-to-noise ratio of the beat frequency is ~35 dB. The beat signal is then amplified and fed into the phase-locked-loop (PLL) in order for the output optical frequency of the DFB laser to be locked to the Rb atomic clock. Finally, a 4 × 1 fiber pigtailed optical switch (Lightbend Mini, Agiltron) is used to select and transfer the frequency-stabilized multiple wavelengths within 10 ms, one by one, to the Twyman-Green interferometer.
3.2. Wavelength conversion for sensitive imaging
Generation of multiple wavelengths at the infra-red telecommunication band near 1.5 µm allows for the use of well-established fiber optic components. However, from the detector’s perspective, conventional Si CCD cameras are not compatible with wavelengths longer than 1.0 µm. Besides, phosphor-coated Si CCD cameras available for infra-red imaging offer low imaging resolutions, slow saturation time, and nonlinear sensitivity, being not adequate for the interferometric use. InGaAs cameras are the potential candidate for wavelengths from 1.0 to 1.7 µm, but they are currently limited in the number of pixels at much higher price. Thus, a novel intermediate scheme of using both the infrared telecommunication-band laser as the light source and the well-established Si CCD as the detector is proposed: the four wavelengths in the telecommunication band are wavelength-converted by use of a nonlinear second harmonic generation (SHG) crystal to the near-infrared (NIR) range at 775 nm, which is well within the sensitivity imaging spectral range of the Si CCD camera (MV-D752-160-CL-8, PhotonFocus) used in our experiments.
The switched output beam is amplified to ~80 mW using an Er-doped fiber amplifier (EDFA) and then focused on a periodically poled lithium niobate crystal (PPLN; SHG-40, Covesion) of a 40 mm length. The converted wavelength yields an average power of ~1 mW at 775 nm, the conversion efficiency being 1.25%. Figure 2 illustrates four NIR wavelengths of 765.266 nm, 765.452 nm, 777.173 nm, and 777.371 nm finally generated after wavelength conversion. The absolute position and uncertainty of the wavelengths were verified using an optical spectrum analyser (OSA; MS9710C, Anritsu) of a 0.07 nm resolution, a wavelength meter (WS-U, HighFinesse) of an accuracy of 30 MHz, and a frequency counter (53131A, Agilent) referenced to the Rb atomic clock. The wavelength uncertainty was evaluated using the relation of [u(fOFG)/fOFG]2 = [u(fr)/fr]2 + [u(fo)/fOFG]2 + [u(fb)/fOFG]2 to 3.44 × 10−12 at an averaging time of 10 s (see Fig. 2(c)) (to 8.21 × 10−11 at an averaging time of 10 ms).
3.3. Twyman-Green interferometer for large step height measurements
The Twyman-Green interferometer used in our experiments provides a large field-of-view (FOV) of 8.7 mm × 6.5 mm. A single-mode fiber is used as the spatial mode filter to generate a perfectly collimated beam. The beam splitter (BS) splits the collimated beam into two paths: one directed towards the reference flat mirror and the other towards the target specimen. Then, the reference mirror with a wavefront error of ~λ/10 is translated six times with a 50 nm displacement by the PZT (P-720, PI) for phase shifting. In order to minimize phase shift errors caused by PZT’s nonlinearity and hysteresis, the arbitrary bucket (A-bucket) algorithm  is adopted which is based on nonlinearity-immune least-square fitting for phase extraction. Resulting interferograms are imaged using the used CCD camera via a macro lens (Micro-NIKKOR 60 mm f/2.8D, Nikon).
To compensate for the fluctuation of the refractive index of air during measurement, the environmental parameters of temperature, relative humidity, and CO2 concentration are monitored in parallel. Their fluctuation levels during a 1-hour period are found to be less than 0.1°C in the ambient temperature, 0.5% in the relative humidity, and 70 ppm in the CO2 concentration. The refractive index of air is corrected by putting the measured environmental parameters into the Ciddor equation .
4. Measurement procedure and data processing
Figure 3 presents the 3-D reconstruction process of the absolute profile of a target surface from 24 consecutive interferograms, which are obtained with four different wavelengths (λi, i = 1, 2, 3, 4) at six different reference mirror positions (δj, j = 1, 2, …, 6) for phase-shifting. First, the A-bucket algorithm determines the relative phase within the range from -π to π at each wavelength (λi) for every CCD pixel; thus, four phase maps are obtained as shown in Fig. 3. Then, the exact fraction method for multi-wavelength interferometry determines the absolute pixel-by-pixel height of the target surface, as shown in Fig. 3(c).
The overall flow diagram of the proposed step-surface reconstruction process is shown in Fig. 4. The left column describes the A-bucket algorithm , which allows us to minimize the uncertainty of excess fractions originating from inaccurate phase-shifting and system nonlinearities. The A-bucket algorithm determines the phase map (ϕ(x,y)) using multi-objective optimization: one objective function determines ϕ(x,y) with phase-shifts (δkj) and the other function determines δk+1j from ϕ(x,y), where k is the iteration number. The least square fit is iterated until the optimization condition of |δk+1j − δkj|≤κ is satisfied, where κ is a predetermined small constant value. The phase map ϕ(x,y) is determined in less than 10 iteration steps.
The exact fraction method for the multi-wavelength interferometry is shown in the right column of Fig. 4. The absolute surface profile is reconstructed using the four phase maps ϕi(x,y) of different wavelengths. To minimize the processing time, the integer number (m) at each pixel (see Eq. (2)) is estimated to be the same with adjacent points (m(x,y-1) or m(x-1, y)) and the error (Ei) is verified to be less than the error bound (Δε) for all wavelengths. If it is not the case, all possible m inside the NAR are tested to satisfy Eq. (2). By repeating this process for all CCD pixels, the absolute 3-D surface profile is finally reconstructed.
5. Experiments and discussions
The wavefront of the frequency-doubled NIR light is spatially distorted by the surface irregularities of the used SHG crystal and also by the nonlinear Kerr-lens effect. This wavefront distortion is suppressed using a single-mode fiber by means of spatial mode filtering. The wavefront error is measured using a Shark-Hartman wavefront sensor (WFA500, Limah Photonics) at two different positions: one before the single-mode filtering (position A in Fig. 1) and the other after the single-mode-filtering (position B in Fig. 1). The resulting wavefronts are as shown in Fig. 5 and the corresponding Seidel coefficients are presented in Table 1. The wavefront error, which has a significant contribution in the combined uncertainty, reduces well from 0.043 λ to 0.009 λ in its root-mean-squared value. The residual error shown in Fig. 5(b) after spatial mode filtering originates from the surface form error of the collimating lens after the single-mode fiber.
Three step-heights of 1800, 500, and 70 μm were measured. Figure 6(a) presents the reconstructed step structures of two grade 0 gauge blocks (gauge block A: 0.5 mm; gauge block B: 1.8 mm) mounted on a base plate by optical contact (see the inset in Fig. 6(a)). The FOV was 8.7 mm × 6.5 mm wide and the lateral pixel resolution was ~13.5 μm. The A-bucket algorithm and exact fraction method were applied to obtain the absolute height at each pixel. Figure 6(b) presents a sectional profile of Fig. 6(a) along line a-a’. The nominal heights of gauge block A and B were measured to be 0.500003 mm and 1.800144 mm, respectively, where the tolerance of the grade 0 gauge blocks was within ± 0.12 μm in the standard laboratory conditions. This demonstrates that large-stepped surfaces up to 1.8 mm can be successfully measured with an absolute uncertainty of less than 15 nm .
A smaller step-height of ~70 µm was also measured, which is in a scale similar to that of industrial specimens in the semiconductors, printed circuit boards, and displays industries. The standard height specimen was manufactured using e-beam lithography with its nominal step height being measured 69.618 μm using a white light scanning interferometer (NanoView, NanoSystems) with a repeatability of 40 nm. The specimen had 25 rod-type bumps at the center and they were surrounded by rectangular flat bumps (see the inset in Fig. 6(c)). The reconstructed surface structure and its sectional profile along line b-b’ are shown in Figs. 6(c) and 6(d). The nominal step height was measured to be 69.624 μm using the proposed method, and it agrees well with the measurements of the white light interferometry.
The measurement repeatability was evaluated by 30 consecutive measurements of a standard gauge block as shown in Fig. 7. The absolute heights of the six sampled points (from point (a) to (f) in Fig. 7(a)) were measured and the results are shown in Figs. 7(b) and 7(c). The height variations of the gauge blocks and variations of the refractive index of air were compensated by measuring the temperature of the base plate and the environmental parameters of the air temperature, pressure, humidity, and CO2 composition in real-time. The temperature variations of the atmosphere and the base plate were less than ± 0.01 °C and ± 0.06 °C during a total measurement time of 480 seconds. The standard deviations were measured to be 3.13 nm and 2.36 nm for gauge blocks A and B, respectively. The maximum peak-to-valley value was less than 10 nm. The overall uncertainty evaluation was made in accordance with the ISO-recommended guidelines as presented in Table 2. The combined uncertainty corresponded to [(1.3 × 10−7∙L)2 + (8.6 nm)2]1/2, for example, 8.6 nm for a 1.8 mm gauge block.
There are three significant factors that determine the measurement speed: the PZT settling time for phase shifting, the frame rate of the CCD camera, and the switching time of the optical switch. In the process of phase shifting, the reference mirror requires a 10 ms settling due to the mechanical resonance of the PZT actuator, which currently limits the measurement speed. The CCD capturing time is 8.3 ms (120 frames per second (fps)) and the wavelength switching takes 10 ms. For complete measurement of a 3-D surface, 24 CCD frames need to be recorded; six phase-shifts for each of four different wavelengths. The total measurement time turns out to be ~400 ms (8.4 ms/frame × 24 frames + 10 ms/switching × 4 switching + 100 ms processing time) in current experiments. The measurement speed can be improved further to tens of milliseconds by use of a high-speed camera together with a field-programmable gate array (FPGA) or a graphic processing unit (GPU) [37,38].
The proposed method of frequency-comb-referenced multi-wavlength interferometry enabled fast, precise absolute measurement of largely stepped surfaces without phase ambiguity. With reference to the frequency comb of an Er-doped femtosecond laser stabilized to the Rb atomic clock, four wavlengths were generated using DFB lasers with an uncertainty of 3.44 × 10−12 at a 10 s averaging time. Then, the generated wavelengths were converted from the NIR to visible range by second harmonic generation, so that the resulting interferogram were observed using a conventional Si CCD camera. The measurement repeatability (one-sigmal) was found 3.13 nm, being direcltly tractable to the time standard. The measurement time was 400 ms. The proposed method is well suited for fast, precise inspection of large industrial specimens such as semiconductor wafers, printed circuit boards, and flat panel displays.
This work was supported by the National Honor Scientist Program, the Space Core Technology Development Program, the Global Research Network Program, and the Basic Science Research Program (2010-0024882) funded by the National Research Foundation of the Republic of Korea.
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