Anal. Chem. 1998, 70, 1346-1351
Order-Sorting Filters for a Grating Spectrometer and Multichannel Detection System: Application to Real-Time Spectroscopic Ellipsometry Yeon-Taik Kim* and Ilsin An†
Department of Chemistry and Atomic-scale Surface Science Research Center, Yonsei University, Seoul, Korea
A multichannel detection system with a grating spectrometer was employed for rapid spectroscopic ellipsometry measurements. Although a grating spectrometer is convenient for dispersing light, it suffers from overlapping orders. Here, we developed novel techniques to eliminate overlapping orders of spectra from a reflection grating. First, an optical filter is properly positioned in the measurement system. Second, a numerical filter is developed. A monochromatic source was used to deduce the exact amount of the overlapping orders, which were determined as the irradiance ratio of the second- to the first-order diffraction. The ratios were found to be increased from 5 to 11% in the 1.5-2.5 eV region. We used the values to correct for overlapping irradiance using the numerical filter. Finally, second-order reflection in real-time rotating element spectroellipsograms were corrected using the filters developed in this work. Grating spectrometers for a multichannel linear array detector have been developed for rapid spectroscopic measurements. In the multichannel detection system, a photodiode array (PDA) or a charged coupled device (CCD) has been employed.1 The array is a multiple of 128 detector pixels spanned over 1 in. in length. Accordingly, the spectrometers are specifically designed for dispersing and focusing the incoming light onto the surface of the detector array without any moving parts. Most of the spectrometers have adopted a diffraction grating to disperse incoming light because a grating has advantages over a prism. It has more dispersing power, and its resolution is easily controlled during the fabrication process.2 In a grating spectrometer, however, incident light is diffracted into several orders, and order-sorting has been a nuisance problem which limits a free spectral range in spectroscopic measurements. In many spectroscopic applications, the overlapping orders have been compensated by the judicious choice of a light source, detector, and filter to solve the problem. However, when we * To whom correspondence should be addressed. Tel.: 82-2-361-2649. Fax: 82-2-364-7050. E-mail:
[email protected]. † Permanent address: Department of Physics and Research Center for Electronic Materials and Components, Hanyang University, Ansan, Korea. (1) (a) Liang, Y. L.; Baker, M. E.; Denton, M. B. Anal. Chem. 1996, 68, 38853891. (b) River, C.; Mermet, J. M. Appl. Spectrosc. 1996, 50, 959-964. (c) Walton, D.; Vanderwal, J. J.; Xia, H.; Zhao, P. Rev. Sci. Instrum. 1996, 67, 2727-2731. (2) Hutley M. C. Diffraction Gratings; Academic Press: San Diego, CA, 1982.
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employ a grating spectrometer and a PDA detection system for real-time spectroscopic ellipsometry (RTSE), the second-order overlap causes significant errors in the ellipsometric data. We constructed a rotating polarizer-type RTSE equipped with a silicon PDA detection system.3 A grating spectrometer4 covering 1.5-6.0 eV was used. A xenon (Xe) light source and the other optical elements limited the spectral range between 1.5 and 5.0 eV. Thus, we focused on solving the second-order diffraction of UV, which overlaps in the 1.5-2.5 eV region. We developed two universal methods to solve the problem of overlapping orders. One is to use a thin optical filter, and the other is to use a numerical filter. In principle, both techniques can be used for grating spectrometers. The latter technique, however, requires information of a diffraction pattern that tells us the relation between the first and higher orders of diffraction. The diffraction pattern for static or slowly varying irradiance can be easily obtained from ratios between the first and higher orders of diffraction. However, if the irradiance is varied during the measurement period such as RTSE, care should be taken to determine the diffraction pattern because data readout from an individual detector element is performed in a serial format. Thus, the time lag between the detector elements makes it impossible to take a snapshot of a concurrent spectrum. The optical or the numerical filter allows us to deconvolute the overlapping orders for both static irradiance and rapidly changing irradiance with a predicted manner. EXPERIMENTAL SECTION Shown in Figure 1 is a rotating element RTSE system developed for monitoring electrochemical processes in real time.3c It consists of a 75-W Xe arc lamp, collimating lens, rotating polarizer with a frequency of 10 Hz, electrode holder, fixed analyzer, grating spectrometer, and multichannel detector. Nonlinearity, image persistence, and stray light in the detection system were corrected by the method developed by An and Collins.3b Unlike a photomultiplier tube (PMT), a PDA integrates photons over the exposure time, which is determined by the optical encoder pulse from the rotating polarizer. The optical (3) (a) Kim, Y.-T.; Collins, R. W.; Vedam, K. Surf. Sci. 1990, 223, 341-350. (b) An, I.; Collins, R. W. Rev. Sci. Instrum. 1991, 62, 1904-1911. (c) An, I.; Cong, Y.; Nguyen, N. V.; Pudliner, B. S.; Collins, R. W. Thin Solid Films 1991, 206, 300-305. (d) Brittain, S. T.; Wurm D. B.; Kim, Y.-T. Presented at the Electrochemical Society Meeting, Reno, NV, May 22-26, 1995. (4) Model CP200 Imaging Quality Spectrograph, Instruments S. A., Inc. Jobin Yvon/Spex Division, Edison, NJ. S0003-2700(97)00923-2 CCC: $15.00
© 1998 American Chemical Society Published on Web 02/28/1998
Figure 1. Schematic of the real-time spectroscopic ellipsometer.
encoder pulse triggers data acquisition. Thus, the rotating speed of the polarizer should be kept constant in order to integrate photons at the same time interval. In our system, an angular frequency of 10 Hz is maintained using a feedback circuit referenced by a quartz clock. The data acquisition speed of RTSE is limited by the readout time of the PDA. In our system, with 128 detector elements grouped by 8 of 1024 pixels, the minimum readout time of the 128 detector elements is 4.5 ms. To construct two ellipsometry parameters, ∆ and Ψ, we need to integrate the irradiance at least three times during an optical cycle (half-mechanical rotation) of the polarizer, which requires a minimum of three optical encoder pulses per optical cycle.5 We use a frequency divider to adjust the number of encoder pulses per optical cycle. All the ellipsometric spectra presented in this article are from four integrations per optical rotation and the average of two mechanical rotations (one mechanical rotation ) two optical rotations). Thus, the data acquisition time for one pair (∆ and Ψ) of ellipsometric spectra is 200 ms with the exposure time of 12.5 ms, which is limited by the angular frequency of 10 Hz. However, data over many optical cycles can be averaged to enhance the signal-to-noise ratio at the expense of time resolution. A. Optical Filter. We used a long-pass filter to remove irradiance from the overlapping orders. In our RTSE system, the irradiance drops rapidly at the high photon energy region (see Figure 2) because of the Xe lamp and the optical response of the system, including the reflectivity of the sample under study. Thus, the contribution of higher orders than the second-order diffraction is negligible at the high photon energy. The practical spectral range lies between 1.5 and 5.0 eV, and our concern is the secondorder diffraction from 3.0 to 5.0 eV, which overlaps between 1.5 and 2.5 eV. A thin long-pass filter was directly attached on the part of the PDA surface where the corresponding energies are between 1.5 and 2.5 eV in order to block the second-order diffraction. This method works well, although it has several problems. First, the detector array is kept in a vacuum in order to prevent water condensation. Thus, it is necessary to break vacuum many times until the exact position of the filter is located by trial and error. It disturbs the alignment of the optical system, including the PDA (5) Erman, M. Ph.D. Thesis, Universite d’Orsay, 1982.
Figure 2. Four spectra of S-integrals measured from gold and c-Si samples. Gold shows much higher reflectance at the low photon energy region. In c-Si, however, the reflectance at the high energy is relatively high.
wavelength calibration. Second, the optical filter causes stray light and multiple reflection. The stray light is generated at the edge of the optical filter. Multiple reflection occurs between the filter and the detector surface, unless the multiple reflection is normal to the surface. Third, we cannot find a long-pass filter with a sharp cutoff edge at 2.5 eV. Fourth, two or more optical filters are required for a wide spectral range application. Thus, we developed a simple numerical method to filter the second or higher orders of irradiance. B. Numerical Filter. We have developed a numerical method to deconvolute overlapping orders. This method is based on the numerical determination of the intensity ratio between the first- and second-order diffraction. Subsequently, the ratio values are used to correct a spectrum in the overlapped photon energies. Two different approaches were made to get the diffraction ratio as a function of photon energy of our system for the numerical filter method. We used an extra spectrometer to generate a quasi-monochromatic source over 3.0-5.0 eV of the PDA. We selected a quasimonochromatic source with a broad bandwidth which covers a few detector elements. It is important to use a broad quasimonochromatic source because of the finite size of the detector element.6 The quasi-monochromatic source was focused on the entrance slit of the spectrograph in RTSE. Figure 3 shows a resulting spectrum obtained with the PDA. A strong peak near 360 nm (or 3.44 eV) is the first-order diffraction of the quasi-monochromatic beam, and the other peak, near 720 nm (or 1.72 eV), is its corresponding second-order diffraction. There exists a simple relation between hν in electronvolts and λ in nanometers, which is hν × λ ) 1238.4. The second to firstorder diffraction ratio, γ(hν), was determined from the area integration of the peaks at hν (second-order term) and 2hν (firstorder term). In other words, γ(hν) ) {Isum(hν)/Isum(2hν)}/2, where Isum(hν) and Isum(2hν) are intensities obtained by the area integration at hν and 2hν, respectively. The division factor of 2 explains that irradiance from the second-order diffraction covers twice the detector area of the first-order diffraction. It can be noticed when the full widths at half-maximum of the two peaks are compared in Figure 3. (6) Talmi, T.; Simpson, R. W. Appl. Opt. 1980, 19, 1401-1414.
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data construction process for the ellipsometer. The irradiance at the detector element k can be expressed as a time-dependent function:3,7
Ik(t) ) I0,k[1 + Rk cos{2(Pk - Ps,k)} + βk sin{2(Pk - Ps,k)}] (2)
Figure 3. Spectrum obtained with a quasi-monochromatic source. Irradiance is proportional to photon counts.
Figure 4. Ratio of the second- to first-order diffraction, γ (hν). Filled circles are obtained with quasi-monochromatic sources, and a line is the best fit to the data. Open circles are obtained from the ellipsometric method using a c-Si sample.
Figure 4 shows the γ(hν) values versus photon energy obtained with several quasi-monochromatic sources (b). A line in the figure indicates the best fitting curve for the data. The positive deviation of the γ(hν) values was determined to be a function of photon energy and the incident angle of dispersed irradiance. The γ(hν) values can also be determined by calculation, as marked by O, if we know the exact ellipsometry spectrum of ∆ and Ψ, (see discussion at the end). Determining γ(hν) values, we correct the overlapping orders for the irradiance at hν, I(hν), in the photon energy between 1.5 and 2.5 eV by the following equation,
I(hν)cor ) I(hν) - γ(hν)I(2hν)
(1)
where subscript “cor” indicates the corrected value. This method, however, is valid only if irradiance over the PDA remains constant during the readout time. Thus, eq 1 is the numerical filter for static or slowly varying measurements to correct the overlapping orders. For instance, the readout time for 128 detector elements in our system is 4.5 ms. If irradiance changes in 4.5 ms, eq 1 cannot be applied to correct for the overlapping orders, because I(hν) and I(2hν) are not a part of the concurrent spectrum anymore. In the RTSE system, the irradiance varies rapidly due to the rotating polarizer. Thus, we further developed the numerical filter in order to correct the RTSE data. C. Data Construction Process in RTSE. Before we discuss the numerical filter for RTSE data, it is necessary to review the 1348 Analytical Chemistry, Vol. 70, No. 7, April 1, 1998
{Rk,βk} are the normalized Fourier coefficients to determine ellipsometric spectra {∆k,Ψk}. Since the PDA detector is an integrating detector, the normalized Fourier coefficients are obtained by Hadamard analysis.5 Pk is an azimuthal angle of the rotating polarizer measured from the position at which the first integration for detector element k starts. Ps,k is an offset angle of the polarizer from the plane of incidence when the first integration starts for detector elements k. Because the detector controller reads one PDA element at a time, Ps,k is a linear function of element number k, and the function is determined during the calibration process.8 That is, Ps,k ) δp,k + δ0. Here, δp is an angle swept by the polarizer during the readout time of one PDA element, and δ0 is a constant offset relative to the plane of incidence. The angle δp is determined from the relation between the polarizer angular frequency and the readout time of the detector. In real experiments, the phase difference, Ps,k, is corrected later with 2Ps,k degree rotational transformation. The PDA integrates four times during one optical cycle (a halfmechanical rotation of the polarizer), triggered by four equallyspaced optical encoder pulses coming from the polarizer shaft. Now we have four S-integrals:
Sjk ) I0k
∫
jπ/4
(j-1) π/4
(1 + R′k cos 2Pk + β′k sin 2Pk) dPk, j ) 1, ..., 4 (3)
where {R′k,β′k} are experimental values which require 2Ps,k phase rotation to get {Rk,βk}. The experimental values of {R′k,β′k} can be determined from the four S-integrals made by the detection system:
R′k ) (π/2){(S1k - S2k - S3k + S4k)/ (S1k + S2k + S3k + S4k)} (4a) β′k ) (π/2){(S1k + S2k - S3k - S4k)/ (S1k + S2k + S3k + S4k)} (4b) I0k ) (1/π)(S1k + S2k + S3k + S4k)
(4c)
Thus, {Rk,βk}, defined in eq 2, can be calculated as a simple rotational transformation of {R′k,β′k}:
( ) (
)( )
Rk cos 2Psk sin 2Psk R′k ) βk -sin 2Psk cos 2Psk β′k
(5)
(7) Gottesfeld, S.; Kim, Y.-T.; Redondo, A. In Physical Electrochemistry: Principles, Methods, and Applications; Rubinstein, I., Ed.; Marcel Dekker: New York, 1995; pp 393-467. (8) (a) Kim, Y.-T. Ph.D. Thesis, The Pennsylvania State University, 1990. (b) Nguyen, N. V.; Pudliner, B. S.; An, I.; Collins, R. W. J. Opt. Soc. Am. 1991, A8, 919-931.
Figure 5. Intensity change as a function of time at detector elements k and k′. Note the phase difference between two detectors due to the data acquisition in series.
Finally, the desired ellipsometric parameters, {∆k,Ψk}, are obtained using {Rk,βk}:
cos ∆k ) βk/(1 - Rk2)1/2
(6a)
tan Ψk ) {(1 + Rk)/(1 - Rk)}1/2 tan A
(6b)
where A is an analyzer azimuth measured with respect to the plane of incidence.8 In addition, a linear function between wavelength (photon energy) and element number k in the PDA was determined using a Hg lamp to assign a wavelength value to each detector element using the equation below:
wavelength (nm) ) -4.963k + 804.382
(7)
D. Numerical Filter for RTSE Data. We describe the numerical filter for rapidly changing irradiance expressed by a known equation such as RTSE. It has been shown that the irradiance is described in eq 2, and the phase delay between two PDA elements k and k′ is (k′ - k)δp, where δp is defined earlier. Let us assume, from now on, the corresponding energies at detector element k and k′ are hν and 2hν, respectively. Here, detector element k suffers from overlapping the second order of detector element k′. The S-integrals at detector elements k and k′ are schematically drawn in Figure 5. As can be seen in Figure 5, the intensity change at detector elements k and k′ is not synchronized owing to the phase difference between two elements. The phase difference is inevitable because the OMA reads the photon integration in a serial fashion. Thus, it is necessary is to reconstruct new S-integrals for element k′ which does not have the phase difference (δp ) 0) from detector element k. This involves the reconstruction of the four S-integrals which are synchronized with element k, and the phase-corrected {R′k′k,β′k′k} is calculated by rotational transformation of {R′k′,β′k′} according to eq 8:
( ) (
)( )
cos 2(k - k′)δp sin 2(k - k′)δp R′k′ R′k′k ) β′k′k -sin 2(k - k′)δp cos 2(k - k′)δp β′k′
(8)
Figure 6. Simulated ellipsometric ∆ spectra with assumed γ values. Constant γ values (0-5%) were assumed over 1.5-2.25 eV.
where {R′k′,β′k′} are the normalized Fourier coefficients corresponding to 2hν, which was obtained using eq 4. The phasecorrected {R′k′k,β′k′k} allows one to calculate new S-integrals, (Sj,k′k, j ) 1, ..., 4), according to eq 3. The I0k′ value, however, is independent of phase. Now, we are ready for the correction of overlapping orders at detector element k. The remaining process is exactly the same as eq 1:
Sj,k, cor ) Sj,k, raw - γkSj,k′k, j ) 1, ..., 4
(9)
where k indicates a detector element whose energy, hν, lies between 1.5 and 2.5 eV, and k′ thus corresponds to 2hν. The subscripts “raw” and “cor” indicate raw and corrected data, respectively. The corrected four S-integrals are now used to construct spectroellipsograms, as discussed above. RESULTS AND DISCUSSION Figure 2 shows raw spectra of four S-integrals for gold and c-Si which are defined in eq 3. The gold spectra show much higher reflectance in the lower energy region than in the higher energy region. In contrast, the c-Si spectra show relatively high photon counts at 3.0 eV or greater. Thus, the effect of overlapping orders is relatively minor in gold compared to that in c-Si. The effect of overlapping orders is simulated for an ellipsometric spectrum (∆) of c-Si with a 20-Å overlayer of SiO2 in the photon energy of 1.5-4.5 eV. As can be seen in Figure 6, we can easily notice that the effect is well correlated with the magnitude of the γ value. Equations 3 and 6 were used to get spectra for four S-integrals (Sj, j ) 1, ..., 4). The simulated spectrum shown in Figure 6 was obtained using the S-integrals with the effect of overlapping orders. That is, Sj,new(hν) ) Sj(hν) + γSj(2hν). Here, the γ value is assumed to be constant for the simple calculation. The effect of overlapping orders was corrected for c-Si and c-GaAs using the optical and numerical filter. Figure 7 shows three spectra obtained from a c-Si sample. A broken line shows the raw spectrum collected without the filters. The magnitude of the effect below 2.5 eV can be clearly identified by comparison with the simulated spectrum shown in Figure 6. An open circle indicates the spectrum obtained with the optical filter, where data are plotted at every fifth point for clarity. Analytical Chemistry, Vol. 70, No. 7, April 1, 1998
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Figure 7. ∆ and Ψ spectra of c-Si measured with (O) and without (- - -) the optical filter. Solid lines show the corrected spectra with the numerical filter. For clarity, the data for open circles are selected from every fifth point of raw data. The optical filter covered detector elements up to 2.2 eV, and the numerical filter was used up to 2.5 eV.
Figure 8. ∆ and Ψ spectra of c-GaAs measured with (O) and without (- - -) the optical filter. Solid lines show the corrected spectra with the numerical filter. For clarity, the data for open circles are selected from every fifth point of raw data. The optical filter covered detector elements up to 2.2 eV, and the numerical filter was used up to 2.5 eV.
It can be easily noticed that the filter is covered up to 2.3 eV because of the discontinuity of the spectrum around 2.3 eV. As mentioned earlier, it is difficult to position the optical filter on 2.5 eV. Furthermore, it was not possible to obtain an optical filter with a sharp cutoff edge at 2.5 eV and to remove scattering light at the filter edge. These effects can be seen in the ∆ spectrum (O) between 2.2 and 2.3 eV in Figure 7. Finally, a line in Figure 7 represents the results with the numerical filter applied to the raw spectra shown as a broken line. The corrected spectrum is free from overlapping orders. A c-GaAs sample was also selected to demonstrate the validity of the filters, and the results are shown in Figure 8. Again, a broken line is for raw data collected without the filters, and an open circle represents spectra obtained with the optical filter. A line in Figure 8 is obtained using the numerical filter. The line spectrum can be easily simulated using a c-GaAs dielectric function. The same plot was made with freshly deposited gold in Figure 9, showing the Ψ spectrum only. Thus, the optical and numerical filters can be applied to all samples suffering from overlapping orders. From the comparison 1350 Analytical Chemistry, Vol. 70, No. 7, April 1, 1998
Figure 9. Ψ spectrum of Au measured with (O) and without (- - -) the optical filter. Solid lines show the corrected spectra with the numerical filter. For clarity, the data for open circles are selected from every fifth point of raw data. The optical filter covered detector elements up to 2.2 eV, and the numerical filter was used up to 2.5 eV.
of Figures 7, 8 and 9, we can see that c-Si is more sensitive to overlapping orders than others. Now we want to go back to Figure 4 in order to explain the data marked as open circles. The γ(hν) values from 1.5 to 2.5 eV was obtained using quasi-monochromatic sources from 3.0 to 5.0 eV. However, the γ(hν) values can also be calculated if we know the exact ellipsometric spectra. c-Si is selected as an example and measured with the spectroscopic ellipsometer to get the (∆,Ψ) spectra. As shown in Figure 7, the spectrum at the low-energy region is degraded because of overlapping orders. The spectrum, however, from 2.5 to 5.0 eV is free from the error associated with overlapping orders, so that we may use the spectrum only in the range of 2.5-5.0 eV to find the best fitting model.9 Obtaining the accurate sample structure through the best fitting model, we can simulate the ellipsometric spectrum (∆sim,Ψsim) in the range of 1.5-2.5 eV, which is also free from overlapping orders. Although we do not have information on the γ (hν) values at this time, we can assume a trial value of γ(hν) and calculate new S-integrals for the 1.5-2.5 eV region according to eq 10:
Sj,new(hν) ) Sj,raw(hν) - γ(hν)Sj(2hν), j ) 1, ..., 4 (10)
With these new S-integrals, we can calculate (∆cal,Ψcal) for the 1.5-2.5 eV region and compare with those simulated values (∆sim,Ψsim). This process will be continued with new trial values of γ(hν) until we get minimum value of [{(∆sim(hν) - ∆cal(hν)}2 - {(Ψsim(hν) - Ψcal(hν)}2]. The resultant γ(hν) values are plotted as O in Figure 4. This method works well up to 1.9 eV in our system with a c-Si sample. However, the γ(hν) values are scattered and deviated between 1.9 and 2.5 eV because of the weak UV reflection in the range of between 3.8 and 5.0 eV. The sensitivity of the method (9) (a) Kim, Y.-T.; Collins, R. W.; Vedam, K.; Allara, D. L. J. Electrochem. Soc. 1991, 138, 3266-3275. (b) Aspnes, D. E. in Optical Characterization Techniques for Semiconductor Technology; So, S. S., Potter, R. F., Eds.; SPIE Conf. Proc. 276; SPIE: Bellingham, WA, 1981; pp 188-195.
can be improved by selecting highly reflecting samples in the UV region and/or changing the angle of incidence. CONCLUSION We developed novel techniques for order sorting in multichannel detection systems integrated with a grating spectrometer. First, an optical filter was inserted in the position to remove overlapping orders of irradiance. Second, we measured the effect using a monochromatic source or an ellipsometric technique in order to remove overlapping orders using the numerical filter. Both filters can be applied to a grating spectrometer with multichannel detection systems. In particular, the numerical filter is very powerful when applied to a broad spectral range and/or
to a case of strong irradiance in the UV region. Although we showed the case of overlapping second orders, the techniques can be applied to higher orders of overlapping. Finally, the computation time for the numerical filter is on the order of microseconds. ACKNOWLEDGMENT Y.T.K. greatly acknowledges KOSEF (971-0305-041-2) and Yonsei University for the financial support. Received for review August 22, 1997. Accepted January 7, 1998. AC970923L
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