Interferometric Imaging Ellipsometer for Characterizing the Physical Parameters of a Grown Oxide Layer

Interferometric and ellipsometric techniques are widely used in object characterization, investigation and testing. Both techniques are crucial for industrial sectors in manufacturing and production. In this work, an interferometric imaging ellipsometry method has been developed to measure the ellipsometric parameters Ψ, Δ and thickness t of native oxide layer formed on a copper thin film at wide angle of incidence 57.8–80.2º instantaneously. In this system, Michelson interferometer is illuminated with 45° polarized laser beam of 30 mm diameter. The produced fringes are split by a polarizing beam splitter (PBS) into p- and s- polarization fringes. The interferograms are captured separately by two CCD sensors at the reference and the measurement states without moving any parts in the optical system for each state. The detected p- and s- interferograms are analyzed using ImageJ software for mathematically calculating the values of Ψ, Δ and t of the oxide layer grown on copper thin film surface. The proposed method avoids the offset errors and the probable misalignment resulting from the beam and the optical components. In addition, it gathers the information of the interference pattern with the state of polarization of light for accurate, precise and real time contrast images. The oxide layer thickness is determined under controlled environment with high accuracy and low uncertainty. The results obtained by the proposed technique are consistent with the spectroscopic ellipsometer measurements.


Introduction
For many decades, thin film thickness and ellipsometric parameters were measured by several techniques such as ellipsometry, SEM (Scanning Electron Microscope), AFM (Atomic Force Microscope) and so forth. Spectroscopic ellipsometer is considered as a contactless non-destructive technique that can characterize thin films with a notable accuracy. It has the advantage of providing fast and real time monitoring of optical and nano measurements [1][2][3]. Rotating analyzer ellipsometer is considered more precise than the other types of spectroscopic ellipsometer for measuring the optical properties and the thickness of materials [4,5]. Imaging ellipsometer, with contribution of an imaging sensor, provides more precise and robust optical measurements for thin film metrology. It is used to determine thickness, roughness, refractive index, transmittance and reflectance. These parameters are based on the measurements of ellipsometric angles D, W which describe the change in the phase and amplitude of the light beam reflected by the thin film, respectively [6][7][8]. Interferometric imaging ellipsometry technique gathers the benefits of interferometric and ellipsometric methods. Interferometric method has the advantage of precise determination of the phase shift between two beams by the fringe analysis [9][10][11], while the imaging ellipsometer enhances the real time contrast of the captured images [12]. Interferometric imaging ellipsometry technique depends on the direct interference between the p-and s-polarization components to measure the change in phase and amplitude D, W which occurred after reflection from the sample surface. This method gathers the information of the interference pattern with the state of polarization of light for accurate, precise, high contrast and real time images.
Interferometric ellipsometry techniques are implemented by generating temporal fringes using the mechanical scanning [13], the beat frequency of a Zeeman split He-Ne laser [14,15] and the wavelength modulation of a semiconductor laser diode [16]. Also, the interferometric ellipsometry can be demonstrated with performed using acousto-optic modulators in a Mach-Zehnder interferometer where the sample is placed outside the interferometer. Then, the ellipsometric parameters can be determined by combining a beam reflected from the sample with the one transmitted through it [17]. Among the developed interferometric ellipsometer techniques, an optical method performed using a polarized light beam reflected from the sample under test at specific angle of incidence. Then the beam enters a modified Michelson type interferometer and the produced fringes are then split by Wollaston prism into p-and s-polarization stated which can be analyzed for phase and amplitude to provide sufficient information on the sample [18,19].
In this work, an interferometric ellipsometry technique based on Michelson interferometer is developed to determine the ellipsometric parameters and the thickness of a naturally grown oxide layer on a copper thin film at wide angle of incidence (57.8-80.28) instantaneously (i.e., under the same conditions) without mechanical adjustment during the experiment. It also can prevent the probable misalignment errors of optical components due to the change of the angle of incident during the experiment. This in turns will increase the accuracy of the measurements and reduce the uncertainty. Consequently, the thin film growth oxide layer thickness can be determined with very small uncertainty.

Experimental Work
As shown in Fig. 1, a Michelson type interferometer is used to acquire the interference fringes. The experiment is performed in two steps: At first: the p-and s-polarization interferograms are captured by placing the object after the interferometer (see Fig. 1a). In the second step: the interferogram of p-and s-polarization states are captured separately without placing the sample. This case is defined as the reference state (see Fig. 1b). In Fig. 1a, a laser beam with wavelength of 632.9 nm passes through a calibrated linear polarizer P [20]. The polarizer axis is adjusted at 458 and the error in its transmission angle is corrected to eliminate the alignment error. The linearly polarized beam is then expanded to 30 mm diameter using a beam expander B. The beam splitter BS splits the expanded beam into two parts toward the mirrors M1 and M2. The two beams are recombined to form two sets of interference fringes. The produced fringes are focused on the sample surface using a convex lens L of a focal length 75.6 mm. The sample S is fixed on a rotating stage and adjusted at angle 69°with respect to the incident beam. The lens L acquires a wide range of angles of incidence (57. 8-80.28) on the sample surface. This wide range of angles is necessary to determine a large set of W, D values for data interpretation. After that, the interferogram reflected from the sample enters a polarizing beam splitter (PBS) which splits the beam into p-and s-interferograms. Each component yields a set of interference fringes that is captured separately by two CCD imaging sensors representing the measurement state (Fig. 1a). The thin film sample is then removed from the experiment and both p-and s-components are recorded separately representing the reference state as illustrated in Fig. 1b.
The polarization direction of the light beam is defined as the direction of E-electric field components. The incident beam is described as E 1 and the reflected beam E 2 [21].
The ratio of the incident beam to the reflected beam is described by symbol v and is equal to where E s1 and E p1 are the electric field components of the s-and p-incident beams and E s2 and E p2 are the same for the reflected beam. They can be described as follows A s1 ; A p1 ; / s1 and / p1 are parameters describe the amplitudes and phases of the incident beam in case of the p-and s-polarization components, A s2 ; A p2 ; / s2 and / p2 are the amplitudes and phases of the p-and scomponents for the reflected beams.
From Eq. (2a) and (2b), the parameters v 1 and v 2 are related to the ellipsometric parameters as where tanW is the amplitude change related to the p-and scomponents of the reflected beams in according to the incident beams.
From these equations, the Fresnel reflection coefficients r p ; r s are the reflected amount of the E-field in the p-and spolarization directions from the sample surface related to the incident amount of the beam [21] A. W. Abdallah et al.
where E p2 ; E s2 are the electric field component of p-and spolarized beams in case of the reflected beam while E p1 ; E s1 of the incident beam.
All E-vector are Jones vector Equation 9 is the general formula of the ellipsometry where the complex reflectance ratio q ¼ r p =r s is represented as a function of the amplitude change W and the phase change D [21][22][23]: To measure the phase change D, four interferograms are recorded. Two images of p-and s-interferograms before placing the sample for describing the reference state (as in Fig. 1(b)) and the other two images are for the p-and smeasured interferograms of the measurement state after placing the sample as represented in Fig. 1a. For each state, the p-and s-interferograms are captured without moving any parts in the experiment. From the analysis of these interferograms, the phase change D can be described as [25]: where / p ; / s describe the phase of p-and s-polarization interferograms. / p2 ; / p1 represent the p-polarization phases of the measured and reference interferograms, while / s2 ; / s1 represent the s-polarization phases of the measured and reference interferograms, respectively. / p is the result of subtraction of the phases of p-component after and before placing the sample and the same calculation is for the phase / s of s-polarization interferograms. The difference in the phase between p-and s-polarization phases play an essential role for measuring D [24]. From Eq. (4), the ellipsometric parameter W is determined from the relation between the amplitudes of the pand s-interferograms in case of the measurement and reference states [21] tan where the fringe amplitude A is defined by I max ; I min are the intensities value at the y-axis corresponding to the peak position (maximum value) and rough position (minimum value) of the measured fringes.
From the ellipsometric parameters, the thickness of the naturally grown oxide layer on copper thin film is determined using the relation [26,27].
where n 1 is the refractive index of the copper oxide layer and h 1 is the refracted angle inside the copper oxide layer and X ¼ e Ài2b , b denotes the phase change due to propagation of the light beam through medium 1 (oxide layer) as represented in Fig. 2. The reflection coefficients r p , r s at each interface and the oxide layer phase factor b are calculated using the index of refraction for each media, the angle of incidence h and the ellipsometric parameters of the layer as described in reference [21]. Figure 2 represents the three-phase optical system air-copper oxide-copper. By substituting n 0 (air) = 1, n 1 (copper oxide layer) = 2.954 and n 2 (copper thin film) = 0.245 ? i 3.408) [24,25], the copper oxide layer thickness can then be extracted. The range of angles of incidence is identified based on the diameter of the collimated beam of 30 mm incident on the lens and the focal length of the convex lens (75.6 mm) used in localizing the fringes on the object surface, which in turns reflects the beams from the object surface toward the PBS in a suitable wide region that enables sufficient data for the calculations. In the present case, as represented in Fig. 3, we supposed that the angle of incidence on the object surface is 0°to determine the range of angles of incidence. The focal length of the convex lens used in the experiment is (75.6 mm), and the focused beam is expected to hit the sample at (h * 22.4°) as in Fig. 3.
The range of angles of incidence are related to the x-axis pixel value of measured p-and s-interferograms. The images analysis relies on comparing the intensity distribution of reference and measured images for both p-and spolarization interferograms. The number of pixels is divided on the range of angles of incidence for defining the relation between the pixel values with the angles of incidence. Then the phase difference can be determined in accordance with pixel value. Figure 4 represents the interferograms and the intensity distribution images of the reference and measured states using interferometric imaging ellipsometer method. The ellipsometric angles W, D are determined by analyzing the captured p-and s-interferograms for both reference and measurement states. The ellipsometric angle D is calculated using Eq. 2 from the values of the phase difference in p-, s-polarization interferograms of the reference and measured states acquired from analysis. The obtained values for D, W and h are then used to determine the thickness of the grown oxide layer on the copper thin film surface at wide range of angles of incidence (57.8-80.28).

Results and Discussion
Our experiment is performed under a controlled environment for high accuracy and stability of the measurements. The ellipsometric parameters and oxide layer thickness of the sample are also determined by the spectroscopic ellipsometer Angstrom PHE-103 using 632.9 nm wavelength at (57.8-80.28) angles of incidence in steps of 1°, and results are so close to those obtained by the interferometric imaging ellipsometer. Figure 5a, b shows the change of W and D obtained by the interferometric ellipsometer and the spectroscopic ellipsometer Angstrom PHE-103, respectively. Figure 6 shows (a) the mean values of the oxide layer thickness by both techniques at different angles of incidence (b) thickness repeatability values using the developed method. We took 5 images of p-and sinterferograms in each state for repeatability calculation. The consistence of the results proves the reliability of the interferometric ellipsometer technique compared to the standard spectroscopic Ellipsometer PHE-103 and confirms the accuracy and precision of the measurements by our developed method. This layout has the advantages of being static since there are no moving parts inside the interferometer. Also, the same optical components are used in both optical arrangements Fig. 1a, b. Therefore, any effect of polarization perturbation of the optical elements on the phase shift measurements is canceled since the phase shift is the result of subtraction of the p-and s-interferograms for the reference and measured states. Additionally, the ellipsometric parameters are measured at a wide range of angles of incidence without readjustment of the experiment so it avoids the probable misalignment errors arising from readjusting the optical components at each angle of incidence. This developed method consumes time, provide accurate and stable measurements. An advantage of ellipsometric technique is that the measurements are relative values not absolute [14,25], so the measured parameters are independent on the intensity distribution of the beam.

Uncertainty Analysis
The uncertainty in measurement is evaluated based on the standard ISO98-3 (2008) [28]. The main sources that contribute to the uncertainty in measuring oxide layer thickness are the repeatability in measuring W and D, the stability in the laser wavelength, the environmental conditions (temperature T, humidity RH, and pressure P), and the resolution of the precision rotating stage. The polarizer presented in the experiment is fixed, so the error in polarizer angle is corrected at the beginning of the experiment and found to be negligible. Thickness measurement repeatability is evaluated from the repeatability of D and W that calculated from a set of images of p-and s- Air refractive index is corrected by monitoring the environment parameters (temperature, pressure and relative humidity) using the modified Edlén formula [29]. The uncertainty of each parameter is evaluated to be considered in the total budget. Table 1 which summarizes the uncertainty evaluation in the experimental measurements using the interferometric imaging ellipsometer.
Finally, the combined uncertainty for oxide layer thickness measurements including all sources can be calculated as follows: The expanded uncertainty U 95% (with k = 2 at level of confidence 95%) in the case of an angle of incidence 70°, measured by the proposed technique (t I ) is found to be:U 95% ¼ 2 Â 0:12 ¼ 0:24nm

Conclusion
The ellipsometric parameters and the thickness of an oxide layer naturally grown on copper thin film are determined using a novel interferometric ellipsometry based on wide angles of incidence technique. The proposed technique provides two set of interference pattern with p-and s-polarization states for both the reference and measurement cases. The instantaneous imaging of the interference pattern avoids temperature and vibration effects that induce additional errors to the measurements. The technique allows accurate measurements of the ellipsometric parameters D; W and the thin film thickness t s at wide angles of incidence within the range 57.8-80. 28. The convex lens used in the setup enables the required range of the angles of incidence without need to a mechanical adjustment of angles of incidence during measurements. The expanded uncertainty in measuring the oxide layer thickness t s is evaluated as ± 0.24 nm. This work aims to measure the thickness of absorptive films in the range of nm and lm. The analysis of the repeated measurements under good control of environment gave a good indication of the developed technique results. The results obtained by the proposed technique are in good consistence with those obtained by the spectroscopic ellipsometer, this in turn confirms the accuracy and stability of our proposed interferometric ellipsometer method.
Funding Open access funding provided by The Science, Technology & Innovation Funding Authority (STDF) in cooperation with The Egyptian Knowledge Bank (EKB). This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.

Declaration
Conflict of interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
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