A non-invasive method for measuring the refractive index, extinction coefficient and film thickness of absorptive thin films using spectral-domain optical coherent tomography is proposed, analyzed and experimentally demonstrated. Such an optical system employing a normal-incident beam of light exhibits a high spatial resolution. There are no mechanical moving parts involved for the measurement except the transversal scanning module for the measurement at various transversal locations. The method was experimentally demonstrated on two absorptive thin-film samples coated on transparent glass substrates. The refractive index and extinction coefficient spectra from 510 to 580 nm wavelength range and film thickness were simultaneously measured. The results are presented and discussed.
© 2014 Optical Society of America
Optical coherence tomography (OCT) is a non-invasive morphological technique based on optical interferometry  involving the employment of a beam of light with a limited coherence length. It provides a micro-scale spatial resolution in both lateral and axial direction, while maintaining a longer scanning depth by using objective lenses of low numerical aperture. The spectroscopic optical coherence tomography further extracts the spectroscopic information from acquired data by analyzing the time-frequency distribution . OCT has been proven to be useful for the characterizing of distribution of materials by analyzing the local spectral attenuation. Quantitative identification is also possible with proper calibration [3–5]. Spectral-domain OCT (SD-OCT) configuration is usually preferred for the spectroscopic measurement, since it provides a better phase stability by removing the mechanical scanning in axial direction and with an improved signal to noise ratio .
The measurement of optical characteristics such as refractive index and extinction coefficient is important for many applications, including imaging technologies and optics involving integrated devices. Various methods were developed for the non-invasive measurement of refractive index, extinction coefficient and film thickness [7–10]. OCT-based techniques for material characterization were also proposed [11–13], as they offer the capability to characterize thick layers. Most of them focus on the measurement of either thickness or group refractive index at a specific wavelength.
We have developed a SD-OCT system which is capable of simultaneous measuring the complex refractive index spectrum in the visible range and film thickness of absorptive thin films. With the normal-incident optical system of OCT, film characterization with SD-OCT has the advantage of spatial resolution. Employing the coherence gating effect, the SD-OCT measurement could eliminate the impact from irrelevant layers of the sample under test. In other words, any stray light resulting from unknown layers away from the layer of interest may not affect the measurement result. The algorithm solves the phase ambiguity problem, making it suitable for samples with film thickness much larger than the half detection wavelength.
2.1 Thin-film sample modeling
OCT is an interferometric technique based on the interference between backscattered (or reflected) light from the sample to be examined and reflected light from a reference plane. For a planar sample with a reflectivity r and a reference plane of a perfect mirror, the spectrum received can be expressed in following form:Fig. 1, the SD-OCT signal consists of the sum of all reflected light from each of the interfaces. By selecting a proper optical path length of the reference arm, the interference signal can be expressed by following equation :Fig. 1. The formula of the reflection coefficient in the square brackets of Eq. (3) can be obtained via the summation of a geometric series of amplitudes of multiple reflections. If we expand the denominator in Eq. (3) into a geometric series, then each of the terms in the series represents a reflection among the sum of multiple reflections.
2.2 Identification and separation of signals
The complex interference spectrum, described in Eq. (3), contains both amplitude and phase information at each wavelength. When considering N discrete wavelengths, we have 2N observables in total, which are less than the 2N + 1 unknowns (i.e. refractive index and extinction coefficient at N wavelengths and the sample thickness). To find a set of unique solution of (n, k, l), it is necessary to extract more information from the data. Equation (3) contains all the reflections and multiple reflections of the sample structure of Fig. 1. If the 1st order reflections ( and) of the two interfaces are isolated to each other and all the other multiple reflections in the temporal domain, the number of observations increases to 4N (2N for each interface), and finding a solution of (n, k, l) becomes possible. To execute such operation, it is important that the signals of each interface is isolated to each other, which requires a better axial resolution for the characterization of thinner films. In addition, system chromatic dispersion compensation via the employment of optical dispersion compensator or digital electronic compensation is needed, since dispersion may also broaden the point spread function. It should be noted that the shape of the light source spectrum is also crucial, since for a spectrum shape away from Gaussian, side lopes arise after the Fourier transform, and prevent the clean separation of signals.
2.3 Model fitting
Once the axial resolution is high enough to separate the first two terms of the multiple reflections due to the front and rear interface of sample, an inverse Fourier transform of the measured interference intensity signal can be performed to each of them. We are interested in the absolute value of these two reflection amplitudes as well as their phase difference. They can be described with following equations:
In other words A is the amplitude spectrum of the front interface reflection, B is the amplitude spectrum of the rear interface reflection, and C is the phase difference between these two amplitudes. Note only the phase difference between the two interface signals are used for the calculation. These three spectral attributes are dependent on the complex refractive index and the film thickness. Equations (4)–(6) must be inverted to obtain these optical parameters of the film. In our experimental investigation, we measured A, B and C at several wavelengths within the range of 510 nm - 580 nm. The optical parameters (n, k, l) of thin film can then be calculated with Gauss-Newton algorithm by fitting the numerical model to the experimental data. The increment vector (δn, δk, δl) of each wavelength can be determined with the following equation:Eq. (7) for each wavelength. With the initial condition described above and an accuracy requirement of, for example, 1%, the result usually converges within 100 iterations.
2.4 Phase ambiguity issue
For optical thickness of films larger than half optical wavelength, phase ambiguity problem may occur. This ambiguity comes from the fact that the phase retrieved with SD-OCT is always within the principal 2π range. A continuous phase spectrum (proportional to the film thickness) can be obtained via the employment of a phase unwrapping method . Unwrapped phase spectrum has a 2πm phase shift from the actual phase, where is an unknown integer, and:16]. For low-coherence interference system, removing the ambiguity by introducing an artificial dispersion in low-coherence optical system was firstly proposed by Hitzenberger . Later Hendargo provided an alternative solution to this problem for SD-OCT system by performing synthetic-wavelength-based phase unwrapping . To solve the phase ambiguity issue with a sample of unknown dispersion properties, a parameter optimization approach was applied. Since the mean square error (MSE) of the model fitting described in Eqs. (4)–(7) increases with incorrect m selection, the ambiguity can be resolved by performing the parameter optimization for different m value. The idea is clear for thin layer with zero dispersion (i.e. air spacing) where the C is a straight line crossing the zero, and for incorrect m, Cexp(m) will not cross the zero and perfect fitting becomes impossible. The idea can be extended to thin film with finite dispersion () and attenuation () if the minimization of the MSE for all three of Aexp, Bexp and Cexp are processed in the optimization process. The MSE to be considered in this case is defined as:
3.1 System setup
In our experimental investigation the light source of the OCT system is a Ce3+:YAG double-clad crystal fiber (DCF) pumped with a 446-nm laser diode [19,20]. It emits a broadband spectrum with a 545-nm center wavelength and a bandwidth of 90 nm, as shown in Fig. 2(a), the corresponding axial resolution is about 1.5 μm in free space. The system is an ordinary SD-OCT setup, as shown in Fig. 2(b). The spectrometer we used is an Ocean Optics USB4000, which provides a 1.5-nm spectral resolution.
The samples are two absorptive polymer films coated on 500-μm aluminosilicate glass substrates. Samples 1 and 2 are visually yellow and green respectively. The transmission spectra measured with a commercial transmission spectrometer are shown in Fig. 3.We note that the transmission peaks of the samples are consistent with their colors. For the SD-OCT measurement, the sample was set in a substrate-incident scheme, which means the light is incident from the substrate (aluminosilicate glass) side, and another 500-μm aluminosilicate glass was put in front of the reference mirror for dispersion compensation. The substrate-incident arrangement offers a more practical demonstration of this technique since in most case the thin film with unknown properties is embedded beneath a layer with a known refractive index. Since the magnitude for the rear interface reflection (B) tend to be smaller because of the extra absorption, and the crosstalk issue described in the previous section is milder if the magnitude of the A and B in Eqs. (4) and (5) are similar or of the same order of magnitude.
4. Result and discussion
In the data analysis of the experimental results, the measured signals of the interference intensity at various wavelengths form an interference spectrum. This spectrum is then transformed into temporal domain via a Fourier transform. Figure 4 shows the results of the Fourier transform (axial scans) and corresponding spectral attributes (A, B, and C versus wavelength). Note the phase spectra C shown here are the continuous phase spectra directly unwrapped from the experimental value. To determine the m0 value for each case, the MSE versus relation were calculated, as shown in Fig. 5.
The MSE versus relation suggests the m0 is 27 for sample 1 and 16 for sample 2, respectively. In both figures asymmetry was found centering m0. With the m0 known via the optimization process using Eq. (9), the phase ambiguity in Eq. (8) is resolved. Using Eqs. (4)-(6) in conjunction with the Gauss-Newton algorithm, we are able to obtain the unknowns n, k and l. A starting condition of (n, k) is set to be (1.6, 0) for all wavelength, and the initial guess of film thickness is the estimated optical thickness of thin film divided by 1.6. The optical thickness was estimated with the distance between coherence spikes in Fig. 4, which are 8.2 μm for sample 1 and 5.4 μm for sample 2, and the corresponding guesses of l are 5.1 μm and 3.4 μm, respectively. The calculation results for (n, k, l) are shown in Fig. 6.The derived refractive indices of both samples show negative dispersion, as expected for this kind of polymer film in the spectral regime where absorption occurs. The calculation precision is discussed by performing the uniqueness test that commonly used for spectroscopic ellipsometry . As shown in Fig. 6, the minima of MSE appear at 4.44 μm and 2.73 μm for sample 1 and sample 2. The corresponding uniqueness range is 6.4 nm and 3.5 nm, respectively.
To further verify the calculated optical properties, the transmittance of the samples is calculated using the measured optical parameters (n, k, l) of the samples with the following equation:Fig. 7.Slight inhomogeneity (film thickness variation) in lateral direction can be seen for both samples, and standard deviation of optical thickness of sample 1 and sample 2 are around 21.5 nm and 10.3 nm, respectively. The characterization process was applied for each sample, and the transmittances were calculated according to Eq. (10). The overall transmission curves were generated by averaging the transmittances, and the comparison to the value measured by a commercial transmission spectrometer is shown in Fig. 8, which shows good agreement for both samples.
We have proposed, analyzed and experimentally demonstrated an OCT technique that is capable of characterizing the refractive index, extinction coefficient and physical thickness of absorptive thin layers. The experimental investigation was demonstrated using two absorptive polymer films coated on transparent glass substrate. The 510~580-nm measurement bandwidth is limited by the Ce:YAG DCF emission spectrum. This technique provides a convenient method for the characterization of film with a micro-scale thickness, with high spatial resolution in both axial and lateral direction, making it useful in imaging technology and integrated optics.
This work was partially supported by the National Science Council.
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