Download Optical Coherence Tomography

Spectroscopic measurement of absorptive thin
films by Spectral-Domain Optical Coherence
Tomography
Tuan-Shu Ho,1 Pochi Yeh,2 Cheng-Chung Tsai,1 Kuang-Yu Hsu,1
and Sheng-Lung Huang1,3,*
2
1
Graduate Institute of Photonics and Optoelectronics, National Taiwan University, Taipei 106, Taiwan
Department of Electrical and Computer Engineering, University of California, Santa Barbara, California 93106,
USA
3
Department of Electrical Engineering, National Taiwan University, Taipei 106, Taiwan
*
[email protected]
Abstract: 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 normalincident 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
OCIS codes: (110.4500) Optical coherence tomography; (310.3840) Materials and process
characterization.
References and links
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Received 24 Jul 2013; revised 22 Feb 2014; accepted 27 Feb 2014; published 5 Mar 2014
10 March 2014 | Vol. 22, No. 5 | DOI:10.1364/OE.22.005675 | OPTICS EXPRESS 5675
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1. Introduction
Optical coherence tomography (OCT) is a non-invasive morphological technique based on
optical interferometry [1] 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
[2]. 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 [6].
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. Method
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:
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Received 24 Jul 2013; revised 22 Feb 2014; accepted 27 Feb 2014; published 5 Mar 2014
10 March 2014 | Vol. 22, No. 5 | DOI:10.1364/OE.22.005675 | OPTICS EXPRESS 5676
S = as I s + ar rI r + 2η as ar I s I r r cos φ
(1)
where as and ar are the attenuation factors (due to optical elements in the beam path, etc.), η is
the interference efficiency, Is and Ir are incident intensities of sample and reference arm,
respectively, φ is the phase related to the optical path difference (assuming zero dispersion of
air, φ ∝ f , where f is the frequency of light). The measured spectrum is real-valued, and by
Fourier transforming, a temporal trace which is symmetric about the zero time delay can be
acquired. And by filtering out the autocorrelation term near the center and the mirror images
appear at negative time delay, followed by inverse Fourier transforming the trace back into
frequency domain, the new spectrum now contains the interference term only, and it becomes
a complex-value function with the phase information recovered:
Si nter = 2η as ar I s I r eiφ r ≡ Gr
(2)
All these factors except the reflection of the sample can be lumped together as the response
function of the interferometer, G. For a sample composed of a thin-film layer surrounded by
two different materials, as shown in 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 [14]:


exp [i 4π (n + ik ) fl c ]
(3)
Sint er = G  rfront + t front t ′front rrear


′ rrear exp [i 4π (n + ik ) fl c ] 
1- rfront

where t is the complex transmission coefficient, r is the complex reflection coefficient
associated with the interfaces (defined by the Fresnel equations), n is the refractive index, k is
the extinction coefficient and l is the thickness of the thin film. The notations of suffix for the
reflection and transmission coefficients are shown in 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.
Fig. 1. The thin film sample scheme.
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 ( Grfront and Gt front t ′front rrear exp[i 4π (n + ik ) fl c] ) 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
#194479 - $15.00 USD
(C) 2014 OSA
Received 24 Jul 2013; revised 22 Feb 2014; accepted 27 Feb 2014; published 5 Mar 2014
10 March 2014 | Vol. 22, No. 5 | DOI:10.1364/OE.22.005675 | OPTICS EXPRESS 5677
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:
(4)
A = G rfront
B = G t front t ′front rrear exp ( −4π kfl c )
(5)
 t front t ′front rrear 
C = ∠
(6)
 + 4π nfl c

rfront


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:
−1
 δ n   ∂A ∂n ∂A ∂k ∂A ∂l   Aexp − A 

  
 
(7)
 δ k  =  ∂B ∂n ∂B ∂k ∂B ∂l   Bexp − B 
 δ l   ∂C ∂n ∂C ∂k ∂C ∂l   C − C 
  
  exp

where Aexp, Bexp and Cexp are the measured value of A, B and C. An appropriate initial guess of
(n, k, l) are (n0, 0, d/ n0), where n0 is the typical refractive index of that sample, d is the optical
thickness of the film estimated by the distance between the coherence spikes. Since the film
thickness l is known to be wavelength-independent, the actual increment of l for each
iteration is the average value of δl derived from 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 [15].
Unwrapped phase spectrum has a 2πm phase shift from the actual phase, where m is an
unknown integer, and:
Cexp (m) = Cunwrapped + 2π m
(8)
where Cunwrapped is the continuous phase spectrum directly generated from the experimental
data. This kind of ambiguity problem generally can be solved by using multiple wavelengths
for the detection [16]. For low-coherence interference system, removing the ambiguity by
introducing an artificial dispersion in low-coherence optical system was firstly proposed by
Hitzenberger [17]. Later Hendargo provided an alternative solution to this problem for SDOCT system by performing synthetic-wavelength-based phase unwrapping [18]. To solve the
phase ambiguity issue with a sample of unknown dispersion properties, a parameter
#194479 - $15.00 USD
(C) 2014 OSA
Received 24 Jul 2013; revised 22 Feb 2014; accepted 27 Feb 2014; published 5 Mar 2014
10 March 2014 | Vol. 22, No. 5 | DOI:10.1364/OE.22.005675 | OPTICS EXPRESS 5678
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 ( dn d λ ≠ 0 ) and attenuation
( k ≠ 0 ) 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:

1
MSE ( m ) = 
N

 A j − A j
 exp

Aexpj
j =1 


N
1
2
  Bexpj − B j
 +
j
 
  Bexp
2
  C expj − C j
 + j
 
  C exp ( m )




2
2
 


(9)
where N is the number of data points in frequency domain, and the superscript j specifies the
discrete frequencies within the light source bandwidth. m0 is defined as the m gives the
minimum MSE, Cexp = Cunwrapped + 2π m0 .
3. Experiment
3.1 System setup
In our experimental investigation the light source of the OCT system is a Ce3+:YAG doubleclad 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.
Fig. 2. (a) Ce:YAG DCF fluorescence spectrum. (b) SD-OCT setup.
3.2 Sample
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 substrateincident 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.
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Received 24 Jul 2013; revised 22 Feb 2014; accepted 27 Feb 2014; published 5 Mar 2014
10 March 2014 | Vol. 22, No. 5 | DOI:10.1364/OE.22.005675 | OPTICS EXPRESS 5679
Fig. 3. Transmission spectra. (a) Sample 1. (b) Sample 2.
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 m relation were calculated, as shown in Fig. 5.
Fig. 4. Axial scan and calculated spectral attributes. (a) Axial scan of Sample 1. (b) Axial scan
of Sample 2. (c) Calculated spectral attribute A, B and C of sample 1. (d) Calculated spectral
attribute A, B and C of sample 2.
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Received 24 Jul 2013; revised 22 Feb 2014; accepted 27 Feb 2014; published 5 Mar 2014
10 March 2014 | Vol. 22, No. 5 | DOI:10.1364/OE.22.005675 | OPTICS EXPRESS 5680
Fig. 5. MSE versus m plot. The m is equal to zero if Cexp is within the principal interval [-π,π]
at 545 nm. The insets show the zoomed portions of the figures around the minimum of MSE.
(a) Sample 1. (b) Sample 2.
Fig. 6. The derived optical properties. From left to right: refractive index, extinction
coefficient, uniqueness test of film thickness. (a) Sample 1. (b) Sample 2.
The MSE versus m 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
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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 [21]. 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:
t front trear exp i 2π ( n + ik ) fl c 
T = tsubstrate
′ rrear exp i 4π ( n + ik ) fl c 
1 − rfront
2
(10)
where tsubstrate is the transmission coefficient of the back interface of substrate. The
transmission spectra of samples within the same spectral range were measured with a
commercial transmission spectrometer, JASCO V670, which uses a halogen lamp as probe to
measure the transmittance within a slit-shaped area. To calculate the transmission spectrum
within that area, cross-sectional tomographic images of samples were measured, as shown in
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.
Fig. 7. Cross-sectional tomographic images of the thin film samples measured with SD-OCT in
the substrate-incident scheme. The figures show tomographic images within 20 μm (axial) X 5
mm (lateral) area. The figures are plotted in logarithmic scale and stretched in the longitudinal
direction. Light is incident from the top of figures. (a) Sample 1. (b) Sample 2.
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Received 24 Jul 2013; revised 22 Feb 2014; accepted 27 Feb 2014; published 5 Mar 2014
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Fig. 8. Comparison of the derived transmittance based on SD-OCT data (with a 50-nm
standard deviation of optical thickness) and the value measured with commercial transmittance
spectrometer. Black solid line: derived from the optical constants measured with the OCT. Red
dash line: measured with transmittance spectrometer. (a) Sample 1. (b) Sample 2.
5. Conclusion
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.
Acknowledgment
This work was partially supported by the National Science Council.
#194479 - $15.00 USD
(C) 2014 OSA
Received 24 Jul 2013; revised 22 Feb 2014; accepted 27 Feb 2014; published 5 Mar 2014
10 March 2014 | Vol. 22, No. 5 | DOI:10.1364/OE.22.005675 | OPTICS EXPRESS 5683