Mueller matrix polarimetry on plasma sprayed thermal barrier coatings for porosity measurement

David A. Luo; Enrique T. Barraza; Michael W. Kudenov

doi:10.1364/AO.56.009770

1. INTRODUCTION

Thermal barrier coatings (TBCs) are widely used in gas turbine engines parts, especially in hot sections, such as blades and combustors in aircraft engines [1,2]. The main coating functions are to thermally insulate the base material (generally metal) to prevent melting, corrosion, and wear [3], and to increase operation efficiency. Yttria-stabilized zirconia (YSZ) is the most popular material for TBCs due to its superior physical properties, including low thermal conductivity, strong adhesion, and good chemical and thermal stability [4]. Thermal plasma spray is the most common spraying technique to deposit such material. It is a complex process that requires many different spraying parameters and conditions (e.g., spraying distance, feedstock powder size, injected particle size and temperature, and so on) to control the final coating’s structure [5]. The porosity, generally expressed in percentage, describes the pores and cracks of the thermal sprayed YSZ coatings formed during the rapid solidification of the melted particles on the substrate when spraying from layer to layer [6]. The porosity and microstructure of the coating play an important role on its physical properties, and studies have demonstrated that the porosity facilitates heat dissipation and increases strain compliance [7]. Due to the need for more efficient engines, producing a physically reproducible TBC is crucial. Therefore, by evaluating the porosity level of the TBCs, consistent control of the aforementioned interrelated spraying parameters can be achieved.

A variety of techniques have been developed to quantify the porosity in order to provide reliable coatings and a lifetime prediction model [8,9]. These techniques can be further classified into destructive and nondestructive methods. Destructive methods include mercury intrusion porosimetry [10], image analysis of a sample [11], and image analysis implemented with a stereology approach [12]. The mercury intrusion porosimetry technique measures a large range of pore sizes based on relating the pressure of the mercury to the pore diameter. Meanwhile, image analysis methods calculate the total porosity level from a two-dimensional (2D) cross-sectional image acquired from a scanning electron microscope; however, the depth of the pores is not quantified using this method [13]. Stereological techniques amend the estimation error from this 2D image analysis by calculating the total porosity based on an interpolated three-dimensional (3D) volume [14]. However, both of the two image analysis-based methods are heavily influenced by the surface preparation after cross section. Small angle neutron scattering (SANS), a nondestructive technique, provides detailed information and high resolution of the sample’s microstructure [15]. However, the SANS instrument is large in size, expensive, and complicated, making it challenging to implement in situ. A final technique includes electrochemical impedance spectroscopy, which measures the ceramic’s capacitance [16]. This is correlated to the porosity level if the ceramic’s thickness remains unchanged. The major limitation of this technique is the costly instrumentation and complex data interpretation [17].

In this study, we use Mueller matrix polarimetry to quantify the polarization properties of the YSZ coatings with different plasma spraying parameters and correlate these measurements to their associated porosity levels as measured by stereology analysis. Mueller matrix and Stokes imaging polarimetry is widely used in the area of biomedical imaging [18], mining [19], and remote sensing [20] to highlight a scene’s structural features (e.g., surface roughness or scattering) and to analyze the polarization states of light reflected or transmitted from or through a sample. Additionally, Mueller matrix spectroscopic ellipsometry has been applied in semiconductor metrology to correlate depolarization to alignment order using block copolymer directed self-assembly methods [21]. Moreover, a recent study of Mueller matrix polarimetry demonstrated a correlation between the depolarization and sub-surface porosity of tablets made from microcrystalline cellulose [22]. A rotating retarder Mueller matrix (RRMM) imaging polarimeter was developed to measure the polarization properties of the plasma sprayed YSZ TBCs with different porosity levels. The first dual-rotating Mueller matrix polarimeter was introduced by Azzam [23]. The polarization properties of the samples were interpreted in the way of polar decomposition from the Lu–Chipman algorithm [24]. In our polarization measurements, we are measuring the depolarization caused by the samples’ surface roughness [25,26], which has been demonstrated to be correlated to porosity level [27]. With the use of a polarimetry technique, a high throughput, full-field, robust, and straightforward analysis of YSZ TBCs can be realized.

In Section 2, we describe the theory of the polarization properties of the sample in terms of Mueller matrices and the Lu–Chipman algorithm. Section 3 contains the theory and experimental result of the RRMM imaging polarimeter calibration, as well as system validation. Section 4 demonstrates the experimental results of the calculated polarization values from the measured Mueller matrix. These are then correlated to the samples’ surface roughness as measured by a contact polarimeter, as well as the total porosity measured from optical micrographs using a destructive stereology-based method.

2. THEORY

A physical sample or surface can be described by either a nondepolarizing or depolarizing Mueller matrix. In reality, most samples can be considered as depolarizing [28–30]. The measured Mueller matrix reveals the complete optical properties of the target and requires prior insight on the physical meaning of the interaction between the light and matter to interpret it. For a depolarizing sample, the assumption is made that the measured Mueller matrix can be further divided into a partial polarizer and depolarizer. In the case of YSZ TBCs, the roughness of the surface leads to a dominance in depolarization.

A depolarizing system can be decomposed into a cascaded diattenuator, retarder, and depolarizer Mueller matrix, and its mathematical form for forward polar decomposition is

M = M_{Δ} M_{R} M_{D},

where

M_{D}

is the diattenuator matrix,

M_{R}

is the retarder matrix,

M_{Δ}

is the depolarizer matrix, and

M

is the measured Mueller matrix of the sample [24,31]. From

M_{D}

,

M_{R}

, and

M_{Δ}

, the total diattenuation (

D

), total retardance (

R

), and depolarization ratio (

Δ

) can be computed. While the detailed derivation of the Mueller matrix decomposition was described by Lu–Chipman in detail [24], we briefly discuss the three decomposed matrices and their corresponding calculated values here. The measured Mueller matrix

M

of the sample can be written as

M = [\begin{matrix} m_{11} & m_{12} & m_{13} & m_{14} \\ m_{21} & m_{22} & m_{23} & m_{24} \\ m_{31} & m_{32} & m_{33} & m_{34} \\ m_{41} & m_{42} & m_{43} & m_{44} \end{matrix}] .

The decomposed diattenuator, depolarizer, and retarder Mueller matrices from the measured $M$ are expressed as

M_{D} = [\begin{matrix} 1 & D^{T} \\ D & m_{D} \end{matrix}],

M_{Δ} = [\begin{matrix} 1 & 0 \\ P_{Δ} & m_{Δ} \end{matrix}],

and

M_{R} = [\begin{matrix} 1 & 0 \\ 0 & m_{R} \end{matrix}],

respectively, where the superscript

T

represents the transpose operator and

D

and

P_{Δ}

are the

1 \times 3

diattenuation vector and polarizance vectors, respectively. The diattenuation

D

can be calculated from

D

by

D = | D | = \frac{1}{m_{11}} \sqrt{m_{12}^{2} + m_{13}^{2} + m_{14}^{2}} .

Moreover,

m_{D}

,

m_{Δ}

, and

m_{R}

are the

3 \times 3

sub-matrices, from

M_{D}

,

M_{Δ}

, and

M_{R}

, respectively.

Δ

is calculated from Eq. (4) and is obtained as

Δ = 1 - \frac{| tr (m_{Δ}) |}{3},

and

R

extracted from Eq. (5) is

R = \cos^{- 1} [\frac{tr (m_{R}) - 1}{2}],

where tr represents the trace of the

3 \times 3 m_{Δ}

and

m_{R}

matrices.

3. SYSTEM CALIBRATION AND ERROR ANALYSIS

The system errors from the electronics, mechanics, and optics are the main factors that influence the reconstruction accuracy of the sample’s Mueller matrix. Here, issues related to calibrating the sensor are discussed along with methods that were used to reduce errors introduced by nonideal attributes of the polarizing optical components. Moreover, the system calibration, data acquisition procedures, and system verification are discussed in here.

A. Experimental Setup and Procedure

A schematic of the system is illustrated in Fig. 1. The LED light source is first collimated by a collimator ( $L_{1}$ ) but slightly diverging and diffused by a ground glass diffuser (D). The light source is a Thorlabs mounted deep red LED (M660L3) with a central wavelength of 660 nm and a full width half-maximum spectral bandwidth of 25 nm. The slightly diverging beam first transmits through the polarization state generator (PSG). The PSG consists of a generating linear polarizer (LP1) and a generating quarter-wave plate (QWP1). The sample reflects the generated polarized light through the polarization state analyzer (PSA). Samples can be tilted at an angle of $θ_{s}$ based on the desired measurement geometry. The PSA’s optical components include an analyzing linear polarizer (LP2) and an analyzing quarter-wave plate (QWP2). The two LPs are thin-film polymer linear polarizers that are laminated between BK7 protective glasses (Thorlabs LPVISE200-A). Additionally, the two QWPs are zero-order polymer achromatic QWPs manufactured by Bolder Vision Optik (AQWP3). All of the four polarized optical components are anti-reflection (AR) coated in the visible light spectrum (400–700 nm).

Fig. 1. Schematic of the automated RRMM polarimeter architecture.

Download Full Size | PDF

After the light emerges from the PSA, an imaging lens ( $L_{2}$ , a Pentax C5028-M) with a 50 mm focal length brings the light into focus onto the focal plane array (FPA). The detector is an AVT Manta G-031 ¼ in. Monochrome CCD Camera with $492 \times 656$ 5.6 μm square pixels. QWP1 and QWP2 are mounted on automated rotation stages (RSs), and the RSs are connected to a stepper motor controller. The automated RS is the NanoRotator (360S) and its corresponding driver is the two-channel APT benchtop stepper motor controller (BSC202) from Thorlabs. The system automation and image acquisition are controlled by MATLAB. Finally, the LED, lens $L_{1}$ , and PSG components are mounted on a rotating bread board while the PSA, lens $L_{2}$ , and FPA are stationary. Hence, different angles ( $θ_{p}$ ) between the source and imaging optics can be achieved in this setup.

The Mueller calculus for an ideal RRMM polarimeter can be expressed as

S_{out} = {AM}_{sample} {PS}_{in}^{T},

where

M_{sample}

is the sample’s Mueller matrix,

S_{out}

is the output Stokes vector (

S_{out} = {[\begin{matrix} S_{0} & S_{1} & S_{2} & S_{3} \end{matrix}]}^{T}

), and

P

and

A

represent the Mueller matrices of the PSG and the PSA, respectively. The Mueller matrices for

P

and

A

can be expressed as

P = M_{QWP 1} M_{LP 1}

and

A = M_{LP 2} M_{QWP 2},

where

S_{in}

is the input Stokes vector, which is assumed a normalized and unpolarized value of

S_{in} = [\begin{matrix} 1 & 0 & 0 & 0 \end{matrix}]

. Finally,

M_{LP 1}

,

M_{LP 2}

,

M_{QWP 1}

, and

M_{QWP 2}

are the Mueller matrices of LP1, LP2, QWP1, and QWP2, respectively. The data reduction technique is used to calculate the Mueller matrix of the sample by rotating the two QWPs with different fast axis orientations to form a calibration matrix [32,33]. The optimal angular rotation ratio between the two QWPs, considering both efficiency and accuracy, is set to be

1 ∶ 5

[23,33,34]. The detected intensity from every pixel at any QWP orientation can be written as

I (θ) = [\begin{matrix} 1 & 0 & 0 & 0 \end{matrix}] \cdot M_{LP 2} (0 °) \cdot M_{QWP 2} (5 θ) \cdot M_{sample} \cdot M_{QWP 1} (θ) \cdot M_{LP 1} (0 °) \cdot S_{in}^{T} .

The RRMM polarimeter is designed to take 37 measurements which are evenly spaced across a 180° rotation of QWP1. Specifically, QWP1 rotates from 0° to 180° with a 5° angular increment, whereas QWP2 rotates 5 times the angular increment of QWP1. Moreover, the QWPs contain small wedge angles due to imperfect lamination, generating a different tilt for each position of the PSA. Hence, image registration is applied to the acquired images to align them to the same spatial location [35].

B. System Error Sources

The error sources consist of both random errors and systematic errors for a rotating Mueller matrix polarimeter [36]. Errors show up in the Mueller matrix elements after reconstruction, if no error compensation were applied during system calibration. Random errors are mainly associated with the light source’s intensity fluctuations and detector noise. In the case of a rotating polarimeter, the positioning accuracy of the automated RSs is also considered a source of random error. However, the random errors are assumed to be negligible due to the stable output of the LED light source and multiple frame averaging (16 frames) during image acquisition to reduce the detector noise. The LED light source has a relative stability of 0.2%, which was calculated as $100 \times (σ_{I} / m_{I})$ , where $m_{I}$ is the mean intensity and $σ_{I}$ is the standard deviation. This data was collected over 10 min. Meanwhile, systematic errors are related to nonrandom positioning error in the RSs, angular alignment errors of the LPs and QWPs, as well as error between the ideal and true retardance of the QWPs. These errors will propagate through the reconstructed Mueller matrix [37]. A corrected reconstruction Mueller matrix can be obtained by mitigating the systematic errors. Here, the errors are estimated by using a least-squares error fitting model. The system is calibrated by setting up the system in transmission and measuring the Mueller matrix of air ( $M_{air}$ ) as an empty polarimeter. The schematic configuration for an empty polarimeter with systematic error sources is illustrated in Fig. 2.

Fig. 2. Schematic configuration for an empty polarimeter.

Download Full Size | PDF

The mathematical form for describing the empty polarimeter with error sources included can be expressed as [33]

I (θ) = [1, 0, 0, 0] \cdot M_{LP 2} (ϵ_{4}) \cdot M_{QWP 2} (ϕ_{2}, 5 θ + ϵ_{3}) \cdot M_{air} \cdot M_{QWP 1} (ϕ_{1}, θ + ϵ_{2}) \cdot M_{LP 1} (ϵ_{1}) \cdot S_{in}^{T},

where

ϕ_{1}

,

ϕ_{2}

,

ϵ_{2}

, and

ϵ_{3}

are the true retardance and alignment errors of QWP1 and QWP2, respectively, and

ϵ_{1}

and

ϵ_{4}

are the alignment errors of LP1 and LP2, respectively. The least-square error fitting model is configured in MATLAB based on Eq. (13). In this experiment, LP1 is aligned with the help of a BK7 glass block whose transmission axes are parallel to the optical table. Experimentally, the collimated red LED light source illuminates the block’s surface with an incident angle below Brewster’s angle. An off-axis camera was then used to image the reflected light’s intensity, through LP1, as it was rotated from 0° to 180° in 10° increments. The measured intensity was then used to identify the orientation of LP1 such that it was parallel to the block’s horizontal eigenvector. LP1 then served as the reference angle for the rest of the RRMM polarimeter’s optics during initial alignment, such that the remaining optical components are approximately aligned parallel to LP1’s transmission axis. To make the numerical optimization of all six variables converge, the estimated orientation errors of

ϵ_{1}

,

ϵ_{2}

, and

ϵ_{3}

are referenced to the transmission axis of LP2. Since LP2 is the system’s reference, its orientation is fixed at 0° (with

ϵ_{4} = 0

) in the least-squares error fitting function. The 37 intensity measurements are each calculated by taking the mean of a

301 \times 301

pixel area centered at a manually selected pixel from every acquired image. Hence, after estimating the errors, the reconstructed and error compensated

M_{air}

is

M_{air} = [\begin{matrix} 1.0000 & 0.0042 & 0.0007 & - 0.0021 \\ 0.0043 & 1.0001 & - 0.0020 & 0.0031 \\ - 0.0010 & 0.0004 & 1.0091 & 0.0006 \\ - 0.0023 & 0.0130 & - 0.0009 & 1.0014 \end{matrix}] .

The largest relative errors for the measured Mueller matrix compared with the theoretical Mueller matrix are located at $m_{33}$ and $m_{42}$ . The mean relative errors at $m_{33}$ and $m_{42}$ are around 0.9% and 1.3%, respectively. There are several factors that could cause the relatively high error in these components compared to the other Mueller matrix elements. For instance, the strain-induced retardance in the LP’s polymer substrate, spatial nonuniformity of the LPs and QWPs [38], and angular dependence in the QWP’s retardance versus field angle (i.e., the angle of incidence). The retardance error caused by the angle of incidence is estimated to be about 0.45 deg given the divergence angle of the LED beam, which is approximately $\pm 2 \deg$ . In this application, the errors of the 16 Mueller matrix elements are in an acceptable range for our studies. The corresponding relative errors of $D$ , $Δ$ , and $R$ in percentage, calculated from Eq. (14), are 0.47%, 0.36%, and 0.16%, respectively.

C. System Verification

The measurement of a known Mueller matrix in reflection was verified by examining the $D$ and $R$ components of a BK7 glass block (BK7-GB) and a BK7 right angle prism (BK7-RAP). The BK7-GB can be modeled as the polarimeter is measuring the reflected light from an air–dielectric interface below Brewster’s angle, where the amplitude of the electric fields change, whereas the BK7-RAP describes the light that undergoes total internal reflection (TIR) at a dielectric–air interface. Using the Fresnel equations and assuming that the $p$ and $s$ polarization lie on the $x$ - and $y$ -axes, propagation along the $z$ -axis forms a right-handed Cartesian coordinate system [39], the diattenuation and the total phase difference of the electric fields can be calculated based on the accurate knowledge of the glass’s refractive indices [40]. The experimental setups for measuring the BK7-GB and BK7-RAP are illustrated in Fig. 3. The angle ( $θ_{p}$ ) between the imaging and source side is approximately 90° for both of the experimental setups, leading to an incident angle of 45° with a $\pm 0.5 °$ positioning error of the BK7 glass elements.

Fig. 3. Experimental layout for the RRMM polarimeter verification by using the (a) BK7-GB and (b) BK7-RP.

Download Full Size | PDF

1. Air–Dielectric Interface

The Mueller matrix for a dielectric material in reflection can be written as [39]

M_{12} = \frac{1}{2} {(\frac{\tan (θ_{-})}{\sin (θ_{+})})}^{2} \cdot [\begin{matrix} p_{p}^{2} + p_{s}^{2} & p_{p}^{2} - p_{s}^{2} & 0 & 0 \\ p_{p}^{2} - p_{s}^{2} & p_{p}^{2} + p_{s}^{2} & 0 & 0 \\ 0 & 0 & - 2 p_{p} p_{s} & 0 \\ 0 & 0 & 0 & - 2 p_{p} p_{s} \end{matrix}],

where

p_{p} = \cos (θ_{+})

,

p_{s} = \cos (θ_{-})

,

θ_{\pm} = θ_{i} \pm θ_{r}

, and

θ_{i}

and

θ_{r}

are the incident and refracted angles, respectively. The theoretical Mueller matrix of a BK7-GB was calculated with an incident angle of 45° including the positioning errors as

M_{t} = [\begin{matrix} 1.0000 & - 0.8255 \pm 0.0133 & 0 & 0 \\ - 0.8255 \pm 0.0133 & 1.0000 & 0 & 0 \\ 0 & 0 & - 0.5644 \pm 0.0195 & 0 \\ 0 & 0 & 0 & - 0.5644 \pm 0.0195 \end{matrix}] .

The errors are yielded by halving the difference of the Mueller matrix elements computed at $θ_{i} = 45.5 °$ and $θ_{i} = 44.5 °$ . The measured Mueller matrix of the BK7-GB is

M_{m} = [\begin{matrix} 1.0000 & - 0.8285 & - 0.0187 & - 0.0023 \\ - 0.8293 & 0.9919 & 0.0346 & 0.0034 \\ - 0.0045 & 0.0260 & - 0.5759 & - 0.0110 \\ - 0.0093 & - 0.0106 & 0.0126 & - 0.5416 \end{matrix}] .

Besides the $m_{32}$ , $m_{12}$ , and $m_{23}$ elements in $M_{m}$ , the relative error of the other MM elements was approximately 1%. The largest error occurs in $m_{32}$ and $m_{23}$ , which was approximately 3%.

2. Dielectric–Air Interface

The Mueller matrix used to describe TIR at a dielectric–air interface can be written as [39]

M_{21} = [\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & \cos (δ) & - \sin (δ) \\ 0 & 0 & \sin (δ) & \cos (δ) \end{matrix}],

where

δ

is the total phase difference of the electric fields as can be calculated from the Fresnel equations [40]. The theoretical Mueller matrix of the BK7-RAP is obtained by assuming an incident angle of 45°, which yields

M_{t} = [\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0.7735 \pm 0.0170 & - 0.6338 \pm 0.0208 \\ 0 & 0 & 0.6338 \pm 0.0208 & 0.7735 \pm 0.0170 \end{matrix}] .

Meanwhile, the measured Mueller matrix of the BK7-RAP was

M_{m} = [\begin{matrix} 1.0000 & - 0.0006 & - 0.0069 & 0.0092 \\ - 0.0002 & 0.9973 & - 0.0142 & 0.0492 \\ - 0.0036 & - 0.0430 & 0.7724 & - 0.6557 \\ - 0.0021 & 0.0415 & 0.6465 & 0.7570 \end{matrix}] .

The relative errors for the 16 Mueller matrix elements are generally higher than that of BK7-GB. Since BK7-RAP is not AR coated, the main reasons could be the double reflections from the BK7-RAP’s entrance and exit faces and the imperfect alignment of the BK7-RAP. The largest relative errors are located at $m_{32}$ , $m_{42}$ , and $m_{24}$ , which generally range from 4% to 5%.

4. EXPERIMENTAL METHOD

In this section, the sample preparation of the thermal plasma sprayed YSZ TBC samples, as well as the polarization values measured by the RRMM imaging polarimeter, are addressed. Meanwhile, to understand how the polarized light reacts with the samples, the surface roughness and porosity level were measured by a contact profilometer and micrographs via stereological analysis, respectively. Lastly, the acquired polarization values are correlated to the surface roughness and the total porosity level.

A. Sample Preparation

There are a total of 18 samples which are classified into six groups based on the porosity level. Each of the groups contains three samples, and they were sprayed at identical times and conditions on 1 in. (25.4 mm) diameter stainless steel buttons. All the samples are produced by the air plasma spraying method with 7–8 percent by weight (wt. %) YSZ powders. A witness sample from each group was cross sectioned to measure the representative porosity level using microscopy. The six groups of coatings fall into three major classes as: (1) YSZ coatings with a porosity level of 4% (YSZ 4%); (2) YSZ coatings with the highest porosity level (YSZ highest); and (3) YSZ with the least porosity level (YSZ least). It is worth mentioning that the first group of coatings with 4% of porosity can be further divided into four groups with slightly different porosity levels based on applying different powder sizes, powder feed rates, and standoff distances [5]. However, they could not be finely classified from the initial witness samples by the stereology method. Moreover, the porosity levels among these four groups are expected to increase from group 1 to group 4, and they are labeled as from YSZ 4%—G1 to YSZ 4%—G4. All of the 18 samples and their associated expected porosity level are summarized in Table 1.

Table 1. Classification for All the 18 Samples Based on the Expected Porosity Level^a

View Table

B. Mueller Matrix Polarimetry for Polarization Measurement

The angle ( $θ_{p}$ ) between the imaging and source sides is 90°. The sample is tilted approximately 22.5° toward the imaging side to both maintain the depth of focus and illuminated area. The resolution is $38 μm / pixel$ and the field of view is approximately 25 mm. A MATLAB script is created to produce the three polarization images, which are the $Δ$ , $D$ , and $R$ images calculated from the measured YSZ samples’ Mueller matrix pixel by pixel by using Eqs. (6)–(8). The acquired images to construct the 37 intensity measurements are obtained with 32 frames averaging. The raw intensity measurement when QWP1 rotates to 0°, and the corresponding $Δ$ , $D$ , and $R$ images are illustrated in Fig. 4. The region of interest from the raw data image can be selected. The mean and standard deviation (STD) of the $Δ$ , $D$ , and $R$ images were calculated from a $51 \times 51$ pixel area (around $1.94 mm \times 1.94 mm$ ) centered from a manually selected pixel among the $Δ$ , $D$ , and $R$ images, respectively. In these experimental results, the pixels were always selected, for post-processing from within the depth of focus.

Fig. 4. Images of the intensity measurement with QWP1 at 0° and the calculated polarization images of $Δ$ , $D$ , and $R$ from one of the measured YSZ least porosity sample’s Mueller matrix.

Download Full Size | PDF

The measured mean retardance, diattenuation, and depolarization of the YSZ samples, color coded by group number, are presented in Fig. 5. All the three polarization values are illustrated in Fig. 5(a) as a 3D scatter plot. Moreover, Figs. 5(b)–5(d) illustrate the other three metrics as 2D scatter plots, which are different projections of the 3D plot.

Fig. 5. (a) 3D scatter plot for the mean of $Δ$ versus $D$ versus $R$ , as well as the 2D scatter plot for the mean of (b) $Δ$ versus $D$ , $Δ$ versus $R$ , and $D$ versus $R$ .

Download Full Size | PDF

As illustrated in Fig. 5(b), $Δ$ and $D$ follow a reciprocal linear relationship. The YSZ samples, with different porosity levels, are clustered together with their own groups and are separated from each other, especially for YSZ highest, YSZ4%—G4, and YSZ least. Due to the reciprocal linear relationship, Fig. 5(d) is a mirror image of Fig. 5(c). Consequently, $Δ$ increases when the porosity level increases; whereas, $D$ decreases when the porosity level increases. There is not a direct relationship between $R$ and the porosity level.

C. Surface Roughness and Porosity Level Measurement

To correlate the measurement of the thermal barrier coating from the RRMM polarimeter, the Veeco Dektak 150 stylus/contact profilometer and stereology analysis are used to measure the surface roughness and internal porosity level, respectively. These nondestructive and destructive (in the case of stereology) measurements were taken after measurement with the RRMM polarimeter. The profilometer scanned three lines across the sample and each line is 10 mm long to statistically represent the overall surface geometry. The distribution of the three scanning lines across the coupon are illustrated in Fig. 6(a).

Fig. 6. Image of the scanning lines on the TBC coupons. (a) Photo of the coupon and the scan lines; (b) direct measurement from one vertical scan of the profilometer—curvature is present; (c) measurement with the curvature in (b) removed.

Download Full Size | PDF

The curvature of the measured samples, illustrated in Fig. 6(b), influences the surface roughness calculation. A high-pass filter was applied on the Fourier transformed measured surface signal to remove the curved geometry of the samples. The leveled down surface signal is illustrated in Fig. 6(c). The root mean square of the surface roughness ( $R_{q}$ ) [41], calculated from the three lines of each sample, was used to describe the roughness in micrometers. After obtaining the $R_{q}$ , the samples were sent back to the manufacturer and cross sectioned to determine the total porosity level from optical micrograph by stereology analysis [42].

D. Polarization Correlation with Surface Roughness and Porosity Level

The relationship between the surface roughness and total porosity level is depicted in Fig. 7. The $R_{q}$ shows a direct linear relationship to the porosity level. However, it is a good estimation on how the surface geometry indicates the internal porosity level. The least porous samples contain the smoothest surface among all the samples with higher porosity levels, whereas the highest porosity samples have the roughest surface.

Fig. 7. Scatter plot of porosity versus $R_{q}$ .

Download Full Size | PDF

The calculated $R_{q}$ and measured porosity level are then compared with the mean and STD of the $Δ$ , $D$ , and $R$ values calculated from Section 4.B. The relationship between the two metrics are presented as scatter plots. In addition, the data are fitted with an exponential regression model by using the built-in MATLAB fit function, which has the form of

y_{o} = a + b e^{c x_{o}},

where

x_{o}

refers to the mean of the polarization values,

y_{o}

indicates the responded

R_{q} / porosity

level, and

a

,

b

, as well as

c

are the fitting coefficients. The goodness of fit (GoF) is evaluated by the relative standard error of estimate (

S

) and adjusted R-squared (

r^{2}

) in percent.

S

is written as [43]

S = \sqrt{\frac{\sum_{n = 1}^{N} {(y_{a} (n) - y_{t} (n))}^{2}}{N - P}},

where

n

is an integer number,

N

is the total number of samples,

P

is the number of fitting coefficients,

y_{a}

is the true value, and

y_{t}

is the estimated value. Smaller

S

indicates better GoF due to the closer geometrical distance between

y_{a}

and

y_{t}

. Conversely, better correlation accuracy is determined when

r^{2}

is closer to 100%. The scatter plots of

Δ

versus

R_{q}

,

D

versus

R_{q}

, and

R

versus

R_{q}

are depicted in Fig. 8. In Figs. 8(a) and 8(b), the polarization values and the

R_{q}

demonstrate a nonlinear relationship versus porosity. In addition, the three main classes of porosity levels are clustered together.

Δ

increases exponentially as the surface roughness increases due to increased optical scattering from the surface. The scatter plot of

D

versus

R_{q}

has an opposite sign of the nonlinear relationship to the

R_{q}

as compared to

Δ

versus

R_{q}

, leading to an exponential decay. Furthermore,

D

decreases when the surface roughness increases. However, there is no strong correlation of the retardance to the surface roughness per Fig. 8(c).

Fig. 8. Scatter plots with error bars for the mean and STD of (a) $Δ$ , (b) $D$ , and (c) $R$ versus $R_{q}$ , respectively.

Download Full Size | PDF

The correlation between the polarization values and the total porosity, calculated via stereological analysis, is depicted in Fig. 9. A nonlinear relationship can be found from the scatter plot of $Δ$ versus porosity and $D$ versus porosity in Figs. 9(a) and 9(b), respectively. The classifications on YSZ 4% G1 to G3 are clustered together; however, YSZ 4%—G4 stands out slightly from the other three YSZ 4% sub-groups. $Δ$ as well as $D$ versus the porosity level share a similar trend as those versus $R_{q}$ . Again, a weak relationship between $R$ and porosity in Fig. 9(c) is observed. The estimated accuracy of porosity level by using either $D$ or $Δ$ is comparable, with $r^{2}$ values of 93.05% and 93.09%, respectively. The standard error $S$ , computed on porosity versus $Δ$ , is 0.3% lower than that acquired from porosity versus $D$ .

Fig. 9. Scatter plots with error bars for the mean and STD of (a) $Δ$ , (b) $D$ , and (c) $R$ versus porosity, respectively.

Download Full Size | PDF

5. CONCLUSION

We demonstrated a correlation between $D$ and porosity of 93.05% and a correlation between $Δ$ and porosity of 93.09%. This is within our maximum experimental error of 1.3%. With this technique, a nondestructive optical method that is capable of full-field and robust measurements can be realized to evaluate the porosity level. Data interpretation is straightforward and system implementation may be more robust than destructive methods. Additionally, the part that is being sprayed can be measured directly at many points across its surface. This offers the possibility sub-surface coating characterization throughout the entire coating process, avoiding the need for witness coupons that offer only one spatial and one temporal sample of coating conditions. Future work will be focusing on decreasing the system errors of the RRMM imaging polarimeter and examining the interaction between polarized light with different internal pore structures.

Funding

U.S. Air Force (USAF) (FA8650-14-C-2434).

Acknowledgment

The authors would like to thank Nicole Hedges for the surface roughness measurement, and this study was performed in part at the NCSU Nanofabrication Facility (NNF), a member of the North Carolina Research Triangle Nanotechnology Network (RTNN), which is supported by the National Science Foundation (NSF) (ECCS-1542015) as part of the National Nanotechnology Coordinated Infrastructure (NNCI).

REFERENCES

1. H. L. Bernstein, “High temperature coatings for industrial gas turbine users,” in 28th Turbomachinery Symposium (1999), pp. 179–188.

2. N. P. Padture, M. Gell, and E. H. Jordan, “Thermal barrier coatings for gas-turbine engine applications,” Science 296, 280–284 (2002). [CrossRef]

3. C. M. Cotell, J. A. Sprague, and F. A. Smidt, ASM Handbook: Surface Engineering (ASM International, 1994), Vol. 5.

4. R. A. Miller, “Current status of thermal barrier coatings — An overview,” Surf. Coat. Technol. 30, 1–11 (1987). [CrossRef]

5. A. Kulkarni, A. Vaidya, A. Goland, S. Sampath, and H. Herman, “Processing effects on porosity-property correlations in plasma sprayed yttria-stabilized zirconia coatings,” Mater. Sci. Eng. A 359, 100–111 (2003). [CrossRef]

6. S. Sampath, U. Schulz, M. O. Jarligo, and S. Kuroda, “Processing science of advanced thermal-barrier systems,” MRS Bull. 37(10), 903–910 (2012). [CrossRef]

7. D. R. Clarke, M. Oechsner, and N. P. Padture, “Thermal-barrier coatings for more efficient gas-turbine engines,” MRS Bull. 37(10), 891–898 (2012). [CrossRef]

8. F. Andreola, C. Leonelli, M. Romagnoli, and P. Miselli, “Techniques used to determine porosity,” Am. Ceram. Soc. Bull. 79(7), 49–52 (2000).

9. S. Deshpande, A. Kulkarni, S. Sampath, and H. Herman, “Application of image analysis for characterization of porosity in thermal spray coatings and correlation with small angle neutron scattering,” Surf. Coat. Technol. 187, 6–16 (2004). [CrossRef]

10. J. Ilavsky, C. C. Berndt, and J. Karthikeyan, “Mercury intrusion porosimetry of plasma-sprayed ceramic,” J. Mater. Sci. 32, 3925–3932 (1997). [CrossRef]

11. K. Mailot, F. Gitzhofer, and M. I. Boulos, “Absolute coating porosity measurement using image analysis,” in 15th International Thermal Spray Conference (1998), Vol. 1, pp. 917–922.

12. S.-H. Leigh and C. C. Berndt, “Quantitative evaluation of void distributions within a plasma-sprayed ceramic,” J. Am. Ceram. Soc. 82, 17–21 (1999). [CrossRef]

13. M. von Bradke, F. Gitzhofer, and R. Henne, “Porosity determination of ceramic materials by digital image analysis--a critical evaluation,” Scanning 27, 132–135 (2005). [CrossRef]

14. J. A. Gan and C. C. Berndt, “Quantification and taxonomy of pores in thermal spray coatings by image analysis and stereology approach,” Metall. Mater. Trans. A 44, 4844–4858 (2013). [CrossRef]

15. A. J. Allen, J. Ilavsky, G. G. Long, J. S. Wallace, C. C. Berndt, and H. Herman, “Microstructural characterization of yttria-stabilized zirconia plasma-sprayed deposits using multiple small-angle neutron scattering,” Acta Mater. 49, 1661–1675 (2001). [CrossRef]

16. J. Zhang and V. Desai, “Evaluation of thickness, porosity and pore shape of plasma sprayed TBC by electrochemical impedance spectroscopy,” Surf. Coat. Technol. 190, 98–109 (2005). [CrossRef]

17. B. A. Boukamp, “Impedance spectroscopy, strength and limitations (Impedanzspektroskopie, Stärken und Grenzen),” tm-Technisches Messen/Sensoren, Geräte, Systeme 71, 454–459 (2004). [CrossRef]

18. S. L. Jacques, J. C. Ramella-Roman, and K. Lee, “Imaging skin pathology with polarized light,” J. Biomed. Opt 7, 329–340 (2002). [CrossRef]

19. J. D. Howe, M. A. Miller, R. V. Blumer, T. E. Petty, M. A. Stevens, D. M. Teale, and M. H. Smith, “Polarization sensing for target acquisition and mine detection,” Proc. SPIE 4133, 202–213 (2000). [CrossRef]

20. J. S. Tyo, D. L. Goldstein, D. B. Chenault, and J. A. Shaw, “Review of passive imaging polarimetry for remote sensing applications,” Appl. Opt. 45, 5453–5469 (2006). [CrossRef]

21. D. J. Dixit, V. Kamineni, R. Farrell, E. R. Hosler, M. Preil, J. Race, B. Peterson, and A. C. Diebold, “Metrology for block copolymer directed self-assembly structures using Mueller matrix-based scatterometry,” J. Micro Nanolithogr. MEMS MOEMS 14, 021102 (2015). [CrossRef]

22. S. Sergey, P. Alexander, O. Yevagen, K. Ivan, S. Pertti, E. Tuomas, K. Jarkko, and P. Kai-Erik, “Characterization of porous media based on the polarimetric matrix models,” J. Biomed. Photon. Eng. 3, 010306 (2017). [CrossRef]

23. R. M. A. Azzam, “Photopolarimetric measurement of the Mueller matrix by Fourier analysis of a single detected signal,” Opt. Lett. 2, 148–150 (1978). [CrossRef]

24. S.-Y. Lu and R. A. Chipman, “Interpretation of Mueller matrices based on polar decomposition,” J. Opt. Soc. Am. A 13, 1106–1113 (1996). [CrossRef]

25. G. Valenzuela, “Depolarization of EM waves by slightly rough surfaces,” IEEE Trans. Antennas Propag. 15, 552–557 (1967). [CrossRef]

26. E. R. Mendez and K. A. O’Donnell, “Observation of depolarization and backscattering enhancement in light scattering from Gaussian random surfaces,” Opt. Commun. 61, 91–95 (1987). [CrossRef]

27. K. Yasuda, S. Suenaga, H. Inagaki, Y. Goto, H. Takeda, and K. Wada, “Relationship between microstructure of plasma-sprayed 8YZ coatings and thermal fatigue life of thermal barrier coatings,” J. Mater. Sci. 35, 317–321 (2000). [CrossRef]

28. W. E. Martin, E. Hesse, J. H. Hough, W. B. Sparks, C. S. Cockell, Z. Ulanowski, T. A. Germer, and P. H. Kaye, “Polarized optical scattering signatures from biological materials,” J. Quant. Spectrosc. Radiat. Transfer 111, 2444–2459 (2010). [CrossRef]

29. N. Ghosh and I. A. Vitkin, “Tissue polarimetry: concepts, challenges, applications, and outlook,” J. Biomed. Opt. 16, 110801 (2011). [CrossRef]

30. S. Krishnan and P. C. Nordine, “Mueller-matrix ellipsometry using the division-of-amplitude photopolarimeter: a study of depolarization effects,” Appl. Opt. 33, 4184–4192 (1994). [CrossRef]

31. R. Ossikovski, A. D. Martino, and S. Guyot, “Forward and reverse product decompositions of depolarizing Mueller matrices,” Opt. Lett. 32, 689–691 (2007). [CrossRef]

32. D. B. Chenault, J. L. Pezzaniti, and R. A. Chipman, “Mueller matrix algorithms,” Proc. SPIE 1746, 231–246 (1992). [CrossRef]

33. D. H. Goldstein, “Mueller matrix dual-rotating retarder polarimeter,” Appl. Opt. 31, 6676–6683 (1992). [CrossRef]

34. M. H. Smith, “Optimization of a dual-rotating-retarder Mueller matrix polarimeter,” Appl. Opt. 41, 2488–2493 (2002). [CrossRef]

35. J. R. Bergen, P. Anandan, K. J. Hanna, and R. Hingorani, “Hierarchical model-based motion estimation,” in 2nd European Conference on Computer Vision (1992), pp. 237–252.

36. K. M. Twietmeyer and R. A. Chipman, “Optimization of Mueller matrix polarimeters in the presence of error sources,” Opt. Express 16, 11589–11603 (2008). [CrossRef]

37. D. H. Goldstein and R. A. Chipman, “Error analysis of a Mueller matrix polarimeter,” J. Opt. Soc. Am. A 7, 693–700 (1990). [CrossRef]

38. L. Broch, A. E. Naciri, and L. Johann, “Systematic errors for a Mueller matrix dual rotating compensator ellipsometer,” Opt. Express 16, 8814–8824 (2008). [CrossRef]

39. D. H. Goldstein, Polarized Light (CRC Press, 2016).

40. M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (CUP Archive, 1999).

41. J. C. Stover, Optical Scattering: Measurement and Analysis (SPIE, 1995), Vol. 2.

42. R. Sobhanverdi and A. Akbari, “Porosity and microstructural features of plasma sprayed yttria stabilized zirconia thermal barrier coatings,” Ceram. Int. 41, 14517–14528 (2015). [CrossRef]

43. R. B. Darlington and A. F. Hayes, Regression Analysis and Linear Models: Concepts, Applications, and Implementation (Guilford Publications, 2016).

Classes	Group	Expected Porosity Level
1	G1	Highest porosity in class 1
	G2	2nd highest porosity in class 1
	G3	3rd highest porosity in class 1
	G4	Least porosity in class 1
2	—	Highest porosity in all classes
3	—	Least porosity in all classes

Mueller matrix polarimetry on plasma sprayed thermal barrier coatings for porosity measurement

Abstract

1. INTRODUCTION

2. THEORY

3. SYSTEM CALIBRATION AND ERROR ANALYSIS

A. Experimental Setup and Procedure

B. System Error Sources

C. System Verification

1. Air–Dielectric Interface

2. Dielectric–Air Interface

4. EXPERIMENTAL METHOD

A. Sample Preparation

B. Mueller Matrix Polarimetry for Polarization Measurement

C. Surface Roughness and Porosity Level Measurement

D. Polarization Correlation with Surface Roughness and Porosity Level

5. CONCLUSION

Funding

Acknowledgment

REFERENCES

Cited By

Figures (9)

Tables (1)

Equations (22)

Applied Optics