*3.2.1. Removing background*

The recorded interferogram contains a background intensity variation which is generated by the spectral distribution of the light source and the spectral response of the CCD camera. By blocking the measurement arm of the interferometer, a reference frame without interference effect can be captured and afterwards removed from each spectral interferogram.

### *3.2.2. Wavelength calibration*

The output of the spectrometer is spectrally decomposed along the rows (or columns) of CCD pixels. For example, if a diffraction grating has space *d*, when a plane wave is incident with an angle *θi* , and the diffraction angle is *θm* to the order *m*, then for a beam with wavelength *λ*, the following equation should be satisfied

$$d\mathfrak{m}\!\!\!/ = d(\sin\theta\_{\!\!\!/} + \sin\theta\_{\!\!\!/} \!\!/ , \tag{10}$$

Grating Eq. (10) shows that the relationship between the dispersed wavelength and pixel number is not linear. Therefore, the exact relationship between the pixel number and the specific wavelength needs to be calibrated. A white light laser source (WhiteLaseTM micro) in conjunction with the acoustic-optical tuneable filter (AOTF) is used for the calibration. The specific wavelength selected by the AOTF is determined by Eq. (11)

$$
\lambda = \Delta n \alpha \text{v}\_a \text{\*} (f\_a)^{-1}. \tag{11}
$$

A second-order polynomial is used to represent the relationship between pixel number and wavelength, as shown in Eq. (12)

$$
\lambda\_p = Ap^2 + Bp + C \tag{12}
$$

where *λP* is the wavelength of pixel p, *C* is the wavelength of pixel 0, *B* is the first-order coefficient (nm/pixel), and *A* is the second-order coefficient (nm/pixel2 ). More than 30 groups of experimental data (spectral lines) have been recorded to calculate the values for *A*, *B* and *C* through the least squares equation. The calibration result is depicted in **Figure 9**.

### *3.2.3. Coordinate transformation*

**3.2. Interferograms analysis**

52 Optical Interferometry

which follows (**Figure 8**).

**Figure 8.** Flowchart of the FFT-based algorithm.

*3.2.1. Removing background*

*3.2.2. Wavelength calibration*

following equation should be satisfied

angle *θi*

Many algorithms have been developed to extract the phase from the spectral interferograms, including the techniques based on fast Fourier transform (FFT) [26], convolution [27] and Hilbert transform [28]. We developed a FFT-based algorithm for the proposed system because it is effective, accurate and insensitive to intensity noise [5]. The developed algorithm for interpreting the captured interferograms contains five sections, as shown in the flowchart

The recorded interferogram contains a background intensity variation which is generated by the spectral distribution of the light source and the spectral response of the CCD camera. By blocking the measurement arm of the interferometer, a reference frame without interference

The output of the spectrometer is spectrally decomposed along the rows (or columns) of CCD pixels. For example, if a diffraction grating has space *d*, when a plane wave is incident with an

Grating Eq. (10) shows that the relationship between the dispersed wavelength and pixel number is not linear. Therefore, the exact relationship between the pixel number and the specific wavelength needs to be calibrated. A white light laser source (WhiteLaseTM micro) in conjunction with the acoustic-optical tuneable filter (AOTF) is used for the calibration. The

= + (sin sin ). *m d*

<sup>1</sup> \*( ) . - = D *a a*

 a  qq

l

specific wavelength selected by the AOTF is determined by Eq. (11)

l

, and the diffraction angle is *θm* to the order *m*, then for a beam with wavelength *λ*, the

*i m* (10)

*nv f* (11)

effect can be captured and afterwards removed from each spectral interferogram.

The original signal obtained is a curve of irradiance with respect to wavelength *λ* (or pixel number). However, the phase variation extracted from the channelled spectrum is linearly related to the wavenumber σ(σ = 1/λ), which makes the coordinate transformation necessary. The original sinusoidal signals are resampled in equal intervals through spline interpolation and finally reconstructed to a wavenumber-related curve.

**Figure 9.** Wavelength calibration of the CCD camera.

#### *3.2.4. Phase calculation*

As mentioned above, we use FFT to extract the phase from the spectral interferogram. The spectral intensity *I*(h, σ) recorded at the output of the spectrometer can be expressed as

$$I(\mathbf{x}, \mathbf{y}; \sigma) = a(\mathbf{x}, \mathbf{y}; \sigma) + b(\mathbf{x}, \mathbf{y}; \sigma) \cos[\varphi(\mathbf{x}, \mathbf{y}; \sigma)] \tag{13}$$

where *I*(*x*, *y*; *σ*) is the normalized intensity with background intensity variation removed, (*x*, *y*) denotes the spatial coordinates of the interferogram, *a*(*x*, *y*; *σ*) and *b*(*x*, *y*; *σ*) represent the background intensity and fringe visibility, respectively, and the phase *ϕ*(*x*, *y*; *σ*) is defined by formula

$$\varphi(\mathbf{x}, \mathbf{y}; \sigma) = 4\pi \mathcal{L}^{-1} \times h(\mathbf{x}, \mathbf{y}) + \varphi\_0 = 4\pi \sigma h(\mathbf{x}, \mathbf{y}) + \varphi\_0 \tag{14}$$

where *h*(*x*, *y*) represents the surface elevation, and *φ0* is the initial phase. Eq. (13) can be written in another form as Eq. (15)

$$I(\mathbf{x}, \mathbf{y}; \sigma) = a(\mathbf{x}, \mathbf{y}; \sigma) + c(\mathbf{x}, \mathbf{y}; \sigma) + c^\bullet(\mathbf{x}, \mathbf{y}; \sigma) \tag{15}$$

with

$$\mathcal{L}\left(\mathbf{x},\mathbf{y};\sigma\right) = \frac{1}{2}b\left(\mathbf{x},\mathbf{y};\sigma\right)\exp[i\ \phi(\mathbf{x},\mathbf{y};\sigma)]\tag{16}$$

where \* denotes a complex conjugate. Fourier transform on the original spatial signals can be expressed in frequency domain as Eq. (17)

$$\tilde{I}\left(\mathbf{x},\mathbf{y};f\right) = A\left(\mathbf{x},\mathbf{y};f\right) + C\left(\mathbf{x},\mathbf{y};f\right) + C\left(\mathbf{x},\mathbf{y};f\right) \tag{17}$$

where the capital letters denote the Fourier spectra, and *f* is the spatial frequency. The unwanted background variation is removed by setting a filtration window, and then, the desired term *C*(*x*, *y*, *f*) is selected to compute the inverse fast Fourier transform (IFFT). Taking the natural logarithm of the IFFT [*C*(*x*, *y*; *f*)], the phase *ϕ*(*x*, *y*; *σ*) of each point is extracted as the imaginary part of the Eq. (18)

$$\log\left\{\frac{1}{2}b(\mathbf{x},\mathbf{y};\sigma)\exp[i\ \boldsymbol{\varphi}(\mathbf{x},\mathbf{y};\sigma)]\right\} = \log\left[\frac{1}{2}b(\mathbf{x},\mathbf{y};\sigma)\right] + \text{ i }\boldsymbol{\varphi}(\mathbf{x},\mathbf{y};\sigma). \tag{18}$$

Finally, the phase unwrapping process, illustrated in detail by Takeda et al. in 1982 [26], is performed to the discontinuities phase distribution.

#### *3.2.5. Height map of a one-dimensional profile*

The vertical axis of the spectral interferogram represents one dimension of lateral resolution, which means each row signal contains the height information of one point. By using the phase slope *S* obtained from the phase calculation process, the height value can be acquired using Eq. (19). After analysis of a series of row signals, the height map of a surface profile can be obtained.

$$h = \frac{S}{4\pi} = \Delta\varphi^\* [4\pi \left(\sigma\_m - \sigma\_n\right)]^{-1} \tag{19}$$

where *σm* and *σn*> are the wavenumbers corresponding to the phase difference Δ*φ*.

### **3.3. Results and discussion**

where *I*(*x*, *y*; *σ*) is the normalized intensity with background intensity variation removed, (*x*, *y*) denotes the spatial coordinates of the interferogram, *a*(*x*, *y*; *σ*) and *b*(*x*, *y*; *σ*) represent the background intensity and fringe visibility, respectively, and the phase *ϕ*(*x*, *y*; *σ*) is de-

where *h*(*x*, *y*) represents the surface elevation, and *φ0* is the initial phase. Eq. (13) can be written

 ps

 js

*I xyf Axyf C xyf C xyf* (17)

s

 js

(18)

 j

 s=++ (15)

= (16)

(14)

( ) ( ) ( ) <sup>1</sup> 0 0 ,; 4 , 4 , -

 *x y* = ´ += + *hxy hxy* j

( ) \* *I xy axy cxy c xy* ,; (,; ) (,; ) (,; )

,; ,; exp[ ( , ; )] <sup>2</sup> *cxy bxy i xy*

( ) ( ) ( ) ( ) \* ,; ,; ,; ,; =++ %

 s

where \* denotes a complex conjugate. Fourier transform on the original spatial signals can be

where the capital letters denote the Fourier spectra, and *f* is the spatial frequency. The unwanted background variation is removed by setting a filtration window, and then, the desired term *C*(*x*, *y*, *f*) is selected to compute the inverse fast Fourier transform (IFFT). Taking the natural logarithm of the IFFT [*C*(*x*, *y*; *f*)], the phase *ϕ*(*x*, *y*; *σ*) of each point is extracted as the imaginary

> , ; exp[ ( , ; )] (,; ) ( , ; ). 2 2 <sup>ì</sup> üé ù í ý = + ê ú <sup>î</sup> þë û *log b x y i x y log b x y i x y*

Finally, the phase unwrapping process, illustrated in detail by Takeda et al. in 1982 [26], is

The vertical axis of the spectral interferogram represents one dimension of lateral resolution, which means each row signal contains the height information of one point. By using the phase slope *S* obtained from the phase calculation process, the height value can be acquired

ss

( ) ( ) <sup>1</sup>

( ) 1 1

 js

s

performed to the discontinuities phase distribution.

*3.2.5. Height map of a one-dimensional profile*

s

fined by formula

54 Optical Interferometry

with

part of the Eq. (18)

in another form as Eq. (15)

j

expressed in frequency domain as Eq. (17)

 s  pl

s

Two surface samples were measured to evaluate the performance of the prototype. First, to verify a good accuracy and measurement repeatability of the system, a standard sample from the National Physical Laboratory (NPL) with a step height of 100.0 nm was measured 24 times as shown in **Figure 10**. The measured average heights were recorded. It was found that the mean value of these 24 values is 96.6 nm, and the standard deviation was then calculated as 8.2 nm using Eq. (20).

**Figure 10.** Measurement result of a micro-fluid chip with 100 µm step height: (a) 3D surface map, (b) cross-sectional profile.

$$\sigma = \left(\sum\_{l=1}^{N} (H\_l - \overline{H})^2 \mathbf{\*} (N-1)^{-1}\right)^{\frac{1}{2}}.\tag{20}$$

The second sample we measured was a multi-layer polyethylene naphthalate (PEN) film manufactured by the Centre for Process Innovation (CPI). This thin film is composed of three layers, namely a PEN substrate layer (125 µm), a planarization layer (3–4 µm) for planarising the pit and spike features on the PEN substrate, and an atomic layer deposition (ALD) barrier (40 nm) for prevention of moisture and oxygen ingress. The results acquired by a Taylor-Hobson Coherence Correlation Interferometry (CCI) instrument (Talysurf CCI 3000) and LSDI are shown in **Figure 11** and **Figure 12**. The 3D surface map shown in **Figure 11** is generated by 1400 profiles, which represents a scanning length of 2.31 mm. The surface topography results shown in **Figure 11** have different forms because different surface tensions are generated when fixing this film sample in these two separate measurements. Both 3D and 2D surface topography results demonstrate that LSDI is capable of detecting most of defects as the CCI does and the relative positions between each defect are well matched.

**Figure 11.** Defects detection on the Al2O3 ALD barrier film surface: (a) 3D surface map result using CCI and (b) result using LSDI.

**Figure 12.** 2D view of defects on the Al2O3 ALD barrier film surface: (a) CCI result and (b) LSDI result.


**Table 1.** Defects' specifications (size, location).

Analysis results of defects specifications for both CCI and LSDI are listed in detail in **Table 1**. The sizes and positions (*x* axis) of the defects correlate well between the CCI measurement of the sample and the measurement performed using LSDI. The slight difference of the relative positions along the *y* axis is due to different surface tensions as mentioned above, which does not affect the assessment of LSDI's performance.

The results above verified that LSDI performs well in defects detection. Only one shot for a surface profile makes it a fast and environmentally robust measurement system.
