**2.2. Phase calculation principle**

ration of the proposed surface measurement system is illustrated in **Figure 1**. The measurement system is composed of two Linnik interferometers that share a common optical path. The measurement interferometer is illuminated by a white light source through an acousto-optic tuneable filter (AOTF) to filter the light from the white light source to the main interferometer. This is to select a specific wavelength for the interferometer, thus producing an interferogram at the CCD sourced only by that specific wavelength. The selected light wavelength is deter-

> = D *<sup>a</sup> a v n f*

 a

where Δn is the birefringence of the crystal used as the diffraction material, α is a complex parameter subject to the design of the AOTF, and *va* and *fa* are the velocity and frequency of the driving acoustic wave, respectively. The wavelength of the light which is selected by the AOFT can therefore be varied just by changing the driving frequency *fa*. Consequently, different wavelengths of light will pass through the AOTF in sequence so a series of interferograms of different wavelengths will be detected by the CCD camera. The absolute optical path difference can be calculated in real time through analyzing interferograms captured by the CCD camera. The reference interferometer, which is illuminated by an inferred superluminescent lightemitting diode (SLED), is used to observe and compensate for the environmental noise, for example, mechanical vibration, temperature drift and air turbulence. Because the two interferometers undergo similar environmental noise, the measurement interferometer will be capable of measuring surface information once the reference interferometer is 'locked' into the

The light beams from the AOTF and the IR SLED are combined by a dichroic mirror that is highly reflective in the inferred wavelength and transmissive in the visible light wavelength range. After passing through the dichroic mirror, the light beam is coupled into an optical fibre patch cable. By separating the light source and AOTF from the interferometers, not only the size and weight of the interferometers have been greatly reduced, but also the thermal

It is well known that surface measurement in the workshop/manufacturing environment has been challenging to achieve using interferometric techniques since they are very sensitive to environmental vibrations, in particular, axial (vertical) vibration [5]. In addition, measurement noise can be induced by air turbulence and temperature drift as well. In this experimental study, the reference interferometer is illuminated by a SLED, which made by EXOLES (EXS2100068–01, 850 nm centre wavelength with 50 nm bandwidth) together with a servo feedback electronic unit to compensate the environmental noise effectively. Output light from the SLED is combined with the measurement light and travels virtually the same optical path as the measurement interferometer. The interference signal of the reference interferometer is picked up by a photodiode after being filtered off by dichroic beamsplitter 2 (Thorlabs, DMSP805 short pass dichroic mirror with 805 nm cut-off wavelength). As a result of a shared optical path, it is intended that if the noise happening in the reference interferometer is

influence from the light source has also been eliminated.

(1)

l

mined by:

44 Optical Interferometry

compensation mode.

Intensities detected by pixel (x, y) of the CCD camera that correspond to one point on the test surface can be expressed by

$$I(\mathbf{x}, \mathbf{y}; k) = a(\mathbf{x}, \mathbf{y}; k) + b(\mathbf{x}, \mathbf{y}; k) \cos(2\pi kh(\mathbf{x}, \mathbf{y})) \tag{2}$$

where *a(x, y; k)* and *b(x, y; k)* are the background intensity and fringe visibility, respectively, *k* is the wavenumber which is the reciprocal of wavelength, *h(x, y)* and is the absolute optical path difference of the interferometer.

The phase of the interference signal *ϕ(x, y; k)* is given by

$$
\phi(\mathbf{x}, \mathbf{y}; k) = 2\pi kh(\mathbf{x}, \mathbf{y}).\tag{3}
$$

The phase shift of the interference signal owing to the wavenumber shift is given by

$$
\Delta\phi(\mathbf{x}, \mathbf{y}; \Delta k) = 2\pi \Delta k h(\mathbf{x}, \mathbf{y}).\tag{4}
$$

The phase change of the interference signal is proportional to the wavenumber k change. Then, the optical path *h(x, y)* difference is given by

$$h(\mathbf{x}, \mathbf{y}) = \frac{\Delta\phi(\mathbf{x}, \mathbf{y}, \Delta k)}{2\pi\Delta k}. \tag{5}$$

The change of *k* can be calibrated by using an optical spectral analyzer, and the key issue here is how to determine the phase change. There are many phase calculation algorithms that may be used in spectral scanning interferometry. These algorithms include phase demodulation by a lock in amplifier [19], and phase calculation by a seven-point method used in classical phaseshifting interferometry [20], extremum position counting [17] and Fourier transform-based techniques [21]. In this paper, we use phase calculation by Fourier transform because it is fast, accurate and insensitive to intensity noise.

In Eq. (2), as mentioned, *a(x, y; k)* and *b(x, y; k)* are slowly variable with respect to due to the response of *k* the CCD camera and the spectrum intensity of the light source. Practically, the optical path difference of the interferometer is adjusted to be large enough with the intention that the frequency of the cosine term is higher than the variation frequency of terms *a(x, y; k)* and *b(x, y; k)* so that they can be easily separated from each other. Eq. (2) can be rewritten as

$$I(\mathbf{x}, \mathbf{y}; k) = a(\mathbf{x}, \mathbf{y}; k) + \frac{1}{2}b(\mathbf{x}, \mathbf{y}; k) \exp[2\pi ikh(\mathbf{x}, \mathbf{y})] + \frac{1}{2}b(\mathbf{x}, \mathbf{y}; k) \exp[-2\pi ikh(\mathbf{x}, \mathbf{y})].\tag{6}$$

The Fourier transform of Eq. (6) with respect to k can then be written as

$$\ddot{I}(\mathbf{x}, \mathbf{y}; \boldsymbol{\xi}) = A(\mathbf{x}, \mathbf{y}; \boldsymbol{\xi}) + B(\mathbf{x}, \mathbf{y}; \boldsymbol{\xi} - h(\mathbf{x}, \mathbf{y})) + B(\mathbf{x}, \mathbf{y}; \boldsymbol{\xi} + h(\mathbf{x}\mathbf{y})) \tag{7}$$

where the uppercase letters denote the Fourier spectra of the signal expressed by the corresponding lower-case letters. If *h*(*x*, *y*) is selected to be higher than the variation of *a*(*x*, *y*, *k*) and *b*(*x*, *y*, *k*), the three spectra can be separated from one another. To retrieve the phase distribution of the fringe, the term B(x, y; ξ−h(x, y)) is selected, and therefore, the background intensities *a*(*x*, *y*, *k*) are filtered out. The inverse Fourier transform of B(x, y; ξ− h(x, y)) is

$$IFFT(B(\mathbf{x}, \mathbf{y}; \tilde{\xi} - h(\mathbf{x}, \mathbf{y}))) = \frac{1}{2} b(\mathbf{x}, \mathbf{y}; k) \exp[ikh(\mathbf{x}, \mathbf{y})].\tag{8}$$

Taking the natural logarithm of this signal yields,

$$\log\left\{\frac{1}{2}b(\mathbf{x},\mathbf{y};k)\exp[2\pi ikh(\mathbf{x},\mathbf{y})]\right\}=\log\left[\frac{1}{2}b(\mathbf{x},\mathbf{y};k)\right]+i\phi(\mathbf{x},\mathbf{y};k)\tag{9}$$

from which the imaginary part of Eq. (9) is exactly the phase distribution to be measured. Following the above processes, the phase distribution of each CCD pixel, as well as the height map of the surface to be measured, can be acquired. Because the main processes here are fast Fourier transform (FFT) and inverse fast Fourier transform (IFFT), the data processing is very fast.

#### **2.3. Experimental results**

The method described in the previous section was used to measure standard height specimens and different types of structured surfaces, for which two examples are presented here. In these experimental studies, the radio frequency applied to the AOTF (Model LSGDN-1, SIPAT Co.) was scanned from 80 to 110 MHz in steps of 10 kHz, equivalent to a wavelength interval of 0.48 nm. This range of radio frequencies provides a range of scanning wavelengths from 680.8 to 529.4 nm. Throughout the wavelength scanning process, 300 interferograms were captured by a high-speed CCD camera (Model OK-AM1131, JoinHope Image Tech. Ltd.) at a frame rate of 100 frames/s. **Figure 2** shows the intensity distribution recorded by one of the CCD pixels (100, 100), and **Figure 2** shows the corresponding retrieved phase of this intensity distribution. It can be seen from **Figure 2** that this result suffers from discontinuities, where the values are in the range from −π to π. These discontinuities have been amended by adding 2π at the discontinuous points to achieve a continuous phase distribution as shown in **Figure 2**. By using the continuous phase distribution, the sample step height of this pixel can be computed.

In Eq. (2), as mentioned, *a(x, y; k)* and *b(x, y; k)* are slowly variable with respect to due to the response of *k* the CCD camera and the spectrum intensity of the light source. Practically, the optical path difference of the interferometer is adjusted to be large enough with the intention that the frequency of the cosine term is higher than the variation frequency of terms *a(x, y; k)* and *b(x, y; k)* so that they can be easily separated from each other. Eq. (2) can be rewritten as

1 1 ( , ; ) ( , ; ) ( , ; )exp[2 ( , )] ( , ; )exp[ 2 ( , )]. 2 2 *I x y k a x y k b x y k ikh x y b x y k ikh x y* =+ + p

*I x y A x y B x y h x y B x y h xy* ( , ; ) ( , ; ) ( , ; ( , )) ( , ; ( ))

<sup>1</sup> ( ( , ; ( , ))) ( , ; )exp[ ( , )]. <sup>2</sup> *IFFT B x y h x y b x y k ikh x y*

1 1 log ( , ; )exp[2 ( , )] log ( , ; ) ( , ; ) 2 2 <sup>ì</sup> üé ù í ý = + ê ú <sup>î</sup> þë û *b x y k ikh x y b x y k i x y k*

from which the imaginary part of Eq. (9) is exactly the phase distribution to be measured. Following the above processes, the phase distribution of each CCD pixel, as well as the height map of the surface to be measured, can be acquired. Because the main processes here are fast Fourier transform (FFT) and inverse fast Fourier transform (IFFT), the data processing is very

The method described in the previous section was used to measure standard height specimens and different types of structured surfaces, for which two examples are presented here. In these experimental studies, the radio frequency applied to the AOTF (Model LSGDN-1, SIPAT Co.) was scanned from 80 to 110 MHz in steps of 10 kHz, equivalent to a wavelength interval of 0.48 nm. This range of radio frequencies provides a range of scanning wavelengths from 680.8 to 529.4 nm. Throughout the wavelength scanning process, 300 interferograms were captured

where the uppercase letters denote the Fourier spectra of the signal expressed by the corresponding lower-case letters. If *h*(*x*, *y*) is selected to be higher than the variation of *a*(*x*, *y*, *k*) and *b*(*x*, *y*, *k*), the three spectra can be separated from one another. To retrieve the phase distribution of the fringe, the term B(x, y; ξ−h(x, y)) is selected, and therefore, the background intensities

The Fourier transform of Eq. (6) with respect to k can then be written as

*a*(*x*, *y*, *k*) are filtered out. The inverse Fourier transform of B(x, y; ξ− h(x, y)) is

xxx

x

p

Taking the natural logarithm of this signal yields,

ˆ

46 Optical Interferometry

fast.

**2.3. Experimental results**

 p

 x= + -+ + (7)


 f

(9)

(6)

**Figure 2.** (a) Intensity distribution for 300 interferograms captured by a CCD pixel (100, 100), (b) retrieved phase discontinuous distribution, (c) phase continuity distribution.

A standard step height specimen with a calibrated step of 2.970 µm, provided by the National Physical Laboratory (NPL), UK, was used as a measurement sample. This specimen has been processed according to the above proposed measurement procedure and an areal surface view obtained as shown in **Figure 3**. The measured average step height is 2.971 µm with a measurement error of 1 nm.

**Figure 3.** Measurement results of a 2.970 µm standard step.

The second measurement example is to investigate the effectiveness of the instrument's vibration compensation as follows:


The difference between the two measured step height values is 13.5 nm. The measurement results are shown in **Figures 4**–**6**, respectively. The observed disturbance attenuation between parts (2) and (3) of the experiment was 12.2 dB, according to the reference interferometer signal output, which is in good agreement with the measured standard sample error.

A standard step height specimen with a calibrated step of 2.970 µm, provided by the National Physical Laboratory (NPL), UK, was used as a measurement sample. This specimen has been processed according to the above proposed measurement procedure and an areal surface view obtained as shown in **Figure 3**. The measured average step height is 2.971 µm with a meas-

The second measurement example is to investigate the effectiveness of the instrument's

**1.** A semiconductor daughterboard sample was measured without inducing mechanical

**2.** A 40 Hz and 400 nm peak-to-peak sinusoidal mechanical disturbance using a PZT was applied to the reference mirror. During the disturbance, the measured surface step height

**3.** When the vibration compensation system is switched on, a decrease in the disturbance of the fringe pattern is clearly observed. A measurement of the sample is carried out with the vibration compensation system on. The data were retrieved as the original measurement and illustrate that the compensation vibration can be used to overcome environ-

The difference between the two measured step height values is 13.5 nm. The measurement results are shown in **Figures 4**–**6**, respectively. The observed disturbance attenuation between parts (2) and (3) of the experiment was 12.2 dB, according to the reference interferometer signal

is 11.711 µm (**Figure 5**). The surface roughness signal is completely distorted;

disturbance. The measured surface step height is 4.756 µm (**Figure 4**);

mental disturbance. The measured step height is 4.743 µm (**Figure 6**).

output, which is in good agreement with the measured standard sample error.

urement error of 1 nm.

48 Optical Interferometry

**Figure 3.** Measurement results of a 2.970 µm standard step.

vibration compensation as follows:

**Figure 4.** Measurement results of a semiconductor daughterboard sample without an induced mechanical disturbance: (a) measured surface and (b) cross-sectional profile.

**Figure 5.** Measurement results of a semiconductor daughterboard sample with a sinusoidal mechanical disturbance of 40 Hz: (a) measured surface and (b) cross-sectional profile.

**Figure 6.** Measurement results of a semiconductor daughterboard sample with vibration compensation: (a) measured surface and (b) cross-sectional profile.

From the above case study, we can conclude that by using a common-path reference interferometer together with an active environmental noise compensation system, a wavelength scanning interferometer can be used for shop floor surface measurement.

Roll-to-roll (R2R) processing is a fast and economic processing method for manufacturers that produce high-volume products using large area foils such as packaging products, photovoltaic films and emerging market sectors such as flexible electronics. However, there is an increased risk of defects forming as the number of interfaces increases in the multi-layer films, and the size and nature of those defects change as the layer thicknesses shrink to the nanoscale [22, 23]. Because of the nature of these practices, the inspection methods have to be in non-contact with the film surfaces. Effective surface inspection is the key for further processes such as applying local repair techniques to eliminate the defects from the film surface.

The above system has been implemented into a R2R system for demonstration of online surface inspection [24]. Nevertheless, a measurement can only be achieved if the measured sample is at a standstill. However, the tested sample surface is in constant movement during inspection, due to the nature of R2R processing. In order to achieve dynamic inspection, all the measurement information must be sampled in just one sample, and the sampling rate should be greater than a few hundred Hz to reduce the effect of mechanical vibration. In this case, a dynamic interferometer [8–10] is the solution. However, these approaches have one drawback for the application of online surface measurement: the 2µ phase ambiguity for phase-shift interferometer, which restricts the vertical measurement range to a few hundred nanometres. It is short of the demands of most surface measurements and inspections. We have introduced a single-shot line-scan dispersive interferometer [25] which is able to perform dynamic surface inspection and has a vertical measurement range over a few hundred micrometers.
