**2.2. Optical configurations**

**Figure 2** illustrates the principle of operation of the present method. A Michelson interferometer is configured with one end mirror replaced by the thin-film specimen. Call this interferometric arm the signal arm, and the other the reference arm. In the signal arm, the specimen is placed with the film side facing the beam splitter. The specimen is oscillated with an acoustic transducer from the rear (substrate) side, and the resultant oscillation of the film surface is detected as the corresponding change in the optical path length relative to the reference path. It is postulated that the elastic modulus of the interface is lower than that of the film or the substrate material, as the right drawing of **Figure 1** illustrates. With this postulate, it follows that the resonant frequency of the interface is lower than that of the film or substrate. In other words, when the acoustic frequency is tuned in a frequency range where the interface can possibly have resonant points, both the film and substrate oscillate as rigid bodies. Therefore, it is possible to detect the differential displacement of the film surface that represents the dynamics of the interface.

**Figure 2.** Michelson interferometer and experimental arrangement.

### **2.3. Optical intensity behind beam splitter**

The light intensity on the image plane behind the beam splitter can be expressed as follows.

$$I\left(t\right) = 2I\_0 + 2I\_0 \cos\left[k\left(l\_{s0} - l\_{r0}\right) + kd\sin\alpha t\right] = 2I\_0 + 2I\_0 \cos\left[\delta\_0 + \delta\sin\alpha t\right] \tag{1}$$

Here *I*0 is the intensity of the reference and signal beams, *k* is the wave number of the laser light in (rad/m), *ls*0 and *lr*0 are respectively, the initial (physical) length of the signal and reference arms, *δ*0 = *ls*0 – *lr*0 is the arm length difference, *δ* = *kd* and *d* are the oscillation amplitude of the film surface in (rad) and (m), and *ω* is the oscillation (driving) angular frequency of the film specimen. Here the reference and signal beams are the noninterfering light beams in the reference and signal arms, respectively; their intensities are assumed to be equal to each other. (In reality, they are not equal to each other but the gist of the argument here is not affected by inequality. The error associated with the inequality is discussed later in this paper.) The second term in the right-hand side of Eq. (1) with cos[*δ*0+ *δ* sin *ωt*] is called the interference term. This term is important as it contains the relative phase change information. *I*(*t*) is captured by a photodetector or an imaging device placed behind the beam splitter.

Two methods are possible to detect the relative optical path change behind the beam splitter. Call them the total-intensity and two-dimensional methods. Both methods have advantages and disadvantages. In the total-intensity method, the total intensity *I*(*t*) is captured by a fast photodetector such as a silicon PIN photodiode. The advantage of this method is that the response time of the detector is fast enough to analyze *I*(*t*) at the same frequency as the acoustic transducer (*ω* in Eq. (1)). A disadvantage of this method is that it detects the interference term representing the entire cross-sectional area of the laser beam reflected off the specimen. It is unable to resolve the intensity over the plane of the specimen. Another disadvantage of this method is that it is vulnerable to unwanted optical path changes due to environmental disturbance.

In the two-dimensional method, an imaging device (an array of photodetector such as a CCD (Charge Coupled Device)) is used for the photodetector behind the beam splitter. (Hereafter, this type of imaging device is referred to as a CCD.) A typical CCD consists of approximately 500 rows and 500 columns of pixels. It is possible to detect the relative phase change two dimensionally on a pixel-by-pixel basis. Another advantage of this method is that by introducing a so-called carrier fringe system, it is possible to reduce the influence of the unwanted optical path length change due to environmental disturbance. The disadvantage of this method is that the frame rate (the sampling rate) of a CCD has normally orders of magnitude lower than the acoustic frequency. Consequently, the detected signal is greatly down-sampled. Below we discuss the two configurations in more detail.

### *2.3.1. Total intensity configuration*

metric arm the signal arm, and the other the reference arm. In the signal arm, the specimen is placed with the film side facing the beam splitter. The specimen is oscillated with an acoustic transducer from the rear (substrate) side, and the resultant oscillation of the film surface is detected as the corresponding change in the optical path length relative to the reference path. It is postulated that the elastic modulus of the interface is lower than that of the film or the substrate material, as the right drawing of **Figure 1** illustrates. With this postulate, it follows that the resonant frequency of the interface is lower than that of the film or substrate. In other words, when the acoustic frequency is tuned in a frequency range where the interface can possibly have resonant points, both the film and substrate oscillate as rigid bodies. Therefore, it is possible to detect the differential displacement of the film surface that represents the

The light intensity on the image plane behind the beam splitter can be expressed as follows.

Here *I*0 is the intensity of the reference and signal beams, *k* is the wave number of the laser light in (rad/m), *ls*0 and *lr*0 are respectively, the initial (physical) length of the signal and reference arms, *δ*0 = *ls*0 – *lr*0 is the arm length difference, *δ* = *kd* and *d* are the oscillation amplitude of the film surface in (rad) and (m), and *ω* is the oscillation (driving) angular frequency of the film specimen. Here the reference and signal beams are the noninterfering light beams in the

ddw

*t* (1)

( ) 0 0 00 ( ) 00 0 =+ -+ =+ + 2 2 cosé ù sin 2 2 cos sin é ù ë û *s r* ë û *I t I I k l l kd t I I* w

dynamics of the interface.

64 Optical Interferometry

**Figure 2.** Michelson interferometer and experimental arrangement.

**2.3. Optical intensity behind beam splitter**

In this configuration, the light beams from the two arms are aligned so that they overlap each other for the entire path they share, i.e., the optical path from the beam splitter to the photodetector. Under these conditions, the film surface displacement due to the acoustic oscillation changes the relative path length difference commonly to all points on the *x*-*y* plane (the plane of the specimen). The photodetector signal is proportional to the total intensity expressed by Eq. (1), where the oscillation comes from the interference term. By measuring the intensity of the signal beam and the reference beam separately, we can evaluate 2*I*0 and express the interference as follows.

$$\cos\left[\delta\_0 + \delta\sin\alpha t\right] = \frac{I\left(t\right) - 2I\_0}{2I\_0} \tag{2}$$

In evaluating the oscillation amplitude *δ* from Eq. (2), the initial phase difference *δ*0 plays an important role. The phase oscillation due to the acoustic transducer occurs around *δ*0. Since the function cos*θ* has the greatest slope at *θ* = 0, the oscillation amplitude of Eq. (2) is maximized when *δ*0 = 0. In other words, the amplitude of the oscillation due to *δ* is maximized when the initial interference is totally destructive. **Figure 3** illustrates this situation for 0 = 0, 4, 2, when the oscillation amplitude *<sup>δ</sup>* is 0.2 as an example. The issue is that when an environmental factor such as changes in the refractive index of the air due to temperature fluctuations vary the value of *δ*0 in a random fashion, it becomes impossible to distinguish whether observed interferometric intensity variation is due to *δ* or *δ*0 in Eq. (2).

**Figure 3.** Interference term with three different initial phases.

### *2.3.2. Two-dimensional configuration*

In this configuration, a CCD is used to capture the intensity represented by Eq. (1). Normally, the frame rate of a CCD is significantly lower than the acoustic frequency. A CCD with a frame rate comparable to the acoustic frequency is available but it is expensive and sometimes the number of pixels is limited in exchange for a higher frame rate. So, here we discuss image analysis for a low frame rate case.

The issue of environmental relative phase fluctuation discussed above applies to the twodimensional configuration as well. However, the introduction of a carrier fringe system in conjunction with frequency domain analysis greatly overcomes this issue. Below we first consider the case when a carrier fringe system is not introduced (called the simple Michelson method) followed by the carrier fringe method.

### (a) Simple Michelson method

In evaluating the oscillation amplitude *δ* from Eq. (2), the initial phase difference *δ*0 plays an important role. The phase oscillation due to the acoustic transducer occurs around *δ*0. Since the function cos*θ* has the greatest slope at *θ* = 0, the oscillation amplitude of Eq. (2) is maximized when *δ*0 = 0. In other words, the amplitude of the oscillation due to *δ* is maximized when the initial interference is totally destructive. **Figure 3** illustrates this situation for

an environmental factor such as changes in the refractive index of the air due to temperature fluctuations vary the value of *δ*0 in a random fashion, it becomes impossible to distinguish

In this configuration, a CCD is used to capture the intensity represented by Eq. (1). Normally, the frame rate of a CCD is significantly lower than the acoustic frequency. A CCD with a frame rate comparable to the acoustic frequency is available but it is expensive and sometimes the number of pixels is limited in exchange for a higher frame rate. So, here we discuss image

The issue of environmental relative phase fluctuation discussed above applies to the twodimensional configuration as well. However, the introduction of a carrier fringe system in conjunction with frequency domain analysis greatly overcomes this issue. Below we first consider the case when a carrier fringe system is not introduced (called the simple Michelson

whether observed interferometric intensity variation is due to *δ* or *δ*0 in Eq. (2).

2, when the oscillation amplitude *<sup>δ</sup>* is 0.2 as an example. The issue is that when

0 = 0,

66 Optical Interferometry

4,

**Figure 3.** Interference term with three different initial phases.

method) followed by the carrier fringe method.

*2.3.2. Two-dimensional configuration*

analysis for a low frame rate case.

Eq. (1) represents the instantaneous intensity observed behind the beam splitter. To discuss the case where the CCD's frame rate is much lower than the signal frequency, it is convenient to rewrite Eq. (1) in terms of Bessel functions of first kind. Using the following identities,

> cos sin (dw d d w d w*tJ J tJ t* ) =+ + + 02 4 ( ) 2 cos2 2 cos4 ( ) ( ) L

sin(dw d w d wsin 2 sin 2 sin3 *t J tJ t* ) =+ + 1 3 ( ) ( ) L

Eq. (1) can be rewritten as follows.

$$\begin{aligned} I\left(t\right) &= 2I\_0 + 2I\_0 \cos\left(\delta\_0 + \delta \sin\alpha t\right) = 2I\_0 + 2I\_0 \cos\delta\_0 \cos\left(\delta \sin\alpha t\right) - \\ 2I\_0 \sin\delta\_0 \sin\left(\delta \sin\alpha t\right) &= 2I\_0 + 2I\_0 \cos\delta\_0 \left\{J\_0\left(\delta\right) + 2J\_2\left(\delta\right)\cos 2\alpha t\right\} + \\ 2J\_4\left(\delta\right) &\cos 4\alpha t + \cdots \right\} - 2I\_0 \sin\delta\_0 \left\{2J\_1\left(\delta\right)\sin\alpha t + 2J\_3\left(\delta\right)\sin 3\alpha t + \cdots \right\} \end{aligned} \tag{3}$$

In the present case, the acoustic frequency is in the range of 1–20 KHz, and the CCD has a frame rate of 30 fps (frames per second), or three orders of magnitude lower than the acoustic frequency. In other words, the data taken by the digital camera is greatly down-sampled. Under these conditions, the output of the CCD can be expressed as follows.

$$\begin{aligned} 2I\_0 \int\_0^\tau \left< 1 + \cos \delta\_0 J\_0 \left( \delta \right) \right> dt + 4I\_0 \cos \delta\_0 \sum\_{N=10}^\infty \int\_{2N} J\_{2N} \left( \delta \right) \cos \left( 2N \alpha t \right) dt - \\ 4I\_0 \sin \delta\_0 \sum\_{N=10}^\infty \int\_{2N-1} J\_{2N-1} \left( \delta \right) \sin(2N-1) \alpha t \end{aligned} \tag{4}$$

Here *τ* is the exposure time of the CCD and *N* is an integer. Of the terms on the right-hand side of Eq. (4), those terms that contain the summation over *N* oscillate. On the other hand, the first integral is constant with respect to time and therefore increases in proportion to *τ*. Consequently, under the condition where the exposure time is much greater than the period of oscillation, i.e., *τ* ≫ 2*π*/*ω*, the signal *S*(*τ*) can be approximated by the first integral.

$$S(\tau) \equiv 2I\_0 \tau \left\{ 1 + \cos \delta\_0 J\_0 \left( \delta \right) \right\} \tag{5}$$

**Figure 4** compares the signal evaluated by Eq. (4) and the approximate signal by Eq. (5) for the driving frequency of 11 KHz as a function of the exposure time. It is seen that for *τ* = 1/30 =33 (ms), the exposure time corresponding to the frame rate of 30 fps (the frame rate used in the present study), the approximation by Eq. (5) is accurate.

**Figure 4.** Comparison of *J*0(*δ*) approximation and complete integral.

In Eq. (5), the first term 2*I*0*τ* is the sum of the optical intensity of the signal and reference arms. Experimentally, these intensities can be easily obtained by blocking one of the arms at a time and using the CCD behind the beam splitter. By subtracting this term from the total signal and dividing the result by 2*I*0*τ*, we can derive expression of the interference term (Eq. (2)) for the total intensity configuration as follows.

$$\cos \delta\_0 J\_0 \left( \delta \right) = \frac{S \left( \tau \right) - 2I\_0 \tau}{2I\_0 \tau} \tag{6}$$

In principle, by knowing the initial relative phase *δ*0 from an independent experiment (such as changing the reference arm length through fringes with the acoustic transducer turned off), we can even estimate the value of *δ* from the known curve of *J*0(*δ*) and *d* = *δ*/*k*. However, in reality, environmental noise causes fluctuations in the optical path length. As will be discussed later, a temperature change of 0.1°C in the air in the beam path can cause a considerable change in the relative phase *δ*0. Also, angular misalignment such as the one due to seismic disturbance reduces the accuracy in the subtraction of the *I*<sup>0</sup> in Eq. (6); since the intensity of the reference and signal beams is measured at different times from the total intensity, any angular misalignment shifts the beam center on the image plane of the CCD, and that reduces the accuracy of Eq. (6).

(b) Carrier fringe method

In this method a linear spatial phase-variation is introduced so that the relative phase changes over several periods of 2*π* across the cross-section of the laser beam. This technique is known as the introduction of carrier fringes and widely used in ESPI (Electronic Speckle–Pattern Interferometry) [7]. Carrier fringes can be introduced by slightly tilting the specimen or the reference mirror, or by inserting a mechanism to introduce a linear phase variation such as an optical wedge. **Figure 2** shows an example where the specimen is slightly tilted.

We can express the CCD's output in this case by replacing cos *δ*0 with cos *αx* as follows.

(ms), the exposure time corresponding to the frame rate of 30 fps (the frame rate used in the

In Eq. (5), the first term 2*I*0*τ* is the sum of the optical intensity of the signal and reference arms. Experimentally, these intensities can be easily obtained by blocking one of the arms at a time and using the CCD behind the beam splitter. By subtracting this term from the total signal and dividing the result by 2*I*0*τ*, we can derive expression of the interference term (Eq. (2)) for the

( ) ( ) <sup>0</sup>

In principle, by knowing the initial relative phase *δ*0 from an independent experiment (such as changing the reference arm length through fringes with the acoustic transducer turned off), we can even estimate the value of *δ* from the known curve of *J*0(*δ*) and *d* = *δ*/*k*. However, in reality, environmental noise causes fluctuations in the optical path length. As will be discussed later, a temperature change of 0.1°C in the air in the beam path can cause a considerable change in the relative phase *δ*0. Also, angular misalignment such as the one due to seismic disturbance reduces the accuracy in the subtraction of the *I*<sup>0</sup> in Eq. (6); since the intensity of the reference and signal beams is measured at different times from the total intensity, any angular misalignment shifts the beam center on the image plane of the CCD, and that reduces the accuracy of

0 2

t

 t

(6)

2 - <sup>=</sup> *S I*

*I* t

0 0

 d

*J*

d

cos

present study), the approximation by Eq. (5) is accurate.

68 Optical Interferometry

**Figure 4.** Comparison of *J*0(*δ*) approximation and complete integral.

total intensity configuration as follows.

Eq. (6).

(b) Carrier fringe method

$$S(\tau) \equiv 2I\_0 \tau \left\{ 1 + \cos \alpha x J\_0(\delta) \right\} \tag{7}$$

Here, *x* is the coordinate axis set up on the specimen's surface and *α* is the angle of tilt.

The advantage of this technique in the present context is as follows. In the simple Michelson method, if environmental disturbance changes the relative phase, it directly affects the signal expressed by Eq. (5). There is no way of knowing whether the change in the signal is due to the oscillation amplitude or the environmental noise unless the sampling rate of the optical detector is higher than the acoustic frequency. When the sampling rate is lower than the acoustic frequency, the detector's signal passes though a number of maxima and minima corresponding to the constructive and destructive interference. On the other hand, if carrier fringes are introduced, it is always possible to capture the constructive and destructive interferences; the former corresponds to the bright fringes and the latter to the dark fringes. An optical path length change causes a shift of the fringe pattern in a lateral direction at the same frequency as the optical path length change. If the optical path length change is due to the acoustic oscillation of the specimen, the fringes move back and forth transversely to the beam (dither) at the acoustic frequency. The CCD cannot resolve this fast dithering motion. Consequently, the fringe contrast is reduced. If the optical path length change is due to an environmental effect, the fringe shift is most likely slower than the sampling rate and can be resolved as a change in the fringe location by the CCD with the fringe contrast unchanged. A slight angular misalignment changes the fringe spacing, and causes some error in the frequency-domain analysis as will be discussed later. However, the error is much smaller than the simple Michelson method.

The carrier fringe method is especially effective if the fringe data is analyzed in the spatialfrequency domain. The reduction in the fringe contrast due to fast dithering is detected in the Fourier spectrum as a reduction in the height of the main peak at the frequency determined by the fringe spacing associated with cos *αx* in Eq. (7). Thus, by forming Fourier spectrum of the optical intensity profile and evaluating the peak height of the spectrum we can evaluate the fringe contrast, and in turn, estimate the oscillation amplitude. The spatial fringe shift due to environmental change in *δ*<sup>0</sup> does not change the Fourier spectrum as it is not a function of *x*. An angular misalignment due to environmental disturbance can change the spectrum peak height, but the effect is relatively small (see below).
