CONTENTS

10.1    Introduction

10.2    Heterodyne Method

10.2.1    High-Precision Measurement: Principle and Defects

10.2.2    Construction of the Cancelable Circuit: An Introduction of Two Signals by Two Beams

10.2.3    Construction of the Cancelable Circuit: Generation and Selection of Two Lights

10.2.4    Construction of the Cancelable Circuit: Symmetrical Layout in the Circuit

10.2.5    Relation between the Doppler Method and the Heterodyne Method

10.3    Accuracy and Noise Reduction

10.3.1    Deterioration of Accuracy Caused by the Environment

10.3.2    Deterioration of Accuracy Caused by Optical Parts

10.3.2.1    AOM and Its RF Driver

10.3.2.2    Frequency-Stabilized HeNe Laser

10.3.2.3    Interference Circuit and Parts

10.3.2.4    Sample Surface and Sample Holder

10.3.2.5    Detection Method and System

10.4    Attention for Static and Dynamic Measurement

10.4.1    Static Measurement

10.4.2    Dynamic Measurement

10.5    Applications

10.5.1    Displacement, Vibration Frequency, and Amplitude

10.5.2    Positioning with Stage

10.5.3    Thick Measurement

10.5.4    Profilometry

10.5.5    Refractometry

10.5.6    Photothermal Interferometry

10.5.7    Thermal Expansion Coefficient

10.5.8    Young?s Modulus of Thin Film

10.5.9    Birefringence Index and Polarimeter

10.5.10  Fiber Sensor

10.5.11  Dynamical Surface Measurement

10.6    Optical Doppler Measurement

10.6.1    Introduction

10.6.2    Principle and Defect

10.6.3    Comparison between Doppler Method and Optical Heterodyne Method

10.6.3.1    Accuracy

10.6.3.2    Measurement of Velocity and Displacement

10.7    Measurement of Transient Phenomena

10.7.1    Application of the Optical Heterodyne Method

10.7.2    Attention to the Design of Transient Measuring System

10.7.3    Transient Measurement Method

10.7.3.1    Signal Processing

10.7.3.2    Photoelectric Conversion Circuit

10.7.3.3    Period Detection

10.7.3.4    Phase Detection and Displacement

10.7.3.5    Analysis of Error Factor

10.7.4    Examples of Transient Measurement

10.7.4.1    Tuning Fork and Cantilever Vibration

10.7.4.2    Concentration Fluctuation

References

10.1    INTRODUCTION

The optical heterodyne method is remarkable as a noncontact method that measures physical values with high precision. To achieve high precision, there are many restraints and conditions. In this chapter, first, the principle of the method is described, and the defects of the method are also shown. Second, the restraints and conditions are explained. These require the introduction of a cancelable optical circuit as a necessary condition. Third, several desirable conditions for optical parts and mechanical parts used in the circuit are given in detail. Finally, some applications are listed.

Utilizing the merits of the heterodyne method, heterodyne technology has also been used in similar fields. This chapter is confined mainly to a description of the basic view of the heterodyne method. Some heterodyne detection techniques and their applications are omitted: for example, OTDR [1], laser radar [2], spectrophotometry [3,4,5], optical CT [6,7,8], nerve bundle measurement [9], gravitational wave detection [10], interferometer in astronomy [11,12], heterodyne speckle interferometer [13,14], optical bistability measurement [15], fiber gyro [16,17], interferometer in atomic physics [18,19], and measurements for many physical variables.

10.2    HETERODYNE METHOD

10.2.1    HIGH-PRECISION MEASUREMENT: PRINCIPLE AND DEFECTS

The optical heterodyne method consists of an interferometer with two laser beams whose wavelengths are shifted by modulators from a laser wavelength.

These beams are represented with the wavelengths of λ_A and λ_B, as follows:

$A\exp \left(i2\pi \frac{ct}{{\lambda }_{\text{A}}}+{\varphi }_{\text{A}}\right)\text{and}B\exp \left(i2\pi \frac{ct}{{\lambda }_{\text{B}}}+{\varphi }_{\text{B}}\right),$

where

A and B are optical strengths

c and t are the light velocity and time, respectively

ϕ_A and ϕ_B are initial phases of two beams

The interference signal of the two beams is represented with the times elapsed from a laser to a detector, t_A and t_B, as follows:

$\begin{array}{l}{\left|A\exp \left(i2\pi \frac{c\left(t-t_{\text{A}}\right)}{{\lambda }_{\text{A}}}+{\varphi }_{\text{A}}\right)+B\exp \left(i2\pi \frac{c\left(t-t_{\text{B}}\right)}{{\lambda }_{\text{B}}}+{\varphi }_{\text{B}}\right)\right|}^2\hfill \\ =\text{const}\text{+}\text{2}AB\text{sin}\left(2\pi ct\left(\frac{1}{{\lambda }_{\text{A}}}-\frac{1}{{\lambda }_{\text{B}}}\right)-2\pi c\left(\frac{t_{\text{A}}}{{\lambda }_{\text{A}}}-\frac{t_{\text{B}}}{{\lambda }_{\text{B}}}\right)+{\varphi }_{\text{A}}-{\varphi }_{\text{B}}\right).\hfill \end{array}$

(10.1)

The frequency of the interference signal (= beat frequency f) is represented as the difference of two wavelengths, 2πc(1/λ_A − 1/λ_B). The phase of the beat signal is described with the sum of two parts: (1) the difference of the initial phases and (2) the difference of optical path lengths in the circuit. This optical path length is given by the relation of optical path length = Σ (the geometric distance of the path × the refractive index of the path). So, when the optical path lengths are L_A (= ct_A) and L_B (= ct_B) as shown in Figure 10.1, the phase of the beat signal changes by

$\phi =2\pi \left(\frac{L_{\text{A}}}{{\lambda }_{\text{A}}}-\frac{L_{\text{B}}}{{\lambda }_{\text{B}}}\right).$

(10.2)

When sample targets move further by ΔL_A and ΔL_B, the phase of the beat signal is

$\phi +\Delta \phi =2\pi \left(\frac{L_{\text{A}}\text{+}\Delta L_{\text{A}}}{{\lambda }_{\text{A}}}-\frac{L_{\text{B}}+\Delta L_{\text{B}}}{{\lambda }_{\text{B}}}\right),\text{that is,}\Delta \phi =2\pi \left(\frac{\Delta L_{\text{A}}}{{\lambda }_{\text{A}}}-\frac{\Delta L_{\text{B}}}{{\lambda }_{\text{B}}}\right).$

(10.3)

Equation 10.2 shows that, under an ordinary condition of L_A, L_B ≫ λ, the value of φ is very large, so it cannot be known; therefore, the value of L cannot be confirmed. However, Equation 10.3 shows that, under a condition ΔL_A, ΔL_B < λ, the value of Δφ can give ΔL (either ΔL_A or ΔL_B) as follows:

$\Delta L=\frac{\Delta \varphi }{2\pi }\lambda .$

(10.4)

This proportional relation between ΔL and λ means (1) the ruler (measuring scale) in the method is the wavelength and (2) the displacement ΔL can be measured in wavelength units. This is the basic relation in the heterodyne method that achieves a high-precision measurement. In addition, Equations 10.1, 10.3, and 10.4 show that two beams must satisfy the following conditions:

FIGURE 10.1  Wavelength and optical path length for one detector.

1.  The difference of the two beams’ frequencies is a small value, which can be detected and handled with an electric circuit. For example, λ₁ = 633.00000 and λ₂ = 633.00001 produce a beat signal of f = 7.5 MHz. And beams need to be a single mode for both a longitudinal mode and a transverse mode.

2.  The wavelength (i.e., frequency f) has to have high stabilization, which is represented with Δf/f ≤ 10⁻⁹, where Δf is a frequency fluctuation (e.g., 0.00001/633 = 1.6 × 10⁻⁸).

These conditions limit the selection of a useful laser. Semiconductor lasers have many longitudinal modes and multi-transverse mode, and frequency stabilization is difficult to achieve with Ar and Kr lasers. HeNe laser has been attracted because its frequency fluctuation in a free-running operation is about 10⁻⁶. No lasers can satisfy the condition in ordinary operations, but by developing a frequency-stabilized HeNe laser, the conditions are almost satisfied. Therefore, the frequency-stabilized HeNe laser is mainly used in the heterodyne measurement, which will be described in a later section.

It should be noted that Equation 10.4 also shows some of the defects in the heterodyne method.

Defect (1): When ΔL is over the wavelength within a sampling time, a correct displacement cannot be known because of its uncertain factor.

Defect (2): The displacement for the direction perpendicular to the beam direction cannot be measured.

Defect (1) can be improved by shortening the sampling time. However, Defect (2) is unescapable. Even if scattering light is gathered, no displacement information can be obtained because the scattering angle distribution cannot be measured. By using one more measurement system, Defect (2) will be improved.

10.2.2    CONSTRUCTION OF THE CANCELABLE CIRCUIT: AN INTRODUCTION OF TWO SIGNALS BY TWO BEAMS

To achieve high-precision measurement with the heterodyne method, a new cancelable optical circuit has been proposed to keep the method’s advantages. In this and the following sections, some essential points are shown.

The interference of two beams makes one ordinarily signal. For only one beat signal, one displacement is calculated with Equation 10.4. Two optical paths (L₁ and L₂) will be fluctuated independently by different causes. Therefore, to obtain a high-precision displacement (ΔL₁ or ΔL₂), the fluctuation has to be suppressed much less than the displacement. For example, the optical length of L₁ = 20 cm demands that the expansion ratio on the path is under 5 × 10⁻⁸ for ΔL₁ = 10 nm resolution. This is very difficult to achieve. To make this strict condition easier, the system introduces two interference signals. As shown in Figure 10.2, both laser beams λ_A and λ_B are divided into two parts, and four optical paths, LA₁, LA₂, LB₁, and LB₂, make two pairs, and two detectors get two beat signals of pairs as written by

$\begin{array}{l}\text{Detector 1 SIG1}\text{=}{C}_1\sin \left(\left({\text{ω}}_{\text{A}}-{\text{ω}}_{\text{B}}\right)t+{\varphi }_{\text{A}}-{\varphi }_{\text{B}}+{\text{ω}}_{\text{A}}\times \frac{{\text{LA}}_1}{c}-{\text{ω}}_{\text{B}}\times \frac{{\text{LB}}_1}{c}\right),\hfill \\ \text{Detector 2 SIG2}\text{=}{C}_2\sin \left(\left({\text{ω}}_{\text{A}}-{\text{ω}}_{\text{B}}\right)t+{\varphi }_{\text{A}}-{\varphi }_{\text{B}}+{\text{ω}}_{\text{A}}\times \frac{{\text{LA}}_2}{c}-{\text{ω}}_{\text{B}}\times \frac{{\text{LB}}_2}{c}\right),\hfill \end{array}$

(10.5)

FIGURE 10.2  Wavelength and optical path length for two detectors.

where

C₁ and C₂ are the strength of two beat signals

ϕ_A and ϕ_B are the initial phase of two beams

ω_A and ω_B are 2πc/λ_A and 2πc/λ_B, respectively

The difference of two signal phases is given as follows:

$\begin{array}{l}\phi +\Delta \phi ={\text{ω}}_{\text{A}}\frac{{\text{LA}}_1+\Delta {\text{LA}}_1-{\text{LA}}_2-\Delta {\text{LA}}_2}{c}-{\text{ω}}_{\text{B}}\frac{{\text{LB}}_1+\Delta {\text{LB}}_1-{\text{LB}}_2-\Delta {\text{LB}}_2}{c}\hfill \\ \therefore \Delta \phi =2\pi \left(\frac{\Delta {\text{LA}}_1-\Delta {\text{LA}}_2}{{\lambda }_{\text{A}}}-\frac{\Delta {\text{LB}}_1-\Delta {\text{LB}}_2}{{\lambda }_{\text{B}}}\right).\hfill \end{array}$

(10.6)

Equation 10.6 is very similar to Equation 10.4. However, in Equation 10.4, ΔL is the displacement of an optical path, while in Equation 10.6, ΔLA₁ − ΔLA₂ is the difference of the displacements of two optical paths. The difference between Equations 10.4 and 10.6 is essential in a cancelable circuit. Equation 10.6 represents that the phase fluctuation can be reduced even if both the optical paths fluctuate. In particular, if two optical paths fluctuate for the same reason, the fluctuations do not affect Δφ at all (i.e., the influences are canceled). In addition, Equation 10.6 also shows that several circuit layouts are possible as listed in Table 10.1.

By discussing these five cases in Table 10.1, the merits of two beat signals will be understood.

For case (a), the following conditions are required:

1.  The optical path lengths, LA₁, LB₁, and LB₂, never vary.

2.  The wavelength never varies.

3.  The distortion of the wave front never appears.

As these conditions are very hard to satisfy, there is no merit.

For the case of (b), the difference, ΔLA₁ − ΔLA₂, is obtained. This case fails one of the conditions, “LA₂ never vary.” When LA₁ and LA₂ include a common path (having nearly the same length), a fluctuation on the common path between LA₁ and LA₂ affects equally and simultaneously. Therefore, the result can exclude or cancel the influence of the fluctuation. This effect is also obtained in a one-beam signal system.

TABLE 10.1
Resolutions for Circuit Layouts

Case	Variable	Constant	Δφ	Requested Result
(a)	LA1	LA2, LB1, LB2	2πΔLA1/λA	ΔLA1
(b)	LA1, LA2	LB1, LB2	2π(ΔLA1 − ΔLA2)/λA	ΔLA1 − ΔLA2
(c)	LA2, LB2	LA2, LB2	2π(ΔLA1/λA − ΔLB1/λB) ≅2π(ΔLA1 − ΔLB1)/λA	ΔLA1 − ΔLB1
(d)	LA1, LB2	LA2, LB1	2π(ΔLA1/λA+ΔLB1/λB) ≅2π(ΔLA1 + ΔLB1)/λA	ΔLA1 + ΔLB1
(e)	LA1, LA2, LB1, LB2	None	2π{(ΔLA1 − ΔLA2)/λA+ (ΔLB1 − ΔLB2)/λB} ≅2π(ΔLA1 − ΔLA2 + ΔLB1 − ΔLB2)/λA	ΔLA1 − ΔLA2 + ΔLB1 − ΔLB2

Notes:  The Δφ values in cases (c) through (e) are approximate. However, the difference between λ_A and λ_B is extremely small, nearly λ_A/λ_B = 1.00000002 or less, so the measurement error is far more below a detectable error. The sign of results is determined by the displacement direction.

For cases (c) through (e), the system of the two beat signals is more effective. Example circuits are illustrated in Figure 10.3 (a sample is set upon a basement) and Figure 10.4. A signal phase of the beam reflected on the sample fluctuates by movements of both the basement and the sample. A signal phase of the beam reflected on the basement fluctuates by only the movement of the basement. Therefore, the operation of ΔLA₁ − ΔLB₁ makes the movement of the basement disappear (cancel) and gives the displacement of the sample only. In the same discussion for Figure 10.4, beams reflected at the front and at the rear of sample cancel a vibration of the sample. Therefore, an operation of ΔLA₁+ΔLB₂ makes the movement of the basement disappear (cancel), and the expansion/contraction of the sample only is measured. For this reason, they are called a cancelable optical circuit in this chapter.

The accuracy of this measurement depends on the degree to which these conditions are satisfied. The relationship between accuracy and these conditions will be described in Section 10.3.

FIGURE 10.3  Cancelable circuit: simultaneous measurement of sample and basement displacement. A fluctuation of basement can be out of consideration.

FIGURE 10.4  Cancelable circuit: thickness variation measurement of sample only.

Why does a cancelable optical circuit achieve better measurement? The answer is very simple. Even if several external noises or error factors occur in φ₁ and φ₂, the influences can be reduced. Two examples are discussed.

Case 1: a frequency fluctuation occurs from ω_A to ω_A+ δω_A.

The phases are modified in Equation 10.6 as follows:

$\begin{array}{ll}{\phi }_1+\text{δ}{\phi }_1\hfill & =\left({\text{ω}}_{\text{A}}+{\text{δω}}_{\text{A}}\right)\frac{{\text{LA}}_1}{c}+{\text{ω}}_B\frac{{\text{LB}}_1}{c}+{\varphi }_{\text{A}}-{\varphi }_{\text{B}},\hfill \\ {\phi }_2+{\text{δφ}}_2\hfill & =\left({\text{ω}}_{\text{A}}+{\text{δω}}_{\text{A}}\right)\frac{{\text{LA}}_2}{c}+{\text{ω}}_B\frac{{\text{LB}}_2}{c}+{\varphi }_{\text{A}}-{\varphi }_{\text{B}}.\hfill \end{array}$

(10.7)

The difference of the phases (= output signal) is given by

$\phi =\left({\phi }_1+\delta {\phi }_1\right)-\left({\phi }_2+\delta {\phi }_2\right)={\phi }_0+\delta {\omega }_{\text{A}}\frac{{\text{LA}}_1-{\text{LA}}_2}{c}.$

(10.8)

In a one-beat signal system, the phase fluctuation is expressed by the relation, δφ = δω× LA/c. Even if δω is very small, a large LA does not result in a small δφ. In a two-beat signal system, the phase fluctuation is expressed with Equation 10.8. Smaller LA₁ − LA₂ results in lower fluctuation and better resolution. Usually, δω_B will occur at one time.

Case 2: an environment fluctuation occurs from LA₁ and LA₂ to LA₁ + δLA₁ and LA₂ + δ LA₂, respectively. The output signal is rewritten as

$\phi ={\phi }_0+{\text{ω}}_A\frac{\left(\delta {\text{LA}}_1-\delta {\text{LA}}_2\right)}{c}.$

(10.9)

Equation 10.9 indicates that the condition δLA₁ = δLA₂ is desired to diminish the fluctuation. To obtain the same fluctuation for two optical paths, all of the paths should be close to place them in the same environmental conditions: temperature, pressure, and humidity.

The cancelable optical circuit and the introduction of two signals are a powerful method for obtaining good measurement results even when many kinds of fluctuations occur.

10.2.3    CONSTRUCTION OF THE CANCELABLE CIRCUIT: GENERATION AND SELECTION OF TWO LIGHTS

In the heterodyne method, ω₂ is very close to ω₂. This condition is required to obtain a beat signal whose frequency can be processed with an electric circuit. A modulator is used to change the optical frequency. An acousto-optic modulator (AOM) [20,21,22] and an electro-optic modulator (EOM) [23] are usually used. AOM and EOM are based on the acousto-optic effect and the electro-optic effect, respectively. Both modulators shift the optical frequency using an input RF signal with a frequency in the range of 30 MHz–100 GHz. As a usage of the modulator in the cancelable optical circuit, there are two schemes:

(S1) Only one beam is modulated.

(S2) Both the beams are modulated by using two modulators.

Obviously, an insertion of the modulator on the path causes several fluctuations and results in imprecision. The selection of the modulator is one of the most significant factors in treating the cancelable optical circuit. So, the characteristics of the modulator have to be ascertained.

AOM is commonly used because of its highly stable performance. To understand why the frequency fluctuation occurs, the principle of AOM will be described briefly as follows and in Figure 10.5. Ferroelectric thin plates (LiNbO₃, etc.) are mounted on an acousto-optic medium (PbMoO₃, TeO₂, etc.) with a metallic cement adherently. An electrode is pasted on the plate by using the vacuum evaporation. When RF power (frequency f₁: 30–400 MHz) is added to the electrode, the plate distorts with a stationary wave form based on the frequency. The distortion propagates into the acousto-optic medium with an ultrasound (nearly) plane wave. In the medium, this compressional wave makes a diffraction grating. An incident optic beam (frequency f₀) is diffracted by the grating. The diffracted beam’s frequency is able to build up theoretically many types of f₀ + nf₁ (n = …, −3, −2, −1, 0, 1, 2, 3, …) (Bragg diffraction or Raman–Nath diffraction). Usually, the diffraction type n = 1 or n = −1 is used. In this diffraction process, the frequency fluctuation arises from the following points:

FIGURE 10.5  Structure of AOM. Ultrasound wave generates and propagates into medium.

1.  The ferroelectric plate is small: about 2 mm × 10 mm. Large-sized plates are difficult to produce because of the plate’s characteristics. This means that only an approximate (not ideal) plane wave only propagates.

2.  The metallic cement and the plate do not always have an ideal uniform thickness. Ideal distortion is not always produced.

3.  The RF power spectrum is broad. A high-frequency power source easily catches external noises from other instruments.

4.  A thermal distribution occurs by adding RF power (about 1 W) to the medium (the temperature of the medium rises by 5°−15°).

The frequency fluctuation of AOM will help selecting (S₁) or (S₂). These four points show that the frequency fluctuation cannot be taken away and cannot be discussed ideally. Therefore, (S₁) is not a desirable condition. To get merits in (S2), the following selection points are requested: (1) use of the same acousto-optic media, (2) use of the same production of a company, (3) use of one electric RF source that branches out two outputs, and (4) use of a cooler for the medium.

EOM is utilized with the Pockels effect or Kerr effect caused in the piezoelectric crystal (KDP, LiNbO₃, LiTaO₃, etc.). The merits are that the modulation frequency extends to 100 GHz, and the distortion of the wave front is smaller than AOM. The demerits are that the modulation is not always stable in a long time measurement, and a modulation rate (= diffraction efficiency) is smaller than AOM. In high-precision measurement, the low beat frequency described in the previous section is desirable. Therefore, EOM is not adopted in this chapter.

10.2.4    CONSTRUCTION OF THE CANCELABLE CIRCUIT: SYMMETRICAL LAYOUT IN THE CIRCUIT

In an optical circuit, several fluctuations also occur as follows: (1) variation of measuring length, (2) disorder of wave front, and (3) fluctuation of beam direction. To fully resolve these issues, the optical circuit has to use a good layout for two beams. In the layout, attention must be paid to the selection of mechanical parts and optical parts because of the following issues. For point (1), these are the following causes:

1.  HeNe laser beam varies its direction slightly and slowly.

2.  Refractive indexes of optical parts and surrounding air have temperature dependence.

3.  The effect of the surface roughness of optical parts depends on the beam diameter.

4.  Mechanical parts expand and contract thermally.

For (2), these are the following causes:

1.  The distortion (including optical rotation) of the optical isolator (which is not always used, but its use is desirable to suppress the frequency fluctuation of the laser) and optical parts.

2.  The medium of the optical parts (usually quartz, fused silica, or BK7 glass) shows rarely irregularity of the refractive index, which has polarization dependence.

3.  A mismatch of the polarization direction between the beam and the optical parts (especially, wave plate and PBS). An oblique incidence and multi-reflection usually appear.

For (3), these are the following causes:

1.  A fluctuation of HeNe laser beam’s direction, but this has been minimized by the manufacturers.

2.  Mechanical parts relax its configuration from the setup position very slowly, so that the reflection points and directions in the layout are shifted slowly.

By understanding that all parts have many defects and do not have ideal characteristics, a suitable means should be conceived to avoid the issues given earlier. In conclusion, the cancelable circuit requires the following points: (1) all parts are used as a pair, (2) a symmetrical arrangement and layout are taken in the circuit, (3) all parts are compact and layout is simple, and (4) optical path length is as short as possible. These are based on the fact that the same parts affect the measurement with the same factor, and the relation ΔLA₁ =ΔLA₂ in Equation 10.6 is the most desirable. In addition, the cancelable circuit is based on an accurate knowledge of optical path length difference (OPD).

10.2.5    RELATION BETWEEN THE DOPPLER METHOD AND THE HETERODYNE METHOD

Reflection light from a moving target shifts its frequency (Doppler effect). In the Doppler method, the displacement can be calculated by integration with the shift value and the measuring time.

Considering the effect, Equation 10.5 is modified and described as a function of time as follows:

$\begin{array}{l}{\text{SIG}}_1\left(t\right)=C_1\sin \left(\left({\text{ω}}_{\text{A1}}\left(t\right)-{\text{ω}}_{\text{B1}}\left(t\right)\right)t+{\varphi }_{\text{A}}-{\varphi }_{\text{B}}\right),\hfill \\ {\text{SIG}}_2\left(t\right)=C_2\sin \left(\left({\text{ω}}_{\text{A2}}\left(t\right)-{\text{ω}}_{\text{B2}}\left(t\right)\right)t+{\varphi }_{\text{A}}-{\varphi }_{\text{B}}\right).\hfill \end{array}$

The phase shift for time variation from t to t + Δt is given within the first-order approximation by

$\begin{array}{l}{\phi }_1\left(t+\Delta t\right)-{\phi }_1\left(t\right)-\left({\phi }_2\left(t+\Delta t\right)-{\phi }_2\left(t\right)\right)\hfill \\ =\left({\text{ω}}_{\text{A1}}\left(t+\Delta t\right)-{\text{ω}}_{\text{B1}}\left(t+\Delta t\right)\right)\left(t+\Delta t\right)-\left({\text{ω}}_{\text{A1}}\left(t\right)-{\text{ω}}_{\text{B1}}\left(t\right)\right)t\hfill \\ -\left({\text{ω}}_{\text{A2}}\left(t+\Delta t\right)-{\text{ω}}_{\text{B2}}\left(t+\Delta t\right)\right)\left(t+\Delta t\right)-\left({\text{ω}}_{\text{A2}}\left(t\right)-{\text{ω}}_{\text{B2}}\left(t\right)\right)t\hfill \\ =\left[{\text{ω}}_{\text{A1}}\left(t\right)-{\text{ω}}_{\text{B1}}\left(t\right)-{\text{ω}}_{\text{A2}}\left(t\right)+{\text{ω}}_{\text{B2}}\left(t\right)\right]\Delta t\hfill \\ +t\left[\frac{\text{d}{\omega }_{\text{A1}}\left(t\right)}{\text{d}t}-\frac{\text{d}{\omega }_{\text{B1}}\left(t\right)}{\text{d}t}-\frac{\text{d}{\omega }_{\text{A2}}\left(t\right)}{\text{d}t}+\frac{d{\omega }_{\text{B2}}\left(t\right)}{\text{d}t}\right]\Delta t\hfill \end{array}$

As a simple example, it is assumed that only LA₁ moves. The total phase shift Φ(t₁, t₂) from t₁ to t₂ is given by

$\Phi \left(t_1,t_2\right)={\displaystyle \underset{t1}{\overset{t2}{\int }}\text{d}t}\left[{\omega }_{\text{A1}}\left(t\right)+t\frac{\text{d}{\omega }_{\text{A1}}\left(t\right)}{\text{d}t}-{\omega }_{\text{A20}}\right]={\omega }_{\text{A1}}\left(t_2\right)t_2-{\omega }_{\text{A1}}\left(t_1\right)t_1-{\omega }_{\text{A20}}\left(t_2-t_1\right),$

where ω_A20 is constant because the target A₂ does not move. When the average speed of the target A₁ is V, the relation, ω_A1(t₂) = c/(c + V) × ω_A1(t₁), is satisfied. And by substituting the initial condition, ω_A1(t₁) = ω_A20, Φ(t₁, t₂) is given as

$\Phi \left(t_1,t_2\right)=2\pi \frac{V\left(t_2-t_1\right)}{{\lambda }_1}=2\pi \frac{\Delta L}{{\lambda }_1}.$

(10.10)

Equation 10.10 coincides with the result based on the Doppler effect.

When considering these relations, it is understood that the heterodyne measurement bases on the difference between the phases at the start time and at the stop time, and it calculates directly the displacement. In the Doppler method, a displacement is indirectly calculated through an integral of the target velocity.

10.3    ACCURACY AND NOISE REDUCTION

It was described earlier that the cancelable circuit is essential in the heterodyne method. In this chapter, several significant factors that affect measuring accuracy are shown, and some countermeasures against these factors are discussed.

It seems that an errorless signal processing and a noiseless phase variation will give the highest accuracy and the highest resolution for measurement. How high can a resolution be achieved in a real circuit? How high can a laser frequency’s stabilization be achieved with commercial instruments? How low can noise be suppressed in a real electric circuit? To obtain better results, the level or grade to which a real system can respond to the demands of these questions must be considered. The answers will reveal the limits of the accuracy, the resolution, and the selection of the optical circuit and the electric circuit.

In this section, several leading factors are discussed. For other factors omitted in this section, other reports have discussed with the miniaturization of the circuit [24], the parts selection [25], the symmetry layout [26], the introduction of the fiber [27], the nonlinearity [28,29,30], the path length [31,32], the structure [33], the three AOM circuit, and the signal processing [34,35,36].

10.3.1    DETERIORATION OF ACCURACY CAUSED BY THE ENVIRONMENT

The refractive index of air depends on temperature, pressure, vapor pressure, constituents, convection of air, etc. When beams propagate in air, the optical path length is changed by the refractive index, that is, by these factors. The refractive index of air has been proposed in some papers [37,38,39,40,41], for example, Edlén’s refractive index formula shown by

$n-1={\left(n-1\right)}_{\text{st}}\frac{0.00138823\times P}{1+0.003671\times T},$

(10.11)

where

n is the refractive index of air

n_st is the refractive index of air on the standard condition

P is the pressure in Torr

T is the temperature in °C

The value of n_st (including 0.03% CO₂) for HeNe laser (wavelength ≃ 633 nm) is 1.0002765. (The dependence on a vapor pressure is discussed in other papers.) Equation 10.11 indicates that variations of temperature and/or pressure change an optical path length and result in virtual displacement as if a target moves. When temperature changes by just 1°C, the refractive index changes only 0.00000038. When a geometric path length is 3 cm, the virtual displacement is 11.5 nm. If high-precision measurement is desired, this serious problem cannot be neglected. The effort of measuring the phase with a high resolution disappears ineffectively. To avoid this problem as effectively as possible, the optical system needs to satisfy the following points:

1.  Optical path length is short to the best of the system’s ability.

2.  Strict control of temperature and humidity in the laboratory. When the weather is bad or unstable, measurement should be postponed until the atomic pressure is stable.

3.  Conditions of the cancelable optical circuit, LA₁ = LA₂ and LB₁ = LB₂ (i.e., OPD = 0), should be satisfied as the first priority. Even if the relation LA₁ − LA₂ = 1 mm is set, it is not sufficient. This is because the change in atomic pressure from 1013 hPa (standard pressure) to 1033 or 980 hPa yields the virtual displacement of 5.4 or −9.1 nm, respectively.

4.  When higher accuracy is desired, the optical circuit has to be set in a vacuum chamber.

10.3.2    DETERIORATION OF ACCURACY CAUSED BY OPTICAL PARTS

In optical circuits, many optical parts are used, but, unexpectedly, some of them bring about desirable results to the circuits. When a designer handles them, he must understand their characteristics satisfactorily. Related to these characteristics, one of the reports shows that nonlinearity in a measurement system occurs also from nonideal polarization effects of the optics [42].

10.3.2.1    AOM and Its RF Driver

The mechanism and the structure of AOM were described briefly in the previous section. The medium of AOM is crystal or glass (the amount supplied is not sufficient) as shown in Table 10.2. PbMoO₄ and TeO₂ are the main mediums used recently, and a GaInSe crystal in the infrared region [43] and a liquid crystal [44] have been proposed.

As the crystal has a crystallographic axis, it does not always have an isotropy for the polarization. This anisotropy results in some limitations for an incident light’s polarization and the other conditions. The angle of diffraction χ satisfies the following relation approximately:

$\chi \cong N\lambda \frac{n{f}_{\text{d}}}{V},$

(10.12)

where N, λ, n, f_d, and V are the order of the diffraction, the wavelength, the refractive index of the medium, the frequency of the RF driver, and the acoustic velocity of the medium, respectively. In two AOM systems that have the RF frequencies f_d1 and f_d2 (≠ f_d1), the designer must recognize that the difference between the angles may yield an unexpected OPD.

Is the sentence of “the interference of two beams produces the beat frequency |f₁ − f₂|” always right? The sentence assumes that (f₀)₁ = (f₀)₂, which means that the laser frequency before the diffraction of the first beam is equal to the laser frequency before the diffraction of the second beam. The frequency of the frequency-stabilized laser (even if Δf/f = 10⁻⁹) fluctuates by Δf = 4.74 × 10¹⁴ × 10⁻⁹ = 474 × 10³ [Hz]: this is not small compared to the RF frequency (= about 80 MHz). When making device arrangements that satisfy this sentence, the following points should be watched and should be observed.

TABLE 10.2
Characteristics of AOM Materials

Notes:  Interaction strength coefficient η in AOM is given by

$\text{η}=\frac{{\pi }^2}{{\cos }^2\text{θ}}{M}_{\text{c}}{P}_{\text{a}}\frac{L}{H}\frac{1}{{\lambda }^2},$

where θ, M_e, P_a, L, H, and λ are Bragg angle, figure of merit, applied acoustic power, transducer length, transducer height, and wavelength, respectively.

Refractive index: No = the index for ordinary ray, Ne = the index for extraordinary ray.

Acoustic velocity: L = longitudinal mode, S = slow shear mode.

LiNbO₃ and LiTaO₃ are hardly used. AOT-40 and AOT-5 were made but are not produced now.

1.  How long are the sampling time and the measuring gate time (= detecting time gated for one signal)? In the case of the gate time interval of 10 ms or more, the fluctuations of HeNe laser and RF driver on the beat signal can be eliminated [45] by an averaged calculation.

2.  Are the beam position (including incident angle), polarization direction, and beam radius that are written in the AOM manufacturers’ manual kept? These points affect the AOM operation.

3.  Is stable cooling designed in the surrounding area of AOMs? Because RF power is about 1 W and the heat generated in AOM changes the refractive index of the medium and distorts the wave front. The change results in the fluctuation of the beam as known from Equation 10.12.

4.  Use one RF driver that makes two RF frequencies simultaneously. Only one oscillator should be used on the driver. For the beat signal, the frequency fluctuation is canceled.

5.  Connect AOMs to RF driver as short as possible with coaxial cables. The cable leaks a high-powered RF frequency noise. The leaks affect the detectors and equipment directly, and a virtual beat signal sometimes happens in spite of no optical signal.

6.  Do not use the beat signal generated by the two RF frequencies as a reference signal. The signals have to be generated through the interference of the optical beams.

10.3.2.2    Frequency-Stabilized HeNe Laser

In the previous sections, it was proven that the laser frequency has to be stabilized. The available products are a solid laser and a frequency-stabilized HeNe laser [46] (including I₂ absorbed type [47]). The latter (not including I₂ absorbed type) has better cost performance than the former. Although attempts have been made to stabilize semiconductor lasers, satisfactory results have not been obtained yet.

A free-running type of HeNe laser has a stability of Δf/f ≈ 10⁻⁶, which is the best stabilization value under a free-running operation. However, the value is not enough for high-precision measurement. The commercial frequency-stabilized HeNe laser has about Δf/f ≈ 2 × 10⁻⁹ [48]. On a manufacturer side, the value is upper limit under a normal control.

Why does the limitation arise? Laser operation occurs within a frequency band induced by a gain curve that is calculated with the energy levels of He and Ne gases. Output beam of HeNe laser has only a few longitudinal modes; the frequencies of these modes are restricted by a stationary wave condition for the laser cavity length. The frequencies of the modes fluctuate with the variations of the cavity length and of the refractive index of He and Ne gases. Under the assumption that the refractive index does not vary and 30 cm cavity length and 632.9913 nm wavelength, the wave numbers of the stationary wave, 47393, 47394, and 47395, are valid. These values correspond to the cavity length 29.999356, 29.999989, and 30.000622 cm, respectively. If the cavity length coincides with one of them, the wavelength is recalculated with the cavity length and the stationary wave condition. A 30.000000 cm cavity and the number of stationary wave 47394 require a wavelength of 632.9915 nm. A shrinkage of only 11 nm cavity length yields a variation of 0.0002 nm in wavelength, which corresponds to Δf/f ≈ 3.16 × 10⁻⁷. Laser makers achieved a value of about 2 × 10⁻⁹ using their original techniques.

Is this stability of 2 × 10⁻⁹ sufficient for high-precision measurement? The answer is no. Therefore, as the second best solution, a countermeasure must be found under the assumption that this unsatisfactory laser is used. A frequency fluctuation is usually random. In spite of the random characteristics, for short gate time t_G (= length of the detector’s opening time in which one signal is outputted), the fluctuation will not always be small by integrating signals because it depends on the detector’s characteristics. So, the sampling time t_S (= integration time of the output signal = time interval of output data) controls the result. A short t_S increases the time resolution but worsens the S/N ratio of the data.

Multiple longitudinal modes are usually simultaneously outputted. Only the largest one is used, and the others must be cut. If the transverse mode is multiple, only TEM00 mode should be utilized because an unexpected interference shifts the beat frequency.

A laser is very weak against a back reflection light. An optical isolator [49] should be inserted perpendicular to the path after the laser. The isolation of 30 dB or more is desirable.

Attention should be paid to the following points:

1.  Maintain the high-frequency stability and do not add any thermal influence. To protect the laser, a cover made of an insulator should be used.

2.  Check the longitudinal mode. When it is multimode, only main mode is used with a suppression ratio of 20 dB or more.

3.  Maintain an extinction ratio of 25 dB or more. The optical isolator should be used to suppress the frequency fluctuation.

10.3.2.3    Interference Circuit and Parts

In optical circuit, the intensity and the polarization of the beam are controlled mainly with several optical parts, mirror, PBS (polarized beam splitter), NPBS (nonpolarized beam splitter), wave plates, corner reflector, optical fiber, polarizer, and lens. These parts change the characteristics of the beam. When designing the circuit, attention should be paid to the following three points:

1.  Suppress the direction fluctuation of the optic axis.

2.  Match the polarization with an extinction ratio of at least 25 dB.

3.  Notice temperature dependences of the refractive index of the parts.

The direction fluctuation occurs mainly from the laser beam’s fluctuation and is sometimes based on the flatness of the parts and a relaxation of the mechanical parts. To suppress the laser beam’s fluctuation, the use of a fiber collimator [27,50] may be a good idea because the output hardly fluctuates.

These optical parts are made mainly from crystal, quartz, fused silica, and glass as the medium [51]. They are usually polished surfaces with a flatness of λ/20–λ/200. The coating film of the parts is mainly a metallic monolayer or a multilayer of a dielectric substance that is manufactured using vacuum evaporation or similar method. The transmission factor of the antireflection coating is about 99% at best, and the maximum reflectivity is about 99%. All parts have a distortion that differs slightly from the ideal form, for example, a perpendicularity of 90°± 1” for the best quality. The refractive index and other characteristics are listed in Table 10.3. Further, quartz has the optical rotatory power. That is to say, no optical parts are ideal. Therefore, for the optical parts, the qualities of the parts should be enhanced carefully as much as possible.

Mechanical parts also have fluctuations in the optical beam direction. The fluctuations are usually larger than that for the optical parts. The main reason for the fluctuations is a heat flow. The heat is produced in active parts: for example, AOM and laser, and is added to every part in the circuit. If the heat does not flow, the influence is too small because a thermal balance rises and disorders stop. Therefore, for the mechanical parts, the suppression of the fluid is the most important scheme for high-precision measurement.

10.3.2.4    Sample Surface and Sample Holder

Only reflected light has information of a target sample. Therefore, the detection method and detection device decide the accuracy and the S/N ratios of the measuring signals. Attention should be paid to the flatness and/or the roughness of the surface because they can easily change the optical path length and result in scattered light. To obtain high S/N ratios, much scattered light must be condensed. To get good condensation rate, attention should be also paid to a kind condensation lens, NA value of the lens, the incident beam radius, and the period of the roughness. Particularly when the roughness is bad, the surface should be improved by repolishing or depositing metal.

TABLE 10.3
Characteristics of Optical Glass

Notes:  Assignment of N-BK7, N-PK52A, N-SF6, N-SF4, and N-LAF3 are based on Schott Glass, Inc. The terms of (c║) and (c ⊥) show that the optical axis is parallel and perpendicular to c axis of the crystal, respectively. In optical isolator, Faraday media are mainly YIG (Yttrium Iron Garnet) or TGG (Terbium Gallium Garnet). Reflection rate at λ = 633 nm of the metal material coated with the vacuum vapor deposition: Al: 90.8%, Au: 95.0%, Ag: 98.3%, Cu: 96.0%.

When the sample is gas or liquid, the material and characteristics of a holder or a cell should be selected carefully to maintain the beam quality. Especially, a thin glass plate brings about multiple reflections from both the front and the rear of the plate.

10.3.2.5    Detection Method and System

The displacement resolution is determined by a phase resolution of the beat signal, but laser noises and electric noises in signal processing (including detector) worsen the quality. The former noises are discussed in the earlier sections. For the latter noises, there are two dominant factors: (1) a photocurrent fluctuation induced by a laser amplitude modulation and (2) characteristics of the electric circuit including memory, processor, amplifier, and filter. To reduce them, (1) the detector should have an extremely wide temporal bandwidth and a highly linear response and (2) the circuit should have a low shot noise and a high S/N ratio with an electric shield. Photodiode (PD) and avalanche photodiode (APD) are usual used as the detector. A larger frequency band needs a smaller detector diameter. To decrease the diameter, a severe condensation of the beam and a small direction fluctuation of light are required. The frequency band of the detector, Fb [MHz], satisfies the following relation with the beat frequency fb [MHz], and the displacement resolution Dr [nm], roughly (HeNe laser use).

$Fb>fb\frac{633}{Dr}.$

(10.13)

This condition gives a limit for Dr. Therefore, the selection of the detector has to take many factors given earlier into consideration. In addition, some detectors depend on polarization. In using them, it is desirable for a wave plate of λ/4 to be inserted in front of the condensed lens.

10.4    ATTENTION FOR STATIC AND DYNAMIC MEASUREMENT

10.4.1    STATIC MEASUREMENT

In this case, the sample hardly moves (almost stationary). All variations drift slowly with a long span. Control of these drifts is absolutely essential. The cancelable optical circuit with zero OPD is required of necessity. Attention must be paid to the following points:

1.  Control of the laboratory’s temperature, pressure, humidity, and sound

2.  Prohibition of all movement (including air convection) near the measurement system

3.  Temperature control of the basement of the optical circuit

4.  Noise reduction in the AOM’s RF driver and in the power source

10.4.2    DYNAMIC MEASUREMENT

In this case, sample vibrates and/or translates with the function of time. However, the defect of the heterodyne method (= defect 1) limits for the sampling time t_S, for which the following relations are required:

$\begin{array}{ll}\lambda >F\left(t+t_{\text{S}}\right)-F\left(t\right)\hfill & \text{for all }t\text{ in}\text{a straight measurement case}\text{.}\hfill \\ \raisebox{1ex}{$\lambda $}\!\left/ \!\raisebox{-1ex}{$2$}\right.>F\left(t+t_{\text{S}}\right)-F\left(t\right)\hfill & \text{for all}t\text{ in}\text{a reflection measurement case}\text{.}\hfill \end{array}$

(10.14)

These cases are classified by the relations between the detector and the sample. The latter case has double the resolution of the former case. Considering “which case is effective” is important for designing the optical circuit and the other factors.

In addition, the relation between the gate time t_G and the sampling time t_S is a key point. Considering three restrictions, (1) Equation 10.14, (2) t_S ≥ t_G, and (3) t_G conditions described in Section 10.2.2, both the time resolution and the displacement resolution are varied complementally. Especially, for short t_S, the following points are required:

1.  High-frequency stability for both the HeNe laser and the RF driver

2.  Small diameter for the detector

3.  High-frequency band in the electric signal processing

10.5    APPLICATIONS

Basically, the physical variable measured in the heterodyne method is a displacement. By considering why the displacement occurs, the displacement result can be applied to the detection of other physical variables.

The displacement does not always originate in the movement of the sample: to be accurate, it originates in a change of the optical path length. This change also occurs because of the variation of the refractive index of the path. So, when a gas, a liquid, a crystal, etc., are set on the path, their characteristics can be investigated such as gaseous pressure, kinds, sort, temperature, disorder degree, anisotropy effect, birefringence index, and optical rotatory power. By extending these ideas, in many fields of thermodynamics, hydromechanics, electrodynamics, magnetohydrodynamics, plasma physics, and statistical mechanics, the heterodyne method will be utilized as one of superior metrology. This method is effective even for dynamic measurement within an extremely short time. For example, with a femto-second pulse laser, a two-photon absorption process is also reported [52].

The focusing of the laser beam with a microscope [53,54,55,56] can be used to measure the displacement in an extremely confined region of 1–30 μm. By accumulating an AOM and an optical path on a crystal, several miniaturized circuits are demonstrated [57,58,59,60,61].

Some companies commercially produce many instruments and systems that base on the heterodyne method: for example, a displacement meter, a vibrometer, a polarimeter, a profilometer, a phasemeter, many typed optical sensors (including fiber sensor), a laser radar, a micro-scanning system, and other interferometers.

10.5.1    DISPLACEMENT, VIBRATION FREQUENCY, AND AMPLITUDE

Displacement measurement is achieved with ultrahigh resolution [62]. Several reports point out a possibility of sub-nm or pm order resolution under laboratory conditions [63,64].

Vibration amplitude and frequency are directly measured and calculated through an integration of the displacement. A combination of heterodyne method and Doppler effect method is effective for a synchronous measurement of variations because the Doppler effect measures the frequency directly. A piezoelectric transducer vibrates within 10 kHz to 10 GHz. For high-frequency vibration, the Doppler method is effective under the heterodyne detection. For low-frequency vibration, the heterodyne method gets superior results.

By utilizing this basic displacement measurement, the temperature of unburned gases is measured by its refractive index variation as a function of a density, a concentration, a type, and a Gladstone–Dale constant of gas [65,66].

With superluminescent diode, 3D measurement is proposed in the full-field heterodyne white-light interferometry and the full-field step-and-scan white-light interferometry.

10.5.2    POSITIONING WITH STAGE

Mechanical stages are utilized to measure the shape and size of a 3D structure sample. By translating and rotating the sample by the stages, a displacement is apparently generated. Therefore, the resolution is determined by qualities of the stages, combination, backlash, lost motion, and electrical feedback control of stages, which seriously affect the reproducibility and linearity. In using a microscope or condensing lenses mounted on the stages, the alignment error between optical axes of the lenses and those of the probe lights has to be compensated for even if the stages move [67]. Also, when the focal length and the focal depth of the lens do not match the surface shape, the signal quality and the resolution get worse. In particular, in the field of the optical lithography and the x-ray lithography, an extremely high-grade stage is required [68,69].

10.5.3    THICK MEASUREMENT

Thin film is usually measured by the ellipsometry. By using the heterodyne method as shown in Figure 10.6, the thickness of the film is measured. The incident angle, thickness, and refractive index of the film decide the OPDs of the two beams that are reflected at the top surface and the bottom surface of the film. By analyzing the relation of the OPD vs. the incident angle, the thickness is obtained. A 300 μm thick amorphous silicon is investigated with a beat frequency of 5 kHz [70].

10.5.4    PROFILOMETRY

Flatness, roughness, and structure on a sample surface (namely, quantitative 2D surface profile) are usually measured by two means: (1) electromechanical beam scanning and (2) camera detection of the expanding beam. The resolution in a direction perpendicular to the beam (transverse resolution) is about 1 μm to 1 mm, which is much larger than the resolution in a direction parallel to the beam (longitudinal resolution). The limitation of the transverse resolution is based on the beam size and the wave front quality.

FIGURE 10.6  Measurement of thickness of filmed sample. Incident lights have different wavelengths for two polarizations. The f₀ light reflects at the sample surface, and the f₀ + f_B light reflects at the base surface. By interfering two beams with two detectors, the sample thickness is analyzed with varying angle θ as a parameter.

For means (1), the laser beam diameter is small: about 0.3–2 mm. (When the microscope is used, the beam waist is about 1–30 μm.) By shifting the beam or moving the target with stages, a displacement distribution in a broad area of the target can be measured. The transverse resolution is limited by the beam diameter and the stage shift control. The performance of the stages and the sensitivity in the shift operation severely affect profile measurement. Several instruments and systems that use with this means are sold commercially.

For means (2), a laser beam is expanded, and a CCD camera is used. The resolution and the measurable region are limited by the size of the CCD’s pixels, the number of pixels, and the focus lens system. A tunable diode laser is also used [71]. A normal photo-detector is used instead of CCD [72].

Other means are also proposed. Detection, scanning, and signal processing have been investigated as a function of beam radius, pixel size, and stage operation [73]. By using a birefringent lens that provides different focal lengths for two orthogonal polarization eigenstates (p and s), the surface profile is reported as a function of phase variations of p and s waves [74]. The surface of an aspherical lens is also investigated with two laser diodes (810 and 830 nm wavelength) [75].

10.5.5    REFRACTOMETRY

An optical path length depends on the refractive index of the path. In particular, the refractive index of air negatively affects high-precision measurement as described in Section 10.2.1. Inversely, by measuring the variation of the length, a refractive index of the sample or several factors hidden by the index, for example, density, pressure, temperature, absorption coefficient, and other physical quantities, are obtained. Therefore, strict conditions must be controlled for whole of measurement system.

For a liquid sample, a refractive index and its thermal coefficient are measured with a phase variation of a probe light passed through a sample. With this principle, the refractive index of the liquid is measured [76].

For a gas sample, more attention is needed. The reason is that the refractive index of gas is 1.0001 (at 1 atm, 15°C) at most, and the refractive index of liquid is at least 1.2. The sample is filled with a cavity, and the cavity is deformed or distorted by a pressure difference between inside and outside. The deformation yields some errors. In particular, the thickness, flatness, and homogeneity of the cavity are important. Therefore, the quality of the cavity should be selected cautiously using the Young modulus and the thermal expansion coefficient of the cavity. Gas density is measured in a high dynamic range of 0.5–700 Torr [77]. As a cavity, Zerodur^® glass is used and investigated [41]. Fiber sensor-typed interferometer is used to measure gas density in an engine [78].

10.5.6    PHOTOTHERMAL INTERFEROMETRY

Thermodynamics shows that heat diffuses until a thermal equilibrium is obtained as a function of the sample’s characteristics and thermal conditions. So, when heat is added to the sample locally, heat propagates in the form of a heat wave. The propagation wave brings about a distortion of the sample. The distortion depends on the diffusion coefficient, viscosity coefficient, compressibility, expansion coefficient, density, etc. Therefore, these physical values can be investigated by measuring the distortion. For example, a high-power “pump” laser (CO₂, Ar, etc.) light and a low-power “probe” laser (HeNe) irradiate to a target sample. This sample heating modulates its characteristics and varies the optical path length of the probe light. By measuring the variation as a function of time, thermal characteristics (absorption, dissipation, energy level, structure, etc.) of the sample can be investigated. In [79], pump laser is Ar laser (514 nm wavelength and 1.5 W max), and a gas sample is confined with a cell of 5 cm in length.

When the target is solid, a thermal acoustic wave (ultrasonic wave) is induced. A longitudinal wave or a surface wave propagates in or on the sample diffusively and distorts the surface. The distortion’s distribution, amplitude, and their time dependence are obtained as a function of the crystal structure, conductivity, diffusivity, density, shear modulus, and Young modulus of the sample [80].

For a semiconductor target, by using a tunable and pulse laser as a pump laser, gain dynamics in a single-quantum-well laser are investigated [81].

10.5.7    THERMAL EXPANSION COEFFICIENT

The expansion coefficient of the sample is calculated by adding heat to the sample homogeneously and measuring the amount of the stretch or shrink of the sample. In particular, for an extremely low coefficient, the utilization of the cancelable optical circuit and a strict control of environmental conditions are indispensable conditions.

To exclude influences of sample pitching and rolling, ω₁ beam reflects on a sample twice and ω₂ beam reflects on a basement twice as shown in Figure 10.7. The OPD variation of the two beams is basically zero as long as the sample and the basement are a rigid body and the layout is symmetrical. The thermal expansion coefficient of a lithographic glass is measured with the resolution of ppb scale under 0.1°C temperature control [82].

For a FZ silicon single crystal oriented in the <111> direction, the coefficient is measured in a range of 300–1300 K [83].

FIGURE 10.7  Thermal expansion detection system. Fiber collimator’s output is divided into two paths. Two beams are modulated by AOMs whose wavelengths are λ_A and λ_B. Beams pass through complex path and finally focus at SIG. detector for P wave and at REF. detector for S wave. On the P wave path, the beam reflects twice at the sample and the basement. Influence of pitching and rolling of them is canceled. And influence of the basement can also be canceled.

10.5.8    YOUNG’s MODULUS OF THIN FILM

By adding pressure to a thin film sample, the film bends. The relation between the pressure and the distortion gives a Young modulus of the film with the relation [84] of

$p=\frac{t\sigma }{r^2}h+\frac{8tE}{3r^4\left(1-\text{ν}\right)}{h}^3,$

(10.15)

where

h is the distortion at the center of the film

p is the difference of the pressures between the front and the back of the film

E, r, t, σ, and ν are the Young modulus, the diameter, the thickness, the internal stress, and the Poisson ratio of the film, respectively

Using a displacement meter and a microscope, the measurement system is constructed as shown in Figure 10.8 [85]. A vacuum chamber connects to a vacuum pump, and a sample is placed on the upper surface of the chamber. The pressure in the chamber decreases slowly, and the sample distorts. The displacement meter measures the distortion at the center of the sample. A curve fitting analysis using Equation 10.15 gives σ and E/(1 − ν).

10.5.9    BIREFRINGENCE INDEX AND POLARIMETER

Some of crystal, liquid, organic compound, sol, gel, and plasma show birefringence. For two orthogonal polarization eigenstates’s lights (p and s waves), their refractive indexes are different. Therefore, the optical path length of p wave is slightly different from that of the s waves. By utilizing the difference, the birefringence of the target is measured as shown in Figure 10.9. Two linear polarization lights (wavelengths λ₁ and λ₂) exactly adjust their polarization directions to the eigenstates’s directions of the sample. The adjustment and the extinction ratio of the lights are the main factors that determine the accuracy of the result [86,87,88]. The analysis of the beat signal’s phase gives the birefringence. This basic principle is applied to several fields, and some commercial instruments have been proposed. Also, the polarization dependence of mode dispersion and power loss has been investigated [89].

FIGURE 10.8  Measuring system for the Young modulus of thin film. A sample is set on a vacuum chamber that connects to a vacuum pump and a vacuum gauge. Slowly, the operation of the pump makes a negative pressure region in the chamber, and the pressure variation adds a stress to the sample. As a result, the sample distorts. A displacement meter on the microscope measures the distortion at the center h. By analyzing the relation of h and the pressure with Equation 10.15, the Young modulus is derived.

FIGURE 10.9  Measurement of birefringence.

Two orthogonal circular polarization lights are used to investigate plasma [90]. A very high extinction ratio of about 10⁻⁹ results in a very high angle resolution of about 10⁻⁷ rad [91]. Birefringence appears on a mirror of high-finesse (38,000) optical cavity; a distortion based on the photorefractive effect is reported [92].

10.5.10    FIBER SENSOR

Fiber sensors are classified into two types: (1) a fiber that is used as a flexible optical guide to a target and (2) a fiber that is used as a long and flexible detector. In the former sensor type, two means are investigated: (1) a sensitive material is set just in front of the fiber and (2) a sensitive material is coated or deposited onto the fiber tip. The selection and construction of the material are key points for obtaining better sensors. In a hydrophone sensor, PVDF, PET [93], and Parylene film [94] are used as the material. By detecting a change in the polarization or a variation in the optical path length, a pressure range of 10 kPa is measured with 25 MHz bandwidth [93]. By detecting a rotation of the polarization of the light passed through the Faraday element [95,96], the current and magnetic field are sensed with a resolution of about 2 G [97]. A near-field fiber-optic probe [98] has also been proposed in which the distance between a sample and the fiber tip is about 1 μm [55].

10.5.11    DYNAMICAL SURFACE MEASUREMENT

For a liquid surface, a longitudinal wave or a surface wave is generated. Hydromechanics can clarify their motions with the equation of continuity, the wave equation, Helmholtz’s equation, and other equations. The group velocity, amplitude, and frequency of the waves depend on sample’s characteristics. Therefore, their characteristics are analyzed by a measurement of the wave’s motion with these equations. For the signal processing, a Fourier-transform analysis is usually used (it is called Fourier-transform heterodyne spectroscopy) [99]. Taylor instability in a rotating fluid is also reported [100].

10.6    OPTICAL DOPPLER MEASUREMENT

10.6.1    INTRODUCTION

The optical Doppler method is a noncontact method based on the Doppler effect. (The effect occurs in all waves, sound, ultrasound, surface wave, electromagnetic wave, and all waves having a finite velocity. But in this chapter, only light is discussed as a subject.) For the light, the effect should be discussed fundamentally with the relativity theories. However, a usual velocity of the sample is extremely less than the light velocity. So, the effect can be analyzed by the classical mechanics for usual system.

There are two patterns in the Doppler shift: (1) a sample itself is a light source and the sample moves relatively against a detector, and (2) a light source is fixed and the light reflects on a sample surface that moves relatively against detector. Many instruments and systems are based on the second pattern.

In this section, the following articles are discussed briefly: the principle and defect of the Doppler method and the comparison with the optical heterodyne method. The Doppler method is utilized for several fields, such as astronomy, flowmetry [101], velocimetry [102], vibrometry, astronomy [103], Doppler lidar [104], medical operation [105], and optical CT [106].

10.6.2    PRINCIPLE AND DEFECT

When a light source, a sample, and a detector are arranged two dimensionally as shown in Figure 10.10 and the sample moves with a velocity V, the Doppler effect shows that the detected light shifts its frequency (i.e., wavelength) from that of the source light. Under the condition of V ≪ c, the wavelength shift Δλ (= Doppler shift) is represented approximately by

$\frac{\Delta \lambda }{\lambda }=\frac{V}{c}\frac{\sin \left(\text{θ}+\varphi \right)}{\sin \varphi }\left(1+\cos \phi \right)$

(10.16)

where

λ, V, c are the wavelength of the source light, the velocity of the sample, and the velocity of the light, respectively

θ, φ, ϕ are angles shown in Figure 10.1

FIGURE 10.10  Doppler effect—the relation of light direction, sample moving direction, and detector position.

The frequency shift Δf is usually observed in the range from 100 [GHz] to 10 [Hz] and gives the velocity V in the range from 100 [km/s] to 10 [μm/s]. By measuring Δf absolute V is obtained as a function of time.

However, Equation 10.16 shows a defect obviously.

Defect: In the cases of (1) θ + φ = 0 or π, (2) φ = 0 or π, and (3) 1 + cos Φ = 0, Δf cannot be detected.

As V can directly represent with Δf, you may misunderstand that a modulator (AOM or EOM) is not used. However, in a real electric circuit, 1/f noise is a critical problem for detection in a low-frequency region. To avoid the problem, a modulator (AOM or EOM) is usually inserted into the optical circuit. As a result of this insertion, a zero velocity gives a nonzero-frequency shift. The fluctuation of the shift mainly determines an influence in the accuracy of the Doppler method. By having the same discussion in the heterodyne method section, the fluctuation in one-AOM system is worse than that in two-AOM system. However, the two-AOM circuit (beat frequency = f₁) limits the measurable region of the velocity for moving that the target comes close to the detector.

10.6.3    COMPARISON BETWEEN DOPPLER METHOD AND OPTICAL HETERODYNE METHOD

10.6.3.1    Accuracy

As the Doppler method directly measures the frequency shift, the accuracy of the measurement is determined by (1) laser frequency fluctuation, (2) modulation fluctuation, (3) frequency resolution in the detection system, (4) arrangement of the sample and the detector (accuracies of θ, ϕ, and Φ), and (5) flatness of the sample.

Even if the laser stabilization is Δf/f = 10⁻⁹, the frequency fluctuation is about 500 kHz. (See the optical heterodyne method section about the frequency-stabilized HeNe laser.) Even if the modulator (case of center frequency ≅ 80 MHz) fluctuation is Δf/f = 10⁻⁵, the frequency fluctuation is about 800 Hz. These fluctuations are too large noises to measure with a high resolution. To solve this problem, attention should be paid to the following points: (1) introduction of a cancelable optical circuit, (2) two detectors, (3) symmetric optical circuit, (4) low-noise electric processing, and (5) averaging of data with a sampling time interval under the assumption that these fluctuations are white noise. Long sampling time makes low noise, but makes low time resolution.

Optical parts and mechanical parts should be selected to obtain high precision by the same reasons mentioned in the optical heterodyne method section.

10.6.3.2    Measurement of Velocity and Displacement

Basically, the physical variable measured in the Doppler method is a velocity. Therefore, the method is applied when the target moves. This point is very attractive in the field of space engineering, automobile industry, high-tech industry, and medical engineering. However, when the target hardly moves, the method is seldom adopted.

In spite of this weak point, by combining with the heterodyne method, an ideal measuring system can be constructed for both a dynamic and a static measurement. For the two methods, many optical parts and electric circuit can be used in common. Therefore, by adding a few parts (including the processing) to the system, an expedient measurement system is built up. By using two methods simultaneously, both velocity and displacement are measured with high resolution.

The detection of the frequency, f, in the Doppler method gives a limitation for the length of the sampling time. The highest time resolution is 1/f. So, the beat frequency has to be selected by considering the time resolution, the maximum velocity of the target, and the optical circuit condition. However, the selection gives a restriction against the heterodyne method. Therefore, the two methods cannot operate simultaneously with the best condition. That is to say, either the heterodyne method or the Doppler method is the main measurement. When displacement is a target physical variable, the heterodyne method should be operated as the main measuring means, and the Doppler method should be treated as the auxiliary means.

10.7    MEASUREMENT OF TRANSIENT PHENOMENA

10.7.1    APPLICATION OF THE OPTICAL HETERODYNE METHOD

All physical phenomena are divided into two types: static phenomena and dynamic phenomena. In the static type, most of physical quantities are described with static equations or partial differential equations in which the variable of time does not include. Most of physical coefficients or physical constants have been investigated under the static condition theoretically and experimentally. Many suitable measurement methods have proposed as systems to observe the static physical response.

In the dynamic type, most of physical quantities are usually described theoretically with the partial differential equation where the time and positional coordinates are the independent variables. The observation methods for the dynamic response have been investigated. In the observations, the transient recording systems have been a barrier against the development of the measurement method. Especially, it was required to realize a measurement method having many characteristics of high-speed, fine variation, wide dynamic range, etc. Recent scientific technology and manufacturing industry technology extend means of the transient measurement method and the transient recording method. For the optical heterodyne measurement, by adding a transient recording function having high time resolution, many dynamic phenomena will be able to be measured with high time and high space resolution performances.

However, an exact measurement for the dynamical response is difficult. The reason is that any physical value interacts complicatedly with the other physical values, and their interactions among the values and their environmental conditions change with the increase in time. The time dependences of the interaction strengths will change the dominant effects operated among physical values. The dynamical response may not connect directly to the transient characteristics of the object. Moreover, the transient measurement system may distort the dynamic phenomena under the situation that several unexpected effects occur. For these problems, the optical heterodyne method can avoid by introducing the relative measurement method described in the previous section.

In the dynamic response, many partial differential equations for the physical quantity (e.g., expansion, diffusion, concentration, pressure, and elasticity) are described with differential coefficient of first or second order of time, so the equation suggests that the decay and/or the vibration appears in the motion of the object. However, all physical quantities are not always observable quantities, which can be appreciated directly. So, as the second best way measured these nonobservable quantities, the following way will be selected. First, search or check several observable quantities relating to the nonobservable quantity theoretically. Second, create or find a measurement method detecting the observable quantity. For example, Young’s modulus is not an observable quantity, but it is expressed with strain and stress. Both the strain and the stress are observable. Therefore, by detecting them transiently, the dynamic characteristic of the Young modulus is obtained.

The transient measurement involves one tough mechanical problem that both the measurement and the analysis must be done continuously at a very short time interval. In addition, the observable quantities must be measured with good signal-to-noise (S/N) ratio even if the spatial variation is very small. Generally, short time interval detection decreases the S/N ratio so that the spatial resolution is poor. However, the heterodyne method is a useful measuring system that keeps high spatial resolution even if the measurement time is very short.

For the transient measurement, the heterodyne method is classified broadly into two detecting ways: (1) detection of the reflected light from the sample and (2) detection of the transmitted light through the sample. Two types have characteristics as shown in Table 10.4. The reflection light detection and transmission light detection are effective to search the property of solid samples and liquid or gaseous samples, respectively.

10.7.2    ATTENTION TO THE DESIGN OF TRANSIENT MEASURING SYSTEM

The transient measuring system is classified roughly into two parts: an optical circuit and a processing electric circuit. Since the optical part is discussed in the previous sections, the processing part is studied in this section. First, to get high time resolution in the system, the following six points must be examined or designed carefully as important factors:

1.  Measuring target property (frequency, velocity, etc.)

2.  Output time interval τ_C: processing operation time on one of output data

3.  Data latched up time interval Δt: latching operation time on one of input data

4.  Measuring point N_F: output data points

5.  Basis beat frequency f_B0: the beat frequency formed by two outputs of the frequency shifter

6.  Recording specifications (the number of bits, memory type, transfer speed, and interface)

These points correlate with each other complicated manner. The designer of the measuring system should pay attention to the following relations (from (A) to (D)) to construct better circuit.

(A) Relation between the measuring target property and the limitation of the measurement

To design better system, the following properties of the target must be estimated correctly: (1) the maximum velocity V_m of the target and (2) the maximum displacement D_m within the output interval τ_C. When the estimation is over the limitation brought with only one essential defect of the heterodyne method, the design must be reconsidered. The defect is written by

<The reflection light detection case>

“When the displacement within τ_C is greater than the half of wavelength of measuring laser light, the displacement is uncertainty”

<The transmission light detection case>

“When the displacement within τ_C is greater than the wavelength of measuring laser light, the displacement is uncertainty”

TABLE 10.4
Characteristics of Reflection Light Detection and Transmission Light Detection

	Reflection Light Detection	Transmission Light Detection
Circuit
	The f1 light reflects at sample. The reflected light interferes with f2 light at detector. Both 1Ch and 2Ch set up the same circuit. But for 2Ch, a sample surface changes to a reference surface.	The f1 light passes through sample. The transmission light interferes with f2 light. Both 1Ch and 2Ch have the same circuit. But for 2Ch, a sample changes to a reference sample or an empty box.
Merit	Direct measurement of the variation on the solid sample surface with high accuracy.	Direct measurement of the variation of the refractive index of a semitransparency sample with high accuracy.
Demerit	Displacement component for the direction perpendicular to the incident light cannot be detected.	Distribution in the path cannot be detected. Large scattering particles become worse S/N ratio.
Example of physical variables	Elastic modulus, pressure, film thickness, vibration, piezoelectric modulus, linear expansion, volume expansion, dielectric constant, sound pressure, etc.	Density, viscosity, compressibility, diffusion, concentration, longitudinal wave strength, dielectric constant, sound velocity, rotatory, solubility, etc.
Example of measurement	Tuning fork and cantilever vibration (Section 10.7.4.1).	Concentration fluctuation (Section 10.7.4.2).

Notes:  The f₁ light and the f₂ light have the frequency of f₁ + f₀ and f₂ + f₀, respectively, where f₀ is the laser frequency, and f₁ and f₂ are shift values produced by AOMs (acousto-optic modulator). PBS, NPBS, and WP are polarizing beam splitter, nonpolarizing beam splitter, and wave plate, respectively. Det is a photo-electric detector whose output is an interference signal having beat frequency. Both f₁ and f₂ lights are linear polarization, but they are perpendicular in the reflection detection case and parallel in the transmission detection case.

These restrictions are expressed by the following relation for the possibility of the measurement:

$D_{\text{m}}<\alpha \lambda \text{and}{V}_{\text{m}}<\frac{\alpha \lambda }{{\tau }_{\text{C}}},$

(10.17)

where α is 0.5 for the reflection light detection case and is 1.0 for the transmission light detection case, and λ is the wavelength of the used laser. When both relations are not satisfied, the value of τ_C must be reduced.

(B) Relation between the measuring time resolution and the measuring point

The output data interval of τ_C is usually so short that the data cannot directly transport to any external memory through any interface. The memory capacity on the circuit board is not limited so far as the designed circuit can carry out completely. However, even though the electric circuit has very high processing speed, the total necessary time transported from the memory to the external personal computer through the interface is so long that the interval of the experiments increases, and the environmental condition changes unexpectedly. Therefore, the number of the data N_F should be set to a minimum necessary amount. In addition, since the number of bit of the data affects the transfer rate considerably, the measuring points N_F should be considered from the number of bit to suppress error including a selection of an analog–digital (A/D) conversion IC.

(C) Correlation among the latch time interval, the resolution time, and the basis beat frequency

The input data latch time interval Δt is limited mechanically by the following points:

1.  The crystal oscillator’s frequency on the processing board

2.  The operating frequency of the analog to digital converter and some other devices on the board

Moreover, the relation of τ_C = N_CΔt (N_C; integer) is required in the program rule of the digital programming device (e.g., FPGA IC). The value of N_C has a restriction; N_C needs several hundreds or more to compute the frequency and the phase of the signals precisely. (These restrictions will be understood in the subsequent section described about the crossing points.)

Since the heterodyne method checks the variations of the frequencies and the phases, the high-resolution system is required to measure them accurately. To get them with high accuracy, it seems that smaller Δt is a more desirable condition. However, small Δt corresponds to small S/N because of small optical power. Therefore, the designer must search an optimum Δt by discussing the requirements and preconditions: laser selection, wavelength (maximum and minimum) optical power, optical circuit, detector, S/N ratio, fluctuation of the optical path, signal processing circuit, measuring time, output time interval, the number of digital signal bit, etc.

When a period of the noise is comparable to Δt, the digital signal occurs as an unexpected component, and the phase detection may bring unnecessary variations. Therefore, some precautions against noises must be planned, and the selection of the crystal oscillator must be done carefully.

(D) Correlation among the processing method, the latch time interval, and the resolution time

In every stage in a series of operations (from the detection of optical signal to the output of analyzed data), many electric circuits are utilized, and they are usually developed and programmed by user originally. As electric IC devices including optical detectors run with very high-frequency performance, the total processing time is controlled dominantly by an execution time of programming ICs. Recently, as the digital programming ICs are improved rapidly, the resolution time becomes available to operate on about 100 ns.

However, for small resolution time, the condition of τ_C = N_CΔt becomes severe restriction. The value of N_C has to be over 100 as described previously. The requirement means that Δt is less than 1 ns, and the processing circuits have to operate with very high frequency. Since the high-speed processing increases noises and gets worse S/N ratio, small Δt is not always useful to get good resolution time. Therefore, the designs in the circuits are an important criterion for the decision of the latch time interval and the resolution time.

10.7.3    TRANSIENT MEASUREMENT METHOD

10.7.3.1    Signal Processing

In the heterodyne method, two optical interference circuits are constructed in the probe(s). (The circuit is described in the previous section.) Two optical beat signals outputted from the circuits, SIG and REF signals, are processed on the system diagram as shown in Figure 10.11. The SIG signal usually contains the information of the sample target. The REF signal contains the information of a reference, for example, a fixed mirror.

FIGURE 10.11  The signal processing system. Two optical signals, SIG and REF, input to a photoelectric conversion circuits independently. Electric signals are changed to digital form with an A/D converter, and they are calculated in a digital circuit to output the displacement and/or the velocity. The outputted data are stored in a memory and transferred to an external computer through an interface. A crystal oscillator is used as a basic clock in these electric circuits.

Signals flow in the following steps. First, the optical signals change to electric signals in the photoelectric conversion circuits. Second, the electric analog signals are converted to digital forms in A/D conversions. Third, the digital signals are processed in a digital programming IC, in which data of displacements and/or velocities are calculated at a constant interval continuously, and they are stored in some external (or internal) memories. Fourth, the data are outputted to an external personal computer through an interface board.

Before discussing the careful points in the four steps, the system has to realize the following points. (1) A synchronous parallel processing mode for SIG and REF signals. (2) A symmetrical and an identical design of electric circuits and electric parts for SIG and REF signals. These points should be introduced to reduce all of errors or noises based on the differences between SIG and REF signals.

In the fourth step, the transfer time is determined mainly by a processing speed and a capacity of the interface board. When the number of measuring points is large, this unexpected time extends the repetition time of the measurement beyond tester’s expectations. This unanticipated long interval may make the reproducibility loss by change in the environmental condition. The other steps are discussed in the following subsections.

10.7.3.2    Photoelectric Conversion Circuit

Optical signals include several kinds of noise as described in the previous sections. The main factor of the noises is based on the direct fluctuations of the optical path length and on some virtual fluctuations of that. As examples of the direct fluctuations, optical polarization rotations, air flows, instruments’ sounds, temperature fluctuations, shaking of the laboratory’s building, testers’ actions, and testers’ conversations are problems awaiting solutions. Examples of virtual fluctuation are weak optical power, damages from sources of electricity, unsymmetrical circuit pattern (time lag needs less than 0.5 ns) for SIG and REF signals, and unsymmetrical electric parts for them. The two types of the fluctuations have to be suppressed by some warrantable ways. For the suppression in the photoelectric conversion circuit, the following points should be considered: (1) use of an APD having a sufficiently wide frequency range, (2) use of a capacitor having a large capacitance for all power sources (0.1 Hz noise needs 0.1 F) to get good smoothing, (3) fixation of all of boards and parts to cut vibrations, (4) use of a shield cover not to detect electromagnetic pulses, (5) use of a symmetrical circuit pattern and layout of symmetrical electric parts, and (6) no action and silence of testers during measurement.

10.7.3.3    Period Detection

The digital signal is expressed by using variables of τ_C and Δt. The number of the digital signal per one output data N_C is determined as described previously. To analyze the data (total output data = N_F), all digital signals are classified with a datum number k (k = 0, 1, 2, …, N_F − 1) and a series number in a datum n (n = 1, 2, …, N_C) in this study. (Total output time = N_Fτ_C). Since all digital signals are defined at Δt intervals, the signal W(t(k, n)) is expressed as a function of the time t(k, n) = kτ_C + nΔt. The following study is discussed with these parameters.

In the heterodyne method, a frequency and a phase of the signal are important factors to calculate the displacement and the velocity. They are computed with crossing times, which are defined as times satisfied the relation W(t) = 0. The time is hardly expressed with integer values of k and n. So, to show the crossing time CR(k) accurately, the following relation is defined as an approximation:

$CR\left(k\right)=t\left(k,n\right)+\frac{\left|W\left(t\left(k,n\right)\right)\right|}{\left|W\left(t\left(k,n+1\right)\right)-W\left(t\left(k,n\right)\right)\right|}\Delta t,$

where the crossing time is assumed to be between t(k, n) and t(k, n + 1), and it is approximated under the assumption that W(t(k, n)) varies linearly in the neighborhood of the crossing time. To realize the approximation for a sine wave, the following relation between Δt and the period is adequate as discussed in Section 10.7.2(C):

$t\left(k,n+1\right)-t\left(k,n\right)=\Delta t<\text{period/}\left(\text{several\_hundereds}\right).$

There are two types for the crossing, a minus-crossing type: signal decreases from plus domain to minus domain, and a plus-crossing type: signal increases from minus domain to plus domain. Several crossing times are appeared within τ_C, that is, the first plus-crossing time (FPC), the second plus-crossing time (SPC), the third plus-crossing time (TPC), the first minus-crossing time (FMC), the second minus-crossing time (SMC), and the third minus-crossing time (TMC). The signals, the period, and the crossing times are understood as shown in Figure 10.12.

For the period, some expressions, SPC(k) − FPC(k), SMC(k) − FMC(k), etc., can be defined within τ_C. However, they do not always coincide because the signal changes subtly by drift or high-frequency noise. Therefore, to obtain more accurate value, the geometric mean of P_f(k) can be defined as the period of k’th data (the first definition). In addition, the difference between the crossing time of k’th data and that of (k + 1)’th data, FPC(k + 1) − FPC(k), etc., must be expressed with an integer multiple of the period. However, the signal variation does not always satisfy this demand because the signals pick up noises and the crossing times shift. Therefore, as the second definition, the period P_S(FPC, k) is proposed by introducing an integer N(FPC, k) for FPC as follows:

$P_S\left(FPC,k\right)=\frac{FPC\left(k+1\right)+{\tau }_C-FPC\left(k\right)}{N\left(FPC,k\right)}.$

The periods for other crossing types are defined with the same rule, but these periods P_S(FPC, k), P_S(FMC, k), and the other values) do not always coincide. Since the period gives the frequency, the period’s definition has an influence upon the precision of the output data. So, as an available judge, the following average value P(k) will be an applicable period:

FIGURE 10.12  Digital signal in the processing. All data are memorized at a constant time interval Δt. In the data curve, two types of the crossing appeared. The data time t(k, n) is expressed with a datum section number k and a serial number n in the section. The crossing time is calculated in a function of W(t(k, n)). The period is defined by using several crossing points (FPC, FMC, SPC, etc.).

$P\left(k\right)=\frac{1}{6}\left(2P_{\text{P}}\left(k\right)+P_{\text{S}}\left(FPC,k\right)+P_{\text{S}}\left(SPC,k\right)+P_{\text{S}}\left(FMC,k\right)+P_{\text{S}}\left(SMC,k\right)\right).$

This relation is too complex that the calculation program may not accept in user’s circuit. However, user must select the most useful and effective definition within a practicable plan.

In the calculation of the period, either SPC or SMC must exist necessarily within τ_C. Therefore, the value of τ_C has the minimum and must satisfy the condition τ_C ≥ 2 × period.

While though long τ_C yields many crossing points and gives more appropriate period, the time resolution decreases. So, the maximum τ_C will be given by the relation ofτ_C < 3 × period because TPC and TMC are not necessary on purpose.

10.7.3.4    Phase Detection and Displacement

In the datum section k, two typed phases are defined as follows:

${\varphi }_{\text{P}}\left(k\right)=\frac{FPC\left(k\right)}{P\left(k\right)}\text{and}{\varphi }_{\text{M}}\left(k\right)=\frac{FMC\left(k\right)}{P\left(k\right)}.$

(10.18)

The phase difference between the phase in the section k and the phase in the section k + 1 has also two types, and they are defined as

$\Delta {\Phi }_{\text{P}}\left(k\right)={\varphi }_{\text{P}}\left(k+1\right)-{\varphi }_{\text{P}}\left(k\right)-N\left(\text{FPC,}k\right)+\frac{{\tau }_{\text{C}}}{P\left(k\right)}\left(\text{for FPC}\text{crossing}\right)$

and ΔΦ_M(k) is represented with the same rule. The phase difference gives the displacement variation ΔR(k) in the section k, which has two types classified by the detecting ways:

$\begin{array}{ll}\Delta R\left(k\right)=\frac{\lambda }{2}\frac{\Delta {\Phi }_{\text{P}}\left(k\right)+\Delta {\Phi }_{\text{M}}\left(k\right)}{2}\hfill & \left(\text{for the reflection measuring case}\right),\hfill \\ \Delta R\left(k\right)=\lambda \frac{\Delta {\Phi }_{\text{P}}\left(k\right)+\Delta {\Phi }_{\text{M}}\left(k\right)}{2}\hfill & \left(\text{for the transmission measuring case}\right),\hfill \end{array}$

where λ is the wavelength of the measuring laser. The total displacement R(N_F) is represented with the sum of the displacement variation terms assigned by the section number

$R\left(N_{\text{F}}\right)={\displaystyle \sum _{k=0}^{N_{\text{F}}-1}\Delta R\left(k\right),}$

(10.19)

where N_F is the number of the output data (the measuring time = N_Fτ_C).

10.7.3.5    Analysis of Error Factor

Earlier description for the digital processing clearly indicates an existence of error factors. That is, the following factors are closely connected with an occurrence of the error:

1.  Absolute value of the laser wavelength

2.  Basis beat frequency (period τ_B0)

3.  Latch time interval (Δt)

4.  Output data interval (τ_C = N_RΔt; N_R: integer)

The effect of these factors for the displacement is investigated by introducing the fluctuations of these factors: λ = λ₀ + Δλ, τ_B = τ_B0 + Δτ_B, and τ_C = τ_C0 + Δτ_C.

In the processing program, the utilized clock time $t\left(k,n\right)\equiv t_n^k$ is represented with the genuine time $T\left(k,n\right)\equiv T_n^k$ as follows:

$t\left(k,n\right)=T\left(k,n\right)+\left(k+\frac{n}{N_{\text{R}}}\right)\times \Delta {\tau }_{\text{C}}.$

This time difference increases (or decreases) monotonously with an increase in k and n. The difference induces the shift of the signal strength W(t(k, n)). Therefore, the functions of CR(k), P(k), ϕ_P(k), ΔR(k), etc., are also changed; finally, the displacement is expressed with these fluctuations. The changes in the functions are expressed approximately as follows:

$\begin{array}{ll}\hfill W\left(t_{n+1}^k\right)-W\left(t_n^k\right)& =W\left(T_{n+1}^k\right)-W\left(T_n^k\right)+\frac{\Delta {\tau }_{\text{C}}}{N_{\text{R}}}\frac{\partial W\left(T_{n+1}^k\right)}{\partial t}+\left(k+\frac{n}{N_{\text{R}}}\right)\frac{{\tau }_{\text{C}}}{N_{\text{R}}}\Delta {\tau }_{\text{C}}\frac{{\partial }^{2}W\left(T_n^k\right)}{\partial t^2}+\cdot \cdot \cdot \hfill \\ \hfill & \cong W\left(T_{n+1}^k\right)-W\left(T_n^k\right)+\frac{\alpha }{N_{\text{R}}}\Delta {\tau }_{\text{C}},\hfill \\ \hfill FPC\left(k\right)& =T_n^k+\left(k+\frac{n}{N_{\text{R}}}\right)\times \Delta {\tau }_{\text{C}}+\frac{-W\left(T_n^k\right)+\left(k+\frac{n}{N_{\text{R}}}\right)\Delta {\tau }_C\frac{\partial W\left(T_n^k\right)}{\partial t}}{W\left(T_{n+1}^k\right)-W\left(T_n^k\right)+\frac{\alpha }{N_{\text{R}}}\Delta {\tau }_{\text{C}}}\hfill \\ \hfill & \cong T_n^k+\frac{-W\left(T_n^k\right)}{W\left(T_{n+1}^k\right)-W\left(T_n^k\right)}+{\gamma }_{\text{P}}\left(k\right)\Delta {\tau }_{\text{C}}\equiv FP{C}_0\left(k\right)+{\gamma }_{\text{P}}\left(k\right)\Delta {\tau }_{\text{C}}\hfill \\ \hfill P_{\text{S}}\left(FPC,k\right)& =\frac{1}{N\left(FPC,k\right)}\left\{FP{C}_0\left(k+1\right)+{\tau }_{\text{C}}-FP{C}_0\left(k\right)+\left({\gamma }_{\text{P}}\left(k+1\right)-{\gamma }_{\text{P}}\left(k\right)\right)\Delta {\tau }_{\text{C}}\right\}+\cdot \cdot \cdot \hfill \\ \hfill & \equiv P_{\text{S0}}\left(FPC,k\right)+{\delta }_{\text{P}}\left(k\right)\Delta {\tau }_{\text{C}}+\cdot \cdot \cdot ,\hfill \\ \hfill {\varphi }_P\left(k\right)& =\frac{FP{C}_0\left(k\right)}{P_0\left(k\right)}+\left(\frac{{\gamma }_P\left(k\right)}{P_0\left(k\right)}-\frac{FP{C}_0\left(k\right)}{P_0^2\left(k\right)}{\text{δ}}_P\left(k\right)\right)\Delta {\tau }_C+\cdot \cdot \cdot \equiv {\varphi }_{P0}\left(k\right)+{\epsilon }_P\left(k\right)\Delta {\tau }_C+\cdot \cdot \cdot ,\hfill \end{array}$

where P₀(k) is given by (P_S0(SPC, k), which are represented with the same rule as P_S0(FPC, k)):

$P_0\left(k\right)=\frac{1}{6}\left(2P_{F0}\left(k\right)+P_{S0}\left(FPC,k\right)+P_{S0}\left(SPC,k\right)+P_{S0}\left(FMC,k\right)+P_{S)}\left(SMC,k\right)\right).$

The phase difference and the displacement variation in the section k are also arranged as follows:

$\begin{array}{ll}\Delta {\Phi }_{\text{P}}\left(k\right)\hfill & ={\varphi }_{\text{P0}}\left(k+1\right)-{\varphi }_{\text{P0}}\left(k\right)-N\left(k\right)+\frac{{\tau }_{\text{C0}}}{{\tau }_{\text{B0}}}+\left({\epsilon }_{\text{P}}\left(k+1\right)-{\epsilon }_{\text{P}}\left(k\right)\right)\Delta {\tau }_{\text{C}}+\frac{{\tau }_{\text{C0}}}{{\tau }_{\text{B0}}}\left(\frac{\Delta {\tau }_{\text{C}}}{{\tau }_{\text{C}}}-\frac{\Delta {\tau }_{\text{B}}}{{\tau }_{\text{B0}}}\right)+\cdot \cdot \cdot \hfill \\ \hfill & \equiv \Delta {\Phi }_{\text{P0}}\left(k\right)+{\xi }_{\text{P}}\left(k\right)\Delta {\tau }_{\text{C}}+\text{η}\Delta {\tau }_{\text{B}}+\cdot \cdot \cdot ,\hfill \end{array}$

where N(FPC, k) =N(SPC, k) = N(FMC, k) = N(SMC, k) = N(k) is used undoubtedly.

$\begin{array}{ll}\Delta R\left(k\right)\hfill & =\frac{{\lambda }_0+\Delta \lambda }{2}\left\{\frac{1}{2}\left(\Delta {\Phi }_{\text{P0}}\left(k\right)+\Delta {\Phi }_{\text{M0}}\left(k\right)\right)+\frac{1}{2}\left({\xi }_{\text{P}}\left(k\right)+{\xi }_{\text{M}}\left(k\right)\right)\Delta {\tau }_{\text{C}}+\text{η}\Delta {\tau }_{\text{B}}+\cdot \cdot \cdot \right\}\hfill \\ \hfill & =\Delta R_0\left(k\right)+\frac{\Delta R_0\left(k\right)}{{\lambda }_0}\Delta \lambda +\frac{\lambda }{4}\left({\xi }_P\left(k\right)+{\xi }_M\left(k\right)\right)\Delta {\tau }_C+\frac{\lambda \text{η}}{2}\Delta {\tau }_B+\cdot \cdot \cdot \hfill \end{array}$

where the zero mark in the subscript of ΔΦ_P0(k) and ΔR₀(k) corresponds to the genuine condition. Therefore, the displacement expressed in Equation 10.19 is arranged with some additional terms as follows:

$R\left(N_{\text{F}}\right)=R_0\left(N_{\text{F}}\right)+{\displaystyle \sum _{k=0}^{N_{\text{F}}}\Delta R_0\left(k\right)\frac{\Delta \lambda }{{\lambda }_0}+\frac{{\lambda }_0}{4}{\displaystyle \sum _{k=0}^{N_{\text{F}}}\left({\xi }_{\text{P}}\left(k\right)+{\xi }_{\text{M}}\left(k\right)\right)\Delta {\tau }_{\text{C}}+\frac{N_{\text{F}}{\lambda }_0\text{η}}{2}\Delta {\tau }_{\text{B}},}}$

(10.20)

where R₀(N_F) is the genuine displacement result. In these relations, parameters of α, γ_P(k), δ_P(k), ε_P(k), _P(k), and η are introduced to shorten the lengthy expressions. Of course, the quantities having subscript of M, γ_M(k), δ_M(k), ε_M(k), and ξ_M(k), are described with the same rule as those having the subscript P.

It is shown clearly that three additional terms in Equation 10.20 generate errors in the displacement results. However, the value of R(N_F) is expressed only as an output of SIG (or REF) channel. When the relative measuring optical system is introduced, the displacement is given with a difference expressed by R_SIG(N_F) − R_REF(N_F). (The subscripts of SIG and REF correspond to SIG and REF channels, respectively.) The relative measuring system brings great noise reduction for three terms as follows:

(First additional term) The factor ΔR₀(k) includes two error sources mainly: the target movement and the fluctuation of the optical path. When the SIG’s optical path is nearly the same to the REF’s optical path (“nearly the same” means that two optical axes are satisfied in the following conditions: parallel, small spacing, the same height from the basement, the same length, the same reflection way for mirrors, the same transmission way for PBS and NPBS, etc.), the fluctuations of two paths are balanced because several causes of the fluctuations including environmental variation induce nearly the same effect on both paths. The balance and the fixation of the REF target make the noises decrease dramatically. Therefore, the displacement of the SIG target can be measured with very small error.

Since the frequency-stabilized HeNe laser is usually kept at the relation Δλ/λ = 3 × 10⁻⁹ [107], even if measurement is made under the worst experimental condition and the measuring point is 10⁶, the measurement result is held down within 0.3% error. (An unstabilized HeNe laser has usually about Δλ/λ = 3 × 10⁻⁶ [108]. The error in 10⁶ measuring points is much more than the genuine value in the worst case. This is the reason the free-running (= unstabilized) HeNe laser cannot be used in the heterodyne method).

(Second additional term) The relation of (ξ_P(k) + ξ_M(k))_SIG ≈ (ξ_P(k) + ξ_M(k))_REF is required to compensate the errors between SIG and REF signals. When the measurement system sets up under “nearly the same” condition, the relation is almost satisfied though some lengthy expansions of the terms are need. Therefore, “nearly the same” condition is one of the necessary conditions to obtain the high-precision results.

(Third additional term) The compensation requires the relation of (η)_SIG ≈ (η)_REF. As the η value does not depend on targets’ movements, this relation is satisfied under “nearly the same” condition.

Three terms suggest that the relative measurement having “nearly the same” condition for SIG and REF paths is essential for the measurement and that the displacement can suppress its errors by working compensations in all of additional terms. In other words, all designers and programmers should pay attention to the following points:

1.  Absolute laser frequency: the frequency-stabilized HeNe laser is indispensable for the high-resolution measurement, because the error is so small even if the worst fluctuation occurs.

2.  Environmental temperature and atmospheric pressure: the refractive index of the air varies largely by the temperature and the atmospheric pressure [109]. Especially, when the laser light transfers through the air space, the optical path length is changed largely by fluctuations of the environmental conditions. Therefore, introductions of the vacuum system and/or the temperature control for the whole optical circuits are acceptable condition to obtain higher precision.

3.  Stability of crystal oscillator: in the digital circuit, the clock time of the crystal oscillator is usually used as a standard time base. Therefore, the fluctuation of the crystal oscillator directly affects the fluctuation of Δt and τ_C. High-quality crystal oscillator should be used as much as possible.

4.  Stability of the frequency shifter: the basis beat frequency is made by the frequency shifter. Therefore, the clock generator of the frequency shifter needs good quality to decrease the fluctuation of the beat frequency. Two absolute frequencies of the shifter do not necessarily keep high quality, but the difference of two output frequencies, the beat frequency f_B, must have high stabilization of about Δf_B/f_B < 1 × 10⁻⁶.

10.7.4    EXAMPLES OF TRANSIENT MEASUREMENT

10.7.4.1    Tuning Fork and Cantilever Vibration

The tuning fork or the cantilever is known as an item vibrating with a constant frequency. Their characteristics have been investigated [110,111,112,113,114,115,116,117,118,119,120,121,122]. An example of the measuring system for them is shown in Figure 10.13. Two lights, f1 light and f2 light, have the frequencies of f₀ + f₁ and f₀ + f₂, respectively. The frequency f₀ is the HeNe laser frequency of about 474 THz. The frequencies f₁ and f₂ are the shift values of about 80 MHz made by the frequency shifter. The f1 light is divided, and two lights are reflected from a sample surface and a reference surface. The f2 light is also divided and comes to optical detectors after reflecting repeatedly. On the optical detectors, f₁ and f₂ lights interfere and yield two electric signals, SIG and REF. Both signals contain all information of the optical path including the sample and the reference surfaces. Especially, the SIG signal includes the displacement information of the sample surface, and the REF signal does that of the reference surface. Therefore, the subtraction between two signals SIG–REF can cancel practically all effects on the optical path excluding the displacement of the sample and the reference surfaces because the optical path length and position of the SIG signal are almost the same as those of the REF signal. (It satisfies “nearly the same” condition.) Therefore, the subtraction value can pick up only the displacement of the sample. (In optical circuit, to suppress the influences of optical parts, mechanical parts, and holders, the same goods are used for both paths.)

FIGURE 10.13  An example of optical circuit in the measurement system for the tuning fork. Two lights (frequency f₁ and f₂) are outputted from the optical fiber and are reflected repeatedly, and make two optical signals (SIG and REF). SIG signals include the displacement information of the tuning fork, while REF signals do that of the reference surface. By measuring under “nearly the same” condition, the difference of two signals, SIG–REF, indicates only the displacement. The “nearly the same” condition means that two optical axes have the following characteristics: parallel, small spacing, the same height from the basement, the same length, the same reflection, and transmission way for optical parts.

An example of results is shown in Figure 10.14a and b [123]. The measurement time is 16 s, the time resolution (τ_C) is 16 μs, and the measuring points are 10⁶. In Figure 10.14a, no sine wave can be observed because the measuring point is too large to draw on this scale. However, in the envelope form, a damping is observed. By multiplying the figure as shown in Figure 10.14b, a sine wave is confirmed apparently. Two figures indicate that the tuning fork vibrates in the form of the damping oscillation. The transient characteristics of the tuning fork can be investigated with three strong points of the heterodyne measuring system: (1) the high displacement resolution (having good S/N ratio for small vibration amplitude of about 300 nm), (2) the high time resolution (having good S/N ratio for small phase detection of about 0.008 rad), and (3) the large measuring points (keeping good S/N ratio for wide amplitude).

By analyzing these data, several dynamic characteristics are obtained for the damping oscillation expressed by Bexp(−γt)sin(2πft + φ). (1) The frequency f increases linearly with a decrease in the square root of the vibration amplitude $\sqrt{B}$ as shown in Figure 10.15. (2) The logarithm of the damping factor log(γ) is proportional to the logarithm of the vibration amplitude log(B) as shown in Figure 10.16. (3) Both f and γ depend on the temperature and the measuring position located on the tuning fork’s tin. (4) The frequency decreases with an increase in temperature as shown in Figure 10.17.

FIGURE 10.14  (a) An example of the transient variation of the tuning fork. Vertical, displacement; horizontal, time. Measurement time, 16 [s]; the time resolution, 16 [μs]; measuring points, 10⁶ points. Any sine wave is not recognized because the number of the data is too large to draw all points in this graph. But the envelope of the curve shows damping. (b) Magnification of (a) within the time region from 7.192 to 7.2 [s]. It is confirmed that vibration form is a sine wave, and its frequency is 489.913 [Hz]. High resolution gives good S/N ratio under very small noise of about 1.4 [nm]. Both figures (a) and (b) show clearly that the tuning fork vibrates undoubtedly in the form of a damping oscillation.

FIGURE 10.15  An example of the relation between f₀ and $\sqrt{B}$, where f₀ is the frequency at zero amplitude and B is the vibration amplitude. Experimental error bars’ lengths for all data are nearly equal to the dot size in this expression. High correlation coefficient confirms a linear relation of $f_0=m\sqrt{B}+b$.

The measurement for the cantilever gives nearly the same results because it has the basic form of the tuning fork. The Bernoulli–Euler equation [124] is the standard equation for investigations of the cantilever vibration. However, the equation can’t explain any damping oscillation transiently. Several modified equations have been proposed and investigated [125,126,127,128,129,130,131,132,133] to understand the transient oscillation. Details are described in these references. One of them is the following form:

FIGURE 10.16  An example of the relation between the log(B) and log(γ) is shown under the condition, X = 15 mm, T = 14.7°C, P = 991.3 hPa, and humidity = 39%. Very high linearity is shown (correlation coefficient is over 0.999). The vibration amplitude B is represented with nm unit. The X, T, and P are the distance from the tuning fork’s tip, the temperature, and the atmospheric pressure, respectively.

FIGURE 10.17  The temperature dependence of the frequency of the tuning fork. The sizes of data points in this figure are comparable to the spread of the error bar. The data distribution seems to form a belt. The width or extension of the belt depends on the vibration amplitude and/or the measuring position on the tuning fork’s tin. The distance between the measurement position and the tip distributes from 5 to 75 mm, the atmospheric pressure does from 981.7 to 1005.1 hPa, and the humidity does from 37% to 68%.

$EI\left(\frac{{\partial }^{4}y}{\partial x^4}+\mu \frac{{\partial }^{5}y}{\partial x^4\partial t}\right)+\text{ρ}A\frac{{\partial }^{2}y}{\partial t^2}+\lambda \frac{\partial y}{\partial t}=0,$

(10.21)

where

E (Young’s modulus)

I (second moment of area),

ρ (density),

A (cross section) are constants given by the material and the size of the tuning fork or the cantilever

y, x, and t are displacement, vibration position, and time, respectively

The damping factors, μ and λ, represent the transient resistance strengths against the shearing stress and the movement, respectively

By analyzing Equation 10.21 and experimental results, it is shown that these damping factors depend on the vibration amplitude, measuring position, and other factors and that they are NOT constant in the damping oscillation of the cantilever.

The modified theoretical equation matches to some of the experimental results [132]. However, some of the dynamic characteristics are hardly understood, especially the relations among f, γ, and vibration amplitude are left as a problem of the transient analyses. In addition, the damping factors have to be investigated transiently including a question of whether they are constants or not.

10.7.4.2    Concentration Fluctuation

Several optical detections for the concentration fluctuation of the fluids have been investigated [134,135,136,137,138]. Generally, when gases or liquids are mixed, a concentration distribution is born and brings about diffusion till it is homogeneous. By measuring the concentration variation transiently, the diffusion as one of dynamic phenomena will be able to be investigated. As one of the optical measurement method for fluid, the concentration dependence of the refractive index of the fluid sample is utilized. Since the concentration varies transiently and spatially in a chamber of packed or filled samples, the variation of the optical path length of the chamber L(t) is measured totally and transiently, which is expressed by the relation

$L\left(t\right)={\displaystyle \underset{0}{\overset{K}{\int }}dyn\left(y,t\right)K\overline{n},}$

where

K is the mechanical length of the chamber

n(y, t) is the refractive index at space variable y and time t

$\overline{n}$ is the average refractive index on the optical path

As the optical heterodyne method measures the value of L(t) − K, any concentration distribution in the chamber is neglected, and only the average value is used in the analysis. Certainly, this method may not be effective to investigate the distribution variation of the concentration. However, this method has a strong point that very low concentration variation can be detected, for example, the refractive index change of 10⁻⁶ in 10 mm chamber length is detectable. Therefore, the high precision of the heterodyne method can be applied to the concentration variation measurement.

FIGURE 10.18  An example of the optical circuit of the concentration measurement. Both Ch1 and Ch2 chambers are filled with pure water. The refined sugar water dropped into the Cha camber only. The difference between Ch1 and Ch2 detectors’ signals corresponds to the variation of the optical path length of Ch1. The variation is generated by a diffusion of the sugar water. Therefore, the diffusion of liquid sample can be observed. To realize a high precision, “nearly the same” condition needs. Especially, the relation of |L₀ − L₁| < 0.5 mm should be kept.

An example of the optical circuits for the concentration variation measurement is shown in Figure 10.18 [139]. A laser light is divided, and both lights are deflected with AOMs. After doing optical interference, two signals are generated. Pure water of 12.8 g is set for both chambers. A refined sugar water (C₁₂H₂₂O₁₁, 0.077 mol/L) of 0.15 cm³ is dropped from 2 cm height above the surface of the pure water for Ch1 chamber only.

An example of the optical path variation (corresponding to the difference between Ch1 chamber and Ch2 chamber) is shown in Figure 10.19. The time resolution is 32 μs, and the data points are 10⁶. The sample is dropped at 5.1 s after the starting time of recording. Though there are noises of about 10 nm width generated by several causes over the whole of data, the optical path length starts to vary slowly, and it repeats the rise and fall and settles into a certain quantity finally. In short, the concentration distribution converges at an equilibrium value after dynamic variation. The variation including overshoot will be understood by the following process. The dropped sample goes straightly into the pure water as a lump of sugar water keeping the potential energy and starts to diffuse slowly by the dissipation of the energy. The lump will start to expand and become hazy gradually. The optical path length depends on the lump’s position and size. Therefore, the movement of the lump (i.e., diffusion) creates the variation of the optical path length including overshoot and/or undershoot.

And the optical path settles down finally into a constant corresponding to a homogeneous distribution in the chamber. As the optical path length is based on the refractive index, the variation of the optical path length ΔL between the starting time (t = 0) and at t = ∞ is expressed with the relation of $\Delta L=K\times \left(\overline{n}\left(t=\infty \right)-\overline{n}\left(t=0\right)\right)$. In the example, as ΔL = 120 nm, the concentration dependence of the refractive index of the sugar water is obtained [139] as follows:

FIGURE 10.19  An example of the transient phenomenon of the concentration diffusion. Measurement time is 32 s. The refined sugar water is dropped at 5.1 s in this graph. Two types of vibration are observed. First type: it decreases monotonously till 11.2 s and repeats to go up and down and finally converges to nearly constant. Second type: it vibrates with an amplitude of 10 nm or less and a period of 0.24 ± 0.07 s after the dropped time. The first type shows the concentration diffusion. The second type may show an impulse wave or a compressed wave.

$n\left(w\right)=n\left(0\right)+0.0065\times w,$

where

w is the concentration in mol/L unit

n(0) is the refractive index of the pure water given by 1.3318

The optical heterodyne measurement system shown in Figure 10.18 can apply to all transparent samples and many semitransparent samples, and it will analyze the characteristics of liquid, vapor, and gas with the transient variations. In addition, this absolute detection of the characteristic variation within very short times will be useful in the observations of various physical quantities, for example, temperature, pressure, force, viscosity, magnetic field, radiation quantity, and current.

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