High dynamic range thickness measurement using a double heterodyne interferometer

Fuma Wakabayashi; Kotaro Kawai

doi:10.1364/OPTCON.470640

1. Introduction

The thickness measurement approach using the wave nature of light waves like interference and polarization has a great benefit. High-precision measurement is feasible, non-contact, and non-destructive; thus, optical measurement is extensively employed in industrial applications. Typical measurement approaches include thin film thickness measurement from sub-wavelength to micrometer order using ellipsometry and thickness measurement from several hundred nm to several hundred micrometers by the peak to valley approach using transmission or reflection spectroscopy. Recently, novel optical measurement approaches have been investigated and reported [1–11]. However, the range of measurable thickness is impeded by the principle of each optical measurement approach. For example, in the basic optical interference measurement using two-beam interference, the retardation caused by sample transmission is computed back from the light intensity of the interference light to generate the thickness information. However, if the object’s thickness is too thick, the retardation exceeds the wavelength of light and cannot be measured. In the peak to valley method using a spectroscopic spectrum, a sample with a relatively extensive thickness range can be determined in principle. However, in a thin film or thick film sample with a sub-wavelength to several hundred micrometers or more, the fringe period of the spectroscopic spectrum is either too wide or narrow to be measurable. In the current thickness measurement by each optical measurement approach, since the range of the measurable thickness is constrained as described above, it is crucial to prepare properly for the measuring device according to the thickness of the measurement target.

In this study, we theoretically suggest a high dynamic range optical interference measurement approach, which can measure thicknesses from sub-wavelength to millimeters. Of great importance that the measuring mechanism can be achieved using only one light source and detector with a minimum and that a wide range of thickness measurements is feasible. Furthermore, since white light is not employed as a light source as in spectroscopic measurement, there is a great benefit in that the influence of the wavelength dispersion of the refractive index of the sample is very small, and correction of the measurement finding is simple. The details of high dynamic range measurement using the double heterodyne interferometer, suggested and shown by theoretical analysis applying the Jones calculus, are described below.

2. Theoretical analysis conditions

Figure 1 depicts the optical system’s analysis condition for the high dynamic range measurement with a double heterodyne four beam interferometer. The supercontinuum (SC) laser beam is split into two light waves using the beam splitter, which is incident on the fiber Bragg grating (FBG). Since the double heterodyne interference method requires optical interference between a reference light and a light wave with a large phase change during transmission through a thick sample as described later, high coherence is required for the light source. Therefore, we adopted SC laser, which has high coherence among various white light sources. The laser beam of arbitrary wavelength λ is reflected from the SC laser beam using FBG. One of the SC laser beams split using the beam splitter is incident on the variable FBG that can regulate the grating period by fiber extension (i.e., the reflection wavelength can be regulated) [12]. The laser light’s wavelength after reflecting the variable FBG differs by Δλ from the wavelength of the other laser beam to become λ+Δλ. The two laser beams with various wavelengths are heterodyne interfered in on-axis by the beam splitter. The light wave obtained by on-axis heterodyne interference is a modulated amplitude wave (called AM wave), and it is crucial to note that its envelope is adequately long compared to the light source’s wavelength. It is also possible to control the wavelength of the envelope of the AM wave by the variable FBG since the wavelength of the envelope of the AM wave depends on the difference Δλ between the wavelengths of the two light waves that cause heterodyne interference. Note that the SC laser’s excitation light pulse width should be picoseconds or more so that the envelope of the AM wave exists in numerous cycles in one pulse. Since the optical system of this study is similar to the pulse interference that has been conventionally conducted [13], it is crucial to accurately adjust the optical path lengths of the optical interference system. The generated AM wave splits into two AM waves, and then one of them passes through the sample, delaying the envelope’s phase. The AM wave that has passed through the sample and AM reference wave interfere with the off-axis. The interference fringes’ intensity distribution generates a long-period light-dark distribution, which reflects the envelope of the AM wave, as depicted in the theoretical analysis findings described later. The position of the intensity distribution of the interference fringes along the grating vector (x-axis) changes according to the relative phase difference between the two beams (between the two AM waves), similar to the normal two-beam interference. Thus, the sample’s thickness can be generated by computing back from the finding of the amount of change in the position of the interference fringes detected using the image sensor. As long as the phase delay (optical distance) owing to sample transmission is less than the period of the envelope of the AM wave since the wavelength difference Δλ can be regulated, it is feasible to measure a wide range of sample thickness. The theoretical analysis and findings that demonstrate the thickness measurement approach using the above-mentioned double heterodyne interferometer are described below.

Fig. 1. Optical system for double heterodyne interferometry. SCL, FBG, M, and BS signify supercontinuum laser, fiber Bragg grating, mirror, and beam splitter, respectively.

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Since the AM waves interfere with each other in this approach, it is crucial to consider the 4-beam polarization interference in the theoretical analysis. The four interfering beams depicted in Fig. 1 are called beam N (N = 1, 2, 3, and 4). The AM wave comprised the heterodyne interference of beam 1 (wavelength: λ), and beam 2 (wavelength: λ+Δλ) is called the reference light that does not pass through the sample. Furthermore, the AM wave comprised the heterodyne interference of beam 3 (wavelength: λ), and beam 4 (wavelength: λ+Δλ) is called the light that passes through the sample and has its phase modulated. The Jones vector of beam N in the xyz-axis coordinate system (Fig. 1) is denoted as

(1)$${{\textbf E}_N} = \left( {\begin{array}{c} {{A_{xN}}\exp ({i{{\textbf k}_N} \cdot {\textbf r} + i{\delta_N}} )}\\ {{A_{yN}}\exp ({i{{\textbf k}_N} \cdot {\textbf r} + i{\delta_N} + i{\psi_N}} )} \end{array}} \right),$$

where A_N, k_N, r, δ_N, and ψ_N denote the amplitude of each electric field component in the xyz coordinate, wave vector, position vector, initial phase, and relative phase between each xy component, respectively. The wave vector k_N is defined as ${k_N}{\left( {\begin{array}{ccc} {\sin {\theta_N},}&{0,}&{\cos {\theta_N}} \end{array}} \right)^\textrm{T}}$; k_N denotes the wave number and is given by 2π/λ in beam 1,3 and 2π/(λ+Δλ) in beam 2,4. The incident angles θ₁ and θ₂ are +θ, and the θ₃ and θ₄ are −θ. The initial phases δ₁ and δ₂ are 0. The initial phases (i.e., the amount of phase change caused by the sample transmission) of Beams 3 and 4 that pass through the sample are given by δ₃ = 2πnd / λ and δ₄ = 2πnd / (λ+Δλ); n and d denote the refractive index and thickness of the sample, respectively. Although the sample’s refractive index dispersion occurs because of the wavelength difference Δλ, as will be described later, the value of Δλ that can be taken is small, and the refractive index dispersion is negligibly small, so it is not considered. For example, in glass, even if Δλ is set by 40 nm for a wavelength of 640 nm, the amount of variation in the refractive index only changes at the refractive index’s third decimal place. The relative phase between each xy component ψN determines the polarization state of beam N. If the polarization states of beam N are linear polarization and circular polarization, the relative phase ψ_N is 0 and ±π/2, respectively. The intensity distributions in the double heterodyne interference electric field are computed using the sum of the Jones vectors (electric field vectors) of all N given in Eq. (1). Since the four beam interference in this study is not on-axis interference, it is crucial to multiply cosθ_N into the amplitude in the perpendicular direction to the wave vector of each beam N to obtain the amplitude A_xN in the x-axis in the image sensor plane. Since the wave vector always exists in the xz plane, the amplitude A_yN in the y-axis direction is used without change.

3. Results and discussion

Figure 2 depicts the computed findings of the interference intensity distribution by double heterodyne interference when all Beam N is 90° linear polarizations. Figures 2(a) and 2(b) show the computation finding when the sample thicknesses d are set to 1 μm and 1 mm. The interference fringe in the upper part of Fig. 2 and the lower part of Fig. 2 are the cases where the sample is uninstalled in the optical system and installed. Owing to the phase delay during sample transmission of the AM wave comprised the on-axis interference of Beams 3 and 4, the position of the interference distribution changes in the x-axis direction between when the sample is or is not installed. The sample thickness is measured from the amount of change in the interference fringe’s position. The details of the approach for deriving the thickness d in this approach will be described later. Since the optical distance nd (n is 1.5 in this study) generated owing to transmission, the sample has to be less than the envelope period of the AM wave described later, Δλ is −40 nm (d = 1 μm) and −0.1 nm (d = 1 mm). The reference wavelength λ is 640 nm. Since the interference fringe period Λ changes depending on the envelope period of the AM wave, for simple observation, the incident angle θ is adjusted, and the incident angle θ is set to 0.1° (d = 1 μm) and 20° (d = 1 mm). A small incident angle of around 0.1° can be attained without any challenge in an actual interference optical experiment [14]. A_x in Eq. (1) is 0 and A_y is 1. As depicted in Fig. 2, the intensity distribution of the interference fringes in this double heterodyne interference produces a long-period light-dark distribution showing the AM wave’s envelope. Since the contrast of the interference fringes reduces when the initial phases of the two AM waves are unequal, the contrast of the interference fringes reduces in the computation finding (lower part of Fig. 2) with the sample inserted. In addition, even if the light intensity differs between the AM waves due to light absorption during sample transmission, the contrast of the interference fringes only decrease, as in normal two-beam interference. The difference in light intensity between the AM waves is independent of the phase of the light waves and therefore does not affect the thickness measurement. Furthermore, a very short-period light-dark distribution is due to the interference of two AM waves (see the upper part of Fig. 2(a) enlarged view). In this approach, the sample thickness on the order of nm can be also determined by blocking the beam with a wavelength of λ+Δλ in the optical system depicted in Fig. 1 and operating it as a normal two-beam interference optical system. From the above, high dynamic range measurement can be achieved by this research’s double heterodyne interference optical system. However, in the interference fringes depicted in Fig. 2, in addition to the long-period intensity distribution due to the AM wave’s envelope, there is also a short-period intensity distribution due to the short amplitude variation of the AM wave. This short-period intensity distribution in the interference fringe hinders detecting the amount of the change of the interference fringes in the actual experiment. Thus, the findings of theoretical investigations on setting the interfering beams to right and left circular polarization to remove the short-period intensity distribution are also described below.

Fig. 2. Calculated intensity distributions of the double heterodyne interference field when the polarization states of the interfering four beams are set to the 90° linear polarization. The sample thicknesses d are set to (a) 1 μm and (b) 1 mm.

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The AM waves comprised the pair of beam 1 and 2 and the pair of beam 3 and 4 after split by the beam splitter closest to the image sensor is incident on the newly added quarter-wave plates to set it circular polarization. The circular polarizations’ rotation directions comprised the pair of beam 1 and 2, and the pair of beam 3 and 4 are opposite directions to each other. The modulated waves obtained by on-axis two-beam heterodyne interference of circularly polarized light with distinct wavelengths and the same rotation direction, like the interference of the Beam1 and 2 and beam 3 and 4, are hereinafter called “circular polarization modulated (CPM) wave.” The period of the intensity (electric field amplitude) of the CPM wave is long, and it has a similar intensity variation period as the period of the envelope of the AM wave generated by the interference of the linear polarizations with distinct wavelengths. The variation period of the intensity of the CPM wave is long since it is determined by the length variation of the composite vector of the electric fields of circularly polarized lights having the same rotation direction with slightly distinct wavelengths. The direction of the electric field vector of the CPM wave rotates in a period on the order of the wavelength of light, like normal circular polarization. The intensity modulation and the direction variation in the electric field vector of the CPM wave will be quantitatively described below. Equation (1) defines the two circular polarizations with slightly different wavelengths, and the CPM wave’s complex amplitude can be generated by adding them together. Since the two light waves that are interfered with in the on-axis are circularly polarized, both A_x and A_y in Eq. (1) are 1 (It is assumed that it is propagating on the z-axis. If the propagation direction is tilted by θ respect to the z-axis, Ax is set cosθ). The initial phase is set to δ = 0 since transmission of the sample does not require to be considered. From the above, the complex amplitude of the CPM wave E_CPM and the intensity I_CPM is

(2)$$\begin{array}{l} {{\textbf E}_{\textrm{CPM}}} = {{\textbf E}_1} + {{\textbf E}_2}\\ = \frac{1}{{\sqrt 2 }}\left( {\begin{array}{c} {\cos \theta }\\ { - i} \end{array}} \right)\left. {\left\{ {\exp \left[ {i\frac{{2\pi }}{\lambda }({x\sin \theta + z\cos \theta } )} \right]} \right. + \exp \left[ {i\frac{{2\pi }}{{\lambda + \Delta \lambda }}({x\sin \theta + z\cos \theta } )} \right]} \right\},\\= \sqrt 2 \left( {\begin{array}{c} {\cos \theta }\\ { - i} \end{array}} \right)\cos \left[ {\frac{{\pi \Delta \lambda }}{{\lambda (\lambda + \Delta \lambda )}}({x\sin \theta + z\cos \theta } )} \right]\exp \left[ {i\frac{{\pi (2\lambda + \Delta \lambda )}}{{\lambda (\lambda + \Delta \lambda )}}({x\sin \theta + z\cos \theta } )} \right]. \end{array}$$

(3)$$\begin{array}{l} {I_{\textrm{CPM}}} = {|{{{\textbf E}_{\textrm{CPM}}}} |^2},\\ = 2\left\{ {1 + \cos \left[ {\frac{{2\pi \varDelta \lambda }}{{\lambda ({\lambda + \varDelta \lambda } )}}({x\sin \theta + z\cos \theta } )} \right]} \right\}. \end{array}$$

Figure 3 depicts the analysis findings of the modulation of the intensity and the variation in the direction of the electric field vector of the CPM wave, which is drawn based on Eqs. (2) and (3). θ is 0°, which assumes that the CPM wave propagates along the z-axis direction only when computing the findings depicted in Fig. 3. In the case of the CPM wave, the short-period intensity modulation is lost, and the concept of envelope disappears, as depicted in Fig. 3(a). The envelope in the AM wave matches the long-period intensity modulation in the CPM wave depicted in Fig. 3(a). Δλ can adjust the period of intensity modulation of the CPM wave. Figure 3(b) shows that the direction of the electric field vector rotates with propagation in the short-period of the wavelength region of the light, as in the case of ordinary circularly polarized light.

Fig. 3. Analysis results of the modulation of (a) the intensity and (b) the variation of the direction in the electric field vector of the CPM wave.

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Figure 4 depicts the computed findings of the interference intensity distribution by double heterodyne interference when the two CPM waves, in which the electric field vectors rotate in the opposite direction, interfered with each other. Other analysis conditions are similar to the analysis of double heterodyne interference using the AM wave depicted in Fig. 2. Due to the CPM wave’s intensity modulation, there is a long-period intensity modulation in the interference field. Additionally, in the double heterodyne interference electric field by the CPM waves, the polarization azimuth varies along the x-axis direction is similar to the usual two-beam right- and left-handed circular polarization interference [15,16]. Therefore, the polarization azimuth’s short-period variation in this interference field does not affect the light intensity distribution. Thus, the short-period intensity modulation due to the double heterodyne interference by the AM waves depicted in Fig. 2 does not appear in the interference field by the CPM waves, and it becomes simple to detect the interference fringe’s position using the image sensor.

Fig. 4. Calculated intensity distributions of the double heterodyne interference field when the polarization states of the interfering four beams are set in the circular polarization. The sample thicknesses d are set to (a) 1 μm and (b) 1 mm.

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Finally, the process of deriving the sample thickness d from the interference fringes’ position by the double heterodyne interference of this research is described below. The wavelength L, which is the electric field modulation period of the CPM wave, is determined as $L = {{2\lambda (\lambda + \Delta \lambda )} / {\Delta \lambda }}$ using the wave number of ${{\pi \Delta \lambda } / {\lambda (\lambda + \Delta \lambda )}}$ in Eq. (2). Furthermore, the ratio of the intensity distribution period Λ of the interference fringes and the amount of variation in the position of the interference fringes ΔΛ is equal to the ratio of the amount of phase difference δ [rad] by the sample and 2π, so that ${{\Delta \Lambda } / {\Lambda = }}{\delta / {2\pi }}$, where the phase difference δ denotes $\delta = {{2\pi nd} / L}$; n denotes the sample’s refractive index. Thus, the sample thickness d can be obtained by

(4)$$d = \frac{{\Delta \Lambda }}{\Lambda }\frac{L}{n} = \frac{{\Delta \Lambda }}{\Lambda }\frac{{2\lambda (\lambda + \Delta \lambda )}}{{n\Delta \lambda }}.$$

4. Conclusions

In conclusion, in this research, we theoretically suggest an approach that can measure the thickness of a high dynamic range from nm to mm using the interference of two modulated waves generated by two-beam on-axis heterodyne interference. Additionally, we also depict that simple measurements can be attained by adopting circularly polarized heterodyne interference modulated waves. We will perform experiments in the future to experimentally show the suggested approach. Furthermore, we will consider the application in on-axis interference in the future and aim to implement a measurement approach that does not depend on the incident angle θ, to implement two-dimensional profile measurement, and to expand the dynamic range further.

Funding

Japan Society for the Promotion of Science (JP22K14622).

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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High dynamic range thickness measurement using a double heterodyne interferometer

Abstract

1. Introduction

2. Theoretical analysis conditions

3. Results and discussion

4. Conclusions

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (4)

Equations (4)

Optics Continuum