As the integration and densification of semiconductors intensify, various associated process technologies are continuously advancing [1]. Consequently, the chemical mechanical polishing (CMP) process, which involves polishing a wafer’s surface, is also continuously researched [2, 3]. The slurry used as an abrasive in this CMP process is composed of various substances such as water, corrosion inhibitor, dispersant, and polishing enhancer, which are carefully blended and managed in appropriate proportions and sizes. The majority of defects that occur in the CMP process are due to improper management of slurry particles, resulting in scratches on the wafer’s surface [4]. For this reason, continuous monitoring of the slurry characteristics is essential in CMP processes, because even slight variations in the size, roughness, presence of agglomerates (uniform distribution), or concentration of slurry particles can significantly affect the high-flatness processing of the wafer in CMP [5, 6].

The current methods for analyzing particle size include scanning electron microscopy [7], transmission electron microscopy [8], fractional creaming [9], dark-field microscopy [10], flow ultramicroscopy [11], electrical sensing zone [12], light scattering [13, 14], sedimentation [15], and hydrodynamic chromatography [16]. Some of these methods may involve complex sample preparation, and these methods generally take a long time to measure and analyze their signals. They also have limitations in measuring particle size and accuracy, depending on the characteristics of the particles. Therefore, a comparative analysis of the advantages and disadvantages should be conducted to determine the suitability of applying these methods to the inline process of CMP [17–19].

Here we select the single-particle optical sizing (SPOS) method, based on the principle of light scattering, owing to its ability to provide short analysis time, wide particle-size analysis range, high precision, and measurement of particle-size distribution over a wide range [17, 19, 20]. Furthermore, the SPOS method enables real-time, high-resolution measurement of the size and quantity of small particles [21].

The SPOS method is a technique that measures the size of particles utilizing either light absorption or scattering. It is classified into two main categories, the partial-detection [22] and full-detection [23] methods, depending on the area of the optical sensing zone (OSZ) within the cell through which the particles flow [24]. Typically, with a liquid particle counter, a prism-shaped flow cell is used [19, 23, 24].

Also, the beams that enter the OSZ used in the SPOS method can be classified into circular Gaussian beams, elliptical Gaussian beams, focusing beams, or collimated beams, depending on their shapes [25]. Each beam shape has different advantages and disadvantages, and commonly SPOS uses focusing or collimated beams [24]. The partial-detection method based on a focusing beam is advantageous, because it measures even small particles through the high irradiance of light at the focal position. However, a disadvantage is that particles outside the OSZ cannot be detected, resulting in reduced counting efficiency for different particle sizes [24]. In contrast, the full-detection method using a collimated beam has the disadvantage of low irradiance of light, making it difficult to measure small particles, but it has the advantage that almost all particles are inside the OSZ, resulting in a uniform counting efficiency for different particle sizes [24, 26].

In this paper we propose a line-beam shape that combines the advantages of the full-detection method using collimated beams and the partial-detection method using focusing beams, while mitigating their respective disadvantages to some extent [27]. In addition, we propose the use of SPOS, which improves the measurement of small particles and counting efficiency by focusing a line beam while irradiating it upon the entire area of the flow cell. Also, to minimize the destabilized signal caused by the internal reflection of the optical system that focuses the laser beam into the flow cell in an improved particle counter with decentered multiple lenses (DMLs), a line-beam optical system based on the theory of paraxial ray tracing is applied to the particle counter without any design modifications, after validating its performance through ray-tracing simulations.

2.1. Theoretical Calculation and Optical Design of a Line Beam Formed by DMLs

In general, the analysis of particle counters is based on Mie scattering theory, established by Gustav Mie [28]. The physical factors that influence the intensity of light scattered from particles are determined by the angle and distance, which are determined by the physical size of the particle counter. Therefore, the optical variables that influence the measured intensity of scattered light are limited to the intensity of the incident light and the size of the measured particles [28, 29].

However, when constructing a particle counter that can measure submicrometer particles using a centered optical system with high intensity of incident light, as shown in Fig. 1(a), a back-reflection beam generated by the flow cell is observed, as shown in Fig. 1(b). When the back-reflection beam returns to the laser it can damage the laser diode, or lead to mode hopping, amplitude modulation, or frequency shifting. As a result, in high-power applications the back-reflection light can result in instability and output spikes [30, 31]. In a particle counter, when particles are not flowing through the flow cell, it may be misunderstood that there is no light incident on the detector. However, due to scratch-dig according to military specification MIL-F-48616, as well as bubbles, inclusions, and additives present on the surface and inside the flow cell, scattered signals exist, with intensity varying in proportion with the unstable laser output. We refer to this as a destabilized signal, as shown in Fig. 1(c). Optical isolators are commonly used to prevent this, but such requires additional space and cost.

Figure 1. Destabilized scattered-light-intensity signals caused by back-reflection of optical components in a no-particle state: (a) Description of the back-reflection beam generated in front of the flow cell, (b) photograph taken at a slightly tilted angle to verify (a), and (c) measured destabilized signal.

Therefore, a multiple-lens line-beam optical system for a particle counter is designed as a decentered type, to reduce space and cost while preventing back-reflection of light. The aforementioned design can be achieved using optical-design software. However, if there are sufficient design factors, the design variables of the DMLs optical system can be theoretically derived. The theoretical approach presented in this paper enables faster design progress, without the need for expensive software. Although a clear analysis of the performance in terms of aberration is not provided, it can be inferred that no significant issues related to aberration will emerge, as the beam diameter of the laser used in the application is mostly below a few millimeters, and the tilt angle of the light source is equal to or less than 5 degrees, with angular differences between sine and radian only on the order of 10⁻⁴.

Table 1 lists the specifications for a particle counter with a line beam. A diode-pumped solid-state (DPSS) laser (CNI Laser, 200 mW; Changchun New Industries Optoelectronics Tech. Co., Ltd., Changchun, China) with a wavelength of 532 nm is used as the light source. It features an input-beam diameter of 1.2 mm and a divergence angle of 1.2 mrad. Using these optical components, the optical system of particle counter with a centered line beam is arranged as shown in Fig. 2. In the case of a long optical axis, it is necessary to reduce the input beam diameter of the DPSS laser to fit adequately into the 400 μm width of the flow cell. This can be achieved by using the spherical Lenses 1 and 2 to reduce the beam size. To ensure that the width of the beam matches the width of the flow cell inside, which is defined by the full width at half maximum (FWHM), the size of the reduced long-axis beam needs to be designed to have a Gaussian beam width of approximately 820 μm. In the case of a short optical axis, to measure standard particles with a diameter of 0.5 μm, a line beam with a width of approximately 28 μm is required. Therefore a cylindrical Lens 3, featuring a relatively short focal length, is necessary.

Figure 2. Layout of the optical system of a particle counter using a centered line beam.

TABLE 1. Design specifications.

Parameter	Value
Source	532 nm DPSS Laser (CNI Laser, 200 mW)
Input Beam Diameter D0 (mm)	1.2
Divergence θ0 (mrad)	1.2
Long-axis Length of Line Beam (μm)	820
Short-axis Length of Line Beam Ds (μm)	28
Flow cell Material	Fused silica
Flow cell Width (μm)	400

In the layout of an optical system of a particle counter using a centered line beam, shown in Fig. 2, the measured signal is affected by destabilized signals generated by the back-reflection beam. To prevent this, each element should be implemented as a decentered type, as shown in Fig. 3, to ensure that the back-reflection beam does not return as a laser. The light source enters obliquely from Lens 1, while Lenses 2 and 3 deviate from the optical axis. To design the DMLs line-beam shown in Fig. 3, the 18 design variables listed below need to be properly integrated and adjusted to meet the specifications. These variables include the angle, distance, lens thickness, radius of curvature of the lens, decentered quantity, and thickness of the flow cell. A detailed explanation of the 18 design variables is as follows:

Figure 3. Layout of a decentered line-beam optical system, to prevent the back-reflection generated by various optical components.

α: Angle of incidence at which the light source is tiltedfrom the optical axis.

h_i (i = 1, 2, 3): Decentered value of the i^th plano-convex lens.

r_i (i = 1, 2, 3): Radius of curvature of the i^th plano-convex lens.

t_i (i = 1, 2, 3): Central thickness of the i^th convex lens.

L_i (i = 1, 2, 3, 4): Specified distance between optical components.

t_a: Distance from the second surface of Lens 3 to the first surface of the flow cell.

t_s: Central front-wall thickness.

t_w: Distance between the surface through which the fluid passes and the focal plane.

L_T: Distance between the center of the light source and the focal plane.

Given that the light source is characterized by a single wavelength, there is no need to consider chromatic aberration. Therefore, all lens materials are N-BK7 (n = 1.519), as shown in Table 1. The material of the flow cell is fused silica (n = 1.461), and the refractive index of water is 1.335. r_i is directly related to the effective focal length (EFL_i), because the shape of lens is plano-convex. In addition, t_s and t_w are variables determined by the specifications of the flow cell, and L_i is a variable related to the effective focal length of each lens within the desired total length L_T, and t_i and t_a allow small design adjustments for the total distance. Once EFL_i is determined, the user can prevent the light reflected from each lens surface, starting from 2.1, from reverting back to the light source for α and h_i.

All equations from Sections 2.2 through 2.4, except for Eq. (3), are calculated based on the coordinates of the meridional rays. Additionally, all of the meridional rays from top to bottom do not return to the source, and because of this the sagittal rays are also affected and do not return.

2.2. Conditions for Lens 1

In conventional lens design, the curvature is formed in the direction of incident parallel light to achieve efficient aberration reduction, as shown in Fig. 4(a). When a curved surface is formed in the direction of incident light, even when an oblique ray is incident, the light regresses to the light source, owing to internal reflection. Therefore, it is preferable to construct the curvature in the direction of the light emitted from the lens, as shown in Fig. 4(b).

Figure 4. Optical path of the reflection beam caused by the curvature direction of the front surface of Lens 1: (a) Front surface with a convex shape, (b) front surface with a flat shape.

In addition, for a ray of light originating from a tilted light source not to return to the light source after being reflected from the first surface of Lens 1, the height of the ray y originating from the top of the laser aperture, as shown in Fig. 5, must be lower than the height of the light source y′ when it is reflected back on the lens surface. To achieve this, the marginal rays starting from the bottom of the laser aperture must pass through the principal-focal-point coordinates of the first lens. When this is arranged in terms of the variables D₀, α, and L₁, the result can be readily expressed by Eq. (1), whereas the result for h₁ is easily expressed by Eq. (2).

Figure 5. Variables related to the angle of incidence on the first
surface of Lens 1.

The condition for the reflection ray not to return to the light source is that the radius of curvature of the second surface of Lens 1 must be greater than the value of the center thickness, as shown in Fig. 6(a). The radius of curvature is generally designed to be larger than the center thickness, owing to the low machinability of the lens with a hemispherical shape. For reference, in cases where the radius of curvature is the same or smaller than the center thickness, the ray will return to the light source, as shown in Figs. 6(b) and 6(c).

Figure 6. Path of the reflection ray caused by the curvature direction of the second surface of Lens 1 : (a) r₁ > t₁; (b) r₁ = t₁; (c) r₁ < t₁.

2.3. Conditions for Lens 2

Figures 7 and 8 show the paths of reflection rays caused by the curvature direction of the first and second surfaces of Lens 2, respectively. First, the focal length of Lens 2 can be calculated based on the focal length of Lens 1 and the magnification of the long-axis beam. Another function of Lens 2 is to transmit collimated light to Lens 3. To set the principal ray parallel to the optical axis, the center of the lens must be at the same height as the focal point of Lens 1. As a result, h₂ can be defined by Eq. (3).

Figure 7. Path of the reflection ray caused by the curvature direction of the first surface of Lens 2.

Figure 8. Path of the reflection ray caused by the curvature direction of the second surface of Lens 2.

To generate marginal rays parallel to the optical axis, the distance from Lens 1 is equal to the sum of BFL₁ and EFL₂. Also, Lens 1 is a plano-convex lens with curvature formed on its rear surface, and BFL₁ and EFL₁ take the same value, which can be easily calculated.

After designing Lens 2, for the reflection ray not to return to the light source, it must be ensured that the marginal rays originating from the first surface center of Lens 1 (which are reflected by Lens 2) do not return to Lens 1. When this ray is reflected from the front surface of Lens 2, the height is closest to that of the optical axis with a negative sign, as shown in Fig. 7. When it is reflected from the rear surface, the height is closest to that of the optical axis with a positive sign, as shown in Fig. 8. The result of the condition established by this definition is expressed by Eqs. (4) and (5), respectively. In the case of Eq. (4), the left-hand side is set to 0 by moving D₁ / 2 to the right-hand side. In the case of Eq. (5), the left-hand side is set to 0 by including h₂ on the right-hand side.

where D₁ and u₀ are the diameter of Lens 1 and the radian value of α, respectively.

where u_t1 is the radian value of the angle formed between the marginal ray emitted from Lens 1 and the optical axis.

2.4. Conditions for Lens 3

Once the size of the short beam is determined according to the design specifications, the focal length of Lens 3 (which is a cylindrical lens) is easily calculated, resulting in the determination of r₃. As shown in Fig. 9, when the vertex of Lens 3 aligns with the extension line below the outer diameter of Lens 2, the light emitted from Lens 2 can be prevented from reflecting on the first surface of Lens 3 and returning to the light source. The mathematical expression for h₃ is given by Eq. (6).

Figure 9. Path of the reflection ray caused by the curvature direction of the first surface of Lens 3.

where D₂ is the diameter of Lens 2.

Furthermore, the value of L₃ must be sufficiently large to prevent the rays reflected from the front and rear surfaces of Lens 3 from returning to Lens 2. To prevent the rays reflected from the front surface of Lens 3 from entering the effective aperture of Lens 2, the distance L_3L must be increased by the amount specified in Eq. (7).

As shown in Fig. 10, the ray reflected from the second surface of Lens 3 is naturally reflected downward, owing to the calculated curvature of the first surface of Lens 3. The expression for the condition for L_3R that prevents this ray from entering the effective aperture of Lens 2 is given by Eq. (8).

Figure 10. Path of the reflection ray caused by the curvature direction of the second surface of Lens 3.

2.5. Conditions for EFL3 and Flow Cell

Once the desired specifications for the beam size are determined, EFL₃ of Lens 3 can be easily calculated using the predetermined magnification of Lenses 1 and 2, as shown in Fig. 11 by using Eq. (9).

Figure 11. Variables related to EFL₃ and the flow cell.

Once the specifications for the flow cell have been determined, the distance from the back of Lens 3 to the front of the flow cell (represented by the variable t_a) can be expressed by Eq. (10).

Then the distance L₄ from the front surface of Lens 3 to the center of the flow cell is given by Eq. (11).

2.6. Design Flow Chart

The flow chart shown in Fig. 12 is constructed and programmed to design optical systems quickly, satisfying the specifications of the optical system according to Eqs. (1)–(11) presented in Sections 2.2 to 2.5. First, the specifications for the light source to be used and the refractive indices of the lenses required to achieve the desired size of the line beam formed within the flow cell are entered; Factors such as overall size and cost are also taken into account. Next, the specifications for the flow cell and desired size of the lens system are entered. Then, EFL₃ and L₄ are calculated from Eqs. (9) and (11), and are used to calculate the magnifications for Lenses 1 and 2.

Figure 12. Design flow chart.

In the following step, when α from the aforementioned section 2 (which considers aberrations) and the initial EFL₁ (which influences the size of most lens systems) are entered, appropriate values for L₂ and L₃ can be set. In the case of L₂, it is determined by EFL₁ and the desired beam size, allowing for immediate assessment of goal achievement.

In the case of L₃, the design is considered complete if L_T meets the desired distance. If the desired distance is not satisfied, it is possible to modify the thickness of the lens to meet specifications, provided that the amount is small (less than a millimeter). If a large modification is needed, it can be realized by modifying EFL₁, and if necessary, α can also be modified to complete the design.

Therefore, this study uses the design flow chart shown in Fig. 12 to simplify the design process for optical designers to easily create an optical system that forms a line beam to prevent back-reflection, using 18 variables. The design is thus simplified so that the optimized solution can be derived by inputting only a few variables.

3.1. Assembly of Optical System with DMLs and Its Performance

The flow chart in Fig. 12 is employed to eliminate random fluctuations in the laser beam caused by back-reflections. This ensures that the internally reflected beam does not regress into the laser, using the algorithm based on the theoretical equations presented in Section 2. Through this process, 13 variables related to lens shape and arrangement listed in Table 2, and 5 variables related to laser incidence angle and flow-cell specifications shown in Table 3, are designed.

TABLE 2. Thirteen variables related to the shape and arrangement of the lenses.

Variables
i	ri (mm)	ti (mm)	hi (mm)	Li (mm)
1	7.792	1.500	0.617	7.056
2	5.195	1.500	−1.312	27.012
3	4.000	2.000	−3.812	9.606
4	-	-	-	8.825

TABLE 3. 5 variables related to the laser incident angle and specifications for the flow cell.

Variables
α[°]	ta (mm)	ts (mm)	tw (mm)	LT (mm)
−5	5.325	1.000	0.500	52.500

Figure 13, obtained using the values from Tables 2 and 3, represents the analysis of rays regressing from each lens surface using the lens-design simulation software (LightTool 2022; Synopsys, CA, USA). Figure 13 shows that none of the rays reflected from 13(a) the front surface of Lens 1, 13(b) the rear surface of Lens 1, 13(c) the front surface of Lens 2, 13(d) the rear surface of Lens 2, 13(e) the front surface of Lens 3, and 13(f) the rear surface of Lens 3 return to the light source.

Figure 13. Simulation results for the ray returning after reflection from each lens surface: (a) Front surface of Lens 1, (b) rear surface of Lens 1, (c) front surface of Lens 2, (d) rear surface of Lens 2, (e) front surface of Lens 3, and (f) rear surface of Lens 3.

Figure 14 shows a photograph produced by combining optical modules assembled with Lens 1, Lens 2, and Lens 3 designed according to the results shown in Fig. 13, and a laser.

Figure 14. Fabricated decentered multiple lenses (DMLs) line-beam optical system.

3.2. Experimental Setup of the Particle Counter

The data obtained by measuring the beam profile of the laser in the optical system shown in Fig. 14, using a beam profiler (SP620; Ophir Optronics Solutions, Inc., Jerusalem, Israel), are analyzed and compared to simulation data.

The Gaussian distribution of the laser beam is measured using a beam profiler, as shown in Fig. 15(a), and applied in the simulation software, as shown in Fig. 15(b). Furthermore, the Gaussian beam implemented as shown in Fig. 15(b) is simulated by using the lens-design software to generate the shape of the beam with the lens system included, as shown in Fig. 16(a), at the central position of the flow cell. Furthermore, when comparing Fig. 16(b) (which shows the measurement of a line beam produced in an optical system designed for a particle counter using a decentered line beam) to the simulation results shown in Fig. 16(a), it can be observed that they are nearly identical. The disconnection of the beam is caused by the step in the flow cell shown in Fig. 16(a). This condition is applied as-is in the simulation. However, the disconnection is not visible in the measurement results shown in Fig. 16(b) because the flow cell is removed and a beam profiler installed in its place.

Figure 15. Gaussian beam of the laser: (a) Measured data using a beam profiler, (b) applied data in simulation.

Figure 16. Gaussian line beam distribution at the focal position: (a) simulation results in the flow cell, (b) line beam measured using a beam profiler after removing the flow cell.

Using an optical system constructed as shown in Fig. 14, a particle counter is fabricated and assembled, as depicted in Fig. 17. According to the schematic diagram shown in Fig. 17(a), the emitted laser beam passes through a decentered laser module, which is constructed similarly to that shown in Fig. 14. The generated line beam enters the flow cell, where scattered light from particles in the flowing water is first transmitted by the scattered-light-detection optical system and detected by an avalanche photodiode (APD; Hamamatsu Photonics, Shizuoka, Japan). The image of the particle counter using a decentered line beam, designed according to this schematic, is shown in Fig. 17(b).

Figure 17. Fabricated particle counter using the decentered multiple lenses (DMLs) line-beam optical system: (a) Schematic diagram; (b) assembled device.

This fabricated optical system is tested according to ISO 21501-2:2019 [32]. The sizes of the particles used in the experiment are 0.5, 0.6, 1.0, and 5.0 μm. These particles are diluted according to predetermined criteria in deionized water. For the control of variables, the flow rate is set at 15 mL/min, and the measurement time is 1 min. The particle size and concentration per milliliter are measured using the light-scattering method, and the raw data are analyzed in detail using the span value of the pulse height distribution (PHD) method. PHD refers to the cumulative variation in electrical pulses that occurs when a particle passes through a passage illuminated by light of constant intensity, as shown in Fig. 18. The value of the span can be calculated using Eq. (12).

Figure 18. Analysis results of pulse height distribution (PHD) for normalized count values and cumulative rate measured by particle size.

where D10, D50, and D90 represent the values of the cumulative distribution at 10%, 50%, and 90% relative to the maximum value, respectively.

By conducting experiments as mentioned in Section 3.2, we are able to obtain a normalized graph of particle counts for the distribution of signal magnitudes (in mV) observed when measuring each standard particle, as shown in Fig. 18. The red curve represents data for particles with a standard size of 0.5 μm, the green curve represents data for particles with a standard size of 0.6 μm, the blue curve represents data for particles with a standard size of 1.0 μm, and the purple curve represents data for particles with a standard size of 5.0 μm. The x-axis of the graph shows the distribution of signal magnitude measured for each particle size, on a logarithmic scale. The y-axis of the graph for the solid lines shows the normalized value of the particle count measured according to the distribution of signal magnitude for each particle. The other y-axis of the graph, for dash-dotted lines, represents the cumulative rate, which is the accumulated sum of the y-axis values of each solid line.

The values of D10, D50, and D90 are calculated from the cumulative rate of particle size represented by each dash-dotted line in Fig. 18. Using these data, the span values for particles of sizes 0.5, 0.6, and 1.0 μm are calculated from Eq. (12) to range from 1.02 to 1.03. For particles of size 5.0 μm, which is ten times larger than those of 0.5 μm particles, the span value is calculated to be 1.35, which is approximately 32% larger. Referring to a high-precision particle counter, it is found that the span value for particles of size 49 μm (which is seven times that of the span value for 7 μm particles) is approximately 35% greater [33]. Therefore, it is confirmed that the span values for the developed particle counter minimize variations in the range of 0.5 μm to 5.0 μm. Additionally, since the measured signal waveform is in a Gaussian pulse shape favorable for signal processing [34], it is confirmed that a high-precision particle counter can be produced using the developed design algorithm, without any design modifications.

To enhance the counting accuracy for particles with sizes ranging from 0.5 to 5.0 μm and improve the accuracy of particle-size measurement, we designed a particle counter with dimensions of 820 μm × 28 μm using a Gaussian line beam formed by DMLs. Additionally, to minimize destabilized signal, measures were taken to prevent the light from returning to the laser light source. Furthermore, several formulas and algorithms have been presented to facilitate the design of the lens system.

After designing and manufacturing the optical system using the aforementioned method, the line beam was measured using beam profiling. The experimental results were notably similar to the simulation results, and confirmed that the light does not return to the light source.

A particle-size analyzer was developed using the aforementioned optical system, and particle sizes were analyzed using the ISO 21501-2:2019 method for sizes of 0.5, 0.6, 1.0, and 5.0 μm. The calculated span values ranged from approximately 1.02 to 1.35, indicating that variations had been minimized. Since the obtained graph exhibited a Gaussian shape, which is highly advantageous for signal processing, it was confirmed that a high-precision particle counter could be produced. In the future, we plan to investigate the measurement of particle sizes down to 0.5 μm or smaller.

This work was supported by the Technology Innovation Program (Grant No. 20004269, Development of LPC Large Particle Counter for the slurry quality monitoring in CMP process with Raman spectroscopy) funded By the Ministry of Trade, Industry and Energy (MOTIE, Korea).

The authors declare no conflict of interest.

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.