Abstract

Polygon mirror scanners are attracting considerable interest owing to their rapid speed and large scanning area. Here, we focused on the back-reflection effect of the polygon scanner. A new polygon scanner system was designed based on a geometric analysis. The final equations for the design express the position of the laser beam source having the largest scanning length without the reflected beam traveling back to the fiber. The proposed system performed a raster scan on an area. Additionally, a paint stripping experiment was conducted to demonstrate the potential use of our scanner in commercial laser cleaning applications.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

Laser cleaning has emerged as a potential method for removing contaminants from surfaces during manufacturing. Examples of contaminants that may be removed include adhered corrosion crusts, stable dirt deposits, aged preservation resins, fire damage, thick nonmetallic paint coatings, and compounds of metal powder with epoxy [16]. The essential requirement of a laser cleaning system is to completely remove contaminants without damaging the substrate. As such, pulse lasers are more attractive than continuous-wave sources because of their lower thermal interaction during processing and superior precision. To date, multiple-hundred-watt lasers have been developed, and powerful systems are now available for efficient ablation. Unfortunately, the velocity of traditional beam delivery techniques, such as a galvanometer mirror scanner or a fixed beam with movable stages, is inadequate for use with laser types that have high pulse repetition rates. As a result, various studies have focused on developing alternative methods to control the deflection of laser beams.

Based on this principle, there are several classes of scanner concepts, including oscillatory (resonant, galvanometric, micro-electromechanical systems) [7,8], rotatable mirrors (polygon mirrors (PMs), pyramidal) [9,10], and refractive devices (with lenses and Risley prisms) [11,12], as well as electro- or acousto-optical systems [13]. Each concept has its own benefits and drawbacks, which are the primary criteria for suitable selection for commercial applications.

Among these, multifaceted mirror (or polygon)-based systems have received much attention. Polygon scanners are not new candidates in the field of laser scanners; The first was released in the mid-1970s in commercially available laser printers [14]. The configuration of a basic polygon scanner system includes a PM, spinning at a constant velocity, and objective lenses. Polygon scanner technology can be observed as a hardware implementation of the raster scan mode, allowing it to deliver much higher scanning speeds [15]. Numerous approaches for polygon scanners have been developed regarding several aspects. System analysis formulas for basic dimensions were introduced by Varughese and Krisha to replace pre-objective scanners with complex F-theta lenses in post-objective scan systems. Post-objective PMs achieved a nearly flat field in convergent beam scanning without F-theta lenses, which can reduce the cost and size of the scanning system [16]. In addition, fundamental factors such as the scanning field, duty cycle, and off-axis defocus in convergent beam scanning were addressed by Li and Karz [17]. In their work, via vector analysis, they highlighted that the scan field is asymmetric to the ray reflected by the polygon at a neutral scan position. In a major advance in 2013, Duma and Podoleanu briefly surveyed the different applications of PMs in biomedical imaging. This approach on PM-based broadband laser sources scanned in frequency was presented, a simple off-axis polygon configuration was analyzed, and some of the characteristic mathematical functions of the PM were inferred [18]. Duma and his group developed the characteristic functions using a MathCad analysis, and then conducted an experiment to demonstrate the validity of the theoretical findings [19]. The development of fast 2D and 3D scanners based on PMs has also gained the attention of many researchers [2022]. In these introductions, the PM was assumed to be part of the fast-scanning axis, and combined with a galvanometer mirror or other scanners as a low-speed axis.

Numerous studies on polygon scanners have been performed; however, many hypotheses regarding back-reflection problems in polygon scanners appear to be ill-defined. Unfortunately, reflected beams can travel back to the laser source and damage the system. Therefore, the optical characteristics of the scan mirror must be understood. Although the main purpose of this study was to design a polygon scanner without back reflection, we also examined the feasibility of applying our polygon scanner for surface cleaning and optimized the scanner parameters to improve the efficiency of laser cleaning.

2. Methods

2.1 Non-back-reflecting polygon scanner design

In this study, the system was described in a 2D local ${O_{jk}}$ system of coordinates, and rotational motion was expressed through rotating coordinates $(\varphi ,\omega )$. The main parameters of the polygon scanner are shown in Fig. 1(a): collimated beam width d, number of polygon scanner head facets $n$, length of facet $a$, length of apothem $R = a/\textrm{cos}({{\pi / n}} )$, angle corresponding to half of a facet $\theta \; = \;{\pi / n}$, and eccentricity of the polygon pivot $O$ with respect to the axis of the incident beam (distance between vertical axis $Ok$ and axis of incident beam). The PM rotates with angular velocity and the rotational angle used to define the position of the PM is denoted by $\varphi $. This is the angle measured from the vertical axis $Ok$ to the apothem $R$ corresponding to the facet hit by the incident beam. At $\varphi = \textrm{ }0$, $R$ is aligned with the vertical axis $Ok$, to which the PM is perpendicular. Figure 1(b) displays three different polygon positions:${\varphi _0}$, ${\varphi _1}$, and ${\varphi _2}$. At ${\varphi _0} = {\pi / 4}$ the polygon is in a neutral position, where the reflected beam is perpendicular to the incident beam. The angles ${\varphi _1}$ and ${\varphi _2}$ correspond to the negative and positive scans, respectively.

 figure: Fig. 1.

Fig. 1. (a) Polygon mirror (PM) scanner. (b) Three different characteristic positions of the PM.

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In the ${O_{jk}}$ system of coordinates, the equation of the polygon facet on which the reflection is produced, as shown in Fig. 1(b), is given by:

$$\begin{aligned} R&\sin \varphi (j - {j_A}) - R\cos \varphi (k - {k_A}) = 0\\ &\Leftrightarrow k = \tan \varphi (j - {j_A}) + {k_A} \end{aligned}$$
Where the normal vector of the reflecting polygon facet is $\overrightarrow {OA} = (R\sin \varphi , - R\cos \varphi )$ and passes through $A = (R\sin \varphi , - R\cos \varphi )$

The equation of the incident beam is:

$$j = e$$
The incident points I are the intersections of the incident ray with the polygon facet.
$$k = \tan \varphi (e - {j_A}) + {k_A}$$
As shown in Fig. 1(a), a collimated beam of width d is illustrated by two different parallel rays. The upper and lower edge rays have different eccentricities with respect to pivot O of ${e_1}$ and ${e_2}$ respectively. In this work, the scanning angular range was defined as the angular range at which the laser beam was entirely reflected by a single facet of the polygon. Specifically, on the MN facet, as shown in Fig. 2, the scanning angular range begins at ${\varphi _{\min }}$ when the lower edge ray hits point N of the facet and ends at ${\varphi _{\max }}$ when the upper ray hits point M.

 figure: Fig. 2.

Fig. 2. Minimum and maximum scanning angles for which facet MN entirely reflects the incident beam.

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At point N:

$$\sin ({\varphi _{\min }} + \theta ) = \frac{{{e_2}\cos \theta }}{R}$$
At point M:
$$\sin ({\varphi _{\max }} - \theta ) = \frac{{{e_1}\cos \theta }}{R}$$
A polygon head with a central ray falling at the midpoint of the facet in the neutral position was designed. However, a serious issue, called back reflection, occurs. When the incident beam moves over a corner, part of the beam hits the mirror at a perpendicular or almost perpendicular angle $(\varphi = 0$ or $\varphi \cong 0)$ and is reflected back to the fiber, as shown in Fig. 3(a). This back reflection damages the fiber or even destroys the system in the case of a laser with high average power. To avoid back reflection, the incident beam should be moved far away from the pivot $O$ of the polygon, thus increasing the eccentricity $e$. However, if the incident beam is far from $O$, the scanning angular field will be reduced. Therefore, the optimal relational position between the polygon and the incident beam must be calculated.

 figure: Fig. 3.

Fig. 3. (a) Back reflection occurs when the incident beam hits the mirror at perpendicular angle. (b) Smallest scan angleφlim at which back reflection does not occur.

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Figure 3(b) shows the limit angle ${\varphi _{\lim }}$ for which the reflected beam does not enter the collimator. At this position, the upper edge rays collide with the corners of the polygon. The collimator radius is denoted as r, and the collimated beam width is d. The collimator was installed at a distance l from the pivot.

Smallest angle:

$$\tan 2{\varphi _{\lim }} = \frac{{r + {d / 2}}}{{l - \frac{a}{{2\sin \theta }}\cos ({\varphi _{\lim }} + \theta )}}$$
Smallest eccentricity:
$${e_1} = (R\tan {\varphi _{\lim }} + \frac{a}{2})\cos {\varphi _{\lim }}$$

2.2 Experimental setup and characterization

In this study, a 1.2-kW Q-switch Nd:YAG laser (Rigel i1200, Powerlase) with the central wavelength of 1,064 nm, pulse duration of 54.4 ns, and repetition rate of 8 kHz was used for the paint stripping process. The 4.2-mm diameter laser beam had a super-Gaussian energy distribution with the laser beam quality factor (M2) of 29.7 horizontally and 30.2 vertically. The maximum average power on the workpiece surface was 1,132 W when all losses in the optical path were considered.

A rotating PM was used to control the beam and deliver it to the sample surface. After collimation, the 12-mm diameter beam was reflected by the surface of the PM and focused with the spot size of 4.2 mm using an F-theta lens with the scan field of 160 mm × 160 mm. The maximum scan speed ${\nu _{sc}}$ that could be achieved was 30.5 m/s. Figure 4 shows a schematic of the polygon scanner setup.

 figure: Fig. 4.

Fig. 4. Schematic of the portable polygon scanner.

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A hexagonal scanning head was created based on the geometry analysis. The details of the system are presented in Table 1. A charge-coupled device camera (Ophir Photonics) was used to measure the scanning length and beam characteristics. The graph in Fig. 5 presents the beam diameter in both the horizontal (x) and vertical (y) directions at different scanning positions. The scanning length was 305 mm; however, the horizontal and vertical beam diameters changed during scanning. They remain almost equal from –75 to 30 mm, but beyond this range, the beam has an elliptical shape and hence is not round. The beam size decreased in the x-direction and increased in the y-direction. This phenomenon may be due to the fact that the F-theta lens does not have a perfectly flat focal plane, and aberrations occur. Another explanation for this is that the beam cannot deliver 100% power when the incident beam travels through the edge of the polygon. Therefore, the beam size in the horizontal direction (x) was trimmed significantly at the end of the scanning line. Moreover, as expected, this investigation also indicated that the scan field is asymmetric to the ray reflected by the polygon at a neutral scan position (x = 0). A significant difference was found in negative scanning and positive scanning up to the length of 95 mm. Clearly, this is the result of satisfying the smallest eccentricity.

 figure: Fig. 5.

Fig. 5. Beam size versus horizontal x position.

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Tables Icon

Table 1. Correlation of the Experimental Results with the Theory.

2.3 Paint stripping experiment

One of the advantages of PM scanners is their stable scanning velocity, unlike galvanometer-based (GB) scanners. In raster scanning, the velocity of the scanning beam in a GB system reaches 0 at the start and end of the scanned lines, producing a higher power than intended. In contrast, the PM scanner has a constant spot overlap and constant power delivery. When the beam traverses the corner between two facet mirrors, the laser is shut off, and the beam position flips to the beginning of the line again.

Normally, rotating scanners scan only in one direction, and a secondary linear motion is required to obtain a 2D scanning system. The direction of the secondary motion is perpendicular to the line scanned by the polygon scanner and is synchronized in speed with the rotational speed of the polygon to obtain a line-by-line scan of the target surface. This type of scanning is often referred to as raster scanning.

Polygon scanners have diverse configurations. They range from prismatic to pyramidal, and from normal to inverse. Among these, prismatic and normal polygon scanners are the most common. The principal advantage of prismatic polygon scanners compared with pyramidal ones is that more facets can be created. In addition, regular shapes have fewer mechanical resistance issues than inverted shapes. Therefore, this was the type of device that was constructed in our study.

Normally, a rotating regular PM spins at a constant speed around the vertical rotation axis. By reflection of identical facets at 90° to the incident beam, a line sweeps repeatedly across the scanning target. To increase the volume ablation rate, polygon scanners have facets set at different angles in the vertical direction, as shown in Fig. 6(a). The first and the fourth sides have no rope cut, but the second and the fifth sides are loft cut from top to bottom, and the third and sixth sides are loft cut from bottom to top. During the operation, our polygon scanning provides parallel sweeping, as shown in Fig. 6(b).

 figure: Fig. 6.

Fig. 6. (a) Six-facet polygonal scanner with varying facet angles. (b) Variation in scan line on the specimen.

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3. Results

3.1 Non-back-reflecting polygon scanner design

A demonstration was conducted to verify the accuracy of the mathematical model. The dimensions of the hexagonal mirror are detailed in Table 1. A laser MGL-H-532-500 mW from TeleOptics with the wavelength of 532 μm was used instead of a high-average-power laser so beam light was visible. A laser with the beam diameter of 7.46 mm and collimator diameter of 8 mm was fixed at the distance of 200 mm from O of the hexagonal mirror. The smallest eccentricity to avoid back-reflecting beam was calculated based on Eq. (7) as 32.95 mm. As shown in Fig. 7, when the incident beam travels through the edge of the hexagon, the reflecting beam does not enter the laser collimator. Thus, even at ${\varphi _{\lim }} = {1.45^0}$, a collimator is warranted.

 figure: Fig. 7.

Fig. 7. Attachment of the non-back reflecting beam system. See Visualization 1

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3.2 Paint stripping result

In this work, high-rate laser paint stripping was investigated using a high average laser power in the range of 395–762 W and a fast polygon scanner. As shown in Fig. 8(a), a painting sample with the width of 6,560 μm was stripped, and the beam focus size was 4,200 μm. The extra ∼2,400 μm or stripped paint represents the increase of 57% in the scanning area. However, this parallel sweeping design results in an uneven fluence, and the paint was not stripped completely, as shown in Fig. 8(b).

 figure: Fig. 8.

Fig. 8. (a) 6,560-μm paint stripping width. (b) Digital image of residual parts after laser cleaning.

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To clean the paint more efficiently, the polygon spinning speed and average laser power were analyzed. Table 2 details the paint-processing speeds of the experimental scanner. The samples were prepared by painting an ASTM-A240M-304 steel specimen, with dimensions of 100 mm × 100 mm, with the black paint from Shinhanart (the main components of the black pigment are particles of carbon and gum Arabic – a natural complex mixture of polysaccharides and hemicellulose). The spinning speed in the range of 100–1,000 rpm and power of 395–762 W were examined. The polygon was controlled to spin for 10 revolutions. The scanning area was analyzed to determine which parameters were the most effective. The target is to completely strip paint in the area through which the laser beam passes. Considerable attention must be paid to the edge of the scanning line. This part is most difficult to remove completely; therefore, this work will be performed if the edge of the laser scanning line is shaped.

Tables Icon

Table 2. Processing Speed of the Polygon Scanning System.

Figure 9 shows the edges of the scanning area. It is apparent that when the hexagon spins faster than 200 rpm and with power less than or equal to 548 W, the edges of the scanning area are unclear. The images also indicate that spinning at high speeds, such as 500 or 1,000 rpm, cannot clean the paint completely. The scanning width was also measured to confirm that the scanning area was smaller than the design under the conditions described above. Although spinning at 100 rpm resulted in a sharp and clear edge, we chose 200 rpm at 652 W as the optimal conditions based on the scanning time and power consumption, and used these settings for the following experiments.

 figure: Fig. 9.

Fig. 9. Influence of spinning speed (rpm) and laser average power (W) on paint stripping efficiency.

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A comparison of the sample before painting and after cleaning is shown in Fig. 10. The original bare sample was painted and then stripped of the paint using our laser at the optimal settings. No significant differences can be seen between the bare substrate and that after cleaning at the laser power of 652 W and polygon spinning speed of 200 rpm. The single most striking observation is that paint contamination did not appear in the laser-scribed part. In addition, the surface morphology implies that the substrate was not melted or damaged after laser scanning.

 figure: Fig. 10.

Fig. 10. Images of (a) bare sample, (b) painting sample, and (c) cleaned sample.

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Figure 11 further strengthens our confidence in paint stripping using our polygon scanner. Figure 11(a) illustrates a well-cleaned part achieved using the laser. Additionally, the SEM images highlight a shaped edge without paint redundancy and no damage to the substrate surface. Figure 11(b) and (c) exhibit the difference between the paint-cleaning results before and after optimization of the laser power and polygon spin speed. Carbon – main component of paint covered iron – main component of substrate. If the carbon was not wiped, the iron could not be revealed. Before the investigation, the carbon was not cleaned completely and it is difficult to determine the part that was cleaned using a laser. However, after finding the best conditions, the carbon was almost lifted off and it was easy to observe the cleaned and non-cleaned parts. These results provide vital evidence for the feasibility of our polygon scanner in surface cleaning applications.

 figure: Fig. 11.

Fig. 11. (a) Laser cleaning under optimal conditions. EDS SEM images of laser scanning edge at (b) 300 rpm and 762 W and (c) 200 rpm and 652 W.

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4. Conclusions

Briefly, this paper introduced an exceptionally simple and rapid fabrication approach for a 1D scanner using a rotating PM and an F-theta lens. A detailed analysis of the scanner head was performed, leading to a design that avoids back reflection, which is an important requirement in the field of short-pulse lasers with high average power. Further, this design produces a scan area with a scanning length of up to 305 mm and width of 6.6 mm. Moreover, the scanning speed can reach 30.5 m/s. Finally, a paint stripping experiment demonstrated the efficacy of the PM for potential use in commercial applications.

Funding

NK 232A.

Acknowledgments

This research was supported by the NK 232A. Also, thank to Dr. Nguyen-Hung Tran for supporting the revision of the manuscript.

Disclosures

The authors declare no conflicts of interest.

Data availability

No data were generated or analyzed in the presented research.

References

1. S. Milles, M. Soldera, T. Kuntze, and A. F. Lasagni, “Characterization of self-cleaning properties on superhydrophobic aluminum surfaces fabricated by direct laser writing and direct laser interference patterning,” Appl. Surf. Sci. 525, 146518 (2020). [CrossRef]  

2. M. K. A. A. Razab, A. Mohamed Noor, M. Suhaimi Jaafar, N. H. Abdullah, F. M. Suhaimi, M. Mohamed, N. Adam, and N. A. A. N. Yusuf, “A review of incorporating Nd:YAG laser cleaning principal in automotive industry,” J. Radiat. Res. Appl. Sci. 11(4), 393–402 (2018). [CrossRef]  

3. S. Zhuang, S. Kainuma, M. Yang, M. Haraguchi, and T. Asano, “Characterizing corrosion properties of carbon steel affected by high-power laser cleaning,” Constr. Build. Mater. 274, 122085 (2021). [CrossRef]  

4. A. C. Tam, W. P. Leung, W. Zapka, and W. Ziemlich, “Laser-cleaning techniques for removal of surface particulates,” J. Appl. Phys. 71(7), 3515 (1992). [CrossRef]  

5. J.-M. Lee and K. Watkins, “Laser removal of oxides and particles from copper surfaces for microelectronic fabrication,” Opt. Express 7(2), 68 (2000). [CrossRef]  

6. K. Liu and E. Garmire, “Paint removal using lasers,” Appl. Opt. 34(21), 4409 (1995). [CrossRef]  

7. J. Montagu, Scanners: Galvanometric and Resonant, in Encyclopedia of Optical and Photonic Engineering, Second Edition (CRC Press, 2015), pp. 1–22.

8. D. C. Adler, Y. Chen, R. Huber, J. Schmitt, J. Connolly, and J. G. Fujimoto, “Three-dimensional endomicroscopy using optical coherence tomography,” Nat. Photonics 1(12), 709 (2007). [CrossRef]  

9. J. Katz and Y. Li, “Laser beam scanning by rotary mirrors. I. Modeling mirror-scanning devices,” Appl. Opt. 34(28), 6403–6416 (1995). [CrossRef]  

10. V.-F. Duma, “Analysis of polygonal mirror scanning heads: from industrial to high-end applications in swept sources for OCT,” Des. Qual. Biomed. Technol. X 10056, 100560P (2017). [CrossRef]  

11. C. A. DiMarzio and C. T. Amirault, “Precision pointing using a dual-wedge scanner,” Appl. Opt. 24(9), 1302–1308 (1985). [CrossRef]  

12. W. C. W. II, S. A. Guerrera, Z. Eastman, and C. A. DiMarzio, “Efficient confocal microscopy with a dual-wedge scanner,” Three-Dimensional Multidimens. Microsc. Image Acquis. Process. XVI 7184, 71840M (2009). [CrossRef]  

13. G. R. B. E. Römer and P. Bechtold, “Electro-optic and acousto-optic laser beam scanners-Invited Paper,” Phys. Procedia 56, 29–39 (2014). [CrossRef]  

14. W. Kasdorf, The Columbia Guide to Digital Publishing (Columbia University Press, 2003).

15. R. De Loor, “Lasers in Manufacturing Conference 2013 Polygon scanner system for ultra short pulsed laser micro-machining applications,” Phys. Procedia 41, 544 (2013). [CrossRef]  

16. K. O. G. Varughese and K. S. R. Krishna, “Flattening the field of postobjective scanners by optimum choice and positioning of polygons,” Appl. Opt. 32(7), 1104–1108 (1993). [CrossRef]  

17. J. Katz and Y. Li, “Asymmetric distribution of the scanned field of a rotating reflective polygon,” Appl. Opt. 36(1), 342–352 (1997). [CrossRef]  

18. V.-F. Duma and A. G. Podoleanu, “Polygon mirror scanners in biomedical imaging: a Review,” (2013).

19. V.-F. Duma and M. Nicolov, “Numerical and experimental study of the characteristic functions of polygon scanners” (2009).

20. W. D. Van Amstel, R. J. Asjes, P. F. A. Van De Goor, and P. Merkelbach, “2 and 3D laser scanners for fast dimensional measurements and inspection,” n.d.

21. J. Schille, L. Schneider, A. Streek, S. Kloetzer, and U. Loeschner, “High-throughput machining using high average power ultrashort pulse lasers and ultrafast polygon scanner,” Opt. Eng. 55(9), 97360R (2016). [CrossRef]  

22. E. S. Lee, M. Kim, K. Moon, I. M. Lee, D. W. Park, J. H. Shin, H. S. Kim, D. H. Choi, K. S. Choi, D. H. Lee, and K. H. Park, “High-Speed and Cost-Effective Reflective Terahertz Imaging System Using a Novel 2D Beam Scanner,” J. Lightwave Technol. 38(16), 4237 (2020). [CrossRef]  

References

  • View by:

  1. S. Milles, M. Soldera, T. Kuntze, and A. F. Lasagni, “Characterization of self-cleaning properties on superhydrophobic aluminum surfaces fabricated by direct laser writing and direct laser interference patterning,” Appl. Surf. Sci. 525, 146518 (2020).
    [Crossref]
  2. M. K. A. A. Razab, A. Mohamed Noor, M. Suhaimi Jaafar, N. H. Abdullah, F. M. Suhaimi, M. Mohamed, N. Adam, and N. A. A. N. Yusuf, “A review of incorporating Nd:YAG laser cleaning principal in automotive industry,” J. Radiat. Res. Appl. Sci. 11(4), 393–402 (2018).
    [Crossref]
  3. S. Zhuang, S. Kainuma, M. Yang, M. Haraguchi, and T. Asano, “Characterizing corrosion properties of carbon steel affected by high-power laser cleaning,” Constr. Build. Mater. 274, 122085 (2021).
    [Crossref]
  4. A. C. Tam, W. P. Leung, W. Zapka, and W. Ziemlich, “Laser-cleaning techniques for removal of surface particulates,” J. Appl. Phys. 71(7), 3515 (1992).
    [Crossref]
  5. J.-M. Lee and K. Watkins, “Laser removal of oxides and particles from copper surfaces for microelectronic fabrication,” Opt. Express 7(2), 68 (2000).
    [Crossref]
  6. K. Liu and E. Garmire, “Paint removal using lasers,” Appl. Opt. 34(21), 4409 (1995).
    [Crossref]
  7. J. Montagu, Scanners: Galvanometric and Resonant, in Encyclopedia of Optical and Photonic Engineering, Second Edition (CRC Press, 2015), pp. 1–22.
  8. D. C. Adler, Y. Chen, R. Huber, J. Schmitt, J. Connolly, and J. G. Fujimoto, “Three-dimensional endomicroscopy using optical coherence tomography,” Nat. Photonics 1(12), 709 (2007).
    [Crossref]
  9. J. Katz and Y. Li, “Laser beam scanning by rotary mirrors. I. Modeling mirror-scanning devices,” Appl. Opt. 34(28), 6403–6416 (1995).
    [Crossref]
  10. V.-F. Duma, “Analysis of polygonal mirror scanning heads: from industrial to high-end applications in swept sources for OCT,” Des. Qual. Biomed. Technol. X 10056, 100560P (2017).
    [Crossref]
  11. C. A. DiMarzio and C. T. Amirault, “Precision pointing using a dual-wedge scanner,” Appl. Opt. 24(9), 1302–1308 (1985).
    [Crossref]
  12. W. C. W. II, S. A. Guerrera, Z. Eastman, and C. A. DiMarzio, “Efficient confocal microscopy with a dual-wedge scanner,” Three-Dimensional Multidimens. Microsc. Image Acquis. Process. XVI 7184, 71840M (2009).
    [Crossref]
  13. G. R. B. E. Römer and P. Bechtold, “Electro-optic and acousto-optic laser beam scanners-Invited Paper,” Phys. Procedia 56, 29–39 (2014).
    [Crossref]
  14. W. Kasdorf, ed., The Columbia Guide to Digital Publishing (Columbia University Press, 2003).
  15. R. De Loor, “Lasers in Manufacturing Conference 2013 Polygon scanner system for ultra short pulsed laser micro-machining applications,” Phys. Procedia 41, 544 (2013).
    [Crossref]
  16. K. O. G. Varughese and K. S. R. Krishna, “Flattening the field of postobjective scanners by optimum choice and positioning of polygons,” Appl. Opt. 32(7), 1104–1108 (1993).
    [Crossref]
  17. J. Katz and Y. Li, “Asymmetric distribution of the scanned field of a rotating reflective polygon,” Appl. Opt. 36(1), 342–352 (1997).
    [Crossref]
  18. V.-F. Duma and A. G. Podoleanu, “Polygon mirror scanners in biomedical imaging: a Review,” (2013).
  19. V.-F. Duma and M. Nicolov, “Numerical and experimental study of the characteristic functions of polygon scanners” (2009).
  20. W. D. Van Amstel, R. J. Asjes, P. F. A. Van De Goor, and P. Merkelbach, “2 and 3D laser scanners for fast dimensional measurements and inspection,” n.d.
  21. J. Schille, L. Schneider, A. Streek, S. Kloetzer, and U. Loeschner, “High-throughput machining using high average power ultrashort pulse lasers and ultrafast polygon scanner,” Opt. Eng. 55(9), 97360R (2016).
    [Crossref]
  22. E. S. Lee, M. Kim, K. Moon, I. M. Lee, D. W. Park, J. H. Shin, H. S. Kim, D. H. Choi, K. S. Choi, D. H. Lee, and K. H. Park, “High-Speed and Cost-Effective Reflective Terahertz Imaging System Using a Novel 2D Beam Scanner,” J. Lightwave Technol. 38(16), 4237 (2020).
    [Crossref]

2021 (1)

S. Zhuang, S. Kainuma, M. Yang, M. Haraguchi, and T. Asano, “Characterizing corrosion properties of carbon steel affected by high-power laser cleaning,” Constr. Build. Mater. 274, 122085 (2021).
[Crossref]

2020 (2)

S. Milles, M. Soldera, T. Kuntze, and A. F. Lasagni, “Characterization of self-cleaning properties on superhydrophobic aluminum surfaces fabricated by direct laser writing and direct laser interference patterning,” Appl. Surf. Sci. 525, 146518 (2020).
[Crossref]

E. S. Lee, M. Kim, K. Moon, I. M. Lee, D. W. Park, J. H. Shin, H. S. Kim, D. H. Choi, K. S. Choi, D. H. Lee, and K. H. Park, “High-Speed and Cost-Effective Reflective Terahertz Imaging System Using a Novel 2D Beam Scanner,” J. Lightwave Technol. 38(16), 4237 (2020).
[Crossref]

2018 (1)

M. K. A. A. Razab, A. Mohamed Noor, M. Suhaimi Jaafar, N. H. Abdullah, F. M. Suhaimi, M. Mohamed, N. Adam, and N. A. A. N. Yusuf, “A review of incorporating Nd:YAG laser cleaning principal in automotive industry,” J. Radiat. Res. Appl. Sci. 11(4), 393–402 (2018).
[Crossref]

2017 (1)

V.-F. Duma, “Analysis of polygonal mirror scanning heads: from industrial to high-end applications in swept sources for OCT,” Des. Qual. Biomed. Technol. X 10056, 100560P (2017).
[Crossref]

2016 (1)

J. Schille, L. Schneider, A. Streek, S. Kloetzer, and U. Loeschner, “High-throughput machining using high average power ultrashort pulse lasers and ultrafast polygon scanner,” Opt. Eng. 55(9), 97360R (2016).
[Crossref]

2014 (1)

G. R. B. E. Römer and P. Bechtold, “Electro-optic and acousto-optic laser beam scanners-Invited Paper,” Phys. Procedia 56, 29–39 (2014).
[Crossref]

2013 (1)

R. De Loor, “Lasers in Manufacturing Conference 2013 Polygon scanner system for ultra short pulsed laser micro-machining applications,” Phys. Procedia 41, 544 (2013).
[Crossref]

2009 (1)

W. C. W. II, S. A. Guerrera, Z. Eastman, and C. A. DiMarzio, “Efficient confocal microscopy with a dual-wedge scanner,” Three-Dimensional Multidimens. Microsc. Image Acquis. Process. XVI 7184, 71840M (2009).
[Crossref]

2007 (1)

D. C. Adler, Y. Chen, R. Huber, J. Schmitt, J. Connolly, and J. G. Fujimoto, “Three-dimensional endomicroscopy using optical coherence tomography,” Nat. Photonics 1(12), 709 (2007).
[Crossref]

2000 (1)

1997 (1)

1995 (2)

1993 (1)

1992 (1)

A. C. Tam, W. P. Leung, W. Zapka, and W. Ziemlich, “Laser-cleaning techniques for removal of surface particulates,” J. Appl. Phys. 71(7), 3515 (1992).
[Crossref]

1985 (1)

Abdullah, N. H.

M. K. A. A. Razab, A. Mohamed Noor, M. Suhaimi Jaafar, N. H. Abdullah, F. M. Suhaimi, M. Mohamed, N. Adam, and N. A. A. N. Yusuf, “A review of incorporating Nd:YAG laser cleaning principal in automotive industry,” J. Radiat. Res. Appl. Sci. 11(4), 393–402 (2018).
[Crossref]

Adam, N.

M. K. A. A. Razab, A. Mohamed Noor, M. Suhaimi Jaafar, N. H. Abdullah, F. M. Suhaimi, M. Mohamed, N. Adam, and N. A. A. N. Yusuf, “A review of incorporating Nd:YAG laser cleaning principal in automotive industry,” J. Radiat. Res. Appl. Sci. 11(4), 393–402 (2018).
[Crossref]

Adler, D. C.

D. C. Adler, Y. Chen, R. Huber, J. Schmitt, J. Connolly, and J. G. Fujimoto, “Three-dimensional endomicroscopy using optical coherence tomography,” Nat. Photonics 1(12), 709 (2007).
[Crossref]

Amirault, C. T.

Asano, T.

S. Zhuang, S. Kainuma, M. Yang, M. Haraguchi, and T. Asano, “Characterizing corrosion properties of carbon steel affected by high-power laser cleaning,” Constr. Build. Mater. 274, 122085 (2021).
[Crossref]

Asjes, R. J.

W. D. Van Amstel, R. J. Asjes, P. F. A. Van De Goor, and P. Merkelbach, “2 and 3D laser scanners for fast dimensional measurements and inspection,” n.d.

Bechtold, P.

G. R. B. E. Römer and P. Bechtold, “Electro-optic and acousto-optic laser beam scanners-Invited Paper,” Phys. Procedia 56, 29–39 (2014).
[Crossref]

Chen, Y.

D. C. Adler, Y. Chen, R. Huber, J. Schmitt, J. Connolly, and J. G. Fujimoto, “Three-dimensional endomicroscopy using optical coherence tomography,” Nat. Photonics 1(12), 709 (2007).
[Crossref]

Choi, D. H.

Choi, K. S.

Connolly, J.

D. C. Adler, Y. Chen, R. Huber, J. Schmitt, J. Connolly, and J. G. Fujimoto, “Three-dimensional endomicroscopy using optical coherence tomography,” Nat. Photonics 1(12), 709 (2007).
[Crossref]

De Loor, R.

R. De Loor, “Lasers in Manufacturing Conference 2013 Polygon scanner system for ultra short pulsed laser micro-machining applications,” Phys. Procedia 41, 544 (2013).
[Crossref]

DiMarzio, C. A.

W. C. W. II, S. A. Guerrera, Z. Eastman, and C. A. DiMarzio, “Efficient confocal microscopy with a dual-wedge scanner,” Three-Dimensional Multidimens. Microsc. Image Acquis. Process. XVI 7184, 71840M (2009).
[Crossref]

C. A. DiMarzio and C. T. Amirault, “Precision pointing using a dual-wedge scanner,” Appl. Opt. 24(9), 1302–1308 (1985).
[Crossref]

Duma, V.-F.

V.-F. Duma, “Analysis of polygonal mirror scanning heads: from industrial to high-end applications in swept sources for OCT,” Des. Qual. Biomed. Technol. X 10056, 100560P (2017).
[Crossref]

V.-F. Duma and M. Nicolov, “Numerical and experimental study of the characteristic functions of polygon scanners” (2009).

V.-F. Duma and A. G. Podoleanu, “Polygon mirror scanners in biomedical imaging: a Review,” (2013).

Eastman, Z.

W. C. W. II, S. A. Guerrera, Z. Eastman, and C. A. DiMarzio, “Efficient confocal microscopy with a dual-wedge scanner,” Three-Dimensional Multidimens. Microsc. Image Acquis. Process. XVI 7184, 71840M (2009).
[Crossref]

Fujimoto, J. G.

D. C. Adler, Y. Chen, R. Huber, J. Schmitt, J. Connolly, and J. G. Fujimoto, “Three-dimensional endomicroscopy using optical coherence tomography,” Nat. Photonics 1(12), 709 (2007).
[Crossref]

Garmire, E.

Guerrera, S. A.

W. C. W. II, S. A. Guerrera, Z. Eastman, and C. A. DiMarzio, “Efficient confocal microscopy with a dual-wedge scanner,” Three-Dimensional Multidimens. Microsc. Image Acquis. Process. XVI 7184, 71840M (2009).
[Crossref]

Haraguchi, M.

S. Zhuang, S. Kainuma, M. Yang, M. Haraguchi, and T. Asano, “Characterizing corrosion properties of carbon steel affected by high-power laser cleaning,” Constr. Build. Mater. 274, 122085 (2021).
[Crossref]

Huber, R.

D. C. Adler, Y. Chen, R. Huber, J. Schmitt, J. Connolly, and J. G. Fujimoto, “Three-dimensional endomicroscopy using optical coherence tomography,” Nat. Photonics 1(12), 709 (2007).
[Crossref]

Kainuma, S.

S. Zhuang, S. Kainuma, M. Yang, M. Haraguchi, and T. Asano, “Characterizing corrosion properties of carbon steel affected by high-power laser cleaning,” Constr. Build. Mater. 274, 122085 (2021).
[Crossref]

Katz, J.

Kim, H. S.

Kim, M.

Kloetzer, S.

J. Schille, L. Schneider, A. Streek, S. Kloetzer, and U. Loeschner, “High-throughput machining using high average power ultrashort pulse lasers and ultrafast polygon scanner,” Opt. Eng. 55(9), 97360R (2016).
[Crossref]

Krishna, K. S. R.

Kuntze, T.

S. Milles, M. Soldera, T. Kuntze, and A. F. Lasagni, “Characterization of self-cleaning properties on superhydrophobic aluminum surfaces fabricated by direct laser writing and direct laser interference patterning,” Appl. Surf. Sci. 525, 146518 (2020).
[Crossref]

Lasagni, A. F.

S. Milles, M. Soldera, T. Kuntze, and A. F. Lasagni, “Characterization of self-cleaning properties on superhydrophobic aluminum surfaces fabricated by direct laser writing and direct laser interference patterning,” Appl. Surf. Sci. 525, 146518 (2020).
[Crossref]

Lee, D. H.

Lee, E. S.

Lee, I. M.

Lee, J.-M.

Leung, W. P.

A. C. Tam, W. P. Leung, W. Zapka, and W. Ziemlich, “Laser-cleaning techniques for removal of surface particulates,” J. Appl. Phys. 71(7), 3515 (1992).
[Crossref]

Li, Y.

Liu, K.

Loeschner, U.

J. Schille, L. Schneider, A. Streek, S. Kloetzer, and U. Loeschner, “High-throughput machining using high average power ultrashort pulse lasers and ultrafast polygon scanner,” Opt. Eng. 55(9), 97360R (2016).
[Crossref]

Merkelbach, P.

W. D. Van Amstel, R. J. Asjes, P. F. A. Van De Goor, and P. Merkelbach, “2 and 3D laser scanners for fast dimensional measurements and inspection,” n.d.

Milles, S.

S. Milles, M. Soldera, T. Kuntze, and A. F. Lasagni, “Characterization of self-cleaning properties on superhydrophobic aluminum surfaces fabricated by direct laser writing and direct laser interference patterning,” Appl. Surf. Sci. 525, 146518 (2020).
[Crossref]

Mohamed, M.

M. K. A. A. Razab, A. Mohamed Noor, M. Suhaimi Jaafar, N. H. Abdullah, F. M. Suhaimi, M. Mohamed, N. Adam, and N. A. A. N. Yusuf, “A review of incorporating Nd:YAG laser cleaning principal in automotive industry,” J. Radiat. Res. Appl. Sci. 11(4), 393–402 (2018).
[Crossref]

Mohamed Noor, A.

M. K. A. A. Razab, A. Mohamed Noor, M. Suhaimi Jaafar, N. H. Abdullah, F. M. Suhaimi, M. Mohamed, N. Adam, and N. A. A. N. Yusuf, “A review of incorporating Nd:YAG laser cleaning principal in automotive industry,” J. Radiat. Res. Appl. Sci. 11(4), 393–402 (2018).
[Crossref]

Montagu, J.

J. Montagu, Scanners: Galvanometric and Resonant, in Encyclopedia of Optical and Photonic Engineering, Second Edition (CRC Press, 2015), pp. 1–22.

Moon, K.

Nicolov, M.

V.-F. Duma and M. Nicolov, “Numerical and experimental study of the characteristic functions of polygon scanners” (2009).

Park, D. W.

Park, K. H.

Podoleanu, A. G.

V.-F. Duma and A. G. Podoleanu, “Polygon mirror scanners in biomedical imaging: a Review,” (2013).

Razab, M. K. A. A.

M. K. A. A. Razab, A. Mohamed Noor, M. Suhaimi Jaafar, N. H. Abdullah, F. M. Suhaimi, M. Mohamed, N. Adam, and N. A. A. N. Yusuf, “A review of incorporating Nd:YAG laser cleaning principal in automotive industry,” J. Radiat. Res. Appl. Sci. 11(4), 393–402 (2018).
[Crossref]

Römer, G. R. B. E.

G. R. B. E. Römer and P. Bechtold, “Electro-optic and acousto-optic laser beam scanners-Invited Paper,” Phys. Procedia 56, 29–39 (2014).
[Crossref]

Schille, J.

J. Schille, L. Schneider, A. Streek, S. Kloetzer, and U. Loeschner, “High-throughput machining using high average power ultrashort pulse lasers and ultrafast polygon scanner,” Opt. Eng. 55(9), 97360R (2016).
[Crossref]

Schmitt, J.

D. C. Adler, Y. Chen, R. Huber, J. Schmitt, J. Connolly, and J. G. Fujimoto, “Three-dimensional endomicroscopy using optical coherence tomography,” Nat. Photonics 1(12), 709 (2007).
[Crossref]

Schneider, L.

J. Schille, L. Schneider, A. Streek, S. Kloetzer, and U. Loeschner, “High-throughput machining using high average power ultrashort pulse lasers and ultrafast polygon scanner,” Opt. Eng. 55(9), 97360R (2016).
[Crossref]

Shin, J. H.

Soldera, M.

S. Milles, M. Soldera, T. Kuntze, and A. F. Lasagni, “Characterization of self-cleaning properties on superhydrophobic aluminum surfaces fabricated by direct laser writing and direct laser interference patterning,” Appl. Surf. Sci. 525, 146518 (2020).
[Crossref]

Streek, A.

J. Schille, L. Schneider, A. Streek, S. Kloetzer, and U. Loeschner, “High-throughput machining using high average power ultrashort pulse lasers and ultrafast polygon scanner,” Opt. Eng. 55(9), 97360R (2016).
[Crossref]

Suhaimi, F. M.

M. K. A. A. Razab, A. Mohamed Noor, M. Suhaimi Jaafar, N. H. Abdullah, F. M. Suhaimi, M. Mohamed, N. Adam, and N. A. A. N. Yusuf, “A review of incorporating Nd:YAG laser cleaning principal in automotive industry,” J. Radiat. Res. Appl. Sci. 11(4), 393–402 (2018).
[Crossref]

Suhaimi Jaafar, M.

M. K. A. A. Razab, A. Mohamed Noor, M. Suhaimi Jaafar, N. H. Abdullah, F. M. Suhaimi, M. Mohamed, N. Adam, and N. A. A. N. Yusuf, “A review of incorporating Nd:YAG laser cleaning principal in automotive industry,” J. Radiat. Res. Appl. Sci. 11(4), 393–402 (2018).
[Crossref]

Tam, A. C.

A. C. Tam, W. P. Leung, W. Zapka, and W. Ziemlich, “Laser-cleaning techniques for removal of surface particulates,” J. Appl. Phys. 71(7), 3515 (1992).
[Crossref]

Van Amstel, W. D.

W. D. Van Amstel, R. J. Asjes, P. F. A. Van De Goor, and P. Merkelbach, “2 and 3D laser scanners for fast dimensional measurements and inspection,” n.d.

Van De Goor, P. F. A.

W. D. Van Amstel, R. J. Asjes, P. F. A. Van De Goor, and P. Merkelbach, “2 and 3D laser scanners for fast dimensional measurements and inspection,” n.d.

Varughese, K. O. G.

Watkins, K.

Yang, M.

S. Zhuang, S. Kainuma, M. Yang, M. Haraguchi, and T. Asano, “Characterizing corrosion properties of carbon steel affected by high-power laser cleaning,” Constr. Build. Mater. 274, 122085 (2021).
[Crossref]

Yusuf, N. A. A. N.

M. K. A. A. Razab, A. Mohamed Noor, M. Suhaimi Jaafar, N. H. Abdullah, F. M. Suhaimi, M. Mohamed, N. Adam, and N. A. A. N. Yusuf, “A review of incorporating Nd:YAG laser cleaning principal in automotive industry,” J. Radiat. Res. Appl. Sci. 11(4), 393–402 (2018).
[Crossref]

Zapka, W.

A. C. Tam, W. P. Leung, W. Zapka, and W. Ziemlich, “Laser-cleaning techniques for removal of surface particulates,” J. Appl. Phys. 71(7), 3515 (1992).
[Crossref]

Zhuang, S.

S. Zhuang, S. Kainuma, M. Yang, M. Haraguchi, and T. Asano, “Characterizing corrosion properties of carbon steel affected by high-power laser cleaning,” Constr. Build. Mater. 274, 122085 (2021).
[Crossref]

Ziemlich, W.

A. C. Tam, W. P. Leung, W. Zapka, and W. Ziemlich, “Laser-cleaning techniques for removal of surface particulates,” J. Appl. Phys. 71(7), 3515 (1992).
[Crossref]

Appl. Opt. (5)

Appl. Surf. Sci. (1)

S. Milles, M. Soldera, T. Kuntze, and A. F. Lasagni, “Characterization of self-cleaning properties on superhydrophobic aluminum surfaces fabricated by direct laser writing and direct laser interference patterning,” Appl. Surf. Sci. 525, 146518 (2020).
[Crossref]

Constr. Build. Mater. (1)

S. Zhuang, S. Kainuma, M. Yang, M. Haraguchi, and T. Asano, “Characterizing corrosion properties of carbon steel affected by high-power laser cleaning,” Constr. Build. Mater. 274, 122085 (2021).
[Crossref]

Des. Qual. Biomed. Technol. X (1)

V.-F. Duma, “Analysis of polygonal mirror scanning heads: from industrial to high-end applications in swept sources for OCT,” Des. Qual. Biomed. Technol. X 10056, 100560P (2017).
[Crossref]

J. Appl. Phys. (1)

A. C. Tam, W. P. Leung, W. Zapka, and W. Ziemlich, “Laser-cleaning techniques for removal of surface particulates,” J. Appl. Phys. 71(7), 3515 (1992).
[Crossref]

J. Lightwave Technol. (1)

J. Radiat. Res. Appl. Sci. (1)

M. K. A. A. Razab, A. Mohamed Noor, M. Suhaimi Jaafar, N. H. Abdullah, F. M. Suhaimi, M. Mohamed, N. Adam, and N. A. A. N. Yusuf, “A review of incorporating Nd:YAG laser cleaning principal in automotive industry,” J. Radiat. Res. Appl. Sci. 11(4), 393–402 (2018).
[Crossref]

Nat. Photonics (1)

D. C. Adler, Y. Chen, R. Huber, J. Schmitt, J. Connolly, and J. G. Fujimoto, “Three-dimensional endomicroscopy using optical coherence tomography,” Nat. Photonics 1(12), 709 (2007).
[Crossref]

Opt. Eng. (1)

J. Schille, L. Schneider, A. Streek, S. Kloetzer, and U. Loeschner, “High-throughput machining using high average power ultrashort pulse lasers and ultrafast polygon scanner,” Opt. Eng. 55(9), 97360R (2016).
[Crossref]

Opt. Express (1)

Phys. Procedia (2)

G. R. B. E. Römer and P. Bechtold, “Electro-optic and acousto-optic laser beam scanners-Invited Paper,” Phys. Procedia 56, 29–39 (2014).
[Crossref]

R. De Loor, “Lasers in Manufacturing Conference 2013 Polygon scanner system for ultra short pulsed laser micro-machining applications,” Phys. Procedia 41, 544 (2013).
[Crossref]

Three-Dimensional Multidimens. Microsc. Image Acquis. Process. XVI (1)

W. C. W. II, S. A. Guerrera, Z. Eastman, and C. A. DiMarzio, “Efficient confocal microscopy with a dual-wedge scanner,” Three-Dimensional Multidimens. Microsc. Image Acquis. Process. XVI 7184, 71840M (2009).
[Crossref]

Other (5)

W. Kasdorf, ed., The Columbia Guide to Digital Publishing (Columbia University Press, 2003).

V.-F. Duma and A. G. Podoleanu, “Polygon mirror scanners in biomedical imaging: a Review,” (2013).

V.-F. Duma and M. Nicolov, “Numerical and experimental study of the characteristic functions of polygon scanners” (2009).

W. D. Van Amstel, R. J. Asjes, P. F. A. Van De Goor, and P. Merkelbach, “2 and 3D laser scanners for fast dimensional measurements and inspection,” n.d.

J. Montagu, Scanners: Galvanometric and Resonant, in Encyclopedia of Optical and Photonic Engineering, Second Edition (CRC Press, 2015), pp. 1–22.

Supplementary Material (1)

NameDescription
Visualization 1       support for Figure 7

Data availability

No data were generated or analyzed in the presented research.

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Figures (11)

Fig. 1.
Fig. 1. (a) Polygon mirror (PM) scanner. (b) Three different characteristic positions of the PM.
Fig. 2.
Fig. 2. Minimum and maximum scanning angles for which facet MN entirely reflects the incident beam.
Fig. 3.
Fig. 3. (a) Back reflection occurs when the incident beam hits the mirror at perpendicular angle. (b) Smallest scan angleφlim at which back reflection does not occur.
Fig. 4.
Fig. 4. Schematic of the portable polygon scanner.
Fig. 5.
Fig. 5. Beam size versus horizontal x position.
Fig. 6.
Fig. 6. (a) Six-facet polygonal scanner with varying facet angles. (b) Variation in scan line on the specimen.
Fig. 7.
Fig. 7. Attachment of the non-back reflecting beam system. See Visualization 1
Fig. 8.
Fig. 8. (a) 6,560-μm paint stripping width. (b) Digital image of residual parts after laser cleaning.
Fig. 9.
Fig. 9. Influence of spinning speed (rpm) and laser average power (W) on paint stripping efficiency.
Fig. 10.
Fig. 10. Images of (a) bare sample, (b) painting sample, and (c) cleaned sample.
Fig. 11.
Fig. 11. (a) Laser cleaning under optimal conditions. EDS SEM images of laser scanning edge at (b) 300 rpm and 762 W and (c) 200 rpm and 652 W.

Tables (2)

Tables Icon

Table 1. Correlation of the Experimental Results with the Theory.

Tables Icon

Table 2. Processing Speed of the Polygon Scanning System.

Equations (7)

Equations on this page are rendered with MathJax. Learn more.

R sin φ ( j j A ) R cos φ ( k k A ) = 0 k = tan φ ( j j A ) + k A
j = e
k = tan φ ( e j A ) + k A
sin ( φ min + θ ) = e 2 cos θ R
sin ( φ max θ ) = e 1 cos θ R
tan 2 φ lim = r + d / 2 l a 2 sin θ cos ( φ lim + θ )
e 1 = ( R tan φ lim + a 2 ) cos φ lim

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