Research on the wear trend analysis model and application method of diffraction grating ruling tools

Hadaqinfu; Hadaqinfu; Shuo Yu; Shuo Yu; Ruipeng Wang; Ruipeng Wang; Jirigalantu; Yilong Wang; Yilong Wang; Bayanheshig

doi:10.1364/OE.516094

1. Introduction

The planar grating is a type of optical element with a periodic relief structure that has optical properties including dispersion, beam splitting, polarization, and phase matching. These gratings are widely used in applications such as spectral analysis, high-precision measurements, astronomy, optical communications, energy modulation, and inertial confinement laser fusion [1–5]. The echelle grating is a special type of large-period, large-slot, deep plane grating. Because this grating operates at high diffraction orders and large blazed angles, and it has unique characteristics that include high dispersion, high resolution, and wideband blazing, it has been used successfully in practical applications [6–9].

The main production method for the echelle grating is the mechanical ruling method [10–12], which involves step-by-step continuous micro-extrusion of a metal film layer on the grating base, where the ruling cutting tool edge of the diffraction grating ruling machine deforms the metal layer and forms a stepped periodic regular groove pattern. During the ruling process, the grating substrate quality remains unchanged, which is unlike methods that increase the substrate mass, e.g., electroforming and electroplating, or methods that reduce the substrate mass, e.g., milling, and ruling is an important equal material manufacturing-type machining method in the manufacturing industry. When compared with other micro-and nano scale processing methods for echelle grating fabrication, mechanical ruling manufacturing technology can achieve large area and high precision operations, but large area ruling requires long stroke operation of the ruling tool, and deep cutting of large grooves forces the tool to face the challenge posed by large-scale damping from the metal film. The low yield for large-area echelle gratings is a long-standing problem worldwide, and tool wear is the main problem. The echelle grating, also known as the reflection stepped grating, is between the small step grating and the step grating. It is different from the blazed grating, not to increase the grating line, but to increase the blazed angle (high spectral order and increase the grating area) to obtain high resolution and high dispersion rate [7]. Cases of ruling failure caused by tool wear have been reported in the literature for many years. For example, Harrison et al. [13,14] tried to rule 12 large-area gratings on the MIT-C machine, but only succeeded with three pieces, and five pieces failed because of tool wear. In China, in recent years, Changchun Institute of Optics, Fine Mechanics and Physics, Chinese Academy of Sciences reported successful development of a large and technologically advanced high-precision diffraction grating ruling machine in 2016, and developed a variety of specifications for 400 mm × 500 mm areas and other areas below the size of the echelle grating for use in optical astronomical telescopes and semiconductor processing [15–17]. However, they also observed some cases of failure due to tool wear. Results have shown that tool wear seriously affects the ruling yield for large-area echelle gratings, and the corresponding relationships between the grating performance indexes and the tool wear remain unclear. Therefore, it is important to study the wear modes and wear trends of grating ruling tools.

Previous studies found that in the ruling process for grating fabrication, when the ruling stroke increased, different degrees of wear would then occur at each of the cutting edges and at the tip of the tool [18–20]. When this wear reaches a certain degree, the grating performance will be affected. The friction coefficient is a good characteristic quantity for analysis of trends in the friction state and the wear of ruling tools. Practical experience of ruling gratings has shown that the tool is most prone to wear at the tip and the main edge, and the radius and the pitch angle of the tool are the most critical factors influencing the wear behavior. However, the lack of a theoretical model that can be used to perform numerical simulation analyses in advance has restricted the development of anti-wear tools and optimization of the ruling technology. In view of this problem, this paper intends to establish the physical and mathematical models required for analysis of the tool friction coefficient based on the tip point and main edge positions, and also discusses the influence and the degree to which four specific parameters contribute to the tool friction coefficient: the cutting edge radius, pitch angle, ruling depth, and tool tip angle. The effect of selection of the ruling edge radius on the diffraction efficiency of a echelle grating with 79 gr/mm is analyzed via rigorous coupled-wave analysis. In addition, using an experimental ruling of a 79 gr/mm echelle grating as an example, the influence of the tool ruling on the success or failure of the grating ruling process is investigated. The aim of performing the research above is to provide a theoretical basis for the design of wear-resistant diamond tools and for optimization of the ruling process parameters required for successful ruling of large-area echelle gratings.

2. Friction coefficient calculation model

The mechanical ruling mode of the grating is illustrated schematically in Fig. 1. Unlike a turning process, which is performed under high temperature and high speed conditions, the ruling grating tool is in contact with the metal film under the action of a load and it moves along the ruling direction at low speeds. The grating ruling tool is composed of a directional surface, a nondirectional surface, a chamfered surface, a directional side edge, a nondirectional side edge, the main edge, and the tip point.

Fig. 1. Grating ruling tool and grating ruling process diagram.

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Figure 2(a) shows the main view of the grating ruling tool. Here, OA represents the main edge, and the cutting edge has a specific cutting edge radius, similar to part of a cylinder set at an angle $\phi $ relative to the horizontal plane. OB and OC are the side edges, which rarely suffer from wear during ruling, and thus they are not analyzed in this paper. The tool and the contact part of the ruling grating are tetrahedral in shape. The Y-axis direction is the ruling direction, the grating grooves are arranged regularly along the X-axis direction, and the Z-axis direction is the direction normal to the grating plane. Previous ruling experience has shown that the positions at which the grating ruling tool is prone to wear appear at the tip point and the main edge. To clarify the friction state of the tool in its working state and predict the friction trends, analysis of the friction behavior of these tool parts is particularly important.

Fig. 2. This diagram shows the three-dimensional diagram of the grating ruling tool, and gives the modeling thinking process of tip point and main edge. (a) Main view of the grating ruling tool. (b) Tip point and main edge structures of the grating ruling tool.

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In this paper, the tip point and the main edge are no longer simply regarded as a point and a line, respectively; here, the tip point is regarded as a sphere and the main edge is considered to be part of a cylinder.Figure 2(b) shows a microstructure diagram of the tip point and the main edge of a ruling tool. AB represents the metal surface to be ruled; $\phi $ is the angle between the main edge and the horizontal plane of the tool, and is called the pitch angle; D represents the sharp angle of the knife, R is the edge radius of the main edge and the tip point; and h is the cylindrical cutting depth of the aluminum film. From a material mechanics perspective, the friction coefficient can be expressed as:

(1)$$\mu = \frac{Q}{P}$$

where Q and P are the tangential force in the ruling direction and the normal force in the grating normal direction, respectively; the forcesacting in these two directions can be determined, and the value of the friction coefficient can then be obtained. In this paper, the ruling tool is divided into three parts: the tip point, the transition zone, and the main edge. Next, mathematical models to solve for the friction coefficient of the interaction between the tool and the aluminum film will be constructed for each of these three parts.

2.1 Tip point model

A spatial rectangular coordinate system is established for the tool at its ruling attitude, and this system can also be regarded as a raster coordinate system, as shown in Fig. 3. The point where $z ={-} R$ will be regarded as the zero point for h, where $h \le R(1 - \cos \phi )$, and only the tip point part and the aluminum film material are in contact; the angle ϕ is the tool pitch angle.

(2)$$\left\{ {\begin{array}{{c}} {d{P_S} = p\cos \alpha d{A_S}}\\ {0 \le \alpha \le \arccos \left( {\frac{{R - h}}{R}} \right)} \end{array}} \right.$$

(3)$$\left\{ {\begin{array}{{c}} {d{Q_S} = (p\sin \alpha \cos \varphi + \tau \sin \varphi )d{A_S}}\\ {0 \le \alpha \le \arccos \left( {\frac{{R - h}}{R}} \right), - \frac{\pi }{2} \le \varphi \le \frac{\pi }{2}} \end{array}} \right.$$

where the surface element dA_S can be expressed as:

(4)$$d{A_S} = {R^2}\sin \alpha d\alpha d\varphi.$$

Here, p and τ represent the positive pressure and the shear stress, respectively, at any point on the tool surface, and α represents the radian angle corresponding to the tip point and the metal contact part when the ruling depth is h. By integrating equations (1) and (2) within the spherical region and substituting the results into equation (3), the expressions for the normal force and the tangential force at the tip point can be obtained as follows.

(5)$${P_S} = 2\mathop \int \nolimits_0^{\frac{\pi }{2}} \mathop \int \nolimits_0^{\arccos \left( {\frac{{R - h}}{R}} \right)} p{R^2}\sin \alpha \cos \alpha d\alpha d\varphi = \frac{\pi }{2}p{R^2}\left[ {1 - {{(\frac{{R - h}}{R})}^2}} \right]$$

(6)$$\begin{array}{c} {Q_S} = 2\int_0^{\frac{\pi }{2}} {\int_0^{\arccos \left( {\frac{{R - h}}{R}} \right)} {({p\sin \alpha \cos \varphi + \tau \sin \varphi } )} } {R^2}\sin \alpha d\alpha d\varphi \\ = p{R^2}\left\{ {\arccos \left( {\frac{{R - h}}{R}} \right) - \frac{{R - h}}{R}} \right.\left. {{{\left[ {1 - {{\left( {\frac{{R - h}}{R}} \right)}^2}} \right]}^{\frac{1}{2}}} + 2\frac{\tau }{p}\frac{h}{R}} \right\} \end{array}.$$

Fig. 3. The state of tip point and main edge in the process of grating ruling, which is also the tool coordinate system in the state of ruling.

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2.2 Transition region model

When the ruling depth h is increased further to R(1−cosϕ)<h ≤ R, the force area at the tip point has been determined and no longer increases, and the transition region then begins to contact the aluminum film. The solution in this region must be integrated on a tilted cylinder, which will increase the complexity of the derivation process. To ease the solution process required here, the coordinate system must be rotated, and the coordinate system transformation process is illustrated in Fig. 4.

Fig. 4. System for ruling coordinate rotation.

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As shown by Fig. 4, the original coordinate system is rotated clockwise about the X-axis by an angle of π/2-ϕ,and the new coordinate system is called the tool coordinate system. At this time, the contact between the main edge and the aluminum film is as shown in Fig. 5(a). In the new coordinate system, the following relationship exists:

(7)$$\left\{ {\begin{array}{{c}} {d{P_C} = p\cos \varphi \cos \phi d{A_C}}\\ { - \frac{\pi }{2} \le \varphi \le \frac{\pi }{2}} \end{array}} \right.$$

(8)$$\left\{ {\begin{array}{{c}} {d{Q_C} = (p\cos \varphi \sin \phi + \tau \sin \varphi )d{A_C}}\\ { - \frac{\pi }{2} \le \varphi \le \frac{\pi }{2}} \end{array}} \right.$$

where the cylindrical panel dA_C can be expressed as:

(9)$$d{A_C} = Rd\varphi dz.$$

Among these parameters, dP_C, dQ_C, and dA_C represent the normal force element, the tangential force element, and the area element, respectively, of the transition region, which is also represented by these three elements as shown in the main edge part below.

Fig. 5. (a) Schematic diagram of the tool coordinate system. (b) Schematic diagram of the process to obtain the integral range of angle φ.

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Next, we determine the integral range of the angle φ. As shown in Fig. 5(b), the angle enclosed by the line segments OD and OE is the integral range of φ at this time. It can be determined from the geometric relations that $OF = {{({R - h} )} / {\cos \phi }}$, and then the angle between OF and OD (set as γ) can be obtained using the following trigonometric function:

(10)$$\gamma = \arccos \frac{{R - h}}{{R\cos \phi }}.$$

After the integral range of φ is determined, the force acting in the transition region can be resolved by applying an integral. Equation (11) represents the intersection curve formed by the horizontal plane and the oblique section of the cylinder, i.e., it gives the expression for $\Gamma $ in Fig. 5(a):

(11)$$\Gamma = \left\{ {\begin{array}{{c}} { - y\cos \phi + \sin \phi \left( {z - \frac{{h - R}}{{\sin \phi }}} \right) = 0}\\ {{x^2} + {y^2} = {R^2},z \ge 0} \end{array}} \right..$$

In this way, the normal force P_C1 and the tangential force Q_C1 of the transition region can be obtained by integrating equations (6) – (10) over the cylinder simultaneously.

(12)$$\begin{array}{c} {P_{C1}} = \int_0^\Gamma {\int_{ - \arccos \frac{{R - h}}{{R\cos \phi }}}^{\arccos \frac{{R - h}}{{R\cos \phi }}} p } \cos \varphi \cos \phi Rd\varphi dz\\ = 2p{R^2}\cot \phi \left( {\frac{{h - R}}{R} + \cos \phi } \right)\sin \left( {\arccos \frac{{R - h}}{{R\cos \phi }}} \right) \end{array}$$

(13)$$\begin{array}{c} {Q_{C1}} = \int_0^\Gamma {\int_{ - \arccos \frac{{R - h}}{{R\cos \phi }}}^{\arccos \frac{{R - h}}{{R\cos \phi }}} {(p} } \cos \varphi \sin \phi + \tau \sin \varphi )Rd\varphi dz\\ = 2p{R^2}\left( {\frac{{h - R}}{R} + \cos \phi } \right)\left[ {\sin \left( {\arccos \frac{{R - h}}{{R\cos \phi }}} \right)} \right.\\ \textrm{ }\left. { + \frac{\tau }{{p\sin \phi }}\left( {1 - \frac{{R - h}}{{R\cos \phi }}} \right)} \right] \end{array}$$

Note here that, at this time, the integral limit of the tip point position changes to $\alpha = \phi $, and the normal force ${P_{S0}}$ and the tangential force ${Q_{S0}}$ acting at the tip point are expressed as:

(14)$${P_{S0}} = \frac{\pi }{2}p{R^2}{\sin ^2}\phi$$

(15)$${Q_{S0}} = p{R^2}\left[ {\phi - \sin \phi \cos \phi + 2\frac{\tau }{p}(1 - \cos \phi )} \right].$$

Therefore, when $R(1 - \cos \phi ) \le h \le R$, the expression for the total normal force and the tangential force acting on the transition region is as follows:

(16)$$\left\{ {\begin{array}{{c}} {P = {P_{C1}} + {P_{S0}}}\\ {Q = {Q_{C1}} + {Q_{S0}}} \end{array}} \right..$$

2.3 Main edge model

When h > R, the transformation integral domain is first processed for equations (12) and (13), and the normal force ${P_{C2}}$ and the tangential force ${Q_{C2}}$ acting on the transition region are obtained as follows:

(17)$${P_{C2}} = \int_0^\Gamma {\int_{ - \frac{\pi }{2}}^{\frac{\pi }{2}} p } \cos \varphi \cos \phi Rd\varphi dz = 2p{R^2}\frac{{{{\cos }^2}\phi }}{{\sin \phi }}$$

(18)$${Q_{C2}} = \int_0^\Gamma {\int_{ - \frac{\pi }{2}}^{\frac{\pi }{2}} {(p\cos \varphi \sin \phi + \tau \sin \varphi )} } Rd\varphi dz = p{R^2}(\cos \phi + \frac{\tau }{p}\cot \phi ).$$

When the integral interval for φ is $( - \frac{\pi }{2},\frac{\pi }{2})$, the normal force and the tangential force acting on the tool transition region can then be expressed.

Next, we find the normal force ${P_{C3}}$ and the tangential force ${Q_{C3}}$ acting on the main edge of the tool:

(19)$${P_{C3}} = \int_{ - \frac{\pi }{2}}^{\frac{\pi }{2}} {p\cos \varphi \cos \phi \frac{{h - R}}{{\sin \phi }}} Rd\varphi = 2pR\cot \phi ({h - R} )$$

(20)$${Q_{C3}} = \int_{ - \frac{\pi }{2}}^{\frac{\pi }{2}} {(p\cos \varphi \sin \phi + \tau \sin \varphi )} \frac{{h - R}}{{\sin \phi }}Rd\varphi = 2pR\csc \phi ({\textrm{h} - R} )\left( {\sin \phi + \frac{\tau }{p}} \right).$$

Equations (18) and (19) above represent the normal force and tangential force acting on the main edge part, respectively. Similarly, the total normal force and total tangential force received by the tool at this time are expressed as the sum of the forces exerted by the three parts of the tool:

(21)$$\left\{ {\begin{array}{{c}} {P = {P_{S0}} + {P_{C2}} + {P_{C3}}}\\ {Q = {Q_{S0}} + {Q_{C2}} + {Q_{C3}}} \end{array}} \right..$$

At this point, the mathematical model required to solve for the tool friction coefficient is complete.

3. Numerical simulation

3.1 Material characteristics analysis

When the force of the grating ruling tool is to be simulated, the attitude and the structural parameters of the tool should be determined first. The tool’s structural parameters include (but are not limited to) the tool tip angle and the cutting edge radius R, which are the only structural parameters discussed in this paper. The attitude parameter of the tool is the angle at which the main edge is lifted by the surface of the ruling metal film, i.e., it represents the pitch angle ϕ of the tool. As mentioned earlier, when faced with the problem of how to select the cutting edge radius and the pitch angle of the tool, we are thus far only able to rely on practical experience, and there is no theoretical basis to follow; this has restricted the development of cutting tool grinding technology and cutting edge radius technology. Therefore, it is essential to study the effects of these two parameters on the tool friction coefficient.

First, it is necessary to determine whether the deformation properties of the metal on the grating surface represent elastic or plastic deformation. Liu et al. [21] proposed a critical value expression for the ruling depth for plastic deformation when materials with different hardness coefficients are in contact.

(22)$$\frac{h}{R} = {(\frac{{\pi KH}}{{2E}})^2}$$

where E is the Hertzian elastic effective modulus, i.e., 1/E = (1−ν12)/ E1 + (1−ν22)/ E2; E1, E2, ν1, and ν2 are the Young's modulus and Poisson's ratio for the two contact materials, respectively; K is the hardness coefficient, which is only related to the Poisson's ratio of the material; and H is the hardness of the softer of the two materials. In the calculations in this paper, it is assumed that the positive pressure p acting on the tool surface is approximately equal to the hardness H of the aluminum film on the grating surface, and τ=H/5.65 is used based on the maximum shear stress criterion of Tresca. To make the elastic-plastic deformation definition curve more general, the material properties of several more commonly used metals on the market were obtained by consulting relevant data sources, with results as shown in Table 1.

Table 1. Material properties of several metals

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The curves defining the elastic-plastic deformation characteristics of the various metals are shown in Fig. 6. As shown in the figure, and based on the assumption that the material deformation is plastic deformation, when the grating surface is the aluminum film, the solution range of ${h / R} > {10^{ - 5}}$ is valid, thus indicating that the current material deformation is plastic deformation. In the case where the surface metal is Au, the solution range is then very generous, and approaches ${h / R} > {10^{ - 7}}$.These results show that the deformation of the metal film on the surface of the grating is almost entirely plastic deformation and that very little elastic deformation occurs during ruling of the grating. In this paper, the deformation of the metal film on the grating surface is thus assumed to be plastic.

Fig. 6. Material properties of common metals and elastic-plastic deformation defining curves defined by ruling depth.

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3.2 Analysis of the influence of cutting edge radius on the friction coefficient of the tool

The blunt radius of the tip point and the main edge radius R of the tool were taken to have values of 100, 200, and 300 nm to analyze the effects of these radii on the friction coefficient at the tip point position and the main edge. At this time, the pitch angle of the tool was taken to be 10 °.

Figure 7 shows a comparison of the friction coefficient curves for the different cutting edge radii at the tip point position. As seen in the figure, with increasing ruling depth, the friction coefficient shows a steady increasing trend. At the same ruling depth position, a smaller cutting edge radius leads to a greater coefficient of friction, i.e., a smaller cutting edge radius produces a greater friction force on the edge point position of the tool. Note here that the maximum value of the interval for the independent variables of each curve ends at $h \le R(1 - \cos \phi )$, which is the depth range of action of the tip point position. Beyond this depth range, the transition region position begins to intervene in the characteristics.

Fig. 7. Friction coefficient characteristics of the tip point position with different cutting edge radius values.

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Figure 8 shows the influence of the different cutting edge radius values on the friction coefficient at the transition region location. As shown in the figure, a small cutting edge radius will still produce a larger friction coefficient, and as the horizontal coordinate increases, the overall slope of the curve is larger for a smaller cutting edge radius, and the friction coefficient rises more quickly. With the same friction coefficient, a larger cutting edge radius means that a larger ruling depth is required; this means that a tool with a large cutting edge radius requires a greater ruling depth to realize the same wear speed as a tool with a smaller cutting edge radius.

Fig. 8. Friction coefficient characteristics in the transition area for different cutting edge radius values.

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Figure 9 shows the friction coefficient curves for different cutting edge radius values at the main edge position. As the figure indicates, the friction coefficient continues to increase with increasing ruling depth. In contrast to the previous two positions, the friction coefficient no longer increases sharply, but approaches a stable value slowly as the depth increases. The influence of the cutting edge radius on the friction coefficient remains the same as that in the previous two positions. As the cutting edge radius decreases, the friction force to which the tool is subjected increases. In addition, with a smaller cutting edge radius, the curve will approach its stable value more quickly, and a larger cutting edge radius will cause the curve to change more gently. The friction coefficients of three types of tool with different cutting edge radii were compared after they reached stability, where the coefficient at R = 100 nm increased by 0.66% and 2.11% when compared with the values at R = 200 nm and R = 300 nm, respectively. The coefficient at R = 200 nm is 1.44% larger than that at R = 300 nm.

Fig. 9. Friction coefficient characteristics at the main edge position for different cutting edge radius values.

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3.3 Analysis of the influence of the pitch angle on the tool friction coefficient

Next, the effect of the pitch angle on the friction coefficient of the tool is analyzed. The main edge pitch angle values are taken to be 3 °, 5 °, 10 °, 30 °, and 45 ° in this case. To analyze the influence of the individual pitch angles separately, a cutting edge radius of R = 300 nm was used to perform the calculations.

Figure 10 shows the friction coefficient curves in the transition region position at the different tool pitch angles. As the figure illustrates, there is a positive correlation between the pitch angle and the friction coefficient. The curves are arranged at pitch angles of 3 °, 5 °, 10 °, 30 °, and 45 ° from bottom to top, and do not intersect with each other, thus indicating that an increase in the pitch angle will produce a greater friction coefficient.

Fig. 10. Friction coefficient characteristics of the transition zone obtained at different pitch angles.

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Figure 11 shows the friction coefficient curves at the different pitch angles at the main edge position. The figure illustrates that a change in friction coefficient also maintains the same trend as that observed at the previous position. A greater pitch angle leads to a higher friction coefficient, and the friction coefficient will still approach a stable value with increasing ruling depth. This shows that when ruling a grating, it should be ensured that the tool pitch angle is not too large because it would result in rapid wear of the tool; additionally, the excessive friction caused may even cause the tool to crawl or lead to the tool jumping phenomenon, which would result in grating ruling failure.

Fig. 11. Friction coefficient characteristics at the main edge position at different pitch angles.

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3.4 Analysis of the influence of the tool tip angle on the tool friction coefficient

In addition to the cutting edge radius and pitch angle parameters discussed above, the knife point angle D of a tool also makes an important contribution to the friction coefficient. The cutting edge radius is related to the shining angle of the grating and is an important tool structural parameter. In the analysis here, four different tool tip angle values of 80 °, 90 °, 100 °, and 110 ° were used in the calculations. To exclude the effects of the other two parameters, the cutting edge radius was taken to be 100 nm and the pitch angle was 3 °. Figure 12 shows the influence of the tool tip angle on the friction coefficient at the tip point position. The results in the figure show that with increasing tool tip angle, the curve for the friction coefficient of the tip point shows an overall translational increase, and larger tool tip angles have a major influence on the friction coefficient at the tool position.

Fig. 12. Influence of different tool tip angles on the friction coefficient at the tip point position.

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Figure 13 shows the influence of the tool tip angle on the friction coefficient at the main edge position. When $R(1 - \cos \phi ) \le h \le R$, i.e., when the ruling depth is within the transition zone, the cutting edge of the tool within the tool tip angle is not entirely in contact with the metal film, and the upper integral limit is smaller than the tool tip angle, although the distribution of the friction coefficient remains unchanged as the ruling depth changes. The influence of the tool tip angle on the friction coefficient at the main edge position maintains the same change law as that at the tip point position, and the tool tip angle still has a more obvious effect on the friction coefficient at that position. A larger tool tip angle leads to a higher friction coefficient and thus to faster tool wear.

Fig. 13. Influence of different tool tip angles on the friction coefficient at the main edge position.

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4. Results and discussion

4.1 Friction coefficient simulation results analysis

Based on a comprehensive consideration of the content of the previous section, the effects of the cutting edge radius, the pitch angle, and the sharp angle of the tool on the friction coefficients of the various parts of a grating ruling tool are summarized here in the form of the maximum friction coefficient characteristics. The friction coefficient at the tip point is obtained by substituting equations (5) and (6) into equation (1); the friction coefficient at the transition region is obtained by substituting equation (16) into equation (1); and the friction coefficient of the main edge is obtained by substituting equation (21) into equation (1). In the following, ${R_1}$, ${R_2}$, and ${R_3}$ represent the 100 nm, 200 nm, and 300 nm cutting edge radii, respectively; ϕ₁, ϕ₂, ϕ₃, ϕ₄, and ϕ₅ represent pitch angles of 3 °, 5 °, 10 °, 30 °, and 45 °, respectively; and D₁, D₂, D₃, and D₄ represent knife angle values of 80 °, 90 °, 100 ° and 110 °, respectively.

4.1.1 Analysis of factors influencing the friction coefficient of the tip point

To reflect the effects of the cutting edge radius and the pitch angle on the friction coefficient at the edge tip position more intuitively, the maximum friction coefficient values when these parameters change are listed in Table 2. As Table 2 indicates, the influence of the cutting edge radius on the maximum friction coefficient in the tip point region is very limited. At the same pitch angle, the maximum friction coefficient values are almost equal, and the effect lies more in the slope of the curve. A smaller cutting edge radius will reach the maximum value in the same region more quickly. The influence of the pitch angle is particularly obvious in this case. Under the condition of a cutting edge radius of 100 nm, the maximum increase in the friction coefficient at a pitch angle of 45 ° is 268.53% when compared with that at 3 °.

Table 2. Maximum friction coefficient values at the tip point corresponding to the different cutting edge radius and pitch angle values

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The tool tip angle is another factor that affects the friction coefficient distribution of the tip point. The friction coefficient at the tip point is positively correlated with the magnitude of the tool tip angle. The corresponding relationship between the tool tip angle, the pitch angle, and the maximum friction coefficient is given in Table 3.

Table 3. Maximum friction coefficient values at the tip point corresponding to the different tool tip angles and pitch angles

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As shown in Table 3, the tool tip angle and the pitch angle have differing degrees of influence on the maximum friction coefficient at the edge point, with the pitch angle having a very obvious influence. At the tool tip angle of 80°, the increase in the coefficient at the pitch angle of 45 ° is 550.27% when compared with the corresponding value at 3 °. At a pitch angle of 3 °, the increase in the maximum friction coefficient at the tool tip angle of 110 ° compared with that at 80 ° is 18.58%. It should be noted that when the tip point is used with different cutting edge radius values, the maximum friction coefficient caused by the change in the knife point angle is completely equal to that caused by a corresponding change in the pitch angle. This indicates that, after the introduction of the tool tip angle, the effects of the pitch angle and the tool tip angle are independent from those of the cutting edge radius, whereas the cutting edge radius is insensitive to the influence of the pitch angle.

By taking these two influencing factors into consideration, and based on the premise of meeting the grating design requirements, the actual ruling cutting tools should be selected to have a large cutting edge radius and a sharp knife angle as far as possible, because these characteristics can reduce the degree of friction damage at the edge point part effectively.

4.1.2 Analysis of the factors influencing the friction coefficient at the transition region

According to Section 3 of this paper, the factors that influence the friction coefficient at the transition area are the cutting edge radius and the pitch angle. The influence of the cutting edge radius on the friction coefficient for this part is similar to that in the case of the tip point part. When three different cutting edge radii are used to perform the calculations, the maximum value of the friction coefficient curve in this region is equal in all cases, and the only difference lies in the speed at which the maximum value is reached. The greater friction coefficient curve slope corresponds to a smaller cutting edge radius. A greater pitch angle will produce a greater friction coefficient, which results in faster tool wear. The corresponding relationship between the pitch angle and the extreme value of the friction coefficient is as shown in Table 4.

Table 4. Relationship between the extreme value of the friction coefficient of the transition area and the pitch angle

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Table 5 shows the corresponding relationship between the maximum friction coefficient of the transition region and the values of the cutting edge radius and the pitch angle. The results show that the maximum friction coefficient in this region is not affected by the cutting edge radius and is only related to the pitch angle. When the pitch angle of 45 ° is used, the maximum friction coefficient increases by 327.13% when compared with that obtained at 3 °.

Table 5. Maximum friction coefficient of the transition region corresponding to the cutting edge radius and the pitch angle

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Based on consideration of the effects of the cutting edge radius and the pitch angle, a large cutting edge radius and a small pitch angle should be selected as appropriate parameters to slow down the speed of wear in this area.

4.1.3 Analysis of the factors influencing the friction coefficient at the main edge

The cutting edge radius, the pitch angle, and the sharp angle of the tool all affect the friction coefficient of the main edge. Therefore, these three factors should be considered comprehensively when selecting the cutting tool parameters. The correspondence between the maximum friction coefficient at the main edge and the cutting edge radius and the pitch angle is shown in Table 6.

Table 6. Maximum friction coefficient of the main edge corresponding to the pitch angle and the cutting edge radius

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Table 6 indicates that the maximum friction coefficient of the main edge is inversely proportional to the cutting edge radius and is directly proportional to the pitch angle. The maximum friction coefficient of cutting edge radius R₃ at a pitch angle of 3 ° decreases by approximately 2.09% when compared with the maximum friction coefficient for R₁; when the cutting edge radius is R₁, the coefficient at the 45 ° pitch angle increases by 248.88% when compared with the corresponding maximum value at 3 °.

When the influence of the tool tip angle on the friction coefficient at the main edge is analyzed, the cutting edge radius value should be limited. The corresponding relationships between the maximum friction coefficient values at the main edge and the tool tip angle, the cutting edge radius, and the pitch angle are shown in Table 7.

Table 7. (a) Relationship between the maximum friction coefficient at the main edge and the tool tip angle and the pitch angle when the cutting edge radius is R₁. (b) Relationship between the maximum friction coefficient at the main edge and the tool tip angle and the pitch angle when the cutting edge radius is R₂. (c) Relationship between the maximum friction coefficient at the main edge and the tool tip angle and the pitch angle when the cutting edge radius is R₃.

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Combination of the contents of Table 7 indicates that the maximum friction coefficient of the main edge of the ruling tool decreases with decreasing tool tip angle, decreasing pitch angle, and increasing cutting edge radius. When the situations at the tip point, the transition region, and the main edge are combined and a 79 gr/mm echelle grating is used as the ruling object, the appropriate cutting edge radius is 300 nm, the appropriate pitch angle is 3 °, and a smaller tool sharp angle is then better based on the premise of meeting the design requirements for the grating diffraction efficiency.

4.2 Influence of the cutting edge radius on the diffraction efficiency of a blazed grating

The previous discussion in this paper is based on the wear trend of the ruling tool, and the diffraction efficiency is used as the main technical index of the grating. To determine the effect of the selected cutting edge radius on the grating diffraction efficiency of a 79 gr/mm echelle grating, the diffractive efficiency was calculated for gratings with different cutting edge radii of 100 nm, 200 nm, and 300 nm via a rigorous coupled-wave analysis [22]. RCWA is a very effective tool to deal with the electromagnetic field problems of periodic structures (especially diffraction gratings). This method is to expand the electromagnetic field and the dielectric constant of the material by Fourier series, and then solve the Maxwell equation by solving the eigenvalue and eigenvector of the matrix.

Figure 14 shows the diffraction efficiency curves for the 79 gr/mm echelle grating at cutting edge radii of R = 100 nm, R = 200 nm, and R = 300 nm, where m represents the diffraction order. The results show that changes in the cutting edge radius have almost no effect on the diffraction efficiency of the 79 gr/mm echelle grating because the groove depth of the 79 gr/mm echelle grating is greater than 4 µm, and the cutting edge radius of the cutting knife differs from this depth by an order of magnitude, which means that it will not affect the effective working area of the blazed surface. Therefore, for the ruling working conditions for a echelle grating at 79 gr/mm, the preferred cutting edge radius is 300 nm, which can increase the working life of the ruling tool effectively.

Fig. 14. Diffraction efficiency curves of a 79 gr/mm echelle grating when the cutting edge radii are R = 100 nm, R = 200 nm, and R = 300 nm.

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5. Echelle grating ruling experiment

5.1 Comparison experiments of ruling of grating at different pitch angles

The production of a mechanically ruled grating is achieved using a grating ruling machine, which is also known as the “king of precision machinery”. During the grating fabrication process, a diamond ruling tool is installed in the ruling tool holder and this tool then moves uniformly with the ruling system. The table of the indexing system carries the grating substrate to perform a unidirectional step movement in a coordinated manner, and this causes the diamond ruling knife to squeeze and wipe the metal coating on the grating substrate under a specific load such that it deforms the coating and forms a stepped groove.

The analysis in the previous section shows that among the three parameters of interest, i.e., the cutting edge radius, the tool tip angle, and the pitch angle, the effects of the pitch angle on the friction coefficient at each tool position are most obvious. On this basis, the same geometric parameters were used for the tool in this experiment; with the exception of the pitch angle, the other installation parameters all remained the same, and the grating ruling process was then analyzed while controlling a single variable. The tool tip angle was 90 ° and the pitch angles were 3 °, 10 °, and 20 °. The linear density of the grating was 79 gr/mm. The grating ruling equipment used in the experiment was the ciomp-5 grating ruling machine, as shown in Fig. 15. This device uses a servo motor to drive the worm gear to drive the screw to move in the indexing direction, and uses a servo motor to drive the crank connecting rod to drive the slide rail to perform reciprocating motion in the ruling direction. At the same time, the whole device is placed on the air-floating vibration isolation system, which has a good vibration isolation effect. During the ruling process, the laboratory temperature is 21 °, and the temperature control accuracy is ± 0.05 °.

Fig. 15. The ciomp-5 grating ruling machine.

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Figure 16 shows the grating grooves observed under a microscope at the three pitch angles. At the lowest pitch angle, the grating grooves are straight and regular, and the grating working face is clear. With increasing pitch angle, the grooves obtained by ruling show increasingly serious ripples. This rippling occurs because the friction coefficient between the tool and the aluminum film increases with increasing tool pitch angle during the grating ruling process. At this time, a creeping phenomenon of extrusion ploughing also appears in the process, and this phenomenon results in the periodic ripples. The experimental results show that establishment of the tool friction coefficient calculation model can guide the grating ruling process effectively, and provides further confirmation of the necessity of establishing the proposed model.

Fig. 16. (a) Grating grooves at a 20° pitch angle. (b) Grating grooves at a 10° pitch angle.(c) Grating grooves at a 3° pitch angle.

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5.2 Feasibility analysis of optimal selection of parameters for the friction coefficient calculation model

Based on the results of the theoretical analysis process described above, the tooling parameters are optimized here. For the selection of the parameters, the tool tip angle is 90 °, the tool cutting edge radius is 300 nm, and the pitch angle is 3 °. The gratings to be ruled have a linear density of 79 gr/mm and a blazed angle of 63 °. The ruling experiment was completed on the ciomp-5 grating ruling machine, and the target grating was fabricated successfully. The grating grooves as observed using an atomic force microscope are shown in Fig. 17.

Fig. 17. Target grating groove as observed under an atomic force microscope.

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Figure 17 shows that the grating ruling produced by the tool after optimization of the tooling parameters is both clear and sharp, the groove shape is regular, there is no zero level plane, and the grating period and the blazed angle meet the design requirements; these results indicate that the calculation model for the tool friction coefficient has practical engineering significance.

6. Conclusion

To improve the yield of large area grating ruling processes, the wear of room temperature and low speed grating ruling tools has been studied in this work. A calculation model for the friction coefficient of the grating ruling tool has been proposed, and a mathematical model to calculate the friction coefficient was established based on the positioning of the tip point and the main edge of the tool. The degrees of influence of the cutting edge radius, the pitch angle, and the knife angle on the friction coefficient were summarized via numerical calculations, and the optimized parameters are presented here. Based on a rigorous coupled-wave analysis, the sensitivity of the diffraction efficiency of a grating with a lineation density of 79 gr/mm to the cutting edge radius was analyzed, and targeted scribing experiments were conducted. The results showed that the friction force of the ruling tool can be reduced greatly by appropriate parameter selection, which will be helpful in the development of anti-wear tools and optimization of the ruling process, and can also provide a theoretical basis for design of the anti-wear tools required for large-area ruling of gratings.

Funding

National Natural Science Foundation of China (62075216); Jilin Province Science & Technology Development Program (20200602051ZP); Jilin Province Science & Technology Development Program (20230505012ZP).

Acknowledgment

This work is supported by National Natural Science Foundation of China (General Program 62075216); Jilin Province Science & Technology Development Program (20230505012ZP); Jilin Province Science & Technology Development Program (20200602051ZP).

Disclosures

The authors declare no competing financial 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.

References

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6. Yifan Li, Xiaotao Mi, Xiangdong Qi, et al., “Echelle gratings tiling method based on lateral shearing interferometry,” Opt. Laser Technol. 145, 107475 (2022). [CrossRef]

7. G. R. Harrison, “The Production of Diffraction Gratings : II;. The Design of Echelle Gratings and Spectrographs,” J. Opt. Soc. Am. 39(7), 522–528 (1949). [CrossRef]

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Materials	Vickers hardness(MPa)	Young's modulus(GPa)	Poisson's ratio
Au	50	91	0.42
Pb	50	14	0.44
Al	150	68	0.35
Cu	500	110	0.34
Ni	750	207	0.31
Mo	2300	324	0.31
W	4800	345	0.28

Influencing factor	$ϕ_{1}$ (◦)	$ϕ_{2}$ (◦)	$ϕ_{3}$ (◦)	$ϕ_{4}$ (◦)	$ϕ_{5}$ (◦)
$R_{1}$ (nm)	0.1344	0.1499	0.1877	0.3514	0.4953
$R_{2}$ (nm)	0.1348	0.1499	0.1878	0.3514	0.4954
$R_{3}$ (nm)	0.135	0.1499	0.1878	0.3514	0.4954

Influencing factor	$ϕ_{1}$ (◦)	$ϕ_{2}$ (◦)	$ϕ_{3}$ (◦)	$ϕ_{4}$ (◦)	$ϕ_{5}$ (◦)
$D_{1}$ (◦)	0.0915	0.113	0.1673	0.3972	0.595
$D_{2}$ (◦)	0.09749	0.1186	0.1717	0.397	0.5912
$D_{3}$ (◦)	0.1032	0.1237	0.1756	0.3957	0.5859
$D_{4}$ (◦)	0.1085	0.1285	0.1789	0.3935	0.5792

Pitch angle	$μ_{min}$	$μ_{max}$
3°	0.07009	0.2296
5°	0.1105	0.2651
10°	0.1876	0.3553
30°	0.3514	0.7319
45°	0.4954	0.9807

Influencing factor	$ϕ_{1}$ (◦)	$ϕ_{2}$ (◦)	$ϕ_{3}$ (◦)	$ϕ_{4}$ (◦)	$ϕ_{5}$ (◦)
$R_{1}$ (nm)	0.2296	0.2651	0.3553	0.7319	0.9807
$R_{2}$ (nm)	0.2296	0.2651	0.3553	0.7319	0.9807
$R_{3}$ (nm)	0.2296	0.2651	0.3553	0.7319	0.9807

Research on the wear trend analysis model and application method of diffraction grating ruling tools

Abstract

1. Introduction

2. Friction coefficient calculation model

2.1 Tip point model

2.2 Transition region model

2.3 Main edge model

3. Numerical simulation

3.1 Material characteristics analysis

3.2 Analysis of the influence of cutting edge radius on the friction coefficient of the tool

3.3 Analysis of the influence of the pitch angle on the tool friction coefficient

3.4 Analysis of the influence of the tool tip angle on the tool friction coefficient

4. Results and discussion

4.1 Friction coefficient simulation results analysis

4.1.1 Analysis of factors influencing the friction coefficient of the tip point

4.1.2 Analysis of the factors influencing the friction coefficient at the transition region

4.1.3 Analysis of the factors influencing the friction coefficient at the main edge

4.2 Influence of the cutting edge radius on the diffraction efficiency of a blazed grating

5. Echelle grating ruling experiment

5.1 Comparison experiments of ruling of grating at different pitch angles

5.2 Feasibility analysis of optimal selection of parameters for the friction coefficient calculation model

6. Conclusion

Funding

Acknowledgment

Disclosures

Data availability

References

Data availability

Cited By

Figures (17)

Tables (7)

Equations (22)

Optics Express

Influencing factor	$ϕ_{1}$ (◦)	$ϕ_{2}$ (◦)	$ϕ_{3}$ (◦)	$ϕ_{4}$ (◦)	$ϕ_{5}$ (◦)
$R_{1}$ (nm)	0.4073	0.4425	0.5325	0.9549	1.421
$R_{2}$ (nm)	0.4051	0.4399	0.529	0.9481	1.412
$R_{3}$ (nm)	0.3988	0.4332	0.5215	0.9372	1.338

(a)	$ϕ_{1}$ (◦)	$ϕ_{2}$ (◦)	$ϕ_{3}$ (◦)	$ϕ_{4}$ (◦)	$ϕ_{5}$ (◦)
$D_{1}$ (◦)	0.1157	0.1506	0.2394	0.6457	1.081
$D_{2}$ (◦)	0.1246	0.1595	0.2483	0.6558	1.093
$D_{3}$ (◦)	0.1337	0.1686	0.2576	0.6664	1.106
$D_{4}$ (◦)	0.1432	0.1782	0.2672	0.6773	1.119
(b)	$ϕ_{1}$ (◦)	$ϕ_{2}$ (◦)	$ϕ_{3}$ (◦)	$ϕ_{4}$ (◦)	$ϕ_{5}$ (◦)
$D_{1}$ (◦)	0.1146	0.1491	0.237	0.6396	1.07
$D_{2}$ (◦)	0.1233	0.1579	0.2458	0.6497	1.082
$D_{3}$ (◦)	0.1324	0.1669	0.255	0.6601	1.095
$D_{4}$ (◦)	0.1418	0.1764	0.2646	0.6709	1.108
(c)	$ϕ_{1}$ (◦)	$ϕ_{2}$ (◦)	$ϕ_{3}$ (◦)	$ϕ_{4}$ (◦)	$ϕ_{5}$ (◦)
$D_{1}$ (◦)	0.1134	0.1476	0.2346	0.6335	1.06
$D_{2}$ (◦)	0.1221	0.1563	0.2433	0.6435	1.072
$D_{3}$ (◦)	0.131	0.1652	0.2524	0.6537	1.084
$D_{4}$ (◦)	0.1403	0.1746	0.2619	0.6644	1.097