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Optik 123 (2012) 1550– 1554

Contents lists available at SciVerse ScienceDirect

Optik

j o ur nal homepage: www.elsev ier .de / i j leo

ew method of circle and arc cutting

in Hua ∗, Sun Cunzhiepartment of Sciences, Shandong University of Technology, Zibo 255049, Shandong, China

r t i c l e i n f o

rticle history:eceived 23 March 2011ccepted 18 August 2011

a b s t r a c t

In practical applications of laser cutting, the quality of the cut surfaces is the critical factor. In this study, anew method for circle and arc cutting is proposed based on optical characteristic of a glass cone. In a laser

eywords:aser cuttingollow light cone

cutting system, we use a glass cone replace beam-focusing optics to direct the beam to the workpiece,and equipment for moving either the laser beam or the workpiece, or both, to obtain the circle or arccutting pattern. Theoretical analysis and simulation studies of a glass cone show that, this method solvesan issue which the focal depth and focal spot cannot be optimized in the meantime, ensures the strictvertical cut surface and the high cutting quality, greatly reduce the workload caused by the motion of thebeam or the workpiece.

ight conversion

. Introduction

Laser cutting is a mostly thermal process in which a focusedaser beam with higher powers is used to melt material in a local-zed area [1]. A co-axial gas jet is used to eject the molten materialrom the cut and leave a clean edge, as shown in Fig. 1. Material toe cut is placed flat on the rotary table, the rotary table is driven tootate with a FM Motor, and at the same time a sensor controlledM Motor is employed for the automatic indexing. A laser focusingevice is placed on a vertical beam fixed under a beam. The beaman travel along a longitudinal rail, and the vertical beam alonghe vertical rail. They all are driven by the stepper motor to com-lete the vertical feed motion and the longitudinal feed motion. Thehole movement depends on a computer-controlled system [2].

f we accurately control the laser output and speed of processingbjects, high-quality laser cutting can be achieved [3,4].

Cutting thicker materials, the light beam with a larger depth ofocus should be adopted to get a better vertical cut surface. How-ver, a larger depth of focus will make spot diameter also increase,nd the radiation power density will decrease, so the cutting speeds reduced. In order to maintain a certain cutting speed, the laserower should be increased. Therefore, it is not possible to givettentions in the meantime to both the focal spot and the depthf focus [5–8].

In order to cut round shapes out of materials the laser focus must

e moving along a circle line, and realized through two methods,ither controlling the work platform to move or controlling theaser output to move. But no matter what kind of movements, it

∗ Corresponding author.E-mail address: zfjqinh@gmail.com (Q. Hua).

030-4026/$ – see front matter © 2011 Elsevier GmbH. All rights reserved.oi:10.1016/j.ijleo.2011.09.013

© 2011 Elsevier GmbH. All rights reserved.

must be done by computer controlled stepper motors, with a largerworkload.

Based upon the above perspectives, a solution was put forwardto conflicts between the focal depth and focal spot, and a largerworkload arising from computer controlled stepper motor drivinga workpiece or the laser output device to move. This solution ismainly dedicated to the processing of circular or arc-shaped materi-als. The program can make the laser processing equipment becomequite simple: the complex motion of a work platform or a laseroutput device with a microcomputer controlled stepper motor isno longer needed; the contradiction which the depth of focus andthe focal spot in the meantime can be solved.

2. Converter of hollow light cone

In this paper, a converter of hollow light cone is used to replacebeam-focusing optics to direct the beam to the workpiece. This con-verter consists of only a transparent cone, so the structure of a lasercutting device is greatly simplified. It is realized based on the totalinternal reflection–refraction principle to convert the optical modefrom a solid light cylinder to a hollow light cone. A parallel beamperpendicular to the bottom surface of a cone passing through thiscone after a number of total internal reflections will exits along itsgeneratrix, a cylindrical beam will be converted into a cone-shapedsurface beam, all energy will be gathered in a conical surface, andits spot is a circular-shaped spot.

2.1. Profile of converter of hollow light cone

The converter of hollow light cone is a transparent glass cone,made of ordinary optical glass materials, as shown in Fig. 2. It

Q. Hua, S. Cunzhi / Optik 12

Fig. 1. The schematic of laser cutting.

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ig. 2. The schematic diagram of a glass cone used as a converter of hollow lightone.

ollects incident rays to its tail depending on the total reflectionf the side wall of cone.

Incident beams normal to the bottom surface of a cone irradi-te the sidewall of a cone, and incident angle is �n(the subscript

= 2,3,. . ., denotes the number of intersection with the side wall ofone). When �n is greater than the critical angle ̋ the total inter-al reflection occurs. ̋ = arcsin(1/�g), where �g is the refractive

ndex of the cone. Let �g = 1.52, then ̋ ≈ 41.1395◦. �n will gradu-lly decrease with the increase of n, when it is less than the criticalngle the light is refracted out of the side wall of the cone.

.2. The total internal reflection of light beam in a glass cone

The reflected ray and the refracted ray are computed in terms ofn incident ray with the vector formula of reflection and refraction,nd completed in two steps. In the first step, deducing the point p2rom the point p1, in the second step, deducing the point p3 fromoint p2, then p4, p5,. . ., and pn.

The geometric position of the incident light can be describedy two vectors: The first is the position vector �P1 representing theoint p1, and the other is the unit vector �Q1 in the direction of the

ight rays. Each vector may be expressed by using three coordinateomponents. Fig. 3 is a diagram showing all these direction vectors

ncident light

{�P1 = x1

�i + y1�j + z1

�k�Q1 = ˛1

�i + ˇ1�j + �1

�k

Fig. 3. The vector of incident ray, refracted ray and reflected ray in a cone.

3 (2012) 1550– 1554 1551

where�i, �j, �k are unit vectors in the directions of the three coordinateaxes, respectively. Because �Q1 is the unit vector, its three coordinatecomponents ˛, ˇ, � is known as its direction cosine. �P2 and �Q2 arethe position vector of the refraction point p2 and the unit vector inthe direction of the refracted light ray, respectively.

Refracted light

{�P2(x2, y2, z2) = x2

�i + y2�j + z2

�k�Q2(

˛2, ˇ2, �2)

= ˛2�i + ˇ2

�j + �2�k

We can get �P2(x2, y2, z2) and �Q2(˛2, ˇ2, �2) in terms of�P1(x1, y1, z1), �Q1(˛1, ˇ1, �1) through the step by step calculation,and the calculation process is as follows: Suppose the point p1 liesin the bottom face of a cone, namely in the YOZ plane, here x1 = 0.The incident light parallel to the x-axis passes through the coni-cal bottom and intersects the internal sidewall of the cone at p2.According to the similarity of triangle, we have

x2

l= h −

√y2

1 + z21

hso

x2 = l

(1 −√

y21 + z2

1

h

)(1)

Using the vector addition formula, we get

�P2 = �P1 + x2�Q1 (2)

In the following, we seek �Q2(˛2, ˇ2, �2) with the law of reflection.The vector form of the law of reflection is

�Q1 × �N2 = �Q2 × �N2

where �N2 is the unit normal vector to a sidewall of cone at the pointp2, which can be derived from calculating the partial derivative ofa cone equation. A cone equation

√z2 + y2/h = (l − x)/l. Manip-

ulating this equation, we can obtain, l2(z2 + y2)/h2 − (l − x)2 = 0, itsnormal vector is �Np2 (2l − 2x2, 2(l2/h2)y2, 2(l2/h2)z2). Turning this

vector into a unit vector: �N2((l − x2)/A, (l2/h2)y2/A, (l2/h2)z2/A),

where A =√

(l − x2)2 + ((l2/h2)y2)2 + ((l2/h2)z2)2.By transposition of the vector form of the law of refraction, we

get ( �Q1 − �Q2) × �N2 = 0. This equation indicates that ( �Q1 − �Q2) and�N2 must be parallel, so the following equation must be satisfied

�Q1 − �Q2 = g �N2 (3)

where g is a coefficient. Now, we dot-multiply both sides of theequation by �N, then we have

( �Q1 − �Q2) · �N2 = g

The dot product of two unit vectors is equal to the cosine of theangle between two vectors, hence we obtain:

g = cos�2 − cos(� − �2) = 2 cos�2 (4)

According to the angular relationship �2 + ̨ = �/2, After its substi-tution into Eq. (4), this equation can be simplified as

g = 2 sin ̨ (5)

Because ̨ is a constant, g is also a constant. By (3) when the totalreflection occurs, we get

�Q2 = �Q1 − g �N2 (6)

In the above section, we have gradually obtained �P2, �Q2 from �P1, �Q1.

In the next section, we will find a recurrence relation for �Pn and �Qn.

Let S denote a distance between two adjacent total reflectionpoints. In the triangle, which consists of the point pn−1, point pn

and cone vertex, applying Sine Theorem we have

1552 Q. Hua, S. Cunzhi / Optik 12

Input ra y (x1, y1) (x2, y2)

(x3, y3)

(x4, y4)θ

θ

Lα

2h

Output ray

y

xo

S

B

P

F

g

w2

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2

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Fig. 4. The sketch of ray-tracing in the cross-section of a glass cone.

Ssin2˛ = (h/sin˛)·((l−xn−1)/l)

sin(�n+�/2) , l = h cot˛Manipulate and obtain

= 2(h cos ̨ − xn−1 sin˛)cos�n

(7)

y the vector addition formula, we have

�n = �Pn−1 + S �Qn−1 (8)

ollowing the derivation process of Eq. (4), we can get

= 2 cos�n (9)

here �n = �/2 − (2n − 3)˛, (n = 2, 3, 4, & . . .), will be given in Section.3.

Following the derivation process of Eq. (6), we can get

�n = �Qn−1 − g �Nn (10)

�n and �Qn may be written as a component form as follows:

xn = xn−1 + S˛(n−1)

yn = yn−1 + Sˇ(n−1)

zn = zn−1 + S�(n−1)

˛n = ˛(n−1) − gNnx

ˇn = ˇ(n−1) − gNny

�n = �(n−1) − gNnz

(11)

here �Nn((l − xn)/B, (l2/h2)yn/B, (l2/h2)zn/B), B =(l − xn)2 + ((l2/h2)yn)2 + ((l2/h2)zn)2

At this point, the recurrence relation for vectors �Pn and �Qn can

e determined by �P1, �Q1g, �N1−→�P2, �Q2, �Pn−1, �Qn−1

g, �N1−→�Pn, �Qn (n > 3). Asong as the initial coordinates (x1, y1, z1) of the incident light andhe size of a cone are given, and by using the recurrence Eq. (11)epeatedly we can get the trajectories of the light inside the cone.

.3. The refraction of light at the sidewall of a cone

Fig. 4 is a cross-section schematic diagram of the glass conehrough its bottom center and vertex. A coordinate system is estab-ished as shown in Fig. 4. The origin of the coordinate systemoincides with the bottom center. L is the height of the cone, and

h is the bottom diameter. Suppose (x1, y1) is the initial coordinatef an incident ray, (xn, yn)(n = 2,3,. . .) are intersection points of lightays with the sidewall of a cone, �n(n = 2,3,. . .) are the angle of inci-ent with respect to the sidewall of a cone, and ̨ is the semiapexngle of a cone. From Fig. 4 we can get �n = �/2 − (2n − 3)˛, (n = 2,, 4, & . . .), (n − 1) is the number of intersections of light rays andhe sidewall, and (n − 2) is the number of the total reflection inside

cone, which is determined by the semiapex angle of a cone. It is

een from this expression that �n will gradually decrease with thencrease of n. When the angle of incidence �n is an angle at whichight does not meet the condition for total internal reflection, light

ill be refracted out of the sidewall.

3 (2012) 1550– 1554

The relationship between the number of total reflection and ˛is as follows:

if(n − 2) = 0 then �n = �

˛< ˝

if(n − 2) > 0, then �n and �n−1, namely

⎧⎨⎩

�

2 − (2n − 3) ̨ < ˝

�

2− (2n − 5)˛ > ˝

(12)

By Eq. (12) we get the relation between ̨ and n as follows

�/2 − ˝

2n − 3< ̨ <

�/2 − ˝

2n − 5(13)

When n = 3, 0.2843 < ̨ < 0.8528

When n = 4, 0.1705 < ̨ < 0.284

When n = 5, 0.1218 < ̨ < 0.1705

And so on

(14)

Table 1 lists ̨ range at different n values, and in the table˛1 = (�/2 − ˝)/(n − 3), ˛2 = (�/2 − ˝)/(2n − 5). It is seen from thistable that, as long as the apex angle of a cone is given the numberof total reflection will be constant.

When the angle of incidence at the sidewall is less than the criti-cal angle, light is refracted out of the sidewall. Suppose the angle ofrefraction and the refractive index of glass are denoted by �′

n and �g,respectively, then the angle of incidence and refraction are relatedin such a way that

�g sin(�n) = sin(�′n) (15)

�′n = arcsin(�g sin(�n)) (n = 2, 3, 4, & . . .).

�′n determines the exit direction of light. From Fig. 4 we may

know that there are two types of exit light distinguished fromanother. Firstly, in the case of �′

n = �/2, the refracted light will exitalong generatrices of the cone; secondly, in the case of �′

n + ̨ = �/2,the exit light are parallel to the incident light, that is to say, exitalong the x-axis direction. Here, our main interest lies in the firstcase. Put �′

n = �/2 and �n = �/2 − (2n − 3)˛ into Eq. (15), we get�n = �/2 − (2n − 3)˛ = arcsin(1/�g), so

˛ = �/2 − arcsin(1/�g)2n − 3

(16)

Compare Eq. (16) with Eq. (13), and we will see that ˛1 in Table 1is just the solution of Eq. (16). Values of ̨ and n which satisfy Eq.(16) are just the number of total reflection, (n − 2), and also ̨ valuesrequired for exiting along generatrices of the cone. It can be seenthat there are an infinite number of such ̨ values.

We also see from Table 1 that, when n = 10, also the numberof total reflection is 8, the required apex angle of a cone is verysmall, at this time if the beam width of incident light is constant, thecorresponding length of the cone must be very long. If the number oftotal reflection is too many, a cone required must have large sizes,and which will bring the inconvenience to use. Therefore, whenusing it we must choose an appropriate value of ̨ depending onthe situation. Taking into account processing and using difficulties,a cone commonly chosen is not too slender.

3. The simulation and analysis of light propagationthrough a cone

By making programs in Matlab according to formulas derivedearlier, we can do some simulation of light propagation. Simula-tion results of light propagation in a cone are shown in Fig. 5. It

Q. Hua, S. Cunzhi / Optik 123 (2012) 1550– 1554 1553

Table 1The relation between the number of total reflection (n − 2) and semiapex angles ̨ (rad).

n 3 4 5 6 7 8 9 10

8 0.07753 0.06556 0.05685 0.0502084 0.0948 0.07753 0.06556 0.05685

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˛1 0.2843 0.1706 0.12184 0.094˛2 0.8528 0.2843 0.1706 0.121

an clearly be seen from Fig. 5 that with the help of selecting theppropriate angle ˛, the incident light perpendicular to the bottomf a cone passing through this cone will be emitted along the coneeneratrices.

Fig. 5 only shows light rays in a cross section of a cone. Dueo symmetry, the transmission characteristics of each conic cross-ection is the same, therefore, we can deduce from the second figuren Fig. 5 that a solid light cylinder parallel to the x-axis passinghrough this cone will be converted to a light beam structured as

hollow cone, called a hollow light cone, and form a circle-typepot on the plane perpendicular to the axis of cone, all its energy isoncentrated in a circle, as shown in the third figure in Fig. 5.

Similarly, a solid light semicylinder parallel to the x-axis passinghrough this cone will be transformed into a hollow light semicone,nd form a semi-circle-type spot on the plane perpendicular to thexis of cone; a solid light third-cylinder parallel to the x-axis passinghrough this cone will be transformed into a hollow light third cone,nd form a one- third -circle-type spot on the plane perpendicularo the axis of cone. A hollow light cone can be transformed into

hollow light cylinder by means of a lens whose focus coincides

ith the cone vertex, so as to only convert a light spot mode and not

hange a light propagation direction, and Fig. 6 shows simulationesults mentioned above.

Fig. 5. The simulation of light passing through a cone.

Fig. 6. A cylindrical or semicylindrical light beam passing through a cone and a lensis transformed into a hollow light cylinder or semicylinder.

4. A new method of circle and arc cutting

We can use the data in Table 1 to produce a glass cone, andsubstitute beam-focusing optics to direct the beam to the work-piece. A schematic diagram of improved laser cutting is shown inFig. 7. In figure, the focus of lens coincides with the cone vertex,and lights passing through this lens are transformed into parallellights. By means of a glass cone and a lens, a solid light cylinder willbe transformed into a hollow light cylinder, the direction of lightrays does not change, and the light energy is completely gatheredin a cylindrical surface. Diameter D of hollow light cylinder and

focal length of lens f are related in such a way that D = 2f*tg(˛),where ̨ is the semiapex angle of cone. When ̨ is constant, solong as we appropriately select lenses with different focal lengths,

Fig. 7. The laser cutting system for processing circle and arc.

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554 Q. Hua, S. Cunzhi / Op

nd the cutting of circle with different diameters will be carriedut.

In the same way, a solid light semicylinder will be transformednto a hollow light semicylinder, and a solid light third cylinder wille transformed into a hollow light third cylinder. Therefore, as longs we control the shape of incident light beam, we can carry out anyhape arc cutting.

. Conclusions

This paper presents a new method for the circle and arc cut-ing. Theoretical study and simulation experiments show that this

ethod is effective.

1) There is no need to implement the laser scanning, so the beamand workpiece complex motion under computer control are nolonger necessary. Therefore, this method helps improve effi-ciency, save the energy consumption and reduce the process,

and make processing, operation, control and using becomesimple.

2) Because the laser beam is perpendicular to the workpiece, theredoes not exist issues which the focal depth and focal spot cannot

[

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3 (2012) 1550– 1554

be optimized in the meantime, therefore, the strict vertical cutsurface and the high cutting quality can be ensured.

(3) When the spot of incidence light beam is a semicircular surface,the spot of emitting light beam becomes a semicircle. Similarly,when the spot of incidence light beam is a third circular surface,the spot of emitting light beam becomes a third circle. So usingthis glass cone in a laser cutting system different types of arccan conveniently be cut.

References

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2] Z.-j. Chen, J. Wu, L.I Xue-gong, Study on laser strengthening on diamond sawbasilar, J. Mech. Eng. (11) (2008) 6–7.

3] N. Rajaram, J. Sheikh-Ahmad, S.H. Cheraghi, CO2 laser cut quality of 4130, Int. J.Machine Tools Manuf. 43 (2003) 351–358.

4] R.B. Tirumala, R. Kaul, P. Tiwari, et al., Inert gas cutting of titanium sheet withpulsed mode CO2 laser, Opt. Lasers Eng. 43 (2005) 1330–1348.

5] Bystronic Operating Manual, Bystronic Maschinen AG Switzerland.6] Y.-j. Deng, The laser incises and its application in cutting the ceramics, Machinery

Vol. 31 (3) (2004) 55–60.7] J. Fieret, M.J. Terry, B.A. Ward, Aerodynamic interactions during laser cutting,

SPIE 668 (1986) 53–62.8] Yangwei, X.-h. Peng, J. Zhang, UV laser cutting technology for siliconwafer, Elec-

tron. Process Technol. Vol. 30 (1) (2009) 37–40.