- Email: [email protected]

Contents lists available at SciVerse ScienceDirect

Optik journal homepage: www.elsevier.de/ijleo

Laser beam shaping with an ellipsoidal lens Daxin Luo ∗ , Baiqin Zhao, Xuelei Chen Institute of Semiconductors, Chinese Academy of Sciences, Beijing, 100190, China

a r t i c l e

i n f o

Article history: Received 10 August 2011 Accepted 16 December 2011

Keywords: Beam shaping Semiconductor laser Ellipsoidal lens Geometrical optics

a b s t r a c t This paper presents a semiconductor laser beam shaping system that can collimate the irradiance proﬁle effectively by using an ellipsoidal lens. Geometrical optics analysis based on the ray tracing method is done and the formulas to calculate the shape of ellipsoidal lens are given. Both the theoretical and experimental result show that the laser beam system works effectively; the divergence angle is reduced to less than 1◦ in the fast-axial direction. By using epoxy resin, this shaper collimates a semiconductor laser beam and packages the laser diode (LD) at the same time, which simpliﬁes the manufacturing process and greatly reduces the LD volume. Because of the small volume, low-cost, high rigidity and easy fabrication, the shaper is of great value in the ﬁeld of semiconductor laser diode applications. © 2012 Elsevier GmbH. All rights reserved.

1. Introduction Because of their low price and small size, semiconductor lasers have been widely used in many ﬁelds. However, the semiconductor laser diodes’ (LDs’) potential applications are limited by their poor beam quality, which is ascribed to the large divergence angle and to the Gaussian irradiance proﬁles. In order to make the light travel far enough and to retain the light intensity in a small area, many ﬁelds, including military and commerce, need the rays emitted from LDs to be parallel. To achieve as small as possible divergence angle beams, experts all over the world have done a great deal of research in beam shaping, and some excellent work and designs have been carried out [1–4]. For single-mode lasers, two of familiar kinds of shaping systems are shown as follows. The ﬁrst method uses a plano-aspheric lens pair to shape the beams with Gaussian irradiance proﬁles, as shown in Fig. 1. The ﬁrst aspheric surface redistributes the rays in such a way as to transform the proﬁle, and the second aspheric surface recollimates the beam [5,6]. This refractive beam shaper can be used to collimate the irradiance proﬁle and get a ﬂat-top beam at the same time, which is substantially superior to a Gaussian beam for illumination applications. Another design uses a hyperboloid cylinder-plane lens for shaping the laser beam [7], as shown in Fig. 2(a). However, it is limited that only one direction can be collimated by using this method. Based on this design, a design has been reported using two hyperboloid cylinder-plane lenses arranged

∗ Corresponding author at: 1#621, Semiconductors Institute, Jia 35, Qinghuadong Load, Haidian district, Beijing, 100083, China. Tel.: +86 10 82304707. E-mail address: [email protected] (D. Luo). 0030-4026/$ – see front matter © 2012 Elsevier GmbH. All rights reserved. doi:10.1016/j.ijleo.2011.12.023

perpendicularly to collimate both the fast and the slow axial directions [8], as shown in Fig. 2(b). In addition, this kind of design is suitable to use for linear light source. Otherwise, there are also some kinds of good methods for laser beam shaping discussed by scientists, for example, Serkan and Kirkici had presented two optical system designs using aspherical lenses for beam circularization, collimation and expansion of semiconductor lasers [9]. Because of the requirement of differing application requirements, no single beam shaping method is suitable for all situations. Speciﬁc laser beam shaping systems need to be developed for speciﬁc applications. When used in the practical industrial and commercial manufacture, both of the methods mentioned above have difﬁculty shaping the diode beams appropriately and being packaged in small volume. The purpose of this paper is to discuss a method for semiconductor lasers beam shaping by putting the lasers into an aspheric lens, so that the divergence angle of the laser beam can be reduced effectively and the LD is packaged at the same time. That makes the package of the LD small and low-cost. 2. Optical model design The optical model is shown in Fig. 3. Since the light source is a point, and this beam shaping method is independent of divergence angle, the ellipsoidal shape in the fast-axial and the slow-axial direction should be the same, so only one direction is discussed here. The semiconductor laser is packaged in an aspheric lens, which is made from epoxy resin or optical glass. This structure is axisymmetric, and the irradiance is emitted from the aspheric side. The package can be produced right after the laser diode is

566

D. Luo et al. / Optik 124 (2013) 565–569

Fig. 4. Two of the optical paths in the shaping system.

Then the equation about the two rays in Fig. 4 is

Fig. 1. Plano-aspheric lens pair used for beam shaping.

nl = n

x2 + y2 + n0 (l − x)

(1)

Here, n and n0 are the refractive indexes for the interior material and exterior material respectively (n0 = 1 in this paper). The equation above can be reduced to

nl − l + x = n

x 2 + y2

where n and l are deﬁned as a n= c l =a+c

(2)

(3) (4)

The equation above can be rewritten like this

2

a − c 2 + xc = a Fig. 2. Hyperboloid lens used for beam shaping.

manufactured, so the laser diode does not need to be encapsulated any more. In this structure, the laser beam is emitted directly into the materials like epoxy resin or optical glass (we use epoxy resin in our experiment), whose refractive index is higher than that of air. When the beam reaches the aspheric side and enters the air, the rays become parallel. The light is emitted from a small source point and transformed in the epoxy resin. Semiconductor lasers have Gaussian irradiance proﬁles, and their divergence angles are approximately 30◦ in the fast axial direction and 15◦ in the slow axial direction. To improve the beam quality, the surface of the lens needs to be designed to refract the rays parallel to the x-axis so they form a parallel beam with a small divergence angle. The shape of the refractive surface of the designed lens can be determined by numerical analysis, but the related constants, such as the refractive index, should be determined ﬁrst, and then we can calculate the radian of the ellipse. According to Fermat’s principle, all the optical paths of any refracted rays from source to target plane should be equal. Here, the light source is supposed to lie in the origin of axis, M(x, y) is any point on the lens, the parameter l is the length between origin and the apex of the lens, as shown in Fig. 4.

Fig. 3. Aspheric lens model used for beam shaping.

x 2 + y2

(5)

Square both sides of the equation (x − c)2 y2 + 2 =1 a2 a − c2

(6)

This is the equation of an ellipse, where the long axis is equal to a, and the short axis is b=

a2 − c 2

(7)

The shape of the aspheric lens is an ellipsoid and the laser lies in one of its foci, as shown in Fig. 5. Since the refractive index of the epoxy resin is known, the shape of the ellipse can be calculated. From the equations above, one parameter of the ellipse equation must be known ﬁrst; that means “a” or “b” or “c” should be determined before starting the calculation. In this paper, “b” is known. In this condition, Eqs. (3) and (7) become a=

bn

c=

b

(8)

n2 − 1

n2 − 1

Fig. 5. Optical path of a bundle of rays through the ellipsoidal lens.

(9)

D. Luo et al. / Optik 124 (2013) 565–569

567

Fig. 6. Instruments used in experiment.

Therefore, the radius of the laser beam shaping system can be accurately calculated by using Eq. (6). 3. Experiments An apparatus has been fabricated and used to demonstrate the beam shaping model described in this paper. The wavelength of the semiconductor laser diode being used is 860 nm and the package material is epoxy resin, whose refractive index is supposed to be 1.45. In this experiment, the laser diode is bonded to a metal matrix. After being ﬁxed, the laser diode is inserted into a plastic mold which is fabricated in an ellipsoidal shape, as shown in Fig. 6. Then epoxy resin is ﬁlled into the mold. The ellipse radian we used in this experiment is x2 y2 + =1 11.91 6.25 The position of the laser diode had been calculated exactly and set up in the plastic mold. The ﬁnished laser diode with beam shaping system is shown in Fig. 7. The size of this package is Ø 5 mm × 10 mm. By using an infrared detector, the divergence angle of the semiconductor laser beam before and after using the shaper designed in this paper is shown in Fig. 8(a) and (b) respectively. These two pictures were gotten from a CRT monitor, which was connected to the infrared detector, and the noise in the pictures was introduced by the monitor. In these pictures, the distances between the LD and screen are 0.5 m, the window size is 0.3 m × 0.2 m. From the pictures above it can be seen that the laser beam shaping system works very well. It collimates the beam into a tiny spot, whose size is reduced from 0.18 m × 0.05 m to nearly 0.009 m × 0.017 m at 0.5 m distance, so the beam can be transmitted effectively with a small divergence angle. In addition, the following sheet has shown the result (Table 1).

Fig. 8. (a) Output beam at 0.5 m distance without using shaper, (b) output beam at 0.5 m after using shaper (window size:0.3 m × 0.2 m).

4. Discussions In this part, what would affect the quality of the output beam after shaped, is discussed. The Fresnel reﬂection in the system would reduce the output light intensity; the sensitivity of the shaping system shows probably ways that affect the experimental result and the fabrication tolerance of this system; the parameter ˇ shows that geometrical optics are limited in some precise systems; and the wavelength is related to the refractive index, which is important in this shaping system. 4.1. Fresnel reﬂection When light moves from a medium (suppose the refractive index is n1 ) into another medium (suppose the refractive index is n2 ), both reﬂection and refraction of the light may occur, as shown in Fig. 9. So in the laser beam shaping system presented here, some of laser would be reﬂected and lost. The Fresnel equations are used to describe the behavior of light in this circumstance, and reﬂectance R is imported to give the fraction of the incident power that is reﬂected from the interface. For linearly polarized light, if it is polarized with the electric ﬁeld of the light perpendicular to the plane of Fig. 8 (s-polarised), the reﬂection coefﬁcient is given by:

⎡ Rs = ⎣

n1 cosi − n2 n1 cosi + n2

1− 1−

n1 /n2 sini

n1 /n2 sini

2 ⎤2 ⎦ 2

Fig. 7. The LD with beam shaper.

Table 1 Divergence angles of the LD before/after shaped. Divergence angle

Before shaped

Horizontal Vertical

10–20◦ 30–40◦

After shaped 1–2◦ 0.5–1◦

Fig. 9. Reﬂection and refraction of light.

(10)

568

D. Luo et al. / Optik 124 (2013) 565–569

Table 2 Divergence angle as affected by refractive index variations. Original divergence angle

Refractive index of the LD

10◦ 15◦ 20◦

1.42

1.43

1.44

n = a/c = 1.45

1.46

1.47

1.48

0.68◦ 1.02◦ 1.39◦

0.45◦ 0.68◦ 0.93◦

0.22◦ 0.34◦ 0.46◦

0◦ 0◦ 0◦

0.22◦ 0.34◦ 0.46◦

0.45◦ 0.68◦ 0.93◦

0.67◦ 1.03◦ 1.40◦

of the packaging material and to the position of the laser diode. During manufacture of the shaping system, these parameters should fabricated very accurately. 4.3. Additional factors Some additional important factors which make the beam shaping problem difﬁcult have been discussed in other papers. These factors mainly include scaling, smoothness and coherence [10]. The book of Laser Beam Shaping: Theory and Techniques discussed that the optical design methods of beam shapers can be divided into two categories: physical optics and geometrical optics. For single-mode Gaussian beams, whether physical or geometrical optics methods should be used can be determined by calculating the parameter ˇ, Fig. 10. The relationship between reﬂectance R and divergence angle.

ˇ=

where i is the incident angle. If the incident light is polarized in the plane of Fig. 8 (p-polarised), the R is given by:

Rp =

n1 n1

2

1 − ((n1 /n2 )sini ) − n2 cosi

(12)

where is the wavelength, r0 is the radius at the 1/e2 point of the input beam, Dout is half-width of the desired output dimension, and f is the focal length of the focusing optic, or the working distance from the optical system to the target plane for systems without deﬁned focusing optics. For simple output geometries, such as circles and rectangles, the following simple rules have been suggested: when ˇ < 4, a beam shaping system will not produce acceptable results; when 4 < ˇ < 32, diffraction effects are signiﬁcant and should be part of the design of beam shaping systems; and when ˇ > 32, geometrical optics should be adequate for the design of beam shaping systems. However, for more complicated output geometries, ˇ may have to be signiﬁcantly higher to produce acceptable results [11,12]. In this laser beam shaping system, while the divergence angle is 30◦ and the size of light spot at 0.5 m is supposed to less than 0.01 m, ˇ = 0.12 can be gotten from Eq. (12). That means geometrical optics are limited when more precision and collimation are demanded from the shaping system. In other words, this proposed laser beam shaping system has been received good results by using geometrical optics, but if the system is wanted to improve and make the rays more parallel and exact, the physical optics method should be considered.

2 (11)

2

√ 2 2r0 Dout f

1 − ((n1 /n2 )sini ) + n2 cosi

Fig. 10 shows their results by using the parameters deﬁned in the experiment, where n1 = 1.45, n2 = 1, and the angles have been translated into the irradiance angles of laser beam. The value of reﬂectance R, which is exactly depended on the direction of the polarized light, is between Rs and Rp. From the ﬁgure above it can be seen that loss is always exist, especially when the laser beam transmits in a large angle. 4.2. Sensitivity of the shaping system In this part, the sensitivity of the shaping system to the ﬂuctuation of the exterior environment variables is discussed, such as the refractive index, the radius of the ellipsoid and the position of the laser diode. In our calculation, divergence angle of the LD before shaped is postulated to be 10◦ , 15◦ or 20◦ separately. Based on the ray tracing method, MATLAB had been used to calculate the irradiance distribution when the refractive index or the position of the laser diode departs from the ideal value, as shown in Tables 2 and 3. The ﬁrst row represents refractive index in Table 2 or distance from focus to LD in Table 3; the ﬁrst column represents the divergence angle before shaping. The results show that, the divergence angle becomes larger quickly when the parameters only change a small amount. That means the shaping system is very sensitive to the refractive index

4.4. Wavelength of laser Finally, one of the factors which affect the beam shaping system is referenced here: the wavelength of the laser. The refractive indexes of different wavelengths of light in the same epoxy resin are not the same, as shown in Fig. 11.

Table 3 Divergence angle as affected by variations in the position of the LD. Original divergence angle

10◦ 15◦ 20◦

Position of the LD departs from the ideal value (on the x-axis) −0.3 mm

−0.2 mm

−0.1 mm

0 mm

0.1 mm

0.2 mm

0.3 mm

0.77◦ 1.18◦ 1.65◦

0.51◦ 0.79◦ 1.10◦

0.26◦ 0.40◦ 0.55◦

0◦ 0◦ 0◦

0.26◦ 0.40◦ 0.56◦

0.51◦ 0.80◦ 1.12◦

0.77◦ 1.20◦ 1.70◦

D. Luo et al. / Optik 124 (2013) 565–569

569

The advantages of this beam shaping system are small volume, low-cost, high reliability and easy fabrication. These factors make the ellipsoidal lens of great value in many areas such as infrared homing missiles, light communication, laser fuses, laser weapons, etc. References

Fig. 11. Dispersion curve of a kind of epoxy resin [13].

5. Conclusions An epoxy ellipsoidal lens is discussed as a way to collimate a semiconductor laser beam, which is packaged by the epoxy resin at the same time. The theoretical analysis is based on geometrical optics. The result shows that rays from the semiconductor laser can be reshaped and collimated perfectly. Some experiments have been done to demonstrate this beam shaping system. They have proved that the focusing system works very well. The divergence angle is reduced to less than 1◦ in fast axis, a major improvement. In addition, the size of the package containing the laser beam shaping system and LD is only Ø 5 mm × 10 mm, which makes applications of the semiconductor LDs much more convenient.

[1] D.L. Shealy, Optical design of laser beam shaping systems, SPIE 4832 (2002) 344–357. [2] S.H. Wang, C.J. Tay, C. Quan, H.M. Shang, Study of collimating laser diode beam by a graded-index optical ﬁbre, Optik 112 (11) (2001) 531–535. [3] X. Zhou, N.K.A. Bryan, S.S. Koh, Single aspherical lens for deastigmatism, collimation, and circularization of a laser beam, Appl. Opt. 39 (7) (2000) 1148–1151. [4] D.L. Shealy, et al., Geometric optics-based design of laser beam, Opt. Eng. 42 (11) (2003) 3123–3138. [5] J.A. Hoffnagle, C.M. Jefferson, Beam shaping with a plano-aspheric lens pair, Opt. Eng. 42 (11) (2003) 3090–3099. [6] Y. Ohtsuka, Y. Arima, Y. Imal, Acoustooptic 2-D proﬁle shaping of a Gaussian laser beam, Appl. Opt. 24 (17) (1985) 2813–2819. [7] Q. Xu, Y. Han, X. Zeng, Y. An, Hyperboloid cylinder-plane lens for shaping laser diode array beam, Optik 121 (2010) 1596–1599. [8] S. Yuan, H. Yang, K. Xie, Design of aspheric collimation system for semiconductor laser beam, Optik 121 (2010) 1708–1711. [9] M. Serkan, H. Kirkici, Optical beam-shaping design based on aspherical lens for circularization, collimation, and expansion of elliptical laser beam, Appl. Opt. 47 (2008) 230–241. [10] L.A. Romero, F.M. Dickey, What makes a beam shaping problem difﬁcult, Proc. SPIE 4095 (2000) 16–25. [11] F.M. Dickey, S.C. Holswade (Eds.), Laser Beam Shaping: Theory and Techniques, Marcel Dekker, New York, 2000, pp. 12–13. [12] D.L. Shealy, S.H. Chao, Geometric optics-based design of laser beam shapers, Opt. Eng. 42 (11) (2003) 3123–3138. [13] W. Shi, C. Fang, X. Yin, Q. Pan, X. Sun, Q. Gu, Z. Qin, Study on refractive index dispersion of organic polymer, Piezoelectrics Acoustooptics 22 (1) (2000), 43 (in Chinese).