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Anamorphic gradient index (GRIN) lens for beam shaping Nisha SoodBiswas a,⁎, Md Asraful Sekh b, Samir Sarkar c, Amitabha Basuray c a b c

Central Glass & Ceramic Research Institute, Kolkata 700032, India Department of Electronics and Communication Engineering, Aliah University, DN-47, Sector-V, Salt Lake, Kolkata 700091, India Department of Applied Optics and Photonics, University College of Science and Technology, Calcutta University, Kolkata 700009, India

a r t i c l e

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Article history: Received 2 February 2011 Received in revised form 10 December 2011 Accepted 14 December 2011 Available online 27 December 2011 Keywords: Anamorphic GRIN rod lens Toric power Beam shaping

a b s t r a c t In this short communication we are reporting a new kind of rod lens with toric power with moderately large power difference. These rods can be directly used in coupling power from a semiconductor laser to optical ﬁber or in free space communication to convert the beam shape. This rod may directly be butt-jointed to the laser, which may attract many application scientists. Moreover, anamorphic power in a GRIN lens can be generated by proper selection of geometry of the substrate for ion exchange. This may lead to a new kind of optical system that needs further exploration. © 2011 Elsevier B.V. All rights reserved.

1. Introduction Anamorphic lenses are speciﬁcally used to transform highly elliptic output beam from a semiconductor laser into a spherical beam in free space optical communication, ﬁber optics communication as well as for beam shaping in data storage and processing. The procedures for achieving such anamorphic transformation were use of prism system [1], binary gratings [2], cylindrical graded index (GRIN) lenses [3,4], or holographic lens [5]. All current techniques for beam shaping start with the use of cylindrical GRIN lens having numerical aperture of the order of 0.5, for collimating it along larger divergence angle [6]. Semiconductor laser diode is divergent in both the orthogonal directions with an aspect ratio typically of 4:1. Two one-dimensional GRIN lenses, arranged in tandem, are used for transformation of an elliptical beam into a spherical one [7]. Carslaw and Jaeger [8] have shown elliptical isotherms in the problem of heat conduction in an elliptical structure. In Ref. [9] it is shown that in the case of diffusion equation in elliptical coordinate may be computed using Mathieu function and some results are being given. Here too the equi-concentration lines are elliptic in nature. Van Burn and Boisvert [10] have conﬁrmed this in their studies of accurate calculation of Mathieu functions. Acosta and others [11] presented a theoretical design for correcting such astigmatic Gaussian beams from laser diode by an anamorphic GRIN lens,

⁎ Corresponding author. Tel.: + 91 33 24837340; fax: + 91 33 24730957. E-mail address: [email protected]ri.res.in (N. SoodBiswas). 0030-4018/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2011.12.061

to obtain a rotationally symmetric Gaussian beam, both in phase and amplitude at its output. Sinzinger and others [12] and Zhou and others [13] tried to make astigmatic micro lenses by exchanging silver for sodium through an elliptic mask, however with not much success. In this communication, we report the development of GRIN rod with elliptic cross section, having toric power with large aspect ratio.

2. GRIN rod fabrication A two-step process comprising exchange of ions at the surface with the cations in the bath and subsequent diffuse of exchanging cations inward is a convenient way of generating GRIN proﬁle in a glass. Advantage of ion-exchange process lies in the fact that the proﬁle generated takes the shape of base glass when accomplished at a temperature that does not distort its shape, which is at or below glass transition temperature. The direction of diffusion is normal to the surface following Fick's Law, ∂C ¼ ∇ðDc ∇C Þ ∂t

ð1Þ

where, ‘C’ is the concentration of dopant in moles and ‘Dc’ is the interdiffusion coefﬁcient at the temperature of exchange and depends upon the concentration of exchanging cations in the glass. Diffusion is circularly symmetric in cylinders with circular cross section since radii are normal to the surface. In cylinders with elliptic cross section, the normal is not always passing through the intersection of the

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symmetric about the center but about x- and y-axes of the ellipse and Eq. (1) takes the form, ∂C ∂ ∂C ∂ ∂C ¼ Dc þ Dc ∂t ∂x ∂x ∂y ∂y

ð2Þ

In reference [14], the authors have shown that in the case of radial diffusion (1) can have a unique and simple solution applying the method of residual solution by deﬁning new variables for diffusion depth and time, which depends upon the radius of the rod, used in exchange process. In the similar fashion we may compute isorefractive index lines by transforming diffusion equation from rectangular coordinates (x,y) to elliptic coordinates (ξ, η) by the formulas x ¼ f cosh ξ cos η;

y ¼ f sinh ξ sin η

ð3Þ

And a solution of the form Ct = C1(ξ)C2(η) is sought; it is found that C1(ξ) and C2(η) must satisfy the equations 2

d C2 dη2 2

Fig. 1. GRIN rods with elliptic cross section and slices made thereof.

major and minor axes that is the center of the ellipse, but along the bisector of the angle formed by the two foci of the ellipse at the point on the surface under consideration. Therefore, the diffusion will not be

d C1 dξ2

! þ ða−2q cos2ηÞC 2 ¼ 0

ð4Þ

−ða−2q cosh2ξÞC 1 ¼ 0

ð5Þ

!

where ‘q = f 2DC2/4’ and ‘a’ are the separation constants arising from the separation of variable method. Eqs. (4) and (5) are known as the ordinary and modiﬁed Mathieu equations, respectively [15]. However, in

Fig. 2. Newton's ring pattern (a) in absence of any sample, (b) with base glass sample (c) in presence of anamorphic elliptic GRIN sample with horizontal major axis and (d) same with vertical major axis.

N. SoodBiswas et al. / Optics Communications 285 (2012) 2607–2610

applications involving Helmholtz equation in elliptic coordinates, equations are better known as angular and radial Mathieu equations [16]. Lytle and Lager [9] have solved diffusion equation in elliptic coordinates and has provided few results. A solution in the line of Gutiérrez-Vega el al. [17] is undertaken and will be communicated separately. Preliminary calculations show that equi-concentration lines are elliptic in nature. Unique rod lenses of elliptical cross section were made from glass of composition 10.5 Na2O–15.0 Li2O–7.5 Al2O3–67.0Al2O3 (in mole percent) melted at 1400 °C. These glass rods are approximately 30 mm long and have cross sections with major axis 3.5 ± 0.1 mm and minor axis being 1.7 ± 0.1 mm. The glass has refractive index ne = 1.5379 and Abbe dispersion value = 58.658. Dimensions of the cross sections deviate slightly from an exact ellipse due to shaping of the rods by hand grinding and polishing. Sodium nitrate bath and vitreous silica crucible were used for the exchange of sodium with lithium ions at 370 °C for 160 h. The glass rod was washed with water and dried after the exchange. Specimens of thickness 1.5 mm have been sliced from the exchanged rod and then polished for further optical analysis. The remaining rod of around 23.1 mm length was also polished for image analysis. A slice of the same thickness was also taken out from the same rod prior to ion exchange for comparing with exchange glass. Some of the samples made are shown in Fig. 1.

Being conﬁrmed that such systems show toric power, rod lens made from the same glass as mentioned above were examined further for gaining knowledge of imaging properties. In the ﬁrst case, the collimated beam after removal of birefringent lens and polarizers in the above set-up is, focused by a normal lens at the input end of the rod lens. The diverging beam coming out from the other end of the rod lens is then focused by another lens. It is observed that there are two focal planes for the horizontal and vertical fan of beams as shown in Fig. 3(a and b) and the saggital focus and tangential focus are widely separated for an axial beam. Here only qualitative power difference between two axial foci is shown. Finally, the rod lens was tested for anamorphic transformation of an elliptic beam into a spherical beam (beam shaping). Once again, we used the birefringent lens set up with an additional cylindrical lens to make the input beam elliptical. Elliptic beam is then focused at the input end of the rod, once with the major axis of the input elliptical beam coincides with the major axis of the cross section of the rod, and in the other the major axis of the elliptic beam coincides with the minor axis. Fig. 4(a, b and c) depicts the incident elliptic beam and the beam modiﬁed in the presence of the grin rod lens with two different conﬁgurations respectively. It is apparent from Fig. 4(c) that output beam is nearly spherical.

3. Results and discussion A deﬂectrometric set-up, developed by this group earlier for measurement of radial GRIN proﬁle [14] is used to study the generated refractive index proﬁle. In this set-up, light from a laser source after collimation using a microscopic objective, a pinhole and a lens, is made incident on a birefringent lens placed between two polarizers. A birefringent lens, having its optical axis perpendicular to its principal axis, has two focal lengths — one corresponding to ordinary vibration and the other corresponding to the extraordinary vibration. This is due to the existence of two discrete refractive index values in a birefringent medium for ordinary and extraordinary rays. If the polarizer is placed at an angle of 45° to the optical axis in front of the birefringent lens, then two orthogonally polarized light beams will be focused at two longitudinally separated points. If an analyzer after the lens is placed in-plane or in-crossed position with respect to the polarizer, the two beams with longitudinal shift between their focus points will form Newton's ring like circular pattern as shown in Fig. 2(a). The sample, for which the refractive index proﬁle is to be measured, is placed in the path in a fashion so that the sample is fully illuminated then fringes will diverge or converge [18] according to the positive or negative gradient of the refractive index at that point, respectively. Therefore, variation in refractive index may be evaluated from the fringe distortion. The fringe distortion for 1.5 mm thick sample of the base glass and the ion-exchanged samples are shown in Fig. 2(b, c and d). Fig. 2(b) is for the base glass with the same elliptic geometry and shows no change in the fringe pattern. The image is noisy due to the presence of scratches in the specimen. However, the noise does not interfere on the image pattern. Fig. 2(c and d) shows the changes in fringe pattern due to refractive index gradient in the case of anamorphic GRIN sample placed with major axis horizontal and vertical, respectively. Changes in fringe spacing in both horizontal and vertical directions prove that refractive index proﬁles in two directions are different. Formation of elliptical fringes converts the system to a multi-focal length lens. This shows that an elliptic variation in refractive index is being generated by exchange of cations. However, the fringes near the central part have not been modiﬁed due to poor diffusion and penetration of ions up to the center. Moreover, as samples are produced by hand processing there is slight deviation of the fringes from an exact ellipse.

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Fig. 3. Lines corresponding to (a) saggital and (b) tangential focus.

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Thus, the development of an anamorphic GRIN rod lens is reported for the ﬁrst time. This rod may be directly used for efﬁcient launching power into the ﬁber or for converting the beam shape in free space optical communication. The major advantage of this type of rod lens is that the lens may be coupled to the laser/ﬁber by butt joint, as the input beam dimension is small enough to be coupled to an elliptic rod lens even with high aspect ratio. This is not always the case when two one-dimensional GRIN lenses are used in tandem. Thus, the developed rod may ease the problem of power coupling in optical communication. 4. Conclusion The results presented in this paper are mostly qualitative in nature and calls for detailed investigation, which includes a complete mathematical analysis of diffusion process involved in such elliptical structure and quantitative evaluation of refractive index proﬁle so generated. To obtain an optimum lens system with elliptic variation in refractive index proﬁle, i.e., with different power in two orthogonal axes of the lens, which does the necessary beam shaping required for a particular laser beam, prior estimation of refractive index proﬁle and aspect ratio of such laser are required. This also includes the choice of proper glass material for ion exchange. The design procedure of proﬁle for correcting astigmatic beam to a spherical one can then be developed properly. This paper is to report the importance of the method with some preliminary results and the detailed analysis is undertaken. Finally, it is also important that with proper selection of geometry, different kinds of refractive index proﬁles can be generated in a GRIN lens system, which might ﬁnd its application in different kinds of laser beam shaping and other applications. These lenses need attention and further investigations. References [1] J.F. Forkner, F.R. Nash, Characteristics of efﬁcient laser diode collimators, Melles Griot Optical Components Division Publication, 1983. [2] W.B. Veldkamp, Applied Optics 21 (1982) 3209. [3] J.M. Stagaman, D. Moore, Applied Optics 23 (1984) 1730. [4] Y. Asahara, H. Sakai, S. Shingaki, S. Ohmi, Applied Optics 24 (1985) 4312. [5] C.C. Aleksoff, K.K. Ellis, B.D. Neagle, Optical Engineering 30 (1991) 537. [6] Y. Asahara, h. Sakai, H. Ohmi, S. Nakayama, Applied Optics 25 (1984) 3384. [7] L. Mi, S.L. Yao, Q. Li, F. Gao, Journal of Physics Conference Series 48 (2006) 785. [8] H.S. Carslaw, J.C. Jaeger, Oxford University Press, 1980, p. 439. [9] R.J. Lytle, D.L. Lager, Solutions of the scalar Helmholtz equation in the elliptic cylinder coordinate system, Mathematical Notes, Note 31, Lawrence Livermore Laboratory, 1973. [10] A.L. Van Burn, J.E. Boisvert, Quarterly of Applied Mathematics 45 (2007) 1. [11] E. Acosta, R.M. Gonzalez, C. Gomez-Reino, Optics Letters 16 (1991) 627. [12] S. Sinzinger, K.H. Brenner, J. Moisel, T. Spick, M. Testorf, Applied Optics 34 (1995) 6626. [13] Z. Zhou, F. Cui, Y. Sun, G. Chunqing, Proceeding SPIE, Vol. 423, 2000, p. 520. [14] N. Sood Biswas, A. Basuray, Physics and Chemistry of Glasses 38 (1997) 33. [15] W. Abramowitz, L. Stegun, Handbook of mathematical functions, Dover, New york, 1964. [16] J.A. Stratton, Electromagnetic theory, McGraw-Hill, New York, 1941. [17] J.C. Gutierrez-Vega, R.M. Rodriguez-Dagnino, M.A. Meneses-Nava, S. Chávez-Cerda, American Journal of Physics 71 (2003) 233. [18] N. Sood Biswas, S. Sarkar, A. Basuray, Optical Engineering 35 (1996) 470.

Fig. 4. (a) Incident elliptic beam, (b) after passing through the sample with major axis vertical and (c) after passing through the sample with minor axis vertical.