The refractive index distribution of even polygonal selfoc lens

Optics & Laser Technology 43 (2011) 674–678

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Optics & Laser Technology journal homepage: www.elsevier.com/locate/optlastec

The refractive index distribution of even polygonal selfoc lens Zigang Zhou n, Li Zhu, Yongjia Yang, Guangchun Sun, Qiang Wang College of Science, Southwest University of Science and Technology, Mianyang 621010, China

a r t i c l e in f o

a b s t r a c t

Article history: Received 4 April 2010 Received in revised form 9 May 2010 Accepted 7 September 2010 Available online 8 October 2010

MATLAB diffusion partial differential toolboxes are used to solve the ion diffusion equation under even polygonal boundary conditions, such as square and hexagon, and obtain the dynamic process ion concentration during ion exchange. The exact solutions of the refractive index distribution are proved by the correctness. The ion concentration equation in even polygonal glass rod is calculated by the variable separation and coordinate transformation method. The computation results are in good agreement with the measured ones in the compound eye system. Crown Copyright & 2010 Published by Elsevier Ltd. All rights reserved.

Keywords: Selfoc lens Refractive index distribution Even polygon

1. Introduction

selfoc lens, we can express ion concentration as C¼ C(x,y,t), and get the corresponding diffusion equation [8]:

With the development of micro-optical systems, the selfoc lens has been widely used in many areas [1]. The lens has good condensing, collimating, and imaging features that can greatly improve the performance of optical systems [2]. The selfoc lens arrays (SLA) are widely used in imaging and image processing equipment [3]. However, the current production and applications mostly use cylindrical or hemispherical lens, and have large pores and leakage information [4]. In order to solve this problem, this paper presents an even polygonal selfoc lens array, such as square and hexagon ones. These arrays can increase the translucent area, reduce the leakage of information between the pores, and thus greatly enhance the transmission of optical information. However, the key is to obtain the refractive index distribution of the lens. At present, generally with numerical solution, the results have not been easily substituted into the formula and do not have good explicability. Therefore, we propose the solution method for variable separation and coordinate transformation to obtain the exact solutions, which will provide a favorable theoretical basis for the imaging system in the compound eye.

! 8 > @Cðx,y,tÞ @2 Cðx,y,tÞ @2 Cðx,y,tÞ > > ¼ D þ > < @t @x2 @y2

2. Regular hexagon ion diffusion equation The ion exchange technology is an important method to manufacture selfoc lens [4, 5]. The refractive index distributions are based on the linear relationship between refractive index and ion concentration [6, 7]. On the side surface of the hexagonal n

Corresponding author. Tel.: + 86 816 6089663; fax: + 86 816 6089662. E-mail address: [email protected] (Z. Zhou).

> CðB,tÞ ¼ C1 x,y Dfthe area surrounded by Bg > > > : Cðx,y,0Þ ¼ C0 t ¼ 0

ð1Þ

where B is the hexagonal boundary, D the diffusion coefficient, and C0 and C1 are, respectively, the initial and boundary values in Fig. 1. By the variable separation method, Eq. (1) was separated to 8 Tu X 00 Y 00 > > > < DT  X  Y ¼ 0 uðB,tÞ ¼ 0 > > > : XðxÞYðyÞTð0Þ ¼ C C 0

x,y Dfthe area surrounded by Bg 1

ð2Þ

t¼0

From Eq. (2) it can be seen that in the boundary conditions, only the upper and lower boundaries can be easily brought in, while the remaining four borders are not easily treated. After analyzing the hexagonal structure and mechanism of proliferation, we found three pairs of parallel lines angled composition. They are combined into the hexagon boundary conditions of the analytical solution in Fig. 2. It will be a good way to solve the boundary conditions. After the separation was coupled with the boundary conditions, we obtained the following: (

Y 00 þuY ¼ 0

pffiffiffi Yð0Þ ¼ 0, Yð 3aÞ ¼ 0

0030-3992/$ - see front matter Crown Copyright & 2010 Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.optlastec.2010.09.008

ð3Þ

Z. Zhou et al. / Optics & Laser Technology 43 (2011) 674–678

675

upper right translated, respectively. Eq. (4) gives ( " #) 8 1  p  p pffiffiffi X > 3a pp > 00 > p ffiffiffi Y x sin  þ y cos  þ ¼ C sin > < 4 3 3 3a p¼0

> > p2 p2 > > :n¼ 3a2

ð8Þ

The constant coefficient C is C¼

2ðcosðppÞ1Þ pp

ð9Þ

The m, n, and p above are integers. Tu þlþuþn ¼ 0   DT 1 TðtÞ ¼ K exp  2 ðm2 þn2 þ p2 Þp2 Dt 3a

ð10Þ ð11Þ

where the coefficient K is the initial condition in Eq. (2). K¼ T (0) ¼C0  C1. By the principle of superposition, and together with Eq. (2), we get   1 1 X 1 X X mp R sin pffiffiffi y C ¼ C0 þ 3a m¼0 n¼0 p¼0 ( " #) p p pffiffiffi np 3a þ y cos þ sin pffiffiffi x sin 4 3 3 3a ( " pffiffiffi #     3a pp p p þy cos  þ sin pffiffiffi x sin  4 3 3 3a

Fig. 1. Hexagonal boundary.

2

eðm

þ n2 þ p2 =3a2 Þp2 Dt

ð12Þ

The constant coefficient R is 2ðcos ðmpÞ1Þ mp 2ðcos ðnpÞ1Þ 2ðcos ðppÞ1Þ   np pp

R ¼ ðC 0 C1 Þ

ð13Þ

Fig. 2. Combination and arrangement of hexagons.

with the solution 8   1 X > mp > > p ffiffiffi y sin Y ¼ A > < 3a m¼0

> > m2 p2 > > :u¼ 3a2 where the coefficients A is   Z pffiffi3a 2 mp 2ðcosðmpÞ1Þ A ¼ pffiffiffi sin pffiffiffi y dy ¼ mp 3a 0 3a

ð4Þ

So the refractive index is [6, 7]   1 1 X 1 X X mp Rusin pffiffiffi y nðx,y,tÞ ¼ n0 þ 3a m¼0 n¼0 p¼0 ( " #) p p pffiffiffi np 3a þ y cos þ sin pffiffiffi x sin 4 3 3 3a ( " #  p  p pffiffiffi pp 3a sin pffiffiffi x sin  þy cos  þ 4 3 3 3a eðm

ð5Þ

2

þ n2 þ p2 =3a2 Þp2 Dt

The constant coefficient R0 is Ru ¼ ðn0 n1 ÞR

Let the origin of the coordinate system be the center of the rotation; we rotate the function Y 601counter-clockwise or clockwise at first. Then, we shift it by a distance of 3a/4 in the direction of the y-axis and obtain the analytic solution of diffusion in the other directions as 8 ( " #) 1 p p pffiffiffi X > np 3a > > Y0 ¼ B p ffiffiffi x sin þ ycos þ sin > < 4 3 3 3a n¼0 ð6Þ > 2 2 > n p > > :l¼ 3a2 where B is B¼

2ðcosðnpÞ1Þ np

ð15Þ

Even in this case, a similar analytic solution of diffusion equation is n ¼ n0 þ

1 X

1 X

m1 ¼ 1 m2 ¼ 1

...

1 X

RN

mN=2 ¼ 1

( N=2 "   pffiffiffi #) Y mk p 2kp 2kp 3 a þ y cos þ  sin pffiffiffi x sin 2 N N 3 a k¼1 ! N=2 1 X 2 2 exp  2 m p Dt 3a k ¼ 1 k

ð7Þ RN ¼ ðn0 n1 Þ

Applying a clockwise rotation by 601 around the origin, the negative and positive directions along the X-axis are lower and

ð14Þ

N=2 Y 2½cosðmk pÞ1 mk p k¼1

where N is a positive even number.

ð16Þ

ð17Þ

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Z. Zhou et al. / Optics & Laser Technology 43 (2011) 674–678

So the analytical solution of refractive index distribution in a square selfoc lens is nðx,y,tÞ ¼ n0 þ

1 X

1 X

m1 ¼ 1 m2 ¼ 1

sm1 m2 sin

m p  1 x a

  m p  m2 þ m2 2 x exp  1 2 2 p2 Dt sin a a where the coefficient sm1 m2 is as follows[9,10]: Z aZ a m p  m p  4 1 2 x sin y dx dy sm1 m2 ¼ ðn0 n1 Þ sin aa a a 0 0

ð18Þ

ð19Þ

3. Mathematical simulation and validation Based on Figs. 3 and 4, we have considered the early stages of the entire diffusion process [11]. As the four corners are irregular,

we will enlarge the scope of the original analog two times, as can be observed in Fig. 5. From the quantitative and numerical validation, the initial refraction n0 is 1.75 and the boundary condition n1 is 1.5. The finite difference equation is as follows: nðxi ,yk Þ ¼ n0 þ

a2 t  nðxi þ 1 ,yk Þ þ nðxi1 ,yk Þ þ nðxi ,yk þ 1 Þ þ nðxi ,yk1 Þ4nðxi ,yk Þ

x

ð20Þ

Analytical and numerical solutions show good arguments of the calculation results very well Fig. 6. This has been quantitatively verified by the analytical solution. For the same boundary condition and initial value, the refractive index of a square lens can be calculated as above, where mp  x Rumn sin a m¼1n¼1   2   np  m n2 y exp  2 þ 2 p2 Dt sin b a b

Cðx,y,tÞ ¼ C1 þ

1 X 1 X

Fig. 3. Regular hexagon boundary conditions.

Fig. 5. Calculated regional expansion for the four times the simulation after the analytical solution.

Fig. 4. Refractive index distribution of three-dimensional simulation.

Fig. 6. Analytical solution and numerical solution for comparison.

Z. Zhou et al. / Optics & Laser Technology 43 (2011) 674–678

Rumn ¼

4 ðc0 c1 Þ ab

Z

b 0

Z

a

sin 0

mp  np  x sin y dx dy a b

The above mentioned solution has been applied to simulate square lens refractive index distribution, which is measured by the interferometer square interference pattern shown in Figs. 7 and 8.We get the solution for refractive index numerically. The refractive index distribution can be studied and modeled by a square law function expression. It can be found that the theoretical result accords with the experimental result very well. The universal solution was proved qualitatively as well as quantitatively. The refractive index measured in the experiment is given in Table 1.

Fig. 7. Square of the refractive index distribution.

Fig. 8. Refractive index distribution of the interference pattern.

677

Table 1 Experimental values compared with the theoretical calculations. Category axis (x, y)

Experiment

Calculation

Relative difference

Relative error (%)

(0.565,0.500) (0.762,0.500) (0.836,0.500) (0.755,0.568) (0.849,0.588) (0.532,0.556) (0.636,0.735) (0.546,0.546) (0.695,0.695) (0.767,0.767)

1.614 1.584 1.561 1.584 1.561 1.614 1.584 1.615 1.584 1.561

1.612 1.580 1.553 1.581 1.556 1.613 1.584 1.613 1.585 1.562

0.002 0.005 0.008 0.002 0.005 0.002 0.001 0.002 0.001 0.001

0.09 0.029 0.051 0.011 0.031 0.087 0.004 0.011 0.063 0.070

Fig. 9. Octagonal boundary conditions for two-dimensional simulation of refractive index distribution.

Fig. 10. Octagonal boundary conditions for refractive index values in the distribution curve.

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Z. Zhou et al. / Optics & Laser Technology 43 (2011) 674–678

Refractive index distribution of the analytical solution under a two-dimensional image is shown in Figs. 9 and 10, which show that the refractive index values are well controlled by the boundary value and initial value (1.5–1.75). The analytical solutions and experiments of the square lens are also consistent with the ones in Refs.[11, 12].

solution of refractive index distribution with the boundary of a regular N-gon (N is even) lens was obtained which will result in more improvement in the study of GRIN lens and similar problems. Hence it has great application and spread value, such as imaging systems and imaging properties of the compound eye.

4. Conclusion

References

Based on the method of variable separation and coordinate transformation, the ion concentration equation is solved exactly for even polygonal class rods. The exact solution of the refractive index distribution is also in good agreement with the the hexagon and square lens experiment [12]. In this paper we obtained the analytic solution of refractive index distribution of regularly hexagonal GRIN lens, and proved its correctness qualitatively and quantitatively. It will be very beneficial for further research of the regularly hexagonal GRIN lens and its array. The solving model that was introduced in this paper can be also used to solve similar problems. The analytic

[1] Horridge G A. Philosophical Transactions of the Royal Society of London B 1978;285(1). [2] Liu DS, Gao YJ. Beijing: Nation Defence Industry Press; 1991. [3] Li YL, He JM, He ZQ. Acta Photonica Sinica 2000;29:302. [4] Tian WJ, Yao SL, Chen RL, et al. Acta Photonica Sinica 2002;31:47. [5] Han YL, Liu DS, Li JY. Acta Photonica Sinica 2006;35:1301. [6] Zhou ZG, Liu DS, Liu XD. Acta Photonica Sinica 1999;28:943. [7] Stan GG, Srinivasan M, Dalczynski J. Applied Optics 1991;30:1695. [8] Liang KM, Bejing: Higher Education Press; 2004. [9] Chen K, Han YL, Zhou ZG. Acta Photonica Sinica 2008;37:1740. [10] Han YL. Chong Qing: Southwest and Normal University Press; 2005. [11] Huang ZY, Que PW. Computer Engineering and Applications 2004;36:196. [12] Zhang R, Zhou ZG, Hu C. Acta Photonica Sinica 2008;37:1346.