A GRIN medium coupler and its application in light

A GRIN medium coupler and its application in light beam spot conversion Ning Wang, Fangkui Sun*, Qingfeng Li, Lixue Chen Dept. of Physics, Harbin Institute of Technology, 92 West Dazhi Street, Harbin, China 150001

ABSTRACT GRIN medium with a lateral sech refractive index variation, can make normal-incident light beam gradually curve to the medium with a larger refractive index, and periodically converge the light beam to a point smoothly and continuously. This property of GRIN medium can be used as a coupl er to realize a mode spot size conversion. This paper mainly discussed the transmission property of Gaussian light beam in a sech GRIN medium by numerical simulation. SOI waveguide is widely used in the photonic integrated circuit. To achieve a higher coupling efficiency between single mode optical fiber and single mode SOI slab waveguide, which suffer a great light beam coupling loss for the mismatch of the spot size. The GRIN medium coupling structures are designed as a coupler, with symmetric refractive index distribution and asymmetric refractive index distribution. The insertion loss calculated in theory are 0.71dB and 1.35dB respectively, which has a significant improvement in coupling loss, compared with the 30dB coupling loss caused by direct butt -joint transmission. Keywords: GRIN medium, SOI waveguide, optical coupling

1. INTRODUCTION Using Gradient refractive index (GRIN) medium as a coupler is an applicable method to accomplish the mode spot size conversion. And with the development of thin-film technologies the GRIN medium has been widely researched. In 2004, A.Delage 1-2 proposed the idea of using amorphous silicon material to create the asymmetric GRIN dielectric film on top of the SOI substrate. In 2005, Kazuo Shiraishi 3-4 designed a spot converter based on the GRIN medium, which was the first spot converter in micro-nano magnitude. In 2007, Rong Sun 5 realized a graded index in the vertical direction by controlling the relative content of nitrogen and oxygen during the deposition progress. In 2009, Atilla Ozgur Cakmak 6-7 proposed the idea of putting the GRIN PC structure onto the input side of the PCWG as a beam coupler based on the focusing effect of Two-dimensional GRIN PC. In 2010, the research team of WANG QIAN 8-9 using an optical thin-film stack to form a micro graded-refractive-index lens. In this paper we discussed the guided wave mode in a sech GRIN medium and the propagation characteristics of Gaussian beam in the GRIN medium. Then we discussed the symmetric and asymmetric coupling structure of GRIN medium. Based on those structures we analyzed the coupling efficiency at last.

2. NUMERICAL ANALYSIS OF THE GUIDED WAVE MODE IN GRIN When the light beam spreads in the hyperbolic secant GRIN medium, the trajectory of the light beam is a sine curve, and the trajectory is periodic. When a light beam incident from the free space to the sech GRIN medium, *[email protected]; phone: 86-451-86266388 Information Optics and Optical Data Storage II, edited by Feijun Song, Hui Li, Xiudong Sun, Francis T. S. Yu, Suganda Jutamulia, Kees A. Schouhamer Immink, Keiji Shono, Proc. of SPIE Vol. 8559 85590Q · © 2012 SPIE · CCC code: 0277-786/12/$18 · doi: 10.1117/12.1000063 Proc. of SPIE Vol. 8559 85590Q-1 Downloaded From: http://proceedings.spiedigitallibrary.org/ on 12/10/2012 Terms of Use: http://spiedl.org/terms

trajectory equation of the light is: y

z

dy



(2-1)

n ( y )  n( y0 ) cos 2  n   1 2

y0

From the literature 10 we can know that the distribution of cylindrical refractive index is:

n( y)  n0 sec h( y)

(2-2)

Where    2F . Substituted (2-2) into the equation (2-1) we can get the geometric solution for light propagation:

cosh( y0 )  cos 2  n sin( z ) cos  n

y When n0  3.2 ,   1/ 5 m1

(2-3)

and the GRIN medium refractive index in the y direction changed as

n( y)  n0 sec h( y) . We simulate the propagation of electric field in the hyperbolic secant GRIN medium by

numerical analysis method. From Figure2.1 we can know that the result of numerical analysis has proved the conclusion of the geometric solution, which the width of the light beam changed from convergence to separation cyclically. This sinusoidal transmission characteristics can concentrate the energy of electromagnetic field in a relatively small area, and can avoid the dispersive loss between different mold which caused by the total reflection at the interface. Y (µm) ÇD

Ói Á Ñ o N A

5.3399

12 10 O1

Op

5

8 E

z T

4

6 4

3

2 2

0

-2

1

4

0

5

10

15

x (µm)

20

25

30

0

T0

Figure 2.1. The distribution of the electric field

Figure 2.2. The distribution of the electric field

mode in the symmetrical sech GRIN medium

mode in the asymmetrical sech GRIN medium

However when the distribution of the refractive index gradually decreased from bottom to top and placing a Ó layer, such as the air medium which has the low refractive index, below the GRIN medium, we can get the asymmetric hyperbolic secant GRIN medium. And in this situation we can also get the cyclical result like the symmetric situation. The result of numerical analysis is showed in Figure 2.2. From Figure 2.2 we can see that the incident beam deviate to the underlying of the GRIN medium where the refractive is larger, and then the light beam converged on the boundary surface of the bottom of the GRIN medium. After that, the converging light beam generates the total internal reflection in the boundary surface and then continues to propagate forward.

3. THE TRANSMISSION CHARACTERISTICS OF GAUSSIAN BEAM IN GRIN MEDIUM Due to the ABCD matrix of light transmission and the q parameter of Gaussian beam when it propagate in the lens-like medium 11-12 , we can get the half-width of Gaussian beam waist at the output surface:

Proc. of SPIE Vol. 8559 85590Q-2 Downloaded From: http://proceedings.spiedigitallibrary.org/ on 12/10/2012 Terms of Use: http://spiedl.org/terms

w0

wz 

(3-1)

cos 2 ( z )  m sin 2 ( z )

where

  n w 2  m 0 0  0  

2

(3-2)

Based on equation（3-1）We can get the changes of the light beam’s half-width. With the variations of the parameter n 0 , α, w0 and the incident wavelength λ 0 , the light beam could have three possible propagation characteristics: cyclical separation, cyclical convergence and remain unchanged collimated transmission. 3.1 The cyclical separation of Gaussian beam When the relation between the parameter n 0 , α, w0 and λ0 satisfies that m （n0 w0  / 0 ) 2 larger than 1, the 2

wz will gradually converge in the range of 0  z   2 . We simulate the

half-width of Gaussian beam waist

propagation characteristics of the Gaussian beam in the GRIN medium when w0  5m ,  =0.2 m1 , 0  1.55m . The result of numerical analysis is showed in Figure 3.1. From Figure 3.1 we can see that the trajectory of the light ray is a cosine curve, and the width of the light beam is cyclically convergent. A 10238

4.6839

A 1.0416

8

8

6

6

6

4.5

4

4

4

3.5

2

4

0.8

3

T

2.5

-2

0.6

0.6

-2

-2

2

-4

0.4

-4

.4

0.4

1.5

-6

.6

-6 0.2

-8

-8

0.5 5

10

15

20

25

30

x(ym)

5

10

2 0965x104

Figure 3.1. The cyclical convergence of

Gaussian beam in GRIN medium

15

20

25

X ({un)

0.2

.8 10

5

30

15

z Ilan)

2.7392x10- 5

20

25

30

0

a

Figure 3.2. The unchanged collimated transmission

Figure 3.3. The cyclical separation of

of Gaussian beam in GRIN medium

Gaussian beam in GRIN medium

3.2 The collimated transmission of Gaussian beam When the relation between the parameter n 0 , α, w0 and λ 0 satisfies that m （n0 w0  / 0 ) 2 equals to 1, the 2


wz will remain unchanged, in other words, the Gaussian beam will keep

collimating transmission. We simulate the propagation characteristics of the Gaussian beam in the GRIN medium when w0  3m ,   0.017 m1 , 0  1.55m , n0  3.2 .The result of numerical analysis is showed in Figure 3.2. From Figure 3.2 we can see that the width of the light beam and the intensity distribution remains unchanged when the light propagating in the GRIN medium. It generates the self-collimation effect. 3.3 The cyclical convergence of Gaussian beam When the relation between the parameter n 0 , α, w0 and λ 0 satisfies that m （n0 w0  / 0 ) 2 smaller than 1，the 2


wz will cyclically converging, and the cycle is 2  . We simulate the

propagation characteristics of the Gaussian beam in the GRIN medium when w0  0.25m ,  =0.2 m1 ,

0  1.55m . The result of numerical analysis is showed in Figure 3.3. From Figure 3.3 we can see that the


trajectory of the light ray is a sine curve, and the width of the light beam increases first and then converges.

4. THE APPLICATION OF GRIN MEDIUM IN LIGHT BEAM SPOT CONVERSION 4.1 Symmetric coupled structure of GRIN medium Figure 4.1 is the structure diagram of symmetric GRIN medium coupled structure. The structure realized the compression of beam and the converged beam can form the stable guided mode in SOI waveguide after the light beam emitted from SMF normally incident onto the surface of GRIN medium. Figure 4.2 is the result of numerical analysis, which showed the distribution of electromagnetic field’s Poynting Vector in x direction all over the structure, when the distribution of refractive index is n( y)  3.45sec h( y 8) and the l ength of the GRIN medium is 12.6μm and the thickness is 12μm. Surface: Power flow, time average, x component (W /m2) 0.1595

7 6 5 4 3 2 SMF

0.14 0.12 0.1

1

0.08 0.06 0.04

0 -1

-2 -3 -4 -5

0.02 0

o

10

5

15

20

-4.0981x10 3

Figure 4.1. The structure diagram of symmetric

Figure 4.2. The distribution of electromagnetic

GRIN medium coupled structure

field’s Poynting Vector in symmetric structure

From Figure 4.2 we can see that the incident beam deviated to the center plane of the GRIN medium where the refractive is the largest, focused on the exit end face of the GRIN medium, which realized the spot conversion. Then the converged beam can form the stable propagation mode in SOI waveguide. And there will aggregate a strong electromagnetic field energy in the waveguide core layer, that is to say the coupling efficiency has been improved. 4.2 Analysis of the coupling efficiency of the symmetric coupled structure of GRIN medium After analyzed the impacts such as the angle between the incident beam and the axis of the GRIN medium , which recorded as  , the focusing parameter of GRIN medium, which recorded as  and the displacement from the axis of the waveguide core to the axis of the GRIN medium, which recorded as y , we can got the result which was showed in Figure 4.3. In Figure 4.3(a), where y  0 ,

  0 ,

the coupling efficiency changed

 . And the coupling efficiency achieved its maximum value when   0.25 m1 . From Figure 4.3(b) and Figure 4.3(c) we can see that the coupling efficiency reduced when y and  increased. However the small with

changes of these two factors will cause a great change of the coupling efficiency, so this is the disadvantage of such coupling structure. Through the analysis above, we know that the optimal coupling condition is

  0.25 m1 , y  0 ,   0 . At this time the maximum coupling efficiency is 84.9% which was a great improve compared with the direct transmission.


b) The relation between rl and Ay

a) The relation between ri and n

0.9

0.9 0.8

0.8

0.6 0.5

0_8

0.7 0.6

0_7

2

3

4

5

6

7

8

9

10

0.5 0.4 0.3

-0_16

e) The relation between n and O

0.7

m. 0.6 0.5 0.4

-0.08

1/«.()+m 1)

0 A YG+m)

0.08

0.16

0.35

-3

-1

1

3

5

O(leg))

Figure 4.3. The coupling efficiency of the symmetric coupled structure of GRIN medium 4.3 Asymmetric coupled structure of GRIN medium Considering the disadvantage of symmetric structure we improved the above-mentioned symmetric coupled structure. We placed the SOI waveguide in the bottom of the GRIN medium, and the incident beam will deviate to the underlying of the GRIN medium where the refractive is larger. Then the converging light beam will generates the total internal reflection at the converging position, and form the stable guided mode in SOI. If we select the parameters like the focus of GRIN media, the length of GRIN medium reasonably, we can avoid the loss which was introduced by the deviation of axis displacement when coupling. i4874

u

to 9 e 7

6 5 43

2 1

o -1

.2 .2 .4

S

6o

10

IS

20

4 iw+rt

Figure 4.4. The structure diagram of

Figure 4.5. The distribution of electromagnetic field’s

asymmetric GRIN medium coupled structure

Poynting Vector in asymmetric structure

Figure 4.4 is the structure diagram of asymmetric GRIN medium coupled structure. The length of GRIN medium was a quarter of the length of the trajectory cycle of the beam which propagated in the GRIN medium, which was  / 2 . Figure 4.5 is the result of numerical analysis of the distribution of electromagnetic field when the distribution of refractive index is n( y )  3.45sec h( y / 9) . From Figure 4.5 we can conclude that this structure can achieve a good conversion of light beam spot, and improved the light beam coupling efficiency effectively. 4.4 Analysis of the coupling efficiency of the asymmetric coupled structure of GRIN medium After analyzed the impacts of the coupling efficiency such as the angle between the incident beam and the axis of the GRIN medium, which recorded as  , the focusing parameter of GRIN medium, which recorded as

 , we

can got the result which was showed in Figure 4.6. In this situation the half-width of Gaussian beam waist is w0  5 m . In Figure 4.6(a), where y  5 m , the coupling efficiency achieved its maximum value when  =1 9 m . In Figure 4.6(b), where  =1 9 m , the coupling efficiency changed with the parameter  . When the half-width of Gaussian beam waist w0  5m , the maximum coupling efficiency is 73.2%, which is smaller than the maximum coupling efficiency of the symmetric coupled structure. However, the asymmetric coupled structure is more easily to realize integrated.


a) The relation between n and a 0_8

0.7 0.6 0_5

0.4 0.3

34567

8 9 10 11 12

b) The relation between n and 8

0.8 0.75 0.7 0.65 0.6 0.55 0.5 -5 -4 -3 -2 -1

11atjarn)

0 1 84(109)

2345

Figure 4.6 The coupling efficiency of the asymmetric coupled structure of GRIN medium

5. CONCLUSION This paper discussed and numerical analyzed the propagation characteristics of Gaussian beam in the sech GRIN medium. Based on that, we discussed a coupling structure which realized the beam coupling between the core of single-mode SOI waveguide and the single-mode fiber. We placing the symmetric and asymmetric coupling structure of GRIN medium between single-mode fiber and SOI waveguide, and discussed the relation between  and the coupling efficiency. For the light beam whose wavelength is 1550nm. The maximum coupling efficiency of two structures is 84.9% and 73.2% respectively. So it is a great improvement compared with the direct transmission.

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