Abstract

A mathematical model for research on the refractive index profile (RIP) of multistep ion exchange processes (IEPs) of gradient refractive index rod lenses (GRINs) is established by the different initial condition and boundary condition, based on the Fickian diffusion equation. GRIN rod lenses have been fabricated using the three-step IEPs. Research results indicate that the experimental deviations of refractive index (DRI) are in good agreement with the theoretical data. The DRI of three-step IEPs is superior to the one- and two-step IEPs and smaller than 105.

© 2011 Optical Society of America

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References

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2010 (1)

2009 (2)

2008 (2)

H. Lv, B. R. Shi, J. J. Wu, L. J. Guo, and A. M. Liu, Opt. Lasers Eng. 46, 252 (2008).
[CrossRef]

H. Lv, B. R. Shi, L. J. Guo, and A. M. Liu, J. Opt. Soc. Am. A 25, 609 (2008).
[CrossRef]

1996 (1)

1990 (2)

Bass, M.

M. Bass, Handbook of Optics (McGraw-Hill Professional, 2000).

Buczynski, R.

Doyle, O.

Fedorov, V. A.

Y. N. Korkishko and V. A. Fedorov, Ion Exchange in Single Crystals for Integrated Optics & Optoelectronics(Cambridge International Science Publishing, 1997).

Galstian, T.

Guo, L. J.

H. Lv, B. R. Shi, J. J. Wu, L. J. Guo, and A. M. Liu, Opt. Lasers Eng. 46, 252 (2008).
[CrossRef]

H. Lv, B. R. Shi, L. J. Guo, and A. M. Liu, J. Opt. Soc. Am. A 25, 609 (2008).
[CrossRef]

Houde-Walter, S. N.

Hudelist, F.

Jun, G. Y.

L. D. Seng and G. Y. Jun, Physics Foundation of Gradient Refractive Index (National Defense Industry Press, 1991).

Kindred, D. S.

Korkishko, Y. N.

Y. N. Korkishko and V. A. Fedorov, Ion Exchange in Single Crystals for Integrated Optics & Optoelectronics(Cambridge International Science Publishing, 1997).

Liu, A. M.

H. Lv, B. R. Shi, L. J. Guo, and A. M. Liu, J. Opt. Soc. Am. A 25, 609 (2008).
[CrossRef]

H. Lv, B. R. Shi, J. J. Wu, L. J. Guo, and A. M. Liu, Opt. Lasers Eng. 46, 252 (2008).
[CrossRef]

Lv, H.

H. Lv, B. R. Shi, J. J. Wu, L. J. Guo, and A. M. Liu, Opt. Lasers Eng. 46, 252 (2008).
[CrossRef]

H. Lv, B. R. Shi, L. J. Guo, and A. M. Liu, J. Opt. Soc. Am. A 25, 609 (2008).
[CrossRef]

McIntyre, B. L.

Messerschmidt, B.

Moore, D. T.

Nowosielski, J. M.

Samuels, J. E.

Seng, L. D.

L. D. Seng and G. Y. Jun, Physics Foundation of Gradient Refractive Index (National Defense Industry Press, 1991).

Shi, B. R.

H. Lv, B. R. Shi, L. J. Guo, and A. M. Liu, J. Opt. Soc. Am. A 25, 609 (2008).
[CrossRef]

H. Lv, B. R. Shi, J. J. Wu, L. J. Guo, and A. M. Liu, Opt. Lasers Eng. 46, 252 (2008).
[CrossRef]

Taghizadeh, M. R.

Takanori, O.

O. Takanori, Foundation of Optical Fiber (Post and Telecom Press, 1980).

Waddie, A. J.

Wu, J. J.

H. Lv, B. R. Shi, J. J. Wu, L. J. Guo, and A. M. Liu, Opt. Lasers Eng. 46, 252 (2008).
[CrossRef]

Appl. Opt. (3)

J. Opt. Soc. Am. A (1)

Opt. Express (2)

Opt. Lasers Eng. (1)

H. Lv, B. R. Shi, J. J. Wu, L. J. Guo, and A. M. Liu, Opt. Lasers Eng. 46, 252 (2008).
[CrossRef]

Opt. Lett. (1)

Other (4)

O. Takanori, Foundation of Optical Fiber (Post and Telecom Press, 1980).

L. D. Seng and G. Y. Jun, Physics Foundation of Gradient Refractive Index (National Defense Industry Press, 1991).

M. Bass, Handbook of Optics (McGraw-Hill Professional, 2000).

Y. N. Korkishko and V. A. Fedorov, Ion Exchange in Single Crystals for Integrated Optics & Optoelectronics(Cambridge International Science Publishing, 1997).

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Figures (2)

Fig. 1
Fig. 1

Laser interferograms in the (a) one-, (b) two-, and (c) three-step IEPs.

Fig. 2
Fig. 2

Effect of the one-, two-, and three-step IEPs on the DRI.

Equations (14)

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C t = 1 r r [ r D C r ] ,
{ C = C 0 ( 0 r < r 0 , t 1 = 0 ) C = k 1 C 0 ( r = r 0 , t 1 > 0 ) .
C 1 ( r , k 1 , t 1 ) = k 1 C 0 + C 0 m = 1 2 ( 1 k 1 ) exp ( β m 2 r 0 2 0 t 1 D ( t 1 ) d t ) β m J 1 ( β m ) J 0 ( β m r r 0 ) .
{ C = C 1 ( r , k 1 , t 1 ) ( 0 r < r 0 , t 2 = 0 ) C = k 2 C 0 ( r = r 0 , t 2 > 0 ) .
C 2 ( r , k 1 , t 1 , k 2 , t 2 ) = k 2 C 0 + C 0 m = 1 2 [ ( k 1 k 2 ) exp ( β m 2 r 0 2 0 t 2 D ( t 2 ) d t ) β m J 1 ( β m ) + ( 1 k 1 ) exp ( β m 2 r 0 2 0 t 1 D ( t 1 ) d t ) exp ( β m 2 r 0 2 0 t 2 D ( t 2 ) d t ) β m J 1 ( β m ) ] J 0 ( β m r r 0 ) .
{ C = C 2 ( r , k 1 , t 1 , k 2 , t 2 ) ( 0 r < r 0 , t 3 = 0 ) C = k 3 C 0 ( r = r 0 , t 2 > 0 ) .
C 3 ( r , k 1 , t 1 , k 2 , t 2 , k 3 , t 3 ) = k 3 C 0 + C 0 m = 1 2 [ ( k 2 k 3 ) exp ( β m 2 r 0 2 0 t 3 D ( t 3 ) d t ) + ( k 1 k 2 ) exp ( β m 2 r 0 2 0 t 2 D ( t 2 ) d t ) exp ( β m 2 r 0 2 0 t 3 D ( t 3 ) d t ) β m J 1 ( β m ) + ( 1 k 1 ) exp ( β m 2 r 0 2 0 t 1 D ( t 1 ) d t ) exp ( β m 2 r 0 2 0 t 2 D ( t 2 ) d t ) exp ( β m 2 r 0 2 0 t 3 D ( t 3 ) d t ) β m J 1 ( β m ) ] J 0 ( β m r r 0 ) .
n ( r ) = n m + ( n 0 n m ) C ( r ) C 0 ,
T j = 1 r 0 2 0 t j D ( t j ) d t j ,
n ( r ) = n m + ( n 0 n m ) [ k 3 + m = 1 2 [ ( k 2 k 3 ) exp ( β m 2 T 3 ) + ( k 1 k 2 ) exp ( β m 2 T 2 ) exp ( β m 2 T 3 ) β m J 1 ( β m ) + ( 1 k 1 ) exp ( β m 2 T 1 ) exp ( β m 2 T 2 ) exp ( β m 2 T 3 ) β m J 1 ( β m ) ] J 0 ( β m r r 0 ) ] .
n m = n 0 ( k m k 0 ) λ / t 0 ,
Z ( r ) = r 0 r r d r { ( N ( r ) n 0 ) 2 + [ 1 ( r 0 r ) 2 ] ( x 0 M 0 y 0 L 0 ) 2 γ 2 } 1 / 2 ,
{ N A ( r ) = n 0 sech ( α r ) N B ( r ) = n 0 [ 1 + ( α r ) 2 ] 1 / 2 ,
N C ( r ) = n 0 [ 1 ( α r ) 2 ] 1 / 2 .

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