Abstract

An equation for the distribution of refractive index of a gradient refractive index square lens has been established, and such a lens has been fabricated using ion exchange. The distributions of refractive indices at different angles of incidence are discussed. Experimental and theoretical data are compared and show good agreement.

© 2009 Optical Society of America

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References

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  6. Y. L. Li, T. H. Li, G. H. Jiao, B. W. Hu, J. M. Huo, and L. L. Wang, “Research on micro-optical lenses fabrication technology,” Optik (Stuttgart) 118, 395-401 (2007).
    [CrossRef]
  7. S. Ilyas and M. Gal, “Gradient refractive index planar microlens in Si using porous silicon,” Appl. Phys. Lett. 89, 211123-211125 (2006).
    [CrossRef]
  8. H. Lv, B. Shi, L. Guo, and A. Liu, “Fabrication of Maxwell fish-eye spherical lenses and research on distribution profiles of gradient refractive index,” J. Opt. Soc. Am. A 25, 609-611 (2008).
    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]

2008 (1)

2007 (1)

Y. L. Li, T. H. Li, G. H. Jiao, B. W. Hu, J. M. Huo, and L. L. Wang, “Research on micro-optical lenses fabrication technology,” Optik (Stuttgart) 118, 395-401 (2007).
[CrossRef]

2006 (1)

S. Ilyas and M. Gal, “Gradient refractive index planar microlens in Si using porous silicon,” Appl. Phys. Lett. 89, 211123-211125 (2006).
[CrossRef]

2003 (1)

Y. L. Li, W. P. Wang, X. D. Zhang, and J. M. Huo, “Preparations on several GRIN lenses,” Acta Opt. Sin. 23, 23-24 (2003) (in Chinese).

1996 (1)

1994 (1)

1980 (2)

Akazawa, N.

Borrell, N. F.

N. F. Borrell, Micro-optics Technology (Marcel Dekker, 1999).

Fujii, K.

Gal, M.

S. Ilyas and M. Gal, “Gradient refractive index planar microlens in Si using porous silicon,” Appl. Phys. Lett. 89, 211123-211125 (2006).
[CrossRef]

Guo, L.

Houde-Walter, S. N.

Hu, B. W.

Y. L. Li, T. H. Li, G. H. Jiao, B. W. Hu, J. M. Huo, and L. L. Wang, “Research on micro-optical lenses fabrication technology,” Optik (Stuttgart) 118, 395-401 (2007).
[CrossRef]

Huo, J. M.

Y. L. Li, T. H. Li, G. H. Jiao, B. W. Hu, J. M. Huo, and L. L. Wang, “Research on micro-optical lenses fabrication technology,” Optik (Stuttgart) 118, 395-401 (2007).
[CrossRef]

Y. L. Li, W. P. Wang, X. D. Zhang, and J. M. Huo, “Preparations on several GRIN lenses,” Acta Opt. Sin. 23, 23-24 (2003) (in Chinese).

Iga, K.

Ilyas, S.

S. Ilyas and M. Gal, “Gradient refractive index planar microlens in Si using porous silicon,” Appl. Phys. Lett. 89, 211123-211125 (2006).
[CrossRef]

Jiao, G. H.

Y. L. Li, T. H. Li, G. H. Jiao, B. W. Hu, J. M. Huo, and L. L. Wang, “Research on micro-optical lenses fabrication technology,” Optik (Stuttgart) 118, 395-401 (2007).
[CrossRef]

Li, T. H.

Y. L. Li, T. H. Li, G. H. Jiao, B. W. Hu, J. M. Huo, and L. L. Wang, “Research on micro-optical lenses fabrication technology,” Optik (Stuttgart) 118, 395-401 (2007).
[CrossRef]

Li, Y. L.

Y. L. Li, T. H. Li, G. H. Jiao, B. W. Hu, J. M. Huo, and L. L. Wang, “Research on micro-optical lenses fabrication technology,” Optik (Stuttgart) 118, 395-401 (2007).
[CrossRef]

Y. L. Li, W. P. Wang, X. D. Zhang, and J. M. Huo, “Preparations on several GRIN lenses,” Acta Opt. Sin. 23, 23-24 (2003) (in Chinese).

Liu, A.

Lv, H.

McIntyre, B. L.

Messerschmidt, B.

Moore, D. T.

Ogi, S.

Shi, B.

Wang, L. L.

Y. L. Li, T. H. Li, G. H. Jiao, B. W. Hu, J. M. Huo, and L. L. Wang, “Research on micro-optical lenses fabrication technology,” Optik (Stuttgart) 118, 395-401 (2007).
[CrossRef]

Wang, W. P.

Y. L. Li, W. P. Wang, X. D. Zhang, and J. M. Huo, “Preparations on several GRIN lenses,” Acta Opt. Sin. 23, 23-24 (2003) (in Chinese).

Wood, R. W.

R. W. Wood, Physical Optics (MacMillan, 1905).

Zhang, X. D.

Y. L. Li, W. P. Wang, X. D. Zhang, and J. M. Huo, “Preparations on several GRIN lenses,” Acta Opt. Sin. 23, 23-24 (2003) (in Chinese).

Acta Opt. Sin. (1)

Y. L. Li, W. P. Wang, X. D. Zhang, and J. M. Huo, “Preparations on several GRIN lenses,” Acta Opt. Sin. 23, 23-24 (2003) (in Chinese).

Appl. Opt. (4)

Appl. Phys. Lett. (1)

S. Ilyas and M. Gal, “Gradient refractive index planar microlens in Si using porous silicon,” Appl. Phys. Lett. 89, 211123-211125 (2006).
[CrossRef]

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

Optik (Stuttgart) (1)

Y. L. Li, T. H. Li, G. H. Jiao, B. W. Hu, J. M. Huo, and L. L. Wang, “Research on micro-optical lenses fabrication technology,” Optik (Stuttgart) 118, 395-401 (2007).
[CrossRef]

Other (2)

R. W. Wood, Physical Optics (MacMillan, 1905).

N. F. Borrell, Micro-optics Technology (Marcel Dekker, 1999).

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Figures (6)

Fig. 1
Fig. 1

Design schematic diagram of gradient refractive index square lens.

Fig. 2
Fig. 2

Schematic diagram of the ion exchange equipment.

Fig. 3
Fig. 3

Interferogram of gradient refractive index square lens.

Fig. 4
Fig. 4

Experimental and fitting data of the distribution of the refractive index at different angles ( θ = 0 , π 12 , π 6 , π 4 ) .

Fig. 5
Fig. 5

Fitting curves of the refractive index at different angles ( θ = 0 , π 12 , π 6 , π 4 ) .

Fig. 6
Fig. 6

Results diagram of Table 2: (a) experimental data, (b) theoretical data, (c) error.

Tables (2)

Tables Icon

Table 1 Optimized Coefficients for the Refractive Index Profile

Tables Icon

Table 2 Theoretical, Experimental Data, and Error of Gradient Refractive Index Square Lens

Equations (27)

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C ( x , y , t ) t = D [ 2 C ( x , y , t ) x 2 + 2 C ( x , y , t ) y 2 ] ,
C ( x , y , 0 ) = C 0 C 1 ,
C ( 0 , y , t ) = C ( a , y , t ) = 0 ,
C ( x , 0 , t ) = C ( x , a , t ) = 0 ,
C ( x , y , t ) = X ( x ) Y ( y ) T ( t ) .
T ( t ) = C m n e ( m 2 + k 2 a 2 ) π 2 D t ,
X ( x ) = A sin ( m π a x ) ,
Y ( y ) = B sin ( k π a y ) ,
C ( x , y , t ) = k = 1 m = 1 C m n sin ( m π a x ) sin ( k π a y ) e ( m 2 + k 2 a 2 ) π 2 D t .
C m k = 4 a 2 0 a 0 a sin ( m π a x ) sin ( k π a y ) d x d y .
Δ C = C ( x , y , t ) C 1 = k = 1 m = 1 C m k sin ( m π a x ) sin ( k π a y ) e ( m 2 + k 2 a 2 ) π 2 D t .
n ( x , y , t ) t = D [ 2 n ( x , y , t ) x 2 + 2 n ( x , y , t ) y 2 ] .
Δ n = n ( x , y , t ) n 1 = k = 1 m = 1 C m k sin ( m π a x ) sin ( k π a y ) e ( m 2 + k 2 a 2 ) π 2 D t ,
C m k = 4 a 2 0 a 0 a sin ( m π a x ) sin ( k π a y ) d x d y ,
n ( r ) = n 0 ( 1 a r 2 ) ,
a = [ A + B 2 + A B 2 cos ( 4 θ ) ] 2 ,
T D T X X Y Y = 0 ,
X ( 0 ) Y ( y ) T ( t ) = X ( a ) Y ( y ) T ( t ) = 0 ,
X ( x ) Y ( 0 ) T ( t ) = X ( x ) Y ( a ) T ( t ) = 0 ,
X ( x ) Y ( y ) T ( 0 ) = C 0 C 1 .
X + E X = 0 ,
Y + F X = 0 .
E = ( m π ) 2 a 2 ,
F = ( k π ) 2 a 2 ,
X ( x ) = A sin ( m π a x ) ,
Y ( y ) = B sin ( k π a y ) .
T ( t ) = C m k e ( m 2 + k 2 a 2 ) π 2 D t .

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