Abstract

By introducing a new kind of Green function, we formulate an improved diffraction integral, which can be used to numerically evaluate the diffracted field of a microlens of plane-convex shape. Analytical expressions for the diffracted field of microlens are derived for the case where the curvature radius of the convex surface is larger than the dimension of the microlens aperture. The validity of the results and the diffracted field of the microlens are illustrated with numerical examples. The focal shifts of the diffracted field are found to depend mainly on the Fresnel number N of the microlens.

© 2005 Optical Society of America

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References

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  1. P. Vukusic, J. R. Sambles, “Photonic structures in biology,” Nature (London) 424, 852–855 (2003).
    [CrossRef]
  2. J. Duparre, P. Dannberg, P. Schreiber, A. Brauer, A. Tunnermann, “Artificial apposition compound eye fabricated by micro-optics technology,” Appl. Opt. 43, 4303–4309 (2004).
    [CrossRef] [PubMed]
  3. J. S. Sanders, C. E. Halford, “Design and analysis of apposition compound eye optical sensors,” Opt. Eng. (Bellingham) 34, 222–235 (1995).
    [CrossRef]
  4. K. Hirayama, E. N. Glytsis, T. K. Gaylord, “Rigorous electromagnetic analysis of diffraction by finite-number-of-periods gratings,” J. Opt. Soc. Am. A 14, 907–917 (1997).
    [CrossRef]
  5. J. M. Bendickon, E. N. Glytsis, T. K. Gaylord, “Scalar integral diffraction methods: unification, accuracy, and comparison with a rigorous boundary element method with application to diffractive cylindrical lenses,” J. Opt. Soc. Am. A 15, 1822–1837 (1998).
    [CrossRef]
  6. R. Dobbs, Electromagnetic Waves (Routledge & Kegan Paul, 1985), Chap. 5.
  7. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1999), Chap. 3.
  8. M. Born, E. Wolf, Principles of Optics (Cambridge U. Press, 1999), Chap. 8.
    [CrossRef]
  9. K. Duan, B. Lu, “Partially coherent nonparaxial beams,” Opt. Lett. 29, 800–802 (2004).
    [CrossRef] [PubMed]
  10. Y. Li, E. Wolf, “Focal shifts in diffracted converging spherical waves,” Opt. Commun. 39, 211–215 (1981).
    [CrossRef]

2004 (2)

2003 (1)

P. Vukusic, J. R. Sambles, “Photonic structures in biology,” Nature (London) 424, 852–855 (2003).
[CrossRef]

1998 (1)

1997 (1)

1995 (1)

J. S. Sanders, C. E. Halford, “Design and analysis of apposition compound eye optical sensors,” Opt. Eng. (Bellingham) 34, 222–235 (1995).
[CrossRef]

1981 (1)

Y. Li, E. Wolf, “Focal shifts in diffracted converging spherical waves,” Opt. Commun. 39, 211–215 (1981).
[CrossRef]

Bendickon, J. M.

Born, M.

M. Born, E. Wolf, Principles of Optics (Cambridge U. Press, 1999), Chap. 8.
[CrossRef]

Brauer, A.

Dannberg, P.

Dobbs, R.

R. Dobbs, Electromagnetic Waves (Routledge & Kegan Paul, 1985), Chap. 5.

Duan, K.

Duparre, J.

Gaylord, T. K.

Glytsis, E. N.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1999), Chap. 3.

Halford, C. E.

J. S. Sanders, C. E. Halford, “Design and analysis of apposition compound eye optical sensors,” Opt. Eng. (Bellingham) 34, 222–235 (1995).
[CrossRef]

Hirayama, K.

Li, Y.

Y. Li, E. Wolf, “Focal shifts in diffracted converging spherical waves,” Opt. Commun. 39, 211–215 (1981).
[CrossRef]

Lu, B.

Sambles, J. R.

P. Vukusic, J. R. Sambles, “Photonic structures in biology,” Nature (London) 424, 852–855 (2003).
[CrossRef]

Sanders, J. S.

J. S. Sanders, C. E. Halford, “Design and analysis of apposition compound eye optical sensors,” Opt. Eng. (Bellingham) 34, 222–235 (1995).
[CrossRef]

Schreiber, P.

Tunnermann, A.

Vukusic, P.

P. Vukusic, J. R. Sambles, “Photonic structures in biology,” Nature (London) 424, 852–855 (2003).
[CrossRef]

Wolf, E.

Y. Li, E. Wolf, “Focal shifts in diffracted converging spherical waves,” Opt. Commun. 39, 211–215 (1981).
[CrossRef]

M. Born, E. Wolf, Principles of Optics (Cambridge U. Press, 1999), Chap. 8.
[CrossRef]

Appl. Opt. (1)

J. Opt. Soc. Am. A (2)

Nature (London) (1)

P. Vukusic, J. R. Sambles, “Photonic structures in biology,” Nature (London) 424, 852–855 (2003).
[CrossRef]

Opt. Commun. (1)

Y. Li, E. Wolf, “Focal shifts in diffracted converging spherical waves,” Opt. Commun. 39, 211–215 (1981).
[CrossRef]

Opt. Eng. (Bellingham) (1)

J. S. Sanders, C. E. Halford, “Design and analysis of apposition compound eye optical sensors,” Opt. Eng. (Bellingham) 34, 222–235 (1995).
[CrossRef]

Opt. Lett. (1)

Other (3)

R. Dobbs, Electromagnetic Waves (Routledge & Kegan Paul, 1985), Chap. 5.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1999), Chap. 3.

M. Born, E. Wolf, Principles of Optics (Cambridge U. Press, 1999), Chap. 8.
[CrossRef]

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Figures (5)

Fig. 1
Fig. 1

Schematic of a plane-convex shape microlens.

Fig. 2
Fig. 2

Illustration of some spatial variables.

Fig. 3
Fig. 3

Normalized transverse intensity distributions at the plane z = 50 μ m . In (a), (b), and (c), solid, circled, and dotted curves represent the results of E x ( x , 0 , z ) 2 obtained by using Eq. (7), of E ( x , 0 , z ) 2 by using Eq. (15), and of E ( x , 0 , z ) 2 by using Eq. (21), respectively. The result of E z ( x , 0 , z ) 2 obtained by using Eq. (7) is shown in (d). The calculation parameters are seen in the text.

Fig. 4
Fig. 4

Axial intensity distribution E ( 0 , 0 , z ) 2 obtained by using Eq. (15).

Fig. 5
Fig. 5

Relative focal shift δ = ( z max f ) f versus the Fresnel number N. The solid curve is for R = 10 D and the circled curve is for R = 50 D .

Equations (28)

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u ( x 0 , y 0 , z 0 ) = 2 n 1 + 1 exp ( i k n 1 z 0 ) i ,
u s = 2 n 1 + 1 sin θ exp ( i k n 1 z 0 ) ,
u p = 2 n 1 + 1 cos θ exp ( i k n 1 z 0 ) .
E x ( x 0 , y 0 , z 0 ) = 2 n 1 + 1 [ t s ( sin θ ) 2 + t p ( cos θ ) 2 cos α 3 ] exp ( i k n 1 z 0 ) ,
E y ( x 0 , y 0 , z 0 ) = 2 n 1 + 1 sin θ cos θ ( t s t p cos α 3 ) exp ( i k n 1 z 0 ) ,
E z ( x 0 , y 0 , z 0 ) = 2 n 1 + 1 t p cos θ sin α 3 exp ( i k n 1 z 0 ) ,
t p = 2 n 1 cos α 1 cos α 1 + n 1 cos α 2 = 2 n 1 [ R 2 ( x 0 2 + y 0 2 ) ] 1 2 [ R 2 ( x 0 2 + y 0 2 ) ] 1 2 + n 1 [ R 2 n 1 2 ( x 0 2 + y 0 2 ) ] 1 2 ,
t p = 2 n 1 cos α 1 n 1 cos α 1 + cos α 2 = 2 n 1 [ R 2 ( x 0 2 + y 0 2 ) ] 1 2 n 1 [ R 2 ( x 0 2 + y 0 2 ) ] 1 2 + [ R 2 n 1 2 ( x 0 2 + y 0 2 ) ] 1 2 .
E ( x , y , z ) = 1 4 π Σ [ E ( x 0 , y 0 , z 0 ) n G E ( x 0 , y 0 , z 0 ) G n ] d s ,
G = 1 2 { [ exp ( i k r 01 ) r 01 + exp ( i k r 02 ) r 02 ] [ exp ( i k r 01 ) r 01 + exp ( i k r 02 ) r 02 ] } ,
E ( x , y , z ) = 1 4 π Σ [ E ( x 0 , y 0 , z 0 ) G n ] d s .
G n = i k 2 { exp ( i k r 01 ) r 01 [ cos ( n , r 01 ) cos ( n , r 01 ) ] + exp ( i k r 02 ) r 02 [ cos ( n , r 02 ) cos ( n , r 02 ) ] } ,
G n = i k R r 01 2 exp ( i k r 01 ) [ R 2 x 0 2 y 0 2 R 2 x 0 2 y 0 2 ( z + R 2 D 2 4 ) ] + i k R r 02 2 exp ( i k r 02 ) [ R 2 x 0 2 y 0 2 + R 2 x 0 2 y 0 2 ( z R 2 D 2 4 ) ] ,
r 0 j = { ( x 0 x ) 2 + ( y 0 y ) 2 + [ z 0 ( 1 ) j 1 z ] 2 } 1 2 .
r 0 j r + ( 1 ) j z D 2 8 R r + 1 2 r ( 1 + ( 1 ) j 1 z R ) ( x 0 2 + y 0 2 ) x x 0 + y y 0 r .
G n i k r 2 exp ( i k a 1 ) ( b 1 + c 1 2 R 2 ρ 2 ) exp [ i k 2 r ( 1 + 2 R ) ρ 2 ] exp ( i k x x 0 + y y 0 r ) + i k r 2 exp ( i k a 2 ) ( b 2 c 2 2 R 2 ρ 2 ) exp [ i k 2 r ( 1 z R ) ρ 2 ] exp ( i k x x 0 + y y 0 r ) ,
ρ = ( x 0 2 + y 0 2 ) 1 2 , a j = r + ( 1 ) j z D 2 ( 8 R r ) ,
b j = z + ( 1 ) j D 2 ( 8 R ) , c j = R + D 2 ( 8 R ) + ( 1 ) j z .
t p 2 n 1 n 1 + 1 + n 1 2 n 1 1 n 1 + 1 x 0 2 + y 0 2 R 2 ,
t s 2 n 1 n 1 + 1 + n 1 n 1 1 n 1 + 1 x 0 2 + y 0 2 R 2 .
E ( x 0 , y 0 , z 0 ) 4 n 1 ( n 1 + 1 ) 2 exp ( i k n 1 D 2 8 R ) exp ( i k n 1 x 0 2 + y 0 2 2 R ) .
E ( x , y , z ) = i k r 2 2 n 1 ( n 1 + 1 ) 2 exp ( i k a 1 ) m = 0 ( 1 ) m d 2 m 2 2 m ( m ! ) 2 [ b 1 Q m 1 + 1 2 R 2 ( b 1 + c 1 ) Q m + 1 1 + c 1 4 R 4 Q m + 2 1 ] i k r 2 2 n 1 ( n 1 + 1 ) 2 exp ( i k a 2 ) m = 0 ( 1 ) m d 2 m 2 2 m ( m ! ) 2 [ b 2 Q m 2 + 1 2 R 2 ( b 2 c 2 ) Q m + 1 2 c 2 4 R 4 Q m + 2 2 ] ,
Q m j = D 2 ( m + 1 ) 2 2 m + 5 { 4 m + 1 F 2 1 [ { m + 1 2 } , { 1 2 , m + 3 2 } , 1 64 g j 2 D 4 ] + i g j D 2 m + 2 F 2 1 [ { m + 2 2 } , { 3 2 , m + 4 2 } , 1 64 g j 2 D 4 ] } ,
g j = k 2 ( 1 r 1 f ( 1 ) j z R r ) ,
0 2 π exp [ i ( a cos θ + b sin θ ) ] d θ = 2 π J 0 ( a 2 + b 2 )
G n 2 i k z r 2 exp ( i k r 0 ) ,
r 0 r + x 0 2 + y 0 2 2 x x 0 2 y y 0 2 r .
E ( x , y , z ) = 4 n 1 ( n 1 + 1 ) 2 i k z r 2 m = 0 ( 1 ) m d 2 m 2 2 m ( m ! ) 2 Q m 0 ,

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