Abstract

Based on the previous Letter [Opt. Lett. 29, 2345 (2004) ], we significantly extend the applications of the improved first Rayleigh–Sommerfeld method (IRSM1) to analyze the focusing performance of cylindrical microlenses for different types of profile (continuous or stepwise), different f-numbers (from f1.5 to f0.75), and different polarizations (the TE or TM). A number of performance measures of the cylindrical microlenses, such as the focal spot size, the diffraction efficiency, the real focal position, and the normalized sidelobe power, are studied in detail. We compare numerical results obtained by the IRSM1, by the original first Rayleigh–Sommerfeld method (ORSM1), and by the rigorous boundary element method (BEM). For continuously refractive lenses, the results calculated by the IRSM1 are quite close to those obtained by the BEM; in contrast, the results calculated by the ORSM1 significantly deviate from those obtained from the rigorous BEM. For multilevel diffractive lenses, the IRSM1 also provides much more accurate results than the ORSM1. In addition, compared with the BEM, a notable advantage of the IRSM1 is much lower computer memory and time consumption in computations.

© 2005 Optical Society of America

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References

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  1. D. A. Pommet, M. G. Moharam, E. B. Grann, “Limits of scalar diffraction theory for diffractive phase elements,” J. Opt. Soc. Am. A 11, 1827–1834 (1994).
    [CrossRef]
  2. J. N. Mait, “Understanding diffractive optic design in the scalar domain,” J. Opt. Soc. Am. A 12, 2145–2158 (1995).
    [CrossRef]
  3. E. N. Glytsis, M. E. Harrigan, K. Hirayama, T. K. Gaylord, “Collimating cylindrical diffractive lenses: rigorous electromagnetic analysis and scalar approximation,” Appl. Opt. 37, 34–43 (1998).
    [CrossRef]
  4. E. N. Glytsis, M. E. Harrigan, T. K. Gaylord, K. Hirayama, “Effects of fabrication errors on the performance of cylindrical diffractive lenses: rigorous boundary-element method and scalar approximation,” Appl. Opt. 37, 6591–6602 (1998).
    [CrossRef]
  5. J. M. Bendickson, 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. K. Hirayama, E. N. Glytsis, T. K. Gaylord, D. W. Wilson, “Rigorous electromagnetic analysis of diffractive cylindrical lenses,” J. Opt. Soc. Am. A 13, 2219–2231 (1996).
    [CrossRef]
  7. 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]
  8. B.-Z. Dong, J.-S. Ye, J. Liu, B.-Y. Gu, G.-Z. Yang, M.-H. Lu, S.-T. Liu, “Analysis of the focal characteristics of cylindrical lenses made of anisotropically dielectric material based on rigorous electromagnetic theory,” J. Mod. Opt. 50, 1195–1208 (2003).
    [CrossRef]
  9. J. Liu, B.-Y. Gu, B.-Z. Dong, G.-Z. Yang, “Interference effect of dual diffractive cylindrical microlenses analyzed by rigorous electromagnetic theory,” J. Opt. Soc. Am. A 18, 526–536 (2001).
    [CrossRef]
  10. J.-S. Ye, B.-Z. Dong, B.-Y. Gu, G.-Z. Yang, S.-T. Liu, “Analysis of a closed-boundary axilens with long focal depth and high transverse resolution based on rigorous electromagnetic theory,” J. Opt. Soc. Am. A 19, 2030–2035 (2002).
    [CrossRef]
  11. J.-S. Ye, B.-Y. Gu, B.-Z. Dong, S.-T. Liu, “Improved first Rayleigh–Sommerfeld method for analysis of cylindrical microlenses with small f-numbers,” Opt. Lett. 29, 2345–2347 (2004).
    [CrossRef] [PubMed]
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    [CrossRef]
  14. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, 1968), Chaps. 3, 4.
  15. J. M. Bendickson, E. N. Glytsis, T. K. Gaylord, “Metallic surface-relief on-axis and off-axis focusing diffractive cylindrical mirrors,” J. Opt. Soc. Am. A 16, 113–130 (1999).
    [CrossRef]
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2004 (1)

2003 (1)

B.-Z. Dong, J.-S. Ye, J. Liu, B.-Y. Gu, G.-Z. Yang, M.-H. Lu, S.-T. Liu, “Analysis of the focal characteristics of cylindrical lenses made of anisotropically dielectric material based on rigorous electromagnetic theory,” J. Mod. Opt. 50, 1195–1208 (2003).
[CrossRef]

2002 (1)

2001 (1)

1999 (1)

1998 (3)

1997 (1)

1996 (1)

1995 (2)

1994 (1)

1989 (1)

Bendickson, J. M.

Born, M.

M. Born, E. Wolf, Principles of Optics (Cambridge U. Press, Cambridge, UK, 1999), chap. 1.
[CrossRef]

Buralli, D. A.

Dong, B.-Z.

Gaylord, T. K.

Glytsis, E. N.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, 1968), Chaps. 3, 4.

Grann, E. B.

Gu, B.-Y.

Harrigan, M. E.

Herzig, H. P.

Hirayama, K.

Koshiba, M.

M. Koshiba, Optical Waveguide Theory by the Finite Element Method (KTK Scientific, Tokyo, 1992), pp. 43–47.

Kunz, R. E.

Liu, J.

B.-Z. Dong, J.-S. Ye, J. Liu, B.-Y. Gu, G.-Z. Yang, M.-H. Lu, S.-T. Liu, “Analysis of the focal characteristics of cylindrical lenses made of anisotropically dielectric material based on rigorous electromagnetic theory,” J. Mod. Opt. 50, 1195–1208 (2003).
[CrossRef]

J. Liu, B.-Y. Gu, B.-Z. Dong, G.-Z. Yang, “Interference effect of dual diffractive cylindrical microlenses analyzed by rigorous electromagnetic theory,” J. Opt. Soc. Am. A 18, 526–536 (2001).
[CrossRef]

Liu, S.-T.

Lu, M.-H.

B.-Z. Dong, J.-S. Ye, J. Liu, B.-Y. Gu, G.-Z. Yang, M.-H. Lu, S.-T. Liu, “Analysis of the focal characteristics of cylindrical lenses made of anisotropically dielectric material based on rigorous electromagnetic theory,” J. Mod. Opt. 50, 1195–1208 (2003).
[CrossRef]

Mait, J. N.

Moharam, M. G.

Morris, G. M.

Pommet, D. A.

Rogers, J. R.

Rossi, M.

Wilson, D. W.

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Cambridge U. Press, Cambridge, UK, 1999), chap. 1.
[CrossRef]

Yang, G.-Z.

Ye, J.-S.

Appl. Opt. (4)

J. Mod. Opt. (1)

B.-Z. Dong, J.-S. Ye, J. Liu, B.-Y. Gu, G.-Z. Yang, M.-H. Lu, S.-T. Liu, “Analysis of the focal characteristics of cylindrical lenses made of anisotropically dielectric material based on rigorous electromagnetic theory,” J. Mod. Opt. 50, 1195–1208 (2003).
[CrossRef]

J. Opt. Soc. Am. A (8)

J. Liu, B.-Y. Gu, B.-Z. Dong, G.-Z. Yang, “Interference effect of dual diffractive cylindrical microlenses analyzed by rigorous electromagnetic theory,” J. Opt. Soc. Am. A 18, 526–536 (2001).
[CrossRef]

J.-S. Ye, B.-Z. Dong, B.-Y. Gu, G.-Z. Yang, S.-T. Liu, “Analysis of a closed-boundary axilens with long focal depth and high transverse resolution based on rigorous electromagnetic theory,” J. Opt. Soc. Am. A 19, 2030–2035 (2002).
[CrossRef]

J. M. Bendickson, 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]

K. Hirayama, E. N. Glytsis, T. K. Gaylord, D. W. Wilson, “Rigorous electromagnetic analysis of diffractive cylindrical lenses,” J. Opt. Soc. Am. A 13, 2219–2231 (1996).
[CrossRef]

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]

D. A. Pommet, M. G. Moharam, E. B. Grann, “Limits of scalar diffraction theory for diffractive phase elements,” J. Opt. Soc. Am. A 11, 1827–1834 (1994).
[CrossRef]

J. N. Mait, “Understanding diffractive optic design in the scalar domain,” J. Opt. Soc. Am. A 12, 2145–2158 (1995).
[CrossRef]

J. M. Bendickson, E. N. Glytsis, T. K. Gaylord, “Metallic surface-relief on-axis and off-axis focusing diffractive cylindrical mirrors,” J. Opt. Soc. Am. A 16, 113–130 (1999).
[CrossRef]

Opt. Lett. (1)

Other (4)

M. Koshiba, Optical Waveguide Theory by the Finite Element Method (KTK Scientific, Tokyo, 1992), pp. 43–47.

M. Born, E. Wolf, Principles of Optics (Cambridge U. Press, Cambridge, UK, 1999), chap. 1.
[CrossRef]

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, 1968), Chaps. 3, 4.

M. Bass, E. W. V. Stryland, D. R. Williams, and W. L. Wolfe, eds., Handbook of Optics, Volume II, Devices, Measurements, and Properties, 2nd ed. (McGraw-Hill New York, 1994), Chap. 33.

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Figures (5)

Fig. 1
Fig. 1

Geometry of a two-dimensional cylindrical refractive microlens.

Fig. 2
Fig. 2

Normalized focal-plane intensity distributions of the f 1.0 cylindrical diffractive microlenses for different quantized levels of profiles in the case of TE polarization. (a) 4-level quantized diffractive lens; (b) 8-level quantized diffractive lens; (c) 16-level quantized diffractive lens; (d) (infinite-level) continuously refractive lens. The solid, the dotted, and the dotted–dashed curves correspond to intensity distributions calculated by the IRSM1, the BEM, and the ORSM1, respectively.

Fig. 3
Fig. 3

Several focusing performance measures calculated by the three methods for the same microlenses as in Fig. 2, including (a) focal spot size d, (b) diffraction efficiency η, and (c) normalized sidelobe power P sl P inc . The solid curve with squares, the dotted curve with circles, and the dotted–dashed curve with stars represent results calculated by the IRSM1, the BEM, and the ORSM1, respectively.

Fig. 4
Fig. 4

Several focusing performance measures calculated by the three methods for the 8-level cylindrical diffractive microlenses with different f-numbers, i.e., f 1.5 , f 1.25 , f 1.0 , and f 0.75 . The focusing performance measures include (a) focal spot size d, (b) diffraction efficiency η, and (c) normalized sidelobe power P sl P inc .

Fig. 5
Fig. 5

Lateral and axial intensity distributions of the f 1.0 cylindrical refractive microlens for TM polarization on (a) the near-field plane y = 0.01 λ , (b) the preset focal plane y = f , and (c) the axial plane x = 0 .

Tables (2)

Tables Icon

Table 1 Focal Performance of f 1.0 Cylindrical Diffractive Microlenses for Different Quantized Levels of Profiles Calculated by the Three Methods

Tables Icon

Table 2 Focal Performance of the Eight-Level Cylindrical Diffractive Microlenses for Different f-Numbers Calculated by the Three Methods

Equations (26)

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ϕ 1 t ( r 1 ) + Γ [ ϕ Γ ( r Γ ) G 1 ( r 1 , r Γ ) n ̂ p 1 G 1 ( r 1 , r Γ ) ϕ Γ ( r Γ ) n ̂ ] d l = ϕ inc ( r 1 ) , r 1 S 1 ,
ϕ 2 t ( r 2 ) + Γ [ ϕ Γ ( r Γ ) G 2 ( r 2 , r Γ ) n ̂ p 2 G 2 ( r 2 , r Γ ) ϕ Γ ( r Γ ) n ̂ ] d l
= 0 , r 2 S 2 ,
( θ Γ 2 π 1 ) ϕ Γ ( r Γ ) + Γ [ ϕ Γ ( r Γ ) G 1 ( r Γ , r Γ ) n ̂ p 1 G 1 ( r Γ , r Γ ) ϕ Γ ( r Γ ) n ̂ ] d l = ϕ inc ( r Γ ) ,
( θ Γ 2 π ) ϕ Γ ( r Γ ) + Γ [ ϕ Γ ( r Γ ) G 2 ( r Γ , r Γ ) n ̂ p 2 G 2 ( r Γ , r Γ ) ϕ Γ ( r Γ ) n ̂ ] d l = 0 ,
ϕ 2 ( r 2 ) = Γ [ ϕ Γ ( r Γ ) G 2 ( r 2 , r Γ ) n ̂ p 2 G 2 ( r 2 , r Γ ) ϕ Γ ( r Γ ) n ̂ ] d l .
ϕ Γ t inc ( r Γ t ) = T 0 ϕ 0 w ( x ) exp [ j δ ( x ) ] ,
G i RS 1 ( r i , r i ) = G i ( r i , r i ) G i ( r i , r j ) ,
G i RS 1 = 0 , G i RS 1 n ̂ t = 2 G i n ̂ t .
ϕ 2 ORSM 1 ( r 2 ) = Γ t [ ϕ Γ t inc ( r Γ t ) G 2 RS 1 ( r 2 , r Γ t ) n ̂ t ] d l ,
ϕ Γ inc ( r Γ ) = T ( x ) ϕ 0 w ( x ) exp [ j Δ ( x ) ] ,
ϕ 2 IRSM 1 ( r 2 ) = Γ [ ϕ Γ inc ( r Γ ) G 2 RS 1 ( r 2 , r Γ ) n ̂ ] d l ,
ASRD = { + [ I 1 ( x ) I 0 ( BEM ) ( x ) ] 2 d x + [ I 0 ( BEM ) ( x ) ] 2 d x } 1 2 ,
E z ( x , y i ) = a 2 ( ρ ) exp [ j ( ρ x β 2 y i ) ] d ρ ,
H x ( x , y i ) = 1 η 2 β 2 k 2 a 2 ( ρ ) exp [ j ( ρ x β 2 y i ) ] d ρ ,
E z ( m Δ x , y i ) = n = M 2 M 2 1 A 2 ( ρ n ) exp ( j ρ n m Δ x ) ,
m = M 2 , , M 2 1 ,
A 2 ( ρ n ) = 1 M m = M 2 M 2 1 E z ( m Δ x , y i ) exp ( j ρ n m Δ x ) ,
n = M 2 , , M 2 1 .
P f = Re { 1 2 d 2 d 2 E z ( x , f ) [ H x ( x , f ) ] * d x } = Re { d 2 η 2 n , m = M 2 M 2 1 β 2 m * k 2 [ A 2 ( ρ m ) ] * A 2 ( ρ n ) sinc [ ( ρ m ρ n ) d 2 ] } ,
P sl = Re { b a 2 η 2 n , m = M 2 M 2 1 exp [ j ( ρ m ρ n ) ( b + a 2 ) ] × β 2 m * k 2 [ A 2 ( ρ m ) ] * A 2 ( ρ n ) sinc [ ( ρ m ρ n ) ( b a 2 ) ] } .
P inc = 1 2 + E z inc ( x ) [ H x inc ( x ) ] * d x = 1 2 η 1 + w 2 ( x ) d x ,
ψ ( x ) = k 0 n 2 ( f f 2 + x 2 ) ,
h ( x ) = ψ ( x ) k 0 ( n 2 n 1 ) = n 2 n 1 n 2 ( f 2 + x 2 f ) .
h q ( x ) = Int [ h ( x ) Δ h ] Δ h ,
w ( x ) = { 1 0 x D 2 l cos 2 ( x D 2 + l 4 l π ) D 2 l < x D 2 + l , 0 D 2 + l x < }

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