Abstract

We investigated the focal characteristics of open-regional cylindrical microlens arrays with long focal depth by using a rigorous boundary-element method (BEM) and three scalar methods, i.e., a Kirchhoff and two Rayleigh-Sommerfeld diffraction integral forms. Numerical analysis clearly shows that the model cylindrical microlens arrays with different f-numbers can generate focusing beams with both long focal depth and high transverse resolution. The performance of the cylindrical microlens arrays, such as extended focal depth, relative extended focal depth, diffraction efficiency, and focal spot size, is appraised and analyzed. From a comparison of the results obtained by the rigorous BEM and by scalar approximations, we found that the results are quite similar when the f-number equals f/1.6; however, they are quite different for f/0.8. We conclude that the BEM should be adopted to analyze the performance of a microlens array system whose f-number is less than f/1.0.

© 2004 Optical Society of America

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References

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  1. N. Davidson, A. A. Friesem, E. Hasman, “Holographic axilens: high resolution and long focal depth,” Opt. Lett. 16, 523–525 (1991).
    [CrossRef] [PubMed]
  2. J. Sochacki, S. Bará, Z. Jaroszewicz, A. Kołodziejczyk, “Phase retardation of the uniform-intensity axilens,” Opt. Lett. 17, 7–9 (1992).
    [CrossRef] [PubMed]
  3. L. R. Staroński, J. Sochacki, Z. Jaroszewicz, A. Kołodziejczyk, “Lateral distribution and flow of energy in uniform-intensity axicons,” J. Opt. Soc. Am. A 9, 2091–2094 (1992).
    [CrossRef]
  4. J. Sochacki, Z. Jaroszewicz, L. R. Staroński, A. Kołodziejczyk, “Annular-aperture logarithmic axicon,” J. Opt. Soc. Am. A 10, 1765–1768 (1993).
    [CrossRef]
  5. Z. Jaroszewicz, J. Sochacki, A. Kołodziejczyk, L. R. Staroński, “Apodized annular-aperture logarithmic axicon: smoothness and uniformity of intensity distributions,” Opt. Lett. 18, 1893–1895 (1993).
    [CrossRef] [PubMed]
  6. J. Sochacki, A. Kołodziejczyk, Z. Jaroszewicz, S. Bará, “Phase retardation of the uniform intensity axilens,” Appl. Opt. 31, 5326–5330 (1992).
    [CrossRef] [PubMed]
  7. B. Z. Dong, G. Z. Yang, B. Y. Gu, “Iterative optimization approach for designing an axicon with long focal depth and high transverse resolution,” J. Opt. Soc. Am. A 13, 97–103 (1996).
    [CrossRef]
  8. B. Z. Dong, J. Liu, B. Y. Gu, G. Z. Yang, J. Wang, “Rigorous electromagnetic analysis of a microcylindrical axilens with long focal depth and high transverse resolution,” J. Opt. Soc. Am. A 18, 1465–1470 (2001).
    [CrossRef]
  9. 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]
  10. K. Hirayama, E. N. Glytsis, T. K. Gaylord, “Rigorous electromagnetic analysis of diffractive cylindrical lenses,” J. Opt. Soc. Am. A 13, 2219–2231 (1996).
    [CrossRef]
  11. 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, 907–914 (1997).
    [CrossRef]
  12. 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]
  13. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, Calif., 1968), Chaps. 3 and 4.

2002 (1)

2001 (1)

1998 (1)

1997 (1)

1996 (2)

1993 (2)

1992 (3)

1991 (1)

Bará, S.

Bendickson, J. M.

Davidson, N.

Dong, B. Z.

Friesem, A. A.

Gaylord, T. K.

Glytsis, E. N.

Goodman, J. W.

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

Gu, B. Y.

Hasman, E.

Hirayama, K.

Jaroszewicz, Z.

Kolodziejczyk, A.

Liu, J.

Liu, S. T.

Sochacki, J.

Staronski, L. R.

Wang, J.

Yang, G. Z.

Ye, J. S.

Appl. Opt. (1)

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

B. Z. Dong, G. Z. Yang, B. Y. Gu, “Iterative optimization approach for designing an axicon with long focal depth and high transverse resolution,” J. Opt. Soc. Am. A 13, 97–103 (1996).
[CrossRef]

K. Hirayama, E. N. Glytsis, T. K. Gaylord, “Rigorous electromagnetic analysis of diffractive cylindrical lenses,” J. Opt. Soc. Am. A 13, 2219–2231 (1996).
[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, “Rigorous electromagnetic analysis of diffraction by finite-number-of-periods gratings,” J. Opt. Soc. Am. A 14, 907–914 (1997).
[CrossRef]

J. Sochacki, Z. Jaroszewicz, L. R. Staroński, A. Kołodziejczyk, “Annular-aperture logarithmic axicon,” J. Opt. Soc. Am. A 10, 1765–1768 (1993).
[CrossRef]

L. R. Staroński, J. Sochacki, Z. Jaroszewicz, A. Kołodziejczyk, “Lateral distribution and flow of energy in uniform-intensity axicons,” J. Opt. Soc. Am. A 9, 2091–2094 (1992).
[CrossRef]

B. Z. Dong, J. Liu, B. Y. Gu, G. Z. Yang, J. Wang, “Rigorous electromagnetic analysis of a microcylindrical axilens with long focal depth and high transverse resolution,” J. Opt. Soc. Am. A 18, 1465–1470 (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]

Opt. Lett. (3)

Other (1)

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

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Figures (5)

Fig. 1
Fig. 1

Schematic of the LFD cylindrical microlens array.

Fig. 2
Fig. 2

Gray-level image representation of the intensity distributions in region S2 for the LFD DCMA with f/# = f/1.0. (a) δf = 0.0 μm, i.e., the conventional DCMA; (b) δf = 10.0 μm; (c) δf = 20.0 μm. The bright (dark) areas mark the region of high (low) intensity.

Fig. 3
Fig. 3

Axial and transverse intensity distributions for the LFD DCMA when f/# = f/1.0. (a) a, b, c, axial intensity distributions for preset focal depths δf of 0.0, 10.0, and 20.0 μm, respectively. (b) Transverse intensity distributions within the LFD region for δf = 20.0 μm. Solid curve, transverse intensity distributions on the real focal plane y = -36.46 μm; long-dashed and dotted-dashed curves, those on the planes y = -31.11 and y = -43.62 μm, respectively.

Fig. 4
Fig. 4

Same as Fig. 2, except for the LFD TCMA. (a) δf = 0.0, (b) δf = 10.0, and (c) δf = 20.0 μm.

Fig. 5
Fig. 5

Same as Fig. 3, except for the LFD TCMA. (a) a, b, c, axial intensity distributions for the preset focal depths δf of 0.0, 10.0, and 20.0 μm, respectively. (b) Transverse intensity distributions within the LFD region for δf = 20.0 μm. Solid curve, transverse intensity distributions on the real focal plane y = -36.33 μm; long-dashed and dotted-dashed curves, those on the planes y = -31.08 and y = -43.77 μm, respectively.

Tables (4)

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Table 1 Extended Focal Depths Δf or Relative Extended Focal Depths Δff 0 (in Micrometers) of LFD DCMAs for Several f-Numbers Calculated by the BEM and by Three Scalar Methods

Tables Icon

Table 2 Diffraction Efficiencies η (%) of LFD DCMAs for Several f-Numbers Calculated by the BEM and by Three Scalar Methods

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Table 3 Focal Spot Sizes d (in Micrometers) of LFD DCMAs for Several f-Numbers Calculated by the BEM and by Three Scalar Methods

Tables Icon

Table 4 Comparison of Focusing Performances of the Single, Dual, and Triple LFD Cylindrical Microlens Systems for the TE Polarization

Equations (26)

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-ϕ1tr1+ΓϕΓrΓG1r1, rΓnˆ-p1G1r1, rΓϕΓrΓnˆdl=-ϕincr1,r1S1,
ϕ2tr2+ΓϕΓrΓG2r2, rΓnˆ-p2G2r2, rΓϕΓrΓnˆdl=0,r2S2,
θΓ2π-1ϕΓrΓ+ Γ ϕΓrΓG1rΓ, rΓnˆ-p1G1rΓ, rΓϕΓrΓnˆdl=-ϕincrΓ,
θΓ2πϕΓrΓ+ Γ ϕΓrΓG2rΓ, rΓnˆ-p2G2rΓ, rΓϕΓrΓnˆdl=0,
ϕ2r2=-ΓϕΓrΓG2r2, rΓnˆ-p2G2r2, rΓϕΓrΓnˆdl.
ϕΓtincrΓt=Tϕ0wxexp-jΔx,
ϕΓtincrΓtnˆt=jk2Tϕ0wxexp-jΔx,
ϕ2Kr2=-ΓtϕΓtincrΓtG2r2, rΓtnˆt-p2G2r2, rΓtϕΓtincrΓtnˆtdl.
GiRSM1ri, ri=Giri, ri-Giri, rj,
GiRSM2ri, ri=Giri, ri+Giri, rj,
GiRSM1=0, GiRSM1nˆt=2 Ginˆt,
GiRSM2=2Gi, GiRSM2nˆt=0.
ϕ2RSM1r2=-ΓtϕΓtincrΓtG2RSM1r2, rΓtnˆtdl,
ϕ2RSM2r2=Γtp2G2RSM2r2, rΓtϕΓtincrΓtnˆtdl.
Ezx, yi=- a2ρexp-jρx-β2yidρ,
Hxx, yi=- 1η2-β2k2 a2ρexp-jρx-β2yidρ,
EzmΔx, yi=n=-M/2M/2-1 A2ρn, yiexp-jρnmΔx,
A2ρn, yi=1Mm=-M/2M/2-1 EzmΔx, yiexpjρnmΔx,
Ptyi=Re- 12-+ Ezx, yiHxx, yi*dx=ReL2η2n=-M/2M/2-1β2n*k2 |A2ρn, yi|2,
Pfyi=-f=kRe- 12xk-dk/2xk+dk/2 Ezx, -f×Hxx, -f*dx=kRedk2η2n=-M/2M/2-1m=-M/2M/2-1β2n*k2A2ρn, -f*×A2ρm, -fsincρn-ρmdk/2,
Pinc=- 12-+ EzincHxinc*dx= 12η1-+ w2xdx,
ψx=k0n2f-f2+x2,
fx=f0+δfx2/R2,
hsx= ψxk0n2-n1= n2n1-n2f2x+x2-fx.
hx= l=1m/2 hs|x|-Dl-1/2×rect|x|D-l-1/2,even ml=1m+1/2 hs|x|-Dl-1×rect|x|D-l-1,odd m,
wx= 10|x|mD2-ξcos2π4ξ|x|-mD2+ξ,mD2-ξ<|x|mD2+ξ0|x|>mD2+ξ,

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