Abstract

We employ a combination of asymptotic methods to speed up the computation of fields in the focal region of a diffractive lens (DL). The DL is treated locally as a linear grating with a slowly varying period and groove orientation. We employ rigorous electromagnetic diffraction theory locally to obtain the field just behind the DL. A simple diffracted-ray formula is derived for the field in the focal region of the DL at observation points that are not in the immediate vicinity of the optical axis. A careful study of the range of validity of this formula is made. For observation points that are not in the immediate vicinity of the optical axis the new algorithm is 3 × 105 times faster than the application of numerical integration to the double integrals involved and approximately 1000–1200 times faster than a recently published algorithm based on using asymptotic theory to replace the double integral with a single integral.

© 2000 Optical Society of America

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References

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  1. N. Sergienko, J. J. Stamnes, V. Kettunen, M. Kuittinen, J. Turunen, P. Vahimaa, A. T. Friberg, “Asymptotic methods for evaluation of diffractive lenses,” J. Pure Appl. Opt. 1, 552–559 (1999).
    [CrossRef]
  2. N. Sergienko, J. J. Stamnes, V. Kettunen, M. Kuittinen, J. Turunen, P. Vahimaa, A. T. Friberg, “Comparison of electromagnetic and scalar methods for evaluation of diffractive lenses,” J. Mod. Opt. 46, 65–82 (1999).
  3. J. J. Stamnes, B. Spjelkavik, H. M. Pedersen, “Evaluation of diffraction integrals using local phase and amplitude approximations,” Opt. Acta 30, 207–222 (1983).
    [CrossRef]
  4. J. J. Stamnes, Waves in Focal Regions (Adam Hilger, Bristol, UK, 1986), Sec. 7.2.
  5. P. Vahimaa, V. Kettunen, M. Kuittinen, J. Turunen, A. T. Friberg, “Electromagnetic analysis of nonparaxial Bessel beams generated by diffractive axicons,” J. Opt. Soc. Am. A 14, 1817–1824 (1997).
    [CrossRef]

1999

N. Sergienko, J. J. Stamnes, V. Kettunen, M. Kuittinen, J. Turunen, P. Vahimaa, A. T. Friberg, “Asymptotic methods for evaluation of diffractive lenses,” J. Pure Appl. Opt. 1, 552–559 (1999).
[CrossRef]

N. Sergienko, J. J. Stamnes, V. Kettunen, M. Kuittinen, J. Turunen, P. Vahimaa, A. T. Friberg, “Comparison of electromagnetic and scalar methods for evaluation of diffractive lenses,” J. Mod. Opt. 46, 65–82 (1999).

1997

1983

J. J. Stamnes, B. Spjelkavik, H. M. Pedersen, “Evaluation of diffraction integrals using local phase and amplitude approximations,” Opt. Acta 30, 207–222 (1983).
[CrossRef]

Friberg, A. T.

N. Sergienko, J. J. Stamnes, V. Kettunen, M. Kuittinen, J. Turunen, P. Vahimaa, A. T. Friberg, “Comparison of electromagnetic and scalar methods for evaluation of diffractive lenses,” J. Mod. Opt. 46, 65–82 (1999).

N. Sergienko, J. J. Stamnes, V. Kettunen, M. Kuittinen, J. Turunen, P. Vahimaa, A. T. Friberg, “Asymptotic methods for evaluation of diffractive lenses,” J. Pure Appl. Opt. 1, 552–559 (1999).
[CrossRef]

P. Vahimaa, V. Kettunen, M. Kuittinen, J. Turunen, A. T. Friberg, “Electromagnetic analysis of nonparaxial Bessel beams generated by diffractive axicons,” J. Opt. Soc. Am. A 14, 1817–1824 (1997).
[CrossRef]

Kettunen, V.

N. Sergienko, J. J. Stamnes, V. Kettunen, M. Kuittinen, J. Turunen, P. Vahimaa, A. T. Friberg, “Asymptotic methods for evaluation of diffractive lenses,” J. Pure Appl. Opt. 1, 552–559 (1999).
[CrossRef]

N. Sergienko, J. J. Stamnes, V. Kettunen, M. Kuittinen, J. Turunen, P. Vahimaa, A. T. Friberg, “Comparison of electromagnetic and scalar methods for evaluation of diffractive lenses,” J. Mod. Opt. 46, 65–82 (1999).

P. Vahimaa, V. Kettunen, M. Kuittinen, J. Turunen, A. T. Friberg, “Electromagnetic analysis of nonparaxial Bessel beams generated by diffractive axicons,” J. Opt. Soc. Am. A 14, 1817–1824 (1997).
[CrossRef]

Kuittinen, M.

N. Sergienko, J. J. Stamnes, V. Kettunen, M. Kuittinen, J. Turunen, P. Vahimaa, A. T. Friberg, “Asymptotic methods for evaluation of diffractive lenses,” J. Pure Appl. Opt. 1, 552–559 (1999).
[CrossRef]

N. Sergienko, J. J. Stamnes, V. Kettunen, M. Kuittinen, J. Turunen, P. Vahimaa, A. T. Friberg, “Comparison of electromagnetic and scalar methods for evaluation of diffractive lenses,” J. Mod. Opt. 46, 65–82 (1999).

P. Vahimaa, V. Kettunen, M. Kuittinen, J. Turunen, A. T. Friberg, “Electromagnetic analysis of nonparaxial Bessel beams generated by diffractive axicons,” J. Opt. Soc. Am. A 14, 1817–1824 (1997).
[CrossRef]

Pedersen, H. M.

J. J. Stamnes, B. Spjelkavik, H. M. Pedersen, “Evaluation of diffraction integrals using local phase and amplitude approximations,” Opt. Acta 30, 207–222 (1983).
[CrossRef]

Sergienko, N.

N. Sergienko, J. J. Stamnes, V. Kettunen, M. Kuittinen, J. Turunen, P. Vahimaa, A. T. Friberg, “Asymptotic methods for evaluation of diffractive lenses,” J. Pure Appl. Opt. 1, 552–559 (1999).
[CrossRef]

N. Sergienko, J. J. Stamnes, V. Kettunen, M. Kuittinen, J. Turunen, P. Vahimaa, A. T. Friberg, “Comparison of electromagnetic and scalar methods for evaluation of diffractive lenses,” J. Mod. Opt. 46, 65–82 (1999).

Spjelkavik, B.

J. J. Stamnes, B. Spjelkavik, H. M. Pedersen, “Evaluation of diffraction integrals using local phase and amplitude approximations,” Opt. Acta 30, 207–222 (1983).
[CrossRef]

Stamnes, J. J.

N. Sergienko, J. J. Stamnes, V. Kettunen, M. Kuittinen, J. Turunen, P. Vahimaa, A. T. Friberg, “Asymptotic methods for evaluation of diffractive lenses,” J. Pure Appl. Opt. 1, 552–559 (1999).
[CrossRef]

N. Sergienko, J. J. Stamnes, V. Kettunen, M. Kuittinen, J. Turunen, P. Vahimaa, A. T. Friberg, “Comparison of electromagnetic and scalar methods for evaluation of diffractive lenses,” J. Mod. Opt. 46, 65–82 (1999).

J. J. Stamnes, B. Spjelkavik, H. M. Pedersen, “Evaluation of diffraction integrals using local phase and amplitude approximations,” Opt. Acta 30, 207–222 (1983).
[CrossRef]

J. J. Stamnes, Waves in Focal Regions (Adam Hilger, Bristol, UK, 1986), Sec. 7.2.

Turunen, J.

N. Sergienko, J. J. Stamnes, V. Kettunen, M. Kuittinen, J. Turunen, P. Vahimaa, A. T. Friberg, “Comparison of electromagnetic and scalar methods for evaluation of diffractive lenses,” J. Mod. Opt. 46, 65–82 (1999).

N. Sergienko, J. J. Stamnes, V. Kettunen, M. Kuittinen, J. Turunen, P. Vahimaa, A. T. Friberg, “Asymptotic methods for evaluation of diffractive lenses,” J. Pure Appl. Opt. 1, 552–559 (1999).
[CrossRef]

P. Vahimaa, V. Kettunen, M. Kuittinen, J. Turunen, A. T. Friberg, “Electromagnetic analysis of nonparaxial Bessel beams generated by diffractive axicons,” J. Opt. Soc. Am. A 14, 1817–1824 (1997).
[CrossRef]

Vahimaa, P.

N. Sergienko, J. J. Stamnes, V. Kettunen, M. Kuittinen, J. Turunen, P. Vahimaa, A. T. Friberg, “Asymptotic methods for evaluation of diffractive lenses,” J. Pure Appl. Opt. 1, 552–559 (1999).
[CrossRef]

N. Sergienko, J. J. Stamnes, V. Kettunen, M. Kuittinen, J. Turunen, P. Vahimaa, A. T. Friberg, “Comparison of electromagnetic and scalar methods for evaluation of diffractive lenses,” J. Mod. Opt. 46, 65–82 (1999).

P. Vahimaa, V. Kettunen, M. Kuittinen, J. Turunen, A. T. Friberg, “Electromagnetic analysis of nonparaxial Bessel beams generated by diffractive axicons,” J. Opt. Soc. Am. A 14, 1817–1824 (1997).
[CrossRef]

J. Mod. Opt.

N. Sergienko, J. J. Stamnes, V. Kettunen, M. Kuittinen, J. Turunen, P. Vahimaa, A. T. Friberg, “Comparison of electromagnetic and scalar methods for evaluation of diffractive lenses,” J. Mod. Opt. 46, 65–82 (1999).

J. Opt. Soc. Am. A

J. Pure Appl. Opt.

N. Sergienko, J. J. Stamnes, V. Kettunen, M. Kuittinen, J. Turunen, P. Vahimaa, A. T. Friberg, “Asymptotic methods for evaluation of diffractive lenses,” J. Pure Appl. Opt. 1, 552–559 (1999).
[CrossRef]

Opt. Acta

J. J. Stamnes, B. Spjelkavik, H. M. Pedersen, “Evaluation of diffraction integrals using local phase and amplitude approximations,” Opt. Acta 30, 207–222 (1983).
[CrossRef]

Other

J. J. Stamnes, Waves in Focal Regions (Adam Hilger, Bristol, UK, 1986), Sec. 7.2.

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Figures (2)

Fig. 1
Fig. 1

Transverse profiles of the electric energy density in the geometrical focal plane for the f = 0.5 lens: solid curve, method (i); crosses, method (ii); stars, method (iii); dashed curve, method (iv).

Fig. 2
Fig. 2

Contour map of the electric energy density in the xz plane of the focal area for the f:0.5 lens. Combined methods (i) and (ii) are used for ρ ≤ ρ0, and combined methods (i) and (iv) are used for ρ > ρ0, where ρ0 = 0.6 µm. All intensity values in the contour map are divided by 106.

Equations (21)

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Eρ, z=xˆAρexpiknz,
uρ, ϕ, 0+=tρ, ϕuρ, ϕ, 0-,
Expρ, ϕ, 0+=AρmPnwmpSmp cos2 ϕ+Tmp sin2 ϕexpi2πmρ-ρp/dp,
Eypρ, ϕ, 0+=Aρsin ϕ cos ϕ mPnwmpSmp-Tmp×expi2πmρ-ρp/dp,
Ezpρ, ϕ, 0+=-Aρnλ/dcos ϕ×mP mSmp expi2πmρ-ρp/dp,
wmp=1-mλ/dp21/2  |m|<d/λ,
dp=ρp+1-ρp.
Am=-1msinπm+1Q sinπm+1/Q sincm/Q,
Expρ, ϕ, 0+=AρnmPsin2 ϕ+wmp cos2 ϕAm×expi2πmρ-ρp/dp,
Eypρ, ϕ, 0+=Aρn sin ϕ cos ϕ mPwmp-1Am×expi2πmρ-ρp/dp,
Ezpρ, ϕ, 0+=-Aρnλ/dcos ϕ mP mAm×expi2πmρ-ρp/dp.
Expρ, ϕ, 0+=AρnmP Am expi2πmρ-ρp/dp
um,pρ, ϕ, z=ρpρp+102π Gm,pρ, ϕ, ρ, ϕ, z×expikFρ, ϕ, ρ, ϕ, zdϕdρ,
Gm,pρ, ϕ, ρ, ϕ, z=-ik2πzρR2 um,pρ, ϕ, 0+×1+ikR,
Fρ, ϕ, ρ, ϕ, z=R=ρ2+ρ2+z2-2ρρ×cosϕ-ϕ1/2.
um,pρ, ϕ, zzλρ exp-iπ/4n=01ρpρp+1 gm,p,nρ×expikfnρdρ,
gm,p,nρ=ρRn1Rn um,pρ, ϕ+nπ, 0+,
fnρ=Rnρ=ρ--1nρ2+z21/2.
um,pρ, ϕ, zz2π02π gm,pρp, ϕexpikFρp, ϕdϕ-02π gm,pρp+1, ϕ×expikFρp+1, ϕdϕ,
gm,pρ, ϕ=um,pρ, ϕ, 0+R1-ρ/ρcosϕ-ϕ.
um,pρ, ϕ, z-z2πλρ expiπ/4×n=01ρRnρum,pρ, ϕ+nπ, 0+ρ--1nρ×expikRnρρ=ρpρ=ρp+1,

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