Abstract

We present an optimization-based synthesis algorithm for the design of diffractive optical elements (DOE’s) that are finite in extent, have subwavelength features, and are aperiodic. The subwavelength nature of the DOE’s precludes the use of scalar diffraction theory, and their finite extent and aperiodic nature prevents the use of coupled-wave analysis. To overcome these limitations, we apply the boundary element method (BEM) as the propagation model in the synthesis algorithm. However, the computational costs associated with the conventional implementation of the BEM prevent the design of realistic DOE’s in reasonable time frames. Consequently, an alternative formulation of the BEM that exploits DOE symmetry is developed and implemented on a parallel computer. Designs of finite extent, subwavelength, and aperiodic DOE’s, such as a lens and a focusing beam splitter, are presented.

© 1998 Optical Society of America

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References

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  1. T. K. Gaylord, M. G. Moharam, “Analysis and applications of optical diffraction by gratings,” Proc. IEEE 73, 894–937 (1985).
    [CrossRef]
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    [CrossRef] [PubMed]
  3. H. Haidner, J. T. Sheridan, J. Schwider, N. Streibl, “Design of a blazed grating consisting of metallic subwavelength binary grooves,” Opt. Commun. 98, 5–10 (1993).
    [CrossRef]
  4. M. Schmitz, R. Brauer, O. Bryngdahl, “Phase gratings with subwavelength structures,” J. Opt. Soc. Am. A 12, 2458–2462 (1995).
    [CrossRef]
  5. Z. Zhou, T. J. Drabik, “Optimized binary, phase-only, diffractive optical element with subwavelength features for 1.55 µm,” J. Opt. Soc. Am. A 12, 1104–1112 (1995).
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  6. E. Noponen, J. Turunen, F. Wyrowski, “Synthesis of paraxial-domain diffractive elements by rigorous electromagnetic theory,” J. Opt. Soc. Am. A 12, 1128–1133 (1995).
    [CrossRef]
  7. D. W. Prather, M. S. Mirotznik, J. N. Mait, “Design of subwavelength diffractive optical elements using a hybrid finite element-boundary element method,” in Diffractive and Holographic Optics Technology III, I. Cindrich, S. H. Lee, eds., Proc. SPIE2689, 14–23 (1996).
    [CrossRef]
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    [CrossRef]
  9. D. H. Raguin, “Subwavelength structured surfaces: theory and applications,” Ph.D. dissertation (University of Rochester, Rochester, N.Y., 1993).
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    [CrossRef]
  11. M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, UK, 1980).
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    [CrossRef]
  13. D. W. Prather, “Analysis and synthesis of finite aperiodic diffractive optical elements using rigorous electromagnetic models,” Ph.D. dissertation (University of Maryland, College Park, Md., 1997).
  14. J. N. Mait, D. W. Prather, M. S. Mirotznik, “Design and optimization of finite aperiodic subwavelength diffractive optical elements having arbitrary phase profiles,” presented at the OSA 1996 Annual Meeting, Rochester, N.Y., Oct. 20–24, 1996.
  15. D. W. Prather, M. S. Mirotznik, J. N. Mait, “Boundary element method for vector modeling diffractive optical elements,” in Diffractive and Holographic Optics Technology II, I. Cindrich, S. H. Lee, eds., Proc. SPIE2404, 28–39 (1995).
    [CrossRef]
  16. 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]
  17. S. Kagami, I. Fukai, “Application of boundary-element method to electromagnetic field problems,” IEEE Trans. Microwave Theory Tech. MTT-32, 455–461 (1984).
    [CrossRef]
  18. S. Kirkpatrick, C. D. Gelatt, M. P. Vecchi, “Optimization by simulated annealing,” Science 220, 671–680 (1983).
    [CrossRef] [PubMed]
  19. M. A. Seldowitz, J. P. Allebach, D. W. Sweeney, “Synthesis of digital holograms by direct binary search,” Appl. Opt. 26, 2788–2798 (1987).
    [CrossRef] [PubMed]

1997 (1)

1996 (2)

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]

P. Lalanne, D. Lamercier-Lalanne, “On the effective medium theory of subwavelength periodic structures,” J. Mod. Opt. 43, 2063–2085 (1996).
[CrossRef]

1995 (4)

1993 (1)

H. Haidner, J. T. Sheridan, J. Schwider, N. Streibl, “Design of a blazed grating consisting of metallic subwavelength binary grooves,” Opt. Commun. 98, 5–10 (1993).
[CrossRef]

1992 (1)

1987 (1)

1985 (1)

T. K. Gaylord, M. G. Moharam, “Analysis and applications of optical diffraction by gratings,” Proc. IEEE 73, 894–937 (1985).
[CrossRef]

1984 (1)

S. Kagami, I. Fukai, “Application of boundary-element method to electromagnetic field problems,” IEEE Trans. Microwave Theory Tech. MTT-32, 455–461 (1984).
[CrossRef]

1983 (1)

S. Kirkpatrick, C. D. Gelatt, M. P. Vecchi, “Optimization by simulated annealing,” Science 220, 671–680 (1983).
[CrossRef] [PubMed]

Allebach, J. P.

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, UK, 1980).

Brauer, R.

Bryngdahl, O.

Drabik, T. J.

Farn, M. W.

Fukai, I.

S. Kagami, I. Fukai, “Application of boundary-element method to electromagnetic field problems,” IEEE Trans. Microwave Theory Tech. MTT-32, 455–461 (1984).
[CrossRef]

Gaylord, T. K.

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]

T. K. Gaylord, M. G. Moharam, “Analysis and applications of optical diffraction by gratings,” Proc. IEEE 73, 894–937 (1985).
[CrossRef]

Gelatt, C. D.

S. Kirkpatrick, C. D. Gelatt, M. P. Vecchi, “Optimization by simulated annealing,” Science 220, 671–680 (1983).
[CrossRef] [PubMed]

Glytsis, E. N.

Haidner, H.

H. Haidner, J. T. Sheridan, J. Schwider, N. Streibl, “Design of a blazed grating consisting of metallic subwavelength binary grooves,” Opt. Commun. 98, 5–10 (1993).
[CrossRef]

Hirayama, K.

Kagami, S.

S. Kagami, I. Fukai, “Application of boundary-element method to electromagnetic field problems,” IEEE Trans. Microwave Theory Tech. MTT-32, 455–461 (1984).
[CrossRef]

Kirkpatrick, S.

S. Kirkpatrick, C. D. Gelatt, M. P. Vecchi, “Optimization by simulated annealing,” Science 220, 671–680 (1983).
[CrossRef] [PubMed]

Lalanne, P.

P. Lalanne, D. Lamercier-Lalanne, “On the effective medium theory of subwavelength periodic structures,” J. Mod. Opt. 43, 2063–2085 (1996).
[CrossRef]

Lamercier-Lalanne, D.

P. Lalanne, D. Lamercier-Lalanne, “On the effective medium theory of subwavelength periodic structures,” J. Mod. Opt. 43, 2063–2085 (1996).
[CrossRef]

Mait, J. N.

D. W. Prather, M. S. Mirotznik, J. N. Mait, “Boundary integral methods applied to the analysis of diffractive optical elements,” J. Opt. Soc. Am. A 14, 34–43 (1997).
[CrossRef]

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

D. W. Prather, M. S. Mirotznik, J. N. Mait, “Boundary element method for vector modeling diffractive optical elements,” in Diffractive and Holographic Optics Technology II, I. Cindrich, S. H. Lee, eds., Proc. SPIE2404, 28–39 (1995).
[CrossRef]

D. W. Prather, M. S. Mirotznik, J. N. Mait, “Design of subwavelength diffractive optical elements using a hybrid finite element-boundary element method,” in Diffractive and Holographic Optics Technology III, I. Cindrich, S. H. Lee, eds., Proc. SPIE2689, 14–23 (1996).
[CrossRef]

J. N. Mait, D. W. Prather, M. S. Mirotznik, “Design and optimization of finite aperiodic subwavelength diffractive optical elements having arbitrary phase profiles,” presented at the OSA 1996 Annual Meeting, Rochester, N.Y., Oct. 20–24, 1996.

Mirotznik, M. S.

D. W. Prather, M. S. Mirotznik, J. N. Mait, “Boundary integral methods applied to the analysis of diffractive optical elements,” J. Opt. Soc. Am. A 14, 34–43 (1997).
[CrossRef]

D. W. Prather, M. S. Mirotznik, J. N. Mait, “Boundary element method for vector modeling diffractive optical elements,” in Diffractive and Holographic Optics Technology II, I. Cindrich, S. H. Lee, eds., Proc. SPIE2404, 28–39 (1995).
[CrossRef]

J. N. Mait, D. W. Prather, M. S. Mirotznik, “Design and optimization of finite aperiodic subwavelength diffractive optical elements having arbitrary phase profiles,” presented at the OSA 1996 Annual Meeting, Rochester, N.Y., Oct. 20–24, 1996.

D. W. Prather, M. S. Mirotznik, J. N. Mait, “Design of subwavelength diffractive optical elements using a hybrid finite element-boundary element method,” in Diffractive and Holographic Optics Technology III, I. Cindrich, S. H. Lee, eds., Proc. SPIE2689, 14–23 (1996).
[CrossRef]

Moharam, M. G.

T. K. Gaylord, M. G. Moharam, “Analysis and applications of optical diffraction by gratings,” Proc. IEEE 73, 894–937 (1985).
[CrossRef]

Noponen, E.

Prather, D. W.

D. W. Prather, M. S. Mirotznik, J. N. Mait, “Boundary integral methods applied to the analysis of diffractive optical elements,” J. Opt. Soc. Am. A 14, 34–43 (1997).
[CrossRef]

D. W. Prather, M. S. Mirotznik, J. N. Mait, “Design of subwavelength diffractive optical elements using a hybrid finite element-boundary element method,” in Diffractive and Holographic Optics Technology III, I. Cindrich, S. H. Lee, eds., Proc. SPIE2689, 14–23 (1996).
[CrossRef]

D. W. Prather, M. S. Mirotznik, J. N. Mait, “Boundary element method for vector modeling diffractive optical elements,” in Diffractive and Holographic Optics Technology II, I. Cindrich, S. H. Lee, eds., Proc. SPIE2404, 28–39 (1995).
[CrossRef]

J. N. Mait, D. W. Prather, M. S. Mirotznik, “Design and optimization of finite aperiodic subwavelength diffractive optical elements having arbitrary phase profiles,” presented at the OSA 1996 Annual Meeting, Rochester, N.Y., Oct. 20–24, 1996.

D. W. Prather, “Analysis and synthesis of finite aperiodic diffractive optical elements using rigorous electromagnetic models,” Ph.D. dissertation (University of Maryland, College Park, Md., 1997).

Raguin, D. H.

D. H. Raguin, “Subwavelength structured surfaces: theory and applications,” Ph.D. dissertation (University of Rochester, Rochester, N.Y., 1993).

Schmitz, M.

Schwider, J.

H. Haidner, J. T. Sheridan, J. Schwider, N. Streibl, “Design of a blazed grating consisting of metallic subwavelength binary grooves,” Opt. Commun. 98, 5–10 (1993).
[CrossRef]

Seldowitz, M. A.

Sheridan, J. T.

H. Haidner, J. T. Sheridan, J. Schwider, N. Streibl, “Design of a blazed grating consisting of metallic subwavelength binary grooves,” Opt. Commun. 98, 5–10 (1993).
[CrossRef]

Streibl, N.

H. Haidner, J. T. Sheridan, J. Schwider, N. Streibl, “Design of a blazed grating consisting of metallic subwavelength binary grooves,” Opt. Commun. 98, 5–10 (1993).
[CrossRef]

Sweeney, D. W.

Turunen, J.

Vecchi, M. P.

S. Kirkpatrick, C. D. Gelatt, M. P. Vecchi, “Optimization by simulated annealing,” Science 220, 671–680 (1983).
[CrossRef] [PubMed]

Wilson, D. W.

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, UK, 1980).

Wyrowski, F.

Zhou, Z.

Appl. Opt. (2)

IEEE Trans. Microwave Theory Tech. (1)

S. Kagami, I. Fukai, “Application of boundary-element method to electromagnetic field problems,” IEEE Trans. Microwave Theory Tech. MTT-32, 455–461 (1984).
[CrossRef]

J. Mod. Opt. (1)

P. Lalanne, D. Lamercier-Lalanne, “On the effective medium theory of subwavelength periodic structures,” J. Mod. Opt. 43, 2063–2085 (1996).
[CrossRef]

J. Opt. Soc. Am. A (6)

Opt. Commun. (1)

H. Haidner, J. T. Sheridan, J. Schwider, N. Streibl, “Design of a blazed grating consisting of metallic subwavelength binary grooves,” Opt. Commun. 98, 5–10 (1993).
[CrossRef]

Proc. IEEE (1)

T. K. Gaylord, M. G. Moharam, “Analysis and applications of optical diffraction by gratings,” Proc. IEEE 73, 894–937 (1985).
[CrossRef]

Science (1)

S. Kirkpatrick, C. D. Gelatt, M. P. Vecchi, “Optimization by simulated annealing,” Science 220, 671–680 (1983).
[CrossRef] [PubMed]

Other (6)

D. W. Prather, M. S. Mirotznik, J. N. Mait, “Design of subwavelength diffractive optical elements using a hybrid finite element-boundary element method,” in Diffractive and Holographic Optics Technology III, I. Cindrich, S. H. Lee, eds., Proc. SPIE2689, 14–23 (1996).
[CrossRef]

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, UK, 1980).

D. W. Prather, “Analysis and synthesis of finite aperiodic diffractive optical elements using rigorous electromagnetic models,” Ph.D. dissertation (University of Maryland, College Park, Md., 1997).

J. N. Mait, D. W. Prather, M. S. Mirotznik, “Design and optimization of finite aperiodic subwavelength diffractive optical elements having arbitrary phase profiles,” presented at the OSA 1996 Annual Meeting, Rochester, N.Y., Oct. 20–24, 1996.

D. W. Prather, M. S. Mirotznik, J. N. Mait, “Boundary element method for vector modeling diffractive optical elements,” in Diffractive and Holographic Optics Technology II, I. Cindrich, S. H. Lee, eds., Proc. SPIE2404, 28–39 (1995).
[CrossRef]

D. H. Raguin, “Subwavelength structured surfaces: theory and applications,” Ph.D. dissertation (University of Rochester, Rochester, N.Y., 1993).

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Figures (9)

Fig. 1
Fig. 1

Diagram of SSBEM-based synthesis algorithm: 1, initial profile; 2, use SSBEM to determine symmetric fields on boundary surface; 3, propagate fields; 4, evaluate performance; 5, modify profile and return to step 2.

Fig. 2
Fig. 2

Subwavelength design of linear blazed grating using Farn’s technique: (a) continuous-phase linear blazed grating, (b) binary subwavelength grating.

Fig. 3
Fig. 3

Subwavelength design of arbitrary phase functions using extension to Farn’s technique: (a) continuous-phase function and (b) its piecewise-linear approximation, (c) binary subwavelength diffractive element and (d) its corresponding effective refractive index.

Fig. 4
Fig. 4

Application of fabrication constraints to SWDOEs: (a) spatial and (b) depth quantization of binary subwavelength profile.

Fig. 5
Fig. 5

Geometry of diffractive structure used in formulation of semi-infinite boundary element method.

Fig. 6
Fig. 6

Terrain of solutions navigated by SA and SQ optimization routines.

Fig. 7
Fig. 7

Continuous-phase lens used in subwavelength design.

Fig. 8
Fig. 8

Subwavelength lens design: (a) initial binary lens profile unconstrained by fabrication, (b) electric field magnitude in space behind lens, (c) electric field intensity in focal plane, (d)–(f) results for profile constrained by fabrication, and (g)–(i) results for final profile.

Fig. 9
Fig. 9

Subwavelength design of 1-to-2 fan-out: (a) Initial profile, (b) electric-field magnitude in space behind fan-out, (c) electric field intensity in detector plane, and (d)–(f) results for final profile.

Equations (31)

Equations on this page are rendered with MathJax. Learn more.

αi=iW/(N+1),
βi=iW/N,
N=W/Δ.
f(x)=k=1Mi=1Nkrectx-(αk,i+βk,i)/2-(i-1)Δ-j=1k-1Wjβk,i-αk,i,
eff,TE(x)=sf(x)+orectxD-f(x) * 1ΔrectxΔ,
eff,TM-1(x)=s-1f(x)+o-1rectxD-f(x) * 1ΔrectxΔ,
0=Esc(rs)1-θ2π+CEsc(r) G1(rs, r)nˆ-G1(rs, r) Esc(r)nˆdl+Einc(rs)1-θ2π+CEinc(r) G1(rs, r)nˆ-G1(rs, r) Einc(r)nˆdl
0=Esc(rs)θ2π+CG2(rs, r) Esc(r)nˆ-Esc(r) G2(rs, r)nˆdl,
Esc[r(ξ)]=n=1NEˆnsc(ξ)=n=1N[Enscϕ1(ξ)+En+1scϕ2(ξ)],
Qsc[r(ξ)]=n=1NQˆnsc(ξ)=n=1N[Qnscϕ1(ξ)+Qn+1scϕ2(ξ)].
ϕ1(ξ)=(1-ξ)/2,
ϕ2(ξ)=(1+ξ)/2,
Esc(r)=CQsc(r)G2(r, r)-Esc(r) G2(r, r)nˆdl,r2,
P(e, T)=exp(-e/T),
0=2u2tot(ρ)+β22u2tot(ρ),ρ2,
-f(ρ)=2u1tot(ρ)+β12u1tot(ρ),ρ1,
u2tot(ρ)=CG2(ρ, ρ)u2tot(ρ)-u2tot(ρ) G2(ρ, ρ)nˆ2dl,ρ2,
u1tot(ρ)=u1inc(ρ)+CG1(ρ, ρ)u1tot(ρ)-u1tot(ρ) G1(ρ, ρ)nˆ1dl,ρ1,
u1inc(ρ)=Cf(ρ)G1(ρ, ρ)dl.
Gi(ρ, ρ)=14jHo(2)(βi|ρ-ρ|)=14jHo(2){βi[(x-x)2+(y-y)2]1/2},
i=1, 2,
u1tot(ρ)=u2tot(ρ)utot(ρ),ρC
1p1u1tot(ρ)nˆ1=-1p2u2tot(ρ)nˆ2utot(ρ),
utot(ρ)=uinc(ρ)+usc(ρ).
0=usc(ρ)+Cusc(ρ) G2(ρ, ρ)nˆ-p2G2(ρ, ρ)usc(ρ)dl+uinc(ρ)+Cuinc(ρ) G2(ρ, ρ)nˆ-p2G2(ρ, ρ)uinc(ρ)dl,ρ2,
0=usc(ρ)+Cp1G1(ρ, ρ)usc(ρ)-usc(ρ) G1(ρ, ρ)nˆdl,ρ1.
Cp1G1(ρ, ρ)vinc(ρ)-uinc(ρ) G1(ρ, ρ)nˆdl=0,
ρ1.
0=usc(ρs)1-θ2π+Cusc(ρ) G2(ρs, ρ)nˆ-p2G2(ρs, ρ)usc(ρ)dl+uinc(ρs)1-θ2π+Cuinc(ρ) G2(ρs, ρ)nˆ-p2G2(ρs, ρ)uinc(ρ)dl,
0=usc(ρs)θ2π+Cp1G1(ρs, ρ)usc(ρ)-usc(ρ) G1tot(ρs, ρ)nˆdl,
utot(ρ)=Cutot(ρ) G1(ρ, ρ)nˆ-p1G1(ρ, ρ)utot(ρ)dl,ρ2.

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