Abstract

A procedure for designing binary diffractive lenses by use of pulse-width-modulated subwavelength features is discussed. The procedure is based on the combination of two approximate theories, effective-medium theory and scalar diffraction theory, and accounts for limitations on feature size and etch depth imposed by fabrication. We use a closed-form expression based on zeroth-order effective-medium theory to map the desired superwavelength phase to the width of a binary subwavelength feature and to examine the requirements imposed by this technique on fabrication and on analysis. Comparisons are also made to more rigorous approaches. In making these comparisons, we show that a trade-off exists between the exactness of the mapping and the fabrication constraints on the minimum feature.

© 1999 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. T. K. Gaylord, M. G. Moharam, “Analysis and applications of optical diffraction by gratings,” Proc. IEEE 73, 894–937 (1985).
    [CrossRef]
  2. M. S. Mirotznik, D. W. Prather, J. N. Mait, “A hybrid finite element-boundary element method for the analysis of diffractive elements,” J. Mod. Opt. 43, 1309–1321 (1996).
    [CrossRef]
  3. 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]
  4. M. S. Mirotznik, J. N. Mait, D. W. Prather, W. A. Beck, “Three-dimensional vector-based analysis of subwavelength diffractive optical elements using the finite-difference-time-domain (FDTD) method,” in Diffractive Optics and Micro-Optics, Vol. 10 of 1998 OSA Technical Digest Series (Optical Society of America, Washington D.C., 1998), pp. 91–93.
  5. F. Nikolajeff, J. Bengtsson, M. Larsson, M. Ekberg, S. Hård, “Measuring and modeling the proximity effect in direct-write electron-beam lithography kinoforms,” Appl. Opt. 34, 897–903 (1995).
    [CrossRef] [PubMed]
  6. W. Stork, N. Streibl, H. Haidner, P. Kipfer, “Artificial distributed-index media fabricated by zero-order gratings,” Opt. Lett. 16, 1921–1923 (1991).
    [CrossRef] [PubMed]
  7. D. H. Raguin, G. M. Morris, “Antireflection structured surfaces for the infrared spectral region,” Appl. Opt. 32, 1154–1167 (1993).
    [CrossRef] [PubMed]
  8. M. W. Farn, “Binary gratings with increased efficiency,” Appl. Opt. 31, 4453–4458 (1992).
    [CrossRef] [PubMed]
  9. H. Haidner, J. T. Sheridan, N. Streibl, “Dielectric binary blazed gratings,” Appl. Opt. 32, 4276–4278 (1993).
    [CrossRef] [PubMed]
  10. 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).
    [CrossRef]
  11. 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]
  12. F. T. Chen, H. G. Craighead, “Diffractive phase elements on two-dimensional artificial dielectrics,” Opt. Lett. 20, 121–123 (1995).
    [CrossRef] [PubMed]
  13. M. E. Warren, R. E. Smith, G. A. Vawter, J. R. Wendt, “High-efficiency subwavelength diffractive optical element in GaAs for 975 nm,” Opt. Lett. 20, 1441–1443 (1995).
    [CrossRef] [PubMed]
  14. J. M. Miller, N. de Beaucoudrey, P. Chavel, E. Cambril, H. Launois, “Synthesis of a subwavelength-pulse-width spatially modulated array illuminator for 0.633 µm,” Opt. Lett. 21, 1399–1401 (1996).
    [CrossRef] [PubMed]
  15. S. Astilean, Ph. Lalanne, P. Chavel, E. Cambril, H. Launois, “High-efficiency subwavelength diffractive element patterned in a high-refractive-index material for 633 nm,” Opt. Lett. 23, 552–554 (1998).
    [CrossRef]
  16. Ph. Lalanne, S. Astilean, P. Chavel, E. Cambril, H. Launois, “Blazed binary subwavelength gratings with efficiencies larger than those of conventional échelette gratings,” Opt. Lett. 23, 1081–1083 (1998).
    [CrossRef]
  17. P. Kipfer, M. Collischon, H. Haidner, J. Schwider, “Subwavelength structures and their use in diffractive optics,” Opt. Eng. 35, 726–731 (1996).
    [CrossRef]
  18. F. T. Chen, H. G. Craighead, “Diffractive lens fabricated with mostly zeroth-order gratings,” Opt. Lett. 21, 177–179 (1996).
    [CrossRef] [PubMed]
  19. M. Schmitz, O. Bryngdahl, “Rigorous concept for the design of diffractive microlenses with high numerical apertures,” J. Opt. Soc. Am. A 14, 901–906 (1997).
    [CrossRef]
  20. D. W. Prather, J. N. Mait, M. S. Mirotznik, J. P. Collins, “Vector-based synthesis of finite, aperiodic subwavelength diffractive optical elements,” J. Opt. Soc. Am. A 15, 1599–1607 (1998).
    [CrossRef]
  21. J. N. Mait, D. W. Prather, M. S. Mirotznik, “Binary subwavelength diffractive lens design,” Opt. Lett. 23, 1343–1345 (1998).
    [CrossRef]
  22. M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980), Chap. 14, pp. 705–708.
  23. P. Lalanne, D. Lemercier-Lalanne, “On the effective medium theory of subwavelength periodic elements,” J. Mod. Opt. 43, 2063–2085 (1996).
    [CrossRef]
  24. L. Li, “New formulation of the Fourier modal method for crossed surface-relief gratings,” J. Opt. Soc. Am. A 14, 2758–2767 (1997).
    [CrossRef]
  25. M. Schmitz, R. Bräuer, O. Bryngdahl, “Phase gratings with subwavelength structures,” J. Opt. Soc. Am. A 12, 2458–2462 (1995).
    [CrossRef]
  26. R. E. Smith, M. E. Warren, J. R. Wendt, G. A. Vawter, “Polarization-sensitive subwavelength antireflection surfaces on a semiconductor for 975 nm,” Opt. Lett. 21, 1201–1203 (1996).
    [CrossRef] [PubMed]
  27. S. Dunn, M. G. Moharam, “Synthesis of high efficiency blazed gratings in two-dimensional binary gratings,” presented at the 1996 Annual Meeting of the Optical Society of America, October 20–25, 1996, Rochester, New York.

1998 (4)

1997 (3)

1996 (6)

M. S. Mirotznik, D. W. Prather, J. N. Mait, “A hybrid finite element-boundary element method for the analysis of diffractive elements,” J. Mod. Opt. 43, 1309–1321 (1996).
[CrossRef]

P. Kipfer, M. Collischon, H. Haidner, J. Schwider, “Subwavelength structures and their use in diffractive optics,” Opt. Eng. 35, 726–731 (1996).
[CrossRef]

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

F. T. Chen, H. G. Craighead, “Diffractive lens fabricated with mostly zeroth-order gratings,” Opt. Lett. 21, 177–179 (1996).
[CrossRef] [PubMed]

R. E. Smith, M. E. Warren, J. R. Wendt, G. A. Vawter, “Polarization-sensitive subwavelength antireflection surfaces on a semiconductor for 975 nm,” Opt. Lett. 21, 1201–1203 (1996).
[CrossRef] [PubMed]

J. M. Miller, N. de Beaucoudrey, P. Chavel, E. Cambril, H. Launois, “Synthesis of a subwavelength-pulse-width spatially modulated array illuminator for 0.633 µm,” Opt. Lett. 21, 1399–1401 (1996).
[CrossRef] [PubMed]

1995 (6)

1993 (2)

1992 (1)

1991 (1)

1985 (1)

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

Astilean, S.

Beck, W. A.

M. S. Mirotznik, J. N. Mait, D. W. Prather, W. A. Beck, “Three-dimensional vector-based analysis of subwavelength diffractive optical elements using the finite-difference-time-domain (FDTD) method,” in Diffractive Optics and Micro-Optics, Vol. 10 of 1998 OSA Technical Digest Series (Optical Society of America, Washington D.C., 1998), pp. 91–93.

Bengtsson, J.

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980), Chap. 14, pp. 705–708.

Bräuer, R.

Bryngdahl, O.

Cambril, E.

Chavel, P.

Chen, F. T.

Collins, J. P.

Collischon, M.

P. Kipfer, M. Collischon, H. Haidner, J. Schwider, “Subwavelength structures and their use in diffractive optics,” Opt. Eng. 35, 726–731 (1996).
[CrossRef]

Craighead, H. G.

de Beaucoudrey, N.

Drabik, T. J.

Dunn, S.

S. Dunn, M. G. Moharam, “Synthesis of high efficiency blazed gratings in two-dimensional binary gratings,” presented at the 1996 Annual Meeting of the Optical Society of America, October 20–25, 1996, Rochester, New York.

Ekberg, M.

Farn, M. W.

Gaylord, T. K.

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

Haidner, H.

Hård, S.

Kipfer, P.

P. Kipfer, M. Collischon, H. Haidner, J. Schwider, “Subwavelength structures and their use in diffractive optics,” Opt. Eng. 35, 726–731 (1996).
[CrossRef]

W. Stork, N. Streibl, H. Haidner, P. Kipfer, “Artificial distributed-index media fabricated by zero-order gratings,” Opt. Lett. 16, 1921–1923 (1991).
[CrossRef] [PubMed]

Lalanne, P.

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

Lalanne, Ph.

Larsson, M.

Launois, H.

Lemercier-Lalanne, D.

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

Li, L.

Mait, J. N.

D. W. Prather, J. N. Mait, M. S. Mirotznik, J. P. Collins, “Vector-based synthesis of finite, aperiodic subwavelength diffractive optical elements,” J. Opt. Soc. Am. A 15, 1599–1607 (1998).
[CrossRef]

J. N. Mait, D. W. Prather, M. S. Mirotznik, “Binary subwavelength diffractive lens design,” Opt. Lett. 23, 1343–1345 (1998).
[CrossRef]

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]

M. S. Mirotznik, D. W. Prather, J. N. Mait, “A hybrid finite element-boundary element method for the analysis of diffractive elements,” J. Mod. Opt. 43, 1309–1321 (1996).
[CrossRef]

M. S. Mirotznik, J. N. Mait, D. W. Prather, W. A. Beck, “Three-dimensional vector-based analysis of subwavelength diffractive optical elements using the finite-difference-time-domain (FDTD) method,” in Diffractive Optics and Micro-Optics, Vol. 10 of 1998 OSA Technical Digest Series (Optical Society of America, Washington D.C., 1998), pp. 91–93.

Miller, J. M.

Mirotznik, M. S.

J. N. Mait, D. W. Prather, M. S. Mirotznik, “Binary subwavelength diffractive lens design,” Opt. Lett. 23, 1343–1345 (1998).
[CrossRef]

D. W. Prather, J. N. Mait, M. S. Mirotznik, J. P. Collins, “Vector-based synthesis of finite, aperiodic subwavelength diffractive optical elements,” J. Opt. Soc. Am. A 15, 1599–1607 (1998).
[CrossRef]

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]

M. S. Mirotznik, D. W. Prather, J. N. Mait, “A hybrid finite element-boundary element method for the analysis of diffractive elements,” J. Mod. Opt. 43, 1309–1321 (1996).
[CrossRef]

M. S. Mirotznik, J. N. Mait, D. W. Prather, W. A. Beck, “Three-dimensional vector-based analysis of subwavelength diffractive optical elements using the finite-difference-time-domain (FDTD) method,” in Diffractive Optics and Micro-Optics, Vol. 10 of 1998 OSA Technical Digest Series (Optical Society of America, Washington D.C., 1998), pp. 91–93.

Moharam, M. G.

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

S. Dunn, M. G. Moharam, “Synthesis of high efficiency blazed gratings in two-dimensional binary gratings,” presented at the 1996 Annual Meeting of the Optical Society of America, October 20–25, 1996, Rochester, New York.

Morris, G. M.

Nikolajeff, F.

Noponen, E.

Prather, D. W.

D. W. Prather, J. N. Mait, M. S. Mirotznik, J. P. Collins, “Vector-based synthesis of finite, aperiodic subwavelength diffractive optical elements,” J. Opt. Soc. Am. A 15, 1599–1607 (1998).
[CrossRef]

J. N. Mait, D. W. Prather, M. S. Mirotznik, “Binary subwavelength diffractive lens design,” Opt. Lett. 23, 1343–1345 (1998).
[CrossRef]

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]

M. S. Mirotznik, D. W. Prather, J. N. Mait, “A hybrid finite element-boundary element method for the analysis of diffractive elements,” J. Mod. Opt. 43, 1309–1321 (1996).
[CrossRef]

M. S. Mirotznik, J. N. Mait, D. W. Prather, W. A. Beck, “Three-dimensional vector-based analysis of subwavelength diffractive optical elements using the finite-difference-time-domain (FDTD) method,” in Diffractive Optics and Micro-Optics, Vol. 10 of 1998 OSA Technical Digest Series (Optical Society of America, Washington D.C., 1998), pp. 91–93.

Raguin, D. H.

Schmitz, M.

Schwider, J.

P. Kipfer, M. Collischon, H. Haidner, J. Schwider, “Subwavelength structures and their use in diffractive optics,” Opt. Eng. 35, 726–731 (1996).
[CrossRef]

Sheridan, J. T.

Smith, R. E.

Stork, W.

Streibl, N.

Turunen, J.

Vawter, G. A.

Warren, M. E.

Wendt, J. R.

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980), Chap. 14, pp. 705–708.

Wyrowski, F.

Zhou, Z.

Appl. Opt. (4)

J. Mod. Opt. (2)

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

M. S. Mirotznik, D. W. Prather, J. N. Mait, “A hybrid finite element-boundary element method for the analysis of diffractive elements,” J. Mod. Opt. 43, 1309–1321 (1996).
[CrossRef]

J. Opt. Soc. Am. A (7)

Opt. Eng. (1)

P. Kipfer, M. Collischon, H. Haidner, J. Schwider, “Subwavelength structures and their use in diffractive optics,” Opt. Eng. 35, 726–731 (1996).
[CrossRef]

Opt. Lett. (9)

F. T. Chen, H. G. Craighead, “Diffractive lens fabricated with mostly zeroth-order gratings,” Opt. Lett. 21, 177–179 (1996).
[CrossRef] [PubMed]

R. E. Smith, M. E. Warren, J. R. Wendt, G. A. Vawter, “Polarization-sensitive subwavelength antireflection surfaces on a semiconductor for 975 nm,” Opt. Lett. 21, 1201–1203 (1996).
[CrossRef] [PubMed]

J. M. Miller, N. de Beaucoudrey, P. Chavel, E. Cambril, H. Launois, “Synthesis of a subwavelength-pulse-width spatially modulated array illuminator for 0.633 µm,” Opt. Lett. 21, 1399–1401 (1996).
[CrossRef] [PubMed]

F. T. Chen, H. G. Craighead, “Diffractive phase elements on two-dimensional artificial dielectrics,” Opt. Lett. 20, 121–123 (1995).
[CrossRef] [PubMed]

M. E. Warren, R. E. Smith, G. A. Vawter, J. R. Wendt, “High-efficiency subwavelength diffractive optical element in GaAs for 975 nm,” Opt. Lett. 20, 1441–1443 (1995).
[CrossRef] [PubMed]

S. Astilean, Ph. Lalanne, P. Chavel, E. Cambril, H. Launois, “High-efficiency subwavelength diffractive element patterned in a high-refractive-index material for 633 nm,” Opt. Lett. 23, 552–554 (1998).
[CrossRef]

Ph. Lalanne, S. Astilean, P. Chavel, E. Cambril, H. Launois, “Blazed binary subwavelength gratings with efficiencies larger than those of conventional échelette gratings,” Opt. Lett. 23, 1081–1083 (1998).
[CrossRef]

J. N. Mait, D. W. Prather, M. S. Mirotznik, “Binary subwavelength diffractive lens design,” Opt. Lett. 23, 1343–1345 (1998).
[CrossRef]

W. Stork, N. Streibl, H. Haidner, P. Kipfer, “Artificial distributed-index media fabricated by zero-order gratings,” Opt. Lett. 16, 1921–1923 (1991).
[CrossRef] [PubMed]

Proc. IEEE (1)

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

Other (3)

S. Dunn, M. G. Moharam, “Synthesis of high efficiency blazed gratings in two-dimensional binary gratings,” presented at the 1996 Annual Meeting of the Optical Society of America, October 20–25, 1996, Rochester, New York.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980), Chap. 14, pp. 705–708.

M. S. Mirotznik, J. N. Mait, D. W. Prather, W. A. Beck, “Three-dimensional vector-based analysis of subwavelength diffractive optical elements using the finite-difference-time-domain (FDTD) method,” in Diffractive Optics and Micro-Optics, Vol. 10 of 1998 OSA Technical Digest Series (Optical Society of America, Washington D.C., 1998), pp. 91–93.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (15)

Fig. 1
Fig. 1

Phase quantization necessary to account for the etch depth.

Fig. 2
Fig. 2

Mappings from the effective index to the index synthesis function: (a) TE polarization, (b) TM polarization. Linear (dashed curve), quadratic (solid curve), and rigorous (dotted curve) mappings are represented.

Fig. 3
Fig. 3

Lens diffraction efficiency as a function of Δ.

Fig. 4
Fig. 4

Binary subwavelength lens design: (a) substrate profile for 2π continuous-phase lens, (b) substrate profile for π continuous-phase lens, (c) index synthesis function, (d) binary subwavelength profile, (e) spatially quantized subwavelength profile.

Fig. 5
Fig. 5

Electric-field amplitude response for the lenses represented in Fig. 4.

Fig. 6
Fig. 6

Determination of minimum feature δ: (a) sampled index synthesis function gm, and encoding in (b) one and (c) two dimensions.

Fig. 7
Fig. 7

Grating deflector design: (a) one-dimensional phase representation, (b) one-dimensional index synthesis function, (c) two-dimensional cylindrical representation of binary, area-modulated element.

Fig. 8
Fig. 8

Same as in Fig. 7, except for linear phase lens design.

Fig. 9
Fig. 9

Same as in Fig. 7, except for quadratic phase lens design.

Fig. 10
Fig. 10

Fabrication requirements for deflectors. Required minimum feature as a function of deflection angle for (a) λ=1 µm, n=1.5; (b) λ=1 µm, n=3; (c) λ=10 µm, n=3. (d)–(f) Space–bandwidth product as a function of deflection angle. The solid curve represents one-dimensional encoding; the dashed curve, two-dimensional encoding.

Fig. 11
Fig. 11

Fabrication requirements for lenses. Required minimum feature as a function of f-number for (a) λ=1 µm, n=1.5; (b) λ=1 µm, n=3; (c) λ=10 µm, n=3. (d)–(f) Space–bandwidth product as a function of f-number. The solid curve represents one-dimensional encoding for a quadratic lens; the dashed curve, one-dimensional encoding for a linear lens; the dotted–dashed curve, two-dimensional encoding for a quadratic lens; the dashed–double-dotted curve, two-dimensional encoding for a linear lens.

Fig. 12
Fig. 12

Spatial quantization necessary to meet fabrication constraints on minimum feature. The original profile is indicated by the dotted curve; the spatially quantized profile, by the solid curve.

Fig. 13
Fig. 13

Sampling surface boundary to generate nodes required for computation.

Fig. 14
Fig. 14

Wavelength density as a function of wavelength.

Fig. 15
Fig. 15

Index synthesis functions for lens designs: curve 1, linear-mapped quadratic lens; curve 2, quadratic-mapped quadratic lens; curve 3, rigorous-mapped quadratic lens; curve 4, linear-mapped linear lens; curve 5, quadratic-mapped linear lens; curve 6, rigorous-mapped linear lens.

Tables (2)

Tables Icon

Table 1 Expressions for Space–Bandwidth Product and Minimum Feature

Tables Icon

Table 2 Binary Subwavelength Lens Design

Equations (52)

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

t(x)=df(x)=dk=1koddKrectx-(zk+1+zk)/2zk+1-zk,
nTE2(x)=(nr2-1)g(x)+1,
nTM-2(x)=(nr-2-1)g(x)+1,
g(x)=f(x) * 1ΔrectxΔ.
nr(x)(nr-1)θ(x)θ0+1,
θ0=2πdn0λ(nr-1).
gTE(x)=nTE2(x)-1nr2-1,
={(nr-1)[θ(x)/θ0]+1}2-1nr2-1,
gTM(x)=nTM-2(x)-1nr-2-1,
={(nr-1)[θ(x)/θ0]+1}-2-1nr-2-1.
gTE(x)=gTM(x)=nr(x)-1nr-1=θ(x)θ0.
f(x)=m=-rectx-mΔ-gmΔ/2gmΔ,
Δsλ=λ2ns,
S=Dδ=MN,
Mmin=ceilDsλ,
Nmin=ceil1dgmin,
dgmin=mminimum|gm-gm-1|,m=[1,  M-1].
L=Mntlnt+Mtlt=MΔ+2Mtd,
J=Lλαns=α2Lsλ,
J<αM2(2d+Δ)sλαD2sλ1+2dsλ=Jub.
σ=JubαD=12sλ1+2dsλ,
δlinδquad=22n0fλ.
θ(x)=2πxW,x=[0, W],
W=λn0 sin ϕ.
gm=2nr+1mM+nr-1nr+1mM2,m=[0, M-1].
dgm=2nr+11M+nr-1nr+12m-1M2,
m=[1, M-1],
dgmin=N-1=2nr+11M+nr-1nr+11M2.
S=MN=nr+12+(nr-1)/MM2
nr+12M2,
δ=WS=2λ(ns+n0)M2 sin ϕ.
Mmin=Wsλ=2nssin ϕ,
S=nr+122nssin ϕ2,
δ=sλ sin ϕns(ns+n0).
θ(x)=2πL+2πfn0λ×1-1+xf2modulo 2π,
x=[-D/2, D/2],
L=ceilfn0λ1+D2 f2-1.
gm=2nr+1L+fn0λ×1-1+mΔf2modulo 1+nr-1nr+1L+fn0λ×1-1+mΔf2modulo 12,
m=[-M/2, M/2],
M=Dsλ.
dgmin=-2nr+1fn0λ1-1+Δf2+nr-1nr+11-1+fn0λ×1-1+Δf22=2 fn0λ1+Δf2-1nrnr+1+nr-1nr+1fn0λ-nr-1nr+1Δn0λ2nrnr+1n0sλ2fλ=sλ(nr+1)2 f.
N=(nr+1)sλ2 f,
S=MN=(nr+1)Dsλ22 f,
δ=DS=sλ2(nr+1)12 f.
θd(x)=2πl=1L1-|x-xl-1|Wlrectx-xl-1Wl.
xl=[2 f(λl/n0)+(λl/n0)2]1/2,l=[0, L],
Wl=xl-xl-1,l=[1, L].
dgmin=2nr+1ΔW1+nr-1nr+1ΔW12,
Nnr+12 W1sλ,
W1=[2 f(λ/n0)+(λ/n0)2]1/2.
S=MN(nr+1)Dsλ2fλ2n0,
δsλ2(nr+1)2n0fλ.

Metrics