Abstract

Multilevel diffractive optical elements are necessary for achieving high-efficiency performance. Here the diffraction efficiency of a multilevel phase-only diffractive lens is analyzed. Approximate, as well as more accurate, approaches are presented. Both plane-wave and Gaussian illumination are discussed. It is shown that for many practical cases the diffraction efficiency can be determined by only a single parameter that takes into account the spatial bandwidth product as well as the focal length of the lens and the illumination wavelength. The analysis is based on the scalar theory and the thin-element approximation. Justification for doing this is presented. The results are valid for lenses with at least F/5.

© 2001 Optical Society of America

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References

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  1. R. W. Wood, Physical Optics (Macmillan, New York, 1934), pp. 37–40.
  2. W. B. Veldkamp, “Wireless focal planes on the road to amacronic sensors,” IEEE J. Quantum Electron. 29, 801–813 (1993).
    [CrossRef]
  3. H. P. Herzig, Micro-optics: Elements, Systems and Applications (Taylor & Francis, London, 1997).
  4. G. J. Swanson, W. B. Veldkamp, “High-efficiency, multilevel, diffractive optical elements,” U.S. Patent#4895790, September21, 1987.
  5. W. H. Welch, J. E. Morris, M. R. Feldman, “Iterative discrete on-axis encoding of radially symmetric computer-generated holograms,” J. Opt. Soc. Am. A 10, 1729–1738 (1993).
    [CrossRef]
  6. M. Kuittinen, H. P. Herzig, “Encoding of efficient diffractive microlenses,” Opt. Lett. 20, 2156–2158 (1995).
    [CrossRef] [PubMed]
  7. J. Fan, D. Zaleta, K. S. Urquhart, S. H. Lee, “Efficient encoding algorithms for computer-aided design of diffractive optical elements by the use of electron-beam fabrication,” Appl. Opt. 34, 2522–2533 (1995).
    [CrossRef] [PubMed]
  8. C. Chen, A. A. Sawchuk, “Nonlinear least-squares and phase-shifting quantization methods for diffractive optical element design,” Appl. Opt. 36, 7297–7306 (1997).
    [CrossRef]
  9. V. Arrizon, S. Sinzinger, “Modified quantization schemes for Fourier-type array generator,” Opt. Commun. 140, 309–315 (1997).
    [CrossRef]
  10. M. G. Moharam, T. K. Gaylord, “Rigorous coupled-wave analysis of planar-grating diffraction,” J. Opt. Soc. Am. 71, 811–818 (1981).
    [CrossRef]
  11. M. G. Moharam, T. K. Gaylord, “Diffraction analysis of dielectric surface-relief gratings,” J. Opt. Soc. Am. 72, 1385–1392 (1982).
    [CrossRef]
  12. E. Noponen, J. Turunen, A. Vasara, “Parametric optimization of multilevel diffractive optical elements by electromagnetic theory,” Appl. Opt. 31, 5910–5912 (1992).
    [CrossRef] [PubMed]

1997 (2)

V. Arrizon, S. Sinzinger, “Modified quantization schemes for Fourier-type array generator,” Opt. Commun. 140, 309–315 (1997).
[CrossRef]

C. Chen, A. A. Sawchuk, “Nonlinear least-squares and phase-shifting quantization methods for diffractive optical element design,” Appl. Opt. 36, 7297–7306 (1997).
[CrossRef]

1995 (2)

1993 (2)

1992 (1)

1982 (1)

1981 (1)

Arrizon, V.

V. Arrizon, S. Sinzinger, “Modified quantization schemes for Fourier-type array generator,” Opt. Commun. 140, 309–315 (1997).
[CrossRef]

Chen, C.

Fan, J.

Feldman, M. R.

Gaylord, T. K.

Herzig, H. P.

M. Kuittinen, H. P. Herzig, “Encoding of efficient diffractive microlenses,” Opt. Lett. 20, 2156–2158 (1995).
[CrossRef] [PubMed]

H. P. Herzig, Micro-optics: Elements, Systems and Applications (Taylor & Francis, London, 1997).

Kuittinen, M.

Lee, S. H.

Moharam, M. G.

Morris, J. E.

Noponen, E.

Sawchuk, A. A.

Sinzinger, S.

V. Arrizon, S. Sinzinger, “Modified quantization schemes for Fourier-type array generator,” Opt. Commun. 140, 309–315 (1997).
[CrossRef]

Swanson, G. J.

G. J. Swanson, W. B. Veldkamp, “High-efficiency, multilevel, diffractive optical elements,” U.S. Patent#4895790, September21, 1987.

Turunen, J.

Urquhart, K. S.

Vasara, A.

Veldkamp, W. B.

W. B. Veldkamp, “Wireless focal planes on the road to amacronic sensors,” IEEE J. Quantum Electron. 29, 801–813 (1993).
[CrossRef]

G. J. Swanson, W. B. Veldkamp, “High-efficiency, multilevel, diffractive optical elements,” U.S. Patent#4895790, September21, 1987.

Welch, W. H.

Wood, R. W.

R. W. Wood, Physical Optics (Macmillan, New York, 1934), pp. 37–40.

Zaleta, D.

Appl. Opt. (3)

IEEE J. Quantum Electron. (1)

W. B. Veldkamp, “Wireless focal planes on the road to amacronic sensors,” IEEE J. Quantum Electron. 29, 801–813 (1993).
[CrossRef]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (1)

Opt. Commun. (1)

V. Arrizon, S. Sinzinger, “Modified quantization schemes for Fourier-type array generator,” Opt. Commun. 140, 309–315 (1997).
[CrossRef]

Opt. Lett. (1)

Other (3)

H. P. Herzig, Micro-optics: Elements, Systems and Applications (Taylor & Francis, London, 1997).

G. J. Swanson, W. B. Veldkamp, “High-efficiency, multilevel, diffractive optical elements,” U.S. Patent#4895790, September21, 1987.

R. W. Wood, Physical Optics (Macmillan, New York, 1934), pp. 37–40.

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Figures (10)

Fig. 1
Fig. 1

Efficiency versus Q based on the approximated approach for a cylindrical lens. Solid curves, complete term; dashed curves, polynomial approximation. (a) Full scale, (b) zoom.

Fig. 2
Fig. 2

Efficiency versus Q based on the approximated approach for a spherical lens. Solid curves, complete term; dashed curves, polynomial approximation. (a) Full scale, (b) zoom.

Fig. 3
Fig. 3

Efficiency versus Q based on the accurate approach for a cylindrical lens (Nmax=4, 8, 16, 32).

Fig. 4
Fig. 4

Efficiency versus Q based on the accurate approach for a spherical lens (Nmax=4, 8, 16, 32).

Fig. 5
Fig. 5

Efficiency versus Q for Gaussian illumination based on the accurate approach for a cylindrical lens (Nmax=4, 8, 16, 32). (a) D/w=0.5, (b) D/w=1, (c) D/w=3, (d) D/w=5.

Fig. 6
Fig. 6

Efficiency versus Q for Gaussian illumination based on the accurate approach for a spherical lens (Nmax=4, 8, 16, 32). (a) D/w=0.5, (b) D/w=1, (c) D/w=3, (d) D/w=5.

Fig. 7
Fig. 7

Error caused by using the approximate approach versus Q and Nmax for a cylindrical lens.

Fig. 8
Fig. 8

Error caused by using the approximate approach versus Q and Nmax for a spherical lens.

Fig. 9
Fig. 9

Diffraction efficiency of a binary grating versus d/λ.

Fig. 10
Fig. 10

TE phase front of the outgoing wave for several values of d/λ. Solid curves, real phase profile; dashed curves, desired binary phase profile.

Equations (31)

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ηN=sin(π/N)π/N2.
Nmax=λ(n-1)R,
ϕ=2πλ [F-(F2-r2)1/2],
|k||sin(θ)|=dϕdr,
sin(θ)=r(F2+r2)1/2rF.
d sin(θ)=λ,
d=λFr.
NmL=dB=λFrB.
ηL(r)=sin(π/NmL)π/NmL2=sin(πBr/λF)πBr/λF2.
η¯=2D0D/2ηL(r)dr=2D0D/2sin(πBr/λF)πBr/λF2dr
η¯=8D20D/2rηL(r)dr=8D20D/2rsin(πBr/λF)πBr/λF2dr
η¯=2λFπBD2cosπBDλF+SiπBDλFπBDλF-1
η¯=2λFπBD2eulergamma-logλFπBD-CiπBDλF
Si(x)=0xsin(t)tdt
Ci(x)=eulergamma+log(x)+0xcos(t)-1tdt.
η¯=2πQ2 [cos(πQ)+Si(πQ)πQ-1]
η¯=2πQ2eulergamma-log1πQ-Ci(πQ).
η¯1-0.27Q2+0.045Q4,
η¯1-0.4Q2+0.068Q4
η¯=NL=NminNmaxγNLηNL,
Nmin=fix2λFBD=fix(2/Q)if2/Q<NmaxNmaxotherwise,
γNL=rminrmaxdr/0D/2dr=2D (rmax-rmin)
γNL=rminrmaxrdr/0D/2rdr=4D2 (rmax2-rmin2)
rmin=0, rmax=D/2, Nmin=NmaxifNL=Nmax2/QλFB(NL+1), rmax=λFBNLif2/Q<NL<NmaxλFB(NL+1), rmax=D/2ifNL2/Q0, rmax=λFBNLotherwise(ifNL=Nmax>2/Q).
γNL=1ifNL=Nmax2/Q(2/Q) 1NL(NL+1)if2/Q<NL<Nmax1-1NL+1 (2/Q)ifNL2/Q1NL (2/Q)otherwise (ifNL=Nmax>2/Q)
γNL=1ifNL=Nmax2/Q(2/Q)22NL+1[NL(NL+1)]2if2/Q<NL<Nmax1-1(NL+1)2 (2/Q)2ifNL2/Q1NL2 (2/Q)2otherwise (ifNL=Nmax>2/Q)
γNL=rminrmaxexp[-2(r/w)2]dr0D/2exp[-2(r/w)2]dr=erf(rmax2/w)-erf(rmin2/w)erf(D/w2),
γNL=rminrmaxr exp[-2(r/w)2]dr0D/2r exp[-2(r/w)2]dr=exp[-2(rmin/w)2]-exp[-2(rmax/w)2]1-exp-12 (D/w)2,
γNL=1ifNL=Nmax2/Qerf2QDw1NL-erf2QDw1NL+1erfDw12if2/Q<NL<Nmax1-erf2QDw1NL+1erfDw12ifNL2/Qerf2QDw1NLerfDw12otherwise(ifNL=Nmax>2/Q),
γNL=1ifNL=Nmax2/Qexp-21QDw1NL+12-exp-21QDw1NL21-exp-12Dw2if2/Q<NL<Nmaxexp-21QDw1NL+12-exp-12Dw21-exp-12Dw2ifNL2/Q1-exp-21QDw1NL21-exp-Dw2otherwise(ifNL=Nmax>2/Q).
error=|ηaccurate-ηapprox|ηaccurate*100.

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