Abstract

Various integral diffraction methods are systematically unified into a single framework, clearly illustrating the interconnections among the numerous scalar and rigorous formulations. This hierarchical depiction of the integral methods makes clear the specific approximations inherent in each integral method. The scalar methods are compared in detail with a rigorous open-region formulation of the boundary element method (BEM). The rigorous BEM provides a reference method for accurately determining the diffracted fields for both TE and TM incidence. The rigorous BEM and the various scalar methods are then applied to the case of focusing of normally incident plane waves by diffractive cylindrical lenses with f-numbers ranging from f/2 to f/0.5. From the diffracted-field calculations, a number of performance metrics are determined including focal spot size, diffraction efficiency, reflected and transmitted powers, and focal-plane sidelobe power. The quantitative evaluation of the performance of the scalar methods with these metrics allows the establishment, for the first time, of the region of validity of the various scalar methods for this application. As expected, the accuracy of the scalar methods decreases as the f-number of the diffractive lenses is reduced. Additionally, some metrics, particularly the focal-plane sidelobe power, appear to be particularly sensitive to the approximations in the scalar methods, and as a result their accuracy is significantly degraded.

© 1998 Optical Society of America

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References

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    [CrossRef]
  18. E. Noponen, J. Turunen, A. Vasara, “Electromagnetic theory and design of diffractive-lens arrays,” J. Opt. Soc. Am. A 10, 434–443 (1993).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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  26. 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).
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  27. 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]
  28. T. Kojima, J. Ido, “Boundary-element method analysis of light-beam scattering and the sum and differential signal output by DRAW-type optical disk models,” Electron. Commun. Jpn., Part 2: Electron. 74, 11–20 (1991).
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1998

1997

1996

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]

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]

1995

1994

1993

1991

1990

M. Totzeck, B. Kuhlow, “Validity of the Kirchhoff approximation for diffraction by weak phase objects,” Opt. Commun. 78, 13–19 (1990).
[CrossRef]

1989

1988

1973

1966

1965

F. Kottler, “Diffraction at a black screen. Part 1: Kirchhoff’s theory,” Prog. Opt. 4, 281–314 (1965).
[CrossRef]

1964

1962

1954

C. J. Bouwkamp, “Diffraction theory,” Rep. Prog. Phys. 17, 35–55 (1954).
[CrossRef]

Bouwkamp, C. J.

C. J. Bouwkamp, “Diffraction theory,” Rep. Prog. Phys. 17, 35–55 (1954).
[CrossRef]

Bryngdahl, O.

Buralli, D. A.

Crosignani, B.

S. Solimeno, B. Crosignani, P. Di Porto, Guiding, Diffraction, and Confinement of Optical Radiation (Academic, Orlando, Fla., 1986), Chap. 4.

Di Porto, P.

S. Solimeno, B. Crosignani, P. Di Porto, Guiding, Diffraction, and Confinement of Optical Radiation (Academic, Orlando, Fla., 1986), Chap. 4.

Feng, D.

Ganci, S.

Gaylord, T. K.

Glytsis, E. N.

Goodman, J. W.

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

Grann, E. B.

Harrigan, M. E.

Heurtley, J. C.

Hirayama, K.

Ido, J.

T. Kojima, J. Ido, “Boundary-element method analysis of light-beam scattering and the sum and differential signal output by DRAW-type optical disk models,” Electron. Commun. Jpn., Part 2: Electron. 74, 11–20 (1991).
[CrossRef]

Ishimaru, A.

A. Ishimaru, Electromagnetic Wave Propagation, Radiation, and Scattering (Prentice-Hall, Englewood Cliffs, N.J., 1991), Chap. 6.

Kojima, T.

T. Kojima, J. Ido, “Boundary-element method analysis of light-beam scattering and the sum and differential signal output by DRAW-type optical disk models,” Electron. Commun. Jpn., Part 2: Electron. 74, 11–20 (1991).
[CrossRef]

Koppelmann, G.

Koshiba, M.

M. Koshiba, Optical Waveguide Theory by the Finite Element Method (KTK Scientific, Tokyo, 1992), pp. 43–47.

Kottler, F.

F. Kottler, “Diffraction at a black screen. Part 1: Kirchhoff’s theory,” Prog. Opt. 4, 281–314 (1965).
[CrossRef]

Kuhlow, B.

M. Totzeck, B. Kuhlow, “Validity of the Kirchhoff approximation for diffraction by weak phase objects,” Opt. Commun. 78, 13–19 (1990).
[CrossRef]

Larochelle, S.

Layet, B.

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]

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]

Marchand, E. W.

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]

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]

Moharam, M. G.

Morris, G. M.

Mukunda, N.

Noponen, E.

Pommet, D. A.

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]

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]

Rogers, J. R.

Schmitz, M.

Sheng, Y.

Smith, G. S.

G. S. Smith, An Introduction to Classical Electromagnetic Radiation (Cambridge U. Press, Port Chester, N.Y., 1997), Chap. 3.

Solimeno, S.

S. Solimeno, B. Crosignani, P. Di Porto, Guiding, Diffraction, and Confinement of Optical Radiation (Academic, Orlando, Fla., 1986), Chap. 4.

Stamnes, J. J.

J. J. Stamnes, Waves in Focal Regions (Hilger, Boston, 1986), Chaps. 4, 5.

Taghizadeh, M. R.

Totzeck, M.

Turunen, J.

Vasara, A.

Wolf, E.

Appl. Opt.

Electron. Commun. Jpn., Part 2: Electron.

T. Kojima, J. Ido, “Boundary-element method analysis of light-beam scattering and the sum and differential signal output by DRAW-type optical disk models,” Electron. Commun. Jpn., Part 2: Electron. 74, 11–20 (1991).
[CrossRef]

J. Mod. Opt.

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.

J. Opt. Soc. Am. A

M. Totzeck, “Validity of the scalar Kirchhoff and Rayleigh–Sommerfeld diffraction theories in the near field of small phase objects,” J. Opt. Soc. Am. A 8, 27–32 (1991).
[CrossRef]

G. Koppelmann, M. Totzeck, “Diffraction near fields of small phase objects: comparison of 3-cm wave measurements with moment-method calculations,” J. Opt. Soc. Am. A 8, 554–558 (1991).
[CrossRef]

D. A. Pommet, M. G. Moharam, E. B. Grann, “Limits of scalar diffraction theory for diffractive phase elements,” J. Opt. Soc. Am. A 11, 1827–1834 (1994).
[CrossRef]

E. Noponen, J. Turunen, A. Vasara, “Electromagnetic theory and design of diffractive-lens arrays,” J. Opt. Soc. Am. A 10, 434–443 (1993).
[CrossRef]

B. Layet, M. R. Taghizadeh, “Electromagnetic analysis of fan-out gratings and diffractive cylindrical lens arrays by field stitching,” J. Opt. Soc. Am. A 14, 1554–1561 (1997).
[CrossRef]

Y. Sheng, D. Feng, S. Larochelle, “Analysis and synthesis of circular diffractive lens with local linear grating model and rigorous coupled-wave theory,” J. Opt. Soc. Am. A 14, 1562–1568 (1997).
[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]

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, 907–917 (1997).
[CrossRef]

S. Ganci, “Equivalence between two consistent formulations of Kirchhoff’s diffraction theory,” J. Opt. Soc. Am. A 5, 1626–1628 (1988).
[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. 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]

Opt. Commun.

M. Totzeck, B. Kuhlow, “Validity of the Kirchhoff approximation for diffraction by weak phase objects,” Opt. Commun. 78, 13–19 (1990).
[CrossRef]

Prog. Opt.

F. Kottler, “Diffraction at a black screen. Part 1: Kirchhoff’s theory,” Prog. Opt. 4, 281–314 (1965).
[CrossRef]

Rep. Prog. Phys.

C. J. Bouwkamp, “Diffraction theory,” Rep. Prog. Phys. 17, 35–55 (1954).
[CrossRef]

Other

G. S. Smith, An Introduction to Classical Electromagnetic Radiation (Cambridge U. Press, Port Chester, N.Y., 1997), Chap. 3.

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

J. J. Stamnes, Waves in Focal Regions (Hilger, Boston, 1986), Chaps. 4, 5.

A. Ishimaru, Electromagnetic Wave Propagation, Radiation, and Scattering (Prentice-Hall, Englewood Cliffs, N.J., 1991), Chap. 6.

S. Solimeno, B. Crosignani, P. Di Porto, Guiding, Diffraction, and Confinement of Optical Radiation (Academic, Orlando, Fla., 1986), Chap. 4.

M. Koshiba, Optical Waveguide Theory by the Finite Element Method (KTK Scientific, Tokyo, 1992), pp. 43–47.

M. Abramowitz, I. E. Stegun, eds., Handbook of Mathematical Functions, Applied Mathematics Series 55 (National Bureau of Standards, Washington, D.C., 1964), p. 364.

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Figures (7)

Fig. 1
Fig. 1

Geometry associated with the open-region integral equation formulation used in this paper. The boundary Γ divides all space into two open, semi-infinite regions S1 and S2 with refractive indices n1 and n2, respectively. A wave is incident from region S1, and nˆ represents the normal to the boundary Γ. The surface-relief profile of the lens is given by h(x), which is a stepwise function for the multilevel lenses considered. The linear boundaries Γr and Γt are the boundaries used in the scalar methods to determine the reflected and the transmitted fields, respectively.

Fig. 2
Fig. 2

Hierarchical diagram of the various rigorous and scalar integral diffraction methods.

Fig. 3
Fig. 3

Near-field normalized intensity and phase profiles for a two-level f/2 lens at 0.01 wavelength from the transmitted side of the lens. One-half of the lens profile is shown, with the dotted–dashed line being its axis of symmetry. The y dimension is exaggerated for clarity. The plots compare the rigorous BEM (for TE incidence) and three scalar methods, all using exact Green’s functions.

Fig. 4
Fig. 4

Same as Fig. 3 but for an eight-level f/2 lens.

Fig. 5
Fig. 5

Focal-plane normalized intensity profiles for eight-level f/1 and f/0.5 lenses. The same methods as those in Figs. 3 and 4 are compared, and again exact Green’s functions are used. The x axis has been scaled by the diffraction-limited spot size d0.

Fig. 6
Fig. 6

Focal-plane normalized intensity profiles for eight-level f/1 and f/0.5 lenses. Dirichlet boundary conditions are used in all methods, and various propagation methods are compared. The solid curves represent propagation with exact Green’s functions, asymptotic Green’s functions, and the plane-wave spectrum, and the dashed curves correspond to Fresnel propagation.

Fig. 7
Fig. 7

Percent error in the diffraction efficiency calculated by scalar methods as compared with the rigorous BEM for TE or TM incidence. The f/2, f/1, and f/0.5 lenses with 2, 4, 8, and 16 levels have been considered. Exact Green’s functions were used in all cases to propagate the fields. The dotted lines indicate the zero-error point.

Tables (5)

Tables Icon

Table 1 Focal Spot Size for Scalar and Rigorous Methods for Lenses with Various f-Numbers and Numbers of Levels

Tables Icon

Table 2 Diffraction Efficiency for Scalar and Rigorous Methods for Lenses with Various f-Numbers and Numbers of Levels

Tables Icon

Table 3 Diffraction Efficiency under Dirichlet Boundary Conditions with Various Propagation Methods for Lenses with Various f-Numbers and Numbers of Levels

Tables Icon

Table 4 Sidelobe Power for Scalar and Rigorous Methods for Lenses with Various f-Numbers and Numbers of Levels

Tables Icon

Table 5 Transmitted Power for Scalar and Rigorous Methods for Lenses with Various f-Numbers and Numbers of Levels

Equations (49)

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

-ϕ1t(r1)+ΓϕΓ(rΓ)G1n(r1, rΓ)-p1G1(r1, rΓ)ψΓ(rΓ)dl=-ϕinc(r1),r1S1,
ϕ2t(r2)+ΓϕΓ(rΓ)G2n(r2, rΓ)-p2G2(r2, rΓ)ψΓ(rΓ)dl=0,r2S2,
Gi(ri, rΓ)=(-j/4)H0(2)(ki|ri-rΓ|)(i=1, 2),
ϕ1t(rΓ)=ϕ2t(rΓ)ϕΓ(rΓ),
1p1ϕ1tn(rΓ)=1p2ϕ2tn(rΓ)ψΓ(rΓ).
θΓ2π-1ϕΓ(rΓ)+ΓϕΓ(rΓ)G1n(rΓ, rΓ)-p1G1(rΓ, rΓ)ψΓ(rΓ)dl=-ϕinc(rΓ),
θΓ2πϕΓ(rΓ)+ΓϕΓ(rΓ)G2n(rΓ, rΓ)-p2G2(rΓ, rΓ)ψΓ(rΓ)dl=0,
Gi=-j4H0(2)(ki|ri-rΓ|),
Gin=-jki4cos(γ)H1(2)(ki|ri-rΓ|);
Gi-j4exp(jπ/4)2πki|ri-rΓ|×exp(-jki|ri-rΓ|),
Ginki4cos(γ)exp(jπ/4)2πki|ri-rΓ|×exp(-jki|ri-rΓ|);
Gi-j4exp(jπ/4)2πki|ri-rΓ| exp(-jki|yi-yΓ|)×exp-jki(xi-xΓ)22|yi-yΓ|,
Ginki4cos(γ)exp(jπ/4)2πki|ri-rΓ|×exp(-jki|yi-yΓ|)×exp-jki(xi-xΓ)22|yi-yΓ|;
Gi-j4exp(jπ/4)2πki|ri-rΓ| exp(-jki|yi-yΓ|)×exp-jkixi22|yi-yΓ|expjkixixΓ|yi-yΓ|,
Ginki4cos(γ)exp(jπ/4)2πki|ri-rΓ|×exp(-jki|yi-yΓ|)×exp-jkixi22|yi-yΓ|expjkixixΓ|yi-yΓ|,
ϕ1K(r1)=ϕ1inc(r1)+ΓrϕΓrinc(rΓr)G1n(r1, rΓr)-p1G1(r1, rΓr)ψΓrinc(rΓr)dl,r1S1,
ϕ2K(r2)=-ΓϕΓtinc(rΓt)G2n(r2, rΓt)-p2G2(r2, rΓt)ψΓtinc(rΓt)dl,r2S2.
ϕΓrinc(rΓr)=Rϕ0w(x)exp[-jδ(x)]ϕΓrincn(rΓr)=-jk1Rϕ0w(x)exp[-jδ(x)],rΓr,
ϕΓtinc(rΓt)=Tϕ0w(x)exp[-jΔ(x)]ϕΓtincn(rΓt)=jk2Tϕ0w(x)exp[-jΔ(x)],rΓt,
δ(x)=-2k0n1h(x),
Δ(x)=k0(n2-n1)h(x).
w(x)=1,0|x|D/2-lcos2|x|-D/2+l4lπ,D/2-l|x|D/2+l,0,D/2+l|x|<
GiRS1(ri, ri)=Gi(ri, ri)-Gi(ri, rj),
GiRS2(ri, ri)=Gi(ri, ri)+Gi(ri, rj),
GiRS1=0,GiRS1n=2GinGiRS2=2Gi,GiRS2n=0forr1=r2=rΓs,
ϕ1RS1(r1)=ϕ1inc(r1)+ΓrϕΓrinc(rΓr)G1RS1n(r1, rΓr)dl,  r1S1,
ϕ2RS1(r2)=-ΓtϕΓtinc(rΓt)G2RS1n(r2, rΓt)dl,
r2S2,
ϕ1RS2(r1)=ϕ1inc(r1)-Γr[p1G1RS2(r1, rΓr)ψΓrinc(rΓr)]dl,
r1S1,
ϕ2RS2(r2)=Γt[p2G2RS2(r2, rΓt)ψΓtinc(rΓt)]dl,
r2S2,
Φ(kx)=-ϕΓsinc(rΓs)exp(jkxx)dx,
ϕ1PW1(r1)=ϕ1inc(r1)+12π-Φ(kx)×exp[-j(kxx+kyy)]dkx,
ϕ2PW1(r2)=12π-Φ(kx)exp[-j(kxx+kyy)]dkx,
ky=±ki2-kx2forkx2ki2jkx2-ki2forkx2>ki2,
Φ(kx)=-jkyΦ(kx),
Φ(kx)=-ϕincn(rΓs)exp(jkxx)dx.
ϕ1PW2(r1)=ϕ1inc(r1)+12π--Φ(kx)jky×exp[-j(kxx+kyy)]dkx,
ϕ2PW2(r2)=12π--Φ(kx)jky×exp[-j(kxx+kyy)]dkx.
Ai(ρn)=1Mm=-M/2M/2-1Ezis(mΔx, yi)exp(jρnmΔx),
Ps=ReL2ηim=-M/2M/2-1βim*ki|Ai(ρm)|2,
Pf=Red2η2m=-M/2M/2-1n=-M/2M/2-1β2m*k2[A2(ρm)]*×A2(ρn)sinc[(ρm-ρn)d/2],
θ(x)=k0n2(f-f2+x2).
h(x)=n2n1-n2(f2+x2-f-mλ2),
xm|x|min(xm+1, D/2),
h(xi)-ihmaxN=0,xm|x|min(xm+1, D/2)
(i=1, 2, , N-1),
AverageValue=2 fn ln12 fn+12 fn2+1,

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