Abstract

Practical collimating diffractive cylindrical lenses of 2, 4, 8, and 16 discrete levels are analyzed with a sequential application of the two-region formulation of the rigorous electromagnetic boundary-element method (BEM). A Gaussian beam of TE or TM polarization is incident upon the finite-thickness lens. F/4, F/2, and F/1.4 lenses are analyzed and near-field electric-field patterns are presented. The near-field wave-front quality is quantified by its mean-square deviation from a planar wave front. This deviation is found to be less than 0.05 free-space wavelengths. The far-field intensity patterns are determined and compared with the ones predicted by the approximate Fraunhofer scalar diffraction analysis. The diffraction efficiencies determined with the rigorous BEM are found to be generally lower than those obtained with the scalar approximation. For comparison, the performance characteristics of the corresponding continuous Fresnel (continuous profile within a zone but discontinuous at zone boundaries) and continuous refractive lenses are determined by the use of both the BEM and the scalar approximation. The diffraction efficiency of the continuous Fresnel lens is found to be similar to that of the 16-level diffractive lens but less than that of the continuous refractive lens. It is shown that the validity of the scalar approximation deteriorates as the lens f-number decreases.

© 1998 Optical Society of America

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References

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1997 (3)

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]

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 (4)

1994 (4)

1993 (1)

1991 (1)

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. Pt. 2 74, 11–20 (1991).
[CrossRef]

1989 (1)

Bryngdahl, O.

Buralli, D. A.

Crosignani, B.

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

Di Porto, A.

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

Fainman, Y.

Gallagher, N. C.

B. Lichtenberg, N. C. Gallagher, “Numerical modeling of diffractive devices using the finite element method,” Opt. Eng. 33, 3518–3526 (1994).
[CrossRef]

Gaylord, T. K.

Glytsis, E. N.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996).

Grann, E. B.

Herzig, H. P.

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. Pt. 2 74, 11–20 (1991).
[CrossRef]

Ishimaru, A.

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

Kingslake, R.

R. Kingslake, Optical System Design (Academic, Orlando, Fla., 1983), p. 124.

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. Pt. 2 74, 11–20 (1991).
[CrossRef]

Koshiba, M.

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

Kunz, R. E.

Lee, S. H.

Lichtenberg, B.

B. Lichtenberg, N. C. Gallagher, “Numerical modeling of diffractive devices using the finite element method,” Opt. Eng. 33, 3518–3526 (1994).
[CrossRef]

Lohmann, A. W.

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 12, 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]

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

Marchand, P.

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 12, 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.

Montiel, F.

Morris, G. M.

Nevière, M.

Nishihara, H.

H. Nishihara, T. Suhara, “Micro Fresnel lenses,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1987), Vol. 24, pp. 1–40.
[CrossRef]

Noponen, E.

Ozaktas, H. M.

Pommet, D. A.

Popelek, J.

J. Popelek, F. Urban, “The vector analysis of the real diffractive optical elements,” in Nonconventional Optical Imaging Elements, J. Nowak, M. Zajac, eds., Proc. SPIE2169, 89–99 (1994).
[CrossRef]

Prata, 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 12, 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.

Rossi, M.

Schmitz, M.

Siegman, A. E.

A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986), Chap. 20.

Solimeno, S.

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

Suhara, T.

H. Nishihara, T. Suhara, “Micro Fresnel lenses,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1987), Vol. 24, pp. 1–40.
[CrossRef]

Turunen, J.

Urban, F.

J. Popelek, F. Urban, “The vector analysis of the real diffractive optical elements,” in Nonconventional Optical Imaging Elements, J. Nowak, M. Zajac, eds., Proc. SPIE2169, 89–99 (1994).
[CrossRef]

Urey, H.

Urquhart, K. S.

Vasara, A.

Wang, A.

Wilson, D. W.

Appl. Opt. (4)

Electron. Commun. Jpn. Pt. 2 (1)

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. Pt. 2 74, 11–20 (1991).
[CrossRef]

J. Mod. Opt. (1)

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 (9)

Opt. Eng. (1)

B. Lichtenberg, N. C. Gallagher, “Numerical modeling of diffractive devices using the finite element method,” Opt. Eng. 33, 3518–3526 (1994).
[CrossRef]

Other (11)

M. G. Moharam, T. K. Gaylord, J. R. Leger, eds., Feature Issue on Diffractive Optics Modeling, J. Opt. Soc. Am. A12, 1026–1169 (1995).

P. K. Banerjee, R. Butterfield, eds., Developments in Boundary Element Methods (Applied Science, London, 1979).

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

J. Popelek, F. Urban, “The vector analysis of the real diffractive optical elements,” in Nonconventional Optical Imaging Elements, J. Nowak, M. Zajac, eds., Proc. SPIE2169, 89–99 (1994).
[CrossRef]

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996).

H. Nishihara, T. Suhara, “Micro Fresnel lenses,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1987), Vol. 24, pp. 1–40.
[CrossRef]

J. R. Leger, M. G. Moharam, T. K. Gaylord, eds., Feature Issue on Diffractive Optics Applications, Appl. Opt.34, 2399–2559 (1995).

R. Kingslake, Optical System Design (Academic, Orlando, Fla., 1983), p. 124.

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

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

A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986), Chap. 20.

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Figures (8)

Fig. 1
Fig. 1

Geometry of a cylindrical multilevel diffractive lens. The incident Gaussian beam is shown with its minimum waist w 0 at y = f where f is the focal length of the lens. The three regions of interest are also shown with their respective refractive indices n i (i = 1, 2, 3). Boundary Γ1 represents the particular lens profile. Distance t 1 represents the thickness of the substrate, h max is the maximum etched depth of the continuous profile Fresnel diffractive lens. The distance t 2 denotes the distance of the observation plane from the flat boundary Γ2, h(x) represents the etched depth of the diffractive lens, and D is its diameter (width). The radii (widths) R 1 and R 2 represent the calculational grid half-widths for the BEM implementation to boundaries Γ1 and Γ2, respectively. The ideal planar wave fronts are shown in region 3, and L represents the slit width in the observation plane.

Fig. 2
Fig. 2

Total field distribution of ϕ t = E z for the eight-level F/4 diffractive lens for a TE-polarized incident cylindrical Gaussian beam of w 0 = 2.2 μm and f = 200 μm. The instantaneous field shown corresponds to the real part of the complex phasor E z . The amplitudes corresponding to the contours are -0.5 and +0.5 relative to the amplitude of the incident field (ϕ0 = 1.0).

Fig. 3
Fig. 3

Same as Fig. 2, but for the eight-level F/2 lens.

Fig. 4
Fig. 4

Same as Fig. 2, but for the eight-level F/1.4 lens.

Fig. 5
Fig. 5

Wave front of the diffracted wave in a region spanning a half-wavelength between -5 - λ0/2 ≤ y ≤ -5 μm away from the Γ2 boundary for the eight-level F/2 lens. The incident Gaussian beam is TE polarized. The wave front is calculated from the (x, y) positions of the peak values of the instantaneous electric field within the calculational grid. The least-squares fits to a planar wave front are shown with the dotted and the dotted–dashed lines for the -30 μm ≤ x ≤ 30-μm and the -25 μm ≤ x ≤ 25-μm ranges.

Fig. 6
Fig. 6

Plot of the instantaneous field intensity as well as the electric-field phase at a distance of -5.22 μm away from the Γ2 boundary for the eight-level F/2 lens. Again the incident Gaussian beam is TE polarized.

Fig. 7
Fig. 7

Electric-field intensity profile at the observation plane (at distance of t 2′ = 49.986 mm away from Γ2 boundary), calculated by use of the BEM, for a TE-polarized incident Gaussian beam and for all the F/2 lens designs. The continuous Fresnel and the 16-level lens intensities are very close and are almost indistinguishable in this figure.

Fig. 8
Fig. 8

Electric-field intensity at the observation plane (at a distance of t 2′ = 49.986 mm away from Γ2 boundary), calculated by use of the BEM (solid curve) and the scalar approximation with y 0′ = 50.0 mm (dashed curve) for a TE-polarized incident Gaussian beam and for the eight-level F/1.4 lens.

Tables (2)

Tables Icon

Table 1 Collimated Wave-Front Quality Parameters for the Eight-Level F/4, F/2, and F/1.4 Lensesa

Tables Icon

Table 2 Diffraction Efficiencies of F/4, F/2, and F/1.4 Lenses of Two-, Four-, Eight-, 16-Levels, Continuous Fresnel, and Continuous Refractive Profiles

Equations (12)

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-ϕ1tr1+Γ1ϕΓ1rΓ1nˆ12·G1r1, rΓ1-p1G1r1, rΓ1ψΓ1rΓ1d=-ϕincr1,
ϕ2tr2+Γ1ϕΓ1rΓ1nˆ12·G2r2, rΓ1-p2G2r2, rΓ1ψΓ1rΓ1d=0,
ϕincr1=ϕincx, Y=ϕ0wfwY1/2ux×exp-x2w2Y-jk1Y-f-12tan-1YY0+φ+k1x22RY,
DEBEM=PdBEMyPinc,
ϕ3Fx=n1λ0|y0|1/2expjπ4-k0n1|y0|iexp-jθi×xixi+1 ϕincx, 0expj 2πn1xxλ0|y0|dx,
DEF=PdFyPinc=a x |ϕ3Fx|2dxPinc,
θx=k0n1f2+x21/2-f.
hx=n1n2-n1f2+x21/2-f-mλ1,xm|x|minxm+1, D/2,
hxi-i hmaxN=0,  xm|xi|minxm+1, D/2i=1, 2,  , N-1,
f-number=D/22+f21/2D.
hx=cx21+1+n22-n12c2x21/2,
wy=0=24.3 μm=w01+f2Y021/2=w01+f2λ02π2n12w041/2,

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