Abstract

We present an analysis of axially symmetric diffractive optical elements illuminated by off-axis or oblique incident plane waves. The analysis is performed with a finite-difference time-domain method that has been formulated to exploit axial symmetry yet accommodate off-axis illumination. This approach is compared with a full three-dimensional formulation and is found to be more efficient in both memory requirements and computational time. Validation and applications of this method are presented.

© 2001 Optical Society of America

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References

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    [CrossRef]
  3. M. S. Mirotznik, D. W. Prather, J. N. Mait, W. A. Beck, S. Shi, X. Gao, “Three-dimensional analysis of subwavelength diffractive optical elements with the finite-difference time-domain method,” Appl. Opt. 39, 2871–1880 (2000).
    [CrossRef]
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    [CrossRef]
  6. M. G. Andreasen, “Scattering from bodies of revolution,” IEEE Trans. Antennas Propag. 13, 303–310 (1965).
    [CrossRef]
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    [CrossRef]
  8. A. A. Kishk, L. Shafai, “On the accuracy limits of different integral-equation formulations for numerical solution of dielectric bodies of revolution,” Can. J. Phys. 63, 1532–1539 (1985).
    [CrossRef]
  9. A. A. Kishk, L. Shafai, “Different formulations for numerical solution of single or multibodies of revolution with mixed boundary conditions,” IEEE Trans. Antennas Propag. 34, 666–673 (1986).
    [CrossRef]
  10. D. W. Prather, S. Shi, “Formulation and application of the finite-difference time-domain method for the analysis of axially symmetric DOEs,” J. Opt. Soc. Am. A 16, 1131–1142 (1999).
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  12. A. Taflove, Computational Electromagnetics: The Finite-Difference Time Domain Method (Artech House, Boston, Mass., 1995).
  13. W. C. Chew, W. H. Weedon, “A 3D perfectly matched medium from modified Maxwell’s equations with stretched coordinates,” Microwave Opt. Technol. Lett. 7, 599–604 (1994).
    [CrossRef]
  14. D. B. Davidson, R. W. Ziolkowski, “Body-of-revolution finite-difference time-domain modeling of space-time focusing by a three-dimensional lens,” J. Opt. Soc. Am. A 11, 1471–1490 (1994).
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  15. J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941).

2000

1999

1994

1993

1986

A. A. Kishk, L. Shafai, “Different formulations for numerical solution of single or multibodies of revolution with mixed boundary conditions,” IEEE Trans. Antennas Propag. 34, 666–673 (1986).
[CrossRef]

1985

A. A. Kishk, L. Shafai, “On the accuracy limits of different integral-equation formulations for numerical solution of dielectric bodies of revolution,” Can. J. Phys. 63, 1532–1539 (1985).
[CrossRef]

1979

M. A. Morgan, K. K. Mei, “Finite-element computation of scattering by inhomogeneous penetrable bodies of revolution,” IEEE Trans. Antennas Propag. 27, 202–214 (1979).
[CrossRef]

1977

T. K. Wu, L. L. Tsai, “Scattering from arbitrarily-shaped lossy dielectric bodies of revolution,” Radio Sci. 12, 709–718 (1977).
[CrossRef]

1969

J. R. Mautz, R. F. Harrington, “Radiation and scattering from bodies of revolution,” Appl. Sci. Res. 20, 405–435 (1969).
[CrossRef]

1965

M. G. Andreasen, “Scattering from bodies of revolution,” IEEE Trans. Antennas Propag. 13, 303–310 (1965).
[CrossRef]

Andreasen, M. G.

M. G. Andreasen, “Scattering from bodies of revolution,” IEEE Trans. Antennas Propag. 13, 303–310 (1965).
[CrossRef]

Beck, W. A.

Chew, W. C.

W. C. Chew, W. H. Weedon, “A 3D perfectly matched medium from modified Maxwell’s equations with stretched coordinates,” Microwave Opt. Technol. Lett. 7, 599–604 (1994).
[CrossRef]

Davidson, D. B.

Gallagher, N. C.

Gao, X.

Grann, E. B.

Gremaux, D. A.

Harrington, R. F.

J. R. Mautz, R. F. Harrington, “Radiation and scattering from bodies of revolution,” Appl. Sci. Res. 20, 405–435 (1969).
[CrossRef]

Kishk, A. A.

A. A. Kishk, L. Shafai, “Different formulations for numerical solution of single or multibodies of revolution with mixed boundary conditions,” IEEE Trans. Antennas Propag. 34, 666–673 (1986).
[CrossRef]

A. A. Kishk, L. Shafai, “On the accuracy limits of different integral-equation formulations for numerical solution of dielectric bodies of revolution,” Can. J. Phys. 63, 1532–1539 (1985).
[CrossRef]

Mait, J. N.

Mautz, J. R.

J. R. Mautz, R. F. Harrington, “Radiation and scattering from bodies of revolution,” Appl. Sci. Res. 20, 405–435 (1969).
[CrossRef]

Mei, K. K.

M. A. Morgan, K. K. Mei, “Finite-element computation of scattering by inhomogeneous penetrable bodies of revolution,” IEEE Trans. Antennas Propag. 27, 202–214 (1979).
[CrossRef]

Mirotznik, M. S.

Moharam, M. G.

Morgan, M. A.

M. A. Morgan, K. K. Mei, “Finite-element computation of scattering by inhomogeneous penetrable bodies of revolution,” IEEE Trans. Antennas Propag. 27, 202–214 (1979).
[CrossRef]

Pommet, D. A.

Prather, D. W.

Shafai, L.

A. A. Kishk, L. Shafai, “Different formulations for numerical solution of single or multibodies of revolution with mixed boundary conditions,” IEEE Trans. Antennas Propag. 34, 666–673 (1986).
[CrossRef]

A. A. Kishk, L. Shafai, “On the accuracy limits of different integral-equation formulations for numerical solution of dielectric bodies of revolution,” Can. J. Phys. 63, 1532–1539 (1985).
[CrossRef]

Shi, S.

Stratton, J. A.

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941).

Taflove, A.

A. Taflove, Computational Electromagnetics: The Finite-Difference Time Domain Method (Artech House, Boston, Mass., 1995).

Tsai, L. L.

T. K. Wu, L. L. Tsai, “Scattering from arbitrarily-shaped lossy dielectric bodies of revolution,” Radio Sci. 12, 709–718 (1977).
[CrossRef]

Weedon, W. H.

W. C. Chew, W. H. Weedon, “A 3D perfectly matched medium from modified Maxwell’s equations with stretched coordinates,” Microwave Opt. Technol. Lett. 7, 599–604 (1994).
[CrossRef]

Wu, T. K.

T. K. Wu, L. L. Tsai, “Scattering from arbitrarily-shaped lossy dielectric bodies of revolution,” Radio Sci. 12, 709–718 (1977).
[CrossRef]

Ziolkowski, R. W.

Appl. Opt.

Appl. Sci. Res.

J. R. Mautz, R. F. Harrington, “Radiation and scattering from bodies of revolution,” Appl. Sci. Res. 20, 405–435 (1969).
[CrossRef]

Can. J. Phys.

A. A. Kishk, L. Shafai, “On the accuracy limits of different integral-equation formulations for numerical solution of dielectric bodies of revolution,” Can. J. Phys. 63, 1532–1539 (1985).
[CrossRef]

IEEE Trans. Antennas Propag.

A. A. Kishk, L. Shafai, “Different formulations for numerical solution of single or multibodies of revolution with mixed boundary conditions,” IEEE Trans. Antennas Propag. 34, 666–673 (1986).
[CrossRef]

M. G. Andreasen, “Scattering from bodies of revolution,” IEEE Trans. Antennas Propag. 13, 303–310 (1965).
[CrossRef]

M. A. Morgan, K. K. Mei, “Finite-element computation of scattering by inhomogeneous penetrable bodies of revolution,” IEEE Trans. Antennas Propag. 27, 202–214 (1979).
[CrossRef]

J. Opt. Soc. Am. A

Microwave Opt. Technol. Lett.

W. C. Chew, W. H. Weedon, “A 3D perfectly matched medium from modified Maxwell’s equations with stretched coordinates,” Microwave Opt. Technol. Lett. 7, 599–604 (1994).
[CrossRef]

Opt. Lett.

Radio Sci.

T. K. Wu, L. L. Tsai, “Scattering from arbitrarily-shaped lossy dielectric bodies of revolution,” Radio Sci. 12, 709–718 (1977).
[CrossRef]

Other

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941).

A. Taflove, Computational Electromagnetics: The Finite-Difference Time Domain Method (Artech House, Boston, Mass., 1995).

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Figures (6)

Fig. 1
Fig. 1

Geometry of the computational space used in the off-axis formulation of the BOR FDTD method of analysis.

Fig. 2
Fig. 2

Number of required cylindrical modes as a function of the diameter of a DOE.

Fig. 3
Fig. 3

Comparison of results of the BOR FDTD formulation and analytic methods for a planar dielectric slab.

Fig. 4
Fig. 4

Profiles of binary, eight-level, and subwavelength diffractive lenses.

Fig. 5
Fig. 5

Line scans of the magnitude of the electric fields in the focal plane: (a) binary, (b) eight level, (c) subwavelength.

Fig. 6
Fig. 6

Two-dimensional plots of the magnitude of the electric field distribution in the x, y plane located a focal length from the lens for (a) eight-level, (b) binary, (c) and subwavelength lenses with an off-axis angle of 10°.

Tables (1)

Tables Icon

Table 1 Tabulation of Diffraction Efficiencies and Computational Times for Diffractive Lenses As a Function of Angle of Incidence

Equations (27)

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Ei=(xˆ cos θi-zˆ sin θi)E0 exp(jωt-jk1kˆ·r)=(ρˆ cos θi cos ϕ-ϕˆ cos θi sin ϕ-zˆ sin θi)E0 exp(jωt-jk1kˆ·r),
Hi=yˆ E0(ρ, ϕ)η1exp(jωt-jk1kˆ·r)=-(ρˆ sin ϕ+ϕˆ cos ϕ)(E0/η1)×exp(jωt-jk1kˆ·r),
kˆ=xˆ sin θi+zˆ cos θi
=ρˆ sin θi cos ϕ-ϕˆ sin θi sin ϕ+zˆ cos θi.
f(t, r)=exp(jωt-jk1kˆ·r)
=exp[jωt-jk1(ρ sin θi cos ϕ+z cos θi)]
=exp(jp)m=- j-mJm(s)exp(jmϕ),
Eρi=E0 cos θi exp[jp-j0.5π(m-1)]m=0cm×[Jm-1(s)-Jm+1(s)]cos mϕ,
Eϕi=-E0 cos θi exp[jp-j0.5π(m-1)]m=0cm×[Jm-1(s)-Jm+1(s)]sin mϕ,
Ezi=-2E0 sin θi exp[jp-j0.5πm]m=0cm×Jm(s)cos mϕ,
Hρi=E0η0 exp[jp+j0.5π(m-1)]m=0cm×[Jm-1(s)-Jm+1(s)]sin mϕ,
Hϕi=E0η0 exp[jp+j0.5π(m-1)]m=0cm×[Jm-1(s)-Jm+1(s)]cos mϕ,
Hzi=0,
cm=0.5m=01m0,
Eρi=E0 exp(jωt-jk1z)cos ϕ,
Eϕi=-E0 exp(jωt-jk1z)sin ϕ,
Ezi=0;
Hρi=E0η0 exp(jωt-jk1z)sin ϕ,
Hϕi=E0η0 exp(jωt-jk1z)cos ϕ,
Hzi=0,
M3.27Dn1 sin θi+6,
T3D(D×D×h×S3)×[h/(cΔt)×n1]
TBOR(D/2×h×S2)×m=0Mh/(cΔtm)×n1
T3DTBOR8DSM(M+1).
R=n1 cos θi-cos θtn1 cos θi+cos θt,T=2n1 cos θin1 cos θi+cos θt,
Er=RE0(-xˆ cos θi-zˆ sin θi)exp(-jk1kˆr·r),
Et=TE0(xˆ cos θt-zˆ sin θt)exp(-jk0kˆt·r),

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