Abstract

The generation of narrow Bessel beams by diffractive axicons for applications such as high-precision alignment requires resonance-domain circular-grating structures. We apply rigorous grating theory locally to determine the amplitudes and phases of all scalar components of the electromagnetic field immediately behind the diffractive element. We then employ the angular spectrum representation to compute far-zone patterns of the diffracted field and the Rayleigh–Sommerfeld formula to determine the field at finite distances in the neighborhood of the optical axis. It is shown that linearly polarized illumination causes considerable circular asymmetry in the diffraction pattern in both cases. This can be avoided by the use of either circularly polarized or unpolarized illumination.

© 1997 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  8. S. R. Mishra, “A vector wave analysis of a Bessel beam,” Opt. Commun. 85, 159–161 (1991).
    [CrossRef]
  9. J. Turunen, A. T. Friberg, “Self-imaging and propagation-invariance in electromagnetic fields,” Pure Appl. Opt. 2, 51–60 (1993).
    [CrossRef]
  10. J. Turunen, A. T. Friberg, “Electromagnetic theory of reflaxicon beams,” Pure Appl. Opt. 2, 539–547 (1993).
    [CrossRef]
  11. Z. Bouchal, M. Olivı́k, “Non-diffractive vector Bessel beams,” J. Mod. Opt. 42, 1555–1566 (1995).
    [CrossRef]
  12. R. Petit, ed., Electromagnetic Theory of Gratings (Springer-Verlag, Berlin, 1980).
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  18. E. Noponen, J. Turunen, A. Vasara, “Parametric optimization of multilevel diffractive optical elements by electromagnetic theory,” Appl. Opt. 31, 5010–5012 (1992).
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    [CrossRef]
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    [CrossRef]
  24. L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U., Cambridge, 1995), Sec. 3.3.3.
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    [CrossRef]

1996 (2)

1995 (1)

Z. Bouchal, M. Olivı́k, “Non-diffractive vector Bessel beams,” J. Mod. Opt. 42, 1555–1566 (1995).
[CrossRef]

1994 (1)

D. F. V. James, “Polarization of light radiated by black-body sources,” Opt. Commun. 109, 209–214 (1994).
[CrossRef]

1993 (2)

J. Turunen, A. T. Friberg, “Self-imaging and propagation-invariance in electromagnetic fields,” Pure Appl. Opt. 2, 51–60 (1993).
[CrossRef]

J. Turunen, A. T. Friberg, “Electromagnetic theory of reflaxicon beams,” Pure Appl. Opt. 2, 539–547 (1993).
[CrossRef]

1992 (3)

1991 (1)

S. R. Mishra, “A vector wave analysis of a Bessel beam,” Opt. Commun. 85, 159–161 (1991).
[CrossRef]

1989 (2)

1988 (1)

1987 (2)

J. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A 4, 651–654 (1987).
[CrossRef]

J. Durnin, J. J. Miceli, J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[CrossRef] [PubMed]

1985 (1)

1980 (1)

1978 (1)

1975 (2)

O. Bryngdahl, “Computer-generated holograms as generalized optical components,” Opt. Eng. 14, 426–435 (1975).
[CrossRef]

M. Lax, W. H. Louisell, W. B. Knight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[CrossRef]

1972 (1)

1954 (1)

Bouchal, Z.

Z. Bouchal, M. Olivı́k, “Non-diffractive vector Bessel beams,” J. Mod. Opt. 42, 1555–1566 (1995).
[CrossRef]

Bryngdahl, O.

O. Bryngdahl, “Computer-generated holograms as generalized optical components,” Opt. Eng. 14, 426–435 (1975).
[CrossRef]

Carter, W. H.

Collins, G.

Durnin, J.

J. Durnin, J. J. Miceli, J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[CrossRef] [PubMed]

J. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A 4, 651–654 (1987).
[CrossRef]

Eberly, J. H.

J. Durnin, J. J. Miceli, J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[CrossRef] [PubMed]

Friberg, A. T.

J. Turunen, A. T. Friberg, “Electromagnetic theory of reflaxicon beams,” Pure Appl. Opt. 2, 539–547 (1993).
[CrossRef]

J. Turunen, A. T. Friberg, “Self-imaging and propagation-invariance in electromagnetic fields,” Pure Appl. Opt. 2, 51–60 (1993).
[CrossRef]

A. T. Friberg, T. Jaakkola, J. Tuovinen, “Electromagnetic Gaussian beam beyond the paraxial regime,” IEEE Trans. Antennas Propag. 40, 984–989 (1992).
[CrossRef]

A. Vasara, J. Turunen, A. T. Friberg, “Realization of general nondiffracting beams with computer-generated holograms,” J. Opt. Soc. Am. A 6, 1748–1754 (1989).
[CrossRef] [PubMed]

J. Turunen, A. Vasara, A. T. Friberg, “Holographic generation of diffraction-free beams,” Appl. Opt. 27, 3959–3962 (1988).
[CrossRef] [PubMed]

Fukumitsu, O.

Granet, G.

Guizal, B.

Ichikawa, H.

Indebetouw, G.

Jaakkola, T.

James, D. F. V.

D. F. V. James, “Polarization of light radiated by black-body sources,” Opt. Commun. 109, 209–214 (1994).
[CrossRef]

Khoo, I.

Knight, W. B.

M. Lax, W. H. Louisell, W. B. Knight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[CrossRef]

Knop, K.

Kuisma, S.

Lalanne, Ph.

Lax, M.

M. Lax, W. H. Louisell, W. B. Knight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[CrossRef]

Leith, E. N.

Louisell, W. H.

M. Lax, W. H. Louisell, W. B. Knight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[CrossRef]

Mandel, L.

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U., Cambridge, 1995), Sec. 3.3.3.

McLeod, J. H.

Miceli, J. J.

J. Durnin, J. J. Miceli, J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[CrossRef] [PubMed]

Miller, J. M.

Mishra, S. R.

S. R. Mishra, “A vector wave analysis of a Bessel beam,” Opt. Commun. 85, 159–161 (1991).
[CrossRef]

Morris, G. M.

Noponen, E.

Olivi´k, M.

Z. Bouchal, M. Olivı́k, “Non-diffractive vector Bessel beams,” J. Mod. Opt. 42, 1555–1566 (1995).
[CrossRef]

Stamnes, J. J.

J. J. Stamnes, Waves in Focal Regions (Hilger, Bristol, 1986).

Taghizadeh, M. R.

Takenaka, T.

Tuovinen, J.

A. T. Friberg, T. Jaakkola, J. Tuovinen, “Electromagnetic Gaussian beam beyond the paraxial regime,” IEEE Trans. Antennas Propag. 40, 984–989 (1992).
[CrossRef]

Turunen, J.

Vasara, A.

Westerholm, J.

Wolf, E.

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U., Cambridge, 1995), Sec. 3.3.3.

Wynn, T.

Yokota, M.

Appl. Opt. (3)

IEEE Trans. Antennas Propag. (1)

A. T. Friberg, T. Jaakkola, J. Tuovinen, “Electromagnetic Gaussian beam beyond the paraxial regime,” IEEE Trans. Antennas Propag. 40, 984–989 (1992).
[CrossRef]

J. Mod. Opt. (1)

Z. Bouchal, M. Olivı́k, “Non-diffractive vector Bessel beams,” J. Mod. Opt. 42, 1555–1566 (1995).
[CrossRef]

J. Opt. Soc. Am. (4)

J. Opt. Soc. Am. A (6)

Opt. Commun. (2)

S. R. Mishra, “A vector wave analysis of a Bessel beam,” Opt. Commun. 85, 159–161 (1991).
[CrossRef]

D. F. V. James, “Polarization of light radiated by black-body sources,” Opt. Commun. 109, 209–214 (1994).
[CrossRef]

Opt. Eng. (1)

O. Bryngdahl, “Computer-generated holograms as generalized optical components,” Opt. Eng. 14, 426–435 (1975).
[CrossRef]

Phys. Rev. A (1)

M. Lax, W. H. Louisell, W. B. Knight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[CrossRef]

Phys. Rev. Lett. (1)

J. Durnin, J. J. Miceli, J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[CrossRef] [PubMed]

Pure Appl. Opt. (2)

J. Turunen, A. T. Friberg, “Self-imaging and propagation-invariance in electromagnetic fields,” Pure Appl. Opt. 2, 51–60 (1993).
[CrossRef]

J. Turunen, A. T. Friberg, “Electromagnetic theory of reflaxicon beams,” Pure Appl. Opt. 2, 539–547 (1993).
[CrossRef]

Other (3)

R. Petit, ed., Electromagnetic Theory of Gratings (Springer-Verlag, Berlin, 1980).

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U., Cambridge, 1995), Sec. 3.3.3.

J. J. Stamnes, Waves in Focal Regions (Hilger, Bristol, 1986).

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Figures (7)

Fig. 1
Fig. 1

Geometrical configuration: generation of a Bessel beam by a diffractive axicon. The propagation-invariant range associated with the mth diffraction order is denoted by Lm.

Fig. 2
Fig. 2

Local coordinate system for application of rigorous diffraction theory to a circular grating. The incident electric field is assumed to point in the x direction.

Fig. 3
Fig. 3

Radiant intensity distributions of fields generated by binary diffractive axicons with (a) d/λ=1.5, (b) d/λ=2.5, and (c) d/λ=5.0. In all cases R1=5λ, R2=40λ, and the axicon is illuminated by a linearly polarized unit-amplitude plane wave. Solid curves, cross section in the x direction (sy=0); dashed curves, cross section in y direction (sx=0); dotted curves, cross section at 45° with respect to the x axis (sx=sy).

Fig. 4
Fig. 4

Same as Fig. 3 but for four-level diffractive axicons with optimized surface profile.

Fig. 5
Fig. 5

Axial profiles of electric energy density generated by diffractive axicons with (a) d/λ=1.5, (b) d/λ=2.5, and (c) d/λ=5.0. Here again R1=5λ, R2=40λ, and linearly polarized illumination is assumed. Solid curves, binary elements; dashed curves, four-level elements.

Fig. 6
Fig. 6

Transverse profiles of electric energy density generated by a diffractive axicon with d/λ=5 across the planes (a) z=40λ, (b) z=100λ, and (c) z=150λ (linear polarization).

Fig. 7
Fig. 7

(a) Transverse profile of electric energy density generated by a diffractive axicon with d/λ=1.5 at z=40λ (linear polarization); (b) cross sections in the x direction (solid curves) and the y direction (dashed curves); (c) cross sections of circularly symmetric fields obtained with circularly polarized or unpolarized illumination (there is no appreciable difference between these two cases).  

Tables (2)

Tables Icon

Table 1 Complex Amplitudes Tm for Some Binary Gratings

Tables Icon

Table 2 Complex Amplitudes Tm for Some Four-Level Gratings

Equations (42)

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Ein(x, y, z)=xˆA(ρ)exp[ikns(z+h)]
zmj=Rj[(d/mλ)2-1]1/2,
ηmTE=ns-1 cos θm|TmTE|2,
ηmTM=ns cos θm|TmTM|2,
Ein(ρ, ϕ, z)=ρˆEρin(ρ, ϕ, z)+ϕˆEϕin(ρ, ϕ, z),
Eρin(ρ, ϕ, z)=A(ρ)cos ϕ exp[ikns(z+h)],
Eϕin(ρ, ϕ, z)=-A(ρ)sin ϕ exp[ikns(z+h)],
Eϕtr(ρ, ϕ, z)=Eϕin(ρ, ϕ, -h) m=-TmTE ×exp(i2πmρ/d)exp(itmz),
tm=[k2-(2πm/d)2]1/2when|m|<d/λi[(2πm/d)2-k2]1/2otherwise.
Hϕin(ρ, ϕ, z)=YnsA(ρ)cos ϕ exp[ikns(z+h)]
Hϕtr(ρ, ϕ, z)=Hϕin(ρ, ϕ, -h)m=-TmTM ×exp(i2πmρ/d)exp(itmz).
Eρtr(ρ, ϕ, z)=(Yk)-1Hϕin(ρ, ϕ, -h)m=-tmTmTM ×exp(i2πmρ/d)exp(itmz),
Eztr(ρ, ϕ, z)=-λ(Yd)-1Hϕin(ρ, ϕ, -h)m=-mTmTM ×exp(i2πmρ/d)exp(itmz).
Extr(ρ, ϕ, z)=Eρtr(ρ, ϕ, z)cos ϕ-Eϕtr(ρ, ϕ, z)sin ϕ,
Eytr(ρ, ϕ, z)=Eρtr(ρ, ϕ, z)sin ϕ+Eϕtr(ρ, ϕ, z)cos ϕ.
Extr(ρ, ϕ, 0)=A(ρ)m=-(k-1nstmTmTM cos2 ϕ+TmTE sin2 ϕ)exp(i2πmρ/d),
Eytr(ρ, ϕ, 0)=A(ρ)sin ϕ cos ϕm=-(k-1nstmTmTM-TmTE)exp(i2πmρ/d),
Eztr(ρ, ϕ, 0)=-A(ρ)nsλd-1 cos ϕm=-mTmTM ×exp(i2πmρ/d),
Hxtr(ρ, ϕ, 0)=A(ρ)Yk-1 sin ϕ cos ϕm=-(tmTmTE-knsTmTM)exp(i2πmρ/d),
Hytr(ρ, ϕ, 0)=A(ρ)Yk-1m=-(tmTmTE sin2 ϕ+knsTmTM cos2 ϕ)exp(i2πmρ/d),
Hztr(ρ, ϕ, 0)=-A(ρ)Yλd-1 sin ϕm=-mTmTE ×exp(i2πmρ/d).
we(r, t)=14 0|E(r)|2
S(r, t)=12 Re[E(r)×H*(r)]
U(x, y, z)=-12π -U(x, y, 0) zR×ik-1R exp(ikR)R dxdy,
Ex(r)=-Ax(p, q)exp[ik(px+qy+tz)]dpdq,
Ey(r)=-Ay(p, q)exp[ik(px+qy+tz)]dpdq,
Ez(r)=--t-1[pAx(p, q)+qAy(p, q)]×exp[ik(px+qy+tz)]dpdq,
Hx(r)=-Y-t-1[pqAx(p, q)+(1-p2)Ay(p, q)]×exp[ik(px+qy+tz)]dpdq,
Hy(r)=Y-t-1[(1-q2)Ax(p, q)+pqAy(p, q)]×exp[ik(px+qy+tz)]dpdq,
Hz(r)=-Y-[qAx(p, q)-pAy(p, q)]×exp[ik(px+qy+tz)]dpdq,
t=(1-p2-q2)1/2whenp2+q21i(p2+q2-1)1/2otherwise,
Ax(p, q)=1λ2 -Ex(x, y, 0)×exp[-ik(px+qy)]dxdy,
Ay(p, q)=1λ2 -Ey(x, y, 0)×exp[-ik(px+qy)]dxdy.
Ex(rs)=Ax(sx, sy)F(sz, r),
Ey(rs)=Ay(sx, sy)F(sz, r),
Ez(rs)=-sz-1[sxAx(sx, sy)+syAy(sx, sy)]F(sz, r),
Hx(rs)=-Ysz-1[sxsyAx(sx, sy)+(1-sx2)Ay(sx, sy)]F(sz, r),
Hy(rs)=Ysz-1[(1-sy2)Ax(sx, sy)+sxsyAy(sx, sy)]F(sz, r),
Hz(rs)=-Y[syAx(sx, sy)-sxAy(sx, sy)]F(sz, r),
F(sz, r)=2πsz exp(ikr)/(ikr).
J(s)=r2|S(rs, t)|whenr,
J(s)=12 Yλ2{sz2[|Ax(sx, sy)|2+|Ay(sx, sy)|2]+|sxAx(sx, sy)+syAy(sx, sy)|2},

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