Abstract

The intensity distribution of a diffraction pattern in oblique illumination is derived analytically. Experimental results are compared with the results of theoretical analyses. Good agreement between them has been obtained.

© 1998 Optical Society of America

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References

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  1. J. H. McLeod, “The axicon: a new type of optical element,” J. Opt. Soc. Am. 44, 592–597 (1954).
    [CrossRef]
  2. S. Fujiwara, “Optical properties of conic surfaces: I. Reflecting cone,” J. Opt. Soc. Am. 52, 287–292 (1962).
    [CrossRef]
  3. J. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A 4, 651–654 (1987).
    [CrossRef]
  4. J. Durnin, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1479–1501 (1987).
    [CrossRef]
  5. 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]
  6. J. A. Davis, J. Guertin, D. M. Cottrell, “Diffraction-free beams generated with programmable spatial light modulators,” Appl. Opt. 32, 6368–6370 (1993).
    [CrossRef] [PubMed]
  7. R. Arimoto, C. Saloma, T. Tanaka, S. Kawata, “Imaging properties of axicon in a scanning optical system,” Appl. Opt. 31, 6653–6657 (1992).
    [CrossRef] [PubMed]
  8. R. M. Herman, T. A. Wiggius, “Production and uses of diffractionless beams,” J. Opt. Soc. Am. A 8, 932–942 (1991).
    [CrossRef]
  9. M. Born, E. Wolf, Principles of Optics, 2nd ed. (Pergamon, New York, 1964), p. 752.

1993

1992

1991

1989

1987

1962

1954

Arimoto, R.

Born, M.

M. Born, E. Wolf, Principles of Optics, 2nd ed. (Pergamon, New York, 1964), p. 752.

Cottrell, D. M.

Davis, J. A.

Durnin, J.

Friberg, A. T.

Fujiwara, S.

Guertin, J.

Herman, R. M.

Kawata, S.

McLeod, J. H.

Saloma, C.

Tanaka, T.

Turunen, J.

Vasara, A.

Wiggius, T. A.

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 2nd ed. (Pergamon, New York, 1964), p. 752.

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Figures (4)

Fig. 1
Fig. 1

Axicon in oblique illumination.

Fig. 2
Fig. 2

Meaning of R and ξ′.

Fig. 3
Fig. 3

Numerical simulation of diffraction pattern.

Fig. 4
Fig. 4

Experimental results of diffraction at z ≈ 3000 mm in a different oblique illumination angle η.

Equations (38)

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t r = exp - ik n - 1 r θ for r < D / 2 0 for r D / 2 ,
E = exp ikx   sin   η = exp ikr   cos   φ   sin   η ,
E = Et r = exp ikr   cos   φ   sin   η exp - ik n - 1 r θ .
E z = exp ikz i λ z     σ   E   exp i   k 2 z x - x 1 2 + y - y 1 2 d x d y = exp ikz i λ z     σ   E   exp i   k 2 z r 2 + r 1 2 - 2 rr 1 cos φ - ξ r d r d φ ,
E z = exp ikz i λ z     σ exp ikr   cos   φ   sin   η × exp - ik n - 1 r θ × exp i   k 2 z r 2 + r 1 2 - 2 rr 1 cos φ - ξ r d r d φ = exp ikz i λ z 0 D / 2   r 0 2 π exp ikf r ,   φ ;   r 1 ,   ξ d φ × exp i   k 2 z r 2 + r 1 2 - ik n - 1 r θ d r ,
f r ,   φ ;   r 1 ,   ξ = r   cos   φ   sin   η - r 1 z   r   cos φ - ξ .
f r ,   φ ;   r 1 ,   ξ = - r z r 1 cos   ξ - z   sin   η cos   φ - r z   r 1 sin   ξ   sin   φ = - r z   R   cos φ - ξ ,
R = r 1 cos   ξ - z   sin   η 2 + r 1 sin   ξ 2 1 / 2 sin   ξ = r 1 R sin   ξ .
0 2 π exp ix   cos φ - φ 0 d φ = 2 π J 0 x ,
E z = exp ikz i λ z 0 D / 2   r 0 2 π exp - ik   r z   R × cos φ - ξ d φ × exp i   k 2 z r 2 + r 1 2 - ik n - 1 r θ d r = 2 π   exp ikz i λ z 0 D / 2   rJ 0 k   r z   R × exp i   k 2 z r 2 + r 1 2 - ik n - 1 r θ d r .
E z π r p i λ z   J 0 k   r p z   R × exp ikz + i   k 2 z   r 1 2 - ik   r p 2 2 z + i   π 4 ,
r p = n - 1 z θ .
r 1 ,   ξ = z   sin   η ,   0
E z = 1 iz λ   σ   E   exp ikL d σ ,
L = z 2 + x - x 1 2 + y - y 1 2 1 / 2 = z 2 + r 2 + r 1 2 - 2 rr 1 cos φ - ξ 1 / 2
1 + x 1 / 2 1 + 1 2 x - 1 8 x 2 .
L z 1 + 1 2 r 2 + r 1 2 - 2 rr 1 cos φ - ξ z 2 - 1 8 r 4 + r 1 4 + 2 r 2 r 1 2 1 + cos 2 φ - 2 ξ + 2 r 2 r 1 2 - 2 r 2 + r 1 2 2 rr 1 cos φ - ξ z 4 ,
E z = exp ik z + r 1 2 / 2 z - r 1 4 / 8 z 3 iz λ 0 D / 2 × exp ik r 2 / 2 z - r 2 r 1 2 / 2 z 3 - ik n - 1 r θ × 0 2 π exp ik r   sin   η   cos   φ - rr 1 z × 1 - r 2 + r 1 2 2 z 2 cos φ - ξ - r 2 r 1 2 4 z 3   × cos   2 φ - 2 ξ d φ r d r .
H r ,   r 1 ,   ξ = 0 2 π exp ikf r ,   φ ;   r 1 ,   ξ d φ ,
f r ,   φ ;   r 1 ,   ξ = r   sin   η   cos   φ - rr 1 z 1 - r 2 + r 1 2 2 z 2 × cos φ - ξ - r 2 r 1 2 4 z 3 cos 2 φ - 2 ξ .
E z = exp ik z + r 1 2 / 2 z - r 1 4 / 8 z 3 iz λ × 0 D / 2   rH r ,   r 1 ,   ξ exp ik r 2 / 2 z - r 2 r 1 2 / 2 z 3 - ik n - 1 r θ d r .
E z H r p ,   r 1 ,   ξ r p 2 i λ z exp ik z - n - 1 2 θ 2 2   z + r 1 2 / 2 z - n - 1 2 θ 2 2 r 1 2 z - r 1 4 / 8 z 3 + i   π 4 ,
r p = n - 1 θ 1 / z - r 1 2 / z 3 n - 1 z θ .
r 2 + r 1 2 / 2 z 2     1 .
f r ,   φ ;   r 1 ,   ξ r   sin   η   cos   φ - rr 1 z cos φ - ξ - r 2 r 1 2 4 z 3 cos 2 φ - 2 ξ .
φ = φ - ξ
f r ,   φ ;   r 1 ,   ξ = f r ,   φ + ξ ;   r 1 ,   ξ = r   sin   η   cos φ + ξ - rr 1 z cos   φ - r 2 r 1 2 2 z 3 cos 2 φ = - r z r 1 - z   sin   η   cos   ξ cos   φ - r   sin   η   sin   ξ   sin   φ - r 2 r 1 2 4 z 3 cos 2 φ = - r z   R   cos φ - ξ - r 2 r 1 2 4 z 3 cos 2 φ .
R = r 1 - z   sin   η   cos   ξ 2 + z   sin   η   sin   ξ 2 1 / 2 , ξ = arctan z   sin   η   sin   ξ r 1 - z   sin   η   cos   ξ .
r 1 ,   ξ = z   sin   η ,   0 .
H r p ,   r 1 ,   ξ = 0 - ξ 2 π - ξ exp ikf r p ,   φ + ξ ;   r 1 ,   ξ d φ = 0 - ξ 2 π - ξ exp - ik   r p z   R   cos φ - ξ - ik   r p 2 r 1 2 4 z 3 cos 2 φ d φ = 0 2 π exp - ik   r p z   R   cos φ - ξ - ik   r p 2 r 1 2 4 z 3 cos 2 φ d φ ,
H r p ,   r 1 ,   ξ 0 2 π exp - ik   r p z   R   cos φ - ξ - ik   r p 2 sin 2   η 4 z cos 2 φ d φ = 0 2 π exp - ik n - 1 θ R   cos φ - ξ - ikz   n - 1 2 θ 2 sin 2   η 4 cos 2 φ d φ .
H r ,   r 1 ,   ξ = 0 π / 2     d φ + π / 2 π     d φ + π 3 π / 2     d φ + 3 π / 2 2 π     d φ = 2   0 π / 2 cos kr p R z cos φ - ξ × exp - ik   r p 2 r 1 2 4 z 3 cos   2 φ d φ + 2 × 0 π / 2 cos kr p R z sin φ - ξ × exp ik   r p 2 r 1 2 4 z 3 cos   2 φ d φ .
H r p ,   R ,   - ξ = H * r p ,   R ,   ξ , H r p ,   R ,   ξ + π = H r p ,   R ,   ξ , H r p ,   R ,   ξ ± π / 2 = H * r p ,   R ,   ξ , H r p ,   R ,   π / 4 - ξ = H * r p ,   R ,   π / 4 + ξ , H r p ,   R ,   3 π / 4 - ξ = H * r p ,   R ,   3 π / 4 + ξ ,
H r p ,   z   sin   η ,   0 = 2 π J 0 kr p 2 r 1 2 4 z 3 = 2 π J 0 kz n - 1 2 θ 2 sin 2   η 4 .
R     r p sin 2   η 4 = z n - 1 θ   sin 2   η 4 ,
H r p ,   r 1 ,   ξ 2 π J 0 kr p R z = 2 π J 0 k n - 1 R θ ,
θ = 0.01 ,   n = 1.5 ,   k = 2 π λ = 2 π 0.6328 × 10 - 3 .
N = kz   n - 1 2 θ 2 sin   η 4 .

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