Abstract

The axial intensity of axicons illuminated by a coherent wave usually exhibits rapid oscillations from diffraction on the sharp edges of the aperture of the element. These oscillations can be suppressed when the diffractive version of the axicon is illuminated from a polychromatic source. This possibility is examined based on the example of the annular-aperture logarithmic axicon. The estimate for the wavelength interval of the illuminating source required for uniformization is obtained with the help of the stationary-phase method. Furthermore the shape of the radial intensity distribution can be maintained almost unchanged. These findings are confirmed by numerical evaluation of the Fresnel diffraction integral.

© 1996 Optical Society of America

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References

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  1. J. H. McLeod, “Axicon: a new type of optical element,” J. Opt. Soc. Am. 44, 592–597 (1954).
  2. A. Sommerfeld, Optics (Academic, New York, 1964), pp. 213–216.
  3. L. M. Soroko, “Axicons and meso-optical imaging devices,” in Progress in Optics, E. Wolf, ed. (Elsevier, New York, 1989), Vol. 27, pp. 109–160.
  4. J. Sochacki, A. Kolodziejczyk, Z. Jaroszewicz, S. Bará, “Nonparaxial designing of generalized axicons,” Appl. Opt. 31, 5326–5330 (1992).
  5. A. J. Cox, J. D’Anna, “Constant-axial-intensity nondiffracting beam,” Opt. Lett. 17, 232–234 (1992).
  6. R. M. Herman, T. A. Wiggins, “Apodization of diffraction-less beams,” Appl. Opt. 31, 5913–5915 (1992).
  7. Z. Jaroszewicz, J. Sochacki, A. Kolodziejczyk, L. R. Staronski, “Apodised annular-aperture logarithmic axicon: smoothess and uniformity of intensity distributions,” Opt. Lett. 18, 1893–1895 (1993).
  8. G. Roy, R. Tremblay, “Influence of the divergence of a laser beam on the axial intensity distribution of an axicon,” Opt. Commun. 34, 1–3 (1980).
  9. A. J. Cox, D. C. Dibble, “Nondiffracting beam from a spatially filtered Fabry–Perot resonator,” J. Opt. Soc. Am.A 9, 282–286 (1992).
  10. J. Sochacki, Z. Jaroszewicz, L. R. Staronski, A. Kolodziejczyk, “Annular-aperture logarithmic axicon,” J. Opt. Soc. Am.A 10, 1765–1768 (1993).
  11. J. Stamnes, Waves in Focal Regions (Hilger, London, 1986), pp. 91–135.
  12. M. V. Pérez, C. Gómez-Reino, J. M. Cuadrado, “Diffraction patterns and zone plates produced by thin linear axicons,” Opt. Acta 33, 1161–1176 (1986).
  13. J. Fujiwara, “Optical properties of conic surfaces. I. Reflecting cone,” J. Opt. Soc. Am. 52, 287–292 (1962).
  14. 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).

1993 (2)

Z. Jaroszewicz, J. Sochacki, A. Kolodziejczyk, L. R. Staronski, “Apodised annular-aperture logarithmic axicon: smoothess and uniformity of intensity distributions,” Opt. Lett. 18, 1893–1895 (1993).

J. Sochacki, Z. Jaroszewicz, L. R. Staronski, A. Kolodziejczyk, “Annular-aperture logarithmic axicon,” J. Opt. Soc. Am.A 10, 1765–1768 (1993).

1992 (4)

1989 (1)

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).

1986 (1)

M. V. Pérez, C. Gómez-Reino, J. M. Cuadrado, “Diffraction patterns and zone plates produced by thin linear axicons,” Opt. Acta 33, 1161–1176 (1986).

1980 (1)

G. Roy, R. Tremblay, “Influence of the divergence of a laser beam on the axial intensity distribution of an axicon,” Opt. Commun. 34, 1–3 (1980).

1962 (1)

1954 (1)

Bará, S.

Cox, A. J.

A. J. Cox, J. D’Anna, “Constant-axial-intensity nondiffracting beam,” Opt. Lett. 17, 232–234 (1992).

A. J. Cox, D. C. Dibble, “Nondiffracting beam from a spatially filtered Fabry–Perot resonator,” J. Opt. Soc. Am.A 9, 282–286 (1992).

Cuadrado, J. M.

M. V. Pérez, C. Gómez-Reino, J. M. Cuadrado, “Diffraction patterns and zone plates produced by thin linear axicons,” Opt. Acta 33, 1161–1176 (1986).

D’Anna, J.

Dibble, D. C.

A. J. Cox, D. C. Dibble, “Nondiffracting beam from a spatially filtered Fabry–Perot resonator,” J. Opt. Soc. Am.A 9, 282–286 (1992).

Friberg, A. T.

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).

Fujiwara, J.

Gómez-Reino, C.

M. V. Pérez, C. Gómez-Reino, J. M. Cuadrado, “Diffraction patterns and zone plates produced by thin linear axicons,” Opt. Acta 33, 1161–1176 (1986).

Herman, R. M.

Jaroszewicz, Z.

Kolodziejczyk, A.

McLeod, J. H.

Pérez, M. V.

M. V. Pérez, C. Gómez-Reino, J. M. Cuadrado, “Diffraction patterns and zone plates produced by thin linear axicons,” Opt. Acta 33, 1161–1176 (1986).

Roy, G.

G. Roy, R. Tremblay, “Influence of the divergence of a laser beam on the axial intensity distribution of an axicon,” Opt. Commun. 34, 1–3 (1980).

Sochacki, J.

Sommerfeld, A.

A. Sommerfeld, Optics (Academic, New York, 1964), pp. 213–216.

Soroko, L. M.

L. M. Soroko, “Axicons and meso-optical imaging devices,” in Progress in Optics, E. Wolf, ed. (Elsevier, New York, 1989), Vol. 27, pp. 109–160.

Stamnes, J.

J. Stamnes, Waves in Focal Regions (Hilger, London, 1986), pp. 91–135.

Staronski, L. R.

J. Sochacki, Z. Jaroszewicz, L. R. Staronski, A. Kolodziejczyk, “Annular-aperture logarithmic axicon,” J. Opt. Soc. Am.A 10, 1765–1768 (1993).

Z. Jaroszewicz, J. Sochacki, A. Kolodziejczyk, L. R. Staronski, “Apodised annular-aperture logarithmic axicon: smoothess and uniformity of intensity distributions,” Opt. Lett. 18, 1893–1895 (1993).

Tremblay, R.

G. Roy, R. Tremblay, “Influence of the divergence of a laser beam on the axial intensity distribution of an axicon,” Opt. Commun. 34, 1–3 (1980).

Turunen, J.

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).

Vasara, A.

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).

Wiggins, T. A.

Appl. Opt. (2)

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am.A (3)

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).

A. J. Cox, D. C. Dibble, “Nondiffracting beam from a spatially filtered Fabry–Perot resonator,” J. Opt. Soc. Am.A 9, 282–286 (1992).

J. Sochacki, Z. Jaroszewicz, L. R. Staronski, A. Kolodziejczyk, “Annular-aperture logarithmic axicon,” J. Opt. Soc. Am.A 10, 1765–1768 (1993).

Opt. Acta (1)

M. V. Pérez, C. Gómez-Reino, J. M. Cuadrado, “Diffraction patterns and zone plates produced by thin linear axicons,” Opt. Acta 33, 1161–1176 (1986).

Opt. Commun. (1)

G. Roy, R. Tremblay, “Influence of the divergence of a laser beam on the axial intensity distribution of an axicon,” Opt. Commun. 34, 1–3 (1980).

Opt. Lett. (2)

Other (3)

A. Sommerfeld, Optics (Academic, New York, 1964), pp. 213–216.

L. M. Soroko, “Axicons and meso-optical imaging devices,” in Progress in Optics, E. Wolf, ed. (Elsevier, New York, 1989), Vol. 27, pp. 109–160.

J. Stamnes, Waves in Focal Regions (Hilger, London, 1986), pp. 91–135.

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Figures (5)

Fig. 1
Fig. 1

Axial intensity of the annular-aperture logarithmic axicon illuminated by a monochromatic source with λ0 = 600 nm: (a) a numerical evaluation of the diffraction integral, (b) a distribution resulting from the stationary-phase method according to Eqs. (6)(9).

Fig. 2
Fig. 2

Axial intensity of the diffractive annular-aperture logarithmic axicon illuminated by a polychromatic source with λ0 = 600 nm: (a) α = 5.5 nm, (b) σ = 9.7 nm [the vertical line denotes z 0 = 170 mm, the point where according to Eq. (12) uniformization ends], (c) σ = 33.9 nm.

Fig. 3
Fig. 3

Axial intensity of the refractive annular-aperture logarithmic axicon illuminated by a polychromatic source with λ0 = 600 nm and σ = 33.9 nm. The dispersion of the axicon material over the spectral range is neglected.

Fig. 4
Fig. 4

Radial intensity of the annular-aperture logarithmic axicon in the plane z 0 = z min = 2(d 1 a R 1 2 ) = 133.3 mm: (a) a diffractive version illuminated by a monochromatic source with λ0 = 600 nm, (b) a diffractive version illuminated by a polychromatic source with λ0 = 600 nm and σ = 33.9 nm, (c) a refractive version illuminated by a polychromatic source with λ0 = 600 nm and σ = 33.9 nm. The dispersion of the axicon material over the spectral range is neglected.

Fig. 5
Fig. 5

Radial intensity of the annular-aperture logarithmic axicon in the plane z 0 = 111.3 mm: (a) a diffractive version illuminated by a monochromatic source with λ0 = 600 nm, (b) a diffractive version illuminated by a polychromatic source with λ0 = 600 nm and σ= 33.9 nm, (c) a refractive version illuminated by a polychromatic source with λ0 = 600 nm and σ = 33.9 nm. The dispersion of the axicon material over the spectral range is neglected.

Equations (23)

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I ( ρ , z , λ ) = S ( λ ) η ( λ ) ( k z ) 2 R 1 R 2 exp { i k [ r 2 2 z + λ λ 0 φ ( r ) ] } × J 0 ( k r ρ z ) r d r 2 ,
φ ( r ) = 1 2 a ln [ d 1 + a ( r 2 R 1 2 ) ] ,   a = d 2 d 1 R 2 2 R 1 2 .
J = R 1 R 2 g ( r ) exp [ i k f ( r ) ] d r ,
J ~ U S + U E ,
U S = { [ 2 π k f ( r s ) ] 1 / 2 g ( r s ) exp ( i { k f ( r s ) + sgn [ f ( r s ) ] π / 4 } )        for           r s ( R 1 , R 2 ) ,                                                            0 otherwise U E = g ( r ) i k f ( r ) exp [ i k f ( r ) ] R 1 R 2        for            r s R 1 R 2 ,
r s = ( z d 1 a + R 1 2 ) 1 / 2 .
I ( z ) I E l + I E u + I S t + 2 ( I E l I E u ) 1 / 2 cos  Φ E l E u   +   2 ( I S t I E l ) 1 / 2 cos  Φ S t E l   +   2 ( I S t I E u ) 1 / 2 cos  Φ S t E u ,
I E u = d 2 2 / ( d 2 z ) 2 ,   I E l = d 1 2 / ( d 1 z ) 2 ,   I S t = π k / a
Φ E u E l = k 2 a [ ln ( d 2 d 1 ) + d 2 d 1 z ] , Φ S t E l = k 2 a [ ln ( z d 1 ) + d 1 z z ] 3 π / 4 , Φ S t E u = k 2 a [ ln ( z d 2 ) + d 2 z z ] 3 π / 4 .
I ( z ) I E l + I E u + 2 ( I E l I E u ) 1 / 2 cos ( Φ E l E u π ) .
Φ S t E u ( z 0 ) Δ + Φ S t E u ( z 0 ) Δ 2 / 2 = 2 π .
δλ λ 0 z 0 = Δ .
δλ = λ 0 2 d 2 z 0 { [ ( z 0 d 2 ) 2 + 4 a λ 0 z 0 ( 2 d 2 z 0 ) ] 1 / 2 + z 0 d 2 } .
z 0 = 150  mm ( the middle of the focal segment ) δλ = 11  nm , z 0 = 170  mm δλ = 19.4  nm , z 0 = 200  mm ( z 0 = d 2 ,  the end of the focal segment ) δλ = 67.9  nm.
I S t ( ρ , z ) = π k a J 0 2 ( k ρ r s / z ) ,
r s = ( λ z / λ 0 d 1 a + R 1 2 ) 1 / 2 .
ρ 0 D = λ 0 h ( λ z / λ 0 ) = ρ 0 M h ( λ z / λ 0 ) / h ( z ) ,
ρ 0 D = ρ 0 M + ρ 0 M δλ λ 0 z z min 2 z z min ,
ρ 0 R = λ h ( z ) = ρ 0 M + ρ 0 M δλ λ 0 .
S ( λ ) = exp [ ( λ λ 0 ) 2 λ 0 2 σ 2 λ 2 ln   2 ] ,
η ( λ ) = sin c 2 ( λ λ 0 λ ) .
I ( z ) = I ( 0 , z , λ ) ,
I ( ρ , z 0 ) = I ( ρ , z 0 , λ ) d λ ,

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