Abstract

Introduction of an annular aperture into an axicon characterized by the logarithmic phase-retardation function, resulted in significant improvement in the uniformity of both lateral resolution and energy flow in comparison with the effect of a full-aperture element. The improved uniformity of the new element was gained at the expense of some efficiency.

© 1993 Optical Society of America

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References

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  1. L. M. Soroko, “Axicons and meso-optical imaging devices,” in Progress in Optics, Vol. XXVII, E. Wolf, ed. (Elsevier, New York, 1989), pp. 109–160.
    [CrossRef]
  2. V. P. Koronkevitch, I. G. Palchikova, “Kinoforms with increased depth of focus,” Optik 87, 91–93 (1991).
  3. J. Sochacki, A. Kołodziejczyk, Z. Jaroszewicz, S. Bará, “Nonparaxial designing of generalized axicons,” Appl. Opt. 31, 5326–5330 (1992).
    [CrossRef] [PubMed]
  4. J. Sochacki, S. Bará, Z. Jaroszewicz, A. Kołodziejczyk, “Phase retardation of the uniform-intensity axilens,” Opt. Lett. 17, 7–9 (1992).
    [CrossRef] [PubMed]
  5. L. R. Staroński, J. Sochacki, Z. Jaroszewicz, A. Kołodziejczyk, “Lateral distribution and flow of energy in uniform-intensity axicons,” J. Opt. Soc. Am. A 9, 2091–2094 (1992).
    [CrossRef]
  6. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), pp. 13 and 63.
  7. M. Abramowitz, I. A. Stegun, eds., Handbook of Mathematical Functions (Dover, New York, 1972), pp. 355–389.
  8. Another interesting coincidence can be noticed for the radii of the relevant Bessel beams. They are equal to rB1′ = 6.06 μm, rB2′ = 7.27 μm, and rB3′ = 9.69 μm for the focal segment lengths equal to d2= 125 mm, 150 mm, and 200 mm, respectively. In all three cases R= 5 mm.
  9. Such an approach has been applied in the analysis of the Bessel beams: see A. J. Cox, J. D’Anna, “Constant-axial-intensity nondiffracting beam,” Opt. Lett. 17, 232–234 (1992).
    [CrossRef] [PubMed]

1992

1991

V. P. Koronkevitch, I. G. Palchikova, “Kinoforms with increased depth of focus,” Optik 87, 91–93 (1991).

Bará, S.

Cox, A. J.

D’Anna, J.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), pp. 13 and 63.

Jaroszewicz, Z.

Kolodziejczyk, A.

Koronkevitch, V. P.

V. P. Koronkevitch, I. G. Palchikova, “Kinoforms with increased depth of focus,” Optik 87, 91–93 (1991).

Palchikova, I. G.

V. P. Koronkevitch, I. G. Palchikova, “Kinoforms with increased depth of focus,” Optik 87, 91–93 (1991).

Sochacki, J.

Soroko, L. M.

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

Staronski, L. R.

Appl. Opt.

J. Opt. Soc. Am. A

Opt. Lett.

Optik

V. P. Koronkevitch, I. G. Palchikova, “Kinoforms with increased depth of focus,” Optik 87, 91–93 (1991).

Other

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

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), pp. 13 and 63.

M. Abramowitz, I. A. Stegun, eds., Handbook of Mathematical Functions (Dover, New York, 1972), pp. 355–389.

Another interesting coincidence can be noticed for the radii of the relevant Bessel beams. They are equal to rB1′ = 6.06 μm, rB2′ = 7.27 μm, and rB3′ = 9.69 μm for the focal segment lengths equal to d2= 125 mm, 150 mm, and 200 mm, respectively. In all three cases R= 5 mm.

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Figures (6)

Fig. 1
Fig. 1

On-axis intensity distribution for annular-aperture logarithmic forward axicons with d2 = 125 mm and ρ = 4 mm (curve 1), d2 = 150 mm and ρ = 3.333 mm (curve 2), and d2 = 200 mm and ρ = 2.5 mm (curve 3). In all three cases R = 5 mm and d1 = 100 mm.

Fig. 2
Fig. 2

Lateral-intensity distribution at the plane z = d1 for the annular-aperture logarithmic forward axicon with R = 5 mm, d1 = 100 mm, d2 = 150 mm, and ρ = 3.333 mm (solid curve) and for the full-aperture logarithmic forward axicon with R = 5 mm, d1 = 100 mm, d2 = 150 mm, and ρ = 0 (dashed curve).

Fig. 3
Fig. 3

Variation of the central spot radii r0′ along z for three annular-aperture logarithmic forward axicons. Average values of these radii are 6.03 μm (curve 1), 7.17 μm (curve 2), and 9.41 μm (curve 3).

Fig. 4
Fig. 4

Energy flow along the focal region of the annular-aperture logarithmic forward axicon with R = 5 mm, ρ = 4 mm, d1 = 100 mm, and d2 = 125 mm, within the cylinder of radius rf1′ 6.04 μm.

Fig. 5
Fig. 5

Energy flow along the focal region of the annular-aperture logarithmic forward axicon with R = 5 mm, ρ = 3.33 mm, d1 = 100 mm, and d2 = 150 mm, within the cylinder of radius rf2′ = 7.22 μm.

Fig. 6
Fig. 6

Energy flow along the focal region of the annular-aperture logarithmic forward axicon with R = 5 mm, ρ = 2.5 mm, d1 = 100 mm, and d2 = mm, within the cylinder of radius rf3′ = 9.50 μm.

Equations (10)

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2 π P σ ρ r r d r = P z d 1 z ( r ) d z ,             r = ( x 2 + y 2 ) 1 / 2             at z = 0 ,
z ( r ) = d 1 - d 2 - d 1 R 2 - ρ 2 ρ 2 + d 2 - d 1 R 2 - ρ 2 r 2 ,             ρ r R ,
d φ ( r ) d r = - sin θ - tan θ = - r / z ( r ) ,             ρ r R .
φ ( r ) = - R 2 - ρ 2 2 ( d 2 - d 1 ) ln ( d 1 - d 2 - d 1 R 2 - ρ 2 ρ 2 + d 2 - d 1 R 2 - ρ 2 r 2 ) + const . ,             ρ r R .
I ( r , z ) = ( 2 π λ z ) 2 | 0 R exp { 2 π i λ [ r 2 2 z + φ ( r ) ] } × J 0 ( 2 π r r / λ z ) r d r | 2 ,
ρ = ( d 1 / d 2 ) R ,
I ( 0 , z ) = ( π λ a z ) 2 | d 1 d 2 exp { π i λ a [ η z - ln ( η ) ] } d η | 2 ,
a = d 2 - d 1 R 2 - ρ 2 .
θ max 1 2 [ ( d 1 d 2 ) 1 / 2 + ( d 2 d 1 ) 1 / 2 ] R d 2 = 1 2 [ ( 1 + d 2 - d 1 d 1 ) - 1 / 2 + ( 1 + d 2 - d 1 d 1 ) 1 / 2 ] R d 2 .
θ max R d 2 [ 1 + ( d 2 - d 1 ) 2 8 d 1 2 ] ,

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