Abstract

A numerical analysis of the Fresnel diffraction integral shows that the axicons described by logarithmic phase retardation functions do not preserve uniformity of the lateral resolution and the energy flow along their focal regions.

© 1992 Optical Society of America

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References

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  1. J. Sochacki, A. Kołodziejczyk, Z. Jaroszewicz, S. Bará, “Nonparaxial designing of generalized axicons,” Appl. Opt. 31, 5326–5330 (1992).
    [CrossRef] [PubMed]
  2. J. Sochacki, S. Bará, Z. Jaroszewicz, A. Kołodziejczyk, “Phase retardation of the uniform-intensity axilens,” Opt. Lett. 17, 7–9 (1992).
    [CrossRef] [PubMed]
  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.
    [CrossRef]
  4. R. Tremblay, Y. d’Astous, G. Roy, M. Blanchard, “Laser plasmas optically pumped by focusing with an axicon a CO2-TEA laser beam in a high-pressure gas,” Opt. Commun. 28, 193–196 (1989).
    [CrossRef]
  5. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), pp. 13, 63.
  6. All diffraction integrals have been evaluated by means of standard procedures. The Bessel function was approximated according to the formulation presented in M. Abramowitz, I. A. Stegun, eds., Handbook of Mathematical Functions (Dover, New York, 1972), pp. 355–389.
  7. J. H. McLeod, “The axicon: a new type of optical element,”J. Opt. Soc. Am. 44, 592–597 (1954).
    [CrossRef]
  8. W. H. Steel, “Axicon means axial image,” Appl. Opt. 18, 2089 (1979).
    [CrossRef] [PubMed]

1992 (2)

1989 (1)

R. Tremblay, Y. d’Astous, G. Roy, M. Blanchard, “Laser plasmas optically pumped by focusing with an axicon a CO2-TEA laser beam in a high-pressure gas,” Opt. Commun. 28, 193–196 (1989).
[CrossRef]

1979 (1)

1954 (1)

Bará, S.

Blanchard, M.

R. Tremblay, Y. d’Astous, G. Roy, M. Blanchard, “Laser plasmas optically pumped by focusing with an axicon a CO2-TEA laser beam in a high-pressure gas,” Opt. Commun. 28, 193–196 (1989).
[CrossRef]

d’Astous, Y.

R. Tremblay, Y. d’Astous, G. Roy, M. Blanchard, “Laser plasmas optically pumped by focusing with an axicon a CO2-TEA laser beam in a high-pressure gas,” Opt. Commun. 28, 193–196 (1989).
[CrossRef]

Goodman, J. W.

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

Jaroszewicz, Z.

Kolodziejczyk, A.

McLeod, J. H.

Roy, G.

R. Tremblay, Y. d’Astous, G. Roy, M. Blanchard, “Laser plasmas optically pumped by focusing with an axicon a CO2-TEA laser beam in a high-pressure gas,” Opt. Commun. 28, 193–196 (1989).
[CrossRef]

Sochacki, J.

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.
[CrossRef]

Steel, W. H.

Tremblay, R.

R. Tremblay, Y. d’Astous, G. Roy, M. Blanchard, “Laser plasmas optically pumped by focusing with an axicon a CO2-TEA laser beam in a high-pressure gas,” Opt. Commun. 28, 193–196 (1989).
[CrossRef]

Appl. Opt. (2)

J. Opt. Soc. Am. (1)

Opt. Commun. (1)

R. Tremblay, Y. d’Astous, G. Roy, M. Blanchard, “Laser plasmas optically pumped by focusing with an axicon a CO2-TEA laser beam in a high-pressure gas,” Opt. Commun. 28, 193–196 (1989).
[CrossRef]

Opt. Lett. (1)

Other (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.
[CrossRef]

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

All diffraction integrals have been evaluated by means of standard procedures. The Bessel function was approximated according to the formulation presented in M. Abramowitz, I. A. Stegun, eds., Handbook of Mathematical Functions (Dover, New York, 1972), pp. 355–389.

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Figures (6)

Fig. 1
Fig. 1

On-axis intensity distribution for axicons with R = 5 mm; d1 = 100 mm; and d2 = 125 (curve 1), 150 (curve 2), and 200 mm (curve 3).

Fig. 2
Fig. 2

Variation of the cutoff lateral position rc′ along z for several axicons. Curves 1, 2, and 3 correspond to the forward axicons with d2 = 125, 150, and 200 mm, respectively; curves 4, 5, and 6 refer to the backward axicons with d2 = 125, 150, and 200 mm, respectively.

Fig. 3
Fig. 3

Energy flow along the focal region of the forward axicon with d1 = 100 mm and d2 = 125 mm.

Fig. 4
Fig. 4

Energy flow along the focal region of the forward axicon with d1 = 100 mm and d2 = 150 mm.

Fig. 5
Fig. 5

Energy flow along the focal region of the forward axicon with d1 = 100 mm and d2 = 200 mm.

Fig. 6
Fig. 6

Energy flow along the focal regions of three backward axicons. Curves 1, 2, and 3 correspond to the axicons with d2 = 125, 150, and 200 mm, respectively.

Equations (6)

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φ ( r ) = - ( 1 / 2 a ) ln ( d 1 + a r 2 ) + const . , r = ( x 2 + y 2 ) 1 / 2 R at z = 0 ,
φ ( r ) = + ( 1 / 2 a ) ln ( d 2 - a r 2 ) + const . , r = ( x 2 + y 2 ) 1 / 2 R at z = 0 ,
a d 2 - d 1 R 2 ,
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 ,
I ( 0 , z ) = ( π λ a z ) 2 | d 1 d 2 exp ± { π i λ a [ η / z - ln ( η ) ] } d η | 2 ,
I ( r 0 , z ) = 2 π 0 r 0 I ( r , z ) r d r ,             d 1 z d 2 ,

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