Abstract

A conical lens is an optical element that produces a continuous focal segment along the optical axis. In this paper we introduce a more general optical device: the fractal conical lens (FCL). As the profile of a FCL is generated using the Cantor function, we show that a classical conical lens is a particular case of these fractal lenses. FCLs are distinguished by the fractal focal segments they produce along the optical axis. The influence of the Fresnel number on the axial irradiance generated by these lenses is investigated.

© 2006 Optical Society of America

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References

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  1. L. M. Soroko, Meso-Optics (World Scientific Publishing, Singapore, 1996).
    [CrossRef]
  2. Z. Jaroszewicz, Axicons, Design and Propagation Properties (SPIE Polish Chapter Research and Development Series, Vol. 5, 1997).
  3. Z. Jaroszewicz, V. Climent, V. Durán, J. Lancis, A. Kolodziejczyk, A. Burvall, and A. T. Friber, "Programmable axicon for variable inclination of the focal segment," J. Mod. Opt. 51, 2185-2190 (2004).
    [CrossRef]
  4. Z. Ding, H. Ren, Y. Zhao, J. S. Nelson, and Z. Chen, "High-resolution optical coherence tomography over a large depth range with an axicon lens," Opt. Lett. 27,243-245 (2002).
    [CrossRef]
  5. Y. F. Xiao, H. H. Chu, H. E. Tsai, C. H. Lee, J. Y. Lin, J. Wang, and S. Y. Chen, "Efficient generation of extended plasma waveguides with the axicon ignitor-heater scheme," Phys. Plasmas 11, L21-L24 (2004).
    [CrossRef]
  6. H. Little, C. T. A. Brown, V. Garcés-Chávez, W. Sibbett, and K. Dholakia, "Optical guiding of microscopic particles in femtosecond and continuous wave Bessel light beams," Opt. Express 12, 2560-2565 (2004), http://www.opticsinfobase.org/abstract.cfm?URI=oe-12-11-2560
    [CrossRef] [PubMed]
  7. J. Y. L. Goh, M. G. Somekh, C. W. See, M. C. Pitter, K. A. Vere, and P. O’Shea, "Two-photon fluorescence surface wave microscopy," J. Microsc. 220, 168-175 (2005).
    [CrossRef] [PubMed]
  8. T. Cizmár, V. Garcés-Chávez, K. Dholakia, and P. Zemánek, "Optical conveyor belt for delivery of submicron objects," Appl. Phys. Lett. 86, 174101 (2005).
    [CrossRef]
  9. D. Zeng, W. P. Latham, and A. Kar, "Temperature distributions due to annular laser beam heating," J. Laser Appl. 17, 256-262 (2005).
    [CrossRef]
  10. J.A. Monsoriu, C.J. Zapata-Rodríguez, and W.D. Furlan, "Fractal axicons," Opt. Commun. 263, 1-5 (2006).
    [CrossRef]
  11. A.D. Jaggard and D.L. Jaggard, "Cantor ring diffractals," Opt. Commun. 158, 141-148 (1998).
    [CrossRef]
  12. M.V. Pérez, C. Gómez-Reino, and J.M. Cuadrado, "Diffraction patterns and zone plates produced by thin linear axicons," Optica Acta 33, 1161-1176 (1986).
    [CrossRef]
  13. J. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1996).
  14. C.J. Zapata-Rodrígez and F.E. Hernández, "Focal squeeze in axicons," Opt. Commun. 254,3-9 (2005).
    [CrossRef]
  15. D.R. Chalice, "A characterization of the Cantor function," Amer. Math. Monthly 98,255-258 (1991).
    [CrossRef]

2006 (1)

J.A. Monsoriu, C.J. Zapata-Rodríguez, and W.D. Furlan, "Fractal axicons," Opt. Commun. 263, 1-5 (2006).
[CrossRef]

2005 (4)

J. Y. L. Goh, M. G. Somekh, C. W. See, M. C. Pitter, K. A. Vere, and P. O’Shea, "Two-photon fluorescence surface wave microscopy," J. Microsc. 220, 168-175 (2005).
[CrossRef] [PubMed]

T. Cizmár, V. Garcés-Chávez, K. Dholakia, and P. Zemánek, "Optical conveyor belt for delivery of submicron objects," Appl. Phys. Lett. 86, 174101 (2005).
[CrossRef]

D. Zeng, W. P. Latham, and A. Kar, "Temperature distributions due to annular laser beam heating," J. Laser Appl. 17, 256-262 (2005).
[CrossRef]

C.J. Zapata-Rodrígez and F.E. Hernández, "Focal squeeze in axicons," Opt. Commun. 254,3-9 (2005).
[CrossRef]

2004 (3)

Z. Jaroszewicz, V. Climent, V. Durán, J. Lancis, A. Kolodziejczyk, A. Burvall, and A. T. Friber, "Programmable axicon for variable inclination of the focal segment," J. Mod. Opt. 51, 2185-2190 (2004).
[CrossRef]

Y. F. Xiao, H. H. Chu, H. E. Tsai, C. H. Lee, J. Y. Lin, J. Wang, and S. Y. Chen, "Efficient generation of extended plasma waveguides with the axicon ignitor-heater scheme," Phys. Plasmas 11, L21-L24 (2004).
[CrossRef]

H. Little, C. T. A. Brown, V. Garcés-Chávez, W. Sibbett, and K. Dholakia, "Optical guiding of microscopic particles in femtosecond and continuous wave Bessel light beams," Opt. Express 12, 2560-2565 (2004), http://www.opticsinfobase.org/abstract.cfm?URI=oe-12-11-2560
[CrossRef] [PubMed]

2002 (1)

1998 (1)

A.D. Jaggard and D.L. Jaggard, "Cantor ring diffractals," Opt. Commun. 158, 141-148 (1998).
[CrossRef]

1991 (1)

D.R. Chalice, "A characterization of the Cantor function," Amer. Math. Monthly 98,255-258 (1991).
[CrossRef]

1986 (1)

M.V. Pérez, C. Gómez-Reino, and J.M. Cuadrado, "Diffraction patterns and zone plates produced by thin linear axicons," Optica Acta 33, 1161-1176 (1986).
[CrossRef]

Brown, C. T. A.

Burvall, A.

Z. Jaroszewicz, V. Climent, V. Durán, J. Lancis, A. Kolodziejczyk, A. Burvall, and A. T. Friber, "Programmable axicon for variable inclination of the focal segment," J. Mod. Opt. 51, 2185-2190 (2004).
[CrossRef]

Chalice, D.R.

D.R. Chalice, "A characterization of the Cantor function," Amer. Math. Monthly 98,255-258 (1991).
[CrossRef]

Chen, S. Y.

Y. F. Xiao, H. H. Chu, H. E. Tsai, C. H. Lee, J. Y. Lin, J. Wang, and S. Y. Chen, "Efficient generation of extended plasma waveguides with the axicon ignitor-heater scheme," Phys. Plasmas 11, L21-L24 (2004).
[CrossRef]

Chen, Z.

Chu, H. H.

Y. F. Xiao, H. H. Chu, H. E. Tsai, C. H. Lee, J. Y. Lin, J. Wang, and S. Y. Chen, "Efficient generation of extended plasma waveguides with the axicon ignitor-heater scheme," Phys. Plasmas 11, L21-L24 (2004).
[CrossRef]

Cizmár, T.

T. Cizmár, V. Garcés-Chávez, K. Dholakia, and P. Zemánek, "Optical conveyor belt for delivery of submicron objects," Appl. Phys. Lett. 86, 174101 (2005).
[CrossRef]

Climent, V.

Z. Jaroszewicz, V. Climent, V. Durán, J. Lancis, A. Kolodziejczyk, A. Burvall, and A. T. Friber, "Programmable axicon for variable inclination of the focal segment," J. Mod. Opt. 51, 2185-2190 (2004).
[CrossRef]

Cuadrado, J.M.

M.V. Pérez, C. Gómez-Reino, and J.M. Cuadrado, "Diffraction patterns and zone plates produced by thin linear axicons," Optica Acta 33, 1161-1176 (1986).
[CrossRef]

Dholakia, K.

Ding, Z.

Durán, V.

Z. Jaroszewicz, V. Climent, V. Durán, J. Lancis, A. Kolodziejczyk, A. Burvall, and A. T. Friber, "Programmable axicon for variable inclination of the focal segment," J. Mod. Opt. 51, 2185-2190 (2004).
[CrossRef]

Friber, A. T.

Z. Jaroszewicz, V. Climent, V. Durán, J. Lancis, A. Kolodziejczyk, A. Burvall, and A. T. Friber, "Programmable axicon for variable inclination of the focal segment," J. Mod. Opt. 51, 2185-2190 (2004).
[CrossRef]

Furlan, W.D.

J.A. Monsoriu, C.J. Zapata-Rodríguez, and W.D. Furlan, "Fractal axicons," Opt. Commun. 263, 1-5 (2006).
[CrossRef]

Garcés-Chávez, V.

Goh, J. Y. L.

J. Y. L. Goh, M. G. Somekh, C. W. See, M. C. Pitter, K. A. Vere, and P. O’Shea, "Two-photon fluorescence surface wave microscopy," J. Microsc. 220, 168-175 (2005).
[CrossRef] [PubMed]

Gómez-Reino, C.

M.V. Pérez, C. Gómez-Reino, and J.M. Cuadrado, "Diffraction patterns and zone plates produced by thin linear axicons," Optica Acta 33, 1161-1176 (1986).
[CrossRef]

Hernández, F.E.

C.J. Zapata-Rodrígez and F.E. Hernández, "Focal squeeze in axicons," Opt. Commun. 254,3-9 (2005).
[CrossRef]

Jaggard, A.D.

A.D. Jaggard and D.L. Jaggard, "Cantor ring diffractals," Opt. Commun. 158, 141-148 (1998).
[CrossRef]

Jaggard, D.L.

A.D. Jaggard and D.L. Jaggard, "Cantor ring diffractals," Opt. Commun. 158, 141-148 (1998).
[CrossRef]

Jaroszewicz, Z.

Z. Jaroszewicz, V. Climent, V. Durán, J. Lancis, A. Kolodziejczyk, A. Burvall, and A. T. Friber, "Programmable axicon for variable inclination of the focal segment," J. Mod. Opt. 51, 2185-2190 (2004).
[CrossRef]

Kar, A.

D. Zeng, W. P. Latham, and A. Kar, "Temperature distributions due to annular laser beam heating," J. Laser Appl. 17, 256-262 (2005).
[CrossRef]

Kolodziejczyk, A.

Z. Jaroszewicz, V. Climent, V. Durán, J. Lancis, A. Kolodziejczyk, A. Burvall, and A. T. Friber, "Programmable axicon for variable inclination of the focal segment," J. Mod. Opt. 51, 2185-2190 (2004).
[CrossRef]

Lancis, J.

Z. Jaroszewicz, V. Climent, V. Durán, J. Lancis, A. Kolodziejczyk, A. Burvall, and A. T. Friber, "Programmable axicon for variable inclination of the focal segment," J. Mod. Opt. 51, 2185-2190 (2004).
[CrossRef]

Latham, W. P.

D. Zeng, W. P. Latham, and A. Kar, "Temperature distributions due to annular laser beam heating," J. Laser Appl. 17, 256-262 (2005).
[CrossRef]

Lee, C. H.

Y. F. Xiao, H. H. Chu, H. E. Tsai, C. H. Lee, J. Y. Lin, J. Wang, and S. Y. Chen, "Efficient generation of extended plasma waveguides with the axicon ignitor-heater scheme," Phys. Plasmas 11, L21-L24 (2004).
[CrossRef]

Lin, J. Y.

Y. F. Xiao, H. H. Chu, H. E. Tsai, C. H. Lee, J. Y. Lin, J. Wang, and S. Y. Chen, "Efficient generation of extended plasma waveguides with the axicon ignitor-heater scheme," Phys. Plasmas 11, L21-L24 (2004).
[CrossRef]

Little, H.

Monsoriu, J.A.

J.A. Monsoriu, C.J. Zapata-Rodríguez, and W.D. Furlan, "Fractal axicons," Opt. Commun. 263, 1-5 (2006).
[CrossRef]

Nelson, J. S.

O’Shea, P.

J. Y. L. Goh, M. G. Somekh, C. W. See, M. C. Pitter, K. A. Vere, and P. O’Shea, "Two-photon fluorescence surface wave microscopy," J. Microsc. 220, 168-175 (2005).
[CrossRef] [PubMed]

Pérez, M.V.

M.V. Pérez, C. Gómez-Reino, and J.M. Cuadrado, "Diffraction patterns and zone plates produced by thin linear axicons," Optica Acta 33, 1161-1176 (1986).
[CrossRef]

Pitter, M. C.

J. Y. L. Goh, M. G. Somekh, C. W. See, M. C. Pitter, K. A. Vere, and P. O’Shea, "Two-photon fluorescence surface wave microscopy," J. Microsc. 220, 168-175 (2005).
[CrossRef] [PubMed]

Ren, H.

See, C. W.

J. Y. L. Goh, M. G. Somekh, C. W. See, M. C. Pitter, K. A. Vere, and P. O’Shea, "Two-photon fluorescence surface wave microscopy," J. Microsc. 220, 168-175 (2005).
[CrossRef] [PubMed]

Sibbett, W.

Somekh, M. G.

J. Y. L. Goh, M. G. Somekh, C. W. See, M. C. Pitter, K. A. Vere, and P. O’Shea, "Two-photon fluorescence surface wave microscopy," J. Microsc. 220, 168-175 (2005).
[CrossRef] [PubMed]

Tsai, H. E.

Y. F. Xiao, H. H. Chu, H. E. Tsai, C. H. Lee, J. Y. Lin, J. Wang, and S. Y. Chen, "Efficient generation of extended plasma waveguides with the axicon ignitor-heater scheme," Phys. Plasmas 11, L21-L24 (2004).
[CrossRef]

Vere, K. A.

J. Y. L. Goh, M. G. Somekh, C. W. See, M. C. Pitter, K. A. Vere, and P. O’Shea, "Two-photon fluorescence surface wave microscopy," J. Microsc. 220, 168-175 (2005).
[CrossRef] [PubMed]

Wang, J.

Y. F. Xiao, H. H. Chu, H. E. Tsai, C. H. Lee, J. Y. Lin, J. Wang, and S. Y. Chen, "Efficient generation of extended plasma waveguides with the axicon ignitor-heater scheme," Phys. Plasmas 11, L21-L24 (2004).
[CrossRef]

Xiao, Y. F.

Y. F. Xiao, H. H. Chu, H. E. Tsai, C. H. Lee, J. Y. Lin, J. Wang, and S. Y. Chen, "Efficient generation of extended plasma waveguides with the axicon ignitor-heater scheme," Phys. Plasmas 11, L21-L24 (2004).
[CrossRef]

Zapata-Rodrígez, C.J.

C.J. Zapata-Rodrígez and F.E. Hernández, "Focal squeeze in axicons," Opt. Commun. 254,3-9 (2005).
[CrossRef]

Zapata-Rodríguez, C.J.

J.A. Monsoriu, C.J. Zapata-Rodríguez, and W.D. Furlan, "Fractal axicons," Opt. Commun. 263, 1-5 (2006).
[CrossRef]

Zemánek, P.

T. Cizmár, V. Garcés-Chávez, K. Dholakia, and P. Zemánek, "Optical conveyor belt for delivery of submicron objects," Appl. Phys. Lett. 86, 174101 (2005).
[CrossRef]

Zeng, D.

D. Zeng, W. P. Latham, and A. Kar, "Temperature distributions due to annular laser beam heating," J. Laser Appl. 17, 256-262 (2005).
[CrossRef]

Zhao, Y.

Amer. Math. Monthly (1)

D.R. Chalice, "A characterization of the Cantor function," Amer. Math. Monthly 98,255-258 (1991).
[CrossRef]

Appl. Phys. Lett. (1)

T. Cizmár, V. Garcés-Chávez, K. Dholakia, and P. Zemánek, "Optical conveyor belt for delivery of submicron objects," Appl. Phys. Lett. 86, 174101 (2005).
[CrossRef]

J. Laser Appl. (1)

D. Zeng, W. P. Latham, and A. Kar, "Temperature distributions due to annular laser beam heating," J. Laser Appl. 17, 256-262 (2005).
[CrossRef]

J. Microsc. (1)

J. Y. L. Goh, M. G. Somekh, C. W. See, M. C. Pitter, K. A. Vere, and P. O’Shea, "Two-photon fluorescence surface wave microscopy," J. Microsc. 220, 168-175 (2005).
[CrossRef] [PubMed]

J. Mod. Opt. (1)

Z. Jaroszewicz, V. Climent, V. Durán, J. Lancis, A. Kolodziejczyk, A. Burvall, and A. T. Friber, "Programmable axicon for variable inclination of the focal segment," J. Mod. Opt. 51, 2185-2190 (2004).
[CrossRef]

Opt. Commun. (3)

C.J. Zapata-Rodrígez and F.E. Hernández, "Focal squeeze in axicons," Opt. Commun. 254,3-9 (2005).
[CrossRef]

J.A. Monsoriu, C.J. Zapata-Rodríguez, and W.D. Furlan, "Fractal axicons," Opt. Commun. 263, 1-5 (2006).
[CrossRef]

A.D. Jaggard and D.L. Jaggard, "Cantor ring diffractals," Opt. Commun. 158, 141-148 (1998).
[CrossRef]

Opt. Express (1)

Opt. Lett. (1)

Optica Acta (1)

M.V. Pérez, C. Gómez-Reino, and J.M. Cuadrado, "Diffraction patterns and zone plates produced by thin linear axicons," Optica Acta 33, 1161-1176 (1986).
[CrossRef]

Phys. Plasmas (1)

Y. F. Xiao, H. H. Chu, H. E. Tsai, C. H. Lee, J. Y. Lin, J. Wang, and S. Y. Chen, "Efficient generation of extended plasma waveguides with the axicon ignitor-heater scheme," Phys. Plasmas 11, L21-L24 (2004).
[CrossRef]

Other (3)

J. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1996).

L. M. Soroko, Meso-Optics (World Scientific Publishing, Singapore, 1996).
[CrossRef]

Z. Jaroszewicz, Axicons, Design and Propagation Properties (SPIE Polish Chapter Research and Development Series, Vol. 5, 1997).

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Figures (4)

Fig. 1.
Fig. 1.

(a) On-axis irradiance produced by a thin CL. The transverse radius of this axicon is a, h 0 is the height, and α is the base angle. (b) Normalized axial irradiances given by a conventional CL for N=700, and the color lines correspond to a focusing geometry of an ideally infinite Fresnel number.

Fig. 2.
Fig. 2.

(a) Triadic Cantor set for S=1, S=2, and S=3. The structure for S=0 is the initiator and the one corresponding to S=1 is the generator. The Cantor function FS (x) is shown under the corresponding Cantor set for S=3. (b) FCLs at stages of growth S=1, S=2, and S=3.

Fig. 3.
Fig. 3.

Normalized axial irradiances given by FCLs for (a) S=1, (b) S=2, and S=3. Solid lines correspond to the case N=700, and the color lines correspond to a focusing geometry of an ideally infinite Fresnel number.

Fig. 4.
Fig. 4.

Correlation function C(N) between the axial irradiance given by a FCL at different stage of growth, S, and the corresponding irradiance with an ideally infinite Fresnel number.

Equations (10)

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T CL ( r ) = exp [ j 2 π ( n 1 ) α λ r ] ,
I CL ( z ) = ( 2 π λ z ) 2 0 a exp [ j 2 π ( n 1 ) α λ r ] exp [ j π λ z r 2 ] r dr 2 .
I CL ( z ̅ , N ) = ( 2 π N z ̅ ) 2 0 1 exp [ j 2 π N ρ ] exp [ j π N z ̅ ρ 2 ] ρ d ρ 2 ,
I CL ( z ̅ , N ) = ( 2 π ) 2 N z ̅ rect [ z ̅ 0.5 ] ,
F S ( x ) = { l 2 S if p S , l x q S , l 1 2 S x q S , l p S , l + 1 q S , l + l 2 S if q S , l x p S , l + 1 ,
T FCL ( ρ , N , S ) = exp [ j 2 π N ( 2 3 ) S F S ( ρ ) ] .
I FCL ( z ̅ , N , S ) = ( 2 π N z ̅ ) 2 0 1 T FCL ( ρ , N , S ) exp [ j π N z ̅ ρ 2 ] ρ d ρ 2 .
I FCL ( z ̅ , N , S ) = ( 2 π ) 2 N z ̅ i = 0 S g ( z ̅ , 2 3 i ) ,
g ( x , Λ ) = rect ( x 0.5 ) rect [ mod ( x + 0.5 Λ 1 , Λ ) Λ ] .
C S ( N ) = 0 I FCL ( z ̅ , N , S ) I FCL ( z ̅ , N , S ) d z ̅ 0 [ I FCL ( z ̅ , N , S ) ] 2 d z ̅ .

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