## Fractal conical lenses

Optics Express, Vol. 14, Issue 20, pp. 9077-9082 (2006)

http://dx.doi.org/10.1364/OE.14.009077

Acrobat PDF (264 KB)

### 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

## 1. Introduction

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

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. Cizmár T., 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]

*fractal conical lens*(FCL), is a CL with a Cantor-like fractal profile. In this text we study the axial irradiance provided by a FCL when illuminated with plane wavefront. In Section 2, we revise the axial behaviour of a conventional CL using the Fresnel diffraction integral and we give the practical limitation to obtain the on-axis irradiance distribution predicted by geometrical optics. In Section 3 we present the procedure we followed for the synthesis of FCL. We perform the numerical analysis of FCLs illuminated by a parallel wavefront. Additionally, the correlation between the axial irradiance provided by the FCLs and the one predicted by geometrical optics is evaluated in terms of the Fresnel number of the focusing geometry. Finally, in Section 4 the main results are outlined and some applications are proposed.

## 2. Axial intensity distribution of a conical lens

*n*. A schematic representation of a CL is shown in Fig. 1(a). The transverse radius of the CL is

*a*, and

*h*

_{0}is the height. Within the paraxial regime, we consider that

*a*>>

*h*

_{0}(thin CL), so the base angle can be approximated to α≈

*h*

_{0}/

*a*. When illuminated by a uniform monochromatic plane wave the CL produces a phase shift of the incident light, which decreases linearly with the radial distance

*r*. Neglecting reflection and transmission losses, the transmission function of the CL may be written as [12

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]

*λ*is the wavelength of the incident light. The parameter α determines the inclination of the beam,

*β*=(

*n*-1)α, with respect to the optical axis, of the conical wavefront leaving the CL. Consequently, the CL produces a continuos focus line of length

*z*≈

_{max}*a*/

*β*.

*ρ*=

*r*/

*a*, and a normalized axial coordinate,

*z̄*=

*z*/

*z*. The result is:

_{max}*N*=

*aβ*/

*λ*[14

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

*N*is the only parameter that appear in Eq. (3) apart from the normalized axial coordinate, and therefore, it characterizes the focusing properties of the CL. For finite Fresnel numbers the theory predicts a lineary increasing axial irradiance modulated by a rapidly oscillating pattern with growing amplitude along the axial coordinate, up to

*z*=

*z*[see Fig. 1(b)]. This modulation also depends on

_{max}*N*[14

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

*x*)=1 for |

*x*|<1/2 and 0 otherwise. Therefore, in this case the on axis irradiance increases linearly, up to the geometrical shadow boundary defined by

*z*. Out of this region the irradiance vanishes [see the red line in Fig. 1(b)]. Next we focus our attention to the FCLs.

_{max}## 3. Cantor-like fractal conical lenses

*initiator*(stage

*S*=0). Next, at stage

*S*=1, the

*generator*of the set is constructed by dividing the segment in three equal parts of length 1/3 and removing the central one. Following this procedure in subsequent stages

*S*=2, 3, … is easy to see that, in general, at stage

*S*there are 2

*segments of length 3*

^{S}^{-S}with 2

*-1 disjoint gaps located at the intervals [*

^{S}*p*,

_{S,l}*q*], with

_{S,l}*l*=1, …, 2

*-1. For example, for*

^{S}*S*=2, the triadic Cantor set presents three gaps at [1/9, 2/9], [3/9, 6/9], and [7/9, 8/9]. In Fig. 2(a), only the three first stages are shown for clarity.

15. D.R. Chalice, “A characterization of the Cantor function,” Amer. Math. Monthly **98**, 255–258 (1991). [CrossRef]

*(*

_{S}*x*), is defined in the interval [0, 1] as,

*l*=1, …, 2

^{S}-1. The lower part of Fig. 2(a) shows the Cantor function F

_{3}(

*x*). On the intervals [1/27, 2/27], [3/27, 6/27], [7/27, 8/27], [9/27, 18/27], [19/27, 20/27], [21/27, 24/27], and [25/27, 26/27], the constant values of F

_{3}(i) are 1/8, 2/8, 3/8, 4/8, 5/8, 6/8, and 7/8, respectively. Outside of these intervals, F

*(*

_{S}*x*) is a continuous increasing linear function.

*S*. In order to maintain a constant value of the base angle, α, at different stages

*S*, the height of the triadic FCL is

*h*=(2/3)

_{S}

^{S}*h*

_{0}, where

*h*

_{0}is the height of a FCL at

*S*=0 corresponding to the conventional CL. Figure 2(b) shows FCLs generated for different values of

*S*. Note that a FCL can be constructed by combining certain sections of a conventional CL. In mathematical terms, a FCL developed up to a certain growing stage

*S*, can be represented by a phase transmission function given by,

*S*=0 Eq. (7) reduces to Eq. (1). Therefore, we have developed a theoretical framework for the FCL in which the classical CL is a particular case.

*S*=1, 2, and 3. The results for a Fresnel number

*N*=700 are shown in Fig. 3 (black lines). The first noticeable feature in these figures is that the axial irradiances increases with distance

*z*, as happens with conventional CL, but in this case it is modulated by the corresponding Cantor set. In other words, the axial irradiance reflects the fractal structure that the lenses has along the radial coordinate. It is important to note that the axial irradiance for a given stage

*S*is a scaled and replicated version of the axial irradiance for the previous stage, as corresponds to a self-similar structure.

*g*(

*x*,Λ) is a Ronchi-type periodic binary function with period Λ that can be written as

*u*,

*v*) gives the remainder on division of

*u*by

*v*. Fig. 3 also shows (color lines) the axial irradiances produced by the FCLs under the geometrical approximation [Eq. (9)].

*C*(

_{S}*N*) for different values of

*S*. Compared with the result obtained for

*S*=1 (red line) the correlation function for increasing fractal level of growth gives lower values. However, in all cases the correlation coefficient is an increasing function which tends to the unity for very high Fresnel number. Note that some oscillations appears for higher values of

*S*which are the result of the lobes in the axial irradiance patters.

## 4. Conclusions

*S*=0. The axial irradiance provided by FCLs illuminated by a monochromatic wavefront has been investigated. We have shown that these lenses produces focal segments distributed along the optical axis in a way that they reproduce the fractal profile of the originating FCL.

*π*phase differences between the different steps in the Cantor function and constant zero phase in the gaps of the Cantor set.

## Acknowledgments

## References and links

1. | L. M. Soroko, |

2. | Z. Jaroszewicz, |

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. |

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. |

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 |

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 |

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. |

8. | Cizmár T., V. Garcés-Chávez, K. Dholakia, and P. Zemánek, “Optical conveyor belt for delivery of submicron objects,” Appl. Phys. Lett. |

9. | D. Zeng, W. P. Latham, and A. Kar, “Temperature distributions due to annular laser beam heating,” J. Laser Appl. |

10. | J.A. Monsoriu, C.J. Zapata-Rodríguez, and W.D. Furlan, “Fractal axicons,” Opt. Commun. |

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 |

13. | J. Goodman, |

14. | C.J. Zapata-Rodrígez and F.E. Hernández, “Focal squeeze in axicons,” Opt. Commun. |

15. | D.R. Chalice, “A characterization of the Cantor function,” Amer. Math. Monthly |

**OCIS Codes**

(050.1940) Diffraction and gratings : Diffraction

(050.1970) Diffraction and gratings : Diffractive optics

(080.3620) Geometric optics : Lens system design

**History**

Original Manuscript: June 2, 2006

Revised Manuscript: June 30, 2006

Manuscript Accepted: June 30, 2006

Published: October 2, 2006

**Citation**

Juan A. Monsoriu, Walter D. Furlan, Pedro Andrés, and Jesús Lancis, "Fractal conical lenses," Opt. Express **14**, 9077-9082 (2006)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-20-9077

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### References

- 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).
- 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]
- 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]
- 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]
- 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]
- 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]
- 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]
- J. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1996).
- C.J. Zapata-Rodrígez and F.E. Hernández, "Focal squeeze in axicons," Opt. Commun. 254,3-9 (2005). [CrossRef]
- D.R. Chalice, "A characterization of the Cantor function," Amer. Math. Monthly 98,255-258 (1991). [CrossRef]

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