## Devil’s lenses

Optics Express, Vol. 15, Issue 21, pp. 13858-13864 (2007)

http://dx.doi.org/10.1364/OE.15.013858

Acrobat PDF (848 KB)

### Abstract

In this paper we present a new kind of kinoform lenses in which the phase distribution is characterized by the “devil’s staircase” function. The focusing properties of these fractal DOEs coined *devil’s lenses* (DLs) are analytically studied and compared with conventional Fresnel kinoform lenses. It is shown that under monochromatic illumination a DL give rise a single fractal focus that axially replicates the self-similarity of the lens. Under broadband illumination the superposition of the different monochromatic foci produces an increase in the depth of focus and also a strong reduction in the chromaticity variation along the optical axis.

© 2007 Optical Society of America

## 1. Introduction

1. J. Courtial and M. J. Padgett, “Monitor-outside-a-monitor effect and self-similar fractal structure in the eigenmodes of unstable optical resonators,” Phys. Rev. Lett. **85**, 5320–5323 (2000). [CrossRef]

5. G. Saavedra, W. D. Furlan, and J. A. Monsoriu, “Fractal zone plates,” Opt. Lett. **28**, 971–973 (2003). [CrossRef] [PubMed]

6. J.A. Monsoriu, G. Saavedra, and W.D. Furlan, “Fractal zone plates with variable lacunarity,” Opt. Express **12**, 4227–4234 (2004). http://www.opticsinfobase.org/abstract.cfm?URI=oe-12-18-4227 [CrossRef] [PubMed]

7. J. A. Davis, L. Ramirez, J. A. Rodrigo Martín-Romo, T. Alieva, and M. L. Calvo, “Focusing properties of fractal zone plates: experimental implementation with a liquid-crystal display,” Opt. Lett. **29**, 1321–1323 (2004). [CrossRef] [PubMed]

8. H.-T. Dai, X. Wang, and K.-S. Xu, “Focusing properties of fractal zone plates with variable lacunarity: experimental studies based on liquid crystal on silicon,” Chin. Phys. Lett. **22**, 2851–2854 (2005). [CrossRef]

9. S. H. Tao, X.-C. Yuan, J. Lin, and R. Burge, “Sequence of focused optical vortices generated by a spiral fractal zone plates,” Appl. Phys. Lett. **89**, 031105 (2006). [CrossRef]

10. C. Martelli and J. Canning, “Fresnel Fibres with Omnidirectional Zone Cross-sections,” Opt. Express **15**, 4281–4286 (2007). [CrossRef] [PubMed]

11. F. Giménez, J. A. Monsoriu, W. D. Furlan, and A. Pons, “Fractal Photon Sieves,” Opt. Express **14**, 11958–11963 (2006). http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-25-11958 [CrossRef] [PubMed]

## 2. Devil’s lenses design

*initiator*(stage

*S*=0). Next, at stage

*S*=1, the

*generator*of the set is constructed by dividing the segment into

*m*equal parts of length 1/

*m*and removing some of them. Then, this procedure is continued at the subsequent stages,

*S*=2, 3. … Without loss of generality, let us consider the triadic CS shown in the upper part of Fig. 1(a). In this case

*m*=3 and it is easy to see that, at stage

*S*there are 2

^{S}segments of length 3

^{-S}with 2

^{S}-1 disjoint gaps intervals [

*p*,

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

_{S,l}*l*=1, …, 2

^{S}-1. Based on this fractal structure, in this case the devil’s staircase Cantor function [13

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

1. J. Courtial and M. J. Padgett, “Monitor-outside-a-monitor effect and self-similar fractal structure in the eigenmodes of unstable optical resonators,” Phys. Rev. Lett. **85**, 5320–5323 (2000). [CrossRef]

*(0)=0 and F*

_{S}*(1)=1. For example, for*

_{S}*S*=3 [see Fig. 1(a)], the triadic Cantor set presents seven gaps limited by the following positions inside the unit length: [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 steps of the devil’s staircase,

*F*(

_{S}*x*), take in the above intervals the constant values

*l*/2

^{3}with

*l*=1, …,7. In between these intervals the continuous function is linear.

*F*(

_{s}*x*) we define the corresponding DL as a circularly symmetric DOE with a phase profile which follows the Cantor function at a given stage,

*S*. At the gap regions defined by the Cantor set the phase shift is -

*l*2

*π*, with

*l*=1, …, 2

^{S}-1. Thus, the convergent DL transmittance is defined by

*a*is the lens radius. Thus, the phase variation is quadratic in each zone of the lens. The surface-relief profile,

*h*(

*r*), of the DL corresponding to the above phase function can be obtained from the relation [17

17. Y. Han, L. N. Hazra, and C. A. Delisle, “Exact surface-relief profile of a kinoform lens from its phase function,” J. Opt. Soc. Am. A **12**, 524–529 (1995). [CrossRef]

_{2π}[

*ϕ*(

*r*)] is the phase function

*ϕ*(

*r*) modulo 2π,

*n*is the refractive index of the optical material used for constructing the lens, and

*λ*is the wavelength of the light.

## 3. Focusing properties of a DL

*p*(

*r*) is given by

_{o}*z*is the axial distance from the pupil plane,

*r*has its origin at the optical axis, and

*λ*is the wavelength of the incident monochromatic plane wave. If the pupil transmittance is defined in terms of the normalized variable in Eq. (3), the irradiance can be expressed as the Hankel transform of the pupil function,

*q*(ζ)=

*p*(

*r*). In Eq. (6),

_{o}*u*=

*a*

^{2}/2

*λz*and

*v*=

*r*/

*a*are the reduced axial and transverse coordinates, respectively. If we focus our attention to the values the irradiance takes along the optical axis, then

*v*=0, and Eq. (6) reads

*q*(ζ). Using Eq. (2) for the transmittance corresponding to a DL and taking into account that one of the features of self-similar structures is that the dimensionality of the structure appears in its power spectrum [5

5. G. Saavedra, W. D. Furlan, and J. A. Monsoriu, “Fractal zone plates,” Opt. Lett. **28**, 971–973 (2003). [CrossRef] [PubMed]

*S*is shown in Fig. 2. The irradiance of the associated kinoform Fresnel lens is shown in the same figure for comparison. Note that the scale in the axial coordinate for each value of

*S*is different. It can be seen that the axial position of the focus of the Fresnel kinoform lens and the central lobe of the DL focus both coincide at the normalized distance

*u*=3

^{s}. Thus, from the change of variables adopted in Eq.(6) the focal length of the DL can be expressed as

*S*becomes larger an increasing number of zeros and maxima are encountered, which are scale invariant over dilations of factor

*γ*=3. The axial intensity distributions corresponding to the ZPs of low level involve the curves of the upper ones. This focalization behavior, which is here demonstrated that DLs satisfy, is, in fact, an exclusive feature of FraZPs and it was called

*the axial scale property*[5

5. G. Saavedra, W. D. Furlan, and J. A. Monsoriu, “Fractal zone plates,” Opt. Lett. **28**, 971–973 (2003). [CrossRef] [PubMed]

*S*=2 and

*S*=3.

## 4. Evolution of the diffraction patterns produced by a DL

*S*=2 is shown in the animated Fig. 3 (left). For comparison, the rigth half of this figure shows the transverse patterns produced by a Fresnel kinoform lens of the same focal length. The counter in this figure shows the value of the normalized distance (

*z/f*) from the lens that corresponds to each transversal pattern. In each frame the diffraction pattern is represented within the range |

*x*/

*a*|<0.5, |

*y*/

*a*|<0.5 of the transverse coordinates and the intensities are normalized to the maximun value at each transverse plane. In this way, the relative intensity at the main focus of both lenses (

*z/f*=1) can be directly compared. From this figure it can be noted that the axial secondary maxima provided by the DL results in an effective increase of the depth of focus. The intensity given by the kinoform Fresnel lens at those far planes is almost zero.

## 5. Polychromatic behavior of DLs

18. M. J. Yzuel and J. Santamaria, “Polychromatic Optical Image.Diffraction Limited System and Influence of the Longitudinal Chromatic Aberration,” Opt. Acta **22**, 673–690 (1975). [CrossRef]

*S*(λ) is the spectral distribution of the source and (

*x*͂ ,

*y*͂,

*z*͂) are the three sensitivity chromatic functions of the detector and (λ

_{1}, λ

_{2}) represent the considered wavelength interval. In particular, in the assessment of visual systems (

*x*͂ ,

*y*͂,

*z*͂) are usually the sensitivity functions of the human eye (CIE 1931) and the axial response is normally expressed in terms of the axial illuminance

*Y*and the axial chromaticity coordinates x, y,

*z/f*<1.3. The open circle in this figure represents the

*z*=

*f*for the design wavelength (λ=550 nm) of both zone plates. The triangles and squares represent

*z/f*=0.8 and

*z/f*=1.3, respectively. It can be observed that there are slow chromaticity variations for the DL and that the whole curve is closer to the point representing the white illuminant.

## 6. Conclusions

## Acknowledgments

## References and Links

1. | J. Courtial and M. J. Padgett, “Monitor-outside-a-monitor effect and self-similar fractal structure in the eigenmodes of unstable optical resonators,” Phys. Rev. Lett. |

2. | O. Trabocchi, S. Granieri, and W. D. Furlan, “Optical propagation of fractal fields. Experimental analysis in a single display,” J. Mod. Opt. |

3. | M. Lehman, “Fractal diffraction gratings built through rectangular domains,” Opt. Commun. |

4. | J. G. Huang, J. M. Christian, and G. S. McDonald “Fresnel diffraction and fractal patterns from polygonal apertures,” J. Opt. Soc. Am. A |

5. | G. Saavedra, W. D. Furlan, and J. A. Monsoriu, “Fractal zone plates,” Opt. Lett. |

6. | J.A. Monsoriu, G. Saavedra, and W.D. Furlan, “Fractal zone plates with variable lacunarity,” Opt. Express |

7. | J. A. Davis, L. Ramirez, J. A. Rodrigo Martín-Romo, T. Alieva, and M. L. Calvo, “Focusing properties of fractal zone plates: experimental implementation with a liquid-crystal display,” Opt. Lett. |

8. | H.-T. Dai, X. Wang, and K.-S. Xu, “Focusing properties of fractal zone plates with variable lacunarity: experimental studies based on liquid crystal on silicon,” Chin. Phys. Lett. |

9. | S. H. Tao, X.-C. Yuan, J. Lin, and R. Burge, “Sequence of focused optical vortices generated by a spiral fractal zone plates,” Appl. Phys. Lett. |

10. | C. Martelli and J. Canning, “Fresnel Fibres with Omnidirectional Zone Cross-sections,” Opt. Express |

11. | F. Giménez, J. A. Monsoriu, W. D. Furlan, and A. Pons, “Fractal Photon Sieves,” Opt. Express |

12. | J. A. Jordan, P. M. Hirsch, L. B. Lesem, and D. L. Van Rooy, “Kinoform lenses,” Appl. Opt. |

13. | D. R. Chalice, “A characterizationof the Cantor function,” Amer. Math. Monthly |

14. | F. Doveil, A. Macor, and Y. Elskens, “Direct observation of a devil’s staircase in wave-particle interaction,” Chaos |

15. | M. Hupalo, J. Schamalian, and M. C. Tringides, “Devil’s staircase in Pb/Si(111) ordered phases,” Phys. Rev. Lett. |

16. | Y. F. Chen, T. H. Lu, K. W. Su, and K. F. Huang, “Devil’s staircase in three-dimensional coherent waves localized on Lissajous parametric surfaces,” Phys. Rev. Lett. |

17. | Y. Han, L. N. Hazra, and C. A. Delisle, “Exact surface-relief profile of a kinoform lens from its phase function,” J. Opt. Soc. Am. A |

18. | M. J. Yzuel and J. Santamaria, “Polychromatic Optical Image.Diffraction Limited System and Influence of the Longitudinal Chromatic Aberration,” Opt. Acta |

**OCIS Codes**

(050.1940) Diffraction and gratings : Diffraction

(050.1970) Diffraction and gratings : Diffractive optics

**ToC Category:**

Fiber Optics and Optical Communications

**History**

Original Manuscript: June 13, 2007

Revised Manuscript: September 13, 2007

Manuscript Accepted: September 17, 2007

Published: October 5, 2007

**Citation**

Juan A. Monsoriu, Walter D. Furlan, Genaro Saavedra, and Fernando Giménez, "Devil’s lenses," Opt. Express **15**, 13858-13864 (2007)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-21-13858

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### References

- J. Courtial and M. J. Padgett, "Monitor-outside-a-monitor effect and self-similar fractal structure in the eigenmodes of unstable optical resonators," Phys. Rev. Lett. 85, 5320-5323 (2000). [CrossRef]
- O. Trabocchi, S. Granieri, and W. D. Furlan, "Optical propagation of fractal fields. Experimental analysis in a single display," J. Mod. Opt. 48, 1247-1253 (2001).
- M. Lehman, "Fractal diffraction gratings built through rectangular domains," Opt. Commun. 195, 11-26 (2001). [CrossRef]
- J. G. Huang, J. M. Christian, and G. S. McDonald "Fresnel diffraction and fractal patterns from polygonal apertures," J. Opt. Soc. Am. A 23, 2768-2774 (2006). [CrossRef]
- G. Saavedra, W. D. Furlan, and J. A. Monsoriu, "Fractal zone plates," Opt. Lett. 28, 971-973 (2003). [CrossRef] [PubMed]
- J.A. Monsoriu, G. Saavedra, and W.D. Furlan, "Fractal zone plates with variable lacunarity," Opt. Express 12, 4227-4234 (2004). http://www.opticsinfobase.org/abstract.cfm?URI=oe-12-18-4227> [CrossRef] [PubMed]
- J. A. Davis, L. Ramirez, J. A. Rodrigo Martín-Romo, T. Alieva, and M. L. Calvo, "Focusing properties of fractal zone plates: experimental implementation with a liquid-crystal display," Opt. Lett. 29, 1321-1323 (2004). [CrossRef] [PubMed]
- H.-T. Dai, X. Wang, K.-S. Xu, "Focusing properties of fractal zone plates with variable lacunarity: experimental studies based on liquid crystal on silicon," Chin. Phys. Lett. 22, 2851-2854 (2005). [CrossRef]
- S. H. Tao, X.-C. Yuan, J. Lin, and R. Burge, "Sequence of focused optical vortices generated by a spiral fractal zone plates," Appl. Phys. Lett. 89, 031105 (2006). [CrossRef]
- C. Martelli and J. Canning, "Fresnel Fibres with Omnidirectional Zone Cross-sections," Opt. Express 15, 4281-4286 (2007). [CrossRef] [PubMed]
- F. Giménez, J. A. Monsoriu, W. D. Furlan, and A. Pons, "Fractal Photon Sieves," Opt. Express 14, 11958-11963 (2006). http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-25-11958> [CrossRef] [PubMed]
- J. A. Jordan, P. M. Hirsch, L. B. Lesem, and D. L. Van Rooy, "Kinoform lenses," Appl. Opt. 9, 1883-1887 (1970) [PubMed]
- D. R. Chalice, "A characterizationof the Cantor function," Amer. Math. Monthly 98, 255-258 (1991). [CrossRef]
- F. Doveil, A. Macor, and Y. Elskens, "Direct observation of a devil’s staircase in wave-particle interaction," Chaos 16, 033130 (2006). [CrossRef]
- M. Hupalo, J. Schamalian, and M. C. Tringides, "Devil’s staircase in Pb/Si(111) ordered phases," Phys. Rev. Lett. 90, 216106 (2003). [CrossRef] [PubMed]
- Y. F. Chen, T. H. Lu, K. W. Su, and K. F. Huang, "Devil’s staircase in three-dimensional coherent waves localized on Lissajous parametric surfaces," Phys. Rev. Lett. 96, 213902 (2006). [CrossRef] [PubMed]
- Y. Han, L. N. Hazra, and C. A. Delisle, "Exact surface-relief profile of a kinoform lens from its phase function," J. Opt. Soc. Am. A 12, 524-529 (1995). [CrossRef]
- M. J. Yzuel and J. Santamaria, "Polychromatic Optical Image. Diffraction limited system and Influence of the Longitudinal Chromatic Aberration," Opt. Acta 22, 673-690 (1975). [CrossRef]

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