## Real time modulable multifocality through annular optical elements

Optics Express, Vol. 16, Issue 7, pp. 5095-5106 (2008)

http://dx.doi.org/10.1364/OE.16.005095

Acrobat PDF (1157 KB)

### Abstract

We present and analyze new multifocal optical elements based on an annular distribution of the transmittance. These elements provide selectable number of foci and can be designed to work between two fixed positions or even to provide extended focal depth. The energy of the foci can be modulated through a single parameter that controls the area of each ring. In our study we analyze the quality of the peaks and also the limit number of foci that can be obtained. The properties shown by these elements make them usable in instrumental optics or in ophthalmic optics, as new intraocular implants, where multifocal elements are required. The implementation has been done on a twisted nematic spatial light modulator, thus allowing real time reconfiguration of the element.

© 2008 Optical Society of America

## 1. Introduction

1. A. Kolodziejczyk, S. Bara, Z. Jaroszewicz, and M. Sypek, “The light sword optical element -a new diffraction structure with extended depth of focus,” J. Mod. Opt. **37**, 1283–1286 (1990). [CrossRef]

6. T. R. M. Sales and G. M. Morris, “Diffractive superresolution elements,” J. Opt. Soc. Am. A **14**, 1637–1646 (1997). [CrossRef]

11. D. M. Cottrell, J. A. Davis, T. R. Hedman, and R. A. Lilly, “Multiple imaging phase-encoded optical elements written as programmable spatial light modulators,” Appl. Opt. **29**, 2505–2509 (1990). [CrossRef] [PubMed]

14. V. F. Canales, J. E. Oti, and M. P. Cagigal, “Three-dimensional control of the focal light intensity distribution by analytically-designed phase masks,” Opt. Commun. **247**, 11–18 (2005). [CrossRef]

*et al.*, proposed and theoretically analyzed a diffractive trifocal lens design with adjustable add powers and light distribution in the foci [15

15. P. J. Valle, J. E. Oti, V. F. Canales, and M. P. Cagigal, “Visual axial PSF of diffractive trifocal lenses,” Opt. Express **13**, 2782–2792 (2005). [CrossRef] [PubMed]

## 2. Theoretical considerations

*n*concentric rings with

*n*>2, each ring being characterized by its optical power

*P*its boundary radial limits

_{m}*a*

_{m-1}and

*a*and an amplitude function

_{m}*A*. Thus, transmittance for the m-ring can be written as:

_{m}*sinc(x)*being the normalized

*sinc*function and λ

_{0}the diffractive optical element design wavelength. This expression is similar to the one obtained in [5

5. V. F. Canales and M. P. Cagigal, “Pupil filter design by using a Bessel functions basis at the image plane,” Opt. Express **14**, 10393–10402 (2006). [CrossRef] [PubMed]

13. C. Iemmi, J. Campos, J. C. Escalera, O. Lopez-Coronado, R. Gimeno, and M. J. Yzuel, “Depth of focus increase by multiplexing programmable diffractive lenses,” Opt. Express **14**, 10207–10217 (2006). [CrossRef] [PubMed]

*g*(

*m*) being a modulation transmittance function of the m-ring. In principle, there is no restriction about the

*g*(

*p*) function, whenever it is continuous in the [1,

*m*-1] interval. Different

*g*(

*p*) functions which determine the power steps between each mask zones can be used. The simplest relation is that resulting from taking

*g*(

*p*)=1, ∀

*p*, which is maintained through the remaining communication.

*a*=

_{n}*R*, we deduce that (see appendix):

_{p}*h*=0 implies that all the rings have the same area. Modifying the size of each ring will have different consequences on the energy distribution around each focus. If we consider the recurrence relation in Eq. (3), previous expression is reduced to:

*a*

^{2}

_{m}≥0, ∀

*m*, provides an upper limit to the parameter

*h*given by (see appendix):

*sinc*function state that as

*b*tends to zero, the function reaches a delta peak at

_{m}*z*=

_{m}*λ*

_{0}/

*P*. Thus, for small values of

_{m}λ*b*the axial distribution presents well defined energy peaks, each one corresponding to a different ring of the MFM. The value of

_{m}*b*increases with the number of rings, being

_{m}*m*=1 the most unfavourable case in all configurations.

*sinc*(ξ)=0 if ξ∊ℤ, the first zero on the right side of the peak number

*m*will be taken into:

*m*+1, can be found at:

*m*assuming fulfillment for

*m*=1. That is, if it is wanted to have separate foci axis, the maximum number of foci is limited by the relation:

*h*=0 are indicated there.

*A*=1) with the following parameters:

_{m}*P*=1/20 mm

_{1}^{-1},

*P*=1/25 mm

_{n}^{-1},

*R*=3.5 mm,

_{p}*h*=0 and with a number of rings from

*n*=2 to

*n*=9. According to what we explained above, we can observe there that as we approach to the limit number of foci (

*n*=8), the quality of the peaks is getting worse. For the case beyond the limit, although we obtain a series of peaks, axial aliasing does not allow multifocal implementations.

*A*in the transmittance function. In Fig. 3, we compare the same phase mask with and without amplitude modulation. In Fig. 3(a), we show the axial irradiance for the only phase case. There, we can see that for the case

_{m}*h*=0 the intensity of the peaks diminishes for larger distances from the element. For the last peak, it is observable an amplitude decay in 40%. The equivalent case in Fig. 3(b), with a factor

*A*=

_{m}*1/P*shows a clear stabilization of the energy of the peaks along the optical axis. Thus, this factor also acts as energy modulator. Different choices in

_{m}*A*have different effects on the axial distribution energy. Nevertheless, optical implementation of amplitude and phase mask is difficult, and it results easier to implement phase only filter. That is why we have introduced the area modulation factor, as we anticipated in Eq. (4). Therefore, it is feasible to find a factor

_{m}*h*that compensates the intensity of each focus. Taking the same parameters of the previous figures, the upper limit for

*h*results

*h*=0.00680556 mm.

_{C}*A*=1, we have represented in Fig. 4 the axial amplitude for a 4 zones MFM for different

_{m}*h*values. As it can be observed, changes in the value of this parameter modify the width and relative height of the peaks. One can also see that as the value of

*h*increases, the further peaks gets higher and narrower, and the contrary happens for the closer peaks. In the limit case

*h*=

*h*-see Eq. (7)- first maximum is completely lost.

_{C}16. D. Mas, J. Pérez, C. Hernández, C. Vázquez, J. J. Miret, and C. Illueca, “Fast numerical calculation of Fresnel patterns in convergent systems,” Opt. Commun. **227**, 245–258 (2003). [CrossRef]

*n*=4 zones, an optical power interval of

*P*

_{1}=50 D,

*P*=40 D, and a pupil diameter of 7.00 mm, at distances where maxima are expected (

_{4}*z*=20.00, 21.43, 23.08 and 25.00 mm) and

*h*=

*h*/9.

_{C}*et al.*, [13

13. C. Iemmi, J. Campos, J. C. Escalera, O. Lopez-Coronado, R. Gimeno, and M. J. Yzuel, “Depth of focus increase by multiplexing programmable diffractive lenses,” Opt. Express **14**, 10207–10217 (2006). [CrossRef] [PubMed]

*et al.*[17

17. G. Mikula, Z. Jaroszewicz, A. Kolodziejczyk, K. Petelczyc, and M. Sypek, “Imaging with extended focal depth by means of lenses with radial and angular modulation,” Opt. Express **15**, 9184–9193 (2007). [CrossRef] [PubMed]

*n*=1000,

*h*=

*h*/3,

_{C}*P*

_{1}=1/20 mm

^{-1}and

*P*=1/25 mm

_{n}^{-1}.

## 3. Experimental results

^{2}, the mask is addressed through a standard VGA output.

*et al.*, [18

18. J. A. Davis, I. Moreno, and P. Tsai, “Polarization Eigenstates for Twisted-Nematic Liquid-Crystal Displays,” Appl. Opt. **37**, 937–945 (1998). [CrossRef]

*D1*and

*D2*and two polarizers

*P1*and

*P2*. Previously, the angles of these four elements were properly fitted so that the phase modulation provided by the modulator was the most efficient one for used wavelength. With the aim of achieving a relatively large phase modulation, it was used an Ar laser,

*F*, of a wavelength of 488 nm together with a spatial filter,

*SF*, used to expand the beam. The lens

*L*with 25 cm of focal length allow to move the object point in order to obtain the images provided by the different ring on a fixed semitransparent screen

*S*, located at 1.25 m from the modulator. The first polarizer was placed against an iris diaphragm,

*ID*, which limits the diameter of the beam to 13.9 mm. Finally, an 8-bit camera,

*C*, captured the image on

*S*. In order to not saturating the image detected by the camera, a grey filter,

*GF*, was located to the left of the screen.

*n*=4) phase mask was implemented in the modulator. In order to conduct the experimental verification, the chosen zone powers (

*P*=1.82 D,

_{1}*P*=0.40 D) followed Eq. (3). Those values were limited by the number and size of the pixels of the used modulator. In Fig. 8 we represent the phase mask implemented in the modulator. The radial limits of the four areas were in this case: 2.32, 3.90, 5.43, 6.95 mm and the area parameter used for modulator was

_{n}*h*=

*h*/1.8=0.00211425 mm.

_{C}*et al.*, in [16

16. D. Mas, J. Pérez, C. Hernández, C. Vázquez, J. J. Miret, and C. Illueca, “Fast numerical calculation of Fresnel patterns in convergent systems,” Opt. Commun. **227**, 245–258 (2003). [CrossRef]

*S*, as we varied the position of the lens

*L*. Maximum intensity at distances,

*z*=

_{i}*1*/

*P*being

_{i}*P*the optical power of the

_{i}*i*-

*th*ring are shown. A simple application of Geometrical Optics laws indicates the required successive distances between the lens

*L*and the spatial filter in order to obtain the different foci on the screen, depending on the powers introduced in the phase modulator. Regarding the value of the maximum peak of each focus, the theoretical calculation shows that the absolute maximum of axial irradiance for the programmed parameters corresponds to the second focus, located 734 mm from the modulator, which has been given a value of 255 (Fig. 9(a)). The grey filter

*GF*has been adjusted to allow maximum response of the camera to this spot. Thus, all other registered intensities must not saturate the camera. As it can be seen in Fig. 9, the relative values between the maximum comply with the theoretical values predicted.

*h*, we introduced a new phase distribution with the same characteristics that those previously described but with

*h*=

*h*(Fig. 10).

_{C}*z*=0.550 m of focal length is lost. In Fig. 11(a) we can see the theoretical axial irradiance for the mask shown in Fig. 10.

*P*=0.873 D (z=1.146 m) marks the absolute maximum. Figure 11(b) shows the experimental radial mean of the intensity found in the three focal planes. Note that the relative values of the maxima are consistent with those theoretically predicted.

_{2}## 4. Conclusions

## Appendix

*m*=3 is used, it can be deduced that:

*a*=

_{n}*R*, we deduce that:

_{p}*g*(

*m*)=1, ∀

*m*, it results:

*a*

^{2}

*≥0, ∀m, provides an upper limit to the parameter*

_{m}*h*given by:

## Acknowledgments

## References and links

1. | A. Kolodziejczyk, S. Bara, Z. Jaroszewicz, and M. Sypek, “The light sword optical element -a new diffraction structure with extended depth of focus,” J. Mod. Opt. |

2. | H. Luo and C. Zhou, “Comparison of superresolution effects with annular phase and amplitude filters,” Appl. Opt. |

3. | J. Monsoriu, W. D. Furlan, P. Andrés, and J. Lancis, “Fractal conical lenses,” Opt. Express |

4. | I. Golub, “Fresnel axicon,” Opt. Lett. |

5. | V. F. Canales and M. P. Cagigal, “Pupil filter design by using a Bessel functions basis at the image plane,” Opt. Express |

6. | T. R. M. Sales and G. M. Morris, “Diffractive superresolution elements,” J. Opt. Soc. Am. A |

7. | A. Burvall, K. Kolacz, Z. Jaroszewicz, and A. Friberg, “Simple lens axicon,” Appl. Opt. |

8. | A. Flores, M. Wang, and J. J. Yang, “Achromatic hybrid refractive-diffractive lens with extended focal length”, Appl. Opt. |

9. | J. A. Davis, C. S. Tuvey, O. López-Coronado, J. Campos, M. J. Yzuel, and C. Iemmi, “Tailoring the depth of focus for optical imaging systems using a Fourier transform approach,” Opt. Lett. |

10. | D. Mas, J. Espinosa, J. Perez, and C. Illueca, “Three dimensional analysis of chromatic aberration in diffractive elements with extended depth of focus,” Opt. Express |

11. | D. M. Cottrell, J. A. Davis, T. R. Hedman, and R. A. Lilly, “Multiple imaging phase-encoded optical elements written as programmable spatial light modulators,” Appl. Opt. |

12. | J. Leach, G. M. Gibson, M. Padgett, E. Exposito, G. McConell, A. J. Wright, and J. M. Girkin, “Generation of achromatic Bessel beams using a compensated spatial light modulator,” Opt. Express |

13. | C. Iemmi, J. Campos, J. C. Escalera, O. Lopez-Coronado, R. Gimeno, and M. J. Yzuel, “Depth of focus increase by multiplexing programmable diffractive lenses,” Opt. Express |

14. | V. F. Canales, J. E. Oti, and M. P. Cagigal, “Three-dimensional control of the focal light intensity distribution by analytically-designed phase masks,” Opt. Commun. |

15. | P. J. Valle, J. E. Oti, V. F. Canales, and M. P. Cagigal, “Visual axial PSF of diffractive trifocal lenses,” Opt. Express |

16. | D. Mas, J. Pérez, C. Hernández, C. Vázquez, J. J. Miret, and C. Illueca, “Fast numerical calculation of Fresnel patterns in convergent systems,” Opt. Commun. |

17. | G. Mikula, Z. Jaroszewicz, A. Kolodziejczyk, K. Petelczyc, and M. Sypek, “Imaging with extended focal depth by means of lenses with radial and angular modulation,” Opt. Express |

18. | J. A. Davis, I. Moreno, and P. Tsai, “Polarization Eigenstates for Twisted-Nematic Liquid-Crystal Displays,” Appl. Opt. |

**OCIS Codes**

(110.0110) Imaging systems : Imaging systems

(110.2990) Imaging systems : Image formation theory

(220.3620) Optical design and fabrication : Lens system design

**ToC Category:**

Imaging Systems

**History**

Original Manuscript: February 14, 2008

Revised Manuscript: March 13, 2008

Manuscript Accepted: March 13, 2008

Published: March 28, 2008

**Virtual Issues**

Vol. 3, Iss. 4 *Virtual Journal for Biomedical Optics*

**Citation**

J. Perez, J. Espinosa, C. Illueca, C. Vázquez, and I. Moreno, "Real time modulable multifocality through annular optical elements," Opt. Express **16**, 5095-5106 (2008)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-7-5095

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### References

- A. Kolodziejczyk, S. Bara, Z. Jaroszewicz, and M. Sypek, "The light sword optical element -a new diffraction structure with extended depth of focus," J. Mod. Opt. 37, 1283-1286 (1990). [CrossRef]
- H. Luo and C. Zhou, "Comparison of superresolution effects with annular phase and amplitude filters," Appl. Opt. 43, 6242-6247 (2004). [CrossRef] [PubMed]
- J. Monsoriu, W. D. Furlan, P. Andrés, and J. Lancis, "Fractal conical lenses," Opt. Express 14, 9077-9082 (2006). [CrossRef] [PubMed]
- I. Golub, "Fresnel axicon," Opt. Lett. 31, 1890-1892 (2006). [CrossRef] [PubMed]
- V. F. Canales and M. P. Cagigal, "Pupil filter design by using a Bessel functions basis at the image plane," Opt. Express 14, 10393-10402 (2006). [CrossRef] [PubMed]
- T. R. M. Sales and G. M. Morris, "Diffractive superresolution elements," J. Opt. Soc. Am. A 14, 1637-1646 (1997). [CrossRef]
- A. Burvall, K. Kolacz, Z. Jaroszewicz, and A. Friberg, "Simple lens axicon," Appl. Opt. 43, 4838-4844 (2004). [CrossRef] [PubMed]
- A. Flores, M. Wang, and J. J. Yang, "Achromatic hybrid refractive-diffractive lens with extended focal length," Appl. Opt. 43, 5618-5630 (2004). [CrossRef] [PubMed]
- J. A. Davis, C. S. Tuvey, O. López-Coronado, J. Campos, M. J. Yzuel, and C. Iemmi, "Tailoring the depth of focus for optical imaging systems using a Fourier transform approach," Opt. Lett. 32, 844-846 (2007). [CrossRef] [PubMed]
- D. Mas, J. Espinosa, J. Perez, and C. Illueca, "Three dimensional analysis of chromatic aberration in diffractive elements with extended depth of focus," Opt. Express 15, 17842-17854 (2007). [CrossRef] [PubMed]
- D. M. Cottrell, J. A. Davis, T. R. Hedman, and R. A. Lilly, "Multiple imaging phase-encoded optical elements written as programmable spatial light modulators," Appl. Opt. 29, 2505-2509 (1990). [CrossRef] [PubMed]
- J. Leach, G. M. Gibson, M. Padgett, E. Exposito, G. McConell, A. J. Wright, and J. M. Girkin, "Generation of achromatic Bessel beams using a compensated spatial light modulator," Opt. Express 14, 5581-5587 (2006). [CrossRef] [PubMed]
- C. Iemmi, J. Campos, J. C. Escalera, O. Lopez-Coronado, R. Gimeno, and M. J. Yzuel, "Depth of focus increase by multiplexing programmable diffractive lenses," Opt. Express 14, 10207-10217 (2006). [CrossRef] [PubMed]
- V. F. Canales, J. E. Oti, and M. P. Cagigal, "Three-dimensional control of the focal light intensity distribution by analytically-designed phase masks," Opt. Commun. 247, 11-18 (2005). [CrossRef]
- P. J. Valle, J. E. Oti, V. F. Canales, and M. P. Cagigal, "Visual axial PSF of diffractive trifocal lenses," Opt. Express 13, 2782-2792 (2005). [CrossRef] [PubMed]
- D. Mas, J. Pérez, C. Hernández, C. Vázquez, J. J. Miret, and C. Illueca, "Fast numerical calculation of Fresnel patterns in convergent systems," Opt. Commun. 227, 245-258 (2003). [CrossRef]
- G. Mikula, Z. Jaroszewicz, A. Kolodziejczyk, K. Petelczyc, and M. Sypek, "Imaging with extended focal depth by means of lenses with radial and angular modulation," Opt. Express 15, 9184-9193 (2007). [CrossRef] [PubMed]
- J. A. Davis, I. Moreno, and P. Tsai, "Polarization Eigenstates for Twisted-Nematic Liquid-Crystal Displays," Appl. Opt. 37, 937-945 (1998). [CrossRef]

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