## Optimization of refractive liquid crystal lenses using an efficient multigrid simulation |

Optics Express, Vol. 20, Issue 10, pp. 11159-11165 (2012)

http://dx.doi.org/10.1364/OE.20.011159

Acrobat PDF (1012 KB)

### Abstract

Abstract: **A multigrid** computational model has been developed to assess the performance of refractive liquid crystal lenses, which is up to 40 times faster than previous techniques. Using this model, the optimum geometries producing an ideal parabolic voltage distribution were deduced for refractive liquid crystal lenses with diameters from 1 to 9 mm. The ratio of insulation thickness to lens diameter was determined to be 1:2 for small diameter lenses, tending to 1:3 for larger lenses. The model is used to propose a new method of lens operation with lower operating voltages needed to induce specific optical powers. The operating voltages are calculated for the induction of optical powers between + 1.00 D and + 3.00 D in a 3 mm diameter lens, with the speed of the simulation facilitating the optimization of the refractive index profile. We demonstrate that the relationship between additional applied voltage and optical power is approximately linear for optical powers under + 3.00 D. The versatility of the computational simulation has also been demonstrated by modeling of in-plane electrode liquid crystal devices.

© 2012 OSA

## 1. Introduction

1. G. Li, D. L. Mathine, P. Valley, P. Ayräs, J. N. Haddock, M. S. Giridhar, G. Williby, J. Schwiegerling, G. R. Meredith, B. Kippelen, S. Honkanen, and N. Peyghambarian, “Switchable electro-optic diffractive lens with high efficiency for ophthalmic applications,” Proc. Natl. Acad. Sci. U.S.A. **103**(16), 6100–6104 (2006). [CrossRef] [PubMed]

2. S. Sato, “Liquid-crystal lens-cells with variable focal length,” Jpn. J. Appl. Phys. **18**(9), 1679–1684 (1979). [CrossRef]

3. H. W. Ren, D. W. Fox, B. Wu, and S. T. Wu, “Liquid crystal lens with large focal length tunability and low operating voltage,” Opt. Express **15**(18), 11328–11335 (2007). [CrossRef] [PubMed]

8. S. Sato, “Applications of liquid crystals to variable-focusing lenses,” Opt. Rev. **6**(6), 471–485 (1999). [CrossRef]

1. G. Li, D. L. Mathine, P. Valley, P. Ayräs, J. N. Haddock, M. S. Giridhar, G. Williby, J. Schwiegerling, G. R. Meredith, B. Kippelen, S. Honkanen, and N. Peyghambarian, “Switchable electro-optic diffractive lens with high efficiency for ophthalmic applications,” Proc. Natl. Acad. Sci. U.S.A. **103**(16), 6100–6104 (2006). [CrossRef] [PubMed]

9. G. Q. Li, P. Valley, P. Ayras, D. L. Mathine, S. Honkanen, and N. Peyghambarian, “High-efficiency switchable flat diffractive ophthalmic lens with three-layer electrode pattern and two-layer via structures,” Appl. Phys. Lett. **90**(11), 111105 (2007). [CrossRef]

*n*) is reduced at the edges of the optical area. This results in a refractive focus along one polarisation vector. A parabolic voltage distribution can be applied using various techniques, including a curved electrode system [3

_{e}3. H. W. Ren, D. W. Fox, B. Wu, and S. T. Wu, “Liquid crystal lens with large focal length tunability and low operating voltage,” Opt. Express **15**(18), 11328–11335 (2007). [CrossRef] [PubMed]

4. H. W. Ren and S. T. Wu, “Adaptive liquid crystal lens with large focal length tunability,” Opt. Express **14**(23), 11292–11298 (2006). [CrossRef] [PubMed]

5. A. F. Naumov, G. D. Love, M. Y. Loktev, and F. L. Vladimirov, “Control optimization of spherical modal liquid crystal lenses,” Opt. Express **4**(9), 344–352 (1999). [CrossRef] [PubMed]

6. M. Ye and S. Sato, “Optical properties of liquid crystal lens of any size,” Jpn. J. Appl. Phys. **41**(Part 2, No. 5B), L571–L573 (2002). [CrossRef]

7. M. Ye, B. Wang, and S. Sato, “Liquid-crystal lens with a focal length that is variable in a wide range,” Appl. Opt. **43**(35), 6407–6412 (2004). [CrossRef] [PubMed]

10. B. Wang, M. Ye, and S. Sato, “Liquid crystal lens with focal length variable from negative to positive values,” IEEE Photon. Technol. Lett. **18**(1), 79–81 (2006). [CrossRef]

1. G. Li, D. L. Mathine, P. Valley, P. Ayräs, J. N. Haddock, M. S. Giridhar, G. Williby, J. Schwiegerling, G. R. Meredith, B. Kippelen, S. Honkanen, and N. Peyghambarian, “Switchable electro-optic diffractive lens with high efficiency for ophthalmic applications,” Proc. Natl. Acad. Sci. U.S.A. **103**(16), 6100–6104 (2006). [CrossRef] [PubMed]

9. G. Q. Li, P. Valley, P. Ayras, D. L. Mathine, S. Honkanen, and N. Peyghambarian, “High-efficiency switchable flat diffractive ophthalmic lens with three-layer electrode pattern and two-layer via structures,” Appl. Phys. Lett. **90**(11), 111105 (2007). [CrossRef]

6. M. Ye and S. Sato, “Optical properties of liquid crystal lens of any size,” Jpn. J. Appl. Phys. **41**(Part 2, No. 5B), L571–L573 (2002). [CrossRef]

## 2. Electric field and potential calculations

*ϕ*denotes the electric potential, ε is the permittivity of the material(s) and x and z are coordinates shown in Fig. 2 . In terms of computational modeling, standard numerical techniques are utilized using MATLAB (R2008A, MathsWorks, Natick, USA), including a successive over-relaxation (SOR) solution and red black ordering. The multigrid algorithm reduces simulation time by initially solving the equation numerically for a coarse grid, with further iterations on progressively finer grids until an adequately accurate solution is realized [11].

## 3. Simulation of RLCLs

*n*) is calculated for each electric potential simulation point across the top of the LC layer, giving an approximation of the refractive profile. In the following simulations, the dielectric and refractive properties of 4-cyano-4-pentylbiphenyl (5CB) are used for modeling the LC layer. While this 1-dimensional model neglects non-linear director field interactions, the approximations are acceptable for providing quick estimates of operating voltages and geometries rather than in-depth calculations. A typical simulation of a 3 mm diameter RLCL is shown in Fig. 3 . The electric potential in the device can be calculated using the multigrid model and the electric potential across the LC layer is deduced. The thickness of the glass and the active and control voltages have been set to produce a parabolic voltage profile.

_{e}6. M. Ye and S. Sato, “Optical properties of liquid crystal lens of any size,” Jpn. J. Appl. Phys. **41**(Part 2, No. 5B), L571–L573 (2002). [CrossRef]

*n*in the centre of the lens, rather than varying from

_{e}*n*in the lens peripheries as used previously. Figure 5(b) shows the calculated operating voltages for a 3 mm diameter RLCL with a 50 µm 5CB LC layer giving optical powers between + 0.50 D and + 3.00 D. This new method of driving the lens reduces the operating voltages for low optical powers. Furthermore, the voltages can be tuned to give the most parabolic refractive profile, hence optimizing the optical quality. The calculation speed is again vital in optimizing the refractive profile due to the number of calculations required. In these lenses, an approximately linear relationship between the variable voltage and the additional optical power was observed, with each additional + 0.50 D corresponding to an increase of approximately 4 V. It was noted that at the peripheries of the lens the potential profile deviated from a parabola as the lens became more powerful, as shown in Fig. 6 . This is expected when a large change in

_{o}*n*is required for lenses of high optical powers, as the relationship between

_{e}*n*and potential is increasingly non-linear at high voltages.

_{e}## 4. Further applications

## 5. Conclusions

*n*in the centre, rather than from

_{e}*n*at the lens periphery. The speed of the multigrid simulation is vital in the optimization of the refractive profile, which maximizes the parabolicity of the refractive index profile. The model shows an approximately linear relationship between active electrode operating voltage and the optical power of the lens. Details of the limitations of RLCL performance are also shown using the model, with director profiles of high powered lenses exhibiting deviation from the parabolic distribution at the peripheries due to the increasingly non-linear relationship between

_{o}*n*and applied voltage at higher voltages. Finally we illustrate the versatility of our model with the simulation of in-plane electrodes systems.

_{e}## Acknowledgments

## References and links

1. | G. Li, D. L. Mathine, P. Valley, P. Ayräs, J. N. Haddock, M. S. Giridhar, G. Williby, J. Schwiegerling, G. R. Meredith, B. Kippelen, S. Honkanen, and N. Peyghambarian, “Switchable electro-optic diffractive lens with high efficiency for ophthalmic applications,” Proc. Natl. Acad. Sci. U.S.A. |

2. | S. Sato, “Liquid-crystal lens-cells with variable focal length,” Jpn. J. Appl. Phys. |

3. | H. W. Ren, D. W. Fox, B. Wu, and S. T. Wu, “Liquid crystal lens with large focal length tunability and low operating voltage,” Opt. Express |

4. | H. W. Ren and S. T. Wu, “Adaptive liquid crystal lens with large focal length tunability,” Opt. Express |

5. | A. F. Naumov, G. D. Love, M. Y. Loktev, and F. L. Vladimirov, “Control optimization of spherical modal liquid crystal lenses,” Opt. Express |

6. | M. Ye and S. Sato, “Optical properties of liquid crystal lens of any size,” Jpn. J. Appl. Phys. |

7. | M. Ye, B. Wang, and S. Sato, “Liquid-crystal lens with a focal length that is variable in a wide range,” Appl. Opt. |

8. | S. Sato, “Applications of liquid crystals to variable-focusing lenses,” Opt. Rev. |

9. | G. Q. Li, P. Valley, P. Ayras, D. L. Mathine, S. Honkanen, and N. Peyghambarian, “High-efficiency switchable flat diffractive ophthalmic lens with three-layer electrode pattern and two-layer via structures,” Appl. Phys. Lett. |

10. | B. Wang, M. Ye, and S. Sato, “Liquid crystal lens with focal length variable from negative to positive values,” IEEE Photon. Technol. Lett. |

11. | U. Trottenberg, |

**OCIS Codes**

(220.3630) Optical design and fabrication : Lenses

(230.3720) Optical devices : Liquid-crystal devices

**ToC Category:**

Optical Design and Fabrication

**History**

Original Manuscript: March 22, 2012

Revised Manuscript: April 27, 2012

Manuscript Accepted: April 27, 2012

Published: April 30, 2012

**Citation**

Harry Milton, Paul Brimicombe, Philip Morgan, Helen Gleeson, and John Clamp, "Optimization of refractive liquid crystal lenses using an efficient multigrid simulation," Opt. Express **20**, 11159-11165 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-10-11159

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### References

- G. Li, D. L. Mathine, P. Valley, P. Ayräs, J. N. Haddock, M. S. Giridhar, G. Williby, J. Schwiegerling, G. R. Meredith, B. Kippelen, S. Honkanen, and N. Peyghambarian, “Switchable electro-optic diffractive lens with high efficiency for ophthalmic applications,” Proc. Natl. Acad. Sci. U.S.A.103(16), 6100–6104 (2006). [CrossRef] [PubMed]
- S. Sato, “Liquid-crystal lens-cells with variable focal length,” Jpn. J. Appl. Phys.18(9), 1679–1684 (1979). [CrossRef]
- H. W. Ren, D. W. Fox, B. Wu, and S. T. Wu, “Liquid crystal lens with large focal length tunability and low operating voltage,” Opt. Express15(18), 11328–11335 (2007). [CrossRef] [PubMed]
- H. W. Ren and S. T. Wu, “Adaptive liquid crystal lens with large focal length tunability,” Opt. Express14(23), 11292–11298 (2006). [CrossRef] [PubMed]
- A. F. Naumov, G. D. Love, M. Y. Loktev, and F. L. Vladimirov, “Control optimization of spherical modal liquid crystal lenses,” Opt. Express4(9), 344–352 (1999). [CrossRef] [PubMed]
- M. Ye and S. Sato, “Optical properties of liquid crystal lens of any size,” Jpn. J. Appl. Phys.41(Part 2, No. 5B), L571–L573 (2002). [CrossRef]
- M. Ye, B. Wang, and S. Sato, “Liquid-crystal lens with a focal length that is variable in a wide range,” Appl. Opt.43(35), 6407–6412 (2004). [CrossRef] [PubMed]
- S. Sato, “Applications of liquid crystals to variable-focusing lenses,” Opt. Rev.6(6), 471–485 (1999). [CrossRef]
- G. Q. Li, P. Valley, P. Ayras, D. L. Mathine, S. Honkanen, and N. Peyghambarian, “High-efficiency switchable flat diffractive ophthalmic lens with three-layer electrode pattern and two-layer via structures,” Appl. Phys. Lett.90(11), 111105 (2007). [CrossRef]
- B. Wang, M. Ye, and S. Sato, “Liquid crystal lens with focal length variable from negative to positive values,” IEEE Photon. Technol. Lett.18(1), 79–81 (2006). [CrossRef]
- U. Trottenberg, Multigrid, 1st ed. (Academic Press, London, 2001).

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