## Control optimization of spherical modal liquid crystal lenses

Optics Express, Vol. 4, Issue 9, pp. 344-352 (1999)

http://dx.doi.org/10.1364/OE.4.000344

Acrobat PDF (399 KB)

### Abstract

Liquid crystal modal lenses are switchable lenses with a continuous phase variation across the lens. A critical issue for such lenses is the minimization of phase aberrations. In this paper we present results of a simulation of control signals that have a range of harmonics. Experimental results using optimal sinusoidal and rectangular voltages are presented. A lack of uniqueness in the specification of the control voltage parameters is explained. The influence of a variable duty cycle of the control voltage on an adaptive lens is investigated. Finally we present experimental results showing a liquid crystal lens varying its focal length.

© Optical Society of America

## 1. Introduction

1. V. Laude, “Twisted-nematic liquid-crystal pixelated active lens,” Opt. Comm. **153**, 134–152 (1998). [CrossRef]

4. C. W. Fowler and E. S. Pateras, “Liquid crystal lens review,” Ophthal. Physiol. Opt. **10**, 186–194 (1990). [CrossRef]

5. S. Masuda, S. Takahashi, T. Nose, S. Sato, and H. Ito, “Liquid-crystal microlens with a beam-steering function,” Appl. Opt. **36**, 4772–4778 (1997). [CrossRef] [PubMed]

7. A. F. Naumov and G. V. Vdovin, “Multichannel LC-based wavefront corrector with modal influence functions,” Opt. Lett. **23**, 1550–1552 (1998). [CrossRef]

9. A. F. Naumov, M. Yu. Loktev, I. R. Guralnik, and G. V. Vdovin, “Liquid crystal adaptive lenses with modal control,” Opt. Lett. **23**, 992–994 (1998). [CrossRef]

## 2. Operation principle and numerical optimization of control voltage parameters

10. G. D. Love, J. V. Major, and A. Purvis, “Liquid-crystal prisms for tip-tilt adaptive optics,” Opt. Lett. **19**, 1170–1172 (1994). [CrossRef] [PubMed]

12. G. D. Love, “Liquid-crystal phase modulator for unpolarized light,” Appl. Opt. **32**, 2222–2223 (1993). [CrossRef] [PubMed]

_{c}and ΔΦ

_{e}are the retardances, measured at the center and edges respectively,

*l*is the MLCL radius, and

*λ*is the wavelength of incident light. The distribution of the rms voltage across the lens is described by Bessel functions [14

14. A. F. Naumov, M. Yu. Loktev, and I. R. Guralnik, “Cylindrical and spherical adaptive liquid crystal lenses,” SPIE **3684**, pp.18–27 (1998). [CrossRef]

*U*) is approximately an inversion logarithmic function. In general, if a voltage of arbitrary magnitude and phase is applied to the cell, then the resulting phase distribution will be far from parabolic, and the lens aberrations will be significant. However, for a certain relationship between the frequency of the applied voltage and the distributed impedance, the retardance distribution is close to a paraboloid.

9. A. F. Naumov, M. Yu. Loktev, I. R. Guralnik, and G. V. Vdovin, “Liquid crystal adaptive lenses with modal control,” Opt. Lett. **23**, 992–994 (1998). [CrossRef]

*m*additional harmonics into the control signal. In this case each harmonic will contribute to the phase distribution across the aperture and by optimizing the strength of each harmonic the total rms phase deviation can be minimized. For the control voltage

*U*

_{0}

*α*

_{k}sin(

*ω*

_{k}

*t*) the computed results obtained by means of a Monte-Carlo method with the descent technique [17

17. B. R. Frieden (ed.), *The Computer in Optical Research. Methods and Applications* (Springer-Verlag, Berlin, Heidelberg, New York, 1980). [CrossRef]

## 3. Lens calibration

*λ*= 0.633 μm), in order to visualize the phase distribution across the lens. The resulting intensity distribution was imaged by an objective lens onto a CCD camera and subsequently processed by a PC. The recovered wavefront was compared with an ideal one, which corresponded to the desired focal length. Using the same descent technique as before we searched for the voltage-frequency values, which produced a phase profile close to parabolic. The experimental apparatus shown in Fig. 4a allowed the calibration of focal lengths from 0.5 m to 1 m. To improve the sensitivity for longer focal lengths (1–4 m) we used a double pass optical set-up, which is shown in Fig. 4b.

*i*and

*j*are matrix indices of the image

*A*

_{ij}. The intensity distribution

*B*

_{j}=

*A*

_{i0j}in the selected section was smoothed by a convolution with the following Gaussian function in order to remove noise caused by the coherence of the laser beam (speckle noise),

_{k}= -2

*πk*/

*z*, where

*z*is a number of passes of the laser beam through the LC layer and

*k*is the fringe order. A continuous phase distribution was built up by spline interpolation. The phase distribution in the center was approximated by a parabola constructed through the 4 minima nearest to the center points. This phase distribution was compared with the ideal parabolic phase profile and rms deviation was calculated. To calibrate each focal length, approximately 200 steps are required with time delays between them (to allow for the finite LC response time). After calibration for 3–5 focal lengths the approximate dependencies of voltage and frequency on focal distance

*U(F)*and

*f(F)*are plotted by means of linear interpolation. However the real signal parameters in the intermediate points are then slightly inaccurate. These values were corrected as follows. The control voltage with the approximate parameters

*U′*and

*f′*for some intermediate focal length

*F′*is applied to the lens. The interferogram obtained in the optical set-up was then recorded and the wavefront was recovered. The real focal length

*F′′*, determined from the recovered wavefront, was used as a calibration parameter for the given values

*U′*and

*f′*.

*U*. For example, to obtain the desired phase profile with depth ΔΦ

_{0}<< ΔΦ

_{max}(see Fig. 7) it is possible to use more than one pair of optimal parameters: voltage - frequency. The voltage applied to the periphery of high-resistance control electrode voltage of

*U*

_{1}with the frequency

*f*

_{1}produces the voltage of

*U*

_{2}at the lens aperture center. The variation of voltages from

*U*

_{1}to

*U*

_{2}gives the phase variation equal to ΔΦ

_{0}in the experimental error limit. The same result can be obtained for the voltage of

*U*

_{3}with the frequency

*f*

_{2}, applied to the angular contact. In this case the distribution of voltages from

*U*

_{3}at the edge decreases to

*U*

_{4}at the center and gives the same phase variation in the measure error limit.

## 4. Lens control by variation of control voltage duty cycle.

*q*=

*τ*/

*T*, where

*τ*is the time during which the voltage is positive, and

*T*is the time period. The variation of

*q*leads to a variation of the control voltage spectrum. Figure 9 shows interferograms obtained with the single-pass experimental set-up using different duty cycles of a control voltage with 9 V amplitude and 4 kHz frequency. The values of

*q*are indicated in the upper left corner on inserts. Because the voltage is bipolar the signal spectrum was varied using

*q*from 0.5 to 1, since values from 0 to 0.5 will give identical results. The focal length and the rms phase deviation from a perfect lens are shown on each interferogram.

*q*parameter in order to obtain minimal values of the rms phase deviation. The dependence of

*q*on

*F*will be essentially nonlinear. Additionally, the variation of

*q*does not produce the focal length variation over the full range. However the duty cycle control method can be used to minimize lens aberrations further, in a lens that has been calibrated by the usual method for voltages and frequencies at

*q*= 0.5.

## 5. Technical parameters

## 6. Conclusion

## Acknowledgements

## References and links

1. | V. Laude, “Twisted-nematic liquid-crystal pixelated active lens,” Opt. Comm. |

2. | W. W. Chan and S. T. Kowel, ”Imaging performance of the liquid-crystal-adaptive lens with conductive ladder meshing,” Appl. Opt. |

3. | J. S. Patel and K. Rastani, “Electrically controlled polarization-independent liquid-crystal Fresnel lens arrays,” Opt. Lett. |

4. | C. W. Fowler and E. S. Pateras, “Liquid crystal lens review,” Ophthal. Physiol. Opt. |

5. | S. Masuda, S. Takahashi, T. Nose, S. Sato, and H. Ito, “Liquid-crystal microlens with a beam-steering function,” Appl. Opt. |

6. | A. F. Naumov, “Modal wavefront correctors,” Proc. of P. N. Lebedev Phys. Inst. |

7. | A. F. Naumov and G. V. Vdovin, “Multichannel LC-based wavefront corrector with modal influence functions,” Opt. Lett. |

8. | E. G. Abramochkin, A. A. Vasiliev, P. V. Vashurin, L. I. Zhmurova, V. A. Ignatov, and A. F. Naumov, “Controlled liquid crystal lens,” preprint of P. N. Lebedev Phys. Inst. |

9. | A. F. Naumov, M. Yu. Loktev, I. R. Guralnik, and G. V. Vdovin, “Liquid crystal adaptive lenses with modal control,” Opt. Lett. |

10. | G. D. Love, J. V. Major, and A. Purvis, “Liquid-crystal prisms for tip-tilt adaptive optics,” Opt. Lett. |

11. | F. L. Vladimirov, I. E. Morichev, L. I. Petrova, and N. I. Pletneva, “Analog indicator based on liquid crystals,” Opto-Mekhanicheskaja Promishlennost |

12. | G. D. Love, “Liquid-crystal phase modulator for unpolarized light,” Appl. Opt. |

13. | J. W. Goodman, |

14. | A. F. Naumov, M. Yu. Loktev, and I. R. Guralnik, “Cylindrical and spherical adaptive liquid crystal lenses,” SPIE |

15. | L. M. Blinov, |

16. | A. F. Naumov, M. Yu. Loktev, I. R. Guralnik, S. V. Sheyenkov, and G. V. Vdovin, ”Modal liquid crystal adaptive lenses,” preprint of P. N. Lebedev Phys. Inst. |

17. | B. R. Frieden (ed.), |

**OCIS Codes**

(010.1080) Atmospheric and oceanic optics : Active or adaptive optics

(230.3720) Optical devices : Liquid-crystal devices

**ToC Category:**

Research Papers

**History**

Original Manuscript: March 31, 1999

Published: April 26, 1999

**Citation**

A. Naumov, Gordon Love, M. Yu. Loktev, and F. Vladimirov, "Control optimization of
spherical modal liquid crystal lenses," Opt. Express **4**, 344-352 (1999)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-4-9-344

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### References

- V. Laude, "Twisted-nematic liquid-crystal pixelated active lens," Opt. Comm. 153, 134-152 (1998). [CrossRef]
- W. W. Chan and S. T. Kowel, "Imaging performance of the liquid-crystal-adaptive lens with conductive ladder meshing," Appl. Opt. 36, 8958-8969 (1997). [CrossRef]
- J. S. Patel and K. Rastani, "Electrically controlled polarization-independent liquid-crystal Fresnel lens arrays," Opt. Lett. 16, 532-534 (1991). [CrossRef] [PubMed]
- C. W. Fowler and E. S. Pateras, "Liquid crystal lens review," Ophthal. Physiol. Opt. 10, 186-194 (1990). [CrossRef]
- S. Masuda, S. Takahashi, T. Nose, S. Sato and H. Ito, "Liquid-crystal microlens with a beam-steering function," Appl. Opt. 36, 4772-4778 (1997). [CrossRef] [PubMed]
- A. F. Naumov, "Modal wavefront correctors," Proc. of P. N. Lebedev Phys. Inst. 217, 177-182 (1993).
- A. F. Naumov, G. V. Vdovin, "Multichannel LC-based wavefront corrector with modal influence functions," Opt. Lett. 23, 1550-1552 (1998). [CrossRef]
- E. G. Abramochkin, A. A. Vasiliev, P. V. Vashurin, L. I. Zhmurova, V. A. Ignatov and A. F. Naumov, "Controlled liquid crystal lens," preprint of P. N. Lebedev Phys. Inst. 194, 18p. (1988).
- A. F. Naumov, M. Yu. Loktev, I. R. Guralnik and G. V. Vdovin, "Liquid crystal adaptive lenses with modal control," Opt. Lett. 23, 992-994 (1998). [CrossRef]
- G. D. Love, J. V. Major and A. Purvis, "Liquid-crystal prisms for tip-tilt adaptive optics," Opt. Lett. 19, 1170-1172 (1994). [CrossRef] [PubMed]
- F. L. Vladimirov, I. E. Morichev, L. I. Petrova and N. I. Pletneva, "Analog indicator based on liquid crystals," Opto-Mekhanicheskaja Promishlennost 3, 27-28 (1987).
- G. D. Love, "Liquid-crystal phase modulator for unpolarized light," Appl. Opt. 32, 2222-2223 (1993). [CrossRef] [PubMed]
- J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill Book Company, New York, 1968).
- A. F. Naumov, M. Yu. Loktev and I. R. Guralnik, "Cylindrical and spherical adaptive liquid crystal lenses," SPIE 3684, pp.18-27 (1998). [CrossRef]
- L. M. Blinov, Electro-Optical and Magneto-Optical Properties of Liquid Crystals (Wiley, New York, 1983).
- A. F. Naumov, M. Yu. Loktev, I. R. Guralnik, S. V. Sheyenkov and G. V. Vdovin, "Modal liquid crystal adaptive lenses," preprint of P. N. Lebedev Phys. Inst. 13, 28p. (1998).
- B. R. Frieden (ed.), The Computer in Optical Research. Methods and Applications (Springer-Verlag, Berlin, Heidelberg, New York, 1980). [CrossRef]

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