## Form birefringence in nanostructured micro-optical devices |

Optical Materials Express, Vol. 1, Issue 7, pp. 1251-1261 (2011)

http://dx.doi.org/10.1364/OME.1.001251

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### Abstract

We present a detailed examination of the design and expected operation of an artificially birefringent material based around the nanostructured stack-and-draw fabrication technique developed recently. The expected degree of birefringence is estimated using a Finite Difference Time Domain simulation of the physical system and is shown to be in agreement with that predicted by a second order effective medium theory treatment of the nanostructured material. The effects of finite device dimensions are studied and an estimate of the required device thickness for a half-wave retardation is made.

© 2011 OSA

## 1. Introduction

1. F. Hudelist, R. Buczynski, A. J. Waddie, and M. R. Taghizadeh, “Design and fabrication of nano-structured gradient index microlenses,” Opt. Express **17**(5), 3255–3263 (2009). [CrossRef] [PubMed]

2. N. Lu, D. Kuang, and G. Mu, “Design of transmission blazed binary gratings for optical limiting with the form-birefringence theory,” Appl. Opt. **47**(21), 3743–3750 (2008). [CrossRef] [PubMed]

## 2. Nanostructured micro-optic element fabrication

*n*) and glass transition temperatures (

_{d}*T*).

_{g}3. L. Li, “New formulation of the Fourier modal method for crossed surface-relief gratings,” J. Opt. Soc. Am. A **14**(10), 2758–2767 (1997). [CrossRef]

4. P. Lalanne, “Improved formulation of the coupled-wave method for two-dimensional gratings,” J. Opt. Soc. Am. A **14**(7), 1592–1598 (1997). [CrossRef]

5. K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propag. **14**, 302–307 (1966). [CrossRef]

3. L. Li, “New formulation of the Fourier modal method for crossed surface-relief gratings,” J. Opt. Soc. Am. A **14**(10), 2758–2767 (1997). [CrossRef]

5. K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propag. **14**, 302–307 (1966). [CrossRef]

*μ*m) focal length microlenses. These optical components, which have been shown to exhibit diffraction limited achromatic performance [7

7. F. Hudelist, J. M. Nowosielski, R. Buczynski, A. J. Waddie, and M. R. Taghizadeh, “Nanostructured elliptical gradient-index microlenses,” Opt. Lett. **35**(2), 130–132 (2010). [CrossRef] [PubMed]

8. J. M. Nowosielski, R. Buczynski, F. Hudelist, A. J. Waddie, and M. R. Taghizadeh, “Nanostructured GRIN microlenses for Gaussian beam focusing,” Opt. Commun. **283**(9), 1938–1944 (2010). [CrossRef]

## 3. Birefringence in one-dimensional nanometre-scale microstructures

_{1}) to the other (refractive index n

_{2}) within a slab of one dimensional nanostructured material, the refractive index of the composite slab can be varied across the full range of refractive indices from n

_{1}to n

_{2}. First order effective medium theory [9

9. A. Sihvola, *Electromagnetic Mixing Formulas and Applications* (IEE, 1999). [CrossRef]

*δ*is the relative proportion of the high index material

*ɛ*. In order to test the validity of this theory, a series of 2D FDTD simulations (with the non-interacting groups of EM field components (

_{i}*H*,

_{x}*H*,

_{y}*E*) and (

_{z}*E*,

_{x}*E*,

_{y}*H*)) of the set of one dimensional arrangements of the two LIC glasses - shown in Fig. 2 - were performed for both transverse electric illumination (TE) - where the electric field component is perpendicular to the index variation in the 1D structure - and transverse magnetic illumination (TM) - where the electric field component is parallel to the 1D structure index variation. The total number of separate inclusions in these simulations was set to be ten - this number adequately demonstrates the desired effect (which requires an inclusion size of <

_{z}*λ*/5 [7

7. F. Hudelist, J. M. Nowosielski, R. Buczynski, A. J. Waddie, and M. R. Taghizadeh, “Nanostructured elliptical gradient-index microlenses,” Opt. Lett. **35**(2), 130–132 (2010). [CrossRef] [PubMed]

*δ*) moves towards 0.5 (corresponding, for the simulations presented here, to 5 high index inclusions). At this point, the first order Maxwell-Garnett formula is no longer valid and instead a 2

*order theory [10*

^{nd}10. I. Richter, P.-C. Sun, F. Xu, and Y. Fainman, “Design considerations of form birefringent microstructures,” Appl. Opt. **34**(14), 2421–2429 (1995). [CrossRef] [PubMed]

*λ*is the wavelength of the incident illumination. It should be noted that the second order effective medium theory is only valid where Λ <

*λ*-when this condition is not satisfied a fully vectorial solution to Maxwell’s curl equations must be used to calculate the effective refractive index of the material.

*πdnt*)/

*λ*between the TE and TM polarisations) over many tens of nanometres can be achieved. The green curve in Fig. 5 is a 1/

*λ*fit to the

*dn*curve for 300nm wavelength bands (i.e. 500–800nm, 800–1100nm, 1100–1400nm, 1400–1700nm and 1700–2000nm). The inverse wavelength fit is excellent in the upper wavelength region (> 800nm) demonstrating that the material is capable of giving constant birefringent operation over several hundred nanometres in the near infra-red.

*λ*/20. It can be seen that for each wavelength of light, there is a definite threshold beyond which the expected level of refractive index difference is significantly reduced. This reduction in dn is due to the structure ceasing to operate as a true effective medium and instead beginning to function as a scalar domain diffraction grating - as shown by the inset figures for an incident wavelength of light of 1000nm. The value of this threshold was determined numerically by fitting a sigmoid-like curve to the simulated dn curves. The fitting curves, shown by the solid lines in Fig. 6, are where Δ

_{Λ}is the large period normalised steady-state dn,

*T*

_{Λ}is the threshold level, Λ is the nanostructure period and

*λ*is the illumination wavelength of light. The variation of the large period normalised steady state dn and the threshold level from the fitting curves (solid lines in Fig. 6) obey a linear trend and are given by, It can be readily seen from Fig. 6 that the large period steady state dn is not in actual fact a constant value, but the fitting curves give an over-estimate of the large period birefringence and therefore this value should be regarded as an upper bound to the expected large period birefringence.

## 4. Conclusions

## Acknowledgments

## References and links

1. | F. Hudelist, R. Buczynski, A. J. Waddie, and M. R. Taghizadeh, “Design and fabrication of nano-structured gradient index microlenses,” Opt. Express |

2. | N. Lu, D. Kuang, and G. Mu, “Design of transmission blazed binary gratings for optical limiting with the form-birefringence theory,” Appl. Opt. |

3. | L. Li, “New formulation of the Fourier modal method for crossed surface-relief gratings,” J. Opt. Soc. Am. A |

4. | P. Lalanne, “Improved formulation of the coupled-wave method for two-dimensional gratings,” J. Opt. Soc. Am. A |

5. | K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propag. |

6. | A. Taflove and S. C. Hagness, |

7. | F. Hudelist, J. M. Nowosielski, R. Buczynski, A. J. Waddie, and M. R. Taghizadeh, “Nanostructured elliptical gradient-index microlenses,” Opt. Lett. |

8. | J. M. Nowosielski, R. Buczynski, F. Hudelist, A. J. Waddie, and M. R. Taghizadeh, “Nanostructured GRIN microlenses for Gaussian beam focusing,” Opt. Commun. |

9. | A. Sihvola, |

10. | I. Richter, P.-C. Sun, F. Xu, and Y. Fainman, “Design considerations of form birefringent microstructures,” Appl. Opt. |

11. | S. D. Gedney, “An anisotropic perfectly matched layer absorbing media for the truncation of FDTD lattices,” IEEE Trans. Antennas Propag. |

**OCIS Codes**

(230.4000) Optical devices : Microstructure fabrication

(050.2065) Diffraction and gratings : Effective medium theory

(050.2555) Diffraction and gratings : Form birefringence

**ToC Category:**

Artificially Engineered Structures

**History**

Original Manuscript: August 10, 2011

Revised Manuscript: October 4, 2011

Manuscript Accepted: October 9, 2011

Published: October 13, 2011

**Citation**

A. J. Waddie, R. Buczynski, F. Hudelist, J. Nowosielski, D. Pysz, R. Stepien, and M. R. Taghizadeh, "Form birefringence in nanostructured micro-optical devices," Opt. Mater. Express **1**, 1251-1261 (2011)

http://www.opticsinfobase.org/ome/abstract.cfm?URI=ome-1-7-1251

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### References

- F. Hudelist, R. Buczynski, A. J. Waddie, and M. R. Taghizadeh, “Design and fabrication of nano-structured gradient index microlenses,” Opt. Express17(5), 3255–3263 (2009). [CrossRef] [PubMed]
- N. Lu, D. Kuang, and G. Mu, “Design of transmission blazed binary gratings for optical limiting with the form-birefringence theory,” Appl. Opt.47(21), 3743–3750 (2008). [CrossRef] [PubMed]
- L. Li, “New formulation of the Fourier modal method for crossed surface-relief gratings,” J. Opt. Soc. Am. A14(10), 2758–2767 (1997). [CrossRef]
- P. Lalanne, “Improved formulation of the coupled-wave method for two-dimensional gratings,” J. Opt. Soc. Am. A14(7), 1592–1598 (1997). [CrossRef]
- K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propag.14, 302–307 (1966). [CrossRef]
- A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite Difference Time-Domain Method, 3rd ed. (Artech House, 2005).
- F. Hudelist, J. M. Nowosielski, R. Buczynski, A. J. Waddie, and M. R. Taghizadeh, “Nanostructured elliptical gradient-index microlenses,” Opt. Lett.35(2), 130–132 (2010). [CrossRef] [PubMed]
- J. M. Nowosielski, R. Buczynski, F. Hudelist, A. J. Waddie, and M. R. Taghizadeh, “Nanostructured GRIN microlenses for Gaussian beam focusing,” Opt. Commun.283(9), 1938–1944 (2010). [CrossRef]
- A. Sihvola, Electromagnetic Mixing Formulas and Applications (IEE, 1999). [CrossRef]
- I. Richter, P.-C. Sun, F. Xu, and Y. Fainman, “Design considerations of form birefringent microstructures,” Appl. Opt.34(14), 2421–2429 (1995). [CrossRef] [PubMed]
- S. D. Gedney, “An anisotropic perfectly matched layer absorbing media for the truncation of FDTD lattices,” IEEE Trans. Antennas Propag.44, 1630–1639 (1996). [CrossRef]

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