## Synthesis of spatially variant lattices |

Optics Express, Vol. 20, Issue 14, pp. 15263-15274 (2012)

http://dx.doi.org/10.1364/OE.20.015263

Acrobat PDF (2469 KB)

### Abstract

It is often desired to functionally grade and/or spatially vary a periodic structure like a photonic crystal or metamaterial, yet no general method for doing this has been offered in the literature. A straightforward procedure is described here that allows many properties of the lattice to be spatially varied at the same time while producing a final lattice that is still smooth and continuous. Properties include unit cell orientation, lattice spacing, fill fraction, and more. This adds many degrees of freedom to a design such as spatially varying the orientation to exploit directional phenomena. The method is not a coordinate transformation technique so it can more easily produce complicated and arbitrary spatial variance. To demonstrate, the algorithm is used to synthesize a spatially variant self-collimating photonic crystal to flow a Gaussian beam around a 90° bend. The performance of the structure was confirmed through simulation and it showed virtually no scattering around the bend that would have arisen if the lattice had defects or discontinuities.

© 2012 OSA

## 1. Introduction

4. J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science **312**(5781), 1780–1782 (2006). [CrossRef] [PubMed]

5. D. H. Spadoti, L. H. Gabrielli, C. B. Poitras, and M. Lipson, “Focusing light in a curved-space,” Opt. Express **18**(3), 3181–3186 (2010). [CrossRef] [PubMed]

9. B. Vasić, G. Isić, R. Gajić, and K. Hingerl, “Controlling electromagnetic fields with graded photonic crystals in metamaterial regime,” Opt. Express **18**(19), 20321–20333 (2010). [CrossRef] [PubMed]

## 2. Synthesis algorithm

### 2.1 Construct input data

### Design the unit cell and compute its spatial harmonics

_{εuc(s→)}is initialized to all zeros and then set to 1 anywhere that dielectric is to be placed. This can be extended to incorporate multiple materials. The actual dielectric constants of the materials are incorporated in a later step.

*M*spatial harmonics for practical implementation on a computer. While the choice of which spatial harmonics are retained is arbitrary, typically all of the lowest order harmonics up to some cutoff are selected.

_{s→}is position,

*a*is the complex amplitude of the

_{m}*m*

^{th}spatial harmonic, and

_{K→m}is the grating vector associated with the

*m*

^{th}spatial harmonic. The complex amplitudes can be calculated using a fast Fourier transform (FFT). The grating vectors associated with the spatial harmonics for orthorhombic symmetries are calculated using Eq. (2) where

*p*,

*q*, and

*r*are integers and Λ

*, Λ*

_{x}*, and Λ*

_{y}*are the unit cell dimensions in the*

_{z}*x*,

*y*, and

*z*directions respectively.

11. L. Z. Cai, X. L. Yang, and Y. R. Wang, “All fourteen Bravais lattices can be formed by interference of four noncoplanar beams,” Opt. Lett. **27**(11), 900–902 (2002). [CrossRef] [PubMed]

### Define the spatially variant parameters

### 2.2 Synthesize the spatially variant lattice

### Step 1: Construct spatially variant K function

*K*function) is computed throughout this grid as illustrated in Fig. 4 .First, the grating vector

_{K→m}associated with the

*m*

^{th}spatial harmonic is computed from Eq. (2). This is then distributed uniformly across the grid to arrive at the uniform

*K*function. Second, the tilt of the orientation function is added to this

*K*function to arrive at an intermediate

*K*function. Third, the magnitude of the

*K*function is divided by the lattice spacing function to produce the final spatially variant

*K*function. In the end, the spatially variant

*K*function describes both the orientation and period of the 1D grating as a function of position throughout the lattice.

### Step 2: Calculate the grating phase function

*K*function. In this work, the finite-difference method [12] was used to approximate the derivatives in Eq. (4). This led to a large set of equations that were written in matrix form as

**Φ**

*is a column vector containing the grating phase at each point on the grid. These values are not known at this point, but will be calculated shortly. The terms*

_{m}**D**

*,*

_{x}**D**

*, and*

_{y}**D**

*are sparse banded matrices that perform derivative operations across the grid using central finite-differences. The terms*

_{z}**K**

*,*

_{x,m}**K**

*, and*

_{y,m}**K**

*are column vectors containing the components of the*

_{z,m}*K*function across the grid.

_{A′}remains a sparse matrix and Eq. (6) should be solved as a sparse system for best efficiency [14].

### Step 3: Compute spatially variant 1D grating

_{ϕm(s→)}and the complex amplitude

*a*of the

_{m}*m*

^{th}spatial harmonic, it is straightforward to compute the corresponding spatially variant 1D grating using Eq. (9). While not discussed here, the amplitude terms can be spatially varied to change the unit cell geometry throughout the lattice. The dielectric function

_{εm(s→)}is in general complex to convey the offset of the grating. While it is not needed at this point, the 1D grating can be visualized by ignoring the imaginary component.

### Step 4: Construct overall lattice

_{ε′(s→)}is constructed from their sum. It is preliminary because the spatially variant fill fraction has not yet been incorporated.

### Step 5: Incorporate spatially variant fill fraction

*ε*

_{1}and

*ε*

_{2}. It is straightforward to extend this approach to handle more than two materials.

*f*of

*ε*

_{2}can be estimated according to

*γ*is lowered, the fraction of the lattice filled by

*ε*

_{2}increases. When many spatial harmonics are used, the resulting dielectric distribution will have high contrast and direct application of Eq. (11) will have little effect. Some control over fill fraction can still be obtained using a grayscale unit cell or by partially blurring the preliminary dielectric function

_{ε′(s→)}before applying Eq. (11).

### 2.3 Notes on implementation

*N*

_{coarse}(4 to 10 is typical) and the lattice constant Λ, the resolution of the coarse grid should be

*L*

_{min}. Given some integer

*N*

_{fine}>10, the resolution of the fine grid should be

## 3. Example – spatially variant self-collimating photonic crystal

15. H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Self-collimating phenomena in photonic crystals,” Appl. Phys. Lett. **74**(9), 1212–1214 (1999). [CrossRef]

18. Z. Lu, S. Shi, J. A. Murakowski, G. J. Schneider, C. A. Schuetz, and D. W. Prather, “Experimental demonstration of self-collimation inside a three-dimensional photonic crystal,” Phys. Rev. Lett. **96**(17), 173902 (2006). [CrossRef] [PubMed]

*k*-space.

*can be engineered so that portions of the isofrequency surfaces are flat over some span of wave vectors*

_{k}ω*k*and over some band of frequencies

*ω*. This is the region where self-collimation is supported. The effect can be designed and optimized by tailoring the symmetry and composition of the unit cell.

*a*

_{1}=

*a*

_{2}= 1.0. The grating vectors associated with the two spatial harmonics were

*γ*= 0.1178. Using a material with dielectric constant 2.5, the center frequency of self-collimation was taken from Fig. 9(b) to be Λ/

*λ*

_{0}≅ 0.7039. Given this, the lattice constant can be chosen to produce self-collimation at whatever wavelength is desired. For operation at

*λ*

_{0}= 1550 nm, the lattice constant should be Λ = 1091 nm.

*λ*

_{0}. The finite-difference time-domain (FDTD) method was used to simulate the lattice using the TM

*mode [21] and the results are shown in Fig. 10 . A uniaxial perfectly matched layer (UPML) [22*

_{z}22. Z. S. Sacks, D. M. Kingsland, R. Lee, and J.-F. Lee, “A Perfectly Matched Anisotropic Absorber for Use as an Absorbing Boundary Condition,” IEEE Trans. Antenn. Propag. **43**(12), 1460–1463 (1995). [CrossRef]

*λ*

_{0}. It is important to reinforce that the beam was turned using spatially variant self-collimation and not waveguiding, band gaps, anisotropy, graded fill fraction, or graded refractive index. That leaves all of these mechanisms as additional degrees of freedom to control other aspects of the field or device.

## 4. Conclusion

## Acknowledgments

## References and links

1. | J. D. Joannopoulos, R. D. Meade, and J. N. Winn, |

2. | H. Benistry, V. Berger, J.-M. Gerard, D. Maystre, and A. Tchelnokov, |

3. | S. A. Ramakrishna and T. M. Grzegorczyk, |

4. | J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science |

5. | D. H. Spadoti, L. H. Gabrielli, C. B. Poitras, and M. Lipson, “Focusing light in a curved-space,” Opt. Express |

6. | E. G. Johnson, M. K. Poutous, Z. A. Roth, P. Srinivasan, A. J. Pung, and Y. O. Yilmaz, “Advanced fabrication methods for 3D meta-optics,” Proc. SPIE |

7. | Z. A. Roth, P. Srinivasan, M. K. Poutous, A. Pung, R. C. Rumpf, and E. G. Johnson, “Azimuthally varying guided mode resonance filters,” Micromachines |

8. | E. Akmansoy, E. Centeno, K. Vynck, D. Cassagne, and J.-M. Lourtioz, “Graded photonic crystals curve the flow of light: An experimental demonstration by the mirage effect,” Appl. Phys. Lett. |

9. | B. Vasić, G. Isić, R. Gajić, and K. Hingerl, “Controlling electromagnetic fields with graded photonic crystals in metamaterial regime,” Opt. Express |

10. | R. C. Rumpf, “Design and optimization of nano-optical elements by coupling fabrication to optical behavior,” Ph.D. Dissertation, University of Central Florida (2006), pp. 171–183. |

11. | L. Z. Cai, X. L. Yang, and Y. R. Wang, “All fourteen Bravais lattices can be formed by interference of four noncoplanar beams,” Opt. Lett. |

12. | S. C. Chapra and R. P. Canale, |

13. | B. Noble and J. W. Daniel, |

14. | Y. Sadd, |

15. | H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Self-collimating phenomena in photonic crystals,” Appl. Phys. Lett. |

16. | R. Iliew, C. Etrich, and F. Lederer, “Self-collimation of light in three-dimensional photonic crystals,” Opt. Express |

17. | J. Shin and S. Fan, “Conditions for self-collimation in three-dimensional photonic crystals,” Opt. Lett. |

18. | Z. Lu, S. Shi, J. A. Murakowski, G. J. Schneider, C. A. Schuetz, and D. W. Prather, “Experimental demonstration of self-collimation inside a three-dimensional photonic crystal,” Phys. Rev. Lett. |

19. | M. Born and E. Wolf, |

20. | R. C. Rumpf, “Design and optimization of nano-optical elements by coupling fabrication to optical behavior,” Ph.D. Dissertation, University of Central Florida (2006), pp. 109–124. |

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

22. | Z. S. Sacks, D. M. Kingsland, R. Lee, and J.-F. Lee, “A Perfectly Matched Anisotropic Absorber for Use as an Absorbing Boundary Condition,” IEEE Trans. Antenn. Propag. |

**OCIS Codes**

(050.0050) Diffraction and gratings : Diffraction and gratings

(130.0130) Integrated optics : Integrated optics

(230.0230) Optical devices : Optical devices

(160.3918) Materials : Metamaterials

(160.5298) Materials : Photonic crystals

**ToC Category:**

Photonic Crystals

**History**

Original Manuscript: April 13, 2012

Revised Manuscript: May 18, 2012

Manuscript Accepted: June 10, 2012

Published: June 22, 2012

**Citation**

Raymond C. Rumpf and Javier Pazos, "Synthesis of spatially variant lattices," Opt. Express **20**, 15263-15274 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-14-15263

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### References

- J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals, Molding the Flow of Light (Princeton University Press, 1995).
- H. Benistry, V. Berger, J.-M. Gerard, D. Maystre, and A. Tchelnokov, Photonic Crystals, Towards Nanoscale Photonic Devices (Springer, 2005).
- S. A. Ramakrishna and T. M. Grzegorczyk, Physics and Applications of Negative Refractive Index Metamaterials, (CRC Press, 2009).
- J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science312(5781), 1780–1782 (2006). [CrossRef] [PubMed]
- D. H. Spadoti, L. H. Gabrielli, C. B. Poitras, and M. Lipson, “Focusing light in a curved-space,” Opt. Express18(3), 3181–3186 (2010). [CrossRef] [PubMed]
- E. G. Johnson, M. K. Poutous, Z. A. Roth, P. Srinivasan, A. J. Pung, and Y. O. Yilmaz, “Advanced fabrication methods for 3D meta-optics,” Proc. SPIE7927, 792706, 792706-7 (2011). [CrossRef]
- Z. A. Roth, P. Srinivasan, M. K. Poutous, A. Pung, R. C. Rumpf, and E. G. Johnson, “Azimuthally varying guided mode resonance filters,” Micromachines3(1), 180–193 (2012). [CrossRef]
- E. Akmansoy, E. Centeno, K. Vynck, D. Cassagne, and J.-M. Lourtioz, “Graded photonic crystals curve the flow of light: An experimental demonstration by the mirage effect,” Appl. Phys. Lett.92(13), 133501 (2008). [CrossRef]
- B. Vasić, G. Isić, R. Gajić, and K. Hingerl, “Controlling electromagnetic fields with graded photonic crystals in metamaterial regime,” Opt. Express18(19), 20321–20333 (2010). [CrossRef] [PubMed]
- R. C. Rumpf, “Design and optimization of nano-optical elements by coupling fabrication to optical behavior,” Ph.D. Dissertation, University of Central Florida (2006), pp. 171–183.
- L. Z. Cai, X. L. Yang, and Y. R. Wang, “All fourteen Bravais lattices can be formed by interference of four noncoplanar beams,” Opt. Lett.27(11), 900–902 (2002). [CrossRef] [PubMed]
- S. C. Chapra and R. P. Canale, Numerical Methods for Engineers with Software and Programming Applications, 4th Ed., 820–856 (McGraw-Hill, 2002).
- B. Noble and J. W. Daniel, Applied Linear Algebra, 3rd ed. (Prentice Hall, 1988), pp. 66–73.
- Y. Sadd, Iterative Methods for Sparse Linear Systems, 2nd ed. (Yousef Sadd, 2000).
- H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Self-collimating phenomena in photonic crystals,” Appl. Phys. Lett.74(9), 1212–1214 (1999). [CrossRef]
- R. Iliew, C. Etrich, and F. Lederer, “Self-collimation of light in three-dimensional photonic crystals,” Opt. Express13(18), 7076–7085 (2005). [CrossRef] [PubMed]
- J. Shin and S. Fan, “Conditions for self-collimation in three-dimensional photonic crystals,” Opt. Lett.30(18), 2397–2399 (2005). [CrossRef] [PubMed]
- Z. Lu, S. Shi, J. A. Murakowski, G. J. Schneider, C. A. Schuetz, and D. W. Prather, “Experimental demonstration of self-collimation inside a three-dimensional photonic crystal,” Phys. Rev. Lett.96(17), 173902 (2006). [CrossRef] [PubMed]
- M. Born and E. Wolf, Principles of Optics, 6th Ed., 673–678 (Cambridge University Press, 1980).
- R. C. Rumpf, “Design and optimization of nano-optical elements by coupling fabrication to optical behavior,” Ph.D. Dissertation, University of Central Florida (2006), pp. 109–124.
- A. Taflove and S. C. Hagness, Computational Electrodynamics, the Finite-Difference Time-Domain Method, 3rd ed. (Artech House, 2005).
- Z. S. Sacks, D. M. Kingsland, R. Lee, and J.-F. Lee, “A Perfectly Matched Anisotropic Absorber for Use as an Absorbing Boundary Condition,” IEEE Trans. Antenn. Propag.43(12), 1460–1463 (1995). [CrossRef]

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