## Transverse wave propagation in photonic crystal based on holographic polymer-dispersed liquid crystal |

Optics Express, Vol. 19, Issue 14, pp. 13428-13435 (2011)

http://dx.doi.org/10.1364/OE.19.013428

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### Abstract

This study investigates the transversely propagating waves in a body-centered tetragonal photonic crystal based on a holographic polymer-dispersed liquid crystal film. Rotating the film reveals three different transverse propagating waves. Degeneracy of optical Bloch waves from reciprocal lattice vectors explains their symmetrical distribution.

© 2011 OSA

## 1. Introduction

## 2. Experiments

15. Z. Peng, T. De-Xing, Z. Jian-Lin, S. Kun, Z. Jian-Bo, L. Bi-Li, and Y. Dong-Sheng, “Light-Induced Array of Three-Dimensional Waveguides in Lithium Niobate by Employing Two-Beam Interference Field,” Chin. Phys. Lett. **21**(8), 1558–1561 (2004). [CrossRef]

17. N. D. Lai, W. P. Liang, J. H. Lin, C. C. Hsu, and C. H. Lin, “Fabrication of two- and three-dimensional periodic structures by multi-exposure of two-beam interference technique,” Opt. Express **13**(23), 9605–9611 (2005). [CrossRef] [PubMed]

^{2}) and object beams (~400 mW/cm

^{2}), that simultaneously illuminate the sample. The former is normally incident while the latter is incident at an angle of

*θ*~39° to the normal. The sample is fixed to a rotation stage revolving in the X-Y plane and is exposed four exposures with the sample rotated 0°, 90°, 180° and 270°.

**k**) of the normal incident beam is labeled

**k**and the other wave vectors in the exposures at 0°, 90°, 180° and 270° are labeled

_{0,}**k**where index n is 1 to 4, respectively. The PDLC mixture consists of ~28 wt% liquid crystal (E7, Merck), ~71 wt% monomer (NOA81, Norland), and 1 wt% photoinitiator dye (rose bengal, Aldrich). Drops of homogeneously mixed compound are sandwiched between two indium-tin-oxide (ITO)-coated glasses separated by ~12 um glass spacers to produce a sample. The exposure area has a diameter of ~1 cm. Figure 2(a) gives the 3D simulated intensity profile of the interference region which is constructed using four exposures. The calculation iswhere

_{n}**k**are the above wave vectors corresponding to four exposures, and

_{n}**G**are the resulting grating vectors. The four exposures establish a 3D BCT; hence, the wave vectors

_{n,0}**G**(

_{n,m}**G**=

_{n,m}**k**-

_{n}**k**) resemble the addition of the reciprocal vector basis (b

_{m}_{1},b

_{2},b

_{3}) as

*p*b

_{1}+

*q*b

_{2}+

*r*b

_{3}(

*p*,

*q*,

*r*are integers) in PCs. Transverse wave propagation was measured by analyzing simplified reciprocal wave vectors (the

**G**group). Figure 2(b) illustrates the unit cell of the recorded film, which is a body-centered tetragonal structure in which

_{n,m}**a**=

**b**≠

**c**. Figures 2(c) and 2(d) are top-view and side-view scanning electron microscope (SEM) images, respectively. The SEM sample is prepared as follows. One glass substrate is split away from an H-PDLC sample. The sample is carefully split to avoid damaging the polymer structure. The H-PDLC film is immersed in n-hexane solution for a week to remove all LCs. The sample is then air dried to evaporate off the n-hexane solvent. Finally, it is coated with a thin gold film to produce a SEM sample. The SEM imaging indicates that the lengths of

**a**and

**c**are 0.8125±2% μm and 3.416±2% μm, respectively. Here, the lattice points (voids) containing LC droplets are approximate spheres, and the diameter of voids is ~0.64±2% μm.

## 3. Results and discussion

*θ*

_{inc}) with the incident probing beam of (a) 0°, (b) 20°, (c) 40°, and (d) 50°, respectively. The probe beam (λ, 532 nm; diameter, ~1 mm) produced by a TE-polarized DPSS laser propagates along the Z-axis in the +Z to –Z direction. Notably, the CCD camera has a fixed position and is not rotated with the sample. Figure 3 shows that some energy of the incident beam is transferred to waves (a series of light spots) propagating transversely outwards from the probed region with the

*θ*

_{inc}≠0°, and energy transfer increases with incident angle. When the incident angle is 20° [Fig. 3(b)], the transversely propagating waves appear in pairs (

*i.e.,*as two beams symmetrical to the X-axis). These two waves [labeled as 1 and 2, in Fig. 3(b)] are directed to the 2nd- and 3rd-quadrants on the X-Y plane. Next, when the incident angle increases to 40°, the transferred waves [labeled as 1-10 in Fig. 3(c)] propagate in more than six directions and most are unable to propagate far from the beam center. These modes are defined as confined local modes. Finally, when the incident angle increases to 50°, the energy transfers into four waves labeled as 1-4 with their direction turning towards the 1st- and 4th-quadrants, respectively, on the X-Y plane [Fig. 3(d)]. In summary, the beam energy is transferred into beams propagating transversely in several different modes, including two-direction, four-direction, and confined local modes.

9. M. S. Li, S. T. Wu, and A. Y.-G. Fuh, “Superprism phenomenon based on holographic polymer dispersed liquid crystal films,” Appl. Phys. Lett. **88**(9), 091109 (2006). [CrossRef]

18. R. Gajić, R. Meisels, F. Kuchar, and K. Hingerl, “Refraction and rightness in photonic crystals,” Opt. Express **13**(21), 8596–8605 (2005). [CrossRef] [PubMed]

19. S. Mahmoodian, A. A. Sukhorukov, S. Ha, A. V. Lavrinenko, C. G. Poulton, K. B. Dossou, L. C. Botten, R. C. McPhedran, and C. M. de Sterke, “Paired modes of heterostructure cavities in photonic crystal waveguides with split band edges,” Opt. Express **18**(25), 25693–25701 (2010). [CrossRef] [PubMed]

12. K. Sakoda, “Group-theoretical classification of eigenmodes in three-dimensional photonic lattices,” Phys. Rev. B **55**(23), 15345–15348 (1997). [CrossRef]

14. S. Foteinopoulou and C. M. Soukoulis, “Electromagnetic wave propagation in two-dimensional photonic crystals: A study of anomalous refractive effects,” Phys. Rev. B **72**(16), 165112 (2005). [CrossRef]

19. S. Mahmoodian, A. A. Sukhorukov, S. Ha, A. V. Lavrinenko, C. G. Poulton, K. B. Dossou, L. C. Botten, R. C. McPhedran, and C. M. de Sterke, “Paired modes of heterostructure cavities in photonic crystal waveguides with split band edges,” Opt. Express **18**(25), 25693–25701 (2010). [CrossRef] [PubMed]

**G**(n,m=0~4 and n≠m) on the X-Y plane to graphically represent optical Bloch waves in the reciprocal space. The transversely propagating wave propagating inside a BCT structure can be simplified as a Bloch mode [7

_{n,m}7. E. Cassan, D. Bernier, G. Maire, D. M. Morini, and L. Vivien, “Bloch wave decomposition for prediction of strong light coupling efficiency into extended planar photonic crystals,” J. Opt. Soc. Am. B **24**(5), 1211–1215 (2007). [CrossRef]

8. B. Lombardet, L. A. Dunbar, R. Ferrini, and R. Houdre, “Bloch wave propagation in two-dimensional photonic crystals: Influence of the polarization,” Opt. Quantum Electron. **37**(1–3), 293–307 (2005). [CrossRef]

20. O. Painter and K. Srinivasan, “Localized defect states in two-dimensional photonic crystal slab waveguides: A simple model based upon symmetry analysis,” Phys. Rev. B **68**(3), 035110 (2003). [CrossRef]

21. G. Sun and A. G. Kirk, “On the relationship between Bloch modes and phase-related refractive index of photonic crystals,” Opt. Express **15**(20), 13149–13154 (2007). [CrossRef] [PubMed]

**u**is the Fourier coefficients of the Bloch field periodic part, the reciprocal lattice vectors

_{n,m}**G**refer from the source grating vectors (

_{n,m}**G**) of interference in multi-exposures in Eq. (1). The transversely propagating waves of the BCT sample are dominated by the reciprocal lattice vectors

_{n,0}**G**with n,m=1~4 (n≠m). Notably, Eq. (2) is expressed with the waves on the basis of grating vectors. The wave vectors in the z component need not be considered when analyzing transverse propagation. Rather it connects closely to Eq. (1) that is relating to the fabrication condition of the sample. Moreover, the reciprocal lattice vector of crystal can be traced back to the relation of wave vectors (

_{n,m}**G**). Further, they are easily separated into the following three parts:

_{n,m}=k_{n}-k_{m}- (i)
**G**: these grating vectors result from the wave vectors_{1,0}, G_{2,0}, G_{3,0}, G_{4,0}**k**and_{0}**k**. They are the basic element in the following two groups (ii) and (iii). Essentially, these elements are in ±X or ±Y-axis, but the wave vector component is in the Z-axis. Notably, each element in this group has wave vector_{n}**k**, but the other 8 wave vectors in the following two groups (ii) and (iii) do not._{0} - (ii)
**G**: the elements in the group evolve from the above (i) grating vectors which can be determined by the wave vectors of oblique record beams_{2,3}, G_{1,2}, G_{3,2}, G_{2,1}**k**between the preceding and the later exposures. For example,_{n}**G**have X- and Y- components simultaneously._{2,3}=G_{2,0}-G_{3,0}=k_{2}-k_{3} - (iii)
**G**: the elements in the group evolve from the above (i) wave vectors, but they only contain ±X or Y- components. The total effect is to double the_{2,4}, G_{4,2}, G_{1,3}, G_{3,1}**G**on X- (or Y-) component (_{l,0}*i.e.,***G**)._{l3}= 2G_{1,0}‧X

*i.e*., the center of the sample) of transversely propagating waves for normal incident light, after propagating 2 μm in the Z-direction. The waves evolve from 8 wave vectors of groups (ii) and (iii); however, they can only propagate a few micro-meters. As shown in Fig. 3(a), the experiment only shows an isolated area. Figures 6(c), 6(d) and 6(e) show the X-Y plane FDTD simulation of light wave at Z= −3 μm in the film for incident angles of 20°, 40° and 50°, respectively. In Fig. 6(c), which is the simulation for an incident angle of 20°, the two refraction lights propagate towards to 2

^{nd}- and 3

^{rd}-quadrants on the X-Y coordinate plane. The propagation direction of the lights is consistent with that observed in the experiment [Fig. 3(b)] and fits the theoretical dominant wave vectors

**G**and

_{2,1}**G**of Eq. (2). Later, in Fig. 6 (d) where the simulated incident angle increases to 40°, the waves transfer in at least six directions [labeled as 1-10, in Fig. 6(d)] which are consistent with the experimental results in Fig. 3(c). Both the wave vectors

_{3,2}**G**,

_{2,1}**G**decompose into two components due to a phenomenon described in reference 10

_{3,2}10. M.-S. Li, S.-Y. Huang, S.-T. Wu, H.-C. Lin, and A. Y.-G. Fuh, “Optical and electro-optical properties of photonic crystals based on polymer-dispersed liquid crystals,” Appl. Phys. B **101**(1–2), 245–252 (2010). [CrossRef]

**G**,

_{2,3}**G**,

_{1,2}**G**, and

_{2,4}**G**, the waves transfer to the right by these four vectors [experimental results in Fig. 3(d)]. The left transverse wave propagation for 50° is substantially weaker. The waves also propagate transversely in pairs (symmetry with X-axis) far from the beam center. Whereas the photos shown in Fig. 3 show the combined intensity profiles of transversely propagating waves at all points along the sample thickness, the profiles in Figs. 6(c)-6(e) (transversely propagating waves at a point z= −3 μm from the center) indicate refraction tendency. Notably, the lattice constants of the PC differ (

_{4,2}**a**=

**b**≠

**c**), so wave propagation at each side differs (

*i.e*., two-, four- directions). For symmetrical wave propagation (

*i.e.,*two-, two- direction at each side), the lattice constants of the PC must be identical (

**a**=

**b**=

**c**).

## 4. Conclusion

## Acknowledgement

## References and links

1. | G. Y. Dong, X. L. Yang, and L. Z. Cai, “Anomalous refractive effects in honeycomb lattice photonic crystals formed by holographic lithography,” Opt. Express |

2. | D. Bernier, X. Le Roux, A. Lupu, D. Marris-Morini, L. Vivien, and E. Cassan, “Compact, low cross-talk CWDM demultiplexer using photonic crystal superprism,” Opt. Express |

3. | R. J. Liu, Z. Y. Li, F. Zhou, and D. Z. Zhang, “Waveguide coupler in three-dimensional photonic crystal,” Opt. Express |

4. | S. T. Wu, M. S. Li, and A. Y. G. Fuh, “Unusual refractions in photonic crystals based on polymer-dispersed liquid crystal films,” Appl. Phys. Lett. |

5. | Y. C. Hsu and L. W. Chen, “Bloch surface wave excitation based on coupling from photonic crystal waveguide,” J. Opt. |

6. | P. Zhang, C. Lou, S. Liu, J. Zhao, J. Xu, and Z. Chen, “Tuning of Bloch modes, diffraction, and refraction by two-dimensional lattice reconfiguration,” Opt. Lett. |

7. | E. Cassan, D. Bernier, G. Maire, D. M. Morini, and L. Vivien, “Bloch wave decomposition for prediction of strong light coupling efficiency into extended planar photonic crystals,” J. Opt. Soc. Am. B |

8. | B. Lombardet, L. A. Dunbar, R. Ferrini, and R. Houdre, “Bloch wave propagation in two-dimensional photonic crystals: Influence of the polarization,” Opt. Quantum Electron. |

9. | M. S. Li, S. T. Wu, and A. Y.-G. Fuh, “Superprism phenomenon based on holographic polymer dispersed liquid crystal films,” Appl. Phys. Lett. |

10. | M.-S. Li, S.-Y. Huang, S.-T. Wu, H.-C. Lin, and A. Y.-G. Fuh, “Optical and electro-optical properties of photonic crystals based on polymer-dispersed liquid crystals,” Appl. Phys. B |

11. | W. Hergert and M. Däne, “Group theoretical investigations of photonic band structures,” Phys. Status Solidi |

12. | K. Sakoda, “Group-theoretical classification of eigenmodes in three-dimensional photonic lattices,” Phys. Rev. B |

13. | P. Zhang, N. K. Efremidis, A. Miller, Y. Hu, and Z. Chen, “Observation of coherent destruction of tunneling and unusual beam dynamics due to negative coupling in three-dimensional photonic lattices,” Opt. Lett. |

14. | S. Foteinopoulou and C. M. Soukoulis, “Electromagnetic wave propagation in two-dimensional photonic crystals: A study of anomalous refractive effects,” Phys. Rev. B |

15. | Z. Peng, T. De-Xing, Z. Jian-Lin, S. Kun, Z. Jian-Bo, L. Bi-Li, and Y. Dong-Sheng, “Light-Induced Array of Three-Dimensional Waveguides in Lithium Niobate by Employing Two-Beam Interference Field,” Chin. Phys. Lett. |

16. | Y. Liu, S. Liu, and X. Zhang, “Fabrication of three-dimensional photonic crystals with two-beam holographic lithography,” Appl. Opt. |

17. | N. D. Lai, W. P. Liang, J. H. Lin, C. C. Hsu, and C. H. Lin, “Fabrication of two- and three-dimensional periodic structures by multi-exposure of two-beam interference technique,” Opt. Express |

18. | R. Gajić, R. Meisels, F. Kuchar, and K. Hingerl, “Refraction and rightness in photonic crystals,” Opt. Express |

19. | S. Mahmoodian, A. A. Sukhorukov, S. Ha, A. V. Lavrinenko, C. G. Poulton, K. B. Dossou, L. C. Botten, R. C. McPhedran, and C. M. de Sterke, “Paired modes of heterostructure cavities in photonic crystal waveguides with split band edges,” Opt. Express |

20. | O. Painter and K. Srinivasan, “Localized defect states in two-dimensional photonic crystal slab waveguides: A simple model based upon symmetry analysis,” Phys. Rev. B |

21. | G. Sun and A. G. Kirk, “On the relationship between Bloch modes and phase-related refractive index of photonic crystals,” Opt. Express |

**OCIS Codes**

(160.3710) Materials : Liquid crystals

(160.5470) Materials : Polymers

(050.5298) Diffraction and gratings : Photonic crystals

**ToC Category:**

Photonic Crystals

**History**

Original Manuscript: March 31, 2011

Revised Manuscript: June 13, 2011

Manuscript Accepted: June 19, 2011

Published: June 27, 2011

**Citation**

Andy Ying-Guey Fuh, Ming Shian Li, and Shing Trong Wu, "Transverse wave propagation in photonic crystal based on holographic polymer-dispersed liquid crystal," Opt. Express **19**, 13428-13435 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-14-13428

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### References

- G. Y. Dong, X. L. Yang, and L. Z. Cai, “Anomalous refractive effects in honeycomb lattice photonic crystals formed by holographic lithography,” Opt. Express 18(16), 16302–16308 (2010). [CrossRef] [PubMed]
- D. Bernier, X. Le Roux, A. Lupu, D. Marris-Morini, L. Vivien, and E. Cassan, “Compact, low cross-talk CWDM demultiplexer using photonic crystal superprism,” Opt. Express 16(22), 17209–17214 (2008). [CrossRef] [PubMed]
- R. J. Liu, Z. Y. Li, F. Zhou, and D. Z. Zhang, “Waveguide coupler in three-dimensional photonic crystal,” Opt. Express 16(8), 5681–5688 (2008). [CrossRef] [PubMed]
- S. T. Wu, M. S. Li, and A. Y. G. Fuh, “Unusual refractions in photonic crystals based on polymer-dispersed liquid crystal films,” Appl. Phys. Lett. 91(25), 251117 (2007). [CrossRef]
- Y. C. Hsu and L. W. Chen, “Bloch surface wave excitation based on coupling from photonic crystal waveguide,” J. Opt. 12(9), 095709 (2010). [CrossRef]
- P. Zhang, C. Lou, S. Liu, J. Zhao, J. Xu, and Z. Chen, “Tuning of Bloch modes, diffraction, and refraction by two-dimensional lattice reconfiguration,” Opt. Lett. 35(6), 892–894 (2010). [CrossRef] [PubMed]
- E. Cassan, D. Bernier, G. Maire, D. M. Morini, and L. Vivien, “Bloch wave decomposition for prediction of strong light coupling efficiency into extended planar photonic crystals,” J. Opt. Soc. Am. B 24(5), 1211–1215 (2007). [CrossRef]
- B. Lombardet, L. A. Dunbar, R. Ferrini, and R. Houdre, “Bloch wave propagation in two-dimensional photonic crystals: Influence of the polarization,” Opt. Quantum Electron. 37(1–3), 293–307 (2005). [CrossRef]
- M. S. Li, S. T. Wu, and A. Y.-G. Fuh, “Superprism phenomenon based on holographic polymer dispersed liquid crystal films,” Appl. Phys. Lett. 88(9), 091109 (2006). [CrossRef]
- M.-S. Li, S.-Y. Huang, S.-T. Wu, H.-C. Lin, and A. Y.-G. Fuh, “Optical and electro-optical properties of photonic crystals based on polymer-dispersed liquid crystals,” Appl. Phys. B 101(1–2), 245–252 (2010). [CrossRef]
- W. Hergert and M. Däne, “Group theoretical investigations of photonic band structures,” Phys. Status Solidi 197(3), 620–634 (2003). [CrossRef]
- K. Sakoda, “Group-theoretical classification of eigenmodes in three-dimensional photonic lattices,” Phys. Rev. B 55(23), 15345–15348 (1997). [CrossRef]
- P. Zhang, N. K. Efremidis, A. Miller, Y. Hu, and Z. Chen, “Observation of coherent destruction of tunneling and unusual beam dynamics due to negative coupling in three-dimensional photonic lattices,” Opt. Lett. 35(19), 3252–3254 (2010). [CrossRef] [PubMed]
- S. Foteinopoulou and C. M. Soukoulis, “Electromagnetic wave propagation in two-dimensional photonic crystals: A study of anomalous refractive effects,” Phys. Rev. B 72(16), 165112 (2005). [CrossRef]
- Z. Peng, T. De-Xing, Z. Jian-Lin, S. Kun, Z. Jian-Bo, L. Bi-Li, and Y. Dong-Sheng, “Light-Induced Array of Three-Dimensional Waveguides in Lithium Niobate by Employing Two-Beam Interference Field,” Chin. Phys. Lett. 21(8), 1558–1561 (2004). [CrossRef]
- Y. Liu, S. Liu, and X. Zhang, “Fabrication of three-dimensional photonic crystals with two-beam holographic lithography,” Appl. Opt. 45(3), 480–483 (2006). [CrossRef] [PubMed]
- N. D. Lai, W. P. Liang, J. H. Lin, C. C. Hsu, and C. H. Lin, “Fabrication of two- and three-dimensional periodic structures by multi-exposure of two-beam interference technique,” Opt. Express 13(23), 9605–9611 (2005). [CrossRef] [PubMed]
- R. Gajić, R. Meisels, F. Kuchar, and K. Hingerl, “Refraction and rightness in photonic crystals,” Opt. Express 13(21), 8596–8605 (2005). [CrossRef] [PubMed]
- S. Mahmoodian, A. A. Sukhorukov, S. Ha, A. V. Lavrinenko, C. G. Poulton, K. B. Dossou, L. C. Botten, R. C. McPhedran, and C. M. de Sterke, “Paired modes of heterostructure cavities in photonic crystal waveguides with split band edges,” Opt. Express 18(25), 25693–25701 (2010). [CrossRef] [PubMed]
- O. Painter and K. Srinivasan, “Localized defect states in two-dimensional photonic crystal slab waveguides: A simple model based upon symmetry analysis,” Phys. Rev. B 68(3), 035110 (2003). [CrossRef]
- G. Sun and A. G. Kirk, “On the relationship between Bloch modes and phase-related refractive index of photonic crystals,” Opt. Express 15(20), 13149–13154 (2007). [CrossRef] [PubMed]

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