## 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

Photonic crystals (PCs) are artificial crystals with dielectric periodic structures, and considered as the semiconductor counterpart for light. Solutions of Maxwell equations for their crystalline structures can be generated by a sum of Bloch modes, which have various anomalous refractive effects such as negative refraction, superprism, and waveguide coupler [1–4]. Analyzing Bloch wave decomposition is a simple and intuitive method for understanding and investigating the propagation and refraction of normal-mode waves. Methods of controlling electromagnetic wave propagation have been studied in both theoretical and experimental investigations [5–8]. Earlier, the authors studied two dimensional (2D) holographic polymer-dispersed liquid crystal (H-PDLC) PCs with square lattice (SL) and hexagonal lattice [4, 9-10]. The superprism and negative refraction that occur at certain incident angles over a range of frequencies are demonstrated and simulated. This study analyzed transversely propagated waves of three dimensional (3D) H-PDLC body-centered tetragonal (BCT) structures. Notably, the top view of the BCT unit array cell surface resembles that of a 2D SL PC. However, the center lattice in the unit cell of BCT determines the propagation modes with the rotation of the sample [3, 11–14]. Specifically, the beam refraction is determined by the center lattice, especially when the BCT is rotated to align the planes of the central lattice with the corner lattices. The present study found that a TE-polarized diode pumped solid state (DPSS) laser is not fully transparent when the beam is at certain incident angles. Instead, part of the beam energy trapped in the cell is transferred to beams propagating transversely along several modes, including two-direction and four-direction, and confined local modes.

## 2. Experiments

Two-beam interference with multi-exposures [15–17] was used for experimental BCT fabrication. Figure 1
depicts the experimental setup. Briefly, a TE-polarized CW DPSS laser beam (Verdi, λ=532 nm) is expanded and then divided by a beam splitter into two beams, the reference (~500 mW/cm^{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°.

The wave vector (**k**) of the normal incident beam is labeled **k _{0,}** and the other wave vectors in the exposures at 0°, 90°, 180° and 270° are labeled

**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 is

_{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

Figures 3(a)
, 3(b), 3(c), and 3(d) are photographs of the sample (on the X-Z plane) taken with a CCD camera when the sample is rotated along the Y-axis to obtain an angle (*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.

Figure 4(a) illustrates the structure and the simulation conditions of the finite-difference time-domain method (FDTD) of the BCT PC. These 3D FDTD calculations are based on the 25 (X)*25 (Y)*10 (Z) lattice array of BCT structure with the SEM parameters indicated in Figs. 2(c) and 2(d). The figures in the next section show the intensity profile from the FDTD calculation at the YZ-, XZ- and XY- plane to analyze transverse wave propagation. The refractive index of the polymer background is ~1.56, and the index modulation between polymer background and LC lattice (Δn) is ~0.2 [9]. The incident beam has a Gaussian-shape intensity profile with a beam diameter of ~1 μm and is located at +5 μm in the Z-axis. The XZ-, YZ- and XY- plane in the simulation appear in yellow, pink and blue, respectively.

Figure 4(b) shows the cut-plane of three BCT unit cells with respect to the laboratory coordinates (XYZ coordinates) when the sample rotates or the incident angle changes. This perspective drawing illustrates how the lattices of three unit cells cut with the XZ-, YZ- and XY-plane when the incident angle of the probe beam is changed. Planes 1 and 2 are the normally incident and obliquely incident conditions, respectively, corresponding to the YZ plane cutting with the BCT lattices. The figure shows that that the latter case (red contour) contains the center lattice.

The upper and lower rows of Fig. 5 show the simulation results for light wave transferred on the X-Z and Y-Z plane of the film, respectively, when using the FDTD method. Figures 5(a), 5(b), and 5(c) (Figs. 5(d), 5(e), and 5(f)) are the intersecting angles of 0°, 20°, and 50°, respectively, on the X-Z (Y-Z) plane. The simulation results for the X-Z plane [Figs. 5(a)-5(c)] show that fewer waves are transferred forward as the film rotates from 0° to 50°. However, most light waves are guided along the lattice vector that is perpendicular to incident [18-19]. At incident angles above 50°, the wave delivers much more energy in transverse direction (X-direction in Fig. 1). Clearly, energy transfer is more symmetry with respect to Z-axis on the Y-Z plane [Figs. 5(d), 5(e), and 5(f)], but incident angles above 50° are more complex in comparison with that on the X-Z plane [12, 14, 19]. In Fig. 5(d), the simulated result shows that most energy of the incident beam transmits through the periodic BCT structure with small fraction being scattered. However, the results in Fig. 5(e) and Fig. 5(f) illustrate that the incident beam refracts into two and even more waves, respectively. These results can be realized from the period change as the BCT structure is rotated [see Figs. 2(b) and 4(b)]. In the case of normal incidence, the Y-Z plane cuts the ABCD plane in laboratory coordinate [Fig. 2(b)]. As the film is rotated, the Y-Z plane may cut the ABC’D’ plane at an incident angle. In the mean time, the central point must be taken into consideration in the periodicity of lattice in the plane of ABC’D’. Because the ratio of lattice constants c/a is ~4.2, the oblique corner lattices (at A, B, C’ and D’) and the center align in a plane parallel to the Y-Z plane as the BCT film is rotated to 14°. Also, the refractive index of polymer is ~1.5. Thus, a ~20° incident angle gives an apparent transverse-component energy transfer according to the Snell’s law. Notably, the FDTD simulation on X-Z and Y-Z planes at 50° incident angle gives similarly unusual transverse energy transfer (results not shown).

Figure 6(a)
shows the projection of wave vectors **G _{n,m}** (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 [7,8, 20,21],

**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

Figure 6(b) shows the simulated intensity profile (X-Y plane) at point X=Y=0 μm (*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 _{2,1}** and

**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. Finally, when the incident angle is 50°, the dominant four wave vectors are

_{3,2}**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

In conclusion, this study analyzed the transverse propagation of light waves entering a BCT crystal film. The ratio of transverse energy increases as incident angle increases (from 0° to 50°). The light waves split into two or four waves in the left and right directions. These waves survive in the film and transfer transversely (X-Y plane) over a long distance. Apparently, the incident light waves bend into the film surface. Therefore, these films have potential use in beam splitters and waveguides.

## Acknowledgement

The authors would like to thank the National Science Council of the Republic of China, (Taiwan) for financially supporting this research under Contract No. NSC 98-2112-M-006-001-MY3 and 98-2923-M-006-001-MY3. Ted Knoy is appreciated for his editorial assistance.

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