1. Introduction

Photonic crystals are periodically structured electromagnetic media in one, two or three dimensions that present photonic band gaps for a certain spectral range and for a given angle of illumination. Volume planar holographic diffraction gratings are one dimensional optical periodic structures that diffract part of the spectrum of the incident beam. They act as a photonic crystal since the resulting spectrum of the transmitted order is the same as that of the initial beam minus the diffracted light. Following this idea, Campbell [1] showed for the first time the possibility of producing periodic microstructures by means of the interference of four non-coplanar laser beams, using a 30 µm thick film of photoresist. He concluded that holographic lithography is a good technique for the fabrication of photonic crystals. Several photonic band-gap materials such as dichromated gelatin [2], polymer with TiO ₂[3], photoresist [4, 5] and sol-gel [6] have recently been used for the fabrication of photonic structures using techniques like lithography or holography. Also, one-step holographic multiplexing techniques have recently been theoretically analyzed [8] showing that different unit cells can be produced.

Ultra fine grain silver halide holographic emulsions have also been used to fabricate photonic crystals [7]. They show better sensitivity than all the recording materials mentioned above, their spatial resolution is higher than 7000 l/mm [9] and they show good index modulation capabilities [10]. In this study we present three dimensional photonic crystals recorded in ultra-fine grain emulsions BB640. For this, two symmetrical holographic transmission gratings and one holographic reflection grating with the appropriate frequencies were multiplexed in a single layer of material to fabricate the three dimensional photonic crystal.

2. Theoretical background

Let us consider the multiplexing of three gratings, each one is obtained by the interference of two plane waves and uses three different wavelengths, λ_v, v=(1, 2, 3). The total complex amplitude of each of the beams used for multiplexing the grating is given by Eq. (1),

where $E_{\mathit{\text{vj}}}^0$ is the real amplitude, the subindex v indicates the order of the multiplexed grating, taking values from 1 to 3, j, with values 1 and 2, is the index for each of the recording plane waves that interferes in the multiplexing step. K⃗_vj is the wave vector given by Eq. (2),

were (l_vj,m_vj,n_vj ) are the cosine directors,r⃗ is the position vector inside the material and ê_vj is the unit polarization vector of the electric field.

After multiplexing, the total intensity will be given by Eq. (3),

where G⃗_v is the grating vector of the i-th multiplexed grating given by Eq. (4),

and χ _v,12=|ê _v1.ê _v2| is the scalar product of the polarization vectors of the plane waves that interfere in each multiplexing step.

Assuming a linear response of the material, the internal points with a maximum value of index modulation of the multiplexed structure will be given by Eq. (5).

Expression (5) generates the equation system 6 as a function of the cosine directors of the gratings ($G_v=\frac{2\pi }{{\lambda }_v}(L_v,M_v,N_v)=\frac{2\pi }{{\lambda }_i}\left(l_{v1}-l_{v2},m_{v1}-m_{i2},n_{v1}-n_{v2}\right)$),

where p_i are integer numbers. Solving the equation system and taking p_i =1 in each of the obtained solutions, we can obtain the i-th vector a⃗_i of the basic periodic structure that results from the multiplexing process.

Then, the crystal basis vectors will be given by Eqs. (7), (8) and (9),

where det[Q] is the determinant of the matrix Q.

$\mathbf{Q}=\left(\begin{array}{ccc}{L}_1& M_1& N_1\\ L_2& M_2& N_2\\ L_3& M_3& N_3\end{array}\right)$

From this analysis we find that the holographic multiplexing technique can be used for fabricating 3D crystal microstructures whose basis vectors are determined by the directions and wavelengths of the six plane waves used in the recording process.

In order to check this theoretical approach, we fabricated a photonic crystal with tetragonal symmetry in photographic emulsion by multiplexing three gratings. Two were volume planar transmission gratings and the third was a volume planar reflection grating. The two transmission gratings had grating vectors given by: G⃗₁=(0.013, 0, 0) nm ^-1 and G⃗₂=(0, 0.013, 0) nm ^-1 respectively. The reflection grating was a symmetric reflection grating with normal incidence on each side of the recording media and a vector given by G⃗₃=(0, 0, 0.031) nm ^-1. Using Eqs. (7), (8), (9), we can obtain the basis vectors of the unit cell (see Fig. 1(a)) given by: a⃗₁=(492,0,0) nm, a⃗₂=(0,492,0) nm and a⃗₃=(0,0,204) nm.

In Fig. 1(b), the simulation of a photonic crystal obtained after multiplexing the above three mentioned gratings recorded with the same wavelength is shown.

The unit cell vectors satisfy the conditions given by Eq. (10),

 

Fig. 1. (a) Unit cell of the photonic crystal showed in Fig. 1(a), with vectors given by: a⃗₁=(492, 0, 0) nm, a⃗₂=(0, 492, 0) nm and a⃗₃=(0, 0, 204) nm. (b) Simulation of the isorefractive index surfaces of photonic crystal formed by multiplexing three diffraction gratings with vectors: G⃗₁=(0.013, 0, 0) nm ^-1, G⃗₂=(0, 0.013, 0) nm ^-1, G⃗₃=(0, 0, 0.031) nm ^-1.

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which implies that the unit cell corresponds to a tetragonal structure and from the analysis of Fig. 1(b), it can be deduced that the symmetry group is P4/mmm. Taking into account this symmetry group, and in order to obtain the band diagram of the photonic crystal, we studied the propagation of the electromagnetic field in all directions of the reciprocal space whose first Brillouin zone is given by the principal directions:

Γ (0, 0, 0), Z (0, 0, 0.5), M (0.5, 0.5, 0.5), A (0.5, 0.5, 0.5), R (0, 0.5, 0.5) and X (0, 0.5, 0)

Assuming a linear response in the material [10], the total intensity after multiplexing (Eq. (3)) generates a refractive index function given by Eq. (11),

where n ₀ is the refractive index of the material, and nv is the refractive index modulation generated in the v-th multiplexing step, the fully-vectorial eigenmodes of Maxwell’s equations with periodic boundary conditions can be computed by preconditioned conjugate-gradient minimization of the block Rayleigh quotient in a plane wave basis, using a freely available software package [11].

Figure 2 shows the band structure of the photonic crystals for all Bloch vectors. As can be seen, a total band gap for all crystal reconstruction directions does not appear, although some partial band gaps for certain wavelengths in the visible spectral range and certain reconstruction directions can be observed. For example, at this low index contrast, two band gaps open in the M-X direction at wavelengths of 690 and 490 nm.

 

Fig. 2. Bands diagram generated by the photonic crystal with unit cell showed in figure 1(b). n ₀=1.579, n_i =0.03, G⃗₁=(0.013, 0, 0) nm ^-1, G⃗₂=(0, 0.013, 0) nm ^-1, G⃗₃=(0, 0, 0.031) nm ^-1

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3. Experimental

One reflection and two transmission planar unslanted volume holographic gratings were multiplexed on a single plate to generate the three-dimensional structure. The chosen recording material chosen was the ultra-fine grain holographic emulsion BB640 manufactured by the British company Colourholographics Ltd., which has a mean grain size of 20 nm and sensitivity in the red region of the spectrum [12]. In all recordings, the plates were pre-sensitised by soaking for 2 min. in a solution of 3% triethanolamine (TEA) in water to increase sensitivity. They were then soaked for 6 min. in deionized water to remove the excess of TEA from the emulsion, dried with a photographic roll and warm air, and left in the exposure room for half an hour in normal laboratory conditions (20°C and 60% relative humidity).

To record the planar unslanted reflection grating, the plate was exposed to a single collimated beam in a Denisyuk configuration [13] using a red He-Ne laser with a wavelength of 632.8 nm [14]. With this configuration the recording spatial frequency for the reflection grating is 4990 l/mm, corresponding to an interference fringe distance of 200 nm., considering a refractive index of 1.579 for the unexposed emulsion. This value may be slightly modified due to changes in the emulsion thickness after the wet processing.

To record the first of the planar unslanted transmission gratings, the plate was exposed to two coplanar collimated beams from a He-Ne laser, in a symmetric configuration, each at an angle of incidence of 40° to each side of the perpendicular to the plate. With this configuration the recording spatial frequency for the transmission grating is 2032 l/mm, corresponding to an interference fringe distance of 492 nm. This frequency remains unaltered after the wet processing. To record the second planar unslanted transmission grating, the plate was rotated 90° and re-exposed in the same setup. The exposure energies for each of the multiplexed recordings were: 450 µJ/cm ² for the reflection grating, and 450 µJ/cm ², 600 µJ/cm ² for transmission gratings, all of them in the linear response region of the material.

Exposed plate was underwent with a standard AAC - R10 processing and was analyzed using a fibre fed spectroradiometer [9]. The crystal was mounted on a three axis rotation platform that permits the analysis of the transmittance spectra in a wide range of crystal orientations.

The experimental transmittance diagram when the direction of illumination is perpendicular to the fabricated 3D crystals is shown in Figs. 3(a). Figure 3(b) shows the theoretical band gaps obtained for this crystal orientation. As can be seen good agreement between theory and experience is obtained. The experimental transmission curves are not null for the wavelength predicted by theory. This effect is due to the low thickness of the material (8 µm) that only generates a few unit cells in the Z direction. It is important to note that a first band gap appears as a consequence of diffraction by (0, 0,1) planes (recorded reflection grating) and another one, diffracted by planes (1, 0, 1) (not related with the directly recorded gratings), which have their origin in the three dimensional multiplexed periodic structure of the crystal.

 

Fig. 3. Experimental (a) and theoretical (b) bands diagrams generated by the photonic crystal with unit cell showed in Fig. 1(b) recorded in a photographic emulsion, with reconstruction direction along z axis.

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Based on this technique we have obtained a band-pass filter for a fixed direction of illumination (Z axis) by rotating the crystal 48° around the Y axis, 9° around the X axis, 56° around the Z axis. Figure 4 shows the transmission spectra of this device. All wavelengths lower than 580 nm shows very low transmission.

 

Fig. 4. (90 KB) Movie of the evolution of the transmission spectra of the band-pass filter starting with normal incidence along the Z axis and ending with the crystal rotated 48° around the Y axis, 9° around the X axis, 56° around the Z axis.

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4. Conclusions

We have demonstrated the possibilities of using multiplexed holographic techniques for fabricating three-dimensional photonic crystals. A tetragonal micro-structure was fabricated and its transmission bands measured, showing the possibilities of the method. Good agreement between theoretical calculations and experimental results have been obtained. At the same time we have successfully tested the use of ultra-fine grain photographic emulsions as recording material for this specific application.