1. Introduction

Photonic crystals (PhC) are refractive-index contrast structures with translational symmetry They exhibit so-called photonic bandgaps. When light transmits through them, peculiar phenomena can happen that have promising prospects for applications in waveguides, optical storage, optical communication, etc. [1] The fabrication of PhC for the visible or near-infrared range is still a challenging topic of research. The principal methods involve chemical etching [2–3], ion beam etching [4], laser micro-fabrication [5], etc. However, all these methods require expensive large-scale equipment. A more convenient approach is to use lasers of good coherent characteristic that permit to create periodic patterns by multi-beams interference with a typical period in the order of the wavelength. A variety of two-dimensional (2D) and three-dimensional (3D) periodic patterns can be obtained by changing the number, polarization state, or propagation directions of the beams [6]. Under special conditions they exhibit translational symmetry and establish a so-called Optical Lattice. The optical method has the advantages of simple one step moulding and has been focused at in recent years. Berger and coworkers [7] utilized the interference of three coherent beams formed by three diffraction gratings with 60 degree crossing angle with respect to each other to fabricate a 2D hexagonal lattice in GaAs in 1997. Kondo and coworkers [8–9] used Femtosecond laser pulses and a diffractive beam splitter to fabricate 2D and 3D microperiodic structures in the photoresist SU-8 by single-photon and multi-photon absorption in 2001 and 2003, respectively. Hu et al. [10] demonstrated an interference technique for arranging TiO2 particles into 2D periodic patterns. Cai et al. [11] proved that all 14 Bravais lattices can be formed by the interference of only four non-coplanar beams.

Splitting a laser beam into several components of equal intensity usually requires a complicated experimental setup and suffers from low stability, so the majority of the research works limits itself to 3–6 beams. Using a diffractive beam splitter one can easily get a multi-beam with more components, but at the disadvantage of low transmittance and of laser mode degradation. High cost is also an issue. In this paper we propose a solution based on pyramids made from optical glass. They have the benefit of simple operation, low cost, ease of integration, good stability, high damage threshold and high transmission, and apply to all kinds of CW and pulse lasers. The crossing angles of the interfering partial beams depend on the refractive index and the roof angle of the pyramid as well as on the direction of the incident beam. In the following we investigate symmetric pyramids with the incident beam aligned along the symmetry axis. 2D-structured patterns are achieved by symmetric acute pyramids, while symmetric flat-topped pyramids are used for 3D structuring. Both cases are simulated numerically. The results are compared with experimental findings.

2. Theory

 

Fig. 1. Multi-beam interference by a four-faceted symmetric acute pyramid (a) and its flattopped analogue (b)

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Symmetric pyramids have an n-fold axis of rotation that defines the optical axis of the system. Figure 1 sketches the experimental situation to be analyzed for the exemplary case n=4, i.e., for four-faceted pyramid and its flat-topped analogue. Incident photons are presupposed to have momentum $\overrightarrow{p}$ =ħ$\overrightarrow{k}$ ₀ directed parallel to this axis. Then refraction from the n differently oriented side faces of the acute pyramid (Fig. 1a) results in a superposition state of n momentum states characterized in the following by the (reduced) momentum vector $\overrightarrow{k}$ _j ,j=1, …, n. For flat-topped pyramids (Fig. 1b) the photon has the additional possibility to evade via the top face, and hence one additional state with momentum vector $\overrightarrow{k}$ ₀ has to be taken into account. Because of energy conservation for the refraction process, the magnitude of all momentums, k=|$\overrightarrow{k}$ _l | with l=0, …, n, is the same.

In principle the amplitude of the contributions undergoing refraction depends on the polarization state, viz. the angular momentum state. Further, reflection and refraction may induce transition of the photon into another polarization state. In order to keep the analysis simple we assume in the following that the crossing angle of the beams is small enough (less than 10 degree), so that the influence of the polarization state on refraction and interference calculation can be neglected, i.e., we consider only superposition of momentum states (so-called scalar-wave approximation).

Suggested by the symmetry of the problem the best choice for an analysis are cylindrical coordinates with z along the symmetry axis, $\rho =\sqrt{x^2+y^2}$ perpendicular to it, and φ being the polar angle with reference to the x axis that is aligned as shown in Fig. 1. We characterize the possibility of the photon to be in a state with momentum $\overrightarrow{k}$ =k _z $\overrightarrow{e}$ _z +k _ρ $\overrightarrow{e}$ _ρ by the probability amplitude

where Eexp(iδ) is the complex amplitude of the partial probability (E, δ being real quantities), $\overrightarrow{r}$ =z$\overrightarrow{e}$ _z +ρ$\overrightarrow{e}$ _ρ is the position vector, $\overrightarrow{e}$ _z is the unit vector in z-direction and $\overrightarrow{e}$ _ρ =$\overrightarrow{e}$ _ρ (φ)=(cos φ, sin φ,0) is the unit vector in the radial direction.

For symmetry reasons all side faces are equivalent and as a consequence E, δ, $k_z=\sqrt{k^2-k_{\rho }^2}$ and k _ρ do not differ for j=1,…, n. Only the direction of the radial unit vector varies with φ=φ _j =2π(j-1)/n. In case on an additional flat-top face the additional component has momentum $\overrightarrow{k}$ ₀=k$\overrightarrow{e}$ _z and as a rule a different value E₀ (but equal δ). As usual the total amplitude ψ is assumed to be properly normalized such that the intensity follows from I=ψψ*, where the asterisk denotes the complex conjugate.

The total probability amplitude to find a photon at the point (z,ρ,φ) in space is now proportional to

and the intensity is given by

Here K=k-k _z , I ₀=$E_0^2$+nE ² is the average intensity, and the common phase δ cancels out.

For E ₀≠0 Eq. (3) describes the intensity distribution resulting from a flat-topped pyramid. By putting E ₀=0 in Eq. (3) we obtain the much simpler expression for the intensity distribution of an acute pyramid. There is one striking difference: For an acute pyramid there is no dependence of the intensity on the z-coordinate anymore, i.e., interference patterns from acute pyramids are structured only in two dimensions.

In both cases the intensity pattern maintains the rotational symmetry group n. This is obvious for the third term in Eq. (3). As for the fourth term, the transformation φ→φ+φ _j for some φ _j =2π(j-1)/n only interchanges the ordering of the cosines in the square bracket and thus may only result in a sign change of the argument that is irrelevant for the outer cosine function. Thus the sum over all terms remains unchanged. By similar reasoning one sees that the intensity function is symmetric with respect to reflection at the symmetry axis, i.e., to a transformation φ→-φ, or equivalently to rotation by 180°, i.e. to φ→φ+π. As a consequence, the patterns resulting from an n-faceted pyramids have rotational symmetry n if n is an even integer and 2n if n is an odd integer.

It is a well-known theorem of crystallography that the rotational symmetry group n is compatible with translational symmetry group only for n=1, 2, 3, 4, and 6. It is interesting to see how this fact can be understood from the structure of Eq. (3): The intensity pattern has no translational symmetry in the x-y plane unless there is one direction (defined by some constant φ) where the intensity is a periodic function of the radial coordinate ρ. For n=3, 4 and 6 this is also sufficient, since a second non-collinear direction with translational symmetry is generated by the symmetry operation of rotation. As we may always choose the x axis along the direction with translational symmetry, we have only to investigate under which conditions $\sum _{i=1}^n\cos \left(k_{\rho }\rho \cos {\phi }_i\right)$ and $\sum _{i=1}^n\sum _{j>1}^n\cos \left(k_{\rho }\rho \left[\cos {\phi }_i-\cos {\phi }_j\right]\right)$ become a periodic function of ρ. Let i ^’ be the index of the smallest non-zero value of all cosφ _i . Then both expressions are periodic if cosφ _i is an integer multiple of cosφ _i ’. A simple inspection of the cosine function shows that this can be fulfilled for n=2, 3, 4, 6, but not for n=5. For n>6 it can be proofed that optical lattices do not exist.

For flat-topped pyramids the intensity is a spatially periodic function in z-direction with periodicity Λ _z =2λ/θ ², where θ is the angle of inclination of the partial beam paths with respect to the optical axis and λ the so-called wavelength of the photon.

3. Experiment

 

Fig. 2. Plane geometrical scheme of a collimated beam passing through a glass pyramid

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As sketched in Fig. 2, a sufficiently expanded laser beam enters through the base of a glass pyramid. The beam is adjusted along the optical axis thus generating partial fields inclined by an angle θ with respect to the optical axis due to refraction. The extension of the resulting interference zone can be estimated by Z _max≈w ₀/tgθ, where w₀ denotes the nominal radius of the beam. In order to have a sufficiently extended interference zone for the investigations, a very small angle is preferable. We used acute and flat-topped pyramids with γ=2° and γ=5°. In this case θ is in good approximation related by θ ≈ (n _r -1)γ with the refractive index n _r and the angle γ between the pyramid base and one of its side faces. Refractive index n _r was 1.5 and wavelength λ was 633nm (He-Ne laser).

The interference pattern produced by multi-beam light is projected directly to the CCD camera by an objective lens. We compared the results of 3- and 4-faceted glass pyramids with numerical simulations. Using MATLAB software, the calculations were performed for the following parameters: λ=633nm, θ=2.5°, ϕ _j=360°*(j-1)/n, E=1δ=0.

4. Results and discussion

Figure 3 gives the simulated intensity distributions in the x-y plane behind n-faceted acute pyramids. They consist essentially of a lattice of point with cylindrical intensity distribution in the z-axis direction. All examples given in Fig. 3 exhibit the expected point group n if n is an even number, and 2n, if n is an odd number. For instance, the pattern of Fig. 3(f) for n=7 exhibits a 14-fold rotational symmetry. In addition, there can be translational symmetry, but only for n=2, 3, 4, and 6. Exclusively in these cases a 2D optical lattice can be created, and the position of the rotational symmetry axis is not anymore uniquely defined.

 

Fig. 3. Simulations of 2D patterns in the x-y plane behind n-faceted acute pyramids (λ=633nm, θ=2.50)

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For comparison with Figs. 3(b) and 3(c), experimentally observed lattices for n=3 and n=4 are shown in Fig. 4. The primitive period is Λ_ρ=9.5µm as observed by a 25X objective and Λ_ρ=33 µm as observed by a 10X objective, respectively. In particular, we note the expected 6-fold symmetry in Fig. 4(a) resulting from a pyramid with 3-fold axis. This doubling does not occur for n=4, and as a consequence, the interference pattern from a 3-faceted pyramid exhibits higher point symmetry than the interference pattern from a 4-faceted pyramid.

 

Fig. 4. Experimental observation of optical lattices formed (a) by a 3-faceted acute pyramid with γ=5° (imaged with a 25X objective) and (b) by a 4-faceted pyramid with γ=2° (imaged with a 10X objective). The scale in the figures is 10µm/div.

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The case n=5 is intriguing, because in the small section shown in Fig. 3(d) the pattern gives at first glance still the illusion of an optical lattice. The traces of minimum intensity are nearly straight and nearly evenly spaced, forming nearly rhombic tiles, if we disregard the fact that the internal pattern of the tiles is of course differing. So by using the pattern to generate a refractive-index structure, the case n=5 could be interesting for the study of photon propagation in a structure that has some resemblance to a quasi-crystal.

For n→∞ the periodicity in the angular variable φ becomes shorter and shorter. For a recording medium of limited resolution this means that in particular for small ρ the response is smearing out, i.e., becomes a circular distribution that is not modulated anymore as a function of φ. Due to the limited resolution of the printer this tendency is nicely reproduced in Figs. 3(g)–3(i). The angularly smoothened intensity pattern in the limiting case n→∞ approaches the intensity pattern of an axicon, i.e., the photon state is essentially described by the zero-order Bessel function J ₀(ρ). Putting E ₀=0 in Eq. (2) we indeed obtain in the limiting case:

The mathematical description of the Bessel beam created by an axicon is usually derived from the Fresnel diffraction integral [12–13]. Our approach shows that an axicon may alternatively be approximated by multi-beam interference from pyramids. In Fig. 5 the experimentally recorded intensity pattern of a glass axicon is compared with the simulated pattern of a 40-faceted acute pyramid. A comparison of the angularly smoothened radial intensity distribution of the simulation with the one of the zero-order Bessel function demonstrates that a 40-faceted acute pyramid is fully sufficient to approximate an axicon. Bessel beams have the characteristic property to be non-diffracting. Therefore they can overcome the limitation of the Rayleigh range of Gaussian beams, which is considered to be useful in manipulating particles in the next generation of optical tweezers [13–15].

 

Fig. 5. Comparison of the experimentally measured intensity pattern of an axicon (a) with the one of the simulated intensity pattern from a 40-faceted acute pyramid (b). In (c) the radial distribution of the simulation is compared with the zero-order Bessel function.

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In Fig. 6 the calculated patterns for flat-topped pyramids are shown, where we used for simplicity E₀=E. For n=3, 4 and 6 three-dimensional lattices are generated, for n=2 a two-dimensional lattice and for all other rotational axes there is only a one-dimensional lattice along the rotational axis.

 

Fig. 6. Simulations of the 3D intensity distribution resulting from flat-topped pyramids (λ=633nm, θ=2.50)

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Experimentally we recorded scans along the z axis with a CCD camera. The videos for a flat-topped prism with n=2 (γ=50), and flat-topped pyramids with n=3 (γ=5°) and n=4 (γ=2°) are shown in Fig. 7(a), 7(b), and 7(c), respectively, and the simulation videos are shown for comparison in Fig. 7(d), 7(e), and 7(f), respectively. One can see that there is a good correspondence between both, in particular for the periodicity along the z axis.

 

Fig. 7. Z-scan videos of the structure of 3D optical lattices using (a) a flat-topped prism with n=2 and γ=5°(imaged with a 25X objective), (b) a flat-topped pyramid with n=3 and γ=5° (imaged with a 25X objective), (c) a flat-topped pyramid with n=4 and γ=2° (imaged with a 10X objective). The corresponding videos of the simulations are shown in (d) for n=2, (e) for n=3 and (f) for n=4, respectively. [Media 1] [Media 2] [Media 3] [Media 4] [Media 5] [Media 6]

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5. Conclusion

2D and 3D optical lattices as well as 1D optical lattices with a rotational symmetry axis can be formed by interference of photon states from acute and flat-topped pyramids. This method has the advantages of simple operation, low cost, ease of integration, good stability, and high transmission. The 2D optical lattice can be used to generate array waveguides. By using shorter wavelengths or larger base angles the 3D optical lattice formed by flat-topped pyramids with n=3, 4, and 6 can be downscaled to fabricate photonic crystals with submicron periods. Since Bessel beams do not exhibit diffraction, multi-beam optical tweezers can be designed that use the gradient force of the extremum points in the optical lattice to trap and manipulate several particles at the same time [13–15]. Because of small size and ease of integration pyramid optical elements can possibly be used for cell sorting and Lab-on-a-chip technology.