## Structuring by multi-beam interference using symmetric pyramids

Optics Express, Vol. 14, Issue 12, pp. 5803-5811 (2006)

http://dx.doi.org/10.1364/OE.14.005803

Acrobat PDF (5346 KB)

### Abstract

A method for producing optical structures using rotationally symmetric pyramids is proposed. Two-dimensional structures can be achieved using acute prisms. They form by multi-beam interference of plane waves that impinge from directions distributed symmetrically around the axis of rotational symmetry. Flat-topped pyramids provide an additional beam along the axis thus generating three-dimensional structures. Experimental results are consistent with the results of numerical simulations. The advantages of the method are simplicity of operation, low cost, ease of integration, good stability, and high transmittance. Possible applications are the fabrication of photonic micro-structures such as photonic crystals or array waveguides as well as multi-beam optical tweezers.

© 2006 Optical Society of America

## 1. Introduction

1. E. Yablonovitch, “Photonic band-gap structures,” J. Opt. Soc. Am. B **10**, 283–295 (1993). [CrossRef]

2. U. Gruning, V. Lehmann, S. Ottow, and K. Busch, “Macroporous silicon with a complete two-dimensional photonic band gap centered at 5 µm,” Appl. Phys. Lett. **68**, 747–749 (1996). [CrossRef]

3. K. Hennessy, A. Badolato, A. Tamboli, P. M. Petroff, E. Hu, M. Atatüre, J. Dreiser, and A. Imamoğlu, “Tuning photonic crystal nanocavity modes by wet chemical digital etching,” Appl. Phys. Lett. **87**, 021108 (2005). [CrossRef]

4. K. Wang, A. Chelnokov, S. Rowson, P. Garoche, and J-M Lourtioz, “Focused-ion-beam etching in macroporous silicon to realize three-dimensional photonic crystals,” J. Phys. D: Appl. Phys. **33**, L119–L123 (2000). [CrossRef]

5. H. Sun, Y. Xu, S. Juodkazis, K. Sun, M. Watanabe, S. Matsuo, H. Misawa, and J. Nishii, “Arbitrary-lattice photonic crystals created by multiphoton microfabrication,” Opt. Lett. **26**, 325–327 (2001). [CrossRef]

6. M. J. Escuti and G. P. Crawford, “Holographic photonic crystals,” Opt. Eng. **43**, 1973–1987 (2004). [CrossRef]

7. V. Berger, O. Gauthier-Lafaye, and E. Costard, “Photonic band gaps and holography,” J. Appl. Phys. **82**, 60–64 (1997). [CrossRef]

8. T. Kondo, S. Matsuo, S. Juodkazis, and H. Misawa, “Femtosecond laser interference technique with diffractive beam splitter for fabrication of three-dimensional photonic crystals,” Appl. Phys. Lett. **79**, 725–727 (2001). [CrossRef]

9. T. Kondo, S. Matsuo, S. Juodkazis, V. Mizeikis, and H. Misawa, “Multiphoton fabrication of periodic structures by multibeam interference of femtosecond pulses,” Appl. Phys. Lett. **82**, 2758–2760 (2003). [CrossRef]

10. W. Hu, H. Li, B. Cheng, J. Yang, Z. Li, J. Xu, and D. Zhang, “Planar optical lattice of TiO2 particles,” Opt. Lett. **20**, 964–966 (1995). [CrossRef] [PubMed]

*et al.*[11

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

## 2. Theory

*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

*ħ*

_{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

_{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

_{0}has to be taken into account. Because of energy conservation for the refraction process, the magnitude of all momentums,

*k*=|

_{l}| with

*l*=

*0*, …,

*n*, is the same.

*z*along the symmetry axis,

*k*

_{z}

_{z}+

*k*

_{ρ}

_{ρ}by the probability amplitude

*E*exp(

*i*δ) is the complex amplitude of the partial probability (E, δ being real quantities),

*z*

_{z}+

*ρ*

_{ρ}is the position vector,

_{z}is the unit vector in z-direction and

_{ρ}=

_{ρ}(φ)=(cos

*φ*, sin

*φ*,0) is the unit vector in the radial direction.

*k*

_{ρ}do not differ for

*j*=1,…,

*n*. Only the direction of the radial unit vector varies with

*φ*=

*φ*

_{j}=2

*π*(

*j*-1)/

*n*. In case on an additional flat-top face the additional component has momentum

_{0}=

*k*

_{z}and as a rule a different value E

_{0}(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.

*z*,

*ρ*,

*φ*) in space is now proportional to

*k*-

*k*

_{z},

*I*

_{0}=

*nE*

^{2}is the average intensity, and the common phase

*δ*cancels out.

*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.

*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}=2

*π*(

*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 2

*n*if

*n*is an odd integer.

*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

*ρ*. Let

*i*

^{’}be the index of the smallest non-zero value of all cos

*φ*

_{i}. Then both expressions are periodic if cos

*φ*

_{i}is an integer multiple of cos

*φ*

_{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.

_{z}=2

*λ*/

*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

*λ*=633nm,

*θ*=2.5°,

*ϕ*

_{j}=360°*(

*j*-1)/

*n*,

*E*=1δ=0.

## 4. Results and discussion

*n*if

*n*is an even number, and 2

*n*, 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.

*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.

*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.

*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:

12. R. M. Herman and T. A. Wiggins, “Production and uses of diffractionless beams,” J. Opt. Soc. Am. A **8**, 932–942 (1991). [CrossRef]

13. J. Arlt, V. Garces-Chavez, W. Sibbett, and K. Dholakia, “Optical micromanipulation using a Bessel light beam,” Opt. Commun. **197**, 239–245 (2001). [CrossRef]

13. J. Arlt, V. Garces-Chavez, W. Sibbett, and K. Dholakia, “Optical micromanipulation using a Bessel light beam,” Opt. Commun. **197**, 239–245 (2001). [CrossRef]

15. M. P. MacDonald, G. C. Spalding, and K. Dholakia, “Microfluidic sorting in an optical lattice,” Nature **426**, 421–424 (2003). [CrossRef] [PubMed]

_{0}=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.

*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.

## 5. Conclusion

*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

13. J. Arlt, V. Garces-Chavez, W. Sibbett, and K. Dholakia, “Optical micromanipulation using a Bessel light beam,” Opt. Commun. **197**, 239–245 (2001). [CrossRef]

15. M. P. MacDonald, G. C. Spalding, and K. Dholakia, “Microfluidic sorting in an optical lattice,” Nature **426**, 421–424 (2003). [CrossRef] [PubMed]

## Acknowledgments

## References and Links

1. | E. Yablonovitch, “Photonic band-gap structures,” J. Opt. Soc. Am. B |

2. | U. Gruning, V. Lehmann, S. Ottow, and K. Busch, “Macroporous silicon with a complete two-dimensional photonic band gap centered at 5 µm,” Appl. Phys. Lett. |

3. | K. Hennessy, A. Badolato, A. Tamboli, P. M. Petroff, E. Hu, M. Atatüre, J. Dreiser, and A. Imamoğlu, “Tuning photonic crystal nanocavity modes by wet chemical digital etching,” Appl. Phys. Lett. |

4. | K. Wang, A. Chelnokov, S. Rowson, P. Garoche, and J-M Lourtioz, “Focused-ion-beam etching in macroporous silicon to realize three-dimensional photonic crystals,” J. Phys. D: Appl. Phys. |

5. | H. Sun, Y. Xu, S. Juodkazis, K. Sun, M. Watanabe, S. Matsuo, H. Misawa, and J. Nishii, “Arbitrary-lattice photonic crystals created by multiphoton microfabrication,” Opt. Lett. |

6. | M. J. Escuti and G. P. Crawford, “Holographic photonic crystals,” Opt. Eng. |

7. | V. Berger, O. Gauthier-Lafaye, and E. Costard, “Photonic band gaps and holography,” J. Appl. Phys. |

8. | T. Kondo, S. Matsuo, S. Juodkazis, and H. Misawa, “Femtosecond laser interference technique with diffractive beam splitter for fabrication of three-dimensional photonic crystals,” Appl. Phys. Lett. |

9. | T. Kondo, S. Matsuo, S. Juodkazis, V. Mizeikis, and H. Misawa, “Multiphoton fabrication of periodic structures by multibeam interference of femtosecond pulses,” Appl. Phys. Lett. |

10. | W. Hu, H. Li, B. Cheng, J. Yang, Z. Li, J. Xu, and D. Zhang, “Planar optical lattice of TiO2 particles,” Opt. Lett. |

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

12. | R. M. Herman and T. A. Wiggins, “Production and uses of diffractionless beams,” J. Opt. Soc. Am. A |

13. | J. Arlt, V. Garces-Chavez, W. Sibbett, and K. Dholakia, “Optical micromanipulation using a Bessel light beam,” Opt. Commun. |

14. | V. Garces-Chavez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, “Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam,” Nature |

15. | M. P. MacDonald, G. C. Spalding, and K. Dholakia, “Microfluidic sorting in an optical lattice,” Nature |

**OCIS Codes**

(170.4520) Medical optics and biotechnology : Optical confinement and manipulation

(220.4000) Optical design and fabrication : Microstructure fabrication

(260.3160) Physical optics : Interference

**ToC Category:**

Trapping

**History**

Original Manuscript: March 9, 2006

Revised Manuscript: May 13, 2006

Manuscript Accepted: May 13, 2006

Published: June 12, 2006

**Virtual Issues**

Vol. 1, Iss. 7 *Virtual Journal for Biomedical Optics*

**Citation**

Ming Lei, Baoli Yao, and Romano A. Rupp, "Structuring by multi-beam interference using symmetric pyramids," Opt. Express **14**, 5803-5811 (2006)

http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-14-12-5803

Sort: Year | Journal | Reset

### References

- E. Yablonovitch, "Photonic band-gap structures," J. Opt. Soc. Am. B 10,283-295 (1993). [CrossRef]
- U. Gruning, V. Lehmann, S. Ottow, and K. Busch, "Macroporous silicon with a complete two-dimensional photonic band gap centered at 5 μm," Appl. Phys. Lett. 68, 747-749 (1996). [CrossRef]
- K. Hennessy, A. Badolato, A. Tamboli, P. M. Petroff, E. Hu, M. Atatüre, J. Dreiser, and A. Imamoðlu, "Tuning photonic crystal nanocavity modes by wet chemical digital etching," Appl. Phys. Lett. 87, 021108 (2005). [CrossRef]
- K. Wang, A. Chelnokov, S. Rowson, P. Garoche, and J-M Lourtioz, "Focused-ion-beam etching in macroporous silicon to realize three-dimensional photonic crystals," J. Phys. D: Appl. Phys. 33, L119-L123 (2000). [CrossRef]
- H. Sun, Y. Xu, S. Juodkazis, K. Sun, M. Watanabe, S. Matsuo, H. Misawa, and J. Nishii, "Arbitrary-lattice photonic crystals created by multiphoton microfabrication," Opt. Lett. 26, 325-327 (2001). [CrossRef]
- M. J. Escuti, and G. P. Crawford, "Holographic photonic crystals," Opt. Eng. 43, 1973-1987 (2004). [CrossRef]
- V. Berger, O. Gauthier-Lafaye, and E. Costard, "Photonic band gaps and holography," J. Appl. Phys. 82, 60-64 (1997). [CrossRef]
- T. Kondo, S. Matsuo, S. Juodkazis, and H. Misawa, "Femtosecond laser interference technique with diffractive beam splitter for fabrication of three-dimensional photonic crystals," Appl. Phys. Lett. 79, 725-727 (2001). [CrossRef]
- T. Kondo, S. Matsuo, S. Juodkazis, V. Mizeikis, and H. Misawa, "Multiphoton fabrication of periodic structures by multibeam interference of femtosecond pulses," Appl. Phys. Lett. 82, 2758-2760 (2003). [CrossRef]
- W. Hu, H. Li, B. Cheng, J. Yang, Z. Li, J. Xu, and D. Zhang, "Planar optical lattice of TiO2 particles," Opt. Lett. 20, 964-966 (1995). [CrossRef] [PubMed]
- L. Cai, X. Yang, and Y. Wang, "All fourteen Bravais lattices can be formed by interference of four noncoplanar beams," Opt. Lett. 27, 900-902 (2002). [CrossRef]
- R. M. Herman, and T. A. Wiggins, "Production and uses of diffractionless beams," J. Opt. Soc. Am. A 8, 932-942 (1991). [CrossRef]
- J. Arlt, V. Garces-Chavez, W. Sibbett, and K. Dholakia, "Optical micromanipulation using a Bessel light beam," Opt. Commun. 197, 239-245 (2001). [CrossRef]
- V. Garces-Chavez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, "Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam," Nature 419, 145-147 (2002). [CrossRef] [PubMed]
- M. P. MacDonald, G. C. Spalding, and K. Dholakia, "Microfluidic sorting in an optical lattice," Nature 426, 421-424 (2003). [CrossRef] [PubMed]

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

### Supplementary Material

» Media 1: AVI (2265 KB)

» Media 2: AVI (2317 KB)

» Media 3: AVI (2402 KB)

» Media 4: AVI (501 KB)

» Media 5: AVI (984 KB)

» Media 6: AVI (925 KB)

« Previous Article | Next Article »

OSA is a member of CrossRef.