## Multiband negative refraction in one-dimensional photonic crystals

Optics Express, Vol. 17, Issue 5, pp. 3042-3051 (2009)

http://dx.doi.org/10.1364/OE.17.003042

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### Abstract

We simulate a lossless one-dimensional photonic crystals (1D-PC) structure and show that negative refraction could be present near the low frequency edge of at least the second, fourth and sixth bandgaps. We experimentally demonstrate for the first time negative refraction in strongly modulated porous silicon 1D-PC in the visible and near infrared regions. This 1D-PC structure may allow the realization of short-focus Veselago lenses in different optical bands. An advantage of our structure is its simplicity allowing for cheap and rapid fabrication of samples.

© 2009 Optical Society of America

## 1. Introduction

1. V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of permittivity and permeability,” Sov. Phys. USPEKHI **10**, 509 (1968). [CrossRef]

2. J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. **85**, 3966–3999 (2000). [CrossRef] [PubMed]

3. D. F. Sievenpiper, M. E. Sickmiller, and E. Yablonovitch, “3D wire mesh photonic crystals,” Phys. Rev. Lett. **76**, 2480–2483 (1996). [CrossRef] [PubMed]

4. A. V. Kavokin, G. Malpuech, and I. Shelykh, “Negative refraction of light in Bragg mirrors made of porous silicon,” Phys. Lett. A **339**, 387–392 (2005). [CrossRef]

5. V. M. Agranovich, Y. R. Shen, R. H. Baughman, and A. A. Zakhidov,”Linear and non linear wave propagation in negative refraction metamaterials,” Phys. Rev. B **69**, 165112 (2004). [CrossRef]

6. P. V. Parimi, W. T. Lu, P. Vodo, J. Sokolo, J. S. Derov, and S. Sridhar, “Negative refraction in 1D photonic crystals,” Phys. Rev. Lett. **92**, 127401 (2004). [CrossRef] [PubMed]

10. H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamura, T. Sato, and S. Kawakami, “Self-collimating phenomensa in photonic crystal,” Appl. Phys. Lett. **74**, 1212 (1999). [CrossRef]

6. P. V. Parimi, W. T. Lu, P. Vodo, J. Sokolo, J. S. Derov, and S. Sridhar, “Negative refraction in 1D photonic crystals,” Phys. Rev. Lett. **92**, 127401 (2004). [CrossRef] [PubMed]

11. E. Cubukcu, K. Aydin, E. Ozbay, S. Foteinopoulou, and C. M. Soukoulis, “Negative refraction by photonic crystals,” Nature **423**, 604 (2003). [CrossRef] [PubMed]

9. C. Luo, S. G. Johnson, J. D. Joannopoulos, and J. B. Pendry,“Subwavelength imaging in photonic crystals,” Phys. Rev. B. **68**, 045115 (2003). [CrossRef]

16. C. Luo, S. G. Johnson, J. D. Joannopoulos, and J. B. Pendry, “Negative refraction without negative index in metallic photonic crystals,” Opt. Express **11**, 746 (2003). [CrossRef] [PubMed]

19. P. Vodo, P. V. Parimi, W. T. Lu, S. Sridhar, and R. Wing, “Microwave photonic crystal with tailor made negative refractive index,” Appl. Phys. Lett. **85**, 1858 (2004). [CrossRef]

22. G. Boedecker and C. Henkel, “All-frequency effective medium theory of a photonic crystal,” Opt. Express **11**, 1590 (2003). [CrossRef] [PubMed]

18. C. Yuan Yuan, H. Z. Ming, S. J. Long, L. C. Fang, and Q. Wang, “Frequency bands of negative refraction in finite one-dimensional photonic crystals,” Chin. Phys. **16**, 173 (2007). [CrossRef]

23. R. Srivastava, B. K. Thapa, S. Pati, and S.P. Ojha, “Negative refraction in 1D photonic crystals,” Solid State Commun. **147**, 157–160 (2008). [CrossRef]

24. J. Yao, Z. Liu, Y. Liu, Y. Wang, C. Sun, G. Bartal, A. M. Stacy, and X. Zhang, “Optical Negative Refraction in Bulk Metamaterials of Nanowires,” Science **321**, 930 (2008). [CrossRef] [PubMed]

4. A. V. Kavokin, G. Malpuech, and I. Shelykh, “Negative refraction of light in Bragg mirrors made of porous silicon,” Phys. Lett. A **339**, 387–392 (2005). [CrossRef]

## 2. Sample fabrication and characterization

^{TM}cell. Anodization begins when a constant current is applied between the c-Si wafer and the electrolyte by means of an electronic circuit controlling the anodization process. To produce the multilayers, current density applied during the electrochemical dissolution was alternated from 3 mA/cm

^{2}(layer a) to 40 mA/cm

^{2}(layer b) and eighty periods (160 layers) were made. Psi samples were partially oxidized at 350 °C for 10 min oxidation induces a blueshift in the peak reflectivity due to the decrease in the refractive indices of the layers, but it is necessary to stabilize the samples.

*a*) and 435 ±11 nm (

*b*). The theoretical photonic bandgaps locations were obtained from the band edge condition given by the following equation:

*ω*is the light frequency and

*c*is the light speed. The best refractive index values we found that fit the experimental photonic bandgap structure are

*n*= 1.1 and

_{a}*n*= 2 (Fig.1(c), green bands). We have experimentally measured the refractive indexes of single Psi layers made with the same electrochemical conditions as for the multilayers [25] and we found that

_{b}*n*= 1.4 and

_{a}*n*= 2.2. Nevertheless, it is known that the refractive index and etching rate for a single layer are modified in the presence of a multilayer structure up to approximately 14%, a phenomenon that has been systematically observed [26

_{b}26. L. Pavesi,“Porous silicon dielectric multilayers and microcavities,” La Rivista del Nuovo Cimento **20**, 1 (1997). [CrossRef]

## 3. Negative refraction condition

18. C. Yuan Yuan, H. Z. Ming, S. J. Long, L. C. Fang, and Q. Wang, “Frequency bands of negative refraction in finite one-dimensional photonic crystals,” Chin. Phys. **16**, 173 (2007). [CrossRef]

23. R. Srivastava, B. K. Thapa, S. Pati, and S.P. Ojha, “Negative refraction in 1D photonic crystals,” Solid State Commun. **147**, 157–160 (2008). [CrossRef]

*V*). They conclude that if light falls obliquely on the y-direction interface (Fig. 1(a)) then when

_{gll}*V*> 0, the wave refracts according to the classical Snell law;

_{gll}*V*= 0 is associated with the perpendicular propagation; but

_{gll}*V*< 0 means the transmitting beam will bend to the ‘wrong’ side, that is the same side as the incident beam resulting in negative refraction if the beam passes through the 1D-PC. Here, light impinges obliquely on the

_{gll}*x*-direction interface with an incidence angle

*α*(Fig. 1(a)), so we expect that to obtain negative refraction we need to look where the group velocity component that is perpendicular to the layers is negative (

*V*

_{g⊥}). In other words, the negative refraction condition in this example is obtained by the fact that if light impinges obliquely with frequency values near the low frequency edge of the second, fourth and sixth bandgap (Fig. 2(a)) the group velocity component that is perpendicular to the layers is negative (Fig. 2(b)) and consequently the effective mass is negative in the same region (Fig. 2(c)). Note that we are not claiming that a negative effective mass implies the negative refraction but under the aforementioned conditions it is true that negative effective mass implies negative refraction. However a simple contradiction for that is when the structure is excited with larger tangential wavevectors. In such cases, it can be seen that both positive and negative refraction are possible for a region with negative effective mass [27

27. W. Q. Zhang and F. Yang, “Negative refraction at various crystal interfaces,” Opt. Commun. **281**, 3081–3086 (2008). [CrossRef]

4. A. V. Kavokin, G. Malpuech, and I. Shelykh, “Negative refraction of light in Bragg mirrors made of porous silicon,” Phys. Lett. A **339**, 387–392 (2005). [CrossRef]

18. C. Yuan Yuan, H. Z. Ming, S. J. Long, L. C. Fang, and Q. Wang, “Frequency bands of negative refraction in finite one-dimensional photonic crystals,” Chin. Phys. **16**, 173 (2007). [CrossRef]

23. R. Srivastava, B. K. Thapa, S. Pati, and S.P. Ojha, “Negative refraction in 1D photonic crystals,” Solid State Commun. **147**, 157–160 (2008). [CrossRef]

**339**, 387–392 (2005). [CrossRef]

*a*(

*ka*/

*π*)

^{2}+

*β*(

*ka*/

*π*) +

*γ*can be rewritten as

*α*

^{*}(

*ka*/

*π*) +

*γ*where

*α*

^{*}=

*α*[1 + (

*β*/

*αk*)]. The effective mass approximation is obtained for small (

*ka*/

*π*), which is equivalent to the condition |

*α*

^{*}|≫|

*α*|. For light with wavelengths of 1350 nm and 633 nm we obtained

*α*

^{*}values of -5.35 and -8.68 respectively. Since the band structure in the y-direction for TE or TM polarization are the same the effective mass approximation is also the same for both polarizations. It is clear that given the periodicity of the band structure in the y-direction we would find the next negative refraction region lying in the sixth allowed band between 418 nm and 446 nm (Fig. 2 shows approximately the three negative refraction regions as yellow bands). The dispersion of photonic modes and the photonic bandgaps in an infinite periodic structure shown in Fig. 3 were calculated by using the well known transfer matrix techniques.

## 4. Finite element simulations

28. S. Foteinopoulou, E. N. Economou, and C. M. Soukoulis, “Refraction in Media with a Negative Refractive Index,” Phys. Rev. Lett. **90**, 107402 (2003). [CrossRef] [PubMed]

*V*

_{g⊥}= 0 and

*V*≠ 0 it is normal to have power propagating in that direction. In a perfect 1D-PC the Bloch waves are evanescent in the bandgap and they decay to zero. However in a finite 1D-PC we should expect power transmissions even inside the bandgap. Figure 5 shows the propagation of three wavelengths with incidence angle

_{gll}*α*= 0 . It is clear that the transmitted power decreases from the first negative refraction band (1350 nm) to the third one (419 nm). These results agree well with the ones presented in [18

**16**, 173 (2007). [CrossRef]

**147**, 157–160 (2008). [CrossRef]

## 5. Experimental results

*β*) but their positions were compensated to account for the positive refraction that the negative refraction beam (Component 7) suffers at the exit of the y-direction interface. This was done by using Snell’s law and an effective refractive index value of 1.6 for the multilayer structure. Once everything was in place we scanned the sample on the z-direction (see component 9) and when a spot image was captured we measured the distance from the x-direction interface to the image (Δ

*x*was corrected due to the sample rotation) by using the micrometer screw of the linear stage. This was possible because our imaging system was capable to give us enough resolution to mark on a screen the x-direction interface and the spot image. The components 9a–9d show four images corresponding to 1350 nm and 633 nm (TE and TM polarizations). The spots are located at a distance of approximately Δ

*x*= 16 microns from the interface (x-direction) which agreed with our numerical simulations.

## 6. Conclusions

**339**, 387–392 (2005). [CrossRef]

## Acknowledgments

## References and links

1. | V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of permittivity and permeability,” Sov. Phys. USPEKHI |

2. | J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. |

3. | D. F. Sievenpiper, M. E. Sickmiller, and E. Yablonovitch, “3D wire mesh photonic crystals,” Phys. Rev. Lett. |

4. | A. V. Kavokin, G. Malpuech, and I. Shelykh, “Negative refraction of light in Bragg mirrors made of porous silicon,” Phys. Lett. A |

5. | V. M. Agranovich, Y. R. Shen, R. H. Baughman, and A. A. Zakhidov,”Linear and non linear wave propagation in negative refraction metamaterials,” Phys. Rev. B |

6. | P. V. Parimi, W. T. Lu, P. Vodo, J. Sokolo, J. S. Derov, and S. Sridhar, “Negative refraction in 1D photonic crystals,” Phys. Rev. Lett. |

7. | H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Superprism phenomena in photonic crystals,” Phys. Rev. B |

8. | E. Cubukcu, K. Aydin, E. Ozbay, and S. Soukoulis, “Subwavelength Resolution in a Two-Dimensional Photonic-Crystal-Based Superlens,” Phys. Rev. Lett. |

9. | C. Luo, S. G. Johnson, J. D. Joannopoulos, and J. B. Pendry,“Subwavelength imaging in photonic crystals,” Phys. Rev. B. |

10. | H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamura, T. Sato, and S. Kawakami, “Self-collimating phenomensa in photonic crystal,” Appl. Phys. Lett. |

11. | E. Cubukcu, K. Aydin, E. Ozbay, S. Foteinopoulou, and C. M. Soukoulis, “Negative refraction by photonic crystals,” Nature |

12. | P. V. Parimi, W. T. Lu, P. Vodo, and S. Sridhar, “Photonic crystals: Imaging by flat lens using negative refraction,” Nature |

13. | A. Berrier, M. Mulot, M. Swillo, M. Qiu, L. Thylín, A. Talneau, and S. Anand, “Negative Refraction at Infrared Wavelengths in a Two-Dimensional Photonic Crystal,” Phys. Rev. Lett. |

14. | E. Schonbrun, Q. Wu, W. Park, T. Yamashita, C. J. Summers, M. Abashin, and Y. Fainman, “Wave front evolution of negatively refracted waves in a photonic crystal,” Appl. Phys. Lett. |

15. | R. Moussa, S. Foteinopoulou, L. Zhang, G. Tuttle, K. Guven, E. Ozbay, and C. M. Soukoulis, “Negative refraction and superlens behavior in a two-dimensional photonic crystal, ” Phys. Rev. B |

16. | C. Luo, S. G. Johnson, J. D. Joannopoulos, and J. B. Pendry, “Negative refraction without negative index in metallic photonic crystals,” Opt. Express |

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

18. | C. Yuan Yuan, H. Z. Ming, S. J. Long, L. C. Fang, and Q. Wang, “Frequency bands of negative refraction in finite one-dimensional photonic crystals,” Chin. Phys. |

19. | P. Vodo, P. V. Parimi, W. T. Lu, S. Sridhar, and R. Wing, “Microwave photonic crystal with tailor made negative refractive index,” Appl. Phys. Lett. |

20. | Z. Feng, X. Zhang, Y. Wang, Z. Y. Li, B. Cheng, and D. Z. Zhang, “Negative Refraction and Imaging Using 12-fold-Symmetry Quasicrystals,” Phys. Rev. Lett. |

21. | M. Notomi, “Negative refraction in photonic crystals,” Opt. Quantum Electron. |

22. | G. Boedecker and C. Henkel, “All-frequency effective medium theory of a photonic crystal,” Opt. Express |

23. | R. Srivastava, B. K. Thapa, S. Pati, and S.P. Ojha, “Negative refraction in 1D photonic crystals,” Solid State Commun. |

24. | J. Yao, Z. Liu, Y. Liu, Y. Wang, C. Sun, G. Bartal, A. M. Stacy, and X. Zhang, “Optical Negative Refraction in Bulk Metamaterials of Nanowires,” Science |

25. | R. Nava, M. B. de la Mora, J. Tagüeña-Martínez, and J. A. del Río, “Refractive index contrast in porous silicon multilayers,” Phys. Status Solidi A (To be published). |

26. | L. Pavesi,“Porous silicon dielectric multilayers and microcavities,” La Rivista del Nuovo Cimento |

27. | W. Q. Zhang and F. Yang, “Negative refraction at various crystal interfaces,” Opt. Commun. |

28. | S. Foteinopoulou, E. N. Economou, and C. M. Soukoulis, “Refraction in Media with a Negative Refractive Index,” Phys. Rev. Lett. |

**OCIS Codes**

(170.3010) Medical optics and biotechnology : Image reconstruction techniques

(260.2510) Physical optics : Fluorescence

**ToC Category:**

Medical Optics and Biotechnology

**History**

Original Manuscript: December 9, 2008

Revised Manuscript: January 21, 2009

Manuscript Accepted: February 9, 2009

Published: February 13, 2009

**Virtual Issues**

Vol. 4, Iss. 5 *Virtual Journal for Biomedical Optics*

**Citation**

Athanasios D. Zacharopoulos, Pontus Svenmarker, Johan Axelsson, Martin Schweiger, Simon R. Arridge, and Stefan Andersson-Engels, "A matrix-free algorithm for multiple wavelength fluorescence tomography," Opt. Express **17**, 3042-3051 (2009)

http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-17-5-3042

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### References

- V. Ntziachristos, J. Ripoll, L. H. V. Wang, and R. Weissleder, "Looking and listening to light: the evolution of whole body photonic imaging," Nat Biotechnol. 23, 313-320 (2005). [CrossRef]
- R. B. Schulz, J. Ripoll, and V. Ntziachristos, "Noncontact optical tomography of turbid media," Opt. Lett. 28, 1701-1703 (2003). [CrossRef] [PubMed]
- H. Meyer, A. Garofalakis, G. Zacharakis, S. Psycharakis, C. Mamalaki, D. Kioussis, E. N. Economou, V. Ntziachristos, and J. Ripoll, "Noncontact optical imaging in mice with full angular coverage and automatic surface extraction," Appl. Opt. 46, 3617-3627 (2007). [CrossRef] [PubMed]
- J. P. Culver, V. Ntziachristos, M. J. Holboke, and A. G. Yodh, "Optimization of optode arrangements for diffuse optical tomography: A singular-value analysis," Opt. Lett. 26, 701-703 (2001). [CrossRef]
- E. Graves, J. Ripoll, R. Weissleder, and V. Ntziachristos, "A submillimeter resolution fluorescence molecular imaging system for small animal imaging," Med. Phys. 30, 901-911 (2003). [CrossRef]
- H. Dehghani, S.C. Davis, S. Jiang, B.W. Pogue, K.D. Paulsen, and M.S. Patterson, "Spectrally resolved bioluminescence optical tomography," Opt. Lett. 31, 365-367 (2006). [CrossRef] [PubMed]
- A. Chaudhari, F. Darvas, J. R. Bading, R. A. Moats, P. S. Conti, D. J. Smith, S. R. Cherry, and R. M. Leahy, "Hyperspectral and multispectral bioluminescence optical tomography for small animal imaging," Phys. Med. Biol. 50, 5421-5441 (2005). [CrossRef]
- J. R. Mansfield, K. W. Gossage, C. C. Hoyt, and R. M. Levenson, "Autofluorescence removal, multiplexing, and automated analysis methods for in-vivo fluorescence imaging," J. Biomed. Opt. 10, 41207 (2005). [CrossRef] [PubMed]
- Z. M. Wang, G. Y. Panasyuk, V. A. Markel, and J. C. Schotland, "Experimental demonstration of an analytic method for image reconstruction in optical diffusion tomography with large data sets," Opt. Lett. 30, 3338-3340 (2005). [CrossRef]
- A. Corlu, T. Durduran, R. Choe, M. Schweiger, E. M. C. Hillman, S. R. Arridge, and A. G. Yodh, "Uniqueness and wavelength optimization in continuous-wave multispectral diffuse optical tomography," Opt. Lett. 28, 2339- 2341 (2003). [CrossRef]
- S. R. Arridge, "Optical tomography in medical imaging," Inverse Problems 15, 41-93 (1999). [CrossRef] [PubMed]
- V. Ntziachristos, and R. Weissleder, "Experimental three-dimensional fluorescence reconstruction of diffuse media by use of a normalized Born approximation," Opt. Lett. 26, 893-895 (2001). [CrossRef] [PubMed]
- H. Du, R. A. Fuh, J. Li, A. Corkan, and J. S. Lindsey, "PhotochemCAD: A computer-aided design and research tool in photochemistry," Photocem. Photobiol. 68, 141-142 (1998). [CrossRef] [PubMed]
- E. Alerstam, S. Andersson-Engels, and T. Svensson, "Improved accuracy in time-resolved diffuse reflectance spectroscopy," Opt. Express 16, 10440-10454 (2008). [CrossRef]
- M. Schweiger, S. Arridge, and I. Nissila, "GaussNewton method for image reconstruction in diffuse optical tomography," Phys. Med. Biol. 50, 2365-2386 (2005). [CrossRef]
- G. Y. Panasyuk, Z. M. Wang, J. C. Schotland, and V. A. Markel, "Fluorescent optical tomography with large data sets," Opt. Lett. 33, 1744-1746 (2008). [CrossRef] [PubMed]
- A. Corlu, R. Choe, T. Durduran, M. A. Rosen, M. Schweiger, S. R. Arridge, M. D. Schnall, and A. G. Yodh, "Three-dimensional in vivo fluorescence diffuse optical tomography of breast cancer in humans," Opt. Express 15, 6696-6716 (2007). [CrossRef]

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