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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 15, Iss. 13 — Jun. 25, 2007
  • pp: 8003–8009
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Negative refractions in two-dimensional photonic crystals formed by holographic lithography

X. X. Shen, X. L. Yang, L. Z. Cai, G. Y. Dong, X. F. Meng, X. F. Xu, and Y. R. Wang  »View Author Affiliations


Optics Express, Vol. 15, Issue 13, pp. 8003-8009 (2007)
http://dx.doi.org/10.1364/OE.15.008003


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Abstract

We present a study on negative refractions in the four lowest bands of two-dimensional (2D) square lattices formed by holographic lithography (HL) and compare these features with those of a lattice of the same kind but with regular dielectric columns. The plane wave calculations and FDTD simulations have shown that in some bands or for some interfaces the negative refraction can only happen in holographic structures, and generally the rightness of holographic structures and regular structures of the same kind may be different.

© 2007 Optical Society of America

1. Introduction

The left-handed materials (LHMs) were first proposed by Veselago and have attracted much attention in recent years for their interesting properties and potential use [1–5

1. V. G. Veselago, “Electrodynamics of substances with simultaneously negative electrical and magnetic permeabilities,” Uspekhi Fiz. Nauk 92, 517–526 (1967).

]. Shelby [2

2. R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292, 77–79 (2001). [CrossRef] [PubMed]

] had an experimental verification of the first LHM structure based on split-ring resonators (μ<0) interconnected with a set of metallic rods (ε<0). Photonic crystals (PhCs) made of periodically modulated dielectric or metallic materials can also show effective negative refraction [6

6. M. Notomi, “Theory of light propagation in strong modulated photonic crystals: Refractionlike behavior in the vicinity of the photonic band gap,” Phys. Rev. B. 62, 10696–10705 (2000). [CrossRef]

]. Unlike the metamaterials made of conductors which have huge losses at higher frequencies, PhCs can exhibit much lower losses even at high frequencies such as in an optical band, since they are composed of insulators and the size and periodicity of their scattering elements are of the order of optical wavelength. Though to define the values of ε or μ of a PhC is impossible, many anomalous effects such as negative refraction and “super prism” [7

7. H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Superprism phenomena in photonic crystals,” Phys. Rev. B 58, R10096–10099 (1998). [CrossRef]

] were observed, and both negative refraction and left-handedness in PhCs were experimentally demonstrated in the microwave range [8–11

8. E. Cubukcu, K. Aydin, E. Ozbay, S. Foteinopoulou, and C. M. Soukoulis, “Negative refraction by photonic crystals,” Nature 423, 604–605 (2003). [CrossRef] [PubMed]

].

2. Structures

Fig. 1. Variation of the shape and size of the cross section of dielectric columns with the same I t = 2.72 and c= 0.16. (a)Holographic inverse structure, FR = 0.416; (b) holographic normal structure, FR = 0.584; and (c) regular structure, r = 0.364a, FR = 0.416.

The intensity distribution of the holographic structure we adopt in analysis is [16

16. L. Z. Cai, G. Y. Dong, C. S. Feng, X. L. Yang, X. X. Shen, and X. F. Meng, “Holographic design of a two-dimensional photonic crystal of square lattice with a large two-dimensional complete bandgap” J. Opt. Soc. Am. B 23, 1708–1711 (2006). [CrossRef]

]

I(x,y)=3+cos(2πxa)+cos(2πxa)c{cos[4π(x+y)a]+cos[4π(xy)a]}.
(1)

With a proper choice of coefficient c and a threshold intensity I t, which is a specific value the region with light intensity below it can be removed and the region above it will remain due to photopolymerization for negative photoresist, we may wash away the region of I < I t to get a normal structure. By filling this structure with a material of high dielectric constant and then removing the template, an inverse structure can be obtained [19

19. M. Campbell, D. N. Sharp, M. T. Harrison, R. G. Denning, and A. J. Turberfield, “Fabrication of photonic crystals for the visible spectrum by holographic lithography,” Nature 404, 53–56 (2000). [CrossRef] [PubMed]

]. In the following discussion we assume ε =11.4 for gallium arsenide as an example and choose c = 0.16 and I t = 2.72 which lead to the largest band gap as we analyzed before [16

16. L. Z. Cai, G. Y. Dong, C. S. Feng, X. L. Yang, X. X. Shen, and X. F. Meng, “Holographic design of a two-dimensional photonic crystal of square lattice with a large two-dimensional complete bandgap” J. Opt. Soc. Am. B 23, 1708–1711 (2006). [CrossRef]

]. To find the difference of propagating property between the holographic structures and the regular ones, we chose several lattices shown in Fig. 1 for investigation, here (a) and (b) are the inverse and normal holographic structures at the same threshold intensity I t = 2.72 respectively, and (c) is a regular structure having the same filling ratio (FR) as (a) but circular columns with r = 0.364a, here r is the radius of column and a the lattice constant.

3. Wave vector diagrams and FDTD results

The propagation of EMWs across both ΓM and ΓX interfaces of the 2D square lattice are investigated. As examples, we took the TE mode with its magnetic field perpendicular to the 2D plane and studied its propagation behavior at four lowest bands. The dimensionless relative frequency f= ωa/2πc is used, where c is the velocity of light in vacuum.

Fig. 2. EFCs plot and wave vector diagram for TE band 1 at the incident angle of 25° across the ΓM interface for the holographic inverse (a) and normal (b) structures and the regular structure (c). (a)f= 0.28 in air and frequency range of 0.15 to 0.30 with 0.01 step; (b)f= 0.24 in air, frequency range of 0.08 to 0.20 with 0.02 step and the range of 0.20 to 0.26 with 0.01 step; and (c)f= 0.28 in air and frequency range of 0.21 to 0.30 with 0.01 step.
Fig. 3. Contour maps of Hz simulated by FDTD showing the propagation wave patterns of TE1 wave for the cases in Figs. 2 (a), 2(b) and 2(c), respectively. The blue, red and yellow arrows denote K i, V g, and the wave vector of transmitted wave, respectively.

For the first band, if we choose ΓX interface as incident plane, the gradients of EFCs are outward and negative refraction cannot happen. On the other hand, for the ΓM interface, the EFCs in air and PhCs are plotted with wave vector diagrams for the incident angle of 25° in Figs. 2 (a), 2(b) and 2(c) for the holographic inverse structure, holographic normal structure, and the regular structure shown in Figs. 1(a), 1(b) and 1(c), respectively. Here and in all the similar figures below the frequency increases when the color changes from blue to red, the black dashed line means the conservation of the parallel components of wave vectors, the blue circle represents the EFC in air, and the blue, green and red arrows denote the directions of incident wave vector K i, refractive wave vector K f and group velocity V g, respectively; and the relative frequency in air is f= 0.28 for (a) and (c), and f= 0.24 for (b). From Figs. 2(a) and 2(b) we can see that in the outside region of the vicinity of M point EFCs are convex having an inward-pointing group velocity V g, but the black dashed line intersects two points at the EFC of photonic lattices, the upper one has V g pointing towards the source, while the bottom one has V g pointing away from the source. This means that only the bottom point contributes to a propagating beam or a transmitted beam. The refractive wave vector K f pointing to the intersect point is on the same side of the normal as K i is, thus right-handed (RH) negative refractions happen with K fV g > 0 and V g pointing to the negative direction [13

13. R. Gajić, R. Meisels, F. Kuchar, and K. Hingerl, “Refraction and rightness in photonic crystals,” Opt. Express 13, 8596–8605 (2005). [CrossRef] [PubMed]

]. Different from (a) and (b), the EFCs with high frequencies around M point in (c) are concave, yielding an outward-pointing V g and positive refractions. In order to get a comprehensive comparison, we have also investigated the regular structures with r/a changing form 0.1 ~ 0.5, since the columns will overlap and not be circles if r/a > 0.5. The frequencies of EFCs reduce with increasing column radius, and we find that negative refractions only happen in one situation of r/a = 0.5 when the columns are just joined together with each other. This means that negative refractions cannot occur in regular structures with separated circular dielectric columns for the first band of TE modes. To demonstrate these phenomena more clearly, we give the FDTD simulated wave patterns of TE 1 in Figs. 3(a), 3(b) and 3(c) for the cases of Figs. 2(a), 2(b) and 2(c), respectively. From Fig. 3 we can see that V g points to the negative direction in (a) and (b) and positive in (c), exactly coincident with the analysis of Fig. 2. The V g in holographic normal structure (b) is the largest, since the corresponding EFCs have the densest distribution and the largest gradient. The EFC plots are sparser for the inverse structure, and sparsest for regular lattices.

Fig. 4. EFCs plot and wave vector diagrams for TE band 2 and an incident angle of 20°. (a), (b) and (c) are for the HL inverse structure, HL normal structure and the structure of regular columns shown in Fig.1 (a), (b) and (c), respectively. (a) f= 0.35 in air and frequency range of 0.29 to 0.36 with 0.01 step; (b) f = 0.32 in air and frequency range of 0.26 to 0.34 with 0.01step; and (c) f= 0.35 in air and frequency range of 0.29 to 0.41 with 0.01step.
Fig. 5. GIF format images showing the propagation wave patterns of TE 2 wave on ΓX interface when incident angle changes from 5° to 60° with 5° step. (a)For inverse HL structure with incident frequency f = 0.35; [Media 1] (b) for normal HL structure and f= 0.32. [Media 2]
Fig. 6. Propagation wave patterns of TE 2 wave at an incidence of 20° on the ΓM interface. (a), (b) and (c) are for the inverse holographic structure, normal holographic structure and regular structure, respectively.

The EFC plots for the second band are shown in Fig. 4; here the incident angle is 20°. Figures 4(a) and 4(b) represent the HL inverse and normal lattices respectively. For these two structures, EFCs become smaller and convex when frequency increases, so negative refractions can be achieved easily when waves of inner higher frequencies are incident upon both ΓX and ΓM interfaces, and in the both cases we can get left-handed (LH) negative refractions with K fV g < 0 with V g pointing to the negative direction. The corresponding FDTD simulations of refractions for these two structures are given in GIF image format with increasing incident angles from 5° to 60° in Figs. 5(a) and 5(b), where the incident frequency is taken as 0.35 and 0.32 respectively. From these images we can see clearly that in almost the whole angular range negative refraction exists in the two structures. On the contrary, for lattices with regular circular columns in Fig. 4(c), there is only one convex EFC of f= 0.36 around the origin, thus negative refraction happens only for the waves incident across ΓM interface; and different from the former two lattices, the negative refraction here is similar as in Figs. 2 (a) and 2(b) with K fV g > 0 and V g is negative. The FDTD simulations given in Fig. 6 are the wave propagation patterns of TE waves at an incidence of 20° on the ΓM interface, (a), (b) and (c) are for the three structures respectively. The consistency of pointing direction of V g in these figures demonstrated our analysis again.

The EFC plots for the third band are four similar parts distributed symmetrically around the origin, and the shape of each part just seems like concave squares with their centers at M point. Different from other bands, the EFC plots and correspondingly the refractive properties here for the three different structures are very similar, so we won’t say more about this band.

Fig. 7. EFCs plot and wave vector diagrams for TE band 4 and incident angle of 45°. (a) For HL inverse structure, f = 0.50 in air and frequency range of 0.44 to 0.53 with 0.01 step; (b) for HL normal structure, f = 0.43 in air and frequency range of 0.38 to 0.46 with 0.01step; and (c) for regular structures, f = 0.50 in air and frequency range of 0.48 to 0.53 with 0.01 step.

For the forth band the EFC plots are shown in Fig. 7 for the same three structures as in Figs. 2 and 4. Now what appear in the four corners are almost circles centered at M with the frequencies increasing towards to the origin. And from calculations we find that there are overlapped band areas between band 3 and band 4 in holographic structures but not in regular lattices, which means that holographic structures have larger band ranges than regular lattices for this band, so the distributions of EFCs around M point are denser in (a) and (b) than in (c). If a wave is incident on ΓX interface at a small angle, there is no intersection point between the dashed conservation line and EFCs. When incident angle increases, intersection points may appear in (a) and (b) but still not in (c), for in the case of (c) there are few EFCs around M point due to their low density. For an incident angle large enough, the coupling efficiency would be strongly affected. Therefore negative refraction happens only in holographic structures. This fact is verified in Fig. 8 using FDTD simulations of TE 4 wave incident on ΓX interface at 45°, the incident relative frequency is 0.50 for (a) and (c), and 0.43 for (b).

At last the propagating properties of the second and forth band for regular structures with changing column radius have also been studied for comparison, and we found that when r/a is large enough the EFCs of regular structures become similar to those of holographic structures, and negative refraction may happen in these situations similarly as in HL structures, but the column radius is restricted within a narrow range.

Fig. 8. Propagation wave patterns of TE 4 wave at an incidence of 45° on the ΓX interface. (a), (b) and (c) are the contour maps of Hz corresponding to Fig. 7 (a), (b) and (c), respectively.

From the analyses above we can conclude that there are some differences in light propagation behavior between the holographic lattices and the lattice of regular columns. For TE1 mode, right-handed refraction happens in the former for ΓM interface, but not in the latter due to its concave EFCs near M point except the case of r/a = 0.5. And for TE2 mode, holographic structures can yield left-handed negative refractions when EMWs are incident either on ΓM or ΓX interface, for EFCs around Γ point are convex and have inward gradients; on the other hand, the regular lattice can only achieve right-handed negative refractions for ΓM interface incident when r/a is not large enough. The most similar properties for these three structures occur in the third band, where LH negative refraction happens in all lattices for a small wave band as the shapes of their EFCs are quite alike. On the contrary, the propagation properties in holographic and regular structures with a small ratio of r/a in band 4 are considerably different. The overlap of bands 3 and 4 in holographic structures leads to dense EFCs and negative refraction in band 4 lacked in regular structure. The negative refraction may appear in regular structures only for a large enough r/a. In table 1 we give all frequency bands that could intricate negative refractions for each structure, from it one can see that negative refraction seems more likely to happen in holographic structures, at least they cover a larger wave band.

Table 1:. The distributions of relative frequencies that can intricate negative refractions

table-icon
View This Table

5. Conclusions

Acknowledgment

This work is supported by the National Science Foundation of China (60477005, 60677026) and the National Science Foundation of Shandong Province (Y2006A09), China.

References and links

1.

V. G. Veselago, “Electrodynamics of substances with simultaneously negative electrical and magnetic permeabilities,” Uspekhi Fiz. Nauk 92, 517–526 (1967).

2.

R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292, 77–79 (2001). [CrossRef] [PubMed]

3.

J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans Microwave Theory Tech 47, 2075–2084 (1999). [CrossRef]

4.

J. Li, L. Zhou, C. T. Chan, and P. Sheng, “Photonic band gap from a stack of positive and negative index materials,” Phys. Rev. Lett. 90, 083901–083904 (2003). [CrossRef] [PubMed]

5.

L. Chen, S. L. He, and L. F. Shen, “Finite-size effects of a left-handed material Slab on the image quality,” Phys. Rev. Lett. 92, 107404-1–4 (2004). [CrossRef] [PubMed]

6.

M. Notomi, “Theory of light propagation in strong modulated photonic crystals: Refractionlike behavior in the vicinity of the photonic band gap,” Phys. Rev. B. 62, 10696–10705 (2000). [CrossRef]

7.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Superprism phenomena in photonic crystals,” Phys. Rev. B 58, R10096–10099 (1998). [CrossRef]

8.

E. Cubukcu, K. Aydin, E. Ozbay, S. Foteinopoulou, and C. M. Soukoulis, “Negative refraction by photonic crystals,” Nature 423, 604–605 (2003). [CrossRef] [PubMed]

9.

R. Gajic, F. Kuchar, R. Meisels, J. Radovanovic, K. Hingerl, J. Zarbakhsh, J. Stampfl, and A. Woesz, “Physical and materials aspects of photonic crystals for microwaves and millimetre waves,” Zeitschrift fuer Metallkunde 95, 618–623 (2004).

10.

K. Guven, K. Aydin, K. B. Alici, C. M. Soukoulis, and E. Ozbay, “Spectral negative refraction and focusing analysis of a two-dimensional left-handed photonic crystal lens,” Phys. Rev. B. 70, 205125-1–5 (2004). [CrossRef]

11.

P. V. Parimi, W. T. Lu, P. Vodo, J. Sokoloff, J. S. Derov, and S. Sridhar, “Negative refraction and lefthanded electromagnetism in microwave photonic crystals,” Phys. Rev. Lett. 92, 127401-1–4 (2004). [CrossRef] [PubMed]

12.

S. Foteinopoulou1 and C. M. Soukoulis “Electromagnetic wave propagation in two-dimensional photonic crystals: A study of anomalous refractive effects,” Phys. Rev. B. 72, 165112-1–20 (2005). [CrossRef]

13.

R. Gajić, R. Meisels, F. Kuchar, and K. Hingerl, “Refraction and rightness in photonic crystals,” Opt. Express 13, 8596–8605 (2005). [CrossRef] [PubMed]

14.

E. Yablonovitch, T. J. Gmitter, and K. M. Leung, “Photonic band structure: The face-centered-cubic case employing nonspherical atoms,” Phys. Rev. Lett. 672295–2298 (1991). [CrossRef] [PubMed]

15.

X. Y. Ao and S. L. He, “Three-dimensional photonic crystal of negative refraction achieved by interference lithography,” Opt. Lett. 29, 2542–2544 (2004). [CrossRef] [PubMed]

16.

L. Z. Cai, G. Y. Dong, C. S. Feng, X. L. Yang, X. X. Shen, and X. F. Meng, “Holographic design of a two-dimensional photonic crystal of square lattice with a large two-dimensional complete bandgap” J. Opt. Soc. Am. B 23, 1708–1711 (2006). [CrossRef]

17.

L. Z. Cai, C. S. Feng, M. Z. He, and X. L. Yang, “Holographic design of a two-dimensional photonic crystal of square lattice with pincushion columns and large complete band gaps” Opt. Express 13, 4325–4330 (2005). [CrossRef] [PubMed]

18.

K. M. Ho, C. T. Chan, and C. M. Soukoulis, “Existence of a photonic gap in periodic dielectric structures,” Phys. Rev. Lett. 653152–3155 (1990). [CrossRef] [PubMed]

19.

M. Campbell, D. N. Sharp, M. T. Harrison, R. G. Denning, and A. J. Turberfield, “Fabrication of photonic crystals for the visible spectrum by holographic lithography,” Nature 404, 53–56 (2000). [CrossRef] [PubMed]

OCIS Codes
(090.2880) Holography : Holographic interferometry
(260.2110) Physical optics : Electromagnetic optics

ToC Category:
Photonic Crystals

History
Original Manuscript: April 13, 2007
Revised Manuscript: May 23, 2007
Manuscript Accepted: May 23, 2007
Published: June 12, 2007

Citation
X. X. Shen, X. L. Yang, L. Z. Cai, G. Y. Dong, X. F. Meng, X. F. Xu, and Y. R. Wang, "Negative refractions in two-dimensional photonic crystals formed by holographic lithography," Opt. Express 15, 8003-8009 (2007)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-13-8003


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References

  1. V. G. Veselago, "Electrodynamics of substances with simultaneously negative electrical and magnetic permeabilities," Usp. Fiz. Nauk 92, 517-526 (1967).
  2. R. A. Shelby, D. R. Smith, and S. Schultz, "Experimental verification of a negative index of refraction," Science 292, 77-79 (2001). [CrossRef] [PubMed]
  3. J. B. Pendry, A. J. Holden, D. J. Robbins, W. J. Stewart, "Magnetism from conductors and enhanced nonlinear phenomena," IEEE Trans Microwave Theory Tech 47, 2075-2084 (1999). [CrossRef]
  4. J. Li, L. Zhou, C. T. Chan, and P. Sheng, "Photonic band gap from a stack of positive and negative index materials," Phys. Rev. Lett. 90, 083901-083904 (2003). [CrossRef] [PubMed]
  5. L. Chen, S. L. He, L. F. Shen, "Finite-size effects of a left-handed material Slab on the image quality," Phys. Rev. Lett. 92, 107404-1-4 (2004). [CrossRef] [PubMed]
  6. M. Notomi, "Theory of light propagation in strong modulated photonic crystals: Refractionlike behavior in the vicinity of the photonic band gap," Phys. Rev. B. 62, 10696-10705 (2000). [CrossRef]
  7. H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, "Superprism phenomena in photonic crystals," Phys. Rev. B 58, R10096-10099 (1998). [CrossRef]
  8. E. Cubukcu, K. Aydin, E. Ozbay, S. Foteinopoulou, and C. M. Soukoulis, "Negative refraction by photonic crystals," Nature 423, 604-605 (2003). [CrossRef] [PubMed]
  9. R. Gajic, F. Kuchar, R. Meisels, J. Radovanovic, K. Hingerl, J. Zarbakhsh, J. Stampfl, and A. Woesz, "Physical and materials aspects of photonic crystals for microwaves and millimetre waves," Z. Metallkd. 95, 618-623 (2004).
  10. K. Guven, K. Aydin, K. B. Alici, C. M. Soukoulis, and E. Ozbay, "Spectral negative refraction and focusing analysis of a two-dimensional left-handed photonic crystal lens," Phys. Rev. B. 70, 205125-1-5 (2004). [CrossRef]
  11. P. V. Parimi, W. T. Lu, P. Vodo, J. Sokoloff, J. S. Derov, and S. Sridhar, "Negative refraction and lefthanded electromagnetism in microwave photonic crystals," Phys. Rev. Lett. 92, 127401-1-4 (2004). [CrossRef] [PubMed]
  12. S. Foteinopoulou1 and C. M. Soukoulis "Electromagnetic wave propagation in two-dimensional photonic crystals: A study of anomalous refractive effects," Phys. Rev. B. 72, 165112-1-20 (2005). [CrossRef]
  13. R. Gajiæ, R. Meisels, F. Kuchar, and K. Hingerl, "Refraction and rightness in photonic crystals," Opt. Express 13, 8596-8605 (2005). [CrossRef] [PubMed]
  14. E. Yablonovitch, T. J. Gmitter, and K. M. Leung, "Photonic band structure: The face-centered-cubic case employing nonspherical atoms," Phys. Rev. Lett. 67, 2295-2298 (1991). [CrossRef] [PubMed]
  15. X. Y. Ao and S. L. He, "Three-dimensional photonic crystal of negative refraction achieved by interference lithography," Opt. Lett. 29, 2542-2544 (2004). [CrossRef] [PubMed]
  16. L. Z. Cai, G. Y. Dong, C. S. Feng, X. L. Yang, X. X. Shen, and X. F. Meng, "Holographic design of a two-dimensional photonic crystal of square lattice with a large two-dimensional complete bandgap" J. Opt. Soc. Am. B 23, 1708-1711 (2006). [CrossRef]
  17. L. Z. Cai, C. S. Feng, M. Z. He, and X. L. Yang, "Holographic design of a two-dimensional photonic crystal of square lattice with pincushion columns and large complete band gaps" Opt. Express 13, 4325-4330 (2005). [CrossRef] [PubMed]
  18. K. M. Ho, C. T. Chan, and C. M. Soukoulis, "Existence of a photonic gap in periodic dielectric structures," Phys. Rev. Lett. 65, 3152-3155 (1990). [CrossRef] [PubMed]
  19. M. Campbell, D. N. Sharp, M. T. Harrison, R. G. Denning, and A. J. Turberfield, "Fabrication of photonic crystals for the visible spectrum by holographic lithography," Nature 404, 53-56 (2000). [CrossRef] [PubMed]

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