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Optics Express

  • Editor: Michael Duncan
  • Vol. 13, Iss. 11 — May. 30, 2005
  • pp: 4325–4330
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Holographic design of a two-dimensional photonic crystal of square lattice with pincushion columns and large complete band gaps

L. Z. Cai, C. S. Feng, M. Z. He, X. L. Yang, X. F. Meng, G. Y. Dong, and X. Q. Yu  »View Author Affiliations


Optics Express, Vol. 13, Issue 11, pp. 4325-4330 (2005)
http://dx.doi.org/10.1364/OPEX.13.004325


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Abstract

In holographic fabrication of photonic crystals the shape and size of the dielectric columns or particles (“atoms”) are determined by the isointensity surfaces of the interference field. Therefore their photonic band gap (PBG) properties are closely related to their fabrication design. As an example, we have investigated the PBGs of a kind of holographically formed two-dimensional (2-D) square lattice with pincushion columns rotated by 45°, and shown that this structure has complete PBGs in a wide range of dielectric contrast comparable to or even larger than those of the same lattice with square columns reported before. The optical design for making this structure is also given. This work may demonstrate the unique feature and advantages of photonic crystals made by holographic method and provide a guideline for their design and experimental fabrication.

© 2005 Optical Society of America

1. Introduction

Photonic crystals (PhCs) have attracted much interest for its potential in controlling and manipulating the propagation of light due to the existence of photonic band gap (PBG) [1

1. E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059–2062 (1987). [CrossRef] [PubMed]

, 2

2. J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals (Princeton University Press, Princeton, 1995).

]. Although three-dimensional (3-D) PhCs seem more versatile, two-dimensional (2-D) PhCs have also been extensively studied, since the latter are usually easier to fabricate and may be employed in many applications [3

3. R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannapoulos, “Existence of a photonic band gap in two dimensions,” Appl. Phys. Lett. 61, 495–497 (1992) [CrossRef]

10

10. Z. Y. Li, B. Y. Gu, and G. Z. Yang, “Large absolute band gaps in two-dimensional anisotropic photonic crystals,” Phys. Rev. Lett. 77, 2574–2977 (1998). [CrossRef]

]. The important task of band gap engineering for 2-D PhCs is to find proper structures having PBG for both TE (or p) and TM (or s) polarization modes and design methods to produce them.

A complete PBG in two dimensions was first demonstrated for a triangular lattice of air columns in dielectric material [3

3. R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannapoulos, “Existence of a photonic band gap in two dimensions,” Appl. Phys. Lett. 61, 495–497 (1992) [CrossRef]

, 4

4. P. R. Villeneuve and M. Piche, “Phonic band gaps in two-dimensional square and hexagonal lattices,” Phys. Rev. B 46, 4969–4972 (1992). [CrossRef]

] and for a square lattice of air columns [4

4. P. R. Villeneuve and M. Piche, “Phonic band gaps in two-dimensional square and hexagonal lattices,” Phys. Rev. B 46, 4969–4972 (1992). [CrossRef]

, 5

5. P. R. Villeneuve and M. Piche, “Photonic band gaps in two-dimensional square lattices: Square and circular lattices,” Phys. Rev. B 46, 4973–4975 (1992). [CrossRef]

]. Since then an extensive investigation has been made in this area [6

6. D. L. Bullock, C. Shih, and R. S. Margulies, “Photonic band structure investigation of two-dimensional Bragg reflector mirrors for semiconductor laser mode control,” J. Opt. Soc. Am. B 10, 399–403 (1993). [CrossRef]

10

10. Z. Y. Li, B. Y. Gu, and G. Z. Yang, “Large absolute band gaps in two-dimensional anisotropic photonic crystals,” Phys. Rev. Lett. 77, 2574–2977 (1998). [CrossRef]

]. It is found that generally a PBG for s polarization is favored in the case of dielectric columns, while a PBG for p polarization is favored if the dielectric regions are connected; and that the PBG property of a lattice is also dependent on the shape and size of the columns. Therefore it is not easy to give simple arguments predicting the existence of a complete PBG for a given lattice. Usually the trial and error method has to be used to find “good” structures [11

11. M. Qiu and S. He, “Optimal design of two-dimensional photonic crystal of square lattice with large complete two-dimensional bandgap,” J. Opt. Soc. Am. A 17, 1027–1030 (2000). [CrossRef]

, 12

12. M. Agio and L. C. Andreanm, “Complete photonic band gap in a two-dimensional chessboard lattice,” Phys. Rev. B 61, 15519–15522 (2000). [CrossRef]

]. An interesting example is a 2-D so-called chessboard lattice, a square lattice with square columns rotated by 45°, which may give quite large a complete PBG for a relatively low dielectric contrast from theoretical simulations, for instance, Δω/ω=8.5% for ε=8.9 (alumina) [12

12. M. Agio and L. C. Andreanm, “Complete photonic band gap in a two-dimensional chessboard lattice,” Phys. Rev. B 61, 15519–15522 (2000). [CrossRef]

]. This is an encouraging result considering that the traditional square lattice of air columns with square cross section requires ε>12.3 for a complete PBG to exist [12

12. M. Agio and L. C. Andreanm, “Complete photonic band gap in a two-dimensional chessboard lattice,” Phys. Rev. B 61, 15519–15522 (2000). [CrossRef]

].

2. Band gap analysis: relation between PBG and concrete structure

To produce a 2-D chessboard-like lattice holographically, we may use a light intensity distribution

I(x,y)=2+cos(2πax)+cos(2πay)+{cos[2πa(x+y)]+cos[2πa(xy)]},
(1)

Obviously, for the special case of c=0 and It=0, we can get a chessboard lattice with filling ratio (FR) f=0.5. This structure can produce a complete band gap for a range of dielectric constant ε>7 [12

12. M. Agio and L. C. Andreanm, “Complete photonic band gap in a two-dimensional chessboard lattice,” Phys. Rev. B 61, 15519–15522 (2000). [CrossRef]

]. But more comprehensive study shows that this is not the optimized case for generating large PBG. In holographic approach there are two adjustable parameters, c in Eq. (1) and It in fabrication process. Both of them have critical influences on PBG and therefore give us more freedom to improve band gap property. In the following we will make a systematic investigation on their effects. In the calculations we use plane wave method [18

18. M. Leung and Y. F. Liu, “Photon band structures: The plane-wave method,” Phys. Rev. B 41, 10188–10190 (1990). [CrossRef]

], the number of plane waves used here are 797 for both s and p polarizations, and the accuracy within 1% is expected.

Fig. 1. The relation between threshold intensity I t and the filling ratio f of dielectric material when c=0.31, where line (I) is for the normal structure and line (II) for the inverse structure.
Fig. 2. Variation of the shape and size of the cross section of dielectric columns with different I t when c=0.31. (a) I t=1.26, f=0.278; (b) I t=1.32, f=0.314; (c) I t=1.37, f=0.347; (d) I t=1.40, f=0.375.

First, we show the effect of I t on the filling ratio, the column shape, and the PBG for a given c, here c is assumed to be 0.31 for calculations (the reason will be explained below). In Fig. 1 we draw the curves of FR versus I t for both the normal and the inverse lattices. Fig. 2 gives the concrete shapes of several inverse lattices formed at different I t. Evidently the shape and FR of the dielectric columns (black in Fig. 2) vary gradually with I t. Generally a smaller It yields smaller dielectric columns with smaller FR, while a greater It gives rise to connected columns with a greater FR. The apparent difference here in holographic fabrication from the ideal chessboard design [12

12. M. Agio and L. C. Andreanm, “Complete photonic band gap in a two-dimensional chessboard lattice,” Phys. Rev. B 61, 15519–15522 (2000). [CrossRef]

] is that usually the shape of dielectric columns now is no longer square but like a pincushion with the middle part of each side shrunken towards the center of the columns. This is characteristic of holographic method in which the shape of columns or atoms is decided by equal-intensity surfaces.

Fig. 3. Variation of relative band gap with intensity threshold I t for the inverse structure in the case of c=0.31 and ε=8.9.
Fig. 4. Gap map for the inversed structure when c=0.31 and ε=8.9.

Naturally the different shapes and filling ratios resulted from different selection of intensity threshold I t will lead to different PBG for a given intensity distribution. In Fig. 3 we give the curve of relative PBG versus It for the inverse structure in the case of c=0.31 and ε=8.9. The corresponding gap map is plotted in Fig. 4. From these two figures we can clearly see the general trend that the s gap is easier to occur for the case of non-overlapping dielectric columns (smaller I t and f), while the p gap is favored in the case of air columns (greater I t and f); and the largest PBG occurs as a compromise of the two factors. However, different from the case of exact square columns, now the columns begin to be overlapped at a smaller f, about 0.355 instead of 0.5, due to the pincushion shape of the columns. Figs. 3 and 4 indicate that the complete PBG can be obtained in the range of I t=1.31 to 1.45 or equally f=0.307 to 0.405; and that the largest relative band gap width appears in It=1.37 when the corresponding FR is f=0.347. In this optimal condition we have Δω/ω=8.8 %, a little larger than the result of best design for square columns [12

12. M. Agio and L. C. Andreanm, “Complete photonic band gap in a two-dimensional chessboard lattice,” Phys. Rev. B 61, 15519–15522 (2000). [CrossRef]

]. The band structure in this condition is shown in Fig. 5.

Fig. 5. The photonic band structure in the optimized case I t=1.37 when c=0.31 and ε=8.9. The solid curves are for the p polarization, and the dotted curves are for the s polarization.
Fig. 6. Different column shape and size of inverse structure of ε=8.9 yielding maximum relative PBG for different values of c. (a) c=0.1, I t=1.78, f=0.443; (b) c=0.2, I t=1.59, f=0.407; (c) c=0.3, I t=1.39, f=0.354; (d) c=0.4, I t=1.20, f=0.293.

Fig. 7. Relation between the value of c and the corresponding maximum relative band width when ε=8.9.
Fig. 8. Optimized column shapes yielding maximum PBG for different ε. (a) ε=12, c=0.24, I t=1.52, f=0.395; (b) ε=16, c=0.20, I t=1.60, f=0.414.

We have also studied the PBG properties of this kind of inverse structures for other dielectric constants and compared our results with those obtained for square columns [12

12. M. Agio and L. C. Andreanm, “Complete photonic band gap in a two-dimensional chessboard lattice,” Phys. Rev. B 61, 15519–15522 (2000). [CrossRef]

]. Our calculations show that the complete PBG occurs when ε is greater than about 6.4 (nearly 1% PBG for c=0.27 and I t=1.51), lower than 7 in the case of square columns [12

12. M. Agio and L. C. Andreanm, “Complete photonic band gap in a two-dimensional chessboard lattice,” Phys. Rev. B 61, 15519–15522 (2000). [CrossRef]

], and the comparison of maximum PBGs for several dielectric constants, namely, 8.9, 12 (GaAs) and 16 (Ge) as used in Ref. [12

12. M. Agio and L. C. Andreanm, “Complete photonic band gap in a two-dimensional chessboard lattice,” Phys. Rev. B 61, 15519–15522 (2000). [CrossRef]

] is listed in table 1. The corresponding optimized column shapes for the latter two dielectric constants are given in Fig. 8. The similar look of the pictures in Fig. 8 has again confirmed the superiority of pincushion columns to the square columns in PBG generation, and the results in table 1 give us an assurance that the holographic structures are usually better than or at least comparable to the square column structures in PBG property in all cases.

Table 1. Comparison of maximum PBGs for two structures with different dielectric constants

table-icon
View This Table

3. Optical design of holographic fabrication

Fig. 9. Optical design of holographic fabrication of the structure expressed by Eq. (1).

Considering the symmetry of Eq. (1), we may use four linearly polarized beams of the same intensity (assumed to be unit here) with the same angle θ with z axis shown in Fig. 9 to fabricate this structure. In this geometry the four wave vectors are

K1=K(sinθ2,sinθ2,cosθ),K2=K(sinθ2,sinθ2,cosθ),
K3=K(sinθ2,sinθ2,cosθ),K4=K(sinθ2,sinθ2,cosθ),
(2)

where K=2π/λ, λ is the wavelength. This symmetric arrangement, which can be realized with the use of a diffraction beam splitter (DBS) [19

19. 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]

], may assure equal optical path of each beam in z axis, and then the intensity distribution of the interference field can be written as

I(x,y)=4+2{(e14+e23)cos(2πxa)+(e12+e34)cos(2πya)
+e13cos[2π(x+y)a]+e24cos[2π(xy)a],
(3)

where a=λ/(21/2sinθ), e ij=e i·e j, e j is the unit polarization vector of the jth plane wave. Comparing the cosine terms in Eqs. (3) and (1), we can find that the varying parts in these two equations have the same relation on the condition

e13=e24=c(e14+e23)=c(e12+e34).
(4)

From this equation and the symmetry of beam geometry, we may assume

e1=(l,m,n),e2=(l,m,n),e3=(p,q,r),e4=(p,q,r),
(5)

where l, m, n, p, q and r are all real numbers. Using normalization condition of each polarization vector, orthogonal condition of e j and K j, and Eq. (4), we have

l2+m2+n2=1,p2+q2+r2=1,
(6)
(l+m)sinθ+2ncosθ=0,(p+q)sinθ2rcosθ=0,
(7)
lp+nr+(1+2c)mq(12c)=0,m2+q22mq/(12c)=1.
(8)

There are six independent equations in Eqs. (6)(8). Solving them for given c and θ we can obtain all the six parameters. A possible solution for c=0.31 and θ=30° is

e1=(0.76253,0.42730,0.48575),e2=(0.76253,0.42730,0.48575),
e3=(0.91602,0.31839,0.24398),e4=(0.91602,0.31839,0.24398).
(9)

Substituting Eq. (9) into Eq. (3), we have

I(x,y)=2.864{1.397+cos(2πxa)+cos(2πya)
+0.31cos[2π(x+y)a]+0.31cos[2π(xy)a]}.
(10)

Comparing this equation with Eq. (1), we can see the present intensity threshold I t’ corresponding to the I t for Eq. (1) to generate the same structure should be modulated as

It=2.864(It0.603).
(11)

For example, the optimized I t=1.37 for ε=8.9 should now become I t’=2.197, readers can verify the same FR f=0.347 as mentioned before.

4. Conclusions

Acknowledgments

This work is supported by the National Natural Science Foundation (64077005), Doctoral Program Foundation of Ministry of Education (20020422047), China, and National Key Lab Foundation, Institute of Crystal Materials, Shandong University, China.

References and links

1.

E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059–2062 (1987). [CrossRef] [PubMed]

2.

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals (Princeton University Press, Princeton, 1995).

3.

R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannapoulos, “Existence of a photonic band gap in two dimensions,” Appl. Phys. Lett. 61, 495–497 (1992) [CrossRef]

4.

P. R. Villeneuve and M. Piche, “Phonic band gaps in two-dimensional square and hexagonal lattices,” Phys. Rev. B 46, 4969–4972 (1992). [CrossRef]

5.

P. R. Villeneuve and M. Piche, “Photonic band gaps in two-dimensional square lattices: Square and circular lattices,” Phys. Rev. B 46, 4973–4975 (1992). [CrossRef]

6.

D. L. Bullock, C. Shih, and R. S. Margulies, “Photonic band structure investigation of two-dimensional Bragg reflector mirrors for semiconductor laser mode control,” J. Opt. Soc. Am. B 10, 399–403 (1993). [CrossRef]

7.

C. M. Anderson and K. P. Giapis, “Larger two-dimensional photonic band gaps,” Phys. Rev. Lett. 77, 2949–2952 (1996). [CrossRef] [PubMed]

8.

J. C. Knight, T. A. Birks, P. St. J. Russell, and D. M. Atkin, “All-silica single-mode fiber with photonic crystal cladding,” Opt. Lett. 21, 1547–1549 (1996). [CrossRef] [PubMed]

9.

S. Y. Lin, G. Arjavalingam, and W. M. Robertson, “Investigation of absolute photonic band-gaps in 2-dimensional dielectric structures,” J. Mod. Opt. 41, 385–393 (1994) [CrossRef]

10.

Z. Y. Li, B. Y. Gu, and G. Z. Yang, “Large absolute band gaps in two-dimensional anisotropic photonic crystals,” Phys. Rev. Lett. 77, 2574–2977 (1998). [CrossRef]

11.

M. Qiu and S. He, “Optimal design of two-dimensional photonic crystal of square lattice with large complete two-dimensional bandgap,” J. Opt. Soc. Am. A 17, 1027–1030 (2000). [CrossRef]

12.

M. Agio and L. C. Andreanm, “Complete photonic band gap in a two-dimensional chessboard lattice,” Phys. Rev. B 61, 15519–15522 (2000). [CrossRef]

13.

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]

14.

L. Z. Cai, X. L. Yang, and Y. R. Wang, “Formation of three-dimensional periodic microstructures by interference of four noncoplanar beams,” J. Opt. Soc. Am. A 19, 2238–2244 (2002). [CrossRef]

15.

Y.A. Vlasov, X. Z. Bo, J. C. Sturm, and D. J. Norris, “On-chip natural assembly of silicon photonic bangap crystals,” Nature 414, 289–293 (2001). [CrossRef] [PubMed]

16.

X. L. Yang, L. Z. Cai, and Q. Liu, “Theoretical bandgap modeling of two-dimensional triangular photonic crystals formed by interference technique of three-noncoplanar beams,” Opt. Express 11, 1050–1055 (2003). [CrossRef] [PubMed]

17.

X. L. Yang, L. Z. Cai, Q. Liu, and H. K. Liu, “Theoretical bandgap modeling of two-dimensional square photonic crystals fabricated by interference technique of three-noncoplanar beams,” J. Opt. Soc. Am. B 21, 1050–1055 (2004). [CrossRef]

18.

M. Leung and Y. F. Liu, “Photon band structures: The plane-wave method,” Phys. Rev. B 41, 10188–10190 (1990). [CrossRef]

19.

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]

OCIS Codes
(090.2880) Holography : Holographic interferometry
(220.4000) Optical design and fabrication : Microstructure fabrication
(260.2110) Physical optics : Electromagnetic optics
(260.3160) Physical optics : Interference
(350.3950) Other areas of optics : Micro-optics

ToC Category:
Research Papers

History
Original Manuscript: March 25, 2005
Revised Manuscript: May 18, 2005
Published: May 30, 2005

Citation
L. Cai, C. S. Feng, M.Z. He, X. L. Yang, X. Meng, G. Y. Dong, and X. Yu, "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)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-13-11-4325


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References

  1. E. Yablonovitch, �??Inhibited spontaneous emission in solid-state physics and electronics,�?? Phys. Rev. Lett. 58, 2059-2062 (1987). [CrossRef] [PubMed]
  2. J. D. Joannopoulos, R. D. Meade, J. N. Winn, Photonic Crystals (Princeton University Press, Princeton, 1995).
  3. R. D. Meade, K. D. Brommer, A. M. Rappe, J. D. Joannapoulos, �??Existence of a photonic band gap in two- dimensions ,�?? Appl. Phys. Lett. 61, 495-497 (1992) [CrossRef]
  4. P. R. Villeneuve, M. Piche, �??Phonic band gaps in two-dimensional square and hexagonal lattices,�?? Phys. Rev. B 46, 4969-4972 (1992). [CrossRef]
  5. P. R. Villeneuve, M. Piche, �??Photonic band gaps in two-dimensional square lattices: Square and circular lattices,�?? Phys. Rev. B 46, 4973-4975 (1992). [CrossRef]
  6. D. L. Bullock, C. Shih, R. S. Margulies, �??Photonic band structure investigation of two-dimensional Bragg reflector mirrors for semiconductor laser mode control,�?? J. Opt. Soc. Am. B 10, 399-403 (1993). [CrossRef]
  7. C. M. Anderson and K. P. Giapis, �??Larger two-dimensional photonic band gaps,�?? Phys. Rev. Lett. 77, 2949-2952 (1996). [CrossRef] [PubMed]
  8. J. C. Knight, T. A. Birks, P. St. J. Russell, D. M. Atkin, �??All-silica single-mode fiber with photonic crystal cladding,�?? Opt. Lett. 21, 1547-1549 (1996). [CrossRef] [PubMed]
  9. S. Y. Lin, G. Arjavalingam, W. M. Robertson, �??Investigation of absolute photonic band-gaps in 2-dimensional dielectric structures,�?? J. Mod. Opt. 41, 385-393 (1994) [CrossRef]
  10. Z. Y. Li, B. Y. Gu, G. Z. Yang, �??Large absolute band gaps in two-dimensional anisotropic photonic crystals,�?? Phys. Rev. Lett. 77, 2574-2977 (1998). [CrossRef]
  11. M. Qiu, S. He, �??Optimal design of two-dimensional photonic crystal of square lattice with large complete two-dimensional bandgap,�?? J. Opt. Soc. Am. A 17, 1027-1030 (2000). [CrossRef]
  12. M. Agio, L. C. Andreanm, �??Complete photonic band gap in a two-dimensional chessboard lattice,�?? Phys. Rev. B 61, 15519-15522 (2000). [CrossRef]
  13. M. Campbell, D. N. Sharp, M. T. Harrison, R. G. Denning, A. J. Turberfield, �??Fabrication of photonic crystals for the visible spectrum by holographic lithography,�?? Nature 404, 53-56 (2000). [CrossRef] [PubMed]
  14. L. Z. Cai, X. L. Yang, Y .R. Wang, �??Formation of three-dimensional periodic microstructures by interference of four noncoplanar beams,�?? J. Opt. Soc. Am. A 19, 2238-2244 (2002). [CrossRef]
  15. Y.A. Vlasov, X. Z. Bo, J. C. Sturm, D. J. Norris, �??On-chip natural assembly of silicon photonic bangap crystals,�?? Nature 414, 289-293 (2001). [CrossRef] [PubMed]
  16. X. L. Yang, L. Z. Cai, Q. Liu, �??Theoretical bandgap modeling of two-dimensional triangular photonic crystals formed by interference technique of three-noncoplanar beams,�?? Opt. Express 11, 1050-1055 (2003). [CrossRef] [PubMed]
  17. X. L. Yang, L. Z. Cai, Q. Liu, H. K. Liu, �??Theoretical bandgap modeling of two-dimensional square photonic crystals fabricated by interference technique of three-noncoplanar beams,�?? J. Opt. Soc. Am. B 21, 1050-1055 (2004). [CrossRef]
  18. M. Leung, Y. F. Liu, �??Photon band structures: The plane-wave method,�?? Phys. Rev. B 41, 10188-10190 (1990). [CrossRef]
  19. T. Kondo, S. Matsuo, S. Juodkazis, 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]

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