Expand this Topic clickable element to expand a topic
Skip to content
Optica Publishing Group
Journal Home About Issues in Progress Current Issue All Issues Feature Issues

Holographic design and band gap evolution of photonic crystals formed with five-beam symmetric umbrella configuration

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

Open Access Open Access

Abstract

We propose a holographic design of five-beam symmetric umbrella configuration, where there are a central beam and four ambient beams symmetrically scattered around the central one with the same apex angle, for fabrication of three-dimensional photonic crystals with tetragonal or cubic symmetries, and systematically analyzed the band gap properties of resultant photonic crystals when the apex angle is continuously increased. Our calculations reveal that large complete photonic band gaps exist in a wide range of apex angle for a relatively low refractive index contrast. Specifically, the face-centered cubic structure with a relative band gap of 25.1% for ε = 11.9 can be obtained with this recording geometry conveniently where all the beams are incident from the same half-space. These results will provide us with more understanding of this important recording geometry and give guidelines to its use in experiments.

©2006 Optical Society of America

Full Article  |  PDF Article
More Like This
Design of holographic structures using genetic algorithms

James W. Rinne and Pierre Wiltzius
Opt. Express 14(21) 9909-9916 (2006)

Characteristics of photonic band gaps in woodpile three-dimensional terahertz photonic crystals

Huan Liu, Jianquan Yao, Degang Xu, and Peng Wang
Opt. Express 15(2) 695-703 (2007)

Distortion of 3D SU8 Photonic Structures Fabricated by Four-beam Holographic Lithography with Umbrella Configuration

Xuelian Zhu, Yongan Xu, and Shu Yang
Opt. Express 15(25) 16546-16560 (2007)

View More...

Supplementary Material (1)

Media 1: GIF (155 KB)     

Cited By

Optica participates in Crossref's Cited-By Linking service. Citing articles from Optica Publishing Group journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (6)
Fig. 1.
Fig. 1. Symmetric umbrella recording geometry and the coordinate system used for calculations.

Download Full Size | PDF

Fig. 2.
Fig. 2. Evolution of irreducible Brillouin zone (Movie, 155K). The irreducible Brillouin zones of tetragonal structures for θ = 80° (a), θ = 90° (b), and θ = 100° (c).

Download Full Size | PDF

Fig. 3.
Fig. 3. (a). The real fcc structure formed by five-beam symmetric umbrella configuration when θ = 70.53° and I t = 1.39 in Eq. (4), the corresponding filling ration is 21.7%; (b) the primitive cell of the fcc structure shown in (a); (c) the rhombohedral structure with fcc symmetry fabricated by four-beam symmetric umbrella configuration when the apex angle is 38.94°; and (d) the primitive cell of the structure shown in (c).

Download Full Size | PDF

Fig. 4.
Fig. 4. Relative band gap of optimized structures as a function of apex angle for 50°< θ < 115° when ε = 11.9. The solid symbols are the date for fcc and bcc structures when θ = 70.53° and θ = 90°, respectively. When the value of θ is near 70.53° and 90° the fct and bct symmetric structures can be obtained.

Download Full Size | PDF

Fig. 5.
Fig. 5. Photonic band structure for the fcc structure with θ = 70.53°. The position of the high symmetry points together with the irreducible Brillouin zone are shown in the inset.

Download Full Size | PDF

Fig. 6.
Fig. 6. Variation of relative band gaps with different filling ratios for different refractive index contrasts for the fcc symmetric structure with the apex angle θ = 70.53°.

Download Full Size | PDF

Equations (12)

Equations on this page are rendered with MathJax. Learn more.

K 1 = ( 2 π λ ) ( − sin θ , 0 , cos θ ) , K 2 = ( 2 π λ ) ( 0 , sin θ , cos θ ) ,
K 3 = ( 2 π λ ) ( sin θ , 0 , cos θ ) , K 4 = ( 2 π λ ) ( 0 , − sin θ , cos θ ) ,
K c = ( 2 π λ ) ( 0 , 0 , 1 ) .
e 1 = e 3 = ( 0 , 1 , 0 ) , e 2 = e 4 = ( 1 , 0 , 0 ) , e c = 1 2 ( 1 , − i , 0 ) .
I ( r ) = 2 E C 2 + 4 E A 2 + 2 E C E A { sin [ 2 π λ ( x sin θ + ( 1 − cos θ ) z ) ]
+ cos [ 2 π λ ( − y sin θ + ( 1 − cos θ ) z ) ]
+ sin [ 2 π λ ( − x sin θ + ( 1 − cos θ ) z ) ]
+ cos [ 2 π λ ( y sin θ + ( 1 − cos θ ) z ) ] } .
Δ I ( r ) = sin [ 2 π λ ( x sin θ + ( 1 − cos θ ) z ) ]
+ cos [ 2 π λ ( − y sin θ + ( 1 − cos θ ) z ) ]
+ sin [ 2 π λ ( − x sin θ + ( 1 − cos θ ) z ) ]
+ cos [ 2 π λ ( y sin θ + ( 1 − cos θ ) z ) ] .
Select as filters
Select Topics Cancel
Top
       
© Copyright 2024 | Optica Publishing Group. All rights reserved, including rights for text and data mining and training of artificial technologies or similar technologies.
Privacy | Terms of Use