Manipulation of Light with Transformation Optics - DiVA Portal

113 Pages·2010·12.01 MB·English

Manipulation of Light with Transformation Optics
WEI YAN
Doctoral Thesis in Microelectronics and Applied Physics
Stockholm, Sweden 2010
KTH School of Information and
TRITA-ICT/MAP AVH Report 2010:09 Communication Technology
ISSN 1653-7610 SE-164 40 Kista
ISRN KTH/ICT-MAP/AVH-2010:09-SE SWEDEN
Akademisk avhandling som med tillstånd av Kungl Tekniska högskolan framlägges
till offentlig granskning för avläggande av teknologie doktorsexamen i datalogi
fredag den 22 November 2010 klockan 10.00 i sal C1, Electrum, Kungl Tekniska
Högskolan, Isafjordsgatan 26, Kista, Stockholm.
© Wei Yan, September 2010
Tryck: Kista Snabbtryck AB
iii
Abstract
Transformation optics, a recently booming area, provides people a new approach to
design optical devices for manipulating light. With transformation optics, a lot of novel
optical devices are proposed, such as invisibility cloaks, optical wormholes, optical black
holes, illusion devices. The present thesis is devoted to investigate transformation optics
for manipulating light.
Firstly, an introduction to transformation optics is given. This part includes: (1) in-
troducing differential geometry as the mathematical preparation; (2) expressing Maxwell’s
equations in an arbitrary coordinate system and introducing the concept of transforma-
tion media as the foundation stone of transformation optics; (3) discussing light from
the geometry perspective as the essence of transformation optics; (4) showing how to use
transformation optics to design optical devices.
For our works on invisibility cloaks, we analyze the properties of arbitrary shaped
invisibility cloaks, and confirm their invisibility abilities. The geometrical perturbations
to cylindrical and spherical shaped cloaks are analyzed in detail. We show that the
cylindrical cloak is more sensitive to the perturbation than a spherical cloak. By imposing
a PEC (PMC) layer at the interior boundary of the cylindrical cloak shell for TM (TE)
wave, the sensitivity can be reduced dramatically. A simplified non-magnetic cylindrical
th
cloak is also designed. We show that the dominant zero order scattering term can be
eliminated by employing an air gap between the cloak and the cloaked region.
We propose a compensated bilayer by a folding coordinate transformation based on
transformation optics. It is pointed out that complementary media, perfect negative index
lens and perfect bilayer lens made of indefinite media are well unified under the scope of
the transformed compensated bilayer. We demonstrate the applications of the compen-
sated bilayer, such as perfect imaging and optical illusion. Arbitrary shaped compensated
bilayers are also analyzed.
Nihility media known as the media with ϵ = µ = 0, are generalized from transforma-
tion optics as transformation media derived from volumeless geometrical elements. The
practical constructions of nihility media by metamaterials are discussed. The eigen fields
in the nihility media are derived. The interactions between an external incident wave and
a slab of nihility media in the free space background are analyzed.
A new type of transformation media called α media is proposed for manipulating light.
Light rays in the α media have a simple displacement or rotation relationship with those
in another media (seed media). Such relationship is named α relationship. The α media
can be designed and simplified to a certain class of diagonal anisotropic media, which are
related to certain isotropic media by the α relationship. Several optical devices based on
the α transformation media are designed. Invisibility cloaks obtained from the coordinate
transformation approach are revisited from a different perspective.
Keywords: transformation optics, Maxwell’s equations, invisibility cloaks, compensated
bilayer, perfect imaging, optical illusion, nihility media, α media
Contents
Contents v
List of Publications ix
List of Symbols and Acronyms xi
Acknowledgements xiii
1 Introduction 1
1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Outline of This Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . 2
2 Basic Introduction to Differential Geometry 3
2.1 Coordinate System and Metric Tensor . . . . . . . . . . . . . . . . . 3
2.2 Covariant and Contravariant Vectors . . . . . . . . . . . . . . . . . . 5
2.3 Coordinate Transformations . . . . . . . . . . . . . . . . . . . . . . . 5
2.4 Covariant Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.5 Properties of Christoffel Symbols and Metric Tensors . . . . . . . . . 8
2.6 Geodesic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.7 Parallel Transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.8 Riemannian Curvature Tensor . . . . . . . . . . . . . . . . . . . . . . 10
3 Maxwell’s Equations in an Arbitrary Coordinate System 13
3.1 Maxwell’s Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3.2 Maxwell’s Equations in an Arbitrary Space Coordinate System . . . 15
3.3 Transformation Media . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.4 Maxwell’s Equations in Tensor Form . . . . . . . . . . . . . . . . . . 19
3.5 Maxwell’s Equations in an Arbitrary Space-time Coordinate System 21
4 Geometry of Light 23
4.1 Geodesic of Light . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
4.2 Reduce Geodesic in Space-time to Geodesic in Space . . . . . . . . . 24
4.3 Geodesic from Maxwell’s Equations . . . . . . . . . . . . . . . . . . . 25
v
vi CONTENTS
5 Controlling Light by Transformation Optics 27
5.1 Design Transformation Media by Coordinate Transformation in a
Flat Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
5.2 Design Transformation Media Based on a Curved Space-Time with
Special Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
5.3 Recent Progresses in Transformation Optics . . . . . . . . . . . . . . 31
6 Results 1: Invisibility Cloaks 33
6.1 Arbitrary Shaped Invisibility Cloaks . . . . . . . . . . . . . . . . . . 33
6.1.1 Outer Boundary . . . . . . . . . . . . . . . . . . . . . . . . . 34
6.1.2 Inner Boundary . . . . . . . . . . . . . . . . . . . . . . . . . . 35
6.2 Geometrical Perturbation Effects to Cylindrical and Spherical Shaped
Invisibility Cloaks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
6.2.1 Cylindrical Cloak . . . . . . . . . . . . . . . . . . . . . . . . . 38
6.2.2 Spherical Cloak . . . . . . . . . . . . . . . . . . . . . . . . . . 41
6.3 Non-magnetic Simplified Cylindrical Cloak . . . . . . . . . . . . . . . 42
7 Results 2: Compensated Bilayer by a Folding Transformation 45
7.1 Compensated Bilayer Structure . . . . . . . . . . . . . . . . . . . . . 45
7.1.1 Geometry of Compensated Bilayer . . . . . . . . . . . . . . . 45
7.1.2 Material Parameters of Compensated Bilayer . . . . . . . . . 46
7.2 Relationship of Fields at Bilayer Boundaries . . . . . . . . . . . . . . 47
7.3 Properties and Applications . . . . . . . . . . . . . . . . . . . . . . . 47
7.4 Explain Some Phenomena Based on Compensated Bilayer . . . . . . 49
7.5 Numerical Analysis for Compensated Bilayer . . . . . . . . . . . . . 50
7.5.1 Slab-shaped Compensated Bilayer of Type A . . . . . . . . . 50
7.5.2 Cylindrical-Shaped Compensated Bilayer of Type A . . . . . 50
7.5.3 Compensated Bilayer of Type B . . . . . . . . . . . . . . . . 53
8 Result 3: Generalized Nihility Media from Transformation Optics 55
8.1 Nihility Media and Zero-Volume Geometrical Elements . . . . . . . . 55
8.2 Eigen Fields in Nihility Media . . . . . . . . . . . . . . . . . . . . . . 57
8.3 Interaction between Nihility Media and External Fields . . . . . . . 59
9 Result 4: Manipulation of Light with α Transformation Media 65
9.1 Transformation Relation and Transformation Media . . . . . . . . . 65
9.2 Light Rays in Seed Media and α Media . . . . . . . . . . . . . . . . 66
9.3 Simplified α Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
9.4 Optical Devices Based on the α Media . . . . . . . . . . . . . . . . . 69
9.5 Revisit Invisibility Cloaks Obtained from Coordinate Transformations 72
10 Conclusion and Future Work 75
11 Summary of Contributions 77
vii
A Transformation Media Expressed in an Arbitrary Space Coor-
dinate System 79
B Newtonian Mechanics Analogy of Light 83
C Basic Introduction to Metamaterials 85
C.1 From Microscopic to Macroscopic Maxwell’s Equations . . . . . . . . 85
C.2 Macroscopic Material Parameters of Media . . . . . . . . . . . . . . 87
C.3 Metamaterials: A Natural Extension from Media in Nature . . . . . 89
Bibliography 93
List of Publications
List of papers included in this thesis:
I. W. Yan, M. Yan, Z. C. Ruan and M. Qiu. Coordinate transformation makes
perfect invisibility cloak with arbitrary shape. New J. Phys., 10:043040, 2008.
II. W. Yan, M. Yan, Z. C. Ruan and M. Qiu. Influence of geometrical perturba-
tion at inner boundaries of invisibility cloaks. J. Opt. Soc. Am. A, 25:968,
2008.
III. W. Yan, M. Yan and M. Qiu. Non-magnetic simplified cylindrical cloak with
suppressed zeroth order scattering. Appl. Phys. Lett., 93:021909, 2008.
IV. W. Yan, M. Yan and M. Qiu. Achieving perfect imaging beyond passive and
active obstacles by a transformed bilayer lens. Phys. Rev. B, 79:161101,
2009.
V. W. Yan, M. Yan and M. Qiu. Generalized compensated bilayer structure
from transformation optics perspective. J. Opt. Soc. Am. B, 26:B39, 2009.
VI. W. Yan, M. Yan and M. Qiu. Generalized nihility media from transformation
optics. accepted by J. Opt., 2010.
VII. W. Yan, M. Yan and M. Qiu. Manipulation of light with α transformation
media. submitted for publication.
List of papers not included in this thesis
VII. M. Yan, W. Yan and M. Qiu. Cylindrical superlens by a coordinate transfor-
mation. Phys. Rev. B, 78:125113, 2008.
VIII. J. Tian, W. Yan, Y. Z. Liu, J. Luo, D. Z. Zhang, Z. Y. Li and M. Qiu. Optical
quality improvement of Si photonic devices fabricated by focused-ion-beam
milling. J. Lightwave Technol., 27:4306, 2009.
ix
x CONTENTS
IX. J. Tian, S. Q. Yu, W. Yan and M. Qiu. Broadband high-efficiency surface-
plasmon-polariton coupler with silicon-metal interface. Appl. Phys. Lett.,
95:013504, 2009.
X. J. Wang, Y. Song, W. Yan and M. Qiu. High-Q photonic crystal surface-mode
cavities on crystalline SOI structures. Opt. Comm., 283:2461, 2010.
XI. J. M. Hao, W. Yan and M. Qiu. Super-reflection and cloaking based on zero
index metamaterial. Appl. Phys. Lett., 96:101109, 2010.
Partial list of refereed conference proceedings:
XV. W. Yan, M. Yan, Z. C. Ruan and M. Qiu. Ideal cylindrical cloak and influence
of tiny perturbation. Quantum Electronics and Laser Science Conference
(QELS), San Jose, CA, USA, 4-9, May 2008.
XVI. W. Yan, M. Yan, Z. C. Ruan and M. Qiu. Perfect invisibility cloaks con-
structed by arbitrary coordinate transformations. NATO Advanced Re-
search Workshop on Metamaterials for Secure Information and Communi-
cation Technologies, Marrakesh, Morocco, 7-10 May 2008.
XVII. W. Yan, M. Qiu and L. F. Shen. High directive antenna based on meta-
material slab with zero permittivity. SPIE international conference on Asia-
Pacific Optical and Wireless Communications (APOC 2008), paper 7135-103,
Hangzhou, October 2008.