Nonlinear Plasmonics

Nonlinear Plasmonics: Theoretical Models and Physical Manipulation

Nonlinear Plasmonics: Theoretical Models and Physical Manipulation Wei E.I. Sha (沙威) College of Information Science & Electronic Engineering, Zhejiang University Department of Electronic & Electrical Engineering, University College London On leave from EEE Department, the University of Hong Kong Email: [email protected] Website: http://www.isee.zju.edu.cn/weisha/ Page 1

Wei E. I. Sha


OUTLINE    

Background for Nonlinear Plasmonics Boundary Element Method Maxwell-Hydrodynamic Model Theoretical Results  Charge Conservation  Energy Conservation  Angular Momentum Conservation  Parity Conservation  Terahertz Generation  Future Perspectives  Conclusion

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NONLINEAR EFFECTS: A QUICK REVIEW P = ε 0  χ (1) ⋅ E + χ ( 2 ) : EE + χ (3) : EEE +  nonlinear responses SHG, SFG, DFG, THG, FWM, Kerr effect … Linear and nonlinear susceptibility:

second-harmonic generation

four-wave mixing

anti-Stokes Raman scattering

Martti Kauranen and Anatoly V. Zayats, Nature Photonics 6: 737–748, 2012. Page 3

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WHY NONLINEAR PLASMONICS? • • • •

Support plasmonic resonance Strong near field enhancement Sensitive to electromagnetic environment Amplify nonlinear processes 𝜔

2𝜔 Control of amplitude, phase, polarization, and wavefront of nonlinear electromagnetic waves by metamaterials and plasmonics is an emerging research direction, which is full of challenges in mathematical modeling and physical designs.

N. I. Zheludev, The Road Ahead for Metamaterials, Science, 328: 582–583, 2010. Page 4

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CHALLENGES  We need to control electromagnetic wave not only in fundamental frequency but also in high harmonic frequencies.  Some fundamental knowledge in linear plasmonics are not valid in nonlinear plasmonics; Some key prerequisites (phase matching) in traditional nonlinear optics are not so important in nonlinear plasmonics.  It is not easy to find commercial software that is capable of modeling nonlinear responses from centro-symmetric metals and graphene.

k radiation pattern of a metallic nanoparticle

E-field distribution of a graphene sheet Page 5

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WHY SHG IS SPECIAL? Second-harmonic generation (SHG) Parametric nonlinear process • charge conservation • energy conservation • angular momentum conservation • parity conservation

Pi =  χ E j Ek (2) ijk

j ,k

Surface SHG in plasmonic nanostructures • • • •

sensitive to particle shape sensitive to environment sensitive to incident beam property strong nonlinear polarization

r → −r

E → −E

P → −P

− P = χ ( 2 ) : ( −E ) 2 = χ ( 2 ) : E 2 = P Page 6

P=0 Wei E. I. Sha

Selection Rules

Centrosymmetric material

Pbulk ≈ 0

Centrosymmetry broken at surfaces

Psurface ≠ 0


APPLICATIONS single nanoparticle detection

ultrasensitive shape characterization

nonlinear plasmonic sensing

𝜔 2𝜔

2𝜔 Butet, Nano Lett.(2012)

Martin, Nano Lett. (2013)

Mesch, Nano Lett. (2016)

𝜔 2𝜔 Butet, ACS Nano (2014)

Zhang, Nano Lett. (2011)

all optical signal processing

OAM wavefront generation

Yu, Science (2012)

new frequency generation 𝟐𝝎

All optical chip, IBM (2011)

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Capasso, Science (2011)

Wei E. I. Sha

Bloch, Phys. Rev. Lett.(2012)


BOUNDARY ELEMENT METHOD Fundamental field

Second harmonic field

nonlinear source for downconversion process, which can be ignored for undepleted pump approximation nonlinear source for upconversion process

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1. Solve fundamental field

boundary conditions

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2. Obtain surface nonlinear sources by surface nonlinear susceptibility

3. Solve second-harmonic field

boundary conditions

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4. Update fundamental field with modified boundary conditions

boundary conditions

self-consistent process (energy conservation law) second-harmonic field

fundamental field

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Nonlinear Electromagnetics in Plasmonic System

HYDRODYNAMIC MODEL When electromagnetic wave strongly interacts with metallic structures, it can couple to free electrons near the metal surface resulting in complex linear and nonlinear responses. The complex motion of electrons within metallic structures resembles motion of fluids.

Maxwell Hydrodynamic

Continuity

The equation is solved by FDTD method with Yee grids. Page 12

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CHARGE CONSERVATION

At each point, the electron density fluctuates. However, the total charge within the sphere is conserved.

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ENERGY CONSERVATION/CONVERSION second-harmonics

third-harmonics

second-harmonics

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third-harmonics

𝐸

2ω ~|𝐸 ω |

𝐸

3ω ~|𝐸 ω |

Wei E. I. Sha


ANGULAR MOMENTUM CONSERVATION (1) •

For a nanostructure with N-fold rotational symmetry The relation between the spin and orbital angular momenta of the incident field and that of the scattered second-harmonic wave is given by 𝑗

𝑗

𝑠 𝑙

𝜎

𝑞𝑁

analogy translational momentum conservation

k 0 sin θ s =sk 0 sin θ i +m

2π

Λ

: total angular momentum of the scattered SH wave

𝜎: spin angular momenta of the incident field (𝜎

1 or 𝜎

1)

𝑙: orbital angular momentum of the incident field 𝑠: order of harmonic generation (s = 1 for linear processes, s = 2 for SH generation) 𝑁: quasi-angular-momentum of the nanostructure with 𝑁-fold rotational symmetry 𝑞: an integer and 𝑞 Page 15

0, 1, 2, … Wei E. I. Sha


ANGULAR MOMENTUM CONSERVATION (2) experimental [1]

numerical N=3

𝑗

2σ

𝑛𝑁

N=1

N→∞

[1] K. Konishi, T. Higuchi, J. Li, J. Larsson, S. Ishii, and M. K.-Gonokami, Phy. Rev. Lett. 112: 135502, 2014. Page 16

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Near-field distribution of a metallic sphere • Radius of the gold sphere: 100 nm • Wavelength of the incident beam: 520 nm • Near-field observation plane: 40 nm away from the back side of the sphere • Laguerre-Gaussian (LG) beam with OAM 𝑙 1, SAM 𝜎 1/ 1.

Fundamental

Second-Harmonic (SH)

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Scattering cross section (SCS) analysis of a silicon sphere • Radius of the silicon sphere: 500 nm • Laguerre-Gaussian (LG) beam with OAM 𝑙 Fundamental SCS, linearly polarized plane wave

Fundamental SCS, Laguerre-Gaussian beam

4, SAM 𝜎

1/ 1.

SH SCS, Laguerre-Gaussian beam

Two SH enhancement mechanisms

Circles: fundamental mode enhancement; Diamonds: SH induced fundamental mode enhancement Page 18

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ANGULAR MOMENTUM CONSERVATION (3) • Chiral cluster with 3-fold rotational symmetry containing 10 identical silicon nanospheres (quasi-angular-momentum 𝑁 3) • Radius of the silicon sphere/cross section: 500 nm • Laguerre-Gaussian (LG) beam with OAM 𝑙 4, SAM 𝜎 1/ 1. • Total angular momentum 𝑗 𝑙 𝜎

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ANGULAR MOMENTUM CONSERVATION (4) 𝜎 FH: 𝑗 𝑗

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𝑙

𝜎

1, 𝑙

4, 𝑁

𝑞𝑁

3

SH: 𝑗

⋯ , 7, 4, 1, 2, 5, 8, …

Wei E. I. Sha

𝑗

2 𝑙

𝜎

𝑞𝑁

⋯ , 8, 5, 2, 1, 4,7,10, … ,


ANGULAR MOMENTUM CONSERVATION (5) Discretization of Hydrodynamic Equation

Spatial interpolation is adopted to force all the physical quantities to be the same locations. The interpolation scheme is crucial to maintain the angular momentum conservation of second-harmonic generation. Page 21

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PARITY CONSERVATION (1) Compact Nonlinear Yagi-Uda Nanoantenna radiation pattern of a metallic nanosphere second-harmonic

fundamental

PNL

symmetry breaking

PNL

phase retardation net

cancelation

PNL

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PNL

net PNL


Designs of Dielectric Reflector and Director Backward Directivity

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Forward Directivity

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Physical Origin of High Directivity (1) multipolar expansion of equivalent electric current for Re(Eθ) at SH frequency

(a, b): feed-director system

(c, d): feed-reflector system

E-dipolar moments of dielectric elements have significant phase shift, induced by metal feed! Page 24

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Physical Origin of High Directivity (2) multipolar expansion of equivalent magnetic current for Re(Eθ) at SH frequency

(a, b): feed-director system

(c, d): feed-reflector system

E-dipolar and M-quadrupolar moments of dielectric elements have phase flip, induced by metal feed! Page 25

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PARITY CONSERVATION (2) Particle-in-cavity (PIC) nanoantenna

symmetry breaking

[1] [2] [1] K. Thyagarajan, J. Butet, O. J. F. Martin, Nano Lett. 13: 1847–1851, 2013 [2] K. Thyagarajan, S. Rivier, A. Lovera, and O. J. F. Martin, Opt. Express 20, 12860–12865, 2012.

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Physical Origin of High Directivity plasmonic coupling and symmetry breaking produce m-dipolar component and more e-dipolar component !

multipole expansion results for sphere, symmetric PIC antenna, and asymmetric PIC antenna Page 27

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twist angle dependent secondharmonic intensity


TERAHERTZ GENERATION (1) Efficient, broadband and tunable THz emission can be generated in a thin layer of metamaterial with tens of nanometers thickness by an infrared laser pumping [1]. Absorption Gaps

[1] L. Luo, I. Chatzakis, J. Wang, F. B. P. Niesler, M. Wegener, etal., Nature Communications, 5: 3055, 2014. Page 28

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TERAHERTZ GENERATION (2) Unit cell Nonlinear hydrodynamic model results subtracted by Drude model ones

linear process

Linear

Rectification SHG,THG … Fourier transform

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Fourier transform SHG

Terahertz


TERAHERTZ GENERATION (3)

tunable terahertz spectra by duration time or bandwidth of incident laser pulse Page 30

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TERAHERTZ GENERATION (4) vertical

horizontal

polarization-dependent terahertz generation Page 31

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FUTURE PERSPECTIVES Maxwell’s Equations Local

Nonlocal

Hydrodynamic Model

Quantum Corrections

nonlinear response from metallic particles with size < 10 nm Page 32

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FUTURE PERSPECTIVES

Vector Potential Equations

nonlinear response from 2D materials

Time-Dependent Density Functional Theory Page 33

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1.

X.Y.Z. Xiong, L.J. Jiang, W.E.I. Sha, Y.H. Lo, and W.C. Chew, “Compact Nonlinear Yagi-Uda Nanoantennas,” Scientific Reports, vol. 6, pp. 18872, 2016.

2.

M. Fang, Z.X. Huang, W.E.I. Sha, X.Y.Z. Xiong, and X.L. Wu, “Full Hydrodynamic Model of Nonlinear Electromagnetic Response in Metallic Metamaterials,” Progress In Electromagnetics Research, vol. 157, 6378, 2016.

3.

X.Y.Z. Xiong, L.J. Jiang, W.E.I. Sha, Y.H. Lo, M. Fang, W.C. Chew, and W.C.H. Choy, “Strongly Enhanced and Directionally Tunable Second-Harmonic Radiation by a Plasmonic Particle-in-Cavity Nanoantenna,” Physical Review A, vol. 94, no. 5, pp. 053825, 2016.

4.

X.Y.Z. Xiong, A. Al-Jarro, L.J. Jiang, N.C. Panoiu, and W.E.I. Sha, “Mixing of Spin and Orbital Angular Momenta via Second-Harmonic Generation in Plasmonic and Dielectric Chiral Nanostructures,” Physical Review B, vol. 95, no. 16, pp. 165432, 2017.

5.

M. Fang, Z.X. Huang, W.E.I. Sha, and X.L. Wu, “Maxwell-Hydrodynamic Model for Simulating Nonlinear Terahertz Generation from Plasmonic Metasurfaces,” IEEE Journal on Multiscale and Multiphysics Computational Techniques, vol. 2, pp. 194-201, 2017.

6.

M. Fang, K.K. Niu, Z.X. Huang, W.E.I. Sha, X.L. Wu, T. Koschny, and C.M. Soukoulis, “Investigation of Broadband Terahertz Generation from Metasurface,” Optics Express, vol. 26, no. 11, pp. 14241-14250, 2018.

7.

N.C. Panoiu, W.E.I. Sha, D.Y. Lei, and G.C. Li, “Nonlinear Optics in Plasmonic Nanostructures,” Journal of Optics, vol. 20, no. 8, pp. 083001, 2018. (Invited Topic Review)

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CONCLUSION

Energy Conservation

Parity Conservation

Angular Momentum Conservation

Nonlinear Plasmonics Page 35

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COLLABORATORS (1)

Li Jun Jiang (HKU)

Xiaoyan Y.Z. Xiong (HKU) Page 36

Zhixiang Huang (AHU)

Nicolae Panoiu (UCL)

Ming Fang (AHU)

Ahmed Al-Jarro (UCL)

Wei E. I. Sha

COLLABORATORS (2)

Costas M. Soukoulis (ISU)

Weng Cho Chew (Purdue)

Thomas Koschny (ISU/AMES Lab) Page 37

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ACKNOWLEDGEMENT Hangzhou

Zhejiang University

Thanks for Your Attention! Page 38

Wei E. I. Sha