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Introductory Chapter: Nonlinear Optical Phenomena in Plasmonics, Nanophotonics and Metamaterials

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Boris I. Lembrikov

Submitted: December 26th, 2021 Reviewed: January 5th, 2022 Published: March 30th, 2022

DOI: 10.5772/intechopen.102498

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1. Introduction

The observation of nonlinear optical effects started with the discovery of the laser in 1960 as a source of the high-intensity coherent radiation [1, 2]. Nonlinear optics is the study of the nonlinear light and matter interaction effects such as the light self-focusing and self-trapping, second harmonic generation (SHG), third-harmonic generation (THG), four-wave mixing (FWM), parametric processes, different types of stimulated light scattering (SLS), soliton generation and propagation [1, 2]. The nonlinear optical phenomena are widely used in modern communication systems for such applications as the generation of ultra-short pulses, all-optical signal processing and ultrafast switching [2, 3]. Recently, new fields of nonlinear optics emerged such as strong-field nano-optics, nonlinear plasmonics, and nonlinear metamaterials [3, 4, 5, 6, 7, 8].

In the case of the strong-field nano-optics the optical field interacts with matter at the wavelength or even at the subwavelength scale [5]. Strong electromagnetic fields of a light wave with the wavelength λ1μminteracting with electrons are confined to a few nanometers which result in a substantial enhancement of the local electric field [4]. Typically, an increase of the nonlinear optical response at a nanoscale is caused by the plasmonic effects, i.e., the coherent oscillations of the conduction electrons near the surface of noble-metal structures [3, 4, 5, 6]. At the extended metal surfaces, the surface plasmon polaritons (SPPs) can occur [3]. SPP is a surface electromagnetic wave propagating at the metal-dielectric interface [3, 4, 5]. In the case of the metal nanoparticles there exist localized surface plasmons (LSPs) with resonances depending on the nanoparticle size and shape [3]. Nonlinear optical effects can be significantly enhanced by plasmonic excitations in two ways: (i) the plasmonic structures provide the optical field enhancement near the metal-dielectric interface due to SPPs or LSPs; (ii) the SPP and LSP parameters are very sensitive to refractive indices of the metal and the surrounding dielectric medium [3, 4, 5, 6]. Plasmonic excitations are characterized by the timescale of several femtoseconds which makes it possible the ultrafast optical signal processing [3].

Metamaterials are artificial materials with desirable properties [7]. For this reason, the magnitude of the metamaterial optical nonlinearity can be substantially increased [7]. Generally, all fundamental nonlinear optical phenomena such as self-action effects, wavelength conversion, nonlinear surface waves, nonlinear guided waves and solitons are possible in nonlinear metamaterials [7]. The nonlinear metamaterials are closely related to plasmonics, active media, and nonlinearity based on liquid crystals (LCs) [7, 8]. LCs possess simultaneously the properties of a liquid and a crystal [8, 9, 10]. Thermotropic LCs self-assemble in a different ordered arrangements of their crystalline axis depending on the temperature [8, 9, 10]. The orientational order of LC may be changed by a moderate external electric field [8, 9, 10]. For this reason, liquid crystalline media are characterized by strong optical nonlinearity [8, 9, 10]. Nematic LCs (NLCs) characterized by the ordering of molecular elongated axes are mainly used in applications [8].

In this chapter, we discuss the fundamentals of plasmonics and the specific features of the nonlinear optical effects in plasmonic nanostructures. The chapter is organized as follows. In Section 2 the interaction of the electromagnetic field with the free electrons in metals is described based on Maxwell’s equations and equation of motion for a free electron in the external electric field. The dielectric function of the free electron gas is obtained based on the Drude model. In Section 3 SPPs at the interface of a metal and a dielectric are investigated. In Section 4 the nonlinear phenomena in the plasmonic structures are briefly discussed. The details of this topic may be found in the corresponding references. Conclusions are presented in Section 5.

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2. Interaction between electromagnetic field and free electrons in metals

The interaction of electromagnetic fields with metals is described by Maxwell’s equations of macroscopic electromagnetism [11]. This approach is valid also for metallic nanostructures characterized by the sizes of several nanometers [11]. The Maxwell’s equations have the form [1, 2, 11].

D=ρextE1
B=0E2
×E=BtE3
×H=Jext+DtE4

Here D,E,Hand Bare the dielectric displacement, the electric field, the magnetic field and the magnetic induction (or magnetic flux density), respectively; ρextand Jextare external charge and current densities, respectively. The four macroscopic fields are related by the following material Equations [11].

D=ε0εEE5
B=μ0μHE6

where ε0and μ0are the electric permittivity and magnetic permeability of vacuum, respectively, εis the relative permittivity and μis the relative permeability of the medium, μ=1for the nonmagnetic medium. The total charge and current densities ρtotand Jtotconsist of the external charge and current densities ρext, Jextand the internal ones ρ,J. They are given by [11]: ρtot=ρext+ρ,Jtot=Jext+J. The internal current density has the form [11].

J=σEE7

where σis the conductivity of the medium. The optical response of metals typically depends on frequency and wave vector in the case of a spatial dispersion [11]. In such a case, Eqs. (5) and (7) can be generalized to the following relationships taking into account the non-locality in time and space [11].

Drt=ε0dtdrεrrttErtE8
Jrt=ε0dtdrσrrttErtE9

The Fourier transform of expressions (8) and (9) results in the following relationships in the Fourier domain [11].

Dkω=ε0εkωEkωE10
Jkω=σkωEkωE11

where kand ωare the wave vector and angular frequency of the field plane-wave components, respectively.

Combining Eqs. (3) and (4) we obtain the wave equation for the electric field Ert[1, 2, 11].

××E=μ02Dt2E12

which takes the form in the frequency domain [11].

kkEk2E=εkωω2c2EE13

where c=ε0μ01is the speed of light in vacuum. There are two types of solutions of Eq. (13): the transverse waves where the electric field vector is perpendicular to the wave propagation direction, and the longitudinal waves where the electric field vector is parallel to the wave propagation direction. The transverse waves are characterized by the condition kE=0and the corresponding dispersion relation

k2=εkωω2c2E14

The dispersion relation for the longitudinal waves can be obtained from Eq. (13).

εkω=0E15

It is seen from Eq. (15) that the longitudinal collective oscillations can exist only for the frequencies ωwhich correspond to the zeros of the dielectric function εkω[11].

Consider a plasma model that explains the optical properties of alkali metals and noble metals for the frequencies up to the ultraviolet ones and to the visible ones, respectively [11, 12]. In the framework of the phenomenological plasma model the metal crystal lattice potential and electron–electron interactions are not taken into account [11]. It is assumed that the details of the metal energy band structure are included into the effective optical mass mof each electron [11]. In such a model, the equation of motion for an electron with the charge eand the effective mass mof the plasma sea in the external electric field Ehas the form [11].

md2xdt2+dxdt=eEE16

where xis the free electron displacement, γ=τ1is a characteristic collision frequency, τis the free electron gas relaxation time. Typically, τ1014sat room temperature which yields γ100THz[11]. For the external field Et=E0expiωtwe obtain from Eq. (16) the following particular solution.

xt=x0expiωt=emω2+iγωEtE17

The macroscopic polarization Pof the medium caused by the displaced electrons is given by [11].

P=nexE18

where nis the free electrons concentration. Substituting expression (17) into Eq. (18) we obtain.

P=ne2mω2+iγωEE19

Taking into account that the dielectric displacement Din a medium is given by [1, 2, 11].

D=ε0E+PE20

and substituting expression (19) into Eq. (20) we obtain.

D=ε01ωp2ω2+iγωEE21

where ωp=ne2/ε0mis the plasma frequency of the free electron gas [4, 11]. Comparing relationships (5), (10) and (21) we obtain the expression of the free electron gas dielectric function εω[4, 11].

εω=1ωp2ω2+iγωE22

The dielectric function εω(22) is known as the Drude model of the optical response of metals [11]. Expression (22) can be divided into real and imaginary components as follows.

Reεω=1ωp2τ21+ω2τ2;Imεω=ωp2τω1+ω2τ2E23

Typically, the frequencies ω<ωpare considered where the metals have the pronounced metallic properties since in such a case Reεω<0[11]. In the high frequencies limiting case ωτ1the imaginary part of the dielectric function (23) can be neglected while expression (22) takes the form [11].

εω1ωp2ω2E24

Comparison of expressions (15) and (24) shows that the longitudinal waves can be excited at the plasma frequency ω=ωp. These longitudinal oscillations are called the volume plasma oscillations, and the quasi-particles of these oscillations are called the volume plasmons [11]. The volume plasmons cannot interact with the transverse electromagnetic waves and can be excited by particle impact [11].

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3. Surface Plasmon Polaritons (SPPs)

Consider now the SPPs at the interface between the metal z<0and the dielectric z>0shown in Figure 1. The metal and the dielectric are characterized by the dielectric function ε1ωdefined by expression (22) and the real dielectric constant ε2>0, respectively. We consider the frequency range ω<ωpsuch that Reε1ω<0. Wave Eq. (12) in such a case takes the form [11, 13].

Figure 1.

Interface between a metal (z<0) and a dielectric (z>0).

2Eεc22Et2=0E25

where it is assumed that the permittivity εin the both media is constant over distances of the order of magnitude of an optical wavelength and D=0since the free external charges are absent [11].

The solution of the wave Eq. (25) in such a case is sought to be {11, 13].

Exzt=EzexpiβxωtE26

Here βis the propagation constant which is complex in the general cases. It can be shown that there exist two types of solution of the Maxwell’s Eqs. (1)-(4) with different polarizations of the propagating waves [11, 13]. The first solution is the transverse magnetic (TM) or p modes with the field components Ex,Ez,Hy; the second solution is the transverse electric (TE) or s modes with the field components Hx,Hz,Ey[11, 13]. We start with the TM modes. The TM electric and magnetic fields in the regions z<0and z>0have the form, respectively [11, 13].

z<0H1y=A1expk1zexpiβxωtE27
E1x=iA1k1ωε0ε1ωexpk1zexpiβxωtE28
E1z=A1βωε0ε1ωexpk1zexpiβxωtE29

and

z>0H2y=A2expk2zexpiβxωtE30
E2x=iA2k1ωε0ε2expk2zexpiβxωtE31
E2z=A2βωε0ε2expk2zexpiβxωtE32

where k1,2are the components of the wave vector in the Z-direction perpendicular to the interface. The value of 1/k1,2is the evanescent decay length of the fields (27)-(32) in the direction perpendicular to the interface. It defines the confinement of the TM waves. The boundary conditions of the continuity of the tangential field components Hyand Exat the interface z=0have the form.

H1yz=0=H2yz=0+;E1xz=0=E2xz=0+E33

Substituting expressions (27), (28), (30) and (31) into Eqs. (33) we obtain [11].

A1=A2;k2k1=ε2ε1ωE34

On the other hand, substituting expressions (28), (29) and (31), (32) into wave Eq. (25) for the medium 1 and medium 2, respectively, we obtain the following dispersion relations [11, 13].

k12=β2k02ε1ωE35
k22=β2k02ε2E36

Here k0=ω/cis the absolute value of the wave vector of the propagating wave in vacuum. Substituting the second expression (34) into Eqs. (35), (36) we obtain the SPP dispersion relation [11, 13].

β=k0ε1ωε2ε1ω+ε2E37

Analysis of TE modes shows that they cannot exist in the form of the surface modes [11, 13]. SPPs can exist only for the TM polarization [11, 13].

Analysis of dispersion relation (37) with the Drude dielectric function (22) shows that in the case of the mid-infrared or lower frequencies the SPP propagation constant βis close to k0[11]. As a result, SPPs can propagate over many wavelengths into the dielectric medium z>0[11]. These SPPs are called Sommerfeld – Zenneck waves.

In the case of large wave vectors, the SPP frequency tends to the surface plasmon frequency ωspgiven by [11].

ωsp=ωp1+ε2E38

The surface plasmon is an electrostatic mode which is a limiting case of SPP for β[11]. In the general case of the essentially complex dielectric function ε1ω(22) the SPP energy attenuation length, or the propagation length L=2Imβ1which is usually between 10 and 100 μm for visible range optical frequencies and different types of the metal/ dielectric interfaces [11]. For instance, the SPPs at a silver/air interface for a vacuum wavelength λ0=450nmare characterized by the propagation length L16μmand the evanescent decay length 1/k2180nm; the corresponding values for λ01.5μmare: L1080μm, 1/k22.6nm[11]. Generally speaking, the lower SPP propagation length corresponds to the better SPP confinement [11]. The confinement of the optical field below the diffraction limit of λ0/2in the dielectric medium can be realized for optical frequencies ωωsp[11].

The SPP excitation at the metal/dielectric interface can be achieved by using the special phase-matching techniques such as a grating or prism coupling for the three-dimensional beams [11].

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4. Nonlinear optical phenomena in Plasmonic nanostructures

In this section, we briefly discuss the peculiarities of the nonlinear optical effects in plasmonic structures. Plasmonics is a new field of photonics related to the interaction of light with matter in metallic nanostructures [12]. Plasmonics combines the capacity of photonics and the miniaturization of electronics because SPs and SPPs can confine light to subwavelength dimensions as it was mentioned above [11, 12, 13]. As a result, the effective nonlinear optical response can be enhanced significantly [3]. SPP crystals and waveguides, nano-antennas and plasmonic metamaterials can be used for the creation of optical responses by using the resonances of the individual units and their electromagnetic coupling [3]. The response of materials to an optical field is determined by its polarization Prtwhich is given by [1, 2, 3, 4, 5].

Pirt=ε0×χij1Ejrt+χijk2EjrtEkrt+χijkl3EjrtEkrtElrt+E39

Here i,j,k,l=1,2,3, χij1,χijk2,χijkl3are the linear, second-order and third-order optical susceptibilities, respectively [1, 2, 3, 4, 5]. In the general case, the optical susceptibility χkis the k+1th–rank tensor [1, 2, 3, 4, 5]. In the centrosymmetric media χ2=0, in homogeneous and isotropic media optical susceptibilities are scalar quantities [1, 2, 3, 4, 5].

The second and third-order nonlinear optical phenomena are the most important for applications [1, 2, 3, 4, 5]. The second-order response results in the wave-mixing effects such as the sum and difference frequency generation and second harmonic generation (SHG) where the incident frequency ωgives rise to the term with the frequency 2ω[1, 2, 3, 4, 5]. The third-order response contains the terms with the incident frequency ω, the new harmonics with the combination frequencies, and the third harmonic with the frequency 3ω[1, 2, 3, 4, 5]. The term with the incident frequency ωis caused by the Kerr effect where the refractive index nis modified by the optical field [1, 2, 3, 4, 5]. It is given by [1, 2, 3, 4, 5].

n=n0+n2IE40

where n0,n2are linear and nonlinear refractive indices of the medium, respectively, and Iis the optical field intensity. The optical Kerr effect results in all-optical switching and light modulation [3]. The local field Elocωrenhancement at the metal/dielectric interface caused by the SPPs or LSPs excitation is characterized by the frequency-dependent local-field factor Lωrgiven by [3].

Lωr=Elocωr/E0ωE41

where ωis the plasmonic excitation frequency and E0ωis the incident field.

The surface–enhanced SHG, third-harmonic generation and FWM in nanoplasmonic structures have been demonstrated experimentally [3]. It has been shown that the SHG signal at the electrochemically roughened silver surfaces was increased by four orders of magnitude for a flat reference surface [3]. However, the SHG in such a case is diffused and incoherent being enhanced by LSP resonances of the nanoscale surface features [3].

The structured plasmonic surfaces for enhanced nonlinear effects represent metal gratings, arrays of different types of nanoparticles and nano-apertures, split-ring resonators [3]. For instance, FWM from a gold grating was 2000 times stronger than FWM from a flat film [3].

Controlling light with light in plasmonic nanostructures is based on LSP and SPP nonlinear effects. In such a case plasmonic structures are used for all-optical modulation or switching due to the enhanced Kerr-type nonlinearities [3]. For instance, the plasmon-enhanced nonlinear materials are metal nanoparticles and bulk materials doped with such nanoparticles [3]. Propagating SPPs can serve as the signal carrier [3]. The modulation and switching of SPPs in plasmonic waveguides can be achieved by changing of the real or imaginary part of the permittivity ε1,2by controlling light in the metal or dielectric at the interface shown in Figure 1 [3]. Optically controlled dispersion in plasmonic waveguides can be realized by the Kerr nonlinearity in dielectric materials or by using the nonlinearity in a metal [3]. It has been shown that under certain conditions the bistability and modulational instability are possible in nonlinear plasmonic waveguides [3]. Plasmonic soliton-like excitation is also possible if the gain and nonlinearity are balanced correspondingly [3].

Consider now the plasmonic metamaterials [3, 7, 8]. They may be used for all-optical switching based on the plasmonic resonances of the split-ring resonators or nanorods and the electromagnetic coupling between these elements [3]. All-optical modulation in metamaterials can be realized by controlling the coupling strength between molecular excitons and plasmonic excitations [3].

Tunable and nonlinear metamaterials can be created by inserting of LCs into a metamaterial structure [7]. The optical fields and the bias electric field are simultaneously applied to a metamaterial with LC influence on LC reorientation [7]. Their interplay demonstrates a specific mechanism of electrically controlled optical nonlinearity in metamaterials [7]. For example, all-optical control of metamaterials with E7 NLC at the telecom wavelength of 1550 nm was investigated experimentally [7]. The integration of highly nonlinear LCs with plasmonic and metamaterials enables active switching and tuning of optical signals with a very low threshold [8]. Three general methods are combining the responses of highly nonlinear NLCs and plasmonics: (i) dissolving of nanoparticles in a bulk NLC cell; (ii) incorporating LC into nanostructures; (iii) chemical synthesis of nanoparticles with LC molecules [8]. Consider the case of a small volume fraction f1of nano-spheres suspended in LC [8]. In such a case the effective optical dielectric constant εcomof the composite material is given by [8].

εcom=1+21εhost,γ=εAuεhostεAu+2εhostE42

Here εAuis the permittivity of the Au nano-spheres described by the Drude model (22) and εhostis the NLC permittivity. It has been demonstrated experimentally that a 100 μmthick sample of NLC L34 containing 0.5% Au nanoparticles provides the enhancement of the nonlinear absorption coefficients by about 250% or even more due to the inclusion of these plasmonic nano-particles into NLC cells [8].

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5. Conclusions

We considered the interaction of the optical waves with free electrons in metals. The theoretical analysis in such a case is based on Maxwell’s equations and the equation of motion of a free electron in an external electric field. As a result, plasmonic excitations emerge. The dielectric function of a free electron gas with plasmonic oscillations is described by the dielectric function (22) known as the Drude model. The longitudinal plasma oscillations excited at the plasma frequency ω=ωpare called the volume plasma oscillations, and the quasi-particles related to these oscillations are called the volume plasmons. SPPs may exist at the interface of the metal described by the Drude dielectric function and the dielectric medium. SPPs are the TM waves. They are characterized by the limited propagation length in the dielectric and the spatial confinement in the direction perpendicular to the propagation direction. The subwavelength confinement of the optical field is possible for SPPs. The nonlinear optical effects in plasmonic structures are significantly enhanced. The nonlinear optical phenomena such as modulation, switching, harmonic generation, FWM are possible in plasmonic nano-structures. Highly efficient all-optical switching and all-optical modulation can be realized in nonlinear plasmonic metamaterials based on nano-particles, split-ring resonators, and LCs.

Plasmonics plays an essential role in modern nonlinear optics. The nonlinear optical effects can be achieved at the reduced optical power due to enhanced effective nonlinearity in plasmonic nanostructures. The size of the nonlinear components can be scaled down which is important for the development of functional nanophotonic circuits. The ultrafast optical signal processing on the femtosecond scale can be realized due to the small response time of the plasmonic excitations.

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Written By

Boris I. Lembrikov

Submitted: December 26th, 2021 Reviewed: January 5th, 2022 Published: March 30th, 2022