Chapter 3 Fundamentals of Surface Plasmons In the last chapter, we introduced the EM properties of materials and discussed the dielectric functions of Drude and Drudeª²Lorentz models. Under certain conditions, the regular electricª²charge oscillation can be induced on the metal surface, driven by external EM fields. Two forms of oscillations are presented: surface plasmon polaritons (SPPs) and localized surface plasmons (LSPs). In this chapter, we will introduced the SPP excitation, propagation mechanism, and dispersion relation in Section 3.1 and the principles of LSP and the optical properties of different nanostructures in Section 3.2. 3.1Surface plasmon polaritons SPPs are EM waveª²like phenomena that propagate along the interface between the dielectric and metal, which are generated by the coupling between photons and free electrons in the metal£Û1ª²2£Ý. SPP properties are defined by the dielectric constants of both the metal and dielectric as well as the frequency of the excitation light, meaning that SPP can be easily manipulated by varying the above conditions£Û3£Ý. Moreover, the EM field of SPP is highly confined to the vicinity of the dielectricª²metal interface, where energy follows an exponentialª²decay relation along the vertical direction of the interface£Û4ª²6£Ý. Interestingly, this confinement leads to a strongly enhanced EM field and is widely used in electronic, chemical, and biological fields. In this section, we highly focus on the fundamental EM properties of the SPPs and the means of exciting them. 3.1.1Properties of surface plasmon polaritons As mentioned above, SPPs are EM waveª²like phenomena on the interface between the dielectric and metal; they can be described as Transverse electric£¨TE£©: E(x,y,z)=Ae¡Àkzzeikxx,£¨3ª²1£© Transverse magnetic£¨TM£©: H(x,y,z)=Ae¡Àkzzeikxx,£¨3ª²2£© where kz and kx are the zª² and xª²components of the wave vector, respectively. Thus, we can use the Maxwell equations with the boundary conditions to obtain each solution in the metal and dielectric. As shown in Fig.3ª²1, we suppose that the SPPs propagate on the interface including an isotropic dielectric with a real and positive dielectric constant ¦Åd. Furthermore, such constant of the Fig.3ª²1SPP excitation and propagation modes (a) Sketch of the SPP excitation and propagation mechanism on a smooth metallic surface; (b) Sketch of the TMª²mode surface wave propagating on the metalª²dielectric interface; (c) Sketch of the TEª²mode surface wave propagating on the metalª²dielectric interface (For colored figure please scan the QR code on page 2) metal is characterized by a frequencyª²dependent, isotropic complex constant of the form ¦Åm(¦Ø)=¦Å¡ä(¦Ø)+i¦Å¡å(¦Ø), where ¦Å¡ä(¦Ø) and ¦Å¡å(¦Ø) represent its real and imaginary parts, respectively. According to the Maxwell equations, we can easily obtain «ý¡ÁE=i¦Ì0¦ØH,£¨3ª²3£© «ý¡ÁH=-i¦Å0¦ÅE.£¨3ª²4£© First of all assume that the SPP is a TM wave propagating along the x direction of the interface. As shown in Fig.3ª²1(b), the magnetic vector of the TM wave is perpendicular to the xzª²plane. Considering the component form of Eqs.(3ª²3) and (3ª²4), we obtain ¡ÀkzEx-i¦ÂEz=i¦Ø¦Ì0Hy,£¨3ª²5£© ¡ÀkzHy=i¦Ø¦Å0¦ÅEx,£¨3ª²6£© i¦ÂHy=-i¦Ø¦Å0¦ÅEz.£¨3ª²7£© In the regions of z>0 and z<0, Eq.(3ª²6) can be described as follows: -kz1Hy1=i¦Ø¦Å0¦Å1Ex1£¨3ª²8£© and kz2Hy2=i¦Ø¦Å0¦Å2Ex2,£¨3ª²9£© respectively. Applying the boundary conditions Hy1=Hy2 and Ex1=Ex2 to Eqs.(3ª²8) and (3ª²9), the relation of the wave vectors with the dielectric constant is afforded: kz1kz2=-¦Å1¦Å2.£¨3ª²10£© For this equation exists, the wavevectors kz1 and kz2 must be real and positive, such that ¦Åm(¦Ø) is negative. From this equation, we can easily see that SPPs are a kind of mixed, waveª²like forms of photons coupled with the collective oscillation of the free electrons on the metal surface. Combining the Eq.(3ª²10) and combining it with the wave equation, we can obtain the dispersion relation of SPPs: ¦Â=k0¦Å1¦Å2¦Å1+¦Å2=¦Øc¦Åm¦Åd¦Åm+¦Åd,£¨3ª²11£© as shown in Fig.3ª²2. It is obvious that at the low frequency range (¦Åm¡ú -¡Þ) the wavevector of the SPPs can be approximatedy to ¦Â=¦Øclim¦Åm¡ú-¡Þ¦Åm¦Åd¦Åm+¦Åd¡Ö¦Øc¦Åd,£¨3ª²12£© which is very close to the wavevector of the light in the dielectric material. Fig.3ª²2Dispersion relation of the SPP and incident light When ¦Åm¡ú¦Åd, ¦Ø approaches a certain value: ¦Øsp=¦Øp1+¦Åd,£¨3ª²13£© where ¦Øp is the plasma frequency, the frequency of bulk plasmon electron oscillation. Now, consider that the TE wave propagates on the interface between the dielectric and metal. As with the solution of the very TM wave and from the Maxwell equations in Eqs.(3ª²3£© and (3ª²4£©, we obtain ¡ÀkzEy=-i¦Ø¦Ì0Hz,£¨3ª²14£© i¦ÂEy=i¦Ø¦Ì0Hz,£¨3ª²15£© ¡ÀkzHx-i¦ÂHz=-i¦Ø¦Ì¦Å0¦ÅEy.£¨3ª²16£© In addition, from Eq.(3ª²6), we obtain: -kz1Ey1=i¦Ø¦Ì0Hx1(z>0),£¨3ª²17£© kz2Ey2=i¦Ø¦Ì0Hx2(z<0).£¨3ª²18£© Combining Eqs.(3ª²17) and (3ª²18) with the boundary conditions Hx1=Hx2 and Ey1=Ey2, we obtain: Ey(kz1+kz2)=0.£¨3ª²19£© If the SPPs can be seen as TE waves, the wavevectors of kz1 and kz2 must be positive in Eq.(3ª²19), meaning that Ey1 and Ey2 should equal 0. Thus, SPPs cannot exist as a TE wave and only the TM mode can be supported. 3.1.2Excitation of surface plasmon polaritons From the dispersion relation discussed in Section 3.1.1, the wavevector of SPPs is clearly mismatched with the light in free space under the same frequency, meaning that the light cannot directly excite SPPs by simply illuminating the smooth surface of a metal. Thus, if we want to excite SPPs using the light in free space, the wavevector of this light should be compensated using other approaches. One way to compensate for the wavevector differences between light and SPPs is using prisms to realize a totalª²internalª²reflection (TIR) system. Fig.3ª²3£¨a£© presents the Kretschmann configuration, comprising a dielectric prism and a thin metal film. When the incident light passes through the prism with an incidence angle of ¦È (larger than the critical angle), the light will have a TIR with an xª²component (evanescent wave) given by kx2=¦Åd2¦Øcsin¦È.£¨3ª²20£© Fig.3ª²3Excitation configuration of SPP (a) SPPs¡¯ excitation configuration of the Kretschmann system; (b) The dispersion relation of SPP and freeª²space light in the prism and dielectric materials (For colored figure please scan the QR code on page 2) Eq.(3ª²20) shows that the wavevector of light increases in the prism because of the larger dielectric constant ¦Åd2>1(¦Åd1), and the wavevector kx,2 can be changed by choosing a different incident angle. As shown in Fig.3ª²3(b), when we choose an appropriate angle ¦È to set kx,2 equivalent to kSPP, the evanescent wave can pass through the thin metal film with a tunneling effect and couple with the free electron in the metal film to achieve SPP excitation. However, the Kretschmann system can only be used for a very thin metal film due to tunneling limitations. For thick or bulk metal films, such system is obviously not suitable for SPP excitation. Here, in Fig.3ª²4, we also show several other means of SPP excitation on the surface of a metal film with no thickness limitation. Fig.3ª²4(a) shows another prism¡¯s TIR system for SPP excitation (Otto configuration). As with the Kretschmann configuration, the Otto system uses the evanescent wave created by the prism for SPP excitation. Nevertheless, in the Otto system, a thin air space is introduced between the prism and metal film to prevent the limitation of the tunneling effect in the metal film. Fig.3ª²4Excitation configuration of SPP (a) Otto excitation system; (b) Periodic grating on the metal surface (For colored figure please scan the QR code on page 2) Another means of inducing SPP excitation on a smooth metal surface is to use periodic grating to compensate the wavevector with a diffraction effect. Fig.3ª²4(b) shows the grating with period d on a metal surface. When the light illuminates the grating, several different diffraction orders will occur. Due to the mismatch between the incidentª²light wavevector and SPP, the light of the regularly propagating diffraction orders cannot directly excite SPP. Rather, only the light from the evanescent diffraction orders can match the SPP with a wavevector of km,x=kinc, x+mK,£¨3ª²21£© where m is the order of diffraction and K is a constant 2¦Ðd. Moreover, light can excite the SPPs on a metal surface with roughness or small particles which can be treated as an abnormal grating. 3.2Localized surface plasmons When light is irradiated upon metallic particles whose size is close to or smaller than the wavelength of light, due to boundary confinement, the chargeª²density wave will be confined to the particle¡¯s surface and form a localized EM field around it, called a localized surface plasmon (LSP). Unlike the SPPs, the LSPs¡¯ dispersion relations are discontinuous in resonant modes with different orders. The resonance of the LSP will generate a significantly enhanced EM field, providing a strong foundation for many fields of surfaceª²enhanced Raman scattering (SERS), surfaceª²enhanced Raman fluorescence, opticalª²force enhancement and manipulation, biosensing, and optical devices. In this section, we mainly discuss the LSP principle and the optical properties of different nanostructures. A nanosphere can be seen as an electric dipole when its diameter is far below the wavelength of the incident light. Under this circumstance, we can use the quasiª²static approximation to solve the resonantª²frequency condition of the LSP. As shown in Fig.3ª²5, under quasiª²static approximation conditions, the entire LSP system can be seen as a metallic sphere of radius a, and the frequencyª²dependent dielectric constant ¦Åm(¦Ø) placed in a uniform zª²polarized planewave. The electric fields inside (E1) and outside (E2) the nanosphere are satisfied with E1,2=-«ý¦¼1,2,£¨3ª²22£© where the potential ¦¼ can be described with Laplace equation «ý2¦¼1=0(ra).£¨3ª²24£© Fig.3ª²5Schematic of the metallic nanosphere in a uniform zª²polarized planewave (For colored figure please scan the QR code on page 2) Combining Eq.(3ª²23) with the boundary conditions ¦¼1=¦¼2(r=a),£¨3ª²25£© ¦Åmªµ¦¼1ªµr=¦Ådªµ¦¼2ªµr(r=a),£¨3ª²26£© limr¡ú¡Þ¦¼2=-E0,£¨3ª²27£© the potential solutions inside and outside of the nanospheres can be obtained: ¦¼1=-E0rcos¦È+¦Åm-¦Åd¦Åm+2¦ÅdE0rcos¦È=-3¦Åd¦Åm+2¦ÅdE0rcos¦È,£¨3ª²28£© ¦¼2=-E0rcos¦È+a3¦Åm-¦Åd¦Åm+2¦ÅdE0cos¦Èr2=-E0rcos¦È+p¡¤r4¦Ð¦Å0¦Ådr3£¬£¨3ª²29£© where p is the dipole momentum, which can also be described with the polarizability ¦Á: p=¦Å0¦Åd¦ÁE0.£¨3ª²30£© Combining Eqs.(3ª²28) with (3ª²29), one can obtain the relation between the dielectric constant and polarizability: ¦Á=4¦Ða3¦Åm-¦Åd¦Åm+2¦Åd£¨3ª²31£© Thus, from Eq.(3ª²31), we can easily determine that the polarizability reaches a maximum value when |¦Å(¦Ø)+2¦Åm| is minimal. This is called the resonantª²enhancement condition: |¦Åm(¦Ø)+2¦Åd|=Minimum,£¨3ª²32£© also known as Fr¢‰hlich condition. Moreover, we can solve the electric field inside and outside the nanosphere (E=-«ý¦¼): E1=3¦Åd¦Åm+2¦ÅdE0,£¨3ª²33£© E2=E0+3n(n¡¤p)-p4¦Ð¦Å0¦Åd1r3.£¨3ª²34£© The above solution of the quasiª²static approximation is only suitable for a nanosphere with a diameter less than 100 nm (visible and inferred excitation). For a particle with a larger size, the quasiª²static approximation is no longer satisfied. However, Mie theory makes the connection between the size and shape parameters and the extinction spectrum: E(¦Ë)=24¦Ð2Na3¦Å32m¦Ëln(10)¦Åi(¦År+¦Ö¦Åm)2+¦Å2i,£¨3ª²35£© where ¦Ö is the shape factor (2 for a sphere,and >2 for a spheroid), ¦Á is the radius of the particle, ¦Åm is external dielectric constant, ¦År is the real metal dielectric constant, and ¦Åi is the imaginary metal dielectric constant. The LSPR effect of the metallic structures strongly depends on the particle size, shape, and dielectric properties of the surrounding medium. Thus, the optical properties of Au nanospheres can be adjusted by changing their size and shape. Based on Mie theory, the efficiencies of the absorption, scattering, and extinction spectra for Au nanospheres of different size have been calculated by Jain et al. (Fig.3ª²6)£Û7£Ý. The figures show that the extinction peaks experience a red shift and the relative contribution of the scattering to extinction also increases with the size of Au nanospheres in the 20~80 nm range. Strong enhancement of absorption and scattering for metal nanoparticles (NPs) is attributed to the wellª²known collective oscillation of surface plasmon electrons on the metal surface. Fig.3ª²6The simulated spectral efficiencies of the absorption (red dashed), scattering (black dotted), and extinction (green solid) spectra of Au nanospheres with diameters of (a) D = 20 nm, (b) D = 40 nm, and (c) D = 80 nm£Û7£Ý (For colored figure please scan the QR code on page 2) References £Û1£ÝKREIBIG U, VOLLMER M. Optical properties of metal clusters£ÛM£Ý.New York: Springerª²Verlag, 1995. £Û2£ÝNOVOTNY L, HECHT B, KELLER O. Principle of nanoª²optics£ÛJ£Ý. Phys Today, 2006, 60: 62. £Û3£ÝPOHL D W.Nearª²field optics and the surface plasmon polariton£ÛM£Ý.Berlin: Springer, 2001. £Û4£ÝRAETHER H. Surface plasmons on smooth and rough surfaces and on gratings£ÛM£Ý. Berlin: Springer, 1988. £Û5£ÝENOCH S, BONOD N. Plasmonics¡ªfrom basics to advanced topics£ÛM£Ý.New York: Springer, 2012. £Û6£ÝGRIFFITH D J, RUPPEINER G. Introduction to electrodynamics£ÛJ£Ý. Am J Phys, 1981, 49: 1188ª²1189. £Û7£ÝJAIN P K, LEE K S, ELª²SAYED I H, et al. Calculated absorption and scattering properties of gold nanoparticles of different size, shape, and composition: Applications in biological imaging and biomedicine£ÛJ£Ý. J Phys Chem B, 2006, 110: 7238ª²7248.