## Abstract

We study far-field angular radiation patterns of second harmonic generation (SHG) from gold nanosphere, nanocube, nanorod, and nanocup illuminated by tightly focused linearly and radially polarized beams, respectively. It is found that under linearly polarized illumination, far-field forward-scattering SHG (FSHG) dominates second harmonic (SH) responses generated by those gold particles. On the contrary, it is amazing that significant backward-scattering SHG (BSHG) can be observed when those gold nanoparticles are excited by a focused radially polarized beam. For the case of gold nanosphere, the effective point dipole systems are developed to reasonably elucidate this interesting difference. Our investigations suggest that for SHG microscopy with backward detection scheme, tightly focused radially polarized beam could be a promising excitation field to improve the backward SH signal.

© 2016 Optical Society of America

## 1. Introduction

Within the electric dipolar approximation of light-matter interaction, second harmonic generation (SHG) is usually forbidden in centrosymmetric materials such as gold and silver [1]. Dipolar SHG is only allowed at the interface where centrosymmetry is broken [2]. Due to localized surface plasmon resonances, strong electromagnetic field enhancement occurs at metal surface, which could reinforce surface SHG from metal nanoparticles at relatively low pump power [3]. Since this nonlinear optical response has a number of potential applications, such as optical imaging [4], frequency converter [5], and biosensing [6], SHG from nanostructured metal particles has attracted a lot of interest in the last decades. Nonlinear optical properties of metal particles can be tailored for specific applications by modifying their geometric parameters including size, length, shape, and gap dimension. Actually, the properties of the excitation field, such as field amplitude, polarization, and phase, also play a crucial role in second harmonic (SH) responses from metal particles. Recently, SH responses from gold nanoparticles excited by different focused fields have been investigated [7–10]. For example, Bautista *et al.* [8] performed SHG imaging of gold nanoparticles with focused cylindrical vector beams and it was found that compared to linear polarization, this novel imaging technique is more sensitive to three-dimensional (3D) orientation and nanoscale morphology of gold nanoparticles.

As a coherent nonlinear optical process, due to the phase-matching constraint SHG is predominantly forward propagation (refers to the emission angles close to the propagation direction, see Eq. (6)) in common. Although in some hyper-Rayleigh scattering experiments SH signal can be detected at right angle [11], for the case of tight focused field illumination, it will become difficult to realize because focusing objectives with high numerical aperture (NA) usually have very short working distance and giant bulk size. Therefore, SH signal is generally collected in the forward direction. For SHG microscopy with forward detection scheme, two objectives are required. One is used to focus the incident fundamental beam and the other is utilized to collect the forward signal [12]. However, because of the short working distance between the two objectives, especially for the case of high NA, only very thin samples can be investigated by forward SHG microscopy. For thick samples in a variety of practical applications, it requires SH signal to be epicollected through the focusing objective due to sample thickness [13]. Consequently, it has a great significance to enhance the backward SH signal for epidetected SHG microscopy.

In previous research we have already studied near-field and far-field SH responses from single gold nanosphere excited by plane wave and focused linear polarization [14]. Here, we further compare far-field angular radiation patterns of SHG from gold nanoparticles with different shapes, including nanosphere, nanocube, nanorod, and nanocup, under tightly focused linearly and radially polarized illuminations, respectively. 3D finite-difference time-domain (FDTD) numerical solution is used to calculate near-field distributions of the fundamental beam and far-field SH angular radiations are obtained by the Green’s function approach. It is indicated that under linear polarization illumination forward-scattering SHG (FSHG) is always predominant whereas backward-scattering SHG (BSHG) could become very strong when particles are excited by a tightly focused radially polarized beam. In order to elucidate this difference, the effective point dipole systems are developed for the case of gold nanosphere. Our investigations suggest that compared to focused linearly polarized beam, radial polarization illumination is more suitable for the backward detection scheme to improve the backward SH signal.

## 2. Calculation model for SHG from gold nanoparticles excited by focused fields

In the frame of vector diffraction theory [15,16], when a *x*-polarized beam or a radially polarized light propagating along the *z* axis is focused by an infinity-corrected objective (see Fig. 1), the electric field at a given point **r*** _{p}* in the focal region can be expressed as

*ϕ*is the azimuthal angle of the point

_{p}**r**

*and*

_{p}*A*represents a constant proportional to the incident field amplitude. The integrals

*I*take the form

_{n}_{L}(

**r**

*) = exp(*

_{p}*ikr*cos

_{p}*θ*cos

*θ*) and M

_{p}_{R}(

**r**

*) = exp[*

_{p}*ikr*cos

_{p}*θ*cos

*θ*−(

_{p}*β*sin

_{0}*θ*/sin

*α*)

^{2}].

*k*is the wavenumber at the fundamental frequency and

*J*is the Bessel function of the first kind with order

_{n}*n*.

*θ*is the polar angle of the point

_{p}**r**

*.*

_{p}*α*is the maximal angle determined by the NA of the focusing objective.

*β*

_{0}is the ratio of the pupil radius to the beam waist. In this paper, the value of

*β*

_{0}is assumed to be 1. Then, the incident field can be determined by its steady value (obtained from Eq. (1)) multiplied with the time-harmonic factor exp(−

*iωt*).

*ω*is the frequency of the fundamental beam.

The presence of gold nanoparticles in the focal region generates scattered field and can significantly disturb the incident beam. As a result, the fundamental field will be not equivalent to the incident field. The analytical method based on Mie theory [17] can be used to obtain the fundamental excitation field for nanoparticles with regular shapes (e.g., nanospheres and nanoshells). However, for other nanoparticles with complicated shapes, the fundamental field has to be evaluated by using numerical solutions [18,19], for example, FDTD [20]. Since the incident field can be obtained analytically, the scattered-field formalism for FDTD [21] will be adopted to reduce the computation burden.

Under the electric dipolar approximation, SHG can only be observed at the surface of metal nanoparticles, which is usually described by a sheet of nonlinear polarization induced by the excitation field at the fundamental frequency. For centrosymmetric materials, the surface nonlinear polarization can be expressed as [22]

**E**(

*ω*)

*ex*

**(r**

*) is the fundamental excitation field at the particle’s surface. The term δ(*

_{s}**r**

*-*

_{s}*h*(

**r**

*)) implies that there is a surface dipole sheet because the centrosymmetry is broken at the particle’s surface*

_{s}*h*(

**r**

*).*

_{s}**χ**(2)

*s*is the surface second-order susceptibility tensor. The surface is assumed to be locally flat. In the case of centrosymmetric medium possessing isotropic with a mirror plane perpendicular to the local surface,**χ**(2)**$$\begin{array}{c}{P}_{\perp}^{(2\omega )}({r}_{s})={\epsilon}_{0}{\chi}_{\perp \perp \perp}\delta ({r}_{s}-h({r}_{s}))[{E}_{x}^{(\omega )}({r}_{s}){E}_{x}^{(\omega )}({r}_{s}){\mathrm{sin}}^{2}{\theta}_{s}{\mathrm{cos}}^{2}{\phi}_{s}\\ +{E}_{y}^{(\omega )}({r}_{s}){E}_{y}^{(\omega )}({r}_{s}){\mathrm{sin}}^{2}{\theta}_{s}{\mathrm{sin}}^{2}{\phi}_{s}+{E}_{z}^{(\omega )}({r}_{s}){E}_{z}^{(\omega )}({r}_{s}){\mathrm{cos}}^{2}{\theta}_{s}\\ +{E}_{x}^{(\omega )}({r}_{s}){E}_{y}^{(\omega )}({r}_{s}){\mathrm{sin}}^{2}{\theta}_{s}\mathrm{sin}2{\phi}_{s}+{E}_{x}^{(\omega )}({r}_{s}){E}_{z}^{(\omega )}({r}_{s})\mathrm{sin}2{\theta}_{s}\mathrm{cos}{\phi}_{s}\\ +{E}_{y}^{(\omega )}({r}_{s}){E}_{z}^{(\omega )}({r}_{s})\mathrm{sin}2{\theta}_{s}\mathrm{sin}{\phi}_{s}],\end{array}$$where $$\begin{array}{l}{E}^{(2\omega )}(R)=\frac{{\mu}_{0}{\omega}^{2}\mathrm{exp}(iKR)}{\pi R}{\displaystyle \underset{V}{\iiint}\text{d}V}\mathrm{exp}(\frac{-iKR\cdot {r}_{s}}{R})\\ \text{}\left[\begin{array}{ccc}0& 0& 0\\ \mathrm{cos}\Theta \mathrm{cos}\Phi & \mathrm{cos}\Theta \mathrm{sin}\Phi & -\mathrm{sin}\Theta \\ -\mathrm{sin}\Phi & \mathrm{cos}\Phi & 0\end{array}\right]\left[\begin{array}{c}{P}_{x}^{(2\omega )}({r}_{s})\\ {P}_{y}^{(2\omega )}({r}_{s})\\ {P}_{z}^{(2\omega )}({r}_{s})\end{array}\right]\end{array}$$where ***s*has three independent nonvanishing elements*χ*_{⊥⊥⊥},*χ*_{⊥||||}, and*χ*_{||⊥||}=*χ*_{||||⊥}. ⊥ and || refer to the directions perpendicular and parallel to the local surface, respectively. The element*χ*_{⊥⊥⊥}generally dominates the surface SH response from gold nanoparticles [23,24] and then the roles of the other elements will be not taken into account in this paper. The coordinate transformation of the excitation field components between the local surface coordinate and the Cartesian coordinate is implemented. Consequently, the surface SH polarization can be shown to be of the form*θ*_{s}and*ϕ*_{s}are the polar and azimuthal angles of the point**r***at the metal surface, respectively. The inhomogeneous wave equation driven by the surface SH polarization can be resolved by using the Green’s function approach. Under the far-field condition (|*_{s}**R**|≫|**r**_{s}|), SH field generated from gold nanoparticles can be obtained by [25,26]*Θ*and*Φ*are the polar and azimuthal angles of the detection point**R**(see Fig. 1), respectively.*K*is the wavenumber at the doubled frequency.## 3. Results and discussion

We investigate SHG from gold nanoparticles in the absence of any metallic layers in the vicinity of the nanoparticle. Gold nanoparticles are dispersed in a homogeneous medium. SHG from surrounding medium is very weak [27,28] and can be negligible. In addition, from the comparison of numerical simulations and experimental measurements, it has been demonstrated that the influence of a glass substrate on far-field SH radiation can be neglected [8]. Near-field distributions of the fundamental excitation fields around nanoparticles are simulated by using 3D FDTD method. The fundamental wavelength is 780nm and the incident beam is focused by a dry objective with NA = 0.9. The optical constants for gold are obtained from Ref [29]. The medium surrounding nanoparticles is assumed to be air. The mesh grid is set to 2.5nm in all directions. The time step ∆*t* is smaller than the Courant time step *t _{c}* and takes the value of 0.9

*t*. We set a convolutional perfect matched layer at all surrounding boundaries.

_{c}Figure 2 shows intensity distributions of various electric field components of the focused beams in the focal plane for linear and radial polarizations, respectively. The intensities have been normalized to the intensity of the total electric field at the focus. From the results in Fig. 2(a), a focused linearly polarized beam has a strong *x*-component *Ex inc,*L with rotational symmetry and the intensity maximum is observed at the focus. In contrast, the intensities of the components *Ey inc,*L and *Ez inc,*L at the focus are zero. Therefore, for small particles around the focus, the role of the component *Ex inc,*L will be dominant. According to Fig. 2(b), in the case of a tightly focused radially polarized beam, the component *Ez inc,*R is much stronger than the other two components and will dominate the optical response from particles that are placed in the focal region.

For a gold nanosphere with 150nm diameter, the near-field distributions of the fundamental field **E**(*ω*) *ex*, the components *E*(*ω*) *ex,x*^{and E(ω) ex,z in the xz plane under linear and radial polarization illuminations are presented by Fig. 3, respectively. As shown in Fig. 3(a), for linear polarization, electromagnetic field enhancement presents two lobes along the x direction. Since the component E(ω) ex,z is much weak, the fundamental field E(ω) ex only exhibits the signatures of the component E(ω) ex,x. For a focused radially polarized beam, the fundamental field E(ω) ex,x presents similar two-lobe distribution except for a 90° rotation (see Fig. 3(b)). The aforementioned results agree well with previous literature [17]. In addition, note that the component E(ω) ex,y is negligible for both linear and radial polarizations.}

Figures 4(a) and 4(b) illustrate 3D far-field SH angular radiation diagrams of single gold nanosphere with 150nm diameter for linear and radial polarizations, respectively. Figure 4(c) displays the corresponding cuts in the *xz* plane. The results for linear and radial polarizations are denoted by the black and red curves, respectively. They have already been normalized with respect to the maximum for the case of linear polarization. Note that those results are taken into account the entire electric field computed from FDTD. From the result in Fig. 4(a) and the black curve in Fig. 4(c), it is found that although SH signal can propagate both in the forward and backward directions when the gold nanosphere is illuminated by a focused linearly polarized beam, the forward signal is obviously predominant. In contrast, strong backward emission is observed under radial polarization illumination while there is only negligible SH signal in the forward direction (see Fig. 4(b) and the red curve in Fig. 4(c)).

Mertz and Moreaux have investigated SHG of inhomogeneously distributed scatterers under focused excitation [30]. It was found that strong BSHG can be generated in certain cases, due to inhomogeneities in the dipole distribution. In our study, a simple interpretation of the distinct difference about angular radiation patterns between the two different focusing conditions can be introduced in terms of the surface SH polarization distribution. According to Fig. 3(a), in the case of linear polarization, the gold nanosphere dominantly interacts with the field component *E*(*ω*) *ex,x* and the impacts of the components *E*(*ω*) *ex,y* and *E*(*ω*) *ex,z* can be neglected. As a result, Eq. (4) is simplified as *P*(2ω)_{⊥ = χ⊥⊥⊥|E(ω) ex,x||2sin2θscos2φs. Because of the term |E(ω) ex,x|2sin2θs, the surface SH polarization is observed around the two spherical caps along the x axis (as shown in Fig. 5(a)), which is similar to the result in another literature [11]. It should be noted that due to the term cos2φs, the nonlinear surface polarization is not rotationally symmetrical about the z axis. From the result in Fig. 5(a), it can also be found that the direction of the surface SH polarization is nearly parallel to the x direction. Far-field SH signal can be viewed as the result from the interference among the dipoles located at different locations within the focal volume whose amplitudes and phases are determined by the surface SH polarization. An effective system composed of four coherent dipolar sources in the xz plane oscillating at the SH frequency (see Fig. 5(b)) is proposed to reproduce the far-field SHG angular radiation pattern for a 150nm gold nanosphere centered at the focus under focused linear illumination. Those dipoles of equal strength are oriented along the positive or negative direction of the x axis. The separations of those dipoles in the x and z directions are d and d/2, respectively. Here, d is the diameter of the gold nanosphere, i.e., 150nm. Additionally, the phase difference ∆ϕ between the dipoles locating at z = ± d/4 is about 0.2π, which is extracted from FDTD simulation. According to the Green’s function approach, the far-field SH power radiated in the two opposite directions along the z axis is proportional to cos2[(kd−∆ϕ)/2]≈cos2(0.09π) in the positive direction, while proportional to cos2[(kd + ∆ϕ)/2] ≈cos2(0.29π) in the negative direction. As a result, the forward SH signal will dominate the nonlinear response from the gold nanosphere. Figure 5(c) gives the far-field SHG angular radiation arising from the interference of those four coherent point dipoles, which is almost identical to the result in Fig. 4(a). It is also confirmed that the component E(ω) ex,x is dominant in SHG from gold nanosphere excited by a focused linearly polarized beam.}

Similarly, as shown in Fig. 3(b), in the case of a focused radially polarized beam, the component *E*(*ω*) *ex,z* plays the dominant role in the nonlinear response and then the surface SH polarization is simplified as *P*(2ω)_{⊥=χ⊥⊥⊥|E(ω) ex,z|2cos2θs. Because of the term |E(ω) ex,z|2cos2θs, the nonlinear surface polarization only occurs around the two spherical caps along the z axis and exhibits full rotational symmetry about the optical axis since there is no dependence on φs (see Fig. 6(a)). In addition, unlike the results for linear polarization, the direction of the surface SH polarization is almost parallel to the z axis. Consequently, the far-field SHG emission pattern corresponding to radial polarization can be accurately represented by a superposition of two interacting dipoles located at the z axis. The two dipoles are oriented along the positive or negative direction of the z axis and separated by a distance of d = 150nm (see Fig. 6(b)). The phase difference ∆φ between the two dipoles is about 0.3π. The far-field SH powers radiated in the positive and negative directions of the z axis are proportional to sin2[(2kd−∆ϕ)/2]≈sin2(0.23π) and sin2[(2kd + ∆ϕ)/2] ≈sin2(0.53π), respectively. Therefore, as shown in Fig. 6(c), due to phase-matching constraint, the corresponding far-field SHG radiation is preferentially backward propagation when the gold nanosphere is excited by a focused radially polarized beam, which agrees well with the result in Fig. 4(b).}

We further investigate SH responses of other gold particles including nanocube, nanorod, and nanocup, illuminated by focused linearly and radially polarized beams, respectively. Here, the edge length of nanocube is 150nm. The length along the *x* direction and the diameter in the *z* direction of nanorod are 200nm and 50nm, respectively. In FDTD simulation, nanocup is represented by a layer of shell formed by a Boolean subtraction of a gold nanosphere and an air nanosphere having the same diameter of 150nm. The centers of these two nanospheres are located at the *z* axis and separated by a distance of 30nm. Figs. 7(a) and 7(b) present the near-field distributions of the fundamental fields surrounding those gold particles in the *xz* plane for linear and radial polarizations, respectively. When illuminated by a focused linearly polarized beam, the fundamental fields surrounding nanocube and nanorod all exhibit two lobes along the *x* direction, which is similar to that for gold nanosphere except for electromagnetic enhancement confined to particle’s edge. Under radial polarization illumination, the orientation of the two lobes of the fundamental fields for nanocube and nanorod rotates by 90° to the *z* direction. In the case of nanocup under linear polarization excitation, electromagnetic enhancement is strongly confined to sharp edge. However, when this nanocup is excited by a focused radially polarized beam, the electric field near its bottom surface is also obviously enhanced, which can be attributed to the role of the dominant component *Ez inc,*R of the incident field. Figs. 7(c) and 7(d) display far-field SHG angular radiation diagrams for nanocube, nanorod and nanocup excited by focused linearly and radially polarized beams, respectively. All of those results are taken into account the entire electric field computed from FDTD. From the results in Figs. 7(c) and 7(d), it is found that similar to nanosphere, FSHG still dominates SH responses when nanocube and nanocup are illuminated by a focused linearly polarized beam. However, in the case of nanorod, FSHG and BSHG become almost equal in intensity. This phenomenon can also be explained by the effective four-point-dipole system shown in Fig. 5(b). Due to the short extension of nanorod in the *z* direction, the phase difference ∆*φ* is so small that there is no distinct difference between the intensities of FSHG and BSHG. When excited by a focused radially polarized beam, significant BSHG is observed for all three kinds of gold nanoparticles. When *Φ* = 0°, SH powers as a function of *Θ* are given by Fig. 7(e), where the black and red curves represent the results for linear and radial polarizations, respectively. The powers are normalized relative to the maximum for the case of linear polarization. The SH power for nanorod under radial polarization illumination has been scaled by a factor of 200. Except for nanorod, the SH power for radial polarization is comparable to or even stronger than that for linear polarization. In the case of nanorod, when the polarization direction of the excitation field is parallel to the nanorod’s orientation [31], the enhancement of electromagnetic field near the metal surface is much stronger than that for radial polarization, which is the reason that in comparison, the far-field SH power for radial polarization is so weak. However, if radial polarization excitation is necessary, for example, for ultrasensitive shape characterization [8], it may be better to collect the backward signal.

Because far-field SHG always propagates in an off-axis direction, in practical applications an objective with high NA is required to efficiently collect SH signal. The intensities of BSHG and FSHG are defined by the integrations of far-field SH power over two cones corresponding to 0 ≤ *Θ* ≤ *β* and π − *β*≤ *Θ* ≤ π, respectively,

*β*is the maximal angle determined by the NA of the collecting objective. Table 1 summarizes the intensity ratios of BSHG to FSHG for four kinds of gold particles under two different focusing conditions. In forward detection scheme, the NA of the collecting objective is 0.9 and then the maximal angle is about 64°. According to Table 1, under radial polarization, the intensity of BSHG can be several times of that of FSHG and even one order of magnitude stronger.

## 4. Conclusions

In summary, we have investigated far-field SHG angular radiation patterns of gold nanosphere, nanocube, nanorod and nanocup excited by focused linearly and radially polarized beams, respectively. The results indicate that under linear polarization illumination FSHG is generally dominant whereas far-field SHG is preferentially backward propagation when those particles are excited by a focused radially polarized beam. For the case of nanosphere, the effective point dipole systems are developed to reasonably explain this difference. It can be found that the distribution of the surface SH polarization and the phase difference play very important roles in determining the ratio of BSHG to FSHG. Those investigations can provide guidelines for optimizing SHG microscopy in order to improve the backward SH signal when the epicollection of BSHG is necessary.

## Acknowledgments

This work was financially supported by the National Natural Science Foundation of China (NSFC) (Grant No. 61378005).

## References and links

**1. **R. W. Boyd, *Nonlinear Optics* (Academic, 2003).

**2. **Y. R. Shen, “Surface properties probed by second-harmonic and sum-frequency generation,” Nature **337**(6207), 519–525 (1989). [CrossRef]

**3. **S. Kim, J. Jin, Y.-J. Kim, I.-Y. Park, Y. Kim, and S.-W. Kim, “High-harmonic generation by resonant plasmon field enhancement,” Nature **453**(7196), 757–760 (2008). [CrossRef] [PubMed]

**4. **X. Huang, W. Qian, I. H. El-Sayed, and M. A. El-Sayed, “The potential use of the enhanced nonlinear properties of gold nanospheres in photothermal cancer therapy,” Lasers Surg. Med. **39**(9), 747–753 (2007). [CrossRef] [PubMed]

**5. **Y. Zhang, N. K. Grady, C. Ayala-Orozco, and N. J. Halas, “Three-dimensional nanostructures as highly efficient generators of second harmonic light,” Nano Lett. **11**(12), 5519–5523 (2011). [CrossRef] [PubMed]

**6. **J. Butet, I. Russier-Antoine, C. Jonin, N. Lascoux, E. Benichou, and P.-F. Brevet, “Sensing with multipolar second harmonic generation from spherical metallic nanoparticles,” Nano Lett. **12**(3), 1697–1701 (2012). [CrossRef] [PubMed]

**7. **G. Bautista, M. J. Huttunen, J. M. Kontio, J. Simonen, and M. Kauranen, “Third- and second-harmonic generation microscopy of individual metal nanocones using cylindrical vector beams,” Opt. Express **21**(19), 21918–21923 (2013). [CrossRef] [PubMed]

**8. **G. Bautista, M. J. Huttunen, J. Mäkitalo, J. M. Kontio, J. Simonen, and M. Kauranen, “Second-harmonic generation imaging of metal nano-objects with cylindrical vector beams,” Nano Lett. **12**(6), 3207–3212 (2012). [CrossRef] [PubMed]

**9. **P. Reichenbach, A. Horneber, D. A. Gollmer, A. Hille, J. Mihaljevic, C. Schäfer, D. P. Kern, A. J. Meixner, D. Zhang, M. Fleischer, and L. M. Eng, “Nonlinear optical point light sources through field enhancement at metallic nanocones,” Opt. Express **22**(13), 15484–15501 (2014). [CrossRef] [PubMed]

**10. **M. A. Tyrk, S. A. Zolotovskaya, W. A. Gillespie, and A. Abdolvand, “Radially and azimuthally polarized laser induced shape transformation of embedded metallic nanoparticles in glass,” Opt. Express **23**(18), 23394–23400 (2015). [CrossRef] [PubMed]

**11. **J. Butet, P.-F. Brevet, and O. J. F. Martin, “Optical second harmonic generation in plasmonic nanostructures: from fundamental principles to advanced applications,” ACS Nano **9**(11), 10545–10562 (2015). [CrossRef] [PubMed]

**12. **A. Volkmer, J. X. Cheng, and X. S. Xie, “Vibrational imaging with high sensitivity via epidetected coherent anti-Stokes Raman scattering microscopy,” Phys. Rev. Lett. **87**(2), 023901 (2001). [CrossRef] [PubMed]

**13. **D. Débarre, N. Olivier, and E. Beaurepaire, “Signal epidetection in third-harmonic generation microscopy of turbid media,” Opt. Express **15**(14), 8913–8924 (2007). [CrossRef] [PubMed]

**14. **J.-W. Sun, X.-H. Wang, S.-J. Chang, M. Zeng, and N. Zhang, “Second harmonic generation of metal nanoparticles under tightly focused illumination,” Chin. Phys. B **25**(3), 037803 (2016). [CrossRef]

**15. **B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system,” Proc. R. Soc. Lond. A Math. Phys. Sci. **253**(1274), 358–379 (1959). [CrossRef]

**16. **K. Youngworth and T. Brown, “Focusing of high numerical aperture cylindrical-vector beams,” Opt. Express **7**(2), 77–87 (2000). [CrossRef] [PubMed]

**17. **K. Şendur, W. Challener, and O. Mryasov, “Interaction of spherical nanoparticles with a highly focused beam of light,” Opt. Express **16**(5), 2874–2886 (2008). [CrossRef] [PubMed]

**18. **K. Kitamura, K. Sakai, and S. Noda, “Finite-difference time-domain (FDTD) analysis on the interaction between a metal block and a radially polarized focused beam,” Opt. Express **19**(15), 13750–13756 (2011). [CrossRef] [PubMed]

**19. **H. Shen, N. Nguyen, D. Gachet, V. Maillard, T. Toury, and S. Brasselet, “Nanoscale optical properties of metal nanoparticles probed by Second Harmonic Generation microscopy,” Opt. Express **21**(10), 12318–12326 (2013). [CrossRef] [PubMed]

**20. **A. Taflove and S. C. Hagness, *Computational Electrodynamics: The Finite-Difference Time-Domain Method* (Artech House, 2000).

**21. **W. Challener, I. Sendur, and C. Peng, “Scattered field formulation of finite difference time domain for a focused light beam in dense media with lossy materials,” Opt. Express **11**(23), 3160–3170 (2003). [CrossRef] [PubMed]

**22. **J. I. Dadap, J. Shan, and T. F. Heinz, “Theory of optical second-harmonic generation from a sphere of centrosymmetric material: small-particle limit,” J. Opt. Soc. Am. B **21**(7), 1328–1347 (2004). [CrossRef]

**23. **G. Bachelier, J. Butet, I. Russier-Antoine, C. Jonin, E. Benichou, and P.-F. Brevet, “Origin of optical second-harmonic generation in spherical gold nanoparticles: Local surface and nonlocal bulk contributions,” Phys. Rev. B **82**(23), 235403 (2010). [CrossRef]

**24. **J. Butet, K. Thyagarajan, and O. J. F. Martin, “Ultrasensitive optical shape characterization of gold nanoantennas using second harmonic generation,” Nano Lett. **13**(4), 1787–1792 (2013). [CrossRef] [PubMed]

**25. **E. Yew and C. Sheppard, “Effects of axial field components on second harmonic generation microscopy,” Opt. Express **14**(3), 1167–1174 (2006). [CrossRef] [PubMed]

**26. **J. X. Cheng and X. S. Xie, “Green’s function formulation for third-harmonic generation microscopy,” J. Opt. Soc. Am. B **19**(7), 1604–1610 (2002). [CrossRef]

**27. **J. Butet, J. Duboisset, G. Bachelier, I. Russier-Antoine, E. Benichou, C. Jonin, and P.-F. Brevet, “Optical second harmonic generation of single metallic nanoparticles embedded in a homogeneous medium,” Nano Lett. **10**(5), 1717–1721 (2010). [CrossRef] [PubMed]

**28. **J. Butet, G. Bachelier, J. Duboisset, F. Bertorelle, I. Russier-Antoine, C. Jonin, E. Benichou, and P.-F. Brevet, “Three-dimensional mapping of single gold nanoparticles embedded in a homogeneous transparent matrix using optical second-harmonic generation,” Opt. Express **18**(21), 22314–22323 (2010). [CrossRef] [PubMed]

**29. **P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B **6**(12), 4370–4379 (1972). [CrossRef]

**30. **J. Mertz and L. Moreaux, “Second-harmonic generation by focused excitation of inhomogeneously distributed scatterers,” Opt. Commun. **196**(1–6), 325–330 (2001). [CrossRef]

**31. **T. Ming, L. Zhao, Z. Yang, H. Chen, L. Sun, J. Wang, and C. Yan, “Strong polarization dependence of plasmon-enhanced fluorescence on single gold nanorods,” Nano Lett. **9**(11), 3896–3903 (2009). [CrossRef] [PubMed]