**3.2 CWLR imaging for characterization of individual gold nanospheres**

Commercial gold nanospheres with a diameter of 50 5 nm (Corpuscular Inc) deposited on 200 nm silicon dioxide (SiO2) films were chosen for CWLR imaging as well. Their confocal white light reflection images were constructed by extracting the light intensity from the corresponding reflection spectra for a selected wavelength range, too. The image at the wavelength of 510-550 nm was shown in Figure 6b as one example. As can be seen from the SEM image shown in Figure 6a, it consists of three isolated single spheres and one dimer where the two spheres are almost in contact with each other. The four dark spots in Figure 6b represent the images of the gold nanospheres, which correspond well to the SEM image. The size of the dark spots is about 410 nm, determined by the spatial resolution of the imaging system. Considering the resolution limitation, it is reasonable that the white light reflection images can not distinguish between single sphere and dimer spheres. For comparison, Figure 6d presented the CWLR imaging results for the spheres by setting D2 to 100 µm while keeping other experimental conditions the same. It can be seen that the images from the spheres severely overlap, confirming the higher spatial resolution by using a 25 µm core diameter collection fiber again.

where *sample I* and *substrate I* refer to the white light reflection intensity of the sample and the substrate, respectively. The result was shown as Figure 5h, which confirms that gold particles always have larger reflection than that of the substrate, as well as the role of SP in

Fig. 5. The CWLR images (a-f) at different wavelength ranges for gold nanoarrays on cover glass fabricated by using 1 µm diameter PS as the lithographic mask while (g) and (h) are the reflected white light spectra and reflection contrast between the gold particle and substrate, respectively. Wavelength ranges for images (a) to (f) correspond to 0.440-0.480 µm, 0.480-0.520 µm, 0.520-0.560 µm, 0.560-0.600 µm, 0.600-0.640 µm and 0.640-0.680 µm,

Commercial gold nanospheres with a diameter of 50 5 nm (Corpuscular Inc) deposited on 200 nm silicon dioxide (SiO2) films were chosen for CWLR imaging as well. Their confocal white light reflection images were constructed by extracting the light intensity from the corresponding reflection spectra for a selected wavelength range, too. The image at the wavelength of 510-550 nm was shown in Figure 6b as one example. As can be seen from the SEM image shown in Figure 6a, it consists of three isolated single spheres and one dimer where the two spheres are almost in contact with each other. The four dark spots in Figure 6b represent the images of the gold nanospheres, which correspond well to the SEM image. The size of the dark spots is about 410 nm, determined by the spatial resolution of the imaging system. Considering the resolution limitation, it is reasonable that the white light reflection images can not distinguish between single sphere and dimer spheres. For comparison, Figure 6d presented the CWLR imaging results for the spheres by setting D2 to 100 µm while keeping other experimental conditions the same. It can be seen that the images from the spheres severely overlap, confirming the higher spatial resolution by using a 25 µm

**3.2 CWLR imaging for characterization of individual gold nanospheres** 

reflection, which is discussed in detail in the next section.

respectively.

core diameter collection fiber again.

*Contrast I I I* ( *sample substrate substrate* ) / (2)

Fig. 6. SEM image (a) and CWLR images at wavelength 510-550 nm of gold nanospheres on 200 nm SiO2 films with collection fiber core diameter 25 µm (b) and 100 µm (d). (c) The contrast spectra for single (black lines) and dimer spheres for incident light parallel (solid lines) and perpendicular (dotted lines) to the dimer axis. (Du et al., 2008).

Figure 6c plotted the contrast spectra for the single and dimer gold nanospheres after locating their positions. Two contrast dips (labelled 1 and 2, respectively) were observed from Figure 6. It is noticed that the position of dip 2 of isolated single spheres (at about 525 nm) coincides with that of the LSP of gold spheres with diameter 50 nm (Dijk et al., 2005). However, the same dip shows red shifts to 548 nm and 542 nm for the incident polarization parallel and perpendicular to the dimer axis, respectively. This results from the coupling effect between the two nanospheres of the dimer (Moores & Goettmann, 2006). The larger red shift for the parallel polarization than that of the perpendicular case is understandable considering the stronger coupling of the dimer for parallel polarization. Figure 6c also reveals a weak dip 1 for the single and dimer located at about 470 nm, which originates from the multi-polar SP excitation of the gold nanospheres. Firstly, SP excitation at about 470 nm has been observed for gold nanospheres with diameters close to 40 nm, although it was ascribed to false spectral lines that arose from using a 488 nm argon laser (Benrezzak et al., 2001). Secondly, similar multi-polar SP excitation has been reported for other isolated metal nanoparticles (Dijk et al., 2005). Moreover, the excitation of LSPs leads to the enhancement of the absorption of the nanospheres, which consequently has the effect of reducing the reflection intensity (Kawata, 2001), making the nanospheres dark in the corresponding images in Figure 6b. The different dip 2 position for the single and dimer spheres also reflects their different dipolar LSP resonant energies, further implying that the CWLR imaging method is capable of resolving the LSP energies for individual noble metal nanoparticles.

Then, to determine the decay length of the electromagnetic (EM) coupling between individual gold nanopspheres and its supporting substrate, different thicknesses (*d*) SiO2 film on Si were obtained by annealing single-crystalline Si (*d* < 20 nm) in air or by RF sputtering SiO2 (*d* > 20 nm) to serve as the substrate.

Confocal White Light Reflection Imaging for Characterization of Nanostructures 293

images at different wavelength regions. The different wavelength regions constructing these contrast images were labelled by the rectangular bars on the left contrast spectra of Figure. 8a. Figure 8a presents an obvious peak at ~530 nm, which originates from the excitation of dipolar LSPs with the peak wavelengths corresponding to the dipolar LSP wavelengths (Du et al., 2008; Okamoto & Yamaguchi, 2003; Abe & Kajikawa, 2006; Pinchuk et al., 2004; Knight et al., 2009). The excitation of LSP enhances the light absorption by the nanosphere and reduces its reflection intensity (Kawata, 2001), then further leads to its higher contrast intensity relative to the substrate in the CWLR contrast images (Figure. 8b). As can be seen, with the selected wavelength region closer to the contrast peak position, the larger image contrast between the Au nanosphere and the substrate is obtained while the maximum contrast is reached at the LSP wavelength. Meanwhile, from the unequivocal one-to-one correspondence between the nanosphere in the contrast images (Figure. 8b) and its SEM

To explore the near-field coupling between individual gold nanospheres and their supporting substrate further, we have measured the contrast spectra between individual Au nanospheres on SiO2/Si substrates with different thicknesses of SiO2 on Si substrate. Several typical spectra were presented in Figure 9, in which the dipolar LSP peaks were guided by the dashed line. The second peak at shorter wavelength for cases of bare Si substrate and *d* = 6 nm corresponds to the multi-polar LSP excitation (Du et al., 2008). In what follows, we will mainly discuss the dipolar LSP (labelled as LSP for short in the following) wavelength (

Fig. 9. Comparative CWLR contrast spectra between individual Au nanospheres on SiO2/Si

Figure 9 reveals that the LSP wavelengths of all the samples show an obvious red-shift compared to the LSP resonant wavelength for an isolated Au nanosphere with diameter about 50 nm in air, which is at about 520 nm (Noguez, 2007). This originates from the different dielectric function of the substrate from that of air and agrees well with the literature reports (Okamoto & Yamaguchi, 2003; Abe & Kajikawa, 2006; Pinchuk et al., 2004). Moreover, the dashed line in Figure 9 demonstrates that the LSP wavelength blue-shifts with *d* increasing. To see this more clearly, we have plotted the function of the resonant wavelength of these LSP modes versus *d* in Figure 10. The experimental data of vs. *d* is noted to can be fitted with a single exponential decay function (the solid line) quite well. All these reflect the near-field EM interaction between the nanosphere and its supporting

substrates (with SiO2 film *d* nm) and on Si substrate. (Du et al., 2010).

)

image (Figure. 8c), we can locate the individual nanosphere we concern.

behaviour as a function of the spacer SiO2 film thickness *d*.

Fig. 7. Schematic diagram of the prepared gold nanospheres on SiO2/Si substrate samples.

Fig. 8. Comparison between the CWLR contrast images (b) of an individual Au nanosphere on a SiO2/Si substrate (with SiO2 film 60 nm) at different selected wavelength regions labelled by the rectangular bars on the corresponding contrast spectra (a) along with the corresponding SEM image of the nanoshpere (c). (Du et al., 2010).

Figure 7 illustrated the schematic diagram of the prepared gold nanospheres on SiO2/Si substrate samples and a typical SEM image was shown as Figure 8c. The distance between different individual nanospheres chosen for study herein was selected purposely to be larger than 1m. Thus the EM coupling between them can be neglected. For convenience, herein the contrast is defined by the following Eq. (3), and hence their contrast images and the CWLR contrast spectra can be obtained.

$$\text{Contrast} = \left( I\_{sub} - I\_{AuNSP} \right) / \left( I\_{sub} \right) \tag{3}$$

Where *AuNSP I* and *sub I* refer to the CWLR intensity from the positions of the Au NP and the substrate, respectively. Adopting Eq. (3), the CWLR contrast spectra for individual gold nanoparticles were obtained. A typical CWLR contrast spectra of an individual Au nanosphere on SiO2/Si substrates was presented as Figure 8a along with its different CWLR

Fig. 7. Schematic diagram of the prepared gold nanospheres on SiO2/Si substrate samples.

Fig. 8. Comparison between the CWLR contrast images (b) of an individual Au nanosphere on a SiO2/Si substrate (with SiO2 film 60 nm) at different selected wavelength regions labelled by the rectangular bars on the corresponding contrast spectra (a) along with the

Figure 7 illustrated the schematic diagram of the prepared gold nanospheres on SiO2/Si substrate samples and a typical SEM image was shown as Figure 8c. The distance between different individual nanospheres chosen for study herein was selected purposely to be larger than 1m. Thus the EM coupling between them can be neglected. For convenience, herein the contrast is defined by the following Eq. (3), and hence their contrast images and

Where *AuNSP I* and *sub I* refer to the CWLR intensity from the positions of the Au NP and the substrate, respectively. Adopting Eq. (3), the CWLR contrast spectra for individual gold nanoparticles were obtained. A typical CWLR contrast spectra of an individual Au nanosphere on SiO2/Si substrates was presented as Figure 8a along with its different CWLR

*Contrast I I I* ( *sub AuNSP sub* ) / (3)

corresponding SEM image of the nanoshpere (c). (Du et al., 2010).

the CWLR contrast spectra can be obtained.

images at different wavelength regions. The different wavelength regions constructing these contrast images were labelled by the rectangular bars on the left contrast spectra of Figure. 8a. Figure 8a presents an obvious peak at ~530 nm, which originates from the excitation of dipolar LSPs with the peak wavelengths corresponding to the dipolar LSP wavelengths (Du et al., 2008; Okamoto & Yamaguchi, 2003; Abe & Kajikawa, 2006; Pinchuk et al., 2004; Knight et al., 2009). The excitation of LSP enhances the light absorption by the nanosphere and reduces its reflection intensity (Kawata, 2001), then further leads to its higher contrast intensity relative to the substrate in the CWLR contrast images (Figure. 8b). As can be seen, with the selected wavelength region closer to the contrast peak position, the larger image contrast between the Au nanosphere and the substrate is obtained while the maximum contrast is reached at the LSP wavelength. Meanwhile, from the unequivocal one-to-one correspondence between the nanosphere in the contrast images (Figure. 8b) and its SEM image (Figure. 8c), we can locate the individual nanosphere we concern.

To explore the near-field coupling between individual gold nanospheres and their supporting substrate further, we have measured the contrast spectra between individual Au nanospheres on SiO2/Si substrates with different thicknesses of SiO2 on Si substrate. Several typical spectra were presented in Figure 9, in which the dipolar LSP peaks were guided by the dashed line. The second peak at shorter wavelength for cases of bare Si substrate and *d* = 6 nm corresponds to the multi-polar LSP excitation (Du et al., 2008). In what follows, we will mainly discuss the dipolar LSP (labelled as LSP for short in the following) wavelength ( ) behaviour as a function of the spacer SiO2 film thickness *d*.

Fig. 9. Comparative CWLR contrast spectra between individual Au nanospheres on SiO2/Si substrates (with SiO2 film *d* nm) and on Si substrate. (Du et al., 2010).

Figure 9 reveals that the LSP wavelengths of all the samples show an obvious red-shift compared to the LSP resonant wavelength for an isolated Au nanosphere with diameter about 50 nm in air, which is at about 520 nm (Noguez, 2007). This originates from the different dielectric function of the substrate from that of air and agrees well with the literature reports (Okamoto & Yamaguchi, 2003; Abe & Kajikawa, 2006; Pinchuk et al., 2004). Moreover, the dashed line in Figure 9 demonstrates that the LSP wavelength blue-shifts with *d* increasing. To see this more clearly, we have plotted the function of the resonant wavelength of these LSP modes versus *d* in Figure 10. The experimental data of vs. *d* is noted to can be fitted with a single exponential decay function (the solid line) quite well. All these reflect the near-field EM interaction between the nanosphere and its supporting

Confocal White Light Reflection Imaging for Characterization of Nanostructures 295

demonstrates that the strength of the near-field EM coupling owing to the substrate effect decreases with *d/R* and then vanishes with further increasing *d/R*. For thicker spacer (*d* > 3*R*), the influence of the image-charges owing to the substrate presence is ignorable and the LSP wavelength approaches to the limit of the individual nanosphere in the background. Meanwhile, the faster decay rate of the shift predicted by the quasi-static theory is resulted from the limitation of the theoretical model, which ignores the retardation effect while such effect is more pronounced for larger nanoparticles (Okamoto & Yamaguchi, 2003; Noguez, 2007). The solid line in the right *y*-axis of Figure 10 presents the single exponential decay

<sup>0</sup> exp( ) *<sup>x</sup>*

to the experimental data with *a*, *t* and *y0* as the fitting parameters. It reveals a decay length *t*  about 0.30 in units of *d/R* and the fitting goodness equals to 0.97. The fitting results are interestingly noted to qualitatively agree with the 'plasmon ruler' scaling theory for the near-field EM coupling between two component noble metal nanoparticles of a dimer (Jain et al., 2007). It points out that the decay length is about 0.20 in units of the 'Gap/Diameter' regardless of the nature of the component metal nanoparticles of the dimer (Jain et al., 2007). The similarity of the decay length in magnitude is owing to that the image-charges induced in the substrate can be just replaced by the actual charges induced in the other particle for the dimeric nanoparticle's case. The deviation between their decay lengths is understandable considering the different polarizabilities between the dimer and our case. Thus, the near-field coupling strength between the individual Au nanosphere and the Si substrate is revealed to exponentially decrease with the spacer (SiO2 film) thickness and the

Silver nanowire samples with a diameter about 100 nm were fabricated by a simple hydrothermal method (Wang et al., 2005) and deposited onto silicon for CWLR imaging. Two typical CWLR images of the silver nanowire were presented as Figures 11a and 11b, which clearly exhibits polarization dependent. By rotating 90 degree of the incident polarization, their contrast reverses from the comparison between Figures 11a and 11b. Even for the same bent nanowire, the different parts exhibits different contrast compared to the substrate. Owing to the larger reflectivity of silver than that of the substrate silicon, it is expected that the nanowires are brighter than the substrate under CWLR imaging system. However, this contradicts with the results of Figure 11, revealing that other factors besides

It is known that the excitation of SPs of sliver nanowires is anisotropic (Schider et al., 2003), which is sensitive to the polarization direction of the incident light. This can account for the polarization dependence of the reflection images of Figure. 11, which also contribute to the different contrast for different parts of the same bent nanowire as well since for different parts of the same nanowire, the only difference lies in their different orientation relative to the incident light polarization direction. As the images shown in Figures 11a and 11b were extracted from the reflections in the range of 600 – 640 nm, it is overlapped with one of the SP mode (500 – 700 nm) (Kim et al., 2003; Mohanty et al., 2007) of the Ag nanowire, which

**3.3 CWLR imaging for characterization of individual silver nanowires** 

material reflectivity contribute to the observed CWLR images.

*<sup>y</sup> <sup>a</sup> <sup>y</sup> <sup>t</sup>* (4)

fitting by Eq. (4)

decay length is about 0.3 in units of *d/R*.

substrate, which dominates the corresponding LSP wavelength shift (Jain et al., 2007; Biring et al., 2008). Accordingly, the near-field EM coupling strength between them determines the magnitude of the dipolar LSP wavelength shift ( <sup>0</sup> ) compared to the LSP wavelength () of the isolated nanosphere case.

Fig. 10. The function of the LSP wavelength versus the spacer thickness *d* (left y-axis and bottom *x*-axis) and the function of the normalized LSP wavelength shift of /0 versus the normalized spacer thickness *d/R* (right y-axis and top *x*-axis): the squared dots, dotted line and solid line correspond to the experimental data, calculated results and the single exponential decay fitting results, respectively). (Du et al., 2010).

By bringing a metal nanoparticle into the vicinity of a flat substrate, nonhomogeneous optical response due to the polarizability of the substrate material is expected. Generally, with an external EM field, a charge polarization on the NP can be induced, which further causes a charge distribution on the substrate. Under the quasi-static approximation, this charge distribution can be seen as the image charge distribution (Okamoto & Yamaguchi, 2006; Noguez, 2007) of the nanoparticle and it can in turn affect the local EM field around the nanoparticle, and further their optical responses. Adopting the method proposed elsewhere (Okamoto & Yamaguchi, 2006; Wind et al., 1987), the extinction spectra of the concerned nanospheres were calculated by including both the dipole and higher multiple image-charge effects along with the Fresnel reflection effect at the nanosphere-substrate interface. The corresponding obtained function of the LSP wavelength () versus the spacer thickness (*d*) was plotted as the dashed line in Figure 10 as well for comparison. The calculation qualitatively verifies the decrease behaviour of the experimental data of versus *d* though it predicts a faster decay rate than the experimental results.

To quantify the near-field EM coupling strength between the individual nanosphere and its supporting substrate, the normalized LSP wavelength shift ( /<sup>0</sup> ) was calculated as well by the quasi-static theory and for the experimental data. Experimentally, 0 is obtained by assuming that it equals to the LSP wavelength of the nanosphere on very thick SiO2 film (take it as 200 nm). Through a simple linear transformation, both the experimental and theoretical data of versus *d* can be transferred into /0 versus *d/R*. The obtained results were scaled according to the right *y*-axis (corresponds to the upper *x*-axis) in Figure 10. It can be seen that /<sup>0</sup> decreases with *d/R* and reaches zero for large enough *d/R*. This

substrate, which dominates the corresponding LSP wavelength shift (Jain et al., 2007; Biring et al., 2008). Accordingly, the near-field EM coupling strength between them determines the

Fig. 10. The function of the LSP wavelength versus the spacer thickness *d* (left y-axis and

the normalized spacer thickness *d/R* (right y-axis and top *x*-axis): the squared dots, dotted line and solid line correspond to the experimental data, calculated results and the single

By bringing a metal nanoparticle into the vicinity of a flat substrate, nonhomogeneous optical response due to the polarizability of the substrate material is expected. Generally, with an external EM field, a charge polarization on the NP can be induced, which further causes a charge distribution on the substrate. Under the quasi-static approximation, this charge distribution can be seen as the image charge distribution (Okamoto & Yamaguchi, 2006; Noguez, 2007) of the nanoparticle and it can in turn affect the local EM field around the nanoparticle, and further their optical responses. Adopting the method proposed elsewhere (Okamoto & Yamaguchi, 2006; Wind et al., 1987), the extinction spectra of the concerned nanospheres were calculated by including both the dipole and higher multiple image-charge effects along with the Fresnel reflection effect at the nanosphere-substrate interface. The corresponding obtained function of the LSP wavelength () versus the spacer thickness (*d*) was plotted as the dashed line in Figure 10 as well for comparison. The calculation qualitatively verifies the decrease behaviour of the experimental data of versus

To quantify the near-field EM coupling strength between the individual nanosphere and its

by the quasi-static theory and for the experimental data. Experimentally, 0 is obtained by assuming that it equals to the LSP wavelength of the nanosphere on very thick SiO2 film (take it as 200 nm). Through a simple linear transformation, both the experimental and

were scaled according to the right *y*-axis (corresponds to the upper *x*-axis) in Figure 10. It

 /

<sup>0</sup> decreases with *d/R* and reaches zero for large enough *d/R*. This

 /

bottom *x*-axis) and the function of the normalized LSP wavelength shift of

exponential decay fitting results, respectively). (Du et al., 2010).

*d* though it predicts a faster decay rate than the experimental results.

supporting substrate, the normalized LSP wavelength shift (

theoretical data of versus *d* can be transferred into

 /

can be seen that

 

<sup>0</sup> ) compared to the LSP

 /

<sup>0</sup> ) was calculated as well

0 versus *d/R*. The obtained results

0 versus

magnitude of the dipolar LSP wavelength shift (

wavelength () of the isolated nanosphere case.

demonstrates that the strength of the near-field EM coupling owing to the substrate effect decreases with *d/R* and then vanishes with further increasing *d/R*. For thicker spacer (*d* > 3*R*), the influence of the image-charges owing to the substrate presence is ignorable and the LSP wavelength approaches to the limit of the individual nanosphere in the background. Meanwhile, the faster decay rate of the shift predicted by the quasi-static theory is resulted from the limitation of the theoretical model, which ignores the retardation effect while such effect is more pronounced for larger nanoparticles (Okamoto & Yamaguchi, 2003; Noguez, 2007). The solid line in the right *y*-axis of Figure 10 presents the single exponential decay fitting by Eq. (4)

$$y = a\* \exp(-\frac{\varkappa}{t}) + y\_0 \tag{4}$$

to the experimental data with *a*, *t* and *y0* as the fitting parameters. It reveals a decay length *t*  about 0.30 in units of *d/R* and the fitting goodness equals to 0.97. The fitting results are interestingly noted to qualitatively agree with the 'plasmon ruler' scaling theory for the near-field EM coupling between two component noble metal nanoparticles of a dimer (Jain et al., 2007). It points out that the decay length is about 0.20 in units of the 'Gap/Diameter' regardless of the nature of the component metal nanoparticles of the dimer (Jain et al., 2007). The similarity of the decay length in magnitude is owing to that the image-charges induced in the substrate can be just replaced by the actual charges induced in the other particle for the dimeric nanoparticle's case. The deviation between their decay lengths is understandable considering the different polarizabilities between the dimer and our case. Thus, the near-field coupling strength between the individual Au nanosphere and the Si substrate is revealed to exponentially decrease with the spacer (SiO2 film) thickness and the decay length is about 0.3 in units of *d/R*.

#### **3.3 CWLR imaging for characterization of individual silver nanowires**

Silver nanowire samples with a diameter about 100 nm were fabricated by a simple hydrothermal method (Wang et al., 2005) and deposited onto silicon for CWLR imaging. Two typical CWLR images of the silver nanowire were presented as Figures 11a and 11b, which clearly exhibits polarization dependent. By rotating 90 degree of the incident polarization, their contrast reverses from the comparison between Figures 11a and 11b. Even for the same bent nanowire, the different parts exhibits different contrast compared to the substrate. Owing to the larger reflectivity of silver than that of the substrate silicon, it is expected that the nanowires are brighter than the substrate under CWLR imaging system. However, this contradicts with the results of Figure 11, revealing that other factors besides material reflectivity contribute to the observed CWLR images.

It is known that the excitation of SPs of sliver nanowires is anisotropic (Schider et al., 2003), which is sensitive to the polarization direction of the incident light. This can account for the polarization dependence of the reflection images of Figure. 11, which also contribute to the different contrast for different parts of the same bent nanowire as well since for different parts of the same nanowire, the only difference lies in their different orientation relative to the incident light polarization direction. As the images shown in Figures 11a and 11b were extracted from the reflections in the range of 600 – 640 nm, it is overlapped with one of the SP mode (500 – 700 nm) (Kim et al., 2003; Mohanty et al., 2007) of the Ag nanowire, which

Confocal White Light Reflection Imaging for Characterization of Nanostructures 297

Fig. 12. The contrast spectra of graphene sheets with different thicknesses, together with the optical image of all the samples. Besides the samples with 1, 2, 3, 4, 7 and 9 layers, samples a, b, c, d, e and f are more than ten layers and the thickness increases from a to f. The arrows in the graph show the trend of curves in terms of the thicknesses of graphene sheets. (Ni et al.,

Fresnel reflection theory. Consider the incident light from air (*n0* = 1) onto a graphene, SiO2, and Si tri-layer system. The reflected light intensity from the tri-layer system can then be

> *R rr* () () ()

> > ( ) *<sup>a</sup> b r*

1 2 1 2 1 2 1 2 () () () () <sup>1</sup> <sup>2</sup> <sup>3</sup> <sup>123</sup> ( ) *iii i ar re re re rrre*

through the media which is determined by the path difference of two neighbouring interfering light beams. The thickness of the graphene sheet can be estimated as *d1* = *NΔd*, where *N* represents the number of layers and *Δd* is the thickness of single layer graphene (*Δd* = 0.335 nm) (Kelly, 1981, Dresselhaus, 1996). The refractive index of graphene is used as a fitting parameter. The thickness of SiO2, *d2*, is 285 nm, with a maximum 5% error. The refractive index of SiO2, *n2*, is wavelength dependent (Palik, 1991). The Si substrate is considered as semi-infinite and the refractive index of Si, *n3*, is also wavelength dependent (Palik, 1991). The reflection from SiO2 background, R0(λ), was calculated by setting *n1* = *n0* =

1 2 1 2 1 2 1 2 () () () () 1 2 1 3 2 3 ( ) *i i ii*

*r* 

> 

> >

(7)

(8)

are the phase difference when the light passes

*r*

*br e rre rre rre*

 

 

 

(5)

(6)

 

 

are the reflection coefficients for different

2007).

described by (Blake et al., 2007; Anders, 1967):

01 12 23 , , *n n n n n n*

1 12 2 2 ,2 *d d*

 *n n* 

*nn nn nn* 

where 0 1 1 2 2 3 123

*rrr*

interfaces and 1 2

1, and *d1* =0.

experiences preferred excitation when the polarization of the incident light is parallel to the nanowire. Thus, this SP mode along the different material reflectivity contributes to the obtained polarization dependent CWLR images. It also demonstrates that the developed CWLR imaging system is able to correlate the polarization dependent CWLR images of single silver nanowires with the nanowire polarization dependent excitation of SP.

Fig. 11. The CWLR images at the wavelength of 600–640 nm for silver nanowires on silicon substrate. The double-direction arrows in the figure indicate the polarization direction of the incident light. (Du et al., 2008).
