Experimental solution for scattered imaging of the interference of plasmonic and photonic mode waves launched by metal nano-slits

Xing Li; Yaru Gao; Shuna Jiang; Li Ma; Chunxiang Liu; Chuanfu Cheng

doi:10.1364/OE.23.003507

1. Introduction

Surface plasmon polaritons (SPPs) are involved in extensive fields of researches ranging from exploring the mechanisms of excitations [1–3] to engineering nanodevices of various functionalities [4–8]. Due to the ability to confine the wavefield in the medium interfaces and to mediate the interaction with surface nano-structures, SPPs have given rise to such interesting applications as microscopy beyond diffraction limit [9,10], light focusing [5,6], light bending [11] and compact plasmonic modulators [12]. The manipulations of wavefields with nano-slit structures on metal films are the important subject in SPP researches, and they cover three key aspects of the wave excitations, the designs of the slit structures and the detections of wavefields. The wave excitations depend highly on polarizations of the incident light: incidence of transverse magnetic(TM) polarization excites plasmonic mode waves, while transverse electric (TE) polarization excites the photonic mode waves [13–15]. Waves of the two modes have different amplitudes and initial phases, and their applications in the design of nano-focusing devices have attracted a lot of investigations [16–18]. With regard to the slit structure designs for manipulating the wavefields, the shapes, widths and orientations of slits are the elements to control the launched wavelets [13,19], with the superposition of the wavelets forming the needed patterns [14, 20, 21]. More generally, the interference of the wavelets can be described comprehensively and intuitively by the Huygens-Fresnel principle for nanoscale metal slits [22–24], and it can facilitate the analysis and implementation of controls over the wavefields [25–28]. The applicability of the principle depends on determination of the amplitudes, the initial phase and the wavevectors of the source wavelets. When the principle is applied to complex slits, the coexistence of the TE and TM excitations due to the decomposed polarizations and the influence of their superposition on the wavefields is obviously an important problem but have rarely been considered in the literature. Pertaining to the detections of plasmonic wavefields, the techniques of scanning near-field microscopy (SNOM) [5, 6, 8, 20, 21] and leakage radiation [29–32] have been developed and are often used. The SNOM is the direct way to achieve the plasmonic wavefields by scanning the tip within a subwavelength distance above the metal surface, yet the complexities and high costs in operations and insensitiveness to the electric component perpendicular to the surface [5, 6, 33] may limit its applications. With the leakage radiation method, both the Fourier images and the real images of the plasmonic waves can be directly obtained [34–36]. The method has been widely investigated and successfully used in the studies of interference imaging and pattern formations of the plasmonic waves [37, 38]. The far-field scattered imaging method has been an important alternative for plasmonic wave detections [39–41]. Small fraction of the wavefield propagating along the metal surface are scattered into the free space from the slightly rough metal surface, and the objective used in the method captures and magnifies the scattered image of pattern on the surface [39, 42, 43]. Due to its simplicity and feasibility, recently the method has been adopted more and more in measurements and observations of SPP regimes [44–49], and related remarkable progress has been achieved in far-field microscopic imaging with subwavelength resolution [50–53].

In this paper, we perform experimental solution for the photonic mode and the plasmonic mode waves in scattered imaging regimes. The quantities needed in the solution including the amplitudes, wavevectors and initial phases of the two waves under the same condition are obtained through the experimental schemes and subsequent data extractions. In the experiment, the scattered pattern imaging setup is embedded in a Mach-Zenhder interferometer and an L-shaped slit on Au film is employed as the sample. The two arms of slit provide the TE and TM incidence simultaneously and to launch the two mode waves independently. The pattern of fringes formed by the interference of the two waves propagating perpendicularly on the surface is imaged by an objective and received by a CCD. From the data of the two sets of fringes, the scattered amplitudes of the two waves are solved and the wavevector components are extracted. The initial phase values of the two waves are directly obtained from the phase map reconstructed with the interference of the scattered image and the reference wave. By introducing the scattering theory under Kirchhoff’s approximation and hybridation of SPP wave and the quasi-cylindrical wave (CW) in plasmonic mode [54, 55], we explain the difference in the scattered wavevector components between the photonic mode and the plasmonic mode waves. This difference also gives rise to an additional phase delay in the two waves and then influences the pattern formation. With these quantities and decomposing the incidence polarization into components parallel and perpendicular to a nano-slit element of arbitrary orientation, we propose for the first time the construction of the elementary secondary source launching both the photonic mode and plasmonic mode waves. This elementary secondary source allows the Huygens-Fresnel principle to be applied directly to nanoslits of complicated shapes. We fabricate a sample of ring-slit on the Au film as an example, and compare experimental pattern with that calculated with the Huygens-Fresnel principle. The high coincidence shows that the experimental solution of wavevector components, amplitudes and initial phases works well in the analysis of the scattered interference patterns.

2. Experimental setup

As shown in Fig. 1(a), a Mach-Zehnder type interferometer is used to record the scattered wavefield launched by metal nanoslit sample S. A linearly polarized light beam from a He-Ne laser $(λ = 632 .8nm)$ is split into two by a beam-splitter (BS1). Two beam attenuators A1 and A2 are used to adjust the intensity of the split beams. The lower arm is used as the reference and the beam is expanded and filtered by spatial pinhole filter (SPF). Its spherical wave is to offset that subjected to the image of the object arm due to large magnification. The upper arm beam illuminates the sample from substrate side. The sample is placed on a three-dimensional translational stage (TS) for fine position adjustment. The light wave scattered from the sample surface is captured by a microscopic objective MO (Nikon, 100x, N.A. = 0.9, WD 1.0mm), and the enlarged image with high magnification is received by a CCD (Cascade-1K, pixels 1004 × 1002, pixel size 8 μm × 8μm, dynamic range 16 bit). To find the exact imaging plane of the sample surface on the CCD, we first use a white light source for the illumination, and acquire clearest image by adjusting the longitudinal position of the sample. Then the laser beam is used instead. The CCD records the interfered intensity pattern of scattered image with the reference beam. The CCD can also record only the scattered wavefield intensity image with the reference beam blocked. The sample is an L-shaped nano-slit milled on an Au film with a focused ion beam (FIB) system. The film is deposited on a fused silica substrate and its thickness is 200nm. The scanning electron microscopy (SEM) image of the sample is shown in Fig. 1(b). The slit consists of a pair of perpendicular linear arms of length 10μm and width 300nm.

Fig. 1 (a) Experimental system with Mach-Zehnder interferometer. The beam from a He-Ne laser is linearly polarized. BS1 and BS2 are beam-splitters. M1 and M2 are mirrors. A1 and A2 are beam attenuators. TS is translational stage. SPF is pinhole spatial filter. The beam of the lower arm is the reference wave. A microscopic imaging setup with the microscopic objective MO (100Х, WD 1.0mm) is embedded in the upper arm. S is the sample of metal nanoslit. The CCD records either the scattered imaging pattern or the interference pattern of the image and reference beam. (b) SEM image of the sample.

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With the recorded intensity pattern, the amplitude and the phase distributions of the scattered wavefield can be reconstructed. We suppose that the scattered wavefield $E (x, y)$ and the reference plane wave $U_{r e f} (x, y)$ are written as:

E (x, y) = A (x, y)exp[i Φ (x, y)] = E_{r e} (x, y) + i E_{i m} (x, y),

U_{r e f} (x, y) = exp[i 2 π (f_{0 x} x + f_{0 y} y)],

where

E_{r e} (x, y)

,

E_{i m} (x, y)

are the real and the imaginary parts of

E (x, y)

, and

A (x, y)

,

Φ (x, y)

are its amplitude and phase, respectively.

f_{0 x}

and

f_{0 y}

are the spatial frequencies of

U_{r e f} (x, y)

. Accordingly, the interference intensity of

E (x, y)

and

U_{r e f} (x, y)

is:

I_{i n t} (x, y) = {| E (x, y) + U_{r e f} (x, y) |}^{2} .

From the principles of interference, the Fourier transform of intensity $I_{i n t} (x, y)$ in the above Equation contains three spatial spectrum spots oriented in the direction perpendicular to the interference fringes. Either of the two-sided spectrum spots contains the Fourier transform of $E (x, y)$ with spectrum coordinate translation. Eliminating the translation of one spot carefully, and performing the inverse Fourier transform, we reconstruct $E (x, y)$ with its real part $E_{r e} (x, y)$ and the imaginary part $E_{i m} (x, y)$ . Then the amplitude $A (x, y)$ of wavefield intensity $I (x, y)$ and the phase distribution $Φ (x, y) = arg[E_{i m} (x, y)/ E_{r e} (x, y)]$ can also be acquired. In addition, in the reconstructed wavefield map, the inherent electrical noises of the CCD can be effectively filtered off.

3. Theoretical basis for scattering of waves generated by metal nano-slit from the film surface

As schematically shown in Fig. 2, a nano-slit milled in a metal film on a dielectric substrate is along y-axis. A TM polarized plane wave with wavelength $λ_{0}$ illuminates the slit from the substrate side. The electric field of the incident wave is written as $E = E_{x} \hat{i}$ , with $\overset{}{\hat{i}}$ the unit vector in x-direction. The wavefield of plasmonic mode [13] is excited by the structure to propagate in the metal film. It contains two components: one is the SPP wave, and the other is the quasi-cylindrical wave (CW) [56]. The magnetic field of the plasmonic mode wave propagating along the air/metal interface is written as [54]:

H_{y} (x)= H_{S P} (x)+ H_{C W} (x)=exp(- i k_{S P} x)+ H_{0 C W} {(x / λ_{0})}^{- 1 / 2} exp(- i k_{0} x),

where

H_{S P} (x)

,

H_{C W} (x)

,

k_{S P}

and

k_{0}

are the SPP wave and the quasi-CW and their wave vectors, respectively.

H_{0 C W}

is ratio of quasi-CW wave to the SPP wave. The plasmonic mode as the combination of SPP wave and the quasi-CW expressed in the above Equation is also called the hybrid wave [55]. In TM incidence, the electric component

E_{z}

of the excited waves is much larger than the component

E_{x}

, and it is main concern in most cases [57]. With the relation of

E_{z} (x) = C H_{y} (x)

and C a complex coefficient,

E_{z} (x)

of the hybrid wave has the same form as the above expression [57]:

E_{z} (x)= E_{z, S P} (x)+ E_{z, C W} (x)=exp(- i k_{S P} x)+ E_{0 C W} {(x / λ_{0})}^{- 1 / 2} exp(- i k_{0} x),

where the subscripts represent the same notations as those for magnetic field

H_{y} (x)

in Eq. (4).

Fig. 2 Schematic of the scattering on an Au film surface. $k$ represents wavevectors. $θ_{2}$ and $θ_{3}$ represent the polar and the azimuthal angles, respectively. The dashed red curve illustrates schematically the scattered intensity.

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During the wavefield propagating on the film surface, the slightly rough surface formed in the film growth will scatter a small fraction of wavefield into the air, and it is utilized in the scattered imaging detection. Fundamentally, the scattering of SPPs is specialized subject, and attentions have been paid mainly to the scattering by defects of invariant heights as grooves and ridges on metal films [54, 58, 59], or by nanoscale topographical film surfaces [60, 61]. Next we will start from the Green’s integral and Kirchhoff approximation in conventional scattering theory, and give a basic analytical description for scattering of SPP mode wavefield from the rough film surface. For the wavefield given in Eq. (5), the scattered wavefield $E_{s} (r)$ in the air space can be written in the Green’s integral form [62]:

E_{s} (r)= \int [E_{z} (r^{'}) \partial G (r, r^{'})/ \partial \hat{n^{'}} - G (r, r^{'}) \partial E_{z} (r^{'})/ \partial \hat{n^{'}}] d S^{'},

where

r = r (x, y, z)

is the position vector of a point in air space, and

r^{'} = r^{'} (x^{'}, y^{'}, z^{'})

and

d S'

are the position vector and the surface element on metal-air interface, respectively.

\hat{n^{'}}

is the surface normal pointing toward the air space, and

G (r, r^{'})

is the Green’s function:

G (r, r^{'})= \frac{exp(i k_{0} | r - r^{'} |)}{4 π | r - r^{'} |} \approx \frac{exp(i k_{0} r)}{4 π r} exp(- i k_{s} \cdot r^{'}) .

From Eq. (5),

E_{z} (r^{'}) = E_{z} (x^{'})

and noticing

\partial / \partial \hat{n^{'}} = \hat{n^{'}} \cdot \nabla

, we obtain:

\partial E_{z} (x^{'})/ \partial \hat{n^{'}} = - i (k_{S P} \hat{i} \cdot \hat{n^{'}})exp(i k_{S P} x^{'}) + \hat{i} \cdot \hat{n^{'}} [- i k_{0} - (1 / 2 x')] E_{0 C W} {(x^{'} / λ_{0})}^{- 1 / 2} exp(- i k_{0} x^{'})

While

\partial G (r, r^{'})/ \partial \hat{n^{'}} \approx - i (\hat{n^{'}} \cdot k_{s}) \frac{exp(i k_{0} r)}{4 π r} exp(- i k_{s} \cdot r^{'}),

where

k_{s}

is the scattering wavevector. Substituting Eqs. (5), (7), (8) and (9) into Eq. (6), we have the scattered wavefield under Kirchhoff approximation:

\begin{array}{l} E_{s} (r) = - i \frac{exp(i k_{0} r)}{4 π r} \int [(k_{s} - k_{S P} \hat{i})] \cdot \hat{n^{'}} exp(- i k_{s} \cdot r^{'})exp(- i k_{S P} x^{'}) d S' \\ - i \frac{exp(i k_{0} r)}{4 π r} E_{0 C W} \int {(x^{'} / λ_{0})}^{- 1 / 2} [(k_{s} - k_{0} \hat{i} - \frac{1}{2 x' i} \hat{i}) \cdot \hat{n^{'}}]exp(- i k_{s} \cdot r^{'})exp(- i k_{0} x^{'}) d S', \end{array}

As shown in Fig. 2,

k_{s}

is expressed with the polar angle

θ_{2}

and azimuthal angle

θ_{3}

:

k_{s} = k_{0} \sin θ_{2} \cos θ_{3} \hat{i} + k_{0} \sin θ_{2} \sin θ_{3} \hat{j} + k_{0} \cos θ_{2} \hat{k},

with

\hat{i}

,

\hat{j}

, and

\hat{k}

the unit vectors. With

h (x', y')

representing the height at point

(x', y')

from a reference plane and using small slope approximation of

\partial h / \partial x' \approx 0

and

\partial h / \partial y' \approx 0

, we have

\hat{n^{'}} \approx \hat{k}

. Equation (10) can be simplified as:

\begin{array}{l} E_{s} (r) = - i \frac{exp(i k_{0} r) k_{0} \cos θ_{2}}{4 π r} \int exp {- i [k_{s x} x^{'} + k_{s y} y' + k_{s z} h (x', y')]} exp(- i k_{S P} x^{'}) d S' \\ - i \frac{exp(i k_{0} r) k_{0} \cos θ_{2}}{4 π r} E_{0 C W} \int {(x^{'} / λ_{0})}^{- 1 / 2} exp {- i [k_{s x} x^{'} + k_{s y} y' + k_{s z} h (x^{'}, y')]} exp(- i k_{0} x^{'}) d S', \end{array}

where

k_{s x} = k_{0} \sin θ_{2} \cos θ_{3}

, and

k_{s z} = k_{0} \cos θ_{2}

. Since the SPP and the quasi-CW at the metal-air interface propagate along x-direction, we may simply consider without loss of generality the case of

k_{s y} = 0

corresponding to

\sin θ_{3} = 0

. By neglecting the spherical wave factor that will not influence the scattered intensity, further simplification can be made for the one-dimensional scattered field given in the vector wave space:

\begin{array}{l} E_{s} (k_{s x}, k_{s z}) = \int exp {- i [k_{s x} x' + k_{s z} h (x', y')]} exp(- i k_{S P} x') d S' \\ - E_{0 C W} \int {(x' / λ_{0})}^{- 1 / 2} exp {- i [k_{s x} x' + k_{s z} h (x', y')]} exp(- i k_{0} x') d S', \end{array}

The two terms in the right-hand side of the above Equation are the scattered fields of the SPP and quasi-CW components of the plasmonic mode launched under TM incidence. While the incidence is TE polarized, the photonic mode wavefield launched by the slit has a similar form to the quasi-CW component in the photonic mode, and its scattered wavefield is similar to the second term in the right-hand side of the above Equation.

4. Experimental solution for scattered imaging of the wavefields with L-shaped metal slit

As schematically illustrated in Fig. 3(a), a linearly polarized light wave illuminates the L-shaped slit, and the polarization is represented by the double-headed arrow. The incidence is TE-polarized for the vertical arm of slit, and is the TM polarized for the horizontal arm. Then the two arms excite waves of photonic mode and plasmonic mode, respectively, with their wavefronts parallel to the corresponding slit arms. The interference of the two perpendicularly propagating waves forms the intensity pattern. Figure 3(b) shows the image pattern excited by the L-shaped slit, and it is obtained with the scattering image detection and with interferometry reconstruction, as depicted in the above section. For comparison, Fig. 3(c) gives the image pattern with the reference beam blocked and directly recorded with the CCD. It can be seen that the reconstructed pattern in Fig. 3(b) is of high fidelity with detrimental electrical noise filtered out. Then we will use the reconstructed pattern in the following context. On the whole, the pattern in Fig. 3(b) appears to be fringes inclining in the direction of about $45^{\circ}$ with respect to both slit arms. In the area close to the right-angle corner of the slit, the pattern has larger brightness. In the opposite upper-right corner, though the pattern appears to be darker, it is still of good visibility with the gray scale of intensity moderately adjusted, as the enlarged view shown in Fig. 3(d) for the labeled part of Fig. 3(b). This indicates damp of the waves with the distance from slit arm sources.

Fig. 3 (a) Diagram for interference of the photonic mode and the plasmonic mode wavefields produced by L-shaped slit. (b) Reconstructed intensity pattern. The white double-headed arrow shows the incident polarization. (c) Intensity pattern directly recorded by the CCD. (d) Enlarged view of the labeled part in (b) with enhanced brightness.

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4.1 The scattered photonic and plasmonic modes

Phonomenologically, the wavefield of the scattered photonic mode produced by the vertical slit can be written as $E_{\exp, v} (x)= A_{v} (x)exp(- i k_{1} x + i Φ_{v 0})$ . While for the scattered wave of plasmonic mode launched by the horizontal slit, though theoretically the wavefield contains two components as expressed in Eq. (5), the fringe pattern appears to be like that formed by a wave with a single wave vector. We then write its scattering wavefield as $E_{\exp, h} (y)= A_{h} (y)exp(- i k_{2} y + i Φ_{h 0})$ . Here in the expressions of $E_{\exp, v} (x)$ and $E_{\exp, h} (y)$ the subscript exp represents experimental quantity, and $A_{v} (x)$ and $A_{h} (y)$ are the amplitudes that damps with the propagation distance x and y, respectively. $Φ_{v 0}$ and $Φ_{h 0}$ are the initial phases of $E_{\exp, v} (x)$ and $E_{\exp, h} (y)$ , respectively. $k_{1}$ and $k_{2}$ are the components of the scattered wavevectors of $E_{\exp, v} (x)$ and $E_{\exp, h} (y)$ . For quantitative analysis, Fig. 4(a) gives a row and a column of the data matrix of the pattern labeled in dot lines AB, and CD in Fig. 3(b), respectively. The black curve in Fig. 4(a) represents the interference intensity of $E_{\exp, v} (x)$ and $E_{\exp, h} (y)$ that takes complex constant $E_{\exp, h} (y_{0})= A_{h} (y_{0})exp(- i k_{2} y_{0} + i Φ_{h 0})$ with $y = y_{0}$ in line AB. The intensity $I_{A B} (x)$ is:

\begin{array}{l} I_{A B} (x)= | E_{\exp, v} (x) + E_{\exp, h} (y_{0}) |^{2} \\ = | A_{v} (x)exp(- i k_{1} x + i Φ_{v 0}) + A_{h} (y_{0})exp(- i k_{2} y_{0} + i Φ_{h 0}) |^{2} \\ = | A_{v} (x) |^{2} + 2 A_{v} (x) A_{h} (y_{0}) \cos (k_{1} x + Φ_{10})+ | A_{h} (y_{0}) |^{2}, \end{array}

where

Φ_{10} = Φ_{v 0} + k_{2} y_{0} - Φ_{h 0}

.With the intensity given by the above Equation where

E_{\exp, h} (y_{0})

acts as a reference wave, the amplitude and phase of

E_{\exp, v} (x)

can be extracted, as will be demonstrated later. On the same reason, the intensity

I_{C D} (y)

in line CD with constant coordinate

x = x_{0}

may be written as:

\begin{array}{l} I_{C D} (y) = | A_{h} (y)exp(- i k_{2} y + i Φ_{h 0}) + A_{v} (x_{0})exp(- i k_{1} x_{0} + i Φ_{v 0}) |^{2} \\ = | A_{h} (y) |^{2} + 2 A_{h} (y) A_{v} (x_{0})cos(k_{2} y + Φ_{20})+ | A_{v} (x_{0}) |^{2}, \end{array}

where

Φ_{20} = Φ_{h 0} + k_{1} x_{0} - Φ_{v 0}

. Also,

A_{v} (x_{0})

acts as the reference wave for the interference expressed by the above Equation, and

A_{h} (y)

can be achieved.

Fig. 4 (a) Curves of the intensity data in dot lines AB and CD labeled in the pattern in Fig. 3(b), respectively. (b) Comparison of the curve $I_{C D}$ in (a) with intensity curve produced by another L-shaped slit with horizontal arm 100nm and vertical arm 300nm.

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4.2 The wavevectors and their difference of the scattered wavefields of the two modes

In Fig. 4(a), the maxima or the minima points of $I_{A B} (x)$ and $I_{C D} (y)$ separate gradually, and this indicates difference in their fringe pitches $p_{x}$ and $p_{y}$ . According to Eqs. (14) and (15), $k_{1} = 2 π / p_{x}$ and $k_{2} = 2 π / p_{y}$ .The fringe pitches are measured to be $p_{x} = 724.70 nm$ and $p_{y} = 709.16 nm$ . Correspondingly, the wavevector components are $k_{1} = 8.67 \times 10^{- 3} n m^{- 1}$ and $k_{2} = 8.86 \times 10^{- 3} n m^{- 1}$ , respectively. The larger value $k_{2}$ of the plasmonic mode is related to its inclusion of the wavevectors $k_{0}$ and $k_{S P}$ of the quasi-CW and the SPP. When the amplitudes of quasi-CW and SPP are comparable [58], which is the case roughly true for the case of the slit-width with 300nm in our experiment, the superposition of the two waves forms a main wave with wavevector $k_{pl, m} = (k_{0} + k_{S P})/ 2$ modulated by a beat envelope with wavevector of $k_{e n v} = (k_{S P} - k_{0})/ 2$ . The envelope has a pitch of about $2.857 \times 10^{4} nm$ , and it is considerably larger than the arm lengths of the slit in discussion. Then the influence of the envelope on the amplitude $A_{h} (y)$ of the plasmonic mode is negligible. As a result, the plasmonic mode is regarded as having the single wavevector $k_{pl, m}$ , and the scattered wavefield has a wavevector component of $k_{2}$ as has been used in Eqs. (14) and (15). However, when the amplitude of one wave in plasmonic mode is much larger than the other, say, that of SPP much larger than quasi-CW, which is the case of slit-width usually smaller than 150nm, the wavevector approaches that of the wave with the larger amplitude. To illustrate this point, we fabricate another L-shaped slit, labeled as sample of 100nm, with horizontal arm width 100nm and vertical arm width 300nm. The horizontal slit arm of this sample produces a plasmonc mode wavefield with the SPP wave dominant over the quasi-CW. The blue curve in Fig. 4(b) is the intensity data on the line perpendicular to the horizontal arm. For comparison, the red curve in Fig. 4(a) for the corresponding data of the original sample is replotted in Fig. 4(b). Obviously, the pitch of the blue curve is smaller than that of the red one. This shows that the plasmonic mode wave produced by the 100nm slit arm has larger scattered wavevector component $k_{2}$ approaching that of SPP.

The fringe pitches $p_{x}$ and $p_{y}$ are larger than the corresponding wavelengths $λ_{0}$ and $λ_{pl, m}$ , or equivalently, the wavevectors $k_{1}$ and $k_{2}$ of the scattered plasmonic and photonic modes are smaller than $k_{0}$ and $k_{pl, m} = (k_{0} + k_{S P})/ 2$ . With Eq. (13) for the scattering of the plasmonic mode waves, the shifts of the wavevectors induced by scattering can be explained. For simplicity, we take the scattering of the quasi-CW, i.e., the second term in the right-hand side in that Equation, as the example for detailed analysis in the following context. In modeling a rough surface, its profile is viewed as a large number of smooth microfacets with random sizes and orientations [63, 64]. The slope $ζ$ of a microfacet is referred to as normal deviation of the microfacet from mean normal of the rough surface. The root-mean-square of the slope $ζ_{r m s}$ is related to roughness $w$ and the correlation length $ξ$ of the surface by $ζ_{r m s} = \sqrt{2} w / ξ$ . We consider an ideal approximation for the film surface, in which an individual microfacet with slope $ζ_{r m s}$ is on the surface. The profile of the microfacet tilting in y-direction is $h (y')= ζ y'$ for $| y' | \leq L / 2$ with $L$ length of the facet and vanishes otherwise. The scattered wavefield of the quasi-CW propagating in y-direction can be obtained by simple calculations of the second right-hand term in Eq. (13):

\begin{array}{l} E_{s, C W} (k_{s y}, k_{s z}) = E_{0 C W} (^{y / λ_{0}) - 1 / 2} {δ (k_{s y} - k_{0}) \\ + (L / 2 π) \sin c [L (k_{s y} + k_{s z} ζ - k_{0})] - (L / 2 π) \sin c [L (k_{s y} - k_{0})]} . \end{array}

The second term in the right-hand side of the above Equation represents wavefield scattered out of the film interface, and it has a central wavevector component

k_{s y, C W} = k_{0} - k_{s z} ζ

, which is regarded as the wavevector component of the scattered quasi-CW:

k_{2, C W} = k_{s y, C W} = k_{0} - k_{s z} ζ .

The first and the third term in the right-hand side of the Eq. (16) represent the quasi-CW that continues to propagate along the surface after the scattering. In the same way, wavevector component of the scattered SPP is

k_{2, S P} = k_{S P} - k_{s z} ζ

. Then as mentioned above, the plasmonic mode as the superposition of quasi-CW and SPP wave has the wavevector

k_{pl, m}

, and the corresponding wavevector scattered into the air space is:

k_{2} = k_{pl, m} - k_{s z} ζ .

The analysis for the scattering of the photonic mode is the same as that of quasi-CW except for its propagation in x-direction. Accordingly, the wavevector component of scattered photonic mode is $k_{1} = k_{0} - k_{s z} ζ$ .The identical shifts $k_{s z} ζ$ give the difference $k_{2} - k_{1} = (k_{S P} - k_{0})/2$ . This also explains the fringe pitch difference in the two directions of the scattered pattern. With the wavevector difference, the pitch difference can be derived by $Δ p^{(t)} = p_{x}^{(t)} - p_{x}^{(t)} \approx (λ_{S P} - λ_{0})/ 2 = 16.50 nm$ with the parenthesized subscript t representing theoretical value. Here $λ_{S P} = 599.8 nm$ is the wavelength of SPP propagating in the Au film. The derived $Δ p^{(t)}$ is very close to the experimental value $Δ p = p_{x}^{} - p_{y}^{} = 724.70 nm - 709.16 nm = 15.54 nm$ . Besides, the wavevector difference $k_{2} - k_{1}$ will result an additional phase difference between the two modes during their propagation. The blue diamonds in Fig. 5(a) show the phase delay $Δ Φ_{d}$ of the photonic mode with respect to the plasmonic mode versus the propagation distance, and the line is the linear fit.

Fig. 5 (a) Phase delay of the photonic mode relative to the plasmonic mode versus the propagation distance. (b) Amplitude curves of the photonic mode (black) and plasmonic mode (red). (c) Phase map reconstructed from the recorded intensity.

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4.3 The solutions of amplitudes and the initial phases

Next, we present the method to extract the amplitudes $A_{v} (x)$ and $A_{h} (y)$ of the photonic and the plasmonic modes from the data $I_{A B} (x)$ and $I_{C D} (y)$ of the curves in Fig. 4(a), respectively. Since $I_{A B} (x)$ is the interference of $A_{v} (x)$ with the constant amplitude $A_{h} (y_{0})$ , at point $P_{1 m}$ of the minimum intensity shown in Fig. 4(a), the fringe contrast reaches the largest value and the amplitudes of the two interfering waves are almost equal. Then we have $A_{h} (y_{0}) = A_{v} (x_{1 m})$ with $x_{1 m}$ the coordinate of $P_{1 m}$ . Based on Eq. (14), the average $I_{f}$ on the intensity data of a single complete fringe centered at $P_{1 m}$ is taken as the intensity of this point:

I_{f, v} = {[A_{h} (y_{0})]}^{2} + {[A_{v} (x_{1 m})]}^{2} = 2 {[A_{h} (y_{0})]}^{2} .

With

A_{h} (y_{0})

obtained from the above Equation, the amplitude distribution of

A_{v} (x)

can be obtained at the points with intensity minima:

A_{v} (x_{j})= A_{h} (y_{0}) \pm \sqrt{I_{f, v j}},

where

I_{f, v j}

is the average on the intensity data of a complete fringe centered at the jth intensity minima with coordinate

x_{j}

. The plus sign is for

x_{j} < x_{1 m}

while the minus for

x_{j} > x_{1 m}

. The obtained curve of

A_{v} (x)

is the amplitude distribution of the photonic mode and it is shown in black diamond in Fig. 5(b). The black curve is the fit of linear function of

x^{- 1}

. The amplitude value at the first minima is used as the initial amplitude of the vertical slit as the secondary source, and it is obtained to be

A_{v 0} = 51.00

. On the same reason,

A_{v} (x_{0})

,

A_{h} (y_{1 m})

and

A_{h} (y_{i})

can be obtained from the data

I_{C D} (y)

. The curve of

A_{h} (y)

is the amplitude distribution of the plasmonic mode and it is also plotted in red diamond in Fig. 5(b). The red curve is also the fit of linear function of

y^{- 1}

. The initial amplitude value at position of the first minima is

A_{h 0} = 63.00

in arbitrary unit. The experimentally solved ratio of the initial amplitudes for the photonic and the plasmonic modes is

η = A_{v 0} : A_{h 0} = 0.81

. Owing to the employment of the L-shaped slit with the advantage of each arm generating one pure mode, the ratio of amplitudes is obtained under completely identical conditions.

The initial phases $Φ_{v 0}$ and $Φ_{h 0}$ of the photonic and the plasmonic modes are also necessary quantities for solutions of the wavefields. They can be directly obtained from the interference pattern recorded by the experimental Mach-Zehnder interferometer setup. In Fig. 5(c), the reconstructed phase map is shown. We read the phase values of $Φ_{v 0}$ and $Φ_{h 0}$ are 1.96 and −1.98 respectively. Though the phase values $Φ_{v 0}$ and $Φ_{h 0}$ are relative to reference beam, the phase difference $Δ Φ_{0} = Φ_{v 0} - Φ_{h 0}$ which determines the patterns formed by interference of the photonic and the plasmonic modes is not affected. We then finally obtain the phase difference $Δ Φ_{0} = - 2.34$ .

Till now the key factors of the amplitude ratio $η$ , the wavevector components $k_{1}$ and $k_{2}$ , and the initial phase difference $Δ Φ_{0}$ have been obtained based on the experimental results. In more general case of arbitrary polarization states of incidence and slit shapes, the illumination can be decomposed into the two components of TE and TM incidence, and the corresponding scattered wavefields $E_{\exp, v} (x)$ and $E_{\exp, h} (y)$ for the photonic and the plasmonic modes can also be constructed with the obtained factors.

It should be noted that the scattered interference pattern in Fig. 3(b) formed in the CCD may be different from the interference pattern existing in the film surface. In principle, amplitude of the wavefield scattered from a point of the film surface is proportional to the SPP wave propagating through the point in the direction of propagation [37]. Since the scattering brings about shift of wavevector component in the propagating direction, it can be deduced that the interference fringe pitch is greater than that in the film. Similar to case of leakage radiation [35, 37, 38], the coherent superposition of scattered wavefields to form the fringes should have the same polarization. Hence the combination of the scattering from the rough surface, the propagation directions and the polarizations of the plasmonic and photonic modes determines the scattered interference pattern.

5. Application of the solved quantities to a ring-slit with Huygens-Fresnel principle

We now consider the scattering wavefield of a ring slit as the verification of the above results. First, for a slit element with its normal in the direction with angle $θ$ to x-direction, as shown in Fig. 6(a), the incident wave $E_{0}$ polarizing in y-direction can be decomposed into $E_{||}$ and $E_{⊥}$ , parallel and perpendicular to the slit element respectively. Hence the TE polarized amplitude of incidence is $E_{||} = E_{0} | \cos θ |$ and the TM polarized amplitude $E_{⊥} = E_{0} | \sin θ |$ , respectively. The initial amplitudes of the scattered wavelets of the photonic mode and plasmonic modes are $d A_{e, v 0} = A_{v 0} E_{0} | \cos θ | d l$ and $d A_{e, h 0} = A_{h 0} E_{0} | \sin θ | d l$ , respectively. Here $A_{v 0}$ and $A_{h 0}$ are experimentally obtained in the above Section. The Huygens-Fresnel principle for metal slit regime [23] can be introduced to describe the scattered wavefield:

\begin{array}{l} d E_{s} = [exp(i π / 4) \cos α / \sqrt{ρ}] [A_{v 0} | \cos θ | exp(- i k_{1} ρ + i Φ_{v 0})+ A_{h 0} | \sin θ | exp(- i k_{2} ρ + i Φ_{h 0})] E_{0} d l \\ = A_{h 0} exp(i Φ_{h 0} - i π / 4)(\cos α / \sqrt{ρ}) [η | \cos θ | exp(- i k_{1} ρ + i Δ Φ_{0})+ | \sin θ | exp(- i k_{2} ρ)] E_{0} d l, \end{array}

where

ρ

is the distance from the slit element to the field point,

d l

is the length of the slit element, and

\cos α

is the inclination factor. For a slit of arbitrary shape, the scattered wavefield on the film surface is written as:

\begin{array}{l} E_{s} (x, y) = A_{h 0} E_{0} exp(i Φ_{h 0} - i π / 4) \\ \int_{l} (\cos α / \sqrt{ρ}) [η | \cos θ | exp(- i k_{1} ρ + i Δ Φ_{0})+ | \sin θ | exp(- i k_{2} ρ)] d l, \end{array}

where the integral is along the slit

l

. For a ring-slit with the coordinates schematically shown in Fig. 6 (b), the above Equation for a field point inside the ring-slit is simplified as:

\begin{array}{l} E_{s} (r, φ) = A_{h 0} E_{0} exp(i Φ_{h 0} - i π / 4) \int_{0}^{2 π} (\cos α / \sqrt{ρ}) \\ [η | \cos θ | exp(- i k_{1} ρ + i Δ Φ_{0})+ | \sin θ | exp(- i k_{2} ρ)] R d φ_{0}, \end{array}

where

R

is the radius of the ring-slit. The above expression of the Huygens-Fresnel principle also reflects the interference of the wavelets from different slit elements. The scattered interference pattern is comprehensively determined by factors of the decomposed amplitudes, the initial phase difference

Δ Φ_{0}

and the phase delay

Δ Φ_{d}

originated from the wavevector difference. In the literature, the Huygens-Fresnel principle only involves the interference of the SPP term propagating along the metal film, and the superposition of the two wavefields and the scattering effect as included in Eqs. (22) or (23) have not been considered. The interference model constructed here may bring about interesting effect in the scattering pattern formation and manipulations.

Fig. 6 (a) Diagram of a slit element. The angle of its normal to x-direction is $θ$ . The incident wave $E_{0}$ in y-direction is decomposed into $E_{⊥}$ and $E_{||}$ . (b) Diagram of the ring-slit. (c) The SEM image of the ring-slit. Intensity patterns of calculation (d) and experiment (e). Phase maps of calculation (f) and experiment (g), corresponding to the enlarged view of squared part in (d) and (e), respectively. The double-headed arrows in the figures represent polarizations.

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We numerically calculate the intensity distribution inside the ring-slit and perform the experimental demonstrations. In the experiment, a ring-slit with inner diameter 6μm, and the slit width 300 nm is milled on the Au film of thickness 200nm. The SEM image of the slit is shown in Fig. 6(c). Here the same slit width and film thickness as those of the L-shaped slit insure the data in the above experiment usable for the ring-slit sample. In the calculations, Eq. (23) is used with the quantities therein set to the values of the experimental solutions. In Fig. 6(d), the calculated intensity pattern inside the ring is given. In the calculations, we have also made trials of discarding both the inclination factor $\cos α$ and the damp factor $\sqrt{ρ}$ to look at their influence, and the results show insignificant changes, especially in the central part of the intensity pattern. Figure 6(e) shows the experimentally reconstructed image pattern obtained with the Mach-Zenhder setup, and we find that the calculated pattern is in very good accordance with the experimental result, and even the details of the two patterns are consistent. In Figs. 6(f) and 6(g), the calculated and experimental phase maps are shown, respectively, and the consistency of calculated pattern with the experimental result is satisfying. We see four vortices in the central area inside the ring. These intensity and phase patterns are formed as a simultaneous effect of the amplitudes and phases of the two terms in the right-hand side of Eq. (23). The slight differences in the experimental and the calculated intensity distributions and phase maps mainly originate, to our practice, from the experimental factors such as aberrations of the objective lens, imperfect recording and reconstruction of the image in the Mach-Zenhder setup, and even errors in sample fabrications. In a former work, we have found four vortex structure in the phase map [65], and proposed an empirical expression for ring-slit to explain the formation of the vortices, in which the scattering effect and the wavevector component difference as well as the corresponding phase delay has not been included. Obviously, the experimental solution in this paper for the scattered imaging of the interference pattern of the two mode wavefields is more rigorous theoretically, clearer in physics and of more general applicability.

6. Summary and Conclusion

We have constructed the model for scattered imaging of the wavefield patterns formed by metal subwavelength slit structures based on the experimental solution. Fundamentally, the issue involves the interference of the photonic mode and the plasmonic mode waves and their scattering from the rough metal film. With the L-shaped slit to achieve the TE and TM incidences under a linearly polarized illumination, the waves of the two modes are launched simultaneously and independently. We record the scattered image of their interference pattern in the Mach-Zenhder interferometer setup. From the fringe data of the pattern, the amplitudes of the two modes are extracted, with the initial ratio $η = A_{v 0} : A_{h 0}$ obtained. The different wavevectors $k_{1}$ and $k_{2}$ of the two scattered wavefields and the corresponding phase delay $Δ Φ_{d}$ are demonstrated. The initial phase difference $Δ Φ_{0} = Φ_{v 0} - Φ_{h 0}$ of the two modes is obtained with the reconstructed phase map of the pattern. The comprehensive influences of these factors on the interference are considered for the first time. Applying the scattering theory under Kirchhoff’s approximation to metal slit regime, we give reasonable explanations for the characteristics of the fringe data and the scattered image pattern. The experimental solution of these quantities allows us to model the slit element as the secondary source by decomposing the incident field into components parallel and perpendicular to the element. Then Huygens-Fresnel principle can be used to slits of arbitrary shapes for the calculations and the manipulations of their scattered wavefields. We fabricate a ring-slit as the sample for calculations and practical measurements of the scattered field inside the ring. The consistency of the calculated and the measured intensity patterns and phase maps demonstrates the accuracy and applicability of the model.

Compared with the well-studied techniques such as leakage radiation [32–38], the measurement in this work needs a detector of higher quality CCD with lower noise due to the weak scattering of the slightly rough film surface or laser of relative higher power for the illumination. The resolution limit is also lower due to the use of the dry objective lens. The N.A. value of 0.9 enables the objective lens to receive light within a cone angle 128° and to have the resolution of 0.61λ/N.A. = 428.9nm with incident laser wavelength 632.8nm. However, measurement of the scattered imaging method is simple and direct in acquiring the observable pattern in the air space formed by excitations and propagations of the plasmonic and the photonic modes. The setup with simple arrangement of the optical elements allows the reference beam to be introduced easily and the phase map to be extracted, which gives the initial phases and makes the comprehensive solution possible. We believe that the solution of the scattered photonic and plasmonic modes would be an advance in facilitating the design of metal slit structure for manipulating the nanoscale wavefields.

Acknowledgments

National Natural Science Foundation of China (Grant No. 10974122), Science and Technology Development Program of Shandong Province, China (Grant No. 2009GG 10001005) and National Natural Science Foundation of China (Grant No. 11404179) are gratefully acknowledged.

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Experimental solution for scattered imaging of the interference of plasmonic and photonic mode waves launched by metal nano-slits

Abstract

1. Introduction

2. Experimental setup

3. Theoretical basis for scattering of waves generated by metal nano-slit from the film surface

4. Experimental solution for scattered imaging of the wavefields with L-shaped metal slit

4.1 The scattered photonic and plasmonic modes

4.2 The wavevectors and their difference of the scattered wavefields of the two modes

4.3 The solutions of amplitudes and the initial phases

5. Application of the solved quantities to a ring-slit with Huygens-Fresnel principle

6. Summary and Conclusion

Acknowledgments

References and links

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Figures (6)

Equations (23)

Optics Express