Azimuthal polarization

Electric field with vibrated direction along azimuthal direction is called azimuthal polarization. Unlike radial polarization, the z-components of one arbitrary point and its symmetrical point at pupil are completely out of phase.

This phenomenon results in a completely destructive interference. The total intensity will be a doughnut-shaped distribution due to there is no longitudinal intensity as shown in the following figure.

Fig. 3-9 Foci of azimuthal polarization with different N.A

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Fig. 3-10Individual component of foci of azimuthal polarization with (a) N.A = 0.2 (b) 1.0

Fig. 3-10 shows the individual intensity of azimuthal polarization.

Compared to the transversal intensity, the longitudinal intensity is insignificant even in the high N.A system, resulting in a doughnut-shaped focus.

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3.3 Summary

We reviewed different polarizations in this chapter. All of that has its individual characteristic. When the incident beam is linear polarization, there is a “squeezing” effect in the high N.A system. This effect leads to asymmetric focus and is not suitable for some applications. Such effect is not found in circular polarization due to the neutralization of transversal components.

However, the focal spot is still too large to modern technology. Fortunately, one of inhomogeneous polarization can provide a small focus by a strongly constructive interference in longitudinal component. The following chapter will use this kind of polarization to a specific application.

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Chapter 4

Surface Plasmon Resonance Sensor

Due to pursuing tiny area and super resolution in modern technology, scientists and engineers try to overcome the natural barrier –“Diffraction limit”.

Typically, the diffraction limit can be given by

in transversal direction and half of the coherence length in longitudinal direction [15].By applying the immersion system and shorter wavelength, the transversal resolution can be increased due to larger N.A. The transversal resolution can be further increased by applying inhomogeneous polarized beam. However, the longitudinal resolution of common system is usually several micrometers. In order to detect smaller area in longitudinal direction, in particular, Surface Plasmon Resonance (SPR) is used to analyze the information which is close to the surface region (in dozens of nanometer). In this chapter, we will introduce the principle of SPR and its applications in recent years. Then we will propose two elements to further enhance the performance of SPR sensor.

4.1 Introduction

In 1902, R. W. Wood firstly observed an uncommon phenomenon which did not obey the diffraction theory of grating when a polarized light upright to the groove of metal grating [16]. He attempted to interpret this interesting phenomenon by oscillation with specific polarization of light and metal grating structure. Until 1941, Fano proposed a new opinion that a new electromagnetic wave along the surface when the polarization of light with electric field upright

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to the groove of metal grating to define this weird phenomenon [17]. Afterward, this weird electromagnetic wave in the interface was so-called Surface Plasmon Resonance (SPR). After several decades, R. H. Ritchie and R. A. Ferrell et al proposed the theoretic model of SPR sequentially [18, 19]. More attention invested in the study of SPR in that it elicited the interests of scientist.

The SPR are collective oscillations of free electrons that can propagate between the metal and dielectric surface. It is a kind of electromagnetic wave which is confined with the sub-wavelength region of surface. As above said, we can find the condition of existence of SPR from Maxwell’s equations due to SPR are electromagnetic wave. In order to characterize the properties of SPR, we consider a metal-insulator interface and look for a homogeneous solution of Maxwell’s equations with s- and p-polarization at the surface as shown in Fig.

4-1 [20].

Fig. 4-1Schematic diagram of incident (a) s- (b) p-polarization waves For s-polarization wave, the field at is

⃑⃑ ( ) [ ] (4.1a), ⃑ ( ) [ ] (4.1b) and the field at is

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⃑⃑ ( ) [ ] (4.1c), ⃑ ( ) [ ] (4.1d) respectively. According to the boundary conditions of electromagnetic wave, the continuity of tangential component Ey and Hx lead to the condition

(4.2a),

(4.2b)

for nonmagnetic materials. The dispersion relation tell us

( ) (4.3).

Comparing with Eq. (4.2a) and Eq. (4.2b), the only solution of this equation is which is contradiction. Hence, there is no surface wave at the interface for s-polarization. In other words, the SPR cannot be excited by the s-polarized incident wave. respectively. Again, the continuity of tangential field results in

(4.5a),

(4.5b).

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Combining Eq. (4.5a), Eq. (4.5b) and Eq. (4.3), we arrive at the central result of this section, the dispersion relation of SPR at the interface

√ (4.6).

This result shows that there is opportunity to excite the SPR when the incident wave is p-polarization. In order to excite the SPR, the permittivity of both media must satisfy the following constraint

(4.7a),

(4.7b)

which means that not only either permittivity of two media must be negative but also the absolute value exceeding the other. Most of metals, in particular, noble metals have large negative real part of dielectric constant. Therefore, the SPR can exist at the interface between noble metal and dielectric when the incident wave is p-polarization.

However, the SPR cannot be excited by common methods due to the wavevector of SPR kSP is always larger than incident wavevector in dielectric k1x

as shown in Fig. 4-2.

Fig. 4-2 Dispersion relations of SPR and dielectric

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The mechanism of Surface Plasmon Polaritons (SPPs) excitation was generally induced by a prism-based coupler, firstly proposed by Kretschmann and Otto in 1968[21, 22].Due to the difficulty of fabrication, Kretschmann’s configuration was more acceptable than the others. Figure 4-3 shows Kretschmann’s configuration and its dispersion relations.

Fig. 4-3 (a) Kretschmann’s configuration and (b) its dispersion relations The phase matching condition in this system is

√ (4.8)

at the intersection point in Fig. 4-3b. By applying angular scanning, we can observe a diagram of the reflectance vs. incident angle as shown in Fig. 4-4.The peak of this curve refers to the total-internal reflection and the dip relates to the incident light to be coupled to the SPR.Although Kretschmann’s configuration can provide us polychromatic information, it remains a problem that the measurement needs angle scanning which may lead to a time consuming issue.

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Fig. 4-4 Angular reflectance of Kretschmann’s configuration

In order to solve this issue, H. Kano then replaced the prism by a collinear objective lens to enable a universal wavevector without the need for angular scanning after several decades[23].Due to the intrinsic characterization of SPR, the incident polarization of Kano’s configuration must be radial polarization and the N.A of objective lens must be as higher as possible. The phase matching condition and the configuration are shown as follower.

(4.9)

Fig. 4-5 (a) Kano’s configuration and (b) reflected beam

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This configuration combining the advantages of radial polarization and SPR, not only provides a super transversal resolution but also in longitudinal.

The necessity of relying wavelength-dependent optical elements to generate radially polarized light has been widely used on RP-SPR sensor [24-32].

However, there are two issues should be considered. Firstly, to our knowledge, there is no synthesized method has been published to generate a polychromatic radial polarization. Besides, the numerical aperture of objective lens limits the maximum incident angle so that we cannot excite surface plasma which requires larger wavenumber.

In this chapter, we propose two elements to improve the performance of objective-based SPR sensor, integrating a polychromatic objective-based SPR sensor with larger sensing range.