Summary

The general solution with homogeneous and inhomogeneous polarization has been discussed in aforementioned statement, respectively. Through the aforesaid discussion, we can find out the homogeneous polarization is not suitable in high numerical aperture system in terms of bigger focal spot size or changed shape of focused fields which results from vector effect. For inhomogeneous polarization, due to the depolarization effect in high numerical aperture, radial polarization could be focused tighter beyond the diffract limits. It can further improve the resolution and be applied to particle acceleration [4,5], particle-trapping [6], lithography [7], and material processing [8]. On the other hand, because of the specific phase distribution of azimuthal polarization, it can generate a sharper focal spot which smaller than the focal spot of linear polarization when it propagates through a vortex 0-2π phase plate [9]. As a consequence, inhomogeneous polarization becomes more important in many application of optics.

23

From the view of wave optics, the defocusing aberration produces finite depth of focus (DoF) because it introduces an additional quadratic phase in the pupil function of the imaging system, resulting in a spatial low-pass filter effect. We will demonstrate in following.

The optical transfer function (OTF) of a single-lens imaging system can be written as a autocorrelation of the pupil function 𝑃(𝑥, 𝑦).

𝐻(𝑓_𝑥, 𝑓_𝑦) =∬ 𝑃(𝑥 +𝜆𝑧_𝑖𝑓_𝑥 aberration is considering, the generalized pupil function will leads to the form

𝑃(𝑥, 𝑦) = |𝑃(𝑥, 𝑦)|𝑒𝑥𝑝,𝑖𝑘𝑊(𝑥, 𝑦)- (3.2) Where k = 2π/λ, and W(x, y) is the aberration function of defocusing, it has the quadratic form

𝑊(𝑥, 𝑦) = 𝑊_𝑚(𝑥²+𝑦²)

𝑏² (3.3) Where b is the radius of the aperture, and the number Wm is a convenient indication of

24

the severity of the defocusing error, is given by 𝑊_𝑚= 𝑏²

2 (1 𝑧_𝑖 + 1

𝑧_𝑜−1

𝑓) (3.4) Where zo is the distance from the object to the lens and f is the focal length of the lens.

When the imaging condition is not fulfilled (W_m is not zero), the OTF distribution is narrower due to the quadratic phase factor arises. For the 1-D case Eq. (3.1) become

𝐻(𝑓_𝑥, 𝑊_𝑚) =∫ 𝑃 (𝑥 +𝜆𝑧_𝑖𝑓_𝑥 of the pupil. Then the OTF can be approximated by

𝐻(𝑓_𝑥) ≈∫ 𝑒𝑥𝑝 [𝑖𝑘𝑊_𝑚2𝜆𝑧_𝑖𝑓_𝑥𝑥 The expression of Eq. (3.7) demonstrates that defocusing aberration results a low-pass filtering effect.

In order to reduce the effect of defocusing, various methods were investigated

25

for overcome this limitation. In 1952, E. H. Linfoot and E. Wolf [10] computed the complete out-of-focus airy pattern for annular apertures and incidentally found out the DoF was increased, then 1960, W. T. Welford [11] investigate the phenomenon in depth; this is the earliest and simplest method to reduce the defocusing aberration.

Due to the DoF is inverse proportion to pupil area, hence, through addition of binary blocking mask in aperture plane, the DoF can be extend. The influence of additional annular ring on total intensity field can be described mathematically

𝐼(𝑧) = 𝑎⁴

Where a is radius of aperture, f is the distance from the exit pupil to the focus, and the ε is obstruction ratio of radius as shown in Fig. 3-1.

Fig. 3-1 The configuration for annular aperture.

We can find out the total intensity field which does not change with different position when ε is tending to unity. It means that DoF will extend to infinite, but unfortunately when ε is larger, the side-lobe intensity of light fields and the obstructive area will arise. Therefore, the system resolution and amount of light reaching the imaging plane is lower.

Another approach uses the refractive elements (Axicon) in the aperture of the imaging

26

system [12,13]. An extended DoF is obtained due to the overlap region of beams being diverted by refractive elements. As illustrated in in Fig. 3-2.

Fig. 3-2 The schematically concept of extending depth of focus by refractive optics.

Recently, because the computer develops vary rapidly; it brings another technique which is called wave-front coding elements to extend depth of focus [14].

This method is not an all-optical approach due to that requires digital post-processing, the idea involves that inserting a basically aberration which is much stronger than the defocusing aberration such that by digital post-processing a sharp image can be reconstructed.

One of the popular elements is the cubic phase element, in normalized coordinates; the pupil function with cubic phase element is given by

𝑃(𝑥) = ,1

⁄√2× 𝑒𝑥𝑝(𝑖𝛼𝑥³) 𝑓𝑜𝑟 |𝑥| ≤ 1

0 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 (3.9) Where α is the coefficient which controls the phase deviation. Then the OTF related to the pupil function can be approximated as

𝐻(𝑓_𝑥, 𝑊_𝑚) ≈ ( 𝜋 12|𝛼𝑓_𝑥|)

1⁄2

exp (𝑖𝛼𝑓_𝑥³

4 ) exp (𝑖𝑘²𝑊_𝑚²𝑓_𝑥

3𝛼 ) , 𝑢 ≠ 0 (3.10)

27

When α is a larger value, the third part of the approximated of the OTF can be ignored, therefore, the OTF will be independent of defocusing. Although this type of solution can increased DoF plenty but rather one that requires digital post-processing, thus it does not fit to ophthalmic. In addition to cubic phase mask, there are still some another mask, i.e. free-form phase mask and exponential phase mask. [15,16]

Then, a novel polarization coding technique will be introduced in following.

3.2 Combination of inhomogeneous beams

After realizing these methods which extending DoF utilize amplitude manipulation or phase manipulation. In this section, we describe a novel inhomogeneous polarization coding aperture to achieve extending DoF. In 2006, Wanli Chi et al purposed this novel technique by combining two independent orthogonal linear polarized lights [17]. As shown in Fig. 3-3, it has been demonstrated the effect which can extend DoF before. But in chapter 2, we already discuss the properties with homogeneous and inhomogeneous light; the inhomogeneous polarized light is superior to linear polarized light in terms of focus spot size or influence of polarized direction in higher numerical aperture. Therefore, maybe we can not only extend DoF but also increase the resolution by combing radial and azimuthal polarized lights.

Fig. 3-3 The polarization coded aperture.

28

In the combination with radially and azimuthally polarized light, there are two kinds of synthesis, as shown in Fig. 3-4. Manipulating the proportion between radially and azimuthally polarized light, we can find out the best ratio to obtain maximum of extending DoF. In Fig. 3-5 (a) to Fig. 3-5 (j), the radially polarized light is in outer ring region and the central circular region is azimuthally polarized light. The opposite arrangement of two polarized light is showed in Fig. 3-6 (a) to Fig. 3-6 (j). The numerical number of system in here is 1.45.

Fig. 3-4 Two types of radial and azimuthal combination.

According to figures 3-4, we can find out the shape of light field is determined by the polarization at outer ring region. The syntheses of two orthogonal polarizations actually have the capability to extend depth of focus. The best combinative ratio of radially polarized light to azimuthally polarized light is 3:7 when radially polarized light is at the outer ring region. On the other hand, when azimuthally polarized light is at the outer ring region, the ratio of radially polarized light to azimuthally polarized light is 7:3; the results can obtain from Fig. 3-5(g) and Fig. 3-6 (g). In Fig. 3-7, we discuss the full width at half maximum (FWHM) in z-axis of the two kind of best combination, Fig. 3-7 (a) and (b) shows the slice of focus fields on z-axis, and FWHM is 1.48 times and 1.7 times comparing combination and non-combination, respectively.

However, although the inhomogeneous polarization coded aperture can extend depth of focus, but one of the aperture cannot obtain more better resolution than linear polarization coded aperture and the extend efficiency of these kinds aperture are not

29

Fig. 3-5 (a) to (j) peak intensity across different z-axis position when radially polarized light is in outer ring region.

30

Fig. 3-6 (a) to (j) peak intensity across different z-axis position when azimuthally polarized light is in outer ring region.

31

Fig. 3-7 The FWHM of (a) radially polarized light and (b) azimuthally polarized light focal fields.

Fig. 3-8 The intensity profile comparison between linear polarization coded aperture and in homogeneous coded aperture on x-y dimension.

good enough, as shown in Fig. 3-8.Therefore, we consider the depolarization effect of radially polarized light again, to obtain the good extended depth of focus due to depolarization effect in higher-order radially polarized light.

32

3.3 Higher-order radially polarized beam

In chapter 2, we already discussed the properties of fundamental mode of radially polarized beam in terms of its strongly longitudinal component which results by depolarization effect forms a tight focal spot in high numerical aperture system.

For this reason, recently, the higher-order radially polarized beam have been attracted much attention due to it can effectively reduce the focal spot size by destructive interference on horizontal components[18], it means that the higher-order radially polarized beam can produce a smaller focal spot size than fundamental mode in high numerical aperture system. Therefore, in this section, we utilized the higher-order radially polarized beam to achieve not only super-resolution but also extending DoF.

Since that very high longitudinal component has been achieved nearly flat top axial distribution in focal volume.

The order number of radially polarized beam depends on how many ring it has.

Single-ring-shaped beam is called fundamental mode radially polarized beam (R-TEM01), and so on, double-ring-shaped beam is the first order of higher-order radially polarized beam which called R-TEM₁₁, as shown in Fig. 3-9. Here, we compare the focus fields on z-axis between fundamental mode and two higher-order modes.

Fig. 3-9 The fundamental mode (R-TEM01) and two higher-order modes (R-TEM₁₁ and R-TEM₂₁) of radially polarized beam.

33

In Fig. 3-10, it shows the ability of extended DoF with higher-order radially polarized beam, compare to fundamental modes (R-TEM01), the FWHM is 1.7 times and 2.11 times in R-TEM₁₁ and R-TEM₂₁, respectively. Of cause, the better results can be expected when the more higher-order to be chosen.

Fig. 3-10 The FWHM of focal fields of different order of radially polarized beam.

Although the higher-order radially polarized beam has many advantages, but it has a vital constraint that the synthesis of higher-order beams is difficult to achieve. Up to now, the strategy to generates higher-order radially polarized beam is nothing more than through spatial light modulator (SLM) [19] or particular laser cavity design [20,21], but these synthesis methods are sensitivity to environment perturbation or precise manufacture. Therefore, the practical application of higher-order still has a barrier to prevent the development.

3.4 Summary

In this chapter, we introduce two recipes of polarization coded to extend DoF;

one is the combination between radial polarization and azimuthal polarization, and the other is the higher-order radially polarized beam. Employing the advantages of

34

cylindrical vector beam, we can achieve high spatial resolution as well as extended DoF in high numerical aperture system.

35

Chapter 4

Application 2 – Chromatic Surface Plasmon Resonance

Recently, surface plasmon resonance (SPR) have been widely used to analyze characteristics of material, providing quantitative and qualitative analyses due to its unique character. In this chapter, the experimental setup for creating the chromatic SPR and the optimized structure for objective-based configuration will be introduced in the following.

4.1 History and Principle

The discovery of SPR begins at 1902, R. W. Wood observed a weird phenomenon that didn’t obey the diffraction theorem of grating when the polarization of light with electric field upright to the groove of metal grating [22]. He attempted to explain the phenomenon by oscillation with specific polarization of light and metal grating structure. Until 1941, Fano purposed a new opinion about the interesting phenomenon, he proposed a new electromagnetic wave along the surface when the polarization of light with electric field upright to the groove of metal grating [23]. The electromagnetic wave is so-called SPR afterward. Then in 1950, R. H. Ritchie and R.

A. Ferrell et al purposed the theoretic model of SPR sequentially [24, 25], From then on, SPR elicited the interests of scientist, more attention invested in this study.

The SPR are collective oscillations of free electrons that can propagate between the metal and dielectric surface. It is a kind of electromagnetic wave and confined within the sub-wavelength of metal surface. Exactly as above said, the SPR are electromagnetic wave, therefore, we can find the condition of existence of SPR from Maxwell’s equation. In order to know the condition, we consider an interface between two media and look for a homogeneous solution of Maxwell’s equations at the surface.

36

The Maxwell’s equation at the surface between two media can be written as

𝛻 × 𝐻⃑⃑ = 𝜀𝑑𝐸⃑

𝑑𝑡 (4.1a)

𝛻 × 𝐸⃑ = −𝑢𝑑𝐻⃑⃑

𝑑𝑡 (4.1b)

𝛻 ∙ 𝜀𝐸⃑ = 0 (4.1c)

𝛻 ∙ 𝐻⃑⃑ = 0 (4.1d) Next considering s-polarization and p-polarization incident waves propagate between two media as shown in Fig. 4-1, for s-polarization incident wave, the wave function is

Fig. 4-1 (a) s-polarization (b) p-polarization waves propagate between two media.

z>0

𝐻⃑⃑ ₁ = (𝐻_𝑥1, 0, 𝐻_𝑧1) 𝑒𝑥𝑝(𝑘_𝑥1𝑥 + 𝑘_𝑧1𝑧 − 𝜔𝑡)𝑖 (4.2a) 𝐸⃑ ₁ = (0, 𝐸_𝑦1, 0) 𝑒𝑥𝑝(𝑘_𝑥1𝑥 + 𝑘_𝑧1𝑧 − 𝜔𝑡)𝑖 (4.2b) z<0

𝐻⃑⃑ ₂ = (𝐻_𝑥2, 0, 𝐻_𝑧2) 𝑒𝑥𝑝(𝑘_𝑥2𝑥 − 𝑘_𝑧2𝑧 − 𝜔𝑡)𝑖 (4.2c) 𝐸⃑ ₂ = (0, 𝐸_𝑦2, 0) 𝑒𝑥𝑝(𝑘_𝑥2𝑥 − 𝑘_𝑧2𝑧 − 𝜔𝑡)𝑖 (4.2d) From Eq. (4.2a) to Eq. (4.2d), these equations must satisfy boundary condition, then electric fields and magnetic fields at the surface are of the form

(a) (b)

37

𝐸_𝑦1 = 𝐸_𝑦2 (4.3a) 𝑢₁𝐻_𝑧1 = 𝑢₂𝐻_𝑧2 (4.3b) 𝐻_𝑥1 = 𝐻_𝑥2 (4.3c) 𝑘_𝑥1 = 𝑘_𝑥2 (4.3d) Substituting Eq. (4.2a) to Eq. (4.2d) into Eq. (4.1b) leads to

𝑘_𝑧1𝐸_𝑦1 = −𝑢₁𝜔𝐻_𝑥1 (4.4a) electron accumulation. Hence, the SPR for s-polarization don’t exist at the surface, in other words, the s-polarization incident wave cannot excite the SPR

For p-polarization incident wave, the wave function is z>0

𝐻⃑⃑ ₁ = (0, 𝐻_𝑦1, 0) 𝑒𝑥𝑝(𝑘_𝑥1𝑥 + 𝑘_𝑧1𝑧 − 𝜔𝑡)𝑖 (4.7a) 𝐸⃑ ₁ = (𝐸_𝑥1, 0, 𝐸_𝑧1) 𝑒𝑥𝑝(𝑘_𝑥1𝑥 + 𝑘_𝑧1𝑧 − 𝜔𝑡)𝑖 (4.7b)

38

z<0

𝐻⃑⃑ ₂ = (0, 𝐻_𝑦2, 0) 𝑒𝑥𝑝(𝑘_𝑥2𝑥 − 𝑘_𝑧2𝑧 − 𝜔𝑡)𝑖 (4.7c) 𝐸⃑ ₂ = (𝐸_𝑥2, 0, 𝐸_𝑧2) 𝑒𝑥𝑝(𝑘_𝑥2𝑥 − 𝑘_𝑧2𝑧 − 𝜔𝑡)𝑖 (4.7d) From Eq. (4.7a) to Eq. (4.7d), these equations must satisfy boundary condition, then electric fields and magnetic fields at the surface are of the form

𝐻_𝑦1 = 𝐻_𝑦2 (4.8a) 𝐸_𝑥1 = 𝐸_𝑥2 (4.8b) 𝜀₁𝐸_𝑧1 = 𝜀₂𝐸_𝑧2 (4.8c) 𝑘_𝑥1 = 𝑘_𝑥2 (4.8d) Due to the symmetric of propagating wave at the interface, 𝐸_𝑧1 = −𝐸_𝑧2, then relation of permittivity between two media is

𝜀₁ = −𝜀₂ (4.9) Finally, from Eq. (4.10e) and Eq. (4.8d) it leads to dispersion relation

𝑘_𝑥1 = 𝑘_𝑠𝑝𝑝(𝜔) =𝜔

𝑐 √ 𝜀₁(𝜔)𝜀₂(𝜔)

𝜀₁(𝜔) + 𝜀₂(𝜔) (4.11a) 𝑘_𝑧𝑖 = √𝜀_𝑖𝑘₀²− 𝑘_𝑥1² 𝑖 = 1,2 (4.11b) In order to excite the SPR, we require that 𝑘_𝑥1 is real and 𝑘_𝑧𝑖 is purely imaginary in

39

both media. Then permittivity of both media only can be

𝜀₁+ 𝜀₂ < 0 (4.12) 𝜀₁∙ 𝜀₂ < 0 (4.13) The significance of Eq. (4.13) and Eq. (4.12) is similar to Eq. (4.9), which means that not only either index of two media must be negative, but also the absolute value of negative index exceeding that of the other. Most of the materials, especially noble metals have large negative real part of dielectric constant. Therefore, the SPR can exist at the interface between a noble metal and a dielectric when the polarization of incident light is p-polarization.

After realized the condition of existence of SPR, we can find out only p-polarization light can excite SPR on the interface between metal and dielectric, so recently, radially polarized light has been wildly used to excite SPR due to the good axial symmetric of p-polarization, and the merits of radially polarized light in terms of a tighter focal spot already confirm in many investigations [26, 27]. But current technology does not allow for a simply frequency sweeping operation system for radially polarized light. These major issues delay the progress on the development of SPR and hinder further study on wavelength dependent material. Therefore, how to simply create the white light radially polarized light and build up a polychromatic radially polarized surface plasmon resonance (PC-RPSPR) sensor will be shown up in following section.

4.2 Experimental Setup

The experimental setup of PC-RPSPR sensor can divide into two principal parts, first part is synthesis of radially polarized white light utilized an approach of spatially varying polarizer (SVP), as shown in Fig. 4-2 (a), and second part is the metal-insulator-metal (MIM) structure for extending sensing range on sensor system.

40

We utilize an unpolarized white LED (model: Luxeon Star/O LXHL-NWE8) as a light source which has advantages of low cost and is speckle free. A collimated unpolarized light was converted to radially polarized light by the use of SVP. The radially polarized white light then relays to the entrance pupil of the commercial immersion objective lens (Olympus PlanApo-N 60x/1.45 Oil). Its corresponding half divergence angle is 75.16°, well beyond the SPR resonant angle SP ~ 45^o at wavelength 0 = 610 nm. After passing through the objective, the white light focus on the MIM structure, Non-coupled reflected light has been collected and guides backward into two different optical paths via the same objective lens. One optical path projected the reflected intensity distribution onto CCD image sensor from the back focal plane of the objective lens. The other optical path records the spectra of reflected beam via a spectrum analyzer. As shown in Fig. 4-2.

Fig. 4-2 Configuration of the PC-RPSPR sensor, where CL: collimated lens, SVP: spatial varying polarizer, RL: relay lens, BS: beam splitter, M: mirror, IL: image lens, MIM: metal-insulator-metal structure. The insets show (a) the photo of SVP, (b) the schematic diagram of MIM structure.

41

4.2.1 Synthesis of polychromatic radially polarized light

The SVP consists of eight pieces of linear polarizer and the transmission axis of every sector aligned to individual principle radial direction, as shown in Fig. 4-3. The SVP is used to convert unpolarized white light into radially polarized white light. In microscopies, nano-optics, and spectroscopy, polychromatic radially polarized beams can supply other perspectives. Based on past reported studies, the common recipe to synthesis or generate radial polarization are designed for a specific working wavelength and rely on the use of phase element, liquid crystal, interference configuration [28, 29, 30, 31]. Those elements are wavelength dependent. This means that it cannot operate universally among different working wavelength in a fixed design. However, only few numbers of devices can generate those kinds of light in recently. The main systems are double conical reflector system which based on geometrical optics [32] and fiber-based system [33]. Unfortunately, the former system crate a discontinuous ring beam shape, which decreases the resolution due to the increment of side-lobe part in focal region and seems difficult to be apply for sensing;

the latter system need the procedure of precise optical alignment to couple incident light into fiber. In other words, these systems are not simply and convenient. On the contrary, proposed SVP is assembled by conventional polarizing element offering wavelength independent properties for polarization conversion and simplify the creative complexity of system. Also, it has a compact size with extremely low cost, but this device exchanges those benefits for power consumption.

Fig. 4-3 The SVP assembly, which is composed of eight sectors.

42 mechanism of SPR, the incidence angle, which provides sufficient wave-vectors to agree phase matching conditions, is greater than the critical angle. The SPR angle is wavelength dependent and is related to the dispersion relation of metal. Its resonance angle shifts up when working wavelength decreased. Furthermore, generally the refractive index of living cells are close to water (SPR, water ~ 77.4 ^o, 0 = 610 nm) which beyond the sensing limit for objective lens with NA = 1.45 (max ~ 75.16°, 0 = 610 nm). Under this circumstance, the SPR dips are outside the observation windows as the wavelength is smaller than 640 nm as shown in Fig. 4-4. Therefore, the MIM structure is purposed,

As the beam focused on the MIM structure, not only cavity resonance (CR) modes but also transformed surface plasmon resonance (T-SPR) modes which have broader sensing range are generated. CR modes are insensitive to the change of the refractive index of sample, but T-SPR mode is. The role of a MIM structure is to transform generated cavity resonance (CR) modes into transformed surface plasmon resonance (T-SPR) modes. The material and thickness of each layer determine the resonance property of both modes. In order to maximum the depth of surface plasmon resonant dips, we kept the symmetry property of MIM structure and set the overall thickness of gold thin film as 40 nm, also, in order to confirm the effect of MIM structure, we chose SiO2 as insulator which is identical to substrate. In our case, when the dip of CR modes exceeded the critical angle, the CR modes would yield abundant

43

Fig. 4-4 Comparison with the different test sample. (a) and (c) are experimental data and (b) and (d) are simulation when test sample is air and water, respectively.(e) and (f) are the slice of (b) and (d), respectively.

The wavelength here is 610 nm.

evanescent waves. Then, the energy of evanescent wave would be transferred to a new SPR which can be excited by smaller propagating constant, because MIM structure [the thickness of insulator should be larger than 100 nm (150 nm) @  = 450 nm

evanescent waves. Then, the energy of evanescent wave would be transferred to a new SPR which can be excited by smaller propagating constant, because MIM structure [the thickness of insulator should be larger than 100 nm (150 nm) @  = 450 nm