Motivation of This Thesis - 光瞳工程應用於表面電漿之研究

Chapter 1 Introduction

1.3 Motivation of This Thesis

1.3 Motivation of this thesis

Due to the dimension of engineering was reduced down to the order comparable with the wavelength, more complete treatment based on vectorial diffraction shall be considered. Polarization status plays an important role in current optical engineering. Different states of polarization would lead to different effects in the matter-light interaction. Therefore, the scalar diffraction theory is no longer suitable anymore and will be replaced with the vector diffraction theory which can fully explain the characteristic of polarization of light. In this thesis, we will discuss the influence of pupil mask in focus fields.

As the consequence, pupil engineering applicable for different purposes was investigated.

The first part of this thesis will discuss the diffraction of lens and the relation between the field distribution at pupil with certain mask and point spread function (PSF). Meanwhile, we will introduce a mathematical tool called Wignerdistribution function (WDF) which represents an optical field in terms of a ray picture [4]. After that, three types of pupil mask will be discussed.

In order to realize the effects of polarization in the focus field, we introduce four kinds of polarization. The focus field with different polarization will be given based on the result in chapter 2.

Then we choose a specific polarization which can be used to excite the Surface Plasmon Resonance (SPR) via objective-based system in the third part of this thesis. Some experimental results will be given.

Finally, we conclude this thesis and provide suggestions for future research toward a continuation of the work presented here.

14 geometrical optics would require a rather lengthy detour. The philosophy of our approach in this chapter is to use wave optics analyses of the systems of interest.

Furthermore, we will describe the optical fields a by means of a Wigner Distribution Function (WDF) when the optical signals and systems can be described by quadratic-phase functions in particular.

The effects of pupil mask are the topic of Sec. 2.2. Different kinds of pupil mask and its characteristics will be given and elucidated with some optical examples. Sec. 2.2.1 is devoted to the amplitude and phase filter which are generally used in current optical systems. The following section treats the polarizer as a filter, and then discusses the vector point spread function and closes this chapter.

2.1 Diffraction by lens

In this section, we will consider the detail of the light field near the focal region of a lens. Considering we have a field distribution in the focal region, i.e.

at , as shown in Fig. 2-1. Suppose that a plane wave with amplitude incident upon the lens. Thus the field distribution in the plane before the lens is

. The lens is just like a modulator which modulates the

wave-front by a complex transmittance

[

] (2.1)

where the quadratic phase factor is the phase transformation of ideal lens. It will be found others phase term if the lens or the pupil has any aberrations.

Fig. 2-1 Schematic diagram of diffraction by a lens Therefore, the field distribution behind the lens is

[

] (2.2).

To find the field distribution in the focal region, the Fresnel diffraction formula is applied. Thus,

⁽⁾

∬ ⁽ ⁾ (2.3) where a constant phase factor has been dropped. Substituting Eq. (2.2) into Eq. (2.3), the quadratic phase factors of the complex transmittance of lens are seen to exactly cancel, resulting

The quadratic phase term preceded the Fourier transform will be eliminated for very special case . Thus the Point Spread Function (PSF) at the focal Point Spread Function (VPSF) in the present section.

The focal field of a polarized pupil was computed in Sec. 1.2.1. In this section we assume that the Jones vector field in the exit pupil stems from the

Substituting Eq. (2.7) into Eq. (1.26), we can get the VPSF

⃑ ∬ ( ) ( ) ⃑ ( ⃑ ⃑ ) (2.8) and the intensity of the VPSF is defined as the total intensity of the individual components of the VPSF

‖ ⃑ ‖ ‖ ⃑ ‖ ‖ ⃑ ‖ ‖ ⃑ ‖ (2.9).

We will show the field distribution at focus with different input polarization in the next chapter via this equation.

2.1.2 Wigner distribution function

Recently, most optical imaging systems which pursued super resolution used coherent or partially coherent illuminations. In this section, we will introduce the Wigner Distribution Function (WDF) which is directly related to the coherent function and is a powerful and visual description of partially coherent light [5].Considering we have an optical wave as shown in Fig. 2-2

Fig. 2-2 Schematic diagram of WDF We have the Wigner distribution function

∬ [ ] (2.10) where is the mutual coherence function defined by

(2.11) where ⁄ ⁄ and and we have this relation and as shown in Fig. 2-2.

The mutual coherence function describes the correlation between the optical fields of two different positions. For example, a well coherent light source such as laser has almost same phase at the same wave-front. That means the correlations between two arbitrary points on this wave-front are strong.

Other source like LED does not have this characteristic. The phase of different point on the same wave-front is very large and will decrease the mutual coherence function[6].

One of the important characteristics of the WDF is describing position and propagation in the same time. Let us simplify Eq. (2.10) to a 1-dimensional question. Considering there is a point source in the position x1 and it emerges light field as spherical wave. Then we can use WDF to describe the behavior of light field emerging from that point source as shown in Fig. 2-3.

Fig. 2-3 WDF of a point source

If there is a plane wave propagating in the free space. The behavior of this plane wave can be described as shown in Fig. 2-4.

Fig. 2-4 WDF for a plane wave

Finally, we use the WDF to describe an imaging system and end this section. We have a point as object and an imaging system with single lens. Then the WDF of this system can be expressed as shown in the following figure.

Fig. 2-5 WDF for an imaging system

Here we just consider an ideal imaging system so that there are no aberrations and diffraction. In real case, the WDF will blur because the effects of geometric aberrations or diffraction.

2.2 Pupil Mask

We have discussed the Fourier relationship between pupil function and point spread function in Sec. 2.1.1. Since the output signal of an imaging system is the convolution of the input signal and system’s point spread function. By the convolution theorem, the spectrum of output signal is the product of the individual spectra of the input signal and point spread function.

{ } { } { } { } (2.12) where f is the input signal, h is system’s point spread function and g is the output signal and their spectra denoted by capitalization. Eq. (2.12) tells us that the output signal can be modified by the system’s point spread function. And we

do know the different pupil function will cause different point spread function.

In this section, we will introduce different kinds of pupil mask. The mechanism of modulation of the input field is also called filtering.

2.2.1 Magnitude and phase mask

The transfer function can be expressed as the magnitude part multiplied the phase part

| | [ ] (2.13).

The magnitude part | | of the system’s transfer function explains the scale factor applied to the amplitude of each sinusoidal component of and is often called the Modulation Transfer Function (MTF) of the system.Typically, filters which only modulate the magnitude part of transfer function are called

“magnitude filter”. On the other hand, filters modulate the phase part of transfer function are called “phase filter” or “allpass filter”[7].

Common categories of the magnitude filter include the lowpass filter, highpass filter and bandpass filter. The origin of each name is the working region in the frequency domain. For example, the lowpass filter attenuates or removes the high frequency and conserves the low frequency. A typical lowpass filter is a rectangular function as shown below

{ | |

(2.14).

Fig. 2-6 Rectangular function

Considering we have an input signal contained different frequency as shown in Fig. 2-7.

Fig. 2-7 Signal contains different frequency

Then we do the Fourier transform to this signal to get the spectrum of this signal as shown in the following figure

Fig. 2-8 Spectrum of the signal shows in figure 2-7

We can apply different magnitude filter to get different signal depends on our requirement. Here we show a simple result of applying different filter

Fig. 2-9 Magnitude filtering

Here is a specific identity worth paying attention. The amplitude of the output signal is obviously smaller than input signal. This phenomenon conforms to the description we mentioned before because the scale factor of

each sinusoidal component is modified by the transfer function. In other words, the filter of left side of Fig. 2-9 constrains the high frequency components and retains the low frequency; the filter of the other side depresses both high and low frequency terms but middle frequency terms.

Above paragraph discussed the magnitude filter whichremoves components with frequencies outside of the specified band. However, this kind of filter modifies the optical power in the same time. That means we will have poor power at the image plane. Another solution is called phase filter which modifies only the phase of the received wave-front and remains magnitude unchanged.In general, phase filter is designed for particular purpose. The most popular design is E. R. Dowski’s Cubic Phase Mask (CPM)[8]. By Goodman’s deducing [1], defocus induces a quadratic phase term in the PSF. Applying the CPM at the pupil plane will cause that the PSF remains nearly unchanged in the focus and out of the focus as shown in Fig. 2-10.

Fig. 2-10 PSF with and without CPM

2.2.2 Polarization mask

Above-mentioned is the common pupil mask and usually used in the low numerical aperture system. Each has specific characterizations and applications.

However, as we mentioned before, current technology focuses on the region smaller than the former. Super-resolution microscopy, lithography and other applications are interesting in a tiny area. Therefore, traditional optical mask had faced a huge challenge and was hard to overcome it. Recently, inhomogeneous polarizations are well developed by different generated methods in particular cases [9-13]. One of that is the radially polarized beam. The schematic diagram of the direction of electric fields is shown in the following figure.

Fig. 2-11 Schematic diagram of radial polarization

In order to further investigate the focusing mechanism of radial polarization, the Vectorial Debye theory was introduced in the Sec. 1.2.1, and the ratio of the longitudinal to transversal component is defined as follows[14]

(2.15)

where the coordinates here and we used in Sec.1.2.1 are same.

Focused radially polarized beams with different L-T ratio are shown in the Fig. 2-12.

Fig. 2-12 Foci of radially polarized beams with different L-T ratio

One can observe that high L-T ratio radially polarized beam produces a smaller focal spot than other polarization as shown in Fig. 2-13.

Fig. 2-13FWHM with different polarizations

The Full Width at Half Maximum (FWHM) of high L-T ratio is almost half wavelength. This phenomenon gives us a window to surmount the diffraction-limit. Many applications such as optical tweezers approach a higher resolution via a strongly focal beam with radiallypolarized beam had already proven that the polarization mask can do what traditional mask cannot do.

However, once again, the numerical aperture will affect the focus, especially in the high NA system. The more discussion of different polarization in high NA system will be introduced in the following chapter.

2.3 Summary

In this chapter, we discussed different type pupil masks. The input signal can be regarded as a combination of sinusoidal wave with different frequencies and scaling factors. Components with frequencies outside of specific band are removed or depressed in magnitude mask so that causes the optical power smaller. The phase mask adds an additional phase term to each component, and will induce different field distribution but remain the optical power unchanged.

The polarization mask provides a window to overcome the natural limit and achieve the super resolution. But it must be noticed that the field distribution will distort because the numerical aperture. We will discuss this effect in the next chapter.

Chapter 3 Polarizations

Polarization means the vibrated direction of the electric field. Different vibrations result in different polarizations. The vibrated direction of electric field can be dominated by the pupil mask. We have discussed the polarization mask in chapter 2. In this chapter, different polarized beams will be introduced. The field distribution in the focus of each polarization will be discussed in high numerical aperture system, too.

3.1 Homogeneous and inhomogeneous polarizations

Generally, homogeneous polarization means that the electric field of every points within the pupil have same direction and inhomogeneous not. The electric field can be described by the two tangential directions (x-y plane) which are perpendicular to the propagated direction (z direction). A y-directional linear polarization means that the electric field of this beam has only y-component. If the electric field has x- and y-component and no phase delay between each component, it is called a 45° linear polarization. The circular polarization represents a beam with 90° phase delay between x- and y-component.

Fig. 3-1 illustrates the difference between homogeneous polarization or not. The arrows represent the vibrated direction of the electric field at each point.

For convenience, we use a square aperture to give a simple illustration. But in real case, circular aperture attracts more attention due to its symmetric feature.

The calculation of a symmetric aperture is easier than asymmetric.

Fig. 3-1 Illustration of (a) homogeneous and (b) inhomogeneous polarization In a focalizing system, the inhomogeneous polarization can be regarded as a linear combination of two eigenmode: s- and p-polarization as shown in Fig.

3-2.

Fig. 3-2 P- and s-polarization for focalizing system

P-wave in focalizing system is an omni-directionally linear polarization along the r-direction in cylindrical coordinate, which is called radial polarization.

Likewise, the s-wave is called the azimuthal polarization because the vibrated direction of electric field is along the -direction.

3.2 Polarized beams at the focus

We had introduced different polarizations in the previous section. This section will discuss the focal field of different polarizations with different numerical aperture.

3.2.1 Linear polarization

Fig. 3-3 Foci of x-linear polarization with different N.A

Fig. 3-3 illustrates the foci of x-linear polarization with different N.A. It is obviously that the field distribution is “squeezed” along the polarized direction.

In order to further discuss this effect, we decompose the total intensity into Ix, Iy

and Iz as we mentioned in Sec. 2.1.1. The intensities of each component are demonstrated in the following figure.

Fig. 3-4 Individual component of foci of x-linear polarization with (a) N.A=0.2 (b) 0.6 (c) 1.0

We can observe that the y- and z-component are too small to influence the total intensity which is dominated by x-component due to the initial polarization is x-linear polarization in low N.A. However, z-component becomes larger and cannot be neglected when the N.A is increasing. This phenomenon results in a squeezing field distribution in high N.A system.

3.2.2 Circular polarization

The phase delay between x- and y-component of the incident electric field is called circular polarization. Due to the incident beam has x- and y-component, we can predict that the “squeezing” effect in linear polarization will be neutralized and the field distribution at the focus will be symmetric. The simulated results indeed demonstrate this prediction as shown in Fig. 3-5.

Fig. 3-5 Foci of circular polarization with different N.A

Fig. 3-6 Individual component of foci of circular polarization with (a) N.A = 0.2 (b) 1.0

The intensities of each component are circular symmetric as shown in the above figure. One can observe that the intensities of x- and y-component are still squeezed but can be neutralized in the total intensity.

3.2.3 Radial polarization

The vibrated direction of electric field along radial direction in cylindrical coordinate is called radial polarization which is an eigenmode of inhomogeneous polarization. As we mentioned before, this kind of polarization will induce a strongly longitudinal component by the in phase interference and lead to a small focus in high N.A system.

Fig. 3-7 Foci of radial polarization with different N.A

Fig. 3-8 Individual component of foci of radial polarization with (a) N.A = 0.2 (b) 1.0

The total intensity is dominated by transversal component in low N.A system due to the longitudinal component is insignificant. As the N.A increasing, the longitudinal component transcends the other and dominates the total intensity as shown in Fig. 3-8.

3.2.4 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

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.

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.

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

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.

在文檔中光瞳工程應用於表面電漿之研究 (頁 23-0)