Fresnel’s equations - Vector diffraction theory

Chapter 1 Introduction

1.2 Vector diffraction theory

1.2.2 Fresnel’s equations

The Fresnel’s equations discussed the reflection and transmission coefficient with different polarization in this situation, a plane wave is incident from medium 1 onto the interface to medium 2 under an incident angle θ_𝑖. If the electric field is tangential to the interface, the electromagnetic wave is called s-polarization; the orthogonal polarization is called p-polarization, as shown in Fig 1-6 (a) and (b).

Fig. 1-6 Geometry of Fresnel’s equations for (a) s-polarization and (b) p-polarization.

(a) (b)

11 Requiring the continuity of fields at the interface, we can obtain

1 + 𝑟_𝑠 = 𝑡_𝑠

𝑛_𝑖𝑐𝑜𝑠𝜃_𝑖 − 𝑟_𝑠𝑛_𝑖𝑐𝑜𝑠𝜃_𝑖 = 𝑡_𝑠𝑛_𝑡𝑐𝑜𝑠𝜃_𝑡 (1.36) Then a non-magnetic medium was assumed. The solution is

𝑟_𝑠 = 𝑛_𝑖𝑐𝑜𝑠𝜃_𝑖 − 𝑛_𝑡𝑐𝑜𝑠𝜃_𝑡

Jones Calculus including matrix and vector operation is a quantitatively mathematical description of polarization state. The polarization states can describe by

a vector, called the Jones vector, and each polarization optical element has associated Jones matrix. Then the optical system can be described by multiplication with the term is ignored and then we denote the Jones vector into normalized forms.

When putting an polarization optical component, the state of polarization transfer from ,𝐸𝑥 𝐸_𝑦- into ,𝐸𝑥′ 𝐸_𝑦^′-, the optical component is represented by Where J is the Jones vector of incident beam, 𝐽^′ is the Jones vector emergent beam, and M is the Jones matrix.

Table 1.1 and Table 1.2 show some general examples of Jones vector and Jones matrices.

Linear at 45°

√20111

Right circular

√20 1

−𝑖1

Left circular 1

√201𝑖1

Table 1-1 The reference between polarization and Jones vector.

Optical element Jones matirx

Neutral element 01 00 11

x-polarizer 01 00 01

y-polarizer 00 00 11

Quarter-wave retarder 0 1 −𝑖

−𝑖 11

Half-wave retarder 00 11 01

Table 1-2 The reference between optical elements and Jones matrices.

1.3 Comparison of two diffraction theory

Through the discussion of scalar and vector diffraction theory, respectively, we can find out some differences as following:

1. The vector diffraction theory is able to completely describe polarization of light,

but scalar diffraction theory cannot.

2. The thickness of obstacle is assumed as zero in scalar diffraction theory, but it must be taken into account in vector diffraction theory.

3. The scalar diffraction theory is only an approximation; therefore, that entails some degree of error.

1.4 Motivation & Organization

Recently, the optical engineering is rapidly developed. More and more parameters of light and material properties should be taken into consideration due to the dimension of interested area is gradually reduced. Therefore, the scalar diffraction theory is not suitable anymore; the vector diffraction theory is the only way to get accurate solution. Polarization is also an important property of light in vector diffraction theory; the different polarizations of light can elicit different effects in material. Through the vector diffraction theory, we can observe the phenomenon of interaction between light and material, and discuss the influence of polarization of light in focus fields of light. Consequently, we hope we can obtain some useful applications utilize the influence of polarization of light in this thesis.

This thesis will be partitioned into three major parts, first part of this thesis talks about the effect of polarization in different numerical aperture in chapter 2, the influence of polarization is more distinct when numerical aperture is bigger. It will be demonstrated in later.

The second part is the first application of polarization which called extended depth of focus; it is achieved through the synthesis of two orthogonal polarizations or higher order of inhomogeneous polarization. It will discuss in chapter 3. The third part is the second application of polarization which is utilized to excited polychromatic surface plasmon resonance, the system and synthesis method will be

introduced in chapter 4.

Finally, the conclusions and future works will be described in chapter 5.

Chapter 2 Polarization Beams

The mode of the electric field vibration is called “polarization”. According to the beam with the same polarization at every point within the pupil plane or not, the polarization beam can divide into homogeneous and inhomogeneous. In this chapter, the focal field properties of two kinds of polarization will be shown in following.

2.1 Homogeneous and inhomogeneous polarization

Homogeneous polarization keep the identical polarization at every point within the pupil plane; we can describe the field distribution by using the function dependent on the ratio of electric field. If the beam only has the x-component of electric field, it is called x-directional linear polarization. If the phase delay between x-component and y-component is 90°, the polarization is called circular polarization.

As the standard operation process is not homogeneous among the pupil, it is inhomogeneous polarization beam. The obvious difference can observe from Fig. 2-1

Fig. 2-1 (a) and (b) illustrates the homogeneous and in homogeneous beam, respectively.

One typical spatially inhomogeneous beams is the cylindrical vector beams,

which attracted much attention due to its symmetric feature. The general solution of the wave equation for cylindrical vector beams includes two solutions: radially polarized beam and azimuthally polarized beam. The orthogonal cylindrical vector beams have the doughnut-shaped irradiance and good symmetry in r-direction or φ-direction. Due to the good symmetry, the radially polarized beam is p-polarization in omni-direction at the incident plane; likewise, the azimuthally polarized is s-polarization, as shown in Fig. 2-2. Then the focused field distribution properties of different polarization with different numerical aperture will be discussed in following.

Fig. 2-2 Intensity distribution of the cylindrical vector beams. (a) Radial (b) Azimuthal polarization.

2.2 Properties of polarized beam

In this section, the polarization states at the entrance pupil are linear, circular, radial, and azimuthal, respectively. Examples of focal distribution with different numerical aperture foci are shown.

2.2.1 Linear polarization

Considering an entrance pupil in which the polarization is x-direction linear polarized and constant amplitude. The focused field distribution is circular symmetry for small numerical aperture, when numerical aperture increases, it become

asymmetrical field which is larger along the direction of polarization due to the electric field of z-components is no longer negligible in high numerical aperture, as shown in Fig. 2-3. Therefore, the focused field is reduced which yields a broader focus when numerical aperture is larger than 0.7. This is a direct result of the vector effect.

Fig. 2-3 Intensity profile of linearly polarized focus as numerical aperture increases from 0.2 to 1.0, where the vectorial feature becomes more apparent with high-NA system.

2.2.2 Circular polarization

Now, considering an entrance pupil in which the polarization is right circular polarized and constant amplitude. The focused field distribution is circular symmetry

no matter small or high numerical aperture, as shown in Fig. 2-4. Because of the electric field of z-component has circular symmetry in the doughnut-shaped and vanished on the optical axis. The x and y component are both asymmetrical but with

orthogonal axes, so the sum of each components is symmetrical.

Fig. 2-4 Intensity profile of circular polarized focus for increasing numerical aperture.

2.2.3 Radial polarization

For aforementioned polarization, their amplitude is constant, but for a radially polarized pupil, amplitude forms a donut-shaped field because of the singularity in the center. When numerical aperture is larger than 0.7, the optical field is transformed from a doughnut-shaped to a flat-topped shape due to the depolarization effect starts to govern the shape of the focused spot. Then the focused field of radially polarized beams has a sharp intensity peak because of the strong depolarization effect which caused the constructive interference of the z-component. Compared with linear polarization and circular polarization, the smaller spot obtained from radial polarization was under the conditions of numerical aperture larger than 0.9 and 0.95, respectively [3], as shown in Fig. 2-5. But it was noted that superior illumination only exists in a high numerical aperture. On the other hand, because of the good polarized symmetry in r-direction, the optical field still maintains circular symmetry for any numerical aperture.

2.2.4 Azimuthal polarization

Azimuthal polarization is similar to radial polarization; the amplitude cannot be constant and forms a donut-shaped field because of the singularity in the center and due to the good polarized symmetry in φ-direction, the optical field also maintains circular symmetry for any numerical aperture. The special focused feature of azimuthal polarization is that yields a doughnut-shaped field even when numerical aperture is larger than 0.7, as shown in Fig. 2-6. The reason of the special focused field is that adjacent parts of this polarization are π-phase shifted with each other, respectively. And another special focused feature is that electric field of z-component is zero, therefore, the azimuthal polarization created an optical cage with the absence of z-component electric field at the vicinity of focal point, as shown in Fig. 2-7.

Fig. 2-5 Intensity profile of radially polarized beams in different numerical aperture. When numerical aperture increased, the optical field is transformed from a doughnut-shaped to a flat-topped shape.

Fig. 2-6 Focused intensity profile of azimuthally polarized beams for high numerical aperture.

Fig. 2-7 Each component of focal fields in azimuthally polarized light, due to the destructive interference of depolarization effect, the electric field of z-component is zero.

2.3 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.

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

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

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

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)

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.

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

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

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

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.

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.

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-TEM₁₁and 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

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

Chapter 4 Application 2 – Chromatic Surface Plasmon Resonance

Recently, surface plasmon resonance (SPR) have been widely used to analyze

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