向列型液晶E7在不同溫度下的光學常數之時析兆赫光譜研究

國

立

交

通

大

學

光電工程研究所

碩

士

論

文

   

向列型液晶 E7 在不同溫度下的光學常數之

時析兆赫光譜研究

 

Terahertz Time-Domain Spectroscopic Studies of

the Temperature-Dependent Optical Constants of

the Eutectic Liquid Crystal E7

       

研 究 生：楊承山

指導教授：潘犀靈 教授

     

中 華 民 國 九 十 八 年 六 月

向列型液晶 E7 在不同溫度下的光學常數之

時析兆赫光譜研究

Terahertz Time-Domain Spectroscopic Studies of the

Temperature-Dependent Optical Constants of the

Eutectic Liquid Crystal E7 

 

研究生：楊承山      Student：Chan-Shan Yang 

指導教授：潘犀靈    教授      Advisor：Prof. Ci-Ling Pan 

國  立  交  通  大  學 

光電工程研究所 

碩  士  論  文 

A Thesis

Submitted to Institute of Electro-Optical Engineering College of Electrical Engineering and Computer Science

National Chiao Tung University In Partial Fulfillment of the Requirements

For the Degree of Master of Engineering In Electro-Optical Engineering

June 2009

Hsinchu, Taiwan, Republic of China

 

i

向列型液晶 E7 在不同溫度下的光學常數之

時析兆赫光譜研究

研究生：楊承山 指導教授：潘犀靈 教授

國立交通大學光電工程研究所

摘 要

兆 赫 波 時 域 光 譜 技 術 是 一 種 研 究 物 質 在 兆 赫 波 段 光 學 性 質 的

方 法 。 本 論 文 是 利 用 兆 赫 波 時 域 光 譜 技 術 來 研 究 一 種 已 經 被 廣

泛 運 用 在 可 見 光 波 段 的 混 合 性 向 列 型 液 晶 E 7 ， 其 在 向 列 相 和 均 相 態

時，在不同溫度(從攝氏 26 度至攝氏 70 度)以及頻率範圍(0.2 THz ~ 1.4

T H z ) 的 光 學 常 數 。 實 驗 結 果 顯 示 此 種 液 晶 在 兆 赫 波 段 下 仍 保 有

正的複折射現象，而且複數折射率的虛數部分皆小於 0.04，也沒有明顯的

吸收譜線。在攝氏 26 度時，E7 的長軸方向與短軸方向折射率分別約為 1.71

與 1.57，即其雙折射約為 0.14。另外，我們從雙折射數據中所得到

的液晶有序參數資訊也與可見光波段者吻合，由此更確定了此次量測的準

確性。另一方面，本研究也成功地驗證液晶折射率溫度相依的 extended

Cauchy equations。本研究成果也將會是液晶元件日後在遠紅外區波段各種

應用的重要依據。

ii

Terahertz Time-Domain Spectroscopic Studies

of the Temperature-Dependent Optical

Constants of the Eutectic Liquid Crystal E7

Student: Chan-Shan Yang Advisor: Prof. Ci-Ling Pan

Institute of Electro-Optical Engineering

College of Electrical Engineering and Computer Science

National Chiao Tung University

Abstract

We have used terahertz (THz) time-domain spectroscopy to investigate the frequency

dependence and temperature dependence of the optical constants of a widely used liquid crystal mixture E7 in both nematic and isotropic phases. The extinction coefficient of E7 at

room temperature is less than 0.04 and without sharp absorption features in the frequency range of 0.2~1.4 THz. The extraordinary and ordinary indices of refraction at 26°C are around 1.71 and 1.57, respectively, giving rise to a birefringence of about 0.14 in this frequency range. The temperature-dependent order parameter extracted from birefringence data agrees

with values in the visible region quite well. On the other hand, the extended Cauchy equations describing the temperature dependence of refractive indices of liquid crystal is also confirmed

iii

Acknowledgements

轉瞬間，時間已是 2009 年的暑日。想起五年前剛進來光電這個大家庭的

青澀模樣，我不禁想要跟所有幫助過我的人、指導我的人和關愛我的人說

一聲「謝謝」。也許這對某些人只是兩個很簡單的音節，但對我來說，卻

灌注了五年中所有的記憶與感動。

其中最要感謝的，就是我從大一到碩士班的指導老師潘犀靈 教授以及在

實驗上給我諸多意見與幫助的趙如蘋 教授。如果沒有他們費心的指導，細

心的提醒以及耐心的包容，我想這一本碩士論文定無法如此順利的完成。

除此之外，課餘間在老師家的師生活動與私下的關心談話，也因為知道後

面一定會有老師的支撐和諒解，進而也構築了我對研究的信心與熱情。

除此之外，實驗室的學長姐 宇泰、怡超、晉瑋、宜貞、卓帆和家任，總

在我最徬徨無助時，給予我最誠懇且受用的建議。在比較熟悉的宇泰、怡

超和晉瑋學長身上，我更看到了一位＂能者＂所應有的風範與態度，一種

我會想要追隨的態度。

我的同學們，晏徵、俞良、阿猛和阿駿，能與你們身處在同一間實驗室

並且一同修課，是我非常難忘記的經歷和感受。你們的優秀，讓我自省本

身的不足；你們的真誠，讓我們相處的方式能夠輕鬆泰然；你們的執著，

讓我學會不再輕言放棄。相信在很久的以後，你們都能找到自己的夢想，

自傲的各立一方。而剛認識的睿茵、聖司和冠儒，也謝謝你們在畢業季節

的時候，對實驗室多番的幫忙及貢獻，更希望你們對日後的研究都能一帆

風順並充滿熱情。

最後，我要對在背後支撐我的父親、母親、弟弟、叔叔、嬸嬸和宜珊說

一聲＂謝謝＂。謝謝你們一直以來的的照顧與包容，若沒有這些元素，我

相信自己在求學的路途上，無法如此的勇敢和堅毅。特別是父親與母親，

你們給的是我生命價值中最重要的一部分，我會用一輩子的時間來證明它。

我是一幅周圍有殘缺的拼圖，曾經我很膚淺的只看到中間完整的部分，

但是因為你們，我慢慢看到並找回不足的地方。也許需要一點時間，不過

我知道自己真的在改變。謝謝~~

Chan-Shan Yang

2009.07 風城~交大

iv

Table of Contents

中文摘要 i Abstract ii Acknowledgements iii Table of Contents iv List of Figures vi List of Tables ix Chapter 1 Introduction 1 1.1 What is Terahertz (THz) 1 1.2 Liquid Crystals (LCs) ₃ 1.2-1 Thermotropic LCs 3 1.2-2 Mixtures 4

1.2-3 Nematic Liquid Crystal (NLC) ₅

1.3 THz Liquid Crystals Optics 8

1.4 Thesis Highlight 9

Chapter 2 Experimental Methods 10

2.1 The Preparation of LC Cell and Reference Cell 10

2.2 Temperature and Humidity Controlled System 16

2.3 THz Time-domain Spectroscopy (THz-TDS) 19

2.3-1 Introduction 19

2.3-2 Theory of Generating THz Radiation 20

2.3-2-1 Optical Rectification 21

2.3-2-2 Photoconductive(PC) Switch 22

2.3-2-3 Current-Surge Model 23

2.3-2-4 Drude-Lorentz model 28

2.3-3 Theory of Detecting THz Radiation 30

2.3-4 Antenna-based Emitter and Detector 31

2.3-5 THz-TDS with Collimated Beam System 33

Chapter 3 Extraction of Optical Constants from THz Time-domain Spectroscopic Measurements

35

3.1 Introduction 35

3.2 Determination of Optical Constants 38

3.3 A Model Describing the Temperature Effect of LC Refractive Indices 41

3.3-1 Effective Medium Theory 41

3.3-1-1 Depolarization Field 41

3.3-1-2 Local Electric Field of Single Atom 42

v

3.3-1-4 Dielectric Constant and Polarizability 44

3.3-2 Extended Cauchy Equations 45

Chapter 4 Results and Discussions 49

4.1 Complex Optical Constants 49

4.2 Temperature Dependence of Refractive Indices 54

4.2-1 Temperature Effect on Real Refractive Indices 54

4.2-2 Temperature Effect on <n2 > and < >n 59

4.3 Temperature Effect on Birefringence and Order Parameter 67

4.4 Temperature Dependence of Gradient of Refractive Indices 72

Chapter 5 Conclusions 75

vi

List of Figures

Fig. 1.1 The diagram shows the name and the applications in different frequency ranges.

2

Fig. 1.2 Molecular alignments of LCs. 4

Fig. 1.3 Molecular structures of the four elements making up the LC E7.

5

Fig. 1.4 The three types of deformation occurring in nematics. 6

Fig. 1.5 The angle between the director of LC molecules and the propagation direction of incident light.

8

Fig. 2.1 Sketches of (a) LC cell and (b) the reference cell. 10

Fig. 2.2 The procedure of preparing the LC cell. 11

Fig. 2.3 The conversion of the original coordinate (x y ) and the new , coordinate (x y ). ', '

13

Fig. 2.4 The pictures of the temperature-controlled sample holder. 16

Fig. 2.5 The pictures of the temperature controller and the probe. 17

Fig. 2.6 The testing result which is set at 26oC of the temperature control system.

17

Fig. 2.7(a) Temporal profiles of THz before and after purging. 18

Fig. 2.7(b) Frequency domain of THz before and after purging. 19

Fig. 2.8 The general setup of the antenna-based time domain spectroscopy system.

20

Fig. 2.9 Optical rectification. 22

Fig. 2.10 Photoconductive switch. 23

Fig. 2.11 Schematic of the THz-TDS. 34

Fig. 3.1 The symbols of deriving the complex refractive index. 40

Fig. 3.2 The depolarization field EP is opposite to P. The fictitious

surface charges are indicate. The field of these charges is E_P within the ellipsoid.

42

Fig. 3.3 The internal electric field on an atom in a crystal is the sum of the external applied field E0

JK

and of the field due to the other atoms in the crystal.

43

Fig. 3.4 Estimation of the field in a spherical fiction in a uniformly polarized medium.

44

Fig. 4.1 Transmitted THz signals (e ray) through the reference (solid line) and LC cell (dashed line) at 26°C for (a) e-ray and (b) o-ray.

49

vii

the reference (solid line) and LC cell (dashed line) at 26°C for (a) o-ray and (b) e-ray.

Fig. 4.3 The real refractive indices of E7 are plotted as fuctions of frequency.

51

Fig. 4.4 The imaginary refractive indices of E7 are plotted as functions of frequency.

52

Fig. 4.5(a) The real refractive indices of fused silica are plotted as fuctions of frequency at 26°C.

53

Fig. 4.5(b) The imaginary refractive indices of E7 are plotted as functions of frequency at 26°C.

53

Fig. 4.6(a) Extraodinary and ordinary refractive indices of E7 are plotted as functions of reduced temperature at frequency of 0.34, 0.41, 0.53, 0.70, 0.80 and 0.90 THz.

56

Fig. 4.6(b) Extraodinary and ordinary refractive indices of E7 are plotted as functions of reduced temperature at frequency of 0.98, 1.10, 1.19, 1.29 and 1.40 THz.

57

Fig. 4.7 Extraodinary and ordinary refractive indices of E7 are plotted as functions of reduced temperature at frequency of 0.34 (Black), 0.70 (Red) and 0.98 (Blue) THz.

58

Fig. 4.8(a) Average refractive indices of E7 are plotted as functions of reduced temperature at frequency of 0.34, 0.41, 0.53, 0.70, 0.80 and 0.90 THz.

60

Fig. 4.8(b) Average refractive indices of E7 are plotted as functions of reduced temperature at frequency of 0.98, 1.10, 1.19, 1.29 and 1.40 THz.

61

Fig. 4.9 Average refractive indices of E7 are plotted as functions of reduced temperature at frequency of 0.70 (Black), 0.89 (Red), 1.10 (Green), and 1.40 (Blue) THz.

62

Fig. 4.10(a) <n2 > of E7 are plotted as functions of reduced temperature at frequency of 0.34, 0.41, 0.53, 0.70, 0.80 and 0.90 THz.

64

Fig. 4.10(b) <n2 > of E7 are plotted as functions of reduced temperature at frequency of 0.98, 1.10, 1.19, 1.29 and 1.40 THz.

65

Fig. 4.11 <n2 > of E7 are plotted as functions of reduced temperature at frequency of 0.70 (Black), 0.89 (Red), 1.10 (Green), and 1.40 (Blue) THz.

67

Fig. 4.12(a) Δn of E7 are plotted as functions of reduced temperature at frequency of 0.34, 0.41, 0.53, 0.709, 0.80 and 0.90 THz.

69

viii

frequency of 0.98, 1.10, 1.19, 1.30 and 1.40 THz. Fig. 4.13 Birefringence of E7 are plotted as functions of reduced

temperature at frequency of 0.34 (Black), 0.80 (Red) and 1.10 (Blue) THz.

71

Fig. 4.14 Birefringence of E7 measured at 26°C are plotted as a function of frequency. 72 Fig. 4.15 _{Temperature-dependent} dne dT of E7 at different frequencies (0.34, 0.41, 0.53, 0.70, 0.80, 0.89, 0.98, 1.10, 1.19, 1.29 and 1.40 THz), respectively. 73 Fig. 4.16 _{Temperature-dependent} dno dT of E7 at different frequencies (0.34, 0.41, 0.53, 0.70, 0.80, 0.89, 0.98, 1.10, 1.19, 1.29 and 1.40 THz), respectively. 74 Fig. 4.17 _{Temperature-dependent} dno dT of E7 at 0.41(THz). 74

ix

List of Tables

Table. 3.1 The references measuring optical constants of E7 in different frequency range.

36

Table. 4.1 The fitting parameters of Eq. (3.33) and (3.34) for frequencies from 0.34 to 1.40 THz.

55

Table. 4.2 The fitting parameters of Eq. (3.31) for frequencies from 0.34 to 1.40THz.

59

Table. 4.3 The fitting parameters of Eq. (3.37) for frequencies from 0.34 to 1.40THz.

66

Table. 4.4 The fitting parameters of Eq. (3.38) for frequencies from 0.34 to 1.40THz.

1

Chapter 1 Introduction

The large birefringence of liquid crystals (LCs) well known can be easily controlled by electric or magnetic field. According to the distinguishing characteristic, application based on LCs have developed widely and rapidly, such as liquid crystal displays which have established a substantial foothold on the market as flat panel displays for computers, transportation, communication and, with increasing importance for TV in the future [1], as well as their extraordinary and ordinary refractive indices (ne and no), are essential for the

modulation of millimeter-wave, infrared, visible region. Among the application of them, phase shifters [2], attenuator, polarizers and wavelength selection filters [3] have been demonstrated. However, the applications and research of LCs should be extended to THz range because of the remarkable development of terahertz (THz) technology [4]. For this purpose, we have to measure accurate refractive indices and the birefringence of LC is necessary for THz applications.

In writing this section, the basic information of THz technology, photoconductive (PC) antenna, THz time-domain-spectroscopy (THz-TDS) and nematic LCs (NLCs) are mentioned, and the structure of the thesis is presented in the last part of this chapter as “Thesis highlight”.

1.1 What is Terahertz (THz)

The THz region of the electromagnetic spectrum where until recently bright sources of light and sensitive means of detection have not existed represents the wavelength range and frequency range between 300μm~1000μm and 100GHz~30THz as shown in Fig. 1.1, respectively [4,5]. On the other hand, this corresponds to wavelengths between 1mm and 0.03mm, so THz is also called sub-millimeter wave. The preceding extreme is located just on

2

the microwave region where satellite dishes and mobile phones operate, and the latter one lies adjacent to infrared region used in devices such as TV remote controllers. However, the efficient and reliable sources and detector of THz radiation are not available easily because not only the conventional microwave source can not work at high enough frequencies, but the thermal effect confines the laser diode sources. Therefore, the development of THz has been held back for a long time. This predicament becomes better in the recent years because of the technology of self-mode-locked operation in Ti:sapphire oscillators [6], there has been a great progress in the generation of ~100 fs pulses. Because the semiconductor irradiated by ultrafast pulses can generate the broadband THz radiation, van Exter et al. demonstrated a dipole antenna based on semiconductor in 1989 [7] and showed the system as a spectroscopy with off-axis paraboloidal mirrors [8]. In 1990, Zhang et al. developed the way of generating subpicosecond pulses from a semiconductor surface [9]. After that, the same author presented free-space electro-optic sampling technology to improve signal to noise ratio (S/N ratio) to 104 and to carry out much large dynamic range in 1996 [10]. Above of all, a spectroscopic technique using pulsed THz radiation has been accomplished and called “THz time-domain spectroscopy (THz-TDS)”.

Figure 1.1 The diagram shows the name and the applications in different frequency ranges. The THz region lies between photonics and electronics regimes.

3

Much of the applications in THz radiation originate from penetrating deep into many organic materials without the damage. Besides, THz radiation is also absorbed by water easily, so that varying water content can discriminate the material from other ones. These characteristics lend themselves to applications in process and quality control as well as biomedical imaging, and some groups have presented THz imaging as a way of monitoring packages at airport. But for further advances in many applications, variety of active and passive THz optical elements are demanded, such as detectors, modulators, phase shifters and polarizers. Unfortunately, these devices in the region are investigated deficiently. Therefore, the development and research of THz optical devices are a burning issue.

1.2 Liquid Crystals (LCs)

1.2-1 Thermotropic LCs

Thermotropic LCs are extensively studied and used widely because of their linear as well as nonlinear optical characteristics. “Thermotropic” means that they exhibit different liquid crystalline phases as a function of temperature. Thermotropic LCs shown as Figure 1.2

includes nemati c phase, smectic phase, chi ral phase . Howe ve r, ne mati c phase is most common investigated; the LCs utilized in our work is all rod-like nematic LCs, where the molecules have no positional order, but they have long-range orientational order.

4

1.2-2 Mixtures

For pure LCs, temperature ranges for the various mesophases and other physical limitations makes lots of disadvantages imposed on application of these material [11]. In order to fixing these shortcomings, eutectic mixtures of two or more LCs are used widely. One of examples is E7 which being a mixture of four liquid crystals, and the molecular structures are shown as Figure 1.3. There are many different points between the individual mixture constituents and E7, such as dielectric anisotropics and optical constants et al.. After making the right mixture, the phase diagram will change, and the range between melting and clearing temperatures will be also larger than the individual constituents.

5

1.2-3 Nematic Liquid Crystal (NLC)

In general, the most common LC phase is the nematic phase. The LCs used in our works is also one of rod-like nematic LCs. The nematic (“thread” in Greek) LC has a high degree of long-range orientational order of the molecules. It differs from the isotropic liquid in that the molecules are spontaneously oriented with a common axis, labeled by a unit vector (or “director”). The states of the director nˆ and - nˆ are indistinguishable in nematic. The director usually varies from point to point in the medium, but a uniformly aligned sample is optically uniaxial with large birefringence. The rotational symmetry of the system can be broken by the particular boundary conditions (ex. Polyimide film, DMOAP coating…etc). The LC molecules around the particularly treated surface tend to orientate with some specific direction, which is the general method to align LCs [12].

6

If the LC molecules don’t orientate parallel to each other, the total energy of the system increases and the deformation exists. Three typical types of the deformation (splay, twist and bend) occur in nematic phase shown in Fig. 1.4. The deformation (or distortion) comes from the perturbations of the boundary condition, external field, chiral dopant …etc. The external field, such as electric field or magnetic field, can be used to re-orientate the LC molecules. Briefly, the LC molecules tend to be parallel or perpendicular (mostly parallel) to the field and the extra energy density due to the field is the function of the direction and magnitude of the field. We now consider a strong boundary case, the LC molecules around the boundary will be aligned and some deformation exists in the bulk due to the configuration of the lowest total energy. If we put the external electric/magnetic field on it, the total energy, F, can be described as

F =

∫

(f_d + f_E/M)dρ. (1.1) Figure 1.4 The three types of deformation occurring in nematics. The figure shows how each type may be obtained separately by suitable glass walls. (Reproduced from P. G. de Gennes and J. Prost, The Physics of Liquid Crystals, 2nd ed. , Oxford, New York, 1983)

7

Where the fd and fE/M are the extra energy from the deformation and the external

electric/magnetic field and ρ is the unit volume. For the parallel case (the LC molecules tend to orientate with the field direction), when the molecules orientate closer to the field direction, the fd increases and the fE/M decreases. The molecules will be aligned again with the request of

the lowest total energy for the stable state. In other words, the deformation term dominates mainly with weak field and the field term dominates for strong field. According to the continuum theory, the director, nˆ , spatially varies slowly and smoothly and the system keeps the long-range order.

As we have mentioned, the LC molecules tend to be parallel to the director such that the aligned nematic forms an optically uniaxial system. The difference between refractive indices measured with polarization parallel or normal to nˆ is quite large. The ordinary index no is

for light with the polarization perpendicular to the director and the extraordinary refractive index ne is for light with the polarization parallel to the director. The no and ne of E7 in visible

range are shown by Shin-Tson Wu [13]. The birefringence, or double refraction of the nematic, is defined as [14]

Δn=n_e−n_o. (1.2) When the incident polarization of light is not parallel to the director, the index of the refraction for the extraordinary ray should be written as

1/2 2 2 2 // 2 ) cos sin ( − ⊥ + = n n n_eff θ θ , (1.3)

where θ is the angle between the propagation direction of the incident light and the optical axis as shown in Fig. 1.5. Therefore, the effective birefringence isΔn=n_eff −n_o.

8

1.3 THz liquid crystals optics

Thanks to its relative large birefringence (Δn = ne - no) from the visible to microwave band,

liquid crystals (LCs), have found a varity of application ranging from display, communication and signal processing, beam steering, as well as their extraordinary and ordinary refractive indices (ne and no), are essential for the modulation of millimeter-wave, infrared, visible

region. Among the application of them, phase shifters [15-17], attenuator, polarizers and wavelength selection filters [18-20] have been demonstrated.

Composed of 4-cyano-4’-n-pentyl-biphenyl (5CB), 4-cyano-4’-nheptyl-biphenyl (7CB), 4-cyano-4’-n-octyloxy-biphenyl (8OCB) and 4-cyano-4’’-n-pentyl-p-terphenyl (5CT), the eutectic liquid crystal (LC) mixture E7, has been widely used in LC devices due to its large birefringence and wide nematic temperature range (-10°C~59°C). In order to pushing THz technology, the electro-optical phase shifters and filters in this region are all in high demand. Recently, E7-based voltage-controlled phase shifter, tunable Solc filter and tunable Lyot filter

Propagation direction

θ

Director of LC

Polarization

Figure 1.5 The angle,θ, between the director of LC molecules and the propagation direction of incident light will affect the effective index of extraordinary wave.

9

in THz range have been reported [17,21,22].

1.4 Thesis highlight

In Chapter 2, the experimental methods and sample preparation of PC antennas are described. For experimental setup, the THz-Time-domain Spectroscopy (THz-TDS), the temperature and humidity controlled system are introduced in detail. The temperature of the E7 sample was controlled with a fluctuation less than 0.1°C. Besides, the basic theories about THz radiation generated from dipole antenna on pulse mode are mentioned.

In Chapter 3, we report the development of measuring LC’s refractive indices in the recent years and the way of extraction of optical constants from THz-TDS we have used.

In Chapter 4, the results are analyzed and discussed, and we also compared the data with the visible range’s in this section. In the last chapter, the conclusions will be mentioned.

10

Chapter 2 Experimental Methods

2.1 The Preparation of LC cell and reference cell

A homogeneously aligned LC cell and a reference cell, schematically shown in Fig. 2.1, were prepared in this work. The LC cell was constructed by sandwiching the LC (E7, Merck) between two fused silica optical-grade windows. Thickness of the liquid crystal layer was controlled with Mylar spacers and measured by subtracting the substrates thicknesses from the total cell thickness. The LC layer thickness in this work is d = 0.552 ± 0.002 mm. We achieved homogeneous alignment of the nematic LC by rubbing t he spin-coated polyimide films on the substrates. The temperature of the liquid crystal cell can be varied and controlled to ± 0.05°C. The reference cell was constructed by two fused silica windows with nominal thickness of 3.0 mm each and in contact to each other.

Fig. 2.1 Sketches of (a) LC cell and (b) the reference cell. The substrates are fused silica and the alignment of LC cell is homogeneous.

11

We showed the procedure of preparing the LC cell as Fig. 2.2:

Fig. 2.2 the procedure of preparing the LC cell

Glass cutting:

1. Put on the gloves in order to avoid being pollute glasses and cut. 2. Choose fused silica glass or quartz glass.

3. Using the diamond knife-edge to cut the glasses.

4. Measure the thickness of the glasses by screw micrometer.

Glass cleaning:

1. Mount the glasses on Teflon foundation of the beaker; add water to surpass the glasses up to, and pour into the cleaner.

12

2. In order to eliminate the dusts and grimes on the glass, put the beaker into the supersonic oscillator and shake it for five minutes. After that, take it out and flush it by the distilled, then use the nitrogen to air-dry.

3. In order to eliminate the organism and oil-slick on the glasses, put the glasses into the beaker with acetone and shake it for five minutes.

4. In order to eliminate particles on the glasses, put the glasses into the beaker with methyl alcohol and shake it for five minutes.

5. Put the glasses into the beaker with pure water and shake it foe five minutes. The function of this step is for eliminating the residual solvent on the glasses.

6. After using nitrogen to air-dry, put the glasses into 120 degrees centigrade ovens in order to dry out the moisture.

Alignment layer coating and Parallel alignment:

1. Put the polyimide in the room temperature to warm for one hour.

2. Put the glasses on the spin coater, and use the burette to trickle the polyimide on the substrate. There are two steps when spining the glasses:

(a) The first step: 2000 rpm (50 sec) (b) The second step: 4000 rpm (50 sec) 3. Put the glasses into 170 degrees centigrade ovens to dry for one hour.

4. Put the substrate coated the polyimide film on the rubbing machine, and make the effect of parallel alignment.

Sealing the cell:

1. In order to remove the pellet and dust on the substrate, use the nitrogen to blow the glasses 2. Assemble two glasses substrate plates and fix the cell with the clips.

3. Spread the AB glue all around in the sample, but leave a gap for filling the LC.

Filling the LC cell and Sealing the gap:

1. The cell gap is enough wide, vacuum is not necessary. Drip LC into the cell directly, and pay attention that do not produce bubbles.

13

2. Seal the gap by using AB glue.

Sample checking:

1. Observe the sample outward appearance, whether the LC leak off. 2. Measure the sample thickness by screw micrometer

3. There is a simple way for sample checking, we put the NLC sample among cross polarizer, if aligned effect is good, we can discover slightly light to shade change when rotating the cell. I will show it as follows:

A set of crossed polarizer is placed before and after the LC sample, respectively. If aligned effect is good, we can discover slightly light to shade change when rotating the cell.

First, we define the original coordinate for the polarization of polarizer and another new coordinate for the optical axis of the LC sample, respectively. Between the two coordinates, there is an angle θ shown as Fig. 2.3:

Fig. 2.3 The conversion of the original coordinate (x y ) and the new coordinate (, _{x y )} '_, ' From the Jones Matrix, we can define the electric field passing through the polarizer is :

0 1 0 x P y _xy E E E E ⎡ ⎤ ⎡ ⎤ =_{⎢ ⎥}= _{⎢ ⎥} ⎣ ⎦ ⎣ ⎦ (2.1) We convert the original coordinate into the new coordinate, and showing as Eq. (2.2):

14

0 0

' '

cos sin 1 cos

E

sin cos 0 sin

LC X Y E θ θ E θ θ θ θ ⎛ ⎞⎛ ⎞ ⎛ ⎞ ′ = _{⎜ ⎟⎜ ⎟}= _{⎜ ⎟} − − ⎝ ⎠⎝ ⎠ ⎝ ⎠ (2.2) After changing the coordinate of reference, the electric field passing through the LC sample can be described as Eq. (2.3):

0 0 cos 0 cos E sin 0 sin e e o o in kd in kd LC in kd in kd e e E E e e θ θ θ θ ⎛ ⎞⎛ ⎞ ⎛ ⎞ ′ = _{⎜ ⎟⎜ ⎟}= _{⎜ ⎟} − − ⎝ ⎠ ⎝ ⎠ ⎝ ⎠

(2.3)

Here, we have to convert the Jones matrix into the original coordinate, and we will derive the

Eq. (2.4):

2 2

0 0

cos sin cos cos sin

E

sin cos sin cos sin sin cos

e e o o e o in kd in kd in kd LC in kd in kd in kd e e e E E e e e θ θ θ θ θ θ θ θ θ θ θ θ − ⎛ ⎞ ⎛ + ⎞ ⎛ ⎞ = _{⎜ ⎟ −⎜ ⎟}= _{⎜ ⎟} − ⎝ ⎠⎝ ⎠ ⎝ ⎠

(2.4)

After passing through the analyzer, the electric field will just exist the y direction, and the result is shown as Eq. (2.5):

2 2

0 0

0 0 cos sin 0

E

0 1 sin cos sin cos sin cos sin cos

e o e o e o in kd in kd A in kd in kd in kd in kd XY e e E E e e e e θ θ θ θ θ θ θ θ θ θ ⎛ + ⎞ ⎛ ⎞ ⎛ ⎞ = _{⎜ ⎟⎜ ⎟}= _{⎜ ⎟} − − ⎝ ⎠⎝ ⎠ ⎝ ⎠

(2.5) If we define the // 2 n n n₌ + ⊥ _and // n n n_⊥

Δ = − , the electric field of y will be derive as

Eq. (2.6):

2 2

0 0

0

ˆ ˆ

E sin cos sin cos y sin cos ( ) y

ˆ

sin cos (2 sin ) y

2 e o n n i kd i kd in kd in kd inkd A inkd E e e E e e e nkd E e i θ θ θ θ θ θ θ θ Δ ₋Δ = − = − Δ = K

(2.6)

From the information of electric field, we can derive the optical intensity as Eq. (2.7):

2 _{2 2} 0 E sin (2 )sin ( ) 2 A I = K =I θ ΔΦ

(2.7) Here, I is the intensity of incident light, ₀ θ is the angle between the polarizer and the optical axis of the LC sample, and ΔΦ is the phase shift caused by passing through the LC

15

sample. From Eq. (2.7), we know that the optical intensity is the highest as θ is 45o_{, and}

16

2.2 Temperature and humidity controlled system

In order to obtain the temperature dependence of the LCs’ refractive indices, an accuracy temperature controlled system is necessary. We put the cell in a copper box, which has two open windows for THz wave passing. The copper box was covered by Teflon with the thickness of 0.5cm for keeping the temperature inside. The copper box also has two small holes with the diameter of 3.0 mm for resistant type temperature probes (YSI 423, Yellow Springs Instrument Co., Inc.) and two larger holes with 1.0-cm-diameter for cylinder current heaters. One of the probe and two heaters were connected to a temperature controller (YSI Model 72, Yellow Springs Instrument Co., Inc.) in order to set the temperature of the cells. Another probe, which directly contacts the cell was connected to a voltage multimeter for probing the actual cell temperature by reading the resistance of the probe. Fig 2.4 shows the pictures of the temperature controlled sample holder. Fig. 2.5 is shown the temperature controller and the probe, respectively. The fluctuation of the temperature of the cell is less than 0.04°C. The testing data is shown in Fig. 2.6. This temperature control system was employed in this work for providing a stable temperature control of the LC cells and the reference cells.

Fig. 2.4 The pictures of the temperature-controlled sample holder, which consists with a copper oven (inside) and a Teflon cover (outside).

17

Fig. 2.6 The testing result which is set at 26oC of the temperature control system.

Fig. 2.5 The pictures of the temperature controller and the probe (YSI 423, Yellow Springs Instrument Co., Inc.)

0 5 10 15 20 25 30 24.0 24.5 25.0 25.5 26.0 26.5

Tem

p

erature(

o

C)

Time (Minute)

20 21 22 23 24 25 25.5 25.6 25.7 25.8 25.9 26.0 26.1 26.2 26.3 Te m p erature( o C) Time (Minute) 26 0.1_± o_C

18

There are several absorption lines of water in the range of 0.2 to 1.5 THz [23]. We found that those water lines will affect the accuracy of the measured results. According to this, a dry environment is necessary for obtaining accurate results. We used acrylic plates to make a box, which covers whole THz system to approach a close system. The THz-TDS was purged with dry nitrogen so that it could maintain at a relative humidity of (4.0 ± 0.5) %. Fig. 2.7(a) and

2.7(b) show the THz temporal profile and the frequency spectra after purging, respectively.

From the Fig. 2.7(b), the determination of Signal-Noise-Ratio can reach 106 for certain.

0 5 10 15 20 25 30 35 40 45 50 -2.0x10-3 -1.0x10-3 0.0 1.0x10-3 2.0x10-3 3.0x10-3 4.0x10-3

TH

z F

ield(a.u.)

Time Delay(ps)

Air(53%)

Purge(4%)

Fig. 2.7(a). Temporal profiles of THz before (solid line) and after (dash line)purging.The relative humidities before and after purging are 53% and 4%, respectively.

19

2.3 THz Time-domain spectroscopy (THz-TDS)

2.3-1 Introduction

Characterization of LCs and LC devices in the THz frequency range are performed using THz time-domain spectroscopy (TDS) [5]. THz-TDS is a powerful technique that allows broadband THz generation and detection of the field amplitude and phase of the THz wave. A schematic representation of our setup is depicted in Figure 2.8. The light source is a mode-lucked femtosecond Ti:sapphire laser (Spectra Physics Tsunami). The pulses are split into two beams; one constitutes the so-called pump or excitation beam, the other is referred as the probe beam. The pump pulses impinged on a low-temperature grown Gallium Arsenide

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 10-20 10-19 10-18 10-17 10-16 10-15 10-14 10-13 10-12 10-11

Power(a.

u

.)

Frequency(THz)

Air(53%)

Purge(4%)

Fig. 2.7(b). Frequency domain of THz before (solid line) and after (dash line)purging.The relative humidities before and after purging are 53% and 4%.

20

(LT-GaAs) photoconductive dipole antenna, acts as a THz emitter. The generated THz pulses are collected and guided by gold-coated parabolic mirrors onto the sample. The incident THz waves on the sample can be also focused by another parabolic gold mirror. The transmitted pulses from the sample are focused by another parabolic gold mirror onto the THz detector, which in our case is also a photoconductive antenna.

2.3-2 Theory of generating THz radiation

Many groups have made great efforts to develop THz radiation sources in both continuous – wave and pulsed way. Free-Space Electron Laser [24,25], and backward wave oscillators [26] are electron beam sources that generate relatively high power signals at the THz region. By injecting a CO2 pump laser light into a cavity filled with a gas that lases, one

21

also can obtain THz radiation [27,28], and the lasing frequency is fixed dictated by the filling gas. In our experimental methods, the optical rectification and resonant THz radiation in photoconductor generated by a pulse laser are used. Here, the two methods will be introduced as follows, respectively.

2.3-2-1 Optical rectification

One of methods to generation THz. is to make use of the Free-Space Electro-Optic Sampling (FSEOS) which based on an emission of optical rectification [29,30]. This effect as shown in Fig. 2.9 arises from the second ordered susceptibility χ of nonlinear crystals. The susceptibility, χ =P/ε₀E measures the degree of polarization which caused by an electric field applied to a dielectric material. Optical rectification in a non-absorbing medium is a process in which a laser pulse creates a time-dependent polarization that radiates an electric field which can be written as

2 2 P E (t) t ∂ ∝ ∂ JK JK

in the far field, where the polarization P follows the pulse intensity envelope. It is called rectification because the rapid oscillations of the electric field of the pulse laser are “rectified” and only the envelope of the oscillations remains. In a material subjected simultaneously to waves of frequencies w and ₁ w , the P ₂ will include the form of cos( )cos( ) cos(wt₁ w t₂ =

[

w w t₁+ ₂) cos(+ w w t₁− ₂) / 2

]

. Thus both sum and difference frequencies as a result of the beats that form between the different frequencies will be generated [31]. The THz pulses consist of a broad range of spectrum, from zero up to the bandwidth of the visible region, and bandwidth as large as 30THz has been obtained using this generation mechanism. On the other hand, FSEOS has typically lower power because the

22

power of THz pulse comes from the incident laser pulses entirely.

2.3-2-2 Photoconductive(PC) switch

Biased PC switch can be used to generate THz radiation. It is generally based on the “current surge model” [32] which will be discussed in detail in the next chapter. According to this model, as the energy of the input laser is higher than the energy of bandgap of photoconductor, then electron-hole pairs are excited and the mechanism to generated THz radiation can occur. Semiconductor such as GaAs (Eg=1.43eV), with the surface bands of a semiconductor lie within the energy produced by input photon and thus Fermi-level pinning occurs, leading to band bending and formation of a depletion region.

In briefly, the electromagnetic field of THz radiation is generated from a transient current which is generated on the surface of the photoconductor. Carriers are generated by the pulse laser instantaneously. In order to accelerate the carriers we add bias across the PC antenna. Then, the resultant transient current, or called current surge, produces an electric field on the surface of the photoconductor. This surface electric field is regarded as the source of the THz radiation. Figure 2.10 shows the setup for this method.

Nonlinear crystal

Laser pulse

THz radiation

23

2.3-2-3 Current-surge model

The first step, there are some time-average parameters which we need to define the THz radiation [33].

Charge density⇒ρ(x,y,z,t) Current density⇒J(x,y,z,t)K

Electric field intensity⇒E(x,y,z,t)JK Magnetic flux density⇒B(x,y,z,t)JK

Then, it is necessary to construct Maxwell’s equation for current-surge model. According to Maxwell’s equation [34]: B E=-t ∂ ∇× ∂ JK JK (Faraday’s Law) (2.8) E=ρ ε ∇⋅JK (Gauss Law) (2.9) D H=J+ t ∂ ∇× ∂ JK JK K (Ampere’s Law) (2.10)

Dual-dipole antenna structure

Laser pulse

bias

THz radiation

24 B=0 ∇⋅JK (2.11) And we know B=∇×A JK JK (2.12) From Eq. (2.8) and (2.12), we can obtain:

(

)

B A A E=- =- A = - E+ =0 t t t t ⎛ ⎞ ⎛ ⎞ ∂ ∂ _⎜ ∂ _{⎟⎟ ⎜} ∂ _⎟⎟ ∇× ∇× ∇×⎜_{⎜ ⎟⎟} ⇒ ∇×⎜_{⎜ ⎟⎟} ⎜ ⎜ ∂ ∂ ⎝ ∂ ⎠ ⎝ ∂ ⎠ JK JK JK JK JK JK (2.13)

Then, we set a non-vector value V and employ that ∇×∇V=0 to substitute the Eq.

(2.13). A - V=E+ t ∂ ∇ ∂ JK JK _A E=- V-t ∂ ⇒ ∇ ∂ JK JK (2.14)

From Eq. (2.10)、(2.12) and H=B μ JK JK 、D= EJK εJK,we obtain B E =J+ t ε μ ∇× ∂ ∂ JK JK K

(

A = J+

)

E t μ⎛⎜ ε∂ ⎟⎞⎟ ⇒ ∇× ∇× ⎜_{⎜ ⎟⎟} ⎜ ∂ ⎝ ⎠ JK JK K (2.15)

Substitute Eq. (2.14) into Eq. (2.15)

(

)

(

)

2 2 2 A A = J+ - V-t t V A A - A= J- -t t μ ε μ με με ⎡ _∂⎛_{⎜ ∂ ⎟}⎞⎤ ⎢ _⎥⎟ ∇× ∇× _⎢ ⎜_⎜ ∇ _⎟⎟⎥ ⎜ ∂ ⎝ ∂ ⎠ ⎢ ⎥ ⎣ ⎦ ∂ ∂ ∇ ∇⋅ ∇ ∇ ∂ ∂ JK JK K JK JK JK K 2 2 2 A V A- =- J+ A+ t t με∂ μ ⎛⎜ με∂ ⎟⎞ ⇒ ∇ _∂ ∇ ∇⋅⎜_{⎜⎝ ∂} ⎟_⎟⎠ JK JK K JK (2.16)

From Eq. (2.9) and D= EJK εJK

( )

E =- V+ A = t ε ⎡⎢ε⎛⎜ ∂ ⎟⎞⎤⎥⎟ ρ ∇⋅ ∇⋅_⎢ ⎜_⎜∇ _⎟⎟⎥ ⎜ ∂ ⎝ ⎠ ⎢ ⎥ ⎣ ⎦ JK JK

25

(

)

2_{V+ A} =-t ρ ε ∂ ⇒ ∇ ∇⋅ ∂ JK (2.17)

Due to Lorentz gauge assumption A+ V=0

t εμ ∂ ∇⋅ ∂ JK , Eq. (2.16) becomes as 2 2 2 A A- =- J t με∂ μ ∇ ∂ JK JK K (2.18) And, Eq. (2.17) can be written as

2 2 2 V V- =-t ρ με ε ∂ ∇ ∂ (2.19)

Eq. (2.18) and (2.19) are the two inhomogeneous wave equations written in terms of

AJKand V. The two wave equations are used to determine a functional and time dependent of the radiated electric field in the far field.

From Eq. (2.10), the continuity equation of the free carriers is obtained.

(

)

D H = J+ = J+ =0 t t ρ ⎛ _∂ ⎞_{⎟ ∂} ⎜ _⎟ ∇⋅ ∇× ∇⋅⎜_{⎜ ⎟⎟} ∇⋅ ⎜ ∂ ∂ ⎝ ⎠ JK JK K K (2.20)

In the fact, the current density of the photoconductor antenna which adds the bias is the

transverse current, parallel the surface of the PC antenna and perpendicular the direction of

propagation. So that

J=0

∇⋅K (2.21) From Eq. (2.20) and (2.21), we can deduce that the charge density does not vary with time and not contribute the time dependent radiated electric field. As the result, Eq. (2.14) becomes

26

( )

E r,t =- A(t) t ∂ ∂ JK K JK (2.22) The solution of the vector potential AJK in Eq. (2.18) leads into the Eq. (2.22) to express the time dependent radiated electric field

E (r,t)

rad

JK

K

at the displacement rKfrom the center of the PC antenna. s rad ₂ 0 r-r' J r',t-c 1 E (r,t)=- da' 4πεc t _r-r' ⎛ _⎞⎟ ⎜ _⎟ ⎜ _⎟ ⎜ _⎟ ⎜ _⎟ ⎜ ⎟ ⎜ ∂ ⎝ ⎠ ∂

∫

K K K K JK K K K (2.23)

where ε is the permittivity of free space, c is the speed of light in vacuum, ₀ Js

K

is the surface current of the PC antenna in the retarded time, and da' is the increment of surface area at the displacement r'K from the center of the PC antenna. The integration is covered with whole the optically illuminated area of the PC antenna. In the far field,

n r' r-r' =r 1- r r ⎛ _{⋅ ⎟}⎞ ⎜ _⎟ ⎜ _⎟≈ ⎜ _⎟⎟ ⎜⎝ ⎠ K  K K (2.24)

The gap between the electrodes of the PC antenna is assumed to be uniformly illuminated by the optical source. Therefore, the surface current Js

K

can be set as a constant in space. Then, the radiated electric field in Eq. (2.23) can be written as

rad ₂ s 0 1 A r E (r,t)=- J t-4πεc r t c ⎛ ⎞ ∂ _{⎜ ⎟} ⎟ ⎜ _⎟ ⎜⎝ ⎠ ∂ JK K K (2.25)

where A is the illuminated area of the PC antenna and _{r= x +y +z .We considered that the} 2 2 2 THz radiation is generated on Z axis (i.e. x = y = 0 ), and let t t- z

c ⎛ ⎞⎟ ⎜

→ _{⎜ ⎟⎜⎝ ⎠}⎟. Thus, The Eq. (2.25) can be written as

( )

rad ₂ s 0 1 A E (r,t)=- J t 4πεc z t ∂ ∂ JK K K (2.26)

27

Beside, we can know the Eq. (2.27) from surface current [35].

0 ( ) J ( ) ( ) 1 1 s b s s t E t t n σ σ η = + + K K (2.27)

where n is the refractive index of photoconductor antenna under the wavelength of mμ . η ₀ is the impedance in free space, and σ_s( )t is the surface conductivity which is shown in the

Eq. (2.28). (1 ) ( ') ( ) t ' ( , ') ( ') exp[ ] s opt car e R t t t dt m t t I t w σ τ −∞ − − − ≡

_∫

= (2.28)

where e is the electric charge, R is the reflectivity of photoconductor antenna, =w is the photon energy, ( , ')m t t is the time-dependent carrier mobility at time t from created carrier at time t', I is the time-dependent of optical intensity, and _opt τ is the carrier lifetime of _car excited carriers. For the present derivation, we assume that carrier mobility is a constant.

( , ')

m t t = (2.29) m And assume that the carrier lifetime is long enough, τ → ∞ . A Gaussian intensity _car profile of the optical beam is assumed:

2 0 2 ' ( ') exp( ) opt t I t I τ − = (2.30) From these assumptions, the surface conductivity becomes

2 0 2 (1 ) ' ( ) t ' exp( ) s e R t t I dt m w σ τ −∞ − − ≡

_∫

= (2.31) From the Eq. (2.29) lead to the Eq. (2.26).

2 2 2 0 0 rad _{2 0 2} 0 (1 ) 1 A (1 ) E (r,t)=- exp( ) 1 exp( ) 4 c z ( 1) t e R I m e R t I m x dx w n w τ η _τ πε τ − −∞ ⎡ ₋ ⎤ − − _{× +⎢ − ⎥} ⎢ ₊ ⎥ ⎣

∫

⎦ JK K = = (2.32)

Comparing with the result from experiment, it is necessary to rewrite the Eq. (2.30) in term of the experimental parameters E which is the bias electric field applied across the _b

28

photoconductor, and F which is the incident optical intensity. _opt

2 0exp( 2 ) 0 opt opt E t F I dt I A π τ τ ∞ −∞ − =

_∫

= ≡ (2.33) Where E is the average optical energy and A is the area of the incident optical beam. Then _opt we define the parameter B and D to simplify the equation.

2 0 (1 ) 4 Ae R m B c z w πε π − = = (2.34) 0 (1 ) ( 1) e R m D n w η π − = + = (2.35) And then, the electric field in the far field can be written as

2 2

2 2

( ) opt exp( ) 1 t exp( )

rad b opt F t E t BE DF τ x dx τ τ − −∞ − ⎡ ⎤ = − × +⎢_⎢ − ⎥_⎥ ⎣

∫

⎦ (2.36)

Beside of current surge model, we can also discuss on the point of the carrier dynamics in semiconductor to analyze the THz generation by Drude-Lorentz model when we need to discuss the factor of material in PC antenna [36,37].

 

2.3-2-4 Drude-Lorentz model

For the calculation of carrier transport and THz radiation in a biased semiconductor, the one-dimensional Drude-Lorentz model is used. When a biased semiconductor is pumped by a laser pulse with photon energies greater than the band gap of the semiconductor, electrons and holes will be created in the conduction band and valence band, respectively. The carrier pumped by ultrashort laser pulse is trapped in the mid-gap states with the time constant of the carrier trapping time. The time-dependence of carrier

29

density is given by the following equation.

c (t) (t) (t) dn n G dt = − τ + (2.37) Where n(t) is the density of the carrier, G(t) is the generation rate of the carrier by the laser pulse, and τ is the carrier trapping time. The generated carriers will be accelerated _c by the bias electric field. The acceleration of electrons (holes) in the electric field is given by , , , , (t) (t) e h e h e h s e h d q E dt m ν ν τ = − + (2.38)

Where ν_{e h,} (t) is the average velocity of the carrier, q is the charge of an electron (a _{e h,} hole), m is the effective mass of the electron (hole), _{e h,} τ is the momentum relaxation _{e h,} time, and E is the local electric field. The subscript e and h represent electron and hole, respectively. The local electric field E is smaller than the applied bias electric field E _b due to the screening effects of the space charges,

b

P

E E

αε

= − (2.39) Where P is the polarization induced by the spatial separation of the electron and hole, ε is the dielectric constant of the substrate and α is the geometrical factor of the PC material. The geometrical factor α is equal to three for an isotropic dielectric material. It is noted that both, the free and trapped carriers contribute to the screening of the electric field. The time dependence of polarization P can be written as

30 dP dt _r P J τ − + ＝ (2.40)

Where τ is the recombination time between an electron and hole. In the Eq. (2.38), J is the _r density of the current contributed by an electron and hole,

h e

J=enν −enν (2.41) Where e is the charge of a proton. The change of electric currents leads to electromagnetic radiation according to Maxwell’s equations. In a simple Hertzian dipole theory, the far-field of the radiation E_THz is given by

THz J E t ∂ ∝ ∂ (2.42) To simplify the following calculations, we introduce a relative speed ν between an

electron and hole,

h e

=

ν ν − (2.43) ν Then the electric field of THz radiation can be expressed as

THz n E e +en t t ν ν∂ ∂ ∝ ∂ ∂ (2.44) The first ter m on the right ha nd si de of t he Eq. (2.42) represents the electromagnetic radiation due to the carrier density change, and the second term represents the electromagnetic radiation which is proportional the acceleration of the carrier under the electric field.

2.3-3 Theory of detecting THz radiation

31

PC antennas is the inversion of the THz generation because the photocarriers detected as a current are accelerated by the electric field of incident THz radiation instead of the bias voltage. The photocurrent response is the convolution of the optical pulse duration and the impulse current of the photoswitch across the photoconductive antenna on pulse mode. The photocurrent at a time delay t is shown as the following Eq. (2.40):

( ) ( ) ( )

J t eμ ∞ E τ N τ t dτ

−∞

=

∫

− (2.40) Where ( )N τ is the number of photocreated carrier, and ( )E τ is the incident electric field of the THz radiation.

There are two factors determine the spectral bandwidth in this detector. One of them is the photocurrent response (i.e. carrier lifetime) and the other one is the frequency dependence of the antenna structure [39,40]. In general, the low frequency cutoff of the detectors results from the collection efficiency of the dipole, while the upper frequency limit is determined by the photocarrier response.

2.3-4 Antenna-based emitter and detector

While the THz transmitter and receiver designs were identical in early THz TDS works, a wide range of different structures have been investigated, and optimized either for maximum signal or maximum bandwidth. Grischkowsky and coworkers pioneered the coplanar strip line for use as a THz emitter. This structure is appealing for both of the simplicity of the design and the extremely broad emission band [41].

32

50μm in length [5]. The collection sensitivity and radiation efficiency of the antenna both vary inversely with the wavelength. Thus, smaller dipole antennas provide a broader bandwidth response; dipoles as small as 30μm have been used. Of course, this 1/λ dependence no longer apply when the radiation wavelength in the substrate becomes comparable to the dipole length, so the details of the high frequency response are more complex. In addition, the low frequency roll-off in these quasi-optical systems is typically limited by diffraction effects due to the finite size of optical elements such as the substrate lenses. As a consequence, the 1/λ dependence may apply over only a rather limited spectral range. A detail analysis of THz-TDS system can accurately predict the measured spectral response, as long as all of these effects are taken into account.

If very small dipole antennas are used for both emission and detection, the high frequency limit is mainly determined by the temporal response of the system [30,42,43]. Another important limitation factor is the response time of the photoconductive material used for substrates of the THz antennas. This limit is more pronounced in the receiver antenna, since one does not expect the highest measured frequency to exceed the inverse of the temporal width of the photo-generated sampling gate, this duration is limited by the carrier trapping time, although a number of schemes have been proposed to avoid this limitation [44,45].

It should be noted that it is possible to detect broadband THz radiation even using a photoconductivity antenna with a slow carrier lifetime. In this case, one relies not on the width of the sampling gate, but merely on its fast-rising edge. If the photocurrent can be modeled as a step function, then the detector operates as a fast sampling gate in an integrating mode, and the measured signal is proportional to the integral of the incident THz field [44]. In this configuration, the bandwidth is limited by the speed of the rising edge of the current, which is determined by the duration of the optical pulse, and by the RC time constant of the

33

antenna [46].

2.3-5 THz-TDS with collimated beam system

For determination of the optical constants, we employ an antenna-based THz-TDS system with a collimated beam at the sample position [47]. A schematic of the experimental setup is shown in Fig. 2.11. Briefly, a mode-locked Ti: Sapphire laser (Spectra Physics, Tsunami, λ = 800 nm) generating 35 fs pulses at a repetition rate of 82 MHz with 400mW average output power is divided into two beams, one constitutes the so-called pump or excitation beam, the

other is the probe beam. The former, with an average power of 35mW irradiated a

low-temperature grown Gallium Arsenide (LT-GaAs) PC dipole antenna that was biased at 5V. The generated THz pulses, are collimated by a gold-coated parabolic mirrors onto the LC sample at normally incidence. The transmitted pulses from the sample are focused onto a detector which in our case is also a PC antenna by the second parabolic gold mirror. In order to make sure the polarization state of THz wave, a pair of parallel wire-grid polarizers was used before and after the LC cell respectively.

34

Fig. 2.11 Schematic of the THz-TDS. PBS: polarization beam splitter. P and A: THz polarizer and analyzer, respectively.

35

Chapter 3 Extraction of Optical Constants

from THz Time-domain Spectroscopic

Measurements

3.1 Introduction

Composed of 4-cyano-4’-n-pentyl-biphenyl (5CB), 4-cyano-4’-nheptyl-biphenyl (7CB), 4-cyano-4’-n-octyloxy-biphenyl (8OCB) and 4-cyano-4’’-n-pentyl-p-terphenyl (5CT), the eutectic liquid crystal (LC) mixture E7, has been widely used in LC devices due to its large birefringence and wide nematic temperature range (-10°C~59°C).

In the visible frequency range, group of Wu et al. has proposed the refractive indices in the range of 450~656 nm [48], and the ne increases from 1.73 to 1.80, while no varies from 1.52 to

1.54. In the near infrared (IR) region, the same author has offered Δn = 0.186 in the 1.55 μm [49]. After that, Brugioni et al. also studied Δn = 0.183 (ne = 1.684, no = 1.500) in the same

wavelength [50]. In the intermediate IR region, Wu et al. has also reported Δn increases from 0.18 to 0.19 in the frequengy range of 52~75 THz [51]. In the far IR region, both of Wu and Brugioni have introduced ne = 1.69 and no =1.49 (Δn = 0.20) in the 10.6 μm [49,52]. In the

microwave region, K. C. Lim et al. has found the Δn of E7 at the 30 GHz was 0.192, and the ne and no were 1.790 and 1,598, respectively [15]. In the recent years, Yang and J. R. Sambles

et al. have also investigated ne =1.800 and no = 1.647 in the frequency range 26.5~40.0 GHz

[53]. After that, ne =1.78 and no = 1.65 were also determined in the 40~75 GHz by the same

author [54,55]. Recently, by photomixing based on log periodic circularly toothed planar antenna, M. Koeberle et al. has reported ne = 1.77, no = 1.64 (Δn = 0.13) at the 300 GHz [56].

36

These references mentioned we list in the Table. 3.1.

Above of all, the physical and chemical properties of E7 have been studied extensively, however, little research has been done between far IR and microwave (0.2 THz~1.4 THz).

Band Wave length

Freq. ne no Bire. Ref.

Microwave 7.5~11.3 (mm) 26.5~40 (GHz) 1.800 1.647 0.153 Yang FZ, Applied Physics Letter 79 (22), 3717-3719, 2001 10 (mm) 30(GHz) 1.79 1.598 0.192 K. C. Lim, Applied Physics Letter 62 (10), 1993 5~7.5 (mm) 40~60 (GHz) 1.78 1.65 0.13 Yang FZ, Liquid Crystals 30 (5), 599-602, 2003 4~6 (mm) 50~75 (GHz) 1.782 1.654 0.128 Yang FZ, Applied Physics Letter 81 (11), 2047-2049, 2002 1 (mm) 300 (GHz) 1.77 1.64 0.13 M. Koeberle, T. Gobel Far IR 10.21~ 10.768 (um) 27.9~ 29.4 (THz) ~ ~ 0.227 Shin-Tson Wu, Physical Review A, Vol 33, Number 2 (1986) 10.6(um) 28.3 (THz) 1.691 1.493 0.198 Shin-Tson Wu, Journal of Applied Physics 97, 073501 (2005) 10.6(um) 28.3 (THz) 1.694 1.495 0.199 S. Brugioni,

Infrared physics & Technology 46,

17-21 (2004) Table. 3.1 The references measuring optical constants of E7 in different frequency

37 Far IR 8.939~ 9.522(um) 31.5~ 33.6 (THz) ~ ~ 0.212 Shin-Tson Wu, Physical Review A, Vol 33, Number 2 (1986) Intermediate IR 5.539~ 5.884(um) 51~ 54.2 (THz) ~ ~ 0.181 Shin-Tson Wu, Physical Review A, Vol 33, Number 2 (1986) 3.628~ 4.317(um) 69.5~ 82.7 (THz) ~ ~ 0.192 Shin-Tson Wu, Physical Review A, Vol 33, Number 2 (1986) 3.511~ 3.993(um) 75.13~ 85.45 (THz) ~ ~ 0.145 N. J. DIORIO JR, Liquid Crystals, Vol. 29, No. 4, 589-596 2.513~ 3.364(um) 89.18~ 119.4 (THz) ~ ~ 0.158 N. J. DIORIO JR, Liquid Crystals, Vol. 29, No. 4, 589-596 Near IR 1.611~ 2.833(um) 106~ 186 (THz) ~ ~ 0.191 Shin-Tson Wu, Physical Review A, Vol 33, Number 2 (1986) 1.55 (um) 193.5 (THz) 1.693 1.507 0.186 Shin-Tson Wu, Journal of Applied Physics 97, 073501 (2005) 1.55 (um) 193.5 (THz) 1.684 1.500 0.183 S. Brugioni,

Infrared physics & Technology 49, 210-212 (2007)

Visible

656(nm) 457 (THz)

1.726 1.518 0.208 Jun Li, Shin-Tson

Wu, IEEE/OSA Journal of display technology, Vol. 1, No. 1(2005) 633(nm) 474 (THz)

1.731 1.519 0.212 Jun Li, Shin-Tson

Wu, IEEE/OSA Journal of display technology, Vol. 1,

38 Visible 633(nm) 474 (THz) ~ ~ 0.22 Shin-Tson Wu, Physical Review A, Vol 33, Number 2 (1986) 589 (nm) 509 (THz) 1.74 1.52 0.22 Shin-Tson Wu, Journal of Applied Physics 97, 073501 (2005) 589 (nm) 509 (THz)

1.739 1.523 0.216 Jun Li, Shin-Tson

Wu, IEEE/OSA Journal of display technology, Vol. 1, No. 1(2005) 546 (nm) 549 (THz)

1.751 1.527 0.224 Jun Li, Shin-Tson

Wu, IEEE/OSA Journal of display technology, Vol. 1, No. 1(2005) 486 (nm) 617 (THz)

1.776 1.535 0.241 Jun Li, Shin-Tson

Wu, IEEE/OSA Journal of display technology, Vol. 1, No. 1(2005) 450 (nm) 667 (THz)

1.801 1.542 0.259 Jun Li, Shin-Tson

Wu, IEEE/OSA Journal of display technology, Vol. 1,

No. 1(2005)

3.2 Determination of optical constants

For the purpose of deriving the complex optical constants without Kramers-Kronig relation, the field amplitude and phase of THz wave transmitted through the reference and the LC cell were used.

Assuming THz wave is a plane wave passing through the cells from one side to another side, and the electric field of the THz goes through the reference cell was given by,