### 國立臺灣大學電機資訊學院電子工程學研究所 碩士論文

### Graduate Institute of Electronics Engineering College of Electrical Engineering & Computer Science

### National Taiwan University master thesis

### 菱形晶格週期性金屬孔洞電漿子熱輻射器和窄頻共振 腔型熱輻射器之特性研究

### The Characteristics of Plasmonic Thermal Emitters with Top Metal Perforated by Hole Array Arranged in Rhombus Lattice and the Study of Narrow Bandwidth

### Cavity Thermal Emitters

### 吳 奕 廷 Wu Yi-Ting

### 指導教授：李嗣涔 博士

### Advisor: Lee Si-Chen, Ph.D.

### 中華民國 98 年 7 月

### July, 2009

**致謝 **

在台大的日子，時光像飛梭般前進，不懂的東西太多，對實驗室的貢獻太少，

好不容易有了那麼一丁點的進展，才知道魔鬼總是藏在細節裡，越是看似簡單的 東西，越需要細心、耐心和提早行動絕不推拖拉的責任心。首先，我要衷心的感 謝我的指導教授 李嗣涔博士。老師開放的心胸和做學問的態度深深影響著我，

老師總是對各種事物保持好奇心和敏銳的洞察力。這兩年在老師的悉心指導下，

獲益良多！老師永遠恭謙的態度、實事求是的精神、精湛的分析力和過人的體 力，深深為我所欽佩；老師在百忙之中時仍能詳細的指引學生探究學術的真理，

並在論文寫作上給予極大的幫助，我深深的感動並受用於指導教授的付出和關 懷，希望老師多注意身體的健康。再來，我要感謝口試委員呂學士教授、管傑雄 教授、林浩雄教授、劉致為教授、張宏鈞教授還有林致廷教授們抽空閱覽本論文 並提出寶貴建議和指導，教授們學有專精的指導提升了本論文的品質和正確性。

除了國中短暫的補習兩年外，從小到大我都不曾補過習，所以學校老師教的 好壞深深影響著我學習的成效，進研究所後我才明白台大之所以成為台大的地的 方：一流師資認真教學、一流同儕認真討論；這兩年修課的老師們都教的相當好，

同儕間也異常的認真，我特別感謝光電所張宏鈞老師、台積電林本堅博士、電子 所郭宇軒、胡振國老師，還有電信所吳宗霖老師，他們紮實而精闢的講解，深深 的補足了我理論方面的不足，浸潤於大師們的丰采是我生命中可以帶走的盛筵。

接著我要感謝交通大學電工所崔秉鉞教授，他讓我明白「態度」的重要性，

做任何事情不是玩家家酒，做任何事都要「玩真的全力以赴」；我還要感謝交通

大學電工所林鴻志教授，教授說：「交大、台大畢業的研究生企業都會搶著要，

可是你要怎麼說服你的老闆十年後繼續用你，而不是用當年交大、台大應屆畢業 的研究生？」這絕對是我人生中最值得拿來警惕的一段話！特別是在這金融海嘯 肆虐的非常年代，我在此表達我對兩位老師高度的敬意和深深的歉意。

接下來我要感謝我大學時的恩師，李佩雯老師、郭明庭老師還有洪志旺老 師，洪老師說你看不懂是因為你看的不夠久；郭老師說你看不懂是因為你看的書 不夠多，一本書看不懂看五六本就會懂，這兩句話真是令我受用無窮！然而我最 感激的還是李佩雯老師，李佩雯老師生動有趣的教學引領我走向奈米電子的道 路，我永遠記得老師「次微米元件物理與技術」這門課，老師出的作業直接翻書 永遠找不到答案，惟有深入的思考和廣泛的查資料才能作答，這樣的啟蒙訓練對

我碩士班的研究提供了莫大的幫助。我也要感謝國小的石玉美老師、國中的蘇媚 慧老師和萬家春校長，謝謝你們的栽培，指引我走向正確的道路。我也要感謝高 中文組班的陳淑娟同學，妳說：「為什麼要認為轉組是浪費半年？轉組是給自己 再一次的選擇機會！」沒有妳我沒有勇氣從文組轉到理組去追尋夢想，也不可能 拿到台灣大學電子工程研究所的碩士學位。

接下來，我要感謝 SP 組的各位：首先是聰哥、昱維、宜函和大慶，謝謝你 們耐心地帶我作實驗以及無私地給予我寶貴的建議和協助；謝謝張議聰學長和葉 宜涵學長，你們無私的幫忙、亦師亦友的督促和討論，是我不放棄的力量；謝謝 江昱維學長和跳級生莊大慶學長(跳級了所以是學長)，雖然我們之間的討論十句 話總有八句是很好笑的垃圾話 XD，但總是有那麼一兩句是切中了問題的要害、

直搗問題的核心，兩年來著實裨益良多；謝謝張沛恩學弟，謝謝你各方面強大的 火力支援，和您的每一次討論都能讓我在研究和分析上得到豁然開朗的快感，很 好很強大！我還要衷心的感謝再感謝蔡尚儒學長，謝謝你在我累的半死曝光曝不 出來的時候及時的發明了濾紙曝光密技，沒有它我可能至今一事也無成！

感謝昱維、獸哥陳鴻欣、宜修、尚儒、尚儒的總機小姐 XD、大慶、吳哲寬、

賴博、恩公、小馬、剛剛、老王、阿肥陳彥廷還有 楊神哲育(我有挪台)，多虧 了你們，研究生涯才能有這麼多的歡樂，那些一起吃飯、一起去台東玩，一起講 垃圾笑話、一起做研究、一起去聯誼的日子，是我永遠也難忘的回憶。感謝 436 真‧傑尼斯事務所的賀軍翔、大師梁為傑、型男伯川、正姐群包含：助理、阿筑、

維珍還有山寨版松島菜菜子林嘿嘿，謝謝你們很辛苦的想把我改造成阮經天，雖 然你們還是失敗了！還有其他熱心且強大的學長：管 group 林士弘學長、本 group 的珈擇、介宏、小胖、昭儒、正暵、tingo、昱帆、旭凱，謝謝你們各方面的幫 助和包容，相信從今以後不會再有人搞爆你們的貴重儀器、一片五萬台幣的光 罩、和耽誤你們的進度了。也祝福恩公、浩菔、楊鈞任、陳逸仁、尚儒、賴博、

小馬、鴻欣、昱維、晨昀、書緯、家銘，願你們的未來如鳳凰花般的燦爛。我還 要感謝台大財金法律雙主修的高材生劉翰芷同學，每次看到您優異的表現總讓我 明白：生命可以更努力也可以更喜樂，永遠不要為遇到的任何的阻礙找藉口。

最後，我要感謝我的爸媽、和以外公、外婆、大舅和大舅媽為首的親戚們，

你們總是在我最需要、最困苦的時候給我最大的幫助和關心、我衷心感謝你們的 陪伴與支持，沒有你們就沒有今天的我，所有的榮耀皆歸於你們和 李嗣涔老師。

**摘要 **

本文的第一個主題是從理論和實驗探討銀薄模上菱形晶格週期性圓型孔洞 的異常性穿透現象及其對電漿子熱幅射器幅射頻譜的影響，當銀/矽模態波長夠 長使得銀/矽模態和銀/空氣模態不耦合時，銀/矽模態的穿透強度由其簡併模態 的數目決定；在銀/矽模態和銀/空氣模態耦合的短波長區，銀/矽模態的峰值將 因藕合而變得不明顯。對於菱形晶格電漿子熱幅射器而言，其幅射峰值的強度為 黑體幅射強度乘上表面金屬的穿透效率，穿透效率正比於簡併模態的數目。本文 的第二個主題是將電漿子熱幅射器的厚度增加到微米等級以分析反射頻譜和幅 射頻譜中的共振腔模態，發現共振腔膜態也會如同表面電漿子般的和表面週期性 孔洞耦合產生布拉格散射共振腔模態；本文提出了隨機孔洞分布型共振腔熱幅射 器打亂表面的週期以消除和週期有關的表面電漿子模態和布拉格散射共振腔模 態以實現窄頻純頻譜的中紅外光共振腔型熱幅射器，並藉由改變孔洞的大小發現 了孔洞的存在將導致了侷域型共振腔模態和 Fabry-Peort 孔洞振盪模態的產 生，雖然小孔洞的隨機孔洞分布型共振腔熱幅射器可以提供純淨的幅射頻譜，其 輸出強度卻受限於低密度的孔洞總面積而相當的弱。據此，本文提出了另一種新 穎的短週期孔洞陣列型共振腔熱幅射器以解決輸出強度的問題：高輸出強度、純 淨的幅射頻譜、可在高溫下穩定的操作、低半高寬的幅射峰值和微弱的非理想效 應效像是侷域型共振腔模態或 Fabry-Perot 孔洞振盪模態可以同時被達成，而 且只要改變共振腔的厚度即可決定不同波長的幅射峰值。

**Abstract**

In this first topic of this thesis, the extraordinary transmission characteristics of silicon substrates with a silver film on top perforated with hole array arranged in a rhombus lattice are investigated in theory and experiment. It is found that the transmissions of Ag/Si modes are approximately linearly dependent on the numbers of degenerated modes in the longer wavelength range where the couplings between Ag/Si and Ag/air modes are weak. In the shorter wavelength range where Ag/Si and Ag/air are coupled together, the transmission peaks of Ag/Si modes are unapparent due to modes coupling. For plasmonic thermal emitters (PTEs) with hole array arranged in rhombus lattice, the peak intensities follow the blackbody radiation curve multiplying transmission efficiency of the top metal film which is dependent on the numbers of degenerated modes. In the second topic of this thesis, the SiO2 thickness of PTEs is increased to the order of μm to investigate cavity modes in the reflection and emission spectra, it is found that cavity modes (CMs) would be scattered by the periodic hole array and result in many Bragg scattered CMs in the spectra. Cavity thermal emitters (CTEs) with randomly distributed hole array (RDHA) are proposed to eliminate Bragg scattered CMs and Ag/SiO2 SPPs modes to realize narrower-band mid-infrared thermal emitters with purer spectra. The influence of hole size to the CMs is also investigated, it is found that larger scattering of light through larger

surface hole array would form the localized CMs (LCMs) and Fabry-Perot hole shape resonance (FP hole) modes. Although CTEs with RDHA can offer pure emission spectra if the hole size is small, their output intensities are very weak due to low density of total hole area. Novel CTEs with short period of hole array (SPHA) are proposed to overcome the intensity problem. High output intensity, pure emission spectra, high temperature operation, narrow full width half maximum of emission peaks and low non-ideal effect such as LCMs and FP-hole modes and could achieve simultaneously. The wavelengths of emission peaks are tunable by the thickness of the cavity.

**Contents**

**Chapter 1 Introduction... 1**

### 1.1 Extraordinary transmission and surface plasmons polaritons...1

### 1.2 Infrared thermal emitters ...4

### 1.3 The motivations of the research in this thesis ...5

### 1.4 Frameworks of this thesis...6

**Chapter 2 The Fundamental theorem... 8**

### 2.1 The fundamentals of surface plasmon polaritons...8

2.1.1 Surface plasmon polaritons at a single smooth interface...8

2.1.2 Surface plasmon polaritons at a smooth metal/dielectric/metal tri-layer structure ...19

### 2.2 Excitation of surface plasmon polaritons ...23

### 2.3 The extraordinary light transmission and infrared thermal emitters ...26

2.3.1 Extraordinary light transmission...26

2.3.2 Infrared thermal emitters...27

### 2.4 Process flow...28

2.4.1 Fabrication processes of samples of metal hole arrays...28

2.4.2 Fabrication processes of infrared thermal emitters...30

### 2.5 Measurement systems...31

2.5.1 Introduction of FTIR...31

2.5.2 Transmission measurement...32

2.5.3 Reflection measurement...33

2.5.4 Thermal emission measurement ...35

**Chapter 3 Extraordinary transmission through a silver ** **film perforated with hole arrays arranged in a ** **rhombus lattice and its application in plasmonic ** **thermal emitters... 37**

### 3.1 Extraordinary transmission through a silver film perforated with hole array arranged in a rhombus lattice...37

3.1.1 Experiments ...37

3.1.2 Results and discussion ...39

### 3.2 Plasmonic thermal emitters with top metal perforated by hole array arranged in rhombus lattice...53

3.2.1 Experiments ...53

3.2.2 Results and discussion ...54

**Chapter 4 The characteristic of cavity modes in the ** **plasmonic thermal emitters and the fabrication ** **of narrow-band cavity thermal emitters... 61**

### 4.1 The characteristic of cavity modes in tri-layer Ag/SiO

_{2}

### /Au plasmonic thermal emitters ...61

4.1.1 Experiments ...61

4.1.2 Results and discussion ...64

### 4.2 Cavity thermal emitters with randomly distributed hole array and the influence of hole size to the cavity modes ...72

4.2.1 Experiments ...73

4.2.2 Results and discussion ...77

### 4.3 Cavity thermal emitters with short period of hole array ...90

4.3.1 Experiments ...91

4.3.2 Results and discussion ...92

**Chapter 5 Conclusions... 101**

**Appendix **

^{ }**Proof of momentum conservation law of grating ** **coupling ... 104**

**References...107**

**Figure Captions **

Fig. 2.1 (a) SPPs at metal/dielectric interface. (b) Electric filed decay rapidly away the interface. (c) The dispersion relation of the SPPs (solid line) and light line

(dash line). ... 11

Fig. 2.2 Decay lengths of SPPs (a) in the dielectric

*δ*

*, (b) in the metal*

_{d}*δ*

*and (c) along the interface (Lsp). The dielectric is SiO2 and the metal is Ag...18*

_{m}Fig. 2.3 (a) The metal/dielectric/metal (MDM) tri-layer structure, the thickness of dielectric is w. The red lines represent the intensity of SPPs of two interfaces and will couple together once Eq. (2.56) is satisfied; (b) the simpler equivalent structure of Fig. 2.3 (a) once Eq. (2.56) is satisfied...22

Fig. 2.4 Parallel polarization plane waves (TM mode) impinge on the one dimensional grating whose period is Λ. The blue lines, green lines and red lines represent the incident lights, reflection lights and transmission lights respectively. ...25

Fig. 2.5 The (a) side and (b) top view of a silicon substrate with a silver film on top
perforated with hole array arranged in a rhombus lattice. ϕ is the incident
and reflective angle, k is the light wave vector component parallel to the _{//}
sample surface along ΓK direction. ...27

Fig. 2.6 (a) The side view and (b) top view of infrared thermal emitters. ...28

Fig. 2.7 The principle of Fourier transform infrared spectrometry...32

Fig. 2.8 The experiment setup of transmission at incline incidence...33

Fig. 2.9 Schematic diagram of reflection measurement in the angleϕfrom 12^{o} to 65^{o}.
...34

Fig. 2.10 The optical path and reflection mirrors of reflection measurement ...34

Fig. 2.11 The thermal emitter chamber, (a) top view (b) side view. ...36

Fig. 3.1 The (a) side and (b) top view of a silicon substrate with a silver film on top
perforated with hole array arranged in a rhombus lattice. a = a =a=5μmJJG_{1} JJG_{2}
and d=2.5μm. ϕ is the incident and transmission angle, k is the light _{//}
wave vector component parallel to the sample surface along ΓK direction. 38
Fig. 3.2 The dispersion relation extracted from transmission spectra for sample (a) A,
(b) B, (c) C, (d) D, (e) E, (f) F and their modes analyses for sample (g) A, (h)
B,(i) C, (j) D, (k) E, (l) F. The green and red dashed lines are the calculated
dispersion curves of Ag/SiO2 and Ag/air modes, respectively. ...42

Fig. 3.3 The transmission spectra of sample C with incident angle φ varying from 0^{o}
to 60^{o} by 10^{o} step. ...44
Fig. 3.4 The transmission spectra for samples (a) A, (b) B, (c) C, (d) D, (e) E and (f) F

in the normal directionϕ=0^{o}. ...48

Fig. 3.5 (a) The schematic cross section and (b) top view of PTEs with top metal
perforated by hole array arranged in rhombus lattice with a = a =a=5μmJJG_{1} JJG_{2}
and d=2.5μm . ...54

Fig. 3.6 Emission spectra in the normal direction of all five PTEs with rhombus lattice.
The lattice angle θ= (a) 50^{o}, (b) 60^{o}, (c) 70^{o}, (d) 80^{o} and (e) 90^{o}. ...59

Fig. 4.1 (a) The side view and (b) top view of PTEs with the thickness of SiO2 tox=0.55, 2μm and 2.6μm for samples A, B and C, respectively. The surface silver films are perforated with hole array in hexagonal lattice with 1 2 a = a =a=3μm JJG JJG and d=1.5μm . ϕ is the incident and reflective angle, k is the parallel component of wave vectors of lights...63 //

Fig. 4.2 Emission spectra of samples (a) A, (b) B and (c) C at temperature 300^{o}C. ...65

Fig. 4.3 The dispersion relation of reflection spectra for samples (a) A, (b) B, (c) C and their modes analysis for samples (d) B and (e) C. The blue dashed lines are the calculated results of Eq. (4.9). ...69

Fig. 4.4 (a) The measured dispersion relation of reflection spectra and (b) corresponding modes analysis for sample B. The blue dashed lines are the calculations results of Eq. (4.13). ...72

Fig. 4.5 (a) The side view and (b) top view of CTEs with RDHA. ...73

Fig. 4.6 The relation between transmission and the distribution of the nearest center-to-center distance of holes for sample I...77

Fig. 4.7 Theoretical dispersion curves of CM for CTEs with RDHA. tox=2μm ...78

Fig. 4.8 The dispersion relation of reflection spectra for sample (a) D, (b) E, (c) F and their modes analysis for sample (d) D, (e) E and (f) F. The blue dashed lines are the calculated results of Eq. (4.9). ...80

Fig. 4.9 The dispersion relation of reflection spectra of samples (a) G, (b) H and (c) I. ...81

Fig. 4.10 The theoretical dispersion curves of CMs and LCMs. ...83

Fig. 4.11 The dispersion relation of reflection spectra of samples (a) J and (b) K. ...84

Fig. 4.12 The relation of the wavelength of FP-hole modes to the hole diameter...85

Fig. 4.13 The reflection spectra of sample L to N with the fixed incident and reflective
angles 12ϕ= * ^{o}* and the fixed diameters of holes d=2μm. ...86

Fig. 4.14 The emission spectra of samples (a) E, (b) H, (c) I and (d) J in the normal
direction (ϕ=0 )* ^{o}* . ...89
Fig. 4.15 The dispersion relation of reflection spectra for (a) sample O and (b) sample

P. The period of hole array for sample O and P are 2.3 μm and 1.7 μm,
respectively ...94
Fig. 4.16 The emission spectra in the normal directionϕ=0^{o} for (a) sample O and (b)

sample P. The period of hole array for sample O and P are 2.3 μm and 1.7 μm, respectively. The thicknesses of SiO2 of both samples are 2 μm. ..96 Fig. 4.17 The emission spectrum for sample Q. The thickness of SiO2 is 1.6 μm. ...97 Fig. 4.18 The calculation of minimum skin depth of subwavelength holes to the

wavelengths of lights guided inside the holes according to Eq. (4.23). ...99
Fig. 4.19 The reflection spectra atϕ=12^{o }for sample E which is a CTE with 15nm top

thin silver film and sample P which is a CTE with SPHA. The period and diameter of hole array of sample P are 1.7 μm and 1 μm, respectively...99

**List of Tables **

Table 2.1 Conditions and purposes of the cleaning solvents ...29 Table 2.2 The photolithography conditions ...30 Table 3.1 The structure parameters of samples A to F. ...39 Table 3.2 The theoretical and measured parameters of EOT in the normal direction for silicon substrates with top silver film perforated with hole array arranged in rhombus lattice. ...49 Table 3.3 The comparisons of normalized transmission intensities for wavelength

larger than 10 μm ...51 Table 3.4 The theoretical and measured parameters of EOT for PTEs with rhombus

lattice. ...60 Table 4.1 The comparisons between theoretical peak wavelengths and measured

results of emission spectra of samples A, B and C in ϕ =0* ^{o}*direction. ...70
Table 4.2 The comparisons between theoretical values and measured results of

emission spectra of samples A to C. ...71 Table 4.3 The structure parameters of samples with randomly distributed hole array, d

denotes the diameters of holes and h denotes the thickness of to silver film.

...74 Table 4.4 The structure parameters and emission peaks of samples O, P and Q, d

denotes the diameters of holes and h denotes the thickness of to silver film.

...91

**Chapter 1 Introduction **

**1.1 ** **Extraordinary transmission and surface ** **plasmons polaritons **

Optical transmission through a single aperture, such as slit and hole in an opaque screen, has been studied for years. In 1944, Hans Bethe [1] derived an analytical expression which tells that for a single hole perforated in the thin metal film with diameter r, the transmission ratio of light normalized to the hole area in the normal direction is

4 4

2

64(kr) r

( ) 374.3

27π λ

*T* λ = ≈ ^{⎛ ⎞}⎜ ⎟⎝ ⎠ (1.1)

where k=2π/λ is the wavector of incident light, λ is the wavelength of incident light.

According to this standard diffraction theory, it has been thought that
subwavelength hole have a very low transmission ratio since the transmission is
proportional to*(r/λ) . However, in 1998, Ebbesen et al. [2-3] had reported that metal *^{4}
films perforated with two-dimensional subwavelength periodic hole arrays exhibit
extraordinary transmission (EOT) which exceeded the prediction of Eq. (1.1) resulting
from resonant excitation of surface plasmons polaritons (SPPs). Even more, the
transmission ratio normalized to the hole area is larger than 1, the flux of photons per
unit area emerging from the hole is larger than the incident flux per unit area of hole.

SPPs, first reported by Ritchie [4-5] in the 1950s, are electromagnetic

excitations that propagate along a dielectric/metal interface. These waves are trapped
*on the interface and decay rapidly away the interface. Grupp et al. [6], provide direct *
evidence that the enhanced transmission through subwavelength apertures is mediated
by SPPs on metal surface.

For circular hole surrounded by concentric grooves and single slit surrounded by periodic parallel grooves in a metallic film, the emitted light from such apertures concentrate in a specific direction with a constrained range of angles instead of diverging in all directions, which supports the highly directional beaming nature of SPPs [7-9]. If the period of the corrugation is appropriate, the SPPs can Bragg reflect and interfere with themselves, which result in an energy gap in the SPPs dispersion relation [10-11]. The momentum conservation of SPPs and light through periodic perforated hole arrays [12] has elucidated the propagation of SPPs. When light impinges on perforated metal films, localized surface plasmons (LSPs) result from the individual hole-shape resonance will shift the wavelength of emission peak and affect the transmittance [13-16]. As the sizes of the perforated apertures become smaller than the thickness of the metal film, a transition from SPP mode to waveguide resonance mode was observed [17]. In addition, the combined effects of SPP and Fabry-Perot resonances determined by the thickness of perforated metal film have been studied in the THz region theoretically and experimentally [18-19]. According to

theoretical analyses, if the perforated metal film has symmetric interfaces, then SPPs
at the top and the bottom interfaces are coupled via evanescent waves [20,21], which
*consistent with the theorem of spoof SPPs what Pendry et al. [22] discovered three *
years later. The theorem of spoof SPPs states that metal films perforated with an array
of subwavelength and sub-period holes can be viewed as an effective metals with a
plasma frequency equal to the cut-off frequency of holes which is much lower than
the plasma frequency of original metals. This spoof low plasma frequency offer
higher confinement of SPPs in the dielectric and longer decay length of SPPs inside
the holes, strong SPPs can exist in such artificial metals This explained why the EOT
can still be observed in the terahertz and microwave region [23, 24] where traditional
theorem of SPPs believe that at such low frequency only highly delocalized
Sommerfeld-Zenneck surface wave [25-27] but SPPs can exist in the metal/dielectric
interface. It solved the long-term question how such weak SPPs in terahertz and
microwave contribute strong extraordinary light transmission.

EOT can be applied in the fields of subwavelength photolithography [28-29], solar cell [30-31], quantum dot infrared photodetector [32], biosensor [33], channel waveguide [34], modulator [35], tunable light sources/filters [36, 37] and plasmonic thermal emitters [38-41].

**1.2 Infrared thermal emitters **

Mid-infrared light source is important in the fields of gas sensing, free space optical communication, healthcare, missile countermeasures and biomedicine [42-43].

Blackboday radiation is a good mid-infrared light source since maximum intensity of
radiation located at the wavelength of 5 μm is in the temperature around 300^{o }C
according to Wien’s law. The physical, chemical and optical properties of most optical
materials in this temperature are stable. A simple optical filter combined with a hot
source can filter out unwanted parts of blackbody radiation and form a stable
mid-infrared light source without complex quantum structures, epitaxy and expensive
cooling equipments which are needed in the infrared semiconductor laser [42].

Infrared thermal light sources with filters structures to tailor the thermal radiation
spectra had been studied for long time from one dimensional to three dimensional
theoretically and experimentally [38-41, 44-50]. Among them, emitters which use
* lossless dielectric distributed Bragg reflectors (DBR) as the filters *[44-46] may be
thought to the best structures since low full width half maximum (FWHM) and high
power output can be achieved simultaneously. However, it is hard to use these
structures in mid-infrared since the emissivity of most optical materials is not small in
the mid-infrared [51]. For example, SiO2, a common optical thin film material, have
large extinction coefficient (k) around 10 μm [52] contributed from strong phonon

vibration [53] makes it impossible to be the material of DBR without strong thermal radiation.

An alternative approach to form the filters without strong emissivity is to use metals perforated with hole arrays whose emissivity are low. This kind of thought was realized by Tsai et al. [38-41] as plasmonic thermal emitters using thin dielectric as the emission source. The emissivity of thin dielectric sandwiched between two metals can be enhanced greatly [54] by the additional dipole radiation in the cavity [55-56].

**1.3 The motivations of the research in this ** **thesis **

The emission spectra of tri-layer Ag/SiO2/Ag plasmonic thermal emitters (PTEs)
with top Ag layer perforated with holes array in square and hexagonal lattice had been
investigated in the previous works [38-41]. However, for two dimensional lattice,
square and hexagonal lattices are the special case of a rhombus lattice with specific
lattice angleθ=90^{o} and 60^{o}. It is interesting to know how emission spectra changes
when θ changes from 90^{o} to 40^{o }by 10^{o} step. Besides, the intensity distribution of
various emission peaks was not known either.

Tsai et al. [41] had point out that once the thickness of SiO2 becomes thicker than 1.3 μm, not only SPPs mode but also parallel-plate waveguide mode (which is also known as Fabry-Perot resonance mode or simply cavity mode) would appear in the

emission spectra. However, not all peaks can be explained satisfactorily by the traditional theorem of surface plasmon polaritons or waveguide modes. Besides, the red shift of emission peaks were also observed but the reason was not clear. These questions are the major goal of research in this thesis.

**1.4 Frameworks of this thesis **

The contents of this thesis are outlined as follows.

In this chapter 2, the basic theorem of SPPs and the excitation of SPPs using grating coupling will be introduced. Next, the basic theorem of extraordinary transmission of light through subwavelength periodic hole arrays and the basic operating principle of PTEs will be introduced. Finally, the fabrication processes and the measurement systems will be elucidated.

In chapter 3, the transmission characteristics of silicon substrates with silver films on top perforated with hole array arranged in a rhombus lattice have been investigated, it is found that the transmission of Ag/Si modes are approximately linearly dependent on the numbers of degenerated modes in the longer wavelength range where the couplings between Ag/Si and Ag/air modes are weak. In the shorter wavelength range where Ag/Si and Ag/air are coupled together, the transmission intensities are approximately constant without apparent peaks in the wavelength range longer than the Ag/air modes and decay rapidly in the wavelength smaller than the

Ag/air modes due to asymmetric slope of the Ag/air mode in the spectra. PTEs with rhombus lattice had been investigated experimentally and theoretically either. Only hexagonal lattice produce the strongest radiation peak due to largest degenerated modes. The peak intensities follow the blackbody radiation curve multiplying transmission efficiency of the top metal film which is dependent on the numbers of degenerated modes.

In chapter 4, the emission spectra of PTEs with oxide thickness of 0.55, 2 and 2.6 μm were investigated, a new model was developed to successfully explain and fit the experimental results which are not able to explain well in the past [41]. According to this new model, CTEs with randomly distributed hole arrays (RDHA) are proposed to offer pure cavity mode radiation spectra with small full width at half maximum (FWHM) which are better than those of the traditional PTEs. However, the low output intensity due to low density of holes is its drawback. The influence of hole size to the emission peaks and reflection spectra was investigated either. In order to overcome the problem of low output intensity, novel cavity thermal emitters with short period were invented to overcome this problem; high output intensity, low FWHM and pure emission spectra can be realized simultaneously.

Finally, the conclusions are given in chapter 5; the mathematical derivations of gratings coupling are given in Appendix.

**Chapter 2 The Fundamental theorem **

In this chapter, the basic theorem of surface plasmon polaritons (SPPs), excitation of SPPs using grating coupling will be introduced. Next, the basic theorem of extraordinary transmission of light through subwavelength periodic hole array and the basic operating principle of plasmonic thermal emitters will be introduced. Finally, the fabrication processes and the measurement system will be elucidated.

**2.1 ** **The fundamentals of surface plasmon ** **polaritons **

**2.1.1 Surface plasmon polaritons at a single smooth ** **interface **

**2.1.1.1 Transverse electric (TE) mode **

First, assume that surface waves of transverse electric (TE) mode can exist in the interface (z=0) of the metal and dielectric and decay rapidly away the interface as shown in Fig. 2.1(a) and (b). The electric fields components in Cartesian coordinates can be described as follow.

For z>0

( )

### ( , , )

^{k z i k x}

^{d}

^{xd}

^{t}*y* *d*

*E x z*

### ω =

*A e*

^{−}

*e*

^{−}

^{ω}(2.1)

and for z<0

( )

### ( , , )

^{k z i k x}

^{m}

^{xm}

^{t}*y* *m*

*E x z*

### ω =

*A e e*

^{−}

^{ω}(2.2)

where

### Re[ ] *k*

*>0 and*

_{d}### Re[ ] *k*

*>0 indicate the fact that surface waves decay rapidly away the interface.*

_{m}*A*

*and*

_{d}*A*

*are constants.*

_{m}*k*

*and*

_{xd}*k*

*denote as*

_{xm}*propagation constants of the traveling waves in dielectric and metal respectively.*

Both

*E*

*and*

_{x}*E*

*are zero in the dielectric and metal for TE mode. Let the materials in Fig 2.1 (a) be homogeneous, that is, the complex relative dielectric constant*

_{z}### ε

*, complex relative permeability*

_{r}### μ

*,*

_{r}*k*

*,*

_{xd}*k*

*,*

_{xm}*k*

*and*

_{d}*k*

*are all independent of positions (x, y, z).*

_{m}Substituting Eqs. (2.1) and (2.2) into Maxwell equation

### ∇ × = − JG *E* *j* ωμ μ

_{0}

_{r}### JJG *H*

, the magnetic fields of both regions can be obtained

For z>0

0

### ( , , ) 1

^{ik x}

^{xd}

^{k z}

^{d}*x* *d* *d*

*rd*

*H x z* ω *iA* *k e e* ωμ μ

### = −

− (2.3)0

### ( , , )

^{xd}

^{ik x}

^{xd}

^{k z}

^{d}*z* *d*

*rd*

*H x z* ω *A* *k* *e e* ωμ μ

### =

− (2.4)and for z<0

0

### ( , , ) 1

^{ik x k z}

^{xm}

^{m}*x* *m* *m*

*rm*

*H x z* ω *iA* *k e* *e*

### = ωμ μ

(2.5)0

### ( , , )

^{xm}

^{ik x k z}

^{xm}

^{m}*z* *m*

*rm*

*H x z* ω *A* *k* *e* *e*

### = ωμ μ

(2.6)where

### μ

_{0}is the permeability in the vacuum;

### μ

*and*

_{rd}### μ

*are the relative permeability of dielectric and metal respectively.*

_{rm}The following boundary conditions of Maxwell equations should be satisfied at the interface (z=0)

Ey continuity

### Eq. (2.1) Eq. (2.2) =

(2.7)Hx continuity

### Eq. (2.3) Eq. (2.5) =

(2.8)Solving Eq. (2.7), Eq. (2.8) and let

### μ

_{rd}### = μ

_{rm}### = 1

for non-magnetic materials, the solutions of above equations are*xd* *xm*

*ik x* *ik x*

*d* *m*

*A e*

### =

*A e*(2.9)

*xd* *xm*

*ik x* *ik x*

*d* *d* *m m*

*A k e* *A k e*

### − =

(2.10)Substituting

### Eq.

(2.9) into### Eq.

(2.10) yields### ( ) ( ) 0

*xm* *xd*

*ik x* *ik x*

*m* *d* *m* *d* *d* *m*

*A e* *k*

### +

*k*

### =

*A e*

*k*

### +

*k*

### =

(2.11)Since

### Re[ ] *k*

*>0 and*

_{d}### Re[ ] *k*

*>0, the only solution of Eq. (2.11) is*

_{m}*A*

_{m}### = *A*

_{d}### = 0

*and the electromagnetic fields intensity from Eq. (2.1) to Eq. (2.6) become all zero. So, *
*there are no surface TE waves existing in the interface of two non-magnetic materials. *

However, it should be noted that surface TE waves can still exist if the interface is formed by magnetic materials or metamaterials where

### μ

_{rd}### ≠ μ

*[56].*

_{rm}(a)

(b)

(c)

Fig. 2.1 (a) SPPs at metal/dielectric interface. (b) Electric filed decay rapidly away the interface. (c) The dispersion relation of the SPPs (solid line) and light line (dash line).

**2.1.1.2 Transverse magnetic (TM) mode **

Next, consider the case of surface transverse magnetic waves (TM mode), the magnetic fields can be presented as follows

For z>0

( )

### ( , , )

^{k z i k x}

^{d}

^{xd}

^{t}*y* *d*

*H x z*

### ω =

*A e*

^{−}

*e*

^{−}

^{ω}(2.12)

and for z<0

( )

### ( , , )

^{k z i k x}

^{m}

^{xm}

^{t}*y* *m*

*H x z*

### ω =

*A e e*

^{−}

^{ω}(2.13)

The definitions and characteristics of

*A*

*,*

_{d}*A*

*,*

_{m}*k*

*,*

_{xd}*k*

*,*

_{xm}*k*

*and*

_{d}*k*

*are identical as what were defined in Sec. 2.1.1.1. Both*

_{m}*H*

*and*

_{x}*H*

*are zero in dielectric and metal for TM mode.*

_{z}Substituting Eq. (2.12) and Eq. (2.13) into Maxwell equation

### ∇ × *H* JJG = *j* ωε ε

_{0 r}*E* JG

, the magnetic fields of both regions can be obtained For z>0

0

### ( , , ) 1

^{ik x}

^{xd}

^{k z}

^{d}*x* *d* *d*

*rd*

*E x z* ω *iA* *k e e* ωε ε

### =

− (2.14)0

### ( , , )

^{xd}

^{ik x}

^{xd}

^{k z}

^{d}*z* *d*

*rd*

*E x z* ω *A* *k* *e e* ωε ε

### = −

− (2.15)and for z<0

0

### ( , , ) 1

^{ik x k z}

^{xm}

^{m}*x* *m* *m*

*rm*

*E x z* ω *iA* *k e* *e*

### = − ωε ε

(2.16)0

### ( , , )

^{xm}

^{ik x k z}

^{xm}

^{m}*z* *m*

*rm*

*E x z* ω *A* *k* *e* *e*

### = − ωε ε

(2.17)where

### ε

_{0}is the permittivity in the vacuum;

### ε

*and*

_{rd}### ε

*are the relative permittivity of dielectric and metal respectively.*

_{rm}The following boundary conditions of Maxwell equations should be satisfied at the interface (z=0)

Hy continuity

### Eq. (2.12) Eq. (2.13) =

(2.18)Ex continuity

### Eq. (2.14) Eq. (2.16) =

(2.19)Dz continuity

### Eq. (2.15) × ε ε

_{0}

_{rd}### = Eq. (2.17) × ε ε

_{0}

*(2.20) Solving*

_{rm}### Eq.

(2.18) - Eq. (2.20) yields*xd* *xm*

*ik x* *ik x*

*d* *m*

*A e*

### =

*A e*(2.21)

### 1

_{ik x}

_{xd}### 1

_{ik x}

_{xm}*d* *d* *m* *m*

*rd* *rm*

*A* *k e* *A* *k e*

### ε ^{= −} ε

^{ (2.22) }

*xd* *xm*

*ik x* *ik x*

*d* *xd* *m xm*

*A k e*

### =

*A k e*(2.23)

Substituting Eq. (2.21) into Eq. (2.22) and Eq. (2.23) yields

### 1 1 1 1

### ( ) ( ) 0

*xd* *xm*

*ik x* *ik x*

*d* *d* *m* *m* *d* *m*

*rd* *rm* *rd* *rm*

*A e* *k* *k* *A e* *k* *k*

### ε ^{+} ε ^{=} ε ^{+} ε ^{=}

^{ (2.24) }

*xd* *xm*

*k* = *k*

(2.25)
Since both

*A*

*and*

_{d}*A*

*can not be zero, the solution of Eqs. (2.24) - (2.25) is*

_{m}### 1 1

### (

_{d}

_{m}### ) 0

*rd* *rm*

*k* *k*

### ε ^{+} ε ^{=}

^{ (2.26) }

Eq. (2.26) can be rearranged as

*d* *rd*

*m* *rm*

*k*
*k*

### ε

### = − ε

(2.27)Since the materials are homogenous, the wave equations Eq. (2.12) and Eq. (2.13) should obey the Helmholtz equation [57]

### ∇

^{2}

*H k* JJG +

_{0}

^{2}

### μ ε

_{r r}*H* JJG = 0

. Substituting Eq.

(2.12) and Eq. (2.13) into Helmholtz equation, the following equations can be obtained

2 2 2

0

### 0

*d* *xd* *rd* *rd*

*k*

### −

*k*

### +

*k*

### μ ε =

(2.28)2 2 2

0

### 0

*m* *xm* *rm rm*

*k*

### −

*k*

### +

*k*

### μ ε =

(2.29)where *k*_{0} =ω

*c* and c is the light speed in the vacuum.

Dividing Eq. (2.28) by Eq. (2.29) gives

2 2 2

0

2 2

0

*d* *xd* *rd* *rd*

*m* *xm* *rm rm*

*k* *k* *k*

*k* *k* *k*

### μ ε μ ε

### ⎛ ⎞ = −

### ⎜ ⎟ −

### ⎝ ⎠

^{ (2.30) }

Substituting Eq. (2.25), Eq. (2.27) into Eq. (2.30) and let

### μ =μ =1

_{rd}

_{rm}for non-magnetic material assumption, the following results are obtained

0

*rd* *rm* *rd* *rm*

*xd* *xm*

*rd* *rm* *rd* *rm*

*k* *k* *k*

*c*

### ε ε ω ε ε

### ε ε ε ε

### × ×

### = = =

### + +

^{ (2.31) }

### ( )

^{2}

0

*rd*
*d*

*rd* *rm*

*k* *k*

### ε

### ε ε

### = −

### +

^{ (2.32) }

### ( )

^{2}

0

*rm*
*m*

*rd* *rm*

*k* *k*

### ε

### ε ε

### = −

### +

^{ (2.33) }

where*k** _{xd}*and

*k*

_{xm}*in the Eq. (2.31) are also known as the dispersion relation of SPPs*

and usually denoted as *k** _{sp}*. Substituting Eq. (2.25) into Eq. (2.23) yields

d m

### A =A =A

(2.34)where A is a non-zero constant. Eqs. (2.31) - (2.34) can now be substituting back into Eqs. (2.12) - (2.17) and the whole electric and magnetic fields distributions in real space can be obtained.

Finally, it should be noted that Eq. (2.27) indicates that

### Re[ε ]

_{rd}and

### Re[ε ]

_{rm}

should have opposite because of

### Re[ ] 0 *k*

_{d}### >

and### Re[ ] 0 *k*

_{m}### >

. Since only metals can offer negative dielectric constants according to Drude model[58, 59]. It is impossible to generate the surface waves on a smooth dielectric/dielectric or most*metal/metal interface whose dielectric constants are both positive or both negative.*

*Surface waves can only exist in the metals/dielectric interface and only TM mode is *
*allowed; these surface TM waves are named as surface plasmon polaritons (SPPs) by *
*Ritchie in 1957 [4]. *

Now, let’s go back to see why it is called surface plasmon polaritons [5]. The
non-zero electric fields inside the metal implies that the free electrons inside metals
will oscillate with these electric fields together as shown in Fig. 2.1(a); the collection
*of free electrons oscillating in certain common frequency are called plasmon [57]. *

Since these only occur near the surface of metal and decay rapidly inside the metal, it
*is more accurate to call it surface plasmon. The coupling of free oscillating electrons *

and surface waves in the dielectric forms a newly kind of virtual particles propagating along the interface and carry with their own energies

### = ω

and momentums### =

*k*

_{sp}*according to the Eq. (2.31) and Fig. 2.1(c). These virtual particles are named as*

*polaritons since only TM mode is allowed for non-magnetic materials. It should be*

note that the momentums of SPPs are always larger than the momentums of the propagation light at the same frequency, as shown in Fig. 2.1 (c) and Eq. (2.31)

0 0 0 0

### 1 1

*rd* *rm*

*sp* *rd* *rd* *d*

*rd* *rm* *rd*

*rm*

*k* *k*

### ε ε ε ε

*k*

### ε ε

*k*

### ε

*k n*

### ε

### = × = × > ≈

### + +

### = = = = =

(2.35)where*n*_{d}

### ≈ ε

*is the refractive index of dielectric and usually >1.*

_{rd}### ε

_{rm}### < 0

for most metals according to Drude mode [58, 59].Finally, the decay lengths of SPPs inside the dielectric and the metal are denoted

as

### δ

_{d}and

### δ ,

_{m}respectively; their values are defined as the distances where the intensity of electric fields drop to 1/e

d

d

### δ = 1

### Re[k ]

^{ (2.36) }

m

m

### δ = 1

### Re[k ]

^{ (2.37) }

where *k** _{d}* and

*k*

*can be calculated from Eq. (2.32) and Eq. (2.33).*

_{m}The decay length of SPPs propagating along the interface is denoted as

### L

_{sp}and is defined as the distances where the energy of SPPs drops to 1/e

sp

xd xm

### 1 1

### L = =

### 2 Im[k ] 2 Im[k ] × ×

^{ (2.38) }

where

### k

_{xd}and

### k

_{xm}can be calculated from Eq. (2.31).

Figs. (2.2) show the examples of calculations results of Eqs. (2.36) - (2.38), the dielectric material is SiO2 and the metal is Ag. All optical constants come from Ref.

[52, 60, 61].

(a)

(b)

(c)

Fig. 2.2 Decay lengths of SPPs (a) in the dielectric

*δ*

*, (b) in the metal*

_{d}*δ*

*and (c) along the interface (Lsp). The dielectric is SiO2 and the metal is Ag.*

_{m}**2.1.2 ** **Surface plasmon polaritons at a smooth ** **metal/dielectric/metal tri-layer structure **

Fig. 2.3 (a) shows a typical symmetric metal/dielectric/metal (MDM) tri-layer structure. There is no TE mode SPPs in the MDM structure [62]. Consider SPPs of TM mode, the magnetic fields distributions are described as follows.

For

### 2

*z*

### ≥

*w*

( )

### ( , , )

^{k z i k x}

^{m}

^{xm}

^{t}*y* *m*

*H x z*

### ω =

*A e*

^{−}

*e*

^{−}

^{ω}(2.39)

For

### 2

*z*

### ≤ −

*w*

( )

### ( , , )

^{k z i k x}

^{m}

^{xm}

^{t}*y* *m*

*H x z*

### ω =

*A e e*

^{−}

^{ω}(2.40)

For

### 2 2

*w* *w*

### − ≤ ≤

*z*

( ) ( )

### ( , , )

^{k z i k x}

^{d}

^{xd}

^{t}

^{k z i k x}

^{d}

^{xd}

^{t}*y* *d* *d*

*H x z*

### ω =

*A e e*

^{−}

^{ω}

### +

*B e*

^{−}

*e*

^{−}

^{ω}(2.41)

The definitions and characteristics of

*H*

*,*

_{x}*H*

*,*

_{z}*k*

*,*

_{xd}*k*

*,*

_{xm}*k*

*and*

_{d}*k*

*are identical as what were defined in Sec. 2.1.1.2.*

_{m}Substituting Eqs. (2.39) - (2.41) into Maxwell equation

### ∇ × *H* JJG = *j* ωε ε

_{0 r}### JG *E*

, the magnetic fields of both regions can be derived

For

### 2

*z*

### ≥

*w*

0

### ( , , ) 1

^{ik x}

^{xm}

^{k z}

^{m}*x* *m* *m*

*rm*

*E x z*

### ω

*iA*

*k e*

*e*

### ωε ε

### =

− (2.42)0

### ( , , )

^{xd}

^{ik x}

^{xm}

^{k z}

^{m}*z* *m*

*rm*

*E x z*

### ω

*A*

*k*

*e*

*e*

### ωε ε

### = −

− (2.43)For

### 2

*z*

### ≤ −

*w*

0

### ( , , ) 1

^{ik x k z}

^{xm}

^{m}*x* *m* *m*

*rm*

*E x z*

### ω

*iA*

*k e*

*e*

### = − ωε ε

(2.44)0

### ( , , )

^{xm}

^{ik x k z}

^{xm}

^{m}*z* *m*

*rm*

*E x z*

### ω

*A*

*k*

*e*

*e*

### = − ωε ε

(2.45)and for

### 2 2

*w* *w*

### − ≤ ≤

*z*

0 0

### 1 1

### ( , , )

^{ik x k z}

^{xd}

^{d}

^{ik x}

^{xd}

^{k z}

^{d}*x* *d* *d* *d* *d*

*rd* *rd*

*E x z*

### ω

*iA*

*k e e*

*iB*

*k e e*

### ωε ε ωε ε

### = − +

− (2.46)0 0

### ( , , )

^{xd}

^{ik x k z}

^{xd}

^{d}

^{xd}

^{ik x}

^{xd}

^{k z}

^{d}*z* *d* *d* *d* *d*

*rd* *rd*

*k* *k*

*E x z*

### ω

*A*

*k e e*

*B*

*k e e*

### ωε ε ωε ε

### = − −

− (2.47)The following boundary conditions of Maxwell equations should be satisfied at the interface (

### 2

*z*

### = ±

*w*respectively)

Hy continuity

### Eq. (2.39) Eq. (2.41) =

(2.48)### Eq. (2.40) Eq. (2.41) =

(2.49)Ex continuity

### Eq. (2.42) Eq. (2.46) =

(2.50)### Eq. (2.44) Eq. (2.46) =

(2.51)Dz continuity

### Eq. (2.43) × ε ε

_{0}

_{rd}### = Eq. (2.47) × ε ε

_{0}

*, (2.52)*

_{rm}0 0

### Eq. (2.45) × ε ε

_{rd}### = Eq. (2.47) × ε ε

*(2.53) Solving Eqs. (2.48) - (2.53) yields*

_{rm}*xd* *xm*

*k* = *k*

(2.54)
### / /

### Exp[ ]

### / /

*d* *rd* *m* *rm*

*d*

*d* *rd* *m* *rm*

*k* *k*

*k w* *k* *k*

### ε ε

### ε ε

### − = +

### −

^{ (2.55) }

*Eq. (2.55) is the dispersion relation of SPPs in symmetric MDM tri-layer structures. *

Theoretically,

*k*

*,*

_{d}*k*

*,*

_{m}*k*

*and*

_{xd}*k*

*can be described in terms of*

_{xm}### ε

*,*

_{rd}### ε

*and w by solving Eq. (2.55) with Eqs. (2.28), (2.29) and (2.54) since four variables can be*

_{rm}solved using four equations. However, it should be note that

### Exp[ − *k w*

_{d}### ]

in Eq.(2.55) is an infinite series so that the solutions of

*k*

*,*

_{d}*k*

*,*

_{m}*k*

*and*

_{xd}*k*

*are infinite series either. These kind of solutions are too complex to use directly in the future sections and chapters.*

_{xm}Fortunately, S. Collin et al. [63] had point out that if

d

### δ 1 w

### Re[ ]

*k*

_{d}### =

(2.56)is satisfied, the SPPs in the two interfaces will couple together tightly as shown as the red lines in Fig. 2.3 (a). Once this tightly coupling occurs, the original tri-layer structure can be approximated to the two layer equivalent structure as shown in Fig.

2.3 (b). The refractive index of this effective dielectric is [63]

1/2

### 1

0### 1

^{rd}*eff* *reff* *rd*

*rm* *rm*

*n* *w*

### λ ε

### ε ε

### π ε ε

### ⎛ ⎞

### = = × + ⎜ ⎜ ⎝ − + − ⎟ ⎟ ⎠

^{ (2.57) }

where

### ε

_{reff}is the effective relative dielectric constant,

### λ

_{0}is the wavelength of SPPs in the vacuum. Now, all formulas derived in Sec. 2.1.1.2 can be applied into Fig.

2.3 (b), and

*k*

*,*

_{xd}*k*

*,*

_{xm}*k*

*and*

_{d}*k*

*of original MDM structure can be obtained easily by substituting Eq.(2.57) into Eqs. (2.31) - (2.33).*

_{m}(a)

(b)

Fig. 2.3 (a) The metal/dielectric/metal (MDM) tri-layer structure, the thickness of dielectric is w. The red lines represent the intensity of SPPs of two interfaces and will couple together once Eq. (2.56) is satisfied; (b) the simpler equivalent structure of Fig. 2.3 (a) once Eq. (2.56) is satisfied.

**2.2 Excitation of surface plasmon polaritons **

The field distributions and dispersion curves of SPPs had been introduced in the previous sections without considering how the waves are generated or excited. Of course, SPPs are excited from some kind of methods [58], such as prisms coupling, near filed scatterings and gratings coupling. Only gratings coupling will be discussed and used in this thesis.

For simplicity, consider y-polarized waves (TM mode) impinge on a one dimensional arbitrary shaped interface whose period is Λ as shown in Fig. 2.4. The coordination (x, z) of interface in space is (x, h(x)) where h(x) is in period of Λ. The blue lines indicate the incident waves with common propagation directions (common kix and kiz); the green lines and red lines represent the reflection and the transmission waves, respectively. Since the propagation directions of reflection and transmission waves may be different at each point of the interface, the final electromagnetic waves in any point of space (x, z) should be the superposition of waves come from all directions. Interference occurs and some of waves are canceled.

The theorem of gratings coupling which are based on linear algebra [Appendix, 64, 65] states that the final distributions of reflection waves and transmission waves can be spanned in the basis of infinite waves whose wavesvectors are discrete

rx rz

+ i(k x + k z)

m=- m

### H (x, z)=

_{r}^{∞}

### R e y

### ∑

∞### JJG G

(2.58)

tx tz

+ i(k x - k z) m=- m

### H (x, z)=

_{t}^{∞}

### T e y

### ∑

∞### JJG G

(2.59)

### m 2

*rx* *tx* *x* *ix*

*k* *k* *k* *k*

### π

### = = = +

### Λ

^{ (2.60) }

where m denotes the order of scattering waves.

Since

2 2 2

0 1 1

*rx* *rz* *r* *r*

*k*

### +

*k*

### =

*k*

### μ ε

(2.61)2 2 2

0 2 2

*tx* *tz* *r* *r*

*k*

### +

*k*

### =

*k*

### μ ε

(2.62)should be obeyed for all kinds of waves, large

### 2 m π

### Λ

will enlarge*k*

_{rx}^{2}and

*k*

_{tx}^{2}in Eq. (2.60). Next, large

*k*

_{rx}^{2}and

*k*

_{tx}^{2}make negative

*k*

_{rz}^{2}and

*k*

_{tz}^{2}in Eq. (2.61) and (2.62). Finally,

*k*

*and*

_{rz}*k*

*become imaginary number, the reflecting waves and transmission waves in Eq. (2.58) and (2.59) become evanescent waves*

_{tz}( ) |Im[ ]|

m m

### H ( , ) R

^{m}

_{r}*x z*

### =

*e*

^{i k x k z}

^{rx}^{+}

^{rz}*y*

### = R

*e*

^{ik x}

^{rx}*e*

^{−}

^{k}

^{rz}

^{z}*y*

### JJJG JG JG

*z h x*

### ≤ ( )

(2.63)( ) |Im[ ]|

m m

### H ( , ) T

^{m}

_{t}*x z*

### =

*e*

^{i k x k z}

^{tx}^{−}

^{rz}*y*

### = T

*e*

^{ik x}

^{tx}*e*

^{+}

^{k}

^{tz}

^{z}*y*

### JJJG JG JG

*z h x*

### ≥ ( )

(2.64)Eq. (2.63) and (2.64) indicate that the scattering waves for large

### 2 m π

### Λ

^{are }

surface waves. These waves can be SPPs if medium 1 is dielectric and medium 2 is metal. Let Eq. (2.60) = Eq. (2.31), the momentum conservation law used to excite SPPs is obtained

### m 2

*sp* *ix*

*k* *k*

### π

### = +

### Λ

^{ }

^{(2.65) }

where _{sp}_{0} ^{rd}^{rm}^{rd}^{rm}

*rd* *rm* *rd* *rm*

*k* *k*

*c*

### ε ε ω ε ε

### ε ε ε ε

### × ×

### = =

### + +

^{. }

Finally, since 2π

Λ is the reciprocal unit vector of 1D gratings according to solid state physics [59], Eq. (2.60) and (2.65) can be extended to the 2D grating in the form

/ / / / / /

*r* *t* *x* *x*

*k* =*k* =*k* +*iG* + *jG*
JJJG JJJG JJG JJG JJG

(2.66)

//

*sp* *x* *x*

*k* =*k* +*iG* + *jG*
JJG JJG JJG JJG

(2.67)
where i, j are any integers, *G*JJG_{x}

and *G*JJG_{y}

are the reciprocal unit vectors of the grating.

//

*k**r*

JJJG, *k*JJJG_{t}_{//}

and *k*JJG_{/ /}

are the parallel component of the wavevector of reflection, transmission and incident light along the interface.

Eq. (2.67) is the momentum conservation law used to excite SPPs which will be used in chapter 3; Eq. (2.66) is the general momentum conservation law which will be utilized and discussed in chapter 4.

Fig. 2.4 Parallel polarization plane waves (TM mode) impinge on the one dimensional grating whose period is Λ. The blue lines, green lines and red lines represent the incident lights, reflection lights and transmission lights respectively.