불균일한 플라스마 및 메타물질에서의 모드변환: 불변 끼워넣기 이론의 적용
Mode conversion in inhomogeneous plasmas and metamaterials: Invariant imbedding theory approach
- 주제(키워드) Mode conversion , Resonant absorption , Wave coupling
- 발행기관 아주대학교
- 지도교수 김기홍
- 발행년도 2012
- 학위수여년월 2012. 2
- 학위명 박사
- 학과 및 전공 일반대학원 에너지시스템학부
- 실제URI http://www.dcollection.net/handler/ajou/000000012590
- 본문언어 영어
- 저작권 아주대학교 논문은 저작권에 의해 보호받습니다.
초록/요약
In homogeneous plasmas wave modes are decoupled and distinct. On the contrary, in inhomogeneous plasmas there can be a linear coupling and associated energy transfer called linear mode conversion between two modes for some special conditions. Mode conversion is one of the most fundamental processes in space and laboratory plasmas. Since the study of mode conversion contains inhomogeneity, one faces great difficulty to understand the physics, so many researches are concentrated on the simple geometries and approximate WKBJ-type methods are often used. In other to understand more deeply about the mode conversion processes in inhomogenoeus plasmas, we use a new kind of invariant imbedding method which gives numerically exact results. This method was successfully applied to various kind of problems including mode conversion in previous researches. We first investigated the effect of periodic density modulation superimposed on linear density profile on mode conversion in a cold, unmagnetized plasma. Most of previous studies are on the linear density profile and universal behavior of mode conversion coefficient with the maximum value of 0.5 is obtained. Superimposing the periodic modulation on the linear density profile, we obtained nonmonotonic behavior of mode conversion such that there appears new mode conversion peak whose value sometimes reaches 1, which depends significantly on the wave number, modulation period, and incident angle. The unusual peak is explained by the formation of cavity near resonance point where the incident field is strongly enhanced. When plotting as a function of frequency multiple peaks are present at certain frequencies with regular pattern. This phenomenon can be useful for enhancing the heating efficiency. Secondly, we studied the effect of periodic electron plasma density variation on mode conversion. It was found that mode conversion band associated with the periodicity appears in some frequency range which is related to transmission. The mode conversion coefficient has a oscillating behavior as a function of frequency. We also found that the mode conversion coefficient has a dip at certain frequency, incident angle, and the number of modulation period, which is associated with the standing wave formation. Thirdly, we investigated the effect of random modulation superimposed on linear electron density profile on mode conversion. Since the electromagnetic waves are mostly reflected when propagating through the random medium, it is expected that the mode conversion coefficient decreases as the strength of randomness increases, which is due to Anderson localization effect. We report additional effect competing with Anderson localization effect. We observed that the mode conversion coefficient becomes reduced at first, increases up to certain maximum value, and then decreases again for strong randomness. The intermediate increase is due to enhanced tunneling between (new) resonance points or between resonance and cutoff points, and the former and the latter decreases are due to multiple scattering, the Anderson localization. In this case both local-range effect near resonance points and long-range effect have significant contributions to mode conversion behavior. All three previous results dealt with direct problems, conversion of electromagnetic wave mode into electrostatic mode, in cold, unmagnetized plasmas. Next, we considered both direct and inverse problems in warm, unmagnetized plasmas where the electromagnetic wave and Langmuir wave are coupled. As in the first case, we investigated linear electron density profile where the ion motion is neglected and found that equal amount of energy is transferred between two modes regardless of the temperature and the reciprocity principle is verified. The temperature-dependent shift of mode conversion coefficient whose maximum is near 0.5 is observed for small temperature range, which is similar to previous results, but the detailed variation is more sensitive than previous result, which means that temperature may play more crucial role in mode conversion process than expected. The mode conversion also occurs outside the atmosphere of the Earth. Outside the ionosphere, there exists magnetosphere and magnetopause in its outer edge. In these areas the plasmas can be treated as a fluid and many kinds of wave exist. The magnetohydrodynamics (MHD) deals with this situations. In cold plasmas two modes exist, compressional mode and transverse (Alfv\'{e}n) mode. The mode conversion from compressional mode into transverse mode is considered at the magnetopause which has an Alfv\'{e}n speed crest where the effect of magnetosheath flow or solar wind is imposed. We studied three cases. The first is that the MHD wave propagates from the magnetosheath into magnetosphere through the magnetopause where the flow is in the magnetosheath. The second case is the reverse of the first case such that the wave propagates from the magnetosphere to the magnetosheath across the magnetopause. Third case is that the wave propagates from a region of magnetosheath to another region of magnetosheath through the magnetosphere. For the first case, when the flow is present there appears a cutoff range in the the mode conversion band and the wave is totally reflected in this range. For the second case, there appears new mode conversion peak in the same frequency range corresponding to cutoff range of the first case. The value and frequency of the the mode conversion peak depend strongly on the flow speed. The third case is very similar to first case except that it has more oscillating behavior. All three cases have ripples in their reflectivity, transmissivity, and mode conversion coefficient curves. Recent theoretical studies provide the possibility of new mode conversion in transition metamaterials where the parameters $\epsilon$ and $\mu$ change arbitrarily from positive to negative values in space. In addition to $p$ wave, mode conversion for $s$ wave can occur when $\mu$ varies continuously from positive to negative values. The mode conversion and resonant absorption similar to $p$ wave case and the interference effect by multiply scattered waves between two mode conversion points were previously reported. Considering that $\epsilon$ and $\mu$ vary from positive to negative values with different ratios, we observed that all the electromagnetic field components can be greatly enhanced at the resonance region, which can give new synergy effect on nonlinear optical uses. In addition, we investigated the mode conversion in transition metamaterials with small gain. The situation is similar to inverse mode conversion where the energy of longitudinal oscillations transforms to that of electromagnetic waves. We observed giant amplification of electromagnetic radiation due to phase coherence when electromagnetic waves are incident on the transition metamaterial with small gain. We also found that the value of amplification and the radiation angle can be controlled by appropriate design of the $\epsilon$ and $\mu$ profiles . It turned out that the interference between two resonance points also significantly affects the radiation intensity. Finally, we applied the invariant imbedding method to a different subject, the propagation of the linear shallow water wave. This application is possible because the equation of $p$ wave has the same form with that of linear shallow water wave identifying the correspondence between the dielectric permittivity and the inverse water depth. By using this relationship we studied the influence of bottom topography on the propagation of linear shallow water wave in three cases. Considering the linear depth profile we have a perfect agreement with the analytical formula such that the field intensity at the shore is greatly enhanced, which is so-called tsunami effect. The wave function has a singular behavior at the shore since the water depth approaches zero. The Bragg scattering competing with the tsunami effect is investigated as a second case, resulting in large reflection in a wide frequency range. Finally, resonant tunneling associated with the symmetric field distribution is observed by examining the ridge-shaped bottom profile.
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Contents
1 General introduction 1
1.1 Mode conversion in natural phenomena and its appli-
cation . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Previous theoretical study of mode conversion . . . . 4
1.3 Contribution of the present study . . . . . . . . . . . 6
2 Theory 11
2.1 Mode conversion . . . . . . . . . . . . . . . . . . . . 11
2.2 The linear mode conversion in cold (warm), unmag-
netized plasmas . . . . . . . . . . . . . . . . . . . . . 16
2.3 Wave equations . . . . . . . . . . . . . . . . . . . . . 22
2.3.1 Single wave equation . . . . . . . . . . . . . . 22
2.3.2 Coupled wave equations: mode conversion in
warm, unmagnetized plasmas . . . . . . . . . 22
2.3.3 Magnetohydrodynamic (MHD) wave equation 26
2.4 Invariant Imbedding Method . . . . . . . . . . . . . . 30
2.4.1 Introduction . . . . . . . . . . . . . . . . . . . 30
2.4.2 Invariant imbedding equations . . . . . . . . . 31
3 Mode conversion in cold, unmagnetized plasmas 39
3.1 Mode conversion due to a periodic density modulation
superposed on a linear electron density profile..... 39
3.1.1 Introduction . . . . . . . . . . . . . . . . . . . 39
3.1.2 Model . . . . . . . . . . . . . . . . . . . . . . 40
3.1.3 Invariant imbedding equations . . . . . . . . . 42
3.1.4 Results . . . . . . . . . . . . . . . . . . . . . . 43
3.1.5 Summary . . . . . . . . . . . . . . . . . . . . 49
3.2 Influence of periodic electron density variation on
mode conversion . . . . . . . . . . . . . . . . . . . . . . . . 50
3.2.1 Introduction . . . . . . . . . . . . . . . . . . . 50
3.2.2 Model . . . . . . . . . . . . . . . . . . . . . . 51
3.2.3 Invariant imbedding equations . . . . . . . . . 52
3.2.4 Results . . . . . . . . . . . . . . . . . . . . . . 54
3.2.5 Summary . . . . . . . . . . . . . . . . . . . . 60
3.3 The effect of a random spatial variation of the elec-
tron plasma density superposed on a linear electron
density profile . . . . . . . . . . . . . . . . . . . . . . 61
3.3.1 Introduction . . . . . . . . . . . . . . . . . . . 61
3.3.2 Model . . . . . . . . . . . . . . . . . . . . . . 63
3.3.3 Results . . . . . . . . . . . . . . . . . . . . . . 65
3.3.4 Summary . . . . . . . . . . . . . . . . . . . . 71
4 Mode conversion in warm, unmagnetized plasmas 73
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . 73
4.2 Wave equations and model . . . . . . . . . . . . . . . 74
4.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . 76
4.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . 78
5 Effect of magnetosheath flow or solar wind on the
magnetohydrodynamic wave mode conversion in the
magnetosphere 81
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . 82
5.2 Model . . . . . . . . . . . . . . . . . . . . . . . . . . 84
5.3 Invariant imbedding equations . . . . . . . . . . . . . 85
5.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . 87
5.4.1 Case 1 . . . . . . . . . . . . . . . . . . . . . . 88
5.4.2 Case 2 . . . . . . . . . . . . . . . . . . . . . . 90
5.4.3 Case 3 . . . . . . . . . . . . . . . . . . . . . . 92
5.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . 95
6 Mode conversion in metamaterials 97
6.1 Metamaterials and mode conversion . . . . . . . . . . 98
6.2 Giant electromagnetic field enhancement and reso-
nant transmission . . . . . . . . . . . . . . . . . . . . 109
6.2.1 Introduction . . . . . . . . . . . . . . . . . . . 109
6.2.2 Model . . . . . . . . . . . . . . . . . . . . . . 110
6.2.3 Invariant imbedding equations . . . . . . . . . 111
6.2.4 Results . . . . . . . . . . . . . . . . . . . . . . 113
6.2.5 Summary . . . . . . . . . . . . . . . . . . . . 117
6.3 Giant amplification of electromagnetic radiation due
to inverse mode conversion in transition metamateri-
als with small gain . . . . . . . . . . . . . . . . . . . 119
6.3.1 Introduction . . . . . . . . . . . . . . . . . . . 119
6.3.2 Model . . . . . . . . . . . . . . . . . . . . . . 122
6.3.3 Invariant imbedding equations . . . . . . . . . 124
6.3.4 Result . . . . . . . . . . . . . . . . . . . . . . 125
6.3.5 Summary . . . . . . . . . . . . . . . . . . . . 128
7 Influence of bottom topography on the propagation
of linear shallow water waves 129
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . 130
7.2 Linear shallow water wave equation . . . . . . . . . . 131
7.3 Models of bottom topography . . . . . . . . . . . . . 134
7.4 Invariant imbedding equations . . . . . . . . . . . . . 136
7.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . 138
7.5.1 Model A . . . . . . . . . . . . . . . . . . . . . 138
7.5.2 Model B . . . . . . . . . . . . . . . . . . . . . 138
7.5.3 Model C . . . . . . . . . . . . . . . . . . . . . 142
7.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . 147
8 Conclusion 149
A Derivation 153
A.1 Derivation of s or p wave equation . . . . . . . . . . . 153
