超連續光譜是由光學中發現的周邊頻譜拓寬現象，當初始波通過傳遞媒介會產生大幅度的頻譜拓寬。朗謬爾波為電漿中具有縱向電場振盪的靜電波，因其傳播行為可以被超連續光譜的控制方程式非線性薛丁格方程式所描述，因此在理論上和模擬中預測其亦可產生超連續光譜的現象。然而，考慮在實驗上產生朗謬爾波超連續光譜時，朗謬爾波的激發成為了困難點。根據著名的朗道線性理論（Landau’s linear theory），在馬克士威分佈的電漿中因為朗道阻尼和相位混合的緣故朗謬爾波將會快速衰減。在本論文中，我嘗試在電漿實驗中產生朗謬爾波超連續光譜。為此，我採用了兩種針對朗道阻尼和相位混合的方式來激發朗謬爾波，其一為配置非馬克士威分佈電子以引發朗謬爾波線性非穩定性，另一種為非線性激發朗謬爾波。

在第一個實驗中，我們展示了透過電子束注入引發電子不穩定性(bump-on-tail)來產生朗謬爾波初始波並激發朗謬爾波超連續光譜。透過在電子分佈函數上建立BOT可以抵消朗道阻尼的影響。當注入的電子束能量落在 10 eV 和 30 eV 之間時，我們驗證了謬爾波初始波的激發和朗謬爾波超連續光譜的生成，其中伴隨了四波混合和連續離子聲波的生成。我們認為產生周邊頻譜導致頻譜拓寬的物理機制為涉及離子聲波產生的的振盪雙流不穩定性 (OTSI)，其引發了波與波之間的耦合。

在第二個實驗中，我們透過金屬網格將微波系統導入電漿當中來提供電位震盪，我們的目標是激發具有顯著振幅的朗謬爾波（BGK模式）和伴隨生成的超連續光譜。當激發條件符合朗謬爾波線性色散關係時，朗謬爾波激發與伴隨生成的超連續光譜即可被驗證。頻譜拓寬範圍與微波功率注入的正比關係為∆f∝〖P_inj〗^0.4，這大致符合電洞模型的理論值∆f∝〖P_inj〗^(1/4)，表示在微波系統模式中朗謬爾波超連續光譜的生成源自於電洞的產生。在電子相空間中的電子捕獲被認為抵消了相位混合的影響。朗謬爾波超連續光譜的產生伴隨了四波混合，但與使用 BOT 不穩定性生成超連續光譜的情況不同，朗謬爾波和離子聲波之間沒有關聯性。在微波系統模式中觀察到的四波混合被認定是只與朗謬爾波相關的調製不穩定性（MI）。

在這兩個實驗中，我們成功使用線性不穩定性和非線性模式激發朗謬爾波超連續光譜。結果驗證了朗道線性理論、BGK 非線性理論和我們對朗謬爾波超連續光譜生成的預測。

為了得到本研究的完整結論，我們還有以下需要完成的事項：在第一個實驗中，為了驗證電子不穩定性我們還需要量測電子速度分佈函數。在第二個實驗中需要更進一步研究波的激發條件，包括在外加電位震盪施加波數變化以及優化量測系統的量測電位等。這些將是未來的研究事項。

Supercontinuum (SC) is a spectral broadening phenomenon originally discovered in the optical physics community. Drastic spectral broadening of seed light is induced as it passes through a nonlinear optical medium. Langmuir waves (LWs), which are electrostatic waves having longitudinal oscillating electrostatic field in plasmas, are theoretically and numerically predicted to exhibit SC generation since they can be described by the nonlinear Schrödinger equation, which is considered as the governing equation for SC generation. However, if we consider experimental generation of Langmuir wave supercontinuum (LWSC), excitation of seed LWs is found to have some difficulties. According to the well- known Landau’s linear theory LWs are always damped in Maxwellian plasma due to Landau damping and phase-mixing. In this thesis, I attempt at generation of SC of LWs in a laboratory plasma experiment. To this end, I employ two methods to overcome Landau damping and phase-mixing to excite seed LWs. One is preparation of non-Maxwellian electrons to induce a linear instability of LWs. The other is nonlinear excitation of LWs.

In the first experiment, we demonstrate the excitation of seed LWs for LWSC generation by a linear instability, called bump-on-tail (BOT) instability, with the use of an electron beam injection (EBI). By constructing a BOT of electron distribution function, it is possible to cancel out the effect of Landau damping. We identified excitation of seed LWs and subsequent LWSC generation when the energy of injected electron beam was in the range between 10 eV and 30 eV, accompanied by coherent four-wave mixings (FWMs) and generation of continuum ion acoustic waves (IAWs). We consider wave-wave couplings through oscillating two-stream instability (OTSI) involving continuum IAWs to be a generation mechanism of the sidebands, leading to spectral broadening.

In the second experiment, we aim at excitation of LWs having large amplitude (electron holes, BGK (Bernstein-Greene-Kruskal) mode) for subsequent LWSC formation by the external potential oscillation with the use of microwave (MW) system through grids immersed in the plasmas. Generation of LWs and subsequent LWSC formation were identified when the excitation condition corresponds to the linear dispersion relation of LWs. The induced spectral broadening ∆f was proportional to the injection power of the microwaves to the power of 0.4 (∆f∝〖P_inj〗^0.4), which roughly agrees with the electron hole-size dependence on the injection power (∆f∝〖P_inj〗^(1/4)) using a simple model of electron holes. This result indicates that LWSC generated in the MW mode stemmed from generation of electron-holes. Creation of trapping electrons in the phase space of the electrons is considered to have canceled the effect of phase mixing. The generation of LWSC was accompanied by coherent FWMs. No correlation between LWs and IAWs was identified unlike in the case of LWSC generation using a BOT instability. The FWMs observed in the MW mode are considered to be modulational instabilities, which involve LWs only.

In the two experiments, LWSC were successfully generated with use of both the linear instability and the nonlinear modes. These results indicate validity of the Landau’s linear theory, BGK’s nonlinear theory, and our prediction to LWSC generation, respectively.

To obtain complete conclusions of this study, there are remaining works as follows: Measurement of velocity distribution functions of electrons is necessary to confirm BOT instabilities as to the first experiment. Further investigation of the excitation method using the externally applied oscillation is needed for the second experiment, including wavenumber scanning of the externally applied oscillation in addition to improvement of measurement method of the wave potential. These are remained as future works.

[1] Lewi Tonks and Irving Langmuir, ‘‘Oscillations in Ionized Gases’’, Phys. Rev. Vol.33, 195 (1929).
[2] D. Bohm and E. P. Gross, ‘‘Theory of Plasma Oscillations. A. Origin of Medium-Like Behavior’’, Phys. Rev. Vol.75, 1851 (1949).
[3] H. Looney Duncan and C. Brown Sanborn, ‘‘The Excitation of Plasma Oscillations’’, Phys. Rev. Letters, Vol.93, 965 (1954).
[4] F. F. Chen, ‘‘Introduction to plasma physics and controlled fusion’’, New York: Plenum press Vol. 1, pp. 84-85 (1984).
[5] L. D. Landau, ‘‘On electron plasma oscillations’’, Sov. Phys. JETP, Vol.16, 574 (1946).
[6] J. H. Malmberg and C. B. Wharton, ‘‘Collisionless damping of electrostatic plasma waves’’, Phys. Rev. Letters, Vol.13, 184 (1964).
[7] Ira B. Bernstein, John M. Greene, and Martin D. Kruskal, ‘‘Exact Nonlinear Plasma Oscillations’’, Phys. Rev. Vol.108, 546 (1957).
[8] M. Trippenbach, Y. B. Band, and P. S. Julienne, ‘‘Four wave mixing in the scattering of Bose-Einstein condensates’’, Opt. Express, Vol.3, 530-537 (1998).
[9] V. E. Zakharov, ‘‘Stability of periodic waves of finite amplitude on the surface of a deep fluid’’, Tech. Phys. Vol.9, 190 (1968).
[10] V. E. Zakharov, ‘‘Collapse of Langmuir waves’’, Sov. Phys. JETP, 35(5), 908-914(1972).
[11] B. D. Fried and Y. H. Ichikawa, ‘‘On the nonlinear Schrödinger equation for Langmuir waves’’, J. Phys. Soc. Jpn, Vol.34, 1073-1082 (1973).
[12] R. R. Alfano and S. L. Shapiro, ‘‘Observation of self-phase modulation and small-scale filaments in crystals and glasses’’, Phys. Rev. Vol.24, 592 (1970).
[13] F. J. McClung and R. W. Hellwarth, ‘‘Giant Optical Pulsations from Ruby’’, JAP Vol. 33, 828 (1962).
[14] A. J. DeMaria and D. A. Stetser, ‘‘Laser pulse-shaping and mode-locking with acoustic waves’’, Phys. Lett. Vol. 7, 71 (1965).
[15] A. Hasegawa, ‘‘Generation of a train of soliton pulses by induced modulational instability in optical fibers’’, Opt. Lett. Vol. 9 (1984).
[16] K. Tai et al., ‘‘Observation of Modulational Instability in Optical Fibers’’, Phys. Rev. Lett. Vol.56, 135 (1986).
[17] A. Chabchoub, N. Hoffmann, M. Onorato, G. Genty, J. M. Dudley, and N. Akhmediev, ‘‘Hydrodynamic supercontinuum’’, Phys. Rev. Vol.111, 054104 (2013).
[18] T. H. Stix, ‘‘Waves in Plasmas’’, AIP-Press (1992)
[19] L. D. Landau, ‘‘On electron plasma oscillations’’, Sov. Phys. JETP, Vol.16, 574 (1946).
[20] K. Case, Ann. ‘‘Plasma oscillations’’, Phys. 7, 349 (1959).
[21] N.G. Van Kampen. ‘‘On the theory of stationary waves in plasmas’’, Physica 21, 949 (1955).
[22] J. H. Malmberg and C. B. Wharton, ‘‘Collisionless damping of electrostatic plasma waves’’, Phys. Rev. Lett., Vol.13, 184 (1964).
[23] Ira B. Bernstein, John M. Greene, and Martin D. Kruskal, ‘‘Exact Nonlinear Plasma Oscillations’’, Phys. Rev. 108, 546 (1957)
[24] K. Saeki, P. Michelsen, H. L. Pécseli, and J. Juul Rasmussen, “Formation and coalescence of electron solitary holes,” Phys. Rev. Lett. Vol.42(8), 501–504 (1979).
[25] J. Pickett, L. Chen, S. Kahler, O. Santolík, D. Gurnett, B. Tsurutani, and A. Balogh, “Isolated electrostatic structures observed throughout the cluster orbit: Relationship to magnetic field strength,” Ann. Geophys. Vol. 22, 2515–2523 (2004).
[26] B. Eliasson and P. K. Shukla, “Dynamics of electron holes in an electron-oxygen-ion plasma,” Phys. Rev. Lett. Vol. 93(4), 045001 (2004).
[27] J. P. Lynov, P. Michelsen, H. L. Pecseli, and J. Juul Rasmussen, “Interaction between electron holes in a strongly magnetized plasma,” Phys. Lett. Vol. 80(1), 23–25 (1980).
[28] J. P. Lynov, P. Michelsen, H. L. Pécseli, J. Juul Rasmussen, and S. H. Sørensen, “Phase-space models of solitary electron holes,” Phys. Scr. Vol. 31(6), 596–605 (1985).
[29] V.E. Zakharov. ‘‘Collapse of Langmuir waves. Sov. Phys. JETP, 35(5) (1972).
[30] G. Thejappa, R. J. MacDowall, and M. Bergamo. ‘‘Evidence for four- and three-wave interactions in solar type III radio emissions’’, ANGEO Vol. 31, No. 8, pp. 1417-1428 (2013).
[31] P. A. Robinson. ‘‘Nonlinear wave collapse and strong turbulence’’, Rev. Mod. Phys., Vol. 69, No. 2 (1997).
[32] V.E. Zakharov and L.A. Ostrovsky. ‘‘Modulation instability: The beginning’’, Physica D Vol. 238 (2009)
[33] G. Thejappa, R. J. MacDowall, and M. Bergamo. ‘‘Evidence for four- and three-wave interactions in solar type III radio emissions’’. ANGEO Vol. 31, No. 8, pp. 1417-1428 (2013).
[34] E. Kawamori. ‘‘Identification of Langmuir wave turbulence-supercontinuum transition by application of von Neumann entropy’’. Physics of Plasmas, Vol. 24, 090701 (2017).
[35] E. Kawamori. ‘‘Generation of Langmuir wave supercontinuum by phase- preserving equilibration of plasmons with irreversible wave–particle interaction’’. The European Physical Journal D, Vol. 72, 63 (2018).
[36] E. Kawamori. ‘‘Thermodynamics of weakly nonlinear Langmuir waves, spontaneous symmetry breaking, and Nambu-Goldstone mode generation’’. J. Phys. Soc. Jpn., Vol. 90 024501 (2021).
[37] Y. Nakamura, M. Nakamura, and T. Itoh. Reception characteristics of monopole antennas for electron plasma waves. IEEE Trans. Plasma Sci., Vol.1(3), 100-106 (1973).
[38] J. M. Beall, Y. C. Kim, and E. J. Powers, ‘‘Estimation of wavenumber and frequency spectra using fixed probe pairs’’, J. Appl. Phys. Vol.53, 3933 (1982).
[39] T. H. Stix, ‘‘Waves in Plasmas’’, AIP-Press (1992)
[40] I. H. Hutchinson, C. B. Haakonsen, and C. Zhou, “Non-linear plasma wake growth of electron holes”, Physics of Plasmas Vol. 22, 032312 (2015).
[41] I. H. Hutchinson, “Electron holes in phase space: What they are and why they matter”, Physics of Plasmas, Vol. 24, 055601 (2017).