磁旋管是一種建立在利用弱相對論效應的機制”電子迴旋脈射(ECM)”下所設計出的高功率同調電磁波源，利用外加磁場使得電子迴旋運動，經由系統性的調控，電子能與電磁波達成同步並交換能量，進而放大電磁波。

磁旋返波震盪器理論首見於蘇俄1960年代中期的文獻[1] ，前人曾研究利用非線性理論預期磁旋返波震盪器(gyro-BWO)之效率將會低於其他形式之磁旋管[2]，也認為TE模之效率會比TM模還來得高[3-4]，隨著可調頻元件需求的增加，能藉著調變磁場和電壓來調頻的磁旋反波振盪器漸漸抬頭，而近期有研究指出TM模適合被使用在這種反波元件上[5] ，而本文的目的在於利用較直觀的二階線性方程式分析當ECM操作在TM模式下會發生哪些特性，並推論如何操作可以提升TM模式ECM之效率，而根據我們的線性方程也看到TM模ECM的確適合操作在反波下，也發現電子在波導管中對於TE模與TM模會有不相同的特性。

在第三章會談到操作非線性程式，嘗試我們在線性理論模型所推測的提升效率之方法。而TM_11與TE_01兩個模式在圓波導中是兩個截止頻率相同的模式，因此適合我們建立在同一基準比較當操作在相同頻率與參數時，兩個模式與電子交互作用的效率，並且之前的研究認為TE_01擁有不差的結果，因此這樣的比較也是有意義的，最後我們也發現在特定的操作下TM_11將會有不錯的結果，這與操作在反波和適當的電子迴旋中心位置有很大的關係。

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