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Terahertz backward wave oscillator circuits

Dissertation
Author: Guillermo Antonio Oviedo Vela
Abstract:
This work focuses on increasing the electric field mode-electron beam interaction in terahertz backward-wave oscillators through increasing the interaction impedance of the slow wave circuit. In a backward wave oscillator (BWO) or a traveling wave tube (TWT), the electric field traveling in the waveguide interacts with an electron beam grazing or piercing the electric field of a slow wave circuit and transfers energy from the beam to the circuit mode. The mechanism of this interaction is analyzed and the traditional model is adapted to provide one that is scalable to terahertz frequencies. The efficiency of the BWO can be enhanced by improving the interaction between the beam and the circuit modes, utilizing beam sources with greater current densities and using larger magnetic fields. This work presents the results on the modeling, fabrication and performance of the mode-piercing and mode-grazing slow wave circuits studied. Meandering folded waveguides and interdigital lines were modeled to evaluate their interaction impedance and electron beam requirements at terahertz frequencies. The models were verified against published results. Because of their low interaction impedance, terahertz meandering waveguides would need to be excited with large current density electron beams which in turn would require large magnets for their control. Interdigital lines were found to be the slow-wave circuit of choice in this work because their high interaction impedance and their ability to operate with low voltage, and low current density electron beams. Finite element numerical calculations were used to design, optimize and scale a terahertz free interdigital line (FIDL) circuit. The designed free interdigital line was fabricated using microfabrication technologies. Characterization of a free interdigital line (FIDL) circuit was performed in a tube body with a dispenser cathode and a three-anode Pierce gun. The FIDL measured output was compared with the calculated result and its performance met the simulated behavior. The resulting vacuum tube had an output of [approximate] 1 μW a total efficiency of [approximate] 0:05% at a center frequency of 1.25 THz. Because of its high interaction impedance, the use of confined electron beams in this FIDL BWO was not necessary. The limit of using magnet-free electron beams in traveling-wave tubes (TWTs) at terahertz frequencies (500-3000 GHz) was explored. The study showed that there is a minimum interaction impedance limit where this approach will not work, due to the diverse losses in the system. This work concludes that the high interaction impedances from beam-grazing circuits should enable a new class of terahertz-frequency tubes for signal sources and amplification. The short high impedance interdigital line demonstrated here enables magnet-free tubes at higher frequencies. Devices realized in this newly-identified design space should empower the advancement of a broad spectrum of fields, from chemical detection, to imaging, to communication, among others.

CONTENTS ABSTRACT..................................................iii CHAPTERS 1.INTRODUCTION.........................................1 1.1 Slow-wave circuits for backward wave oscillators.................1 1.2 Work and thesis content...................................3 1.2.1 Backward wave oscillator and traveling wave tube theory.....................................3 1.2.2 Numerical study of slow-wave circuit interaction impedances................................4 1.2.3 Fabrication,testing and modeling of a FIDL................4 1.2.4 Magnet-free BWOs...................................4 1.2.5 Contributions and conclusions...........................5 2.BACKGROUND...........................................8 2.1 History of backward wave oscillators..........................8 2.2 BWO device theory.......................................9 2.2.1 The electronic equation................................10 2.2.2 The circuit equation..................................11 2.2.3 The determinantal equation and its solution................13 2.2.4 Current condition for oscillation.........................15 2.2.5 Gain and efficiency...................................18 2.2.6 Tuning and spatial harmonics...........................21 2.3 State of the art..........................................23 2.3.1 Commercial BWOs and TWTs..........................23 2.3.2 Current research.....................................24 2.4 Competing terahertz source technologies.......................24 3.DEVELOPMENT..........................................31 3.1 Backward wave oscillator design,scaling and modeling............................................32 3.1.1 BWO parameter design- complete overlap case.........................................32 3.1.2 BWO parameter design- partial overlap case.........................................34 3.1.3 Numerical modeling...................................35 3.1.4 Scaling.............................................36

3.2 Slow-wave circuit design...................................37 3.2.1 Meandering rectangular waveguide circuits.................37 3.2.2 Interdigital line circuits................................38 3.3 Discussion..............................................40 4.FABRICATION,TESTING AND MODELING...............54 4.1 THz BWO slow-wave circuit fabrication.......................55 4.1.1 Fabrication processes..................................55 4.1.2 Vacuum tube assembly................................56 4.1.3 Testing hardware setup................................56 4.2 THz BWO testing........................................57 4.2.1 Power output measurements............................58 4.2.2 Power input measurements.............................58 4.2.3 Analysis and modeling.................................58 5.SUMMARY,CONCLUSIONS AND FUTURE WORK........77 5.1 Summary of work and results...............................77 5.2 Conclusions and discussion.................................78 5.3 Future work.............................................79 APPENDIX:MAGNET FREE ELECTRON BEAMS FOR TERAHERTZ BWOS.................................80 REFERENCES................................................89 vii

CHAPTER 1 INTRODUCTION 1.1 Slow-wave circuits for backward wave oscillators Backward wave oscillators (BWOs) and traveling wave tubes (TWTs) rely on slow-wave circuits for the generation and amplification of signals.This generation and amplification is accomplished by coupling energy from an electron beam to the slow-wave circuit.Slow-wave circuit designs for terahertz frequency backward- wave oscillators (BWOs) encounter several complications,notably practical limits on cathode current densities,large magnetic fields for beam confinement,energy density [1],and beam alignment.A prominent design goal entails maximizing the electron-beam–circuit-mode overlap,characterized by the interaction impedance Z i . For devices where the beam pierces the circuit mode,as in the folded waveguide circuit [2] shown in Fig.1.1,beam current density constraints yield a minimum beam tunnel which disturbs the waveguide modes.For devices where the beam grazes circuit modes extending from the surface,as the interdigital lines shown in Fig.1.2 or helices [3],the beam must be brought and kept within microns of the surface and a complete overlap of the circuit electric field and the beam is not possible. The slow-wave circuit dimensions involved at terahertz frequencies makes them ideal candidates for realization utilizing lithographic microfabrication techniques [4]. These techniques involve subtractive processes such as deep reactive ion etching (DRIE),LIGA,electron discharge machining,chemical etching and additive processes such as polymer spinning,chemical vapor deposition,sputtering,evaporation,among others.While the fabrication processes are well established,their application on the fabrication of terahertz slow-wave circuits poses several challenges.The three- dimensional characteristics of mode-piercing circuits make process parameters such as

2 surface roughness,aspect ratios and alignment particularly challenging.Such devices have been demonstrated to work at frequencies up to 650 GHz [5].The nature of mode-grazing circuits can make their fabrication less dependent on three-dimensional features,but substrate materials,supporting structures and electric field distributions are important factors to consider. The terahertz frequency spectrum is typically considered to be between 300 GHz (1000 µm,1 meV,10 cm −1 ) to 10 THz (10 µm,41 meV or 333 cm −1 ) [6].Tera- hertz sources capable of higher power outputs,larger tuning ranges and small form factors are highly desirable for use in applications such as chemical sensing [7,8] and imaging [9–11].Distinct identification of molecules is possible through terahertz spectroscopy.In biological samples this identification is specially desirable,but is difficult and time consuming [8,12] as the high water content in the sample absorbs the radiation and dehydration is not desirable as it changes the structures being observed. Consider the study done by Globus,et al.[8] where very thin DNA samples were prepared to overcome the small amount of radiation available to them.Furthermore, changes in the preparation procedures produced variations in the measurements [8]. The team was finally able to compare and successfully differentiate between the DNA of salmon and herring using complicated sample preparation techniques.This study illustrates how a brighter source of terahertz radiation would facilitate the process of obtaining valuable data from biological samples. Terahertz radiation is being used to image,identify and treat cancerous cells [10, 11].The non-ionizing nature of terahertz radiation makes it a good alternative to x-rays in applications where reducing the energy delivered is desirable.Terahertz frequencies are also used to analyze material surfaces and to perform quality control of products such as with medicine pills and capsules [13].Space exploration would also benefit from having a more efficient and compact source of radiation that can be used to amplify the black body radiation originating from distant stars [14].

3 1.2 Work and thesis content The work presented in this thesis focused on optimizing the interaction impedance of slow wave circuits in the terahertz frequency range.The interaction impedance is a parameter that arises from the field equations that describe the electron beam and circuit fields.Several circuits were numerically investigated,with interdigital lines providing the best interaction impedance at terahertz frequencies.Fabrication and testing of the preferred circuit for 1.5 THz,a freestanding interdigital line (FIDL), gave interaction impedances consistent with the numerical design estimates.These measurements also demonstrated that high impedance circuits scaled to terahertz frequencies can be short enough to not require magnetically-confined electron beams. Emphasizing the interaction impedance has helped open a circuit design space for terahertz frequency BWOs and TWTs which should help widen the field of practical applications. 1.2.1 Backward wave oscillator and traveling wave tube theory Chapter 2 gives a brief overview of the history of backward wave oscillators as radiation sources.The traveling wave tube was invented during the second world war in in Britain by Rudolf Kompfner,an architect.Kompfner worked in the development of the backward wave oscillator in his free time,working during the day on the high power klystrons necessary to make RADAR scanners for the war effort.To date, backward-wave oscillators can reach powers on the order of a few hundred watts with voltage-tunable bands greater than one octave.The background chapter presents a survey of the BWO state of the art,current research and competing technologies as they relate to reaching the terahertz boundary. Chapter 2 describes the fundamental derivation of the equations describing the behavior of the backward wave oscillator.In a backward wave oscillator the electric field traveling in a transmission structure interacts with the space charge of an electron beam and transfers energy from the beam to the circuit mode.The solution of

4 the equation system allows for the development of highly efficient,widely tunable backward wave oscillators. 1.2.2 Numerical study of slow-wave circuit interaction impedances In Chapter 3 the typical backward wave oscillator theory was adapted to fit the constraints encountered at terahertz frequencies.A generalized backward wave oscillation and interaction model is described assuming complete overlap between the circuit mode and the electron beam.The interaction impedance of terahertz rectangular folded waveguide circuits was calculated with this model.To account for the partial overlap in terahertz mode-grazing circuits,a model to find design parameters was developed.This numerical model was then compared with published measurements on a substrate interdigital line for validation.Two new circuit concepts are introduced to increase the interaction impedance of mode-grazing circuits namely a membrane interdigital line and a free standing interdigital line.Of all the circuits compared the interaction impedances of interdigital lines were found to be more appropriate for terahertz BWOs. 1.2.3 Fabrication,testing and modeling of a FIDL Chapter 4 presents the fabrication and testing a FIDL circuit.A FIDL circuit was designed per the procedures presented in Chapter 3.The FIDL circuit was fabricated using lithographic microfabrication techniques.The results of the fabrication process are presented.The slow wave circuit was mounted in a vacuum tube to test it. The testing setup and device performance are described.The result of the testing is compared with the simulation results and models. 1.2.4 Magnet-free BWOs Terahertz traveling wave tubes and backward wave oscillators with electron beams that do not require magnetic confinement are explored in the Appendix.The size of the slow-wave circuits at terahertz frequencies allows them to interact with electron

5 beams before they diverge due to space charge and thermalization of the electrons. The interaction impedance limit at which this approach becomes practical is explored. 1.2.5 Contributions and conclusions This work contributes new designs of slow-wave circuits for implementation in terahertz BWOs and provides insights how to make BWO devices more powerful and compact.Two types of slow wave circuits with improved beam-interaction character- istics over the state of the art are contributed in this work.The membrane interdigital line (MIDL) analyzed in this work increases the interaction of the electron beam with the circuit modes by increasing the electron beam overlap with the circuit mode.The freely supported interdigital line (FIDL) circuit analyzed,fabricated and tested in this work increases the interaction of the electron beam beyond the capabilities of the MIDL circuit.This work suggests that the magnetic field requirements of the electron gun in BWOs may be reduced because the distances over which the electron beamhas to be maintained linear are reduced.The reduced magnetic field requirements may make terahertz BWO devices more compact while the use of circuits with improved interaction mechanisms may make them more powerful.

6 Figure 1.1.Serpentine rectangular waveguide slow-wave circuit.In this mode-pierc- ing circuit the electron beam interacts with the mode’s electric field as it passes through a beam tunnel piercing the waveguide as shown in this figure.

7 Figure 1.2.A free interdigital line.In this mode-grazing circuit the electron beam interacts with the fringing electric field as it grazes the surface of the circuit.

CHAPTER 2 BACKGROUND This chapter gives a brief overview of the history of backward wave oscillators as radiation sources.The basic derivation of the equations describing the behavior of the backward wave oscillator is summarized.A survey of the state of the art in BWOs and competing terahertz sources is presented. 2.1 History of backward wave oscillators The backward-wave oscillator (BWO) is a special case of the traveling wave tube [15] and was originally developed by Kompfner [16] in England.Later,J. R.Pierce [17],from Bell labs,improved on the theory.The wide band and high power produced by traveling wave tubes made them ideal to develop vacuum tube amplifiers.Kompfner’s backward wave oscillator came about from making traveling wave tubes oscillate to take advantage of their wide band tunability with voltage up to 11 GHz [16]. Since Kompfner’s 11 GHz BWO,improvements on the theory of interaction,slow wave circuit design and improved fabrication techniques have made it possible to increase the operation frequency of those devices up to 1.42 THz at power levels as high as 200 µW[18–24]. The theories developed since the BWOfirst inception allowed for the improvement of power output by engineering the electron beam space charge,circuit loss,beam thickness,velocity spread and circuit mismatches [18,23].Chapter 3 in this work modifies the available theory to the special case where partial beam-mode interaction occurs.

9 Slow wave circuits such as serpentine waveguides [2,25],vanes [22] and interdigital lines [19,20] among others [21,26] have complemented Kompfner’s original helical slow-wave circuit.These circuits have increased the power output,beam-mode in- teraction,operational frequency and tunability of BWOs.Advances in fabrication methods such as semiconductor and MEMS microfabrication techniques have en- abled the scaling of conventional circuits into the terahertz region of the frequency spectrum [25,27]. 2.2 BWO device theory When a BWO oscillates at the resonance condition,charge density waves in the electron beam couple with,and transfer energy to,the periodic field modes of the slow-wave circuit.The resonant condition occurs when the frequency and propagation constant of the electron beam match the frequency and propagation constant of the circuit mode.The frequency of oscillation can therefore be varied by changing the beam voltage.Oscillations start when the electronic gain per circuit period exceeds the losses per period.The BWO electronic gain is given by the amount of coherent, positive,amplifying feedback to the noise-coupled wave as it propagates backwards from the downstream end of the circuit.The BWO electronic gain increases with beam current to the critical value,defining the start current.The tunable band of the device is typically defined by the band of frequencies where the beam interacts with the first backward harmonic of the circuit.Interaction with higher harmonics is possible;however,the output of the device would also contain the lower harmonic components and power would be distributed among all of the harmonic components of the resulting radiation. This section describes the backward wave interaction following the derivations in refs.[17,18,23,28].The derivation involves three fundamental equations namely the electronic equation,the circuit equation and the determinantal equation to describe a slow-wave and beam system similar to Fig.1.1.The electronic equation developed in Section 2.2.1 relates the current waves on the electron beam to the ac electric field along the beam.The circuit equation in Section 2.2.2 relates the ac current

10 in the slow-wave transmission line circuit to the slow-wave circuit electric fields. Resonant conditions can then be obtained by equating the electric fields from these two equations resulting in the BWO determinantal equation in Section 2.2.3.The four solutions to the determinantal equation describe the damped and propagating waves in the system as explained in Section 2.2.4. 2.2.1 The electronic equation This derivation of the electronic equation is taken from [17,29].This derivation assumes a one-dimensional beam and that second-order effects are negligible. The total electron beam current is the product of the total charge density and the total charge velocity.The total current,the total charge density and the total velocity are each constituted by ac and dc terms, I 0 +i = (ρ 0 +ρ)(ν 0 +ν) = ρ 0 ν 0 +ρ 0 ν +ν 0 ρ +ρν, (2.1) I 0 ,ρ 0 and ν 0 are the dc current,charge density and velocity,and i,ρ and ν are the corresponding ac variables,respectively.From equation 2.1 it can be noted that: I 0 = ρ 0 ν 0 , (2.2) and i = ρ 0 ν +ν 0 ρ +ρν. (2.3) The term ρν can be neglected assuming it is much smaller than the other quantities. The continuity equation ∇· i + ∂ρ ∂t = 0 (2.4) can be reduced to ∂i ∂z = −jωρ (2.5)

11 by the one-dimensional assumption that the ac field is only varying in the z direction and that it has a sinusoidal time dependence e jωt where ω is the angular frequency. Applying this set of assumptions,the Lorentz force equation becomes through the chain rule d(ν 0 +ν) dt = ν 0 ∂ν ∂z + ∂ν ∂t = qE m e . (2.6) In this equation,m e is the free electron mass.Equation 2.6 assumes that ν 0 is independent of both time and distance and dz/dt is composed of only the dc velocity because the ac velocity is negligible.Assuming that the ac quantities vary as e jωt then equation 2.6 becomes: ν 0 ∂ν ∂z +jων = qE m e . (2.7) Using equations 2.3,2.5 and 2.7 to eliminate the ac quantities ν and ρ gives an expression for the total electric field in the region occupied by the electron beam. ∂ 2 i ∂z 2 +2jβ e ∂i ∂z −β 2 e i = jβ e ρ 0 q m e ν 0 E. (2.8) This equation is the electronic equation,in which the ac beam velocity ν has been replaced by the beam propagation constant β e . 2.2.2 The circuit equation The following derivation is taken from [29] as it was derived by [17].This deriva- tion augments the transmission line equations with a displacement current term yielding the following expressions: ∂V ∂z = −ZI (2.9) ∂I ∂z = −Y V +I D , (2.10)

12 where Z is the line impedance in the z direction,Y is the line reactance,I and V are the voltage and current and I D is the displacement current induced by the beam in the z direction.I D may be found taking into account Maxwell’s curl equation of a beam of crossectional area σ. ∇×H = i + ∂D ∂t (2.11) the divergence of which yields ∇· (i + ∂D ∂t ) = ∇· ∇×H = 0. (2.12) Integrating the above equation by means of Gauss’ theorem,

V ∇· (i + ∂D ∂t dV ) =

S (i + ∂D ∂t ) · dS = 0, (2.13) gives the displacement current I D = −σ ∂i ∂z . (2.14) Turning equation 2.10 into ∂I ∂z = −Y V −σ ∂i ∂z , (2.15) taking the partial derivative of equation 2.9 with respect to z and combining it with equation 2.15 one obtains

13 ∂ 2 V ∂z 2 −ZY V = Zσ ∂i ∂z (2.16) which can be written in terms of electric fields since E = −∇V assuming all the quantities vary only in the negative z direction, ∂ 2 E ∂z 2 −ZY E = Zσ ∂ 2 i ∂z 2 (2.17) The circuit equation is then obtained by further substituting Γ 0 2 = ZY and K 0 2 = Z Y into equation 2.17, ∂ 2 E ∂z 2 −Γ 2 0 E = Γ 0 K 0 σ ∂ 2 i ∂z 2 . (2.18) This equation was first given in this form by Pierce and an alternate solution also given by Bernier [17,28,30]. 2.2.3 The determinantal equation and its solution The interaction between the beam and the circuit mode can now be calculated by equating the electric fields given by the electronic and the circuit equation to give the determinantal equation.The solutions to the determinantal equation describe the waves in the system.The derivation of the determinantal equation presented in this work is taken from [28].By using the Laplace transform with respect to z of the electronic equation 2.8 it takes the form

Γ +jβ e

2 i = jβ e I 0 E 2σV 0 . (2.19) Doing the same with the circuit equation it becomes

14

Γ 2 −Γ 2 0

E = Γ 0 K 0 σΓ 2 i. (2.20) Solving for the electric field in equation 2.20 and equation 2.19 and equating the resulting expressions gives the following equation

Γ +jβ e

2

Γ 2 −Γ 2 0

= jβ e I 0 K 0 Γ 0 Γ 2 2V 0 . (2.21) This expression can be simplified by defining the interaction parameter C C 3 = K 0 I 0 4V 0 , (2.22) turning equation 2.23 into the determinantal equation

Γ +jβ e

2

Γ 2 −Γ 2 0

= j2βeC 3 Γ 0 Γ 2 . (2.23) The four particular solutions to this system yield three forward resonant propagation constants Γ Γ 1 = Γ 2 = Γ 3 = −jβ e , (2.24) and a backward propagation constant Γ 4 = jβ e . (2.25) These solutions assume no loss,no space charge and a perfect synchronism condition (i.e.,Γ 0 = jB e .)

15 Using these values of Γ to solve for the electric field with respect to z the electric field expression becomes: E(z) = E 1 e Γ 1 z +E 2 e Γ 2 z +E 3 e Γ 3 z +E 4 e Γ 4 z . (2.26) This electric field expression will be used in Section 2.2.4 to deduce a current condition for oscillation,in Section 2.2.5 to deduce the gain and efficiency and in Section 2.2.6 to analyze the tuning and space harmonics of the system. 2.2.4 Current condition for oscillation Under the ideal conditions explained in Section 2.2.3 the systemwould oscillate at any electron beam current.However,[18,22,23,31–34] explained that space charge, circuit loss and beamvelocity spread are factors that determine the level of interaction between the beamand the circuit mode and establish a minimumI 0 threshold required for oscillations. The effects of the space charge in these papers are taken into account by adding the propagation constant of the space charge β q to the electronic equation 2.19 as follows: [

Γ +jβ e

2 +β 2 q ]i = jβ e I 0 E 2σV 0 , (2.27) which in turn makes the determinantal equation take the form of [

Γ +jβ e

2 +β 2 q ]

Γ 2 −Γ 2 0

= j2βeC 3 Γ 0 Γ 2 (2.28) where β 2 q = 4QC 3 β 2 e with QC 3 =

ω q /ω

accounting for the space charge parame- ter [28],ω q being the cyclotron frequency of the electrons in a beam of radius r given by ω 2 q = ηρ 0 2πr 2

0 . (2.29)

16 The circuit loss is factored in by introducing a loss parameter d d = 20 ln

L db CN

, (2.30) where L db is the loss in dB per the amount of interaction in a circuit period CN. Because the electron beampropagation constant spread β e and the phase propagation constant spread in the circuit β p are not equal,a synchronism parameter is also introduced.The synchronism parameter b is defined as the ratio of the difference in propagation spreads and the beam spread such that b = β p −β e Cβ e (2.31) The loss parameter and the synchronism parameter modify the cold circuit propa- gation constant by introducing an attenuation term Cd and a propagation term Cb such that Γ 0 = jβ e +jβ e Cb +β e Cd. (2.32) Further accounting for the synchronismdeviation in the systems propagation constant an extra term is introduced to its expected value from equation 2.24 such that Γ = −jβ e +βeCδ. (2.33) Applying the conditions from equations 2.32 and from 2.33 to the determinantal equation results in the equation: δ 2 +4QC = 2j(j +jCb +Cd)(1 −2jCδ +C 2 δ 2 ) (δ +jb +d)(−2j +cδ +jCb −Cd) . (2.34)

17 The right side of this equation can be further simplified by realizing that the Cδ and Cb terms are much smaller than unity and can be neglected [17,23,29] giving the final expression for the modified determinantal equation as follows: (δ 2 +4QC)(−b −jd +jδ) = −1 (2.35) with solutions for a critically damped forward traveling propagation constant: Γ 1 = −jβ e +jβ e Cδ 1 , (2.36) δ 1 =

√ 3 2 − j 2

, (2.37) a forward damping propagation constant: Γ 2 = −jβ e +β e Cδ 2 , (2.38) δ 2 =

− √ 3 2 − j 2

, (2.39) a forward traveling propagation constant: Γ 3 = −jβ e +jβ e Cδ 3 , (2.40) δ 3 = j, (2.41) and a backward traveling propagation constant:

18 Γ 4 = jβ e − jβ e C 3 4 . (2.42) When the circuit oscillates,the magnitude of the electric field at z = L is equal to zero and thus the oscillation condition becomes: 0 = (δ 2 1 +4QC) (δ 1 −δ 2 )(δ 1 −δ 3 ) e 2πCNδ 1 + (δ 2 2 +4QC) (δ 2 −δ 1 )(δ 2 −δ 3 ) e 2πCNδ 2 + (δ 3 2 +4QC) (δ 3 −δ 1 )(δ 3 −δ 2 ) e 2πCNδ 3 . (2.43) This equation can be numerically analyzed for particular values of b,d,QC and CN. Values for the space charge parameter QC and the loss parameter d are typically dependent on design requirements.Then the frequency spread parameter b and the interaction parameter CN are simultaneously solved.The resulting values of b and CN where the system starts oscillation are thus defined to be CN st and b st .Prior to the widespread use of computer simulation tools these four parameters were solved through the use of engineering curves.The interested reader may refer to the literature [29,35,36] for more information. By substituting the value of CN st into equation 2.22,an initial value for the current required for the starting of oscillations can be obtained from equation 2.44 I 0st = 4V 0 (CN st ) 3 K 0 (2.44) 2.2.5 Gain and efficiency The solutions that account for the losses and inefficiencies of the coupling system can now be fully applied to an expression for the system gain by substituting the propagation constants into either the circuit or the electronic equation to obtain values for the electric field at the beginning of the tube and at the end of the tube. The gain relationship is then given by the ratio between the electric field at the input of the tube E(L) and the output E(0).

Full document contains 103 pages
Abstract: This work focuses on increasing the electric field mode-electron beam interaction in terahertz backward-wave oscillators through increasing the interaction impedance of the slow wave circuit. In a backward wave oscillator (BWO) or a traveling wave tube (TWT), the electric field traveling in the waveguide interacts with an electron beam grazing or piercing the electric field of a slow wave circuit and transfers energy from the beam to the circuit mode. The mechanism of this interaction is analyzed and the traditional model is adapted to provide one that is scalable to terahertz frequencies. The efficiency of the BWO can be enhanced by improving the interaction between the beam and the circuit modes, utilizing beam sources with greater current densities and using larger magnetic fields. This work presents the results on the modeling, fabrication and performance of the mode-piercing and mode-grazing slow wave circuits studied. Meandering folded waveguides and interdigital lines were modeled to evaluate their interaction impedance and electron beam requirements at terahertz frequencies. The models were verified against published results. Because of their low interaction impedance, terahertz meandering waveguides would need to be excited with large current density electron beams which in turn would require large magnets for their control. Interdigital lines were found to be the slow-wave circuit of choice in this work because their high interaction impedance and their ability to operate with low voltage, and low current density electron beams. Finite element numerical calculations were used to design, optimize and scale a terahertz free interdigital line (FIDL) circuit. The designed free interdigital line was fabricated using microfabrication technologies. Characterization of a free interdigital line (FIDL) circuit was performed in a tube body with a dispenser cathode and a three-anode Pierce gun. The FIDL measured output was compared with the calculated result and its performance met the simulated behavior. The resulting vacuum tube had an output of [approximate] 1 μW a total efficiency of [approximate] 0:05% at a center frequency of 1.25 THz. Because of its high interaction impedance, the use of confined electron beams in this FIDL BWO was not necessary. The limit of using magnet-free electron beams in traveling-wave tubes (TWTs) at terahertz frequencies (500-3000 GHz) was explored. The study showed that there is a minimum interaction impedance limit where this approach will not work, due to the diverse losses in the system. This work concludes that the high interaction impedances from beam-grazing circuits should enable a new class of terahertz-frequency tubes for signal sources and amplification. The short high impedance interdigital line demonstrated here enables magnet-free tubes at higher frequencies. Devices realized in this newly-identified design space should empower the advancement of a broad spectrum of fields, from chemical detection, to imaging, to communication, among others.