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SECTION I

INTRODUCTION

THE TRAVELING wave tube (TWT) is a key element in telecommunication systems, satellite-based transmitters, military radar, electronic countermeasures, and communication data links Formula$[1-4]$. Variations in performance due to finite fabrication tolerances in the manufacturing process can lower the fraction of TWTs that meet specifications and drive up the cost of manufacturing [5], [6]. These errors produce proportionately larger perturbations to the circuit as the circuit size is reduced.Their effects on the small signal gain and output phase have been studied by Pengvanich et al. [7] who considered the evolution of the three forward waves in a TWT in which the Pierce parameters vary randomly along the tube axis. A peculiar feature of the results in [7] is that, in the statistical evaluation of a large number of samples with random errors in the circuit phase velocity, a significant number of these samples show an output gain that is higher than the corresponding error-free tube. It is this intriguing feature that prompted us to analyze the expectation values of the gain and phase reported in this paper. As we shall see shortly, we provide an explanation of this statistical feature in this paper. We also extend [7] to include AC space charge effects and non-synchronous interactions. We shall ignore the effects of the reverse propagating circuit wave, which we recently analyzed [8]. In [8], we found that reflections may significantly increase the statistical effects on the gain and the output phase. Effects on the TWT backward-wave mode [9] by random manufacturing errors were also recently analyzed [10].

The standard deviations in the gain and in the output phase, which were analytically calculated in [7], required only an account of the first order effects of random errors. The expected mean of the output gain and phase, which is our focus here, requires consideration of the second order effects of random errors, and is therefore more difficult to evaluate. Since deviation from the mean (a second order effect) is much less than the standard deviation (a first order effect), a significant number of the samples in a statistical analysis would naturally show an output gain that is higher than the corresponding error-free tube, as observed in Pengvanich et al. [7]. We use three approaches to analyze this problem. The first approach is analytical where we apply successive perturbations on all three forward waves. The second approach is also analytical where we include only the dominant, growing mode in the analysis. The third approach is purely numerical where we numerically integrate the governing differential equation (at least) 5000 times using as many random samples in the coefficients that represent random axial variations in the circuit phase velocity. Comparison of these three approaches is presented.

This paper is organized as follows. Section II presents the model to evalute the gain and phase of a traveling wave tube for a general ratio of circuit phase velocity to beam velocity, including the effects of space charge. Section III presents the analytic expressions for the expected gain and phase, and for the standard deviations. The details of the two analytical formulations are given in Appendicies A and B. Section IV presents comparison of numerical results from the three different approaches. An example of a 210 GHz, G-band TWT is presented. Section Vsummarizes our results.

SECTION II

MODEL

We follow the model of [7] which is based on Pierce's theory except that the assumption of axial uniformity in the circuit parameters has been relaxed. Assuming Formula$e^{j\omega t}$ dependence, the linearized force law, including the “AC space charge effects,” reads, Formula TeX Source $$\left[\left({{\partial}\over{\partial z}}+j\beta_{e}\right)^{2}+\beta_{q}^{2}\right] s=a,\eqno{\hbox{(1)}}$$ where Formula$s$ is the electronic displacement caused by the normalized circuit electric field Formula$a$, Formula$\beta_{e}=\omega/v_{0}$, Formula$\beta_{q}\equiv C\beta_{e}\sqrt{4QC}$, Formula$v_{0}$ is the streaming velocity of the electron beam, Formula$C$ is Pierce's dimensionless gain parameter, and Formula$QC$ is Pierce's space charge parameter. In the absence of AC space charge effects, Formula$QC=0$ and (1) is identical to (1) of [7]. The slow-wave circuit equation is unchanged by the presence of AC space charge, Formula TeX Source $$\left({{\partial}\over{\partial z}}+j\beta_{p}+\beta_{e}Cd\right)a=-j\left(\beta_{e}C\right)^{3}s,\eqno{\hbox{(2)}}$$ which is the corrected form of (2) of [7]. (The third term on the left hand side of (2) should read Formula$\beta_{e}Cd$ instead of Formula$\beta_{p}Cd$, a typo in [7]that has propagated through the literature.) In (2), Formula$\beta_{p}=\omega/v_{p}$, where Formula$v_{p}$ is the phase velocity of the slow wave in the absence of the beam, and Formula$d$ is the normalized cold tube circuit loss rate. For an error-free tube in which Formula$\beta_{p}$, Formula$C$, and Formula$d$ are constants, (1) and (2) yield the Pierce dispersion relation, Formula TeX Source $$\big (\delta^{2}+4QC\big)\big (\delta+jb+d\big)=-j,\eqno{\hbox{(3)}}$$ assuming Formula$e^{j\omega t-j\beta z}$ dependence, where Formula$\delta=-j(\beta-\beta_{e})/C\beta_{e}$, and Formula$b=(v_{0}/v_{p}-1)/C$ is the mismatch between the beam and circuit velocities. When the quantities Formula$\beta_{p}$, Formula$C$, or Formula$d$ are allowed to vary axially, (3) is no longer applicable, and we combine (1) and (2) to yield, Formula TeX Source $$\eqalignno{&{{d^{3}f(x)}\over{dx^{3}}}+jC\left(b-jd\right){{d^{2}f(x)}\over{dx^{2}}}+4QC^{3}{{df(x)}\over{dx}}\cr&\qquad\qquad\quad~\,+jC\left(4QC^{3}\left(b-jd\right)+C^{2}\right)f(x)=0, &{\hbox{(4)}}}$$ where Formula$x=\beta_{e}z$ is the normalized axial distance, and Formula$f(x)=e^{jx}s(x)$ represents Pierce's three-wave solution to the third order ordinary differential (5). In the absence of AC space charge effects, Formula$QC=0$ and (4) reduces to (5) of [7]. We solve (4) subject to the initial conditions at the TWT input Formula$(x=0)$, Formula TeX Source $$f(0)=0, f^{\prime}(0)=0, f^{\prime\prime}(0)=1,\eqno{\hbox{(5)}}$$ which represent, respectively, zero AC current, zero AC velocity, and unit input electric field. The change in the amplitude gain, Formula$G_{1}$ in Formula$e$-folds, and in the phase, Formula$\theta_{1}$ in radians, due to random errors is given by, Formula TeX Source $$e^{G_{1}+j\theta_{1}}={{f^{\prime\prime}(x)+4QC^{3}f(x)}\over{f_{0}^{\prime\prime}(x)+4QC^{3}f_{0}(x)}},\eqno{\hbox{(6)}}$$ where Formula$f_{0}$ represents the solution to (4) for an error-free tube and the prime denotes differentiation with respect to Formula$x$. From (1), we see that (6) is simply Formula$a(x)/a_{0}(x)$, where Formula$a_{0}(x)$ is the error-free solution of Formula$a(x)$.

SECTION III

PERTURBATIVE AND RICCATI APPROACHES

The random manufacturing errors enter the Pierce parameters Formula$b$, Formula$d$, and Formula$C$. It has been shown that the effects of random errors in the velocity parameter, Formula$b$, dominates those of random errors in Formula$d$ and Formula$C$ [7], so we only consider random errors in Formula$b$ in this paper. Random errors are assigned to Formula$b(x)$ as a set of Gaussian random variables uniformly spaced in Formula$x$, each with a mean of Formula$b_{0}$ and a specified standard deviation, Formula$\sigma_{b}$, as illustrated in Fig. 1. We define the correlation length as Formula$\Delta=L/N$, where Formula$N$ is the number of uniformly spaced nodes over the normalized length Formula$(L)$ of the TWT.

Figure 1
Fig. 1. Sample velocity mismatch profile with a mean value of Formula$b_{0}$ of a TWT of length Formula$L$.

Our work differs from the previous work of Pengvanich et al\.by applying the random Gaussian errors directly to the parameter Formula$b$, instead of to the circuit phase velocity Formula$v_{p}$ through a dimensionless quantity Formula$q(x)=(v_{p}(x)-v_{p0})/v_{p0}$, where Formula$v_{p0}$ is the unperturbed circuit phase velocity. The random function Formula$q(x)$ had a specified standard deviation Formula$\sigma_{q}$. The velocity parameter Formula$b$ is related to Formula$q$ by Formula$b(x)=(1/C)[Cb_{0}-q(x)]/(1+q(x))$ and the standard deviations in Formula$b$ and Formula$q$ are approximately related by Formula$\sigma_{b}=(\sigma_{q}/C)(1+Cb_{0})$. Due to the non-linear relationship between Formula$b$ and Formula$q$, a Gaussian random error profile assigned to Formula$q(x)$ is no longer Gaussian for Formula$b(x)$. Our numerical integration of (4) over many 5000-sample calculations shows that this subtle difference led to quantitatively different results. One reason is that the mean deviation is a second order effect in the random error, as we have already mentioned, and this subtle difference is important. In this work, all random errors are characterized by a Gaussian distribution in Formula$b(x)$ with a standard deviation of Formula$\sigma_{b}$.

Pengvanich et al. showed analytically that the standard deviations in the gain and in the output phase from an error-free TWT are first order in Formula$\sigma_{b}$ (cf. Equation (9a), (9b) below). In this work, we need to carry out the analysis to second order in the effects of the random errors. With only perturbations in Formula$b$, we show in Appendix A [cf. Equation (A16)], Formula TeX Source $$\eqalignno{&\big<G_{1}(x)+j\theta_{1}(x)\big>=-{{1}\over{2}}C^{2}\sigma_{b}^{2}\Delta\Bigg\{\bigg[4QC^{3}\sum_{l=1}^{3}{{\tau_{l}}\over{C\delta_{l}}}\cr&\qquad\quad+\sum_{k=1}^{3}\tau_{k}C\delta_{k}\bigg]\int\limits_{0}^{x}{{Q_{1}(x,s)ds}\over{a_{0}(x)}}+\sum_{l=1}^{3}\sum_{k=1}^{3}\cr&\qquad\quad~\left(\tau_{l}C\delta_{l}\right)\left(\tau_{k}C\delta_{k}\right)e^{C\left(\delta_{l}+\delta_{k}\right)x}\int\limits_{0}^{x}{{Q_{2}(x,s)ds}\over{a_{0}^{2}(x)}}\Bigg\},&{\hbox{(7)}}}$$ where Formula$\left<G_{1}(x)+j\theta_{1}(x)\right>$ is the ensemble-average deviation in gain and in phase from the error-free tube due to random errors, Formula$\delta_{k}~(k=1, 2, 3)$ are the three roots to the Pierce dispersion relation (3), Formula$\tau_{k}~(k=1, 2, 3)$ which depends only on Formula$\delta_{k}$, is defined by (A5) of [7], and Formula$Q_{1}(x, s)$, Formula$Q_{2}(x, s)$ depend only the error-free, three-wave solution. The expressions for Formula$Q_{1}(x, s)$, Formula$Q_{2}(x, s)$ are given in Appendix A in (A17a), (A17b). Use of (7) will be referred to as the “perturbation” method.

The second analytical method calculates Formula$\left<G_{1}(x)+j\theta_{1}(x)\right>$ using a Riccati formulation of the complex wave number for a single wave. This formulation yields, [see Appendix B, (B33)], Formula TeX Source $$\left<G_{1}(x)+j\theta_{1}(x)\right>=-{{\lambda}\over{2}}\left({{C}\over{1+Cb_{0}}}\right)^{2}\sigma_{b}^{2}x\Delta,\eqno{\hbox{(8)}}$$ where Formula$\lambda$ is a complex value that is determined by the value of the mismatch parameter, Formula$b_{0}$. This method will be referred to as the “Riccati” method.

Finally, we revise the standard deviation of gain and phase variations calculated in Ref. [7] to include the space charge effects Formula$(QC\ne 0)$. In terms of the standard deviation of Formula$b$, Formula$\sigma_{b}$, the standard deviation in the gain Formula$G_{1}$ and in the phase Formula$\theta_{1}$ is given by, respectively, Formula TeX Source $$\sigma_{Gb}=S_{Gb}\sigma_{b},\,\, S_{Gb}=\sqrt{{x}\over{N}}\sqrt{\int\limits_{0}^{x}ds\,\Big\vert g_{br}(x, s)\Big\vert^{2}}\eqno{\hbox{(9a)}}$$ and Formula TeX Source $$\sigma_{\theta b}=S_{\theta b}\sigma_{b},\,\, S_{\theta b}=\sqrt{{x}\over{N}}\sqrt{\int\limits_{0}^{x}ds\,\Big\vert g_{bi}(x, s)\Big\vert^{2}},\eqno{\hbox{(9b)}}$$ where Formula$g_{br}$ and Formula$g_{bi}$ are the real and imaginary parts, respectively, of Formula$g_{b}$, given by Formula TeX Source $$g_{b}(x,s)=-jC\left(4QC^{3}f_{0}(s)+a_{0}(s)\right)a_{0}(x-s)/a_{0}(x).\eqno\hbox{(10)}$$

In the absence of space charge, i.e., Formula$QC=0$, (10) reduces to (A15) of [7], whose (A4) defines the error-free solutions Formula$f_{0}$ and Formula$a_{0}$.

Equation (9a) and (9b) show that the standard deviations in the gain and phase are linear in Formula$\sigma_{b}$. Equation (7) and (8) show that Formula$\left<G_{1}(x)\right>$ and Formula$\left<\theta_{1}(x)\right>$ are both quadratic in Formula$\sigma_{b}$, and their magnitudes are therefore much less than the standard deviations.This contrast between the standard deviation, and the deviation in the mean from the error-free tubes, was also apparent inFig. 8 of [8].

SECTION IV

RESULTS

We start with the TWT base case with length Formula$x=100$ where Formula$b_{0}=d=QC=0$, and Formula$C=0.05$. Equation (4) yields an error-free gain of 28.1 dB and an output phase of-5872°. Random errors are then introduced into the velocity parameter, Formula$b$, as shown in Fig. 1. The value at each node is an independent Gaussian random variable with a mean of Formula$b_{0}$ and a specified standard deviation, Formula$\sigma_{b}$. A correlation length of Formula$\Delta=1$ has been used in all calculations, meaning that each node of the Gaussian random error profile would correspond to Formula$x=1, 2,\ldots, 100$ in the TWT. For a specified value of Formula$\sigma_{b}$, we integrate (4) numerically 5000 times. Previous work [7] showed that performing only 500 integrations would provide sufficient results. That work, however, was focused on calculating the standard deviation in gain and phase and not the mean. While 500 integrations is sufficient to calculate these standard deviations, significantly more are required to calculate the mean accurately, since the mean is second order in Formula$\sigma_{b}$. We have checked that integrating (4) up to 25,000 times does not provide a significantly different answer, even for phase variations as small as a fraction of a degree given in Figs. 2(b) and 7(c). Calculations performed in this manner will be designated as “numerical”. One important note is that this numerical calculation is strongly dependent on the random number seed used in these calculations. Different seed values do not produce a difference in the mean gain output, however, the exact values for the output phase will be different albeit of the same order. In all of the following calculations, the seed used for the random number sequence has been fixed.

Fig. 2(a) shows the gain variations for the numerical, perturbation, and Riccati methods. All three methods show good agreement. The phase calculation is shown in Fig. 2(b). The perturbation method shows good agreement with the numerical results. It should be noted that in this case Formula$\left<\theta_{1}(x)\right>=0$ for the Riccati method (cf. the last sentence in Appendix B). This result is consistent with those from the perturbative analysis and the numerical solution to (4), in that the phase variations Formula$\left<\theta_{1}(x)\right>$ due to random errors, measured in radians, is found to be negligible compared with the amplitude variations Formula$\left<G_{1}(x)\right>$, measured in Formula$e$-folds, in this case.This case is identical to the one considered by [7].

Figure 2
Fig. 2. (a) Mean values of the power and (b) phase at the output relative to the unperturbed values at the output for a synchronous beam velocity, Formula$b_{0}=0$. The points are the results of numerically integrating (4). The solid and dashed lines show the perturbation and Riccati formulas, (7) and (8), respectively. Here, Formula$x=100$, Formula$C=0.05$, Formula$\Delta=1$, Formula$d=0$, and Formula$QC=0$.

Fig. 3 shows two cases where the the velocity mismatch is nonzero. For Formula$C=0.05$, Formula$b_{0}=\pm 1$ corresponds to a difference of Formula${\pm}{5\%}$ between the beam velocity and circuit phase velocity. All three methods are in agreement even when the velocity mismatch is allowed to be nonzero. Looking at the phase output it appears that the perturbation method is more accurate than the Riccati method.

Figure 3
Fig. 3. (a) Mean values of the power and (b) phase at the output relative to the unperturbed values at the output for non-synchronous beam velocities of Formula${\pm}0.05v_{p}~(b_{0}=\pm 1)$. The points are the results of numerically integrating (4). The solid and dashed lines show the perturbation and Riccati formulas, (7) and (8), respectively. Here, Formula$x=100$, Formula$C=0.05$, Formula$\Delta=1$, Formula$d=0$, and Formula$QC=0$.

Figs. 4 and 5 show how the gain and phase are affected by the inclusion of the Formula$QC$ term, increasing it from 0 to 0.35 for the synchronous case, Formula$b_{0}=0$. When Formula$QC\ne 0$, both the perturbation and Riccati methods predict a larger variation in gain and phase than shown by the numerical analysis. The Riccati method tends to predict smaller variations than the perturbation method, but neither prediction shows agreement with the numerical data. Figs. 6 and 7 show the gain and phase variations for Formula$QC$ again increasing from 0 to 0.35, this time for the Formula$b_{0}=1$ case. In this case, the Riccati method shows good agreement with the numerical data for gain. Neither analytical method shows agreement with the numerical phase data in this case. The Formula$b_{0}=-1$ case could not be calculated reliably because the TWT would not amplify for any significant values of Formula$QC$.

Figure 4
Fig. 4. (a) Mean values of the power at the output relative to the unperturbed values for Formula$QC=0$, (b) Formula$QC=0.15$, (c) Formula$QC=0.25$, and (d) Formula$QC=0.35$ for the synchronous velocity case, Formula$b_{0}=0$. The points are the results of numerically integrating (4). The solid and dashed lines show the perturbation and Riccati formulas, (7) and (8), respectively. Here, Formula$x=100, C\,{=}\,0.05,\Delta\,{=}\,1$, and Formula$d\,{=}\,0$.
Figure 5
Fig. 5. (a) Mean values of the phase at the output relative to the unperturbed values for Formula$QC=0$, (b) Formula$QC=0.15$, (c) Formula$QC=0.25$, and (d) Formula$QC=0.35$ for the synchronous velocity case, Formula$b_{0}=0$. The points are the results of numerically integrating (4). The solid and dashed lines show the perturbation and Riccati formulas, (7) and (8), respectively. Here, Formula$x\,{=}\,100, C\,{=}\,0.05,\Delta\,{=}\,1$, and Formula$d\,{=}\,0$.
Figure 6
Fig. 6. (a) Mean values of the power at the output relative to the unperturbed values for Formula$QC=0$, (b) Formula$QC=0.15$, (c) Formula$QC=0.25$, and (d) Formula$QC=0.35$ for the non-synchronous velocity case, Formula$b_{0}=1$. The points are the results of numerically integrating (4). The solid and dashed lines show the perturbation and Riccati formulas, (7) and (8), respectively. Here, Formula$x\,{=}\,100, C\,{=}\,0.05,\Delta\,{=}\,1$, and Formula$d\,{=}\,0$.
Figure 7
Fig. 7. (a) Mean values of the phase at the output relative to the unperturbed values for Formula$QC=0$, (b) Formula$QC=0.15$, (c) Formula$QC=0.25$, and (d) Formula$QC=0.35$ for the non-synchronous velocity case, Formula$b_{0}=1$. The points are the results of numerically integrating (4). The solid and dashed lines show the perturbation and Riccati formulas, (7) and (8), respectively. Here, Formula$x=100, C=0.05,\Delta=1$, and Formula$d=0$.

Fig. 8 shows how the analytical standard deviation calculation from (9) compares to the statistical standard deviation as calculated from the numerical integration of (4) for a non-synchronous beam velocity. Both calculations are in agreement over a range of non-synchronous beam velocities. Fig. 9 shows the analytical standard deviation as calculated by (9) with the space charge modified expression for Formula$g_{b}$ from (10), as well as the statistical standard deviation calculation. With the inclusion of the space charge term, Formula$QC$, (9) is no longer in agreement with the statistical calculation. The difference between the two increases with increasing values of Formula$QC$.

Figure 8
Fig. 8. Mean values and standard deviation of the (a) gain and (b) phase at the output relative to the unperturbed values for Formula$QC=0$ for the non-synchronous velocity case, Formula$b_{0}=-1$. The circles are the results of numerically integrating (4). The diamonds are the standard deviation results from numerically integrating (4). The dashed line is the analytic standard deviation as calculated from (9). Here, Formula$x=100$, Formula$C=0.05$, Formula$\Delta=1$, and Formula$d=0$.
Figure 9
Fig. 9. Mean values and standard deviation of the (a) gain and (b) phase at the output relative to the unperturbed values for Formula$QC=0.25$ for the non-synchronous velocity case, Formula$b_{0}=1$. The circles are the results of numerically integrating (4). The diamonds are the standard deviation results from numerically integrating (4). The dashed line is the analytic standard deviation as calculated from (9). Here, Formula$x=100$, Formula$C=0.05$, Formula$\Delta=1$, and Formula$d=0$.

Finally, as a concrete example, we consider the G-band (210 GHz) folded waveguide TWT previously studied [8] with a beam voltage of 11.7 kV, a beam current of 120 mA, a length of 1.2 cm, and an average circuit pitch of 0.02 cm. This corresponds to a normalized length of Formula$x=240$, and we take a correlation length of Formula$\Delta=4$. For this example we consider the specific case with Formula$C=0.0197$, Formula$QC=0$, and Formula$b_{0}=0.36$, using Figs.11 and 12 of [8]. Fig. 10 shows both the gain and phase variation of this G-Band-like TWT accurately predicted by both the perturbation and Riccati methods. The statistical standard deviations in gain and phase and analytic formula are also presented in Fig. 10, showing good agreement as well. Results for the standard deviation using the Riccati approach are not yet available.

Figure 10
Fig. 10. Mean values and standard deviation of the (a) gain and (b) phase at the output relative to the unperturbed values for a G-band-like TWT. Results from the statistical, perturbation, and Riccati calculations for mean as well as analytic and statistical results for standard deviation are plotted. Here, Formula$x=240$, Formula$\Delta=4$, Formula$C=0.0197$, Formula$b_{0}=0.36$, and Formula$QC=d=0$.
SECTION V

SUMMARY AND CONCLUSION

Two different formulas were derived to predict the deviations in gain and phase in a traveling wave tube in the presence of random axial errors: a second-order perturbation analysis that accounts for all three forward propagating waves and a Riccati analysis that includes only the amplifying wave. We have compared both of these models against a numerical integration of the governing, third-order linear differential equation for cases with nonzero Formula$b$ and the inclusion of AC space charge effects. We have found that the perturbation analytic model shows good agreement with the numerical analysis for non-synchronous beam velocity, i.e., nonzero Formula$b$, in the absence of space charge. We have also found that the analytic models do not accurately predict the TWT behavior in the presence of AC space charge. A possible explanation is that a nonzero Formula$QC$ would enlarge the range of Formula$b$ in which the amplifying wave would have a reduced or even zero gain, in which case all three waves would have comparable amplitudes.

Since we have shown in this paper that the standard deviation is much larger than the deviation in the mean from an error-free tube, we have essentially solved the puzzle as to why random variations in Formula$b(x)$, presumably caused by manufacturing errors, could lead to a higher gain in a significant fraction of the samples simulated [7]. Identification of the types of random errors that would lead to higher gain awaits further study.

APPENDIX A

SECOND-ORDER SMALL-SIGNAL SOLUTION IN THE PRESENCE OF RANDOM ERRORS

We derive the second-order perturbative solution to (4) when the Pierce parameters Formula$C$ and Formula$d$ are constant and the parameter Formula$b$ contains small random perturbations denoted as Formula$b_{1}(x)$. We re-write (6)as, Formula TeX Source $$a=a_{0}e^{G_{1}+j\theta_{1}},\eqno{\hbox{(A1)}}$$ where Formula$a$ is the normalized electric field and Formula$a_{0}(x)$ is the solution in the error-free tube given by (A4) of [7]. To second order, we write Formula$a(x)=a_{0}(x)+a_{10}(x)+a_{11}(x)$. The first and second order perturbations are Formula$a_{10}(x)$ and Formula$a_{11}(x)$, respectively. Expanding Formula$a(x)$ in (A1) yields an expression for the modification of amplitude and phase of Formula TeX Source $$G_{1}+j\theta_{1}={{a_{10}+a_{11}}\over{a_{0}}}-{{1}\over{2}}{{a_{10}^{2}}\over{a_{0}^{2}}}.\eqno{\hbox{(A2)}}$$

This equation can be solved for the gain and phase change when the expressions for Formula$a_{10}$ and Formula$a_{11}$ are substituted into (A2). These quantities are to be derived in this appendix.

Equation (4) can be written as three coupled first-order differential equations expressed in matrix notation as Formula TeX Source $${{d{\bf Y}}\over{dx}}=\left({\bf M}+{\bf M}_{1}\right){\bf Y},\eqno{\hbox{(A3)}}$$ in the presence of random variation Formula$b_{1}(x)$, where Formula TeX Source $$\eqalignno{&{\bf M}=\left[\matrix{0 & 1 & 0\cr 0 & 0 & 1\cr\scriptstyle-jC\left(4QC^{3}\left(b-jd\right)+C^{2}\right) &\scriptstyle-4QC^{3}&\scriptstyle-jC\left(b-jd\right)}\right], &{\hbox{(A4)}}\cr&\qquad\qquad~{\bf M}_{1}=\left[\matrix{0 & 0& 0\cr 0 & 0 & 0\cr m_{31}(x) & 0 & m_{33}(x)}\right],&{\hbox{(A5)}}}$$ where Formula$m_{31}(x)=-jC\left(4QC^{3}\right)b_{1}(x)$ and Formula$m_{33}(x)=-jCb_{1}(x)$. Equation (A4) is a constant matrix containing error-free tube parameters. Equation (A5) is a matrix containing the random perturbations. We assume that there are no losses, i.e., Formula$d=0$. We write Formula TeX Source $${\bf Y}={\bf Y}_{0}(x)+{\bf Y}_{1}(x)\equiv\left[\matrix{f_{0}(x)+f_{1}(x)\cr v_{0}(x)+v_{1}(x)\cr a_{0}(x)+a_{1}(x)}\right],\eqno{\hbox{(A6)}}$$ where quantities with subscript 1 are due only to random Formula$b_{1}(x)$. The error-free solutions are Formula$f_{0}$, Formula$v_{0}$, and Formula$a_{0}$ and are given by (A4) of [7]. Combining (A6) with (A3) yields, to second order Formula TeX Source $${{d{\bf Y}_{1}(x)}\over{dx}}-{\bf MY}_{1}(x)={\bf M}_{1}{\bf Y}_{0}(x)+{\bf M}_{1}{\bf Y}_{1}(x).\eqno{\hbox{(A7)}}$$ Ignoring the second order term Formula${\bf M}_{1}{\bf Y}_{1}(x)$, the solution to (A7) is Formula${\bf Y}_{1}(x)={\bf Y}_{10}(x)$, whose solution is given by (A10) of [7]. Next, let us approximate Formula${\bf M}_{1}{\bf Y}_{1}(x)$ in (A7) as Formula${\bf M}_{1}{\bf Y}_{10}(x)$, and write Formula TeX Source $${\bf Y}_{1}={\bf Y}_{10}(x)+{\bf Y}_{11}(x)\equiv\left[\matrix{f_{10}(x)+f_{11}(x)\cr v_{10}(x)+v_{11}(x)\cr a_{10}(x)+a_{11}(x)}\right].\eqno{\hbox{(A8)}}$$

Equation (A7)then becomes Formula TeX Source $${{d{\bf Y}_{11}(x)}\over{dx}}-{\bf MY}_{11}(x)\cong{\bf M}_{1}(x){\bf Y}_{10}(x),\eqno{\hbox{(A9)}}$$ which is of the same form as (A7) of [7]. We may then express Formula${\bf Y}_{11}(x)$ as (A10) from [7] to obtain Formula TeX Source $$a_{11}(x)=\int\limits_{0}^{x}ds\,\big (m_{31}(s)f_{10}(s)+m_{33}(s)a_{10}(s)\big)P_{3}(x,s),\eqno{\hbox{(A10)}}$$ where Formula TeX Source $$P_{3}(x,s)=\Psi_{31}(x)\Psi_{13}^{-1}(s)+\Psi_{32}(x)\Psi_{23}^{-1}(s)+\Psi_{33}(x)\Psi_{33}^{-1}(s),\eqno{\hbox{(A11)}}$$ and Formula$\Psi_{ij}(x)~(i, j=1, 2, 3)$ is defined by (A3) of [7]. The first order perturbations Formula$f_{10}(x)$ and Formula$a_{10}(x)$ are given by (A11) of [7], where the expression for Formula$V_{k}(s)$ now contains the AC space charge term in Formula$m_{31}(x)$. Substituting these into (A10) yields Formula TeX Source $$\eqalignno{&a_{11}(x)=\int\limits_{0}^{x}ds\, P_{3}(x,s)\Bigg\{\sum_{l=1}^{3}{{\tau_{l}}\over{C\delta_{l}}}e^{C\delta_{l}s}\int\limits_{0}^{s}ds^{\prime}\,e^{-C\delta_{l}s^{\prime}}\cr&\quad~\times\big [f_{0}(s^{\prime})m_{31}(s^{\prime})m_{31}(s)+a_{0}(s^{\prime})m_{33}(s^{\prime})m_{31}(s)\big]\cr&\quad+\sum_{k=1}^{3}\tau_{k}C\delta_{k}e^{C\delta_{k}s}\int\limits_{0}^{s}ds^{\prime}\,e^{-C\delta_{k}s^{\prime}}\cr&\quad~\times\big [f_{0}(s^{\prime})m_{31}(s^{\prime})m_{33}(s)+a_{0}(s^{\prime})m_{33}(s^{\prime})m_{33}(s)\big]\Bigg\},&{\hbox{(A12)}}}$$ where Formula$\delta_{k}~(k=1, 2, 3)$ are the three roots to the Pierce dispersion relation (3), and Formula$\tau_{k}~(k=1, 2, 3)$ which depends only on Formula$\delta_{k}$, is defined by (A5) of [7].

We next take the ensemble-average of (A12), assuming that Formula$\left<b_{1}(s)b_{1}(s^{\prime})\right>=\left<b_{1}^{2}\right>\Delta\delta (s-s^{\prime})$, where Formula$\Delta$ is the correlation length and Formula$\delta$ is the Dirac delta function. With Formula$\left<b_{1}^{2}\right>=\sigma_{b}^{2}$, we obtain Formula TeX Source $$\eqalignno{&\big<a_{11}(x)\big>=-C^{2}\sigma_{b}^{2}\Delta\int\limits_{0}^{x}ds\, P_{3}(x,s)\cr&\quad~~\times\Bigg\{\left(4QC^{3}\right)\sum_{l=1}^{3}{{\tau_{l}}\over{C\delta_{l}}}\left[4QC^{3}{{f_{0}(s)}\over{2}}+{{a_{0}(s)}\over{2}}\right]\cr&\qquad\quad~+\sum_{k=1}^{3}\tau_{k}C\delta_{k}\left[4QC^{3}{{f_{0}(s)}\over{2}}+{{a_{0}(s)}\over{2}}\right]\Bigg\}. &{\hbox{(A13)}}}$$

Similarly, squaring Formula$a_{10}(x)$ and taking the ensemble-average yields Formula TeX Source $$\eqalignno{&\big<a_{10}^{2}(x)\big>=-C^{2}\sigma_{b}^{2}\Delta\sum_{k=1}^{3}\sum_{l=1}^{3}\left(\tau_{k}C\delta_{k}\right)\left(\tau_{l}C\delta_{l}\right)e^{C\left(\delta_{k}+\delta_{l}\right)x}\cr&\qquad\qquad\qquad\quad\times\int\limits_{0}^{x}ds\, e^{-C\left(\delta_{k}+\delta_{l}\right)s}\big[\left(4QC^{3}\right)^{2}f_{0}^{2}(s)\cr&\qquad\qquad\qquad\quad+2\left(4QC^{3}\right) f_{0}(s)a_{0}(s)+a_{0}^{2}(s)\big].&{\hbox{(A14)}}}$$

Note that Formula$\left<a_{10}(x)\right>=0$ since Formula${\bf M}_{1}$ is linear in Formula$b_{1}(x)$ and therefore Formula$\left<{\bf Y}_{10}(x)\right>=0$ (cf. Equation (A10) of [7]). With this result, we obtain from (A2), Formula TeX Source $$\left<G_{1}(x)+j\theta_{1}(x)\right>={{\left<a_{11}(x)\right>}\over{a_{0}(x)}}-{{1}\over{2}}{{\left<a_{10}^{2}(x)\right>}\over{a_{0}^{2}(x)}},\eqno{\hbox{(A15)}}$$ where Formula$\left<a_{11}(x)\right>$ is given by (A13), Formula$a_{0}(x)$ by (A4) of [7], and Formula$\left<a_{10}^{2}(x)\right>$ by (A14). This can be written in the form Formula TeX Source $$\eqalignno{&\big<G_{1}(x)+j\theta_{1}(x)\big>=-{{1}\over{2}}C^{2}\sigma_{b}^{2}\Delta\Bigg\{\bigg[4QC^{3}\sum_{l=1}^{3}{{\tau_{l}}\over{C\delta_{l}}}+\cr&\quad~\,\sum_{k=1}^{3}\tau_{k}C\delta_{k}\bigg]\int\limits_{0}^{x}{{Q_{1}(x,s)ds}\over{a_{0}(x)}}+\sum_{l=1}^{3}\sum_{k=1}^{3}\left(\tau_{l}C\delta_{l}\right)\cr&\quad~\,\left(\tau_{k}C\delta_{k}\right)e^{C\left(\delta_{l}+\delta_{k}\right)x}\int\limits_{0}^{x}{{Q_{2}(x,s)ds}\over{a_{0}^{2}(x)}}\Bigg\}, &{\hbox{(A16)}}}$$ which is (7) of the text. In (A16), Formula$Q_{1}(x,s)$, Formula$Q_{2}(x,s)$ are given by, with the substitution Formula$\lambda_{i}=C\delta_{i}$, Formula TeX Source $$\eqalignno{&~Q_{1}(x,s)=\Big\{\big [\lambda_{1}^{2}\left(\lambda_{2}-\lambda_{3}\right)e^{\lambda_{1}\left(x-s\right)}+\lambda_{2}^{2}\left(\lambda_{1}-\lambda_{3}\right)e^{\lambda_{2}\left(x-s\right)}\cr&+\lambda_{3}^{2}\left(\lambda_{1}-\lambda_{2}\right)e^{\lambda_{3}\left(x-s\right)}\big]\big [\sum_{i=1}^{3}\lambda_{i}\tau_{i}e^{\lambda_{i}s}+4QC^{3}\cr&\times\sum_{j=1}^{3}{{\tau_{j}}\over{\lambda_{j}}}e^{\lambda_{j}s}\big]\Big\}\Big/\left[\left(\lambda_{1}-\lambda_{2}\right)\left(\lambda_{1}-\lambda_{3}\right)\left(\lambda_{2}-\lambda_{3}\right)\right],&{\hbox{(A17a)}}\cr&Q_{2}(x,s)=\left({{1}\over{\lambda_{1}\lambda_{2}\lambda_{3}}}\right)^{2}e^{-\left(\lambda_{k}+\lambda_{l}\right)}\cr&\quad~\times\big [\tau_{1}\lambda_{2}\lambda_{3}\big (4QC^{3}+\lambda_{1}^{2}\big)e^{\lambda_{1}s}+\tau_{2}\lambda_{1}\lambda_{3}\left(4QC^{3}+\lambda_{2}^{2}\right)e^{\lambda_{2}s}\cr&\hskip9.9em+\tau_{3}\lambda_{1}\lambda_{2}\left(4QC^{3}+\lambda_{3}^{2}\right)e^{\lambda_{3}s}\big].&{\hbox{(A17b)}}}$$

In the limit of zero space charge effects, (A16) reduces to Formula TeX Source $$\left<G_{1}(x)+j\theta_{1}(x)\right>=-{{1}\over{2}}C^{2}\sigma_{b}^{2}\Delta\int\limits_{0}^{x}ds\,A(x,s),\eqno{\hbox{(A18)}}$$ where Formula TeX Source $$\eqalignno{&A(x,s)=\sum_{j=1}^{3}\tau_{j}C\delta_{j}P_{3}(x,s){{a_{0}(s)}\over{a_{0}(x)}}+\sum_{k=1}^{3}\sum_{l=1}^{3}\left(\tau_{k}C\delta_{k}\right)\left(\tau_{l}C\delta_{l}\right)\cr&\hskip7.5em\times e^{C\left(\delta_{k}+\delta_{l}\right)x}e^{-C\left(\delta_{k}+\delta_{l}\right)s}{{a_{0}^{2}(s)}\over{a_{0}^{2}(x)}}.&{\hbox{(A19)}}}$$

APPENDIX B

RICCATI FORMULATION OF WAVENUMBER FOR SINGLE WAVE

To study the effect on the small-signal gain and phase in a TWT due to small random errors, we begin with the following equations, Formula TeX Source $$\eqalignno{&\quad\left[\left({{d}\over{dz}}-i{{\omega}\over{v_{z}}}\right)^{2}+k_{p}^{2}\right]n=a,&{\hbox{(B1)}}\cr&\left[{{d^{2}}\over{dz^{2}}}+k_{s}^{2}(z)\right]a(z)=-2k_{0}k_{g}^{3}n(z),&{\hbox{(B2)}}}$$ where Formula$a(z)$ represents the field amplitude, Formula$n(z)$ represents the perturbed beam density, and we have assumed Formula$e^{-i\omega t}$ dependence for these quantities. The coefficients in (B1) and (B2) represent the following: Formula$k_{s}(z)$ is the axially varying wave number for the structure. The beam plasma wavenumber is Formula$k_{p}$. The nominal gain rate is Formula$k_{g}=Ck_{0}$, where Formula$C$ is the Pierce parameter. The unperturbed beam velocity is Formula$v_{z}$. It is assumed that Formula$k_{s}(z)$ is close to some reference value such that, Formula TeX Source $$k_{s}^{2}(z)=k_{0}^{2}+\delta k^{2}(z),\eqno{\hbox{(B3)}}$$ where Formula$k_{0}$ is defined such that the expectation of the deviation vanishes, Formula$\left<\delta k^{2}(z)\right>=0$. Equations (B1) and (B2} are equivalent to (1) and (2) of the main text if we ignore the reverse propagating mode in (B2), change the sign of Formula$\omega$, and set Formula$n=s$, Formula$\omega/v_{z}=\beta_{e}$, Formula$k_{s}^{2}=\beta_{p}^{2}$, Formula$k_{p}^{2}=\beta_{q}^{2}$, Formula$k_{0}=\beta_{p0}(={\rm error}-{\rm free~value})$, Formula$k_{g}=Ck_{0}$, and Formula$i=j=\sqrt{-1}$.

Equation (B2) describes both the forward and backward structure wave. In the presence of random variations in wavenumber there will be a coupling of the forward and backward waves. This is the same coupling that gives rise to Anderson localization in condensed matter physics. If the forward wave is growing in Formula$z$, the lowest order effect of the conversion of forward wave power into backward wave power is to create an effective attenuation of the forward wave. In other words, we can neglect the conversion of backward wave power back into forward wave power, and assume that the backward wave power is effectively lost. We will not calculate this effect here, except to assume that if we wish, we can add an attenuation to the forward wave at the end of the calculation.

The next step is to write the field amplitude as a sum of forward and backward waves, Formula TeX Source $$a(z)=a_{+}(z)e^{ik_{0}z}+a{\_}(z)e^{-ik_{0}z},\eqno{\hbox{(B4)}}$$ where we choose Formula TeX Source $$0=a_{+}^{\prime}(z)e^{ik_{0}z}+a{\_}^{\prime}(z)e^{-ik_{0}z},\eqno{\hbox{(B5)}}$$ and the prime denotes differentiation with respect to Formula$z$. We insert (B4) into (B2), use our constraint (B5), drop the backward wave, and introduce the revised density perturbation, Formula${\mathhat{n}}=ne^{-ik_{0}z}$, to obtain the third order system describing the coupled forward wave and beam space charge waves, Formula TeX Source $$\eqalignno{&2ik_{0}{{da_{+}}\over{dz}}=-\delta k^{2}(z)a_{+}-2k_{0}k_{g}^{3}{\mathhat{n}}&{\hbox{(B6)}}\cr&~\left[\left({{d}\over{dz}}+i\Delta k\right)^{2}+k_{p}^{2}\right]{\mathhat{n}}=a_{+},&{\hbox{(B7)}}}$$ where Formula$\Delta k=k_{0}-\omega/v_{z}$ is the mismatch wavenumber between the structure mode and the beam mode.

We expect the solutions of (B6) and (B7) to correspond on average to exponentially growing waves. The average spatial growth rate will be affected (reduced) by the random variations in structure wavenumber. To calculate this effect we will recast (B6) and (B7) as a nonlinear system of equations for the complex rate of exponentiation of the relevant quantities. Specifically we introduce Formula TeX Source $$\mu_{+}(z)={{1}\over{a_{+}}}{{da_{+}}\over{dz}},\eqno{\hbox{(B8)}}$$ and Formula TeX Source $$\mu_{n}(z)={{1}\over{\mathhat{n}}}{{d{\mathhat{n}}}\over{dz}},\eqno{\hbox{(B9)}}$$ and rewrite (B6) and (B7) Formula TeX Source $$\eqalignno{&\quad 2ik_{0}\mu_{+}=-\delta k^{2}(z)-2k_{0}k_{g}^{3}\rho (z)&{\hbox{(B10)}}\cr&\left[\left(\mu_{n}+i\Delta k\right)^{2}+k_{p}^{2}\right]+{{d}\over{dz}}\mu_{n}=\rho^{-1}.&{\hbox{(B11)}}}$$ Here the quantity Formula$\rho (z)={\mathhat{n}}/a_{+}$ gives the spatially varying ratio of density perturbation to field amplitude. It satisfies a differential equation, Formula TeX Source $${{1}\over{\rho}}{{d\rho}\over{dz}}=\mu_{n}-\mu_{+}.\eqno{\hbox{(B12)}}$$

It is important to point out that (B10)(B12) are equivalent to (B6) and (B7). That is, the only approximation we have made is to drop the backward structure wave.

We now separate the dynamical variables into mean and fluctuating parts; specifically Formula$\mu_{+}=\bar{\mu}_{+}+\delta\mu_{+}$, Formula$\mu_{n}=\bar{\mu}_{n}+\delta\mu_{n}$, and Formula$\rho=\bar{\rho}+\delta\rho$, where the overbar denotes the ensemble mean, which is the same mean that we have previously denoted with angular brackets. We then take the mean of (B10)(B12), Formula TeX Source $$\eqalignno{&\qquad\qquad\quad~\,2ik_{0}\bar{\mu}_{+}=-2k_{0}k_{g}^{3}\left<\rho\right>,&{\hbox{(B13)}}\cr&\left[\left(\bar{\mu}_{n}+i\Delta k\right)^{2}+\left<\left(\delta\mu_{n}\right)^{2}\right>+k_{p}^{2}\right]=\left<\rho^{-1}\right>,&{\hbox{(B14)}}}$$ and Formula TeX Source $$\left<{{1}\over{\rho}}{{d\rho}\over{dz}}\right>={{d}\over{dz}}\left<\ln\rho\right>=\bar{\mu}_{n}-\bar{\mu}_{+}=0.\eqno{\hbox{(B15)}}$$

Thus, from (B15) we see that the field amplitude and density perturbation must grow on average at the same rate, Formula$\bar{\mu}_{n}=\bar{\mu}_{+}\equiv\bar{\mu}$. Evaluation of Formula$\left<\rho^{\pm 1}\right>$ in (B13) and (B14)requires integration of (B12). Specifically, we write Formula$\rho=\rho_{0}\exp[\delta\Gamma (z)]$ where Formula$\rho_{0}$ is a constant to be determined, and Formula TeX Source $${{d}\over{dz}}\delta\Gamma (z)=\delta\mu_{n}-\delta\mu_{+}.\eqno{\hbox{(B16)}}$$

We then expand Formula$\rho$ and its inverse under the assumption of small fluctuations in Formula$\delta\Gamma$, Formula TeX Source $$\rho^{\pm 1}=\rho_{0}^{\pm 1}\left(1\pm\delta\Gamma+{{1}\over{2}}\delta\Gamma^{2}+\ldots\right).\eqno{\hbox{(B17)}}$$

We will terminate the series after the third term. We thus have, Formula TeX Source $$\left<\rho^{\pm 1}\right>=\rho^{\pm 1}_{0}\left(1+{{1}\over{2}}\left<\delta\Gamma^{2}\right>\right).\eqno{\hbox{(B18)}}$$

We now insert (B18) in (B13) and (B14) and take the product to eliminate Formula$\rho_{0}$, giving us a dispersion relation for the common mean rate of change of the exponent, Formula$\bar{\mu}$, Formula TeX Source $$D(\bar{\mu})=D_{b}(\bar{\mu})+D_{g}(\bar{\mu})=-\left<\delta\mu_{n}^{2}\right>-D_{g}\left<\delta\Gamma^{2}\right>,\eqno{\hbox{(B19)}}$$ where Formula$D_{b}(\bar{\mu})=[(\bar{\mu}+i\Delta k)^{2}+k_{p}^{2}]$, and Formula$D_{g}(\bar{\mu})=-ik_{g}^{3}/\bar{\mu}$. The dispersion relation in the absence of errors is Formula$D(\bar{\mu}_{0})=0$ and is third order in Formula$\bar{\mu}$. In the presence of errors, the right side of (B19) will be nonzero and there will be a shift in wavenumber, Formula$\bar{\mu}\simeq\bar{\mu}_{0}+\bar{\mu}_{1}$ where Formula$\bar{\mu}_{0}$ is the error-free wavenumber and Formula TeX Source $$\bar{\mu}_{1}=-\left[\left<\delta\mu_{n}^{2}\right>+D_{g}\left<\delta\Gamma^{2}\right>\right]/D^{\prime}(\bar{\mu}),\eqno{\hbox{(B20)}}$$ with Formula$D^{\prime}(\bar{\mu})=dD/d\bar{\mu}$. We may now solve for the constant Formula$\rho_{0}$ using the lowest order version of (B13), Formula$\rho_{0}=-D_{g}^{-1}$.

To evaluate the right side of (B20) we linearize (B10) and (B11) for the fluctuating quantities, Formula TeX Source $$\delta\mu_{+}=-{{\delta k^{2}(z)}\over{2ik_{0}}}+ik_{g}^{3}\rho_{0}\delta\Gamma,\eqno{\hbox{(B21a)}}$$ and Formula TeX Source $$2\left(\mu_{n}+i\Delta k\right)\delta\mu_{n}+{{d}\over{dz}}\delta\mu_{n}=-\rho_{0}^{-1}\delta\Gamma,\eqno{\hbox{(B21b)}}$$ which along with (B16) constitute a second order system of linear differential equations for the fluctuating quantities. Using the notation of (B19) we write this system, Formula TeX Source $$\left({{d}\over{dz}}+D_{b}^{\prime}\right)\delta\mu_{n}=D_{g}\delta\Gamma,\eqno{\hbox{(B22a)}}$$ and Formula TeX Source $$\left({{d}\over{dz}}+\bar{\mu}\right)\delta\Gamma=\delta\mu_{n}+{{\delta k^{2}(z)}\over{2ik_{0}}},\eqno{\hbox{(B22b)}}$$ where Formula$D_{b}^{\prime}(\bar{\mu})=dD_{b}/d\bar{\mu}$. We then write a formal solution to (B22) in terms of Green's functions, Formula TeX Source $$\eqalignno{\delta\Gamma (z)=&\,\int\limits_{-\infty}^{z}dz^{\prime}G_{\Gamma}(z-z^{\prime}){{\delta k^{2}(z^{\prime})}\over{2ik_{0}}}&{\hbox{(B23a)}}\cr\delta\mu_{n}(z)=&\,\int\limits_{-\infty}^{z}dz^{\prime}G_{n}(z-z^{\prime}){{\delta k^{2}(z^{\prime})}\over{2ik_{0}}}.&{\hbox{(B23b)}}}$$ The Green's functions satisfy equations similar to (B22) but with the source replaced by a delta function, Formula TeX Source $$\left({{d}\over{dz}}+D_{b}^{\prime}\right)G_{n}=D_{g}G_{\Gamma},\eqno{\hbox{(B24a)}}$$ and Formula TeX Source $$\left({{d}\over{dz}}+\bar{\mu}\right)G_{\Gamma}=G_{n}+\delta (z).\eqno{\hbox{(B24b)}}$$

The initial conditions for (B24) are that both Green's functions vanish for Formula$z<0$. Alternatively, we can solve Eqs. (B24) for Formula$z>0$, without the delta function source, if we take as initial conditions, Formula$G_{n}(0)=0$, and Formula$G_{\Gamma}(0)=1$. In principle (B24a) and (B24b) can be solved and the solution expressed as a pair of exponentials. We will postpone this step because what will ultimately be needed are integrals of the squares of the Green's functions, and these can be obtained by other means.

Calculation of the growth rate correction due to random errors in (B24) requires evaluation of the square of the fluctuating quantities such as, Formula TeX Source $$\eqalignno{&\left<\delta\mu_{n}^{2}\right>=-{{\left<\left(\delta k^{2}\right)^{2}\right>}\over{4k_{0}^{2}}}\int\limits_{-\infty}^{z}dz^{\prime}G_{n}(z-z^{\prime})\cr&\qquad~\,\times\int\limits_{-\infty}^{z}dz^{\prime\prime}G_{n}(z-z^{\prime\prime})C(\vert z^{\prime}-z^{\prime\prime}\vert).&{\hbox{(B25)}}}$$ Here we have assumed that the fluctuations in wavenumber are statistically homogeneous and characterized by a correlation function Formula$C$, where Formula$\left<\delta k^{2}(z^{\prime})\delta k^{2}(z^{\prime\prime})\right>=\left<(\delta k^{2})^{2}\right>C(\vert z^{\prime}-z^{\prime\prime}\vert)$.

We assume that the correlation function is localized to a small range compared with the characteristic growth lengths. A typical correlation length might be one period of a coupled cavity structure. Thus, the integrand in the double integral in (B25) is peaked at Formula$z^{\prime}=z^{\prime\prime}$, and can be turned into a single integral of the form, Formula TeX Source $$\left<\delta\mu_{n}^{2}\right>=-{{\left<\left(\delta k^{2}\right)^{2}\right>L_{c}}\over{4k_{0}^{2}}}I_{nn},\eqno{\hbox{(B26a)}}$$ where the correlation length is given by Formula$L_{c}=\int_{-\infty}^{\infty}dz\,C(\vert z\vert)$, and the integral Formula$I_{nn}=\int_{0}^{\infty}dz\,G_{n}^{2}(z)$. Similar analysis gives Formula TeX Source $$\left<\delta\Gamma^{2}\right>=-{{\left<\left(\delta k^{2}\right)^{2}\right>L_{c}}\over{4k_{0}^{2}}}I_{\Gamma\Gamma},\eqno{\hbox{(B26b)}}$$ where the integral Formula$I_{\Gamma\Gamma}=\int_{0}^{\infty}dz\,G_{\Gamma}^{2}(z)$.

We now turn to the evaluation of the integrals Formula$I_{nn}$ and Formula$I_{\Gamma\Gamma}$. Rather than solve explicitly for the Green's functions and integrate over Formula$z$, we multiply (B24a) by Formula$G_{n}$ and (B24b) by Formula$G_{\Gamma}$ and integrate from Formula$z=0^{+}$ to infinity. We use the initial conditions Formula$G_{n}(0)=0$ and Formula$G_{\Gamma}(0)=1$, and we assume Formula$G_{n}, G_{\Gamma}\rightarrow 0$ as Formula$z\rightarrow\infty$ to evaluate the end point contributions of the integrals. The result is Formula TeX Source $$D_{b}^{\prime}I_{nn}=D_{g}I_{\Gamma n}\eqno{\hbox{(B27a)}}$$ and Formula TeX Source $$\bar{\mu}I_{\Gamma\Gamma}-{{1}\over{2}}=I_{\Gamma n},\eqno{\hbox{(B27b)}}$$ where Formula$I_{\Gamma n}=\int_{0}^{\infty}dz\,G_{\Gamma}(z)G_{n}(z)$. Next, we multiply (B24a) by Formula$G_{\Gamma}$ and (B24b) by Formula$G_{n}$, integrate from Formula$z=0^{+}$ to infinity, and add. This result is Formula TeX Source $$\left(D_{b}^{\prime}+\bar{\mu}\right)I_{\Gamma n}=D_{g}I_{\Gamma\Gamma}+I_{nn}.\eqno{\hbox{(B27c)}}$$ Equations (B27) are a linear system that can be solved for the integrals, Formula$I_{nn}$, Formula$I_{\Gamma n}$, and Formula$I_{\Gamma\Gamma}$. Then we find, after a little algebra Formula TeX Source $$\bar{\mu}_{1}z=-{{\left<\left(\delta k^{2}\right)^{2}\right>L_{c}z}\over{8k_{0}^{2}}}{{D_{b}^{\prime}D_{g}^{\prime}}\over{D^{\prime 2}}}.\eqno{\hbox{(B28)}}$$ Next we wish to cast (B28) in terms of Formula$\sigma_{b}$. Substituting Formula$q(x)=(v_{p}(x)-v_{p0})/v_{p0}$ into (B3) it can be shown that Formula TeX Source $$\delta k^{2}(z)=-2k_{0}^{2}q(x).\eqno{\hbox{(B29)}}$$

The relation Formula$\sigma_{b}=(\sigma_{q}/C)(1+Cb_{0})$ allows us to write that Formula TeX Source $$\left<q^{2}(x)\right>=\sigma_{q}^{2}=\left[{{C\sigma_{b}}\over{1+Cb_{0}}}\right]^{2}.\eqno{\hbox{(B30)}}$$ Taking the derivative of (B19) with respect to Formula$\bar{\mu}$ we define Formula TeX Source $$\lambda (\gamma)\equiv{{D_{b}^{\prime}D_{g}^{\prime}}\over{D^{\prime 2}}}={{2\gamma^{2}\left({{\Delta k}\over{k_{g}}}-i\gamma\right)}\over{\left[1+2\gamma^{2}\left({{\Delta k}\over{k_{g}}}-i\gamma\right)\right]^{2}}},\eqno{\hbox{(B31)}}$$ where Formula$\Delta k/k_{g}=b_{0}/(1+Cb_{0})$ and Formula$\gamma\equiv\bar{\mu}/k_{g}$. The value of Formula$\gamma$ is determined by solving the dispersion relationship Formula$D(\bar{\mu}_{0})=0$, which after substituting Formula$k_{p}^{2}/k_{g}^{2}=4QC$ can be written in terms of Formula$\gamma$ to read Formula TeX Source $$\gamma\left[\left(\gamma+{{ib_{0}}\over{1+Cb_{0}}}\right)^{2}+4QC\right]=i.\eqno{\hbox{(B32)}}$$ Combining (B29)(B31) with (B28) and substituting the correlation length in terms of Formula$x=k_{0}z$ as Formula$k_{0}L_{c}=x/N=\Delta$ yields Formula TeX Source $$\bar{\mu}_{1}z=\left<G_{1}(x)+j\theta_{1}(x)\right>=-{{\lambda (\gamma)}\over{2}}\left({{C}\over{1+Cb_{0}}}\right)^{2}\sigma_{b}^{2}x\Delta,\eqno{\hbox{(B33)}}$$ where Formula$\lambda$ is given by (B31)and the value of Formula$\gamma$ is determined by (B32). Equation (B33) is (8) of the text. Note that if Formula$b_{0}=0$ and Formula$QC=0$, then Formula$\Delta k=0$, Formula$\gamma^{3}=i$, and Formula$\lambda$ is real by (B31), in which case Formula$\left<\theta_{1}\right>=0$ by (B33), as in the example in Fig. 2(b).

Footnotes

This work was supported in part by the Air Force Office of Scientific Research, the Office of Naval Research, and the L-3 Communications Electron Device Division. The review of this paper was arranged by Editor M. Anwar.

I. M. Rittersdorf is with the University of Michigan, Ann Arbor, MI 48108 USA(e-mail: ianrit@umich.edu).

T. M. Antonsen, Jr. is with the University of Maryland, College Park, MD 20742 USA (e-mail: antonsen@glue.umd.edu).

D. Chernin is with Science Applications International Corporation, McLean, VA 22102 USA (e-mail: david.chernin@saic.com).

Y. Y. Lau is with the University of Michigan, Ann Arbor, MI 48108 USA, and also with the Naval Research Laboratory, Washington, DC 20375 USA (e-mail: yylau@umich.edu).

Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org.

References

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Authors

Ian M. Rittersdorf

Ian M. Rittersdorf

Ian M. Rittersdorf (S'11) received the B.S.E. and M.S.E. degrees in nuclear engineering and radiological sciences from the University of Michigan, Ann Arbor, MI, USA, in 2008 and 2010, respectively, where he is currently pursuing the Ph.D. degree.

His current research interests include general theory of oscillators, theory of magneto-Rayleigh-Taylor instability, and tube-based microwave sources.

Mr. Rittersdorf is a student member of the American Physical Society. He was a recipient of the Michigan Institute for Plasma Science and Engineering Fellowship in 2011.

Thomas M. Antonsen, Jr.

Thomas M. Antonsen, Jr.

Thomas M. Antonsen, Jr. (F'11) was born in Hackensack, NJ, USA, in 1950. He received the B.S. degree in electrical engineering and the M.S. and Ph.D. degrees from Cornell University, Ithaca, NY, USA, in 1973, 1976, and 1977, respectively.

From 1976 to 1977, he was a National Research Council Post-Doctoral Fellow with the Naval Research Laboratory, Washington, DC, USA. From 1977 to 1980, he was a Research Scientist with the Research Laboratory of Electronics, Massachusetts Institute of Technology (MIT), Cambridge, MA, USA. In 1980, he was with the University of Maryland, College Park, MD, USA, where he joined the faculty of the Department of Electrical Engineering and the Department of Physics in 1984, was the Acting Director of the Institute for Plasma Research, University of Maryland from 1998 to 2000, and is currently a Professor of Physics and Electrical and Computer Engineering. He has held visiting appointments with the Kavli Institute for Theoretical Physics, University of California, Santa Barbara, CA, USA, the Ecole Polytechnique Fédérale de Lausanne, Lausanne, Switzerland, and the Centre de Physique Théorique, Ecole Polytechnique, Palaiseau, France. He is the author or a co-author of more than 300 journal articles and a co-author of the book, Principles of Free-Electron Lasers. His current research interests include the theory of magnetically confined plasmas, the theory and design of high-power sources of coherent radiation, nonlinear dynamics in fluids, and the theory of the interaction of intense laser pulses and plasmas. Dr. Antonsen was a fellow in 1986 of the Division of Plasma Physics, American Physical Society, for which he served as the Chair in 2010. He has served on the Editorial Board of Physical Review Letters, The Physics of Fluids, and Comments on Plasma Physics. He was a co-recipient of the 1999 Robert L. Woods Award for Excellence in Vacuum Electronics Technology and was the recipient of the IEEE Plasma Science and Applications Award in 2003 and the Outstanding Faculty Research Award from the Clark School of Engineering in 2004.

David Chernin

David Chernin

David Chernin received the B.A. and Ph.D. degrees in applied mathematics from Harvard University, Cambridge, MA, USA, in 1971 and 1976, respectively. From 1976 to 1978, he was a member of the Institute for Advanced Study, Princeton, NJ, USA, where he focused on problems in magnetic confinement fusion. From 1978 to 1981, he was a Senior Scientist with Maxwell Laboratories, San Diego, CA, USA, where he focused on the design and analysis of excimer lasers and high-power X-ray sources. Since 1984, he has been with Science Applications International Corporation (SAIC), McLean, VA, USA, where he has conducted research on the theory and simulation of beam-wave interactions in particle accelerators and on the design, simulation, and analysis of vacuum electron devices in collaboration with the Vacuum Electronics Branch, Naval Research Laboratory, Washington, DC, USA. From 2005 to 2008, he served as the Chief Scientist with the Technology and Advanced Systems Business Unit, SAIC.

Dr. Chernin is a member of the American Physical Society.

Y. Y. Lau

Y. Y. Lau

Y. Y. Lau (M'98–SM'06–F'08) received the B.S., M.S., and Ph.D. degrees in electrical engineering from the Massachusetts Institute of Technology (MIT), Cambridge, MA, USA, in 1968, 1970, and 1973, respectively.

From 1973 to 1979, he was an Instructor and an Assistant Professor of applied mathematics with MIT. From 1980 to 1983, he was with Science Applications International Corporation, McLean, VA, USA, as a Research Physicist. From 1983 to 1992, he was a Research Physicist with the Naval Research Laboratory, Washington, DC, USA. In 1992, he joined the University of Michigan, Ann Arbor, MI, USA, where he is currently a Professor with the Department of Nuclear Engineering and Radiological Sciences and the Applied Physics Program. He is a Faculty Appointee at the Naval Research Laboratory. He has been involved in electron beams, coherent radiation sources, plasmas, and discharges. He is the author or a co-author of more than 190 refereed publications. He holds ten patents. His current research interests include electrical contacts, heating phenomenology, high-power microwave sources, and magneto-Rayleigh-Taylor instabilities.

Dr. Lau is a fellow of the American Physical Society. He received the 1989 Sigma-Xi Scientific Society Applied Science Award and the 1999 IEEE Plasma Science and Applications Award. He served three terms from 1994 to 2005 as an Associate Editor for Physics of Plasmas.

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