By Topic

• Abstract

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 When the quantities $\beta_{p}$, $C$, or $d$ are allowed to vary axially, (3) is no longer applicable, and we combine (1) and (2) to yield, 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

SECTION III

## PERTURBATIVE AND RICCATI APPROACHES

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

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

Our work differs from the previous work of Pengvanich et al\.by applying the random Gaussian errors directly to the parameter $b$, instead of to the circuit phase velocity $v_{p}$ through a dimensionless quantity $q(x)=(v_{p}(x)-v_{p0})/v_{p0}$, where $v_{p0}$ is the unperturbed circuit phase velocity. The random function $q(x)$ had a specified standard deviation $\sigma_{q}$. The velocity parameter $b$ is related to $q$ by $b(x)=(1/C)[Cb_{0}-q(x)]/(1+q(x))$ and the standard deviations in $b$ and $q$ are approximately related by $\sigma_{b}=(\sigma_{q}/C)(1+Cb_{0})$. Due to the non-linear relationship between $b$ and $q$, a Gaussian random error profile assigned to $q(x)$ is no longer Gaussian for $b(x)$. 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 $b(x)$ with a standard deviation of $\sigma_{b}$.

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 $\sigma_{b}$ (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 $b$, we show in Appendix A [cf. Equation (A16)], 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 $\left<G_{1}(x)+j\theta_{1}(x)\right>$ is the ensemble-average deviation in gain and in phase from the error-free tube due to random errors, $\delta_{k}~(k=1, 2, 3)$ are the three roots to the Pierce dispersion relation (3), $\tau_{k}~(k=1, 2, 3)$ which depends only on $\delta_{k}$, is defined by (A5) of [7], and $Q_{1}(x, s)$, $Q_{2}(x, s)$ depend only the error-free, three-wave solution. The expressions for $Q_{1}(x, s)$, $Q_{2}(x, s)$ are given in Appendix A in (A17a), (A17b). Use of (7) will be referred to as the “perturbation” method.

The second analytical method calculates $\left<G_{1}(x)+j\theta_{1}(x)\right>$ using a Riccati formulation of the complex wave number for a single wave. This formulation yields, [see Appendix B, (B33)], 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 $\lambda$ is a complex value that is determined by the value of the mismatch parameter, $b_{0}$. 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 $(QC\ne 0)$. In terms of the standard deviation of $b$, $\sigma_{b}$, the standard deviation in the gain $G_{1}$ and in the phase $\theta_{1}$ is given by, respectively, 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 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 $g_{br}$ and $g_{bi}$ are the real and imaginary parts, respectively, of $g_{b}$, given by 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., $QC=0$, (10) reduces to (A15) of [7], whose (A4) defines the error-free solutions $f_{0}$ and $a_{0}$.

Equation (9a) and (9b) show that the standard deviations in the gain and phase are linear in $\sigma_{b}$. Equation (7) and (8) show that $\left<G_{1}(x)\right>$ and $\left<\theta_{1}(x)\right>$ are both quadratic in $\sigma_{b}$, 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 $x=100$ where $b_{0}=d=QC=0$, and $C=0.05$. 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, $b$, as shown in Fig. 1. The value at each node is an independent Gaussian random variable with a mean of $b_{0}$ and a specified standard deviation, $\sigma_{b}$. A correlation length of $\Delta=1$ has been used in all calculations, meaning that each node of the Gaussian random error profile would correspond to $x=1, 2,\ldots, 100$ in the TWT. For a specified value of $\sigma_{b}$, 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 $\sigma_{b}$. 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 $\left<\theta_{1}(x)\right>=0$ 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 $\left<\theta_{1}(x)\right>$ due to random errors, measured in radians, is found to be negligible compared with the amplitude variations $\left<G_{1}(x)\right>$, measured in $e$-folds, in this case.This case is identical to the one considered by [7].

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, $b_{0}=0$. 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, $x=100$, $C=0.05$, $\Delta=1$, $d=0$, and $QC=0$.

Fig. 3 shows two cases where the the velocity mismatch is nonzero. For $C=0.05$, $b_{0}=\pm 1$ corresponds to a difference of ${\pm}{5\%}$ 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.

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 ${\pm}0.05v_{p}~(b_{0}=\pm 1)$. 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, $x=100$, $C=0.05$, $\Delta=1$, $d=0$, and $QC=0$.

Figs. 4 and 5 show how the gain and phase are affected by the inclusion of the $QC$ term, increasing it from 0 to 0.35 for the synchronous case, $b_{0}=0$. When $QC\ne 0$, 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 $QC$ again increasing from 0 to 0.35, this time for the $b_{0}=1$ 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 $b_{0}=-1$ case could not be calculated reliably because the TWT would not amplify for any significant values of $QC$.

Fig. 4. (a) Mean values of the power at the output relative to the unperturbed values for $QC=0$, (b) $QC=0.15$, (c) $QC=0.25$, and (d) $QC=0.35$ for the synchronous velocity case, $b_{0}=0$. 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, $x=100, C\,{=}\,0.05,\Delta\,{=}\,1$, and $d\,{=}\,0$.
Fig. 5. (a) Mean values of the phase at the output relative to the unperturbed values for $QC=0$, (b) $QC=0.15$, (c) $QC=0.25$, and (d) $QC=0.35$ for the synchronous velocity case, $b_{0}=0$. 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, $x\,{=}\,100, C\,{=}\,0.05,\Delta\,{=}\,1$, and $d\,{=}\,0$.
Fig. 6. (a) Mean values of the power at the output relative to the unperturbed values for $QC=0$, (b) $QC=0.15$, (c) $QC=0.25$, and (d) $QC=0.35$ for the non-synchronous velocity case, $b_{0}=1$. 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, $x\,{=}\,100, C\,{=}\,0.05,\Delta\,{=}\,1$, and $d\,{=}\,0$.
Fig. 7. (a) Mean values of the phase at the output relative to the unperturbed values for $QC=0$, (b) $QC=0.15$, (c) $QC=0.25$, and (d) $QC=0.35$ for the non-synchronous velocity case, $b_{0}=1$. 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, $x=100, C=0.05,\Delta=1$, and $d=0$.

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 $g_{b}$ from (10), as well as the statistical standard deviation calculation. With the inclusion of the space charge term, $QC$, (9) is no longer in agreement with the statistical calculation. The difference between the two increases with increasing values of $QC$.

Fig. 8. Mean values and standard deviation of the (a) gain and (b) phase at the output relative to the unperturbed values for $QC=0$ for the non-synchronous velocity case, $b_{0}=-1$. 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, $x=100$, $C=0.05$, $\Delta=1$, and $d=0$.
Fig. 9. Mean values and standard deviation of the (a) gain and (b) phase at the output relative to the unperturbed values for $QC=0.25$ for the non-synchronous velocity case, $b_{0}=1$. 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, $x=100$, $C=0.05$, $\Delta=1$, and $d=0$.

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 $x=240$, and we take a correlation length of $\Delta=4$. For this example we consider the specific case with $C=0.0197$, $QC=0$, and $b_{0}=0.36$, 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.

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, $x=240$, $\Delta=4$, $C=0.0197$, $b_{0}=0.36$, and $QC=d=0$.
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 $b$ 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 $b$, 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 $QC$ would enlarge the range of $b$ 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 $b(x)$, 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 $C$ and $d$ are constant and the parameter $b$ contains small random perturbations denoted as $b_{1}(x)$. We re-write (6)as, TeX Source $$a=a_{0}e^{G_{1}+j\theta_{1}},\eqno{\hbox{(A1)}}$$ where $a$ is the normalized electric field and $a_{0}(x)$ is the solution in the error-free tube given by (A4) of [7]. To second order, we write $a(x)=a_{0}(x)+a_{10}(x)+a_{11}(x)$. The first and second order perturbations are $a_{10}(x)$ and $a_{11}(x)$, respectively. Expanding $a(x)$ in (A1) yields an expression for the modification of amplitude and phase of 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 $a_{10}$ and $a_{11}$ 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 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 $b_{1}(x)$, where 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 $m_{31}(x)=-jC\left(4QC^{3}\right)b_{1}(x)$ and $m_{33}(x)=-jCb_{1}(x)$. 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., $d=0$. We write 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 $b_{1}(x)$. The error-free solutions are $f_{0}$, $v_{0}$, and $a_{0}$ and are given by (A4) of [7]. Combining (A6) with (A3) yields, to second order 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 ${\bf M}_{1}{\bf Y}_{1}(x)$, the solution to (A7) is ${\bf Y}_{1}(x)={\bf Y}_{10}(x)$, whose solution is given by (A10) of [7]. Next, let us approximate ${\bf M}_{1}{\bf Y}_{1}(x)$ in (A7) as ${\bf M}_{1}{\bf Y}_{10}(x)$, and write 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 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 ${\bf Y}_{11}(x)$ as (A10) from [7] to obtain 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 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 $\Psi_{ij}(x)~(i, j=1, 2, 3)$ is defined by (A3) of [7]. The first order perturbations $f_{10}(x)$ and $a_{10}(x)$ are given by (A11) of [7], where the expression for $V_{k}(s)$ now contains the AC space charge term in $m_{31}(x)$. Substituting these into (A10) yields 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 $\delta_{k}~(k=1, 2, 3)$ are the three roots to the Pierce dispersion relation (3), and $\tau_{k}~(k=1, 2, 3)$ which depends only on $\delta_{k}$, is defined by (A5) of [7].

We next take the ensemble-average of (A12), assuming that $\left<b_{1}(s)b_{1}(s^{\prime})\right>=\left<b_{1}^{2}\right>\Delta\delta (s-s^{\prime})$, where $\Delta$ is the correlation length and $\delta$ is the Dirac delta function. With $\left<b_{1}^{2}\right>=\sigma_{b}^{2}$, we obtain 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 $a_{10}(x)$ and taking the ensemble-average yields 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 $\left<a_{10}(x)\right>=0$ since ${\bf M}_{1}$ is linear in $b_{1}(x)$ and therefore $\left<{\bf Y}_{10}(x)\right>=0$ (cf. Equation (A10) of [7]). With this result, we obtain from (A2), 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 $\left<a_{11}(x)\right>$ is given by (A13), $a_{0}(x)$ by (A4) of [7], and $\left<a_{10}^{2}(x)\right>$ by (A14). This can be written in the form 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), $Q_{1}(x,s)$, $Q_{2}(x,s)$ are given by, with the substitution $\lambda_{i}=C\delta_{i}$, 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 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 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, 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 $a(z)$ represents the field amplitude, $n(z)$ represents the perturbed beam density, and we have assumed $e^{-i\omega t}$ dependence for these quantities. The coefficients in (B1) and (B2) represent the following: $k_{s}(z)$ is the axially varying wave number for the structure. The beam plasma wavenumber is $k_{p}$. The nominal gain rate is $k_{g}=Ck_{0}$, where $C$ is the Pierce parameter. The unperturbed beam velocity is $v_{z}$. It is assumed that $k_{s}(z)$ is close to some reference value such that, TeX Source $$k_{s}^{2}(z)=k_{0}^{2}+\delta k^{2}(z),\eqno{\hbox{(B3)}}$$ where $k_{0}$ is defined such that the expectation of the deviation vanishes, $\left<\delta k^{2}(z)\right>=0$. 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 $\omega$, and set $n=s$, $\omega/v_{z}=\beta_{e}$, $k_{s}^{2}=\beta_{p}^{2}$, $k_{p}^{2}=\beta_{q}^{2}$, $k_{0}=\beta_{p0}(={\rm error}-{\rm free~value})$, $k_{g}=Ck_{0}$, and $i=j=\sqrt{-1}$.

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 $z$, 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, TeX Source $$a(z)=a_{+}(z)e^{ik_{0}z}+a{\_}(z)e^{-ik_{0}z},\eqno{\hbox{(B4)}}$$ where we choose 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 $z$. We insert (B4) into (B2), use our constraint (B5), drop the backward wave, and introduce the revised density perturbation, ${\mathhat{n}}=ne^{-ik_{0}z}$, to obtain the third order system describing the coupled forward wave and beam space charge waves, 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 $\Delta k=k_{0}-\omega/v_{z}$ 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 TeX Source $$\mu_{+}(z)={{1}\over{a_{+}}}{{da_{+}}\over{dz}},\eqno{\hbox{(B8)}}$$ and TeX Source $$\mu_{n}(z)={{1}\over{\mathhat{n}}}{{d{\mathhat{n}}}\over{dz}},\eqno{\hbox{(B9)}}$$ and rewrite (B6) and (B7) 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 $\rho (z)={\mathhat{n}}/a_{+}$ gives the spatially varying ratio of density perturbation to field amplitude. It satisfies a differential equation, 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 $\mu_{+}=\bar{\mu}_{+}+\delta\mu_{+}$, $\mu_{n}=\bar{\mu}_{n}+\delta\mu_{n}$, and $\rho=\bar{\rho}+\delta\rho$, 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), 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 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, $\bar{\mu}_{n}=\bar{\mu}_{+}\equiv\bar{\mu}$. Evaluation of $\left<\rho^{\pm 1}\right>$ in (B13) and (B14)requires integration of (B12). Specifically, we write $\rho=\rho_{0}\exp[\delta\Gamma (z)]$ where $\rho_{0}$ is a constant to be determined, and TeX Source $${{d}\over{dz}}\delta\Gamma (z)=\delta\mu_{n}-\delta\mu_{+}.\eqno{\hbox{(B16)}}$$

We then expand $\rho$ and its inverse under the assumption of small fluctuations in $\delta\Gamma$, 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, 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 $\rho_{0}$, giving us a dispersion relation for the common mean rate of change of the exponent, $\bar{\mu}$, 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 $D_{b}(\bar{\mu})=[(\bar{\mu}+i\Delta k)^{2}+k_{p}^{2}]$, and $D_{g}(\bar{\mu})=-ik_{g}^{3}/\bar{\mu}$. The dispersion relation in the absence of errors is $D(\bar{\mu}_{0})=0$ and is third order in $\bar{\mu}$. In the presence of errors, the right side of (B19) will be nonzero and there will be a shift in wavenumber, $\bar{\mu}\simeq\bar{\mu}_{0}+\bar{\mu}_{1}$ where $\bar{\mu}_{0}$ is the error-free wavenumber and 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 $D^{\prime}(\bar{\mu})=dD/d\bar{\mu}$. We may now solve for the constant $\rho_{0}$ using the lowest order version of (B13), $\rho_{0}=-D_{g}^{-1}$.

To evaluate the right side of (B20) we linearize (B10) and (B11) for the fluctuating quantities, TeX Source $$\delta\mu_{+}=-{{\delta k^{2}(z)}\over{2ik_{0}}}+ik_{g}^{3}\rho_{0}\delta\Gamma,\eqno{\hbox{(B21a)}}$$ and 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, TeX Source $$\left({{d}\over{dz}}+D_{b}^{\prime}\right)\delta\mu_{n}=D_{g}\delta\Gamma,\eqno{\hbox{(B22a)}}$$ and 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 $D_{b}^{\prime}(\bar{\mu})=dD_{b}/d\bar{\mu}$. We then write a formal solution to (B22) in terms of Green's functions, 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, TeX Source $$\left({{d}\over{dz}}+D_{b}^{\prime}\right)G_{n}=D_{g}G_{\Gamma},\eqno{\hbox{(B24a)}}$$ and 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 $z<0$. Alternatively, we can solve Eqs. (B24) for $z>0$, without the delta function source, if we take as initial conditions, $G_{n}(0)=0$, and $G_{\Gamma}(0)=1$. 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, 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 $C$, where $\left<\delta k^{2}(z^{\prime})\delta k^{2}(z^{\prime\prime})\right>=\left<(\delta k^{2})^{2}\right>C(\vert z^{\prime}-z^{\prime\prime}\vert)$.

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 $z^{\prime}=z^{\prime\prime}$, and can be turned into a single integral of the form, 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 $L_{c}=\int_{-\infty}^{\infty}dz\,C(\vert z\vert)$, and the integral $I_{nn}=\int_{0}^{\infty}dz\,G_{n}^{2}(z)$. Similar analysis gives 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 $I_{\Gamma\Gamma}=\int_{0}^{\infty}dz\,G_{\Gamma}^{2}(z)$.

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

The relation $\sigma_{b}=(\sigma_{q}/C)(1+Cb_{0})$ allows us to write that 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 $\bar{\mu}$ we define 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 $\Delta k/k_{g}=b_{0}/(1+Cb_{0})$ and $\gamma\equiv\bar{\mu}/k_{g}$. The value of $\gamma$ is determined by solving the dispersion relationship $D(\bar{\mu}_{0})=0$, which after substituting $k_{p}^{2}/k_{g}^{2}=4QC$ can be written in terms of $\gamma$ to read 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 $x=k_{0}z$ as $k_{0}L_{c}=x/N=\Delta$ yields 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 $\lambda$ is given by (B31)and the value of $\gamma$ is determined by (B32). Equation (B33) is (8) of the text. Note that if $b_{0}=0$ and $QC=0$, then $\Delta k=0$, $\gamma^{3}=i$, and $\lambda$ is real by (B31), in which case $\left<\theta_{1}\right>=0$ 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

No Data Available

## Cited By

No Data Available

None

## Multimedia

No Data Available
This paper appears in:
No Data Available
Issue Date:
No Data Available
On page(s):
No Data Available
ISSN:
None
INSPEC Accession Number:
None
Digital Object Identifier:
None
Date of Current Version:
No Data Available
Date of Original Publication:
No Data Available