# Lecture #7

Echoes

## Summary of Lecture #6

1). We have derived the [ponderomotive force]{.underline}, which arises
from the nonlocal character of the particle motion in a finite amplitude
field.

2). We have examined so-called [nonlinear Landau damping]{.underline},
which is the long time behavior of the damping process when a
significant number of particles become trapped in the wave.

3). We have discussed two types of solitons that can exist, arising from
the KdV equation of the Zakharov equations. Both types represent a wave
which results as a nonlinear force is balanced by dispersive effects.

Overview of Lecture #7

Longitudinal beam echoes

-   free-streaming model

-   Vlasov formalism

-   collisional effects -- F-P equation

-   experimental results

Bunched-beam echoes

In this lecture we turn to a novel application of (weakly) nonlinear
waves that has found application in both plasmas and beams: echoes.
Although, much of the mathematical formalism evolved from plasma
physics, echoes occur in many systems throughout physics, and are a
classical feature of many-body systems obeying the Vlasov Equation.

When echoes were first observed in plasmas, they were taken as physical
evidence of Landau damping, and a physical manifestation of the Vlasov
model. As it turns out, they do confirm the physics of Landau damping,
but also can give insight into the microscopic collisional processes
that affect the reversibility of the Vlasov equation. Moreover, they are
related to three-wave coupling, as viewed in the time domain.

Reference for this entire lecture: Spentzouris thesis.

What is an echo?

Suppose in a uniform plasma or in a beam, we make a short duration
perturbation. According to Landau theory, the perturbation decays away
in a characteristic time that depends on the thermal spread of the
particle motion.

For definiteness, we choose an unbunched, coasting beam, and make a
perturbation at harmonic m

$\text{exp}\left( \text{im}\left\lbrack \omega_{0} + k_{0}\varepsilon \right\rbrack t \right)$

where $\in$ is the energy deviation from the synchronous particle, and
$k_{0}$ is the frequency dispersion given by

$k_{0} = - \frac{\text{ηω}_{0}}{\beta^{2}E_{0}}$

The damping time in this case is

$1\tau_{\text{damp}}\text{\textasciitilde}\text{mk}_{0}\sigma_{E}$

Now after the initial perturbation has died away, we make a second
perturbation at harmonic n, with m\<n,

$\text{exp}\left\{ \text{in}\left( \omega_{0} + k_{0}\varepsilon \right)(t + \Delta t) \right\}$

Provided there is a means for mixing of these two waves, a response will
occur at the sum and difference frequencies. In particular for the
difference frequency, a phase term of the form

$\text{exp}\left\{ i\lbrack n - m\rbrack\left\lbrack \omega_{0} + k_{0}\varepsilon \right\rbrack\lbrack t - \Delta t\rbrack - \text{in}\left\lbrack \omega_{0} + k_{0}\varepsilon \right\rbrack\Delta t \right\}$

will appear. It is noted that at a particular time given by

$t_{\text{echo}} = \frac{n\Delta t}{n - m}$

the energy terms in the exponential cancel out. Thus, [at this one
time]{.underline}, the effects of damping are erased, and some portion
of the original disturbance can reappear. This is the basis of an echo.

The interesting feature of such an echo is that the
[accumulated]{.underline} effect of any decorrelatoin mechanism, such as
collisions, has an effect on the ability of the medium to reconstruct
the original disturbance. By measuring changes in the echo over time, it
is possible to gain an extremely sensitive measure of the decorrelatoin
mechanisms.

In the remainder of this lecture, we will derive the analytic formulas
for echo generation and decay in both unbunched and bunched beams and
use the theoretical models to compare to experimental results. The
analysis in a plasma is identical, so in this lecture we will focus only
on the beam case.

We begin by considering echoes in an unbunched beam (our favorite plasma
analogue) where [wakefields can be considered negligible.]{.underline}
This has the effect of making the dielectric constant of the beam
exactly equal to unity. There are virtually no other media in Nature
that have this property.

For a particle traveling around a storage ring, we have already seen
that the phase evolution is given by

$\theta = \theta_{i} + \left( \omega_{0} + k_{0}\varepsilon \right)t$

where $k_{0} = - \frac{\text{ηω}_{0}}{\beta^{2}E_{0}}$ is the frequency
dispersion. We note that the beam can respond resonantly if we excite it
at any one of an infinite series of harmonics of the revolution
frequency, or $\omega = {n\omega}_{0}$.

If we now make a kick in the form of a short burst of rf at a given
harmonic, we can find an expression for the beam response by examining
the phase variation of a single particle, then averaging over the
particle distribution.

The phase evolution after one kick is given by

$\theta(t) = \theta_{i} + \left( \omega_{0} + k_{0}\varepsilon \right)t + k_{0}{\Delta\varepsilon}_{}t$

where the energy gain is given by

${\Delta\varepsilon}_{} = \frac{{e\omega}_{0}}{2\pi}V_{0}{\Delta T}_{\rho}\text{cos}{k\theta}_{i}$

To find the perturbed density, we average this phase factor over the
particle distribution function as

$n_{m} = \int_{}^{}{d\theta\int_{}^{}{f(\theta,\varepsilon,t)}}e^{- \text{im}\theta(t)}d\varepsilon$

which yields

$n_{m} = \int_{}^{}{d\theta}_{i}\int_{}^{}f_{0}(\varepsilon)e^{- \text{im}\left( \theta_{i} + \left\lbrack \omega_{0} + k_{0}\varepsilon \right\rbrack t + k_{0}{\Delta\varepsilon}_{}t \right)_{}}$

Using the following Bessel function identity, we can rewrite the above
expression

$e^{- \text{iA}\text{cos}\theta} = \sum_{l}^{}( - i)^{l}J_{e}(A)e^{- \text{il}\theta}$

Then the instantaneous current [after the first kick only]{.underline}
can be written as

$I_{m}(\theta) = \text{iI}_{0}J_{}\left( m\frac{{e\omega}_{0}}{2\pi}k_{0}V_{0}\Delta t \right)e^{- \text{im}\omega_{0}t}e^{- 12\left( {m\omega}_{0}k_{0}\sigma_{E} \right)t^{2}}$

which exhibits the 'Landau damping' time given by

$\frac{1}{\tau} = \left( \frac{{m\omega}_{0}k_{0}\sigma_{E}}{\sqrt{2}} \right)^{12}$

Now we make a second kick after a delay $\Delta t$ and follow the same
analysis. The perturbed density is given by

$n_{m} = \int_{}^{}{d\theta}_{i}\int_{}^{}{d\varepsilon}f_{0}(\varepsilon)e^{- \text{im}\left( \theta_{i} + \left\lbrack \omega_{0} + k_{0}\varepsilon \right\rbrack t \right)}e^{- \text{imk}_{0}t\text{cos}{k\theta}_{i}}e^{- \text{imk}_{0}(t - \Delta t)A\text{cos}n\theta}$

And using the Bessel function identity as before, the perturbed density
can be written as

$$\begin{array}{r}
n_{m} = \frac{N}{(2\pi)^{23}\sigma_{E}}\sum_{l_{2}}^{}{( - i)^{l_{2}}J_{l_{2}}}\left( \text{mk}_{0}A\lbrack t - \Delta t\rbrack \right)\sum_{l_{}}^{}( - i)^{l_{}}J_{l_{}}\left( k_{0}A\left\lbrack \text{mt} + l_{2}n\Delta t \right\rbrack \right) \\
\int_{}^{}e^{- i\left( m + l_{2}n + l_{}k \right)\theta_{i}}e^{- {i\omega}_{0}\left( \text{mt} + \text{nl}_{2}\Delta t \right)}{d\theta}_{i}\int_{}^{}{e^{- \text{ik}_{0}\varepsilon\left( \text{mt} + l_{2}n\Delta t \right)}e^{- \text{ik}_{0}\varepsilon\left( \text{mt} + l_{2}n\Delta t \right)}}e^{- 12\left( \frac{\varepsilon}{\sigma_{E}} \right)^{2}}d\varepsilon
\end{array}$$

It is noted that the response averages to zero unless the phase factor
vanishes, namely when

$m + l_{}k + l_{2}n = 0$

(reminiscent of the three-wave coupling!) And the time of the echo can
be found from the condition that the second phase factor vanishes,
namely when

$t_{\text{echo}}\text{==}\frac{n\Delta t}{m}$; $l_{} = l_{2} = 1$

Thus, the instantaneous current [after the second kick]{.underline}
shows a delayed response, modulated by a form factor, which is the echo

$$\begin{array}{r}
I_{m} = \left( \frac{\text{eN}\omega_{0}}{2\pi} \right)J_{}\left( \frac{{e\omega}_{0}}{2\pi}k_{0}V_{0}{\Delta T}_{p}\left\lbrack \text{mt} - m\Delta t \right\rbrack \right) \cdot J_{}\frac{{e\omega}_{0}}{2\pi}k_{0}V_{0}{\Delta T}_{p}\left\lbrack \text{mt} + n\Delta t \right\rbrack \\
\text{exp}\left( - {i\omega}_{0}\left\lbrack \text{mt} + n\Delta t \right\rbrack \right)\text{exp}\left\{ - 12\left( k_{0}\sigma_{\in} \right)^{2}\left( \text{mt} + n\Delta t \right)^{2} \right\}
\end{array}$$

We note that the form of the echo [in time]{.underline} is essentially
that of the original distribution function [in energy]{.underline},
modulated by J~/~. Thus a measurement of the echo shape gives a direct
measurement of the distribution function.

![](media/image1.jpeg){width="4.875in" height="2.6375in"}

This is essentially the free-streaming result where we did not have to
take any collective effects into account.

Wakefields can also affect the echo shape, and that feature has not been
utilized to date. But the most important application is through the
effect of collisions on the echo shape, as it leads to a sensitive
diagnostic for determining collision rates experimentally.

![](media/image2.jpeg){width="5.99375in" height="7.240972222222222in"}

![](media/image3.jpeg){width="5.999305555555556in" height="4.69375in"}

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height="4.446527777777778in"}

![](media/image6.jpeg){width="5.998611111111111in"
height="4.331944444444445in"}

![](media/image7.jpeg){width="5.740277777777778in"
height="6.1402777777777775in"}

![](media/image8.jpeg){width="5.999305555555556in"
height="4.6409722222222225in"}

![](media/image9.jpeg){width="5.997916666666667in"
height="3.9881944444444444in"}

![](media/image10.jpeg){width="5.99375in" height="5.06875in"}

Echoes with Wakefields and Collisions

In this case, we cannot carry out an exact analysis, but can find an
approximate form by a perturbation expansion of the Boltzmann equation.

In the longitudinal coordinates, the Boltzmann equation is given by

$\begin{array}{r}
\frac{\partial f_{n - m}}{\partial t} + i(n - m)\left( \omega_{0} + k_{0}\varepsilon \right)f_{n - m} - \frac{{e\omega}_{0}}{2\pi}\frac{\partial f_{0}}{\partial\varepsilon}U_{n - m} \\
 + \frac{{e\omega}_{0}}{2\pi}\sum_{n^{'}}^{}U_{- n^{'} + n - m}\frac{\partial f_{n^{'}}}{\partial\varepsilon} = \upsilon\left( \frac{\partial}{\partial p}\left\lbrack \text{pf}_{n - m} \right\rbrack + p_{0}^{2}\frac{\partial^{2}f_{n - m}}{\partial p^{2}} \right)
\end{array}$

where $p = \frac{\in}{\beta c}$, and we have spatially Fourier analyzed
the Boltzmann equation. (The reason for choosing the n-m^th^ harmonic
will become evident shortly.)

The potential charge per turn in the ring due to wakefields is written
in terms of the longitudinal impedance

$U_{n - m} = \frac{{e\omega}_{0}Z_{n - m}}{2\pi}\int_{- \infty}^{\infty}{{d\varepsilon f}_{n - m}(\varepsilon,t)} + V_{n - m}^{\text{ext}}$

We now drop the first term (drag term) on the RHS of the Boltzmann
equation since this only makes a small change in the linear dispersion
relation.

The second term on the RHS causes phase diffusion, which is the dominant
effect of collisions on the echo dynamics.

Lastly we use a Laplace transform on the Boltzmann equation as

$L\left\{ f(t) \right\} = \int_{0}^{\infty}e^{- \text{st}}f(t)\text{dt} \equiv \overset{\sim}{f}(s)$

We arrive at the following transformed equation

$$\begin{array}{r}
\left\{ s + i(n - m)\left( \omega_{0} + k_{0}\varepsilon \right) \right\}{\overset{\sim}{f}}_{n - m}(s,\varepsilon) = \frac{{e\omega}_{0}}{2\pi}{\overset{\sim}{U}}_{n - m}(s)\frac{\partial f_{0}}{\partial\varepsilon} + \text{υε}_{0}^{2}\frac{\partial^{2}{\overset{\sim}{f}}_{n - m}(s,\varepsilon)}{\partial\varepsilon^{2}} \\
 + \frac{{e\omega}_{0}}{2\pi}\int_{\sigma - i\infty}^{\sigma + i\infty}{ds^{'}}\frac{1}{2\pi i}\sum_{n^{'}}^{}{\overset{\sim}{U}}_{- n^{'} + m - n}\left( s - s^{'} \right)\frac{\partial f_{n^{'}}\left( s^{'},\varepsilon \right)}{\partial\varepsilon}
\end{array}$$

We can solve this by a perturbation approach. First ignore the nonlinear
(last) term in the above equation, and solve for
${\overset{\sim}{f}}_{n - m}$, ${\overset{\sim}{U}}_{n - m}$.

The first-order part of ${\overset{\sim}{f}}_{n - m}$ satisfies the
following equation

$\text{υε}_{0}^{2}\frac{\partial^{2}{\overset{\sim}{f}}_{n - m}}{\partial\varepsilon^{2}} + \left( s + i\lbrack n - m\rbrack\left\lbrack \omega_{0} + k_{0}\varepsilon \right\rbrack \right){\overset{\sim}{f}}_{n - m} = \frac{{e\omega}_{0}}{2\pi}\frac{\partial f_{0}}{\partial\varepsilon}{\overset{\sim}{U}}_{n - m}$

To solve this, we use a second Laplace transform (in energy) for the
form

$\Im_{n - m}(\varepsilon) = \int_{}^{}{d\tau e}^{- \text{τε}}{\hat{\Im}}_{n - m}(\tau)$

The transformed equation becomes

$$\begin{array}{r}
\text{υε}_{0}^{2}\tau^{2}{\hat{\Im}}_{n - m}(\tau) + \left( s + i\lbrack n - m\rbrack\omega_{0} \right){\hat{\Im}}_{n - m}(\tau) + i(n - m)k_{0}\frac{\partial{\hat{\Im}}_{n - m}(\tau)}{\partial\tau} = \frac{{e\omega}_{0}}{2\pi}\frac{\partial f_{0}}{\partial\varepsilon}|_{\varepsilon_{0}}{\hat{U}}_{n - m} \\
 \Rightarrow \Im_{n - m}(\varepsilon) = \frac{{e\omega}_{0}}{2\pi}\frac{\partial f_{0}}{\partial\varepsilon}|_{\varepsilon_{0}}\frac{{\overset{\sim}{U}}_{n - m}}{\text{imk}_{0}}\int_{0}^{\infty}{d\tau\text{exp}\left\{ \frac{s + i(n - m)\left( \omega_{0} + k_{0}\varepsilon \right)}{i(n - m)k_{0}}\tau - \upsilon\frac{\varepsilon_{0}^{2}\tau^{3}}{3i(n - m)k_{0}} \right\}}
\end{array}$$

And ${\overset{\sim}{U}}_{n - m}$ is found from

${\overset{\sim}{U}}_{n - m}(s) = Z_{n - m}\frac{{e\omega}_{0}}{2\pi}\int_{- \infty}^{\infty}{{\overset{\sim}{\Im}}_{n - m}(s,\varepsilon)}d\varepsilon + V_{n - m}^{\text{ext}}$

We now substitute these first-order expressions for
${\overset{\sim}{f}}_{n - m}$ and ${\overset{\sim}{U}}_{n - m}$ into the
Boltzmann equation, keeping only second-order terms, and integrating
again to find the second-order current according to

$I_{m}^{(2)}(s) = \frac{{e\omega}_{0}}{2\pi}\int_{- \infty}^{\infty}{d\varepsilon{\overset{\sim}{f}}_{m}^{(2)}}(\varepsilon,s)$

The result is

$$\begin{array}{r}
I_{m}^{(2)}(s) = - \left( \frac{{e\omega}_{0}}{2\pi} \right)^{3}\sum_{n^{'}}^{}\frac{1}{(n - m)n^{'}k_{0}^{2}} \\
\int_{\sigma - i\infty}^{\sigma + i\infty}\frac{ds^{'}}{2\pi i}\frac{V_{- n^{'} + n - m}^{\text{ext}}\left( s - s^{'} \right)V_{n^{'}}^{\text{ext}}\left( s^{'} \right)}{D_{- n^{'} + n - m}\left( s - s^{'} \right)D_{n^{'}}\left( s^{'} \right)D_{n - m}(s)}\int_{- \aleph}^{\infty}{d\varepsilon{\overset{\sim}{\Im}}_{n - m}}\frac{\partial}{\partial\varepsilon}\left\lbrack \frac{\partial f_{0}}{\partial\varepsilon}{\overset{\sim}{\Im}}_{n^{'}} \right\rbrack
\end{array}$$

D~n-m~(s) is the dispersion relation for longitudinal mode n-m given by

$D_{n - m}(s) = 1 - \left( \frac{{e\omega}_{0}}{2\pi} \right)^{2}Z_{n - m}\int_{- \infty}^{\infty}{d\varepsilon}\frac{\frac{\partial f_{0}}{\partial\varepsilon}}{s + i(n - m)\left( \omega_{0} + k_{0}\varepsilon \right)}$

As the factors D~k~ (s,ś) occur in the denominator, they represent the
dielectric response of the beam at each of the three frequencies
involved in producing the second-order current. This is exactly the form
of the response we found associated with parametric coupling (for good
reason).

This is a general, but complicated, expression that nonetheless has been
shown to produce an echo response (in plasmas). A major simplification
can be made in cases where wakefields can be considered negligible, as
then $D_{k} \rightarrow 1$.

In this limit, the integrations can be explicitly carried out, yielding

$$\begin{array}{r}
I_{m}^{(2)}(t) = \frac{i}{\text{nk}_{0}}(t - \Delta t)\left( \frac{{e\omega}_{0}}{2\pi} \right)^{3}\int_{- \infty}^{\infty}{d\varepsilon\frac{\partial f_{0}}{\partial\varepsilon}} \\
\text{exp}\left\{ - i(n - m)\left( \omega + k_{0}\varepsilon \right)(t - \Delta t) - \text{in}\left( \omega_{0} + k_{0}\varepsilon \right)\Delta t - \frac{{\upsilon k}_{0}^{2}\varepsilon_{0}^{2}}{3}\left( \lbrack n - m\rbrack^{2}\lbrack t - \Delta t\rbrack^{3} - n^{2}{\Delta t}^{2} \right) \right\}
\end{array}$$

where we have assumed

> $V_{n - m}^{\text{ext}} = 1$ ; $V_{n}^{\text{ext}} = e^{- s\Delta t}$

We note that the integral vanishes [except]{.underline} where the
exponential phase factor vanishes, namely where

$t_{\text{echo}} = \frac{m\Delta t}{m - n}$

The echo occurs at $t_{\text{echo}}$ and has a profile in time
proportional to the energy derivative of the distribution.

The echo amplitude decays with a time constant proportional to
$\upsilon^{13}$ and hence [a measurement of the echo decay can provide a
direct measurement of the collision frequency]{.underline}, $\upsilon$.

We note that in this [free-streaming limit]{.underline} (no wakefield),
we did not need to use the perturbation expansion approach. We may
include the effect of collisions directly in the previous free-streaming
echo model, resulting in the same echo form multiplied by the decay term
just derived.

The [echo amplitude]{.underline} goes as

$I_{m} = \text{AJ}_{}\left( 2{m\delta k}_{0}\varepsilon_{0}\Delta t \right)\text{exp}\left( - {\upsilon k}_{0}^{2}\varepsilon_{0}^{2}\frac{n^{2}\lbrack n - m\rbrack^{2}}{3m^{2}}t^{3} \right)$

> $\upsilon$ is found by a fit to this standard form.

Bunched-Beam Echoes

We may carry out the same type of analysis in a bunched beam, and we
will sketch that analysis here. We will ignore wakefields, but include
collisional effects. There is much work yet to be done here.

We restrict ourselves to longitudinal motion only and adopt the
action-angle coordinate system

![](media/image11.jpeg){width="2.3743055555555554in"
height="2.2493055555555554in"} $\begin{array}{r}
J = \tau^{2} + \left( \overset{˙}{\tau}\omega_{s} \right)^{2} \\
\tau = \sqrt{J}\text{cos}\varphi \\
\overset{˙}{\tau}\omega_{s} = \sqrt{J}\text{sin}\varphi
\end{array}$

The motion of the particles in the rf well, at least near the bottom of
the well, is periodic, with the action being a conserved quantity. Thus
the Vlasov equation can be written as

$\frac{\partial\psi}{\partial t} + \omega_{s}(J)\frac{\partial\psi}{\partial\varphi} = \frac{\partial}{\partial J}\left( D\lbrack J\rbrack J\frac{\partial\psi}{\partial J} \right)$

where
$\omega_{s}(J) = \omega_{s0} + \frac{\partial\omega_{s0}}{\partial J}J$

We now transform to the rotating frame

$\begin{array}{r}
u = \varphi -_{s}(J)t \\
t^{'} = t \\
J^{'} = J \\
\frac{\partial\psi}{\partial t^{'}} = \frac{\partial\psi}{\partial t} + \omega_{s}\frac{\partial\psi}{\partial\varphi};\frac{\partial\psi}{\partial J} = \frac{\partial\psi}{\partial J^{'}} - t\frac{\partial\omega_{s}}{\partial J}\frac{\partial\psi}{\partial u}
\end{array}$

Combining and dropping primes, gives

$\frac{\partial\psi}{\partial t} = \left( \frac{\partial}{\partial J} - t\frac{\partial\omega_{s}}{\partial J}\frac{\partial}{\partial u} \right)\left( \text{DJ}\left\lbrack \frac{\partial\psi}{\partial J} - t\frac{\partial\omega_{s}}{\partial J}\frac{\partial\psi}{\partial u} \right\rbrack \right)$

We consider only times long with respect to a Landau damping time

$\begin{array}{r}
\left| t\frac{\partial\omega_{s}}{\partial J}J \right|\text{>>}1 \\
 \Rightarrow \frac{\partial\psi}{\partial t} \simeq \left\lbrack \frac{\partial\omega_{s}}{\partial J}t \right\rbrack^{2}\text{DJ}\frac{\partial^{2}\psi}{\partial u^{2}}
\end{array}$

Assume we make a [dipole]{.underline} kick in energy, and expand the
perturbed distribution frequency in terms of the unperturbed state

$\psi_{} = \psi_{0} + \frac{\partial\psi_{0}}{\partial J}\Delta J + \frac{\partial\psi_{0}}{\partial\varphi}\Delta\varphi$

The distribution function just after the energy kick is given by

$\psi_{i} = \psi_{0} + 2\sqrt{J}\frac{\partial\psi_{0}}{\partial J}\Delta_{}\text{sin}\varphi$

And the solution of the above approximate equation for is

$\psi_{} = \psi_{0} + 2\sqrt{J}\Delta_{}\frac{\partial\psi_{0}}{\partial J}\left( \text{sin}u \right)e^{- \frac{\text{JD}}{3}\left( \frac{\partial\omega_{s}}{\partial J} \right)^{2}t^{3}}$

This exhibits a Landau damping (phase mixing) time of

$t_{\text{Landau}} = \left( \frac{\partial\omega_{s0}}{\partial J}J_{0} \right)^{- 1}$

Now apply a [quadrupolar]{.underline} kick at $t = t_{}$

$\psi_{2i} = \psi_{}\left( t = t_{},\varphi \right) + \Delta_{2}\sqrt{J}\text{cos}\varphi\left( 2\sqrt{J}\text{sin}\varphi\frac{\partial\psi_{}}{\partial J} + \frac{\text{cos}\varphi}{\sqrt{J}}\frac{\partial\psi_{}}{\partial\varphi} \right)$

$$\begin{array}{r}
\frac{\partial\psi_{}}{\partial J} = \frac{\partial\psi_{0}}{\partial J} + \Delta_{}\text{sin}\text{ue}^{- \text{JD}\left( \frac{\partial\omega_{s}}{\partial J} \right)^{2}\frac{t_{}^{3}}{3}} \\
\left( \frac{1}{\sqrt{J}}\frac{\partial\psi_{0}}{\partial J} + 2\sqrt{J}\frac{\partial^{2}\psi_{0}}{\partial J^{2}} - 2\sqrt{J}\frac{\partial\psi_{0}}{\partial J}\left\lbrack D\left( \frac{\partial\omega_{s}}{\partial J} \right)^{2} + J\frac{\partial D}{\partial J}\left( \frac{\partial\omega_{s}}{\partial J} \right)^{2} + 2\text{JD}\frac{\partial\omega_{s}}{\partial J}\frac{\partial^{2}\omega_{s}}{\partial J^{2}} \right\rbrack\frac{t_{}^{3}}{3} \right) \\
 - 2t_{}\sqrt{J}\Delta_{}\frac{\partial\psi_{0}}{\partial J}\frac{\partial\omega_{s}}{\partial J}e^{- \text{JD}\left( \frac{\partial\omega_{s}}{\partial J} \right)^{2}t_{}^{3}3}\text{cos}u \\
\frac{\partial\psi_{}}{\partial\varphi} = 2\sqrt{J}\frac{\partial\psi_{0}}{\partial J}\text{cos}\text{ue}^{- \text{JD}\left( \frac{\partial\omega_{s}}{\partial J} \right)^{2}\frac{t_{}^{3}}{3}}
\end{array}$$

Keeping only the dominant terms after damping of this kick has occurred,
we find for the distribution after the second kick

$\psi_{2i} \simeq 2t_{}\Delta_{}\Delta_{2}J^{23}\text{sin}2\varphi\frac{\partial\omega_{s}}{\partial J}\frac{\partial\psi_{0}}{\partial J}\text{cos}\text{ue}^{- \text{JD}\left( \frac{\partial\omega_{s}}{\partial\text{GJ}} \right)^{2}\frac{t_{}^{3}}{3}}$

As before, we make a change of variable

$\begin{array}{r}
v = \varphi - \omega_{s}\left( t - t_{} \right) \\
\frac{\partial\psi}{\partial t} = \left( \frac{\partial}{\partial J} - \frac{\partial\omega_{s}}{\partial J}\left\lbrack t - t_{} \right\rbrack\frac{\partial}{\partial v} \right) \cdot \left( \text{DJ}\left\lbrack \frac{\partial\psi}{\partial J} - \frac{\partial\omega_{s}}{\partial J}\left( t_{} - t_{} \right)\frac{\partial\psi}{\partial v} \right\rbrack \right)
\end{array}$

And again assume the second kick has Landau damped

$\begin{array}{r}
\frac{\partial\omega_{s}}{\partial J}J\left( t - t_{} \right)\text{>>}1 \\
 \Rightarrow \frac{\partial\psi}{\partial t} \simeq \left( \frac{\partial\omega_{s}}{\partial J}\left\lbrack t - t_{} \right\rbrack \right)^{2}\text{DJ}\frac{\partial^{2}\psi}{\partial v^{2}}
\end{array}$

Finally, we arrive at the solution for $\psi_{{}^{2}}$

$$\psi_{{}^{2}} = - \frac{1}{2}t_{}\Delta_{}\Delta_{2}J^{32}\text{sin}2v\frac{\partial\omega_{s}}{\partial J}\frac{\partial\psi_{0}}{\partial J}\text{cos}u \cdot e^{- 4\text{JD}\left( \frac{\partial\omega_{s}}{\partial J} \right)^{2}\left( t_{}^{3}3 + \frac{\left\lbrack t - t_{} \right\rbrack^{3}}{3} \right)}$$

We find an echo will occur as a dipole oscillation (note that the
three-wave resonance conditions must be met among the interacting
waves). We can find the dipolar response by constructing

$\left\langle \tau \right\rangle\begin{array}{r}
\text{echo}
\end{array}_{\text{dipole}} = \frac{1}{2}\int_{0}^{\infty}{\text{dJ}\int_{0}^{2\pi}{d\varphi\sqrt{J}}}\text{cos}\text{φψ}_{2}\left( J,\varphi,t,t_{} \right)$

Carrying out the φ integration, we find

$\left\langle \tau \right\rangle_{\text{echo}} = - \frac{t_{}\Delta_{}\Delta_{2}\pi}{2}\int_{0}^{\infty}\text{dJJ}^{2}\frac{\partial\omega_{s}}{\partial J}\frac{\partial\psi_{0}}{\partial J}e^{- \delta\left( t_{}^{3}3 + \frac{\left\lbrack t - t_{} \right\rbrack^{3}}{3} \right)} \cdot \text{sin}\omega_{s}\left( t - 2t_{} \right)$

Now let $t \simeq 2t_{}$;
$\delta = 4\text{JD}_{0}\left( \frac{\partial\omega_{s0}}{\partial J} \right)^{2}$

$\psi_{0} = \frac{1}{2{\pi J}_{0}}e^{- JJ_{0}}$ ;
$\omega_{s} = \omega_{s0} + \frac{\partial\omega_{s0}}{\partial J}J$

This finally gives

$\begin{array}{r}
\left\langle \tau \right\rangle_{\text{echo}} = \frac{t_{}\Delta_{}\Delta_{2}\frac{\partial\omega_{s0}}{\partial J}J_{0}}{4}\frac{2\alpha\left( \alpha^{2} - 3\beta^{2} \right)}{\left( \alpha^{2} + \beta^{2} \right)^{3}}\text{sin}\omega_{s0}\left( t - 2t_{} \right) \\
\alpha = 1 + \frac{8}{3}D_{0}\left( \frac{\partial\omega_{s0}}{\partial J} \right)^{2}J_{0}t_{}^{3} \\
\beta = \frac{\partial\omega_{s0}}{\partial J}J_{0}\left( t - 2t_{} \right)
\end{array}$

The echo envelope follows the following form

$\left\langle \tau \right\rangle\begin{array}{r}
\text{echo}
\end{array}_{\text{enveolope}} = \frac{\Delta_{}\Delta_{2}\frac{\partial\omega_{s0}}{\partial J}J_{0}}{2}t_{}\left( 1 + \frac{8}{3}D_{0}\left\lbrack \frac{\partial\omega_{s}}{\partial J} \right\rbrack^{2}J_{0}t_{}^{3} \right)^{- 3}$

![](media/image12.jpeg){width="4.875in" height="2.0097222222222224in"}

Since we have

$\begin{array}{r}
\frac{1}{\tau_{\text{diff}}} \simeq \frac{D_{0}}{J_{0}} \\
\frac{1}{\tau_{\text{decoherence}}} \simeq \frac{\partial\omega_{s0}}{\partial J}J_{0}
\end{array}$

We can finally arrive at the following relationship with measured
parameters of the echo envelope.

$\tau_{\text{diff}}\tau_{\text{decoh}}^{2} = \frac{\text{64}}{3}\left( \frac{9}{8} \right)^{9}\left( \frac{E_{\text{max}}}{s} \right)^{3}$

We note that transverse echoes have also been predicted (but never
observed), as well as a unique echo crossing transition, which is not
possible in any other medium. Echoes have proven to be a very useful
means of measuring weak diffusion processes in a beam.

![](media/image13.jpeg){width="6.003472222222222in"
height="4.009722222222222in"}

![](media/image14.jpeg){width="5.559722222222222in"
height="4.129861111111111in"}.

Ref: Assadi et. Al.

Summary of Lecture #7

1.  We have derived analytical formulas for one-dimensional longitudinal
    echoes in both bunched and unbunched beams.

2.  Echoes are seen to be related to Landau damping, as well as the
    three-wave interaction among various longitudinal modes of a beam.

3.  Due to weak diffusion processes, the information contained in the
    echo is dissipated slowly, permitting the collision rate to be
    directly measured from the echo decay rate. There is the possibility
    that higher-order correlations may be observed experimentally, owing
    to the extreme sensitivity of this technique.

4.  There are many unexplored applications of echoes in high-energy
    beams, including transverse echoes and transition echoes.

References:\
\
1. L.K. Spentzouris, Coherent Nonlinear Phenomena in

Unbunched Synchrotron Beams, Ph.D. Thesis,

Northwestern University, 1996

2.  E. Lifshitz and L. Pitaevskii, Phyiscal Kinetics, Pergamon Press,
    1981

3.  R.W. Gould, et. al., Phys. Rev. Lett., **19**, 5, 1967

4.  C.H. Su and C. Oberman, Phys. Rev. Lett, **20**, 9, 1968

5.  T. M. O'Neil and R. W. Gould, Phys. Fluids, **11**, 1, 1968

6.  L.K. Spenztouris, et. al., Phys. Rev. Lett, **76**, 4, 1996