Basic Linear Dynamics
---------------------

Define emittance =1, then for an upright ellipse:

\theta_o=\sqrt{1 \over \beta_o}
a_o = \sqrt{beta_o}

Drift
-----

after a drift of length z, the top of the ellipse has moved by $\zeta$:

\zeta=z \ \theta_o

The maximum width of the ellipse is the rms sum of $\zeta$ and $a_o$:

a'=\sqrt{a_o^2 + \zeta^2}

and thus

\beta'=a_o^2 + z^2 theta_o^2
\beta'=beta_o + {z^2 \over \beta_o}

Defining \alpha_o=0 at z=0, when the elipse is upright, and

\alpha'=\alpha_o +{z \over beta_o}

then

\beta'=beta_o (1 + \alpha'^2)


The angle $\eta'$ of the point of maximum size can be obtained by symmetry 
considerations:

{\eta' \over \theta_o}={\zeta' \over a'}

so

\eta' = {\zeta' \theta_o \over a'}

The ratio $\kappa'$ of the angle $\eta'$ to its amplidude a' is a measure of 
the divergence of the beam. It gives, for instance, the strength of a focus 
system that would set the ellipse upright again: 

\kappa' = {\eta' \over a'}={\theta_o^2 z \over (a')^2}={\alpha' \over \beta'}

Thin Lens
---------

At a thin lens, the size a', and thus $\beta'$, cannot change, but the angles 
change in proportion to the focus strength $k$ times the amplitude a'. For a 
paricle at the maximum amplitude $a'$

\eta''=\eta' + k \ a'
\kappa'' = \kappa' + k
\alpha'' = \alpha' +  {k \beta'}
\beta'' = \beta'


Another Drift
-------------

As before, we can give $\beta$ in the following drift by:

\beta'''= beta'''_o (1 + (\alpha''')^2)

where

\alpha'''=\alpha'' + {z \over beta'''_o}

and $\beta'''_o$ is the beta at the following waist, if focussed, or at an 
imaginary waist before the thin lens; it can be obtained from the values of 
$\alpha''$ and $\beta''$ after the thin lens:

\beta'''_o = {\beta'' \over (1 + (\alpha'')^2)}


Momentum Dependences
--------------------

At the end of a matched solenoid, i.e. one in which the beta function remains 
constant,

\beta_o (solenoid) \propto p

but at the end of a matched lithium lens:

\beta_o (lens) \propto \sqrt{p}

The strength of a thin solenoid:

k_{solenoid} \propto p^2

But for a thin lithium lens or horn:

k_{lens} \propto p


Matching
--------

Using the above formulae and momentum dependences we can ask if solutions are 
possible where we go from one matche slement, such as a solenoid, into 
another, in such a way that matching is achieved both on momentum, and for 
first order deviations from that momentum. I have not solved th equations 
explicitly, but there seems no fundamental law that prohibits this, even in 
the absence of chromatic correction elements (e.g. sextupoles in dispersion). 
We are NOT correcting chromaticity. The phase advance through the system will 
NOT be independent of momentum. But this we do not require. 

EXAMPLES
========

I will consider a number of examples, the first 3 of which are motivated by 
the need for a match between alternating solenoids in a cooling channel. Where 
the field reverses, it must go to zero at one point, and be low over a 
distance comparable with the beam pipe diameter. So in the following examples, 
I approximate this situation by requiring a finite gap between a central pair 
of thin solenoids. 

I am given the initial $\beta$, initial ${d \beta/ dp}$, $\alpha=0$ and 
$d\alpha/dp=0$; and require all four to have the same value at the entry to 
the following solenoid: 4 constraints.

But ignoring the energy changes in the hydrogen and 
reaccelerators, I can assume symmetry about the middle of the gap. 
In this case the matching requirement reduces to just two 
requirements at the symmetry plane: 
        

\alpha= 0

{d \alpha \over dp} = 0


0) Simple solenoid-solenoid 
---------------------------

In the first example place a thin lens at the exit of the solenoid, 
followed by the gap. 

In the solenoid, alpha=0 and the $\beta$'s are constant. 
The thin lens drives $\alpha$ negative and the $\beta$ falls to a minimum and 
then rises to meat the thin lens at the entrance to the opposite solenoid. 
For a fixed gap I have only one variable (the thin solenoid strength). These
thin solenoid strengths can be chosen to put the minimum at the center for one 
momentum, and at this momentum a good match is achieved. But {d \alpha \over 
dp} can not be set to zero, and at other momenta the match is bad. 

see figures Match0

1) Simple solenoid after a gap
------------------------------

Again I match from one long solenoid to another, but now with gaps between the 
end of the solenoids and the thin lenses (two thin lenses and three gaps 
altogether)

Now I have two variables (the first gaps and the thin solenoid strengths, and 
should thus be able to satisfy both constraints.
In fact, I looked for, and found, solutions with perfect matching at two 
separated momenta - probably a best solution for wide momentum acceptance. 
Clearly I could have satisfied the two other constraints.

See figures Match1 

  a: tracings at relative momenta of  .6 to 1.4;  

  b: $\alpha$ vs. relative p

2) solenoid-solenoid with tapers
--------------------------------

The above solution is nice because it tells us that solutions are possible, 
but the length of the match and is constrained and would not accept the 
required acceleration.

So in the second example I keep the symmetry and again match between solenoids, 
but I introduce a sequence of lenses to allow the $\beta$ to grow to a much 
larger value at the symmetry point. This is close to the solution required for 
alternating solenoids.

Again, I looked for, and found, solutions with perfect matching at two 
separated momenta.

See Figures Match2

    ---------------------------------------------------
The second example will be the same, but will match between lithium lenses