10.86 RFTM110—Tracks through a TM110-mode (deflecting) rf cavity with all magnetic and electric field components. NOT RECOMMENDED—See below.

Tracks through a TM110-mode (deflecting) rf cavity with all magnetic and electric field components. NOT RECOMMENDED—See below.
Parallel capable? : yes
GPU capable? : no
Back-tracking capable? : no






Parameter Name UnitsType Default

Description






PHASE DEGdouble 0.0

phase






TILT RADdouble 0.0

rotation about longitudinal axis






FREQUENCY HZ double 2856000000

frequency






VOLTAGE V double 0.0

peak deflecting voltage






PHASE_REFERENCE long 0

phase reference number (to link with other time-dependent elements)






VOLTAGE_WAVEFORM STRINGNULL

<filename>=<x>+<y> form specification of input file giving voltage waveform factor vs time






VOLTAGE_PERIODIC short 0

If non-zero, voltage waveform is periodic with period given by time span.






ALIGN_WAVEFORMS short 0

If non-zero, waveforms’ t=0 is aligned with first bunch arrival time.






VOLTAGE_NOISE double 0.0

Rms fractional noise level for voltage.






PHASE_NOISE DEGdouble 0.0

Rms noise level for phase.






GROUP_VOLTAGE_NOISE double 0.0

Rms fractional noise level for voltage linked to group.






GROUP_PHASE_NOISE DEGdouble 0.0

Rms noise level for phase linked to group.






VOLTAGE_NOISE_GROUP long 0

Group number for voltage noise.






PHASE_NOISE_GROUP long 0

Group number for phase noise.






START_PASS long -1

If non-negative, pass on which to start modeling cavity.






END_PASS long -1

If non-negative, pass on which to end modeling cavity.






GROUP string NULL

Optionally used to assign an element to a group, with a user-defined name. Group names will appear in the parameter output file in the column ElementGroup






NB: Although this element is correct insofar as it uses the fields for a pure TM110 mode, it is recommended that the RFDF element be used instead. In a real deflecting cavity with entrance and exit tubes, the deflecting mode is a hybrid TE/TM mode, in which the deflection has no dependence on the radial coordinate.

To derive the field expansion, we start with some results from Jackson[17], section 8.7. The longitudinal electric field for a TM mode is just

                     (    )
Ez = - 2iE0Ψ (ρ,ϕ)cos  pπz- e-iωt,
                        d
(137)

where p is an integer, d is the length of the cavity, and we use cylindrical coordinates (ρ,ϕ,z). The factor of -2i represents a choice of sign and phase convention. We are interested in the TM110 mode, so we set p = 0. In this case, we have

Ex = Ey = 0
(138)

and (using CGS units)

⃗         iϵω        - iωt
H = - 2iE0ck2zˆ× ∇ Ψe    .
(139)

For a cylindrical cavity, the function Ψ for the m = 1 aximuthal mode is

Ψ (ρ,ϕ) = J1(k ρ)cosϕ,
(140)

where k = x11∕R, x11 is the first zero of J1(x), and R is the cavity radius. We don’t need to know the cavity radius, since k = ω∕c, where ω is the resonant frequency. By choosing cosϕ for the aximuthal dependence, we’ll get a magnetic field primarily in the vertical direction.

In MKS units, the magnetic field is

             (                             )
⃗    2E0-- iωt   J1(kρ)       ˆ     ∂J1-(kρ)
B =  kc e      ˆρ  ρ    sinϕ + ϕ cosϕ   ∂ρ     .
(141)

Using mathematica, we expanded these expressions to sixth order in k *ρ. Here, we present only the expressions to second order. Taking the real parts only, we now have

  Ez  ≈  E0k(ρcos ϕsinω)t                                  (142)
                  k2ρ2-
 cB ρ  ≈  E0   1-   8    sin ϕcosωt                         (143)
            (       2 2)
cB ϕ  ≈  E0   1-  3k-ρ-- cosϕcos ωt                       (144)
                    8
The Cartesian components of ⃗B can be computed easily
cBx   =  cB ρcosϕ - cBϕ sin ϕ                                 (145)
          E0-2 2
      =   4 ρ k  cosϕsinϕ cosωt                              (146)
cBy   =  cB ρsin ϕ + cBϕ cosϕ                                 (147)
            (      2 2    2       )
      =  E0   1- k--ρ(2-cos-ϕ+--1) cos ωt                    (148)
                         8

The Lorentz force on an electron is F = -eEz - ecβ⃗ ×B⃗, giving

Fx ∕e  =  βzcBy                                         (149)
 F ∕e  =  - β cB                                        (150)
  y          z  x
 Fz∕e  =  - Ez - βxcBy + βycBx                          (151)
We see that for ρ 0, we have Ez = 0, Bx = 0, and
cBy = E0 cosωt.
(152)

Hence, for ωt = 0 and E0 > 0 we have Fx > 0. This explains our choice of sign and phase convention above. Indeed, owing to the factor of 2, we have a peak deflection of eE0L∕E, where L is the cavity length and E the beam energy. Thus, if V = E0L is specified in volts, and the beam energy expressed in electron volts, the deflection is simply the ratio of the two. As a result, we’ve chosen to parametrize the deflection strength simply by referring to the “deflecting voltage,” V .

Explanation of <filename>=<x>+<y> format: Several elements in elegant make use of data from external files to provide input waveforms. The external files are SDDS files, which may have many columns. In order to provide a convenient way to specify both the filename and the columns to use, we frequently employ <filename>=<x>+<y> format for the parameter value. For example, if the parameter value is waveform.sdds=t+A, then it means that columns t and A will be taken from file waveform.sdds. The first column is always the independent variable (e.g., time, position, or frequency), while the second column is the dependent quantity.

RFTMEZ0