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


PHASE DEGdouble 0.0


TILT RADdouble 0.0

rotation about longitudinal axis

FREQUENCY HZ double 2856000000


VOLTAGE V double 0.0

peak deflecting voltage


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


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


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


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

VOLTAGE_NOISE double 0.0

Rms fractional noise level for voltage.


Rms noise level for phase.


Rms fractional noise level for voltage linked to group.


Rms noise level for phase linked to group.


Group number for voltage noise.


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.


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,

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

and (using CGS units)

⃗         iϵω        - iωt
H = - 2iE0ck2zˆ× ∇ Ψe    .

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

Ψ (ρ,ϕ) = J1(k ρ)cosϕ,

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     .

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)
 cB ρ  ≈  E0   1-   8    sin ϕcosωt                         (143)
            (       2 2)
cB ϕ  ≈  E0   1-  3k-ρ-- cosϕcos ωt                       (144)
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)

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.

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.