Tutorial 2: couplers and waveguide ports

Real cavities have couplers. A coupler turns the lossless eigenproblem into a complex one: energy leaves through the port, the eigenfrequency acquires an imaginary part, and the external Q follows. This tutorial adds a coax coupler and then a waveguide coupler to the pillbox from Tutorial 1, and ends with a two-port case that requires the nonlinear eigensolver.

Part A: coax coupler

Example directory: examples/omega3p/pillbox-rtop+coax

Coax dimension	Value
Inner conductor radius	8.7 mm
Outer conductor radius	20 mm
Axial position	z = 100 mm
Inner conductor tip	30 mm from beam axis
Tip rounding	5 mm
Outer-to-beampipe rounding	7 mm

Build in steps

Complex models are easiest to build incrementally. Reuse the full pillbox saved in Tutorial 1, add the coax, then cut to a half model. Only one symmetry plane survives because the coupler breaks the other.

step2-ADD-coax.jou

reset
create Cylinder height 120 radius 20
create Cylinder height 120 radius 8.7
body 1 2 rotate 90 about y
body 1 2 move -60 0 100
body 2 move -30 0 0
import acis "pillbox-rtop0.sat"
unite body 1 3
compress ids
subtract body 2 from 1
compress ids
modify curve 10 blend radius 7
modify curve 12 blend radius 5
export acis "pillbox-rtop+coax.sat" overwrite
webcut volume 1 with plane yplane offset 0
delete body 1
compress ids
export acis "pillbox-rtop+coax2.sat" overwrite

Mesh with a port sideset

step3-Mesh-pillbox-rtop+coax.jou

reset
import acis "pillbox-rtop+coax2.sat"
volume all scheme Tetmesh
volume all sizing function type skeleton scale 7 time_accuracy_level 2 min_size auto max_size 15 max_gradient 1.5
mesh volume all
Sideset 2 surface 1            # symmetry plane
Sideset 3 surface 3            # beam pipe end
Sideset 4 surface 2            # beam pipe end
Sideset 7 surface 17           # coax port
Sideset 6 surface all except 1 3 2 17
block 1 volume all
block 1 element type tetra10
volume all scale 0.001
export Genesis "pillbox-rtop+coax2.gen" block all overwrite

The coax end surface gets its own sideset, 7, so it can be declared a port. The beam pipe ends are 3 and 4.

Input with a port

pillbox-rtop+coax.omega3p

ModelInfo : {
  File: ./pillbox-rtop+coax2.ncdf
  BoundaryCondition : {
    Magnetic:  2
    Electric:  3, 4          // beam pipe ends, below cutoff
    Exterior:  6
    Waveguide: 7             // coax port
  }
  SurfaceMaterial : {
    ReferenceNumber: 6
    Sigma: 5.8e7
  }
}

FiniteElement: {
  Order:           2
  CurvedSurfaces: on
  Tolerance: 1e-7
}

EigenSolver : {
  NumEigenvalues: 1
  FrequencyShift:  1.0e9
}

Port : {
  ReferenceNumber : 7
  NumberOfModes : 1          // the coax TEM mode
}

PostProcess : {
  Toggle: on
  ModeFile: mode
}

The beam pipe ends are closed with Electric. That is fine here because the pipe is below cutoff at 1.3 GHz, so the field there has decayed to nothing. The coax port is Waveguide with one mode loaded: a coax has a TEM mode with zero cutoff, and nothing else propagates at this frequency.

Run and read

Convert, run, and post-process exactly as in Tutorial 1. The mode summary now has a complex frequency and an external Q. From the example's stored results:

Mode : {
    TotalEnergy : 4.4270939088102e-12 , 0
    ExternalQ : 1024235.9558753
    QualityFactor : 28815.235456204
    File : omega3p_results/omega3p.l0.m0000.R1.3137561e+09.I6.4133469e+02.mod
    PowerLoss : 1.2682100135575e-06
    Frequency : 1313756106.8639 , 641.33469408486
}

ExternalQ is the coupler Q_ext, here about 1.0 million: with the inner conductor tip 30 mm from the axis, this coax barely couples to TM₀₁₀. QualityFactor is still the wall Q₀, essentially unchanged from the closed cavity. The loaded Q is \(1/Q_L = 1/Q_0 + 1/Q_{ext}\), and the imaginary part of the frequency is \(f_r / 2 Q_L\). Pushing the probe deeper toward the axis lowers Q_ext rapidly; the hands-on exercise in Part B does the equivalent for a waveguide iris.

Port modes in ParaView

Omega3P writes the 2D port mode it computed on the port surface to omega3p_results/port7/mode_1.00000GHz_000.vtu. Load the 3D mesh with the SLAC toolbar, then File, Open the .vtu file and color by Ereal. Switch to wireframe if the port surface hides behind the wall.

Part B: rectangular waveguide coupler

Example directory: examples/omega3p/pillbox+recWG

A waveguide coupler attaches a rectangular guide to the cavity through an iris. The journal step2-ADD-recWG-to-pillbox.jou parameterizes the guide width wguide, height hguide, iris width wiris, and iris depth diris, builds the guide as two bricks, unites with the full pillbox, rounds the iris corners, and cuts to a quarter model. The waveguide end becomes sideset 7.

The input is the same shape as the coax case:

BoundaryCondition : {
  Magnetic:  1, 2
  Electric:  3, 4
  Exterior:  6
  Waveguide: 7
}
Port : {
  ReferenceNumber : 7
  NumberOfModes : 1          // TE10 of the rectangular guide
}

A rectangular guide, unlike a coax, has a nonzero cutoff. Omega3P computes the guide's modes on the port surface numerically and writes them to 2dports.log. Check that the TE₁₀ cutoff is below the cavity frequency, otherwise the port is not actually propagating.

Absorbing boundary alternative

If modal accuracy at the port does not matter, a first-order absorbing boundary is simpler:

BoundaryCondition : {
  ...
  Absorbing: 7
}
Port : {
  ReferenceNumber : 7
  CutoffFrequency : 1e9      // phase velocity from this cutoff
}

Without a Port block, the absorbing boundary assumes a phase velocity of \(c\). With CutoffFrequency, it uses the guide's phase velocity at the eigenfrequency. The absorbing boundary is less accurate than a waveguide port with modal expansion and should be reserved for beam-pipe truncation.

Hands-on

Change wiris in the journal, remesh, rerun, and find the iris width that gives Q_ext of about 4000. Two or three iterations usually bracket it.

Part C: two ports with different cutoffs

Example directory: examples/omega3p/pillbox+recWG+cirWG

Add a circular waveguide on the beam axis to the Part B model. Now there are two ports, each with several modes near the cavity frequency and with different cutoffs. The waveguide boundary term depends on frequency through \(\sqrt{\omega^2 - \omega_c^2}\) for each loaded mode, and with more than one distinct cutoff the eigenproblem is no longer quadratic. Omega3P solves it with CORK, a rational Krylov method.

in1.omega3p (EigenSolver and Port blocks)

EigenSolver : {
  NumEigenvalues:  3
  Tolerance:       1e-10
  FrequencyShift:  1.0e9
  SolverMode:      CORK
  CORK : {
    TargetFrequencyMin:   1.0e9    // search window, real part
    TargetFrequencyMax:   2.4e9
    TargetFrequencyImag:  5.0e8    // search window, imaginary part
    MaxIterations:        500
    ApproxTolerance:      1e-12    // rational approximation accuracy
    ApproxMaxDegree:      100
  }
}

Port : {
  ReferenceNumber : 4            // circular guide
  NumberOfModes : 10
}
Port : {
  ReferenceNumber : 7            // rectangular guide
  NumberOfModes : 10
}

CORK searches a rectangle in the complex frequency plane rather than expanding around a single shift, so you give it a window instead of a target. Start by loading generously many port modes, here 10 per port, so no propagating mode is missed.

Check the port modes

After the run, 2dports.log lists each port's modes with cutoff frequency and type. For this geometry:

Mode	Port 4 (circular)	Port 7 (rectangular)
1	TM 1.043 GHz	TE 0.908 GHz
2	TE 1.325 GHz	TE 1.897 GHz
3	TM 2.228 GHz	TM 1.897 GHz
4	TE 2.307 GHz	TE 2.726 GHz
5	TM 2.395 GHz	TE 3.194 GHz

The first run found three eigenmodes at 1.176, 2.062, and 2.197 GHz. Compare these against the cutoffs. Port 4 has TM modes with cutoffs at 2.228 and 2.395 GHz, very close to eigenmodes 2 and 3. A loaded mode whose cutoff sits almost on top of an eigenfrequency has \(\sqrt{\omega^2-\omega_c^2}\) near zero, which makes the rational approximation poorly conditioned and contaminates the solution.

Rerun with only the propagating modes

Load exactly the modes that propagate at the eigenfrequencies of interest:

in2.omega3p

Port : {
  ReferenceNumber : 4
  NumberOfModes : 2
}
Port : {
  ReferenceNumber : 7
  NumberOfModes : 3
}

Mode	in1 f (GHz)	in1 Qext	in2 f (GHz)	in2 Qext
1	1.176	267	1.176	271
2	2.062	496	2.057	273
3	2.197	749	2.196	693

Mode 1 is unaffected since no cutoff is near it. Modes 2 and 3 change noticeably, and the second run is the trustworthy one. The lesson generalizes: load all propagating port modes and no evanescent mode whose cutoff is close to an eigenfrequency. Run once with many modes to learn the cutoffs, then run again with the right count.

Complex electric fields of the three modes from the second run.