Eigenmode Source

This example demonstrates using the eigenmode-source to couple exclusively to a single waveguide mode. The structure, shown in the schematic below, is a 2d dielectric waveguide with ε=12, width =1 μm, and out-of-plane electric field Ez. The dispersion relation ω(k) for index-guided modes with even mirror symmetry in the y-direction is computed using MPB and shown in blue. The light cone which denotes radiative modes is the green section. Using this waveguide configuration, we will investigate two different frequency regimes: (1) single mode (normalized frequency of 0.15) and (2) multi mode (normalized frequency of 0.35), both shown as dotted lines in the figure. We will use the eigenmode source to excite a specific mode in each case — labeled A and B in the band diagram — and compare the results to using a constant-amplitude source for straight and rotated waveguides. Finally, we will demonstrate that a monitor plane in the y-direction is sufficient for computing the Poynting flux in a waveguide with arbitrary orientation.

The simulation script is in examples/oblique-source.py.

The simulation consists of two parts: (1) computing the flux, and (2) plotting the field profile. The field profile is generated via compute-flux=false. For the single-mode case, a constant-amplitude current source (eig-src=false) excites both the waveguide mode and radiating fields in both directions. This is shown in the main inset of the figure above. The eigenmode-source excites only the right-going waveguide mode A as shown in the smaller inset. Exciting this mode requires setting eig-src=true, fsrc=0.15, kx=0.4, and bnum=1. Note that eigenmode-source is a line centered at the origin extending the length of the entire cell. The constant-amplitude source is a line that is slightly larger than the waveguide width. The parameter rot-angle specifies the rotation angle of the waveguide axis and is initially 0 (i.e., straight or horizontal orientation). This enables eig-parity to include EVEN-Y in addition to ODD-Z and the cell to include a mirror symmetry plane in the y-direction.

For the multi-mode case, a constant-amplitude current source excites a superposition of the two waveguide modes in addition to the radiating field. This is again shown in the main inset of the figure above. The eigenmode-source excites only a given mode: A (fsrc=0.35, kx=0.4, bnum=2) or B (fsrc=0.35, kx=1.2, bnum=1) as shown in the smaller insets.

(set-param! resolution 50) ; pixels/μm

(set! geometry-lattice (make lattice (size 14 14 no-size)))

(set! pml-layers (list (make pml (thickness 2))))

(define-param rot-angle 0) ; rotation angle (in degrees) of waveguide, CCW around z-axis
(set! rot-angle (deg->rad rot-angle))

(set! geometry (list (make block
                       (center 0 0 0)
                       (size infinity 1 infinity)
                       (e1 (rotate-vector3 (vector3 0 0 1) rot-angle (vector3 1 0 0)))
                       (e2 (rotate-vector3 (vector3 0 0 1) rot-angle (vector3 0 1 0)))
                       (material (make medium (epsilon 12))))))

(define-param fsrc 0.15) ; frequency of eigenmode or constant-amplitude source
(define-param kx 0.4)    ; initial guess for wavevector in x-direction of eigenmode
(define-param bnum 1)    ; band number of eigenmode

(define-param compute-flux? true) ; compute flux (true) or output the field profile (false)

(define-param eig-src? true)      ; eigenmode (true) or constant-amplitude (false) source

(set! sources (list
               (if eig-src?
                   (make eigenmode-source
                     (src (if compute-flux? (make gaussian-src (frequency fsrc) (fwidth (* 0.2 fsrc))) (make continuous-src (frequency fsrc))))
                     (center 0 0 0)
                     (size 0 14 0)
                     (direction (if (= rot-angle 0) AUTOMATIC NO-DIRECTION))
                     (eig-kpoint (rotate-vector3 (vector3 0 0 1) rot-angle (vector3 kx 0 0)))
                     (eig-band bnum)
                     (eig-parity (if (= rot-angle 0) (+ EVEN-Y ODD-Z) ODD-Z))
                     (eig-match-freq? true))
                   (make source
                     (src (if compute-flux? (make gaussian-src (frequency fsrc) (fwidth (* 0.2 fsrc))) (make continuous-src (frequency fsrc))))
                     (center 0 0 0)
                     (size 0 2 0)
                     (component Ez)))))

(if (= rot-angle 0)
    (set! symmetries (list (make mirror-sym (direction Y)))))

(if compute-flux?
    (let ((tran (add-flux fsrc 0 1 (make flux-region (center 5 0 0) (size 0 14 0)))))
      (run-sources+ 50)
      (display-fluxes tran))
    (run-until 100 (in-volume (volume (center 0 0 0) (size 10 10 0))
                              (at-beginning output-epsilon)
                              (at-end output-efield-z))))

Note that in the eigenmode source, direction must be set to NO-DIRECTION for a non-zero eig-kpoint which specifies the waveguide axis.

Additionally, we can demonstrate the eigenmode source for a rotated waveguide. The results are shown in the two figures below for the single- and multi-mode case. There is one subtlety: for mode A in the multi-mode case, the bnum parameter is set to 3 rather than 2. This is because a non-zero rotation angle breaks the symmetry in the y-direction which therefore precludes the use of EVEN-Y in eig-parity. Without any parity specified for the y-direction, the second band corresponds to odd modes. This is why we must select the third band which contains even modes. An oblique waveguide also leads to a breakdown in the PML. A simple workaround for mitigating reflections from the boundary layers is to increase its length: the PML length has been doubled from 1 to 2.

There are numerical dispersion artifacts due to the FDTD spatial and temporal discretizations which create negligible backwards (leftward-propagating) wave artifacts by the eigenmode current source, carrying approximately 10-5 of the power of the desired rightward-propagating mode. These can be seen as artifacts in the field profiles.

Finally, we demonstrate that the power flux through a waveguide with an arbitrary orientation can be computed by a single flux plane oriented along the y direction: thanks to Poynting's theorem, the flux through any plane crossing a lossless waveguide is the same, regardless of whether the plane is oriented perpendicular to the waveguide. Furthermore, the eigenmode source is normalized in such a way as to produce the same power regardless of the waveguide orientation — in consequence, the flux values for mode A of the single-mode case for rotation angles of 0°, 20°, and 40° are 77.000266, 76.879339, and 76.817850, within 0.2% (discretization error) of one another.