Meep Tutorial/Optical forces

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Here we will demonstrate Meep's ability to compute classical forces using the Maxwell stress tensor (MST) as well as the eigenmode source feature which integrates the frequency-domain functionality of our other open-source tool MPB. Our example consists of two identical dielectric waveguides supporting the same mode while separated in space. Due to the symmetry and orientation of the waveguide profile, the two modes can be chosen to be either symmetric or anti-symmetric with respect to a mirror plane between them. As the two waveguides are brought closer and closer together, the individual guided modes interact and give rise to an optical gradient force that is transverse to the waveguide axis (this is to be contrasted with radiation pressure that involves momentum exchange between photons and is longitudinal in nature). An interesting phenomena that occurs is that the force can be tuned to be either attractive or repulsive depending on the relative phase of the two modes which we will demonstrate in this tutorial.

The optical gradient force arises from the evanescent coupling of the modes of two adjacent structures and can be computed using the following analytic expression:

F=-\frac{1}{\omega}\frac{d\omega}{ds}\Bigg\vert_\vec{k}U,

where ω is the eigenmode frequency of the coupled-waveguide system, s is the separation distance between the parallel waveguides, k is the conserved wave vector and U is the total energy of the electromagnetic fields. By convention, negative and positive values correspond to attractive and repulsive forces, respectively. This expression has been shown to be mathematically equivalent to the Maxwell stress tensor method and in this tutorial we will verify this result for the parallel-waveguide example. It is convenient to properly normalize the force to eliminate units and since we know that the total power transmitted through the waveguide is P = vgU / L (vg is the group velocity, L is the waveguide length and U is defined as before), we use the dimensionless quantity force per unit length and power (F / L)(ac / P) (a is an arbitrary unit length and c is the speed of light).

We can therefore compute the optical gradient force in two ways, one indirect and the other direct: 1) in MPB, we compute the eigenmode frequency and group velocity for a mode with a fixed propagating wavevector and then use a finite-difference scheme to evaluate the above expression and 2) in Meep, we compute the force directly using the MST as well as the power transmitted through the waveguide.

First, let us set up the two-dimensional computational cell (although since we are interested in a propagating mode of the structure which requires an axial wavevector, technically we will have a three-dimensional simulation)

 (set-param! resolution 20)
 (define-param nSi 3.45)
 (define Si (make medium (index nSi)))
 (define-param dpml 5.0)
 (define-param sx 20)
 (define-param sy 5)
 (set! geometry-lattice 
     (make lattice (size (+ sx (* 2 dpml)) (+ sy (* 2 dpml)) no-size)))
 (set! pml-layers (list (make pml (thickness dpml))))
 (define-param sw 1.0) ; waveguide width
 (define-param d 1.0)  ; waveguide separation distance
 (set! geometry (list 

(make block (center (* -0.5 (+ d sw)) 0) (size sw sw infinity) (material Si)) (make block (center (* 0.5 (+ d sw)) 0) (size sw sw infinity) (material Si))))

There are two mirror symmetries that we can exploit to reduce the simulation size by a factor of four.

 (define-param xodd? true)
 (define-param yodd? true)
 (set! symmetries (list 

(make mirror-sym (direction X) (phase (if xodd? -1 +1))) (make mirror-sym (direction Y) (phase (if yodd? -1 +1)))))

Next, we set the Bloch-periodic boundary condition in order to excite the specific guided mode of the waveguide system

 (define-param beta 0.5)
 (set! k-point (vector3 0 0 beta))

Since we do not know a-priori what the eigenmode frequency will be for the given separation distance, we first use a point-dipole source positioned in the middle of the waveguides with a Gaussian profile and then use harminv to determine the resonant mode frequency.

 (define-param fcen 0.2)
 (define-param df 0.1)
 (set! sources (list 

(make source (src (make gaussian-src (frequency fcen) (fwidth df)))

   (component Ey) (center (* -0.5 (+ d sw)) 0))

(make source (src (make gaussian-src (frequency fcen) (fwidth df)))

   (component Ey) (center (* 0.5 (+ d sw)) 0) (amplitude (if xodd? -1.0 1.0)))))
 (define-param runtime 300)
 (run-sources+ runtime 
     (after-sources (harminv Ey (vector3 (* 0.5 (+ d sw)) 0) fcen df)))
 (define f (harminv-freq-re (car harminv-results)))
 (print "freq:, " d ", " f "\n")

Once we have determined what the eigenmode frequency is, we can then use this value to precisely excite the mode of interest using the the eigenmode-source feature and also compute the force and the transmitted power only at this value.

 (reset-meep)
 (change-sources! (list
    (make eigenmode-source
      (src (make gaussian-src (frequency f) (fwidth df)))
      (component Ey)
      (size sw sw 0)
      (center (* -0.5 (+ d sw)) 0)
      (eig-kpoint k-point)
      (eig-match-freq? true)
      (eig-parity (if yodd? ODD-Y EVEN-Y)))
    (make eigenmode-source
      (src (make gaussian-src (frequency f) (fwidth df)))
      (component Ey)
      (size sw sw 0)
      (center (* 0.5 (+ d sw)) 0)
      (eig-kpoint k-point)
      (eig-match-freq? true)
      (eig-parity (if yodd? ODD-Y EVEN-Y))
      (amplitude (if xodd? -1.0 1.0)))))
 (define wvg-pwr (add-flux f 0 1
    (make flux-region (direction Z) (center 0 0) 
 	   (size (* 1.2 (+ (* 2 sw) d)) (* 1.2 sw) 0))))
 (define-param dpad 0.1)
 (define wvg-force (add-force f 0 1
   (make force-region (direction X) (weight +1.0) 

(center (- (* 0.5 d) dpad) 0) (size 0 sy))

   (make force-region (direction X) (weight -1.0) 

(center (+ (* 0.5 d) sw dpad) 0) (size 0 sy))))

 (run-sources+ runtime)
 (display-fluxes wvg-pwr)
 (display-forces wvg-force)


Normalized force per unit length and input power versus waveguide separation of two parallel waveguides.
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