Coupled mode theory
Given an optical resonator (e.g. a photonic crystal cavity),
consider one of its quasinormal modes
with frequency and decay rate .
Its complex amplitude is governed by:
We choose to normalize so that
is the total energy inside the resonator at time .
Suppose that waveguides are now “connected” to this resonator,
meaning that the resonator mode and the outgoing waveguide mode
overlap sufficiently for to leak into at a rate .
Conversely, the incoming mode brings energy to .
Therefore, we can write up the following general set of equations:
Where and are unknown coupling constants,
and represents reflection.
so that is the power flowing towards at time ,
and likewise for .
Note that we have made a subtle approximation here:
by adding new damping mechanisms,
we are in fact modifying ;
see the harmonic oscillator for a demonstration.
However, the frequency shift is second-order in the decay rate,
so by assuming that all are large,
we only need to keep the first-order terms, as we did.
This is called weak coupling.
If we also assume that is large
(its effect is already included in ),
then we can treat the decay mechanisms separately:
to analyze the decay into a certain waveguide ,
it is first-order accurate to neglect all other waveguides and :
To determine , we use energy conservation.
If all ,
then the energy in decays as:
Since all other mechanisms are neglected,
all this energy must go into , meaning:
Taking the square root, we clearly see that .
Because the phase of is arbitrarily defined,
need not be complex, so we choose .
Next, to find , we exploit the time-reversal symmetry
of Maxwell’s equations,
which govern the light in the resonator and the waveguides.
In the above calculation of , evolved as follows,
with the lost energy ending up in :
After reversing time, evolves like so,
where we have taken the complex conjugate
to preserve the meanings of the symbols
, , and :
We insert this expression for into its original differential equation, yielding:
Isolating this for leads us to the following power balance equation:
But thanks to energy conservation,
all power delivered by ends up in , so we know:
To reconcile the two equations above,
we need .
Discarding the phase thanks to our choice of ,
we conclude that .
Finally, can also be determined using energy conservation.
Again using our weak coupling assumption,
if energy is only entering and leaving through waveguide , we have:
Meanwhile, using the differential equation for ,
we find the following relation:
By isolating both of the above relations for
and equating them, we arrive at:
We insert the definition of and ,
and use :
This equation is only satisfied if .
Combined with ,
the coupled-mode equations take the following form:
By connecting multiple resonators with waveguides,
optical networks can be created,
whose dynamics are described by these equations.
The coupled-mode equations are extremely general,
since we have only used weak coupling,
conservation of energy, and time-reversal symmetry.
Even if the decay rates are quite large,
coupled mode theory still tends to give qualitatively correct answers.
- H.A. Haus,
Waves and fields in optoelectronics,
- J.D. Joannopoulos, S.G. Johnson, J.N. Winn, R.D. Meade,
Photonic crystals: molding the flow of light,
2nd edition, Princeton.