Side Coupling with Doubly Degenerate Modes

Cavity

4.4 Critical Coupling in Coupled Waveguide-Resonator Systems

4.4.2 Side Coupling with Doubly Degenerate Modes

or

re.

In reality, it is difficult to fabricate a dielectric structure with perfrct mirror reflection symmetry and there will always be some small difference between

r~

^and

f::_. \Vith symmetry broken, the reflection coefficient R and transmission coeflicient Tare

, 4p·

re

R

=

lr!2

= -

+

~W2 +([Cl+ f':_ + [~~)2

• ~(.(}2 - -

(ro + re -- re

)2

T

=

ftl2

=

~w2 + (ro +

r~

+ 1<)2 .

(4.62)

These results show that the general reflection and transmission features of the system is not significantly changed. v\'e can still achieve zero resonant (,0,.w

=

0) transmission

by

^tuning

r

⁰

= r: - r::_.

The unity transmission can also be achieved by dl(Josing

r

⁰

= -r::_.

lk>

..

le>

x=O lk>

...

IO>

x=O

X

(a)

10 E

""

·u ~ 8

"'

u 0 C 0 6

·;;;

<fl

.E

<fl C: 4

@ f-

E "' ^C: ₀ 2

<fl

"'

cc

0 0 2

(c)

I-k>

... .J

1-k>

...

8 10

10

E 8

"'

·u ~

8 ⁶

C:

·;;; 0

-~ 4

<fl C:

@ f- 2

lk>

(b)

1-k>

r⁰/r⁰ = -0.!

f'O/rc = 0 f'O/rc = 1.0

r⁰Jr⁰ = 5.0

OL_ _ _ _

----=-::,,,,..~~=---~

-5 0 5

LI.wire

(d)

Figure 4.4: (a) A waveguide side coupled with a cavity supporting two degenerate modes. The cavity is symmetric with respect to x

=

0 plane and support two degen- erate modes with opposite parity under mirror reflection. The even mode is le) and the odd mode is lo). (b) The side coupling geometry with two degenerate traveling wave modes in the cavity. The mode traveling in the clockwise direction is

I+)

and the mode traveling in the counterclockwise direction is

I-). (

c) The resonant trans- mission coefficient as a function of

r

⁰

/re. (

d) The transmission spectrum for different values of

r

⁰

/re.

(1.67) where

Fi.c

represents the coupling between the incident wave ik) and the even cavity mode le), and 11.:,o represents the coupling between lk) and the odd cavity mode Jo).

Following Ref. [75] and [76], vve assume that the waveguide mode jk) couples equally strong with the even mode and the odd mode, i.e.

(4.68) Consequently, from Eq. (4.66) we obtain

(-1.69) where

re

^{is p:}

= Lll1.:,el

²

/v

₉^. \Ve notice the remarkable· result that the reflection coefficient R remains O for all the frequencies. This is a direct consequence of the destructive interference between the reflected waves due to the two degenerate cavity modes, as was pointed out in Ref. [75]. In fact, this side coupling geometry can be regarded as half of the photonic crystal add-drop filters studied in Rd. [7i:i] to [77].

Here the coupling to the second waveguid<' is represented by the '·intrinsic'' cavity decay rate

r

^0.

In addition to the condition of frequency degeneracy and equal mode decay rate, Eq. ( 4.68) must also be strictly satisfied to eliminate reflection. It is very difficult to simultmwously realize these requirements dming the fabrication processes. In practice, it is easier to fabricate semiconductor ring or disk resonators [84]-[86] and dielectric microspheres [80]-[8:3], which support two counter-propagating modes, as shown in Fig. ( 4.4b). In the• follmving analysis, we shmv that thE' reflection and transmission ccwfficients of a waveguidP coupled to this type of resonators are also described by Eq. ( 4.69).

If thP waveguide mode and the traveling wave mode in the resonator are phase- matched, it is safe to assume that the waw~guide mode can only induce the travding

wave circulating in one direction, due to the requirement of phase matching. As shown in Fig. ( 4.4b), we denote the clockwise circulating mode as

I+),

and the coun- terclockwise mode as 1-). Csing these notations, the condition fcfr phaHe-matched coupling iH li,-

=

0, and l'_k,+

=

0. Furthermore, using Eq. (4.7) and the time reversal symmetry, we find

h .. =

V:k,-. With these conditions, from Eq. (¹1.66) we have

(4.71) (4.72) The above results arc the same as Eq. (4.69).

In Fig. (4.4c), the resonant (.6.w

=

0) transmission coefficient was plotted as a funct.ion of

r

⁰

/re.

:\"oticc that at

r

⁰

/re =

^1, T is always equal to zero. This phenomenon is the principle behind many add-drop filters studied in the literature [75]-'.77:, [82]-[86;, and was named "critical coupling" in Ref. '.87]. The transmission spectrum is shown in Fig. (4.4d) for different values of

r

⁰ /P. \Ve notice that when the lasing threshold is approached (f⁰

/rc---+

-1), the optical wave is amplified and the resonance width is narrowed.

The ''critical coupling'· condition depends on the assumption that the waveg- uide mode couples to only one of the traveling wave modes, and the two counter- propagating traveling wave modes are independent. Under these t,vo conditions, the transmitted wave is the supperposition of two scattering waves: One due to the direct scattering -where the photons propagatC' through the waveguide without interacting with the resonator, the other due to the indirect scattering where thC' photons are first coupled into the resonator, travel in the resonator while experiencing loss or gain, and arc coupled back into the waveguide. At ~w

=

0 and

r

⁰

=

P, the scattering am- plitudes due to the direct scattering and the indirect scattering have equal amplitude but opposite sign, and cancel each other exactly.

In reality, the surface of the dielectric rnicroring, microdisk. or rnicrospheres would

not be ideally smooth and can cause coupling between the counter-propagating cavity modes [94, 95]. It is also conceivable that a Bragg grating is deliberately introduced into the resonator to couple the two counter-propagating modes together. In either scenario, it becomes essential to take the coupling between the resonator modes into account. Using the expression in Eq. (4.30), we can introduce a phenomenological parameter r,, such that the :E matrix can be written as

( 4. 73)

where

w

representing the difference between the "bare" resonant frequency and the

"renormalized" resonant frequency. In reaching Eq. (4.73), we also keep the as::mmp- tion of

Vi,+=

v_:k,-l

Vi,-= v-k,+ =

0, and

re

is given by Eq. (4.72). With the relative phase of the two counter-propagating modes carefully chosen, r,, can be assumed to be a positive number. To diagonalize the :E matrix, we use a new representation

( 4.74)

12)= ~[1-)-1+)]

^(4.75)

In this representation, the :E matrix becomes

(4.76)

From this expression and Eq. (4.29), we can read out the resonant frequencies of mode