Methods - DIRAC SOLITONS IN OPTICAL MICRORESONATORS

DIRAC SOLITONS IN OPTICAL MICRORESONATORS

6.7 Methods

Solving the conservative coupled LLE

We copy the conservative coupled LLE here for convenience:

𝜕 𝐸₁

𝜕 𝑡

=−𝑖 𝛿𝜔 𝐸₁+𝑖𝑔_c𝐸₂−𝛿 𝐷₁

𝜕 𝐸₁

𝜕 𝜃

+𝑖(𝑔₁₁|𝐸₁|²𝐸₁+𝑔₁₂|𝐸₂|²𝐸₁), (6.11)

𝜕 𝐸₂

𝜕 𝑡

=−𝑖 𝛿𝜔 𝐸₂+𝑖𝑔_c𝐸₁+𝛿 𝐷₁

𝜕 𝐸₂

𝜕 𝜃

+𝑖(𝑔₂₂|𝐸₂|²𝐸₂+𝑔₁₂|𝐸₁|²𝐸₂). (6.12) We seek soliton solutions in the form of 𝐸₁_,₂(𝜃 − 𝑣𝑡), where 𝑣 is the repetition rate shift in the symmetric co-moving frame, which reduces the partial differential

equations to ordinary differential equations:

(𝛿 𝐷₁−𝑣)𝜕_𝜃𝐸₁ =−𝑖 𝛿𝜔 𝐸₁+𝑖𝑔_c𝐸₂+𝑖(𝑔₁₁|𝐸₁|²𝐸₁+𝑔₁₂|𝐸₂|²𝐸₁), (6.13)

−(𝛿 𝐷₁+𝑣)𝜕_𝜃𝐸₂ =−𝑖 𝛿𝜔 𝐸₂+𝑖𝑔_c𝐸₁+𝑖(𝑔₂₂|𝐸₂|²𝐸₂+𝑔₁₂|𝐸₁|²𝐸₂). (6.14) Continuous symmetries of the system result in conservation laws [74], which can reduce the dimensions of the system. As the system is conservative, we expect the equations will have a Hamiltonian structure. Indeed, the following quantity is conserved when𝜃 is viewed as an evolution coordinate [46]:

𝐻 =−𝛿𝜔(|𝐸₁|²+ |𝐸₂|²) +𝑔_c(𝐸^∗

1𝐸₂+𝐸^∗

2𝐸₁) + 1

𝑔₁₁|𝐸₁|⁴+𝑔₂₂|𝐸₂|⁴+2𝑔₁₂|𝐸₁|²|𝐸₂|²

. (6.15)

The conservation of ¯𝐻 can be verified by rewriting (𝛿 𝐷₁− 𝑣)𝜕_𝜃𝐸₁ = 𝑖 𝜕_𝐸∗ 1

¯ 𝐻 and

−(𝛿 𝐷₁+𝑣)𝜕_𝜃𝐸₂ =𝑖 𝜕_𝐸∗ 2

¯ 𝐻.

Another quantity that is conserved is the photon number flow along the𝜃-axis:

𝑁 =(𝛿 𝐷₁−𝑣) |𝐸₁|²− (𝛿 𝐷₁+𝑣) |𝐸₂|². (6.16) The conservation of ¯𝑁 can be verified by observing that all the nonlinear terms do not change the individual numbers of particles, while the coupling terms do not change the total number of particles.

For soliton solutions, these two conserved quantities can be determined as ¯𝐻 =

𝑁 = 0 since the solution should vanish exponentially as 𝜃 → ∞ without periodic boundary conditions. This determination leads to the following amplitude-phase parametrization of the solutions:

𝐸₁ = 1

√

𝛿 𝐷₁−𝑣

𝜓exp(𝑖 𝜒₁), 𝐸₂=− 1

√

𝛿 𝐷₁+𝑣

𝜓exp(𝑖 𝜒₂), (6.17) 𝜓 =p

𝛿 𝐷₁−𝑣|𝐸₁|=p

𝛿 𝐷₁+𝑣|𝐸₂|, 𝜒₁_,₂= 1

2𝑖 ln𝐸₁_,₂ 𝐸^∗

1,2

, (6.18)

which automatically satisfies the ¯𝑁conservation (the negative sign is added for later convenience). The ¯𝐻 conservation reads as:

0=− 2𝛿 𝐷₁𝛿𝜔 𝛿 𝐷²

1−𝑣²

𝜓²− 2𝑔_c q

𝛿 𝐷²

1−𝑣²

𝜓²cos(𝜒₂− 𝜒₁)

𝑔₁₁

2(𝛿 𝐷₁−𝑣)² + 𝑔₂₂

2(𝛿 𝐷₁+𝑣)² + 𝑔₁₂ 𝛿 𝐷²

1−𝑣²

𝜓⁴, (6.19)

from which the cosine of the phase difference 𝜒₂−𝜒₁can be solved as:

cos(𝜒₂− 𝜒₁)= 𝐺 𝜓²−2𝛿 𝐷₁𝛿𝜔 2𝑔_c

q 𝛿 𝐷²

1−𝑣²

, (6.20)

where for convenience, we defined a combined nonlinear coefficient:

𝐺 = 𝛿 𝐷₁+𝑣 𝛿 𝐷₁−𝑣

𝑔₁₁

2 + 𝛿 𝐷₁−𝑣 𝛿 𝐷₁+𝑣

𝑔₂₂

2 +𝑔₁₂. (6.21)

Turning back to the original equations of evolution along𝜃, we substitute 𝐸₁_,₂with the parametrization and split the real and imaginary parts:

𝜕 𝜓²

𝜕 𝜃

= 2𝑔_c q

𝛿 𝐷²

1−𝑣²

𝜓²sin(𝜒₂− 𝜒₁), (6.22)

𝜕 𝜒₁

𝜕 𝜃

=− 𝛿𝜔 𝛿 𝐷₁−𝑣

− 𝑔_c

q 𝛿 𝐷²

1−𝑣²

cos(𝜒₂− 𝜒₁) + 𝑔₁₁

(𝛿 𝐷₁−𝑣)² + 𝑔₁₂ 𝛿 𝐷²

1−𝑣²

! 𝜓²,

(6.23)

−𝜕 𝜒₂

𝜕 𝜃

=− 𝛿𝜔 𝛿 𝐷₁+𝑣

− 𝑔_c

q 𝛿 𝐷²

1−𝑣²

cos(𝜒₂− 𝜒₁) + 𝑔₂₂

(𝛿 𝐷₁+𝑣)² + 𝑔₁₂ 𝛿 𝐷²

1−𝑣²

! 𝜓².

(6.24) For the differential equation for𝜓², expressing sin(𝜒₂− 𝜒₁)in terms of𝜓²gives:

𝜕 𝜓²

𝜕 𝜃

=± 2𝑔_c q

𝛿 𝐷²

1−𝑣² 𝜓²

vu uu uu t

1−©

𝐺 𝜓²−2𝛿 𝐷₁𝛿𝜔 2𝑔_c

q 𝛿 𝐷²

1−𝑣² ª

, (6.25)

which can be integrated (with the boundary condition𝜓² →0 as𝜃 → ∞) in terms of elementary functions:

𝜓²= 𝑔_c

q 𝛿 𝐷²

1−𝑣² 𝐺

2(1−𝜉˜²) cosh

1−𝜉˜²𝜃˜

−𝜉˜

, (6.26)

where the pulse center is chosen as𝜃 =0 without loss of generality and the reduced detuning and coordinate are defined as:

𝜉 = 𝛿 𝐷₁ q

𝛿 𝐷²

1−𝑣² 𝛿𝜔

𝑔_c

, (6.27)

𝜃 = 𝑔_c q

𝛿 𝐷²

1−𝑣²

𝜃 , (6.28)

As an aside, we also obtain that:

cos(𝜒₂−𝜒₁) =

1−𝜉˜cosh 2p

1−𝜉˜²𝜃˜

cosh 2p

1−𝜉˜²𝜃˜

−𝜉˜

. (6.29)

The differential equation for 𝜒₁_,₂ can be integrated after substitution of the above solution for𝜓²and cos(𝜒₂−𝜒₁). Because the equation has global phase symmetry (𝐸₁_,₂→ 𝑒^{𝑖 𝜙}𝐸₁_,₂, where𝜙is an arbitrary constant phase), we can fix 𝜒₁(𝜃 =0) =0, which also forces𝜒₂=0 through cos(𝜒₁− 𝜒₂) |_𝜃=0=1. We obtain:

𝜒₁=− 𝑣 𝛿 𝐷₁

𝜉˜𝜃˜

+ 1

𝐺

𝛿 𝐷₁+𝑣 𝛿 𝐷₁−𝑣

𝑔₁₁− 𝛿 𝐷₁−𝑣 𝛿 𝐷₁+𝑣

𝑔₂₂

arctan





 s

1+𝜉˜ 1−𝜉˜

tanh q

1−𝜉˜²𝜃˜ 



 , (6.30) 𝜒₂=− 𝑣

𝛿 𝐷₁

˜ 𝜉𝜃˜

− 1

𝐺

𝛿 𝐷₁−𝑣 𝛿 𝐷₁+𝑣

𝑔₂₂− 𝛿 𝐷₁+𝑣 𝛿 𝐷₁−𝑣

𝑔₁₁

arctan





 s

1+𝜉˜ 1−𝜉˜

tanh q

1−𝜉˜²𝜃˜ 



 . (6.31) With these results, the soliton solutions can be expressed as:

𝐸₁=+ r2𝑔_c

𝐺 q

1−𝜉˜²(𝛿 𝐷₁+𝑣 𝛿 𝐷₁−𝑣

)^1/4

h cosh

1−𝜉˜²𝜃˜

−𝜉˜ i^𝛾/2

hp1−𝜉˜coshp

1−𝜉˜²𝜃˜

−𝑖

p1+𝜉˜sinhp

1−𝜉˜²𝜃˜

i1+𝛾 exp

−𝑖 𝑣 𝛿 𝐷₁

˜ 𝜉𝜃˜

(6.32) 𝐸₂=−

r2𝑔_c 𝐺

1−𝜉˜²(𝛿 𝐷₁−𝑣 𝛿 𝐷₁+𝑣

)^1/4

h cosh

1−𝜉˜²𝜃˜

−𝜉˜ i−𝛾/2

hp1−𝜉˜coshp

1−𝜉˜²𝜃˜

+𝑖

p1+𝜉˜sinhp

1−𝜉˜²𝜃˜

i1−𝛾 exp

−𝑖 𝑣 𝛿 𝐷₁

˜ 𝜉𝜃˜

(6.33)

where we introduce the phase exponent:

𝛾 = 1 𝐺

𝛿 𝐷₁+𝑣 𝛿 𝐷₁−𝑣

𝑔₁₁− 𝛿 𝐷₁−𝑣 𝛿 𝐷₁+𝑣

𝑔₂₂

. (6.34)

Although we have not been very rigorous for multivalued functions encountered in the calculations, direct substitution shows that the 𝐸₁_,₂obtained above is indeed a solution to the original conservative LLE when principal branches are used.

Resonance line and the band gap

The general bright soliton solution includes the square root of 1−𝜉˜², which requires that|𝜉˜| < 1. Expanded with resonator parameters, this gives:

|𝛿𝜔| ≤ q

𝛿 𝐷²

1−𝑣² 𝛿 𝐷₁

𝑔_c. (6.35)

For a fixed 𝑣, the inequality gives the detuning range where the solution is well defined. A quick plot of the range (Fig. 6.2b) shows that the boundaries are tangent to the mode spectrum curves. Indeed, using coupled mode theory, the frequencies can be described as:

𝜔_± =∓ q

𝛿 𝐷²

1𝑘²+𝑔²_c, (6.36)

where𝜔₊ (𝜔₋) is the eigenfrequency for the lower symmetric branch (upper anti- symmetric branch) and𝑘is the wavenumber. The tangent lines for the upper branch with slope𝑣satisfy:

𝑣= 𝜕 𝜔₋

𝜕 𝑘

𝛿 𝐷²

1𝑘 q

𝛿 𝐷²

1𝑘²+𝑔²

. (6.37)

Eliminating 𝑘 recovers the previous boundaries. Thus, the soliton resonance lines can stay only in the band gap and cannot cut through the band curves.

We note that in dissipative cases,𝑣is not fixed but depends on the pumping details (as discussed in the main text), so this point should not be understood as a limitation on detuning when pumping the soliton. Instead, the detuning range should be determined from the momentum constraints imposed on the soliton at fixed 𝑘 (the longitudinal mode being pumped).

Reduction of a DS to a KS

Following the above discussions on resonance lines, we focus on the case in which

𝜉 → −1⁺, where the resonance line is almost tangent to the upper branch of the

mode spectrum. Taking the limits and expanding the reduced quantities results in 𝐸₁_,₂=±

r2𝑔_c 𝐺

1+𝜉˜(𝛿 𝐷₁±𝑣 𝛿 𝐷₁∓𝑣

)^1/4sech q

2(1+𝜉˜)𝜃˜

exp

𝑖 𝑣 𝛿 𝐷₁

˜ 𝜃

, (6.38)

𝐸₁_,₂=± s

2𝛿 𝐷₁(𝛿𝜔−𝛿𝜔_min)

𝐺(𝛿 𝐷₁∓𝑣) sech p

2(𝛿𝜔−𝛿𝜔_min) s

𝑔_c𝛿 𝐷₁ (𝛿 𝐷²

1−𝑣²)³^/² 𝜃

×exp©

« 𝑖

𝑔_c𝑣 𝛿 𝐷₁

q 𝛿 𝐷²

1−𝑣² 𝜃ª

, (6.39)

where𝛿𝜔_min =−𝑔_c q

𝛿 𝐷²

1−𝑣²/𝛿 𝐷₁. The hyperbolic secant form is now apparent, and to complete the reduction, we explicitly calculate the local quantities of the mode spectrum.

When the resonance line is tangent to the mode spectrum, the wavenumber 𝑘 can be solved from the previous section:

𝑘 = 𝑔_c𝑣 𝛿 𝐷₁

q 𝛿 𝐷²

1−𝑣²

, (6.40)

which matches the exponential term. The minimum detuning that can be achieved at this particular𝑚also matches𝛿𝜔_min. The local second-order dispersion is given by:

𝜕²𝜔₋

𝜕 𝑘²

𝑔²

c𝛿 𝐷²

(𝛿 𝐷²

1𝑘²+𝑔_c²)^3/2 = (𝛿 𝐷²

1−𝑣²)^3/2 𝑔_c𝛿 𝐷₁

, (6.41)

where we have eliminated𝑘 using𝑣and it matches the dispersion term. The mode composition can be found using coupled mode theory and can be found as:

𝐸₁ 𝐸₂

=−𝜔₋+𝛿 𝐷₁𝑘 𝑔_c

√

𝛿 𝐷₁+𝑣

√

𝛿 𝐷₁−𝑣

, (6.42)

which agrees with the prefactors in𝐸₁_,₂and is also consistent with the conservation of ¯𝑁. Finally, the effective nonlinear coefficient 𝐺(𝛿 𝐷²

1 − 𝑣²)/(2𝛿 𝐷²

1) can be calculated as a weighted average of the nonlinear coefficients, the weight being the power proportions on each mode derived above. It matches the nonlinear coefficient except for the extra factor (𝛿 𝐷₁± 𝑣)/(2𝛿 𝐷₁), which is the power ratio of each mode component to the total power and is a result of expressing the solution using components rather than the hybridized field.

To complete the discussion of reducing a DS to a KS, we also present a perturbative approach that is explicitly based on the hybridized field. We begin by defining the following auxiliary fields:

𝜓₋ = r

𝛿 𝐷₁−𝑣 2𝛿 𝐷₁

𝐸₁− r

𝛿 𝐷₁+𝑣 2𝛿 𝐷₁

𝐸₂

! exp

𝑖

𝑣 𝛿 𝐷₁

˜ 𝜉𝜃˜

, (6.43)

𝜓₊ = r

𝛿 𝐷₁−𝑣 2𝛿 𝐷₁

𝐸₁+ r

𝛿 𝐷₁+𝑣 2𝛿 𝐷₁

𝐸₂

! exp

𝑖

𝑣 𝛿 𝐷₁

˜ 𝜉𝜃˜

. (6.44)

We note that while the 𝜓₊ component is the normalized linear eigenstate of the lower branch at the wavenumber corresponding to𝑣, the𝜓₋ term defined here is, in general, not the eigenstate of the upper branch, and𝜓₋ and𝜓₊ are not orthogonal (although in the special case 𝜓₊ = 0, 𝜓₋ becomes proportional to the true field amplitude). Rewriting the conservative coupled LLE in terms of𝜓_±results in:

𝜕 𝜓₊

𝜕𝜃˜

=−𝑖(1+𝜉˜)𝜓₋+ 𝑖 𝛿 𝐷₁ 2𝑔_c

q 𝛿 𝐷²

1−𝑣²

𝛿 𝐷₁+𝑣 𝛿 𝐷₁−𝑣

𝑔₁₁

2 |𝜓₊+𝜓₋|²(𝜓₊+𝜓₋)

−𝑔₁₂(𝜓₊²−𝜓₋²)𝜓^∗₋

−𝛿 𝐷₁−𝑣 𝛿 𝐷₁+𝑣

𝑔₂₂

2 |𝜓₊−𝜓₋|²(𝜓₊−𝜓₋)

, (6.45)

𝜕 𝜓₋

𝜕𝜃˜

=𝑖(1−𝜉˜)𝜓₊+ 𝑖 𝛿 𝐷₁ 2𝑔_c

q 𝛿 𝐷²

1−𝑣²

𝛿 𝐷₁+𝑣 𝛿 𝐷₁−𝑣

𝑔₁₁

2 |𝜓₊+𝜓₋|²(𝜓₊+𝜓₋) +𝑔₁₂(𝜓²₊−𝜓²₋)𝜓₊^∗

+𝛿 𝐷₁−𝑣 𝛿 𝐷₁+𝑣

𝑔₂₂

2 |𝜓₊−𝜓₋|²(𝜓₊−𝜓₋)

, (6.46) where we have substituted𝛿𝜔and𝜃with ˜𝜉and ˜𝜃, respectively, for later convenience.

Based on the structure of the above equation, we seek the following solution near

𝜉 → −1⁺:

𝜓₋ ∼𝑂(1+𝜉˜)^1/2, 𝜓₊ ∼ 𝑂(1+𝜉˜), (6.47) Keeping the lowest-order terms gives:

𝜕 𝜓₊

𝜕𝜃˜

=−𝑖(1+𝜉˜)𝜓₋+ 𝑖 𝛿 𝐷₁ 2𝑔_c

q 𝛿 𝐷²

1−𝑣²

𝐺|𝜓₋|²𝜓₋, (6.48)

𝜕 𝜓₋

𝜕𝜃˜

=2𝑖𝜓₊. (6.49)

Combining gives:

1 2

𝜕²𝜓₋

𝜕𝜃˜²

− (1+𝜉˜)𝜓₋+ 𝛿 𝐷₁ 2𝑔_c

q 𝛿 𝐷²

1−𝑣²

𝐺|𝜓₋|²𝜓₋ =0, (6.50) which is the steady-state single-mode LLE and its solution is the same as the limit of𝐸₁_,₂as derived above.

Repetition rate shifts in the DS

We copy the dissipative coupled LLE here for convenience:

𝜕 𝐸₁

𝜕 𝑡

=−𝑖 𝛿𝜔 𝐸₁+𝑖𝑔_c𝐸₂−𝛿 𝐷₁

𝜕 𝐸₁

𝜕 𝜃

+𝑖(𝑔₁₁|𝐸₁|²𝐸₁+𝑔₁₂|𝐸₂|²𝐸₁) − 𝜅₁

2 𝐸₁+ 𝑓₁, (6.51)

𝜕 𝐸₂

𝜕 𝑡

=−𝑖 𝛿𝜔 𝐸₂+𝑖𝑔_c𝐸₁+𝛿 𝐷₁

𝜕 𝐸₂

𝜕 𝜃

+𝑖(𝑔₂₂|𝐸₂|²𝐸₂+𝑔₁₂|𝐸₁|²𝐸₂) − 𝜅₂

2𝐸₂+ 𝑓₂. (6.52) We define the following momentum integral in the hybrid system:

𝑃 =

∫ 𝐸^∗

−𝑖

𝜕 𝐸₁

𝜕 𝜃

+𝐸^∗

−𝑖

𝜕 𝐸₂

𝜕 𝜃

𝑑𝜃 . (6.53)

For a steady-state solution, 𝑃 should be a constant in time. We thus calculate the first derivative of𝑃with respect to𝑡:

0= 𝜕 𝑃

𝜕 𝑡

∫

−𝑖

𝜕 𝐸^∗

𝜕 𝑡

𝜕 𝐸₁

𝜕 𝜃 +𝑖

𝜕 𝐸₁

𝜕 𝑡

𝜕 𝐸^∗

𝜕 𝜃

−𝑖

𝜕 𝐸^∗

𝜕 𝑡

𝜕 𝐸₂

𝜕 𝜃 +𝑖

𝜕 𝐸₂

𝜕 𝑡

𝜕 𝐸^∗

𝜕 𝜃

𝑑𝜃 , (6.54) where we have used integration by parts to move the spatial derivatives to the conjugated field. After plugging the equations of motion into the integral, all the conservative terms cancel each other out and the pumping terms vanish by integration by parts. We are left with:

𝜅₁ 2

∫ 𝑖 𝐸^∗

𝜕 𝐸₁

𝜕 𝜃

−𝑖 𝐸₁

𝜕 𝐸^∗

𝜕 𝜃

𝑑𝜃+ 𝜅₂ 2

∫ 𝑖 𝐸^∗

𝜕 𝐸₂

𝜕 𝜃

−𝑖 𝐸₂

𝜕 𝐸^∗

𝜕 𝜃

𝑑𝜃=0. (6.55) Rewriting the above equation using arguments gives:

𝜅₁

∫

|𝐸₁|²𝜕arg𝐸₁

𝜕 𝜃

𝑑𝜃+𝜅₂

∫

|𝐸₂|²𝜕arg𝐸₂

𝜕 𝜃

𝑑𝜃 =0. (6.56) To proceed further, we take the soliton ansatz as the exact solution of the DS derived earlier. In this case, the integration can be carried out analytically:

∫

|𝐸₁_,₂|²𝜕arg𝐸₁_,₂

𝜕 𝜃 𝑑𝜃

=2𝑔_c 𝐺

𝛿 𝐷₁±𝑣 𝛿 𝐷₁∓𝑣

− 𝑣 𝛿 𝐷₁

+𝛾±1

(𝜋−arccos ˜𝜉)𝜉˜+ (𝛾±1) q

1−𝜉˜²

. (6.57)

All the quantities can be explicitly expressed in𝑣, and the resulting equation can be solved numerically.

In the special case of𝜅₁=𝜅₂and𝛿𝜔 =0, we have ˜𝜉 =0 independent of𝑣, and the criterion is greatly simplified:

𝑣 𝛿 𝐷₁

=−𝛾 . (6.58)

Expanding𝛾 gives a cubic equation in𝑣and is used in the plot of Fig. 6.3a.

First-order perturbation calculation of mode coupling in wedge resonators Here, using first-order degeneracy perturbation theory and the integral form of the propagation constant, we derive the mode coupling in wedge resonators as an overlap integral of the unperturbed modes.

For a circular waveguide, the angular momentum number (angular propagation constant) of a mode can be expressed as [61]:

𝑚= 𝜔₀ 2𝑐

∫ 𝑛²(𝐸^∗

𝑟𝐸_𝑟 +𝐸^∗

𝑧𝐸_𝑧−𝐸^∗

𝜃𝐸_𝜃) +𝑐²(𝐵^∗

𝑟𝐵_𝑟 +𝐵^∗

𝑧𝐵_𝑧−𝐵^∗

𝜃𝐵_𝜃) 𝑟 𝑑𝑟 𝑑 𝑧

∫

𝑐(𝐸_𝑟𝐵^∗_𝑧−𝐸_𝑧𝐵^∗_𝑟)𝑑𝑟 𝑑 𝑧

, (6.59) where 𝜔₀ is the angular frequency of the light and 𝐸_𝑟, 𝐸_𝑧, 𝐸_𝜃 (𝐵_𝑟, 𝐵_𝑧, 𝐵_𝜃) are the mode electric field (magnetic flux density) components (the coordinate system in use is shown in Fig. 6.6). The linear propagation of a field (with fixed 𝜔₀) is described by𝑑𝐸₁/𝑑𝜃 =𝑖 𝑚 𝐸₁, where𝐸₁ is the field amplitude at different angular positions. If the mode profile in another waveguide with a slightly different shape is nearly identical to the current waveguide, which is usually true up to the first order of the geometry differences, then the same integral can be used to calculate the propagation constant using the known field profile and the perturbed refractive index profile.

For a pair of nearly degenerate modes, the propagation constant generalizes into a matrix:

𝑑 𝑑𝜃

𝐸₁ 𝐸₂

=𝑖

𝑚₁₁ 𝑚₁₂ 𝑚₂₁ 𝑚₂₂

! 𝐸₁ 𝐸₂

, (6.60)

𝑚_{𝑖 𝑗}= 𝜔₀ 2𝑐

∫ h 𝑛²(𝐸^∗

𝑟 ,𝑖𝐸_{𝑟 , 𝑗}+𝐸^∗

𝑧 ,𝑖𝐸_{𝑧 , 𝑗} −𝐸^∗

𝜃 ,𝑖𝐸_{𝜃 , 𝑗}) +𝑐²(𝐵^∗

𝑟 ,𝑖𝐵_{𝑟 , 𝑗}+𝐵^∗

𝑧 ,𝑖𝐵_{𝑧 , 𝑗} −𝐵^∗

𝜃 ,𝑖𝐵_{𝜃 , 𝑗})i 𝑟 𝑑𝑟 𝑑 𝑧 q∫

𝑐(𝐸_{𝑟 ,𝑖}𝐵^∗

𝑧 ,𝑖−𝐸_{𝑧 ,𝑖}𝐵^∗

𝑟 ,𝑖)𝑑𝑟 𝑑 𝑧 q∫

𝑐(𝐸_{𝑟 , 𝑗}𝐵^∗

𝑧 , 𝑗−𝐸_{𝑧 , 𝑗}𝐵^∗

𝑟 , 𝑗)𝑑𝑟 𝑑 𝑧

. (6.61)

The off-diagonal elements 𝑚₁₂ and𝑚₂₁ have an overlap integral structure and are proportional to𝑔_c. Since the modes are orthogonal in the original waveguide, only the changes in refractive index induce coupling:

𝑚_{𝑖 𝑗} = 𝜔₀ 2𝑐

∫ Δ(𝑛²) (𝐸^∗

𝑟 ,𝑖𝐸_{𝑟 , 𝑗}+𝐸^∗

𝑧,𝑖𝐸_{𝑧, 𝑗} −𝐸^∗

𝜃 ,𝑖𝐸_{𝜃 , 𝑗})𝑟 𝑑𝑟 𝑑 𝑧 q∫

𝑐(𝐸_{𝑟 ,𝑖}𝐵^∗

𝑧,𝑖−𝐸_𝑧,𝑖𝐵^∗

𝑟 ,𝑖)𝑑𝑟 𝑑 𝑧 q∫

𝑐(𝐸_{𝑟 , 𝑗}𝐵^∗

𝑧, 𝑗 −𝐸_{𝑧, 𝑗}𝐵^∗

𝑟 , 𝑗)𝑑𝑟 𝑑 𝑧

, (6.62)

whereΔ(𝑛²)is the change in𝑛²of the perturbed waveguide compared to the original waveguide.

The introduction of the wedge angle adds a dielectric triangle to the lower-right part and subtracts a dielectric triangle to the upper-right part (Fig. 6.6). As these are the only areas in which the refractive index changes, the overlap integral is effectively restricted to the triangles. If 𝜋/2−𝛼is small, we can further replace all the fields by their values on the vertical boundary of the wedge. This replacement results in:

𝑚₁₂≈ 𝜔₀ 2𝑐

(𝑛²

M−1)

−∫𝑡/2

−𝑡/2(𝐸^∗

𝑟 ,1𝐸_{𝑟 ,}₂+𝐸^∗

𝑧,1𝐸_𝑧,₂−𝐸^∗

𝜃 ,1𝐸_{𝜃 ,}₂) (𝐷/2) (𝜋/2−𝛼)𝑧 𝑑 𝑧 q∫

𝑐(𝐸_{𝑟 ,}₁𝐵^∗

𝑧,1−𝐸_𝑧,₁𝐵^∗

𝑟 ,1)𝑑𝑟 𝑑 𝑧 q∫

𝑐(𝐸_{𝑟 ,}₂𝐵^∗

𝑧,2−𝐸_𝑧,₂𝐵^∗

𝑟 ,2)𝑑𝑟 𝑑 𝑧 , (6.63) where𝑛_Mis the dielectric index. The integral can be further reduced by symmetry, using 𝐸_{𝑟 ,}₁(𝑧) = 𝐸_{𝑟 ,}₁(−𝑧), 𝐸_𝑧,₁(𝑧) = −𝐸_𝑧,₁(−𝑧) and 𝐸_{𝜃 ,}₁(𝑧) = 𝐸_{𝜃 ,}₁(−𝑧) for the TE mode and𝐸_{𝑟 ,}₂(𝑧) =−𝐸_{𝑟 ,}₂(−𝑧),𝐸_𝑧,₂(𝑧) =𝐸_𝑧,₂(−𝑧)and𝐸_{𝜃 ,}₂(𝑧)=−𝐸_{𝜃 ,}₂(−𝑧)for the TM mode. This process reduces the integration limits by half:

𝑚₁₂ ≈ 𝜔₀ 𝑐

𝐷 2(𝑛²

M−1)

−∫𝑡/2 0 [(𝑛²

M+1)𝐷^∗

𝑟 ,1𝐷_{𝑟 ,}₂/(2𝑛²

M𝜀²

0) +𝐸^∗

𝑧,1𝐸_𝑧,₂−𝐸^∗

𝜃 ,1𝐸_{𝜃 ,}₂]𝑟=𝐷/2𝑧 𝑑 𝑧 q∫

𝑐(𝐸_{𝑟 ,}₁𝐵^∗

𝑧,1−𝐸_𝑧,₁𝐵^∗

𝑟 ,1)𝑑𝑟 𝑑 𝑧 q∫

𝑐(𝐸_{𝑟 ,}₂𝐵^∗

𝑧,2−𝐸_𝑧,₂𝐵^∗

𝑟 ,2)𝑑𝑟 𝑑 𝑧 (𝜋

2 −𝛼), (6.64) where the radial electric field is replaced by the electric displacement field 𝐷_𝑟 to prevent ambiguities across the dielectric boundary;𝜀₀is the vacuum permittivity.

Finally, 𝑚₁₂ can be converted to 𝑔_c in the same way that the effective index is

z r θ

r = 0 z = t/2

z = -t/2

r = D/2

Figure 6.6: Illustration of the perturbation induced by the wedge angle in the wedge resonator. The light grey area indicates the dielectric removed compared to a symmetric resonator, while the dark grey area indicates the dielectric added. The cylindrical coordinates used to describe the resonator are also shown.

converted to the mode spectrum:

𝑔_c = 2𝑐 𝑛_eff𝐷

|𝑚₁₂|

≈ 𝜔₀ 𝑛_eff

(𝑛²

M−1)

∫𝑡/2 0 [(𝑛²

M+1)𝐷^∗

𝑟 ,1𝐷_{𝑟 ,}₂/(2𝑛²

M𝜀²

0) +𝐸^∗

𝑧,1𝐸_𝑧,₂−𝐸^∗

𝜃 ,1𝐸_{𝜃 ,}₂]𝑟=𝐷/2𝑧 𝑑 𝑧 q∫

𝑐(𝐸_{𝑟 ,}₁𝐵^∗

𝑧,1−𝐸_𝑧,₁𝐵^∗

𝑟 ,1)𝑑𝑟 𝑑 𝑧 q∫

𝑐(𝐸_{𝑟 ,}₂𝐵^∗

𝑧,2−𝐸_𝑧,₂𝐵^∗

𝑟 ,2)𝑑𝑟 𝑑 𝑧 (𝜋

2 −𝛼). (6.65) Given a target hybridization wavelength, modes in symmetric resonators with different thicknesses can be simulated to find the degeneracy point, and the mode coupling in the asymmetric case can be estimated from the above overlap integral without actually simulating the asymmetric resonators. For 780-nm-wavelength silica resonators (using𝑛_M=1.454), the prefactor in𝑔_cis 15.78 GHz, or 0.275 GHz per degree angle. This estimate agrees with the full simulation results for wedge resonators for𝛼close to 90^◦(Fig. 6.4e).

Dalam dokumen Kerr Solitons and Brillouin Lasers in Optical Microresonators (Halaman 97-107)