Implementation of Dirac solitons

DIRAC SOLITONS IN OPTICAL MICRORESONATORS

6.4 Implementation of Dirac solitons

The wedge resonator [59] is used to induce mode hybridization and Dirac soliton formation. This resonator offers very high-quality factors [60] and the independent control over key parameters during the fabrication process (Fig. 6.4a). A wedge is entirely characterized by three geometric parameters: the diameter 𝐷, which

Silica

Silicon a

t

D/2

α

1.425 1.43 1.435

740 820

TE1 TM1 TE2 TE3

TE1

TM1 TE2 TE3

b c

-20 20

-500 500

400

760 780 800 Wavelength (nm) Frequency (THz)

neff

390 380 370

Offset frequency (GHz)

Relative mode number0 -10

0 10

-20 20

-500 500

Offset frequency (GHz)

Relative mode number0 -10

0 10

0 1

1.0 1.5

500 1000 1500

Wavelength (nm)

Anomalous

Normal

2.0 2.5

t (µm)

SilicaBulk d

TE1

e 15 10 5 0 gc/2π (GHz)

α (degree) 90 80 70 60 50 40 30

f

β2 (ps2/km) 100

0 -100 -200 -300

Normal

Anomalous BulkTE2 TM1Hybrid, 30°

Hybrid, 60°

200 300 400 500 600 700

Wavelength (nm)

1500 1000 800 700 600 500 400

Frequency (THz) 1200

Figure 6.4: Implementation of mode hybridization. (a) Cross-sectional view of a silica wedge resonator on a silicon pillar (not to scale). The parameters that define the wedge geometry are also shown. (b) Plot of 𝑛_eff for the first four modes (TE1, TM1, TE2, and TE3) versus wavelength (740 to 820 nm) for a wedge resonator. The parameters are𝑡 =1.47µm and𝛼=90^◦. (c) Left panel: mode spectrum plot for the boxed region in (b). The insets are simulated mode profiles (electric field norm).

Right panel: same as left panel but with𝛼 = 30^◦. (d) Relationship between 𝑡 and 𝜆_X(black curve). Additionally, the zero-dispersion wavelength of bulk silica (green dashed line) and the zero-dispersion boundary for the TE1 mode (purple curve) are shown. (e) Coupling 𝑔_c versus wedge angle 𝛼 at𝜆_X = 778 nm. The dashed line is the result from perturbation theory (see Methods) and is tangent to the 𝑔_ccurve at 𝛼 = 90^◦. (f) Effective GVD 𝛽₂ that can be achieved using mode hybridization across the infrared and visible spectra. The parameters are 𝛼 = 30^◦ and𝛼 = 60^◦. The dispersion of bulk silica, TE2, and TM1 modes is also shown for comparison.

The color bar shows the approximate color of light in the visible band.

depends on the lithographic pattern; the thickness𝑡, which depends on the oxidation growth time of the silicon wafer; and the wedge angle 𝛼, which depends on the adhesion between silica and the photoresist used for patterning. In the following we will fix the resonator diameter as 𝐷 = 3.2 mm (corresponding to a resonator FSR of approximately 20 GHz at approximately 1550 nm), but we note that this can be readily generalized to resonators of other sizes.

For a symmetrical wedge resonator (𝛼 = 90^◦), the typical simulated effective re- fractive index𝑛_effversus wavelength is shown in Fig. 6.4b. At shorter wavelengths, TE1 and TM1 have the highest indices, followed by TE2, TE3, and other high-order modes. Since the electrical fields of the TM modes are along the thickness direction,

their indices are more sensitive to changes in the wavelength scale, and the index of TM1 decreases faster than TE2 as the wavelength increases. Eventually, TM1 and TE2 cross, and their relative positions are interchanged at longer wavelengths.

However, for 𝛼 = 90^◦, no hybridization occurs, as the reflection symmetry pro- hibits interactions between modes of different parities. On the other hand, if we explicitly break the reflection symmetry of the resonator by decreasing the wedge angle (𝛼 <90^◦), the original modes will see an asymmetric change in the refractive index profile, which causes them to couple. Such couplings lift the degeneracy, leading to avoided crossing. The two cases are compared in Fig. 6.4c, where 𝑛_eff is first converted to the mode number 𝑚 via𝑚 = 𝑛_eff𝐷 𝜔_𝑚/(2𝑐), where𝜔_𝑚 is the resonance (angular) frequency, and then plotted as offset (angular) frequencies 𝜔_off = 𝜔_𝑚 −𝜔_X− (𝑚−𝑚_X)𝐷¯₁ versus the relative mode number𝑚−𝑚_X, where the subscript X indicates the quantity at the degeneracy point. We note that the relative mode number has the same role as the relative wavenumber 𝑘 in the theo- retical analyses, except that it is restricted to integer values for periodic boundary conditions.

In view of perturbation theory [61], the wedge part of the resonator perturbs the underlying symmetrical structure and induces polarization coupling similar to the coupling obtained in directional couplers [62]. Therefore, we expect that the center wavelength of hybridization 𝜆_X is determined by the thickness𝑡, while the wedge angle controls the coupling strength𝑔_c. A plot of𝜆_X versus𝑡is shown as the black curve in Fig. 6.4d. As𝑡is the only geometry scale close to optical wavelengths in the system, we expect that𝜆_Xwill scale linearly with𝑡, which can be visually verified in the plot. This scaling allows for hybridization to occur at short wavelengths where the dispersion of the original modes (for example, the TE1 mode shown in the figure) is typically normal. A plot of𝑔_cversus𝛼is shown in Fig. 6.4e. While only a particular wavelength (778 nm) is shown,𝑔_cdepends on the wavelength very weakly, varying less than 5% from wavelengths of 400 to 1600 nm. The coupling strength scales linearly with 𝛼 near 𝛼 = 90^◦, which can also be independently verified by first-order perturbation theory (see Methods), but the coupling effect eventually saturates at shallow wedge angles because mode profiles cannot “squeeze” into the wedge tip as 𝛼 decreases. The calculated GVD 𝛽₂ is shown in Fig. 6.4f, which is related to 𝐷₂ via 𝛽₂ = −𝑛 𝐷₂/(𝑐 𝐷²

1). Using suitably designed thicknesses and wedge angles greater than 30^◦, an anomalous dispersion window can be created all the way down to the blue side of the visible spectrum, where simple geometrical dispersion fails to compensate for normal material dispersion.