BACKGROUND

Nonlinear optics is the field of studying systems that can generate new frequencies (De Angelis 2021). In optics, this often looks like sending light of a specific frequency (ω) into a material and observing different frequencies exiting the material. The polarization (P ) of the material can be described by a Taylor Expansion

P = ε₀χ₁E + ε₀χ₂E² + ε₀χ₃E³ + ...

This relationship between the polarization and electric field determines the materials optical response to the incident light (Pedrotti et al. 2017). When a material behaves nonlinearly, the optical properties of the material, like the refractive index, can no longer be approximated as linear but become functions of the incident light's electric field (E). The nonlinear third order term produces nonlinear effects such as frequency mixing. Frequency mixing occurs when two or more frequency beams interact in the medium and produce new frequencies of 2ω₁, 2ω₂, ω₁ + ω₂, ω₁ − ω₂, and so on (Pedrotti et al. 2017).

It was well known in the scientific community that materials had nonlinear properties and that our description of linear behavior in materials was an approximation valid only at low light intensities. The existence of nonlinear behavior was interesting, but unexplored due to the lack of a light source with a strong enough intensity to produce these optical nonlinearities. This all changed with the invention of the laser in 1960. In 1961, Franken demonstrated second-harmonic generation for the first time using a red laser and crystalline quartz, which served as the beginning of the study of nonlinear optics (Franken et al. 1961).

Light and matter interactions are so short lived that it was difficult to stimulate efficient nonlinear optical effects. Optical waveguides, a common example of which is the optical fiber (Stolen et al. 1974), helped solve this problem by directing a light wave by total internal reflection. This greatly increased the interaction time between the light and the waveguide material. A ring resonator is an optical instrument that contains at least one straight waveguide and a ring waveguide very close together. These waveguides are commonly made of silicon due to silicon's high refractive index contrast (Bogaerts et al. 2012). When light is sent into this structure, the light travels from the straight waveguide into the ring via evanescent coupling (Mansuripur 2009). If the light is sent in at the correct wavelength, the ring will be in resonance, allowing for light intensity to grow inside the ring, instead of over an inconveniently long length of waveguide. Because the resonance of the ring is wavelength specific, any frequencies that were generated due to the nonlinear behavior of the material will not remain in resonance in the ring and instead will couple back out to the straight waveguide.

METHODS

Simulation Setup — **Figure 1.**View of the designed ring resonator in the XY plane. The grey rectangle is a base made of silicon dioxide of refractive index n = 1.44 and the red structures are waveguides made of silicon nitride. The yellow objects are frequency-domain field and power monitors. The orange boxes are the simulation region and override mesh regions.

We began with ring of radius r = 3.1 um made of linear silicon nitride with a refractive index of n = 2.1¹. The basic steps taken to optimize ring resonator design to see nonlinearity are:

1. Contain the mode within the silicon nitride waveguide to achieve coupling from the source to the waveguide

2. Increase Q factor

3. Achieve critical coupling by adjusting gap width

4. Add a nonlinear χ₃ Raman Kerr material with χ₃, α, ωRaman, and δRaman values of silicon nitride

5. Send a low power broadband source to determine
resonance frequencies

6. Send light at a resonance frequency with a spectrally
narrow span and increased power to see nonlinear behavior

Fundamental Mode — **Figure 2.**Vertical cross section of the straight waveguide in the YZ plane from the mode solver. Pictured is the real fundamental mode that the mode source is injecting the waveguide with. The mode solver considers the wavelength and frequency of the injected light, as well as the refractive index, n, of the material. The color bar indicates the electric field intensity of the modal field. Most of the time, as in this case, it is preferred to chose to inject a single mode and to design the structure to completely enclose the mode of choice. Leakage of the modal field outside of the waveguide could be a result of the low index contrast between the silicon nitride waveguide and the silicon dioxide base material.

RESULTS

Linear Resonance

Figure 3. Frequency-domain field and power monitor image of electric field intensity in the ring resonator from a broadband source with a wavelength range of 1.5 to 1.6 um at an amplitude 1.

Linear Through Port Spectrum — **Figure 4.** Through port transmission spectrum of the broadband source of 1.5 to 1.6 um at an amplitude 1. Three resonance dips can be seen but show a low Q factor. The Q factor can be estimated by looking at the width of the resonance dips. The Q factor is indicative of the number of trips around the ring that light completes before the energy is lost. A high Q factor is preferred. The deepest of the resonance dips only reaches about 0.1, indicating that not all the light from the straight waveguide is coupling to the ring at the resonance wavelengths.

High-Q Linear Resonance

Figure 5. Frequency-domain field and power monitor image of electric field intensity from a broadband source with a wavelength range 1.5 to 1.6 um and amplitude 1.

Nonlinear material Through Port Spectrum — **Figure 6.** Through port transmission spectrum of the broadband source of 1.5 to 1.6 um at an amplitude 1. 6 Resonance dips can be seen. The dips are deeper and much narrower than the dips shown in Figure 4, indicating a sufficiently high Q factor and coupling efficiency.

The material of the ring resonator was then changed to nonlinear silicon nitride. A χ₃ Raman Kerr material was added to the Ansys Lumerical material database with values of χ₃ = 4.83686e-21 m²/V², α = 0.957 , ωRaman = 7e+13 Hz, and δRaman = 3.125e+13 Hz. The α value controls the relative weighting between the Kerr and Raman terms. The ωRaman term is the non-linear Raman angular frequency and the δRaman term is the linewidth of the resonance. The χ₃ and ωRaman values for silicon nitride are known and used, but the α and δRaman values for silicon were used.

To increase the Q factor, the radius of the ring was increased from r = 3.1 um to r = 6.0 um. This decreased bend loss in the ring, which increased the Q factor. In order to reach critical coupling, which is when 100% of incident light is coupled to the ring, the gap between the straight waveguide and the ring was increased in steps of 50 nm until the depth of the resonance dips reached a maximum, at a gap width of 0.45 um.

Nonlinear Frequency Generation

Lumerical Incorrect Spectrum — **Figure 7.** Log-linear plot of the through port transmission from a smaller broadband source with a center wavelength of 1.5 um and a 0.05 um span. The vertical red line indicates the source frequency. The source amplitude was increased to 3e06 to cause nonlinear effects. The graph shows a lack of the predicted nonlinear frequency peaks around the source frequencies as well as strange, powerful peaks at unexpected frequencies, such as around 1e-06. These results point to a possible problem with Lumerical's FFT correction.

Raman Generation — **Figure 8.** Graph of through port field intensity from a source wavelength of 1.46651 um with a span of 0.00027 um and amplitude 3e06. This graph was generated via an FFT script that uses the time-domain data from Lumerical in order to avoid any problems with Lumerical's FFT calculation and resulting transmission spectrum. The span width was determined by the width of the input wavelength's resonance peak in Figure 6. We expect to see resulting peaks near 1.13 um, or 2.63e14 Hz, which is the sum of the laser frequency 1.93e14 Hz, and the Raman frequency of 7e13 Hz. The vertical red line indicates the source frequency, and the vertical blue line indicates the beginning of a generated peak due to the nonlinearity of the material.

Four-Wave Mixing — **Figure 9.** Generated frequency peaks, shown by blue vertical lines, as a result of nonlinear four-wave mixing. The two source frequencies are shown by red vertical lines. 'Pump' source (ω₁) at wavelength 1.53 um and amplitude 5. 'Signal' source (ω₂) at wavelength 1.54 um and amplitude 1.5. Four-wave mixing generates frequencies at 2ω₁- ω₂ and 2ω₂- ω₁, seen to the immediate left and right of the source frequencies respectively, as well as other frequency combinations. The pulses we increased and offset in order to better observe nonlinear results

Applications

These nonlinear effects can create an optical frequency comb. The incredibly accurate and evenly spaced generated frequency peaks form a comb shape which can be used to measure light with extreme precision and accuracy. These optical frequency combs are helping in the search for exoplanets, improving technology for radio astronomy, and improving distance measurements in lidar. The next hurdle for frequency combs is shrinking them to microchip size, something that microring resonators could make possible.^2,3

References

[1] C. Krückel, A. Fülöp, T. Klintberg, J. Bengtsson, P. Andrekson, and V. Torres-Company, "Linear and nonlinear characterization of low-stress high-confinement silicon-rich nitride waveguides," Opt. Express 23, 25827-25837 (2015).
[2] R. McCracken, J. Charsley, and D. Reid, "A decade of astrocombs: recent advances in frequency combs for astronomy [Invited]," Opt. Express 25, 15058-15078 (2017).
[3] Optical frequency combs. NIST. (2022, April 5). https://www.nist.gov/topics/physics/optical-frequency-combs

ABOUT ME

My name is Emma VanderKooi and I am a rising junior at Wheaton College as a Physics major and Math minor. I am interested in astrophysics and astronomy, and in addition to classes I enjoy working as Lead TA of my school's observatory! This summer I worked with Dr. Hooman Mohseni's lab at Northwestern University's McCormick School of Engineering to simulate nonlinear optical behavior in silicon nitride ring resonators. Apart from doing research, I love watching soccer, reading, and spending time with my friends and family!

Contact: emma.vanderkooi@my.wheaton.edu or LinkedIn