Home Assignment 3
Problem 1
The pharmaceutical company PhotonicSensing AB is designing a photonic chip sensor for de-tection of antibodies from saliva samples. It has developed a chemical that has very interesting properties: antibodies can adhere with high efficiency to the material, resulting in a refractive index change in proportion to the antibody concentration, following a simple linear relation
n = n0 + αN, (1)
where n0 is the refractive index of the material in absence of antibodies, N is the number of antibodies that have been adhered, and α = 10-12 is a proportionality constant. As an example, if 1 000 000 antibodies bind to the material, the refractive index changes by 10-6 . For simplicity, we will assume that n0 takes the same value as that of silica.
The working function of the sensor is described in Figure 1 below. The material sensitive to the antibodies is used as cladding in a buried waveguide structure. This waveguide is used to build an optical microresonator designed to operate under critical coupling condition. A laser at 1300 nm is tuned to one longitudinal mode and the chip is impregnated with the saliva from a patient. The antibodies cause a change in refractive index, resulting in a shift of the longitudinal mode, which can be detected as an increase in output power. The intrinsic quality factor is measured to be Qi = 106 .
Figure 1 : Cross-section geometry (left) of buried waveguide geometry with the chemical for antibody adhesion used as cladding. When this waveguide is used in a critically coupled resonator (middle) it gen- erates a set of longitudinal modes. A laser tuned to one of the longitudinal modes will become attenuated in power (right) . However, the presence of antibodies will modify the refractive index of the material, resulting in a phase shift and consequently a change in the frequency location of the longitudinal mode. The laser will thus experience less attenuation. The changes in power can be mapped to the number of antibodies that have been bound to the resonator.
a) Engineer the cross-section of the waveguide core geometry of the ring so that it provides the maximum change in effective index for the fundamental mode when the antibodies bind. Consider the range of waveguide widths [500 - 1000] nm and the waveguide height [100 - 300] nm. The layer of the sensitive material is 2 µm, and the lower cladding 3 µm. Justify your choice of geometry and discuss your results in terms of the optical confinement of the fundamental mode.
Notes:
• The two cladding layers are so thick that they can be assumed to fill the simulation area in Lumerical.
• There is no need to perform any calculation on the ring geometry for this part of the problem.
b) Assume that the resonator is designed with with radius r = 200µm. What order is the mode that is closest to the laser frequency at 1300nm?
c) How much will the resonance frequency of the mode shift if one million antibodies bind to the chemical on top of the resonator?
d) Suggest at least two ways to improve the sensitivity of the antibody sensor. Explain why and how those improvements work.
Problem 2
Mach-Zehnder interferometers (MZIs) can be adapted to operate as wavelength multiplexers and demultiplexers. Figure 2 shows a schematic of a simple 1x2 wavelength demultiplexer. Ignore the dispersion of the bulk materials and the waveguide.
a) Design the cross-section geometry of the waveguide, the length difference ∆L between the arms, and the 2x2 couplers (i.e., the gap between the waveguides and the coupler length) so that the MZI works as a 2-λ demultiplexer with frequency spacing of 100GHz for wavelengths in the region of 1300nm. Show the transfer function for either port as a function of optical frequency, i.e., plot the normalized transmitted power for either the top or bottom port as a function of input frequency.
b) Silicon nitride, which is used as the core material of the waveguide, has a thermo-optic coefficient of 2 × 10-6 K-1 . This means that the refractive index changes by two parts per million for a variation in temperature of one Kelvin. How much will the frequency spacing change if the above device is placed close to a computer chip in a datacenter, where the temperature can change from 25 。C to 80 。C? By how much does the transmitted wavelength λ 1 change when the temperature increases from 25 。C to 80 。C?
Figure 2: Cross-section of the waveguide (left) and a Mach-Zehnder interferometer (right) operating as a demultiplexer for light with wavelengths λ1 , λ2 .
Problem 3
a) Using Lumerical MODE, design an inverse taper coupler with more than 80% coupling efficiency when coupling light from a standard single-mode optical fiber SMF-28 at 1500nm into a single-mode silicon waveguide core with height h = 380nm and width w = 380nm. Assume the waveguide has silica cladding.
b) How much can the waveguide and fiber be mismatched before the coupling efficiency drops below 70%? Plot the coupling efficiency vs. fiber misalignment (choose one direction for the misalignment).