Summary of "Lecture 6: Interferometers with Y Splitter and Directional Coupler (Part 2)"

Summary of Lecture 6: Interferometers with Y Splitter and Directional Coupler (Part 2)

This lecture continues the analysis of interferometers, focusing on the use of Y splitters/combiners and directional couplers in Mach-Zehnder interferometer (MZI) configurations. The instructor explains the practical and theoretical aspects of these components, their influence on output signals, and how to optimize interferometer performance for sensing applications.

Main Ideas and Concepts

Recap of Previous Lecture and Setup
- The interferometer is constructed using Y splitters (input) and Y combiners (output).
- Input light splits into two arms (needle and back arms), then recombines to produce interference at the output.
- The output intensity depends on the phase difference between the two arms.
Practical Issues in Interferometer Construction
- Unexpected noise and losses appear in the transmission spectrum.
- Losses primarily arise from mismatches in waveguide dimensions (e.g., 400 nm waveguide branches connected to 500 nm waveguides).
- Direct connection between waveguides of different widths causes coupling losses.
Solution to Waveguide Mismatch
- Use of tapered waveguides to gradually transition between different widths (400 nm to 500 nm), reducing coupling losses.
- This tapering improves mode matching and overall device performance.
Directional Coupler vs. Y Splitter/Combiner
- The interferometer can be built using directional couplers instead of Y splitters/combiners.
- Directional couplers split and combine light based on coupling coefficients (T for transmission, K for coupling).
- The directional coupler is characterized by an S-matrix, a unitary matrix ensuring power conservation.
- Phase shifts (notably a -90° phase shift) occur during coupling and must be accounted for in the analysis.
Mathematical Modeling
- Output electric fields are expressed as combinations of inputs multiplied by transmission and coupling coefficients, including phase factors.
- The total output intensity is the square of the magnitude of the complex electric field.
- Analytical expressions allow prediction of output intensities at each port.
Phase Shifters and Interference Control
- Phase difference between the two arms controls constructive or destructive interference at output ports.
- Phase shifters (e.g., thermal heaters) can induce phase shifts (e.g., 90° or 180°), enabling control over output power distribution.
- By adjusting the phase, all power can be directed to one output port or split equally between ports.
Experimental and Simulation Results
- Directional couplers with balanced arms (equal length ~100 μm) show expected 50/50 power splitting (approx. -3 dB loss per output).
- Introducing a path length difference (ΔL) between arms creates phase difference, shifting output power between ports.
- Large path differences (~25 μm) result in narrowband operation, limiting broadband applicability.
- The interferometer’s performance depends on wavelength, coupling length, and phase difference.
Applications in Sensing
- The Mach-Zehnder interferometer is suitable for sensing applications (e.g., detecting glucose concentration or cancer markers in blood).
- Changes in refractive index of the sample cause shifts in the interference pattern, enabling detection.
- The interferometer can be designed as an open interface for direct sample interaction.

Methodology / Instructions Highlighted

To minimize losses in waveguide connections: Use tapered waveguides to connect waveguides of different widths.
To analyze directional coupler behavior:
- Model the coupler using an S-matrix with transmission (T) and coupling (K) coefficients.
- Include phase shifts (especially -90° for cross coupling) in calculations.
- Ensure the S-matrix is unitary for power conservation.
To control output power distribution:
- Apply phase shifts to one arm of the interferometer using phase shifters (thermal or otherwise).
- Calculate output intensities using complex amplitude expressions and square magnitudes.
To design for sensing applications:
- Adjust arm lengths to create desired phase differences.
- Use the interferometer to detect refractive index changes via shifts in interference patterns.

Key Equations and Concepts

Output field at port 1: [ E_{\text{out}1} = T \cdot E_1 - K \cdot e^{-j\phi} \cdot E_2 ]
Output field at port 2: [ E_{\text{out}2} = T \cdot E_2 - K \cdot e^{-j\phi} \cdot E_1 ]
Power conservation: [ T^2 + K^2 = 1 ]
Phase shift (φ):
- Typically (-90^\circ) (or (-\pi/2)) for cross coupling in directional couplers.
- Phase shifters can add adjustable phase differences to control interference.
Phase difference related to path length difference (ΔL): [ \Delta \phi = \frac{2\pi}{\lambda} \Delta L \cdot n_{\text{eff}} ]

Speakers / Sources

Primary Speaker: The lecturer (unnamed), presumably a university professor or instructor specializing in photonics or optical engineering.

Summary Conclusion

This lecture deepens understanding of interferometer design using Y splitters and directional couplers, emphasizing practical issues like waveguide mismatches and losses. It introduces tapered waveguides as a solution and rigorously models directional couplers with phase shifts using S-matrix formalism. The role of phase shifters in controlling interference and output power distribution is highlighted, with direct applications in sensing technologies such as biosensing. The lecture combines theoretical derivations with experimental and simulation insights to provide a comprehensive view of Mach-Zehnder interferometer operation and optimization.