Mach-Zehnder interferometer: classical optics approach
As an independent researcher in the field of optical physics, I find the Mach-Zehnder interferometer particularly intriguing due to its exquisite sensitivity to phase variations. This tool, fundamental in many advanced physics experiments, uses two beam splitters and mirrors to manipulate and measure the coherence of light.
Introduction
The journey of light through a Mach-Zehnder interferometer starts when a coherent beam, typically from a laser, enters the device at mode (1). The beam first encounters a beam splitter that divides the light into two separate paths, directing beams along paths (3) and (4). This initial split is crucial as it sets the stage for the intricate interplay of light that follows.
Path and mirrors
Each beam travels along its path, interacting with mirrors \text{M}_3 and \text{M}_4. These mirrors are precisely positioned to reflect the beams towards a second beam splitter. The exact alignment of the mirrors is vital as any deviation can significantly affect the interference outcome. The paths (3) and (4) are typically adjusted using a piezo-electric transducer (\text{PZT}), which allows for precise control over path length differences down to sub-nanometer accuracy.
Recombination and interference
The two beams recombine at the second beam splitter, where they interfere with each other. The resulting interference pattern is highly sensitive to the phase differences introduced along the paths. These differences are often a consequence of minute physical changes or experimental conditions affecting one of the beams.
Mathematical framework
To quantify the behavior of light in the interferometer, consider the electric fields at different points in the setup:
- At the first beam splitter at point O:
\begin{aligned} \mathbf{E}_3^{(+)} & = r \mathbf{E}_1^{(+)} \\ \mathbf{E}_4^{(+)} & = t \mathbf{E}_1^{(+)} \end{aligned}
- After traveling distances L_3 and L_4, respectively:
\begin{aligned} \mathbf{E}_3^{(+)}(O') & = e^{ikL_3}\mathbf{E}_3^{(+)} \\ \mathbf{E}_4^{(+)}(O') & = e^{ikL_4}\mathbf{E}_4^{(+)} \end{aligned}
- At the second beam splitter at point O’:
\begin{aligned} \mathbf{E}_5^{(+)} & = -r \mathbf{E}_3^{(+)}(O') + t \mathbf{E}_4^{(+)}(O') \\ \mathbf{E}_6^{(+)} & = t \mathbf{E}_3^{(+)}(O') + r \mathbf{E}_4^{(+)}(O') \end{aligned}
The intensities at detectors D_5 and D_6 are then given by:
\begin{aligned} \left|\mathbf{E}_5^{(+)}\right|^2 & = \left|\mathbf{E}_1^{(+)}\right|^2 \left|-r^2 e^{ikL_3} + t^2 e^{ikL_4}\right|^2 \\ \left|\mathbf{E}_6^{(+)}\right|^2 & = \left|\mathbf{E}_1^{(+)}\right|^2 \left|t^2 e^{ikL_3} + r^2 e^{ikL_4}\right|^2 \end{aligned}
These equations reveal how the phase differences transform into measurable intensity variations at the output, allowing me to derive precise information about the phase shifts.
Conclusion
Through my investigations with the Mach-Zehnder interferometer, I have been able to measure extremely small phase changes, which are critical in many areas of physics research. The ability to control and measure such minute differences makes this tool indispensable in the realm of experimental physics.
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