For this project, we attempted to build an interferometric wavemeter with a resolution of <1pm, which typically cost $15,000 to $50,000 from commercial vendors. The design of this wavemeter is based on the [Talbot effect](https://en.wikipedia.org/wiki/Talbot_effect), which is an optical phenomenon that occurs when a plane wave is incident upon a periodic diffraction grating so that the image of the grating is repeated at regular distances away from the grating plane: ![[Pasted image 20240520115003.png]] source: [Wikipedia](https://en.wikipedia.org/wiki/Talbot_effect) This phenomenon is a consequence of Fresnel diffraction and the length of the spatial period of the pattern is called the Talbot length, which can be expressed as: $z_T = \frac{\lambda}{1 - \sqrt{1 - \frac{\lambda^2}{d^2}}}$ where $d$ is the period of the grating and $\lambda$ is the wavelength of the incident light. Therefore, if we can measure $z_T$, [we can determine the wavelength](https://arxiv.org/abs/1907.11072) as: $\lambda = \frac{2z_T d^2}{z_T^2 + d^2}$ If we tilt an image sensor by an angle $\theta$ from the grating normal (about the axis defined by the grating periodicity), the sensor can sample different values of $z$ while keeping $x$ constant along a single pixel column, so that each column captures the periodic intensity $z_T$. ![[Pasted image 20240520115913.png]] ## Modeling We can model the Talbot patterns incident on the sensor by some plane wave optics calculations. The intensity profile is given by: $\begin{aligned} I(x, z) & \propto A_0^2+A_1^2+A_{-1}^2+2 A_1 A_{-1} \cos (4 \pi x / d) \\ + & 2 A_0\left(A_1+A_{-1}\right) \cos (2 \pi x / d) \cos \left(2 \pi z / z_T\right) \\ & +2 A_0\left(A_1-A_{-1}\right) \sin (2 \pi x / d) \sin \left(2 \pi z / z_T\right) \end{aligned}$ It becomes clear that if $d$ is a multiple of $p$, the sensor pixel pitch, the pattern is washed out: ![[Pasted image 20240520120642.png]] The periodicity of the pattern in the x-direction is washed out entirely by aliasing. Therefore, we need to choose a sensor with a pixel pitch as close to a half-multiple of the grating pitch as possible. We use an Arducam with an IMX298 sensor, which has a 1.38um pixel pitch with a grating that has 0.8um pitch, which for $\lambda = 632.8$nm (HeNe emission line), has a simulated pattern of the following at $\theta = 20$: ![[Pasted image 20240520121337.png]] ## Construction ![[Pasted image 20240520131338.png]] ![[Pasted image 20240520131349.png]] ![[Pasted image 20240520131413.png]] ## Analysis ![[Pasted image 20240520132216.png]] ![[Pasted image 20240520131520.png]] ![[Pasted image 20240520131423.png]] ![[Pasted image 20240520131501.png]] ![[Pasted image 20240520131445.png]]