A library to compute diffraction patterns in 2D, and to undistort diffraction patterns in presence of defects in diffraction gratings.

Background

In the special case when the grating grooves/lines are perpendicular to the plane of incidence, the diffraction grating equation is expressed as $$\sin\theta_m + \sin\theta_i = m\frac{\lambda}{d}, \quad m=0, \pm 1, \pm 2, \ldots$$ where $m$ is the order of the diffracted wave, $\theta_i$ is the angle of incidence, $\lambda$ is the wavelength, and $d$ is the period of the grating. For this situation all of the diffracted orders lie in the plane of incidence. Harvey and Vernold provide an elegant explanation of diffraction for the general case by introducing grating equations in the direction cosine space: $$\alpha_m + \alpha_i = m\frac{\lambda}{d}, \quad \beta_m + \beta_i = 0,$$ where $(\alpha_i,\ \beta_i)$ are the direction cosines, so $\alpha_m = \sin\theta_m \cos\phi_0$, $\alpha_i = \sin\theta_0 \cos\phi_0$, and $\beta_i = -\sin\phi_0$. To emphasize, these equations apply to the general case, including the non-paraxial ("conical") diffraction grating case where the incident beams are oblique.

Intuitive as these are, the equations still make use of trigonometrical functions, which introduce numerical instability, and furthermore are not conducive to vectorization. This library avoids altogether the use of trigonometric functions in computing diffraction patterns.

Assumptions and notations

In a lab we captured diffraction patterns induced by monochromatic lasers using the wide-angle lens of an Android camera that had diffraction gratings affixed. The goals were to determine the wavelength of the lasers, and undistort(rectilinearize) the captured images.

Assumptions:

1. The camera behaves like a perfect pin-hole camera (in particular images present no distortions apart from the diffraction distortion)
2. The camera's sensor is perfectly symmetric

In the figure above, red, green, and blue arrows represent the $x$-, $y$-, and $z$-axis respectively. The $xy$ plane in this scheme represents the direction cosine space, which we denote as $Q$ space. The camera's sensor, which floats above and is parallel to $Q$, is assumed to be distance $f$ away from $Q$, where $f$ is the focal length of the camera. We denote that plane as $R$. Finally, the set of points on the unit hemisphere will be labeled as $S$.

The figure shows 9 points (red) $R_1, \ldots, R_9 \in R$, which are part of a diffraction pattern. The points ${R_7,\ R_8,\ R_9,\ R_6,\ R_3 }$ can be the $m$th order diffraction of a laser, in which case the set ${R_4,\ R_5,\ R_2}$ can be considered the $(m-1)$ th order, and ${R_1}$ the $(m-2)$ th order diffraction. These points are mapped to $S$ by dividing them by their 3D vector norms. Viz., $R_i \mapsto R_i/||R_i|| \in S$ (green dots). The mapping $S\rightarrow Q$ in turn is a simple projection: $S_i = (x_i, y_i, z_i) \mapsto (x_i, y_i)$ (blue dots).

In theory, if $R_i, i=1,\ldots 9$ were from diffraction, their corresponding points $Q_i, i=1,\ldots 9$ must form a perfectly rectilinear grid in $Q$, with spacing $\lambda/d$, where $\lambda$ is the wavelength of the laser. The mapping $Q\rightarrow R$ simply reverses these steps.

It is no coincidence that the math involved resembles that of camera matrix computations using homogeneous coordinates.

Imperfect gratings

In practice there are imperfections. The captured diffraction patterns exhibited characteristics that were results of rotation, shearing, and translation. My initial attempts in finding the right transformation was sequential - for instance finding the rotation first and then the remaining transformations. It turned out that errors from each step accumulated. Thus in the end the correct method was to find a single projective transformation that modeled all the deformations in $Q$ space at once.

Modules in this package can be used to undistort (convert) the first image below to the second.

For details of the procedure, see application2 live script. (live script is matlab's equivalent to jupyter notebook)

To generate basic diffraction patterns, see basic_usage.m.

References

Description of Diffraction Grating Behavior in Direction Cosine Space:

@article{Harvey1998,
  doi = {10.1364/ao.37.008158},
  url = {https://doi.org/10.1364/ao.37.008158},
  year = {1998},
  publisher = {Optica Publishing Group},
  volume = {37},
  number = {34},
  pages = {8158-8159},
  author = {James E Harvey and Cynthia L Vernold},
  title = {Description of Diffraction Grating Behavior in Direction Cosine Space},
  journal = {Applied Optics}
}

Diffraction grating equation (wikipedia)