7. Examples.
7.1. Interference examples.
When two or more waves overlap each other in space a more or less complicated pattern of fringes can be observed on a screen placed in the overlap area. These fringes are the result of superposition of the fields, which cause local enhancement or decrement of the intensity depending on the phase and amplitude distribution of the waves in the observation regio. In this chapter we will demonstrate a number of interference examples.
7.2. Diffraction examples.
7.3. Non-diffracting Airy beams.
An Airy beam will propagate without diffraction and keeps its shape
during long propagation distances. That makes it interesting for a
large number of applications.
A special feature of the Airy beam is that it accellerates in the
transverse direction by its self, although the center of
gravity propagates as a straight line as it should be.
From LightPipes for Python version 2.1.4. there are two new commands which
substitude an Airy beam into the field. A one- and a two-dimensional
Airy beam: AiryBeam1D and AiryBeam2D
For further reading we recommend the
open access review article by Yiqi Zhang et al. 1
Below we demonstrate how an Airy beam can be generated using a Gaussian laser beam, a spatial light modulator (SLM) and a Fourier transform with a postive lens. The SLM substitutes a cubic phase into the field. A non-diffracting 2D Airy beam will exist along a long distance from the focus of the lens.
References:
- 1
Yiqi Zhang, Hua Zhong, Milivoj R., Belić and Yanpeng Zhang, Appl. Sci. 2017, 7(4), 341; https://doi.org/10.3390/app7040341
7.4. Non-diffractive Bessel beam.
A Bessel beam has the interesting property that it does not diffract and that it keeps its shape over large distances. Several meters, depending on parameters, can be realized. Applications of Bessel beams take advantage of the very large size of the focus, which cannot be obtained using lenses or mirrors. For example generation of a long narrow plasma channel can be realized using a high-power laser beam converted into a Bessel beam by an axicon lens.
Besides an axicon, a combination of an annular slit and a positive lens or concave mirror can be used in stead 2. In the following example a Poisson spot is generated by illuminating a circular disk by a plane mono-chromatic beam. The disk is positioned in the primary focus of a positive lens so that the waves originating from the edge of teh disk will be collimated. By blocking the rest of the incoming beam only the edge waves remain which results in a non-diffracting Bessel beam.
Reference:
- 2
J.Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory.”, JOSA A, Vol. 4, Issue 4, pp. 651-654 (1987)
7.5. Laser examples.
- 7.5.1. Hermite Gauss modes.
- 7.5.2. Laguerre Gauss modes.
- 7.5.3. Laguerre doughnut modes.
- 7.5.4. Laser simulation, stable laser resonator.
- 7.5.5. Unstable laser resonator.
- 7.5.6. Transformation of a fundamental Gauss mode into a doughnut mode with a spiral phase plate.
- 7.5.7. Transformation of high order Gauss modes.
- 7.5.8. Gauss TEM00 mode described with geometric optics.
7.6. Phase recovery.
7.7. Zernike aberration.
Any aberration in a circle can be decomposed over a sum of Zernike polynomials. The Zernike command accepts four arguments: 1. The radial order n 2. The azimuthal order m. 3. The radius, R 4. The amplitude of the aberration.