The following plot is the result of the calculation as a function of x normalized by the wavelength. The following definitions apply: w(x) = w_0\sqrt \left ( (\partial^2 A/ \partial x^2) / (2ik \partial A/\partial x) \right ) \right ). Where w(x), R(x), and \eta(x) are the beam radius as a function of x, the radius of curvature of the wavefront, and the Gouy phase, respectively. Assuming a certain polarization, it further reduces to a scalar Helmholtz equation, which is written in 2D for the out-of-plane electric field for simplicity: The time-harmonic assumption (the wave oscillates at a single frequency in time) changes the Maxwell equations to the frequency domain from the time domain, resulting in the monochromatic (single wavelength) Helmholtz equation. This is the first important element to note, while the other portions of our discussion will focus on how the formula is derived and what types of assumptions are made from it.īecause the laser beam is an electromagnetic beam, it satisfies the Maxwell equations. The paraxial Gaussian beam formula is an approximation to the Helmholtz equation derived from Maxwell’s equations. Deriving the Paraxial Gaussian Beam Formula When we use the term “Gaussian beam” here, it always means a “focusing” or “propagating” Gaussian beam, which includes the amplitude and the phase.
![cut off wavelength formula cut off wavelength formula](https://www.teamwavelength.com/wp-content/uploads/fourier-1024x535.png)
Note: The term “Gaussian beam” can sometimes be used to describe a beam with a “Gaussian profile” or “Gaussian distribution”. In a future blog post, we will discuss ways to simulate Gaussian beams more accurately for the remainder of this post, we will focus exclusively on the paraxial Gaussian beam.Ī schematic illustrating the converging, focusing, and diverging of a Gaussian beam. In other words, the formula becomes less accurate when trying to observe the most beneficial feature of the Gaussian beam in simulation. The limitation appears when you are trying to describe a Gaussian beam with a spot size near its wavelength. However, there is a limitation attributed to using this formula. There is a formula that predicts real Gaussian beams in experiments very well and is convenient to apply in simulation studies. To obtain the tightest possible focus, most commercial lasers are designed to operate in the lowest transverse mode, called the Gaussian beam.Īs such, it would be reasonable to want to simulate a Gaussian beam with the smallest spot size. These qualities are why lasers are such attractive light sources. Gaussian Beam: The Most Useful Light Source and Its Formulaīecause they can be focused to the smallest spot size of all electromagnetic beams, Gaussian beams can deliver the highest resolution for imaging, as well as the highest power density for a fixed incident power, which can be important in fields such as material processing. In a later blog post, we’ll provide solutions to the limitations discussed here. We’ll also provide further detail into a potential cause of error when utilizing this formula. Today, we’ll learn about this formula, including its limitations, by using the Electromagnetic Waves, Frequency Domain interface in the COMSOL Multiphysics® software. To describe the Gaussian beam, there is a mathematical formula called the paraxial Gaussian beam formula. The Gaussian beam is recognized as one of the most useful light sources.