The Nonparaxial Gaussian Beam Formula for Simulating Wave Optics
In a previous blog post, we discussed the paraxial Gaussian beam formula. Today, we’ll talk about a more accurate formulation for Gaussian beams, available as of version 5.3a of the COMSOL® software. This formulation based on a plane wave expansion can handle nonparaxial Gaussian beams more accurately than the conventional paraxial formulation.
Paraxiality of Gaussian Beams
The well-known Gaussian beam formula is only valid for paraxial Gaussian beams. Paraxial means that the beam mainly propagates along the optical axis. There are several papers that talk about paraxiality in a quantitative sense (see Ref. 1).
Roughly speaking, if the beam waist size is near the wavelength, the beam propagates at a higher angle to a focus. Therefore, the paraxiality assumption breaks down and the formulation is no longer accurate. To alleviate this problem and to provide you with a more general and accurate formulation for general Gaussian beams, we introduced a nonpariaxial Gaussian beam formulation. In the user interface this is referred to as Plane wave expansion.
Angular Spectrum of Plane Waves
Let’s briefly review the paraxial Gaussian beam formula in 2D (for the sake of better visuals and understanding).
We start from Maxwell’s equations assuming time-harmonic fields, from which we get the following Helmholtz’s equation for the out-of-plane electric field with the wavelength \lambda for our choice of polarization:
where k=2 \pi/\lambda.
The angular spectrum of plane waves is based on the following simple fact: an arbitrary field that satisfies the above Helmholtz equation can be expressed as the following plane wave expansion:
where A(k_y) is an arbitrary function.
The integration path is a circle of radius k for real k_x and k_y. (For complex k_x and k_y, the integration domain extends to a complex plane.) The function A(k_y) is called the angular spectrum function. One can prove that this E_z satisfies Helmholtz’s equation by direct substitution.
Now that we know that this formulation always gives exact solutions to Helmholtz’s equation, let’s try to understand it visually. From the constraint, k_x^2+k_y^2=k^2, we can set k_x=k cos(\varphi) and k_y=k sin(\varphi) and rewrite the above equation as:
The meaning of the above formula is that it constructs a wave as a sum, or integral, consisting of many waves propagating in various directions, all with the same wave number k. This is shown in the following figure.
Visualization of the angular spectrum of plane waves.
Back to k_y representation, when actually solving a problem using this formula, all you have to do is find the angular spectrum function A(k_y) that satisfies the boundary conditions. By assuming that the profile of the transverse field (perpendicular to the propagating direction, i.e., optical axis) is also a Gaussian shape (see Ref. 4), one can derive that A(k_y) = \exp(-k_y^2 / w_0^2) , where w_0 is the spectrum width.
By some more mathematical manipulations, we get a relationship between the spectrum width w_0 and the beam waist radius. For example, for a slow Gaussian beam, the angular spectrum is narrow. A plane wave, on the other hand, is the extreme case where the angular spectrum function is a delta function. For a fast Gaussian beam, the angular spectrum is wider, and vice versa.
This was a quick summary of the underlying theory for nonparaxial Gaussian beams. To recap what we have shown so far, let’s rewrite the formula once more by using polar coordinates, x=r \cos \theta, \ y = r \sin \theta:
This is the formulation that Born and Wolf (Ref. 2) use in their book.
The 3D formula is more complicated and looks different due to polarization, but the basic idea is the same as seen in the references mentioned above. It can also look different depending on whether or not you consider evanescent waves. The Plane Wave Expansion method used in the Wave Optics Module and the RF Module, although based on the angular spectrum theory, is adapted for numerical computations.
Plane Wave Expansion: Settings and Results
Let’s compare the new feature, Plane wave expansion, with the previously available feature, Paraxial approximation. The Settings window covering both methods is shown below.
The Plane Wave Expansion feature settings.
With the new feature, you have two options if the Automatic setting doesn’t give you a satisfactory approximation:
- Wave vector count
- Maximum transverse wave number
The first option determines the number of discretization levels, depending on how fine you want to represent the Gaussian beam. The more plane waves, the finer it gets. The second option is related to the integral bound in the previous equation; i.e., -\pi/2 \le \varphi \le \pi/2. This integral bound can be the maximum \pi/2 for the smallest possible spot size and can be more shallow for slower beams, depending on how fast the Gaussian beam is. You need more angled plane waves with a larger transverse wave number to represent faster (more focused) beams.
The following results compare the two formulas for the case where the spot radius is \lambda/2, which is considerably nonparaxial. As in the previous blog post, the simulation is done with the Scattered Field formulation and the domain is surrounded by a perfectly matched layer (PML). This way, the scattered field represents the error from the exact Helmholtz solution.
The left images below show the new feature, while the images on the right show the paraxial approximation. The top images show the norm of the computed Gaussian beam background field,
ewfd.Ebz, while the bottom images show the scattered field norm,
ewfd.relEz, which represents the error from the exact Helmholtz solution. Obviously, the error from the Helmholtz solution is greatly reduced in the nonparaxial method.
We have discussed the theory and results for an approximation method for nonparaxial Gaussian beams using the new plane wave expansion option. Remember that this formulation is extremely accurate, but is still an approximation under assumptions. First, we have made an assumption for the field shape in the focal plane. Second, we assume that the evanescent field is zero. If you are interested in the field coupling to some nanostructure near the focal region in a fast Gaussian beam, you may need to calculate the evanescent field.
Learn more about the formulations and features available for modeling optically large problems in the COMSOL® software by clicking the button below:
Note: This functionality can also be found in the RF Module.
- P. Vaveliuk, “Limits of the paraxial approximation in laser beams”, Optics Letters, vol. 32, no. 8, 2007.
- M. Born and E. Wolf, Principles of Optics, ed. 7, Cambridge University Press, 1999.
- J. W. Goodman, Fourier Optics.
- G. P. Agrawal and M. Lax, “Free-space wave propagation beyond the paraxial approximation”, Phys. Rev. a. 27, pp. 1693–1695, 1983.
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