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plane wave expansion method
Posted 31 oct. 2020, 13:22 UTC−4 Wave Optics Version 5.4 2 Replies
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Hello, I'd like to ask a few questions about plane wave expansion method in the wave optics module and some general questions.
How does Comsol deal with the paraxial Gaussian beam when the beam waist is very small? According to the equations in the User's guide (pages 154-155), there is an explicit result for the amplitude of the electric field. However when I examine the E field profile in the focal plane, the Comsol result does not always match this explicit result. The disagreement becomes worse when the beam waist parameter, wo, becomes less than the medium's wavelength. Is Comsol "correcting" the paraxial Gaussian beam? If so, how?
Are the Comsol Multiphysics files of version 5.4 is compatible in version 5.5?
According to a Comsol blog post (www.comsol.com/blogs/evanescent-component-of-the-nonparaxial-gaussian-beam/), the author shows the evanescent contribution only when z>0 (beyond the focal plane). What is done for z<0?
When I looked for the explicit equations of the plane wave expansion method in the User's guide, the Comsol blogs and reference (Chaumet's paper), they all seem different. What equations in the plane wave expansion method are used for both the electric and magnetic fields? The (sole) component of the electric field in the focal plane is specified by the parameters Eo and wo. Is the functional form of this component in the focal plane preserved by the plane wave expansion method?
It is expected that the beam waist of a focused Gaussian beam cannot be smaller than a diffraction limit, but I don't find how Comsol deals with this limitation. Is there a definite limit on beam size? If so, what constraint does this impose on the choice of the parameter wo?
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Dear Suhun Jo,
Below I will answer your questions:
- The paraxial background field is given by the analytical formulas provided in the User's guide. The formulas are different for 2D and 3D. If you plot the background field variables, you should find that it matches those analytic formulas. However, the paraxial wave equation is just an approximation to the full vector Helmholz equation. This approximation gets worse the smaller the spot size is compared to the wavelength. Since COMSOL solves the full vector Helmholz equation, the full field will deviate more and more from the background field the smaller the spot size is compared to the wavelength. COMSOL is not "correcting" the paraxial beam background fields.
- Yes. COMSOL 5.5 adds handling of evanescent waves, but the functionality from COMSOL 5.4 still works.
- Evanescent waves are launched from a boundary. As you move away from that boundary the evanescent waves decays exponentially with a different decay constant depending on the transverse (tangential to the boundary) spatial frequency content. Thus, a few decay constants into the medium it is only the propagating plane waves that are left in the medium. Would you take this mathematical solution for z > 0 and use it for z < 0, the evanescent waves will increase exponentially as z gets smaller. Thus, as z gets smaller, the evanescent waves will be dominating. However, the domain z < 0 is not a physical domain. As said, the evanescent waves are created at a boundary, like at a tiny aperture.
- The expressions used in COMSOL for the plane wave expansion method are found at the end of the Gaussian Beams as Background Fields section in the User's guide. Notice that the plane wave expansion method is just a method to fit the Gaussian field distribution for the tangential components of the electric field in the focal plane. To fit the Gaussian distribution, you need a sufficient number of plane waves and they should also have suitable tangential wave vectors to be able to fit and resolve the Gaussian distribution.
- When using the paraxial Gaussian analytic field distribution, as this is an approximation to the full vector wave equation, this should not be used when the spot size start to approach the wavelength. As a safety margin, the default w0 value is 10 times the wavelength. However, even that w0 value could be too large, depending on your application (i.e. depending on how weak your scattered field is). When using the plane wave expansion and including evanescent waves, you could use spot size values smaller than the wavelength, if you put the focal plane at your input boundary. As already discussed, it doesn't make sense locating the focal plane in the middle of a domain, if you use spot sizes smaller than the wavelength. In this case your field will be totally dominated by the evanescent field decaying towards the focal plane. So to answer your question, there is no definite minimum limit for the spot size in COMSOL.
Best regards,
Ulf Olin
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Thank you VERY mush Ulf :) It helps a lot!