Do evanescent waves span a vector space?

In summary, the conversation discusses the differences between propagating and evanescent waves in the context of wave equations such as Poisson, Laplace, and Schrodinger. The completeness and orthogonality relations for propagating waves are well-known, but there is uncertainty about how evanescent waves fit into this framework. The definition of evanescent waves and the appropriate scalar product play a role in determining whether they can satisfy the completeness condition. There is also a discussion about the symmetry between propagating and evanescent waves and the possibility of expanding any function on [0, oo> that goes to zero at oo in the evanescent wave basis.
  • #1
Heirot
151
0
In dealing with Poisson, Laplace, Schrodinger and other wave equations, one has to deal with propagating and evanescent waves. We know all about the propagating waves - orthogonality and completeness relations, but what about evanescent waves? Do they form a vector space with corresponding identities?
 
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  • #2
I wonder how evanescent waves could satisfy the completeness condition. It seems quite unlikely.
 
  • #3
That depends on your precise definition of evanescent waves, then on the appropriate definition of the scalar product, which will be rather a different one than the usual one. You will get a pre-Hilbert space and, after the completion - a Hilbert space. Because of the different scalar product - the identities will be different.
 
  • #4
By evanescent waves, I mean Exp[-k x] with both k and x real.
 
  • #5
Just "real", or "positive"?
 
  • #6
For now, I'm going to go with "real". I want to allow all possibilities.
 
  • #7
Are you in a finite box or in an infinite space? When you say "othogonality" relations for waves, what do you mean by these. Waves are not square integrable in infinite space.
 
  • #8
I have a problem with the lack of symmetry between propagating and evanescent waves. Propagating waves can be treated in infinite space in the distributional sense, so that they span the "plane wave" basis. Why can't I expand any function on [0, oo> that goes to zero at oo in the evanescent wave basis?
 

1. What are evanescent waves?

Evanescent waves are a type of electromagnetic wave that propagates parallel to a surface, but decays exponentially as it moves away from the surface. They are often associated with total internal reflection and are commonly found in optical systems.

2. Do evanescent waves span a vector space?

Yes, evanescent waves do span a vector space. This means that they can be represented as a linear combination of other vectors within the space. In the case of evanescent waves, the vectors represent the electric and magnetic fields of the wave.

3. How are evanescent waves different from propagating waves?

Evanescent waves and propagating waves differ in their behavior as they move away from a surface. While propagating waves continue to propagate and carry energy, evanescent waves decay exponentially and do not transport energy. Additionally, evanescent waves have a shorter wavelength and higher intensity near the surface compared to propagating waves.

4. What are some practical applications of evanescent waves?

Evanescent waves have a variety of applications in optics, such as in near-field microscopy, optical sensing, and data storage. They are also used in wireless power transfer, where the near-field component of evanescent waves can transfer energy to a nearby receiver.

5. Can evanescent waves be manipulated or controlled?

Yes, evanescent waves can be manipulated and controlled through various methods, including changing the angle of incidence, using materials with different refractive indices, and applying external fields. This allows for the control of the intensity, wavelength, and direction of evanescent waves, making them useful in a variety of applications.

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