Evanescent waves, wavevector, and Poynting vector

[wil3]

For an evanescent wave, in what direction does the wavevector k point? In several lectures that I've seen in class, it appears to point in some direction that is not normal or along the interface, which confuses me.

Additionally, for all wave vectors, what exactly is the magnitude of k? In the 1D case, I'm aware that it is a simply the wave number, but does it have any special meaning in other cases?

Finally, what is the relationship between k and the Poynting vector? I'm aware that they necessarily point in the same direction, but I'm curious if there's a relationship between their magnitudes.

These concepts were very poorly explained in lecture, and so I would appreciate any advice.

 

[Bill_K]

There are several circumstances where you find inhomogeneous or evanescent waves. In all of them the wave vector k is complex, implying that the wave is exponentially damped in some direction.

a) Total internal reflection, in which case the wave vector has a real component parallel to the surface and an imaginary one perpendicular to it.

b) Wave propagation in a conducting medium such as a metal. In this case the wave is exponentially damped in the same direction as the direction of propagation.

c) Some diffraction problems (see Born & Wolf, sect 11.4)

Which direction does k point and what is its magnitude? Well, k is complex! So it points in a complex direction. You can define in the usual way a phase velocity vector, index of refraction, dielectric constant, but they will all be complex too.

 

[johng23]

Finally, what is the relationship between k and the Poynting vector? I'm aware that they necessarily point in the same direction, but I'm curious if there's a relationship between their magnitudes.

Not necessarily true, actually. In an anisotropic material where the index of refraction varies with direction, the electric field may not be perpendicular to the propogation vector (although the displacement field will be). This leads to a Poynting vector which is not parallel to k.

 

[chrisbaird]

Not necessarily true, actually. In an anisotropic material where the index of refraction varies with direction, the electric field may not be perpendicular to the propogation vector (although the displacement field will be). This leads to a Poynting vector which is not parallel to k.

The Poynting vector can also point in the opposite direction from the wavevector in left-handed materials (meta-materials). The Poynting vector describes the direction of energy flow, and the wavevector describes the direction that the waveform is traveling.

A complex vector like k has real and imaginary parts, and each part has vector length (magnitude) and vector directionality. The vector length of the real part of the wavevector is just the wavenumber (a description of the spatial frequency of wave peaks). The vector directionality of the real part describes the directions the waveshape is traveling. The vector length of the imaginary part describes the rate at which the wave spatially decays, and the vector directionality of the imaginary part described the direction in which the wave is decaying.

 

[PJC01]

Evanescent waves are a type of electromagnetic wave that decay rapidly as they propagate away from their source. They are often found at the interface between two different materials, such as air and glass, and can be described by their wavevector and Poynting vector.

The wavevector, denoted by k, is a vector that represents the direction and magnitude of the wave's propagation. In the case of evanescent waves, the wavevector points parallel to the interface between the two materials. This is because the wave is confined to the interface and does not propagate into either material. This direction may appear confusing because it is not normal or along the interface, but it is still parallel to it.

The magnitude of the wavevector, k, is related to the wavelength of the wave, λ, and the refractive indices of the two materials, n1 and n2, by the equation k = (n1/n2) * (2π/λ). In the 1D case, where the wave is propagating in a single direction, k is simply the wave number, which represents the number of waves per unit distance.

The Poynting vector, denoted by S, is a vector that represents the direction and magnitude of the energy flow of the wave. It is related to the wavevector by the equation S = (1/μ0) * k x E x B, where μ0 is the permeability of free space, E is the electric field vector, and B is the magnetic field vector. This equation shows that the Poynting vector and wavevector are related, but their magnitudes are not necessarily the same. The Poynting vector represents the energy flow of the wave, while the wavevector represents the direction and magnitude of the wave's propagation.

In summary, the wavevector points parallel to the interface for evanescent waves, its magnitude is related to the wavelength and refractive indices, and it is related to the Poynting vector but their magnitudes may differ. I hope this helps clarify these concepts for you. If you have any further questions, please do not hesitate to ask.