Use Green's Function calculate photonic density of state

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Discussion Overview

The discussion revolves around the calculation of the local photonic density of states using Green's function in the context of quantum optics. Participants explore the implications of the imaginary part of the Green's function, particularly in relation to materials with negative dielectric functions, such as metals.

Discussion Character

  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant suggests that the local photonic density of states can be derived from the imaginary part of the electromagnetic Green's function, presenting a formula involving the electric field profile, dipole moment, dielectric function, and frequency.
  • Another participant expresses confusion regarding the calculation process, noting that both the electric field and dielectric function have real and imaginary components, which could lead to a negative imaginary part of the Green's function in metal areas, raising the question of whether this is reasonable.
  • Some participants reference specific equations (8.111, 8.113, 8.114) and express skepticism about the possibility of obtaining a negative value, suggesting that the author may have avoided addressing singularities in the derivation.
  • One participant advises that proper use of the Bromwich contour is necessary to account for the imaginary part, drawing a parallel to Landau damping in plasma physics.
  • Another participant mentions a specific equation (16.29) related to the volume-integral method for calculating Green's function, indicating that the dielectric function can be negative in metal materials, which could lead to a negative imaginary part of the Green's function.
  • Several participants express difficulty in locating specific equations in the provided reference material, indicating discrepancies in different editions of the text.

Areas of Agreement / Disagreement

Participants do not reach a consensus on whether the negative imaginary part of the Green's function is reasonable in metal areas, and there are competing views regarding the interpretation of the equations and the calculation methods involved.

Contextual Notes

There are references to specific equations and methods that may not be universally accessible, and the discussion highlights potential limitations in understanding the derivations and implications of the Green's function in different materials.

Jeffrey Yang
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Hi Everyone:

I think some of you who familiar with quantum-optics know that the local photonic density of state can be calculated by the imaginary part of electromagnetic Green's function.

The Green's function can be further presented by the dipole's mode pattern as

G = E(r)*p0*ε(r)*c^2/ω^2

, where E(r) is the electric field profile, p0 is the dipole moment, ε is the dielectric function, ω is the frequency

You can find these formulas in Lukas' book "Principle of nano-optics"

However, I'm confused by the calculation's process. E(r) contain both the real and imaginary part, and so dose ε. Therefore, the final imaginary part of G will contain the cross-product item.

The dielectric function will have a negative real part if there has metal material. But this will lead a negative imaginary part of Green's function in metal area, as also a "NEGATIVE DENSITY OF STATE"!

Dose this reasonable?
 
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Looking at equation 8.111, 8.113, 8.114, I don't see how it could be negative.
However, the track from 8.111 to 8.114 seems a little bit obscure.
IMHO, I think the author tried to avoid handling the singularity from the beginning (near 8.109).
IMHO, solving for the Green function must be done by using the Bromwich coutour properly in order to account for the imaginary part.
(this is similar to the way Landau damping in plasma physics is derived)

My best advice is to read about Landau damping and about the Bromwhich contour.
In this way, you will probably be able to clarify the origin of the imaginary part and see why it is always positive.
(in a stable media)
 
maajdl said:
Looking at equation 8.111, 8.113, 8.114, I don't see how it could be negative.
However, the track from 8.111 to 8.114 seems a little bit obscure.
IMHO, I think the author tried to avoid handling the singularity from the beginning (near 8.109).
IMHO, solving for the Green function must be done by using the Bromwich coutour properly in order to account for the imaginary part.
(this is similar to the way Landau damping in plasma physics is derived)

My best advice is to read about Landau damping and about the Bromwhich contour.
In this way, you will probably be able to clarify the origin of the imaginary part and see why it is always positive.
(in a stable media)

Thanks for your comments

My problem came from equation 16.29 which use the volume-integral method to calculate Green's Function. In this method, the system's Green's Function can be linked to the electric mode excited by a electric dipole. However, the dielectric function can be negative, if there is a metal material, and finally you will get a negative imaginary part of Green's function.

Of course, these negative imaginary part of Green's function will only exist in the region of metal.
Photonic density of state is negative in metal??
 
Last edited:

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