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Laser: Second harmonic generation

  1. Jan 23, 2010 #1
    Hi guys

    I'm reading a section in a book about non-linear optics. They have an expression for the polarization for the second harmonic field. They say that this expression has a linear and a non-linear contribution at frequency 2ω, but they don't explain this any further.

    I am a little confused about this statement, because we have light at frequency ω going into the medium. This light rocks the electrons at ω (linear) and at 2ω (non-linear). So when we write an expression for the polarization of the electrons oscillating at 2ω, then where does the linear term come from?

    I'm really confused about this, so I would be very happy, if you could solve this mystery.

  2. jcsd
  3. Jan 29, 2010 #2
    Let me check if I understand: You have incoming light at frequency [tex]\omega[/tex]. It interacts with electrons in some matter structure. Since the electrons are not free, their vibrations are not purely sinusoidal/harmonic. So you choose to approximate this as a motion that can be represented as a Fourier sum of two terms, one component with frequency [tex]\omega[/tex] and one component with frequency [tex]2\omega[/tex]? This would be the first approximation of non-harmonic motion for the electrons.

    The radiation coming from those vibrating electrons will generate the outgoing non-linearized scattered light which is what you are interested in? Since the movement of the electrons can be written as a sum of two terms, I'm guessing that the resulting radiation from them can also be written as a sum of two contributions, at least as an approximation? Thus the result is a sum of contributions at frequency [tex]\omega[/tex] and [tex]2\omega[/tex].

    I think I may have misunderstood something, though :-)

  4. Jan 29, 2010 #3


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    Maybe the linear term is for the 2 omega in -> 2 omega out, i.e., the usual linear relation between a field E(2 omega) and P(2 omega)?
  5. Jan 29, 2010 #4
  6. Jan 29, 2010 #5
    Second harmonic generation occurs with varying levels of efficiency.
    I used to look at SHG generated on surfaces to help characterize the surfaces... in particular metal surfaces (which are a bit more efficient than non-metals). Generally, you think of hitting a mirror with a light source (like a laser) and you see reflected light off the surface of the same frequency. This is the linear term -- and it's the most efficient. However, if the light is very high-amplitude (like a pulsed laser) you can excite the electrons in a material (especially those at the surface) to have higher frequency components to their motion, and hence (like little antennas) reradiate out light of that higher frequency (subject to boundary constraints, and material efficiencies).

    Of course this can occur in the bulk as well... the linear term is just your transmission term for the linear light (the common phenomena)... and some materials are again better than others for getting out higher frequency conversion terms.
  7. Jan 29, 2010 #6
    I wonder why the default case is not third harmonic. Even harmonics only occur when there is something non-symmetric in the restoring force. For example, a high-amplitude pendulum would resolve into only odd harmonics because it is non-linear but symmetric.
  8. Jan 29, 2010 #7
    Ok, what they describe in the book is just that when the second harmonic is generated, it itself makes the electrons rock at 2ω, and this is the linear term.

    Thanks for participating.
  9. Feb 1, 2010 #8

    Claude Bile

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    What the book may be saying is that the polarisation at 2ω is due to the nonlinear contribution of the ω wave component and the linear contribution of the 2ω wave component.

    i.e. P(2ω) = C*E(ω)*E(ω) + D*E(2ω).

    @Conway; You are right, most materials do not possess 2nd-order nonlinearities, but all materials possess a 3rd-order nonlinearity. Only crystals with non-centrosymmetric geometries possess an appreciable 2nd-order nonlinearity.

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