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In Ward & New (1969), the expression for second harmonic generation (SHG) and third harmonic generation (THG) intensity is derived for the focal volume of a strongly focused gaussian beam (axially thick medium).

The predicted value of the J-integral term in the expression for intensity of SHG under perfect phase matching conditions ($\Delta k=0$) is found to be π on page 184.

While for THG, the value of the J-integral term in the expression for intensity under perfect phase matching conditions ($\Delta k=0$) is found to be zero.

For THG, things seem to make sense. Third harmonic light generated before focus is born with a phase that is π radians out of phase with third harmonic light that is born after focus.

Since there is no phase slippage ($\Delta k$) of the funamental pump wave before and after focus except for that caused by the Gouy phase shift, the third harmonic light destructively interferes since it's π radians out of phase.

I don't understand why if this logic works with THG, it doesn't work for SHG. What am I missing?

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  • $\begingroup$ Consider to spell out acronyms. $\endgroup$
    – Qmechanic ♦
    Commented Apr 3, 2023 at 4:11
  • $\begingroup$ Please clarify your specific problem or provide additional details to highlight exactly what you need. As it's currently written, it's hard to tell exactly what you're asking. $\endgroup$
    – Community Bot
    Commented Apr 3, 2023 at 4:57
  • $\begingroup$ Welcome to Physics! I've edited your post to explain the acronyms and include MathJaX coding. One question, though: is the intensity really π for SHG under perfect matching, or is it something else? (I'm not well-versed enough in non-linear optics to know, but it seems weird that this is a value without units.) If that's actually a typo, please edit your question appropriately to correct it. $\endgroup$
    – Michael Seifert
    Commented Apr 3, 2023 at 13:55
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    $\begingroup$ Thats because propagation is $\omega t - k r$ and then odd harmonics will have odd multiples of $\pi$ as a phase shift due to the gouy phase while even harmonics do not. Also consider looking into Boyd where he further shows for a negative phase mismatch (or was it positive?, whichever the refractive index of $3\omega$ is lower than for $\omega$) you again have a signal. Only in the case for positive up to 0 $\Delta k$ do you have perfect cancellation. $\endgroup$
    – José Andrade
    Commented Apr 25, 2023 at 13:36

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