Second-harmonic generation, usually abbreviated SHG, is one of the most important processes in nonlinear optics.
In SHG, light at angular frequency \(\omega\) interacts with a nonlinear crystal and generates light at angular frequency \(2\omega\).
Since frequency doubles, the wavelength is halved:
\(\lambda_{2\omega}=\lambda_\omega/2\).
This is the basic mechanism behind frequency doubling, such as converting infrared laser light into visible green output.
The physical origin of SHG is the second-order nonlinear polarization,
\[
P^{(2)}(2\omega)=\epsilon_0\chi^{(2)}E(\omega)^2,
\]
where \(\chi^{(2)}\) is the second-order nonlinear susceptibility.
This nonlinear polarization acts as a source term that drives radiation at twice the original frequency.
The larger the fundamental electric field, the stronger the generated second-harmonic field.
Because optical intensity is proportional to the square of the field amplitude, the SHG signal scales quadratically with the input intensity.
In the simplest undepleted-pump, plane-wave model, the second-harmonic intensity obeys
\[
I_{2\omega}\propto I_\omega^2L^2\operatorname{sinc}^2\!\left(\frac{\Delta kL}{2}\right),
\]
where \(I_\omega\) is the fundamental intensity, \(L\) is the crystal length, and \(\Delta k\) is the phase mismatch.
The function
\(\operatorname{sinc}(x)=\sin x/x\)
describes the reduction in conversion efficiency caused by imperfect phase matching.
The phase mismatch is typically written as
\[
\Delta k = k_{2\omega}-2k_\omega,
\]
where \(k_\omega\) and \(k_{2\omega}\) are the wave numbers of the fundamental and second-harmonic waves inside the crystal.
If \(\Delta k=0\), the process is perfectly phase matched, and the generated second-harmonic wave builds up coherently along the entire crystal.
In that ideal case the sinc factor becomes 1, so the scaling simplifies to
\[
I_{2\omega}\propto I_\omega^2L^2.
\]
This shows two central features immediately:
doubling the input intensity increases the SHG signal by a factor of four, and doubling the crystal length increases the ideal signal by a factor of four as well.
Under this simple model, the conversion can therefore rise very quickly when both intensity and interaction length are increased.
If \(\Delta k\neq 0\), the nonlinear polarization and the generated second-harmonic wave gradually slip out of phase.
Then the signal no longer grows perfectly as \(L^2\), and the sinc² factor suppresses conversion.
A useful quantity in this context is the coherence length,
\[
L_c=\frac{\pi}{|\Delta k|},
\]
which represents the distance over which the generated wave stays approximately in phase with the nonlinear driving polarization before beginning to reverse its growth.
In this calculator, the sample case uses
\(I_\omega=1\,\text{GW/cm}^2\),
\(L=1\,\text{cm}\),
and
\(\Delta k=0\).
Then
\[
\operatorname{sinc}^2\!\left(\frac{\Delta kL}{2}\right)
=
\operatorname{sinc}^2(0)=1,
\]
so the normalized second-harmonic output is simply proportional to
\[
I_\omega^2L^2=(1)^2(1)^2=1.
\]
The relative conversion factor is therefore proportional to
\(I_\omega L^2\),
exactly as expected for perfect phase matching in this simplified treatment.
It is important to note that this calculator reports a normalized SHG signal, not an absolute laboratory power.
To predict an actual output intensity, one would also need the effective nonlinear coefficient, refractive indices, focusing conditions, beam area, losses, and a more complete coupled-wave model.
Still, the present formula is extremely useful because it isolates the main dependences on intensity, crystal length, and phase mismatch.
At a more advanced university level, SHG theory includes birefringent phase matching, quasi-phase matching, pump depletion, beam walk-off, group-velocity mismatch, and focused Gaussian beams.
But the simple preview model already captures the essential message:
strong SHG requires high intensity, long interaction length, and phase matching as close to \(\Delta k=0\) as possible.
\[
I_{2\omega}\propto I_\omega^2L^2\operatorname{sinc}^2\!\left(\frac{\Delta kL}{2}\right),
\qquad
\Delta k=0 \Rightarrow I_{2\omega}\propto I_\omega^2L^2.
\]
These formulas let you estimate how strongly SHG should grow, how much phase mismatch suppresses it, and why frequency-doubling crystals are so sensitive to phase-matching conditions.
