If I use non-zero initial amplitudes for all three frequencies (say w1,w2,w3) then real parts of the solutions appear with changing amplitudes for all (as expected). But when I use zero initial amplitude for any one of w (say w1) and non-zero for other two (say w2,w3), the real part of solutions for w1 remains zero while that for w2 and w3 is non-zero. However, the imaginary parts for w1 are non-zero causing changes in amplitudes for w2 and w3!
I expect that field amplitude must be real not imaginary to say that a new frequency is generated. Hence, based on my results these equations rather describe only the interaction of existing frequencies and not their generation (from zero, of course generation is there for non-zero). Am I correct or not?
Equations used:
c=0;
dy(1) =1i *conj(y(2)) * y(3) * exp(1i*c*t);
dy(2) =1i *conj(y(1)) * y(3) * exp(1i*c*t);
dy(3) =1i *y(1)*y(2) * exp(-1i*c*t);
Solver ode45 (from MATLAB)
Why do you discount the imaginary part of the complex field amplitude?
You do not provide details of how your equations are derived. The usual convention is that a complex amplitude coefficient describes both magnitude and phase of the field. The real part is the field component in phase with a reference at the same frequency, and the imaginary part is in quadrature (i.e. with 90 degrees phase difference to the reference). Intensity is proportional to the square of the absolute value [ y * conj(y) ].
Does 't' represent time, propagation distance, or some normalised propagation quantity in a moving frame of reference - presumably normalised to the SHG non-linear coefficient?
You wish to start with real y2, y3, and with y1 initially zero. While the absolute value of y1 remains small, its imaginary part will grow, the phase of y2 will increase with little initial change in the magnitude of y2, and power will be transferred from y3 to y1. As the phase of y2 increases, and the magnitude of y1 grows, there will be an increasing rate of power transfer from y3 to y2.
Regards,
Alan.
As per the definition [ y * conj(y) ].for intensity, it seems right (to me now, probably i ignored this while trying to visualize only real field amplitudes) that a new frequency is generated, so my reasoning in second para (of question description) is incorrect.
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