Differential Equations of Complex Variables

 Several years ago there was a question on the Igor Mail List about solving in Igor ODEs having complex variables. John Weeks replied that you have to decompose the system into 2N real variables. I have just had the need to do this, and found that a systematic approach does not require full decomposition by the user, and Igor complex number functions can reduce the burden of doing so. Here is a brief code example showing how to maintain much of the complex nature of the internal variables in the ODE derivative routine, with only minimal additional burden. The example happens to be for phase-matched optical parametric amplification including pump depletion (not generally treated in texts). All units and variables are normalized.

Function OPA(pw, zz, yw, dydz)
	Wave	pw      // pw[0] = imK1, pw[1] =imK2, pw[2] = imK3
	Variable  zz      // z value at which to calculate derivatives
	Wave 	yw      // yw[0]-yw[5] containing real and imag parts of 3 complex variables
	Wave 	dydz   // wave to receive d reE1/dz, d ImE1/dz etc. (output)

	//             idler                      signal                              pump	
	variable/C E1 = cmplx(yw[0],yw[1]) , E2 = cmplx(yw[2],yw[3]), E3 = cmplx(yw[4],yw[5])
	variable/C K1 = cmplx(0,pw[0]),	     K2 = cmplx(0,pw[1]),     K3 = cmplx(0,pw[2])
	
	dydz[0] =  real( K1 * conj(E2) * E3 )     //   idler		
	dydz[1] =  imag( K1 * conj(E2) * E3 )	

	dydz[2] =  real( K2 * conj(E1) * E3 )     //   signal
	dydz[3] =  imag( K2 * conj(E1) * E3 )

	dydz[4] =  real( K3 * E1 * E2 )           //   pump
	dydz[5] =  imag( K3 * E1 * E2 )
	return 0
End
//---------------------------------------------------------------------------------------------------------------------------------------
Function testOPA()

	Make/D/O/N=(400,6) wopa
	setscale/P x, 0, 0.005, wopa
	wopa[0][0]  =  0.0	   //	real idler	   //  initial conditions at z=0
	wopa[0][1]  =	  0.0   //	imag idler
	wopa[0][2]  =	  0.2   //	real signal
	wopa[0][3]  =	  0.0   //	imag signal
	wopa[0][4]  =	  5     //	real pump
	wopa[0][5]  =	  0.0   //	imag pump
	make/D/O wparm = {(1/3), 1.000, (4/3)}             // ratios of inverse wavelengths
	IntegrateODE/M=1/E=1e-6 OPA, wparm, wopa  // derivative function, parameters, results
end

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