Haus Theory of Mode-Locking: How Does Equation 2 Relate to Equation 4?

[Spectrum81]

This question is related to the 1974 Hermann Haus paper "Theory of mode-locking with a slow saturable absorber"
It is not clear how he passes from equation 2 to equation 4 (I'm using the same references as in the paper).

Homework Statement

$$\omega_{k}$$ is the varying frequency, $$\omega_{0}$$ is the spectrum peak frequency
v2($$\omega_{k}$$-$$\omega_{0}$$) is the pulse envelope spectrum v1($$\omega_{k}$$-$$\omega_{0}$$) after time T2+T3 and passing through a bandwidth limiting element which transfer function is:

H($$\omega_{k}$$)=exp[-($$\omega_{0}$$*Tr/4Q)(1+(($$\omega_{k}$$-$$\omega_{0}$$)/$$\omega_{c}$$)$$^{2}$$)]

Tr is the cavity round trip time
Q is the cavity Q factor
$$\omega_{c}$$ is the width of the loss "well"
j$$^{2}$$=-1

Homework Equations

equation 2:
v2($$\omega_{k}$$-$$\omega_{0}$$)=exp[-j($$\omega_{k}$$-$$\omega_{0}$$)(T2+T3)].H($$\omega_{k}$$).v1($$\omega_{k}$$-$$\omega_{0}$$)

He then says that multiplication of v($$\omega_{k}$$-$$\omega_{0}$$) by j($$\omega_{k}$$-$$\omega_{0}$$) corresponds to operation by (d/dt -j$$\omega_{0}$$) on v(t)exp(j$$\omega_{0}$$*t) which is true.

But then he says that equation 2 in the time domain is:

equation 4:
v2(t)=exp[(-$$\omega_{0}$$*Tr/4Q)(1-(1/$$\omega_{c}$$$$^{2}$$)(d$$^{2}$$/dt$$^{2}$$))]v1(t-T2-T3)

the d$$^{2}$$/dt$$^{2}$$ argument origin is not explained.

The Attempt at a Solution

It is maybe a matter of approximation

 

[lippyka]

of H(\omega_{k}) by a Taylor series around \omega_{0}.But I don't get the coefficient of the second order and where it comes from.