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Haus theory of mode-locking

  1. Sep 7, 2009 #1
    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).

    1. The problem statement, all variables and given/known data

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

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

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

    2. Relevant equations

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

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

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

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

    the d[tex]^{2}[/tex]/dt[tex]^{2}[/tex] argument origin is not explained.

    3. The attempt at a solution
    It is maybe a matter of approximation
     
    Last edited: Sep 7, 2009
  2. jcsd
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