DAZZLER
TM
system manual Part I : installation & operation 2.2
and extraordinary polarizations travel with different group velocity, each frequency will see a
different group delay.
The group delay τ applied to the diffracted pulse can be expressed by [1]:
τ(ω) = n
g
o
(ω)/c ∗ z(ω) + n
g
e
(ω)/c ∗ (L − z(ω)) (2.4)
where n
g
o
and n
g
e
are respectively the ordinary and extraordinary group indexes along the
propagation direction and L the crystal length.
Controlling for each optical frequency ω the position z(ω) where ω is diffracted enables to
control the pulse group delay. The amplitude of the output pulse is controlled by the acoustic
power at position z(ω).
Explanation in terms of time convolution
For low diffraction efficiency ( 100%), the optical output complex electric field E
diff
(t) is
proportional to the convolution of the optical input complex electric field E
in
(t) with the elec-
tric signal S(t/α) where α is the ratio between optical and acoustic frequencies (Eq. Equa-
tion 2.3)[1].
E
diff
(t) = E
in
(t) ⊗ S(t/α) (2.5)
In the frequency domain, this convolution relation can be written:
E
diff
(ω) = E
in
(ω) · S(αω) = E
in
(ω) · S(ω
ac
) (2.6)
The spectral phase of the diffracted optical pulse ϕ
diff
(ω) can be written:
ϕ
diff
(ω) = ϕ
ac
(ω
ac
) + ϕ
in
(ω) (2.7)
where ϕ
ac
and ϕ
in
are respectively the spectral phase of the acoustic and optical input waves.
This relationship shows that the spectral shaping is performed via a phase transfer from the
acoustic wave to the optical input one.
Let’s consider H(ω), the AOPDF optical transfer function defined by:
E
diff
(ω) = H(ω)E
in
(ω) (2.8)
and:
H(ω) =
p
η(ω) exp[iφ(ω)] (2.9)
where η(ω) is the AOPDF diffraction efficiency ( subsection 2.2.1) and φ(ω) the spectral phase
programmed in the software and applied to the input optical pulse.
The spectral phase of the diffracted optical pulse ϕ
diff
(ω) can be written:
ϕ
diff
(ω) = φ(ω) + ϕ
in
(ω) (2.10)
In the time domain, the optical complex electric fields E
in
(t) and E
diff
(t) can be linked by:
E
diff
(t) = E
in
(t) ⊗ h(t) (2.11)
where h is the Fourier-Transform of AOPDF optical transfer function H. It is the AOPDF
optical impulse time response.
Note: Due to the finite length of the crystal, the function h(t) is inherently clipped. For
this reason, the actual diffraction spectrum may differ from a programmed spectrum. This is
visualized by the black (programmed) and red (actual) curves in the spectrum window.
V3.00 - 8
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April 2019 (ContentsTable) (FiguresTable) 9/94
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