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 [
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.
E
dif f
(
t
) =
E
in
(
t
)
⊗
S
(
t/α
)
(2.5)
In the frequency domain, this convolution relation can be written:
E
dif f
(
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
dif f
(
ω
) =
H
(
ω
)
E
in
(
ω
)
(2.8)
and:
H
(
ω
) =
p
η
(
ω
) exp[i
φ
(
ω
)]
(2.9)
where
η
(
ω
) is the AOPDF diffraction efficiency (
) 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
dif f
(
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
th
April 2019
