P
5
ð Þ
9 ðx 1 ; x 2 Þ ¼ À
Nn
4
UV n VIS l rÃr
j
j
4 l pÃp
2
h
5
X
mnjkluvw
B m n j m
h
i n j j
h
i k j j
h
i l j m
h
i w j u
h
i w j v
h
i;
 L rÃn;rm x UV
ð
ÞD rj;rm x 1
ð ÞL rÃk;rm x UV
ð
ÞD pu;pv x 2
ð ÞL pÃw;pv x t
ð Þ
ð18Þ
P
5
ð Þ
10 ðx 1 ; x 2 Þ ¼ À
Nn
4
UV n VIS l rÃr
j
j
4 l pÃp
2
h
5
X
mnjkluvw
B m n j m
h
i n j j
h
i k j m
h
i l j j
h i w j u
h
i w j v
h
i;
 L rm;rÃn Àx UV
ð
ÞD rm;rj x 1
ð ÞL rÃk;rj x UV
ð
ÞD pu;pv x 2
ð ÞL pÃw;pv x t
ð Þ
ð19Þ
P
5
ð Þ
11 ðx 1 ; x 2 Þ ¼ À
Nn
4
UV n VIS l rÃr
j
j
4 l pÃp
2
h
5
X
mnjkluvw
B m n j m
h
i n j j
h
i k j m
h
i l j j
h i w j u
h
i w j v
h
i;
 L rÃn;rm x UV
ð
ÞD rj;rm x 1
ð ÞL rj;rÃk Àx UV
ð
ÞD pu;pv x 2
ð ÞL pÃw;pv x t
ð Þ
ð20Þ
P
5
ð Þ
12 ðx 1 ; x 2 Þ ¼ À
Nn
4
UV n VIS l rÃr
j
j
4 l pÃp
2
h
5
X
mnjkluvw
B m n j m
h
i n j j
h
i k j j
h
i l j m
h
i w j u
h
i w j v
h
i:
 L rm;rÃn Àx UV
ð
ÞD rm;rj x 1
ð ÞL rm;rÃk Àx UV
ð
ÞD pu;pv x 2
ð ÞL pÃw;pv x t
ð Þ
ð21Þ
In the above polarization components, laser pulses with the subscripts UV and
VIS are taken to interact with triiodide and diiodide, respectively.
For convenience, we further group the terms into three classes of signal fields
under the assumption of perfect phase-matching conditions
E
5
ð Þ
r;r ðx 1 ; x 2 Þ ¼
ix t l
2e 0 n x t
ð Þc
X 4
m¼1
P
5
ð Þ
m ðx 1 ; x 2 Þ;
ð22Þ
E
5
ð Þ
p;p ðx 1 ; x 2 Þ ¼
ix t l
2e 0 n x t
ð Þc
X 8
m¼5
P
5
ð Þ
m ðx 1 ; x 2 Þ ;
ð23Þ
and
E
5
ð Þ
r;p ðx 1 ; x 2 Þ ¼
ix t l
2e 0 n x t
ð Þc
X 12
m¼9
P
5
ð Þ
m ðx 1 ; x 2 Þ :
ð24Þ
Here, the two subscripts of the signal fields represent sensitivity to the triiodide
reactant (subscript r) and diiodide product (subscript p) in the two frequency
dimensions, x 1 and x 2 .
Top Curr Chem (Z) (2017) 375:87
123
271
Reprinted from the journal