Resonance generation of sum harmonic in static electric field
Using the group theory considerations, we investigate the nonlinear effect tensor (NET) for a crystal in a static electric field when a harmonic or the initial emission frequency approaches that of the excitonic absorption band. The dependence of additional terms in NET (resulting from the electric...
Gespeichert in:
| Veröffentlicht in: | Semiconductor Physics Quantum Electronics & Optoelectronics |
|---|---|
| Datum: | 1999 |
| Hauptverfasser: | , , |
| Format: | Artikel |
| Sprache: | English |
| Veröffentlicht: |
Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України
1999
|
| Online Zugang: | https://nasplib.isofts.kiev.ua/handle/123456789/119876 |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Zitieren: | Resonance generation of sum harmonic in static electric field / E.F. Venger, V.M. Griban, A.V. Melnichuk // Semiconductor Physics Quantum Electronics & Optoelectronics. — 1999. — Т. 2, № 3. — С. 98-102. — Бібліогр.: 4 назв. — англ. |
Institution
Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
nasplib_isofts_kiev_ua-123456789-119876 |
|---|---|
| record_format |
dspace |
| spelling |
Venger, E.F. Griban, V.M. Melnichuk, A.V. 2017-06-10T08:08:17Z 2017-06-10T08:08:17Z 1999 Resonance generation of sum harmonic in static electric field / E.F. Venger, V.M. Griban, A.V. Melnichuk // Semiconductor Physics Quantum Electronics & Optoelectronics. — 1999. — Т. 2, № 3. — С. 98-102. — Бібліогр.: 4 назв. — англ. 1560-8034 PACS: 78.20.Jq https://nasplib.isofts.kiev.ua/handle/123456789/119876 Using the group theory considerations, we investigate the nonlinear effect tensor (NET) for a crystal in a static electric field when a harmonic or the initial emission frequency approaches that of the excitonic absorption band. The dependence of additional terms in NET (resulting from the electric field applied) on the electric field components is found. Their form is shown to differ from that of usual NET. The results obtained are illustrated by considering the case of the D₂d crystal symmetry. en Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України Semiconductor Physics Quantum Electronics & Optoelectronics Resonance generation of sum harmonic in static electric field Article published earlier |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| title |
Resonance generation of sum harmonic in static electric field |
| spellingShingle |
Resonance generation of sum harmonic in static electric field Venger, E.F. Griban, V.M. Melnichuk, A.V. |
| title_short |
Resonance generation of sum harmonic in static electric field |
| title_full |
Resonance generation of sum harmonic in static electric field |
| title_fullStr |
Resonance generation of sum harmonic in static electric field |
| title_full_unstemmed |
Resonance generation of sum harmonic in static electric field |
| title_sort |
resonance generation of sum harmonic in static electric field |
| author |
Venger, E.F. Griban, V.M. Melnichuk, A.V. |
| author_facet |
Venger, E.F. Griban, V.M. Melnichuk, A.V. |
| publishDate |
1999 |
| language |
English |
| container_title |
Semiconductor Physics Quantum Electronics & Optoelectronics |
| publisher |
Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України |
| format |
Article |
| description |
Using the group theory considerations, we investigate the nonlinear effect tensor (NET) for a crystal in a static electric field when a harmonic or the initial emission frequency approaches that of the excitonic absorption band. The dependence of additional terms in NET (resulting from the electric field applied) on the electric field components is found. Their form is shown to differ from that of usual NET. The results obtained are illustrated by considering the case of the D₂d crystal symmetry.
|
| issn |
1560-8034 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/119876 |
| citation_txt |
Resonance generation of sum harmonic in static electric field / E.F. Venger, V.M. Griban, A.V. Melnichuk // Semiconductor Physics Quantum Electronics & Optoelectronics. — 1999. — Т. 2, № 3. — С. 98-102. — Бібліогр.: 4 назв. — англ. |
| work_keys_str_mv |
AT vengeref resonancegenerationofsumharmonicinstaticelectricfield AT gribanvm resonancegenerationofsumharmonicinstaticelectricfield AT melnichukav resonancegenerationofsumharmonicinstaticelectricfield |
| first_indexed |
2025-11-25T23:28:33Z |
| last_indexed |
2025-11-25T23:28:33Z |
| _version_ |
1850581058795339776 |
| fulltext |
98 © 1999, Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
Semiconductor Physics, Quantum Electronics & Optoelectronics. 1999. V. 2, N 3. P. 98-102.
1. Introduction
The sum harmonic generation is among the nonlinear opti-
cal effects in crystals. It involves emission at a sum fre-
quency, ω1(k1) + ω2(k2), due to emission at frequencies
ω1(k1) and ω2(k2). The well -known frequency doubling,
i.e. the second harmonic generation, is a special case of the
above effect.
When theoretically investigating the sum harmonic gen-
eration, it is convenient to use the nonlinear effect tensor
(NET) α. NET is a factor in the expression for the matrix
element corresponding to an elementary act of the above
process. The generation intensity is proportional to the NET
modulus squared.
The NET α form was determined in the paper [1] using
the polariton theory. The latter takes into account the
retardation of interaction. This enables one to correctly
analyze resonance phenomena that occur when an emission
frequency approaches the excitonic absorption band fre-
quency. The NET form in crystals of various symmetries
was also investigated in the above paper.
Using the technique developed in the paper [1], one of
the authors of the present paper has found the NET form
for crystals in a static electric field E [2]. It was shown that
the effect of the electric field results in appearance of an
additional term α(E) in the NET α. The form of this addi-
tional term, α(E), differs from that of α. This means that, at
PACS: 78.20.Jq
Resonance generation of sum harmonic
in static electric field
E.F. Venger*, V.M. Griban, A.V. Melnichuk
*Institute of Semiconductor Physics, NAS of Ukraine, 45, prospekt Nauki, Kyiv, 252028, Ukraine
Phone: (38044) 265 25 93; Fax: (38044) 265 83 42; E-mail: mickle@semicond.kiev.ua
Mykola Gogol State Pedagogic University, 2 Kropyv�yans�kogo Str., Nizhin, 251200, Ukraine
Abstract. Using the group theory considerations, we investigate the nonlinear effect tensor (NET) for a
crystal in a static electric field when a harmonic or the initial emission frequency approaches that of the
excitonic absorption band. The dependence of additional terms in NET (resulting from the electric field
applied) on the electric field components is found. Their form is shown to differ from that of usual NET.
The results obtained are illustrated by considering the case of the D2d crystal symmetry.
Keywords: sum harmonic, nonlinear effect tensor, group theory considerations.
Paper received 29.01.99; revised manuscript received 01.06.99; accepted for publication 11.10.99.
some emission polarization, the harmonic generation is due
to the field action only.
The tensor α(E) was found in the first order of the per-
turbation theory, so in general case the intensity of the sum
harmonic generation due to it has to be much less than that
due to the tensor α. The tensor α(E), however, has resonance
terms in denominators. This means that, when an emission
frequency approaches the resonance, the effect of an elec-
tric field is to grow abruptly. Hence it follows that in the
case of the resonance it is of both interest and importance to
study the structure of the tensor α(E) and its transformation
properties. The same is also true for the dependence of the
tensor α(E) form on the symmetry type of the exciton state
whose frequency is resonant. It is the above problems the
present paper deals with.
2. Results and discussion
Let us give, first of all, the explicit form of the tensor
α(E). According to the paper [2], it is a product of the vec-
tor E and a tensor β of rank four:
∑=
i
ijjjijjj E .2121
)( βα E (1)
The tensor β has 16 rather complicated components.
For the following analysis it will be sufficient to give ex-
plicitly only one of them:
E.F. Venger et al.: Resonance generation of sum harmonic...
99SQO, 2(3), 1999
[ ]∑
×
−
−=
',',, '2
2,22
21 2
21
)'()(
)',')(','0(
k kk
kkk
µµµ µµ
µµµ
β
EE
pj
N i
ijjj
[ ][ ] .
)()()(
)0,)(,,(
212122
211211122
12
⋅⋅⋅+
+−+−
++
×
ωωω
µµµ
µµ hh kkk
kkkkk
EE
jj
(2)
Here N is the number of crystal unit cells; Εµ(k), |µ, k) are,
respectively, the energy and wave function of a Coulomb
exciton with wave vector k that belongs to the µ-th band;
|0) is the wave function for the ground state of the crystal; j,
j1, j2 index the projections of polarization vectors for the
harmonic and initial emission (they may take the values x, y,
z); pi(i = x, y, z) is the i-th component of the electric dipole
moment for a unit cell; (µ1, k1|jn| µ2, k2) is the matrix element
of the operator nj
I
m
e
(where e, m, I are, respectively, the
electron charge, mass and momentum operator). The other
terms in expression (2) differ from the above one by the
place of the matrix element for the component pi and the
differences (sums) in the denominators. The frequencies ω1
and ω2 can be considered as interchangeable, i.e., the right
hand side of expression (2) is symmetric about interchang-
ing the indices 1 and 2.
The form of the tensor α(E) depends on that of the ten-
sor β. When determining the latter form, one has to keep in
mind that the exciton states |µ, k) are classified according
to the irreducible representations of the point group of crystal
symmetry, while the expressions of the type
.
)(
,)(,∑
±µ µ
µ
ω
µ
hk
kk
E
transform according to the entire symmetry representations
[2]. So it follows from expression (2) that the tensor β is
isomorphic to (jj1j2i)0:
02121
)( ijjjijjj →β . (3)
Here index 0 means that the product of coordinates, jj1j2i,
is projected onto the entire symmetry representation of the
point symmetry group. Basing on expression (3), the form
of the tensor α(E) was determined in the paper [2] for
crystals of various symmetries.
Now let us consider the form of the tensor α(E) when
the sum frequency, ω1 + ω2, approaches the excitonic band
absorption frequency, 1/h ⋅Εµ(k1+ k2). (The case h (ω1+
ω2) = Εµ(k1+ k2) is, however, excluded. Such a two-photon
absorption will be considered elsewhere.)
First of all, it should be noted that generally the terms
with different µ1 make different contributions to different
components of tensor β, depending on the |µ1, k1 + k2) state
symmetry. To demonstrate this, let us separate out those
terms in expression (2) for which µ1 = µ:
[ ]∑
×
−
−=
',', '2
2,22)(
2 2
21 )'()(
)',')(','0(
k kk
kkk
µµ µµ
µ µµµ
β
EE
pj
N i
ijjj
[ ][ ] .
)()()(
)0,)(,,(
212122
2121122
2
⋅⋅⋅+
+−+−
++
×
ωωω
µµµ
µµ hh kkk
kkkkk
EE
jj
(4)
There is no summation over µ here, and hence the quantity
[ ].)()(
,)(,
2121
2121
ωω
µµ
µ +−+
++
hkk
kkkk
E
is not of full symmetry. This means that the term given in
an explicit form in (4) is isomorphic not to (jj1j2i)0 but to
the product (j2ij1)µ(j)µ where index µ signifies projection
onto the irreducible representation to which the |µ1, k1 +
k2) state belongs. An analysis made for the rest of terms in
(4) (denoted by dots) shows that tensor β(µ) is isomorphic
to the following sum:
.)()()()( 2112
)(
21 µµµµ
µβ jjijijjjijjj +→ (5)
After summing over µ one gets the relation (3) (as it was to
be expected).
Identifying the index µ with designation of the corre-
sponding irreducible representation, one can determine the
factors in the right-hand side of expression (5) and obtain
first the tensor β(µ) and then the tensor α(µ)(E):
.)( )()(
2121 ∑=
i
ijjjijjj E µµ βα E (6)
After summing the corresponding tensor α(µ)(E)
components over all the irreducible representations, one can
get the tensor α(E). Its form is to be the same as that fol-
lowing from expression (3).
Now let us return to the resonance conditions when
h (ω1+ ω2)→Εµ(k1+ k2). In this case the terms in β(µ) with
µ = µ1 will prevail. As a result, the components α(µ)(E) of
the tensor α(E) will prevail. Thus under resonance
conditions one can write down:
( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( ) ( )
β β β β
α α α α
µ
µ
µ
µ µ
µ
µ µ µ
µ µµ
= = +
= = +
∑ ∑
∑∑
≠
≠
1
1
1
1
1 1
11
,
.E E E E
According to the above considerations, the non-
resonance components may be neglected and therefore
α(E) ≈ α(µ)(E). This means that under resonance conditions
the tensor α(E) retains its form; α(µ)(E), however, become
the governing components. Similar reasoning was used in
paper [3] to analyze the Raman scattering tensor under
resonance conditions.
E.F. Venger et al.: Resonance generation of sum harmonic...
100 SQO, 2(3), 1999
To illustrate the above results, let us consider a crystal
of D2d symmetry. For it the usual NET has the following
non-zero components: αxyz = αxzy = αyxz = αyzx, αzxy = αzyx.
In a static electric field the tensor α(E) appears whose form
follows from expression (3). It is given in Table 2 - see the
row designated by
µ
∑ .
Let us determine the tensor α(µ)(E). The D2d group has
four one-dimensional (1D) irreducible representations, A1,
A2, B1, B2, and a 2D irreducible representation, E, with two
rows, E(x) and E(y), after which the coordinates x, y trans-
form.
Using expression (5), one can find the form of the ten-
sor β(µ). To this end one has to determine the irreducible
representations of the D2d group after which the coordinates
x, y, z, as well as their double and triple products, trans-
form. The corresponding technique is discussed in the pa-
pers [3,4]. The results of our calculations are given in Table 1.
Let us consider the case µ = A1, i.e. the wave function of
the intermediate state, |µ1, k1+k2), corresponds to the
irreducible representation A1. One can conclude from Table 1
that projections of coordinates x, y, z onto the representation
A1 are zero. Therefore, the first term in the right-hand side of
expression (5) drops out. For the second term the non-zero
projections of double products are (x1x2)A1 = (y1y2)A1 and
(z1z2)A1. Thus, according to expression (5), the non-zero
components of the tensor β(µ) are the following:
1
)1()1()1( aA
yyxx
A
xxyy
A
xxxx === βββ , 2
)1( aA
zzzz =β ,
3
)1()1( aA
yyzz
A
xxzz == ββ , 4
)1()1( aA
zzyy
A
zzxx == ββ . (7)
Thereafter, using (6), one can determine the form of the
tensor ( )E)( 1Aα :
xEaA
xxx 1)()1( =Eα , yEaA
yyy 1)()1( =Eα , zEaA
zzz 2)()1( =Eα ,
yEaA
yxx 1)()1( =Eα , xEaA
xyy 1)()1( =Eα , zEaA
zxx 3)()1( =Eα ,
z
A
zyy Ea3
)( )(1 =Eα , xEaA
xzz 4)()1( =Eα , yEaA
yzz 4)()1( =Eα . (8)
All the other components are zero.
The same procedure can be used to find the tensor α(µ)(E)
components for other irreducible representations of the D2d
group. The corresponding results are given in Table 2. (There
the numbers 1, 2, ... are used instead of a1, a2, ..., and the
relations between components correspond to the only
representation.)
After performing summation of the corresponding
components over all the irreducible representations, one can
obtain the tensor form in the case when there is no resonance
(it is given in the row designated with ∑
µ
).
Now let us consider the case when the frequency ω2
approaches the frequency of the excitonic absorption band.
One can show that an isomorphism at the initial emission
frequency follows from expression (2):
.)()()()( 1212
)(
21 µµµµ
µβ jijjjjjjijjj +→ (9)
If µ = A2, then, basing on expressions (9) and (1) and Table
1, we get:
yEaA
xyx 1)()2( =Eα , yEaA
yxx 1)()2( −=Eα ,
xEaA
yxy 1)()2( =Eα , xEaA
xyy 1)()2( −=Eα . (10)
All the other components are zero.
The form of the tensor α(µ)(E) in this case is given in
Table 3. This tensor is non-symmetric in respect to the indi-
ces j1, j2, because there is no symmetry now between the
frequencies ω1 and ω2: ω2 is the resonance frequency, while
ω1 is not.
When the resonance occurs at a frequency ω1, then
one has to interchange the components )(
21
µα jjj and )(
12
µα jjj
in Table 3. If frequency is doubled (ω1 = ω2 = ω), then one
has to take )()(
1221
µµ αα jjjjjj = .
µ Coordinates and their products
A1 x1x2 + y1y2 z1z2 x1y2z3 + y1x2z3 x1z2y3+ y1z2x3 z1x2y3 + z1y2x3
A2 x1y2 - y1x2 x1x2z3 - y1y2z3 x1z2x3 - y1z2y3 z1x2x3 - z1y2y3
B1 x1x2 - y1y2 x1y2z3 - y1x2z3 x1z2y3 - y1z2x3 z1x2y3 - z1y2x3
B2 z z1z2z3 x1x2z3 + y1y2z3 x1z2x3 + y1z2y3 z1x2x3 + z1y2y3
E(x) x y1z2 z1y2 x1x2x3 y1y2x3 y1x2y3 x1y2y3 z1z2x3 z1x2z3 x1z2z3
E(y) y x1z2 z1x2 y1y2y3 x1x2y3 x1y2x3 y1x2x3 z1z2y3 z1y2z3 y1z2z3
Table 1. Coordinates and their products that transform according to the irreducible representations of the D2d group.
E.F. Venger et al.: Resonance generation of sum harmonic...
101SQO, 2(3), 1999
jj 1j2
µ xxx yyy zzz xxz xzx zxx zzy zyz yzz
xxy xyx yxx yyz yzy zyy xyz xzy zxy
yyx yxy xyy zzx zxz xzz yxz yzx zyx
1Ex 1Ey 2Ez 0 0 3Ez 0 0 4Ez
A1 0 0 1Ey 0 0 3Ez 0 0 0
0 0 1Ex 0 0 4Ex 0 0 0
0 0 4Ey 0 0 0 0 0 0
A2 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0
1Ex 1Ey 0 0 0 0 0 0 0
B1 0 0 -1Ey 0 0 0 0 0 0
0 0 -1Ex 0 0 0 0 0 0
0 0 1Ez 0 0 3Ez 4Ey 4Ey 0
B2 2Ey 2Ey 0 0 0 3Ez 0 0 0
2Ex 2Ex 0 4Ex 4Ex 0 0 0 0
1Ex 0 0 4Ez 4Ez 0 7Ey 7Ey 0
E(x) 2Ey 2Ey 0 5Ez 5Ez 0 0 0 0
0 0 3Ex 0 0 6Ex 0 0 0
0 1Ey 0 4Ez 4Ez 0 0 0 7Ey
E(y) 0 0 2Ey 5Ez 5Ez 0 0 0 0
3Ex 3Ex 0 6Ex 6Ex 0 0 0 0
1Ex 2Ey 3Ez 8Ez 8Ez 9Ez 14Ey 14Ey 15Ey
∑
µ
4Ey
6Ex
4Ey
6Ex
5Ey
7Ex
10Ez
12Ex
10Ez
12Ex
11Ez
13Ex
0
0
0
0
0
0
Table 3. Form of the tensor α(µ)(E) for the resonance frequency ω2.
µ jj 1j2
xxx yyy zzz xxz xzx zxx zzy zyz yzz
xxy xyx yxx yyz yzy zyy xyz xzy zxy
yyx yxy xyy zzx zxz xzz yxz yzx zyx
1Ex 1Ey 2Ez 0 3Ez 0 0 0 0
A1 0 1Ey 0 0 3Ez 0 0 0 0
0 1Ex 0 0 4Ex 0 0 0 0
0 0 0 0 0 0 0 0 0
A2 1Ey 0 -1Ey 0 0 0 0 0 0
1Ex 0 -1Ex 0 0 0 0 0 0
1Ex 1Ey 0 0 0 0 0 0 0
B1 0 -1Ey 0 0 0 0 0 0 0
0 -1Ex 0 0 0 0 0 0 0
0 0 1Ez 0 3Ez 0 4Ey 0 5Ey
B2 2Ey 0 0 0 3Ez 0 0 0 0
2Ex 0 0 4Ex 0 5Ex 0 0 0
1Ex 0 0 5Ez 0 6Ez 10Ey 0 11Ey
E(x) 2Ey 0 3Ey 7Ez 0 8Ez 0 0 0
0 4Ex 0 0 9Ex 0 0 0 0
0 1Ey 0 5Ez 0 6Ez 0 11Ey 0
E(y) 0 2Ey 0 7Ez 0 8Ez 0 0 0
3Ex 0 4Ex 9Ex 0 10Ex 0 0 0
Table 2. Form of the tensor α(µ)(E) for the resonance frequency ω1 +ω2.
E.F. Venger et al.: Resonance generation of sum harmonic...
102 SQO, 2(3), 1999
3. Conclusions
From the above results it follows that, under the resonance
generation of the sum harmonic in a static electric field, the
nonlinear effect tensor involves a term α(µ)(E) that depends
on both the field E and the symmetry type of the exciton
state whose frequency is the resonance one. The form of
the tensor α(µ)(E) differs from that of the non-resonance
nonlinear effect tensor. Besides, it is different at the fre-
quencies of the harmonic and initial emission.
References
1. L.N. Ovander, Nonlinear optical effects in crystals // Uspekhi Fiz.
Nauk 86 (1), pp. 1-39 (1965) (in Russian).
2. V.M. Griban, On sum harmonic generation in crystals with inversion
center (in Russian) // Izv. Vuzov SSSR. Fizika N 6, pp.107-110 (1969).
3. L.D. Landau, E.M. Lifshits, Quantum Mechanics, Nauka, Moscow,
p.443 (1989) (in Russian).
4. M.I. Petrashen, E.D. Trifonov, Group Theory Applications in
Quantum Mechanics, Nauka, Moscow (1967) (in Russian).
|