Investigation of β-CdP₂ crystals by laser spectroscopy methods
β-CdP₂ CdP₂ single crystal of tetragonal modification is investigated by methods of lasermodulated spectroscopy at 293 K. The spectra of coherent two-photon absorption (TPA) have been measured and their theoretical analysis has been carried out. Double resonance of TPA at the total energy of two p...
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Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України
2004
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irk-123456789-1181112017-05-29T03:03:00Z Investigation of β-CdP₂ crystals by laser spectroscopy methods Patskun, I.I. Slipukhina, I.A. β-CdP₂ CdP₂ single crystal of tetragonal modification is investigated by methods of lasermodulated spectroscopy at 293 K. The spectra of coherent two-photon absorption (TPA) have been measured and their theoretical analysis has been carried out. Double resonance of TPA at the total energy of two photons 2.60 eV is observed which occur through impurity level d3 in the gap at energy Ec - 0.86 eV (Ec is the energy of the conduction band edge). The time of transverse relaxation of electrons is found to be equal to 4.3*10⁻¹⁴ s. The dependence is observed of single resonance on mutual orientation of vector of linear polarized light and crystallographic axis enabling us to conclude that the single TPA resonance in CdP₂ occurs between the states Г₇(Г₄) → Г₇(Г₃), Г₇(Г₅) → Г₇(Г₃), Г₆(Г₅) → Г₆(Г₁) in the case of the total energy of two photons is ranged from 2.5 to 3.11 eV. The energies of the states mentioned above are obtained. Remove selected 2004 Article Investigation of β-CdP₂ crystals by laser spectroscopy methods / I.I. Patskun, I.A. Slipukhina // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2004. — Т. 7, № 1. — С. 31-35. — Бібліогр.: 10 назв. — англ. 1560-8034 PACS: 42.62.Fi, 78.40.-q http://dspace.nbuv.gov.ua/handle/123456789/118111 en Semiconductor Physics Quantum Electronics & Optoelectronics Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України |
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β-CdP₂
CdP₂ single crystal of tetragonal modification is investigated by methods of lasermodulated spectroscopy at 293 K. The spectra of coherent two-photon absorption (TPA) have been measured and their theoretical analysis has been carried out. Double resonance of TPA at the total energy of two photons 2.60 eV is observed which occur through impurity level d3 in the gap at energy Ec - 0.86 eV (Ec is the energy of the conduction band edge). The time of transverse relaxation of electrons is found to be equal to 4.3*10⁻¹⁴ s. The dependence is observed of single resonance on mutual orientation of vector of linear polarized light and crystallographic axis enabling us to conclude that the single TPA resonance in CdP₂ occurs between the states Г₇(Г₄) → Г₇(Г₃), Г₇(Г₅) → Г₇(Г₃), Г₆(Г₅) → Г₆(Г₁) in the case of the total energy of two photons is ranged from 2.5 to 3.11 eV. The energies of the states mentioned above are obtained.
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Patskun, I.I. Slipukhina, I.A. Investigation of β-CdP₂ crystals by laser spectroscopy methods Semiconductor Physics Quantum Electronics & Optoelectronics |
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Investigation of β-CdP₂ crystals by laser spectroscopy methods |
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Investigation of β-CdP₂ crystals by laser spectroscopy methods |
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Investigation of β-CdP₂ crystals by laser spectroscopy methods |
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Investigation of β-CdP₂ crystals by laser spectroscopy methods |
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Investigation of β-CdP₂ crystals by laser spectroscopy methods |
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investigation of β-cdp₂ crystals by laser spectroscopy methods |
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Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України |
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2004 |
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Investigation of β-CdP₂ crystals by laser spectroscopy methods / I.I. Patskun, I.A. Slipukhina // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2004. — Т. 7, № 1. — С. 31-35. — Бібліогр.: 10 назв. — англ. |
| series |
Semiconductor Physics Quantum Electronics & Optoelectronics |
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AT patskunii investigationofbcdp2crystalsbylaserspectroscopymethods AT slipukhinaia investigationofbcdp2crystalsbylaserspectroscopymethods |
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1837085213379788800 |
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31© 2004, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
Semiconductor Physics, Quantum Electronics & Optoelectronics. 2004. V. 7, N 1. P. 31-35.
PACS: 42.62.Fi, 78.40.-q
²nvestigation of βββββ-CdP2 crystals
by laser spectroscopy methods
I.I. Patskun*, I.A. Slipukhina**
*The Dragomanov National Pedagogical University, 9, Pirogova str., Kyiv Ukraine, Phone: +380 (44) 2393063;
**National Aircraft University, Ukraine, Kiev, prospect of the spaceman Komarov, 1, tel. +380 (44) 484-9166,
E-mail: ira47@mail.ru
Abstract. CdP2 single crystal of tetragonal modification is investigated by methods of laser-
modulated spectroscopy at 293 K. The spectra of coherent two-photon absorption (TPA) have
been measured and their theoretical analysis has been carried out. Double resonance of TPA
at the total energy of two photons 2.60 eV is observed which occur through impurity level d3
in the gap at energy Ec � 0.86 eV (Ec is the energy of the conduction band edge). The time of
transverse relaxation of electrons is found to be equal to 4.3⋅10�14 s. The dependence is ob-
served of single resonance on mutual orientation of vector of linear polarized light and
crystallographic axis enabling us to conclude that the single TPA resonance in CdP2 occurs
between the states Ã7(Ã4) → Ã7(Ã3), Ã7(Ã5) → Ã7(Ã3), Ã6(Ã5) → Ã6(Ã1) in the case of the total
energy of two photons is ranged from 2.5 to 3.11 eV. The energies of the states mentioned above
are obtained.
Keywords: laser spectroscopy methods, band structure, two-photon absorption.
Paper received 18.11.03; accepted for publication 30.03.04.
1. Introduction
The results of a few studies of CdP2 of tetragonal modifi-
cation give foundations to make conclusion about the
potentialities of this semiconductor compound in the field
of optoelectronics and non-linear optics. From this the
urgency is also raised of the investigation of two-photon
absorption in CdP2 by method of laser-modulated spect-
roscopy.
2. Experimental and discussion
The crystals investigated were grown by vapor phase tech-
nique in two-zone furnace. Pure cadmium and phospho-
rus constituents were taken in stoichiometric ratio for
synthesis of single crystals. Samples had the configura-
tion of rectangular bar with typical dimensions of
4×4×4 mm3 with optical axis oriented along Z-axis. Ex-
perimental set-up was the modification of that one de-
scribed previously [l]. Unlike the set-up used earlier, in
this set-up the portion of probe beam is directed into sup-
plementary monochromator, after transmission of which
it falls on the cathode of photo sensor. It allows us to
determine the intensity of linear polarized radiation of
probe beam in the case of the analyzer of polarized light
is mounted before the monochromator.
In the spectroscopic method used information is ex-
tracted from measurements of intensity and polarization
of probe wave with frequency ω2 at excitation of media
by pumping wave with frequency ω1. Radiation of Q-
switched Nd:YAG laser with the half-width of giant pulse
τ1 = 15 ns was used as pumping wave, whereas radiation
of powerful pulsed xenon lamp with pulse duration
τ2 = 150 ns was used as probe wave. Both waves are propa-
gated at the same direction. Due to high excitation of
media by pumping wave the susceptibility of it is changed
on the value of ∆K. This change is detected on frequency
using probe beam. The quantities ∆K contain noncoherent
and coherent contributions. Noncoherent contribution
is attributed to resonant redistribution of state popula-
tion, as coherent one is attributed to medium polariza-
32
SQO, 7(1), 2004
I.I. Patskun, I.A. Slipukhina: ²nvestigation of βββββ-CdP
2
crystals by laser spectroscopy methods
tion. If impurity levels f with concentration Nf exist in the
gap, then using the formalism of density matrix one can
show that probe wave is affected by changing of level
population via resonant susceptibility
( )
,
2
2
2
2
2
∆
Γ+−
⋅
+
∆
Γ+−
⋅
=
vf
fvfv
fc
cfcf
f
R
i
vref
i
frecN
ρ
ωω
ρ
ωω
ωχ
r
r
h
(1)
where ccfffc ρρρ −=∆ and ffvvvf ρρρ −=∆ is the diffe-
rence between population of states f and c , v and
f , respectively. Due to fast relaxation of electrons and
holes in c- and v-bands one obtains 1,0 == ρρcc and,
consequently, fffc ρρ =∆ , ffvf ρρ −=∆ 1 . In accordance
with [1]
( ) ,exp 11
1
)0(
fcvf
vffcvf
fcvf
vf
ffff
I
σσ
σ
τω
ω
σσ
σσ
σ
ρρ
′+′
′
+
⋅⋅
′+′
−×
×
′+′
′
−=
h
(2)
where I(ω1) is the intensity of pumping wave, )0(
ffρ is the
population of impurity level f before pumping, σ′ is the
cross-section of absorption of pumping wave. It follows
from equations (1) and (2) that ( )2ωχR exponentially
depends on pumping intensity.
Coherent two-photon contribution is attributed to
four-wave mixing caused by nonlinear susceptibility of
the third order )3(χ . Microscopic expression for )3(χ can
be received by means of the perturbation theory. Cubic
susceptibility may be written as the sum of resonant and
non-resonant parts: )3()3()3(
RNR χχχ += . Coherent two-pho-
ton contribution is attributed to coherent resonant inter-
action caused by )3(χ . In consequence of such interac-
tion two photons with frequencies )( 11 k
r
ω and )( 22 k
r
ω are
absorbed coherently, whereas crystal is passed from ini-
tial v to final c states. This is the coherent TPA. TPA
expressed in terms of altering of the intensity of probe
wave ( )2ωI∆ is proportional to the imaginary part of
cubic susceptibility Imχ(3). ( )2ωI∆ corresponds to chang-
ing of absorption coefficient of probe wave
( ) ( )
( ) ( )22
2
2 ln
1
ωω
ωω
II
I
d
K
∆−
=∆ , (3)
where I(ω2) is the intensity of probe wave, d is the thick-
ness of sample.
In consequence of four-wave mixing double refrac-
tion of probe beam is induced. This caused the rotation
of polarization plane, and at the output of analyzer,
crossed with the polarization plane of probe wave, inten-
sity of probe beam increases [2]
( ) ( ) ( )21
22)3()3(
2 ωωχχω xxyyxxyxyxy III +∝∆ (4)
in the case of linear polarization of the pumping wave,
polarization plane of which is situated at angle of 45°
with respect to the polarization plane of probe wave; and
( ) ( ) ( )21
22)3()3(
2 ωωχχω xyyxxyxyxy III −∝∆ (5)
in the case of circular polarization of the pumping wave.
The angle θ of rotation of the polarization plane is de-
fined by formula
( )
( )2
22sin
ω
ω
θ
x
y
I
I∆
= . (6)
In the case of both beams are propagated along the
crystal symmetry axis for non-resonant part the ratio of
symmetry is fulfilled ( ) ( )
yyxxNRyxyxNR
)3()3( χχ = , however
( ) ( )yyxxRyxyxR
)3()3( χχ ≠ that is why one can receive
( ) ( ) ( ) ( ) ( )21
2
2)3()3(
2 ωωχχω xxyyxxRyxyxRy III −∝∆ . (7)
The important advantage of the method used consist
of the fact, that special measures on maintenance of phase
synchronism are not necessary; the condition
1122 kkkk
rrrr
−+= is carried out identically. Therefore the
direction of propagation of a probe beam with respect to
a pumping beam can be arbitrary.
∆K(ω2) consists of coherent and noncoherent parts.
Noncoherent part ∆K(1)(ω2) is attributed to the amplitude
modulation of impurity single-photon absorption of probe
wave, and coherent one equals to two-photon absorption
coefficient ∆K(2)(ω2):
( ) ( ) ( ) ( ) ( )( )
( ) ( )( ) ( ) ( )2
)2(
2
)1(
2
)2(
02
)1(
0
2
)2(
2
)1(
2022
ωωωω
ωωωωω
KKKK
KKKKK
+∆=+−
−+=−=∆
(8)
because of ( ) 02
)2(
0 =ωK .
( ) ( ) ( )2
)1(
02
)1(
2
2
2
)1( ImIm4 ωχωχωπω −=∆
cn
K ,(9)
( ) ( ) ( )2
12112
)3(
2
2
2
)2( Im4 ωωωωωχωπω E
cn
K
r
+−==
(10)
Accounting for in equation (9) expressions (1) and (2)
one obtains exponential dependence of ∆K(1)(ω2) on I(ω1).
K(2)(ω2) can belong to single- and double-resonance of
TPA. At ordinary resonance the condition vcωωω =+ 21
is fulfilled, where ωvc is the frequency of transition bet-
ween v- and c-bands. In theory of such resonance virtual
states are considered as intermediate. In the case of double
resonance vfωω =1 and cfωω =2 , or (and) cfωω =1 and
fvωω =2 and real f levels are considered to be the in-
I.I. Patskun, I.A. Slipukhina: ²nvestigation of βββββ-CdP
2
crystals by laser spectroscopy methods
33SQO, 7(1), 2004
termediate states. Thus, )()()( 2
)2(
22
)2(
12
)2( ωωω KKK += ,
where )( 2
)2(
1 ωK and )( 2
)2(
2 ωK are the coefficients of
single and double resonance of TPA respectively, and
)()()()( 2
)2(
22
)2(
12
)1(
2 ωωωω KKKK ++∆=∆ . (11)
Each of these three spectroscopic components con-
tain important spectroscopic information and they de-
mand of particular analysis. For their separation kinetic,
spectral, intensity, polarization and angle characteris-
tics of ∆K(ω2) were investigated, analogous to those ones
investigated in ZnP2 previously [1].
In Fig. 1 the spectral dependencies ∆K(ω2) are shown
at variable mutual orientation of the vectors of polariza-
tion and crystallographic axis. Also shown is kinetic char-
acteristic ∆I(ω2), which is typical for long- and short-
wavelength bands of spectra. The curves K(2)(ω2) were
extracted from the dependencies ( ))()( 12 ωω IfK =∆ us-
ing extrapolation procedure [1] (by drawing straight lines
through zero point in parallel to high-intensity parts of
the dependencies analyzed). Resulting spectra K(2)(ω2)
thus obtained are shown in Fig. 2. Above the dependence
of K(2)(ω2) is shown on angle between the plain in which
electric vectors 1e
r
and 2e
r
are confined and Z-axis.
At propagation of beams along the optical axis c
r
of a
crystal a spectrum sin2θ of polarized modulation
spectroscopy was also measured. Its structure is similar
to that one shown in Fig. 2 (curve number 6). Thus, the
most convincing confirmation of the fact that the spectra
shown in Fig. 2 belong to two-photon absorption phe-
nomena is received. The spectra ∆K(ω2) (see Fig.1) contains several peaks
with maximum at energies of photons of probe wave
2ωh =1.34; 1.43; 1.50; 1.60; 1.69; 1.77; and 1.84 eV.
Their intensity much depend on mutual orientation of
vectors 1e
r
, 2e
r
and c
r
. Comparing spectra ∆K(ω2) shown
in Fig. 1 with spectra of TPA K(2)(ω2) shown in Fig. 2 one
can see, that these peaks, except peak at 1.43 eV, belong
to ∆K(1)(ω2). In work [3] it has been shown, that peaks at
energies 1.50; 1.60; 1,69 and 1.77eV are caused by opti-
cal transitions of electrons in donor-acceptor complexes,
in which vacancies of Cd are acceptors and atoms of re-
placement of phosphorus, being in the 4-th, 3-rd, 2-th
and 1-th coordination spheres are donors. The peak
1.84 eV was investigated in work [4] and is referred to
intra-center transitions.
At propagation of beams along c
r
-axis, it has been
found that the dependencies K(2)(ω2) on angle ϕ between
1e
r
and 2e
r
vectors as well as on the angle of rotation of a
crystal around of its optical axis are absent. In the case
of the propagation of beams is perpendicular to c
r
-axis
distinct angular dependence of on angle between vectors
1e
r
and 2e
r
and dependence on angle between 1e
r
|| 2e
r
and
vector is observed (Fig. 2).
The peak number 4 shown on spectral dependence
with a maximum at )( 21 ωω +h = 2.60 eV has the Lorenz
contour, which is characteristic for double two-photon
resonance. Coefficients of double TPA resonance
)( 2
)2(
2 ωK on deep impurity level f correspond to permit-
permitted (p-p) two-photon transitions, as the deep local
0.025
1.3 1.5 1.7 1.9
0.050
0.075
0.100
D
K
(
),
c
m
�
1
2
w
1, 2
3 4
5
6
2
2
h , w eV
0.025
2.5 2.7 2.9 3.1
6
4
2
0.050
0.075
0.03 0.03 0.03
0.06 0.06 0.06
0
80 80 8090 90 90
0 0
0.100
a = a , de g1 2
K
(
),
c
m
(2
)
–1
2
w
K
(
),
c
m
(2
)
–1
2
w
Fig. 1. Spectral dependencies of ∆K(ω2) in β-CdP2:
;1 2121 eecqq
rrrrr
⊥− ;2 2121 eecqq
rrrrr
⊥⊥− ;3 2121 eecqq
rrrrr
⊥⊥−
;4 2121 eecqq
rrrrr ⊥− ;5 2121 eecqq
rrrrr
⊥⊥⊥− ;6 2121 eecqq
rrrrr
⊥⊥−
21,qq
rr
� wave vectors of laser and probe radiation; 21, ee
rr
� princi-
ple vectors of electrical polarization of laser and probe radiation
accordingly. Below the continuous lines represented are typical
dependencies of ∆I(ω2) in long- and shortwave regions of the
spectrum. The spectra correspond to points of maximum of ki-
netics. The dotted lines represent pulses of laser radiation. Inten-
sity of laser radiation is 4 MW cm�2.
Fig. 2. Spectral dependencies of K(2)(ω2) in β-CdP2. On mutual
orientation of vectors the numbering of spectra corresponds to
numbering in Fig. 1. Within measurement error limits, the spec-
tra 1 and 3 not shown in this figure coincide with the spectrum 2,
as well as the spectrum 5 with spectrum 6. Shown above are
dependencies of K(2)(ω2) on the angle ce
rr
,11 =α 2121 , eecqq
rrrr
⊥ at
geometry of the experiment 21 ωω hh + =2.54; 2.75; 3.01eV.
34
SQO, 7(1), 2004
I.I. Patskun, I.A. Slipukhina: ²nvestigation of βββββ-CdP
2
crystals by laser spectroscopy methods
electron states in the gap have not certain parity and tran-
sitions between them and v- and c-zones are dipole-per-
mitted. For such transitions, in accordance with [1], in
the case of 0,0 ≠′=′ vffc σσ ,
( ) ( )
( ) ( ) .exp1 11
1
)0(
1,22
)2(
,2
××
′
−−×
×=
τω
ω
σ
ρ
ωβω
I
IK
vf
ff
m
fcfc
h
(12)
Here m
fc,2β is the constant of double resonance TPA,
proportional to the maximum number Nf of intermediate
levels participating in absorption:
( )[ ]2
21
,2
cvcv
fcvm
fc
N
Γ+−+
Γ
∝
ωωω
β
. (13)
cvΓ is attenuation constant. If 0,0 =′≠′ vffc σσ , then
( ) ( )
( ) ( ) .exp1 11
1
)0(
1,22
)2(
,2
××
′
−−×
×=
τω
ω
σ
ρ
ωβω
I
IK
fc
ff
m
vfvf
h
(14)
Where equation (13) is carried out for m
fv,2β , too. It is
seen from (13) that the spectrum p-p of a double TPA
resonance is represented by the narrow Lorenz line.
For uniaxial crystals such as β-CdP2 with point-sym-
metry group C4v, at propagation of beams along with c
r
-
axis )( 2
)2(
2 ωK depends on the angle between vectors 1e
r
and 2e
r
of linear polarization, and also on the type of
beam polarization. In the case of linear polarization
),cos()( 212
)2(
,2 eeK pp
rr
∝− ω . For circular polarization with
opposite directed rotation 0)( 2
)2(
,2 ≠− ωppK , and for circu-
lar polarization with identically directed rotation
0)( 2
)2(
,2 =− ωppK . These characteristics are typical for peak
at 1.43 eV. )( 2
)2(
,2 − ωppK in β-CdP2 is permitted-forbidden
(p-f) type of transition [5], so angular and polarization
dependencies obtained are not typical for it. The half-
width of the resonant peak is cvΓh = 0.29 eV. From this
value, the transverse relaxation time of electrons at a dou-
ble TPA resonance
cv
T
Γ
=
1
2 is obtained to be 4.3⋅10�14 s.
Intensity dependence of this peak is in accordance with
Eq. (14), which corresponds to fulfilling the condition
0,0 =′≠′ vffc σσ . It seems to be an evidence of impurity
level of double resonance located at the depth more than
1.17 eV from the edge of valence band, instead of more
than 1.17 eV from the edge of conductivity band and no
more than 1.43 eV from the edge of the valence band. In
this range of energy, the known level d3 (Ec � 0,86) eV is
located [6].
)( 2
)2(
1 ωK as well as appropriate cubic susceptibility
χ(3) in general case is a tensor. The selection rule for them
can be deduced from the theory of groups. They were
received by Inoue and Toyozawa [7] for 32-point groups
of crystal symmetry. The general view of the angular de-
pendence )( 2
)2(
1 ωK is determined by group G of crystal
symmetry transformations. In a general case, the prob-
ability of TPA W(2) can be expanded on linearly inde-
pendent invariants concerning symmetry transformations
of group G, which are non-zero real values and which
can be constructed of two vectors 1e
r
, 2e
r
, and two vectors
1e
r
* and 2e
r
*. Also ( ) ( )
212
2
1
2 )()( ωωω hIKW ×= .
Such expansion of W(2) for cubic symmetry was ob-
tained in the work [8], and for hexagonal symmetry in
[9]. Similarly, in agreement with the results of the work
[7], it is possible to write down expansion of W(2) for crys-
tals of tetragonal symmetry (point-symmetry group D4)
in the case of two linearly polarized beams of light:
( ) ( ) ( )[ ]
( )[ ] ( )[ ]
( )[ ]
( )[ ] .~
~
2
21
*
2215
2
21214
2
21213
2
2
*
11
2
212211
2
⋅×+⋅+
+⋅+⋅+
+⋅−⋅+×+
+×+×=
⊥⊥⊥
⊥⊥
eeaeea
eeeea
eeeeaeea
eeaeeaW
zz
yyxxx
zzz
rrrr
rr
rr
rrrr
(15)
Here Z-axis is directed along with axis C4 and values
(ai)2, which are proportional to intensities ( ) ( )21 , ωω II ,
depend on light frequencies 21,ωω as well as parameters
of crystals, and do not depend on vectors 1e
r
, 2e
r
. The
vectors cannot be determined by methods of the group
theory. They can be calculated by the methods of the
second order perturbation theory taking into account the
light absorption probability for a real band structure of
crystals.
The right part of the equation (15) contains a sum of
five squares, which at symmetry transformation of crys-
tal can be expanded on irreducible representations of
group D4: Ã1, Ã2, Ã3, Ã4, Ã5.These are the representations
of transitions. They are expanded in direct products of
irreducible representations of initial and final states of
transitions. At linear polarization of light the transitions
between states with different projections of a complete
angular momentum of electrons on an axis of
quantization, oriented along a pulse direction, are for-
bidden [9]. That is, transitions Ã6
± → Ã6
±, Ã7
± → Ã7
±,
Ã5 → Ã6
±, Ã5 → Ã7
±, with amplitudes of dipole moments
(equal to matrix elements of transitions), are transformed
on irreducible representations Ã1, and Ã5, at symmetry
transformation of group D4. Therefore in the case of lin-
ear polarization of beams coefficients 431 ,,~ aaa in (15)
are equal to zero as well as 5221 ,~,, aaaa , are not.
3. Summary
The received theoretical results are shown in Table 1.
They are in accordance with experimental results shown
in Fig. 2 provided that . It is possible to explain equality
by the fact of absence of transitions between states Ã6
(Ã1, Ã2) and in the Ã6 (Ã1, Ã2) spectral region used, be-
cause of such states is absent, too [10].
I.I. Patskun, I.A. Slipukhina: ²nvestigation of βββββ-CdP
2
crystals by laser spectroscopy methods
35SQO, 7(1), 2004
Table 1.
Irreducible Irreducible [ ]00121 qq
rr [ ]01021 qq
rr
representations representations ( ),0,sin,cos 111 ϕϕe
r ( ),cos,0,sin 111 ααe
r
of dipole of initial and ( ),0,sin,cos 222 ϕϕe
r
( ),cos,0,sin 222 ααe
r
momentum final states 2211 ,,, exe
rr
== ϕϕ 212211 ,,,,, eezeze
rrrr
=== ααα
of two-photon of two-photon 21,,, eex
rr
=ϕ
transitions transitions α1, degree 0 30 60 90 0 90
ϕ 0° 90° α2, degree 0 30 60 90 90 0
α3, degree 0 0 0 0 90 90
( ) ( )1616 ΓΓ−ΓΓ ++
Ã1 ( ) ( )2626 ΓΓ−ΓΓ −− ϕ22
2 sina 0 2
2a 0 0 0 0 0 0 0
( ) ( )3737 ΓΓ−ΓΓ ++
( ) ( )4747 ΓΓ−ΓΓ −+ 0 0 ( )2211 coscos ααa 2
1a 2
1
16
9
a 2
1
16
1
a 0 0 0
Ã2 ( ) ( )2616 ΓΓ−ΓΓ −+ 0 0 0 0 0 0 0 0 0 0
( ) ( )4737 ΓΓ−ΓΓ −+
Ã3 ( ) ( )1637 ΓΓ−ΓΓ ++ 0 0 0 0 0 0 0 0 0 0
( ) ( )2647 ΓΓ−ΓΓ −−
Ã4 ( ) ( )1647 ΓΓ−ΓΓ +− 0 0 0 0 0 0
( ) ( )2637 ΓΓ−ΓΓ −+
Ã5 ( ) ( )216765 ,ΓΓΓ−Γ+ΓΓ ±±± 2
5a 2
5a 2
5a
+
+
α
αα
2
2
215
sin~
sinsin
a
a 0
2
516
1
a 2
5
16
9
a 2
5a 0 0
( ) ( )−±±±+ ΓΓΓ−Γ+ΓΓ 437735 ,
0 0 0 0 0 0 0
2
2
~a 2
2a
Number of the spectrum 1 2 4 3 6 5
The diagram of TWA appropriate to received results
in is represented in Fig. 3, where and are amplitudes of
TPA probabilities of an unary resonance, and d3 is the
impurity level, on which double resonance comes true.
References
1. I.I. Pazkun, Kvantovaya elektronika, Kiev: Naukova dumka,
1993. 45. p. 3-30 (in Russian).
2. I.P. Shen. Principi nelinejnoi optiki, Moskva: Nauka, 1989
(in Russian).
3. I.I. Pazkun, I.I. Tichina, I.A. Kolesnik // Ukr. Fiz. Zhurnal,
¹9, p.1110-1115 (1997).
4. G.A. Grishenko et al. // Optika i spektroskopija, 69, p.115-
119 (1990) (in Russian).
5. P.E. Mozol et al. // Fiz. Tehnika Polupr., 14, p. 902-907 (1980)
(in Russian).
6. I.S. Gorban et al. // Ibid, 15, p. 424-427 (1981).
7. M. Inoue, Y. Tayozawa // J. Phys. Soc. Japan, 20, p. 363
(1965).
8. E.L. Ivchenko // Fiz. Tverdogo Tela, 14, p. 3489-3497 (1972)
(in Russian).
9. E.V. Beregulin, et al. // Fiz. Technika Polupr., 9, p. 876-885
(1975) (in Russian).
10. Gavrilenko V.I. et al. Opticheskiye svoistva poluprovodnikov,
Kiev: Naukova dumka, 1987.
2
2
~a 2
1a
2
5a d3
0.46
2.02
Ã7 Ñ2
Ã6 Ñ1
Ã7 V1
Ã6 (Ã5) V2
Ã7 (Ã5) V3
0.35 0.33
hω
1
+
h
ω
2,
å
V
Fig. 3. The diagram of TPA for plane-polarized light in β-CdP2.
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