The crossflow piezo-electrooptic effect in crystals. Example of lithium tantalate
In the present paper the thermodynamic and phenomenological descriptions of the crossflow piezo-electrooptic effect in crystals have been made. For example the necessary experimental measurements of this effect have been carried out in the lithium tantalate crystals. For these crystals at first the P₃...
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| Zitieren: | The crossflow piezo-electrooptic effect in crystals. Example of lithium tantalate / A.S. Andrushchak, I.V. Sydoryk, M.V. Kaidan, R.O. Vlokh // Condensed Matter Physics. — 2000. — Т. 3, № 4(24). — С. 857-862. — Бібліогр.: 8 назв. — англ. |
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irk-123456789-1209872017-06-14T03:03:53Z The crossflow piezo-electrooptic effect in crystals. Example of lithium tantalate Andrushchak, A.S. Sydoryk, I.V. Kaidan, M.V. Vlokh, R.O. In the present paper the thermodynamic and phenomenological descriptions of the crossflow piezo-electrooptic effect in crystals have been made. For example the necessary experimental measurements of this effect have been carried out in the lithium tantalate crystals. For these crystals at first the P₃₃₁₁₃ - P₁₁₁₁₃ = −4, 6 • 10⁻¹⁹ m³/ N•V absolute coefficients difference of crossflow piezo-electrooptic effect were determined by two measurement methods. У даній роботі було здійснено термодинамічний і феноменологічний описи перехресного п’єзо-електрооптичного ефекту в кристалах. Для прикладу були проведені необхідні експериментальні вимірювання вказаного ефекту в кристалах танталату літію. Для цих кристалів на основі двох різних методів вимірювання вперше було визначено різницю абсолютних коефіцієнтів перехресного п’єзо-електрооптичного ефекту P₃₃₁₁₃ - P₁₁₁₁₃ = −4, 6 · 10⁻¹⁹ м³/ Н·В. 2000 Article The crossflow piezo-electrooptic effect in crystals. Example of lithium tantalate / A.S. Andrushchak, I.V. Sydoryk, M.V. Kaidan, R.O. Vlokh // Condensed Matter Physics. — 2000. — Т. 3, № 4(24). — С. 857-862. — Бібліогр.: 8 назв. — англ. 1607-324X DOI:10.5488/CMP.3.4.857 PACS: 77.84.Dc, 78.20.Hp, 78.20.Fm http://dspace.nbuv.gov.ua/handle/123456789/120987 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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In the present paper the thermodynamic and phenomenological descriptions of the crossflow piezo-electrooptic effect in crystals have been made. For example the necessary experimental measurements of this effect have been carried out in the lithium tantalate crystals. For these crystals at first the P₃₃₁₁₃ - P₁₁₁₁₃ = −4, 6 • 10⁻¹⁹ m³/ N•V absolute coefficients difference of crossflow piezo-electrooptic effect were determined by two measurement methods. |
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Andrushchak, A.S. Sydoryk, I.V. Kaidan, M.V. Vlokh, R.O. |
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Andrushchak, A.S. Sydoryk, I.V. Kaidan, M.V. Vlokh, R.O. The crossflow piezo-electrooptic effect in crystals. Example of lithium tantalate Condensed Matter Physics |
| author_facet |
Andrushchak, A.S. Sydoryk, I.V. Kaidan, M.V. Vlokh, R.O. |
| author_sort |
Andrushchak, A.S. |
| title |
The crossflow piezo-electrooptic effect in crystals. Example of lithium tantalate |
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The crossflow piezo-electrooptic effect in crystals. Example of lithium tantalate |
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The crossflow piezo-electrooptic effect in crystals. Example of lithium tantalate |
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The crossflow piezo-electrooptic effect in crystals. Example of lithium tantalate |
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The crossflow piezo-electrooptic effect in crystals. Example of lithium tantalate |
| title_sort |
crossflow piezo-electrooptic effect in crystals. example of lithium tantalate |
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Інститут фізики конденсованих систем НАН України |
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2000 |
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The crossflow piezo-electrooptic effect in crystals. Example of lithium tantalate / A.S. Andrushchak, I.V. Sydoryk, M.V. Kaidan, R.O. Vlokh // Condensed Matter Physics. — 2000. — Т. 3, № 4(24). — С. 857-862. — Бібліогр.: 8 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
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2025-07-08T18:58:33Z |
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1837106321900437504 |
| fulltext |
Condensed Matter Physics, 2000, Vol. 3, No. 4(24), pp. 857–862
The crossflow piezo-electrooptic effect
in crystals. Example of lithium tantalate
A.S.Andrushchak, I.V.Sydoryk, M.V.Kaidan, R.O.Vlokh
Institute of Physical Optics, 23 Dragomanov Str., 79005 Lviv, Ukraine
Received February 16, 2000, in final form September 26, 2000
In the present paper the thermodynamic and phenomenological descrip-
tions of the crossflow piezo-electrooptic effect in crystals have been made.
For example the necessary experimental measurements of this effect have
been carried out in the lithium tantalate crystals. For these crystals at first
the P33113−P11113 = −4.6·10−19 m3/ N·V absolute coefficients difference of
crossflow piezo-electrooptic effect were determined by two measurement
methods.
Key words: thermodynamic and phenomenological descriptions,
crossflow piezo-electrooptic effect, lithium tantalate crystals, induced
birefringence
PACS: 77.84.Dc, 78.20.Hp, 78.20.Fm
1. Introduction
The crossflow piezo-electrooptic effect, resulting frommutual interaction of piezo-
and electrooptic effects in crystal materials, is poorly investigated because of its small
values [1] on the background of a direct action of the piezo- or electrooptic effects.
The aim of the present work is to carry out thermodynamic and phenomenolog-
ical description of the crossflow piezo-electrooptic effect and make necessary exper-
imental measurements for its verification.
2. Thermodynamic description
According to the general thermodynamic theory [2–4], a crystal appears as a
thermodynamic system the monocrystal state of which is determined by certain
numbers of variables. Mechanical stress σij , electrical field strength Ei and temper-
ature T are chosen as independent variables. Herein Gibbs free energy G is assumed
as thermodynamic function for which [1,5]:
G = U − εσ −E(D /4π)− TS, (1)
c© A.S.Andrushchak, I.V.Sydoryk, M.V.Kaidan, R.O.Vlokh 857
A.S.Andrushchak et al.
where U is the inner energy. Then, the components of the mechanical deformation
εij, electrical field induction Di and entropy S will be functions of independent
variables. Let’s spread in the MacLaurin’s series the εij and Di up to the third order
of smallness (no thermal effects are taken into account) because their meanings of the
second and the third order of smallness are insignificant [6]). Setting for convenience
δm = Dm/4π, we will have:
εkl =
∂εkl
∂σij
σij +
∂εkl
∂En
En +
1
2
[
∂2εkl
∂σij∂σqr
σijσqr + 2
∂2εkl
∂σij∂En
σijEn +
∂2εkl
∂En∂Eo
EnEo
]
+
1
6
[
∂3εkl
∂σij∂σqr∂σst
σijσqrσst + 3
∂3εkl
∂σij∂σqr∂En
σijσqrEn
+3
∂3εkl
∂σij∂En∂Eo
σijEnEo +
∂3εkl
∂En∂Eo∂Ep
EnEoEp
]
, (2)
δm =
∂δm
∂σij
σij +
∂δm
∂En
En +
1
2
[
∂2δm
∂σij∂σqr
σijσqr + 2
∂2δm
∂σij∂En
σijEn +
∂2δm
∂En∂Eo
EnEo
]
+
1
6
[
∂3δm
∂σij∂σqr∂σst
σijσqrσst + 3
∂3δm
∂σij∂σqr∂En
σijσqrEn
+3
∂3δm
∂σij∂En∂Eo
σijEnEo +
∂3δm
∂En∂Eo∂Ep
EnEoEp
]
. (3)
Besides, the following equations are valid [5]:
∂G
∂σkl
= −εkl,
∂G
∂Em
= −δm,
∂δm
∂σkl
=
∂εkl
∂Em
. (4)
Considerations of the physical sense of the partial derivatives are given more
in full in [1]. We will take into consideration only components describing crossflow
effects. They are:
Aklijn =
∂2εkl
∂σij∂En
= −
∂3G
∂σij∂σkl∂En
=
∂Sklij
∂En
≡
∂2δn
∂σij∂σkl
=
∂dnij
∂σkl
. (5)
It is a correction part to the elastic compliance constant Sklij, which is connected
with the electrical field En acting in the crystal, or the same, correction part to the
reverse piezoelectric effect constant dnij under the action of the mechanical stress
σij . It is fifth-rank tensor, which describes a crossflow piezo-electrostrictive effect
caused in the crystal by mutual interaction of the electrical field and the mechanical
stress. Furthermore:
Bmnijo =
∂3δm
∂σij∂En∂Eo
= −
∂4G
∂σij∂Em∂En∂Eo
=
∂µmnij
∂Eo
≡
∂
∂Eo
(
∂kmn
∂σij
)
=
∂
∂σij
(
∂kmn
∂Eo
)
=
∂ρmno
∂σij
. (6)
858
The crossflow piezo-electrooptic effect in crystals
Here kmn – permittivity tensor and Bmnijo – a tensor of crossflow piezo-electroop-
tic effect, which determine the change of the piezooptic module µmnij = ∂kmn/∂σij
under the action of the electrical field Eo, or the same, the change of the liner
electrooptic effect tensor ρmno = ∂kmn/∂Eo under the action of the mechanical
stress σij .
Then, as parts of the second order of smallness are neglected, the simplified
correlations (2) and (3) are:
εkl = Sklijσij + dnklEn + AklijnσijEn, (7)
Dm = 4πdmijσij + En[kmn + µmnijσij + ρmnoEo/2 +BmnijoσijEo/2], (8)
where dmij is a tensor of piezoelectric effect.
3. Phenomenological description
In the given work, the phenomenological description of a crossflow effect needed
for the explanation of our experimental measurements is carried out.
As is generally known, the external action (be it the mechanical stress σmn or
the external electric field El) exerted on the crystal sample results in the change of
birefringence δ(∆nk) = δni − δnj (or refractive indices δni, δnj) of the sample as
well as its length δtk in the direction k of light propagation, which are registered by
polarization-optic method through the change of an optical path difference δ∆k of
this sample:
δ∆k = δ(∆nktk) = tkδ(∆nk) + ∆nkδtk. (9)
The value δ(∆nk) or δni, δnj can be determined from the tensor for polarization
constants aij. Beside the coefficients of piezo- πijmn and linear electrooptic rijl ef-
fects, this tensor in the first approximation for acentric crystals also contains the
coefficients of their crossflow effect [1], analogous to (8):
∆aij = πijmnσmn + rijlEl + Pijmnl(σmnEl), (10)
where Pijmnl = ∂πijmn/∂El = ∂rijl/∂σmn is the tensor of crossflow piezo-electrooptic
effect (tensor of absolute coefficients), σmnEl = gmnl is the 3-rank tensor, which
equals the product of the 2-rank tensor σmn and vector El. From a thermodynamic
description of crossflow piezo-electrooptic effect Pmnijl (see formula (6)), one can
describe symmetrical properties of this tensor while replacing the indexes P ijmnl =
Pjimnl = Pijnml = Pijlmn. The complete form of this tensor is given in [7].
Analogously, the value δtk is determined from the deformation tensor εkl (see
formula (7)), which beside the coefficients of elastic Sklij and piezoelectric dnkl effects
also contains the coefficients of their crossflow Aklijn effect. This crossflow Aklijn
effect is named “false” effect in our case.
To derive the necessary operating correlations we use the matrix interpretation
of the processed tensors. For a crystal with small initial birefringence the correlation
(9) will be simplified to δ∆k = tkδ(∆nk), where the δ(∆nk) value is equal to [5,8]:
859
A.S.Andrushchak et al.
δ(∆nk) = −π∗
kmσm/2. (11)
Here π∗
km = πimn
3
i − πjmn
3
j is a known piezooptic coefficient of the induced birefrin-
gence. For half-wave stresses σo
m, when δ∆k = λ/2 (λ is a length of light wave), one
can obtain:
λ/2 = −π∗
kmσ
o
mtk/2. (12)
After applying the electric field El, for repeatedly measured half-wave stresses
σo′
m, we obtain:
λ/2 = −π∗
kmσ
o′
mtk/2− P ∗
kmlσ
o′
mEltk/2. (13)
Analogously P ∗
kml = Pimln
3
i −Pjmln
3
j is the crossflow piezo-electrooptic coefficient
of the induced birefringence. Mutual solution of (12) and (13) gives us the value of
this coefficient to be searched for:
P ∗
kml = λ[(σo′
m/σ
o
m)− 1]/(σo′
mEltk). (14)
A similar formula can be obtained at measuring the magnitudes of half-wave
electric fields Eo
l and Eo′
l for σm = 0 and σm 6= 0 accordingly.
4. Experimental results
Figure 1. The dependences of bire-
fringence change δ(∆n2) for lithium
tantalate crystals on the normal me-
chanical stress σ11 for different mag-
nitudes of electrical fields E3 (for light
wave λ = 0, 6328 µm and temperature
T=293 K).
The measurements of crossflow piezo-
electrooptic effect were carried out us-
ing the polarisation-interferometrical tech-
nique by Senarmont method and half-wave
stresses method [5].
To exclude a possible error, connected
with the measurement of “false” crossflow
effect, the LiTaO3 crystals with a small
initial birefringence (∆nk = 0, 005) were
used. For these crystals, the elastic compo-
nent in (9) and therefore “false” crossflow
effect could be neglected.
We have determined the crossflow
piezo-electrooptic effect P ∗
22113
component
on the sample of direct cut, when k ‖ Y,
σm ‖ X and El ‖ Z. In the figures the de-
pendences for the δ(∆n2) value under the
normal mechanical stress σ11 for different magnitudes of E3 (figure 1) are shown as
well as the δ(∆n2) value under the electrical field E3 for different magnitudes of σ11
(figure 2). From the changes of angular coefficients for linear interpolation of these
behaviours we have calculated the average magnitudes of P ∗
22113
= −5.0 ·10−18 m3/N
· V (from figure 1) and P ∗
22113
= −4.4 · 10−18 m3/N · V (from figure 2).
860
The crossflow piezo-electrooptic effect in crystals
Figure 2. Analogous to figure 1 the
dependences of δ(∆n2) on the electri-
cal fields E3 for different magnitudes
of mechanical stress σ11.
The difference of these coefficients can
be explained by the induced rotation of the
optical indicatrix around the X axis under
the action of the mechanical stress σ11. It
leads to the change of extraordinary index
ne of light wave according to the known
equation:
δne = ne(1−cos(π41σ11/(n
−2
e −n−2
o )). (15)
Taking this into account, we have found
the new magnitude of P ∗
22113
= −4.7 ·10−18
m3/N · V from figure 1.
This coefficient P ∗
22113
has also been de-
termined from the method of half-wave
stresses. Having determined the half-wave mechanical stress σ o
m for El = 0 and
σo′
m upon the action of electrical field El 6= 0, we have calculated the magnitude
P ∗
kkmml according to formula (14), which was equal P ∗
22113
= −5.1 · 10−18 m3/N · V.
It is also noted that from P ∗
22113
one can calculate the absolute coefficients com-
bination of crossflow piezo-electrooptic effect (taking into account that no ≈ ne =
2.175): P33113 − P11113 ≈ P ∗
22113
/n3
o = −4.6 · 10−19 m3/N · V.
Besides, according to (6): P ∗
22113
= ∂π∗
2211
/∂E3 ≈ ∆π∗
2211
/∆E3, and therefore at
the electrical field change of ∆E3 = 2.3 ·105 V/m the change of piezooptic coefficient
is equal ∆π∗
2211
= P ∗
22113
∆E3 = 1.1 · 10−12 m2/N. For comparison, the magnitude of
coefficient π∗
2211
from our measurements is equal to 10.4 · 10−12 m2/N.
5. Conclusions
1. The thermodynamic descriptions of crossflow piezo-electrooptic effect in crys-
tals and the necessary phenomenological descriptions for anisotropic materials with
small initial birefringence have been made in the paper.
2. For lithium tantalate crystals, at first the P33113−P11113 = −4.6·10−19m3/N · V
absolute coefficients difference of crossflow piezo-electrooptic effect were determined.
For these crystals, the change of ∆π∗
2211
because of crossflow effect corresponds to
∼ 10% of coefficient π∗
2211
at electrical field change of ∆E3 = 2.3 · 105 V/m.
3. The equality (within the limits of experimental error) of the P ∗
22113
coefficient,
obtained from different measurement methods, demonstrated the correctness of our
results and showed the possibility of measuring the crossflow piezo-electrooptic effect
in anisotropic crystals with small initial birefringence using these methods.
References
1. Sirotin Yu.I., Shaskolskya M.P. Essentials of Crystallophysics. Moscow, Nauka, 1979
(in Russian).
861
A.S.Andrushchak et al.
2. Mason W. Crystal Physics of Interaction Processes. New–York, Lond. Acad. Press,
1966.
3. Mason W. // Bell. Techn. J., 1950, vo1. 29, p. 161.
4. Nye J. Physical Properties of Crystals. Moscow, Mir, 1967 (in Russian).
5. Sonin A.S., Vasilevskaya A.S. Electrooptical Crystals. Moscow, Atomizdat, 1971 (in
Russian).
6. Mason W. Piezooptical Crystals and their Application in Acoustics. Moscow, Izd.
inostran. liter., 1952 (in Russian).
7. Koptsyk V.A. Shubnikov’s Groups. Moscow, Izd. MGU, 1966 (in Russian).
8. Mytsyk B.G., Pryriz Ya.V., Andrushchak A.S. The lithium niobate piezooptical fea-
tures. // Cryst. Res. Technol., 1991, vol. 26, No. 7, p. 931–940.
Перехресний п’єзо-електрооптичний ефект на
прикладі кристалів танталату літію
А.С.Андрущак, І.В.Сидорик, М.В.Кайдан, Р.О.Влох
Інститут фізичної оптики,
79005 Львів, вул.Драгоманова 23
Отримано 16 лютого 2000 р., в остаточному вигляді –
26 вересня 2000 р.
У даній роботі було здійснено термодинамічний і феноменологічний
описи перехресного п’єзо-електрооптичного ефекту в кристалах.
Для прикладу були проведені необхідні експериментальні вимірю-
вання вказаного ефекту в кристалах танталату літію. Для цих кри-
сталів на основі двох різних методів вимірювання вперше було ви-
значено різницю абсолютних коефіцієнтів перехресного п’єзо-елек-
трооптичного ефекту P33113 − P11113 = −4, 6 · 10−19 м3/ Н·В.
Ключові слова: термодинамічний та феноменологічний описи,
перехресний п’єзо-електрооптичний ефект, кристал танталату
літію, індуковане двозаломлення
PACS: 77.84.Dc, 78.20.Hp, 78.20.Fm
862
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