Calculation of stretching factor for optical pulse stretcher
A task of temporal stretching of pulses in an optical pulse stretcher of CPA-laser systems is considered. The output pulse profiles are obtained and the stretching factors are calculated on the foundation of the solution of the wave equation for a passage through an optical stretcher of the ultrasho...
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
2015
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| Cite this: | Calculation of stretching factor for optical pulse stretcher / V.P. Leshchenko, A.I. Povrozin, S.Y. Karelin // Вопросы атомной науки и техники. — 2015. — № 3. — С. 132-135. — Бібліогр.: 16 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1859720409570607104 |
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| author | Leshchenko, V.P. Povrozin, A.I. Karelin, S.Y. |
| author_facet | Leshchenko, V.P. Povrozin, A.I. Karelin, S.Y. |
| citation_txt | Calculation of stretching factor for optical pulse stretcher / V.P. Leshchenko, A.I. Povrozin, S.Y. Karelin // Вопросы атомной науки и техники. — 2015. — № 3. — С. 132-135. — Бібліогр.: 16 назв. — англ. |
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| container_title | Вопросы атомной науки и техники |
| description | A task of temporal stretching of pulses in an optical pulse stretcher of CPA-laser systems is considered. The output pulse profiles are obtained and the stretching factors are calculated on the foundation of the solution of the wave equation for a passage through an optical stretcher of the ultrashort pulses with the different profiles. The calculations for the optical stretcher of Offner triplet type had made.
Розглянуто задачу про часове розширення імпульсів в оптичному розширювачі імпульсів CPA-лазерної системи. На основі рішення хвильового рівняння для проходження надкоротких вхiдних імпульсів різного профiлю отримано їх вихідні профілі та розраховано коефіцієнт розширення. Розрахунки виконанo для оптичного розширювача типу триплетa Оффнера.
Рассмотрена задача о временном расширении импульсов в оптическом расширителе импульсов CPA-лазерной системы. На основе решения волнового уравнения для прохождения сверхкоротких импульсов различного профиля через оптический расширитель импульсов получены их выходные профили и рассчитаны коэффициенты расширения. Расчеты выполнены для оптического расширителя типа триплета Оффнера.
|
| first_indexed | 2025-12-01T09:22:18Z |
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CALCULATION OF STRETCHING FACTOR FOR OPTICAL
PULSE STRETCHER
V.P.Leshchenko, A. I.Povrozin∗, S.Y.Karelin
National Science Center ”Kharkov Institute of Physics and Technology”, 61108, Kharkov, Ukraine
(Received April 24, 2014)
A task of temporal stretching of pulses in an optical pulse stretcher of CPA-laser systems is considered. The output
pulse profiles are obtained and the stretching factors are calculated on the foundation of the solution of the wave
equation for a passage through an optical stretcher of the ultrashort pulses with the different profiles. The calculations
for the optical stretcher of Offner triplet type had made.
PACS: 42.25.Bs;42.25.Md;42.60.By;42.60.Fe;42.60.Mi
1. INTRODUCTION
An optical stretcher of pulses that stretches pulses
of a driving oscillator from the tens of femtoseconds
to as rule the hundreds of picoseconds [1] is an im-
portant structural part of the CPA-laser systems.
The stretching of the femtosecond pulses performs in
CPA-laser systems due to the phase modulation. A
long-wavelength component of the pulse passes a path
short-er than short- wavelength one due to this mod-
ulation and thus an output pulse duration increases.
This effect was achieved in the first stretchers, con-
sisting of two antiparallel diffraction gratings with a
telescopic system between them, introducing an ad-
ditional dispersion in the entire optical system [2-4].
Therefore the CPA-laser systems with the stretchers,
using only reflective optics, were created [5-9]. These
stretchers include the Offner triplet [10], which in cre-
ating of the CPA-laser systems was often preferable.
The calculation method presented below is applica-
ble not only to the Offner triplet, but also for other
stretcher schemes. The consideration at an example
of the above type stretcher made to allow com-parison
of the obtaining results with published data [9].
Fig.1. An optical scheme of the Offner triplet
An optical scheme of the Offner triplet includes the
minimum elements (Fig.1): a diffraction grating 1, a
concave mirror M1, a convex mirror M2, a reflector
M3 and a return mirror R.
An important task in a determining of the max-
imum value for the stretching factor of the driving
oscillator pulse is to establish the connection between
the stretcher parameters and the temporal profile of
this pulse. How-ever, this problem has not considered
in a literature.
2. ANALITICAL RESULTS
The aim of this work is to create of a mathemati-
cal model for the passage of the pulse through the
stretcher that is suitable for the engineering calcula-
tions of the pulse stretching factors with the different
time profiles. An influence of the third and fourth
orders dispersion in the stretching of the pulse not
previously considered in a literature takes into ac-
count in this case. The wave equation for complex
electric field amplitude A(z, t) of the pulse passing
through the stretcher with taking into account of this
influence is in the form [11]:
∂A
∂z
− i
2
K2
∂2A
∂t2
− 1
6
K3
∂3A
∂t3
+
i
24
K4
∂4A
∂t4
= 0 , (1)
where K = 2π/λ – a wave vector, λ – a wave length,
K2 = ∂2K
∂ω2 – a dispersion of a group velocity of the
wave packet,
K3 =
∂3K
∂ω3
+
3
K
∂2K
∂ω2
∂K
∂ω
,
K4 =
∂4K
∂ω4
+
4
K
∂2K
∂ω2
∂K
∂ω
+
3
K
(
∂2K
∂ω2
)2
,
ω – a frequency of a radiation. Here and hereafter
the digital codes ”2” and higher ones corresponds to
the order dispersion. The values of the wave vector
and its derivatives had taken at the average frequency
of the wave packet.
The solution of the equation (1) gives the oppor-
tunity to compare the influence on the pulse width
and shape of the second, third and fourth orders
dispersion. First, we consider the influence on the
stretching factor of only second order of the disper-
sion. The solution of the equation (1) in such a repre-
sentation founded by the Fourier integral transforms
∗Corresponding author E-mail address: pai40@kipt.kharkov.ua
132 ISSN 1562-6016. PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY, 2015, N3(97).
Series: Nuclear Physics Investigations (64), p.132-135.
method [12] is on the form:
A(t, z) =
∫ ∞
−∞
A0(t1)G(t− t1, z)dt1 , (2)
where A0(t1) = A(0, t1) – the electric field amplitude
at the input to the stretcher, G(t− t1, z) – influence
function:
G(t− t1, z) =
exp
[
i(t−t1)
2K2z
]
√
2πiK2z
. (3)
We assume, that at the input of the stretcher the
pulse electric field amplitude has a Gaussian tempo-
ral profile:
AG(t1) = A0G exp
[
−2 ln 2
(
t1
τ0G
)2
]
, (4)
where A0G – the maximum amplitude of the elec-
tric field, the index ”G” means ”Gaussian”, τ0G – a
pulse duration determined at a level J = 1
2 |A0G|2,
that determines the specific form of the expression
(4).
When the pulse passes through the stretcher, a
spatial coordinate z equals its effective length L. The
second order dispersion is on the form:
Φ2 =
∂
∂2Φω2
= K2L , (5)
where K2 is given by expression [13]:
K2 =
λ3
0
2πc2d2 cos2 θ0
, (6)
where λ0 – the average wavelength of the wave
packet, θ0 –the diffraction angle at a wavelength λ0,
d – a constant of the grating, c – a velocity of a light
in a vacuo.
The expression for the effective length of the
Offner triplet obtained on the basis of the theory of
the optical systems [14] and is on the form:
L =
3R− 8r + 2R r
S
+ S
(
R
r
+ 8 r
R
− 6
)
6 + 4r
(
1
S
− 2
R
)
− R
r
− 1.5R
S
, (7)
where R – a radius of the concave mirror, r – a ra-
dius of the convex mirror, S – distance between the
center of the concave mirror and the middle of the
diffraction grating.
Substituting (3), (4) and (5) in (2) with followed
squaring a number gives an expression for the inten-
sity profile of the output stretcher pulse:
J2G(t, Φ2) =
A2
0G
V0G
exp
[
−4 ln 2
V 2
0G
(
t
τ0G
)2
]
, (8)
where V0G – stretching factor of the pulse with a
Gaussian intensity distribution:
V0G =
√
1 +
(
4 ln 2Φ2
τ2
0G
)2
. (9)
As it is clear from a literature the driving oscillator
of the pulses with Gaussian temporal intensity pro-
file and the profile proportional to the square of the
hyperbolic secant are in practice most often in the
CPA-laser systems. The electric field amplitude of
this pulse is on the form:
AS(t) = A0Ssech
1.763t
τ0S
, (10)
where A0S - a peak amplitude of the electric field,
τ0S – the pulse duration on the level 1
2 |A0S |2. Index
”S” means a belonging to the function ”sech”.
The solution of the equation(1) taking into ac-
count only the second order dispersion for the pulses
with an amplitude of electric field(10) at the stretcher
input is on the form:
A2S(t, Φ2) =
∫ ∞
−∞
A0S exp
[
i(t−t1)
2
2Φ2
]
√
2πiΦ2
sech
1.763t1
τ0S
dt1 .
(11)
The squaring a number (11) gives the expression for
the intensity profile of the stretcher output:
J2S(t, Φ2) =
∣∣∣∣∣∣
∫ ∞
−∞
A0S exp
[
i(t−t1)
2
2Φ2
]
√
2πiΦ2
sech
1.763t1
τ0S
dt1
∣∣∣∣∣∣
2
.
(12)
The stretching factor in this case is defined: VOS =
Φ2s/Φ0s, where Φ2s - is determined by measuring at
the level of half J2s(t,Φ2).
A consistent taking into account of the influence
of the second and third orders dispersion and then the
second, third and fourth orders dispersion for output
pulse with the input Gaussian profile (4) gives a so-
lution of the wave equation (1) on the form:
A4G(t, Φ0 Φ1 Φ2 Φ3) = A0G
∫ ∞
−∞
exp {−i [ωt+
Φ2ω
2
2
− 1
6
(Φ3 +
3
Φ0
Φ1Φ2)ω
3
]
− τ2
0Gω
2
8 ln 2
}
dω , (13)
A3G(t, Φ0 Φ1 Φ2 Φ3) = A0G
∫ ∞
−∞
exp {−i [ωt+
Φ2ω
2
2
− 1
6
(Φ3 +
3
Φ0
Φ1Φ2)ω
3 +
+
1
24
(Φ4 +
4
Φ0
Φ1Φ3 +
3
Φ0
Φ2
2)ω
4
]
− τ2
0Gω
2
8 ln 2
}
dω . (14)
Here Φ0 is the phase shift acquired in the stretcher.
It is on the form[15]:
Φ0 = −ω0L
c
cos θ0(cos θ0 + cos γ) , (15)
where ω0 – average frequency of the wave packet, θ0
– an angle of a diffraction at the frequency ω0, γ –
an angle of incidence of the pulse to the diffraction
grating. Φ1 = ∂Φ
∂ω – a first derivative with respect to
frequency of the phase. It is on the form [15]:
Φ1 = −L
c
[1 + cos(γ − θ0)] , (16)
Φ3 = ∂3Φ
∂ω3 and Φ4 = ∂4Φ
∂ω4 – the third and fourth or-
ders dispersion, respectively. According to [13] the
expressions for them are on the forms:
Φ3 = − 3λ4
0L
4π2c3d2 cos2 θ0
[
1 +
λ0
d
sin θ0
cos2 θ0
]
, (17)
133
Φ4 =
3λ5
0L
4π3c4d2 cos2 θ0
{
4 + 8
sin θ0
cos2 θ0
+
+
λ2
0
d2
[
1 + tg2θ0(6 + 5tg2θ0)
]}
. (18)
The calculations Φ0 Φ1 Φ2 Φ3 Φ4 performed for the
Offner triplet with specific operating parameters:
R = 1024mm, r = 512mm, S = 774mm, λ0 =
800nm, θ0 = 20, 14◦, d = 1/1200mm ( for the for-
mulas (5-7)), taken from [8]. The calculations of the
integrals (13) and (14) performed numerically.
The solutions of the equation (1) for the input
pulse having a temporal profile of the input field (10)
pre-pared similarly. The field amplitude of this pulse
with taking into account the second and third orders
is on the form:
A3S(t, Φ0 Φ1 Φ2 Φ3) = A0S
∫ ∞
−∞
sech
πτ0Sω
3.526
×
exp
{
−i
[
ωt+
Φ2ω
2
2
− 1
6
(Φ3 +
3
Φ0
Φ1Φ2)ω
3
]}
dω. (19)
And the field amplitude of the same pulse with
taking into account the second, third and fourth or-
ders is on the form:
A4CS(t, Φ0 Φ1 Φ2 Φ3 Φ4) = A0S
∫ ∞
−∞
sech
πτ0Sω
3.526
×
exp
{
−i
[
ωt+
Φ2ω
2
2
− 1
6
(Φ3 +
3
Φ0
Φ1Φ2)ω
3+
+
1
24
(Φ4 +
4
Φ0
Φ1Φ3 +
3
Φ0
Φ2
2ω
4
]}
dω . (20)
3. NUMERICAL RESULTS
Fig.2 shows the profiles of the pulses at the stretch-
er output with taking into account the second order
dispersion for the input Gaussian pulse (8) and the
input pulses with a hyperbolic secant profile (12).
Expression (12) has been calculated numerically.
Fig.2. The profiles of the pulses at the stretcher
output
The results of calculations of Gaussian pulses pro-
files at the stretcher output with taking into account
the third (13) and fourth (14) orders dispersion have
coincided with the output Gaussian pulse profiles (8)
calculated with taking into account only the second
order dispersion with the accuracy 0.6%. A similar
coincidence of the pulse profiles calculations with
taking into account the third (19) and fourth (20)
orders dispersion received in the comparison theirs
with the pulse profile (12) with taking into account
only the second order dispersion for the case of the
hyperbolic secant. Duration of the input pulses in
the calculations taken τ0G = τ0S = 30 fs. As can be
seen from Fig.2 the duration of the stretched input
Gaussian pulse on the half intensity level is 273 ps,
and of the stretched input pulse with the profile of the
hyperbolic secant is 195 ps. According to the formula
(9) and Fig.2 the stretching factor was V0 = 9118.
This value is 9.7% lower than the experimental one,
obtained under the same parameters of stretcher [9].
This discrepancy may be cause of the fact that the
calculations do not take into account the phase shift
caused by the spatial structure of the beam [16]. The
value of the stretching factor for the input pulse with
the profile of the hyperbolic secant is equal to 6500.
4. CONCLUSIONS
The following conclusions from the obtained solutions
of the wave equation for the passage of the pulses
with the input Gaussian profile and the input profile
of the hyperbolic secant through the stretcher tak-
ing into account the second , third and fourth orders
dispersion have been made:
1. The negligible influence on the results of the
calculations of the profiles pulses passing through the
stretcher of the third and fourth orders dispersion al-
lows to simplify the calculation of the stretching fac-
tor. The calculation of the stretching factor may be
performed with taking into account only the second
order dispersion.
2. The discrepancy between the value calculation
of the stretching factor for the Gaussian shape pulse
and the value of the stretching factor experimentally
deter-mined in the range of 10% is acceptable and
demonstrates the possible applicability of the pre-
sented methods of the calculations for the practical
purposes.
3. To achieve the largest stretching factor for the
same duration of the input pulses with the different
pro-files preferable to use a pulse with the Gaussian
profile, as in this case, the stretching factor in 1.4
times larger than that of the pulse with the profile of
hyperbolic secant.
4. The maximum stretching factor of the output
pulse with the Gaussian profile at the input stretcher
is determined analytically and may be obtained by
reducing its input duration and the increasing of the
second order dispersion of the stretcher. The max-
imum stretch-ing factor of the pulse with temporal
profile of the hyperbolic secant can be defined nu-
merically by selection of the specific initial data.
ACKNOWLEDGEMENTS
The authors are pleased to acknowledge assistance
in this work the doctor of physical and mathemati-
cal sciences Onishchenko I.N. and useful discussion of
134
the presented material with the candidate of physical
and mathematical sciences Sinitsyn V.G.
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ÐÀÑ×ÅÒ ÊÎÝÔÔÈÖÈÅÍÒÀ ÐÀÑØÈÐÅÍÈß ÄËß ÎÏÒÈ×ÅÑÊÎÃÎ
ÐÀÑØÈÐÈÒÅËß ÈÌÏÓËÜÑÎÂ
Â.Ï.Ëåùåíêî, À.È.Ïîâðîçèí, Ñ.Þ.Êàðåëèí
Ðàññìîòðåíà çàäà÷à î âðåìåííîì ðàñøèðåíèè èìïóëüñîâ â îïòè÷åñêîì ðàñøèðèòåëå èìïóëüñîâ CPA-
ëàçåðíîé ñèñòåìû. Íà îñíîâå ðåøåíèÿ âîëíîâîãî óðàâíåíèÿ äëÿ ïðîõîæäåíèÿ ñâåðõêîðîòêèõ èìïóëü-
ñîâ ðàçëè÷íîãî ïðîôèëÿ ÷åðåç îïòè÷åñêèé ðàñøèðèòåëü èìïóëüñîâ ïîëó÷åíû èõ âûõîäíûå ïðîôèëè è
ðàññ÷èòàíû êîýôôèöèåíòû ðàñøèðåíèÿ. Ðàñ÷åòû âûïîëíåíû äëÿ îïòè÷åñêîãî ðàñøèðèòåëÿ òèïà òðè-
ïëåòà Îôôíåðà.
ÐÎÇÐÀÕÓÍÎÊ ÊÎÅÔIÖI�ÍÒÀ ÐÎÇØÈÐÅÍÍß ÄËß ÎÏÒÈ×ÍÎÃÎ
ÐÎÇØÈÐÞÂÀ×À IÌÏÓËÜÑIÂ
Â.Ï.Ëåùåíêî, À. I.Ïîâðîçií, Ñ.Þ.Êàðåëií
Ðîçãëÿíóòî çàäà÷ó ïðî ÷àñîâå ðîçøèðåííÿ iìïóëüñiâ â îïòè÷íîìó ðîçøèðþâà÷i iìïóëüñiâ CPA-ëàçåðíî¨
ñèñòåìè. Íà îñíîâi ðiøåííÿ õâèëüîâîãî ðiâíÿííÿ äëÿ ïðîõîäæåííÿ íàäêîðîòêèõ âõiäíèõ iìïóëüñiâ ðiç-
íîãî ïðîôiëþ îòðèìàíî ¨õ âèõiäíi ïðîôiëi òà ðîçðàõîâàíî êîåôiöi¹íò ðîçøèðåííÿ. Ðîçðàõóíêè âèêî-
íàío äëÿ îïòè÷íîãî ðîçøèðþâà÷à òèïó òðèïëåòó Îôôíåðà.
135
|
| id | nasplib_isofts_kiev_ua-123456789-112101 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1562-6016 |
| language | English |
| last_indexed | 2025-12-01T09:22:18Z |
| publishDate | 2015 |
| publisher | Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| record_format | dspace |
| spelling | Leshchenko, V.P. Povrozin, A.I. Karelin, S.Y. 2017-01-17T16:06:09Z 2017-01-17T16:06:09Z 2015 Calculation of stretching factor for optical pulse stretcher / V.P. Leshchenko, A.I. Povrozin, S.Y. Karelin // Вопросы атомной науки и техники. — 2015. — № 3. — С. 132-135. — Бібліогр.: 16 назв. — англ. 1562-6016 PACS: 42.25.Bs;42.25.Md;42.60.By;42.60.Fe;42.60.Mi https://nasplib.isofts.kiev.ua/handle/123456789/112101 A task of temporal stretching of pulses in an optical pulse stretcher of CPA-laser systems is considered. The output pulse profiles are obtained and the stretching factors are calculated on the foundation of the solution of the wave equation for a passage through an optical stretcher of the ultrashort pulses with the different profiles. The calculations for the optical stretcher of Offner triplet type had made. Розглянуто задачу про часове розширення імпульсів в оптичному розширювачі імпульсів CPA-лазерної системи. На основі рішення хвильового рівняння для проходження надкоротких вхiдних імпульсів різного профiлю отримано їх вихідні профілі та розраховано коефіцієнт розширення. Розрахунки виконанo для оптичного розширювача типу триплетa Оффнера. Рассмотрена задача о временном расширении импульсов в оптическом расширителе импульсов CPA-лазерной системы. На основе решения волнового уравнения для прохождения сверхкоротких импульсов различного профиля через оптический расширитель импульсов получены их выходные профили и рассчитаны коэффициенты расширения. Расчеты выполнены для оптического расширителя типа триплета Оффнера. The authors are pleased to acknowledge assistance in this work the doctor of physical and mathematical sciences Onishchenko I.N. and useful discussion of the presented material with the candidate of physical and mathematical sciences Sinitsyn V.G. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Теория и техника ускорения частиц Calculation of stretching factor for optical pulse stretcher Розрахунок коефiцiэнта розширення для оптичного розширювача iмпульсiв Расчет коэффициента расширения для оптического расширителя импульсов Article published earlier |
| spellingShingle | Calculation of stretching factor for optical pulse stretcher Leshchenko, V.P. Povrozin, A.I. Karelin, S.Y. Теория и техника ускорения частиц |
| title | Calculation of stretching factor for optical pulse stretcher |
| title_alt | Розрахунок коефiцiэнта розширення для оптичного розширювача iмпульсiв Расчет коэффициента расширения для оптического расширителя импульсов |
| title_full | Calculation of stretching factor for optical pulse stretcher |
| title_fullStr | Calculation of stretching factor for optical pulse stretcher |
| title_full_unstemmed | Calculation of stretching factor for optical pulse stretcher |
| title_short | Calculation of stretching factor for optical pulse stretcher |
| title_sort | calculation of stretching factor for optical pulse stretcher |
| topic | Теория и техника ускорения частиц |
| topic_facet | Теория и техника ускорения частиц |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/112101 |
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