THz GENERATION BY OPTICAL RECTIFICATION FOR A NOVEL SHOT TO SHOT SYNCHRONIZATION SYSTEM BETWEEN ELECTRON BUNCHES AND FEMTOSECOND LASER PULSES IN A PLASMA WAKEFIELD ACCELERATOR ∗ Longitudinal diagnostics and synchronisation

We investigate the influence of the optical properties and of the theoretical description of the THz generation on the conversion efficiency of the generation of short THz pulses. The application is a feedback system for SINBAD with a time resolution of less than 1 fs for the synchronization of the electron bunch and of the plasma wake field in a laser driven plasma particle accelerator. Here stable THz pulses are generated by optical rectification of a fraction of the plasma generating high energy laser pulses in a nonlinear lithium niobate crystal. Then the generated THz pulses will energy modulate the electron bunches shot to shot before the plasma to achieve the required time resolution. In this contribution we compare different approximations for the modeling of the generation dynamics using second order or first order equations as well as considering pump depletion effects. Additionally, the dependence of the efficiency of the THz generation on the choice of the dielectric function has been investigated.


hoice of the
ielectric function has been investigated.

INTRODUCTION

Achieving new discoveries, e.g. the Higgs boson or the strong interacting Quark Gluon Plasma, or cancer treatment with the high-energy particle beams are two of several application fields of particle accelerators.Because of their extremely large accelerating electric fields [1], plasma-based particle accelerators driven by laser beam can overcome the limit of the standard accelerators given by the physicalchemical properties of the material used for the construction as well as by the huge size and the financial costs.In fact, the acceleration gradients of conventional linear accelerators are limited to 10 MVm −1 [2], whereas for laser-driven particle accelerators values in the order of 1 TVm −1 can be achieved.Nevertheless, the period of the fields in this method known as plasma wakefield acceleration (PWA) is in the range of 10 fs.Consequently, an optimal acceleration requires stable synchronization of the electron bunch and of th plasma wakefield in the range of few femtoseconds.

For this purpose, we are developing a new shot to shot feedback system for SINBAD with a time resolution of less than 1 fs.We plane to generate stable THz pulses by optical rectification of a fraction of the plasma generating high energy laser pulses in a periodically poled lithium niobate crystal (LiNbO 3 ) (PPLN).These pulses allow to energy modulate the electron bunches shot to shot before the p asma in order to achieve the time resolution [3].

This paper focuses on the generation of THz pulses.We present our investigation of the influence of the optical properties as well as the theoretical description of the THz generation on the conversion efficiency as well on the optimum crystal leng h for the generation length of short THz pulses.

The paper is organized as follows.First, we derive the general equations for the description of the THz generation and then we introduce two different methods in order to include the effects of the laser pump on the crystal.Therefore we compare the corresponding results for the efficiency as well as for optimum crystal length of the THz

the effects of the whole disper
ion relation.

For the THz region we use a physical motivated description for ( T ) based on the oscillator model given in [13].


INFLUENCE OF THE FREE CARRIERS

An important effect is given by the free carries (FC), which are generated by the pump adsorption in the material [5,9,10,14] and lead to a d

reasing of the pump intensity in the crystal
In this manuscript we follow two different strategies in order to systematically consider the influence of the FC.


(a) Modification of the Dielectric Function

In this first approach (a) we decoupling the differential equat ify the dielectric function, which encodes the FC contributions to the optical properties as following [5,11,14,15]
𝜀 tot (𝜔 T ) = 𝜀 osc (𝜔 T ) − 𝜔 2 pl 𝜔 2 T + 𝑖𝜔 T /𝜏 sc ,(8)
The contribution of the free carries encoded the second term of  tot ( T ) are implemented along the line of a Drude model with electron scattering length  sc [10] and with rge carries  FC .In our previous works [5,11,14,15] we developed a phenomenological description for  FC in a PPLN,
𝜌 FC (𝐹 L ) = ⎧ { ⎨ { ⎩ 𝜌 3PA (𝐹 L ) 𝐹 L ≤ 𝐹 0 𝜌 s − 𝐴 e −𝑎(𝐹 L −𝐹 0 ) 𝐹 L > 𝐹 0 ,(9)
For fluences smaller than a transition fluence  0 we describe the FC densi

ere the parameter  (2)
eff = 336 pm V −1 is the effective second order nonlinear susceptibility and Λ = 237.74μm is the quasi-phase-matching orientationreversal period.We consider a Gaussian laser beam pulse with se duration at full width of half-maximum  FWHM = 25 fs


Conversion Efficiency

Because for the optimization of the synchronization an efficient and stable generation of THz pulses is a fundamental task, the conversion efficiency of the THz generation, defined as [3,4]
𝜂 = 𝜋𝜖 0 𝑐 ∫ ∞ 0 d𝜔 T 𝑛(𝜔 T )|𝐸 T (𝜔 T , 𝑧)| 2 𝐹 L , (14)
where  L indicates the pump fluence, has to be maximized.In order to understand the dependence of  on the optical properties we calculate the efficiency by solving directly the equation of motion as differential equation of the second order, i.e.Eq. ( 1) as well as using the SVA, i.e.Eq. 3. In Fig. 1 we compare the results for the conversion efficiency  for a fixed pump fluence  L = 5 mJ cm −2 as function of the crystal length  between the second order calculation (solid lines) and the SVA (dashed lines).All results in this figure are obtained using the approach (a) for the description of FC, because at this fluence the results with the strategy (b) does not present significative deviations.In the linear approximation is used, see.Eq. ( 7), we note a saturation of  for large crystal lengths as well as deviations between the second order and SVA calculations.

Using the Sellmeier equation, the conversion efficiency shows a local maximum about around  ≈ 10 mm and is a decreasing function of the larger crystal length.We note effects of the FC contributions as well as larger deviations between the results coming between from s cond order calculation and SVA.Therefore, at small fluences the second order dynamics effects are stronger than the contribution of the free charge carries.However, at large laser pump intensities this finding changes and the contribution of FC plays a key role, whereas the effects of the second order can be neglected [5,15].

Therefore, for the investigation of the  at higher pump fluence we can restrict ourselves to the SVA calculation and we consider the efficiency as function of  L at fixed crystal length using both approaches for the inclusion of the FC contributions.In Fig. 2 we compare  calculated obtained within a (minimal) depleted calcula ion (b) for a crystal length of  = 40 mm as function of  L as well as the efficiency  calculated in SVA within the linear approximation within the approach (a) for the different parameterizations of the FC density.For comparison, the results for the efficiency calculated at vanishing FC are plotted, which shows a linear behavior.

In the approach (a), for unsaturated density of FC, the  holds a non-monotonic behavior with a local maximum around 10 m cm −2 and then it decreases to an asymptotic value for large fluences, whereas for saturated  FC with  0 = 9 mJ cm −2 exhibits a monotonic increasing behavior.Nevertheless, a change of the slope of this behavior occurs at the value of the fluence, where the local maximum of the efficiency for the unsaturated FC density is located.

Usi

the strategy (b) the d
pleted calculation shows a deviation from the linear dependence on  L .Furthermore, the asymptotic behavior for large fluences follows a power law  ∝   L with  = 0.79, whereas the efficiency calculated including the FC contribution parameterized by  0 = 9 mJ cm −2 shows a linear asymptotic functional dependence.


Optimum Crys -back system a estimation of the optimum crystal length  0 for the generation is a fundamental task.For this purpose, we restrict ourselves on the SVA and we firstly consider the approach (a).Under these assumptions, the absolute value of the field squared can be factorized as [5,7,15]
|𝐴 T (𝜔 T , 𝑧)| 2 = |𝐹(𝜔 T )| 2 |𝐿 Gen (𝜔 T , 𝑧)| 2 , (15 T , 𝑙 0 )|, (20)
is a frequency dependent measure for the highest possible conversion efficiency, whereas iciency over the full spectrum.

In general  max ( T ) and  0 ( T ) as functions of the frequency exhibit a maximum at the critical frequency  cr , where t e mismatching vanishes, i.e.Δ( cr ) = 0.According on the using of the linear approximation or the complete Sellmeier equation and on the choice for treatment of the influence of the free carries, i.e. approach (a) or (b), following features can be summarized: i) In the approach (a) without the FC contribution the  max ( T ) is independent from the laser fluence.

ii) In the approa h (a) with non-vanishing FC contribution this scaling behavior is broken and we observe a shift of the c itical frequency and a dropping of the corresponding maximum  max ( cr ) by increasing of the fluence.As example we show in the panel (A) of the Fig. 3  max () for vanishing and unsaturated FC density using the complete Sellmeier equation at different fluencies.

iii) In the minimal depleted calculation (b) only the decreasing of  max ( cr ) as function of  L occurs.

iv) The optimum crystal length  0 in the approach (a) using the linear approximation diverges for  T →  cr , whereas the maximum od the generation length is finite and given by  max ( cr ) = 2/ T ( cr ).The dependence of  0 ( cr ) on the fluence  L is shown in the panel (B) of the Fig. 3 for the depleted calculation (b) as well as for different FC density within the approach (a).The comparison suggests that the linear approximation for the polarization in the depleted calculation (b) leads to an overestimation of the optimum crystal lengt .In general we note the importance of the details of the dynamics, i.e. the choice the FC density within the approach (a), for the determination of  0 .


CONCLUSION

The influence of the optical properties of the lithium niobate crystal on the conversion efficiency as well as on the optimum crystal length of the generation of THz pulses has been investigated.We compare different approximation for the modeling of the generation dynamic (SVA vs. second order calculation) and of the influence

f the free
arries and we present a (minimal) depleted calculation.

We clearly show the importance of a consistent description of the optical properties as well as a detailed generation dynamics in order to develop the planned shot to shot feedback system, which have to perform the synchronization between electron bunch and the ultrashort laser with a time resolution of less than 1 fs.

0

) or Eq.(3) with = T has to be Content from this work may be used under the terms of the CC BY 3.0 licence (© 2019).Any distribution of this work must maintain attribution to the author(s), title of the work, publisher, and DOI


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[3-5, 8, 10,  11, 14, 15].Content from this work may be used under the terms of the CC BY 3.0 licence (