## Abstract

We consider the propagation of powerful laser radiation in an anisotropic medium with natural birefringence and cubic nonlinearity. By the example of a quarter-wave plate, we show theoretically and experimentally that, under the simultaneous influence of linear birefringence and nonlinearity, the accuracy of polarization transformation decreases in proportion to squared *В*-integral.

©2013 Optical Society of America

## 1. Introduction

Powerful laser systems generating petawatt power femtosecond pulses having intensities of 10^{20} W/cm^{2} and higher open a way to the rapidly advancing trend in laser physics – creation of compact accelerators of high-energy particles, such as electrons, protons (ions) and neutrons. The principle of operation of these accelerators is based on the interaction of a powerful laser pulse with a thin solid target (e.g., aluminum or gold foil). During several femtoseconds the target is ionized so that free electrons may be accelerated under the action of ponderomotive force of the laser pulse, giving rise to strong charge separation into electrons and heavier, hence, immobile ions. The resulting ambipolar electric field leads to acceleration of protons in the target plasma in different directions.

Of major interest are mechanisms of proton acceleration as proton beams with energies from several tens to hundreds of MeV may be used for proton magnetic resonance imaging of small-scale objects [1], hadron therapy in oncology [2], and fast ignition of a thermonuclear target [3]. According to the works by [4–6] laser radiation polarization is an important parameter affecting the process of proton acceleration. Investigation of the intense laser pulse interaction with a thin foil demonstrated that proton acceleration efficiency with the use of circularly polarized radiation is higher than with linearly polarized radiation [7]. In the case of circular polarization, ambipolar electrostatic field has positive projection onto the direction of laser pulse propagation; hence, the protons are accelerated only forward. In the case of linear polarization, ponderomotive force has an oscillatory longitudinal component providing both forward and backward proton acceleration. Moreover, it was found that the use of circularly polarized radiation allows more efficient proton beam collimation, as compared to linearly polarized radiation [7].

Thus, it is necessary to produce circularly polarized radiation with the intensity of about 10^{20} W/cm^{2}. Consider the most broadly used method – the transformation of linear polarization to a circular one by means of a quarter-wave plate. In this case, it is important that polarization transformation should be accomplished in the region of intense radiation. Indeed, the CPA (Chirp Pulse Amplification) as well as OPCPA (Optical Parametric Chirped-Pulse Amplification) technology used for laser radiation amplification requires linear polarization, as diffraction gratings that are inherent components of the system operate with maximum efficiency with polarization of this type. Consequently, it is interesting to investigate the simultaneous influence of the properties of the wave plate material (natural birefringence and nonlinearity) on laser radiation polarization.

We will be interested in wave plates cut from crystalline quartz, possessing both quadratic and cubic nonlinearity. In this work we consider plates for polarization rather than frequency transformation. The wave plates position relative to the direction of radiation propagation is such that the phase synchronism condition responsible for the effects related to quadratic nonlinearity is not fulfilled a fortiori. Therefore, we will neglect nonlinearity of this type.

Effect of polarization ellipse rotation caused by cubic nonlinearity is well-known [8–12]. In paper [13] authors investigated the isotropic cubic nonlinearity influence on accuracy of polarization transformation. Reduction of polarization transformation accuracy for the *В-*integral more than unity was shown. In this paper we considered the anisotropic cubic nonlinearity because crystalline quartz is positive uniaxial crystal possessing trigonal crystalline structure. Therefore, we had to take into account not only diagonal elements of fourth-order nonlinear susceptibility tensor *χ _{ijkl}* but also the nondiagonal ones. This correction was very important because we proofed experimentally that the effect of depolarization growing is stronger than we calculated using model of isotropic nonlinearity. In other words, according to calculation within model of isotropic nonlinearity the 1% depolarization degree achieved for higher intensity (

*I*~4 TW/cm

^{2}) then in case of the anisotropic nonlinearity (

*I*~700 GW/cm

^{2}).

In the second and third sections of the paper we will consider nonlinear optical properties of crystalline quartz and propose a system of differential equations describing laser radiation propagation in an anisotropic medium with cubic nonlinearity. Results of theoretical and experimental investigation of the influence of cubic nonlinearity on accuracy of polarization transformation by means of a quarter-wave plate will be presented in Sec. 4, where it will be shown that the degree of depolarization (energy fraction in the radiation with polarization orthogonal to the polarization in the absence of nonlinearity) and the ellipticity of output radiation polarization at the output of the quarter-wave plate is proportional to squared intensity (*В*-integral).

## 2. Nonlinear optical properties of crystalline quartz

Crystalline quartz belongs to the group of positive uniaxial crystals. At room temperature, quartz consists of *α*-modification crystals possessing trigonal crystalline structure [14]. The permittivity and gyration tensors for a structure of this type are written in the form

*k*= 2

*π*/

*λ*, and

*λ*is the wavelength of light in vacuum. The expression Eq. (2) permits characterizing the principal properties of quartz. For example, if a quartz plate is cut parallel to the optical axis, then the birefringent properties of quartz are shown due to the difference between

*ε*

_{11}and

*ε*

_{22}. If the plate is cut normally to the optical axis, then quartz possesses gyrotropic properties, which is used in manufacturing 90° polarization rotators. In the present work we will be interested in plates cut parallel to the optical axis of the crystal –

*Oz*.

To reduce the influence of refractive index dispersion on radiation polarization, zero-order plates are used. In *λ*/4, *λ*/2, *λ* zero-order plates the phase difference between the extraordinary and ordinary waves is *π*/2, *π*, and 2*π,* respectively. However, the thickness of the *λ* plate would then be *L* = 90 μm (at *λ* = 795 nm), which is hard to achieve in practice. Hence, two multiple-order wave plates are used, the difference in thicknesses of which produces path difference of about 1/4 or 1/2 of wavelength, depending on the plate type. Optical axes of these multiple-order plates make an angle of 90° with each other (Fig. 1
).This is the type of wave plate of interest to us.

The influence of cubic nonlinearity in a wave plate of crystalline quartz is specified by nonzero components of the fourth-order nonlinear susceptibility tensor *χ _{ijkl}*:

*z*corresponds to the optical axis of the crystal. The direction of the optical axis has been fixed, so now we have to define two other directions. From the expression Eq. (3) it follows that nonlinear properties of the crystal also depend on the way a wave plate is cut relative to the

*Ох*and

*Оу*axes. Note that there is no symmetry in tensor

*χ*with respect to subscripts

*x*and

*y*. For instance, the tensor component with subscripts

*yyyz*has no pair component with subscript

*xxxz.*Consequently, if the plate is cut along the

*Оу*axis, components with indices

*y*and

*z*play an important role in the system of differential equations. Quite the contrary, if the plate is cut along the

*Ox*axis, the components with indices

*х*and

*z*are of primary importance. It is worth mentioning that the choice of crystal cutting along the

*Оу*and

*Ox*axes does not affect the birefringent properties of the plate, which complicates identification of the direction cutting.

As a measure of nonlinearity researchers usually use *В-*integral – phase incursion related to the nonlinear part of refractive index *n*. For example, in an isotropic medium with cubic nonlinearity, *n* linearly depends on intensity *n*(*I*) *= n _{0} + γ_{NL}I*, where

*n*is a linear refractive index and

_{0}*γ*is the characteristic of the nonlinear medium. In this case, for a medium with length

_{NL}*L*, the magnitude of

*В*-integral is defined by

The magnitude of *В* for an anisotropic medium depends on polarization and direction of radiation propagation. Hence, in the present work we will determine *В* for radiation with linear polarization coinciding with the polarization of an ordinary or extraordinary wave of crystalline quartz. The magnitudes of *γ _{NL}* corresponding to eigen modes of the crystal, according to [15], differ less than by 4%; so we set the largest

*γ*= 3.2⋅10

_{NL}^{−7}cm

^{2}/GW.

## 3. System of differential equations

With the crystal cut parallel to the *Ох* axis, the *Оу* axis will be taken to be the direction of radiation propagation and *Ох* and *Оz* will be directed along the optical axes in the first and second parts of a composite wave plate (Fig. 1). Then, the impact of cubic nonlinearity will be described by the *χ _{ijkl}* components, where

*i*,

*j*,

*k*,

*l*are

*х*and

*z*. Whereas with the crystal cut parallel to the

*Oy*axis, formal substitution of

*x*for

*y*should be done in the equations taking into account the new terms containing components of permittivity tensor Eq. (3) with indices

*i*,

*j*,

*k*,

*l*are

*y*and

*z*.

The change of the parameters of radiation is described through vector $\overrightarrow{E}$ projections onto the principal directions of the plate for the two cases of plate cutting:

*n*=

_{x}*n*= (

_{y}*ε*

_{11})

^{½}is the refractive index of an ordinary wave and

*n*= (

_{z}*ε*

_{22})

^{½}is the refractive index of an extraordinary wave. Dispersion characteristics of these waves of crystal quartz were found by [16,17].

Our estimates show that the length of pulse broadening due to the difference in group velocities of the “blue” and “red” wavelengths that limit the radiation spectrum used in our experiment is much larger than the thickness of the considered wave plate. Consequently, neglecting dispersion and diffraction, a steady-state system of differential equations for electric field vector in the case of the radiation propagating in a nonlinear medium with a trigonal crystal lattice will be written as

*δ*(

*y*) =

*k*(

*n*–

_{z}*n*)

_{x}*y*in Eq. (6а) and

*δ*(

*x*) =

*k*(

*n*–

_{z}*n*)

_{y}*x*in Eq. (6b) are the phase differences of the extraordinary and ordinary waves. System Eq. (6а) corresponds to the case of the plate cut parallel to the

*Ох*axis, and Eq. (6b) parallel to the

*Оу*axis. From [15] it follows that

*χ*= 1.89⋅10

_{zzzz}^{−14}esu,

*χ*=

_{yyyy}*χ*= 1.82⋅10

_{xxxx}^{−14}esu. The magnitudes of

*χ*,

_{xxzz}*χ*and

_{yyzz}*χ*are unknown to the best of our knowledge.

_{yyyz}We remind the readers that in the present work we consider the transformation of linear polarization to the circular one using a zero-order quarter-wave plate. Assume that the vector of the electric field $\overrightarrow{E}$ incident on the plate makes an angle of 45° with the optical axes of the plate (Fig. 1). In this case projections of vector $\overrightarrow{E}$ onto the plate axes are equal. Since *χ _{zzzz}* ≅□

*χ*=

_{yyyy}*χ*, from previous remark and system Eq. (6) it follows that it is the terms containing nondiagonal elements of tensor

_{xxxx}*χ*that will determine the change of polarization.

The system Eq. (6а) or Eq. (6b) allows calculating energy redistribution between the *Е _{х}* and

*Е*components as well as

_{у}*Е*at the output of the plate with preset

_{z}*δ*and

*В*. Such a change will be characterized by the degree of local (at each point of the cross-section) depolarization – the fraction in the output radiation

**E**

_{out}(

*B*≠ 0) of the intensity with polarization orthogonal to the polarization in the absence of nonlinearity

**E**

_{out⊥}(

*B*= 0) (i.e., circular polarization but with a different direction of rotation):

*В*-integral. Experimental verification of this result is presented in Section 4.

Calculation of Γ allows us finding out the ellipticity Σ of the polarization at the output of the wave plate

**E**

_{+},

**E**radiation components with clockwise and counterclockwise polarization respectively. Notice, that Σ

_{−}*=*−1 ( + 1) corresponds to the clockwise (counterclockwise) circular polarized radiation, Σ

*=*0 corresponds to the linear polarization. We assume that the vector

*E*of the linearly polarized incident radiation on the quarter-wave plate makes an angle of 45° with the optical axes of the plate. Therefore, the ellipticity of output radiation is defined by following expression:

## 4. Experimental investigation of the influence of cubic nonlinearity on accuracy of transformation polarization by means of a quarter-wave plate

We measured radiation polarization ellipticity of a terawatt-power titanium-sapphire laser with central wavelength *λ* = 795 nm [18]. In other words, we determined depolarization of the laser system specified by accuracy of obtaining linear polarization. The depolarization degree was calculated as a fraction of the energy contained in the radiation component with polarization corresponding to the small semi-axis of the polarization ellipse. After that, we experimentally investigated the dependence of the energy fraction of this component in the output radiation on the magnitude of *В*-integral.

#### 4.1 Schematic of the experiment

The schematic of the experimental setup is shown in Fig. 2
(а). On reflection from mirror **1** the main portion of the radiation passed through Galilean telescope **2**-**3** with magnification 1:2.5. This telescope permits increasing intensity up to the calculated level (~700 GW/cm^{2}) at which the influence of cubic nonlinearity on polarization transformation in quarter-wave plate **4** was calculated to be high enough. One more plate **6** placed in lower-intensity radiation reflected from glass wedge **5** was used to increase accuracy of depolarization measurements. Hence, cubic nonlinearity could not affect polarization transformation in plate **6**. After that the principal radiation component with polarization corresponding to a large semi-axis of the polarization ellipse and the depolarized component with orthogonal polarization were separated in space by calcite wedge **7**. The calcite wedge was adjusted so that its principal axes coincided with the principal axes of the polarization ellipse at *В* = 0. The image was transferred from the plane of wave plate **4** to the matrix of CCD-camera **9** by lens **8**. Thanks to the additional wave plate **6** radiation polarization at the output of the experimental setup at *В* = 0 coincides with the initial polarization. In this case, two-dimensional intensity distributions of the principal (bright spot) and depolarized (spot at the level of CCD-camera noise) components of radiation were recorded on the CCD-camera (Fig. 2(b), 2(c)). The brightness of the spot corresponding to the depolarized component increased at *В* > 1 (Fig. 2(e)).

The transverse intensity distribution at the input had a Gaussian profile ~*exp*(−*r*/*R*_{0})^{2}, where *R*_{0} = 0.24 cm. The radiation time distribution is described well by the function 1/*ch*^{2}(*t*/*t*_{0}), where *t*_{0} = 28 fs, which corresponds to the pulse duration of 75 fs FWHM. Pulse energy was measured within the 1-7 mJ range.

#### 4.2 Experimental results

Results of the experiments are presented in Figs. 3(a)
, 4(a)
(the same experimental points are used in these plots). The local depolarization degree Γ and ellipticity Σ calculated by Eq. (9) during processing of two-dimensional intensity distributions depicted in Figs. 2(с), 2(е) were calculated in the central region of the beam. The fraction of energy contained in the depolarized radiation component as a function of radiation intensity without wave plates is plotted by blue dots. In this fashion we determined ellipticity of the initial polarization, i.e. its difference from the linear polarization. One can readily see that the level of cold/initial depolarization is 0.2%. Red triangles were used to plot analogous dependences after wave plates **4** and **6** were added to the scheme. In this case the dependence of depolarization on intensity *I* (*В*-integral) is nonlinear. The value of Γ(*B* = 0) equal to 0.1% is less than the analogous value in the absence of wave plates **4** and **6**, which may be attributed to reduced polarization ellipticity due to the plates. The maximum intensity magnitude of 700 GW/cm^{2} achieved in experiment corresponds to Г = 1.1% and Σ = −0.87.

For comparison of the obtained experimental and theoretical data it is necessary to determine the region of possible values of components of nonlinear susceptibility tensor *χ* unknown for crystalline quartz. Let us take *I* = 700 GW/cm^{2} and plot a one-dimensional Γ(*χ _{xxzz}*) and a two-dimensional Γ(

*χ*,

_{yyzz}*χ*) functions for each type of wave plate cutting, respectively (Figs. 3(b), 4(b)). The resulting dependences will be used to determine values of the sought components of tensor

_{yyyz}*χ*, for which Г(

*I*= 700 GW/cm

^{2}) = 1.1%. Further, we will search among the selected values of

*χ*components the ones at which the theoretical and experimental curves for Г(

*I*) coincide at all points (Fig. 3(a), 4(a)). Note, we assume that the both parts (multiple-order plates) of the quarter-wave plates used in experiments have the identical direction cutting.

Two possible values of *χ*: *χ _{xxzz}* = 0.76

*χ*and

_{xxxx}*χ*= −0.28

_{xxzz}*χ*were found for the plate cut parallel to the

_{xxxx}*Ох*axis (Fig. 3(b)). The curves in Fig. 3(b) are plotted for different values of radiation polarization ellipticity at the input, i.e., for the depolarization Г(

*В*= 0) = 0%, 0.01% and 0.1%. Thus, we have found that the uncertainty of determining Г(

*В*= 0) in the given range of values leads to 8% error in calculations of

*χ*. Note that the obtained values of

_{xxzz}*χ*demonstrate a good agreement of experimental and theoretical data for the clockwise polarized radiation (black solid and dashed curves in Fig. 3(а)). For the case of the counterclockwise polarized radiation, good agreement between experiment and theory is attained only at one point

_{xxzz}*I*= 700 GW/cm

^{2}(Fig. 3(а), grey solid line).

For a plate cut parallel to the *Оу* axis, similarly to the considered case of clock- and counterclockwise polarized radiation, there exist several sets of pairs *χ _{yyzz}*,

*χ*, at which experiment and theory agree well. Values of tensor

_{yyyz}*χ*components corresponding to the clockwise polarized radiation are shown by the solid curve in Fig. 4(b), where the dashed curve mark pairs

*χ*,

_{yyzz}*χ*for the counterclockwise polarization.

_{yyyz}Note that the principal goal of the current work is experimental verification of the supposition that accuracy of polarization transformation by means of wave plates reduces at *В*-integral larger than unity, rather than determination of unknown components of tensor *χ*. We have found the region of possible values of parameters *χ _{xxzz}*,

*χ*and

_{yyzz}*χ*at which the experimental data agree with the theoretical calculations.

_{yyyz}To conclude this section, we will make some estimates demonstrating why in our theoretical analysis we neglected spectrum broadening caused by cubic nonlinearity as the pulse was passing through the quartz zero-order quarter-wave plate. Let a laser pulse at the input to a plate having thickness *L* = 0.18 cm have the following shape *А*_{0}(*t*) = 1/*ch*^{2}(*t*/*t*_{0}), where *t*_{0} = 28 fs. For the radiation with linear polarization coinciding with one of the eigen polarizations of the plate, a phase additive appears at the plate output defined by the nonlinear part of the refractive index: *А*(*t*) = 1/*ch*^{2}(*t*/*t*_{0})⋅exp(– *ikγ _{NL}I*

_{0}(

*А*(

_{0}*t*))

^{2}

*L*), where

*I*

_{0}= 700 GW/cm

^{2}is the maximum value obtained in the experiment. The spectra corresponding to the pulses

*А*

_{0}(

*t*) and

*А*(

*t*) are plotted in Fig. 5 .

We also plotted depolarization degree Γ_{0} versus *λ* specified by *δ*(*λ*) = *k*(*n _{z}*(

*λ*) –

*n*(

_{y}*λ*))

*L*. This function crosses zero at the point

*λ*= 795 nm as the plate is fabricated to for radiation at this central wavelength. Integral of depolarization Γ

_{0}(

*λ*) over the spectrum

*А*(

_{λ}*λ*) is much less than the Γ measured in the experiment (0.1-1%). Hence, the radiation spectrum broadening due to cubic nonlinearity may be neglected even at

*I*

_{0}= 700 GW/cm

^{2}.

## 5. Conclusion

The method of obtaining the circular polarization by means of a quarter-wave plate for a beam intensity at the plate 1 TW/cm^{2} and higher was investigated. A specific feature in this case is that, despite the small thickness of the wave plate, we have to consider the influence of ellipse polarization rotation caused by cubic nonlinearity on the polarization transformation.

Within the framework of the formulated problem we proposed a system of steady-state differential equations describing the propagation of laser radiation with arbitrary polarization in an anisotropic medium with cubic nonlinearity. Using this system of equations we calculated local depolarization degree Γ (fraction of energy in the polarization orthogonal to the polarization in the absence of nonlinearity) and polarization ellipticity at the output of a quarter-wave plate of crystalline quartz in the presence of cubic nonlinearity. Experiment on measuring depolarization at the output of a zero-order *λ*/4 plate 0.18 cm thick was conducted. It was found that the depolarization degree Γ is proportional to squared *В*-integral (nonlinear phase incursion). The maximum measured value of Γ was 1.1% at the laser radiation intensity of 700 GW/cm^{2}. Good agreement between the experimental and the theoretical data was obtained.

Most promising for practical applications are wave plates made of KDP (or DKDP) crystals, especially for powerful lasers. According to [15,19], values of the components of nonlinear susceptibility tensor in these crystals are smaller than in crystalline quartz. Moreover, in contrast to quartz these crystals possess no gyrotropy, hence, whole *λ*/4 plates of arbitrary thickness may be cut from them. Study of the specific features of using such wave plates will be the next step in the work on this problem.

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