We consider a problem of laser radiation propagating in a medium with birefringence of two types: linear birefringence independent of intensity and polarization, and intensity and polarization dependent circular birefringence caused by cubic nonlinearity. It is shown theoretically and experimentally that the efficiency of the broadly employed method of linear depolarization compensation by means of a 90° polarization rotator decreases with increasing В-integral (nonlinear phase incursion induced by cubic nonlinearity). The accuracy of polarization transformation by means of a half-wave and a quarter-wave plate also decreases if В > 1. By the example of a λ/4 plate it is shown that this parasitic effect may be suppressed considerably by choosing an optimal angle of inclination of the optical axis of the plate.
©2011 Optical Society of America
One of the promising trends in laser physics is creation of laser systems possessing high peak power and high average power simultaneously. One of the key problems to be solved in construction of such lasers is suppression of thermal effects restricting average power and of parasitic nonlinear optical effects limiting peak power. The range of laser parameters and, hence, of applications is extremely wide: from femto- and picosecond lasers with pulse power of fractions of joule and repetition rate of 1 kHz to nanosecond Nd-glass lasers with energies of hundreds of joules and pulse repetition rate of tens and hundreds of fractions of Hertz. The latters are an integral part of the currently constructed petawatt and multipetawatt laser complexes in which Nd-glass amplifiers are used either as an active medium for Chirped Pulse Amplification (CPA), or as a pump for Ti:SA chirped pulse amplifier, or as a pump of DKDP crystal parametric amplifier [1,2].
In this connection of great importance is investigation of the mutual (simultaneous) influence of thermal and nonlinear optical effects. In particular, it is interesting to study the propagation of light through a medium with birefringence induced by two effects simultaneously: thermoelastic stress and cubic nonlinearity. The contribution of the enumerated effects is nonadditive in principle, as thermally induced birefringence depends neither on intensity, nor on laser field polarization, whereas anisotropy induced by cubic nonlinearity is a function of intensity and polarization.
The main cause of thermoelastic stresses and the consequent thermally induced birefringence is generation of heat produced by pump radiation absorption in active elements [3, 4]. Thermally induced eigen polarizations are linear and orthogonal to each other, but they have different orientations at different points of cross-section. In glass active elements they are oriented along and across the temperature gradient [5, 6], whereas in crystals and crystal ceramics the arrangement is more complicated [7–12]. The phase difference (value of birefringence) is also a function of transverse coordinates. As a result, on passing the birefringent medium the initially polarized laser radiation becomes depolarized. Under depolarized radiation we understand the radiation whose polarization is constant in time, but changes from point to point of the cross-section. Consequently, depolarization is transformation of polarized radiation to depolarized one. The negative consequence of thermally induced birefringence is amplitude and phase modulation of radiation after passage through the polarizer (e.g., Maltese cross and astigmatism, respectively). This reduces the power in the initial single-mode beam due to polarization amplitude and phase losses . One of the basic causes limiting the peak power of modern solid-state pulse lasers is cubic nonlinearity, that is the dependence of the index of refraction on intensity n(I) = n0 + γNLI, where n0 is the linear index of refraction and γNL is the characteristic of the nonlinear medium. A conventional measure of nonlinearity is the В-integral – nonlinear phase incursion in a medium having length L:14–16].
In addition to self-focusing, field induced anisotropy appears in a medium with cubic nonlinearity. The arising difference in the refractive indices of circularly polarized components of radiation leads to the phase difference between them and, as a consequence, to rotation of the polarization ellipse [16–20]. Thus, both cubic nonlinearity and thermally induced birefringence introduce polarization distortions in the laser beam, giving rise to depolarization, the contributions of both the effects being nonadditive.
In laser systems with high average power, two identical active elements and a 90° polarization rotator placed in between are broadly used for minimization of thermally induced depolarization in active elements . For laser systems with high average and peak power, the efficiency of this method reduces significantly because of the polarization ellipse rotates linearly increasing in the course of propagation along the medium with cubic nonlinearity. Consequently, combined influence of cubic nonlinearity and thermally induced birefringence is one of the basic restricting factors in development of laser systems with high average and high peak power.
The system of differential equations describing the propagation of laser radiation of arbitrary polarization in a medium with cubic nonlinearity in the presence of birefringence is analyzed by the authors in Sec. 2 of the paper . The obtained system of differential equations allows determining the degree of radiation depolarization in optical elements (hereinafter referred to as nonlinear elements) with thermally induced birefringence in the presence of cubic nonlinearity. In Sec. 3 of  we investigated the efficiency of the broadly used method of depolarization compensation by means of two identical nonlinear elements (NE) and a 90° polarization rotator placed between them . It is shown that the efficiency of compensation of thermally induced depolarization by means of a 90° polarization rotator decreases with increasing В-integral.
The current work is concerned with experimental confirmation of mentioned theoretical results. The method of thermally induced depolarization compensating at different values of В-integral is investigated experimentally in Sections two and three. The increase of depolarization at the output of the compensation scheme is demonstrated for nonzero В-integral. Experiments are in a good agreement with the theoretical data obtained by solving the system Eq. (6) in  for considered compensation scheme.
Of special interest are studies of the impact of cubic nonlinearity on media with natural birefringence, such as, for instance, wave plates. Indeed, almost all lasers use λ/2 and λ/4 wave plates. As follows from the said above, accuracy of polarization transformation by means of such plates decreases with increasing B-integral. This is particularly important for femtosecond pulses, for which the В-integral may be order of unity, even in thin plates, because of high pulse intensities. This effect is investigated in Section 4, where we show on an example of a λ/4 plate that it may be suppressed substantially by controlling the angle of inclination of the wave plate axes.
2. Laser radiation propagation in a medium with cubic nonlinearity in the presence of thermally induced birefringence
In this work we neglect diffraction, hence all results obtained below will hold only in this approximation. Equations for dimensionless complex amplitudes of right- and left-polarized waves22]:23]. The Kerr effect is almost absent in solids and the electron mechanism of nonlinearity stipulated by anharmonic oscillatory motion of electrons is brought to the foreground for nanosecond (and shorter) pulses. In this case, β = 1 . The effect of thermally induced birefringence manifests itself in solid lasers, hence we choose in the present work β = 1.
Notice, in this work only initially isotropic solid medium is considered, therefore, we neglect the influence of quadratic nonlinearity on radiation propagation. Indeed, permittivity third-rank tensor responsible for quadratic nonlinearity does not exist in the medium with a central symmetry. The impact of quadratic optical nonlinearity caused by thermoelastic stress is the higher order effect in comparison with thermally induced birefringence and electron Kerr effect.
For cylindrically shaped nonlinear elements (NE) we will choose the corresponding frame of reference (r, φ, z). Induced birefringence at the point (r, φ, z) is usually characterized by the phase acquired by linearly polarized eigen waves passing through a depolarizing medium, as well as by the angle of inclination of eigen polarizations θ(r, φ, z). Hereinafter we assume θ = φ, i.e., the eigen polarizations coincide with the radial and tangential directions. This is true for glass NEs [5, 6], cubic crystals with  orientation [5, 7], and ceramics in some approximation [11, 12]. For cubic crystals with orientation different from , θ ≠ φ [6–10], but this case will be considered elsewhere.
Let us introduce a quantity characterizing depolarization: local (at each point of the cross-section) polarization degree Γ, that is a fraction in output radiation Eout of intensity with polarization orthogonal to the reference one Eref⊥:
For the case of linearly polarized radiation, Eref⊥, we can obtain the following expression for Γ at the output of one element in the absence of cubic nonlinearity:
It is clear from this expression that maximum depolarization is attained at δ = π and φ = 45°. In order to obtain Γ(Β), one has to find polarization at the output of a linear element by solving the system Eq. (3). Following the work  we reduced this system to one differential equation and solved it numerically.
The function Γ(δ) at the output of the scheme of two elements and a 90° polarization rotator is plotted in Fig. 1 for different values of В-integral and rectangular pulse shape when φ = 45°. This function was found by solving the system of Eq. (3) step-by-step in the first and second NEs. Note that the values of δ and В correspond to the values of phase and to the В-integral in one element. Note also that there exists a value of δcr = 2π/5 independent of В that determines the first and highest maximum of function Γ(δ); Γ nonmonotonically but rapidly decreases with increasing δ.
The presence of maximum of function Γ(δ) may be physically explained as follows. For δ << 1, linear birefringence very weakly changes radiation polarization that persists to be close to a linear one. There is no birefringence induced by cubic nonlinearity either; hence, when δ << 1, Γ is close to zero. On the other hand, when δ >>1, linear birefringence is so high that the cubic nonlinearity additive is no longer important. Hence, in this case, the 90°-polarization rotator effectively compensates depolarization and Γ tends to zero.
3. Experimental study of the cubic nonlinearity impact on accuracy of compensation of thermally induced birefringence
According to the theoretical analysis given above, the presence of cubic nonlinearity will lead to reduced efficiency of compensation of thermally induced depolarization in the scheme with two identical NEs and a 90° polarization rotator placed in between. In this Section we will report results of the experimental study of the considered depolarization compensation scheme efficiency for nonzero B-integral.
3.1. Schematic of the experiment
The schematic of the experiment is shown in Fig. 2 . Thermoelastic stress was produced in NEs 3 by a flash of pump lamps arranged along the laser Nd:glass rod (NE) 33 cm long with n0 = 1.534 and γNL = 3.2·10−7 cm2/GW. The NEs were air cooled convectively during characteristic time of about 1 hour . During this time, the temperature gradient changed; consequently, the value of thermally induced birefringence δ at a point with coordinates (r, φ) through which the probe laser beam passed changed too. Thus, by changing the probe beam delay we made measurements at different values of δ.
A flat-top probing laser beam 0.2 сm in diameter passed through a NE 10 сm in diameter. At the NE output the radiation contained a component with polarization orthogonal to the initial one, the value of which varied at different points of the cross-section, as a result the radiation was depolarized. The polarization ellipse was turned by means of 90° quartz rotator 4. The local depolarization degree was measured by a sequence of elements 9, 10 (Fig. 2). Calcite wedge 9 provided space separation of the components with orthogonal polarizations. Images of radiation components with initial and orthogonal polarizations were relayed to the same CCD camera 10. The energy of the pulse was measured by pirodetector 7. The software processing of the obtained transverse distributions of the orthogonally polarized components in the CCD-camera allowed measuring the local degree of depolarization Γ. As the probe laser radiation we used in experiments 20 ns and 1 ns pulses which have a near Gaussian temporal profile with energies in the range of 10 mJ – 2 J coming from the Nd:YLF laser described in . The 20 ns pulses correspond to low intensity and, consequently, to zero B-integral. Contrary to this case, the 1 ns pulses correspond to experiments with nonzero B-integral. Three types of measurements were made for two sets of coordinates of the points of laser beam probing in the first and second NE: (а) r1 = r2 = 4 cm, φ1 = φ2 = 45°; (b) r1 = r2 = 4 cm, φ1 = 42°, φ2 = 48°. The inclination angle of the intrinsic polarization φ in each NEs is counted off from the horizontal.
The first set of coordinates of the points of laser beam probing (а) r1 = r2 = 4 cm, φ1 = φ2 = 45° was chosen because of the following reasons. According to the Eq. (5) depolarization obtains its maximum when phase difference acquired by linearly polarized eigen waves δ = π and the inclination angle of eigen polarizations φ = 45° so that in our experiments we tried to achieve a close values for these parameters. In the work  was shown that the largest thermoelastic stresses and the consequent thermally induced birefringence were observed at the edge of the laser Nd:glass rod. After a flash of pump lamps the value of thermally induced birefringence was less than π in our experiments. So, we decided to pass probing laser beam as close to the rod edge as possible for more contrast measurements. The second set of coordinates of the points of laser beam probing in the first and second NE (b) was chosen randomly because the theoretical analysis we worked out can be hold for all possible values of δ and φ.
In the first part of the experiment, we measured the time dependence of local depolarization degree separately in the first and second NE for the B-integral equal to zero. These measurements allowed assessing the birefringent properties of the NEs used in the scheme in Fig. 2 and, consequently, the value of the residual depolarization degree in the scheme of depolarization compensation with В = 0. Using the measured function Γ(t), from the Eq. (5) we found the magnitude of the thermally induced birefringence δ(t) (Fig. 3 ). Note that for r1 = r2 = 4 cm, φ1 = φ2 = 45°, starting with t > 5 min the functions δ(t) behave identically in each element (Fig. 3а). Therefore, the method of depolarization compensation by means of the 90° polarization rotator must be very efficient. In the case r1 = r2 = 4 cm, φ1 = 42°, φ2 = 48°, experimental curves for δ(t) point to appreciable difference in the magnitudes of thermally induced birefringence in the studied samples (Fig. 3b).
In the second part of the experiment, we studied the negative impact of cubic nonlinearity. Therefore, the local degree of depolarization at the output of the compensation system (Fig. 2) at zero В-integral and В ~1 was measured at maximally close time instants (less than 30 s). This experiment allows considering the value of thermally induced birefringence in each NE to be constant for В = 0 and В ~1. Two-dimensional distributions of the radiation component with initial polarization and of the depolarized component at В = 0 and В ~1 are shown in Fig. 4 а and Fig. 4b, respectively.
Measurements of thermally induced phase difference in each NE revealed residual depolarization in the compensation system, even at В = 0. This difference is appreciably higher for the case when φ1 = 42°, φ2 = 48°. An increase of depolarization induced by cubic nonlinearity relative to the zero level corresponding to В = 0 was observed in the experiment that will be described in Section 3.2.
In the third part of the experiment we attempted to conduct more contrast measurements of residual depolarization at В ~1. Towards this end, we dynamically adjusted the 90° polarization rotator so that at В = 0 the depolarized radiation component was not registered against the background noise of the CCD camera. This experiment will be described in Section 3.3.
3.2. The impact of cubic nonlinearity at inaccurate depolarization compensation by a 90° rotator
Consider results of measurements of residual depolarization in the case when the 90° rotator was adjusted so that the radiation reflected from the front edge of the rotator propagated exactly backwards. This means that 90° rotator made from the crystalline quartz has only circular birefringence. Such rotator realized 90° polarization rotation which in theory is needed for full depolarization compensation by means of the scheme of two absolutely identical birefringence media. The rotator may be considered to be ideal in this case. However, in practice even at В = 0, the magnitude of Γ is nonzero because of different values of δ in the two elements (Fig. 3) and by virtue of φ1 ≠□φ2. This fact is illustrated in Fig. 5 . One can see that the experimental and theoretical data are in the very good agreement when B = 0. Therefore, we can conclude about the right determination of the δ for each t. Differences between experimental and theoretical results when B ≠ 0 in Fig. 5 can be explained by accuracy of the energy and pulse duration measurements. Indeed, in each series we know the values of these parameters accurate to 15-20%. Notice that the cubic nonlinearity influence on rotator was neglected because of rotator small optical thickness. Indeed, we had В ~1 in the 33 cm long NE. For the same intensity the B-integral in 1 cm long rotator was 33 times less.
Note that in the first series of experiments the maximum of the function Γ(t, В = 0) corresponds to the difference of the thermoelastic stresses induced in the elements within the time interval t < 5 min. The value of Γ(t, В = 0) at t > 5 min does not exceed 0.1% (Fig. 5a). In the second series of experiments the maximum of Γ(t, В = 0) was higher by virtue of φ1 ≠□φ2, but did not exceed 0.8% (Fig. 5b).
Contrary to this case, when B-integral is about unity in each NE the residual depolarization increases up to 0.9% and 3%, with the experimental data being in good agreement with the theoretical predictions (Fig. 5). Thus, we proved experimentally that the operation efficiency of the 90° rotator reduces in the presence of cubic nonlinearity.
As was mentioned above, the magnitude of residual local depolarization Γ depends on both the В-integral and the value of δ at a given moment of time. One can see in Fig. 1 that Γ grows in magnitude as δ approaches δcr = 2π/5. According to Fig. 3, δ rapidly grows up to δmax < δcr during the first ten minutes, and then decreases. Therefore, for В ~1, Γ first increases together with δ and then reduces with the decrease of δ. This is especially pronounced in Fig. 5b. Note that the values of Γ(δ, В) in the plots in Fig. 1 and Fig. 5 do not coincide exactly, as in the calculations of depolarization degree (Fig. 5) averaging was done over a Gaussian shape pulse and the mentioned value of В-integral corresponds to the pulse maximal over time.
3.3. The impact of cubic nonlinearity at accurate depolarization compensation by a 90° rotator
For more exact verification of the increase of residual depolarization degree at В ~1 we decreased Γ(В = 0) to the noise level of the CCD camera by turning the 90° rotator around the vertical axis on angle Ф. The efficiency of depolarization compensation by this method depends on the equality of thermally induced phase differences in the elements δ, as well as on rotator adjustment. As follows from the plots in Fig. 3, thermally induced birefringence is different in different elements. In addition, the conditions of the experiment on measuring δ(t) in an NE may vary in different series.
When a plane wave is propagating in crystal quartz (or another crystal with the same symmetry) at angle Ф to the optical axis, there occurs superposition of linear and circular birefringence. A change of angle Ф results in additional linear birefringence. We fitted angle Ф until at В = 0 the depolarized component ceased to be recorded against the background noise of the CCD camera, which corresponds to Γ < 0.1%. After each adjustment we measured local depolarization degree at В-integral of order 1. The results of the measurements are presented in Fig. 6 .
We remind the reader that for the case r1 = r2 = 4 cm, φ1 = φ2 = 45° the values of δ differ only slightly in different NEs, and starting with t > 5 min they coincide (Fig. 3a). Hence, up to this time the polarization rotator must be adjusted, and rather fast too, which greatly impeded our measurements at В ~1 within the time interval 0 < t < 5. For t > 5 min dynamic adjustment of the polarization rotator was not required, and angle Ф was 2.4°.
For the case r1 = r2 = 4 cm, φ1 = 42°, φ2 = 48° the values of δ weakly differ from each other throughout the time interval (Fig. 3b), which results in a smooth variation of the rotator angle within 0.5° - 1°.
With allowance for adjustment of the quartz polarization rotator, we obtained a very good agreement of experimental and theoretical data (Fig. 6) that demonstrates the influence of cubic nonlinearity on the compensation of birefringence. In the case r1 = r2 = 4 cm, φ1 = φ2 = 45° at В ~1, depolarization increases up to 0.7%, and in the case r1 = r2 = 4 cm, φ1 = 42°, φ2 = 48° up to 1.4%.
Since in experiments angle Ф didn’t exceed 2.4°, the phase difference of circular birefringence didn’t change and polarization was rotated on 90°. As for the phase difference of linear birefringence, its value was enough to compensate the difference in thermally induced birefringence in two NEs.
4. The influence of cubic nonlinearity on accuracy of polarization conversion by means of wave plates
λ/2 and λ/4 plates are an integral part of many lasers. In powerful femtosecond lasers, even on plates 1 mm thick and less, the magnitude of В-integral is sufficient for cubic nonlinearity to affect polarization formation.
Using the theoretical analysis made in the work  we will investigate the impact of cubic nonlinearity on an example of a quarter-wave plate. Let radiation with vertical linear polarization be incident on such a plate. According to the notation introduced in Section 2, the λ/4 plate is described by the following parameters: natural birefringence for all r equal to δ = π/2, and the inclination angle of the plate eigen polarizations φ = 45°. In other words, to obtain the circular polarization from incident vertical linear polarization we have to set the inclination angle of the plate eigen polarizations φ = 45° in case of zero B-integral. The value of the parameter characterizing the magnitude of nonlinearity in the plate is taken to be γNL = 3.2 10−7 cm2/GW. The computations are made for radiation with the wavelength λ = 1054 nm.
For В = 0, the λ/4 plate transforms the linear polarization to the circular one (left-handed for definiteness). Depolarization Γ (the part of energy in the right-handed circular polarization at the output of the λ/4 plate) as a function of В-integral is plotted in Fig. 7 . The plots are constructed for a rectangular pulse shape (Fig. 7a) and a Gaussian pulse (Fig. 7b). It is clear from Fig. 7a that for B ≈3 the local depolarization degree is 4%. As follows from the plot, the residual polarization degree may be reduced to 0.1% and lower by choosing an optimal inclination of the plate axes: φopt = 34°.
The maximum value of В-integral during the Gaussian pulse (Fig. 7b) is laid off along the х-axis. Consequently, the impact of cubic nonlinearity is less, as the pulse average value of В is less. In addition, the efficiency of compensating this effect by turning the axes of the λ/4 plate is also lower, as angle φopt is different for each value of В-integral.
The main aim of the proposed new values of the inclination angle of the wave plate eigen polarizations, e.g. φopt = 34°, is to decrease the depolarization degree (the part of energy in the right-handed circular polarization at the output of the λ/4 plate) in case when B-integral is not equal to zero. Contrary to this case, the choice of wave plate angle φ diverse from 45° will certainly increase the depolarization when B = 0. Therefore, for different application we choose the particular optimal value of polarization rotation angle.
Deterioration of the accuracy of polarization conversion by means of the λ/4 and λ/2 wave plates caused by cubic nonlinearity and suppression of this parasitic effect by turning the plate axis may be used in powerful femtosecond lasers.
The principal goal of the work was experimental confirmation of the earlier predicted  effect of reducing compensation accuracy of thermally induced depolarization in a system of two identical depolarizing nonlinear elements and a 90° polarization rotator.
Two series of experiments were conducted. Two elements with identically oriented axes of thermally induced birefringence were used in the first series, and in the second one directions of the axes differed by 6°. In the first series, compensation at В = 0 was more accurate than in the second one, but not ideal still: Γ(В = 0) ≠ 0.
The measurements of depolarization degree at the output of the NE – 90° polarization rotator – NE system at В = 0 and В ~1 (Fig. 5) confirmed reduction of the efficiency of using the considered depolarization compensation method in the presence of cubic nonlinearity. The best agreement between theoretical and experimental data was observed when Γ(В = 0) was reduced to the noise level by turning the 90° rotator around the vertical axis (Fig. 6).
The carried out experiments confirm that the theoretical model developed in  may be a useful tool for calculating depolarization degree in media with cubic nonlinearity and birefringence. The model was used to study the impact of cubic nonlinearity on the operation of wave plates. Reduction of polarization transformation accuracy for the В-integral more than unity was shown by an example of λ/4 plate. It was found that by rotating the plate around the axis of radiation propagation it is possible to reduce the described negative effect significantly. For example, for В = 3 the depolarization degree decreased from 4% to 0.1% in the case of a rectangular pulse shape (Fig. 7a), and the pulse average depolarization degree for a Gaussian pulse decreased from 0.7% to 0.02% (Fig. 7b).
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