## Theoretical and experimental study of laser radiation propagating in a medium with thermally induced birefringence and cubic nonlinearity |

Optics Express, Vol. 19, Issue 22, pp. 21977-21988 (2011)

http://dx.doi.org/10.1364/OE.19.021977

Acrobat PDF (1388 KB)

### Abstract

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 OSA

## 1. Introduction

1. E. A. Khazanov and A. M. Sergeev, “Petawatt laser based on optical parametric amplifiers: their state and prospects,” Sov. Phys. Usp. **51**(9), 969–974 (2008). [CrossRef]

2. A. V. Korzhimanov, A. A. Gonoskov, E. A. Khazanov, and A. M. Sergeev, “Horizons of petawatt laser technology,” Sov. Phys. Usp. **54**(1), 9–28 (2011). [CrossRef]

5. W. Koechner and D. K. Rice, “Birefringence of YAG:Nd laser rods as a function of growth direction,” J. Opt. Soc. Am. **61**(6), 758–766 (1971). [CrossRef]

6. L. N. Soms and A. A. Tarasov, “Thermal deformation in color-center laser active elements. 1.Theory,” Sov. J. Quantum Electron. **9**(12), 1506–1509 (1979). [CrossRef]

7. L. N. Soms, A. A. Tarasov, and V. V. Shashkin, “On the problem of depolarization of linearly polarized light by a YAG:Nd3+ laser rod under conditions of thermally induced birefringence,” Sov. J. Quantum Electron. **10**(3), 350–351 (1980). [CrossRef]

12. M. A. Kagan and E. A. Khazanov, “Compensation for thermally induced birefringence in polycrystalline ceramic active elements,” Quantum Electron. **33**(10), 876–882 (2003). [CrossRef]

13. E. A. Khazanov, N. F. Andreev, A. N. Mal'shakov, O. V. Palashov, A. K. Poteomkin, A. M. Sergeev, A. A. Shaykin, V. V. Zelenogorsky, I. Ivanov, R. S. Amin, G. Mueller, D. B. Tanner, and D. H. Reitze, “Compensation of thermally induced modal distortions in Faraday isolators,” IEEE J. Quantum Electron. **40**(10), 1500–1510 (2004). [CrossRef]

*n*(

*I*)

*= n*, where

_{0}+ γ_{NL}I*n*is the linear index of refraction and

_{0}*γ*is the characteristic of the nonlinear medium. A conventional measure of nonlinearity is the

_{NL}*В-*integral – nonlinear phase incursion in a medium having length

*L*:where

*λ*is the wavelength of light in vacuum. For

*В*> 3, there occurs small-scale self-focusing of light that destroys laser elements. In this sense, it is usually said that self-focusing limits the power of solid-state pulse lasers [14

14. A. K. Potemkin, E. V. Katin, A. V. Kirsanov, G. A. Luchinin, A. N. Mal’shakov, M. A. Mart’yanov, A. Z. Matveev, O. V. Palashov, E. A. Khazanov, and A. A. Shaikin, “Compact neodymium phosphate glass laser emitting 100-J, 100-GW pulses for pumping a parametric amplifier of chirped pulses,” Quantum Electron. **35**(4), 302–310 (2005). [CrossRef]

16. S. N. Vlasov, V. I. Kryzhanovskiĭ, and V. E. Yashin, “Use of circularly polarized optical beams to suppress selffocusing instability in a nonlinear cubic medium with repeaters,” Sov. J. Quantum Electron. **12**(1), 7–10 (1982). [CrossRef]

16. S. N. Vlasov, V. I. Kryzhanovskiĭ, and V. E. Yashin, “Use of circularly polarized optical beams to suppress selffocusing instability in a nonlinear cubic medium with repeaters,” Sov. J. Quantum Electron. **12**(1), 7–10 (1982). [CrossRef]

20. D. Auric and A. Labadens, “On the use of circulary polarized beam to reduce the self-focusing effect in a glass rod amplifier,” Opt. Commun. **21**(2), 241–242 (1977). [CrossRef]

21. W. C. Scott and M. de Wit, “Birefringence compensation and TEM00 mode enhancement in a Nd:YAG laser,” Appl. Phys. Lett. **18**(1), 3–4 (1971). [CrossRef]

*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 [22

22. M. S. Kochetkova, M. A. Martyanov, A. K. Poteomkin, and E. A. Khazanov, “Propagation of laser radiation in a medium with thermally induced birefringence and cubic nonlinearity,” Opt. Express **18**(12), 12839–12851 (2010). [CrossRef] [PubMed]

*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

22. M. S. Kochetkova, M. A. Martyanov, A. K. Poteomkin, and E. A. Khazanov, “Propagation of laser radiation in a medium with thermally induced birefringence and cubic nonlinearity,” Opt. Express **18**(12), 12839–12851 (2010). [CrossRef] [PubMed]

*E*and

_{x},*E*are the transverse Cartesian components of electric field vector,

_{y}*k*= 2

*πn*/

_{0}*λ*is wave vector,

*β*characterizes the type of nonlinearity. The nonlinearity caused by the Kerr orientation effect is essential for liquids and gases. It arises in media with anisotropically polarized molecules and is, actually, preferred orientation of the axes of molecules’ highest polarizability in the direction of electric field. In this case,

*β*= 6 [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 [23]. The effect of thermally induced birefringence manifests itself in solid lasers, hence we choose in the present work

*β*= 1.

*r, φ, z*). Induced birefringence at the point (

*r, φ, z*) is usually characterized by the phase

*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

5. W. Koechner and D. K. Rice, “Birefringence of YAG:Nd laser rods as a function of growth direction,” J. Opt. Soc. Am. **61**(6), 758–766 (1971). [CrossRef]

6. L. N. Soms and A. A. Tarasov, “Thermal deformation in color-center laser active elements. 1.Theory,” Sov. J. Quantum Electron. **9**(12), 1506–1509 (1979). [CrossRef]

5. W. Koechner and D. K. Rice, “Birefringence of YAG:Nd laser rods as a function of growth direction,” J. Opt. Soc. Am. **61**(6), 758–766 (1971). [CrossRef]

7. L. N. Soms, A. A. Tarasov, and V. V. Shashkin, “On the problem of depolarization of linearly polarized light by a YAG:Nd3+ laser rod under conditions of thermally induced birefringence,” Sov. J. Quantum Electron. **10**(3), 350–351 (1980). [CrossRef]

11. E. A. Khazanov, “Thermally induced birefringence in Nd:YAG ceramics,” Opt. Lett. **27**(9), 716–718 (2002). [CrossRef] [PubMed]

12. M. A. Kagan and E. A. Khazanov, “Compensation for thermally induced birefringence in polycrystalline ceramic active elements,” Quantum Electron. **33**(10), 876–882 (2003). [CrossRef]

*θ*≠

*φ*[6

6. L. N. Soms and A. A. Tarasov, “Thermal deformation in color-center laser active elements. 1.Theory,” Sov. J. Quantum Electron. **9**(12), 1506–1509 (1979). [CrossRef]

10. E. Khazanov, N. Andreev, O. Palashov, A. Poteomkin, A. Sergeev, O. Mehl, and D. H. Reitze, “Effect of terbium gallium garnet crystal orientation on the isolation ratio of a Faraday isolator at high average power,” Appl. Opt. **41**(3), 483–492 (2002). [CrossRef] [PubMed]

**E**

*of intensity with polarization orthogonal to the reference one*

_{out}**E**

_{ref}_{⊥}:The reference polarization

**E**

_{ref}_{⊥}is output polarization in the absence of birefringence, i.e., at

*δ*= 0 and Γ = 0.

**E**

_{ref}_{⊥}, we can obtain the following expression for Γ at the output of one element in the absence of cubic nonlinearity:

*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 [22

22. M. S. Kochetkova, M. A. Martyanov, A. K. Poteomkin, and E. A. Khazanov, “Propagation of laser radiation in a medium with thermally induced birefringence and cubic nonlinearity,” Opt. Express **18**(12), 12839–12851 (2010). [CrossRef] [PubMed]

*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

*δ*2

_{cr}=*π*/5 independent of

*В*that determines the first and highest maximum of function Γ(

*δ*); Γ nonmonotonically but rapidly decreases with increasing

*δ*.

*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

*B*-integral.

### 3.1. Schematic of the experiment

**3**by a flash of pump lamps arranged along the laser Nd:glass rod (NE) 33 cm long with

*n*= 1.534 and

_{0}*γ*3.2·10

_{NL}=^{−7}cm

^{2}/GW. The NEs were air cooled convectively during characteristic time of about 1 hour [24

24. A. A. Kuzmin, E. A. Khazanov, and A. A. Shaykin, “Large-aperture Nd:glass laser amplifiers with high pulse repetition rate,” Opt. Express **19**(15), 14223–14232 (2011). [CrossRef] [PubMed]

*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*.

**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 [25

25. A. K. Poteomkin, E. A. Khazanov, M. A. Martyanov, A. V. Kirsanov, and A. A. Shaykin, “Compact 300 J/ 300 GW frequency doubled neodimium glass laser. Part II: Description of laser setup,” IEEE J. Quantum Electron. **45**(7), 854–863 (2009). [CrossRef]

*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)

*r*=

_{1}*r*4 cm,

_{2}=*φ*= 45°; (b)

_{1}= φ_{2}*r*=

_{1}*r*4 cm,

_{2}=*φ*= 42°,

_{1}*φ*= 48°. The inclination angle of the intrinsic polarization

_{2}*φ*in each NEs is counted off from the horizontal.

*r*=

_{1}*r*4 cm,

_{2}=*φ*= 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

_{1}= φ_{2}*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 [24

24. A. A. Kuzmin, E. A. Khazanov, and A. A. Shaykin, “Large-aperture Nd:glass laser amplifiers with high pulse repetition rate,” Opt. Express **19**(15), 14223–14232 (2011). [CrossRef] [PubMed]

*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*.

*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

*r*=

_{1}*r*4 cm,

_{2}=*φ*= 45°, starting with

_{1}= φ_{2}*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

*r*=

_{1}*r*4 cm,

_{2}=*φ*= 42°,

_{1}*φ*= 48°, experimental curves for

_{2}*δ*(

*t*) point to appreciable difference in the magnitudes of thermally induced birefringence in the studied samples (Fig. 3b).

*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.

*В*= 0. This difference is appreciably higher for the case when

*φ*= 42°,

_{1}*φ*= 48°. An increase of depolarization induced by cubic nonlinearity relative to the zero level corresponding to

_{2}*В*= 0 was observed in the experiment that will be described in Section 3.2.

*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

*В*= 0, the magnitude of Γ is nonzero because of different values of

*δ*in the two elements (Fig. 3) and by virtue of

*φ*≠□

_{1}*φ*This fact is illustrated in Fig. 5 . One can see that the experimental and theoretical data are in the very good agreement when

_{2}.*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.

*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

*φ*≠□

_{1}*φ*but did not exceed 0.8% (Fig. 5b).

_{2},*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.

*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

*δ*2

_{cr}=*π*/5. According to Fig. 3,

*δ*rapidly grows up to

*δ*during the first ten minutes, and then decreases. Therefore, for

_{max}< δ_{cr}*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

*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.

*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 .

*r*=

_{1}*r*4 cm,

_{2}=*φ*= 45° the values of

_{1}= φ_{2}*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°.

*r*=

_{1}*r*4 cm,

_{2}=*φ*= 42°,

_{1}*φ*= 48° the values of

_{2}*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°.

*r*=

_{1}*r*4 cm,

_{2}=*φ*= 45° at

_{1}= φ_{2}*В*~1, depolarization increases up to 0.7%, and in the case

*r*=

_{1}*r*4 cm,

_{2}=*φ*= 42°,

_{1}*φ*= 48° up to 1.4%.

_{2}28. A. A. Jaecklin and M. Lietz, “Elimination of disturbing birefringence effects on faraday rotation,” Appl. Opt. **11**(3), 617–621 (1972). [CrossRef] [PubMed]

*δ*, that is the phase difference between circular eigen polarizations, and linear birefringence by

_{c}*δ*, that is the phase difference between linear eigen polarizations. The phase difference of circular birefringence

_{l}*δ*at

_{c}*Ф*<< 1 weakly depends on

*Ф*. The phase difference of linear birefringence

*δ*depends on

_{l}*Ф*; for

*Ф*<< 1 this dependence is square. In the case under consideration, for the introduced quantities the following expressions are valid:where

*n*

_{o,}

*are ordinary and extraordinary refractive index of crystalline quartz.*

_{e}*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.

**18**(12), 12839–12851 (2010). [CrossRef] [PubMed]

*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

*γ*3.2 10

_{NL}=^{−7}cm

^{2}/GW. The computations are made for radiation with the wavelength

*λ*= 1054 nm.

*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:

*φ*34°.

_{opt}=*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

*φ*is different for each value of

_{opt}*В*-integral.

*φ*34°, is to decrease the depolarization degree (the part of energy in the right-handed circular polarization at the output of the

_{opt}=*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.

*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.

## 5. Conclusion

**18**(12), 12839–12851 (2010). [CrossRef] [PubMed]

*В*= 0 was more accurate than in the second one, but not ideal still: Γ(

*В*= 0) ≠ 0.

*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).

**18**(12), 12839–12851 (2010). [CrossRef] [PubMed]

*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).

## References and links

1. | E. A. Khazanov and A. M. Sergeev, “Petawatt laser based on optical parametric amplifiers: their state and prospects,” Sov. Phys. Usp. |

2. | A. V. Korzhimanov, A. A. Gonoskov, E. A. Khazanov, and A. M. Sergeev, “Horizons of petawatt laser technology,” Sov. Phys. Usp. |

3. | W. Koechner, |

4. | A. V. Mezenov, L. N. Soms, and A. I. Stepanov, |

5. | W. Koechner and D. K. Rice, “Birefringence of YAG:Nd laser rods as a function of growth direction,” J. Opt. Soc. Am. |

6. | L. N. Soms and A. A. Tarasov, “Thermal deformation in color-center laser active elements. 1.Theory,” Sov. J. Quantum Electron. |

7. | L. N. Soms, A. A. Tarasov, and V. V. Shashkin, “On the problem of depolarization of linearly polarized light by a YAG:Nd3+ laser rod under conditions of thermally induced birefringence,” Sov. J. Quantum Electron. |

8. | I. Shoji and T. Taira, “Intrinsic reduction of the depolarization loss in solid-state lasers by use of a (110)-cut Y3Al5O12 crystal,” Appl. Phys. Lett. |

9. | I. B. Mukhin, O. V. Palashov, E. A. Khazanov, and I. A. Ivanov, “Influence of the orientation of a crystal on thermal polarization effects in high-power solid-state lasers,” JETP Lett. |

10. | E. Khazanov, N. Andreev, O. Palashov, A. Poteomkin, A. Sergeev, O. Mehl, and D. H. Reitze, “Effect of terbium gallium garnet crystal orientation on the isolation ratio of a Faraday isolator at high average power,” Appl. Opt. |

11. | E. A. Khazanov, “Thermally induced birefringence in Nd:YAG ceramics,” Opt. Lett. |

12. | M. A. Kagan and E. A. Khazanov, “Compensation for thermally induced birefringence in polycrystalline ceramic active elements,” Quantum Electron. |

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17. | Y. B. Zel'dovich and Y. P. Raizer, “Self-focusing of light. Role of Kerr effect and striction,” JETP Lett. |

18. | P. D. Maker, R. W. Terhune, and C. M. Savage, “Intensity-dependent changes in the refractive index of liquids,” Phys. Rev. Lett. |

19. | A. L. Berkhoer and V. E. Zakharov, “Self excitation of waves with different polarizations in nonlinear media,” Sov. Phys. JETP |

20. | D. Auric and A. Labadens, “On the use of circulary polarized beam to reduce the self-focusing effect in a glass rod amplifier,” Opt. Commun. |

21. | W. C. Scott and M. de Wit, “Birefringence compensation and TEM00 mode enhancement in a Nd:YAG laser,” Appl. Phys. Lett. |

22. | M. S. Kochetkova, M. A. Martyanov, A. K. Poteomkin, and E. A. Khazanov, “Propagation of laser radiation in a medium with thermally induced birefringence and cubic nonlinearity,” Opt. Express |

23. | G. Fibich and B. Ilan, “Self-focusing of circularly polarized beams,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. |

24. | A. A. Kuzmin, E. A. Khazanov, and A. A. Shaykin, “Large-aperture Nd:glass laser amplifiers with high pulse repetition rate,” Opt. Express |

25. | A. K. Poteomkin, E. A. Khazanov, M. A. Martyanov, A. V. Kirsanov, and A. A. Shaykin, “Compact 300 J/ 300 GW frequency doubled neodimium glass laser. Part II: Description of laser setup,” IEEE J. Quantum Electron. |

26. | A. P. Voitovich and V. N. Severikov, |

27. | W. J. Tabor and F. S. Chen, “Electromagnetic propagation through materials possessing both Faraday rotation and birefringence: experiments with ytterbium orthoferrite,” J. Appl. Phys. |

28. | A. A. Jaecklin and M. Lietz, “Elimination of disturbing birefringence effects on faraday rotation,” Appl. Opt. |

**OCIS Codes**

(140.6810) Lasers and laser optics : Thermal effects

(190.0190) Nonlinear optics : Nonlinear optics

**ToC Category:**

Nonlinear Optics

**History**

Original Manuscript: August 18, 2011

Revised Manuscript: September 26, 2011

Manuscript Accepted: September 27, 2011

Published: October 21, 2011

**Citation**

M. S. Kuzmina, M. A. Martyanov, A. K. Poteomkin, E. A. Khazanov, and A. A. Shaykin, "Theoretical and experimental study of laser radiation propagating in a medium with thermally induced birefringence and cubic nonlinearity," Opt. Express **19**, 21977-21988 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-22-21977

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### References

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