## Compensation of thermally induced depolarization in Faraday isolators for high average power lasers |

Optics Express, Vol. 19, Issue 7, pp. 6366-6376 (2011)

http://dx.doi.org/10.1364/OE.19.006366

Acrobat PDF (1224 KB)

### Abstract

A compensation scheme for thermally induced birefringence in Faraday isolators is proposed. With the use of this scheme a 36-fold increase of the isolation degree was attained in experiment. A comparative analysis of the considered scheme and the earlier Faraday isolator schemes with high average radiation power is performed. A method for optimizing the earlier Faraday isolator scheme with birefringence compensation is developed.

© 2011 OSA

## 1. Introduction

^{−3}cm

^{−3}) of the magnetooptical material used in it [1

1. G. Mueller, R. S. Amin, D. Guagliardo, D. McFeron, R. Lundock, D. H. Reitze, and D. B. Tanner, “Method for compensation of thermally induced modal distortions in the input optical components of gravitational wave interferometers,” Class. Quantum Gravity **19**(7), 1793–1801 (2002). [CrossRef]

5. N. P. Barnes and L. P. Petway, “Variation of the Verdet constant with temperature of terbium gallium garnet,” J. Opt. Soc. Am. B **9**, 1912–1915 (1992). [CrossRef]

6. R. Yasuhara, S. Tokita, J. Kawanaka, T. Kawashima, H. Kan, H. Yagi, H. Nozawa, T. Yanagitani, Y. Fujimoto, H. Yoshida, and M. Nakatsuka, “Cryogenic temperature characteristics of Verdet constant on terbium gallium garnet ceramics,” Opt. Express **15**(18), 11255–11261 (2007). [CrossRef] [PubMed]

7. E. A. Khazanov, O. V. Kulagin, S. Yoshida, D. Tanner, and D. Reitze, “Investigation of self-induced depolarization of laser radiation in terbium gallium garnet,” IEEE J. Quantum Electron. **35**(8), 1116–1122 (1999). [CrossRef]

8. D. S. Zheleznov, A. V. Voitovich, I. B. Mukhin, O. V. Palashov, and E. A. Khazanov, “Considerable reduction of thermooptical distortions in Faraday isolators cooled to 77 K,” Quantum Electron. **36**(4), 383–388 (2006). [CrossRef]

9. D. S. Zheleznov, V. V. Zelenogorskii, E. V. Katin, I. B. Mukhin, O. V. Palashov, and E. A. Khazanov, “Cryogenic Faraday isolator,” Quantum Electron. **40**(3), 276–281 (2010). [CrossRef]

10. E. A. Khazanov, “Compensation of thermally induced polarization distortions in Faraday isolators,” Quantum Electron. **29**(1), 59–64 (1999). [CrossRef]

11. E. Khazanov, N. Andreev, A. Babin, A. Kiselev, O. Palashov, and D. Reitze, “Suppression of self-induced depolarization of high-power laser radiation in glass-based Faraday isolators,” J. Opt. Soc. Am. B **17**(1), 99–102 (2000). [CrossRef]

13. A. V. Voitovich, E. V. Katin, I. B. Mukhin, O. V. Palashov, and E. A. Khazanov, “Wide-aperture Faraday isolator for kilowatt average radiation powers,” Quantum Electron. **37**(5), 471–474 (2007). [CrossRef]

## 2. Analysis and optimization of birefringence compensation

### 2.1. Determining thermally induced depolarization

*x-*axis) in the absence of thermal effects. In the reverse passage (from А to В) the polarization changes to the vertical one (parallel to the

*y*-axis) and the radiation is reflected by polarizer 1. Due to heat absorption in MOEs the birefringence induced by the photoelastic effect leads to appearance at point В of radiation with a horizontal component that passes through polarizer 1.

**x**

_{0}is the unit vector in the direction of the x-axis and

*r*is the characteristic transverse size of the field. The local thermally induced depolarization at point В is defined bywhere

_{h}**E**

_{B}is the complex amplitude of the field at a point with the same transverse coordinates but in the plane passing through point В. Of great interest is the thermally induced FI depolarization integral over the beam section that is defined by

*I*[dB] =

_{c}*10 log(1/γ)*.

**E**

_{B}is found using the Jones matrix formalism [14

14. R. C. Jones, “A new calculus for the treatment of optical systems,” J. Opt. Soc. Am. **31**(7), 488–503 (1941). [CrossRef]

*θ*and plate

_{r}*λ*/2 are described, respectively, by matriceswhere

*θ*is the angle of slope of the plate’s optical axis relative to the x-axis. With allowance for the linear birefringence caused by the photoelastic effect in addition to the circular birefringence, the MOE is described by the Jones matrix [15

_{pl}15. M. J. Tabor and F. S. Chen, “Electromagnetic propagation through materials possessing both Faraday rotation and birefringence: experiments with ytterbium orthoferrite,” J. Appl. Phys. **40**(7), 2760–2765 (1969). [CrossRef]

*δ*are the phase differences in the case of purely circular (in the absence of linear) and purely linear (in the absence of circular) birefringence, respectively; and Ψ is the angle of slope of eigenpolarization relative to the x-axis. Note that the angle by which the Faraday rotator turns the polarization plane is

_{с}, δ_{lin}*Ф = δ*/2. Only the linear birefringence caused by the photoelastic effect is induced in AOE [7 in Fig. 1(c)]; therefore, the Jones matrix for it is found from Eqs. (5) and (6) at

_{с}*δ*= 0.

_{с}*δ*and Ψ are written in the form where

_{lin}*φ*is polar angle,

*θ*is the angle between one of the crystallographic axes lying in the (

*x,y*)-plane and the

*x*-axis (Fig. 1);

*p*,

*h,*and

*ξ*are the normalized power of heat generation, temperature distribution integral, and optical anisotropy parameter, respectively, that are defined by

*P*is the total power of heating radiation,

_{in}*λ*is radiation wavelength,

*κ*is heat conductivity,

*r*is polar radius;

*p*are the elements of photoelasticity tensor in the two-index Nye form [16];

_{ij}*α*is thermal expansion coefficient;

_{T}*n*, and

_{0}*α*are the index of refraction and the absorption coefficient at the wavelength

_{0}*λ*;

*ν*is Poisson’s ratio, and

*L*is the length of the optical element. The expressions Eq. (10) were obtained assuming axial symmetry of the problem and rod sample geometry, i.e., either

*L>>r*, or the heat sink from the ends is infinitesimal.

_{0}17. 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]

*δ*and Ψ, hence, all the formulas following from them for the [111] orientation may be obtained from Eqs. (7) and (8) by substituting

_{lin}**Е**

_{В}for any scheme presented in Fig. 1. For the FI schemes shown in Fig. 1, we will obtain

### 2.2. The case of small birefringence

*θ*is fulfilled. Then, by substituting Eq. (12) into Eq. (2) and expanding

_{1}= θ_{2}*δ*into a series, for the schemes in Fig. 1 we will obtain expressions for local depolarization: where

_{lin}*ξ*,

_{1}= ξ_{2}*Q*,

_{1}= Q_{2}*κ*) and have identical orientation (

_{1}= κ_{2}*θ*), from Eq. (9) and Eq. (17) one can readily obtain

_{1}= θ_{2}*G = D = L*. From Eqs. (15) and (16) it follows that for

_{2}/L_{1}*Г*and

_{in}*Г*are proportional to

_{out}*Г*is always proportional to

_{0}10. E. A. Khazanov, “Compensation of thermally induced polarization distortions in Faraday isolators,” Quantum Electron. **29**(1), 59–64 (1999). [CrossRef]

*γ:*where In this order of smallness the integral depolarizations

*γ*and

_{in}*γ*do not depend on

_{out}*θ*, i.e., on crystal orientation relative to the polarization of the radiation incident on the FI; whereas

_{1}= θ_{2}= θ*γ*depends on this quantity and at

_{0}### 2.3. Comparison of schemes with internal and external compensation

*γ versus p*plotted by Eqs. (23)–(25) are presented in Fig. 2 (а) and by Eqs. (23)–(25) with the substitution of Eq. (11) in Fig. 2(b) (dashed lines). It is clear from Fig. 2 that schemes with compensation are superior to the traditional FI scheme [Fig. 1(а)]. It follows from comparison of Eqs. (24) and (25) that, at fixed power of incident radiation in the scheme with compensation inside magnetic field at weak linear birefringence, the integral depolarization is 13.3 times less than in the scheme with compensation outside magnetic field. In practice, the integral depolarization ratio for the two compared schemes will be less than 13.3 times because for using a scheme with compensation a quartz rotator must be placed between two MOEs in the magnetic field, hence each MOE will be shifted closer to the edge of the magnetic system, where the magnetic field is weaker. Magnetic field weakening leads to ~15% increase of MOE length in the scheme in Fig. 1(b), hence, the integral depolarization will be only 7.6 times less than in the scheme in Fig. 1(c).

*dn/dT*having opposite sign it is possible to partially compensate not only thermally induced depolarization but thermal lens too. If AOE material is chosen with optical anisotropy parameter ξ<0, the polarization rotator may be omitted from the proposed scheme. Other optical elements of the laser system, such as active element (AE), polarizer (the experiment that confirms this statement will be described below), and others may also act as AOE. In the third place, the scheme allows fabricating analogous compensators for elements of powerful laser systems other than Faraday isolators.

### 2.4. Numerical optimization of FI parameters

*θ*. Therefore, we carried out numerical computations.

_{1}≠θ_{2}*γ*in schemes with compensation [Figs. 1(b) and 1(c)] depends on four parameters:

*θ*,

_{r}*D*,

*θ*, and

_{1}*θ*for the [001] orientation and on two parameters:

_{2}*θ*and

_{r}*D*for the [111] orientation. Optimization of these parameters may provide minimal integral thermally induced depolarization

*γ*at a given power of incident radiation and chosen crystal orientation.

*p*which, in turn, may be recalculated to the maximum admissible laser power

_{max}*P*using Eq. (9). For a TGG crystal with parameters

_{max}*Q*= 17∙10

^{−7}K

^{−1},

*κ*= 5 W∙K

^{−1}∙m

^{−1},

*L/λ*= 2∙10

^{4},

*α*= 3∙10

^{−3}cm

^{−1}[13

13. A. V. Voitovich, E. V. Katin, I. B. Mukhin, O. V. Palashov, and E. A. Khazanov, “Wide-aperture Faraday isolator for kilowatt average radiation powers,” Quantum Electron. **37**(5), 471–474 (2007). [CrossRef]

*P*= 520∙

_{max}*p*, and with

_{max}*Q*= 17∙10

^{−7}K

^{−1},

*κ*= 5 W∙K

^{−1}∙m

^{−1},

*L/λ*= 2∙10

^{4},

*α*= 7∙10

^{−4}cm

^{−1}[18] we have

*P*= 2100∙

_{max}*p*. Here

_{max}*P*is the incident radiation power below which the integral depolarization

_{max}*γ*is less than 0.001, i.e., the FI isolation ratio is more than 30 dB.

*p*of the FI with [111] orientation from 0.15 to 0.95, i.e., 6.3-fold (12-fold for the scheme with internal compensation [Fig. 1(b)]), and with [001] orientation from 0.25 to 1.7, i.e., 6.8-fold (13.2-fold). For crystals with [001] orientation, at p>5 the integral depolarization does not depend on relative position of the crystals in schemes with compensation, whereas at

_{max}*p*<5 there exists optimal crystal position

*θ*at which

_{1}≠ θ_{2}*P*of the Faraday isolator is 1.7 times more than at

_{max}*θ*.

_{1}= θ_{2}*θ*≈67.5°,

_{r}*D*≈1; for the scheme with external compensation,

*θ*≈67.5°,

_{r}*D*≈0.9. By setting

*θ*= 67.5°,

_{r}*D*= 1 (or

*θ*= 67.5°,

_{r}*D*= 0.9) we obtain deviation of integral depolarization from its optimum value no more than 15% throughout the power range.

*θ*and

_{1}*θ*allows compensating thermally induced birefringence two orders of magnitude better than in crystals with [111] orientation, on the one hand, but complicates scheme tuning on the other hand. Optimal parameters are functions of laser radiation power that take on values close to

_{2}*θ*≈73.18°,

_{r}*D*≈0.964,

*θ*≈20°, and

_{1}*θ*≈15.3° in the scheme with internal compensation and

_{2}*θ*≈73.5°,

_{r}*D*≈0.908,

*θ*≈27.2°, and

_{1}*θ*≈22.3° in the scheme with external compensation. The ratio of the rms deviation to the average value for

_{2}*θ*and

_{r}*D*is less than 1%, and for

*θ*and

_{1}*θ*is larger, amounting to ~5%. By fixing these four parameters at the values presented above we find that the integral depolarization deviation from its value at optimal parameters for

_{2}*p*in the 0.1-1.5 interval is not more than 17%.

*θ*and

_{r}*D*[Figs. 3(a) and 3(b)] and as a function of

*θ*and

_{1}*θ*(c,d) is plotted in Fig. 3 for schemes with internal (а,с) and external (b,d) compensation for the same value of normalized heat generation power

_{1}-θ_{2}*p*= 0.8 and [001] crystal orientation. Two parameters were varied, and two more fixed at the values presented above. The region of parameters at which

*γ*is less than 0.001 is shown by the contours in Fig. 3. In order to remain inside this region at normalized power

*p*= 0.8, the deviation from the optimum values must not exceed 21.6% for

*θ*and 28.1% for

_{r}*D*for the scheme with compensation inside magnetic field (9.8% for

*θ*and 15% for

_{r}*D*for the scheme with compensation outside magnetic field).

*γ*depends to a greater extent on the difference of angles

*θ*and

_{1}*θ*and, at a certain power, for any value of

_{2}*θ*it is possible to find

_{1}*θ*at which

_{2}*γ*will not exceed 0.001.

## 3. Results of experiments

*θ*= 67.5°, diameter 10.3 mm and length 10.7 mm was used as polarization rotator 4. A TGG crystal having orientation [001], diameter 20 mm, and length 18 mm was taken as AOE 5. In the absence of elements 3, 4 and 5, the calcite wedge and Glan prism gave contrast of order 3∙10

_{r}^{−5}. In the presence of elements 3-5, depolarized radiation appeared in them due to thermally induced birefringence, which passed through Glan prism 7 and then to the CCD camera 8 where intensity distribution was registered. The magnitude of the depolarized component depended on the power of incident radiation. The intensity distribution of depolarized and polarized radiation was integrated over the cross-section, and their ratio gave integral depolarization

*γ*.

*γ*on the incident radiation power was measured in experiment for only FI 3 (in the absence of elements 4 and 5) (Fig. 4) and for the scheme with compensator 4, 5 presented in Fig. 5 .

*θ*= 67.5°,

_{r}*D*= 0.75;

*θ*= 22.5°,

_{1}*θ*= 22.5° implemented in the experiment. The calculated curve in the scheme with compensation is almost parallel to the calculated curve in the scheme without compensation, which indicates that the integral depolarization is proportional to squared power, rather than to the 4th power predicted by the theory [see Eq. (25)]. This behavior is explained by a pronounced difference between the experimental and optimal parameters. Nevertheless, a 36-fold decrease of integral polarization was attained with the use of compensator at maximum laser power. The integral polarization with optimal parameters (green curve) is given for comparison in Fig. 5. The computations show that the effect of thermally induced depolarization could have been observed in experiment at optimal parameters with available “cold” depolarization and scheme contrast of about 6∙10

_{2}^{−5}, if the power were three times maximum power of the available laser, and

*P*were 1.6 kW for the crystals used in the experiment.

_{max}7. E. A. Khazanov, O. V. Kulagin, S. Yoshida, D. Tanner, and D. Reitze, “Investigation of self-induced depolarization of laser radiation in terbium gallium garnet,” IEEE J. Quantum Electron. **35**(8), 1116–1122 (1999). [CrossRef]

*γ*= 10

_{v}^{−3}

*γ*, that for maximum of the available laser power corresponds to a

_{0}*γ*equal 10

_{v}^{−5}. As can be seen, the contribution is 6 times less “cold” depolarization and contrast schemes. The contribution of the effect of temperature dependence of Verdet constant increases only in proportion to the squared power, while

*γ*the 4th power. Therefore, at low laser power

_{out}*γ*be neglected in comparison with “cold” depolarization, while at large - in comparison with

_{v}*γ*.

_{out}*γ*on

*θ*was obtained in experiment for fixed

_{2}*θ*(see Fig. 6 ) of 22.5 and −22.5 degrees, corresponding to the maximum and minimum integral depolarization of FI without compensation [Fig. 1(a)]. These values of

_{1}*θ*may be attained in experiment to a high accuracy. Angle

_{1}*θ*was varied in the interval from −30 to + 60 degrees. The incident radiation power was 96 W.

_{2}*y-*axis (Fig. 1). A TGG crystal with [001] orientation was used as an AOE. With a smooth turn of the AOE about the

*y-*axis the integral depolarization of

*γ*decreased (the compensation became better). The compensation was improved primarily due to the increased path length of light inside the crystal during its turn, which resulted in an increase of the absorbed power in the AOE and, consequently, in an increase of parameter

*D*which approached its optimum value. Thus, the turn of AOE allows a smooth increase of parameter

*D*, hence providing an additional potential for improving compensation. If the initial value of

*D*is larger than the optimum one, then the AOE turn relative to the

*y*-axis will only impair the compensation.

## 4. Conclusion

## References and links

1. | G. Mueller, R. S. Amin, D. Guagliardo, D. McFeron, R. Lundock, D. H. Reitze, and D. B. Tanner, “Method for compensation of thermally induced modal distortions in the input optical components of gravitational wave interferometers,” Class. Quantum Gravity |

2. | 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. |

3. | I. B. Mukhin, A. V. Voitovich, O. V. Palashov, and E. A. Khazanov, “2.1 tesla permanent -magnet Faraday isolator for subkilowatt average power lasers,” Opt. Commun. |

4. | T. V. Zarubina and G. T. Petrovsky, “Magnetooptical glasses made in Russia,” Opticheskii Zhurnal |

5. | N. P. Barnes and L. P. Petway, “Variation of the Verdet constant with temperature of terbium gallium garnet,” J. Opt. Soc. Am. B |

6. | R. Yasuhara, S. Tokita, J. Kawanaka, T. Kawashima, H. Kan, H. Yagi, H. Nozawa, T. Yanagitani, Y. Fujimoto, H. Yoshida, and M. Nakatsuka, “Cryogenic temperature characteristics of Verdet constant on terbium gallium garnet ceramics,” Opt. Express |

7. | E. A. Khazanov, O. V. Kulagin, S. Yoshida, D. Tanner, and D. Reitze, “Investigation of self-induced depolarization of laser radiation in terbium gallium garnet,” IEEE J. Quantum Electron. |

8. | D. S. Zheleznov, A. V. Voitovich, I. B. Mukhin, O. V. Palashov, and E. A. Khazanov, “Considerable reduction of thermooptical distortions in Faraday isolators cooled to 77 K,” Quantum Electron. |

9. | D. S. Zheleznov, V. V. Zelenogorskii, E. V. Katin, I. B. Mukhin, O. V. Palashov, and E. A. Khazanov, “Cryogenic Faraday isolator,” Quantum Electron. |

10. | E. A. Khazanov, “Compensation of thermally induced polarization distortions in Faraday isolators,” Quantum Electron. |

11. | E. Khazanov, N. Andreev, A. Babin, A. Kiselev, O. Palashov, and D. Reitze, “Suppression of self-induced depolarization of high-power laser radiation in glass-based Faraday isolators,” J. Opt. Soc. Am. B |

12. | N. F. Andreev, O. V. Palashov, A. K. Potemkin, D. H. Reitze, A. M. Sergeev, and E. A. Khazanov, “45-dB Faraday isolator for 100-W average radiation power,” Quantum Electron. |

13. | A. V. Voitovich, E. V. Katin, I. B. Mukhin, O. V. Palashov, and E. A. Khazanov, “Wide-aperture Faraday isolator for kilowatt average radiation powers,” Quantum Electron. |

14. | R. C. Jones, “A new calculus for the treatment of optical systems,” J. Opt. Soc. Am. |

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

16. | J. F. Nye, |

17. | 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. |

18. | A. V. Starobor, D. S. Zheleznov, O. V. Palashov, and E. A. Khazanov, “Novel magnetooptical mediums for cryogenic Faraday isolator,” in |

**OCIS Codes**

(140.6810) Lasers and laser optics : Thermal effects

(230.2240) Optical devices : Faraday effect

(260.1440) Physical optics : Birefringence

**ToC Category:**

Lasers and Laser Optics

**History**

Original Manuscript: January 10, 2011

Revised Manuscript: March 4, 2011

Manuscript Accepted: March 6, 2011

Published: March 21, 2011

**Citation**

Ilya Snetkov, Ivan Mukhin, Oleg Palashov, and Efim Khazanov, "Compensation of thermally induced depolarization in Faraday isolators for high average power lasers," Opt. Express **19**, 6366-6376 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-7-6366

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### References

- G. Mueller, R. S. Amin, D. Guagliardo, D. McFeron, R. Lundock, D. H. Reitze, and D. B. Tanner, “Method for compensation of thermally induced modal distortions in the input optical components of gravitational wave interferometers,” Class. Quantum Gravity 19(7), 1793–1801 (2002). [CrossRef]
- 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]
- I. B. Mukhin, A. V. Voitovich, O. V. Palashov, and E. A. Khazanov, “2.1 tesla permanent -magnet Faraday isolator for subkilowatt average power lasers,” Opt. Commun. 282(10), 1969–1972 (2009). [CrossRef]
- T. V. Zarubina and G. T. Petrovsky, “Magnetooptical glasses made in Russia,” Opticheskii Zhurnal 59, 48–52 (1992).
- N. P. Barnes and L. P. Petway, “Variation of the Verdet constant with temperature of terbium gallium garnet,” J. Opt. Soc. Am. B 9, 1912–1915 (1992). [CrossRef]
- R. Yasuhara, S. Tokita, J. Kawanaka, T. Kawashima, H. Kan, H. Yagi, H. Nozawa, T. Yanagitani, Y. Fujimoto, H. Yoshida, and M. Nakatsuka, “Cryogenic temperature characteristics of Verdet constant on terbium gallium garnet ceramics,” Opt. Express 15(18), 11255–11261 (2007). [CrossRef] [PubMed]
- E. A. Khazanov, O. V. Kulagin, S. Yoshida, D. Tanner, and D. Reitze, “Investigation of self-induced depolarization of laser radiation in terbium gallium garnet,” IEEE J. Quantum Electron. 35(8), 1116–1122 (1999). [CrossRef]
- D. S. Zheleznov, A. V. Voitovich, I. B. Mukhin, O. V. Palashov, and E. A. Khazanov, “Considerable reduction of thermooptical distortions in Faraday isolators cooled to 77 K,” Quantum Electron. 36(4), 383–388 (2006). [CrossRef]
- D. S. Zheleznov, V. V. Zelenogorskii, E. V. Katin, I. B. Mukhin, O. V. Palashov, and E. A. Khazanov, “Cryogenic Faraday isolator,” Quantum Electron. 40(3), 276–281 (2010). [CrossRef]
- E. A. Khazanov, “Compensation of thermally induced polarization distortions in Faraday isolators,” Quantum Electron. 29(1), 59–64 (1999). [CrossRef]
- E. Khazanov, N. Andreev, A. Babin, A. Kiselev, O. Palashov, and D. Reitze, “Suppression of self-induced depolarization of high-power laser radiation in glass-based Faraday isolators,” J. Opt. Soc. Am. B 17(1), 99–102 (2000). [CrossRef]
- N. F. Andreev, O. V. Palashov, A. K. Potemkin, D. H. Reitze, A. M. Sergeev, and E. A. Khazanov, “45-dB Faraday isolator for 100-W average radiation power,” Quantum Electron. 30(12), 1107–1108 (2000). [CrossRef]
- A. V. Voitovich, E. V. Katin, I. B. Mukhin, O. V. Palashov, and E. A. Khazanov, “Wide-aperture Faraday isolator for kilowatt average radiation powers,” Quantum Electron. 37(5), 471–474 (2007). [CrossRef]
- R. C. Jones, “A new calculus for the treatment of optical systems,” J. Opt. Soc. Am. 31(7), 488–503 (1941). [CrossRef]
- M. J. Tabor and F. S. Chen, “Electromagnetic propagation through materials possessing both Faraday rotation and birefringence: experiments with ytterbium orthoferrite,” J. Appl. Phys. 40(7), 2760–2765 (1969). [CrossRef]
- J. F. Nye, Physical Properties of Crystals (Oxford University Press, 1964).
- 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]
- A. V. Starobor, D. S. Zheleznov, O. V. Palashov, and E. A. Khazanov, “Novel magnetooptical mediums for cryogenic Faraday isolator,” in ICONO/LAT 2010 (2010), LTuL23.

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