Compensation of thermally induced depolarization ... - OSA Publishing

Compensation of thermally induced depolarization in Faraday isolators for high average power lasers Ilya Snetkov,* Ivan Mukhin, Oleg Palashov, and Efim Khazanov Institute of Applied Physics of the Russian Academy of Sciences, Nizhny Novgorod, Russia *[email protected]

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 Optical Society of America OCIS codes: (140.6810) Thermal effects; (260.1440) Birefringence; (230.2240) Faraday effect.

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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). 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). 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). 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). 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). 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). 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). 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). E. A. Khazanov, “Compensation of thermally induced polarization distortions in Faraday isolators,” Quantum Electron. 29(1), 59–64 (1999). 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). 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). 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). R. C. Jones, “A new calculus for the treatment of optical systems,” J. Opt. Soc. Am. 31(7), 488–503 (1941). 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). J. F. Nye, Physical Properties of Crystals (Oxford University Press, 1964).

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Received 10 Jan 2011; revised 4 Mar 2011; accepted 6 Mar 2011; published 21 Mar 2011

28 March 2011 / Vol. 19, No. 7 / OPTICS EXPRESS 6366

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). 18. 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.

1. Introduction The continuous growth of average radiation power of CW and pulse-repetition lasers makes study of the thermal effects caused by laser radiation absorption in the bulk of optical elements increasingly more important. The Faraday isolator (FI) is one of the elements strongly subject to thermal self-action because of the relatively strong absorption (~10 3 cm3) of the magnetooptical material used in it [1–4]. The radiation absorption in the bulk of an FI optical element leads to inhomogeneous temperature distribution, resulting in inhomogeneous distribution of all temperature dependent optical characteristics, such as index of refraction, heat conductivity, Verdet constant, and others. The temperature gradient also gives rise to internal stresses and thermally induced birefringence produced by the photoelastic effect. The inhomogeneous distribution of refraction index and changes in the geometrical size of the optical element result in wave front distortions referred to as “thermal lens”. The dependence of the Verdet constant on transverse coordinates leads to a path-length difference between two circular eigenpolarizations without changing the latters [5,6]. Thermally induced birefringence at each point of the cross-section changes both, the pathlength difference between the eigenpolarizations and the eigenpolarizations themselves that become elliptical. Both these effects result in inhomogeneous change of the polarization plane. The contribution of the photoelastic effect, as was shown in [7], is much greater. Hence, we should look for ways to suppress the thermally induced birefringence produced by mechanical stresses due to the temperature gradient. The birefringence may be suppressed if we eliminate causes of its occurrence (i) by choosing a proper material or cooling to nitrogen temperature [8,9] in order to decrease heat release inside the sample, or (ii) by choosing a method of cooling or a profile of heating radiation so as to decrease a radial component of the temperature gradient. Another option is to compensate the birefringence induced in one element by means of thermally induced birefringence in the other. Such a scheme of polarization distortions compensation was first proposed in [10] and was successfully implemented in [11–13]. The main idea was to replace one 45° Faraday rotator [Fig. 1(а)] by two identical 22.5° rotators and a 67.5° reciprocal polarization rotator placed between them [Fig. 1(b)]. This allowed partial compensation of the thermally induced birefringence, if the magnetooptical elements (MOEs) and quartz rotator rotate the polarization plane in one direction. All optical elements in such a scheme are inside the magnetic system, the MOEs are identical and their crystallographic axes are oriented identically.

Fig. 1. FI schematic: (a) traditional, (b) internal compensation, (c) external compensation. 1,4 – polarizers; 2 – λ/2 plate; 3 – 45° MOE; 5 – 22.5° MOE; 6 – polarization rotator; 7 – additional optical element.

In the presented work we proposed and investigated in experiment a new FI scheme with compensation of thermal depolarization. Our idea was to add to the 45° Faraday rotator [Fig.

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1(а)] outside its magnetic field a compensator comprising two optical elements: polarization rotator 6 and additional optical element 7 (AOE) made of material with thermo-optical properties close to those of MOE [Fig. 1(c)]. In this case, the MOE induced depolarization is partially compensated in the AOE. Using this scheme it is possible to modify the traditional FIs, thus increasing the degree of their isolation. We have found optimal parameters of polarization rotator and AOE for the two described schemes, provided that the crystals have different orientations of their crystallographic axes and are made of different optical materials. 2. Analysis and optimization of birefringence compensation 2.1. Determining thermally induced depolarization Consider the principle of operation of the FI depicted in Fig. 1. Thanks to a half-wave plate, the polarization of the radiation in direct passage (from В to А, see Fig. 1) through polarizer 4 persists to be horizontal (parallel to the 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. We assume that the complex amplitude of the field in the section crossing point A has the form E A  E0 x 0 exp   r 2 / rh2  ,

(1)

where x0 is the unit vector in the direction of the x-axis and rh is the characteristic transverse size of the field. The local thermally induced depolarization at point В is defined by Г

EBx 0 EA

2

2

,

(2)

where EB 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 2





 d  Г  r,   exp( r 0

0 2



0

0

2

/ rh2 )rdr .

(3)

2 2  d  exp( r / rh )rdr

We assume that the FI light diameter is such that aperture losses may be neglected and integrate Eq. (3) over polar radius to infinity. The isolation degree in dB is found from the expression Ic[dB] = 10 log(1/γ). The magnitude of EB is found using the Jones matrix formalism [14]. The polarization plate rotator by angle θr and plate λ/2 are described, respectively, by matrices  cos  r R ( r )    sin  r

 sin  r  , cos  r 

 cos 2 pl L( pl )    sin 2 pl

sin 2 pl  ,  cos 2 pl 

(4)

where θpl 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]:

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      cot  i lin cos 2  c  i lin sin 2      2    F  Ф  ,  lin ,    sin  ,  lin   lin 2 2  c     i sin 2  cot  i cos 2     2    

(5)

 2   lin2   c2 ,

(6)

c

where

δс, δlin 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 Ф = δс/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 δс = 0. Consider two most frequently used cubic crystal orientations [001] and [111]. In the case [001] expressions for δlin and Ψ are written in the form 1   2 tan 2  2  2  , 1  tan 2  2  2 

(7)

tan  2  2    tan  2  2  ,

(8)

 lin  ph

where 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

h

 r / rh 

2

 exp   r 2 / rh2   1

 r / rh 

2

n 1  Q  T  p11  p12  , 4 1  3 0

,

QPh



,

(9)



2 p44 , p11  p12

(10)

Ph  (1  exp   0 L ) Pin   0 LPin ,

where Pin is the total power of heating radiation, λ is radiation wavelength, κ is heat conductivity, r is polar radius; pij are the elements of photoelasticity tensor in the two-index Nye form [16]; αT is thermal expansion coefficient; n0, and α0 are the index of refraction and the absorption coefficient at the wavelength 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>>r0, or the heat sink from the ends is infinitesimal. As was shown in [17], the expressions for δlin and Ψ, hence, all the formulas following from them for the [111] orientation may be obtained from Eqs. (7) and (8) by substituting 1  2 , (11) 3 Knowing the Jones matrix for all optical elements one can readily find field ЕВ for any scheme presented in Fig. 1. For the FI schemes shown in Fig. 1, we will obtain

  1,

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p p



EB 0  L  2 pl   / 4   F Ф   / 4,  lin1 , 1   E A , EBin  L  2 pl   / 4   r   F Ф   / 8,  lin 2 ,  2   R  r   F Ф   / 8,  lin1 , 1   E A , (12) EBout  L  2 pl   / 4   r   F Ф  0, lin 2 ,  2   R  r   F Ф   / 4,  lin1 , 1   E A .

By substituting Eqs. (7) and (8) into Eq. (5) and the result together with Eq. (4) into Eq. (12) and then into Eq. (2) and Eq. (3) we will obtain, respectively, local Г and integral γ FI thermal depolarization for the schemes presented in Fig. 1. 2.2. The case of small birefringence Consider the case when thermally induced linear birefringence is small, i.e.,

 lin  1.

(13) We assume that MOE and AOE are single crystals with [001] orientation and the condition θ1 = θ2 is fulfilled. Then, by substituting Eq. (12) into Eq. (2) and expanding δlin into a series, for the schemes in Fig. 1 we will obtain expressions for local depolarization:

Г0  Г in 

4 lin2 l

2

Г out 

2sin2  2   / 4 

2

2  2  sin 2 



 lin2  O  lin4  ,

(14) 2

r



3   3   2 G  cos  2    O  lin4 , 8   8 

 

2  lin2 1    4 cos 2   2  G  cos 2   sin 2        r   O  lin  ,  2  2 

(15)

(16)

where

G

 lin 2 D  lin1

1   1  

2 2

2 1

tan 2  2  2   tan 2  2  2  

,

(17)

(18) D  p2 / p1. When MOE and AOE are made of the same material (ξ1 = ξ2, Q1 = Q2, κ1 = κ2) and have identical orientation (θ1 = θ2), from Eq. (9) and Eq. (17) one can readily obtain G = D = L2/L1. From Eqs. (15) and (16) it follows that for opt rin  rin  3 / 8   m,

Gin  Ginopt  1,

 r out   ropt out  3 / 8   m,

opt Gout  Gout  8 /.

(19) (20)

Гin and Гout are proportional to  lin4 , whereas Г0 is always proportional to  lin2 . The substitution of  ropt and Gopt into Eqs. (15) and (16) yields Г inopt

Г

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  2 2  4 4

opt out



  2  4 4

2

2

 lin4  O  lin6  ,

 lin4  O  lin6  .

(21)

(22)



Equation (21) fully agrees with the results obtained in the work [10]. By substituting Eqs. (7) and (8) into Eqs. (14), (21) and (22) and then into Eq. (3) we obtain expressions for the integral depolarization γ:

A    0       p2 12  0.014 p2 , 8    in  ropt ,G opt   p 4

A2 3

 4 32

 out ropt , Gopt   p4

  2 2   2



4

(23)

2 2      2  1  0.4  10 5  4   2  1 p 4 , (24) 3 3   

A2 3 2 2 2   2   4   2  1  5.3 105  4   2  1 p4 , (25) 4  32 3 3    

where   y  exp   y   1  dy A1     0.137,  y  exp  y  0 

(26)

  y  exp y   1 dy A2     exp y   0.042. y  0 

(27)

2

4

In this order of smallness the integral depolarizations γin and γout do not depend on θ1 = θ2 = θ, i.e., on crystal orientation relative to the polarization of the radiation incident on the FI; whereas γ0 depends on this quantity and at  0opt   / 8 takes on the minimum value Eq. (23). Equations (23) and (25) p is the normalized power of heat generated in MOE 3 having length L [Fig. 1(а) and 1(c)], and in Eq. (24) p is the normalized power of heat generated in two elements 5 [Fig. 1(b)], each having length L/2; p is found from Eq. (9). As was said above the expressions for the [111] orientation are obtained from Eqs. (23)– (25) by the substitution of Eq. (11). 2.3. Comparison of schemes with internal and external compensation The curves for integral depolarization 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). The proposed scheme of external compensation of thermally induced birefringence presented in Fig. 1(c) has advantages over the schemes with internal compensation. Firstly, it allows modernizing (by adding two optical elements) traditional FIs, thus enhancing Pinmax to the values below which the isolation degree is less than the preset one. Secondly, AOE may be made of material other than MOE. By choosing a material with the value of 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 ξ5 the integral depolarization does not depend on relative position of the crystals in schemes with compensation, whereas at p