## 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

## 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 path-length 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.

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. 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

where**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

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 *I _{c}*[dB] =

*10 log(1/γ)*.

The magnitude of **E**
_{B} is found using the Jones matrix formalism [14]. The polarization plate rotator by angle *θ _{r}* and plate

*λ*/2 are described, respectively, by matrices

*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}*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}Consider two most frequently used cubic crystal orientations [001] and [111]. In the case [001] expressions for *δ _{lin}* and Ψ are written in the form

*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}As was shown in [17], the expressions for *δ _{lin}* and Ψ, hence, all the formulas following from them for the [111] orientation may be obtained from Eqs. (7) and (8) by substituting

Knowing the Jones matrix for all optical elements one can readily find field **Е**
_{В} for any scheme presented in Fig. 1. For the FI schemes shown in Fig. 1, we will obtain

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.,

We assume that MOE and AOE are single crystals with [001] orientation and the condition *θ _{1} = θ_{2}* is fulfilled. Then, by substituting Eq. (12) into Eq. (2) and expanding

*δ*into a series, for the schemes in Fig. 1 we will obtain expressions for local depolarization:

_{lin}When MOE and AOE are made of the same material (*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 ${\delta}_{lin}^{4}$, whereas

_{out}*Г*is always proportional to ${\delta}_{lin}^{2}$. The substitution of ${\theta}_{r}^{opt}$ and ${G}_{opt}$ into Eqs. (15) and (16) yields

_{0}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:*

*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 ${\theta}_{0}^{opt}=-\pi /8$ takes on the minimum value Eq. (23).

_{0}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 ${P}_{in}^{\mathrm{max}}$ 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 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

Analytical expressions for integral depolarization cannot be obtained in the case of strong birefringence, when Eq. (13) is not fulfilled, or in the case of *θ _{1}≠θ_{2}*. Therefore, we carried out numerical computations.

At fixed radiation power, the integral depolarization *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.

Optimal parameters for both compensation schemes [Figs. 1(b) and 1(c)] and two most frequently used orientations [001] and [111] of TGG crystals have been found numerically for a wide power range. The corresponding dependences are plotted in Fig. 2.

A characteristic value of maximum admissible integral depolarization for FI is 0.001. It corresponds to the maximum admissible normalized heat generation power *p _{max}* which, in turn, may be recalculated to the maximum admissible laser power

*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] we obtain

*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.

From the plots in Fig. 2 it follows that the scheme with external compensation [Fig. 1(c)] with optimum parameters enables enhancing *p _{max}* 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

*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}Our calculations demonstrated that, when both the crystals have [111] orientation, optimal parameters weakly (<1%) depend on the power of incident radiation. For the scheme with internal compensation, optimal parameters take on the value *θ _{r}* ≈67.5°,

*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.

For the [001] orientation, addition of two independent parameters *θ _{1}* and

*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%.

Let us consider how much integral depolarization is sensitive to parameter variation. The integral depolarization as a function of *θ _{r}* and

*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).

One can see in Figs. 3(c) and 3(d) that the minimum of *γ* depends to a greater extent on the difference of angles *θ _{1}* and

*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

For verification of the obtained results we staged an experiment shown schematically in Fig. 4 . A 300 W fiber laser operating at the wavelength of 1076 nm was used as a source of CW linearly polarized radiation. The intensity distribution in the beam cross-section had a Gaussian profile. Calcite wedge 1 ensured linear polarization. Wedges 6 of fused quartz were used to attenuate radiation. Glan prism 7 was adjusted to a minimum transmitted signal whose intensity distribution was measured by CCD camera 8.

A [001] oriented TGG crystal 20 mm in diameter and 18 mm long, rotating the polarization plane of incident radiation by 45° was used as MOE 3. Crystal quartz with *θ _{r}* = 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

^{−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*.

The dependence of *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
.

The theoretical curves were plotted for the parameters *θ _{r}* = 67.5°,

*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}Using the expressions obtained in [7], we can estimate the contribution of the temperature dependence of Verdet constant in the integral depolarization for external compensation scheme. For our TGG crystal and parameters of laser radiation the contribution of effect is *γ _{v}* = 10

^{−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}The dependence of *γ* on *θ _{2}* was obtained in experiment for fixed

*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}Figure 6 demonstrates a rather good agreement between the experimental data and the numerical computations, with the rms being 7% or less.

Note that the compensation effect persists, even if the radiation is incident on an AOE not normally to its surface. This was confirmed in the experiment on the compensation of thermally induced birefringence with AOE crystal turn relative to the *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

A new scheme of thermally induced depolarization compensation in Faraday isolators is proposed [Fig. 1(c)]. The scheme is based on using an additional compensating element outside magnetic field. In contrast to the currently used scheme depicted in Fig. 1(b), the proposed scheme allows compensating depolarization, leaving the magnetic system and the magnetooptical element of the Faraday rotator unchanged. Results of the computations were verified in experiments; for the laser radiation power of 300 W, the isolation degree was enhanced from 20 dB to 35 dB.

It was shown that for both compensation schemes [(Figs. 1(b) and 1(c)] with [001] crystal orientation depolarization compensation may be improved by orienting crystallographic axes not parallel to each other. In this case, given optimal parameters and using available crystals, it is possible to create a Faraday isolator having isolation degree higher than 30 dB for laser radiation power up to 1.6 kW.

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