Self-compensation of thermally induced ... - OSA Publishing

Self-compensation of thermally induced depolarization in CaF2 and definite cubic single crystals Anton G. Vyatkin,* Ilya L. Snetkov, Oleg V. Palashov, and Efim A. Khazanov Institute of Applied Physics of the Russian Academy of Sciences, 46 Ulyanova street, Nizhny Novgorod 603950, Russia * [email protected]

Abstract: Compensation of thermally induced depolarization in laser active elements at small birefringence without additional phase elements was proposed and observed experimentally. Requirements to the crystals were formulated. An order of magnitude reduction of depolarization degree was obtained experimentally. A further modification of the scheme was developed. ©2013 Optical Society of America OCIS codes: (140.6810) Thermal effects; (140.3580) Lasers, solid-state.

References and links 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19.

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#191777 - $15.00 USD Received 6 Jun 2013; accepted 26 Jul 2013; published 16 Sep 2013 (C) 2013 OSA 23 September 2013 | Vol. 21, No. 19 | DOI:10.1364/OE.21.022338 | OPTICS EXPRESS 22338

20. N. F. Andreev, N. G. Bondarenko, I. V. Eremina, S. V. Kuznetsov, O. V. Palashov, G. A. Pasmanik, and E. A. Khazanov, “Single-mode YAG:Nd laser with a stimulated Brillouin scattering mirror and conversion of radiation to the second and fourth harmonics,” Soviet J. Quantum Electron. 21(10), 1045–1051 (1991). 21. E. A. Khazanov, “A new Faraday rotator for high average power lasers,” Quantum Electron. 31(4), 351–356 (2001). 22. W. A. Clarkson, N. S. Felgate, and D. C. Hanna, “Simple method for reducing the depolarization loss resulting from thermally induced birefringence in solid-state lasers,” Opt. Lett. 24(12), 820–822 (1999). 23. E. Khazanov, A. Poteomkin, and E. Katin, “Compensating for birefringence in active elements of solid-state lasers: novel method,” J. Opt. Soc. Am. B 19(4), 667–671 (2002). 24. E. Khazanov, “Use of parallel axicon for compensation of birefringence in active elements of solid-state lasers,” Proc. SPIE 4632, 155–163 (2002) (Laser and Beam Control Technologies, ed. S. Basu and J. F. Riker). 25. R. C. Jones, “A new calculus for the treatment of optical systems,” J. Opt. Soc. Am. 31(7), 488–503 (1941). 26. S. Timoshenko and J. N. Goodier, Theory of Elasticity (McGraw-Hill, 1951). 27. F. W. Quelle, Jr., “Thermal distortion of diffraction-limited optical elements,” Appl. Opt. 5(4), 633–637 (1966). 28. M. J. Weber, Handbook of optical materials (CRC Press, 2003). 29. K. Veerabhadra Rao and T. S. Narasimhamurty, “Photoelastic constants of CaF2 and BaF2,” J. Phys. Chem. Solids 31, 876–878 (1969). 30. R. W. Dixon, “Photoelastic properties of selected materials and their relevance for applications to acoustic light modulators and scanners,” Appl. Phys. (Berl.) 38, 5149–5153 (1967). 31. R. E. Joiner, J. Marburger, and W. H. Steier, “Elimination of stress-induced birefringence effects in singlecrystal high-power laser windows,” Appl. Phys. Lett. 30(9), 485–486 (1977). 32. W. Martienssen and H. Warlimont, Handbook of Condensed Matter and Materials Data (Springer, 2006). 33. 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). 34. 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(22), 21977–21988 (2011).

1. Introduction Thermally induced depolarization is one of the parasitic thermal effects limiting the output power of solid-state lasers [1]. Its main sources are laser active elements (AEs). Heat dissipation in AE bulk produces thermal gradients which give rise to elastic stresses due to inhomogeneous thermal expansion. The photoelastic effect leads to variation of dielectric permeability proportionally to the stress tensor and makes all materials biaxial regardless of their initial symmetry, with the eigen polarizations varying with coordinates [2]. In cubic crystals the induced birefringence results in phase distortions and change of polarization. In this study the depolarized radiation is understood as the radiation whose polarization is constant in time but varies in the beam cross-section. Let Eout be the electric field at the output of the optical system and E⊥ out be its component polarized orthogonally to the radiation at the same point at zero heat generation power. Then the local Γ and the integral γ depolarization degrees are defined as Γ=

E⊥ out Eout

2 2

,

 ΓE γ =  E S

S

2 out

out

2

dS

dS

,

(1)

where the integration is performed over the beam cross-section. Thermal depolarization in cubic crystals of symmetry groups 432, 43m and m3m (e. g., for garnets, fluorides, etc.) was investigated for the [111] [3–7] and [001] [8,9] orientations back in the 1970s. Later, specificity of the [011] orientation was noticed ([10‒12]). The depolarization at arbitrary crystal orientation was studied in detail in [11,13‒15] using the method proposed in [16]. The depolarization in cubic crystals of symmetry groups 23 and m3 (the latter includes the sesquioxides Sc2O3, Lu2O3, Y2O3, etc.) was investigated recently [17]. Those works showed that both, the profile and the power of the depolarized beam depend on crystal orientation, crystal rotation around the wavevector relative to the incident


polarization, photoelastic coefficients, power of heat release, as well as on heating radius and profile. Several schemes for linear compensation of depolarization have been proposed to date. Thermal depolarization in AEs can be fully eliminated using two identical elements with a 90° rotator between them [18] or a Faraday mirror [19] and advanced schemes [20,21]. There are also a number of schemes that allow compensation of depolarization only at weak birefringence: a λ/4 plate with the axis parallel to the initial polarization and a mirror [22], an equivalent scheme that requires two identical AEs with a λ/2 plate between them, and others [23,24]. All these schemes require application of additional phase elements. In Section 2 of the current paper we propose a scheme for compensating depolarization in the AEs at weak birefringence that does not contain any phase elements apart from AEs and demonstrate the requirements to the crystal anisotropy parameters and orientation for this method. The experimental results on reduction of depolarization degree are presented in Section 3. In Section 4 we give a visual criterion for the possibility of depolarization compensation in the scheme suggested. A possibility of a further decrease of the depolarization degree in some of the schemes that normally compensate only its quadratic term of the power series is considered in Section 5. 2. Compensation of thermally induced depolarization at weak birefringence in the scheme with counterrotation 2.1. Thermally induced depolarization in a single crystal The photoelastic effect results in dielectric impermeability tensor increment ΔB = πσ , where π is the piezooptic tensor and σ is the stress tensor. The additive to the dielectric tensor ε = B−1 is Δε = −ε 0 2 ΔB , where ε 0 is the “cold” dielectric constant. It is convenient to describe the change of the beam polarization in the transmission through birefringent elements by the Jones matrix formalism [25]. We introduce the laboratory frame of reference with Cartesian ( x, y, z ) and cylindrical ( r , ϕ , z ) coordinates in which z is the axis of the cylindrical AE. We suppose that the stress tensor within the AE does not depend on z. This condition is fulfilled, for example, for radially cooled long rod and thin disk [26,27]. In this case, for the media without circular birefringence the AE’s Jones matrix has the form J i = sin

δ i  cot (δ i 2 ) + j cos 2Ψ i  2

j sin 2Ψ i

j sin 2Ψ i  , cot (δ i 2 ) − j cos 2Ψ i 

(2)

where Ψi is the angle of inclination of eigenpolarizations, δi is the difference of their phase incursions, i numbers AEs, and j is the imaginary unit. The Jones matrix of the system of several optical elements is equal to the product of the Jones matrices of separate elements taken in the reverse order: J Σ = J1,2,,( N −1), N = J N J N −1  J 2 J1 .

(3)

We assume that the incident radiation has vertical or horizontal polarization. The local depolarization degree Γ of the transmitted beam is equal in this case to the squared nondiagonal element of the Jones matrix: 2

2

Γ = J Σ 12 = J Σ 21 .

(4)

The depolarization degree at weak birefringence ( δ i  1 ) will be designated by Γsmall. In a general case, it is defined by the quadratic term in the Taylor power series Γ(2):


Γsmall ≈ Γ (2) =

1 4

(

)

N

2

δ sin 2Ψ i . i =1 i

(5)

Consider the i-th AE made of a cubic crystal cut in the direction specified relative to the crystallographic axes ( a, b, c ) by the first and second Euler angles (α i , β i ) [13] (the third Euler angle will be introduced in the next subsection). Let us introduce a reference frame related to the element and its cylindrical ( r0 , ϕ0 , z ) and Cartesian ( x1 , x2 , z ) coordinates. In

this frame of reference, by diagonalizing ΔB in the (x1, x2) plane, one can obtain tan 2Ψ 0i = Di Ci , δ i = −δ 0 i Di

( sin 2Ψ 0i ) ,

δ 0 i = 12 k0 Li n0 i 3 ,

(6)

where Li is the length of the AE, n0 i = ε0 i1/2 is its “cold” refractive index, k0 is the wavenumber in free space, and Ci = ΔB11 − ΔB22 = A1 (ξ , ξ d ; α i , βi ) Σ ( r ) + +  A2 (ξ ; α i , β i ) cos 2ϕ0 + A3 (ξ , ξ d ; α i , β i ) sin 2ϕ0  Δ ( r ) ,

Di = 2ΔB12 = B1 (ξ , ξ d ; α i , β i ) Σ ( r ) + +  B2 (ξ , ξ d ; α i , β i ) cos 2ϕ0 + B3 (ξ ; α i , βi ) sin 2ϕ0  Δ ( r ) ,

(7)

with Σ ( r ) = π S  12 (σ rr + σ ϕϕ ) − σ zz  ,

Δ ( r ) = π S (σ rr − σ ϕϕ ) ,

π S = π aaaa − 12 (π aabb + π bbaa ) .

(8)

The coefficients Am, Bm in Eq. (7) are written in the form A1 (α , β ) = (1 − ξ ) a1 (α , β ) + ξ d a2 (α , β ) ,

A2 (α , β ) = ξ + (1 − ξ ) a3 (α , β ) ,

A3 (α , β ) = (1 − ξ ) c1 (α , β ) − ξ d c2 (α , β ) ,

B1 (α , β ) = (1 − ξ ) b1 (α , β ) + ξ d b2 (α , β ) ,

B2 (α , β ) = (1 − ξ ) c1 (α , β ) + ξ d c2 (α , β ) , B3 (α , β ) = ξ + (1 − ξ ) b3 (α , β ) ,

(9)

where

ξ = 2π abab π S , ξ d = (π aabb − π bbaa ) π S

(10)

are the first and the second parameters of photoelastic anisotropy [17]. The subscript “i” in material constants ξ, ξd, πS, and π is omitted for brevity, although they also generally differ for different optical elements. The coefficients a1 = sin 2 2α (1 − cos 4 β ) − sin 2 2 β  2, a2 = cos 2α cos 2β ,

b1 = 14 sin β sin 2β sin 4α , b2 = sin 2α cos β (1 − 3cos 2 β ) 2,

a3 = 1 − ( sin 2 2α ) 4  sin 4 β + cos 2 2α cos 2 β , b3 = sin 2 2α cos 2 β , c1 = − sin 4α cos β (1 + cos 2 β ) 4,

c2 = 34 sin 2α cos β si n 2 β

(11)

define the dependence of tensor ΔB on crystal orientation. In crystals of the symmetry groups 432, 43m and m3m, π aabb = π bbaa and the expressions are simpler: ξ d = 0 , π S = π aaaa − π aabb .


The stress tensor for a long rod and a thin disk can be found, for instance, in [26,27]. We will not present these expressions here and only note that in the linear approximation stresses in the AE are proportional to dimensionless heat release power p defined in the disk and rod, respectively, by pdisk = α T EPΣ k0 n03π S ( 8πκ ) ,

prod = pdisk (1 −ν ) ,

(12)

where PΣ is the power of heat release in the whole AE, αT is the linear coefficient of thermal expansion, κ is thermal conductivity, E is Young’s modulus, and ν is Poisson’s ratio. From Eqs. (5)–(8) it follows that Ψ0 does not depend on p, δ ∝ p, and Γ(2) ∝ p2.

Fig. 1. Initial polarization (a) and calculated depolarized beam profiles for CaF2 (ξ = −0.47) with [001] orientation (α = β = 0) for χ = 0 (b) and χ = 22.5° (c), and with [011] orientation (α = 0, β = π/4) for χ = 6° (d) and χ = 18° (e).

The calculations [11,17] show that, on transmission of radiation through a single optical element the depolarized beam can have a profile of a Maltese cross [Figs. 1(b) and 1(c)] or a more complicated one [Figs. 1(d) and 1(e)]. 2.2. Scheme with counterrotation Let the optical elements be turned around their z-axes by angles χi (the third Euler angles are equal to −χi). Тhen

ϕ = ϕ0 + χ i , Ψ i = Ψ 0i + χ i ,

(13)

and Eq. (5) takes on the form 2

N Γ (2) = 14   i =1 δ 0 iVi  , Vi = Di cos 2 χ i + Ci sin 2 χ i . (14)   Let two cylindrically shaped coaxial AEs be made of the same cubic crystal, have the same length and orientations defined by the Euler angles (α1, β1) = (α, β) and (α2, β2) = (π + α, π−β), respectively, as if the crystal [Fig. 2(a)] were cut into two halves one of which were rotated back-to-front. We will rotate both crystals around their common geometrical axis z simultaneously in the opposite directions by the same angle χ. In this case χ1 = χ, χ2 = −χ [Fig. 2(b)]. We will refer to this configuration as to the scheme with counterrotation. Then, Eq. (14) takes on the form

Γ (2) =

δ 02 4

(V1 + V2 )

2

{

,

}

V1 + V2 = Δ ( r ) sin 2ϕ 2ξ + (1 − ξ ) ( a3 + b3 ) + ( b3 − a3 ) cos 4 χ + 2c1 sin 4 χ  . (15)

This expression has the following characteristic features. First, the transverse profile of the depolarized beam is always a Maltese cross oriented along the field polarization [Fig. 1(b)] and does not depend on ξ and crystal orientation. Second, Eq. (15) has the same form for all cubic crystals as it does not contain the second parameter of photoelastic anisotropy ξd. Third, it can be shown that, if the condition


2

 ξ   ξ  2   −  ( a3 + b3 ) + ( a3b3 − c1 ) ≤ 0 1 1 ξ ξ − −    

(16)

is fulfilled, there exists an angle χ, i.e. a crystal pair position such that Γ(2) = 0. Zeroing of the quadratic term Γ(2) means that Γsmall reduces appreciably and, in a general case, becomes proportional to p4.

Fig. 2. Single element with double length (a) and schemes for compensation of depolarization with counterrotation (b) and with λ/2 plates (c). The rightmost λ/2 plate in (c) is presented for simplicity of the further statement.

2.3. Conditions of compensating thermally induced depolarization in the scheme with counterrotation We will now investigate the cases when compensation takes place. In the [0MN] orientations (α = 0), Γ(2) may turn to zero only at

ξ  4 + (ξ − 1) sin 2 2 β  ≤ 0.

(17)

It can be readily shown that at positive ξ Eq. (17) has no solutions and compensation is impossible. Although ξ > 0 for many popular cubic crystals (YAG, TGG, GGG, etc.), some crystals with negative ξ are used in optics: CaF2, SrF2, and KCl [28]; BaF2 [29], YIG [30], etc. The region of the parameters (β, ξ) at which compensation occurs is shown in Fig. 3(a). For example, in the [001] orientation (α = β = 0) the compensation is possible for any negative ξ, whereas in the [011] orientation (α = 0, β = π/4; following the works [11–15,17], we do not distinguish physically equivalent orientations) the compensation takes place only if −3 < ξ < 0.


Fig. 3. Set of [0MN] [α = 0; (a)] and [MMN] [α = π/4; (b)] orientations of the crystals with depolarization compensation in the scheme with counterrotation (grey).

Fig. 4. Domain of existence of compensation in the scheme with counterrotation (all shades of blue for any ξd) and of profile rotation of the depolarized beam for ξd = D(ξ‒1): all shades of blue for D = 0, medium blue and dark blue for D = 1, dark blue for D = 2 on a cube in crystallographic axes (a,b,c) at different ξ (Media 1). a) ξ = −0.001, b) ξ = −0.1, c) ξ = −0.3, d) ξ = −0.47 (CaF2), e) ξ = −0.8, f) ξ = −2, g) ξ = −3, h) ξ = −5, i) ξ = −30. Markers denote simplest orientations: [001] (filled squares), [011] (filled circles), [111] (filled triangles), as well as [[C]] orientation for D = 0 (open diamonds with dots), D = 1 (open circles with dots), and D = 2 (crosses). [[C]] orientation is close to [011] and therefore is not shown in (a) and does not exist for D = 1 and D = 2 in (h) and (i).

In the [MMN] (α = π/4) orientations, compensation occurs at 1 − (1 − ξ ) sin 2 β  ξ + 34 (1 − ξ ) sin 4 β  ≤ 0

(18)

[see Fig. 3(b)]. As before, ξ < 0 is the necessary condition for depolarization compensation. One can see that compensation cannot take place in the [111] (β ≈0.304π) orientation. The #191777 - $15.00 USD Received 6 Jun 2013; accepted 26 Jul 2013; published 16 Sep 2013 (C) 2013 OSA 23 September 2013 | Vol. 21, No. 19 | DOI:10.1364/OE.21.022338 | OPTICS EXPRESS 22344

degenerate point ξ = −½ is characterized by the absence of birefringence (and therefore depolarization) in this orientation [15]. The full set of orientations in which compensation occurs is demonstrated in Fig. 4. For positive ξ, no compensation can occur in any orientation. At small negative ξ compensation takes place in the orientations close to [0MN] [Fig. 4(a)]. At ξ ≈ −½, compensation is possible in all orientations, except the ones close to [111] [Fig. 4(d)]. At large negative ξ, compensation can take place only close to [001] [Fig. 4(i)]. 2.4. Numerical computation of depolarization degree in the scheme with counterrotation The integral depolarization degree γ and its quadratic term γ(2) are plotted in Fig. 5 as a function of angle χ at p = 5 in CaF2 (ξ = −0.47 by the data of [15]) with [001] orientation for a single AE of double length [Fig. 2(a)] and in the scheme with counterrotation [Fig. 2(b)]. It is clear from the figure that the depolarization may be reduced substantially in the scheme with counterrotation as compared to the minimal depolarization in a single element.

Fig. 5. Integral depolarization degree γ (solid curves) and its quadratic term γ(2) (dotted curves) as a function of angle χ at p = 5 in CaF2 (ξ = −0.47) with [001] orientation for a single element of double length (black curves) and in the scheme with counterrotation (red curves).

Note that usually in crystals where compensation is possible in the scheme with counterrotation (i.e., ξ < 0), there exists the [[C]] orientation (Fig. 4) in which depolarization may be fully eliminated at plane-stress and plane-strain [15,17,31], i.e. in the approximations adopted in our computations. Exceptions are crystals of symmetry groups 23 and m3 with ξ < −3 and rather high ξd in which there is no [[C]] orientation [17] [Figs. 4(h) and 4(i)]. Despite the presence of the [[C]] orientation, the proposed compensation scheme may prove to be preferable in a number of cases. For example, it may be used in AEs that have already been made, as well as when crystal orientation has been specified for some other reason, or several AEs are made from one boule with unknown orientation. 2.5. Comparison of the quality of depolarization compensation in the schemes with counterrotation and with λ/2 plates The integral depolarization degree as a function of p is plotted in Fig. 6 for a) BaF2 (ξ = −0.192; π was obtained by the formula pc−1 from the elastooptic tensor p [32] and the elastic stiffness tensor c [28]), b) SrF2 (ξ = −0.252; obtained in the same way by data from [28]), c) CaF2 (ξ = −0.47 [15]), d) KCl (ξ = −1.2 [28]) with prevalent orientations [001] and [111] for a single double-length AE, in the scheme with λ/2 plates and in the scheme with counterrotation (see Fig. 2). At p  1 in crystals with −2 < ξ < − 1 5 the depolarization degree without compensation is less in the [111] orientation than in the [001] orientation; in the case of CaF2


it is much less because of closeness of [111] to the [[C]] orientation [15]. In the [111] orientation the depolarization compensation in the scheme with counterrotation is impossible, and in the [001] orientation both the schemes in this crystal are greatly inferior to the scheme with λ/2 plates in the [111] orientation and even to a single element of this orientation.

Fig. 6. Integral depolarization degree γ as a function of dimensionless heat power in a) BaF2, b) SrF2, c) CaF2, d) KCl with [001] orientation (solid curves) and [111] orientation (dashed curves) for a single element of double length [Fig. 2(a), black curves] and in schemes with λ/2 plates [Fig. 2(b), red curves] and with counterrotation [Fig. 2(c), blue curves]. The horizontal line is a guide for the eye indicating a typical value of “cold” depolarization degree.

By substituting Eqs. (6)–(13) into the exact expression for Γ [Eqs. (2)–(4)] rather than into Eq. (5) for Γ(2) one can show that there exists a specific value ξ = −1 at which the scheme with counterrotation in the [001] orientation allows compensating depolarization not only at weak birefringence. Numerical computations demonstrate that at ξ, close but not equal to −1, the scheme with counterrotation provides appreciable reduction of depolarization at sufficiently large values of p ~20–40 also [see Fig. 6(d)]. For example, at p = 30 the depolarization degree reduces more than twice relative to the average value at p  1 for a single element at −1.3 < ξ < −0.75 and for the scheme with λ/2 plates at −1.27 < ξ < −0.77. Note that at p  1 in crystals of the simplest orientations the depolarization degree in the scheme with λ/2 plates is, as a rule, less than in the scheme with counterrotation. The numerical computations showed that the depolarization degree in the scheme with counterrotation in crystals with [001] orientation is less than in the scheme with λ/2 plates in crystals of the same orientation for ξ in a rather narrow range (−1.75, −0.58) and less than in the scheme with λ/2 plates in crystals with [111] orientation for ξ in a still narrower range (−1.085, −0.945). However, the advantage of the scheme presented in Fig. 2(b) is absence of plates, which simplifies the scheme and may be significant, for instance in systems with high average and peak power, for reducing the total B-integral. In these cases, however, the distortions introduced by the photoelastic effect and cubic nonlinearity should be considered consistently [33,34]. Besides, the scheme with counterrotation may be implemented in the form of a single AE comprising two halves in optical contact. It is also worth noting that for p  1 the integral depolarization degree γ in the scheme with counterrotation is worse than for a single element. In this power range γ depends only on the angle of inclination of its eigen polarizations Ψ in the cross-section of the active element #191777 - $15.00 USD Received 6 Jun 2013; accepted 26 Jul 2013; published 16 Sep 2013 (C) 2013 OSA 23 September 2013 | Vol. 21, No. 19 | DOI:10.1364/OE.21.022338 | OPTICS EXPRESS 22346

and does not depend on the phase difference δ [11]. Thus, the minimum depolarization degree is attained at the maximum uniform distribution of Ψ over the cross-section and by fitting incident polarization. However, in the scheme with counterrotation at an arbitrary point of the cross-section these angles are different in elements “1” and “2”, which increases depolarization degree. 3. Experiment on compensation of thermally induced depolarization in the scheme with counterrotation

To test the proposed compensation method a model experiment was carried out (see the schematic in Fig. 7). A 300 W cw Yb-fiber laser with a wavelength of 1076 nm was used as a source of linearly polarized laser radiation. The intensity distribution in the beam crosssection had a Gaussian profile 3 mm in diameter. The laser radiation was used for both, heating and probing. Calcite wedge 1 provided linearity of the input polarization. Wedges 5 made of fused quartz were used for radiation attenuation. Glan prism 6 was adjusted to minimum power of the transmitted signal whose intensity distribution was measured by CCD camera 7.

Fig. 7. Schematic of the experiment: 1 – calcite wedge, 2 – beam absorber, 3, 4 – two halves of investigated sample, 5 – quartz wedge, 6 – Glan prism, 7 – CCD camera.

The studied samples 3 and 4 were two elements obtained by cutting the sample CaF2 single crystal of arbitrary unknown orientation of the crystallographic axes into two halves, preserving their orientation relative to each other. In this way we could model the behavior of a crystal of double length and implement the scheme with counterrotation as well.

Fig. 8. a) Integral depolarization degree in CaF2 versus angle χ for the transmitted radiation power of 70 W for a double-length crystal – black circles, and in the scheme with counterrotation – red squares: experiment (symbols) and theory for [001] orientation (curves). b) Experimental plots for minimal integral depolarization degree versus transmitted radiation power (symbols) and their approximation by power functions (lines) in the same cases.

For the laser radiation power of 70 W we measured the integral depolarization degree as a function of the angle of turn of crystals 3 (χ1) and 4 (χ2) relative to the radiation propagation, both in one direction [Fig. 2(a)] and in opposite directions in the scheme with counterrotation [Fig. 2(b)]. The results are presented in Fig. 8(a) along with the calculated dependences for crystals with [001] orientation and the heat power fitting the experiment. The shapes of the


experimental and theoretical plots are in a qualitative agreement, despite probable noncoincidence of crystal orientations. The spread of experimental data is caused primarily by the poor quality of the used CaF2 crystals and insufficient accuracy of crystal alignment relative to each other during rotation. Note that, if the crystals have no marks specifying the origin of angles χi, for implementation of the scheme with counterrotation they should be rotated asynchronously until the depolarized beam acquires the shape of an upright Maltese cross [see Fig. 1(b)]. We also measured the integral depolarization degree γ as a function of laser radiation power for two cases: minimal γ for a double-length crystal and minimal integral depolarization degree in the scheme with counterrotation [Fig. 8(b)]. The use of the scheme with counterrotation permitted reducing γ in CaF2 samples by more than an order of magnitude. In this case, the dependence of the integral depolarization degree on the laser radiation power changed from quadratic to the one proportional to the power of four. Note that, when angle χ is close to its optimal value, Γsmall is determined not only by the quadratic term Γ(2) but by higher order terms too. In this case, the depolarized beam no longer repeats the shape of Γ(2). It is clear from Fig. 9 that in the scheme with counterrotation, as angle χ is approaching the optimal value the transverse distribution of the depolarized field component changes: the upright Maltese cross [Fig. 9(d)] turns by 45 degrees [Fig. 9(f)]. This effect was observed in our experiments and is illustrated in Figs. 9(a)–9(c).

Fig. 9. Experimental [(a)–(c)] and calculated at p = 1 [(d)–(i), Media 2] depolarized beam profiles in the scheme with counterrotation for CaF2 (ξ = −0.47), a) χ = 30°, b) χ = 12°, c) χ = 17° (the angle is close to optimum); d) χ = 13° (γ = 7.4·10−4), e) χ = 15° (γ = 2.6·10−4), f) χ = 17° (γ = 5·10−5); g)–i) same as (d)–(f) but both crystals are additionally tilted by 1° to one side. (a)–(c) are scaled to their maximum values. White dots in (d)–(i) are marks on the crystals.

4. Relationship between depolarization compensation in the scheme with counterrotation and the behavior of the profile of the depolarized beam during rotation of a single active element

It was shown in [15] that in the case of weak birefringence when a crystal with positive ξ and [001] orientation is rotating around its axis the Maltese cross oscillates (nonharmonically in a general case) with a period corresponding to the crystal turn by 90°. In crystals with negative ξ of the same orientation, the Maltese cross rotates nonuniformly, making a full revolution in #191777 - $15.00 USD Received 6 Jun 2013; accepted 26 Jul 2013; published 16 Sep 2013 (C) 2013 OSA 23 September 2013 | Vol. 21, No. 19 | DOI:10.1364/OE.21.022338 | OPTICS EXPRESS 22348

half a turn of the crystal. Let us generalize this result to the case of arbitrary crystal orientation.

Fig. 10. Depolarized beam profiles as a function of crystal angle of turn χ for a single element of double length [(a),(c),(e)] and in the scheme with counterrotation [(b),(d),(f)] for CaF2 (ξ = −0.47) in [001] [(a),(b), Media 3] and [011] [(c),(d), Media 4] orientations, and for a crystal with ξ = + 0.47 in [001] [(e),(f), Media 5] orientation. White dots are marks on the crystals.

The points ϕ of local maxima of the quadratic term of the local depolarization degree Γ(2) (and thus of the depolarized beam as the incident beam is considered to be axially symmetric) after passing one AE [Eq. (14) at N = 1] as a function of angle φ (i.e., the points of local extremums of V1) are defined by tan 2ϕ =

( A2 + B3 ) + ( B3 − A2 ) cos 4 χ + ( A3 + B2 ) sin 4 χ . ( B2 − A3 ) + ( A3 + B2 ) cos 4 χ + ( B3 − A2 ) sin 4 χ

(19)

Note that ϕ does not depend on r. The profile of the depolarized beam rotates if and only if ϕ is a monotonic function of χ. The monotony condition may be written in the form 2

2

 ξ   ξ   ξ  2 c2 2  d  +   −  ( a3 + b3 ) + ( a3b3 − c1 ) ≤ 0. − − − 1 1 1 ξ ξ ξ      

(20)

Comparison of Eqs. (20) and (16) shows that rotation of the profile of the depolarized beam as a single AE is rotating around its axis is the necessary and sufficient condition for depolarization compensation in the scheme with counterrotation for crystals with ξd = 0 (crystals of 432, 43m and m3m symmetry groups) and a sufficient condition for compensation at ξd ≠ 0 (crystals of 23 and m3 symmetry groups). The boundary orientations between oscillation and rotation of the depolarized beam are shown in Fig. 4 (in the [111] orientation the Maltese cross is motionless, which is a particular case of zero-amplitude oscillations). The continuous transition from oscillation to rotation is possible because in boundary orientations there exists a crystal position (angle χ) in which Γ(2) #191777 - $15.00 USD Received 6 Jun 2013; accepted 26 Jul 2013; published 16 Sep 2013 (C) 2013 OSA 23 September 2013 | Vol. 21, No. 19 | DOI:10.1364/OE.21.022338 | OPTICS EXPRESS 22349

is axially symmetric and angle ϕ , hence ∂ϕ ∂χ , are not defined. In the region of oscillation, the amplitude grows up to 45° when approaching the boundary and the range of angles χ corresponding to the motion of the depolarized beam profile in the direction opposite to that of crystal rotation reduces to zero. The depolarized beam profiles for a single double-length AE and in the scheme with counterrotation are shown in Fig. 10 for different χ. The profiles are plotted for CaF2 (ξ = −0.47) that allows depolarization compensation in a wide variety of orientations [see Fig. 4(d)] in the [001] and [011] orientations and for a hypothetic crystal with ξ = + 0.47, for which compensation is impossible, in the [001] orientation. The figure is a bright illustration of the fact that compensation is accompanied by rotation of the depolarized beam profile. 5. Compensation of higher-orders of depolarization degree with increasing number of identical elements

There is a certain analogy between the schemes of depolarization compensation with λ/2 plates and with counterrotation. They both compensate γ(2) only and can be treated as a sequence of two elements “1” and “2”, each having a unit Jones matrix in the absence of heating [Figs. 2(b) and 2(c)]. If element “1” in the scheme in Fig. 2(c) has Jones matrix J [1] = sin

δ  cot (δ 2 ) + j cos 2Ψ  2

j sin 2Ψ

 , cot (δ 2 ) − j cos 2Ψ  j sin 2Ψ

(21)

then the Jones matrix of element “2” is  − j sin 2Ψ  −1 0   −1 0  δ  cot (δ 2 ) + j cos 2Ψ J [ 2] =   . (22)  J [1]   = sin  − j sin 2Ψ cot (δ 2 ) − j cos 2Ψ  2  0 1  0 1

Fig. 11. Modifications of the depolarization compensation scheme with λ/2 plates. The following power orders are compensated: a) second, b) second and fourth, с) from second to sixth, d) from second to eighth (see also Fig. 12). Redundant plates are omitted.

By obtaining from Eq. (3) the Jones matrices of the schemes “1-2”, “1-2-2-1”, “1-2-2-1 2-1-1-2”, “1-2-2-1 - 2-1-1-2 - 2-1-1-2 - 1-2-2-1” [Figs. 11(a)–11(d), respectively], etc. from Eq. (4) one can show that with the use of 2N elements local Γsmall and integral γsmall depolarization degrees at p  1 are proportional to p2N+2. In practical applications, the scheme scalability will be restricted by nonrigorous identity of AEs and of distribution of heat release power in AEs, by beam diffraction, as well as by the presence of “cold” depolarization due to internal stresses in the medium, and by other factors. Simple analytical expressions for doubling the number of AEs in the scheme with counterrotation have not been obtained, but its similarity to the scheme with λ/2 plates hints at the improvement of compensation.


Fig. 12. The depolarization degree γ as a function of dimensionless heat power p in the cases of a single active element (black curves) and compensation schemes with λ/2 plates [Fig. 2(b), red curves] and with counterrotation [Fig. 2(c), blue curves]; a) CaF2 (ξ = −0.47), b) KCl (ξ = −1.2); crystal orientation is [001] in all cases. The number of active elements is indicated in circles. The horizontal line is a guide for the eye indicating a typical value of “cold” depolarization degree.

We calculated numerically the depolarization degree at the output of 4, 8, and 16 AEs for both schemes of depolarization compensation in a wide range of p (see Fig. 12; the total length of the AEs was the same in all cases). Precise calculation indicates that the compensation of higher orders of depolarization in the scheme with counterrotation is not full. For p → 0 , γsmall ∝ p6 for 8 and more AEs as well as for 4 AEs [Fig. 12(b)], though a sharper dependence γ(p) is also possible in the region above the threshold of cold depolarization interesting for applications [Fig. 12(a)]. Nevertheless, the decrease in γ is still substantial. Note that the compensation scheme with a 90-degree rotator [18] may also be represented as a sequence of elements “1” and “2” if we change the phase elements in Fig. 2(c) correspondingly. However, as in the classical formulation of the problem this scheme allows depolarization to be fully compensated, there is no need to consider 4 and more AEs as a unified system. Nevertheless, it was shown in [33] that compensation will not be complete in the case of essential self-focusing, i.e., B-integral ~1. Depolarization compensation was investigated in [33] in a scheme with four AEs and the scheme analogous to the one in Fig. 11(b) proved to be the best. In that work, however, the birefringence was not small and reduction of depolarization on splitting AEs was not considered. Thus, we proposed a method for further reduction of depolarization degree that may be used in some schemes allowing incomplete depolarization compensation. 6. Conclusion

We have proposed and implemented in experiment the scheme of depolarization compensation in single crystals at weak birefringence without phase elements and called it the scheme with counterrotation. In this scheme compensation is attained by rotating identical active elements around their axes in opposite directions. The conditions for the parameter of photoelastic anisotropy and orientation of a crystal under which compensation occurs have been found. An illustrative criterion for experimental determination of a possibility of compensation in available active elements has been proposed. It was demonstrated in experiment that the depolarization degree in CaF2 reduces in the scheme with counterrotation by more than an order of magnitude compared to a single element of double length. The measured dependences of the integral depolarization degree on the laser radiation power and crystal position relative to laser radiation polarization are in a qualitative agreement with the theoretical calculations. We have proposed a method of further reduction of depolarization degree in the schemes with counterrotation and with λ/2 plates that allow incomplete depolarization compensation. It #191777 - $15.00 USD Received 6 Jun 2013; accepted 26 Jul 2013; published 16 Sep 2013 (C) 2013 OSA 23 September 2013 | Vol. 21, No. 19 | DOI:10.1364/OE.21.022338 | OPTICS EXPRESS 22351

was shown analytically that doubling of the number of active elements in the scheme with λ/2 plates permits increasing the order of the power dependence of depolarization degree at weak birefringence by two. The numerical computations demonstrated that in the scheme with counterrotation the coefficient of the sixth power does not turn to zero, nevertheless the depolarization reduces.
