Self-compensation of thermally induced depolarization in CaF<sub>2</sub> and definite cubic single crystals

Anton G. Vyatkin; Ilya L. Snetkov; Oleg V. Palashov; Efim A. Khazanov

doi:10.1364/OE.21.022338

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 E_out 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

\begin{matrix} Γ = \frac{{| E_{⊥ out} |}^{2}}{{| E_{out} |}^{2}}, & γ = \frac{\int_{S} Γ {| E_{out} |}^{2} d S}{\int_{S} {| E_{out} |}^{2} d S} \end{matrix},

where the integration is performed over the beam cross-section.

Thermal depolarization in cubic crystals of symmetry groups 432, $\bar{4} 3 m$ 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 $Δ B = π σ$ , where π is the piezooptic tensor and σ is the stress tensor. The additive to the dielectric tensor ε = B⁻¹ is $Δ ε = - ε_{0}^{2} Δ B$ , 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 \frac{δ_{i}}{2} (\begin{matrix} \cot (δ_{i} / 2) + j \cos 2 Ψ_{i} & j \sin 2 Ψ_{i} \\ j \sin 2 Ψ_{i} & \cot (δ_{i} / 2) - j \cos 2 Ψ_{i} \end{matrix}),

where Ψ_i is the angle of inclination of eigenpolarizations, δ_i 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} = J_{1, 2, \dots, (N - 1), N} = J_{N} J_{N - 1} \dots J_{2} J_{1} .

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 non-diagonal element of the Jones matrix:

Γ = {| J_{Σ 12} |}^{2} = {| J_{Σ 21} |}^{2} .

The depolarization degree at weak birefringence (

δ_{i} ≪ 1

) will be designated by Γ_small. In a general case, it is defined by the quadratic term in the Taylor power series Γ₍₂₎:

Γ_{small} \approx Γ_{(2)} = \frac{1}{4} {(\sum_{i = 1}^{N} δ_{i} \sin 2 Ψ_{i})}^{2} .

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 $(r_{0}, φ_{0}, z)$ and Cartesian $(x_{1}, x_{2}, z)$ coordinates. In this frame of reference, by diagonalizing ΔB in the (x₁, x₂) plane, one can obtain

\begin{matrix} \tan 2 Ψ^{0}_{i} = D_{i} / C_{i}, & δ_{i} = - δ_{0 i} D_{i} / (\sin 2 Ψ^{0}_{i}), & δ_{0 i} = \frac{1}{2} k_{0} L_{i} n_{0 i}^{3}, \end{matrix}

where L_i is the length of the AE, n₀_i = ε₀_i^1/2 is its “cold” refractive index, k₀ is the wavenumber in free space, and

\begin{matrix} C_{i} = Δ B_{11} - Δ B_{22} = A_{1} (ξ, ξ_{d}; α_{i}, β_{i}) Σ (r) + \\ + [A_{2} (ξ; α_{i}, β_{i}) \cos 2 φ_{0} + A_{3} (ξ, ξ_{d}; α_{i}, β_{i}) \sin 2 φ_{0}] Δ (r), \\ D_{i} = 2 Δ B_{12} = B_{1} (ξ, ξ_{d}; α_{i}, β_{i}) Σ (r) + \\ + [B_{2} (ξ, ξ_{d}; α_{i}, β_{i}) \cos 2 φ_{0} + B_{3} (ξ; α_{i}, β_{i}) \sin 2 φ_{0}] Δ (r), \end{matrix}

with

\begin{matrix} Σ (r) = π_{S} [\frac{1}{2} (σ_{r r} + σ_{φ φ}) - σ_{z z}], \\ Δ (r) = π_{S} (σ_{r r} - σ_{φ φ}), \\ π_{S} = π_{a a a a} - \frac{1}{2} (π_{a a b b} + π_{b b a a}) . \end{matrix}

The coefficients A_m, B_m in Eq. (7) are written in the form

\begin{matrix} A_{1} (α, β) = (1 - ξ) a_{1} (α, β) + ξ_{d} a_{2} (α, β), & B_{1} (α, β) = (1 - ξ) b_{1} (α, β) + ξ_{d} b_{2} (α, β), \\ A_{2} (α, β) = ξ + (1 - ξ) a_{3} (α, β), & B_{2} (α, β) = (1 - ξ) c_{1} (α, β) + ξ_{d} c_{2} (α, β), \\ A_{3} (α, β) = (1 - ξ) c_{1} (α, β) - ξ_{d} c_{2} (α, β), & B_{3} (α, β) = ξ + (1 - ξ) b_{3} (α, β), \end{matrix}

where

\begin{matrix} ξ = 2 π_{a b a b} / π_{S}, & ξ_{d} = (π_{a a b b} - π_{b b a a}) / π_{S} \end{matrix}

are the first and the second parameters of photoelastic anisotropy [17]. The subscript “i” in material constants ξ, ξ_d, π_S, and π is omitted for brevity, although they also generally differ for different optical elements. The coefficients

\begin{matrix} a_{1} = [\sin^{2} 2 α (1 - \cos^{4} β) - \sin^{2} 2 β] / 2, & b_{1} = \frac{1}{4} \sin β \sin 2 β \sin 4 α, \\ a_{2} = \cos 2 α \cos 2 β, & b_{2} = \sin 2 α \cos β (1 - 3 \cos^{2} β) / 2, \\ a_{3} = [1 - (\sin^{2} 2 α) / 4] \sin^{4} β + \cos^{2} 2 α \cos^{2} β, & b_{3} = \sin^{2} 2 α \cos^{2} β, \\ c_{1} = - \sin 4 α \cos β (1 + \cos^{2} β) / 4, & c_{2} = \frac{3}{4} \sin 2 α \cos β si n^{2} β \end{matrix}

define the dependence of tensor

Δ B

on crystal orientation. In crystals of the symmetry groups 432,

\bar{4} 3 m

and m3m,

π_{a a b b} = π_{b b a a}

and the expressions are simpler:

ξ_{d} = 0

,

π_{S} = π_{a a a a} - π_{a a b b}

.

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

\begin{matrix} p_{disk} = α_{T} E P_{Σ} k_{0} n_{0}^{3} π_{S} / (8 π κ), & p_{rod} = p_{disk} / (1 - ν) \end{matrix},

where P_Σ is the power of heat release in the whole AE, α_T 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 $\propto$ p, and Γ₍₂₎ $\propto$ p².

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)].

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).

Download Full Size | PDF

2.2. Scheme with counterrotation

Let the optical elements be turned around their z-axes by angles χ_i (the third Euler angles are equal to −χ_i). Тhen

\begin{matrix} φ = φ_{0} + χ_{i}, & Ψ_{i} = Ψ^{0}_{i} + χ_{i}, \end{matrix}

and Eq. (5) takes on the form

\begin{matrix} Γ_{(2)} = \frac{1}{4} {[\sum_{i = 1}^{N} δ_{0 i} V_{i}]}^{2}, & V_{i} = D_{i} \cos 2 χ_{i} + C_{i} \sin 2 χ_{i} \end{matrix} .

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

\begin{matrix} Γ_{(2)} = \frac{δ_{0}^{2}}{4} {(V_{1} + V_{2})}^{2}, \\ V_{1} + V_{2} = Δ (r) \sin 2 φ {2 ξ + (1 - ξ) [(a_{3} + b_{3}) + (b_{3} - a_{3}) \cos 4 χ + 2 c_{1} \sin 4 χ]} \end{matrix} .

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 ξ_d. Third, it can be shown that, if the condition

{(\frac{ξ}{ξ - 1})}^{2} - (\frac{ξ}{ξ - 1}) (a_{3} + b_{3}) + (a_{3} b_{3} - c_{1}^{2}) \leq 0

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 Γ_small 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.

Download Full Size | PDF

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] \leq 0.

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).

Download Full Size | PDF

In the [MMN] (α = π/4) orientations, compensation occurs at

[1 - (1 - ξ) \sin^{2} β] [ξ + \frac{3}{4} (1 - ξ) \sin^{4} β] \leq 0

[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 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)].

Fig. 4 Domain of existence of compensation in the scheme with counterrotation (all shades of blue for any ξ_d) and of profile rotation of the depolarized beam for ξ_d = 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).

Download Full Size | PDF

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).

Download Full Size | PDF

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 ξ_d 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 $- 2 < ξ < - \frac{1}{5}$ 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.

Download Full Size | PDF

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 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 cross-section 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.

Download Full Size | PDF

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.

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.

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.

Download Full Size | PDF

Note that, if the crystals have no marks specifying the origin of angles χ_i, 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, Γ_small 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.

Download Full Size | PDF

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

The points $\bar{φ}$ 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 2 \bar{φ} = \frac{(A_{2} + B_{3}) + (B_{3} - A_{2}) \cos 4 χ + (A_{3} + B_{2}) \sin 4 χ}{(B_{2} - A_{3}) + (A_{3} + B_{2}) \cos 4 χ + (B_{3} - A_{2}) \sin 4 χ} .

Note that

\bar{φ}

does not depend on r. The profile of the depolarized beam rotates if and only if

\bar{φ}

is a monotonic function of χ. The monotony condition may be written in the form

c_{2}^{2} {(\frac{ξ_{d}}{ξ - 1})}^{2} + {(\frac{ξ}{ξ - 1})}^{2} - (\frac{ξ}{ξ - 1}) (a_{3} + b_{3}) + (a_{3} b_{3} - c_{1}^{2}) \leq 0.

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 ξ_d = 0 (crystals of 432,

\bar{4} 3 m

and m3m symmetry groups) and a sufficient condition for compensation at ξ_d ≠ 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₎ is axially symmetric and angle $\bar{φ}$ , hence $\partial \bar{φ} / \partial χ$ , 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.

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.

Download Full Size | PDF

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 \frac{δ}{2} (\begin{matrix} \cot (δ / 2) + j \cos 2 Ψ & j \sin 2 Ψ \\ j \sin 2 Ψ & \cot (δ / 2) - j \cos 2 Ψ \end{matrix}),

then the Jones matrix of element “2” is

J [2] = (\begin{matrix} - 1 & 0 \\ 0 & 1 \end{matrix}) J [1] (\begin{matrix} - 1 & 0 \\ 0 & 1 \end{matrix}) = \sin \frac{δ}{2} (\begin{matrix} \cot (δ / 2) + j \cos 2 Ψ & - j \sin 2 Ψ \\ - j \sin 2 Ψ & \cot (δ / 2) - j \cos 2 Ψ \end{matrix}) .

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 2^N elements local Γ_small and integral γ_small depolarization degrees at p $≪$ 1 are proportional to p²^N⁺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.

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.

Download Full Size | PDF

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.

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 $p \to 0$ , γ_small $\propto$ 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.

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.

Download Full Size | PDF

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 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.

References and links

1. W. Koechner, Solid-State Laser Engineering (Springer-Verlag, 1999).

2. J. F. Nye, Physical Properties of Crystals (Oxford University Press, 1964).

3. J. D. Foster and L. M. Osterink, “Thermal effects in a Nd:YAG laser,” J. Appl. Phys. 41(9), 3656–3663 (1970). [CrossRef]

4. G. A. Massey, “Criterion for selection of cw laser host materials to increase available power in the fundamental mode,” Appl. Phys. Lett. 17(5), 213–215 (1970). [CrossRef]

5. W. Koechner, “Absorbed pump power, thermal profile and stresses in a cw pumped Nd:YAG crystal,” Appl. Opt. 9(6), 1429–1434 (1970). [CrossRef] [PubMed]

6. W. Koechner and D. K. Rice, “Effect of birefringence on the performance of linearly polarized YAG:Nd lasers,” IEEE J. Quantum Electron. 6(9), 557–566 (1970). [CrossRef]

7. M. A. Karr, “Nd:YAlG laser cavity loss due to an internal Brewster polarizer,” Appl. Opt. 10(4), 893–895 (1971). [CrossRef] [PubMed]

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

9. L. N. Soms and A. A. Tarasov, “Thermal deformation in color-center laser active elements,” Soviet J. Quantum Electron. 9(12), 1506–1508 (1979). [CrossRef]

10. I. Shoji and T. Taira, “Intrinsic reduction of the depolarization loss in solid-state lasers by use of a (110)-cut Y₃Al₅O₁₂ crystal,” Appl. Phys. Lett. 80(17), 3048–3050 (2002). [CrossRef]

11. I. B. Mukhin, O. V. Palashov, E. A. Khazanov, and I. A. Ivanov, “Influence of the orientation of a crystal on thermal polarization effects in high-power solid-state lasers,” JETP Lett. 81(3), 90–94 (2005). [CrossRef]

12. I. B. Mukhin, O. V. Palashov, and E. A. Khazanov, “Reduction of thermally induced depolarization of laser radiation in [110] oriented cubic crystals,” Opt. Express 17(7), 5496–5501 (2009). [CrossRef] [PubMed]

13. 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]

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

15. I. Snetkov, A. Vyatkin, O. Palashov, and E. Khazanov, “Drastic reduction of thermally induced depolarization in CaF₂ crystals with [111] orientation,” Opt. Express 20(12), 13357–13367 (2012). [CrossRef] [PubMed]

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

17. A. G. Vyatkin and E. A. Khazanov, “Thermally induced depolarization in sesquioxide class m3 single crystals,” J. Opt. Soc. Am. B 28(4), 805–811 (2011). [CrossRef]

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

19. G. Giuliani and P. Ristori, “Polarization flip cavities: a new approach to laser resonators,” Opt. Commun. 35(1), 109–112 (1980). [CrossRef]

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). [CrossRef]

21. E. A. Khazanov, “A new Faraday rotator for high average power lasers,” Quantum Electron. 31(4), 351–356 (2001). [CrossRef]

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). [CrossRef] [PubMed]

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). [CrossRef]

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). [CrossRef]

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). [CrossRef] [PubMed]

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 single-crystal high-power laser windows,” Appl. Phys. Lett. 30(9), 485–486 (1977). [CrossRef]

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). [CrossRef] [PubMed]

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). [CrossRef] [PubMed]

Self-compensation of thermally induced depolarization in CaF₂ and definite cubic single crystals

Abstract

1. Introduction

2. Compensation of thermally induced depolarization at weak birefringence in the scheme with counterrotation

2.1. Thermally induced depolarization in a single crystal

2.2. Scheme with counterrotation

2.3. Conditions of compensating thermally induced depolarization in the scheme with counterrotation

2.4. Numerical computation of depolarization degree in the scheme with counterrotation

2.5. Comparison of the quality of depolarization compensation in the schemes with counterrotation and with λ/2 plates

3. Experiment on compensation of thermally induced depolarization in the scheme with counterrotation

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

5. Compensation of higher-orders of depolarization degree with increasing number of identical elements

6. Conclusion

References and links

Supplementary Material (5)

Cited By

Figures (12)

Equations (22)

Optics Express