Effect of full compensation of thermally induced ... - OSA Publishing

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Letter

Vol. 41, No. 10 / May 15 2016 / Optics Letters

Effect of full compensation of thermally induced depolarization in two nonidentical laser elements ILYA L. SNETKOV,1,* VITALY V. DOROFEEV,2

AND

OLEG V. PALASHOV1

1

Institute of Applied Physics of the Russian Academy of Sciences, 46 Uljanov Street, Nizhny Novgorod 603950, Russia G.G. Devyatykh Institute of Chemistry of High-Purity Substances of the Russian Academy of Sciences, 49 Tropinin Street, Nizhny Novgorod 603950, Russia *Corresponding author: [email protected]‑nnov.ru

2

Received 15 March 2016; accepted 21 April 2016; posted 26 April 2016 (Doc. ID 261259); published 13 May 2016

Thermally induced depolarization of radiation introduced by a system of two optical elements separated by a quartz rotator has been analyzed. The conditions of full compensation of thermally induced depolarization for two nonidentical optical elements have been found. The model experiment has demonstrated that full compensation in two optical elements of different materials is possible without a quartz rotator between them. © 2016 Optical Society of America OCIS codes: (140.6810) Thermal effects; (260.1440) Birefringence.

dedicated to the theoretical analysis of thermally induced depolarization in a system of two nonidentical elements separated by a quartz rotator in a steady-state regime. Depolarized radiation is an orthogonal component of radiation constant in time, with nonuniform distribution over the cross section, arising in polarized radiation during transmission through a thermally loaded element: R 2 jE⊥out j2 S ΓjEout j dS R Γ ; (1) ; γ 2 jEout j2 S jEout j dS

http://dx.doi.org/10.1364/OL.41.002374

where the field at the output of the element is determined by the field at the input and by the Jones matrix of the optical element [6],

An increase in the average power of laser radiation leads to a more pronounced role of parasitic thermal effects arising in laser optical elements that should be taken into consideration, mitigated, and compensated for. One of the principal parasitic thermal effects is thermally induced depolarization arising as a result of changes in the refraction index of the medium caused by temperature stress [1]. Studies of thermally induced depolarization in laser optical elements were started in the 1960s [2–4] and are continued until now. Thermally induced depolarization in laser optical elements leads to the power loss of polarized radiation equal to the depolarization rate and, more importantly, to changes in the transverse mode structure due to the amplitude and phase modulation caused by depolarization nonuniformity, thereby introducing additional diffraction losses and impeding usage of elements operating with polarized light (e.g., Pockels cells, frequency doublers, and parametric crystals). Thus, thermally induced depolarization of radiation appreciably limits the maximum average power of single-mode lasers generating radiation with specified polarization. One of the well-known and widely used methods of compensating for thermally induced depolarization is to use two identical elements (i.e., made of the same material and having identical orientation and quality) that are separated by a 90° polarization rotator [5]. According to the theory, the thermally induced depolarization arising in one element in this scheme is fully compensated for by the specified polarization during transmission through the second element. This Letter is

Eout J · Ein ; i sin 2Ψ δ cotδl ∕2 i cos 2Ψ : J sin l 2 i sin 2Ψ cotδl ∕2 − i cos 2Ψ

0146-9592/16/102374-04 Journal © 2016 Optical Society of America

(2)

Here, δl is the phase incursion difference between two eigenpolarizations, and Ψ is the angle of inclination of eigenpolarizations relative to the laboratory frame of reference. The field at the input is regarded to be linearly polarized along the abscissa axis Ein Er · x0 , where Er specifies transverse distribution of the signal wave, and I r jErj2 is its intensity. The expressions for δl and Ψ for a single crystal with arbitrary orientation of crystallographic axes and with a crystal lattice symmetry of m3m, 432, and 43m can be found in [7], and with 23 and m3 symmetry in [8]. The dependence of δl and Ψ on transverse coordinates leads to the appearance of radiation depolarization. In the case of several optical elements, their Jones matrices are multiplied. For two consecutive optical elements separated by a quartz rotator rotating the polarization plane by angle θr , using Eqs. (1) and (2), in a general case, one can obtain for Γ the following expression: δ δ Γ sin2 l 1 sin2 l 2 sin2 2Ψ1 − 2Ψ2 2θr 2 2 δ δ sin2Ψ2 − 2θr cos l 1 sin l 2 2 2 2 δ δ ; (3) sin2Ψ1 sin l 1 cos l 2 2 2


Letter from which follows the condition 2Ψ1 − 2Ψ2 2θr πm : δl 2 ∕δl 1 −1m1

(4)

(5) from which it follows that thermally induced depolarization is fully compensated for with the use of a 90° quartz rotator (θr π∕2) [5]. Let us address the question of whether compensation is possible in the case of nonidentical optical elements (different materials, orientations of crystallographic axes, etc.). Optical elements may be made of glass, single crystal with different orientations of crystallographic axes, or ceramics. Crystal optical elements with [001] and [111] orientations of crystallographic axes are used most widely. We will consider in more detail all the above mentioned cases for axial symmetry of heat release source and geometry of the elements. The expressions for δl and Ψ in the case of glass elements will be written as [3,7] Ψn φ.

(6)

In addition, in the case of single crystal elements, with [111] orientation as [7,9,10] 1 2ξn ; Ψn φ: (7) 3 For ceramic elements, taking into consideration that from the viewpoint of thermally induced depolarization ceramics behaves on the average as a single crystal with [111] orientation with effective parameter of optical anisotropy, the expressions for δl and Ψ will be written in the form [11,12] δ l n pn h

2 3ξn ; Ψn φ. (8) 5 In addition, for a single crystal with [001] orientation, in the form [7,13] sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ξ2n tan2 2θn − 2φ δ l n pn h ; 1 tan2 2θn − 2φ δ l n pn h

tan2Ψn − 2θn ξn tan2φ − 2θn :

Z z Z QP h 1 u ; hu r 2 ∕r 2h dz F h ζdζ; (10) λκ u 0 0 where Q is the thermo-optical characteristic of material [15]; P h is heat release power; λ is a wavelength; κ is thermal conductivity; r h and F h are the characteristic transverse size and shape of the heat release source distribution. In the case of passive optical elements, when the transmitted radiation is a source of heat release itself, I r I 0 · F h r; in the case of active elements, F h is determined by pump radiation and may differ significantly from the shape of the signal wave. For disk geometry of optical elements, the expression Eq. (10) take on a different form [16,17], which, however, does not affect the conclusions that will be made later. The expressions Eqs. (6)–(9) assume that the cross-sectional distribution of heat release sources F h and the signal wave have identical shapes in each element, which leads to identical integrals h. If the size of the crystal, and the transverse sizes of the heat release sources and of the signal wave change proportionally, h and hence Γ do not change their forms. All material constants except optical anisotropy parameter are included in p, the normalized heat release power. It is important to note that p has its own sign for each material that is determined by the sign of thermooptical characteristic Q which, in turn, is fully determined by the sign of the piezo-optical coefficient difference (π 11 − π 12 ). This sign greatly affects the compensation of thermally induced depolarization [18] and may be found in experiment [12]. Let us consider the case when the first and second optical elements are made of glass, single crystal with [111] orientation, or ceramics. The condition Ψ1 Ψ2 is fulfilled automatically; as in these cases, they are both equal to the polar angle φ. Then the expression Eq. (3) is rewritten as δ δ Γ sin2 l 1 sin2 l 2 sin2 2θr 2 2 δ δ sin2φ − 2θr cos l 1 sin l 2 2 2 2 δ δ ; (11) sin2φ sin l 1 cos l 2 2 2 p

If the relations in Eq. (4) are fulfilled by choosing appropriate materials of the elements, heat release power in the elements, and angle of rotation θr in the quartz rotator, the expression Eq. (3) becomes identically zero, which corresponds to full compensation of thermally induced depolarization. In the case of two identical optical elements with identical heat release, the conditions δl 1 δl 2 , Ψ1 Ψ2 are met automatically and the expression Eq. (3) is rewritten in the form δ Γ sin4 l 1 sin2 2θr 2 δl 1 δl 1 2 2 2 cos sin2Ψ1 −2θr sin2Ψ1 ; sin 2 2

δl n pn h;

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(9)

In these expressions, n is the number of the optical element; r, φ are polar coordinates; ξ is the optical anisotropy parameter of the material [14]; θn is the angle between one of the crystallographic axes and the x axis of the laboratory reference frame; p is the normalized heat release power; and h is the shape function of heat release. For the elements of rod geometry,

and the condition of full compensation Eq. (4) is rewritten as 2θr πm; (12) D −1m1 : Here, we have introduced the variable D δl 2 ∕δl 1 whose sign depends on the used materials. When Eq. (12) is fulfilled, thermally induced depolarization is fully compensated for, as in [5]; in this case, for D > 0, a 90°quartz rotator should be used (θr π∕2) and, for D < 0, there is no need for a quartz rotator (θr 0). As all the dependence on transverse coordinates is contained in the integral h, the equality jD j 1 is fulfilled by choosing the heat release power ratio P h2 ∕P h1 in the optical elements. For passive optical elements at weak absorption P h ≈ α0 LP laser (α0 is a linear absorption coefficient, L is the length of the element, and P laser is laser radiation power), and the condition may be met by varying the lengths of the first and second elements. For the active elements, where heat release power depends mainly on pump power, the condition may be fulfilled by selecting pump power ratio. Let us now analyze a possibility of compensation without a quartz rotator. In this case, it is necessary to meet the condition

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D −1, and the sign in this condition imposes significant limitations on the used materials. In the case of glass optical elements, D p2 ∕p1 < 0 may be fulfilled only for materials having different signs of differences of piezo-optical coefficients (π 11 − π 12 ). In the case of single crystal optical elements with [111] orientation, the value and sign of optical anisotropy parameter of materials are important for meeting the condition D p2 1 2ξ2 ∕p1 1 2ξ1 < 0. Full compensation of thermally induced depolarization is possible for materials with the same sign of difference of piezo-optical coefficients (π 11 − π 12 ) whose optical anisotropy parameters lie on different sides of ξ −1∕2 or for the materials with different signs of (π 11 − π 12 ), but whose optical anisotropy parameters lie on the same side of ξ . The case of ceramics is analogous to that of single crystals with [111] orientation; the only differences are the value of ξ −2∕3 and the presence of small-scale polarization distortions that are not compensated for [19]. All possible combinations: glass—[111] single crystal, glass— ceramics and [111] single crystal—ceramics may be analyzed analogously. Note that for all materials for which optical anisotropy parameter lies in the range between −2∕3 < ξ < −1∕2, thermally induced depolarization in a single crystal with [111] orientation may be compensated for by ceramics of the same material. In the case of [001] single crystals [the expression Eq. (9)], the angle of inclination of eigenpolarizations Ψ depends on the polar angle φ, the optical anisotropy parameter ξ, and on the rotation of the optical element around the radiation propagation axis θ, which greatly complicates meeting the first condition in Eq. (4). It is fulfilled when ξ2 ξ1 ; then θ2 θ1 , D p2 ∕p1 −1m1 , 2θr πm, and m 0 or 1 depending on the sign of D ; when ξ2 1∕ξ1 , then θ2 θ1 π∕4, D p2 ∕ξ1 p1 −1m1 , 2θr πm, and m 0 or 1 depending on the sign of D . In a general case of arbitrary materials of two optical elements (arbitrary ξi and signs of differences π 11 − π 12 i ), full compensation of thermally induced depolarization is not observed. However, in the case of weak birefringence (δl ≪ 1), by expanding integral depolarization into a Taylor series in terms of δl it is possible to zero the first term in the expansion ∼δ2l for an arbitrary two materials of optical elements ∼δ2l [18,20]. The level of thermally induced depolarization will then reduce significantly, and its dependence on heat release power will change from ∼P 2h to ∼P 4h . To confirm the results of our studies, we conducted a model experiment on the compensation of thermally induced depolarization in two glass elements that allow full compensation without quartz rotator. According to the theoretical predictions, these glasses should have different signs of differences of piezooptical coefficients (π 11 − π 12 ). One of the elements was made of magneto-optical glass MOC-33 π 11 − π 12 < 0, and the other of tungstate–tellurite glass π 11 − π 12 > 0 [21]. We chose those materials because they are readily available and meet the condition of full compensation. The majority of laser glasses (phosphate, silicate, and others) and crystals have π 11 − π 12 < 0 like the MOC used in our experiment and, tellurite glasses doped with active ions are used as laser material [22–24]. The difference of the (π 11 − π 12 ) signs was verified experimentally employing the technique described in [12]. Sample lengths were chosen so as to provide close magnitudes of thermally induced depolarizations in the samples. The dependence of thermally induced depolarization on laser radiation power in

Letter

Fig. 1. (a) Integral thermally induced depolarization versus laser radiation power: for tungstate–tellurite glass—diamond, for magnetooptical glass MOC-33—triangle, and for both elements simultaneously (compensation case)—circles; distribution of local depolarization at 200 W (b) for tungstate–tellurite glass, (c) for MOC-33, and (d) for the compensation case.

crossed polarizers was measured in each sample, as well as in both samples simultaneously [Fig. 1(a)]. Radiation from a cw Yb-fiber laser with a maximum radiation power of 300 W was used as heating and reading of thermally induced birefringence radiation simultaneously. Almost full compensation of thermally induced depolarization was observed in the experiment during radiation transmission through two consecutive optical elements. At the power of 200 W, the magnitude of radiation depolarization reduced by more than a factor of 4.6, as compared to the case of transmission through one of these elements. No further laser power increase was undertaken to avoid the risk of optical element breakdown. The magnitude of residual depolarization was determined by “cold” depolarization specified by the quality of the used samples. The lengths of the samples were chosen roughly (giving D −0.83), which resulted in formation against the background of “cold” depolarization of a characteristic transverse distribution of thermally induced depolarization in the form of a “Maltese cross” [Fig. 1(d)]. To conclude, we have theoretically and experimentally considered the possibility of compensating thermally induced depolarization in a system of two nonidentical optical elements. An original method of full compensation of thermally induced depolarization using optical elements of glass, single crystal with [111] orientation, or ceramics has been proposed. This method may be a useful tool for developing lasers with high average power. The use of different materials may give an additional advantage, such as partial compensation of thermal lens that is another parasitic thermal effect. This may be done if the materials have different signs of the temperature dependence of refractive index [18,25–27]. The use of active elements of different materials will allow an additional increase of gain bandwidth, thereby enabling generation and amplification of shorter pulses. For instance, with the use of Yb3 :Sc2 O3 and Yb3 :Y 2 O3 ceramic active elements in one cavity, pulses with a duration of 53 fs and an average power of 1 W were generated [28]. Another possible application of the proposed scheme is in high-power multi-element lasers of transmissive optics of different materials. Most optical elements are, as a rule, made of fused silica π 11 − π 12 < 0, and their number in a laser scheme may be very large. During passage of powerful

Letter laser radiation through these elements, thermally induced depolarization will accumulate and may reach significant values. By using part of transmissive optics made of glass, [111] single crystal, or ceramics for which D < 0, it will be possible to decrease the magnitude of radiation depolarization significantly and, thus, to reduce losses and improve the quality of generated radiation. Funding. Government of the Russian Federation (14. B25.31.0024); Russian Science Foundation (RSF) (1512-30021). Acknowledgment. The authors highly appreciate the contribution of Prof. Ken-Ichi Ueda in analyzing the results and discussion of this Letter. REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11.

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