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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 18, Iss. 19 — Sep. 13, 2010
  • pp: 20461–20474
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Intrinsic reduction of the depolarization in Nd:YAG crystals

Oliver Puncken, Henrik Tünnermann, James J. Morehead, Peter Weßels, Maik Frede, Jörg Neumann, and Dietmar Kracht  »View Author Affiliations


Optics Express, Vol. 18, Issue 19, pp. 20461-20474 (2010)
http://dx.doi.org/10.1364/OE.18.020461


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Abstract

The output power of linearly polarized Nd:YAG lasers is typically limited by thermally induced birefringence, which causes depolarization. However, this effect can be reduced either by use of some kind of depolarization compensation or by use of crystals which are cut in [110]- and [100]-direction, instead of the common [111]-direction. Investigations of the intrinsic reduction of the depolarization by use of these crystals are presented. To our knowledge, this is the first probe beam-experiment describing a comparison between [100]-, [110]- and [111]-cut Nd:YAG crystals in a pump power regime between 100 and 200 W. It is demonstrated that the depolarization can be reduced by a factor of 6 in [100]-cut crystals. The simulations reveal that a reduction of depolarization by use of a [110]-cut crystal in comparison with a [100]-cut crystal only becomes possible at pump powers in the kW region. Analysis also shows that the bifocusing for [100]-cut is slightly smaller and more asymmetrical than for [111]-cut.

© 2010 OSA

1. Introduction

In optically pumped laser rods, a transverse thermal gradient causes stress, which leads to a refractive index change inside the crystals via the photoelastic effect. This thermally induced birefringence results in depolarization, which is defined as the ratio of depolarized power to the initially linearly polarized power, and bifocusing, since the refractive power of the thermal lens depends on the polarization. Both effects limit the output power of linearly polarized Nd:YAG lasers [1

1. M. P. Murdough and C. A. Denman, “Mode-volume and pump-power limitations in injection-locked TEM00 Nd:YAG rod lasers,” Appl. Opt. 35(30), 5925–5936 (1996). [CrossRef] [PubMed]

]. To compensate for the thermally induced birefringence in a two head resonator, a 90° quartz rotator and an imaging optics between the crystals can be applied [2

2. Q. Lü, N. Kugler, H. Weber, S. Dong, N. Müller, and U. Wittrock, “A novel approach for compensation of birefringence in cylindrical Nd: YAG rods,” Opt. Quantum Electron. 28(1), 57–69 (1996). [CrossRef]

]. Simpler schemes to partly compensate for moderate thermal depolarization have been presented by Clarkson et al. [3

3. 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]

] and Morehead [4

4. J. J. Morehead, “Compensation of laser thermal depolarization using free space,” IEEE J. Sel. Top. Quantum Electron. 13(3), 498–501 (2007). [CrossRef]

]. Koechner and Rice [5

5. 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]

] have shown that for Nd:YAG crystals the depolarization depends on the growth direction of the crystal, but because of a mistake they concluded that depolarization is the same for all crystal cuts in the high power limit. Soms et al. [6

6. L. N. Soms, A. A. Tarasov, and V. V. Shashkin, “Problem of depolarization of linearly polarized light by a YAG:Nd3+ laser-active element under thermally induced birefringence conditions,” Sov. J. Quantum Electron. 10(3), 350–351 (1980). [CrossRef]

] derived the correct results for the [100] case, whereas Shoji and Taira [7

7. 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]

] concentrate on high pump power levels and the [110]-cut. In this paper, different crystal cuts are theoretically and experimentally investigated in the low pump power regime in order to intrinsically reduce the depolarization in this operating regime.

2. Theoretical background

YAG has a cubic crystal structure and is thus optically isotropic. Stress (from nonuniform thermal expansion, for example) breaks this isotropy and induces birefringence. The inverse of the dielectric tensor can be expressed as I/n02 + B, where I is the 3-dimensional identity matrix, n0 is the unstressed index of refraction, and B implies the change of index with stress. It is usually expressed in terms of the elastic strain ε, B = p·ε, where B and ε are 3 × 3 matrices (rank-2 tensors) and the elasto-optic tensor p is rank-4. For YAG’s symmetry class m3m there are only three independent elements: p11 = −0.029, p12 = 0.009, and p44 = −0.0615 [8

8. R. W. Dixon, “Photoelastic properties of selected materials and their relevance for applications to acoustic light modulators and scanners,” J. Appl. Phys. 38(13), 5149–5153 (1967). [CrossRef]

] (here, the Nye-notation [9

9. J. F. Nye, Physical properties of crystals (Oxford University Press, 1957, 1985).

] is used, see appendix). The principal indices of refraction are approximately [10

10. 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]

]
n1,2n01/2n03B1,2,
(1)
where B 1,2 are the eigenvalues of the transverse B-matrix, which is
(BxxBxyBxyByy)
(2)
with the beam’s direction as z. The matrix’s eigendirections are the principal polarizations.

The power depolarization in propagating a distance L is
D=sin2[2(θγ)]sin2(ψ/2),
(3)
where θ is the angle between principal polarization and the x-axis, γ is the angle of the input polarization to the x-axis, and the phase shift is

ψ=2πλL(n1n2).
(4)

For a uniform cylinder uniformly heated, following Shoji and Taira [7

7. 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]

] one can write this phase shift as
ψ=Ω4PheatPdepol(rr0)2,Pdepol=32κ(1ν)λ/α,
(5)
where r 0 is the cylinder’s radius, κ is the thermal conductivity (0.014 W/(mm°C) for YAG), ν is Poisson’s ratio (0.25), and α is the coefficient of thermal expansion (7.5 × 10−6 / °C). Shoji & Taira [7

7. 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]

] list and our appendix derives the principal directions and the birefringence for the three major orientations. If ϕ is the angular coordinate with respect to the x-axis, the principal directions are
tan(2θ)={tan(2φ)[111]2p44p11p12tan(2φ)[100]
(6)
and for [110],

tan(2θ)=8p44sin(2φ)[3(p11p12)+2p44]cos(2φ)+(p11p122p44)(2r02/r2).
(7)

For [100] the x-axis is taken along crystal axis and for [110] it is along a 45° diagonal (e.g. [1¯  ​10]) with y along a crystal axis. For [111] symmetry implies these results are independent of the direction of the x-axis.

The birefringence coefficient is
Ω={1/3n03(1+ν)(p11p12+4p44)=0.69[111]n03(1+ν)(p11p12)2cos2(2φ)+(2p44)2sin2(2φ)[100]
(8)
and for [110]

Ω=n03(1+ν)116{[3(p11p12)+2p44]cos(2φ)(p11p122p44)(2r02/r2)}2+(2p44)2sin2(2φ)
(9)

First, the case of low depolarization is considered, for which the phase shift ψ is much less than one cycle. In Fig. 2
Fig. 2 Depolarization of all three investigated crystal-cuts in dependence of the pump power.
we will see this is less than approximately 400 W absorbed pump for our crystals. The appendix shows that if we choose the x-direction along the input polarization (γ = 0), then the depolarization of a ray is approximately
D=(πLn03/λ)2Bxy2,
(10)
involving only the off-diagonal element of the matrix, which is the coupling between the two polarizations. The most interesting polarization orientations are along the crystal axes for all three crystal cuts and polarization at 45° to crystal axes in [100]-cut case. For these orientations B xy = c·εxy, the off-diagonal index element is proportional to the off-diagonal element of strain. As derived in the appendix [Eqs. (24) and (25)], the constants of proportionality are

c111=13(p11p12+4p44),c1000=2p44,c10045=p11p12,c1100=2p44.
(11)

Hence, for a plane wave, the ratio of depolarizations for [100] and [111] is
D10045D111=f2=0.161,D1000D111=D1100D111=(fk)2=1.69,
(12)
where
f=3(p11p12)p11p12+4p44=0.402,k=p11p122p44=0.31.
(13)
[100]-oriented YAG with the polarization at 45° has 6 times less depolarization than [111]. [100] with polarization along a crystal axis has 1.69 times more depolarization than [111], as does [110] with polarization along the axis. [110]-oriented crystals with polarization at 45° have much higher depolarization and it depends on more than εxy.

Also, by studying the diagonal element B xx, the appendix shows that for low depolarization, [100]-oriented YAG with polarization at 45° has a stress-induced lens which adds 17% to the thermal lens in the polarization direction and subtracts 13.6% in the perpendicular direction. This is somewhat smaller and more asymmetric than the familiar [111], which adds 20% to the polarization direction and subtracts 3% from the other [11

11. W. Koechner, Solid-State Laser Engineering (2nd ed., Springer, 1988).

].

In the general case, the depolarization of a given ray [Eq. (3)] can be integrated over the cross section to find a beam’s depolarization. We consider a Gaussian beam of radius r g. In that case three of the four orientations of interest can be integrated in closed form (see appendix).

D¯111=d21+4d2,     d=Ω111PheatPdepol(rgr0)2D¯10045=2b21+(2b)2[1+(2b)2+1+(2b/k)2],b=fdD¯1000=2(b/k)21+(2b/k)2[1+(2b)2+1+(2b/k)2]
(14)

These have the low-depolarization ratios of (12) and the high-depolarization asymptotes of

D¯11114,D¯10045k2(1+k)=0.118,D¯100012(1+k)=0.382.
(15)

3. Numerical simulations

In contrast to the case of a [111]-cut crystal, the depolarization of [110]-cut and [100]-cut crystals depends on the orientation of the linearly polarized incident light with respect to the crystal orientation.

The comparison of Bxy for the different crystal-cuts is sufficient to find the differences in the depolarization [Eq. (10)]. Figure 2 shows the evolution of depolarization with increasing pump power for polarization states with minimal and maximal depolarization. Low power pumping behavior as given in Eq. (12) is demonstrated on Fig. 2(a). At pump powers below 2 kW, the [100]-cut crystal with 45° between the incident polarization and the crystallographic axes yields the lowest depolarization. At higher pump powers [Fig. 2(b)] the depolarization converges to constant values for all crystals.

The depolarization also depends on the ratio of probe beam radius to crystal radius. For [111]- and [100]-cuts, Eq. (14) shows that the depolarization depends on the squared heating power Pheat, which is scaled by the squared ratio rg/r0. Therefore, the depolarization of a [111]-cut crystal saturates at the same value of 25%, independently of the probe beam size rg, although the point of saturation will be shifted to higher pump powers for smaller probe beam radii [Fig. 3(a)
Fig. 3 Depolarization in dependence of heating power and probe size for an [111]-cut crystal (a), an [100]-cut crystal (b) and an [110]-cut crystal (c).
].

The curve for the depolarization as a function of thermal power for [100]-cut crystals looks similar, although the minimal depolarization is lower by at least a factor of 2.12 [Eq. (15), Fig. 3(b)].

The saturation of the depolarization in a [110]-cut crystal occurs at lower powers [Fig. 3(c)]. Here, the depolarization saturation value depends on the probe size.

For the simulations in Fig. 3, a uniform pumping case has been assumed. However, in order to maintain a good overlap between pump- and laser light, variations in the beam radius rg must be accompanied by variations in the pump spot size. For [111]-cut and [110]-cut crystals, the depolarization increases when decreasing the pump spot radius rpump and reaches a maximum at rpump = rg. In case of the [100]-cut, there is no local maximum, but the depolarization increases further when decreasing the pump spot. Furthermore, in case of [111]-cut and [100]-cut crystals the depolarization remains at a constant value for rg = rpump. The actual depolarization value depends on the crystal configuration. The depolarization of a [110]-cut crystal decreases with the size of the probe- and pump beam radius.

4. Single pass experiments

All three crystal-cuts were used in a single-pass setup with a linearly polarized Nd:YLF probe laser source with an output power of approximately 1 W at a wavelength of 1053 nm as shown in Fig. 4
Fig. 4 Single pass setup. A linearly polarized Nd:YLF probe beam propagates through a pumped Nd:YAG crystal. The polarization can be rotated with a half-wave plate (HWP). The depolarization can be analyzed behind a polarizing beamsplitter (PBS) with a camera (CCD) and a photodiode.
. This wavelength was chosen, because it will not be amplified in the pumped Nd:YAG crystals. On the other hand it is close to the Nd:YAG laser wavelength and therefore only small errors compared to 1064 nm occur, whereas the functionality of the AR-coatings on the crystal end faces and the HR-coated mirrors is ensured. The Nd:YAG test crystals consist of a 40 mm 0.1 at.% doped section and 7 mm long undoped endcaps at both ends to reduce surface stresses due to thermal expansion. With a highly reflective coating for the pump light at the backside, a smoothed longitudinal temperature distribution can be achieved, while 96% of the pump light is absorbed. The crystals were optically pumped at 808 nm by a fiber coupled OEM module with a maximum output power of 210 W. The input polarization was varied using a half-wave plate (HWP 1) in front of the pumped crystal. A second half-wave plate (HWP 2) was installed behind the crystal to rotate the polarization back. Both waveplates were rotated by the same angle. Alternatively, one could rotate the crystal itself, but this would be mechanically much more imprecise. The depolarization ratio was measured with a photodiode behind a polarizing beam splitter (PBS) cube. The probe beam power of the unpumped crystal was compared with the probe beam power under pumped conditions. To investigate the spatial structure of the depolarization, a CCD camera was used. The beam profile, as it occurs at the end face of the crystal, was imaged via a telescope between the crystal and the polarizing beam splitter on the CCD.

The general shape of the depolarization pattern did not depend on the pump- and probe beam sizes for [111]- and [100]-cut crystals. However, we used a pump spot diameter of 2 mm and a probe beam diameter of 1300 µm, since these parameters are the optimized dimensions for laser operation.

The [111]-cut crystal produces a cloverleaf-like depolarization pattern, as shown in the inset close to the CCD in Fig. 4. In these crystals, the axes of the local birefringence are orientated in radial and tangential direction. Therefore, the depolarization pattern is independent of the probe beam polarization angle, but the orientation of the pattern follows the orientation of the input polarization.

The [100]-beam pattern [Fig. 5(a)
Fig. 5 Depolarization patterns of [100]- and [110]-cut crystals. The probe beam is linearly polarized in each case in different orientations with respect to the crystal orientation (see scale on the left). The exposure time for the CCD images has been adapted for each picture.
] also shows this cloverleaf-like form, but its shape changes with the probe beam polarization, since the principal axes of the refractive index ellipsoides are no longer oriented in radial and tangential direction. It can be seen that near a 45° angle between the incident polarization and the crystallographic axes the “leaves” are thinner than for example at 0° and, thus, the depolarization is reduced.

The [110]-cut crystal [Fig. 5(b)] produces an elliptical polarization pattern for an input polarization of 45°. The principal axes for this crystal are given by Eq. (7). Note the dependence on beam radius r in this case, which causes a different depolarization pattern for various beam sizes. For r0>>rg, the last term of the denominator in Eq. (7) becomes large, which means that the principal axes are almost orthogonal with respect to the [1¯  ​10]- and the [001]-axis in the center of the crystal. Thus, the change of the polarization is large near the center in this case, while for the other crystal types no depolarization occurs in the center. The experimental observations can be well explained by our theoretical simulations as demonstrated in Fig. 5.

The depolarization ratio has been measured for different input polarizations at beam radii of rg = 150 µm, rg = 350 µm, rg = 650 µm, and rg = 900 µm. The graphs shown in Fig. 6
Fig. 6 Measured and simulated depolarization in dependence of probe beam polarization for a [111]-cut crystal (left), calculated phase shift for 180 W pump power (middle) and calculated local orientation of the birefringent axes (right).
, 7
Fig. 7 Depolarization in dependence of probe beam polarization for a [100]-cut crystal (left), phase shift (middle) and local orientation of the birefringent axes (right).
, and 8
Fig. 8 Depolarization in dependence of probe beam polarization for a [110]-cut crystal (left), phase shift (middle) and local orientation of the birefringent axes (right).
for the [111]-, [100]-, and [110]-cut crystals have been measured at a pump power of 140 W. The [111]-cut crystal (Fig. 6) did not show any change in depolarization with variation of the input polarization angle. Small beam sizes led to less depolarization, as one can expect from the radial phase shift distribution with a gradient from low to higher values in radial direction.

In [100]-cut crystals the depolarization depends on the input polarization (Fig. 7). The minimum has been found for all probe beam sizes at 45° between probe beam polarization and [010]- and [001]-axis, respectively. This is caused by the fact that the principal axes are rather oriented towards the diagonal directions than towards the radial and tangential directions (Fig. 7, right). Therefore, the geometric factor sin2 (2(θ-π/4)) = cos2 (2θ) in Eq. (3) becomes smaller. Where the geometric factor is not small (along the crystal axes), the phase shifts and thus the birefringence is small.

In [110]-cut crystals (Fig. 8) the minimum depolarization can be found at an incident polarization of 0°, if rg < 0.5r0. At rg ≥ 0.5r0 this behavior is inverted and minimum depolarization occurs at 45°, which can be understood from the depolarization pattern (Fig. 5). 0° corresponds to the common clover leaf form with no depolarization in the center, and hence low depolarization for small beams. The depolarization figure at 45° corresponds to an ellipse with the depolarization maximum in the center of the rod, but less depolarization in the outer regions. Therefore, less depolarization requires large probe laser beam sizes.

Again, the experimental results are in good agreement with the theory for all crystal types. The numerical simulations in Fig. 68 include the fact that the crystals are end pumped, while the analytical results presented in section 2 are derived for homogeneous pumping. However, homogeneous pumping is already a good approximation.

Figure 9
Fig. 9 Measured minimum and maximum depolarization as a function of pump power.
shows the measured depolarization against the pump power for the polarization orientations leading to the most and the least depolarization for a probe beam with a diameter of 650 µm. As expected, the depolarization of the [100]-cut crystal can be reduced to about 1/6 of the [111]-cut crystal depolarization. In the worst case, the depolarization is about 1.6 times the depolarization for the [111]-cut crystal. The best results we achieved with the [110]-cut crystal are comparable with the worst case of the [100]-cut crystals, and worse than all the other crystals with worst polarization orientation. The fraction of depolarization still increases almost linearly with the pump power, confirming the low power pumping case. The comparison of the experimental curves in Fig. 9 with the theoretical curves in Fig. 2 (left) shows a good agreement in terms of the qualitative differences between the crystals. Since the calculations had been done with a homogeneous pump light distribution assumed, the simulated numbers are lower than the measured ones. According to Fig. 2 (right) the lowest depolarization can be expected for the [110]-cut at pump power levels above 2 kW. Due to limited pump power, this regime was not accessible in our experiments. However, an experimental verification of reduction of depolarization by use of a [110]-cut crystal has been recently reported by Mukhin et al. [13

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

].

5. Conclusion

In the theory section analytical results for the depolarization of a Gaussian beam for [111]-crystals and for favorable and disadvantageous orientations of [100]-crystals under different pumping conditions were derived. We deduced an equation that makes comparisons of the depolarization of the three orientations easy, once the stress-optic tensor is known, since it mainly depends on the off-diagonal elements Bxy of the impermeability matrix B. We also found the stress-induced lens for [100]-oriented crystals.

The depolarization behavior in [111]-, [100]-, and [110]-cut Nd:YAG rods was experimentally investigated by a probe beam. An excellent agreement between theory and experiment was found. For less than 2 kW pump power, the least depolarization has been achieved in a [100] cut crystal. Using this crystal, a decrease of depolarization by a factor of 6 has been demonstrated.

We expect also that the depolarization in linearly polarized laser operation can be reduced by use of non-conventionally cut crystals. While for [111]-cut crystals the polarization direction does not matter, it will be necessary to take care of the orientation of [110]-cut or [100]-cut crystals with respect to the laser’s polarization direction to achieve minimal depolarization.

Appendix

This appendix provides results presented in Section 2: the form of the strain, the principal directions and values of the birefringence matrix, the form of the elasto-optic tensor in the four orientations of interest, the value of the stress-induced thermal lens for 45°-oriented, [100]-cut crystal, and the average of the depolarization over a Gaussian beam for this case.

For a rod of radius r0 and length L, which is uniformly heated by power Pheat, the principal directions of the elastic strain are radial, angular, and longitudinal with magnitudes [10

10. 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]

]:

εr=S[3ν1+(17ν)r2/r02],εφ=S[3ν1+(35ν)r2/r02],εz=S[2(1ν)r2/r02],
(16)
S=αPheat16πLκ(1ν),

with α the coefficient of thermal expansion, κ the thermal conductivity and ν Poisson’s ratio. Translating these to transverse Cartesian coordinates, with ϕ the angle to the x-axis,

(εxxεxyεxyεyy)=(εrcos2φ+εφsin2φ(εrεφ)sinφcosφ(εrεφ)sinφcosφεrsin2φ+εφcos2φ)
(17)

Substituting in Eq. (16), the Cartesian components are

εxx=3ν1+(17ν)x2/r02+(35ν)y2/r02,εyy=3ν1+(35ν)x2/r02+(17ν)y2/r02,εxy=2(1+ν)xy/r02
(18)

in units of S.

The same process can be used in reverse with the birefringence tensor. Call its principal values B 1 and B 2 and say principal direction 1 makes angle θ with the x-axis. Then assuming the form Eq. (17) and using double-angle formulas, we find

B1+B2=Bxx+Byy,(B1B2)sin(2θ)=2Bxy,(B1B2)cos(2θ)=BxxByy
(19)

which imply principal directions and values

tan(2θ)=2BxyBxxByy,B1,2=12(Bxx+Byy)±[12(BxxByy)]2+Bxy2
(20)

These relations allow for the simplification of the depolarization [Eq. (3)] for small birefringence. From Eq. (19) we have that (B 1-B 2)sin(2θ) = 2B xy. Using this and Eqs. (1) and (4) we can approximate for small ψ

D=sin2(2θ)sin2(ψ/2)(1/2ψsin(2θ))2(πLn03/λ)2Bxy2,
(21)

which is reasonable since B xy is the coupling between polarizations.

The elasto-optic tensor p is rank-4. The two rank-2 tensors it relates are symmetric and three-dimensional, so they have six independent components each. Thus p can be written as a 6 × 6 matrix. Following Nye [9

9. J. F. Nye, Physical properties of crystals (Oxford University Press, 1957, 1985).

], to account for double-counting the symmetric strain, its off-diagonal elements get a factor 2. That is, we call the vectors

B=(BxxByyBzzByzBzxBxy),ε=(εxxεyyεzz2εyz2εzx2εxy).
(22)

YAG is a cubic crystal of symmetry class m3m. Nye [9

9. J. F. Nye, Physical properties of crystals (Oxford University Press, 1957, 1985).

] shows that the form of the birefringence tensor is

p=(p11p12p12000p12p11p12000p12p12p11000000p44000000p44000000p44).
(23)

For isotropic materials (e.g., glass), p 44=(p 11-p 12)/2. This provides a good check of all nontrivial formulas derived.

To find the form of the elasto-optic tensor in orientations not along the crystal axes we can rotate p as a rank-4 tensor. Alternately, knowing how vectors B and ε rotate, Eqs. (17) and (22), we can directly rotate the 6 × 6 matrix p. Write the transverse part of the birefringence tensor as

(BxxByyBxy)=P(εxxεyy2εxy)+p  εzz
(24)

in terms of a matrix P and a vector p. The results for the four orientations of interest are

P1000=(p11p120p12p11000p44),p1000=(p12p120)P10045=12(p11+p12+2p44p11+p122p440p11+p122p44p11+p12+2p44000p11p12),p10045=(p12p120)P1100=((1/2)(p11+p12)+p44p120p12p11000p44),p1100=((1/2)(p11+p12)p44p120)P111=16(3p11+3p12+6p44p11+5p122p440p11+5p122p443p11+3p12+6p44000p11p12+4p44),p111=16(2p11+4p124p442p11+4p124p440)
(25)

For [110], y is along a crystal axis and x is at 45°. These forms, along with Eqs. (18) and (20) produce the principal directions and birefringences, Eqs. (6)(9).

Diagonal elements B xx and B yy describe the focusing of x- and y-polarized light, respectively. Quadratic terms are the lens power. Substituting Eq. (18) into Eq. (24) we can express, for x-polarized light, B xx = (S/r 0 2)(c xx x 2 + c xy y 2). For [111] and for [100]-45° we find

cxx111=13[(717ν)p11+(1731ν)p128(1+ν)p44]=0.206cxy111=13[(915ν)p11+(1533ν)p12]=0.03cxx100=2[(13ν)p11+(35ν)p12(1+ν)p44]=0.171cxy100=2[(13ν)p11+(35ν)p12+(1+ν)p44]=0.136.
(26)

Compare this with the direct thermal lens. The two lenses are

nthermallens=n0dndTQ4κ(x2+y2),nstress=n012n03B,

where Q =P heat/(π r 0 2 L) is the heat density and dn/dT = 7 × 10−6 /°C. Thus in each direction the ratio of stress to thermal lens is

1/2n03S/r02dn/dTQ/(4κ)c=n03α8(1ν)dn/dTc
(27)

By a handy coincidence, the coefficient of c equals 1 for YAG. So the coefficients c [Eq. (26)] describe the strength of the stress-induced lens relative to the direct thermal lens. Diagonal terms (e.g., c xx) are lens in the polarization direction and off-diagonal are lens in the opposite spatial direction. The results for [111] are long known (see, for example [11

11. W. Koechner, Solid-State Laser Engineering (2nd ed., Springer, 1988).

]). The stress lens for [100]-45° is smaller and more asymmetric than for [111].

Finally, we average the depolarization of [100]-45° over a Gaussian beam to derive the result given in Eq. (14). The calculation for [100]-0° is similar and for [111] is easier. In the depolarization product [Eq. (3)] the geometrical factor is

cos2(2θ)=k2cos2(2φ)sin2(2φ)+k2cos2(2φ),
(28)

using the expression in Eq. (6) and the definitions in Eq. (13). The argument of the evolution factor in Eq. (3) can be written as

ψ/2=2bksin2(2φ)+k2cos2(2φ)(rrg)2,
(29)

using Eqs. (5), (8), and the definitions in Eqs. (13) and (14). Averaging this depolarization product over a normalized Gaussian,

D¯10045=2πrrg202πk2cos2(2φ)dφsin2(2φ)+k2cos2(2φ)Ir,
(30)

where the radial integral

Ir=0drrexp(2r2/rg2)sin2(2bksin2(2φ)+k2cos2(2φ)(rrg)2).
(31)

This radial integral can be expressed as the sum of three Gaussian integrals and evaluated to yield

D¯10045=14π(2bk)202πk2cos2(2φ)dφ1+(2b/k)2sin2(2φ)+(2b)2cos2(2φ).
(32)

Using a special case of Eq. (3.615.1) of [14

14. I. S. Gradshteyn, and I. M. Ryzhik, Table of Integrals, Series, and Products (7th ed., Academic Press, 2007)

],

02πcos2(ψ)dψ1+a2sin2(ψ)=2π1+a2+1,
(33)

we obtain the final result, Eq. (14).

Acknowledgement

The authors would like to thank Gerald Mitchell from Precise Light Surgical, who encouraged this work and also introduced us to start this collaboration. This work was supported by the German Volkswagen Stiftung.

References and links

1.

M. P. Murdough and C. A. Denman, “Mode-volume and pump-power limitations in injection-locked TEM00 Nd:YAG rod lasers,” Appl. Opt. 35(30), 5925–5936 (1996). [CrossRef] [PubMed]

2.

Q. Lü, N. Kugler, H. Weber, S. Dong, N. Müller, and U. Wittrock, “A novel approach for compensation of birefringence in cylindrical Nd: YAG rods,” Opt. Quantum Electron. 28(1), 57–69 (1996). [CrossRef]

3.

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]

4.

J. J. Morehead, “Compensation of laser thermal depolarization using free space,” IEEE J. Sel. Top. Quantum Electron. 13(3), 498–501 (2007). [CrossRef]

5.

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]

6.

L. N. Soms, A. A. Tarasov, and V. V. Shashkin, “Problem of depolarization of linearly polarized light by a YAG:Nd3+ laser-active element under thermally induced birefringence conditions,” Sov. J. Quantum Electron. 10(3), 350–351 (1980). [CrossRef]

7.

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]

8.

R. W. Dixon, “Photoelastic properties of selected materials and their relevance for applications to acoustic light modulators and scanners,” J. Appl. Phys. 38(13), 5149–5153 (1967). [CrossRef]

9.

J. F. Nye, Physical properties of crystals (Oxford University Press, 1957, 1985).

10.

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]

11.

W. Koechner, Solid-State Laser Engineering (2nd ed., Springer, 1988).

12.

R. Wilhelm, D. Freiburg, M. Frede, D. Kracht, and C. Fallnich, “Design and comparison of composite rod crystals for power scaling of diode end-pumped Nd:YAG lasers,” Opt. Express 17(10), 8229–8236 (2009). [CrossRef] [PubMed]

13.

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

14.

I. S. Gradshteyn, and I. M. Ryzhik, Table of Integrals, Series, and Products (7th ed., Academic Press, 2007)

OCIS Codes
(140.3530) Lasers and laser optics : Lasers, neodymium
(140.6810) Lasers and laser optics : Thermal effects

ToC Category:
Lasers and Laser Optics

History
Original Manuscript: July 14, 2010
Revised Manuscript: September 2, 2010
Manuscript Accepted: September 5, 2010
Published: September 10, 2010

Citation
Oliver Puncken, Henrik Tünnermann, James J. Morehead, Peter Weßels, Maik Frede, Jörg Neumann, and Dietmar Kracht, "Intrinsic reduction of the depolarization in Nd:YAG crystals," Opt. Express 18, 20461-20474 (2010)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-19-20461


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References

  1. M. P. Murdough and C. A. Denman, “Mode-volume and pump-power limitations in injection-locked TEM00 Nd:YAG rod lasers,” Appl. Opt. 35(30), 5925–5936 (1996). [CrossRef] [PubMed]
  2. Q. Lü, N. Kugler, H. Weber, S. Dong, N. Müller, and U. Wittrock, “A novel approach for compensation of birefringence in cylindrical Nd: YAG rods,” Opt. Quantum Electron. 28(1), 57–69 (1996). [CrossRef]
  3. 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]
  4. J. J. Morehead, “Compensation of laser thermal depolarization using free space,” IEEE J. Sel. Top. Quantum Electron. 13(3), 498–501 (2007). [CrossRef]
  5. 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]
  6. L. N. Soms, A. A. Tarasov, and V. V. Shashkin, “Problem of depolarization of linearly polarized light by a YAG:Nd3+ laser-active element under thermally induced birefringence conditions,” Sov. J. Quantum Electron. 10(3), 350–351 (1980). [CrossRef]
  7. 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]
  8. R. W. Dixon, “Photoelastic properties of selected materials and their relevance for applications to acoustic light modulators and scanners,” J. Appl. Phys. 38(13), 5149–5153 (1967). [CrossRef]
  9. J. F. Nye, Physical properties of crystals (Oxford University Press, 1957, 1985).
  10. 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]
  11. W. Koechner, Solid-State Laser Engineering (2nd ed., Springer, 1988).
  12. R. Wilhelm, D. Freiburg, M. Frede, D. Kracht, and C. Fallnich, “Design and comparison of composite rod crystals for power scaling of diode end-pumped Nd:YAG lasers,” Opt. Express 17(10), 8229–8236 (2009). [CrossRef] [PubMed]
  13. I. Mukhin, O. Palashov, and E. Khazanov, “Reduction of thermally induced depolarization of laser radiation in [110] oriented cubic crystals,” Opt. Express 17(7), 5496–5501 (2009). [CrossRef] [PubMed]
  14. I. S. Gradshteyn, and I. M. Ryzhik, Table of Integrals, Series, and Products (7th ed., Academic Press, 2007)

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