## Intrinsic reduction of the depolarization in Nd:YAG crystals |

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

1. M. P. Murdough and C. A. Denman, “Mode-volume and pump-power limitations in injection-locked TEM_{00} 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:Nd^{3+} 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 Y_{3}Al_{5}O_{12} crystal,” Appl. Phys. Lett. **80**(17), 3048–3050 (2002). [CrossRef]

## 2. Theoretical background

*I/n*, where

_{0}^{2}+ B*I*is the 3-dimensional identity matrix,

*n*is the unstressed index of refraction, and

_{0}*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:

*p*= −0.029,

_{11}*p*= 0.009, and

_{12}*p*= −0.0615 [8

_{44}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]

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]

*B*

_{1,2}are the eigenvalues of the transverse

*B*-matrix, which iswith the beam’s direction as

*z*. The matrix’s eigendirections are the principal polarizations.

*L*iswhere

*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

7. I. Shoji and T. Taira, “Intrinsic reduction of the depolarization loss in solid state lasers by use of a (110)-cut Y_{3}Al_{5}O_{12} crystal,” Appl. Phys. Lett. **80**(17), 3048–3050 (2002). [CrossRef]

*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 Y_{3}Al_{5}O_{12} crystal,” Appl. Phys. Lett. **80**(17), 3048–3050 (2002). [CrossRef]

*ϕ*is the angular coordinate with respect to the x-axis, the principal directions areand for [110],

*γ*= 0), then the depolarization of a ray is approximatelyinvolving 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*·ε

_{xy}, 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

_{xy}.

*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].

*r*

_{g}. In that case three of the four orientations of interest can be integrated in closed form (see appendix).

## 3. Numerical simulations

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]

*B*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.

_{xy}*P*, which is scaled by the squared ratio

_{heat}*r*. Therefore, the depolarization of a [111]-cut crystal saturates at the same value of 25%, independently of the probe beam size

_{g}/r_{0}*r*, although the point of saturation will be shifted to higher pump powers for smaller probe beam radii [Fig. 3(a) ].

_{g}*r*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

_{g}*r*and reaches a maximum at

_{pump}*r*. 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

_{pump}= r_{g}*r*=

_{g}*r*. 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.

_{pump}## 4. Single pass experiments

*r*in this case, which causes a different depolarization pattern for various beam sizes. For

*r*, the last term of the denominator in Eq. (7) becomes large, which means that the principal axes are almost orthogonal with respect to the [

_{0}>>r_{g}*r*= 150 µm,

_{g}*r*= 350 µm,

_{g}*r*= 650 µm, and

_{g}*r*= 900 µm. The graphs shown in Fig. 6 , 7 , and 8 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.

_{g}^{2}

*(2(θ-π/4)) =*cos

^{2}

*(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.

*r*< 0.5

_{g}*r*. At

_{0}*r*≥ 0.5

_{g}*r*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.

_{0}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

*B*of the impermeability matrix

_{xy}*B.*We also found the stress-induced lens for [100]-oriented crystals.

## Appendix

_{0}and length L, which is uniformly heated by power P

_{heat}, 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]

*ϕ*the angle to the x-axis,

*S*.

*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

*B*

_{1}-

*B*

_{2})sin(2

*θ*) = 2

*B*

_{xy}. Using this and Eqs. (1) and (4) we can approximate for small ψ

*B*

_{xy}is the coupling between polarizations.

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

*p*

_{44}=(

*p*

_{11}-

*p*

_{12})/2. This provides a good check of all nontrivial formulas derived.

*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

*P*and a vector

**p**. The results for the four orientations of interest are

*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

*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

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

## Acknowledgement

## References and links

1. | M. P. Murdough and C. A. Denman, “Mode-volume and pump-power limitations in injection-locked TEM |

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. |

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. |

4. | J. J. Morehead, “Compensation of laser thermal depolarization using free space,” IEEE J. Sel. Top. Quantum Electron. |

5. | W. Koechner and D. K. Rice, “Birefringence of YAG:Nd Laser Rods as a Function of Growth Direction,” J. Opt. Soc. Am. |

6. | L. N. Soms, A. A. Tarasov, and V. V. Shashkin, “Problem of depolarization of linearly polarized light by a YAG:Nd |

7. | I. Shoji and T. Taira, “Intrinsic reduction of the depolarization loss in solid state lasers by use of a (110)-cut Y |

8. | R. W. Dixon, “Photoelastic properties of selected materials and their relevance for applications to acoustic light modulators and scanners,” J. Appl. Phys. |

9. | J. F. Nye, |

10. | W. Koechner and D. K. Rice, “Effect of Birefringence on the Performance of Linearly Polarized YAG:Nd Lasers,” IEEE J. Quantum Electron. |

11. | W. Koechner, |

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 |

13. | I. Mukhin, O. Palashov, and E. Khazanov, “Reduction of thermally induced depolarization of laser radiation in [110] oriented cubic crystals,” Opt. Express |

14. | I. S. Gradshteyn, and I. M. Ryzhik, |

**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

- 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]
- 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]
- 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]
- J. J. Morehead, “Compensation of laser thermal depolarization using free space,” IEEE J. Sel. Top. Quantum Electron. 13(3), 498–501 (2007). [CrossRef]
- 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]
- 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]
- 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]
- 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]
- J. F. Nye, Physical properties of crystals (Oxford University Press, 1957, 1985).
- 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]
- W. Koechner, Solid-State Laser Engineering (2nd ed., Springer, 1988).
- 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]
- 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]
- I. S. Gradshteyn, and I. M. Ryzhik, Table of Integrals, Series, and Products (7th ed., Academic Press, 2007)

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