1. Introduction

The generation of a high power laser pulse using the chirped-pulse amplification (CPA) technique [1–3] has given rise to secondary radiation sources such as a high-energy particle beam, a coherent femto-second X-ray, and a laser Compton scattering gamma ray [4–7]. Such secondary radiation sources have contributed to many research fields such as compact laser acceleration, medical applications, nuclear power, and radiation applications [8, 9]. This research was once only possible in a research facility but progress in the development of the high-power laser has made it possible in a laboratory and this paves the way for new industrial applications [10].

The low repetition rate of current high power lasers limits its use in the aforementioned applications. A high energy and high repetition rate laser is required as a pumping source. Some of these pumping lasers have been developed by the HiLASE project [11] and the Mercury laser project [12], and cause 100 times more thermal load inside the Ti:sapphire laser gain medium than inside this medium in the existing peta-watt class CPA laser systems. The existing peta-watt class CPA laser system could generate a few pulses per minute [2, 3]. The thermal load causes a significant thermal stress, without cryogenic cooling [13]. The thermal stress causes a thermal lens effect and birefringence that distort the spatio-temporal beam shape. The depolarization and thermal lensing effect degenerates the beam spatially. These spatial distortions make the temporal distortion due to the birefringence of medium. This distortion occurs when the beam moves through the Ti:sapphire gain medium and could cause detrimental effects. Therefore, it is essential to analyze the thermal stress and the spatio-temporal distortion in order to prevent these adverse effects [14, 15]. The stress and distortion have been studied in an yttrium aluminum garnet (YAG) laser system which is in an isotropic crystal [16–19]. However, there has been no study until now on the thermal stress and birefringence in the Ti:sapphire gain medium, which is an anisotropic and trigonal crystal [20–22].

In this paper, mathematical modeling of the Ti:sapphire medium classified as an anisotropic crystal was performed for the analysis of thermally induced distortion in a CPA system of a particular geometry. Natural birefringence, thermal birefringence and the amplification with a polarization dependency are simultaneously considered by using the differential Jones calculus developed in this work [23]. It is difficult to express the exact variation of the beam in the medium with the conventional Jones matrix. To obtain the distortion of the ultra-intense laser beam and its continuous change, a differential equation model with the Jones matrix was firstly developed. The solution of the equation was found using a matrix decomposition [24].

2. Thermally induced birefringence

To derive the thermally induced birefringence in the Ti:sapphire crystal, its characteristics need to be considered. It has a trigonal structure (D_3d) [21], and its elastic compliance and elasto-optic tensor express the relation between the crystal structure and the refractive index [22]. The relation is shown in Eq. (1). ∆B_m means the variation of a relative dielectric impermeability induced by the stress (σ_n). The tensors in Eq. (1)p_mn and s_rn, contain the relation between the stress in medium and optical characteristic. We have assumed a particular condition for our simulation. The crystal has a diameter of 7 cm and is 2-cm-thick. The pulse energy of the pumping beam is 100 J. The repetition rate of the pumping beam is 10 Hz. We referred to the research conducted within the HiLASE project for these assumptions and through simulations [11].

We assume that the residual heat in the gain medium is uniformly distributed and thus, that the orientation of the temperature gradient is along the radial direction without a dependency of θ. Actually, temperature gradient of sapphire can be a radial oriented since its thermal conductivity has anisotropic [21]. The medium is expanded non-uniformly owing to the non-uniform temperature distribution. A thermal stress is derived by this distribution. The thermal stress in each orientation is shown in Fig. 1. The principal axis of the thermal stress is represented by Fig. 2(a). If the crystal is isotropic, then the axis aligns with the radial direction. However, in the case of the Ti:sapphire medium, the thermal stress principal axis is not in line with the radial direction because the Ti:sapphire is anisotropic.

 

Fig. 1 Illustration of the Ti:sapphire medium and the coordinate used in this paper. θ is a position angle of the polar coordinate. p₁ and p₂ are the principal axis of the stress, and α is the angle between the principal axis and the geometric axis. We also have assumed for perfect linear polarized (x-axis oriented polarization) laser pulse as an incidence pulse for the amplification.

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Fig. 2 Thermal stress analysis: (a) the principal axis of thermal stress, (b) thermal stress along the x-axis, (c) y-axis, and (d) shear stress [Appendix A]

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The derivation of the thermal birefringence from the thermal stress requires the use of the dielectric impermeability tensor (∆B). The principal axis and the disturbance of the dielectric impermeability are shown in Eqs. (2) and (3). The details of the derivation could be found in the appendix of this paper (Appendix A). In Eq. (5), the matrix R means a matrix which rotates angle α(r,θ).

The thermal birefringence (∆k_t(r,θ)) induced by thermal stress is represented in Eq. (4). α is the angle between the principal axis and x-axis. The subscripts, p₁(r,θ) and p₂(r,θ), represent each principal axis of the thermal stress as shown in Fig. 1(a). A_t is a Jones matrix which could explain the variation of E-field by thermal birefringence. The expression shows only the change of E-field after it propagates through the medium caused by the thermal birefringence. In the equation, the propagating length is represented by z. Since the E-field changes continuously and the sapphire crystal has natural birefringence, the differential equation form that contains these effects was used.

3. Complex Jones matrix modeling

To describe the distortion of a laser medium, the changes of the beam which is going through the medium should be defined firstly. In this paper, we consider three effects: the amplification, the natural birefringence, and the thermal birefringence. Since sapphire is a negative uniaxial crystal, there is a natural birefringence [21] (∆k_c = 2π (n_e − n_o)/λ). The thermal birefringence was derived in Section 2. The changes in the beam on going through the medium can be described with these three effects. An amplification (A_g, g means the gain of medium), natural birefringence (A_c), and thermal birefringence (A_t) are described in Eq. (6), respectively. According to the Jones [23], the differential form(N) of a matrix(M) could be shown as N ≡ (dM/dz)M⁻¹. From this Jones expression, we can derive the matrices in Eq. (7). However, since the laser beam experiences all of these effects simultaneously and dependently, the matrix demonstrating laser medium should prove its simultaneity and dependency. We can guess that the N-matrix of laser medium becomes N_Ag +N_Ac +N_At, but there is no explanation on simultaneity and dependency of the laser medium.

To resolve this problem, we used a polynomial summation of Eq. (6). The polynomial summation is Eq. (8). A variable, z in Eq. (8) means the propagation length. We suppose the laser medium system matrix, A_medium, to A_gA_cA_t. Then, A_medium is shown as Eq. (9). In that equation, the communicative law is not satisfied on the high order terms.

Therefore, when the z is a very small value, the communicative law is satisfied. This process seems to be messy, but it is necessary to validate Eq. (10). The change of the E-field can be found as a differential equation with a 2-by-2 matrix. Furthermore, $N_{A_{medium}}$ becomes identical with $N_{A_g}+N_{A_c}+N_{A_t}$. Equation (10) is the first order differential equation system. Thus, the solution can be found easily by decomposing the matrix. [24] It is, to the best of our knowledge, the only model of a laser amplification medium, which has simultaneously considered polarization dependent gain (g_x,g_y), the natural birefringence (∆k_c), and thermal birefringence(∆k_t) of the continuously changing E-field. To prove its reliability, the result was compared with a well-known and commonly accepted result [20].

In the case of YAG medium, an E-field of the depolarized component is shown by Eq. (12), $-i{E}_o{e}^{gz/2}sin\left(2\alpha \right)sin\left(\mathrm{\Delta }{k}_{t}z/2\right)$, with our method, and it is analytically identical with conventional equation of that, even the deriving methods are totally different [20]. As it turned out, both of the mathematical model and the solution are acceptable. We decomposed the system matrix into the eigenmatrix and the eigenvectors in order to find the solution efficiently [Appendix B]. The solutions of YAG and sapphire model are Eqs. (11)–(12) and Eqs. (13)–(14), respectively. In Eqs. (13) and (14), ∆k_n means $\sqrt{\mathrm{\Delta }{k}_c^2+\mathrm{\Delta }{k}_t^2+2\mathrm{\Delta }{k}_c\mathrm{\Delta }{k}_t\cos 2\alpha }$. If we consider the amplification, the analytic solution of Ti:sapphire case is too complex to write down here. That is the reason why we consider only birefringence effects. The solution considering whole effects is introduced in Appendix C.

Thermally induced birefringence is much smaller than the natural birefringence in the case of sapphire. In our simulation conditions, the natural birefringence(∆k_c) is about 93 times larger than the thermal birefringence(∆k_t). Hence, we could make the depolarized component in Eq. (14), $E_y^{sap}$, to iE_o∆k_t/∆k_c sin(2α)sin(∆k_tz/2). From this, we could know that the large natural birefringence suppresses the depolarization loss [25].

4. Simulation analysis

Figure 3 and 4 show the simulation results of the Ti:sapphire and YAG laser medium under the same thermal condition(Q = 4.35W/cm³). The geometrical details are here. In the case of Nd:YAG, Fig. 3(a) and 3(c), the medium has a diameter of 2-cm and a thickness is 8.5-cm. The Ti:sapphire medium(Fig. 3(b) and 3(d)) has a diameter of 7-cm and 2-cm-thickness. We supposed that the geometry of the crystals is nonidentical in order to reflect the actual use of each crystal in the real situation. The depolarization loss means ${\displaystyle \iint {\left|E_y\left(x,y\right)/E_o\left(x,y\right)\right|}^{2}dxdy}$, the two-dimensional integration of the normalized intensity. The loss of linear polarization component increases when the pulse energy of the pumping beam increases. With a pumping beam of which the pulse energy is 100 J and the repetition rate is 10 Hz, the single-pass depolarization loss of the YAG medium is about 24.6 %. Under the same pumping conditions, that of Ti:sapphire is 0.0017 %, Fig. 3(a) and 3(b). The thermally induced focal length is also given in Fig. 4(b). The focal lengths of the thermal lens in Ti:sapphire are about 39 m and 33 m with each orientation. In the case of a four-pass amplification, thermal depolarization and the focal length of the thermal lens are about 0.004 % and 9 m, respectively. We make notes on the analytic details in the Appendix C.

 

Fig. 3 The y-polarized electric field amplitude distribution of YAG, $\left|E_x^{YAG}\right|$ (a) and sapphire, $\left|E_x^{Sappa}\right|$ (b). The wavefront distortion map of YAG (c) and sapphire (d) under the same thermal load condition. (Q = 4.35W/cm³).

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The YAG and Ti:sapphire crystals should have similar thermal characteristics. Even though the each axis has a different level of conductivity, it cannot explain the difference shown in Fig. 3. This difference is because of the natural birefringence in the Ti:sapphire crystal. On the other, the YAG medium has no natural birefringence. In the case of YAG medium, the depolarization loss is significant. In contrast, in the case of the Ti:sapphire crystal, the thermally depolarized component is much smaller because the depolarized component is separated from the main beam due to its natural birefringence. This means that the medium has enough natural birefringence to suppress the depolarization loss [25]. We calculated and made a figure of the loss in Fig. 4(a). In the figure, the loss means the 2-dimensional integration of the normalized intensity of the horizontal-oriented-polarization component which is mentioned in the previous paragraph.

 

Fig. 4 Simulation sapphire vs. power of the pumping beam: (a) Depolarization loss (∫∫|E_y(x,y)/E_o(x,y)|²dxdy), and (b) focal length of thermally-induced lensing effect (Red: focal length of x-axis, Blue: focal length of y-axis).

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There is another thermal effect that is the thermally-induced lensing effect. From wavefront shape, we could calculate the thermally-induced focal length as shown in Fig. 4(b). The focal length is derived from the second order coefficient of wavefront polynomials [25–27]. We focused on not only the thermal lens focal length itself also its astigmatism aberration. The astigmatism in Ti:sapphire medium is already introduced and the reason of this astigmatism has been known as the difference between thermal conductivities of each orientations [26, 27]. Using our simulation, we could calculate the focal length of thermal lensing effect and the focal length of a sagittal orientation and a tangential orientation are about 39 m and 33 m, respectively. The thermally induced wavefront distortion also could be one of the reasons for the astigmatism of thermal lensing not only the difference of the thermal conductivity along the orientations. Furthermore, this thermally induced distortion also could cause temporal distortion. If the post pulse is generated by a medium with a high thermal load, a pre-pulse is generated in the procedure of multi-pass amplification, which is a significant issue for an experiment because pre-pulse arrives at the target before the main pulse [27, 29]. A further research of this paper will cover this temporal distortion.

5. Conclusion

The deformation due to thermal stress in a Ti:sapphire gain medium is derived by a differential Jones calculus. A differential equation model is derived to include effects of the amplification, the natural birefringence, and the thermal birefringence. We have checked the validity of the solution of the differential equation model by comparing it with the well-known results. With a pumping beam pulse energy of 100 J and a repetition rate of 10 Hz, the maximum depolarization loss is 0.0017 %, and the focal lengths of the thermal lens are about 39 m along the sagittal orientation and 33 m along the tangential orientation, respectively. In the case of four-pass simulation, the depolarization loss is 0.004 %. A further research will be carried out to derive temporal distortions from the thermal distortion and to find a method preventing the temporal distortion.

Appendix

This paper contains many equations to describe the thermal birefringence and the distortion of the beam. This appendix provides the details of the mathematical deriving referred in this paper. In the subsection A, we represent the thermally-induced birefringence as the Jones matrix, and then we introduce the matrix decomposition for solving the differential equation in the subsection B. The last subsection contains the analytic solutions of the differential equation system what we can not handle because it is too complex to write in main text. In the following sections, we assume that the Ti:sapphire laser medium has the radius, r_o of 3.5 cm, thickness L in 2 cm, the pumping beam energy in 100 J, the repetition rate in 10 Hz, and the pumping beam is flat-top shape. The Cartesian coordinate is used for the sub-letters. The illustration in Fig. 1 is helpful for your understanding.

A. Thermal birefringence

In cylindrical geometry, the thermal stress is known as Eqs. (15)–(17) [20].

(15)$${\sigma }_r\left(r\right)=QS\left(r^2-r_o^2\right)$$

(16)$${\sigma }_{\theta }\left(r\right)=QS\left(3r^2-r_o^2\right)$$

(17)$${\sigma }_z\left(r\right)=2QS\left(2r^2-r_o^2\right)$$

(18)$$Q=P_{heat}/\pi r_o^{2}L$$

(19)$$S=\alpha E/16\kappa \left(1-\nu \right)$$

P_heat is a power generated by the pumping beam, E is an elastic modulus, α is a thermal expansion coefficient, κ is a thermal conductivity, and ν is a poisson’s ratio of the sapphire. The following equations, Eqs. (20)–(22) are the Cartesian form of cylinderical equations, Equations. (15)–(17).

(20)$${\sigma }_{xx}\left(r,\theta \right)={\sigma }_r\left(r\right){\cos }^2\left(\theta \right)+{\sigma }_{\theta }\left(r\right){\sin }^2\left(\theta \right)$$

(21)$${\sigma }_{yy}\left(r,\theta \right)={\sigma }_r\left(r\right){\sin }^2\left(\theta \right)+{\sigma }_{\theta }\left(r\right){\cos }^2\left(\theta \right)$$

(22)$${\sigma }_{xy}\left(r,\theta \right)=\left({\sigma }_r\left(r\right)-{\sigma }_{\theta }\left(r\right)\right)\cos \left(\theta \right)\sin \left(\theta \right)$$

∆B is a variation of a relative dielectric impermeability. The variation of refractive index could be derived by the variation of the relative dielectric impermeability.

(23)$$B_1{x}_1^2+B_2{x}_2^2+B_3{x}_3^2=1$$

(24)$$\frac{x_1^2}{n_1^2}+\frac{x_2^2}{n_2^2}+\frac{x_3^2}{n_3^2}=1$$

The subscripts 1,2, and 3 in the above equations mean the principal axis of the indicatrix for the crystal. Therefore, we should find both of the principal axis and the variation to find the thermally-induced birefringence. In the our geometrical assumption, the axis of thermal stress is expressed in Eq. (27). The p_mr is an elasto-optics coefficient, and s_rn is an elasto compliance coefficient. These two coefficients matrix are given in Fig. 5.

(25)$$\mathrm{\Delta }{B}_m={\pi }_{mm}{\sigma }_n$$

(26)$${\pi }_{mn}=p_{mr}\cdot S_{rn}$$

(27)$$\left(\begin{array}{c}\mathrm{\Delta }{B}_{zz}\\ \mathrm{\Delta }{B}_{xx}\\ \mathrm{\Delta }{B}_{yy}\\ \mathrm{\Delta }{B}_{xy}\\ \mathrm{\Delta }{B}_{yz}\\ \mathrm{\Delta }{B}_{zx}\end{array}\right)=p_{mr}\cdot s_{rn}\cdot {\sigma }_n={\pi }_{mn}\cdot \left(\begin{array}{c}{\sigma }_{zz}\\ {\sigma }_{xx}\\ {\sigma }_{yy}\\ {\sigma }_{xy}\\ {\sigma }_{yz}\\ {\sigma }_{zx}\end{array}\right)$$

 

Fig. 5 Form of the photo-electric matrices, p_mn of YAG and sapphire. The form of sapphire is more complex than that of YAG.

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Then, we find the angle, α, Equation (28), and the variation of the relative dielectric impermeability, Equation (29), and thermally-induced refractive index, Equation (30), that we have introduced in Section 1, Equations. (2)–(4).

(28)$$\alpha =\frac{1}{2}{\tan }^{-1}\frac{\mathrm{\Delta }{B}_{xy}}{\left(\mathrm{\Delta }{B}_{yy}-\mathrm{\Delta }{B}_{xx}\right)/2}$$

(29)$$\mathrm{\Delta }{B}_{p1}-\mathrm{\Delta }{B}_{p2}=\sqrt{{\left(\left(\mathrm{\Delta }{B}_{yy}-\mathrm{\Delta }{B}_{xx}\right)/2\right)}^2+\mathrm{\Delta }{B}_{xy}^2}$$

(30)$$\mathrm{\Delta }n=-n_o^3\left(\mathrm{\Delta }{B}_{p1}-\mathrm{\Delta }{B}_{p2}\right)/2$$

B. Matrix decomposition

Matrix decomposition is also called the eigen decomposition or the eigen diagonalization. It is useful method for solving a differential equation system. In our case, Equation (10) is the first order differential equation system. We used this method to find the solution efficiently, and now, we introduce about the matrix decomposition method.

We can think of a n-by-n matrix A which has n eigenvectors. Then, we also can assume that a matrix, S of which columns are the eigenvectors of the A. S⁻¹AS becomes the diagonal matrix(Λ) which has a eigenvalue(λ) of the matrix A as its diagonal components [24]. We used this decomposition for solving the differential equation. Our equation has a 2-by-2 matrix as the system matrix.

(31)$$S^{-1}AS=\mathrm{\Lambda }=\left(\begin{array}{ccc}{\lambda }_1& \cdots & 0\\ ⋮& \ddots & ⋮\\ 0& \cdots & {\lambda }_n\end{array}\right)$$

Equation (32) is a differential equation with the n-th order system matrix, A. Now, we will show the process of finding the solution.

(32)$$\frac{d\overrightarrow{X}}{dz}=A\overrightarrow{X}$$

The vector $\overrightarrow{X}$ is expressed by a combination of the eigenvectors. Then,

(33)$$\overrightarrow{X}=S\overrightarrow{Y}$$

(34)$$\frac{d\overrightarrow{X}}{dz}=S\frac{d\overrightarrow{Y}}{dz}$$

(35)$$AS\overrightarrow{Y}=S\mathrm{\Lambda }\overrightarrow{Y}$$

Form Eqs. (33)–(35), we can make a differential equation of $\overrightarrow{Y}$, Equation (36). Because the matrix is a diagonal matrix, the solution of Eq. (36) becomes Eq. (37). ${\overrightarrow{Y}}_o$ means an initial vector.

(36)$$\frac{d\overrightarrow{Y}}{dz}=\mathrm{\Lambda }\overrightarrow{Y}$$

(37)$$\overrightarrow{Y}=e^{\mathrm{\Lambda }z}{\overrightarrow{Y}}_o$$

The ${\overrightarrow{X}}_o$ is the initial value we have already known. Hence, the solution of Eq. (32) becomes Eq. (38). Therefore, the only thing we should do for solving the differential equation is that we multiply the three matrices.

(38)$$\frac{d\overrightarrow{X}}{dz}=S\left(\begin{array}{cc}{e}^{{\lambda }_{x}z}& 0\\ 0& e^{{\lambda }_{y}z}\end{array}\right)S^{-1}{\overrightarrow{X}}_o$$

C. Analytic solutions

In this subsection, we will introduce the analytic solutions of each case. For this calculation, we use the method what we have shown in this paper. In the case of YAG which is one of the isotropic crystals, the analytic solutions are in Eqs. (39) and (40). The subscript x means the vertical orientation, and the subscript y means the horizontal orientation. Equations (42) and (43) are the solutions on the anisotropic crystal.

(39)$$E_x^{YAG}=E_o{e}^{\left(g-i\mathrm{\Delta }{k}_t\right)z/2}\left(e^{i\mathrm{\Delta }{k}_{t}z}{\cos }^2\alpha +{\sin }^2\alpha \right)$$

(40)$$E_y^{YAG}=-i{E}_o{e}^{gz/2}\sin \left(2\alpha \right)\sin \left(\mathrm{\Delta }{k}_{t}z/2\right)$$

(41)$$\mathrm{\Delta }{K}^2=\left({\left(g_x-g_y+2i\mathrm{\Delta }{k}_c\right)}^2-4i\left(g_x-g_y+2i\mathrm{\Delta }{k}_c\right)\mathrm{\Delta }{k}_t\cos \left(2\alpha \right)\right)/4$$

(42)$$E_x^{Sappa}=E_o{e}^{\left(g_x+g_y\right)z/4}\frac{\cosh \left(\mathrm{\Delta }Kz/2\right)+\left(g_x-g_y+2i\left(\mathrm{\Delta }{k}_c+\mathrm{\Delta }{k}_t\cos \left(2\alpha \right)\right)\right)\sinh \left(\mathrm{\Delta }Kz/2\right)}{2\mathrm{\Delta }K}$$

(43)$$E_y^{Sappa}=-i{E}_o{e}^{\left(g_x+g_y\right)z/4}\left(\mathrm{\Delta }{k}_t/\mathrm{\Delta }K\right)\sin \left(2\alpha \right)\sinh \left(\frac{g_x-g_y+2i\mathrm{\Delta }{k}_c}{4}\right)$$

If the crystal natural birefringence is larger enough than the thermal birefringence, then the depolarization loss in Eq. (43) could be written as Eq. (44).

(44)$$E_y^{Sappa}=-i{E}_o{e}^{\left(g_x+g_y\right)z/4}\left(\frac{2\mathrm{\Delta }{k}_t}{g_x-g_y+2i\mathrm{\Delta }{k}_c}\right)\sin \left(2\alpha \right)\sinh \left(\frac{g_x-g_y+2i\mathrm{\Delta }{k}_c}{4}\right)$$

Acknowledgments

This work was supported by the industrial strategic technology development program, 10048964, Development of 125 J·Hz laser system for laser peening funded by the Ministry of Trade, Industry & Energy (MI, Rep. of Korea).

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