1. Introduction

Since passively Q-switched microchip laser sources enables over the MW-class laser output [1], the concept of “laser ignition” has been considered realistic due to the possibility of the ubiquitous laser plug [2]. In reality, the driving of automobile that equipped the heat engine ignited by microchip laser plug was already demonstrated [3], and the output of high-power Q-switched microchip lasers reached sub-GW-level [4]. Without Cr⁴⁺-doped Y₃Al₅O₁₂ (Cr⁴⁺:YAG) as an inorganic saturable absorber that can handle the high power photon energy, this success of passively Q-switched microchip laser would have never been realized. Although passive Q-switching was discovered over 50 years ago, the generation of giant-pulses by this method had been not practical due to poor durability of saturable absorbers. Thus, Cr⁴⁺:YAG has been interested and deeply evaluated.

As a result of plenty studies concentrated into the evaluation of Cr⁴⁺:YAG [5–8], the saturable absorption in Cr⁴⁺:YAG and its possibility for the saturable absorber became well-known. During the development of giant-pulsed microchip laser sources with Cr⁴⁺:YAG, the anisotropy of the saturable absorption in Cr⁴⁺:YAG with the isometric crystal structure has been pointed [8]. Although the mechanism about the appearance of anisotropy have been still unknown, it was found that this anisotropy can produce the modulation of the pulse duration [9] and single-polarized giant pulses without other polarization elements in the microchip laser cavity [10, 11]. Because the polarization control is quite useful for various applications such as the wavelength conversion including THz generation [12], it is necessary for the improvement of the extinction ratio of giant pulses from microchip lasers to explain of the polarization dependence in Cr⁴⁺:YAG.

In reality, currently we can never request the designed concentration of Cr⁴⁺-ion when we purchase a commercial Cr⁴⁺:YAG crystal. We can appoint only one specification of the thickness and the initial transmittance T₀. Thus, it is not realistic to use the exact value of absorption cross sections. In addition, although we can order 2-didits for T₀ of commercial Cr⁴⁺:YAG crystals only one digit is a significant figure. It means that the practical solution of the saturable absorption in Cr⁴⁺:YAG using minimum parameters based on physically appropriate approximations has been truly desired.

Before we consider the anisotropy of Cr⁴⁺:YAG, we define the cylindrical coordination (r, θ, z) that pump beam with the intensity of I propagate inside Cr⁴⁺:YAG along z-axis. Here the direction of the beam radius and the pump polarization angle can be directly described by use of coordinates r and of θ. The relation between the direction of the pump polarization and crystal axes of Cr⁴⁺:YAG crystal is defined by angles (Θ, Φ) as illustrated in Fig. 1.

 

Fig. 1 Representation of the polarization by use of the polar coordinate with angles Θ and Φ.

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In this work, we investigated the polarization dependence of the transmittance in Cr⁴⁺:YAG, and found how to describe it by means of the polarization angles Θ and Φ. Especially also the formula with the polarization angle θ in several specific planes of (100), (110), and (111) are proposed. We also confirmed this model experimentally by use of CW pump sources.

2. Model for polarization dependence

It is found that the expression of the saturable absorption coefficient α(I, Θ, Φ) in Cr⁴⁺:YAG under the steady-state condition can be expressed by use of only three parameters of the initial (unsaturated) absorption coefficient α₀, the parameter β, and the saturation intensity I_s as (See Appendix 1)

Here we consider the effective absorption coefficient for representative incident planes for Cr⁴⁺:YAG. In the case with (100)-cut Cr⁴⁺:YAG crystals, θ should be the angle between the polarization direction and the [010]-axis as shown in Fig. 2. Under this definition, Θ equals to $\pi /2-\theta$ and Φ equals to π/2. Thus, Eq. (1) brings the effective absorption coefficient α₁₀₀(I, θ) for (100)-cut Cr:YAG crystals as

 

Fig. 2 Definition of θ for (100)-cut Cr⁴⁺:YAG crystals. (a) The relation between θ and crystal axes described with the view direction of the [100]-axis. The square and tetrahedrons indicate the unit cell of YAG crystal and S₄ site in the unit cell, respectively. (b) The relation between θ, Θ, and Φ. The gray surface indicates the (100)-plane. r- axis and z-axis are parallel to the direction of the polarization and the direction of [$\overline{1}00$], respectively.

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Next, we consider the saturable absorption against (110)-cut Cr⁴⁺:YAG crystals. Figure 3 shows that θ should be the angle between the polarization direction and the [001]-axis, and Θ equals to θ and Φ equals to -π/4. Thus, Eq. (1) brings the effective absorption coefficient α₁₁₀(I, θ) for (110)-cut Cr:YAG crystals as

 

Fig. 3 Definition of θ for (110)-cut Cr⁴⁺:YAG crystals. (a) The relation between θ and crystal axes described with the view direction of the [110]-axis. The square and tetrahedrons indicate the unit cell of YAG crystal and S₄ site in the unit cell, respectively. (b) The relation between θ, Θ, and Φ. The gray surface indicates the (110)-plane. r- axis and z-axis are parallel to the direction of the polarization and the direction of [$\overline{1}\overline{1}0$], respectively.

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We also show the notation for the saturable absorption against (111)-cut Cr⁴⁺:YAG crystals. Although relations between θ, Θ, and Φ are not so simple as previous 2-planes, Fig. 4 indicates that θ should be the angle between the polarization direction and the $[11\overline{2}]-\text{axis}$. In this case, Θ and Φ are given by

 

Fig. 4 Definition of θ for (111)-cut Cr⁴⁺:YAG crystals. (a) The relation between θ and crystal axes described with the view direction of the [111]-axis. The square and tetrahedrons indicate the unit cell of YAG crystal and S₄ site in the unit cell, respectively. (b) The relation between θ, Θ, and Φ. The gray surface indicates the (111)-plane. r- axis and z-axis are parallel to the direction of the polarization and the direction of [$\overline{1}\overline{1}\overline{1}$], respectively.

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Figure 5 shows the calculated α₁₀₀(I, θ), α₁₁₀(I, θ), and α₁₁₁(I, θ) for Cr⁴⁺:YAG with T₀ of 75% and the thickness l of 1 mm. According to Lambert’s law, α₀ is calculated to be 2.8 cm⁻¹. In this calculation β and the ratio between I and I_s are assumed to be 0.28 and 27.6, respectively. Since I_s can be calculated to be 4.3 kW/cm² by use of previously reported σ_abs and τ [13], this calculation relates to the condition where pump with the power of 1W is focused into the spot with radius of 25 μm. It was found that the saturation is caused not by the direction of wavevectors but by only the polarization of the pump source. In other words, anisotropy comes from the nonlinearity: when Cr⁴⁺ ions are affected by the allowed pump with the intensity I with the electric field E is parallel to the [100] axis, by use of the same pump Cr⁴⁺ ions affected by I/2 and I/3 under E parallel to [110] and E parallel to [111], respectively. The result of this calculation is consistent to the previously reported anisotropy [5, 10].

 

Fig. 5 Estimated polarized absorption coefficients of (100)-, (110)-, and (111)-cut Cr⁴⁺:YAG with α₀ of 2.8 cm⁻¹. β and the ratio between I and I_s are assumed to be 0.28 and 27.6, respectively.

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3. Experimental setup and result

We chose a (110)-cut Cr⁴⁺:YAG with the thickness l of 1.0 mm and T₀ of 75% (Scientific Materials) for the experimental confirmation of the saturable absorption in Cr⁴⁺:YAG, because the (110)-plane in cubic system include all important represented vectors such as [100], [110], [111], [211], and so on. Figure 6 shows the experimental setup for the evaluation of the saturable absorption. The probe source was 1-W single-mode Nd:YAG laser with the wavelength of 1064 nm (Mephisto, Coherent), and its polarization was controlled by a λ/2-plate (WPH05M, Thorlabs). This probe beam was chopped to 25 Hz with the duration of 10 ms (25% duty cycle) in order to prevent the excess heating of the Cr⁴⁺:YAG crystal. The beam was focused onto the sample by use of an aspheric lens with focusing length of 50 cm, and the radius was evaluated by knife-edge method to be 18 μm on the sample surface, and the mean radius w inside Cr⁴⁺:YAG was 25 μm (with the minimum spot radius of 18 μm and the confocal length of 1.2 mm). The transmitted power through Cr⁴⁺:YAG was detected by the photo-energy sensor (PE25BF-C, Ophir).

 

Fig. 6 The schematic diagram of experimental setup.

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Measured transmittance T and the calculations of T using Eq. (4) and Lambert’s law are shown in Fig. 7. Even though the evaluated dependence on the polarization angle was similar to calculations, the difference of the modulation depth in T was not negligible. Since the reliability of the beam radius evaluated by the knife-edge method is high, we have to conclude it was very difficult to fit the evaluated transmittance to Eq. (4) using Lambert’s law and the estimated I_s from the previous report. In following section, we tried to explain this difference between Eq. (4) and the experimental result.

 

Fig. 7 Transmittance of a (110)-cut Cr⁴⁺:YAG with T₀ of 75% under 1-W pumping focused into the spot with the radius of 25 μm. Black marker is the experimental value, and red line, blue line, and green lines are the calculation by use of Eq. (4) with β of 0.3, 0.4, and 0.5, respectively.

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4. Discussions

Influence of pump depletion

In order to describe the experimental transmittance by Eq. (4), we have to consider also the beam mode and the influence of pump depletion inside Cr⁴⁺:YAG, because performances categorized nonlinear phenomena severely depend on the intensity of probe sources. We can assume that power of pump beam along z-axis P(z) is the function of z, and it varies during the beam propagation according to (See Appendix 2):

 

Fig. 8 Transmittance of a (110)-cut Cr⁴⁺:YAG crystal with T₀ of 75% unde 1-W pumping focused into the spot with the radius of 25 μm. Red marker is the experimental value, and blue line is the calculation by use of Eqs. (4) and (7) with β of 0.28.

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Saturable absorption by giant pulse

We have to consider also the transient saturation for pulsed optical sources. In recent high power microchip lasers that can generate giant pulses [1–4], the photon intensity inside the laser cavity is much larger than I_s. This indicates that the rise time of these giant pulses is of the order of below 10 ps, because the ratio between this rise time and the radiative decay time of the meta-stable level in Cr⁴⁺:YAG τ is comparable to the ratio of I_s and I_i. For example, the time constant of the absorption saturation in single crystalline Cr⁴⁺:YAG for E//[100] and E //[110] are three and two times faster than that for E //[111]. This difference in the time constant enables to control the polarization of the oscillation mode by means of [110]-cut Cr⁴⁺:YAG crystals for saturable absorbers. The rise time with the order of several 10ps is short enough compared to the pulse-duration. Thus, it should be possible to apply our model based on the continuous pumping to the lasing of the passive-Q switched lasers that can generate giant pulses. In future we will examine the anisotropy of Cr⁴⁺:YAG in passive-Q switched laser cavities as in Ref. [13].

5. Summary

The model of the polarization dependence in Cr⁴⁺:YAG was established, where the notation of the saturable absorption by numerical functions of the angle θ between representative axes of incident surfaces and the polarization of pump source well described with only three spectroscopic parameters of α₀ (or T₀), β, and I_s. Our novel model for the anisotropy in saturable absorption should be useful to design the extinction ratio and the build-up time of the pulse generation in the passively Q-switched microchip lasers.

Appendix 1 Derivation of Eq.(1)

The site symmetry in crystals is well described in the authoritative tabulations [14]. Figure 9(a) shows the crystal structure of YAG crystal, where Cr⁴⁺-ions can be substituted Al³⁺-ions in YAG is located at 24d Wyckoff Position (WP) with site symmetry of the fourfold inversion (S₄) [15]. WP 24d is surrounded by four O^2--ions that form a tetrahedron. There are 24 S₄-sites in the unit cell, and each of [100]-, [010]-, and [001]-axis is parallel to 8 S₄-axes in these sites as shown in Fig. 9(b).

 

Fig. 9 Unit cell in the crystal structure of YAG. (a) The position of all ions in unit cell, which contains 24 Y³⁺-ions with site symmetry of D₂ (gray spheres), 16 Al³⁺-ions with site symmetry of S₆ (spheres in hexahedrons), 24 Al³⁺-ions with the site symmetry of S₄ (spheres in tetrahedrons), and 96 O^2--ions with the site symmetry of C₁ (red spheres). (b) The projection of 24 Al³⁺-ions with site symmetry of S₄ from the [100]-direction.

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If we set z-axis along the direction of one of S₄–axis, electrons at WP 24d have the following wave-function ϕ with the symmetry of

(8)$$\varphi \left(x,y,z\right)=\varphi \left(y,-x,-z\right)=\varphi \left(-x,-y,z\right)=\varphi \left(-y,x,-z\right).$$

As a result, the transition dipole moments along x- axis, y-axes, and z-axis become zero, zero, and non-zero, respectively. Thus we can treat transitions in Cr⁴⁺-ions with the polarization parallel to S₄-axis is allowed, while transitions with the polarization perpendicular to S₄-axis is forbidden. Authors’ group already observed this situation experimentally, where the cross section for allowed polarization is two orders of magnitude greater than forbidden polarization [13]. It means that optical pumping with the polarization parallel to [100]-axis can excite one third of Cr⁴⁺-ions doped into YAG crystals.

Here we describe the mechanism of anisotropy of the saturable absorption in Cr⁴⁺:YAG. Of course, the absorption cross section itself is a second-ordered tensor that can show no polarization dependence in the cubic crystalline system. On the contrary, the cascade process between the absorption and the excited state absorption include two second-ordered tensors and it can perform anisotropy even in cubic systems such as the garnet structure. According to the representative energy diagram in Cr⁴⁺:YAG [8] as shown in Fig. 10, we can set up the rate equation as following:

(9)$$\frac{d{n}_2^i}{dt}=\frac{1}{\tau }\frac{I_i}{I_s}\frac{n_{tot}}{3}-\frac{1}{\tau }\left(1+\frac{I_i}{I_s}\right)n_2^i,$$

where I_i, n_tot, and n₂ⁱ are the pump intensity with the polarization parallel to i-axis of Cr⁴⁺:YAG, the number density of Cr⁴⁺-ions in Cr⁴⁺:YAG, and the population density of the Cr⁴⁺-ions in the meta-stable state with S₄-axis parallel to i-axis, respectively. The suffix i indicates one of [100]-, [010]-, and [001]-axes.

 

Fig. 10 Energy diagram of Cr⁴⁺:YAG, where A, B, and E are irreducible representations under S₄ symmetry. Related transitions with cross sections (σ_abs, σ_esa), radiative the decay time (τ), and the population density (n) are also indicated.

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By use of angles (Θ, Φ), the pump intensity I_i can be expressed by

(10a)$$I_{[100]}=I{\cos }^2\Phi {\sin }^2\Theta ,$$

(10b)$$I_{[010]}=I{\sin }^2\Phi {\sin }^2\Theta ,$$

(10c)$$I_{[001]}=I{\cos }^2\Theta .$$

Under the assumption that there is an excited state absorption (ESA) from the meta-stable state with the cross section of σ_esa and the much faster decay of ESA upper level, the pump absorption in Cr⁴⁺:YAG with the initial transmittance T₀ and the beam-path length l can be given by

(11)$$\frac{d{I}_i}{dz}=-\left(\frac{n_{tot}}{3}-n_2^i\right){\sigma }_{abs}{I}_i-n_2^i{\sigma }_{esa}{I}_i=-{\alpha }_0{I}_i\left[1-\left(1-\beta \right)\frac{3n_2^i}{n_{tot}}\right].$$

where the initial (unsaturated) absorption coefficient α₀ and β are expressed by

(12a)$${{\rm T}}_0=\exp \left(-{\alpha }_{0}l\right)=\exp \left(-\frac{n_{tot}}{3}{\sigma }_{abs}l\right),$$

(12b)$$\beta =\frac{{\sigma }_{esa}}{{\sigma }_{abs}}.$$

We can obtain the expression of the saturable absorption coefficient α(n₂ⁱ, I_i) from Eq. (11) as

(13)$$\alpha \left(n_2^i,I_i\right)={\alpha }_0\left[1-\left(1-\beta \right)\frac{3}{n_{tot}}{\displaystyle \sum _i{n}_2^i\frac{I_i}{I}}\right].$$

Especially under the steady-state condition, Eq. (9) gives the notation of n₂ⁱ as

(14)$$n_2^i=\frac{n_{tot}}{3}\frac{I_i}{I_i+I_s}.$$

From Eqs. (10), (13), and (14) the expression of the saturable absorption coefficient α(I, Θ, Φ) under the steady-state condition is found to be Eq. (1).

Appendix 2 Derivation of Eq.(7)

If the beam is a Gaussian beam with the radius of w, the pump intensity I can be expressed by

(15)$$I\left(z,r\right)=\frac{2P\left(z\right)}{\pi w^2}\exp \left(-\frac{2r^2}{w^2}\right).$$

We can describe the pump depletion inside Cr⁴⁺:YAG as a differential equation by

(16)$$\frac{dP\left(z\right)}{dz}=-{\displaystyle {\int }_0^{\infty }2\pi r{\alpha }_j\left(I,\theta \right)I\left(z,r\right)dr},$$

where j is a subscript that indicates incident direction of pump beam. We can derive Eq. (7) applying the following substitution:

(17)$$k=\exp \left(-\frac{2r^2}{w^2}\right).$$

In addition, we can express the transmittance T as an explicit function of the pump input power P_in and θ of the pump polarization angle by use of the integral equation form from

(18)$${\displaystyle {\int }_0^{l}dz}=-{\displaystyle {\int }_{P\left(0\right)}^{P\left(l\right)}\frac{dP}{P{\displaystyle {\int }_0^{1}dk{\alpha }_j\left(\frac{2kP}{\pi w^2},\theta \right)}}},$$

where $z=0$ at the beam incident surface and $z=l$ at the beam transmitted surface of the sample. Finally we obtain

(19)$$l={\displaystyle {\int }_{T\left(P_{in},\theta \right)P_{in}}^{P_{in}}\frac{dP}{P{\displaystyle {\int }_0^1{\alpha }_j\left(\frac{2kP}{\pi w^2},\theta \right)dk}}}.$$

Funding

This work is partially supported by Genesis Research Institute, Inc. and New Energy and Industrial Technology Development Organization (NEDO).

Acknowledgments

Authors thank Prof. B. Boulanger for his comment. Authors also thank Dr. R. Bhandari for his check of the manuscript.

References and links

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