Modified Frantz-Nodvik equation and numerical simulation of a high-power Innoslab picosecond laser amplifier

Bingnan Shi; Jiatong Li; Shuai Ye; Hongkun Nie; Kejian Yang; Kejian Yang; Jingliang He; Jingliang He; Jingliang He; Tao Li; Baitao Zhang; Baitao Zhang

doi:10.1364/OE.450145

1. Introduction

High power picosecond laser sources have attracted much attention and play significant roles in the applications such as industrial production, scientific research, medical treatment, and military defense, etc [1–4]. In the last two decades, laser architectures like fiber, thin-disk, and partially end-pumped slab (Innoslab) are used to obtain such laser sources due to the advantages of effectively alleviating thermal effect, the great progress of laser crystals, and the ongoing improvements in power and brightness of commercial laser diodes. Compared with other techniques, the master oscillator power amplifier (MOPA) configuration with picosecond fiber laser as the seed and Innoslab as the laser amplifier ensures considerable compactness and good beam quality while obtaining high power output [5]. Besides, it has much less nonlinearity than the fiber amplifier and more single-pass gain than the thin-disk amplifier [6]. Thus, the Innoslab picosecond amplifier is getting more and more attention presently [7–11]. However, all the reports are about experimental results. Therefore, it is essential to develop a theoretical model for designing and optimizing the high power and high efficiency Innoslab picosecond amplifier.

There are some difficulties in establishing this theoretical model. Firstly, the optical path of Innoslab amplifier is folded and tilted, thus the Frantz-Nodvik (F-N) equation [12] derived for a straight-through optical path cannot be applied directly to simulate the amplified output energy of Innoslab amplifier. Besides, the anisotropy of the stimulated emission cross-section in laser crystals [13] should be considered in the tilted optical path. Secondly, the cross-section of the laser beam increases continuously while passing through the gain medium every roundtrip. As a result, the spatial overlap between the seed and pump laser should be taken into account. Thirdly, the absorption coefficient should be considered to change with different positions in the gain medium because of the nonnegligible pump absorption saturation effect [14].

To solve the above-summarized problems, a simple one-dimensional unfolded slicing model for simulating the Innoslab picosecond amplifier is established for the first time. In the beginning subject, the F-N equation is modified to include the anisotropic stimulated emission cross-section of the laser crystal and the influence of the tilted optical path. Next, the slice model is developed by unfolding the Innoslab amplifier into a stack of amplifiers and using the finite element method to calculate the spatial overlap between the seed and pump laser and the pump absorption saturation effect. Finally, based on the as-established theory, a Nd:YVO₄ Innoslab picosecond laser amplifier is designed and optimized. The output powers as high as 76.2 W, 81.4 W, and 85.5 W are obtained with 4-, 6- and 8-passes schemes. The experimental results agree well with the simulated ones, demonstrating that our model is a powerful tool for designing and optimizing high-power Innoslab picosecond laser amplifiers.

2. Modeling

2.1 Modified F-N equation for Innoslab amplifiers

The schematic of Innoslab amplifier is shown in Fig. 1(a). In the fast axis (perpendicular to the pump line), it is a stable cavity and the equality of the seed and pump beam size ensures good mode-matching. However, in the slow axis (in the plane of the pump line), it is a non-stable cavity and the seed beam diameter gradually increases in each roundtrip. There are two methods to achieve this purpose. One is to use a concave and a convex cylindrical mirror to form an off-axis positive confocal unstable cavity in the slow axis [5,6], and the other is to use a simpler plane–plane mirror configuration [7–11]. Compared with the former, the latter utilizes the inherent divergence of the Gaussian seed beam, which can be easily adjusted by the cylindrical lens. Therefore, considering the simplicity and popularity, we chose the latter for modeling and simulating. The slice model of a typical 4-pass Innoslab amplifier using plane-plane mirror is shown in Fig. 1(b). The length Dz of these elementary slices is set so small that the absorption coefficient, the beam size of pump laser and seed laser can be regarded as independent of it (in our simulation, Dz is set to 0.01 mm and the variations of these three parameters in one elementary slice are all less than 0.02%). The magnified view shows the part of one micro-slice, where the seed beam size can be considered to remain constant. It can be seen that the seed laser has a propagation vector at an angle θ from z-axis.

Fig. 1. (a) The schematic of Innoslab amplifier. (b) The slice model of the Innoslab amplifier and on the right is the magnified view.

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Most high-power Innoslab laser amplifiers for picosecond laser pulse with spectral FWHM of ∼1 nm [7–9,11] select Nd:YVO₄ crystal as the gain medium because of its large product of stimulated emission cross-section and fluorescence lifetime. Besides, compared with the Nd:YAG crystal, there is no thermally induced birefringent depolarization loss so that even in the regime of high pump power, the output power of Nd:YVO₄ amplifiers will not significantly drop. Thus, here we take Nd:YVO₄ as an example to make the derivation process. Due to the heat dissipation characteristics of Innoslab structure (heat can transfer homogeneously and efficiently through the large sides of the slab crystal) and the polarization-dependent absorption characteristics of Nd:YVO₄, the c-axis (optical axis) of the slab crystal is parallel to the x-axis. The polarization of the seed laser is also in x-z plane for high gain. It means that the seed laser belongs to extraordinary light (e light). According to [13], the stimulated emission cross-section σ_e (at 1064 nm) of e light has the following relationship with the propagating directions:

(1)$${\sigma _e}(\theta ) = {\sigma _{e,\sigma }}{\sin ^2}\theta + {\sigma _{e,\pi }}{\cos ^2}\theta$$

where σ_e,σ and σ_e,π is the stimulated emission cross-section of σ-polarization (E⊥c-axis) and the π-polarization (E ‖ c-axis), respectively. It should be noticed that the angle θ here is the complementary angle of that in [13].

For the tilted optical path, the photon-transport equation can be given by [15]:

(2)$$\frac{{\partial \phi }}{{\partial t}} + c\frac{{\partial \phi }}{{\partial z}}\cos \theta = cn{\sigma _e}(\theta )\phi$$

where $\phi$ is the photon density, c is the velocity of light in the medium, and n is the population inversion density. For picosecond pulse amplification, the pulse width is much shorter than the spontaneous emission time and the pumping rate so that the population inversion can be given by ignoring the effect of fluorescence and pumping during the pulse duration:

(3)$$\frac{{\partial n}}{{\partial t}} ={-} cn{\sigma _e}(\theta )\phi$$

Then, the amplifier output fluence J^out in an elementary slice can be obtained by solving the coupled Eqs. (2) and (3):

(4)$${J^{\textrm{out}}} = {J_s}\ln \{ 1 + [\exp (\frac{{{J^{in}}}}{{{J_s}}}) - 1]\exp (\frac{{n{\sigma _e}(\theta )Dz}}{{\cos (\theta )}})\}$$

where Jⁱⁿ is the fluence of the input pulse, J_s is the saturation fluence and defined by:

(5)$${J_s} = \frac{{hv}}{{{\sigma _e}(\theta )}}$$

h is the Planck’s constant, v is the laser frequency.

Finally, the output pulse energy from the Innoslab amplifier can be obtained by applying the one-dimensional finite element method.

2.2 Geometrical optics analysis and modeling

Here, two key approximations are made in this section. Approximation I is that the seed and pump beam in the fast axis have good mode-matching and the divergence of the seed laser can be ignored in this direction. One-dimensional heat flow in the slab establishes a homogeneous cylindrical thermal lens in the fast axis which can reproduce the laser mode on every roundtrip between the mirrors [6]. Thus, we only need to concentrate on the variety of the seed beam size in the slow axis. Approximation II is that the effect of the non-collinear beam overlap in the slow axis of the Innoslab amplifier is ignored in the model. In comparison to the active mirror (AM) [16], zigzag [17] geometries, the overlapping area (inside the crystal) of the beam in Innoslab geometry is much smaller. In Innoslab geometry, the seed laser is reflected by the mirrors instead of the side of the slab crystal. It is the distance between the mirrors and the slab crystal that makes most of the overlapping area outside the gain medium. Therefore, the effect of the beam overlap can be ignored.

The coordinate system used in the simulation is shown in Fig. 2(a). The coordinate origin O is in the center of the slab’s front end-face and the direction of the pump laser is opposite to the z-axis (denoted as ^-P). First, the inversion population distribution caused by the pump laser in the slab crystal is simulated. According to the Ref. [9,18], the coupling system composed of a planar waveguide, several cylindrical and spherical lenses can shape the pump laser into a homogenous line with the size equal to the width of the crystal. In the slow-axis direction, the pump laser density is homogeneous and can be considered to have a flat-top beam profile. In the fast-axis direction, it is nearly Gaussian distribution. As a result, the pump laser’s normalized intensity distribution of the kth slice can be expressed as:

(6)$${r_k}(x,y) = \frac{{2\sqrt {2\pi } }}{{\pi CW{\omega _{py}}(k)}}\exp \{ - 2{[\frac{{2x}}{W}]^{SG}} - 2{[\frac{y}{{{\omega _{py}}(k)}}]^2}\}$$

where W is the width of the slab crystal, SG is the super-Gaussian exponent, C is the normalization constant and given by:

(7)$$C = {2^{1 - 1/SG}}\Gamma (1 + 1/SG)$$

Fig. 2. (a) The coordinate system for slab crystal. (b) Top view of 4-pass Innoslab amplifier (in y = 0 plane) (c) The unfolded 4-pass Innoslab amplifier.

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Γ is the Gamma function, ω_py(k) is the pump beam radius of the y-direction in the kth slice and given by:

(8)$${\omega _{py}}(k) = \sqrt {[\omega _{py0}^2 + {{((kDz - {z_{y0}}){\theta _y})}^2}]} ,\textrm{ }\textrm{ }Dz = L/m,\textrm{ }z = kDz,\textrm{ }0 \le k \le m$$

ω_py0 is the pump beam waist radius in the y-direction, z_y0 is the waist location, θ_y is the divergence angle in the y-direction.

As shown in Fig. 2(c), the Innoslab amplifier is unfolded into a stack of amplifiers to simplify the analysis. N_m is the one-dimensional matrix of inverted population density from the whole slab crystal. The number m of elements in the matrix N_m is equal to the length of the crystal L divided by the length Dz of the elementary slice, namely, m = L/Dz and 1 ≤ k ≤ m. The operator Flip_n means to flip the order of elements and the subscript n means the number of flips (n is less than or equal to the pass number minus one and for example, in Fig. 2(b), n≤3). For the convenience of calculation, we overturn the pump laser direction and make it along the z-axis (denoted as ⁺P). Here, N_m is the Flip₁(N_m) (or Flip₃(N_m)) in fact. Thus, the intensity distribution of inverted population density N_k(x,y) in the kth slice is solved from the rate equation for four-level laser system [19]:

(9)$${N_k}(x,y) = \frac{{{W_k}{n_{tot}}}}{{{W_k} + 1/{\tau _f}}}$$

where n_tot is the total population density, τ_f is the fluorescence lifetime, W_k is the pumping rate given by:

(10)$${W_k} = \frac{{P_k^a\sigma _a^a + P_k^c\sigma _a^c}}{{h{\nu _p}}}{r_k}(x,y)$$

${P_k^a}$ and ${P_k^c}$ are the pump powers of the kth slice for the polarization direction parallel to a-axis and c-axis, respectively (their initial powers are equal to half of the total pump power), ν_p is the frequency of pump laser, ${\sigma_a^a}$and ${\sigma_a^c}$are the absorption cross-sections at the pump wavelength for each polarization.

On account of the pump absorption saturation effect [20], the absorption coefficient of the kth slice is given by:

(11)$$\alpha _k^h(x,y) = \sigma _a^h[{n_{tot}} - {N_k}(x,y)],h = a\,\,or\,\,c$$

Thus, the pump power in the (k+1)th slice can be given by:

(12)$$P_{k + 1}^h = \int\!\!\!\int {P_k^h{r_k}({x,y} )} \exp [ - \alpha _k^h(x,y)Dz]dxdy,h = a\textrm{ or }c,Dz = L/m,\textrm{ }z = kDz,\textrm{ }0 \le k \le m$$

Then, we need to consider the spatial overlap of the seed laser and population inversion. The normalized intensity distributions of the seed laser in the kth slice of the nth flip (the (n+1)th pass) can be written as:

(13)$$s_{n,k}^{}({x,y} )= \frac{2}{{\pi {\omega _{lx}}({n,k} ){\omega _{ly}}}}\exp \left\{ { - \frac{{2{x^2}}}{{{{[{{\omega_{lx}}({n,k} )} ]}^2}}} - \frac{{2{y^2}}}{{\omega_{ly}^2}}} \right\}$$

According to Approximation I, ω_ly can be considered a constant, which is equal to the pump beam waist radius ω_py0. In y = 0 plane, supposing the seed beam waist location in x-direction is (x₀, 0, z₀), the seed laser radius ω_lx in the kth slice of the nth flip can be given by:

(14)$${\omega _{lx}}({n,k} )= {\omega _{lx0}}{\left\{ {1 + {{\left[ {\frac{{M_l^2{\lambda_l}}}{{\pi \omega_{lx0}^2}}\left( {k\frac{{Dz}}{{{n_l}\cos \beta }} + n(\frac{L}{{{n_l}\cos \beta }} + \frac{{2D}}{{\cos \alpha }}) - \frac{{{z_0}}}{{\cos \alpha }}} \right)} \right]}^2}} \right\}^{1/2}},n \le 3$$

where ω_lx0 is the seed beam waist radius in x-direction, ${M_l^2}$is the beam quality of the seed laser, λ_l is the wavelength of the seed laser, n_l is the refractive index of crystal at λ_l, L is the length of the crystal, D is the distance between M1 (or M2) and the crystal, α (or β) is the angle of the seed laser in the air (or crystal) to the z-axis. It is important to note that the seed laser is e light which does not obey the law of refraction. Therefore, the value of β should be obtained by [21].

By applying the overlap integral, the effective population inversion density in the kth slice of the nth flip can be defined as:

(15)$$N_{n,k}^{eff} = \int\!\!\!\int {s_{n,k}^{}(x,y)} N_{n,k}^{}({x,y} )dxdy$$

For amplification in pulsed, high-frequency regime (f ≥1/τ_f), the initial gain which depends on the pulse repetition frequency f should be used in the F-N equation. Hence, the initial small-signal gain ${g_{n,k}^i}$ in the kth slice of the nth flip can be given by [22]:

(16)$$g_{n,k}^i = g_{n,k}^\infty - (g_{n,k}^\infty - g_{n,k}^f)\exp ( - \frac{1}{{f{\tau _f}}})$$

where

(17)$$g_{n,k}^\infty = \frac{{{\sigma _e}(\beta )N_{n,k}^{eff}Dz}}{{\cos \beta }}$$

and ${g_{n,k}^f}$ is the final small-signal gain just after the amplified pulse.

Supposing ${E_{n,k}^{in}}$ is the input pulse energy in the kth slice of the nth flip, the input pulse fluence ${J_{n,k}^{in}}$ can be given by:

(18)$$J_{n,k}^{in} = \frac{{E_{n,k}^{in}}}{{\pi {\omega _{lx}}(n,k){\omega _{ly}}}}$$

According to Eq. (4), the amplified output fluence ${J_{n,k}^{out}}$ can be written as:

(19)$$J_{n,k}^{out} = {J_s}\ln \{ 1 + [\exp (\frac{{J_{n,k}^{in}}}{{{J_s}}}) - 1]\exp (g_{n,k}^i)\}$$

Besides, ${g_{n,k}^i}$ and ${g_{n,k}^f}$ also satisfy the following relation [22]:

(20)$$g_{n,k}^i - g_{n,k}^f = \frac{{J_{n,k}^{out} - J_{n,k}^{in}}}{{{J_s}}}$$

The energy of the input pulse in the (k+1)th slice of the nth flip can be given by:

(21)$$E_{n,k + 1}^{in} = J_{n,k}^{out}\pi {\omega _{lx}}(n,k){\omega _{ly}}$$

Based on the above-mentioned Eqs. (6–21), the amplified output energy of a Nd:YVO₄ Innoslab amplifier can be obtained.

3. Experiments and results

3.1 Experimental setup

Based on the above theory and model, a Nd:YVO₄ Innoslab picosecond laser amplifier is designed. The experimental setup is shown in Fig. 3. The seed source is a home-made picosecond MOPA system with fiber laser as the seed and two-stage preamplifier, which is similar to the structure in our previous work [23]. The output power is about 13 W with a pulse width of ∼12 ps, the pulse repetition rate (PRR) of 1MHz, and the M² factor of 1.2. The seed laser passes through an optical isolation system and enters into the Innoslab amplifier. The pump source is a 300 W laser diode array (LDA) with six collimated bars and a central wavelength around 808 nm. The LDA is fixed on a water-cooled copper heat sink, which is kept at 22°C. The pump beam is shaped into a ∼14 mm × 0.4 mm homogenous line by the coupling system described in the Refs. [9,18], which has a total transmittance of 85%. The gain medium is a 0.3 at. % doped Nd:YVO₄ slab crystal with the size of 14 mm(W) × 1 mm(H) × 10(L) mm. The transmitting surfaces (W×H) are wide-angle wideband anti-reflection (AR) coated at 800∼1100 nm and 0°∼60°. The top and bottom surfaces of the slab crystal are welded tightly with the water-cooled copper heat sinks and its temperature is kept at 20°C.

Fig. 3. (a) Setup of the Nd:YVO₄ Innoslab picosecond laser amplifier (the subpicture is the stereoscopic view of f_x1 and f_y1). The c-axis of the crystals is parallel to the ground. λ/2: half-wave plates at 1064 nm; f_x₁, f_x₂: plano-convex cylindrical lenses (horizontal direction); f_x3: plano-concave cylindrical lens (horizontal direction); f_y₁, f_y₂: plano-convex cylindrical lenses (vertical direction); M₁, M₂: dichroic mirrors (HR 1064 nm and HT 808 nm at 0°); HM: 45° high reflectivity mirrors at 1064 nm. (b) Single-pass amplification of seed laser at different incident angles. BHM: Broadband high reflectivity mirror.

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The seed laser is incident on the slab crystal at different angles for single-pass amplification at first. A plano-convex cylindrical lens with a horizontal focal length of 200 mm is used to collimate the seed laser so that its horizontal diameter is 1.5 mm. In the vertical direction, the seed laser is focused by the cylindrical lens f_y1 with a 150 mm focal length to match its waist radius with the pump beam waist radius. Since the Rayleigh length (∼100 mm) of the seed laser in the vertical direction is much larger than the length of the slab crystal, the seed beam size in this direction can also be considered to be constant. As shown in Fig. 3(b), a broadband high reflectivity mirror (BHM) which is high-reflection (HR) coated at 750∼1100 nm and insensitive to laser incidence angle is rotated to adjust the angle of the seed laser through the slab crystal.

Then, the multi-pass amplification structure is established by replacing the BHM with a pair of plane mirrors M₁ and M₂. And in the horizontal direction, the cylindrical lens for collimation is also replaced by a plano-convex cylindrical lens f_x1 with focal length of 80 mm, which can focus the seed laser to a small spot. By adjusting the distance from f_x1 to the seed source, the focal spot size and location can be tuned to change the beam magnification on every roundtrip. The angle of the seed laser incident on slab crystal can be adjusted by tilting the 45° mirror located behind the lens f_y1, thus changing the pass number of the amplifier. During the experiment, the 4-, 6-, and 8-pass schemes are compared while keeping the overlapping area of the seed and pump lasers constant (about 80% of the total pump area). According to the above experimental setup, the simulation parameters are summarized in Table 1.

Table 1. Parameters in the simulation

View Table

3.2 Results and discussion

Figure 4(a) shows the simulated and experimental results of the single-pass Innoslab amplifier under the maximum pump power when the seed laser incidents at different angles. To observe the influence caused by the anisotropy of the stimulated emission cross-section σ_e more directly, the incident angle α is converted to the corresponding refraction angle β in the crystal. And the blue dashed line shows that the simulated output power without considering anisotropic σ_e gradually increases with refraction angle β increasing. This is because the tilted optical path makes the length L_s of the gain crystal where the seed laser passes increase with β, and they satisfy the relationship of L_s= L/cosβ. However, considering the anisotropy of σ_e, the simulated output power decreases with the increase of β, as shown by the red dashed line. This demonstrates that for the gain, the influence of anisotropic σ_e is greater than that of the increase of L_s. And their gap becomes increasingly apparent as β grows. The experiment data also support this argument. The simulated and experimental output powers vs. the incident pump power in the 4-, 6-, and 8-pass schemes are shown in Fig. 4(b). We can see the agreement between the experimental results and the simulated ones. And the subtle difference between them may be caused by the temperature-dependent stimulated cross-section of Nd:YVO₄ [24]. This value decreases with temperature rise caused by the thermal effect. As shown in the experimental data, at the pump power of 256 W after the coupling system, the output powers of the 4-, 6-, and 8-pass Innoslab amplifiers are 76.2 W, 81.4 W, and 85.5 W, respectively. And the slope efficiencies are 22.4%, 24.4%, and 25.9%. It can be noted that even if the size of overlapping areas is the same, the output power and slope efficiency still increase with the pass number. This indicates that the amplifier structure with more passes is more conducive to extracting the energy stored in the slab crystal. The reason is that the incident angle in the structure with a higher pass number is smaller so that this structure can provide a larger σ_e according to Eq. (1), finally causing a higher gain.

Fig. 4. (a) The simulation (with or without using anisotropic σ_e) and experiment data of the output power vs. the angle β in the single-pass amplification at the maximum pump power of LDA. (b) The simulated and experimental output powers of the 4-, 6- and 8-pass structure.

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Figure 5(a) shows the spectrum of the fiber seed laser in the home-made MOPA system and the output spectrum of 8-pass Nd:YVO₄ Innoslab amplifier. The self-phase modulation (SPM) in the fiber seed laser broadens its spectrum into a multi-peaked structure. Limited by the finite gain band of Nd:YVO₄, only part of the seed light whose spectrum coincides with the gain spectrum is amplified. The temporal characteristics of the seed and output pulses are measured by a commercial autocorrelator (APE, Pulse Check 150). As presented in Fig. 5(b), the pulse duration (FWHM) of the fiber seed laser is 12.4 ps, while the output pulse duration is 13 ps. After amplification, the elliptical output laser is converted to a near-circular beam by using a vertical plano-convex cylindrical lens and a horizontal cylindrical telescope. Figure 5(c) shows that the beam quality of the 8-pass Innoslab amplifier is ${M_x^2}$= 1.48 and ${M_y^2}$= 1.41 in the orthogonal directions.

Fig. 5. (a) The spectrum of the fiber seed laser in the home-made picosecond MOPA system and the output spectrum of 8-pass Nd:YVO₄ Innoslab amplifier. (b) Intensity autocorrelation traces of the fiber seed laser and amplified output. And the solid curves represent Gaussian fit values. (c) The beam quality of the 8-pass Innoslab amplifier.

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4. Conclusions

In summary, we modify the F-N equation and develop a simple one-dimensional unfolded slicing model for simulating the Innoslab picosecond laser amplifier for the first time. By introducing the anisotropic stimulated emission cross-section of the laser crystal, the modified F-N equation is developed to consider the influence of the tilted optical path. Then, we unfold the Innoslab amplifier into a stack of slab amplifiers and divide them into many thin slices to accurately simulate the spatial overlap between the seed laser and pump laser and the pump absorption saturation effect. Finally, based on the above theory and model, 4-, 6- and 8-pass Nd:YVO₄ Innoslab picosecond amplifiers are designed and established with output powers of 76.2 W, 81.4 W, and 85.5 W, respectively. The simulated and experimental results indicate that the incident angle is a significant influence factor on the Innoslab amplifier because of the anisotropic σ_e. Therefore, the structure with a higher pass number is more favorable to energy extraction. The agreement between the simulated and experimental results proves that our model is useful for designing and optimizing high-power Innoslab picosecond amplifier.

Funding

This work was supported by National Key Research and Development Program of China (2017YFB0405204), National Research Foundation of China (Grant No. 61975097, 61975095), the Youth Cross Innovation Group of Shandong University (Grant No.2020QNQT), the Financial Support from Qilu Young Scholar of Shandong University (2019JZZY020206).

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

References

1. J. Kleinbauer, R. Knappe, and R. Wallenstein, “A powerful diode-pumped laser source for micro-machining with ps pulses in the infrared, the visible and the ultraviolet,” Appl. Phys. B 80(3), 315–320 (2005). [CrossRef]

2. P. Zhu, D. Li, Q. Liu, J. Chen, S. Fu, P. Shi, K. Du, and P. Loosen, “39.1 μJ picosecond ultraviolet pulses at 355 nm with 1 MHz repeat rate,” Opt. Lett. 38(22), 4716 (2013). [CrossRef]

3. R. J. Gray, X. H. Yuan, D. C. Carroll, C. M. Brenner, M. Coury, M. N. Quinn, O. Tresca, B. Zielbauer, B. Aurand, V. Bagnoud, J. Fils, T. Kühl, X. X. Lin, C. Li, Y. T. Li, M. Roth, D. Neely, and P. McKenna, “Surface transport of energetic electrons in intense picosecond laser-foil interactions,” Appl. Phys. Lett. 99(17), 171502 (2011). [CrossRef]

4. P. M. Paul, “Observation of a Train of Attosecond Pulses from High Harmonic Generation,” Science 292(5522), 1689–1692 (2001). [CrossRef]

5. G. Schmidt, C. Schnitzler, J. Giesekus, R. Wester, K.- Du, P. Loosen, and R. Poprawe, “Scaling the output power of a diode end pumped Nd:YAG slab laser with a hybrid resonator,” in Advanced Solid State Lasers, OSA Technical Digest Series (Optica Publishing Group, 2000), paper MA3

6. P. Russbueldt, T. Mans, G. Rotarius, J. Weitenberg, H. D. Hoffmann, and R. Poprawe, “400W Yb:YAG Innoslab fs-Amplifier,” Opt. Express 17(15), 12230 (2009). [CrossRef]

7. C. Heese, A. E. Oehler, L. Gallmann, and U. Keller, “High-energy picosecond Nd:YVO₄ slab amplifier for OPCPA pumping,” Appl. Phys. B 103(1), 5–8 (2011). [CrossRef]

8. H. Lin, J. Li, and X. Liang, “105 W, <10 ps, TEM₀₀ laser output based on an in-band pumped Nd:YVO₄ Innoslab amplifier,” Opt. Lett. 37(13), 2634 (2012). [CrossRef]

9. L. Xu, H. Zhang, Y. Mao, Y. Yan, Z. Fan, and J. Xin, “High-average-power and high-beam-quality Innoslab picosecond laser amplifier,” Appl. Opt. 51(27), 6669 (2012). [CrossRef]

10. J. Guo, H. Lin, J. Li, P. Gao, and X. Liang, “High power TEM₀₀ picosecond output based on a Nd:GdVO₄ discrete path Innoslab amplifier,” Opt. Lett. 41(12), 2875 (2016). [CrossRef]

11. W. Chen, B. Liu, Y. Song, L. Chai, Q. Cui, Q. Liu, C. Wang, and M. Hu, “High pulse energy fiber/solid-slab hybrid picosecond pulse system for material processing on polycrystalline diamonds,” High Pow. Laser Sci. Eng. 6, e18 (2018). [CrossRef]

12. L. M. Frantz and J. S. Nodvik, “Theory of Pulse Propagation in a Laser Amplifier,” J. Appl. Phys. 34(8), 2346–2349 (1963). [CrossRef]

13. R. Guo, Y. Shen, Y. Meng, and M. Gong, “Anisotropic stimulated emission cross-section measurement in Nd:YVO₄,” Chinese Phys. B 28(4), 044204 (2019). [CrossRef]

14. D. C. Brown, T. M. Bruno, and V. Vitali, “Saturated Absorption Effects in CW-Pumped Solid-State Lasers,” IEEE J. Quantum Electron. 46(12), 1717–1725 (2010). [CrossRef]

15. J. M. Eggleston, L. M. Frantz, and H. Injeyan, “Deviation of the Frantz-Nodvik equation for zig-zag optical path, slab geometry laser amplifiers,” IEEE J. Quantum Electron. 25(8), 1855–1862 (1989). [CrossRef]

16. T. Liu, Q. Liu, Z. Sui, M. Gong, and X. Fu, “Spatiotemporal characterization of laser pulse amplification in double-pass active mirror geometry,” High Pow Laser Sci Eng 8, e30 (2020). [CrossRef]

17. A. K. Sridharan, S. Saraf, S. Sinha, and R. L. Byer, “Zigzag slabs for solid-state laser amplifiers: batch fabrication and parasitic oscillation suppression,” Appl. Opt. 45(14), 3340 (2006). [CrossRef]

18. M. Traub, H.D. Hoffmann, H.D. Plum, K. Wieching, P. Loosen, and R. Poprawe, “Homogenization of high power diode laser beams for pumping and direct applications,” Proc. SPIE 6104, 61040Q (2006). [CrossRef]

19. W. Koechner, Solid-State Laser Engineering, 6th ed. (Springer, 2006).

20. X. Délen, F. Balembois, O. Musset, and P. Georges, “Characteristics of laser operation at 1064 nm in Nd:YVO₄ under diode pumping at 808 and 914 nm,” J. Opt. Soc. Am. B 28(1), 52 (2011). [CrossRef]

21. E. Cojocaru, “Direction cosines and vectorial relations for extraordinary-wave propagation in uniaxial media,” Appl. Opt. 36(1), 302 (1997). [CrossRef]

22. M. Grishin, ed., Advances in Solid State Lasers Development and Applications (InTech, 2010).

23. B. Shi, G. He, J. Mao, F. Wang, K. Yang, B. Zhang, and J. He, “Simulation and experimental results of a high-gain two-stage and double-pass off-axis Nd:YVO₄ picosecond laser amplifier,” Appl. Opt. 60(1), 186 (2021). [CrossRef]

24. X. Délen, F. Balembois, and P. Georges, “Design of a high gain single stage and single pass Nd:YVO₄ passive picosecond amplifier,” J. Opt. Soc. Am. B 29(9), 2339 (2012). [CrossRef]

Parameter	Value	Parameter	Value
W (mm)	14	ω_lx0 (µm)	35 (4-pass)
H (mm)	1		45 (6-pass)
L (mm)	10		65 (8-pass)
D (mm)	5	z₀ (mm)	-100 (4-pass)
ω_py0 (µm)	200		-80 (6-pass)
z_y0 (mm)	5		-70 (8-pass)
θ_y (mrad)	10	α (mrad)	200 (4-pass)
SG	10		140 (6-pass)
ω_ly (µm)	200		105 (8-pass)
M2 l	1.2	E in (µJ)	13
σ_e,σ [13]	6.17×10⁻¹⁹ cm²	σ_e,π [13]	14.43×10⁻¹⁹ cm²

Modified Frantz-Nodvik equation and numerical simulation of a high-power Innoslab picosecond laser amplifier

Abstract

1. Introduction

2. Modeling

2.1 Modified F-N equation for Innoslab amplifiers

2.2 Geometrical optics analysis and modeling

3. Experiments and results

3.1 Experimental setup

3.2 Results and discussion

4. Conclusions

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (5)

Tables (1)

Equations (21)

Optics Express