3D simulation and beam quality improvement of a pulsed Cr,Tm,Ho:YAG laser

Ramon Springer; Christoph Pflaum

doi:10.1364/OE.27.022898

1. Introduction

Efficient laser sources operating at 2 μm have drawn considerable attention in industry and science. Possible applications are sensing, communication, material processing and healthcare. Among other elements of the lanthanide series, Holmium is preferred to conduct stimulated emission due to its long upper state lifetime and large emission cross section at 2090 nm [1–3]. However, the absorption coefficient of Holmium in the visible range is rather low. Depending on the choice of pumping source and experimental setup the incorporation of additional ion species in the crystal lattice is performed to increase the pump absorption [4,5]. The broad absorption of Cr⁺³ ions in the visible range makes Cr,Tm,Ho:YAG a suitable crystal for flashlamp pumping [6]. The introduction of Thulium in the lattice structure secures the energy transfer from Cr⁺³ to Tm⁺³ and from Tm⁺³ to Ho⁺³ [7, 8]. Besides the crystal composition the development of a pulsed CTH-Laser requires an efficient pump cavity and resonator design to generate pulses in the Joule regime at low repetition rates. Additionally, an improved beam quality is desirable since many applications require the subsequent incoupling of generated pulses in an optical fiber. The experimental development of such systems is time intense and complicated without the usage of simulation tools. Unfortunately, an accurate simulation technique for extended state systems of multiple doped gain media, which includes mode dependent amplification has not been developed at this point, to the best of our knowledge.

Here, the experimental improvement of a pulsed CTH-Laser is presented, which is based upon the accurate numerical simulation of the entire system. To this end, a novel method called gDMA is suggested, which is able to simulate both complex excitation dynamics and mode competition during pulse generation. This method includes DMA [9] and is combined with an abstract formalism to describe arbitrary sets of rate equations. In order to determine the resonator stability during pulse formation the proposed gDMA is coupled to a FEM heat model. As a result of this new simulation technique, dominant interionic mechanisms in Cr,Tm,Ho:YAG are identified. Additionally, the time dependent beam quality of pulses is calculated. Based on the simulated results the experimental improvement to the CTH-Laser is conducted by reducing the crystal diameter. Measurement of pulse energy and beam quality are in good agreement with the results from simulation.

2. Experimental setup of the CTH-Laser

The system consists of a 129 mm-long-Cr,Tm,Ho:YAG rod with a diameter of 4 mm. The rod is pumped by a flashlamp and operated at 10 Hz repetition rate, which can be controlled by flashlamp pulses with the same repetition rate and the laser is free-running. The rod and the flashlamp are located within a mirrored pump cavity, which surrounds the two components in order to maximize the absorbed pump light. Moreover, the rod is placed within a resonator with a total length of 253 mm, including a convex mirror and a plane output coupler with a reflectivity of 80% at 2090 nm. The curvature of the convex mirror is 1000 mm and and the crystal is located in the center of the resonator. The pump cavity has a radial curvature, which can be seen from its reconstruction in Fig. 1.

Fig. 1 Reconstructed pump cavity in simulation. The Flashlamp is located at the left hand side and is represented with the yellow cylinder. The Cr,Tm,Ho:YAG rod is placed parallel to the flashlamp axis.

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In simulation the pump absorption of Cr,Tm,Ho:YAG crystal is calculated with a ray tracing model. Fig. 2 visualizes that the spectrum of the flashlamp emission overlaps well with the absorption spectrum of the Cr³⁺ ions. Moreover, the pump pulse is approximated with a square pulse, which is in agreement with the experimental measurement.

Fig. 2 The absorption spectrum of Cr³⁺ ions [7] matches the spectral range of to the measured flashlamp emission spectrum.

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3. State system of Cr,Tm,Ho:YAG

The state system of Cr,Tm,Ho:YAG is displayed in Fig. 3. As reported in previous work, the excitation dynamics are strongly influenced by interionic mechanisms involving Cr³⁺, Tm³⁺ and Ho³⁺ ions [10, 11]. The excitation of the upper lasing state ⁵I₇ in Holmium in the CTH state system origins from the pump absorption by Cr³⁺ ions and the subsequent energy transfers from Chromium to Thulium and from Thulium to Holmium, respectively [12]. It is important to point out that each of the mentioned transfer mechanism competes with a back transfer from Holmium to Thulium and from Thulium to Chromium. Among those transfer mechanisms, interionic mechanisms between ions of the same species occur such as upconversion and cross relaxation [13,14]. As a consequence, the excitation of Thulium and Holmium ions is influenced by these mechanisms in addition to the respective energy transfers. Even though each individual interionic mechanism can be precisely described, it is not clear, which are the dominant interionic mechanisms that mainly determine the excitation of the upper lasing state ⁵I₇ in Holmium during pulsed operation of the CTH-Laser. Therefore, the excitation dynamics in Cr,Tm,Ho:YAG are analyzed in Section 5.4 and the most dominant loss sources are identified.

Fig. 3 Excitation and relaxation mechanisms in Cr,Tm,Ho:YAG: a) Pump absorption P_abs b) Net (effective) Energy Transfer E_13,12,1,11 c) Upconversion U₂₁₂₄ d) Upconversion U₂₁₂₃ e) Cross Relaxation C₄₂₁₂ f)Energy Transfer E_4,1,6,10 g) Energy Transfer E₂₁₆₈ h) Energy Transfer E₂₁₅₆ i) Energy Transfer E₆₅₁₂ j) Upconversion U₆₅₆₈ k) Stimulated Emission at 2090 nm. The blue color highlights interionic mechanism that involve ions of the same species. Spontaneous emission is included in all calculations but not displayed here.

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4. Model description

The simulation technique gDMA enables the mode dependent simulation of gain media, which mathematical description clearly exceeds the simplified representation by quasi 3-level or 4-level systems. The essential part of the proposed gDMA is an abstract formalism to describe arbitrary sets of rate equations including interionic mechanisms. The introduced generalization is required to define and solve arbitrary sets of coupled rate equations in an automated way (Section 4.1). Based upon that novel approach a graphical user interface was implemented, which enables the fast definition and solving of large sets of coupled differential equations [15]. Moreover, thermal lensing was included in all calculations by coupling gDMA to a FEM thermal analysis of the Cr,Tm,Ho:YAG crystal, which is pointed out in Section 4.3. In order to calculate the large number of modes a grouping algorithm is applied, which significantly reduces the computational effort (Section 4.4).

4.1. The gDMA approach for arbitrary rate equations

State systems that describe hosts with multiple doping species can be expressed with numerous rate equations. Each rate equations might contain terms to describe stimulated emission, fluorescent decay, interionic mechanisms or other excitation and decay channels. Here, a general formalism is presented that allows to define an arbitrary set of rate equations, where each equation may have individual excitation and decay mechanisms. The abstract formalism is applied to Eq. (1), which exhibits the structure of an ordinary laser rate equation:

\frac{\partial N_{i}}{\partial t} = F_{i} - \sum_{\begin{array}{c} s \in {1, \dots, S} \\ i_{s} = i \end{array}} σ_{i_{s}, j_{s}}^{s} N_{i} + \sum_{\begin{array}{c} s \in {1, \dots, S} \\ j_{s} = i \end{array}} σ_{i_{s}, j_{s}}^{s} N_{i} for i = 0, \dots, L - 1 .

In Eq. (1) the parameter S represents the number of transitions, F_i pumping, and

σ_{i_{s}, j_{s}}^{s}

a non-linear function, which can describe any kind of excitation or decay mechanism between the states N_i and N_j. Here, N_i and N_j are space dependent but the spatial coordinates are omitted for brevity. In case of spontaneous emmission,

σ_{i_{s}, j_{s}}^{s}

represents the decay rate

\frac{1}{τ}

. Moreover, the effective mode volume is defined by

V_{eff} = \int_{Ω_{resonator}} n (z) {| u_{m} |}^{2} d V = \int_{0}^{L} n (z) d z,

where n(z) is the index of refraction [9]. Furthermore, it is supposed that the transition for stimulated emission takes place between the levels i and j. Then, DMA contains the following rate equation for the number of photons corresponding to each mode m = 1, ..., M:

\frac{\partial Φ_{m}}{\partial t} = \frac{c}{V_{eff}} Φ_{m} \int_{Ω_{crystal}} (σ_{e} N_{i} - σ_{a} N_{j}) {| u_{m} |}^{2} d V - \frac{Φ_{m}}{τ_{c}},

where τ_c is the decay rate inside the cavity and mainly depends on the intracavity losses and the reflectivity of the output coupler. The mode shape |u_m| describes the light field of a specific mode in three spatial dimensions in the resonator. In our calculations the mode shape |u_m| is equivalent to the group shape |siTEM(x⃗)| that results from the mode grouping procedure introduced in Chapter 4.4. In this context, the index m relates to the one specific siTEM group. In Chapter 4.4 the siTEM groups are indexed with i and j, which is required to relate each siTEM group to consecutive summarized Hermite-Gauss modes dependent on the transversal directions x and y. Moreover, the population inversion can be calculated with Δ := N_i − (σ_a/σ_e)N_j [16]. To simplify the further derivations, the following abbreviations are applied:

{\tilde{σ}}_{e} : = \frac{σ_{e} c}{V_{eff}}, {\tilde{σ}}_{a} : = \frac{σ_{a} c}{V_{eff}} .

It will be demonstrated in the subsequent examples that each mechanism can be described by Eq. (1). In addition, it will be shown that any kind of excitation or decay mechanism can be formulated by the appropriate definition of the transition function

σ_{i_{s}, j_{s}}^{s}

. This results in the generalized version of the former DMA. As a consequence the generalization enables the coupling of DMA to arbitrary sets of rate equations. Both the abstract formalism of arbitrary rate equations and the former DMA defines the introduced method called gDMA (Fig. 4).

Fig. 4 The proposed gDMA method is obtained by coupling the former DMA with the suggested abstract formalism for arbitrary rate equations. The gDMA is required to compute mode structure, mode competition and beam quality of the CTH-Laser.

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4.1.1. Abstract formulation of stimulated emission

By applying the general formalism from Eqs. (1) to (3), the following rate equations are obtained, which describes the upper and lower state for stimulated emission.

\frac{\partial N_{i}}{\partial t} = \dots + \sum^{m} {| u_{m} |}^{2} Φ_{m} (- {\tilde{σ}}_{e} N_{i} + {\tilde{σ}}_{a} N_{j}) = \dots - σ_{i, j}^{e} N_{i} + σ_{i, j}^{a} N_{j}

\frac{\partial N_{j}}{\partial t} = \dots + \sum^{m} {| u_{m} |}^{2} Φ_{m} ({\tilde{σ}}_{e} N_{i} - {\tilde{σ}}_{a} N_{j}) = \dots + σ_{i, j}^{e} N_{i} - σ_{i, j}^{a} N_{j} .

The population of each state can be calculated by a 3-dimensional function. Here, the application of the transition function

σ_{i_{s}, j_{s}}^{s}

results in a general description for stimulated emission involving two states. Since those states have a general description (Eq. (4) and Eq. (5)) they can be coupled to sets of rate equations with arbitrary size.

4.1.2. Abstract formulation of nonlinear interionic transitions

Energy transfer is an interionic mechanism that involves two ions of different species where energy is transfered from dopant A to dopant B. Level i and l are levels corresponding to dopant A while j and k denote the respective levels of dopant B. The states i and j are the states from which the mechanisms originates. The overall transfer rate is determined by the state population of i and j and the rate coefficent W_E. As a consequence, energy transfer results in the excitation of state l of dopant B and the relaxation of state k in dopant A. The mechanism can be formulated by the following ordinary differential equations:

\frac{\partial N_{k}}{\partial t} = \dots + W_{E} N_{j} N_{i} = \dots + σ_{i, k} N_{j} where σ_{i, k} = W_{E} N_{i}

\frac{\partial N_{i}}{\partial t} = \dots - W_{E} N_{j} N_{i} = \dots - σ_{i, k} N_{j} where σ_{i, k} = W_{E} N_{i}

\frac{\partial N_{j}}{\partial t} = \dots - W_{E} N_{i} N_{j} = \dots - σ_{j, l} N_{i} where σ_{j, l} = W_{E} N_{j}

\frac{\partial N_{l}}{\partial t} = \dots + W_{E} N_{i} N_{j} = \dots + σ_{j, l} N_{i} where σ_{j, l} = W_{E} N_{j} .

Here, it can be observed that the energy transfer mechanism can be described by the transition functions σ_i,k and σ_j,l. When comparing to stimulated emission it is interesting to see that both mechanisms have the same generalized formalism. The present example describes the proposed abstract formulation to energy transfer. Other nonlinear transitions such as upconversion and cross relaxation can be described in an analog way. This points out the applicability of the gDMA to arbitrary rate equations. Moreover, the general description of the contributions to individual rate equations enables the automated generation of rate equations.

4.2. Definition of the CTH state system

The CTH state system, which is visualized in Fig. 3, is defined by Eqs. (10)–(22). Here, the indexing of states was chosen based upon a previous study on Tm,Ho:YAG [17]. Without the automated generation of rate equations, the straight implementation of a system with this size is time intense and the probability to include errors is high.

\frac{\partial^{2} E_{1}}{\partial t} : = \frac{\partial N_{13}}{\partial t} = R_{pump} - k_{13, 12, 1, 11} N_{13} N_{1} - \frac{N_{13}}{τ_{13}}

\frac{\partial^{4} A_{2}}{\partial t} : = \frac{\partial N_{12}}{\partial t} = - R_{pump} + k_{13, 12, 1, 11} N_{13} N_{1} + \frac{N_{13}}{τ_{13}}

\frac{\partial^{3} F_{2, 3}}{\partial t} : = \frac{\partial N_{11}}{\partial t} = k_{13, 12, 1, 11} N_{13} N_{1} - \frac{N_{11}}{τ_{11}}

\frac{\partial^{5} F_{2}}{\partial t} : = \frac{\partial N_{10}}{\partial t} = k_{4, 1, 6, 10} N_{4} N_{6} + \frac{N_{11}}{τ_{11}} - \frac{N_{10}}{τ_{10}}

\frac{\partial^{5} F_{5}}{\partial t} : = \frac{\partial N_{9}}{\partial t} = \frac{N_{10}}{τ_{10}} - \frac{N_{9}}{τ_{9}}

\frac{\partial^{5} I_{5}}{\partial t} : = \frac{\partial N_{8}}{\partial t} = k_{2168} N_{2} N_{6} + k_{6568} {N_{6}}^{2} + \frac{N_{9}}{τ_{9}} - \frac{N_{8}}{τ_{8}}

\frac{\partial^{5} I_{6}}{\partial t} : = \frac{\partial N_{7}}{\partial t} = \frac{N_{8}}{τ_{8}} - \frac{N_{7}}{τ_{7}}

\begin{array}{l} \frac{\partial^{5} I_{7}}{\partial t} : = \frac{\partial N_{6}}{\partial t} & = & - σ_{e} c n N_{6} + σ_{a} c n N_{5} - k_{4, 1, 6, 10} N_{4} N_{6} + k_{2156} N_{2} N_{5} \\ - k_{6512} N_{6} N_{1} - k_{2168} N_{2} N_{6} - 2 k_{6568} {N_{6}}^{2} + \frac{N_{7}}{τ_{7}} - \frac{N_{6}}{τ_{6}} \end{array}

\begin{array}{l} \frac{\partial^{5} I_{8}}{\partial t} : = \frac{\partial N_{5}}{\partial t} & = & σ_{e} c n N_{6} - σ_{a} c n N_{5} - k_{2156} N_{2} N_{5} + k_{6512} N_{6} N_{1} \\ k_{6568} N_{6}^{2} + \frac{N_{6}}{τ_{6}} \end{array}

\frac{\partial^{3} H_{4}}{\partial t} : = \frac{\partial N_{4}}{\partial t} = - k_{4212} N_{4} N_{1} - k_{4, 1, 6, 10} N_{4} N_{10} + k_{2124} {N_{2}}^{2} - \frac{N_{4}}{τ_{4}}

\frac{\partial^{3} H_{5}}{\partial t} : = \frac{\partial N_{3}}{\partial t} = k_{2123} {N_{2}}^{2} + \frac{N_{4}}{τ_{4}} - \frac{N_{3}}{τ_{3}}

\begin{array}{l} \frac{\partial^{3} F_{4}}{\partial t} : = \frac{\partial N_{2}}{\partial t} & = & - 2 k_{2124} {N_{2}}^{2} - 2 k_{2123} {N_{2}}^{2} + 2 k_{4212} N_{4} N_{1} - k_{2156} N_{2} N_{5} \\ + k_{6512} N_{1} N_{6} - k_{2168} N_{6} N_{2} + \frac{N_{3}}{τ_{3}} - \frac{N_{2}}{τ_{2}} \end{array}

\begin{array}{l} \frac{\partial^{3} H_{6}}{\partial t} : = \frac{\partial N_{1}}{\partial t} & = & k_{2124} {N_{2}}^{2} + k_{2123} {N_{2}}^{2} - k_{4212} N_{4} N_{1} + k_{2156} N_{2} N_{5} \\ - k_{6512} N_{1} N_{6} + k_{4, 1, 6, 10} N_{4} N_{6} + k_{2168} N_{2} N_{6} \\ - k_{13, 12, 1, 11} N_{13} N_{1} + \frac{N_{2}}{τ_{2}} . \end{array}

Here, τ represents the fluorescent lifetime and k_x a mechanism specific constant, which depend on the crystal composition and doping concentration. All rate equations are solved with an explicit Finite Difference method. The chosen values for simulation can be found in Tables 1 and 3.

Table 1. Coefficients of Interionic Mechanisms in Cr,Tm,Ho:YAG

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Table 2. Fluorescent (Fl.) lifetimes that are not listed have values in the μs range.

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Table 3. Calculation Module and chosen Simulation Parameters

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Equations (10) to (12) and (22) contain the net energy transfer from Chromium to Thulium, which includes already the respective back transfer from Thulium to Holmium. Using the formulas introduced in Section 4.1 we can describe this set of rate equations by the mechanism coefficients W. As shown in Fig. 3 the CTH state system exhibits several interionic mechanisms. The respective mechanism coefficients for Energy Transfer (E), Cross Relaxation (C), Upconversion (U) and Decay (D) are listed here:

W_E = [k_13,12,1,11, k₆₅₁₂, k₂₁₅₆, k₂₁₆₈, k_4,1,6,10],
W_U = [k₂₁₂₄, k₂₁₂₃],
W_C = [k₄₂₁₂, k₆₅₆₈],
W_D = [τ₂⁻¹, τ₃⁻¹, . . . , τ₁₃⁻¹].

As a consequence, a straight implementation of the CTH state system is not required. Instead the entire rate equation system can be generated computationally with the abstract formalism by using the mechanism coefficients W as input parameters. An object oriented programming approach was applied to implement the automatic generation of Eqs. (10) to (22).

4.3. Finite element heat model

Since the CTH crystal is pumped by a flash lamp a suitable heat model is required. The heat source P_heat is calculated with Eq. (23), which includes the ratio of pump and lasing wavelength. The heat efficiency factor η_heat,eff was chosen to be 20%, since ion relaxation processes lead not only to an increase of temperature but also to spontaneous emission. The exact determination of η_heat,eff is a non-trivial task and requires elaborated modelling. In Cr,Tm,Ho:YAG radiative decay represents a major loss source due to spontaneous emission from the origin states of energy transfer between Chromium, Thulium and Holmium. Therefore, those decay channels do not contribute to heat generation in Cr,Tm,Ho:YAG. However, the applied value of η_heat,eff was determined heuristically by comparing the simulated laser pulse energy with experimental measurements. The following equation is considered to be suitable to model heat generation by flashlamp pumping:

P_{heat} = η_{heat, eff} \sum_{λ_{pump} \leq λ_{laser}} P_{abs} (λ_{pump}) (1 - \frac{λ_{pump}}{λ_{laser}}) + \sum_{λ_{pump} > λ_{laser}} P_{abs} (λ_{pump}) .

Contributing to Eqs. (10) and (11), the absorbed pump rate R_pump is calculated by

R_{pump} = \sum_{λ \leq λ_{laser}} P_{abs} (λ) / E^{photon} (λ)

where P_abs(λ) expresses the absorbed pump power and E^photon(λ) represents the energy of a photon with respect to the wavelength. The calculated heat source P_heat is then integrated into the FEM heat model, which was implemented with trilinear finite elements on a block structured grid. Moreover, a multigrid algorithm related to the finite element discretization is applied for efficient solving of the heat equation in three dimensions. A listed overview of all thermo-optic parameters that were used in simulation can be found in Table 3.

In order to calculate the pulse energy we apply a time independent FEA calculation of the thermal lense. This is less computational intensive than a transient simulation. In Section 5.2 the results of the obtained focal length are compared to a time-dependent simulation and experimental measurements. This comparison shows that a time independent approximation leads to relative small approximation error of the thermal lense. After calculating the temperature with the FEA heat model a parabolic fit of the temperature distribution is performed in each crystal slice (x-y-plane). This parabolic fit is required to determine the q-parameter from a ABCD matrix calculation. Every slice is considered as gaussian duct, where the respective q parameter updates the mode shape |u_m|, which is then inserted to Eqs. (2) and (3). The detailed procedure of temperature calculation (FEA), parabolic fit, ABCD matrix calculation and coupling to DMA was orginially published by Wohlmuth et al. and is not extensively described here. [9]

4.4. Super-Imposed-Gauss-Hermite modes

The presented version of DMA has the advantage to calculate the mode specific output power and the multimode content of the generated pulse from the CTH-Laser. Furthermore, the time dependent beam quality factor M² can be computed by the output power and the beam quality of each mode i with following equations:

M_{x}^{2} (t) = \sum_{i} M_{x, i}^{2} \frac{P_{out, i} (t)}{P_{out} (t)},

M_{y}^{2} (t) = \sum_{i} M_{y, i}^{2} \frac{P_{out, i} (t)}{P_{out} (t)} .

For Hermite-Gaussian modes the beam quality factors for each mode can be calculated with

M_{x, p}^{2} = 2 p + 1,

M_{y, q}^{2} = 2 q + 1

where p and q determines the mode order in x and y direction, respectively. From the Eqs. (27) and (28) it can be seen that the simulation of an overall beam quality factor of 36 requires a minimum of 73 modes. In order to decrease the enormous computational effort a generalized mode structure is applied, which splits the entire set of eigenmodes into subsets of M_α eigenmodes, where α denotes the index of the subset. An elaborate description of the application of super imposed Hermite-Gaussian modes (siTEM) has been published in the past by Wohlmuth et al. [21]. This concept is based on the idea to combine modes with similar oscillation behavior. As a requirement to conduct mode grouping one of the following properties need to be fullfilled: I) The output power must be distributed equally among contained modes. II) The mode overlap with the gain profile is similiar for the contained modes in one group in a temporal average. The requirements I) and II) are met by grouping consecutive Hermite Gaussian modes (TEM_k). The field distribution in the x–z plane is defined as

{| u_{i}^{g_{i}, l_{i}} (x, z) |}^{2} = \frac{1}{g_{i}} \sum_{k = l_{i}}^{l_{i} + g_{i} - 1} {| {TEM}_{k} (x, z) |}^{2}

where g_i denotes the group order, which defines the number of consecutive modes in x- or y-direction. Moreover, l_i defines the first mode when summarizing until the mode number l_i + g_i − 1. Since all Hermite-Gaussian modes are orthogonal the tensor product of modes with a spatial distribution in x- and y-direction gives all possible superpositions in each x–y plane. Therefore, arbitrary field distributions can be constructed from the following tensor product, which represents the definition of a general siTEM group.

{| {siTEM}_{i j} (\vec{x}) |}^{2} = | u_{i}^{g_{i}, l_{i}} (x, z) | | u_{j}^{g_{j}, l_{j}} (y, z) |

One example is |siTEM₀₄(x⃗)|², which can be seen in Fig. 7(b). This siTEM group contains 100 modes and is calculated with Eq. (30) from 10 Hermite-Gaussian modes in each transversal direction. These modes are numbered with 0 to 9 in x-direction and 36 to 44 in y-direction, respectively. Therefore, Eq. (29) can be written with respect to x and y in the following way:

{| u_{0}^{10, 0} (x, z) |}^{2} = \frac{1}{10} \sum_{k = 0}^{0 + 10 - 1} {| {TEM}_{k} (x, z) |}^{2},

{| u_{4}^{10, 36} (y, z) |}^{2} = \frac{1}{10} \sum_{k = 36}^{36 + 10 - 1} {| {TEM}_{k} (y, z) |}^{2},

In our calculations 81 siTEM groups were used containing 100 modes each (assuming g_i = 10), which translates to a total number of 8100 Hermite-Gaussian modes. Consequently, the application of siTEM groups reduces the computational effort significantly from 8100 to 81 modes. In this context, one siTEM group is treated as one mode. Since siTEM groups consist of Hermite-Gaussian modes the calculation of

M_{x, y}^{2}

is enabled, which is not uniform in the transversal directions (x and y). A non-radial symmetric beam shape is expected from the chosen pump geometry and flashlamp pumping. As a consequence different values for

M_{x}^{2}

and

M_{y}^{2}

are obtained as it can be seen in Section 5.1. In contrast, the choice of Laguerre modes would not be suitable to simulate the CTH-Laser due to their cylindrical mode shape. In that case, values of

M_{x, y}^{2}

that are different in x- and y-direction could not be calculated.

Fig. 5 The simulation of the CTH-Laser with a 4 mm diameter rod exhibits a close match to the experimental measurements. However, the exact cross sections for stimulated emission and absorption are not known and reported differently in literature [7,20].

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Fig. 6 Simulated Data: An improved beam quality is obtained by decreasing the diameter of the Cr,Tm,Ho:YAG crystal from 4 mm (blue line) to 3.5 mm (orange line). At the same time, the pulse power remains constant.

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Fig. 7 Figures 7(a) to 7(f) display simulated siTEM modes in the Cr,Tm,Ho:YAG crystal. Each siTEM mode represents a superposition of 100 Hermite-Gauss modes. Modes of higher order exhibit strong intensity regions close to the jacket surface of the crystal.

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5. Results

5.1. Simulation of the CTH-Laser with 4 mm rod diameter

The simulation was conducted with varying pump pulse energy from 80 J to 180 J for a constant rod diameter of 4 mm. Figure 5 shows the obtained simulation results and the experimental measurement of the generated pulse energy from the CTH-Laser.

It can be seen that all curves exhibit a linear increase in pulse energy over the range of applied pump pulse energy. For a flashlamp pulse energy larger than 100 J the simulation with σ_a = 1.3 · 10⁻²¹ cm² exhibits a close match to the experimental curve with respect to slope and value, which points out the accuracy of the presented DMA approach coupled to the CTH state system shown in Fig. 3. However, the lasing threshold couldn’t be determined to be in agreement with the experiment with the chosen cross section of σ_a = 1.3 · 10⁻²¹ cm² [7]. By simulating with σ_a = 2.6 · 10⁻²¹ cm² [20] the threshold could be approximated better but then the pulse energy above a lamp energy of 100 J exhibits a larger discrepancy. The reported values in literature differ significantly. It would be necessary to measure the cross sections for the identical crystal. More accurate results are expected from the exact values of both the emission and absorption cross for the identical crystal that was employed in the CTH-Laser. Additional uncertainty comes with the coefficients for energy transfer between the dopants, which were also taken from literature (Table 1). However, the simulation with σ_a = 1.3 · 10⁻²¹ cm² shows a close match to the experimental curve. Therefore, this value was taken for all following simulations. Transient thermal lensing is not considered to be the reason for the mismatched threshold. This can be observed in Table 4 where an accurate approximation of thermal lensing is demonstrated. Additionally, the simulation of $M_{x, y}^{2}$ over the pulse duration is performed, which can be seen in Fig. 6. Observe that $M_{x, y}^{2}$ is not constant but changes rapidly until 200 μs. After 200 μs $M_{x, y}^{2}$ starts to stabilize and reaches a value of around 32. Even though the pulse power drops from 550 μs to 1 ms the $M_{x, y}^{2}$ value remains constant. Table 5 contains the simulated values for $M_{x, y}^{2}$ . In experiment a $M_{x, y}^{2}$ value of (39.6/31.6) and (26.7/27.4) was measured for 4 mm and 3.5 mm crystal diameter, respectively. The simulated values of $M_{x, y}^{2}$ represent a close approximation of the measured values.

Table 4. The simulated focal length from thermal lensing is in good agreement with the experimental measurement. All focal lengths are given in mm with respect to x- and y-direction (x,y).

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Table 5. A crystal diameter less than 3 mm results in an instable resonator. In experiment a $M_{x, y}^{2}$ value of (39.6/31.6) and (26.7/27.4) was measured for 4 mm and 3.5 mm crystal diameter, respectively.

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The fluctuation of $M_{x, y}^{2}$ during the pulse generation can be explained by mode superposition and mode competition. In the resonator of the CTH-Laser, different transverse modes simultaneously overlap with the spatial gain distribution in the Cr,Tm,Ho:YAG crystal. The fluctuation stabilizes once the most dominant modes have emerged from that competition. This highly dynamic behaviour is shown in Visualization 1 (“siTEM Modes in Cr,Tm,Ho:YAG Resonator”), which shows the pulse generation from 90 μs to 480 μs. The video corresponds to the time axis in Fig. 6, where the Power and $M_{x, y}^{2}$ of the same pulse is demonstrated. In order to analyze which modes arise from that competition the profile of the individual modes are plotted in Figs. 7(a) to 7(f) in the Cr,Tm,Ho:YAG crystal with 4 mm diameter. All modes have been normalized to the transverse field shape, which means that the integral of |u_m|² over the transverse directions is equal to 1. Here, it can be seen that modes with increasing order exhibit a transversal field profile that drifts from the center to the surface area of the crystal. For example in Fig. 7(a) the mode propagates in the inner volume of the crystal. In contrast, the siTEM mode in Figs. 7(e) and 7(f) require more space in the x–y plane for propagation. Additionally, those high order modes exhibit a field profile that is most intense close to the jacket surface of the crystal.

As the Figs. 7(a) to 7(f) demonstrate, the amplification of higher order modes could be avoided by decreasing the crystal diameter. As a consequence, the suppression of higher order modes in the cavity leads to an improvement of $M_{x, y}^{2}$ , which can be seen in Fig. 6. In this case the pulse energy is almost constant, even though additional reflections are required for full pump pulse absorbance. The following Section further investigates this approach by simulation and experiment.

5.2. Transient thermal lensing

Due to the low repetition rate of 10 Hz, the time dependency of the thermal lens is analyzed. Figure 8 visualizes the temporal progress of the crystal’s focal length that results from the periodic pumping process. Observe that the focal length converges towards a range exhibiting a maximum and minimum value, which is shown in Table 4. Furthermore, it can be observed that the measured average thermal focal length of 122 mm is located in the range from the time-dependent simulation. The time-dependent FEA is computationally very expensive in combination with the full gDMA simulation. Therefore, the calculated pulse energies in Fig. 5 were simulated based upon a time independent calculation of the thermal lense, where an average focal length in between these extreme values of the heat source was assumed. Still, the results from the time-independent simulation are in close agreement with the experimental measurements and demonstrate a suitable approximation for the CTH-Laser.

Fig. 8 Results from the time-dependent simulation of the crystal’s focal length for 182 J flashlamp pulse energy.

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5.3. Beam quality improvement with decreased rod diameter

In order to improve the value of M² the simulation of the CTH-Laser is conducted with decreasing crystal diameter from 4 mm to 2.5 mm in steps of 0.5 mm. Table 5 summarizes the obtained results. From the simulated results it can be seen that the values for $M_{x}^{2} / M_{y}^{2}$ have significantly improved from 32.0/28.0 to 18.1/15.0 by decreasing the crystal diameter from 4 mm to 3 mm. At the same time, the pulse energy experiences only a minor decrease from 4.65 J to 4.32 J. This relates to the decrease in geometrical-optical efficiency (geo. opt. eff.) of the pump absorption, which can be explained by the increasing amount of reflections in the pump cavity, which are necessary for full absorption by Cr³⁺ ions in the crystal. Additionally, it is also observed that the decrease of the crystal diameter has a limit. At a diameter below 3 mm the resonator gets unstable, which is a consequence of the thermal lensing in the crystal. Due to the water cooling of the crystal the temperature on the surface is considered to be constant at 25 °C even though the crystal diameter changes. It is important to mention that the rotational axis of the crystal is parallel to the rotational axis of the flashlamp. Each axis is located on the respective focal axis of the pump cavity. As a consequence, the center of the crystal experiences the most pump absorption and consequently, exhibits a strong heat source. By decreasing the crystal diameter the temperature difference between center and surface does not change drastically but the temperature gradient increases. Figure 9 demonstrates that a reduced crystal diameter results in an increased temperature gradient. The increase in temperature gradient results in a stronger thermal lensing effect, which finally provokes the outcoupling of modes from the resonator. Therefore, decreasing the diameter of the crystal to improve $M_{x, y}^{2}$ is limited by the stability of the resonator.

Fig. 9 The absolute temperature gradient increases with decreasing crystal diameter, which leads to stronger thermal lensing. Reducing the crystal diameter improves the beam quality but thermal lensing gets more dominant. This leads to an unstable resonator below a diameter of 3 mm

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The temperature profile in Fig. 9 is taken from position z = 64.5 mm. This position represents a cut through the middle of the rod since it has a total length of 129 mm. The temperature profile along the z-axis, which is not shown, is uniform without significant gradients. This stems from the flashlamp pumping and the pump cavity. Both the crystal and the flashlamp are oriented along the two focal axis in the pump cavity, which leads to a uniform absorption in z-direction. This can be seen from the population inversion in Visualization 1.

5.4. Identification of dominant excitation mechanisms in Cr,Tm,Ho:YAG

The state population during pumping and laser pulse generation is plotted in Fig. 10. The excitation dynamics in Cr,Tm,Ho:YAG are mainly determined by the pump absorption in Chromium at state ²E₁ (N₁₃), state ³F₄ (N₂) in Thulium and the upper lasing state ⁵I₇ (N₆) in Holmium. It can be seen, that the excitation of Cr³⁺ ions is transferred to Thulium, and from there to Holmium. Since the pump pulse has a duration of 550 μs, the excited state population in Chromium and Thulium drops from this point, due to fluorescent decay. Interestingly, the state population of the upper lasing state in Holmium ⁵I₇ (N₆) remains constant, even though the excitation of ³F₄ (N₂) in Thulium decays exponentially. This can be explained by a stationary state that includes I) the decaying rate of stimulated emission and II) the decaying rate of the ongoing energy transfer from Tm to Ho. Before 550 μs the stationary state of ⁵I₇ (N₆) in Holmium was a result of two processes. First, the continous excitation of ³F₄ (N₂) in Thulium that results from pump absorption in Chromium and the subsequent transfer to the upper lasing state ⁵I₇ (N₆) in Holmium. Secondly, the constant stimulated emission rate from 200 μs to 550 μs, which can be seen in Fig. 6. As a consequence of the stationary state above 550 μs, which is determined by I) and II), the population in Holmium remains constant, even though the population in Chromium and Thulium decays.

Fig. 10 a) State excitation is shifted from Cr³⁺ via Tm³⁺ to Ho³⁺.The population on each state has been normalized to the doping density of the respective ion species. b) The transfer from Tm³⁺ to Ho³⁺ is the dominant mechanism for the excitation of the upper lasing state in Ho³⁺.

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In order to compare the strength of the energy transfer mechanisms, the metric k_xN_iN_j is introduced. This product appears in the rate equations and includes the rate coefficient k_x and the respective population of the origin states N_i and N_j. In Fig. 10(b) it can be observed that the strongest mechanisms is clearly the transfer from Thulium to Holmium. The strength of the mechanism k₂₁₅₆N₂N₅ increases as the population of N₂ rises (Fig. 10(a)). Moreover, it reaches a value that is 5 times larger than the competing back transfer from Holmium to Thulium k₆₅₁₂N₆N₁. After the pump process, which ends at 550 μs the transfer rate from Thulium to Holmium drops while the back transfer k₆₅₁₂N₆N₁ remains constant. This can be explained by the constant population of the lasing state ⁵I₇ (N₆) in Holmium, which was addressed before. Finally, the shifting of excitation between Thulium and Holmium, which is conducted by the mechanisms k₂₁₅₆N₂N₅ and k₆₅₁₂N₆N₁, approaches a state of equilibrium at 1.1 ms. In this state of equilibrium the net energy transfer from Thulium to Holmium is zero, which can also be seen in Fig. 10.

Additionally, the impact of upconversion, cross relaxation and other energy transfer mechanisms was analyzed. The corresponding mechanisms are displayed in Fig. 3. Depending on the mechanism the product k_xN_iN_j exhibits a value that is between one and two orders of magnitude lower than the back transfer k₆₅₁₂N₆N₁ from Holmium to Thulium. The weakness of those mechanisms are either due to a low rate coefficient k_x and/or low populated origin states. One example is the state ³H₄ (N₄) in Thulium, which has a lifetime of 7.9 · 10⁻⁷ s. The conduction of the cross relaxation mechanism k₄₂₁₂N₄N₁ requires two origin states, which are the ground state of Thulium and ³H₄ (N₄). Since the state ³H₄ (N₄) experiences a rapid fluorescent decay, the population of this state is rather low compared to ⁵I₇ (N₆), which has a lifetime of 7 ms and initiates the back transfer from Holmium to Thulium. Therefore, the back transfer k₆₅₁₂N₆N₁ is considered to be the most dominant loss source in Cr(0.89 at %), Tm(6.29 at %), Ho(0.36 at %):YAG with respect to other interionic mechanisms that are present.

6. Conclusion

Within this work, the novel method gDMA was introduced, which includes an abstract formalism for rate equations and the former DMA. With gDMA the simulation of a pulsed Cr,Tm,Ho:YAG laser has been performed. With the suggested approach the simulated pulse energy precisely matches the experimental measurements above a lamp energy of 100 J while the mismatch of the lasing threshold is explained with not exactly determined emission and absorption cross section for the employed crystal. Moreover, the beam quality factor $M_{x, y}^{2}$ could be simulated by combining the presented gDMA method with super-imposed Hermite-Gaussian modes. The calculated values for $M_{x, y}^{2}$ approximate well the measured value. In detail, a simulated $M_{x, y}^{2}$ value of (32/28) was obtained for a crystal diameter of 4 mm. In order to improve the beam quality factor, the reduction of the crystal diameter was suggested. This procedure was analyzed comprehensively in simulation, where it could be observed that thermal lensing leads to instability of the resonator below a diameter of 3 mm and therefore, sets a limit to the minimum obtainable $M_{x, y}^{2}$ . In experiment the measurement of $M_{x, y}^{2}$ resulting from a 3.5 mm crystal matches well the simulated value (26.9/24.1). Additionally, individual super-imposed Hermite-Gaussian modes have been visualized and a video has been generated to give insights in the mode competition during pulse generation. Finally, the excitation dynamics in Cr,Tm,Ho:YAG were analyzed during pulse generation. Here, it could be determined that the back transfer from Holmium to Thulium represents the main loss source of excitation for the upper lasing state ⁵I₇ in Holmium. In summary, the presented gDMA represents a novel modelling technique to simulate the coupling of complex excitation dynamics and mode dependent amplification in multiple doped gain media.

Funding

Deutsche Forschungsgemeinschaft (DFG) (PF 372/10-1)

Acknowledgments

The authors gratefully acknowledge support by the Erlangen Graduate School in Advanced Optical Technologies (SAOT).

References

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10. K. H. Kim, Y. S. Choi, N. P. Barnes, R. V. Hess, C. H. Bair, and P. Brockman, “Investigation of 2.1-μm lasing properties of Ho:Tm:Cr:YAG crystals under flash-lamp pumping at various operating conditions,” Appl. Opt. 32, 2066–2074 (1993). [CrossRef] [PubMed]

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14. G. Rustad and K. Stenersen, “Modeling of laser-pumped Tm and Ho lasers accounting for upconversion and ground-state depletion,” IEEE J. Quantum Electron. 32, 1645–1656 (1996). [CrossRef]

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Mechanism	Ion Interaction	Constant	Value in 10⁻¹⁷ $\frac{{cm}^{3}}{s}$	Source States	Reference

E_13,12,1,11	Cr³⁺ → Tm³⁺	k_13,12,1,11	2.0	N₁₂, N₁	[12]
E₆₅₁₂	Ho³⁺ → Tm³⁺	k₆₅₁₂	2.6	N₆, N₁	[18]
E₂₁₅₆	Tm³⁺ → Ho³⁺	k₂₁₅₆	20	N₂, N₅	[18]
E₂₁₆₈	Tm³⁺ → Ho³⁺	k₂₁₆₈	2.4	N₂, N₆	[14]
E_4,1,6,10	Tm³⁺ → Ho³⁺	k_4,1,6,10	3.0	N₄, N₆	[18]
U₂₁₂₄	Tm³⁺ → Tm³⁺	k₂₁₂₄	0.1	N₂	[14]
U₂₁₂₃	Tm³⁺ → Tm³⁺	k₂₁₂₃	3.0	N₂	[14]
C₄₂₁₂	Tm³⁺ → Tm³⁺	k₄₂₁₂	10	N₄, N₁	[18]
U₆₅₆₈	Ho³⁺ → Ho³⁺	k₆₅₆₈	0.1	N₆	[1]

Parameter	Description	Value	Unit	Reference

τ₁₃	Fl. lifetime of state N₁₃ in Cr	1	ms	[12]
τ₂	Fl. lifetime of state N₂ in Tm	11	ms	[11]
τ₆	Fl. lifetime of state N₆ in Ho	8	ms	[19]
σ_e	Emission cross section in Ho at 2090 nm	12 · 10⁻²¹	cm²	[20]
σ_a	Absorption cross section in Ho at 2090 nm	1.3 · 10⁻²¹	cm²	[7]
Cr³⁺	Doping concentration Cr in Cr,Tm,Ho:YAG	0.89, 1.21 · 10²⁰	at%, $\frac{1}{{cm}^{3}}$
Tm³⁺	Doping concentration Tm in Cr,Tm,Ho:YAG	6.29, 8.55 · 10²⁰	at%, $\frac{1}{{cm}^{3}}$
Ho³⁺	Doping concentration Ho in Cr,Tm,Ho:YAG	0.36, 0.49 · 10²⁰	at%, $\frac{1}{{cm}^{3}}$

Module (Method)	Simulation Parameter	Value	Unit

Temperature (Finite Element Method)	Number of Gridpoints	86609	/
	Meshsize in z-Direction	0.502	mm
	Multigrid iterations	10	/
	Thermal conductivity (isotropic)	1.03 · 10⁻²	$\frac{W}{m m K}$
	Thermal expansion coefficient (isotropic)	6.90 · 10⁻⁶	$\frac{1}{K}$
	Density of YAG	4.56 · 10⁻³	$\frac{g}{m m^{3}}$
	Specific heat capacity	5.9 · 10⁻¹	$\frac{J}{g K}$
	Refractive index YAG (2090 nm)	1.7991	/
	Temp. dep. refractive index (isotropic)	9.86 · 10⁻⁶	$\frac{1}{K}$

Mode Calculation (gDMA)	Number of Gridpoints in z-Direction	5	/
	Number of Gridpoints in x- and y-Direction	20	/
	Meshsize in z Direction	25.8	mm

Rate Equations (Finite Difference)	Timestep	60 · 10⁻⁹	s

siTEM Mode Grouping	Total number of Hermite-Gauss modes	8100	/
	group order g_i	10	/
	Number of siTEM Groups	81	/

E_Lamp	56 J	88 J	129 J	182 J

Simulation - time independent (x, y)	395, 371	255, 240	178, 168	129, 123
Simulation - time dependent max (x, y)	456, 440	289, 284	201, 198	147, 144
Simulation - time dependent min (x, y)	388, 346	244, 227	170, 157	127, 116
Experimental measurement	470	266	175	122

Diameter	E_Lamp	E_Pulse	Simulation $M_{x}^{2} / M_{y}^{2}$	geo. opt. eff.

4 mm	158 J	4.65	32.0 / 28.0	86.7 %
3.5 mm	158 J	4.45	26.9 / 24.1	84.0 %
3.0 mm	158 J	4.32	18.1 / 15.0	80.4 %
2.5 mm	158 J	instable resonator	/	/

3D simulation and beam quality improvement of a pulsed Cr,Tm,Ho:YAG laser

Abstract

1. Introduction

2. Experimental setup of the CTH-Laser

3. State system of Cr,Tm,Ho:YAG

4. Model description

4.1. The gDMA approach for arbitrary rate equations

4.1.1. Abstract formulation of stimulated emission

4.1.2. Abstract formulation of nonlinear interionic transitions

4.2. Definition of the CTH state system

4.3. Finite element heat model

4.4. Super-Imposed-Gauss-Hermite modes

5. Results

5.1. Simulation of the CTH-Laser with 4 mm rod diameter

5.2. Transient thermal lensing

5.3. Beam quality improvement with decreased rod diameter

5.4. Identification of dominant excitation mechanisms in Cr,Tm,Ho:YAG

6. Conclusion

Funding

Acknowledgments

References

Supplementary Material (1)

Cited By

Figures (10)

Tables (5)

Equations (33)

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