Two-dimensional modeling of transient gain gratings in saturable gain media

Robert Elsner; Roland Ullmann; Axel Heuer; Ralf Menzel; Martin Ostermeyer

doi:10.1364/OE.20.006887

1. Introduction

Transient volume Bragg gratings inside saturable gain media, also known as gain gratings, have been shown to possess numerous desirable attributes for enabling novel laser designs. Hereby, the optical interference pattern of two or more coherent beams, intersecting inside a laser amplifier is transferred into the spatial distribution of the gain coefficient by means of spatial hole burning. Exploiting this effect in a degenerate four-wave mixing (DFWM) geometry, self-pumped phase conjugate (SPPC) ring resonators with excellent beam quality and effective correction of intracavity phase distortions have been experimentally demonstrated [1–8].

The gain grating simultaneously realizes a passive Q-switch, acts as a narrow-band spectral filter [9,10] and as a phase-conjugating element in these resonators. Recently, a novel approach to coherent beam combination using gain gratings has been reported [11].

These applications motivated several analytical and numerical descriptions of DFWM inside saturable gain media [9, 10, 12–15]. To the best of our knowledge all of these descriptions are limited to one spatial dimension along the principal propagation direction of the pump beams. Consequently the existing models do not take into account the effects of transverse features of the intersecting beams such as their focus size or transverse position inside the gain medium. Neither can the influence of inhomogeneous gain distributions be modeled that arise in SPPC resonators employing bounce amplifier geometries [6].

In this paper, a transient numerical model featuring two spatial dimensions is presented. This allows for a quantitative description of the transverse features of DFWM in saturable gain media. A numerical simulation based on a one-dimensional model was implemented as a reference. Comparisons to the existing results of one-dimensional simulations reveal sensitive dependencies of the phase conjugate reflectivity on the transverse shape and position of the beam. Due to the transient nature of the gain depletion, this treatment is different from other treatments of DFWM in classical χ⁽³⁾ media.

2. Model

A transient numerical description of the DFWM interaction in one dimension has been presented in [16] and was extended to include vector phase conjugation by [17]. This model describes the operation of short pulse lasers with characteristic timescales of less than 1 μs. Therefore the much slower dynamics of spontaneous emission and inversion build up due to pumping are neglected in this treatment.

The DFWM geometry is shown in Fig. 1. The model describes the experimental situation where three Gaussian pulses (Both with respect to their temporal and transverse profiles) A₁ to A₃ enter a saturable gain medium with a given population inversion. Depending on the polarization states of the beams A₁ to A₄, any combination of the gratings τ, ρ, Δ, δ might be present simultaneously. Generally the beams A₁ and A₂ are called the pump beams with A₃ and A₄ the probe and conjugate respectively. The positive z-direction is along the direction of propagation of the A₁ pump beam.

Fig. 1 Left: The four-wave mixing geometry inside the gain medium. In the most general case, four different gratings are written by the beams A₁ to A₄. From top to bottom: Transmission grating (τ), reflection grating (ρ), pump-pump grating (Δ) and probe-conjugate grating (δ). The blue arrows indicate the polarization. Right: Detailed geometry of the problem. θ is the angle between the z-axis and the wave vector k₃.

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The temporal dynamics of the four amplitudes are now considered in the x–z plane. The electric fields are assumed to be monochromatic with

E_{j} (r, t) = A_{j} e^{i (k_{j} \cdot r - ω t)}

and the waves A₁, A₂ and A₃, A₄ are counter-propagating with the wave vectors given by

k_{1} = | k | e_{z}, k_{3} = | k | [cos (θ) e_{z} + sin (θ) e_{x}]

Starting from the nonlinear wave Eq. (3) together with the expression Eq. (4) for the amplitude gain coefficient α in the transient regime and exploiting the slowly-varying envelope approximation where appropriate, yields a system of four coupled wave Eq. (5). To arrive at that result, the periodic modulation of the amplitude gain coefficient was conveniently expanded into a Fourier cosine series [17]. The relation between α and (3) is readily established via the complex refractive index n = n′ + in″ with n″ = −α/|k₀| [18].

(Δ - \frac{{n^{'}}^{2}}{c_{0}^{2}} \partial_{t}^{2}) E = - \frac{2 i n^{'}}{c_{0}^{2} | k_{0} |} \partial_{t}^{2} (α E)

α = α_{0} exp (\frac{1}{τ_{l} I_{S}} \int_{0}^{t} I_{T} (r, t^{'}) d t^{'})

\begin{array}{l} (\frac{- i}{2 | k |} Δ \pm sin (θ_{j}) \partial_{x} \pm cos (θ_{j}) \partial_{z} + \frac{n^{'}}{c_{0}} \partial_{t}) A_{j} (x, z, t) \\ = F_{j} (A, t) \end{array}

with j = 1...4 and the plus sign for the waves A₁, A₃. I_T is the total intensity, I_S the saturation intensity, τ_l the upper state lifetime of the gain medium and

Δ = \partial_{x}^{2} + \partial_{z}^{2}

. Formally this equation can be regarded as a transport equation with

\frac{- i}{2 | k |} Δ

the diffusion term, the ∂_x and ∂_z terms describing advection and a nonlinear source term F_j(A,t). The right-hand sides F_j(A,t) are given by

F_{1} (A, t) = γ A_{1} + κ_{Δ} A_{2} + κ_{τ} A_{3} + κ_{ρ} A_{4}

F_{2} (A, t) = κ_{Δ}^{*} A_{1} + γ A_{2} + κ_{ρ}^{*} A_{3} + κ_{τ}^{*} A_{4}

F_{3} (A, t) = κ_{τ}^{*} A_{1} + κ_{ρ} A_{2} + γ A_{3} + κ_{δ} A_{4}

F_{4} (A, t) = κ_{ρ}^{*} A_{1} + κ_{τ} A_{2} + κ_{δ}^{*} A_{3} + γ A_{4}

where γ can be interpreted as the average gain and κ_τ, κ_ρ, κ_Δ, κ_δ as the grating strength of the respective τ, ρ, Δ, δ gratings. Note that these equations describe only the right-hand sides of the two-dimensional Eq. (5) with A_i = A_i(x, z,t). In the limit θ = 0 and neglecting the second order diffusion terms, this expression reduces to the well-known one-dimensional model [17] with A_i = A_i(z,t). For the sake of brevity, the complex definition of the coupling coefficients γ, κ_i has been omitted. A detailed derivation is given in [17]. Note that the treatment given there is directly applicable to the two-dimensional model because the transverse spatial dependency of the amplitudes A_i(x, z,t) automatically results in the correct coupling coefficient γ = γ(x,z,t) and κ_i = κ_i(x, z,t).

While the one-dimensional model is easily solved using the method of characteristics or equivalently a coordinate transformation to a moving frame of reference, the situation is more difficult in the two-dimensional case. Due to two distinguished directions of propagation compared to one in the one-dimensional case, the characteristics of the problem do not fit well with common spatial discretization schemes. Furthermore, the presence of a diffusion term $\partial_{x}^{2}$ which can not be neglected requires implicit time stepping. The problem is solved using an operator split-stepping technique with separate operators for the advection, diffusion and nonlinear parts of the equation. Accuracy and convergence of the simulation is checked by monitoring energy conservation of identical calculations on different spatio-temporal grid spacings. With two spatial and one temporal dimension, the model is computationally expensive. The necessity to employ relatively fine grids prohibits a straightforward extension to three spatial dimensions at the moment. This shortcoming might be remedied by employing spectral approximations to the spatial discretizations, which is currently under investigation.

In this paper, the derived model is applied to an Nd:YAG rod as gain material with 5mm diameter and 150mm length. The upper state lifetime is τ_l = 230μs. The rod has a homogenous gain distribution with a small-signal amplitude gain coefficient α₀ = 14.6. The saturation fluence used for the calculations in this paper is U_sat = 5835 J/m².

3. Sensitivity to spatial features

The one-dimensional model is unable to predict the influence of transverse effects. Because easily accessible by experiment, this treatment concentrates on three transverse beam features - the 1/e beam radius w₀ at the focus, the crossing angle θ between pump and probe and the transverse position Δx of the pump beams with respect to the center of the rod. This model describes the physical situation in which three laser pulses A₁ – A₃ enter the gain medium simultaneously, focused at the rod’s center x = 0,z = L/2. The pulses have a Gaussian temporal intensity profile with a FWHM width of 10ns. Unless noted otherwise, all simulations are performed with respect to a transmission (τ) grating with the default values w₀ = 0.5mm, θ = 1° and Δx = 0mm. The normalized pump fluence Û_P = Û₁ = Û₂ is fixed to Û_P = 10⁻¹ and the normalized probe fluence to Û₃ = 10⁻³. Normalization is with respect to the saturation fluence of Nd:YAG with Û = U/U_sat. The definition given in Eq. (12) is used for the two-dimensional fluence. The colored contour plots showing the spatial distribution of the κ_τ coefficient (Figs. 3, 4, and 5) are temporal snapshots taken at the end of the simulation run. Because the model neglects spontaneous emission and pumping, these images show the final state of the κ_τ distribution after the pulses have passed through the active medium.

Fig. 2 The phase conjugate reflectivities of the transmission and reflection gratings for both the one-dimensional and two-dimensional models. The normalized fluence of the probe beam A₃ is Û₃ = 10⁻³ while U_P = U₁ = U₂.

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Fig. 3 Top: The phase conjugate reflectivity R_U of the FWM interaction in dependence of the transverse displacement of the pump beam. Bottom: The spatial coupling strength distribution of the gain grating for three different transverse positions of the pump beams. From left to right: Δx = 0.8 mm, Δx = 0.0 mm, Δx = −0.9 mm. The images show the magnitude of κ_τ, which is directly related to the grating strength. The beams A₁, A₃ enter at z = 0. A displacement of zero indicates that the pump beams enter the laser rod at the center. All three images share the same color scale. Blue indicates a weak, red a strong grating amplitude.

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Fig. 4 Top: The phase conjugate reflectivity R_U of the FWM interaction in dependence of the waist size w₀ of all three input beams. The fluences stay constant at their default values. Bottom: Spatial distribution of the grating strength for different beam waist sizes w₀. From left to right: w₀ = 0.1 mm, w₀ = 0.5 mm, w₀ = 0.9 mm. The images show the magnitude of κ_τ and share a common color scale. Blue indicates a weak, red a strong grating amplitude.

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Fig. 5 Top: The phase conjugate reflectivity R_U of the FWM interaction in dependence of the angle θ between the k₃-vector and the z-axis. Bottom: Spatial distribution of the grating strength for different crossing angles. From left to right (in degree): θ = 1.2, θ = 0.83, θ = 0.5. The images show the magnitude of κ_τ and share a common color scale. Blue indicates a weak, red a strong grating amplitude.

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The efficiency of SPPC laser resonators [2, 3] employing gain gratings critically depends on the reflectivity of the gain grating with the phase conjugate reflectivity defined as the ratio of the fluences of the beams A₄ and A₃ at the left face (z=0) of the gain medium.

R_{U} = \frac{U_{4} (z = 0)}{U_{3} (z = 0)}

U (z) = \int I (z, t) d t

This definition of the fluence is unequivocal for a one-dimensional model because those models imply a flat-top transverse intensity profile of the beam with a constant fluence anywhere within the beam. In contrast this definition is not unequivocal in the two-dimensional case, because a two-dimensional simulation generates one fluence value for each grid point on the x-axis. Again the two-dimensional treatment implies a constant intensity profile of the beam in the y-direction which is the neglected spatial dimension. To be able to compare the fluences of the one- and two-dimensional simulations, it is necessary to introduce an equivalent quantity for the two-dimensional case. With the 1/e beam radius w_spot (z) in x-direction given at position z by the results of the simulation, the fluence of a flat-top beam with the same spot size and power is

U_{2 D} (z) = \frac{1}{2 w_{spot} (z)} \iint I (x, z, t) d t d x

This corresponds to an equivalent physical situation as described by the one-dimensional model, allowing for a meaningful comparison of U(z) and U₂_D(z). Equation (10) can then be written as

R_{U} = U_{2 D}^{(4)} (z = 0) / U_{2 D}^{(3)} (z = 0)

.

A plot of the fluence reflectivity for the reflection and transmission gratings at different pump fluences is shown in Fig. 2. As can be seen, the reflection configuration is less effective when compared to the transmission case. Both the one-dimensional and the two-dimensional models confirm this characteristic behavior which is consistent with previously published results [13, 16]. In the two-dimensional simulation the difference between transmission and reflection grating is less pronounced. The distinctive maximum of the reflectivity results from the excessive gain depletion that is caused by the pump beams past a certain pump fluence.

The two-dimensional simulation reveals that a considerably higher pump fluence is required to obtain the highest possible reflectivity for a given probe fluence. The difference is one order of magnitude. This is due to the limited interaction volume that is governed by the spatial overlap of the beams. In contrast, the one-dimensional model unrealistically implies complete beam overlap over the whole length of the gain medium. Simultaneously, the smaller interaction volume limits the achievable peak reflectivity of the grating which decreases by half.

The laser rod geometry limits θ to approximately 1.5 degrees assuming a waist radius of the beams A₃, A₄ of w₀ = 0.5mm. At the same time a sufficient beam separation behind the laser rod requires a minimum angle not smaller than 0.5 degrees. This essentially fixes the paths of the beams A₃ and A₄ while the beams A₁ and A₂ enjoy a relative freedom with respect to their transverse position within the laser rod. This freedom can be exploited to optimize the grating efficiency by shifting the interaction region to a favorable area inside the gain medium. Figure 3 shows the position of the interaction volume for three different transverse positions of A₁ and A₂. The reflectivity changes by a factor of four between the neutral position Δx = 0 and the optimal position Δx ≈ −0.9mm (Fig. 3).

By shifting the interaction region to the far end of the laser rod at z = L, both the intensities of A₁ and A₃ increase due to a longer path through the gain medium, resulting in a deeper modulation of the grating. This in turn leads to an improved reflectivity. Eventually the interaction region is clipped at the rod’s end faces, counterbalancing the positional improvement and resulting in an overall decrease of the reflectivity. Additionally the stronger writing beams at the far end of the rod might contribute significantly to the depletion of the available gain and thus erasing the grating.

To exploit the available gain inside G_A as efficiently as possible, it is advantageous to employ an aperture-limited spot-size. On the other hand, with four-wave mixing being a third-order nonlinear effect, it is not obvious if the reduced beam peak intensities at bigger spot sizes lead to a reduced reflectivity or not. A simulation with all three beams having identical waist sizes w₀ ranging from 0.1mm to 1mm at a constant average fluence is shown in Fig. 4. A small focus size leads to a small grating volume and the reduced overlap between the beams A₁ and A₃ shifts the center of gravity of the grating towards the center of the rod with z = L/2, decreasing reflectivity as explained above. Simultaneously the higher transverse peak intensities quickly deplete the available gain, erasing the grating in the process. Consequently, the grating itself is comparably weak. The processes leading to the increased reflectivity are identical to those above, emphasized by the qualitative similarity between plots 3 and 4. Therefore for a given beam fluence, a bigger beam spot size improves reflectivity. The situation might be different if the pulse energy is kept constant. Nonetheless the same competing processes are bound to limit the reflectivity of the grating to the extent that a small focus does not imply a maximum reflectivity.

One might expect from the difference between the one- and two-dimensional simulations, that the reflectivity will assume its maximum in the limit θ → 0 due to the bigger effective interaction volume. Simulations show, that this is not the case. The reflectivity depends critically on the size and the position of the interaction volume. In turn the optimal size and position depends sensitively on the beam fluences and the gain factor. Figure 5 shows a steady increase in reflectivity with decreasing angle up to θ = 0.55°. At small angles - limited by the necessity of separating the beams behind the laser rod - of around θ = 0.5° the reflectivity remains constant or even decreases slightly. Figure 5 shows that both the volume and the z-position of the grating changes with θ. A bigger grating volume at comparable grating strengths equals a higher reflectivity while the effects of a change in position has been elaborated above. This is a typical example - for the chosen beam parameters - where the interaction volume gets shifted to an unfavorable region with decreasing crossing angles smaller than θ = 0.5°. This is due to a premature extinction of the grating because of gain depletion caused by the respective intensity distributions. Therefore the smallest possible crossing angle obtainable in the laboratory might not necessarily yield an optimal reflectivity of the grating.

Excessive gain depletion and small spatial overlap of the beams are the major factors limiting the reflection efficiency of the gain grating. Therefore the naive approach to use a minimum crossing angle and a maximal displacement into the negative x-direction might yield suboptimal reflectivities. Due to the high dimensionality of the parameter space, the optimal geometrical parameters for a given set of beam parameters must be calculated on a case by case basis.

4. Conclusions

A novel two-dimensional model, describing degenerate four-wave mixing in saturable gain media has been developed. This model enables new insights into the effects of the transverse structure of the interacting beams onto the wave mixing process. A reimplementation of a one-dimensional numerical model served as a reference to compare the results to numerical results obtained in other groups. Features demonstrated before by experiments [1, 2] and one-dimensional models were confirmed qualitatively by the two-dimensional model. The quantitative differences to one-dimensional models is substantial though. It was shown that the phase conjugate reflectivity of the gain grating can be drastically increased by careful tuning of easily accessible beam parameters such as spot size, orientation and transverse position. A transverse shift in the position of the pump beams can yield a threefold improvement of the reflectivity. Though not considered here, this model also enables the simulation of gain gratings inside gain media with inhomogeneous transverse gain profiles that are encountered in high gain bounce amplifier geometries. In conclusion the new model is a greatly improved tool to assess and optimize the performance of gain gratings and consequently the associated self-pumped phase conjugate laser resonators.

Acknowledgments

This research was funded by the DFG, project grant no. OS 194/8-1. Robert Elsner acknowledges the support of Volker Gustavs by providing technical support and computing time on the cluster resources at the University of Potsdam. Furthermore Mindaugas Radziunas from the Weierstrass Institute for Applied Analysis and Stochastics contributed to the numerical simulations with fruitful discussions and advice.

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Two-dimensional modeling of transient gain gratings in saturable gain media

Abstract

1. Introduction

2. Model

3. Sensitivity to spatial features

4. Conclusions

Acknowledgments

References and links

Cited By

Figures (5)

Equations (12)

Optics Express