Combined guiding effect in the end-pumped laser resonator

Xingpeng Yan; Qiang Liu; Dongsheng Wang; Mali Gong

doi:10.1364/OE.19.006883

1. Introduction

The transverse mode formation mechanism of diode pumped laser has attracted many researchers’ interest [1–5]. In most quasi three-level lasers (e.g., Yb³⁺ doped gain medium), the inherent absorption of the laser in the unpumped region of the gain medium works equivalently to a soft transverse gain aperture [6]. This aperture guiding effect dominates the transverse mode formation in the laser resonator. In contrast, in the Nd³⁺ based four-level laser, the un-pumped region has very low absorption at the laser wavelength so that the aperture guiding can be neglected. However, the thermal induced refractive index guiding (or called the thermal guiding) [7] and gain guiding effect [8] are found important for the formation of the transverse mode especially for plano cavity with high power and high intensity pumping.

The thermal guiding [7,9] and gain guiding [5,8,10,11] have been researched respectively in previous works. The combined guiding effect has been observed and studied in the microchip lasers [12–15]. This is because that, the gain medium with high Nd³⁺ ion doping concentration is always used in microchip laser to achieve effective pump absorption, and this can lead to significant thermal induced refractive index guiding and gain guiding. The quadratic index profile is always assumed in the gain medium, and the Gaussian gain profile is used in previous works. Meanwhile, these profiles are always assumed constant along the optical axis in the gain medium. These approximations are only worked for microchip lasers. In particular, the high power fiber-coupled diode pumped laser can get high gain as well as large temperature gradient in the gain medium, and the former approximation on the temperature- and gain- distribution is not enough accurate anymore. Therefore, the combined guiding effect in a general diode pumped laser should be investigated. However, the combined guiding effect in a general diode pumped laser but not in microchip laser has not been theoretically and experimentally investigated in detail.

In this work, the effect of the combined guiding mechanism on the transverse mode formation of an end pumped laser is investigated theoretically and experimentally. Both the thermal induced refractive index guiding and gain guiding effect are taken into account. The wave equation of the laser field in the gain medium is established with the combined guiding effect considered. The exact temperature distribution but not the quadratic index approximation is calculated numerically, and the actual gain model but not the constant Gaussian gain is used. In addition, transverse mode of the resonator is calculated using the split step Fourier method (SSFM) but not the traditional diffraction integral method [1, 16]. The effect of combined guiding effect on the formation of transverse mode profiles is also observed and researched experimentally which shows well agree with the prediction of our theoretical model.

The outline of this paper is as following. In Section 2 the theoretical model is presented. The cell-train model of a laser resonator is given, and then the nonlinear Schrödinger-type wave equation in the gain medium is derived in which the combined guiding effect is considered. The step-split Fourier method is presented to solve the wave equation. In Section 3, the experimental setup of an end-pumped Nd:YVO₄ laser resonator is introduced, and the pump absorption model in the experiment is presented. In Section 4, the temperature and the gain distribution in the gain medium is calculated, and then the profiles of the transverse mode is calculated from our wave equation. The theoretical results are compared with the experimental results. Finally, we summarize and conclude in Section 5.

2. Theoretical model

2.1. The cell-train model of laser resonator

The configuration of a typical laser resonator is shown in Fig. 1. The laser resonator often has two cavity-mirrors, one is high reflection (HR) at the laser wave, and the other one is with partial transmissivity T called as the output coupler (OC). The gain medium is placed into the cavity with the position parameters of L ₁ and L ₂. The laser oscillates in the resonator when the gain from the gain medium is higher than the loss from the OC, geometrical diffraction, aperture diffraction and so on.

Fig. 1 The sketch of a general laser oscillator. HR: high reflectivity mirror, OC: output coupler, E ^±: the forward- and backward- propagating waves respectively, and L _1,2: the cavity length parameters.

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According to the physical essence of the laser resonation, the resonator can be unfolded along the optical axis of the resonator, and then the periodic cell-train topological model can be obtained as shown in Fig. 2. One of the cell represents the twice transmission of the laser field trough the gain medium in one round-trip. The junction of the two cell is with the loss of T for the laser intensity. The steady state oscillation in the resonator builds up from the random spontaneous emission and then oscillates in the resonator until a steady state laser transverse mode or called as the self-reproducing field pattern is obtained. In the equivalent periodic cell-train model, the initial field at the junction 0 (J ₀) transmits through the cells one after another. The self-reproducing field pattern is obtained if the field at J_k ₋₁ has the same field pattern as that at J_k. Using the iteration method, the self-reproducing field pattern of the laser resonator can be calculated. It should be noticed that, the forward direction in the unfolded cell-train model is defined as the forward of the unfolded optical axis. This definition for the forward-and backward- direction is a relative direction while that in Fig. 1 is a absolute direction. The gain in the gain medium is share with the forward propagating waves E ⁺ and the backward propagating waves E ⁻, i.e, the spatial hole burning is considered which is neglected in previous works [4].

Fig. 2 The periodic cell-train model of the laser resonator. C_k: the k–th cell, J_k: the junction between the cell C_k and C_k ₊₁ while J ₀ is the initial reference, $E_{k}^{1 \pm}$ : the forward- and backward- propagating waves for the first pass through of the gain medium in the unfolded model, and $E_{k}^{2 \pm}$ denotes that of the second pass.

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According to the physical essence of the unfold model as shown in Fig. 2, in the iterative calculation, the two backward field in C_k can be represented recursively as following

\begin{array}{l} E_{k}^{1 -} = E_{k - 1}^{2 +} \\ E_{k}^{2 -} = E_{k}^{1 +}, \end{array}

with

E_{1}^{1 -} = 0

. The gain medium has specific temperature and gain distribution because of the pumping. When the wave E propagates in the gain medium, it suffers the thermal induced refractive index Δn_T and the gain Δn_G as shown in Fig. 3. The propagation of the wave is guided by the combined action of Δn_T and Δn_G, ie., the combined guiding. Thus, the wave equation included the combined guiding effect should be derived.

Fig. 3 The sketch for the combined guiding in the laser gain medium. Δn_T : the profile of the refractive index induced by the temperature gradient(the real refractive index), Δn_G: the profile of the gain induced refractive index(the imaginary refractive index).

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2.2. Wave equation in the gain medium

The field propagating in the gain medium can be derived from the Maxwell equation. For the conventional laser gain medium, the medium are always dielectric. Within the paraxial approximation, the field propagating is described by the wave equation

(\nabla^{2} - ɛ μ \frac{\partial^{2}}{{\partial t}^{2}}) \vec{E} (x, y, z, ω, t) = μ \frac{\partial^{2}}{{\partial t}^{2}} \vec{P} (x, y, z, ω, t),

where E⃗ is the electric field intensity, P⃗ is the polarization intensity, ε and μ is the permittivity and permeability, respectively, of the gain medium. μ = μ ₀ μ_r ≈ μ ₀ for the nonmagnetic material where μ_r ≈ 1 is the relative permeability, and μ ₀ is the permittivity of free space, or called the electric constant. ω is the resonant frequency of the light field in the laser resonator. (x,y,z) denotes the position coordinates, and t denotes the time. The operator ∇² is the Laplacian,

\nabla^{2} = \frac{\partial^{2}}{{\partial x}^{2}} + \frac{\partial^{2}}{{\partial y}^{2}} + \frac{\partial^{2}}{{\partial z}^{2}}

.

Considering a linear polarized field with frequency ω and with transverse polarized vector ê, we can write the propagating electric field with separating out the time-dependent phase variations in the field as following

\begin{array}{l} \vec{E} (x, y, z, ω, t) & = & \frac{1}{2} \hat{e} [E (x, y, z) e^{i (k z - ω t)} + c . c .] \\ = & \frac{1}{2} \hat{e} [E^{†} (x, y, z, ω) e^{- i ω t} + c . c .], \end{array}

and

\vec{P} (x, y, z, ω, t) = \frac{1}{2} \hat{e} [P^{†} (x, y, z, ω) e^{- i ω t} + c . c .],

where E ^† is the complex amplitude of the electric field with an approximate time dependence of e^−iωt. Substitute Eq. (3) and Eq. (4) into Eq. (2), then we can get the Helmholtz equation

(\nabla^{2} + k_{0}^{2}) E^{†} (x, y, z, ω) = - {μ ω}^{2} \frac{\partial^{2}}{{\partial t}^{2}} P^{†} (x, y, z, ω),

where the relationship of

k = \frac{2 π}{λ}

and

c = \frac{1}{\sqrt{ɛ μ}}

was used. Separating out the rapid on-axis phase variations (i.e., the z-dependent phase variations) in the field, we have

E^{†} (x, y, z, ω) \sim E (x, y, z, ω) e^{{ik}_{0} \int^{z} n_{b} (z^{'}) d z^{'}} .

Assume that the nonlinear effect of the electric field in the dielectric can be ignored, and then the linear polarization response approximation is considered,

P^{†} (x, y, z, ω) \sim ɛ_{0} χ (x, y, z, ω) E (x, y, z, ω) e^{{ik}_{0} \int^{z} n_{b} (z^{'}) d z^{'}},

where χ is a complex quantity denoted the electric susceptibility of the dielectric. The complex refraction index of the dielectric can be represented with the electric susceptibility as following,

n^{2} (x, y, z, ω) = 1 + χ (x, y, z, ω) .

The refractive index at frequency ω in the gain medium can be described as

n (x, y, z, ω) = n_{b} (z) + Δ n (x, y, z, ω),

where n_b(z) is the base refractive index of the medium in the absence of guiding given by k = n_b(z)k ₀, where Δn(x,y,z,ω) = n(x,y,z,ω) – n_b(z). Substituting Eq. (9) into Eq. (8) and applying the first-order approximation when Δn ≪ n_b

χ (x, y, z, ω) \approx n_{b}^{2} (z) + 2 n_{b} (z) Δ n (x, y, z, ω) - 1.

Then from Eqs. (6), (7), and (10), the Helmholtz equation changed into

[\nabla^{2} + 2 {ik}_{0} n_{b} \frac{\partial}{\partial z} + k_{0}^{2} - k_{0}^{2} n_{b}^{2}] E (x, y, z, ω) \approx - k_{0}^{2} [n_{b}^{2} + 2 n_{b} Δ n (x, y, z, ω) - 1] E (x, y, z, ω) .

Applying the slowly varying envelope approximation

| k_{0}^{2} n_{b}^{2} E | ≫ | 2 k_{0} n_{b} \frac{\partial E}{\partial z} | ≫ | \frac{\partial^{2} E}{\partial z^{2}} |,

and defined the transverse Laplacian as

\nabla_{⊥}^{2} = \frac{\partial^{2}}{{\partial x}^{2}} + \frac{\partial^{2}}{{\partial y}^{2}}

, then Eq. (11) changed into

[\nabla_{⊥}^{2} + 2 {ik}_{0} n_{b} \frac{\partial}{\partial z} + 2 k_{0}^{2} n_{b} Δ n (x, y, z, ω)] E (x, y, z, ω) = 0.

The complex-valued guiding refractive index Δn can be written in an explicit form,

Δ n (x, y, z, ω) = Δ n_{T} (x, y, z) + Δ n_{G} (x, y, z, ω),

Here Δn_T is the thermal induced guiding refractive index which is a real one, and Δn_G is the gain induced guiding refractive index which is a imaginary one.

The thermal induced guiding refractive index Δn_T is mainly determined by the thermo-optic effect of the gain medium. In diode-pumped solid-state lasers, the temperature gradient in the gain medium arises from the heat generation from unconverted pump power that manifests itself as heat and phonon energy. This temperature gradient, when coupled with the refractive indices temperature coefficient(dn/dT) can give a distribution of the refractive index in the material. The thermal induced guiding refractive index Δn_T is given as following

Δ n_{T} (x, y, z) = \frac{d n}{d T} (T (x, y, z) - T_{r}),

where T(x,y,z) is the steady-state temperature field in the gain medium volume and T_r is the room temperature. The steady-state temperature field can be calculated from the following heat diffusion equation with certain boundary conditions,

- [K_{x} \frac{\partial^{2}}{{\partial x}^{2}} + K_{y} \frac{\partial^{2}}{{\partial y}^{2}} + K_{z} \frac{\partial^{2}}{\partial z^{2}}] T (x, y, z) = η_{h} P_{abs} (x, y, z),

where K_i(i = x, y, z) is the thermal conductivity of the gain medium, η_h is the fraction of absorbed pump light converted to heat, and P _abs is the absorbed power density in the crystal. Δn_T can be then obtained by solving the equation with certain boundary conditions.

For the homogeneously broadened gain medium, the gain guiding refractive index Δn_G is

Δ n_{G} (x, y, z, ω) = - \frac{G (x, y, z, ω)}{2 k} \frac{i - Δ ω / Ω}{1 + {(Δ ω / Ω)}^{2}},

where G is the amplitude modal gain coefficient of the gain medium, Ω is the FWHM of the gain spectra, and Δω is the detuning of the frequency. In our model, we assume that the laser frequency ω oscillates at the line center of the gain medium ω_c, and then the frequency detuning Δω = 0, therefore, gain guiding refractive index changes into

Δ n_{G} (x, y, z) = - i \frac{G (x, y, z, ω_{c})}{2 k} .

For most high power laser worked well above threshold, the gain saturation effect must be taken in account [17],

G (x, y, z) = \frac{G_{0} (x, y, z, ω_{c})}{1 + \frac{I (x, y, z, ω_{c})}{I_{sat}}},

where G ₀ is the small signal gain coefficient and I _sat is the saturation intensity. If ground state depletion is neglected in the four level laser, the small signal gain coefficient G ₀ can be represented as following [18],

G_{0} (x, y, z) = P_{abs} (x, y, z) \frac{λ_{p} σ_{21} τ_{f}}{h c},

where σ ₂₁ is the stimulated emission cross section of the laser transition and τ_f is the upper laser level life time, h is the Plank constant. The saturation intensity I _sat was given by [22]

I_{sat} = (W_{r} + γ) \frac{h c}{λ_{l} σ_{21} τ_{f}},

where W_r is the pump rate, and γ is the level degeneracy. For four level laser, W_r ≪ τ_f and γ ≈ 1.

The actual resonant internal laser intensity is saturated down by the forward and back ward traveling electric fields E ⁺ and E⁻. Therefore, using the relationship $I = \frac{1}{2} \sqrt{\frac{ɛ}{μ}} | E |^{2} \approx \frac{c ɛ_{0} \sqrt{ɛ_{r}}}{2} | E |^{2}$ , the effective gain coefficient in the gain medium can be represented as following,

G (x, y, z) = \frac{P_{abs} (x, y, z)}{\frac{h c}{λ_{p} σ_{21} τ_{f}} + \frac{λ_{l} c ɛ_{0} \sqrt{ɛ_{r}}}{2 λ_{p}} | E + (x, y, z) + E - (x, y, z) |^{2}} .

The wave equation of the field E^± then can be rewritten as a nonlinear Schrödinger-type wave equation,

\frac{\partial}{\partial z} E^{\pm} (x, y, z) = [\frac{i}{2 k_{0} n_{b}} \nabla_{⊥}^{2} + {ik}_{0} Δ n_{T} (x, y, z) + \frac{G (x, y, z)}{2}] E^{\pm} (x, y, z),

where Δn_T and G are real function, and Δn_T is represented by Eqs. (14) and (15) while G is represented by Eq. (21). The first term on the right side of the equation expresses the space diffraction of the electric field in the dielectric; the second term means that the phase modulation of the electric field by the thermal induced refractive index profile, in other words, the thermal guiding term; and the third term means that the amplitude modulation of the electric field by the gain profile, the gain guiding effect, and this is a nonlinear term according to Eq. (21). The gain profile G is related to the electric field E ⁺ and E ⁻ this made the wave equation is a coupled nonlinear 2-order partial differential equation. This coupled wave function can be numerically solved with the iteration method.

2.3. SSFM for the wave equation

The nonlinear wave equation (Eq. (22)) can be rewritten as the following simple type,

i \frac{\partial ψ}{\partial z} = (\hat{D} + \hat{N}) ψ,

where ψ = E ⁺(x,y,z) is the wave function, and D̂ is the linear differential operator which denotes the spatial diffraction effect,

\hat{D} = \frac{- \nabla_{⊥}^{2}}{2 k_{0} n_{b}} = \frac{- 1}{2 k_{0} n_{b}} (\frac{\partial^{2}}{{\partial x}^{2}} + \frac{\partial^{2}}{{\partial y}^{2}}),

and N̂ is the nonlinear operator denoted the combined guiding effect

\hat{N} = - k_{0} Δ n_{T} (x, y, z) + i \frac{G (x, y, z)}{2} .

The formal solution of Eq. (23) can be represented as following,

ψ (x, y, z + Δ z) = e^{- i (\hat{D} + \hat{N}) Δ z} ψ (x, y, z) .

According to the Baker-Campbell-Hausdorff (BCH) formula

ψ (x, y, z + Δ z) \approx e^{- i (\hat{D} Δ z)} e^{- i (\hat{N} Δ z)} ψ (x, y, z) .

This means in physically that, the simultaneous action of the spatial diffraction effect D̂ and the combined guiding effect N̂ act on the wave function ψ, is approximately equivalent to the independent and separately action that N̂ act first and then D̂ act when the step length Δz is short enough.

Only considering the spatial diffraction effect D̂ on the ψ

ψ (x, y, z + Δ z) \approx e^{- i (\hat{D} Δ z)} ψ (x, y, z) .

According to the definition of the inverse fourier transform,

exp (- i \hat{D} Δ z) ψ (x, y, z) = \int \int exp (- i \hat{D} Δ z) exp [i (ω_{x} x + ω_{x} y)] \tilde{ψ} (ω_{x}, ω_{y}, z) d ω_{x} d ω_{y},

\tilde{ψ}

= 𝒡 (ψ) is the Fourier transform of ψ where 𝒡 is the Fourier transform operator, and ω_x,y are the spatial angular frequencies in the Fourier domain. The operator 𝒡⁻¹ is defined as the inverse Fourier transform operator.

Using the Maclaurin’s expansion,

\begin{array}{l} exp (- i \hat{D} Δ z) exp [i (ω_{x} x + ω_{x} y)] \\ = & {\sum_{n} \frac{1}{n!} {[\frac{i Δ z}{2 k_{0} n_{b}} (\frac{\partial^{2}}{{\partial x}^{2}})]}^{n} exp (i ω_{x} x)} {\sum_{n} \frac{1}{n!} {[\frac{- i Δ z}{2 k_{0} n_{b}} (\frac{\partial^{2}}{{\partial y}^{2}})]}^{n} exp (i ω_{y} y)} \\ = & [exp (i ω_{x} x) \sum_{n} \frac{1}{n!} {(\frac{- i Δ z}{2 k_{0} n_{b}} ω_{x}^{2})}^{n}] [exp (i ω_{y} y) \sum_{n} \frac{1}{n!} {(\frac{- i Δ z}{2 k_{0} n_{b}} ω_{y}^{2})}^{n}] \\ = & exp [\frac{- i Δ z}{2 k_{0} n_{b}} (ω_{x}^{2} + ω_{y}^{2})] exp [i (ω_{x} x + ω_{x} y)], \end{array}

then

\begin{array}{l} ψ (x, y, z + Δ z) \\ = & \int \int exp [\frac{- i Δ z}{2 k_{0} n_{b}} (ω_{x}^{2} + ω_{y}^{2})] exp [i (ω_{x} x + ω_{x} y)] \tilde{ψ} (ω_{x}, ω_{y}, z) d ω_{x} d ω_{y} \\ = & ℱ^{- 1} {ℱ [ψ (x, y, z)] exp [\frac{- i Δ z}{2 k_{0} n_{b}} (ω_{x}^{2} + ω_{y}^{2})]} . \end{array}

It means that, the calculation of the differential operator is replaced with the calculation of Fourier transform. To solve the differential equation problem, the Fourier transform is with faster calculation speed than the traditional finite difference method to achieve the same calculation accuracy, especially the faster Fourier transform (FFT) is used.

When only considers the combined guiding effect N̂ on the ψ,

ψ (x, y, z + Δ z) \approx e^{- i (\hat{N} Δ z)} ψ (x, y, z) .

The equation can be directly integrated and yields to

ψ (x, y, z + Δ z) \approx exp [- i \int_{z}^{z + Δ z} \hat{N} (x, y, z^{'}) d z^{'}] ψ (x, y, z) .

When the step size Δz short enough, the trapezoidal rule can be used to estimate the integral

\int_{z}^{z + Δ z} \hat{N} (x, y, z^{'}) d z^{'} \approx \frac{Δ z}{2} [\hat{N} (x, y, z) + \hat{N} (x, y, z + Δ z)] .

Therefore, according to Eq. (27), the solving of Eq. (26) turns to the calculation of diffraction effect (Eq. (31)) and the combined guiding effect (Eq. (34)) respectively. The above method is called as the Split-Step Fourier Method (SSFM). The SSFM is an effective method used to solve the Schrödinger equation [19–21]. From the BCH formula, the approximation of formula (27) has the accuracy of order Δz ² since the operators D̂ and N̂ are noncommutative. If the operators are rearranged into a symmetric form, and then we have the new approximation

exp [- i (\hat{D} + \hat{N}) Δ z] \approx exp [- i \hat{D} Δ z / 2] exp [- i \hat{N} Δ z] exp [- i \hat{D} Δ z / 2] .

The new approximation is with the accuracy up to order Δz ³ which increased an order compared to the former one, Eq. (27).

Since the nonlinear operator is as the function of the field E, therefore, the value of N̂ at z+Δz is unknown because the field E(z + Δz) is what we want to obtained. However, we can first assume that N̂(z + Δz) ≈ N̂(z), and then we can calculate the approximate value of E(z + Δz). Then, the more accurate value of N̂(z + Δz) can be obtained from the calculated E(z + Δz). If we do this again, the more accurate value of both N̂(z + Δz) and E(z + Δz) will be achieved. This inner-iterative method can improve the accuracy of the calculation while the inner-iterative method increases the calculation complexity. However, the inner-iteration is always with fast convergence within several cyclical iteration. Meanwhile, since the inner-iterative method can improve the accuracy, the step size Δz can be appropriately increased to improve the calculation speed. In our calculation, the new approximation called symmetric split-step Fourier Method was used.

The accuracy of the SSFM depends on the grid size in the transverse direction (Δx,y) and the step size in the axial direction (Δz). The FFT method requires N = 2^m (m is positive integer) sampling points in the transverse direction while the spatial resolution should be high enough. The low resolution may lead to the aliasing and overlapping of the spatial angular frequencies in the Fourier domain. The electric field of laser is spatial bandlimited since its high spatial coherence. Therefore, according to the Nyquist-Shannon sampling theorem, the transverse spatial step size Δx,y should be at least twice of reciprocal of the spatial angular frequency, ie., Δx, $y < \frac{1}{2 Δ ω_{x, y}}$ , where Δω_x,y is the bandwidth of the angular frequencies. The axial step size Δz should be decreased until the enough computational accuracy can be obtained.

3. Experimental setup

In our experiment, a composite a-cut Nd:YVO₄ crystal was used as the gain medium. For the four level laser emission when being pumped at λ_p =808 nm (⁴ I _9/2 → ⁴ F _5/2) and lasing at λ_l =1064 nm (⁴ F _3/2 →⁴ I _11/2), Nd:YVO₄ crystal has a large effective stimulated absorption cross section (σ ₁₂ ≈ 2.7 × 10⁻²³m²@ 1 at.%) and effective stimulated emission cross section (σ ₂₁ ≈ 15.6 × 10⁻²³m²@ 1 at.%) [22, 23], therefore, Nd:YVO₄ can provide high gain. The composite crystal was also used, and two updoped YVO₄ end caps were thermally bonded on the both ends of the 0.3 at. % Nd³⁺ ion concentrated Nd:YVO₄ crystal. The end caps were with the dimension of 3 mm× 3 mm×2 mm while that of the doped crystal was 3 mm× 3 mm×16 mm. Thus, the size of the composite crystal was w × h × l = 3 mm × 3 mm × 20 mm. One reason using the composite crystal is that the thermally bonded undoped end caps can prevent the thermal fracture. The thermal fracture results from the temperature gradients and the consequent thermal stress, especially when the crystal was intensely pumped to get high gain. Meanwhile, the thermal stress also can make the so called end-effect, the deformation and the bulge of the end-face of the gain medium. The curvature on the end-face acts as a built-in index guide which can cover up the thermal index guiding. So the thermally bonding technique is used to reduce the end-effect significantly. The composite crystal was dual end pumped. The 45 W pump light from the laser diode (LD) was coupled into a 400 μm diameter, 0.22 numerical aperture fiber, and then imaged using a four-lens imaging system and then the focused pump beam was delivered into the gain medium. The imaging system can be adjusted conveniently to adjust the waist size of focused pump as well as the location of the beam waist in the gain medium. The temperature of the laser diode was controlled using the thermoelectric cooling module (TECM), and the center wavelength of the laser diode could be temperature-tuned by adjusting the temperature of the LD with TECM. The wavelength tuning coefficient of the LD was about 0.3 nm/°C as measured. From Eqs. (15) and (19), we can see that the profile of the thermal index and the gain distribution can be changed conveniently by tuning the temperature of the LD.

A “Z”-type planar-planar cavity was used in the experiment. The 22.5º dichroic mirrors were antireflection coated at 808 nm and high reflection coated at 1064 nm. The cavity lengths were L ₁= 80 mm and L ₂= 50 mm. The experimental setup is shown in Fig. 4.

Fig. 4 The experimental setup of the dual-end pumped composite Nd:YVO₄ laser.

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Figure 5 shows the pump model of the dual-end-pumped geometry. For the fiber coupled end-pumped laser, the pump distribution in the x – y plane is an N-order super-Gaussian function in radial, and the pump power is absorbed in an exponential function along the longitudinal direction z. The pump saturation is neglected since the low doping concentration crystal was used. Therefore, the pump intensity distribution in the active medium can be represented as following, when the crystal is only pumped from left,

I_{P}^{+} (x, y, z) = \frac{C_{0}}{π ω_{p}^{2} (z)} exp [\frac{- 2 {(\sqrt{x^{2} + y^{2}})}^{N}}{ω_{p}^{N} (z)}] exp (- α_{eff} z),

where ω_p(z) is the radius of the pump mode at position z, and N is the order of super-Gaussian function which was measured as N ∼ 4 in our experiment using a 90/10 knife-edge method. α _eff is the effective absorption coefficient at the effective pump wavelength λ _eff of the pump laser in the active medium, and it is an averaged absorption coefficient measured in the experiment, considering the average of the both absorption coefficients along a- axis and c- axis, and also averaged on the absorption of the emitting spectrum of LD. The α _eff is always lower than the peak absorption coefficient because the effective absorption coefficient is a parameter averaged over the absorption spectrum width of the diode laser, and the effective pump wavelength is defined by λ _eff = ∫ ρ(λ′)(λ′)dλ′/ ∫ ρ(λ′)dλ′, and ρ(λ′) is the power spectral density of the pump source. The effective pump wavelength was measured with an optical spectrum analyzer and an integrating-sphere photometer. The constant C ₀ was determined by Eq. (37)

C_{0} = \frac{P_{0}}{\iint_{z = 0} \frac{1}{π ω_{p}^{2} (z)} exp [\frac{- 2 {(\sqrt{x^{2} + y^{2}})}^{N}}{ω_{p}^{N} (z)}] exp (- α_{eff} z) d x d y},

where P ₀ is the pump power of each end. Suppose the waist radius of the pump mode is ω_p ₀ and the waist is located at a distance from the bonded surface of the Nd:YVO₄ crystal of z ₀, i.e., ω_p ₀ = ω_p(z ₀). The radius of pump mode at location z can be represented as

ω_{p} (z) = ω_{p 0} \sqrt{1 + {[\frac{θ_{p} (z - z_{0})}{ω_{p 0}}]}^{2},}

In Eq. (38) θ_p is the the far-field divergence (half angle) of the pump mode in the crystal.

Fig. 5 The pump absorption model of the gain medium. z ₀: the location of the pump waist, ω_p(z): the pump beam radius at the location z, and ω_p ₀: the pump waist radius at the pump waist location of z ₀, i.e., ω_p ₀=ω_p(z0). θ_p: the half far-field divergence angle of the pump beam. $I_{p}^{\pm}$ : the forward- and backward- pump beam.

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For the symmetric dual-end pumped geometry, the pump intensity distribution in the active medium can be represented as Eq. (39),

I_{p} (x, y, z) = I_{P}^{+} (x, y, z) + I_{P}^{-} (x, y, z),

where

I_{P}^{-} (x, y, z)

is pump intensity in the crystal when only being pumped from right with a similar expression as

I_{P}^{+} (x, y, z)

. The absorbed pump distribution per unit volume (i.e., the absorbed pump density) in the active medium can be represented as following,

P_{abs} (x, y, z) = \frac{- \partial I_{p} (x, y, z)}{\partial z} \approx α_{eff} I_{P} (x, y, z) .

4. Results and discussion

4.1. Temperature and gain

The temperature distribution in the crystal can be obtained by the solving of the heat diffusion Eq. (15). For convenient representation, here we placed the origin of the coordinate at the center of the bulk crystal. The boundary condition then can be represented as following,

\begin{array}{l} {\frac{\partial T (x, y, z)}{\partial x} |}_{x = \pm \frac{w}{2}} = \frac{h_{c}}{K_{x}} [T_{c} - T (\pm \frac{w}{2}, y, z)] \\ {\frac{\partial T (x, y, z)}{\partial y} |}_{y = \pm \frac{h}{2}} = \frac{h_{c}}{K_{y}} [T_{c} - T (x, \pm \frac{h}{2}, z)] \\ {\frac{\partial T (x, y, z)}{\partial z} |}_{z = \pm \frac{l}{2}} = \frac{h_{a}}{K_{z}} [T_{r} - T (x, y, \pm \frac{l}{2})], \end{array}

where K_y = K_z = 5.23 W/mK and K_x = 5.10 W/mK are the anisotropic thermal conductivities of the a-cut crystal; h_c is the forced convection heat transfer coefficient on the cooling face, which is 10000 W/m²K for the water-cooled cooper heat sink; h_a is the free convection heat transfer coefficient between the air and the end faces, while those faces are assumed to be heat insulated, i.e., h_a = 0; T_r = 20°C is the room temperature; and T_c = 10°C is the coolant temperature of the heat sink.

The value of the fraction of absorbed pump light converted to heat, η_h in Eq. (15), is a very important parameter for the solving of the temperature distribution. In diode-pumped solid-state lasers, the major sources of heat production are quantum defect, non-radiative transition [24], fluorimetric quenching induced by impurity and defect [25], and the Auger recombination [26]. The heat conversion fraction caused by the quantum defect is η_h = 1 − λ_p/λ_l ≈ 0.24 when pumped at 808 nm and lasing at 1064 nm. Additional heat load produced by above nonlinear processes, rather than the quantum defect of the pumping, aroused more serious heat generation, especially for high power intensity pumping. For the neodymium doped vanadate crystals and glass pumped at 808 nm (⁴ I _9/2 → ⁴ F _5/2) and lasing at 1064 nm (⁴ F _3/2 → ⁴ I _11/2), many researchers have taken the total heat conversion fraction around 0.3 [24, 27, 28]. Therefore, the compromise and relative reasonable value of η_h = 0.28 is used as the total heat conversion fraction in our model for the high doping concentration crystal.

The energy budget method is used to establish the difference equation from the heat diffusion Eq. (15) with the boundary condition (41). The physical essence of the energy budget method is the energy conservation. The crystal is divided into many small bulk units. The heat dissipating capacity from the six side faces of each bulk unit equals to the quantity of heat production in the bulk unit, and then the difference equation can be established. Using the finite difference method, the temperature distribution in the gain medium is calculated while the Gauss–Seidel iteration method is employed to accelerate the convergence. Figure 6 shows the 3D temperature profile of the 1/8 crystal volume when the gain medium is pumped at 45 W pump power with the effective absorption coefficient α _eff = 2.0 cm⁻¹, and the value of z ₀ and ω_p ₀ are 2 mm and 0.4 mm respectively. If not special specified in the following calculation and discussion, the pump power of each end is fixed at P ₀ = 45 W, and the pump waist radius is ω_p ₀ = 0.4mm located at z ₀ = 2mm as shown in the coordinate of Fig. 5.

Fig. 6 3D profile of the temperature distribution of the gain medium pumped at P ₀ = 45 W with ω_p ₀ = 0.4mm, z ₀ = 2mm, and α _eff =2.0 cm⁻¹.

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Figure 7 shows the temperature on the optical axis in the gain medium at different effective absorption coefficient. From the calculated results we can see that, the thermally bonded end caps can reduced the temperature gradient effectively near the end-faces. Thus, the deformation and the bulge of the end-face can be reduced significantly so that the curvature on the end face which acts as a built-in index guide will not cover up the thermal index guiding. The highest temperature locates near the pump beam waist, and increase with the increment of effective absorption coefficient. Meanwhile, the temperature gradient is smaller at lower effective absorption coefficient. The thermal induced guiding refractive index Δn_T can be calculated easily using Eq. (14).

Fig. 7 The temperature distribution on the optical axis in the gain medium varies with the effective absorption coefficient α _eff increased from 1.0 cm^–1 to 6.0 cm⁻¹ with the increment of Δα _eff = 0.5 cm⁻¹.

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The small signal gain coefficient can be calculated from Eq. (19) and Eqs. (36)–(40). Figure 8 shows the 3D small signal gain coefficient profile with α _eff =2.0 cm⁻¹. The gain saturation effect must be considered in high power four level lasers well above threshold. As the signal intensity increases and the pump rate is constant, the inversion level will reduce and thereby the gain is reduced. Figure 9 shows the total gain profile in one pass through the gain medium, $Γ (x, y) = \int_{0}^{l} G (x, y, z) d z$ , with the small signal gain coefficient and with the saturated gain coefficient where the initial signal intensity $I_{0} (x, y) = I_{sat} exp [- 2 (x^{2} + y^{2}) / ω_{p 0}^{2}]$ and $I_{0} (x, y) = 4 I_{sat} exp [- 2 (x^{2} + y^{2}) / ω_{p 0}^{2}]$ . It shows that the gain is weaken seriously when the gain saturation works. Especially, the effective gain is depressed near the optical axis where the gain is weakened more seriously by the gain saturation.

Fig. 8 3D small signal gain profile of the gain medium pumped at P ₀ = 45 W with ω_p ₀ = 0.4mm, z ₀ = 2mm, and α _eff =2.0 cm⁻¹.

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Fig. 9 small signal gain (solid line) and the saturated gain with $I_{0} = I_{sat} exp [- 2 (x^{2} + y^{2}) / ω_{p 0}^{2}]$ (dash-dot line) and $I_{0} = 4 I_{sat} exp [- 2 (x^{2} + y^{2}) / ω_{p 0}^{2}]$ (dash line) for one pass through the gain medium, P ₀ = 45 W, z ₀ = 2mm, ω_p ₀ = 0.4mm, and α _eff =2.0 cm⁻¹.

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4.2. Transverse mode with the combined guiding

The profile of the transverse mode of the laser resonator was detected by alternately imaging the mode profile on the output coupler onto a high resolution 1/1.8” GRAS20 CCD camera. The size of the CCD camera chip was 1600 × 1200 pixels, and the physical pixel size of each pixel was 4.4 μm × 4.4 μm. The glass window of the CCD camera was removed to avoid the optical interference effect aroused from the glass surfaces of the window. The high precision optical beam splitters were used to reduce the laser intensity incident onto the CCD chip. The experimental transverse modes were found to have good circular symmetry so that the cross section was taken along the long axis (1600 pixel) of the CCD camera to compare conveniently with the calculated transverse mode profile.

The Fox-Li iteration method is used to calculate the transverse mode in the resonator. Using the calculated temperature and small signal gain coefficient, the nonlinear Schrödinger-type wave Eq. (22) is solved by the split-step Fourier method. Figure 10 shows the transverse mode formation in the typical iterative process. The initial field, iterated field pattern and the converged iteration which means the self-reproducing field pattern are shown in the iteration process, form which we can see that the iterative process shows the performance of fast convergence.

Fig. 10 The transverse mode profile varied with the number of iteration cycle of the Fox-Li iteration.

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As mentioned in Section 3, the thermal induced refractive index and the gain distribution can be adjusted conveniently by changing the effective absorption coefficient α _eff, and α _eff can be adjusted by shitting the effective pump wavelength λ _eff. In the experiment, the temperature of LD was shifted to tune the effective pump wavelength λ _eff. For the low doping concentration 0.3% at. Nd:YVO₄ crystal, the effective absorption coefficient α _eff can be changed conveniently from 1.5 cm⁻¹ to 7.0 cm⁻¹. Figure 11 shows the experimental- and theoretical- profiles of the transverse mode while the α _eff increased from 1.5 cm⁻¹ to 7.0 cm⁻¹ with the increment of 0.5 cm⁻¹. The theoretical profiles include the following three cases: (1) the combined guiding (CG) effect considered, (2) only the gain guiding (GG) effect considered, ie., Δn_T = 0, and (3) only the thermal induced refractive index guiding (IG) effect considered, ie., Δn_G = 0. The theoretical results of case 2 give similar Gaussian profiles. This is because that the pump beam in our experiment has the 4-order super-Gaussian profile, and this yields to a top-hat one pass small signal gain profile (see Fig. 9) which is not sensitive to the variation of α _eff. The theoretical results of case 3 show that the transverse mode varied acutely with α _eff, and this is because that the temperature distribution is sensitive to the variation of α _eff. In case 1, the theoretical profiles with the combined guiding considered agree with the experiment reasonably well. This agreement shows that in the high power laser resonator, the formation of the transverse mode was dominated by the both the thermal induced refractive index guiding and gain guiding, ie., the combined guiding effect, as shown in the wave equation (Eq. (22)). The mode size in case 1 is smaller than that in case 2 but bigger than that in case 3.

Fig. 11 The experimental- and theoretical- transverse mode profiles for α _eff increased from 1.5 cm⁻¹ to 7.0 cm⁻¹. Exp. denotes the experimental results(blue solid line), CG denotes the theoretical results with the combined guiding effect considered(red solid line), GG and IG denote theoretical results with only gain guiding effect/thermal induced refraction index guiding effect considered respectively (dash dot line/dash line). Mode i corresponds to the transverse mode with α _eff = 1.5 + (i − 1)×0.5 cm⁻¹. P ₀ = 45 W, ω_p ₀ = 0.4mm, z ₀ = 2mm.

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It should be noticed that, when the α _eff is higher than 5.5 cm⁻¹, the experimental- and theoretical- profiles of case 1 did not agreed very well compared with the formers. This can be explained that, the absorption saturation of the pump light in the gain medium is not considered in our modeling which is reasonable for the low absorption coefficient. The absorption saturation occurred and became important for the case of the high absorption coefficient. Thus, the absorbed intensity P _abs, and the dependent thermal refractive index as well as the gain distribution, deviate from the exponential absorption in our modeling. Therefore, the deviations between the experimental- and theoretical- profiles at high α _eff is resulted from the pump saturation effect.

The beam quality became better and better from Mode 1 to Mode 6 and then became worse and worse. The beam quality factors of Mode 4 - Mode 8 were measured less than 1.3 with output power around 40 W which means that the high power TEM₀₀ mode output was obtained when α _eff between 3.0 cm⁻¹ and 5.0 cm⁻¹.

In the traditional geometric design criterion of the TEM₀₀ mode resonator, the mode size of TEM₀₀ and TEM₀₁ and the equivalent pump radius ω _equ are only considered, since higher-order transverse modes always can not oscillate because of their larger geometric- and thermally induced- diffraction loss and poorer gain. The equivalent pump radius ω _equ of the pump beam in the gain medium can be defined as the weighted average of ω _eff(z),

ω_{equ} = \frac{∭ P_{abs} (x, y, z) ω_{eff} (z) d x d y d z}{∭ P_{abs} (x, y, z) d x d y d z},

where, the ω _eff(z) denotes the effective radius of the pump mode at position z which can be defined as

\frac{\iint_{x^{2} + y^{2} \leq ω_{eff}^{2} (z)} P_{abs} (x, y, z) d x d y}{\iint P_{abs} (x, y, z) d x d y} = 1 - \frac{1}{e^{2}} \approx 0.865.

To achieve TEM₀₀ mode operating, the equivalent pump radius ω _equ should be larger than the mode radius of the TEM₀₀ mode in the gain medium, but be smaller than the mode radius of the TEM₀₁ mode. In this case, the TEM₀₀ mode can get higher gain than TEM₀₀ mode and oscillates. In addition, the phase aberration always occurs in the wing of the pump region where thermal-induced refractive index and temperature-gradient-induced thermal stress are higher. This thermal phase aberration leads to higher diffractive loss for TEM₀₁ and inhibits its oscillation. For the fiber-coupled end-pumped laser, the pump mode is circularly symmetrical. Therefore, the mode size can be described as Laguerre-Gaussian mode TEM₀ _n approximately. The v mode radius ratio between TEM₀₀ mode and TEM_pl mode is $ω_{{TEM}_{00}} / ω_{{TEM}_{p l}} = 1 / \sqrt{p + 2 l + 1}$ . Therefore, we have ω _TEM₀₁ ≈ 1.73ω _TEM₀₀. For the geometric design criterion of the TEM₀₀ mode resonator, the equivalent pump radius must satisfy the following criterion,

1 < \frac{ω_{equ}}{ω_{{TEM}_{00}}} < 1.73.

For the design theory of high power TEM₀₀ mode laser resonator, the size ratio of ω_L/ω _equ was always designed between 0.7 − 0.8 to afford enough gain and prevent thermal aberration for the TEM₀₀ mode [29–31], suppressing the oscillation of the high order mode, especially the TEM₀₁ mode. The region of (0.7 − 0.8)ω _equ can be called as the geometrical TEM₀₀ mode region.

The size of the TEM₀₀ mode in the resonator can be calculated from the ABCD matrix using the geometrical theory of the diffraction optics (GTDO) in which the gain medium was considered as an thermal lens. Figure 12 shows the the equivalent pump radius ω _equ in the experiment, the TEM₀₀ mode radius ωTEM₀₀ calculated by GTDO, the laser radius ω_L calculated by our nonlinear Schrödinger-type wave equation (NStWE). The geometrical TEM₀₀ mode region is also shown in the figure. From Fig. 12 we can see that the ω _TEM00 locates outside the geometrical TEM₀₀ mode region when the effective absorption coefficient increased from 1.5 cm⁻¹ to 7.5 cm⁻¹. According to the traditional geometric design criterion of the TEM₀₀ mode resonator, the resonator should be operated in multi transverse mode output. But actually, the laser resonator was with near diffraction limit output when the effective absorption coefficient increased from 3.0 cm⁻¹ to 5.5 cm⁻¹, both for the experimental results and the theoretical results with combined guiding effect considered (see Mode 4-Mode 8 in Fig. 11).

Fig. 12 the equivalent pump radius (solid line), the radius of TEM₀₀ mode calculated by the geometrical theory of the diffraction optics (GTDO) (dot line), and the transverse mode size calculated by the nonlinear Schrödinger-type wave equation (NStWE) (dash line). The dash-dot lines show the geometrical TEM₀₀ mode region. P ₀ = 45 W, ω_p ₀ = 0.4mm, z ₀ = 2mm.

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The failure of the traditional geometric design criterion of the TEM₀₀ mode resonator was caused by that the passive cavity is considered. In the passive cavity, the resonator operated only with the spatial diffraction effect but without the complex gain in which the thermal induced refractive index acts as the real refractive index and the gain acts as the imaginary refractive index. Unfortunately, for the most actual laser resonators especially pumped at high power, the combined guiding mechanism becomes important, even dominate the transverse mode formation in the laser resonator. From the experimental- and the theoretical- results, when α _eff increased from 3.0 cm⁻¹ to 4.5 cm⁻¹, even though the laser radius ω_L in the gain medium does not satisfied geometric design criterion, the resonator can still operate in ”near” TEM₀₀ mode. Here we called the transverse mode as ”near” TEM₀₀ mode is because of that it is not ”true” TEM₀₀ mode, rather ”single lobe lowest order mode”. In this case, the propagation of light field in the gain medium is guided by the combined guiding effect, and only the near TEM₀₀ mode can oscillate but TEM₀₁ and higher order modes can’t, even though ω _TEM₀₀ is much smaller than ω _equ. Therefore, the combined guiding mechanism must be taken into account when predicating the transverse mode of high power laser.

5. Summary and conclusions

We have presented a theoretical model for the transverse mode formation in diode pumped laser with the combined guiding effect considered, and an experimental verification in an end-pumped Nd:YVO₄ laser is also presented.

In the theoretical model, the laser resonator is unfolded along the optical axis and a cell-train model for the resonator is presented. In this model, the foreword- and backward- propagating waves (ie., the spatial hole burning) is taken into account. From the Maxwell’s equation, we consider the thermal induce refractive index guiding and gain guiding effect and derive the nonlinear Schrödinger-type wave equation in the gain medium. The gain saturation is also introduced in the model. The numerical solution for our nonlinear wave equation - the split step Fourier method - is then derived.

An end-pumped Nd:YVO₄ laser is built up to validate the theoretical model. The pump absorption of the gain medium in the experiment is presented. We calculate and discuss the temperature- and gain- distribution in the gain medium. Then we calculate the theoretical transverse mode profiles from the established nonlinear wave equation. The theoretical results well agree with the experimental ones. The geometric design theory of the TEM₀₀ mode laser is also compared with our wave theory, and the experimental- and theoretical- results shows that our wave theory with the combined guiding mechanism should be considered for the design of the high power TEM₀₀ mode laser.

In future, an intensive investigation on the combined guiding effect will be carry on, and the effect of pump saturation and gain saturation on the transverse formation in end-pumped Nd:YVO₄ resonators and amplifiers will be researched theoretically and experimentally.

References and links

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Combined guiding effect in the end-pumped laser resonator

Abstract

1. Introduction

2. Theoretical model

2.1. The cell-train model of laser resonator

2.2. Wave equation in the gain medium

2.3. SSFM for the wave equation

3. Experimental setup

4. Results and discussion

4.1. Temperature and gain

4.2. Transverse mode with the combined guiding

5. Summary and conclusions

References and links

Cited By

Figures (12)

Equations (45)

Optics Express