## Abstract

A theoretical model as well as the experimental verification of the combined guiding mechanism for the transverse mode formation in the end-pumped laser resonator are investigated. The nonlinear Schrödinger-type wave equation in the gain medium is derived, in which the combined guiding mechanism: the thermal induced refractive index guiding effect as well as the gain guiding effect, is taken into account. The gain saturation and spatial hole burning are considered. The split step Fourier method is used to solve the nonlinear wave equation. A high power end-pumped Nd:YVO_{4} laser resonator is built up. After establishing the pump absorption model of our laser resonator, the temperature distribution in the gain medium is obtained by the numerical solving of the heat diffusion equation. The combined guiding effect is first observed in the end-pumped Nd:YVO_{4} laser resonator, and the experimental transverse mode profiles well agree with the theoretical prediction from the derived nonlinear Schrödinger-type wave equation. The geometric design criterion of the TEM_{00} mode laser is compared with our wave theory. The experimental- and theoretical- results show that our wave theory with the combined guiding mechanism dominates the transverse mode formation in high power end-pumped laser resonator.

© 2011 OSA

## 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., Yb^{3+} 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 Nd^{3+} 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 Nd^{3+} 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:YVO_{4} 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.

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*. 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

_{k}*E*

^{+}and the backward propagating waves

*E*

^{−}, i.e, the spatial hole burning is considered which is neglected in previous works [4].

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

*E*propagates in the gain medium, it suffers the thermal induced refractive index Δ

*n*and the gain Δ

_{T}*n*as shown in Fig. 3. The propagation of the wave is guided by the combined action of Δ

_{G}*n*and Δ

_{T}*n*, ie., the combined guiding. Thus, the wave equation included the combined guiding effect should be derived.

_{G}#### 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

**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*≈ 1 is the relative permeability, and

_{r}*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

*E*

^{†}is the complex amplitude of the electric field with an approximate time dependence of

*e*. Substitute Eq. (3) and Eq. (4) into Eq. (2), then we can get the Helmholtz equation

^{−iωt}*z*-dependent phase variations) in the field, we have

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

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

*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 Δ

*n*(

*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 Δ

*n*≪

*n*

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

Applying the slowly varying envelope approximation

The complex-valued guiding refractive index Δ*n* can be written in an explicit form,

*n*is the thermal induced guiding refractive index which is a real one, and Δ

_{T}*n*is the gain induced guiding refractive index which is a imaginary one.

_{G}The thermal induced guiding refractive index Δ*n _{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(d

*n*/d

*T*) can give a distribution of the refractive index in the material. The thermal induced guiding refractive index Δ

*n*is given as following

_{T}*T*(

*x,y,z*) is the steady-state temperature field in the gain medium volume and

*T*is the room temperature. The steady-state temperature field can be calculated from the following heat diffusion equation with certain boundary conditions,

_{r}*K*(

_{i}*i*=

*x, y, z*) is the thermal conductivity of the gain medium,

*η*is the fraction of absorbed pump light converted to heat, and

_{h}*P*

_{abs}is the absorbed power density in the crystal. Δ

*n*can be then obtained by solving the equation with certain boundary conditions.

_{T}For the homogeneously broadened gain medium, the gain guiding refractive index Δ*n _{G}* is

*G*is the amplitude modal gain coefficient of the gain medium, Ω is the FWHM of the gain spectra, and Δ

*ω*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*, and then the frequency detuning Δ

_{c}*ω*= 0, therefore, gain guiding refractive index changes into

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

*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],

*σ*

_{21}is the stimulated emission cross section of the laser transition and

*τ*is the upper laser level life time,

_{f}*h*is the Plank constant. The saturation intensity

*I*

_{sat}was given by [22] where

*W*is the pump rate, and

_{r}*γ*is the level degeneracy. For four level laser,

*W*≪

_{r}*τ*and

_{f}*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
$I=\frac{1}{2}\sqrt{\frac{\varepsilon}{\mu}}|E{|}^{2}\approx \frac{c{\varepsilon}_{0}\sqrt{{\varepsilon}_{r}}}{2}|E{|}^{2}$, the effective gain coefficient in the gain medium can be represented as following,

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

*n*and

_{T}*G*are real function, and Δ

*n*is represented by Eqs. (14) and (15) while

_{T}*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,

where*ψ*=

*E*

^{+}(

*x,y,z*) is the wave function, and

*D̂*is the linear differential operator which denotes the spatial diffraction effect,

*N̂*is the nonlinear operator denoted the combined guiding effect

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

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

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 Δ*z* is short enough.

Only considering the spatial diffraction effect *D̂* on the *ψ*

According to the definition of the inverse fourier transform,

*are the spatial angular frequencies in the Fourier domain. The operator *

_{x,y}^{−1}is defined as the inverse Fourier transform operator.

Using the Maclaurin’s expansion,

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*,

The equation can be directly integrated and yields to

When the step size Δ*z* short enough, the trapezoidal rule can be used to estimate the integral

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 Δ*z*
^{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

The new approximation is with the accuracy up to order Δ*z*
^{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*+Δ*z* is unknown because the field *E*(*z* + Δ*z*) is what we want to obtained. However, we can first assume that *N̂*(*z* + Δ*z*) ≈ *N̂*(*z*), and then we can calculate the approximate value of *E*(*z* + Δ*z*). Then, the more accurate value of *N̂*(*z* + Δ*z*) can be obtained from the calculated *E*(*z* + Δ*z*). If we do this again, the more accurate value of both *N̂*(*z* + Δ*z*) and *E*(*z* + Δ*z*) 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 Δ*z* 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 (Δ*x,y*) and the step size in the axial direction (Δ*z*). 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 Δ

*x,y*should be at least twice of reciprocal of the spatial angular frequency, ie., Δ

*x*, $y<\frac{1}{2\Delta {\omega}_{x,y}}$, where Δ

*ω*is the bandwidth of the angular frequencies. The axial step size Δ

_{x,y}*z*should be decreased until the enough computational accuracy can be obtained.

## 3. Experimental setup

In our experiment, a composite *a*-cut Nd:YVO_{4} 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*=1064 nm (

_{l}^{4}

*F*

_{3/2}→

^{4}

*I*

_{11/2}), Nd:YVO

_{4}crystal has a large effective stimulated absorption cross section (

*σ*

_{12}≈ 2.7

*×*10

^{−23}m

^{2}@ 1 at.%) and effective stimulated emission cross section (

*σ*

_{21}≈ 15.6

*×*10

^{−23}m

^{2}@ 1 at.%) [22, 23], therefore, Nd:YVO

_{4}can provide high gain. The composite crystal was also used, and two updoped YVO

_{4}end caps were thermally bonded on the both ends of the 0.3 at. % Nd

^{3+}ion concentrated Nd:YVO

_{4}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

*μ*m 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.

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,

*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*′)d

*λ*′

*/*∫

*ρ*(

*λ*′)d

*λ*′, 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)

*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:YVO

_{4}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

*θ*is the the far-field divergence (half angle) of the pump mode in the crystal.

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

## 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,

*K*=

_{y}*K*= 5.23 W/mK and

_{z}*K*= 5.10 W/mK are the anisotropic thermal conductivities of the

_{x}*a*-cut crystal;

*h*is the forced convection heat transfer coefficient on the cooling face, which is 10000 W/m

_{c}^{2}K for the water-cooled cooper heat sink;

*h*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.,

_{a}*h*= 0;

_{a}*T*= 20°C is the room temperature; and

_{r}*T*= 10°C is the coolant temperature of the heat sink.

_{c}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*= 1 −

_{h}*λ*/

_{p}*λ*≈ 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 (

_{l}^{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*= 0.28 is used as the total heat conversion fraction in our model for the high doping concentration crystal.

_{h}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.

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 Δ*n _{T}* can be calculated easily using Eq. (14).

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,
$\Gamma (x,\hspace{0.17em}y)={\int}_{0}^{l}G(x,\hspace{0.17em}y,\hspace{0.17em}z)\text{d}z$, with the small signal gain coefficient and with the saturated gain coefficient where the initial signal intensity
${I}_{0}(x,\hspace{0.17em}y)={I}_{\text{sat}}\text{exp}[-2({x}^{2}+{y}^{2})/{\omega}_{p0}^{2}]$ and
${I}_{0}(x,\hspace{0.17em}y)=4{I}_{\text{sat}}\text{exp}[-2({x}^{2}+{y}^{2})/{\omega}_{p0}^{2}]$. 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.

#### 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 *μ*m *×* 4.4 *μ*m. 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.

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:YVO_{4} 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., Δ*n _{T}* = 0, and (3) only the thermal induced refractive index guiding (IG) effect considered, ie., Δ

*n*= 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

_{G}*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.

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 TEM_{00} 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 TEM_{00} mode resonator, the mode size of TEM_{00} and TEM_{01} 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*

_{eff}(

*z*) denotes the effective radius of the pump mode at position

*z*which can be defined as

To achieve TEM_{00} mode operating, the equivalent pump radius *ω*
_{equ} should be larger than the mode radius of the TEM_{00} mode in the gain medium, but be smaller than the mode radius of the TEM_{01} mode. In this case, the TEM_{00} mode can get higher gain than TEM_{00} 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 TEM_{01} 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 TEM_{0}
* _{n}* approximately. The

*v*mode radius ratio between TEM

_{00}mode and TEM

*mode is ${\omega}_{{\text{TEM}}_{00}}/{\omega}_{{\text{TEM}}_{pl}}=1/\sqrt{p+2l+1}$. Therefore, we have*

_{pl}*ω*

_{TEM01}≈ 1.73

*ω*

_{TEM00}. For the geometric design criterion of the TEM

_{00}mode resonator, the equivalent pump radius must satisfy the following criterion,

For the design theory of high power TEM_{00} 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 TEM

_{00}mode [29–31], suppressing the oscillation of the high order mode, especially the TEM

_{01}mode. The region of (0.7 − 0.8)

*ω*

_{equ}can be called as the geometrical TEM

_{00}mode region.

The size of the TEM_{00} 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 TEM_{00} mode radius *ω*TEM_{00} calculated by GTDO, the laser radius *ω _{L}* calculated by our nonlinear Schrödinger-type wave equation (NStWE). The geometrical TEM

_{00}mode region is also shown in the figure. From Fig. 12 we can see that the

*ω*

_{TEM00}locates outside the geometrical TEM

_{00}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 TEM

_{00}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).

The failure of the traditional geometric design criterion of the TEM_{00} 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” TEM

_{00}mode. Here we called the transverse mode as ”near” TEM

_{00}mode is because of that it is not ”true” TEM

_{00}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 TEM

_{00}mode can oscillate but TEM

_{01}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:YVO_{4} 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:YVO_{4} 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 TEM_{00} 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 TEM_{00} 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:YVO_{4} resonators and amplifiers will be researched theoretically and experimentally.

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