## Abstract

Mode characteristics in the solid-state planar waveguide (PWG) laser amplifiers are investigated theoretically, in consideration of the temperature gradient generated by cooling across the thickness and by pumping inhomogeneity along the width direction. When variation of the refractive index along the width direction is dominated by the lower spatial frequencies, the vector wave equation is solved analytically by means of the perturbation method. It is similar to the zigzag slab amplifier in which the phase aberration depending on the width coordinate plays the most important role to cause degradation of the beam quality. The crossing mode distortions owing to two dimension nature of the index variations are illustrated, and that mode profile is varied by the index variation along both the thickness and the width directions. Modes in the single-mode or the few-mode PWGs are shown to suffer weaker thermal-induced distortion across the thickness than those in the multi-mode PWGs.

© 2013 Optical Society of America

## 1. Introduction

Diode pumped laser amplifier with solid-state planar waveguide (PWG) has been proposed to be an efficient way to achieve high power output [1–3]. This is because in principle it inherits advantages of the fiber lasers, such as compactness and high optical efficiency, while it avoids the limitation by the nonlinear optical processes. A typical PWG possesses one-dimensional step-index waveguide structure that may be completely similar to the double-cladding step-index fiber (DC-SIF) to confine the laser beam and pumping beam respectively [4]. Another unguided dimension, namely the width of PWG, may be large enough to scale output power to high power. PWG offers a large flat surface providing efficient cooling and its waveguide nature suppresses the optical distortion caused by the temperature gradient along the thickness [5].

One of the central problems on high power PWG laser is to keep a better beam quality. For instance, the technique of self-imaging has been used to keep the single-mode propagation in multi-mode PWG oscillators [6] or amplifiers [7]. On the other hand, lessons from fiber lasers tell us that the index variation along the guided dimension causes the mode distortion in PWG, such as mode-field shrink and higher order mode (HOM) excitation [8], as the heat load is high enough. This kind of mode distortions is known to be dependent on the local transverse distribution of the refractive indices only and is not effected by the beam propagation. It is essentially different from traditional bulk solid-state lasers in which the thermal-induced phase aberrations (e.g., the thermal lens) are proportional to length of the optical path in the medium. Therefore decrease of the beam quality due to the thermal-induced mode distortion in waveguides is much smaller than that in traditional bulk solid-state lasers. Unfortunately, it is similar to the zigzag slab lasers [9] where the most important optical distortion in high power PWG laser should come from temperature gradient in the unguided dimension caused by the pumping inhomogeneity. This inhomogeneity is especially serious in the side-pumping scheme [10], which can be reduced to be about ten percent only in the end-pumping scheme [2, 11]. An interesting physical problem is whether there is crossing perturbations, that is whether the index variation in the transverse unguided dimension can cause the mode distortion in the guided dimension, and whether the guiding mode nature may reduce the optical distortion in the unguided dimension. In this paper we will use analytical perturbation method to investigate the guiding mode characteristics in presence of two-dimensional transverse thermal-induced index variations. It was found that the vector waveguide equation can be solved exactly if the transverse index variations in two orthonormal directions are independent, which is in general true due to one-dimensional thermal conduction in the very thin core of PWGs. Therefore thermal-induced two dimensional beam distortion in high power PWG amplifiers can be fully understood.

## 2. Theoretical model

#### 2.1. Mode fields in ideal PWG

This subsection is devoted to a brief review on the mode field solution in the ideal step-index PWGs. In Fig. 1 we label the transverse guided dimension, the transverse unguided dimension and the beam propagation dimension as *x*, *y* and *z* axes respectively. The electromagnetic guided wave is in the form of

**e**and

**h**satisfy the following vector wave eqation:

*n*=

*n*(

*x*) is the refractive index,

*x̂*and

*ẑ*are the unit vector along the

*x*- and

*z*-axes. Solutions of Eq. (2) could be symmetric (even modes) or anti-symmetric (odd modes) with respect to the

*x*-axis

*ϕ*denotes two independent components of the electromagnetic field, 2

*ρ*is the core thickness,

*U*and

*W*are defined as usual:

*k*= 2

*π*/

*λ*is wave vector with

*λ*the wavelength in the vacuum,

*n*

_{co}and

*n*

_{cl}are the refractive indices of the core and of the inner cladding, respectively. For the transverse electric (TE) modes, one has

*e*=

_{z}*e*=

_{x}*h*= 0 and

_{y}*e*can be assigned to satisfy the Eq. (3), while for the transverse magnetic (TM) modes,

_{y}*h*=

_{z}*h*=

_{x}*e*= 0 is held and it is convenient to set that

_{y}*h*has the form of the Eq. (3). Other electromagnetic components can be solved from

_{y}*e*or

_{y}*h*by the following Maxwell’s equations:

_{y}*μ*

_{0}and

*ε*

_{0}the dielectric constant and the magnetic permittivity in the vacuum respectively. The propagation constant

*β*or the effective refractive index

*n*

_{eff}for various modes are determined by the following boundary conditions:

*n*

_{co}≃

*n*

_{cl}) with lower numerical aperture (NA). Finally, it is well known that the single mode condition is governed by the convenient

*V*-parameter

#### 2.2. Guiding equations in presence of two-dimensional index variations

Real high power PWG amplifiers apparently suffer mode distortion due to the thermal-induced index variations. Since most of the laser energy is confined within regions of several times of the core size, the heat conduction equation in an active cooling scheme is approximately one-dimensional across the *x*-axis:

*κ*is the thermal conductivity,

*χ*(

*y*) depicts the pumping profile in the

*y*-axis. For the sake of convenience, an average value of unit is set for

*χ*(

*y*) so that the constant

*q*denotes an average power density of heat load inside the core. The temperature by solving Eq. (8) is in the form of The explicit expression of

*A*(

*x*) depends on the structure of PWGs and the cooling schemes. An example for double clad PWG will be given in Sect. 4. Thus the thermal-induced index variations along the

*x*and

*y*directions are independent each other, i.e.,

*n̄*is original refractive index in the unheated waveguide,

*ψ*(

*x*) =

*n̄A*(

*x*) denotes the index profile in the

*x*-axis and is yielded by the active cooling. Recalling that the thermal-optic coefficient

*dn/dT*has a small value, change of the mode field should be small as well in the most of situations. We can assume that the mode fields in the heated PWG possess the following perturbation formulation:

*i*=

*x*,

*y*,

*z*,

*ē*and

*h̄*denote the background mode fields defined on the unheated PWG,

*δe*and

*δh*denote the perturbations of mode fields.

The finite width of PWG indicates that *χ*(*y*) can always be expanded in cosine series

*w*the width of PWG. Equation (12) together with vector wave equations inspire the following expansion for the perturbed fields

Substituting Eqs. (11)–(13) into Eq. (2), it is straightforward to obtain the first order perturbation equations for the TE mode excitations:

*n̄*

_{co}≃

*n̄*

_{cl}, is used again in derivation of the Eq. (15).

Equations (14) and (15) indicate that perturbed fields can be thought as the excitations by the thermal-induced index gradient where the background fields serve as the sources. Because source terms in the right hand side of Eqs. (14) and (15) are known, all equations in (14) and (15) can be solved analytically. Other components of the perturbed fields can be obtained by solving the Maxwell’s equation (5). It is obvious that all components, including vanished longitude component *e _{z}* or

*h*in the original TE or TM modes, are excited unless

_{z}*a*= 0 for

_{m}*m*> 0. This is the first example on crossing perturbations that the

*y*-dependent index variation forbids the presence of pure transverse electromagnetic modes. Moreover, the perturbations of the TE modes and of the TM modes are degenerated as well. Apparently, if we set ${\overline{h}}_{y}^{\text{TM}}={\overline{e}}_{y}^{\text{TE}}$, the degeneracy of other background components and perturbed fields is given by the following identifications,

## 3. Guiding modes under two-dimensional index variations

Usually we are concerned with the energy flux along the direction of beam propagation in these kind of waveguide structures. Up to the first order perturbation, the energy flux has the form of *S _{z}* ∼

*ē*+

_{y}h̄_{x}*ē*+

_{y}δh_{x}*h̄*+ ...; hence we consider the components of

_{x}δe_{y}*G*and

_{y}*F*only in the following. On the other hand, in most situations the index variation order

_{x}*m*along the

*y*direction is not so large that the following approximations would be available,

*Ū*and

*W̄*are defined by Eq. (5) with the unheated refractive indices. Then all explicitly

*y*-depending terms in Eq. (14) and (15) can be ignored so that one has

*x*-axis. Together with Eqs. (14) and (15) it indicates that the perturbed fields possess the same symmetry as the background mode fields. For instance, solution of

*F*component on the background of the even TE modes must be in the form of

_{z}*d*

_{1}and

*d*

_{2}are real integration constants,

*M*(

*x*) is a special solution of Eq. (14),

In the perturbation method, the background solutions are always assumed to have kept all of their original properties, including the boundary conditions in particular. Thus the perturbed fields should satisfy the electromagnetic boundary conditions independently. Recalling the background boundary conditions in Eq. (6), continuum of *G _{y}* and

*F*at boundary of

_{z}*x*=

*ρ*yields that

The above equation has no non-vanishing solution in general. However, it should be noticed that the background mode fields can always be defined on a global heated PWG instead of original unheated one. It results to shift both *n*_{co} and *n*_{cl} with a constant. Thus we may subtract a constant *ψ*_{0} from *ψ*(*x*). Meanwhile, the thermal-induced index shift by *ψ*_{0} is added into the unheated refractive indices to yield a new one-dimensional step-index waveguide structure, which is used to define the background mode fields here. Let *ψ*(*x*) = *ψ*_{0} + Δ*ψ*(*x*), the constant *ψ*_{0} is chosen so that Eq. (21) allows a nonvanishing solution by replacing *ψ*(*x*) with Δ*ψ*(*x*), i.e.,

It should be stressed that the change of the *V*-parameter caused by a global but small index shift can usually be ignored. For instance, recalling *dn/dT* = 8 × 10^{−6} for Nd:YAG, relative change of the *V*-parameter is about 4×10^{−4} only for a temperature rise of 200K. Consequently this kind of slight index shifts does not affect the mode properties except to add a constant to the propagation constant *β*. However, it is extremely important that the shift of *ψ*(*x*) with a constant does not correspond to the global index shift actually, otherwise it will produce a *y*-dependent index shift so that the background wave equation should be rewritten as

Again we assume that the order of the transverse index variations, namely *m*, is not so large that *y*-derivatives in Eq. (23) can be ignored according to the approximations in (17). The solutions of Eq. (23) keep the form of (1) and (3) except to introduce a *y*-dependent shift in the original propagation constant *β̄*:

However, we should worry about that some explicit *z*-dependent terms appear by substituting Eq. (24) and (1) into (23). Physically these terms describe the diffraction induced by the transverse index variation. According to Eq. (17) we have

*L*denotes the length of PWG,

*γ*=

*ψ*

_{0}(

*χ*

_{max}−

*χ*

_{min})

*dn/dT*,

*f*

_{1}and

*f*

_{2}are functions of order one. So that influence of these

*z*-dependent terms to the background mode fields can be ignored if and only if In some cases of high power performance, this condition can not be satisfied very well, e.g.,

*kρ*> 300,

*L/w*> 10 and

*ψ*

_{0}(

*χ*

_{max}−

*χ*

_{min}) >20K. This problem can be solved by introducing a

*z*-dependent second-order term in Eq. (24) to cancel the

*L*

^{2}-term in Eq. (25), that is to let

*Lγ/w*≪ 1 is held, it is not hard to check that the first-order contribution in the remaining terms is

*g*

_{1}and

*g*

_{2}are functions of order one as well. Equation (28) is not only significantly smaller than the background term,

*k*

^{2}

*n̄*

^{2}−

*β̄*

^{2}, but also apparently smaller than perturbation term of order

*k*

^{2}Δ

*T*

_{co}(

*χ*

_{max}−

*χ*

_{min})

*dn/dT*with Δ

*T*

_{co}the temperature difference in the core. That perturbation term is used to determine distortion of the mode amplitude according to Eq. (14) and (15).

The integration constant *d*_{1} and *d*_{2} can not be fixed yet even though *ψ*(*x*) in Eq. (21) is replaced by Δ*ψ*(*x*) given by Eq. (22). Recalling that the perturbation fields are excited on the thermal-induced index gradient by the background fields, condition of the energy conservation should be imposed, i.e.,

Equation (29) together with Eq. (21) will determine the integration constant *d*_{1} and *d*_{2}, and Eqs. (18), (19) and (27) will describe fully the mode characteristics of heated PWG in presence of two dimensional transverse index variations. Physically, Eq. (27) is very important as it depicts the thermal-induced phase aberration in the width direction. This phase aberration is obviously accumulated with the beam propagation: The first-order correction term in Eq. (27) corresponds to the optical path difference (OPD) in geometric optic approximation, while the second-order term describes the diffraction correction because the beam propagates in an inhomogeneous media. Moreover, it should be pointed out that the phase aberration depends on both the background field and the thermal load through the constant *ψ*_{0} given by Eq. (22). Since right hand side of Eq. (19) does not depend on the transverse order “*m*” of the index variation, the mode distortion along the *x*-axis is the same for all values of “*m*”, thus profile of the perturbed fields along the *x*-axis is independent of that along the *y*-axis. For instance, the complete expression of *E _{y}* is given by

*U*-parameter solved from Eq. (6).

## 4. Thermally induced mode distortion in PWG amplifiers

In this section we will show the thermally induced mode distortion in PWG explicitly according to two examples of double clad Nd:YAG PWG amplifiers. Because thermal conduction close to the core region approximates to one-dimension, the temperature distribution along the *x*-axis is given by [12]

*T*

_{cl}and Δ

*T*

_{co}are respectively the temperature rise at boundary of the core and the temperature difference between center and boundary of the core. In a double clad PWG they are explicitly obtained by solving the heat conduction equation (8),

*q*is a constant power density of the heat load,

*h*is the surface heat transfer coefficient,

*D*

_{PWG}and

*D*

_{in}are the thickness of PWG and of inner cladding respectively,

*κ*

_{out}and

*κ*

_{YAG}are the thermal conductivity of the outer cladding and of the doped/undoped YAG respectively. Introducing

*a*=

*n̄*(Δ

*T*

_{cl}+ Δ

*T*

_{co}) −

*ψ*

_{0}and

*b*=

*n̄*Δ

*T*

_{co}and inserting Eqs. (3) and (31) into Eq. (20), we obtain the perturbation function

*M*(

*x*) for the even modes:

*ψ*

_{0}is obtained by inserting Eqs. (33) or (34) into (22):

*ψ*

_{0}to Δ

*T*

_{co}depends on the background mode fields, but is independent of the heat load; hence it can be treated as a structure parameter to govern the mode properties in the heated PWG.

Both the two double clad PWGs considered in the section consist of a Nd:YAG core (*n*_{co} = 1.818), an inner cladding of undoped YAG and an outer cladding of Al_{2}O_{3}. Thicknesses of the inner cladding and outer cladding are 400*μ*m and 2mm, respectively. The width of two PWGs and the surface heat transfer coefficient are chosen as *w* = 2cm and *h* =5W/K·cm^{2} respectively. The pump uniformity along the *y*-axis is assumed to be 90% where the inhomogeneity is yielded according to Eq. (12) by a random distribution of {*a _{m}*} with

*m*< 20.

In order to study the mode distortion under various pumping loads, three different heat load power per unit length, 260W/cm, 520W/cm and 1.04kW/cm, are considered for both the two PWGs. Moreover, it is interesting to explore the difference on the mode distortions for those PWGs with different mode contents. The following parameters are used: *ρ* =50*μ*m, *n*_{cl} = 1.8172 for the first PWG; and *ρ* =20*μ*m, *n*_{cl} = 1.8176 for the second PWG.

The first PWG supports six even and five odd TE modes (*V* = 15.9227). The parameters determining characteristics of various modes, namely *U* and *ψ*_{0}/*n̄*Δ*T*_{co}, are given by Fig. 2. Three different heat load correspond to Δ*T*_{co} = 1.5K, 3K and 6K,respectively. In Fig. 3 we show that the field profiles of several modes along the *x*-axis under different heat loads, where *χ* = 1 is used and the modes under different heated structures are normalized by energy conservation condition (30). It should be noticed that the mode distortion of TE_{10} mode given by Fig. 3(f) is not right for Δ*T*_{co} =6K. We present it here because it provides a helpful example to illustrate a case in which the above perturbation method fails. Precisely, Eq. (21) does not allow any solutions of *d*_{1} and *d*_{2} to keep field distortions in the core and in the inner cladding as perturbations simultaneously. To ignore this special case, we conclude that higher-order modes are more robust on the thermal effect than lower-order modes.

The second PWG with the core thicknesses of 40*μ*m supports one even TE mode and one odd TE mode only. In this case, three heat load powers correspond to Δ*T*_{co} = 0.6K, 1.2K and 2.4K, respectively. The corresponding mode field profiles in the *x*-axis is given by Fig. 4. Comparing the Fig. 4(a) with the Fig. 2(a), we can see that the profile deformation in the *x*-axis in the second PWG is much smaller than that in the first PWG. In particular, that profile deformation in the Fig. 4(a) with Δ*T*_{co} =2.4K is smaller than that in the Fig. 2(a) with Δ*T*_{co} =1.5K. Therefore, we may conclude that the second PWG is advantaged to resist the thermal-induced mode distortion in the *x*-axis. It is not only because of a lower temperature rise in the second PWG, but also because of tighter confinement of the fundamental mode in the second PWG.

It can be checked that *ψ*_{0}/*n̄*Δ*T*_{co} ≃ 60 in the second PWG. Recalling that the value of Δ*T*_{co} in the first PWG is 2.5 times higher than Δ*T*_{co} in the second PWG with the same heat load power, the value of *ψ*_{0} in the two PWGs is almost equal. It indicates that the two PWGs have the almost same phase aberrations in the *y*-axis. In Fig. 5, we show an example of how the *y*-axis phase aberration of the fundamental mode is accumulated with the beam propagation. Here the phase aberration is generated by a pumping non-uniformity of 10% given by Fig. 5(a). Two different heat loads corresponding to Δ*T*_{co}=1.5K and 3K are considered in Fig. 5(b) and 5(c), respectively. Obviously this pumping non-uniformity results a serious phase aberration for longer optical path and a nonlinear aberration evolution due to the diffraction nature in beam propagation.

The phase aberration in the *y*-axis will remarkably impact on the performance of the PWG amplifiers by worsening the beam quality. In Fig. 6, we investigate the far field pattern and the beam quality of the fundamental mode in the first PWG. The similar results can be obtained for the second PWG according to the previous discussions. Here the heat load per unit length is taken as 260W/cm that corresponds to Δ*T*_{co}=1.5K. The beam quality (BQ) definition used here is:

_{R}and PIB

_{DL}are the powers contained within a far-field bucket with a radius of

*λ*/(

*D*)

_{x}D_{y}^{1/2}for the beam with the

*y*-axis phase aberration, and a diffraction-limited beam with the same near-field dimensions and wavelength respectively.

*D*and

_{x}*D*are, respectively, the

_{y}*x*and

*y*dimensions of the beam near-field. For the sake of convenience, in our calculation the

*x*dimensions of the beam near-field is assumed to be magnified to 5mm. From Fig. 6(b) to 6(d) we can see that even a pumping non-uniformity of 5% in the y-axis may caused apparent distortion of the far-field pattern. This distortion is ever serious with the power scaling by increasing the optical length inside the PWG amplifiers, or by increasing pumping load per unit length. Figure 6(e) further shows the degradation of beam quality with the increase of pumping non-uniformity. Obviously this phase aberration must be corrected in high power performance to achieve a better beam quality, e.g.,

*BQ*< 2.

Although the above examples are both on double clad PWG with Nd:YAG core, the method and conclusions elucidated by the present paper should be valid for either PWG with the Yb:YAG core [13] or other PWG structures [14].

## 5. Conclusion

In conclusion, thermally induced mode distortions in solid-state planar waveguide amplifier is investigated theoretically. Two dimensional transverse refractive index variations are considered, where the temperature gradient along the thickness direction is generated by active cooling and that along the width direction is assumed to be caused by the pumping inhomogeneity. Under the approximations that thermal conduction is one-dimensional and lower spatial frequency perturbations are dominant along the width direction, the vector wave equation is solved rigorously by means of the perturbation method. The thermal-induced two dimensional mode distortions in PWG amplifiers are obtained explicitly. The crossing mode perturbations are confirmed not only by the results that the spatial distribution of the mode amplitude is changed by the index variation along the width, but also by the observations that the vanished longitude components in the original modes are excited by the transverse index variation. The degeneracy between TE modes and TM modes is shown for both of the background fields and the perturbed fields.

The consistence between the solution of wave-guide equation and the electromagnetic boundary conditions requires that the background mode fields are defined on a global heated step-index PWG instead of original unheated one. Although mode field profiles in the *x*-axis in this heated structure remain unchanged, an additional *y*-dependent phase factor is required to be added. It yields the thermal-induced phase aberration along the width direction in the presence of the transverse pumping inhomogeneity. Similar to the zigzag slab amplifier, this phase aberration is accumulated with laser propagation and amplification. In Eq. (27) we work out the first two order corrections on this phase aberration. The first order correction corresponds to the OPD in geometric optics, and the second order one denotes certain diffractive contributions. According to two examples on high power performance of the PWG amplifiers, it is shown that this phase aberration is the most important ingredient to cause degradation of the beam quality. Moreover, modes in the single-mode or few-mode PWGs suffer weaker thermal-induced distortion across the thickness than those modes in the multi-mode PWGs.

## Acknowledgments

This work was partly supported by Key Laboratory of Science and Technology on High Energy Laser, CAEP, under Grant No. LJG2012-07.

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