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Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 21, Iss. 15 — Jul. 29, 2013
  • pp: 17999–18010
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Thermal-induced two dimensional beam distortion in planar waveguide amplifiers

Xiao-Jun Wang, Wei-Wei Ke, and Hua Su  »View Author Affiliations


Optics Express, Vol. 21, Issue 15, pp. 17999-18010 (2013)
http://dx.doi.org/10.1364/OE.21.017999


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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 OSA

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

1. N. Hodgson, V. V. Ter-Mikirtychev, H. J. Hoffman, and W. Jordan, “Diode-pumped, 220W ultra-thin slab Nd:YAG laser with near-diffraction limited beam quality,” in Advanced Solid-State Lasers, M. Fermann and L. Marshall, eds., Vl. 68 of Trends in Optics and Photonics Series (Optical Society of America, 2002), paper PD2.

3

3. C. Grivas, “Optically pumped planar waveguide lasers, Part I: Fundamentals and Fabrication techniques,” Prog. Quantum Electron. 35, 159–239 (2011) [CrossRef] .

]. 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

4. C. L. Bonner, T. Bhutta, D. P. Shepherd, and A. C. Tropper, “Double-clad structures and proximity coupling for diode-bar-pumped planar waveguide lasers,” IEEE J. Quantum Electron. 36, 236–242 (2000) [CrossRef] .

]. 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

5. J. I. Mackenzie, “Dielectric solid-state planar waveguide lasers: A review,” IEEE J. Sel. Topics Quantum Electron. 13, 626–637 (2007) [CrossRef]

].

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

6. H. J. Baker, J. R. Lee, and D. R Hall, “Self-imaging and high-beam-quality operation in multi-mode planar waveguide optical amplifiers,” Opt. Express 10, 297–302 (2002) [CrossRef] [PubMed] .

] or amplifiers [7

7. G. J. Wagner, B. E. Callicoatt, G. T. Bennett, M. Tartaglia, L. Rubin, S. Field, A. I. R. Malm, and C. Ryan, “375W, 20kHz, 1.5ns Nd:YAG planar waveguide MOPA,” in Conference on Lasers and Electro-Optics (2011), paper PDPB1.

]. 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

8. K. R. Hansen, T. T. Alkeskjold, J. Broeng, and J. Lagsgaard, “Thermo-optical effects in high-power Ytterbium-doped fiber amplifiers,” Opt. Express 19, 23965–23980 (2011) [CrossRef] [PubMed] .

], 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

9. S. Redmond, S. McNaught, J. Zamel, L. Iwaki, S. Bammert, R. Simpson, S. B. Weiss, J. Szot, B. Flegal, T. Lee, H. Komine, and H. Injeyan, “15 kW near-diffraction-limited single-frequency Nd:YAG laser,” in Conference on Lasers and Electro-Optics (2007), paper CTuHH5.

] 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

10. L. Xiao, X. J. Cheng, and J. Q. Xu, “High-power Nd:YAG planar waveguide laser with YAG and Al2O3 claddings,” Opt. Commun. 281, 3781–3785 (2008) [CrossRef] .

], which can be reduced to be about ten percent only in the end-pumping scheme [2

2. D. Filgas, D. Rockwell, and K. Spariosu, “Next-generation lasers for advanced active EO systems,” Raytheon Tech. Today 1, 9–13 (2008).

, 11

11. J. I. Mackenzie, C. Li, and D. P. Shepherd, “Multi-watt, high efficiency, diffraction limited Nd:YAG planar waveguide laser,” IEEE J. Quantum Electron. 39, 493–500 (2003) [CrossRef] .

]. 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(x,t)=e(x)eiωt+iβz,H(x,t)=h(x)eiωt+iβz,
(1)
where e and h satisfy the following vector wave eqation:
{(2x2+k2n¯2β2)e=(x^x+iβz^)(exlnn2x),(2x2+k2n¯2β2)h=lnn2x[(x^x+iβz^)×h]×x^,
(2)
where n = n(x) is the refractive index, and 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(x)={cos(Ux/ρ)/cosUexp[W(|x|/ρ1)]or{sin(Ux/ρ)/sinU|x|ρx|x|exp[W(|x|/ρ1)]|x|>ρ
(3)
In the above equation, ϕ denotes two independent components of the electromagnetic field, 2ρ is the core thickness, U and W are defined as usual:
U=kρnco2neff2,W=kρneff2ncl2,neff=β/k,
(4)
where k = 2π/λ is wave vector with λ the wavelength in the vacuum, nco and ncl are the refractive indices of the core and of the inner cladding, respectively. For the transverse electric (TE) modes, one has ez = ex = hy = 0 and ey can be assigned to satisfy the Eq. (3), while for the transverse magnetic (TM) modes, hz = hx = ey = 0 is held and it is convenient to set that hy has the form of the Eq. (3). Other electromagnetic components can be solved from ey or hy by the following Maxwell’s equations:
E=iμ0ε01kn2×H,H=iε0μ01k×E,
(5)
with μ0 and ε0 the dielectric constant and the magnetic permittivity in the vacuum respectively. The propagation constant β or the effective refractive index neff for various modes are determined by the following boundary conditions:
tanU={W/U(EvenTE)orU/W(OddTE),nco2W/ncl2U(EvenTM)ornco2W/ncl2U(OddTM).
(6)
We can find that TM modes are almost degenerated with TE modes for weak guiding case (nconcl) with lower numerical aperture (NA). Finally, it is well known that the single mode condition is governed by the convenient V-parameter
V=kρnco2ncl2π/2.
(7)

Fig. 1 Planar optical waveguide.

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:
κ2Tx2+Qχ(y)=0,Q={q,|x|ρ0,|x|>ρ,
(8)
where 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
T(x,y)=T0+A(x)χ(y).
(9)
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.,
n2(x,y)=(n¯+dndTΔT(x,y))2n¯2+2dndTψ(x)χ(y),
(10)
where 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:
ei=e¯i+2dndTδei,hi=h¯i+2dndTδhi,
(11)
where i = x, y, z, ē and 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
χ(y)=mamcosmπyw,
(12)
with w the width of PWG. Equation (12) together with vector wave equations inspire the following expansion for the perturbed fields
{δez(x,y)=mGzm(x)sinmπyw,δhz(x,y)=mFzm(x)cosmπyw,forTEmodesδez(x,y)=mGzm(x)cosmπyw,δhz(x,y)=mFzm(x)sinmπyw,forTMmodes
(13)

Substituting Eqs. (11)(13) into Eq. (2), it is straightforward to obtain the first order perturbation equations for the TE mode excitations:
{(2x2+k2n¯2β2m2π2w2)Gzm=iamβn¯2mπwψe¯y,(2x2+k2n¯2β2m2π2w2)Fzm=iε0μ0kam(ψe¯y)x,
(14)
and for the TM mode excitations:
{(2x2+k2n¯2β2m2π2w2)Gzm=iμ0ε0iamkn¯2(ψh¯y)x,(2x2+k2n¯2β2m2π2w2)Fzm=iamβn¯2mπwψh¯y.
(15)
The weak guiding condition, cocl, 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 ez or hz in the original TE or TM modes, are excited unless am = 0 for 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 h¯yTM=e¯yTE, the degeneracy of other background components and perturbed fields is given by the following identifications,
e¯x,zTM=1n¯2μ0ε0h¯x,zTE,GiTM=1n¯2μ0ε0FiTE,FiTM=GiTE.
(16)
Therefore, in the rest part of this paper we focus our attentions on the thermally induced distortions of TE modes only.

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 Szēyx + ēyδhx + xδey + ...; hence we consider the components of Gy and Fx only in the following. On the other hand, in most situations the index variation order m along the y direction is not so large that the following approximations would be available,
k2n¯2β2~{U¯2ρ2orW¯2ρ2}m2π2w2,andmπwx~1ρ,
(17)
where Ū and 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
Gymkβμ0ε0Fxmikk2n¯2β2μ0ε0Fzmxk2amk2n¯2β2ψe¯y.
(18)
In this paper we are interested in the case that PWG is cooled symmetrically with respect to 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 Fz component on the background of the even TE modes must be in the form of
Fzmikamμ0ε0={Mco(x)+d1sin(U¯x/ρ)/sinU¯,|x|ρ,Mcl(x)+d2x|x|exp[W¯(|x|/ρ1)],|x|>ρ
(19)
where d1 and d2 are real integration constants, M(x) is a special solution of Eq. (14),
Mco(x)=Re{eiU¯x/ρdx[e2iU¯x/ρdxeiU¯x/ρ(ψe¯y,co)x]},Mcl(x)=eW¯x/ρdx[e2W¯x/ρdxeW¯x/ρ(ψe¯y,cl)x],
(20)
where “Re” means to take the real part.

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 Gy and Fz at boundary of x = ρ yields that
{d1+Mco(ρ)=d2+Mcl(ρ),d1+ρW¯U2(Mcoxψ)||x|=ρ=d2ρW(Mclxψ)||x|=ρ
(21)

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 nco and ncl with a constant. Thus we may subtract a constant ψ0 from ψ(x). Meanwhile, the thermal-induced index shift by ψ0 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) = ψ0 + Δψ(x), the constant ψ0 is chosen so that Eq. (21) allows a nonvanishing solution by replacing ψ(x) with Δψ(x), i.e.,
Δψ(ρ)=ψ(ρ)ψ0=(W¯2V2Mcox+U¯2V2Mclx)||x|=ρU¯2W¯ρV2[Mco(ρ)Mcl(ρ)].
(22)

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
(2x2+2y2+2z2+k2n¯2+2k2dndTψ0χ(y))(EH)=0.
(23)

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:
ββ(y)=β¯(1+dndTψ0n¯2χ(y)).
(24)

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
1k2n¯2β22eiβ(y)zy2~ρ2w2kLγ[f1(y)+kLγf2(y)],
(25)
where L denotes the length of PWG, γ = ψ0(χmaxχmin)dn/dT, f1 and f2 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
kργLw1.
(26)
In some cases of high power performance, this condition can not be satisfied very well, e.g., > 300, L/w > 10 and ψ0(χmaxχmin) >20K. This problem can be solved by introducing a z-dependent second-order term in Eq. (24) to cancel the L2-term in Eq. (25), that is to let
ββ(y,z)=β¯[1+dndTψ0n¯2χ(y)z26(dndTψ0n¯2χy)2].
(27)
When condition Lγ/w ≪ 1 is held, it is not hard to check that the first-order contribution in the remaining terms is
(2y2+2z2+k2n¯2+2k2dndTψ0χ(y))eiβ(y,z)z~k2γ3L4w4g1(y)+k2γ2g2(y).
(28)
where g1 and g2 are functions of order one as well. Equation (28) is not only significantly smaller than the background term, k22β̄2, but also apparently smaller than perturbation term of order k2ΔTco(χmaxχmin)dn/dT with ΔTco 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 d1 and d2 can not be fixed yet even though ψ(x) in Eq. (21) is replaced by Δψ(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.,
dx|e¯y|2=dx|e¯y+2dndTδey|2dxe¯yGym=0.
(29)

Equation (29) together with Eq. (21) will determine the integration constant d1 and d2, 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 ψ0 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 Ey is given by
Ey(x,t)=exp[iωt+izβ(y,z)]×{e¯y+2dndTGy(x)χ(y)},
(30)
where Gym=amGy. Therefore, the independent modes with two-dimensional index variations are still labeled 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

12. J. M. Eggleston, T. J. Kane, K. Kuhn, J. Unternahrer, and R. L. Byer, “The slab geometry laser I: Theory,” IEEE J. Quantum Electron. 20, 289–301 (1984) [CrossRef] .

]
ψ(x)={n¯ΔTcl+n¯ΔTco(1x2/ρ2),|x|ρ,n¯ΔTcl+2n¯ΔTco(1|x|/ρ),|x|>ρ.
(31)
Here ΔTcl and ΔTco 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),
ΔTcl=qρ(1h+DPWGDin2κout+Din2ρ2κYAG),ΔTco=qρ22κYAG,
(32)
where q is a constant power density of the heat load, h is the surface heat transfer coefficient, DPWG and Din are the thickness of PWG and of inner cladding respectively, κout and κYAG are the thermal conductivity of the outer cladding and of the doped/undoped YAG respectively. Introducing a = Tcl + ΔTco) −ψ0 and b = ΔTco and inserting Eqs. (3) and (31) into Eq. (20), we obtain the perturbation function M(x) for the even modes:
Mco(x)=(a2b4U2bx26ρ2)xcos(U¯x/ρ)cosU(3a32b16U2+bx24ρ2)ρsin(U¯x/ρ)UcosU,Mcl(x)=(aW¯+b(W¯+1)4W2(ρ+2W¯|x|bx22ρ)x|x|exp[W¯(|x|/ρ1)],
(33)
and for the odd modes:
Mco(x)=(a2b4U2bx26ρ2)xsin(U¯x/ρ)sinU+(11a32b16U2+bx24ρ2)ρcos(U¯x/ρ)UsinU,Mcl(x)=(aW¯+b(W¯+1)4W2(ρ+2W¯|x|bx22ρ)exp[W¯(|x|/ρ1)].
(34)
The constant ψ0 is obtained by inserting Eqs. (33) or (34) into (22):
ψ0=n¯ΔTcl+n¯ΔTcoW¯2(4U¯2+3+3W¯/V¯2)6U¯26U2W(1+W).
(35)
This result is held for both even and odd modes. Clearly the ratio of ψ0 to ΔTco 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 (nco = 1.818), an inner cladding of undoped YAG and an outer cladding of Al2O3. Thicknesses of the inner cladding and outer cladding are 400μm and 2mm, respectively. The width of two PWGs and the surface heat transfer coefficient are chosen as w = 2cm and h =5W/K·cm2 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 {am} 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 =50μm, ncl = 1.8172 for the first PWG; and ρ =20μm, ncl = 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 ψ0/ΔTco, are given by Fig. 2. Three different heat load correspond to ΔTco = 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 TE10 mode given by Fig. 3(f) is not right for ΔTco =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 d1 and d2 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.

Fig. 2 Parameter U and ψ0/ΔTco for various modes in first example.
Fig. 3 Field profiles of several modes in the x-axis for the first PWG with ΔTco = 1.5K, 3K and 6K.

The second PWG with the core thicknesses of 40μm supports one even TE mode and one odd TE mode only. In this case, three heat load powers correspond to ΔTco = 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 ΔTco =2.4K is smaller than that in the Fig. 2(a) with ΔTco =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.

Fig. 4 Mode field profiles in the x-axis for the second PWG with ΔTco = 0.6K, 1.2K and 2.4K.

It can be checked that ψ0/ΔTco ≃ 60 in the second PWG. Recalling that the value of ΔTco in the first PWG is 2.5 times higher than ΔTco in the second PWG with the same heat load power, the value of ψ0 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 ΔTco=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.

Fig. 5 Phase difference in the width direction vs propagation distance with (b) ΔTco=1.5K and (c) ΔTco=3K. Here (a) denotes a pumping distribution with 90% uniformity in central region of 90% of width.

Fig. 6 Beam quality degradation of the fundamental mode caused by the y-axis phase aberration in the first PWG with ΔTco=1.5K. (a) Far field pattern without the phase aberration. (b)–(d) Far field patterns for the optical length L inside the PWG amplifiers, in consideration of the phase aberration induced by a pumping non-uniformity of 5% in y-axis. (e) BQ vs the pumping non-uniformity with different optical lengths.

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

13. K. Sueda, H. Takahashi, S. Kawato, and T. Kobayashi, “High-efficiency laser-diode-pumped microthickness Yb:Y3Al5O12 slab laser,” Appl. Phys. Lett. 87, 151110-1–151110-3 (2005) [CrossRef] .

] or other PWG structures [14

14. C. F. Wang, T. H. Her, L. Zhao, X. Y. Ao, L. W. Casperson, C. H. Lai, and H. C. Chang, “Gain saturation and output characteristics of index-antiguided planar waveguide amplifiers with homogeneous broadening,” J. Lightwave Technol. 29, 1958–1964 (2011) [CrossRef] .

].

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.

References and links

1.

N. Hodgson, V. V. Ter-Mikirtychev, H. J. Hoffman, and W. Jordan, “Diode-pumped, 220W ultra-thin slab Nd:YAG laser with near-diffraction limited beam quality,” in Advanced Solid-State Lasers, M. Fermann and L. Marshall, eds., Vl. 68 of Trends in Optics and Photonics Series (Optical Society of America, 2002), paper PD2.

2.

D. Filgas, D. Rockwell, and K. Spariosu, “Next-generation lasers for advanced active EO systems,” Raytheon Tech. Today 1, 9–13 (2008).

3.

C. Grivas, “Optically pumped planar waveguide lasers, Part I: Fundamentals and Fabrication techniques,” Prog. Quantum Electron. 35, 159–239 (2011) [CrossRef] .

4.

C. L. Bonner, T. Bhutta, D. P. Shepherd, and A. C. Tropper, “Double-clad structures and proximity coupling for diode-bar-pumped planar waveguide lasers,” IEEE J. Quantum Electron. 36, 236–242 (2000) [CrossRef] .

5.

J. I. Mackenzie, “Dielectric solid-state planar waveguide lasers: A review,” IEEE J. Sel. Topics Quantum Electron. 13, 626–637 (2007) [CrossRef]

6.

H. J. Baker, J. R. Lee, and D. R Hall, “Self-imaging and high-beam-quality operation in multi-mode planar waveguide optical amplifiers,” Opt. Express 10, 297–302 (2002) [CrossRef] [PubMed] .

7.

G. J. Wagner, B. E. Callicoatt, G. T. Bennett, M. Tartaglia, L. Rubin, S. Field, A. I. R. Malm, and C. Ryan, “375W, 20kHz, 1.5ns Nd:YAG planar waveguide MOPA,” in Conference on Lasers and Electro-Optics (2011), paper PDPB1.

8.

K. R. Hansen, T. T. Alkeskjold, J. Broeng, and J. Lagsgaard, “Thermo-optical effects in high-power Ytterbium-doped fiber amplifiers,” Opt. Express 19, 23965–23980 (2011) [CrossRef] [PubMed] .

9.

S. Redmond, S. McNaught, J. Zamel, L. Iwaki, S. Bammert, R. Simpson, S. B. Weiss, J. Szot, B. Flegal, T. Lee, H. Komine, and H. Injeyan, “15 kW near-diffraction-limited single-frequency Nd:YAG laser,” in Conference on Lasers and Electro-Optics (2007), paper CTuHH5.

10.

L. Xiao, X. J. Cheng, and J. Q. Xu, “High-power Nd:YAG planar waveguide laser with YAG and Al2O3 claddings,” Opt. Commun. 281, 3781–3785 (2008) [CrossRef] .

11.

J. I. Mackenzie, C. Li, and D. P. Shepherd, “Multi-watt, high efficiency, diffraction limited Nd:YAG planar waveguide laser,” IEEE J. Quantum Electron. 39, 493–500 (2003) [CrossRef] .

12.

J. M. Eggleston, T. J. Kane, K. Kuhn, J. Unternahrer, and R. L. Byer, “The slab geometry laser I: Theory,” IEEE J. Quantum Electron. 20, 289–301 (1984) [CrossRef] .

13.

K. Sueda, H. Takahashi, S. Kawato, and T. Kobayashi, “High-efficiency laser-diode-pumped microthickness Yb:Y3Al5O12 slab laser,” Appl. Phys. Lett. 87, 151110-1–151110-3 (2005) [CrossRef] .

14.

C. F. Wang, T. H. Her, L. Zhao, X. Y. Ao, L. W. Casperson, C. H. Lai, and H. C. Chang, “Gain saturation and output characteristics of index-antiguided planar waveguide amplifiers with homogeneous broadening,” J. Lightwave Technol. 29, 1958–1964 (2011) [CrossRef] .

OCIS Codes
(140.3580) Lasers and laser optics : Lasers, solid-state
(140.6810) Lasers and laser optics : Thermal effects
(230.7390) Optical devices : Waveguides, planar
(140.3295) Lasers and laser optics : Laser beam characterization

ToC Category:
Lasers and Laser Optics

History
Original Manuscript: April 22, 2013
Revised Manuscript: July 8, 2013
Manuscript Accepted: July 11, 2013
Published: July 19, 2013

Citation
Xiao-Jun Wang, Wei-Wei Ke, and Hua Su, "Thermal-induced two dimensional beam distortion in planar waveguide amplifiers," Opt. Express 21, 17999-18010 (2013)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-15-17999


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References

  1. N. Hodgson, V. V. Ter-Mikirtychev, H. J. Hoffman, and W. Jordan, “Diode-pumped, 220W ultra-thin slab Nd:YAG laser with near-diffraction limited beam quality,” in Advanced Solid-State Lasers, M. Fermann and L. Marshall, eds., Vl. 68 of Trends in Optics and Photonics Series (Optical Society of America, 2002), paper PD2.
  2. D. Filgas, D. Rockwell, and K. Spariosu, “Next-generation lasers for advanced active EO systems,” Raytheon Tech. Today1, 9–13 (2008).
  3. C. Grivas, “Optically pumped planar waveguide lasers, Part I: Fundamentals and Fabrication techniques,” Prog. Quantum Electron.35, 159–239 (2011). [CrossRef]
  4. C. L. Bonner, T. Bhutta, D. P. Shepherd, and A. C. Tropper, “Double-clad structures and proximity coupling for diode-bar-pumped planar waveguide lasers,” IEEE J. Quantum Electron.36, 236–242 (2000). [CrossRef]
  5. J. I. Mackenzie, “Dielectric solid-state planar waveguide lasers: A review,” IEEE J. Sel. Topics Quantum Electron.13, 626–637 (2007) [CrossRef]
  6. H. J. Baker, J. R. Lee, and D. R Hall, “Self-imaging and high-beam-quality operation in multi-mode planar waveguide optical amplifiers,” Opt. Express10, 297–302 (2002). [CrossRef] [PubMed]
  7. G. J. Wagner, B. E. Callicoatt, G. T. Bennett, M. Tartaglia, L. Rubin, S. Field, A. I. R. Malm, and C. Ryan, “375W, 20kHz, 1.5ns Nd:YAG planar waveguide MOPA,” in Conference on Lasers and Electro-Optics (2011), paper PDPB1.
  8. K. R. Hansen, T. T. Alkeskjold, J. Broeng, and J. Lagsgaard, “Thermo-optical effects in high-power Ytterbium-doped fiber amplifiers,” Opt. Express19, 23965–23980 (2011). [CrossRef] [PubMed]
  9. S. Redmond, S. McNaught, J. Zamel, L. Iwaki, S. Bammert, R. Simpson, S. B. Weiss, J. Szot, B. Flegal, T. Lee, H. Komine, and H. Injeyan, “15 kW near-diffraction-limited single-frequency Nd:YAG laser,” in Conference on Lasers and Electro-Optics (2007), paper CTuHH5.
  10. L. Xiao, X. J. Cheng, and J. Q. Xu, “High-power Nd:YAG planar waveguide laser with YAG and Al2O3 claddings,” Opt. Commun.281, 3781–3785 (2008). [CrossRef]
  11. J. I. Mackenzie, C. Li, and D. P. Shepherd, “Multi-watt, high efficiency, diffraction limited Nd:YAG planar waveguide laser,” IEEE J. Quantum Electron.39, 493–500 (2003). [CrossRef]
  12. J. M. Eggleston, T. J. Kane, K. Kuhn, J. Unternahrer, and R. L. Byer, “The slab geometry laser I: Theory,” IEEE J. Quantum Electron.20, 289–301 (1984). [CrossRef]
  13. K. Sueda, H. Takahashi, S. Kawato, and T. Kobayashi, “High-efficiency laser-diode-pumped microthickness Yb:Y3Al5O12 slab laser,” Appl. Phys. Lett.87, 151110-1–151110-3 (2005). [CrossRef]
  14. C. F. Wang, T. H. Her, L. Zhao, X. Y. Ao, L. W. Casperson, C. H. Lai, and H. C. Chang, “Gain saturation and output characteristics of index-antiguided planar waveguide amplifiers with homogeneous broadening,” J. Lightwave Technol.29, 1958–1964 (2011). [CrossRef]

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