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Optics Express

Optics Express

  • Editor: C. Martijin de Sterke
  • Vol. 19, Iss. 7 — Mar. 28, 2011
  • pp: 6883–6902
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Combined guiding effect in the end-pumped laser resonator

Xingpeng Yan, Qiang Liu, Dongsheng Wang, and Mali Gong  »View Author Affiliations


Optics Express, Vol. 19, Issue 7, pp. 6883-6902 (2011)
http://dx.doi.org/10.1364/OE.19.006883


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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:YVO4 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:YVO4 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 TEM00 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

1. A. Fox and Li Tingye, “Computation of optical resonator modes by the method of resonance excitation,” IEEE J. Quantum Electron . 4, 460–465 (1968). [CrossRef]

5

5. C. F. Maes and E. M. Wright, “Mode properties of an external-cavity laser with Gaussian gain,” Opt. Lett. 29, 229–231 (2004). [CrossRef] [PubMed]

]. In most quasi three-level lasers (e.g., Yb3+ doped gain medium), the inherent absorption of the laser in the unpumped region of the gain medium works equivalently to a soft transverse gain aperture [6

6. T. Y. Fan, “Aperture guiding in quasi-three-level lasers,” Opt. Lett. 19, 554–556 (1994). [CrossRef] [PubMed]

]. This aperture guiding effect dominates the transverse mode formation in the laser resonator. In contrast, in the Nd3+ based four-level laser, the un-pumped region has very low absorption at the laser wavelength so that the aperture guiding can be neglected. However, the thermal induced refractive index guiding (or called the thermal guiding) [7

7. J. J. Zayhowski, “Thermal Guiding in Microchip Lasers,” in Advanced Solid State Lasers, G. Dube, ed., Vol. 6 of OSA Proceedings Series (Optical Society of America, 1990), paper DPL3.

] and gain guiding effect [8

8. G. K. Harkness and W. J. Firth, “Transverse modes of microchip solid state lasers,” J. Mod. Opt. 39, 2023–2037 (1992). [CrossRef]

] are found important for the formation of the transverse mode especially for plano cavity with high power and high intensity pumping.

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

1. A. Fox and Li Tingye, “Computation of optical resonator modes by the method of resonance excitation,” IEEE J. Quantum Electron . 4, 460–465 (1968). [CrossRef]

, 16

16. H. Kogelnik and T. Li, “Laser beams and resonators,” Appl. Opt. 5, 1550–1567 (1966). [CrossRef] [PubMed]

]. The effect of combined guiding effect on the formation of transverse mode profiles is also observed and researched experimentally which shows well agree with the prediction of our theoretical model.

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

2. Theoretical model

2.1. The cell-train model of laser resonator

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

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

Fig. 2 The periodic cell-train model of the laser resonator. Ck: the k–th cell, Jk: the junction between the cell Ck and Ck +1 while J 0 is the initial reference, Ek1±: the forward- and backward- propagating waves for the first pass through of the gain medium in the unfolded model, and Ek2± denotes that of the second pass.

According to the physical essence of the unfold model as shown in Fig. 2, in the iterative calculation, the two backward field in Ck can be represented recursively as following
Ek1=Ek12+Ek2=Ek1+,
(1)
with E11=0. The gain medium has specific temperature and gain distribution because of the pumping. When the wave E propagates in the gain medium, it suffers the thermal induced refractive index ΔnT and the gain ΔnG as shown in Fig. 3. The propagation of the wave is guided by the combined action of ΔnT and ΔnG, ie., the combined guiding. Thus, the wave equation included the combined guiding effect should be derived.

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

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
(2ɛμ2t2)E(x,y,z,ω,t)=μ2t2P(x,y,z,ω,t),
(2)
where E⃗ is the electric field intensity, P⃗ is the polarization intensity, ε and μ is the permittivity and permeability, respectively, of the gain medium. μ = μ 0 μrμ 0 for the nonmagnetic material where μr ≈ 1 is the relative permeability, and μ 0 is the permittivity of free space, or called the electric constant. ω is the resonant frequency of the light field in the laser resonator. (x,y,z) denotes the position coordinates, and t denotes the time. The operator ∇2 is the Laplacian, 2=2x2+2y2+2z2.

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(x,y,z,ω,t)=12e^[E(x,y,z)ei(kzωt)+c.c.]=12e^[E(x,y,z,ω)eiωt+c.c.],
(3)
and
P(x,y,z,ω,t)=12e^[P(x,y,z,ω)eiωt+c.c.],
(4)
where E is the complex amplitude of the electric field with an approximate time dependence of e−iωt. Substitute Eq. (3) and Eq. (4) into Eq. (2), then we can get the Helmholtz equation
(2+k02)E(x,y,z,ω)=μω22t2P(x,y,z,ω),
(5)
where the relationship of k=2πλ and c=1ɛμ was used. Separating out the rapid on-axis phase variations (i.e., the z-dependent phase variations) in the field, we have
E(x,y,z,ω)E(x,y,z,ω)eik0znb(z)dz.
(6)

Assume that the nonlinear effect of the electric field in the dielectric can be ignored, and then the linear polarization response approximation is considered,
P(x,y,z,ω)ɛ0χ(x,y,z,ω)E(x,y,z,ω)eik0znb(z)dz,
(7)
where χ is a complex quantity denoted the electric susceptibility of the dielectric. The complex refraction index of the dielectric can be represented with the electric susceptibility as following,
n2(x,y,z,ω)=1+χ(x,y,z,ω).
(8)

The refractive index at frequency ω in the gain medium can be described as
n(x,y,z,ω)=nb(z)+Δn(x,y,z,ω),
(9)
where nb(z) is the base refractive index of the medium in the absence of guiding given by k = nb(z)k 0, where Δn(x,y,z,ω) = n(x,y,z,ω) – nb(z). Substituting Eq. (9) into Eq. (8) and applying the first-order approximation when Δnnb
χ(x,y,z,ω)nb2(z)+2nb(z)Δn(x,y,z,ω)1.
(10)

Then from Eqs. (6), (7), and (10), the Helmholtz equation changed into
[2+2ik0nbz+k02k02nb2]E(x,y,z,ω)k02[nb2+2nbΔn(x,y,z,ω)1]E(x,y,z,ω).
(11)

Applying the slowly varying envelope approximation
|k02nb2E||2k0nbEz||2Ez2|,
(12)
and defined the transverse Laplacian as 2=2x2+2y2, then Eq. (11) changed into
[2+2ik0nbz+2k02nbΔn(x,y,z,ω)]E(x,y,z,ω)=0.
(13)

The complex-valued guiding refractive index Δn can be written in an explicit form,
Δn(x,y,z,ω)=ΔnT(x,y,z)+ΔnG(x,y,z,ω),
Here ΔnT is the thermal induced guiding refractive index which is a real one, and ΔnG is the gain induced guiding refractive index which is a imaginary one.

The thermal induced guiding refractive index ΔnT is mainly determined by the thermo-optic effect of the gain medium. In diode-pumped solid-state lasers, the temperature gradient in the gain medium arises from the heat generation from unconverted pump power that manifests itself as heat and phonon energy. This temperature gradient, when coupled with the refractive indices temperature coefficient(dn/dT) can give a distribution of the refractive index in the material. The thermal induced guiding refractive index ΔnT is given as following
ΔnT(x,y,z)=dndT(T(x,y,z)Tr),
(14)
where T(x,y,z) is the steady-state temperature field in the gain medium volume and Tr is the room temperature. The steady-state temperature field can be calculated from the following heat diffusion equation with certain boundary conditions,
[Kx2x2+Ky2y2+Kz2z2]T(x,y,z)=ηhPabs(x,y,z),
(15)
where Ki(i = x, y, z) is the thermal conductivity of the gain medium, ηh is the fraction of absorbed pump light converted to heat, and P abs is the absorbed power density in the crystal. ΔnT can be then obtained by solving the equation with certain boundary conditions.

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

17. A. E. Siegman, Lasers, (Univ. Sci. Books, 1986), pp. 323.

],
G(x,y,z)=G0(x,y,z,ωc)1+I(x,y,z,ωc)Isat,
(18)
where G 0 is the small signal gain coefficient and I sat is the saturation intensity. If ground state depletion is neglected in the four level laser, the small signal gain coefficient G 0 can be represented as following [18

18. J. Frauchiger, P. Albers, and H. P. Weber, “Modeling of thermal lensing and higher order ring mode oscillation in end-pumped C-W Nd:YAG lasers,” IEEE J. Quantum Electron. 28, 1046–1056 (1992). [CrossRef]

],
G0(x,y,z)=Pabs(x,y,z)λpσ21τfhc,
(19)
where σ 21 is the stimulated emission cross section of the laser transition and τf is the upper laser level life time, h is the Plank constant. The saturation intensity I sat was given by [22

22. W. Koechner, Solid-tate Laser Engineering, 6th ed. (Springer-Verlag Publications, 2006).

]
Isat=(Wr+γ)hcλlσ21τf,
(20)
where Wr is the pump rate, and γ is the level degeneracy. For four level laser, Wrτf and γ ≈ 1.

The actual resonant internal laser intensity is saturated down by the forward and back ward traveling electric fields E + and E. Therefore, using the relationship I=12ɛμ|E|2cɛ0ɛr2|E|2, the effective gain coefficient in the gain medium can be represented as following,
G(x,y,z)=Pabs(x,y,z)hcλpσ21τf+λlcɛ0ɛr2λp|E+(x,y,z)+E(x,y,z)|2.
(21)

2.3. SSFM for the wave equation

The nonlinear wave equation (Eq. (22)) can be rewritten as the following simple type,
iψz=(D^+N^)ψ,
(23)
where ψ = E +(x,y,z) is the wave function, and is the linear differential operator which denotes the spatial diffraction effect,
D^=22k0nb=12k0nb(2x2+2y2),
(24)
and is the nonlinear operator denoted the combined guiding effect
N^=k0ΔnT(x,y,z)+iG(x,y,z)2.
(25)

The formal solution of Eq. (23) can be represented as following,
ψ(x,y,z+Δz)=ei(D^+N^)Δzψ(x,y,z).
(26)

According to the Baker-Campbell-Hausdorff (BCH) formula
ψ(x,y,z+Δz)ei(D^Δz)ei(N^Δz)ψ(x,y,z).
(27)

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

Only considering the spatial diffraction effect on the ψ
ψ(x,y,z+Δz)ei(D^Δz)ψ(x,y,z).
(28)

According to the definition of the inverse fourier transform,
exp(iD^Δz)ψ(x,y,z)=exp(iD^Δz)exp[i(ωxx+ωxy)]ψ˜(ωx,ωy,z)dωxdωy,
(29)
ψ˜ = 𝒡 (ψ) is the Fourier transform of ψ where 𝒡 is the Fourier transform operator, and ωx,y are the spatial angular frequencies in the Fourier domain. The operator 𝒡−1 is defined as the inverse Fourier transform operator.

Using the Maclaurin’s expansion,
exp(iD^Δz)exp[i(ωxx+ωxy)]={n1n![iΔz2k0nb(2x2)]nexp(iωxx)}{n1n![iΔz2k0nb(2y2)]nexp(iωyy)}=[exp(iωxx)n1n!(iΔz2k0nbωx2)n][exp(iωyy)n1n!(iΔz2k0nbωy2)n]=exp[iΔz2k0nb(ωx2+ωy2)]exp[i(ωxx+ωxy)],
(30)
then
ψ(x,y,z+Δz)=exp[iΔz2k0nb(ωx2+ωy2)]exp[i(ωxx+ωxy)]ψ˜(ωx,ωy,z)dωxdωy=1{[ψ(x,y,z)]exp[iΔz2k0nb(ωx2+ωy2)]}.
(31)

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 on the ψ,
ψ(x,y,z+Δz)ei(N^Δz)ψ(x,y,z).
(32)

The equation can be directly integrated and yields to
ψ(x,y,z+Δz)exp[izz+ΔzN^(x,y,z)dz]ψ(x,y,z).
(33)

When the step size Δz short enough, the trapezoidal rule can be used to estimate the integral
zz+ΔzN^(x,y,z)dzΔz2[N^(x,y,z)+N^(x,y,z+Δz)].
(34)

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

19. G. M. Muslu and H. A. Erbay, “Higher-order split-step Fourier schemes for the generalized nonlinear Schrodinger equation,” Math. Comput. Simulat. 67, 581–595 (2005). [CrossRef]

21

21. Y. L. Bogomolov and A. D. Yunakovsky, “Split-step Fourier method for nonlinear Schrodinger equation,” in International Conference Days on Diffraction 2006 , Proceedings of the International Conference ’Days on Diffraction’ 2006, DD (Inst. of Elec. and Elec. Eng. Computer Society, 2006), 34–42. [CrossRef]

]. From the BCH formula, the approximation of formula (27) has the accuracy of order Δz 2 since the operators and are noncommutative. If the operators are rearranged into a symmetric form, and then we have the new approximation
exp[i(D^+N^)Δz]exp[iD^Δz/2]exp[iN^Δz]exp[iD^Δz/2].
(35)

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

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

The accuracy of the SSFM depends on the grid size in the transverse direction (Δx,y) and the step size in the axial direction (Δz). The FFT method requires N = 2m (m is positive integer) sampling points in the transverse direction while the spatial resolution should be high enough. The low resolution may lead to the aliasing and overlapping of the spatial angular frequencies in the Fourier domain. The electric field of laser is spatial bandlimited since its high spatial coherence. Therefore, according to the Nyquist-Shannon sampling theorem, the transverse spatial step size Δx,y should be at least twice of reciprocal of the spatial angular frequency, ie., Δx, y<12Δωx,y, where Δωx,y is the bandwidth of the angular frequencies. The axial step size Δz should be decreased until the enough computational accuracy can be obtained.

3. Experimental setup

In our experiment, a composite a-cut Nd:YVO4 crystal was used as the gain medium. For the four level laser emission when being pumped at λp =808 nm (4 I 9/24 F 5/2) and lasing at λl =1064 nm (4 F 3/24 I 11/2), Nd:YVO4 crystal has a large effective stimulated absorption cross section (σ 12 ≈ 2.7 × 10−23m2@ 1 at.%) and effective stimulated emission cross section (σ 21 ≈ 15.6 × 10−23m2@ 1 at.%) [22

22. W. Koechner, Solid-tate Laser Engineering, 6th ed. (Springer-Verlag Publications, 2006).

, 23

23. D. G. Matthews, J. R. Boon, R. S. Conroy, and B. D. Sinclair, “Comparative study of diode pumped microchip laser materials: Nd-doped YVO4, YOS, SFAP and SVAP,” J. Mod. Opt. 43, 1079–1087 (1996). [CrossRef]

], therefore, Nd:YVO4 can provide high gain. The composite crystal was also used, and two updoped YVO4 end caps were thermally bonded on the both ends of the 0.3 at. % Nd3+ ion concentrated Nd:YVO4 crystal. The end caps were with the dimension of 3 mm× 3 mm×2 mm while that of the doped crystal was 3 mm× 3 mm×16 mm. Thus, the size of the composite crystal was w × h × l = 3 mm × 3 mm × 20 mm. One reason using the composite crystal is that the thermally bonded undoped end caps can prevent the thermal fracture. The thermal fracture results from the temperature gradients and the consequent thermal stress, especially when the crystal was intensely pumped to get high gain. Meanwhile, the thermal stress also can make the so called end-effect, the deformation and the bulge of the end-face of the gain medium. The curvature on the end-face acts as a built-in index guide which can cover up the thermal index guiding. So the thermally bonding technique is used to reduce the end-effect significantly. The composite crystal was dual end pumped. The 45 W pump light from the laser diode (LD) was coupled into a 400 μm diameter, 0.22 numerical aperture fiber, and then imaged using a four-lens imaging system and then the focused pump beam was delivered into the gain medium. The imaging system can be adjusted conveniently to adjust the waist size of focused pump as well as the location of the beam waist in the gain medium. The temperature of the laser diode was controlled using the thermoelectric cooling module (TECM), and the center wavelength of the laser diode could be temperature-tuned by adjusting the temperature of the LD with TECM. The wavelength tuning coefficient of the LD was about 0.3 nm/°C as measured. From Eqs. (15) and (19), we can see that the profile of the thermal index and the gain distribution can be changed conveniently by tuning the temperature of the LD.

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

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

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,
Fig. 5 The pump absorption model of the gain medium. z 0: the location of the pump waist, ωp(z): the pump beam radius at the location z, and ωp 0: the pump waist radius at the pump waist location of z 0, i.e., ωp 0=ωp(z0). θp: the half far-field divergence angle of the pump beam. Ip±: the forward- and backward- pump beam.
IP+(x,y,z)=C0πωp2(z)exp[2(x2+y2)NωpN(z)]exp(αeffz),
(36)
where ωp(z) is the radius of the pump mode at position z, and N is the order of super-Gaussian function which was measured as N ∼ 4 in our experiment using a 90/10 knife-edge method. α eff is the effective absorption coefficient at the effective pump wavelength λ eff of the pump laser in the active medium, and it is an averaged absorption coefficient measured in the experiment, considering the average of the both absorption coefficients along a- axis and c- axis, and also averaged on the absorption of the emitting spectrum of LD. The α eff is always lower than the peak absorption coefficient because the effective absorption coefficient is a parameter averaged over the absorption spectrum width of the diode laser, and the effective pump wavelength is defined by λ eff = ∫ ρ(λ′)(λ′)dλ/ρ(λ′)dλ′, and ρ(λ′) is the power spectral density of the pump source. The effective pump wavelength was measured with an optical spectrum analyzer and an integrating-sphere photometer. The constant C 0 was determined by Eq. (37)
C0=P0z=01πωp2(z)exp[2(x2+y2)NωpN(z)]exp(αeffz)dxdy,
(37)
where P 0 is the pump power of each end. Suppose the waist radius of the pump mode is ωp 0 and the waist is located at a distance from the bonded surface of the Nd:YVO4 crystal of z 0, i.e., ωp 0 = ωp(z 0). The radius of pump mode at location z can be represented as
ωp(z)=ωp01+[θp(zz0)ωp0]2,
(38)
In Eq. (38) θp is the the far-field divergence (half angle) of the pump mode in the crystal.

For the symmetric dual-end pumped geometry, the pump intensity distribution in the active medium can be represented as Eq. (39),
Ip(x,y,z)=IP+(x,y,z)+IP(x,y,z),
(39)
where IP(x,y,z) is pump intensity in the crystal when only being pumped from right with a similar expression as IP+(x,y,z). The absorbed pump distribution per unit volume (i.e., the absorbed pump density) in the active medium can be represented as following,
Pabs(x,y,z)=Ip(x,y,z)zαeffIP(x,y,z).
(40)

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,
T(x,y,z)x|x=±w2=hcKx[TcT(±w2,y,z)]T(x,y,z)y|y=±h2=hcKy[TcT(x,±h2,z)]T(x,y,z)z|z=±l2=haKz[TrT(x,y,±l2)],
(41)
where Ky = Kz = 5.23 W/mK and Kx = 5.10 W/mK are the anisotropic thermal conductivities of the a-cut crystal; hc is the forced convection heat transfer coefficient on the cooling face, which is 10000 W/m2K for the water-cooled cooper heat sink; ha 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., ha = 0; Tr = 20°C is the room temperature; and Tc = 10°C is the coolant temperature of the heat sink.

The value of the fraction of absorbed pump light converted to heat, ηh 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

24. T. Y. Fan, “Heat generation in Nd:YAG and Yb:YAG,” IEEE J. Quantum Electron. 29, 1457–1459 (1993). [CrossRef]

], fluorimetric quenching induced by impurity and defect [25

25. H. G. Danielmeyer, M. Blatte, and P. Balmer, “Fluorescence quenching in Nd:YAG,” Appl. Phys. A 1, 269–274 (1973).

], and the Auger recombination [26

26. J. L. Blows, T. Omatsu, J. Dawes, H. Pask, and M. Tateda, “Heat generation in Nd:YVO4 with and without laser action,” IEEE Photon. Technol. Lett. 10, 1727–1729 (1998). [CrossRef]

]. The heat conversion fraction caused by the quantum defect is ηh = 1 − λp/λl ≈ 0.24 when pumped at 808 nm and lasing at 1064 nm. Additional heat load produced by above nonlinear processes, rather than the quantum defect of the pumping, aroused more serious heat generation, especially for high power intensity pumping. For the neodymium doped vanadate crystals and glass pumped at 808 nm (4 I 9/24 F 5/2) and lasing at 1064 nm (4 F 3/24 I 11/2), many researchers have taken the total heat conversion fraction around 0.3 [24

24. T. Y. Fan, “Heat generation in Nd:YAG and Yb:YAG,” IEEE J. Quantum Electron. 29, 1457–1459 (1993). [CrossRef]

, 27

27. P. Xiaoyuan, X. Lei, and A. Asundi, “Power scaling of diode-pumped Nd:YVO4 lasers,” IEEE J. Quantum Electron. 38, 1291–1299 (2002). [CrossRef]

, 28

28. B. Comaskey, B. D. Moran, G. F. Albrecht, and R. J. Beach, “Characterization of the heat loading of Nd-doped YAG, YOS, YLF, and GGG excited at diode pumping wavelengths,” IEEE J. Quantum Electron. 31, 1261–1264 (1995). [CrossRef]

]. Therefore, the compromise and relative reasonable value of ηh = 0.28 is used as the total heat conversion fraction in our model for the high doping concentration crystal.

The energy budget method is used to establish the difference equation from the heat diffusion Eq. (15) with the boundary condition (41). The physical essence of the energy budget method is the energy conservation. The crystal is divided into many small bulk units. The heat dissipating capacity from the six side faces of each bulk unit equals to the quantity of heat production in the bulk unit, and then the difference equation can be established. Using the finite difference method, the temperature distribution in the gain medium is calculated while the Gauss–Seidel iteration method is employed to accelerate the convergence. Figure 6 shows the 3D temperature profile of the 1/8 crystal volume when the gain medium is pumped at 45 W pump power with the effective absorption coefficient α eff = 2.0 cm−1, and the value of z 0 and ωp 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 ωp 0 = 0.4mm located at z 0 = 2mm as shown in the coordinate of Fig. 5.

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

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 ΔnT can be calculated easily using Eq. (14).

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

The small signal gain coefficient can be calculated from Eq. (19) and Eqs. (36)(40). Figure 8 shows the 3D small signal gain coefficient profile with α eff =2.0 cm−1. The gain saturation effect must be considered in high power four level lasers well above threshold. As the signal intensity increases and the pump rate is constant, the inversion level will reduce and thereby the gain is reduced. Figure 9 shows the total gain profile in one pass through the gain medium, Γ(x,y)=0lG(x,y,z)dz, with the small signal gain coefficient and with the saturated gain coefficient where the initial signal intensity I0(x,y)=Isatexp[2(x2+y2)/ωp02] and I0(x,y)=4Isatexp[2(x2+y2)/ωp02]. It shows that the gain is weaken seriously when the gain saturation works. Especially, the effective gain is depressed near the optical axis where the gain is weakened more seriously by the gain saturation.

Fig. 8 3D small signal gain profile of the gain medium pumped at P 0 = 45 W with ωp 0 = 0.4mm, z 0 = 2mm, and α eff =2.0 cm−1.
Fig. 9 small signal gain (solid line) and the saturated gain with I0=Isatexp[2(x2+y2)/ωp02] (dash-dot line) and I0=4Isatexp[2(x2+y2)/ωp02] (dash line) for one pass through the gain medium, P 0 = 45 W, z 0 = 2mm, ωp 0 = 0.4mm, and α eff =2.0 cm−1.

4.2. Transverse mode with the combined guiding

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

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

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

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

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

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

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

In the traditional geometric design criterion of the TEM00 mode resonator, the mode size of TEM00 and TEM01 and the equivalent pump radius ω equ are only considered, since higher-order transverse modes always can not oscillate because of their larger geometric- and thermally induced- diffraction loss and poorer gain. The equivalent pump radius ω equ of the pump beam in the gain medium can be defined as the weighted average of ω eff(z),
ωequ=Pabs(x,y,z)ωeff(z)dxdydzPabs(x,y,z)dxdydz,
(42)
where, the ω eff(z) denotes the effective radius of the pump mode at position z which can be defined as
x2+y2ωeff2(z)Pabs(x,y,z)dxdyPabs(x,y,z)dxdy=11e20.865.
(43)

To achieve TEM00 mode operating, the equivalent pump radius ω equ should be larger than the mode radius of the TEM00 mode in the gain medium, but be smaller than the mode radius of the TEM01 mode. In this case, the TEM00 mode can get higher gain than TEM00 mode and oscillates. In addition, the phase aberration always occurs in the wing of the pump region where thermal-induced refractive index and temperature-gradient-induced thermal stress are higher. This thermal phase aberration leads to higher diffractive loss for TEM01 and inhibits its oscillation. For the fiber-coupled end-pumped laser, the pump mode is circularly symmetrical. Therefore, the mode size can be described as Laguerre-Gaussian mode TEM0 n approximately. The v mode radius ratio between TEM00 mode and TEMpl mode is ωTEM00/ωTEMpl=1/p+2l+1. Therefore, we have ω TEM01 ≈ 1.73ω TEM00. For the geometric design criterion of the TEM00 mode resonator, the equivalent pump radius must satisfy the following criterion,
1<ωequωTEM00<1.73.
(44)

For the design theory of high power TEM00 mode laser resonator, the size ratio of ωL/ω equ was always designed between 0.7 − 0.8 to afford enough gain and prevent thermal aberration for the TEM00 mode [29

29. P. Laporta and M. Brussard, “Design criteria for mode size optimization in diode-pumped solid-state lasers,” IEEE J. Quantum Electron. 27, 2319–2326 (1991). [CrossRef]

31

31. W. A. Clarkson, “Thermal effects and their mitigation in end-pumped solid-state lasers,” J. Phys. D 34, 2381–2395 (2001). [CrossRef]

], suppressing the oscillation of the high order mode, especially the TEM01 mode. The region of (0.7 − 0.8)ω equ can be called as the geometrical TEM00 mode region.

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

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

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

5. Summary and conclusions

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

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

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

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

References and links

1.

A. Fox and Li Tingye, “Computation of optical resonator modes by the method of resonance excitation,” IEEE J. Quantum Electron . 4, 460–465 (1968). [CrossRef]

2.

B. N. Perry, P. Rabinowitz, and M. Newstein, “Exact solution of the scalar wave equation with focused gaussian gain,” Phys. Rev. Lett. 49, 1921–1924 (1982). [CrossRef]

3.

B. N. Perry, P. Rabinowitz, and M. Newstein, “Wave propagation in media with focused gain,” Phys. Rev. A 27, 1989–2002 (1983). [CrossRef]

4.

C. A. Schrama, D. Bouwmeester, G. Nienhuis, and J. P. Woerdman, “Mode dynamics in optical cavities,” Phys. Rev. A 51, 641–645 (1995) [CrossRef] [PubMed]

5.

C. F. Maes and E. M. Wright, “Mode properties of an external-cavity laser with Gaussian gain,” Opt. Lett. 29, 229–231 (2004). [CrossRef] [PubMed]

6.

T. Y. Fan, “Aperture guiding in quasi-three-level lasers,” Opt. Lett. 19, 554–556 (1994). [CrossRef] [PubMed]

7.

J. J. Zayhowski, “Thermal Guiding in Microchip Lasers,” in Advanced Solid State Lasers, G. Dube, ed., Vol. 6 of OSA Proceedings Series (Optical Society of America, 1990), paper DPL3.

8.

G. K. Harkness and W. J. Firth, “Transverse modes of microchip solid state lasers,” J. Mod. Opt. 39, 2023–2037 (1992). [CrossRef]

9.

J. K. Jabczynski, J. Kwiatkowski, and W. Zendzian, “Gain and thermal guiding effects in diode-pumped lasers,” SPIE 5120, 164(2003) [CrossRef]

10.

F. Salin and J. Squier, “Gain guiding in solid-state lasers,” Opt. Lett. 17, 1352–1354 (1992). [CrossRef] [PubMed]

11.

O. Denchev, S. Kurtev, and P. Petrov, “Experimental investigation of saturable gain-guided modes,” Appl. Opt. 41, 1677–1684 (2002). [CrossRef] [PubMed]

12.

N. J. Druten, S. S. R. Oemrawsingh, Y. Lien, C. Serrat, M. P. van Exter, and J. P. Woerdman, “Observation of transverse modes in a microchip laser with combined gain and index guiding,” J. Opt. Soc. Am. B 18, 1793–1804 (2001). [CrossRef]

13.

A. J. Kemp, R. S. Conroy, G. J. Friel, and B. D. Sinclair, “Guiding effects in Nd:YVO4 microchip lasers operating well above threshold,” IEEE J. Quantum Electron. 35, 675–681 (1999). [CrossRef]

14.

S. Longhi, G. Cerullo, S. Taccheo, V. Magni, and P. Laporta, “Experimental observation of transverse effects in microchip solid-state lasers,” Appl. Phys. Lett. 65, 3042–3044 (1994). [CrossRef]

15.

C. Serrat, M. P. Exter, N. J. Druten, and J. P. Woerdman, “Transverse mode formation in microlasers by combined gain- and index-guiding,” IEEE J. Quantum Electron. 35, 1314–1321 (1999). [CrossRef]

16.

H. Kogelnik and T. Li, “Laser beams and resonators,” Appl. Opt. 5, 1550–1567 (1966). [CrossRef] [PubMed]

17.

A. E. Siegman, Lasers, (Univ. Sci. Books, 1986), pp. 323.

18.

J. Frauchiger, P. Albers, and H. P. Weber, “Modeling of thermal lensing and higher order ring mode oscillation in end-pumped C-W Nd:YAG lasers,” IEEE J. Quantum Electron. 28, 1046–1056 (1992). [CrossRef]

19.

G. M. Muslu and H. A. Erbay, “Higher-order split-step Fourier schemes for the generalized nonlinear Schrodinger equation,” Math. Comput. Simulat. 67, 581–595 (2005). [CrossRef]

20.

T. R. Taha and X. Xiangming, “Parallel split-step Fourier methods for the coupled nonlinear Schrodinger type equations,” J. Supercomput. 32, 5–23 (2005). [CrossRef]

21.

Y. L. Bogomolov and A. D. Yunakovsky, “Split-step Fourier method for nonlinear Schrodinger equation,” in International Conference Days on Diffraction 2006 , Proceedings of the International Conference ’Days on Diffraction’ 2006, DD (Inst. of Elec. and Elec. Eng. Computer Society, 2006), 34–42. [CrossRef]

22.

W. Koechner, Solid-tate Laser Engineering, 6th ed. (Springer-Verlag Publications, 2006).

23.

D. G. Matthews, J. R. Boon, R. S. Conroy, and B. D. Sinclair, “Comparative study of diode pumped microchip laser materials: Nd-doped YVO4, YOS, SFAP and SVAP,” J. Mod. Opt. 43, 1079–1087 (1996). [CrossRef]

24.

T. Y. Fan, “Heat generation in Nd:YAG and Yb:YAG,” IEEE J. Quantum Electron. 29, 1457–1459 (1993). [CrossRef]

25.

H. G. Danielmeyer, M. Blatte, and P. Balmer, “Fluorescence quenching in Nd:YAG,” Appl. Phys. A 1, 269–274 (1973).

26.

J. L. Blows, T. Omatsu, J. Dawes, H. Pask, and M. Tateda, “Heat generation in Nd:YVO4 with and without laser action,” IEEE Photon. Technol. Lett. 10, 1727–1729 (1998). [CrossRef]

27.

P. Xiaoyuan, X. Lei, and A. Asundi, “Power scaling of diode-pumped Nd:YVO4 lasers,” IEEE J. Quantum Electron. 38, 1291–1299 (2002). [CrossRef]

28.

B. Comaskey, B. D. Moran, G. F. Albrecht, and R. J. Beach, “Characterization of the heat loading of Nd-doped YAG, YOS, YLF, and GGG excited at diode pumping wavelengths,” IEEE J. Quantum Electron. 31, 1261–1264 (1995). [CrossRef]

29.

P. Laporta and M. Brussard, “Design criteria for mode size optimization in diode-pumped solid-state lasers,” IEEE J. Quantum Electron. 27, 2319–2326 (1991). [CrossRef]

30.

Y. F. Chen, T. S. Liao, C. F. Kao, T. M. Huang, K. H. Lin, and S. C. Wang, “Optimization of fiber-coupled laser-diode end-pumped lasers: influence of pump-beam quality,” IEEE J. Quantum Electron. 322010–2016 (1996). [CrossRef]

31.

W. A. Clarkson, “Thermal effects and their mitigation in end-pumped solid-state lasers,” J. Phys. D 34, 2381–2395 (2001). [CrossRef]

OCIS Codes
(140.3410) Lasers and laser optics : Laser resonators
(140.3430) Lasers and laser optics : Laser theory
(140.3480) Lasers and laser optics : Lasers, diode-pumped

ToC Category:
Lasers and Laser Optics

History
Original Manuscript: January 5, 2011
Revised Manuscript: February 24, 2011
Manuscript Accepted: March 14, 2011
Published: March 25, 2011

Citation
Xingpeng Yan, Qiang Liu, Dongsheng Wang, and Mali Gong, "Combined guiding effect in the end-pumped laser resonator," Opt. Express 19, 6883-6902 (2011)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-7-6883


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References

  1. A. Fox and Li Tingye, “Computation of optical resonator modes by the method of resonance excitation,” IEEE J. Quantum Electron . 4, 460–465 (1968). [CrossRef]
  2. B. N. Perry, P. Rabinowitz, and M. Newstein, “Exact solution of the scalar wave equation with focused gaussian gain,” Phys. Rev. Lett. 49, 1921–1924 (1982). [CrossRef]
  3. B. N. Perry, P. Rabinowitz, and M. Newstein, “Wave propagation in media with focused gain,” Phys. Rev. A 27, 1989–2002 (1983). [CrossRef]
  4. C. A. Schrama, D. Bouwmeester, G. Nienhuis, and J. P. Woerdman, “Mode dynamics in optical cavities,” Phys. Rev. A 51, 641–645 (1995) [CrossRef] [PubMed]
  5. C. F. Maes and E. M. Wright, “Mode properties of an external-cavity laser with Gaussian gain,” Opt. Lett. 29, 229–231 (2004). [CrossRef] [PubMed]
  6. T. Y. Fan, “Aperture guiding in quasi-three-level lasers,” Opt. Lett. 19, 554–556 (1994). [CrossRef] [PubMed]
  7. J. J. Zayhowski, “Thermal Guiding in Microchip Lasers,” in Advanced Solid State Lasers , G. Dube, ed., Vol. 6 of OSA Proceedings Series (Optical Society of America, 1990), paper DPL3.
  8. G. K. Harkness and W. J. Firth, “Transverse modes of microchip solid state lasers,” J. Mod. Opt. 39, 2023–2037 (1992). [CrossRef]
  9. J. K. Jabczynski, J. Kwiatkowski, and W. Zendzian, “Gain and thermal guiding effects in diode-pumped lasers,” SPIE 5120, 164(2003) [CrossRef]
  10. F. Salin and J. Squier, “Gain guiding in solid-state lasers,” Opt. Lett. 17, 1352–1354 (1992). [CrossRef] [PubMed]
  11. O. Denchev, S. Kurtev, and P. Petrov, “Experimental investigation of saturable gain-guided modes,” Appl. Opt. 41, 1677–1684 (2002). [CrossRef] [PubMed]
  12. N. J. Druten, S. S. R. Oemrawsingh, Y. Lien, C. Serrat, M. P. van Exter, and J. P. Woerdman, “Observation of transverse modes in a microchip laser with combined gain and index guiding,” J. Opt. Soc. Am. B 18, 1793–1804 (2001). [CrossRef]
  13. A. J. Kemp, R. S. Conroy, G. J. Friel, and B. D. Sinclair, “Guiding effects in Nd:YVO4 microchip lasers operating well above threshold,” IEEE J. Quantum Electron. 35, 675–681 (1999). [CrossRef]
  14. S. Longhi, G. Cerullo, S. Taccheo, V. Magni, and P. Laporta, “Experimental observation of transverse effects in microchip solid-state lasers,” Appl. Phys. Lett. 65, 3042–3044 (1994). [CrossRef]
  15. C. Serrat, M. P. Exter, N. J. Druten, and J. P. Woerdman, “Transverse mode formation in microlasers by combined gain- and index-guiding,” IEEE J. Quantum Electron. 35, 1314–1321 (1999). [CrossRef]
  16. H. Kogelnik and T. Li, “Laser beams and resonators,” Appl. Opt. 5, 1550–1567 (1966). [CrossRef] [PubMed]
  17. A. E. Siegman, Lasers , (Univ. Sci. Books, 1986), pp. 323.
  18. J. Frauchiger, P. Albers, and H. P. Weber, “Modeling of thermal lensing and higher order ring mode oscillation in end-pumped C-W Nd:YAG lasers,” IEEE J. Quantum Electron. 28, 1046–1056 (1992). [CrossRef]
  19. G. M. Muslu and H. A. Erbay, “Higher-order split-step Fourier schemes for the generalized nonlinear Schrodinger equation,” Math. Comput. Simulat. 67, 581–595 (2005). [CrossRef]
  20. T. R. Taha and X. Xiangming, “Parallel split-step Fourier methods for the coupled nonlinear Schrodinger type equations,” J. Supercomput. 32, 5–23 (2005). [CrossRef]
  21. Y. L. Bogomolov and A. D. Yunakovsky, “Split-step Fourier method for nonlinear Schrodinger equation,” in International Conference Days on Diffraction 2006 , Proceedings of the International Conference ’Days on Diffraction’ 2006, DD (Inst. of Elec. and Elec. Eng. Computer Society, 2006), 34–42. [CrossRef]
  22. W. Koechner, Solid-tate Laser Engineering , 6th ed. (Springer-Verlag Publications, 2006).
  23. D. G. Matthews, J. R. Boon, R. S. Conroy, and B. D. Sinclair, “Comparative study of diode pumped microchip laser materials: Nd-doped YVO4, YOS, SFAP and SVAP,” J. Mod. Opt. 43, 1079–1087 (1996). [CrossRef]
  24. T. Y. Fan, “Heat generation in Nd:YAG and Yb:YAG,” IEEE J. Quantum Electron. 29, 1457–1459 (1993). [CrossRef]
  25. H. G. Danielmeyer, M. Blatte, and P. Balmer, “Fluorescence quenching in Nd:YAG,” Appl. Phys. A 1, 269–274 (1973).
  26. J. L. Blows, T. Omatsu, J. Dawes, H. Pask, and M. Tateda, “Heat generation in Nd:YVO4 with and without laser action,” IEEE Photon. Technol. Lett. 10, 1727–1729 (1998). [CrossRef]
  27. P. Xiaoyuan, X. Lei, and A. Asundi, “Power scaling of diode-pumped Nd:YVO4 lasers,” IEEE J. Quantum Electron. 38, 1291–1299 (2002). [CrossRef]
  28. B. Comaskey, B. D. Moran, G. F. Albrecht, and R. J. Beach, “Characterization of the heat loading of Nd-doped YAG, YOS, YLF, and GGG excited at diode pumping wavelengths,” IEEE J. Quantum Electron. 31, 1261–1264 (1995). [CrossRef]
  29. P. Laporta and M. Brussard, “Design criteria for mode size optimization in diode-pumped solid-state lasers,” IEEE J. Quantum Electron. 27, 2319–2326 (1991). [CrossRef]
  30. Y. F. Chen, T. S. Liao, C. F. Kao, T. M. Huang, K. H. Lin, and S. C. Wang, “Optimization of fiber-coupled laser-diode end-pumped lasers: influence of pump-beam quality,” IEEE J. Quantum Electron. 322010–2016 (1996). [CrossRef]
  31. W. A. Clarkson, “Thermal effects and their mitigation in end-pumped solid-state lasers,” J. Phys. D 34, 2381–2395 (2001). [CrossRef]

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