Appl. Phys. B 72, 167–170 (2001) / Digital Object Identifier (DOI) 10.1007/s003400000433
Applied Physics B
Lasers and Optics
Generation of Laguerre–Gaussian modes in fiber-coupled laser diode
end-pumped lasers
Y.F. Chen1,∗, Y.P. Lan2, S.C. Wang2
1Department of Electrophysics, National Chiao Tung University, Hsinchu, Taiwan, Republic of China 2Institute of Electro-Optical Engineering, National Chiao Tung University, Hsinchu, Taiwan, Republic of China
Received: 8 June 2000/Revised version: 2 July 2000/Published online: 20 September 2000 –Springer-Verlag 2000
Abstract. A technique has been developed for the generation
of Laguerre–Gaussian (LG0,l) modes in a fiber-coupled laser diode end-pumped microchip laser. A theoretical model is also developed to predict how the oscillation of LG0,l modes is affected by the pump position, the pump size and the cavity mode size. With a 1-W fiber-coupled diode, the highest-order LG0,lmode that can be generated is l= 23.
PACS: 42.55; 42.60
The high-order Laguerre–Gaussian (LGp,l) mode exhibits in-teresting physics and has the potential for technological ap-plications [1–4], where p and l are the radial and azimuthal indices of the LG mode. A pure high-order LG0,l-mode has been reported in electrically pumped [5, 6] and optically pumped [7] vertical-cavity surface-emitting semiconductor lasers (VCSELs). However, the main difficulty of the emis-sion of high-order LG modes in VCSELs is that the processed wafer is in need of extraordinary homogeneity.
Diode-pumped solid-state lasers offer the advantages of high efficiency, compactness and reliability, especially in an end-pumping configuration [8]. Fiber delivery of the pump power enables us to keep the laser resonator apart from the pump source, so that the laser resonator can be isolated from disturbances of the pump source [9]. Recently, we have gener-ated high-order Hermite–Gaussian modes by a fiber-coupled diode end-pumped Nd:YAG laser [10, 11]. In this letter, we report a technique for the generation of the pure LG modes with p= 0 and specified values of l in a fiber-coupled diode end-pumped microchip laser. The key novelty is to produce a doughnut-shaped pump profile by defocusing a standard fiber-coupled diode. We also develop a theoretical model to analyze how the focal position of the pump beam in the laser crystal influences the oscillation of LG0,lmodes. Experimen-tal results have shown a fairly good agreement with theoret-ical predictions.
∗Corresponding author.
(Fax: +866-35/729-134, E-mail: [email protected])
For a multi-mode fiber-coupled diode laser beam pass-ing through a focuspass-ing lens, the profile at the focal plane is like a top-hat distribution; however, away from the fo-cal plane it is like a doughnut-shaped distribution, as shown in Fig. 1a. With this property, we can defocus a standard fiber-coupled diode to result in a good overlap with the high-order LG0,l mode and generate it purely. Figure 1b shows the schematic of a fiber-coupled laser diode end-pumped Nd:YVO4 laser considered in this work. In our experiments, we have used a plano-concave cavity that consists of one planar Nd:YVO4 surface, high-reflection coated at 1064 nm and high-transmission coated at 809 nm for the pump light to enter the laser crystal, and a spherical output mirror. The second surface of the Nd:YVO4 crystal (1-mm length) is anti-reflection-coated at 1064 nm. A mirror with a reflectance
Fig. 1. a A typical beam profile of a fiber-coupled laser diode away from the
focal plane. b Schematic of a fiber-coupled diode end-pumped solid-state laser
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of R= 98.5% and a radius of curvature of 25 cm was used in the resonator to couple the output power. For a 1-cm resonator length, the waist of the fundamental mode was around 0.24 mm. The fiber-coupled laser diode (Coherent, F-81-800C-100) has a 0.1-mm core diameter and was focused into the Nd:YVO4crystal by using a focusing lens with 0.57 magnification.
Figure 2 shows a sketch of the position dependence of the beam profile for a multi-mode fiber-coupled laser-diode beam passing through a focusing lens. According to the property of the beam profile, the normalized pumping distribution in the laser crystal is approximated as
Rp(r, z) = 1 πω2 p αe−αz 1− e−αLΘ r− θp|z − z0| × Θ(θp(z − z0))2+ ω2p−r , (1)
whereωp is the radius at the waist,α is the absorption coef-ficient at the pump wavelength, L is the length of the laser crystal,θp is the far-field half-angle, the point z= 0 is taken to be at the incident surface of the gain medium, z0is the fo-cal position of the pump beam in the laser crystal andΘ() is the Heaviside step function. The functional form of (1) comes from the fact that the pump profile is like a top-hat distribu-tion at the focal plane and like a ring-type distribudistribu-tion away from the focal plane.
The expected preferred laser modes in the ring shape of the pump beam are the LG0,l modes, since these show opti-mum overlap with the ring pump. For an end-pumped solid-state laser, the threshold pump power for a single LG0,lmode oscillation is given by [10] Pth 0,l=γIsat ηpL 1 s0,l(r, φ, z)Rp(r, z)dV , (2) whereγ is the total logarithmic loss per pass, Isatis the sat-uration intensity, L is the length of the active medium, and
ηp= ηtηa(vl/vp) where ηt is the optical transfer efficiency (ratio between optical power incident on the active medium and that emitted by the pump source),ηais the absorption ef-ficiency (ratio between power absorbed in the active medium and that entering the rod) andvp andvl are pump and laser
frequencies, respectively. s0,l(r, φ, z) is the normalized cav-ity mode intenscav-ity distribution. Equation (2) indicates that the LG0,l mode with the biggest overlap with the gain structure has the minimum threshold and will dominate in the laser output.
r
z
z = z
θp
Fig. 2. A sketch of the position dependence of the beam profile for a
multi-mode fiber-coupled laser-diode beam passing through a focusing lens
Considering a single LG0,lmode, s0,l(r, φ, z) is given by s0,l(r, φ, z) = 4 (1 + δ0,l)l! 1 πω2 0L ×(cos2lφ) 2r2 ω2 0 l exp −2r2 ω2 0 , (3)
where the z-dependent variation in s0,l(r, φ, z) is neglected
and the spot radius of the laser beamω0is approximated to be constant along the laser axis in the laser crystal.
Substituting (1) and (3) into (2) and carrying out the inte-grations over r andφ, we obtain
Pth 0,l(z0) =πγIsat ηp 1− e−αL α ω2 p L 0 Ql(z0) exp(−αz)dz , (4) where Ql(z0) = l j=0 1 (l − j)! × l− jexp(−) − +2ω 2 p ω2 0 l− j exp − +2ω 2 p ω2 0 , (5) and = 2 θp(z − z 0) ω0 2 . (6)
With the parameters of the present cavity: ω0= 0.24 mm,
ωp= 0.028 mm, θp= 0.34 rad, γ = 0.016, ηp= 0.8, L =
1 mm, Isat= 13.87 W/mm2 andα = 3 mm−1, the threshold power for different LG0,lmodes was calculated as a function of the focal position of the pump beam in the laser crystal. Figure 3 shows the dependence of the threshold power on the focal position of the pump beam for the first six LG0,l
modes. The results show that the LG0,0 mode has the min-imum threshold power for the focal position in the range between z0= −0.3 and z0= 1.0. Farther away from the su-perior range of the LG0,0mode, the higher-order LG0,lmode has the minimum threshold power. This result indicates that adjusting the focal position of the pump beam can lead to the generation of the LG0,l modes with specified values of
l. Since the transverse mode with the minimum threshold
power can break into oscillation at first, the dependence of the threshold power on the focal position is given by Pth min(z0) = min{Pth 0,l(z0)}.
Experimental results for the threshold power versus the focal position of the pump beam z0 are shown in Fig. 4. It can be seen that experimental data agree very well with the calculations of the theoretical analysis. Figure 5 shows the beam profiles with different LG0,l-mode distributions, meas-ured with the CCD camera (Coherent, Beam-Code), in the fourteen positions. The relationship between the dominant LG0,l mode and the focal position of the pump beam is also consistent with the theoretical predictions as shown in Fig. 4. In our experiment, the highest-order LG0,l mode that can be obtained with a 1-W fiber-coupled diode is l= 23, as shown
169 zo (mm) -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.5 Pth 0, l (zo ) (mW) 101 102 103 104 105 106 l = 0 1 2 3 4 5
Fig. 3. The dependence of the threshold power on the focal position of the
pump beam for the first six LG0,l modes
zo (mm) -2 -1 0 1 2 3 P th m in (z o ) (mW) 0 200 400 600 800 1000 1200
Fig. 4. A plot of experimental results (symbols) and theoretical results (solid line) of the threshold power versus the focal position of the pump beam z0
in Fig. 6. It is worth mentioning that the present analysis does not take saturation effects into account. Saturation of gain by the first mode to reach threshold can raise the threshold of the other modes by a factor of about 1.2 ∼ 2. Experimental results show that the times-above-threshold factor before ad-ditional modes to oscillate depends on the mode number, i.e. the diameter of the pumping doughnut. A theoretical model for the saturation effect is currently under way.
We have generated the pure LG0,l modes in an end-pumped microchip laser by defocusing a standard fiber-coupled diode to produce a doughnut-shaped pump profile. A theoretical analysis has also been provided to calculate the threshold pump power of LG0,l modes as a function of the
Zo=0.5 (mm) Zo=1.10 Zo=1.23 Zo=1.39
Zo=1.49 Zo=1.62 Zo=1.73 Zo=1.83
Zo=1.92 Zo=2.01 Zo=2.10
Zo=2.17 Zo=2.25 Zo=2.32
Fig. 5. Beam profiles with different LG0,l-mode distributions, measured with the CCD camera, in the fourteen positions
Fig. 6. The beam profile of the highest-order LG0,lmode with l= 23
laser-diode beam quality, the focal position of the pump beam and the cavity parameters. Experimental results have shown a fairly good agreement with the predictions of the present analysis.
References
1. S.L. McCall, A.F.J. Levi, R.E. Slusher, S.J. Pearton, R.A. Logan: Appl. Phys. Lett. 60, 289 (1992)
2. D.Y. Chu, M.K. Chin, W.G. Bi, H.Q. Hou, C.W. Tu, S.T. Ho: Appl. Phys. Lett. 65, 3167 (1994)
3. M.J. Ries, E.I. Chen, N. Holonyak Jr.: Appl. Phys. Lett. 68, 2035 (1996)
170
4. L. Dijaloshinski, M. Orenstein: Opt. Lett. 23, 364 (1998) 5. C. Degen, I. Fischer, W. Elsäßer: Opt. Exp. 5, 38 (1999) 6. Q. Deng, H. Deng, D.G. Deppe: Opt. Lett. 22, 463 (1997)
7. S.F. Pereira, M.B. Willemsen, M.P. van Exter, J.P. Woerdman: Appl. Phys. Lett. 73, 2239 (1998)
8. J. Berger, D.F. Welch, D.R. Scifres, W. Streifer, P. Cross: Appl. Phys. Lett. 51, 1212 (1987); D.L. Sipes: Appl. Phys. Lett. 47, 74 (1985)
9. H. Hemmati, J.R. Lesh: Opt. Lett. 19, 1322 (1994); Y. Kaneda, M. Oka, H. Masuda, S. Kubota: Opt. Lett. 17, 1003 (1992); J. Berger, D.F. Welch, W. Streifer, D.R. Scifres, N.J. Hoffman, J.J. Smith, D. Radecki: Opt. Lett. 13, 306 (1988)
10. Y.F. Chen, T.M. Huang, C.F. Kao, C.L. Wang, S.C. Wang: IEEE J. Quantum Electron. QE-33, 1025 (1997)
