Appl. Phys. B 73, 11–14 (2001) / Digital Object Identifier (DOI) 10.1007/s003400100566
Applied Physics B
Lasers and Optics
Dynamics of helical-wave emission in a fiber-coupled diode
end-pumped solid-state laser
Y.F. Chen1,∗, Y.P. Lan2
1Department of Electrophysics, National Chiao Tung University, 1001 TA Hsueh Road, Hsinchu 30050, Taiwan, Republic of China 2Institute of Electro-Optical Engineering, National Chiao Tung University Hsinchu, Taiwan, Republic of China
Received: 14 November 2000/Revised version: 22 January 2001/Published online: 23 May 2001 – Springer-Verlag 2001
Abstract. We have generated TEM∗0,l modes in an end-pumped microchip laser using a standard fiber-coupled diode. A rich set of dynamic behaviors, such as periodic and quasi-periodic self-modulation, chaotic pulsing and frequency
lock-ing was observed in the generated TEM∗0,lmodes.
Experimen-tal results confirm the theoretical predictions that the locking occurs as a subcritical bifurcation and that a region of coexist-ing locked and unlocked states exists.
PACS: 42.55.Rz; 42.60.Mi; 42.65.Sf
Optical vortices in lasers have been observed in low-order transverse modes containing one or several phase singular-ities [1–4]. The simplest examples are Laguerre-Gaussian
fields TEM∗p,l with p= 0, the doughnut modes of charge l,
which consist of l helical wave fronts winding about each other, around the cavity axis. Here p and l are the radial and azimuthal indices of the LG mode. These have circularly symmetric doughnut-shaped intensity distributions. In recent
years the mechanical and optical effects of TEM∗0,l modes
have attracted a great deal of interest because they possess well-defined angular momentum along the optical axis when
l is not zero [5]. These modes are also important in laser
cooling and trapping experiments [6]. Therefore, study of the
generation of LG TEM∗0,l modes in solid-state lasers is of
great interest.
Recently, we reported a technique for the generation of
cylindrically symmetric LG modes with p= 0, and specified
values of l in a fiber-coupled diode end-pumped solid-state laser [7]. The key novelty was to produce a doughnut-shaped pump-profile by defocusing a standard fiber-coupled diode. Experimental results demonstrate that the stable
transverse-mode pattern near the pump threshold is usually a LG TEM0,l
mode with the distribution cos2lθ (or sin2lθ) in the azimuthal
angle, having 2l nodes in azimuth. Even though the geometry is of cylindrical symmetry, there is still a certain astigmatism in the cavity due to the thermal lensing effect and anisotropic properties of the gain medium. This is the reason why sine
∗Corresponding author.
(Fax: +886-35/729-134, E-mail: [email protected])
or cosine LG modes were generated near the pump threshold,
instead of doughnut modes. A similar high-order LG TEM0,l
mode has been reported in electrically pumped [8] and op-tically pumped [9] vertical-cavity surface-emitting semicon-ductor lasers (VCSELs). However, the main difficulty associ-ated with the emission of high-order LG modes in VCSELs is that the processed wafer needs to be of extraordinary homo-geneity.
Slightly above the pump threshold, a LG TEM∗0,l or
TEM∗0,−lmode, having a circle of constant intensity in the
ra-dial direction, can be generated by the superposition of two
like TEM0,lmodes, which have a fixed relative phase equal to
π/2. Astigmatism-induced splitting of the like mode
frequen-cies has a significant influence on laser dynamics. Temporal instabilities and chaotic emission caused by the non-linear interaction of transverse modes in a class-A laser have been reported by Tamm [10], who experimentally confirmed the existence of a “cooperative frequency locking” state [11]
for the nearly degenerate TEM0,1 and TEM1,0 modes of
a helium-neon laser. However, the dynamic characteristics of a solid-state laser are those of an oscillator with an inertial (non-instantaneous) non-linearity. In the case of such oscilla-tors, the perturbations exhibit oscillatory relaxation. Because of relaxation oscillations, the two-mode locking in a class-B laser occurs as a subcritical bifurcation [12, 13], unlike that in class-A lasers where the locking is a supercritical bifurca-tion. Recently, Zehnl´e [14] analytically studied the dynamic behavior of class A and B lasers operating in two transverse modes. His analysis provides evidence of the qualitative dif-ferences in the stationary as well as periodic behaviors for class A and B bimodal lasers. Even so, to date not much has been done to observe these differences.
In this work, we perform an experimental investigation of the relaxation oscillations in a solid-state laser with helical wave emission. Experimental results show that the relaxation oscillations play an important role not only in locking pro-cesses, but also under stationary state conditions. A rich set of dynamic behaviors, such as periodic and quasi-periodic self-modulation, chaotic pulsing, and frequency locking was
ob-served experimentally in the generated TEM∗0,l hybrid mode.
It was found that the experimental data exhibited a generally satisfactory agreement with the theoretical predictions.
12
1 Results and discussion
A non-linear system of the Maxwell–Bloch equations, given in terms of partial derivatives, is usually reduced to a sys-tem of ordinary differential equations by expanding in terms of the empty-cavity modes for the laser system with a small number of excited transverse modes [12–16]. The dynamics of generating two transverse modes in a class-B laser was investigated theoretically [12, 13]. The main results of the cal-culations can be summarized as follows.
When the frequency difference between two like TEM0,l
modes,∆Ω is greater than the relaxation frequency ωr, the
total intensity exhibits a modulation oscillation. The self-modulation oscillation can be periodic or quasi-periodic,
de-pending on the magnitude of the frequency difference∆Ω.
When ∆Ω ωr the total intensity in the self-modulation
regime is a periodic oscillation. However, when∆Ω is close
to the relaxation frequency in the self-modulation regimes, the total intensity is like to a quasi-periodic oscillation. In addition to the self-modulation regimes characterized by peri-odic or quasi-periperi-odic modulation, dynamic chaos can appear
under the conditions ofωr/2 < ∆Ω < ωr. Note that the
sys-tem of equations for the dynamics of a class B laser operating
in two LG modes with opposite angular indices±l is like to
the system describing the generation of a counter-propagating wave (CPW) in a bidirectional ring class-B laser, as discussed in [17–19]. Therefore, the condition for chaotic emission is also predicted in a bidirectional ring class-B laser [18]. When
∆Ω < ωr/2, the intensity represents the frequency-locked
transverse mode. In class-B lasers the locking occurs at a
sig-nificantly smaller mode frequency difference, ∆Ω < ωr/2.
Thus it is more difficult to obtain frequency locking in class-B than in class-A lasers. The relaxation oscillation plays an important role, not only in transient processes but also under conditions of steady operation of a solid-state laser (class-B laser). The build-up of the relaxation oscillation is one of the mechanisms resulting in the appearance of dynamic chaos in solid-state lasers. Using the system of equations given in [13], we find that if the loss difference is introduced near the lock-ing region in a class B laser, the total intensity may exhibit a self-pulsing train. The total intensity under the self-pulsing
region is almost like a LG TEM0,lmode with the distribution
cos2lθ (or sin2lθ) in azimuthal angle because of loss
differ-ence. The self-pulsing phenomenon is like to the recent result concerning the dynamics of a solid-state laser sustaining the oscillation of two orthogonally-polarized eigenstates [20].
Figure 1 shows the schematic diagram of the
fiber-coupled laser diode end-pumped Nd:YVO4 laser considered
in this work. We used a plano-concave cavity that consisted
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:YVO4crystal (1 mm in
length) was anti-reflection coated at 1064 nm. A mirror with
a reflectance of R= 97% 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
(Coher-ent, F-81-800C-100) had a core diameter of 0.1 mm and was
focused into the Nd:YVO4crystal using a focusing lens with
a magnification of 0.57.
θ
Fig. 1. Schematic diagram of a fiber-coupled diode end-pumped laser; a typical beam profile of a fiber-coupled laser diode away from the focal plane
Fig. 2. Beam profiles with different LG TEM0,l mode distributions, meas-ured with the CCD camera, in 14 positions
For the beam of a multi-mode fiber-coupled diode laser passing through a focusing lens, the profile at the focal plane is like a top hat-shaped distribution; however, away from the focal plane it is like a doughnut-shaped distribution, as de-picted in Fig. 1. With this property, we can defocus a standard fiber-coupled diode beam, resulting in a good overlap with
the high-order LG TEM0,l-mode and generate it purely. From
the characteristics of the pump beam profile, the radius of maximum pump intensity amplitude can be approximately described by rp(z) = θp| z − zo|, where θpis the far-field
half-13
Fig. 3. a Power intensity spectra of laser emission for one TEM∗0,l mode. Spectra b–f are recorded when the laser simultaneously oscillates in the two first-order transverse modes, showing the transition from the self-modulation state (b) and (c) to chaotic puls-ing (d) and the frequency-locked state (e), and the bifurcation of the locking to regular self-pulsing (f). Vertical scale: 10 dB/div; ho-rizontal scale: 1 MHz/div. Average transverse intensity distribution and time dependence are shown in the insets
angle, the point z= 0 is taken to be at the incident surface
of the gain medium and zo is the focal position of the pump
beam in the laser crystal. The average radius of maximum
pump intensity inside the gain medium, rpa, is calculated by
L
0 rp(z)e−αzdz/ L
0 e−αzdz, whereα is the absorption
coef-ficient at the pump wavelength and L is the length of the laser
crystal. Carrying out the integration and using e−αL → 0,
the average radius of maximum pump intensity is given by
rpa= θp[zo+ (2e−αzo− 1)/α].
For a single LG TEM0,l mode, the normalized cavity
mode distribution is given by
s0,l(r, φ, z) = 4 1+ δ0,ll! 1 πω2 oL cos2lφ 2r 2 ω2 o l × exp −2rω22 o , (1)
where the z-dependent variation in s0,l(r, φ, z) is neglected
and the spot radius of the laser beamωo is approximated to
be constant along the laser axis in the laser crystal. From (1), the radius of maximum mode intensity amplitude is trivially
r0,l= ωo√l/2. Since the cavity mode with the biggest over-lap with the gain structure has the minimum threshold, we
can obtain a LG TEM0,l-mode output by adjusting the focal
position zo to achieve rpa= r0.lfor the best overlap. Figure 2
shows the experimental results for the output beam profiles with different transverse-mode distributions, measured with a CCD camera (Coherent, Beam-Code), in fourteen positions. The relation between transverse-modes and pump positions is consistent with the prediction of pump-to-mode matching.
Typically, the free-running one-mode class-B laser dis-plays relaxation oscillations, as shown in Fig. 3a. Relaxation oscillations play an important role in the locking of two-mode class-B lasers. With a pump power slightly larger than the
pump threshold, the laser is operated in LG TEM∗0,l
dough-nut mode, which is a linear combination of two like TEM0,l
modes that have a fixed relative phase equal to π/2. Since
astigmatism lifts the degeneracy of the two like LG TEM0,l
modes, a perfectly circular pattern is usually an “unlocked doughnut” as can be confirmed by the observation of a peri-odic oscillation shown in Fig. 3b. From the state of the per-fectly circular pattern, the frequency difference between the two nearly-degenerate modes can be decreased by a slight adjustment in the output coupler to result in quasi-periodic os-cillation, as shown in Fig. 3c. It is found that the experimental results in the self-modulation regimes are consistent with the-oretical predictions. Upon further decreasing the frequency
14
Fig. 4a–d. Power intensity spectra of laser emission for TEM∗0,6 mode. Spectra a–d show the transition from the self-modulation state (a) and (b) to chaotic pulsing (c), and the bifurcation of the locking to regular self-pulsing (d). Vertical scale: 10 dB/div; horizontal scale: 0.5 MHz/div
difference, the appearance of chaotic generation regimes was observed, as shown in Fig. 3d. This result confirms the fact that there is a chaotic set of solutions when the frequency difference is of the order of magnitude of the relaxation fre-quency. Upon decreasing the frequency difference below the locking threshold, the two like modes eventually lock to the same frequency, as shown in Fig. 3e. We observed that the locking state has a strong tendency to jump to states of chaotic pulsing if subjected to a small perturbation. The bistable re-gion of coexistence between locked and unlocked modes was observed, which confirms the fact that the locking in a class-B laser is a subcritical bifurcation, as opposed to that in a class-A laser [12]. In the vicinity of the locking point, slightly ad-justing the output coupler frequently may lead to very stable, regular self-pulsing operation of the laser, as shown in Fig. 3f. Note that there are two obvious intensity maxima superposed
on the background of the doughnut intensity distribution in the pulsing operation. We found that the regular self-pulsing state can hold for several hours, like a passively Q-switched laser. In addition, it is striking to notice that the rep-etition rate of the self-pulsing is slightly below the relaxation oscillation frequency. Moreover, we experimentally observed that the repetition rate increases with the pumping rate of the laser. The self-pulsed regime is experimentally obtained only in the vicinity of the locking point. This result proves that this self-pulsed regime is due to both the frequency locking of the two nearly-degenerate modes and the existence of relax-ation oscillrelax-ations, i.e., to the fact that the populrelax-ation inversion cannot be eliminated adiabatically in class-B lasers [12, 13]. Finally, we also found a dynamic-like behavior similar to
a TEM∗0,1mode for other higher TEM∗0,l modes, i.e. periodic
and quasi-periodic self-modulation (Fig. 4a and b), chaotic pulsing (Fig. 4c), and self-pulsing (Fig. 4d).
2 Summary
We have generated the LG TEM0,l and TEM∗0,l modes in an
end-pumped microchip laser by defocusing a standard fiber-coupled diode to produce a doughnut-shaped pump profile. From the observations of the locking phenomena of the first order family, it was found that the locking occurs as a sub-critical bifurcation and a region of coexisting locked and un-locked states exists. These observations are consistent with the interesting theoretical predictions for class-B lasers. We believe that our experimental method gives a convenient way of furthering the investigation of the non-linear dynamics of
LG TEM∗0,l modes.
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