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We have demonstrated highly efficient Laguerre Gaussian beam generation by using a ring self-pumped phase conjugate mirror in the pico-second regime for the first time. The phase conjugate reflectivity was typically ~55%. We have also investigated the conservation of optical angular momentum.
©2006 Optical Society of America
1. Introduction
A Laguerre-Gaussian (LG) beam, a solution of the paraxial propagation electro-magnetic equation, exhibits orbital angular momentum of lħ per photon, because of its helical wavefront as well as an on-axis phase singularity (optical vortex) [1, 2]. To date, the existence of orbital angular momentum of the LG beam has been experimentally verified by several demonstrations such as the rotation of trapped micro-particles in optical tweezers [3–5].
Recently, high intensity fields produced by pico-second or femto-second lasers have been attracting a lot of interest in high field physics including coherent extra ultra-violet generation, above threshold ionization, chemical quantum control, and atto-second physics, etc.. If high intensity and high quality LG beams are created efficiently, the orbital angular momentum of the LG beam will open up new applications in high field laser physics [6, 7].
However, distortions such as thermo-optical effects in a high intensity laser amplifier system frequently induce degradation of beam quality and, in the worst case, destroy the optical vortex. The use of a phase conjugate mirror (PCM) is one method for correcting distortions in laser systems and producing high power and high quality output. Until now, high quality master oscillator power amplifiers (MOPA) utilizing a PCM based on a photorefractive rhodium doped barium titanate (Rh:BaTiO3) have been reported to generate >10W diffraction-limited pico-second output [8–10].
We aim to produce high quality and high power pico-second LG beams using a MOPA with a PCM.
In this paper, we report the first step towards this goal, that of producing efficient phase conjugation of a 1 μm pico-second LG beam. A photorefractive BaTiO3 crystal with 1000 ppm rhodium (Rh) doping was used as a phase conjugate mirror. A phase conjugate reflectivity of >50% was obtained. We also investigated experimentally the conservation of the optical vortex in the phase conjugation using an interferometic technique.
2. Experiment
The experimental setup is shown in Fig. 1(a). We used a CW mode-locked Nd:YVO4 pico-second laser having an output wavelength of 1064 nm. It had a pulse duration of 6.6 ps and a pulse repetition rate of 100 MHz. A reflection-type spatial light modulator Holoeye R-2500 (SLM), on which a computer-generated hologram (Fig. 1(b)) was displayed, was used for generating the LG beams. The maximum phase shift of the SLM was about 1.5π, resulting in a 1st-order diffraction efficiency of about 15%. The 1st-order diffraction beam was used as a probe beam for the phase conjugate mirror.
Fig. 1. (a) Experimental geometry of the ring self-pumped phase-conjugator. SLM, spatial light modulator; HWP, half wave plate; PBS, polarized beam splitter; H.M, half mirror; L1, L2, spherical lenses(focal length 200 mm).
The BaTiO3 crystal with 1000 ppm Rh ion doping was cut at 0° relative to the normal of the c-axis, and its dimensions were 8 mm × 7 mm × 8 mm. The crystal surfaces were AR-coated for 1 μm radiation. A self-pumped, phase conjugate mirror was formed by the Rh:BaTiO3 crystal and an external loop cavity made up of 4f imaging optics. The crystal location coincides with the conjugate points of the imaging optics. The probe beam was incident to the crystal at an external incidence angle of ~16° with respect to the normal of the +c-face. The angle between the probe and forward-pump beams was ~16°. With this setup, the experimental two-wave mixing gain κl and transmission loss of the external loop cavity were 3.8 and 0.1, respectively. According to conventional coupled mode theory, a phase conjugate reflectivity of >50% can be expected theoretically [11].
With self-pumped geometry with the external loop cavity, the fidelity of phase conjugation in the direction perpendicular to the intersection is relatively poor in comparison with that in the intersection plane. To improve the fidelity of the phase conjugation, a dove prism was used to rotate the wavefronts of the pump beam counter-propagating in the external loop cavity by 90° [12, 13]. The intensity profile of the phase conjugate LG beam was investigated using a charge-coupled-device (CCD) camera. In this system, the coherence length of the pico-second laser was considerably shorter than the length of the external loop, so that the formation of the reflection and 2K gratings was negligible and only the transmission grating contributed to the generation of the self-pump phase conjugation.
3. Results and discussion
Figures 2(a) and 3(a) show the spatial forms of the probe and the phase conjugate beams with azimuthal indexes of l = 1 and l = 2. The phase conjugation showed an annular intensity profile with a central hole, and no significant difference between the probe and phase conjugate beams was observed. When the dove prism was removed (see Fig. 2(c)), the central hole was not observed clearly in the spatial form of the phase conjugation. This degradation of the fidelity is mainly due to the lack of non-collinearity of the interacting beams [12]. In order to confirm that the phase conjugation contains an optical vortex, we also investigated the wavefront fidelity of the phase conjugation by analyzing the interferometric fringes formed by the phase conjugation and a spherical reference beam. This technique was initially proposed by Bazhenov et al.[14]. Figures 2(b) and 3(b)show interferograms obtained for the vortex in the phase conjugation. The phase conjugate beams of the 1st - and 2nd - order LG modes exhibit single- and double-arm spirals, respectively. And the direction of the spiral and the number of arms in the phase conjugate beams were consistent with those in the probe beams. These show that the phase conjugation of the LG beams can be produced efficiently without any loss in the orbital angular momentum.
Fig. 2. (a) Probe light of LG beam and phase conjugation with dove prism, (b) Conservation of topological charge, and (c) phase conjugation without dove prism at l=1.
Fig. 3. (a) Probe light of LG beam and phase conjugation with dove prism and (b) Conservation of topological charge at l=2.
Experimental phase conjugate reflectivity defined as the phase conjugate power divided by the incident probe beam power is shown in Fig. 4. There was no significant difference in the phase conjugate reflectivity between the 1st - and 2nd -order LG beams. Experimental phase conjugate reflectivity of ~55% was typically obtained at any incident power.
Finally we measured the autocorrelation trace of the phase conjugation by second harmonic generation with a 5 mm KTP crystal. Experimental intensity autocorrelation traces are shown in the Fig. 5. The phase conjugate pulse exhibited a full width at half maximum (FWHM) of 6.5 ps, and no significant pulse broadening was observed. As described in our previous paper [9], the wavelength selectivity by refractive index gratings formed in the PCM is given by,
where κ is the two-wave mixing gain coefficient of the BaTiO3 crystal, n is the refractive index of the crystal, θb is Bragg’s angle of the refractive grating, and ʌ (=λ/2sinθb) is the grating period. Using experimental parameters (κ= 3.8/0.7 = 5.4 cm-1, ʌ ~ 3 μm, n ~ 2.4, and θb ~ 4°), the wavelength selectivity Δλ was estimated to be ~7.4 nm. This value is sufficiently broader than the spectrum bandwidth of the probe pulse (0.35 nm). Consequently, no pulse broadening effect was seen.
The photorefractive effect results from the optically induced redistribution of electrons and holes. And thus, the temporal response is typically slow. This PCM is useful specially for high repetitive pico-second lasers with the repetition rate of > 1kHz.
4. Conclusions
We have demonstrated highly efficient phase conjugation of 1 μm pico-second LG beams by using a ring self-pumped phase conjugate mirror based on a Rh:BaTiO3 crystal. The phase conjugate reflectivity was typically ~55%.
We also experimentally confirmed the existence of an optical vortex in the phase conjugation by an interferometic technique. We believe that this work can be extended to produce high power, high quality pico-second LG beam by using a master oscillator and power amplifier system in conjunction with a phase conjugate mirror. High intensity and high quality LG beams with orbital angular momentum have the potential to be used in many applications including high field laser physics.
Fig. 5. Intensity autocorrelation traces of the probe pulse (red) and phase conjugation pulse (blue).
Acknowledgments
The authors acknowledge support from a scientific research grant-in-aid (11555010, 15035202) from the Ministry of Education, Science and Culture of Japan and the Japan Society for the Promotion of Science. T. Omatsu’s e-mail address is omatsu@faculty.chiba-u.jp.
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