Creation, doubling, and splitting, of vortices in intracavity second harmonic generation

We demonstrate generation and frequency doubling of unit charge vortices in a linear astigmatic resonator. Topological instability of the double charge harmonic vortices leads to well separated vortex cores that are shown to rotate, and become anisotropic, as the resonator is tuned across resonance. Creation, doubling, and splitting, of vortices in intracavity second harmonic generation2 Optical vortices are topological objects whose transformation properties under propagation in linear and nonlinear optical media have been the subject of much recent work[1]. The vortex charge of a beam, defined as the closed loop contour integral of the wave phase modulo 2π, is generally a conserved quantity under linear propagation in free space. Optical vortices occur naturally in speckle fields[2], and can be generated in a controlled fashion using diffraction from holographic plates[3, 4]. They can also be generated in lasers[5, 6] and cavities with nonlinear elements[7], while high order vortex modes have been observed in active cavities with field rotating elements[8, 9]. Astigmatic optical elements as well as nonlinear wave interactions can be used to change the vortex charge of a beam. For example in the weak pump depletion regime of second harmonic generation the amplitude of the envelope of the harmonic field at frequency ω2 can be written as A2 ∼ A1. Thus an input field with charge m of the form A1 ∼ e generates an output field A2 ∼ e with twice the charge. This effect has been demonstrated experimentally by several groups using optical vortices created by diffraction from a hologram, and then allowing the beam to pass through a frequency doubling crystal[10, 11, 12, 13, 14]. In this work we describe a different approach to the generation of vortices in second harmonic generation that is based on frequency doubling of resonator modes with a vortical structure. Consider an empty resonator with the pump beam mode matched to the lowest order transverse resonator mode (TEM00 mode), which has a slowly varying amplitude at the cavity waist given by u00 ∼ exp(−r2/w2 c), with ~r = xx̂ + yŷ and wc the cavity waist. By changing the cavity tuning, and slightly tilting and displacing the pump beam, we can couple to higher order transverse modes of the cavity. By appropriate alignment of the pump beam it is possible to couple to a single higher order transverse mode, such as u10 ∼ x exp(−r2/w2 c ) or u01 ∼ y exp(−r2/w2 c) which have edge dislocations. Following the cavity by an astigmatic mode-converter[15] the HermiteGauss modes can be efficiently converted into azimuthally symmetric Laguerre-Gauss modes with non-zero vortex charge, as was demonstrated by Snadden et al.[16]. As we show here it is also possible to generate a vortex mode directly, without using an astigmatic mode converter, by aligning the pump beam to give the desired superposition of u10 and u01 modes. Let the pump beam be a displaced and tilted Gaussian. Ignoring unimportant constant amplitudes as well as any overall phase we have up = exp (−|~r − ~rp|/w p) exp (i~q · ~r). Here wp is the pump beam waist, ~rp = xpx̂ + ypŷ is the transverse displacement of the pump beam, and ~q = qxx̂ + qy ŷ is the transverse wavevector that is proportional to the pump beam tilt in the x, y plane. The lowest order odd cavity mode that the pump couples to can be written as u = o10a(ν − ν10)u10 + o01a(ν − ν01)u01 (1) where omn ∼ ∫ dxdy upumn is an overlap integral, and a(ν − νmn) is a complex coefficient that depends on the difference between the pump frequency ν and the resonant frequency of the mode νmn. The amplitude and phase of the overlap coefficients can be independently varied by adjusting the pump beam[17]. Performing the integrals Creation, doubling, and splitting, of vortices in intracavity second harmonic generation3 Figure 1. Experimental setup. we find o10 = h(2xp/wp−ikxwp) and o01 = h(2yp/wp−ikywp), where h is an unimportant common factor. By adjusting the pump beam we can obtain o01 = o10 exp (iπ/2) which results in vortex generation when a(ν − ν10) = a(ν − ν01). Vortices were generated in this way using the experimental setup shown in Fig. 1. A Ti:Sapphire laser at 858 nm generates a continuous wave fundamental beam with a power of up to 300 mW incident on the freqency doubling cavity. The beam is mode matched to a linear cavity with two R = 25 mm end mirrors that contains a 1 cm long a-cut KNbO3 crystal with anti-reflection coated ends. The input and output mirrors had T858nm = 4.8%, 0.04% and T429nm = 92% so that only the fundamental field was resonant in the cavity. Phase matching was controlled by varying the temperature of the crystal. With the distance between the mirrors set to Lcf = R + nc−1 nc Lc = 30.7 mm for confocal operation (nc, Lc are the crystal refractive index and length), and the crystal temperature tuned for large phase mismatch so no harmonic beam was generated, a cavity finesse of about 80 was measured. This agrees well with the theoretical value of F = 84 that was calculated using measured values of the crystal losses. With the crystal temperature tuned for optimum phase matching up to about 60 mW of 429 nm light was generated in a TEM00 mode. The cavity length was then reduced by 1.63 mm which resulted in the appearance of higher order transverse modes in the cavity transmission spectrum as seen in Fig. 2. Using the measured free spectral range as a scaling parameter the theoretically calculated frequencies of the first few higher order (q, m, n) modes have been indicated in the figure. The observed resonance frequencies agree to within a few percent with the calculated values. The cavity was locked to a (q, 1, 0) resonance using the rf sideband technique[19] which resulted in stable generation of a unit charge vortex mode as seen Creation, doubling, and splitting, of vortices in intracavity second harmonic generation4 Figure 2. Scan of resonator transmission with theoretical positions of the modes marked by arrows. q denotes the axial mode index. The upper line is the linear ramp voltage applied to the cavity piezo,and the inset shows the far-field structure of the fundamental field at the odd mode resonance. in the inset of Fig. 2. The phase structure of the harmonic field was observed by interference with a TEM00 beam generated in a second doubling cavity. As expected the second harmonic beam contained a doubly charged vortex as seen in Fig. 3. The detailed structure of the beam had a sensitive dependance on resonator alignment. Careful alignment of the crystal position resulted in observation of a doubly charged core region, although there was an apparent tendency for the vortices to repel each other so that a small vortex separation remained as seen in Fig. 3. We attribute the splitting to the topological instability of m > 1 vortices[18]. Adjustments to the crystal and/or pump beam alignment resulted in the core splitting into two well separated singly charged vortices, with a controllable relative orientation and separation. It was also found that when the resonator was aligned so that the cores were well separated, the relative orientation angle of the cores rotated in a repeatable fashion as the resonator was tuned across the (q, 1, 0) resonance, as seen in Fig. 4. The observations of vortex rotation can be explained by taking account of the cavity astigmatism due to crystal birefringence. Let the fundamental beam propagate along z (a-axis of KNbO3) and be polarized along y (b-axis). Beams propagating in the x − z plane correspond to ordinary polarized rays with an index nb = 2.279 at 858 nm and the temperature set for phase matching. We will label the x − z plane modes with the first transverse index m (the x axis is horizontal in the figures). Beams propagating in the y−z plane correspond to extraordinary polarized rays with an index n(θ) = nb(1 + tan 2 θ)/(1 + (nb/na) 2 tan θ) where θ is the angle of the ray with respect to the y axis and na = 2.238 We will label the y − z plane modes with the Creation, doubling, and splitting, of vortices in intracavity second harmonic generation5 Figure 3. Near field images of the harmonic charge 2 vortex. The central frame is overexposed to reveal the vortex splitting, as verified by the far-field interferogram on the right. second transverse index n. The effective crystal thickness for x−z plane modes is Lc/nb while the effective thickness for y − z plane modes is Lc/ñ, where ñ = na/nb is the effective index[20]. Since ñ < nb the crystal is effectively longer in the y − z plane and these modes have lower resonance frequencies. A short calculation shows that the frequency splitting can be written as