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Vortices in the microcavity optical parametric oscillator
Superlattices and Microstructures 41 (2007) 297–300 www.elsevier.com/locate/superlattices
Vortices in the microcavity optical parametric oscillator D.M. Whittaker Department of Physics and Astronomy, The University of Sheffield, Sheffield, S3 7RH, UK Available online 23 April 2007
Abstract A theoretical treatment is presented for a novel from of optical vortex state, in a microcavity optical parametric oscillator (OPO). The state comprises a vortex/anti-vortex pair in the signal and idler beams, with a Gaussian pump. A numerical solution of the microcavity OPO equations shows that such a vortex can be stable. c 2007 Elsevier Ltd. All rights reserved.
Keywords: Optical parametric oscillator; Optical vortex; Theoretical modelling
The phenomenon of the microcavity optical parametric oscillator (OPO) is well established: when a planar microcavity is pumped at a finite angle (10◦ –20◦ ) to the surface normal, coherent signal and idler beams are generated, with one, defined as the signal, always close to the normal direction, and the idler on the high angle side of the pump [1,2]. This is a rather unusual type of OPO, in that the underlying excitonic non-linearity is χ (3) rather than χ (2) , so the phase matching conditions on the frequencies and wave-vectors are 2ω p = ωs + ωi and 2k p = ks + ki . Vortices can be formed in many systems with two-dimensional scalar fields; they are states where the phase of the field winds about a point, the vortex core. Typical systems where vortices are found include bosonic condensates, such as dilute gases, liquid helium and superconductors. However, they also occur in classical systems, particularly electromagnetic fields, with examples including optical vortex beams [3], and vertical cavity lasers [4]. This latter case is probably the closest to the present study, as it corresponds to a non-linear dissipative system, though it is electrically, not optically, driven, and involves only one coherent field. The vortices in the microcavity OPO are thus more complicated, in that they involve three coherent fields; the pump
E-mail address: [email protected]. c 2007 Elsevier Ltd. All rights reserved. 0749-6036/$ - see front matter doi:10.1016/j.spmi.2007.03.032
298
D.M. Whittaker / Superlattices and Microstructures 41 (2007) 297–300
Fig. 1. Schematic diagram of the microcavity OPO dispersion, showing the proposed method for generating a vortex, using a Gauss–Laguerre probe beam to ‘stir’ the signal. The corresponding anti-vortex should appear in the idler. The probe could be continuous wave or pulsed.
is in a state of zero angular momentum, m p = 0, while the signal and idler carry a vortex–antivortex pair, m s = −m i , so that angular momentum is conserved in the non-linear scattering process: 2m p = m s + m i . The microcavity OPO has been treated theoretically both in a microscopic quantum [5] model and as a classical [6,7] non-linear optical effect, though in the former case, mean field approximations mean that the equations developed are essentially the same in the two descriptions. Details of the numerical treatment of these equations have been described elsewhere [8]. Briefly, the numerical solution is obtained in the time domain on a twodimensional spatial grid, using an alternating-direction implicit algorithm. For the results presented here, the grid comprised 29 × 29 points. Time resolved images of the pump, signal and idler polariton populations are obtained by filtering short time sequences of data in the frequency domain to separate the different modes. The proposed method of introducing vortices into the OPO, shown in Fig. 1, is to use a probe pulse to ‘stir’ the signal state. The pulse is chosen to have an energy and angle matching those of the signal, and has a Gauss–Laguerre spatial profile with finite angular momentum, m s . Driving the system like this causes it to behave as a parametric amplifier, generating an idler containing an anti-vortex with m i = −m s , which conserves angular momentum in the scattering process. The probe pulse could be continuous, in which case a steady vortex would be obtained, or it could be a short pulse, in order to investigate issues such as the stability and dynamics of the vortex state. Fig. 2 shows a vortex created in the OPO state using the stirring method described above. Comparison of the phase plots for the signal and idler demonstrates that m i = −m s , as expected. The image corresponds to a time ∼1 ns after the end of the probe pulse. This is a very long time compared with the characteristic timescales of the system (∼ps), so it is reasonable to claim that the vortex is, within the model, a stable state. It should be noted that it is not always the case that a stable vortex is obtained. In the transient period, just after the probe has been removed, there is a tendency for the vortex core to drift around the spot before settling down. Sometimes, and particularly at powers close to threshold, the core drifts right out of the spot and the vortex disappears. There are, of course, other processes which could destroy the vortex, which are not
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Fig. 2. Spatial images of signal ((a), (b)) and idler ((c), (d)) vortex/anti-vortex pair following ‘stirring’ with a Gauss–Laguerre beam. (a) and (c) are the intensities of the photon component, while (b) and (d) are the phases. Lines of constant phase spiral out from the vortex core because there is a radial flow out of the centre of the spot as well as the circulation of the vortex. The spirals are in opposite directions for the signal and idler. The side of each image is 33 µm long.
included in the model. These include the effects of the finite sample temperature, incoherent scattering of polaritons, and noise or mode-hopping in the pump laser. Introducing a vortex by stirring with a probe is a very controlled way of investigating the state. However, it is also possible that vortices could occur spontaneously with just a simple Gaussian pump beam. At high powers and/or larger pump spot sizes, when the simulated signal becomes unstable, the phase maps contain many singular points, but they evolve rapidly on a picosecond timescale. In the stable regime, the fields always maintain the bilateral symmetry defined by the pump direction, which precludes vortex formation. However, if this symmetry is broken, by a strong ‘wedge’ of the cavity in the perpendicular direction, stable vortices can appear spontaneously. The simulations presented in this paper have shown that it is possible to obtain stable vortex solutions to a model of the microcavity OPO. These states are novel, in that they involve three coherent fields, the pump, signal and idler, with vortices of equal and opposite angular momentum in the signal and idler. It could be argued that the vortex state consists of a supercurrent of polaritons circulating the core. However, it is equally reasonable to describe the state as simply a spatial variation in the classical phase of the excitonic polarisation around the
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core. Perhaps the key difference between the two types of system is that a measurable quantity, charge or mass, is being transported by the persistent current in a superconductor or liquid helium system, while polaritons are uncharged and carry only a pseudo-momentum. Thus in a superconducting vortex the circulating charge generates a detectable magnetic field, while 4 He vortices can be excited in the ‘rotating bucket’ experiment. In the polariton system, by contrast, an optical phase measurement would appear to be the only means of demonstrating the circulation. References [1] R.M. Stevenson, V.N. Astratov, M.S. Skolnick, D.M. Whittaker, M. Emam-Ismail, A.I. Tartakovskii, P.G. Savvidis, J.J. Baumberg, J.S. Roberts, Phys. Rev. Lett. 85 (2000) 3680. [2] J.J. Baumberg, P.G. Savvidis, R.M. Stevenson, A.I. Tartakovskii, M.S. Skolnick, D.M. Whittaker, J.S. Roberts, Phys. Rev. B 62 (2000) R16247. [3] L. Allen, M.W. Beijersbergen, R.J.C. Spreeuw, J.P. Woerdman, Phys. Rev. A 45 (1992) 8185. [4] J. Scheuer, M. Orenstein, Science 285 (1999) 230. [5] C. Ciuti, P. Schwendimann, B. Deveaud, A. Quattropani, Phys. Rev. B 62 (2000) R4825. [6] D.M. Whittaker, Phys. Rev. B 63 (2001) 193305. [7] N.A. Gippius, S.G. Tikhodeev, V.D. Kulakovskii, D.N. Krizhanovskii, A.I. Tartakovskii, Europhys. Lett. 67 (2004) 997. [8] D.M. Whittaker, Phys. Status Solidi C 2 (2005) 733.
Vortices in the microcavity optical parametric oscillator D.M. Whittaker Department of Physics and Astronomy, The University of Sheffield, Sheffield, S3 7RH, UK Available online 23 April 2007
Abstract A theoretical treatment is presented for a novel from of optical vortex state, in a microcavity optical parametric oscillator (OPO). The state comprises a vortex/anti-vortex pair in the signal and idler beams, with a Gaussian pump. A numerical solution of the microcavity OPO equations shows that such a vortex can be stable. c 2007 Elsevier Ltd. All rights reserved.
Keywords: Optical parametric oscillator; Optical vortex; Theoretical modelling
The phenomenon of the microcavity optical parametric oscillator (OPO) is well established: when a planar microcavity is pumped at a finite angle (10◦ –20◦ ) to the surface normal, coherent signal and idler beams are generated, with one, defined as the signal, always close to the normal direction, and the idler on the high angle side of the pump [1,2]. This is a rather unusual type of OPO, in that the underlying excitonic non-linearity is χ (3) rather than χ (2) , so the phase matching conditions on the frequencies and wave-vectors are 2ω p = ωs + ωi and 2k p = ks + ki . Vortices can be formed in many systems with two-dimensional scalar fields; they are states where the phase of the field winds about a point, the vortex core. Typical systems where vortices are found include bosonic condensates, such as dilute gases, liquid helium and superconductors. However, they also occur in classical systems, particularly electromagnetic fields, with examples including optical vortex beams [3], and vertical cavity lasers [4]. This latter case is probably the closest to the present study, as it corresponds to a non-linear dissipative system, though it is electrically, not optically, driven, and involves only one coherent field. The vortices in the microcavity OPO are thus more complicated, in that they involve three coherent fields; the pump
E-mail address: [email protected]. c 2007 Elsevier Ltd. All rights reserved. 0749-6036/$ - see front matter doi:10.1016/j.spmi.2007.03.032
298
D.M. Whittaker / Superlattices and Microstructures 41 (2007) 297–300
Fig. 1. Schematic diagram of the microcavity OPO dispersion, showing the proposed method for generating a vortex, using a Gauss–Laguerre probe beam to ‘stir’ the signal. The corresponding anti-vortex should appear in the idler. The probe could be continuous wave or pulsed.
is in a state of zero angular momentum, m p = 0, while the signal and idler carry a vortex–antivortex pair, m s = −m i , so that angular momentum is conserved in the non-linear scattering process: 2m p = m s + m i . The microcavity OPO has been treated theoretically both in a microscopic quantum [5] model and as a classical [6,7] non-linear optical effect, though in the former case, mean field approximations mean that the equations developed are essentially the same in the two descriptions. Details of the numerical treatment of these equations have been described elsewhere [8]. Briefly, the numerical solution is obtained in the time domain on a twodimensional spatial grid, using an alternating-direction implicit algorithm. For the results presented here, the grid comprised 29 × 29 points. Time resolved images of the pump, signal and idler polariton populations are obtained by filtering short time sequences of data in the frequency domain to separate the different modes. The proposed method of introducing vortices into the OPO, shown in Fig. 1, is to use a probe pulse to ‘stir’ the signal state. The pulse is chosen to have an energy and angle matching those of the signal, and has a Gauss–Laguerre spatial profile with finite angular momentum, m s . Driving the system like this causes it to behave as a parametric amplifier, generating an idler containing an anti-vortex with m i = −m s , which conserves angular momentum in the scattering process. The probe pulse could be continuous, in which case a steady vortex would be obtained, or it could be a short pulse, in order to investigate issues such as the stability and dynamics of the vortex state. Fig. 2 shows a vortex created in the OPO state using the stirring method described above. Comparison of the phase plots for the signal and idler demonstrates that m i = −m s , as expected. The image corresponds to a time ∼1 ns after the end of the probe pulse. This is a very long time compared with the characteristic timescales of the system (∼ps), so it is reasonable to claim that the vortex is, within the model, a stable state. It should be noted that it is not always the case that a stable vortex is obtained. In the transient period, just after the probe has been removed, there is a tendency for the vortex core to drift around the spot before settling down. Sometimes, and particularly at powers close to threshold, the core drifts right out of the spot and the vortex disappears. There are, of course, other processes which could destroy the vortex, which are not
D.M. Whittaker / Superlattices and Microstructures 41 (2007) 297–300
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Fig. 2. Spatial images of signal ((a), (b)) and idler ((c), (d)) vortex/anti-vortex pair following ‘stirring’ with a Gauss–Laguerre beam. (a) and (c) are the intensities of the photon component, while (b) and (d) are the phases. Lines of constant phase spiral out from the vortex core because there is a radial flow out of the centre of the spot as well as the circulation of the vortex. The spirals are in opposite directions for the signal and idler. The side of each image is 33 µm long.
included in the model. These include the effects of the finite sample temperature, incoherent scattering of polaritons, and noise or mode-hopping in the pump laser. Introducing a vortex by stirring with a probe is a very controlled way of investigating the state. However, it is also possible that vortices could occur spontaneously with just a simple Gaussian pump beam. At high powers and/or larger pump spot sizes, when the simulated signal becomes unstable, the phase maps contain many singular points, but they evolve rapidly on a picosecond timescale. In the stable regime, the fields always maintain the bilateral symmetry defined by the pump direction, which precludes vortex formation. However, if this symmetry is broken, by a strong ‘wedge’ of the cavity in the perpendicular direction, stable vortices can appear spontaneously. The simulations presented in this paper have shown that it is possible to obtain stable vortex solutions to a model of the microcavity OPO. These states are novel, in that they involve three coherent fields, the pump, signal and idler, with vortices of equal and opposite angular momentum in the signal and idler. It could be argued that the vortex state consists of a supercurrent of polaritons circulating the core. However, it is equally reasonable to describe the state as simply a spatial variation in the classical phase of the excitonic polarisation around the
300
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core. Perhaps the key difference between the two types of system is that a measurable quantity, charge or mass, is being transported by the persistent current in a superconductor or liquid helium system, while polaritons are uncharged and carry only a pseudo-momentum. Thus in a superconducting vortex the circulating charge generates a detectable magnetic field, while 4 He vortices can be excited in the ‘rotating bucket’ experiment. In the polariton system, by contrast, an optical phase measurement would appear to be the only means of demonstrating the circulation. References [1] R.M. Stevenson, V.N. Astratov, M.S. Skolnick, D.M. Whittaker, M. Emam-Ismail, A.I. Tartakovskii, P.G. Savvidis, J.J. Baumberg, J.S. Roberts, Phys. Rev. Lett. 85 (2000) 3680. [2] J.J. Baumberg, P.G. Savvidis, R.M. Stevenson, A.I. Tartakovskii, M.S. Skolnick, D.M. Whittaker, J.S. Roberts, Phys. Rev. B 62 (2000) R16247. [3] L. Allen, M.W. Beijersbergen, R.J.C. Spreeuw, J.P. Woerdman, Phys. Rev. A 45 (1992) 8185. [4] J. Scheuer, M. Orenstein, Science 285 (1999) 230. [5] C. Ciuti, P. Schwendimann, B. Deveaud, A. Quattropani, Phys. Rev. B 62 (2000) R4825. [6] D.M. Whittaker, Phys. Rev. B 63 (2001) 193305. [7] N.A. Gippius, S.G. Tikhodeev, V.D. Kulakovskii, D.N. Krizhanovskii, A.I. Tartakovskii, Europhys. Lett. 67 (2004) 997. [8] D.M. Whittaker, Phys. Status Solidi C 2 (2005) 733.