Spatial Characterization Of Two-Photon States - GAP-Optique

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3. <strong>Spatial</strong> correlations and OAM transfer Configuration Weight 1 0 -4 4 0 <strong>Spatial</strong> -4 4 distribution OAM content 1 Mode Figure 3.2: In a collinear configuration a pump beam with oam equal to 1 transfers its oam into the signal photon when the idler is projected into a Gaussian mode. The figure shows the pump in blue and the signal and idler in pink. Additionally, it shows the spatial distribution and the oam content of the signal photon for co-propagating and counter-propagating configurations. 3.3 OAM transfer in collinear configurations The aperture of the down-conversion cone is given by the angle of emission of the photons. In the collinear configuration this angle is zero, and therefore the cone collapses onto its axis, as shown in figure 1.2 (d). As each photon propagates in the same direction as all the other photons, equation 3.18 is valid. To analyze the transfer of oam from the pump beam to the generated photons, we will calculate the signal oam content, assuming a lg1 pump beam so that lp = 1, and an idler photon projected into a Gaussian state li = 0. We will consider two kinds of collinear configurations: co-propagating and counterpropagating, both shown in figure 3.2. The collinear counter-propagating generation of photons requires fulfilling a special phase-matching condition, which can be achieved by quasi-phase-matching in a periodic structure. Such counterpropagating generation has been demonstrated in second harmonic generation [53], parametric fluorescence [54], and spdc in fibers [55]. Chapter 5 explores another way to generate counter-propagating two-photon states: Raman scattering. Figure 3.2 shows the calculated signal spatial distribution and oam content in two cases. The first row shows the collinear co-propagating case. The second row shows the collinear counterpropagating case. According to figure 3.2, the collinear case fulfills the selection rule, except for a minus sign in the counterpropagating case. The phase appears because the photon’s oam content is evaluated with respect to the pump propagation direction, as figure 3.3 shows. To take this effect into account equation 3.18 34
Phase front 3.3. OAM transfer in collinear configurations OAM respect to z direction Figure 3.3: The direction of rotation of the phase shift defines the sign of the mode’s oam content. For a Laguerre-Gaussian mode corresponding to positive oam content, the phase shift rotates clockwise with respect to the propagation direction. If the same mode propagates backwards, the phase shift rotates anticlockwise with respect to the original propagation direction, and the mode has a negative oam content. can be generalized to lp = dsls + d1li +2 -2 -2 (3.19) where ds,i = ±1 corresponds to forward and backward propagation of the corresponding photon. Conclusion The oam content of the pump is completely transferred to the signal and idler photons, emitted all over a cone. This makes it possible to introduce a selection rule that always holds if all possible emission directions are considered, but not necessarily if only a small part is detected. This result clarifies the apparent contradiction between several works that support the selection rule [7, 28, 56], and those that report that it is not valid [32, 31, 57]. When only a portion of all generated photons are considered there is no simple relationship between lp, ls and li. The next chapter studies the oam transfer in those cases, describing the effect of different spdc parameters on the amount of oam that is transferred to the subset of the photons considered. 35
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DOCTORAL THESIS IN PHOTONICS ICFO B
Page 5: Spatial Characterization Of Two-Pho
Page 9 and 10: Contents Contents vii Acknowledgeme
Page 11 and 12: Acknowledgements This thesis compil
Page 13 and 14: Abstract In the same way that elect
Page 15 and 16: Resumen De la misma manera que la e
Page 17 and 18: Introduction The role of photons in
Page 19: This thesis is based on the followi
Page 22 and 23: 1. General description of two-photo
Page 35 and 36: CHAPTER 2 Correlations and entangle
Page 37 and 38: 2.1. The purity as a correlation in
Page 39 and 40: 2.2. Correlations between space and
Page 41 and 42: 2.2. Correlations between space and
Page 43 and 44: 2.3. Correlations between signal an
Page 45 and 46: 2.3. Correlations between signal an
Page 47 and 48: Signal purity 1 0 (a) 0.1 3 2.3. Co
Page 49 and 50: CHAPTER 3 Spatial correlations and
Page 51 and 52: 3.2. OAM transfer in general SPDC c
Page 53: 3.2. OAM transfer in general SPDC c
Page 58 and 59: 4. OAM transfer in noncollinear con
Page 72 and 73: 5. Spatial correlations in Raman tr
Page 81 and 82: CHAPTER 6 Summary This thesis chara
Page 83 and 84: APPENDIX A The matrix form of the m
Page 85 and 86: v =γ 2 L 2 Np sin ϕi − γ 2 L 2
Page 87: The matrix C is given by C = 1 ⎛
Page 90 and 91: B. Integrals of the matrix mode fun
Page 92 and 93: C. Methods for OAM measurements Inp
Page 94 and 95: Bibliography [13] M. Barbieri, C. C
Page 96 and 97: Bibliography [41] C. I. Osorio, A.
Page 98: Bibliography [68] S. Chen, Y.-A. Ch
Page 103: Spatial Characterization Of Two-Pho
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A Luz Stella y Luis Alfonso, mis pa
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Contents B Integrals of the matrix
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When he [Kepler] found that his lon
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Abstract The matrix notation, intro
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Abstract La notación matricial int
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Introduction the transfer of oam fr
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CHAPTER 1 General description of Tw
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y y x z z pump s i (a) signal idle
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1.3. Approximations and other consi
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x 1.3. Approximations and other con
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x Wave fronts 1.3. Approximations a
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1 0 -0.2 1.3. Approximations and ot
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1.4. The mode function in matrix fo
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2. Correlations and entanglement (a
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2. Correlations and entanglement ch
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2. Correlations and entanglement th
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2. Correlations and entanglement Fi
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2. Correlations and entanglement q
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2. Correlations and entanglement Si
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2. Correlations and entanglement ri
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3. Spatial correlations and OAM tra
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CHAPTER 4 OAM transfer in noncollin
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4.2. Effect of the pump beam waist
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1.0 0.0 4.2. Effect of the pump bea
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Coincidences 800 0 -4 4.2. Effect o
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Weight 1 0 4.3. Effect of the Poynt
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4.3. Effect of the Poynting vector
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Before the crystal 4.3. Effect of t
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CHAPTER 5 Spatial correlations in R
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x y z 5.1. The quantum state of Sto
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where 5.2. Orbital angular momentum
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weight 1 0.6 -180º -90º 0º 90º
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5.3. Spatial entanglement 5.20 is s
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6. Summary Experiments that explain
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A. The matrix form of the mode func
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A. The matrix form of the mode func
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APPENDIX B Integrals of the matrix
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APPENDIX C Methods for OAM measurem
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Bibliography [1] M. A. Nielsen and
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Bibliography [27] J. P. Torres, A.
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Bibliography [55] M. C. Booth, M. A
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