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OPTICS LETTERS / Vol. 38, No. 20 / October 15, 2013
Walsh modes and radial quantum correlations of spatially entangled photons D. Geelen and W. Löffler* Huygens Laboratory, Leiden University, P.O. Box 9504, 2300 RA Leiden, The Netherlands *Corresponding author: [email protected] Received August 1, 2013; revised September 13, 2013; accepted September 17, 2013; posted September 19, 2013 (Doc. ID 194872); published October 9, 2013 Orthogonal sets of 2D transverse modes are key to controlling the spatial degree of freedom of light in a classical or quantum context. In contrast to the azimuthal part, which is easily accessible using orbital angular momentum modes, control of the radial part is more difficult. We show here that simple sets of orthogonal binary sequences, the Walsh functions, provide a workable solution for exploration of the radial space with phase-only spatial light modulation. We demonstrate this by measuring “sequency” quantum correlations between different radial Walsh modes of spatially entangled photon pairs and for numerically optimized versions thereof. © 2013 Optical Society of America OCIS codes: (030.4070) Modes; (270.0270) Quantum optics. http://dx.doi.org/10.1364/OL.38.004108
Photon pairs produced by spontaneous parametric downconversion (SPDC) show quantum correlations in a number of degrees of freedom. These differ fundamentally in the amount of information that is carried per photon: the polarization degree of freedom is twodimensional; time, energy, and momentum are infinite dimensional. We study the last case, which gives rise to spatial correlation of the photon fields [1,2]. We can choose from a number of possible bases of transverse fields, many of which allow for an intuitive explanation of the quantum correlations: linear momentum entanglement [3], position (“pixel”) entanglement [4], and entanglement in sets of transverse modes. In case of transverse modes, the “natural” set of orthogonal modes, or fields, is obtained via the Schmidt decomposition. For a Gaussian pump beam, and in the thin-crystal approximation, Laguerre–Gauss (LG) modes are Schmidt modes if mode parameters are chosen appropriately [5]. For LG modes, the azimuthal rotation symmetry gives rise to quantized orbital angular momentum (OAM) correlations [3,6,7]. OAM modes are “phase-only” modes ∼ expilϕ and can therefore easily be detected using a combination of phase modulation on a spatial light modulator (SLM), and fundamental-mode selection with a singlemode fiber. The effective number of available entangled modes is given by the Schmidt number K [5,8]. However, the usable number of modes depends on the experimental details, and in particular on the choice of basis modes course not [9,10]. Via OAM modes alone, we can of p access all K modes, but only approximately K modes, because the radial degree of freedom is not used. Control of the radial degree of freedom is experimentally more challenging: SPDC phase-matching and the pump beam envelope depend on the radial coordinate; in general, amplitude- and phase-sensitive detection is needed. Amplitude modulation using phase-only SLMs is complicated and usually inefficient [11,12]. One solution is to fine-tune the radial LG mode parameters [13–15] to avoid amplitude modulation, but this approach is not very generic. Here, we show a different route to access the radial modes by phase-only discretization of the radial mode 0146-9592/13/204108-04$15.00/0
space using Walsh functions [16]. These are complete sets of orthogonal functions on the unit interval that have a value of 1 or −1 only, and transitions between these values occur on N fixed intervals, where N 2S for integer S. An example of the Walsh functions W N k x for N 4 (k 1…N is the sequence index and x is the position coordinate) is shown in Fig. 1(a). They can be obtained in a number of ways; for instance, they appear as rows in the sequency-ordered N-dimensional Hadamard matrix H N . Sequency is associated with the number of value changes of a binary-valued function, very similar to the frequency of continuous signals. Because they are orthogonal and complete, Walsh functions are ideally suited in signal analysis, filtering, and communications [17–21]. The Walsh sequences form a complete system as do the sine and cosine functions in Fourier frequency analysis, and the Walsh–Fourier analysis is often more
Fig. 1. Comparison of Walsh function and radial Walsh modes. The left panel (a) shows W 4k x, the original Walsh functions defined in the interval [0,1]. On the right (b), the radial part of the radial Walsh modes f N k r are shown, where r 0…∞. Here, pump- and detection-mode waists were set to wp w 1. It is easy to see that (a) and (b) describe orthogonal functions in 1D and 2D, respectively. © 2013 Optical Society of America
October 15, 2013 / Vol. 38, No. 20 / OPTICS LETTERS
easily implemented than conventional Fourier analysis. In optics, Walsh functions have been studied in aberration analysis [22], image processing and filtering [23], optical signal processing [24], wavefront sensing [25], and wavefront synthesis [26]. In quantum optics, however, Walsh modes are largely unexplored. Here, we show first results on using Walsh functions to obtain a set of orthogonal detection modes to study radial quantum correlations of spatially entangled photons. We start with the biphoton amplitude C [5], which gives the probability P jCj2 that the photons of a pair produced in SPDC are detected in the transverse fields us and ui . In the thin-crystal approximation for collinear SPDC, we obtain the simple spatial integral Z (1) C dϕrdrup rus rui r; where up is the pump beam profile that is assumed to be Gaussian with waist wp : up r exp−r 2 ∕w2p . Now, we separate the detection modes us;i in a radial and an azimuthal part ur; ϕ f k reilϕ , where k is a radial mode number and l corresponds to the OAM mode number. We ignore the azimuthal part from now on. Mind that, in ordinary LG modes, the radial modes depend also on the azimuthal mode number. We assume that f k are realized by phase-only spatial light modulation with phase Φk r in combination with projection onto the Gaussian mode of a single-mode fiber with waist w (mea2 sured at the SLM): f k eiΦk r − er2∕w . The idea is to use Walsh functions for eiΦk r to obtain a set of orthogonal radial modes. StartingR from the orthogonality relation of N 1D Walsh functions, 01 W N ks xW ki xdx ∝ δks ;ki , we need to find a coordinate transformation that allows us to express the radial part of the 2D fields in Eq. (1) by Walsh functions, while preserving orthogonality. We find x → hr 1 − exp−r 2 1∕w2p 2∕w2 , and with this we can write the radial Walsh-mode detection fields N −r2∕w2 . We insert this into Eq. (1), as f N k r W k hre ignore prefactors, and for the coincidence amplitude for the signal and idler photon detected in the Walsh modes with indices ks and ki , respectively, we get Z∞ 2 2 N C∝ rdre−r ∕wp f N (2) ks rf ki r ∝ δks ;ki :
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Fig. 2. Experimental setup. The photon pairs produced by SPDC in a PPKTP crystal are imaged in 4f configuration with L1 and L2 onto the SLMs (7.5× magnification). A wide pinhole (PH) is used to select the first diffraction order of the SLM. The SLM surface is again far-field imaged onto the single-mode fiber (SMF) with 10× microscopy objectives (MO). Photons are counted with avalanche photo diodes (APD), and photons which did belong to a single entangled pair are post-selected by coincide detection.
desired Walsh-mode patterns W N ks;i hr (with an additional blaze of 3 mrad). The SLMs are then far-field imaged onto the core of single-mode fibers with microscopy objectives. The relevant mode waists have to be determined at the SLM; we have here wp 3.6 mm for the pump and w 1.55 mm for the single-mode fiber detection waist. The single-mode fibers are connected to single-photon detectors (count rate typically 105 s−1 , dark counts
OPTICS LETTERS / Vol. 38, No. 20 / October 15, 2013
Walsh modes and radial quantum correlations of spatially entangled photons D. Geelen and W. Löffler* Huygens Laboratory, Leiden University, P.O. Box 9504, 2300 RA Leiden, The Netherlands *Corresponding author: [email protected] Received August 1, 2013; revised September 13, 2013; accepted September 17, 2013; posted September 19, 2013 (Doc. ID 194872); published October 9, 2013 Orthogonal sets of 2D transverse modes are key to controlling the spatial degree of freedom of light in a classical or quantum context. In contrast to the azimuthal part, which is easily accessible using orbital angular momentum modes, control of the radial part is more difficult. We show here that simple sets of orthogonal binary sequences, the Walsh functions, provide a workable solution for exploration of the radial space with phase-only spatial light modulation. We demonstrate this by measuring “sequency” quantum correlations between different radial Walsh modes of spatially entangled photon pairs and for numerically optimized versions thereof. © 2013 Optical Society of America OCIS codes: (030.4070) Modes; (270.0270) Quantum optics. http://dx.doi.org/10.1364/OL.38.004108
Photon pairs produced by spontaneous parametric downconversion (SPDC) show quantum correlations in a number of degrees of freedom. These differ fundamentally in the amount of information that is carried per photon: the polarization degree of freedom is twodimensional; time, energy, and momentum are infinite dimensional. We study the last case, which gives rise to spatial correlation of the photon fields [1,2]. We can choose from a number of possible bases of transverse fields, many of which allow for an intuitive explanation of the quantum correlations: linear momentum entanglement [3], position (“pixel”) entanglement [4], and entanglement in sets of transverse modes. In case of transverse modes, the “natural” set of orthogonal modes, or fields, is obtained via the Schmidt decomposition. For a Gaussian pump beam, and in the thin-crystal approximation, Laguerre–Gauss (LG) modes are Schmidt modes if mode parameters are chosen appropriately [5]. For LG modes, the azimuthal rotation symmetry gives rise to quantized orbital angular momentum (OAM) correlations [3,6,7]. OAM modes are “phase-only” modes ∼ expilϕ and can therefore easily be detected using a combination of phase modulation on a spatial light modulator (SLM), and fundamental-mode selection with a singlemode fiber. The effective number of available entangled modes is given by the Schmidt number K [5,8]. However, the usable number of modes depends on the experimental details, and in particular on the choice of basis modes course not [9,10]. Via OAM modes alone, we can of p access all K modes, but only approximately K modes, because the radial degree of freedom is not used. Control of the radial degree of freedom is experimentally more challenging: SPDC phase-matching and the pump beam envelope depend on the radial coordinate; in general, amplitude- and phase-sensitive detection is needed. Amplitude modulation using phase-only SLMs is complicated and usually inefficient [11,12]. One solution is to fine-tune the radial LG mode parameters [13–15] to avoid amplitude modulation, but this approach is not very generic. Here, we show a different route to access the radial modes by phase-only discretization of the radial mode 0146-9592/13/204108-04$15.00/0
space using Walsh functions [16]. These are complete sets of orthogonal functions on the unit interval that have a value of 1 or −1 only, and transitions between these values occur on N fixed intervals, where N 2S for integer S. An example of the Walsh functions W N k x for N 4 (k 1…N is the sequence index and x is the position coordinate) is shown in Fig. 1(a). They can be obtained in a number of ways; for instance, they appear as rows in the sequency-ordered N-dimensional Hadamard matrix H N . Sequency is associated with the number of value changes of a binary-valued function, very similar to the frequency of continuous signals. Because they are orthogonal and complete, Walsh functions are ideally suited in signal analysis, filtering, and communications [17–21]. The Walsh sequences form a complete system as do the sine and cosine functions in Fourier frequency analysis, and the Walsh–Fourier analysis is often more
Fig. 1. Comparison of Walsh function and radial Walsh modes. The left panel (a) shows W 4k x, the original Walsh functions defined in the interval [0,1]. On the right (b), the radial part of the radial Walsh modes f N k r are shown, where r 0…∞. Here, pump- and detection-mode waists were set to wp w 1. It is easy to see that (a) and (b) describe orthogonal functions in 1D and 2D, respectively. © 2013 Optical Society of America
October 15, 2013 / Vol. 38, No. 20 / OPTICS LETTERS
easily implemented than conventional Fourier analysis. In optics, Walsh functions have been studied in aberration analysis [22], image processing and filtering [23], optical signal processing [24], wavefront sensing [25], and wavefront synthesis [26]. In quantum optics, however, Walsh modes are largely unexplored. Here, we show first results on using Walsh functions to obtain a set of orthogonal detection modes to study radial quantum correlations of spatially entangled photons. We start with the biphoton amplitude C [5], which gives the probability P jCj2 that the photons of a pair produced in SPDC are detected in the transverse fields us and ui . In the thin-crystal approximation for collinear SPDC, we obtain the simple spatial integral Z (1) C dϕrdrup rus rui r; where up is the pump beam profile that is assumed to be Gaussian with waist wp : up r exp−r 2 ∕w2p . Now, we separate the detection modes us;i in a radial and an azimuthal part ur; ϕ f k reilϕ , where k is a radial mode number and l corresponds to the OAM mode number. We ignore the azimuthal part from now on. Mind that, in ordinary LG modes, the radial modes depend also on the azimuthal mode number. We assume that f k are realized by phase-only spatial light modulation with phase Φk r in combination with projection onto the Gaussian mode of a single-mode fiber with waist w (mea2 sured at the SLM): f k eiΦk r − er2∕w . The idea is to use Walsh functions for eiΦk r to obtain a set of orthogonal radial modes. StartingR from the orthogonality relation of N 1D Walsh functions, 01 W N ks xW ki xdx ∝ δks ;ki , we need to find a coordinate transformation that allows us to express the radial part of the 2D fields in Eq. (1) by Walsh functions, while preserving orthogonality. We find x → hr 1 − exp−r 2 1∕w2p 2∕w2 , and with this we can write the radial Walsh-mode detection fields N −r2∕w2 . We insert this into Eq. (1), as f N k r W k hre ignore prefactors, and for the coincidence amplitude for the signal and idler photon detected in the Walsh modes with indices ks and ki , respectively, we get Z∞ 2 2 N C∝ rdre−r ∕wp f N (2) ks rf ki r ∝ δks ;ki :
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Fig. 2. Experimental setup. The photon pairs produced by SPDC in a PPKTP crystal are imaged in 4f configuration with L1 and L2 onto the SLMs (7.5× magnification). A wide pinhole (PH) is used to select the first diffraction order of the SLM. The SLM surface is again far-field imaged onto the single-mode fiber (SMF) with 10× microscopy objectives (MO). Photons are counted with avalanche photo diodes (APD), and photons which did belong to a single entangled pair are post-selected by coincide detection.
desired Walsh-mode patterns W N ks;i hr (with an additional blaze of 3 mrad). The SLMs are then far-field imaged onto the core of single-mode fibers with microscopy objectives. The relevant mode waists have to be determined at the SLM; we have here wp 3.6 mm for the pump and w 1.55 mm for the single-mode fiber detection waist. The single-mode fibers are connected to single-photon detectors (count rate typically 105 s−1 , dark counts
