Angular Diffraction - SPIE Digital Library

Invited Paper

Angular Diffraction S. Franke-Arnold, B. Jack, J. Leach, M. J. Padgett Department of Physics and Astronomy, SUPA, University of Glasgow, Glasgow, UK ABSTRACT The angular profile and the orbital angular momentum of a light mode are related by Fourier transform. Any modification of the angular distribution, e. g. via diffraction off a suitably programmed spatial light modulator, influences the orbital angular momentum spectrum of the light. This holds true even at the single photon level. We observe the influence of various angular masks on the orbital angular momentum spectrum, both in the near and the far field, and describe the resulting patterns in terms of angular diffraction. If photons are entangled in their orbital angular momentum, diffraction of one photon affects the orbital angular momentum spectrum of its partner photon, and angular ghost diffraction can be measured in the coincidence counts. We highlight the role of the angular Fourier relationship for these effects.

1. INTRODUCTION Light can carry energy, as well as linear and angular momentum, and the latter can be separated into spin and orbital angular momentum (OAM). The last two decades have seen research on the OAM in a wider and wider context, reaching from first fundamental studies1–3 to applications in quantum information,4–6 light-matter interaction,7, 8 free-space communication with large data capacity9 and potential applications in astronomy10 to name but a few. As the field is moving towards technological applications, the ability to create, manipulate, and characterize OAM states is becoming increasingly important. Light carries OAM if its phase fronts are not perpendicular to its direction of propagation. In this case the local Poynting vectors, and therefore the direction of energy flow, have an azimuthal component. Light modes with an azimuthal phase dependence exp(iφ) correspond to an OAM of ¯h per photon. Such light can be generated via spiral phase plates11 or holograms with fork dislocations,12 which conveniently can be displayed on spatial light modulators (SLMs). OAM also occurs naturally in speckle patterns,13 and OAM sidebands can be generated by aperture masks.14 In 2001, the OAM of a single photon was measured for the first time in experiments that investigated the OAM entanglement of photon pairs generated in parametric down-conversion.15 The Fourier relation16 that links the angular distribution of light to its OAM is relevant for many fundamental properties of optical OAM and at the same time offers a useful tool to control the OAM spectrum of light modes. The Fourier relation defines angular diffraction,14 and it is the classical analogue of the angular uncertainty relation17 which connects the spread of an angle measurement with the spread of the related OAM distribution. The Fourier relation, in connection with azimuthal phase matching is responsible for the conservation of OAM during down-conversion processes,18 and therefore for entanglement between the OAM of the generated photon pairs.15 Angle and OAM represent conjugate variables and various experiments have investigated the entangled nature of angle and OAM.20–22 This paper will describe some effects of the angular Fourier relationship and related angular diffraction.

2. ANGULAR FOURIER RELATION The conjugate variable to the OAM is the angular position,23 and both are linked by a Fourier relationship.20, 21 We can therefore write the amplitudes of the OAM states, A , and the azimuthal dependence of the corresponding Further author information: (Send correspondence to S. Franke-Arnold) E-mail: [email protected]

Complex Light and Optical Forces III, edited by Enrique J. Galvez, David L. Andrews, Jesper Glückstad, Proc. of SPIE Vol. 7227, 72270I · © 2009 SPIE · CCC code: 0277-786X/09/$18 · doi: 10.1117/12.813564

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complex beam amplitude, Ψ(φ), as generating functions of each other:  π 1 Ψ(φ) exp(−iφ)dφ, A = √ 2π −π ∞ 1  A exp(iφ). Ψ(φ) = √ 2π =−∞

(1) (2)

The second equation simply states that any wavefunction can be expressed in terms of its spiral harmonics exp(−iφ). For situations where the radial profile is unimportant, this is more convenient than decomposing a wavefunction into its Laguerre-Gauss (LG) contributions. One should note, however, that the spiral harmonics, unlike the LG modes, are not eigenfunctions of free-space propagation. In order to describe propagation dynamics the full set of LG modes needs to be considered, illustrated in section 5. It is a consequence of the Fourier relationship that any modification of the angular profile of a light mode influences its OAM spectrum. Generally, a narrow angular distribution generates many OAM sidebands, whereas a pure OAM state corresponds to a uniform angular distribution. The fact that the angle is 2π periodic causes a natural quantization of the OAM. In the Fourier relations (1,2) this is manifest in the Fourier sum over the various OAM components and in the bounds of the angle integration. Depending on the angular profile of a mode, several OAM modes are present, or in other words, the OAM spectrum of a light beam can be altered by diffraction off a mask with an angular variation in amplitude or phase, see Fig. 1. A forked hologram, or a waveplate with angularly increasing thickness shifts the OAM of the incoming light beam (a), a phase mask with n-fold symmetry generates OAM sidebands at  = ±n and higher order sidebands (b), a sinusoidal angle mask generates a superposition of exactly two OAM states (c) and a mask with a Gaussian angular dependance generates a Gaussian enveloped OAM spectrum (d). The Fourier relationship is valid not only for classical light beams but survives on the single-photon level. The operator relationship24 ˆ z = −i d L (3) dϕ can be interpreted as the quantum mechanical equivalent of the Fourier relationship (1). The corresponding angle operator is then a multiplicative operator that denotes ϕ within a 2π radian range (θ0 , θ0 + 2π]. Just like the commutation relation between linear position and momentum, or their Fourier relation, is associated with Heisenberg’s uncertainty relation ΔxΔp ≥ ¯h/2, an angular uncertainty relation can be derived for OAM and angular position. The angular uncertainty relation23 is ΔLz Δϕ ≥

1 |1 − |Ψ(θ0 )|2 | 2

(4)

where |Ψ(θ0 )|2 is the probability density for finding the system at the edge of the observation interval at θ0 . Unlike Heisenberg’s uncertainty, this uncertainty relation is not bounded by a a constant but the lower bound of ΔΔφ depends on the chosen angular range in which the width of the angular profile is expressed, a direct consequence of the periodic nature of the angle variable. The intelligent states,17 i. e. states that obey the equality in the uncertainty relation, are Gaussian angle distributions truncated to a 2π radian range. For a small spread of angular position, with vanishing probability |Ψ(θ0 )|2 , the uncertainty relation simplifies to ΔLz Δϕ ≥ 1/2. In this case, the intelligent states coincide with the minimum uncertainty states? and are characterized by a Gaussian angle distribution, shown in Fig. 1d). The Fourier relationship dictates that the corresponding OAM spectrum also has a Gaussian envelope. This example also serves to illustrates the difference between the spiral harmonics and the Laguerre Gauss modes. The OAM spectrum expressed in terms of the spiral harmonics (i. e. the sum over all p-indices of the relevant Laguerre Gauss modes) shows the Gaussian envelope, whereas the OAM spectrum for any given order of p may look very different. For the case shown in Fig 1d) the OAM spectrum for Laguerre Gauss modes of order p = 0 for example has a dip for  = 0, and for order p = 1 a spike. The exact modal distribution depends on the choice of the waist for the decomposition, or in an experiment on how the different modes couple to the detector. For mathematical simplicity, we will concentrate in the next two sections on a description in terms of spiral harmonics.

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S

S .1! x

Figure 1. The OAM spectrum of light is modified by a) an  = 2 hologram, b) a phase mask with two-fold symmetry, and c) an amplitude mask with a two-fold sinusoidal angle variation, d) an amplitude mask with Gaussian angular variation ∝ exp(−φ2 /σφ 2 ) where σp hi = 2π/10. Shown is the modal decomposition in Laguerre-Gauss modes characterized by the indices p and  on the left, and the phase and amplitude of the light just after interaction with the masks on the right. For the decomposition the waist of the LG modes was set to be identical to that of the incoming fundamental Gaussian mode LG00 . The modal decomposition shows the contributions of LGp for || ≤ 10 and p = 0 → 4, and additionally a decomposition in terms of spiral harmonics, averaged over all p values, indicated by the row index ’sum’. The OAM spectrum in terms of the spiral harmonics is given by the Fourier decomposition of the aperture function.

In the classical regime, the Fourier relation can be illustrated by angular diffraction (section 3), when the OAM spectrum of light that has passed through an aperture with an azimuthal varying transmission function can be interpreted as a diffraction pattern of the angular grating. Extending angular diffraction to down-converted photon pairs that are entangled in their OAM means that setting the angular mode of one photon determines the OAM distribution of the other photon. This allows the observation of angular ghost diffraction (section 4).

3. ANGULAR DIFFRACTION It can be shown that the OAM spectrum of light passing through an aperture with an amplitude M (φ) is given by the Fourier spectrum of the aperture,  π 1 A = dφM (φ)e−iφ ψ0 . (5) 2π −π For a mask with transmission only within a sector of angle β the OAM spectrum simplifies to14 |A |2 = |ψ0 |2



β 2π

2

sinc2



 β  , 2

(6)

reminiscent of the diffraction pattern of a conventional single slit experiment. However, in the angular case, the sinc function is the envelope for the discrete OAM modes. The angular analogue of a double slit is a mask with two symmetrically placed opening angles, and a grating of N slits corresponds to an angle mask with N-fold rotational symmetry. The opening angle β of the individual sectors corresponds to the width of the slits, whereas the repetition angle α corresponds to the separation of the

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photo diode

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Figure 2. Left: Setup of the angular diffraction experiment. A laser beam is subjected to a spatial light modulator (SLM) that is displaying an angular aperture. The generated OAM sidebands are detected by scanning through  forked holograms and detecting the on-axis intensity via photo detectors that are coupled to single-mode fibres. For simplicity the angular aperture and the -forked holograms are displayed on the same SLM. Right: angular diffraction patterns measured for a) no aperture, b) a single slit aperture with sector width β = π, c) a single slit aperture with sector width β = 2π/3 and d) a three-fold symmetric aperture with sector width β = π/3 and α = 2π/3.

the slits or the grating constant. The sinc2 envelope of the individual sector is multiplied with a function that describes interference between the different sectors, 2

2

|A | = |ψ0 |



β 2π

2

2

sinc



 sin2 (N α2 ) β ,  2 sin2 ( α2 )

(7)

The main difference to normal multiple slit interference is the fact that the repetition angle α is directly linked to the number of opening angles N by α = 2πN. This means that an OAM mode is present in the OAM spectrum of the transmitted light only if its symmetry matches the mask symmetry. We have confirmed this experimentally by shining a laser beam onto a spatial light modulator (SLM) that displayed an angular aperture corresponding to single or multiple slit diffraction gratings.14 We have detected the resulting OAM spectrum by subjecting the transmitted light to an -forked hologram and subsequent on-axis detection with photo detectors. For convenience, and without restricting generality, the angular aperture and the -holograms are phase multiplied and displayed on the same SLM. The detector then registers the OAM spectrum as a function of . The observed spectrum shows the predicted angular diffraction pattern, with an envelope determined by the angular width of the single segment, and a sinusoidal variation of the probability for the different  modes determined by the repetition angle, shown in Fig. 2.

4. ANGULAR GHOST DIFFRACTION The Fourier relationship between angle and OAM applies not only to classical light but holds also for correlations between down-converted photon pairs. This means that defining the angular profile of one photon sets the OAM spectrum of its entangled partner photon, if measured in coincidence.22 This phenomenon is exactly analogous to the better known ordinary ghost diffraction,19, 20 where a linear grating is inserted into one of the arms of the

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down-converted photons, and the resulting interference pattern becomes visible in the counts in the other arm, if measured in coincidence. It is important to note that each of the down-converted light beams is incoherent,25 so that diffraction off a linear or angular diffraction grating, does not result in visible interference fringes in the single counts. Also the OAM spectrum of the down-converted light beams has a broad distribution, centered around  = 0 for a Gaussian pump beam, i.e. each of the down-converted light beams contains a range of OAM states, with a probability amplitude set by the overlap between the OAM state and the pump beam, so that the count rate decreases for larger absolute values of ||. The conservation of angular momentum in the down-conversion process means, however, that a photon with a given OAM of  is always paired with a photon with − : +∞  |Ψ = cn |n| − n. (8) n=−∞

If one photon is transmitted through an angular aperture, thereby changing its OAM decomposition according to the Fourier transform, this automatically affects the OAM modes in which its partner photon can be measured. The coincidence counts therefore show the characteristic sinc2 envelope encountered also in the classical angular diffraction experiment. Experimentally we have demonstrated this using the experimental apparatus shown in Fig. 3. In order to obtain high diffraction rates we have opted for angular phase apertures rather than amplitude apertures. Instead of obscuring the light within given angular segments, these masks shift the phase of the transmitted light by π within the angular segments. The OAM decomposition of such phase and amplitude masks is identical apart from the central contribution at  = 0. The central contribution is decreased by destructive interference between the mode areas at π and 0 phase, leading to a complete suppression for an equal mark ratio. coincidence 400

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Figure 3. Left: Setup of the quantum experiment. A frequency tripled ND:YAG laser at 355nm, incident on a BBO crystal, generates entangled photon pairs at 710nm. Signal and idler photons are coupled using single mode fibre to avalanche photodiodes. Both single and coincidence counts are recorded as functions of the hologram design displayed on the SLMs. Right: Observed OAM spectra. a) Coincidence counts without aperture show correlations predominantly if the signal hologram matches the idler hologram, i.e.  = 0. b) The single channel counts do not depend on the inserted aperture and reflect the conversion efficiency for different  modes. c) Inserting an aperture with two-fold rotational symmetry suppresses all odd OAM values in the coincidence counts. d) A two-fold symmetry with a smaller opening angle increases the width of the sinc2 envelope.

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A frequency-tripled, mode-locked Nd:YAG laser (pulse repetition at 100 MHz, average power 150mW, 355nm) pumps a 3mm long BBO crystal cut for type I phase matching. Signal and idler photons emerge as a cone with a half angle of 4◦ . Signal and idler photons are incident on spatial light modulators (SLMs) that can modify phase and amplitude of the light. In one of the arms (arm B in Fig. 3) the SLM displays a sector aperture that sets the angular profile of the light and alters its OAM contributions. Detection of the various OAM contributions happens by a fork-dislocation hologram followed by detection of the on-axis intensity via singlemode fibre coupled to avalanche photo detectors (60% quantum efficiency) in arm (A). The single counts at each detector are detected independently while scanning the OAM selection holograms, and their coincident counts are monitored simultaneously. Without aperture, the single counts in channel (A) reflect the conversion efficiency of different OAM modes, dependent on the overlap of pump mode and detected signal mode. The efficiency therefore decreases with ||, Fig. 3a). The single count rate in channel (B) of course does not depend on the SLM in channel (A) and remains constant at around 25,000/s. If no aperture is inserted into arm (B), coincidences are detected predominantly if the SLM in arm (A) is set to  = 0, (Fig. 3b)) registering approximately 750/s corresponding to a quantum efficiency of 3%. The gate time of the detection electronics was set to 25ns which kept accidental coincidence counts at roughly 3/s. By inserting sector apertures into the path (B) and scanning the  forked holograms in path (A) the OAM spectrum could be determined. The coincidence counts show the characteristic angular diffraction pattern, known from the classical experiment described in section 3, i.e. angular ghost diffraction. Fig. 3c) and d) show the coincidence and single counts for an angular mask with twofold rotational symmetry. We clearly see a suppression of odd OAM values due to the two-fold symmetry and an sinc2 envelope with a width related to the segment opening angle β, as predicted by the Fourier relationship. This establishes that the Fourier relationship (1) holds for entangled photon pairs.

5. PROPAGATION OF ANGULAR MODE PROFILES When discussing the Fourier relationship and its consequences we have developed the light mode in terms of the spiral harmonics exp(−iφ). These modes describe the angular profile of light modes that are associated with OAM. Any radial dependence of the modes is incorporated in the mode amplitudes A and ψ in Eq. (1,2). Similarly, the holograms used in most set-ups concentrate on defining the azimuthal profile of the light modes and assume a uniform radial distribution. Of course, the spiral harmonics are not eigenmodes of free-space propagation as they are not solutions of Maxwell’s equations. In order to describe propagation dynamics, it is therefore more appropriate to express the light field in terms of the Laguerre-Gauss modes LGp, :  LGp, (r, ϕ, z)

=

  √ || r 2 2p! 1 π(|| + p)! w(z) w(z)   r2 ) −ikz(1+ 2r2 2(z 2 +z 2 ) || R e e−i(2p+||+1)ArcTan(z/zR ) e−iϕ , ·Lp 2 w(z)

(9)

2 is the waist at propagation distance z and z = πw 2 /λ is the Rayleigh range. The where w(z) = w0 1 + z 2 /zR R Laguerre-Gauss modes contain an azimuthal factor exp(−iφ) characterised by the OAM quantum number . Their radial dependence is associated with both the indices p, and : the number of √ concentric intensity rings is given by p + 1, and the radius of the central bright intensity ring increases with . The LG modes defined in (9) have their narrowest waist at z = 0. Compared to a plane wave, the phase of the LG modes differs by the Guoy phase26, 27 (2p + || + 1)ArcTan(z/zR ), so that at the Rayleigh range they have gained a phase of (2p + || + 1)π/4 and in the far field for z → ∞ a phase of (2p + || + 1)π/2. The mode-specific Gouy phase means that the interference between the various LGp, modes varies with propagation distance. As the LG-modes form a complete basis set, any light field can be decomposed into its LGp, contributions. In order to determine the propagation of a mode after a certain aperture, the coefficients of each LGp, mode have to be evaluated and each mode then propagates freely. The total light field is the sum over all propagated LG modes. Fig. 4 shows the propagation of light after an  = 2 hologram and a mask with a Gaussian angle distribution, and Fig. 5 the propagation after a sinusoidal mask and a segment mask with two-fold symmetry. The waist of the decomposition modes is chosen such that most of the light is in p = 0. Although the LG modes form a complete basis set, depending on the aperture and the matching of the waists, the sum over the different

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Figure 4. Propagation of light after an  = 2 hologram (a) and a mask with Gaussian angle distribution (b), corresponding to Fig. 1 a) and b). The amplitude distribution is displayed as a graylevel plot and the phase profile in hue colors. The beam profile is shown directly after the mask (top panel), at the Rayleigh range (middle panel), and in the far field, at 10 times the Rayleigh range (lower panel). For each propagation distance the contribution is shown of the first 3 p-values, p = 0, 1, 2 and the total field up to p = 30. The additional phase contribution due to the curvature of the phasefronts as well as due to propagation is subtracted. Each square has a size of 3 times the waist at the respective propagation distance. Other than in Fig. 1 the waist of the mode decomposition was chosen to be 0.57 times the waist of the input Gaussian, giving the largest contribution in p = 0.

contributions may converge only slowly. This is visible for the -hologram (in Fig. 4a) where due to an upper limit of p = 30 residual rings are visible in the near field. In an experiment, the size of the aperture itself, and the coupling of the resulting light field into single mode fibre determine the waist of the decomposition and set an upper limit to the modes that are considered by the measurement process. In all simulations we include -values up to || = 10. For the  = 2 hologram of Fig. 4a) the only contributing modes are LGp,=2 . The phaseshift induced by the Gouy phase between adjacent p modes is 0 at the beam focus, π/2 at the Rayleigh range and π in the far field. This causes the far field phase and amplitude to resemble the near field phase and amplitude. For an aperture with Gaussian angle distribution shown in Fig. 4b) the OAM spectrum has a Gaussian envelope. The simulations show a narrow angular intensity standard deviation of 18◦ , reflected in the near field intensity profile. The narrow width of the angular distribution corresponds, according to the angular uncertainty relation, to a large width of the OAM distribution, that in the far field becomes a large angular spread. This effect is the exact analogy of the relation between linear position and momentum from near to far field. Light transmitted through a mask with sinusoidal amplitude variation ∝ sin(2φ) (Fig. 5a)) contains only con-

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Figure 5. Propagation of light after a sinusoidal mask (a) or a segment mask with two-fold symmetry, corresponding to Fig. 1 c) and d). The amplitude distribution is shown as a graylevel plot and the phase profile in hue colors. The beam profile is shown directly after the mask (top panel), at the Rayleigh range (middle panel), and in the far field, at 9 times the Rayleigh range (lower panel). For each propagation distance the contribution is shown of the first 3 p-values, and the total field. The additional phase contribution due to the curvature of the phasefronts as well as due to propagation is subtracted. Each square has a size of 3 times the waist at the respective propagation distance.

tributions of LGp,=±2 . As a consequence, just like for the −forked hologram, the intensity profile remains similar from near to far field. A phase mask with segments alternating by π (Fig. 5b)) shows the same phase pattern as the sinusoidal mask, but a different intensity. It contains additional mode contributions of LGp,=±6 , LGp,=±10 ... These higher order sidebands cause additional interference, especially in the near field. The Gouy phase obviously plays an important role in determining the angular diffraction of apertures in the near and far field.

6. CONCLUSIONS We have investigated the Fourier relationship between angular position and OAM both in classical experiments and for entangled photon pairs produced by a down-conversion source. The Fourier relationship provides a convenient tool to link the effect of angular masks on the generated OAM spectrum. Experiments on angular diffraction and angular ghost diffraction confirm the Fourier relationship both in the classical and quantum regime, and thereby establish that angle and OAM are conjugate variables. This has implications for the validity of the angular Heisenberg uncertainty relationship,17 the quantum description of the azimuthal coordinate23 and

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entanglement between angle and OAM.28 We have illustrated that the propagation of light subjected to angular segment apertures is influenced by the Gouy phase between different Laguerre-Gauss modes. Interference between contributions with different Gouy phase leads to interference patterns, particularly in the near field. We are grateful to Bob Boyd, Anand Jha, Steve Barnett and Jacquiline Romero for their important experimental and theoretical contributions to the work presented in this paper. SF-A is a RC-UK research Fellow. This work was supported by the UK EPSRC, the Royal Society, the Wolfson Foundation and the Leverhulme Trust.

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