ISSN 1054660X, Laser Physics, 2010, Vol. 20, No. 2, pp. 325–333. © Pleiades Publishing, Ltd., 2010. Original Russian Text © Astro, Ltd., 2010.
Manifestation of SpinOrbital Interaction of a Photon N. D. Kundikova Institute of Electrophysics, Ural Division of Russian Academy of Sciences, 106 Amundsen str., Yekaterinburg, 620016 Russia SouthUral State University, 76 Lenin Ave., Chelyabinsk, 454080 Russia email:
[email protected] Received June 15, 2009
Abstract—The review of the effects which can be considered as manifestation of spinorbital interaction of a photon (mutual influence of light polarization and propagation) has been presented. The effect of polariza tion influence on propagation and the effect of the propagation influence on polarization under light propa gation through locally isotropic inhomogeneous medium have been described. The notion of spin and angu lar momentum of a photon and their conversion into mechanical momentum have been discussed. The experimental conditions which are suitable for investigation of spinorbital interaction under light propaga tion through optical fiber have been described. DOI: 10.1134/S1054660X10040092
The notion of the spinorbital interaction of pho ton was introduced in [1]. The effect of circularity on the propagation of radiation in optical fiber was pre dicted in [2] and was demonstrated in [1]. In accor dance with a concept based on the similarity of light waves and de Broglie waves, the effect is interpreted as the interaction of the spin angular momentum of pho ton (polarization degree of freedom) with the projec tion of its orbital angular momentum along the axis of optical fiber. A mathematical representation of this statement [3] was derived due to the simulation of the propagation of polarized light in an optical fiber with a step profile of the refractive index and a detailed experimental study of the effect of circularity on the propagation of radiation in optical fiber with the step refractive index [4]. In terms of classical optics, the spinorbital interaction of photon can be considered as the interaction of polarization and propagation. In this work, we review the optical effects that can be considered as the results of the spinorbital interaction of photon.
number of photons in the beam. Such a concept was employed in the experiment performed in 1936 [5]. In the experiment, the passage of the circularly polarized radiation through a quartz halfwave retardation plate suspended in vacuum on a thin quartz thread (Fig. 1) results in the reversal of circularity. The plate is rotated owing to the conversion of the spin angular momen tum of photon into the torque of the plate, such that the direction and angle of rotation are in agreement with the theoretical predictions. Note also that the experimental curve of the angle of rotation of the half wave retardation plate versus the incident intensity is in agreement with the theoretical curve. Thus, the experiment from [5] proves the existence of the spin angular momentum of photon and its trans fer to the medium. In the framework of classical optics, the orbital angular momentum of photon is related to propaga
First, we separately consider the spin and orbital angular momenta of photon. As was mentioned, the spin angular momentum is related to the polarization of light. The rightcircularly polarized radiation can be considered as a flux of photons with a spin of +1 and a spin angular momentum of +ប, whereas the leftcir cularly polarized radiation can be represented as a flux of photons with a spin of –1 and a spin angular momentum of –ប. The linearly polarized radiation (a superposition of the right and leftcircularly polar ized beams) represents a photon flux with equal num bers of the photons with spins of +1 and –1. A medium in which the leftcircularly polarized radia tion is transformed into the rightcircularly polarized radiation must acquire a torque of +2nប, where n is the 325
+ប
–ប
Fig. 1. Optical scheme illustrating the experiment from [5].
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KUNDIKOVA x
equation in the paraxial approximation for the propa gation along the z axis is written as
s l
l
2
l C r 2 2r u pl ( r, φ, z ) = L p 2 2 1/2 w ( z ) 2 w (z) ( 1 + z /z R )
r(l) z
2
2
–r – ikr z exp exp ( – ilφ ) × exp 2 2 2 w (z) 2 ( z + zR )
y
(3)
–1 z . × exp i ( 2p + l + 1 ) tan zR
Fig. 2. Beam trajectory in an optically inhomogeneous medium.
l
tion of light. A beam trajectory (Fig. 2) is given by the known equations dr = s, dl
(1)
ds = ∇ ln n – s ( s ⋅ ∇ ln n ), dl where s is a unit vector of the tangent to the trajectory, l is the length of trajectory, r is the radius vector of a point on trajectory, and n(x, y, z) is the refractive index of the optically inhomogeneous medium. The equa tions show that the shape of the beam trajectory is determined by the inhomogeneity of the refractive index. On the assumption that the beam represents a photon flux, we can define the angular momentum of photon as M = r × p,
(2)
where p = ប(ω/c)n(r), ប is the Planck constant, c is the velocity of light, and ω is the radiation frequency. In the framework of wave optics, we can define the angular momentum of photon using the approach from [6]. We consider the Laguerre–Gaussian beams, for which complex scalar function of the distribution of field amplitude upl(r, φ, z) that satisfies the wave
l = +1
Here, r, φ, and z are cylindrical coordinates; L p is the Laguerre polynomial; l is the azimuthal number; p is the radial number; zR is the Rayleigh length; w(z) is the beam radius at distance z; k = 2π/λ is the wave num ber; λ is the radiation wavelength; C is constant; and the beam waist is located at z = 0. In expression (3), phase term exp(–ilφ) describes the helical wave front, such that the sign of the exponent determines the direction of rotation of the wave front and number l is often called topological charge [7]. Note that the amplitude of a Bessel beam also exhibits the azimuthal angular dependence determined by the phase term exp(–ilφ). Figure 3 demonstrates the experimentally measured interference patterns of the Bessel beams of the first order with topological charges of different signs (l = ±1) and a divergent Gaussian beam [8]. The phase term exp(–ilφ) determines the spiral interfer ence pattern, such that the direction of spiral rotation depends on the sign of topological charge l. For the polarized Laguerre–Gaussian beam, the expression for the z component of the total angular momentum per unit power is given by [6] 2
l 2 σr ∂ u M z = u + . ω 2ω ∂r 2
(4)
2
Here, u ≡ u ( r, φ, z ) , σ = ±1 for the right and left circularly polarized light, σ = 0 for the linearly polar ized light, and ω = 2πc/λ is the radiation frequency.
l = −1
Fig. 3. Interference patterns of a divergent Gaussian beam and the Bessel beams with different signs of helicity. LASER PHYSICS
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Expression (4) shows that, for the polarized light, the z component of the total angular momentum (the pro jection of the total angular momentum along the direction of propagation) contains the term that is related to the azimuthal number l and the term that is determined by the polarization, so that the orbital angular momentum of photon depends on its spin angular momentum. Evidently, the orbital angular momentum of pho ton can be transferred to the medium. An experimen tal scheme for the observation of such a transfer was proposed in [6] (Fig. 4). The main element of the experimental setup is the astigmatic system that pro vides the sign reversal of the topological charge of the Laguerre–Gaussian beam. The sign reversal of the topological charge must lead to the conversion of the orbital angular momentum of the light beam into the torque transferred to the optical system consisting of two cylindrical lenses. The astigmatic optical system suspended on a thread in vacuum must be rotated at a certain angle owing to the conversion of the orbital angular momentum into the torque. To the best of our knowledge, the experiment was not performed and the transfer of the orbital angular momentum (as well as the spin angular momentum) to the medium was demonstrated in the experiments on the manipulation of microscopic objects using focused laser beams. For the first time, the trapping of trans parent particles with sizes ranging from 10 μm to 25 nm in the focus of laser beam in water was demon strated in 1986 [9]. The trapping is related to the radi ation pressure, which is sufficient for manipulations with micro and nanoparticles. The transfer of the angular momentum (spin and orbital) to an absorbing particle was demonstrated in several works [10–13]. Comprehensive data can be found in [14]. The inde pendent transfer of the spin and orbital angular momenta of photon was also proven. The transfer of the spin angular momentum to a microparticle results in the rotation of particle around its axis, and the transfer of the orbital angular momentum causes the rotation of particle around the beam axis [12]. Note that the orbital angular momentum of a single photon was experimentally measured in [15]. Based on the possibility of the transfer of the spin and orbital angular momenta of photons to the medium, we must admit a possibility of the reverse process lying in the transfer of torque to a photon. The simplest implementation of such a process involves the transfer of the spin angular momentum of photon to the medium and, then, the conversion of torque into the orbital angular momentum of photon or the trans fer of the orbital angular momentum of photon to the medium and, then, the conversion of torque into the spin angular momentum of photon. In the first case, the circularly polarized radiation with the plane wave front is incident on the medium, and the output radi ation represents the linearly polarized radiation with LASER PHYSICS
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+l ប
–l ប
Fig. 4. The experimental optical scheme proposed in [6] for the demonstration of the transfer of the orbital angular momentum of photon to the medium.
the spiral wave front. The experiment from [8] can be interpreted in accordance with such a scenario. In the experiment, the circularly polarized radiation with the known direction of rotation is delivered to an optical fiber. The output radiation having passed through a polarization system represents the linearly polarized radiation with the spiral wave front (l = ±1) such that the sign of circularity depends on the sign of the topo logical charge. The conversion of the orbital angular momentum of photon into the spin angular momen tum in an optically inhomogeneous anisotropic medium is presented in [16]. In the experiment, the circularly polarized radiation with the known direc tion of rotation, which has a certain spin angular momentum, is transmitted through a liquidcrystal cell with an inhomogeneous distribution of the direc tions of local optical axes. The transmitted light acquires the nonzero topological charge l = ±2, and the sign of the topological charge depends on the sign of the spin angular momentum [16]. Thus the laser beams can be characterized by spin and orbital angular momenta, which can be trans ferred to the medium and can be mutually converted. In the framework of classical optics, the spin orbital interaction of photon can be considered as the interaction of polarization and propagation. Consider the effects of beam trajectory on the polarization state and the effects of polarization on the propagation. In the paraxial approximation, the propagation of relatively narrow beams in an optically homogeneous medium is described using field E(r, t) represented as E ( r, t ) = ( c 1 e x + c 2 e y )A ( x, y, z )e
– i ( ω ( k 0 )t – k 0 z )
.
(5)
Here, (c1ex + c2ey) is the vector that determines the polarization state of the beam; ex and ey are unit vector in the x and y directions, respectively; c1 and c2 are complex numbers that determine the polarization state; r ≡ (x, y, z), A(x, y, z) is the complex amplitude of the beam; and k0 is the vector along the propagation direction, whose magnitude is |k0| ≡ k0 = 2π/λ. Expres
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KUNDIKOVA Lens Fuber Screen Laser Polarization system
Fig. 5. Optical scheme of the setup for the first observation of the optical Magnus effect.
sion (5) describes the field for which the polarization is independent of the amplitude. The simplest example of the refraction of light at an interface of two media shows that the polarization of light and its propagation are not independent in the optically inhomogeneous medium. The propagation of the linearly polarized light along a nonplanar trajectory was studied in [17], and the scenario in which the tangents to the trajectory at the start and end points are parallel was analyzed in [18]. In these cases, the polarization rotation takes place and the angle of rotation is numerically equal to the solid angle formed on a unit sphere by the tangent to the trajectory. This result was later reproduced in [19] using the adiabatic quantummechanical Berry theorem [20]. In the experiments, the rotation of the polarization plane was observed upon the propagation of the linearly polarized light in the singlemode opti cal fiber twisted as a helix [21]. For the first time, a possibility of the effect of polar ization on the trajectory upon the total reflection was discussed in [22, 23]. The longitudinal shift of the lin early polarized light was demonstrated, and the polar izationdependent shift was on the order of magnitude of the radiation wavelength. Such a shift was measured in the experiments on the propagation of the linearly polarized beams with different azimuths in a planar waveguide [24, 25]. The penetration of light into the medium with a smaller refractive index was experi mentally demonstrated in [26, 27], and the geometri cal interpretation of the longitudinal shift was pro posed in [28]. For the total reflection of the circularly polarized light, a transverse shift of the beam center takes place. Such a transverse shift is comparable with the radia tion wavelength, and the direction of the shift depends on the sign of the circular polarization [29, 30]. In the experiments, the transverse shift is observed when a beam with the nonuniform polarization over the cross section passes through a polyhedral prism. One half of the beam has the righthanded circular polarization, and another half of the beam has the left
handed circular polarization. In the output beam, the two parts are shifted relative to each other owing to multiple total reflections inside the prism. The direc tion and value of the shift are in qualitative agreement with the predictions from [31–33]. Note that the lin early polarized beam, which represents a superposi tion of the left and rightcircularly polarized beams, must be divided into two parts. However, the shifts from [30–32] were insufficient for such splitting. The study of the longitudinal shift upon the total reflection is virtually terminated but the theoretical analysis of the transverse shift is continued [34–40]. The transverse shift upon the propagation of a circu larly polarized beam from a medium with a lower refractive index to a medium with a higher refractive index is demonstrated in [40]. Note opposite shifts for the refracted and reflected beams. The effect in photo nic crystals has gathered much recent interest [40]. Before 1990, the works were independent, the interaction of polarization and propagation was not taken into account, and the spinorbital interaction of photons was disregarded. On the assumption that the effect of trajectory on polarization must be supplemented with the effect of polarization on trajectory, Zel’dovich and Liberman [2] analyzed the effect of circular polarization on the propagation of circularly polarized light in optical fiber. The authors indicate the twisting of the originally plane trajectory of the circularly polarized meridian beam in the optical waveguide with an infinite para bolic profile of refractive index. However, the resulting expressions yield the rotation of the speckle pattern at the exit of the fiber upon the sign reversal of the circu lar polarization. For the fiber with the infinite para bolic profile of the refractive index, the speckle pattern remains unchanged and the rotation angle of the speckle pattern is given by δn λz, δφ = 2 2 πρ n co LASER PHYSICS
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σ = −1
329
σ = +1
Fig. 6. Photographs of the fragments of the speckle patterns for the radiation having passed through the optical fiber with a step profile of refractive index on the screen with a mesh on polar coordinates.
where ρ is the radius of the fiber core, δn = nco – ncl, nco is the refractive index at the axis of the fiber, ncl is the refractive index at the interface of the core, and z is the fiber length. The effect was experimentally demonstrated for an optical fiber with a step profile of refractive index [1, 4]. It was called the optical Magnus effect and was interpreted as a result of the spinorbital interaction of photon. The name comes from the Magnus effect known in mechanics since 1852. In accordance with the mechanical Magnus effect, a force that is perpen dicular to the rotation axis is exerted on an object rotating around an intrinsic axis and the direction of force depends on the direction of rotation. Such a force results in a deviation from the trajectory that cor responds to the motion of a nonrotating object. Figure 5 shows the optical setup that was used for the first observation of the optical Magnus effect [1]. The linearly polarized radiation of a helium–neon laser passes through a polarization system that pro vides the conversion of the linearly polarized radiation to the circularly polarized radiation with a variable sign of the circular polarization. The circularly polar ized radiation is focused by an objective on the entrance of a multimode optical fiber with a step pro file of the refractive index. The speckle pattern of the radiation having passed through the optical fiber can be observed on the screen. The sign reversal of the cir cular polarization causes the rotation and distortions of the speckle pattern. The numerical simulation of the propagation of the circularly polarized radiation in the optical fiber with a step profile of refractive index [4] shows that, in contrast to the result from [2], the rotation of the speckle pattern must be supplemented with distortions. The calculations from [41] for the optical fiber with a bounded parabolic profile of refractive index show that the rotation of the speckle pattern must be supplemented with the distortions of the speckle pattern. However, the observations of the optical Magnus effect in such a fiber are missing. Figure 6 demonstrates two fragments of speckle patterns for different circular polarizations. We clearly LASER PHYSICS
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observe the distortions of speckle patterns. The arrow shows one of the spots with a developed shift relative to the mesh on the screen. The experimental angle of rotation of the speckle pattern (+1.4 ± 0.5) is in good agreement with the result of the numerical simulation (+1.5 ± 0.5) [4]. On the assumptions of transverse electromagnetic waves, a relatively small optical inhomogeneity (Δn Ⰶ n), and the absence of reflections, (t|| = t⊥ = 1)Liber man and Zel’dovich [3] derived a phenomenological expression for the transverse displacement of the tra jectory related to the circularity of polarization. With allowance for the polarization effect, the modified equations for the beam trajectory are written as c ( s × ∇ ln n ), dr = s – σ dl ωn ds = ∇ ln n – s ( s ⋅ ∇ ln n ). dl
(7)
The first equation shows that the circular polarization leads to the deflection of the trajectory in the direction perpendicular to the vector of tangent to the trajectory s and the direction of deflection depends on the sign of circularity σ (σ = ±1). A transition from the equations for the beam trajec tory to the canonical equations with Hamiltonian Ᏼ is also presented in [3]. In the Hamiltonian, instanta neous coordinates of the trajectory r and the complex 3D vector e related to the polarization serve as coordi nates and vector momentum of photon p and complex 3D vector u serve as momenta. Along the entire trajec tory, coordinates (r, e) and momenta (p, u) satisfy the following conditions: p = ប⎛ ω ⎞ n ( r ), ⎝ c⎠ u = iបe*,
e ⋅ p = 0,
e ⋅ e* = 1,
u ⋅ p = 0, 2
u ⋅ u* = ប ,
if the expressions were valid at the initial moment. Such momenta and coordinates provide only two complex polarization degrees of freedom and the
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KUNDIKOVA
ρ
b
Fig. 7. Fictitious trajectory of oblique (sagittal) beams in a multimode optical fiber with a step profile of refractive index.
transverse polarization of electromagnetic waves. Hamiltonian Ᏼ is represented as
Ᏼ ( p, u, r, e ) c c = p – p ⋅ [ ( e × u ) × ∇ ln n ( r ) ], n(r) n ( r )p
(8)
where p ≡ (pp)1/2. Using Hamiltonian Ᏼ, we can rep resent canonical equations ∂Ᏼ dr = , dt ∂p
∂Ᏼ dp = – , dt ∂r
∂Ᏼ de = – , dt ∂u
∂Ᏼ du = dt ∂e
as dr c [ ( e × u ) × ∇ ln n ( r ) ] = c p – dt n p np c + p { p ⋅ [ ( e × u ) × ∇ ln n ] }, 2 np dp cp ∂ ln n c = – [ ( e × u ) × ∇ ln n ( r ) ] k dt i n ∂x i np 2
∂ ln n ∂ ln n ∂ ln n × – , ∂x i ∂x k ∂x i ∂x k
(9)
de c ( p ⋅ e )∇ ln n, = – c p ( e ⋅ ∇ ln n ) + dt np np du = – c p ( u ⋅ ∇ ln n ) + c ( p ⋅ u )∇ ln n. dt np np Eliminating time, we derive the following equations for the beam trajectory: dr c c = s – σ ( s × ∇ ln n ) , dt n ωn ds = cp [ ∇ ln n – s ( s ⋅ ∇ ln n ) ], dt n
(10)
de = – c s ( e ⋅ ∇ ln n ). dt n Here, we employ the notation s = p/p and σ = (e × e*) · (p/p). The first equation is similar to the phenomeno logical equation and describes the optical Magnus effect. The second equation coincides with the known
equation of trajectory. The third equation describes a variation in polarization of the linearly polarized beam upon a variation in the trajectory (Rytov rotation). For an optical fiber with an infinite parabolic profile of refractive index, a correction to the Hamiltonian that takes into account the spinorbital interaction of pho ton is written as 2
c 1 dn δ Ᏼ = – 3 ( Σ ⋅ M ). បωn r ⊥ dr ⊥
(11)
Here, Σ = e × u is the spin of photon and M = r × p is the orbital angular momentum of photon. The spin orbital interaction of photon is determined by the product of spin angular momentum of photon Σ and orbital angular momentum M. Such a representation of the correction to the Hamiltonian makes it possible to describe the optical Magnus effect and to derive an expression for the angle of rotation of the speckle pat tern for the radiation propagating in the optical fiber with an infinite parabolic profile of refractive index upon the sign reversal of the circular polarization: δn λz, δφ = 2 2 πρ n co
(12)
which coincides with expression (6) derived in [2]. The effect of trajectory on polarization was observed in an optical fiber twisted as a helix [21] whereas the optical Magnus effect (the effect of polar ization on trajectory) was observed in a rectilinear fiber [1, 4]. Note that mutually reversed effects must be observed under identical conditions. Consider the propagation of oblique (sagittal) beams in a rectilinear optical fiber. Such beams are excited when a narrow laser beam is directed at a certain angle relative to the fiber axis to the point that is located at a certain (impact) distance from the fiber axis. In a conven tional waveguide, a notion of beam is applicable only at distances l < 0.1 cm, since the beam decays to the waveguide modes, which propagate at different veloc ities and yield the speckle pattern due to the interfer ence. The propagation of such a beam in the waveguide is manifested as a ring structure of the out put speckle pattern, such that the radius of the speckle pattern depends on the angle between the direction of the incident beam and the waveguide axis. We may LASER PHYSICS
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331
Screen
Po
Polarization system
l ar i ze r
Fiber Laser Lens
Fig. 8. Optical scheme of the experimental setup for the observation of the polarization rotation upon the propagation of light along a nonplanar trajectory and the optical Magnus effect.
L2 L1 2 2 Io
8
10 4
I'0
1 Ipc
9 6
7
5
Is0
3
1
Fig. 9. Experimental scheme for the observation of the splitting of the polarized beam into two beams with orthogonal polariza tions: (1) semitransparent mirror, (2) and (7) totally reflecting mirrors, (3) lens, (4) objective, (5) and (6) quarterwave retardation plates, (8) BNN crystal, (9) optical fiber, (10) CCD array, and (L1) and (L2) He–Ne lasers.
assume that the trajectory in Fig. 7 is the beam trajec tory in spite of the fact that the notion of beam is meaningless in the optical waveguide. It is seen that the trajectory is similar to a helix, and the pitch of helix depends on the angle of incidence of the incoming beam. Such a representation prompts the polarization rotation of the linearly polarized incident radiation. Figure 8 shows the optical scheme of the setup for the observation of the optical Magnus effect and the polarization rotation upon the propagation of light along a nonplanar trajectory (intrafiber polarization rotation) [42, 43]. The linearly polarized radiation of a helium–neon laser passes through a polarization system, which pro vides the linearly polarized radiation with the given polarization azimuth or the circularly polarized radia LASER PHYSICS
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tion with the given sign of circularity. The radiation is focused by an objective to the entrance of the fiber at a certain angle ψ. Using a microscope and a side mirror, we control the point at which the radiation is delivered to the fiber. The ringshaped speckle pattern is observed on the screen that is placed at a certain dis tance from the fiber end. Angle of incidence ψ is found from the diameter of the speckle pattern and the dis tance from the output end to the screen. In the case of the linearly polarized incident radiation, the output radiation is predominantly linearly polarized and the polarization plane is rotated by the angle depending on input angle ψ. The azimuth of the linear polarization at the exit of the fiber is determined using a polariza tion analyzer that is placed in front of the screen. The direction of rotation depends on the point of inci dence at the entrance of the fiber (i.e., the direction of
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KUNDIKOVA
Fig. 10. Image of the beam splitting into two circularly polarized beams.
rotation of the helical trajectory). The experimental results are in good agreement with the results of the calculations in the geometricoptical approximation [42] and the data from [44]. The same experimental setup is used for the obser vation of the optical Magnus effect [43]. When a nar row beam with the circular polarization is incident at a certain angle relative to the fiber axis, the speckle pat tern also represents a ring and the sign reversal of the circular polarization causes the rotation of the speckle pattern, such that the angle of rotation increases with an increase in the diameter of the speckle pattern, which depends on angle of incidence ψ. The above analysis shows that the leftcircularly polarized radiation and the rightcircularly polarized radiation propagate in the optical fiber along different trajectories although the notion of trajectory becomes meaningless. In the case of the linearly polarized inci dent radiation (a superposition of the left and right circularly polarized radiation), the output speckle pat terns are overlapped and cannot be separated. When the incident radiation corresponds to the speckle pat tern with the amplitude and phase distributions observed for the output radiation, the optical Magnus effect must give rise to two output beams with orthog onal circular polarizations. Such a splitting of one beam into two circularly polarized beams can be observed in the experiments on the setup presented in Fig. 9 [45–47]. Two helium–neon lasers serve as the radiation sources. The linearly polarized radiation of laser L1 is divided into reference (I0) and signal (Is0) beams using the semitransparent mirror. Signal beam Is0 passes through quarterwave retardation plate 5, acquires cir cular polarization, and is incident on the entrance of the rectilinear fiber with step profile of refractive index 9. The radiation transmitted by the optical fiber is col lected by objective 4 and is incident on photorefractive crystal 8. Quarterwave retardation plate 6 sets the needed polarization state of signal beam Is0 before its incidence on the photorefractive crystal. The refer ence and signal beams with linear polarizations in the
plane of the table record a dynamic diffraction grating in the photorefractive crystal. Radiation of the second laser I 0' is oppositely directed relative to the reference beam and is diffracted by the recorded dynamic holo graphic grating. This gives rise to the phaseconju gated light wave Ipc whose phase and amplitude distri butions are identical to the corresponding distribu tions of wave Is0. Note that the phase conjugated and signal waves propagate in opposite directions. Thus, the fourwave mixing in the photorefractive crystal yields a phaseconjugation mirror that provides the phase conjugation of the wave having passed through the optical fiber. A variation in the orientation of the quarterwave retardation plate allows a variation in the polarization state of the phaseconjugated wave. In the case of the linearly polarized conjugated wave at the entrance of the fiber, the leftcircularly polarized radi ation and the rightcircularly polarized radiation propagate in the fiber along different trajectories and we observe two circularly polarized beams with equal intensities at the exit of the fiber. Figure 10 demon strates the beams imaged by the CCD array. Such a method for the beam splitting makes it possible to determine the dependence of the rotation angle of speckle pattern on its radius [46]. A possibility of the measurement of a relatively small ellipticity is ana lyzed in [47]. Thus, the experiments prove that a beam with an arbitrary ellipticity can be split into two circularly polarized beams with the intensity ratio depending on the ellipticity of the original beam. The splitting of a linearly polarized beam into two circularly polarized beams was later demonstrated in [48] under different experimental conditions. The reviewed optical effects can be considered as manifestations of the spinorbital interaction of photon. REFERENCES 1. A. V. Dugin, B. Ya. Zel’dovich, N. D. Kundikova, and V. S. Liberman, JETP Lett. 53, 197 (1991). 2. B. Ya. Zel’dovich and V. S. Liberman, Quantum Elec tron. 20, 427 (1990). 3. V. S. Liberman and B. Ya. Zel’dovich, Phys. Rev. A 46, 5199 (1992). 4. A. V. Dooghin, N. D. Kundikova, V. S. Liberman, and B. Ya. Zel’dovich, Phys. Rev. A 45, 8204 (1992). 5. R. A. Beth, Phys. Rev. 50, 115 (1936). 6. L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, Phys. Rev. A. 45, 8185 (1992). 7. M. S. Soskin, V. N. Gorshkov, and M. V. Vasnetsov, Phys. Rev. A 56, 4064 (1997). 8. M. Ya. Darsht, I. V. Kataevskaya, N. D. Kundikova, and B. Ya. Zel’dovich, Sov. Phys. JETP. 80, 817 (1995). 9. A. Ashkin, J. M. Dziedzic, J. E. Bjorkhom, and S. Chu, Opt. Lett. 11, 288 (1986). 10. S. M. Barnett and L. Allen, Opt. Commun. 110, 670 (1994). LASER PHYSICS
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MANIFESTATION OF SPINORBITAL INTERACTION OF PHOTON 11. N. B. Simpson, K. Dholakia, L. Allen, and M. J. Pad gett, Opt.Lett. 22, 52 (1997). 12. T. O’Neil, I. MacVicar, L. Allen, and M. J. Padgett, Phys. Rev. Lett. 88, 053601 (2002). 13. V. G. Volostnikov, S. P. Kotova, N. N. Losevskii, and M. A. Rakhmatulin, Sov. J. Quantum Electron. 32, 565 (2002). 14. L. Allen, S.M. Barnett, and M.J. Padgett, Optical Angu lar Momentum (Institute of Physics Publishing, Lon don, 2003). 15. J. Leach, M. J. Padgett, S. M. Barnett, S. Franke Amold, and J. Courtial, Phys.Rev.Lett. 88, 257901 (2002). 16. L. Marrucci, C. Manzo, and D. Paparo, Phys. Rev. Lett. 96, 163905 (2006). 17. S. M. Rytov, Dokl. Akad. Nayk. 18, 2 (1938). 18. V. V. Vladimirsky, Dokl. Akad. Nayk. 21, 222 (1941). 19. R. Y. Chiao and Y.S. Wu, Phys. Rev. Lett. 57, 933 (1986). 20. M. V. Berry, Proc. Roy. Soc. A. 392, 45 (1984). 21. A. Tomita and R. Y. Chiao, Phys. Rev. Lett. 57, 936 (1986). 22. J. Picht, Ann. Physik 3, 433 (1929). 23. J. Picht, Physik Z. 30, 905 (1929). 24. F. Goos and H. Hanchen, Ann. Physik. 1, 333 (1947). 25. F. Goos and H. Hanchen, Ann. Physik. 5, 251 (1949). 26. P. Acloque and C. GuiHemet. Compt. Rebd. 250, 4328 (1960). 27. H. Osterberg and L. W. Smith, J. Opt. Soc. Am. 54, 1078 (1964). 28. C. A. Risset and J. M. Vigoureux, Optics Comm. 91, 155 (1992).
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29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45. 46. 47. 48.
333
F. I. Fedorov, Dokl. Akad. Nayk 105, 465 (1955). N. Kristofel, Proc. Tartu Univ. 42, 94 (1956). C. Imbert, Phys. Lett. A 31, 337 (1970). C. Imbert, Phys. Rev. D 5, 787 (1972). O. Costa de Beauregard and C. Imbert, Phys. Rev. Lett. 28, 1211 (1972). V. G. Fedoseev, Optics and Spectroscopy 58, 296 (1985). V. G. Fedoseev, J. Opt. Soc. Am. A 3, 826 (1986). V. G. Fedoseev, Optics and Spectrosc. 71, 483 (1991). V. G. Fedoseev, Optics and Spectrosc. 71, 570 (1991). V. G. Fedoseyev, J. Phys. A 21, 2045 (1988). V. G. Fedoseyev, Opt. Commun. 193, 9 (2001). M. Onoda, S. Murakami, and N. Nagaosa, Phys. Rev. Lett. 93, 083901 (2004). N. D. Kundikova and G. V. Chaptsova, J. Opt. A: Pure Appl. Opt. 1, 341 (1999). B. Ya. Zel’dovich and N. D. Kundikova, Quantum Electron. 25, 172 (1995). B. Ya. Zel’dovich, I. V. Kataevskaya, and N. D. Kundi kova, Quantum Electron. 26, 87 (1996). V. S. Liberman and B. Ya. Zel’dovich, Pure Appl. Opt. 2, 367 (1993). M. Ya. Darscht, B. Ya. Zel’dovich, R. Kowarschik, and N. D. Kundikova, Proc. Chel. Sci. Centr. 2, 10 (2003). S. A. Assel’born, M. V. Bol’shakov, N. D. Kundikova, and I. I. Naumova, Proc. Chel. Sci. Centr. 3, 6 (2003). S. A. Asselborn, N. D. Kundikova, and I. V. Novikov, Proc. SPIE D 6024, 60240 (2006). K. Y. Bliokh, A. Niv, V. Kleiner, and E. Hasman, Nature Photonics 2, 748 (2008).
