714 J. Opt. Soc. Am. B / Vol. 29, No. 4 / April 2012 A. Okulov Rotational Doppler shift of a phase-conjugated photon A. Yu. Okulov Russian Academy of Sciences, Moscow, Russia (alexey.okulov@gmail.com) Received September 27, 2011; revised December 12, 2011; accepted December 29, 2011; posted January 6, 2012 (Doc. ID 155277); published March 20, 2012 The rotational Doppler shift of a photon with orbital angular momentum ℓℏ is shown to be an even multiple of the angular frequency Ω of the reference frame rotation when the photon is reflected from the phase-conjugating mirror. The one-arm phase-conjugating interferometer is considered. It contains N Dove prisms or other angularmomentum-altering elements rotating in opposite directions. When such interferometer is placed in the rotating vehicle, the δω ˆ 4…N ‡ 1∕2†ℓ · Ω rotational Doppler shift appears. As a result, the helical interference pattern will rotate with angular frequency δω∕2ℓ. The accumulation of angular Doppler shift via successive passages through the N image-inverting prisms is due to the phase conjugation; for a conventional parabolic retroreflector, the accumulation is absent. The features of such a vortex phase-conjugating interferometry at the single-photon level are discussed. © 2012 Optical Society of America OCIS codes: 020.7010, 030.6140, 050.4865, 070.5040, 140.3560, 160.1585. 1. INTRODUCTION Single-photon interferometry utilizes the superposition of mutually coherent quantum states Ψj [1] to which a photon belongs simultaneously. The interference pattern depends on a method of Ψj preparation. A double-slit Young interferometer creates two free-space wave functions, Ψ1 and Ψ2, whose interference pattern, produced by detection of the individual photons, is recorded by an array of detectors or a photographic plate located in the near or far field. In a Mach– Zehnder configuration [2,3], two wave functions separated by an entrance beam splitter recombine at the output beam splitter. The Michelson interferometer recombines at the input beam splitter two retroflected quantum states, provided these states are phase locked and their path difference δL is smaller than the coherence length Lc. Thus, the interference pattern is simply ∼‰1 ‡ V …δL† · cos…k · δL†Š, where V …δL† is a visibility or second-order correlation function and k ˆ 2π∕λ. When retroreflection is accompanied by wavefront reversal realized with phase-conjugating mirrors (PCMs) [4] based upon stimulated Brillouin scattering [5,6], photorefractivity [7,8], or holographic PCMs, the optical path δL difference is almost entirely compensated due to phase conjugation (PC). Noteworthy is the observable phase lag via the relatively small frequency shift δω ˆ ωf − ωb arising due to the excitation of internal waves inside the PCM volume [9], where ωf and ωb are the carrier frequencies of incident and PC-reflected photons, respectively. This leads to the frequency shift modulated interference term 1 ‡ V …δL† · cos…δk · δL†, where δk ˆ δω∕c [6]. The aim of this article is to present the other nontrivial property of the phase-conjugated optical vortex fields and their single photons. The predicted effect is in the accumulation of the small Doppler frequency shifts [3,10–14] caused by very slow rotations of the phase-conjugated interferometer as a whole and the optical components therein. This effect takes place for the photon in the optical vortex quantum state [2,3] with topological charge ℓ, where the angular momentum Lz ˆ ℓ · ℏ [15] is due to the phase singularity located at propagation axis z. Hereafter, the spin component of angular momen- tum [16] is supposed to be zero due to the linear polarization. It is convenient to use the single-photon wave functions that coincide with the positivepfrequency component of the electric field envelope jΨi ˆ 2ϵ0 · E…t; r~† [17]. The square modulus of Ψ is proportional to the energy density of the continuous wave (CW) laser beams and to the photon count rate in different fringes of the interference pattern for the single-photon experiments [18]. We will assume that Ψ is a Laguerre–Gaussian (LG) beam with ℓℏ orbital angular momentum (OAM) per photon [9], but any other isolated vortex solutions, e.g., Bessel vortices [19,20] will led to the same results: Ψ…f ;b†…z; r; θ; t† ∼ …Er0…∕fD;b†0p†jℓ2jϵe0x p· ex−pD‰−0i2ω…1……f1;‡br†‡2tizi∕ziz∕kRz…†Rf;†b;†z  iℓθŠ zR ˆ k…f ;b†D02; (1) where the cylindrical coordinates …z; r; θ† are used, D0 is the vortex radius, zR is the Rayleigh range, Ψf and Ef stand for the forward wave, propagating in the positive Z direction, and Ψb and Eb stand for the conjugated wave propagating in the opposite direction. Of special interest is the subhertz-order frequency splitting δω∕2π ˆ c…kf − kb†∕2π, which appears due to the slow mechanical rotation of setup [3,21]. It was already shown that rotation of the λ∕2 wave plate with angular frequency Ω ∼ 2π…1–100† rad∕s in one arm of a Mach–Zehnder interferometer induces the rotational Doppler shift (RDS) δω ˆ 2Ωℓ for circularly polarized broadband CW with linewidth Δω∕2π ≃ 1010 Hz. In this configuration, the broadband spectrum was shifted as a whole via mechanical rotation (by the angular Doppler effect) at δω∕2π ˆ 2 · 7 Hz and the beats at the output mirror induced an appropriate rotation of the interference pattern [22]. 0740-3224/12/040714-05$15.00/0 © 2012 Optical Society of America A. Okulov 2. PHASE-CONJUGATING MIRROR IN A REST FRAME Let us consider first the single-arm phase-conjugating vortex interferometer (PCVI) when the PCM is in the rest frame. Because of the reflection from the PCM, the helical photon with a linear polarization proves to be in a superposition of the two counterpropagating quantum states Ψf ;b [Fig. 1]. Currently the best candidate for the ideal single-photon PCM is a thick hologram written with sufficiently high diffraction efficiency (R ∼ 0.9) for the ℓ-charged optical vortex [3,23,24]. In such a case, the amplitudes of the forward and backward fields are close to each other and visibility of the interference pattern V …δL† is close to 1, provided that coherence length Lc ∼ 2πc∕Δω is bigger than the doubled length of the PCVI. The ideal PCM ensures the perfect coincidence of the helical phase surfaces of the counterpropagating optical vortices Ψf ;b and zeros of their electric field amplitudes on the z axis. In contrast to the speckle fields whose interference pattern is composed of intertwined Archimedean screws [25], in a PCVI the isolated Archimedean screw pattern appears both for the single photon with an LG wave function and for CW. This results in the intensity profile Itw composed of 2ℓ twisted fringes [7,26,27]: z0 ˆ z − zpc; jΨj2 ˆ jΨf ‡ Ψbj2 ∼ Itw…z0; r; θ; t† ˆ 2ϵ0cjE…f ;b†j22…jℓj‡1†…r∕D0†2jℓj πℓ!D02…1 ‡ z02∕zR2† · exp  − D02…1 2r2  ‡ z02∕zR2† × ‰1 ‡ R2 ‡ 2R · cos‰δω · t − …kf ‡ kb†z0 ‡ 2ℓ芊; (2) where zpc is the location of the PCM entrance window. Geometrically, the infinitesimal rotation of the Dove prism at the angle dθ leads to the appropriate phase change of dθ · 2ℓ [28,29]. Thus, the angular speed of pattern rotation θ_ ˆ δω∕2ℓ is given by differentiation of the self-similar argument 2θ…t† · ℓ ‡ δω · t − …kf ‡ kb†z0 [22] versus time t. Consider the origin of RDS δω [3,10–12,14] for the photon with topological charge ℓ after the double passage through a Dove prism rotat- ing with angular velocity Ω~ and reflection from the PCM. OAM projection on the propagation axis is hΨf ;bjL^zjΨf ;bi ˆ ℓℏ, where L^z ˆ −iℏ · ∂∕∂θ. The RDS occurs because the optical torque on a slowly ro- tating element changes the angular momentum of the prism Fig. 1. (Color online) Additivity of RDS for the PCM-reflected photon in the rest frame. Rotation of the Dove prism (positive Ω) decreases frequency to −2ℓ · Ω of the corotating incident photon with Lz ˆ ‡ℓℏ. Reflection from the helical grating inside the PCM [7,8] alters Lz projection to the opposite one, and counterclockwise rotation of the Dove prism (as seen to backward photon) again decreases the frequency of the corotating photon to −2ℓ · Ω. The helical interference pattern is static between the prism and the PCM (:where δω ˆ 0) and rotates before the prism with angular velocity θ ˆ −2Ω having opposite handedness. Vol. 29, No. 4 / April 2012 / J. Opt. Soc. Am. B 715 [9,16]. In turn, this changes the prism’s angular velocity Ω~ and such a change requires an energy supply. Because typical optical elements, including prisms, are macroscopic classical objects having a continuous spectrum of energies, the energy ℏωf ;b and the frequency of the photon may be changed continuously [22]. In the following phase-conjugating optical interferometer, the photon’s OAM direction is altered, as well [9,30] (Fig. 1). Let the optical vortex Ef …t; r~† of charge ℓ pass through a rotating Dove prism and to be reflected with Eb…t; r~† from some ideal PCM. The nonrotating PCM is supposed to produce no frequency shift, as it happens in some cases in photorefrac- tive crystals [7,8], degenerate four-wave mixing [5,30], and holographic PC couplers [3,23]. Note that a small 10−1–10 Hz frequency shift in a BaTiO3 photorefractive PCM may mask the RDS. These additional frequency shifts caused by slow in- ternal charge waves and filamentation effects were reported in the early 1980s [31,32]. Because space is homogeneous and isotropic, the conser- vation of the momentum and angular momentum is expected [33]. Following [11,22], we have the conservation laws for the projection of the angular momenta Lz on the z axis and the energies of the incident photons and those transmitted through the Dove prism when the latter rotates with the angular velocity Ω~ : I zz · Ω ‡ Lz ˆ I zz · Ω0 ‡ L0zℏωf ‡ I zz Ω2 2 ˆ ℏω0 ‡ I zz Ω0 2 2 ; (3) where Izz is the moment of inertia around the z axis; the lefthand sides of this system correspond to the incident photon and the right-hand sides correspond to the transmitted one. For the incident Lz ˆ ‡ℓℏ and passed L0z ˆ −ℓℏ, Eq .(3) gives the difference of the angular velocities of the prism before and after the photon passage: Ω − Ω0 ˆ − 2ℓ · ℏ : (4) I zz This means that corotation increases the angular velocity of the prism, because the energy is transmitted to the prism by virtue of the optical torque jT~ j ˆ 2ℓ · P∕ωf , where P is the total power carried by LG [3] and, hence, Doppler frequency shift for the photon ω0 − ωf is negative: δω ˆ ω0 − ωf ˆ I zz 2ℏ …Ω − Ω0 †…Ω0 ‡ Ω† ˆ −2ℓ · Ω − 2ℓ · ℏ : I zz (5) Obviously, in the counterrotating case, when projections of Ω~ and Lz are in the opposite directions, the rotational Doppler shift is positive. The net RDS during total forward–backward passage is additive due to the PCM and this results in the net OAM change δω ˆ 4ℓ · Ω. The frequency shift δω is zero between the Dove prism and the PCM and the helical pattern is static there. In the region before the Dove prism, the frequency shift causes the clockwise (δω ˆ −4ℓ · Ω) or counterclockwise (δω ˆ ‡4ℓ · Ω) rotation of the helix pattern [9] with the angular speed θ_ ˆ 2Ω [Eq. (2)]. Because of the angular momentum conservation, the PCM feels the rotational recoil, which is proportional to 716 J. Opt. Soc. Am. B / Vol. 29, No. 4 / April 2012 Rto0∞pjoElof ;gbji2cradl r charge is the ℓ: jT~ power pcj ˆ ℓ · P…ω−f 1 ‡ ω−b1†, transferred by the LG where beam. P∼ The Dove prism feels doubled torque : jT~ Dovej ˆ 2ℓ · P…ω−f 1‡ ω−b1†. Note that Lebedev radiation pressure force on the PCM is always directed in the positive Z direction and jF~ pcj ˆ 2 · P∕c is independent of ℓ [34]. In the absence of the true PCM when the forward beam is retroreflected by spherical mirror M (Fig. 2) without altering the angular momentum, the interference pattern is a toroidal one Itor [7,9,19,35]: jΨj2 ˆ jΨf ‡ Ψbj2 ∼ Itor…z0; r; θ; t† ˆ 2ϵ0cjE…f ;b†j22…jℓj‡1†…r∕D0†2jℓj · expπℓ!−DD020…21…1‡‡2zr02z2∕02z∕Rz2R†2  ‰1 † ‡ R2 ‡ 2R · cos‰δω · t − …kf ‡ kb†z0ŠŠ: (6) The RDS is not accumulated here (the Doppler shifts for the forward and backward photons cancel each other δω ˆ 0) because true PC is absent; hence, the toroidal inter- ference pattern is static. The mechanical torques on prism T~ induced by OAM alternation will cancel each other, too: jT~ j ˆ 2ℓ · P…ω−f 1 − ω−b1† ≅ 0. 3. PHASE-CONJUGATING MIRROR IN A ROTATING FRAME The more fundamental case is a rotation of the setup as a whole around some axis and this general case is relevant to the detection of the slow rotations of the reference frame [1]. In contrast to a Sagnac interferometer, where the frequency splitting and running interference pattern appear in the active loop only, the PCVI produces frequency splitting δω without optical gain (Fig. 3). Apparently, the RDS δω will be the same when a CW laser source is placed in both the inertial (rest) frame and in the noninertial frame associated with the rotating vehicle. For the simplest case, when the sole PCM rotates around propagation axis z of a twisted photon with charge ℓ, the RDS appears due to the alternation of the photon’s OAM. Again Eqs. (3) give the frequency shift δω ˆ ωb − ωf due to the reflection from the rotating PCM: δω ˆ ωb − ωf ˆ 2ℓ · Ω ‡ 2ℓ · ℏ …I zz †PCM : (7) Fig. 2. (Color online) Mutual cancellation of RDS for a retroreflected photon. Rotation of the Dove prism again decreases frequency of the corotating forward photon with Lz ˆ ‡ℓℏ by −2ℓ · Ω. In backward propagation, the Dove prism is counterrotating with respect to the photon. Backward RDS is positive; thus, the resulting δω is zero and the toroidal interference pattern is static for all Z. A. Okulov Fig. 3. (Color online) Additivity of RDS in PCVI inside the rotating vehicle. The sign of δω is positive when optical torque T~ produced by a bunch of the rotating photons upon the mirror has the opposite direction compared to the rotation frequency of the PCM Ω~. When the Dove prism rotates in the opposite direction versus the PCM, the net RDS reaches the sixfold value δω ˆ 6ℓ · Ω due to the additional OAM alternation. The sequence of N counterrotating OAM-altering elements (including helical wave plates and cylindrical lenses) will produce net RDS of δω ˆ 4ℓ · Ω…N ‡ 1∕2† value. jΨ1;2;3;4i designates the antibunching of photons, which belongs to the two (for ℓ ˆ 1) helical wave functions, separated by a λ∕2 interval, deflected by the entrance beam splitter [7]. The second term in the right-hand side of Eq. (7) is negligible for typical masses (m ∼ g) and sizes (r ∼ cm) of prisms and mirrors ℏ∕Izz ∼ ℏ∕…m · r2† ≅ 10−27 Hz, as in Beth’s [16] and Dholakia and co-worker’s [22] experiments for the interaction of circularly polarized photons with a macroscopic object (half-wavelength plate). The angular speed of rotation of the interference pattern proves to be θ_ ˆ δω∕2ℓ ˆ Ω; thus, the pattern rotates synchronously with the reference frame. The helical interference pattern outside the PCM will be dragged by the helical diffraction grating [9] within the PCM. Consequently, the sole PCM cannot detect frame rotation. Note that conventional photorefractive [7,8] or static holography mechanisms [24] are sufficient to observe this drag effect and the atomic coherence is not necessary in our case, as compared to [36]. Nevertheless, there exists a possibility to accumulate the RDS by means of a chain of OAM-alternating elements. To achieve the accumulation of RDS, the adjacent components of the PCVI must rotate in opposite directions Ω~n ˆ …−1†nΩ~, where n ˆ 0 stands for the PCM, n ˆ 1 for the Dove prism adjacent to the PCM, and n ˆ N for the last Dove prism near the beam splitter. This is necessary because OAM is altered after the passage of each Dove prism and the mutual orientation of the angular momenta of the photon and the next prism should be maintained throughout the chain. When even elements (the PCM itself and N∕2 Dove prisms) of the PCVI are fixed in the Ω~ rotating frame, the other N∕2 odd elements ought to rotate with the bias angular velocity −2Ω~. The chain of the N rotating OAM-alternating elements will produce the net rotational Doppler shift amounting to δω ˆ 4ℓ · Ω…N ‡ 1∕2†. Thus, the PCVI interference pattern (Fig. 3) will revolve with enhanced angular speed θ_ ˆ 2 · …N ‡ 1∕2† · Ω. In such a way, the PCVI (Fig. 3) may be used for demonstration of the possibility of detection of the subHertzian rotation of the reference frame with the Earth (Ω⊕ ∼ 2π∕86;400). The helical interference pattern will rotate much faster than Earth itself. Namely, the equation 4ℓ · …N ‡ 1∕2† ˆ 24 has only one solution for integer ℓ, N (ℓ ˆ 4, N ˆ 1). Hence, the optical vortex with charge ℓ ˆ 4 passed twice through the single Dove prism rotating with Ω~⊕ and A. Okulov the PCM rotating in the opposite direction −Ω~ ⊕ will produce at the entrance beam splitter BS the 2 · ℓ spots of the interference pattern revolving once per hour. The further accumulation of RDS in the PCVI might be achieved by means of installing the N ˆ 60 counterrotating image-inverting elements and ℓ ˆ 6 optical vortex. In such configuration, the 2 · ℓ helices of the interference pattern [Eq. (2)] will produce 2ℓ spots at the PCVI beam splitter with one pass through the detector window within approximately each 60 s. For this purpose, the even components (the PCM and N∕2 Dove prisms) may be fixed at setup to rotate with velocity Ω~ ⊕, while the other N∕2 prism should rotate with bias angular velocity −2Ω~⊕ with respect to the rotating setup (e.g., rotating table). This enhancement will model the detection of the Earth’s rotation, and this will alter the ℓℏ OAM of each photon 121 times during one passage through the PCVI. Apart from the Dove prisms, the other OAM-altering helical wave plates and cylindrical lenses may be used [11,15,23]. The re- quired helical interference pattern within the PCVI might also be written by means of atomic coherence effects in a solid-state resonant medium [36]. Note that, in the proposed measurement of the Earth’s rotation, δω ˆ 4ℓ · …N ‡ 1∕2†Ω⊕ cos…ϕ† will show dependence on geographical latitude ϕ as it is known for the Foucault pendulum [37]: on the poles, δω will be equal to the maximum value when the angle ϕ between normal and the PCVI axis is 0 or π, while at the equator, δω might reach the maximum value when the PCVI axis is parallel to Earth’s rotation axis. The preliminary analysis has shown that the laser linewidth of the order Δω∕2π ∼ 103 Hz might be sufficient for Earth rotation detection by the PCVI (Fig. 3). 4. SINGLE-PHOTON OPERATION OF THE PHASE-CONJUGATING VORTEX INTERFEROMETER Single-photon operation [18] is based upon the superposition of the forward and backward quantum states with ℓℏ OAM: jΨiH ˆ p1 2 …jΨℓℏ if ‡ jΨ∓ℓℏib† ˆ p1 2ℓ X jΨjh jh i: (8) The detection of this superposition is not a trivial two-detector procedure, because the interference pattern is composed of 2ℓ twisted helices jΨjh i. The entrance beam splitter will reflect both upward and downward the interference pattern [7,9] composed of the 2ℓ spots located on an ellipse, rather than independent forward jΨℓℏif and backward jΨ∓ℓℏib photon states. For the simplest case ℓ ˆ 1, the photon will be in the superpositional state of the two helical wave functions de- signated by appropriate colors at Fig. 3: jΨiH ˆ p1 2 …jΨBluei ‡ jΨYellow i†: (9) This means that two detectors (for jΨ1i and jΨ2i) placed above the entrance beam splitter [7] and two detectors located below the beam splitter (for jΨ3i and jΨ4i) can indicate the antibunching of the photons [1,18], belonging to either of the two helices composing the interference pattern. As in a double-slit Young interference experiment, the attempt of the eavesdropping [1] the which way photon moves (the for- Vol. 29, No. 4 / April 2012 / J. Opt. Soc. Am. B 717 ward or backward one) will reduce the visibility of the helical interference pattern. On the other hand, when the single- photon quantum state is prepared as a toroidal pattern (Fig. 2), the photon belongs to the sequence of the equidistantly spaced toroidal Wannier wave functions jΨjtor i spaced by λ∕2 intervals: jΨitor ˆ p1 2 …jΨℓℏif ‡ jΨℓℏib† ˆ p1 N tor X Ψjtor jtor : (10) 5. CONCLUSION In summary, we analyzed the PCVI for the single-photon [18] and CW laser output [7,9]. Because of the alignment of all the optical components along the photon propagation axis z, the PCVI looks promising from the point of view of rotation sens- ing [1]. The self-adjusting property of the PCM [6] (also referred to as time reversal) will provide the automatic com- pensation of the beam displacements caused by rotations of Dove prisms [3] or rotating lenses [11]. 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