Optical beam steering based on electromagnetically induced transparency

Optical beam steering base on electromagnetically inuce transparency Qingqing Sun, Yuri V. Rostovtsev, an M. Suhail Zubairy Department of Physics an Institute of Quantum Stuies, Texas A&M University, College Station, Texas 77843-4242, USA Receive 9 December 2005; publishe 26 September 2006 We propose a scheme that provies all-optically-controlle steering of light beam. The system is base on steep ispersion of a coherently riven meium. Using the eikonal equation, we stuy the steering angle, the sprea of the optical beam, an the limits set by resiual absorption of the meium uner conitions of electromagnetically inuce transparency. Implementation of another scheme for ultrashort pulses is also iscusse. DOI: 10.1103/PhysRevA.74.033819 PACS number s : 42.50.Gy, 42.50.Nn I. INTRODUCTION Optical beam eflection is an important technology in moern optics. It has applications in the fiels of raar, optical imaging, laser machining, an free space communication. Many physical mechanisms have been use to obtain the eflection 1,2. Among them mechanical motion 3 5 is the simplest an most convenient way, since it only mechanically moves or rotates the eflector mirror, grating, etc.. A thermal graient 6, acousto-optical interaction 7 9, an the electro-optic effect 10 12 all can inuce a refractive inex graient which eflects the light. The electro-optic eflectors are faster than their acousto-optic counterparts as compare in Ref. 13. Nowaays with the evelopment of new materials an evices, attention has been focuse on using photonic crystals 14,15 an phase arrays 16 18 to get fast beam steering. For a recent review of electro-optical systems, see Ref. 19. Light can also change the propagation irection of another beam of light through interaction with matter. Beam eflections have been reporte in soium vapor via optical pumping 20 an in rubiium vapor via saturate absorption an hyperfine pumping 21. Electromagnetically inuce transparency EIT provies another mechanism since the refractive inex can change near the transparency center 22 24. Moseley et al. first observe the electromagnetically inuce focusing an efocusing effects in a rubiium vapor 25,26, which come from the spatial Gaussian istribution of the pump fiel. These authors inclue both refraction an absorption moification in the numerical calculation an foun a qualitative accorance with experiment. Mitsunaga et al. observe absorption imaging in col soium atoms 27. The probe is mostly absorbe after the atom clou except for the focal point of the pump beam. Transmission through this EIT point can reach almost 200%, which is obviously a focusing feature. Another example base on EIT is electromagnetically inuce waveguiing 28 30 which uses the riving fiel as a fiber to confine the probe fiel. Raman spatial solitons have been emonstrate by Walker et al. 31 33.Asan alternative to the near-resonant configuration use in the above papers, the pump an probe can both be etune to the blue sie an remain in two-photon resonance. Inuce focusing an waveguiing in such a system have been iscusse uner EIT 34 an coherent population trapping CPT conitions 35. Shpaisman et al. 36 have iscusse both these cases an have extene the metho to ouble- systems, where one system can inuce waveguiing in the other. The maximum Rabi frequency of the pump was chosen near the EIT or CPT threshol to ensure low absorption an significant focusing. In this paper we explore the possibilities an limits of beam eflection through the EIT effect. An inhomogeneous pump fiel intensity prouces a refractive inex graient for the probe. Ray optics is aopte to analyze the steering angle an absorption for each probe ray. Uner optimal pump istribution rays of the same frequency can be eflecte at the same angle, unaffecte by the starting position. For a single frequency we can also obtain exact focusing. Finally we show that even a whole beam with spatial an spectral with can be eflecte together using a specific setup. II. BEAM PROPAGATION IN AN INHOMOGENEOUS MEDIUM The iea of all-optical steering of an electromagnetic wave is as follows. Consier a pulse of central frequency propagating through a three-level EIT meium as shown in Fig. 1. We assume that the spectral with of the pulse lies well within the EIT winow such that the inequalities, are satisfie. Here is the Rabi frequency of the riving fiel resonant with an a-c transition, is the atomic ecay rate, an = ab is the etuning of the probe frequency with the atomic transition a-b. Now in orer to steer the incient pulse in a ifferent irection, we introuce a phase shift for the ifferent transverse positions x of the pulse at the output by moulating the Rabi frequency of the riving fiel. Thus the whole pulse has a ifferent irection while coming out of the EIT meium. Here we erive a simple expression for the beam steering angle an the corresponing losses. We assume that we have a highly ispersive meium. For a large enough, the inex of refraction for the probe fiel can be written as n 1+ 1+, 1 2 where N is the atomic ensity an =N 2 /2 0 2 with being the atomic ipole moment on the a-b transition. Inhomogeneous x leas to inhomogeneous n x. 1050-2947/2006/74 3 /033819 6 033819-1 2006 The American Physical Society

SUN, ROSTOVTSEV, AND ZUBAIRY a c p x b z (c) FIG. 2. Color online For ifferent istributions of the riving fiel, ifferent regimes of probe light propagation take place. a The light eflection angle is the same for all the light rays; b the probe light turns but spreas; c focusing of the probe. s n s X = n x an s n s z = n z. The equation escribing the amplitue of the electromagnetic fiel can be obtaine in a similar manner. It follows from the imaginary part of Eq. 2 that 6 FIG. 1. Color online a The schematics of the three-level atomic system with the fiels. b A slab of the three-level atomic meium turns the probe light via an inhomogeneous riving fiel. c The real 1 an imaginary 2 parts of the atomic susceptibility, an the spectrum of probe pulse 3 vs probe frequency. The trajectory of the light rays propagating in an inhomogeneous meium can be foun by using the eikonal approximation 37. We start with Maxwell s equation, which escribes the propagation of the electromagnetic waves, 2 E 1 2 E c 2 t 2 = 2 P 0 t 2. 2 We can expan the fiel an the polarization in terms of the slowly varying amplitues E an P an the eikonal as E = E e i t+ik, P = P e i t+ik. Here k= /c. The polarization of the meium is relate to the fiel intensity as P = 0 E, where the susceptibility is = +i. If we neglect the secon-orer erivative over coorinates for the amplitue E, the eikonal equation is given by 2 =1+ n 2. 4 So we can write own =n R/s an obtain the geometrical optics ifferential equation in the vector form s n R n, 5 s = where R is the point of the ray. Here R x,z =X z xˆ +zẑ an xˆ,ẑ are unit vectors along the axes. Then, for the x an z components, 3 2k E + k 2 E = 2 c 2 E. 7 The solution of the above equation has the following form E = E s 2 0 n exp s 1 2n c s. 8 In the next section we will iscuss several inhomogeneous riving fiel istributions an their steering effects. III. DISCUSSION A. Single-frequency eflection For the first case assume that x = 0 / 1+ x, 9 where is the parameter that etermines the inhomogeneous istribution of the riving fiel. The refractive inex is therefore of the form n x 1+ 0 1+ x with 0 =N 2 /2 0 2 0. This form leas to constant n for a single frequency. In the following we show that, for such a situation, we obtain the same steering angle for all the transverse positions x as shown in Fig. 2 a. If the epenence of the riving fiel has a ifferent form then there will be a sprea of the optical rays as shown in Fig. 2 b. InEq. 6 using s= X z 2 +z 2 an n =1+ 0 1+ x, we obtain s n s X = 0 an s n s z =0. 10 The orinary ifferential equation to escribe the ray trajectory is given by 2 X z z 2 an its solution is = 1+ 0 1+ X z 1+ 0 1+ X 0 2 0 0, 11 033819-2

OPTICAL BEAM STEERING BASED ON (c) () FIG. 4. Color online Numerical simulation for beam propagation. Here the etuning =3.3 MHz. FIG. 3. Color online Deflection angle an transmission E /E 0 2 for x = 0 / 1+ x. In a, b the etuning is fixe at =3.3 MHz; the angle is constant when X 0 varies from 0 to 0.2 mm. In c, X 0 =0 while changes from 4 to 4 MHz; the angle follows the etuning linearly. X z X 0 + 0 z 2. 12 2 The light turning angle can be foun from X z /z =tan. The resulting angle is 0 L = N 2 2 2 0 L. 13 0 A calculation base on the full expression of 38 shows that these simple estimations Eqs. 12 an 13 are vali for small eflections. With the parameters =5000 m 1, L =1 cm, 0 =50 MHz, N=10 15 cm 3, =0.5 m, ab = ra =2 MHz, cb =1 khz we get Fig. 3. A probe beam with etuning =3.3 MHz Figs. 3 a an 3 b will experience the same eflection angle 5.76, although the ray starting position X 0 ranges from 0 to 0.2 mm. The transmission ecreases at larger X 0 because there is smaller. Figures 3 c an 3 show the linear epenence of the eflection angle on etuning. Now X 0 is fixe at 0 an varies from 4 to 4 MHz. The transmission at negative is larger because the ray goes to the negative x irection where is larger. In this paper we have assume the meium to be homogeneously broaene. For an inhomogeneously broaene meium the transmission woul be many orers smaller. However, the steering angle coul still be the same. Another factor that influences the transmission is cb. This ecay rate can be much larger uner high ensity. If cb =10 4 Hz then the transmission will be ten times smaller while the angle will remain unaffecte. On further increasing cb the transmission will ecrease much faster. To stuy the behavior of a beam with a finite iameter we perform some numerical simulation. Consier a monochromatic electromagnetic wave with frequency ; the Helmholtz equation is given by 2 E +2 E ln n + 2 c 2 n2 E = 0, 14 where n=n +in = 1+. For this inhomogeneous meium, n 0 =1 an n r =n r n 0, k 0 = n 0 /c, r= r,z, E r,t =exp ik 0 z i t A r,z. We assume that the fiel has linear polarization set to the y irection, an at the entrance it is given by A x,0 = A 0 exp x x 0 w 0 2 ŷ, 15 where w 0 =0.1 mm is the beam waist. x 0 =0.1 mm is the beam center at the entrance. To get an accurate result we go beyon the paraxial limit, following the metho of Refs. 39,40. Separating the fiel into transverse an longituinal components an expaning the equation in the small parameters /w 0 an n, we obtain i A y z = 1 2n 0 k 0 2 n x 2 A y +2 n A y x x + A y n 2 x 2 1 2 A y 2k 0 x 2 k 0 n n 0 A y. 16 It is then straightforwar to get the beam istribution at z 0. To compare with the result in Figs. 3 a an 3 b we use the same parameters. The only ifference is now it is a beam instea of a ray. The simulation result is shown in Fig. 4. It clearly shows the beam being eflecte an absorbe as it propagates along the z axis. The peak of the beam goes to X 1 =0.556 mm. This is a little smaller than the value X 1 =0.600 mm of a ray starting from X 0 =0.1 mm, which is reasonable since the absorption increases with x an thus lowers the peak position. The peak intensity of the beam is A y /A 0 2 =0.0664, slightly higher than the ray transmission 0.0555. The eflection angle from the simulation is 5.75, matching the former result precisely. This angle is almost a constant for the whole beam espite its extremely slow increase with z. From this example we see that the eflection angle coul be as large as 033819-3

SUN, ROSTOVTSEV, AND ZUBAIRY (c) () FIG. 5. Color online Deflection angle for x = 0 1+ x. The etuning is fixe at =3.3 MHz. The angle is not a constant for varying X 0. 0.1 ra, 17 with afforable losses, which shows a potential for alloptical light steering. Here the eflection angle is only constant for a single frequency, but this is not a big problem for a probe pulse with narrow banwith. As we have mentione before, other riving fiel istributions will lea to sprea of the probe pulse because the refractive inex graient epens on the spatial coorinates see Fig. 2 b. Consier the simplest case x = 0 1 + x, which gives n x =1+ 0 / 1+ x 2. =1500 m 1 an all the other parameters are the same as in the first case. Now probe rays with =3.3 MHz starting from ifferent X 0 have ifferent eflection angles, as shown in Fig. 5. There is spreaing even for a single frequency, so this is not suitable for beam steering. B. Focusing an efocusing Next we show that, by controlling the spatial epenence of the Rabi frequency of the riving fiel, we can have a focusing or efocusing coherent meium, which as aitional flexibility to hanle the probe fiel. Thus the coherent meium not only can act as an effective beam eflector but also can be transforme into a lens with controllable focal istance. We recall that both Moseley et al. 25 an Mitsunaga et al. 27 are using Gaussian coupling beams so the focusing is not very intense. Here we assume that the Rabi frequency of the riving fiel epens on the transverse coorinate via x = 0 1 x 2. 18 The space-epenent refractive inex is then given by n x =1+ 0 1 x 2. This creates the ray structure shown in Fig. 2 c. To see this, we consier a ray that starts at X 0. As a simple estimate, it goes out of the cell at X 1 =X L X 0 1 0 L 2 an the eflection angle is 2 0 LX 0. This ray passes the axis at the istance FIG. 6. Color online Focusing effect for x = 0 / 1 x 2. In a, b the etuning is fixe at =3.3 MHz; the focal istance is constant when X 0 varies from 0.6 to 0.6 mm. In c, X 0 =0 while changes from 0.1 to 3 MHz; the angle follows Eq. 19. F = X 1 1 0 L 2 2 0 L, 19 which is inepenent of the initial position of the ray, X 0. This represents a lens with a focal istance F. A calculation base on the full expression also supports these estimates. Consier a system with =3.3 MHz, =10 6 m 2 an all the other parameters the same as in the first case. It is obvious from Fig. 6 a that F oes not change after X 0. So the focal istance F is well efine for a single frequency. However, this istance changes for ifferent frequencies see Fig. 6 c as in Eq. 19. It ecreases for large etuning which is easy to unerstan since large etuning has a rapi refractive inex change an the beam eflects quickly. The focal istance F can also be controlle by varying 0 or. Both smaller 0 an larger can lea to smaller focal istance. The reason is the same as above. Finally, by using a negative or negative we can obtain goo efocusing effect. If both an are negative we get focusing again. This is more applicable for experiment since the highest riving fiel require is 0 at the center. Note that here the riving fiel epens on x so this is only a two-imensional focusing. To simulate a real lens it shoul epen on r. C. Short-pulse eflection Up to now our iscussion applies to the propagation of continuous waves an pulses that have time uration long enough to fit the EIT winow, i.e., 2 T/ 1, where T is the pulse with. The problem with shorter pulses is that they have a broa spectrum that may not fit the EIT winow, leaing to substantial absorption an a nonlinear ispersion. As a result, we may encounter strong reshaping an absorption while the pulse propagates through the EIT meium. Recently, we propose a solution to the problem of broaban pulse propagation through an EIT meium 41. The 033819-4

OPTICAL BEAM STEERING BASED ON probe pump (c) EIT Anti-Helmholtz () FIG. 7. Color online A scheme of optical steering for broaban pulses using a magnetic fiel graient create by a pair of anti-helmholtz coils. main iea is that we can exten the absorptionless linear ispersion of the EIT three-level system using an inhomogeneous meium where we maintain the resonance with the optical fiel by shifting the atomic level via an inhomogeneous magnetic fiel. In our scheme, we have a graient of the inex of refraction ue to the graient of the magnetic fiel, because the etuning is position epenent. The scheme is epicte in Fig. 7. The system consists of a set of prisms or iffraction gratings with total ispersion equal to zero. The first prism isperses the probe pulse into a ivergent beam. The secon prism transforms the beam into a parallel beam, with ifferent frequencies shifte in space. When the beam goes into the EIT cell with col atoms, an inhomogeneous magnetic fiel moves the level b only so that each frequency is resonant with the local a b transition. At the same time, the constant riving fiel propagates along the x irection, which is resonant with the a c transition. Now a single-frequency ray at ifferent x positions will experience ifferent etunings an refractive inexes because ab changes with x ue to the applie inhomogeneous magnetic fiel. Such a refractive inex graient will cause eflection. For an ieal system, the frequency istribution after the secon prism is linear, i.e., /x is a constant. Also ab /x shoul be the same to match the fiel. We assume that a ray of frequency enters the EIT cell at x 0 position, an efine x=x x c where x c is the position at which the ray is resonant, i.e., = ab x c. So the etuning at position x is x = ab x = ab x c ab x = x ab x. 20 The refractive inex will be n x 1+ 0 x =1 0 x x. 21 The refractive graient n = 0 /x xˆ is a constant. It follows from Eqs. 12 an 13 that x 1 x 0 n L 2 /2 an sin n L for small eflection. They o not epen on x 0 or. So all the rays go through parallel paths. This is crucial for the recombination of the pulse. Thus a probe beam with finite banwith an finite iameter can be eflecte perfectly. FIG. 8. Color online Depenence of the light eflection angle a, c an the probe transmission b, on the position of the probe ray, x 0, an on the magnitue of the Rabi frequency. The above is only a simple estimate. In Fig. 8 we show some numerical results for /x= 10 11 Hz/m, L=1 mm an all the other parameters the same as in the first case. In Figs. 8 a an 8 b we can see the rays of frequency 2 6 10 14 Hz starting from ifferent x 0. The eflection angles are aroun 3.47 although a little larger for nonzero x 0. The transmission for x 0 =0 is the highest 0.98 but it ecreases quickly for other starting positions. This is goo enough since we can always put the central rays, which are the main part for an orinary beam, at x 0 =0. Unlike the first case, now the frequency oes not influence the eflection angle. A calculation for varying by 10 10 Hz still gives the same curve, which is reasonable because only local ecays are slightly change. As a result the whole beam, espite its iameter an banwith, eflects at the same angle. If the iameter is small enough to fit within x 0 = ±0.05 mm, the whole pulse will be eflecte with transmission 1. We can control the angle by varying the rive Rabi frequency, as shown in Figs. 8 c an 8. The parameters are still the same, =5000 Å, x 0 =0, only varying from 2 1.25 10 7 Hz to 2 3 10 7 Hz. It is easy to fin from the graph that a larger eflection angle accompanies smaller transmission. Angles greater than 10 are even achievable at small riving fiel. But then the sie rays nonzero x 0 suffer substantial absorption. For large riving fiel the system goes to the limit angle 0 an transmission 1. Since the level b can be move by a magnetic fiel, it must be a magnetic sublevel with M 0 an have some nonegenerate magnetic sublevels nearby. If we inclue them in the calculation, the population in level b will ecrease, which effectively ecreases the optical ensity. The resiual absorption ue to these off-resonance levels can be avoie by using a circularly polarize probe to interact with only level b. IV. CONCLUSION We have shown that for a single-frequency or a narrowbanwith probe fiel, high-quality eflection an focusing 033819-5

SUN, ROSTOVTSEV, AND ZUBAIRY can be achieve uner optimal pump istribution. The practical ifficulty is how to generate such a istribution. One possible way is to put a screen with the esire transmission function as in Eqs. 9 an 18 behin the riving fiel. Pump iffraction may not be so severe if the propagation istance is short. Then we can control the eflection angle or focal istance by varying only the input riving fiel intensity 0. This metho is convenient an continuous. However, the scan spee is limite by the EIT establishment time, which is about the raiative lifetime 1/ ra. A short pulse can also be eflecte to the same angle in the propose scheme. The key point is here that the magnetic graient provies the same refractive inex graient for all the frequency components. The eflection angle can also be controlle by the riving intensity. This metho is promising ue to its broaban ability. Here the problem is that the lowest-frequency graient of a real prism system is 10 14 Hz/m an it is only approximately constant. We nee a better system to provie larger spatial ispersion. In both of the eflection schemes the maximum eflection angle without significant absorption is 0.1 ra. Larger eflection always comes with larger absorption. This is etermine by the relation between refractive inex an absorption coefficient in EIT. Note that 0.1 ra is alreay goo enough for some applications, an aitional evices like multiple birefringent prisms or holographic glass with multiple holograms can further increase the angle. ACKNOWLEDGMENTS We gratefully acknowlege the support from the Air Force Office of Scientific Research. 1 V. J. Fowler an J. Schlafer, Proc. IEEE 54, 1437 1966. 2 M. Gottlieb, C. L. M. Irelan, an J. M. Ley, Electro-Optic an Acousto-Optic Scanning an Deflection Dekker, New York, 1983. 3 I. Cinrich, Appl. Opt. 6, 1531 1967. 4 D. H. McMahon, A. R. Franklin, an J. B. 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