Bessel & Laguerre-Gauss Beam Generation using SLM as a Reconfigurable Diffractive Optical Element

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1 Bessel & Laguerre-Gauss Beam Generation using SLM as a Reconfigurable Diffractive Optical Element Sendhil Raja S., Rijuparna Chakraborty*, L.N.Hazra*, A.G.Bhujle Laser Instrumentation Section, Instrumentation & Control Division, Centre for Advanced Technology, Indore-4513, India *Department of Applied Physics, University College of Science & Technology, 9, A.P.C. Road, Calcutta -79 ABSTRACT Bessel and LG beam of different orders are generated using Spatial Light Modulator (SLM and also rotating Bessel and LG beam with programmable speed using the same SLM are produced, which needs no extra component or mechanical motion and can be used to rotate the micro particles with controlled speed, with a beam of particular mode, i.e., with a particular radius beam. The radius of a particular mode can also be changed if necessary. Keywords: Bessel Beam, LG Beam, optical tweezers 1. INTRODUCTION With the advent of laser tweezers generation of laser beams with zero central intensity, such as Bessel beams and higher order Laguerre-Gaussian beams is gaining importance due to their unique physical properties, such as barrel shaped intensity distribution, helical wave-front, centre phase singularity, ability to carry spin and orbital angular momentum, and spatial propagation invariance. They can be used as optical pipes, optical tweezers and optical spanners, and hence become a powerful tool in the manipulation and control of microscopic particles. Bessel beam and Laguerre-Gauss (LG beam of different orders can be generated using a Spatial Light Modulator (SLM which acts as a reconfigurable Diffractive Optical Element. Moreover the generated Bessel and LG beam can be rotated with programmable speed using the same SLM, which will need no extra component or mechanical mot ion. This can be used to rotate microscopic particles with controlled speed. 1.1 Experimental set-up The experimental set -up is as shown in figure 1. The expanded beam of a diode laser (636 nm., 15 mv, is used to illuminate the SVGA SLM, i.e., the reconfigurable Diffractive Element. The far-field diffraction pattern is captured by a Pulnix 11 Digital Camera, using IMAQ Vision software and displayed on the PC monitor. The patterns generated using MATLAB are downloaded to the drive electronics of the SVGA SLM. polarizer SVGA SLM Pulnix 11 Digital Camera 63 nm 1 mv Diode Laser 5x beam expander PCI image card Drive electronics Software: MATLAB Software: IMAQ Vision VGA out Figure 1: The experimental set-up..

2 . Bessel Beam Exact solution of wave equation shows that there are solutions for beamlike fields in free space which are not subject to transverse spreading (diffraction after the plane where the beam is formed. These solutions are nonsingular and, like plane waves, have finite energy density rather than finite energy. Most importantly, they can have sharply defined intensity distributions as small as several wavelengths in every transverse plane, independent of propagation distance. The wave equation for free space is 1 E( r, t = where, r = x, y, z c t Using cylindrical polar coordinate we can write d E 1 de + + ( k k E = z d d This is the differential equation for the Bessel functions of order. So that E ( = J ( k where, = x + y J is the zero order Bessel function of first kind. So the scalar field propagating into the source-free region z>= is π i k z ω t i k ( x cos φ + y sin φ where, E [ ], e dφ ( z ( x y, z ; t = e A( φ k k k z + k z = k = k sin α = k cosα ω = c A(f is an arbitrary complex function of f a is the angle between wave-vector and optical axis Therefore, we get E ( ( r t e i k z t J ( k ( A z ω, = Κ Κ Κ Κ Κ Κ Κ Κ Κ Κ Κ Κ Κ Κ Κ Κ Κ when k z is real, we get a class of fields that are nondiffracting in the sense that the time averaged intensity profile at z=, I 1 ( x, y, z = E( r, t = I ( x, y, z =, is exactly reproduced for all z> in every plane normal to the z axis. When k? =, the solution is simply a plane wave, but for < k? < k, the solution is a nondiffracting beam whose intensity profile decays at a rate inversely proportional to k??. The effective width of the beam is determined by k?, and when k? =k=?/c=p/? (the maximum possible value for a nonevanescent field the central spot assumes it minimum possible diameter ~ 3?/4. Since the intensity distribution of J beam decays as 1/?, it is not square integrable. In fact even though the intensity profile is sharply peaked, the amount of energy in each ring is approximately equal to that contained in the central maximum. It would therefore require infinite amount of energy to create a J beam over an entire plane. One can, however, create such a beam over a finite area, but with smaller depth of field. There is a simple yet accurate method for finding the range of a J beam of finite aperture. From equation (A we see that the J beam is a superposition of plane waves, all having the same amplitude travelling at the same angle a = sin -1 (k??/p relative to the z -axis but having different azimuthal angles ranging from to p radians. Such a beam can be produced by using a thin annular aperture of radius? 1 <?<? placed in the back focal plane (i.e., the Fourier plane of a lens of focal length f, having the effect of producing a wave with a narrow range of spatial

3 frequencies in between p? 1/?f and p? /?f. This entails severe loss of light, so instead of a annular aperture we can use a diffractive optical element. Such a diffractive element of radius R is characterized by complex amplitude function iπ / t(, φ = A( φ e = when R when > R The term p/? gives the radial spatial frequency of the Bessel wave. The expression shows that the phase must uniformly vary with the radius of the DOE to achieve the conical wave-front. The parameter? controls the size of the dark core of the Bessel beam. As the value of? increases, the size of the dark core increases as well. Now, J (x can be expressed as J 1 π π ( x = this comes from the more general form J or, alternatively, from n 1 ( x = J n π e ixsin φ π i ( xsinφ nφ e n π i π ( x = e dφ dφ i( x cosφ+ nφ dφ i.e., the higher order Bessel beam can be achieved with an additional phase factor exp(-inf or exp(inf. This additional exp(±inf phase factor inside the integrals can be achieved by adding a circular inφ wedged phase plate to Durnin s annular slit configuration, or by introducing A ( φ = e in the DOE configuration. Then, t(?,f becomes a phase function of the form t(, φ = e ψ iψ (, φ (, φ = nφ + π/, where n is the charge of the phase singularity of the Bessel beam from which the phase profile originates. This charge or winding number of the singularity is proportional to the amount of optical angular momentum around the singularity. The charge n of the Bessel beam is defined by a closed counterclockwise contour on the phase mask and is equal to the net number of phase discontinuities it crosses. By convention the sign of the discontinuity is taken to be positive when the phase jumps from p to and negative when the phase jumps the other way. To steer the beam in desired direction using the DOE, we can use an interference pattern of the required field and an off-axis reference beam on the DOE. We define the parameter that specifies the diffraction angle by projecting the reference beam at an angle to the surface of DOE onto the normal to the surface of DOE. This phase is denoted as π k. r = sin γ cosφ λ The total phase profile is then ψ, where, k = wave number π (, φ = n φ + π + sin γ cosφ λ? = angle between reference beam and optical axis giving a structure of dislocated grating. For on axis reference beam? =, and we get a spiral pattern..3 Laguerre-Gaussian Beam The circularly symmetric Laguerre-Gaussian (LG laser modes form a complete basis set for paraxial light beams. A given mode is usually denoted by m ( n m / LG, where m and n are two integer indices satisfying n = m, m +, m +4,..that describe the modes. m refers to the number of p phase cycles around the circumference of the mode, i.e., the number of p phase jumps about the centre point on a transverse plane determining the order of phase

4 singularity at the centre of the beam. The number of radial zeros is fixed by the number of zeros of the generalized Laguerre polynomial (n - m / in addition to the central vortex that appears when m?. Hence the total number of bright lobes (including the central one, if applicable is 1 + (n - m /. The LG beam modes can be represented by U r = G ˆ, zˆ R ˆ Φ φ Z zˆ, and nm G R Φ Z ( ( ( ( ( nm m n where, ˆ = / ω( zˆ, the radial co ordinate scaled by Gaussian spot size ω( zˆ 1 / ω( zˆ = ω [ 1+ zˆ ] ( ˆ, zˆ nm n ( zˆ zˆ = z / z, z = πω / λ the longitudinal iψ ( zˆ m m ( ˆ = ( ˆ L( ( ˆ, imι ( φ = e, inψ ( zˆ ( zˆ = e m where, ω = ω. e. e 1 ψ ( zˆ = tan ( zˆ, is the Gouy phase The function G(, zˆ e ˆ iˆ zˆ n m /, co ordinate scsled by Rayleighlength z ˆ is common to all modes and includes the radial Gaussian envelope of the beam, a quadratic phase, and the Gouy phase, If we record the interference pattern between the required field and a reference field on a DOE and illuminate it by the reference field, then at the output we get the required field, similar to the case of holography. By this method we effectively modulate the required field by a carrier frequency. In case of LG mode the interference pattern takes the form of a forked diffraction grating with m dislocations, when the reference beam is not collinear. For a collinear reference beam, i.e., when we want to get an on-axis LG mode we use the formula (, φ = exp[ i( ˆ. c ( m + n + 1 ψ ( zˆ mφ ] E + where c is a constant. When the two interfering beams are inclined to an angle?, then an extra term is added to the equation resulting in E (, φ = exp[ i( ˆ. c ( m + n + 1 ψ ( zˆ + mφ + k sin γ cos φ ] giving an off-axis LG beam. Both the functions generates a screw dislocation on the beam axis that gives us the characteristic exp(-imf phase structure of these beams and also an annular intensity pattern (for m > in the far field. For a given m, the beam has m intertwined helical phase fronts with a phase singularity at the beam centre, which necessitates zero intensity there, and the annular intensity pattern which persists no matter how tightly the beam is focused..4 Results and discussions The computed phase profiles where downloaded to the SLM and the far field diffraction pattern captured using a digital CCD camera. The figure below shows the theoretical and experimental beam patterns for the Bessel beam of order zero, two and six and the corresponding patterns used to generate them on the SLM.

5 Figure : Experimental output of Bessel of order zero Figure 3: Simulated Bessel beam of order zero Figure 4: Mask on SLM for Bessel of order zero Figure 5: Experimental output of Bessel of order two Figure 6: Simulated Bessel beam of order two

6 Figure 7: Mask on SLM for Bessel of order two Figure 8: Experimental output of Bessel of order six Figure 9: Simulated Bessel beam of order six Figure 1: Mask on SLM for Bessel of order six The radius of the central core depends on the constant term?, so for a particular order it can be changed. As the value of? increases, the size of the dark core increases as well. The diameter of the dark core of the beam (except the zeroth order increases with the order of t he beam. From the images acquired with the help of CCD camera we see the central spot diameter of Bessel zero order beam is.5mm. The diameter of the dark core of the Bessel beam of order two and six are. and.5 mm respectively. And the diameters of the 1st annular rings are.46 and.93 respectively. In each case the diameter of the central core and the spacing between the neighbor rings change very little over a long distance of propagation, ~ 5 meter. So it can overcome the limitation of Rayleigh range of Gaussian beam and can trap particles in a longer propagation zone without dispersion after using an inverse telescope in a optical tweezer.

7 The formulation for the off-axis beam also produces similar modal patterns. Here are some examples of patterns for generating of axis beams. Figure 11: For order Figure 1: For order 1 The figure below shows the theoretical and experimental beam patterns for the Laguerre-Gaussian beam of order one, Figure 13: Experimental output of LG of order one Figure 14: Simulated LG beam of order one Figure 15: Mask on SLM for LG of order one

8 seven and twenty and the corresponding patterns used to generate them with the SLM. Figure 16: Experimental output of LG of order seven Figure 17: Simulated LG beam of order seven Figure 18: Mask on SLM for LG of order seven

9 Figure 19: Experimental output of LG of order twenty Figure : Simulated LG beam of order twenty Figure 1: Mask on SLM for LG of order twenty The radius of the central core depends on the constant term c, so for a particular order it can be changed. The diameter of the dark core of the beam (except the zeroth order increases with the order of the beam. From the images acquired with the help of CCD camera we see the central core diameter of LG first order beam is. mm. The diameter of the dark core of the LG beam of order seven and twenty are.87 and 1.31 mm respectively. And the diameters of the 1st annular rings are 1.9 and 1.78 mm respectively. In each case the pattern of the beam change very little over a long distance of propagation, ~ 1 metre. The formulation for the off-axis beam also produces similar modal patterns. Figure and 3 show examples of patterns for generating off axis beams. The maximum deflection angle in 3D is limited by the pixel number and pixel size or the space bandwidth product of the SLM. In our case the SLM has 6 8 pixels with each pixel having a size of 33µm 33µm. So the maximum deflection angle is? = sin -1 ((1/33 wavelength(mm which is 1.1 degree (half angle.

10 Figure : For order Figure 3: For order In the results some noise are produced due to the phase error generated by the four-step voltage level of the SLM. The total angular momentum of any electromagnetic field is the sum of its spin and orbital momentum contributions. In free space, the Poynting vector, which gives the direction and magnitude of the momentum flow, is simply the vector product of the electric and magnetic field intensities. For helical phase fronts, the Poynting vector has an azim uthal component, which produces an orbital angular momentum parallel to the beam axis. Because the momentum circulates about the beam axis, such beams are said to contain an optical vortex. The centre of the vortex (with zero intensity carries neither linear, nor angular momentum; instead the angular momentum is associated with the region of high intensity, i.e., the bright annular ring. This property of the beam is used to rotate microscopic particles in an optical tweezer, which works by the transfer of momentum from a tightly focused laser to the particles. The particles refract and scatter the light and distort the profile of the beam. The intensity gradient force generated by this process, traps the particles at the location of its maximum intensity, when it balance the scattering force and also the forces due to gravitation and Brownian motion. When the linear momentum associated with the Poynting vector, which governs the direction of the force, has an azimuthal component, the particles rotate. The magnitude of this angular momentum is very small. An additional angular momentum can be imposed by interfering this beam with a beam of rotating circular polarization, which can be generated by a polarizer and a rotating quarter-wave plate. Instead, we have demonstrated a rotating LG or Bessel beam with controlable speed using SLM, which will need no extra component or mechanical motion. REFERENCES 1. N.Chattrapiban, E.A.Rogers, D.Cofield, W.T.Hill, R.Roy; Optics Letters, vol. 8 ( M.A.Clifford, J.Arlt, J.Courtial, K.Dholakia; Optics Communications, vo1.156 ( N.R.Heckenberg, R.McDuff, C.P.Smith, A.G.White; Optics Letters, vol. 17 ( *sendhil@cat.ernet.in; phone ; fax

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