Home Search Collections Journals About Contact us My IOPscience On the relationship between artificial Kerr nonlinearities and the photorefractive effect This content has been downloaded from IOPscience. Please scroll down to see the full text. 2008 J. Phys. D: Appl. Phys. 41 224001 (http://iopscience.iop.org/0022-3727/41/22/224001) View the table of contents for this issue, or go to the journal homepage for more Download details: IP Address: 130.64.11.153 This content was downloaded on 28/03/2016 at 12:04 Please note that terms and conditions apply. IOP PUBLISHING JOURNAL OF PHYSICS D: APPLIED PHYSICS J. Phys. D: Appl. Phys. 41 (2008) 224001 (5pp) doi:10.1088/0022-3727/41/22/224001 On the relationship between artificial Kerr nonlinearities and the photorefractive effect Mark Cronin-Golomb Department of Biomedical Engineering, Tufts University, Medford, MA 02155, USA Received 12 May 2008, in final form 12 June 2008 Published 24 October 2008 Online at stacks.iop.org/JPhysD/41/224001 Abstract The photorefractive effect and artificial Kerr effect are both drift and diffusion based nonlinearities. This paper treats them in a unified fashion and develops a formalism for the analysis of general nonlinearities of this type. The intensity dependence of the steady state and response time is discussed. 1. Introduction It is the combination of the two components that results in the nonlinearity. The sphere moves in response to light The photorefractive rate equations [1], now 30 years old, in the fluid medium and causes a change in the spatial have been integral in supporting the growth of the field distribution of refractive index, just as spatial variations in of photorefractive nonlinear optics. They have been used optical intensity can drive the distribution or orientation of to model the strength and response time of the phase molecules in molecular orientational or electrostrictive Kerr gratings responsible for nonlinear effects such as optical phase effects. This is the basis of the artificial Kerr medium, which conjugation, image processing and spatial solitons. Over was studied extensively in the early 1980s [4, 5]. These include time, several researchers in the photorefractive field have particle orientational nonlinearities [6] and nonlinearities of begun to display an interest in another optically mediated optomechanical arrays realized by beads rolling on transparent effect: optical tweezers. It is may not be such a coincidence substrates [7]. Similar effects have been observed and studied that Arthur Ashkin, one of the founders of the field of in critical microemulsions [8, 9], phospholipid vesicles [10] photorefractive nonlinear optics [2], also helped to found and even phototactic microorganisms [11]. It is the aim of the field of optical tweezers [3]. Both of these effects are this paper to compare the Kukhtarev and hopping models for mediated by photoinduced motion of small particles: the first the photorefractive effect and the theory of optical tweezers or by charge migration induced by spatial gradients in the optical optophoretic nonlinearities. intensity, the second by forces induced by spatial gradients of optical intensity. Although the optical tweezers effect is 2. Artificial Kerr medium usually thought of in terms of manipulation of microscopic particles such as silica or polystyrene microspheres and The equations of motion in an artificial Kerr medium consisting biological microorganisms and cells, it can also be thought of a suspension of transparent nanoparticles include both drift of as a nonlinear optical effect. Many of the methods of and diffusion terms. The drift term may be found from models measuring forces on trapped particles rely on measurement for the viscous flow of particles under the influence of optical of the displacement of the trapped particle from the axis gradient forces. Suppose the suspension is illuminated with an of the trapping beam. Commonly, this is done by using optical field propagating in the z direction: a split photodiode to monitor the deviation of the trapping E(r,t)= E(r,t)cos(kz − ωt). (1) light after passage through the trapped microsphere acting as a ball lens. It can be thought of as a case of light Then the optical gradient force in the dipole approximation for interacting with light via a medium with a nonlinear optical small particles in the direction x transverse to the propagation effect. Where does this nonlinearity come from? After direction z is all, neither the trapped microsphere nor the surrounding ∂E α ∂E2 ∂I ∂ϕ F = αE = = αZ =− opt , (2) fluid medium has any appreciable nonlinearity by themselves. ∂x 2 ∂x ∂x ∂x 0022-3727/08/224001+05$30.00 1 © 2008 IOP Publishing Ltd Printed in the UK J. Phys. D: Appl. Phys. 41 (2008) 224001 M Cronin-Golomb where Z is the impedance of free space, α is the particle photoexcited electron (number density n) results in an empty + polarizability and ϕopt is the potential energy of the particle trap (number density ND) with net charge e. The total number in the optical electromagnetic field. density of traps (both empty and occupied) is ND, and the The polarizability α of a microsphere with volume Vp and average number density of traps ionized in the dark is NA. permittivity εA in a surrounding material with permittivity εB The term β represents the thermally excited background of is [12] electrons in the conduction band. It can be thought of as − εA εB resulting from an effective dark intensity I . The parameter s is α = 3ε0εB Vp. (3) d εA +2εB the photoionization cross section and γR is the recombination The velocity v of a particle with radius a moving at low constant. Reynolds number in a Newtonian viscous fluid with viscosity η The first equation in (9) describes optical excitation and under the influence of a force F is given by recombination, the second equation is for current density and number density conservation, the third equation is Gauss’s law F v = (4) relating charge density to space charge electric field Esc, and 6πηa the final equation represents the current J of photoexcited particles (electrons). The first term in the current equation so that the drift term for current density is accounts for drift and the second term accounts for diffusion. ∂ϕ The last equation in (9) may be written as = = αZρ ∂I =− ρ opt Jdrift ρv , (5) 6πηa ∂x 6πηa ∂x ∂ k T J =−µen ϕ + B ln n , (10) where ρ is the number density of particles. The Stokes– ∂x e Einstein diffusion current density is where ϕ is the electrostatic potential. The model can be kBT ∂ρ simplified if we make the approximations Jdiffusion =− . (6) ∂N+ 6πηa ∂x ∂n D (1) ∂t ∂t since the number density of electrons is much The total current density equation is thus less than the number density of empty traps + (2) ND ND: the empty trap density is much less than the ρ ∂ total trap density J =− (ϕ + k T ln ρ). (7) 6πηa ∂x opt B (3) The number density of electrons can be regarded as quasi- steady state Current conservation implies that the resulting equation for the = (sI + β)ND complete model is n + . (11) γRND ∂ρ ∂J ∂ ρ ∂ Writing the combined intensity term sI + β as I ∗ we have =− = (ϕ + k T ln ρ). (8) ∂t ∂x ∂x 6πηa ∂x opt B µeI ∗N ∂ k T I ∗ J =− D ϕ + B log (12) γ N + ∂x e N + 3. Kukhtarev rate equations R D D so that the full model, taking current conservation into In the photorefractive effect, the optical field of equation (1) account, is excites charge in an electro-optic crystal such as barium titanate. These charges move under the influence of drift ∂N+ ∂ µI ∗N ∂ k T I ∗ D =− D ϕ + B log . (13) and diffusion to form a refractive index distribution via the + + ∂t ∂x γRND ∂x e ND electro-optic effect. The charge migration and resulting refractive index grating can be successfully modeled by the 4. The hopping model photorefractive rate equations first introduced by Kukhtarev et al [1]: In 1980, Feinberg et al proposed a hopping model for the ∂N+ photorefractive effect in which the probability for a charge D = − + − + (sI + β)(ND N ) γRnN , carrier to hop from one site to another is proportional to the ∂t D D local light intensity In. Suppose the total number density of ∂n ∂N+ 1 ∂J = D − , sites is N. The probability Wn that a charge carrier with charge ∂t ∂t e ∂x q occupies the nth site at position x is written as (9) n ∂E (n + N − N +)e sc = A D dWn βφnm βφmn , =− D W I exp −W I exp , ∂x ε dt mn n n 2 m m 2 m k T ∂ ln n J = µen E − B . (14) sc e ∂x where Dmn measures the tendency for a carrier to hop from site In these equations it is assumed that it is electrons that m to site n, β = 1/(kBT)and φnm = q(ϕn −ϕm) is the electric are photoexcited by light with intensity I from traps to the potential difference between sites n and m so that βφnm is the conduction band where they can move with mobility µ. Each ratio of the electrostatic potential energy difference between 2 J. Phys. D: Appl. Phys. 41 (2008) 224001 M Cronin-Golomb Table 1. Differential equations for artificial Kerr medium, and Kukhtarev and hopping photorefractive effect. ∂ρ 1 ∂ ∂ Artificial Kerr = ρ (ϕ + k T ln ρ) ϕ =−αZI ∂t 6πηa ∂x ∂x opt B opt ∂N+ µN ∂ I ∗ ∂ k T I ∗ ∂2ϕ e Kukhtarev D =− D ϕ + B ln = (N + − N ) + + 2 D A ∂t γR ∂x N ∂x e N ∂x ε D D ∂W Dqd2 ∂ ∂ k T ∂2ϕ Nq Hopping = WI ϕ + B ln WI = (W − W ) 2 0 ∂t kBT ∂x ∂x q ∂x ε sites n and m to the thermal energy kBT .