On the Relationship Between Artificial Kerr Nonlinearities and the Photorefractive Effect

$On the Relationship Between Artificial Kerr Nonlinearities and the Photorefractive Effect$

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On the relationship between artificial Kerr nonlinearities and the photorefractive effect

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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 . It can be shown that Table 2. Linearized differential equations for nonlinearities driven this equation reduces to by sinusoidal fringe patterns with visibility m. 2 Linearized equation ∂W = Dqd ∂ ∂ kBT WI ϕ + ln WI , dρ k Tk2 k2αZρ mI ∂t kBT ∂x ∂x q Artificial Kerr 1 + B ρ = 0 0 (15) dt 6πηa 1 6πηa 2 2 ∂ ϕ Nq + = (W − W ) dN (sI0 + β)ND Ed 2 0 D + =− ∂x ε Kukhtarev + (Ep + Ed)ND smI0ND dt NAEµ Eµ assuming that D = D = D, the trapping centres are k Tk eN γ N nm mn E = B ; E = A ; E = R A evenly spaced by d and only nearest neighbour hopping is d e p εk µ µk included. The hopping current can be read from the first dW DI (kd)2 Hopping 1 + 0 (E + E )W =−D(kd)2W mI equation in (15)as dt E p d 1 0 0 d 2 Dqd ∂ kBT ∂ Jhop =− WI ϕ + ln WI . (16) kBT ∂x q ∂x with the observation that the initial (time zero) slope is inversely proportional to intensity and the steady-state first 5. Comparison of models order particle density is proportional to intensity. The time taken to reach steady state is proportional to the product of the Table 1 shows the partial differential equations for the three initial slope and the steady-state particle density is independent models side by side for comparison. of intensity. On the other hand, the photorefractive risetimes The hopping model differs from the Kukhtarev model in are inversely proportional to intensity, while the steady-state that the particle density multiplies the intensity in the former response is intensity independent, if the dark conductivity case, but divides it in the latter case. However after application factor β is neglected. In each case, the sensitivity is of first order perturbation theory, as described below, the two independent of intensity. One of the main differences between models become identical in form. the photorefractive and artificial Kerr nonlinearities is that Each of these models is in the form the artificial Kerr nonlinearity is washed out by thermal ∂ρ ∂ ∂ fluctuations, whereas the photorefractive nonlinearity depends = A g(ρ,F) (U + k T ln g(ρ, F )), (17) ∂t ∂x ∂t B on thermal fluctuations. The photorefractive effect is diffusion driven, so while the optical excitation of charge carriers where A is a constant, ρ is a generalized particle density, g is depends on the intensity of the light, they will not become a function of the particle density and optical energy source F and U is a potential energy density. When redistributed to form a space charge in the absence of thermally activated diffusion. The optical artificial Kerr nonlinearity is (1) optical excitation by a low visibility sinusoidal fringe directly optically driven. pattern is assumed: I = I0 + mI0 cos(kx) and (2) the model equations are linearized by taking first order 6. Refractive index distribution and optical perturbations in the variables, for example ρ → ρ0 + ρ1 where ρ1 is small, of order m, nonlinearity they become even more similar, as shown in table 2. Table 3 Once the particle distribution is known, the refractive index shows the resulting parameters for the three linearized models: distribution can be determined. In the artificial Kerr medium steady-state response, time constant, sensitivity and refractive case, we can use Bruggeman’s theory when the suspended index grating amplitude. The first result from this comparison microparticles have a dimension much less than the wavelength that can be read from table 3 is a relation between the hopping of the light being used [13]. This theory shows that the coefficient D and the Kukhtarev parameters refractive index of a suspension of microspheres with volume s k TµN fraction f = ρV microspheres of volume V is given in terms 2 = B D p p Dd . (18) of the dielectric constants by the relation γR eNA NA

The counterintuitive result that the exponential risetime for εA − ε εB − ε f + (1 − f) = 0, (19) artiﬁcial Kerr media is independent of intensity is consistent εA +2ε εB +2ε

3 J. Phys. D: Appl. Phys. 41 (2008) 224001 M Cronin-Golomb Table 3. Characteristic parameters for the three models. Artiﬁcial Kerr Kukhtarev Hopping ρ αZmI N + smI E W E Steady-state particle distribution 1 = 0 D1 =− 0 d 1 =−m d ρ0 kBT NA sI0 + β Ed + Ep W0 Ed + Ep 6πηa 1 N E 1 E Time constant τ A µ d 2 2 kBTk sI0 + β ND Ed + Ep DI0(kd) Ed + Ep

αZ 2 s kBTµND 2 2 2 Sensitivity (steady state/(mI0 τ)) k k Dd k 6πηa γR eNA NA 2 − 2 2 mI ρ V 2 3 (nA nB) 0 0 p 1 3 smI0 iEpEd 1 3 imEpEd Steady-state refractive index perturbation n1 2 2 n0reff n0reff 2 nA +2nB ckBT 2 sI0 + β Ep + Ed 2 Ep + Ed

= 2 using the following characteristic fields: diffusion field where εA nA is the relative dielectric constant of the = 2 microspheres and εB nB is the relative dielectric constant of kBTk the fluid in which the microspheres are suspended. A Taylor Ed = (25) expansion of equation (19) shows that the linear approximation e

ε = fεA + (1 − f)εB = εB + f(εA − εB) and limiting space charge field = − − εB + ρ0Vp(εA εB) + ρVp(εA εB) (20) eN E = A . (26) is in error by at most 0.2% for polystyrene beads in water. p εk It results in the following expression for refractive index ◦ perturbation n : The space charge field is 90 out of phase with respect to the 1 optical interference pattern. The phase shift is characteristic 1 (εA − εB) of the diffusion dominated photorefractive effect and is n1 = ρ1Vp. (21) 2 εB responsible for two beam coupling gain, an effect that is not seen in degenerate Kerr nonlinearities. Table 3 shows the Combining the expression for polarizability, equation (3), the refractive index perturbation for each case. expression for the first order particle distribution (table 3) and We are now in a position to explain the intensity equation (21)wefind dependence of these nonlinearities. Why is it that the steady- 2 2 2 2 2 2 2 3 (n − n ) mI0ρ0V 3 (n − n ) U state photorefractive effect is independent of average intensity n = A B p = A B fm E , 1 2 2 2 2 (neglecting the dark conductivity term β in the Kukhtarev 2 n +2n ckBT 2 n +2n kBT/Vp A B A B model), while the steady-state artificial Kerr nonlinearity is (22) proportional to intensity? Consider first the Kukhtarev model. where UE is the energy density of the optical field. The A doubling of the driving intensity does lead to a doubling magnitude of the refractive index grating is proportional to of the concentration n of electrons in the conduction band. = the filling factor f ρ0Vp, the fringe visibility m and the ratio The steady-state current, equation (10) is zero. This implies of the optical energy density to the thermal energy density. If that ϕ + (kBT/e)ln n is constant. Doubling of n leads merely we choose microspheres of a size, say 20 nm, such that the to the addition of a constant to the electrostatic potential ϕ. scattering coefficient µs is negligible then the thermal energy This makes no difference to the space charge field, since it is −3 density is about 1 kJ m . For 100 µm diameter beams with a proportional to the gradient of the potential. This result can ∼ × −5 power of 1 W, we find n 5 10 resulting in a four wave be read right away from the Kukhtarev differential equation mixing phase conjugate reflectivity of the order of 0.1% for model in table 1. This equation also shows that the speed of coupling lengths of the order 1 cm. Use of larger microspheres, response is linearly proportional to the driving intensity. The more tightly focused beams, and lower temperatures will result same result can also be read from the differential equations in in higher reflectivities. However the first two methods for table 1 for the hopping model. The key observation is that the increasing reflectivity will also increase scattering noise. intensity appears in the logarithmic terms of both the hopping In both the Kukhtarev and hopping models of the and Kukhtarev equations. Since the spatial derivatives of these photorefractive effect, the refractive index perturbation n1 is logarithms appear, the doubling of intensity has no effect on the derived from the space charge field Esc through the effective steady state. In contrast, the driving intensity does not appear electro-optic coefficient reff : in the logarithmic term in the artificial Kerr model of table 1, but rather directly in the potential ϕ . In this case, doubling n =−1 n3r E . (23) opt 1 2 0 eff sc of the intensity leads to a doubling of the first order density The space charge field is given in terms of the first order empty perturbation and hence a doubling of the optical nonlinearity. trap density by This equation also shows that the response time is independent of intensity. + ND1 EpEd Finally, we note that the formalism developed here can Esc = Ep =−m (24) NA Ep + Ed be used to quickly analyse alternative nonlinearities. For

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