Enhancing the photorefractive effect

Optics Communications 101 (1993) 397-402 North-Holland

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Enhancing the photorefractive effect G a l e n C. D u r e e Jr., J o h n L. Shultz, N i a n y u Bei, G r e g o r y J. S a l a m o University of Arkansas, PhysicsDepartment, Fayetteville,AR 72701, USA

E d w a r d J. Sharp, G a r y L. W o o d Army Research Laboratory, Fort Belvoir, VA 22060-5028, USA

Steven M o n t g o m e r y NavalAcademy, Annapolis, MD, USA and

R a t n a k a r R. N e u r g a o n k a r Rockwell International Science Center, Thousand Oaks, CA 91360, USA

Received 2 March 1993

The transient response for beam fanning in photorefractivetungsten bronze crystalswith an applied electric field and a focused laser beam is studied. Response times on the order of 1 ms for an incident power of I mW observed for incident beams focused t o a 30 ttm diameter in the crystal. This improved responsetime is accomplishedwithout the significant reduction in the magnitude of the photorefractiveeffect normally observed for focusedbeams.

1. Introduction Photorefractive crystals have potential for applications in the fields of dynamic holography and optical phase conjugation. This is basically due to the fact that these crystals exhibit large changes in their index of refraction with milliwatts of incident laser power. At the same time, however, these crystals suffer from a reputation of having a response time that is considered very slow. In this paper, we demonstrate that milliwatt lasers can in fact produce large index changes with fast response times in many photorefractive crystals available today. Our experimental results are compared with theory and show some salient features not previously observed. Given a laser power of I m W in a 1 m m diameter beam incident on a tungsten bronze photorefractive crystal, the typical photorefractive response time is on the order of I to 10 s. This response time is considered relatively slow for such applications as op-

tical limiting or image processing. In order to improve the response time, two techniques can be used. The first takes advantage of the fact that the photorefractive response time goes inversely with the incident intensity. By focusing the laser beam to a small area in the crystal, the incident intensity can be increased and consequently a dramatic improvement in the response time may be expected. However, when the incident laser light is focused to a small spot, the magnitude of the photorefractive effect decreases and nearly vanishes [ 1 ]. The physical reason for the decrease in the strength of the effect is due to the fact that the gain-length product for a 30 Ixm diameter beam is extremely small when compared to that of a millimeter beam. As a result, while focusing the incident laser light is attractive, it proves to be impractical. One method of overcoming the gain-length reduction is the use of a cylindrical lens for focusing the incident light into the crystal [2]. In this case, the intensity is increased by focusing only in one

0030-4018/93/$06.00 © 1993 Elsevier Science Publishers B.V. All fights reserved.

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plane and the gain-length product is maintained by allowing the incident beam to be unfocused in the corresponding orthogonal plane. While the cylindrical lens certainly provides an improvement in response time, it is limited to one to two orders of magnitude producing times on the order of 100 ms for an incident power of 1 mW, a time still considered too long by many. A second method for improving the response time is the use of an applied electric field. With no applied field, photorefractive charge transport in tungsten bronze crystals is dictated by the diffusion of excited carriers. With an applied field of 5 to 10 k V / c m the transient response is driven by the drift of carriers and can be considerably faster. Typically, for an incident power of I m W in a 1 m m diameter beam, the response time can be improved from 1 to 10 s to 0.1 to 1 s, or one order of magnitude. While both techniques, focusing and applying an electric field, improve the photorefractive response time, they are limited to improvements of about two orders of magnitude. In this paper we report the use of a spherical lens to produce a small spot, on the order of 30 ~tm, in a photorefractive crystal with an applied electric field of about 6 kV/cm. While the use of the small focused spot normally shows near zero gain-length and consequently a vanishing photorefractive effect, we demonstrate for the first time that simultaneous application of an applied dc electric field increases the gain-length product and therefore makes possible the full advantage in speed that can be gained by focusing. In fact, we report a response time of 1 ms for an incident power of 1 mW with a focused diameter of 30 lam and an applied field of 6 kV/cm. This four-order-of-magnitude improvement in response time is achieved without a significant reduction in the gain-length product or the magnitude of the photorefractive effect.

2. Theory Two different theoretical approaches have been used to describe the behavior of photorefractive materials. The first is a transport model [ 3 ] while the second is a hopping model [ 4 ]. In this paper, we will follow the transport model first proposed by Ku398

1 September 1993

khtarev [ 1 ] and recently discussed in detail by Valley and Klein [ 5 ]. In these papers, the transport equations are solved for the time dependence of the electric field in the crystal using three approximations. First, the spatial variation of all quantities is assumed to be described by a constant zeroth-order term and a first-order term, i.e. the space charge field can be written as E (t) = Eo + mE~ (t) exp (ikx). Second, only terms to first order are kept, and finally, the free carrier density is assumed to be small. Under these assumptions, a second-order differential equation in time can be written for the density of ionized donors, the current density, the excited charge carrier density, and the space charge field. For example, the equation for the space charge field is

d2E~/dt2 + A dEs¢/dt + BEs¢ = C,

( 1)

where A, B and C are complex. The solution to this equation has a damped oscillatory behavior and is given by E ~ ( t ) = E ~ [ - c o s ( k g z + 0) +e-t/~cos(kgz-o~t+O)] .

(2)

For the experiment reported here, rR 1. In this case the damping part of the solution is characterized by z, the photorefractive response time r ~ rz(rE/rR) 2 ,

(3)

while the oscillatory part of the solution is characterized by a period T = 2n/to, T ~ ri(rE/rR)2Zt •

(4)

In these expressions, rR is the carder recombination time, rD the material characteristic diffusion time over a grating wavelength, rE the characteristic electric field driven drift time over, roughly, a grating wavelength, rl the carrier excitation time and Tdi the dielectric relaxation time. the physical interpretation of eq. (4) is that rE represents the time to transport the equilibrium value of charge, no, in the conduction band over a distance of approximately one grating wavelength, ;t s. The term rl/rR is the factor needed to bring the transported charge up to the value Neff= (rI/zR) no, needed to produce the equilibrium space charge field. Meanwhile, eq. (3) can be under-

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stood as the number of periods T required before relaxation is complete, z= T(ZE/ZR). In addition to the transient character of the solution, there is also the steady-state character described by the magnitude of the equilibrium space charge field Es~. "

2

2

--1/2

and the equilibrium phase shift between the intensity spatial pattern and the index spatial pattern tan ¢ =

1+ ~

+ EDEq/"

(6)

In these expressions, Eo is the applied static electric field, ED the effective field generating diffusion, and E~ the maximum space charge field limited by the maximum charge separation that could possibly be produced without an applied field. The characteristic transient times z and T expressed in eqs. (3) and (4) are for the space charge field. The expression for gain for two-beam coupling is given by

?=E~kn3r~cos 0,

(7)

where k is the vacuum wave number, n is the index of refraction, reff is the effective electro-optic coefficient, and 0 is the internal crossing angle. As a result, the time dependence of the gain coefficient follows the expression for the space charge field.

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1 September 1993

However, the behavior of the energy exchange taking place in two-beam coupling is given by I1(0) Io I x ( t ) = I , ( 0 ) + [Io--I1(0) ] e r='

(8)

where Io = 11+ 12= constant. In other words, the expected behavior is not that of a simple exponential as long as the gain-length is large. However, the general features of decay and oscillation do carry over and are expected to be present even in the case of large gain-length.

2. Experimental apparatus Our experimental apparatus is shown in fig. 1. An argon-ion laser beam is focused into a photorefracrive crystal. The spot size in the crystal is on the order of 30 gm. With a waist of this size in the crystal, beam fanning is minimal [6]. The physical reason for the relative absence of the beam fanning effect is that, with the beam diameter in the crystal as sma'll as 30 ttm, the interaction length between scattered light and the incident beam is very small except in the forward direction. Unfortunately, in the forward direction, where the interaction length remains large, the coupling coefficient tends to zero since the space charge field tends to zero, as predicted by eq. (5). As a result, the gain-length product for the coupling of energy from the incident beam into the scattered light, or fan, is minimal. However, by applying an

APPLIED

ARGON'ION[J!I

[C ° T u t e r ~ - ~ " Digital Signal[

i ,/o'=r

l

Fig. 1. Experimentalapparatusfor observingtransmittedand fannedlight. 399

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1 September 1993

1.130 [INCID~rr P O W ~ = 0.042 m w I

~ 0.75

D

In(C)= In(C) - Aln(I) C = 0.00245159

4 i~O.~

~ 5 0.0

0:5

1.0 TIME(s)

--"11| JJ.__._ " ......... . / ! .5 2.0

-5

-4 -3 -2 -1 0 NATURAL LOG OF THE INCIDENT POWER in(milliwatts)

1'751

T i

1.25

0%

In(Z)= In(C) - AIn(V)

/ 20

4o

6'.o

8:0

~6.o

TIME(ms)

Fig. 2. (a) Typical behavior of the transmitted beam as s e e n by detector D2. (b) Typical behavior of the beam fan as seen by d e t e c t o r DI. electric field, Eo, the fan is observed to be greatly enhanced [ 7 ]. This is due to the fact that the applied field is particularly effective for small grating wave vectors, ks. Since the focused beam only allows light scattered at small angles to have large interaction lengths, it is dominantly the light at small angles, or correspondingly small ks, that is responsible for the observed enhanced beam fan and the large reduction in transmitted light. In fact, the range o f k , for which the scattered fan light can experience a significant gain-length product is small. The angle formed between the incident beam and scattered light over a 6 m m path is only 0.3 °. This angle only increases to 0.9 ° over a 2 m m path. Consequently, these small angles restrict the enhanced coupling to a small range of grating wave vectors. In our experiment, the grating wave vector contributing to oscillations in the transmitted beam is further restricted by the choice of aperture A. For example, with an aperture of I m m in diameter, the 400

'~'

1

NATURAL LOG OF THE APPLIED VOLTAGE In(kilovolts) Fig. 3. These log-log plots show that the observed photorefractire response time ~ has a 1/I dependence and 1/ V 2 dependence as seen by D2. Identical behavior is observed on Dr.

detector D~ which monitors the transmitted beam also collects all scattered light at an angle less than about 0.6 °. As a result, the range o f wave vectors which can significantly contribute to energy exchange is sharply limited to a range of crossing anDes from, at most, 0.9 ° to 0.6 ° . Consequently, it should not be surprising that observations in our experiment tend to follow predictions based on twobeam coupling. It is important to note that our results apply to the transient region. That is, during the transient period the phase of the space charge field relative to the intensity pattern is oscillating so that the energy exchange also experiences an oscillatory behavior. At equilibrium, however, coupling at small angles fo. which Eq > Eo yields an equilibrium phase difference, given by eq. (6), which is close to 0 °. This means that at equilibrium little or no light is fanned at small angles.

0!

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1 September 1993

O

In(T) = ln(C) - Ain(I) C = 0.0021355

In(T) ffi In(C) - Aln(1) C = 0.0016232

-8 NATURAL LOG OF THE INCIDENT POWER ln(milliwatts)

NATURAL LOG OF THE INCIDENT FOWER In(milliwalts)

-1 lnfT) = in(C) - AIn(V)

m[-

~8

,

~

c = 58.0288

-'<2. 6'.0

63

7'.0

7'.5

810

ln(T) = In(C) - AIn(V) C ffi 224.8717

8 4

815

NATURAL LOG OF THE APPLIED VOLTAGE In(volts) Fig. 4. T h e s e l o g - l o g p l o t s s h o w t h a t t h e p e r i o d T o f t h e o b s e r v e d o s c i l l a t i o n s in t h e t r a n s m i t t e d s i g n a l h a s a 1 / I d e p e n d e n c e a n d 1 / V d e p e n d e n c e a s s e e n b y D2.

6'.0

63

710

73

810

s'.5

NATURAL LOG OF THE APPLIED VOLTAGE In(volts)

Fig. 5. These log-log plots show that the period Tofthe observed oscillations in the beam fan signal has a l/1 dependence and 1/ Vdependence as seen by D~.

3. Experimental results Figure 2a shows the transmitted intensity as a function of time for an input power of about 0.04 mW. As can be seen, the characteristic response time z is on the order of 800 ms while the characteristic frequency f o r period, T = l/f, is shorter and on the order of 200 ms. A typical beam fan curve is shown in fig. 2b. The observed response time is now on the order of l ms for an incident power of 1 mW. Two characteristic features of eq. (3) are that the formation time z of the space charge field should behave inversely proportional to the incident intensity and to the square of the applied electric field. This predicted behavior is observed and demonstrated in fig. 3. In addition, eq. (4) shows that the period T should be inversely proportional to both the intensity I o of the incident light and to the applied field Eo. Figure 4 shows that this is exactly what is observed. Identical behavior was observed for the re-

sponse time and oscillation period in the beam fan curves and is shown in fig. 5.

4. Conclusion In conclusion, we have demonstrated that with powers of 1 mW, response times on the order of 1 ms are possible with large index changes. Careful measurements of the transient behavior observed or oscillatory frequency co as a function of intensity or applied field can yield values of the product of the retrapping time and the mobility over a small probing region of the crystal and will be the subject of further work. In addition, the enhanced response time achieved via focusing and applying an electric field can yield the production of a phase conjugate signal with rapid response times using self-pumped phase 401

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c o n j u g a t i o n or d o u b l e phase conjugation. This is also the subject o f f u r t h e r work.

References [ 1 ] M. Segev, Y. Ophir and B. Fischer, Optics Comm. 77 (1990) 265. [2] G.J. Salamo, B.D. Monson, W.W. Clark III, G.L. Wood, E.J. Sharp and R.R. Neurgaonkar, Appl. Optics 30 ( 1991 ) 1847.

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[3] N.V. Kukhtarev, Soy. Tech. Phys. Lett. 2 (1976) 438; N.V. Kuldatarcv, V.B. Markov, S.G. Odulov, M.S. Soskin and V.L. Vinetskii, Ferroclectrics 22 (1979) 949; G.C. Valley, IEEE J. Quantum Electron. QE-19 (1983) 1637. [4]J. Feinberg, D. Heiman, A.R. Tanguary Jr., and R.W. HeUwarth, J. Appl. Phys. 51 (1980) 1297. [ 5 ] G.C. Valley and M.B. Klein, Opt. Eng. 22 (1983) 704. [6] G.J. Salamo, B. Monson, W.W. Clark III, G.L. Wood, E.J. Sharp and R.R. Neurgaonkar, Appl. Optics 30 ( 1991 ) 1847. [ 7 ] W.W. Clark, III, G.L. Wood, M.J. Miller, E.J. Sharp and G.J. Salamo, Appl. Optics 29 (1990) 1249; K. Sayano, A. Yariv and R.R. Neurgaonkar, Optics Lett. 15 (1990) 9.