given to the effect of finite slew rate, uneven mark-space ratio, and other approximations to square ac waveforms. It is shown that the ideal square waveform ...
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J. Opt. Soc. Am. B/Vol. 7, No. 3/March 1990
Walshet al.
Techniques for the enhancement of space-charge fields in
photorefractive materials K. Walsh, A. K. Powell, C. Stace, and T. J. Hall
Department of Physics, King's College, University of London, Strand, London WC2R 2LS, England Received July 19, 1988; accepted October 16, 1989
Solutions to the band-transport equations are developedto include effects on the amplitude, dynamics,and spatialfrequency response of the space-charge field when periodic or dc electric fields are applied to a photorefractive material. Analysis of the effects of both idealized sinusoidal and square ac waveforms are given and compared with the case of resonant space-charge field enhancement with applied dc fields and moving fringes. Consideration is
given to the effect of finite slew rate, uneven mark-space ratio, and other approximations to square ac waveforms. It is shown that the ideal square waveform provides the greatest enhancement of the space-charge field. It is also shown that in absorbing materials resonant enhancement with dc fields and moving fringes is localized to thin regions within the material, as the resonance condition is intensity dependent. This reduces the effective enhancement. Illustrative experimental results are presented for Bi12SiO20 and GaAs.
INTRODUCTION
Two theoretical approaches have been adopted. Step-
Much interest has been generated by the potential applications of photorefractive
media in low-power image process-
ing. Proposed applications range from coherent image amplification to implementation of reconfigurable interconnections. For any proposed application to be implemented efficiently it is necessary to achieve a high two-wave mixing gain or a high four-wave mixing diffraction efficiency. In BaTiO 3 and Sr0.6Ba 0 .4 Nb 2O 6 large electro-optic coefficients
ensure that a large phase modulation is created for moderate photoinduced space-charge fields. In other photorefractive materials, such as those of the Bi12SiO20 family (Bi12SiO20, Bi12GeO20, Bi12TiO2 0) and the semiconductor materials (GaAs, InP, CdTe), the electro-optic coefficients are much smaller, and it is necessary to produce much larger internal space-charge fields to generate phase modulations similar to those achieved in BaTiO3. Generation of these higher space-charge fields requires enhancement techniques involving the use of applied electric fields. Despite this additional complication, there is much interest in the smaller electro-optic coefficient materials and enhancement techniques. This is due to their faster response times and, in the case of semiconductor materials, their sensitivity at technologicallyimportant spectral regions in the near infrared. The most widely used enhancement technique has been to apply a dc electric field and to frequency detune one of the beams so that the intensity pattern moves through the crystal.' An alternative technique, due to Stepanov and Petrov,2 is to apply an alternating electric field. In a recent paper they presented theoretical results for both square and sine waveforms, showing that the square waveform is considerably more effective in enhancing the space-charge field.
They also reported experimental results in Bi12TiO20 . Experimental observations of this enhancement have also been reported in GaAs (Ref. 3) and in Bi 2 SiO2 0 .4 Enhancement
techniques in this context mean techniques that increase the magnitude of the space-charge field over and above that which would result from the application of a dc field of the same magnitude, alone. 0740-3224/90/030288-16$02.00
anov 2 adopted a technique of averaging over the period of
the electric field to determine the behavior of the spacecharge field. This technique forms the basis of the analysis in this paper. A simple proof of this technique for a square waveform is discussed in the text, with a more formal discussion for arbitrary waveformsbeing presented in Appendix A. A second approach for a sinusoidal waveform has been de-
veloped by Kumar et al. 5 The result obtained for a sinusoidal waveform is slightly different in the two cases, for reasons discussed in Appendix A. The time-averaging approach is also computationally much simpler and more easily extended to other waveforms. In this paper dc and ac techniques are compared. The effect of absorption on the technique of dc field and moving
fringes is considered. It is shown that the use of this technique may be of limited value in the case of semiconductor materials. For the ac field technique we examine the sensitivity of enhancement to the details of the waveform and address the question of how square a waveform must be. This becomes particularly important when one is trying to generate a square waveform at the high frequencies required for semiconductor materials at which, owingto the high slew rate required, experimental waveforms deviate from the perfect square. The easiest way of producing a high-voltage ac field is to use a sinusoidal wave form. This then permits the use of a driven resonant ac circuit to produce the voltages required. It is for this reason that we also study the effect of sinusoidal applied field wave forms on the photorefractive effect. The frequency of the applied field is not a critical factor to the enhancement method, although maximum and minimum frequency limits apply. On the long time scale, the applied field must have a period that is shorter than the grating relaxation time in the material. The space-charge field may then be considered stationary and does not require implementation of moving fringe techiques to produce enhancement. On the short time scale, the period of the applied field must be longer than the free-carrier lifetime in the material. If this were not the case,the charge carriers would ©1990 Optical Society of America
289
Vol. 7, No. 3/March 1990/J. Opt. Soc. Am. B
Walshet al. experience only a time-averaged
field and thus not move
appreciably under the influence of the field. This would then prevent the buildup of a large space-charge field.
NA
is the number density of acceptors. (It is assumed
that these are all ionized.)
As
the grating relaxation time is light-intensity dependent, the applied field frequency must be high enough to allow for the
For ease of manipulation these equations are normalized with respect to key parameters. To do this it is necessary to
shortest grating relaxation time but still have a period longer than the free-carrier lifetime. In Bi12Si020 the grating relaxation time can typically be of
define some new parameters, which are now discussed.
the order of milliseconds to seconds, depending on the level
occupation W is given by
The first of these parameters is the probability of occupation of a donor site. Under illumination, the probability of
of illumination. The free-carrier lifetime is of the order of tens of nanoseconds. 6 Applied field frequencies in the range of 100 Hz to 10 MHz are thus suitable for Bi12 SiO2 0. In the
case of gallium arsenide, the grating relaxation time is typically in the range 10 msec to 10 jusec, with carrier recombina-
tion times again in the nanosecond range, thus requiring an applied field in the frequency region 100 KHz to 10 MHz. Enhancement of the photorefractive effect with ac fields is
In the
case of gallium arsenide, response times greater than television frame rates should be achievable, making the crystal suitable for image processing applications.
ization. If this level of intensity is Ic, then the probability of occupation WCat this intensity is W
= ND
is assumed that n, >1 but low absorption or a thin interaction region so that L
. .. . .
....
10
lengths, but this is at the cost of a narrower bandwidth. For image-processing applications a flat spatial-frequency response is often the ideal case. If this is of crucial impor-
sine wave
----
1o 8
.
10
. 0
10
5
Spatial Frequency K (cm- 1) Comparison of enhancement
sensitivity
of the space-
charge field with spatial frequency for sinusoidal and square applied waveforms. Bi 12SiO2 0 -
1++P
P0 . This figure illustrates the way in which this enhancement technique produces an enhanced field only for a narrow band of spatial frequencies in an image. The enhance-
160.4
Fig. 19.
(115)
1+ [Po¢(1+ P 2) - Po(l+p2)2
10*4
301
The gain
UL shown is a theoretical
prediction for
tance, the use of a dc field and a moving fringe pattern may
be not so suitable as the use of ac enhancement techniques. If gain can be sacrificed for image fidelity, a cosinusoidal waveform or even a dc field may be preferable to a square waveform.
DISCUSSION S
s
-
i1W
1 + iogrg
1 + i (cog -
(108)
co),Tg(18
The spatial frequency for which w = cogis defined as K 0 , and spatial frequency is normalized to this value by r
= K/Ko.
(109)
In addition, a new parameter Po is defined as Po = LEKO.
(110)
When we make the approximation that the spatial frequency is small enough that LD 2K 2
