The temporal dynamics of photorefractive and absorptive gains during two-wave mixing in Stark geometry photorefractive quantum wells are investigated using moving gratings to break the symmetry of the photorefractive diodes to achieve nonreciprocal energy transfer between two coherent laser beams.
©1998 Optical Society of America
Photorefractive quantum wells [1,2] are a new class of optically addressed spatial light modulators that combine the advantages of large excitonic electroabsorption of quantum-confined excitons with large carrier mobilities to produce high-sensitivity holographic devices operable at extremely low light intensities.  The longitudinal-field Stark geometry, [4–8] operates with photocarrier transport perpendicular to the grating vector, and requires insulating cladding layers to isolate the electro-optic quantum-well layer from the electrical contacts. The longitudinal-geometry operates only under transient electrical excitation, during which charge separates within the electro-optic layer (which is also the photoconductive layer) and moves to the interfaces between the electro-optic layer and the cladding layer. All-semiconductor photorefractive p-i-n quantum well diodes were developed [7,8] with low-temperature-grown (LTG) multiple quantum wells which eliminate the need for Cr-doping  during growth and any post-growth processing. The LTG quantum wells have ultrafast lifetimes that extend down to picoseconds and retain excellent excitonic properties.  The fundamental operation of the device is not affected when all defects are removed from the buffers in which the multiple quantum well layer fulfills all three functions of photoconductive layer, electro-optic layer and charge trapping layer. 
Two-wave mixing occurs when each of the zero order transmitted beams interferes with one of the first diffracted orders of the other beam. This interference can cause nonreciprocal energy transfer between the two laser beams if a spatial shift between the refractive index grating and the incident intensity pattern can be achieved. This has been well established in bulk photorefractive crystals  as well as in thin films operating in the transverse Franz-Keldysh geometry [3,11] with the electric field in the plane of the quantum wells. Energy transfer requires broken symmetry between the two mixing light fields to define an energy transfer direction. Non-reciprocal energy transfer is normally forbidden by symmetry in longitudinal Stark-Effect photorefractive devices (electric field applied perpendicular to the quantum wells) because the carrier transport is perpendicular to the holographic grating vector, and all optical changes respond locally to the illumination. A spatial shift between the incident intensity pattern and the resulting space-charge formation can be achieved by using a well known technique of moving gratings. [12,13] An applied electric field produces a space-charge field which is in phase with the intensity pattern. By introducing a running grating/intensity pattern which moves on the time scale of the formation of the photoinduced space-charge, it is possible to achieve a condition in which the space-charge lags behind the intensity pattern. The moving gratings break the symmetry and make it possible to observe two-wave mixing in longitudinal Stark-geometry photorefractive p-i-n diode structures. Under optimal grating velocities, a (π/2)-phase shift can be introduced, which produces large photorefractive gains.  Large photorefractive gains are one of the unique features of photorefractive phenomena that distinguish them from other optical nonlinearities and make them candidates for many optical applications.
The samples used in our experiments were photorefractive diodes grown by molecular beam epitaxy (MBE) using a flux of As4. The structures were LTG MBE Al0.3Ga0.7As/GaAs multiple quantum wells (MQW) grown on n+ GaAs substrates. Contact and stop-etch layers of n-type materials were grown on a n+ GaAs substrate at 600 °C, followed by a LTG (320 °C) MQW layer consisting of a 120.5 period superlattice of 10 nm GaAs wells and 3.5 nm Al0.3Ga0.7As barriers. The low temperature growth at 320 °C results in approximately 0.1% excess arsenic in the MQW. The LTG superlattice was sandwiched between 500 nm Al0.5Ga0.5As cladding layers, grown at 320 °C. A 200 nm p- Al0.3Ga0.7As (1×1018 cm-3) layer followed by a 200 nm top p-GaAs (1×1019 cm-3) layer were grown at 450 °C on top of the LTG layers. Gold contacts were deposited on the top p-GaAs layer. The samples were epoxied to glass and the substrate was removed using standard techniques. A final gold contact was made to the exposed n- Al0.5Ga0.5As stop-etch layer after substrate removal.
We performed nearly degenerate two-wave and four-wave mixing using a CW Ti:Sapphire laser tuned near the low-energy zero-crossing of the electroabsorption of the device, with a fringe/grating spacing of Λ = 30 μm. Two acousto-optic modulators controlled the frequency difference, Ω, between the two equal-intensity writing laser beams. The moving interference pattern was created by keeping one of the acousto-optic modulators at a fixed frequency (f = 80 MHz) while varying the frequency of the other modulator, f + Ω. The electric field across the photorefractive quantum well was modulated using a single-sided reversed biased square pulse. The transmitted zero-order beams and the diffracted first-order signals were detected using silicon photodiodes with 650 nm long pass filters. The transmitted and diffracted signals were recorded as a function of time on a digital storage oscilloscope.
Figure 1 shows the temporal response of the photorefractive diode upon crossing the relative polarization of the two interfering laser beams. The electric field applied to the device was a 200 μs reverse-biased 7.5 V/μm square pulse at a repetition rate of 1 kHz with a detuning of 3 kHz between the two laser beams. The yellow shaded regions in fig. 1. show where the electric field is on. When the two laser beams are cross-polarized, the transient response reflects the transient electroabsorption of the device. This is identical for the two equal-intensity laser beams. When the two coherent laser beams are co-polarized, nonreciprocal energy transfer is demonstrated as one laser beam is amplified at the expense of the other. The direction of the energy exchange changes sign several times over a single period of the applied voltage, indicating that the space charge oscillates between leading and lagging the incident moving intensity pattern.
The photorefractive gain, Γn is defined by the asymmetry in the transmitted intensity divided by the total transmitted intensity and the active crystal length when both beams are co-polarized.
where I∥ is the transmitted intensity when the two writing beams are co-polarized, I⊥ is the transmitted intensity when the two writing beams are cross-polarized, I0 is the transmitted intensity when the electric field is turned off, L is the active crystal length, and the subscripts 1 and 2 refer to the two writing beams. The absorptive gain, Γα is given by the symmetric component of the transmitted intensities divided by the same factor.
The temporal response of the photorefractive gain is shown in figure 2 for increasing frequency offsets Ω introduced by changing the frequency of the variable-frequency acousto-optic modulator.  In figure 2 the oscillations persist after the electric field has been turned off, which is a consequence of the beating of the moving interference fringes against the quasi-static strobed grating created in the photorefractive diode by the electrical pulse. These oscillations are caused by the time-varying argument of the photorefractive phase shift  describing the shift between the intensity pattern and the space-charge field. The decay of the oscillation amplitude is due to erasure of the index (and absorption) gratings in the photorefractive diode as the interference fringes move across the device, after the electric field is turned off. The decay is determined by the dielectric relaxation rate of the photorefractive diode.
The oscillations present in figure 2 after the electric field has been turned off have characteristic frequencies and amplitudes. The applied electric field determines the strength of the index and absorption gratings  for a given laser intensity, while the frequency offset between the laser beams dictates the temporal dynamics. The inset in figure 2 shows the frequency of the oscillatory gain as a function of the frequency difference, Ω, between the two laser beams. The oscillatory frequency is equal to the frequency difference between the two laser beams.
The contribution to the transmitted intensity from the index grating is asymmetric with respect to the photorefractive phase shift  while the contribution from the absorption gratings is symmetric. This feature allows us to measure the contributions from the index and absorption gratings separately. Figure 3(a) shows the transient response of the photorefractive and absorptive gains respectively of the photorefractive diode as a function of the electric field. The photorefractive diode was operated with a 300 μs reverse biased square pulse at a repetition rate of 1 kHz with a fringe spacing of 30 μm. The incident laser intensity was 25 mW/cm2 tuned at a wavelength of 852 nm (close to the low-energy zero-crossing of the electroabsorption) with a detuning of 7 kHz between the two laser beams. The two contributions define a complex gain given by
The temporal response of the magnitude of the complex gain is shown in figure 3(b) for the same electric fields. The inset in the figures 3(a,b) show the respective peak gains as a function of the electric field. At low electric fields the contribution due to the index gratings dominate. At higher electric fields the absorptive gratings dominate, probably because of a significant Stark shift at higher fields, which results in a larger contribution to the absorptive gratings. Although the peak contribution for both the photorefractive and absorptive gains approach 1000 cm-1 the magnitude of the complex gain only approaches 1200 cm-1. This is because the two individual contributions act in quadrature.
In conclusion we have presented the transient response of a photorefractive diode during two-wave mixing using moving gratings. The oscillatory nature of the photorefractive gain exhibits a simple linear relation to the detuning between the two laser beams, making these devices viable candidates for optical vibration analysis and laser velocimetry.
This research is supported in part by the Materials Research Science and Engineering Center of the National Science Foundation (NSF) under award DMR-9400415. D. D. Nolte acknowledges support by NSF grant ECS-9708230. M. R. Melloch acknowledges support from U. S. Air Force Office of Scientific Research. L. J. Pyrak-Nolte acknowledges support from the NSF Young Investigator Award. I. Lahiri would like to thank W. S. Rabinovich and M. Duncan for their help in creating the animations.
References and links
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