V. Prokofiev
 
Department of Physics, University of Joensuu, Finland

Abstract

Based on the fanning effect self-pumped and mutually pumped phase conjugation is demonstrated in photorefractive BTO-crystals with applied electric ac-fields. We used fiber-like crystals that enable a long interaction region and high applied electric fields. We realized three kinds of phase conjugating mirrors, self-pumped, externally self-pumped, and mutually pumped. Experimental results are given characterizing the spatial and temporal behavior of the phase conjugated wave and the transmission (imaging) of intensity and phase patterns.

Phase conjugation at an phase conjugate mirror (PCM) means that a wave is reflected back while its wave front is inverted with respect to its propagation direction. For the wave equation (1) this means that the wave vector and the phase term change their signs (= signal wave, = phase conjugated wave).

and (1)

Figure 1 shows the reflection of a spherical divergent wave from a point source.

Figure 1: Comparison

If the wave is reflected at a conventional mirror it follows the normal reflection law and keeps divergent. If the wave is reflected at a PCM it becomes convergent and propagates back to the source.

Phase-conjugation (PC) can be realized using Four-Wave-Mixing (FWM) in photorefractive crystals. Two waves, a signal and a pump wave, interfere inside the medium and create an intensity distribution. According to this distribution charge carriers are exited and transported by diffusion. An electric field structure builds up and creates a refractive index grating via the Pockels effect. This grating is a hologram and is read out by a second pump wave . The diffracted part of this pump wave is the phase conjugated wave of the signal wave (Fig. 2).

A self-pumped PCM can be realized in media with strong beam coupling efficiency, as for instance in barium titanate. The two pump waves are created from the signal wave by fanning. This is an asymmetrically amplified scattering of light. The second pump wave is built by the fanout reflected at the crystal corner. The beam path inside the crystal is called loop (right side of fig.3).

Figure 2: Four-wave mixing

Figure 3: Principle of the Self-Pumped Phase Conjugate Mirror (SPPCM)

Figure 4: Fiber-like BTO-crystals

We used fiber-like crystals of BTO (fig.4) where the dimension along the propagation direction is much longer then the other dimensions. These fiber-like photorefractive materials have some special properties compared to bulk materials [1,2]. They enable high interaction efficiency because of the large interaction length of the beams inside the crystal. Because of the small distance of the electrodes high electric fields can be applied using a moderate voltage of a few kV.

We studied three kinds of PCM. The first kind is the SPPCM (fig.5). The fanout is created by the fanning process. The gratings is written by the signal wave and its own fanout. The fanout that is reflected at the rear side of the crystal acts as the reading pump wave for the four-wave mixing. The self-pumped phase conjugation (SPPC) could be shown for the fiber-like BTO crystal. Because of the low reflectivity at the back face and the absorption inside the crystal the reading pump wave is much weaker then the signal wave. Therefore, the pc-reflectivity in the used SPPCM configuration with BTO is small (in the range of 1%).

Figure 5: Self-pumped phase conjugate mirror (SPPCM)

Another PCM is the "external" SPPCM (ESPPCM, fig.6). The arrangement is nearly the same as for the SPPCM, but the fanout is reflected at an external curved mirror. With this ESPPCM higher reflectivities can be reached.

Figure 6: External self-pumped phase conjugate mirror (ESPPCM)

Figure 7: Mutually pumped phase conjugate mirror (MPPCM)

The third kind of SPPCM is the "mutually" SPPCM. Mutually pumped PC (MPPC) is a principle where two incident signal waves get phase conjugated at the same time (fig.7). The waves do not need to be mutually coherent. Both waves build up their fanning. Every signal wave interferes with its own fanning building a grating. Parts of these gratings are jointly used. Every signal wave acts as the reading pump wave for the phase conjugation of the other one.

Some experimental results are presented in order to characterize these kinds of PCM concerning to temporal behavior, reflectivity, and quality of the phase conjugation.

First, the temporal behavior of the ESPPCM and the MPPCM is shown and compared in fig.8. We worked with relative low intensities of 0.75 mW/mm² and 1.7 mW/mm² and an applied ac-field of 1.5 kV/mm. The graphs on the left show the buildup of the pc-reflection. The buildup time is in the range of some seconds, what is qualitatively in the same range as known for barium titanate. The reflectivity is in the range of some percent. With the right graphs the long time stability is studied. The ESPPCM shows a high stability, whereas the reflectivity of the MPPCM strongly decreases and shows fluctuations as also known for barium titanate.

Next we studied how an intensity structure is transferred or imaged by an ESPPCM (fig.9, upper row). The left picture shows the pc-wave if the signal wave has a homogenous intensity. Then we introduced an amplitude picture into the signal wave what lead to the image on the right side. The picture could be imaged. Now this reflection is compared to the reflection at an conventional mirror (fig.9, lower line). The left picture shows once more the image using a ESPPCM in colors and the right the same with an conventional mirror. In this case the image quality is lower because of diffraction, what can especially be seen in the center.

Figure 8: Temporal behavior and reflectivity of the ESPPCM and of the MPPCM

Figure 9: Imaging with ESPPCM

We investigated the image properties of the MPPCM. Fig.10 shows the intensity distribution of one of the reflected waves if it is coupled out by a beam splitter and projected on a screen. A disadvantage of this method gets obvious. The intensity of the reflected signal builds a part of a ring structure, instead of a point that is expected from an unexpanded laser beam. The reason is the structure of the fanning in space, it builds a cone. Therefore, there are additional gratings that are not built by the interference of signal and fanning but by the interference of different parts of the fanning with one another. This system of gratings is read out by the fanning of the other wave that has the shape of a cone too. This results in an output signal that consists of waves that lie on the surface of a cone and therefore build the intensity distribution shown in Fig.10. MPPC could be shown in BaTiO₃ too. The coupling in this medium is very anisotropic because of the very different coefficients of the electrooptic tensor. By a self-organizing process, the additional gratings are suppressed and after a certain time the intensity is concentrated very near to the ideal point. In BTO this is not the case. Therefore, the applicability of MPPC in BTO could be limited.

Figure 10: Intensity distribution of a reflected wave on the screen

Figure 11: Quality of phase conjugation

Now we evaluated the quality of phase conjugation. We used a special feature of the pc-reflection. If a signal wave with plane wave front passes a phase distortion then is reflected at a PCM and passes the same distortion once more it gets plane again and the distortion is repaired. Fig. 11 shows a Michelson interferometer where one mirror is a PCM. Because of the above mentioned feature the phase distortion by a lens in the pc-arm should be corrected. At the exit of the interferometer we got parallel fringes as the interference pattern of two plane waves, what means that the phase conjugation works fine and with high quality.

A second feature of such an interferometer is that one can measure directly the phase distribution of an input wave with double sensibility. We measured a spherical wave produced by a lens. As can be seen, our measurement gave the right result, what once more confirms the good quality of the phase conjugation in BTO.

We showed the realizability of self-pumped (SPPCM), externally self-pumped (ESPPCM), and mutually pumped (MPPCM) phase conjugation in a fiber-like BTO crystal with applied elecctric ac field and with moderate intensities. This is remarkable because these effects normally only occur in media with very high coupling, like BaTiO₃. Best results were obtained with externally self pumped and mutually pumped phase conjugation. The advantage of using BTO is that the crystals are easier to grow and because of that they are much cheaper. Intensity structures can be phase conjugated (lensless imaging) and phase structures can be conjugated with a high quality.

This research has been partially supported by the Deutsche Forschungsgemeinschaft (DFG) within the Innovationskolleg "Optische Informationstechnik" and within the SFB225 and by the DAAD.

1. Kamshilin, V.V. Prokofiev, and T. Jaaskelainen: "Beam fanning and double phase conjugation in a fiber-like photorefractive sample", IEEE J. Quantum Electron. 31, 1642 (1995)
2. E. Raita, A.A. Kamshilin, V.V. Prokofiev, and T. Jaaskelainen: "Fast mutually pumped phase conjugation using transient photorefractive coupling", Appl. Phys. Lett. 70, 1641 (1997)