Gain and stability in photorefractive two-wave mixing Ivan de Oliveira and Jaime Frejlich Laborato´rio de O ´ ptica, IFGW, UNICAMP, Campinas, Sa ˜ o Paulo, Brazil ~Received 27 October 2000; published 13 August 2001! We demonstrate that the negative amplitude gain in a photorefractive two-wave-mixing experiment under applied electric field measurably reduces the characteristic instability of the recorded hologram. In this sense photorefractive materials behave like electronic amplifiers with feedback. We analyze the case of stationary and running holograms both under an externally applied electric field. A continuous phase-modulation method is used to simultaneously measure diffraction efficiency and phase shift. Measurements carried out on a Bi 12 TiO 20 crystal at 514.5 nm wavelength confirm the occurrence of continuous oscillations in both the diffraction efficiency and the phase shift. The perturbations in the diffraction efficiency increase considerably with increasing applied field and are reduced when energy is transferred from the weaker to the stronger beam ~negative gain!. Our results indicate that the perturbations in our experiments are probably due to resonantly excited transient effects. DOI: 10.1103/PhysRevA.64.033806 PACS number~s!: 42.60.Lh, 42.65.Hw, 42.40.Ht, 42.40.Pa I. INTRODUCTION Photorefractives are photoconductive and electro-optic materials where electrons ~and/or holes! may be excited, by the action of light, from photoactive centers into the conduc- tion ~valence! band. These electrons move by diffusion or by the action of an applied electric field and are retrapped some- where else. If a nonuniform pattern of light is projected onto the sample a correspondingly nonuniform spatial distribution of charges arises with an associated space-charge electric field, which gives rise to an index-of-refraction modulation via the electro-optic effect. In this way an index-of-refraction modulation is produced when a pattern of fringes of light of the form I 5u R 1S u 2 5~ I R 1I S !@ 1 1u mu cos~ Kx 1f !# , ~1! arising from the interference of two coherent beams of com- plex amplitudes S 5u S u e 2i c S and R 5u R u e 2i c R , is projected onto a photorefractive material @1–4#. Here I R 5u R u 2 and I S 5u S u 2 are the corresponding irradiances of the incident beams, K 52 p / D is the value of the vector K W , directed along the x coordinate, with D being the fringes’ spatial period, m 52 S * R /( u S u 2 1u R u 2 ) is the pattern-of-fringes complex modulation depth, and f 5c S 2c R . The resulting index-of- refraction modulation has the same K W but is f p phase shifted from the pattern of light @4,5#. This index-of-refraction modulation in the whole volume represents a reversible vol- ume phase hologram which produces amplitude and phase coupling between the interfering recording beams @6,7#. Such a coupling results in a feedback between the pattern of fringes and the reversibly recorded hologram, a process known as self-diffraction. The amplitude coupling does in fact represent a transfer of energy from one beam to the other and is described by the equations @4,6# I S ~ d ! 5I S ~ 0 ! 1 1b 2 1 1b 2 e 2Gd /2 , ~2! I R ~ d ! 5I R ~ 0 ! 1 1b 2 b 2 1e Gd /2 , ~3! where G and g are @4,5# G} Im$ k % , g } Re$ k % . ~4! Here ~0! and ~d! indicate the input ( z 50) and output ( z 5d ) positions inside the crystal with b 2 5I R (0)/ I S (0), Im$% and Re$% stand for the imaginary and real parts, respectively, and k is the coupling constant as defined in the dynamic coupled-wave theory @4,5#. This k depends on the externally applied field E 0 as well as on material and experimental parameters and fully characterizes the nature of the dynamic hologram being recorded, including the value of f p that is computed from @4,5# tan f p 5 Im$ k % Re$ k % 5 G g . ~5! The coupling between the phases of the interfering beams produces a bending or tilting of the hologram and is de- scribed by @4,5# c S ~ d ! 2c R ~ d ! 5c S ~ 0 ! 2c R ~ 0 ! 1 1 2 tan f p ln ~ b 2 1e Gd ! 2 ~ 1 1b 2 ! 2 e Gd . ~6! The diffraction efficiency h and the phase shift w between the transmitted and diffracted beams along any one of the two directions behind the crystal are formulated, respec- tively, by @4,5# h 5 2 b 2 1 1b 2 cosh~ G d /2! 2cos~ g d /2! b 2 e 2Gd /2 1e Gd /2 , ~7! PHYSICAL REVIEW A, VOLUME 64, 033806 1050-2947/2001/64~3!/033806~7!/$20.00 ©2001 The American Physical Society 64 033806-1