Файл: Weber H., Herziger G., Poprawe R. (eds.) Laser Fundamentals. Part 1 (Springer 2005)(263s) PEo .pdf

Table 4.4.2. SBS threshold, max. reflectivity, far-field fidelity, M 2-limit, and power limit for di erent fiber phase conjugators, coherence length 1.5 m. The reflectivity is corrected with respect to Fresnel and coupling losses.

shows the energy reflectivity of carbon disulfide as a function of the input power at 1 µm wavelength. Carbon disulphide shows one of the smallest power thresholds for liquids of about 18 kW. Applying gases as SBS media, the power thresholds are about one order of magnitude higher. A saturation of energy reflectivity close to 80 % is a typical value for liquid SBS media, although reflectivities up to 96 % had been demonstrated [91Cro].

For high-power input pulses bulk solid-state media like quartz are investigated as SBS media, too [97Yos]. To reduce the power threshold of SBS a waveguide geometry can be applied [95Jac]. The beam intensity inside the waveguide is high within a long interaction length resulting in low power thresholds. To avoid toxic liquids and gases under high pressure multimode quartz fibers can be used [97Eic]. The lower Brillouin gain of quartz glass compared to suitable SBS gases and liquids can be overcome using fibers with lengths of several meters resulting in SBS thresholds down to 200 W peak power [98Eic].

The power threshold Pth can be estimated from (4.4.8), where Ae is the e ective mode field area inside the fiber core, Le the e ective interaction length, which depends on the coherence length, and g is the Brillouin gain coe cient; for quartz g is about 2.4 cm/GW [89Agr].

Table 4.4.2 shows the power threshold, the maximum energy reflectivity, the far-field fidelity, the M 2-limit (see below), and an approximated power limit of fiber phase conjugators with di erent core diameters. The used quartz-quartz fibers had a step-index geometry and a numerical aperture of 0.22. They were investigated with an Nd:YAG oscillator amplifier system generating pulses of 30 ns (FWHM) at 1.06 µm wavelength. Regarding applications it is important to couple also spatially aberrated beams into the fiber. The upper limit for the beam parameter product is due to the finite numerical aperture and the core diameter of the fiber. This can by expressed by a “times di raction limit value” M 2, see Chap. 2.2 for further information about beam characterization. The upper power limit is approximated assuming a damage threshold above 500 MW/cm2 for ns pulses.

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An important feature of a fiber phase conjugator is the threshold behavior for di erent M 2- values of the incoming beam. In case of a fiber the SBS threshold is nearly independent of the incoming beam quality. This is caused by mode conversion inside the fiber resulting in homogeneous illumination and therefore in constant SBS reflectivity. In case of a Brillouin cell the reflectivity depends on the far-field distribution of the incoming beam. Here phase distortions result in amplitude fluctuations in the focal region. A comparison between a di raction-limited beam (M 2 = 1.0) and a highly distorted beam (M 2 = 10) showed an increase of the SBS threshold of 300 % in case of the Brillouin cell. For the fiber phase conjugator no remarkable changes of the power threshold were observed [97Eic].

Practically, the reproduction of the initial wavefront is not perfect after phase conjugation. To characterize the deviation with respect to the reference wave the term fidelity F is introduced [77Zel]:

The fidelity equals unity in case of perfect wavefront reproduction and is smaller than unity for practical cases. To calculate the fidelity, the electric field distribution of the incident signal Ein and the not perfectly phase-conjugated wave Ep – the perfectly phase-conjugated wave is denoted Epc in Sect. 4.4.2 – has to be known. The determination requires sophisticated measurement equipment. In contrary, the far-field fidelity can be measured with less e ort and is therefore often used. The transmission through an aperture of the phase-conjugate signal is compared with the transmission of the input signal. The ratio is called far-field fidelity, because the aperture is placed in the focal plane of a focusing lens.

4.4.5 Applications of SBS phase conjugation

Phase conjugation generates a wave which retraces the incoming wave in a time-reversed way. Thereby it is possible to eliminate phase distortions in optical systems. For example, in a solidstate laser amplifier, the incoming beam is not only amplified but su ers also from phase distortions due to thermal refractive-index changes in the laser crystal. After passing this amplifier crystal, the beam is reflected by a phase conjugator and passes the crystal a second time. As the wavefronts are inverted with respect to the propagation direction, the refractive-index changes reduce the phase distortions and after the second passage, these distortions disappear so that the beam quality of the incoming wave is reproduced. In Fig. 4.4.5 a double-pass scheme with phase-conjugate mirror to compensate for phase distortions is shown.

Typically, phase conjugators are applied in Master Oscillator Power Amplifier (MOPA) setups, where a nearly di raction-limited master oscillator beam is increased in power within an amplifier

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Fig. 4.4.7. Far-field intensity distributions of the oscillator beam, the distorted beam after single-pass amplification, and the highly amplified beam after double-pass amplification with phase conjugation.

arrangement, see Fig. 4.4.6. After the first amplification pass the beam quality is reduced due to thermally induced phase distortions. The spatial-distorted beam enters the SBS mirror and becomes phase-conjugated. The initial beam quality of the master oscillator can be roughly reproduced after the second amplification pass. The amplified beam is extracted with an optical isolation, which consists in this case of a Faraday rotator and a polarizer.

Figure 4.4.6 shows a MOPA system producing up to 210 W average output power at 2 kHz average repetition rate (1.08 µm wavelength). The system is part of an advanced setup yielding up to 520 W average output power [99Eic]. The oscillator beam has a nearly di raction-limited

beam quality (M 2 < 1.2) which is already reduced in front of the first amplifier (M 2 1.5). This

=

results from optical components between oscillator and amplifier which introduce phase distortions.

After single-pass amplification the beam quality decreases to M 2 5 due to phase distortions

=

introduced by both pumped amplifier rods at 6.5 kW pumping power for each amplifier. After phase conjugation and double-pass amplification the initial beam quality can be nearly reproduced (M 2 < 1.9). Di erences between the initial and final beam quality are caused by a fidelity smaller than unity and di raction at several apertures in the amplifier chain.

The performance of the phase-conjugate mirror can be illustrated by far-field intensity profiles recorded at di erent positions in the setup. In Fig. 4.4.7 the oscillator output beam exhibits a smooth Gaussian profile corresponding to the nearly di raction-limited beam quality. After singlepass amplification the reduction of beam quality is confirmed by a strongly aberrated far-field profile. After phase conjugation and double-pass amplification the initial intensity distribution can be nearly reproduced. In this example the average power of the master oscillator beam (approx. 1 W) was increased to 130 W after double-pass amplification.

Presently, phase distortion elimination in double or multipass laser amplifiers is the most often application of phase conjugation. In addition phase conjugators are useful as mirrors in laser oscillators replacing one of the conventional mirrors. Again, the phase conjugator eliminates phase distortions in the laser medium induced by optical or discharge pumping. For recent advances and applications of SBS-phase-conjugation see [02Eic, 03Rie, 04Rie].

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4.4.6 Photorefraction

The photorefractive e ect belongs to the nonlinear optical e ects with the highest sensitivity for operation at low optical intensity levels. Photorefractive phase conjugators are able to operate using intensities of only mW/cm2. The price paid of the low intensity performance is diminished speed. The response times of recent photorefractive phase conjugators span in the range of milliseconds to several minutes.

The photorefractive e ect describes light-induced refractive-index changes in the material when the incident light is spatially nonuniform [88Gue, 93Yeh, 95Nol, 96Sol]. The spatial nonuniformity distinguishes the photorefractive e ect from other common nonlinear optical e ects that occur under spatial uniform intensity. The maximum refractive-index change induced in a photorefractive material does not occur necessarily locally where the light intensity is a maximum. The nonlocal response occurs because electric charges move and are stored inside the material. In case of classical nonorganic bulk photorefractive materials, such as ferroelectric oxides (BaTiO3, LiNbO3, KNbO3), sillenites (Bi12SiO20, Bi12TiO20, Bi12GeO20) or semi-insulating semiconductors (GaAs, InP, CdTe), electrons (or holes) are photoexcited from localized impurity centers or defect sites, which are energetically located deep in the band gap of the material, into the conduction (or valence) band.

The energy of the exciting photons is smaller than the band-gap energy. Free carriers excited in bright crystal regions move due to di usion and drift into dark crystal regions where they are trapped by empty defect sites, see Fig. 4.4.8. As a consequence of separated and trapped electric charges the formation of space-charge electric fields occurs. These electric fields change the refractive index of the material by electrooptics e ects, usually the Pockels e ect.

Nonuniform illumination occurs when two coherent laser beams interfere in the crystal. The intersecting beams create a periodical interference pattern. The formation of a photorefractive index grating due to a sinusoidal intensity pattern is shown in Fig. 4.4.9. When di usion is the main e ect for the transport of the excited charge carriers (there is no external electric field applied on the crystal) the electric-field maxima are shifted by a quarter fringe spacing relative to the intensity maxima. This π/2 phase shift of the induced index grating plays a fundamental role in photorefractive non-linear optical wave mixing. It allows for an energy transfer between the two beams writing the grating in a process called two-wave mixing. One of the beams (called signal beam) is amplified at the expense of the other beam (called pump beam).

A phase-conjugate beam can be created by four-wave mixing processes. In this case the two-wave mixing arrangement is extended with a second pump beam which counterpropagates with respect to the first pump beam, see Fig. 4.4.3. In case of external pump beams, the phase-conjugation

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