Lock, external phase

Let us consider an optical system with two modes at the frequencies oo and 2oo interacting through a nonlinear crystal with second-order susceptibility placed within a Fabry-Perot interferometer. In a general case, both modes are damped and driven with external phase-locked driving fields. The input external fields have the frequencies (0/, and 2(0/,. The classical equations describing second-harmonic generation are [104,105] ... 

Function q is 27r-periodic, and in the simplest case q -) = sin( ) Eq. (13.3) is called the Adler equation. One can easily see that on the plane of the parameters of the external forcing (cu, e) there exist a region eqmin < oo loq < eqmax, where Eq. (13.3) has a stable stationary solution. This solution corresponds to the conditions of phase locking (the phase 0 just follows the phase of the force, i.e. (f> = uit + constant) and frequency entrainment (the observed frequency of the oscillator Cl = (0) exactly coincides with the forcing frequency tu brackets () denote time averaging). 

Fig. 6. b) Heterodyne signal of two phase-locked external cavity lasers. Linear vertical scale 10 Hz/div, 10 Hz resolution band-... 

Historically, the first and most important capacitance method is the vibrating capacitor approach implemented by Lord Kelvin in 1897. In this technique (now called the Kelvin probe), the reference plate moves relative to the sample surface at some constant frequency and tlie capacitance changes as tlie interelectrode separation changes. An AC current thus flows in the external circuit. Upon reduction of the electric field to zero, the AC current is also reduced to zero. Originally, Kelvin detected the zero point manually using his quadrant electrometer. Nowadays, there are many elegant and sensitive versions of this technique. A piezoceramic foil can be used to vibrate the reference plate. To minimize noise and maximize sensitivity, a phase-locked... 

The proton noise-decoupled 13c-nmr spectra were obtained on a Bruker WH-90 Fourier transform spectrometer operating at 22.63 MHz. The other spectrometer systems used were a Bruker Model HFX-90 and a Varian XL-100. Tetramethylsilane (TMS) was used as internal reference, and all chemical shifts are reported downfield from TMS. Field-frequency stabilization was maintained by deuterium lock on external or internal perdeuterated nitromethane. Quantitative spectral intensities were obtained by gated decoupling and a pulse delay of 10 seconds. Accumulation of 1000 pulses with phase alternating pulse sequence was generally used. For "relative" spectral intensities no pulse delay was used, and accumulation of 200 pulses was found to give adequate signal-to-noise ratios for quantitative data collection. 

Figure 6.5 Dynamics of a classical electric dipole induced and driven on resonance by a sequence of two phase-locked ultrashort laser pulses, The driving laser field is shown as gray solid lines in all frames, In addition, the top frames show the induced dipole oscillation as a black dashed line. The instantaneous interaction energy V(t) of the induced dipole in the external driving field is shown in the bottom frames as a black dotted line. Bold black lines display the time average of the interaction energy In Figure 6,5a, the phase relation between both pulses is designed such that the second pulse couples in antiphase to...

As mentioned above, for finite forcing amplitudes one already has periodic A solutions in the range e < 0. But, when the system is quenched into the two-phase region with e > 0, where one may choose for reasons of simplicity e = 1, the spin-odal decomposition sets in and the late stage of the phase separation process depends on the forcing amplitude a. It is an interesting question, for which parameter combinations the systems ends up in a A solution that is locked to the periodicity of the external forcing, independent of the initial conditions before the quench. 

As was shown before, if phase separation is forced by a stationary and spatially periodic temperature modulation, the coarsening dynamics is interrupted above some critical value of the forcing amplitude a and it is locked to the periodicity of the external forcing. However, if this forcing is pulled by a velocity v 0, the traveling periodic solutions of (61) exist only in a certain range of v depending on a. 

Previously, sudden changes of amplitudes of magnetic satellites observable below 2 K [7] were interpreted in terms of lock-in transitions into phases with commensurate propagation vectors. Within the framework of the thermodynamic potential (5) it is impossible to describe a low-temperature transition from an incommensurate phase into a commensurate phase at zero external magnetic field. 

Phase-locked electronics are used to record the dynamic IR spectral change in-phase and out of phase with the applied mechanical field. The method provides insight to the response to the external stress on the molecular and submolecular scale. Rheooptical FTIR spectroscopy is one of the few techniques providing data on the crystallisation, orientation and conformational changes of a polymer during mechanical treatment (83). Two dimensional correlation analysis is utilised to enhance the information derived from the vibrational spectroscopic data. 

However, application of the above technique can be problematic when strong solution-phase absorption obscures weak bands of a surface species. To overcome this limitation, the phase rotation approach [263] can be used. Phase-sensitive detection such as with a lock-in amplifier (LIA) provides two signals the signal that is in phase IP) and the signal that is out of phase (the quadrature, Q) with the external perturbation [264]. These quantities can be represented at each... 

Figure 1 (A) 32.44 MHz spectrum of the alcoholysis products of P4S10 with methanol, ethanol, and isopropanol. The sample is dissolved in toluene, with external CeDe as the field frequency lock. (B) The P APT spectrum of the same mixture, showing the encoding of functional group and multiplicity information into the intensity and phase of the resonances. (Reprinted from Jancke H, Radeglia R, Neels J, and Porzel A (1984) Application of the attached proton test technique in P NMR spectroscopy. Organic Ma etic Resonance 22 376-378 Wiley.)...

Active mode locking utilizes a time-varying element within the laser cavity, such as an integrated electro-optic amplitude or phase modulator driven by an external frequency reference, to promote phase locking of the cavity modes. The advantage of active mode locking is that pulses are generated at a stable repetition rate. The schematic of a typical actively mode locked fiber laser is shown in Fig. 9. 

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