Axial modes

Figure 9.3 Doppler limited laser line with twelve axial modes within the line width...

The variety of possible axial modes is generally of greater consequence. The various frequencies possible are given by Equation (9.2) so that the separation Av of the axial modes is given by... 

For a cavity length of, for example, 50 cm the axial mode separation is 300 MFIz... 

In practice the laser can operate only when n, in Equation (9.2), takes values such that the corresponding resonant frequency v lies within the line width of the transition between the two energy levels involved. If the active medium is a gas this line width may be the Doppler line width (see Section 2.3.2). Figure 9.3 shows a case where there are twelve axial modes within the Doppler profile. The number of modes in the actual laser beam depends on how much radiation is allowed to leak out of the cavity. In the example in Figure 9.3 the output level has been adjusted so that the so-called threshold condition allows six axial modes in the beam. The gain, or the degree of amplification, achieved in the laser is a measure of the intensity. 

Each axial mode has its own characteristic pahem of nodal planes and the frequency separation Av between modes is given by Equation (9.4). If the radiation in the cavity can be modulated at a frequency of cjld then the modes of the cavity are locked both in amplitude and phase since t, the time for the radiation to make one round-trip of the cavity (a distance 2d), is given by... 

Lateral vibrations result from the coupling with axial mode of buckling. The lateral vibrations may also result from a coupling with the torsional vibrations. 

The classic way that we perform force versus deformation measurements is to deform a sample at a constant rate, while we record the force induced within it. We normally carry out such tests in one of three configurations tensile, compressive, or flexural, which are illustrated in Fig. 8.1. We can also test samples in torsion or in a combination of two or more loading configurations. For the sake of simplicity, most tests are uni-axial in nature, but we can employ bi-axial or multi-axial modes when needed,... 

Different axial modes (optical standing waves) can be set up in a cavity of length L provided that the frequency v fulfills the condition... 

EXAMPLE 2.5 For a Nd YAG laser crystal located in a cavity of length 0.1 m, determine (a) the frequency separation among the axial modes and (b) the number of axial modes. Assume that the laser is operating in a laser line at 1.06 fxm, whose full width, Sv, is 1.1 x 10 Hz. 

Sirignano L. Crocco, "A Shock Wave Model of Unstable Rocket Combustors , AIAA 2, 1285(1964) I) B.T. Zinn, "A Theoretical Study of Nonlinear Transverse Combustion Instability in Liquid Propellant Rocket Motors (PhD Thesis)> Princeton Univ, May 1966 J) C.E. Mitchell, "Axial Mode Shock Wave Combustion Instability in Liquid Propellant Rocket Engines (PhD Thesis), Princeton Univ, NASA CR 72229(1967) K) E.L. 

Capener et al, "Driving Processes of Finite-Amplitude Axial-Mode Instability in Solid Propellant Rockets , AIAA, 5, p 938(May 1967) L) H. Krier et al, "Nonsteady Burning Phenomena of Solid Propellants Theory and Experiments AIAA 6, 278(1968)... 

N) E.W. Price, "Recent Advances in Solid Propellant Combustion Instability , Ibid, pp 101-113 O) G.A. Marxman C.E. Wooldridge, "Finite-Amplitude Axial Instability in Solid-Rocket Combustion , Ibid, pp 115-27 P) W.A. Sirignano, "A Theory of Axial-Mode Shock-Wave Oscillations in a Solid-Rocket Combustor ,Ibid, pp 129-37 Q) B.T.Zinn C.T. Saveli, "A Theoretical Study of Three-Dimensional Combustion Instability in Liquid-Propellant Rocket Engines , Ibid, pp 139-47 R) R.J. Priem E.J. Rice, "Combustion Instability with Finite Mach Number Flow and Acoustic Liners , Ibid, pp 149-59 S) M.W. Thring, "Combustion Oscillations in Industrial Combustion Chambers , Ibid, pp 163-68... 

Sirignano, "A Theory of Axial-Mode Shock-Wave Oscillations in a Solid-Rocket Combustor , 12thSympCombstn (1968), pp 129-37... 

The commercially available stand-alone LITs, marketed under the name LTQ, are made of four hyperbolic cross-sectional rods (Fig. 1.25). Since ions are trapped in an axial mode as opposed to central trapping on 3D ion traps, LTQs have been successfully coupled with Orbitrap and FTICR for achieving high-resolution capabilities (Peterman et al., 2005 Sanders et al., 2006) (Chapter 5). Functional improvements in 2D traps over 3D traps include 15 times increase in ion storage capacity, 3 times faster scanning, and over 50% improvement in detection efficiency and trapping efficiency. 

In practice this condition may be fulfilled not only in excitation, e.g. by means of a pulsed laser or a continuous dye laser with insufficient frequency selectivity, but also by means of fines from a continuous gas laser working in simultaneous axial mode u>i (multimode) generation regime see Fig. 3.10(a). Let Au>i = u>i+1 — uii = itc/L denote the mode separation in a laser, L being the resonator length. Then, as pointed out in [110, 127, 231], broad line approximation works if Awj is smaller than the width of the Bennet holes r en [268, 320] in the absorption contour see Fig. 3.10(6). The positions of the Bennet holes are determined by the condition ujq — w/ + kv = 0, where luq is the central transition frequency, k is the wave vector and v is the velocity of the absorbing particle. The broad fine approximation is valid if the following conditions are fulfilled (see Fig. 3.10) ... 

The data from Table 3.7 make it possible to make a few estimates. Let us consider a concrete example. If the length of the resonator is L = 2 m, the Doppler contour, say of a K2 molecule in the BlTlu state, contains about 11 axial modes, the distance between them being Au>i = ttc/L m 4 108 s-1. If we assume that the width of the Bennet dip is equal to the homogeneous line width, rBen T = 0.86 108s 1 (see... 

The elements Al, Mn, and Sr were determined by means of a Perkin-Elmer Optima 4300DV inductively coupled plasma emission spectrometry (ICP-AES) instrument (axial mode), equipped with an AS-90 Plus autosampler, a cross-flow nebulizer, and a Scott-type spray chamber in Ryton. The instrumental operating parameters are listed in Table 10.1. 

The phenomenon of unstable combustion results from a self-amplifying interaction between combustion processes and the. acoustic oscillations of the gas within the rocket motor. The unexpected appearance of combustion instability in any rocket generally terminates its mission thru motor case rupture from overpressure, disruption of guidance systems by severe vibration, or thrust malalignment. Both axial mode and transverse mode instabilities are observed (Ref 45). In the case of the transverse mode the characteristic wave time is usually that required to travel radially around the proplnt cavity whereas the characteristic time for the axial mode is the time for the wave to travel from end to end in the combustion chamber. Double-base proplnts predominantly are prone to transverse wave instabilities and infrequently to those in the axial mode, while composite proplnts appear to go unstable mostly in the axial mode. In the case of transverse instability chamber pressures have been known o double whereas in axial mode instabilities artificially induced by pulsing the chamber pressure at lOOOpsi, the pressure excursion may reach 300—400psi. A review of recent theoretical combustion modeling for combustion instability has been made by Price (Ref 47)... 

Figures 7.23 and 7.24 show the results of scans for metal content of dietary supplements using the radial and axial modes, respectively.

Table 7.11 Results of analysis of dietary supplements for metal content of diluted filtered solutions containing high levels of metals using the radial mode of the plasma torch and axial mode for undiluted filtered solutions for the lower levels of metals. (The results show excellent agreement with label quantity)...

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