Принципы нанометрологии (1027623), страница 11
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Additionally, the method does not require thecalculation of model sensitivity coefficients since the only interaction withthe model is to evaluate the model for values of the input quantities.However, there are also some practical issues that arise in the applicationof a Monte Carlo method. The degree of numerical approximation obtainedfor the distribution for Y is controlled by the number M of trials, and a largevalue of M (perhaps 105 or 106 or even greater) may sometimes be required.One issue, therefore, is that the calculation for large values of M may not bepracticable, particularly when a (single) model evaluation takes an appreciable amount of time. Another issue is that the ability to make randomdraws from the distributions for the Xi is central, and the use of high-qualityalgorithms for random-number generation gives confidence that reliableresults are provided by an implementation of the method.
In this regard, theability to draw pseudo-random numbers from a rectangular distribution isfundamental in its own right, and also as the basis for making random drawsfrom other distributions using appropriate algorithms or formulae.The laser2.9 The laserThe invention of the laser in 1960 has had a significant impact on metrology.The realization of the definition of the metre (see section 2.3) involves theuse of a frequency-stabilized laser and many commercial interferometersystems use a laser source. The most common form of laser in the metrologyarea is the helium-neon laser, although solid-state lasers are becoming morewidespread.2.9.1 Theory of the helium-neon laserThe tube of a continuous-wave helium-neon (He-Ne) gas laser containsa mixture of approximately eight parts of helium to one part of neon at a totalpressure of a few millibars.
The laser consists of an optical cavity, similar tothat of a Fabry-Pérot etalon (see section 4.4.4), formed by a plasma tube withoptical-quality mirrors (one of which is semi-transparent) at both ends. Thegas in the tube is excited by a high-voltage discharge of approximately 1.5 kVto 2.5 kV, at a current of approximately 5 mA to 6 mA. The discharge createsa plasma in the tube that emits radiation at various wavelengths corresponding to the multitude of allowed transitions in the helium and neonatoms.The coherent radiation emitted by the He-Ne laser at approximately632.8 nm wavelength corresponds to the 3s2 – 2p4 atomic transition in neon[30].
The excited 3s2 level is pumped by energetic 2s0 helium atoms collidingwith the neon atoms; the 2s0 helium energy level is similar in energy to the3s2 level of neon and the lighter helium atoms are easily excited into the 2s0level by the plasma discharge (see Figure 2.5). The excess energy of thecollision is approximately thermal, i.e., it is easily removed by the atoms inthe plasma as kinetic energy.The collisional pumping of the 3s2 level in neon produces the selectiveexcitation or population inversion that is required for lasing action.
The 2pneon state decays in 108 seconds to the 1s state, maintaining the populationinversion. This state relaxes to the ground state by collision with the walls ofthe plasma tube. The laser gain is relatively small and so losses at the end ofthe mirrors must be minimised by using a high-reflectance coating, typically99.9%. The output power is limited by the fact that the upper lasing statereaches saturation at quite low discharge powers, whereas the lower stateincreases its population more slowly. After a certain discharge power isreached, further increase in the power leads to a decrease in the populationinversion, and hence lower light power output.2324C H A P T ER 2 : Some basics of measurementFIGURE 2.5 Energy levels in the He-Ne gas laser for 632.8 nm radiation.The 632.8 nm operating wavelength is selected by the spacing of the endmirrors, i.e. by the total length of the optical cavity, lc.
The length of thecavity must be such that the waves reflected by the two end mirrors are inphase for stimulated emission to occur. The wavelengths of successive axialmodes are then given by2lc ¼ ml:(2.16)These modes are separated in wavenumber byDs ¼12lc(2.17)Dn ¼c2lc(2.18)or in terms of frequencywhere c is the speed of light in a vacuum. This would lead to a series ofnarrow lines of similar intensity in the spectrum, if it were not for the effectsof Doppler broadening and the Gaussian distribution of atoms available forstimulated emission.When a particular mode is oscillating, there is a selective depopulation ofatoms with specific velocities (laser cooling) that leads to a dip in the gainprofile.
For modes oscillating away from the centre of the gain curve theatomic populations for the two opposite directions of propagation aredifferent due to the equal but opposite velocities. For modes oscillating at theThe lasercentre of the gain curve, the two populations become a single population ofeffectively stationary atoms. Thus a dip in the gain profile occurs at thecentre of the gain curve – the so-called Lamb dip. The position of the Lambdip is dependent on other parameters of the laser such as the position of thegain curve and can be unstable.For early lasers with typical cavity lengths of 1 m the mode spacing was0.5 m1, with a gain profile width of approximately 5.5 m1. Thus severalaxial modes were present in the gain profile with gains sufficient for laseraction, and so two or more modes would operate simultaneously, making thelaser unsuitable for coherent interferometry.
By using a shorter tube and thencarefully lowering the power of the discharge and hence lowering the gaincurve, it is possible to achieve single-mode operation.2.9.2 Single-mode laser wavelength stabilization schemesTo allow a laser to be used in interferometry with coherence lengths abovea few millimetres (see section 4.3.4) it must operate in a single mode andthere have been many proposed schemes for laser stabilization.The Lamb dip, mentioned above, was used in an early stabilizationscheme.
Here the intensity of the output beam was monitored as the lengthof the cavity was modulated, for example by piezoelectric actuators (PZTs).Alternatively, mirrors external to the laser cavity are used that could bemodulated – the output intensity being monitored and the laser locked tothe centre of the Lamb dip.
The reproducibility of lasers locked to the Lambdip is limited by shift of the Lamb dip centre as the pressure of the gasinside the laser tube varies and also by a discharge current dependent shift.The large width of the Lamb dip itself (about 5 107 of the laserfrequency) also limits the frequency stability obtainable from thistechnique.Use has also been made of tuneable Fabry-Pérot etalons in a similarsystem. Other groups have locked the output of one laser to the frequency ofa second stabilized laser. Others have used neon discharge absorption cellswhere the laser was locked to the absorption spectrum of neon in an externaltube, the theory being that the unexcited neon would have a narrower linewidth than the neon in the laser discharge.2.9.3 Laser frequency-stabilization using saturated absorptionThe technique with the greatest stability is used in the Primary Referencelasers which realize the NMI’s Primary Standard of Length and involvescontrolling the length of the laser cavity to alter the wavelength, and lockingthe wavelength to an absorption line in saturated iodine vapour [30].
This is2526C H A P T ER 2 : Some basics of measurementa very stable technique since the absorption takes place from a thermallypopulated energy level that is free from the perturbing effects of the electricdischarge in the laser tube.If the output beam from a laser is passed straight through an absorptioncell, then absorption takes place over a Doppler broadened transition.However, if the cell is placed in a standing-wave optical field the highintensity laser field saturates the absorption and a narrow dip appears at thecentre of the absorption line corresponding to molecules that are stationaryor moving perpendicular to the direction of beam propagation.
This dipproduces an increase in the laser power in the region of the absorption line.The absorption line is reproducible and insensitive to perturbations. Thelinewidth is dependent on the absorber pressure, laser power and energy levellifetime. Saturated absorption linewidths are typically less than 1 108 ofthe laser wavelength.In a practical application an evacuated quartz cell containing a smalliodine crystal is placed in the laser cavity and temperature controlled to15 C. As the iodine partly solidifies at this temperature, this guaranteesa constant iodine gas pressure.
The laser mirrors are mounted on PZTs andthe end plates are separated by low thermal expansion bars to ensurea thermally stable cavity. A small frequency modulation is then applied toone of the PZTs. This leads to an amplitude modulation in the output powerthat is detected using a phase-sensitive detector and fed back to the otherPZT as a correction signal. The frequency control system employs a photodiode, low noise amplifier, coherent filter and phase-sensitive detector followed by an integrating filter. Figure 2.6 is a schema of the iodine-stabilizedHe-Ne instrumentation.Detection of the absorption signal at the laser modulation frequencyresults in a first derivative scan that shows the hyperfine componentssuperimposed on the sloping background of the neon gain curve.
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