【正文】 the entrance slit.The exit pupil of the entrance optics is AS itself seen axially from the entrance slit of the spectrometer.The exit pupil of the spectrometer is the image of the grating seen axially through M2 from the exit slit. Aperture Ratio (f/value, f/Number), and Numerical Aperture (NA)The light gathering power of an optic is rigorously characterized by Numerical Aperture (NA).Numerical Aperture is expressed by:where μ is the refractive index (μ = 1 in air) (22)and f/value by:(23)Table 2: Relationship between f/value, halfangle, and numerical aperturef/valuef/2f/3f/5f/7f/10f/15n (degrees)NA f/value of a Lens Systemf/value is also given by the ratio of either the image or object distance to the diameter of the pupil. When, for example, a lens is working with finite conjugates such as in Fig. 12, there is an effective f/value from the source to L1 (with diameter AS) given by:(24)and from L1 to the entrance slit by:(25)In the sections that follow f/value will always be calculated assuming that the entrance or exit pupils are equivalent to the aperture stop for the lens or grating and the distances are measured to the center of the lens or grating.When the f/value is calculated in this way for f/2 or greater (. f/3, f/4, etc.), then sin w is ~ tan w and the approximation is good. However, if an active optic is to function at an f/value significantly less than f/2, then the f/value should be determined by first calculating Numerical Aperture from the halfangle. f/value of a SpectrometerBecause the angle of incidence alpha is always different in either sign or value from the angle of diffraction beta (except in Littrow), the projected size of the grating varies with the wavelength and is different depending on whether it is viewed from the entrance or exit slits. In Figures 13a and 13b, the widths W39。 and W39。39。 are the projections of the grating width as perceived at the entrance and exit slits, respectively.Figure 13. Projection of the grating width on (a) the Entrance and (b) the Exit.To determine the f/value of a spectrometer with a rectangular grating, it is first necessary to calculate the equivalent diameter, D39。, as seen from the entrance slit and D as seen from the exit slit. This is achieved by equating the projected area of the grating to that of a circular disc and then calculating the diameter D39。 or D.(26)(27)In a spectrometer, therefore, the f/valuein will not equal the f/valueout.(28)(29)where, for a rectangular grating, D39。 and D are given by:(210)(211)where, for a circular grating, D39。 and D are given by:(212)(213)Table 3 shows how the f/value changes with wavelength.Table 3 Calculated values for f/valuein and f/valueout for a CzernyTurner configuration with 68 x 68 mm, 1800 g/mm grating and LA = LB = F = 320 nm. Dv = 24176。.l(nm)abf/valueinf/valueout200320500680800 Magnification and Flux DensityIn any spectrometer system a light source should be imaged onto an entrance slit (aperture) which is then imaged onto the exit slit and so on to the detector, sample, etc. This process inevitably results in the magnification or demagnification of one or more of the images of the light source. Magnification may be determined by the following expansions, taking as an example the source imaged by lens L1 in Figure 12 onto the entrance slit:(214)Similarly, flux density is determined by the area that the photons in an image occupy, so changes in magnification are important if a flux density sensitive detector or sample are present. Changes in the flux density in an image may be characterized by the ratio of the area of the object, S, to the area of the image, S39。, from which the following expressions may be derived:(215)These relationships show that the area occupied by an image is determined by the ratio of the square of the f/values. Consequently, it is the EXIT f/value that determines the flux density in the image of an object. Those using photographic film as a detector will recognize these relationships in determining the exposure time necessary to obtain a certain signaltonoise ratio. Exit Slit Width and AnamorphismAnamorphic optics are those optics that magnify (or demagnify) a source by different factors in the vertical and horizontal planes (see Figure 14).Figure 14. (a) Vertical and (b) Horizontal MagnificationIn the case of a diffraction gratingbased instrument, the image of the entrance slit is NOT imaged 1:1 in the exit plane (except in Littrow and perpendicular to the dispersion plane assuming LA = LB).This means that in virtually all mercial instruments the tradition of maintaining equal entrance and exit slit widths may not always be appropriate.Geometric horizontal magnification depends on the ratio of the cosines of the angle of incidence, alpha, and the angle of diffraction, beta, and the LB/LA ratio (Equation 216). Magnification may change substantially with wavelength (see Table 4).(216)Table 4 illustrates the relationship between alpha, beta, dispersion, horizontal magnification of entrance slit image, and bandpass.Table 4 Relationship Between Dispersion, Horizontal Magnification, and Bandpass in a CzernyTurner Monochromator. LA = 320 mm, LB = 320 mm, Dv = 24176。, n = 1800 g/mm Entrance slit width = 1 mmWavelength(nm)a (degrees)b (degrees)dispersion (nm/mm)horizontal magnificationbandpass* (nm)200260320380440500560620680740800Exit slit width matched to image of entrance slit.*As the inclination of the grating bees increasingly large, a in the system will increase. Consequently, in spite of the fact that the bandpass at 800 nm is superior to that at 200 nm, it is unlikely that the full improvement will be