Spectral resolving power

The spectral resolving power discussed for the different instruments in the previous sections can be expressed in a more general way, which applies to all 

With the maximum time difference ATm = As /c for traversing the two paths with the path difference A -m, we obtain with v = c/X from (4.102) for the minimum resolvable interval Av = —(c/X )AX, 

The product of the minimum resolvable frequency interval Av and the maximum difference in transit times through the spectral apparatus is equal to 1. 

The upper limit for the resolving power is therefore, according to (4.102), 

For m =1 and N = 10 this gives R = 2 xlO, or AX = 5 xl0 t. Because of diffraction, which depends on the size of the grating (Sect. 4.1.3), the realizable resolving power is 2—3 times lower. This means that at A = 500 nm, two lines with AX 10 nm can still be resolved. 

The spectral resolving power of any dispersing instrument is defined by the 

Lord Rayleigh has introduced a criterion of resolution for diffraction limited line profiles, where two lines are considered to be Just resolved if the central diffraction maximum of the profile Ii(A-Ai) coincides with the first minimum of l2(A-A2) [4.3].- 

Let us consider the attainable spectral resolving power of a spectrometer. When passing the dispersing element (prism or grating), a parallel beam composed of two monochromatic waves with wavelengths A and A+AA is split into two partial beams with the angular deviations 6 and 0+A6 from their initial direction (Fig. 4.8). The angular separation is 

The factor dx/dA is called linear dispersion of the instrument. It is generally measured in mm/A. In order to resolve two lines at A and A+AA, the separation Ax2 in (4.5) has to be at least the sum 5x3 (A) + 5x3 (A+AA) of the widths of the two slit images. Since the width 6x2 is related to the width Sx of the entrance slit according to geometrical optics by 

When a parallel light beam passes a limiting aperture with diameter a, a Fraunhofer diffraction pattern is produced in the plane of the focussing lens L3 (Fig.4.9). The intensity distribution as a function of the angle 0 with the optical axis of the system is given by the well-known formula [4.3] 

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The analytical resolving power is applied in several analytical fields in form of well-known expressions such as, e.g., spectral resolving power Rx = X/AX or mass resolving power RM = M/AM. 

Shoosmith, 1956) shows two main feamres at 215.7nm and 216.3 nm but does not show any significant rotational structure as a result of the rapid dissociation of the CH3 excited electronic level. Childs et al. (1992) were able to resolve the two features in the CH3 absorption spectrum with a spectral resolution of 0.12nm, which was about 3 times the theoretical resolution limit of 0.04 nm for their instrument. For most measurements, however, Childs et al. (1992) used the spectrometer at a lower spectral resolving power with the input slits opened up and with the averaging of neighboring pixels, since the width of the CH3 absorption feature at 216.3 nm is 1.2nm. Although the two features of the CH3... 

This chapter is devoted to a discussion of instruments and techniques that are of fundamental importance for the measurements of wavelengths and line profiles, or for the sensitive detection of radiation. The optimum selection of proper equipment or the application of a new technique is often decisive for the success of an experimental investigation. Since the development of spectroscopic instrumentation has shown great progress in recent years, it is most important for any spectroscopist to be informed about the state-of-the-art regarding sensitivity, spectral resolving power, and signal-to-noise ratios attainable with modern equipment. 

The theoretical spectral resolving power is the product of the diffraction order m with the total number N of illuminated grooves. If the finite slit width b[ and the diffraction at limiting aperatures are taken into account, the practically achievable resolving power according to (4.13) is about 2—3 times lower. 

The spectral resolving power X/AX of the Michelson interferometer equals the maximum path difference As/X measured in units of the wavelength X. 

If the central plane of the near-confocal FPI is imaged by a lens onto a circular aperture with sufficiently small radius b < (Ar ) / only the central interference order is transmitted to the detector while all other orders are stopped. Because of the large radial dispersion for small p one obtains a high spectral resolving power. With this arrangement not only spectral line profiles but also the instrumental bandwidth can be measured, when an incident monochromatic wave (from a stabilized single-mode laser) is used. The mirror separation d = r - is varied by the small amount e and the power... 

The optimum choice for the radius of the aperture is based on a compromise between spectral resolution and transmitted intensity. When the interferometer has the finesse F, the spectral halfwidth of the transmission peak is Sv/F, see (4.53b), and the maximum spectral resolving power becomes F A /A (4.56). For the radius b = (Px/F y of the aperture, which is just (F )1/4 iiYnes the radius p of a fringe with p = 1 in (4.77), the spectral resolving power is reduced to about 70% of its maximum value. This can be verified by inserting this value of b into (4.79) and calculating the halfwidth of the transmission peak P(X, F, 6). 

For a given finesse F, the etendue of the confocal FPI increases with the mirror separation d — r. The spectral resolving power... 

Inserting the value of d given by this equation into the spectral resolving power v/Ay = IdF""/X, we obtain... 

While the spectral resolving power is proportional to U for the confocal FPI, it is inversely proportional to U for the plane FPL This is because the etendue increases with the mirror separation d for the confocal FPI but decreases proportional to I/d for the plane FPI. For a mirror radius r > /Ad, the etendue of the confocal FPI is larger than that of a plane FPI with equal spectral resolution. 

This example shows that for a given light-gathering power, the confocal FPI can have a much higher spectral resolving power than the plane FPI. 

The spectral resolving power discussed for the different instruments in the previous sections can be expressed in a more general way, which applies to all devices with spectral dispersion based on interference effects. Let As be the maximum path difference between interfering waves in the instrument, e.g., between the rays from the first and the last groove of a grating (Fig. 4.62a) or between the direct beam and a beam reflected m times in a Fabry-Perot interferometer (Fig. 4.62b). Two wavelengths X and X2 = + AX can still be... 

Fig. 4.62a,b. Maximum optical path difference and spectral resolving power (a) in a grating spectrometer (b) in a Fabry-Perot interferometer... 

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