The advantage of this distribution is that it allows the relatively large proportion of noise from the small signal value to be separated from the noise proportion of larger signal values (larger than approximately 10% of the maximum signal). This improves the result within the framework of an acceptable error level of measuring data. Ocean Optics, for instance, shows the signal-to-noise relation using the full signal, which can lead to 50% lower numerical values, but does not allow conclusions to be drawn about the noise proportion of the smaller signal values. Noise power increases significantly as signal strength decreases. As a result of this limitation, it is only worthwhile to provide noise power figures for signal values over a certain strength threshold. In the example, the relative noise power mean value for signals greater than 10% of the maximum signal is N[%] = (0.91 N 0.38)%.

LaserMethod

In the laser method a monochromatic light source, e.g. a HeNe laser with a 543.5 nm or 632.8 nm wavelength is used. Here the signal from the laser wavelength is compared with the signal which can be detected using the eightfold bandwidth distance. There is, however, no binding convention for the bandwidth distance, which leads to differing results. At the same time, increasing bandwidth results in an increase in the proportion of stray light. Experience shows that this method delivers fundamentally higher values than the filter method. A disadvantage to the laser method is that the Blooming behavior of the detector line differs, and this influences the signal strength of the neighboring pixel.

First, the emission spectrum of a deuterium halogen lamp is measured with the edge filter and the ND filter with an ND = 0.1 transmission. The dark spectrum is not included. Exposure time should be chosen to maximize usage of the detector measuring area. Finally the same spectrum is measured without the ND filter (illustration 15). The signal in the unblocked range is corrected with the transmission from the ND filter. (ND (ND 1.0) = 0.1).

NEP

With regard to the sensitivity of the spectrometer, the details of the minimal provable light output are significant. After compensation has been made for all the systematic influences and the statistical deviation is known, the borderline is the Noise Equivalent Power (NEP), that is, the particular radiant power which causes a signal to noise ratio of 1.

Root Mean Square

RMS is increasingly being used for the details about noise power. To use it, one must be sure that the systematic proportions which appear as offset in the measuring signal are eliminated. For capital N this is included in the standard deviation SD which takes the Offset into account.

RMS = ( 1/N Σ x_i² )^1/2

In this context, the expression for detectivity is also found, defined as

Det = 1 / NEP

Spectral Resolution

The resolution power in a grating spectrometer is determined by the grating characteristics. These include: the geometrical dimensions of the light path, the entrance slit, and the detector characteristics, in this case, a line detector. The following expressions are of interest for the closer examination of the resolution power. Angle Dispersion and Linear Dispersion The angle dispersion of the grating describes the dependency of the reflection angle β_m on the wavelength. It is dependent on the grating characteristics. It is defined as follows:

(dβ_m) / (dλ) = m / (a_Gitter ⋅ cos β_m) with a_Grating = 1 / L

Here, m is the order of diffraction of the reflection angle being observed and aGrating the distance between two lines of the grating given by the number of grooves per unit length. The order of diffraction can have both positive and negative values. Normally, just one order of diffraction is evaluated, that is, the detection of further orders of diffraction is suppressed.

The linear dispersion at the exit from the spectrometer is dependent on the angle of dispersion and the focal length f. Normally the focal length is defined by the distance between the focusing mirror in the spectrometer and its exit slit. The following results from the dispersion angle and the focal length:
(dl) / (dλ) = f ⋅ (d β_m) / (d λ) = (f ⋅ m) / (a_Grating ⋅ cos β_m )

Linear dispersion defines the degree of the spectral widening at the exit slit of the line detector in the spectrometer. This value is commonly used as reciprocal linear dispersion dλ/dl to characterize the resolution power of the monochromator. With a focal length of 100 mm the reciprocal linear dispersion for a grating with L = 1200 Str/mm depending on the angle is about dλ/dl = 8 nm/mm.

Bandwidth and Resolution Power

The bandwidth Δλ is the spectral width of the light which travels through the spectrometer. By reducing the slit width, the bandwidth can be reduced, but there exists a practical limit to the amount by which this width can be reduced. The bandwidth defines the resolution power λ/Δλ of the spectrometer, which is therefore always connected to an observed wavelength. In practice, the bandwidth figures are used alone to characterize the resolution power. One way to determine the bandwidth is to measure known spectral lines (Hg, Xe, Kr). The transitions of the noble gases have a line-width of less than 0.001 nm and, as a result, fall below the order of magnitude of the bandwidth which needs to be determined. All of the spreading effects can be assigned to the spectrometer and as a result have a predictable statistical nature. The homogenous spread line characteristics can be described by the Gaussian distribution G(Δλ). Fitting this function to the measured data provides the amplitude factor A, the peak wavelength λ_Peak and the width of the standard deviation σ.

The spectral width of the whole line (2σ) is at about 60% of the y-maximum of the Gaussian distribution and consequently is less than the full width at half maximum Δλ_FWHM which is defined as the spectral width at 50% maximum intensity. So the value for Δλ_FWHM is about 18% larger than 2σ. Giving the resolution details in a Δλ_FWHM form is favored by, for example, Ocean Optics, Laser2000, Yobin Yvon, Oriel Acton, Hamamatsu, and Spectral Products.

Pixel Resolution

An important parameter for representing the optical resolution is the pixel resolution. Pixel resolution is calculated by dividing the spectral width by the number of pixels used. This measurement is of little significance. For known resolutions it shows how many pixels the spectrometer uses for the representation. From these figures the most precise possible resolution can also be calculated.

To determine the peak wavelength and the bandwidth of the spectral line to be measured, there need to be at least three values in the representation. This is the minimum number of values for which a Gaussian function can be employed. Wavelength calibration involves assigning the pixel number p to the corresponding wavelength. Along with the previously mentioned spectral lamps, holmium perchlorate or holmium oxide glass (e.g. Hellma) are also used to determine the wavelength. The rare-earth ions holmium and praseodymium have well-defined electron transitions in the visible spectral range. The emission spectra of NeHg and Xe spectral lamps are measured. The lines which show sufficient intensity and are individual spectral lines are used for the evaluation.

The Hg lines at 576.96 nm and 579.97 nm cannot be distinguished as separate with an optical resolution of Δλ  0.5 nm. They do, however, clearly show the resolution which has been reached. Krypton lines can also be evaluated, but they do not supply any additional spectral information, in particular, not in the gap between 600 nm and 800 nm. The marked spectral lines are individual lines to which the Gaussian function can be adjusted. In performing this adjustment, it is important to avoid an offset which could lead to erroneous results. One advantage of this method is its independence from the relative position of the distribution in the pixels of a line spectrometer. The result from this calculation is the peak wavelength λPeak and the width of distribution 2σ from which the bandwidth ΔλFWHM can be determined using the above formula. Comparison of the actual value and the target value of the peak wavelength provides the spectral wavelength precision λacc of the system.

Result

Bandwidth Δλ_FWHM = 1,18 · 2σ

Wavelength accuracy Δλ_acc = SD (λ_expected – λ_actual)

Evaluation of Measuring Errors

Adjusting the Gaussian function from the origin provides the error of the peak wavelength Δ(λ_Peak) and the error in the bandwidth Δ(2σ) depending on the adjustment precision. Experience shows, however, that this error is 10 times smaller than the error resulting from averaging the bandwidth when adjusting to various spectral lines.

For simplicity, the individual pixels are treated as equivalent to one another. The statistical behavior of the entirety of the pixels is equivalent, therefore, to the chronological statistic of an individual pixel. The standard deviation of the dark current gives the noise power N for calculating the dynamics according to the following equations. Since this calculation involves more than 2000 data points, the noise power calculation using the Standard Deviation (SD) instead of the Root Mean Square (RMS) is justified. Result ( RAD - Sdark ) / N Dynamics N = SD (Sdark)