The equations listed in the previous sections are those that are needed to estimate the groove densities of diffraction gratings, focal length, changes in wavelength dispersion, theoretical bandpass, magnification of the pinhole, and the optical geometry (Dv angles). All we need to know in advance is the observed wavelengthrange over a known width of an array detector.
Nikon C1Si spectral imaging system with an IPMT linear array detector.
Let us take as an example the Nikon C1Si spectral imaging system. From published literature, we are informed that a user can sample a spectrum in 2.5, 5, or 10 nm increments, in what Nikon refers to “wavelength resolution,” by exchanging three diffraction gratings. It is more accurate to refer to these settings as the “wavelength sampling increment” (WSI). The use of three gratings implies that the wavelength dispersion will increase by a factor of two and four, based on the 10 nm condition. From Eq. (3), we know that to change dispersion we must either change the focal length (Lb) or the groove density (n). Consequently, we know that it is the groove density that will change because Lb is fixed. For this exercise, we took the pinhole to be 100 μm in diameter.
Observed wavelength scans acquired with a 2.5 nm WSI places the wavelength range of 531–607 nm across a 32 element, imaging photomultiplier tube (IPMT), linear array detector (Hamamatsu Corp, Bridgewater, NJ) in a single shot. The elements are on 1 mm centers with an active width of 800 μm, 7 mm in height, with 200 μm deadspace between elements, and each spectrum will be characterized by up to 32 wavelength data points (WDP).
(Note: As described the term “wavelength resolution” is really the WSI. The actual resolution, calculated by measuring the FWHM of a monochromatic emission line, will be between 2.5, 5, or 10 nm only when an image of the entrance pinhole strikes the center of an IPMT detector element.)
Therefore, the wavelength dispersion is 75.6 nm (607–531.4) spread over 31 mm (allowing 2 × 0.5 mm from centertocenter) for an average wavelength dispersion of 2.44 nm/mm, (76/31). From diffraction grating catalogs, we note that offtheshelf gratings are available in 150, 300, 600, 1,200, 1,800, and 2,400 g/mm, blazed at a variety of wavelengths. (Horiba/Jobin Yvon Edison NJ, Newport CorpRichardson Gratings, Rochester NY).
The easiest way to estimate the geometric parameters is to construct Table 1 in an Excel spreadsheet. Then, select a catalog diffraction grating, vary α, and calculate the precise value of Lb using 2.44 nm/mm dispersion at the center of the chip at 569 nm (the wavelength that falls on the center detector element), using Eq. (3).
Table 1. Estimated Geometric Parameters for a WSI of 2.5 nmObserved (F) wavelength range (nm)  Eq. (1) (C) β (°)  Eq. (2) (C) Dv (°)  Eq. (12)(C) Pinhole (mm)  Eq. (3) (C) Disp (nm/mm) 


531  11.04  −15.46  0.091  2.46 
569  13.70  −12.80  0.092  2.44 
607  16.39  −10.11  0.093  2.41 
This process will display the Dv values at the extremes and the center of the wavelength range. The Dv values are then fixed and will not change when we select alternative groove densities to obtain the 5 and 10 nm WSI values. We find by iteration that a 1,200 g/mm grating provides the reasonable solution shown in Table 1. We also note that the image of the pinhole appears to demagnify in the dispersion plane; however, in these calculations, we assume that the entrance and exit arm lengths are the same, and that the pinhole will change in size only as a function of the cosines of the angles of incidence and diffraction. In an actual instrument, unequal arm lengths are possible and are certainly different across the array. Changing values of Lb with wavelength will contribute to the pinhole magnification.
To summarize the results:
Lb (focal length) = 332 mm.
α (the angle of incidence) = of 26.5° for the wavelength range from 531 to 607 nm.
Diffraction grating = 1,200 g/mm
To select an alternate wavelength range, the grating would be rotated to change the angle of incidence while keeping Lb and the Dv angles constant.
To determine α for a 10 nm WSI, the groove density of the diffraction grating must be four times less than the 1,200 g/mm grating used earlier. By using a 300 g/mm grating, and keeping Lb and the Dv angles constant, we vary α until we reach the desired wavelength range shown in Table 2. Here, the wavelength range from 406 to 717 nm (311 nm) is acquired in a single shot in 10 nm increments with α = 11.25°. The results appear to be consistent with observations.
Table 2. Estimated Geometric Parameters for a WSI of 10 nmEq. (1) (D) wavelength range (nm)  Eq. (2) (C) β (°)  (F) Absolute Dv [From Table 1]  Eq. (12) (C) Pinhole (mm)  Eq. (3) Disp (nm/mm) 


406  −4.21  −15.46  0.098  10.01 
560  −1.55  −12.80  0.098  10.04 
717  1.14  −10.11  0.098  10.04 
To keep astigmatism and coma to a minimum, the NA of the system should be kept to a minimum. Astigmatism varies with the square of the NA and coma with the cube of the NA. The projected NA of a 63× objective (NA = 1.32) is 0.02 (1.32/63), and a 10× (0.3 NA) projects at 0.03. If we used a 30mm wide diffraction grating, the NA of the spectrometer would be 0.05 (F/10) and would accommodate the NA of both the high and lowmagnification microscope objectives. In relative terms, an NA of 0.05 is low enough to keep aberrations to a minimum, and the large size of the detector elements will mask whatever residual aberrations remain.
Figure 8 shows an acquisition of the MIDL wavelength calibration lamp with a 10nm WSI taken in a single shot (see section titled “Wavelength Calibration of Spectral Systems” for details). The emission maxima match the absolute values very well considering that each acquisition was 10 nm wide. The scan also illustrates that, when an image of the pinhole strikes a single detector element, the bandpass measured at the FWHM of the 545 nm peak is indeed 10 nm. However, the 611 nm line clearly falls between two detector elements reducing the apparent FWHM and peak intensity by a factor of two. This is aliasing and is most noticeable when the WSI is large. Although expected, this effect can cause unreliable intensity ratios between wavelengths. Changing the initial wavelength of the scan can push a particular wavelength from one detector to another and change intensity ratios and FWHM values. This is a hazard or an opportunity depending in whether the instrument operator is aware of the consequences of changing a scan starting wavelength.
Note that although we derived a plausible solution there are many variables. For example, Nikon may have used custom diffraction gratings and/or focusing optics; consequently, the actual instrument may have an alternative geometry.