Understanding
Second Order Diffraction

Second Order Diffraction
Most Hyperspectral imaging systems operate as a spectrograph where the angle of incidence is fixed. Therefore, the general diffraction grating equation simplifies to (See "Imaging Spectrometer Fundamentals.pdf )

Spectrograph diffraction grating equation =		(1)

Where:
Beta = angle of diffraction
k = order
n = groove density of the diffraction grating
Lambda = wavelength

We see that as we expect the angle of diffraction varies with wavelength. However we can also expect that if 800 nm appears at an angle of diffraction of, let us say, 15 degrees; then all light at 400 nm will co-mingle with light at 800 nm.

This results in two major problems:

1. The efficiency of the grating drops: Almost all spectrographs are used in first order. When light appears in second order first order efficiency drops.
   
   For example, if most your work is between 400 and 600 nm then you have to share the light in first order with the same same wavelength range appearing in second order.
2. Second order pollutes first order! When working in the red in first order, light in the blue will be superimposed on top of the red.
   
   In the "best" case this results in enhanced background, in the worst case superimposed spectra. For those working in brightfield this is a disaster.
   
   Using second order filters will eliminate first order light from appearing in second order, but the overall efficiency of the system will be lessened.

Second order diffraction observed
Figure 1 shows the spectrum of an MIDL lamp acquired with a transmission grating based instrument. As the lamp warms up all argon lines above 650 nm vanish.( If you would like to see a video of this in real-time click here). The spectrum in Figure 1 is after all red lines above 650 nm have faded.

However, we can still observe emission lines above 650 nm. So what is going on?

This is most easily explained by considering the wavelength dispersion equation (2). This is derived from the general diffraction grating equation that is described in ImagingSpectrometerFundamentals.pdf

Linear wavelength dispersion =		(2)

Where:
Beta = angle of diffraction
k = order
n = groove density
F = focal length

From the equation (2) we note that wavelength dispersion varies with the cosine of the angle of diffraction, the order and focal length. Figure 1 shows that indeed the distance between two calibration emission lines in first order is doubled in second order and the line width (FWHM) in second order is double that in first order.

Because wavelength dispersion is doubled, and the slit width is the same in both first and second order, second order resolution, or bandpass, is twice that in first order. (Bandpass = slit width x wavelength dispersion.)

Figure 2 is the same spectrum acquired with a prism based instrument in which light only appears in "first" order. In fact, light is only refracted through a prism in a single "order;" so all wavelengths that can pass though the prism material will be available to the detector without polluting any other wavelength region.

Figure 3. A Hg spectrum emitted by a MIDL wavelength calibration lamp acquired by a PARISS Hyperspectral Imaging System. All wavelengths were acquired simultaneously in 30 ms.

Figure 3 shows a full spectrum of a pure Hg lamp acquired with the PARISS prism based Hyperspectral Imaging system. The acquisition was taken within a few seconds from ignition to ensure that all argon lines above 650 nm would be recorded.

If second order had been present then the Argon lines would have been contaminated with first order lines.

Second order greatly reduces first order efficiency:
Figure 4 shows how the efficiency of a diffraction grating is split between first and second order. The dotted yellow lines mark the peak height at 436 nm in both first in second order.

Note that the 436 nm line in second order is about half the intensity in first. If we keep in mind that the wavelength dispersion in second order is double that in first order then light at 436 nm is equally split between first and second order!

Figure 4. The peak height of the 436 nm line in first order is about 50% that in first order. If we consider that the spectrum is spread over double the distance first order at 436 nm light is equally split between first and second order

In this case the maximum possible diffraction efficiency at 436 nm is 50%. (In reality is unlikely to exceed 25%.)

Conclusions
Figure 2 shows an MIDL spectrum acquired on a prism based instrument. Note that above 650 nm there is absolutely NO spectral features. Argon lines that emit above 650 nm were allowed to fade with time before acquiring this spectrum. For a detailed explanation click here

Figure 3 shows the same MIDL spectrum before the Argon lines had an opportunity to fade.

Note that if this spectrum had been acquired on a diffraction grating based instrument all the spectral features, and increased background, observed above 650 nm in Figure 1 would be superimposed on the argon lines.

Figure 4 shows how light in the blue wavelengths is split between first and second order dramatically reducing the efficiency of the diffraction grating.

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