1.3 The Effects of Modulation

The detector and the atmosphere are two important sources of noise for interferometric observations in the infrared. Noise from the former source is present in all observations and intensity variations due to changes in atmospheric transparency occur on many nights. Fluctuations of the refractive index of the atmosphere along the light path between the source and the detector, the so-called scintillation noise, can dominate in bright objects and in the thermal regime.

While the ubiquitous photon noise is characterized by its standard deviation being proportional to the square root of the total intensity, scintillation noise has the interesting property of being linearly proportional to the source intensity. This is the reason it becomes increasingly important for bright stars and in wavelength regions where the sky background becomes very large. It is generally not a factor in traditional grating spectroscopy because each detection element `sees' radiation originating from only a narrow wavelength region, and as a result the intensities at the detector are quite low and photon noise continues to dominate. However, scintillation noise is an important consideration in interferometry where the detector is always measuring radiation multiplexed from a wide range of wavelengths. In fact, scintillation can introduce a multiplex disadvantage when observing bright stars.

Detector and scintillation noise both have a frequency dependence. This suggests that they might both be suppressed by modulating the signal so that it is sampled at high frequencies. In infrared photometry this is usually accomplished by use of a chopping secondary mirror, which is an example of amplitude modulation. Although chopping can also be used for interferometric observations, the same result can be achieved more efficiently with high frequency, low amplitude modulation of the path difference, a technique referred to as phase or internal modulation.

The CFHT FTS uses internal modulation and as a consequence the nature of the recorded interferogram is changed. In the case of a monochromatic source, the measured signal now represents the change of the signal given by 1.2 occurring over a small increment of retardation. As a result, the constant component of the interferogram, , and low frequency variations in this term are not modulated and can be totally eliminated providing the frequency of the modulation is chosen to be high enough. For the CFHT FTS, modulation frequencies of 20, 150, and 300 Hz are available at the turn of a switch (see Error Signal/Servo Control Panel).

Generalizing to the broadband case, if the path difference is modulated with an amplitude about some mean path difference , the signal recorded by a synchronous detector will be (from 1.8)

Trigonometric identities can be used to reduce this to the form

It is apparent from this expression that the spectral distribution can be recovered when modulation is applied, although a sine transform must be used. Moreover, the result should be divided by (). Besides suppressing noise, modulation also makes the location of ZPD easy to identify because it now occurs as a zero crossing, instead of a maximum in signal intensity, which can be more difficult to locate accurately.

The optimum amplitude for modulation occurs when () () is maximized. This occurs when

where is the wavenumber at the middle of the spectral region being studied and is the optimum amplitude of the modulation. This condition is satisfied when

where is an integer. In other words, modulation is most efficient when its amplitude is an odd multiple of the central wavelength of the region being observed divided by 4.

Please send comments and suggestions to: veillet@cfht.hawaii.edu