Optical Shop Testing

Chapter 2.3.2 - Temporal Coherence

The OPD (θ) given by Eq. (2.5) also imposes some minimum requirements on the
monochromaticity of the light source. Considering first the case of an interferometer
that is uncompensated because of the lack of a compensating plate or the presence of
an optical system with lenses or prisms in one of the arms, from Eq. (2.5) we can
write

 

and, using again the Rayleigh criterion (ΔOPD ≤ λ/4),

 

Therefore, since the glass dispersion (d n/dλ) is never zero, and tn is also nonzero,
the bandwidth Δλ must not have a very large value if the interferometer is not
compensated. If the interferometer is exactly compensated, white light fringes can be
observed when OPD is nearly zero; otherwise, a highly monochromatic light source
such as a low vapor pressure lamp or (even better) a gas laser must be used.

If many different kinds of glasses are present in both arms of the interferometer,
we may take a more general approach by considering that the interferometer is
compensated for the bandwidth Δλ of the light, if the phase difference for the light
following the two paths in the interferometer is independent of the wavelength.
According to Steel (1962), we can say that, if each arm of the interferometer contains
a series of optical components of thickness t and refractive index n, the phase
difference for the two arms is

 

This relative phase is independent of the wavelength when dφ/dλ = 0, thus giving

 

where ñ is the ‘‘group refractive index,’’ defined by

 

Thus, the interferometer is compensated for the bandwidth Δλ when the ‘‘group
optical path’’ for both arms is the same. Steel (1962) pointed out that the compensation
for the bandwidth of the light source can be examined by looking at the fringes
formed by a white light source through a spectroscope with its slit perpendicular to
the fringes. The spectrum is crossed by the fringes and their inclination shows the
change of fringe position with wavelength. The fringes will be straight along the
direction of dispersion if the bandwidth compensation is perfect. Otherwise, the glass
optical paths can be adjusted until the fringes show a maximum (zero slope) at the
wavelength to be used. If the bandwidth of the light source is very large, a detailed
balancing of the glass paths has to be made by using the same types and thicknesses
of the glass on both arms.

If the OPD(0) is very large, as in the unequal-path interferometer (described in
Section 2.5), the last term in Eq. 2.5 dominates, and we can write

 

but from the Rayleigh criterion, the order number m should not change from one end
of the wavelength bandwidth to that of the other by more than 1/4; thus

 

where Δλ is the maximum allowed bandwidth. Thus we can write the approximation

 

Since the length of a train of waves with bandwidth Δλ is equal to λ2Δλ, this
condition is equivalent to saying that the OPD(0o) should be smaller than one fourth
of the length of the wavetrain (or wavelength of the modulation). In uncompensated
interferometers, this condition is incompatible with the condition for an extended
light source.

A very interesting and practical case occurs when the light source is a gas laser, but
this discussion is left to Section 2.5 on the unequal-path interferometer.

 

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