  In the near infrared where detectors are limited by read-out noise, most
interferometers have been operated in wide band in order to benefit from larger
photon rates. We analyze in this paper the biases caused by instrumental and
turbulent effects to $V^2$ estimators for both narrow and wide band cases.
Visibilities are estimated from samples of the interferogram using two
different estimators, $V^{2}_1$ which is the classical sum of the squared
modulus of Fourier components and a new estimator $V^{2}_2$ for which complex
Fourier components are summed prior to taking the square. We present an
approach for systematically evaluating the performance and limits of each
estimator, and to optimizing observing parameters for each. We include the
effects of spectral bandwidth, chromatic dispersion, scan length, and
differential piston. We also establish the expression of the Signal-to-Noise
Ratio of the two estimators with respect to detector and photon noise. The
$V^{2}_1$ estimator is insensitive to dispersion and is always more sensitive
than the $V^{2}_2$ estimator. However, the latter allows to reach better
accuracies when detection is differential piston noise limited. Biases and
noise directly impact the dynamic range of reconstructed images. Very high
dynamic ranges are required for direct exoplanet detection by interferometric
techniques thus requiring estimators to be bias-free or biases to be accurately
calibrated. We discuss which estimator and which conditions are optimum for
astronomical applications especially when high accuracy visibilities are
required. We show that there is no theoretical limit to measuring visibilities
with accuracies as good as $10^{-5}$ which is important in the prospect of
detecting faint exoplanets with interferometers.
