Young’s interference experiment with light of any state of coherence and of polarization
Introduction
Since the publication of a classic paper by Zernike [1] in which he introduced a quantitative measure of coherence by the use of the Young’s interference experiment, the experiment has played a central role in optical coherence theory. Zernike’s analysis was based on a scalar model of the optical field. Recently, it was generalized to random electromagnetic fields [2], [3]. However, the role of polarization of the field in the formation of the interference pattern has not been elucidated so far.
In the present paper, we revisit Young’s interference experiment when performed with a random electromagnetic beam. We first derive an expression for the 2 × 2 electric cross-spectral density matrix of the electric field in the observation plane in terms of the cross-spectral density matrix of the electric field at the pinholes. We then examine the consequences of this relation for the important case when the field incident on the pinholes is an electromagnetic Gaussian Schell-model beam. We find that the degree of polarization of the light in the interference pattern differs, in general, from the degree of polarization of the light at each pinhole, the difference depending on the degree of coherence of light at the pinholes. This result brings into evidence an intimate but subtle relationship which exists between coherence and polarization of random electromagnetic fields.
Section snippets
Propagation of the electric cross-spectral density matrix of a random electromagnetic beam
We consider a random, statistically stationary, electromagnetic beam propagating close to the z-axis. Let {E(r, ω)} be a statistical ensemble of the fluctuating component of frequency ω of the electric field of the beam at a point P(r). Suppose that an opaque screen is placed across the plane z = 0, with small openings at points Q1(ρ1) and Q1(ρ2) and that measurements are made at points P1(r1) and P1(r2) in the observation plane which is parallel to (see Fig. 1). The correlation properties
An example
In order to elucidate the subtle relation which exists between the spectral degree of coherence and the spectral degree of polarization, we will consider an electromagnetic Gaussian Schell-model beam [4], [5], [6], propagating close to the z-axis which is incident on the pinholes in the plane . For such a beam, the elements of the cross-spectral density matrix of the field at the pinholes are given bywhere (i = x, y; j = x
Effect of the degree of coherence of light at the pinholes on the degree of polarization in the plane of observation
Fig. 2 shows the behavior of the spectral degree of polarization on the beam axis in the plane of observation , when the spectral degree of coherence η(0) (ρ1, −ρ1, ω) at the pinholes is changed, while the degree of polarization at each pinhole is kept constant. The curves correspond to different values of the parameter δxy subject to the constraints (11).
As can be seen from Eq. (11), when the field incident on the pinholes is fully coherent (δ → ∞), the parameter δxy → ∞ also. Hence for a
Acknowledgments
We are obliged to Dr. Mircea Mujat for some helpful comments on the analysis presented in this paper. This research was supported by the US Air Force Office of Scientific Research under Grant No. F4920-03-1-0138, by the Engineering Research Program of the Office of Basic Energy Sciences at the US Department of Energy under Grant No. DE-FG02-2DE 45992 and by the Air Force research Laboratory (AFRC) under contract FA9451-04-C-0296.
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