**1. Introduction**

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The original meaning of coherence was attributed to the ability of light to exhibit interference phenomena. There are two types of coherence: temporal coherence and spatial coherence. Temporal coherence measures the ability of two relatively delayed beams to form interference fringe. Interference in a Michelson interferometer refers to temporal coherence. However, spatial coherence reflects the ability of a beam to interfere with a spatially shifted (but not delayed) version across the beam. Young's double-slit experiment is an example which concerns spatial coherence.

Since the first experimental realization of Young's double-slit interference, it has been known that the observation of interference-diffraction pattern of an object requires a spatially coherent source. The waves emitted from positions outside the coherent area possess independent irregular phases which may degrade the interference pattern. In the early days when coherent sources were unavailable, interference experiments were carried out with an extended thermal source restricted by a pinhole aperture, which can improve the spatial coherence. Holography is one of the most important applications of spatial interference. In the first holography experiment, Dennis Gabor stated that (Gabor, 1972) ''The best compromise between coherence and intensity was offered by the high-pressure mercury lamp, which had a coherence length of only 0.1 millimeter, ... But in order to achieve spatial coherence, we had to illuminate, with one mercury line, a pinhole 3 microns in diameter. '' The pinhole eventually reduced the power of the source and thus impeded the potential application of optical interferometric techniques such as holography. This barrier was overcome with the invention of the laser, whose intense and coherent beam was ideal for performing interference.

Coherent sources are obtainable within a certain range of optical frequencies. However, various holographic techniques have been developed using incoherent sources such as Xrays, electrons, and radiation. To improve the coherence one has to pay the cost of decreasing the intensity of the source, as already pointed out in relation to Gabor's experiment. A challenging question would be whether coherence is absolutely necessary in holography, or can we bypass the coherence requirement? As a matter of fact, in the early days of holography, a technique using incoherent illumination was first proposed by Mertz and Young (Mertz & Young, 1963), and then extended by Lohmann (Lohmann, 1965), Stroke and Restrick (Stroke & Restrick, 1965), and Cochran (Cochran, 1966). The strategy is based on the fact that each point of a spatially incoherent object produces, through interference of

Incoherent Holographic Interferometry 139

where h�(x, x�) and h�(x, x�) are the impulse response functions for the object and reference paths, t� and t� are the times taken for the source fields to propagate to the recording plane through the object and reference paths, respectively. x� (j = 1,2) is the transverse coordinate

Fig. 1. A holographic interferometer consisting of two beamsplitters, BS1 and BS2, and two

The temporal coherence of the source field is characterized by the coherence time . If

However, spatial coherence requires that the beam has a well-defined wave-front, such as a field emitted from a point source or a laser beam. When the source field satisfies both

We now consider the case when the source field satisfies temporal coherence (i.e. |t� − t�| < �) but not spatial coherence. The first-order correlation function of the source field is written

where is an arbitrary phase. For simplicity, the intensity distribution I� is assumed to be homogeneous. Equation (6) describes complete spatial incoherence. The wave-front of the beam fluctuates randomly and any two positions across the beam are statistically

∗(x)E�(x)〉 = I�e�� � T(x�)h�(x, x�)h�

mirrors, M1 and M2; S is a source and O an object; RP is the recording plane.

∗(x�, t�)E�(x�, t�)〉 = E�

which contains the object information. This is what has been known before.

∗(x, t)E�(x, t)〉 = E�

∗(x, x�)〈E�

in the source field. Hence the interference term is obtained to be

∗(x, t)E�(x, t)〉 = � T(x�)h�(x, x�)h�

〈E�


temporal and spatial coherence, then

as

〈E�

〈E�

〈E�

independent. Substituting Eq. (6) into Eq. (3), we obtain

〈E�

Thus coherent interferometry occurs in the form of

E�(x, t) = � h�(x, x�)E�(x�, t�)dx�, (2b)

∗(x�, t�)E�(x�, t�)〉 = 0 and interference never occurs.

∗(x�, t�)E�(x�, t�)〉 = I�e��δ(x� − x�), (6)

∗(x�, t�)E�(x�, t�). (4)

∗(x, t)E�(x, t), (5)

∗(x, x�)dx�. (7)

∗(x�, t�)E�(x�, t�)〉dx�dx�. (3)

its wave fronts, a stationary interference pattern which uniquely encodes the position and intensity of the object point. However, the method has various weaknesses that significantly limit its application. The schemes proposed for incoherent holography can only record the intensity distribution of a fluorescent object. Moreover, each elementary fringe pattern is formed by two extremely tiny portions of the light, and the summation of many weak interference patterns results in a very large bias level in the hologram, much larger than that in coherent holography. Hence incoherent holography is appropriate only for objects with a low number of resolution elements (Goodman, 1996).

In this chapter we introduce an incoherent interference mechanism which seems to contradict our existing knowledge of the interference requirements. Consider a holographic interferometer in which an object is placed in one arm; the object wave then interferes with the reference wave in the other arm, resulting in an interference pattern which records the object information. To obtain well-defined interference fringes in an ordinary balanced interferometer where both arms have the same path length, the optical field illuminating the interferometer must have spatial coherence, that is, its transverse coherence length must be large enough. We find that the requirement of spatial coherence is due to the particular geometry of the balanced interferometer. If the two fields to be interfered travel through different lengths or different diffraction configurations, then spatial coherence is no longer necessary for spatial interference. Thus we have proposed several unbalanced interferometers where the two beams travel different path lengths or have different diffraction configurations (Zhang et al., 2009a, 2009b). These interferometers are capable of exhibiting interference using either coherent or spatially incoherent light, but their interference fringes are different, signifying that they have different origins.

The chapter is organized as follows. In Section 2 we first analyze why both temporal and spatial coherence conditions are necessary in an ordinary holographic interferometer. Then we propose an unbalanced holographic interferometer with different path lengths and demonstrate that it can exhibit interference using a light source with spatial incoherence. The setup is capable of performing holography. In Section 3 we further suggest several types of unbalanced interferometers which are able to realize incoherent interferometry. In particular, some of the schemes can reduce the requirement of temporal coherence and employ a true incoherent source such as a lamp with an extended illumination area. Moreover, we show that phase reversal diffraction can occur in the incoherent interferometer. Finally, a brief summary is given in Section 4.
