Spatial coherence properties and directionality of partially coherent cosh-Gaussian beams
Introduction
Partially coherent beams are a topic that has been of considerable theoretical and practical interest. In 1978 Collett and Wolf predicted that partially coherent beams, like Gaussian Schell-model (GSM) beams, may produce the same far-field radiant intensity distribution as a Gaussian laser beam [1], [2]. It means that full spatial coherence is not a necessary condition for the generation of highly directional light beams. The prediction was confirmed experimentally [3], [4]. A further study of spatial coherence properties of GSM beams in the far field was made recently in [5]. The highly directional character of GSM beams under certain conditions is, of course, useful for some practical applications. On the other hand, apart from GSM beams, there are a variety of partially coherent beams which can be produced in experiments [6], [7], [8], [9]. Therefore, a question arises: if there is other type of partially coherent beams which may have the same directionality as a fully coherent laser beam? In this paper, based on the theory of optical coherence [10], we take the partially coherent cosh-Gaussian (ChG) beam as an example and deal with the spatial coherence properties and directionality of such a type of beams and compare the results with [1], [2], [5].
Section snippets
Partially coherent cosh-Gaussian beams and their spatial coherence properties in the far field
To simplify the mathematical formulation, our analysis is restricted to the two-dimensional case (x, z) [11], [12]. The field of a spatially fully coherent two-dimensional ChG beam at the source plane z = 0 is expressed aswhere w0 is the waist width of the Gaussian amplitude distribution, Ω0 is a parameter associated with the cosh part and A is the amplitude at the position x = z = 0, ω is the frequency. The ChG beam can be realized experimentally, for example, by
Equivalent conditions for partially coherent cosh-Gaussian beams
From Eq. (13) it is readily seen that two ChG beams will generate the same far-field radiant intensity distribution, if their beam parameters α, β, w0 and A satisfy the following conditions:
Conditions (17), (18), (19), (20) indicate that: (I) There does not exist a fully coherent ChG beam (β = 1) whose far-field radiant intensity distribution is the same as that of partially coherent ChG beams due to the restrictions of Eqs. (18), (20). (II) There do not exist
Numerical calculation results and analysis
To illustrate the above theoretical results, numerical calculations were performed and some typical examples are compiled in Fig. 2, Fig. 3, Fig. 4 and the corresponding parameters are listed in Table 1, Table 2, Table 3. In our calculations two symmetric points with respect to the z-axis are chosen, thus s1x = −s2x = sinθ, and Eq. (8) simplifies to
Fig. 2a gives spectral densities S(0)(x, ω) (left) and spectral
Conclusions
In this paper the spatial coherence properties and directionality of partially coherent ChG beams have been studied in detail. It has been shown that unlike GSM beams, in the strict sense there do not exist partially coherent ChG beams which may generate the same far-field radiant intensity distribution as a fully coherent laser beam. However, under the condition , we can find two partially coherent ChG beams which generate the same far-field radiant intensity distribution, and
Acknowledgements
This work was supported by the National Natural Science Foundation of China under Grant 10574097. One of the authors (B. Lü) is grateful to Prof. Dr. E. Wolf for sending us their recent publications relating to spatial coherence and polarization of partially coherent beams which stimulate us to deal with the subject of this paper.
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