## Abstract

In this paper, we investigate the Fraunhofer diffraction of a class of partially coherent light diffracted by a circular aperture. It is shown that by the illumination of partially coherent light of the special spatial correlation function, the anomalous behaviors of the diffraction patterns are found. We find that the decrease of the spatial coherence of the light in the aperture leads to the drastic changes of the diffraction pattern. Specifically, when the light in the aperture is fully coherent, the diffraction pattern is just an Airy disc. However, as the coherence decreases, the diffraction pattern becomes an annulus, and the radius of the annulus increases with the decrease of the coherence. Flattened annuli can be achieved, when the parameters characterizing the correlation of the partially coherent light are chosen with suitable values. Potential applications of modulating the coherence to achieve desired diffraction patterns are discussed.

©2003 Optical Society of America

## 1. Introduction

The diffraction and propagation of partially coherent light have been extensively studied for over thirty years 1–4. This is due to the fact that in some applications partially coherent light possesses some advantages over completely coherent light. For example, in the propagation of partially coherent light through turbulence media, it has been found that partially coherent light may be less susceptible to distortions than completely coherent light. Regarding the diffraction of partially coherent light, there have been several papers concerning the influence of the correlation function of the illuminating partially coherent light on diffraction patterns.1–2 Som and his coworkers investigated the diffraction properties of partially coherent light of Besinc correlation.1 Asakura studied the diffraction patterns of partially coherent light of Besinc correlation, *or* exponential correlation, *or* Schell-model etc.2 It has been shown that, for partially coherent light of a given correlation, the decrease of the coherence of the light in the diffraction aperture does result in the spreading of the central diffraction disc. It is well known that, in the limit of complete coherence, the central diffraction disc is called an Airy disc. In this paper, we study the diffraction of a class of partially coherent light by a circular aperture. The anomalous behavior of the Fraunhofer diffraction patterns is found. Specifically, when the light in the aperture is completely coherent, the diffraction pattern is just an Airy disc; as the coherence decreases, the diffraction pattern becomes an annulus, and the radius of the annulus increases with the decrease of the coherence. A flattened annulus can be achieved at a specific coherence level. Potential applications of modulating the coherence to achieve desired diffraction patterns are also discussed.

## 2. Theory

Consider a spatially partially coherent light of wavelength λ incident normally upon a circular aperture ** A** of radius

*a*(see Fig. 1). The cross-spectral density of the partially coherent light in the aperture is assumed to be

where

is the degree of coherence, and *S*
_{0} is intensity of the light in the aperture. *k* = 2π/λ = ω/*c* is the wave number associated with the frequency ω; ρ¯_{1} and ρ¯_{2} are two position vectors in the aperture. Besinc(*x*) 2*J*
_{1}(*x*)/*x*, and J_{1}(*x*) is the Bessel function of order unity; ε* _{s}* and

*b*are two positive constants (here, 0 ≤ ε

_{s}<1). The partially coherent light whose cross-spectral density and degree of coherence are given by Eqs. (1)–(2) has been discussed in Refs. [5–7], in which the spectral switches in the case of polychromatic partially coherent illumination were found. As shown in Eq. (2), ε

_{s}and

*b*are two important parameters in determining the coherence of the light in the aperture.

It follows from propagation law of the cross-spectral density of partially coherent light that the intensity distribution in the far field of the aperture ** A** is given by

Here *r⃗* is the position vector in the far-field observation plane, and *z* is the distance between the aperture *A* and the far-field observation plane.

Upon substituting Eqs. (1) and (2) into Eq. (3), and by use of Appendix of Ref. 8, one can rewrite Eq. (3) as

$$\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\frac{{\epsilon}_{s}^{2}}{1-{\epsilon}_{s}^{2}}{\int}_{0}^{1}C\left(u\right)\mathrm{Besinc}\left[7.664\left(\frac{a}{\overline{L}}\right){\epsilon}_{s}u\right]{J}_{0}\left(7.664\mathit{uv}\right)\mathit{udu}\}$$

where *J*
_{0} is the Bessel function of order zero, and

Here *L̄* is a quality characterizing the coherence of the light in the aperture ** A**, which has been defined as 8

The variable *v* in Eq. (4) is given by

where

is the diffraction angle in the far field.

According to Eq.(4), we can calculate numerically the Fraunhofer diffraction pattern. In the next Section, we will give some results to show the effect of the coherence on the diffraction pattern.

## 3. Fraunhofer diffraction patterns: Numerical results

As indicated above, *L̄* is a quality characterizing the coherence of the light in the aperture. It is readily found that in the limiting case *L̄* → ∞, the light in the aperture is completely coherent; and the other limiting *L̄* → 0 corresponds to the incoherent case. This indicates that the larger *L̄* means the higher coherence of the light in the aperture. To show the characteristics of the partially coherent light related to ε_{s}, we plot the correlation µ(Δρ) as a function of Δρ in the case of *a* /*L̄* = 1, ε_{s} = 0.9, 0.6, and 0.4, respectively (see Fig. 2). Here Δρ = |ρ¯_{1}–ρ¯_{2}|. As shown in Fig. 2, when ε_{s} is very large, for example, ε_{s} = 0.9, the oscillation of µ(Δρ) against Δρ prolongs so long, i.e., the oscillation of µ(Δρ) continues even when Δρ is very large. But for the smaller ε_{s}, the oscillation of Δρ against Δρ seems vanishing when Δρ is larger than eight.

In Fig. 3, we plot the diffraction pattern for the illumination of completely coherent light, i.e., *a*/*L̄* → 0 (or *L̄* → ∞). The left side of the figure is the diffraction pattern and the right side is the corresponding intensity distribution *I*(*v*) as a function of the diffraction angle *v*. It is readily found that the intensity distribution is just an Airy disc. Now we consider the decrease of the coherence on the diffraction pattern. In Fig. 4, the coherence of the light in the aperture decreases to *a* / *L̄* = 1. We find that at this coherence the diffraction pattern becomes an annulus. In Figs.5–6, the coherence of the light in the aperture is reduced to *a* / *L̄* = 2 and *a* / *L̄* = 4, respectively. It is shown from Figs. 4–6 that the radius of the annulus increases with the increment of *a* / *L̄* (i.e. the radius of the annulus increases with the decrease of the coherence). This indicates that when *ka* is fixed, the larger *a* / *L̄* is, the larger radius of the diffracted annulus is. It is also shown that the maximum intensity of the annulus decreases with the increment of *a* / *L̄* (see Figs. 3–6). This anomalous behavior of the diffraction patterns is quite different from that of the existing findings. Generally the lower coherence of the illuminating partially coherent light makes the diffraction pattern more spreading and smoother. However in this study, it is found that, for the illuminating partially coherent light whose correlation function is given by the Eq. (2), the lower coherence makes the diffraction pattern be an annulus. In other word, the change of the coherence has made the diffraction light propagate along the desired direction. This anomalous behavior of the diffraction pattern may be used in optical switches, in which the coherence can be modulated.

Another interesting phenomenon of this paper is the generation of a flattened annulus (see Figs. 7–8). In Fig. 7, we choose the parameters characterizing the coherence of the incident partially coherent light as ε_{s} = 0.4 and *a* / *L̄* = 4. It is found that a flattened annulus is achieved at this coherence. By choosing larger value of *a* / *L̄* (*e.g. a* / *L̄* = 5, in Fig. 8), a more flattened annulus is generated. This indicates that the larger *a* / *L̄* is, the more flattened is the annulus. It is also found that *a* / *L̄* and ε_{s} are two critical parameters for achieving the flattened annulus.

## 4. Conclusion and discussion

In this paper, we have studied the Fraunhofer diffraction patterns for a particular class of partially coherent light diffracted by a circular aperture. It is shown that by the illumination of partially coherent light of the particular correlation function, anomalous behavior of the diffraction pattern is found, this is, a decrease of the spatial coherence of the light in the aperture leads to drastic changes in diffraction pattern. Specifically, when the light in the aperture is fully coherent, the diffraction pattern is just an Airy disc. However as the coherence decreases, the diffraction pattern becomes an annulus, and the radius of the annulus increases with the decrease of the coherence. This anomalous behavior in the diffraction patterns may be used in optical switches, in which the coherence of the incident partially coherent light is varied by some method (for example, acousto-optic technology 9–11 etc). When the coherence of the incident light is quite high, the diffraction pattern is nearly an Airy disc, *i.e.*, the most of diffracted light energy propagates along the *z* axis (*i.e.*, θ ≈ 0 in Fig. 1). If the output port of an optical switch is placed at the direction θ = 0, the output port is *on* in this coherence; when the coherence decreases, the diffraction pattern becomes an annulus, *i.e.*, the output port at the direction θ = 0 is *off*, but the other output ports, which are located along the annulus, are *on*.

We also show that flattened annuli can be achieved at some coherence. This shows that by modulating the coherence of a light beam in the aperture, we can make the light beam a specific shape in the diffraction field.11 Further study on this topic will be carried out.

## Acknowledgement

This work is supported by Fujian Natural Science Foundation of China.

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