Analytical models for random sources producing far fields with frame-like intensity profiles are introduced. The frames can have polar and Cartesian symmetry and adjustable sharpness of the inner and outer edges. The frames are shape invariant throughout the far zone but expand due to diffraction with growing distance from the source. The generalization to multiple nested frames is also discussed. The applications of the frames are envisioned in material surface processing and particle trapping.
© 2014 Optical Society of America
Random sources radiating beam-like fields with prescribed intensity distributions can be efficiently modeled with the help of a reciprocity relation of the second-order coherence theory . Namely, the far-zone spatial intensity distribution of any statistically stationary field is proportional to the two-dimensional Fourier transform of the degree of coherence of the field in the source plane, provided that the r.m.s. correlation width is much smaller than the r.m.s. intensity distribution width. The use of this reciprocity relation has led to the families of the ring-shaped beams [2–4], non-uniformly correlated beams , flat-topped beams having circular [6,7] and rectangular [8,9] symmetry and to some other beamlets .
The purpose of this paper is to establish the analytical model for the source degree of coherence which allows radiation of optical frames with polar and Cartesian symmetries, i.e. circular/elliptical and square/rectangular frames. Such frames are to have both inner and outer edges with adjustable sharpness and to be shape-invariant everywhere in the far zone of the source. We note that for our modeling of the frames the specification of the source intensity distribution is not necessary, since the far-zone intensity distribution is independent from it. There are at least two general methods for practical realization of the random sources with prescribed correlations, one based on the field modulation by a spatial light modulator  and the other on the modulation of the intensity in the Fourier plane .
The paper is organized as follows: the sources for the flat-top optical frames with polar and Cartesian symmetries are introduced in sections 2 and 3, respectively; section 4 discusses numerical examples, section 5 briefly goes over possible generalizations to nested frames and section 6 summarizes the obtained results.
2. Modeling of the random frames
In order to introduce the mathematical models for the optical frames we will first briefly review the sufficient conditions that are to be met by the cross-spectral density function of the optical source. The cross-spectral density of the field at the source plane is defined as a two-point correlation function 13]Eq. (3) into Eq. (2) leads to the expression
In the limiting case when varies much faster compared to the cross-spectral density (4) can be approximated by the product Eq. (6) at once that in the quasi-homogeneous limit the far-field spectral density distribution differs from by a trivial geometrical factor. We will hence confine our attention to the behavior of μ and alone.
3. Optical frames with polar symmetry
In order to generate a far-field with circular/elliptical ring profile one may employ the following form ofEquations (8) and (9) are slightly more general distributions than given in , the difference accounting for anisotropy of correlation functions along x and y directions. In order for to be nonnegative [13,14] it suffices to set
On taking the Fourier transform of Eq. (12) we obtain the formula for the source degree of coherence:
The degree of coherence must also satisfy other important conditions :
Figures 1 and 2 present typical density plots of frames with circular symmetry. They show the spatial distribution of the absolute value of the source degree of coherence, , as a function of (a), (b) and (c), and the corresponding spatial distribution of function p varying with : (d), (e), and (f). Figure 1 corresponds to M = 1, and Fig. 2 corresponds to M = 40. Here and in all subsequent figures the horizontal and vertical axes are given in meters for and in meters−1 for p. While for all figures mm, (a), (d), are plotted for mm; (b), (e) formm; (c), (f) for mm. Unlike for distributions for M = 1, the ones for M = 40 are capable of having adjustable edge sharpness and adjustable frame thickness the latter being fully controllable by the difference in the r.m.s. correlation widths.
Figure 3 presents several elliptical frames with M = 40. It shows the spatial distribution of the absolute value of the source degree of coherence, as a function of (a), (b), (c), and the corresponding spatial distribution of function p varying with: (d), (e), (f). While for all plots mm, mm, mm and mm for (a), (d); mm and mm for (b), (e) and mm and mm for (c), (f). It is clear from the plots that assigning different correlations and their differences along the x and y directions it is possible to achieve practically any anisotropic features of the elliptic frames.
4. Optical frames with Cartesian symmetry
In order to introduce optical frames with Cartesian symmetry we may choose in the form8]:
On taking the Fourier transform of Eq. (17) we obtain the formula for the source degree of coherence that is responsible for generation of the frames with rectangular symmetry:
Figure 4 illustrates the absolute value of the degree of coherence, (a)–(c) and the corresponding distribution for p (d)–(f) for the frames with Cartesian symmetry, M = 40 and the same set of r.m.s. correlations as in Fig. 2: while for all figures mm, (a), (d) are plotted for mm; (b), (e) for mm; (c), (f) for mm.
Similarly to the circular case, while the frame sharpness is adjusted by the value of M, its size and thickness is controlled by the outer and inner correlation coefficients, respectively. We note that for thin frames the corners become somewhat highlighted.
Figure 5 shows the frames with rectangular shape, M = 40 and the same r.m.s. correlations as in Fig. 2, i.e., mm, mm (for all plots); mm and mm for (a), (d); mm and mm for (b), (e) and mm and mm for (c), (f). Figure 5 illustrates that as in the case of the elliptic frames, any anisotropic properties can be modeled into the rectangular frames, such as different size and thickness along the x and y directions.
5. Frame combinations
Linear superpositions of the degrees of coherence introduced in the two preceding sections may lead to optical fields being the nested intensity combinations of the individual frames. In particular, the addition of the degrees of coherence with different weights lead to radiation of several frames with desired intensities. In the case when the intensity profile is the same for all contributions the addition of the spectral densities reduces to the addition of the spectral degrees of coherence, i.e.
Figure 6 illustrates some of these possibilities for the case when two spectral degrees of coherence in rectangular frames (21) are superposed. For the inner frame with the r.m.s. correlations are mm mm; for the outer frame for they are mm and mm, respectively. The outer frame is produced with sources of smaller correlation and hence shows larger diffraction. The weights are chosen as and (a) and (b) and illustrate the possibility of highlighting the individual frames in the nested sequences.
Figure 7 shows some combinations of frames with different symmetries. For Fig. 7(a) three spectral degrees of coherence with Cartesian symmetry are chosen. For two inner frames with , the r.m.s. correlations are mm, mm, mm, mm and mm, mm, mm, mm, respectively; for the outer frame with they are mm and mm. For two inner frames with elliptical symmetry on Fig. 7(b) with , the r.m.s. correlations are mm, mm, mm, mm and mm, mm, mm, mm, respectively; for the outer frame with they are mm and mm. The weights for all cases are chosen as 1.
We have modeled random optical sources for generation of novel type of far fields, termed optical frames, as linear superpositions of recently introduced Multi-Gaussian Schell-model sources with either polar or Cartesian symmetries. A single frame is constructed as a difference of two Multi-Gaussian distributions with the same on-axis intensities but different source r.m.s. correlation widths. The upper index of the Multi-Gaussian functions allows for fine tuning of the sharpness of the inner and outer frame edges.
Our analytic model is also capable of forming the nested (about the optical axis) sequence of the frame-like far fields, with any shape, thickness and orientation of each individual frame. Moreover the relative intensities of the individual frames can be readily manipulated by assigning different weighing coefficients.
The random optical frames can be experimentally generated with the help of the nematic spatial light modulators or the holograms. They might find uses in material surface processing, optical particle manipulation, active imaging and communications.
O. Korotkova’s research is supported by ONR (N00189-12-T-0136) and AFOSR (FA9550-12-1-0449), E. Shchepakina’s is funded by the Russian Foundation for Basic Research (grants 13-01-97002 p, 14-01-97018 p).
References and links
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