Random sources for optical frames

Olga Korotkova; Elena Shchepakina

doi:10.1364/OE.22.010622

Random sources for optical frames

Olga Korotkova and Elena Shchepakina

Optics Express
Vol. 22,
Issue 9,
pp. 10622-10633
(2014)
•https://doi.org/10.1364/OE.22.010622

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Olga Korotkova and Elena Shchepakina, "Random sources for optical frames," Opt. Express 22, 10622-10633 (2014)

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Table of Contents Category
- Coherence and Statistical Optics
Optics & Photonics Topics
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The topics in this list come from the OSA Optics and Photonics Topics applied to this article.
Previously assigned OCIS codes
- Coherence (030.1640)
- Propagation (350.5500)

About this Article
History
- Original Manuscript: March 26, 2014
- Manuscript Accepted: April 3, 2014
- Published: April 24, 2014

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Abstract

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.

1. Introduction

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 [1]. 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 [5], flat-topped beams having circular [6,7] and rectangular [8,9] symmetry and to some other beamlets [10].

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 [11] and the other on the modulation of the intensity in the Fourier plane [12].

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 [1]

W_{}^{(0)} ({ρ^{'}}_{1}, {ρ^{'}}_{2}; ω) = 〈 E_{}^{*} ({ρ^{'}}_{1}; ω) E_{} ({ρ^{'}}_{2}; ω) 〉,

where

E

denotes the field fluctuating in a direction perpendicular to the z axis, and the angular brackets denote an ensemble average. In what follows the angular frequency dependence of all the quantities of interest will be omitted but implied. A genuine cross-spectral density function is limited by the constraint of non-negative definiteness. This condition is fulfilled if the cross-spectral density can be written as a superposition integral of the form [13]

W_{}^{(0)} ({ρ^{'}}_{1}, {ρ^{'}}_{2}) = \int p (v) H_{}^{*} ({ρ^{'}}_{1}, v) H ({ρ^{'}}_{2}, v) d^{2} v,

where

p (v)

is an arbitrary non-negative weight function, and

H (ρ^{'}, v)

is an arbitrary kernel. A simple and significant class of cross-spectral densities, the Schell-model class, can be obtained by having

H (ρ^{'}, v)

in a Fourier-like form, i.e.

H (ρ^{'}, v) = τ (ρ^{'}) \exp (- 2 π i v \cdot ρ^{'}),

where

τ (ρ^{'})

is an amplitude profile function. Then, substitution from Eq. (3) into Eq. (2) leads to the expression

W_{}^{(0)} ({ρ^{'}}_{1}, {ρ^{'}}_{2}) = τ^{*} ({ρ^{'}}_{1}) τ ({ρ^{'}}_{2}) μ ({ρ^{'}}_{1} - {ρ^{'}}_{2}),

where

μ = \tilde{p},

tilde symbol denotes the Fourier transform. The choice of

p (v)

defines a family of sources with different correlation functions. For our purposes the choice of

τ (ρ^{'})

is arbitrary, and may be for instance simply a Gaussian function.

In the limiting case when $τ (ρ^{'})$ varies much faster compared to $μ ({ρ^{'}}_{1} - {ρ^{'}}_{2})$ the cross-spectral density (4) can be approximated by the product [1]

W_{}^{(0)} ({ρ^{'}}_{1}, {ρ^{'}}_{2}) \approx τ^{2} (\frac{{ρ^{'}}_{1} + {ρ^{'}}_{2}}{2}) μ ({ρ^{'}}_{1} - {ρ^{'}}_{2}) .

Such model source, known in the literature as a quasi-homogeneous source, possesses one property that is of utmost importance for our analysis: the far-field spectral density of such a source is proportional to the Fourier transform of the source degree of coherence, i.e.

S_{}^{(\infty)} (k s_{⊥}) \approx \frac{{(2 π k)}^{2}}{r^{2}} s_{z}^{2} \tilde{μ} (k s_{⊥}),

where

k = 2 π / λ

is the wave number,

s = r / r = (s_{x}, s_{y}, s_{z})

is the unit vector collinear to vector

r

, while

s_{⊥} = (s_{x}, s_{y})

is its projection onto the source plane and

δ_{o x} = 2.0

δ_{o x}, δ_{o y}, δ_{i x}, δ_{i y}

θ

being the angle between vector

r

and the direction of propagation z. Since

p

and

μ

are the Fourier transform pair it follows from Eq. (6) at once that in the quasi-homogeneous limit the far-field spectral density distribution

S_{}^{(\infty)}

differs from

p

by a trivial geometrical factor. We will hence confine our attention to the behavior of μ and

p

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 of $p (v) :$

p (v) = p_{o}^{(p)} (v) - p_{i}^{(p)} (v),

\begin{matrix} p_{o}^{(p)} (v) = \frac{A}{C_{p}} \sum_{m = 1}^{M} {(- 1)}^{m - 1} (\begin{matrix} M \\ m \end{matrix}) \exp [- \frac{m}{2} (δ_{o x}^{2} v_{x}^{2} + δ_{o y}^{2} v_{y}^{2})] \\ = \frac{A}{C_{p}} (1 - {1 - \exp [- \frac{1}{2} (δ_{o x}^{2} v_{x}^{2} + δ_{o y}^{2} v_{y}^{2})]}^{M}), \end{matrix}

\begin{matrix} p_{i}^{(p)} (v) = \frac{A}{C_{p}} \sum_{m = 1}^{M} {(- 1)}^{m - 1} (\begin{matrix} M \\ m \end{matrix}) \exp [- \frac{m}{2} (δ_{i x}^{2} v_{x}^{2} + δ_{i y}^{2} v_{y}^{2})] \\ = \frac{A}{C_{p}} (1 - {1 - \exp [- \frac{1}{2} (δ_{i x}^{2} v_{x}^{2} + δ_{i y}^{2} v_{y}^{2})]}^{M}), \end{matrix}

(\begin{matrix} M \\ m \end{matrix})

is the binomial coefficient,

C_{p}

is the normalization factor:

C_{p} = \sum_{m = 1}^{M} \frac{{(- 1)}^{m - 1}}{m} (\begin{matrix} M \\ m \end{matrix}),

and the value of parameter A is discussed below. Equations (8) and (9) are slightly more general distributions than given in [6], the difference accounting for anisotropy of correlation functions along x and y directions. In order for

p (v)

to be nonnegative [13,14] it suffices to set

δ_{i x} > δ_{o x} and δ_{i y} > δ_{o y} .

On simplifying (7)-(9) we get

\begin{matrix} p (v) = p_{o}^{(p)} (v) - p_{i}^{(p)} (v) \\ = \frac{A}{C_{p}} \sum_{m = 1}^{M} {(- 1)}^{m - 1} (\begin{matrix} M \\ m \end{matrix}) {\exp [- \frac{m}{2} (δ_{o x}^{2} v_{x}^{2} + δ_{o y}^{2} v_{y}^{2})] - \exp [- \frac{m}{2} (δ_{i x}^{2} v_{x}^{2} + δ_{i y}^{2} v_{y}^{2})]} . \end{matrix}

On taking the Fourier transform of Eq. (12) we obtain the formula for the source degree of coherence:

\begin{array}{l} μ ({ρ^{'}}_{d}) = \frac{A}{C_{p}} \sum_{m = 1}^{M} {(- 1)}^{m - 1} (\begin{matrix} M \\ m \end{matrix}) \\ \times {\frac{1}{m δ_{o x}^{} δ_{o y}^{}} \exp [- \frac{1}{2 m} (\frac{ρ'_{d x}^{2}}{δ_{o x}^{2}} + \frac{ρ'_{d y}^{2}}{δ_{o y}^{2}})] - \frac{1}{m δ_{i x}^{} δ_{i y}^{}} \exp [- \frac{1}{2 m} (\frac{ρ'_{d x}^{2}}{δ_{i x}^{2}} + \frac{ρ'_{d y}^{2}}{δ_{i y}^{2}})]}, \end{array}

where

ρ'_{d} = (ρ'_{d x}, ρ'_{d y}), ρ'_{d x} = ρ'_{o x} - ρ'_{i x}, ρ'_{d y} = ρ'_{o y} - ρ'_{i y} .

The degree of coherence must also satisfy other important conditions [1]:

μ (0) = 1, μ (\infty) = 0, and | μ ({ρ^{'}}_{d}) | \leq 1.

While the last two conditions follow from inequalities (11) the first requires that, in addition, one must set

\frac{1}{δ_{o x} δ_{o y}} - \frac{1}{δ_{i x} δ_{i y}} = 1.

Since in practice such restriction seems rather inconvenient one can set:

A = {(\frac{1}{δ_{o x} δ_{o y}} - \frac{1}{δ_{i x} δ_{i y}})}^{- 1} .

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, $μ$ , as a function of $ρ'_{d} :$ (a), (b) and (c), and the corresponding spatial distribution of function p varying with $v$ : (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 $μ$ and in meters⁻¹ for p. While for all figures $δ_{o x} = δ_{o y} = 1.0$ mm, (a), (d), are plotted for $δ_{i x} = δ_{i y} = 1.1$ mm; (b), (e) for $δ_{i x} = δ_{i y} = 2.0$ mm; (c), (f) for $δ_{i x} = δ_{i y} = 8.0$ 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.

Fig. 1 Illustration of the frames with circular symmetry. The absolute value of the degree of coherence μ and its Fourier transform p for several values of $δ_{o x}, δ_{o y}, δ_{i x}, δ_{i y}$ and M = 1.

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Fig. 2 Illustration of the frames with circular symmetry. The absolute value of the degree of coherence μ and its Fourier transform p for several values of $δ_{o x}, δ_{o y}, δ_{i x}, δ_{i y}$ and M = 40.

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Figure 3 presents several elliptical frames with M = 40. It shows the spatial distribution of the absolute value of the source degree of coherence, $μ,$ as a function of $ρ'_{d} :$ (a), (b), (c), and the corresponding spatial distribution of function p varying with $v$ : (d), (e), (f). While for all plots $δ_{o x} = 2.0$ mm, $δ_{o y} = 1.0$ mm, $δ_{i x} = 2.1$ mm and $δ_{i y} = 1.1$ mm for (a), (d); $δ_{i x} = 4.0$ mm and $δ_{i y} = 2.0$ mm for (b), (e) and $δ_{i x} = 16.0$ mm and $δ_{i y} = 8.0$ 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.

Fig. 3 Illustration of the frames with elliptical symmetry. The absolute value of the degree of coherence μ and its Fourier transform p for several values of $δ_{o x}, δ_{o y}, δ_{i x}, δ_{i y}$ and M = 40.

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4. Optical frames with Cartesian symmetry

In order to introduce optical frames with Cartesian symmetry we may choose $p (v)$ in the form

p (v) = p_{o}^{(r)} (v) - p_{i}^{(r)} (v),

where the outer and inner distributions were introduced in [8]:

\begin{matrix} p_{o}^{(r)} (v) = \frac{A}{C_{r}^{2}} \sum_{m = 1}^{M} {(- 1)}^{m - 1} (\begin{matrix} M \\ m \end{matrix}) \exp [- \frac{m δ_{o x}^{2} v_{x}^{2}}{2}] \sum_{m = 1}^{M} {(- 1)}^{m - 1} (\begin{matrix} M \\ m \end{matrix}) \exp [- \frac{m δ_{o y}^{2} v_{y}^{2}}{2}] \\ = \frac{1}{C_{r}^{2}} {1 - {(1 - \exp [- \frac{δ_{o x}^{2} v_{x}^{2}}{2}])}^{M}} {1 - {(1 - \exp [- \frac{δ_{o y}^{2} v_{y}^{2}}{2}])}^{M}}, \end{matrix}

\begin{matrix} p_{i}^{(r)} (v) = \frac{A}{C_{r}^{2}} \sum_{m = 1}^{M} {(- 1)}^{m - 1} (\begin{matrix} M \\ m \end{matrix}) \exp [- \frac{m δ_{i x}^{2} v_{x}^{2}}{2}] \sum_{m = 1}^{M} {(- 1)}^{m - 1} (\begin{matrix} M \\ m \end{matrix}) \exp [- \frac{m δ_{i y}^{2} v_{y}^{2}}{2}] \\ = \frac{1}{C_{r}^{2}} {1 - {(1 - \exp [- \frac{δ_{i x}^{2} v_{x}^{2}}{2}])}^{M}} {1 - {(1 - \exp [- \frac{δ_{i y}^{2} v_{y}^{2}}{2}])}^{M}}, \end{matrix}

C_{r} = \sum_{m = 1}^{M} \frac{{(- 1)}^{m - 1}}{\sqrt{m}} (\begin{matrix} M \\ m \end{matrix}) .

In order for the function in (17) be satisfy non-negative it suffices to require that the same set of inequalities as in the polar case holds, i.e. that

δ_{i x} > δ_{o x}

and

δ_{i y} > δ_{o y} .

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:

\begin{matrix} μ (ρ'_{d}) = \frac{A}{C_{r}^{2} δ_{o x}^{} δ_{o y}^{}} \sum_{m = 1}^{M} (\begin{matrix} M \\ m \end{matrix}) \frac{{(- 1)}^{m - 1}}{\sqrt{m}} \exp [- \frac{1}{2 m} \frac{ρ'_{d x}^{2}}{δ_{o x}^{2}}] \sum_{m = 1}^{M} (\begin{matrix} M \\ m \end{matrix}) \frac{{(- 1)}^{m - 1}}{\sqrt{m}} \exp [- \frac{1}{2 m} \frac{ρ'_{d y}^{2}}{δ_{o y}^{2}}] \\ - \frac{A}{C_{r}^{2} δ_{i x}^{} δ_{i y}^{}} \sum_{m = 1}^{M} (\begin{matrix} M \\ m \end{matrix}) \frac{{(- 1)}^{m - 1}}{\sqrt{m}} \exp [- \frac{1}{2 m} \frac{ρ'_{d x}^{2}}{δ_{i x}^{2}}] \sum_{m = 1}^{M} (\begin{matrix} M \\ m \end{matrix}) \frac{{(- 1)}^{m - 1}}{\sqrt{m}} \exp [- \frac{1}{2 m} \frac{ρ'_{d y}^{2}}{δ_{i y}^{2}}] . \end{matrix}

Figure 4 illustrates the absolute value of the degree of coherence, $μ,$ (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 $δ_{o x} = δ_{o y} = 1.0$ mm, (a), (d) are plotted for $δ_{i x} = δ_{i y} = 1.1$ mm; (b), (e) for $δ_{i x} = δ_{i y} = 2.0$ mm; (c), (f) for $δ_{i x} = δ_{i y} = 8.0$ mm.

Fig. 4 Illustration of the frames with square symmetry. The absolute value of the degree of coherence μ and its Fourier transform p for several values of $δ_{o x}, δ_{o y}, δ_{i x}, δ_{i y}$ and M = 40.

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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., $δ_{o x} = 2.0$ mm, $δ_{o y} = 1.0$ mm (for all plots); $δ_{i x} = 2.1$ mm and $δ_{i y} = 1.1$ mm for (a), (d); $δ_{i x} = 4.0$ mm and $δ_{i y} = 2.0$ mm for (b), (e) and $δ_{i x} = 16.0$ mm and $δ_{i y} = 8.0$ 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.

Fig. 5 Illustration of the frames with rectangular symmetry. The absolute value of the degree of coherence μ and its Fourier transform p for several values of $δ_{o x}, δ_{o y}, δ_{i x}, δ_{i y}$ and M = 40.

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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.

μ (ρ'_{d}) = B \sum_{n = 1}^{N} a_{n} μ_{n} (ρ'_{d}), B = {(\sum_{n = 1}^{N} a_{n})}^{- 1},

where factor B is introduced to meet the first of conditions (11),

μ_{n}

are of the form (13) or (21) starting summation, say, from the most inner frame,

a_{n}

are the weights.

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 $μ_{1}$ the r.m.s. correlations are $δ_{o x} = δ_{o y} = 1.0$ mm $δ_{i x} = δ_{i y} = 1.1$ mm; for the outer frame for $μ_{2}$ they are $δ_{o x} = δ_{o y} = 0.50$ mm and $δ_{i x} = δ_{i y} = 0.51$ mm, respectively. The outer frame is produced with sources of smaller correlation and hence shows larger diffraction. The weights are chosen as $a_{1} = 1$ and (a) $a_{2} = 2;$ and (b) $a_{2} = 10$ and illustrate the possibility of highlighting the individual frames in the nested sequences.

Fig. 6 Illustration of the square frame combination.

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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 $μ_{1}$ , $μ_{2}$ the r.m.s. correlations are $δ_{o x} = 2.1$ mm, $δ_{o y} = 0.9$ mm, $δ_{i x} = 2.5$ mm, $δ_{i y} = 1.0$ mm and $δ_{o x} = 0.9$ mm, $δ_{o y} = 2.1$ mm, $δ_{i x} = 1.0$ mm, $δ_{i y} = 2.5$ mm, respectively; for the outer frame with $μ_{3}$ they are $δ_{o x} = δ_{o y} = 0.8$ mm and $δ_{i x} = δ_{i y} = 0.9$ mm. For two inner frames with elliptical symmetry on Fig. 7(b) with $μ_{1}$ , $μ_{2}$ the r.m.s. correlations are $δ_{o x} = 2.0$ mm, $δ_{o y} = 1.0$ mm, $δ_{i x} = 2.1$ mm, $δ_{i y} = 1.1$ mm and $δ_{o x} = 1.0$ mm, $δ_{o y} = 2.0$ mm, $δ_{i x} = 1.1$ mm, $δ_{i y} = 2.1$ mm, respectively; for the outer frame with $μ_{3}$ they are $δ_{o x} = δ_{o y} = 0.8$ mm and $δ_{i x} = δ_{i y} = 0.9$ mm. The weights for all cases are chosen as 1.

Fig. 7 Illustration of the combination of frames with different symmetries.

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6. Summary

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.

Acknowledgments

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

1. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, Cambridge, 1995).

2. F. Gori, G. Guattari, and C. Padovani, “Modal expansion for J₀-correlated Schell-model sources,” Opt. Commun. 64(4), 311–316 (1987). [CrossRef]

3. Z. Mei and O. Korotkova, “Random sources generating ring-shaped beams,” Opt. Lett. 38(2), 91–93 (2013). [CrossRef] [PubMed]

4. Z. Mei and O. Korotkova, “Cosine-Gaussian Schell-model sources,” Opt. Lett. 38(14), 2578–2580 (2013). [CrossRef] [PubMed]

5. H. Lajunen and T. Saastamoinen, “Propagation characteristics of partially coherent beams with spatially varying correlations,” Opt. Lett. 36(20), 4104–4106 (2011). [CrossRef] [PubMed]

6. S. Sahin and O. Korotkova, “Light sources generating far fields with tunable flat profiles,” Opt. Lett. 37(14), 2970–2972 (2012). [CrossRef] [PubMed]

7. O. Korotkova, S. Sahin, and E. Shchepakina, “Multi-Gaussian Schell-model beams,” J. Opt. Soc. Am. A 29(10), 2159–2164 (2012). [CrossRef] [PubMed]

8. O. Korotkova, “Random sources for rectangular far fields,” Opt. Lett. 39(1), 64–67 (2014). [CrossRef] [PubMed]

9. O. Korotkova and E. Shchepakina, “Rectangular multi-Gaussian Schell-model beams in free space and atmospheric turbulence,” J. Opt. 16(4), 045704 (2014). [CrossRef]

10. C. Liang, F. Wang, X. Liu, Y. Cai, and O. Korotkova, “Experimental generation of Cosine-Gaussian-correlated Schell-model beams with rectangular symmetry,” Opt. Lett. 39(4), 769–772 (2014). [CrossRef] [PubMed]

11. S. Avramov-Zamurovic, O. Korotkova, C. Nelson, O. Korotkova, and R. Malek-Madani, “The dependence of the intensity PDF of a random beam propagating in the maritime atmosphere on source coherence,” Waves in Complex and Random Media. 24(1), 69–82 (2014). [CrossRef]

12. F. Wang, X. Liu, Y. Yuan, and Y. Cai, “Experimental generation of partially coherent beams with different complex degrees of coherence,” Opt. Lett. 38(11), 1814–1816 (2013). [CrossRef] [PubMed]

13. F. Gori, V. R. Sanchez, M. Santarsiero, and T. Shirai, “On genuine cross-spectral density matrices,” J. Opt. A, Pure Appl. Opt. 11(8), 085706 (2009). [CrossRef]

14. M. Santarsiero, G. Piquero, J. C. G. de Sande, and F. Gori, “Difference of cross-spectral densities,” Opt. Lett. 39(7), 1713–1716 (2014). [CrossRef] [PubMed]

References

View by:
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|
Year
|
Author
|
Publication

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, Cambridge, 1995).
F. Gori, G. Guattari, and C. Padovani, “Modal expansion for J0-correlated Schell-model sources,” Opt. Commun. 64(4), 311–316 (1987).
[Crossref]
Z. Mei and O. Korotkova, “Random sources generating ring-shaped beams,” Opt. Lett. 38(2), 91–93 (2013).
[Crossref] [PubMed]
Z. Mei and O. Korotkova, “Cosine-Gaussian Schell-model sources,” Opt. Lett. 38(14), 2578–2580 (2013).
[Crossref] [PubMed]
H. Lajunen and T. Saastamoinen, “Propagation characteristics of partially coherent beams with spatially varying correlations,” Opt. Lett. 36(20), 4104–4106 (2011).
[Crossref] [PubMed]
S. Sahin and O. Korotkova, “Light sources generating far fields with tunable flat profiles,” Opt. Lett. 37(14), 2970–2972 (2012).
[Crossref] [PubMed]
O. Korotkova, S. Sahin, and E. Shchepakina, “Multi-Gaussian Schell-model beams,” J. Opt. Soc. Am. A 29(10), 2159–2164 (2012).
[Crossref] [PubMed]
O. Korotkova, “Random sources for rectangular far fields,” Opt. Lett. 39(1), 64–67 (2014).
[Crossref] [PubMed]
O. Korotkova and E. Shchepakina, “Rectangular multi-Gaussian Schell-model beams in free space and atmospheric turbulence,” J. Opt. 16(4), 045704 (2014).
[Crossref]
C. Liang, F. Wang, X. Liu, Y. Cai, and O. Korotkova, “Experimental generation of Cosine-Gaussian-correlated Schell-model beams with rectangular symmetry,” Opt. Lett. 39(4), 769–772 (2014).
[Crossref] [PubMed]
S. Avramov-Zamurovic, O. Korotkova, C. Nelson, O. Korotkova, and R. Malek-Madani, “The dependence of the intensity PDF of a random beam propagating in the maritime atmosphere on source coherence,” Waves in Complex and Random Media. 24(1), 69–82 (2014).
[Crossref]
F. Wang, X. Liu, Y. Yuan, and Y. Cai, “Experimental generation of partially coherent beams with different complex degrees of coherence,” Opt. Lett. 38(11), 1814–1816 (2013).
[Crossref] [PubMed]
F. Gori, V. R. Sanchez, M. Santarsiero, and T. Shirai, “On genuine cross-spectral density matrices,” J. Opt. A, Pure Appl. Opt. 11(8), 085706 (2009).
[Crossref]
M. Santarsiero, G. Piquero, J. C. G. de Sande, and F. Gori, “Difference of cross-spectral densities,” Opt. Lett. 39(7), 1713–1716 (2014).
[Crossref] [PubMed]

2014 (5)

O. Korotkova, “Random sources for rectangular far fields,” Opt. Lett. 39(1), 64–67 (2014).
[Crossref] [PubMed]

O. Korotkova and E. Shchepakina, “Rectangular multi-Gaussian Schell-model beams in free space and atmospheric turbulence,” J. Opt. 16(4), 045704 (2014).
[Crossref]

C. Liang, F. Wang, X. Liu, Y. Cai, and O. Korotkova, “Experimental generation of Cosine-Gaussian-correlated Schell-model beams with rectangular symmetry,” Opt. Lett. 39(4), 769–772 (2014).
[Crossref] [PubMed]

S. Avramov-Zamurovic, O. Korotkova, C. Nelson, O. Korotkova, and R. Malek-Madani, “The dependence of the intensity PDF of a random beam propagating in the maritime atmosphere on source coherence,” Waves in Complex and Random Media. 24(1), 69–82 (2014).
[Crossref]

M. Santarsiero, G. Piquero, J. C. G. de Sande, and F. Gori, “Difference of cross-spectral densities,” Opt. Lett. 39(7), 1713–1716 (2014).
[Crossref] [PubMed]

2009 (1)

F. Gori, V. R. Sanchez, M. Santarsiero, and T. Shirai, “On genuine cross-spectral density matrices,” J. Opt. A, Pure Appl. Opt. 11(8), 085706 (2009).
[Crossref]

1987 (1)

F. Gori, G. Guattari, and C. Padovani, “Modal expansion for J0-correlated Schell-model sources,” Opt. Commun. 64(4), 311–316 (1987).
[Crossref]

Avramov-Zamurovic, S.

Cai, Y.

F. Wang, X. Liu, Y. Yuan, and Y. Cai, “Experimental generation of partially coherent beams with different complex degrees of coherence,” Opt. Lett. 38(11), 1814–1816 (2013).
[Crossref] [PubMed]

de Sande, J. C. G.

M. Santarsiero, G. Piquero, J. C. G. de Sande, and F. Gori, “Difference of cross-spectral densities,” Opt. Lett. 39(7), 1713–1716 (2014).
[Crossref] [PubMed]

Gori, F.

M. Santarsiero, G. Piquero, J. C. G. de Sande, and F. Gori, “Difference of cross-spectral densities,” Opt. Lett. 39(7), 1713–1716 (2014).
[Crossref] [PubMed]

F. Gori, V. R. Sanchez, M. Santarsiero, and T. Shirai, “On genuine cross-spectral density matrices,” J. Opt. A, Pure Appl. Opt. 11(8), 085706 (2009).
[Crossref]

F. Gori, G. Guattari, and C. Padovani, “Modal expansion for J0-correlated Schell-model sources,” Opt. Commun. 64(4), 311–316 (1987).
[Crossref]

Guattari, G.

F. Gori, G. Guattari, and C. Padovani, “Modal expansion for J0-correlated Schell-model sources,” Opt. Commun. 64(4), 311–316 (1987).
[Crossref]

Korotkova, O.

O. Korotkova, “Random sources for rectangular far fields,” Opt. Lett. 39(1), 64–67 (2014).
[Crossref] [PubMed]

O. Korotkova and E. Shchepakina, “Rectangular multi-Gaussian Schell-model beams in free space and atmospheric turbulence,” J. Opt. 16(4), 045704 (2014).
[Crossref]

Z. Mei and O. Korotkova, “Random sources generating ring-shaped beams,” Opt. Lett. 38(2), 91–93 (2013).
[Crossref] [PubMed]

Z. Mei and O. Korotkova, “Cosine-Gaussian Schell-model sources,” Opt. Lett. 38(14), 2578–2580 (2013).
[Crossref] [PubMed]

S. Sahin and O. Korotkova, “Light sources generating far fields with tunable flat profiles,” Opt. Lett. 37(14), 2970–2972 (2012).
[Crossref] [PubMed]

O. Korotkova, S. Sahin, and E. Shchepakina, “Multi-Gaussian Schell-model beams,” J. Opt. Soc. Am. A 29(10), 2159–2164 (2012).
[Crossref] [PubMed]

Lajunen, H.

H. Lajunen and T. Saastamoinen, “Propagation characteristics of partially coherent beams with spatially varying correlations,” Opt. Lett. 36(20), 4104–4106 (2011).
[Crossref] [PubMed]

Liang, C.

Liu, X.

F. Wang, X. Liu, Y. Yuan, and Y. Cai, “Experimental generation of partially coherent beams with different complex degrees of coherence,” Opt. Lett. 38(11), 1814–1816 (2013).
[Crossref] [PubMed]

Malek-Madani, R.

Mei, Z.

Z. Mei and O. Korotkova, “Random sources generating ring-shaped beams,” Opt. Lett. 38(2), 91–93 (2013).
[Crossref] [PubMed]

Z. Mei and O. Korotkova, “Cosine-Gaussian Schell-model sources,” Opt. Lett. 38(14), 2578–2580 (2013).
[Crossref] [PubMed]

Nelson, C.

Padovani, C.

F. Gori, G. Guattari, and C. Padovani, “Modal expansion for J0-correlated Schell-model sources,” Opt. Commun. 64(4), 311–316 (1987).
[Crossref]

Piquero, G.

M. Santarsiero, G. Piquero, J. C. G. de Sande, and F. Gori, “Difference of cross-spectral densities,” Opt. Lett. 39(7), 1713–1716 (2014).
[Crossref] [PubMed]

Saastamoinen, T.

H. Lajunen and T. Saastamoinen, “Propagation characteristics of partially coherent beams with spatially varying correlations,” Opt. Lett. 36(20), 4104–4106 (2011).
[Crossref] [PubMed]

Sahin, S.

S. Sahin and O. Korotkova, “Light sources generating far fields with tunable flat profiles,” Opt. Lett. 37(14), 2970–2972 (2012).
[Crossref] [PubMed]

O. Korotkova, S. Sahin, and E. Shchepakina, “Multi-Gaussian Schell-model beams,” J. Opt. Soc. Am. A 29(10), 2159–2164 (2012).
[Crossref] [PubMed]

Sanchez, V. R.

F. Gori, V. R. Sanchez, M. Santarsiero, and T. Shirai, “On genuine cross-spectral density matrices,” J. Opt. A, Pure Appl. Opt. 11(8), 085706 (2009).
[Crossref]

Santarsiero, M.

M. Santarsiero, G. Piquero, J. C. G. de Sande, and F. Gori, “Difference of cross-spectral densities,” Opt. Lett. 39(7), 1713–1716 (2014).
[Crossref] [PubMed]

F. Gori, V. R. Sanchez, M. Santarsiero, and T. Shirai, “On genuine cross-spectral density matrices,” J. Opt. A, Pure Appl. Opt. 11(8), 085706 (2009).
[Crossref]

Shchepakina, E.

O. Korotkova and E. Shchepakina, “Rectangular multi-Gaussian Schell-model beams in free space and atmospheric turbulence,” J. Opt. 16(4), 045704 (2014).
[Crossref]

O. Korotkova, S. Sahin, and E. Shchepakina, “Multi-Gaussian Schell-model beams,” J. Opt. Soc. Am. A 29(10), 2159–2164 (2012).
[Crossref] [PubMed]

Shirai, T.

F. Gori, V. R. Sanchez, M. Santarsiero, and T. Shirai, “On genuine cross-spectral density matrices,” J. Opt. A, Pure Appl. Opt. 11(8), 085706 (2009).
[Crossref]

Wang, F.

F. Wang, X. Liu, Y. Yuan, and Y. Cai, “Experimental generation of partially coherent beams with different complex degrees of coherence,” Opt. Lett. 38(11), 1814–1816 (2013).
[Crossref] [PubMed]

Yuan, Y.

F. Wang, X. Liu, Y. Yuan, and Y. Cai, “Experimental generation of partially coherent beams with different complex degrees of coherence,” Opt. Lett. 38(11), 1814–1816 (2013).
[Crossref] [PubMed]

J. Opt. (1)

O. Korotkova and E. Shchepakina, “Rectangular multi-Gaussian Schell-model beams in free space and atmospheric turbulence,” J. Opt. 16(4), 045704 (2014).
[Crossref]

J. Opt. A, Pure Appl. Opt. (1)

F. Gori, V. R. Sanchez, M. Santarsiero, and T. Shirai, “On genuine cross-spectral density matrices,” J. Opt. A, Pure Appl. Opt. 11(8), 085706 (2009).
[Crossref]

J. Opt. Soc. Am. A (1)

O. Korotkova, S. Sahin, and E. Shchepakina, “Multi-Gaussian Schell-model beams,” J. Opt. Soc. Am. A 29(10), 2159–2164 (2012).
[Crossref] [PubMed]

Opt. Commun. (1)

F. Gori, G. Guattari, and C. Padovani, “Modal expansion for J0-correlated Schell-model sources,” Opt. Commun. 64(4), 311–316 (1987).
[Crossref]

Opt. Lett. (8)

Z. Mei and O. Korotkova, “Random sources generating ring-shaped beams,” Opt. Lett. 38(2), 91–93 (2013).
[Crossref] [PubMed]

Z. Mei and O. Korotkova, “Cosine-Gaussian Schell-model sources,” Opt. Lett. 38(14), 2578–2580 (2013).
[Crossref] [PubMed]

H. Lajunen and T. Saastamoinen, “Propagation characteristics of partially coherent beams with spatially varying correlations,” Opt. Lett. 36(20), 4104–4106 (2011).
[Crossref] [PubMed]

S. Sahin and O. Korotkova, “Light sources generating far fields with tunable flat profiles,” Opt. Lett. 37(14), 2970–2972 (2012).
[Crossref] [PubMed]

O. Korotkova, “Random sources for rectangular far fields,” Opt. Lett. 39(1), 64–67 (2014).
[Crossref] [PubMed]

M. Santarsiero, G. Piquero, J. C. G. de Sande, and F. Gori, “Difference of cross-spectral densities,” Opt. Lett. 39(7), 1713–1716 (2014).
[Crossref] [PubMed]

F. Wang, X. Liu, Y. Yuan, and Y. Cai, “Experimental generation of partially coherent beams with different complex degrees of coherence,” Opt. Lett. 38(11), 1814–1816 (2013).
[Crossref] [PubMed]

Waves in Complex and Random Media. (1)

Other (1)

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, Cambridge, 1995).

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Fig. 1 Illustration of the frames with circular symmetry. The absolute value of the degree of coherence μ and its Fourier transform p for several values of

δ_{o x}, δ_{o y}, δ_{i x}, δ_{i y}

and M = 1.

View in Article | Download Full Size | PPT Slide | PDF

Fig. 2 Illustration of the frames with circular symmetry. The absolute value of the degree of coherence μ and its Fourier transform p for several values of

δ_{o x}, δ_{o y}, δ_{i x}, δ_{i y}

and M = 40.

View in Article | Download Full Size | PPT Slide | PDF

Fig. 3 Illustration of the frames with elliptical symmetry. The absolute value of the degree of coherence μ and its Fourier transform p for several values of

δ_{o x}, δ_{o y}, δ_{i x}, δ_{i y}

and M = 40.

View in Article | Download Full Size | PPT Slide | PDF

Fig. 4 Illustration of the frames with square symmetry. The absolute value of the degree of coherence μ and its Fourier transform p for several values of

δ_{o x}, δ_{o y}, δ_{i x}, δ_{i y}

and M = 40.

View in Article | Download Full Size | PPT Slide | PDF

Fig. 5 Illustration of the frames with rectangular symmetry. The absolute value of the degree of coherence μ and its Fourier transform p for several values of

δ_{o x}, δ_{o y}, δ_{i x}, δ_{i y}

and M = 40.

View in Article | Download Full Size | PPT Slide | PDF

Fig. 6 Illustration of the square frame combination.

View in Article | Download Full Size | PPT Slide | PDF

Fig. 7 Illustration of the combination of frames with different symmetries.

View in Article | Download Full Size | PPT Slide | PDF

Equations (22)

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W_{}^{(0)} ({ρ^{'}}_{1}, {ρ^{'}}_{2}; ω) = 〈 E_{}^{*} ({ρ^{'}}_{1}; ω) E_{} ({ρ^{'}}_{2}; ω) 〉,

W_{}^{(0)} ({ρ^{'}}_{1}, {ρ^{'}}_{2}) = \int p (v) H_{}^{*} ({ρ^{'}}_{1}, v) H ({ρ^{'}}_{2}, v) d^{2} v,

H (ρ^{'}, v) = τ (ρ^{'}) \exp (- 2 π i v \cdot ρ^{'}),

W_{}^{(0)} ({ρ^{'}}_{1}, {ρ^{'}}_{2}) = τ^{*} ({ρ^{'}}_{1}) τ ({ρ^{'}}_{2}) μ ({ρ^{'}}_{1} - {ρ^{'}}_{2}),

W_{}^{(0)} ({ρ^{'}}_{1}, {ρ^{'}}_{2}) \approx τ^{2} (\frac{{ρ^{'}}_{1} + {ρ^{'}}_{2}}{2}) μ ({ρ^{'}}_{1} - {ρ^{'}}_{2}) .

S_{}^{(\infty)} (k s_{⊥}) \approx \frac{{(2 π k)}^{2}}{r^{2}} s_{z}^{2} \tilde{μ} (k s_{⊥}),

p (v) = p_{o}^{(p)} (v) - p_{i}^{(p)} (v),

\begin{matrix} p_{o}^{(p)} (v) = \frac{A}{C_{p}} \sum_{m = 1}^{M} {(- 1)}^{m - 1} (\begin{matrix} M \\ m \end{matrix}) \exp [- \frac{m}{2} (δ_{o x}^{2} v_{x}^{2} + δ_{o y}^{2} v_{y}^{2})] \\ = \frac{A}{C_{p}} (1 - {1 - \exp [- \frac{1}{2} (δ_{o x}^{2} v_{x}^{2} + δ_{o y}^{2} v_{y}^{2})]}^{M}), \end{matrix}

\begin{matrix} p_{i}^{(p)} (v) = \frac{A}{C_{p}} \sum_{m = 1}^{M} {(- 1)}^{m - 1} (\begin{matrix} M \\ m \end{matrix}) \exp [- \frac{m}{2} (δ_{i x}^{2} v_{x}^{2} + δ_{i y}^{2} v_{y}^{2})] \\ = \frac{A}{C_{p}} (1 - {1 - \exp [- \frac{1}{2} (δ_{i x}^{2} v_{x}^{2} + δ_{i y}^{2} v_{y}^{2})]}^{M}), \end{matrix}

C_{p} = \sum_{m = 1}^{M} \frac{{(- 1)}^{m - 1}}{m} (\begin{matrix} M \\ m \end{matrix}),

δ_{i x} > δ_{o x} and δ_{i y} > δ_{o y} .

\begin{matrix} p (v) = p_{o}^{(p)} (v) - p_{i}^{(p)} (v) \\ = \frac{A}{C_{p}} \sum_{m = 1}^{M} {(- 1)}^{m - 1} (\begin{matrix} M \\ m \end{matrix}) {\exp [- \frac{m}{2} (δ_{o x}^{2} v_{x}^{2} + δ_{o y}^{2} v_{y}^{2})] - \exp [- \frac{m}{2} (δ_{i x}^{2} v_{x}^{2} + δ_{i y}^{2} v_{y}^{2})]} . \end{matrix}

\begin{array}{l} μ ({ρ^{'}}_{d}) = \frac{A}{C_{p}} \sum_{m = 1}^{M} {(- 1)}^{m - 1} (\begin{matrix} M \\ m \end{matrix}) \\ \times {\frac{1}{m δ_{o x}^{} δ_{o y}^{}} \exp [- \frac{1}{2 m} (\frac{ρ'_{d x}^{2}}{δ_{o x}^{2}} + \frac{ρ'_{d y}^{2}}{δ_{o y}^{2}})] - \frac{1}{m δ_{i x}^{} δ_{i y}^{}} \exp [- \frac{1}{2 m} (\frac{ρ'_{d x}^{2}}{δ_{i x}^{2}} + \frac{ρ'_{d y}^{2}}{δ_{i y}^{2}})]}, \end{array}

μ (0) = 1, μ (\infty) = 0, and | μ ({ρ^{'}}_{d}) | \leq 1.

\frac{1}{δ_{o x} δ_{o y}} - \frac{1}{δ_{i x} δ_{i y}} = 1.

A = {(\frac{1}{δ_{o x} δ_{o y}} - \frac{1}{δ_{i x} δ_{i y}})}^{- 1} .

p (v) = p_{o}^{(r)} (v) - p_{i}^{(r)} (v),

\begin{matrix} p_{o}^{(r)} (v) = \frac{A}{C_{r}^{2}} \sum_{m = 1}^{M} {(- 1)}^{m - 1} (\begin{matrix} M \\ m \end{matrix}) \exp [- \frac{m δ_{o x}^{2} v_{x}^{2}}{2}] \sum_{m = 1}^{M} {(- 1)}^{m - 1} (\begin{matrix} M \\ m \end{matrix}) \exp [- \frac{m δ_{o y}^{2} v_{y}^{2}}{2}] \\ = \frac{1}{C_{r}^{2}} {1 - {(1 - \exp [- \frac{δ_{o x}^{2} v_{x}^{2}}{2}])}^{M}} {1 - {(1 - \exp [- \frac{δ_{o y}^{2} v_{y}^{2}}{2}])}^{M}}, \end{matrix}

\begin{matrix} p_{i}^{(r)} (v) = \frac{A}{C_{r}^{2}} \sum_{m = 1}^{M} {(- 1)}^{m - 1} (\begin{matrix} M \\ m \end{matrix}) \exp [- \frac{m δ_{i x}^{2} v_{x}^{2}}{2}] \sum_{m = 1}^{M} {(- 1)}^{m - 1} (\begin{matrix} M \\ m \end{matrix}) \exp [- \frac{m δ_{i y}^{2} v_{y}^{2}}{2}] \\ = \frac{1}{C_{r}^{2}} {1 - {(1 - \exp [- \frac{δ_{i x}^{2} v_{x}^{2}}{2}])}^{M}} {1 - {(1 - \exp [- \frac{δ_{i y}^{2} v_{y}^{2}}{2}])}^{M}}, \end{matrix}

C_{r} = \sum_{m = 1}^{M} \frac{{(- 1)}^{m - 1}}{\sqrt{m}} (\begin{matrix} M \\ m \end{matrix}) .

\begin{matrix} μ (ρ'_{d}) = \frac{A}{C_{r}^{2} δ_{o x}^{} δ_{o y}^{}} \sum_{m = 1}^{M} (\begin{matrix} M \\ m \end{matrix}) \frac{{(- 1)}^{m - 1}}{\sqrt{m}} \exp [- \frac{1}{2 m} \frac{ρ'_{d x}^{2}}{δ_{o x}^{2}}] \sum_{m = 1}^{M} (\begin{matrix} M \\ m \end{matrix}) \frac{{(- 1)}^{m - 1}}{\sqrt{m}} \exp [- \frac{1}{2 m} \frac{ρ'_{d y}^{2}}{δ_{o y}^{2}}] \\ - \frac{A}{C_{r}^{2} δ_{i x}^{} δ_{i y}^{}} \sum_{m = 1}^{M} (\begin{matrix} M \\ m \end{matrix}) \frac{{(- 1)}^{m - 1}}{\sqrt{m}} \exp [- \frac{1}{2 m} \frac{ρ'_{d x}^{2}}{δ_{i x}^{2}}] \sum_{m = 1}^{M} (\begin{matrix} M \\ m \end{matrix}) \frac{{(- 1)}^{m - 1}}{\sqrt{m}} \exp [- \frac{1}{2 m} \frac{ρ'_{d y}^{2}}{δ_{i y}^{2}}] . \end{matrix}

μ (ρ'_{d}) = B \sum_{n = 1}^{N} a_{n} μ_{n} (ρ'_{d}), B = {(\sum_{n = 1}^{N} a_{n})}^{- 1},

Abstract

1. Introduction

2. Modeling of the random frames

3. Optical frames with polar symmetry

4. Optical frames with Cartesian symmetry

5. Frame combinations

6. Summary

Acknowledgments

References and links

References

2014 (5)

2013 (3)

2012 (2)

2011 (1)

2009 (1)

1987 (1)

Avramov-Zamurovic, S.

Cai, Y.

de Sande, J. C. G.

Gori, F.

Guattari, G.

Korotkova, O.

Lajunen, H.

Liang, C.

Liu, X.

Malek-Madani, R.

Mei, Z.

Nelson, C.

Padovani, C.

Piquero, G.

Saastamoinen, T.

Sahin, S.

Sanchez, V. R.

Santarsiero, M.

Shchepakina, E.

Shirai, T.

Wang, F.

Yuan, Y.

J. Opt. (1)

J. Opt. A, Pure Appl. Opt. (1)

J. Opt. Soc. Am. A (1)

Opt. Commun. (1)

Opt. Lett. (8)

Waves in Complex and Random Media. (1)

Other (1)

Cited By

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Equations (22)

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