Optical coherence grids and their propagation characteristics

Lipeng Wan; Daomu Zhao

doi:10.1364/OE.26.002168

1. Introduction

In second-order coherence theory, the reciprocity theorem indicates that far-field spectral density depends on the Fourier transform of the spatial correlation of the light distribution across the source, provided that the source is quasi-homogeneous. Whereas early research is limited to the spatial correlation function in a simplex form [1], patterns radiated by these sources thus are confined. However, thanks to the mathematical description of devising genuine special correlation functions which has been demonstrated by Gori and Santarsiero [2], several interesting sources are proposed in theory and experiment in recent years. Examples of these sources include cosine Gaussian Schell-model beam with dark-hollow and four-beamlet array profile [3,4], Laguerre-Gaussian Schell-model with ring-shape profile [5,6], Hermite-Gaussian Schell-model beam with self-splitting properties [7], sinc Schell-model beams [8], nonuniform correlated beams with self-focusing properties [9,10], optical frame source [11], optical coherence lattices [12–14], twisted solitons and beams with a position-dependent phase [15,16], crescent-like beams [17] and circular coherent sources [18]. The propagation characteristics, such as spectral density, spatial coherence and polarization of these sources in free space and random media have been investigated [19–29]. It turns out that these sources possess untrivial characteristics compared to those with traditional correlation functions and thus is expected to open new age in optical manipulation and communication.

On the other hand, light field of spatial periodic structure is central to modern optics because it can provide mass of degrees of control over multiple particles. Normally the periodic structure is lattices-like, which has been widely used in quantum optics and atomic physics as a way to trap and cool ultracold atoms [30,31], and in optical tweezers for controlling micro-particles [32]. However, researchers recently find that in partially coherent domain, engineering the source spatial correlation can lead to the generation of optical lattices in the far field [12,13], which termed optical coherence lattices (OCLs) beams in the literature. Not long after their theoretical introductions, the experimental generation of OCLs is realized by synthesis of multiple partially coherent Schell-model beams [14]. Due to OCLs beams possess tunable structure and shape invariant features, it has attracted considerable interest since it is proposed [23,26].

In this work, we present a model for planar sources, for which the degree of coherence is unconventional, such a source with arbitrary intensity distribution, can naturally evolve into a network of hollow cages on propagation. We refer to this class of sources as optical coherence grids. Further, it is interesting that the optical grids structure remains stable after it propagates over a certain distance and the periodicity of optical grids can be flexibly tuned by changing the correlation parameters of the source field or by performing convolution of degree of coherence. Indeed, both OCLs and OCGs produce multiple Gaussian patterns in the far field, which are perfectly periodic in transverse dimensions. Therefore, the OCGs, in general, can be seen as a better controlled OCLs and may act as the versatile potential field to trap particles.

Organization of the paper is as follows: we initial with introducing optical coherence grids based on mathematical description of a genuine spatial correlation function and reciprocity theorem in section 2; this is followed by evaluating its behavior on free-space propagation in section 3; in section 4, several special cases are shown by changing the source correlation parameters, proving that the structure of optical grids is tunable; due to limitation of OCGs source, in section 5, we introduce POCG by performing convolution of degree of coherence; in section 6, we review the OCGs and POCG sources and discuss their potential applications.

2. Model for the optical coherence grids source

Let us first recall that in the space-frequency domain, second order coherence properties of a stationary partially coherent beam in source plane (z = 0) are characterized by cross-spectral density (CSD) function

W^{(0)} (ρ_{1}, ρ_{2}, ω) = 〈 U^{*} (ρ_{1}, ω) U (ρ_{2}, ω) 〉,

where ρ₁ = (s₁_x,s₁_y)_, ρ₂ = (s₂_x,s₂_y) are two spatial points in source plane, U(ρ) represents the fluctuating field at point ρ. Asterisk is complex conjugate and bracket is ensemble average. Since the angular frequency ω is assumed to be fixed, it would be omitted in this paper. To be a physically genuine CSD, W⁽⁰⁾(ρ₁, ρ₂) has to meet a necessary and sufficient non-negative criteria, this is satisfied while CSD can be written as the integral form [2]

W^{(0)} (ρ_{1}, ρ_{2}) = \iint p (v) H_{0}^{*} (ρ_{1}, v) H_{0} (ρ_{2}, v) d^{2} v,

where H₀(ρ, v) is an arbitrary kernel and p(v) must be a nonnegative function. If H₀(ρ, v) is chosen as a Fourier-like structure

H_{0} (ρ, v) = τ (ρ) \exp (- 2 π i v \cdot ρ),

here τ(ρ) is the profile function. On substituting from Eq. (3) into Eq. (2) and assuring p(v) is a Fourier transformable function, we can at once rewrite the expression of CSD function

W^{(0)} (ρ_{1}, ρ_{2}) = τ^{*} (ρ_{1}) τ (ρ_{2}) \tilde{p} (ρ_{1} - ρ_{2}),

the tilde symbol denotes the Fourier transformation. Recalling that the cross-spectral density of Schell-model source is defined as follows

W^{(0)} (ρ_{1}, ρ_{2}) = \sqrt{S (ρ_{1})} \sqrt{S (ρ_{2})} μ (ρ_{1} - ρ_{2}),

where S(ρ) is the spectral density at source point ρ, μ is the degree of coherence that depends only on difference of two spatial source points. By comparing Eq. (4) and Eq. (5), one can directly notice that source defined by Eq. (4) is so-called Schell-model, as well as that function p and source coherence μ are a Fourier transform pair. Whereas, we already know that in quasi-homogeneous case, the Fourier transform of μ is proportional to far-field spectral density S^∞. Therefore, S^∞ depends on the spatially reversed version of function p, while is independent on spectral density of the source field. Based on this result, it is feasible to generate optical grids in the far field by choosing p(v) described as follows

\begin{array}{l} p (v_{x}, v_{y}) = \frac{δ_{x}}{N M L \sqrt{2}} \exp (- \frac{v_{y}^{2}}{L^{2}}) \sum_{n_{x} = - P}^{P} [\exp (- 2 π^{2} {(δ_{x} v_{x} + n_{x} R_{x})}^{2}) + \exp (- 2 π^{2} (^{δ_{x} v_{x} - n_{x} R_{x}) 2})], \\ + \frac{δ_{y}}{N M L \sqrt{2}} \exp (- \frac{v_{x}^{2}}{L^{2}}) \sum_{n_{y} = - Q}^{Q} [\exp (- 2 π^{2} {(δ_{y} v_{y} + n_{y} R_{y})}^{2}) + \exp (- 2 π^{2} (^{δ_{y} v_{y} - n_{y} R_{y}) 2})] \end{array}

where P = (N-1)/2 and Q = (M-1)/2, δ_x, δ_y, R_x, R_y, N, M and L are correlation parameters in the source field. On substituting from Eq. (6) into Eq. (4) and considering the profile function is in the Gaussian form τ(ρ) = exp(-ρ²/4σ₀²) with σ₀ being the r.m.s source width, we derive the CSD function in the source plane:

\begin{array}{l} W^{(0)} (ρ_{1}, ρ_{2}) = \frac{1}{N M} \exp (- \frac{ρ_{1}^{2} + ρ_{2}^{2}}{4 σ_{0}^{2}}) [\exp (- \frac{{(s_{1 x} - s_{2 x})}^{2}}{2 δ_{x}^{2}}) \exp (- L^{2} π^{2} {(s_{1 y} - s_{2 y})}^{2}) \sum_{n_{x} = - P}^{P} \cos (\frac{2 π n_{x} R_{x} (s_{1 x} - s_{2 x})}{δ_{x}}) . \\ + \exp (- \frac{{(s_{1 y} - s_{2 y})}^{2}}{2 δ_{y}^{2}}) \exp (- L^{2} π^{2} {(s_{1 x} - s_{2 x})}^{2}) \sum_{n_{y} = - Q}^{Q} \cos (\frac{2 π n_{y} R_{y} (s_{1 y} - s_{2 y})}{δ_{y}})] \end{array}

In Eq. (7), provided that N = M = 1, and

L = \sqrt{1 / 2 π^{2} δ_{x}^{2}} = \sqrt{1 / 2 π^{2} δ_{y}^{2}}

, OCGs beam reduces to a traditional Gaussian Schell-model source. Source described by Eq. (7) is termed OCGs beam, whereas two restrictions on parameters should be imposed:

(a) the value of L should be larger than L_min, $L_{\min} = \max {\frac{(N - 1) R_{x}}{δ_{x} 2 \sqrt{\ln 2}}, \frac{(M - 1) R_{y}}{δ_{y} 2 \sqrt{\ln 2}}} .$
This constraint is associated with that the far-field pattern is grids-like, which will be demonstrated in section 3.
(b) the value of parameters should be restricted by beam conditions.
To generates an OCGs beam, the far-field spectral density S^∞(r)≈0 unless the radiant intensity lies within a narrow solid angle around z-axis. The far-field spectral density S^∞(r), characterized by position vector r = ru (u² = 1), is given by following formula [1]:
$S^{\infty} (r) = {(2 π k / r)}^{2} {\tilde{W}}^{(0)} (- k u_{⊥}, k u_{⊥}) \cos^{2} θ,$
here k is the wavelength number, $u_{⊥}$ is the projection of the three-dimensional unit vector u onto the source plane, and θ stands for the angle that u makes with z-axis. The tilde symbol denotes the Fourier transform of W⁽⁰⁾, that is,
$W^{%}^{(0)} (f_{1}, f_{2}) = {(2 π)}^{- 4} \int \int W^{(0)} (ρ_{1}, ρ_{2}) \exp [- i (f_{1} \cdot ρ_{1} + f_{2} \cdot ρ_{2})] d^{2} ρ_{1} d^{2} ρ_{2} .$
On substituting from Eqs. (7) and (10) into Eq. (9), we then obtain the following expression for the far-field spectral density:
$\begin{array}{l} S^{\infty} (r) = \frac{2 σ_{0}^{2} k^{2} \cos^{2} θ}{r^{2}} {\frac{1}{\sqrt{a_{x} b}} \exp (- \frac{2 k^{2} u_{y}^{2}}{b}) \exp (- \frac{k^{2} u_{x}^{2}}{2 a_{x}}) \\ \times \sum_{n_{x} = - P}^{P} \exp (- \frac{c_{n x}^{2}}{2 a_{x}}) \cos h (\frac{c_{n x}^{2} k u_{x}}{a_{x}}) + \frac{1}{\sqrt{a_{y} b}} \exp (- \frac{2 k^{2} u_{x}^{2}}{b}) \\ \times \exp (- \frac{k^{2} u_{y}^{2}}{2 a_{y}}) \sum_{n_{y} = - Q}^{Q} \exp (- \frac{c_{n y}^{2}}{2 a_{y}}) \cos h (\frac{c_{n y}^{2} k u_{y}}{a_{y}})}, \end{array}$
where cosh is the hyperbolic function and
$a_{j} = \frac{1}{4 σ_{0}^{2}} + \frac{1}{δ_{j}^{2}}; b = \frac{1}{σ_{0}^{2}} + 8 π^{2} L^{2}; c_{_{n j}} = \frac{2 π n_{j} R_{j}}{δ_{j}},$
with j = x,y. Since hyperbolic function has its minimum value of 1, and cosθ is approximately equal to 1 for a beam propagating close to z-axis, therefore, to satisfy beam condition, we must have
$k^{2} > > 2 a_{j}; 2 k^{2} > > b .$
Substituting for a_j and b from Eq. (12) and using the relation k equals to 2π/λ with λ being wavelength, we obtain the following necessary and sufficient condition for a OCGs source to generate a beam
$\begin{array}{l} \frac{1}{4 σ_{0}^{2}} + \frac{1}{δ_{j}^{2}} < < \frac{2 π^{2}}{λ^{2}}; \\ \frac{1}{4 σ_{0}^{2}} +2 π^{2} L^{2} < < \frac{2 π^{2}}{λ^{2}} . \end{array}$
shows the modulus of degree of coherence of the source plane for parameters N = M = 2, 3 and 4. The other source parameters are set as R_x = R_y = 2, σ₀ = δ_x = δ_y = 2 mm, and L = L_min. In Figs. 1(a)-1(c), degrees of coherence are evaluated as a function of s_dx = s₁_x-s₂_x and s_dy = s₁_y-s₂_y. For clarity, we also present it along s_dx-axis by setting s_dy = 0, as shown in Figs. 1(d)-1(f). One can see that the pattern of degree of coherence becomes periodic as N and M increases.

Figure 1

Fig. 1 Degree of coherence of OCGs in the source plane for (a) and (d) N = M = 2, (b) and (e) N = M = 3, (c) and (f) N = M = 4, while s_dy is set to zero for (d)-(f).

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3. Propagation characteristics of optical coherence grids

In the paraxial approximation, propagation characteristics of partially coherent light in free space are depicted by CSD function W(r₁,r₂, z) at two positions r₁ = (x₁,y₁) and r₂ = (x₂,y₂) in observing plane, this can be calculated with the help of formula [1]:

\begin{array}{l} W (x_{1}, y_{1}, x_{2}, y_{2}, z) = \iint \iint d s_{1 x} d s_{2 x} d s_{1 y} d s_{2 y} W^{(0)} (ρ_{1}, ρ_{2}) \\ \times \exp {- \frac{i k}{2 z} [{(x_{1} - s_{1 x})}^{2} + {(y_{1} - s_{1 y})}^{2} - {(x_{2} - s_{2 x})}^{2} - {(y_{2} - s_{2 y})}^{2}]} \end{array}

On substituting from Eq. (7) into Eq. (15), then we can derive the following analytic expression

\begin{array}{l} W (r_{1}, r_{2}, z) = \frac{k^{2} σ_{0}^{2}}{4 z^{2} N M \sqrt{g}} \exp [- \frac{i k}{2 z} (r_{1}^{2} - r_{2}^{2})] {\frac{1}{\sqrt{β_{x}}} \exp [- α {(x_{1} - x_{2})}^{2}] \\ \times \exp [- \frac{η_{y}}{2 g}] \times \sum_{n_{x} = - P}^{P} [\exp (\frac{ξ_{x +}^{2}}{β_{x}}) + \exp (\frac{ξ_{x -}^{2}}{β_{x}})] + \frac{1}{\sqrt{β_{y}}} \exp [- α (^{y_{1} - y_{2}) 2}] \\ \times \exp [- \frac{η_{x}}{2 g}] \times \sum_{n_{y} = - Q}^{Q} [\exp (\frac{ξ_{y +}^{2}}{β_{y}}) + \exp (\frac{ξ_{y -}^{2}}{β_{y}})]}, \end{array}

where

\begin{array}{l} α = \frac{k^{2} σ_{0}^{2}}{2 z^{2}}, β_{j} = \frac{1}{8 σ_{0}^{2}} + \frac{1}{2 δ_{j}} + α, \\ g = \frac{1}{8 σ_{0}^{2}} + π^{2} L^{2} + α, \\ η_{j} = \frac{k^{2}}{4 σ_{0}^{2} z^{2}} (j_{1}^{2} + j_{2}^{2}) + \frac{π^{2} L^{2} k^{2}}{z^{2}} {(j_{1} - j_{2})}^{2} - \frac{i k^{3}}{2 z^{3}} (j_{1}^{2} - j_{2}^{2}), \\ ξ_{j \pm} = (\frac{3 k^{2} σ_{0}^{2}}{4 z^{2}} - \frac{α}{2}) (j_{1} - j_{2}) + \frac{i k}{4 z} (j_{1} + j_{2}) \pm \frac{i π n_{j} R_{j}}{δ_{j}}; j = x, y, \end{array}

Equations (16) and (17) give the CSD function at a certain propagation distance z. Thus, the spectral density at a point r = (x,y) in the observing plane can be calculated by the expression S(x,y,z) = W(r,r,z).

Our main result is displayed in Fig. 2, showing the transverse spectral density of OCGs beam on propagation in free space. Being quite different from the Gaussian Schell-model beam that has an unchanged Gaussian beam profile as it propagates, the initial Gaussian pattern of OCGs beam gradually evolves into grids-like field on propagation, as well as that rotational symmetry of source field transform into rectangle symmetry. Further, such optical-grids structure remains stable after it propagates over a certain distance, while the size of pattern expands due to free-space diffraction. The feature of optical grids can be clearly seen in Fig. 2(g), that light distributes in multiple crossed lines and thereby depicts a (N-1) × (M-1) array of optical cages. Therefore, such structure may be useful for trapping particles.

Fig. 2 Spectral density of OCGs propagating in free space. (a) z = 0, (b) z = 10m, (c) z = 20m, (d) z = 30m, (e) z = 100m and (f) z = 1000m. Parameters are set as N = M = 6, R_x = R_y = 3, δ_x = δ_y = 3 mm, σ₀ = 6 mm, L = 6 mm⁻¹ and λ is fixed at 632.8 nm henceforth. (g) Stereograms of optical grids that corresponding to (e).

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Evolution of coherence can be calculated by the following formula

μ (r_{1}, r_{2}, z) = \frac{W (r_{1}, r_{2}, z)}{\sqrt{W (r_{1}, r_{1}, z)} \sqrt{W (r_{2}, r_{2}, z)}} .

In Fig. 3(a) we illustrate the behavior of the degree of coherence μ of OCGs beam for propagation in free space with increasing distance from the source plane. The values of the parameters are chosen the same as in Fig. 2 except for N = M = 3. For clarity, we only consider the degree of coherence along x_d-axis and set y₁ = y₂, the behavior along y_d-axis is in the same form according to symmetry. The figure shows that for propagation in free space, the degree of coherence transforms from an oscillation distribution to a Gaussian pattern. Figure 3(b) shows the influence of the parameter L on the degree of coherence of the source field. We set L = 0.1,1,100 mm⁻¹, whereas other parameters are unchanged. It turns out that the periodic pattern degenerates for a small value of L, yet a large value of L would hardly affect the pattern of degree of coherence.

Fig. 3 Degree of coherence of OCGs beams for (a) different propagation distance z and (b) different parameter L in the source plane.

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Since the source degree of coherence is affected by the parameter L. We then discuss its influence on the far-field spectral density with the same values of parameters as used in Fig. 2. The spectral density distributions in transverse plane z = 100 m for various value of L are shown in Fig. 4. An interesting phenomenon can be noticed that, a small value of L gives rise to a dozen of optical lattices, however, as the value of L increases, optical lattices gradually stretch out, and eventually evolve into grid lines. Therefore, to generate optical coherence grids rather than optical lattices, the parameter L should be large enough. This is why the restriction, depicted by Eq. (8), must be imposed on L.

Fig. 4 Spectral density distribution of OCGs beam in transverse plane z = 100 m for different values of L. (a) L = 0.1, (b) L = 0.5, (c) L = 1, (d) L = 2, (e) L = 5 and (f) L = 20 with units being mm⁻¹. Sizes of pictures (a)-(f) are identical and thus omitted.

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The beam propagation factor, or so-called M² factor, is an important indicator that evaluates the quality of light beam. In partially coherent domain, the definition of M² factor is introduced by Gori et al [7,33], which is in following form along x and y directions

M_{x}^{2} = 4 π σ_{x 0} σ_{x \infty}; M_{y}^{2} = 4 π σ_{y 0} σ_{y \infty},

where

σ_{j 0}

and

σ_{j \infty}

with j = x,y are the r.m.s widths of the transverse intensity profiles at the source plane and in the far field, viz.,

\begin{array}{l} σ_{x 0}^{2} = \frac{1}{I_{t o t a l}} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} s_{x}^{2} W^{(0)} (s_{x}, s_{y}, s_{x}, s_{y}) d s_{x} d s_{y}; \\ σ_{y 0}^{2} = \frac{1}{I_{t o t a l}} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} s_{y}^{2} W^{(0)} (s_{x}, s_{y}, s_{x}, s_{y}) d s_{x} d s_{y}, \end{array}

\begin{array}{l} σ_{x \infty}^{2} = \frac{1}{4 π^{2} I_{t o t a l}} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} [{\frac{\partial W^{(0)} (s_{1 x}, s_{1 y}, s_{2 x}, s_{2 y})}{\partial s_{1 x} \partial s_{2 x}} |}_{\begin{array}{l} s_{1 x} = s_{2 x} = s_{x} \\ s_{1 y} = s_{2 y} = s_{y} \end{array}}] d s_{x} d s_{y}; \\ σ_{y \infty}^{2} = \frac{1}{4 π^{2} I_{t o t a l}} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} [{\frac{\partial W^{(0)} (s_{1 x}, s_{1 y}, s_{2 x}, s_{2 y})}{\partial s_{1 y} \partial s_{2 y}} |}_{\begin{array}{l} s_{1 x} = s_{2 x} = s_{x} \\ s_{1 y} = s_{2 y} = s_{y} \end{array}}] d s_{x} d s_{y}, \end{array}

where I_total represent the total power of the beam:

I_{t o t a l} = \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} W^{(0)} (s_{x}, s_{y}, s_{x}, s_{y}) d s_{x} d s_{y} .

Then, on substituting from Eq. (7) for source CSD function into Eqs. (20)-(23), we arrive at M ² factor of OCGs beam along x and y directions

\begin{array}{l} M_{x}^{2} = \sqrt{\frac{2}{N + M} {\frac{2 σ_{0}^{2}}{δ_{x}^{2}} \sum_{n_{x} = - P}^{P} [1 + 2 π^{2} (δ_{x}^{2} L^{2} + 2 n_{x}^{2} R_{x}^{2})] + 1}}; \\ M_{y}^{2} = \sqrt{\frac{2}{N + M} {\frac{2 σ_{0}^{2}}{δ_{y}^{2}} \sum_{n_{y} = - Q}^{Q} [1 + 2 π^{2} (δ_{y}^{2} L^{2} + 2 n_{y}^{2} R_{y}^{2})] + 1}} . \end{array}

Provided that N = M = 1, and $L = \sqrt{1 / 2 π^{2} δ_{x}^{2}} = \sqrt{1 / 2 π^{2} δ_{y}^{2}}$ , Eq. (24) reduces to the M ² factor of Gaussian Schell-model source. Curves of M_x² as a function of N for δ_x = 1 mm, 2 mm and 3 mm are shown in Fig. 5. Parameters are set as L = L_min, N = M, R_x = 1 and σ₀ = 0.5 mm. One finds that M ² factor increases as the number of gridding lines N increases or as δ_x decreases.

Fig. 5 Propagation factor M_x² of OCGs beams versus the number of periodicity N for different values of δ_x.

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4. Diversity of optical coherence grids

Optical grids produced above are squares, however, the structure of optical grids, such as length-width ratio, grid-line’s thickness, and periodicity can be flexibly modulated by changing the correlation parameters of the source. For example, the shape of cage can be rectangle rather than square by setting parameters N = 6, M = 12, R_x = 6, R_y = 3, δ_x = δ_y = 3 mm, σ₀ = 6 mm, and L = 8 mm⁻¹, in which the length-width ratio of a cage is 2. The radiant intensity distribution in the far field for this situation is exhibited in Fig. 6(a). Moreover, the periodic structure can degenerate into a perfect single cage by setting N = M = 2, R_x = R_y = 3, δ_x = δ_y = 3 mm, σ₀ = 6 mm, L = L_min, as shown in Fig. 6(b).

Fig. 6 (a) Spectral density distribution of rectangle optical grids at z = 60 m. (b) Spectral density distribution of single cage at z = 60 m.

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We note one extreme case that δ_y→0. Figure 7 shows corresponding spectral density distribution at different propagation distances for parameters N = M = 10, R_x = R_y = 1, δ_x = 1 mm, σ₀ = 1 mm, L = L_min. One finds that on free-space propagation, a Gaussian pattern naturally evolves into a series of alternating bright and dark bands. Notice that the series of intensity bands is like the interfering fringe of Young’s double-slit experiment, such interesting source thus can be referred to as Young’s correlated source.

Fig. 7 Spectral density distribution in several transverse planes when δ_y→0. (a) z = 0, (b) z = 1 m, (c) z = 3 m, (d) z = 5 m, (e) z = 15 m and (e) z = 30 m.

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5. Perfect optical coherence grids

In this section, we demonstrate a perfect optical coherence grid (POCG) source. Compared to the OCGs source, the grids pattern of POCG is in a fully controllable fashion, besides that, the POCG beam is insensitive to the value of parameter L. This is achieved by introducing convolution operation of degree of coherence [34]. The source degree of coherence of POCG is defined as the convolution of two legitimate degrees of coherence, i.e., the sinc-like degree of coherence and source degree of coherence of OCGs:

\begin{array}{l} μ_{_{POCG}}^{^{(0)}} (ρ_{1} - ρ_{2}) = {\frac{1}{N + M} \exp (- \frac{{(s_{1 x} - s_{2 x})}^{2}}{2 δ_{x}^{2}}) \exp (- L^{2} π^{2} {(s_{1 y} - s_{2 y})}^{2}) \sum_{n_{x} = - P}^{P} \cos (\frac{2 π n_{x} R_{x} (s_{1 x} - s_{2 x})}{δ_{x}})} \otimes A_{y} si nc (A_{y} (s_{1 y} - s_{2 y})), \\ + {\frac{1}{N + M} \exp (- \frac{{(s_{1 y} - s_{2 y})}^{2}}{2 δ_{y}^{2}}) \exp (- L^{2} π^{2} {(s_{1 x} - s_{2 x})}^{2}) \sum_{n_{y} = - Q}^{Q} \cos (\frac{2 π n_{y} R_{y} (s_{1 y} - s_{2 y})}{δ_{y}})} \otimes A_{x} si nc (A_{x} (s_{1 x} - s_{2 x})) \end{array}

where A_x = (N-1)R_x/(2δ_x) and A_y = (M-1)R_y/(2δ_y), ⊗ denotes convolution operation. Figure 8 shows the transverse coherence of POCG sources. For comparison, the coherence of OCGs sources is presented in the right side of the figure. Parameters are set as N = M = 3, R_x = R_y = 3, δ_x = δ_y = 0.6 mm, σ₀ = 6 mm while with L = 50 mm⁻¹ for (a)-(b) and L = 1 mm⁻¹ for (c)-(d).

Fig. 8 Degree of coherence in the source plane. (a) POCG and (b) OCGs for parameter L = 50 mm⁻¹; (c) POCG and (d) OCGs for parameter L = 1 mm⁻¹.

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Since the main structure of far-field intensity distribution S^∞ can be given by Eqs. (9) and (10). By using convolution theorem that the Fourier transform of a convolution of two functions is equal to the product of the Fourier transforms of each function, far-field intensity distribution can be simply derived

\begin{array}{l} S_{_{POCG}}^{^{\infty}} (r) = \frac{2 σ_{0}^{2} k^{2} \cos^{2} θ}{r^{2}} {\frac{1}{\sqrt{a_{x} b}} \exp (- \frac{2 k^{2} u_{y}^{2}}{b}) \exp (- \frac{k^{2} u_{x}^{2}}{2 a_{x}}) rect (\frac{k u_{y}}{A_{y}}) \sum_{n_{x} = - P}^{P} \exp (- \frac{c_{n x}^{2}}{2 a_{x}}) \cos h (\frac{c_{n x}^{2} k u_{x}}{a_{x}}) \\ + \frac{1}{\sqrt{a_{y} b}} \exp (- \frac{2 k^{2} u_{x}^{2}}{b}) \exp (- \frac{k^{2} u_{y}^{2}}{2 a_{x}}) rect (\frac{k u_{x}}{A_{y}}) \sum_{n_{y} = - Q}^{Q} \exp (- \frac{c_{n y}^{2}}{2 a_{y}}) \cos h (\frac{c_{n y}^{2} k u_{y}}{a_{y}})}, \end{array}

where a_j_, b, c_nj (j = x,y) are defined as Eq. (12). Figure 9 shows the structure of the far-field spectral density distribution for various values of N and M with the other parameters are chosen the same as in Fig. 8(a). One sees that the generated optical grids are rather satisfactory, since the outstretched components which surround optical grids have been vanished. In practical, such convolutions of the degrees of coherence can be achieved by use of the spatial light modulators (SLM).

Fig. 9 Radiant spectral density of POCG in far field for (a) N = M = 8 and (b) N = 6, M = 12.

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6. Conclusions

In this paper, we have modeled random light sources for generation of optical grids with stable and tunable characteristics. The concept of optical grids, as an extension of the optical lattices, may be useful for trapping, sorting and assembling particles due to their periodic potential field. The propagation properties of it in free space, including spectral density, degree of coherence and M ² factor are investigated in detail. Our results show that grids-like pattern is produced in the far field, and such structure remains stable on free-space propagation, implies that it is expected to have applications in optical communication. Furthermore, various types of OCGs beams with interesting structure, for example, rectangle OCGs, single OCGs and fringe pattern are achieved by active control the correlation parameters of the source field. To make up for the shortcomings of OCGs source and ensure that OCGs is in a fully controllable fashion, we proposed POCG sources based on convolution of degree of coherence. Finally, we discuss the possible experimental methods aimed at producing OCGs sources. One is that OCGs may be achieved by imposing a random phase mask with prescribed correlation on SLM [10,35]. And the synthesis of multiple partially coherent Schell-model beams, which has been used for the generation of OCLs [14], may serve as another alternative scheme.

Funding

National Natural Science Foundation of China (NSFC) (11474253 and 11274273); Fundamental Research Funds for the Central Universities (2017FZA3005).

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Optical coherence grids and their propagation characteristics

Abstract

1. Introduction

2. Model for the optical coherence grids source

3. Propagation characteristics of optical coherence grids

4. Diversity of optical coherence grids

5. Perfect optical coherence grids

6. Conclusions

Funding

References and links

Cited By

Figures (9)

Equations (25)

Optics Express