Source coherence-based far-field intensity filtering

Zhangrong Mei; Olga Korotkova; Yonghua Mao

doi:10.1364/OE.23.024748

1. Introduction

Modulation and control of the spatial profiles of laser beams is an important topic of current interest because of the attractive possibility of affordable improvement of optical systems. It is well known in statistical optics that the spatial coherence state of an optical field in the source plane is closely related to the intensity distribution of its far field [1]. Therefore, it is possible to generate desired far-field beam patterns by customizing the spatial correlation properties of the source field [2]. For example, it was shown that the spatial coherence effects can produce a local minimum of intensity at the geometrical focus on passing through a high-Fresnel-number focusing system [3], the principles of amplitude and phase modulation of the spatial coherence of the optical field can be implemented for designing specific shapes of power distributions [4], the interference of radiation produced by a pair of mutually correlated Schell-model sources can lead to a beam in the far field with an intensity distribution that displays a narrow bright or dark dot of a small radius at its center [5], the spatial coherence lattices can generate lattice-like radiation patterns composed of highly directional individual lobes [6], etc.

Recently, a general representation for partially coherent sources ensuring validity of new forms of spatial correlation functions was introduced by Gori and Santarsiero [7]. It provided new impetus for achieving prescribed far-field intensity distributions by modeling new classes of partially coherent sources, such as the non-uniformly correlated sources leading to self-focus and laterally shifted intensity maxima [8], the multi-Gaussian Schell-model and sinc-Schell model sources generating far fields with tunable flat profiles [9], the cosine-Gaussian Schell-model sources generating a dark-hollow profile [10,11], random sources producing far fields with frame-like intensity profiles [12]. Based on these models, some research was carried out on analyzing the experimental generation and the measurement of the statistical properties of the light fields in free space and various media [13–20]. Moreover, some legitimate operations for correlation functions, including differences, products and powers, have also been shown to lead to novel partially coherent sources and fields radiated by them [21–23]. We have also recently extended the calculus of the source correlation functions to alternation series operation of N correlation functions and by doing so we introduced a novel class of multi-sinc Schell-model (MSSM) source [24]. It was demonstrated that far fields produced by the MSSM sources carry interesting characteristics, being adjustable multi-rings and lattice patterns. We have also confirmed that the convolution of two degrees of coherence represents the novel legitimate degree of coherence for Schell-like sources [25].

The purpose of this paper, is to derive the inherent relationships between far-field beam pattern properties and the convolution of any two legitimate degrees of coherence in the source plane based on the weighted superposition rule. For illustration of these relationships, we show how to construct new classes of partially coherent sources by the convolution operation of the Gaussian Schell-model correlation function and the multi-sinc Schell-model correlation function, respectively in the polar and the Cartesian symmetries, which lead to useful far-field intensity patterns.

2. Theory

Consider a fluctuating planar source, located in the plane $z = 0$ and characterized by a wide-sense stationary statistical ensemble. The second-order correlation properties of light emerging from two source points specified by position vectors ${ρ^{'}}_{1} = ({x^{'}}_{1}, {y^{'}}_{1})$ and ${ρ^{'}}_{2} = ({x^{'}}_{2}, {y^{'}}_{2})$ can be characterized by the CSD function $W^{(0)} ({ρ^{'}}_{1}, {ρ^{'}}_{2})$ , where the explicit dependence on the temporal frequency is omitted. For a Schell-Model (SM) source, $W^{(0)} ({ρ^{'}}_{1}, {ρ^{'}}_{2})$ can be expressed in the form [1]

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

where

S^{(0)} (ρ^{'})

is the spectral density at position

ρ^{'}

and

μ

is the spectral degree of coherence of the source. The CSD function has to be a nonnegative definite [1], which is fulfilled if the function can be written in form [7]

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

where

H_{0}

is an arbitrary kernel and

p (v)

is a nonnegative function. For SM sources,

H_{0}

has the Fourier-like structure

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

where

τ (ρ^{'})

is a possible complex profile function. Then the SM CSD takes on the form

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

where the tilde symbol denotes the two-dimensional Fourier transform. Comparing Eq. (1) and (4), we can see that

μ ({ρ^{'}}_{1} - {ρ^{'}}_{2}) = \tilde{p} ({ρ^{'}}_{1} - {ρ^{'}}_{2}) .

Let us now suppose that a new degree of coherence $μ$ is a legitimate convolution of two degrees of coherence, $μ_{1}$ and $μ_{2}$ , i.e [25],

μ ({ρ^{'}}_{1} - {ρ^{'}}_{2}) = A μ_{1} ({ρ^{'}}_{1} - {ρ^{'}}_{2}) \otimes μ_{2} ({ρ^{'}}_{1} - {ρ^{'}}_{2}_{2}),

where the symbol

\otimes

stands for the convolution operation,

A

is a normalization factor. Since the Fourier transform of the convolution of two integrable functions is given by the product of their Fourier transforms, then kernel

p (v)

corresponding to the new degree of coherence

μ

can be expressed as the product of two nonnegative functions,

p_{1} (v)

and

p_{2} (v)

, i.e.,

p (v) = p_{1} (v) p_{2} (v),

where

p_{1} (v)

and

p_{2} (v)

are the Fourier transforms of

μ_{1}

and

μ_{2}

, respectively.

The spectral density of the partially coherent field propagating in the half-space $z > 0$ can be constructed at any plane based on the weighted superposition as [8]

S (ρ, z) = \int p (v) {| H (ρ, z, v) |}^{2} d^{2} v,

where

\begin{array}{l} {| H (ρ, z, v) |}^{2} = \frac{k^{2}}{4 π^{2} z^{2}} \iint H_{0}^{*} ({ρ^{'}}_{1}, v) H_{0} ({ρ^{'}}_{2}, v) \\ \times \exp [- i k \frac{{(ρ - {ρ^{'}}_{1})}^{2} - {(ρ - {ρ^{'}}_{2})}^{2}}{2 z}] d^{2} {ρ^{'}}_{1} d^{2} {ρ^{'}}_{2} . \end{array}

On substituting from Eq. (7) into Eq. (8), we find that

S (ρ, z) = \int p_{1} (v) p_{2} (v) {| H (ρ, z, v) |}^{2} d^{2} v .

Equation (10) can be regarded as the spectral density of a field corresponding to $p_{2} (v)$ is modulated by $p_{1} (v)$ . For instance, $p_{2} (v)$ can be of the desired class and $p_{1} (v)$ is a soft-adged aperture. The technique can be applied in intensity modulation, laser beam shaping and other fields.

3. Examples: Gaussian modulated multi-sinc Schell-model sources and the corresponding far-field patterns

3.1 The polar symmetry case

Let us now demonstrate application of Eq. (10) with an example, relating to fields with polar symmetry. We employ the Gaussian Schell-model correlation [1] and multi-sinc Schell-model correlation functions [24] for $μ_{1}$ and $μ_{2}$ , i.e.,

μ_{1} ({ρ^{'}}_{1} - {ρ^{'}}_{2}) = μ_{g} ({ρ^{'}}_{1} - {ρ^{'}}_{2}) = \exp [- \frac{{({ρ^{'}}_{1} - {ρ^{'}}_{2})}^{2}}{2 δ_{g}^{2}}],

μ_{2} ({ρ^{'}}_{1} - {ρ^{'}}_{2}) = μ_{s} ({ρ^{'}}_{1} - {ρ^{'}}_{2}) = \frac{1}{B} \sum_{n = 1}^{N} \frac{{(- 1)}^{n - 1}}{C_{n}} \sin c (\frac{{ρ^{'}}_{1} - {ρ^{'}}_{2}}{C_{n} δ_{s}}),

with normalization factor

B = \sum_{n = 1}^{N} {(- 1)}^{n} / C_{n}

, where

C_{n} = \sqrt[m]{(2 N - 1) / [2^{m} (2 N - 2 n + 1)]}

,

m

being an arbitrary positive real number.

Then, the new degree of coherence can be written as

\begin{array}{l} μ_{c} ({ρ^{'}}_{1} - ρ^{'}) = A μ_{g} ({ρ^{'}}_{1} - ρ^{'}) \otimes μ_{s} ({ρ^{'}}_{1} - {ρ^{'}}_{2}) \\ = A \exp [- \frac{{({ρ^{'}}_{1} - ρ^{'})}^{2}}{2 δ_{g}^{2}}] \otimes \frac{1}{B} \sum_{n = 1}^{N} \frac{{(- 1)}^{n - 1}}{C_{n}} \sin c (\frac{{ρ^{'}}_{1} - {ρ^{'}}_{2}}{C_{n} δ_{s}}) . \end{array}

Figures 1 and 2 illustrate the degree of coherence

μ_{g}

,

μ_{s}

and their convolution

μ_{c}

for

δ_{g} = 0.04 mm

,

δ_{s} = 0.1 m m

and several sets of values of

N

and

m

. It is clearly seen from the figures that the convolution of Gaussian and Multi-sinc degrees of coherence generate new correlation features of the source field, while the degree of coherence

μ_{c}

changes with variation of the values of

N

and

m

.

Fig. 1 The degree of coherence $μ_{c}$ derived from the convolution of $μ_{g}$ with $δ_{g} = 0.04 mm$ and $μ_{s}$ with $δ_{s} = 0.1 mm$ , $m = 1$ , $N = 10$ (upper row) and $N = 11$ (lower row).

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Fig. 2 The same as Fig. 1 but for .

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The modulation to the degree of coherence in the source field can realize the modulation to intensity distribution in the far field. Indeed, evaluating the corresponding function $p_{c} (v)$ of the degree of coherence $μ_{c}$ we get

\begin{array}{l} p_{c} (v) = p_{g} (v) p_{s} (v) \\ = \sqrt{2 π δ_{g}^{2}} \exp (- 2 π^{2} δ_{g}^{2} v^{2}) \frac{δ_{s}}{B} \sum_{n = 1}^{N} {(- 1)}^{n - 1} r e c t (C_{n} δ_{s} v) . \end{array}

Let us now set the Gaussian profile with the r.m.s. source width $σ$ for the amplitude function of Eq. (3):

τ (ρ^{'}) = \exp [- {ρ^{'}}^{2} / (2 σ^{2})],

On substituting from Eq. (15) into (3) and then into Eq. (9) we arrive at

{| H (ρ, z, v) |}^{2} = \frac{σ^{2}}{w^{2} (z)} \exp [- {(ρ + 2 π z v / k)}^{2} / w^{2} (z)],

w^{2} (z) = σ^{2} + z^{2} / (k^{2} σ^{2}) .

Further, on substituting from Eqs. (14), (16) and (17) into Eq. (10), the far-field spectral density can be evaluated numerically. Figures 3 and 4 illustrate the spectral densities generated by the sources with the same Gaussian intensity distribution defined as in Eq. (14) with $σ = 1 mm$ and different degrees of coherence $μ_{g}$ , $μ_{s}$ and $μ_{c}$ shown in Figs. 1 and 2. One clearly sees that the far-field intensities of the new beams have the multi-sinc Schell-model transverse distribution with a Gaussian envelope. It is equivalent to the multi-sinc Schell-model beams passed through a soft-edge Gaussian filter in the far field.

Fig. 3 Far field spectral densities generated by the sources with degrees of coherence $μ_{g}$ , $μ_{s}$ and $μ_{c}$ shown in Fig. 1.

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Fig. 4 Far field spectral densities generated by the sources with degrees of coherence $μ_{g}$ , $μ_{s}$ and $μ_{c}$ shown in Fig. 2.

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Figure 5 indicates how by choosing different $μ_{g}$ the desirable structural features of the multi-sinc Schell-model beams pass through the Gaussian filter, while the undesirable features are blocked. This implies that the convolution of Gaussian and multi-sinc Schell-model degrees of coherence represents a novel legitimate degree of coherence and can be used for realizing the Gaussian modulation of the multi-sinc Schell-model beams.

Fig. 5 Far field spectral density $S_{c}$ for the circular symmetry MSSM beams with different $N$ (upper row $N = 10$ and low row $N = 11$ ) are filtered by Gaussian beams with different coherence length $μ_{g}$ . (a) and (d) $μ_{g} = 0.02 mm$ ; (b) and (e) $μ_{g} = 0.06 mm$ ; (c) and (f) $μ_{g} = 0.15 mm$ .

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3.2 The Cartesian symmetry case

Let us now turn to the case of the Cartesian symmetry, by setting the CSD of a beam field ensemble at a pair of points ${ρ^{'}}_{1} = ({x^{'}}_{1}, {y^{'}}_{1})$ and ${ρ^{'}}_{2} = ({x^{'}}_{2}, {y^{'}}_{2})$ in the source plane in a factorized form

\begin{array}{l} W^{(0)} ({ρ^{'}}_{1}, {ρ^{'}}_{2}) = \prod_{t = x^{'}, y^{'}} W (t_{1}, t_{2}) \\ = \prod_{t = x^{'}, y^{'}} \int p (v_{t}) H_{0}^{*} (t_{1}, v_{t}) H_{0} (t_{2}, v_{t}) d v_{t} . \end{array}

For the Schell-model correlations, Eq. (18) takes on the general form [17]

W^{(0)} ({ρ^{'}}_{1}, {ρ^{'}}_{2}) = τ^{*} ({ρ^{'}}_{1}) τ ({ρ^{'}}_{2}) \prod_{t = x^{'}, y^{'}} {\tilde{p}}_{t} ({t^{'}}_{1} - {t^{'}}_{2}) .

Here, the tilde denotes one-dimensional Fourier transform. Comparing Eq. (1) and (19), we can see that the degree of coherence of the source becomes

μ ({ρ^{'}}_{1} - {ρ^{'}}_{2}) = \prod_{t = x^{'}, y^{'}} {\tilde{p}}_{t} ({t^{'}}_{1} - {t^{'}}_{2}) .

So, if the degree of coherence of each direction in the Cartesian coordinate system is a legitimate convolution of two degrees of coherence

μ_{1} ({t^{'}}_{1} - {t^{'}}_{2})

and

μ_{2} ({t^{'}}_{1} - {t^{'}}_{2})

, i.e.,

μ ({ρ^{'}}_{1} - {ρ^{'}}_{2}) = A \prod_{t = x^{'}, y^{'}} μ_{1} ({t^{'}}_{1} - {t^{'}}_{2}) \otimes μ_{2} ({t^{'}}_{1} - {t^{'}}_{2}),

then, the corresponding function

p (v)

can be expressed as

p (v) = \prod_{t = x^{'}, y^{'}} p (v_{t}) = \prod_{t = x^{'}, y^{'}} p_{1} (v_{t}) p_{2} (v_{t}) .

where

p_{1} (v_{t})

and

p_{2} (v_{t})

are the inverse Fourier transform of function

μ_{1}

and

μ_{2}

, respectively.

Since the CSD of source (18) is separable in the Cartesian coordinate system, the paraxial free-space propagation law for the spectral density of a field to point $(ρ, z)$ of the half-space $z > 0$ can also be expressed as the product of two 1D integral representations, i.e.,

\begin{array}{l} S (ρ, z) = \prod_{t = x, y} \int p (v_{t}) {| H_{t} (t, z, v_{t}) |}^{2} d v_{t} \\ = \prod_{t = x, y} \int p_{1} (v_{t}) p_{2} (v_{t}) {| H_{t} (t, z, v_{t}) |}^{2} d v_{t}, \end{array}

Let us also set the Gaussian profile for the amplitude function of source field. We then can easily arrive at

{| H_{t} (t, z, v_{t}) |}^{2} = \frac{σ}{w (z)} \exp [- {(t + 2 π z v_{t} / k)}^{2} / w^{2} (z)] .

The use of the Gaussian Schell-model correlation function and multi-sinc Schell-model correlation function for $μ_{1}$ and $μ_{2}$ in convolution leads to expression

\begin{array}{l} μ_{c} ({ρ^{'}}_{1} - {ρ^{'}}_{2}) = A \prod_{t = x^{'}, y^{'}} μ_{g t} ({t^{'}}_{1} - {t^{'}}_{2}) \otimes μ_{s t} ({t^{'}}_{1} - {t^{'}}_{2}) \\ = A \prod_{t = x^{'}, y^{'}} \exp [- \frac{{({t^{'}}_{1} - {t^{'}}_{2})}^{2}}{2 δ_{g t}^{2}}] \otimes \frac{1}{B} \sum_{n = 1}^{N} \frac{{(- 1)}^{n - 1}}{C_{n}} \sin c (\frac{{t^{'}}_{1} - {t^{'}}_{2}}{C_{n} δ_{s t}}) . \end{array}

Evaluating its Fourier transform $p_{c} (v)$ we get

\begin{array}{l} p_{c} (v) = \prod_{t = x^{'}, y^{'}} p_{1} (v_{t}) p_{2} (v_{t}) \\ = \prod_{t = x^{'}, y^{'}} \sqrt{2 π δ_{g t}^{2}} \exp (- 2 π^{2} δ_{g t}^{2} v_{t}^{2}) \frac{δ_{s t}}{B} \sum_{n = 1}^{N} {(- 1)}^{n - 1} r e c t (C_{n} δ_{s t} v_{t}) . \end{array}

To illustrate the typical behavior of the spectral degree of coherence, we display in Fig. 6, $μ_{c}$ derived from the convolution of $μ_{g}$ with $δ_{g x} = δ_{g y} = 0.04 mm$ and $μ_{s}$ with $δ_{s x} = δ_{s y} = 0.1 mm$ and different parameters $N$ and $m$ . The convolution of two degrees of coherence generates new coherent state which in its turn forms a new far-field intensity distribution.

Fig. 6 The degree of coherence $μ_{c}$ derived from the convolution of Gaussian degree of coherence $μ_{g}$ with $δ_{g x} = δ_{g y} = 0.04 mm$ and Cartesian symmetry MSSM degree of coherence $μ_{s}$ with $δ_{s x} = δ_{s y} = 0.1 mm$ , (a) $N = 10$ , $m = 1$ ; (b) $N = 11$ , $m = 1$ ; (c) $N = 10$ , $m = 1.5$ ; (d) $N = 11$ , $m = 1.5$ .

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Figure 7 displays the far-field spectral density distributions $S_{c}$ generated by the new sources with degrees of coherence $μ_{c}$ shown in Fig. 6. We can see from Fig. 7 that the far-field spectral density of rectangular multi-sinc Schell-model beams are filtered by a soft-edge Gaussian aperture and shape the rectangular lattice patterns with Gaussian envelope. Figure 8 illustrate the far-field spectral density for the Cartesian symmetry MSSM beams with $m = 1.5$ and different $N$ that are filtered by Gaussian beams with different coherence states. Since the coherence length of the source field determines the width of the far-field radiation pattern, one can adjust the width of soft-edge Gaussian filter by changing the degree of coherence $μ_{g}$ .

Fig. 7 Far-field spectral density $S_{c}$ generated by the new sources with degrees of coherence $μ_{c}$ shown in Fig. 6.

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Fig. 8 Far field spectral density $S_{c}$ for the Cartesian symmetry MSSM beams with $m = 1.5$ and different $N$ (upper row $N = 10$ and low row $N = 11$ ) are filtered by Gaussian beams with different coherence length $μ_{g}$ , (a) and (d) $δ_{g x} = δ_{g y} = 0.02 mm$ ; (b) and (e) $δ_{g x} = 0.02 mm$ , $δ_{g y} = 0.06 mm$ ; (c) and (f) $δ_{g x} = δ_{g y} = 0.15 mm$ .

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4. Summary

We have examined the intrinsic relationships between far-field beam intensity pattern properties and the convolution of any two legitimate degrees of coherence in the source plane. Two examples for partially coherent sources formed with the help of convolution operation of the Gaussian Schell-model correlation function and the multi-sinc Schell-model correlation function in the polar and Cartesian symmetries cases have been provided, respectively, and their far-field intensity features produced by the new sources have been explored by using the established relationships. The results showed that the far-field intensity patterns of the new sources have multi-sinc Schell-model transverse distribution and the Gaussian envelope. The result resembles the multi-sinc Schell-model beams through a soft-edge Gaussian filter. The width of the Gaussian filter can be adjusted by changing the degree of coherence of the source field, thereby it can yield structurally different intensity patterns. The results establish a significant tool for modeling new classes of partially coherent sources and implementing far-field intensity-like filter by convolution operation of the source field correlation functions. The capability of the proposed filter-like process employed in the source plane is of importance in applications where far fields cannot be reached directly.

Acknowledgments

Z. Mei’s research is supported by the National Natural Science Foundation of China (NSFC) (11247004). O. Korotkova’s research is supported by US AFOSR (FA9550-12-1-0449).

References and links

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

2. E. Collett and E. Wolf, “New equivalence theorems for planar sources that generate the same distributions of radiant intensity,” J. Opt. Soc. Am. 69(7), 942–950 (1979). [CrossRef]

3. G. Gbur and T. D. Visser, “Can spatial coherence effects produce a local minimum of intensity at focus?” Opt. Lett. 28(18), 1627–1629 (2003). [CrossRef] [PubMed]

4. R. Betancur and R. Castañeda, “Spatial coherence modulation,” J. Opt. Soc. Am. A 26(1), 147–155 (2009). [CrossRef] [PubMed]

5. E. E. García-Guerrero, E. R. Méndez, Z.-H. Gu, T. A. Leskova, and A. A. Maradudin, “Interference of a pair of symmetric partially coherent beams,” Opt. Express 18(5), 4816–4828 (2010). [CrossRef] [PubMed]

6. L. Ma and S. A. Ponomarenko, “Optical coherence gratings and lattices,” Opt. Lett. 39(23), 6656–6659 (2014). [CrossRef] [PubMed]

7. F. Gori and M. Santarsiero, “Devising genuine spatial correlation functions,” Opt. Lett. 32(24), 3531–3533 (2007). [CrossRef] [PubMed]

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

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

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

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

12. O. Korotkova and E. Shchepakina, “Random sources for optical frames,” Opt. Express 22(9), 10622–10633 (2014). [CrossRef] [PubMed]

13. Z. Tong and O. Korotkova, “Electromagnetic nonuniformly correlated beams,” J. Opt. Soc. Am. A 29(10), 2154–2158 (2012). [CrossRef] [PubMed]

14. Z. Tong and O. Korotkova, “Nonuniformly correlated light beams in uniformly correlated media,” Opt. Lett. 37(15), 3240–3242 (2012). [CrossRef] [PubMed]

15. S. G. Reddy, A. Kumar, S. Prabhakar, and R. P. Singh, “Experimental generation of ring-shaped beams with random sources,” Opt. Lett. 38(21), 4441–4444 (2013). [CrossRef] [PubMed]

16. F. Wang, C. Liang, Y. Yuan, and Y. Cai, “Generalized multi-Gaussian correlated Schell-model beam: from theory to experiment,” Opt. Express 22(19), 23456–23464 (2014). [CrossRef] [PubMed]

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

18. X. Liu and D. Zhao, “Electromagnetic random source for circular optical frame and its statistical properties,” Opt. Express 23(13), 16702–16714 (2015). [CrossRef] [PubMed]

19. C. Ding, O. Korotkova, Y. Zhang, and L. Pan, “Cosine-Gaussian correlated Schell-model pulsed beams,” Opt. Express 22(1), 931–942 (2014). [CrossRef] [PubMed]

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

21. 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]

22. Z. Mei, O. Korotkova, and Y. Mao, “Products of Schell-model cross-spectral densities,” Opt. Lett. 39(24), 6879–6882 (2014). [CrossRef] [PubMed]

23. Z. Mei, O. Korotkova, and Y. Mao, “Powers of the degree of coherence,” Opt. Express 23(7), 8519–8531 (2015). [CrossRef] [PubMed]

24. Z. Mei and O. Korotkova, “Alternating series of cross-spectral densities,” Opt. Lett. 40(11), 2473–2476 (2015). [CrossRef] [PubMed]

25. O. Korotkova and Z. Mei, “Convolution of degrees of coherence,” Opt. Lett. 40(13), 3073–3076 (2015). [CrossRef] [PubMed]

Source coherence-based far-field intensity filtering

Abstract

1. Introduction

2. Theory

3. Examples: Gaussian modulated multi-sinc Schell-model sources and the corresponding far-field patterns

3.1 The polar symmetry case

3.2 The Cartesian symmetry case

4. Summary

Acknowledgments

References and links

Cited By

Figures (8)

Equations (26)

Optics Express