Powers of the degree of coherence

Zhangrong Mei; Olga Korotkova; Yonghua Mao

doi:10.1364/OE.23.008519

1. Introduction

In the scalar treatment the second-order spatial coherence properties of a wide-sense statistically stationary light field can be described by a cross-spectral density (CSD) function [1]. The choice of the mathematical model for the CSD is restricted in several ways, including integrability conditions over space and frequency variables, as well as hermiticity and non-negative-definiteness conditions [1]. The latter restriction being the most difficult to work with, has a much simpler alternative, expressed via an integral with a kernel having certain properties [2]. The integral alternative of [2] has turned out to be a very convenient tool for analyzing the structure of sources of any state of coherence and also the fields that the sources generate. Moreover, with its help a slew of partially coherent fields were devised, including non-uniformly correlated sources [3–5], Bessel- and Laguerre-Gaussian Schell-model sources [6], multi-Gaussian Schell-model sources with circular and rectangular symmetry [7–10], cosine-Gaussian Schell-model sources with circular and rectangular symmetry [11–13], optical frames [14], sinc Schell-model sources [15], etc. Furthermore, the superposition rule has also played an important role in establishing legitimate operations on the CSD that can lead to whole new classes of random beams. For example, Santarsiero et al. [16] furnished a sufficient condition ensuring that a difference of two CSDs generates a new, genuine CSD. In particular, Gori et al. [17] discussed the conditions under which the difference of two CSDs of Gaussian type constitutes a new valid CSD. We also recently considered the product operation for two Schell-like CSDs of the same and different types and found out when it leads to a novel valid CSD [18]. The question then arises under what conditions the product operation can be extended from two to N CSDs.

The purpose of this paper is to discuss the most general case of product of multiple CSDs and furnish the criterion to assess whether the result is a valid CSD. We will focus our attention to products of N CSDs of the same type, i.e. we will consider the operation of raising the CSD to the N-th power. We will, hence, show how one can use the already established CSD model and power operation for developing novel sources. The paper is organized as follows: the condition under which a product of N CSDs constitutes a valid CSD is established in Section 2; Section 3 contains application of the theory by means of several typical examples of sources constructed with the product (power) operation; Section 4 includes investigation of evolution of scalar beams generated by such new sources with the help of the weighted superposition method and Section 5 summarizes the obtained results.

2. Proposition

Suppose that a random beam-like field is generated by a source located in the plane $z = 0$ and propagates along the positive $z$ direction. For the sake of simplicity, we shall refer to cases in which the fields depend only on one transverse dimension, say $x$ , by noting that the generalization of the results to symmetric two-dimensional cases is straightforward. The CSD at two typical points, ${x^{'}}_{1}$ and ${x^{'}}_{2}$ , of a planar source will be denoted by $W^{(0)} ({x^{'}}_{1}, {x^{'}}_{2})$ , where the explicit dependence on frequency is omitted. According to the superposition rule [2], $W^{(0)} ({x^{'}}_{1}, {x^{'}}_{2})$ is a genuine CSD if it can be expressed in the form

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

where

p (v)

is an arbitrary non-negative weighting function and

H_{0} (x^{'}, v)

is an arbitrary kernel. The functions in the kernel may have different forms leading to distinct classes of the CSD functions. A simple and significant class of the CSD functions is obtained by giving

H_{0} (x^{'}, v)

a Fourier-like structure, i.e.,

H_{0} (x^{'}, v) = F (x^{'}) \exp [- 2 π i v g (x^{'})],

where

g (x^{'})

is an arbitrary real function and

F (x^{'})

is a (possibly) complex profile function. Then, the use of any arbitrary nonnegative, Fourier transformable function

p (v)

, puts Eq. (1) to the form

\begin{array}{l} W^{(0)} ({x^{'}}_{1}, {x^{'}}_{2}) = F^{*} ({x^{'}}_{1}) F ({x^{'}}_{2}) \tilde{p} [g ({x^{'}}_{1}) - g ({x^{'}}_{2})] \\ = F^{*} ({x^{'}}_{1}) F ({x^{'}}_{2}) μ [g ({x^{'}}_{1}) - g ({x^{'}}_{2})], \end{array}

where

μ = \tilde{p}

denotes the degree of coherence of the source, tilde denoting the Fourier transform. The choices of

g (x^{'})

give rise to a much wider set of correlation functions. If

g (x^{'}) = x^{'}

, Eq. (2) becomes

H_{0} (x^{'}, v) = F (x^{'}) \exp (- 2 π i v x^{'}),

in which case Eq. (3) reduces to the conventional Schell-model sources. Alternatively, we let

g (x^{'}) = {(x^{'} - x_{0})}^{2}

, and hence Eq. (2) becomes

H_{0} (x^{'}, v) = F (x^{'}) \exp [- 2 π i v {(x^{'} - x_{0})}^{2}],

where

x_{0}

is a real constant. In this case, Eq. (3) represents a non-uniformly correlated source [3].

Once a kernel $H_{0}$ is chosen, the weighting function $p (v)$ controls the state of coherence of the field, i.e. different CSD functions may be obtained by varying the weighting function. All of these CSDs will be said to belong to that class with respect to the kernel $H_{0}$ .

Let us now assume that the weighting function is defined as the convolution of $N$ valid nonnegative weighting functions, i.e.,

p_{N} (v) = p_{1} (v) \otimes p_{2} (v) \otimes \dots \otimes p_{N} (v) .

On substituting from Eq. (6) into Eq. (3), and employing the convolution theorem, we find that the corresponding CSD takes form

\begin{array}{l} W^{(0)} ({x^{'}}_{1}, {x^{'}}_{2}) = F^{*} ({x^{'}}_{1}) F ({x^{'}}_{2}) \prod_{i = 1}^{N} {\tilde{p}}_{i} [g ({x^{'}}_{1}) - g ({x^{'}}_{2})] \\ = F^{*} ({x^{'}}_{1}) F ({x^{'}}_{2}) \prod_{i = 1}^{N} μ_{i} [g ({x^{'}}_{1}) - g ({x^{'}}_{2})] . \end{array}

According to the preceding analysis, we arrive at the following proposition: For the source degree of coherence being a product of N (valid) degrees of coherence corresponding to the same kernel function with Fourier-like structure, to be legitimate, it suffices to require that their Fourier transforms are non-negative functions. Such product defines a new degree of coherence belonging to the same class.

This result establishes a new approach to novel classes of correlation functions. In the following, some classes of CSDs obtained through this method will be examined. Since for our purposes the choice of the amplitude profile function is arbitrary, we simply set it to be Gaussian with the r.m.s. source width $σ$ for all examples, i.e.,

F (x^{'}) = e x p [- {x^{'}}^{2} / (2 σ^{2})] .

3. Examples

3.1 Gaussian-correlated models

Let us first refer to the Gaussian-correlated sources. We recall that the weighting function $p (v)$ for this type of sources has the form

p_{g} (v) = \sqrt{2 π δ^{2}} \exp (- 2 π^{2} δ^{2} v^{2}) .

On substituting from Eqs. (8) and (9) into Eq. (3) with

g (x^{'}) = x^{'}

, we find that the CSD takes on the form

W^{(0)} ({x^{'}}_{1}, {x^{'}}_{2}) = \exp (- \frac{{x^{'}}_{1}^{2} + {x^{'}}_{2}^{2}}{2 σ^{2}}) \exp [- \frac{{({x^{'}}_{1} - {x^{'}}_{2})}^{2}}{2 δ^{2}}],

being a GSM source. If we choose Eq. (5) as the kernel and Eq. (9) as the weighting function, the CSD takes on the form

W^{(0)} ({x^{'}}_{1}, {x^{'}}_{2}) = \exp (- \frac{{x^{'}}_{1}^{2} + {x^{'}}_{2}^{2}}{2 σ^{2}}) \exp {- \frac{{[{({x^{'}}_{2} - x_{0})}^{2} - {({x^{'}}_{1} - x_{0})}^{2}]}^{2}}{δ^{4}}},

being a non-uniformly Gaussian-correlated (NUGSM) source [3].

We now consider a new weighting function defined by the convolution of N functions $p_{g} (v)$ , i.e.,

p_{g N} (v) = \underset{N}{\underset{︸}{p_{g} (v) \otimes p_{g} (v) \otimes \dots \otimes p_{g} (v)}} .

Performing the convolution operation specified in Eq. (12), we find that the new weighting function turns out to be

p_{g N} (v) = \sqrt{2 π δ^{2} / N} \exp (- 2 π^{2} δ^{2} v^{2} / N) .

On using Eq. (4) as the kernel and Eq. (13) as the weighting function and evaluating the corresponding Schell-model CSD through Eq. (7), we easily obtain the expression

W^{(0)} ({x^{'}}_{1}, {x^{'}}_{2}) = \exp (- \frac{{x^{'}}_{1}^{2} + {x^{'}}_{2}^{2}}{2 σ^{2}}) \exp [- \frac{N {({x^{'}}_{1} - {x^{'}}_{2})}^{2}}{2 δ^{2}}] .

Equation (14) represents the Nth order modified GSM source.

As a second choice, using Eq. (5) as the kernel weighted by the function Eq. (13), we find that the CSD for the non-uniformly correlated source has the form

W^{(0)} ({x^{'}}_{1}, {x^{'}}_{2}) = \exp (- \frac{{x^{'}}_{1}^{2} + {x^{'}}_{2}^{2}}{2 σ^{2}}) \exp {- \frac{N {[{({x^{'}}_{2} - x_{0})}^{2} - {({x^{'}}_{1} - x_{0})}^{2}]}^{2}}{δ^{4}}} .

Equation (15) represents the Nth order modified NUGSM source.

Figure 1 shows the degree of coherence of the CSDs given by Eqs. (14) and (15) with $σ = 3 mm$ , $δ = 0.1 σ$ , $x_{0} = 0.7 σ$ for several values of parameter N. It can be seen that the state of coherence of the modified GSM source is independent of the lateral coordinate, while for the modified NUGSM source it depends on the lateral coordinate. For N = 1, it corresponds to the conventional GSM and NUGSM sources, respectively. With increasing N, the coherence length becomes shorter and the width of the curve of the degree of coherence narrows down.

Fig. 1 Illustration of the degree of coherence, the top row corresponding to the CSD of the Nth order modified GSM source and the bottom row corresponding to the CSD of the Nth order modified NUGSM source for several values of parameter N, N = 1 on the left hand side, N = 2 in the middle column and N = 6 on the right hand side.

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3.2 Sinc-correlated models

In order to generate sinc-function correlation models one may employ the following form for the weighting function [15]

p_{s} (v) = \frac{1}{c} r e c t (\frac{v}{c}) = {\begin{matrix} 1 / c, | v | \leq c / 2, \\ 0, | v | > c / 2, \end{matrix}

where

c

is a positive constant and

r e c t (x)

is the rectangular function with width

c

. On using Eq. (4) as the kernel weighted by the function Eq. (16), the CSD for sinc Schell-model (SSM) sources has the form [15]

W^{(0)} ({x^{'}}_{1}, {x^{'}}_{2}) = \exp (- \frac{{x^{'}}_{1}^{2} + {x^{'}}_{2}^{2}}{2 σ^{2}}) \sin c (c {x^{'}}_{1} - c {x^{'}}_{2}) .

On using Eq. (5) as the kernel weighted by the function Eq. (16), the CSD takes on the form

W^{(0)} ({x^{'}}_{1}, {x^{'}}_{2}) = \exp (- \frac{{x^{'}}_{1}^{2} + {x^{'}}_{2}^{2}}{2 σ^{2}}) \sin c [c {({x^{'}}_{1} - x_{0})}^{2} - c {({x^{'}}_{2} - x_{0})}^{2}],

which represents a non-uniformly sinc-correlated (NUSSM) source.

Now, we suppose all of the weighting functions in Eq. (6) assume the rectangular shape given by function in Eq. (16):

p_{1} (v) = p_{2} (v) = \dots = p_{N} (v) \equiv p_{s} (v) .

The corresponding weighting function can be written, with reference to Eq. (6), as the Nth-order self-convolution of

p_{s} (v)

, i.e.,

P_{s N} (v) = \underset{N}{\underset{︸}{p_{s} (v) \otimes p_{s} (v) \otimes \dots \otimes p_{s} (v)}} .

On substituting from Eq. (16) into Eq. (20) and performing the convolution calculation, we find that the expressions of several weighting functions

P_{s N} (v)

are given by the following equations [19]

\begin{array}{l} P_{s 2} (v) = p_{s} (v) \otimes p_{s} (v) \\ = \frac{1}{c} T r i (v / c) = \frac{1}{c} {\begin{matrix} 0, | v | \geq c, \\ 1 - | v | / c, | v | < c, \end{matrix} \end{array}

\begin{array}{l} P_{s 3} (v) = p_{s} (v) \otimes p_{s} (v) \otimes p_{s} (v) \\ = \frac{1}{2! c} {\begin{matrix} 0, | v | \geq 3 c / 2, \\ {(3 / 2 - | v | / c)}^{2}, c / 2 \leq | v | < 3 c / 2, \\ {(3 / 2 - | v | / c)}^{2} - 3 {(1 / 2 - | v | / c)}^{2}, 0 \leq | v | < c / 2. \end{matrix} \end{array}

\begin{array}{l} P_{s 6} (v) = p_{s} (v) \otimes p_{s} (v) \otimes p_{s} (v) \otimes p_{s} (v) \otimes p_{s} (v) \otimes p_{s} (v) \\ = \frac{1}{5! c} {\begin{matrix} 0, | v | \geq 3 c, \\ {(3 - | v | / c)}^{5}, 2 c \leq | v | < 3 c, \\ {(3 - | v | / c)}^{5} - 6 {(2 - | v | / c)}^{5}, c \leq | v | < 2 c, \\ \begin{array}{l} 9 (^{1 - | v | / c) 5} - 2 (^{2 - | v | / c) 5} - 4 (^{| v | / c) 5} \\ + (^{1 + | v | / c) 5} + 5! (1 - | v | / c), | v | < c . \end{array} \end{matrix} \end{array}

On substituting from Eq. (16) into Eq. (20) then into Eq. (7) and choosing Eq. (4) as the kernel function, we obtain

W^{(0)} ({x^{'}}_{1}, {x^{'}}_{2}) = \exp (- \frac{{x^{'}}_{1}^{2} + {x^{'}}_{2}^{2}}{2 σ^{2}}) \sin c^{N} (c {x^{'}}_{1} - c {x^{'}}_{2}),

Equation (24) represents the Nth order modified SSM sources.

Furthermore, we choose Eq. (5) as the kernel function together with the weighting function given by Eq. (20). Hence the CSD takes the form

W^{(0)} ({x^{'}}_{1}, {x^{'}}_{2}) = \exp (- \frac{{x^{'}}_{1}^{2} + {x^{'}}_{2}^{2}}{2 σ^{2}}) \sin c^{N} [c {({x^{'}}_{1} - x_{0})}^{2} - c {({x^{'}}_{2} - x_{0})}^{2}],

Equation (25) representing the Nth order modified NUSSM source.

Figure 2 shows the degree of coherence of the CSDs given by Eqs. (24) and (25) with the same parameters with Fig. 1. Similarly, the state of coherence of the modified SSM source is independent of the lateral coordinate, while for the modified NUSSM sources it depends on the lateral coordinate. When N = 1, the models correspond to the conventional SSM and NUSSM fields, some sidelobes appear in the edge of the coherence patterns. With increasing N, these side-lobes gradually disappear similarly to the Gaussian correlated models.

Fig. 2 Illustration of the degree of coherence, the top row corresponding to the CSD of the Nth order modified SSM sources and the bottom row corresponding to the CSD of the Nth modified NUSSM sources for several values of parameter N, N = 1 on the left hand side, N = 2 in the middle column and N = 6 on the right hand side.

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3.3 Cosine-Gaussian correlated models

Let us now consider another example for the cosine-Gaussian correlated model by taking the following weighting function [11]

p_{c} (v) = \sqrt{2 π δ^{2}} \cos h [a {(2 π)}^{3 / 2} δ v] \exp (- 2 π^{2} δ^{2} v^{2} - a^{2} π),

where

δ

and a are two positive real constants. Using the Eq. (4), the CSD for cosine-Gaussian Schell-model (CGSM) sources has the form [11]

W^{(0)} ({x^{'}}_{1}, {x^{'}}_{2}) = \exp (- \frac{{x^{'}}_{1}^{2} + {x^{'}}_{2}^{2}}{2 σ^{2}}) \cos [\frac{a \sqrt{2 π} ({x^{'}}_{1} - {x^{'}}_{2})}{δ}] \exp [- \frac{{({x^{'}}_{1} - {x^{'}}_{2})}^{2}}{2 δ^{2}}] .

Using the Eq. (5), the CSD for non-uniformly cosine-Gaussian correlated source (NUCG) takes on the form

\begin{array}{l} W^{(0)} ({x^{'}}_{1}, {x^{'}}_{2}) = \exp (- \frac{{x^{'}}_{1}^{2} + {x^{'}}_{2}^{2}}{2 σ^{2}}) \cos {\frac{2 a \sqrt{π} [{({x^{'}}_{2} - x_{0})}^{2} - {({x^{'}}_{1} - x_{0})}^{2}]}{δ^{2}}} \\ \times \exp {- \frac{{[{({x^{'}}_{2} - x_{0})}^{2} - {({x^{'}}_{1} - x_{0})}^{2}]}^{2}}{δ^{4}}} . \end{array}

Similarly, the new weighting functions can be obtained by performing the Nth-order self-convolution of

p_{c} (v)

, i.e.,

P_{c N} (v) = \underset{N}{\underset{︸}{p_{c} (v) \otimes p_{c} (v) \otimes \dots \otimes p_{c} (v)}} .

Then, according to Eq. (7), the CSD for the Nth-order modified CGSM and NUCG sources can be expressed as

W^{(0)} ({x^{'}}_{1}, {x^{'}}_{2}) = \exp (- \frac{{x^{'}}_{1}^{2} + {x^{'}}_{2}^{2}}{2 σ^{2}}) \cos^{N} [\frac{a \sqrt{2 π} ({x^{'}}_{1} - {x^{'}}_{2})}{δ}] \exp [- \frac{N {({x^{'}}_{1} - {x^{'}}_{2})}^{2}}{2 δ^{2}}],

and

\begin{array}{l} W^{(0)} ({x^{'}}_{1}, {x^{'}}_{2}) = \exp (- \frac{{x^{'}}_{1}^{2} + {x^{'}}_{2}^{2}}{2 σ^{2}}) \cos^{N} {\frac{2 a \sqrt{π} [{({x^{'}}_{2} - x_{0})}^{2} - {({x^{'}}_{1} - x_{0})}^{2}]}{δ^{2}}} \\ \times \exp {- \frac{N {[{({x^{'}}_{2} - x_{0})}^{2} - {({x^{'}}_{1} - x_{0})}^{2}]}^{2}}{δ^{4}}} . \end{array}

Figure 3 represents the degree of coherence of the CSDs given by Eqs. (30) and (31) with the same parameters with Fig. 1 except for a = 3. Clearly, the states of coherence of the sources for different order N have different distributions. These differences in source-correlated properties lead to different far-field spectral density distributions that will be analyzed in the following section.

Fig. 3 Illustration of the degree of coherence, the top row corresponding to the CSD of the Nth order modified CGSM sources and the bottom row corresponding to the CSD of the Nth modified NUCG sources for several values of parameter N, N = 1 on the left hand side, N = 2 in the middle column and N = 6 on the right hand side.

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4. Propagation effects

The CSD of the generated field at a pair of points $(x_{1}, z)$ and $(x_{2}, z)$ in any transverse plane $z = c o n s t > 0$ is related, under the paraxial approximation, to that in the source plane through the following transform [20]

W (x_{1}, x_{2}, z) = \iint W^{(0)} ({x^{'}}_{1}, {x^{'}}_{2}) K_{z}^{*} (x_{1}, {x^{'}}_{1}) K_{z} (x_{2}, {x^{'}}_{2}) d {x^{'}}_{1} d {x^{'}}_{2},

where the free propagation kernel

K_{z}

is given by expression

K_{z} = - \sqrt{\frac{k}{2 π z}} \exp [\frac{i k}{2 z} {(x - x^{'})}^{2}],

k

being the wave number of light. On substituting from Eqs. (1) and (33) into Eq. (32) we obtain, after interchanging the orders of integrals, the expression

W (x_{1}, x_{2}, z) = \int p (v) H_{z}^{*} (x_{1}, v) H_{z} (x_{2}, v) d v,

where

H_{z} (x, v) = \int H_{0} (x^{'}, v) K_{z} (x, x^{'}) d x^{'} .

Equations (34) and (35) illustrate that the propagation of the fields can be studied by using the weighted superposition method. Hence one may consider the propagation of the kernel function separately. On substituting form Eq. (4) into Eq. (34), we obtain the formula

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

where

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

On substituting form Eq. (5) into Eq. (35), we arrive at the result

{| H_{z} (x, v) |}^{2} = \frac{σ}{w (z, v)} \exp [- \frac{{(x - 4 π v z x_{0} / k)}^{2}}{w^{2} (z, v)}],

where

w^{2} (z, v) = σ^{2} {(1 - 4 π z v / k)}^{2} + z^{2} / (k^{2} σ^{2}) .

Equations (36) and (38) are the intensity of the modes corresponding to two different types defined by Eqs. (4) and (5). The spectral density of the partially coherent field at any point $(x, z)$ can then be calculated from Eq. (34) as

S (x, z) = \int p (v) {| H_{z} (x, v) |}^{2} d v .

Based on this weighting superposition, we will study the propagation-induced intensity changes of newly defined partially coherent field as described by Eqs. (14), (15), (24), (25), (30) and (31). The values of parameter $k = 10^{7} m^{- 1}$ and other field parameters are same as in Fig. 1.

Figure 4 shows the longitudinal distribution (left) and corresponding contour graphs (middle) of the spectral density S of the Nth order modified GSM beams with different values of parameter N, as well as the transverse distribution (right) at several selected distances. One clearly sees that the modified GSM beams of different orders present the similar propagation characteristics with the conventional GSM beam. But the r.m.s. width increases and the height of centre intensity decrease with increasing values of N at the same propagating distance. The changes in the on-axis spectral density S of the modified GSM beams of different orders are shown in Fig. 5(a).

Fig. 4 Evolution of the spectral intensity S of the Nth order modified GSM beam with several values of parameter N on propagation in free space, the top row corresponding to N = 2 and the bottom row corresponding to N = 6.

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Fig. 5 Changes on-axis spectral intensity S of the Nth order modified GSM beam (a) and the Nth order modified NUGSM beam (b) with x₀ = 0 for several values of parameter N.

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Figure 6 shows the evolution of the spectral density of the Nth order modified NUGSM beams with N = 2 and N = 6 on propagation in free space. Two features are worth noting. First, the maximum intensity laterally shifted from the on-axis to off-axis positions for the modified NUGSM beam of any order. Second, the longitudinal distributions of the spectral density have self-focusing effects, and that the focal lengths become shorter with increasing order N, as shown in Fig. 5(b).

Fig. 6 Evolution of the spectral intensity S of the Nth order modified NUGSM beam with several values of parameter N on propagation in free space, the top row corresponding to N = 2 and the bottom row corresponding to N = 6.

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Figure 7 shows the longitudinal distribution (left) and the corresponding contour graphs (middle) of the spectral density S of the Nth order modified SSM beams with different values of parameter N, as well as the transverse distribution (right) at several selected distances. We have already known that the far fields of the SSM beams (N = 1) have a flat-top intensity profile [15], and the 2th order modified SSM beams have a peaked cap intensity profile [18], as shown in the top row of Fig. 7. While for the larger value of N, we find that the spectral intensity of Nth order modified SSM and GSM beams have a similar evolution properties, this is due to the coherence distribution of high order modified SSM beams similar to the Gaussian correlated models. The r.m.s. width increases and the on-axis intensity decrease with increasing values of N in the same propagating plane, as shown in Fig. 8(a).

Fig. 7 Evolution of the spectral intensity S of the Nth order modified SSM beam with several values of parameter N on propagation in free space, the top row corresponding to N = 2 and the bottom row corresponding to N = 6.

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Fig. 8 Changes on-axis spectral intensity S of the Nth order modified SSM beam (a) and the Nth order modified NUSSM beam (b) with x₀ = 0 for several values of parameter N.

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Figure 9 illustrates typical evolution of the spectral density of the Nth order modified NUSSM beams with N = 2 and N = 6 on propagation in free space. It is shown that such light fields also have self-focusing effects and laterally shifted intensity maxima. The high order NUSSM beams are sharper than the low order beams before the focal plane. After the focal plane, the high order beams spread faster than the low order beams and the height of centre intensity decreases with the increasing values of N, as shown in Fig. 8(b).

Fig. 9 Evolution of the spectral intensity S of the Nth order modified NUSSM beam with several values of parameter N on propagation in free space, the top row corresponding to N = 2 and the bottom row corresponding to N = 6.

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Figure 10 shows the longitudinal distribution (left) and corresponding contour graphs (middle) of the spectral density S of the Nth order modified CGSM beams with different values of parameter N, as well as the transverse distribution (right) at several selected distances. One clearly sees that the different-order modified CGSM beams present completely different propagation characteristics and exhibit different transverse intensity profiles in the far-field.

Fig. 10 Evolution of the spectral intensity S of the Nth order modified CGSM beam with several values of parameter N on propagation in free space, the top row corresponding to N = 1 and the bottom row corresponding to N = 2.

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Figure 11 illustrates typical evolution of the spectral density of the different-order modified NUCG beams propagating in free space. Similarly, the modified NUCG beams also have self-focusing effects and laterally shifted intensity maxima, but those exhibit strikingly different propagation characteristics for different value of order N.

Fig. 11 Evolution of the spectral intensity S of the Nth order modified NUCG beam with several values of parameter N on propagation in free space, the top row corresponding to N = 1 and the bottom row corresponding to N = 2.

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5. Concluding remarks

In this article, we have examined the problem of determining in which cases the N-th power of the degree of coherence furnishes a physically realizable beam-like field. On using the the superposition rule for partially coherent fields we have established a new approach for designing novel random sources. With the help of the proven proposition, we introduced four new types of partially coherent sources with the Nth order uniform and non-uniform GSM and SSM correlation properties. The free-space propagation-induced spectral density changes of scalar beams generated by these four types of sources are investigated with the help of the weighted superposition method and numerical examples.

Thus, this paper is in line with several recent other works on a novel of how one can mathematically model novel cross-spectral densities (CSD) and on how to legitimately manipulate them. It is well known that it is not a trivial task to develop a new (and simple) mathematical model for CSD. The alternative approach is to use certain algebraic operations such as linear superposition and product of known CSD. The operation of raising of the CSD to a power is not a trivial one in the class of CSD but, as we have shown, might lead to novel and finely adjustable random fields. The results show that this method is useful for modeling the partially coherent sources and suggest a significant tool to study the new classes of CSDs.

The comparison between the Gaussian-correlated, sinc-correlated, and the cos-correlated sources implies that the power operation might lead only to size modification, as in the former case or to both size and shape modification, as in the latter two cases, with increasing values of power N. In fact, we believe, that a Gaussian-correlated source is the only one that is shape-invariant with respect to power operation.

As we have shown, power operation applied to a certain class of CSD with adjustable N can be employed for fine adjustment of the shape of the generated intensity distribution. For instance, as is evident from our example on sinc-correlated field, the triangle-like distributions with adjustable solid angle can be formed as N increases from 2 to 6 (see Fig. 7). Here we considered a scalar treatment only, but interesting phenomena can be expected when they are extended to a vectorial realm.

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).

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Powers of the degree of coherence

Abstract

1. Introduction

2. Proposition

3. Examples

3.1 Gaussian-correlated models

3.2 Sinc-correlated models

3.3 Cosine-Gaussian correlated models

4. Propagation effects

5. Concluding remarks

Acknowledgments

References and links

Cited By

Figures (11)

Equations (40)

Optics Express