Propagation factor of a stochastic electromagnetic Gaussian Schell-model beam

Shijun Zhu; Yangjian Cai; Olga Korotkova

doi:10.1364/OE.18.012587

1. Introduction

In 1994 it was found that the degree of polarization of a stochastic electromagnetic beam may change on propagation in vacuum, and such changes depend on the coherence properties of the source of the beam [1]. Electromagnetic Gaussian Schell-model (EGSM) beam [2–4] was introduced as an extension of scalar GSM beam [5–8]. Due to its importance in the theories of coherence and polarization of light, numerous theoretical and experimental papers relating to stochastic electromagnetic beams have been published in the past several years [9–24].

Over the past several decades, many works have been carried out concerning the propagation of various laser beams through the turbulent atmosphere due to their wide applications, e.g. in free-space optical communication, laser radar, atmospheric imaging systems and remote sensing, and it has been found that the behavior of a laser beam in a turbulent atmosphere is closely related to its initial beam profile, coherence and polarization properties [25–46]. Behavior of the statistical properties including the averaging intensity, coherence, degree of polarization, state of polarization and scintillation index of an EGSM beam, propagating in turbulent atmosphere has been studied in details [40–48]. It was found that the EGSM beams may have reduced levels of scintillations compared to the scalar GSM beams [47], which is useful for free-space optical communications and laser radar systems (LIDARs) [48]. To our knowledge no results have been reported up until now on the propagation factor of EGSM beams passing through the turbulent atmosphere.

The propagation factor (also known as the $M^{2}$ -factor) proposed by Siegman is a particularly useful property of an optical laser beam, and plays an important role in the characterization of beam behavior on propagation. Several methods have been developed to obtain the propagation factors of the laser beams in free space [49–51]. The definition of propagation factor in terms of second-order moments of the Wigner distribution function has been adopted widely for characterizing laser beams [52–54]. Recently, this method was extended for the analysis of the propagation factor of a laser beam traveling in the turbulent atmosphere [55–57]. The purpose of this paper is to investigate, based on this recently extended method, the M²-factor of an EGSM beam propagating in the turbulent atmosphere, by deriving the explicit expression for the propagation factor of an EGSM beam for both free space propagation and atmospheric propagation, and exploring them comparatively.

2. Theory

The second-order statistical properties of an EGSM beam can be characterized by the $2 \times 2$ cross-spectral density matrix $\overset{\leftrightarrow}{W} (ρ_{1}^{'}, ρ_{2}^{'}, 0)$ specified at any two points with position vectors $ρ_{1}^{'}$ and $ρ_{2}^{'}$ in the source plane with elements [2–4]

W_{α β}^{} (ρ_{1}^{'}, ρ_{2}^{'}; 0) = A_{α} A_{β} B_{α β} \exp [- \frac{ρ_{1}^{' 2}}{4 σ_{a}^{2}} - \frac{ρ_{2}^{' 2}}{4 σ_{β}^{2}} - \frac{{(ρ_{1}^{'} - ρ_{2}^{'})}^{2}}{2 δ_{α β}^{2}}], (α = x, y; β = x, y)

where

σ_{α}

is the r.m.s width of the spectral density along α direction;

δ_{x x}

,

δ_{y y}

and

δ_{x y}

are the r.m.s. widths of auto-correlation functions of the x component of the field, of the y component of the field and of the mutual correlation function of x and y field components, respectively,

B_{x y}

is the complex correlation coefficient between the x and y components of the electric field; parameters

A_{α}

,

B_{α β} = | B_{α β} | \exp (i ϕ_{α β}) = B_{β α}^{*}

,

σ_{α}

and

δ_{α β}

are independent of position and, in our analysis, of frequency. The nine real parameters

A_{x}

,

A_{y}

,

σ_{x}

,

σ_{y}

,

| B_{x y} |

,

ϕ_{x y}

,

δ_{x x}

,

δ_{y y}

and

δ_{x y}

entering the general model are shown to satisfy several intrinsic constraints and obey some simplifying assumptions (e.g. the phase difference between the x- an y-components of the field is removable, i.e.

ϕ_{α β} = 0

) [9,14,18]. The trace of the cross-spectral density matrix of an EGSM beam is expressed as [2–4,9,10]

W_{tr} (ρ_{1}^{'}, ρ_{2}^{'}; 0) = Tr \overset{\leftrightarrow}{W} (ρ_{1}^{'}, ρ_{2}^{'}, 0) = W_{x x} (ρ_{1}^{'}, ρ_{2}^{'}; 0) + W_{y y} (ρ_{1}^{'}, ρ_{2}^{'}; 0) .

Within the validity of the paraxial approximation, the propagation of the trace of the cross-spectral density matrix of an EGSM beam in the turbulent atmosphere can be studied with the help of the following generalized Huygens-Fresnel integral [26,27]

\begin{array}{l} W_{tr} (ρ, ρ_{d}; z) = {(\frac{k}{2 π z})}^{2} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} W_{tr} (ρ^{'}, ρ_{d}^{'}; 0) \\ \times \exp [\frac{i k}{z} (ρ - ρ^{'}) \cdot (ρ_{d} - ρ_{d}^{'}) - H (ρ_{d}, ρ_{d}^{'}; z)] d^{2} ρ^{'} d^{2} ρ_{d}^{'}, \end{array}

where

k = 2 π / λ

is the wave number withλ being the wavelength. In Eq. (3) we have used the following sum and difference vector notation

ρ^{'} = \frac{(ρ_{1}^{'} + ρ_{2}^{'})}{2}, ρ_{d}^{'} = ρ_{1}^{'} - ρ_{2}^{'}, ρ = \frac{(ρ_{1} + ρ_{2})}{2}, ρ_{d} = ρ_{1} - ρ_{2},

where

ρ_{1}, ρ_{2}

are two arbitrary points in the receiver plane, perpendicular to the direction of propagation of the beam and

W_{tr} (ρ^{'}, ρ_{d}^{'}; 0) = W_{tr} (ρ_{1}^{'}, ρ_{2}^{'}; 0) = W_{tr} (ρ^{'} + \frac{ρ_{d}^{'}}{2}, ρ^{'} - \frac{ρ_{d}^{'}}{2}; 0) .

The term

H (ρ_{d}, ρ_{d}^{'}, z)

in Eq. (3) is the contribution from the atmospheric turbulence expressed as

H (ρ_{d}, ρ_{d}^{'}; z) = 4 π^{2} k^{2} z \int_{0}^{1} d ξ \int_{0}^{\infty} [1 - J_{0} (κ | ρ_{d}^{'} ξ + (1 - ξ) ρ_{d} |)] Φ_{n} (κ) κ d κ,

where

J_{0}

is the Bessel function of zero order,

Φ_{n}

represents the one-dimensional power spectrum of the index-of-refraction fluctuations [27].

Equation (3) can be expressed in the following alternative form [55–57]

\begin{array}{l} W_{tr} (ρ, ρ_{d}; z) = {(\frac{1}{2 π})}^{2} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} W_{tr} (ρ'', ρ_{d} + \frac{z}{k} κ_{d}; 0) \\ \times \exp [- i ρ \cdot κ_{d} + i ρ'' \cdot κ_{d} - H (ρ_{d}, ρ_{d} + \frac{z}{k} κ_{d}; z)] d^{2} ρ'' d^{2} κ_{d}, \end{array}

where

κ_{d} \equiv (\begin{matrix} κ_{d x} & κ_{d y} \end{matrix})

is the position vector in spatial-frequency domain. We can express

W_{tr} (ρ'', ρ_{d} + \frac{z}{k} κ_{d}; 0)

of an EGSM beam as

\begin{array}{l} W_{tr} (ρ'', ρ_{d} + \frac{z}{k} κ_{d}; 0) = A_{x}^{2} \exp [- \frac{1}{4 σ_{x}^{2}} {(ρ^{"} + \frac{ρ_{d} + \frac{z}{k} κ_{d}}{2})}^{2} - \frac{1}{4 σ_{x}^{2}} {(ρ^{"} - \frac{ρ_{d} + \frac{z}{k} κ_{d}}{2})}^{2} - \frac{{(ρ_{d} + \frac{z}{k} κ_{d})}^{2}}{2 δ_{x x}^{2}}] \\ + A_{y}^{2} \exp [- \frac{1}{4 σ_{y}^{2}} {(ρ^{"} + \frac{ρ_{d} + \frac{z}{k} κ_{d}}{2})}^{2} - \frac{1}{4 σ_{y}^{2}} {(ρ^{"} - \frac{ρ_{d} + \frac{z}{k} κ_{d}}{2})}^{2} - \frac{{(ρ_{d} + \frac{z}{k} κ_{d})}^{2}}{2 δ_{x x}^{2}}] . \end{array}

The Wigner distribution function of an EGSM beam in turbulent atmosphere can be expressed in terms of the trace of the cross-spectral density matrix by the formula [55]

h_{tr} (ρ, θ; z) = {(\frac{k}{2 π})}^{2} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} W_{tr} (ρ, ρ_{d}; z) \exp (- i k θ \cdot ρ_{d}) d^{2} ρ_{d},

where

θ \equiv (θ_{x}, θ_{y})

denotes an angle which the vector of interest makes with the z-direction,

k θ_{x}

and

k θ_{y}

are the wave vector components along the x-axis and y-axis, respectively.

Substituting from Eqs. (6) and (7) into Eq. (8) and applying following integral formula,

\int_{- \infty}^{\infty} \exp (- s^{2} x^{2} \pm q x) d x = \frac{\sqrt{π}}{s} \exp (\frac{q^{2}}{4 s^{2}}), (s > 0),

we obtain (after tedious integration) the expression

\begin{array}{l} h_{tr} (ρ, θ, z) = h_{x x} (ρ, θ, z) + h_{y y} (ρ, θ, z) \\ = A_{x}^{2} \frac{σ_{x}^{2} k^{2}}{8 π^{3}} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} \exp [- a_{x x} κ_{d}^{2} - \frac{2 z}{k} b_{x x} ρ_{d} \cdot κ_{d} - i ρ \cdot κ_{d} - b_{x x} ρ_{d}^{2} - i k θ \cdot ρ_{d} - H (ρ_{d}, ρ_{d} + \frac{z}{k} κ_{d}, z)] d^{2} κ_{d} d^{2} ρ_{d} \\ + A_{y}^{2} \frac{σ_{y}^{2} k^{2}}{8 π^{3}} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} \exp [- a_{y y} κ_{d}^{2} - \frac{2 z}{k} b_{y y} ρ_{d} \cdot κ_{d} - i ρ \cdot κ_{d} - b_{y y} ρ_{d}^{2} - i k θ \cdot ρ_{d} - H (ρ_{d}, ρ_{d} + \frac{z}{k} κ_{d}, z)] d^{2} κ_{d} d^{2} ρ_{d}, \end{array}

with

a_{α α} = (\frac{1}{8 σ_{α}^{2}} + \frac{1}{2 δ_{α α}^{2}}) \frac{z^{2}}{k^{2}} + \frac{σ_{α}^{2}}{2}

,

b_{α α} = \frac{1}{8 σ_{a}^{2}} + \frac{1}{2 δ_{a a}^{2}}

, (

α = x, y

).

Based on the second-order moments of the Wigner distribution function, the $M^{2}$ -factor of an EGSM beam can be defined as follows [49–57]

\begin{array}{l} M^{2} (z) = k {(〈 ρ^{2} 〉 〈 θ^{2} 〉 - {〈 ρ \cdot θ 〉}^{2})}^{1 / 2} \\ = k {[(〈 x^{2} 〉 + 〈 y^{2} 〉) (〈 θ_{x}^{2} 〉 + 〈 θ_{y}^{2} 〉) - {(〈 x θ_{x} 〉 + 〈 y θ_{y} 〉)}^{2}]}^{1 / 2}, \end{array}

where

< x^{n_{1}} y^{n_{2}} θ_{x}^{m_{1}} θ_{y}^{m_{2}} > = \frac{1}{P} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} x^{n_{1}} y^{n_{2}} θ_{x}^{m_{1}} θ_{y}^{m_{2}} h_{tr} (ρ, θ, z) d^{2} ρ d^{2} θ,

P = \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} h_{tr} (ρ, θ, z) d^{2} ρ d^{2} θ .

Substituting Eq. (10) into Eqs. (12) and (13), we obtain (after tedious integration) the formulas for the moments

P = 2 π (A_{x}^{2} σ_{x}^{2} + A_{y}^{2} σ_{y}^{2}),

〈 ρ^{2} 〉 = 2 π A_{x}^{2} \frac{σ_{x}^{2}}{P} (4 a_{x x} + \frac{4}{3} π^{2} T z^{3}) + 2 π A_{y}^{2} \frac{σ_{y}^{2}}{P} (4 a_{y y} + \frac{4}{3} π^{2} T z^{3}),

〈 θ^{2} 〉 = 2 π A_{x}^{2} \frac{σ_{x}^{2}}{P} (\frac{4}{k^{2}} b_{x x} + 4 π^{2} T z) + 2 π A_{y}^{2} \frac{σ_{y}^{2}}{P} (\frac{4}{k^{2}} b_{y y} + 4 π^{2} T z),

〈 ρ \cdot θ 〉 = 2 π A_{x}^{2} \frac{σ_{x}^{2}}{P} (\frac{4 z}{k^{2}} b_{x x} + 2 π^{2} z^{2} T) + 2 π A_{y}^{2} \frac{σ_{y}^{2}}{P} (\frac{4 z}{k^{2}} b_{y y} + 2 π^{2} z^{2} T) .

with

T = \int_{0}^{\infty} Φ_{n} (κ) κ^{3} d κ .

After substituting from Eqs. (14)-(17) into Eq. (11) we obtain the following expression for the $M^{2}$ -factor of an EGSM beam travelling in turbulent atmosphere

\begin{array}{l} M^{2} (z) = k {(〈 ρ^{2} 〉 〈 θ^{2} 〉 - {〈 ρ \cdot θ 〉}^{2})}^{1 / 2} \\ = k {[2 π A_{x}^{2} \frac{σ_{x}^{2}}{P} (4 a_{x x} + \frac{4}{3} π^{2} T z^{3}) + 2 π A_{y}^{2} \frac{σ_{y}^{2}}{P} (4 a_{y y} + \frac{4}{3} π^{2} T z^{3})] \\ \times [2 π A_{x}^{2} \frac{σ_{x}^{2}}{P} (\frac{4}{k^{2}} b_{x x} + 4 π^{2} T z) + 2 π A_{y}^{2} \frac{σ_{y}^{2}}{P} (\frac{4}{k^{2}} b_{y y} + 4 π^{2} T z)] \\ {- {[2 π A_{x}^{2} \frac{σ_{x}^{2}}{P} (\frac{4 z}{k^{2}} b_{x x} + 2 π^{2} z^{2} T) + 2 π A_{y}^{2} \frac{σ_{y}^{2}}{P} (\frac{4 z}{k^{2}} b_{y y} + 2 π^{2} z^{2} T)]}^{2}}}^{1 / 2} . \end{array}

Under the condition of $Φ_{n} (κ) = 0$ , Eq. (19) reduces to following expression for the free space $M^{2}$ -factor of an EGSM beam

\begin{array}{l} M^{2} (z) = k {(〈 ρ^{2} 〉 〈 θ^{2} 〉 - {〈 ρ \cdot θ 〉}^{2})}^{1 / 2} \\ = {\frac{(A_{x}^{2} σ_{x}^{4} + A_{y}^{2} σ_{y}^{4}) [4 A_{x}^{2} δ_{y y}^{2} σ_{x}^{2} + δ_{x x}^{2} (A_{x}^{2} δ_{y y}^{2} + A_{y}^{2} δ_{y y}^{2} + 4 A_{y}^{2} σ_{y}^{2})]}{δ_{x x}^{2} δ_{y y}^{2} {(A_{x}^{2} σ_{x}^{2} + A_{y}^{2} σ_{y}^{2})}^{2}}}^{1 / 2} . \end{array}

One finds from Eq. (20) that the

M^{2}

-factor of an EGSM beam in free space is independent of z, so its value remains invariant on propagation, while being closely determined by parameters

A_{α}, σ_{α} and δ_{α α} .

Under the condition of

A_{x} = 0

or

A_{y} = 0

, Eq. (20) reduces to the following expression for the

M^{2}

-factor of a scalar GSM beam (i.e., fully polarized GSM beam)

M^{2} (z) = {(1 + \frac{4 σ_{α}^{2}}{δ_{α α}^{2}})}^{1 / 2}, (α = x, y)

Equation (21) agrees well with the result reported in [51]. Under the condition of

δ_{α α} = \infty

, the right side of Eq. (21) reduces to unity, coinciding with the

M^{2}

-factor of a coherent scalar Gaussian beam propagating in free space [49].

3. Numerical examples

In this section we study the $M^{2}$ -factor of an EGSM beam in free space and in turbulent atmosphere numerically. In the following examples, we choose the Tatarskii spectrum for the spectral density of the index-of-refraction fluctuations, which is expressed as [27]

Φ_{n} (κ) = 0.033 C_{n}^{2} κ^{- 11 / 3} \exp (- \frac{κ^{2}}{κ_{m}^{2}}),

where

C_{n}^{2}

is the structure constant of the turbulent atmosphere,

κ_{m} = 5.92 / l_{0}

with

l_{0}

being the inner scale of the turbulence. Substituting from Eq. (22) into Eq. (18), we obtain the formula

T = \int_{0}^{\infty} Φ_{n} (κ) κ^{3} d κ = 0.1661 C_{n}^{2} l_{0}^{- 1 / 3} .

Substituting Eq. (23) into Eq. (19), we now can calculate the

M^{2}

-factor of an EGSM beam numerically.

For the convenience of analysis, we only consider the EGSM beam that is generated by an EGSM source whose cross-spectral density matrix is diagonal, i.e. of the form

\overset{\leftrightarrow}{W} (ρ_{1}^{'}, ρ_{2}^{'}; 0) = (\begin{matrix} W_{x x} (ρ_{1}^{'}, ρ_{2}^{'}; 0) & 0 \\ 0 & W_{y y} (ρ_{1}^{'}, ρ_{2}^{'}; 0) \end{matrix}) .

The degree of polarization of the initial source beam at point

ρ^{'}

can be expressed as follows [40–48]

P_{0} (ρ^{'}; 0) = \sqrt{1 - \frac{4 D e t \overset{\leftrightarrow}{W} (ρ^{'}, ρ^{'}; 0)}{{[T r \overset{\leftrightarrow}{W} (ρ^{'}, ρ^{'}; 0)]}^{2}}} .

Under the condition of

W_{x x} (ρ^{'}, ρ^{'}; 0) = 0

or

W_{y y} (ρ^{'}, ρ^{'}; 0) = 0

, the EGSM beam reduces to a scalar GSM beam with

P_{0} (ρ^{'}; 0) = 1

.

In the following numerical examples, we set $σ_{x} = σ_{y} = 0.02 m$ and $A_{x} = 1$ unless stated otherwise. In this case, the polarization properties are uniform across the source plane with $P_{0} (r; ω) = | \frac{A_{x}^{2} - A_{y}^{2}}{A_{x}^{2} + A_{y}^{2}} | .$ Figure 1 shows the dependence of the degree of polarization in the source plane on $A_{y}$ . It is clear from Fig. 1 that the degree of polarization in the source plane varies as the value of $A_{y}$ changes, any nonzero $P_{0}$ can be achieved either for $A_{y} < A_{x}$ or for $A_{y} > A_{x}$ .

Fig. 1 Dependence of the degree of polarization at source plane on $A_{y}$ with $A_{x} = 1$

Download Full Size | PDF

First we study the properties of the $M^{2}$ -factor of an EGSM beam in free space. We calculate in Fig. 2 the dependence of the $M^{2}$ -factor of an EGSM beam on its degree of polarization in the source plane (z = 0) for two different cases with $δ_{x x} = 0.004 m, δ_{y y} = 0.01 m$ . One finds from Fig. 2 that the $M^{2}$ -factor of an EGSM beam is closely determined by its degree of polarization at z = 0. Its value increases as the degree of polarization increases for the case of $A_{y} < A_{x}$ or decreases with increase of the degree of polarization for the case of $A_{y} > A_{x}$ . This is caused by the fact that the contribution of the element $W_{x x} (ρ_{1}^{'}, ρ_{2}^{'}; 0)$ to the $M^{2}$ -factor dominates that of the element $W_{y y} (ρ_{1}^{'}, ρ_{2}^{'}; 0)$ for the case of $A_{y} < A_{x}$ , and the contribution of the element $W_{y y} (ρ_{1}^{'}, ρ_{2}^{'}; 0)$ plays a dominant role otherwise.

Fig. 2 Dependence of the $M^{2}$ -factor of an EGSM beam in free space on its degree of polarization at source plane for two different cases (a) $A_{y} < A_{x}$ , (b) $A_{y} > A_{x}$

Download Full Size | PDF

Figure 3 shows the dependence of the $M^{2}$ -factor of an EGSM beam in free space on the r.m.s width ( $σ_{x}$ ) of the spectral density along x direction with $A_{y} < A_{x}$ , $P_{0} = 0.6$ , $σ_{y} = 0.5 σ_{x}$ , $δ_{x x} = 0.004 m$ , $δ_{y y} = 0.01 m$ . From Fig. 3 it is clearly seen that the value of the $M^{2}$ -factor of an EGSM beam in free space increases as its r.m.s. width of the spectral density increases. Figure 4 shows the dependence of the $M^{2}$ -factor of an EGSM beam in free space on the r.m.s width ( $δ_{x x}$ ) of auto-correlation functions of the x component of the field with $A_{y} < A_{x}$ , $P_{0} = 0.6$ , $δ_{y y} = 0.5 δ_{x x}$ , $σ_{x} = 0.025 m, σ_{y} = 0.01 m$ . One finds from Fig. 4 that the value of the $M^{2}$ -factor of an EGSM beam decreases as the correlation factors $δ_{x x}$ and $δ_{y y}$ decrease, due to the fact that the spectral degree of coherence $\overset{\leftrightarrow}{W} (ρ_{1}^{'}, ρ_{2}^{'}; 0) = Tr \overset{\leftrightarrow}{W} (ρ_{1}^{'}, ρ_{2}^{'}; 0) / \sqrt{Tr \overset{\leftrightarrow}{W} (ρ_{1}^{'}, ρ_{1}^{'}; 0) Tr \overset{\leftrightarrow}{W} (ρ_{2}^{'}, ρ_{2}^{'}; 0)}$ decreases with the decrease of the correlation factors.

Fig. 3 Dependence of the $M^{2}$ -factor of an EGSM beam in free space on the r.m.s width ( $σ_{x}$ ) of the spectral density along x direction

Download Full Size | PDF

Fig. 4 Dependence of the $M^{2}$ -factor of an EGSM beam in free space on the r.m.s width ( $δ_{x x}$ ) of auto-correlation functions of the x component of the field

Download Full Size | PDF

In what follows we study the properties of the normalized $M^{2}$ -factor of an EGSM beam on propagation in turbulent atmosphere. It is clear from Eq. (19) that the $M^{2}$ -factor of an EGSM beam is now determined by both the parameters of the beam and of the turbulence together. We calculate in Fig. 5 the normalized $M^{2}$ -factor of an EGSM beam on propagation in turbulent atmosphere for different values of the initial degree of polarization with $δ_{x x} = 0.004 m$ , $δ_{y y} = 0.01 m$ , $λ = 632.8 n m$ , $C_{n}^{2} = 10^{- 14} m^{- 2 / 3}$ , $l_{0} = 0.01 m$ . As seen from Fig. 5, unlike its propagation-invariant properties in free-space, the normalized $M^{2}$ -factor increases on propagation in turbulent atmosphere, which means the beam quality of an EGSM beam degrades. It is clear from Fig. 5(a) that the value of the normalized $M^{2}$ -factor increases more rapidly as the initial degree of polarization $P_{0}$ decreases for the case of $A_{y} < A_{x}$ , which means that the beam quality of a scalar GSM beam is less affected by the atmospheric turbulence than that of an EGSM beam. From Fig. 5(b) we conclude that for the case of $A_{y} > A_{x}$ , the beam quality of an EGSM beam with low initial degree of polarization is less affected by the atmospheric turbulence than that of an EGSM beam with large degree of polarization or that of a scalar GSM beam.

Fig. 5 Normalized $M^{2}$ -factor of an EGSM beam on propagation in turbulent atmosphere for different values of the initial degree of polarization

Download Full Size | PDF

Figure 6 shows the normalized $M^{2}$ -factor of an EGSM beam on propagation in turbulent atmosphere for different initial r.m.s. correlations $δ_{x x}$ and $δ_{y y}$ with $A_{y} < A_{x}$ , $P_{0} = 0.6$ , $λ = 632.8 n m$ , $C_{n}^{2} = 10^{- 14} m^{- 2 / 3}$ , $l_{0} = 0.01 m$ . One finds from Fig. 6 that the value of the normalized $M^{2}$ -factor increases slower on propagation as the initial correlation factors decreases, implying that an EGSM beam with low initial spectral degree of coherence is less affected by the atmospheric turbulence. Figure 7 shows the normalized $M^{2}$ -factor of an EGSM beam on propagation in turbulent atmosphere for different initial r.m.s. widths $σ_{x}$ and $σ_{y}$ of the spectral densities with $A_{y} < A_{x}$ , $P_{0} = 0.6$ , $δ_{x x} = 0.004 m$ , $δ_{y y} = 0.01 m$ , $λ = 632.8 n m$ , $C_{n}^{2} = 10^{- 14} m^{- 2 / 3}$ , $l_{0} = 0.01 m$ . Figure 8 shows the normalized $M^{2}$ -factor of an EGSM beam on propagation in turbulent atmosphere for different values of wavelengthλ with $A_{y} < A_{x}$ , $P_{0} = 0.6$ , $σ_{x} = σ_{y} = 0.02 m$ , $δ_{x x} = 0.004 m$ , $δ_{y y} = 0.01 m$ , $C_{n}^{2} = 10^{- 14} m^{- 2 / 3}$ , $l_{0} = 0.01 m$ . The results in Fig. 7 and Fig. 8 lead to the conclusion that the EGSM beam with larger r.m.s. width of the spectral densities and longer wavelength λ is less affected by the atmospheric turbulence.

Fig. 6 Normalized $M^{2}$ -factor of an EGSM beam on propagation in turbulent atmosphere for different initial correlation factors $δ_{x x}$ and $δ_{y y}$

Download Full Size | PDF

Fig. 7 Normalized $M^{2}$ -factor of an EGSM beam on propagation in turbulent atmosphere for different initial r.m.s widths $σ_{x}$ and $σ_{y}$ of the spectral densities

Download Full Size | PDF

Fig. 8 Normalized $M^{2}$ -factor of an EGSM beam on propagation in turbulent atmosphere for different values of wavelengthλ

Download Full Size | PDF

The parameters of turbulence also affect the evolution properties of the normalized $M^{2}$ -factor in turbulent atmosphere. We calculate in Fig. 9 the normalized $M^{2}$ -factor of an EGSM beam on propagation using different values of the structure constant ( $C_{n}^{2}$ ) of the turbulent atmosphere with $A_{y} < A_{x}$ , $P_{0} = 0.6$ , $δ_{x x} = 0.004 m$ , $δ_{y y} = 0.01 m$ , $λ = 632.8 n m$ and $l_{0} = 0.01 m$ . Figure 10 shows the normalized $M^{2}$ -factor of an EGSM beam on propagation in turbulent atmosphere for different values of inner scale of the turbulence ( $l_{0}$ ) with $A_{y} < A_{x}$ , $P_{0} = 0.6$ , $δ_{x x} = 0.004 m$ , $δ_{y y} = 0.01 m$ , $λ = 632.8 n m$ and $C_{n}^{2} = 10^{- 14} m^{- 2 / 3}$ . As seen in Fig. 9 and Fig. 10, the normalized $M^{2}$ -factor increases more rapidly as $C_{n}^{2}$ increases or $l_{0}$ decreases.

Fig. 9 Normalized $M^{2}$ -factor of an EGSM beam on propagation using different values of the structure constant ( $C_{n}^{2}$ ) of the turbulent atmosphere

Download Full Size | PDF

Fig. 10 Normalized $M^{2}$ -factor of an EGSM beam on propagation in turbulent atmosphere for different values of inner scale of the turbulence ( $l_{0}$ )

Download Full Size | PDF

4. Conclusion

We have derived the analytical formula for the $M^{2}$ -factor of an EGSM beam valid for both free space and atmospheric propagation by means of the extended Huygens-Fresnel integral and the second-order moments of the Wigner distribution function. Our numerical results show that the initial degree of polarization, the r.m.s. width of the spectral densities and the r.m.s. correlation coefficients together determine the $M^{2}$ -factor of an EGSM beam in free space, whose value is independent of the propagation distance. In turbulent atmosphere, the parameters of the turbulence also affect the $M^{2}$ -factor of this class of beam on propagation. Furthermore, the EGSM beam with lower degree of polarization, lower correlation factors, larger r.m.s. widths of the spectral densities and longer wavelength is less affected by the atmospheric turbulence under suitable conditions. Our results may find applications in long-distance free-space optical communications.

Acknowledgments

Yangjian Cai acknowledges the support by the National Natural Science Foundation of China under Grant No. 10904102, the Foundation for the Author of National Excellent Doctoral Dissertation of PR China under Grant No. 200928, the Natural Science of Jiangsu Province under Grant No. BK2009114 and the Huo Ying Dong Education Foundation of China under Grant No. 121009. O. Korotkova's research is funded by the US AFOSR (grant FA 95500810102) and US ONR (grant N0018909P1903).

References and links

1. D. F. V. James, “Changes of polarization of light beams on propagation in free space,” J. Opt. Soc. Am. A 11(5), 1641–1643 (1994). [CrossRef]

2. F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello, and R. Simon, “Partially polarized Gaussian schell-model beams,” J. Opt. A, Pure Appl. Opt. 3(1), 1–9 (2001). [CrossRef]

3. G. Piquero, F. Gori, P. Romanini, M. Santarsiero, R. Borghi, and A. Mondello, “Synthesis of partially polarized Gaussian Schell-model sources,” Opt. Commun. 208(1-3), 9–16 (2002). [CrossRef]

4. E. Wolf, “Unified theory of coherence and polarization of random electromagnetic beams,” Phys. Lett. A 312(5-6), 263–267 (2003). [CrossRef]

5. E. Wolf and E. Collett, “Partially coherent sources which produce same far-field intensity distribution as a laser,”Opt. Commun . 25(3), 293–296 (1978). [CrossRef]

6. F. Gori, “Collet-Wolf sources and multimode lasers,” Opt. Commun. 34(3), 301–305 (1980). [CrossRef]

7. A. T. Friberg and R. J. Sudol, “Propagation parameters of Gaussian Schell-model beams,” Opt. Commun. 41(6), 383–387 (1982). [CrossRef]

8. F. Wang and Y. Cai, “Experimental observation of fractional Fourier transform for a partially coherent optical beam with Gaussian statistics,” J. Opt. Soc. Am. A 24(7), 1937–1944 (2007). [CrossRef]

9. E. Wolf, Introduction to the theory of coherence and polarization of light (Cambridge U. Press, 2007).

10. F. Gori, “Matrix treatment for partially polarized, partially coherent beams,” Opt. Lett. 23(4), 241–243 (1998). [CrossRef]

11. G. P. Agrawal and E. Wolf, “Propagation-induced polarization changes in partially coherent optical beams,” J. Opt. Soc. Am. A 17(11), 2019–2023 (2000). [CrossRef]

12. Y. Cai, D. Ge, and Q. Lin, “Fractional Fourier transform for partially coherent and partially polarized Gaussian Schell-model beams,” J. Opt. A, Pure Appl. Opt. 5(5), 453–459 (2003). [CrossRef]

13. T. Shirai, O. Korotkova, and E. Wolf, “A method of generating electromagnetic Gaussian Schell-model beams,” J. Opt. A, Pure Appl. Opt. 7(5), 232–237 (2005). [CrossRef]

14. F. Gori, M. Santarsiero, R. Borghi, and V. Ramírez-Sánchez, “Realizability condition for electromagnetic Schell-model sources,” J. Opt. Soc. Am. A 25(5), 1016–1021 (2008). [CrossRef]

15. B. Kanseri and H. C. Kandpal, “Experimental determination of electric cross-spectral density matrix and generalized Stokes parameters for a laser beam,” Opt. Lett. 33(20), 2410–2412 (2008). [CrossRef] [PubMed]

16. H. Wang, X. Wang, A. Zeng, and K. Yang, “Effects of coherence on anisotropic electromagnetic Gaussian-Schell model beams on propagation,” Opt. Lett. 32(15), 2215–2217 (2007). [CrossRef] [PubMed]

17. D. Ge, Y. Cai, and Q. Lin, “Propagation of partially polarized Gaussian Schell-model beams through aligned and misaligned optical systems,” Chin. Phys. 14(1), 128–132 (2005). [CrossRef]

18. O. Korotkova, M. Salem, and E. Wolf, “Beam conditions for radiation generated by an electromagnetic Gaussian Schell-model source,” Opt. Lett. 29(11), 1173–1175 (2004). [CrossRef] [PubMed]

19. M. Yao, Y. Cai, H. T. Eyyuboğlu, Y. Baykal, and O. Korotkova, “Evolution of the degree of polarization of an electromagnetic Gaussian Schell-model beam in a Gaussian cavity,” Opt. Lett. 33(19), 2266–2268 (2008). [CrossRef] [PubMed]

20. O. Korotkova, M. Yao, Y. Cai, H. T. Eyyuboğlu, and Y. Baykal, “The state of polarization of a stochastic electromagnetic beam in an optical resonator,” J. Opt. Soc. Am. A 25(11), 2710–2720 (2008). [CrossRef]

21. Z. Tong, O. Korotkova, Y. Cai, H. T. Eyyuboglu, and Y. Baykal, “Correlation properties of random electromagnetic beams in laser resonators,” Appl. Phys. B 97(4), 849–857 (2009). [CrossRef]

22. Y. Cai and O. Korotkova, “Twist phase-induced polarization changes in electromagnetic Gaussian Schell-model beams,” Appl. Phys. B 96(2-3), 499–507 (2009). [CrossRef]

23. C. Zhao, Y. Cai, and O. Korotkova, “Radiation force of scalar and electromagnetic twisted Gaussian Schell-model beams,” Opt. Express 17(24), 21472–21487 (2009). [CrossRef] [PubMed]

24. S. Zhu and Y. Cai, “Spectral shift of a twisted electromagnetic Gaussian Schell-model beam focused by a thin lens,” Appl. Phys. B 99(1-2), 317–323 (2010). [CrossRef]

25. A. Ishimaru, Wave Propagation and Scattering in Random Media, Vol. 2, (Academic Press, New York, 1978).

26. L. C. Andrews, R. L. Phillips, and C. Y. Hopen, Laser Beam Scintillation with Applications (SPIE Press, Washington, 2001).

27. L. C. Andrews, and R. L. Phillips, Laser beam propagation in the turbulent atmosphere, 2nd edition, SPIE press, Bellington, 2005.

28. J. C. Ricklin and F. M. Davidson, “Atmospheric turbulence effects on a partially coherent Gaussian beam: implications for free-space laser communication,” J. Opt. Soc. Am. A 19(9), 1794–1802 (2002). [CrossRef]

29. H. T. Eyyuboğlu and Y. Baykal, “Analysis of reciprocity of cos-Gaussian and cosh- Gaussian laser beams in a turbulent atmosphere,” Opt. Express 12(20), 4659–4674 (2004). [CrossRef] [PubMed]

30. H. T. Eyyuboğlu, C. Arpali, and Y. K. Baykal, “Flat topped beams and their characteristics in turbulent media,” Opt. Express 14(10), 4196–4207 (2006). [CrossRef] [PubMed]

31. Y. Cai and S. He, “Propagation of a partially coherent twisted anisotropic Gaussian Schell-model beam in a turbulent atmosphere,” Appl. Phys. Lett. 89(4), 041117 (2006). [CrossRef]

32. R. J. Noriega-Manez and J. C. Gutiérrez-Vega, “Rytov theory for Helmholtz-Gauss beams in turbulent atmosphere,” Opt. Express 15(25), 16328–16341 (2007). [CrossRef] [PubMed]

33. M. Alavinejad, B. Ghafary, and F. D. Kashani, “Analysis of the propagation of flat-topped beam with various beam orders through turbulent atmosphere,” Opt. Lasers Eng. 46(1), 1–5 (2008). [CrossRef]

34. Y. Cai and S. He, “Propagation of various dark hollow beams in a turbulent atmosphere,” Opt. Express 14(4), 1353–1367 (2006). [CrossRef] [PubMed]

35. Y. Cai and S. He, “Average intensity and spreading of an elliptical gaussian beam propagating in a turbulent atmosphere,” Opt. Lett. 31(5), 568–570 (2006). [CrossRef] [PubMed]

36. Y. Cai, Y. Chen, H. T. Eyyuboğlu, and Y. Baykal, “Scintillation index of elliptical Gaussian beam in turbulent atmosphere,” Opt. Lett. 32(16), 2405–2407 (2007). [CrossRef] [PubMed]

37. H. Lin and J. Pu, “Propagation properties of partially coherent radially polarized beam in a turbulent atmosphere,” J. Mod. Opt. 56(11), 1296–1303 (2009). [CrossRef]

38. Y. Cai, Q. Lin, H. T. Eyyuboğlu, and Y. Baykal, “Average irradiance and polarization properties of a radially or azimuthally polarized beam in a turbulent atmosphere,” Opt. Express 16(11), 7665–7673 (2008). [CrossRef] [PubMed]

39. Y. Yuan, Y. Cai, J. Qu, H. T. Eyyuboğlu, and Y. Baykal, “Average intensity and spreading of an elegant Hermite-Gaussian beam in turbulent atmosphere,” Opt. Express 17(13), 11130–11139 (2009). [CrossRef] [PubMed]

40. O. Korotkova, M. Salem, and E. Wolf, “The far-zone behavior of the degree of polarization of electromagnetic beams propagating through atmospheric turbulence,” Opt. Commun. 233(4-6), 225–230 (2004). [CrossRef]

41. H. Roychowdhury, S. A. Ponomarenko, and E. Wolf, “Change in the polarization of partially coherent electromagnetic beams propagating through the turbulent atmosphere,” J. Mod. Opt. 52(11), 1611–1618 (2005). [CrossRef]

42. O. Korotkova, M. Salem, A. Dogariu, and E. Wolf, “Changes in the polarization ellipse of random electromagnetic beams propagating through the turbulent atmosphere,” Waves Random Complex Media 15(3), 353–364 (2005). [CrossRef]

43. Z. Chen and J. Pu, “Propagation characteristics of aberrant stochastic electromagnetic beams in a turbulent atmosphere,” J. Opt. A, Pure Appl. Opt. 9(12), 1123–1130 (2007). [CrossRef]

44. H. Wang, X. Wang, A. Zeng, and W. Lu, “Changes in the coherence of quasi-monochromatic electromagnetic Gaussian Schell-model beams propagating through turbulent atmosphere,” Opt. Commun. 276(2), 218–221 (2007). [CrossRef]

45. H. T. Eyyuboğlu, Y. Baykal, and Y. Cai, “Degree of polarization for partially coherent general beams in turbulent atmosphere,” Appl. Phys. B 89(1), 91–97 (2007). [CrossRef]

46. Y. Cai, O. Korotkova, H. T. Eyyuboğlu, and Y. Baykal, “Active laser radar systems with stochastic electromagnetic beams in turbulent atmosphere,” Opt. Express 16(20), 15834–15846 (2008). [CrossRef] [PubMed]

47. O. Korotkova, “Scintillation index of a stochastic electromagnetic beam propagating in random media,” Opt. Commun. 281(9), 2342–2348 (2008).

48. O. Korotkova, Y. Cai, and E. Watson, “Stochastic electromagnetic beams for radar systems operating through turbulent atmosphere,” Appl. Phys. B 94(4), 681–690 (2009). [CrossRef]

49. A. E. Siegman, “New developments in laser resonators,” in Optical Resonators, D. A. Holmes, ed., Proc. SPIE 1224, 2C14 (1990).

50. R. Martinez-Herrero and P. M. Mejias, “Second-order spatial characterization of hard-edge diffracted beams,” Opt. Lett. 18(19), 1669–1671 (1993). [CrossRef] [PubMed]

51. M. Santarsiero, F. Gori, R. Borghi, G. Cincotti, and P. Vahimaa, “Spreading properties of beams radiated by partially coherent Schell-model sources,” J. Opt. Soc. Am. A 16(1), 106–112 (1999). [CrossRef]

52. B. Zhang, X. Chu, and Q. Li, “Generalized beam-propagation factor of partially coherent beams propagating through hard-edged apertures,” J. Opt. Soc. Am. A 19(7), 1370–1375 (2002). [CrossRef]

53. X. Ji, T. Zhang, and X. Jia, “Beam propagation factor of partially coherent Hermite–Gaussian array beams,” J. Opt. A, Pure Appl. Opt. 11(10), 105705 (2009). [CrossRef]

54. G. Zhou, “Generalzied M² factors of truncated partially coherent Lorentz and Lorentz-Gauss beams,” J. Opt. A, Pure Appl. Opt. 12(1), 015701 (2010).

55. Y. Dan and B. Zhang, “Beam propagation factor of partially coherent flat-topped beams in a turbulent atmosphere,” Opt. Express 16(20), 15563–15575 (2008). [CrossRef] [PubMed]

56. Y. Dan and B. Zhang, “Second moments of partially coherent beams in atmospheric turbulence,” Opt. Lett. 34(5), 563–565 (2009). [CrossRef] [PubMed]

57. Y. Yuan, Y. Cai, J. Qu, H. T. Eyyuboğlu, Y. Baykal, and O. Korotkova, “M²-factor of coherent and partially coherent dark hollow beams propagating in turbulent atmosphere,” Opt. Express 17(20), 17344–17356 (2009). [CrossRef] [PubMed]

Propagation factor of a stochastic electromagnetic Gaussian Schell-model beam

Abstract

1. Introduction

2. Theory

3. Numerical examples

4. Conclusion

Acknowledgments

References and links

Cited By

Figures (10)

Equations (25)

Optics Express