Coherence and polarization properties of a radially polarized beam with variable spatial coherence

Gaofeng Wu; Fei Wang; Yangjian Cai

doi:10.1364/OE.20.028301

1. Introduction

In the past decades, as a typical kind of cylindrical vector beam with spatially non-unifrom state of polarization [1], radially polarized beam has been studied extensively in both theory and experiment due to its interesting and unique focusing properties, and has been found wide applications in microscopy, lithography, free space optical communications, electron acceleration, proton acceleration, particle trapping, material processing, optical data storage, high-resolution metrology, super-resolution imaging, plasmonic focusing, and laser machining [1–15]. Different methods have been developed to generate radially polarized beam [1]. Paraxial and nonparaxial propagation properties of radially polarized beam have been explored in detail [16–20]. Most of previous literatures on cylindrical vector beam or RP beam have been confined to coherent beam.

Coherence is one of the important characteristics of light beams [21]. Recently, partially coherent vector beam attracts more and more attentions [22–40]. In 2003, Wolf proposed a unified theory of coherence and polarization for partially coherent vector beam [23], which can be applied to study the changes of the statistical properties such as degree of coherence, degree of polarization and sate of polarization of such beam upon propagation. Partially coherent vector beam with spatially uniform state of polarization usually is called stochastic electromagnetic beam in spatial frequency domain or partially coherent and partially polarized beam in space-time domain [22–24]. Characterization, generation and propagation of stochastic electromagnetic beam have been studied extensively due to its important applications in free-space optical communications, optical imaging, active laser radar systems and remote sensing [25–51].

Partially coherent vector beam with spatially non-uniform state of polarization called cylindrical vector partially coherent beam was introduced recently [52, 53]. Partially coherent RP beam can be regarded as a special case of cylindrical vector partially coherent beam [54–57]. More recently, partially coherent RP beam with variable spatial coherence was generated in experiment [57], and it was found that we can shape the beam profile of the focused RP beam by varying its initial spatial coherence, which is useful for material thermal processing and particle trapping.

For a stochastic electromagnetic beam, it has been shown that its spectral degree of coherence, spectral degree of polarization and state of polarization change on propagation [22–48]. Coherence-induced polarization changes of a stochastic electromagnetic beam were demonstrated both theoretically [49] and experimentally [50]. More importantly, it was found that the stochastic electromagnetic beam could be used for sensing semi-rough target by comparing coherence and polarization properties of the source beam and of the return beam in turbulent atmosphere [51]. In this paper, our aim is to explore the coherence and polarization properties of a partially coherent RP beam both theoretically and experimentally. Our results show that the degree of coherence and the degree of polarization of a focused partially coherent RP beam varies on propagation, while the state of polarization of the completely polarized part of such beam remains invariant.

2. Theory

We first outline briefly the theoretical model for a partially coherent RP beam, then we study its coherence and polarization properties. The vector electric field of a coherent RP beam can be expressed as the superposition of orthogonally polarized Hermite Gaussian HG₁₀ and HG₀₁ modes [1]

\vec{E} (x, y) = E_{x} (x, y) {\vec{e}}_{x} + E_{y} (x, y) {\vec{e}}_{y} = \frac{x}{ω_{0}} \exp (- \frac{x^{2} + y^{2}}{ω_{0}^{2}}) {\vec{e}}_{x} + \frac{y}{ω_{0}} \exp (- \frac{x^{2} + y^{2}}{ω_{0}^{2}}) {\vec{e}}_{y},

where

ω_{0}

denotes the beam waist size of a fundamental Gaussian mode.

{\vec{e}}_{x}

and

{\vec{e}}_{y}

represent the unit vectors in the x and y directions, respectively.

Based on the unified theory of coherence and polarization, the second-order correlation properties of a partially coherent vector beam at z = 0, in space–frequency domain, can be characterized by the $2 \times 2$ cross-spectral density (CSD) matrix of the electric field, defined by the formula [22, 23]

\overset{\leftrightarrow}{W} (x_{1}, y_{1}, x_{2}, y_{2}, 0) = (\begin{array}{l} W_{x x} (x_{1}, y_{1}, x_{2}, y_{2}, 0) W_{x y} (x_{1}, y_{1}, x_{2}, y_{2}, 0) \\ W_{y x} (x_{1}, y_{1}, x_{2}, y_{2}, 0) W_{y y} (x_{1}, y_{1}, x_{2}, y_{2}, 0) \end{array}),

where

W_{α β} (x_{1}, y_{1}, x_{2}, y_{2}, 0) = 〈 E_{α} (x_{1}, y_{1}, 0) E_{β}^{*} (x_{2}, y_{2}, 0) 〉 . (α = x, y; β = x, y)

Here

(x_{1}, y_{1})

and

(x_{2}, y_{2})

denote the coordinates of two arbitrary points at the source plane, E_x and E_y denote the components of the random electric vector, along two mutually orthogonal x and y directions perpendicular to the z-axis. The angular brackets denote ensemble average and the asterisk denotes the complex conjugate.

The element of the CSD matrix of a partially coherent RP beam radiated from a Schell-model source [21,22] can be expressed as [52,53]

W_{x x} (x_{1}, y_{1}, x_{2}, y_{2}, 0) = \frac{x_{1} x_{2}}{ω_{0}^{2}} \exp [- \frac{x_{1}^{2} + y_{1}^{2} + x_{2}^{2} + y_{2}^{2}}{ω_{0}^{2}} - \frac{{(x_{1} - x_{2})}^{2} + {(y_{1} - y_{2})}^{2}}{2 δ_{x x}^{2}}],

W_{x y} (x_{1}, y_{1}, x_{2}, y_{2}, 0) = B_{x y} \frac{x_{1} y_{2}}{ω_{0}^{2}} \exp [- \frac{x_{1}^{2} + y_{1}^{2} + x_{2}^{2} + y_{2}^{2}}{ω_{0}^{2}} - \frac{{(x_{1} - x_{2})}^{2} + {(y_{1} - y_{2})}^{2}}{2 δ_{x y}^{2}}],

W_{y x} (x_{1}, y_{1}, x_{2}, y_{2}, 0) = W_{x y}^{*} (x_{2}, y_{2}, x_{1}, y_{1}, 0),

W_{y y} (x_{1}, y_{1}, x_{2}, y_{2}, 0) = \frac{y_{1} y_{2}}{ω_{0}^{2}} \exp [- \frac{x_{1}^{2} + y_{1}^{2} + x_{2}^{2} + y_{2}^{2}}{ω_{0}^{2}} - \frac{{(x_{1} - x_{2})}^{2} + {(y_{1} - y_{2})}^{2}}{2 δ_{y y}^{2}}],

where

B_{x y} = | B_{x y} | \exp (i ϕ)

is the correlation coefficient between the E_x and E_y field components,

ϕ

is the phase difference between the x-and y-components of the field,

δ_{x x}

,

δ_{y y}

and

δ_{x y}

are the 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. The parameters

B_{x y}

,

δ_{x x}

,

δ_{y y}

and

δ_{x y}

satisfy the realizability conditions

B_{x y} = 1

,

δ_{x x} = δ_{y y} = δ_{x y}

(see Appendix A).

Within the validity of the paraxial approximation, propagation of the elements of the CSD matrix of a partially coherent vector beam through ABCD optical system can be studied with the help of the following generalized Collins formula [58]

\begin{array}{l} W_{α β} (u_{1}, v_{1}, u_{2}, v_{2}, z) = {(\frac{1}{λ | B |})}^{2} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} W_{α β} (x_{1}, y_{1}, x_{2}, y_{2}, 0) \\ \times \exp [- \frac{i k}{2 B} (A x_{1}^{2} - 2 x_{1} u_{1} + D u_{1}^{2}) - \frac{i k}{2 B} (A y_{1}^{2} - 2 y_{1} v_{1} + D v_{1}^{2})] \\ \times \exp [\frac{i k}{2 B^{*}} (A^{*} x_{2}^{2} - 2 x_{2} u_{2} + D^{*} u_{2}^{2}) + \frac{i k}{2 B^{*}} (A^{*} y_{2}^{2} - 2 y_{2} v_{2} + D^{*} y_{2}^{2})] d x_{1} d x_{2} d y_{1} d y_{2}, \end{array}

where

k = 2 π / λ

is the wave number with

λ

being the wavelength, A, B, C and D are the transfer matrix elements of optical system and the asterisk is required for a general optical system with loss or gain, although it does not appear in Eq. (13) of [58]. Equation (8) is also valid for the propagation of the elements of the CSD matrix of an electromagnetic Gaussian Schell-model (GSM) beam. With the help of Eq. (8), the propagation properties of an electromagnetic GSM beam through paraxial ABCD optical system, resonator and turbulent atmosphere have been studied in [42, 43, 47, 48]. In fact, Eq. (8) also can be used to treat the propagation properties of an electromagnetic GSM beam in free space by setting

A = 1, B = z, C = 0, D = 1

in Eq. (8) [28]. It has been found that the degree of polarization of the electromagnetic GSM beam varies on propagation even in free space [28, 42, 43, 47, 48].

Substituting from Eqs. (4)-(7) into Eq. (8), after some tedious integration and operations, we obtain the following expressions for the elements of the CSD matrix of a partially coherent RP beam after propagating through ABCD optical system

\begin{array}{l} W_{x x} (u_{1}, v_{1}, u_{2}, v_{2}, z) = = \frac{π^{2}}{8 δ_{x x}^{2} M_{1 x x}^{2} M_{2 x x}^{3} ω_{0}^{2}} {(\frac{1}{λ | B |})}^{2} \exp [\frac{i k D}{2 B} (u_{1}^{2} + v_{1}^{2}) - \frac{i k D^{*}}{2 B^{*}} (u_{2}^{2} + v_{2}^{2})] \\ \times \exp [- \frac{k^{2} (u_{1}^{2} + v_{1}^{2})}{4 M_{1 x x} B^{2}} - \frac{4 k^{2} M_{1 x x}^{2} B^{2} δ_{x x}^{4} (u_{2}^{2} + v_{2}^{2}) + k^{2} {(B^{*})}^{2} (u_{1}^{2} + v_{1}^{2}) - 4 k^{2} M_{1 x x} {| B |}^{2} δ_{x x}^{2} (u_{1} u_{2} + v_{1} v_{2})}{16 M_{1 x x}^{2} M_{2 x x} {| B |}^{4} δ_{x x}^{4}}] \\ \times [- k^{2} {(\frac{2 M_{1 x x} B δ_{x x}^{2} u_{2} - B^{*} u_{1}}{2 M_{1 x x} {| B |}^{2} δ_{x x}^{2}})}^{2} + \frac{2 M_{2 x x} k^{2} δ_{x x}^{2} u_{1}}{B} (\frac{2 M_{1 x x} B δ_{x x}^{2} u_{2} - B^{*} u_{1}}{2 M_{1 x x} {| B |}^{2} δ_{x x}^{2}}) + 2 M_{2 x x}], \end{array}

\begin{array}{l} W_{x y} (u_{1}, v_{1}, u_{2}, v_{2}, z) = \frac{B_{x y} k^{2} π^{2}}{8 M_{1 x y}^{2} M_{2 x y}^{3} δ_{x y}^{2} w_{0}^{2}} {(\frac{1}{λ | B |})}^{2} \exp [\frac{i k D}{2 B} (u_{1}^{2} + v_{1}^{2}) - \frac{i k D^{*}}{2 B^{*}} (u_{2}^{2} + v_{2}^{2})] \\ \times \exp [- \frac{k^{2} (u_{2}^{2} + v_{2}^{2})}{4 M_{1 x y} B^{*}^{2}} - \frac{4 k^{2} M_{1 x y}^{2} B^{*}^{2} δ_{x y}^{4} (u_{1}^{2} + v_{1}^{2}) + k^{2} B^{2} (u_{2}^{2} + v_{2}^{2}) - 4 k^{2} M_{1 x y} {| B |}^{2} δ_{x y}^{2} (u_{1} u_{2} + v_{1} v_{2})}{16 M_{1 x y}^{2} M_{2 x y} {| B |}^{4} δ_{x y}^{4}}] \\ \times (\frac{u_{2}}{2 M_{1 x y} B^{*} δ_{x y}^{2}} - \frac{u_{1}}{B}) (\frac{v_{1}}{B} - \frac{1 + 4 M_{1 x y} M_{2 x y} δ_{x y}^{4}}{2 M_{1 x y} B^{*} δ_{x y}^{2}} v_{2}), \end{array}

W_{y x} (u_{1}, v_{1}, u_{2}, v_{2}, z) = W_{x y}^{*} (u_{2}, v_{2}, u_{1}, v_{1}, z),

\begin{array}{l} W_{y y} (u_{1}, v_{1}, u_{2}, v_{2}, z) = \frac{π^{2}}{8 δ_{y y}^{2} M_{1 y y}^{2} M_{2 y y}^{3} ω_{0}^{2}} {(\frac{1}{λ | B |})}^{2} \exp [\frac{i k D}{2 B} (u_{1}^{2} + v_{1}^{2}) - \frac{i k D^{*}}{2 B^{*}} (u_{2}^{2} + v_{2}^{2})] \\ \times \exp [- \frac{k^{2} (u_{1}^{2} + v_{1}^{2})}{4 M_{1 y y} B^{2}} - \frac{4 k^{2} M_{1 y y}^{2} B^{2} δ_{y y}^{4} (u_{2}^{2} + v_{2}^{2}) + k^{2} {(B^{*})}^{2} (u_{1}^{2} + v_{1}^{2}) - 4 k^{2} M_{1 y y} {| B |}^{2} δ_{y y}^{2} (u_{1} u_{2} + v_{1} v_{2})}{16 M_{1 y y}^{2} M_{2 y y} {| B |}^{4} δ_{y y}^{4}}] \\ \times [- k^{2} {(\frac{2 M_{1 y y} B δ_{y y}^{2} v_{2} - B^{*} v_{1}}{2 M_{1 y y} {| B |}^{2} δ_{y y}^{2}})}^{2} + \frac{2 M_{2 y y} k^{2} δ_{y y}^{2} v_{1}}{B} (\frac{2 M_{1 y y} B δ_{y y}^{2} v_{2} - B^{*} v_{1}}{2 M_{1 y y} {| B |}^{2} δ_{y y}^{2}}) + 2 M_{2 y y}], \end{array}

Where

M_{1 x x} = 1 / ω_{0}^{2} + 1 / (2 δ_{x x}^{2}) - i k A / (2 B), M_{2 x x} = 1 / ω_{0}^{2} + 1 / (2 δ_{x x}^{2}) + i k A^{*} / (2 B^{*}) - 1 / (4 M_{1 x x} δ_{x x}^{4}),

M_{1 x y} = 1 / ω_{0}^{2} + 1 / (2 δ_{x y}^{2}) + i k A^{*} / (2 B^{*}), M_{2 x y} = 1 / ω_{0}^{2} + 1 / (2 δ_{x y}^{2}) - i k A / (2 B) - 1 / (4 M_{1 x y} δ_{x y}^{4}),

M_{1 y y} = 1 / ω_{0}^{2} + 1 / (2 δ_{y y}^{2}) - i k A / (2 B), M_{2 y y} = 1 / ω_{0}^{2} + 1 / (2 δ_{y y}^{2}) + i k A^{*} / (2 B^{*}) - 1 / (4 M_{1 y y} δ_{y y}^{4}) .

The average intensity of the partially coherent RP beam at the output plane is expressed as [22]

〈 I (u, v, z) 〉 = Tr \overset{\leftrightarrow}{W} (u_{1}, v_{1}, u_{2}, v_{2}, z) = W_{x x} (u, v, u, v, z) + W_{y y} (u, v, u, v, z),

where Tr denotes the trace of the matrix.

The degree of polarization of the partially coherent vector beam at point $(u v)$ is defined by the expression [22]

P (u, v, z) = \sqrt{1 - \frac{4 D e t \overset{\leftrightarrow}{W} (u, v, u, v, z)}{{[T r \overset{\leftrightarrow}{W} (u, v, u, v, z)]}^{2}}},

where Det stands for the determinant of the matrix.

It was shown in [52] that a completely polarized cylindrical vector partially coherent beam is depolarized (i.e., becomes partially polarized) after propagation, thus it is important to analyze its state of polarization. The CSD matrix of a partially polarized partially coherent vector beam at point $(u v)$ can be locally represented as a sum of completely polarized beam and a completely unpolarized beam [22, 28]

\overset{\leftrightarrow}{W} (u, v, u, v, z) = {\overset{\leftrightarrow}{W}}^{(u)} (u, v, u, v, z) + {\overset{\leftrightarrow}{W}}^{(p)} (u, v, u, v, z),

where

{\overset{\leftrightarrow}{W}}^{(u)} (u, v, u, v, z) = (\begin{matrix} A (u, v, u, v, z) & 0 \\ 0 & A (u, v, u, v, z) \end{matrix}),

{\overset{\leftrightarrow}{W}}^{(p)} (u, v, u, v, z) = (\begin{matrix} B (u, v, u, v, z) & D (u, v, u, v, z) \\ D^{*} (u, v, u, v, z) & C (u, v, u, v, z) \end{matrix}),

With

\begin{array}{l} A (u, v, u, v, z) = \frac{1}{2} [W_{x x} (u, v, u, v, z) + W_{y y} (u, v, u, v, z) \\ - \sqrt{{[W_{x x} (u, v, u, v, z) - W_{y y} (u, v, u, v, z)]}^{2} + 4 {| W_{x y} (u, v, u, v, z) |}^{2}}], \end{array}

\begin{array}{l} B (u, v, u, v, z) = \frac{1}{2} [W_{x x} (u, v, u, v, z) - W_{y y} (u, v, u, v, z) \\ + \sqrt{{[W_{x x} (u, v, u, v, z) - W_{y y} (u, v, u, v, z)]}^{2} + 4 {| W_{x y} (u, v, u, v, z) |}^{2}}, \end{array}

\begin{array}{l} C (u, v, u, v, z) = \frac{1}{2} [W_{y y} (u, v, u, v, z) - W_{x x} (u, v, u, v, z) \\ + \sqrt{{[W_{x x} (u, v, u, v, z) - W_{y y} (u, v, u, v, z)]}^{2} + 4 {| W_{x y} (u, v, u, v, z) |}^{2}}], \end{array}

D (u, v, u, v, z) = W_{x y} (u, v, u, v, z) .

The state of polarization of the completely polarized beam can be characterized by the polarization ellipse. The major and minor semi-axes of the polarization ellipse, $A_{1}$ and $A_{2}$ , as well as its degree of ellipticity, ε, and its orientation angle, θ, are related to the elements of the CSD matrix $\overset{\leftrightarrow}{W}$ by the following formulas [28]

\begin{array}{l} A_{1, 2} (u, v, z) = \frac{1}{\sqrt{2}} [\sqrt{{[W_{x x} (u, v, u, v, z) - W_{y y} (u, v, u, v, z)]}^{2} + 4 {| W_{x y} (u, v, u, v, z) |}^{2}} \\ {\pm \sqrt{{[W_{x x} (u, v, u, v, z) - W_{y y} (u, v, u, v, z)]}^{2} + 4 {[Re W_{x y} (u, v, u, v, z)]}^{2}}]}^{1 / 2}, \end{array}

ε (u, v, z) = \frac{A_{2} (u, v, u, v, z)}{A_{1} (u, v, u, v, z)},

θ (u, v, z) = \frac{1}{2} arc \tan [\frac{2 Re W_{x y} (u, v, u, v, z)}{W_{x x} (u, v, u, v, z) - W_{y y} (u, v, u, v, z)}] .

In Eq. (25) signs “+” and “

-

” between the two square roots correspond to

A_{1}

and

A_{2}

, respectively. Re stands for taking the real part.

The average intensity of the completely unpolarized beam and the intensity of the completely polarized beam are expressed as

〈 I^{(u)} (u, v, z) 〉 = Tr {\overset{\leftrightarrow}{W}}^{(u)} (u, v, u, v, z), 〈 I^{(p)} (u, v, z) 〉 = Tr {\overset{\leftrightarrow}{W}}^{(p)} (u, v, u, v, z) .

There are two definitions of the degree of coherence for a paraxial partially coherent vector beam [22–24, 30]. For the convenience of experimental measurement, in this paper we adopt the definition introduced by Tervo et al., according to [30], the degree of coherence $μ$ of the partially coherent vector beam at a pair of transverse points with position vectors $(u_{1} v_{1})$ and $(u_{2} v_{2})$ is defined as

μ^{2} (u_{1}, v_{1}, u_{2}, v_{2}, z) = \frac{Tr [\overset{\leftrightarrow}{W} (u_{1}, v_{1}, u_{2}, v_{2}, z) \overset{\leftrightarrow}{W} (u_{2}, v_{2}, u_{1}, v_{1}, z)]}{〈 I (u_{1}, v_{1}, z) 〉 〈 I (u_{2}, v_{2}, z) 〉} .

where

〈 I (u_{i}, v_{i}, z) 〉

is the average intensity of the beam at point

(u_{i} v_{i})

.

If the partially coherent vector beam obeys the Gaussian statistics [21], the numerator in right hand of Eq. (29) can be simplified as [30]

Tr [\overset{\leftrightarrow}{W} (u_{1}, v_{1}, u_{2}, v_{2}, z) \overset{\leftrightarrow}{W} (u_{2}, v_{2}, u_{1}, v_{1}, z)] = {\sum_{α, β} | W_{α β} (u_{1}, v_{1}, u_{2}, v_{2}, z) |}^{2} . (α, β = x, y)

Furthermore, the fourth-order correlation function between two points

(u_{1} v_{1})

and

(u_{2} v_{2})

can be expanded as [45]

G^{(2)} (u_{1}, v_{1}, u_{2}, v_{2}, z) = 〈 I (u_{1}, v_{1}, z) 〉 〈 I (u_{2}, v_{2}, z) 〉 + {\sum_{α, β} | W_{α β} (u_{1}, v_{1}, u_{2}, v_{2}, z) |}^{2} .

Then the normalized fourth-order correlation function can be expressed in terms of the degree of coherence as follows

g^{(2)} (u_{1}, v_{1}, u_{2}, v_{2}, z) = 1 + μ^{2} (u_{1}, v_{1}, u_{2}, v_{2}, z) .

Thus, we can obtain the information of the degree of coherence by measuring the fourth-order correlation function in experiment as shown later in section 3.

Now applying the above derived formulas, we study the polarization and coherence properties of a partially coherent RP beam after passing through a thin lens with focal length $f = 400 mm$ which is located at z = 0. The transfer matrix between the source plane at z = 0 and the output plane at z reads as

(\begin{matrix} A & B \\ C & D \end{matrix}) = (\begin{matrix} 1 & z \\ 0 & 1 \end{matrix}) (\begin{matrix} 1 & 0 \\ - 1 / f & 1 \end{matrix}) = (\begin{matrix} 1 - z / f & z \\ - 1 / f & 1 \end{matrix}) .

Figure 1 shows the degree of polarization of a focused partially coherent RP beam at point $(0 .05mm 0 .05mm)$ versus the propagation distance z for different values of the correlation radii $δ_{x x}, δ_{y y}, δ_{x y}$ with $λ = 632.8 nm, ω_{0} = 1.05 mm .$ From Fig. 1, one finds that the degree of polarization of a coherent RP beam remains invariant on propagation, while the degree of polarization of a partially coherent RP beam decreases with the propagation distance increases, which means the completely polarized partially coherent RP beam is depolarized on propagation as expected [52]. Furthermore, the variation of the degree of polarization on propagation is closely determined by the correlation radii, and the degree of polarization decreases more rapidly as the correlation radii decreases (i.e., the coherence of the beam decreases). We may physically explain this phenomenon by the fact that the degree of polarization denotes the ratio of the intensity of the polarized part of the beam to the total intensity of the beam for a fixed point, while the correlation radii represent the strength of the correlations of the statistical field. Decreasing the values of correlation radii (i.e., decreasing the coherence of the beam) will enhance the relative intensity of the unpolarized part for the fixed point on propagation.

Fig. 1 Degree of polarization of a focused partially coherent RP beam at point $(0 .05mm 0 .05mm)$ versus the propagation distance z for different values of the correlation radii $δ_{x x}, δ_{y y}, δ_{x y} .$

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Now we analyze the state of polarization of a focused partially coherent RP beam. Substituting Eq. (33) into Eqs. (9)-(12) and (25)-(27), we obtain

A_{2} (u, v, u, v, z) = 0, ε (u, v, u, v, z) = 0, θ (u, v, u, v, z) = arc \tan (v / u) .

From Eq. (34), we can come to the conclusion that the completely polarized part of the partially coherent RP beam remains radially polarized on propagation although the completely unpolarized part appears, in other words, the state of polarization of the completely polarized part of the partially coherent RP beam remains invariant on propagation.

We calculate in Fig. 2 and Fig. 3 the intensity distributions of a partially coherent RP beam, its completely polarized part and its completely unpolarized part for $δ_{x x} = δ_{y y} = δ_{x y} = 1 mm$ and $δ_{x x} = δ_{y y} = δ_{x y} = 0.20 mm$ , respectively. One finds from Figs. 2 and 3 that the dark hollow beam profile of a focused partially coherent RP beam disappears on propagation as expected [57], and the dark hollow beam profile disappears more rapidly as the correlation radii decreases. The completely polarized part keeps its dark hollow beam profile invariant on propagation, while the completely unpolarized part gradually appears on propagation and has a Gaussian beam profile. Furthermore, the contribution of the completely unpolarized part to the total intensity becomes larger on propagation, thus leading to the change of the degree of polarization of the partially coherent RP beam.

Fig. 2 Intensity distributions of a focused partially coherent RP beam, its completely polarized part and its completely unpolarized part with $δ_{x x} = δ_{y y} = δ_{x y} = 1 mm .$

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Fig. 3 Intensity distributions of a focused partially coherent RP beam, its completely polarized part and its completely unpolarized part with . $δ_{x x} = δ_{y y} = δ_{x y} = 0.20 mm$

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To learn more about the power transition from the completely polarized part to the completely unpolarized part, we now study variation of the normalized power of the completely polarized part or unpolarized part, which is defined as

η^{(l)} = \frac{\int_{- \infty}^{\infty} \int_{- \infty}^{\infty} 〈 I^{(l)} (u, v, z) 〉 d u d v}{\int_{- \infty}^{\infty} \int_{- \infty}^{\infty} 〈 I (u, v, z) 〉 d u d v}, (l = p, u)

where

〈 I^{(u)} (u, v, z) 〉

and

〈 I^{(p)} (u, v, z) 〉

are given by Eq. (28). Figure 4(a) shows the variation of the normalized powers of the completely polarized part and the completely unpolarized part of a focused partially coherent RP beam versus the propagation distance with

δ_{x x} = δ_{y y} = δ_{x y} = 0.20 mm

. Figure 4(b) shows the variation of the normalized powers at the focal plane versus the correlation coefficient

δ_{x x}

with

δ_{x x} = δ_{y y} = δ_{x y}

. One finds from Fig. 4(a) that the normalized power

η^{(p)}

of the completely polarized part decreases on propagation, while the normalized power

η^{(u)}

the completely unpolarized part increases. At the focal plane,

η^{(u)}

approaches to 0.99, which means the partially coherent RP beam almost becomes completely unpolarized at the focal plane. One finds from Fig. 4(b) that

η^{(p)}

and

η^{(u)}

at the focal plane are closely determined by the correlation radii. With the decrease of the correlation radii,

η^{(u)}

increases while

η^{(p)}

decreases. For the case of

δ_{x x} < 0.1 mm

, the partially coherent RP beam at the focal plane can be regarded as a completely unpolarized beam. Furthermore, we also find an interesting phenomenon that when

η^{(u)} = η^{(p)}

, the total intensity distribution of the focused partially coherent RP beam has a flat-topped beam profile (see Fig. 5 ).

Fig. 4 Variation of the normalized powers of the completely polarized part and the completely unpolarized part of a focused partially coherent RP beam (a) versus the propagation distance with $δ_{x x} = δ_{y y} = δ_{x y} = 0.20 mm$ , (b) versus the correlation coefficient $δ_{x x}$ with $δ_{x x} = δ_{y y} = δ_{x y}$ at the focal plane z = 400mm.

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Fig. 5 Normalized intensity (cross line, v = 0) of a focused partially coherent RP beam under the condition of $η^{(u)} = η^{(p)}$ .

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To learn about the coherence properties of a partially coherent RP beam, we calculate in Fig. 6 the square of the degree of coherence of a focused partially coherent RP beam at several propagation distances with $δ_{x x} = δ_{y y} = δ_{x y} = 0.20 mm$ . Furthermore, we calculate in Fig. 7 the square of the degree of coherence of a focused partially coherent RP beam at the focal plane z = 400mm for different values of the correlation radii $δ_{x x}, δ_{y y}, δ_{x y} .$ One finds from Fig. 6 that the degree of coherence of the partially coherent RP beam at z = 0 has a Gaussian profile, while it varies on propagation and it gradually becomes of non-Gaussian profile. The distribution of the degree of coherence is also closely related with the correlation radii $δ_{x x}, δ_{y y}, δ_{x y},$ and the side robes become more significant as the correlation radii increases.

Fig. 6 Square of the degree of coherence of a focused partially coherent RP beam at several propagation distances with $δ_{x x} = δ_{y y} = δ_{x y} = 0.20 mm$ .

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Fig. 7 Square of the degree of coherence of a focused partially coherent RP beam at the focal plane z = 400mm for different values of the correlation radii $δ_{x x}, δ_{y y}, δ_{x y} .$

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3. Experimental results

In this section, we carry out experimental study of the coherence and polarization properties of a focused partially coherent RP beam. Figure 8 shows the experimental setup for generating and measuring the coherence properties of a partially coherent RP beam. Similar to Ref [57], a linearly polarized He-Ne laser beam focused by the thin lens L1 is reflected by a reflecting mirror and then illuminates a rotating ground-glass plate (RGGP), producing a partially coherent beam with Gaussian statistics [59, 60]. After passing through a collimation thin lens L2 and a Gaussian amplitude filter (GAF), the transmitted beam becomes a linearly polarized Gaussian Schell-model (GSM) beam characterized by the cross-spectral density $W (x_{1}, y_{1}, x_{2}, y_{2}) = \exp [- (x_{1}^{2} + y_{1}^{2} + x_{2}^{2} + y_{2}^{2}) / ω_{0}^{2} - {(x_{1} - x_{2})}^{2} / 2 δ^{2} - {(y_{1} - y_{2})}^{2} / 2 δ^{2}],$ with $ω_{0}$ and $δ$ being the beam waist size and the transverse coherence width. The radial polarization converter (RPC) located just behind the GAF converts the generated GSM beam into a partially coherent RP beam. The RPC just alter the polarization state of the GSM beam, while it does not alter its spatial coherence and beam spot size, thus the beam waist size of the partially coherent RP beam is approximately equal to that of the GSM beam, and the correlation radii of the partially coherent RP beam are approximated as $δ_{x x} = δ_{y y} = δ_{x y} = δ$ . The transmission function of the GAF determines the value of $ω_{0}$ , and $ω_{0}$ is equal to 1.05mm in our experiment. The transverse coherence width $δ$ is determined by the focused beam spot size on the RGGP and the roughness of the RGGP together. In our experiment, the roughness of the RGGP is fixed and we mainly modulate the value of $δ$ by varying the focused beam spot on the RGGP (i.e., the distance between L₁ and RGGP). We adopt the method proposed in [59] to measure the coherence width $δ$ of the GSM beam. In our experiment, we choose two different values of $δ$ ( $δ = 0.20 mm,0 .55mm$ ) to generate a RP beam with variable spatial coherence. Figure 9 shows the experimental results (dotted curves) of the modulus of the square of the degree of coherence of the generated GSM beam and the corresponding Gaussian fit (solid curves) for two different focused beam spot sizes on the RGGP. One finds from Fig. 9 that the degree of coherence of the beam produced from the RGGP indeed satisfy Gaussian distribution as expected.

Fig. 8 Experimental setup for generating a partially coherent RP beam and measuring its coherence properties. L₁, L₂, L₃, thin lenses; RM, reflecting mirror; RGGP, rotating ground-glass plate; GAF, Gaussian amplitude filter; RPC, radial polarization converter; beam splitter; D₁, D₂, single photon detectors.

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Fig. 9 Experimental results (dotted curves) of the modulus of the square of the degree of coherence of the generated GSM beam and the corresponding Gaussian fit (solid curves) for two different focused beamspot sizes on the RGGP.

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After passing through a thin lens L₃ located just behind the RPC, the focused partially coherent RP beam is split into two beams by a beam splitter. The transmitted and reflected beams arrive at D₁ and D₂, which scan the transverse plane u₁ and u₂, respectively. Both the distances from the L₃ to D₁ and from L₃ to D₂ are f. The electronic coincidence circuit is used to measure the fourth-order correlation function (i.e., intensity correlation function) between two detectors. By measuring the fourth-order correlation function, we can obtain the information of the degree of coherence of the focused partially coherent RP beam at the focal plane through the relation given by Eq. (32). Figure 10 shows our experimental results of the modulus of the square of the degree of coherence of a focused partially coherent RP beam at the focal plane for two different values of the coherence width $δ$ . For the convenience of comparison, the corresponding results calculated by the theoretical formulas with the measured values of $ω_{0}$ and $δ$ are also shown in Fig. 10. One finds from Fig. 10 that the degree of coherence of the focused partially coherent RP beam at the focal plane indeed becomes of non-Gaussian distribution, and our experimental results agree well with the theoretical results.

Fig. 10 Experimental results (dotted curves) of the modulus of the square of the degree of coherence of a focused partially coherent RP beam at the focal plane for two different values of the coherence width $δ$ . The solid curves are calculated by theoretical formulas.

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In order to obtain the information of the degree of polarization of a focused partially coherent RP beam, we need to measure the elements of its cross-spectral density matrix. Here we use the method proposed in [61] to measure the degree of polarization of a focused partially coherent RP beam, and this method has been adopted successfully to measure the degree of polarization of an electromagnetic GSM beam [62]. To measure the elements $W_{x x}$ and $W_{y y}$ , we put a linear polarizer whose transmission axis forms an angle $θ$ with the x-axis just before the thin lens L₃ (see Fig. 11 ), and the charge-coupled device (CCD) is used to measure the average intensity at the focal plane. The average intensity distribution of the partially coherent RP beam at the focal plane is written as

\begin{array}{l} 〈 I_{θ} (u, v, f) 〉 = \cos^{2} θ W_{x x} (u, v, u, v, f) + \sin^{2} θ W_{y y} (u, v, u, v, f) \\ + [W_{x y} (u, v, u, v, f) + W_{y x} (u, v, u, v, f)] \sin θ \cos θ . \end{array}

For the case of

θ = 0

, we obtain

W_{x x} (u, v, u, v, f) = 〈 I_{0} (u, v, f) 〉 .

For the case of

θ = π / 2

, we obtain

Fig. 11 Experimental setup for measuring the degree of polarization of a focused partially coherent RP beam at the focal plane. LP, linear polarizer; QWP, quarter-wave plate; CCD, charge-coupled device.

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W_{y y} (u, v, u, v, f) = 〈 I_{π / 2} (u, v, f) 〉 .

To measure the elements $W_{x y}$ and $W_{y x}$ , we put a quarter-wave plate and a linear polarizer together just before the thin lens L₃ (see Fig. 10), then the CCD is used to measure the average intensity at the focal plane. The fast axis and slow axis are along y-axis and x-axis, respectively. In this case, the average intensity distribution at the focal plane is written as

\begin{array}{l} 〈 I_{θ, ϕ} (u, v, f) 〉 = \sin^{2} θ W_{y y} (u, v, u, v, f) + \cos^{2} θ W_{x x} (u, v, u, v, f) \\ + \exp (- i ϕ) \sin θ [W_{x y} (u, v, u, v, f) + \exp (2 i ϕ) W_{y x} (u, v, u, v, f)] \cos θ, \end{array}

where

ϕ = π / 2

represents the phase difference between the x-component and y component of the vector field induced by the quarter-wave plate. In Eq. (39),

ϕ = 0

means the quarter-wave plate is removed. From Eq. (39), we can obtain the following expressions for the elements

W_{x y}

and

W_{y x}

\begin{array}{l} W_{x y} (u, v, u, v, f) = \frac{1}{2} [〈 I_{π / 4, 0} (u, v, f) 〉 - 〈 I_{3 π / 4, 0} (u, v, f) 〉] \\ + i \frac{1}{2} [〈 I_{π / 4, π / 2} (u, v, f) 〉 - 〈 I_{3 π / 4, π / 2} (u, v, f) 〉], \end{array}

W_{y x} (u, v, u, v, f) = W_{x y}^{*} (u, v, u, v, f) .

Thus by measuring the average intensities

〈 I_{π / 4, 0} 〉

,

〈 I_{3 π / 4, 0} 〉

,

〈 I_{π / 4, π / 2} 〉

,

〈 I_{3 π / 4, π / 2} 〉

at the focal plane, we can obtain the information of

W_{x y}

and

W_{y x}

.

Figure 12 shows our experimental results of the degree of polarization of a focused partially coherent RP beam at the focal plane versus the transverse coordinate u (v = 0) for two different values of the coherence width $δ$ . For the convenience of comparison, the corresponding results calculated by the theoretical formulas are also shown in Fig. 12. One finds from Fig. 12 that the focused partially coherent RP beam at the focal plane indeed is depolarized, and the depolarization is more serious as $δ$ decreases. Furthermore, with the increase of the transverse coordinate u, the degree of polarization increases. Our experimental results are also consistent with the theoretical predictions. Note that with the increase of the transverse coordinate, the degree of polarization decrease at about u = 0.2mm, which may be due to the unexpected fluctuation from the beam source.

Fig. 12 Experimental results of the degree of polarization of a focused partially coherent RP beam at the focal plane versus the transverse coordinate u (v = 0) for two different values of the coherence width $δ$ . The solid curves are calculated by theoretical formulas.

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4. Summary

We have studied the coherence and polarization properties of a new class of partially coherent vector beam with spatially non-uniform state of polarization named partially coherent radially polarized beam both theoretically and experimentally. Our results show that the degree of coherence of focused partially coherent RP beam becomes of non-Gaussian distribution, and the focused partially coherent RP beam is depolarized, while the state of polarization of its completely polarized part remains invariant, which is much different from that of an electromagnetic GSM beam. We have found that the changes of the degree of coherence and the degree of polarization of a partially coherent RP beam on propagation are controlled by the correlation radii of the source (i.e., degree of coherence of source field), which means that we can modulate the coherence and polarization properties of a partially coherent RP beam by varying its spatial coherence. Our experimental results agree well with theoretical results.

Appendix A. Derivation of the realizability conditions for a partially coherent radially polarized beam

It is known that a partially coherent vector beam generated by a Schell-model source should satisfy the semi-positive definiteness condition [21, 22]. Based on this condition, Gori et al. derived the following general realizability conditions for the beam parameters [33]

\sqrt{\frac{δ_{x x}^{2} + δ_{y y}^{2}}{2}} \leq δ_{x y} \leq \sqrt{\frac{δ_{x x} δ_{y y}}{| B_{x y} |}}, | B_{x y} | \leq \frac{2}{δ_{x x} / δ_{y y} + δ_{y y} / δ_{x x}} .

For a partially coherent RP beam generated by a Schell-model source, besides the restrictions on the parameters $B_{x y}, δ_{x x}, δ_{y y}, δ_{x y}$ shown in Eq. (A1), following two additional conditions should be satisfied

(1) Any point of the beam in the source plane is linearly polarized.
(2) The orientation angle of the polarization at any point in the source plane should satisfy $θ (x, y, 0) = arc \tan (y / x)$ .

According to Eqs. (25)-(27) in the text, the major and minor semi-axes of the polarization ellipse, $A_{1}$ and $A_{2}$ , as well as its degree of ellipticity, ε, and its orientation angle, θ, of the completely polarized part of the partially coherent RP beam in the source plane are expressed as

\begin{array}{l} A_{1, 2} (x, y, 0) = \frac{1}{\sqrt{2}} [\sqrt{{[W_{x x} (x, y, x, y, 0) - W_{y y} (x, y, x, y, 0)]}^{2} + 4 {| W_{x y} (x, y, x, y, 0) |}^{2}} \\ {\pm \sqrt{{[W_{x x} (x, y, x, y, 0) - W_{y y} (x, y, x, y, 0)]}^{2} + 4 {[Re W_{x y} (x, y, x, y, 0)]}^{2}}]}^{1 / 2}, \end{array}

ε (x, y, 0) = \frac{A_{2} (x, y, x, y, 0)}{A_{1} (x, y, x, y, 0)},

θ (x, y, 0) = \frac{1}{2} arc \tan [\frac{2 Re W_{x y} (x, y, x, y, 0)}{W_{x x} (x, y, x, y, 0) - W_{y y} (x, y, x, y, 0)}] .

The additional condition (1) requires that $A_{2}$ equals to zero. With the requirements of the additional conditions (1) and (2), and applying Eqs. (4)-(7), one can obtain the following restriction on $B_{x y}$ ,

Im [B_{x y}] = 0, Re [B_{x y}] = 1,

where Im and Re denote the imaginary and real parts of

B_{x y}

, respectively. From Eq. (A5), one can easily obtain

B_{x y} = 1

. Under the condition of

B_{x y} = 1

, one can easily obtain from Eq. (A1) that

δ_{x x} = δ_{y y} = δ_{x y}

. Thus, the final realizability conditions for a partially coherent RP beam should be

B_{x y} = 1

and

δ_{x x} = δ_{y y} = δ_{x y}

.

Acknowledgments

This work is supported by the National Natural Science Foundation of China under Grant Nos. 11274005 & 10904102 &11104195, the Foundation for the Author of National Excellent Doctoral Dissertation of PR China under Grant No. 200928, the Natural Science Foundation of Jiangsu Province under Grant No. BK2009114, the Huo Ying Dong Education Foundation of China under Grant No. 121009, the Key Project of Chinese Ministry of Education under Grant No. 210081, the Universities Natural Science Research Project of Jiangsu Province under grant 11KJB140007, and the Project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions.

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Coherence and polarization properties of a radially polarized beam with variable spatial coherence

Abstract

1. Introduction

2. Theory

3. Experimental results

4. Summary

Appendix A. Derivation of the realizability conditions for a partially coherent radially polarized beam

Acknowledgments

References and links

Cited By

Figures (12)

Equations (46)

Optics Express