Statistical properties in Young’s interference pattern formed with a radially polarized beam with controllable spatial coherence

Shijun Zhu; Fei Wang; Yahong Chen; Zhenhua Li; Yangjian Cai

doi:10.1364/OE.22.028697

1. Introduction

Coherence is one important property of a light beam [1], and partially coherent beam has advantage over coherent beam in many applications, such as free-space optical communications [2–4], optical imaging [5, 6], atom cooling [7], inertial confinement fusion [8], particle trapping [9, 10], optical scattering [11, 12] and second-harmonic generation [13, 14]. In the past decades, Generation and propagation of a scalar partially coherent beam has been studied extensively both in theory and in experiment [1, 15, 16].

Polarization is another important property of a light beam [17]. Vector beams can be classified as the ones with a spatially uniform state of polarization (e.g., linearly, circularly, and elliptically polarized beams) and with a spatially nonuniform state of polarization [e.g., radially polarized (RP), azimuthally polarized (AP), and cylindrical vector beams] [18]. When a RP beam is focused by a high numerical aperture (NA) lens, a strong longitudinal electric field appears and the tightly focused beam spot is much smaller than that of a linearly polarized beam [19]. Due to its unique focusing properties, RP beam is useful in many applications, such as in microscopy, lithography, electron acceleration, proton acceleration, material processing, optical data storage, high-resolution metrology, super-resolution imaging, plasmonic focusing and laser machining [18–26].

Recently, more and more attention is being paid to partially coherent vector beams [27, 28]. It was revealed that coherence and polarization are interrelated, and the degree of polarization of a partially coherent vector beam may change on propagation [27–31]. Partially coherent vector beams can also be classified as the ones with spatially uniform and spatially non-uniform state of polarization [32]. Based on the unified theory of coherence and polarization for partially coherent vector beam [29], one can study the statistical properties such as the degree of coherence, the degree of polarization and the state of polarization, of such beam on propagation. Electromagnetic Gaussian Schell-model (GSM) beam is a typical kind of partially coherent vector beam with spatially uniform state of polarization [33–35]. Partially coherent vector beams with spatially non-uniform state of polarization, such as partially coherent cylindrical vector beam, partially coherent RP beam and partially coherent azimuthally polarized beam, were introduced in theory and generated in experiment [36–42] recently. It was revealed in [40] that a partially coherent RP beam has advantage over a linearly polarized partially coherent beam for reducing turbulence-induced scintillation, which will be useful in free-space optical communications.

At the beginning of the nineteenth century, Thomas Young performed Young’s two-slit interference experiment [43]. Since then, Young’s interference experiment has played an important role both in classical and in quantum optics. The statistical properties in Young’s interference pattern formed with scalar partially coherent beam [44–50], partially coherent vector beam with spatially uniform state of polarization [51–57], coherent cylindrical vector beam [58] and radially polarized vortex beam [59] have been explored in detail both theoretically and experimentally, and it was revealed that one can modulate the statistical properties in the observation plane by varying the coherence and polarization properties of the incident beam. To our knowledge no results have been reported up until now on statistical properties in Young’s interference pattern formed with a partially coherent vector beam with spatially non-uniform state of polarization. In this paper, our aim is to study the statistical properties in Young’s interference pattern formed with a partially coherent RP beam both in theory and in experiment, and illustrate the effect of initial spatial coherence on the statistical properties.

2. Statistical properties in Young’s two-slit interference pattern formed with a partially coherent radially polarized beam: Theory

Based on the unified theory of coherence and polarization [27, 32], the second-order statistical properties of a partially coherent vector beam can be characterized by the cross-spectral density (CSD) matrix of the electric field, defined by the formula

\overset{\leftrightarrow}{W} (r_{1}, r_{2}) = (\begin{matrix} W_{x x} (r_{1}, r_{2}) & W_{x y} (r_{1}, r_{2}) \\ W_{y x} (r_{1}, r_{2}) & W_{y y} (r_{1}, r_{2}) \end{matrix}),

with elements

W_{α β} (r_{1}, r_{2}) = 〈 E_{α}^{*} (r_{1}) E_{β} (r_{2}) 〉, (α = x, y; β = x, y),

where

r_{1} \equiv (\begin{matrix} x_{1} & y_{1} \end{matrix})

and

r_{2} \equiv (\begin{matrix} x_{2} & y_{2} \end{matrix})

are transverse position vectors,

E_{x}

and

E_{y}

denote the components of the random electric vector, with respect to two mutually orthogonal, x and y directions, perpendicular to the z-axis. The asterisk denotes the complex conjugate and the angular brackets denote ensemble average.

For a partially coherent RP beam, the elements of the CSD matrix radiated from a Schell-model source are expressed as [38–40]

W_{α β} (r_{1}, r_{2}) = \frac{α_{1} β_{2}}{σ_{0}^{2}} \exp [- \frac{r_{1}^{2} + r_{2}^{2}}{σ_{0}^{2}} - \frac{{(r_{1} - r_{2})}^{2}}{2 δ_{0}^{2}}], (α, β = x, y),

where

σ_{0}

and

δ_{0}

represent the transverse beam size and spatial coherence width, respectively.

Figure 1 shows the illustration of the notation relating to Young’s two-slit interference experiment with a partially coherent beam. A two-slit with slit width b and separation of the slits 2a + b is placed in the x-y plane and parallel to the y direction. The distance between the two-slit plane and the observation plane is z. The transmission function of the two-slit is expressed as $H (x, y) = 1$ for $a \leq | x | \leq a + b$ and $- a - b \leq | x | \leq - a$ , and 0 otherwise. For the convenience of calculations and analysis, we expand the transmission function of the two slit as a finite sum of Gaussian modes

H (x, y) = \sum_{m = 1}^{M} A_{m} {\exp [- \frac{B_{m}}{{(b / 2)}^{2}} {(x - a - \frac{b}{2})}^{2}] + \exp [- \frac{B_{m}}{{(b / 2)}^{2}} {(x + a + \frac{b}{2})}^{2}]},

where

A_{m}

and

B_{m}

are the expansion and Gaussian coefficients, which can be obtained by optimization computation directly, and a table of

A_{m}

and

B_{m}

can be found in [60, 61]. The simulation accuracy improves as M increases. For a hard aperture, M = 10 assures a good accuracy [60–62].

Fig. 1 Illustration of the notation relating to Young’s two-slit interference experiment with a partially coherent beam.

Download Full Size | PDF

When a partially coherent RP beam illuminates the two-slit, the elements of the CSD of the beam just behind the two-slit plane are can be expressed as

\begin{array}{l} W_{1 α β} (r_{1}, r_{2}) = W_{α β} (r_{1}, r_{2}) H (x_{1}, y_{1}) H^{*} (x_{2}, y_{2}) = \frac{α_{1} β_{2}}{σ_{0}^{2}} \exp (- \frac{r_{1}^{2} + r_{2}^{2}}{σ_{0}^{2}}) \exp (- \frac{{(r_{1} - r_{2})}^{2}}{2 δ_{0}^{2}}) \\ \times \sum_{m = 1}^{M} \sum_{i = 1}^{M} A_{m} A_{i}^{*} {\exp [- \frac{B_{m}}{{(b / 2)}^{2}} {(x_{1} - a - \frac{b}{2})}^{2}] + \exp [- \frac{B_{m}}{{(b / 2)}^{2}} {(x_{1} + a + \frac{b}{2})}^{2}]} \\ \times {\exp [- \frac{B_{i}^{*}}{{(b / 2)}^{2}} {(x_{2} - a - \frac{b}{2})}^{2}] + \exp [- \frac{B_{i}^{*}}{{(b / 2)}^{2}} {(x_{2} + a + \frac{b}{2})}^{2}]} . \end{array}

Paraxial propagation of the partially coherent RP beam from the two-slit to the observation plane can be studied with help of the following generalized Huygens-Fresnel integral [63]

\begin{array}{l} W_{α β} (u_{1}, u_{2}) = \frac{1}{λ^{2} z^{2}} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} W_{1 α β} (r_{1}, r_{2}) \\ \times \exp [- \frac{i k}{2 z} (r_{1}^{2} - r_{2}^{2}) + \frac{i k}{z} (r_{1} \cdot u_{1} - r_{2} \cdot u_{2}) - \frac{i k}{2 z} (u_{1}^{2} - u_{2}^{2})] d^{2} r_{1} d^{2} r_{2}, \end{array}

where

k = 2 π / λ

is the wave number with

λ

being the wavelength,

u_{1} \equiv (\begin{matrix} u_{1} & v_{1} \end{matrix})

and

u_{2} \equiv (\begin{matrix} u_{2} & v_{2} \end{matrix})

are transverse position vectors in the observation plane.

Substituting Eq. (5) into Eq. (6), we obtain (after tedious integration and operation) the following expressions for the elements of the CSD matrix of the field in the observation plane

\begin{array}{l} W_{x x} (u_{1}, u_{2}) = \frac{k^{2}}{16 σ_{0}^{2} z^{2} M_{1}^{1 / 2} Π_{0}^{1 / 2}} \sum_{m = 1}^{M} \sum_{i = 1}^{M} \frac{A_{m} A_{i}^{*}}{M_{1 B}^{3 / 2} Ω_{B}^{3 / 2}} \exp [- \frac{k^{2}}{4 z^{2} Π_{0}} {(\frac{δ_{0}^{- 2} v_{1}}{2 M_{1}} - v_{2})}^{2}] \\ \times {(δ_{0}^{- 2} + N_{1 m} Δ_{11} + \frac{δ_{0}^{- 2} Δ_{11}^{2}}{2 Ω_{B}}) \exp [\frac{Δ_{11}^{2}}{4 Ω_{B}} + \frac{N_{1 m}^{2}}{4 M_{1 B}}] + (δ_{0}^{- 2} + N_{2 m} Π_{21} + \frac{δ_{0}^{- 2} Π_{21}^{2}}{2 Ω_{B}}) \\ \times \exp [\frac{Π_{21}^{2}}{4 Ω_{B}} + \frac{N_{2 m}^{2}}{4 M_{1 B}}] + (δ_{0}^{- 2} + N_{1 m} Δ_{12} + \frac{δ_{0}^{- 2} Δ_{12}^{2}}{2 Ω_{B}}) \exp [\frac{Δ_{12}^{2}}{4 Ω_{B}} + \frac{N_{1 m}^{2}}{4 M_{1 B}}] \\ + (δ_{0}^{- 2} + N_{2 m} Π_{22} + \frac{δ_{0}^{- 2} Π_{22}^{2}}{2 Ω_{B}}) \times \exp [\frac{Π_{22}^{2}}{4 Ω_{B}} + \frac{N_{2 m}^{2}}{4 M_{1 B}}]} \exp [- \frac{i k}{2 z} (u_{1}^{2} - u_{2}^{2})] \\ \times \exp [- \frac{k^{2} v_{1}^{2}}{4 z^{2} M_{1}} - \frac{O^{2}}{{(b / 2)}^{2}} (B_{m} + B_{i}^{*})], \end{array}

\begin{array}{l} W_{y y} (u_{1}, u_{2}) = \frac{k^{2}}{16 σ_{0}^{2} z^{2} M_{1}^{3 / 2} Π_{0}^{3 / 2}} \sum_{m = 1}^{M} \sum_{i = 1}^{M} \frac{A_{m} A_{i}^{*}}{M_{1 B}^{1 / 2} Ω_{B}^{1 / 2}} \exp [- \frac{k^{2}}{4 z^{2} Π_{0}} {(\frac{δ_{0}^{- 2} v_{1}}{2 M_{1}} - v_{2})}^{2}] \\ \times [δ_{0}^{- 2} - \frac{k^{2}}{z^{2}} (\frac{δ_{0}^{- 2} v_{1}}{2 M_{1}} - v_{2}) v_{1} - \frac{k^{2} δ_{0}^{- 2}}{2 z^{2} Π_{0}} {(\frac{δ_{0}^{- 2} v_{1}}{2 M_{1}} - v_{2})}^{2}] \exp [- \frac{i k}{2 z} (u_{1}^{2} - u_{2}^{2})] \\ \times {\exp [\frac{Δ_{11}^{2}}{4 Ω_{B}} + \frac{N_{1 m}^{2}}{4 M_{1 B}}] + \exp [\frac{Π_{21}^{2}}{4 Ω_{B}} + \frac{N_{2 m}^{2}}{4 M_{1 B}}] + \exp [\frac{Δ_{12}^{2}}{4 Ω_{B}} + \frac{N_{1 m}^{2}}{4 M_{1 B}}] \\ + \exp [\frac{Π_{22}^{2}}{4 Ω_{B}} + \frac{N_{2 m}^{2}}{4 M_{1 B}}]} \exp [- \frac{k^{2} v_{1}^{2}}{4 z^{2} M_{1}} - \frac{O^{2}}{{(b / 2)}^{2}} (B_{m} + B_{i}^{*})], \end{array}

\begin{array}{l} W_{x y} (u_{1}, u_{2}) = \frac{i k^{3}}{32 σ_{0}^{2} z^{3} M_{1}^{1 / 2} M_{1 B}^{3 / 2} Π_{0}^{3 / 2} Ω_{B}^{3 / 2}} (\frac{δ_{0}^{- 2} v_{1}}{2 M_{1}} - v_{2}) \sum_{m = 1}^{M} \sum_{i = 1}^{M} A_{m} A_{i}^{*} \exp [- \frac{i k}{2 z} (u_{1}^{2} - u_{2}^{2})] \\ \times {(2 N_{1 m} Ω_{B} + \frac{Δ_{11}}{δ_{0}^{2}}) \exp [\frac{Δ_{11}^{2}}{4 Ω_{B}} + \frac{N_{1 m}^{2}}{4 M_{1 B}}] + (- 2 N_{2 m} Ω_{B} - \frac{Π_{21}}{δ_{0}^{2}}) \exp [\frac{Π_{21}^{2}}{4 Ω_{B}} + \frac{N_{2 m}^{2}}{4 M_{1 B}}] \\ + (2 N_{1 m} Ω_{B} + \frac{Δ_{12}}{δ_{0}^{2}}) \exp [\frac{Δ_{12}^{2}}{4 Ω_{B}} + \frac{N_{1 m}^{2}}{4 M_{1 B}}] + (- 2 N_{2 m} Ω_{B} - \frac{Π_{22}}{δ_{0}^{2}}) \exp [\frac{Π_{22}^{2}}{4 Ω_{B}} + \frac{N_{2 m}^{2}}{4 M_{1 B}}]} \\ \times \exp [- \frac{k^{2}}{4 z^{2} Π_{0}} {(\frac{δ_{0}^{- 2} v_{1}}{2 M_{1}} - v_{2})}^{2} - \frac{k^{2}}{4 M_{1} z^{2}} v_{1}^{2} - \frac{O^{2}}{{(b / 2)}^{2}} (B_{m} + B_{i}^{*})], \end{array}

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

where

\begin{array}{l} O = a + \frac{b}{2}, M_{1} = \frac{1}{σ_{0}^{2}} + \frac{1}{2 δ_{0}^{2}} + \frac{i k}{2 z}, M_{2} = \frac{1}{σ_{0}^{2}} + \frac{1}{2 δ_{0}^{2}} - \frac{i k}{2 z}, M_{1 B} = \frac{B_{m}}{{(b / 2)}^{2}} + M_{1}, \\ N_{1 m} = \frac{2 B_{m} O}{{(b / 2)}^{2}} + \frac{i k u_{1}}{z}, N_{2 m} = \frac{2 B_{m} O}{{(b / 2)}^{2}} - \frac{i k u_{1}}{z}, Ω_{B} = M_{2} + \frac{B_{i}^{*}}{{(b / 2)}^{2}} - \frac{δ_{0}^{- 4}}{4 M_{1 B}}, \\ Π_{0} = M_{2} - \frac{δ_{0}^{- 4}}{4 M_{1}}, Δ_{11} = \frac{2 B_{i}^{*} O}{{(b / 2)}^{2}} - \frac{i k u_{2}}{z} + \frac{δ_{0}^{- 2} N_{1 m}}{2 M_{1 B}}, Δ_{12} = - \frac{2 B_{i}^{*} O}{{(b / 2)}^{2}} - \frac{i k u_{2}}{z} + \frac{δ_{0}^{- 2} N_{1 m}}{2 M_{1 B}}, \\ Π_{21} = - \frac{2 B_{i}^{*} O}{{(b / 2)}^{2}} + \frac{i k u_{2}}{z} + \frac{δ_{0}^{- 2} N_{2 m}}{2 M_{1 B}}, Π_{22} = \frac{2 B_{i}^{*} O}{{(b / 2)}^{2}} + \frac{i k u_{2}}{z} + \frac{δ_{0}^{- 2} N_{2 m}}{2 M_{1 B}} . \end{array}

The average intensity in the observation plane is given as

I (u) = T r \overset{\leftrightarrow}{W} (u, u) = W_{x x} (u, u) + W_{y y} (u, u) = I_{x} (u) + I_{y} (u) .

where Tr represents the trace of matrix.

There are two definitions of the degree of coherence for partially coherent vector beam [32, 64]. For the convenience of experimental measurement, we adopt the definition introduced in [64], where the degree of coherence $μ (u_{1}, u_{2})$ of a partially coherent vector beam at a pair of transverse points with position vectors $u_{1}$ and $u_{2}$ are expressed as

μ^{2} (u_{1}, u_{2}) = \frac{T r [{\overset{\leftrightarrow}{W}}^{†} (u_{1}, u_{2}) \overset{\leftrightarrow}{W} (u_{1}, u_{2})]}{T r [\overset{\leftrightarrow}{W} (u_{1}, u_{1})] T r [\overset{\leftrightarrow}{W} (u_{2}, u_{2})]},

where the symbol ‘

†

’ denotes Hermitian adjoint. In our case, due to the fact that the partially coherent vector beam obeys the Gaussian statistics [1], the numerator in right hand of Eq. (13) can be expanded as [64]

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

What is more, applying the Gaussian moment theorem [1], the fourth-order correlation function between two points $u_{1}$ and $u_{2}$ can be expanded as

F^{(4)} (u_{1}, u_{2}) = T r [\overset{\leftrightarrow}{W} (u_{1}, u_{1})] T r [\overset{\leftrightarrow}{W} (u_{2}, u_{2})] + \sum_{α, β} {| W_{α β} (u_{1}, u_{2}) |}^{2} .

The normalized fourth-order correlation function is expressed in terms of

μ (u_{1}, u_{2})

as follows

f^{(4)} (u_{1}, u_{2}) = 1 + μ^{2} (u_{1}, u_{2}) .

Thus, one can obtain the information of the degree of coherence through measuring the fourth-order correlation function in experiment.

According to [27], the degree of polarization at point u is defined as

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

Applying Eqs. (7)-(13) and (17), we can study the statistical properties in Young’s interference pattern formed with a partially coherent RP beam numerically.

Figure 2 shows the intensity distributions (contour graphs) $I (u) / I_{\max} (u)$ , $I_{x} (u) / I_{\max} (u)$ , $I_{y} (u) / I_{\max} (u)$ of the Young’s interference pattern formed with a partially coherent RP beam at z = 700mm for two different values of the transverse spatial coherence width $δ_{0}$ with $a = b = 0.2 mm,$ $σ_{0} = 1.16 mm$ and $λ = 632.8 nm$ . One finds from Fig. 2 that the intensity distribution of the interference pattern formed with a partially coherent RP beam is quite different from that formed with a scalar or linearly polarized partially coherent beam as shown in [44–50], and is strongly affected by the spatial coherence width. Here the intensity distribution $I (u)$ displays spatial structure both in the u direction and in the v direction, which is caused by that fact $I_{y} (u)$ exhibits spatial structure both in the u direction and in the v direction. For large value of $δ_{0}$ , the intensity [see (a)-(a2)] shows a similar distribution with that formed with a coherent RP beam as shown in [58]. With the decrease of $δ_{0}$ , the spatial structure gradually becomes blurred. We can explain this phenomenon as follows. From the expression of $I_{y} (u)$ of the incident beam (i.e., beam in the source plane), one sees that the distribution of $I_{y} (u)$ displays a two-beamlet distribution along v direction. After passing through the two-slit, the distribution of $I_{y} (u)$ displays spatial structure in u direction due to the interference effect of the two-slit, and the distribution of $I_{y} (u)$ displays spatial structure in v direction due to the fact that the initial two-beamlet distribution of the incident beam along v direction isn’t affected by the two-slit because the two-slit has no confinement along v direction.

Fig. 2 Normalized intensity distributions (contour graphs) $I (u) / I_{\max} (u),$ $I_{x} (u) / I_{\max} (u)$ , $I_{y} (u) / I_{\max} (u)$ of the Young’s interference pattern formed with a partially coherent RP beam at z = 700mm for two different values of the transverse spatial coherence width $δ_{0}$ . (a)-(a2) $δ_{0} = 0.99 mm$ ; (b)-(b2) $δ_{0} = 0.26 mm$ .

Download Full Size | PDF

Figure 3 shows the square of the degree of coherence $μ^{2} (u, 0)$ (contour graph) and the corresponding cross line (v = 0) of the Young’s interference pattern formed with a partially coherent RP beam at z = 700mm for two different values of the transverse spatial coherence width $δ_{0}$ . Figure 4 shows the degree of polarization (contour graph) and the corresponding cross line (v = 0) of the Young’s interference pattern formed with a partially coherent RP beam at z = 700mm for two different values of the transverse spatial coherence width $δ_{0}$ . The other parameters are set as $a = b = 0.2 mm,$ $σ_{0} = 1.16 mm$ and $λ = 632.8 nm$ . One sees from Figs. 3 and 4 that both the degree of coherence and the degree of polarization of the interference pattern differ from those of the incident partially RP beam, and display spatial structure both in the u direction and in the v direction, which are modulated by the transverse spatial coherence width $δ_{0}$ . In the u direction, both the degree of coherence and the degree of polarization show a lattice-like structure, and they share the same periodicity. The spatial structures in the degree of coherence and in the degree of polarization of the interference pattern are caused by the fact that the degree of coherence and the degree of polarization are determined by the components of the CSD matrix of the field in the observation plane and the components of the CSD matrix display spatial structures in both u and v directions. The cause for the difference compared to other types of beam is the nonuniform state of polarization and the interdependence of coherence and polarization of a partially coherent RP beam. Thus, one can modulate the statistical properties in Young’s two-slit interference pattern formed with a RP beam by varying its transverse spatial coherence width.

Fig. 3 Square of the degree of coherence $μ^{2} (u, 0)$ (contour graph) and the corresponding cross line (v = 0) of the Young’s interference pattern formed with a partially coherent RP beam at z = 700mm for two different values of the transverse spatial coherence width $δ_{0}$ .

Download Full Size | PDF

Fig. 4 Degree of polarization (contour graph) and the corresponding cross line (v = 0) of the Young’s interference pattern formed with a partially coherent RP beam at z = 700mm for two different values of the transverse spatial coherence width $δ_{0}$ .

Download Full Size | PDF

In this paper, we mainly consider the influence of the spatial coherence on the statistical properties of the Young’s interference pattern formed with a partially coherent RP beam. Our numerical results (not present here to save space) show that the intensity, the degree of coherence and the degree of polarization of the Young’s interference pattern at different z have similar distributions if z is not very large. Furthermore, our numerical results (not present here to save space) also show that the intensity, the degree of coherence and the degree of polarization in the interference pattern for different values of slit width b also have similar distributions if b is much smaller than the beam width of the incident beam.

3. Statistical properties in Young’s two-slit interference pattern formed with a partially coherent radially polarized beam: Experiment

In this section, we carry out experimental study of the statistical properties in Young’s two-slit interference pattern formed with a partially coherent RP beam.

Figure 5 shows our experimental setup for generating a partially coherent RP beam and measuring the intensity and the degree of coherence in Young’s two-slit interference pattern formed with the generated beam. A linearly polarized beam reflected by a mirror is focused by a thin lens (L₁) with focal length $f = 10 c m$ , and then illuminates a rotating ground-glass disk (RGGD). The transmitted beam from the RGGD becomes a partially coherent beam with Gaussian statistics. Here the linear polarizer (LP) is used to covert the He-Ne laser beam into a linearly polarized beam. After passing through a collimation lens L₂ with focal length $f = 15 c m$ and a Gaussian amplitude filter (GAF), the transmitted beam becomes a linearly polarized GSM beam, whose CSD is expressed as

W (r_{1}, r_{2}) = C_{0} \exp [- (r_{1}^{2} + r_{2}^{2}) / σ^{2} - {(r_{1} - r_{2})}^{2} / 2 δ^{2}] .

Here

C_{0}

is a constant,

σ

and

δ

denote the transverse beam size and spatial coherence width, respectively. The transverse beam size is determined by the transmission function of the GAF, and

δ

is mainly determined by the beam spot size on the RGGD, which is controlled through varying the distance between L₁ and RGGD (i.e., the distance l) [38]. After passing through a radial polarization converter (RPC), the linearly polarized GSM beam becomes a partially coherent RP beam with

δ_{0} \approx δ

and

σ_{0} \approx σ

due to the fact that RPC only changes the state of polarization of the incident beam. The generated partially coherent RP beam illuminates the two-slit with

a = b = 0.2 m m

, producing Young’s two-slit interference pattern in the observation plane. The interference field from the two-slit is split into two fields by the 50:50 nonpolarization beam splitter (BS). The transmitted field arrives at the beam profile analyzer (BPA), which is used to measure the intensity

I (u)

of the interference pattern. In order to measure the composition components

I_{x} (u)

and

I_{y} (u)

, one can put a linear polarizer whose transmission axis forms an angel

θ

with the x-axis just before the BPA. For the case of

θ = 0

or

θ = π / 2

,

I_{x} (u)

or

I_{y} (u)

is measured by the BPA. The reflected field arrives at a charge-coupled device (CCD), which is used to measure the degree of coherence of the interference pattern. The detailed principle and the measuring process can be found in [65, 66]. If we remove the two-slit and insert a thin lens L₃ with focal length

f_{3}

just before the BS, and both the distances from the RPC to L3 and from L3 to BPA and CCD are

2 f_{3}

(i.e., 2f-imaging system), then the BPA and CCD measure the intensity and degree of coherence of the generated partially coherent RP beam in the output plane of RPC.

Fig. 5 Experimental setup for generating a partially coherent RP beam and measuring the intensity and the degree of coherence in Young’s two-slit interference pattern formed with the generated beam. LP, linear polarizer; M, mirror; L₁, L₂, thin lenses; RGGD, rotating ground-glass disk; GAF, Gaussian amplitude filter; RPC, radial polarization converter; T-S, two-slit; BS, 50:50 nonpolarization beam splitter; BPA, beam profile analyzer; CCD, charge-coupled devices; PC, personal computers.

Download Full Size | PDF

Figure 6 shows our experimental results of the intensity distribution and corresponding cross line (y = 0, dotted curve) of the generated partially coherent RP beam in the output plane of RPC. Through theoretical fit of the experimental data, we obtain $σ_{0} = 1.16 mm$ . Figure 7 shows our experimental results of the square of the degree of coherence $μ^{2} (r_{1}, r_{2} = 0)$ and the corresponding cross line (y₁ = 0, dotted curve) of the generated partially coherent RP beam in the output plane of RPC for two different values of the spatial coherence width. Through theoretical fit of the experimental data, we obtain $δ_{0} = 0.99 mm$ and $δ_{0} = 0.26 mm$ .

Fig. 6 Experimental results of the intensity distribution and the corresponding cross line (y = 0, dotted curve) of the generated partially coherent RP beam in the output plane of RPC. The solid curve denotes the theoretical fit of the experimental data.

Download Full Size | PDF

Fig. 7 Experimental results of the square of the degree of coherence $μ^{2} (r_{1}, r_{2} = 0)$ and the corresponding cross line (y₁ = 0, dotted curve) of the generated partially coherent RP beam in the output plane of RPC for two different values of the spatial coherence width. The solid curve denotes the theoretical fit of the experimental data.

Download Full Size | PDF

Figure 8 shows our experimental results of the intensity distributions $I (u)$ , $I_{x} (u)$ , $I_{y} (u)$ of the Young’s two-slit interference pattern formed with the generated partially coherent RP beam at z = 700mm for two different values of the transverse spatial coherence width $δ_{0}$ . Figure 9 shows our experimental results of the square of the degree of coherence $μ^{2} (u, 0)$ of the Young’s interference pattern formed with a partially coherent RP beam at z = 700mm for two different values of the transverse spatial coherence width $δ_{0}$ . The degree of polarization can be measured through the procedure adopted in [39]. Figure 10 shows our experimental results of the degree of polarization (cross line v = 0) of the Young’s interference pattern formed with the generated partially coherent RP beam at z = 700mm for two different values of the transverse spatial coherence width $δ_{0}$ . For the convenience of comparison, the corresponding theoretical results calculated by Eq. (17) with the beam parameters measured in experiment are also shown in Fig. 10. It is found that our experimental results also agree reasonably well with theoretical results, while slight differences between the experimental and numerical results exist, especially in the degree of coherence and in the degree of polarization, because their values are not measured directly but obtained through measuring the components of the CSD matrix and processed by the computer. Furthermore, the differences between the experimental and numerical results also can be caused by the non-ideal optical elements and the resolution limits of the CCD or BPA.

Fig. 8 Experimental results of the intensity distributions $I (u)$ , $I_{x} (u)$ , $I_{y} (u)$ of the Young’s two-slit interference pattern formed with the generated partially coherent RP beam at z = 700mm for two different values of the transverse spatial coherence width $δ_{0}$ . (a)-(a2) $δ_{0} = 0.99 mm$ ; (b)-(b2) $δ_{0} = 0.26 mm$ .

Download Full Size | PDF

Fig. 9 Experimental results of the square of the degree of coherence $μ^{2} (u, 0)$ of the Young’s interference pattern formed with a partially coherent RP beam at z = 700mm for two different values of the transverse spatial coherence width $δ_{0}$ . (a) $δ_{0} = 0.99 mm$ ; (b) $δ_{0} = 0.26 mm$ .

Download Full Size | PDF

Fig. 10 Experimental results of the degree of polarization (cross line v = 0) of the Young’s interference pattern formed with the generated partially coherent RP beam at z = 700mm for two different values of the transverse spatial coherence width $δ_{0}$ . The solid lines denote theoretical results calculated by Eq. (17) with the beam parameters measured in experiment..

Download Full Size | PDF

4. Summary

We have carried out theoretical and experimental studies of the statistical properties in Young’s two-slit interference pattern formed with a partially coherent RP beam based on the unified theory of coherence and polarization, and we have found that the statistical properties in Young’s two-slit interference pattern, such as the intensity, the degree of coherence and the degree of polarization, are closely related with the transverse spatial coherence of the incident RP beam. Furthermore, we have found that the statistical properties in Young’s interference pattern formed with a partially coherent RP beam are different from those formed with a scalar or linearly polarized partially coherent beam. Thus we can modulate the statistical properties through varying the spatial coherence of the incident RP beam, which may be useful in some application, such as particle or atom trapping, where light field with special statistical properties are required.

Acknowledgments

This work is supported by the National Natural Science Foundation of China under Grant Nos. 11474213&61371167&11274005&11104195, the Project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions, the Universities Natural Science Research Project of Jiangsu Province under Grant No. 11KJB140007, the Innovation Plan for Graduate Students in the Universities of Jiangsu Province under Grant No. KYLX_1218, and the Project Sponsored by the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry.

References and links

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

2. 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] [PubMed]

3. X. Liu, Y. Shen, L. Liu, F. Wang, and Y. Cai, “Experimental demonstration of vortex phase-induced reduction in scintillation of a partially coherent beam,” Opt. Lett. 38(24), 5323–5326 (2013). [CrossRef] [PubMed]

4. X. Liu, F. Wang, C. Wei, and Y. Cai, “Experimental study of turbulence-induced beam wander and deformation of a partially coherent beam,” Opt. Lett. 39(11), 3336–3339 (2014). [CrossRef] [PubMed]

5. T. E. Gureyev, D. M. Paganin, A. W. Stevenson, S. C. Mayo, and S. W. Wilkins, “Generalized eikonal of partially coherent beams and its use in quantitative imaging,” Phys. Rev. Lett. 93(6), 068103 (2004). [CrossRef] [PubMed]

6. Y. Cai and S. Y. Zhu, “Ghost imaging with incoherent and partially coherent light radiation,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 71(5 Pt 2), 056607 (2005). [CrossRef] [PubMed]

7. J. Zhang, Z. Wang, B. Cheng, Q. Wang, B. Wu, X. Shen, L. Zheng, Y. Xu, and Q. Lin, “Atom cooling by partially spatially coherent lasers,” Phys. Rev. A 88(2), 023416 (2013). [CrossRef]

8. Y. Kato, K. Mima, N. Miyanaga, S. Arinaga, Y. Kitagawa, M. Nakatsuka, and C. Yamanaka, “Random phasing of high-power lasers for uniform target acceleration and plasma-instability suppression,” Phys. Rev. Lett. 53(11), 1057–1060 (1984). [CrossRef]

9. C. Zhao, Y. Cai, X. Lu, and H. T. Eyyuboğlu, “Radiation force of coherent and partially coherent flat-topped beams on a Rayleigh particle,” Opt. Express 17(3), 1753–1765 (2009). [CrossRef] [PubMed]

10. C. Zhao and Y. Cai, “Trapping two types of particles using a focused partially coherent elegant Laguerre-Gaussian beam,” Opt. Lett. 36(12), 2251–2253 (2011). [CrossRef] [PubMed]

11. T. van Dijk, D. G. Fischer, T. D. Visser, and E. Wolf, “Effects of spatial coherence on the angular distribution of radiant intensity generated by scattering on a sphere,” Phys. Rev. Lett. 104(17), 173902 (2010). [CrossRef] [PubMed]

12. C. Ding, Y. Cai, O. Korotkova, Y. Zhang, and L. Pan, “Scattering-induced changes in the temporal coherence length and the pulse duration of a partially coherent plane-wave pulse,” Opt. Lett. 36(4), 517–519 (2011). [CrossRef] [PubMed]

13. M. S. Zubairy and J. K. McIver, “Second-harmonic generation by a partially coherent beam,” Phys. Rev. A 36(1), 202–206 (1987). [CrossRef] [PubMed]

14. Y. Cai and U. Peschel, “Second-harmonic generation by an astigmatic partially coherent beam,” Opt. Express 15(23), 15480–15492 (2007). [CrossRef] [PubMed]

15. Y. Cai and F. Wang, “Tensor method for treating the propagation of scalar and electromagnetic Gaussian Schell-model beams: a review,” Open Opt. J. 4(1), 1–20 (2010). [CrossRef]

16. Y. Cai, “Generation of various partially coherent beams and their propagation properties in turbulent atmosphere: a review,” Proc. SPIE 7924, 792402 (2011). [CrossRef]

17. C. Brosseau, Fundamentals of Polarized Light: A Statistical Approach (Wiley, 1998).

18. Q. Zhan, “Cylindrical vector beams: from mathematical concepts to applications,” Adv. Opt. Photon. 1(1), 1–57 (2009). [CrossRef]

19. R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91(23), 233901 (2003). [CrossRef] [PubMed]

20. P. Wróbel, J. Pniewski, T. J. Antosiewicz, and T. Szoplik, “Focusing radially polarized light by a concentrically corrugated silver film without a hole,” Phys. Rev. Lett. 102(18), 183902 (2009). [CrossRef] [PubMed]

21. L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett. 86(23), 5251–5254 (2001). [CrossRef] [PubMed]

22. Q. Zhan, “Trapping metallic Rayleigh particles with radial polarization,” Opt. Express 12(15), 3377–3382 (2004). [CrossRef] [PubMed]

23. H. Wang, L. Shi, B. Lukyanchuk, C. J. R. Sheppard, and C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics 2(8), 501–505 (2008). [CrossRef]

24. W. Chen, D. C. Abeysinghe, R. L. Nelson, and Q. Zhan, “Plasmonic lens made of multiple concentric metallic rings under radially polarized illumination,” Nano Lett. 9(12), 4320–4325 (2009). [CrossRef] [PubMed]

25. K. P. Singh and M. Kumar, “Electron acceleration by a radially polarized laser pulse during ionization of low density gases,” Phys. Rev. ST Accel. Beams 14(3), 030401 (2011). [CrossRef]

26. J. Li, Y. Salamin, B. J. Galow, and C. Keitel, “Acceleration of proton bunches by petawatt chirped radially polarized laser pulses,” Phys. Rev. A 85(6), 063832 (2012). [CrossRef]

27. E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge University, 2007).

28. Y. Cai, F. Wang, C. Zhao, S. Zhu, G. Wu, and Y. Dong, “Partially coherent vector beams: from theory to experiment,” in Vectorial Optical Fields: Fundamentals and Applications, Q. Zhen, ed. (World Scientific, 2013), Chap. 7, pp. 221–273.

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

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

31. 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] [PubMed]

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

33. 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]

34. 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]

35. F. Wang, G. Wu, X. Liu, S. Zhu, and Y. Cai, “Experimental measurement of the beam parameters of an electromagnetic Gaussian Schell-model source,” Opt. Lett. 36(14), 2722–2724 (2011). [CrossRef] [PubMed]

36. Y. Dong, Y. Cai, C. Zhao, and M. Yao, “Statistics properties of a cylindrical vector partially coherent beam,” Opt. Express 19(7), 5979–5992 (2011). [CrossRef] [PubMed]

37. Y. Dong, F. Feng, Y. Chen, C. Zhao, and Y. Cai, “Statistical properties of a nonparaxial cylindrical vector partially coherent field in free space,” Opt. Express 20(14), 15908–15927 (2012). [CrossRef] [PubMed]

38. F. Wang, Y. Cai, Y. Dong, and O. Korotkova, “Experimental generation of a radially polarized beam with controllable spatial coherence,” Appl. Phys. Lett. 100(5), 051108 (2012). [CrossRef]

39. G. Wu, F. Wang, and Y. Cai, “Coherence and polarization properties of a radially polarized beam with variable spatial coherence,” Opt. Express 20(27), 28301–28318 (2012). [CrossRef] [PubMed]

40. F. Wang, X. Liu, L. Liu, Y. Yuan, and Y. Cai, “Experimental study of the scintillation index of a radially polarized beam with controllable spatial coherence,” Appl. Phys. Lett. 103(9), 091102 (2013). [CrossRef]

41. S. Zhu, X. Zhu, L. Liu, F. Wang, and Y. Cai, “Theoretical and experimental studies of the spectral changes of a polychromatic partially coherent radially polarized beam,” Opt. Express 21(23), 27682–27696 (2013). [CrossRef] [PubMed]

42. Y. Dong, F. Wang, C. Zhao, and Y. Cai, “Effect of spatial coherence on propagation, tight focusing and radiation forces of an azimuthally polarized beam,” Phys. Rev. A 86(1), 013840 (2012). [CrossRef]

43. T. Young, “The bakerian lecture: on the theory of light and colours,” Philos. Trans. R. Soc. Lond. 92(0), 12–48 (1802). [CrossRef]

44. E. Wolf, “Young’s interference fringes with narrow-band light,” Opt. Lett. 8(5), 250–252 (1983). [CrossRef] [PubMed]

45. D. F. V. James and E. Wolf, “Spectral changes produced in Young’s interference experiment,” Opt. Commun. 81(3–4), 150–154 (1991). [CrossRef]

46. S. A. Ponomarenko and E. Wolf, “Coherence properties of light in Young’s interference pattern formed with partially coherent light,” Opt. Commun. 170(1–3), 1–8 (1999). [CrossRef]

47. H. C. Kandpal and J. S. Vaishya, “Experimental study of coherence properties of light fields in the region of superposition in Young’s interference experiment,” Opt. Commun. 186(1–3), 15–20 (2000). [CrossRef]

48. H. F. Schouten, T. D. Visser, and E. Wolf, “New effects in Young’s interference experiment with partially coherent light,” Opt. Lett. 28(14), 1182–1184 (2003). [CrossRef] [PubMed]

49. C. H. Gan, G. Gbur, and T. D. Visser, “Surface plasmons modulate the spatial coherence of light in Young’s interference experiment,” Phys. Rev. Lett. 98(4), 043908 (2007). [CrossRef] [PubMed]

50. H. F. Schouten, G. Gbur, T. D. Visser, and E. Wolf, “Phase singularities of the coherence functions in Young’s interference pattern,” Opt. Lett. 28(12), 968–970 (2003). [CrossRef] [PubMed]

51. H. Roychowdhury and E. Wolf, “Young’s interference experiment with light of any state coherence and of polarization,” Opt. Commun. 252(4–6), 268–274 (2005). [CrossRef]

52. G. S. Agarwal, A. Dogariu, T. D. Visser, and E. Wolf, “Generation of complete coherence in Young’s interference experiment with random mutually uncorrelated electromagnetic beams,” Opt. Lett. 30(2), 120–122 (2005). [CrossRef] [PubMed]

53. F. Gori, M. Santarsiero, R. Borghi, and E. Wolf, “Effects of coherence on the degree of polarization in a young interference pattern,” Opt. Lett. 31(6), 688–690 (2006). [CrossRef] [PubMed]

54. T. Setälä, J. Tervo, and A. T. Friberg, “Contrasts of Stokes parameters in Young’s interference experiment and electromagnetic degree of coherence,” Opt. Lett. 31(18), 2669–2671 (2006). [CrossRef] [PubMed]

55. T. Setälä, J. Tervo, and A. T. Friberg, “Stokes parameters and polarization contrasts in Young’s interference experiment,” Opt. Lett. 31(14), 2208–2210 (2006). [CrossRef] [PubMed]

56. Y. Li, H. Lee, and E. Wolf, “Spectra, coherence and polarization in Young’s interference pattern formed by stochastic electromagnetic beams,” Opt. Commun. 265(1), 63–72 (2006). [CrossRef]

57. A. Luis, “Modulation of coherence of vectorial electromagnetic waves in the Young interferometer,” Opt. Lett. 33(13), 1497–1499 (2008). [CrossRef] [PubMed]

58. Y. Li, X. L. Wang, H. Zhao, L. J. Kong, K. Lou, B. Gu, C. Tu, and H. T. Wang, “Young’s two-slit interference of vector light fields,” Opt. Lett. 37(11), 1790–1792 (2012). [CrossRef] [PubMed]

59. J. Qi, W. Wang, X. Li, X. Wang, W. Sun, J. Liao, and Y. Nie, “Double-slit interference of radially polarized vortex beams,” Opt. Eng. 53(4), 044107 (2014). [CrossRef]

60. J. J. Wen and M. A. Breazeale, “A diffraction beam field expressed as the superposition of Gaussian beams,” J. Acoust. Soc. Am. 83(5), 1752–1756 (1988). [CrossRef]

61. D. Ding and X. Liu, “Approximate description of Bessel, Bessel–Gauss, and Gaussian beams with finite aperture,” J. Opt. Soc. Am. A 16(6), 1286–1293 (1999). [CrossRef]

62. Y. Cai and L. Hu, “Propagation of partially coherent twisted anisotropic Gaussian Schell-model beams through an apertured astigmatic optical system,” Opt. Lett. 31(6), 685–687 (2006). [CrossRef] [PubMed]

63. Q. Lin and Y. Cai, “Tensor ABCD law for partially coherent twisted anisotropic Gaussian-Schell model beams,” Opt. Lett. 27(4), 216–218 (2002). [CrossRef] [PubMed]

64. J. Tervo, T. Setälä, and A. T. Friberg, “Degree of coherence for electromagnetic fields,” Opt. Express 11(10), 1137–1143 (2003). [CrossRef] [PubMed]

65. Y. Chen, F. Wang, L. Liu, C. Zhao, Y. Cai, and O. Korotkova, “Generation and propagation of a partially coherent vector beam with special correlation functions,” Phys. Rev. A 89(1), 013801 (2014). [CrossRef]

66. Y. Cai, Y. Chen, and F. Wang, “Generation and propagation of partially coherent beams with non-conventional correlation functions: a review,” J. Opt. Soc. Am. A 31(9), 2083–2096 (2014). [CrossRef]

Statistical properties in Young’s interference pattern formed with a radially polarized beam with controllable spatial coherence

Abstract

1. Introduction

2. Statistical properties in Young’s two-slit interference pattern formed with a partially coherent radially polarized beam: Theory

3. Statistical properties in Young’s two-slit interference pattern formed with a partially coherent radially polarized beam: Experiment

4. Summary

Acknowledgments

References and links

Cited By

Figures (10)

Equations (18)

Optics Express