Theoretical and experimental studies of the spectral changes of a polychromatic partially coherent radially polarized beam

Shijun Zhu; Xianglong Zhu; Lin Liu; Fei Wang; Yangjian Cai

doi:10.1364/OE.21.027682

1. Introduction

In 1986, it is revealed by Wolf that the spectrum of a partially coherent beam generally changes on propagation, even in free space, unless the degree of coherence satisfies a certain scaling law [1]. This effect known as Wolf effect is induced by the source correlations, which is different from that associated with the Doppler effects, which also can cause the changes of the spectrum of light on propagation [2–4]. Since then, numerous efforts have been paid to the spectral changes of partially coherent beams both theoretically and experimentally [5–21], and a review on this subject can be found in [5]. It was demonstrated that Wolf effect has important applications in determining the angular diameter of the stars [6], determining the angular separation of a double star [7], determining the intensity distribution across a distant source [8, 9], spectroradiometry [10], optical signal processing [11, 12], solving the inverse problems of scattering [13], information encoding and exchange [14, 15]. All literatures mentioned above are confined to the spectral changes of scalar partially coherent beams.

Recently, more and more attention is being paid to partially coherent vector beams [22–54]. Partially coherent vector beam with spatially uniform state of polarization (e.g., elliptically polarized beam and circularly polarized beam) usually is called stochastic (i.e., random) electromagnetic beam or partially coherent and partially polarized beam [22, 23]. Since Wolf proposed the unified theory of coherence and polarization [22], stochastic electromagnetic beam has been studied extensively both in theory and experiment [24–39]. Spectral changes of a stochastic electromagnetic beam on propagation have been investigated in [32–39].

Partially coherent vector beam with spatially non-uniform state of polarization is called partially coherent cylindrical vector beam [40], which was introduced recently as a natural extension of coherent cylindrical vector beam [41–48]. Nonparaxial propagation properties of a partially coherent cylindrical vector beam were explored in [49]. Statistical properties of a partially coherent cylindrical vector beam in turbulent atmosphere were reported in [50].Two typical kinds of partially coherent cylindrical vector beam named partially coherent radially polarized (RP) beam and partially coherent azimuthally polarized (AP) beam were generated experimentally and their statistical properties were measured in [51–53], and it was found that such beams were useful for material thermal processing and particle trapping. More recently, Wang et al. reported experimental study of the scintillation index of a partially coherent RP beam propagating through thermally induced turbulence and it was found 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 [54]. The partially coherent RP or AP beam reported in [51–54] is a monochromatic beam, and the spectral changes of a polychromatic partially coherent cylindrical vector beam haven’t been reported up until now. In this paper, our aim is to study the spectral changes of a polychromatic partially coherent RP beam focused by a thin lens both theoretically and experimentally. Some interesting results are found.

2. Spectral changes of a polychromatic partially coherent RP beam: Theory

We first outline briefly the theoretical model for a polychromatic partially coherent RP beam, then we analyze the spectral changes of such beam focused by a thin lens. Based on the unified theory of coherence and polarization, in space-frequency domain, the second-order correlation 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}, ω) = (\begin{matrix} W_{x x} (r_{1}, r_{2}, ω) & W_{x y} (r_{1}, r_{2}, ω) \\ W_{y x} (r_{1}, r_{2}, ω) & W_{y y} (r_{1}, r_{2}, ω) \end{matrix}),

with elements

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

where

E_{x}

and

E_{y}

denote the components of the random electric vector, at frequency

ω

, 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.

We assume the initial spectrum of a polychromatic partially coherent RP beam is of the Lorentz type with $ω_{0}$ being the central frequency and $Γ_{0}$ being the half-width at half-maximum, then according to [51, 52], we can express the elements of the CSD matrix of the polychromatic partially coherent RP beam as follows

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

where

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

is the position vector in the source plane,

σ_{0}

and

δ_{0}

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

Within the validity of the paraxial approximation, propagation of the elements of the CSD matrix of a partially coherent vector beam through a real stigmatic ABCD optical system can be studied with the help of the following generalized Collins formula [55]

\begin{array}{l} W_{α β} (u_{1}, u_{2}, ω) = \frac{ω^{2}}{4 π^{2} c^{2} B^{2}} \int \int W_{α β} (r_{1}, r_{2}, ω) \exp [- \frac{i k A}{2 B} (r_{1}^{2} - r_{2}^{2})] \\ \times \exp [\frac{i k}{B} (r_{1} \cdot u_{1} - r_{2} \cdot u_{2}) - \frac{i k D}{2 B} (u_{1}^{2} - u_{2}^{2})] d r_{1}^{2} d r_{2}^{2}, \end{array}

where

u \equiv (\begin{matrix} u_{x} & u_{y} \end{matrix})

is the position vector in the output plane, A, B, C and D are the elements of the transfer matrix for the real optical system,

k = ω / c

with c being the speed of light in vacuum. Note Eq. (4) represents a special case of Eq. (10) in Ref [55], which represents the generalized Collins formula for treating the propagation of a partially coherent beam through a real astigmatic ABCD optical system.

Substituting Eq. (3) into Eq. (4), we can obtain the following expressions for the elements of the CSD matrix of a polychromatic partially coherent RP beam in the output plane

W_{α α} (u_{1}, u_{2}, ω) = \frac{Γ_{0}^{2} V (u_{1}, u_{2})}{16 σ_{0}^{2} [{(ω - ω_{0})}^{2} + Γ_{0}^{2}]} [δ_{0}^{- 2} + \frac{k^{2}}{B^{2}} (u_{α 2} - \frac{u_{α 1}}{2 Δ δ_{0}^{2}}) (u_{α 1} + \frac{u_{α 1}}{4 Δ Π δ_{0}^{4}} - \frac{u_{α 2}}{2 Π δ_{0}^{2}})],

\begin{array}{l} W_{α β} (u_{1}, u_{2}, ω) = \frac{k^{2} Γ_{0}^{2} V (u_{1}, u_{2})}{16 σ_{0}^{2} B^{2} [{(ω - ω_{0})}^{2} + Γ_{0}^{2}]} (u_{β 2} - \frac{u_{β 1}}{2 Δ δ_{0}^{2}}) (u_{α 1} + \frac{u_{α 1}}{4 Δ Π δ_{0}^{4}} - \frac{u_{α 2}}{2 Π δ_{0}^{2}}), \\ (α \neq β, α, β = x, y), \end{array}

where

V (u_{1}, u_{2}) = \frac{k^{2}}{Δ^{2} Π^{2} B^{2}} \exp [- \frac{k^{2} u_{1}^{2}}{4 Δ B^{2}} - \frac{k^{2}}{4 Π B^{2}} {(u_{2} - \frac{u_{1}}{2 Δ δ_{0}^{2}})}^{2} - \frac{i k D}{2 B} (u_{1}^{2} - u_{2}^{2})],

with

Δ = \frac{1}{σ_{0}^{2}} + \frac{1}{2 δ_{0}^{2}} + \frac{i k A}{2 B}, Π = \frac{1}{σ_{0}^{2}} + \frac{1}{2 δ_{0}^{2}} - \frac{i k A}{2 B} - \frac{1}{4 Δ δ_{0}^{4}} .

The spectral intensity of the output polychromatic partially coherent RP beam at the point

u

is then given as follows [22, 23]

S (u, ω) = W_{x x} (u, u, ω) + W_{y y} (u, u, ω) .

Applying Eqs. (5)–(9), we can study the spectral changes of a polychromatic partially coherent RP beam propagating through a stigmatic ABCD optical system conveniently. Note the expressions Eqs. (5) and (6) are a little different from Eqs. (9)–(12) in Ref [52], which represent the expressions for the elements of the CSD matrix of a monchromatic partially coherent RP beam in the output plane and can be used to treat the propagation of such beam through a complex stigmatic ABCD optical system with loss or gain.

For the convenience of comparison with the spectral changes of a typical kind of scalar partially coherent beam named scalar polychromatic Gaussian Schell-model (GSM) beam, we also outline briefly the propagation formula for such beam. The second-order correlation properties of a scalar polychromatic GSM beam is characterized by the CSD given by [16]

W (r_{1}, r_{2}, ω) = \frac{Γ_{0}^{2}}{{(ω - ω_{0})}^{2} + Γ_{0}^{2}} \exp [- \frac{r_{1}^{2} + r_{2}^{2}}{σ_{0}^{2}} - \frac{{(r_{1} - r_{2})}^{2}}{2 δ_{0}^{2}}] .

Here we have assumed the initial spectrum is of the Lorentz type.

Equation (10) can be expressed in the following tensor form [55]

W (\tilde{r}, ω) = \frac{Γ_{0}^{2}}{{(ω - ω_{0})}^{2} + Γ_{0}^{2}} \exp (- \frac{i k}{2} {\tilde{r}}^{T} M_{0}^{- 1} \tilde{r}),

where

{\tilde{r}}^{T} = (\begin{matrix} r_{1}^{T} & r_{2}^{T} \end{matrix}) = (\begin{matrix} x_{1} & y_{1} & x_{2} & y_{2} \end{matrix})

,

M_{0}^{- 1}

is a

4 \times 4

matrix called partially coherent complex curvature tensor, and it takes the following form

M_{0}^{- 1} = (\begin{matrix} (- \frac{i}{2 k σ_{0}^{2}} - \frac{i}{k δ_{0}^{2}}) I & \frac{i}{k δ_{0}^{2}} I \\ \frac{i}{k δ_{0}^{2}} I & (- \frac{i}{2 k σ_{0}^{2}} - \frac{i}{k δ_{0}^{2}}) I \end{matrix}),

where I is a

2 \times 2

unit matrix. After propagating through a stigmatic ABCD optical system, the CSD of the scalar polychromatic GSM beam in the output plane is obtained as follows [55]

\begin{array}{l} W (\tilde{u}, ω) = \frac{Γ_{0}^{2}}{[{(ω - ω_{0})}^{2} + Γ_{0}^{2}] {[\det (\tilde{A} + \tilde{B} M_{0}^{- 1})]}^{1 / 2}} \\ \times [- \frac{i k}{2} {\tilde{u}}^{T} (\tilde{C} + \tilde{D} M_{0}^{- 1}) {(\tilde{A} + \tilde{B} M_{0}^{- 1})}^{- 1} \tilde{u}], \end{array}

where det stands for the determinant of a matrix,

{\tilde{u}}^{T} = (\begin{matrix} u_{1}^{T} & u_{2}^{T} \end{matrix})

with

u_{1}

and

u_{2}

being two arbitrary position vectors in the output plane,

\tilde{A}, \tilde{B}, \tilde{C} and \tilde{D}

are expressed as follows

\tilde{A} = (\begin{matrix} A I & 0 I \\ 0 I & A I \end{matrix}), \tilde{B} = (\begin{matrix} B I & 0 I \\ 0 I & - B I \end{matrix}), \tilde{C} = (\begin{matrix} C I & 0 I \\ 0 I & - C I \end{matrix}), \tilde{D} = (\begin{matrix} D I & 0 I \\ 0 I & D I \end{matrix}) .

The spectral intensity of the output polychromatic GSM beam at the point

u

is obtained as

S (u) = W (u, u)

.

Now we study the spectral changes of a polychromatic partially coherent RP beam focused by a thin lens, and compare with that of a scalar polychromatic GSM beam. The parameters used in the following examples are set as $σ_{0} = 1.35 mm,$ $δ_{0} = 0.06 mm,$ $f = 100 mm,$ $Γ_{0} = 1.4163 \times 10^{14}$ , $ω_{0} = 3.61685 \times 10^{15}$ . The focusing geometry is shown in Fig. 1, where the thin lens with focal length f is located at z = f and the exit plane is located at z. The transfer matrix for the optical system between the source plane and the exit plane reads as

Fig. 1 Focusing geometry.

Download Full Size | PDF

(\begin{matrix} A & B \\ C & D \end{matrix}) = (\begin{matrix} 1 & z - f \\ 0 & 1 \end{matrix}) (\begin{matrix} 1 & 0 \\ - 1 / f & 1 \end{matrix}) (\begin{matrix} 1 & f \\ 0 & 1 \end{matrix}) = (\begin{matrix} 2 - z / f & f \\ - 1 / f & 0 \end{matrix}) .

We suppose that the focused polychromatic partially coherent RP beam passes through a linear polarizer (LP) whose transmission axis forms an angle $θ$ with the x-axis. The spectral intensity of the transmitted beam reads as

S_{θ} (u, ω) = W_{x x} (u, u, ω) \cos^{2} θ + W_{y y} (u, u, ω) \sin^{2} θ + W_{x y} (u, u, ω) \sin 2 θ .

For the case of

θ = 0

, the transmitted beam represents a polychromatic partially coherent TEM₁₀ beam. For the case of

θ = π / 2

, the transmitted beam represents a polychromatic partially coherent TEM₀₁ beam. Thus the spectral intensity of a polychromatic partially coherent RP beam can be expressed as the superposition of those of orthogonally polarized partially coherent TEM₁₀ and TEM₀₁ beams (see Eq. (9)).

Figure 2 shows the spectral intensities of a focused polychromatic partially coherent RP beam and its corresponding composition components $W_{x x} (u, u, ω)$ and $W_{y y} (u, u, ω)$ at several propagation distances. One finds from Fig. 2 that the evolution properties of the intensity pattern of the focused polychromatic partially coherent RP beam are similar to that of a focused monchromatic partially coherent RP beam as reported in [51], i.e., the dark hollow beam profile disappears on propagation and a Gaussian beam profile can be formed in the focal plane. Furthermore, the evolution properties of the intensity patterns of the composition components $W_{x x} (u, u)$ and $W_{y y} (u, u)$ of the focused polychromatic partially coherent RP beam are also similar to the corresponding results of a monochromatic partially coherent RP beam as reported in [51].

Fig. 2 Spectral intensities (contour graphs) of a focused polychromatic partially coherent RP beam and its corresponding composition components $W_{x x} (u, u, ω)$ and $W_{y y} (u, u, ω)$ at several propagation distances.

Download Full Size | PDF

Figure 3 show the normalized on-axis spectrum $S (u = 0, ω) / S {(u = 0, ω)}_{\max}$ of a focused polychromatic partially coherent RP beam in the focal plane (z = 2f). For the convenience of comparison, the normalized spectrum in the source plane (z = 0) is also shown in Fig. 3. Note that the spectral intensity of the beam center of the polychromatic partially coherent RP beam at z = 0 is zero, thus the dark solid curve in Fig. 3 in fact denotes the off-axis normalized spectrum, which is independent of the transverse position across the source plane. One finds from Fig. 3 that the normalized on-axis spectrum of the focused polychromatic partially coherent RP beam is similar to the normalized off-axis spectrum at z = 0, but its peak position is blue-shifted, which is similar to that of the focused scalar polychromatic GSM beam as reported in [16].

Fig. 3 Normalized on-axis spectrum of a focused polychromatic partially coherent RP beam in the focal plane (z = 2f) and the normalized spectrum of such beam in the source plane (z = 0).

Download Full Size | PDF

Now we study the variation of the relative spectral shift of a focused polychromatic partially coherent RP beam on propagation, and compare with that of a focused scalar polychromatic GSM beam. The spectral shift $Δ ω$ is the difference between the peak frequency $ω_{m}$ of the spectrum of the field after propagation and the central frequency $ω_{0}$ of the source spectrum. A positive value of $Δ ω$ denotes a blue shift, while a negative value represents a red shift. The relative spectral shift is defined as

η = (ω_{m} - ω_{0}) / ω_{0} .

Figure 4 shows the on-axis relative spectral shift of a focused polychromatic partially coherent RP beam versus the propagation distance. Figure 5 shows the on-axis relative spectral shift of a focused scalar polychromatic GSM beam versus the propagation distance. One finds from Figs. 4 and 5 that the dependence of the on-axis relative spectral shift of a focused polychromatic partially coherent RP beam on the propagation distance is different from that of a focused scalar polychromatic GSM beam. For the focused polychromatic partially coherent RP beam, its on-axis spectral shift just behind the focal length is red shift, and the red shift decreases gradually with the increase of the propagation distance, and there is no shift at all at certain propagation distance. With the further increase of the propagation distance, blue shift appears and the maximum blue shift occurs in the focal plane. After the focal plane, the blue-shift gradually decreases with the increase of the propagation distance, and finally read shift appears again. For the focused scalar polychromatic GSM beam, the on-axis spectral shift is always blue shift, and the maximum blue shift occurs in the focal plane.

Fig. 4 On-axis relative spectral shift of a focused polychromatic partially coherent RP beam versus the propagation distance.

Download Full Size | PDF

Fig. 5 On-axis relative spectral shift of a focused scalar polychromatic GSM beam versus the propagation distance.

Download Full Size | PDF

To learn about the variation of the off-axis spectral shift on propagation, we calculate in Fig. 6 the relative spectral shift of a focused polychromatic partially coherent RP beam versus the transverse coordinate $u_{x}$ with $u_{y} = 0$ at several propagation distances and in Fig. 7 the relative spectral shift of a focused polychromatic partially coherent RP beam versus the propagation distance and the transverse coordinate $u_{x}$ with $u_{y} = 0$ . For the convenience of comparison, we calculate in Figs. 8 and 9 the corresponding results of a focused scalar polychromatic GSM beam. From Figs. 6–9, one sees that the dependence of the spectral shift of the focused polychromatic partially coherent RP beam near the focal plane on the transverse coordinate $u_{x}$ is similar to that of the focused scalar polychromatic GSM beam, i.e., the blue shift gradually decreases as $u_{x}$ increases, and the red shift can be observed when $u_{x}$ is large enough, no spectral shift occurs for certain value of $u_{x}$ . At the plane far away from the focal plane, difference between the spectral shifts of above two beams appears. For a focused polychromatic partially coherent RP beam, the red shift gradually decreases with the increase of $u_{x}$ , and blue shift appears when $u_{x}$ reaches certain value. After the blue shift reaches its maximum value, the relative spectral shift decreases gradually with the further increase of $u_{x}$ , and finally red shift appears again when $u_{x}$ is large enough. For a focused scalar polychromatic GSM beam, the dependence of the relative spectral shift on $u_{x}$ at the plane far away from the focal plane is similar to the case near the focal plane, while the absolute value of the relative spectral shift is quite small.

Fig. 6 Relative spectral shift of a focused polychromatic partially coherent RP beam versus the transverse coordinate $u_{x}$ with $u_{y} = 0$ at several propagation distances.

Download Full Size | PDF

Fig. 7 Relative spectral shift of a focused polychromatic partially coherent RP beam versus the propagation distance and the transverse coordinate $u_{x}$ with $u_{y} = 0$ .

Download Full Size | PDF

Fig. 8 Relative spectral shift of a focused scalar polychromatic GSM beam versus the transverse coordinate $u_{x}$ with $u_{y} = 0$ at several propagation distances.

Download Full Size | PDF

Fig. 9 Relative spectral shift of a focused scalar polychromatic GSM beam versus the propagation distance and the transverse coordinate $u_{x}$ with $u_{y} = 0$ .

Download Full Size | PDF

3. Spectral changes of a polychromatic partially coherent RP beam: Experiment

In this section, we carry out experimental study of the spectral changes of a polychromatic partially coherent RP beam. Part I of Fig. 10 shows our experimental setup for generating a polychromatic partially coherent RP beam. The polychromatic light with extremely low coherence emitted from a light-emitting diode passes through a circular aperture (CA₁) with radius $a_{1} = 1 mm$ , which is used to select a portion of the diverging light. The transmitted beam is reflected by a reflecting mirror and then passes through the second circular aperture (CA₂) with radius $a_{2} = 0.5 mm$ , which is used to improve the coherence of the polychromatic beam. The transmitted beam from the CA₂ can be regarded as a polychromatic partially coherent beam, and it becomes a polychromatic GSM beam after passing through the collimation lens L₁ and the Gaussian amplitude filter (GAF). After passing through a beam expander and a linear polarizer, the generated polychromatic GSM beam becomes linearly polarized, and then it illuminates a radial polarization converter (RPC) produced by the company Arcoptix which is used to convert a linearly polarized polychromatic GSM beam into a polychromatic partially coherent RP beam. The transmitted beam just behind the RPC is regarded as the light source for a polychromatic partially coherent RP beam. Part II of Fig. 10 shows our setup for measuring the spectral intensity and the spectrum of the generated polychromatic partially coherent RP beam focused by the thin lens L₂ with focal length f = 100mm. In our experiment, we use a charge-coupled device to measure the spectral intensity and a spectrometer to measure the spectrum. The distance between the RPC and the L₂ equals to f, and the transfer matrix of the optical system between the source plane and the receiver plane is given by Eq. (15).

Fig. 10 Experimental setup for generating a polychromatic partially coherent RP beam and measuring its focused spectral intensity and spectrum. LED, light-emitting diode; CA₁, CA₂, circular apertures; L₁, L₂, thin lenses; RM, reflecting mirror; GAF, Gaussian amplitude filter; BE, beam expander; LP, linear polarizer; RPC, radial polarization converter; CCD, charge-coupled device; PC, personal computer.

Download Full Size | PDF

Figure 11 shows our experimental results of the spectral intensity of the generated polychromatic GSM beam just behind the LP and its normalized spectrum for different values of the transverse coordinate x with y = 0. One finds from Fig. 11 (a) that the spectral intensity of the generated polychromatic GSM beam has a Gaussian beam profile as expected. From Fig. 11(b), we find that the normalized on-axis spectrum and the normalized off-axis spectrum is almost the same.

Fig. 11 Experimental results of (a) the spectral intensity of the generated polychromatic GSM beam just behind the LP and (b) its normalized spectrum for different values of the transverse coordinate x with y = 0.

Download Full Size | PDF

Figure 12 shows our experimental results of the spectral intensity and the corresponding cross line (y = 0, dotted curve) of the generated polychromatic partially coherent RP beam just behind the RPC. The corresponding theoretical fit (solid curve) of the experimental data is also shown in Fig. 12(b). One finds that the spectral intensity of the generated partially coherent RP beam can be approximately characterized by Eq. (3) with $σ_{0} = 0.68 mm$ .

Fig. 12 Experimental results of (a) the spectral intensity and (b) the corresponding cross line (y = 0, dotted curve) of the generated polychromatic partially coherent RP beam just behind the RPC. The solid curve denotes the theoretical fit of the experimental data with $σ_{0} = 0.68$ .

Download Full Size | PDF

As regards the coherence properties of the generated polychromatic partially coherent RP beam, we would like to say it’s hard for us to measure the coherence function and the coherence width $δ_{0}$ with the existing experimental setup in our lab. In Ref [51], we have measured the coherence function and the coherence width $δ_{0}$ of the generated monochromatic partially coherent RP beam through measuring the fourth-order correlation function of the beam by the electronic coincidence circuit, while the response time of the electronic coincidence circuit should be smaller than the characteristic time of the intensity fluctuation of the beam. Because the characteristic time of the intensity fluctuation of the polychromatic beam is much smaller than the response time of the electronic coincidence circuit, it’s impossible for us to measure the coherence function with the existing experimental setup in our lab. While it is reasonable to assume that the coherence function of the generated polychromatic partially coherent RP beam has a quasi-Gaussian distribution and can be approximately characterized by Eq. (3) because the coherence function of most random light has a quasi-Gaussian profile even for sunlight. As shown in [56], the coherence function of sunlight has a quasi-Gaussian distribution. Furthermore, the circular aperture may change the coherence function of the generated beam when the radius of the aperture is much smaller than the beam width. While when the radius of the aperture is comparable (or larger than) to the width of the beam, the coherence function of the transmitted beam is still of quasi-Gaussian distribution (see Ref [57].). In our paper, the radius of the aperture CA₂ is chosen to be comparable to the width of the incident beam, thus the coherence function of the generated polychromatic partially coherent RP beam can be approximately characterized by Eq. (3).

The central spectral intensity of the generated polychromatic partially coherent RP beam is nearly zero just behind the RPC and we can’t measure the spectrum of the beam center. Thus, we show in Fig. 13 our experimental results of the normalized spectrum of the generated polychromatic partially coherent RP beam at z = 2cm for different values of the transverse coordinate x with y = 0. From Fig. 13, we find that the profile of the normalized spectrum of the generated polychromatic partially coherent RP beam is not as smooth as that of the generated polychromatic GSM beam due to the influence of the RPC, while the peak position of the normalized on-axis or off-axis spectrum is the same with that of the generated polychromatic GSM beam. Furthermore, the difference between the normalized on-axis spectrum and the normalized off-axis spectrum is quite small.

Fig. 13 Experimental results of the normalized spectrum of the generated polychromatic partially coherent RP beam at z = 2cm for different values of the transverse coordinate x with y = 0.

Download Full Size | PDF

Figure 14 shows our experimental results of the spectral intensities and its corresponding composition components $W_{x x} (u, u, ω)$ and $W_{y y} (u, u, ω)$ of the generated polychromatic partially coherent RP beam focused by a thin lens at several propagation distances. Comparing Figs. 2 and 14, one finds that our experimental results of the evolution properties of the spectral intensities and its corresponding composition components $W_{x x} (u, u, ω)$ and $W_{y y} (u, u, ω)$ on propagation are consistent with the theoretical predictions.

Fig. 14 Experimental results of the spectral intensities and its corresponding composition components $W_{x x} (u, u, ω)$ and $W_{y y} (u, u, ω)$ of the generated polychromatic partially coherent RP beam focused by a thin lens at several propagation distances.

Download Full Size | PDF

Figure 15 shows our experimental results of the normalized on-axis spectrum of the generated polychromatic partially coherent RP beam focused by a thin lens at two propagation distances. One finds that the peak position of the normalized spectrum indeed is shifted on propagation as expected by Fig. 3. Figure 16 shows our experimental results of the relative spectral shift of the generated polychromatic partially coherent RP beam focused by a thin lens versus the transverse coordinate $u_{x}$ with $u_{y} = 0$ at several propagation distances. Comparing Figs. 6 and 16, one sees that our experimental results about the variation of the relative spectral shift on propagation are also consistent with the theoretical predictions.

Fig. 15 Experimental results of the normalized on-axis spectrum of the generated polychromatic partially coherent RP beam focused by a thin lens at two propagation distances.

Download Full Size | PDF

Fig. 16 Experimental results of the relative spectral shift of the generated polychromatic partially coherent RP beam focused by a thin lens versus the transverse coordinate $u_{x}$ with $u_{y} = 0$ at several propagation distances.

Download Full Size | PDF

4. Summary

We have carried out theoretical and experimental studies of the spectral changes of a polychromatic partially coherent RP beam focused by a thin lens for the first time. It is found that the variation properties of the spectral changes of a polychromatic partially coherent RP beam on propagation are much different from that of a scalar polychromatic GSM beam. For a focused polychromatic partially coherent RP beam, its on-axis spectral shift is red shift at the distances far away from the focal plane, and its on-axis spectral shift is blue shift at the distances near the focal plane. For a focused scalar polychromatic GSM beam, its on-axis shift is always blue shift. Our experimental results are consistent with the theoretical predictions.

Acknowledgments

This research is supported by the National Natural Science Foundation of China under Grant Nos. 11274005 and 11104195, 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 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. CXLX12_0780, and the Project Sponsored by the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry.

References and links

1. E. Wolf, “Invariance of the spectrum of light on propagation,” Phys. Rev. Lett. 56(13), 1370–1372 (1986). [CrossRef] [PubMed]

2. E. Wolf, “Non-cosmological redshifts of spectral lines,” Nature 326(6111), 363–365 (1987). [CrossRef]

3. E. Wolf, “Redshifts and blueshifts of spectral lines caused by source correlations,” Opt. Commun. 62(1), 12–16 (1987). [CrossRef]

4. E. Wolf, “Red shifts and blue shifts of spectral lines emitted by two correlated sources,” Phys. Rev. Lett. 58(25), 2646–2648 (1987). [CrossRef] [PubMed]

5. E. Wolf and D. F. James, “Correlation-induced spectral changes,” Rep. Prog. Phys. 59(6), 771–818 (1996). [CrossRef]

6. H. C. Kandpal, A. Wasan, J. S. Vaishya, E. S. R. Gopal, M. Singh, B. B. Sanwal, and R. Sagar, “Application of spatial coherence spectroscopy for determining the angular diameters of stars: feasibility experiment,” Indian J. Pure Appl. Phys. 36, 665–674 (1988).

7. D. F. James, H. C. Kandpal, and E. Wolf, “A new method for determining the angular separation of double stars,” Astrophys. J. 445, 406–410 (1995). [CrossRef]

8. D. F. V. James and E. Wolf, “Determination of field corrections from spectral measurements with application to synthetic aperture imaging,” Radio Sci. 26(5), 1239–1243 (1991). [CrossRef]

9. H. C. Kandpal, J. S. Vaishya, K. Saxena, D. S. Mehta, and K. C. Joshi, “Intensity distribution across a source from spectral measurements,” J. Mod. Opt. 42(2), 455–464 (1995). [CrossRef]

10. H. C. Kandpal, J. S. Vaishya, and K. C. Joshi, “Wolf shift and its application in spectroradiometry,” Opt. Commun. 73(3), 169–172 (1989). [CrossRef]

11. E. Wolf, T. Shirai, H. Chen, and W. Wang, “Coherence filters and their uses: 1. Basic theory and examples,” J. Mod. Opt. 44, 1345–1353 (1997).

12. T. Shirai, E. Wolf, H. Chen, and W. Wang, “Coherence filters and their uses: 2. One-dimensional realizations,” J. Mod. Opt. 45(4), 799–816 (1998). [CrossRef]

13. D. Zhao, O. Korotkova, and E. Wolf, “Application of correlation-induced spectral changes to inverse scattering,” Opt. Lett. 32(24), 3483–3485 (2007). [CrossRef] [PubMed]

14. B. K. Yadav, S. A. M. Rizvi, S. Raman, R. Mehrotra, and H. C. Kandpal, “Information encoding by spectral anamolies of spatially coherent light diffracted by an annular aperture,” Opt. Commun. 269(2), 253–260 (2007). [CrossRef]

15. F. Gori, G. L. Marcopoli, and M. Santarsiero, “Spectrum invariance on paraxial propagation,” Opt. Commun. 81(1–2), 123–130 (1991). [CrossRef]

16. C. Palma, G. Cincotti, and G. Guattari, “Spectral shift of a Gaussian Schell-model beam beyond a thin lens,” IEEE J. Quantum Electron. 34(2), 378–383 (1998). [CrossRef]

17. J. Pu and S. Nemoto, “Spectral shifts and spectral switches in diffraction of partially coherent light by a circular aperture,” IEEE J. Quantum Electron. 36(12), 1407–1411 (2000). [CrossRef]

18. L. Pan and B. Lu, “The spectral switch of partially coherent light in Young’s experiment,” IEEE J. Quantum Electron. 36, 1407–1411 (2001).

19. Y. Cai, Y. Huan, and Q. Lin, “Spectral shift of partially coherent twisted anisotropic Gaussian–Schell-model beams focused by a thin lens,” J. Opt. A, Pure Appl. Opt. 5(4), 397–401 (2003). [CrossRef]

20. O. Korotkova and E. Shchepakina, “Color changes in stochastic light fields propagating in non-Kolmogorov turbulence,” Opt. Lett. 35(22), 3772–3774 (2010). [CrossRef] [PubMed]

21. Z. Tong and O. Korotkova, “Spectral shifts and switches in random fields upon interaction with negative-phase materials,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 82, 013829 (2010).

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

23. 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]

24. 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]

25. 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] [PubMed]

26. O. Korotkova and E. Wolf, “Generalized Stokes parameters of random electromagnetic beams,” Opt. Lett. 30(2), 198–200 (2005). [CrossRef] [PubMed]

27. A. Luis, “Degree of polarization for three-dimensional fields as a distance between correlation matrices,” Opt. Commun. 253(1–3), 10–14 (2005). [CrossRef]

28. T. Setälä, K. Lindfors, and A. T. Friberg, “Degree of polarization in 3D optical fields generated from a partially polarized plane wave,” Opt. Lett. 34(21), 3394–3396 (2009). [CrossRef] [PubMed]

29. 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]

30. 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]

31. 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]

32. J. Pu, O. Korotkova, and E. Wolf, “Polarization-induced spectral changes on propagation of stochastic electromagnetic beams,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 75(5), 056610 (2007). [CrossRef] [PubMed]

33. J. Pu, O. Korotkova, and E. Wolf, “Invariance and noninvariance of the spectra of stochastic electromagnetic beams on propogation,” Opt. Lett. 31(14), 2097–2099 (2006). [CrossRef] [PubMed]

34. O. Korotkova, J. Pu, and E. Wolf, “Spectral changes in electromagnetic stochastic beams propagating through turbulent atmosphere,” J. Mod. Opt. 55(8), 1199–1208 (2008). [CrossRef]

35. F. Zhou, S. Zhu, and Y. Cai, “Spectral shift of an electromagnetic Gaussian Schell-model beam propagating through tissue,” J. Mod. Opt. 58(1), 38–44 (2011). [CrossRef]

36. L. Pan, Z. Zhao, C. Ding, and B. Lu, “Effect of polarization on spectral switches in the diffraction of stochastic electromagnetic beams,” Appl. Phys. Lett. 95(18), 181112 (2009). [CrossRef]

37. 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]

38. M. Yao, Y. Cai, and O. Korotkova, “Spectral shift of a stochastic electromagnetic Gaussian Schell-model beam in a Gaussian cavity,” Opt. Commun. 283(22), 4505–4511 (2010). [CrossRef]

39. S. Joshi, B. K. Yadav, M. Verma, M. S. Khan, and H. C. Kandpal, “Effect of polarization on spectral anomalies of diffracted stochastic electromagnetic beams,” J. Opt. 15(3), 035405 (2013). [CrossRef]

40. 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]

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

42. K. S. Youngworth and T. G. Brown, “Focusing of high numerical aperture cylindrical-vector beams,” Opt. Express 7(2), 77–87 (2000). [CrossRef] [PubMed]

43. D. P. Biss and T. G. Brown, “Cylindrical vector beam focusing through a dielectric interface,” Opt. Express 9(10), 490–497 (2001). [CrossRef] [PubMed]

44. Q. Zhan and J. R. Leger, “Focus shaping using cylindrical vector beams,” Opt. Express 10(7), 324–331 (2002). [CrossRef] [PubMed]

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

46. 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]

47. 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]

48. 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]

49. 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]

50. R. Chen, Y. Dong, F. Wang, and Y. Cai, “Statistical properties of a cylindrical vector partially coherent beam in turbulent atmosphere,” Appl. Phys. B 112(2), 247–259 (2013). [CrossRef]

51. 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]

52. 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]

53. 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]

54. 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]

55. 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]

56. H. Mashaal, A. Goldstein, D. Feuermann, and J. M. Gordon, “First direct measurement of the spatial coherence of sunlight,” Opt. Lett. 37(17), 3516–3518 (2012). [CrossRef] [PubMed]

57. F. Wang, Y. Cai, and Q. Lin, “Experimental observation of truncated fractional Fourier transform for a partially coherent Gaussian Schell-model beam,” J. Opt. Soc. Am. A 25(8), 2001–2010 (2008). [CrossRef] [PubMed]

Theoretical and experimental studies of the spectral changes of a polychromatic partially coherent radially polarized beam

Abstract

1. Introduction

2. Spectral changes of a polychromatic partially coherent RP beam: Theory

3. Spectral changes of a polychromatic partially coherent RP beam: Experiment

4. Summary

Acknowledgments

References and links

Cited By

Figures (16)

Equations (17)

Optics Express