Random sources for beams with azimuthally varying polarization properties

Fei Wang; Olga Korotkova

doi:10.1364/OE.24.015446

1. Introduction

Scalar, spatially variable optical fields may possess various types of phase singularities, such as zeros, wave dislocations, and vortices [1–3]. In particular, optical phase vortex is a point of zero amplitude in the neighborhood of which phase increases monotonically with the increasing polar angle. In electromagnetic domain, in addition to disclinations, i.e. points where the local vector field vanishes, one classifies singularities according to the shape of polarization ellipse: C points (or regions) where the polarization ellipse degenerates to circular state and L points (curves) along which it degenerates to linear state [4–6].

Mathematical models of deterministic scalar beam-like fields possessing singularities have also been developed, the most famous being the Laguerre-Gauss beam, carrying phase vortex on the axis [7]. The deterministic optical beams with polarization properties varying across the transverse cross-sections in a prescribed manner, known as vectorial or cylindrical beams, have been thoroughly studied in the literature as well (see [8] and references wherein). Typical examples of such beams are radially and azimuthally linearly polarized beams, i.e. beams in which the orientation of polarization direction coincides with rays $θ = c o n s t$ , and perpendicular to $θ = c o n s t$ , respectively, where $θ$ is phase angle in polar coordinates. More complex types of vector beams can be obtained by superposition techniques [9].

In random scalar optical beam-like fields in addition to vortex singularities in separable phase [10], the singularities of the degree of coherence can be defined [11]. In random electromagnetic fields, in addition to coherence and polarization state singularities one can define the singularity in the degree of polarization (DOP), being the ratio of polarized to total intensities. For instance, in Ref [12]. the degree of polarization singularities, the so-called P and U points, have been defined as those at which the DOP takes on values 1 and 0, respectively.

Only few models for random electromagnetic beams have been introduced so far analytically [13–20] and realized experimentally [18–22]. In such beams the distributions of the average intensity and of the polarization properties, which are typically mutually dependent, have been recently shown to be made independent or partially correlated [23].

In this paper we first introduce a general class of statistically stationary (with respect to temporal and spatial coherence as well as polarization) electromagnetic beams in which the average intensity and the polarization properties can have arbitrary dependence on radial distance and polar angle. Our goal is to extend the results of Ref [24]. which were concerned with scalar sources radiating far fields with azimuthally dependent average intensity to electromagnetic domain. Then, using this model, we construct beams whose DOP distribution is vortex-like in the far field, which means that in the transverse cross-section at off-axis positions of the beam the value of the DOP increases from value nearly 0 to value nearly 1 around the polar angle for l times with l being the topological charge. In addition, we carry out the experiment for generating such class of the electromagnetic beam using Mach-Zehnder interferometer with two spatial light modulators (SLMs) and confirm that the radiated far fields agree well with theoretical predictions.

2. Electromagnetic azimuthally variable beams

Let us consider a wide-sense statistically stationary electromagnetic beam-like field propagating close to z-axis. In space-frequency domain such a beam may be characterized by $2 \times 2$ cross-spectral density (CSD) matrix ${\overset{\leftrightarrow}{W}}^{(0)} (r_{1}, r_{2})$ in the source plane (z = 0) having elements [25]

W_{α β}^{(0)} (r_{1}, r_{2}, ω) = 〈 E_{α}^{*} (r_{1}, ω) E_{β} (r_{2}, ω) 〉, (α, β = x, y),

where

r_{1} \equiv (x_{1}, y_{1})

and

r_{2} \equiv (x_{2}, y_{2})

are two arbitrary position vectors in the source plane;

E_{x}

and

E_{y}

denote mutually orthogonal components of the electric field, perpendicular to z-axis; the angular brackets stand for the statistical average over the source realizations;

ω

is the angular frequency of the beam. For brevity, the dependence of the derived quantities on

ω

will be omitted.

The elements of the CSD matrix of the source have been shown to have alternative representations in the form [26, 27]

W_{α β}^{(0)} (r_{1}, r_{2}) = \int p_{α β} (v) H_{α}^{*} (r_{1}, v) H_{β} (r_{2}, v) d^{2} v,

where

α, β = x, y

;

H_{x}

and

H_{y}

are two arbitrary kernel functions;

p_{α β} (v)

are the elements of the

2 \times 2

weighting matrix

\overset{\leftrightarrow}{p} (v)

. While the kernel functions define the class of correlation, such as Schell-like or non-uniform, the weighting matrix elements can be shown to be proportional to the elements of the single-position CSD matrix in the far field, and hence determine its polarization state. In order to guarantee the non-negative definiteness of the CSD matrix

p_{x x} (v)

,

p_{y y} (v)

and the determinant of

\overset{\leftrightarrow}{p} (v)

must all be non-negative for any value of vector v [27].

We will be interested in modeling with $\overset{\leftrightarrow}{p} (v)$ elements separable in radial and azimuthal variables, i.e.,

p_{α β} (v) = p_{α β}^{r} (v) p_{α β}^{θ} (θ) .

While in the general case when the x and y electric field components are correlated the conditions in [27] stipulate that

\begin{array}{l} p_{α α}^{r} (v) \geq 0, p_{α α}^{θ} (θ) \geq 0, α = x, y \\ p_{x x}^{r} (v) p_{x x}^{θ} (θ) p_{y y}^{r} (v) p_{y y}^{θ} (θ) - [^{p_{x y}^{r} (v) p_{x y}^{θ}] 2} \geq 0 \end{array}

where

v

and

θ

are the radial and azimuthal coordinates of vector

v

_, for the case of uncorrelated field components, i.e.

p_{x y} (v) = p_{y x} (v) = 0

, which we will discuss below, only two first inequalities must hold.

3. Electromagnetic beams with azimuthally varying degree of polarization

As an example, let us suppose that the on-diagonal elements of the weighting matrix are separable with respect to radial and azimuthal polar coordinates in the form and off-diagonal elements vanish:

p_{x x} (v) = \frac{k^{2} δ_{x x}^{2}}{2 π^{2}} \exp (- \frac{k^{2} δ_{x x}^{2} v^{2}}{2}) m o d (l_{x} θ, 2 π),

p_{y y} (v) = \frac{k^{2} δ_{y y}^{2}}{3 π^{2}} \exp (- \frac{k^{2} δ_{y y}^{2} v^{2}}{2}) (2 π - \frac{m o d (l_{y} θ, 2 π)}{2}),

p_{x y} (v) = p_{y x} (v) = 0,

where

l_{x}

and

l_{y}

are positive integers; function mod returns value

l_{α} θ - 2 p π

where

p

is the integer part of

l_{α} θ / 2 π

;

k = 2 π / λ

is a wavenumber with

λ

being the wavelength;

δ_{x x}

,

δ_{y y}

are the r.m.s. widths of auto-correlation functions of the x and y components of the field, respectively. It is evident from Eqs. (5) and (6) that

p_{x x} (v)

and

p_{y y} (v)

are non-negative for any v. Note that such choices for the coefficients k²/2π² and k²/3π² in

p_{x x} (v)

and

p_{y y} (v)

are to ensure that the degrees of self-correlations

μ_{x x}^{(0)}

and

μ_{y y}^{(0)}

do satisfy the constraints

| μ_{x x}^{(0)} | \leq 1

and

| μ_{y y}^{(0)} | \leq 1

in the following discussion.

For our modeling we will involve the Schell-model beam [25] being the most fundamental model of a random optical beam in the classical statistical optics that can be readily realized by means of diffusers of spatial light modulators. For such beams the kernel function in Eq. (2) takes form:

H_{α} (r, v) = A_{α} \exp (- \frac{r^{2}}{2 σ_{0}^{2}}) \exp (- i k r • v), α = x, y

In Eq. (8)

σ_{0}

is the r.m.s source width;

A_{x}

and

A_{y}

are the amplitudes of the field components. On substituting from Eqs. (5)-(8) into Eq. (2), we obtain for the elements of the CSD matrix in polar coordinates the expressions:

W_{α α}^{(0)} (r_{1}, r_{2}) = A_{α}^{2} \exp (- \frac{r_{1}^{2} + r_{2}^{2}}{2 σ_{0}^{2}}) μ_{α α}^{(0)} (r_{2} - r_{1}),

W_{x y}^{(0)} (r_{1}, r_{2}) = W_{y x}^{(0)} (r_{1}, r_{2}) = 0,

with

\begin{array}{l} μ_{α α}^{(0)} (r_{2} - r_{1}) = \exp (\frac{- r_{d}^{2}}{2 δ_{α α}^{2}}) \mp \frac{1}{C_{α}} \sqrt{\frac{1}{2 π}} \frac{r_{d}}{δ_{α α}} \exp (\frac{- r_{d}^{2}}{4 δ_{α α}^{2}}) \\ \times \sum_{m = 1}^{\infty} \frac{{(- i)}^{m l_{α}}}{m} [I_{(m l_{α} - 1) / 2} (\frac{r_{d}^{2}}{4 δ_{α α}^{2}}) - I_{(m l_{α} + 1) / 2} (\frac{r_{d}^{2}}{4 δ_{α α}^{2}})] \\ \times s i n (m l_{α} φ_{d}), (α = x, y) . \end{array}

Here C_x = 1, C_y = 3; the minus and plus signs before

C_{α}

corresponds to the x-x and y-y component of correlation function, respectively;

I_{m}

is the modified Bessel function of order m;

r_{d} = | r_{2} - r_{1} |

is the distance in the radial direction and

φ_{d} = \arctan [(y_{2} - y_{1}) / (x_{2} - x_{1})] .

The Spectral Density (SD) and the DOP in the source plane can be calculated by the expressions

S^{(0)} (r) = T r \overset{\leftrightarrow}{W} (r, r) = (A_{x}^{2} + A_{y}^{2}) \exp (- r^{2} / σ_{0}^{2}),

P^{(0)} (r) = \sqrt{1 - \frac{4 D e t \overset{\leftrightarrow}{W} (r, r)}{{[T r W (r, r)]}^{2}}} = \frac{| A_{x}^{2} - A_{y}^{2} |}{A_{x}^{2} + A_{y}^{2}},

where Tr and Det denote the trace and the determinant of the CSD matrix. One finds from Eqs. (12) and (13) that the SD of the random source has Gaussian profile and the DOP across the source plane is uniform.

Within the validity of paraxial approximation, propagation of a random beam from the source plane to the output plane of an ABCD optical system can be studied by the extended Collins integral formula [28, 29]. In this paper, we are only interested in the CSD matrix at a single position, i.e., $ρ = ρ_{1} = ρ_{2}$ . In this situation, the elements of the CSD matrix in the output plane can be expressed as

\begin{array}{l} W_{α β} (ρ, ρ, z) = \frac{1}{λ^{2} B^{2}} \int \int W_{α β}^{(0)} (r_{1}, r_{2}) \exp (- \frac{i k A}{2 B} r_{1}^{2} + \frac{i k}{B} r • ρ) \\ \times \exp (\frac{i k A}{2 B} r_{2}^{2} - \frac{i k}{B} r_{2} • ρ) d^{2} r_{1} d^{2} r_{2} . \end{array}

where

ρ

is the position vector in the output plane and A, B are the elements of the transfer matrix for the ABCD optical system. To evaluate Eq. (14), we first substitute from Eqs. (2) and (8) into Eq. (14) and define a new function

f (r) = \exp [- (\frac{1}{2 σ_{0}^{2}} - \frac{i k A}{2 B}) r^{2}] .

Next, we introduce the “sum” and “difference” coordinates:

r_{s} = (r_{1} + r_{2}) / 2

,

r_{d} = r_{2} - r_{1}

, and write functions

f

and

f^{*}

in terms of their Fourier transforms,

f (r) = F^{- 1} [\tilde{f} (u)]

. After some straightforward manipulations and simplifications, Eq. (14) can be represented in the following convolution form (see also [30])

\begin{array}{l} W_{α β} (ρ, ρ, z) = A_{α} A_{β} \int \int S (- ρ / B - v) p_{α β} (v) d^{2} v \\ = A_{α} A_{β} S (- ρ / B) \otimes p_{α β} (- ρ / B) \\ = A_{α} A_{β} F^{- 1} [\tilde{S} (f) μ_{α β} (f)] (- ρ / B) \end{array}

where

S (u) = {| \tilde{f} (- u) |}^{2} / λ^{2} B^{2}

and

\tilde{S}

denotes the Fourier transform of

S

. Further,

S (u)

has the analytical form

S (u) = \exp (- B^{2} u^{2} / σ_{0}^{2} F (B)) / F (B)

with

F (B) = A^{2} + B^{2} / k^{2} σ_{0}^{4}

.

It is of importance to note that in the quasi-homogeneous limit, i.e. when $δ_{x x} < < σ_{0}$ and $δ_{y y} < < σ_{0}$ , Eq. (16) reduces to the simple form with a good approximation

W_{α β} (ρ, ρ, z) \approx A_{α} A_{β} p_{α β} (- ρ / B)

And the far-field average intensity of CSD matrix components becomes approximately separable in radial and azimuthal variables.

Equation (16) can be further simplified to obtain the analytical expression for the elements of the CSD matrix if one substitutes from Eqs. (11) and (17) into Eq. (16) and performs the inverse Fourier transform. In addition, it is also convenient to numerically calculate Eq. (16) with the help of the fast Fourier Transform (FFT) algorithm. The SD in the output plane is

S_{t} (ρ, z) = S_{x x} (ρ, z) + S_{y y} (ρ, z),

where

S_{α α} (ρ, z) = W_{α α} (ρ, ρ, z), (α = x, y)

denote the field component

α

of the spectral density. For uncorrelated x and y field components, the DOP becomes

P (ρ, z) = | S_{x x} (ρ, z) - S_{y y} (ρ, z) | / S_{t} (ρ, z) .

Combining Eqs. (16), (19) and (20) one can analyze the evolution of the SD and DOP of the beams propagating through the ABCD optical system.

Under the condition $δ_{0} = δ_{x x} = δ_{y y}$ and $l = l_{x x} = l_{y y}$ , it is interesting to find that the value of the DOP in the central point ( $ρ =0$ ) depends only on the ratio of the amplitudes A_y/A_x and remains invariant on propagation. This can be shown with the help of Eq. (16). By inserting Eq. (16) into Eq. (20) we obtain the formula

P (ρ, z) = \frac{S (- ρ / B) \otimes | A_{x}^{2} p_{x x} (- ρ / B) - A_{y}^{2} p_{y y} (- ρ / B) |}{S (- ρ / B) \otimes (A_{x}^{2} p_{x x} (- ρ / B) + A_{y}^{2} p_{y y} (- ρ / B))} .

On applying the expressions for p_xx and p_yy in Eq. (21) one finds that

P (ρ, z) = \frac{S (u) \otimes [| \frac{2 a π}{3} - (\frac{1}{2} + \frac{a}{6}) m o d (l θ, 2 π) | \exp (- \frac{k^{2} δ_{0}^{2} u^{2}}{2})]}{S (u) \otimes [(\frac{2 a π}{3} + (\frac{1}{2} - \frac{a}{6}) m o d (l θ, 2 π)) \exp (- \frac{k^{2} δ_{0}^{2} u^{2}}{2})]},

with

a = A_{y}^{2} / A_{x}^{2}, u = - ρ / B

. It is evident from Eq. (22) that the value of the DOP is closely related the parameter a. On the beam axis we find that the convolution results in

S (u) \otimes [\mod (l θ, 2 π) \exp (- \frac{k^{2} δ_{0}^{2} u^{2}}{2})] = π S (u) \otimes \exp (- \frac{k^{2} δ_{0}^{2} u^{2}}{2}), (u = 0) .

Thus, if we only consider the on-axis values of the DOP after some mathematical manipulations, Eq. (22) can be written a simple form by applying Eq. (23) as

P (0, z) = | A_{x}^{2} - A_{y}^{2} | / (A_{x}^{2} + A_{y}^{2}) .

It is evident that this expression has the same form as that in the source plane [see Eq. (13)], implying that the values of the on-axis values of the DOP are invariant on propagation.

In Fig. 1 the theoretical results of the normalized SD and the DOP distribution are shown with A_y/A_x = 1.732 and l = l_x = l_y in the focal plane. It is seen that the SDs in the focal plane are identical Gaussian profiles for l = 1,2,3 and the DOP distributions are vortex-like patterns with topological charges l = 1,2,3 respectively. The parameters in the calculation are chosen to be $σ_{0} = 1.2 m m,$ $δ_{x x} = δ_{y y} = 0.08 m m$ , $λ =532nm$ , A = 0, B = f and f = 500mm. The DOP increases linearly from $θ = π^{-}$ to $θ = π^{+}$ along clockwise direction for l = 1 when the radial coordinate is $ρ = 0.2 m m$ . The theoretical values of DOP are 0.06 for $θ = π^{-}$ and 0.95 for $θ = π^{+}$ .

Fig. 1 (a)-(c), theoretical results of the normalized SD in the focal plane with l = 1,2,3. (d)-(f), the corresponding distributions of the DOP in the focal plane.

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4. Experimental setup for generating random Electromagnetic beams

The experimental setup for generation of electromagnetic sources with uncorrelated field components and measurement of the far-field intensity and DOP distribution is shown in Fig. 2.

Fig. 2 Schematic diagram of the experimental setup for generating the electromagnetic beam and measuring its intensity distribution in the far field (focal plane). LD: laser diode; BS: intensity beam splitter; M: reflected mirror; NDF: neutral density filter; SLM₁,SLM₂: spatial light modulators; LP₁,LP₂: linear polarizers; HWP: half wave plate; PBS: polarization beam splitter; L₁,L₂,L₃: single lenses; D: detector; PC: personal computer.

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A x-linearly polarized green laser beam ( $λ =532nm$ ) leaves a laser diode (LD) and is split along the transmission and reflection paths by an intensity beam splitter (BS). A variable neutral density filter (NDF) is inserted in the reflection path to control the ratio of the light amplitudes in two paths. In both paths, the light is incident on the two identical spatial light modulators (SLMs). The SLMs (Holoeye, LC2012) used here are based on transmission liquid crystal micro-display with $1024 \times 768$ pixel resolution (pixel size: $36 u m \times 36 u m$ ). Two sequences of random phases with prescribed correlation functions being $μ_{x x}^{(0)}$ and $μ_{y y}^{(0)}$ are commanded onto SLM₁ and SLM₂, respectively, which are used to produce the corresponding x and y components of the CSD in Eq. (9). The procedure for producing the phase screens belongs to [31].

After the SLMs, the light in both paths passes through linear polarizers (LPs) whose transmission angles are along the x-axis to obtain the x polarized component of the light. In transmission path, a half-wave plate (HWP) is placed after the LP₁ to transform x linear polarization to y linear polarization. Two orthogonal random lights are superimposed together at the output of the polarization beam splitter (PBS). After traveling from the PBS, the electromagnetic light enters a two-lens system with lenses of focal lengths 250mm. An iris is located in the focal plane of L₁ to remove the unwanted diffraction orders from the two SLMs. We recognize the plane of L₂ as the effective source plane. Finally, the generated electromagnetic source is focused by lens L₃ with focal length f = 500mm, and a detector is placed in the focal plane of L₃ to measure the average intensity and DOP distribution of the beam. Both distances from L₂ to L₃ and from L₃ to the detector are f. Thus the elements of the transfer matrix from L₂ to D can be expressed as A = 0, B = f, C = −1/f and D = 1.

5. Experimental results and discussion

Figures 3(a)-3(c) show the experimental results of the normalized SD in the source plane, x and y field components of the normalized SD in the focal plane, respectively when l_x = l_y = 1. In the experiment, we insert another LP in front of L₃ and rotate the transmission axis of the LP along the x and y direction to measure the x and y component of the SD in the focal plane, respectively. It is shown in Fig. 3(a) that the SD of the random beam is nearly a Gaussian distribution in the source plane, while in the focal plane, the x or y field components of the SD vary with the azimuthal angle, exhibiting vortex-like distribution. For comparison, the corresponding numerical calculation of the normalized SD in the source and in the focal plane is illustrated in Figs. 3(d)-3(f). One finds that the experimental results agree well with the theoretical predictions, subject to some background noises leading to slightly inhomogeneous SD distribution.

Fig. 3 (a)-(c) Experimental results of the normalized total SD in the source plane and of the normalized SD of the x and y field components in the focal plane; (d)-(f) the corresponding theoretical results.

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Let us now turn to examination of the DOP distribution in the focal plane. In the experiment, we first measure the x and y field components of SD with the help of the LP inserted in front of the L₃, and then, calculate the DOP with the help of Eq. (20). By adjusting the ratios of the amplitudes A_y/A_x controlled by the variable NDF and the parameter l = l_x = l_y controlled by the SLMs, we could obtain vortex-like distribution of the DOP in the focal plane.

Figure 4 gives the experimental results of the normalized SD of the random beam in the focal plane with three different l = 1,2,3 and the corresponding contour graph of the DOP. Such experimental results correspond to the theoretical distributions shown in Fig. 1. The ratio of amplitude A_y/A_x is about 1.7 in the experiment. It is indeed that the SDs are all nearly Gaussian for three different values of l, while the distributions of the DOP depend on the azimuthal angle and form the vortex-like profiles with topological charge l = 1, 2, and 3. The value of the DOP for l = 1 increases monotonously from nearly 0 for $θ = π^{-}$ to about 0.88 with $θ = π^{+}$ at the fixed $ρ = 0.2 m m$ in the experiment. Therefore, one can achieve the vortex-like DOP pattern in the focal plane (far field) through devising the correlation structure in Eq. (11) in the source plane and by appropriately choosing the ratio of amplitudes. In Figs. 4(d)-4(f) the DOP distribution is plotted only for the area $1.0 m m \times 1.0 m m$ around the optical axis, where the SD stands out well from the background noise, which makes the experimental results for DOP accurate enough to agree with the theory.

Fig. 4 (a)-(c) Normalized SD of the random beam in the focal plane with three different $l = l_{x} = l_{y} = 1, 2, 3$ when $A_{y} / A_{x} \approx 1.7$ ; (d)-(f) The corresponding experimental results of the distribution of the DOP.

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Through modulating the ratio A_y/A_x, we could obtain other interesting SD and DOP patterns in the far field. Figure 5 shows the experimental results of the SD density and the corresponding DOP in the focal plane for different l with A_y/A_x = 1. In this situation, the distribution of the SD varies with the changes of the azimuthal angle. The number of petals in the SD pattern equals to the parameter l. The DOP is also angularly-dependent but exhibits different patterns compared to the case $A_{y} / A_{x} \approx 1.7$ .

Fig. 5 (a)-(c) Experimental results of the normalized SD in the focal plane for three different values l = 1,2,3 with A_y/A_x = 1 . (d)-(f) the corresponding measured DOP in the focal plane.

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6. Summary

In summary, we have introduced a class of random electromagnetic beams having angular-dependence of its DOP distribution. By suitably choosing the ratio of the field amplitudes between x and y components, a vortex-like distribution of the DOP is formed in the far field. We have successfully generated such class of random beams using Mach-Zehnder interferometer and two transmissive phase-only SLMs. The obtained experimental results agree reasonably well with the theoretical results. The azimuthal control of polarization properties is of importance in microscopy [32–34], free-space optical communications [35], Raman spectroscopy [36], optical tweezers [37], etc. However in these applications only the fully polarized beams have been used so far. With the capability of our new beams to provide azimuthal control of the degree of polarization may open a new dimension to these applications.

Acknowledgment

This work is supported by National Natural Science Foundation of China (NSFC) (11474213), and US AFOSR (FA9550-121-0449).

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Random sources for beams with azimuthally varying polarization properties

Abstract

1. Introduction

2. Electromagnetic azimuthally variable beams

3. Electromagnetic beams with azimuthally varying degree of polarization

4. Experimental setup for generating random Electromagnetic beams

5. Experimental results and discussion

6. Summary

Acknowledgment

References and links

Cited By

Figures (5)

Equations (24)

Optics Express