Statistical properties of electromagnetic twisted Gaussian Schell-model array beams during propagation

Yujie Zhou; Daomu Zhao

doi:10.1364/OE.27.019624

1. Introduction

Statistical properties of coherence and polarization of light waves have been treated separately for many years. Until 2003, Wolf developed the unified theory of coherence and polarization for random partially coherent electromagnetic beams [1, 2], which made it evident that coherence and polarization of light, whilst distinct phenomena, were just two related aspects of statistical optics. In particular, the spectral density, degree of coherence (DOC) and degree of polarization (DOP) of such an electromagnetic beam can be expressed by a 2 × 2 cross- spectral density matrix (CSDM). With the help of this theory, propagation statistical characteristics of light waves in different optical systems or media have been investigated in detail [3–12]. Coherent properties of the source can cause the spectrum of light change on propagation. The degree of polarization as beam propagates generally also changes, which is caused by the difference in the correlation coefficients between the x and y components of the electrical field vector at the two source points [13]. Besides, the shape and the orientation of the polarization ellipse of the polarization portion of the beam also change on propagation [14]. Light beams whose degree of polarization remains constant on propagation was also given, which corresponded to a linear polarization along the x direction [15].

Compared with the single beam, the coherent array beam has higher output power, and it has been used to multiple fields, including photonic lithography [16, 17], atomic capture and cooling [18], microfluidic sorting [19]. Besides, partially coherent beams endowed with twist have also aroused much attention from researchers [20–25]. Hence in the present paper, we extend the concept of partially coherent twisted array beams to the electromagnetic (vector) domain by using a generalized complex Gaussian representation for the CSDM. Moreover, propagation characteristics of such electromagnetic twisted Gaussian Schell-model array (EM TGSMA) beam in free space are analyzed. In particular, we demonstrated the twist effects of DOC and DOP during propagation in detail.

2. Propagation of EM TGSMA beam in free space

For a random statistically stationary electromagnetic beam propagating along the positive z-axis, the second-order correlation property of the light field in space-frequency can be presented by a 2 × 2 CSDM, as following [2]:

W_{i j} (r_{1}, r_{2}, z, ω) = 〈 E_{i}^{*} (r_{1}, z, ω) E_{j} (r_{2}, z, ω) 〉, (i, j = x, y),

where

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

and

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

are two arbitrary transverse position vectors; E_x and E_y denote two mutually orthogonal components of the random electric vector, perpendicular to the z-axis. The angular brackets stand for the ensemble average and the asterisk denotes the complex conjugate. For brevity, we ignore the dependence of the angular frequency.

Assume the field taken across the z=0 plane. When the beam axis is tilted, field distribution would become $E (r, z, ω) \exp [i (k_{x} x + k_{y} y)]$ , where k_x and k_y are the transverse components of the mean wave vector $k_{i j}$ . By giving an inclination $k_{i j} (r_{0}) = - u_{i j} (y_{0}, - x_{0})$ , with real u_ij, the corresponding element of CSDM $W_{i j}^{0} (r_{1}, r_{2}, r_{0})$ , is [26]

W_{i j} (r_{1}, r_{2}, r_{0}) = W_{i j}^{0} (r_{1} - r_{0}, r_{2} - r_{0}) exp (2 π i υ_{i j} \cdot r_{0}),

with vector

υ_{i j} = - \frac{u_{i j}}{2 π} (y_{2} - y_{1}, x_{1} - x_{2}),

which denotes a rotated version of

D = r_{1} - r_{2}

. Assuming an incoherent superposition of such mutually uncorrelated tilted fields

W_{i j} (r_{1}, r_{2}, r_{0}),

with a non-negative weight function

p (r_{0})

, and introducing a variable change

r_{0} = r_{m} - r

, where

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

, a bona fide CSDM is produced, as following [27]:

W_{i j} (r_{1}, r_{2}) = R_{i j} (r_{1}, r_{2}) exp (2 π i υ_{i j} \cdot r_{m}),

with function R_ij referred as the reminder:

R_{i j} (r_{1}, r_{2}) = \int p (r_{0}) W_{i j}^{0} (r + D / 2, r - D / 2) exp (- 2 π i υ_{i j} \cdot r) d^{2} r .

For simplicity, we shall put $p (r_{0}) = 1$ . With a Gaussian Schell-model array model $W_{i j}^{0}$ , one can obtain a rotating EM Gaussian array pattern as following [28]:

\begin{array}{l} W_{i j} (r_{1}, r_{2}) = \exp [- \frac{{(x_{1} - x_{2})}^{2}}{2 δ_{1 i j}^{2}}] \exp [- \frac{{(y_{1} - y_{2})}^{2}}{2 δ_{2 i j}^{2}}] \\ \times \underset{n_{1} = - P}{\sum^{P}} \cos [C_{1 i j} (x_{1} - x_{2})] \underset{n_{2} = - Q}{\sum^{Q}} \cos [C_{2 i j} (y_{1} - y_{2})] \\ \times \exp (- i u_{i j} x_{1} y_{2}) \exp (i u_{i j} x_{2} y_{1}) . \end{array}

In order to impose restriction on the amplitude of the source, we place a proper amplitude mask $τ (r)$ as a spatial limitation of the actual source, as following [25]:

τ (r) = exp (- \frac{x^{2}}{4 σ_{1 i j}^{2}}) exp (- \frac{y^{2}}{4 σ_{2 i j}^{2}}),

here

C_{1 i j} = 2 π n_{1} R_{1} / δ_{1 i j}

,

C_{2 i j} = 2 π n_{2} R_{2} / δ_{2 i j}, R_{1}, R_{2}

and

δ_{1 i j}

,

δ_{2 i j}

are coherence parameters, and u_ij denotes the twist strength.

σ_{1 i j}

and

σ_{2 i j}

are the beam widths of Gaussian intensity distribution.

P = (N_{1} - 1) / 2

and

Q = (N_{2} - 1) / 2

, which N₁ and N₂ are positive integrals that determine the number of lobes of the array. Hence, the element of CSDM becomes:

\begin{array}{l} W_{i j} (r_{1}, r_{2}) = \exp (- \frac{x_{1}^{2} + x_{2}^{2}}{4 σ_{1 i j}^{2}}) \exp (- \frac{y_{1}^{2} + y_{2}^{2}}{4 σ_{2 i j}^{2}}) \\ \times \exp [- \frac{{(x_{1} - x_{2})}^{2}}{2 δ_{1 i j}^{2}}] \exp [- \frac{{(y_{1} - y_{2})}^{2}}{2 δ_{2 i j}^{2}}] \\ \times \underset{n_{1} = - P}{\sum^{P}} \cos [C_{1 i j} (x_{1} - x_{2})] \underset{n_{2} = - Q}{\sum^{Q}} \cos [C_{2 i j} (y_{1} - y_{2})] \\ \times \exp (- i u_{i j} x_{1} y_{2}) \exp (i u_{i j} x_{2} y_{1}) . \end{array}

According to the extended Huygens-Fresnel principle in the paraxial form, the element of CSDM of the EM TGSMA beams during propagation can be presented as following [29]:

W_{i j} (ρ_{1}, ρ_{2}, z) = \iint W_{i j} (r_{1}, r_{2}) H_{z}^{*} (ρ_{1}, r_{1}) H_{z} (ρ_{2}, r_{2}) d^{2} r_{1} d^{2} r_{2},

here the function H_z called the propagation kernal that characterizes propagation in a domain outside the source, given by the following form for free-space propagation:

H_{z} (ρ, r) = - i k {(2 π z)}^{- 1} exp [i k (ρ - r)^{2} / (2 z)] .

Substituting Eqs. (7) and (9) into Eq. (8), we obtain the element of CSDM in free space as following:

\begin{array}{l} W_{i j} (ρ_{1}, ρ_{2}, z) = E_{i j} \overset{P}{\sum_{n_{1} = - P}} \underset{n_{2} = - Q}{\sum^{Q}} \frac{{(k π σ_{1 i j})}^{2}}{\sqrt{A_{i j} B_{i j} D_{i j}} z^{2}} \exp [- \frac{i k}{2 z} (ρ_{1}^{2} - ρ_{2}^{2})] \\ \times \exp [- \frac{k^{2} σ_{1 i j}^{2}}{2 z^{2}} {({x^{'}}_{1} - {x^{'}}_{2})}^{2}] \\ \times {\exp [\frac{1}{4 D_{i j}} {(2 m_{i j 1 +} + \frac{i k}{z} ({y^{'}}_{1} - {y^{'}}_{2}))}^{2} + \frac{1}{4 A_{i j}} γ_{i j + +}^{2} + \frac{1}{4 B_{i j}} β_{i j +}^{2}] \\ + \exp [\frac{1}{4 D_{i j}} {(2 m_{i j 1 +} + \frac{i k}{z} (y_{1}' - y_{2}'))}^{2} + \frac{1}{4 A_{i j}} γ_{i j + -}^{2} + \frac{1}{4 B_{i j}} β_{i j +}^{2}] \\ + \exp [\frac{1}{4 D_{i j}} {(2 m_{i j 1 -} + \frac{i k}{z} (y_{1}' - y_{2}'))}^{2} + \frac{1}{4 A_{i j}} γ_{i j - +}^{2} + \frac{1}{4 B_{i j}} β_{i j -}^{2}] \\ + \exp [\frac{1}{4 D_{i j}} {(2 m_{i j 1 -} + \frac{i k}{z} (y_{1}' - y_{2}'))}^{2} + \frac{1}{4 A_{i j}} γ_{i j - -}^{2} + \frac{1}{4 B_{i j}} β_{i j -}^{2}]}, \end{array}

setting:

\begin{array}{l} E_{i j} = 1 (i = j), E_{i j} = 0 (i \neq j); \\ γ_{i j \pm \mp} = \frac{i k}{2 z} (y_{1}' + y_{2}') - m_{i j 2 \pm} + \frac{g_{i j \pm}}{4 D_{i j}} \mp i C_{2 i j;} \\ β_{i j \pm} = \frac{i k}{2 z} (x_{1}' + x_{2}') + \frac{k^{2} σ_{1 i j}^{2}}{z^{2}} (x_{1}' - x_{2}') \pm i C_{1 i j;} \\ m_{i j 1 \pm} = \frac{- i u_{i j}}{4 B_{i j}} β_{i j \pm;} m_{i j 2 \pm} = - \frac{u_{i j} k σ_{1 i j}^{2}}{z} [\frac{β_{i j \pm;}}{2 B_{i j}} - (x_{1}' - x_{2}')]; \\ A_{i j} = \frac{1}{α_{2 i j}} - \frac{u_{i j}^{2} k^{2} σ_{1 i j}^{4}}{4 B_{i j} z^{2}} + \frac{u_{i j}^{2} σ_{1 i j}^{2}}{2} - \frac{1}{4 D_{i j}} {(\frac{i u_{i j}^{2} k σ_{1 i j}^{2}}{2 B_{i j} z} + \frac{i k}{z})}^{2}; \\ g_{i j \pm} = - 2 (\frac{i u_{i j}^{2} k σ_{1 i j}^{2}}{2 B_{i j} z} + \frac{i k}{z}) [2 m_{i j 1 \pm} + i k (y_{1}' - y_{2}') / z]; \\ B_{i j} = \frac{1}{α_{1 i j}} + \frac{k^{2} σ_{1 i j}^{2}}{2 z^{2}}; D_{i j} = \frac{1}{2 σ_{2 i j}^{2}} + \frac{u_{i j}^{2}}{4 B_{i j}}; \\ \frac{1}{α_{1 i j}} = \frac{1}{8 σ_{1 i j}^{2}} + \frac{1}{2 δ_{1 i j}^{2}}; \frac{1}{α_{2 i j}} = \frac{1}{8 σ_{2 i j}^{2}} + \frac{1}{2 δ_{2 i j}^{2}} . \end{array}

Equations (10) and (11) can be considered as the general analytical expression of the CSDM of the beam generated by an EM TGSMA source. When $N_{1} = N_{2} = 1,$ it would reduce to the propagated CSDM for EM TGSM beams. Applying Eqs. (10) and (11), one may ready to study the statistical properties of such an EM TGSMA beam on propagation. Setting $ρ_{1} = ρ_{2} = ρ$ , the spectral density can be defined as:

S (ρ, z) = T r W (ρ, ρ, z) = W_{x x} (ρ, ρ, z) + W_{y y} (ρ, ρ, z);

and the DOP of the beam with uncorrelated beam components can be calculated by the expression:

P (ρ, z) = \frac{| W_{x x} (ρ, ρ, z) - W_{y y} (ρ, ρ, z) |}{W_{x x} (ρ, ρ, z) + W_{y y} (ρ, ρ, z)} .

The DOC at a pair of points ρ₁ and ρ₂ generally is given by:

μ (ρ_{1}, ρ_{2}, z) = \frac{W_{x x} (ρ_{1}, ρ_{2}, z) + W_{y y} (ρ_{1}, ρ_{2}, z)}{\sqrt{S (ρ_{1}, z) S (ρ_{2}, z)}} .

3. Numerical results and discussions

In this section, we will demonstrate the evolution of statistical properties of EM TGSMA beam during propagation in free space. If no other explanation is made, those parameters are invariable in the following simulation: $λ = 632.8 nm, R_{1} = 2 R_{2} = 3 mm, N_{1} = 2, N_{2} = 1.$

Fig. 1 Average intensity generated by an EM TGSMA source at several propagation distances in free space and setting: u_xx = −u_yy (mm⁻²); rows 1 (a1-a4) and 2 (b1-b4): σ₁_xx = σ₁_yy = 1mm, σ₂_xx = σ₂_yy = 0.3mm. δ₁_xx = δ₁_yy = 1mm, δ₂_xx = δ₂_yy = 0.3mm; rows 3 (c1-c4) and 4 (d1-d4): σ₁_xx = δ₁_xx = 0.4mm, σ₂_xx = δ₂_xx = 0.5mm. σ₁_yy = δ₁_yy = 0.5mm,; σ₂_yy = δ₂_yy = 0.4mm.

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A stack of images taken in Fig. 1 illustrates the typical evolution of the transverse spectral density as an EM TGSMA beam propagates in free space. In rows 1 and 2 of Fig. 1, we show the intensity profiles with two different twist strength, $u_{x x} = - u_{y y} = - 3 {mm}^{- 2}$ and $u_{x x} = - u_{y y} = - 2 {mm}^{- 2}$ respectively. One can see that initial spots would progressively split into an array with propagation. Besides, the evolutions of intensity profiles during propagation are similar, both transformed from a horizontal ellipse to a vertical ellipse. Two other cases of beam propagation are given in rows 3 and 4 with $u_{x x} = - u_{y y} = - 3 {mm}^{- 2}$ and $u_{x x} = - u_{y y} = 3 {mm}^{- 2}$ respectively. What need to emphasize that evolutions of spectral distribution in rows 1 and 2 are corresponding to the parameters setting for $σ_{1 x x} / σ_{1 y y} = σ_{2 x x} / σ_{2 y y}$ and $\dot{e} l t a_{1 x x} / δ_{1 y y} = δ_{2 x x} / δ_{2 y y}$ , and evolutions in rows 3 and 4 are for $σ_{1 x x} / σ_{2 y y} = σ_{2 x x} / σ_{1 y y}$ and $δ_{1 x x} / δ_{2 y y} = δ_{2 x x} / δ_{1 y y}$ .

Fig. 2 DOP generated by an EM TGSMA source at several propagation distances in free space with parameters as in Fig. 1.

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Figure 2 presents the modulus of DOP of the field generated by the same source as in Fig. 1. In row 1 of Fig. 2, DOP transforms from a certain cross-like shape into two stable cross shapes arranged side by side in the far field without rotating. Then it remains shape-invariant (spreading with distance) as the beam propagates. When only changing twist strength, as seen in row 2, the evolution of DOP does not show the twist effect, either. Then in rows 3 and 4 of Fig. 2, we set $σ_{1 x x} / σ_{2 y y} = σ_{2 x x} / σ_{1 y y}$ and $δ_{1 x x} / δ_{2 y y} = δ_{2 x x} / δ_{1 y y}$ . It shows that DOP rotates counterclockwise around the beam center upon propagation in row 3, yet rotates clockwise in row 4, which are caused by the difference setting of twist strength. In addition, profiles of DOP in rows 3 and 4 rotate at the same pace during propagation.

Fig. 3 DOC generated by an EM TGSMA source at several propagation distances in free space with parameters as in Fig. 1.

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In Fig. 3, we show the evolution of DOC of the field generated by the same source as in Fig. 2. As seen in rows 1 and 2, profiles of DOC transform from a horizontal ellipse to a vertical ellipse and have no twist effects. Then in rows 3 and 4, one can find that DOC rotates clockwise upon propagation in row 3, yet rotates counterclockwise in row 4.

Thus, one can conclude that twist effects of coherence distribution and polarization distribution during propagation can be influenced by the parameters $σ_{1 i j}$ , $σ_{2 i j}$ and $δ_{1 i j}$ , $δ_{2 i j}$ . Senses of rotation can also be controlled by adjusting the parameters of u_ij. In particular, for rows 1 and 2 of Figs. 2 and 3, coherent lengths for the element W_xx of CSDM are identical to the element W_yy, and beam widths are also the same. For $u_{x x} = - 3 {mm}^{- 2}$ , profiles of W_xx would rotate counterclockwise on propagation. Yet, the evolution of W_yy would rotate clockwise, for $u_{y y} = 3 {mm}^{- 2}$ . Both rotate at a same pace. Hence, evolutions of the DOP and DOC for such EM beam propagating in free space do not show the twist effects. What’s more, an interesting feature from this rotating field is that the twist phase causes DOC to rotate in an opposite direction of DOP, as seen by comparing the rotation in Figs. 2 and 3.

Fig. 4 The DOP on the axis of an EM TGSMA beam as a function of z for σ₁_xx = δ₁_xx = 0.4mm, σ₂_xx = δ₂_xx = 0.5mm. σ₁_yy = δ₁_yy = 0.5mm, σ₂_yy = δ₂_yy = 0.4mm. (a) N₁ = N₂ = 2, (b) and (c) u_xx = −u_yy = −3mm⁻².

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Let us now consider the on-axis behaviors of DOP of the EM TGSMA beam propagating in free space from the source plane, for selected values of the parameters. Figure 4(a) shows that for propagation in free space, DOP acquires a particular value after propagating a certain distance and it retains this value as the beam propagates further. Furthermore, twist strengths can also affect the behavior of beam statistics. The larger the value of u_ij, the greater the final stable value of the DOP. The evolutions of curves, corresponding to $u_{x x} = - u_{y y} = - 3 {mm}^{- 2}$ and $u_{x x} = - u_{y y} = 3 {mm}^{- 2}$ , are completely coincident. Figures 4(b) and 4(c) show the influence of N₁ in the source plane on the DOP for EM TGSMA beam propagating in free space. When $N_{1} = N_{2}$ are odd, 3,5,7 or 9, curves of DOP present similar evolutionary. After two oscillations, the polarization value tends to zero. For the case where $N_{1} = N_{2}$ are even, 4,6,8 or 10, the polarization value tends to a certain value but not zero. Hence, the on-axis behavior of DOP during propagation can be beneficial to determine the number parity of such array.

It is also of interest to consider the transverse coherent distribution of such beam at distance z=5m and plots of this variation, as given in Fig. 5. One can find that twist strength can have effect on the coherent distribution, as shown in Fig. 5(a). Evolutions of curves, corresponding to $u_{x x} = - u_{y y} = - 3 {mm}^{- 2}$ and $u_{x x} = - u_{y y} = 3 {mm}^{- 2}$ respectively, are also totally coincident. Coherence curve can present oscillation and as the propagation distance increases, the oscillation will gradually weaken. The decay rate of the case N1=3,5,7 or 9 is much greater than the decay rate of N1=2,4,6 or 10. For the case where $N_{1} = N_{2}$ are odd, 3,5,7 or 9, coherent curves present a Gaussian-like distribution. When $N_{1} = N_{2}$ are even, 4,6,8 or 10, DOP for EM TGSMA beams performs a central peak with two secondary peaks structure. It seems that the value of $N_{1} (N_{1} = N_{2})$ does not have a great impact on the trend of curves.

Fig. 5 The DOC of EM TGSMA beam at z = 5m and y_d = 0 with parameters as in Fig. 4.

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4. Conclusions

In summary, we introduced an analytic model for an EM TGSMA beam. We have derived the analytic expression for CSDM of such beam propagating in free space and investigated the spectral density, DOP and DOC in detail. Particularly, we investigated the twist effects of DOP and DOC of such beam propagating in free space. We have found that twist effects of polarization distribution and coherent distribution on propagation can be influenced by beam width and coherent length. Senses of rotation can be controlled by adjusting the parameters of u_ij. Moreover, the twist phase always causes DOC to rotate in an opposite direction compared with that of DOP. In addition, DOP on its axis during propagation would tend to a particular value. The larger the value of u_ij, the greater the final stable value of the DOP. Transverse coherent distribution at a certain distance are useful to determine the array’s parity. Typical behavior of DOP propagation on its axis can be beneficial to judge the parity of such array. Our results might be crucial for operation of communication, imaging, and sensing systems.

Funding

National Natural Science Foundation of China (NSFC) (11874321); Fundamental Research Funds for the Central Universities (2018FZA3005).

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Statistical properties of electromagnetic twisted Gaussian Schell-model array beams during propagation

Abstract

1. Introduction

2. Propagation of EM TGSMA beam in free space

3. Numerical results and discussions

4. Conclusions

Funding

References

Cited By

Figures (5)

Equations (14)

Optics Express