Hyperbolic sine-correlated beams

Zhangrong Mei

doi:10.1364/OE.27.007491

1. Introduction

Spatial correlation properties of random sources are closely related to the propagation characteristics of optics field generated by sources [1,2]. Starting from the first seminal papers of Wolf’s modal theory of coherence [3,4], there has been a growing interest in modeling partially coherent sources with different coherent state and also analyzing the fields generated by the sources [5–14]. The superposition rule [15], ensuring the cross-spectral density (CSD) function to be genuine, played a significant role in recent studies on partially coherent fields [16–26]. However, most of these studies are restricted on considering fields with so-called Schell-model correlations, where the correlation function depends on the distance between two spatial points [1]. The far-field intensity distribution of such fields is proportional to the Fourier transform of the source correlation function [27]. The distribution of the radiant intensity is therefore independent of the shape of the source.

Dark hollow beams with zero central intensity are of particular interest for applications as optical tweezers and optical wrenches in laser trapping and manipulation of microparticles [28–30]. Many investigations in both theory and experiments have been carried out to generate such beams and their propagation properties [31–35]. Generally, the hollow shape of light intensity is not invariant on propagation. For instance, if a hollow beam is generated by a deterministic source, the hollow shape is formed only in the near field and transforms to a Gaussian shape in free-space propagation [36]. And the hollow beams generated by random sources is only formed in the far field [19,37]. In this connection, it is interesting to ask whether one can construct partially coherent beams in which a radiated hollow intensity pattern remains invariant in shape from the source field to the far field. In this paper, a new class of partially coherent beam with such properties is introduced. We shall evaluate a random source generating these beams can be represented as a hyperbolic sine correlated mode. We then study propagation properties of this new family of beams.

2. Hyperbolic sine correlated sources and beam conditions

We consider a quasi-monochromatic partially coherent field in the space-frequency domain. The CSD function across a scalar partially coherent source can be written [1]

W_{0} ({x^{'}}_{1}, {x^{'}}_{2}) = 〈 E^{*} ({x^{'}}_{1}) E ({x^{'}}_{2}) 〉,

where

E

is the optical disturbance, the asterisk stands for complex conjugate, the angular brackets denote ensemble average,

{x^{'}}_{1}

and

{x^{'}}_{2}

are the position coordinates of two points in the source plane. For brevity, we limited the analysis to fields that depend on only one transverse dimension x, but the results can be straightforwardly generalized to a symmetric two-dimensional case. A genuine CSD function is limited by the constraint of non-negative definiteness [1], which is fulfilled if the function can be written as a superposition integral of the form [15]

W_{0} ({x^{'}}_{1}, {x^{'}}_{2}) = \int p (v) H^{*} ({x^{'}}_{1}, v) H ({x^{'}}_{2}, v) d v,

where H is an arbitrary kernel and p is a non-negative function. Equation (2) can be interpreted to define a partially coherent field consisting of an incoherent superposition of elementary fields H weighted by the function p [16]. The choice of the kernel H defines the correlation class of a light field. If the kernel H is chosen as a Fourier-like form, a huge variety of genuine CSD belonging to Schell-model correlation class can be obtained by varying the weight function.

In order to define different classes of sources, let us consider a kernel of the Fourier-sine form, namely,

H (x^{'}, v) = A \exp (- {x^{'}}^{2} / σ^{2}) \sin (x^{'} v),

where A and σ are two positive constants. Let us further choose a weight function as follows:

p (v) = δ / (2 \sqrt{π}) \exp (- δ^{2} v^{2} / 4),

where δ is the spatial coherence length. Evaluating the corresponding CSD through Eq. (2), we readily obtain the expression

W_{0} ({x^{'}}_{1}, {x^{'}}_{2}) = A^{2} \exp [- ({x^{'}}_{1}^{2} + {x^{'}}_{2}^{2}) / w_{0}^{2}] \sin h (2 {x^{'}}_{1} {x^{'}}_{2} / δ^{2}),

where

w_{0}^{- 2} = σ^{- 2} + δ^{- 2},

_{$w_{0}$} is the beam width. Equation (5) is not of the Schell-model type anymore, which will be termed the hyperbolic sine correlated (HSC) sources.

The degree of coherence (DOC) is defined by the expression [1]

μ ({x^{'}}_{1}, {x^{'}}_{2}) = W_{0} ({x^{'}}_{1}, {x^{'}}_{2}) / \sqrt{W_{0} ({x^{'}}_{1}, {x^{'}}_{1}) W_{0} ({x^{'}}_{2}, {x^{'}}_{2})} .

On substituting from Eq. (5) into Eq. (6), the SDOC turns out to be

μ ({x^{'}}_{1}, {x^{'}}_{2}) = \sin h (2 {x^{'}}_{1} {x^{'}}_{2} / δ^{2}) / \sqrt{\sin h (2 {x^{'}}_{1}^{2} / δ^{2}) \sin h (2 {x^{'}}_{2}^{2} / δ^{2})} .

The absolute value of the DOC at a pair of points in the source plane is displayed in Fig. 1 for several values of $δ$ . It can be seen that the modulus of the DOC exhibit a skew symmetric distribution of 45 degree angle with the coordinate axis. The values of DOC attain the maximum at the points with the skew symmetric axis $| {x^{'}}_{1} | = | {x^{'}}_{2} |$ and that the greater value of $δ$ , the slower the decrease of $| μ |$ with the increase of the location interval $| | {x^{'}}_{1} | - | {x^{'}}_{2} | |$ between the two points.

Fig. 1 Modulus of the degree of coherence given by Eq. (8) for different values of the spatial coherence length δ, (a) δ = 0.25mm; (b) δ = 1.00mm.

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In order for the function $W_{0} ({x^{'}}_{1}, {x^{'}}_{2})$ to generate a beam, certain restrictions on the values of source parameters must be imposed. For this purpose we first solve the Fourier transform of $W_{0}$ by the following formula [1]

{\tilde{W}}_{0} (f_{1}, f_{2}) = {(2 π)}^{- 1} \iint W_{0} ({x^{'}}_{1}, {x^{'}}_{2}) \exp [- i (f_{1} {x^{'}}_{1} + f_{2} {x^{'}}_{2})] d {x^{'}}_{1} d {x^{'}}_{2} .

On substituting from Eq. (5) into Eq. (8) we obtain the following expression:

{\tilde{W}}_{0} (f_{1}, f_{2}) = (A^{2} w_{0} β / 2) \exp [- β^{2} (f_{1}^{2} + f_{2}^{2}) / 4] \sin h [(w_{0}^{2} β^{2} f_{1} f_{2}) / (2 δ^{2})],

where

β^{- 2} = w_{0}^{- 2} - w_{0}^{2} δ^{- 4}

. Equation (9) is of the same functional form as the source field (5), so the HSC is a self-Fourier transforming field [38,39].

The required condition can be stated as [1]

| {\tilde{W}}_{0} (- f, f) | \approx 0 u n l e s s f^{2} < < k^{2},

where

k = 2 π / λ

is the wave number,

λ

being the wave-length. Due the fact that

\sinh (x) \neq 0

except when

x = 0

, we can cast condition (10) into the following form

\exp (- β^{2} f^{2} / 2) \approx 0 u n l e s s f^{2} < < k^{2} .

Hence we obtain the following necessary and sufficient condition for a HSC source to generate a beam:

1 / σ^{2} + 1 / (σ^{2} + δ^{2}) < < 2 π^{2} / λ .

3. Propagation of hyperbolic sine correlated beams

The CSD of a beam generating by source (5) at a pair of points $(x_{1}, z)$ and $(x_{2}, z)$ in any transverse plane of the half-space $z > 0$ is related, in paraxial domain, to those in the source plane through the Fresnel transform [1]:

W (x_{1}, x_{2}, z) = k / (2 π z) \iint W_{0} ({x^{'}}_{1}, {x^{'}}_{2}) \exp {i k [{(x_{1} - {x^{'}}_{1})}^{2} - {(x_{2} - {x^{'}}_{2})}^{2}] / (2 z)} d {x^{'}}_{1} d {x^{'}}_{2} .

On substituting from Eq. (5) into Eq. (13), the CSD takes the form

W (x_{1}, x_{2}, z) = \frac{A^{2}}{b_{z}} \exp [- \frac{i k}{2 z} (\frac{1}{b_{z}^{2}} - 1) (x_{1}^{2} - x_{2}^{2})] \exp (- \frac{x_{1}^{2} + x_{2}^{2}}{w_{0}^{2} b_{z}^{2}}) \sin h (\frac{2 x_{1} x_{2}}{δ^{2} b_{z}^{2}}),

where the spreading coefficients is given by the expression

b_{z}^{2} = (4 z^{2}) / (k^{2} w_{0}^{4}) - (4 z^{2}) / (k^{2} δ^{4}) + 1.

Equation (14) satisfies

W (- x_{1}, x_{2}, z) = - W (x_{1}, x_{2}, z)

, so it is an antispecular CSD [40,41].

The spectral degree of coherence between the two spatial points $x_{1}$ and $x_{2}$ in any plane is defined by the expression [1]

μ (x_{1}, x_{2}, z) = W (x_{1}, x_{2}, z) / \sqrt{W (x_{1}, x_{1}, z) W (x_{2}, x_{2}, z)} .

One finds, with the help of Eq. (14), that

μ (x_{1}, x_{2}, z) = \exp [- \frac{i k}{2 z} (\frac{1}{b_{z}^{2}} - 1) (x_{1}^{2} - x_{2}^{2})] \frac{\sin h [2 x_{1} x_{2} / (δ^{2} b_{z}^{2})]}{\sqrt{\sin h [2 x_{1}^{2} / (δ^{2} b_{z}^{2})]} \sqrt{\sin h [2 x_{2}^{2} / (δ^{2} b_{z}^{2})]}} .

Figure 2 presents the modulus of the degree of coherence on the z = 2m plane calculated by Eq. (17). Without loss of generality, the values of the model source are chosen to be σ = 0.5mm, k = 10⁴mm⁻¹ and δ is the same as Fig. 1. It can be seen from Fig. 2 that the HSC beams maintains hyperbolic sine correlation invariance. But relative to the case of source field, the modulus of the degree of coherence decrease more slowly with the location interval between the two spatial points.

Fig. 2 Modulus of the degree of coherence on the z = 2m plane corresponding to Fig. 1.

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From the cross-spectral function (14), the spectral density of the HSC beams at any point (x, z) within the cross section of the beam can be calculated by the expression

S (x, z) = W (x, x, z) = (A^{2} / b_{z}) \exp [- 2 x^{2} / (w_{0}^{2} b_{z}^{2})] \sin h [2 x^{2} / (δ^{2} b_{z}^{2})] .

Figure 3 shows the evolution of the spectral density S on the x-z plane and the transverse distribution at several selected distances of the HSC beams with different spatial coherence length. Two significant features are worth noting from Fig. 3. First, unlike the existing dark beams, the hollow intensity distribution appears only near field or far field, the transverse intensity distribution of this new beam always retains a dark hollow pattern throughout the propagating process. Second, like other partially coherent beams, the divergence of the beams decreases with the increase of the spatial coherence length.

Fig. 3 The evolution of the spectral density S on the x-z plane of a HSC beam with σ = 0.5mm, and (a) δ = 0.25mm or (b) δ = 1.00mm. The lateral distribution for selected propagation distances is also shown in figure.

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To describe more fully the directionality of partially coherent beams, one may use the mean-squared beam width, which is defined as [42]

ω^{2} (z) = 4 \int_{- \infty}^{\infty} x^{2} S (x, z) d x / \int_{- \infty}^{\infty} S (x, z) d x .

Substituting from Eq. (18) into Eq. (19) and calculating the resulting integral we obtain

ω^{2} (z) = (σ^{2} + σ γ + γ^{2}) b_{z}^{2},

where

γ^{- 2} = σ^{- 2} + 2 δ^{- 2}

. The results are presented in Fig. 4, the mean-squared beam width versus is plotted as a function of the propagation distance z for the HSC beams with different spatial coherence length. It can be clearly seen from the figure that the greater spatial coherence length of the source field, the better directivity of the beam.

Fig. 4 The change of the mean-squared beam width w(z) with the propagation distance z for difference values of the spatial coherence length δ.

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4. Concluding remarks

In this article we have introduced a new type of partially coherent sources with hyperbolic sine correlated correlation properties and confirmed that such sources are physically genuine. The analytical formula for the CSD function of the beams generated by the new sources on propagation in free space is derived and used to explore the evolution of the spectral density and the degree of coherence. It was shown that the propagation formula of the CSD function of the new class of beams is of the same functional form as the source field, that is, this is a self-Fourier transforming field. This property enables such beams maintain the invariance of correlation distribution and intensity shape, but not size. Unlike the existing light field, the hollow intensity shape can only be maintained in the near field or far field, the spectral density of the new beams hold a hollow shape in any transverse plane from the source field to the far field. This remarkable property makes them particularly suitable for applications involving in optical trapping and tweezers to manipulate particles at any path.

Funding

Zhejiang Provincial Natural Science Foundation of China (LY16F050007).

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Hyperbolic sine-correlated beams

Abstract

1. Introduction

2. Hyperbolic sine correlated sources and beam conditions

3. Propagation of hyperbolic sine correlated beams

4. Concluding remarks

Funding

References

Cited By

Figures (4)

Equations (20)

Optics Express