Random sources for cusped beams

Jia Li; Fei Wang; Olga Korotkova

doi:10.1364/OE.24.017779

1. Introduction

In the past few decades, partially coherent beams with specially designed correlation functions have attracted considerable research interest due to their unique features exhibited on propagation in free space, through optical systems and on interacting with deterministic and random media [1–13]. The statistical properties of a propagating partially coherent beam, including its spectral density, its Degree of Coherence (DOC) and its degree and state of polarization, strongly depend on its source’s correlation profile and type (uniform or Schell-model and non-uniform) [14–21]. The Gaussian Schell-Model (GSM) beam, the J₀-correlated Schell-model beam, and the I₀-correlated Schell-model beam were the first three beams introduced in coherence theory [22–24]. After Gori et al. suggested a convenient integral representation for the correlation function of a partially coherent scalar [25] and electromagnetic [26–28] sources it became possible to substantially extend the variety of random model sources and beams they radiate. Moreover, the calculus of the correlation functions, including the operations of difference, product, convolution, powers, etc. has been recently developed [29–34]. In particular, individual classes of random sources were devised to produce far fields with prescribed spectral densities (average intensities) including the non-uniformly correlated source [35–38], the multi-Gaussian correlated Schell-model source [39–41], the cosine-Gaussian correlated Schell-model source [42,43], the Hermite-Gaussian correlated Schell-model source [44,45], the Gaussian Schell arrays and others. Besides, some Schell-model beams have been generated experimentally, for example, the multi-Gaussian correlated Schell-model beam [46], the partially coherent flat-topped beam [47], the Laguerre-Gaussian correlated Schell-model vortex beam [48], the vector cosine-Gaussian correlated beam [49,50], etc. All these partially coherent model beams exhibit interesting phenomena on free-space propagation, e.g. they can form the flat-topped intensity distribution, the multi-ring profile, the self-focusing pattern and the lateral shifted peak, to name a few. These extraordinary beam properties are important to beam shaping, free-space optical communications and guiding of small particles.

In this regard, the only issue that has not been satisfactorily addressed so far is the mathematical modeling of random sources that produce beams with spectral densities narrower than Gaussian in the central part of the beam. Experimental solution to this problem has been recently proposed by Xiao et al. [51] by employing the super-Gaussian distribution for the far-field spectral density. However, analytical formula for the corresponding source DOC is not known. In this paper, we aim to develop two analytical source DOC models that lead to highly spiked far-field on-axis spectral densities. In other words, we will introduce two classes of beams whose source degrees of coherence are prescribed by the rectangular Lorentz-correlated Schell-Model (LSM) and the rectangular Factional Multi-Gaussian-correlated Schell-Model (FMGSM) functions, respectively. We show that both beams form cusp-like intensity profiles when they propagate to the far field or to the focal plane of a thin lens. Beams belonging to both classes are experimentally produced with the help of the Spatial Light Modulator (SLM). As shown, the spectral densities of experimental beams are in good agreement with theoretical curves. Furthermore, the obtained results are compared with those of the GSM beam to illustrate the enhanced directionality about the optical axis.

2. Lorentz-correlated Schell-model (LSM) source

In this section, we introduce the theoretical model for the LSM source and study propagation of the beam it radiates through a stigmatic ABCD optical system, which is primarily needed to describe a single-lens focusing for access of the far field. Suppose a beam-like field, which is generated by a planar source in the plane z = 0, propagates into the half space z>0. Its second-order statistical properties in the source plane can be described by the two-point Cross-Spectral Density (CSD) function specified at position vectors $ρ_{1}' = (x_{1}', y_{1}')$ and $ρ_{2}' = (x_{2}', y_{2}')$ [1,2]

W^{(0)} (ρ_{1}', ρ_{2}'; ω) = \sqrt{S (ρ_{1}'; ω)} \sqrt{S (ρ_{2}'; ω)} μ (ρ_{1}', ρ_{2}'; ω),

where

S (ρ_{j}'; ω) = W^{(0)} (ρ_{j}', ρ_{j}'; ω)

(j = 1, 2) is the spectral density,

μ (ρ_{1}', ρ_{2}'; ω)

is the spectral DOC of the source. The genuine CSD of a partially coherent source can be represented by the following superposition rule [25,26]

W^{(0)} (ρ_{1}', ρ_{2}') = \int p (v) H_{0}^{*} (ρ_{1}', v) H_{0} (ρ_{2}', v) d^{2} v,

where p(v) is a non-negative weight function, H₀(ρ_j, v) is an arbitrary kernel function, and the asterisk denotes complex conjugate. Starting from Eq. (2) and in what follows, we omit the dependence of all quantities of interest on frequency ω. Furthermore, if we consider the simplest class of the CSD function, i.e. the Schell-model class, the kernel function H₀ can be expressed in the Fourier-like form [25,26]:

H_{0} (ρ_{j}', v) = f (ρ_{j}') \exp (- 2 π i ρ_{j}' \cdot v) . (j = 1, 2) .

As a consequence, the CSD function of the source can be written as the product

W^{(0)} (ρ_{1}', ρ_{2}') = f^{*} (ρ_{1}') f (ρ_{2}') \tilde{p} (ρ_{2}' - ρ_{1}'),

where

\tilde{p} (ρ_{2}' - ρ_{1}') = \int p (v) \exp [- 2 π i (ρ_{2}' - ρ_{1}') \cdot v],

is the two-dimensional (2D) Fourier transform of p(v). Suppose that p(v) satisfies the non-negative exponential function

p (v) = π δ^{2} \exp [- 2 π δ (| v_{x} | + | v_{y} |)],

then the DOC of the source can be obtained by substituting from Eq. (6) into Eq. (5) and using Eqs. (1) and (4). As a result, the DOC is of the rectangular Lorentz type

μ (x_{d} {,y}_{d}) = \frac{δ^{4}}{(x_{d}^{2} + δ^{2}) (y_{d}^{2} + δ^{2})},

where

x_{d} = x_{2}' - x_{1}',

y_{d} = y_{2}' - y_{1}',

and

μ (0, 0) = 1

must hold at coinciding arguments. Let us also term δ in Eqs. (6) and (7) the coherence length of the source. The Schell-model source with the DOC given by Eq. (7) may be called the rectangular LSM source. It is readily seen that such a beam can always be practically realized, since p(v) in Eq. (6) is a non-negative function everywhere. Figure 1(a)-1(c) shows the DOCs of the rectangular LSM beam for different coherence lengths, i.e. δ = 0.1mm, 1mm and 2mm, respectively. We notice that the distribution of the DOC broadens with the increase of the coherence length. For a relatively small coherence length (δ = 0.1mm), the DOC becomes an extremely sharp apex, which is also shown by the solid curve in Fig. 1(d).

Fig. 1 Density plots of the DOC of the rectangular LSM beam for different coherence lengths. (a) δ = 0.1mm, (b) δ = 1mm, (c) δ = 2mm, (d) comparison of cross-section profiles of (a), (b) and (c) at y_d = 0.

Download Full Size | PDF

Next, we assume that the LSM source beam has the Gaussian intensity distribution. Hence, the source CSD function is given by expression

W^{(0)} (ρ_{1}', ρ_{2}') = \exp (- \frac{ρ'_{1}^{2} + ρ'_{2}^{2}}{4 σ_{0}^{2}}) \frac{δ^{4}}{[{(x_{2} - x_{1})}^{2} + δ^{2}] [{(y_{2} - y_{1})}^{2} + δ^{2}]},

where σ₀ is the effective width of the source. Based on Eq. (8), the paraxial propagation of the LSM beam through a stigmatic ABCD optical system can be analyzed by using the generalized Collins formula [52,53]

\begin{array}{l} W (ρ_{1}, ρ_{2}, z) = \frac{k^{2}}{4 π^{2} B^{2}} \int_{- \infty}^{+ \infty} \int_{- \infty}^{+ \infty} \int_{- \infty}^{+ \infty} \int_{- \infty}^{+ \infty} W^{(0)} (ρ_{1}', ρ_{2}') \exp [- \frac{i k}{2 B} (A ρ_{1}'^{2} - 2 ρ_{1}' \cdot ρ_{1} + D ρ_{1}^{2}) \\ + \frac{i k}{2 B} (A ρ_{2}'^{2} - 2 ρ_{2}' \cdot ρ_{2} + D ρ_{2}^{2})] d x_{1}' d x_{2}' d y_{1}' d y_{2}', \end{array}

where

ρ_{j} = (x_{j}, y_{j})

(j = 1, 2) is the position vector in the receiver plane, and A, B, C and D are the elements of the transfer matrix of the optical system, k = 2π/λ with λ being the wavelength. Upon substituting from Eq. (8) into Eq. (9) and changing the integral variables

u_{-} = x_{1}' - x_{2}', u_{+} = x_{1}' + x_{2}', v_{-} = y_{1}' - y_{2}', v_{+} = y_{1}' + y_{2}',

the intensity distribution of the propagating LSM beam in the receiver plane can be obtained by setting

ρ_{1} = ρ_{2} = ρ

in the CSD function, such that

\begin{array}{l} I (ρ, z) = W (ρ, ρ, z) = \frac{k^{2} δ^{4}}{16 π^{2} B^{2}} \int_{- \infty}^{+ \infty} \int_{- \infty}^{+ \infty} \int_{- \infty}^{+ \infty} \int_{- \infty}^{+ \infty} \exp (- \frac{u_{+}^{2} + u_{-}^{2} + v_{+}^{2} + v_{-}^{2}}{8 σ_{0}^{2}}) \frac{1}{(u_{-}^{2} + δ^{2}) (v_{-}^{2} + δ^{2})} \\ \times \exp [- \frac{i k}{2 B} A u_{+} u_{-} - \frac{i k}{2 B} A v_{+} v_{-} + \frac{i k}{B} u_{-} x + \frac{i k}{B} v_{-} y] d u_{+} d u_{-} d v_{+} d v_{-} . \end{array}

If we utilize the convolution theorem of the Fourier transform [54]

F_{1} (τ) \otimes F_{2} (τ) = \int_{- \infty}^{+ \infty} f_{1} (κ) f_{2} (κ) \exp (- i τ κ) d κ,

where

f_{1} (k)

and

f_{2} (k)

are the Fourier transforms of

F_{1} (τ)

and

F_{2} (τ)

, respectively. Then Eq. (11) can be rewritten in the following alternative form

I (ρ, z) = \frac{k^{2} δ^{4}}{2 π B^{2}} σ_{0}^{2} \prod_{j = x, y} [F_{1} (\frac{k}{B} j) \otimes F_{2} (\frac{k}{B} j)],

with

\begin{matrix} F_{1} (τ) = \frac{π}{δ} \exp [- δ | τ |], & F_{2} (τ) = \sqrt{\frac{2 π}{\frac{k^{2} A^{2}}{B^{2}} σ_{0}^{2} + \frac{1}{4 σ_{0}^{2}}}} \exp [- \frac{τ^{2}}{\frac{2 k^{2} A^{2}}{B^{2}} σ_{0}^{2} + \frac{1}{2 σ_{0}^{2}}}] . \end{matrix}

In Eq. (13), the convolution can be computed by using the formula

F_{1} (τ) \otimes F_{2} (τ) = \int_{- \infty}^{+ \infty} F_{1} (ξ) F_{2} (τ - ξ) d ξ .

After tedious but straightforward integral calculations, the intensity distribution of the beam propagating through the ABCD optical system results in the analytic formula

\begin{array}{l} I (ρ, z) = \frac{π^{3} k^{2} δ^{2} σ_{0}^{2}}{2 B^{2}} \exp (Q δ^{2}) {\exp (- \frac{k δ}{B} x) e r f c {\sqrt{\frac{1}{2} Q} [δ - \frac{k x}{B Q}]} \\ + \exp (\frac{k δ}{B} x) e r f c {\sqrt{\frac{1}{2} Q} [δ + \frac{k x}{B Q}]} \cdot {\exp (- \frac{k δ}{B} y) e r f c {\sqrt{\frac{1}{2} Q} [δ - \frac{k y}{B Q}]} \\ + \exp (\frac{k δ}{B} y) e r f c {\sqrt{\frac{1}{2} Q} [δ + \frac{k y}{B Q}]}}, \end{array}

where

Q = \frac{k^{2} A^{2}}{B^{2}} σ_{0}^{2} + \frac{1}{4 σ_{0}^{2}},

e r f c (u) = 1 - \frac{2}{\sqrt{π}} \int_{0}^{u} \exp (- t^{2}) d t,

is the complementary error function [54]. We are primarily interested in the properties of the LSM beam passing through a thin lens with the focal length f, which helps to access the far-field distribution. Assume that the distance from the source plane to the lens is f, and the distance between the lens and the receiver plane is z. Then the transfer matrix of such lens system takes form

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

Based on Eqs. (16)-(19), we will perform numerical calculations to show the intensity distributions of the propagating LSM beam in the focal plane. The numerical parameters are chosen as λ = 532nm, f = 500mm, σ₀ = 2mm, unless specified otherwise.

Figure 2 illustrates the intensity distributions of the propagating LSM beam at the focal plane z = f. We notice that the propagating beam maintains the rectangular distribution when the coherence length is relatively small, e.g. δ = 0.1mm but gradually loses it and finally converts to a circular, Gaussian-like pattern when the coherence length reaches a certain value, e.g. δ = 2mm. According to the reciprocity relations of partially coherent beams, the propagating intensity distribution must be strongly dependent on the DOC of the source, so the variation of the coherence length induces considerable effect on the intensity of the beam. As the beam produces a highly focused intensity peak in the focal plane, we may classify the LSM beam as a typical highly focused partially coherent beam.

Fig. 2 Density plots of the intensity distributions of the LSM beam in the focus plane with different coherence lengths. (a) δ = 0.1mm, (b) δ = 0.5mm, (c) δ = 1mm, (d) δ = 2mm.

Download Full Size | PDF

Figure 3 shows the focused intensity distributions of the LSM and GSM beams. For comparison, the focused intensities for both beams are plotted by choosing the equivalent coherence length, which enables the source DOC to be 1/e the half maximum. By using this criterion, the coherence length of the GSM beam is chosen as δ_GSM = 1mm, while the coherence length of the LSM beam is δ_LSM = (e^1/2-1)^1/2 mm. One can notice that the focused intensity profile of the LSM beam is much sharper than that of the GSM beam, even though both beams have the equivalent coherence length. Due to substantial cusping, the LSM beam may be more suitable than the GSM beam for optical systems operating in the presence of random media.

Fig. 3 Focused intensity distributions of the LSM and GSM beams at the cross-section y = 0.

Download Full Size | PDF

3. Fractional multi-Gaussian-correlated Schell-model (FMGSM) source

In this section, we will introduce another beam type that also exhibits a highly focused intensity peak when it propagates to the far field or focal plane. To model such a beam, we first assume that p(v) takes the form of the rectangular FMGSM function [55]

p (v) = {1 - {[1 - \exp (- 2 π^{2} δ^{2} v_{x}^{2})]}^{1 / M}} \cdot {1 - {[1 - \exp (- 2 π^{2} δ^{2} v_{y}^{2})]}^{1 / M}},

where δ has the same meaning as it appears in Eq. (6), and M is a non-negative integer. It is readily observed that p(v) is non-negative everywhere. If in Eq. (20), the term (1-q)^1/M is expanded into the infinite Taylor series, p(v) can be rewritten as

p (v) = {\sum_{n = 1}^{\infty} \frac{{(- 1)}^{n} \prod_{m = 1}^{n} [1 - (m - 1) M]}{n! M^{n}} \exp (- 2 n π^{2} δ^{2} v_{x}^{2})} \cdot {\sum_{n' = 1}^{\infty} \frac{{(- 1)}^{n'} \prod_{m' = 1}^{n'} [1 - (m' - 1) M]}{n'! M^{n'}} \exp (- 2 n π^{2} δ^{2} v_{y}^{2})} .

Similarly to the derivation of Eq. (7) the DOC of such a source can be obtained by substituting from Eq. (20) into Eq. (5) and using Eqs. (1) and (4)

\begin{array}{l} μ (x_{d}, y_{d}) = \frac{1}{C_{0}^{2}} {\sum_{n = 1}^{\infty} \frac{{(- 1)}^{n + 1} \prod_{m = 1}^{n} [1 - (m - 1) M]}{\sqrt{n} n! M^{n}} \exp (- \frac{x_{d}^{2}}{2 n δ^{2}})} \\ \times {\sum_{n' = 1}^{\infty} \frac{{(- 1)}^{n' + 1} \prod_{m' = 1}^{n'} [1 - (m' - 1) M]}{\sqrt{n'} n'! M^{n'}} \exp (- \frac{y_{d}^{2}}{2 n' δ^{2}})}, \end{array}

where

C_{0} = \sum_{n = 1}^{\infty} \frac{{(- 1)}^{n} \prod_{m = 1}^{n} [1 - (m - 1) M]}{\sqrt{n} n! M^{n}},

is the normalization factor. In what follows, we define the source whose DOC obeys Eq. (22) as the rectangular FMGSM beam.

In Fig. 4, the contour plots (a)-(c) illustrate the DOC of the FMGSM beam for M = 2, and the coherence length δ = 2.25mm, 0.225mm and 0.113mm, respectively. It is shown that the DOC distribution becomes much narrower when the beam has a smaller coherence length. Figure 4(d)-4(f) presents the cross-section profiles of the DOC of the beam for M = 2, 5 and 10, respectively, and the same coherence length, δ = 0.225mm, for all the plots. For comparison, we utilize Eq. (22) to plot the DOC of the beam by setting up two truncated numbers for the Taylor series, i.e. N = 300 (dash curve) and 1000 (dash-dotted curve), respectively. As an alternative approach, we also substitute Eq. (20) into Eq. (5) and perform the numerical integration to plot the DOC (solid curve). For M = 2 and M = 5, the DOC calculated from Eq. (22) by using N = 300 and N = 1000 shows a good approximation to that resulted from the direct numerical integration. However, it is noticed that the tails of the DOCs calculated from Eq. (22) slightly differ from those resulted from the direct integration.

Fig. 4 DOC of the rectangular FMGSM beam in the initial plane. (a)-(c) are the contour plots of the DOC (M = 2) with different coherence lengths: δ = 2.25mm, 0.225mm and 0.113mm, respectively. (d)-(f) are the cross-section profiles of the DOC for M = 2, 5 and 10, respectively, while the coherence length remains the same, i.e. δ = 0.225mm.

Download Full Size | PDF

Now let us study the propagation properties of the FMGSM beam through a stigmatic ABCD optical system. Suppose that the source has the Gaussian intensity distribution, so the source CSD function is given by expression

\begin{array}{l} W^{(0)} (ρ_{1}', ρ_{2}') = \frac{1}{C_{0}^{2}} \exp (- \frac{ρ'_{1}^{2} + ρ'_{2}^{2}}{4 σ_{0}^{2}}) {\sum_{n = 1}^{\infty} \frac{{(- 1)}^{n + 1} \prod_{m = 1}^{n} [1 - (m - 1) M]}{\sqrt{n} n! M^{n}} \exp (- \frac{x_{d}^{2}}{2 n δ^{2}})} \\ \times {\sum_{n' = 1}^{\infty} \frac{{(- 1)}^{n' + 1} \prod_{m' = 1}^{n'} [1 - (m' - 1) M]}{\sqrt{n'} n'! M^{n'}} \exp (- \frac{y_{d}^{2}}{2 n' δ^{2}})} . \end{array}

Upon substituting from Eq. (24) into Eq. (9) and performing tedious but straightforward integrations, the CSD function of the FMGSM beam yields the following expression

\begin{array}{l} W (x_{1}, y_{1}, x_{2}, y_{2}, z) = \frac{1}{C_{0}^{2}} {\sum_{n = 1}^{\infty} \frac{{(- 1)}^{n + 1} \prod_{m = 1}^{n} [1 - (m - 1) M]}{\sqrt{n Q'} n! M^{n}} \exp [- \frac{{(x_{1} + x_{2})}^{2}}{8 σ_{0}^{2} Q'}] \exp [- \frac{k^{2} σ_{0}^{2}}{2 B^{2}} (1 - \frac{A^{2}}{Q'}) {(x_{2} - x_{1})}^{2}] \\ \times \exp [\frac{i k}{2 B} (D - \frac{A}{Q'}) (x_{2}^{2} - x_{1}^{2})]} \cdot {\sum_{n' = 1}^{\infty} \frac{{(- 1)}^{n' + 1} \prod_{m' = 1}^{n'} [1 - (m' - 1) M]}{\sqrt{n' Q'} n'! M^{n'}} \exp [- \frac{{(y_{1} + y_{2})}^{2}}{8 σ_{0}^{2} Q'}] \\ \times \exp [- \frac{k^{2} σ_{0}^{2}}{2 B^{2}} (1 - \frac{A^{2}}{Q'}) {(y_{2} - y_{1})}^{2}] \exp [\frac{i k}{2 B} (D - \frac{A}{Q'}) (y_{2}^{2} - y_{1}^{2})]}, \end{array}

with

Q' = A^{2} + \frac{B^{2}}{4 k^{2} σ_{0}^{4}} + \frac{B^{2}}{n k^{2} σ_{0}^{2} δ^{2}} .

By setting x₁ = x₂ = x and y₁ = y₂ = y in Eq. (25), the intensity distribution of the propagating beam becomes

I (x, y, z) = \frac{1}{C_{0}^{2}} {\sum_{n = 1}^{\infty} \frac{{(- 1)}^{n + 1} \prod_{m = 1}^{n} [1 - (m - 1) M]}{\sqrt{n Q'} n! M^{n}} \exp (- \frac{x^{2}}{2 σ_{0}^{2} Q'})} \cdot {\sum_{n' = 1}^{\infty} \frac{{(- 1)}^{n' + 1} \prod_{m' = 1}^{n'} [1 - (m' - 1) M]}{\sqrt{n' Q'} n'! M^{n'}} \exp (- \frac{y^{2}}{2 σ_{0}^{2} Q'})} .

Based on Eq. (27), we plot the spectral density distribution of the FMGSM beam propagating through a thin lens in the focal plane. From Fig. 5 it is found that the FMGSM beam exhibits a highly focused intensity peak in the focal plane. Compared with the focused intensity produced by the GSM beam, the FMGSM beam has a better capacity for generating a sharper intensity maximum at the focal point. With the increase of M the intensity peak value is greatly enhanced. When M = 10, the focused beam pattern looks like an extremely sharp “needle” apex in the center [see Fig. 5(c)]. Besides, the peak intensity value of the FMGSM beam in the focal plane is much larger than that produced by the GSM. It implies that the FMGSM beam might be a very suitable candidate for generating a highly focused intensity apex.

Fig. 5 Spectral density of the FMGSM beam at the focal plane. The density plot (a) shows the focused intensity profile produced by the GSM beam, (b)-(c) show the focused intensity of the FMGSM beam for M = 5 and M = 10, respectively. For comparison, the cross-section profiles (d)-(f) show the intensity distributions for M = 2, 5, 10 and the GSM beam, respectively. As the numerical parameter, the coherence length remains as a fixed value of δ = 1.13mm.

Download Full Size | PDF

4. Experimental generation of the highly focused partially coherent beams

In this section, we demonstrate the experimental generation of the LSM and the FMGSM beams and measure their focused intensity distributions on passing through a thin lens. Figure 6 illustrates the schematic diagram of the experimental setup for generating the two beams. The laser beam generated from a Laser Diode (LD, λ = 532nm) first passes through a neutral density filter (NDF) which is used to control the power of the output beam, and then transmits through a linear polarizer (LP). The transmitted beam from LP₁ passes through a Spatial Light Modulator (SLM, HOLOEYE LC2012), controlled by a personal computer (PC₁) which acts as a random phase screen to generate the LSM or the FMGSM beam. Methods for synthesis of the random phase screens can be found in Refs [51,56–59]. The second linear polarizer (LP₂) whose transmission axis coincides with LP₁ that is placed behind the SLM is to block other polarization component of the scattered light from the SLM. Lenses L₁ and L₂ constitute a 2f-imaging system with each focal length f = 250mm, in which a circular aperture (CA) is used to select out the first-order diffraction order. The focusing lens L₃ is inserted after L₂ to focus the generated partially coherent beam. In the focal plane, a detector (DCC1545M-GL) is placed to capture the focused intensity patterns and export the experimental data to the personal computer (PC₂).

Fig. 6 Experimental setup for generating the LSM and the FMGSM beams, and measuring their focused intensity distributions. NDF: neutral density filter; LP₁ and LP₂: linear polarizers; CA: circular aperture; SLM: spatial light modulator; L₁, L₂ and L₃: thin lenses; PC₁ and PC₂: personal computers.

Download Full Size | PDF

Figure 7 shows the focused intensity distributions of the LSM beam measured from the experiments, where the coherence length for each figure is obtained by the theoretical fit. It is found that the focused beam exhibits a rectangular intensity distribution when the coherence length is relatively small, e.g. δ = 0.4mm. However, the focused intensity gradually loses the rectangular symmetry with the increase of the coherence length. For a relatively large coherence length, e.g. δ = 1.1mm, the focused intensity converts into a circular, Gaussian-like profile, as shown in Fig. 7(c). These results are well consistent with those concluded from Fig. 2. Figure 7(d)-7(e) compares the experimental data with the theoretical fit results for the cross-section intensities of the focused beam. The solid curves are generated from the theoretical calculations, and the dash-dotted curves are the experimental data. Through the theoretical fit, we obtain the coherence length for each subplot. It shows that the experimental data correspond well to the theoretical fit results, despite a slight discrepancy is distinguished in the tails of the transversal intensities when δ = 0.4mm.

Fig. 7 Experimental intensity distributions of the LSM beam. The contour plots (a)-(c) show the intensity profiles for δ = 0.4mm, 0.73mm and 1.1mm, respectively. The cross-section profiles (d)-(e) exhibit the comparisons between the experimental data and the theoretical fit results at the plane y = 0.

Download Full Size | PDF

Figure 8(a)-8(c) shows the contour plots of the focused FMGSM beam with different values of M. The coherence length for each plot is obtained by the theoretical fit. It is shown that the focused intensity profile of the beam has a very sharp apex in the center. Such result is consistent with those shown in Fig. 5(b)-5(c). Furthermore, we also compare the experimental data with the theoretical fit results in Fig. 8(d)-8(f). Despite a slight discrepancy between the two results when M = 20, the experimental data shows a very good agreement with the theoretical fit for M = 5 and 10. For a relatively large M, e.g. M = 20, the discrepancy between the two results is obvious, and is due to the limitation of the pixel resolution of the SLM. It well explains why the focused intensity pattern from the experiments [Fig. 7(b)] is not analogous to the focused “needle” apex obtained from the theory [Fig. 5(c)].

Fig. 8 Experimental intensity distributions of the FMGSM beam. The contour plots (a)-(c) show the intensity distributions of the beam for M = 5, 10 and 20, respectively. The cross-section profiles (d)-(f) present the comparisons between the experimental data with the theoretical fit results at the plane y = 0.

Download Full Size | PDF

In order to highlight the importance of the obtained results, we make the comparison between the FMGSM beam and the GSM beam for the focused intensity profiles. In Fig. 9, the dashed and the solid curves represent the FMGSM beam with M = 5 and 10, respectively. Meanwhile, the focused intensity pattern of the GSM beam is also plotted (dotted curve). It is interesting to note that the focused intensity of the FMGSM beam has sharp apex that is much higher than that of the GSM beam with the same coherence length. In addition, Fig. 9 also corresponds well to the theoretical results presented in Fig. 5(e)-5(f). The outstanding self-focusing capacity of the FMGSM beam shows that this beam is a better candidate than the GSM beam to be used in some applications, e.g. the material thermal process and the beam shaping in the focused region.

Fig. 9 Experimental intensity distributions of the focused FMGSM beam at the cross-section y = 0. Comparisons are made between the focused intensities of the FMGSM beam with M = 5, 10 and those produced by the GSM beam.

Download Full Size | PDF

5. Conclusion

In summary, we introduced two partially coherent beam families whose spectral DOCs satisfy the rectangular Lorentz and the Factional Multi-Gaussian functions. Propagation properties of the two beams through a stigmatic ABCD optical system are derived analytically. It is shown that both beams can form highly focused intensity patterns in the far field. For the FMGSM beam, the self-focused intensity profile is analogous to an extremely sharp “needle” apex. Experimental results are presented for the focused intensity distributions of the beams and shown to agree well with the theoretical predictions. The proposed cusped partially coherent beams, especially the FMGSM beam, have potential applications to the material thermal process and the beam shaping in the focused region.

Funding

O. Korotkova and J. Li acknowledge the support from the Air Force Office of Scientific Research (AFOSR) (FA9550-121-0449). F. Wang’s research is supported by the National Natural Science Foundation of China (NSFC) (11474213).

References and Links

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

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

3. Y. Baykal, M. A. Plonus, and S. J. Wang, “The scintillations for weak atmospheric turbulence using a partially coherent source,” Radio Sci. 18(4), 551–556 (1983). [CrossRef]

4. P. S. Carney and E. Wolf, “Power-excitation diffraction tomography with partially coherent light,” Opt. Lett. 26(22), 1770–1772 (2001). [CrossRef] [PubMed]

5. G. Gbur and E. Wolf, “Spreading of partially coherent beams in random media,” J. Opt. Soc. Am. A 19(8), 1592–1598 (2002). [CrossRef] [PubMed]

6. G. Gbur, T. D. Visser, and E. Wolf, “‘Hidden’ singularities in partially coherent wavefields,” J. Opt. A, Pure Appl. Opt. 6(5), S239–S242 (2004). [CrossRef]

7. H. Roychowdhury, G. P. Agrawal, and E. Wolf, “Changes in the spectrum, in the spectral degree of polarization, and in the spectral degree of coherence of a partially coherent beam propagating through a gradient-index fiber,” J. Opt. Soc. Am. A 23(4), 940–948 (2006). [CrossRef] [PubMed]

8. M. Lahiri and E. Wolf, “Negative refraction of a partially coherent electromagnetic beam,” Opt. Lett. 38(9), 1407–1409 (2013). [CrossRef] [PubMed]

9. M. Singh, J. Tervo, and J. Turunen, “Elementary-field analysis of partially coherent beam shaping,” J. Opt. Soc. Am. A 30(12), 2611–2617 (2013). [CrossRef] [PubMed]

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

11. G. Gbur, “Partially coherent beam propagation in atmospheric turbulence [invited],” J. Opt. Soc. Am. A 31(9), 2038–2045 (2014). [CrossRef] [PubMed]

12. G. Wu and T. D. Visser, “Hanbury Brown-Twiss effect with partially coherent electromagnetic beams,” Opt. Lett. 39(9), 2561–2564 (2014). [CrossRef] [PubMed]

13. M. W. Hyde IV, S. Basu, X. Xiao, and D. G. Voelz, “Producing any desired far-field mean irradiance pattern using a partially-coherent Schell-model source,” J. Opt. 17(5), 055607 (2015). [CrossRef]

14. Y. Cai and S. Y. Zhu, “Ghost interference with partially coherent radiation,” Opt. Lett. 29(23), 2716–2718 (2004). [CrossRef] [PubMed]

15. K. Drexler and M. Roggemann, “Use of a partially coherent transmitter beam to improve the statistics of received power in a free-space optical communication system: theory and experimental results,” Opt. Eng. 50(2), 025002 (2011). [CrossRef]

16. E. E. García-Guerrero, E. R. Méndez, Z. H. Gu, T. A. Leskova, and A. A. Maradudin, “Interference of a pair of symmetric partially coherent beams,” Opt. Express 18(5), 4816–4828 (2010). [CrossRef] [PubMed]

17. H. Lajunen and T. Saastamoinen, “Propagation characteristics of partially coherent beams with spatially varying correlations,” Opt. Lett. 36(20), 4104–4106 (2011). [CrossRef] [PubMed]

18. 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]

19. R. Zhang, X. Wang, and X. Cheng, “Far-zone polarization distribution properties of partially coherent beams with non-uniform source polarization distributions in turbulent atmosphere,” Opt. Express 20(2), 1421–1435 (2012). [CrossRef] [PubMed]

20. M. Lahiri and E. Wolf, “Theory of refraction and reflection with partially coherent electromagnetic beams,” Phys. Rev. A 86(4), 043815 (2012). [CrossRef]

21. S. Zhu and Z. Li, “Theoretical and experimental studies of the spectral changes of a focused polychromatic partially coherent flat-topped beam,” Appl. Phys. B 118(3), 481–487 (2015). [CrossRef]

22. E. Wolf and E. Collett, “Partially coherent sources which produce the same far-field intensity distribution as a laser,” Opt. Commun. 25(3), 293–296 (1978). [CrossRef]

23. F. Gori, G. Guattari, and C. Padovani, “Modal expansion for J₀-correlated Schell-model sources,” Opt. Commun. 64(4), 311–316 (1987). [CrossRef]

24. S. A. Ponomarenko, “A class of partially coherent beams carrying optical vortices,” J. Opt. Soc. Am. A 18(1), 150–156 (2001). [CrossRef] [PubMed]

25. F. Gori and M. Santarsiero, “Devising genuine spatial correlation functions,” Opt. Lett. 32(24), 3531–3533 (2007). [CrossRef] [PubMed]

26. F. Gori, V. Ramírez-Sánchez, M. Santarsiero, and T. Shirai, “On genuine cross-spectral density matrices,” J. Opt. A, Pure Appl. Opt. 11(8), 085706 (2009). [CrossRef]

27. H. Roychowdhury and O. Korotkova, “Realizability conditions for electromagnetic Gaussian Schell-model sources,” Opt. Commun. 249(4-6), 379–385 (2005). [CrossRef]

28. 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]

29. M. Santarsiero, G. Piquero, J. C. G. de Sande, and F. Gori, “Difference of cross-spectral densities,” Opt. Lett. 39(7), 1713–1716 (2014). [CrossRef] [PubMed]

30. Z. Mei, O. Korotkova, and Y. Mao, “Products of Schell-model cross-spectral densities,” Opt. Lett. 39(24), 6879–6882 (2014). [CrossRef] [PubMed]

31. O. Korotkova and Z. Mei, “Convolution of degrees of coherence,” Opt. Lett. 40(13), 3073–3076 (2015). [CrossRef] [PubMed]

32. Z. Mei, O. Korotkova, and Y. Mao, “Powers of the degree of coherence,” Opt. Express 23(7), 8519–8531 (2015). [CrossRef] [PubMed]

33. Z. Mei and O. Korotkova, “Alternating series of cross-spectral densities,” Opt. Lett. 40(11), 2473–2476 (2015). [CrossRef] [PubMed]

34. Z. Mei, D. Zhao, O. Korotkova, and Y. Mao, “Gaussian Schell-model arrays,” Opt. Lett. 40(23), 5662–5665 (2015). [CrossRef] [PubMed]

35. H. Lajunen and T. Saastamoinen, “Propagation characteristics of partially coherent beams with spatially varying correlations,” Opt. Lett. 36(20), 4104–4106 (2011). [CrossRef] [PubMed]

36. H. Lajunen and T. Saastamoinen, “Non-uniformly correlated partially coherent pulses,” Opt. Express 21(1), 190–195 (2013). [CrossRef] [PubMed]

37. Z. Tong and O. Korotkova, “Nonuniformly correlated light beams in uniformly correlated media,” Opt. Lett. 37(15), 3240–3242 (2012). [CrossRef] [PubMed]

38. Z. Mei, Z. Tong, and O. Korotkova, “Electromagnetic non-uniformly correlated beams in turbulent atmosphere,” Opt. Express 20(24), 26458–26463 (2012). [CrossRef] [PubMed]

39. S. Sahin and O. Korotkova, “Light sources generating far fields with tunable flat profiles,” Opt. Lett. 37(14), 2970–2972 (2012). [CrossRef] [PubMed]

40. O. Korotkova, S. Sahin, and E. Shchepakina, “Multi-Gaussian Schell-model beams,” J. Opt. Soc. Am. A 29(10), 2159–2164 (2012). [CrossRef] [PubMed]

41. Z. Mei, O. Korotkova, and E. Shchepakina, “Electromagnetic multi-Gaussian Schell-model beams,” J. Opt. 15(2), 025705 (2013). [CrossRef]

42. Z. Mei and O. Korotkova, “Cosine-Gaussian Schell-model sources,” Opt. Lett. 38(14), 2578–2580 (2013). [CrossRef] [PubMed]

43. Z. Mei, E. Schchepakina, and O. Korotkova, “Propagation of cosine-Gaussian-correlated Schell-model beams in atmospheric turbulence,” Opt. Express 21(15), 17512–17519 (2013). [CrossRef] [PubMed]

44. Y. Chen, J. Gu, F. Wang, and Y. Cai, “Self-splitting properties of a Hermite-Gaussian correlated Schell-model beam,” Phys. Rev. A 91(1), 013823 (2015). [CrossRef]

45. Z. Song, Z. Liu, K. Zhou, Q. Sun, and S. Liu, “Propagation characteristics of a non-uniformly Hermite-Gaussian correlated beam,” J. Opt. 18(1), 015606 (2016). [CrossRef]

46. F. Wang, C. Liang, Y. Yuan, and Y. Cai, “Generalized multi-Gaussian correlated Schell-model beam: from theory to experiment,” Opt. Express 22(19), 23456–23464 (2014). [CrossRef] [PubMed]

47. S. Zhu and Z. Li, “Theoretical and experimental studies of the spectral changes of a focused polychromatic partially coherent flat-topped beam,” Appl. Phys. B 118(3), 481–487 (2015). [CrossRef]

48. Y. Chen, F. Wang, C. Zhao, and Y. Cai, “Experimental demonstration of a Laguerre-Gaussian correlated Schell-model vortex beam,” Opt. Express 22(5), 5826–5838 (2014). [CrossRef] [PubMed]

49. C. Liang, F. Wang, X. Liu, Y. Cai, and O. Korotkova, “Experimental generation of cosine-Gaussian-correlated Schell-model beams with rectangular symmetry,” Opt. Lett. 39(4), 769–772 (2014). [CrossRef] [PubMed]

50. S. Zhu, Y. Chen, J. Wang, H. Wang, Z. Li, and Y. Cai, “Generation and propagation of a vector cosine-Gaussian correlated beam with radial polarization,” Opt. Express 23(26), 33099–33115 (2015). [CrossRef] [PubMed]

51. X. Xiao, O. Korotkova, and D. G. Voelz, “Laboratory implementation of partially coherent beams with Super-Gaussian distribution,” Proc. SPIE 9224, 92240N (2014). [CrossRef]

52. S. A. Collins, “Lens-system diffraction integral written in terms of matrix optics,” J. Opt. Soc. Am. 60(9), 1168–1177 (1970). [CrossRef]

53. 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]

54. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Dover, 1972).

55. Y. Li, “Flat-topped beam with non-circular cross-sections,” J. Mod. Opt. 50(13), 1957–1966 (2003). [CrossRef]

56. S. Basu, M. W. Hyde IV, X. Xiao, D. G. Voelz, and O. Korotkova, “Computational approaches for generating electromagnetic Gaussian Schell-model sources,” Opt. Express 22(26), 31691–31707 (2014). [CrossRef] [PubMed]

57. M. W. Hyde IV, S. Basu, D. G. Voelz, and X. Xiao, “Generating partially coherent Schell-model sources using a modified phase screen approach,” Opt. Eng. 54(12), 120501 (2015). [CrossRef]

58. M. V. Hyde IV, S. Basu, D. G. Voelz, and X. Xiao, “Experimentally generating any desired partially coherent Schell-model source using phase-only control,” J. Appl. Phys. 118(9), 093102 (2015). [CrossRef]

59. M. V. Hyde IV, S. Basu, X. Xiao, and D. G. Voelz, “Producing any desired far-field mean irradiance pattern using a partially-coherent Schell-model source,” J. Opt. 17(5), 055607 (2015). [CrossRef]

Random sources for cusped beams

Abstract

1. Introduction

2. Lorentz-correlated Schell-model (LSM) source

3. Fractional multi-Gaussian-correlated Schell-model (FMGSM) source

4. Experimental generation of the highly focused partially coherent beams

5. Conclusion

Funding

References and Links

Cited By

Figures (9)

Equations (27)

Optics Express