Paraxial propagation of a class of Bessel-correlated fields

Atri Halder; Matias Koivurova; Henri Partanen; Jari Turunen

doi:10.1364/OE.26.011055

1. Introduction

Partially spatially coherent model sources and light beams generated by them have attracted a great deal of interest over the past few decades. Apart from classical Gaussian Schell-model (GSM) sources and beams [1], a large number of fields with more unconventional correlation properties have been introduced and demonstrated, which exhibit a wide variety of interesting propagation properties and applications [2–4]. Among these, Bessel-correlated fields originally introduced by Gori [5] are particularly interesting [6–10]. This type of fields can be generated, e.g., by collimating light emitted by a spatially incoherent ring source. They can be (approximately) propagation-invariant in the sense that the spatial coherence coherence properties remain nearly unchanged as the field traverses forward over considerable distances [11]. As such, they can be viewed as generalizations of fully coherent Bessel fields [12] into the domain of partially coherent optics.

While no fields with a finite spatial extent can possess the property of strict propagation-invariance, a less strict condition of shape-invariant propagation can sometimes be satisfied. By this we refer to fields that experience only a scale change of the intensity distribution upon propagation. An even less strict condition is that the field at the source plane is of the same functional form as the radiation field in the far zone; we will call fields that behave in this matter as being self-Fourier-transforming. Hermite–Gaussian and Laguerre–Gaussian laser beams are well-known examples of fully coherent fields that are both shape-invariant and self-Fourier-transforming. Partially coherent fields that satisfy these criteria are also known. These include standard GSM fields, which can be thought of as superpositions of either laser modes [13–15] or ‘elementary’ spatially or angularly displaced Gaussian beams [16–18]. Such superpositions can also lead to spatially partially coherent fields with anisotropic shape-invariant intensity distributions extending from the source plane to the far zone, which is not possible with fully coherent fields [19]. Moreover, partially coherent beams with properties not shared by coherent beams, such as twist phase [20], can also be described using coherent-mode superpositions [21]. It is therefore safe to state that partial spatial coherence adds new degrees of freedom to optical beam formation, which motivates further studies on the subject.

We have recently introduced a class of Bessel-correlated fields, which can be generated by particularly simple experimental techniques: passing a coherent light beam through a rotating tilted glass plate or wedge [22]. Under certain conditions the cross-spectral density function at the source plane and the angular correlation function were found to have the same functional form. In this sense the fields introduced in [22, 23], behave like self-Fourier-transforming fields. In this paper we consider the paraxial propagation of this class of Bessel-correlated fields both analytically end experimentally.

The paper is structures as follows. In Sect. 2 we summarize briefly the main results of [22]. The propagation of these fields in free space and in first-order optical systems is discussed in Sects. 3–5, with particular attention to self-Fourier-transforming fields in Sect. 4. It is shown, e.g., that the absolute value of the complex degree of spatial coherence of these beams is a shape-invariant quantity if the condition for self-Fourier-transformation is satisfied. However, the beam intensity distribution is not strictly shape-invariant, and in this sense the behavior of the model fields considered here is different from those radiated by GSM sources [17]. Experimental verifications of the main results are provided in Sect. 6, and conclusions are drawn in Sect. 7.

2. Source model

Figure 1 illustrates the geometry to be considered in our experiments, and it represents a simple method to generate model fields discussed below. Here a coherent Gaussian beam with its waist at the plane z = 0 is incident on a rotating prism with axial thickness d, wedge angle α, and refractive index n. When the wedge rotates about the z axis, the axial ray draws a circle of radius R = d tan (α – α′), with α′ = arcsin (sin α/n), on the rear face of the wedge (at z = 0) and propagates at an angle β = arcsin [n sin (α – α′)] with respect to the z axis. At each instant of time a Gaussian beam propagating in the direction of this ray emerges; upon rotation of the wedge a partially spatially coherent field consisting of all such ‘elementary’ fields is generated in time-averaged sense. We describe it with a cross-spectral density function, considering the CSD in the sense of ensemble averaging with the elementary fields acting as realizations with equal probability of occurrence.

Fig. 1 Geometry. A Gaussian beam with 1/e² waist radius w₀ is incident on a rotating prism with wedge angle α, central thickness d, and refractive index n. In the paraxial domain (small values of α) the output beam at z = 0 is a spatially displaced [by R ≈ dα (1−1/n)] and angularly deflected [by β ≈ α(n – 1)] Gaussian beam.

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The cross-spectral density function (CSD) of the field that emerges from the wedge in the geometry of Fig. 1 can be described (at any plane z ≥ 0) by an elementary-field superposition [22]

W (x_{1}, y_{1}, x_{2}, y_{2}, z) = \frac{1}{2 π} \int_{0}^{2 π} e^{*} (x_{1}, y_{1}, z; ϕ) e (x_{2}, y_{2}, z; ϕ) d ϕ,

where a single elementary-field contribution e(x, y, z; ϕ) depends on the (time-varying) azimuth angle ϕ. When the wedge is illuminated by a coherent Gaussian beam, the emerging Gaussian elementary field at the exit plane z = 0 of the wedge can be expressed as

e (x, y, 0; ϕ) = e_{0} \exp [- \frac{{(x - R \cos ϕ)}^{2} + {(y - R \sin ϕ)}^{2}}{w_{0}^{2}}] \exp [i k_{0} \sin β (x \cos ϕ + y \sin ϕ - R)],

where k₀ = 2π/λ₀ is the vacuum wave number. At the plane z = 0 the CSD then becomes

W (x_{1}, y_{1}, x_{2}, y_{2}, 0) = S_{0} \exp (- \frac{x_{1}^{2} + x_{2}^{2} + y_{1}^{2} + y_{2}^{2} + 2 R^{2}}{w_{0}^{2}}) I_{0} [a (x_{1}, y_{1}, x_{2}, y_{2}, 0)],

where S₀ = |e₀|², I₀ is the zeroth-order modified Bessel function,

a (x_{1}, y_{1}, x_{2}, y_{2}, 0) = \frac{4 R}{w_{0}^{2}} {[{(\bar{x} - i \frac{z_{R} \sin β}{R} \frac{Δ x}{2})}^{2} + {(\bar{y} - i \frac{z_{R} \sin β}{R} \frac{Δ y}{2})}^{2}]}^{1 / 2},

and

z_{R} = k_{0} w_{0}^{2} / 2

is the Rayleigh range of the incident Gaussian beam. In writing Eq. (4) we have introduced average spatial coordinates

\bar{x} = \frac{1}{2} (x_{1} + x_{2})

,

\bar{y} = \frac{1}{2} (y_{1} + y_{2})

as well as difference spatial coordinates Δx = x₂ − x₁, Δy = y₂ − y₁. The intensity distribution at z = 0 is given by

S (x, y, 0) = W (x, x, y, y, 0) = S_{0} \exp [- \frac{2 (x^{2} + y^{2} + R^{2})}{w_{0}^{2}}] I_{0} [\frac{4 R}{w_{0}^{2}} \sqrt{x^{2} + y^{2}}]

and the complex degree of spatial coherence takes the form

\begin{array}{l} μ (x_{1}, x_{2}, y_{1}, y_{2}, 0) = \frac{W (x_{1}, x_{2}, y_{1}, y_{2}, 0)}{\sqrt{S (x_{1}, y_{1}, 0) S (x_{2}, y_{2}, 0)}} \\ = \frac{I_{0} [a (x_{1}, y_{1}, x_{2}, y_{2}, 0)]}{\sqrt{I_{0} [a (x_{1}, y_{1}, x_{1}, y_{1}, 0)] I_{0} [a (x_{2}, y_{2}, x_{2}, y_{2}, 0)]}} . \end{array}

This is the class of Bessel-correlated model sources introduced in [22], where also the limiting cases R → 0 and β → 0 were considered in detail.

3. Propagation in free space

By inserting from Eq. (2) into the standard Fresnel formula

\begin{array}{l} e (x, y, z; ϕ) = \frac{k_{0}}{i 2 π z} \exp (i k_{0} z) \exp [\frac{i k_{0}}{2 z} (x^{2} + y^{2})] \\ \times \int \int_{- \infty}^{\infty} e (x^{'}, y^{'}, 0; ϕ) \exp [\frac{i k_{0}}{2 z} ({x^{'}}^{2} + {y^{'}}^{2})] \exp [- \frac{i k_{0}}{z} (x x^{'} + y y^{'})] d x^{'} d y^{'} \end{array}

for free-space propagation and carrying out the integrations, we obtain an expression for the elementary field at any propagation distance in the form

\begin{array}{l} e (x, y, z; ϕ) = \frac{e_{0}}{1 + i z / z_{R}} \exp [i k_{0} z (1 - \frac{1}{2} \frac{\sin^{2} β}{1 + i z / z_{R}})] \exp [- \frac{{(x - R \cos ϕ)}^{2} + {(y - R \sin ϕ)}^{2}}{w_{0}^{2} (1 + i z / z_{R})}] \\ \times \exp [\frac{i k_{0} \sin β}{1 + i z / z_{R}} (x \cos ϕ + y \sin ϕ - R)] . \end{array}

On inserting this expression into Eq. (1) we arrive at

\begin{array}{l} W (x_{1}, y_{1}, x_{2}, y_{2}, z) = \frac{{| e_{0} |}^{2}}{1 + {(z / z_{R})}^{2}} \\ \times \exp [- k_{0} z_{R} \sin^{2} β \frac{{(z / z_{R})}^{2}}{1 + {(z / z_{R})}^{2}}] \exp [- \frac{x_{1}^{2} + y_{1}^{2} + R^{2}}{w_{0}^{2} (1 - i z / z_{R})}] \exp [- \frac{x_{2}^{2} + y_{2}^{2} + R^{2}}{w_{0}^{2} (1 + i z / z_{R})}] \\ \times \frac{1}{2 π} \int_{0}^{2 π} \exp {[c (z) \bar{x} - i d (z) Δ x / 2] \cos ϕ + [c (z) \bar{y} - i d (z) Δ y / 2] \sin ϕ} d ϕ, \end{array}

where

c (z) = \frac{4}{w_{0}^{2}} \frac{R + z \sin β}{1 + {(z / z_{R})}^{2}} = \frac{4 R}{w^{2} (z)} (1 + \frac{z}{R} \sin β),

d (z) = 2 k_{0} \frac{R z / z_{R}^{2} + \sin β}{1 + {(z / z_{R})}^{2}} = \frac{4 R}{w^{2} (z)} (\frac{z}{z_{R}} + \frac{z_{R}}{R} \sin β),

and

w (z) = w_{0} {[1 + {(z / z_{R})}^{2}]}^{1 / 2} .

The integration in Eq. (9) then gives

\begin{array}{l} W (x_{1}, y_{1}, x_{2}, y_{2}, z) = S_{0} \frac{w_{0}^{2}}{w^{2} (z)} \exp [- \frac{x_{1}^{2} + x_{2}^{2} + y_{1}^{2} + y_{2}^{2} + 2 {(R - z \sin β)}^{2}}{w^{2} (z)}] \\ \times I_{0} [a (x_{1}, y_{1}, x_{2}, y_{2}, z)] \exp [\frac{i k_{0}}{2 R (z)} (x_{2}^{2} - x_{1}^{2} + y_{2}^{2} - y_{1}^{2})], \end{array}

where

\begin{array}{l} a (x_{1}, y_{1}, x_{2}, y_{2}, z) = {{[c (z) \bar{x} - i d (z) Δ x / 2]}^{2} + {[c (z) \bar{y} - i d (z) Δ y / 2]}^{2}}^{1 / 2} \\ = \frac{4 R}{w^{2} (z)} {{[(1 + \frac{z}{R} \sin β) \bar{x} - i (\frac{z}{z_{R}} + \frac{z_{R}}{R} \sin β) \frac{Δ x}{2}]}^{2} \\ {+ {[(1 + \frac{z}{R} \sin β) \bar{y} - i (\frac{z}{z_{R}} + \frac{z_{R}}{R} \sin β) \frac{Δ y}{2}]}^{2}}}^{1 / 2} \end{array}

and

R (z) = z + z_{R}^{2} / z .

The intensity distribution S(x, y, z) = W(x, y, x, y, z) of the field reads as

\begin{array}{l} S (x, y, z) = \\ S_{0} \frac{w_{0}^{2}}{w^{2} (z)} \exp {- \frac{2}{w^{2} (z)} [x^{2} + y^{2} + {(R - z \sin β)}^{2}]} I_{0} [\frac{4 R}{w^{2} (z)} (1 + \frac{z}{R} \sin β) \sqrt{x^{2} + y^{2}}] . \end{array}

and the complex degree of coherence has the form

\begin{array}{l} μ (x_{1}, x_{2}, y_{1}, y_{2}, z) = \\ \frac{I_{0} [a (x_{1}, y_{1}, x_{2}, y_{2}, z)]}{\sqrt{I_{0} [a (x_{1}, y_{1}, x_{1}, y_{1}, z)] I_{0} [a (x_{2}, y_{2}, x_{2}, y_{2}, z)]}} \exp [\frac{i k_{0}}{2 R (z)} (x_{2}^{2} - x_{1}^{2} + y_{2}^{2} - y_{1}^{2})] . \end{array}

Clearly, w(z) and R(z) are the standard propagation parameters (beam width and radius of wavefront curvature) of the incident Gaussian beam at propagation distance z, which are determined by the initial beam width w₀. In addition, the parameters R and β related to the rotating wedge appear in the expressions of the CSD and the transverse intensity profile.

In the far zone z ≫ z_R we may approximate, as usual in the theory of Gaussian beams, w(z) → w₀z/z_R = 2z/k₀w₀ and R(z) → z. Then the CSD takes the asymptotic form

\begin{array}{l} W^{(\infty)} (x_{1}, y_{1}, x_{2}, y_{2}, z) = S_{0} {(\frac{z_{R}}{z})}^{2} \exp [- {(\frac{z_{R}}{z})}^{2} \frac{x_{1}^{2} + x_{2}^{2} + y_{1}^{2} + y_{2}^{2} + 2 {(R - z \sin β)}^{2}}{w_{0}^{2}}] \\ \times I_{0} [a^{(\infty)} (x_{1}, y_{1}, x_{2}, y_{2}, z)] \exp [\frac{i k_{0}}{2 z} (x_{2}^{2} - x_{1}^{2} + y_{2}^{2} - y_{1}^{2})] \end{array}

with

a^{(\infty)} (x_{1}, y_{1}, x_{2}, y_{2}, z) = \frac{2 R k_{0}}{z} {[{(\frac{z_{R}}{R} \sin β \bar{x} - i \frac{Δ x}{2})}^{2} + {(\frac{z_{R}}{R} \sin β \bar{y} - i \frac{Δ y}{2})}^{2}]}^{1 / 2} .

The intensity distribution in the far zone is

S^{(\infty)} (x, y, z) = S_{0} {(\frac{z_{R}}{z})}^{2} \exp [- 2 {(\frac{z_{R}}{z})}^{2} \frac{x^{2} + y^{2} + z^{2} \sin^{2} β}{w_{0}^{2}}] I_{0} [2 k_{0} \sin β \frac{z_{R}}{z} \sqrt{x^{2} + y^{2}}]

and the far-zone complex degree of coherence has the form

\begin{array}{l} μ^{(\infty)} (x_{1}, y_{1}, x_{2}, y_{2}, z) = \\ \frac{I_{0} [a^{(\infty)} (x_{1}, y_{1}, x_{2}, y_{2}, z)]}{\sqrt{I_{0} [a^{(\infty)} (x_{1}, y_{1}, x_{1}, y_{1}, z)] I_{0} [a^{(\infty)} (x_{2}, y_{2}, x_{2}, y_{2}, z)]}} \exp [\frac{i k_{0}}{2 z} (x_{2}^{2} - x_{1}^{2} + y_{2}^{2} - y_{1}^{2})] . \end{array}

These results are obtained also by inserting Eqs. (20) and (21) of [22], into the paraxial forms of the far-field formulas for the CSD presented, e.g., in Sect. 5.3 of [1].

4. Propagation of self-Fourier-transforming fields

Expressions (13)–(19) are valid for any combination of the parameters R and β. However, as demonstrated in [22], a particularly interesting class of fields is obtained if the condition

z_{R} \sin β = R

holds. This condition means that the backward-continuation of the local optical axis of the elementary Gaussian beam crosses the z axes at a distance z = −z_R ≈ −d/n, implying that

w_{0} \approx \sqrt{d λ_{0} / π n}

.

If condition (22) is satisfied, the field is self-Fourier-transforming, i.e., the functional forms of the CSD at the plane z = 0 and the angular correlation function are precisely the same [22]. At z = 0 we have, from Eq. (4),

a (x_{1}, y_{1}, x_{2}, y_{2}, 0) = \frac{4 R}{w_{0}^{2}} {[{(\bar{x} - i Δ x / 2)}^{2} + {(\bar{y} - i Δ y / 2)}^{2}]}^{1 / 2} .

In the far zone Eq. (19) reduces to

a^{(\infty)} (x_{1}, y_{1}, x_{2}, y_{2}, z) = \frac{2 R k_{0}}{z} {[{(\bar{x} - i Δ x / 2)}^{2} + {(\bar{y} - i Δ y / 2)}^{2}]}^{1 / 2} .

Let us next introduce normalized transverse coordinates x(z) = xz_R/z, y(z) = yz_R/z similarly similarly for $\bar{x} (z)$ , $\bar{y} (z)$ , Δx(z), and Δy(z). With these conventions we can write Eq. (19) as

a^{(\infty)} (x_{1}, y_{1}, x_{2}, y_{2}, z) = \frac{4 R}{w_{0}^{2}} {{[\bar{x} (z) - i Δ x (x) / 2]}^{2} + {[\bar{y} (z) - i Δ y (z) / 2]}^{2}}^{1 / 2} .

This result is identical in functional form with Eq. (4). Therefore the absolute value of the complex degree of spatial coherence in the far zone has the same shape as it has in the plane z = 0. In the scaled coordinates Eq. (20) reduces to

S^{(\infty)} (x, y, z) = S_{0} {(\frac{z_{R}}{z})}^{2} \exp {- \frac{2 [x^{2} (z) + y^{2} (z) + R^{2}]}{w_{0}^{2}}} I_{0} [\frac{4 R}{w_{0}^{2}} \sqrt{x^{2} (z) + y^{2} (z)}] .

Hence also the intensity distributions S(x, y, 0) and S^(∞) (x, y, z) are of the same form.

Let us next examine the implications of condition (22) at an arbitrary propagation distance. If this condition holds, Eq. (14) reduces to

a (x_{1}, y_{1}, x_{2}, y_{2}, z) = \frac{4 R}{w^{2} (z)} (1 + \frac{z}{z_{R}}) {[{(\bar{x} - i Δ x / 2)}^{2} + {(\bar{y} - i Δ y / 2)}^{2}]}^{1 / 2} .

It follows from Eq. (16) that the transverse intensity profile of the field takes the form

S (x, y, z) = S_{0} \exp {- \frac{2}{w^{2} (z)} [x^{2} + y^{2} + R^{2} {(1 - \frac{z}{z_{R}})}^{2}]} I_{0} [\frac{4 R}{w^{2} (z)} (1 + \frac{z}{z_{R}}) \sqrt{x^{2} + y^{2}}] .

The complex degree of spatial coherence has the same form as in Eq. (17), with a (x₁, y₁, x₂, y₂, z) given by Eq. (27). Hence its absolute value depend on the function a(x₁, y₁, x₂, y₂, z) alone. Let us next generalize the definition of the scaled transverse coordinates into the form

x (z) = x \frac{1 + z / z_{R}}{1 + {(z / z_{R})}^{2}}, y (z) = y \frac{1 + z / z_{R}}{1 + {(z / z_{R})}^{2}},

and analogously for

\bar{x} (z)

,

\bar{y} (z)

, Δx(z), and Δy(z). We note that, in the far zone, these definitions reduce to the previously defined scaled coordinates, and at z = 0 to the actual spatial coordinates at that plane. We then find that

a (x_{1}, y_{1}, x_{2}, x_{2}, z) = \frac{4 R}{w_{0}^{2}} {{[\bar{x} (z) - i Δ x (z) / 2]}^{2} + {[\bar{y} (z) - i Δ y (z) / 2]}^{2}}^{1 / 2}

and therefore |µ(x₁, y₁, x₂, y₂, z)| is a shape-invariant quantity at all propagation distances. The only propagation-induced change in µ(x₁, y₁, x₂, y₂, z) is the quadratic phase factor.

The condition for shape-invariant propagation does not extend to the transverse intensity distribution. Using the modified scaled coordinates introduce in Eq. (29) we can cast Eq. (28) in the form

\begin{array}{l} S (x, y, z) = S_{0} \frac{w_{0}^{2}}{w^{2} (z)} \exp (- \frac{2}{w_{0}^{2}} \frac{1 + {(z / z_{R})}^{2}}{{(1 + z / z_{R})}^{2}} {x^{2} (z) + y^{2} (z) + R^{2} {[\frac{1 - {(z / z_{R})}^{2}}{1 + {(z / z_{R})}^{2}}]}^{2}}) \\ \times I_{0} [\frac{4 R}{w_{0}^{2}} \sqrt{x^{2} (z) + y^{2} (z)}], \end{array}

which implies that the Bessel contribution to the intensity profile is shape-invariant but the Gaussian contribution is not. Figure 2 illustrates the scaling of the transverse spatial coordinates given by Eq. (29) in comparison with the scaling

x (z) = x {[1 + {(z / z_{R})}^{2}]}^{- 1}, y (z) = y {[1 + {(z / z_{R})}^{2}]}^{- 1}

of the coordinates of the incident Gaussian beam required to transform it (mathematically) into a propagation-invariant form. While the latter is a monotonously decreasing function of z, the former initially increases, reaching a peak at

z / z_{R} = \sqrt{2} - 1

, where

x (z) = x (1 / 2 + 1 / \sqrt{2}) \approx 1.21

.

Fig. 2 Evolution of scaled coordinates for the complex degree of coherence of self-Fourier-transforming Bessel-correlated fields (blue solid line) and the incident Gaussian beam (red dashed line).

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Propagation-invariant distributions of the absolute value of the complex degree of spatial coherence |µ(x₁, 0, x₂, 0, z)| are illustrated in Fig. 3. These distributions show strong variations of the antidiagonal width as a function of the average (diagonal) coordinate $\bar{x} (z)$ , indicating that the fields are not of the Schell-model form (the coherence width depends on $\bar{x}$ ). A ‘waist’ of the coherence width is seen at $\bar{x} (z) = 0$ , and the width of this waist narrows down as the normalized ratio r = R/w₀ increases.

Fig. 3 Absolute values of the complex degree of spatial coherence of self-Fourier-transforming Bessel-correlated fields for various values of the ratio r = R/w₀. (a) r = 0.5. (b) r = 0.75. (c) r = 1. (d) r = 2.

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Figure 4 shows intensity distributions self-Fourier-transforming fields in scaled coordinates. The dependence of the intensity profile at z = 0 on r = R/w₀ is illustrated in Fig. 4(a). When r exceeds ~ 0.75, a dip appears in the center of the intensity profile. This dip deepens with increasing r, and the beam takes on a doughnut shape as the Bessel contribution becomes dominant. The propagation of a beam with r = 0.5 is illustrated in Fig. 4(b). Since the Gaussian contribution still affects the intensity profile significantly, the shape varies with propagation distance. However, when r > 0.75, these changes become nearly indistinguishable (at least within the plotting accuracy of figures) and hence also the spatial profile becomes virtually shape-invariant.

Fig. 4 Intensity profiles of self-Fourier-transforming Bessel-correlated fields. (a) Profiles at z = 0 for various values of r = R/w₀. Black: r = 0.5. Red: r = 0.75. Green: r = 1. Blue: r = 2. (b) Profiles with r = 0.5 at distances z = 0 (black), z = z_R (red), z = 2z_R (green), z = 4z_R (blue), and z = 8z_R (purple). For clarity the profiles have been normalized to their maximum values.

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5. Propagation in paraxial systems

Wave propagation of the elementary field in a rotationally symmetric paraxial systems characterized by a 2 × 2 ABCD matrix is governed by the generalized Fresnel formula [24]

\begin{array}{l} e (x, y, L; ϕ) = \frac{k_{0}}{i 2 π B} \exp (i k_{0} L) \exp [\frac{i k_{0} D}{2 B} (x^{2} + y^{2})] \\ \times \int \int_{- \infty}^{\infty} e (x^{'}, y^{'}, 0; ϕ) \exp [\frac{i k_{0} A}{2 B} ({x^{'}}^{2} + {y^{'}}^{2})] \exp [- \frac{i k_{0}}{B} (x x^{'} + y y^{'})] d x^{'} d y^{'}, \end{array}

where L is the axial path length through the system. By inserting from Eq. (2), performing the integrations, and simplifying we obtain

\begin{array}{l} e (x, y, L; ϕ) = \frac{e_{0}}{A + i B / z_{R}} \exp (i k_{0} L) \exp (- \frac{i k_{0}}{2} \frac{B \sin^{2} β}{A + i B / z_{R}}) \exp (- \frac{R^{2}}{w_{0}^{2}} \frac{A}{A + i B / z_{R}}) \\ \times \exp (- i k_{0} R \sin β \frac{A}{A + i B / z_{R}}) \exp [\frac{i k_{0}}{2} \frac{C + i D / z_{R}}{A + i B / z_{R}} (x^{2} + y^{2})] \\ \times \exp [i k_{0} \frac{\sin β - i R / z_{R}}{A + i B / z_{R}} (x \cos ϕ + y \sin ϕ)], \end{array}

where the ABCD matrix determinant relation AD − BC = 1 was used.

It is useful, at this stage, to introduce the propagation parameters of Gaussian beams in rotationally symmetric ABCD systems:

w_{L} = w_{0} {[A^{2} + {(B / z_{R})}^{2}]}^{1 / 2}

and

R_{L} = \frac{A^{2} + {(B / z_{R})}^{2}}{A C + B D / z_{R}^{2}} .

These expressions give the beam width and radius of curvature at the output plane of an ABCD system for axial Gaussian beams if a beam waist is located at the input plane of the system. They apply to Gaussian Schell-model beams as well, with an appropriate extension of the definition of the Rayleigh range [25]. They are also extensions of the free-space propagation parameters w (z) and R(z) in Eqs. (12) and (15).

If we insert from Eq. (34) into Eq. (2) and make use of the propagation paramters just introduced, we can cast the CSD at the output plane of the ABCD system in the form

\begin{array}{l} W (x_{1}, y_{1}, x_{2}, y_{2}, z) = S_{0} \frac{w_{0}^{2}}{w_{L}^{2}} \exp [- \frac{x_{1}^{2} + x_{2}^{2} + y_{1}^{2} + y_{2}^{2} + 2 {(A R - B \sin β)}^{2}}{w_{L}^{2}}] \\ \times I_{0} [a (x_{1}, y_{1}, x_{2}, y_{2}, L)] \exp [\frac{i k_{0}}{2 R_{L}} (x_{1}^{2} - x_{2}^{2} + y_{1}^{2} - y_{2}^{2})], \end{array}

where

a (x_{1}, y_{1}, x_{2}, y_{2}, L) = {[{(c_{L} \bar{x} - i d_{L} Δ x / 2)}^{2} + {(c_{L} \bar{y} - i d_{L} Δ y / 2)}^{2}]}^{1 / 2},

c_{L} = \frac{4 R}{w_{L}^{2}} (1 + \frac{B}{R} \sin β),

d_{L} = \frac{4 R}{w_{L}^{2}} (\frac{B}{z_{R}} + \frac{A z_{R}}{R} \sin β) .

These expressions reduce to the free-space propagation formulas derived in Sect. 3 in the special case A = D = 1, B = L = z, and C = 0.

6. Experimental verification

In this section we demonstrate two examples of Bessel-correlated fields: in the first example the condition (22) is approximately satisfied, while in the second example it is clearly violated. In both cases we measure both the degree of spatial coherence and the intensity profile across several planes and compare the results with theoretical simulations.

The experimental setup is presented schematically in Fig. 5. The light source is a single-mode HeNe laser operating at 633 nm. A beam forming system BE consisting of a zoom camera lens and an achromatic fixed focal length (8.57 mm) lens was aligned in such a way that it reduces the size of the beam waist according to our requirements. The system BF is closely afocal, and the demagnification could be varied from ~ 2 − 10.5 by changing the focal length of the zoom lens.

Fig. 5 Experimental setup: A is a single-mode HeNe laser source, BF a beam forming system, W the rotating wedge, L an imaging lens, DMD a digital micro-mirror device, and P is a plane at a (variable) distance z from the exit plane of the wedge.

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A CCD sensor was used to find the axial position of the beam waist and to the measure its size w₀. The wedge W was then placed in the beam path so that the waist position formed by BE coincides with its exit plane. A positive lens L with 40 mm focal length was used to produce a magnified image of the plane P of interest to the DMD plane, with a magnification factor of ~5. The detailed working principle of the DMD can be found in [26]. The wedge was rotated around the optical axis with a reasonable angular speed compared to the integration time of the recording device. Figure. 6 shows the simulated and measured two-point coherence functions, respectively, for the case when the condition (22) is nearly satisfied, and Fig. 7 shows cross-sectional beam intensity profiles at several distances.

Fig. 6 Simulated (top row) and measured (bottom row) absolute value of the complex degree of spatial coherence in the (x₁, x₂) co-ordinate system when Eq. (22) is nearly satisfied: (a,e) z = 0 mm. (b,f) z = 2 mm. (c,g) z = 4 mm. (d,h) z = 6 mm. w₀ ≈ 40 µm.

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Fig. 7 Intensity cross-sections at y = 0 µm when (22) is nearly satisfied. Blue: z = 0 mm. Black: z = 2 mm. Red: z = 4 mm. Pink: z = 6 mm. Green: z = 8 mm. Maroon: z = 10 mm. w₀ ≈ 40 µm.

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We used a 6.5 mm thick glass wedge with a prism angle of 1°, which deviates the beam by ~ 0.5° from its path. With these parameters the desired beam waist size, according to Eq. (22), is w₀ ~ 29 µm (R ≈ 39 µm). However, in the experimental setup w₀ ≈ 40 µm was the smallest attainable waist size. Hence condition (22) could not be satisfied exactly. However, in view of the simulations in Fig. 6, carried out with the actual experimental parameters, the absolute value of the complex degree of spatial coherence is nearly shape-invariant.

The two-point degree of spatial coherence was measured at four planes, including the source plane z = 0. With the chosen parameters z_R ≈ 4 mm, and the measurements extended to a distance of z = 1.5z_R = 6 mm from the source plane. Intensity measurements were made up to a distance z = 10 mm. Another set of experiments is illustrated in Figs. 8 and 9. Here we used a wedge thickness of 3.5 mm (with the same prism angle as above) and chose w₀ = 100 µm, which gives z_R = 49 mm. With these parameters the condition (22) is violated strongly and, as we move away from the source plane by a distance that is only a small fraction of the Rayleigh range, the spatial coherence and intensity profiles already change drastically in both simulations and experiments. That clearly confirms that close satisfaction of (22) is a necessary condition for obtaining an essentially shape-invariant spatial coherence function.

Fig. 8 Same as Fig. 6, but under experimental conditions where Eq. (22) is not satisfied: (a) z = 0 mm. (b) z = 2 mm. (c) z = 4 mm. (d) z = 6 mm. w₀ ≈ 100 µm.

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Fig. 9 Intensity cross-sections when (22) is not satisfied. Blue: z = 0 mm. Black: z = 4 mm. Red: z = 8 mm. Pink: z = 10 mm. w₀ ≈ 100 µm.

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The agreement between the simulated and experimental results is quite reasonable (though not perfect). In particular, several side lobes of the spatial coherence profile |γ(x₁, x₂)| (in the antidiagonal direction) at short propagation distances are seen in Fig. 8 (also experimentally.) The departure of |γ(x₁, x₂)| from the Schell-model form at larger propagation distances is likewise observed. The measurements were disturbed somewhat by interference effects due to back-reflections from the glass plate protecting the CCD sensor, which could not be completely eliminated. As we can see some asymmetry in case of Fig 9, was due to the rotational speed of the wedge glass plate was not very high, so while capturing the image for a particular plane of interest the intensity was not necessarily distributed uniformly through the whole ring shape. Additionally, reflections between the CCD surface and the protective glass created independent ring like interference pattern, which contributes to the asymmetry. This type of effects would be greatly reduced if a source with more limited temporal coherence were used for the experiments, such as a superluminescent diode. Small deviations between simulations and experiments also undoubtedly arise from difficulties in measuring the beam width w₀ at a high precision, and placing the beam waist exactly at the exit plane of the wedge.

7. Conclusions

In conclusion, we have developed an analytical model for propagation of a novel class of spatially partially coherent, Bessel-correlated, fields in free space and paraxial optical systems. Here we considered only monochromatic fields, but the analysis can be extended to polychromatic stationary field and to pulsed fields. We have also considered only a generation scheme based on rotating wedges. Other types of rotating elements, in particular custom-designed diffractive elements, should provide considerable extra freedom in customizing the spatial coherence and may lead to a wider class of experimentally realizable fields with shape-invariant spatial coherence properties.

Funding

Academy of Finland (285880); Strategic funding of the University of Eastern Finland (930350).

References and links

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Paraxial propagation of a class of Bessel-correlated fields

Abstract

1. Introduction

2. Source model

3. Propagation in free space

4. Propagation of self-Fourier-transforming fields

5. Propagation in paraxial systems

6. Experimental verification

7. Conclusions

Funding

References and links

Cited By

Figures (9)

Equations (40)

Optics Express