Lensing effects in a random inhomogeneous medium

Austin McDaniel; Alex Mahalov

doi:10.1364/OE.25.028157

1. Introduction

The dynamic and multiscale variations in the atmospheric medium significantly affect many technologies that rely on electromagnetic wave propagation through various regions of the atmosphere. In the troposphere, tropopause, and lower stratosphere, turbulence mixes layers of different temperatures and densities [1], can be anisotropic (see, e.g., [2,3]), and is characterized by an outer scale of 100 m to several kilometers and an inner scale of 1 cm [4] or smaller [5]. Also in these regions, very strong temperature gradients can occur within “sheets” with thicknesses on the order of meters [6]. In the ionosphere, plasma instabilities result in turbulent structures of scales that range from larger than 10⁵ m to smaller than 0.1 m [7,8], and ionospheric turbulence affects, e.g., Global Positioning System (GPS) microwave signals and spaceborne synthetic aperture radar [9]. Larger-scale mean variations in atmospheric properties can often be resolved; for example, the decreases in pressure, temperature, and humidity over altitude increases on the order of kilometers cause the average radiowave refractivity to decrease exponentially with altitude in the troposphere [10,11]. However, small-scale variations due to atmospheric turbulence cannot be accounted for by deterministic models. Moreover, clouds, fog, and dust often give rise to an effectively random medium that can cause reduced resolution in imaging or a loss of bandwidth in optical communication [12]. Therefore, stochastic models are often used to statistically describe these features and study their effects on propagation.

Variations in the atmospheric medium, and therefore in the refractive index, can in certain situations lead to focusing of light rays or a propagating beam. Ripples in an atmospheric layer caused by local meteorological conditions, or a cold front, consisting of a layer of dense air, may focus a beam in the same way as a simple optical lens [13,14]. Furthermore, a local maximum in the vertical refractive index profile can act as a gradient-index lens that focuses a beam propagating around the height of this maximum; conversely, a local minimum spreads a beam (Fig. 1). Such a local maximum can arise, for example, near a weather front, where it results from the slope of the front and the presence of smaller vertical refractive index gradients on the cold side than on the warm side [15]. More generally, focusing or spreading can result from not only local extrema in the refractive index profile, but also any segment of the profile that has nonzero curvature.

Fig. 1 Lensing effect: An illustration of ray paths in two different layered mediums. Darker shades of gray represent larger values of the refractive index. In (a), a local maximum (in the x-direction) of the refractive index gives rise to a focal point at which rays converge. In (b), a local minimum of the refractive index creates a negative lens, causing rays to diverge from each other. For the case (a) of a local maximum of the mean refractive index, we show that the focal length associated with propagation in the random medium is smaller than in the deterministic setting.

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Such focusing, which is sometimes referred to as atmospheric lensing, can have detrimental effects on imaging applications. The curvature of a section of the average vertical refractive index profile can cause an apparant stretching or compression of features in photographic images taken at a long range. For example, a building can appear taller in an image than it actually is, an effect that is most pronounced when the camera is at the height of the curved part of the profile [16,17]. The angle formed at the camera plane between two rays originating from different heights along the building is larger when these rays are focused by the medium than when the medium is homogeneous. Therefore, in the presence of a focusing medium, the distance between, e.g., two levels of the building in the image is greater than the distance corresponding to the actual height difference between the levels. Another effect that is believed to be the result of atmospheric lensing is a certain type of degradation of synthetic aperture radar (SAR) images referred to as the “moving-hill phenomenon” (see [9,18] for the mathematical foundations of SAR). This phenomenon is characterized by light and dark bands that appear in a SAR image, but that do not repeat in the same location from image to image. The proposed explanation for this is that an atmospheric feature creates a focal spot that moves across the scene being imaged as the SAR platform (airplane) moves. The resulting bright and dark bands in the image give the artificial appearance of hills in the terrain [13,19–21].

In this paper, we consider the problem of describing the lensing due to a curved segment of the mean refractive index profile in the presence of stochastic fluctuations in the medium. In free space, each component of time-harmonic electric and magnetic fields satisfies the scalar Helmholtz equation. This equation is often a good model and used for describing wave propagation in situations where the medium is inhomogeneous (the exact description in this case is, of course, given by Maxwell’s equations, see, e.g., [22]). When the refractive index n varies in space, the scalar Helmholtz equation is

Δ u + k_{0}^{2} n^{2} (r) u = 0

where k₀ is the wavenumber. It is common and useful to look for solutions of the form

u (r) = 𝒜 (r) e^{i k_{0} ψ (r)}

where the amplitude (or envelope) 𝒜 and phase Ψ = k₀ψ are real-valued. In many propagation scenarios of interest, the wavenumber is large compared to the other characteristic scales in the problem, and so it is useful to match terms of the same order in k₀ in the equation obtained after substituting (2) into (1). The leading

O (k_{0}^{2})

equation involves only the function ψ, called the eikonal, and not the amplitude 𝒜, and is therefore called the eikonal equation:

{| \nabla ψ |}^{2} = n^{2}

Finding the field u by first solving (3) and then solving the next order equation, which depends on ψ, for the amplitude 𝒜 is called the geometrical optics approximation. The characteristics of the first-order equation (3) are called rays, as they correspond to the notion of light rays. From (3), we see that a ray is orthogonal to the level surfaces of the solution of the eikonal equation, i.e., under the geometrical optics approximation a ray is perpendicular to the wavefronts of the solution of the Helmholtz equation. Letting s denote arclength, a ray r(s) = (x(s), y(s), z(s)) satisfies the ray equation

\frac{d}{d s} [n (r) \frac{d r}{d s}] = \nabla n (r)

Let ψ_g.o. denote the solution of (3) so that k₀ψ_g.o. is the geometrical optics approximation of the phase of the solution of the Helmholtz equation. The difference in ψ_g.o. between two points r_A and r_B can be found by integrating the refractive index along the connecting ray (see, e.g., [23]):

ψ_{g . o .} (r_{B}) = ψ_{g . o .} (r_{A}) + \int_{s_{A}}^{s_{B}} n (r (s)) d s

Here r is the ray connecting r_A and r_B, s is arclength along this ray, r(s_A) = r_A, and r(s_B) = r_B.

The main results of this paper are as follows. First, we derive formulas for the mean ray paths in the presence of both curvature and stochastic fluctuations in the refractive index profile. For the case of a local maximum of the mean refractive index, we then derive an expression for the focal length of the mean trajectories, which is seen to be smaller than in the deterministic setting. We begin by describing the refractive index model that we will use in (4).

2. Refractive index model

In order to describe a ray’s trajectory in any specific sense, one needs a model for the refractive index n of the medium. In models of the refractive index in the troposphere, the refractive index is commonly taken to depend only on altitude [10, 11]. We therefore consider a layered medium, where the refractive index depends only on a single spatial variable. Most models for the troposphere take the refractive index to monotonically decrease with altitude, and at low heights, e.g. less than a kilometer from the ground, a linear approximation is often used [10,11]. By its nature, such a linear model cannot account for local curvature or extrema in the refractive index profile and therefore cannot describe or predict atmospheric lensing. When measurements are used to generate plots of the refractive index in the troposphere as a function of altitude on a 1 km scale, the profile may exhibit multiple local minima and maxima [13,24]. In addition, there is evidence that in certain situations the curvature of some part of the refractive index profile is a robust feature that can be predicted and accounted for. In [16,17], the value of the curvature of the vertical refractive index profile within about ten meters of the ground was deduced from measurements of the resulting apparant stretching of a building in time-lapse photographic images. While the profile changed significantly over the course of a day, the largest stretch for each day typically occurred at the same time of day, and the corresponding value of the curvature parameter was consistent over a five day period.

We focus on describing propagation near such a curved segment or extremum of the mean refractive index profile. Near an extremum, the mean refractive index function can be Taylor-expanded locally about the critical point. In this expansion, the linear term is equal to zero, and the quadratic term is sufficient to capture the lensing effect. Furthermore, a quadratic term in the refractive index model is sufficient to describe the ray paths locally near a curved segment of the refractive index profile. The coefficient of the quadratic term can be tuned as a parameter to match the curvature, while the ray’s initial position in (4) can be used as another parameter to match the first derivative of the actual profile near the ray. Therefore, we consider a refractive index model that consists of a reference value, a quadratic term, and a stochastic term that describes random fluctuations (a linear term results in the mirage phenomenon, and has been considered previously within a random medium [25]).

In the troposphere the refractive index, which is a nondimensional quantity, differs from a value of one by a very small amount. Because of this, it is common to express changes in the refractive index in terms of the refractivity N = 10⁶(n − 1). As an example of the size of refractive index variations in the troposphere, a representative average vertical gradient of the radiowave refractivity at midlatitudes in the first kilometer is −40 km⁻¹ [11]. Therefore, we take the variations in the refractive index to be small in magnitude compared to a reference value n₀, and model the refractive index n by

n (x) = n_{0} [1 + δ β x^{2} + ∊ σ η (\frac{x}{L_{c}})]

where η is a mean-zero stochastic process. Here, δ and ∊ are dimensionless parameters that control the order of magnitude of the curvature of the mean profile and that of the stochastic fluctuations, respectively. We assume 0 ≤ ∊, δ ≪ 1, and we will discuss below the values of these parameters in the atmosphere in certain situations. We use the dimensional parameter β and the dimensionless parameter σ ≥ 0 to control more precisely the sizes of these terms. The parameter L_c represents the correlation length of the refractive index, so that the dimensionless process η has a correlation length of one. A negative value of β corresponds to a segment of the mean refractive index profile with negative curvature, e.g. a local maximum, which focuses a propagating beam. Conversely, β > 0 corresponds to a segment with positive curvature that gives rise to negative lensing (defocusing). We assume that η is a stationary process with variance equal to one, so that its autocovariance function K_η(x̃, x̃′) is of the form

E [η (\tilde{x}) η ({\tilde{x}}^{'})] = K_{η} (\tilde{x}, ˜^{x'}) = R (| \tilde{x}, {\tilde{x}}^{'} |)

where R(0) = 1 and |R(d)| ≤ 1 for all d > 0. We assume that R is twice differentiable, and we make the physically realistic assumption R′(0) = 0. Note that the variance of the refractive index is given by

∊^{2} σ^{2} n_{0}^{2}

.

Because, in (6), δ and ∊ are small compared to the reference value of one, which reflects the small magnitude of the variations of the refractive index in, e.g., the troposphere, it is natural to use perturbation methods to describe a ray’s trajectory in this medium. Such a method requires specifying the sizes of the small dimensionless parameters δ and ∊ in relation to each other. We therefore turn now to a discussion of the sizes of these and the other parameters in (6) in the atmosphere. As previously mentioned, in [16, 17] the curvature of the refractive index profile within about ten meters of the ground was deduced from measurements of the stretching of a building in photographic images. This curved portion of the refractive index profile was modeled with a quadratic term whose coefficient was calculated for the profile corresponding to the largest measured stretch during a day. This calculated value, which was about the same each day over a five day period, corresponds to a value of around δβ ≈ −1.5 × 10⁻⁹ m⁻² in (6). This amount of curvature of the profile resulted in an apparant stretch of about 5 m of a 33 m tall building in images taken at a distance of about 15 km. Moving on to the stochastic term in (6), the correlation length L_c of the refractive index is usually identified with the outer scale L₀ of optical turbulence.

Within the atmospheric boundary layer, the outer scale is commonly taken as L₀ = 0.4h , where h is height above the ground. This is in contrast to the free atmosphere, where the outer scale is about 100 m in the vertical direction and a few kilometers in the horizontal direction [4]. The magnitude of the refractive index fluctuations due to turbulence is usually given in terms of the structure constant $C_{n}^{2}$ . This is a highly variable quantity, where values on the order of 10⁻¹³ m^−2/3 might be representative of relatively strong turbulence. Values on this order were measured, for example, close to the ground during a typical winter day in Central Florida. Averaged over a horizontal path about 150 m long at a height of 1.5 m above the ground, early afternoon values of $C_{n}^{2}$ for optical frequencies were found to be around 5 × 10⁻¹³ m^−2/3 [26]. Radiosonde data suggest that $C_{n}^{2}$ can be close to 10⁻¹⁰ m^−2/3 in turbulent layers located at the top boundaries of clouds [27]. At locations close to the ground at which the outer scale of turbulence is on the order of meters, the order of magnitude of the variance of the refractive index can be taken to be the order of magnitude of the structure constant $C_{n}^{2}$ as measured in units of m^−2/3.

To illustrate possible values in the atmosphere of the parameters of the refractive index model (6), we take as an example the aforementioned curvature value δβ ≈ −1.5 × 10⁻⁹ m⁻² from [16,17] and structure constant value $C_{n}^{2} = 5 \times 10^{- 13} m^{- 2 / 3}$ from [26]. Both of these values were obtained at locations close to the ground and so, in particular, we take the variance of the refractive index to be equal to this value of $C_{n}^{2}$ . Then taking δ = 10⁻⁹ and ∊ = 10⁻⁶, it follows that β is an order one parameter with dimension m⁻², L_c is an order one parameter with a dimension of meters, and σ is a dimensionless order one parameter. Thus we see from this example that two perturbation regimes that are relevant and important for describing propagation in the atmosphere are those given by taking δ = ∊² and δ = ∊ in (6). We therefore turn now to the problem of describing the ray trajectories in each of these regimes.

3. Mean ray path and focal length

In order to determine a specific ray’s trajectory using (4), the ray’s initial position and direction must be specified. As previously noted, the ray’s initial x-coordinate, which we denote by b, can be used as a parameter to control the first derivative of the refractive index profile (6) near the ray. Therefore, together with (4), we consider the following initial conditions:

r (0) = b e_{1}, \frac{d r}{d s} (0) = u

where e₁ = (1, 0, 0) is the unit vector in the positive x-direction. Note that ‖u‖ = 1 since s is arclength. We let θ denote the angle between u and the yz-plane. We restrict our attention to the case of rays that are normally-incident with respect to the refractive index variations, that is, rays for which θ = 0. This corresponds to, for example, horizontally-directed rays propagating through a region in the troposphere modeled by a vertical refractive index profile. In the calculations that follow, however, it is convenient to initially treat the angle θ as an arbitrary parameter, and in doing so we use the convention that θ is positive when u · e₁ > 0.

We first consider the case δ = ∊² in (6). We derive the mean ray path to order ∊² by following the method of Keller [28]. By viewing r as a function of both s and ∊, we Taylor-expand r in the variable ∊ about ∊ = 0. In this way, we can express r(s) as the following expansion in powers of ∊:

r (s) = r_{0} (s) + ∊ r_{1} (s) + ∊^{2} r_{2} (s) + \dots

where

r_{m} (s) = \frac{1}{m!} {\frac{\partial^{m} r}{\partial ∊^{m}} (s, ∊) |}_{∊ = 0}

We substitute (9) into (4), with the refractive index n defined by (6). We equate terms of the same order in ∊, thereby obtaining a sequence of equations to be solved recursively for r₀, r₁, r₂, ... By solving the zeroth order in ∊ equation subject to the initial conditions (8) we get r₀, which is simply the deterministic, straight-line trajectory of a ray traveling in a medium with constant refractive index:

r_{0} (s) = (x_{0} (s), y_{0} (s), z_{0} (s)) = s u + b e_{1}

Next, we solve in succession the first order in ∊ equation for r₁ in terms of r₀ and the second order in ∊ equation for r₂ in terms of r₁ and r₀. We then calculate the expectations of r₁(s) and r₂(s) in order to find the expectation of r(s) to second order in ∊. First, we find

r_{1} (s) = (e_{1} - sin (θ) u) \int_{0}^{s} (s - t) \frac{σ}{L_{c}} η^{'} (\frac{x_{0} (t)}{L_{c}}) d t

Taking the expectation and interchanging the order of the expectation and the derivative, we get E[r₁(s)] = 0 since

\frac{d}{d \tilde{x}} E [η] = 0

. Both the quadratic and the stochastic terms in the refractive index give rise to second order in ∊ contributions to the mean trajectory that appear in the formula for the expectation of r₂. The contribution to E[r₂(s)] due to the stochastic term involves the quantity

E [\frac{σ}{L_{c}} η^{'} (\frac{x_{0} (t_{2})}{L_{c}}) \frac{σ}{L_{c}} η^{'} (\frac{x_{0} (t_{1})}{L_{c}})]

for t₁ ≤ t₂ ≤ s, as well as similar expressions. To calculate this quantity, we define the auxiliary functions ζ(ṽ, w̃) = η(ṽ) and ξ(ṽ, w̃) = η(w̃). Then the expression (11) can be written as

\begin{array}{l} \frac{σ^{2}}{L_{c}^{2}} E [\frac{\partial}{\partial \tilde{w}} \frac{\partial}{\partial \tilde{v}} (ξ ζ) (\frac{x_{0} (t_{1})}{L_{c}}, \frac{x_{0} (t_{2})}{L_{c}})] & = \frac{σ^{2}}{L_{c}^{2}} \frac{\partial}{\partial \tilde{w}} \frac{\partial}{\partial \tilde{v}} E [ξ ζ] (\frac{x_{0} (t_{1})}{L_{c}}, \frac{x_{0} (t_{2})}{L_{c}}) \\ = \frac{σ^{2}}{L_{c}^{2}} {[\frac{\partial}{\partial \tilde{w}} \frac{\partial}{\partial \tilde{v}} R (| \tilde{w} - \tilde{v} |)]}_{\tilde{w} = x_{0} (t_{2}) / L_{c}, \tilde{v} = x_{0} (t_{1}) / L_{c}} \\ = - \frac{σ^{2}}{L_{c}^{2}} R^{″} (\frac{| x_{0} (t_{2}) - x_{0} (t_{1}) |}{L_{c}}) \end{array}

For an arbitrary initial angle θ, the expression for the expectation of r₂(s) is lengthy and, in particular, involves an integral of the autocorrelation function R. However, for rays that are normally-incident with respect to the refractive index variations, the expression simplifies considerably. Using (10) and (12) we find that for θ = 0, to second order in ∊, the expected position of the endpoint of a ray of length s is given by

(θ = 0) : E [r (s)] = b e_{1} + s u + ∊^{2} b β s^{2} e_{1} + \frac{1}{6} ∊^{2} σ^{2} L_{c}^{- 2} R^{″} (0) s^{3} u

On the right-hand side of (13), the first two terms consist of the zeroth order in ∊ straight-line trajectory r₀. The third term describes the lensing effect that results from the quadratic term in the refractive index. For β < 0, this term causes a ray to bend in the x-direction toward the plane x = 0, by an amount that depends on the ray’s initial distance |b| from this plane. For β > 0, rays are bent away from the plane x = 0, resulting in negative lensing. The last term describes, to second order in ∊, the effect of the stochastic fluctuations on the mean ray path. Since R″(0) ≤ 0, this term reduces the component of E [r(s)] in the initial direction u. Note that the only information about the stochastic fluctuations that enters into (13) is the second derivative at zero of the covariance function of the refractive index (divided by

n_{0}^{2}

).

To estimate the magnitude of the correction to the mean ray path that is due to the stochastic fluctuations, it is useful to relate this term to the standard parameters of optical turbulence. Let B_n denote the covariance function of the refractive index, which is related to the covariance function R of the process η by

R (d) = {(n_{0} ∊ σ)}^{- 2} B_{n} (L_{c} d)

Here we set the correlation length of the refractive index equal to the outer scale of turbulence: L_c = L₀. We use the usual expression for the variance of the refractive index in terms of the structure constant and outer scale, setting this variance

∊^{2} σ^{2} n_{0}^{2}

as given by (6) equal to

\frac{1}{2} C_{n}^{2} L_{0}^{2 / 3}

. We also assume as usual that the correlation (normalized covariance) function of the refractive index does not depend on the structure constant

C_{n}^{2}

. Finally, we assume that B_n is quadratic locally near zero, and that the coefficient of the quadratic term depends on the scale l₀ of the smallest random inhomogeneities. These assumptions give the following expression for B_n near zero, where the power of l₀ is determined from dimensional analysis:

B_{n} (d) = \frac{1}{2} C_{n}^{2} (L_{0}^{2 / 3} - l_{0}^{- 4 / 3} d^{2}), 0 \leq d ≪ l_{0}

Note that the above expression is consistent with our previous assumption that R′(0) = 0. From (14) and (15) we have

R^{″} (0) = - 2 l_{0}^{- 4 / 3} L_{0}^{4 / 3}

. The magnitude of the last term in (13) can then be expressed as

- \frac{1}{6 n_{0}^{2}} C_{n}^{2} l_{0}^{- 4 / 3} s^{3}

where the negative sign indicates that the stochastic fluctuations reduce the component of the mean ray path in the initial propagation direction. The smaller l₀ is, the larger this reduction, since rapid medium fluctuations cause the direction of a given realization of the ray trajectory to vary quickly. Note in particular that the expression (16) does not depend on the outer scale L₀, and that its magnitude increases with propagation distance. As an example of the size of this correction, for the value

C_{n}^{2} = 5 \times 10^{- 13} m^{- 2 / 3}

from the scenario in [26], a value of l₀ = 1 m, and a propagation distance on the order of tens of kilometers (along with n₀ ≈ 1 in the atmosphere), (16) is on the order of meters.

We now use the mean ray path (13) to further describe the lensing that results from a local maximum of the mean refractive index. In the following we assume β < 0 in (6), so that the mean profile corresponds to a gradient-index lensing layer. This refractive index profile gives rise to a focal point r̄_f of the mean paths of normally-incident rays having the same initial direction u and different initial x-coordinates b. Let L̄_f denote the focal length associated with these mean paths, so that r̄_f = L̄_fu. From (13) we find

{\bar{L}}_{f} = \frac{1}{∊ \sqrt{| β |}} (1 + \frac{1}{6 | β |} σ^{2} L_{c}^{- 2} R^{″} (0))

We note that this formula is valid for

σ^{2} L_{c}^{- 2} | R^{″} (0) | ≪ 6 | β |

, since (17) gives L̄_f = 0 when

σ^{2} L_{c}^{- 2} | R^{″} (0) | = 6 | β |

. Since R″(0) ≤ 0, the focal length of the mean paths is smaller than that of the ray paths in the medium with no stochastic fluctuations. Setting σ = 0 in (6) corresponds to a deterministic medium for which the focal length associated with the local maximum of the refractive index is

{(∊ \sqrt{| β |})}^{- 1}

. When stochastic fluctuations as described in (6) and (7) are introduced into this medium, the focal length of the mean trajectories is reduced by the amount

{(6 ∊)}^{- 1} {| β |}^{- 3 / 2} σ^{2} L_{c}^{- 2} | R^{″} (0) |

.

Next we consider the phase difference, at the focal point, associated with the mean trajectories corresponding to the two different initial x-coordinates x = 0 and x = b. In (5), in place of the ray r we substitute the mean ray path r̄ ≡ E[r] as given by (13), and we use the mean refractive index n̄(x) ≡ E[n(x)] = n₀(1 + ∊²βx²). It is important here to note that while we have defined s to be arclength along the exact solution of (4) for a given realization of the medium (i.e., refractive index), the parameter s does not correspond to arclength along the approximate mean trajectory r̄ given by (13). We therefore introduce the parameter τ to denote arclength along the trajectory r̄. We then define

\tilde{Ψ} ({\bar{r}}_{B}) = Ψ (r_{0}) + k_{0} \int_{0}^{τ_{B}} \bar{n} (\bar{r} (τ)) d τ

where r̄_B = r̄(τ_B), r₀ = r(0) = r̄(0), and Ψ(r₀) is assumed to be known. In order to use (13) to evaluate the integral in (18), one needs the relationship between the two parameterizations. This is given by, with an error of order ∊⁴ at the focal point r̄_f,

\frac{d τ}{d s} (s) = 1 + \frac{1}{2} ∊^{2} σ^{2} L_{c}^{- 2} R^{″} (0) s^{2} + \frac{2 ∊^{4} b^{2} β^{2} s^{2}}{1 + \frac{1}{2} ∊^{2} σ^{2} L_{c}^{- 2} R^{″} (0) s^{2}}

This expression is valid for

σ^{2} L_{c}^{- 2} | R^{″} (0) | ≪ 2 ∊^{- 2} s^{- 2}

, so in order to evaluate (18) at r̄_f we assume

σ^{2} L_{c}^{- 2} | R^{″} (0) | ≪ 2 | β |

. Let Ψ̃₀(r̄_f and Ψ̃_b(r̄_f) denote Ψ̃(r̄_f) as defined in (18) with r₀ equal to the origin 0 and be₁, respectively. We calculate the difference Ψ̃_b(r̄_f) − Ψ̃₀(r̄_f) assuming Ψ(0) = Ψ(be₁) (this is the case, for example, for a plane wave that is normally-incident with respect to the refractive index variations). Using (13) and (19) in (18), we find

\begin{array}{l} {\tilde{Ψ}}_{b} ({\bar{r}}_{f}) - {\tilde{Ψ}}_{0} ({\bar{r}}_{f}) = & - \frac{8}{15} ∊ b^{2} {| β |}^{1 / 2} n_{0} k_{0} - \frac{4}{105} ∊ b^{2} {| β |}^{- 1 / 2} L_{c}^{- 2} R^{″} (0) n_{0} k_{0} \\ \begin{array}{l} + \frac{4 ∊ b^{2} β^{2} L_{c}^{3} n_{0} k_{0}}{σ^{3} {| R^{″} (0) |}^{3 / 2}} [- {| β |}^{- 1 / 2} σ L_{c}^{- 1} \sqrt{| R^{″} (0) |} \\ + \frac{\sqrt{2}}{2} ln (\frac{\sqrt{2} + {| β |}^{- 1 / 2} σ L_{c}^{- 1} \sqrt{| R^{″} (0) |}}{\sqrt{2} - {| β |}^{- 1 / 2} σ L_{c}^{- 1} \sqrt{| R^{″} (0) |}})] \end{array} \end{array}

All of the calculations of this section so far have been for the regime given by δ = ∊² in (6). Now we briefly consider the different case δ = ∊. We again consider rays that are normally-incident with respect to the refractive index variations, i.e., rays with an initial angle θ = 0. By a similar calculation as before, we find that in this regime the mean ray path to second order in ∊ is given by

\begin{array}{l} (θ = 0) : E [r (s)] = b e_{1} + s u + ∊ b β s^{2} e_{1} + ∊^{2} β^{2} b s^{2} (\frac{1}{6} s^{2} - b^{2}) e_{1} \\ - \frac{2}{3} ∊^{2} β^{2} b^{2} s^{3} u + \frac{1}{6} ∊^{2} σ^{2} L_{c}^{- 2} R^{″} (0) s^{3} u \end{array}

The first two terms and last term are the same as in (13). The third term differs from that in (13) by a factor of ∊, and the remaining terms are higher-order corrections due to the quadratic term in the refractive index. We note that it is immediately seen from (21) that, as before, the focal length of the mean trajectories is smaller than that of the ray paths in the medium with no stochastic fluctuations.

4. Conclusion

We have considered the problem of describing propagation in the presence of both curvature and stochastic fluctuations in the refractive index profile. Often the troposphere is modeled as a layered medium in which the refractive index decreases with altitude. Linear models used at low altitudes cannot, however, account for curvature or local extrema of the refractive index profile, both of which can cause atmospheric lensing. The model (6) can describe locally such sections of the refractive index profile, while at the same time taking into account stochastic fluctuations due to atmospheric turbulence. The formulas (13) and (21) that we derived for the mean ray paths describe propagation within these regions.

The formulas (13) and (21) give the mean ray paths in two different regimes that are differentiated by the size of the effects of the curvature of the mean refractive index profile compared to those of the stochastic fluctuations. Values of local curvature and turbulence parameters that have been obtained from measurements show that both of these regimes are important for characterizing propagation in the atmosphere. Both formulas describe the bending of rays due to the curvature of the profile, as well as the reduction of the component of the mean trajectories in the initial direction that is due to the stochastic fluctuations. Rays with different initial x-coordinates are bent by different amounts, which gives rise to the focusing of rays for β < 0 in (6) and negative lensing (defocusing) for β > 0. The formulas (13) and (21) describe in a robust way the effects on the mean ray path of stochastic fluctuations in a layered medium, since we have only made very general assumptions about the stochastic process that models such fluctuations.

For the case of a local maximum of the mean refractive index profile, we have derived the formula (17) for the focal length associated with the mean trajectories. The focal length of this gradient-index lensing layer is smaller in the presence of stochastic fluctuations than in the deterministic setting. The corrections due to the stochastic fluctuations that appear in the formulas (13) and (21) for the mean ray paths and the formula (17) for the focal length involve the second derivative at zero of the covariance function of the refractive index. Using usual assumptions about the local form of the covariance function near zero, one can relate these corrections to the standard parameters of optical turbulence, as we did in (16). The simultaneous consideration in this paper of both mean gradients and random medium fluctuations is important for describing propagation through the atmosphere, where larger-scale mean variations and smaller-scale stochastic fluctuations are both present.

Funding

Air Force Office of Scientific Research (AFOSR) (FA9550-15-1-0096).

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Lensing effects in a random inhomogeneous medium

Abstract

1. Introduction

2. Refractive index model

3. Mean ray path and focal length

4. Conclusion

Funding

References and links

Cited By

Figures (1)

Equations (23)

Optics Express