2.1.

Chromatic Dispersion in the Atmosphere

The trajectory of an optical beam propagating through the atmosphere is fundamentally dependent on the dispersion relationship (wavelength dependency of the refractive index) of the air. If the refractive index field for a particular volume of air is known, then the propagation path of an optical beam through the volume can be traced by applying Snell’s law of refraction. The atmospheric refractive index profile for dry air can be expressed as¹

2.2.

Standard Atmospheric Models

For numeric simulation of optical ray trajectory through the inhomogeneous atmosphere for a specific wavelength, the altitude-dependent (vertical) refractive index profile is required. From Eq. (1), the vertical refractive index can be obtained from altitude-dependent pressure and temperature profiles. We note that at optical frequencies, the humidity-dependent contribution to the refractive index is very small and can be ignored.³ The temperature and pressure profiles of the atmosphere are commonly modeled based on geographic reference atmospheric models. These models are described by a sequence of spherical layers in the lower troposphere with each layer having a constant vertical temperature gradient.

The altitude-dependent atmospheric properties of six commonly cited reference atmospheric models are listed in a tabular form in an Air Force Geophysics Laboratory technical report by Anderson et al.,⁴ which is publicly available. These layered atmospheric models assume constant temperature gradients for the vertical temperature profile for each layer. In our simulations, we applied the 1976 U.S. Standard Atmosphere and Subarctic Winter Atmosphere reference models. The Subarctic Winter Atmosphere has the largest temperature gradient among the six reference atmospheric models and hence tends to cause the largest bending of the propagating optical signals. For these atmospheric models, the temperature profile in a given layer in the lower troposphere is modeled as

Inserting Eqs. (3) and (4) into Eq. (1), we get the following expression for the atmospheric refractive index profile:

Considering the reduced refractivity $A_D(\lambda )$ to be independent of the altitude, an expression for the refractive index gradient can be obtained analytically, and it is given as

2.3.

Refractivity, Effect of Earth Curvature, Modified Refractivity, and Refractive Conditions

The value of the refractive index of the atmosphere is close to unity, and its value varies slowly in the vertical direction for the reference atmospheric models. Therefore, the refractive index at optical frequencies only changes in the fifth to sixth decimal places over the visible spectra. Hence, it is convenient to use the quantity:

For long horizontal or low-angle optical propagation, the refractivity model should include the curvature of the Earth. Optical signals traveling at low grazing angles tend to move away from the Earth’s surface due to the Earth’s curvature. To account for this effect and simplify the analyses, the modified refractivity ($M$) is used, where it is given as

In numerical modeling, the gradient of the modified refractivity is used to simulate actual ray trajectory for horizontal and low-angle optical signal propagation. Based on the modified refractivity gradient values, the refractive conditions can be divided into four categories (Fig. 1).^7,8 For radar or optical ray tracing, when $\partial M/\partial h>157$ $M$-units/km, rays will propagate away from the Earth’s surface, and this is referred to as the subrefraction condition. When $157M\text{-}\text{units}/\mathrm{km}\ge \partial M/\partial h>118M\text{-}\text{units}/\mathrm{km}$, standard wave propagation occurs, and this is known as the standard condition. With $118M\text{-}\text{units}/\mathrm{km}\ge \partial M/\partial h>0M\text{-}\text{units}/\mathrm{km}$, rays bend downward, and this condition is termed as super-refraction. In ducting/trapping condition where $\partial M/\partial h\le 0$, the rays (particularly for radar) can become trapped between the ground and the ducting layer. As a point of comparison, for the standard condition, the radio refractivity gradient $dN/dh$ is $\sim -40\mathrm{N}\text{-units}/\mathrm{km}$ near the Earth’s surface whereas for the visible spectra the value is $\sim -39\mathrm{N}\text{-units}/\mathrm{km}$.

For our study, we are particularly interested in the super-refraction and ducting region as the temperature inversion layer creates strong negative refractivity gradients for optical frequencies. The gradients in the ducting layer or temperature inversion layers can give rise to chromatic dispersion and significantly impact the propagation path of optical signals. The inversion layer parameters for this study were chosen based on experimental results from time-lapse camera measurements that were carried out in Las Cruces, New Mexico.⁹ Generally, the U.S. Standard Atmosphere with the modeled temperature inversion layer produces the super-refraction condition for our simulations. To study extreme atmospheric refractive conditions, we also slightly modified and applied the aforementioned temperature inversion profile to the Subarctic Winter Atmosphere. Under certain conditions, ducting of optical signals in the vicinity of the inversion layer was observed in our subarctic winter results.

2.4.

Temperature Inversion, Atmospheric Ducts and Webster Duct Model

In layered tropospheric models, such as 1976 U.S. Standard or Subarctic Winter models, the temperature is usually highest near the ground and generally decreases with altitude. Temperature inversion occurs when this trend is broken and the temperature increases with altitude. A hot layer of air can become trapped between two relatively cold layers to form a temperature inversion layer, and such a structure can extend over hundreds of kilometers along the ground. Conditions such as rapid cooling of the Earth’s surface and the adjacent air early in the night or adiabatic compression of sinking air over the land due to certain regional geographic features can create these sandwiched temperature layers. The denser (cool) air usually has a higher refractive index than the rarer (hot) layer of air in the troposphere. A sufficiently high gradient can trap optical/electromagnetic signals by creating a propagation channel where the rays swing up and down within the atmosphere between the ducting layer and Earth’s surface. This phenomenon is known as ducting or trapping. The presence of temperature inversion layers over long distances can lead to propagation channels or ducts for optical/electromagnetic signals and could increase the range of radio signals to over thousand miles with very low path loss.

Based on their formation process and modified refractivity profiles, the atmospheric ducts are commonly classified into four types: evaporation ducts, surface-based ducts, elevated ducts, and surface ducts.⁸ Generally, the modified refractivity for evaporation ducts is modeled with log-linear function whereas surface-based and elevated ducts are modeled as a trilinear curve and the surface ducts are modeled with a bilinear curve. Evaporation ducts are particularly important in radio refractivity as radio signals interact with the polarity of water vapor molecules. Thus, radio refractivity is a strong function of relative humidity.^3,10 For optical signals, humidity has a negligible effect, so evaporation ducts are ignored for our study.³ Figure 2 shows representative refractivity profiles of three duct types, excluding the evaporation duct.

Fig. 1

Illustration of common atmospheric refractive conditions (after Zeng et al.⁷).

Fig. 2

Modified refractivity profiles for (a) surface-based duct, (b) surface duct, and (c) elevated duct for a simplified atmospheric model with linear vertical refractive index gradient (after Dinc and Akan⁸).

There are several detailed analytical models available in the literature describing the form of the inversion profile for the duct types mentioned above. For our study, we chose the Webster duct profile¹¹ and applied this profile in the surface-based, elevated, and surface duct types to examine their effect on chromatic dispersion. The Webster duct model, introduced by Webster,¹¹ was later modified by Grabner and Kvicera,¹² where the refractivity profile is defined by the following equation:

Taking the Earth’s curvature into consideration, the wavelength- and altitude-dependent modified refractivity then becomes

For the simulations results presented here, we used a parameterization of the Webster duct model published by Wijerathna et al.⁹ These authors analyzed time-lapse imagery, and the results suggested that a temperature inversion-type of layer was present in the early morning at their location in New Mexico. Parameters for the Webster duct model were obtained by a fitting approach associated with the image shift results. For five consecutive mornings in late November, 2014, these authors found the mean of the calculated duct width to be $\mathrm{\Delta }h=48.5\mathrm{m}$ and the duct height was found to be $h_0=10.8\mathrm{m}$. The parameters were calculated for the time of the day where the maximum image stretching (distortion) was observed and the authors assumed $\mathrm{\Delta }N$ to be $-2$ N-units. This profile represents a surface-based temperature inversion profile. The three temperature inversion profiles (surface, surface-based, and elevated) were modeled in our simulations by changing the position of the duct height $h_0$.

3.1.

Ray Tracing Method and Algorithm

When refraction effects are much larger than diffractive effects, light propagation through an inhomogeneous media can be simulated using ray tracing methods. This condition generally holds for the propagation path and refractive features that we are studying. The ray trajectory is usually described by the solution of the eikonal equation.^13–17 A commonly used solution involves the application of a linear transfer equation coupled with a bending equation. The bending equation determines the change in ray trajectory and is a function of change in refractive index between two layers. Starting with Snell’s law of refraction,

The small-angle approximation holds for our numeric simulations as the rays travel in a horizontal/near-horizontal path, with a total angular change of $\sim 20\mathrm{mrad}$ over a 150-km path along the ground. The bending equation between layers is obtained from Eq. (14) as

With index $j$ for discrete layers, the bending equation for the numeric simulation is

The rays are assumed to follow a linear path between planes with the linear transfer equation given as

Our problem of interest involves refractive index gradients defined in the vertical direction and ray trajectories at low angles relative to the horizontal. For this situation, a second-order ray tracing algorithm¹⁸ was developed that relaxes the step size constraint and allows for faster but accurate ray trajectory calculation. We implemented this second-order ray tracing method in our simulation, where a quadratic correction term is introduced to the transfer equation. This term comes from the solution of a 2-D form of the eikonal equation describing the ray trajectory in an inhomogeneous media. The eikonal equation is given as

For numeric simulations, the transfer equation [Eq. (19)] and bending equation [Eq. (20)] can be rewritten as follows:

3.2.

Chromatic Dispersion Simulation Results

The chromatic dispersion between rays with wavelength 532 and 1550 nm was simulated using the second-order ray tracing algorithm for several practical atmospheric conditions for a distance of 150 km along the ground. Figure 3 provides an example of how the presence of an elevated inversion layer, described by the Webster duct model, impacts the ray trajectory and chromatic divergence.

Fig. 3

Ray trajectories of three pairs of 532 nm (blue) and 1550 nm (red) rays where the rays within a pair have the same initial launch angle of $-9.75$, $-9.425$, or $-9.1\mathrm{mrad}$: (a) standard atmosphere and (c) with the temperature inversion. Color bars show the refractivity gradient $dN/dh$ in ${\mathrm{km}}^{-1}$ at 532 nm. The separation between ray pairs as a function of the distance along the Earth’s surface for (b) standard atmosphere and (d) with the inversion layer.

Figure 3(a) shows ray trajectories through the U.S. Standard Atmosphere gradient profile. Three pairs of 532- and 1550-nm rays are traced from left to right, and the apparent upward trajectory of the rays beyond the 100-km distance is an effect related to the curvature of the Earth’s surface. Each pair is launched at a slightly different downward initial angle and as the rays traverse the atmosphere, the 532-nm wavelength rays bend more toward the ground than the 1550-nm rays. Because the refractivity gradient is altitude dependent (indicated by the color bar), the overall amount of ray bending is primarily dependent on the source altitude. Figure 3(b) shows the separation between rays within a pair ($h_{\mathrm{diff}}$) as a function of distance along the Earth’s surface. Regardless of the slight launch angle difference, the rays for this case diverge about 5 m within each pair at the 150-km distance along the Earth’s surface in the standard atmosphere. In general, we find for the standard atmosphere case that $h_{\mathrm{diff}}$ is also primarily a function of source altitude, where the separation becomes smaller as the source is moved higher in altitude.

However, if the source is located above the inversion layer [Fig. 3(c)], then the shorter wavelength ray encounters the layer feature earlier (in terms of distance along the Earth’s surface) than the longer wavelength. This can separate the rays with different colors further or even bring them closer to their standard atmospheric prediction, depending on several factors. Figure 3(d) shows chromatic separation between ray pairs launched at the same angles as in Fig. 3(a) but in the presence of the inversion layer. The ray separation depends on the relative source position with respect to the inversion layer, launch angle, and wavelength of the source. For the case shown in Fig. 3, the separation can be as large as 23 m, which is significantly larger than the $\sim 5\text{-}\mathrm{m}$ prediction for the standard atmosphere.

Figure 4 presents ray separation results for packets of 101 pairs of rays, where each pair is aimed at a slightly different angle toward the inversion layer. Results are shown for source locations below, inside, and above the inversion layer. For a specific source height ($h_{\mathrm{s}}$), ray packets were launched over an angular range of a few milliradians, such that the rays would encounter the inversion layer. Figure 4 shows ray separation as a function of distance along the Earth’s surface for four source altitudes. For the case in Fig. 4(a) with the source below the layer, the result for the ray packet is essentially the same as Fig. 3(b) without the layer. The other cases illustrate that the largest separation effects occur when the source is above the layer and the rays are launched at larger initial angles.

Fig. 4

With the same index gradient profile as Fig. 3(c), the ray separation $h_{\mathrm{diff}}$ for 101 pair of rays of 532- and 1550-nm wavelength with four different source locations are shown as a function of distance along the Earth’s surface. The rays within each pair have the same launch angle. The source height $h_s$ and launch angle range are (a) 10 m, $-1$ to 1 mrad; (b) 50 m, $-1$ to 1 mrad; (c) 100 m, $-3.5$ to $-2.5\mathrm{mrad}$; and (d) 200 m, $-6.5$ to $-5.5\mathrm{mrad}$.

For all of the results in Sec. 3.2, the elevated temperature inversion layer was modeled using the following Webster duct parameter values: $\mathrm{\Delta }N=-2$ $N$ units, $\mathrm{\Delta }h=50\mathrm{m}$, $h_0=50\mathrm{m}$. The value of the duct width $\mathrm{\Delta }h$ was 50 m (average value is 48.5 m in the work by Wijerathna et al.⁹) and the duct height value $h_0=50\mathrm{m}$ (average value is 10.8 m in the work by Wijerathna et al.⁹) was used to model an elevated inversion layer to ensure maximum possible interaction of the propagating rays with the inversion layer. This profile coupled with U.S. Standard Atmosphere does not produce a strong enough gradient for ray trapping or ducting but rather corresponds to a super-refraction refractive condition. We ran simulations with different $h_0$ values to model surface and surface-based duct type inversion layers as well. Our simulations show that the ray separation $h_{\mathrm{diff}}$ can be considered independent of the duct type but strongly dependent on the relative distance between the source height ($h_{\mathrm{s}}$), duct height ($h_0$), and launch angle.

For the source located above the inversion layer, we ran simulations with the 532-nm wavelength rays to study the ray AOA at the 150-km distance as a function of launch angle. For the standard atmosphere, the AOA increases essentially linearly with increasing launch angle for a particular source height. However, in the presence of an inversion layer, this general trend is broken and the AOA can decrease with increasing launch angle. Figure 5(a) illustrates the dependence of the AOA on launch angle in the standard atmosphere, and Fig. 5(b) shows the effect of the elevated inversion layer (as defined above) on the AOA with the source located at $h_{\mathrm{s}}=200\mathrm{m}$ and rays launched at angles between $-6.5$ and $-5.5\mathrm{mrad}$.

Fig. 5

AOA at the detector location as a function of launch angle for 532-nm source (a) in the U.S. Standard Atmosphere and (b) in the presence of the inversion layer.

3.3.

Ducting and Dispersion Simulation Results Under Extreme Conditions

Observation of ducting of radio signals goes back as early as World War II, and such anomalous refraction is linked with temperature inversion and the evaporation duct.¹⁹ Hence, it is a curious question to ask whether ducting of optical signals is a possibility. As humidity has a negligible effect on optical signal propagation, we found that a large temperature gradient along with a large temperature inversion is required for optical ducting. The subarctic winter atmospheric profile exhibits a large temperature gradient with height, and literature suggests that large temperature inversion profiles can exist in the subarctic regions.^20,21 Thus, the subarctic region could produce atmospheric conditions that support optical ducting. To examine this possibility, we applied our simulation approach to the Subarctic Winter Atmosphere model with an embedded temperature inversion. Currently, we do not have data for the parametrization of duct profiles for the subarctic atmosphere and we did not find suitable analytic subarctic inversion models for use in our approach. Hence, we applied a temperature inversion layer identical to the one described in Sec. 3.2 to the Subarctic Winter Atmosphere with a change in the parameter $\mathrm{\Delta }N=-3\mathrm{N}$ units to simulate a stronger inversion. This inversion layer along with the subarctic winter refractive index profile created conditions that cause actual ducting of the optical signal. Figure 6 shows the ducting effect for 101 green (532 nm) rays launched near the center ($h_s=40\mathrm{m}$) of the ducting layer (duct height = 50 m) with initial launch angles ranging from 0 to 0.4 mrad. The ducting of rays limits the chromatic divergence as long as the rays of different wavelength are all trapped. However, our simulation studies show that for specific source altitudes there can exist an angular launch range over which rays with one color can become trapped by the ducting layer and the rays of the other color may not meet the ducting condition. The chromatic separation between 101 pair of 532- and 1550-nm rays is shown in Fig. 7 for the ducting case described above, with different launch angle ranges and the source located at 40 m above the ground. Chromatic separation is small after the 150-km propagation distance when both colors are ducted [Fig. 7(a)], but for larger launch angles, the 1550-nm rays leave the duct and chromatic separations up to 280 m are demonstrated [Fig. 7(b)]. Although the temperature inversion layer for this example is ad hoc, it illustrates the parameters required to produce optical ducting and extreme chromatic separation.

Fig. 6

Ducting of green (532 nm) rays in the subarctic winter atmosphere in the presence of the Webster-type inversion layer (as described in Sec. 3.2 but with $\mathrm{\Delta }N=-3\mathrm{N}$ units) with the source located at 40 m height and rays launched at 0- to 0.4-mrad angle.

Fig. 7

Chromatic separation between 101 pair of rays of 532- and 1550-nm wavelength as a function of distance along the Earth’s surface with the source located at 40-m height in the presence of the inversion layer (as described in Sec. 3.2 but with $\mathrm{\Delta }N=-3\mathrm{N}$ units) and the subarctic winter atmosphere. Ray launch angles are between (a) 0 to 0.4 mrad, showing ducting of both wavelengths, and (b) 0.3 to 0.6 mrad, where the 1550-nm light leaves the duct for the larger angles.